ELASTIC-PLASTIC

FRACTURE

A symposium
sponsored by ASTM
Committee E-24 on
Fracture Testing of Metals
AMERICAN SOCIETY FOR
TESTING AND MATERIALS
Atlanta, Ga., 16-18 Nov. 1977

ASTM SPECIAL TECHNICAL PUBLICATION 668

J. D. Landes, Westinghouse R&D Center
J. A. Begley, The Ohio State University
G. A. Clarke, Westinghouse R&D Center
editors

04-668000-30

AMERICAN SOCIETY FOR TESTING AND MATERIALS

1916 Race Street, Philadelphia, Pa. 19103
#
Copyright © by AMERICAN SOCIETY FOR TESTING AND MATERIALS 1979
Library of Congress Catalog Card Number: 78-72514

NOTE
The Society is not responsible, as a body,
for the statements and opinions
advanced in this publication.

Printed in Tallahassee, Fla.

October 1972
Second Printing, July 1981
Baltimore, Md.
 Ken Lynn

Dedication
It was with great sorrow and disbelief that we all learned of the
sudden and untimely death of Ken Lynn during the summer of 1978.
We have lost an imaginative and competent practitioner of the art of
fracture mechanics who was able to cut through the many details of
a problem and get to the essence of it to seek the practical solution.
We have also lost a great friend who was intensely interested in the
lives and achievements of his co-workers and contemporaries. It is
with sincere appreciation for his fruitful technical life and his uplifting
personal outreach that we dedicate this ASTM fracture mechanics
symposium volume to his memory.
Ken grew up near Pittsburgh and in Florida; he received his B.S.
in Mechanical Engineering in 1946 and M.S. in Engineering Me-
chanics in 1947 from Pennsylvania State University. His first employ-
ment was with the U.S. Steel Corporation, both in Kearny, New
Jersey, and in Cleveland, Ohio, where he worked on brittle crack
initiation and propagation in steels—a subject to which he would
devote much of his efforts later in life. He was always proud of the
fact that, while at U.S. Steel, he had established the strength of the
cables which still support the original Delaware Memorial Bridge. In
March of 1955, he joined the Lockheed Aircraft Corporation, and
was employed at both the Burbank, California, and Marietta, Georgia,
facilities. As a senior research engineer, he was in charge of struc-
tural materials research on the nuclear-powered bomber project as
well as fatigue life prediction of aircraft wing structures. In August of
 1957, he moved to the Rocketdyne Division of North American Rock-
well Corporation in Canoga Park, California, where he began his
serious development as a practitioner of fracture mechanics. Through
a series of increasingly challenging assignments in experimental stress
analysis and fracture mechanics evaluations, he became a lead con-
sultant on structural problems and fracture mechanics for Rocketdyne
hardware. A key responsibility of Ken's was for development of the
fracture control plan for several critical Rocketdyne structures. It was
at Rocketdyne that Ken became actively involved with ASTM, and
with Committee E-24 in particular. He quickly recognized the con-
sensus agreement value of the ASTM system and strongly promoted
it. Ken's approach to ASTM was not to seek leadership, but rather to
stay "down in the trenches" at the technical working level. He main-
tained this philosophy throughout his association with ASTM,
especially in later years as he came to rely on ASTM E-24 more and
more for consensus agreement. Ken next became intrigued by the
technical challenges presented by the field of nuclear power generation.
So, in January of 1971, he joined the Westinghouse PWR Division
where he became deeply involved in applying advanced fracture
mechanics techniques to the analysis of pressurized water com-
ponents, mainly reactor pressure vessels. Because the nuclear industry
was then in the process of upgrading safety analysis in terms of
fracture mechanics, he eagerly helped promote the standardization of
LEFM testing and analysis through ASTM. His Westinghouse ex-
perience led him to join the Atomic Energy Commission in August of
1972. At AEC he worked on applying fracture mechanics to thermal
shock analysis problems and to flaw evaluation procedures which
later were incorporated into the ASME Boiler and Pressure Vessel
Code, Section XI. Recognizing greater opportunity for development
and application of fracture mechanics. Ken joined the Division of
Reactor Safety Research—now part of the Nuclear Regulatory
Commission—whereupon he took over management of a series of
research programs all directed at ensuring the safety of structures in
the primary system of light water power reactors. Full of energy.
Ken made many contributions to the understanding and application
of fracture mechanics principles for the evaluation and solution of
problems faced in primary system integrity. Included among these
were thermal shock, crack arrest, crack growth rates, irradiation
effects, and linear elastic and elastic-plastic analysis of vessels under
overpressurization. With NRC, Ken undertook a front-line leadership
of grounding technical advancements in fracture mechanics through
ASTM Standards. His commitment to the ASTM E-24 Committee,
and their efforts, was complete. He was especially looking forward
to the ASTM standardization of test specimens and methods for both
crack arrest and for J-R curve testing of ductile steels, and personally
assured that all work done under his direction was aimed at this goal.
Because of his position as a program manager. Ken did not write
many technical papers; he always felt that the individual researcher
should take credit for the work, not himself. However, the technical
literature today is filled with articles based on his understanding and
direction of research and application in the field of fracture mechanics,
and many acknowledgments and technical directions can be found in
these papers. Because of his experience and competence in fracture
mechanics. Ken was often asked to organize meetings and to chair
some of the sessions. His summaries of the information presented
and his conclusions and suggested directions were looked forward to,
as we knew that if we did not understand what had happened, or
what was truly significant. Ken usually did, and his evaluation would
help to clarify the situation.
Ken was deeply devoted to his wife, Lois, and was thoroughly
enjoying the experience of his two grandchildren, by his son David,
who lives in Denver; and by his daughter, June Mesnik, who lives in
Los Angeles. He was quite proud of his other daughter, Carol, and
thoroughly enjoyed competing against his two younger sons, Gordon
and Jerry, at golf or pool. In both his technical and personal life,
Ken always strove for perfection and always challenged himself and
his family to the same end. One of the true joys of his last few years
was to be able to take Lois with him on several business trips to
Europe, where they renewed many acquaintances they had made
with Ken's contemporaries, who looked to him for technical leadership
in fracture analysis of reactors, and also for good times after the job
was done. At the time of his death. Ken was planning for several
ASTM Meetings where crack arrest, fracture toughness, and crack
growth rates were approaching, to his great satisfaction, true national
and international standardization. We will no longer have the benefit
of his contributions to his chosen discipline, and we will miss them.
But most of all, we will miss Ken himself.
Foreword

The symposium on Elastic-Plastic Fracture was held in Atlanta, Georgia,

16-18 Nov. 1977. The symposium was sponsored by ASTM Committee
E-24 on Fracture Testing of Metals. J. D. Landes, Westinghouse Research
and Development Center, J. A. Begley, The Ohio State University, and
G. A. Clarke, Westinghouse Research and Development Center, presided
as symposium chairmen. They are also editors of this publication.
 Related
ASTM Publications

Cyclic Stress-Strain and Plastic Deformation Aspects of Fatigue Crack

Growth, STP 637 (1977), $25.00, 04-637000-30

Use of Computers in the Fatigue Laboratory, STP 613 (1976), $20.00,

04-613000-30

Handbook of Fatigue Testing, STP 566 (1974), $17.25, 04-566000-30

References on Fatigue, 1965-1966, STP 9P (1968), $11.00, 04-0009160-30

 A Note of Appreciation
to Reviewers
This publication is made possible by the authors and, also, the un-
heralded efforts of the reviewers. This body of technical experts whose
dedication, sacrifice of time and effort, and collective wisdom in reviewing
the papers must be acknowledged. The quality level of ASTM publications
is a direct function of their respected opinions. On behalf of ASTM we
acknowledge with appreciation their contribution.

ASTM Committee on Publications

 Editorial Staff
Jane B. Wheeler, Managing Editor
Helen M. Hoersch, Associate Editor
Ellen J. McGlinchey, Senior Assistant Editor
Helen Mahy, Assistant Editor
Contents

Introduction 1

ELASTIC-PLASTIC FRACTURE CRITERIA AND ANALYSIS

Instability of the Tearing Mode of Elastic-Plastic Crack Growth—

p . C. PARIS, H. T A D A , Z . Z A H O O R , AND H. ERNST 5
The Theory of Stability Analysis of/-Controlled Crack Growth—
J. W. HUTCHINSON AND P . C. PARIS 37
Studies on Crack Initiation and Stable Crack Growth—c. F. SHIH,
H. G. DELORENZI, AND W. R. ANDREWS 65
Elastic-Plastic Fracture Mechanics for Two-Dimensional Stable
Crack Growth and Instability Problems—M. F. KANNINEN,
E. F. RYBICKI, R. B. STONESIFER, D. BROEK,
A. R. ROSENFIELD, C. W. MARSCHALL, AND G. T. HAHN 121
A Numerical Investigation of Plane Strain Stable Crack Growth
Under Small-Scale Yielding Conditions—E. P. SORENSON 151
On Criteria for /-Dominance of Crack-Tip Fields in Large-Scale
Yielding—R. M. MCMEEKING AND D. M. PARKS 175
A Finite-Element Analysis of Stable Crack Growth—I—M. NAKAGAKI,
W. H. CHEN, AND S. N. ATLURI 195
A Comparison of Elastic-Plastic Fracture Parameters in Biaxial Stress
S t a t e s — K . J. MILLER AND A. P . KFOURI 214
Numerical Study of Initiation, Stable Crack Growth, and Maximum
Load, with a Ductile Fracture Criterion Based on the Growth
of Holes—Y. D'ESCATHA AND J. C. DEVAUX 229

EXPERIMENTAL TEST TECHNIQUES AND FRACTURE TOUGHNESS DATA

An Initial Experimental Investigation of the Tearing Instability

Theory—P. C. PARIS, H. TADA, A. ZAHOOR, AND H. ERNST 251
Evaluation of Estimation Procedures Used in /-Integral Testing—
J. D. LANDES, H. WALKER, A N D G. A. CLARKE 266
An Evaluation of Elastic-Plastic Methods Applied to Crack Growth
Resistance Measurements—D. E. MCCABE AND J. D. LANDES 288
Elastic-Plastic Fracture Toughness Based on the COD and /-Contour
Integral Concepts—M. G. DAWES 306
J-Integral Determinations and Analyses for Small Test Specimens and
Their Usefulness for Estimating Fracture Toughness—
I. ROYER, J. M . T I S S O T , A. PELISSIER-TANON, P. LE POAC,
AND D. MIANNAY 334
Effect of Size on t h e / Fracture Criterion—i. MILNE AND G. G. CHELL 358
Determination of Fracture Toughness with Linear-Elastic and
Elastic-Plastic Methods—c. BERGER, H. P. KELLER, AND
D. MUNZ 378
Minimum Specimen Size for the Application of Linear-Elastic
Fracture Mechanics—D. MUNZ 406
Thickness and Side-Groove Effects on /• and S-Resistance Curves for
Steel at 93°C—w. R. ANDREWS AND C. F. SHIH 426
Computer Interactive 7^ Testing of Naval Alloys—j. A. JOYCE AND
I. p. GUDAS 451
Characterization of Plate Steel Quality Using Various Toughness
Measurement Techniques—^^A. D. WILSON 469
Static and Dynamic Fibrous Initiation Toughness Results for Nine
Pressure Vessel Materials—w. L. SERVER 493
Dynamic Fracture Toughness of ASME SA508 Class 2a Base and
Heat-Affected-Zone Material—w. A. LOGSDON 515
Tensile and Fracture Behavior of a Nitrogen-Strengthened,
Chromium-Nickel-Manganese Stainless Steel at Cryogenic
Temperatures—R. L. TOBLER AND R. P. REED 537
Fracture Behavior of Stainless Steel—w. H. BAMFORD AND A. J. BUSH 553
APPLICATIONS OF ELASTIC-PLASTIC METHODOLOGY

A Procedure for Incorporating Thermal and Residual Stresses into

the Concept of a Failure Assessment Diagram—G. G. CHELL 581
The COD Approach and Its Application to Welded Structures—
J. D . HARRISON, M. G. DAWES, G. L. ARCHER, AND
M. S. K A M A T H 606
Fracture Mechanics Analysis of Pipeline Girthwelds—H. I. MCHENRY,
R. T. READ, AND I. A. BEGLEY 632
An Elastic-Plastic R-Curve Description of Fracture in Zr-2.5Nb
Pressure Tube Alloy—L. A. SIMPSON AND C. F. CLARKE 643
Correlation of Structural Steel Fractures Involving Massive
Plasticity—B. D. MACDONALD 663
An Approximate Method of Elastic-Plastic Fracture Analysis for
Nozzle Comer Cracks—J. G. MERKLE 674
Elastic-Plastic Fracture Mechanics Analyses of Notches—
M. M. HAMMOUDA AND K. J. MILLER 703
Size Effects on the Fatigue Crack Growth Rate of Type 304 Stainless
S t e e l — w . R. BROSE A N D N . E . D O W L I N G 720
Use of a Compact-Type Strip Specimen for Fatigue Crack Growth
Rate Testing in the High-Rate Regime—D. F. MOWBRAY 736

SUMMARY
Summary 755
Index 767
 STP668-EB/Jan. 1979

Introduction

Interest in elastic-plastic fracture mechanics grew as a natural extension

of linear elastic fracture mechanics (LEFM) concepts when it became
obvious that LEFM methods were not adequate to handle many problems
in the design and reliability analysis of structural components. Some of the
early elastic-plastic fracture parameters and crack-tip analyses were de-
veloped in the 1960's; in the early 1970's, however, work on elastic-plastic
fracture characterization was greatly expanded. Many new parameters and
methods of fracture prediction were introduced and interest in this topic
became widespread.
This publication represents papers presented at the ASTM Committee
E-24 sponsored Symposium on Elastic-Plastic Fracture held in Atlanta,
Ga., in November 1977. The symposium was organized to provide a forum
for presenting current work in this rapidly developing field. No single
approach was taken in the papers presented; rather, a variety of parame-
ters and methodologies was presented. For the most part the papers
presented new approaches and new data; some of the papers presented
summaries and applications of existing approaches.
The symposium was very successful in that a good cross section of
workers presently engaged in elastic-plastic fracture studies was repre-
sented. Most of the methods presently being used were discussed. The
work presented herein is a fairly accurate account of the present status of
the elastic-plastic fracture field, which status is that work is progressing at
a fairly rapid pace, new ideas are frequently introduced, different approaches
are being attempted, and to date no single method has been adopted by all
of the workers in this field.
The contents of this publication will be particularly useful to persons
working in the elastic-plastic fracture field. This would include researchers
involved in material property studies and structural analysis, designers,
and persons concerned with safety and licensing. The contents of the book
do not so much represent an end product in the development of elastic-
plastic fracture; rather, they represent a step in the development of this
field which should be followed by other important publications on the
subject. Some of the papers may become dated as the technology advances
and present techniques are discarded for new ones, while other papers may
have more permanent value, marking the first introduction of a significant
new concept.
Three major areas were covered in the symposium: fracture criteria and
analysis; experimental evaluation and toughness testing; and applications

Copyright 1979 b y AS FM International www.astm.org

2 ELASTIC-PLASTIC FRACTURE

of elastic-plastic methodology, including the application of elastic-plastic

fracture concepts to fatigue crack growth analysis. The analysis papers
dealt mainly with the assessment of new and existing criteria. The present
emphasis is on extending fracture prediction based on an initiation cri-
terion to include the characterization of stable crack growth and ductile
instability in the fracture process. Some of the criteria are mainly empirical
while others are based on the postulation of a fracture mechanism. Finite-
element analysis remains the most popular method for evaluating crack-tip
behavior in the elastic-plastic regime.
Fracture toughness test results were directed at determining properties of
materials for specific applications and at evaluating present fracture
criteria. Many of the materials evaluated were steels used in the nuclear
industry. Of particular interest were pressure vessel steels tested under
dynamic loading and stainless steels. Experimental evaluation of existing
fracture criteria dealt with the evaluation of test specimen size, the evalua-
tion of analysis methods, and the use of advanced testing methods such as
computer-based data acquisition and reduction systems.
The application of elastic-plastic techniques to the evaluation of struc-
tural components is directed toward fracture problems in pressure vessels,
pipelines, and other structural members. Specific areas of given structures
were often considered, generally areas of high stress concentration such as
nozzle comers and notches. Special note was given to the application of
elastic-plastic techniques to fatigue crack growth studies, particularly in
the high-strain low-cycle regime.
The variety of topics covered should be of interest to a large number of
researchers working in the elastic-plastic area. This publication represents
the first major collection of papers devoted solely to the topic of elastic-
plastic fracture.

J. D. Landes
Westinghouse Electric Corp. Research and
Development Center, Pittsburgh, Pa.; co-
editor.
Elastic-Plastic Fracture Criteria
and Analysis
p. C. Paris,' H. Tada,' A. Zahoor,' and H. Ernst^

The Theory of Instability of the

Tearing Mode of Elastic-Plastic
Crack Growth

REFERENCE: Paris, P. C, Tada, H., Zahoor, A., and Ernst, H., "The Theory of In-
stabili^ of the Tearing Mode of Elastic-Plastic Cracli Growth," Elastic-Plastic Fracture.
ASTMSTP 668, J. D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society
for Testing and Materials, 1979, pp. 5-36.

ABSTRACT: This paper presents a new approach to the subject of crack instability
based on the J-integral R-curve approach to characterizing a material's resistance to frac-
ture. The results are presented in the chronological order of their development (including
Appendices I and II).
First, a new nondimensional material parameter, T, the "tearing modulus," is de-
fined. For fully plastic (nonhardening) conditions, instability relationships are de-
veloped for various configurations, including some common test piece configurations,
the surface flaw, and microflaws. Appendix 1 generalizes these results for the fully plastic
case and Appendix II treats confmed yielding cases.
The results are presented for plane-strain crack-tip and slip field conditions, but may
be modified for plane-stress slip fields in most cases by merely adjusting constants.
Moreover, an accounted-for compliance of loading system is included in the analysis.
Finally, Appendix III is a compilation of tearing modulus, T, properties of materials
from the literature for convenience in comparing the other results with experience.

KEY WORDS: crack instability, tearing instability, tearing modulus, elastic-plastic

fracture, J-integral fracture mechanics, crack propagation

It has become common to characterize a material's static crack extension

behavior under monotonically increasing deformation using a J-integral
R-curve [1-4].^ However, unlike the situation with linear-elastic (/if-type)
R-curves [5], crack instability phenomena have yet to be analyzed in terms of
J-integral R-curves.
In low-strength steels, a common material for characterization via
J-integral R-curves, there appears to be two distinct possibilities for instabil-
ity. First is the so-called "cleavage" instability which is normally attributed
to a local material instability on a microscopic scale (such as inclusion spac-
' Professor of mechanics, senior research associate, and graduate research assistants, respec-
tively, Washington University, St. Louis, Mo.
^The italic numbers in brackets refer to the list of references appended to this paper.

Copyright 1979 b y AS FM International www.astm.org

6 ELASTIC-PLASTIC FRACTURE

ing). The second type of instability is associated with the global conditions in
a test specimen or component and loading arrangement providing the driving
force to cause continuous crack extension by a so-called "tearing"
mechanism. Cleavage is associated with very flat fracture on crystalline
planes whereas tearing is normally associated with dimpled rupture
mechanisms on a microscopic level.
Moreover, in testing compact tension specimens to produce J-integral
R-curves (for example, see Ref 2), cleavage is associated with a sudden in-
stability where the crack jumps ahead, severing the test piece almost instan-
taneously. At low temperature this occurs prior to the beginning of tearing.
At higher temperatures, just above transition, steady tearing commences
first, followed after some amount of stable tearing by the sudden cleavage in-
stability. At yet higher temperatures, much more extensive stable tearing oc-
curs prior to cleavage, if the sudden cleavage occurs at all. (These patterns of
behavior are more fully described in a later work [6].)
Indeed, the patterns of instability behavior versus stable tearing do not
seem to be well understood, although attempts by Rice and co-workers [7,8]
have provided some analysis of the mechanics of the situation. Herein a sim-
ple approach is taken to the mechanics of potential instability associated with
the steady tearing portion of J-integral R-curves. The analysis is developed
from simple examples of structural component (or test specimen) configura-
tions with cracks, examining their instability possibilities individually, in
order to draw more general conclusions about elastic-plastic cracking in-
stability as contrasted to linear-elastic behavior. Finally, an attempt is
made to model a more local cleavage-like instability for material in the frac-
ture process zone just ahead of a crack tip.

The J-Integral R-Curve

It has been noted in many recent works t h a t / may be interpreted as the in-
tensity of the elastic-plastic deformation and stress field surrounding a crack
tip (for example see Refs 3 or 4). A s / is increased for a given crack situation,
the response to increasing/ is crack extension. Following Fig. 1, the crack
extension first takes on the form of some minor lengthening due to flow and
blunting of the tip until the conditions for tearing develop, whereupon in-
crements of tearing extension, da, proceed approximately linearly with
added increments of/, or dJ. As noted previously, sudden cleavage may oc-
cur either before or after the commencement of stable tearing, or not at all,
depending on the temperature and material.
Here, at least initially, attention shall be directed to the stable tearing por-
tion of the J-integral R-curve. It has been noted that plotting this R-curve,
using//ffo instead of/ as the ordinate, results in a blunting and stable tearing
behavior that is reasonably independent of temperature [6,9] for a given
 PARIS ET AL O N INSTABILITY OF T H E T E A R I N G MODE

no interceding
cleavage instability
thigh temperature)

da ) dJ/do = constant
V — — interceding cleavage
^ instability after start of stable
tearing (transition temperature)
beginning of stable teoring (J. )

ding cleavage instability before stoble

tearing (at low temperature)

extension due to blunting

shorp crack (prior to loading)

blunting prior to tearing

tearing after blunting to commencement

of stable tearing (4^- = constont)
da

Ao

FIG. 1—J-integral R-curve (with some diagrammatic details following Rice [4]).

material and plane-strain conditions. This is simply schematically illustrated

n Fig. 2.
Plane-strain conditions are, as usual [1,3,4], assumed to be present as long
IS the test specimen (or component) thickness and remaining ligament are
large enough, by the criterion

size > 25J/ao (1)

If this condition is met, then the J-integral R-curve is size independent [2]
and is also reasonably independent of configuration with some reservations
110].
8 ELASTIC-PLASTIC FRACTURE

&a
FIG. 2—Temperature-independent plot of the blunting and stable tearing portion of the
J-integral R-curves of a given material.

Therefore, for present purposes, we may characterize a material's stable

tearing properties by
dJ
—— = constant (temperature dependent)
da

or by (2)

dJ 1
—- X — = constant (temperature independent)
da (To

but most importantly, as shall be observed later, let

^ dJ ^ E
(3)
T = ^da X ^ffo" = constant
where T shall be termed the tearing modulus of the material. See Appendix
III for the values of these material characteristics. Refer to the J-integral
R-curves, plotted such as Figs. 1 or 2 from actual test data, to provide the
values. Now it is acknowledged that the straight-line stable tearing portion of
the R-curves in Figs. 1 and 2 involves perhaps some idealization, but that
view is sufficient herein.
Moreover, although that portion of the R-curve is termed "stable tearing,"
it shall later be noted inherently as "stable" only for tests involving substan-
tial bending loading imposed on a remaining ligament, such as deeply
cracked compact tension or bend specimens. Indeed, the discussion now
turns to other configurations where instability may be possible on the tearing
portion of the R-curve; thus, it may simply be termed "tearing" or "steady
tearing" rather than "stable tearing."
 PARIS ET AL ON INSTABILITY OF THE TEARING MODE

An Analysis of Possible Tearing Instabilities

Consider a center-cracked strip with crack length, 2a, width, W,
thickness, B, and length, L, as illustrated by Fig. 3. It is presumed that fully
plastic behavior occurs, that is, the slip lines develop, prior to crack extension
by the tearing behavior discussed previously. The appropriate slip lines are
simply 45-deg slips from the crack tips to the edge, giving a limit load, PL, by

PL = a^W- 2a)B (4)

where for this case the flow stress, a/, on the remaining ligament is simply the
flow stress for simple tension, ao.
The increased crack opening stretch, 6r, at each crack can be viewed as
contributed directly by the slips as they operate, increasing the length of the
specimen, L and AZ,, by a like amount. Incrementally, that is

d(ALpiastic) — £?(6r) (5)

an increase in 6r implies a corresponding increase in/by the usual relation-

ship [J, 4,7]

dJ
(6)
6T — oiJ/ao or dbr = a

where, for convenience, a will be taken as approximately 1 (a is more nearly

0.7 for plane strain).
Combining Eqs 5 and 6

d{J)
c/(ALplastic) — (7)
Co

crock

slip lines

FIG. 3—Fully-plastic center-cracked strip.

10 ELASTIC-PLASTIC FRACTURE

Now the increment diJ), increase in 7, would imply a crack extension da from
Eq 2 or Eq 3. As a consequence, from Eq 4 the limit load would be reduced
by
dPi = -laodaB (8)
and this load reduction would further imply an increment of elastic shorten-
ing of the length of the specimen^

MAT . _ dPtL _ -lapdoL

d(ALe,as«c) - ^ ^ - w^£ (9)

Now, if the specimen were being tested in a rigid machine (fixed grips or end
displacements) instability would ensue if the magnitude of elastic shorten-
ing exceeded the corresponding plastic lengthening required for crack ex-
tension. Equating Eqs 7 and 9 leads to the criterion of

™ _ _dJ_ ^ _ ^ ^ ^ ^ (for instability of a center- .

da ffo^ ~ W cracked strip in tension)

for instability. The left-hand side of Eq 10 depends only on material proper-

ties, with dJ/da to be supplied from the tearing slope of the J-integral
R-curve as implied in Eq 2 or Eq 3. The right-hand side of Eq 10 is a non-
dimensional configuration parameter. Before discussing the detailed im-
plications of this instability criterion, Eq 10, further, consider first other con-
figurations.
For instability of a fully plastic double-edge notched strip (see Fig. 4) the
analysis proceeds identically except that the flow stress at the cracked sec-
tion. Of, is elevated by the nature of the slip field (implied constraint) and is
about three times the flow stress in a simple tension test, ao [7,8,11]; thus
since

Of = 3ffo (double edge cracked strip)

and for the rigid-plastic velocity field of a double-edge cracked strip

rf(ALpu«ic) = V2d8, (11)

resubstituting as appropriate in the analysis preceding Eq 10 leads to

j,__dJ_ _E_ 12L (for instability of a double-edge cracked strip in ...

da ao^ ~ W tension)
•'Neglecting for the moment the elastic shortening due to the crack, which would be small here
at any rate.
 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 11

FIG. 4—Double-edge cracked strip.

It is noted that for the double-edge cracked strip the notches must be deep
enough to induce the sUpfieldshown in Fig. 4, if the analysis leading to Eqs
11 and 12 is to be correct. This simply requires a/W » Vs.
Moreover, with both of the previous illustrations, center-cracked and
double-edge cracked strips, tearing is likely to proceed in an unsymmetrical
manner, causing bending and further undermining the stability of the situa-
tion. Thus in Eqs 10 and 12 the numerical coefficient on the right-hand side
should be regarded as a lower limit for commencement of tearing instability.
It is estimated that, due to bending, this numerical coefficient might be as
much as a factor of 2 larger.

Approximate Analysis of Tearing Stability for a Deep Surface Flaw

A case of considerable interest in pressure vessels and other applications is
that of a deep surface flaw where yielding ensues over the remaining liga-
ment. An approximate analysis based on some methods first used by Irwin
[12] is attempted here. Let the surfaceflawbe described as having a depth, a,
and length, /, in a plate of thickness, t, subject to an applied stress, a, as il-
lustrated in Fig. 5. The stress, a, is presumed to be below the yield stress, but
at the remaining ligament, t-a, behind the crack it is presumed to induce
flow, as noted by the shaded area in the Fig. 5 plan view. That is to say, a
condition for the analysis is

oAt-a)
< a < oo (13)

where again, due to the nature of the slip field (see the side view in Fig. 5),
12 ELASTIC-PLASTIC FRACTURE

Plan View -shaded area = yielding

FIG. 5—Deep surface flaw.

theflowstress, a/, is just the yield strength in simple tension,CTO.For a deep

flaw this is not an unrealistic restriction.
Now the front view of Fig. 5 may be regarded as an elastic through-the-
thickness crack with an average (through-the-thickness averaged) closure
stress, a', assisting in holding back the opening displacement, 6, at the
center, where tearing instability shall be examined. Equilibrium gives

a't = ao(t — a) (14)

From elastic analysis, the opening displacement, 5, is

8= ^(a-a') (15)
E

Combining Eqs 14 and 15 and noticing that for a deep crack {a/t > Vi) the
displacement 6 is a conservative estimate of the crack opening stretch

5, = 6 - -gT I a - ao 1 - - ^ (16)

Again, recalling the relationship of 6, and J or

(17)
 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 13

and examining for instability under a crack extension, da, from the cor-
responding, dJ, by combining Eqs 16 and 17, and differentiating, leads to^
dJ E 21
T= X < (for tearing instability of a deep surface flaw) (18)

Note the similarity of this result, Eq 18, with the previous results, Eqs 10
and 12. A first observation is that

T= X
da ffo^

keeps reappearing as the left-hand member of the equation and may be inter-
preted as characterizing the material's resistance to tearing instability. On
the other hand, the right-hand sides of these instability criteria are depen-
dent on a nondimensional geometrical factor for the configuration plus a
numerical factor which in part depends on the geometrical character of the
flowfieldthat develops (that is, the constraint).
Also very important and curious to note is that none of these instability
criteria, Eqs 10, 12, and 18, contain the crack size, a. This is very unlike
linear-elastic fracture mechanics instability criteria, where for the same
geometrical configurations the crack size is strongly present! This rather
striking difference will be further interpreted later, especially where potential
explanations of fracture triggered by tinyflawsin a necked tensile bar or im-
mediately ahead of a crack or notch are concerned.
The instability criterion for the surface flaw, Eq 18, has been formulated in
a way that is conservative for assuring stability. First, the crack opening
stretch, 5,, was estimated, Eqs 16 and 17, only at the center of the crack
border of the surface flaw; for deepflawsthe estimate was conservative (too
high) when considerations of bending due to the crack were made. Further,
incompatibility of the slip fields at the ends of the crack assures a slight
underestimate of the effective a' in Eqs 14-16. Finally, the applied stress, a,
would be likely to diminish as crack growth occurs or, at most, remain con-
stant. Thus, because of this nature, Eq 18 is put forward as a reasonable
estimate for assuring against tearing instability for fully plastic surface flaws.
On the other hand, the instability criteria formulated for center-cracked
and double-edge cracked strips, Eqs 10 and 12, assumed a rigid loading ap-
paratus (fixed grips), and, unlike the surface flaw, they have a higher pro-
pensity for instability if the loading apparatus is flexible. This effect of the
testing machine stiffness could be included as an additional term on the
right-hand sides of Eqs 10 and 12; see Appendix I.

^An increment dl can simultaneously be considered but does not add significantly to results.
14 ELASTIC-PLASTIC FRACTURE

Tearing Instability in Bending

The 3-point bend specimen or its equivalent, the deeply cracked double-
cantilever beam, shall now be considered; see Fig. 6. The remaining liga-
ment, b, at the crack section is basically subject to pure bending and is also
assumed to be small enough that all plasticity is confined to that region. This
implies b/W < 0.68 for 3-point bending and for the double-cantilever beam.
The limit bending moment. ML , is assumed to occur at the remaining liga-
ment prior to crack extension, and from Green and Hundy [13] it is

ML=PLS = OJSaoBb^ (19)

With crack extension, da (or —db), the limit load, PL, diminishes, differen-
tiating Eq 19, by

0.35aoB
dPL = {-2bda) (20)

Again assuming a rigid loading apparatus during crack extension, simple

beam theory shall be used for analysis of flexing of the elastic ends of the
specimens. The reduction of beam bending due to diminishing load, dPi,
while the load-point displacements are fixed implies a further rotation, dd, to
be absorbed at the plastic sections, where

2dA IdPiS^
dd = (21)
3EI

plasticity

measured to
approximate
"hinge point"

FIG. 6—Three-point bend and deeply cracked double-cantilever specimens.

 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 15

where

1 =
12

Now an increased rotation while at limit load further implies an increment of

/ , that is, dJ, by the usual Rice et al {[14] or [J]) pure bending analysis. While
crack extension, da, implies a decrease

2
dJ = -— Mide - eaoda (22)
Bb

Simply combining Eqs 19-22 leads to

_ ^ _£^ 4b ^S BE (for tearing instability in .« .

da ffo^ ~ W^ ffo 3-point bending)

Now it is interesting here to notice that the remaining ligament size, b, comes
into the instability criterion, Eq 23. Indeed, if instability occurs, h will then
diminish with crack extension, and stability is regained. (The term 6E/ao is a
small adjustment in the effect.) This is often observed in bending type (in-
cluding deeply cracked compact tension) tests. Instability occurs and the
crack runs rapidly but arrests before severing the ligament completely. This
could occur with the tearing mechanism (not cleavage) in the following way.
Referring to Fig. 1, suppose that full plasticity at the ligament develops
while on the initial blunting part of the R-curve. Deformation, that is, in-
creasing / , continues without instability until the beginning of stable tearing
is reached. At that point, the slope dj/da suddenly changes to the value for
"stable tearing," but if Eq 23 now predicts instability, sudden unstable crack
growth should occur by tearing, until b becomes small enough to regain
stability. However, if b were small enough in the first place, the situation
would have remained stable throughout and the specimen would have been
severed by slow stable tearing as loading (deformation) progressed.

Initial Note on Tearing Versus Cleavage Instability and Other Effects

Considerations that unstable tearing, implying enormous strain rates at
the crack tip, might trigger cleavage in rate-sensitive materials are self-
evident. But referring to the preceding paragraph, this possibility is shown to
be ligament size, b, dependent in bending. The implications here are vast
and it is seen especially that many past conclusions about the nature of
cleavage may be in error. Indeed, sudden fracture might often be a result of a
tearing instability, which rapidly changes to cleavage, so that cleavage might
16 ELASTIC-PLASTIC FRACTURE

be an effect, not the cause of the instability. Indeed, cleavage fractures in-
itially bordered by dimpled rupture are often observed [15].
Moreover, up to this point, the tearing portion of the R-curve in Fig. 1 has
been regarded as a straight line. However, many experimentally determined
R-curves appear to be curved concave downward as tearing progresses exten-
sively; see Fig. 7 [16]. This implies that at least in some cases, dJ/da, the
tearing slope, may diminish with crack extension, Aa. If so, then considering
the instability criterion for bending, Eq 23, if dJ/da diminishes faster than

0.0005 0.001
AQ (in.)

FIG. 7—R-curve for 5083 aluminum alloy [16] (a) large scale (b) small scale.
 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 17

b^, continuing instability would ensue in spite of the tendency for bend tests
to regain stability. For such material behavior, advanced tearing instability
considerations are appropriate.^

Approximate Tearing Instability Analysis of the Deeply Cracked Compact

Specimen
In contrast to the previous analysis of 3-point bending, where the elastic
bending of the beam provides the driving mechanism for instability, the
deeply cracked compact tension specimen will now be considered. Indeed, if
the specimen is deeply cracked, all of the significant deformations, both
elastic and plastic, occur at the remaining ligament or very nearby. Referring
to Fig. 8, the load-point displacement. A, is caused by rotation, 6, centered
at about the middle of the remaining ligament or

A = e(W-b/2) (24)

Now the angle change 0 is made up of its elastic, dei, and plastic, 6PL , parts,
or Eq 24 becomes

A = (6EL + epL){W- b/2) (25)

It is again assumed that the test machine and fixtures are rigid, or, examin-
ing for instability by presuming an increment of crack extension da (or
—db), during that increment

dA dA
dA = 0=—-de + --db
de db

or (26)

0 = (dOEL + ddpi) ( ^ - y ) + i^EL + BPL) ( - -y-

Substituting from Eq 25 into Eq 26 and rearranging

Assuming pure bending of the remaining ligament, the elastic deformation is

[17]

^Indeed 5083 aluminum alloy is being used for liquefied natural gas storage tanks in large
ships, where the hazards are enormous, but with minimal consideration of possible crack in-
stabilities.
18 ELASTIC-PLASTIC FRACTURE

16M
7£i (28)
EBb^

which is assumed to be occurring during limit load conditions, or the limit

moment, ML, is (as before)

M = ML = 0.35 ffoi?£>2 (29)

Combining Eqs 28 and 29

„ _ 16(0.35) go
OEL — ::; (30)

which is a constant unaffected by ligament changes, db, or

(31)
Now reexamining Eq 27, noting Eq 31, and that A and {W — A/2) are
positive, but that db is negative, the conclusion is
ddpL — negative (32)

This would imply a reduction in angle of plastic deformation or, conse-

quently, a reduction in/. Indeed this shows that the situation is always stable
when considering tearing instability of a very deeply cracked compact
specimen in a rigid testing machine.

Tearing Instability Analysis of the Notched Round Bar

An instability analysis of the notched (cracked) round bar can proceed in
the same manner for either internal or external notches; see Fig. 9. As

shaded area
assumed rigid
^ \ ^ \ \ \ V ^ > V f < 1(compared to
ligoment region)

\ \ \ \ \ \H-b-
::^ \^ \ \ ^ \ \ v^ V

FIG. 8—Deeply cracked compact specimen.

 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 19

Section

:^ ^ H
—d— UJ
h— D

p ' p

FIG. 9—Notched round bar.

before, a rigid test machine and fixtures are assumed. The analysis proceeds
very much as with Eqs 4-10. Here the change in limit load with respect to
crack size is

dPi = Ofirdda (33)

or the elastic shortening is

dPJ.
dApi = (34)

and, with no total shortening, the plastic lengthening is converted to crack

opening stretch or

/3cf6, = d^PL = —dAsL (35)

As usual the change in 8, implies an increment of J by

dJ = aoddi (36)

Combining and rearranging

_, _ jdJ_ _E_ ^ 4_ / g/\ Ld (for tearing instability of notched ,,_.

~ da ao^ ~ /3 \ ao / D^ round bars)
20 ELASTIC-PLASTIC FRACTURE

For internal notches

0 ao
whereas for external notches
1 Of ^ (^ d 1
- > 6 for — < —

due to the constraint implied at the plastic section for each of these types of
specimens. Of course a less than rigid test machine and fixtures, and bend-
ing and unsymmetrical cracking, all contribute to increasing the possibilities
for instability in the notched round bar. Therefore, it is doubtful that, for ex-
ternal notches, the diminishing of d on the right-hand side of Eq 37 will in
reality restore tearing stability under actual test conditions.

A Note on Instability of Microcraclcs at the Neclted Section of a Tensile Bar

Consider Eq 37 or modifications of it for possible application to instability
of tiny flaws at the necked section of a tensile bar at fracture. For tiny flaws,
it is evident that the flow fields will be virtually undisturbed by the flaws
while extreme deformation of the material takes place, especially at the neck.
Initially, well before necking, the flow stress, o/, is by definition, ao, but as
intense deformation occurs, it is evident in both Eq 33 and Eq 36 that it
becomes the true stress at fracture, Ot fracture. Meanwhile, the tearing
resistance dJ/da might well diminish in the highly deformed material. Both
of these considerations drive Eq 37 toward instability.

Consideration of the Possible Tearing Instability of Microflaws Ahead of a

Crack Tip
In the plane-strain region ahead of a crack tip, many authors have
depicted the slip field; for the most extensive analysis, see Rice \IS\, McClin-
tock \19\, or Kachanov [ii], and, for effects of large deformation and
hardening, see McMeeking \20\. For purposes of simplicity, the elementary
slip field is shown here in Fig. 10. In Fig. 10 the stress conditions near the
crack surface and ahead of the crack are depicted. The flow parameter, K, is
ao/V3 or ao/2 for the Mises or Tresca criteria, so we see that the maximum
normal stress ahead of the crack is about 3ao. McMeeking \20\ suggests that,
with accounting for hardening and blunting, the stress ahead of the crack
can easily reach 4ao.
The situation for a microcrack ahead of the main crack is depicted in
Fig. 11. Taking the view that the main crack'sflowfielddominates the sur-
 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 21

(t+tt)K 'l2+n)K
Stresses
(a) near crack surface (b) ahead of the crock

FIG. 10—Elementary plane-strain slip field near the tip of a crack.

roundingflowfieldleads to the stress, Fig. life, whereas above and below the
microcrack the stresses are relaxed, Fig. Ha, for the shaded diamond-
shaped region. (Note that this model for the relaxation of stresses above and
below the microcrack has some objectionable features, but the discussion is
continued as a model for its dimensional features.) Now computing the
elastic strains over the height, a, for both fields of stress gives

K
€.<"> = [ - 3 M ]

and (38)

e,('')=[(2 + ^ ) - ( l + 2 7 r ) , x ] | -

which implies a reduction from (b) to (a) in elastic strains over the height, a,

K ^0
Stresses
(a) near the microcrack (b) the surrounding
flow field
FIG. 11—Microcrack within the flow field ahead of a crack.
22 ELASTIC-PLASTIC FRACTURE

which would be turned into crack opening stretch, 6„ at the tips of the micro-
crack, or

6, = W^ - €/'")2a = [2 +IT+ 2^- 2irix]^2a

E
For /i = 0.3 and K = ao/2, this gives approximately

6, = ^ (39)

Again taking the usual relationship

6r = — (40)

and combining Eqs 39 and 40 with an increment of crack extension, gives

_ d/ E (for a model of tearing instability

~ da a(? " ofamicroflaw ahead of a crack) ^ '

Again, as with the discussion of the tensile bar, undoubtedly deformation

would increase the flow stress, ao, and reduce the tearing resistance, dj/da,
promoting instability. It is noted that the instability criterion resulting from
this model does not contain the crack size, a, which may be a reasonable ex-
planation for a local tearing instability from microflaws. The conclusions
may be relevant even if the model is not numerically correct but is only di-
mensionally correct.
However, further exploration of this prospect should be pursued with a
more realistic plasticity model such as those of Hutchinson and Shih [21,22],

Discussion
The preceding analysis has presented an approach to tearing instability
criteria which is indeed proper from a basic mechanics viewpoint, hinged
only on the concept that the J-integral R-curve is an appropriate representa-
tion of a material's tearing resistance.* In fact, the R-curve need not be either
configuration or size independent or contain straight-line segments; it need
only be appropriate for the particular situation for which tearing stability is
being examined. However, the R-curve in reality is at least reasonably un-
varying with size, configuration, etc., which adds to the value of this analysis,
since simple laboratory tests for the R-curve characteristic may be used to
predict behavior of other components, etc.
The methods of plasticity, that is, slip field analysis, for displacement rela-

*An alternative crack opening stretch, ST, approach can be used which is equivalent but
not any further enlightening.
 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 23

tionships and limit loads used herein are only approximations but very
reasonable tools from at least a dimensional viewpoint. Therefore, the in-
stability relationships presented are to be regarded as at least dimensionally
correct but numerically approximate.
The weakest assumptions of the analysis are that the plasticity solutions
are formed for constant geometry conditions and then subjected to crack
length changes, that is, differentiated with respect to crack length. This
causes "nonradial loading" of elements near the crack tip and other
discrepancies. However, if these analyses are only reasonable approxima-
tions, increment by increment, as crack length changes are occurring, then
they are dimensionally correct and only numerically approximate. Moreover,
since local crack-tip "error" will occur both in developing the R-curve and in
its companion application, a strong tendency for compensating error effects
will occur within crack-tip fields. This compensation is indeed relied upon in
currently widely accepted elastic R-curve analysis, when considering distur-
bances due to the crack-tip plastic zone in elastic analysis. It is desirable to
be equally optimistic here. Thus if the instability criteria presented herein are
not corrected for large crack length changes, they are at least good approx-
imations for small increments of crack extension and, in any event, are
regarded as dimensionally correct.
The material's resistance to tearing instability is clearly identified here as
the tearing modulus, T, where T = dJ/da X E/aa^ depends only on the slope
of the J-integral R-curve and other well-known properties, theflowstress, ao,
in simple tension, and the elastic modulus, E, Appendix III has, for com-
parative purposes, a brief table of these properties for some rotor steels and a
cast steel. It is noted that some materials have a tearing modulus, T, of
over 100 (nondimensional), implying a very high degree of stability against
tearing mechanisms for all crack configurations. However, other materials
have a tearing modulus, T, below 10, which virtually ensures tearing in-
stability in some configurations (such as double-edge cracked, see Eq 12) as
soon as/ic and limit load are reached. Therefore, tearing instability is a real-
ity to be dealt with in such materials, and the size and geometry effects for
tearing instability of various crack configurations become of practical in-
terest for such materials.
The situation where tearing instability may trigger cleavage in rate-
sensitive materials is a reality more than an important possibility and bears
further study. Past observations have almost always led to conclusions that
cleavage instability was a cause any time fractures were rapid and displayed a
high amount of cleavage on the fracture surface. Clearly different cause,
tearing instability, is possible. Indeed, many perplexing cracking instability
situations, unexplained previously, seem perfectly logical with a little study
of the possibilities for tearing instability. The possibilities seem endless for
reconsidering everything from K^ test behavior observed, the effects on
material property changes (in T) for irradiation-damaged nuclear materials,
24 ELASTIC-PLASTIC FRACTURE

and cutting and machining problems, to cracking under forming and other
large scale plastic flow problems. It would be too much to consider all the
immediate possibilities here.
Continuation of the detailed development of other aspects of tearing in-
stability criteria also seems relevant. In Appendix I an attempt has been
made to generalize the preceding analysis of the tearing instability criteria for
various configurations. The generalization is developed for any situation
where the elastic components of deformation are linear, where limit load oc-
curs, etc. The tearing modulus, T, again reappears as the key material
parameter. It is shown that testing machine stiffness can easily be taken into
account. The effect of changes in geometrical aspects of instability due to
deformation itself appears, in the term in Eq 51 involving J as a measure of
deformation. Further studies should account for such effects as work harden-
ing, elastic nonlinearity, and geometric nonlinearity (large deformation),
which seem a bit out of place in this first discussion of the concept of tearing
instability.
Finally, it is evident that to date no systematic experimental programs
have been performed to explore tearing instability; this is perhaps the first
thing that should be done. Previous J-integral fracture testing provides many
examples of tearing instability (though not interpreted as such at the time)
which should be reexamined. But most of these previous tests were on bend-
ing configurations, that is, those of the most natural stability; thus the more
unstable types of test configurations should be employed, and a wide variety
(extremes) of a material's tearing moduli, T, should be included. Again, the
possibilities seem endless, and it is evident that judiciously chosen critical ex-
perimentation is needed.

Acknowledgments
This work was made possible through a contract from the U.S. Nuclear
Regulatory Commission (NRC) with Brown University, Providence, R.I.,
during the summer of 1976 and a later contract from NRC with Washington
University (St. Louis, Mo.). In addition to the financial support, the con-
tinued encouragement of the regulatory staff, and especially of Messrs.
W. Hazelton and R. Gamble, is gratefully acknowledged.
The special efforts and encouragement of Professor J. R. Rice (upon whose
work and methods [4,7,18] much of this current work is based) during the
first author's visit to Brown University (1974-1976) are due special acknowl-
edgment and thanks. The continued assistance of this work by Professor
Rice and Professor J. W. Hutchinson of Harvard University, as consultants
to the current NRC contract at Washington University, is also gratefully
acknowledged.
 PAIRS ET AL ON INStABILITY OF THE TEARING MODE 25

APPENDIX I
A General Analysis of Teiuing Instability, Including the Effect of Testing Machine
Stiffness for the FuUy Plastic Case
The preceding descriptions of tearing instability criteria for various crack con-
figurations are all based on relaxation of "global" contributions to reduction in elastic
displacements which causes increases in plastic displacements which drive the crack
ahead. (The only exception is the analysis of the microcrack in the flow field ahead of
a large crack.) For a general analysis, consider the arbitrary configuration shown in
Fig. 12.
Now, during an examination of stability, the displacement of the loading train re-
mains constant; thus

Ap = AEL + ApL -H AM = constant (42)

where AEL and APL are the elastic and plastic components of the specimen
displacements and AM is the testing machine displacement. During crack extension
then

CIAEL + dApL + dAM — 0 (43)

Elastic displacements are (normally) linearly proportional to load and have the form:
P /a B L
— , — . — , etc. (44)
AEL- BE ^AW' W' W
where/( ) is a nondimensional function of specimen dimensions. The plastic displace-
ment, ApL, presuming we are at limit load for the cracked section, will have a linear
relationship to the crack opening stretch, S,, or

ApL^b,Xg[^, | . | . etc. (45)

(displacement control)

elastic

plastic

W, B, L, etc. = ottier
characteristic dimensions
(test mochine (constants)
stiffness)

FIG. 12—An arbitrary configuration.

26 ELASTIC-PLASTIC FRACTURE

and finally the testing machine will be regarded as a linear spring or

AM - -^ (46)
AM

where KM is the spring constant. The load, P, is assumed to be at limit load, PL,
which will depend linearly on the flow stress, ao, and thickness, B; therefore

P = P, = ooBWh(-^, ^- j ^ . etc.] (47)

and the usual relationship between 7 and 6, is assumed with a constant, a, adjustable
approximately to suit the configuration, or

6, = a— (48)
ao
Now during an increment of crack extension, da, the variables in the foregoing expres-
sions which change are, a, P, 8,, and / . These cause changes in the displacements;
therefore, writing the differentials of Eqs 44-48 gives

dP P df{ )
dAEi = - ^ X / ( ) + - ^ ^ ' ^ ^ (44fl)
BE BE da
de( )
dApL = d6,g( ) + 5,——da (45a)
da

dAM = - ^ (46«)

dP = dPi = aoBW^^da (47a)

da
and

d8, = a— (48a)
ao
Now substituting Eqs 47, 48, 47a, and 48a in the right-hand sides of Eqs 44a, 45a, and
46a, so that the remaining differentials there are da and dJ, gives

aoW d
dAEL = —r-[h( ) X / ( )]da (44A)
E da
dJ aJ dg( )
dApL = a—Xg( )+ •— da (45b)
ao Of da
aoBW dh{ )
dAM = — 7—da (46i)
KM da
Finally, substituting Eqs 44b-46b into Eq 43 and rearranging gives
 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 27

"" "• ' ' ' w ) x / ( , , + ^^*"

da ffo^ ag( ) i^da KM da
(49)
otJE dg{ )
+ Wao' da
But Eq 49 can be further simplified by noting that (with signs chosen for usual
behavior)

W dh{) ^ (± A A
g( ) da ^\W' W"W
and
W dgO (a B L
* = r —. —. —. etc. (50)
g() da \W W W
Therefore, substituting Eq 50 into Eq 49, the simplification leads to

dJ E \ \ , .,EB , . JE
da atf, ^ 3aJ C
p ( ) + KM
— 9 ( ) - - S 7VKao'
7 T ' - ( )( (51)

as the general form for the tearing instability criterion for any specimen under fully
plastic conditions.
All of the tearing instability criteria given previously herein fit this general form,
Eq 51. In the previous criteria a rigid testing machine was assumed, that is. KM — °°,
so the term with q( ) did not enter. Moreover, considering Eqs 45 and 50 for most
configurations, g( ) did not contain crack size; thus r( ) was zero. (The exception is
the deeply cracked compact specimen, where h( ) X / ( ), which did not contain
crack size; thus p{ ) was zero, and r( ) turned out positive, thus assuring an always
stable situation).
Moreover, even though the microcrack within the fiow field ahead of the crack does
not fit the physical model here as depicted by Fig. 12, the criterion which resulted still
fits the form of Eq 51.
Finally, all of these tearing instability criteria have T = (dJ/da) X (,E/ao^) as a key
material parameter in resisting tearing instability. Also note that the term with K )
sometimes enters as a partly geometrical term assisting resistance to tearing insta-
bility.

APPENDIX II
Tearing Instability of the General Small-Scale Yielding Case^
All of the preceding discussion addresses tearing instability which occurs following
development of full plasticity on the remaining ligament at the cracked section. Here a

^Attention to this topic was drawn at the suggestion of James R. Rice of Brown University.
28 ELASTIC-PLASTIC FRACTURE

general analysis of tearing instability is done where the remaining ligament is

predominately elastic; that is to say, only small-scale yielding at the crack tip is pres-
ent. The effects of elasticity of the loading apparatus, for example, testing machine
stiffness, are also considered.
The analysis begins by following the notation and arrangement as shown in Figure
13. For an arbitrary configuration, such as Fig. 13, the form of the crack-tip stress in-
tensity factor, K, is always

(52)

Assuming small-scale yielding or / = Q, then"

J = Q = — = (53)
W,

Taking differentials associated with a crack length change, da, results in

2Pa „ , / a \ .„ , P^
dJ ^ H [-] da (54)
EB^W^ W/ ^^ "^ EB'^W^

where'

H Y^ + 2
W W ^ \W ^'
Now during the increment of crack extension, the total displacement, Ap, will be
constant. Again referring to Fig. 13, this may be written

Ap = '^specimen + AM = COHStant

(displacement control)

small scale yielding at

the crock tip

W, B, L and ottier
ctiaracterlstic dimensions

K|^(te$ting machine stiffness)

FIG. 13—An arbitrary configuration.

*The function Y(a/W, L/W, B/W, etc.), which is a nondimensionat function of configura-
tion dimensions, will be denoted Y(a/W) for simplicity in the analysis to follow. Similar
simplifications will be taken here for other functions, such as/( ).
 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 29

or again writing differentials

d^p = d^,p + dAM == 0 (55)

For an elastic small-scale yielding situation, the specimen displacement Ajp has the
form

A. = : ^ / (^) (56)

and the testing machine behavior is expressed by

AM = / - (57)

Differentiating and substituting Eqs 56 and 57 into Eq 55 gives

dP (a\ P (a\ dP

or rearranging Eq 58 gives

P^ \W,

dP = ^~— da (59)

Equation 59 gives the load change, dP, associated with a crack length change, da,
when examining for instability. It may therefore be substituted in Eq 54, which
results in

ri-
P2 W
E2 B^W^ a JaY
\W) \W,
la_
^\w) P^ la'
<^ = ^7-^ '^'^ ^i^w^^Kw)''' (^°>

Now let

/a\ a JaY \W
W W \WJ J a
30 ELASTIC-PLASTIC FRACTURE

and recall

H
W. ^ w^\w) ^ \w.
where Y(a/W) a n d / ( a / W ) are configuration factors in the K and Asp formulas.
Finally, Eq 60 can be rearranged as the instability criterion,

2G
^ dJ E
+H (61)
da ao EB
1+
wu^
where in addition it is noted that:

BW

In Eq 61 it is noted that the H(a/W) and the stress (a/ao)^ combine to drive the
situation toward tearing instability and that H(a/W) depends only on the form for
the stress-intensity factor solution, Eq 52. The other term with G{a/W) involves the
relative stiffness of the testing machine (or loading arrangement); note that if KM =
0, this term is zero, and that as loading arrangement stiffness. KM, is added, the
negative sign implies adding to the stability of the situation. It is easy to apply the
criterion.

Plastic Zone Correction

The instability criteria for small-scale yielding may be reformulated using the usual
plastic zone correction in the analysis. Adding the plastic zone correction to the crack
length gives

(62)
where
JE
-(-)"
Differentiating Eq 62 with respect to crack size gives

da eft - da (63)
^aa ao / yir

For correcting the instability criterion, a and da in Eq 60 should be replaced by a^

and da ^ff from Eqs 62 and 63, which gives

T =
dJ E
da ao^
^ni (64)

yic ' \ao.

 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 31

where

a eft
- 2G
W "_eff
{ } = + H
EB
I +
K.fi^
However, the second term in the denominator of the right-hand side of Eq 64 is nor-
mally small compared to one, so that within it a may be used instead of a ^ff with no ap-
preciable error. Nevertheless, a^f( would be retained in the numerator. Thus Eq 64
represents the plastic zone-corrected instability criterion in the usual linear elastic
fracture mechanics spirit.

Dugdale Strip- Yield Zone Correction

Although the Dugdale strip-yield zone methodology is most appropriate for "plane
stress" conditions, the tearing instability criterion can be appropriately formulated
for the mixed mode and here may be restricted to plane stress if necessary. The
analysis proceeds as follows.
The form of crack opening stretch, 6r, solutions from strip-yield models is

. _ ooa la a B L
(65)

As in earlier analysis

ao
Differentiating and rearranging gives

da ao^ ~ a \ao' W W W ^ '^'

(66)
A- - — f I — — — II
a da-' Vo' W W W^*"'
From here, further details follow in the manner of Eq 54 and subsequent develop-
ment. But the form is already clear with Eq 66 and is not developed further herein.
32 ELASTIC-PLASTIC FRACTURE

X ^ OOOOOOlOOO^OT^•^a^CT^O^^^O^rO'^'«lnO
•a -a -L

^^ £ :xxxxxxxxxxxxxxxxxxxxxx

. O O r ^ r o o ^ O u o — -,w,^.
S S '. -^ IT)
:K V ^ ^ <

a;^^U^^^OoC>000^^0^/)OOOOOO^n^O(N^/>

h I u 5 u^ i/> u^ O <
I ^ ( S <N PO t
' I

H H
O
o

I,
3

B 1 •S

aQ 1
Z
i
<i;
>
M *« ii?5i 6
'I- a A ^ ij"?
^ ^ < «i ^ < « i
-< <^« 2
•s gc S
^ < < <
H
<
 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 33

^H -CH l7^ ^H lO ^H OS

^'b-fc-b'bij^'Sj-b ^_ O
O O O O O ^ O
oO ,'h'h'h'h 'aO- -v ^ fM
O O O O

X X X X X X X X X X X X X X X X X xxxxxxxxxx
^CO^^^r^Ir^(N*^a^^^ <*)<N<N-H^^rM-H ^'«r-Hi/>ir)fs^^^r^

O ^ ro ro m <N < ' <N O O lO "^ ro ; o o 00 m I

\0 ^ fO f*l i-< O^ < • in **5 <N (N a> ^
00 ^ Tf fO 00 *« > I ^ f- 1^ r- -H <N

a ^ o » n o o o o o o o o « o s ' ^ o o o r ^ o r o o o
ro ^ i/i p*^ f s r^ t^' t^' o

' O I/) O Q O <N t/)

' l o r^ i/i o I/) i/> t^
' (N r^ lO TT
(NU^(N(NrSfSrsIfN(N(N
• » • » • < » • • < » • • * • * • > » • • < » • • < > •
I
I I I I I 1 1 I I

H H
(J
H H H H
U O OU

<
2, ^OS a
<
>
5 H H > t<n • f:

— c
^ Q Q < > acs

a
c

lily
-2 s *^ Bi
s_ 2 "<
< 3 « jj < ji 2
<s S " .S "s .S 1"
if-' ^.1 an
< <
34 ELASTIC-PLASTIC FRACTURE

la

• in 00 (T> o o ' ^
•<3 p s -L

-&-b-&-i Is-&•£.•& ts ^ ^ ' i 1=

x x x x x x x x x x x x x

n'-HfN^^<Ni^r'^i-ii-H,-ii/)r*i

f N 0 Q O i O i 0 . j 3 ^ * N O O r - ^ ( S i
TrrO<N^t-^fv}'<1"00iO^r^'-HQ5

r- vo m »-* OS "^' o^ lO '-H r^ \o ^ I--*

(N <NI (N (N (N (N I/) <S ( S

i/J l O l O t o in iTi t-- in in
TT Tf T T ^
I I I I
5 I I I I

1^ u O

<

r Qi

SGfe£Gooo£>

»i o< •<> f-^

S <5: • 5 « H < fe a

Im
«

U 1/5
<
" H "^

I
 PARIS ET AL ON INSTABILITY OF THE TEARING MODE 35

References (for Appendix III only)

[/] Logsdon, W. A. and Begley, J. A., Engineering Fracture Mechanics, Vol. 9, 1977, pp.
461-470.
[2] Logsdon, W. A. in Mechanics of Crack Growth, ASTM STP 590, American Society for
Testing and Materials, 1976, pp. 43-61.
[3] Begley, i. A., Logsdon, W. A., and Landes, J. D. in Flaw Growth and Fracture, ASTM STP
631. American Society for Testing and Materials, 1977, pp. 112-120.
[4] Wells, J. M., Logsdon, W. A., Kossowsky, R., and Daniel, M. R., "Structural Materials for
Cryogenic Applications," Research Report 76-9D9-CRYMT-R1, Westinghouse Research
Laboratories, Pittsburgh, Pa., 1976.
[5] Logsdon, W. K., Advances in Cryogenic Engineering, Vol. 22, 1977, pp. 47-58.
[6] Logsdon, W. A., Wells, J. M., and Kossowsky, R. in Proceedings, Second International
Conference on Mechanical Behavior of Materials, Boston, Mass., Aug. 1976, pp.
1283-1289.
[7] Clarke, G. A., Andrews, W. R., Schmidt, D. W., and Paris, P. C. in Mechanics of Crack
Growth. ASTM STP 590, American Society for Testing and Materials, 1976, pp. 27-43.

References
[/] Begley, J. A. and Landes, J. D. in Fracture Analysis. ASTM STP 560, American Society
for Testing and Materials, 1974, pp. 170-186.
[2] Clarke, G. A., Andrews, W. R., Schmidt, D. W., and Paris, P. C. in Mechanics of Crack
Growth, ASTM STP 590, American Society for Testing and Materials, 1976, pp. 27-42.
[3] Paris, P. C. in Flaw Growth and Fracture, ASTM STP 631, American Society for Testing
and Materials, 1976, pp. 3-27.
[4] Rice, J. R., "Elastic Plastic Fracture Mechanics," The Mechanics of Fracture, F. Er-
dogan, Ed., American Society of Mechanical Engineers, 1976.
[5] Fracture Toughness Evaluation by R-Curve Methods, ASTM STP 527 (papers on linear-
elastic R-curve analysis), American Society for Testing and Materials, 1974.
[6] Paris, P. C. and Clarke, G. A., "Observations of Variation in Fracture Characteristics with
Temperature Using a J-Integral Approach," submitted to the Symposium on Elastic-
Plastic Fracture, Atlanta, Ga., American Society for Testing and Materials, 1977.
[7] Rice, J. R., "Elastic-Plastic Models for Stable Crack Growth," Mechanics and
Mechanisms of Crack Growth, British Steel Corp., 1973.
[8] Ritchie, F. O., Knott, J. F., and Rice, J. R., Journal of the Mechanics and Physics of
Solids, Vol. 21, 1973, pp. 395-410.
[9] Private communication on fracture test results, Westinghouse Research Laboratories,
Fracture Mechanics Group, E. T. Wessel, Manager, Pittsburgh, Pa., 1976-1977.
[10] Begley, J. A. and Landes, J. D., InternationalJournal of Fracture Mechanics, Vol. 12,
No. 5, Oct. 1976, pp. 764-766.
[//] Kachanov, L. M. in Foundations of the Theory of Plasticity, H. A. Lauwener and W. M.
Koiter, Eds., North Holland Publishing Co., 1971.
[12] Irwin, G. R., private communication on ligament flow (6), 1969.
[13] Green, A. P. and Hundy, B. B., Journal of the Mechanics and Physics of Solids, Vol. 4,
1956, pp. 128-144.
[14] Rice, I. R., Paris, P. C , and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing, ASTM STP 536, American Society for Testing and Materials, 1973,
pp. 231-245.
[15] Private communication of observations by E. T. Wessel, Manager, the Fracture Mechanics
Group, Westinghouse Research Laboratories, Pittsburgh, Pa., 1976-1977.
[16] Argy, G., Paris, P. C , and Shaw, F. in Properties of Materials for Liquefied Natural
Gas Tankage. ASTM STP 579, American Society for Testing and Materials, 1975, pp.
96-137.
[17] Tada, H., Paris, P. C , and Irwin, G. R., The Stress Analysis of Cracks Handbook, Del
Research Corp., 226 Woodboume Dr., St. Louis, Mo., 1973.
36 ELASTIC-PLASTIC FRACTURE

[18] Rice, J. R., "Mathematical Aspects of Fracture," Fracture, Academic Press, New York,
VoL 2, 1968.
[19] McClintock, F. A., "Plasticity Aspects of Fracture," Fracture, Academic Press, New York,
Vol. 3, 1968.
[20] McMeeking, R. M., "Large Plastic Deformation and Initiation of Fracture at the Tip of a
Crack in Plane Strain," Brown University report. Providence, R. L, Dec. 1976.
[21] Shih, C. F. in Mechanics of Crack Growth, ASTM STP 590, American Society for Testing
and Materials, 1976, pp. 3-26.
[22] Shih, C. F. and Hutchinson, J. W., "Fully Plastic Solutions and Large-Scale Yielding
Estimates for Plane Stress Crack Problems," Harvard University, Rep-deap.-S-14, Cam-
bridge, Mass., July 197S.
[23] Goldman, N. L. and Hutchinson, J. W., InternationalJournal of Solids and Structures,
Vol. 11, 1975, pp. 575-591.
/. W. Hutchinson^ and P. C. Paris^

Stability Analysis of J-Controlled

Crack Growth

REFERENCE: Hutchinson, I. W. and Paris, P. C, "Stobillty Analysis of/-Controlled

Crack Growth," Elastic-Plastic Fracture, ASTM STP 668, J. D. Landes, J. A. Begley,
and G. A. Clarke, Eds., American Society for Testing and Materials, 1979, pp. 37-64.

ABSTRACT; The theoretical basis for use of the J-integral in crack growth analysis is
discussed and conditions for/-controlled growth are obtained. Calculations related to the
stability of crack growth are carried out for several deeply cracked specimen configura-
tions. Relatively simple formulas are obtained which, in certain cases, permit an assess-
ment of stability using data from a single load-displacement record. Numerical results
for a bend specimen and for a center-cracked specimen illustrate the influence of strain-
hardening and system compliance on stability.

KEY WORDS: crack propagation, fracture (materials), plastic deformation, stable

crack growth

This paper builds upon the report of Paris et al [/]' which promulgates an
approach to the stability analysis of crack growth based on the concept of a
J-integral resistance curve. We start by presenting a theoretical justification
for use of the J-integral of the deformation theory of plasticity in the analysis
of crack growth. Restrictions on such use are discussed in detail with par-
ticular emphasis on application in the large-scale yielding range.
When applicable, the approach of Ref 1 and the present paper is the
natural extension of Irwin's resistance curve analysis (for example, see Ref 2)
for small-scale yielding based on the elastic stress intensity factor K. In a
sense this approach is less fundamental, and less ambitious, than studies
based on flow (that is, incremental) theories of plasticity which attempt to
identify and calculate a single near-tip parameter governing the initiation
and continuation of crack growth. Studies along such lines [3,4] have at-
tempted to discuss the source of stable crack growth but they have not
cleared the way for much progress in its analysis. In part, this is because
there is not yet agreement on a suitable near-tip growth criterion; but it is
' Professor of applied mechanics, Harvard University, Cambridge, Mass. 02138.
^Professor of mechanics, Washington University, St. Louis, Mo. 63130.
^The italic numbers in brackets refer to the list of references appended to this paper.

37

Copyright 1979 b y AS FM International www.astm.org

38 ELASTIC-PLASTIC FRACTURE

also due to the difficulties of carrying out crack growth calculations using a
flow theory of plasticity. Deformation theory has distinct computational ad-
vantages leading in some instances to closed-form solutions which would
otherwise be unobtainable. Illustrations of this point will be found within the
present paper. Moreover, the conditions for /-controlled growth are derived
to ensure essentially identical results from deformation theory and flow
theory on the topics herein.
Following the discussion of the applicability of / to analyzing crack
growth, we discuss the stability of crack growth with emphasis on the role of
the compliance of the entire system under given prescribed loading condi-
tions. An analysis of a deeply cracked bend specimen is carried out. Rela-
tively simple formulas are obtained for assessing stability. Numerical results
for a bending specimen in plane-stress conditions are presented to illustrate
the influence of strain hardening and compliance on stability. A possible
scheme for measuring the resistance curve experimentally is mentioned. The
final sections of the paper deal with the analyses of deeply cracked edge and
center-notched specimens.

Applicability of J-Integral to Analysis of Crack Growth

Crack growth invariably involves some elastic unloading and distinctly
nonproportional plastic deformation near the crack tip. The J-integral [5] is
based on the deformation theory of plasticity which inadequately models
both of these aspects of plastic behavior. At just a glance it would appear that
use of/ must be restricted to the analysis of stationary cracks. In the follov/-
ing, a rationale is given for use of / to analyze crack growth and stability
under conditions which will be called J-controlled growth. The argument
relies on the fact that many metals sustain only very small amounts of crack
growth relative to other dimensions for overall deformations well beyond in-
itiation of growth. If/-controlled growth is to exist, it is essential (and by im-
plication sufficient) that nearly proportional plastic deformation occurs
everywhere but in a small neighborhood of the crack tip. This results from
the fact that when nearly proportional deformation occurs, the differences
between a deformation theory of plasticity and the corresponding flow (in-
cremental) theory become essentially negligible [6], and both theories are
surely physically realistic.
In small-scale yielding it is widely accepted that either the elastic stress in-
tensity factor K or J (J = /iC^/modulus) can be used in a resistance curve
analysis of stable crack growth, at least for amounts of growth which are
small compared to all other relevant geometric lengths. Under limiting con-
ditions of plane stress or plane strain the increase in crack length Aa has a
unique functional relationship to A" or / which is otherwise configuration-
independent. In small-scale yielding the existence of such a relationship rests
on the fact that K or J do uniquely measure the intensity of the fields sur-
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 39

rounding the immediate vicinity of the crack tip. In this paper we shall be
concerned mainly with growth under large-scale yielding, including fully
plastic situations, where extra conditions must be met for 7 to be meaningful
and for the relation of Aa to / to be configuration-independent. When these
conditions are met, a /-resistance curve analysis is the appropriate
generalization of small-scale yielding resistance cur\'e analysis.
Figure 1 displays a schematic sketch of / versus Aa for a typical
intermediate-strength steel under nominally plane-strain conditions as ob-
tained by large-scale yielding testing techniques [7], Leaving aside for the
moment the issue of the validity of experimentally measured /-values used to
generate such data, the main feature of importance to our argument in the
following is the relatively large increase in/above the initiation value/ic (that
is, an increase which can be as much as several times/u) needed to produce
an increase in crack length of, say, only several millimetres. Emphasis in this
discussion is on this range of small growth with its attendant large increases
in /. We look for conditions under which it can be expected that the domi-
nant singularity crack-tip fields of deformation theory, with amplitude / ,
continue to be relevant in the presence of small amounts of growth as
depicted in Fig. 2.
Consider a material with a strain hardening index n such that the plastic
strain is proportional to the stress to the «th power well into the plastic
range. The strain field of the dominant singularity at the crack tip according
to deformation theory is [8,9]

ey = A;„/''/(" + 'V-"/(''+"6-^(0) (1)

where r, 9 = planar-polar coordinates centered at the tip. The 0 variation,

iij, depends on n and on whether plane strain or plane stress pertains; k„ is a
dimensional constant not needed in the present discussion. Let R denote the
characteristic radius of the region controlled by Eq 1 in the deformation
theory solution. In small-scale yielding, R will be some fraction of the plastic

Aa

FIG. 1—Material ^-resistance curve for small amounts of crack growth.

40 ELASTIC-PLASTIC FRACTURE

NEARLY PROPORTIONAL
LOADING CONTROLLED
BY DEFORMATION
THEORY SINGULARITY
NON-PROPORTIONAL
PLASTIC LOADING FIELDS

ELASTIC ^ /
UNLOADING

FIG. 2—Schematic of crack tip conditions for J-controlled crack growth.

zone size/ while in a fully yielded specimen R will be some fraction of the un-
cracked ligament. Since the wake of elastic unloading and the region of
distinctly nonproportional plastic loading will be of the order Aa in length,
one condition for/-controlled growth is

Aa <s: R (2)

Next we examine the strain increments determined from deformation

theory under a simultaneous increase in / and crack length. The crack lies
along the x-axis and is assumed to advance by an amount da in the
jc-direction. For deformation theory, the strain field, Eq 1, continues to hold
in the presence of growth with r and d centered at the current tip location.
From Eq 1, the strain increments are

deij — ;t^-i/(„+i) dJr-""^'' + ^>eij{d)

n + 1
(3)
A:„/"^("+" da ^ [r-"/<" + '>€(,(«)]

Using

d - d sin ^ a
T- = cos ^ -7 TT
dx dr r dd

^Adjusting Eq 1 to represent alternatively the elastic field, R would be substantially larger

than the plastic zone size for small-scale yielding.
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 41

Eq 3 becomes

dtij = k Jn/(n-lr\)j.-n/(n + l) " *'^ r.. + "*"

^« (4)

where
n (f\\ — -1 1 1^ - 1 1111 rt -
Pij\^) *- ^ _|_ j^ COS e'€y -h sin ti ^„ e,>

The first term in the braces in Eq 4 corresponds to a proportional loading in-

crement (that is, dtij - ea), while the second term is not proportional. Since
€# and 0ij are of comparable magnitude, the first term in the braces will over-
whelm the second term if

^ ^ f (5)
r J
That is, predominantly proportional loading will occur in the dominant
singularity region where Eq 5 holds.
Define a material-based length quantity D as

D- daJ ^^^

where for records such as those in Fig. 1, just beyond initiation, D can be in-
terpreted approximately as the crack advance associated with a doubling of 7
above Ju • Equation 5 can be restated as

D <s^ r (7)
If
D <s: R (8)

then there exists an annular region

D <t: r < R (9)

in which plastic loading is predominantly proportional and the singularity
field, Eqs 1 or 4, is dominant. In other words, if Eq 8 is satisfied, it can be ex-
pected that there will be little difference between the strain fields predicted
byflowtheory and deformation theory for r » D and, most importantly, /
uniquely measures, or physically controls, the fields in the Eq 9 region sur-
rounding the tip.
42 ELASTIC-PLASTIC FRACTURE

To summarize, we note that the foregoing argument is somewhat

analogous to that made for the relevance of the elastic analysis for K in the
presence of small-scale yielding. Here, however, the argument has dealt with
two factors: (1) small-scale growth requiring Eq 2 and (2) applicability of
deformation theory and / requiring Eq 8. The foregoing argument relies on
the existence of some strain hardening since the ^-variation, eij, of the strain
field of the dominant singularity is not unique for an ideally plastic material
but will, in general, depend on the overall geometry. In effect, R diminishes
to zero with vanishing strain hardening. Tliis same point is at issue in the use
of / in analyzing initiation [10] and has been addressed experimentally by
Landes and Begley [11] using inherently different specimen types. In making
the foregoing argument, we have also tacitly assumed that, if predominantly
proportional loading occurs throughout most of the dominant singularity
region, it will occur outside (that is, r > R) this region as well. This can be
expected for the same reasons which apply to the configuration with a sta-
tionary crack under monotonic loading.
For a specimen or configuration which has fully yielded, R will be some
fraction of the relevant uncracked ligament, b (or other characteristic
distance from the crack tip to a boundary or loading point if smaller than b).
Introduce a nondimensional parameter as defined by

Thus, for a fully yielded specimen, the condition for/-controlled growth, Eq

8, can be restated
w» 1 (11)
together with Eq 2.
This Eq 11 requirement for proportionality of strain in the singularity field
is different than an earlier J-integral test specimen size requirement ([7] and
discussion to Ref / / ) . The earlier requirement may be stated as

^ » 1 (12)

where oo isflowstress. This requirement, Eq 12, may be interpreted as keep-

ing the crack opening displacement small compared to the ligament dimen-
sion, b, and is regarded as applying both to the initiation and growth phases
of cracking. Indeed, perhaps both requirements must be met to maintain a
proper singularity field during crack growth; however, the later one, Eq 12,
will not be discussed further here.
One consequence of/-controlled growth, which follows immediately from
the foregoing arguments, is that / will be approximately independent of the
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 43

integration path when calculated using the standard line integral definition
in conjunction with a flow (incremental) theory of plasticity as long as the
points on the path satisfy (7). Shih et al [12] have found less than 5 percent
variation in/over a wide range of paths for as much as a 5 percent increase in
crack length in their finite-element analysis of a fully yielded, plane-strain
compact tension specimen of A533-B steel. Using their values for b,Jic, and
the initial slope dJ/da following initiation, we find that w = 40 for their
specimen. Shih et al [12] also found that their computed values of/were in
good agreement with /-values obtained using the experimentally measured
load-deflection curve and a deformation theory formula for a deeply cracked
specimen. These two facts strongly suggest that /-controlled growth is in
evidence in their specimens. An important question which remains to be
answered is, What is the smallest value of w which will guarantee/-controlled
growth? To a certain extent the answer will depend on specimen configura-
tion and on strain hardening. Closely related is the need for a systematic
study of the amount of growth allowable under/-controlled conditions.
An additional consequence of /-controlled growth is that the material
resistance curve of/versus Aa obtained under large-scale yielding conditions
must coincide with that obtained under small-scale yielding conditions,
assuming that the same plane-strain or plane-stress conditions prevail in
both instances. [As discussed earlier, a relation of/ (or K) versus Aa in
small-scale yielding can be meaningful independent of the condition
ofEqS.]

Stability of/-Controlled Growth

Elaborating further in the development of Paris et al [1], we derive some
general expressions related to growth and stability based on a /-resistance
curve analysis.
Consider a two-dimensional specimen (in plane strain or plane stress) with
a straight crack of length a and of a material characterized by deformation
theory. As depicted in Fig. 3, let P be the generalized load acting on the
specimen and let A be the load point displacement of the specimen through
which P works. The specimen is loaded in series with a linear spring with
compliance CM (which can, if desired, be identified with the testing machine
compliance) such that the total load point displacement is

Ar = CMP + A (13)

L e t / have the usual deformation theory definition [5], equivalently as the

path-independent line integral, or as

J= [^ dP= - \ \rr: dA (14)

44 ELASTIC-PUVSTIC FRACTURE

p. A T

CM

P. A

nrh
FIG. 3—Typical specimen geometry.

It will be important to draw a distinction between applied values of/ and the
values of/ on the material resistance curve such as that in Fig. 1. For this
purpose, values of/falling on the material resistance curve will be denoted
by /mat and will be regarded as a function solely of the increase in crack
length Aa. For given material properties and overall specimen geometry, the
"applied" / in Eq 14 can be regarded as a function of P and current crack
length a = ao + Aa, where ao is the initial crack length. At any P and cur-
rent length, a, equilibrium based on the resistance curve data requires

J{a,P) = /„.,(Aa) (15)

Stability will be considered with the total load point displacement AT held
fixed. (Then, note that CM = 0 corresponds to a rigid test machine while
CM = 00 corresponds to a dead-load machine.) Stability of the equilibrium
state, Eq 15, will be ensured if

f) < dJda (16)

The Eq 15 state is assumed to be unstable if

dJ„ (17)
da/AT da
and the onset of instability is associated with equality in Eq 16 or 17. Follow-
ing Ref / we introduce nondimensional quantities

(18)
da

where E is Young's modulus and ao is an appropriate flow stress. In terms of

these quantities, Eqs 16 and 17 become
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 45

T < r „ . , (stability) (19)

T > r „ „ (instability) (20)

A general expression for {dJ/da)^^, to be used in later sections, will now

be derived by regarding / and A as functions of a and P. For arbitrary da
and dP

dJ =
i-a - - iu - (21)

But with AT held fixed, from Eq 13

' dA\
dAr = CM dP + , I da + h f 5 dP = 0

and thus

'dA\ dA_
dP = - da CM + (22)
dPjp dP

Combining Eqs 21 and 22 gives

(23)
[daj^r \dc)p \9p)a \da)p r" "^ \dp).

If CM is identified with the compliance of the testing machine, then

specialization to the two limiting cases noted in the foregoing can be made.
With CM = 0, Eq 23 applies to a rigid test machine. If CM — oo, the test
machine applies a dead load and Eq 23 reduces to

(24)
da /Aj- da IP

Analysis of Deeply Cracked Bend Specimen

In the spirit of Rice, Paris, and Merkle \13\, a formula will be derived for
idJ/da)AT for a deeply cracked bend specimen which, up to initiation, in-
volves quantities that can be taken from a single experimental test record.
First, however, we derive a more general result for / than that given in Ref
13, which allows for determination of/under changing crack length. In the
configuration of Fig. 4a, M is the applied moment per unit thickness, a is
the current crack length, and OQ is the initial crack length. The specimen
46 ELASTIC-PLASTIC FRACTURE

• M.ST

I CM

M,e
1^
/P. A
= pw-«- •
a b CM P,AT

w
(a) (b)
77777777'
M
FIG. 4—Bend specimen (a) ami three-point bend specimen (b).

load-point rotation 6 at any given M can be decomposed into two parts

according to

e = + {-25)

where ^„c, by definition, is the rotation of the uncracked specimen under the
same M, and 6c is the remainder (that is, the contribution due to the
presence of the crack). It is assumed that the current ligament length b = W
— a is sufficiently small compared with W such that 6c at a given M de-
pends only on b and not onZ or W. From dimensional considerations it then
follows that 6c must be a function of the combination M/b^, that is

6c = fiM/b^) or M = b^F(6c) (26)

In Ref 13 it is noted that Eq 26 implies

d6\ 'd6c\ ^ 2M[ ddc]

(27)
da /M da/M b XdMja

Using Eq 27 in the definition of/ from the first form of Eq 14, that is
(•M
d6\
J = dM (28)
da J A

gives

-I Md6c = 2b F(6c) d6, (29)

However, it is noted here that using the second form of Eq 14 with the second
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 47

form of Eq 26 leads to this same result (Eq 29) more directly. The definition
of/ in Eq 28 and the derivation leading to Eq 29 require a to be heldfixedin
the integration. Nevertheless, by virtue of its deformation theory definition,
J (a, 0 c) is independent of the history, giving rise to the current values of a
and dc- (Equivalently, / can be regarded as a function of a and M, but for
present purposes $c is a more convenient independent variable.) For ar-
bitrary increments in a and Oc it follows from Eq 29 that
rOc
dJ = IbddcFiec) - Ida F(e) dOc
(30)
-yddc--^ da

Since dJ is a perfect differential, the general expression for / from Eq 30,

that is

J = 2 y ^ dSc - \ ^da (31)

holds for any history of a and dc leading to the current values and is neces-
sarily independent of the history. With no change in crack length, Eq 31
reduces to Eq 29. In the presence of growth, the second term represents a
correction to Eq 29 which should be but is not currently used to determine/
from experimental data [7,11], For small amounts of growth, this correction
is usually small but not necessarily negligible. In addition, b should take on
its variable value in the first term in Eq 31.
As in the general discussion in the previous section, let CM be the com-
pliance of a linear spring in series with the bend specimen. The total load
point rotation is given by

OT = CMM + e = CMM + e^c + Oc (32)

To obtain an expression for {dJ/da)gj^, it is first necessary to evaluate

(dJ/da)M. From Eq 28
dj\ r fd^e
. , , >. 2/ dM (33)

Using Eq 26, it is readily verified that

/ M ^ (d^dc] ^ 6M (deA , 4Mf (d^\ ,»>.

\da'jM \da^jM b' \dMj. ^ b' \dM')a ^"^^^

from which it follows that

48 ELASTIC-PLASTIC FRACTURE

M2 /a^e.
¥) =' 0
^ de. + 4
*> JO b^ \dM^
dM (35)

Here again it is important to note that {dJ/da)M depends only on the current
values of a and $c or M . In Eq 35 it is understood that the integrals are to be
evaluated with a held fixed at the current value. The first term in Eq 35 is
3J/b. The second term can be integrated by parts and combined with the
first to give

'dj\ ^ _ J , 4M^ (ddc

(36)
.da/M b b^ \dM,

For the bend specimen, the general expression, Eq 23, translates to

'dj\ ^ /dj\ dJ\ (deA CM + (37)

This expression can be substantially simplified without approximation using

Eqs 27 and 36

(38)
BML b \dMJa
and

de_ de_
= c„. + m)a (39)
dM + dM

Here C^^ is the compliance of the uncracked specimen. In general, C„c may
be a function of M, but for a deeply cracked specimen it will usually be the
elastic compliance since M will seldom exceed initial yield of the uncracked
specimen. The resulting exact reduction of Eq 37 is

'dJ\ J 4M2 r c 1 (40)

3a) 6T b b^
[ -^m
where C is the combined compliance

C = CM + C, (41)

In a rigid machine. CM = 0 and Eq 40 applies with C = C„c. In a dead-

load machine. CM — oo and Eq 40 reduces to the expression for {dJ/da)M in
Eq 36. For a fully yielded, elastic-perfectly plastic specimen, M is the limit
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 49

moment which is independent of dc for fixed a, so that Eq 40 becomes very

simply just

'a/\ ^ _ J ^ 4M^c (42)

.dajej- b b^

The analysis of the three-point bend specimen of Fig. 4b is essentially

identical to that carried out in the foregoing. Now AT is the total load point
displacement through which P works and Ac is the displacement of the
specimen due to the presence of the crack defined similarly to Eq 25. With C
as the combined compliance, now as

C = CM + = CM + C„

one finds

'dJ\ J 4Pl
(43)
b b2 dP
1 + C
dAc

Both Eqs 40 and 43 are exact for deeply cracked specimens. It is now
possible to make contact with the simpler approximate analysis of a fully
yielded, elastic-perfectly plastic three-point bend specimen given by Paris et
al [1]. At the limit state, M = PL/4 = Aaob^, where ao is the tensile yield
stress and A = 0.35 for plane strain and A = 0.27 for plane stress. In a
rigid testing machine, C = C„c = Ly(4EW^). Under these circumstances,
Eq 43 immediately leads to

EJ
T =
ao' ¥) =16^^ hIL. ao'b
(44)
pa/AT
The first term in Eq 44 is the same as that obtained in Ref / , while the
second term has been approximated in the analysis of Ref/.
Prior to the initiation of crack growth, Eqs 40 and 43 involve quantities
which can all be obtained from a single test record. Furthermore, all quan-
tities on the right-hand side of Eqs 40 and 43 are continuous across the initia-
tion point and, thus, so are ldJ/da)fj. and {dJ/da)^j. It follows that
stability of crack growth initiation can be assessed using Eqs 40 or 43 from
quantities obtained directly from the experimental test record just prior to in-
itiation. For a fully yielded specimen with little strain hardening, it may be
possible in some instances to neglect C{dM/d0c)a in Eq 40 (or the analogous
term in Eq 43 and thereby use Eq 42). When this term is not negligible, it will
50 ELASTIC-PLASTIC FRACTURE

be necessary to estimate (dM/d$c)a in order to assess stability using Eq 40

beyond the initiation of crack growth.

Nnmerical Results for a Deeply Cracked Bend Specimen in Plane Stress

The estimation procedure of Shih and Hutchinson [14] will be used to
relate / and 6^ to M for the deeply cracked, plane stress bend specimen of
Fig. 4a. These relations are then sufficient to calculate {dJ/da)ef using Eq
40. The material is assumed to be governed by a Ramberg-Osgood stress-
strain curve in uniaxial tension, that is

e/eo = <^/<ro + a(ff/ffo)" (45)

where ao is the effective yield stress and fo = ao/E. The estimation pro-
cedure uses the linear elastic solution and a fully plastic power-law solution,
which is given in Ref 14, to interpolate over the entire range from small-scale
yielding to fully yielded conditions.
The results of the procedure as applied to the deeply cracked plane stress
bend specimen are

To='•'''^' © + « ^ ^ ( « ^ %y (^^>
where Mo = Aaob^ and A = 0.27. The plasticity adjustment for the effective
crack length in the low M range is incorporated through the ^ factor [14],
which for the deeply-cracked specimen is given by

|=i^=.-O..8O6(^-^)(0 M.M, m

= 1 - 0.1806 r ~ j ) , M^Mo (49)

The numerical coefficients in the first terms in Eqs 46 and 47 and in Eq 48

are from the appropriate deeply cracked limits for the linear elastic problems
which can be found in Ref 15. For hi{n) and hiin) we have used the
numerical values presented in Table 2 of Ref 14 which were computed for
b/W — 1/2. (These are listed again in Table 1 of the present paper.) As
discussed in Ref i4, these values are expected to be very close to the deeply-
cracked limit for b/W ~ 0 when « > 3.
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 51

TABLE 1—Numerical values for h ;(n) and h j(n), taken from Ref 14.

n = 2 n = 3 n = 5 n = 10

Bend
Al(n) 0.957 0.851 0.717 0.551
A3(n) 2.36 2.03 1.59 1.12
Center-cracked
Ai(/i) 1.09 0.906 0.717
hi(n) 1.93 1.35 0.88

From Eqs 47 and 48

Mo (dOA
\=,t^l - 4.238\^2 + 3.062vt^ (" , ] M_y
€o \aM/a \M + 1 Mo)

+ anhAn) [j^J

for M < Mo. For M > Mo we take

f m l = ^-238^^ + 3.062,3 ( ^

+ anh,(n) ^ ^ j

The second term in Eq 51 would be absent if we had used Eq 49. It is in-

cluded to ensure continuity of {d$c/dM)a at M = Mo- Both this quantity
and {dJ/dM)a are discontinuous across M = Mo according to the estima-
tion procedures of Ref 14, leading to Eqs 46-49, where in fact these two
quantities should be continuous. In this paper, where we are primarily in-
terested in displaying the trends due to strain hardening, the effect of in-
cluding or omitting the second term in Eq 51 makes relatively little differ-
ence. For future work, however, the estimation procedure will need to be'
improved upon in this regard.
Equation 40 can be written in nondimensional form as

dJ EJ
T = ffo^ \da/e

-"' m ^ Mo V 90c J a
(52)

where C = Eb^C = Eb^iCu + C^) is the nondimensional combined com-

52 ELASTIC-PLASTIC FRACTURE

r> M.e^

4-

0 "T

E J

FIG. 5—Numerical results for T versus EI/{aff^h) for a deeply cracked plane-stress bend
specimen for various values of the strain hardening index n. Nondimensional combined com-
pliance is C — 20, corresponding to a typically dimensioned specimen in a rigid test machine
(see text).

pliance. Values of T and£7/(ffo^i) for values of M/Mo are generated using

Eqs 46-52. We have taken a = 3/7 for n > 1 and for « = 1 we have used the
linear elastic formulas. In Figs. 5 and 6 we have cross-plotted J as a function
of EJ/{ao^b) for various values of the hardening index n. The curve in Fig. 5
labeled « = oo is obtained from the fully yielded result, Eq 42, for the elastic-
perfectly plastic specimen, that is

EJ + 4.42c (53)
T = -
ao'b

Fully yielded conditions set in when M > Mo and this occurs when
EJ/ioo^b) is a bit larger than 2 for essentially all the higher n-values.
For the bend specimen of Fig. 4a

Eb^C„. = ULb^/W^ (54)

The results in Fig. 5 are for a combined nondimensional compliance C — 20.

In a rigid test machine, this corresponds to a specimen w i t h i i ^ / ^ s = 5/3^
which is representative of a typical test specimen. For « > 5, it can be seen
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 53

FIG. 6—Influence of combined compliance C on "X for plane-stress bend specimen for strain
hardening index n = 5.

from Fig. 5 that T will always be less than about 4 in plane stress. Fully
plastic power-law solutions for bend specimens are not yet available for plane
strain. However, judging from Eqs 52 and 53, we can perhaps expect
J-values to be larger in plane strain by a factor of as much as 2, due to the
presence ofA^. Thus it is reasonable to expect that T will always be less than
some number around 10 in plane strain when C — 20 and when n > 5. In
contrast, T^at spans the range 1 < T^^t < 200 for a wide range of steels
under nominally plane-strain conditions just following initiation (Ref/, Ap-
pendix II). Thus, only those specimens of steels in the low end of the range
of T^^f will display unstable crack growth upon crack growth initiation in a
bend specimen rigid test machine (compare Eqs 18 and 19).
The effect of increasing the combined compliance C is seen in Fig. 6 for «
= 5. The lowest curve is from Fig. 5. The uppermost curve is for C = oo, cor-
responding to a specimen loaded under dead (constant) moment. Paris et al
([/], Part II) varied the combined compliance in their test program by in-
54 ELASTIC-PLASTIC FRACTURE

eluding an extra bar in series with a three-point bend specimen. In this way
they were able to achieve a tenfold increase in compliance with the associated
large increase in T.
The parameter w in Eq 10 related to the validity of/-controlled growth in
the fully yielded range can be expressed in the revealing form

w -
= (2ik 1
I-PV-) (55)
EJ
involving only the ordinate and abscissa of Figs. 5 and 6. For the results of
Fig. 5 for C = 20, w does not exceed unity. Consequently, as a result of Eq
11, it is unlikely that the conditions for/-controlled growth are satisfied for
an increment of crack growth with dr fixed under these low compliance con-
ditions. This should be no cause for concern if 7^^, » T since then crack
growth is almost certainly highly stable anyway. The parameter T in Eq 55
increases with increasing compliance. At the compliance associated with the
onset of instability

c. = ( ^ ) T^. (56)

In the plane-strain tests of Paris et al ([/], Part II), r^a, = 36 and w = 15,
using Eq 56 with / = /,£.

Two-Specimen Method for Detenuining {dJ^,/da) and {dM/ddc)a in

/•Controlled Growth
Here we note a method which, in principle, can be used to measure
{dJ^f/da) without recourse to direct measurement of crack length changes.
It will permit an experimental determination of (dM/ddc)a once growth
starts.
Consider two deeply cracked bend specimens of identical material,
denoted by A and B, with differing initial ligament lengths 6^° ^nd b^°.
Equation 26 applies to both specimens, that is

MA = bA^ FiOc) and M^ = b^'FiOc) (57)

where b /^ and bg are the current ligaments. The F{dc) in each of Eqs 57 are
the same sinceF(dc) depends only on material properties. Thus at the same
value of 6c, from Eq 29

/A//B = *A/6B (58)

Consequently, initiation, that is, / = /,c, will occur at a larger value of ^c for
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 55

Specimen A than for Specimen B, lib/P < fee". With b^P < b^°. Speci-
men A can be used to measure F{fic) from

F^dc) = MA/{b^of (59)

in the range of Sc prior to initiation in A, as depicted in Fig. 7.

In Fig. 7 we have also depicted the curve of MB/(Z>B'')^ versus 6c, where
( i s ' ' ) is the initial ligament of B. Prior to initiation in B, the two curves in
Fig. 7 must coincide, assuming both specimens have the same material and
same degree of plane-strain constraint. From the second equation in Eq 57

dMe = 2*8 db^F + b^^F' ddc

where F ' = dF/dOc. With da = —db^ the foregoing equation can be rear-
ranged to give

F'iOc) cfM„
. . = ' - d6c - (60)
Fide) M„
This equation provides a relation for indirectly obtaining increments in crack
length in Specimen B in terms of the measured relation between MB and dc
and from F(0c) determined by using A. The associated change in/B> that is,
dJ^ = dJ^t, is given by Eq 30. Combining Eqs 60 and 30 gives

dJ„ F'idc) 4M|

4F(ec) (61)
da Fide) MB ddc b„

These results, Eqs 60 and 61, may be used to assess crack length changes,
da, and dJ^^/da up to values of ©c where initiation occurs in Specimen A, or

ONSET OF GROWTH
IN SPECIMEN A
bo^

ONSET OF GROWTH IN SPECIMEN B

FIG. 7—Curves of normalized moment as a function of $c for two deeply cracked bend
specimens with differing initial uncracked ligaments (h^^ < bfl").
56 ELASTIC-PLASTIC FRACTURE

/A = Jic • This limit can be assessed from the onset of growth in Specimen B,
where J^ = /,,., and using Eqs 29 and 58.
The practicality of using Eqs 60 and 61 together with experimental records
must await further work.* One desirable feature of these relations is that the
resistance curve data, dJ^i/da versus Aa, are generated without having to
specify a precise definition of initiation.
Using Eq 57 we also note that

^i^). " *B'^'(^C) (62)

Thus F ' (0c) obtained from Specimen A also provides the one term in the
general expression, Eq 40, for (dJ/da)eT which cannot be obtained from
Specimen B itself. For small amounts of growth, the replacement offeghy its
initial value in Eqs 60-62 will introduce little error.

Analysis of Deeply Cracked Center-Notched and Edge-Notched

Specimens Under Tension
In this section, expressions are obtained for {dJ/da)^j. for deeply cracked
center and edge-notched specimens in plane strain or plane stress and for
the deeply cracked round bar. First the two-dimensional plane specimens in
Fig. 8a and b will be considered. In each case we now write the load-point
displacement of the specimen A as the sum of the elastic part Ae and the
plastic part Ap according to

A = Ae + Ap (63)

For b/w « : 1, dimensional analysis implies the general functional

dependence

Ap = bf{P/b) (64)

where P is the load per unit thickness carried by each ligament. Let

dAe
J' = \ ( TdaT )jp ^^ (65)
denote the value of/ for an elastic specimen at P. Then

^ = i (i), * = ^- + [ (t), ^
^Although these methods were used in Ref/, Part II, for data reduction, their full applica-
tion and limitations to practical testing procedures are yet to be explored.
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 57

2P,AT 2P,Ai

2P.A > Y C M

2b 2b

(a) 2W (C)
V
FIG. 8—Center-cracked specimen (a), edge-cracked specimen (b), and edge-cracked round
bar specimen (c).

Using Eq 64 and following the development in Ref/J, one can show that

/ = /e + 6 - ' 2 PdAp - PA/. (67)

Next using Eqs 64 and 66 we find

dj\ _ (dJe
, , , + 1 - (^^^ dP (68)

But for deeply cracked specimens

/e = kPy{2bE) (69)
where
*: = (! — >'^)87r/(7r^ — 4) (plane strain, center-notched)
= 87r/(ir^ — 4) (plane stress, center-notched)
= (1 — »'^)8/7r (plane strain, edge-notched)
= 8/IT (plane stress, edge-notched)

and where v is Poisson's ratio.

So, {dJe/da)p = Je/b. Integrating the second term in Eq 68 by parts and
using Eq 66 gives

dA ^ J . 2Je
da)p b b b^ b^ \dP a ^ '

With 1/2 CM as the compliance of a linear spring in series with the

specimen, the total load point displacement is

AT = CMP + A = CMP + A^ + A/ (71)

58 ELASTIC-PLASTIC FRACTURE

To reduce the general expression, Eq 23, for {dj/da)t^-p, we note the follow-
ing relations

^^p\ ^ _ Ap , P (dAp
da Jp b ^ b \dP

dAe] ^ kP ^ 2Je
da Jp Eb P

Let Ct = (dAe/dP)a be the compliance of the cracked elastic half-specimen

and denote by C the combined elastic compliance

C — Ce + CM (72)

Then Eq 23 can be reduced by algebraic manipulation without approxima-

tion to

2 Je_ PAp P2 /^A^N -]

'dj\
=
-i- b i2 ^ b^\ dP / J
X

(73)
C
-i- Ap
P
X
[- - (^)J -'

In a dead-load machine, C = <x> and Eq 73 reduces to {dJ/da)p. For a

fully yielded, elastic-perfectly plastic specimen, P is at the limit load and
idP/dAp)a = 0 so that Eq 73 becomes

(74)
da/AT b * V E P

The same comments made in connection with the analogous formulas for the
bend specimen apply here; namely, Eq 73 allows determination of
(dJ/da)^j. from a single experimental record prior to initiation. Further-
more, idJ/da)^j. is continuous across the point of initiation. Beyond initia-
tion, {dP/dAp)a must be estimated or perhaps neglected, as in Eq 74, if the
specimen is fully yielded and strain hardening is not significant.
For the deeply cracked edge-notched round bar of Fig. 8c we write, copy-
ing the procedure for the bend specimen

AT = CMP + A = CMP + A„, + Ac (75)

Now, with b as the radius of the circular ligament

A, = bf(P/b^) (76)
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 59

where P is the total load. Omitting details, we find

•Ac
/ = 2-Kb^ 3 I PcfAc - PAc (77)
o

bajp b irb^ -xb^ -m. (78)

b irfe3 "^ 2iri3 4P^C + 4PAc

(79)
'dP\ ap
X 1 + C
dA.
where

(80)

For a fully yielded, elastic-perfectly plastic specimen, Eq 79 reduces to

fa/)
fa/") = _ Z + : ? ^ + 2P^C (81)

Numerical Results for a Deeply Cracked Center-Notched Specimen

in Plane Stress
We again employ the-estimation procedure of Ref 14 together with the
general formula, Eq 73, to generate numerical results for T for the deeply
cracked center-notched specimen of Fig. 8a. The Ramberg-Osgood stress-
strain curve, Eq 45, is used. Of the quantities needed in Eq 73, Je has been
given previously while / and Ap are

^=i,0.„M<r (82)

Ap _ . , , fP\ . . . s fPV (83)

^^^=kln^[^J+ahAn) ^^^

where Po = oob and

(84)
= 1 - f ( ^ ^ ) , P ^ Po
4ir \n + 1/
60 ELASTIC-PLASTIC FRACTURE

The linear elastic solution for the limit of a deeply cracked specimen from
Ref 15 has been used to arrive at the first terms in Eq 82 and i/-; the first term
in Eq 83 follows from the same limiting solution plus the definition of A/>
in Eq 63. From Eqs 83 and 84

Po dAp\ _
A; In 1^ + 2^(1/- - 1) + anhjin) {^j ' (85)
eob \dP

To ensure continuity at Po, this same expression is used for P > Po-
Values of Ai and A3 for the power-law solution are listed in Table 1. These
values are converted from Table 1 of Ref 14 using hi — (a/w)gi and

hi = {a/w)gi/(\ — a/w) for a/w = 3/4

Equation 73 can be expressed in nondimensional form for T, involving

only the foregoing quantities and the nondimensional combined compliance

C = EC = E{Ce + CM) (86)

Curves of T as a function ofEJ/iao^b) are shown in Figs. 9 and 10. For the
deeply cracked specimen

ECe = L/W + k \n (W/b) (87)

so the value C = 10 in Fig. 9 can be regarded as being fairly representative of

a test specimen with typical dimensions in a rigid test machine. In contrast to
the bend specimen of Fig. 5 in the same range oiEJ/{ao^b), a decrease in
strain hardening increases T, that is, decreases stability. In both cases the
effect of strain hardening is quite significant. Only for the bend specimen
does T decrease at sufficiently large increasing / ; this is most evident from
the formula for the perfectly plastic case, Eq 42. The effect of increasing
the combined compliance C is seen in Fig. 10 for n = 5. Here the trends
are very similar to those for the bend specimen.

Discussion
Two open questions are the maximum allowable amount of crack growth
and the minimum admissible value of the nondimensional parameter co to en-
sure /-controlled growth to a reasonable approximation. For a fully yielded
specimen with ligament b, the most relevant material-based estimate of co
just following initiation is Eq 56, that is

^ b_dJ ^ (ao^b\
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 61

1 ^•A T
C = 10
J
f
b
-
0
E / aJ
{-±±-\
111111111
'^^ / ^ ^
^ . - ^ ^ n =3

—•""''^ n =1 (elastic)
-

1 1 1 1 1 1
10 12 14
EJ
0-^2 b

FIG. 9—Curves ofTas a function ofBJ/{ao^h)for various levels of strain hardening index n
for a deeply center-cracked plane-stress tension specimen. Nondimensional combined com-
pliance is C = 10, corresponding to a typically dimensioned specimen in a rigid test machine
(see text).

The range of this parameter for the steels listed by Paris et al ([1] Appendix
II) is roughly 0.1 < w < 100 with most entries satisfying « > 10. Thus it
does seem likely that there will be an important class of metals whose proper-
ties are such that a limited amount of growth can be analyzed, both for
equilibrium and stability, using the deformation theory/. However, Landes
and Begley ([16] and private communications) have noted that the J-integral
resistance curves for compact or bend specimens have a different slope,
dJmu/da, than those for center-cracked specimens. On the other hand,
their curves were plotted with /-values which were uncorrected for effects of
crack length changes, such as is illustrated by the second term in Eq 31.
Moreover, in their tests co was not evaluated and their ligaments sizes, b, did
not quite meet the other requirements, as illustrated by Eq 12. In addition,
they report unsymmetrical crack extension for the two crack tips in their
center-crack specimens. Nevertheless, this perplexing point cannot be
dismissed; thus further exploration is warranted until a reasonable explana-
tion is found.
It should be most interesting to compare results from deformation theory
calculations and flow theory calculations for precisely the same prescribed
growth conditions. In this way it should be possible to learn more about the
62 ELASTIC-PLASTIC FRACTURE

FIG. 10—Influence of combined compliance on T for deeply center-cracked plane-stress ten-

sion specimen for strain-hardening index n = 5.

influence of strain hardening and configuration on the minimum permissible

value of oj. It is worth bearing in mind that the simplest flow theories of
plasticity, based on a smooth yield surface, have limitations which may also
be important in the analysis of crack growth. In particular, certain of the in-
cremental moduli tend to be overestimated by the simplest flow theories,
leading in some problems to unrealistically high resistance to plastic defor-
mation. In this connection, we also note that our argument requiring w » 1
may be relaxed somewhat by appealing to the total loading concept for justi-
fying deformation theory as discussed by Budiansky [6].
In this paper we have emphasized the analysis of deeply cracked configura-
tion because relatively simple formulas for T can be obtained which, in some
instances, are directly applicable to test specimens. Work is underway [17\
using estimation procedures based on the power-law solutions to predict T
for arbitrary crack lengths. At the moment, however, power-law solutions are
available only for a very few configurations and, in plane strain, only for the
center-cracked strip [18].
 HUTCHINSON AND PARIS ON J-CONTROLLED CRACK GROWTH 63

Finally, within this work, suggestions have been made for methods of
analysis of load-displacement records which permit establishing a material's
J-integral R-curve without direct measurements of crack length changes.
Though a special case of this approach was used with success in earlier tests
[1], the method is unexplored, but holds great promise for simplifying
testing. Indeed, those methods can be extended to eliminate the requirement
for deeply cracked specimens, but that will be a topic for subsequent discus-
sions. Furthermore, since those methods determine dJ^^^/da from load-
displacement relations as influenced by crack growth, they are a most
natural way to assess material parameters affecting stability. That is true
because stability itself depends directly on the influence of crack growth on
load-displacement behavior, as is observed throughout the analysis herein.

Acknowledgments
The first author (J. W. H.) acknowledges support of this work by the Na-
tional Science Foundation, Grant No. ENG76-04019, and in part by the
Division of Applied Sciences of Harvard University. The second author (P.
C. P.) acknowledges the support of this work by the United States Nuclear
Regulatory Commission, Contract No. NRC-03-77-029 with Washington
University. The work was also significantly assisted by many stimulating
discussions with several co-workers, including J. R. Rice, H. Tada, A.
Zahoor, H. Ernst, C. F. Shih, and R. Gamble.

References
[1] Paris, P., Tada, H., Zahoor, A., and Ernst, H., "A Treatment of the Subject of Tearing
Instability," U. S. Nuclear Regulatory Commission Report NUREG-0311, Aug. 1977. (See
also papers in this publication by these authors.)
[2] Fracture Toughness Evaluation by R-Curve Methods, ASTM STP 527, American Society
for Testing and Materials, 1974.
[J] McClintock, F. and Irwin, G. R. in Fracture Toughness Testing and Its Applications.
ASTM STP 381, American Society for Testing and Materials, 1965, pp. 84-113.
[4\ Rice, J. R. in Mechanics and Mechanisms of Crack Growth, British Steel Corp., 1973.
[5] Rice, J. R. in Fracture. Vol. 2, Academic Press, New York, 1968.
[6\ Budiansky, B.,Joumal ofApplied Mechanics, Vol. 26, 1959, pp. 259-264.
[7] Begley, I. A. and Landes, J. D. in Fracture Analysis, ASTM STP 560, American Society
for Testing and Materials, 1974, pp. 170-186.
[8] Hutchinson, J. W., Journal of the Mechanics and Physics of Solids, Vol. 16, 1968, pp.
13-31.
[9] Rice, J. R. and Rosengren, G. F., Journal of the Mechanics and Physics of Solids, Vol.
16, 1968, pp. 1-12.
[10] McClintock, F. A. in Fracture, Vol. 3, Academic Press, New York, 1971.
[;/] Landes, J. D. and Begley, J. A. in Fracture Toughness, ASTM STP 514, American Society
for Testing and Materials, 1972, pp. 1-20 and 24-39.
[12] Shih, C. F., de Lorenzi, H. G., Andrews, W. R., Van Stone, R. H., and Wilkinson, J. P.
D., "Methodology for Plastic Fracture," Fourth Quarterly Report by General Electric Co.
to Electric Power Research Institute, 6 June 1977.
[13] Rice, J., Paris, P., and Merkle, J. in Progress in Flaw Growth and Fracture Toughness
Testing, ASTM STP 536, American Society for Testing and Materials, 1973, pp. 231-245.
64 ELASTIC-PLASTIC FRACTURE

[14] Shih, C. F. and Hutchinson, J. W., Journal ofEngineering Materials and Technology, Vol.
98, 1976, pp. 289-295.
[15] Tada, H., Paris, P. C , and Irwin, G. R., The Stress Analysis of Cracks Handbook, Del
Research Corp., 226 Woodboume Drive, St. Louis, Mo., 1973.
[16] Begley, J. A. andLandes, J. V>.,IntemationalJoumalof Fracture Mechanics, Vol. 12, No.
5, Oct. 1976.
[17] Zahoor, A., work in progress.
[18] Goldman, N. L. and Hutchinson, J. W., IntemationalJoumal of Solids and Structures,
Vol. 2, 1975.
 C. F. Shih,^ H. G. deLorenzi,^ and W. R. Andrews^

Studies on Crack Initiation and

Stable Crack Growth

REFERENCE: Shih, C. F., deLorenzi, H. G., and Andrews, W. R., "Studies on

Crack Initiation and Stable Cracli Growtli," Elastic-Plastic Fracture, ASTM STP 668,
J. D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and
Materials, 1979, pp. 65-120.

ABSTRACT: Experimental results are presented which suggest that parameters based
on the J-integral and the crack opening tip displacement 8 are viable characterizations
of crack initiation and stable crack growth. Observations based on some theoretical
studies and finite-element investigations of the extending crack revealed that / and 6
when appropriately employed do indeed characterize the near-field deformation. In
particular, the analytical and experimental studies show that crack initiation is
characterizable by the critical value of J or 5, and stable crack growth is characterizable
in terms of the 7 or 5 resistance curves. The crack opening angle, d8/da, appears to
be relatively constant over a significant range of crack growth. Thus, appropriate
measures of the material toughness associated with initiation are Jic and 5ic, and
measures of material toughness associated with stable crack growth are given by the
dimensionless parameters Tj [= (E/a,^)(dJ/da)\ and Tj [= (E/ao)(d8/da)\. The two-
parameter characterization of fracture behavior by/jc and Tj or 5ic and Ti is analogous
to the characterization of deformation behavior by the yield stress and strain hardening
exponent.

KEY WORDS: fracture, cracks, crack initiation, crack growth, crack opening dis-
placement, crack opening angle, J-integral, resistance curve, plastic deformation,
elastic properties, plastic properties, fracture toughness, tearing modulus, crack
propagation

The progress of ductile fracture from an existing sharp-tipped flaw may

be separated into four regimes: the blunting of the initially sharp crack tip,
initial crack growth, stable crack growth, and unstable crack propagation.
An illustration of these regimes is given in Fig. 1. For low-toughness
materials, there is relatively little crack-tip blunting, and essentially no
stable growth when fracture proceeds under plane-strain conditions. For
high-toughness materials on the upper shelf, there is significant crack-tip

' Mechanical engineers. Corporate Research and Development, and metallurgical engineer.
Materials and Processes Laboratory, respectively. General Electric Co., Schenectady, N.Y.

65

Copyright 1979 b y A S T M International www.astm.org

66 ELASTIC-PLASTIC FRACTURE

UNSTABLE GROWTH

(FOR LOAD CONTROLLED

SYSTEM)

CRACK EXTENSION ( O - O Q I

FIG. 1—Stages of crack extension, showing crack tip blunting, initiation, and growth.

blunting and substantial stable crack growth. More importantly, the

nominal 'resistance' of the material to further crack extension increases
with increasing crack growth. In this case, the conventional assessment
of the margin of safety of flawed structures based on the onset of crack
extension is conservative, since the nominal or measured 'toughness' of the
material may increase significantly with relatively small crack extension.
Stable crack growth can occur under small-scale yielding conditions
when the plastic zone size is small compared to crack length, and under
large-scale yielding conditions when the plastic zone extends across the
remaining ligament [1-4].^ Under small-scale yielding conditions, the
characterization of material resistance to crack growth by the resistance
curve procedure based on the stress intensity factor K has received con-
siderable attention [4]. Extensive investigations on the characterization of
crack growth by /-resistance curves and on approximate methods for pre-
dicting fracture instability are reported in Ref 5.
In the fracture of ductile materials, the central portion of the crack
typically advances in a thumbnail fashion, and eventually the outer edges
fail in shear. The stress and deformation state ahead of the crack that
extends in this manner is rather complex, and it is unlikely that the crack
^The italic numbers in brackets refer to the list of references appended to this paper.
 SHIH ET AL ON CRACK INITIATION AND GROWTH 67

extension can be characterized by a single parameter valid for the changing

modes of fracture. This may be argued from the completely different strain
energy density fields associated with plane-strain and plane-stress conditions
at the crack tip. However, some theoretical and experimental results
suggest that a single-parameter characterization may apply if crack exten-
sion occurs by a single mode of fracture, that is, when it is either strictly
flat or shear fracture under either plane-strain or plane-stress conditions
[1-5,16,29]. The critical value of the parameter will of course be different
for the two different modes of fracture.
In this investigation, we seek characterizing parameters for macroscopi-
cally flat fracture (initiation and growth) under large-scale yielding condi-
tions. Throughout the investigation, a close coupling was maintained
between testing and analysis. The basic material employed in the experi-
mental program is A533B steels. Most tests were carried out at upper-shelf
temperatures, where the mode of fracture is dimpled rupture.

Potential Fracture Criteria

In this section we discuss the background for the potential fracture
criteria that have been studied during our investigations. In particular,
the J-integral and the crack opening displacement (COD) are discussed in
detail, because, as is demonstrated in this paper, they seem to have con-
siderable potential for predicting the behavior of growing cracks.
The works of Hutchinson [6] and Rice and Rosengren [7] revealed that,
for stationary cracks, the stresses and strains in the vicinity of the crack tip
under both small-scale yielding and fully plastic conditions may be repre-
sented by

!/(/,+1).

(1)
ffo/ EJ \ «/(B+1)
^''~ ~^\ 2T eij{0,n)

where
/ = J-integral defined by Rice [8],
ffo = yield stress,
E = elastic modulus,
r = radial distance from crack tip,
iij = known dimensionless functions of the circumferential position B
and the hardening exponent n, and
/„ = constant which is a function only of « [6,9].
68 ELASTIC-PLASTIC FRACTURE

In Eq 1, the J-integral is the ampHtude of the stress and strain singularity. ^

For an ideally plastic material {n = oo), the strain fields exhibit a l / r
singularity. In this case the foregoing expressions may also be rewritten in
terms of the crack tip opening displacement 8t by exploiting the relation-
ship [8.10]

J = a Oodi (2)

where a is a parameter of the order unity. Thus

ay = ffo dij ($, n = oo)

and

a 8t
iij id, « = oo) (3)
Inr
The functions dij {$, n = oo) and e,> (d, n = oo) are the stress and strain
variations associated with the Prandtl field [6-10], which represents the
nonhardening limit of Eq 1. Expressions for strain hardening materials
where 6 appears as the amplitude of the singular fields have been discussed
by Tracey [11]. Equations 1 and 3 are known as the Hutchinson-Rice-
Rosengren (HRR) singularities or fields, and / and 8 are the HRR field
parameters. When the HRR field encompasses the fracture process zone
(and there is evidence that it is indeed the case for both small- and large-
scale yielding conditions at the onset of crack growth for certain crack
configurations), the HRR parameters, / and 8, are natural candidates for
characterizing fracture.
In crack-growth situations, the near-tip field is far more complex than
in the stationary case. To date there is no complete description of the
stress and strain fields ahead of an extending crack. Some features have
emerged from studies due to Rice [12], Chitaley and McClintock [13], and
Amazigo and Hutchinson [14]. In general these studies revealed a milder
singularity—an ln(l/r) strain singularity for elastic-perfectly plastic
materials. The studies by Rice, based on a 72flowtheory of plasticity for an
ideally plastic material (« = oo), showed that the incremental strains in the
immediate vicinity of the crack are related to an increase of the crack

'The J-integral characterizes the crack tip field in the same spirit that the elastic stress-
intensity factor characterizes the elastic singular field under small-scale yielding conditions
[33]. Expressions similar to Eq 1 may also be written for mixed-mode situations (cracks
subjected to combined tensile and shear loadings). The crack-tip fields are now characterized
by two parameters, the J-integral, which is again the amplitude of the singularity, and a
parameter M'' which is a measure of the relative strength of the tensile and shear stress
directly ahead of the crack tip [9].
 SHIH ET AL ON CRACK INITIATION AND GROWTH 69

opening displacement dd, and the increment of the crack extension da,
through the relationship [12]

J _ '^^ ^ //i\ 1 Ooda . R(d) ,a\ /j\

dey = yfij (6) + -^-y In - ^ gy (6) (4)

In terms of the rate of change of the strain field with crack growth, we
can rewrite Eq 4 in the form

- ^ - r'da-f'^^^^^ Er^'' r ^^^^^ ^^^

Here, R{d) is a measure of the distance to the elastic-plastic boundary and

gij is a dimensionless function of order unity. The first term of Eq 4 repre-
sents the additional strain due to crack-tip blunting if the crack did not
advance during load/displacement increments, and /,> is related to the
stationary crack expression (Eq 3) by the relationship//, = a £,> {d)/In, and
is of order unity. The second term in Eq 4 represents the additional plastic
strain induced by the incompatible elastic strain increments caused by the
advance of the stress field through the material.''
For ductile metals (A533B steels are an example), subsequent results will
show that the first term in Eqs 4 and 5 will be dominant terms over a
significant interval (compared to dd except right at the crack tip, where the
(1/r) In (R/r) singularity will dominate. In other words, the strains at the
crack tip are uniquely characterized by the crack-tip opening angle,
dS/da,'•if

^ » ^ in f « ) (6)
da E \ r J
For r varying between 0.1 /? and /?, Eq 6 expresses the condition that the
crack opening angle must be large compared with the yield strain aJE.
An expression for the incremental strains during crack growth has
recently been derived by Hutchinson and Paris [75] on the basis of Ji
deformation theory of plasticity. For an ideally plastic material, this ex-
pression reduces to

de, = ^--f,{e)+ ^-J~ ^, m (7)

a Oo r a Oo r^
^For a perfectly plastic or low hardening material, the stress field in the immediate vicinity
of the crack tip is essentially determined by the yield condition and the stress equilibrium
equations. The stress field moving with the crack tip therefore generates an elastic strain
increment which is not derivable from a set of displacement increments; the incompatible
elastic strain increment in turn induces additional plastic strain at the crack tip.
*For a Cartesian coordinate system fixed in space, where x is parallel to the crack plane,
db/ da = — 96/dx. Thus the crack opening angle reflects the actual slopes of the crack faces.
70 ELASTIC-PLASTIC FRACTURE

where /3/, (d) is a dimensionless quantity of order unity. Rewriting Eq 7 in

terms of the rate of change of strain with respect to crack extension, we get

dey
da = ( '

V aoTo ,
)7f/.<«>-
•

( '
)^fc(«)
'
(8)

Hutchinson and Paris argued that / uniquely characterizes the near field if
the first term in Eq 7 is the dominant term, that is, if*

f » ^ (9)
da r
We note that while Eqs 5 and 8 are derived from distinctly different
approaches (that is, J2 flow theory and J2 deformation theory, respectively),
they have a similar structure. Their first terms represent proportional
increments in the strain fields due to the increase in size or strength of the
HRR singularity, while the second terms represent the nonproportional
strain increments due to the advance of the HRR field with the extending
crack. Therefore, if the HRR field increases in size more rapidly than it
advances, then the crack opening angle, d8/da, and dJ/da describe the
crack-tip environment for an extending crack. When the fracture process
zone is enclosed in the region dominated by dh/da or dJ/da, Eqs 5 and 8
coupled with the respective conditions (Eqs 6 and 9), provide the basis for
a COD-based or a/-based resistance approach for stable crack growth.
Other fracture parameters were also examined in this investigation.
They include the work density W in a process zone of size / [16] and the
energy separation rate [17.18]. These parameters were found to be rather
dependent on the finite-element mesh size and the crack-tip mesh con-
figuration used in the analysis, and involve the introduction of an additional
length parameter. Details of these parameters are given in Refs 16-18. In
this discussion, we have focused our attention on the /-based and COD-
based approaches.

Strategy
In order to ascertain which parameters are viable fracture criteria, a
philosophy was followed whereby the results of tests were analyzed with

*The characterization of the singular strain fields by the J-integral assumes some amount
of strain hardening since the 9-variation of the strains is not unique for perfectly plastic
material [34]. We employ the perfectly plastic idealization to simplify the structure of Eqs 7
and 8 and for comparison with Eqs 4 and 5.
 SHIH ET AL ON CRACK INITIATION AND GROWTH 71

detailed finite-element computations. The tests were done on compact

specimens, and center-cracked panels with varying remaining ligaments.
Finite-element calculations in plane strain were carried out using the
ADINA code [19] suitably modified to allow crack-tip blunting and growth
and the computation of appropriate fracture parameters [20]. The analytical
investigations were carried out in two phases and these are discussed in the
following.

Phase I—Initial Filter

The objective of this first analytical phase is to observe the behavior
of potential fracture criteria when crack growth in the finite-element model
is forced to follow the measured relationship between load-line displacement
(LLD) and crack extension. From experimental measurements, whether
they be made by heat tinting, by the unloading compliance technique, or
by other procedures, it is possible to determine the crack extension,
a — flo, as a function of the measured load line displacement (LLD). This
information is the basic input to the finite-element crack-growth simulation
calculations. In the finite-element model, the rate of crack growth is
prescribed to follow the experimentally measured LLD — (a — ao) rela-
tionship. Thus at an applied displacement of, say, LLD,, the crack is
extended by an amount a, — Oo (determined from the experimental data)
as illustrated in Fig. 2. The process is repeated for the entire finite-element
crack growth calculations. The various potential fracture parameters, / and
cU/da at near-field and remote contours, COD and crack opening angle,
energy separation rates, work density over a process zone, etc., are com-
puted during the crack-growth simulation. The calculations are repeated
for different test configurations. Here again, the experimentally measured
LLD — (a — Co) relationship is employed to control the crack-growth rate
in the corresponding finite-element model. In this phase of study, the
compact specimen and the center-cracked panel which exhibit different
deformation fields in the fully plastic state are employed; these are shown
in Fig. 3. The parameters which appear to be viable are retained for
analysis in the subsequent evaluation phase. The viability of the criteria is
determined by a number of requirements that are discussed in the following.
Certain checks are carried out during the filter phase of the study. The
calculated load-deflection relationship is compared with the experimental
load-deflection record. The degree of agreement is a direct measure of the
capability of the finite-element model and the crack extension technique
for modeling the complete range of crack extension.

Phase II—Evaluation
In the evaluation phase, the analytical process is reversed. The selected
72 ELASTIC-PLASTIC FRACTURE

- 0 4 .S

4 8 12 16 20 24 28
CRACK EXTENSIOM (mm)

FIG. 2—Load-line displacement versus crack extension for 4T compact specimen T52.

viable parameters and their critical values (or their resistance curve values),
which are obtained in Phase I, are now employed in finite-element crack-
growth calculations for a variety of cracked configurations. In these calcu-
lations, the crack growth is governed by the fracture parameter itself. The
results and, in particular, the LLD-crack extension relationship and the
LLD-load relationship, are compared with the experimental measurements
for the corresponding crack configurations. Based on these evaluations,
final fracture criteria can be selected.

Requirements for Viable Fracture Parameters

In the two phases of the investigation, the fracture parameters are
 SHIH ET AL ON CRACK INITIATION AND GROWTH 73

I - 1

- 2 Q — ^

DEFORiWION ZONE

A) CENTER CRACKED PANEL

SLIP LINE FIELD DEFORMATION ZONE

B) SINGLE EDGE CRACK

FIG. 3—Global deformationfieldsunderfiillyplastic conditions.

assessed on the basis of several requirements which a viable fracture

criterion should satisfy:
1. The parameter must be a measure of the stress and deformation state
in the vicinity of the crack tip.
2. The critical value of the parameter must be independent of initial
crack length and specimen geometry.
3. The parameter must be employable in instability analyses.
4. The parameter should be applicable to three-dimensional crack geom-
etries, for example, to an elliptic surface flaw.
5. The parameter should be generalizable to mixed-mode fracture,
though the critical value will depend on the relative ratio of shear to flat
fracture.
74 ELASTIC-PLASTIC FRACTURE

6. The parameter should remain constant during crack extension. This

is an attractive but not an essential feature. In an R-curve approach, the
resistance parameter will increase with crack extension.
From the computational viewpoint, the viable parameter should possess
the following additional properties:
7. The parameter must be relatively insensitive to finite-element model-
ing, to mesh size, process zone size, and load/displacement increment
size.
8. The parameter should be computable within reasonable computational
cost.
From the experimental viewpoint the viable parameter should have an
additional feature:
9. The parameter should be obtainable from global measurements
remote from the crack, but, if this is not possible, it must be obtainable
from local measurements near the crack tip.
These requirements are the basis of the subsequent discussions on
experimental and analytical results.

Finite Element Modeling of Blunting and Crack Growth

From a micromechanics viewpoint, ductile fracture involves three
distinct processes—void nucleation, void growth, and void sheet coalescence
[21]. Attempts to quantify these processes using rather simple models are
discussed in Refs 22-24. However, the development of a quantitative
fracture methodology based on microstructural aspects of dimpled rupture
is a long way off. The alternative quantification of fracture at a macro-
scopic scale is adopted in this investigation. At this level, fracture occurs
by crack-tip blunting followed by crack advance. This process has been
observed using three separate techniques—rubber infiltration of a crack
specimen, metallographic study of interrupted crack propagation tests,
and macrofractograph of fractured specimens [16]. The crack-tip profiles
observed in A533B steels tested at the upper shelf are shown in Figs. 4-6.
These figures reveal a macroscopically flat but a microscopically tortuous
crack path.
Our effort is focused on the identification of parameters that characterize
the stress and strain state in a region (with an extent of several times the
crack-tip opening displacement) that encompasses the fracture process
zone. At this scale, a schematic of the crack tip profile is shown in Fig. 7.
A finite-element model of the crack tip region using 8-noded isoparametric
elements is shown in Fig. 8; in this figure, only the corner nodes are
indicated. The remaining body is also modeled by 8-noded elements. In
our two-dimensional studies, the original sharp crack is modeled with
degenerate elements in which the two corner nodes and the mid-side node
are initially collapsed to a common point—the crack tip—giving a 1/r
 SHIH ET AL ON CRACK INITIATION AND GROWTH 75

mm
1
FIG. 4—Optical micrograph showing a blunted crack in an interrupted A533B compact
specimen tested at 93°C (200°F).

strain singularity as discussed by Barsoum [25]. As the load is increased,

the crack-tip blunting is modeled by the separating of the nodes at the
common point, as shown in Fig. %b. Crack extension is modeled by shifting
the current crack tip node as discussed by Shih et al [26] and is illustrated
in Fig. 8c.' The adjacent mid-side nodes are also shifted so that they
remain midway between the corner nodes. This shifting continues at each
load or displacement increment as long as a controlling parameter for
growth (for example, crack opening angle) is at a critical value; shifting
ceases when its value is less than this critical value. When the ratio of the
remaining element ligament to the overall element size reaches a critical
ratio, the crack tip node is released together with the corresponding
mid-side node. The shifting process is repeated with the nodes of the
next element.

' A S the crack extends, the element ahead of the crack is distorted, which causes the
integration points (Gauss points) to be "dragged" along by the extending tip. Our studies
revealed that if the crack tip extends by small increments (compared to the element size), the
shift in the Gauss points at each increment is small, and the stresses associated with the
Gauss points could be "dragged" along since the error incurred is small compared to the
stress redistribution due to crack growth and additional external loading. Typically, the
crack extends through an element in about 30 increments. For larger increments, the stresses
associated with the new location of the Gauss points are obtained by linear interpolation.
76 ELASTIC-PLASTIC FRACTURE

mm

FIG. 5—Optical micrograph showing crack growth in an interrupted compact specimen of

A533B at g3°C (200°F).

Eight-noded isoparametric elements are employed for several reasons.

Nagtegaal, Parks, and Rice [27\ have shown that conventional 4-noded
isoparametric elements are not appropriate for analyses in the fully plastic
range. Numerical studies by deLorenzi and Shih [28] showed that the
8-noded isoparametric element is ideally suited for fully plastic analyses.
Furthermore, the latter element allows the modeling of crack tip blunting
by the rather simple technique discussed in the foregoing, and does not
require the degree of mesh refinement associated with constant-strain
elements. Accurate procedures for evaluating the J-integral by contour
integration and further details on crack-tip studies in the elastic and
fully plastic range are discussed in Refs 16 and 20.

Fracture Investigations
The basic material employed in our investigation is A533B steels, which
are employed in the fabrication of pressure vessels. Because of the high
toughness of A533B in the upper-shelf region, compact specimens of the
size IT to 4T invariably exhibit the formation of shear lips. The goal of
this investigation is to identify parameters that will characterize flat frac-
ture. Consequently, compact specimens of varying thickness and side
 SHIH ET AL ON CRACK INITIATION AND GROWTH 77

FIG. 6—Siticone rubber replica of crack profile in 4T Specimen T71 at 93°C (200°F).

groove depth were tested at 93°C (200°F). A description of these tests,

showing the effect of thickness and side grooving on fracture toughness
measurements, has been given by Andrews and Shih [29]. The experiments
showed that side grooves 12.5 percent or deeper promoted flat fracture
surfaces with straight crack fronts as illustrated in Fig. 9. Some lateral
contraction occurred for 12.5 percent grooves, but 25 and 50 percent
grooves produced no appreciable lateral contraction. Three-dimensional
elastic analysis of grooved and smooth compact specimens were carried out
by Shih et al [30]. The analyses revealed that while 25 percent side grooves
were sufficient to promote a uniform plane-strain condition across the
crack front, its effect on the stress-intensity factor and compliance is
minimal. In addition, the linear crack front allows the crack extension
to be measured without ambiguity. Consequently, crack configurations
with 25 percent side grooves were employed in the experimental program
and the corresponding finite-element calculations assumed plane-strain
conditions.
78 ELASTIC-PLASTIC FRACTURE

SHARP CRACK

ONSET OF CRACK
'oi I EXTENSION

CRACK EXTENSION

•i:: RIGID- PLASTIC

MATERIAL

CRACK EXTENSION
ELASTIC-PLASTIC
MATERIAL

FIG. 7—Schematic of crack extension illustrating COD and two definitions of angle
between separated surfaces.

Experimental Data for Compact Specimens

Comprehensive crack growth experiments have been carried out on 4T
compact specimens of A533B steel at 93 and 260°C (200 and SOCF).
Details of material properties, specimen geometries, remaining ligaments,
and experimental procedures are given in Refs 16 and 29. Two different
heats were used: one for the thickness and side groove study (Material 1),
and one for the comprehensive crack growth experiments (Material 2).
Measurements were made on the load-line displacement, load, crack
opening displacement, J-integral, and the unloading compliance using the
techniques described in Refs 16 and 29. Test results for the J-integral and
COD as a function of crack extension are summarized in Figs. 10 through
15. The thickness and side groove studies reported in Figs. 10 and 11 show
that the side grooving procedure did not affect the fracture behavior of
the plate tested at 93 °C (200 °F) as long as the crack extension is measured
at the midsection, where presumably plane-strain conditions prevail. Based
on these results, comprehensive crack growth experiments were performed
on 4T specimens with side grooves either 12.5 or 25 percent deep, since
the crack extension can be measured unambiguously by the compliance
technique. The heat-tint technique was also employed to check some of the
measurements. The variations of J and COD (measured at the original
crack tip) with crack extension are relatively independent of initial crack
length for a/W ranging from 0.5 to 0.85. These are shown in Figs. 12 and
 SHIH ET AL ON CRACK INITIATION AND GROWTH 79

-rrTTTrrrr'•fmjiiniimnniiinnnn/irn
NODES l - »
o) ORIGINAL MESH

b) MESH AT CRACK INITIATION

c) MESH AFTER CRACK INITIATION

FIG. 8—Finite-element model for crack tip blunting and growth.

13. Similar studies were also conducted at higher temperatures—260 °C

(500°F)—and the test results are given in Figs. 14 and 15. The good
agreement between the test results at different temperatures and for
specimens with widely different initial crack lengths suggests that the
J-integral or the COD possesses features suited for characterizing crack
initiation and growth. Further discussion of the experimental investigation
is given in Refs 16 and 29.

Analytical Studies of Compact Specimens

Extensive analytical studies were carried out for all the compact speci-
mens tested. The following discussion focuses on studies based on Tests
T52 and T61. These 4T specimens have remaining ligaments {W — ao) of
85.98 and 40.46 mm (3.385 and 1.593 in.), respectively. All the other
specimens had ligaments between these two limits and, in general, values
80 ELASTIC-PLASTIC FRACTURE

THICKNESS
(mm)
25.4

63.5

101.6

101.6

101.6

25.4 (50%)
101.6

SIDE-GROOVE DEPTH
mm (%)
FIG. 9—Fractured compact specimens of A533B steel tested at 93°C (200°F)—various
thickness and side grooves.

of the fracture parameters calculated from these specimens fall between

the range of results reported for T52 and T61.
The finite-element mesh employed for Specimen T52 is shown in Fig. 16;
that employed for T61 had a finer mesh near the crack tip. The size of the
elements at the crack tip ranges between 2.5 and 5.0 mm (0.1 and 0.2 in.);
this is about 5 to 10 times the blunted crack-tip diameter 6 at initiation.
The stress-strain curve employed for all the calculations is given in Fig. 17.
In the initial filter phase, the crack extension in the finite-element model
is prescribed to follow the load-line displacement versus crack-extension
data obtained from experiments. For Test T52, the basic input to the
finite-element calculations and the relationship reproduced by the finite-
element simulation are in good agreement; they are shown in Fig. 2. Two
sets of calculations were performed, one based on Ji flow theory and the
other employing an incremental form of/z deformation theory of plasticity.
 SHIH ET AL ON CRACK INITIATION AND GROWTH 81

01 02 03 04
—r- T-
SPECIMEN.
1.8
30319 - 3 I in.
O 10

o 30321 - 2 50%S6.
30322- 2
0 5 0 % S.G.

A 30322 - I 4 in. (SIDE MOUNTED GAGE)

30321 - I 4 In

1.4 Q 30319 - I 2 5 in
A 30319 - 4 2 ^ in
0 30XXX-2 2 5 % SG.
0 30XXX- I 2 5 % SG

- 6
1.0

- S

0.6 - O

06
- 3
O

0.4 i-
- 2

0.2

i3'
_L _L
4 6 •i s*"
mm
CRACK EXTENSION

FIG. 10—hresistance curves for A533B Material 1 tested at 93°C (200°F)—4T-plan com-
pact specimens.
82 ELASTIC-PLASTIC FRACTURE

03 04

009

- ooe

I- 0 07
z
2
15 O 0 06

0.05 5

O SPECIMEN
A
O 30319 I in. 0 04

D 30321 5 0 % SG.
30322 • 5 0 % S.G.
CtoG
0 O03
A 30322- I 4 m* "
A 30321 - I 4 in

o 30319 - I z'/ain _l
c\
ao2
30319 - 4
0 30XXX-2 2 5 % SG

0 30XXX - I 2 5 % SG - 0 01
• SIDE MOUNTED GAGE

__l I
4 6 10
CRACK EXTENSION, mm

FIG. n—COD-resistance curves for A533B Material 1 tested at 93°C (20O''F)—4T-plan

compact specimens.
 SHIH ET AL ON CRACK INITIATION AND GROWTH 83

ai 0.2 03 04

I 1 1
In
i.a
A 10
0
1.6 9

A
K 8
1.4
o a

o X 0 7
12 0 o
a
cy o 0 6
10
E o
i A
•t o 5
o 0
SPECIMEN
as
o T-52 Ck
0 4
o T-71 0
as o 0 T-32 O T"
T-21 0
T-31 3
T-22 0
0.4 ""i^
T>0 T-51 A
T-«l 0 2
Ji
0 HEAT TINT X
(T-41 )
0.21
I) J
ic
- 1

1 1 ..,_,.l 1
mm
CRACK EXTENSION

FIG. 12—hresistance curves for A533B Material 2 tested at 93°C {200°F)—4T side-grooved
compact specimens.
84 ELASTIC-PLASTIC FRACTURE

01 0,2 03
2J5 _ —I—

0.09

0.08
2D

0.0 T

1.5 0.06
"5
o 0
009

SPECIMEN

LO T-52 O a 04
T-71 O
T-32 o
T-21
T-31
0 - 0.03
A
T-22 0
^0 T-51 /i
OS
-J: T-61
HEAT TINT
(T-41]
0
X
- 0.02

4 6 10
CRACK EXTENSION, mm

FIG. i3—COD-resistance for A533B Material 2 tested at 93°C (20O''F)—4T side-grooved

compact specimens.
 SHIH ET AL ON CRACK INITIATION AND GROWTH 85

mm
CRACK EXTENSION

FIG. 14—J-resistance curves for A533B Material 2 tested at 260°C (500°F)—4T side-
grooved compact specimens.
86 ELASTIC-PLASTIC FRACTURE

02 &3
2.5

009

2.0 008

1.5 0.06

A O
0 05

A
O 0 04
Q. 10
SPEC IMEN
A
A •'•-82
O 0.03

cf O T-ll

Q5 - O A — Q02
CA

_ 0 01
:»

2 4 6
CRACK EXTENSION, mm

FIG. IS—COD-resistance curves for A533B Material 2 tested at 260°C (500°F)—4T side-
grooved compact specimens.

The calculated load versus load-line deflection based on the two theories of
plasticity are almost identical, and are also in excellent agreement with the
experimental data. They are shown in Fig. 18; here the total load was
computed on basis of the net thickness, which is 76.2 mm (3.0 in.). The
good agreement strongly suggests that test specimen net thickness is the
appropriate value to employ in calculations based on the plane-strain
assumption.
The foregoing calculations were repeated for the deeply cracked con-
figuration based on Test T61. The experimentally determined relationship
which governed the crack growth simulation is shown in Fig. 19. The
calculated applied load versus load-line displacement and the experimental
results are compared in Fig. 20; the calculated curve is slightly lower than
the experimental results, but the trend is in complete agreement.
 SHIH ET AL ON CRACK INITIATION AND GROWTH 87

CRACK TIP

NODE-RELEASE
SPRINGS

FIG. 16—Finite-element model for 4T Compact Specimen T52.

1 1 1
ISO 900
120 - •
800
StlO
• •5
awo TOO 1
!^ A5338-EPRt PUTE
Si 90
QUARTER THICKNESS - 600M
i» LONGITUDINAL
93'CBOO'F)
soof
70

CO 400

() 02 04 06 OS ID
PLASTIC STRAW

FIG. n—Stress-strain curve for A533B plate {Material 2) at 93°C (,200''F).

88 ELASTIC-PLASTIC FRACTURE

(in.)
0 0,1 0.2 0.3 0.4 0.5 0.6 0.7
15001 1 1 1 1 —\ 1

300

1000-

200

. A A A TEST POINTS

500-
100

4 6 8 10 12 14
LOAD LINE DISPLACEMENT (mm)

FIG. 18—Applied load versus load-line displacement for 4T Compact Specimen T52, 25
percent side-grooved; W — ao = 56 mm {3.385 in.).

Crack Opening Displacement and Angle Criteria—The computed

relationship between the crack tip opening displacement 60 defined at the
original crack tip (given by the nodal displacement for Node 5 in Fig. 8)
and the experimental measurements is shown in Fig. 21 for the T52 con-
figuration. The calculated curves are slightly lower than the experimental
values but follow the complete trend of the test data. Interestingly, the
COD obtained from a deformation theory calculation is slightly larger
than that from a flow theory and is in better agreement with the experi-
mental results. This is in accord with the expectation that the deformation
theory of plasticity or nonlinear elasticity will in fact lead to a larger COD
[12]. The calculated and measured COD for Test T61 is shown in Fig. 22
and similarly good agreement is noted.
Two definitions of the crack opening angle (COA) have been employed
in our work. The definition of the average COA, «„, is based on the crack
extension a — Uo measured from the original crack tip and COD, 6„, mea-
sured at the original crack tip. The local COA a; is based on the opening
 SHIH ET AL ON CRACK INITIATION AND GROWTH 89

(in)
0.05 0.10 0.15 0.20
20 1—

0.7
SIMULATED // TEST POINTS
CURVE~

0.6

0.5

0.4

- 0.3

- 0.2

- 0.1

2 3 4 5 6
CRACK EXTENSION (mm)

FIG. 19—Load-lute displacement versus crack extension for 4T Compact Specimen T61.

displacement 6i measured at a fixed distance, Aa, behind the current crack

tip. Thus, as shown in Fig. 8

tto = (6o ~ doi)/(a — Co) cti = 8i/Aa (10)«

where doi denote the opening displacement at initiation.

it was noted in the Potential Fracture Criteria section that the plastic
strain increments for an extending crack have a (1/r) In (1/r) singularity
which gives rise to a crack opening profile that has the form r In (1/r) [12].
Thus while the opening displacement vanishes at the tip, the opening
profile has a vertical tangent at the crack tip corresponding to an opening
angle of x radians. The angle cannot be defined unambiguously near the
^The local and average COA are computational measures, at slightiy different scales, of the
angle between the fractured surfaces behind the advancing crack tip.
90 ELASTIC-PLASTIC FRACTURE

(in.I
0.1 0.2 0.3 0.4 0.5 0.6 0.7
150

TEST POINTS 30
A A A A AA
\. A

DEFORMATION
AND FLO* THEORY

20

10

4 6 8 10 12
LOAD LINE DISPLACEMENT (mml

FIG. 20—Applied load versus load-line displacement for 4T Compact Specimen 161, 25
percent side-grooved; VI — a.o = 40 mm (1.593 in.).

crack tip; however, at a small but finite distance away fi-om the tip, a
meaningful definition is possible as our subsequent discussion shows.
The calculated COA for Test T52 defined by Eq 10 from both deforma-
tion and flow theory analyses is shown in Fig. 23. The COA varies con-
siderably during the initial stages of crack extension but appears to ap-
proach a constant value with further growth. As expected, the angle
computed from the deformation analysis is slightly larger. The crack-tip
element for the T52 configuration has lengths of about 5 mm (0.2 in.),
which is about 10 tim^s the COD at initiation. Perhaps this level of repre-
sentation is not fine ent^ugh to capture an adequate description of the near-
tip deformation. Thus, based on these calculationsf, the COA's approach
a value of about 0.21 rad after 3 mm (0.12 in.) of crack extension. With a
finer mesh, the angles would presumably approach a constant after smaller
extensions. This expectation is in fact borne out in the calculations for
T61; here the crack tip element has size of the order of 2.5 mm (0.1 in.).
In this case the angles approach about 0.23 rad after 1.5 mm (0.06 in.) of
crack extension as shown in Fig. 24. The computed angles for the T61
specimen are slightly larger than the corresponding angles for the T52
 SHIH ET AL ON CRACK INITIATION AND GROWTH 91

(in.)
0.1 0.2 0.3 0.4 0.5
0.30

0.25

0.20

0.15 .E

o 3
0.10

JjFLOW THEORY - 0.05

4 6 -8 ID
CRACK EXTENSION (mm)

FIG. 21—Crack opening displacement versus crack extension for Specimen T52.

specimen, and are consistent with the slight mesh sensitivity associated
with the CO A. Calculations for T61 with larger near-tip elements gave
angles that are in closer agreement with the values for T52. On the basis of
these results, the COA appears to be a viable toughness parameter for
crack growth; higher values of the angle correspond to higher resistance
to crack growth.
J-Integral and dJ/da Criteria—Crack-growth calculations based on a h
flow theory, for the T52 configuration, showed that the J-integral com-
puted from a remote contour, J«, and that computed using the Merkle-
Corten expression [31], JUQ, are in excellent agreement for significant
intervals of crack extension. Subsequent calculations employing a Ji
deformation theory gave values of / that are essentially identical to those
values obtained on the basis of/i flow theory. The variation of/ with load-
line displacement for deformation and flow theories and test measure-
ments is given in Fig. 25; the experimental / is determined from the
measured load-displacement record using the Merkle-Corten expression
based on net thickness. Crack-growth studies for the T61 configuration
confirmed the foregoing observations; the results are shown in Fig. 26.
These observations suggest that the highly nonproportional deformation
due to crack growth and to the elastic unloading at the wake of the ad-
92 ELASTIC-PLASTIC FRACTURE

(in)
0.05 0.10 0.15 0.20
1.8 —I 1 1 1 =10.07

1.6 -
0.06

0.05

0.04

JzFLOW THEORY

0.03

0.02

0.01

1 2 3 4 5
CRACK EXTENSION (mm)

FIG. 22—Crack opening displacement versus crack extension for Specimen T61.

vancing crack is rather localized (of the order of the crack extension), and
does not appear to appreciably influence the region at distances greater
than several times the blunted tip diameter away from the crack tip.
Finite-strain studies, based on &h flow theory by McMeeking [32], showed
that the J-integral is path dependent at distances less than 5 So from the
tip. Based on these observations, it will be convenient to distinguish two
regions in the vicinity of the crack tip. At distances less than 5 6o, the',
deformation is highly nonproportional and will not be characterizable
by the HRR field; this will be defined as the crack-tip field. Beyond this
tip region, the deformation is characterizable by the HRR singulaiity if
certain conditions are met [15] (also discussed earlier in the Potential Frac-
ture Criteria section); this region will be defined as the near field. The size
of the near field will in general depend on specimen geometry, material
strain hardening, the plastic zone size, and the amount of crack growth.
The region at large distances from the crack is called the far field. An
illustration of these regions is given in Fig. 27.
To distinguish these different regions, a typical mesh configuration in
the vicinity of the crack is shown in Fig. 28. The characteristic element
 SHIH ET AL ON CRACK INITIATION AND GROWTH 93

(in.)
0.1 02 0.3 0.4 0.5 0.6

0.3

\
\ a D
a a a
A A

Q2 o -
- - -O- - 9 _ ^ _ - ? " ?
_
O O
"Xx X X ><

EXPERIMENT
01
FLOW THEORY

DEFORMATION
THEORY

0 2 4 6 8 10 12 14
CRACK EXTENSION (mm)

FIG. 23—Crack opening angle versus crack extension for Specimen T52.

size, /, is typically about 5 5o. Thus the J-integral evaluated in the tip
field (< 5 6o) is termed /dp and that evaluated in the near field (> 5 6o)
is termed /„f; /ff is evaluated along contours remotefi-omthe crack tip.
The variation of the Ts, evaluated along the different contours, with
crack extension is shown in Figs. 29 and 30 for the T52 and T61 configura-
tions, respectively, /dp deviates from path independence almost immediately
and the ratio Jtip/Jn approaches zero after some crack growth. However,
/nf is in good agreement with the Ts evaluated along remote contours and
the / computed from the Merkle-Corten expression for crack extensions up
to 6 percent of the remaining ligament. The path independence of the
near-field /-integral suggests that the deformation in the near field is
essentially proportional and that the J-integral would be an appropriate
parameter for characterizing the near-tip deformation during crack growth.
To obtain a direct comparison between the predictions of deformation
and flow theories of plasticity, the foregoing calculations were repeated
with an incremental form of /a deformation theory. The far-field Ts are
essentially identical with those summarized in Figs. 29 and 30 for flow
theory. The near-field Ts are in good agreement up to about 6 percent of
94 ELASTIC-PLASTIC FRACTURE

(in.)
0.05 OK) 0.15 O20 025

03 - »

O N
a a
X O 9 9 9
X 0 8 ^ o Ox
xxxxx x - x - i - -
02

&

— EXPERIMENT
X Qo •)
FLOW
o a^ I THEORY

DEFORMATION
THEORY
H2

0 1 2 3 4 5 6 7
CRACK EXTENSION (mm)

FIG. 24—Crack opening angle versus crack extension for Specimen T61.

crack growth. Beyond this range, the Ts from deformation theory calcula-
tions remain, for practical purposes, path independent. The slight deviation
from path independence is probably due to finite-element computational
and discretization errors. A direct comparison of the predictions of de-
formation and flow theory for Specimen T52 is given in Fig. 31. The
near-field and far-field ys are evaluated along contours that advance at
the same rate as the crack tip. While the ^ s from deformation theory show
slight path dependence, /„f (flow theory) deviated from Jn and /MC by about
10 percent at 5-mm (0.2 in.) crack extension. At 8-mm (0.32 in.) crack
extension the deviation of /„f (flow theory) from the remote / s exceeds
20 percent. The former and latter correspond to crack extension of 6 and
10 percent, respectively, of the remaining ligament.
The foregoing calculations demonstrate that the predictions of deforma-
tion and flow theories of plasticity are in agreement for a limited range of
crack growth. Therefore, the near-field environment during initiation and
some amount of growth is characterizable by the J-integral. One of the
requirements for a/-controlled growth concerns the slope of the/ resistance
curve. The variation of dJ/da with crack extension for the T52 and T61
 SHIH ET AL ON CRACK INITIATION AND GROWTH 95

(in.)
3 0.1 0.2 0.3 0.4 0,5 0.6
3Q000
1 1 1 1 1 1
5_
X J„
I FLOW THEORY
OJMC -— 25000

4—
DEFORMATION THEORY
•JMC _
- 20JD00

3
- 15000

2
- 10000

1
- 5000
X
|-« START OF CRACK EXTENSION
ft
I
,-f'' 2
1
4 6
1 1
8
1
10
1
12 14 16
LOAD LINE DISPLACMENT (mm)
FIG. 25—J-integral versus load-line deflection for Specimen T52.

(in.)
0.1 0.2 0.3 0.4 05 0.6 0.7
2.0
1 1 1 1 1 1 1
X J„ _
- 10000
FLOW THEORY
° JMC
" "Iff
i
1 DEFORMATION THEORY ,3 - 8000

~ •"•" EXrERlMLNTAL ^

- 6000 7
I 1.0 —
X
-
y
X -~ 4000

_ X. ' - ^ - 2000
X
^^ START OF CRACK EXTENSION

6^T
.^^ 1 1 1 1 1 1 1 1
4 6 8 10 12 14 16 18 20
LOAD LINE DISPLACEMENT (mm)

FIG. 2f>—J-integral versus load-line deflection for Spedtnen T61.

96 ELASTIC-PLASTIC FRACTURE

•NEAR FIELD CHARACTERISED BY J„,

•FAR FIELD CHARACTERISED BY J „
•TIP FIELD; J , i p — 0 AS r — 0

FAR FIELD

/'
/
/ Ao
ELASTIC U J NLAK H L L U \
UNIJOADING.. n r r ^
/UNLOADING (GOVERNED BY HRR \
( SINGULARITY )

v^...\ TIP FIELD,(NON-PROPORTIONAI/

PLASTIC LOADING) /

- 5 8^ 0 58.

FIG. 27—Schematic of the fields surrounding a growing crack.

FIG. 28—Paths for J-integral evaluations.

 SHIH ET AL ON CRACK INITIATION AND GROWTH 97

(In.)
0.1 0.2 0.3 0.4
2.0 T" 1— I

1.8
- 10000

1.4 8000

L A
- 6000 e
S. i.oh

0.8
+ EXPERIMENT
4000
0 Jmc
0.6
• J6
X J4
0.4 ==0.06-
W-a„ • J3 2000
A Jz
0.2,

4 6 8 10 12
CRACK EXTENSION (mm)

FIG. 79—J-mtegral evaluated at different contours for Compact Specimen T52.

configurations is shown in Figs. 32 and 33. Although the two configura-

tions have very diff'erent crack lengths, the trends of the curves are very
similar. In particular, (dJ/da)nt rapidly drops to zero after about 6 percent
crack extension while {dJ/da)tf appears to level off. The results also
suggest that the dJ/da measured from experiments [5] is a meaningful
tearing modulus parameter only for a restricted range of crack growth.
Although the measured dJ/da may remain finite or constant after this
range of growrth, (dj/da)^ rapidly declines to zero and this violates the
requirements of a/-controlled growth [15].
Sensitivity of Mesh and Step Size—In a series of calculations, the mesh-
size and the crack propagation step-size were varied for the T52 crack
98 ELASTIC-PLASTIC FRACTURE

(in.)

0.05 0.10 0.15 0.20

~ ^ 10,000

1.6-

4- 8000

•9 '
_ 1.2- • 9
6000;
- ; 1-0
AAA A A
»AA
0.8 9 5:-"
4000
+ EXPERIMENT
0.6
1 0 Jmc
• J6
0.4
X J4 2000
= 0.06-
J* w-o. O Jj
0.2 A J2
/

0 1 2 3 4 5
CRACK EXTENSION (mm)
FIG. 30—J-integral evaluated at different contours for Compact Specimen T61.

configuration. The coarse mesh typically has dimensions that are twice as
large as the regular mesh while the fine mesh is typically twice as fine as
the regular mesh; the large step-size is about twice as large as the regular
step-size. The load-deflection relationship is only slightly sensitive to the
mesh variation and the step-size variation; this is shown in Fig. 34. The
COA measured at the original crack tip and consequently the average
COA, Oo, are also relatively insensitive to mesh and step-size variation;
however, the local COA, ai, is moderately sensitive. The sensitivity of these
COD-based parameters is summarized in Figs. 35 and 36. For the same
mesh and step-size variations, the relationship between / (evaluated dis-
tances of the order of 10 So) and crack extension is shown in Fig. 37. It is
apparent that the J-integral is only slightly affected by mesh and step-size
variation. Thus from a finite-element modeling viewpoint, the J-integral
 SHIH ET AL ON CRACK INITIATION AND GROWTH 99

(in.)
0,2 03 04
1.8
10,000

L6 X
/.o''^
1.41- //y 8000

1.2
/ •
6000
10

_. 0.8 / •
<i
ac ,«
o - 4000
LiJ

FLOW THEORY
^ 06 r=0 375 m.
oo r =0.625 In
2000
DEFORMATION THEORY
0.4
r = 0 375, 0 625 m
JMC * N D J AT REMOTE
PATHS FOR FLOW AND
0.2 DEFORMATION THEORY
_1_ I _L _L J_
0 2 4 6 8 10 12
o-flj (mm)

FIG. 31—J-integral evaluatedfor contours advancing with crack tip. Compact Specimen T52.

and the COD (or average COA) defined at the original crack tip are
attractive fracture parameters.
The Evaluation Phase—In the next series of calculations, the COD-
based criterion was employed to govern the crack extension for Configura-
tion T61. The crack initiates at a critical value of the COD, 6o,, and its
propagation is determined by the critical value of the average angle ao.
From the initial filter phase, the upper and lower bounds of the COD at
initiation are 0.508 and 0.417 mm (0.02 and 0.0164 in.), respectively. The
angles range from 0.21 to 0.33 rad. Three crack-growth calculations were
carried out using these limiting values to control the rate of crack extension.
An intermediate value of 6o< = 0.508 mm (0.02 in.) and oo = 0.21 rad
was also used. The prescribed relationship between COD (measured at the
original crack tip) and the crack extension is illustrated in Fig. 38. The
100 ELASTIC-PLASTIC FRACTURE

(in.)
0.1 0.2 0.3 0.4 0.5
500 — I —
70
X- -X FAR FIELD

o- -O NEAR FIELD

EXPERIMENT
60
400

50

300 - Av X X

40

30
200

20

10

4 6 8 10 12
CRACK EXTENSION ( m m l

FIG. 32—dJ/da for Compact Specimen T52, a/W = 0.577.

calculated load-displacement relationship and the experimental measure-

ments are compared in Fig. 39. Curves I and II reach maximum load at
12.7 mm (0.5 in.) of crack extension, and follow the trend of the experi-
mental results while Curve III continues to rise. This is not unexpected
since the average angle, ao, for the third calculation was deliberately fixed
at a value much higher than the values typically encountered in the initial
filter phase.
In another series of crack growth analyses, the crack extension in Speci-
men T52 was governed by the /-resistance curve. The / (far field) versus
crack extension (a — a a) obtained in the initial filter was employed to
control the rate of crack extension. The governing curve is given by/MC in
Fig. 29. A comparison of the calculated load-displacement relationship and
the test measurements is given in Fig. 40; the agreement is remarkably
 SHIH ET AL ON CRACK INITIATION AND GROWTH 101

(in)
0.05 0.10 0.15 0.20
500
- 70

60
400-

50

_ 300
40

50
200

20
X X FAR FIELD

O—G NEAR FIELD

A EXPERIMENT

2 3 4 "j 6
CRACK EXTENSION(mm)

FIG. 33—dJ/dSL for Compact Specimen T61, a/W = 0.50/.

good. These computations were repeated using a COD resistance criterion;

the same general agreement with experiment was observed, as shown in
Fig. 40.

Analytical and Experimental Investigations for Center-Cracked Panels

Experiments were carried out for center-cracked panel geometries, with
some interesting results. It was found that for A533B panels with short
remaining ligament [about 25 mm (1 in.)], the failure mode in the fully
plastic state is apt to be due to slip along shear bands and subsequent
fracture along 45-deg lines as opposed to crack growth ahead of the crack.
Side-grooved and smooth center-cracked panels made from A533B steel
(Material 2) were tested at 93°C (200°F). The specimens have a total
width, 2W, of 150 mm (6 in.) and a thickness of 50 mm (2 in.). Some
102 ELASTIC-PLASTIC FRACTURE

(in.)
ai 02 03 0.4 05 06
1500
T T
A TEST POINTS
REFINED MESH 300
CRUDE MESH
X REFINED MESH, LARGE STEP

250

1000

200

150

500
100

- 50

0 2 4 6 8 10 12 14 16
LOAD LINE DISPLACEMENT (mm)

FIG. 34—Sensitivity of load-deflection to mesh and step size for Specimen T52.

specimens were side-grooved to 25 percent, and the total remaining liga-

ments, 2iW — a), ranged from 25 to 50 mm (1 to 2 in.). In all the tests,
intense shear deformation developed along approximately 45-deg bands,
emanating from the original crack tip as shown in Fig. 41. Examination of
the fracture surfaces and of silicone rubber castings of the crack tips
revealed that initiation of cracking occurred after the crack tip had opened
into a three-cornered profile as illustrated by the insert in Fig. 41. Exten-
sion at mid-thickness occurred at a COA in excess of the COA observed in
compact specimens. At the surface of smooth-faced specimens, separation
had occurred in the direction of the shear bands. Apparently the shear
deformation controlled both crack initiation and extension. The value of
the J-integral as determined from the load-displacement record is approxi-
mately 1.8 MJ/m^ (10 000 in.-lb/in.^) at the point shown in the figure,
where the experiment was terminated. Finite-element calculations for the
identical configuration indeed showed that the deformation is concentrated
on narrow shear bands emanating from the crack tip at 45 deg to the crack
plane; these deformations are shown in Fig. 42. An examination of the
stress fields and the crack-tip profile revealed that crack-tip fields do not
 SHIH ET AL ON CRACK INITIATION AND GROWTH 103

(in.)
0.1 0.2 0.3 0.4 05 0.6

0.30
— REFINED MESH
•-CRUDE MESH
X REFINED MESH, LARGE STEP SIZE
0.25
E
6 6

0.20

5 4
- 0.15

- 0.10

- 0.05

0 2 4 6 6 10 12 14 16
CRACK EXTENSION (mm)

FIG. 35—Sensitivity of COD measured at original crack tip to mesh and step size in
Specimen T52.

approach the HRR fields, but appear to be strongly influenced by the

global shear deformation. It is clear that the minimum size requirements
for fracture were not met in the center-cracked panel and this is discussed
further in the next section.

Size Requirements for Fracture Toughness Testing in the Fully Plastic

Range
The success of a characterizing parameter approach for fracture based
on / or 5 requires that the Hutchinson-Ride-Rosengren (HRR) singular
field govern over the fracture process zone. In the case of small-scale
yielding, the HRR field is embedded in the elastic singular A'-field [33],
and consequently / and K are related by [8]

J = K^/E' (11)

where E' = E for plane-stress and E' = E/{1 — v^) for plane-strain
conditions. For small-scale yielding, the invariance of the critical stress
intensity factor, Kic, to crack length and specimen geometry strongly
104 ELASTIC-PLASTIC FRACTURE

c 0.1 0.2 0.3 0.4 0.5 0.6

1 1 1 1 1 1

0.4 „ REFINED MESH

xxxx CRUDE MESH
oooo REFINED MESH, LARGE STEP SIZE

X FLOW THEORY

1°
0.3 - I X

\x
\\ °* \
sA,x '* X X X X X
Xl O O o ° ' } .
0.2
— }••

n 1 1 1 1 1 1 1 1
0 2 4 6 8 10 12 14
CRACK EXTENSION (mm)

FIG. 36—Sensitivity of average and local crack opening angles to mesh and step size in
Specimen T52.

suggests that the HRR field governs the deformation state in the fracture
process zone. In this instance, Kic is a valid fracture toughness parameter
if the plastic zone size, r?, is small compared to crack length, thickness,
and remaining ligament. Since the crack-tip fields can also be represented
in terms of J and 6 via Eqs 1 and 3, the Ki^ criterion would also imply a
Jic or 6ic fracture criterion. It also follows from Eq 11 that

Ju — Kic^/E' (12)

In small-scale yielding, there is no intrinsic advantage in characterizing

toughness in terms of/k or 6ic since Ku is an adequate toughness parameter,
and elastic solutions to numerous crack configurations are available in
terms of K. However, if toughness characterization beyond the regime of
small-scale yielding is considered, as is the case in this study, then charac-
terization ifr4erms of Ju or 6ic becomes advantageous as the following
discussion will show.
 SHIH ET AL ON CRACK INITIATION AND GROWTH 105

in I
r 01 0 2 03 04 05 0.6
30,000
1 1 1 1 1 1
5 J-CONTOUR (NEAR FIELD)
• REFINED MESH
—^— -CRUDE MESH 25,000
X REFINED MESH, LARGE STEP SIZE
4

- 20,000
—3
S
- 3 „00^
_l -
OH x-^,'-^ -
- 15,000 a
o y ^ ^
UJ
y ^ ^
z y ^ ^
—3
• 2
.^y^ - 10,000

1,
-5000

n 1
6 8 10 14 16
CRACK EXTENSION (mm)

FIG. 37—Sensitivity ofj-integral to mesh and step size in Specimen T52.

In the fully plastic range, the HRR singularity is embedded in a plastic

zone that extends throughout the remaining ligament. This situation is
illustrated in Fig. 43. There is no elastic ^-field in this case, and any
toughness characterization must now be made in terms of HRR field
parameters and the J-integral or 5 < are therefore natural candidates. Now
some important points regarding the size of the HRR fields must be made.
In small-scale yielding, the size of the plastic zone rp, and therefore
of the HRR field To, is governed by the ratio of the applied load to the load
corresponding to net section yield and is otherwise relatively independent of
the specimen geometry and the strain-hardening properties. The size
requirements [ASTM Test for Plane-Strain Fracture Toughness of Metallic
Materials (E 399-74)] for a valid Ku test can therefore be stated in terms of
the ratio of (Ku/ao)^. In contrast, our experimental results suggest that
the size of the HRR field beyond the regime of small-scale plasticity is
influenced by specimen geometry and strain-hardening properties. This has
also been alluded to by McClintock [34] and Rice [35]. Thus the specimen
size requirements for a valid toughness test in the fully plastic range will
differ for different material properties and crack geometry.
Under general yield or fully plastic conditions, the only characteristic
length is the COD. The size requirements may be stated in terms of this
106 ELASTIC-PLASTIC FRACTURE

(mils)
50 100 150
1.6
60

1.4 - 8=20+0.2lxAfl

50

40

30.-=

20
INPUT CURVES
CALCULATED CURVES
- 10
01

0 1 2 3 4
CRACK EXTENSION (mm)

FIG. 38—Various COD-resistance curves usedfor controlling crack growth in Specimen T61.

characteristic length 6,. For the HRR parameters (namely, / or di) to

characterize the near-tip field, the crack length, remaining ligament, and
thickness must be large compared to the crack tip opening displacement.
Just how large the three dimensions must be compared to the crack opening
displacement 6, appears to depend on the material strain-hardening
properties and specimen geometry. In the fully plastic center-cracked
panel, the deformation is concentrated along slip planes at 45 deg from
the crack plane. For this configuration, the global deformation will domi-
nate the deformation state over a considerable interval in the vicinity of the
crack tip. The situation with the single-edge crack subjected to bending is
quite .different. Here the deformation in the fully plastic state closely
resembles the Green and Hundy slipline field solution [36]. In the vicinity
of the crack tip, the imposed deformation, due to the Green and Hundy
field, is very similar to the HRR field. Thus one expects that the HRR field
will govern over relatively larger intervals in the bend specimen, compared
with the center-cracked panel at similar levels of loading. The global
deformation fields for the two geometries are illustrated in Fig. 3.
To explore these issues more precisely, finite-element calculations were
carried out for the two geometries with a/ W ratio of 0.5 and a ligament of
 SHIH ET AL ON CRACK INITIATION AND GROWTH 107

(in.)
c 0.1 02 03 04 0.5 as
180 1 1 1 1 1 1- 40/)00

160 -

m
140 - TEST POINTS
30,000
\
>-^^^ n T
120 —
A
A ^— I
ty"^

~ 100 - J
20,000
O
o 80

1 I- CODj - 16.4 mils, a = 0.21 rod

60 - / TL- CODj - 20 mils, 0:= 0.21 rod
/ M- CODj = 20 mils , a = 0.33 rod
- 10,000
40

20

n1 1 1 1 1 1 1 1
4 6 8 10 12 14 16
LOAD LINE DISPLACEMENT (mm)
FIG. 39—Comparison of load-deflection relationships generated by crack-growth simulation
based on COD-resistance curve for Specimen T61.

26 mm (1 in.). Calculations based on elastic-perfectly plastic idealization

attained limit loads that are 2 to 3 percent higher than the theoretical limit
loads for the center-cracked panel and the single-edge cracked bend bar;
the details are given in Fig. 44. The variation of the tensile stress Oyy, or
the 'crack opening stress' across the remaining ligament under fully plastic
conditions at the same levelof / is shown in Fig. 45. The Prandtl field,
which represents the limit of the HRR field for elastic-perfectly plastic
materials, requires the tensile stress to reach a maximum of 2.97 Oa ahead
of the crack. The HRR or Prandtl field is attained in the bend bar. For
the center cracked panel, however, the stress at a distance of 0.02 {W —
flo) ahead of the crack is only approximately twice the yield stress. For the
latter configuration, the Prandtl field is attained over a rather small
interval. The introduction of strain hardening affects the stress field some-
108 ELASTIC-PLASTIC FRACTURE

01 02 03 0.4 05 06
1500

300
X BASED ON J
BASED ON COD
250

1000 -

200

150

TEST P O I N T S ^ - ^ ^ A "*^
500
/
100

-START OF CRACK EXTENSION

- 50

0 2 4 6 8 10 12 14 16
LOAD LINE DISPLACEMENT (mm)

FIG. 40—Comparison of toad-deflection relationships generated by crack-growth simulations

based on J and COD-resistance curves for Specimen T52.

what. For a power-law curve corresponding to « = 10, the variation of the

tensile stress for the two geometries at approximately about the same level
of applied / is also shown in Fig. 45. It is clear that, between the two
configurations, the HRR field associated with the bend bar dominates over
a comparatively larger interval. The calculations are repeated for the
compact specimen and the center-cracked panel, both with a/ W ratios of
0.7 and a remaining ligament of 19 mm (0.75 in.), using the stress-strain
curve for A533B steels. The stress fields are shown in Fig. 46.
These calculations for stationary cracks show that, when the uncracked
ligament is subjected primarily to bending, the HRR field dominates over a
comparatively large interval. If the loading in the ligament is primarily
tensile, the HRR field is attained over comparatively small intervals and
the triaxial stress state is relatively low. Thus the minimum size require-
ments for fracture toughness testing to evaluate/ic or Sic will be significantly
larger for the center-cracked panel. We may also infer from the stationary
crack solutions that the size requirements for obtaining J or COD resistance
curves will vary with crack configuration and material properties.
 SHIH ET AL ON CRACK INITIATION AND GROWTH 109

g^'!?«^~
FIG. 41—View of deformed crack tip in center-cracked panel specimen, showing shear
bands emanating from crack tip at 45 deg; W — ao = 19.6 mm (0.784 in.). Top photo is
profile of silicon rubber casting of crack tip.
110 ELASTIC-PLASTIC FRACTURE

FIG. 42—Effective strain contours from finite-element solution for center-cracked panel
shown in Fig. 41 at 1 = 1.75 MJ/m^ (10 000 in. -lb/in. ^).

REGION DOMINATED BY ZONE DOMINATED BY

ELASTIC SINGULARITY HRR SINGULARITY

SMALL-SCALE YIELDING LARGE-SCALE PLASTICITY

FIG. 43—Crack tip fields.

It may be noted that in the early stages of these same calculations the
HRR field was attained in all the configurations under conditions of small-
scale plasticity.' These results are consistent with that absolute size re-
quirement (independent of specimen geometry) placed on valid Kic tests.

Assessment of Fracture Parameters

The requirements that viable fracture criteria must satisfy have been
discussed already in the Strategy section. A comparison of the contending
' i n the center-cracked panel, the crack opening stress oyy gradually 'unloads' due to the
stress redistribution caused by the intense formation of the intense shear band as the limit
load is approached.
 SHIH ET AL ON CRACK INITIATION AND GROWTH 111

3.0

/ 'DOUBLE EDGE CRACK

2.5

2.0
^NET

/ ^FOUR-POINT BEND BAR

LS

/ ^CENTER-CRACKED PANEL

1.0

0.5

n ' 1 1 1 1 1
0 10 20 30 40 50
NORMALIZED DISPLACEMENTS (mils)

FIG. 44—Limit loads for cracked bend bar, center-cracked panel, and double-edge cracked
panel.

fracture parameters on the basis of these requirements is summarized in

Table 1. Most of the parameters satisfy the first six requirements. This is
not unexpected since all these parameters, when appropriately defined and
employed, characterize the crack near-field. However, in terms of the last
four requirements, the trend is clear. From the computation and measure-
ment point of view, the viable candidates are the /-integral and the COD
and COA.
The results from our analytical and experimental investigations support
the conclusions reached by Begley and Landes [/] that the onset of plane-
strain flat fracture under small- or large-scale yielding conditions is charac-
terizable by/ic or equivalently by 6ic. Thus our experimental and analytical
investigations gave values ofJu ranging from 0.175 to 0.263 MJ/m^ (1000
to 1500 in.-lb/in.2) and 6ic ranging from 0.25 to 0.40 mm (0.010 to 0.016
in.) for A533B steels at 93 °C (200°F). These values are consistent with
J^ic toughness numbers reported elsewhere [37\. In addition, our studies
112 ELASTIC-PLASTIC FRACTURE

yy/(^o

4-POINT BEND BAR

-2.0 _L
0.2 0.4 0.6 0.8 1.0
X/lW-Oo)

FIG. 45—Variation of tensile stress across remaining ligament for 4-point bend bar and
center-cracked panel for elastic-perfectfy plastic and strain-hardening materials.

showed that some amount of stable crack growth is characterizable by /

or COD resistance curves of the COA [16,29]. Paris et al [5] proposed to
characterize resistance to crack growth by the tearing modulus [T = (JE/a^)
•(dJ/da)]. In subsequent paragraphs we discuss the requirements for a / or
COD characterization of crack growth and possible limitations.
In arguments based on considerations involving a deformation theory of
plasticity, Hutchinson and Paris [15] identified the requirements for J-
controlled crack growth. The first requires that crack extension, Aa, be
small compared to the characteristic radius, R, of the region controlled
by the HRR singularity. In addition, dJ/da must be large compared to
J/r (see the Potential Fracture Criteria section for details). The latter
requirement may be stated in terms of a material-based length quantity D
defined by
 SHIH ET AL ON CRACK INITIATION AND GROWTH 113

J_ dJ 1
(13)
D da J

Thus the two requirements for a /-controlled growth are

Aa « R (13a)

and

Z) « r < /? (13*)

The deformation field ahead of an extending crack was derived by Rice

[10] using a J2 flow theory of plasticity. The study revealed that the near
field of an extending crack may be characterized by the local COA (see
the Potential Fracture Criteria section for more details). On the basis of
Rice's work, Shih et al [16] suggested the use of COA as a stable growth

5.0

>
y X

4.0 --

3.0 -
COMPACT /
SPECIMEN " X ^ / j
yy't^o

2.0 -
CENTER-CRACKED
PANEL \

1.0

1 1 / l 1
0.2 0.4 0.6 0.8 1.0
X/(W-Oo)
FIG. 46—Variation of tensile stress across remaining ligament for compact specimen and
center-cracked panel for A533B steels.
114 ELASTIC-PLASTIC FRACTURE

•8

".^Idi

.i o o
rt B B

If B

"ll I
nQ
o
•O

u B

O
o
go
11
si
IJll -.;
B

r§i -.11.
Ill
m I a
t!
a 5b

II
inii
HP If ill
.§ 8 S E
5S

liili1! m §1
 SHIH ET AL ON CRACK INITIATION AND GROWTH 115

parameter. The requirement for a COA-controlIed growth is, as shown in

the Potential Fracture Criteria section

The crack opening profile has a vertical tangent at the crack tip (correspond-
ing to a COA of IT rad at the tip) and thus the angle cannot be defined in
any meaningful manner close to the current crack tip. If the requirement
given by Eq 14 is satisfied, however, then the crack profile exhibits a well-
lefined angle at a small but finite distance away from the crack tip.
The requirements for COA-controlled growth can be restated in terms of
a tearing modulus Ti based on the COA

Ta = f f » 1 (15)
da <To

In other words, the crack opening angle, db/da, must be large compared
to the yield stress divided by the elastic modulus (or the yield strain). For
A533B steels on the upper shelf, direct measurements by rubber infiltra-
tion [16\ and finite-element crack growth calculations reported in this
paper showed COA's of the order of 0.2 to 0.3 rad. This is significantly
larger than Oo/E, which is about 0.002.
From Eq 13A, the requirement for a/-controlled growth can be restated
as
dJE
Tj = - - » 1 (16)
da ot
By exploiting deformation theory for crack growth, namely

dJ db,
— ^= Ot CTo (17)
da da

it may be argued that the inequalities given by Eqs 15 and 16 are equivalent
when the near field is governed by the HRR field. For A533B steels at the
upper shelf, J j ranges from 100 to 300 and T5 ranges from 100 to ISO.'"
Thus Tj or Tt, can be viewed as toughness parameters for stable crack
growth; J j is more appealing because it is a more fundamental quantity
and is relatively more constant for a given material. How large Ti or Tj
must be for a /-controlled or COD-controlled growth is yet to be explored.
This will be the subject of further investigations.
'"jT/ generally falls between 20 and 150 for a wide variety of materials reported in Ref 5.
116 ELASTIC-PLASTIC FRACTURE

An additional limitation placed on the J resistance approach is expressed

by Eq 13a. Our finite-element investigations based on actual experimental
data suggest that, for A533B steels in the upper shelf, the / resistance ap-
proach will be valid for crack growth up to 6 percent of the remaining
ligament when the mode of loading is primarily bending. The amount of
crack extension where the / resistance approach is valid is expected to
depend on specimen geometry and material properties, and in particular
on the strain hardening. A similar restriction on the COA's does not
appear to be necessary.
It is instructive to compare the / resistance and the COD resistance
approaches with the onset of growth and stable growth. In terms of tMfe
J-integral, therfe are basically two quantities, /„f and J«- Jnt is evaluated in
the region dominated by the HRR singularity while /« is evaluated along a
remote contour. The derivatives of these quantities are (dj/da)nf and
{dJ/da)fr. Our analytical studies show that Jft is the quantity measured
in experimental investigations [1-3,5,16.29]. Similarly, the dJ/da referred
to in Ref 5 is in this context {dj/da)n. There are also two possible defini-
tions of the COD. One definition is based on the opening displacement at
the original crack tip, bo, while the other is based on the opening dis-
placement at a fixed distance behind the current crack tip 6/. The average
crack opening angle, Uo, is determined from the first definition while the
local crack opening angle, ai, is determined from the latter.
A typical variation of J, dJ/da, b, and a with crack extension is illus-
trated in Fig. 47. These figures show that while /« continues to increase,
J at begins to level off after some crack extension. Thus (dJ/da) „f falls to zero
and violates one of the requkements for /-controlled growth expressed by
Eq 16. On the other hand, both the crack opening angles, ao and ai, after
an initial transient remain constant for a considerable range of crack growth.
This satisfies the requirement for COA-controUed growth specified by
Eq 15. It also appears that beyond the initial stage of crack growth, a
constant critical value of the average or local COA may be employed to
characterize stable growth. This is not unexpected since the COA is
derived from more fundamental considerations.
We also note that, while the local crack opening angle ai is the funda-
mental measure of the crack growth fields, it is not an easily measured
quantity. However, the average crack opening angle ao is easily obtained
from linear variable differential transformer (LVDT) measurements over
the entire range of crack extension [29]. Our calculations show that both
angles are essentially identical for crack extension up to 10 times the
blunted tip opening, boi, at initiation. Therefore, even though the average
angle cto is not a fundamental measure of the crack field, it is an attractive
quantity from a practical viewpoint. Similarly {dJ/da)fs is a parameter
readily determined from experimental measurements, while {dJ/da)at can
be obtained only through theoretical and numerical crack-growth studies.
 SHIH ET AL ON CRACK INITIATION AND GROWTH 117

A) J-RESISTANCE CURVES FOR B) COD-RESISTANCE CURVES BASED

NEAR AND FAR FIELDS. ON OPENING AT ORIGINAL CRACK
TIP (80 ) AND AT A FIXED
DISTANCE BEHIND CURRENT TIP
(8/).

^ For Field
do

Ntor Field \

Aa
c) dJ/do DERIVED FROM NEAR D) COA'S Be AND a^, DERIVED
AND FAR FIELD J . FROM AND 8 /

FIG. 47—Typical behavior of J and COD-based parameters during crack growth.

This study shows that it is appropriate to characterize the toughness

properties of ductile metals in terms of initiation and growth. The material
toughness associated with initiation is characterizable by Ju or 6ic, while
the material toughness associated with crack growth is characterizable by
the dimensionless parameters [Tj = (E/oo^KdJ/da)] and [Ts = (E/oo).
(dS/da)]. The two-parameter characterization of fracture toughness
properties by Ju and Tj or Sic and Ts is analogous to the characterization of
material deformation properties by the yield stress and strain-hardening
exponent. In fact the ambiguities inherent in the definition of the yield
stress are present in the definition of Ju or 6ic. Thus the range of variation
in/ic and 6ic for any given material is significantly larger than the variation
in TjundTi [16,29].
118 ELASTIC-PLASTIC FRACTURE

Conclusions
This experimental and analytical investigation was directed toward the
identification of viable criteria for the characterization of flat fracture
under essentially plane-strain conditions in the large-scale yielding range.
The following summarizes our studies:
1. Macroscopically flat fracture surfaces with a straight leading edge can
be produced by employing side grooves on test specimens. Side grooves 25
percent of the specimen thickness are recommended, since they promote an
essentially uniform plane-strain constraint along the crack front while
producing minimal effect on specimen compliance and stress-intensity
factor.
2. The experimentally determined / and COD (measured at the original
crack tip) resistance curves appears to be independent of specimen size
and initial crack length when plane-strain flat fracture occurs and if
certain minimal size requirements are met.
3. Analytical investigations also reveal that J and COD resistance curve
can be employed to characterize crack initiation and growth. The slope of
the/-resistance curve {dJ/da) appears to be constant for a relatively short
interval of crack extension, while both the local and average crack opening
angle remain constant over the entire range of crack extension explored in
our experimental and analytical investigations. The /-based criteria appear
to be valid for limited amounts of crack growth. For A533B steel on the
upper shelf, this amount is about 6 percent of the original remaining liga-
ment for test specimens subjected to bending. The range of validity will
depend on the strain-hardening exponent and specimen geometry. The
COD-based criteria appear to be valid for larger amounts of crack growth.
4. The tearing modulus [Tj — (E/al) {dJ/da)] proposed by Paris and
co-workers as a measure of material toughness during stable growth is
constant over relatively short intervals of growth. Our investigations
suggest that a tearing modulus based on the COA [Ts — {E/oo) {db/da)]
is an attractive alternative. The latter modulus is measurable directly and
appears to be constant over the entire range of stable growth. Fracture
toughness associated with crack initiation is measured by /ic or 6ic, while
material resistance associated with crack growth is measured by Tj or Tj.
The two-parameter characterization of fracture properties by /k and Tj or
6ic and Ts is analogous to the characterization of material deformation
properties by the yield stress and the strain-hardening exponent.
5. Certain size requirements must be met for fracture toughness testing
in the fully plastic range. These requirements are analogous to the size
requirements for valid Kic testing in linear elastic fracture mechanics. For
the /-based or COD-based parameters to govern over size scales that
encompass the fracture process zone, the remaining ligament, crack
length, and specimen thickness should be large compared to the crack tip
 SHIH ET AL ON CRACK INITIATION AND GROWTH 119

Opening displacement. The precise magnitudes of these quantities with

respect to the COD appear to depend on material strain hardening and
specimen geometry.

Acknowledgments
The authors wish to acknowledge helpful discussions with J. W. Hutchin-
son of Harvard University and J. R. Rice of Brown University. We are
grateful for the assistance rendered by J. P. D. Wilkinson, R. H. VanStone,
M. D. German, S. Yukawa, and D. F. Mowbray of the General Electric
Co. Some of the analyses presented were carried out by R. H. Dean of
Harvard University and Figs. 4 and 5 were kindly provided by R. H.
VanStone. Discussions with G. T. Hahn, M. F. Kanninen, and E. F.
Rybicki of Battelle Columbus Laboratories who are engaged in a similar
program are gratefully acknowledged. This work was sponsored by the
Electric Power Research Institute, Palo Alto, Calif, and we wish to thank
R. E. Smith and T. U. Marston for their encouragement.

References
[1] Begley, J. A. and Landes, J. D. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 1-23 and pp. 24-39.
[2] Clarke, G. A., Andrews, W. R., Paris, P. C , and Schmidt, D. W. in Mechanics of
Crack Growth. ASTM STP 590, American Society for Testing and Materials, 1976,
pp. 27-42.
[3] Griffis, C. A. and Yoder, G. R., Transactions, American Society of Mechanical
Engineers, Journal ofEngineering Materiah and Technology, Vol. 98, 1976, pp. 152-158.
[4] Fracture Toughness Evaluation by R-Curve Methods, ASTM STP 527, (papers on
linear-elastic R-curve analysis), American Society for Testing and Materials, 1973.
[5] Paris, P. C , Tada, H., Zahoor, A., and Ernst, H., this publication, pp. 5-36.
[6] Hutchinson, J. W., Journal of the Mechanics and Physics of Solids, Vol. 16, 1968,
pp. 13-31 and pp. 337-347.
[7] Rice, J. R. and Rosengren, G. F., Journal of the Mechanics and Physics of Solids,
Vol. 16, 1968, pp. 1-12.
[8] Rice, J. R., Journal of Applied Mechanics, Vol. 35, 1968, pp. 379-386.
[9] Shih, C. F. in Fracture Analysis, ASTM STP 560, American Society for Testing and
Materials, 1974, pp. 187-210.
[10] Rice, J. R. and Tracey, D. M. in Numerical and Computer Methods in Structural
Mechanics, S. J. Fenves et al, Eds., Academic Press, New York, 1973, pp. 585-623.
[11] Tracey, D. M., Transactions, American Society of Mechanical Engineers, Journal of
Engineering Materials and Technology, Vol. 98, 1976, pp. 146-151.
[12] Rice, J. R. in Mechanics and Mechanisms of Crack Growth (Proceedings, Conference
at Cambridge, England, April 1973), M. J. May, Ed., British Steel Corporation
Physical Metallurgy Centre Publication, 1975, pp. 14-39.
[13] Chitaley, A. D. and McCUntock, F. A., Journal of the Mechanics and Physics of Solids,
Vol. 19, 1971, pp. 147-163.
[14] Amazigo, J. C. and Hutchinson, J. W., Journal of the Mechanics and Physics of Solids,
Vol. 25, 1977, pp. 81-97.
[15] Hutchinson, J. W. and Paris, P. C , this publication, pp. 37-64.
[16] Shih, C. F. et al, "Methodology for Plastic Fracture," 1st through 5th Quarterly Reports,
General Electric Co., Schenectady, N. Y., Sept. 1976-Dec. 1977.
120 ELASTIC-PLASTIC FRACTURE

[77] Hahn, G. T. et al, "Methodology for Plastic Fracture," 1st through 5th Quarterly
Reports, Battelle Columbus Laboratories, Columbus, Ohio, Sept. 1976-Dec. 1977.
[18] Kfouri, A. P. and Rice, J. R., "Elastic/Plastic Separation Energy Rate for Crack
Advance in Finite Growth Steps" in Fracture 1977 {Proceedings, 4th International
Congress on Fracture, Waterloo, Ont., Canada, 1977), D. M. R. Taplin, Ed., Vol. 1,
1977.
[19] Bathe, K. J., "ADINA—A Finite Element Program for Automatic Incremental Non-
linear Analysis," Report No. 82448-1, Massachusetts Institute of Technology, Cambridge,
Mass., 1975.
[20] deLorenzi, H. G. and Shih, C. F., "Application of ADINA to Elastic-Plastic Fracture
Problems" in Proceedings, ADINA User's Conference, Report No. 82448-6, Cambridge,
Mass., 1977.
[21] Cox, T. B. and Low, J. R., Jr., Metallurgical Transactions, Vol. 5, 1974, pp. 1457-1470.
[22] Argon, A. S. and Im, J., Metallurgical Transactions, Vol. 6A, 1975, pp. 839-851.
[23] Rice, J. R. and Tracey, D. M., Journal of the Mechanics and Physics of Solids, Vol. 17,
1969, pp. 201-217.
[24] McClintock, F. A., Journal of Applied Mechanics, Vol. 35, 1968, pp. 363-371.
[25] Barsoum, R. S., International Journal for Numerical Methods in Engineering, Vol. 11,
1977, pp. 85-98.
[26] Shih, C. F., deLorenzi, H. G., and German, M. D., International Journal of Fracture
Mechanics, Vol. 12, 1976, pp. 647-651.
[27] Nagtegaal, J. C , Parks, D. M., and Rice, J. R., Computer Methods in Applied
Mechanics and Engineering, Vol. 4, 1974, pp. 153-177.
[28] deLorenzi, H. G. and Shih, C. F., International Journal of Fracture Mechanics, Vol. 13,
1977, pp. 507-511.
[29] Andrews, W. R. and Shih, C. F., this publication, pp. 426-450.
[30] Shih, C. F., deLorenzi, H. G., and Andrews, W. R., International Journal of Fracture,
Vol. 13, 1977, pp. 544-548.
[31] Merkle, J. G. and Corten, H. T., Transactions, American Society of Mechanical
Engineers, Journal of Pressure Vessel Technology, 1974, pp. 286-292.
[32] McMeeking, R. M. in Flaw Growth and Fracture, ASTM STP 631 American Society
for Testing and Materials, 1977, pp. 28-41.
[33] Irwin, G. K.., Journal of Applied Mechanics, Vol. 24, 1957, pp. 361-364.
[34] McClintock, F. A. in Fracture: An Advanced Treatise, H. Leibowitz, Ed., Vol. 3,
Academic Press, New York, 1971, pp. 47-225.
[35] Rice, J. R. in The Mechanics of Fracture, F. Erdogan, Ed., Applied Mechanics
Division, American Society of Mechanical Engineers, Vol. 19, 1976, pp. 23-53.
[36] Green, A. P. and Hundy, B. B., Journal of the Mechanics and Physics of Solids, Vol. 4,
1956, pp. 128-144.
[37] Shabbits, W. O., Pryle, W. H., and Wessel, E. T., "Heavy Section Fracture Toughness
Properties of A533 Grade B Class 1 Steel Plate and Submerged Arc Weldment,"
WCAP-74I4, Westinghouse Electric Corp., Pressurized Water Reactor Systems Division,
Pittsburgh, Pa., Dec. 1969 (also available as HSSTP-TR-6).
M. F. Kanninen,^ E. F. Rybicki,^ R. B. Stonesifer,^
D. Broek,^A. R. Rosenfield,^ C. W. Marschall,^
and G. T. Hahn^

Elastic-Plastic Fracture Mechanics

for Two-Dimenslonal Stable Crack
Growth and Instability Problems

REFERENCE: Kanninen, M. F., Rybicki, E. F., Stonesifer, R. B., Broek, D., Rosen-
field, A. R., Marschall, C. W., and Hahn, G. T., "Elastic-Plastic Fracture Mechanics
for TVo-Dimensloiial Stable Crack Growth and Instability Problems," Elastic-Plastic
Fracture, ASTMSTP668, J. D. Landes, J. A. Begley, and G. A. Clarke, Eds., Ameri-
can Society for Testing and Materials, 1979, pp. 121-150.
ABSTRACT: An elastic-plastic fracture mechanics methodology for treating two-
dimensional stable crack growth and instability problems is described. The paper
draws on "generation-phase" analyses in which the experimentally observed applied-
load (or displacement) stable crack growth behavior is reproduced in a finite-element
model. In these calculations a number of candidate stable crack growth parameters
are calculated for the material tested. The quality of the predictions that can be made
with these parameters is tested with "application-phase" analyses. Here, the finite-
element model is used to predict stable crack growth and instability for a different
geometry, with a previously evaluated parameter serving as the criterion for stable
growth. These analyses are applied to and compared with measurements of crack
growth and instability in center-cracked panels and compact tension specimens of
the 2219-T87 aluminum alloy and the A533-B grade of steel.
The work shows that the crack growth parameters (COA)c, Jc, dJc/da, and the linear
elastic fracture mechanics (LEFM)-R, which sample large portions of the elastic-plastic
strain field, vary monotonically with stable crack extension. However, the parameters
(CTOA)c, (R, So, and Fc, which reflect the state of the crack tip process zone, are
essentially independent of the amount of stable growth when the mode of fracture
does not change. Useful, stable growth criteria can therefore be evaluated from the
crack tip state at the onset of crack extension and do not have to be continuously
measured during stable crack growth. The possibility of making accurate predictions
for the extent of stable crack growth and the load level at instability is demonstrated
using only the value of/c at the onset of crack extension.

KEY WORDS: plastic fracture mechanics,finite-elementmodels, stable crack growth,

crack instability, J-integral, J-integral derivative, generalized energy release rate,
process zone, computational process zone energy, crack opening angle, node force,
linear elastic fracture mechanics resistance curve, center-cracked panel, compact
tension specimen, aluminums, steels, crack propagation

' The authors are members of the staff, Battelle Columbus Laboratories, Columbus, Ohio.

121

Copyright 1979 b y A S T M International www.astm.org

122 ELASTIC-PLASTIC FRACTURE

Nomenclatuie

a Crack length
Initial crack length
Crack length at crack growth instability
Aa Crack growth increment
b Plate thickness
COA Crack opening angle evaluated from crack opening at position
of initial crack tip
iCOA), Critical value of COA for stable crack growth
COD Crack opening displacement
CTOA Crack opening angle evaluated from slope of the crack faces at
the crack tip
{CTOA)c Critical value of CTOA for stable crack growth
E Elastic modulus
F Crack tip node force in finite-element model of crack growth
process
Fc Critical value ofF for stable crack growth
G Energy release rate based on LEFM concepts
Generalized energy release rate based on computational process
9 zone concept
Work of separating crack faces per unit area of crack growth
So Critical value of 9o for stable crack growth
Energy change in computational process zone per unit of crack
9. growth
Critical value of 9^ for stable crack growth
Value of the J-integral evaluated on a contour remote from the
crack tip
Jc Critical value of/for stable crack growth
Critical values of / for initiation of crack grovrth under plane
strain and plane stress, respectively
dJ/da Rate of change of/ with crack growth
dj/duc Rate of change of/c with crack growth
Kic Fracture toughness for initiation of crack growth
LEFM Linear elastic fracture mechanics
Pi Scaling parameter = (Kic/ary/b
R Critical value of G for stable crack growth
(R Critical value of 9 for stable crack growth
T Surface traction
u Displacement
w Plate width
W Strain energy density
V Volume of the computational process zone
e Strain
 KANNINEN ET AL ON INSTABILITY PROBLEMS 123

rContour for evaluation of/

a
Stress
ffr
Yield stress
(To
Applied stress at initiation of stable crack growth
Oc
Applied stress at crack growth instability
N Number of load increments used in finite-element solution pro-
cedure during a crack growth step
P Applied load
AP Increment of applied load

Existing linear elastic fracture mechanics (LEFM) and J-integral analyses

are well suited for safety assessments of high-strength/low-toughness ma-
terials. These analyses apply only to the onset of crack growth, which is
usually tantamount to crack instability and structural failure in that class
of materials. However, this is not the case for the lower-strength, higher-
toughness grades when crack instability may be preceded by extensive
stable crack growth under rising load. Here, a substantial margin of safety
may exist even when the onset of crack growth is imminent. Attempts at
accurate calculation of this margin of safety by extending LEFM or J-
analyses into the stable growth regime are precluded by their inability to
treat the inelastic effects arising from large-scale plasticity and material
unloading [1].^ Consequently, improved methods are needed.
This paper describes research leading to a plastic fracture mechanics
methodology designed to treat two-dimensional large-scale yielding and
stable crack growth problems. While the intended applications are the steels
and failure modes encountered in nuclear pressure vessels, the approach
has greater generality. The research draws on elastic-plastic finite-element
analyses. Although some closed-form solutions exist [2,3] the virtual im-
possibility of obtaining closed-form treatments for the conditions of interest
necessitate numerical analyses. A special "toughness-scaled" aluminum
alloy was used to avoid the cost of full-scale experiments on reactor pressure
vessel steel (A533-B). This alloy approximates the flaw size/plastic zone
size/structural size relations of the steel vessel wall in thinner, more man-
ageable test pieces. The experiments employed both center-cracked panels
and compact tension specimens and are compared with results for the
A533-B steel generated by a concurrent and closely related program at the
General Electric Co. [4-6].
A key element of the research is the "generation-phase" analysis pro-
cedure in which the experimentally observed applied-load (or displacement)
stable crack growth behavior is reproduced in a finite-element model. In
these calculations, each of a number of candidate stable growth parameters
is calculated for the material tested. The quality of the predictions that can
^The italic numbers in brackets refer to the list of references appended to this paper.
124 ELASTIC-PLASTIC FRACTURE

be made with these parameters is tested with "application-phase" analyses.

Here, the finite-element model is used to predict stable crack growth and
instability for a different geometry, with a previously evaluated parameter
serving as the criterion for stable growth. The feasibility, economy, and
accuracy of the application-phase calculations offer a basis for appraising
- the various candidate criteria.
One finding of this work is that parameters truly reflecting the state of
the crack tip process zone are not a function of the extent of stable crack
growth when the mode of fracture (full shear or flat) remains fixed. The
possibility exists, therefore, that useful, stable growth parameters can be
evaluated from .the state of the crack tip at the onset of crack extension.
Consequently, they will not have to be measured separately. The possibility
of making accurate predictions of stable crack growth and crack instability
directly from /c, in this way is demonstrated.

Background Discussion
The fracture criteria examined in this program include the J-integral,
its rate of change during crack growth dJ/da, the crack tip opening angle
{CTOA), and the average crack opening angle (,COA). In addition, two
new candidates are examined. One is the generalized energy release rate g.
This corresponds to the energyflowinginto a computational process zone sur-
rounding the tip of the extending crack per unit area of crack extension.
The other candidate is the crack tip force F which acts at the crack tip
nodes in afinite-elementmodel during the stable crack growth process.
The J-integral—more specifically, its derivative with crack length dJ/da—
has been proposed by Paris et al [7] as a geometry-independent material
parameter for a limited amount of stable crack growth. The crack opening
angle has also been proposed [8,9]. It should be recognized, however, that
there are two distinct definitions of the COA that have been used. De
Koning [8\ uses the angle (JCTOA) that refiects the actual slopes of the
crack faces at the crack tip. Green and Knott [9] use an average value
(COA) based on the COD at the original crack tip position. These are
appealing because of their readily grasped physical significance and the
opportunity offered for direct measurement. Garwood and Turner [10]
have apparently been successful in extending infiltration techniques to the
tip of a stably growing crack. The average COA can be obtained with a
displacement gage mounted near the original crack tip and a measurement
of the crack length.
Energy release rate concepts have been examined by a number of investi-
gators [11-17], The inherent difficulty caused by the dependence on the
computational model, originally pointed out by Rice [18], has not yet been
completely resolved. But, as argued by Kfouri and Rice [19], for example,
this can be circumvented by appealing to micromechanical considerations.
 KANNINEN ET AL ON INSTABILITY PROBLEMS 125

A possibly more satisfactory alternative approach is described in the next

section of this paper.

Analytical Approach

Finite-Element Analysis
The finite-element program being used in this study utilizes constant-
strain triangular elements and quadrilateral elements that are composed of
four triangular elements. The use of these simple elements allows crack
growth to be accommodated and permits various candidate fracture pa-
rameters to be calculated readily. The finite-element program satisfies two
further important requirements. These are the ability to model strain
hardening plasticity and elastic unloading. The use of more sophisticated
elements could possibly increase the accuracy/cost ratio for the types of
analyses presented here, but the use of higher-order elements would also
greatly complicate the manner in which the crack growth is simulated.
Several methods for modeling crack extension are in the literature.
These are based on uncoupling nodal points ahead of the crack tip by
relaxing the forces holding them together. Two methods for releasing nodes
are common. Kobayashi et al [20], de Koning [6], and Light et al [21] first
apply forces to the nodes that are equal to, but opposite in direction to,
those holding the nodes together; then, these are generally relaxed.
Andersson [22,23] and Newman and Armen [24], in contrast, reduce the
stiffness associated with coupling the nodes together. While both of these
approaches are conceptually similar, they are procedurally different. In the
work reported here, the first technique was adopted.
These two methods for modeling crack extension can be used when the
nodal spacing along the path of crack growth is small relative to the total
crack extension. If one uses higher-order elements, however, the nodal
spacing will generally be increased and could possibly become comparable
to the total crack extension. In this case, a more sophisticated scheme for
modeling the crack extension is required. One such scheme is to shift the
crack tip node along the crack growth path. This method is currently being
used by Shih [6].

Computational Procedure
The finite-element method is utilized in the fracture analysis procedure
in two conceptually different ways. In the first, the finite-element analysis
is used to further analyze data from fracture experiments on simple
geometries. In this role, experimentally measured load (or displacement)
versus crack growth records are used as input to the analysis. The outputs
of the analysis are the candidate fracture criteria and their dependence on
126 ELASTIC-PLASTIC FRACTURE

crack extension. This mode of analysis is called a generation-phase analysis,

that is, generation of fracture parameters from experimental measurements.
In the second mode of analysis, the role of the finite-element analysis is
reversed. The input is a selected fracture criterion and its dependence on
crack growth. The output is the extent of stable crack growth, and the load
level giving crack growth instability. This type of analysis is called an
application-phase analysis, that is, application of a fracture criterion to
determine the structure's response.
In the generation-phase analysis, the relation between the applied load
and the crack length is known beforehand. Therefore, it is possible to
increase the external load simultaneously with the release of the crack tip
nodes. This approach results in a piecewise linear approximation to the
experimental load versus crack length curve. Another approach is to main-
tain a constant load during the crack-tip-node release increments. How-
ever, this approach results in a stepwise approximation to the experimental
curve and, therefore, is not as representative of the physical situation for
finite crack growth increments.
In an application-phase analysis, where the load/crack growth relation is
the desired output, one must either iterate to determine the increase in
load for the given increment in crack growth or be satisfied with a stepwise
approximation. Since the iteration procedure would be as much as three
times as expensive as the stepwise approach in the application mode, the
latter choice is more attractive. However, if the less expensive option is
used only for the application phase, consistency between the generation
phase and the application phase will be lost. Since the computed fracture
parameters could be sensitive to differences in the manner of load applica-
tion for the finite increments of crack growth considered here, this should
be avoided. For analyses presented here, the generation and application
phases have been conducted in a consistent manner using the piecewise
linear approach and iteration in the applications phase. The one exception
to this is the application-phase analysis which used the critical node tip
force, Fc
The application-phase analysis using iv was conducted using a stepwise
application of load and therefore did not require the additional expense
involved in iteration. Since F^ is determined during the application analysis,
no generation-phase analysis is required. Therefore, the question of
consistency does not exist. As will be made more clear in discussing the
applications-phase calculations, this is a very attractive feature, unique (at
least at present) to this particular fracture criterion.

Computation of Candidate Fracture Criteria

The parameters evaluated in this program can all be calculated without
recourse to a special crack tip element. The J-integral calculation is ac-
 KANNINEN ET AL ON INSTABILITY PROBLEMS 127

complished by evaluating an integral over a closed path containing the

crack tip. The expression is

/ =
= f fvKdy + T--^ds\ (1)

where T denotes a counterclockwise path surrounding the crack tip, W is

the strain energy density given by

W=\aij deij (2)

and T and u are the surface traction and displacement vectors, respectively.
Note that a path remote from the crack tip is used.
The crack opening angle values are obtained in an obvious way from the
node point displacements. The average CO A is obtained from the displace-
ment at the initial crack tip position while the crack tip value {CTOA) is
obtained from the displacement of the nodes nearest the crack tip. The
parameter F, of course, is obtained directly. Thus, only the generalized
energy release rate needs further elaboration. This is as follows.
The generalized energy-release rate is intended as a direct extension of
the basic energy-balance concept that has proven itself in LEFM. Two gen-
eralizations are included in this approach. First, a small region surround-
ing the crack tip is identified which contains the three-dimensional hetero-
geneous processes that must be excluded from continuum-mechanics
considerations. Second, a direct computation is made of the plastic-energy
dissipation rate for the material outside the excluded region. Two key
assumptions are then required. The first is that the energy dissipation rate
in the excluded region can be taken as a material property, independent of
crack length and other dimensions of the body containing the crack. The
second assumption is that 9> the energy flow rate to the excluded region,
is unaffected by the details of the deformation occurring within it. Thus,
the computation can be made entirely by two-dimensional continuum
mechanics techniques.
The basic parameter involved in the approach is the critical compu-
tational process zone energy-dissipation rate (R: the energy dissipation
accompanying ductile crack extension from processes such as hole growth
and coalescence that occur within a small region surrounding the crack
tip.^ The approach will be successful if (R-values can be found that are
independent of the structural geometry and of the crack length. This
hinges on a third key assumption—that the geometry-dependent portion of
the energy dissipation rate accompanying crack extension can be accurately
^ h e word "rate" is used here, and throughout this paper, just as in conventional LEFM,
to mean per unit area of crack growth.
128 ELASTIC-PLASTIC FRACTURE

calculated by applying a continuum plasticity formulation to the material

outside the computational process zone. Then, the geometry dependence
can be separated from the material dependence. Taken together, these
assumptions will lead to an approach that has two primary virtues: (1)
recognition is given to the micromechanical nature of the crack growth
process, and (2) at the same time, computations can be carried out con-
veniently, using continuum-mechanics techniques, for example, the finite-
element method.
Computationally, the generalized energy-release rate is the sum of two
terms. That is, S = So + 8z where 9o is related to the work done in sepa-
rating the crack faces according to

9o = ^ J _ "riiO Uii^a - |)rf? (3)

and Sz is related to the change in the energy contained in the computa-

tional process zone, from prior to crack extension—State 1—to following
crack extension—State 2—according to

8z = r T 7 f [ f <'odMv (4)

where
b = plate thickness,
Aa = increment of crack growth,
Ti = tractions holding the crack tip closed,
Ui = crack-opening displacements behind the crack tip, and
V = volume of the computational process zone.
Crack growth then proceeds such that

« = 8 ^ So + 8z (5)

in this approach. Crack instability (fracture) will then occur when S > ^
for the prescribed loads or displacements at some crack length.
It should be emphasized that this analysis scheme is not based on what
might be termed a "recoverable energy" criterion. Although the approach
is based on the idea of an energy balance which does include recoverable
elastic energy, there are fundamental differences which circumvent the
pitfalls of a technique based solely on this idea. In particular, as Rice [16]
has shown, for a material that saturates at large plastic strain (for example,
an elastic-perfectly plastic material), the elastic energy rate supplied to the
crack tip is exactly equal to the plastic energy dissipation rate. Hence, in
this case the crack-driving force is identically zero for all load levels. The
 KANNINEN ET AL ON INSTABILITY PROBLEMS 129

feature of the approach described here that eliminates this fundamental

objection is the use of a crack tip computational process (CP) zone.
While the size and shape of the CP-zone must be somewhat arbitrary,
there are some physical considerations that guide the choices. Most impor-
tantly, for a through crack in a thin plate, the 45 deg through-thickness
mode of plastic relaxation will dominate. In order to cope with this three-
dimensional effect within the bounds of a two-dimensional analysis, in
plane-stress conditions, the dimensions of the CP-zone were taken to be
roughly equal to the plate thickness. Specifically, for plate thicknesses of
6.35 mm, calculations of Qz are made using a computational process zone
having a height of 10 mm in the direction normal to the crack plane, and
a length of 4.5 mm in the direction along the crack plane. In thicker plates
where plane-strain conditions are approached, a smaller CP-zone height
probably could be used but with attendant higher computation costs.
The effect of the CP-zone size has not yet been thoroughly explored. How-
ever, it is clear that (R will depend on the size of the zone, so that these
dimensions must be known in order to apply the result.

Experimental Verification
Toughness-Scaled Materials
The finite-element analyses are based upon and verified by systematic
measurements of load extension curves, COD, COA, stable crack growth,
and instability. The verification task is greatly simplified (1) by limiting
both the finite-element models and the experiments to essentially two-
dimensional events, and (2) by reducing the scale of the experiments
relative to actual vessels. This was done by using 6.35-mm-thick panels of
2219-T87 aluminum, a material that matches the flaw size/plastic zone
size/structural size relations of the full-scale vessel.
Toughness scaling, accomplished by preserving the relation between the
plane-strain plastic zone size and the plate thickness of a nuclear pressure
vessel, is expressed by the scaling parameter Pi = (,l/b){Kic/ary. The
scaling parameter has a value Pi = 1.42 for a 200-mm-thick plate of
A533-B steel {Ku = 220 MPam'''', ar = 413 MPa); see Refs 4-6. The
same value of the scaling parameter is obtained with 2219-T87 aluminum
Ku = 36 MPam'''', ar = 379 MPa) with a panel thickness * = 6.35 mm.
The aluminum plates can thus be regarded as 1/32-scale models of much
larger steel plates.
The cracks produced in the aluminum panels actually extended by the
full shear mode. Thus, these experiments model full-scale (200 mm-thick)
steel plate loaded in the same fashion that also fail with a full-shear mode.
Failures with some amount of shear are expected since the thickness re-
quirement for a flat plane-strain fracture is b = 710 mm for the A533-B
steel.
130 ELASTIC-PLASTIC FRACTURE

Verification for the flat (ductile) plane-strain mode of crack extension,

and for the toughness scaling concept in general, was obtained by analyzing
measurements on an A533-B steel test piece performed by Shih et al [5,6\,
The measurements included load, load point displacement, and crack
opening displacement as a function of stable crack growth for a 4T com-
pact tension specimen with 25 percent side grooves tested at 93 °C. More
information on this experiment (No. T52) and on the test material is given
in Refs 5 and 6.
A piecewise-linear approximation of the stress-strain curve derived from
specimens of the 2219-T87 aluminum panels is shown in Fig. 1. The mate-
rial exhibited essentially no anisotropy. A similar representation of the
stress strain curve for the A533-B steel, reported by Shih et al [6], is also
given in Fig. 1.

Experimental Details
Experiments were performed on center-cracked panels as well as on
compact tension specimens. The center-cracked panels were 6.35-mm-thick
2219-T87 aluminum, either 305 mm wide by 1016 mm long with three
different initial crack lengths, loo = 25.4, 102, and 204 mm, and 152 mm
wide by 813 mm long with 2a„ = 102 mm. The experiments were per-
formed in a closed-loop electrohydraulic testing machine of 2.3-MN
dynamic capacity (3 MN static). The specimens had a central 6.24-mm-
diameter hole with a 2-mm-wide milled slit at both sides of the hole.
The slits were fatigue cracked at a cyclic load equal to one-third or less

90
_- A533B..,

80

- E2I9 _J8J. 70

, 400 f.
22W T37 A535B 5 0 «;
c(Wk) .(J) a(MPA) At) (7)
40 c
V2 0.52 '415 0.21 —
m 2.00 1)36 1.51
'°l
3. SO 5148 i|.31
i4.Q3 8.26
- 2 0 ^
160 600

1(83 9.26 6114 11.00 10

1 : 1 1 1 0
D 2 4 6 8 iO i:
Equivalen Strain perce nt
FIG. 1—Piecewise linear stress-strain curves used for analysis of 2219-T87 and A533-B
steel fracture specimens.
 KANNINEN ET AL ON INSTABILITY PROBLEMS 131

of the expected failure load. Antibuckling guides were applied. These

consisted of four steel angle sections, two by two bolted together outside
the specimen. The specimens were pulled to fracture in times varying
between 60 and 120 s.
Three clip gages were used, one mounted in the central hole, one at the
end of the slit, and one at the end of the fatigue crack. They were each
supported by 2 spring pins 1.6 mm in diameter mounted in holes of this
size spaced 12.5 mm apart and located symmetrically with respect to the
notch. In addition, the compliances of the specimens were determined
through measurement of the displacement of the specimen end by means
of a linear variable differential transformer (LVDT) induction coil. The
three clip gages and the LVDT were recorded as a function of the load on
four separate X-Y recorders. A digital load recorder was placed in front
of the specimen. Moving pictures at 20 frames per second were made of
each experiment to obtain a record of crack size versus load.
The compact tension specimens complied with ASTM proportions,
except for the thickness. They were 127-mm-wide, 122-mm-high, and
6.35-mm-thick 2219-T87 aluminum. The tests were performed in an
Instron testing machine. Cover plates were applied as antibuckling guides
with a porthole to observe crack growth by means of an optical microscope.
Simultaneouly, crack growth measurements were made ultrasonically,
using a probe at the edge of the specimen opposite the crack.
The most relevant data from the center-cracked panel tests are given
in Table 1. Actual load-crack extension curves for the three panels are
reproduced in Fig. 2. Some results of the tests on compact tension speci-
mens are presented in a later section where they are compared with ana-
lytical results.
The average COA was obtained from the experimental data in the
following way. A clip gage was mounted over the tip of the original fatigue
crack. The displacement at this point was determined at the onset of
slow stable crack growth. Any further displacements were assumed to be
entirely due to crack growth. Thus the additional displacement was divided
by the instantaneous amount of slow crack grovrth to give COA.

Computatioiuil Results
This section describes the results of several generation-phase and appli-
cation-phase analyses for center-cracked panels and compact tension
specimens. Figures 3 and 4 show typical finite-element grids used for the
two geometries. The grid for the center-cracked specimens represents one
quadrant of the panel. Typical models contain approximately 325 elements
and 700 degrees of freedom. The compact tension specimen shown is a 2T
specimen with an initial crack length of 40 mm. It contains approximately
the same number of elements and degrees of freedom as the center-cracked
132 ELASTIC-PLASTIC FRACTURE

00 ;^ r<N- <N
00
—I •If 0 0 TT
o oO O

s 0^ <N ^
<*> <N <N
o o o o

r-~ o 00 —

O vO 00 O
rt •V OO Cl

I
«§ii§

I S^SS5

</l M t/] M
is S o o </> V) l/i V)

« <N • * lO
'H . ^ CT; <S
«S > 0 o d TT
^

0)
<2 o r- 00
>o <»> rs S
< ^ <S <-i

n <N <N. K —I 00 •»

•S5

ssi s
r
1/1
uoou
<S «S <N <N
i^
 KANNINEN ET AL ON INSTABILITY PROBLEMS 133

Crack Growth, inches

0 1 02 03 0.4 OS 06
120

26 mm
initial crack

100.

50

01 . I lO
0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15
Crack Growth(a-ao) .mm

FIG. 2—Experimental crack growth data for three 2219-T87 aluminum center-crack panels
(2w = 305 mm).

panels. In all cases, the stress-strain curve was represented as being piece
wise linear. Incremental plasticity relations based on the Prandtl-Reuss
equations were used in all calculations. The aluminum center-cracked
panels and compact tension specimens were all 6.35 mm thick and there-
fore were modeled under the assumption of plane stress. The A533-B
specimen, however, was nominally 102 mm thick (76 mm at root of side
groove) and therefore was modeled under the assumption of plane strain.
The hardening rule used for the analyses assumed that the yield strength
of the material upon reloading is unchanged from initial yield. Since there
is little reloading in the analyses due to the simple geometries and loading,
the type of hardening rule is expected to be of secondary importance.
The program simulates a crack growth step by releasing the nodal force
at the pair of nodes representing the crack tip. The nodal forces are re-
leased incrementally. For an initial semicrack length a^, the external load
P is increased by an amount AP in each of N steps such that NAP =
Pi — Pi. Because it is possible only to match the curve at discrete points, a
piecewise linear representation of the experimental curve is actually used.
During each load step, the crack-tip nodal constraint force F is relaxed
and the free crack surface extended one element ahead. This stepwise pro-
cess is continued until the onset of unstable crack growth is reached.
Usually, five relaxation increments were employed to release the nodal
 134 ELASTIC-PLASTIC FRACTURE

Symmetry

51 mm (2 in.)

FIG. 3—Finite-element model of a center-crack panel.

force at the crack tip; that is, N = 5. For the 1.5-mm crack growth incre-
j ment used in the majority of the analysis, approximately 50 incremental
! solutions were required for a stable growth of 15 mm.

Generation-Phase Computations
Generation-phase computations were made for three aluminum center-
cracked panels, an aluminum compact tension specimen, and a steel com-
pact tension specimen. Computational results for the different fracture
parameters during stable crack growth are presented in Fig. 6. The quan-
tities that reflect the toughness of the material in the locale of the crack tip,
Goc, (R, Szc, iCTOA)c, and Fa are relatively invariant during stable crack
 KANNINEN ET AL ON INSTABILITY PROBLEMS 135

-^ _ _- 127 mm -
15 ^
,

61 mm
(2.4 m l /r
u
25 mm P
( 1 in.) 1

AS ^<^
/ A \ \
ZN A, A A /\i\
f\
B=^
Jf-- mm ffmffl
Symmetry
~K- ++ i—1
' i Lft 1
_ "o
1 4 0 mm
(1.57 in)
( 4 in.)

FIG. 4—Finite-element model of a 2T compact tension specimen.

Loadline Displacement, in
0.02 0.04 006 008 OiO

06

o
0 4(5

Loadline Displacement, mm

FIG, 5—Calculated load and crack growth for a 2219-T87 aluminum 2T specimen (w =
102 mm, dLo = 40 mm, b = 6.35 mm).

growth. The parameter dJJda, which might be expected to be vaUd only

for limited amounts of stable crack growth, shows some systematic crack
growth dependence. Of these quantities, Fc and the (,CTOA)c appear to be
most nearly constant. All of the local quantities, and also dJc/da, reflect a
loss in crack growth resistance in the first 2 mm of crack extension, but are
then constant. In contrast, Jc, the LEFM-/?,^ and (COA)c, which can re-
""These /J-values do not contain a plastic zone size adjustment. The adjusted R values
136 ELASTIC-PLASTIC FRACTURE

Crack G r o w t h , inches
0,1 0.2 0.3 0.4 0.5
?i^

liOO
-•• ^1200 2400
O
- 1100 2200

* 8 " 900
1000 2000
1800
150 " •
• 800 . 1600

i TOO •< 1400

1200
600 - .
,100
- » 500 ^ 1000;
31«6 B - too
d i j , 2no 800
50 jL A A 2 6 m m initial crack 300 600
• o
• D
l02mmini}ialcrock
2 0 4 m m initial c r a c k
- SOO 400
100 - 200
0 .
3 4 5 6
1 •, • 1 !

12 13 14 15 16
I >
Jo
Crack Growthta-Oo) ,mm

(a)

C r a c k G r o w t h ,inch6S Crack Growtti,inches

Ol 0.2 0.3 0.4 0.5 0.6 0.2 0.3 04
(CoaiciCToav Zoj, 120 TOO
A 2 6 m m initial crock ' A 26mminitiol cracK
HO • lOZmmmitiQl crock
1 0 2 m m initial crock
100 •• 0
0 2 0 4 m m initial crock 100 I • 2 0 4 mm initial crack
600

90

ao
_5 | e *> „ S B « ° 500 S

4
o 0

9 g g S 8 °
' - TO 400 :
TO

60 •l i
» f 60
1 =0 300l
>°
50
! ! ! ! • • • ~3? ^ 40
40
~ 30 I 1 1i I 1
J 30
- 20
20
- _ iO
10
0
- i -U 1 J U 1 1 1 1 1 1 1 1 ..
0
4 5 6 7 B 9 10 II 12 13 14 15 16
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Crack6fowth(a-a(j) ,mfT
Crack Growth(a-ao),mm

(b) (c)

Crock Growth,inches Crock Growtti.mcties

0.1 02 0.3 04 0.5 0.6 0.2 0.3 04 0.5

A
2an
2 6 m m i n i t i a l crock
2200
2a
9 ' (?00
9 " 0
- 0
102 m m initial c r o c k
2 0 4 m m initial c r o c k
- 200C 2 6 m m initial crock
O 1 0 2 m m i n i t i o ! crack - 1100
1800 a 2 0 4 m m i n i t i a l crock 1000
O -

i^ g 8 a a 8 fi H 0 0 1600 0 ° 900
150 0
1400 o 800

z ,6
1200- z 0 S
D TOO
10001'
u 5
CE"100 8 600
800
4 0 u A ^ -^ \ 500
3
- - 600 £i 400
2
- 400 i k\-/^ "(f^f
300
200
R sec
; 200
1 1 ' , :
0 • !
2w = 3 0 5 mm 100
3 4 5 6 7 8 9 10 II 12 13 14 15 16
±. 1 ; ; 1 1 0
Crock Growth (o-Oo) >mm 2 3 4 5 6 7 8 9 10 I! 12 13 14 15 16
Crock Growth(a-Oo) ,mm

(d) (e)

FIG. 6—Calculated values of candidatefracture criteria for three 2219-T87 aluminum center-
crack panels (2w = 305 mm, b = 6.35 mm).
 KANNINEN ET AL ON INSTABILITY PROBLEMS 137

Loadllne Displacement, in.

0.1 02 0.3 04 0.5

- 160

140

80 Q
General Electric
Battel le Columbus

40

20

0 5 10 15
Leadline Displacement, mm
FIG. 1^Load-displacement curves calculated for an A533-B steel 4T specimen with 25 per-
cent fide groovesi(v/ — 203 mm, An — 163 mm, b = 76 mm).

fleet changes in the character of the strain field remote from the crack tip,
vary strongly but systematically. With the exception of the LEFM-/?, all of
the parameters were relatively insensitive to the initial flaw size, with Fc
and Jc showing the least dependence.
Figuries 5 and 7 contain the load line displacement versus crack length
measurements that serve as inputs for the generation-phase calculations for
the aluminum and steel compact specimens. These figures also compare
the calculated load-displacement curves with the measurements. The latter
results illustrate that the finite-element model for the aluminum compact
specimen is 7 to 15 percent stiffer than the actual test piece. For this reason
the calculations for the aluminum test piece were repeated using load ver-
sus crack length as the input. This established a lower bound for the dif-
ferent fracture parameters.
The results of the generation-phase analysis of the aluminum compact
specimen are presented in Figs 8 and 9, where they are~compared with the
values for the center-cracked panels.* The results for the A533-B steel com-
pact tension specimen are given in Fig. 10. Qualitatively the results for the
steel and the aluminum compact specimen are very similar despite the
*As reflected by the different /cj-values for the compact tension specimen and the center-
cracked panel (see Fig. 8a), there were some metallurgical differences in the materials used.
138 ELASTIC-PLASTIC FRACTURE

Crack Growth, inch

a2 0.3 0.4

600

i P vs a
^ ^ R a n g e of center crock panels
( P vs a )

6 e 10 12 14
Crack Growth, mm
(a)

Crock Growth, inch

O.i 0.2 0.3 0.4 0.5 0.6
I I I r
" 8 vs a
' P vs a 30
LRonge of center crock panels
( P vs 0 :
25
'S 15

J L _L
6 8 10 12
Crock Growth, mm

FIG. S—Calculated values ofJc and Hc/Adifor a 2219-T87 aluminum 2T compact tension
specimen (w = 102 mm, SLO = 40 mm, b = 6.35 mm).

fact that both the modes of fracture (ductile shear for the 2219-T87, ductile
flat for the A533-B) and the absolute values of the toughness parameters
differ greatly. The parameters that reflect the deformation in the crack tip
process zone, iCTOA)c, (R, andiv, are essentially independent of the extent
of stable crack growth. In contrast, the parameters that sample larger por-
tions of the elastic and plastic strain field, (COA)c, Jc and dJc/da, vary
 KANNINEN ET AL ON INSTABILITY PROBLEMS 139

monotonically with stable crack extension. The variation of (COAX is

confirmed by actual measurements as shown in Fig 9c. Within the precision
of the analyses, comparison of the 2219-T87 center-cracked panel and
compact specimen results indicates that (R, (.CTOA)c and possibly the
iCOA)c are independent of geometry. The quantities/c and dJc/da show
some geometry dependence after about 4 mm of stable crack growth (6
percent of the remaining ligament).
Figure 11 summarize results of generation-phase analyses of a 2219-T87
center-cracked panel (2w = 305 mm, 2a„ = 102 mm) for three different
mesh sizes in the crack growth region. The calculations indicate that J^,
{COA)c, and (R are not sensitive to mesh size. The {CTOA)c shows some
dependence. The quantities goc and F^ are strongly mesh size dependent,
approaching zero as the mesh size approaches zero, as reported by Rice [18\.

Application-Phase Analyses
Application-phase analyses are shown in Figs. 12 and 13. These were
carried out on two center-cracked panels: 2Cm2.04, a load-controlled
experiment, and 2Cs2.04, a displacement-controlled experiment (see
Table 1). The first analysis employed / = /,. as the initiation criterion
and 8 = (R as the stable crack growth criterion; the second employed
/ = /;, for initiation and F = Fc for stable growth. However, an even more
important distinction is that the first calculation relied on two separately
measured toughness values (/r, and (R) which were obtained by averaging
the results of three previously performed generation-phase analyses. The
second calculation relied on a single toughness value as input: the value
of Jci at the onset of crack extension to characterize both initiation and
stable growth. This was made possible by virtue of the fact that Fc is essen-
tially constant during stable crack growth (see Figs. 5, 9, and 10). Its
value is determined by the finite-element at the load level producing / = 7^,
(that is, Fc was calculated from /<., at the onset of growth). The results of
the two application-phase computations presented in Figs. 12 and 13 show
that the model predicted, with quite good accuracy, the load versus stable
crack growth behavior of the test pieces, including the maximum load and
corresponding crack length, and the instability condition.

Discussion of Results
The present findings illuminate the basic cause of stable growth in elastic-
plastic materials. In the cases analyzed here, crack stability cannot be
attributed to an increase with crack growth of the toughness of the material
in the process zone. This is supported by the constancy of the CP-zone
energy and the CTOA with the unchanging fracture mode in the 2219-T87
and A533-B test pieces. The constancy of (R coupled with increasing load
140 ELASTIC-PLASTIC FRACTURE

Crack Growth, inch

02 0.3 0,4

300
40 -

° Svs a
20 » P vs a
^ ^ R a n g e of center crack panels
^^(Pvsa)
J_ J_ _L
6 8 10
Crack Growth, mm
Ca)

Crack Growth, inch

0.2 0.3 0.4 0.5
1 1 I I I 1

28 -
A
-3"°

<
o ° 8 vs a
» P vs a
^ f e Range of center crack panels _
( P vs a )

1 1 I I 1 1 1
6 8 10
Crock Growth, mm
 KANNINEN ET AL ON INSTABILITY PROBLEMS 141

Crack Growth, inch

0.2 0.3 0.4

4 6 8 10 12
Crack Growth (a-a„), mm

(C)

Crack Growth, in.

0.2 03 04

^////////////////y/z^m.

57
g 8 o
X

o 8 vs a
" P vs a
Range ot center crack panels
(P vs a)

4 6 8 10 12
Crock Growth (a-Oj), mm
(d)
FIG. 9—Calculated values of candidate fracture criteria for a 2219-T87 aluminum 2T com-
pact tension specimen (w = 102 mm. ao = 40 mm. b = 6.35 mm).
142 ELASTIC-PLASTIC FRACTURE

Crock Growth, In
01 02 03 01 05 06 07 08 09
1 1 1 1 1 1 1 1
dJc
- 18
3
—. — Calculated _ 16
• Experiment
_^^^ - 14
fO g
_ s - 12'O
\ 4 X
K
E \ '° E
•s.

z
\
- 8 S
.3£
u
u 6 ^
-» 1 2 •or"
£ St
- 4
—'^""-.
2

1 —L. ... 1 1
0 -•o
6 10 15
Crack Growth, mm

(3)

Creek Growth, in.

01 02 03 04 05 06 0.7 08
I I I
30
— Computed (CTOA)^
--Computed (COA)(
25 • Experimental (COA),.''

c 20

. 15

O
10
5 <S
u
i/COD-CODj \
<
O
(COA)j = TAN" O
u

4 6 8 10 12 14 16 20
Crack Growth (o-Oo), mm

(b)
 KANNINEN ET AL ON INSTABILITY PROBLEMS 143

Crack Growth, in.

01 0 2 03 04 0.5 0.6 0.7 08
- 5

-I I I I I L.
0 2 4 6 8 10 12 14 16 IS 20
Crock Growth (o-Qo), mm
(C)

Loodline Displocement, in.

0.1 0.2 0.3 0.4 O.S 06
r—

20

- 15 U

5E
- 10
o Critical node force (F^)
( ) Node release number
- 5

_L
5 10
Loodline Displacement, mm

FIG. 10—Calculated values of candidate fracture criteria for an A533-B steel, 4T, 25
percent side-grooved compact tension specimen (w = 203 mm, Ha — 134 mm, b = 102 mm).
144 ELASTIC-PLASTIC FRACTURE

Crack Growth, inches

0.1 0.2 03 04 0 5 06

o g 1200
200
o 1100
D
O
1000
0 900
150
800
700
100 600
6 500
MESH SIZE 400
1 d 0.75 mm
J O 1.50mm
300
200
Q 3.00mm
100
2 3 4 5 6 7 8 9 10 II 12 13 14 15 16
0
Crack Growth ( a - a g ) , m m

(a)

Crock Growth .inches

0.1 0.2 03 04 05 06

120 (COAJc ICTOAJc MESH SIZE

A A 0.75 mm
1 10 • O 1.50 mm
o mm a 3.00 mm
100
qn
^ 80 ^ ' ^ ^ ^^ A A ^ ^
o
1-0 a 0
8 ° g
•<i 60
o
1-
so
• • 1 • •
<-> 40
c
-" 30
^
1 >
20
10
0
2 3 4 5 6 7 8 9 10 II 12 13 14 15 16
Crack Growth(a-QQ),mm

(b)
 KANNINEN ET AL ON INSTABILITY PROBLEMS 145

Crack Growth,inches
01 0 2 03 0 4 0 5 06
I 1
120 ^ -fif Mesh size TOO
A 4 0.75 mm
110
A 0 • 1 5 0 mm 600
100 0 D • 300mm
„ 0
Q O O 0
500 c
^ 80 ^Cto

- 70 o 400 -
V 60 • • ^°
1 50
. 300|

^ 40 • ^
30 A 200

20
100
10
0
3
.
4 5 6 7 8 9 10 II 12 13 14 15
0

Crock 6rowth(a-ao) . " ' " i

(c)

Crock Growth Increment, in

0 02 004 006 0 06 010 012
I

075 1 15 2
Crock Growth Increment, mm

(d)

FIG. 11—Mesh size dependence of candidate fracture criteria for a 22I9-T87 aluminum
center-crack panel (2w = 305 mm, 2io — 102mm, b = 6.35 mm).
146 ELASTIC-PLASTIC FRACTURE

Crack Growth, in.

01 0.2 0.3 04
-I 1 1 1 1 1 1 P

100 -
20
80-

60
m
O

540
Predicted (Jj. = 33 kN/m, F = F^ = 813 kN/m)
• Predicted onset of unstable growth (large growth!.
20 with negligible increase in applied displacement)
• Experiment
1 1 1 I l_
4 6 8 10
Crock Growth (a-Og), mm

FIG. 12—Application-phase analysis of a 2219-T87 aluminum center-crack panel using (R

as the fracture criterion (2w = 305 mm. 2io = 102 mm, b = 6.55 mm).

Crack Growth, in
0.2 0.3 0.4

- 80^
-o"
Predicted o
150 - Predicted onset of unstable growth
(moximum load) -40 °

100 - • Experimental -130

50 - 20

10
6 8 10 12
Crock Growth (a-Oo), mm

FIG. 13—Application-phase analysis of a 2219-T87 aluminum center-crack panel using Fc

s the fracture criterion (2w = 152 mm. 2ao = 102 mm. b = 6.35 mm).
KANNINEN ET AL ON INSTABILITY PROBLEMS 147

a o M A M

g g s,g

I ^iR

1
s
1 g s.g>i II
1a o
^
o
§
1
<
io §£.Rg.E
^
' • " '

Ik.

.2
M M M rt n
4> U U 2 S
S-» Si, si» c c

I
I J i.g g s

I "ti O O O
^ c c e

e
3>

c
•g
a
•c
148 ELASTIC-PLASTIC FRACTURE

during stable crack growth means that the portion of the energy flow
reaching the crack tip region diminishes with crack extension. The reduced
energy flow can be thought to result from the "screening" action of the
plastic zone accompanying the growing crack, as described by Broberg [9].
It is possible that the toughness increases in the case of a mixed-mode frac-
ture with an increasing shear component. But, where this condition is not
attained, the interpretation of the so-called "/-resistance" curve as a mani-
festation of increasing toughness of material of the process zone is funda-
mentally incorrect. This point is treated more fully by Hutchinson and
Paris [25] and by Shih et al [27] elsewhere in this publication. They also
conclude that there are departures from the Hutchinson-Rice-Rosengren
singularity and, hence, the meaning of J, in a zone at the tip of a growing
crack. This zone, which expands with crack extension, is "/-controlled" in
the same sense as the small-scale yielding plastic zone of a stationary crack
is " AT-controlled." That is, there is a direct relation between the /-value
measured remotely and the deformation state of the crack tip. However,
this relation changes as crack growth proceeds. The rising Jc curve is a
consequence of this changing relation, not of a change in the local tough-
ness. Also, as Hutchinson and Paris have shown, the relation becomes
invalid after some small amount of stable crack growth.
The present work also shows that the LEFM energy release rate concept
can be generalized to elastic-plastic materials by enlarging the energy sink
to include a finite computational process zone. The generalized energy
release concept has several attractive features. The critical energy release
rate has a well-defined physical meaning. It does not require elaborate
modeling and is insensitive to mesh spacing and specimen geometry. It can
reduce three-dimensional plane stress and mixed-mode crack extension to
a two-dimensional problems. Also, it appears to be essentially constant
from the beginning of stable crack growth. This latter feature provides the
basis for the application-phase calculation, described in the previous sec-
tion, which illustrates that a single constant value of (R can predict stable
crack growth and instability with precision.
The deduction that the process zone is essentially invariant during fixed-
mode stable growth is important because it serves to validate other criteria,
such as 9oc, Fc, and (CTOA)c, which may be computationally more con-
venient. It follows that any parameter reflecting the state of the process
zone may be independent of crack growth. More important, such param-
eters would not have to be measured separately. Instead, process zone
parameters that are invariant with crack growth can be evaluated from a
finite-element model of the state of the crack tip region at the onset of
crack extension. This is illustrated by a second application-phase calculation
in the previous section. Here the crack growth parameter Fc, which is
determined by the value of /<;„ serves only an operational function in the
calculation. This calculation demonstrates the feasibility of characterizing
 KANNINEN ET AL ON INSTABILITY PROBLEMS 149

the onset of crack extension, stable crack growth, and instability for a
large-scale yielding problem using a single toughness value as input.
Recently, nine different requirements for an acceptable plastic fracture
criterion were identified [26]. These include that it be (1) well suited for
models of three-dimensional crack fronts, (2) valid for both small and large
stable crack extensions, (3) geometry independent, (4) computer model
independent, (5) economical to use, (6) measurable with small test pieces,
(7) able to predict crack initiation, (8) able to predict crack instability, and
(9) valid for fully plastic behavior. An appraisal based on these require-
ments is summarized in Table 2. Shih et al [27] have examined the same
group of candidate parameters (except for (R). After subjecting them to
essentially the same requirements as those just given, they have concluded
that the Jc- and (COD)c-curves are the most promising. In our view, how-
ever, such a conclusion is premature.
It should be clear from the foregoing discussion that the requirements
for geometry independence, model independence, and the ability to measure
the parameter are all redundant when the parameter is constant during
stable crack growth. Of the parameters examined so far, the (CTOA)c, Fc,
and So appear to be both constant and computationally convenient oper-
ational criteria. These can be related to the value of a crack initiation
parameter such as Ju (or/„) or possibly (COD)c, thereby satisfying require-
ments 2, 4, and 8 regarding initiation, stable growth, and instability. Fully
plastic behavior has not posed special problems, but more work is needed
to establish the utility of this approach for three-dimensional crack fronts.

Acknowledgment
This work was supported by the Electric Power Research Institute (EPRI),
Palo Alto, California. The authors would like to express their appreciation
to T. U. Marston and R. E. Smith of EPRI for their help and encourage-
ment of the work. They are also indebted to John Fox of Battelle's Colum-
bus Laboratories for his nondestructive evaluation work in this program
and to F. Shih and W. Andrews of General Electric for making some of
their results available. Many useful and stimulating discussions with the
EPRI plastic fracture analysis group should also be acknowledged.

References
[1] Rice, J. R. in The Mechanics of Fracture, F. Erdogan, Ed., American Society of Me-
chanical Engineers Publication AMD, Vol. 19, 1976, pp. 23-53.
[2] Shih, C, F. and Hutchinson, J. W., Journal of Engineering Materials and Technology,
Vol. 98, 1976, pp. 289-295.
[3] Amazigo, J. C. and Hutchinson, J. W., Journal of the Mechanics and Physics of Solids,
Vol. 25, 1977, pp. 81-97.
[4] Wilkinson, J. P. D., Hahn, G. T., and Smith, R. E., "Methodology for Plastic Frac-
150 ELASTIC-PLASTIC FRACTURE

ture—A Progress Report" in Proceedings, 4th International Conference on Structural

Mechanics and Reactor Technology, Aug. 15-19, 1977.
[5] Shih, C. F., de Lorenzi, H. G., Yukawa, S., Andrews, W. R., Van Stone, R. H., and
Wilkinson, J. P. D., "Methodology for Plastic Fracture," Third Quarterly Report to the
Electric Power Research Institute, Palo Alto, Calif., 1 Nov. 1976 to 31 Jan. 1977.
[6] Shih, C. F., de Lorenzi, H. G., Andrews, W. R., Van Stone, R. H., and Wilkinson,
J. P. D., "Methodology for Plastic Fracture," Fourth Quarterly Report to the Electric
Power Research Institute, Palo Alto, Calif., 1 Feb. 1977 to 30 April 1977.
[7] Paris, P. C. et al, "A Treatment of the Subject of Tearing Instability," Washington
University Report to the Nuclear Regulatory Commission, NUREG-0311, 1977.
[5] de Koning, A. U., "A Contribution to the Analysis of Slow Stable Crack Growth," The
Netherlands National Aerospace Laboratory Report NLR MP 75035U, 1975.
[9] Green, G. and Knott, J. F., Journal of the Mechanics and Physics of Solids, Vol. 23,
1975, pp. 167-183.
[10] Garwood, S. and Turner, C. E., unpublished work. Imperial College, Department of
Mechanical Engineering, London, U. K., 1977.
[11] Broberg, K. B., Journal of the Mechanics and Physics of Solids, Vol. 23, 1975, pp.
215-237.
[12] de Koning, A. U. in Proceedings, 4th International Conference on Fracture, University of
Waterloo Press, 1977, Vol. 3, pp. 25-31.
[13] Kfouri, A. P. and Miller, K. J. in Proceedings, Institution of Mechanical Engineers,
Vol. 190, No. 48/87, 1976, pp. 571-584.
[14] Hellan, K.., Engineering Fracture Mechanics, Vol. 8, 1976, pp. 501-506.
[15] Carlsson, A. J., "Progress in Nonlinear Fracture Mechanics," Presentation at the 14th
International Congress of Theoretical and Applied Mechanics, Delft, The Netherlands,
30 Aug. to 4 Sept. 1976.
[16] Cotterell, B. and Reddel, J. K., International Journal of Fracture, Vol. 13, 1977, pp.
267-278.
[17] Andrews, E. H. and Billington, E. W., Journal of Materials Science, Vol. 11, 1976,
pp. 1354-1361.
[18] Rice, J. R. in Proceedings, 1st International Conference on Fracture, Japanese Society
for Strength and Fracture, Tokyo, Vol. 1, 1966, pp. 309-340.
[19] Kfouri, A. P. and Rice, J. R. in Proceedings. 4th International Conference on Fracture,
University of Waterloo Press, Vol. 1, 1977, pp. 43-60.
[20] Kobayashi, A. S., Chiu, S. T., and Beeuwkes, R., Engineering Fracture Mechanics,
Vol. 5, 1973, pp. 293-305.
[21] Light, M. F., Luxmoore, A. R., and Evans, E. T., International Journal of Fracture,
Vol. 11,1975, pp. 1045-1046.
[22] Andersson, H., Journal of the Mechanics and Physics of Solids, Vol. 2, 1974, pp. 285-308.
[23] Andersson, H. in Computational Fracture Mechanics, E. F. Rybicki and S. E. Benzley,
Eds., American Society of Mechanical Engineers Special Publication, 1975, pp. 185-198.
[24] Newman, T. C. and Armen, H., Jr., "Elastic-Plastic Analysis of a Propagating Crack
Under Cyclic Loading," American Institute of Aeronautics and Astronautics Paper No.
74-366, AIAA/ASME/SAE Conference, Las Vegas, Nev., 1974.
[25] Hutchinson, J. W. and Paris, P. C , this publication, pp. 37-64.
[26] Private communication. Plastic Fracture Analysis Group, T. U. Marston and R. E.
Smith, chairmen. Electric Power Research Institute, Palo Alto, Calif., 12 Aug. 1977.
[27] Shih, C. F., de Lorenzi, H. G., and Andrews, W. R., this publication, pp. 65-120.
E. P. Sorensen^

A Numerical Investigation of Plane

Strain Stable Crack Growth Under
Small-Scale Yielding Conditions

REFERENCE: Sorensen, E. P., "A Numerical Investigation of Plane Strain Stable

Crack Growth Under SmaU-Scale Yielding Conditions," Elastic-Plastic Fracture,
ASTMSTP 668. J. D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society
for Testing and Materials, 1979, pp. 151-174.

ABSTRACT: Plane strain crack advance under small-scale yielding conditions in

elastic-perfectly plastic and power-law hardening materials is investigated numerically
via the finite element method. Results indicate that the stress distribution ahead of
a growing crack is essentially the same as that ahead of a stationary crack, and that the
numerically evaluated steady-state crack tip profiles reflect a vertical tangent at the
extending crack tip which corresponds to the theoretically predicted outline. It is found
that the increment dSi in crack tip opening, when loads are increased at fixed crack
length, seems to be uniquely related to dJ/og irrespective of the amount of previous
crack growth, and for increments dl of crack advance at constant external load, the
incremental crack tip opening appears related to In (J/oot) dl when evaluated at distance
r from the tip. A discussion of proposed fracture parameters for continued crack
growth (as opposed to growth initiation) is included.

KEY WORDS: stable crack growth, small-scale yielding, nonhardening and power-law
hardening materials, fracture criteria for continued crack advance, crack propagation

In contrast to the extensive literature pertinent to stationary cracks, for

example, [1-4],'^ a paucity of solutions for the growing crack case is evident.
One obvious reason for this is the added mathematical complexity inherent
in a continuum formulation of the growing crack.
Analytic investigations of quasi-statically extending cracks under Mode III
conditions in elastic-perfectly plastic materials are presented in [5-7], and
Rice [7] extends the discussion to the form of the solution for a growing
crack subject to plane strain, Mode I conditions. One conclusion of these
studies is that the strain field ahead of an extending crack is dominated by
'Research assistant. Division of Engineering, Brown University, Providence, R.I. 02912.
Current affiliation: Mathematics Department, General Motors Research Laboratories,
Warren, Mich. 48090.
^The italic numbers in brackets refer to the list of references appended to this paper.

151

Copyright 1979 b y A S T M International www.astm.org

152 ELASTIC-PLASTIC FRACTURE

a logarithmic singularity which is weaker than the 1/r singularity experienced

at the tip of a stationary crack (where r is the distance measured from the
crack tip). The weaker strain singularity is due to the crack extending into
material that has deformed plastically so that complete refocusing of the
strain field at the tip of the extended crack is prevented. This reduced
crack-tip strain concentration is a primary reason for stable crack growth
in elastic-plastic materials.
Various aspects of the incremental solution to crack growth problems
are considered by Rice [8]. His work reveals an incompatible elastic strain
increment (one not derivable from a displacement field) caused by a Prandtl
stress distribution traveling with the crack tip. The incompatible elastic
strain increment induces plastic strain during an increment of crack advance
and the additional straining promotes crack growth. Another consequence
of the elastic incompatibility is that, contrary to the rigid-plastic case in
which the crack advances with a finite crack tip opening angle (COA),
the elastic-plastic incremental formulation results in a crack face profile
exhibiting a vertical tangent at the crack tip and a corresponding ill-defined
crack tip opening angle. Both the rigid-plastic case and the elastic-perfectly
plastic case predict a zero crack opening displacement (COD) at the tip
of an extending crack. In this respect, the Mode III asymptotic analysis
presented by Chitaley and McClintock [9] is incorrect in its prediction
of a nonzero COD at the tip of a steadily extending crack, and the difficulty
seems to arise from their approximate numerical evaluation of an integral
which should have given zero for a result [S\.
Experimental observations of stable crack growth have been reported by
several investigators, for example, Refs 10-12. Green and Knott [12] observe
a constant increase in the nominal COD per increment of crack growth
and conclude that the crack face profile associated with an extending crack
tip in a ductile metal is constant. J-integral experimentation due to Clarke
et al [13] and Griffis and Yoder [14] indicates a constant change of / with
change of crack length following the blunting of an initially sharp crack.
The implication is that the advancing crack tip experiences constant sur-
rounding fields. These results are analogous should / and 6, the crack-tip
opening displacement, remain linearly related as they are in the stationary
crack case.
Finite element solutions to extending crack problems include the work
of deKoning [15], Andersson [16], and Sorensen [17]. k. key objective of
the numerical solutions, aside from the illumination of the stress and
strain distributions accompanying growing cracks, is the investigation of
possible macroscopic parameters which may be correlated with the "state"
at the growing crack tip. Although the meaning of the J-integral is unclear
in the growing-crack case, / is known to rise monotonically, for example,
Ref 13, and its use as a fracture predictor in the extending crack case is a
possibility. Other proposed parameters are the crack-tip opening angle [18]
 SORENSEN ON PLANE STRAIN STABLE CRACK GROWTH 153

and the Griffith-like separation energy rate associated with a finite crack
advance step [19]. Recent work by Shih et al [20] examines and evaluates
various proposed criteria for continuing fracture.
The present paper investigates the distributions of stress and deformation
associated with an extending crack tip in hardening and nonhardening
materials under plane strain, small-scale yielding conditions. This is accom-
plished via large-scale finite element calculations. The results indicate the
existence of a Prandtl stress distribution traveling with the crack tip, and
the crack face profiles are consistent with the logarithmic dependence noted
by Rice [8]. Discussion of various proposed fracture criteria is provided
and a relation characterizing continuing fracture is sketched from con-
siderations presented in Ref 5 and the numerical results.

Numerical Considerations

Finite Element Equations

A form of the virtual work equation, valid for incremental small-strain
theory in which all integrals are carried out over the reference volume and
surface, is used in the derivation of the governing finite-element equations

OijSuj,/ dV = I Ti bill dS

where
M, = displacement vector,
Ti = traction vector, and
Oij = Cauchy stress tensor.
Superimposed dots denote rates.
Let [N] denote the shape functions used to represent variations of dis-
placement within an element as interpolated from nodal displacement
values, [u\, so that [^]{«) represents the displacement field. The incre-
mental strain-displacement relation is (e) = [5]{M ), where [B] is composed
of the appropriate derivatives of [N]. The constitutive matrix is denoted
by [C] such that the incremental stress is related to the incremental
strain by [a] = [C]{ej. Substituting the foregoing matrix relations into
the governing variational equation and recognizing that arbitrary variations
may not influence the resulting equilibrium equations, one obtains the
well-known tangent stiffness equations

I [BY[C][B]dV [u] = J [W{T]dS

154 ELASTIC-PLASTIC FRACTURE

where integrals are carried out over all elements and over all externally
loaded surfaces. In conventional finite-element notation this equation is
written [/iT] {ii) = [P] with [K] termed the master stiffness matrix and [P]
the forcing function or right-hand side.

Constitutive Relations
The material constitutive behavior is modeled as isotropic, elastic-perfectly
plastic, and elastic power-law hardening together with the Mises yield
condition and the associated Prandtl-Reuss flow law [21]. The power-law
hardening relation is that used by Tracey [22], namely

a/a, = (a/oo + ZGif/ao)"

where
N = hardening exponent,
G ~ shear modulus,
Oa = yield stress in tension
- - ll ,. .
Sij = deviatoric Cauchy stress tensor,
i"'' = L e/e/, and
e.j'" = plastic portion of the deviatoric strain tensor.
This power-law hardening expression is obtained from the relation ^/TO =
iy/yo)'^ for pure shear used by Rice and Rosengren [23] through the con-
versions T — ff/V3, y'' = VSe'', and 7 = 7 ' + y''. No account is taken
of the Bauschinger effect or possible vertex development of the yield surface
during the nonradial loading experienced by material points during crack
growth. These omissions must be kept in mind during the interpretation
of the present results. The nonlinear problem is linearized by specifying
small load increments and iterating within each increment for convergence
to the best representative plastic constitutive matrices. The constitutive
matrix [C] at any point in the loading history may be written as

[C] =m[C'^ + (l-m)[C"-p']

with 0 < m < 1. This partial-stiffness approach is due to Marcal and King
[24]. [C'P'] is determined from the normal to the yield surface and m de-
pends on the amount of elastic response an element undergoes during a
load increment (that is, w = 1 for totally elastic response and w = 0 for
totally plastic response). The normal used in these expressions is chosen
in the manner of Rice and Tracey [25] such that resulting stress states
precisely satisfy the yield criterion in the elastic-perfectly plastic case and
 SORENSEN ON PLANE STRAIN STABLE CRACK GROWTH 155

approximately satisfy the yield criterion in the power-law hardening case

[22]. Typically two to three iterations are required per loading increment
for convergence to an appropriate constitutive representation. Reassembly
and redecomposition of the master stiffness matrix are accomplished in a
cost-minimizing manner, using the efficient procedure discussed by Yang
[26] and Sorensen [27] and various in- and out-of-core procedures as re-
quired [28].

Crack Growth Simulation

A nodal release technique is implemented to simulate crack advance-
ment through the finite element mesh. This technique is used by Andersson
[18] and by Kfouri and Miller [19]. As applied here, the technique pro-
ceeds as follows. Upon satisfaction of a chosen fracture criterion or speci-
fied load level, the crack is deemed ready to propagate and the boundary
condition at the crack tip passes from displacement controlled to trac-
tion controlled. The reaction force corresponding to the zero displace-
ment condition at the crack tip node is calculated and relaxed to zero in
five equal increments as more steps provide minimal differences in results
at significant computational expense. Following this procedure, the crack
tip has advanced by one element length. The present analyses hold external
loads constant during the nodal relaxation procedure. It is anticipated
that due to the history dependence of the strain distribution the process
of nodal force relaxation under increasing external load might result in a
somewhat different strain rate ahead of the crack tip from that obtained
under constant external load. The present results are interpreted as repre-
sentative of crack advance under constant load or perhaps under slight
increases of external load.

Element Modeling
The element used in the present analyses is the constant-strain triangle.
Quadrilaterals are formed from four of these elements in the manner of
Nagtegaal et al [29] to accommodate the possibility of nearly incompressible
straining, and the degrees of freedom associated with the internal node
are eliminated from the stiffness equations [30]. This configuration in no
way accounts for the mathematical singularities encountered at the crack
tip, but useful results are obtained by sufficient mesh refinement (see dis-
cussion of results). Due to the nodal release technique employed in the
analyses, no use of special crack-tip singular elements, for example, Tracey
[31] and Barsoum [32], is made since the crack advance would require
a procedure for refocusing the mesh at the tip of the extended crack. In
this context a Eulerian finite element formulation holds much promise,
for then a mesh remains focused at the crack tip, and singularity elements
156 ELASTIC-PLASTIC FRACTURE

may be employed; however, such a formulation presents other difficulties

such as the convective terms which require spatial derivatives of field quanti-
ties, and an economically feasible formulation has not yet been found.

Numerical Procedure
Following an elastic increment in which the highest stressed element
is scaled to cause incipient yielding, various increments of load equal to
10 or 20 percent of that in the initial solution are carried out. The nodal
release procedure is implemented upon achievement of the static similarity
solution of Tracey [22] and further loading is applied at the new crack
length. Various steps of crack advance and external loading at constant
crack length are performed. Displacement boundary conditions correspond-
ing to the elastic singular strain dominant at the crack tip are specified
on a radius which is 224 times the smallest element size and 20 times the
maximum extent which the plastic zone acquires in the course of the compu-
tation. These ratios insure an appropriate boundary-layer formulation of
the small-scale yielding situation [7]. The next term beyond the inverse
square-root singularity in the surrounding elastic field, namely, a tension
T parallel to the crack, is taken as zero. Figure 1 presents the load histories
relevant to the analyses presented here; in this figure, Ka is the stress in-
tensity factor at the first load increment, ffo is the yield stress in tension,
and / — /o is the difference between the current and initial crack lengths.
These "staircase" load histories represent hypothetical cases which might

0.5 1.0 1.5 2.0 2.5

FIG. 1—Load histories applied to the present analyses.

 SORENSEN ON PLANE STRAIN STABLE CRACK GROWTH 157

be found in service and reflect the different K levels required to obtain

appropriate similarity solutions prior to crack advance.

Mesh Configuration and Material Properties

The finite-element grid used in these analyses is indicated in Fig. 2 with
details of the refined mesh surrounding the crack tip presented in Fig. 3.
A total of 1660 elements is used together with 865 nodes and 1730 degrees
of freedom. Use of static condensation reduces the number of active de-
grees of freedom to 946. The radius of the outer ring in Fig. 3 divided
by the smallest element length is 28. The radii of the rings in Fig. 2 divided
by the inner element length are 28, 34, 42, 52, 64, 80, 104, 136, 176, and
224. The radial lines are spaced at 10-deg intervals. Material properties
are v = 0.3 and E/oo = 1000 where E is Young's modulus and CTQ the
tensile yield strength. The analysis corresponding to the ideal plasticity
case was carried out on the IBM 360/67 available at Brown University
Computing Laboratory. The two hardening analyses were performed using
the IBM 370/168 available at the Massachusetts Institute of Technology.

Results and Discussion

Crack Face Profiles

Figure 4 presents crack face profiles following the load incrementation
procedure for the stationary crack and following the final crack advance

Asymptotic Displacements Prescribed

u. ^ K i y r g. {S,v)

^TRACTION-FREE CRACK FACE SYMMETRY DISPLACEMENT-

CONDITION
FIG. 2—Schematic representation of finite element grid and boundary conditions.
158 ELASTIC-PLASTIC FRACTURE

element t\^ | element C

original
igmai crock
crock tip
tip --^ element B

FIG. 3—Arrangement of elements in fine-mesh region.

step for the cases N = 0.0, 0.1, and 0.2. The profiles following the final
crack advance step are considered representative of steady-state conditions
in the vicinity of the crack tip, but not overall, as away from the crack
tip the crack faces experience continuing deformation. Direct comparisons
of the stationary crack profiles with those of Tracey [22] indicate maximum
deviations of 6, 5, and 3 percent for N = 0.0, 0.1, and 0.2, respectively.
Since the present analyses do not employ special singularity elements like
those of Tracey [31] and do not include finite geometry changes, the crack
tip opening displacements, 8, are estimated by extrapolation. For the non-
hardening case, a value of 0.66 is obtained for 5 nondimensionalized by
J/oo, where/ is taken equal to (1 ~ v'^)K\^/E, corresponding to the small-
scale yielding situation. Tracey predicts a value of 0.54 for this ratio, but
Parks [33] suggests that this number should be 0.65 due to the artificial
path dependence of/ that seems (through comparison with a corresponding
"deformation theory" solution based on Tracey's mesh, leading to similar
path-dependence) to be directly traceable to Tracey's nonhardening sin-
gularity element. For nonhardening blunting solutions, McMeeking [3] re-
ports values for the nondimensionalized COD between 0.55 and 0.67, de-
pending on the point of measurement. The larger of these two numbers
is representative of larger values of Oo/E, on the order of 1/100. For Oo/E
equal to 1/300 and N — 0.1, McMeeking reports values of the nondimen-
sionalized COD between 0.41 and 0.44, and for N = 0.2 values between
0.27 and 0.30 are reported, although higher values result when measured
at the elastic-plastic boundary. The present hardening analyses predict
a value of 0.54 for the nondimensionalized COD when iV = 0.1 and 0.44
when the hardening exponent equals 0.2. The good agreement of the extrap-
 SORENSEN ON PLANE STRAIN STABLE CRACK GROWTH 159

K i = K i FOR FINAL STATIC SOLUTION

r MEASURED FROM ORIGINAL
CRACK TIP
I I 1 1
-.16 -.04
r
(KI/O-Q)

FIG. 4—Crack surface displacements, stationary and steady-state advancing crack solutions
/ o r N = 0.0, 0.1, and 0.2.

olated values of 5 with the work of McMeeking and others lends confidence
to the results of the present analyses as regards the prediction of the crack
face profiles.
Figure 5 presents crack face profiles at key points in the evolution of
the prescribed load history for the nonhardening case. This analysis, which
models crack growth at constant load between equidistant nodal points
in rate-independent materials, results in a crack face profile which as the
crack advances becomes less angular with distance from the crack tip
(where the "angle" is measured clockwise from a horizontal line behind the
crack tip). This is also true for the hardening cases as indicated by the
final crack profiles shown in Fig. 4. Rice [7,8], for quasi-static crack ad-
vance in a nonhardening material, derives a displacement distribution pro-
portional to r In r (where r is the radial distance measured from the crack
tip) which implies a vertical tangent at r = 0. Nodal displacements from
160 ELASTIC-PLASTIC FRACTURE

K i = K i FOR FINAL STATIC SOLUTION

r MEASURED FROM ORIGINAL
CRACK TIP
I -L«- -^t^!!.
.16 .12 -.08 -.04 .00 .04 .08 .12
r

FIG. 5—Evolution of crack surface displacements through the loading history of the non-
hardening case.

the present analyses permit curve fairing, which exhibit the vertical tangent
required by the analytic solution. This infinite slope is a local phenomenon
and may not overly influence the effective definition of a crack tip opening
angle, defined here as the total angle between the separating crack faces
behind the extending crack tip. However, this remains an open issue in
need of further study. Due to the linear interpolation functions used in the
present analysis, this angle is evaluated at the node immediately behind
the crack tip, and resulting values are presented in Table 1. The trend
toward a steady-state value is anticipated from the prescribed loading, and
the numbers in Table 1 indicate the material dependent nature of the CO A.
As the final displacement distributions in Fig. 4 suggest, the angles would
be less if based on elements farther back from the crack tip, and the param-
eter seems to be meaningless according to theoretical considerations in the
limit of r approaching zero. To further clarify the role of the COA and its
 SORENSEN ON PLANE STRAIN STABLE CRACK GROWTH 161

TABLE 1 --Crack-tip opening angles. rad.

Release 1 Release 2 Release 3 Release 4 Release 5

N= 0.0 0.015 0.016 0,017 0.017 0.017

N= 0.1 0.016 0.017 0.018 0.018
N=0.2 0.018 0.019 0.020 0.020

relation to continuing fracture, a finite strain analysis in the spirit of

McMeeking [3] is desirable for the region in the vicinity of the crack tip.
Prior to any crack advance, the COD is related to the external loading
according to 6 = aJ/a„, where a is dependent on material properties and
hardening exponent. As previously remarked, values of a for the stationary
portion of the present analyses are 0.66, 0.54, and 0.44, which agree well
with published values. Following crack advance, the load histories prescribe
increments of external load at constant crack length, and a relation be-
tween incremental COD and increments in external loads is

dJ
d6 = a— (1)
oo

Values of a in this relation are presented in Table 2; increments in COD

are measured at the node immediately behind the current crack tip and
the original crack tip position. The numbers in Table 2 indicate that the
nominal crack tip opening displacement continues to be effectively char-
acterized by increments in J for external loading at fixed crack length fol-
lowing crack advance steps. For each of these analyses, a seems to be
constant, namely, 0.66,0.54, and 0.44 foriV = 0.0,0.1, and 0.2, respectively.
The present analyses prescribe crack advance at constant external load,
and here the resulting incremental crack surface displacements are related
to the crack advance step. Rice [8] obtains a relation between incremental
displacements resulting from an increment of crack advance, dl, and the
increment of crack advance. This relation may be written as

dui - — Am + 5i(0) In — dl
E r

wherevl,(0) and Bi(d) are dimensionless functions and /? is a characteristic

length of the plastic zone. Noting that for small-scale yielding, R scales
with EJ/a^, and specializing the above expression to the crack surface
(B = TT), the following is obtained

dh = ^&\J\^^dl (2)
E \ ffoW
162 ELASTIC-PLASTIC FRACTURE

«
3i
<it
.£
"oS
c ci
.2
•t;
u
a>
•5 *2
<
a
.& (N
H U
M
Ctt
« a>
s
t0
19 4:
•o <
u
is
sMCQ a>
M
U A
s Q:S
bi
V

«
<

a:

•a:

8 »

;2

a OS o d o
<

o ^ <N
o d d

:2;S;S;
 SORENSEN ON PLANE STRAIN STABLE CRACK GROWTH 163

Here, the proportionality constants have been lumped into |8 and X, and
d6 is the COD increment. Equation 2 may be integrated to obtain an ex-
pression for the additional COD resulting at a fixed point X as the crack
advances under constant / , from h to ^2, two crack lengths such that
/z > /i > X. The integration results in

5(/2, X) - 5(/,, X) = 0^ ih - X) In
a^Hh - X)
XeEJ
(Zi -X)\n (3)
OoHh - X)

For the node immediately behind the advancing crack tip, h = X and the
second term on both the left side and right side of Eq 3 is zero. Figure 6
presents a plot of crack surface displacement values for the node immediately
behind the crack tip taken from the present hardening and nonhardening
analyses. The plot indicates a unique pair (/3, X) which satisfies Eq 3,
namely, /3 = 9.5 and X = 0.04. These values of j3 and X do not correctly
predict the incremental COD due to crack advance at the other nodes
behind the advancing tip, indicating that Eq 2 may apply only within a
certain distance from the crack tip. This latter point is currently under
further investigation. However, the good correlation provided by Eq 2 for
incremental crack opening near the advancing crack tip due to an increment
in crack advance at constant external load indicates the possibility of using
Eq 2 in conjunction with Eq 1 to provide a relationship for the characteriza-
tion of incremental COD's with increments of external loads as well.
That is

dd = a — + 0^\n ^ d l (4)

This equation appears useful in the study of continuing fracture, and its
use in developing a fracture criterion based on near-tip crack opening is
more fully explored in Ref 40.

Stress Distributions Ahead of the Crack Tip

The stress fields associated with hardening and nonhardening plane
strain stationary cracks under small-scale yielding conditions are provided
by Tracey [22]. Figure 7 presents curves for the opening stress ahead of
a stationary crack together with corresponding centroidal stress values of
elements directly ahead of the crack tip taken from the present analyses.
The distance from the crack tip to the centroid is used for the position on
the abscissa of the plot. Maximum deviations of the present results from
164 ELASTIC-PLASTIC FRACTURE

0 y
20

'"'C — y
/
18
• N= 0 . 0
- / X N = 0.1
ONrO.Z
16
;8=9.5
X= 0 . 0 4
• /

14

i'j.^A 1 1 1
4.8 5.0 5.2 5.4
eEJ
JU

FIG. 6—Correlation between increments of crack opening displacement and I.

4.0

3,5 \ \ / TR ACEY N = 0.2

3.0

2.5

2.0

STATIONARY STEADY STATE

N = 0.0
N = 0.1

1 N = 0.2

.01
I
.02 .03 .04 .05 .06
r

FIG. 7—022 stress distributions ahead of the crack tip.

 SORENSEN ON PLANE STRAIN STABLE CRACK GROWTH 165

Tracey's curves are 5, 5, and 7 percent for iV = 0.0, 0.1, and 0.2, respec-
tively. The good agreement of these values is a consequence of the fine
mesh employed in the solution.
As the stress gradients near the crack tip become steeper with increasing
hardening exponent, the deviations of the present results from those of
Tracey are expected to increase since these analyses make no use of singular
elements but rely on fine-mesh gradation to capture the appropriate stress
distributions. This formulation provides little information on the angular
stress distribution as r approaches zero and does not precisely obtain the
factor of 2.97 in an stress elevation overCTOas the Prandtl solution demands
in the nonhardening case.
Figure 7 also indicates points corresponding to the apparently steady-
state stress distribution predicted in the present analyses. Following the
final crack advance step, there are minor elevations of on ahead of the
tip with maximum deviations |^rom Tracey's results for a stationary crack
of 4, 3, and 7 percent for N = 0.0, 0.1, and 0.2, respectively. Similar
plots of the 022 stress distribution ahead of the crack tip, following inter-
mediate crack advance steps, reveal points between the static and steady-
state points presented in Fig. 7. The conclusion is that under small-scale
yielding conditions for both hardening and nonhardening materials, the
(T22 stress distribution ahead of a growing crack is effectively the same as
the corresponding stress distribution for a stationary crack.

Material Stress Histories

Three material point stress histories are now described. The points under
scrutiny are the element immediately behind the initial crack tip, an element
ahead of the initial crack tip and on the prospective fracture plane, and
an element ahead of the initial crack tip but removed from the fracture
plane. These material points are designated Elements A, B, and C, re-
spectively, and are indicated in Fig. 3. The stresses described are centroidal
stresses associated with the constant-strain triangles used in the analyses.
Element A—Following the final static load incrementation step, values
offfii/(7oare 1.14, 1.29, and 1.51; values of a22/<^o are 1.27, 1.44, and 1.69;
and values of on/oo are - 0 . 5 7 , - 0 . 6 9 , and - 0 . 8 6 for N = 0.0, 0.1, and
0.2, respectively. Upon subsequent crack advance and load incrementation
steps, both ffi2 andCT22become small in contrast withCTH,which dominates
the later stress history. After the last crack advance step, nondimensionalized
values of an are 1.07, 1.13, and 1.06, respectively, for the three analyses.
The final value ofCTHseems to represent a residual tensile stress in a plastic
wake region. For the nonhardening case there is some continuing plastic
flow in this wake, but not in the hardening cases. This is due to the isotropic
model of strain hardening used in these analyses and underlines the sensi-
tivity of the results to the constitutive model used. The Prandtl stress
166 ELASTIC-PLASTIC FRACTURE

distribution suggests that an = 1.15CTOon the crack surfaces immediately

behind the crack tip, and the final value of an in the nonhardening case
is 7 percent below this number.
Element B—This material point lies ahead of the initial crack tip but
behind the final crack tip position in these analyses. This material experiences
continued increases in an and 022 during the first two nodal release steps.
Values of 022/0^ following the second nodal release step are 2.74, 3.03,
and 3.45 for JV = 0.0, 0.1, and 0.2, respectively. Then, during the remaining
nodal release steps,CT22drops to a small value and ow becomes the dominant
stress at the point. It is not surprising thatCT22should drop drastically,
because the traction-free boundary condition imposed on the open crack
face requires that 1x22 be zero there (assuming negligible deviation of the
surface normal from the X2 direction). Also, following the third nodal
release step during which this element becomes part of the material behind
the crack tip, values of ffn/uo are —0.56, —0.67, and —0.80, respectively,
for the three hardening exponents. For the nonhardening case, an and
a22 are nearly equal and this is consistent with the notion that at this stage
of its history the element is part of a centered fan above the crack tip,
requiring a hydrostatic stress state coupled with plastic shearing. Then, as
the crack advances farther, this element passes from the centered fan region
to a residually stressed wake region as discussed above.
Element C—This element is farther removed from the plane of fracture
than Element B. Figure 8a presents a plot of its stress history versus crack
advance step for the nonhardening case. This plot is also representative of
the hardening results but with an appropriate shift of the vertical axis.
Figure 8a also presents stresses as predicted from a Prandtl stress distribution
traveling with the crack; the Prandtl slipline construction is indicated
in Fig. 86. The material point under discussion is imagined enveloped by
Region 1 in Fig. 8£> following Release 1, in the centered fan of Region 2
following Releases 2 and 3, and in constant-state Region 3 following the
final two steps of crack advance. For the stress values plotted in Fig. 8a,
6 is taken as the angle between a horizontal line and a line joining the
centroid of Element C to the actual crack tip. The numerically calculated
stress distribution reflects a pattern similar to that predicted by the Prandtl
field. However, the fuzzy crack-tip phenomenon is also evident. This ter-
minology describes the consequences of a finite element discretization
which cannot exactly reproduce appropriate strain singularities at the
crack tip so that elements surrounding the crack tip respond to an ill-defined
crack tip with a corresponding vagueness in the definition of B and r. This
effect is minimized as the mesh is refined.
These brief descriptions of material stress-point histories reinforce the
previously made point that the steady-state a22 stress distribution is effectively
the same as that prior to any crack extension. Additional conclusions are
as follows. (1) Following crack advance steps, points positionally similar
 SORENSEN ON PLANE STRAIN STABLE CRACK GROWTH 167

a-./ 3 Prondtl field

Element Length

FIG. 8a—Stress history of Element C plotted versus crack advance for the nonhardening
case.

(l + ^ - 2 5 ) r o

(2 + TT) TO

(l + Tr)To

FIG. 8fc—Prandtl slipline field stress distribution.

with respect to the crack tip experience similar stress histories of loading
and unloading, which suggests that a steady state prevails near the advancing
crack tip. Positionally similar stress histories are a tacit assumption of the
ideally plastic theoretical analysis of plastic strain singularities for growing
cracks [7,8] discussed earlier. (2) Prior to any given nodal release step,
the elements surrounding the crack tip experience essentially the same
stress field as those surrounding a stationary crack. This stress field, within
the limitations of the present analyses, resembles the Prandtl slipline
solution for the nonhardening situation. (3) The wake material is dominated
by a residual, tensile an stress which results in continued yielding in the
168 ELASTIC-PLASTIC FRACTURE

nonhardening case, but not in the hardening cases due to the isotropic
hardening model used.

Plastic Zone Shapes

The shapes of the active plastic zones, corresponding to the final static
load increment and the final step of crack advance, nondimensionalized
by the similarity quantity {Ki/ooY, are presented in Fig. 9a-c for the
hardening exponents 0.0, 0.1, and 0.2, respectively. The plastic zone
shapes corresponding to the static crack solutions are in good agreement
with appropriate cases documented in Refs 22, 25, and 34. The effect of
the crack advance on the shape of the plastic zone is to constrict its width
and to tilt the inclination of the zone toward the symmetry axis (X2 = 0).
Between the nodal release steps and depending on the amount of load
incrementation, the plastic zone attempts to restore the butterfly shape
familiar from static crack analysis. The achievement of a steady-state
solution is particularly evident from results of the nonhardening case,
which includes two consecutive nodal release steps; there is negligible
difference between the plastic zone shapes following these crack advance
steps. The small, actively plastic wake region in the nonhardening case
contrasts sharply with the lack of this region in the hardening cases. As
has been remarked, this is attributed to the isotropic constitutive theory
used in these analyses. Also, the angular tilt and plastic zone constriction
are reminiscent of similar growth effects encountered in Mode III analyses,
for example, Refs 9 and / 7.

Material Strain Histories

The equivalent plastic strain history of a material point in the slipline
fan above the crack tip before crack advance indicates a high amount
of straining prior to crack advance, further straining during the first nodal
release step, and negligible further straining. The strain incurred during
the first crack advance step is due to the fan region sweeping by the material
point. A material point positionally similar with respect to the crack tip
before the third crack advance step is not plastically strained until the
crack has advanced sufficiently to engulf this point with its accompanying
plastic zone. This element is then strained irreversibly, but less than the
corresponding element in the stationary crack case. Ratios of the plastic
strain at the second material point prior to Nodal Release 3 divided by
the plastic strain at the positionally similar material point prior to the
first step of crack advance are 0.95, 0.92, and 0.87 for N — 0.0, 0.1, and
0.2, respectively. These values corroborate the observation of Rice \S\ that
the strain field associated with an extending crack sustains a weaker
singularity than that associated with a stationary crack, r~' versus In r for
 SORENSEN ON PLANE STRAIN STABLE CRACK GROWTH 169

X = 0 AT O R I G I N A L CRACK TIP
t INDICATES F I N A L CRACK TIP POSITION
0 20

0. I 5

0. 10

0.05

X = 0 AT O R I G I N A L CRACK TIP
t I N D I C A T E S F I N A L CRACK T I P P O S I T I O N

^ I .'I I L
0.00 ' 0.10 0.20
(Kj/cT^)'^

X = 0 AT O R I G I N A L CRACK TIP

(Kj/a-o) t INDICATES F I N A L CRACK TIP P O S I T I O N

0.15

0. 10

0,05

,1 . ' I I U
0.00 ' 0.10 0.20
(Kj/CTo)^

FIG. 9—Stationary and steady-state plastic zone shapes for (a) N = 0.0, (b) N = 0.1. and
(c) N = 0.2

the nonhardening case. The foregoing ratios also imply that, at least for
strain-controlled ductile rupture mechanisms, the extent of stable crack
growth is greater for a hardening material than for a nonhardening material.

Separation Energy Rates

Each of the nodal release steps provides a record of vertical displacement
versus reaction force. Calculating the area under this curve and dividing
by the finite crack advance step, one obtains a separation energy rate, G^
in the notation of Kfouri and Rice [35]. It is the finite value of the crack
advance step that renders this calculation nontrivial, for in the limit of
growth step approaching zero and for materials which exhibit a finite
170 ELASTIC-PLASTIC FRACTURE

stress level at the crack tip, such a calculation yields zero for G^ [36].
G^ values taken from the work of Kfouri and Miller [19] and McMeeking [3]
together with values from the present analyses are plotted in Fig. 10. The
points of Kfouri and Miller result from the plane strain analysis of the
tensile and equibiaxial loading of a finite plate containing a crack. The
ratio of crack length to plate width is 0.125 and the ratio of Young's
modulus to initial yield stress is 667.7. Their analyses, model the material
as linear hardening with a tangent modulus equal to 0.023 times the elastic
modulus and Poisson's ratio equal to 0.3. The points of McMeeking are
taken from separation energy rate calculations for the small-scale yielding
analysis of a blunted notch; material properties are E/oo = 300, Poisson's
ratio = 0.3, and a power-law hardening exponent N equal to 0.1. Mc-
Meeking's work includes finite strain effects at the blunted notch and
employs crack-growth steps on the order of the crack opening displacement,
whereas the growth steps employed by Kfouri and Miller and the author
are much larger. The "steady state" point of McMeeking corresponds to
the final growth step calculated and it is presented for completeness although
his analysis does not indicate that steady-state conditions are achieved.
The explanation for the separate pattern of points due to Kfouri and
Miller is thought to be the "T effect", which is explored by Larsson and
Carlsson [34] and Rice [37]. The origin of the effect is the presence of
nonvanishing, nonsingular terms in the eigenvalue expansion of the elastic

®
J
.50
O

®
.40
A
o

.30 + o
•
• INITIAL STEADY STATE
.20 ° Kfouri Uniaxial • 0

Kfouri Biaxial * ®
• McMeeking • n
.10 • fN = 0.0 + s
Sorensen ( H- 0.1 A A
•
1 N = 0.2 T
n -r 1 1 i 1 I. 1 1 1
.01 .02 .03 .04 .05 .06 .07

FIG. 10—Separation energy rates correlated with J and plotted versus growth step.
 SORENSEN ON PLANE STRAIN STABLE CRACK GROWTH 171

stress tensor at the crack tip in plane strain. The points corresponding
to the equibiaxial tensile loading case of Kfouri and Miller provide the best
fit with points from the present analyses. This is because the present
formulation has T = 0 and, for an infinite plate under equibiaxial loading,
T = 0 (for a finite plate T = 0). Kfouri and Rice [35] present a relation
between / and G^ for the tensile load case in an effort to correlate the
two quantities, but a subsequent communication with Kfouri has indicated
a different relation for the equibiaxial loading case. The conclusion is that
the use of / as a correlator of the separation energy during a finite growth
step of the size they explored is sensitive to the nonzero, nonsingular
stress terms present at the crack tip in plane strain. Figure 10 also indicates
that the value of G'^ is sensitive to the degree of strain hardening. But,
these observations are made for points corresponding to growth step sizes
far in excess of values comparable to the COD. Since it is in a region of
linear extent on the order of COD that ductile fracture mechanisms such
as void coalescence and localization of shear dominate, it would seem to be
step sizes of this order that are of greatest interest. By using such small
crack advance steps it may be investigated whether / correlates with G^
independently of T, the value of N, and the extent of yielding. That is,
do all the curves which are different for larger crack advance steps merge
into a single curve for step sizes on the order of the COD? At present,
only the values reported by McMeeking have grovrth steps in this range.
Of course, in such analyses, finite-strain considerations must be properly
treated by McMeeking and Rice [38]. Due to the elastic unloading that
occurs during crack advance, / should be correlated with G^ values for the
initial nodal release; similar correlations with C" values for subsequent
nodal releases must be interpreted carefully due to the clouded meaning of
/ following crack extension.
Finally, a discussion of the validity of the G^ quantity is warranted.
The nodal reaction force to be relaxed to zero is related to the stress field
surrounding the crack tip. As this force is relaxed to zero, the appropriate
nodal displacement is monitored so the work expended in the relaxation
process may be calculated. The elastic unloading of the body and the
accumulated strain at the crack tip influence this displacement. However,
no size scale is inherent in this calculation except that imposed by the
finite-element mesh, and, as such, the fundamental significance of G'^ is
obfuscated unless a direct correlation of the step of crack advance may be
made to a microstructurally significant dimension such as the crack opening
displacement. Although their model involves failure by cleavage, Ritchie
et al [39] emphasize the necessity of an appropriate size scale in a fracture
criterion and correlate the fracture toughness of mild steel with the achieve-
ment of a critical tensile stress over a distance on the order of the spacing
of crack nucleating carbides.
172 ELASTIC-PLASTIC FRACTURE

Conclusions
The following conclusions are drawn from the present analyses:
1. The advancing crack profiles are consistent with the theoretical
result of a vertical tangent at the crack tip. Since this is a local effect,
it may yet be possible to sensibly define a crack opening angle.
2. Extending cracks in hardening and nonhardening materials are
subject to effectively the same stress distributions as geometrically similar
stationary cracks. The strains accumulated ahead of a moving crack tip are
less than those of a corresponding stationary crack in corroboration of the
analytical work of Rice [7,8].
3. The active plastic zone ahead of a growing crack constricts and
tilts, paralleling the behavior predicted from analytic and numerical in-
vestigations of Mode III cracks.
4. For separation energy rates calculated for crack growth steps much
greater than the nominal crack opening displacement, the use of / as a
correlator is highly sensitive to strain-hardening properties and the details
of external loading.
5. The incremental opening at the crack tip, due to load increase at
fixed crack length, seems to be given hy dd = a dJ/oo; the value of a
depends on material properties but is the same regardless of the extent of
crack growth. Increments in crack surface displacement may be correlated
with increments of crack growth at constant external load through the
expression

The parameters & and X are constant in these analyses.

Considerations which remain to be addressed are the effects of crack
advance under increasing external load, the effects of different plasticity
models on the present small-scale yielding solutions, and the effects of
large-scale, fully plastic specimen behavior. Also, the role of the finite
strain at the root of a blunted crack should be further investigated in the
extending-crack situation.

Acknowledgments
This study was supported by the Energy Research and Development
Agency under Contract EY-76-S-02-3084 and by the National Science
Foundation Materials Research Laboratory at Brown University. The
author expresses his gratitude to Professor James R. Rice for his guidance
in this study and his patience in reviewing this manuscript.
 SORENSEN OF PLANE STRAIN STABLE CRACK GROWTH 173

References

(/] Rice, J. R. in The Mechanics of Fracture, F. Erdogan, Ed., American Society of Me-
chanical Engineers, AMD-Vol. 19, 1976, pp. 23-53.
[2] Rice, J. R. and Johnson, M. A. in Inelastic Behavior of Solids, M. F. Kanninen et al,
Eds., McGraw-Hill, New York, 1970, pp. 641-672.
[3] McMeeking, R. M., Journal of the Mechanics and Physics of Solids, Vol. 25, 1977,
pp. 357-381.
[4] Shih, C. F. in Fracture Analysis. ASTM STP 560, American Society for Testing and
Materials, 1974, pp. 187-210.
[5] McClintock, F. A., Journal of Applied Mechanics, Vol. 25, 1958, pp. 582-588.
[6] McClintock, F. A. and Irwin, G. R. in Fracture Toughness Testing and Its Applications,
ASTM STP 381, American Society for Testing and Materials, 1965, pp. 84-113.
[7] Rice, J. R. in Fracture: An Advanced Treatise, H. Liebowitz, Ed., Vol. 2, Academic
Press, New York, 1968, pp. 191-311.
[8] Rice, I. R. in Mechanics and Mechanisms of Crack Growth (Proceedings, Conference at
Cambridge, England, April 1973), M. J. May, Ed., British Steel Corporation Physical
Metallurgy Centre Publication, 1975, pp. 14-39.
[9] Chitaley, A. D. and McQintock, F. k.. Journal of the Mechanics and Physics of Solids,
Vol. 19, 1971, pp. 147-163.
[10] Broek, D., International Journal of Fracture Mechanics, Vol. 4, 1%8, pp. 19-29.
[7/] Green, G., Smith, R. F., and Knott, J. F. in Mechanics and Mechanisms of Crack
Growth (Proceedings, Conference at Cambridge, England, April 1973), M. J. May, Ed.,
British Steel Corporation Physical Metallurgy Centre Publication, 1975, pp. 40-54.
[12] Green, G. and Knott, J. F., Journal of the Mechanics and Physics of Solids, Vol. 23,
1975, pp. 167-183.
[13] Clarke, G. A., Andrews, W. R., Paris, P. C , and Schmidt, D. W. in Mechanics of Crack
Growth, ASTM STP 590, American Society for Testing and Materials, 1976, pp. 27-42.
[14] GrifRs, C. A. and Yoder, G. R., Transactions, American Society of Mechanical Engineers,
Journal of Engineering Materials and Technology, Vol. 98, 1976, pp. 152-158.
[15] de Koning, A. U., "A Contribution to the Analysis of Slow Stable Crack Growth,"
presented at the 14th International Congress of Theoretical and Applied Mechanics,
Delft (also Report NLR MP 75035 U, National Aerospace Laboratory NLR, Amsterdam),
The Netheriands, 1976.
[16] Andersson, H.,Joumal of the Mechanics and Physics of Solids. Vol. 22,1974, pp. 285-308.
[17] Sorensen, E. P., International Journal ofFracture, Vol. 14, 1978, pp. 485-500.
[18] Andersson, H., Journal of the Mechanics and Physics of Solids, Vol. 21, 1973, pp. 337-
356.
[19] Kfouri, A. P. and Miller, K. J. in Proceedings, Institution of Mechanical Engineers,
Vol. 190, 1976, pp. 571-584.
[20] Shih, C. F., de Lorenzi, H. G., and Andrews, W. R., this publication, pp. 65-120.
[21] Hill, R., The Mathematical Theory of Plasticity, Oxford University Press, Oxford,
England, 1950.
[22] Tracey, D. M., Transactions, American Society of Mechanical Engineers, Journal of
Engineering Materials and Technology, Vol. 98, 1976, pp. 146-151.
[23] Rice, I. R. and Rosengren, G. F.,Joumalof the Mechanics and Physics of Solids, Vol. 16,
1968, pp. 1-12.
[24] Marcal, P. V. and King, I. P., International Journal of Mechanical Sciences, Vol. 9,
1%7, pp. 143-155.
[25] Rice, J. R. and Tracey, D. M. m Numerical and Computer Methods in Structural
Mechanics, S. J. Fenves et al, Eds., Academic Press, New York, 1973, pp. 585-623.
[26] Yang, W. H., Computer Methods in Applied Mechanics and Engineering, Vol. 12, 1977,
pp. 281-288.
[27] Sorensen, E. P., Computer Methods in Applied Mechanics and Engineering, Vol. 13,
1978, pp. 89-93.
[28] Sorensen, E. P., "Some Numerical Studies of Stable Crack Growth," Ph.D. dissertation.
Brown University, Providence, R.I., 1977.
174 ELASTIC-PLASTIC FRACTURE

[29] Nagtegaal, J. C , Parks, D. M., and Rice, J. R., Computer Methods in Applied Mechanics
and Engineering, Vol. 4, 1974, pp. 153-177.
[30] Guyan, R. J., Journal of the American Institute of Aeronautics and Astronautics, Vol. 3,
1965, p. 380.
[31] Tracey, D. M., Engineering Fracture Mechanics, Vol. 3, 1971, pp. 255-265.
[32] Barsoum, R. S., IntemationalJoumal for Numerical Methods in Engineering, Vol. 11,
1977, pp. 85-98.
[33] Parks, D. M., "Some Problems in Elastic-Plastic Finite Element Analysis of Cracks,"
Ph.D. dissertation. Brown University, Providence, R.I., Chapter 3, 1975.
[34] Larsson, S. G. and Carlsson, A. I,, Journal of the Mechanics and Physics of Solids,
Vol. 21, 1973, pp. 263-277.
[35] Kfouri, A. P. and Rice, J. R. in Fracture 1977, D. M. R. Taplin et al, Eds., Solid
Mechanics Division Publication, University of Waterloo Press, Waterloo, Ont., Canada,
Vol. 1, 1977, pp. 43-59.
[36] Rice, J. R. in Proceedings, 1st International Congress on Fracture, Sendai, Japan,
T. Yokobori et al, Eds., Japanese Society for Strength and Fracture, Vol. 1, 1%5,
pp. 309-340.
[37] Rice, J. R., Journal of the Mechanics and Physics of Solids, Vol. 22, 1974, pp. 17-26.
[38] McMeeking, R. M. and Rice, J. R., International Journal of Solids and Structures,
Vol. 11, 1965, pp. 601-616.
[39] Ritchie, R. O., Knott, J. P., and Rice, I. R., Journal of the Mechanics and Physics of
Solids, Vol. 21, 1973, pp. 395-410.
[40] Rice, J. R. and Sorensen, E. P., Journal of the Mechanics and Physics of Solids. Vol. 26,
1978, pp. 163-186.
R. M. McMeeking^ and D. M. Parks^

On Criteria for J-Dominance of

Crack-Tip Fields In Large-Scale
Yielding

REFERENCE: McMeeking, R. M. and Parks, D. M., "On Criteria for/-Dominance

of Crack-Tip Fields in Large-Scaie Yielding," Elastic-Plastic Fracture. ASTM STP
668, J. D. Landes, J. A. Begley, and G. A. Qarke, Eds., American Society for Testing
and Materials, 1979, pp. 175-194.

ABSTRACT: Very detailed finite-strain/finite-element analyses of deeply cracked

plane-strain center-notch panel and single-edge crack bend specimens were generated
using nonhardening and power-law-hardening constitutive laws. The deformation
was followed from small-scale yielding into the fully plastic range. The objective
was to provide insight as to the minimum specimen size limitations, relative to the
characteristic crack-tip opening dimension J/oo, necessary to assure a /-based domi-
nance of the crack-tip region. The criterion used to judge the degree of dominance
was the extent of agreement of the present stress and deformation fields at the blunted
crack tips with those calculated by McMeeking for small-scale yielding. For deeply
cracked bend specimens, we find very close agreement of the near-tip fields with those
of small-scale yielding up to / values of aoX/25, where Z represents the remaining
uncracked ligament (and in the deeply cracked case, the only pertinent specimen
dimension). This value is consistent with previously proposed / testing size limitations.
However, we find that quite detectable deviation from the small-scale yielding fields
occurs in both hardening and nonhardening center-crack specimens at considerably
smaller / values relative to ligament dimension. This suggests that minimum specimen
size requirements necessary to ensure a /-based characterization of the crack tip
region may well be more stringent for center-crack or other low plastic constraint
configurations than in bend-type specimens. A perhaps overly conservative value
of 200 is proposed as the minimum ligament-to-//ao ratio which ensures a sensible
/-based characterization of the crack-tip region in center-crack specimens of materials
exhibiting moderate to low strain hardening.

KEY WORDS: crack propagation, J-integral, plasticity, large-scale yielding, fracture

(materials), fracture toughness testing, tip field dominance

'Formerly, acting assistant professor. Division of Applied Mechanics, Stanford University,

Stanford, Calif. 94305; currently, assistant professor. Department of Theoretical and Applied
Mechanics, University of Illinois, Urbana, III. 61801.
^Formerly, assistant professor, Department of Engineering and Applied Science, Yale
University, New Haven, Conn. 06520; currently, assistant professor. Department of Mechanical
Engineering, Massachussetts Institute of Technology, Cambridge, Mass. 02139.

175

Copyright 1979 b y A S T M International www.astm.org

176 ELASTIC-PLASTIC FRACTURE

In 1971 Begley and Landes [1,2Y and, independently, Broberg [J],

proposed that the J-integral [4] could be used as a ductile fracture crite-
rion, subject to certain not-well-defined limitations. In addition, Begley
and Landes provided experimental evidence suggesting that fi-acture initia-
tion could be correlated with attainment of a critical / value in a wide,
but not unlimited, range of specimen sizes and configurations.
In the original papers, Begley and Landes discussed the possible influence
of specimen geometry on the suitability of a one-parameter ductile fracture
criterion. They noted McClintock's [5] observations concerning the widely
varying crack-tip stress and deformation states, as calculated from non-
hardening plane-strain slipline solutions, in different specimen (or struc-
tural) geometries. For example, deep double-edge notched (DEN) tension
and edge-cracked bend (ECB) type specimens exhibit high triaxial tension
on the plane ahead of the crack, while the center-cracked panel (CCP)
develops no such elevation of triaxiality, as straight 45-deg sliplines pro-
ceed from the crack tips to the free surfaces. McClintock noted that the
radically different near-tip stress states in these specimens could presumably
affect the microstructural mechanisms of ductile cracking, notably void
growth and coalescence. The nonuniqueness in crack-tip fields determined
from small strain formulations is a consequence of the nonhardening
idealization. When any strain hardening is included, asymptotic analysis
[6,7] leads to crack-tip singular fields which are unique to within a scalar
amplitude factor, and the J-integral serves as a measure of this amplitude.
Begley and Landes argued that since virtually all materials exhibit some
strain hardening, / should characterize the near-tip fields at least up to the
inception of crack extension.
However, Rice's [8] complete solutions in antiplane strain for large-scale,
but contained, yielding in power-law hardening materials (a oc e'' in the
plastic range) show that the size over which the /-controlled term actually
dominates higher-order terms in the crack-tip region is a decreasing func-
tion of strain hardening exponent N, and vanishes in the nonhardening
limit of N going to zero. If the unique Hutchinson-Rice-Rosengren (HRR)
[6,7] fields are to dominate the crack tip fields over a physical size scale
relevant to microstructural fracture processes, it is evident that the degree
of hardening is important as well.
Begley and Landes also argued that the finite geometry changes associ-
ated with crack-tip plastic blunting should contribute to a /-characterized
uniqueness. However, as the plastic deformation leading to the blunted
configuration is itself a response to the changing crack-tip fields, it is not
clear that blunting per se should be assigned a casual role as to maintain-
ing a /-characterized uniqueness. Furthermore, the degree of crack-tip
uniqueness associated with blunting must itself depend upon strain harden-

^The italic numbers in brackets refer to the list of references appended to this paper.
 MCMEEKING AND PARKS ON CRACK-TIP FIELDS 177

ing. For example, in the nonhardening case, the blunted crack configura-
tion may be smooth [9] or contain any number of sharp vertex features [10].
The finite geometry changes associated with blunting, however, do
provide a framework for assessing the degree to which a single parameter
characterization of the crack tip is appropriate. The reason is that the
blunted crack opening, given roughly by J/aaow, where CF now is a representa-
tive tensile stress level for plastic deformation, sets the local size scale over
which large strain and high triaxiality develop [9,11] and, consequently,
the size scale on which microscopic ductile fracture processes may be
presumed to act. Indeed, for ductile fracture initiation in small-scale
yielding, calculated crack opening displacements correlate well with the
spacing of void-nucleating second-phase particles [9,11].
For this blunted region to be uniquely characterized by a single param-
eter such as /, but otherwise independent of geometry and loading, it
is evident that its size must be small compared to any other specimen
dimension. Or, to put it the other way, for operative microstructural frac-
ture processes to be embedded within a blunted region characterized
by /, a specimen must meet certain minimum size requirements. Paris
[12] suggested that, in addition to generating plane-strain constraint along
the crack front, all specimen dimensions in a valid / test be chosen to
exceed some multiple M ofJu/aoow, where/ic is the value of/ identified
with the initiation of stable crack growth. Values of 25 or 50 for the co-
efficient M have been found to give rise to essentially size-independent
/ic and/-Aa curves in bend or compact tension (CT) specimens [13]. How-
ever, Begley and Landes [14] report that the resistance curve determined
for a CCP specimen was quite different from that of a CT specimen, even
though the remaining ligament of the CCP did exceed 25 times the inferred
/ic/ffnow. In fact, the direction of macroscopic crack growth in this CCP
specimen followed the 45-deg sliplines of the nonhardening idealization.
This suggests the likelihood of a breakdown, due to the remote plastic flow
field, of a /-dominated crack-tip region in this specimen at a ratio of liga-
ment L over //ffnow somewhat greater than 25. In discussing Ref 14, Rice
[15] suggested that the size, relative to remaining ligament, over which /
dominates crack-tip fields may well be considerably smaller in fully plastic
CCP specimens than in bend configurations. Consequently, the numerical
factor M in the minimum-size requirement may be considerably larger
than 50 in the case of CCP specimens.
In view of the uncertainties just noted regarding the limits of validity of
a /-characterized fracture process zone, the present work was undertaken.
The objective was to provide some insight into the specimen geometry and
strain-hardening dependence of the scalar M which, for fixed specimen
dimensions, defines the limit of deformation (as measured by / ) at which
uniqueness of the crack-tip fields breaks down.
In the following sections, the basic computational procedures are out-
178 ELASTIC-PLASTIC FRACTURE

lined and certain of the results are presented. Finally, the results are dis-
cussed with special consideration of the possible implications for future
experimental work in Jic and / resistance curve testing.

Details of the Computer Analysis

The updated Lagrangian finite-deformation finite-element method sug-
gested by McMeeking and Rice [16], modified according to Appendix 2 of
Nagtegaal, Parks, and Rice [17] to free the mesh of artificial constraint,
was used to analyze plane-strain precracke^ specimens. The method is
based on Hill's [18] variational principle phrased in terms of a current
reference state. In yielded elements the partial stiffness approach of Marcal
and King [19], as modified by Rice and Tracey [20] and Tracey [21,22],
is used to calculate the tangent stiffness, based on the Prandtl-Reuss equa-
tions for isotropically hardening materials as in Hill [23, pp. 15-39], but
generalized to account for material spin.
In some of the calculations a power law for hardening of the uniaxial
Kirchhoff stress 7 [16] versus logarithmic plastic strain Ip curve was used
which had the form

(r/ffo)''^ = r/ao + SGe'/ffo (1)

whereCTOis the tensile stress and G is the elastic shear modulus. In other
calculations a nonhardening law for ? versus plastic strain was used which
will be designated N = 0.
An undeformed finite-element mesh representing one quarter of a
center-cracked panel in tension or one half of an edge-cracked bend speci-
men is shown in Fig. 1, with the detail of the undeformed near-tip mesh
in Fig. 2. The mesh shown has an a/w ratio of 0.9 (see Fig. 3) where a is
measured to the center of the semicircular notch tip, h/w — 3, and the
ratio of undeformed notch width to ligament, bo/L, is 2 X 10"^. There
were also two other meshes, one with a/w = 0.9 and bo/L = 2 X 10"''
and one with a/w = 0.5 and bo/L = 2 X lO"**. All elements were 4-node
quadrilaterals. To model the precracked specimens, traction-free boundary
conditions were applied on the notch surface, while the nodes ahead of the
tip on the crack line were restrained to remain on the crack line. For the
center-cracked panel in tension, the appropriate nodes on the vertical axis
of symmetry were restrained to remain on the axis of symmetry and uniform
vertical displacements were applied across the top of the specimen. In the
case of the edge-cracked bend specimen, traction-free conditions were ap-
plied on the sides while the nodes across the top were constrained to lie on
a straight line rotating around the center of the top of the specimen. This
was done in such a way as to assure that there was no constraint of nodes
parallel to the straight line and that the sum of nodal force increments
 MCMEEKING AND PARKS ON CRACK-TIP FIELDS 179

FIG. 1—Finite-element mesh in undeformed configuration.

FIG. 2—Crack-tip detail of finite element mesh.

normal to the line was zero. Table 1 summarizes the specimen configurations
and material properties.
The crack tips were blunted out to maximum openings between 25 and
55 times their undeformed openings. These openings were accommodated
partly by arranging the near-tip mesh in a way that would lead to length-
ening of the short sides of elements as the tip opened up. The elements on
the crack-tip surface ultimately became extremely long in the circumferential
direction and, through plastic incompressibility, extremely thin in the
radial direction. The results were considered to be sufficiently accurate
because the blunted crack tip was always defined by 13 fairly regularly
spaced nodes and the region within a few crack-tip openings of a crack tip
was always composed of elements quite small compared to the current crack
180 ELASTIC-PLASTIC FRACTURE

M,e

2h

Center-Cracked Edge-Crocked
Panel in Tension Beam in Bending

FIG. 3—Schematic representation of specimen geometries and loadings considered.

tip Opening. Furthermore, the long thin elements are oriented in a way that
is quite favorable for modeling the deformation near the tip surface, which
is predominantly one that stretches fibers parallel to the tip surface.

Results
The current crack-tip opening to undeformed crack-tip opening ratio
St/bo has been plotted versus J/(aobo) in Fig. 4 for all seven specimens
analyzed. The current crack opening 6, was arbitrarily measured in all

TABLE 1—Summary of finite-element solutions generated.

Specimen a/w bo/L N

ECB» 0.9 2 X 10-" 0

ECB 0.9 2 X 10-^ 0
ECB 0.9 2 X 10-" 0.1
CCP* 0.9 2 X 10-" 0
CCP 0.9 2 X 10-" 0.1
CCP 0.5 2 X 10-" 0
CCP 0.5 2 X 10"" 0.1

"ECB = edge-cracked bend specimen.

*CCP = center-cracked panel in tension.
 MCMEEKING AND PARKS ON CRACK-TIP FIELDS 181

o/w N bo/L
60
CC P Q9 0 2x10-*
5t/bo C C P 0.9 0.1 ZxlO"*
CC P 0.5 0 2x10-^
50
C CP 0.5 01 2x10-*
E CB 09 0 2x10"*
E C B 0.9 0 2x10-5
40
E CB 0.9 O.I ZxlO"*

30

20

10

60 70 80
j/(aobo)
FIG. 4—Crack-tip opening 5t versus applied J value for all solutions generated. Crack-tip
opening is normalized with respect to undeformed notch root diameter bo and J is normalized
with respect to aobo.

cases as the distance between the nodes that, in the undeformed configura-
tion, lay at the intersection of the straight flank and the semicircular tip as
shown in the inset of Fig. 4. The value of / was determined from a load
deflection curve of the specimen through the formulas of Rice, Paris, and
Merkle [24]. In addition, the virtual crack extension (VCE) method of
Parks [25] was used to obtain a numerical analog of the contour value of/.
The contour integral definition of J appropriate to finite deformation was
given by Eshelby [26] and has been discussed by McMeeking [11,27], For
constant-strain triangular elements, the VCE method exactly computes the
contour integral / value on the path connecting midpoints of the sides of
the distorted elements and, for isoparametric elements, agrees closely with
line integral values [25], For each analog contour remote from the crack
tip, the VCE method gave results for / in agreement with the load-deflection
curve method of Rice et al [24], However, the VCE method showed that/
decreases as the contour on which it is computed approaches the crack
tip to within a few current crack-tip openings both in small-scale and large-
scale yielding. This is in agreement with line integral calculations near
blunted crack tips for both small-scale yielding in hardening and non-
hardening elastic-plastic materials [11] and the fully plastic deformation of
a nonhardening DEN specimen [27],
182 ELASTIC-PLASTIC FRACTURE

Returning to Fig. 4, we note that each specimen is subject to three dif-

ferent stages of deformation. At first, the plastic zone is small, or com-
parable in size to bo and the ratio 6,/Ao is not much greater than unity.
Later, small-scale yielding occurs as the plastic zone develops into a size
that is very much larger than bi, and 5, is only a few times larger than bo.
During small-scale yielding, the slope of bt/bo versus J/{oobo) is close to
constant in each specimen with the value of constant depending only
on material properties rather than on specimen configuration. The values
of the constants are in agreement with those found earlier for small-scale
yielding of a sharp crack by McMeeking [11], whose work indicates that
the near-tip fields calculated here for a crack of width b, are very close to
that around an initially sharp crack in the same material blunted to the
same width. Since the influence of original notch geometry is lost as far as
both db,/dJ and the near-tip stress and deformation are concerned, the
curves of 6, versus J/oo for an initially sharp crack may be obtained from
Fig. 4. First, regard bo as an arbitrary length measure, then extrapolate
from the small-scale yielding regime a straight line down and to the left,
with the appropriate small-scale yielding slope db,/dJ for the material as
in Ref//. For an initially sharp crack, the value ofj/oo in arbitrary length
units bo for a crack-tip opening 6, in arbitrary length units bo is then mea-
sured from the intercept of the extrapolated line with bt = 0. This value
will be quoted in future references to / .
In large-scale yielding, the value of db,/dJ depends not only on material
properties but also on specimen configuration, as can be seen in Fig. 4.
This has been pointed out for nonhardening materials by Rice [28], who
noted that the center-cracked panel in fully plastic tension has a larger
value of dbt/dJ than the fully plastic bend specimen. The nonhardening
specimens were all deformed until the loads reached those corresponding
to slipline solutions. In particular, at very large deformations, the moment
on the nonhardening bend specimen attained a value 25 percent in excess of
the limit load for the initial geometry. In the final increments of computation,
however, the relationship of end rotation rate to rate of change of bt was ap-
propriate to the deformation of the fully plastic specimen [28]. It may be that a
part of the excess load can be attributed to having too few elements across the
ligament, although the precise meaning of a limit load is unclear when
finite geometry change is considered.
The near-tip tensile stress normal to the crack line for the nonhardening
edge-cracked bend specimen has been plotted in Fig. 5 against the position
R of the material in the undeformed configuration for material points lying
along the crack line ahead of the tip. The stress is normalized by yield
stress and the position is normalized by J/oo- The results are taken from
the later stages of deformation as can be seen from the key, where rp is the
maximum extent of the plastic zone from the crack tip (—indicates that
the zone has reached the specimen edge). The mesh with bo/L = 2 X 10""
 MCMEEKING AND PARKS ON CRACK-TIP FIELDS 183

5 -- | , ECB a/w = 0.9 N =0 -

-1.25
|» CTo/E = 1/300 L
i'
^ (J/Oc) L/St rp/L
4 _
- f 0« 89 190 .27 - .20
IA am 26 49 —
1 o • 16 32 —
1 ii A 1 1 24 —
\ ^^ o QO» 9 21 —

"^F"^^^
3 - 0 ,15

^ TTR^N'} SMALL SCALE YIELDING

,10

7 \ 0 .
o
/ \ \
I - ,05

- —«
1 1 1 1 1

R/(J/Oo)
FIG. 5—Normal stress distribution on the plane ahead of the blunting crack tip versus
distance R of material points ahead of the notch root in the undeformed configuration for
the nonhardening edge-cracked bend specimen. Distance axis is normalized by 3/ao. Solid
curve is McMeeking's [11] solution for small-scale yielding. Also shown is equivalent plastic
strain inversus distance from crack tip for material points at 45 degfrom the plane ahead of
the crack tip in the undeformed configuration. Dashed line is this strain distribution is small-
scale yielding [11],

was used for the hexagons and these are in fact the results at the largest
J/oa for this specimen. The remaining points in Fig. 5 were taken from the
result for the mesh with bo/L = 2 X lO"^, and it can be seen that much
larger values of 6, and//ffo, measured in terms of L, were achieved in this
solution. The full line is the small-scale yielding result of McMeeking [11],
who determined the near-tip deformations and stresses around a blunting
crack in small-scale yielding by enforcing at a distance remote from the
tip an asymptotic dependence of the deformation on the singular term of
the crack-tip elastic displacements. Note that this last-mentioned result
is self-similar when lengths are measured in terms of 8, or J/oo, since these
two quantities are proportional in small-scale yielding. Remarkably, the
normal stress distribution on the plane ahead of the crack in the ECB
specimen is nearly identical to that of small-scale yielding even for large /
values. For L/(J/ao) = 26, the stress state agrees closely over a distance
of eight blunted openings. At the three largest / values shown, the stress
points farthest from the blunted crack tip in Fig. 5 lie considerably below
the small-scale yielding curves at distances from six to nine blunted openings
away. These can be explained by noting that for these points the blunted
opening 8, corresponds to between 3 and 5 percent of the uncracked liga-
184 ELASTIC-PLASTIC FRACTURE

5 -- , . ECB a/w = 0 . 9 N = 0 -
H,25
|» CTo/E = 1/300 L
6"
§f - 1 0 »
(J/Qo) L/8t
89 190
rp/L
.27 - .20
A D • 26 49 —
0 • 16 32 —
1 ^ A 1 1 24 —
1 JM n nO* 9 21 -
3 - 15
' \f^
Q.l/5
0

- ™ 7 S ^ 1ALL SCALE YIELDING

of ^ "• -.10
0 .
7 \ O
/ \%
I - .05

1 1 1 1 1

R/(j/a„)
FIG. b—Normal stress on plane ahead of crack tip versus distance for edge-cracked bend
with strain hardening exponent N = 0.1. Also shown is equivalent plastic strain versus dis-
tance from crack tip at 45 degfrom plane ahead of crack tip.

ment. Since there is no net force on the ligament, the normal stress must
go into compression at a point somewhat nearer to the crack tip than to
the back face, and the three low data points are in accord with this. Fig-
ure 5 also shows the equivalent plastic strain at 45 deg from the plane
ahead of the crack in this specimen and in small-scale yielding [11] (dashed
line). The fully plastic bend specimen shows somewhat higher strains than
small-scale yielding, but the data appear to be rather closely clustered over
an order of magnitude range of/.
Figure 6 shows the normal stress distribution in the edge-cracked beam
with N = 0.1 and a/w = 0.9 at three deformation levels in the fully plastic
range. Also shown is the small-scale yielding result which, for N = 0.1,
reaches a maximum value of Oyy approximately equal to 3.8ao as opposed to
Sffo for the nonhardening case shown in Fig. 5. Again, at values ofLAJ/oo)
as small as 50, the characteristic near-tip field is as in small-scale yielding.
The plastic strains at 45 deg are also shown in Fig. 6, and are again close
to but somewhat larger than those in the small-scale yielding.
The equivalent plastic strain distribution on the plane ahead of the crack
is fairly insensitive to specimen geometry or hardening behavior. This is in
accord with the results of McMeeking [11], who found that the plastic
strain distribution ahead of the blunted crack was virtually identical for
strain hardening exponents N = 0,0.1, and 0.2.
Figure 7 shows the normal stress acting on the plane ahead of the crack
tip for the nonhardening CCP specimen with a/w = 0.9, and the solid line
 MCMEEKING AND PARKS ON CRACK-TIP FIELDS 185

5r CCP a/w = 0.9 N = 0

a/E =1/300 L
* (J/Go) L/8t rp/L
o 454 822 .09
o 193 349 .27
a 1 1 1 194 .79
X 56 82

SMALL SCALE YIELDING

0 1 2 3 4 5
R/U/0„)
FIG. 7—Normal stress on plane ahead of crack tip versus distance for the nonhardening
center-crack panel, a/w = 0.9. Note the fallqff of maximum attained triaxiality at the larger
J values.

is again for small-scale yielding. Data from a wider range of J values, from
contained plasticity to fully plastic conditions, are shown. As can be seen,
the stress distribution deviates sharply from the small-scale yielding result
as fully plastic conditions are approached. AtL/{J/ao) = 56, the maximum
stress ahead of the crack is only 2.3ffo as opposed to 3ffo for contained
yielding.
The stress state at the smallest / shown lies somewhat below the curve,
and it may be significant to consider this fact. We may roughly identify an
ASTM-like limit for a valid fracture toughness test in this very deeply cracked
specimen as Ki — 0.6CTO VZ7 or by using the small-scale yield relationship
y = (1 - v^)Ki^/E, i/(//a„) « 900 for v = 0.3, and E/a. = 300. Thus
the smallest / value shown is well beyond small-scale yielding. Furthermore,
Larsson and Carlsson [29] showed that center-cracked specimens exhibited
larger overall plastic zones at the ASTM limit than do other geometries.
They attribute the specimen dependence of overall plastic zone size, and of
stress-state variations within the plastic zone, to the effect of the normal
stress parallel to the crack plane. For center-cracked geometries, because
this term is negative. Rice [30] noted that the expected effect should be to
reduce triaxial tension, and hence maximum normal stress, ahead of the
crack tip. Indeed, Larsson and Carlsson noted considerably lower stresses
on the plane ahead of crack tip at the ASTM limit in the center-cracked
geometry than in bend, double-edge notched or compact-tension specimens.
Their mesh was not nearly so detailed as in the present investigation, how-
186 ELASTIC-PLASTIC FRACTURE

ever, and our observed effects at very small / levels of order WoX/lOOO in-
dicated a maximum normal stress ahead of the crack of value 2.89ao, which
is within 3 percent of the expected Prandtl value. This may well account for
the somewhat lower contained yielding stress distributions shown in Fig. 7
and later in Figs. 9 and 11. This subject could be further investigated by ap-
plying the modified boundary-layer analysis reported by Larsson and Carls-
son [29] to the small-scale yielding blunting solution method used by
McMeeking [//].
Figure 8 shows the equivalent plastic strain at 45 deg from the plane
ahead of the crack tip for the nonhardening CCP specimen with a/w = 0.9.
For well-contained plasticity the results are virtually identical to small-scale
yielding values. It is quite apparent, however, that for larger deformations
crack-tip plastic strain on this ray is considerably larger than for contained
yielding. Furthermore, although not shown in Fig. 8, the results are con-
sistently drifting farther from the solid curve at each of the last few defor-
mation increments computed. This means that the intense global deforma-
tion on the 45-deg sliplines is intruding on the near tip field, substantially
amplifying the deformation on this ray far beyond the /-controlled value.
The crack tip characterizing property of J is breaking down.
One may reasonably argue that the dramatic decrease of triaxiality ahead
of the crack, and amplification of plastic deformation at 45 deg at general
yield shown in Figs. 7 and 8, could be expected from the nonhardening

.25
CCP a/w = 0.9 N =0
e' L
(j/aj L/St rp/L
.20 - a 4 54 822 09
0 193 349 .27
e, 1 1 1 194 .79
.15 X 56 82 —

X
YIELDIIMC
.10

.05

2 3
R/U/CTo)

FIG. 8—Equivalent plastic strain at 45 deg from plane ahead of crack tip versus distance
for the nonhardening center-cracked panel with a/w = 0.9. Note the amplification of plastic
deformation for these points at the higher J values shown.
 MCMEEKING AND PARKS ON CRACK-TIP FIELDS 187

idealization. However, the same trend is shown in Figs. 9 and 10 for the
CCP specimen with a/w = 0.9 and strain hardening exponent N = 0.1.
Again, the maximum small-scale yield triaxiality of 3.8 is not quite reached
in well-contained yielding, and the stress distribution for moderately con-
tained plasticity again lies somewhat below the reference curve. However,
it is apparent that maximum achieved triaxiality is steadily dropping as
large plastic deformation ensues. Furthermore, Fig. 10 shows that the
plastic strain at 45 deg is also drifting away from the /-characterized field.
Although calculations were terminated L/{J/ao) = 72, it would seem that
the trend indicated in Figs. 9 and 10 would result in crack-tip fields con-
siderably far from /-dominance if extrapolated to values of L/(J/ao) equal
to 50 or 25.
It is realized that the extreme a/w value of 0.9 used in the previous CCP
calculations is likely to be of little value in actual / testing because of dif-
ficulties in machining, fatigue sharpening, instrumentation, etc. Rather,
they were performed so as to isolate a single characteristic specimen dimen-
sion, the ligament, for comparison with J/oo. As was noted earlier, the
CCP specimen was also solved with a/w = 0.5 so that a=L = w/1.
Figures 11 and 12 show the stress and plastic strain distributions, respec-
tively, plotted for the CCP specimen with a/w = 0.5 and iV = 0.1. Again,
a substantial deviation from /-dominance occurs as the macroscopic plastic
deformation field impinges on the blunting crack-tip region. A similar

5r CCP a /w =0.9 N = 0.1

gk Qo/E =1/300

r / i O D ^

o a
X

- L
(J/Qo) L/8t rp/L
D 439 1092 .09
o 213 553 .27
- i 1 13 2 99 .46
72 180
Al 1 Qr'Ai c v i c i rtiM/n
1 1 1 1

R/(j/aj
FIG. 9—Normal stress on plane ahead of crack tip versus distance for center-cracked
panel with a/w = 0.9 and N = 0.7. Note that the maximum attained normal stress is de-
creasing at the larger J values.
188 ELASTIC-PLASTIC FRACTURE

• 25r
CCP a/w = 0.9 N =0.1
L
(J/Oo) L/8t rp/L
.20 -
0 D 439 1092 .09
& O 213 55 3 .27
a 1 1 3 299 .46
.15 — 1 X X 72 180 -

YIELDING
.10 - \ A
\ ^
X
.05 - X""
\sP
o A X

1 1 1
° n -1
0 1 2 3 4
RAJ/Oo)
FIG. 10—Equivalent plastic strain at 45 deg from plane ahead of crack tip versus distance
for the center-cracked panel with a/w = 0.9 and N = 0.1. Plastic deformation on this
plane is intensifying at the larger J values.

CCP a/w= 0.5 N = 0.1

-"yy
0 : / E = 1/300

3 -]

L
(J/a,) L/8t fp/L
a 537 1321 .09
o 286 710 .24
I - t^ 177 429 .55
X 64 148 —
•SMALL SCALE YIELDING
_J I L
I 2
R/(J/CTo)
FIG. 11—Normal stress distribution on plane ahead of crack tip versus distance for center-
cracked panel with a/w = 0.5 and N =0.1.
 MCMEEKING AND PARKS ON CRACK-TIP FIELDS 189

.25r- w = 0.5 N = O.I

CCP
L
(J/Qo) L/8t rp/L
D 537 1321 .09
O 286 710 .24
A 1 77 429 .55
X 64 148 —

SMALL SCALE YIELDING

R/(J/Q;)
FIG. 12—Equivalent plastic strain at 45 degfrom plane ahead of crack tip versus distance
for center-cracked panel with a/w = 0.5 and N =0.1.

marked deviation from /-dominance also occurred in the nonhardening

solution of this specimen.

Discussion
We believe that the calculations presented in the previous section have
considerable relevance to the development of elastic-plastic fracture testing
and evaluation. In this section tentative conclusions drawn from the nu-
merical results are discussed, as well as factors which may tend to modify
these conclusions. The utility of this type of investigation in assessing the
validity of experimental results is noted, as well as a brief consideration of
the implications for micro-mechanical modeling.
A major result of this work is the demonstration in fully plastic bend
specimens of unique fields in the blunted crack-tip region, when scaled by
J/oo. Indeed, Fig. 5 shows stress distributions ahead of the crack at/values
as large as OoL/lS which are virtually identical to small-scale yielding within
a region of eight blunted openings. Similarly, the plastic strain distributions
are also close to those in small-scale yielding, as shown in Figs 5 and 6.
This strongly suggests that current suggested minimum specimen size
requirements of 25 to 50 J/oo reasonably assure /-dominance of the crack-
tip region in bend specimens.
On the other hand, calculations for the CCP geometry, even with a mod-
erate amount of strain hardening, suggest that more stringent size require-
190 ELASTIC-PLASTIC FRACTURE

ments may be necessary to assure a sensible / characterization of the crack

tip region. Indeed, the maximum tensile stress ahead of the crack tip for
N = 0.1 has decreased by 15 percent from its contained yield maximum at a /
level of roughly aoL/70 in both the CCP problems of a/w = 0.5 and 0.9.
Further, the marked intensification of plastic straining at 45 deg to the
crack tip, as compared to that in small-scale yielding, is quite apparent at
this / level, as shown in Figs. 10 and 12. While these results show that
near-tip fields are drifting from those of well-contained yielding, they do
not provide a clear-cut indication of when to declare the end of/-dominance.
It seems likely that any such explicit criterion must involve a certain degree
of arbitrariness, although perhaps consideration of criteria for driving the
microscopic fracture mechanisms may provide a rationale for imposing
restrictions. In any event, it would appear from the present work that a
value of M = 200 would be a (perhaps unduly) conservative estimate for
minimum specimen size as compared with J/oo.
As an example of the potential utility of such calculations, we can con-
sider the experiments of Markstrom [31] and Begley and Landes [14].
Markstrom considered two alloys in CT, CCP, and DEN configurations.
He reported essentially constant values of / at crack initiation in each
specimen geometry for both materials. Because the CT specimen has a
fully plastic flow field similar to that of pure bending, it is presumed that
/c values determined by this test were valid, since specimen dimensions
exceeded 25 Jc/oo, where <TO is the 0.2 percent offset yield strength. Let us
now ask if the more stringent size requirements we propose for a valid /c
test in the CCP specimens were met. The material Domex 400 had tensile
yield and ultimate tensile strength values of 460 and 620 MN/m^, respec-
tively. Although no explicit value of a strain-hardening exponent is given,
it is inferred from the ratio of the previous numbers to be near N = 0.1.
The reported/c value for this material is roughly 0.1 MN/m, so that/c/ffo =
0.22 mm. Although test-by-test data are not reported, the minimum liga-
ment size over all tests was 200 mm. Consequently, the minimum ratio of
L/{Jc/ao) is about 910. This corresponds to well-contained plasticity in the
numerical solution with a/w = 0.5 and N = 0.1, so we would expect
/-dominance at the tip, and hence a specimen-independent/c. For the other
material, Ox 802, the tensile yield and ultimate tensile strengths were 800
and 836 MN/m^, respectively, and /c is roughly 0.2 MN/m. This gives a
minimum ratio of X/(/c/ao) = 800. For a/w = 0.5, ^ = 0, t h i s / level
also corresponds to contained yielding, and again a /-characterized crack
tip would be expected. As a contrasting interpretation, we may cite the
tests of Begley and Landes [14]. They infer a specimen-independent /ic
value at crack extension of 0.105 MN/m in a nickel-chromium-molybdenum-
vanadium steel at 394 K. The yield and ultimate strengths are 855 and
957 MN/m^, respectively. The CCP specimen had an a/w ratio of 0.6 and
a remaining ligament dimension L of 5.08 mm, corresponding to L/(Ju/ao)
 MCMEEKING AND PARKS ON CRACK-TIP FIELDS 191

= 41. Our calculations for N^ = 0.1, a/w = 0.5 strongly suggest that /
characterization of the crack tip has broken down by this point, so a ge-
ometry-independent Jic would not necessarily be expected. Although Begley
and Landes report specimen-independent Jic values, obtained by extrapola-
tion to zero crack extension on the/ — Aa curve, the initial stable growth in
the CCP specimen was evidently influenced by the remoteflowfieldas noted
earlier. We take this as indirect evidence supporting our prediction that
/-dominance of the crack-tip fields has broken down.
We return to the point that even in this CCP specimen, the inferred /
at the initiation of crack growth was apparently the same as in the CT con-
figuration, even though our calculations would suggest rather different
crack-tip states. Of course, the agreement could be merely coincidental. On
the other hand, some rationalization of the consistency of the results may
be attempted. In the earlier stages of loading, the CCP specimen exhibited
high triaxiality at the crack tip, and, if this constraint was generated over
a significant microstructural dimension, it may be that voids were at least
initiated by decohesion or cracking of second-phase particles. As defor-
mation increased, however, the high triaxiality relaxed. This relaxation, by
itself, would be expected to decrease the void growth rate. However, there
is another factor to consider. As seen in Fig. 4, the CCP specimens tend,
in the fully plastic range, to exhibit larger crack opening displacements
than do bend specimens at equivalent / values. Consequently, the larger
overall deformations in CCP, as opposed to bend specimens, could tend
to offset the effect of reduced triaxiality on void growth and coalescence.
While this discussion has been of a rather qualitative (and speculative)
nature, it does seem to emphasize the importance of obtaining a clearer
understanding of the microscopic processes of ductile fracture in developing
rational macroscopic criteria which characterize crack extension.
Another area which requires further study is the area of constitutive
equations. We used a straightforward generalization of the isotropic hard-
ening Prandtl-Reuss equations based on a smooth yield surface. Clearly this
idealization does not precisely model certain aspects of the constitutive
behavior of polycrystalline materials at large deformation, so it is reasonable
to ask if our constitutive law contributes substantially to the observed
breakdown of a /-dominated crack tip in fully plastic CCP specimens. In
the nonhardening limit N = 0, this must be true, because in some sense
no hardening brings us back to slipline theory, regardless of the proper
treatment of changing geometry. Consequently, we feel that the dramatic
loss of triaxiality and amplification of plastic deformation at 45 deg in the
fully plastic nonhardening CCP specimens is indeed a direct consequence
of the constitutive law. On the other hand, the less dramatic, but quite
observable, breakdown with N = 0.1 is not so readily attributable to a
constitutive inadequacy.
There are at least two important features of stress/strain behavior which
192 ELASTIC-PLASTIC FRACTURE

have not been modeled here, but they would probably have opposing effects
regarding the continuation of uniqueness at the crack tip. The first ne-
glected feature is the possibility that subsequent yield surfaces may develop
vertices, or at least very high local curvatures, at the current stress state on
the yield surface in stress space. The effect of such features is to make the
material behave a bit more like a nonlinear elastic material, as in deforma-
tion theory plasticity, thus tending to promote a continued crack-tip unique-
ness, as discussed by Rice [32]. On the other hand, for nonzero strain
hardening exponent N, the present stress/strain law (Eq 1) never saturates
in terms of Kirchhoff stress for arbitrarily large deformation. If, say, the
Kirchhoff stress for continuing plasticflowin a particular material saturates
after some finite amount of deformation, perhaps identifiable with an equi-
librium of dislocation generation and annihilation, then our proposed
limitations for /-dominance should be conservative since, as discussed
previously, this type of behavior does not tend to promote continuing near-
tip uniqueness. These observations suggest that further consideration should
be given to the influence of constitutive equations on computed crack-tip
fields in both contained and large-scale yielding.
Although the present calculations were not directed toward the problem
of stable crack growth, the procedures can possibly be modified to assess
the likelihood of/-dominance of crack growth over small distances, perhaps
of the order of a few times the blunted crack opening at initiation [JJ].

Conclusions
On the basis of the finite-element solutions generated here, the following
conclusions are drawn:
1. The proposed specimen size limitations [13] for / testing requiring all
specimen dimensions to exceed MJ/oo, where M is typically 25 to 50, sen-
sibly assure a /-based characterization of the crack-tip region at the initia-
tion of crack extension in pure bending specimens. Because of their similar
fully plastic flow fields, this conclusion is presumed to apply to compact
tension and three-point bend specimens as well.
2. More stringent specimen size limitations seem necessary to assure a
similar /-based characterization of the crack-tip fields in center-cracked
panel test configurations, at least in lightly to moderately strain-hardening
materials. Based on the present calculations, we would propose a conserva-
tive estimate of M = 200 as a size limitation which seems to assure the
validity of/-characterized crack tip fields.
3. Because the loss of/-dominance of the crack-tip fields in the center-
crack geometries is gradual rather than abrupt, there is an arbitrariness in
the imposition of size or deformation limitations beyond which /, or other
single-parameter characterizations of the crack-tip region, should be deemed
invalid. This arbitrariness can be removed, and rational guidelines adopted,
 MCMEEKING AND PARKS ON CRACK-TIP FIELDS 193

only from the results of careful and systematic experimental investigations

of these geometries.
4. Even when a substantial body of such experimental results does be-
come available (there is currently nothing comparable to the extensive
results in bend-type specimens) it may be that necessary size limitations to
obtain Ju results consistent with those obtained in bend-type geometries
will vary from material to material depending upon aspects of the micro-
structural ductile fracture mechanisms involved.

Acknowledgment
This work was supported by the National Science Foundation's Center
for Materials Research at Stanford University. We are pleased to acknowl-
edge the encouragement received from Professor W. D. Nix in pursuing
this topic.

References
[/] Begley, J. A. and Landes, J. D. in Fracture Toughness. ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 1-23.
[2] Landes, J. D. and Begley, J. A. in Fracture Toughness, ASTM STP 514. American
Society for Testing and Materials, 1972, pp. 24-39.
[3] Broberg, K. B., Journal of the Mechanics and Physics of Solids, Vol. 19, 1971, pp.
407-418.
[4\ Rice, J. 'R., Journal of Applied Mechanics, Vol. 35, 1968, pp. 379-386.
[5] McClintock, F. A. in Fracture: An Advanced Treatise, H. Leibowitz, Ed., Vol. 3,
Academic Press, New York, 1971, pp. 47-225.
[6] Hutchinson, J. W., Journal of the Mechanics and Physics of Solids, Vol. 16, 1968,
pp. 13-31.
[7] Rice, J. R. and Rosengren, G. F.,Journal of the Mechanics and Physics of Solids, Vol. 16,
1968, pp. 1-12.
[8] Rice, J. R., Journal of Applied Mechanics. Vol. 34, 1967, pp. 287-298.
[9] Rice, J. R. and Johnson, M. A. in Inelastic Behavior of Solids, M. F. Kanninen et al,
Eds., McGraw-Hill, New York, 1970, pp. 641-671.
[10] McMeeking, R. M., Transactions, American Society of Mechanical Engineers, Journal
of Engineering Materials Technology, Vol. 99, 1977, pp. 290-297.
[//] McMeeking, R. M., Journal of the Mechanics and Physics of Solids, Vol. 25, 1977,
pp. 357-381.
[12\ Paris, P. C , discussion in Fracture Toughness, ASTM STP 514, American Society for
Testing and Materials, 1972, pp. 21-22.
[13] Landes, J. D. and Begley, J. A. in Fracture Analysis, ASTM STP 560, American Society
for Testing and Materials, 1974, pp. 170-186.
[14] Begley, J. A. and Landes, J. D., International Jourrml of Fracture Mechanics, Vol. 12,
1976, pp. 764-766.
[15] Rice, J. R. in The Mechanics of Fracture, F. Erdogan, Ed., Applied Mechanics Division,
American Society of Mechanical Engineers, Vol. 19, 1976, pp. 23-53.
[16] McMeeking, R. M. and Rice, J. R., International Journal of Solids and Structures,
Vol. 11, 1975, pp. 601-616.
[17] Nagtegaal, I. C , Parks, D. M., and Rice, J. R., Computer Methods in Applied Mechanics
and Engineering, Vol. 4, 1974, pp. 153-177.
[18] Hill, R., Journal of the Mechanics and Physics of Solids, Vol. 7, 1959, pp. 209-225.
[19] Marcal, P. V. and King, I. P., International Journal of Mechanical Sciences, Vol. 9,
1967, pp. 143-155.
194 ELASTIC-PLASTIC FRACTURE

[20] Rice, i. R. and Tracey, D. M. in Numerical and Computer Methods in Structural Me-
chanics, S. J. Fenves et al, Eds., Academic Press, New York, 1973, pp. 585-623.
[21] Tracey, D. M., "On the Fracture Mechanics Analysis of Elastic-Plastic Materials Usi^g
the Finite Element Method," Ph.D. dissertation. Brown University, Providence, R.I.,
1973.
[22] Tracey, D. M., Transactions ASME, Journal of Engineering Materials Technology,
Vol. 98, 1976, pp. 146-151.
[23] Hill, R., The Mathematical Theory of Plasticity, Oxford University Press, London,
U.K., 1950.
[24] Rice, J. R., Paris, P. C , and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing, ASTM STP 536, American Society for Testing and Materials, 1973,
pp. 231-245.
[25] Parks, D. M., Computer Methods in Applied Mechanics and Engineering, Vol. 12,
1977, pp. 353-364.
[26] Eshelby, J. D. in Inelastic Behavior of Solids, M. F. Kanninen et al, Eds., McGraw-
Hill, New York, 1970, pp. 77-115.
[27] McMeeking, R. M. in Flaw Growth and Fracture, ASTM STP 631, American Society
for Testing and Materials, 1977, pp. 28-41.
[28] Rice, J. R. in Mechanics and Mechanisms of Crack Growth (Proceedings, Conference
at Cambridge, England, April 1973), M. J. May, Ed., British Steel Corporation Physical
Metallurgy Centre Publication, 1975, pp. 14-39.
[29] Larsson, S. G. and Carlsson, A. J., Journal of the Mechanics and Physics of Solids,
Vol. 21, 1973, pp. 263-277.
[30] Rice, } . R., Journal of the Mechanics and Physics of Solids, Vol. 22, 1974, pp. 17-26.
[31] MSrkstrom, K.., Engineering Fracture Mechanics, Vol. 9, 1977, pp. 637-646.
[32] Rice, J. R. in Numerical Methods in Fracture Mechanics, A. R. Luxmoore and D. R. J.
Owen, Eds., Proceedings, International Symposium on Numerical Methods in Fracture
Mechanics, Swansea, Wales, Jan. 1978.
[33] Hutchinson, J. W. and Paris, P. C , this publication, pp. 37-64.
M. Nakagaki,' W. H. Chen,' and S. N. Atluri'

A Finite-Element Analysis of Stable

Crack Growth—I

REFERENCE: Nakagaki, M., Chen, W. H., and Atluri, S. N., "A Finlte-Elemeiit
Analysis of Stable Crack Growth—I," Elastic-Plastic Fracture, ASTM STP 668,
J. D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing
and Materials, 1979, pp. 195-213.

ABSTRACT: A finite-element methodology is developed to study the phenomenon of

stable crack growth in two-dimensional problems involving ductile materials. Crack
growth is simulated by (1) the translation in steps, of a core of sector elements, with
embedded singularities of Hutchinson-Rice-Rosengran type by an arbitrary amount,
Aa in each step, in the desired direction; (2) reinterpolation of the requisite data in
the new finite-element mesh; and (3) incremental relaxation of tractions in order to
create a new crack face of length Aa. Steps 1 and 2 were followed by corrective equilib-
rium-check iterations. A finite deformation analysis based on the incremental updated
Lagrangian formulation of the hybrid-displacement finite-element method is used.
The present procedure is used to simulate available experimental data on stable
crack growth, and thus to study the variation during crack growth of certain physical
parameters that may govern the stability of such growth and the subsequent onset of
rapid fracture. Attention is focused in this study on the following parameters: G*^,
the energy release to the crack tip per unit crack growth, for growth in finite steps
Aa, calculated from global energy balance considerations; Gpz^ the energy release to
a finite "process zone" near the crack tip per unit crack growth, for growth in finite
steps Aa, calculated again from global energy balance considerations; and the crack
opening angles. However, the work reported here is limited to the first phase of our
study, that is, to simulation of available experimental data.

KEY WORDS: ductile fracture, stable crack growth, translation of singularities,

finite-element method, J-integral, crack-tip energy release rate, process zone energy
release rate, crack opening angle, aluminums, steels, crack propagation.

Considerable research has been reported in recent literature on elastic-

plastic fracture mechanics that deals with the use of the characteristic
parameters such as the J-integral and crack opening displacement (COD)
to define the conditions of incipient growth of preexisting cracks in ductile
materials. Only currently are efforts underway to identify characteristic
'Post-doctoral fellow, graduate student (presently, associate professor. National Tsing Hua
University, Taiwan), and professor, respectively. School of Engineering Science and Mechanics,
Georgia Institute of Technology, Atlanta, Ga. 30332.

195

Copyright 1979 b y A S T M International www.astm.org

196 ELASTIC-PLASTIC FRACTURE

parameters, if any, to deal with the often-observed phenomenon of stable

crack growth under rising load, prior to fmal fracture, in ductile materials.
In the present paper we describe attempts to analyze such crack growth.
The study is conducted in two phases: the first is a finite-element simula-
tion of available experimental data to understand the variation of several
parameters of interest during stable crack growth; and the second phase
involves application of criteria chosen from the first phase to predict, using
finite-element methodology, stable crack growth and final failure in dif-
ferent cases and to check the accuracy of the prediction against available
experimental data. The work reported herein, however, is limited to the
first phase of our study.

Scope of Analysis
We consider a two-dimensional stable crack growth situation in an elastic-
plastic material and consider an instant of time during which a crack
extension by an increment Aa is taking place. During this incremental
growth, let AW/ be the work performed by the forces applied to the struc-
ture; A We be the change in elastic internal energy of the structure, A Wp
be the change in plastically dissipated energy in the structure; AT the
change in kinetic energy, if any, in the structure; and AWc be the work
done in quasi-statically and proportionally erasing the tractions (hold-
ing the length Aa of the crack together) in order to create a new crack
surface of length Aa. Neglecting any thermal input to the structure, one
can write an energy balance equation for the entire structure as

AWf _ AW, + AWp + AWc + A r

(la)
Aa Aa
or
AWf - AWe - AWp _ AWe + AT
Aa Aa (lb)
or more conveniently

AjWf- We- Wp) ^ A ^ , _^ AT

(2)
Aa Aa Aa
It is noted to start with, that the increment of crack growth, Aa is postu-
lated to be finite; and in the following we discuss the ramifications when
the limit Aa — 0 is considered and accordingly the incremental symbol
A is replaced by the symbol d, denoting partial differentiation. The left-
hand-side term of Eq 2 is the rate of energy release per unit crack growth
(for growth in a finite increment of Aa) from the structure to the crack tip
and will be denoted as G*^. Thus
 NAKAGAKI ET AL ON STABLE CRACK GROWTH—I 197

Aa
Thus G*'^ can be interpreted as the rate of energy "available" to create a
new crack surface. By simulating a stable crack extension of amount Aa
in the finite-element computations, G*^ is computed in the present study
by directly computing the terms AW/, and AWe and AWp, in the entire
body during such crack extension. It is noted that it has been indicated
by Rice [1]^, for nonhardening materials, that G*'^ ^ 0 as Aa ^ 0, and
thus, according to Ref /
lim A{Wf-W.-W,) _
Ao-o = 0 (.4;
Aa

Later in the present paper, the validity of Eq 4 is numerically examined.

If Eq 4 is valid, then the dependence of G* ^ on Aa must be studied, and
possible guidelines for selecting "finite" growth step size Aa in finite-
element simulations of stable growth must be arrived at. These issues are
addressed later in the present paper.
The first term on the right-hand side of Eq 2, on the other hand, is the
rate of work "needed" to quasi-statically and proportionally release the
cohesive tractions holding the crack surfaces of length Aa, and this term
is denoted by G'^. Thus

°- = ^ (5,
Aa
Calculations of G'^ for a center-cracked plate in plane strain were presented
recently by Kfouri and Miller [2] using an elastic-plastic finite-element
analysis. In the analysis of Ref 2, the crack-growth increment, of a finite
size Aa, is in fact the distance between two neighboringfinite-elementnodes
on the crack axis. Thus, let Node A be the "current" crack tip, and let
the next immediate node, located at distance Aa from A, be Node B. Then
crack growth is simulated in the procedure of Ref 2 by proportionally
reducing to zero the restraining "nodal force" at Node A. The work done
in this nodal force release process is considered to be AWc, and G^ was
computed from Eq 5. In the study of Ref 2, the center-cracked panel was
loaded to different levels of far-field tensile stress and, at each load level,
a crack-growth increment of Aa was simulated and the attendant G^ com-
puted. However, it should be noted that the growth increment Aa was
considered from the same virgin crack length ao at all load levels; and,
since the finite-element mesh was kept constant in each load-level case,
^ The italic numbers in brackets refer to the list of references appended to this paper.
198 ELASTIC-PLASTIC FRACTURE

the growth increment Aa itself was of constant magnitude at each load

level. Kfouri and Rice [3] later concluded that the results of Ref 2 show
that, at the same value of the applied load, G^ decreases and eventually
vanishes as a certain parameter S — {ay/Ki)^Aa (where Oy is the uniaxial
yield stress and Ki is the Mode I linear elastic stress-intensity factor for
crack of length ao at the applied load) tends to zero. This conclusion in
Ref J is examined in some detail in the present study.
In the present study, as shown later, crack growth is simulated by trans-
lating the finite elements near the crack tip in the direction of growth by
an arbitrary amount Aa which is not related to the 'distance' between two
adjacent nodes; thus the "new" crack tip need not be a 'node' in the original
finite-element mesh prior to translation. Thus, the dependence of G* "^ and
G^ on Aa at the same load level is more easily studied. Knowing this
dependence and postulating afinite-growthincrement Aa to be about three
to five times the COD at incipient growth condition, available experimental
data for fracture test specimens in the form of load versus crack growth
or J-integral versus crack growth are simulated in thefinite-elementmodel.
Thus the variation of G*^ and G^ during stable crack growth and at the
point of unstable fracture is obtained.
Also in the present study, attention is focused on the rate of energy flow,
per unit crack growth (for growth in finite increments), into a "process
zone" from the rest of the structure. Even though the dimensions of this
process zone are arbitrary, several choices can be made. In the present
study, the process zone is considered to be a circle F of radius equal to
Ofl/lO.O, where ao is the initial crack length, and, for the problem treated,
this size is roughly 40 percent of the plastic zone size at the onset of stable
growth. This energy flow rate to the process zone, denoted as Gr*^, is
given by the relation

_ AVy, (AW^ + AW,)\

^' ~ Aa Aa ""'" ^^^
In Eq 6 thv hanges of elastic and plastic energies, (AW^)J a-r and (AWp)
J a-T, are evaluated in the rest of the structure, excluding the process zone
(Q is the domain of the structure and T is the domain of the process zone).
Equivalently, GT*^ can be evaluated as

Gr*^= 1 Ti — ddV (7)

Jar ^«

where T, are the tractions at the boundary dV of the process zone F, and
AM, are increments of displacements of dT. The variations of Gr*^ during
finite-element simulation of experimental data of stable crack growth in a
test specimen are also studied.
 NAKAGAKI ET AL ON STABLE CRACK GROWTH—I 199

Further, the crack-tip opening angle (CTOA), which reflects the angle
between crack faces at the tip of the advancing crack, and an average value,
designated CO A, based on the crack opening displacement at the original
crack tip position, are studied during the simulated stable crack growth.
Finally, it is perhaps worth noting that, in the present finite-element
formulation, the well-known Hutchinson-Rice-Rosengren (HRR) singular-
ities for stresses/strains are built into the elements near the crack tip while
analyzing a stationary crack. However, during the translation of the near-
tip singular elements to simulate crack growth, the same singularities are
assumed to be present at the new crack-tip location. Notice is taken of
existing attempts in the literature to obtain analytical solutions to the
problem of "steadily" moving cracks in perfect-plastic materials [4,5]
which show a logarithmic strain-singularity at the crack tip. However, one
of these solutions [4] has been recently argued by Broberg [6] to be in
error. For this reason, and for lack of criteria to define "steady"-state
conditions, a priori, the HRR singularities are allowed, in the present
simulation, to be translated with the advancing crack tip. This may, how-
ever, be viewed as an approximation in the general context of the finite-
element method.

Brief Description of the Finite-Element Method

To account for the previously mentioned plastic singularities of the HRR
type near the crack tip, several special elements called "sector-core"
elements or "singularity" elements are designed to model the crack tip
region. Each of the core elements has six or eight nodes with the apex
node being the crack tip. A hybrid displacement finite-element model is
considered only in these core elements while a conventional displacement
model is assumed in the rest of the domain, where eight-noded quadrilateral
elements are used. Displacement compatibility and traction reciprocity
between these two regions are still maintained through the variational
principle governing the hybrid displacement model.
In dealing with the geometric as well as material nonlinearities, the
present finite-element procedure is based on a large deformation theory,
using an updated Lagrangian coordinate formulation, and a tangent
modulus incremental plasticity approach with iterative equilibrium cor-
rections. In each step or iterative loading a constitutive relation between
the incremental stress A*Sij and the incremental strain A*e{, is assumed as

A*Sij = Eijt,' {T „„^) A*e„ (8)

where Eyti' is the current elastic-plastic constitutive property and is a func-

tion of current true stress T^/' in Nth state of increment.
For defining the yield function / , the Huber-Mises-Hencky type initial
200 ELASTIC-PLASTIC FRACTURE

yield criterion is considered together with the kinematic hardening rule,

modified by Ziegler, which best describes Bauschinger effects. Thus the
yield surface in the general condition may be written as

| ( r ^ - a , ) ( r ' , - a ^ ) = ay2 (9)

where
an = translation of the yield surface in the stress space,
Oy = yield strength, and
T'g = deviatoric part of the stress.

The stress increment A*5{, in Eq 8 is added to the true stress of the

previous state to update the Kirchhoff-Trifftz stress

^ij(N)r' = T/ + A*S, (10)

Then 5,j(w)^+' are converted to the Euler true stress of the current state
by the relation

d x,^+' d x/+'
S,nsr' (11)
D d Xk" d xf

where D is the determinant of the Jacobian

r dxr' ^

and Xf is the updated material coordinate at the Mh state.

The variational functional -KHD Euler-Lagrange equations derived from
the principle BAITHD = 0, and the details of the finite-element formulations
leading to the incremental (tangent-modulus type) stiffness relations for the
cracked structure, are elaborated in Ref 7. For purposes of present interest,
we briefly discuss the relevant field assumptions in the sector-shaped
"core" (singularity) elements. The three field variables in each sector
elenient are assumed as

Au = f//3; Av = lAq; TL = Ra (12)

Where /3 and a are undetermined independent parameters, and Aq are

incremental nodal displacements. The interpolation function for the dis-
 NAKAGAKI ET AL ON STABLE CRACK G R O W T H - 1 201

placements, U in Eq 12 includes regular polynomial modes and a singular-

ity mode such as

r'^("+')(/3,+ /32e+ (83^2) (13)

Where (r, d) are polar coordinates at the crack tip and n is an exponent
coefficient in the strain hardening law. The functions L, which interpolate
the displacements at the interelement radial boundary between two sector
elements in terms of nodal displacements, also contain the singularity
behavior of the type

air'^c+^ + a z r + 03 (14)

whereas at the interface between the singular sector elements and the far-
field regular elements, they contain a variation of the type

fl,(?2 + ajfl + 03 (15)

The traction interpolation function R on the element boundary is derived

from a self-equilibrated stress field derived from a set of Airy stress func-
tions. R consists of regular polynomials as well as

r-''<"+'>(ai + ai6 + aid') (16)

type singularity behavior.
At the end of each incremental load step, the equilibrium of the total
structure is examined in such a way that the Euler stresses are converted
to equivalent forces concentrated at the element nodes, to check whether
a total norm of the resultant residual force vector at the nodes, except those
on constrained or loaded boundary, is small enough or not. Otherwise,
the unequilibrated nodal forces are reapplied using an updated stiffness
matrix. The Newton-Raphson type iteration is continued until the residual
norm reaches below a certain specified tolerance level.
In each element, 5 by 5 (7 by 7 for a singularity element) product
Gaussian integration points are used, with data of loading history necessary
to determine the current state of deformation and plasticity being stored
at each of these Gaussian points. Therefore, a more precise distribution of
plasticity conditions is defined than by using constant-stress triangular
elements. This makes it possible to use comparatively large quadrilateral
elements even adjacent to the core elements as can be seen in Fig. 1. Once
the effective stress on the Gaussian point reaches the yield strength, the
point is claimed as yielded. The subsequent yield condition is determined
by checking {df/dTij)-dTy as loading, neutral loading, or unloading de-
pending on whether it is positive, zero, or negative, respectively. However,
202 ELASTIC-PLASTIC FRACTURE

CRACK JflQ L:NEW CR/CK TIP

FIG. 1—Scheme of translation of core elements and reinterpolation of data.

when an elastic or unloaded part of the domain which is about to yield

becomes plastic, it is difficult to follow a uniaxial stress-strain curve with
sharp transition point. This difficulty is overcome by using a "knee-cor-
rection" procedure, the mathematical details of which are omitted here.

Finite-Element Modeling of Crack Growth

The finite-element simulation of crack growth may be described as (1)
geometrical change in the crack surface boundary, (2) translation of the
crack tip singularities to the advanced crack tip, and (3) release of surface
tractions on the newly created crack surface.
The change in the crack surface boundary is made by translating the
whole set of crack tip core elements, as shown in Fig. 1, by an arbitrary
distance Aa in the direction of intended crack extension; thus the new
crack tip node, which is designated by the center of the sector-shaped core
elements, need not be coincident with any previously existing finite-element
node before extension. Thus for the Mode I case, even though the fixed
boundary in the uncracked ligament of the structure is changed, the con-
straining condition of the nodes need not be altered. Elements immediately
adjacent to the core must be readjusted to fit the translated core. This
process of translating the core mesh also moves the embedded singularity
in the elements to the new crack tip area, leaving no singularities but
 NAKAGAKI ET AL ON STABLE CRACK GROWTH-I 203

large deformations and strains in the wake of the advanced crack tip. All
the 7 by 7 Gaussian data points in each of the translated core elements
(5 by 5 points for the conventional elements) may generally not coincide
with those before translation, for which plastic history data such as current
stresses, plastic strains, plastically dissipated work, and yield surface
translation are available. Therefore the data at points in the new mesh are
estimated by linearly interpolating data on four Gaussian points in the
old mesh that are nearest to the point under question in the new mesh.
For the sake of brevity, the mathematical details of this interpolation and
smoothing process are omitted here. With the fitted plastic data and the
new element geometry, element stiffness matrices are recalculated for the
core elements as well as for the surrounding rearranged elements, and the
global stiffness is appropriately modified. Subsequent equilibrium check
iterations using the new stiffness of the structure correct fitting errors, if
any, of the plastic data in the new mesh. At the same time, the tractions
over the distance AB (Aa as shown in Fig. 1) are incrementally removed,
with equilibrium check iterations at each step, to create a new traction-
free crack surface of length Aa. The finite-element simulation of crack
extension is now completed. The mathematical details of the steps just
described are omitted here. During the foregoing extension process, incre-
ments for externally supplied energy, elastic strain energy, and plastically
dissipated energy in the structure and energy flow into the process zone
are calculated to estimate the previously defined quantities G*'^ and Gr*'^.
The work done in releasing the preexisting tractions to create a new crack
surface of length Aa is computed, and labeled C^. The angles COA and
CTOA are computed according to the previously cited definitions.

Problems and Results

First, to understand the dependence of G*'^ and C^ on Aa, the problem
of a center-cracked plate under plane strain, identical to the case solved
by Kfouri and Miller [2], is analyzed. As in Ref 2, the plate is dimensioned
as 40.6 by 40.6 mm and the original half-crack length ao is 2.60 mm.
Material properties are described as Young's modulus, E = 207 KN/mm^;
Poisson's ratio, v = 0.3; yield stress, Oy = 310 N/mm^; and a linear strain
hardening with a tangent modulus of 4830 N/mm^. In the present finite-
element analysis, a total of 51 elements and a total number of 148 deg of
freedom are used. The hybrid displacement model is used only in the six
6-noded sector-shaped core elements surrounding the crack tip, while the
rest of the domain is divided into 8-noded quadrilateral elements wherein
a conventional displacement model is used. Three J-integral paths encircling
the crack tip, as indicated by broken lines in Fig. 2, are used. The average
of / values integrated over the three paths is used in the present analysis,'
where differences between the paths are within ±0.4 percent. Keeping
204 ELASTIC-PLASTIC FRACTURE

Op
i
mnt t t t t f i f
A i

W = L

E
E

LroAri^
-CRACK TIP
TIP I

FIG. 2—Finite-element idealization of center-cracked square plate under uniaxial tension.

the crack extension Aa as a constant (5 percent of the original crack length

ao), the global energy release rate G*^ is estimated for single-step exten-
sions at various load levels and is plotted in Fig. 3 against the parameter
S, which is the ratio of Aa divided by the plastic zone size. At about S =
0.54, discrepancies among G*^, G, and / are negligible, representing a
small-scale yielding stage of the loaded specimen, where G is a strain-energy
release rate defined by G = [(1 — v^)Ki^/E\. (Note that the normalizing
factor G used in Fig. 3 is not a constant, but varies with the load level as
per the given definition.) However, remarkable discrepancies are noticed
between/ and G*^ under large-scale yield conditions, as shown in the Fig. 3,
wherein J/G*^ attains large values as S tends to zero. This result indicates
that unless the yielded region is so small that a linear elastic analysis
is approximately applicable, / cannot be used to characterize the growth
process even at the beginning of the stable crack growth. This would con-
tradict Brogerg's hypothesis, which states that the energy flow # to the
end region per unit of crack growth is represented by

* = / [ « + ( ! - « ) exp{ -0aHa -aoVEJ]]

 NAKAGAKl ET AL ON STABLE CRACK GROWTH—1 205

where a and |8 are some constants; therefore, * = / at the start of the

crack growth.
Also plotted in Fig. 3 is the variation of G*^ normalized by the variable
G, which is compared with Kfouri and Miller's result for the crack surface
energy release rate G\ It is noted that Kfouri and Miller obtained G^ at
various load levels for a constant Aa = 0.05 ao, as in the present case.
The present results are 5 to 10 percent higher than the G^ value reported
in Ref 2 up to the value of S = 0.02. Even though no numerical results
for C/G were obtained by Kfouri and Miller [2] for values of 0 < 5 < 0.02,
it was surmised in Refs 2,5 that C'/G may tend to zero as 5 — 0. As
shown in Fig. 3, however, the present results indicate that in fact G*^IG
does not monotonically tend to zero as 5 — 0, but reverses its trend and
raises again. The results of Kfouri and Miller [2] for G^IG up to 5 — 0.02
were in fact considered to be numerical evidence for the paradox that the
global energy release rate G*^ tends to zero as Aa — 0, as originally pre-
dicted by Rice [/]. To understand this further, the absolute value of G*^,
for constant value of growth increment (Aa = 0.05ao), is shown in Fig 4
for a single-step growth at various values of applied stress, ap. It is seen

JS
imi
J/G*A
TTTTTl 13
\ G 1.4

AND 2

G'A
%

.8

.6 ' G ^ / G BY KFOURI AND MILLER

.4 fla/ao= .05

.2 I-

0 .1 .2 .3 .4 .5

FIG. 3—Crack separation energy release rates: constant Aa at various load levels.
206 ELASTIC-PLASTIC FRACTURE

from Fig. 4 that G*^ in fact increases monotonically with increasing Op,
at least for the case of Aa/ao = 0.05, even though the normalized values
G*'^/G (with G as shown in Fig. 4) may decrease with ap.
Thus, to numerically study the original hypothesis of Rice [i] (that
G*^ _ 0 as Afl -» 0), the calculations were repeated for various values of
Aa, ranging from Aa/ao = 0.07 to 0.001, while keeping the load constant
at Op = 245 N/mm^, at which load large-scale yielding conditions are
numerically found to exist. Precisely speaking, keeping the load constant,
single-step growth increments were simulated for various values of growth
increments, Aa. The results are shown in Fig. 5, where G*^ is normalized
with respect to constant G = (1 — v^)Ki^/E, Ki being the elastic intensity
factor at Op = 245 N/mm^. It is seen from Fig. 5 that even for the present
slightly hardening material, G*^ tends to zero as Aa — 0 while the load is
kept constant. The result in Fig. 5 may then be considered as a direct
numerical proof of Rice's original hypothesis [/].
Thus, even though G*^ — 0 as Aa — 0 at constant load, as seen from
Fig. 5, it is finite for all finite growth step values of Aa. Thus, postulating
a finite growth step to be of the order of Aa/a = 0.01 ~ 0.02 (to avoid
possible numerical difficulties in the region Aa/ao < 0.005 as in Fig. 5),

400
LOAD
a .1 "^"^ "P
FIG. 4—Crack separation energy release rate: constant Aa at various load levels.
 NAKAGAKI ET AL ON STABLE CRACK GROWTH—I 207

FIG. 5—Crack separation energy release rate: for various values of An at constant load level
corresponding to large-scale yielding.

one may meaningfully simulate experimental data on stable crack growth

to study the behavior of G*^, G\ Gr*'^, during such growth. This is done
in the next problem.
Using the foregoing concepts, experimental data on stable crack growth
in two 3-point bend specimens are simulated using the present finite-ele-
ment procedure. Such experimental data are reported by Griffis and Yoder
[8]. The geometry of the first specimen is: width W = 1.91 cm, thickness
B = 0.64 cm, original crack length ao = 0.89 cm, and S/W = 4, where
S is the roll-span of the specimen. The initial crack length in the second
specimen was ao = 1.14 cm and all the other dimensions were identical
to those of the first specimen. The material of the specimen is an inter-
mediate-strength aluminum alloy, 2024-T351, and the properties are:
Young's modulus £• = 80 300 MPa, yield strength Oy = 338 MPa, ultimate
tensile strength Ou = 492 MPa, and elongation 21.5 percent. This uniaxial
data are fitted by a Ramberg-Osgood type strain hardening law, e = (a/£^
+ (a/Bo)", where the hardening coefficients are Bo = 768.6 MPa and
n = 7.57. Afinite-elementmesh breakdown analogous to that of the cen-
trally cracked square plate is considered for both of the present specimens,
with a total of 43 elements and a total number of 143 degrees of freedom.
This present two-dimensional analysis does involve the thickness of the
specimen, with only an "either-or" choice of plane stress versus plane
strain. Based on earlier experience [9], a plane-stress condition was invoked.
A constant crack growth step of Aa = 0.128 mm is assumed, which is
roughly four times the crack opening displacement and about i/is of the
plastic zone radius at the loading level corresponding to crack-growth
208 ELASTIC-PLASTIC FRACTURE

initiation in both specimens. The crack is extended in a total of nine equal

steps so that the overall crack extension is about 10 to 13 percent of the
original crack length in the specimens. The crack tip is kept unopened
until the externally applied load reaches to the level that crack growth
initiation is observed in Griffis and Yoder's experiment, that is, at P/2B =
0.2125 MN/m for the first specimen and P/W = 0.1365 MN/m for the
second. Thereupon, a load increment AP corresponding to Aa as in the
experimental P versus a (load versus crack growth) record is applied to
the structure with an extended crack length a + Aa. It is noted therefore
that the incremental loading and incremental crack opening are carried out
simultaneously in thefinite-elementanalysis; thus the present finite analysis
results in a piecewise linear approximation to the actual experimental
data rather than in a "staircase"-type approximation. This incremental
crack extension process is continued to about nine steps (with crack ex-
tension of about 13 percent in the first specimen and about 10 percent in
the second), when the experimental load records terminate. During this
process, the load-point displacement (6) is computed, and the computed
load versus 6-curve is plotted in Fig. 6 for the first specimen (ao = 0.89 cm),
where Griffis and Yoder's experimental data are also included. The excel-
lent correlation of the present results with the cited experimental data
suggests the validity of the present finite-element procedure and the cor-
rectness of assuming plane-stress conditions in analyzing the problem.
Similar correlation for the second specimen has also been noticed but, for
want of space, the corresponding curve is not included here. The averages
of the three /-contour integral values obtained on three paths for both
specimens are plotted against the crack extension (a — ao) in Fig. 7 along

INCIPIENT
CRACK GROWTH

EXPERIMENT BY GRIFFIS
AND YODER

o PRESENT FEM

.01 .02 .03 .04

6/W
FIG. 6—Load versus load-point displacement crack growth curve.
 NAKAGAKI ET AL ON STABLE CRACK GROWTH—I 209

EXPERIMENT
• ao=8.9mm
° ao=ll.4mm

^ 8
I 7
Q.
5 6
<3 A

.16
.14
.12

.2 .4 .6 .8 1.0 1.2
a - Qo (mm)
FIG. 1—Variation of J, G*^, and P/2B during crack growth.

with the experimental / obtained by Griffis and Yoder [8] using a method
established by Begley and Landes [10]. At incipient crack growth in the
first specimen, the presently computer J-integral average value is 12.4
kPa-m and 12.6 kPa-m in the second, whereas that reported by Griffis
and Yoder is / ^ = 13.7 kPa-m. Again, a good correlation is observed
between the present numerical J-integral values and the experimental /
results at growth initiation. Crack surface profiles in the crack tip region
for each step of extension are demonstrated in Fig. 8, showing the char-
acteristic shape of the extended crack surface.
From Fig. 8 it can be seen that at growth initiation the crack tip is
blunted, thus reflecting the behavior of asymptotic crack surface deforma-
tions of the HRR type that are embedded in the core elements in the
present analysis. It is also seen from Fig. 8 that the crack profile becomes
much sharper after crack extension; this suggests the possibility of a change
in the order and nature of strain singularities at the tip of an extending
210 ELASTIC-PLASTIC FRACTURE

a (mm)

FIG. 8—Crack surface profile during crack growth.

crack. It is interesting to note that in the present analysis the HRR singu-
larities are translated with the extending crack tip without any further
modification. The fact that in spite of this the crack profile tends to be-
come sharper after extension is surprising. However, if the nature of singu-
larities near the advancing crack tip both in the "transitory" as well as in
the "steady"-state conditions is clarified analytically, it is, in principle,
possible to effect the appropriate changes in the finite-element modeling.
For the present, in view of the foregoing observations concerning Fig. 8,
it appears that the assumption of HRR singularities at the tip of an ad-
vancing crack may be viewed as an "approximation" in the general con-
text of the finite-element method, in the sense that the hypothetical "exact"
solution is approximated by a set of assumed basis functions.
Also shown in Fig. 7 are the variations of G*^ during crack growth, for
both specimens. It is noted that, for both specimens, G '^ exhibits marked
variations during the crack extension process, from about 4 kPa-m at
initiation to about 7 kPa-m at final fracture. Further, G*'^ is seen to in-
crease almost monotonically during the crack extension process, except
for a slight dip near the point of unstable crack propagation, for both test
cases. Also shown in Fig. 7 are the load versus crack-growth curves for
both specimens which, of course, are also the experimental curves used
in the present simulation.
In Fig. 9, the rate of energy flow to the process zone T for finite growth
steps, designated as Gr*'^, is plotted for both of the simulated cases. Once
again it is seen that, for both cases, Gt*^ increases monotonically almost
up to the point of final fracture. The rate of energy dissipated in the
process zone, which is the difference between Gr*^ and G*^, is also shown
 NAKAGAKI ET AL ON STABLE CRACK G R O W T H - I 211

in Fig. 9 for both test cases. Once again this energy dissipation rate in the
process zone, which in the present analysis is completely embedded in
the plastic zone near the crack tip, is seen to increase monotonically during
crack extension, but is seen to level off or start decreasing near the point
of unstable fracture as observed in the experiment.
Finally, the variations of the crack tip opening angles during crack ex-
tension, for both of the simulated cases, are shown in Fig. 10. It is observed
that this variation of CTOA is analogous to that of G*^ as shovm in Fig. 7.

Conclosions
It is recognized that formulating any criterion or criteria governing the
loss of stability of crack growth, based on numerical simulation of a few
experimental data, is, at best, a risky proposition. Thus, we defer any
conclusions regarding such criteria until the completion of the second phase
of our research. The results reported herein, however, lead to the following
conclusions that may be germane to the problem of stable crack growth
in ductile materials.
1. A direct numerical proof is provided for the original hypothesis of
Rice [1] that G*^ ^ 0 as Aa — 0 for those materials for which the flow
stress saturates at a finite value of large strain. Thus, for any meaningful
numerical study of stable crack growth, a finite growth step must be
postulated. Results similar to those in Fig. 5 may be useful in providing
guidelines for choosing Ac such that the numerically computed G*^ is not

50- • * • Go = 8.9 mm
oA« Qo = 11.4mm
4C - - - ENERGY DISSIPATED ,A
IN PROCESS '
ZONE
< 30
> •
<s> 20-
a:
UJ
z
10 G*-a

.2 .4 .6 .8 1.0
a-Qo (mm)
FIG. 9—Variation of Gi* ^, G* \ and energy dissipation in process zone during crack
growth.
212 ELASTIC-PLASTIC FRACTURE

.14
. ^ - ^ ^ ^ ^ ^
" i .12
' -•- a© = 8.9 mm
< .10 —o- QQ = 11.4 mm
o

^.08

.06
•-•

.2 .4 .6 .8 1.0 \2
0- Qo ( mm )
FIG. 10—Variation of crack-tip opening angle during crack growth.

sensitive to the errors inherent in numerical processes such as in the finite-

element method.
2. Since the magnitudes of G*^ and Gr*'^ are clearly shown to depend
on the postulated magnitudes of "finite" growth steps Aa, in the finite-
element modeling it is clear that any criteria governing the loss of stability
of crack growth cannot be based on the absolute magnitudes of these
quantities. Any such criteria can be based only on the relative qualitative
behavior of G*"^ and Gr*^ for the postulated growth step Aa.
3. In view of the previous observation and the fact that G*^ and Gr*^
vary substantially during the crack-extension process, as in Figs. 7 and 9,
it is clear that the generalizations of Griffis's approach, in the sense that a
ductile material has some characteristic work of separation per unit of new
crack area, and that this is to be equated at the critical condition to the
rate of surplus work done on the material, cannot be made in situations of
stable growth under large-scale yield conditions.
4. The only discernible trend near the points of fracture, as observed
experimentally for both of the cases studied, is the marked change in the
behavior of G*^, CTOA, and to an extent in Gr*'^, near these points, when
these quantities reverse their monotonically increasing trend during the
prior extension process. While a theoretical argument explaining this is
lacking, it remains to be seen whether these observations can be used to
numerically predict loss of stability of growth in different specimens of the
same material. This is the object of our work in progress.
 NAKAGAKI ET AL ON STABLE CRACK GROWTH-I 213

Acknowledgment
The results presented here were obtained during the course of an inves-
tigation sponsored by Air Force Office of Scientific Research under Grant
AFOSR-74-2667 and by the National Science Foundation under Grant
NSF-ENG-74-21346. These and the supplemental support from the Georgia
Institute of Technology are gratefully acknowledged. The authors also
appreciate the thoughtful comments offered by the reviewers of this manu-
script.

References
[/] Rice, J. R. in Proceedings, 1st International Congress on Fracture, T. Yokobori et al,
Eds., Sendai, Japan, 1965, Japanese Society for Strength and Fracture, Tokyo, Vol. 1,
1966, pp. 309-340.
[2] Kfouri, A. P. and Miller, K. J. in Proceeding, Institution of Mechanical Engineers,
London, U. K., Vol. 190, 1976, pp. 571-586.
[3] Kfouri, A. P. and Rice, J. R. in Proceedings, 4th International Conference on Fracture,
D. M. R. Taplin, Ed., Waterloo, Ont., Canada, June 1977.
[4] Chitaley, A. D. and McClintock, F. A., Journal of the Mechanics and Physics of Solids,
Vol. 19, 1971, pp. 147-163.
[5] Rice, J. R. in Mechanics and Mechanisms of Crack Growth (Proceedings, Conference at
Cambridge, England, April 1973), M. J. May, Ed., British Steel Corporation Physical
Metallurgy Center Publication, 1975, pp. 14-39.
[6] Broberg, K. B., Journal of the Mechanics and Physics of Solids, Vol. 23, 1975, pp.
215-237.
[7\ Atluri, S. N., Nakagaki, M., and Chen W. H. in Flaw Growth and Fracture, ASTM STP
631, American Society for Testing and Materials, 1977, pp. 42-61.
[5] GrifFis, C. A. andYoder, G. R., Transactions, American Society of Mechanical Engineers,
Journal of Engineering Materials and Technology, Vol. 98, 1976, pp. 152-158.
[9] Atluri, S. N. and Nakagaki, M., American Institute of Aeronautics and Astrormutics
Journal, Vol. 15, No. 7, 1977, pp. 923-931.
[10] Begley, J. A. and Landes, J. D. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 1-20.
K. J. Miller' and A. P. Kfouri'

A Comparison of Elastic-Plastic
Fracture Parameters in Biaxial
Stress States

REFERENCE: Miller, K. J. and Kfouri, A. P., "A Comparison of Elastic-Plastic Frac-

ture Parameters in Biaxial Stress Stotes," Elastic-Plastic Fracture, ASTM STP 668, J. D.
Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and
Materials, 1979, pp. 214-228.

ABSTRACT! Parameters used in fracture predictions in elastic-plastic materials will dif-

fer from those used when the material is elastic. However, the first set of parameters may
reduce to the latter in the limiting case of brittle behavior involving minimal plasticity.
Analyses of the stress and plastic strain fields in the region of the crack tip and evalua-
tions of energy release rates are therefore relevant to studies on fracture processes in
engineering materials. It is becoming generally recognized that more than one parameter
is needed in the formulation of a realistic fracture criterion applicable to elastic-plastic
materials. In particular, such a criterion must take into account the possible biaxial
nature of the applied stress.
This paper presents some of the results of extensive elastic-plastic finite-element
analyses on a center-cracked plate. Information is provided, and comparisons made, on
such features as crack-tip plastic zone sizes, intensities of plastic strain near the tip, the
major principal stress in the crack-tip region, crack opening displacements, values of the
J contour integral, and crack separation energy rates C^—all corresponding to different
biaxial stress states.

KEY WORDS: biaxiality, center-cracked plate, compact tension specimen, crack

growth step, crack-tip opening displacement, crack-tip plasticity, crack tip plastic zone
sizes, crack-tip stresses and strains, crack separation energy rate, elastic-plastic fracture
mechanics, finite-element method, Griffith's energy release rate, incremental-load
initial-stress elastic-plastic finite-element analysis, / contour integral, small-scale yield-
ing, crack propagation

Nomenclature
A Crack-tip plastic zone size factor
a Half crack letigth of center-cracked plate
E Modulus of elasticity
G Griffith's energy release rate for a linear elastic material
' Professor and research fellow, respectively. Faculty of Engineering, University of Sheffield,
Sheffield, U. K.

214

Copyright 1979 b y AS FM International www.astm.org

 MILLER AND KFOURI ON BIAXIAL STRESS STATES 215

Go Value of G at incipient yielding in a plane-strain uniaxial finite-

element analysis of the center-cracked plate
G^ Crack separation energy rate
H Tangent modulus in linear hardening stress-strain law
/ Rice's contour integral for a path through unyielded material
JQ GO
K\ Irwin's Mode I stress intensity factor
Ki* Non-dimensional Mode I stress intensity factor
rp Maximum dimension of the crack-tip plastic zone
VA Displacement normal to the crack plane of Node A on the crack sur-
face, nearest to the crack-tip node
Aa Crack growth step = side of leading element at the tip of the crack
AW Work absorbed during the stress release for the growth step Aa
7o Calculated equivalent strain at the crack-tip node before crack exten-
sion
€1 Calculated equivalent strain at the crack-tip node after the consecutive
release of three crack-tip nodes
r/o 20 000 Vx/Aa before crack extension
iji 20 000 Vx/Aa after the consecutive release of three crack-tip nodes
\l/ Load parameter = G/Go
X Biaxiality parameter = OP/OQ
v Poisson's ratio for the material in the elastic state
V* Effective value of Poisson's ratio for the elastic-plastic material, v < v*
< 0.5
ffp Applied stress normal to the crack plane, on the boundaries of the
center-cracked plate parallel to the crack plane
ffpo Value of ap at the start of the incremental load elastic-plastic analysis
OQ Applied stress parallel to the crack plane, on the boundaries of the
center-cracked plate normal to the crack plane
aio Major principal stress at the center of the leading element ahead of the
crack tip before crack extension
CTii Major principal stress at the center of the leading element ahead of the
crack tip after the consecutive release of three crack tip nodes
Oy Yield stress

Defects in engineering structures are usually situated in a complex stress

field and so it is of interest to ascertain the extent to which currently used
fracture parameters are affected by the biaxiality of the mode of loading.
Several fracture parameters are known to be dependent on the size of the
crack-tip plastic zone. The size and, to a lesser extent, the shape and orienta-
tion of the crack-tip plastic zone depend on the biaxiality of the applied load
[/]^ or on the type of specimen used [2]. The suggestion has also been made

^The italic numbers in brackets refer to the list of references appended to this paper.
216 ELASTIC-PLASTIC FRACTURE

[3,4] that even for small-scale yielding it is not sufficient to consider only the
first singular term in a series expansion for determining the crack tip stresses
in a boundary-layer approach. A second nonsingular term reflecting an addi-
tional load parallel to the crack must be taken into account to obtain a
realistic assessment of the shape and size of the plastic enclave. This is not
surprising if one considers that at the elastic-plastic boundary the equivalent
stress in a von Mises material must be equal to the yield stress and therefore
stresses of the order of the yield stress must influence the exact location of the
boundary.
This paper collates unpublished data obtained from elastic-plastic finite-
element analyses on a center-cracked plate [5] in order to assess and compare
the effect of load biaxiality on crack-tip plastic zone size, crack-tip opening
displacement, crack-tip stresses and strains, Rice's J-integral, and the crack
separation energy rate, G^.

The Analyses
Thefinite-elementanalyses were carried out on the center-cracked plate of
a von Mises material in plane strain of which thefinite-elementidealization
of one quadrant is shown in Fig. 1. The idealization of the region near the tip
of the crack is shown in Fig. 2. Material properties were E = 207 GN/m^,
Poisson's ratio v (elastic) = 0.3, yield stress Oy = 310 MN/m^, and linear
strain hardening with tangent modulus H of 4830 MN/m^. The elastic-
plastic finite-element program is based on the initial stress approach and is
described elsewhere [5-7]. Simple isoparametric quadrilateral elements with
four corner nodes are used. The effective in-plane Poisson ratio v* varies
from 0.3 on the elastic-plastic boundary to a maximum of 0.5 in regions
which have incurred substantial plastic flow, the actual value being dictated
by the extent of the plastic flow and the plane-strain constraint. A conse-
quence of this is that the plastic zone size is somewhat larger than would be
the case if a constant value of v* equal to 0.5 were used throughout the
plastic region.
A measure of the load applied to the crack-tip region is given as G equal to
Ki^(l — v^)/E, that is, equal to the Griffith energy release rate for the
equivalent unyielding elastic material. KereKi is Irwin's Mode I elastic stress
intensity factor. At incipient yielding, the largest equivalent stress occurring
at any node of any element of the structure is equal to Oy. The load G will
often be normalized with respect to Go, that is, ^ = G/Go, where Go is the
value of G required to cause incipient yielding in thefinite-elementanalysis
for uniaxial loading. Note that Go is proportional to Oy^ Aa. This follows from
the expression ai = Kiilirr)'"^ for the normal stress at a point situated at a
small distance r ahead of the crack tip in the plane of the crack in a linear
elastic material. Alternatively, as a load characterization parameter. Rice's
J-integral, calculated along a contour running entirely through elastic
 MILLER AND KFOURI ON BIAXIAL STRESS STATES 217

f t \ NODE No.
221
, 13
n 12 168 180
ELEMENT No.
192

12

RICE J
INTEGRAL
PATH 1
- ^ a
n
c
c si

-^

10
THIS SEGM ENT
9 /ENLARGED FIG l b
9
8
B ^
7 7
f X
f 1 t 170 183 196 209
CRA CK TIP
NOD E No. 53
"*
FIG. 1—Finite-element idealization of top right quadrant of center-cracked plate.

8 20 32 140 152

7
7
\ /
r.
6
\ / ^ ^^
5 V /
^
3 -4i --"
3 5
_ .-fT ^
2 1 2 S _, rf B597 121 133 145 X
1
fl • 1
27 " 13 1 1!57 |-' 0 ••
— 2 !)4mm 0 • 2 7 mm
CRACK TIP
NODE No.53.

FIG. 2—Nodes, element numbers, and J-contours in the neighborhood of the crack tip.
218 ELASTIC-PLASTIC FRACTURE

material, is used and normalized with respect to Jo equal to Go. The crack
growth step Aa is equal to the side of the leading element ahead of the crack
tip, that is, 0.127 mm in the mesh shown in Fig. 2. For convenience the load
biaxiality parameter X = OQ/OP is used where ap and OQ are the applied
boundary stresses normal and parallel to the crack, respectively. Values of X
corresponding to the uniaxial, equibiaxial, and shear modes are 0, 1, and
— 1, respectively.

Incremental Load Analyses and Crack-Tip Node Releases

. Details of the incremental load elastic-plastic finite-element analyses and
of the crack-tip node release technique used in the determination of the crack
separation energy rate can be found in Refs 5, 8, and 9. For each loading
mode the analyses were carried out as follows:
1. An initial elastic analysis adjusted the load ap to a value apo equal to 95
percent of the value required for incipient yielding.
2. For the incremental load elastic-plastic analysis the load was increased
from apo in steps of 0.08 apo, causing plastic zones to develop at the crack tip,
without crack extension. Now at each of the loads ap equal to 1.56 apo, 2.12
apo, 2.68 apo, 3.24 apo, 3.80 apo • • • corresponding to different sizes of crack-
tip plastic zones, four crack-tip nodes were released consecutively by the
following procedure.
3. The equivalent nodal reaction at the crack-tip node is calculated. This
force is in a direction normal to the plane of the crack.
4. The constraint on the corresponding displacement of the crack-tip node
is relaxed and an external equivalent nodal force equal to the reaction is
substituted. Thus the crack is extended, but not opened to the next node
along the plane of the crack.
5. Maintaining ap constant, the equivalent nodal force at the released
node is gradually reduced to zero in six incremental release steps and the
displacement of the released node is calculated for each release. This enables
the evaluation of the work absorbed during the release, AW, to be carried
out, and G^ is determined by dividing AWhy Aa.
6. Steps 3, 4, and 5 of the foregoing are repeated on the new crack tip
node until four nodes have been released.

Results
The results of this work are given in Table 1 for ease of reference.

Crack-Tip Plastic Zone Sizes

In small-scale yielding, the maximum dimension of the crack-tip plastic
zone is usually related to the applied load and yield stress by
 MILLER AND KFOURI ON BIAXIAL STRESS STATES 219

where A is assumed to take a constant value approximately equal to 0.175

from an analysis based on the boundary-layer approach and small-scale
plasticity \10\. In fact the finite-element results reveal that only in the
equibiaxial mode can>l be taken as a constant. Note that the boundary-layer
approach approximates the equibiaxial mode. Values of A equal to
rp/{Ki/ayy against ^ for the different biaxial modes of loading are shown in
Fig. 3. For the equibiaxial mode, A takes the constant value of 0.19. In the
uniaxial mode, A appears to increase linearly with xp from an initial value of
approximately 0.24. The variation of y4 is quite large in the case of the shear
mode. Also shown in Fig. 3 are some spot checks carried out in the uniaxial
mode on a finer mesh having tip elements of side equal to 0.0635 mm, that is,
one half of that for the mesh of Fig. 1. The points referring to the finer mesh
are marked by triangles in Fig. 3 and by asterisks in Table 1. Recent work,
reported later in this paper, gives a value of ^4 equal to 0.149 for the compact
tension specimen.
In Fig. 4, 4/ and J/Jo are plotted against the plastic zone size normalized
with respect to Aa. For a given value of i/-, the value ofrp/Aa is smallest for
the equibiaxial mode and largest for the shear mode while the value cor-
responding to the uniaxial mode occupies an intermediate position. Note that
for small values of rp/Aa the values of / are approximately equal to those
ofG.
Since G and Ki* depend only on a? and not on OQ, that is

Ki = apKi*y/^ (2)

(where Ki*, approximately equal to VTT, is the non-dimensional stress inten-

sity factor and a is the half crack length), G and/ do not depend on the mode
of biaxiality of the load in small-scale yielding situations. Note that 0 is ap-
proximately equal to 0.51 Aa~' {Ki/oyy; that is, ^ is equal to rp/Aa if the
constant^ in Eq 1 assumes the same value of 0.51 for all modes of biaxiality.
This means that Fig. 3 is a plot of actual plastic zone size against that
predicted by Eq 1.

Crack Tip Opening Displacement (CTOD)

A nondimensional measure of the CTOD is obtained from the value of T;
equal to IVA/AO X lO"* where VA is the distance from the crack plane to the
node A on the crack surface immediately before the crack tip. As the load is
applied without crack extension, the crack-tip region incurs permanent
plastic deformation and attains a value 770. A distinction is made between
220 ELASTIC-PLASTIC FRACTURE

:l«
d d

. r-i . o . »n
. rn . t^ , o S :§
• ^ • ^ • r>i

O .00 .00 .00

! s5 ! r- cN , -o .0; . •-; . fo

a.

;a
•#

•^

. 0 <?v
;(G : S^ -.9,
• 0 • 0 • *-i
dd • 0 0 • 0 0 •d • d

orsirtr-i—i^a^i/jioi/iO^oofNoq ioaviO(Nf^^*0(N»ooq
D k? fS(N<sH'^^'i/i»cc^*o^dHi/ir-'a^ r-i <N -^ in r-' 00 d ri ^' «*

.o .a* .ON . *-l . fO

• d • (-^ * o * f-i ; >«
 MILLER AND KFOURI ON BIAXIAL STRESS STATES 221

50 , ^ 100 150

FIG. 3—Variation of crack-tip plastic zone size factor with applied load for different biaxial
loading modes.

the case mentioned in the foregoing and that of crack extension under cons-
tant load by the gradual and consecutive release of successive crack-tip
nodes. The value of r; after three tip nodes have been released will be referred
to here as r;i. In the idealization shown, r)o will therefore refer to the vertical
displacement of Node 40 and rji to the vertical displacement of Node 79; see
Fig. 5. A change in crack profile is known to occur as the crack begins to ex-
tend during the initial stages of subcritical crack growth [//]. In Fig. 6 the
top three curves give the values of rjo and the bottom three those of rj i, for the
three biaxial modes of loading. The values of rjo diverge considerably with
different values of X, the values for the shear mode being greater than those
for the other two modes. The values of tfi are all smaller than those of JJO
but the order of the relative magnitudes is reversed in the case of the shear
and equibiaxial modes. This is probably due to the residual stress pattern
developed in the wake of the crack tip which is an effect of the size, shape,
and position of the plastic zone prior to crack extension, all of which are
functions of stress biaxiality [/]. In all cases the variation of rj with ^ seems to
stabilize into a near linear relationship with increasing \p but the slope is of
course different for each mode.

Crack-Tip Strains and Stresses Ahead of Tip

Figure 7 shows the calculated equivalent strains at the crack tip plotted
222 ELASTIC-PLASTIC FRACTURE

•S-

e
o
O

•a

I
X
/£
"9/
puO /C
Si
 MILLER AND KFOURI ON BIAXIAL STRESS STATES 223

BEFORE CRACK EXTENSION

53 66 79 92 105

AFTER THIRD NODE RELEASE

^/////^/^,,/,,/,/
105

7), = 2tQn0ix1o''

FIG. 5—Node numbers on the crack profile and beyond the crack tip before and after the
release of the crack-tip nodes.

against 0 for the same situations as described for CTOD, that is, Fo applying
to a stationary crack being loaded and Fi applying to the equivalent strain
after the release of three tip nodes. The crack-tip node is 53 in the case of io
and 92 in the case of Fi. The results are of only qualitative interest since the
exact strain at the crack tip cannot be known with any precision, but they do
give some indication of the relative strains incurred in the tip region under
different biaxial modes of loading. The general pattern in Fig. 7 is not very
different from that of Fig. 6. However, the grouping of the lower three curves
giving ei is much closer than the corresponding grouping in Fig. 6 for r/i.
This suggests that plasticity intensity at the tip of a growing crack is not
greatly influenced by load biaxiality.
Figure 8 gives the magnitude of the main principal stress at the center of
the leading element ahead of the crack tip, normalized with respect to the
yield stress for the three modes of load biaxiality. The broken lines refer to
the case of the stationary crack and the solid curves to the extended crack
after three tip nodes have been released. The element number is 49 in the
first case and 85 in the second. The stresses are highest for the equibiaxial
mode and lowest for the shear mode. The stresses for the stationary crack are
lower than those occurring after the release of the three tip nodes. The dif-
ference between the three modes is attributed to the hydrostatic component,
which is highest in the case of the equibiaxial mode and lowest in the case of
the shear mode, and also to the different plastic zone sizes. On the whole the
curves in Fig. 8 seem to follow a somewhat similar pattern to the r/i curves in
Fig. 6. Distributions of normal principal stresses ahead of the crack tip have
also been given in Ref 12.
224 ELASTIC-PLASTIC FRACTURE

^1
 MILLER AND KFOURI ON BIAXIAL STRESS STATES 225

FIG. 7—Calculated equivalent strains at the crack tip before crack extension (to) and after
three tip nodes have been released (?i) against applied load for different biaxial loading modes.

Crack Separation Energy Rate

For a finite growth step Aa, let the stresses holding the crack surfaces
together, over the distance Aa beyond the crack tip, be quasi-statically and
proportionally reduced to zero, causing the surfaces to separate. Calling AW
the work absorbed during the release of the stresses, then the crack separa-
tion energy rate, G^, is defined as AW/Aa.
The effect of load biaxiality on C^ has already been presented in Ref 12,
giving values of G^/G and J/G against i/* for different modes of load
biaxiality. The values of G^/G converge from unity, when G equals Go, to
zero for large values of G. However, the paths are different for different modes
of biaxiality, the largest value of G^ corresponding to the equibiaxial mode. On
the same figure of Ref 12 is shown the lack of dependence of J/G on load
biaxiality for small-scale yielding as previously indicated.
Figure 9 shows that when G'^/G is plotted against actual normalized
plastic zone size (rp/Aa), the dependence of C'/G on the mode of load biax-
iality is very much reduced. Hence G'^/G appears to be a function of plastic
zone size only and independent of X. As Fig. 4 shows, however, rp/Aa
depends upon the mode of biaxiality X.
226 ELASTIC-PLASTIC FRACTURE

200

I 00

KEY
Before crock extension
After third tip node release

50 100 150
v.
FIG. 8—Major principal stress at the center of the element ahead of the crack tip against ap-
plied load for different biaxial loading modes.

Discussion
Most of the quantities investigated here show a more or less marked
dependence on the biaxial mode of loading. Exceptions are the crack-tip
equivalent strain for the moving crack Fi and the relation between C^/G and
rp/Aa. Very little is known about the growth step Aa and its dependence on
biaxiality. If Aa is dependent on the intensity of plasticity in the crack-tip
region, it is plausible to suppose that Aa is not affected by the mode of load
biaxiality.
When considering parameters such as rj and 7tip, a distinction must be
made between initiation and propagation. Generally for a stationary crack rjo
and the crack-tip plastic strain, eo would appear to be more relevant to the in-
itiation stage than to incipient unstable crack propagation, while C^ is in-
tended as a propagation parameter. The CTOD for the moving crack rji
would appear to be more relevant to propagation than initiation.
If Aa can be taken to be independent of the mode of biaxiality. Fig. 9
shows that the crack-tip plastic zone size is almost uniquely related to G^ and
can therefore be used as a propagation parameter independent of X.
However, it must be noted that actual values of Tp depend on X. It follows
that crack propagation cannot be uniquely determined by/. The values of G'^
used here are those corresponding to the third release of the crack-tip node.
 MILLER AND KFOURI ON BIAXIAL STRESS STATES 227

00
• - C T S specimen

0-50

100 200
^AQ

FIG. 9—Variation of crack separation energy rate with crack-tip plastic zone size for different
biaxial loading modes.

Now cracks in engineering components may have any orientation. Our

results show that in symmetric cases, where the sizes and shapes of the crack-
tip plastic zones above and below the crack tip are equal (hence no ambiguity
exists about the direction of crack propagation in isotropic materials), an ap-
proximate evaluation of G^ can be obtained from a knowledge of rp. For the
record, some recent elastic-plasticfinite-elementanalyses on a CTS specimen
are included in Fig. 9 [13].
In the case of inclined cracks, a more comprehensive analysis is required
since the symmetry is lost on account of the applied shear load component
additional to ap and OQ. This problem is currently being studied and is not a
simple matter since elastic-plasticfinite-elementanalyses show that unequal
plastic zones are produced at the tip and the direction of crack propagation
must be ascertained for each case. Furthermore, there may be some signifi-
cant interaction between Mode I and Mode II propagation in some cases.
Finally, it is interesting to note that fatigue growth studies under biaxial
load conditions of \ equal to — 1, 0, and + 1 indicate that crack growth rates
could be correlated on a basis of similar plastic zone sizes albeit that elastic
stress intensity factors are different [14]. Reference 14 concluded with a
i,|itatement that a two-parameter description for fatigue fracture was required.
Likewise, the present work suggests that a two-parameter relationship, that
is, / and X, is necessary to describe brittle fracture in real engineering situa-
tions.
228 ELASTIC-PLASTIC FRACTURE

Conclusion
Brittle crack propagation for various degrees of load biaxiality can be cor-
related for a given material on a basis of similar plastic zone size, which is a
function of applied stress and loading mode.

References
[/] Miller, K. J. and Kfouri, A. P., International Journal of Fracture. Vol. 10, No. 3, 1974,
pp. 393-404.
[2] Larsson, S. G. and Carlsson, A. J., Journal of the Mechanics and Physics of Solids, Vol.
21, 1973. pp. 263-277.
[3] Rice, J. ^., Journal of the Mechanics and Physics of Solids, Vol. 22, 1974, pp. 17-26.
[4] Eftis, J., Subramonian, J., and Liebowitz, H., Engineering Fracture Mechanics, Vol. 9,
1977, pp. 189-210.
[5] Kfouri, A. P. and Miller, K. J. in Proceedings, Institution of Mechanical Engineers, Vol.
190, No. 48/76, 1976, pp. 571-584.
[6] Owen, D. R. J., Nayak, G. C , Kfouri, A. P., and Griffiths, J. R., International Journalfor
Numerical Methods in Engineering, Vol. 8, 1973, pp. 63-73.
[7] Hellen, T. K., Galluzzo, N. G. and Kfouri, A. P., International Journal of Mechanical
Sciences, Vol. 19, 1977, pp. 209-221.
[S\ Kfouri, A. P. and Miller, K. J., International Journal of Pressure Vessels and Piping, Vol.
2,1974, pp. 179-191.
[9] Kfoun, A. P. and Miller, K. J., "Separation Energy Rates in Elastic Plastic Fracture
Mechanics," Technical Report CUED/C-MAT/TR18, Engineering Department, Univer-
sity of Cambridge, Cambridge, England, 1974.
[10] Rice, J. R. and Johnson, M. A. in Inelastic Behavior of Solids, M. F. Kanninen, W. F.
Adler, A. R. Rosenfield, and R. I. Yafee, Eds., McGraw-Hill, New York, 1970, pp.
641-672.
[11] Rice, J. R. in Proceedings, Conference on the Mechanics and Mechanisms of Crack
Growth, April 1973; British Steel Corp. Physical Metallurgy Centre Report, M. J. May,
Ed., 1975, pp. 14-39.
[12] Kfouri, A. P. and Miller, K. J., Fracture, Vol. 3, ICF 4, Waterloo, Canada, 19-24 June
1977.
[13] Kfouri, A. P., "An Elastic-Plastic Finite Element Evaluation of G^ in a Compact Tension
Specimen," to be published.
[14] Miller, K. J., "Fatigue Under Complex Stress" in Proceedings, "Fatigue 1977" Con-
ference, Cambridge, England; Metal Science Journal. Vol. 11, Nos. 8 and 9, 1977, pp.
432-438.
Y. d'Escatha^ and J. C. Devaux^

Numerical Study of Initiation, Stable

Crack Growth, and Maximum Load,
with a Ductile Fracture Criterion
Based on the Growth of Holes

REFERENCE: d'Escatha, Y. and Devaux, J. C , "Numerical Study of Initiation,

Stable Crack Growth, and Maximum Load, with a Ductile Fracture Criterion Based
on the Growth of Holes," Elastic-Plastic Fracture, ASTM STP 668, J. D. Landes,
J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and Materials,
1979, pp. 229-248.

ABSTRACT: Considering a material at the ductile plateau, fracture tests on small

specimens in generalized yielding conditions are common. We need to extract from
them adequate information to characterize the fracture resistance properties of the
material. We also need to predict initiation, crack growth, and maximum load for a
crack found in a ductile structure. But the real problems are three-dimensional; for
instance, semi-elliptical surface cracks or through-cracks in "small" thicknesses
(tunneling and mixed-mode fracture). Moreover, they are not only in the symmetrical
Mode I case (angled crack extension).
The common denominator of all these phenomena is the ductile fracture processes
in the material at the crack border; these micromechanisms extend over some charac-
teristic length which needs to be introduced at a crack tip because of the very intense
strain gradient. We thus need a ductile fracture damage function belonging to the
continuum mechanics frame and related to the history of stresses and strains averaged
over such a characteristic volume. In this numerical feasibility study, using elastic-
plastic finite-element computations and guided by a ductile fracture model in three
stages—void nucleation, void growth, and coalescence—we tried such a differential
damage history, in a most simplified form. We integrated this during the whole stress
and strain history in each finite element along the crack path, and we studied the
influence of the mechanical and numerical parameters playing a role in this method-
ology. We describe herein the evolution, which results from this criterion, of some
parameters used in the literature as initiation and crack growth criteria.

KEY WORDS: ductile fracture, void growth, finite elements, elastic-plastic deforma-

'Maitre de Conferences a I'Ecole Polytechnique et a I'Ecole des Mines, Laboratoire de

Me'canique des Solides, Ecole Polytechnique, 91128 Palaiseau Cedex, France.
^Adjoint au Chef du D^partement Calcul de la Division des Fabrications, Framatome,
B.P. 13, 71380 Saint Marcel, France.

Copyright 1979 b y AS FM International www.astm.org

230 ELASTIC-PLASTIC FRACTURE

tion, generalized yielding, crack initiation, stable crack growth, instability, crack
propagation

The necessity to develop physical and numerical models of ductile frac-

ture arises from a twofold need:
1. We must learn to read deeply into fracture tests, at the ductile pla-
teau, of small specimens which are in more or less extensive yielding condi-
tions, in order to extract adequate information about the fracture resis-
tance of the studied materials, and thereby to characterize them. In effect,
tests on small specimens which are in generalized yielding conditions are
common, and as these tests seek the fracture resistance properties of the
material, we must develop the ability to characterize the properties from
the observation of initiation, crack growth, and maximum load (mea-
suring load, displacement, crack opening, and crack growth).
2. We must develop the ability to predict, at least approximately, initia-
tion, stable crack growth, and maximum load for a real flaw found in a
ductile structure, in order to estimate safety margins.
Industry is now being urged to answer these questions. But the real
problems in the desired applications are, as a matter of fact, very compli-
cated. Most of them are essentially three-dimensional—for instance, semi-
elliptical surface cracks or through-cracks in "small" thicknesses when
the stress and strain state at the crack border depends on the through-
thickness coordinate. Moreover, the real problems may be in complex
loading conditions, that is to say, not only in the symmetrical (Mode I)
case.
It does not seem that the elastic-plastic fracture mechanics criteria
proposed today in the literature can be suitably extended to account for
initiation, stable crack growth, and maximum load in three-dimensional
cases or in complex loading cases. That is why we looked toward the only
common denominator of all these various situations, namely, the damage
corresponding to the ductile fracture mechanisms, undergone during the
loading history by the material situated at the crack tip.
Initiation, crack growth, change in shape of a semi-elliptical surface
crack, tunneling, mixed-mode fracture (flat fracture and shear lips) in
"small" thicknesses, and angled crack extension in complex loading—all
these phenomena can be thought of as resulting from the distribution of
stresses and strains, and of the damage they create in the material situated
at the crack tip, along the successive positions of the crack border, during
the loading history and the crack growth.
In the case of ductile fracture, this damage can be imagined, generally
speaking, as the following three stages [/,2]'. First, included particles

^The italic numbers in brackets refer to the list of references appended to this paper.
 D'ESCATHA AND DEVAUX ON INITIATION, GROWTH, AND LOAD 231

either break or separate from the matrix, thus nucleating voids [3,4]. Then
comes a void growth stage [5,6] until finally coalescence of voids occurs,
meaning that localized internal necking between the voids happens. These
micromechanisms of ductile fracture depend on the material and extend
over a characteristic length scale, for instance, the mean distance between
void nucleation sites. This characteristic length scale gives the size effect.
If we consider the ductile fracture damage as it develops in the material
situated at a crack tip, where the strain gradient is very intense, we recog-
nize the necessity to consider the damage developing globally in a volume
of material situated at the crack tip and having dimensions of the order of
magnitude of the characteristic length of the microscopic ductile fracture
processes [7-13],
We are thus looking for a macroscopic damage function which we want
to belong to the frame of continuum mechanics and whose evolution would
be related to the history of stresses and strains averaged over such charac-
teristic volumes of material representative of the length scale of the ductile
fracture micromechanisms in the considered material.
To avoid being completely arbitrary in the choice of this continuum
mechanics damage history, we can find some guidance by considering the
foregoing three general stages of ductile fracture. Therefore, this model is
restricted to fracture in the fully ductile range of temperature when purely
ductile fracture is caused by micromechanisms of the general type just
described. In particular, there should be no cleavage.
Ignoring completely whether this way of handling the problem could
produce the general trends of the usual experimental observations, and
could open onto an industrially compatible numerical tool, we began a
simple and rapid feasibility study. We chose the two-dimensional plane-
strain case, in symmetrical conditions (pure Mode I), taking three-point
bend specimens. We used a most simplified damage history model [12],
added to classical incremental elastic-plastic finite-element computations,
made with the TITUS program, in the "small geometry changes" approxi-
mation, using the initial stress and tangent stiffness method, with the Von
Mises yield criterion, Hill's Maximum Work Principle, and the perfectly
plastic case. This ductile fracture methodology is thus "noninteraction" in
the sense that the void growth does not alter the element stiffness.

Present Ductile Fracture Model

Considering a characteristic volume of material, we assume that very
early in its stress and strain history, included particles either break or
separate from the matrix. We thus neglect for this material the nucleatioft
stage. Then comes a void growth stage, and we use the void growth rate
232 ELASTIC-PLASTIC FRACTURE

formula obtained by Rice and Tracey [6] for a single spherical void in an
infinite rigid-perfectly plastic material

d(iog^\ =0.28(signa.-)d6„-expCl.5i^^ (1)

where
Ro = initial void radius,
R = present mean radius,

Om" = = mean stress at infinity, and

2
ddtq" = (-r-dea" dcy")^'^ = Von Mises equivalent incremental stram at
infinity [with de/j = deij — (detJt/3) 5;,].
This formula does not take into account void interaction or work-hardening.
Here the values of stress and strain at infinity must be understood as the
averaged values over the characteristic volume.
The finite elements are there only to calculate an approximate solution
of the partial differential equations, and then this approximate solution
must be averaged over the characteristic volumes. Here we simplify by
taking as the characteristic volume one finite element and we use the mean
values of stresses and strains in the element.
By symmetry about the crack plane, we have only to study one half of
the three-point bend specimens and, in the finite-element mesh, we put
along the crack path (symmetry axis), which is known a priori, a layer of
identical quadrilateral elements which will be the successive characteristic
material volumes at the crack tip during the crack growth.
Here we chose to represent the characteristic volume of material in front
of the crack tip by a square element of side Aa = 0.2 mm. We also tried
once Aa = 0.4 mm for comparison.
We thus follow, in each characteristic element along the crack path, the
stress and strain history and the corresponding evolution of the R/Ro ratio
by integrating Eq 1 step by step during the elastic-plastic incremental
process.
We assume that initially there is a constant distance /o between void
centers, the same in the x, y, and z directions, for instance, the x, y, and
z axes being defined in Fig. 1, and we assume that these distances change
with the mean strains in the elements along the crack path according to

.- = exp (e,) i = XX, yy, zz

D'ESCATHA AND DEVAUX ON INITIATION. GROWTH, AND LOAD 233

NB : NOTE THE EXPANDED

Specimen U 140 ( 8 nodes, Uy imposed, A a = 0.2 mm ) ORDINATE SCALE

0 ^ ( symmetry axis )

Relaxations

! -•— onset of relaxation

!•-- end of relaxation

FIG. 1—ffxx/ay along y-axis, during crack growth, Case 1.

234 ELASTIC-PLASTIC FRACTURE

and we define

/ = min li
i

About coalescence, here we simply assume that it occurs when

R _

Thus the very simplified present criterion for ductile fracture of the
characteristic volume at the crack tip, giving separation and thus an ele-
mental crack growth Aa along the a priori known crack path, is (defining
Uo = Ro/lo)

R_
Ro ou .
= — = given material property (2)

Of course, it would be very easy to incorporate in this model and numeri-

cal process any void nucleation criterion operating on continuum me-
chanics stresses and strains averaged over the characteristic volume of
material at the crack tip, if such a criterion is known with enough reliability
for the considered material (for instance, void nucleation at a manganese
sulfide inclusion by decohesion of the inclusion/matrix interface). Likewise,
void interaction and work-hardening could be included in the void growth
rate [5,14,15], and a more realistic coalescence criterion should be intro-
duced to characterize the localized internal necking which takes place be-
tween some voids and the crack tip and which leads to separation and
incremental crack growth (for instance, void coalescence by void sheet
formation, where a second population of smaller voids nucleates at carbides
within a band of intense shear between two inclusion nucleated larger voids,
giving a duplex distribution of dimple sizes on the fracture surfaces). In
the same way, an interaction between void growth and the constitutive law
of the material at the crack tip should be introduced.
For us, initiation is thus defined as the moment when the criterion, Eq 2,
is met in the first element at the crack tip. We then simulate separation at
the crack tip by relaxing step by step the corresponding nodal force from
/max to 0 for four-node quadrilateral elements. The normal displacement of
this node then grows from 0 to Vmax (Fig. 2). Note that one must take care
to do this relaxation with small enough steps, especially at the end of the
process.
 D'ESCATHA AND DEVAUX ON INITIATfON, GROWTH, AND LOAD 235

N/ran
900

700

500

400

300

O W 60 (4n, Uy irtp. , A a = 0.2niti )

jgfW60 (4n, F imp., Aa = 0.2iun))
200

Aa ntn

FIG. 2—Nodal force f at the onset of the relaxations. Cases 3-5. plotted versus correspond-
ing crack growth.
236 ELASTIC-PLASTIC FRACTURE

During this relaxation process, the R/Ro ratio in the following element
increases and l/lo decreases, so that we have to check whether the criterion
has been reached at the end of the relaxation process in this new crack tip
element: if it has been, we proceed to relax the nodal force of the new crack
tip, and if not, we proceed to load further the specimen until the criterion
is reached again, and so on.
For eight-node quadrilateral elements, the principle is the same except
that we have two nodes to relax when the criterion is reached in the crack
tip element: the corner node at the tip and the mid-side node (Fig. 3).
Here we relax step by step the forces at these two nodes simultaneously and
proportionally. We also made one try relaxing them one after the other,
which is somewhat daring, to compare.
We studied the three-point bend specimens in the displacement-controlled
case, imposing upon them a growing displacement Uy of the roller and
keeping this displacement constant during the relaxations. We also made
one try with the load-controlled case (imposing a growing load F and keep-
ing it constant during the relaxations).
The foregoing oversimplified model can be used in the present Mode I
case because we do not need any directionality in the criteria since in this
case the crack is known a priori to grow in its own plane.
The present approach to the problems presented in the introduction is
attractive because it can be incorporated very easily in any elastic-plastic
finite-element program, and because it could be applied to the important
three-dimensional, symmetrical (pure Mode I) cases—semi-elliptical sur-
face cracks, or through-cracks in "small" thicknesses—to predict initiation,
stable crack growth, and maximum load.
By an adequate adaptation of the mesh and of the criteria (directionality),
the angled crack extension problem in complex loading could, it is hoped,
be treated.
The calculations of the feasibility study are made in the elastic-perfectly
plastic case (yield stress ay = 520 MPa, Von Mises yield criterion), on
three-point bend specimens of various widths W, but with the same a/W
ratio (0.475); they are in-plane homothetic. The (F/BWOY, Uy/W) normal-
ized load-displacement curves are the same for these various specimens
until initiation, but the initiation point and crack growth behavior are
different when the size of the specimen is changed (size effect) (Fig. 4). A
large enough specimen will be in small-scale yielding conditions when
initiation occurs, whereas a small enough specimen will be in generalized
yielding conditions. This, of course, comes from the characteristic volume
of material over which the ductile fracture micromechanisms extend. Since
the damage results from the history of stresses and strains averaged over
this characteristic volume (independent of specimen size), and since the
stresses and strains are identical within the homothesis acting upon the
in-plane coordinates, it is obvious that at homologous loads the damage
 D'ESCATHA AND DEVAUX ON INITIATION, GROWTH, ANH LOAD 237

i Specimen W =140 mm
f '
8 node elenents, Uy imposed.^e = 0.2mm )
N/nm j ^ : '
F i r s t relaxation

Xi

. < ^ ^ ^ ^
i^^v/^
i
160
\\ L\\ r T
\\ \ \ '^/'y^.
1 ^ ^ ^
T
T
1
1
140
\\ \ \

\ \^ L^"'
?!
>-
\ \ \
W ""\
i^ '
120

W \
\
\
100

A - 2 " kind o f relaxation

80 ( one node after the other )
^ simultaneous and proportional
relaxation
\ \ ^
60 _ \ \ \
\ '^ ^
\
40 N

\ ^ >\ ^ ' ^X
20 -

\
0 1 11 \ " ^ ^ 1 ,.l ^ 2 , , ^
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7^10"'
VI m m

FIG. 3—Relaxation curves. Case 1.

will be lower in the smaller specimen, so that its initiation point will be
later.

Results
The first calculations of the feasibility study showed that this model,
though very simplified, reproduced the general trends of the usual experi-
mental observations and gave no results in contradiction to them. More-
238 ELASTIC-PLASTIC FRACTURE

I
F/B '

"'^o.n
L _-
pir^y=l
0.10 A^' *
0.09

0.08
Uj

0.07

0.06

0.05

^ W 140

0.04 • W 60

( 8 N, U i r a p . , 4 a =0.2nim
I Relaxation
0.03
V W 140 z" kind of
relaxation
L initiation
0.02

0.01

2.,i10-^
u/w

FIG. 4—Normalized load-displacement curves. Cases I and 2.

 D'ESCATHA AND DEVAUX ON INITIATION, GROWTH, AND LOAD 239

over, an industrial tool was built. As these results had only a general
trend value because the whole numerical treatment was not refined enough,
and as they were very encouraging, we made the present new calculations,
reported here, with the necessary refinements to study the influence of the
various parameters playing a role in this methodology: mesh, size of ele-
ments, type of elements, convergence precision, steps in the incremental
process (especially in the node relaxations), type of relaxation, and precision
in the criterion fulfillment. We get smooth solutions, with no oscillations,
the principal results being given in the following, and showing in particular
the behavior, which results from the present criterion, of some parameters
used in the literature [7,16-20] as initiation and crack growth criteria.
The results reported here deal with the following cases:
1. W = 140 mm, 8-node elements, Aa = 0.2 mm, displacement-con-
trolled process, nodal forces relaxed simultaneously and proportionally.
In order to test the influence of the relaxation technique for the 8-node
elements, however, we made one try for this specimen where we relaxed the
two nodes one after the other. Starting again from the solution obtained
at the initiation point, we relaxed the two nodes one after the other, and
then we loaded again the specimen until the criterion was met in the new
crack tip element. This calculation is labelled "2nd kind of relaxation" in
the figures, which show that, as expected, the effect is much smaller on
global quantities (Figs. 4, 5, 6) than on local quantities (Figs. 7, 8, 3,
and 9).
2. W = 60 mm, 8-node elements, Aa = 0.2 mm, displacement-controlled
process, nodal forces relaxed simultaneously and proportionally.
3. W = 60 mm, 4-node elements, Aa = 0.2 mm, displacement-controlled
process.
4. W = 60 mm, 4-node elements, Aa = 0.2 mm, load-controlled process.
5. W — 60 mm, 4-node elements, Aa = 0.4 mm, displacement-controlled
process.
In the first four cases, where Aa = 0.2 mm, we took the same critical
value in the criterion (1.286). In the fifth case, where Aa = 0.4 mm, we
took a lower critical value (1.184) chosen to give initiation at approximately
the same value of J-integral.
The normalized load-displacement curves are shown for Cases 1 and 2
on Fig. 4; as expected, they are found identical before crack growth initi-
ation for the large and for the small specimen, and initiation (i) occurs
"later" for the small one. The arrows point out the relaxations (made here
with Uy kept constant). Since we were interested here in initiation, stable
crack growth, and maximum load, we stopped the calculations when the
load began to decrease.
The details of initiation, stable crack growth, and maximum load are
shown in the same way for Cases 3-5 in Fig. 10. In the load-controlled
case (No. 4), we have at initiation two successive relaxations, and it can be
240 ELASTIC-PLASTIC FRACTURE

A a = 0.2nni )

60 {4n, I' imp., i a = 0.2imi )

W 60 (8n, U imp., Aa = 0.2nm ]

A w 140 2 kind of relaxation

^a

FIG. S—Load at the onset of the relaxations, plotted versus corresponding crack growth.
 D'ESCATHA AND DEVAUX ON INITIATION, GROWTH, AND LOAD 241

J e acKial s>at»
i-acnial srar*

0. 0-2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

FIG. 6—J-integral at the onset of the relaxations, plotted versus corresponding crack growth.
242 ELASTIC-PLASTIC FRACTURE

"P /ic l'" ( 4 node elements

jmax ^2iT)ax
t^. d v i + ^ ^ J t^. dV2 ( 8 node elements

W 60/ mean value for

the successive
(fi) w 60,^a= 0.4 I two relaxations

A W 140 (8n, Uy imp. , A a = 0.2ntn

60 (4n, i n p . , A a = 0.2mn )
)gl W 60 (4n, F lnj)., Aa = 0.2iiiii )
• W 60 (8n, U ijnp. ,,^a=0.2imi )

kind of relaxation

0 0.2 0.4 0.6 0.8 1.0

Aa ntn

FIG. 7—7p/or the relaxations, plotted versus corresponding crack growth (yp = relaxation
work per unit crack extension area).
 D'ESCATHA AND DEVAUX ON INITIATION, GROWTH, AND LOAD 243

(0.2)tgp

W 140 (8n, U i n p . , A a = 0 . a t i n )
W 60 (4n, U„ imp.-,^a = 0.2im )
W 60 (4n, F ijip., A a = 0.2ittn )
7 X 10
W 60 (8n, Uy Ijip., Aa=0.2imi )
W 60 (4n, U imp.,Aa=0.4niti )

kind of relaxation

A a

F I G . 8—Conventional crack opening angle ft given by (0.2) tgft, ram, at the onset of the
relaxations, plotted versus corresponding crack growth.

seen that, though they correspond to two different loading histories, the
load-controlled case (No. 4) and the displacement-controlled case (No. 3)
give quite close results. In Case 5, where Aa = 0.4 mm, we have also two
successive relaxations at initiation, giving a crack growth of 0.8 mm, which
is larger than the crack growth to maximum load obtained with Aa = 0.2
mm. This can be paralleled with the fact that the load obtained at the next
relaxation of Case 5 is lower than the initiation one.
In Fig. 5, the load F/BW at the onset of each relaxation is plotted
versus the corresponding crack growth Aa; the upper group of points
represents the corresponding behavior from initiation to maximum load
244 ELASTIC-PLASTIC FRACTURE

MB : NOTE THE VERY EXPANDED

(I « m • , ORDINATE SCALE

2 1
« -
N/mm

200

125

^a nm

FIG. 9—Nodal forces f at the onset of the relaxations. Cases 1. 2, plotted versus corre-
sponding crack growth.

for the small specimen. The lower curve represents this behavior for the
large specimen.
Figure 7 gives the relaxation work per unit crack extension area yp for
the relaxations, plotted versus corresponding crack growth, yp is defined on
the figure for 4-node elements and 8-node elements. We note that, in the
8-node element cases (1 and 2), 7p is found almost constant during crack
growth, initiation included, and almost the same for the large and for the
small specimen. This result is interesting since the criterion used here to
recognize whether a crack tip element has reached a critical state with
 D'ESCATHA AND DEVAUX ON INITIATION, GROWTH, AND LOAD 245

HB : NOTE THE EXPANDED SCALES

F/B

0.09

FIG. 10—Normalized load-displacement curves (upper part), Cases 3-5.

respect to ductile fracture is a criterion which integrates the whole growth

process all along the elastic-plastic incremental stress and strain history,
and thus is not directly linked to this energetic criterion. When we had two
successive relaxations (Cases 4 and 5), we plotted the jp value correspond-
ing to the first relaxation and also the mean value on the two successive
relaxations.
We note, among all parameters which play a role in the approximate
numerical solution of this elastic-plastic incremental problem combining
specimen loading and node relaxation, differences in behavior of the 4-node
and 8-node elements around the crack tip, where the mesh is the same.
These differences appear on figures representing local quantities and local
effects at the crack tip, for instance, Figs. 7 and 8. Recall that there is one
246 ELASTIC-PLASTIC FRACTURE

nodal force to relax for 4-node elements and two nodal forces for 8-node
elements, and that in this case the way of relaxing these two forces has a
local effect. Note that there are in the present solution, which is mathe-
matically singular, high plastic strains and high strain gradients, and that
the specimens are loaded up to the fully plastic range. Effects of mesh,
size and type of elements on global quantities appear in Figs. 4,10, 5, and 6.
Figure 8 shows the crack opening angle, which is conventionally defined
in the figure, at the onset of the relaxations, plotted versus the correspond-
ing crack growth. We note that it decreases after initiation with a tendency
to stabilization with crack growth and comparable trends for the large and
for the small specimen.
Figure 6 shows the J-integral, recalled on the figure, at the onset of the
relaxation, plotted versus corresponding crack growth. / was computed on
paths distant from the crack tip, where it was found almost path-inde-
pendent (within a few percent). We note an important effect of the mesh,
size, and type of elements (see second-last paragraph in the foregoing).
Figure 3 compares the relaxation curves obtained at initiation in Case 1
for the two kinds of relaxation. Besides, we noticed that, for Cases 1 and 2
(8-node elements), the relaxation curves obtained at initiation for the large
and for the small specimen were almost identical, though the large speci-
men is then in very contained yielding conditions and the small one in
generalized yielding conditions.
Figures 9 and 2 show the nodal forces at the onset of the relaxations,
plotted versus corresponding crack growth, in Cases 1 and 2 and 3-5. We
note a tendency to stabilization with crack growth and comparable trends
for the large and for the small specimen (Fig. 9). Besides, we noticed in
Cases 1 and 2 that the nodal normal displacements v at the end of the
relaxations were almost constant during crack growth and almost the same
for the large and for the small specimen. There is thus, for the relaxation
curves, a tendency to stabilization with crack growth and comparable
trends for the large and for the small specimen.
Figure 1 shows the opening normal stress (over yield stress) ahead of
the crack tip during crack growth, in Case 1. We note that, in this speci-
men, the stable crack growth takes place under a high triaxiality stress
field.

Further Developments
We think that the results of this feasibility study are quite encouraging.
Moreover, we are now equipped with a completely automatic tool whose
parameter dependence and sensibility have been studied. Thus, we pro-
ceeded recently to the most important stage—the comparison between
calculation predictions and experiments. Tests are being made on compact
tension specimens, on round bars with an external circular notch in tension.
 D'ESCATHA AND DEVAUX ON INITIATION, GROWTH, AND LOAD 247

and on single-edge notched specimens in tension and in bending, with

different values of W, of a/W, of the notch angle, and of the notch root
radius, in order to have, for the ductile fracture processes, different con-
ditions of stress triaxiality in the characteristic volume at the notch root.
The material is A 508 CI 3 steel in fully ductile conditions. Adjusting the
various parameters once and for all, we shall see to what extent this model
and methodology can reproduce the experimental results for load, dis-
placement, crack opening, and crack growth, for these very different con-
ditions of ductile fracture, especially for stress triaxiality.

Acknowledgments
We wish to acknowledge the financial support of the French "Service
Central de Surete des Installations Nucleaires," and J. Devaux, G. Mottet,
and C. Vouillon for their precious help with the calculations.

References
[/] Bluhm, 1.1, and Morrissey, R. J., Transactions, 1st International Conference on Fracture,
Sendai, Japan, Vol. 3, 1966, pp. 1739-1780.
[2] Hodgson, D. E., "An Experimental Investigation of Deformation and Fracture Mecha-
nisms in Spheroidized Carbon Steels," Ph.D. Thesis, Stanford University, Stanford,
Calif., 1972.
{3] Argon, A. S., Im, J., and Safoglu, R., Metallurgical Transactions, Series A, Vol. 6A,
April 1975, pp. 825-837.
[4] Tanaka, K., Mori, T., and Nakamura, T., Transactions, Iron and Steel Institute of
Japan, Vol. 11, 1971, pp. 383-389.
[5] McClintock, F. A., Transactions, American Society of Mechanical Engineers, Journal
of Applied Mechanics, June 1968, pp. 363-371.
[6] Rice, J. R. and Tracey, D. M., Journal of the Mechanics and Physics of Solids, Vol. 17,
1969, pp. 201-217.
[7] McClintock, F. A., Transactions, American Society of Mechanical Engineers, Journal
of Applied Mechanics, Dec. 1958, pp. 582-588.
[8] Rice, J. R. in Fracture, H. A. Liebowitz, Ed., Vol. 2, Academic Press, New York, 1968.
[9] Rice, J. R. and Johnson, M. A. in Inelastic Behavior of Solids, M. F. Kanninen, W, F.
Adler, A. R. Rosenfield, and R. I. Jaffee, Eds., McGraw-Hill, New York, 1970, pp.
641-672.
[10] McMeeking, R. M., "Finite Deformation Analysis of Crack Tip Opening in Elastic-
Plastic Materials and Implications for Fracture Initiation," Technical Report COO-
3084/44, Division of Engineering, Brown University, Providence, R. I., May 1976.
[//] Ritchie, R. O., Knott, J. F., and Rice, J. R., Journal of the Mechanics and Physics of
Solids, Vol. 21, 1973, pp. 395-410.
[12] d'Escatha, Y. and Devaux, J. C , Transactions, 4th Structural Mechanics in Reactor
Technology Conference, San Francisco, Calif., Vol. G, Paper G2/4, Aug. 1977.
113] Mackenzie, A. C , Hancock, J. W., and Brown, D. K., Engineering Fracture Me-
chanics, Vol. 9, 1977, pp. 167-188.
[14] Tracey, D. }A., Engineering Fracture Mechanics, Vol. 3, 1971, pp. 301-315.
[15] Needleman, A., Transactions, American Society of Mechanical Engineers, Journal of
Applied Mechanics, Dec. 1972, pp. 964-970.
[16] Andersson, H., Journal of the Mechanics and Physics of Solids, Vol. 21, 1973, pp.
337-356.
248 ELASTIC-PLASTIC FRACTURE

[17] Kobayashi, A. S., Chiu, S. T., and Beeuwkes, R., Engineering Fracture Mechanics, Vol.
5, 1973. pp. 293-305.
[18] Kfouri, A. and Miller, K. J., "Separation Energy Rates in Elastic-Plastic Fracture
Mechanics," CUED/C. MAT/TR 18, Department of Engineering, University of Cam-
bridge, Cambridge, U.K., Dec. 1974.
[19] Rousselier, G., "Croissance Subcritique de Fissure et Crit^res de Rupture: Une Ap-
proche Numerique," Transactions, 4th International Conference on Fracture, Waterloo,
Canada, June 1977.
[20] Wilkinson, J. P. D., Hahn, G. T., and Smith, R. E. E., Transactions, 4th Structural
Mechanics in Reactor Technology Conference, San Francisco, U.S.A., Vol. G, Paper
G2/2, Aug. 1977.
Experimental Test Techniques and
Fracture Toughness Data
p. C. Paris,' H. Tada, * H. Ernst,' and A. Zahoor^

Initial Experimental Investigation of

Tearing Instability Theory

REFERENCE: Paris, P. C , Tada, H., Ernst, H., andZahoor, A., "InMalExperlmeiital

Investigation of Tearing Instability Theoiy," Elastic-Plastic Fracture, ASTM STP 668. I.
D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and
Materials, 1979, pp. 251-265.

ABSTRACT: An initial experimental investigation was conducted to confirm the theory

of tearing instability developed in previous work. A simple testing program was selected
which employed 3-point bend specimens with various crack size to specimen depth ratios
and an additional spring bar to easily adjust effective specimen span (or loading system
compliance).
AU stable-unstable behaviors observed in the tests are in good agreement with those
predicted by the theory. Thus the present study demonstrates the appropriateness of the
tearing instability analysis, presenting guidelines for its further development.

KEY WORDS: tearing instability, crack stability, experimental fracture mechanics,

J-integral, 3-point bend tests, crack propagation

In previous work the authors have developed an analysis of tearing in-

stability ^ for application to cracking instability predictions based on
J-integral R-curve representation of material characteristics. In this previous
work, emphasis was placed on applications to fully plastic cracked ligament
and plane-strain conditions. However, it was noted that the theory is quite
general and is applicable from small-scale yielding through the fully plastic
range for plane-strain through plane-stress conditions with work hardening,
etc. For a fuller understanding, one is referred to the previous report. ^
Since application of this new cracking instability theory seems relevant for
certain fully plastic plane-strain Situations of practical interest (such as reac-
tor vessels at operating temperatures with low upper-shelf Charpy materials),
it is of immediate and special importance to experimentally verify the
analysis for those situations where no other instability analysis has as yet

' Professor of mechanics, senior research associate, and graduate research assistants, respec-
tively, Washington University, St. Louis.
^Paris, P., Tada, H., Zahoor, A., and Ernst, H., this publication, pp. 5-36.

251

Copyright 1979 b y A S T M International www.astm.org

252 ELASTIC-PLASTIC FRACTURE

been formulated. For that reason, a simple testing program was developed
which could be performed quickly using a material which was pedigreed and
would exhibit fully plastic plane-strain behavior (in the J-integral tearing
sense) in reasonably sized specimens.
Therefore, the objective herein is to present results of a testing program
which clearly demonstrates the appropriateness of the tearing instability
analysis and which illustrates its broad potential for future application, as
well as presenting guidelines for its further development.

Testing Program
The material selected was nickel-chromium-molybdenum-vanadium
(NiCrMoV) rotor steel supplied by Westinghouse Research. This materi^ll
was previously subjected to extensive testing by Westinghouse; for example,
as reported by Logsdon.^ The material has flow properties and J-integral
R-curve properties (Ju and dJ/da for tearing) for temperatures well above the
transition temperature, which make it quite convenient for crack instability
tests. It is very convenient to be able to select reasonable test specimen pro-
portions requiring moderate test loads and the usual instrumentation while
being able to change from stable to unstable results from test to test due to
simple modifications to the test variables.
The test specimen configuration was selected to be a 3-point bend speci-
men of a full span, L, of 8 in. with a specimen depth, W, of 1 in. and thickness,
B, of Vi in. Specimens were notched and fatigue precracked to various crack
size to specimen depth ratios, a/W. A schematic diagram of the test con-
figuration is shown in Fig. 1.
In the tests, stability was affected mainly by varying the a/W of the test
specimen or the effective (or equivalent) elastic span of the test specimen, or
both. The method of adjusting the effective elastic span was by inserting in
the test arrangement an elastic spring bar of adjustable span for a variable
spring constant. Analysis details for this arrangement will be presented
subsequently.
The testing arrangement as shown in Fig. 1 also permitted measuring
load, from a load cell, and displacement, from the ram displacement, in a
standard MTS servo-hydraulic testing machine to produce a load displace-
ment record for analysis of the stability of the situation. An important
feature of the arrangement was the ability to remove various components in-
dividually (spring bar, test specimen, and appropriate rollers) in order to
make direct elastic compliance calibrations of the various components of the
test arrangement, including the test machine itself. These compliance

^Logsdon, V*^. A. in Mechanics of Crack Growth, ASTM STP 590. American Society for
Testing and Materials, 1976, pp. 43-61.
 PARIS ET AL ON TEARING INSTABILITY THEORY 253

test macliine frame

rigid bar

L£Q QT_J
•test specimen

-t- spring bor

H Ezr TZIE
V
spacers for ralle rs
ii- JV
^ rigid bar

^test mact)ine rom

(displacement m<
by L V O T in ram )

FIG. 1—Testing arrangement.

calibrations will be seen to be an important feature of the analysis of the test

results.

Tearing Instability Analysis of the FoDy Plastic 3-Point Bend Specimen Test
The formula for instability of a 3-point bend specimen where the remain-
ing uncracked ligament, b{orW — a), is fully plastic was given in the earlier
paper (footnote 2). It is

2b^L dcE
•'mat — J X < = T applied (1)
da (To ffo

That is to say, if the right-hand side, Jappiied, exceeds the left hand side, Tmat,
instability will occur when the uncracked ligament becomes fully plastic and
Jjc is exceeded so that tearing begins. This formula assumes a rigid testing
machine (fixed displacements) and rigid test fixtures. On the other hand, the
driving energy or force for instability comes from the elastic unloading of the
bend specimen as the limit load diminishes due to crack extension. Thus
^applied contains these influences through specimen proportions L, b, and W,
but in addition it is affected by the bend angle. Be, of the uncracked ligament
section of the test specimen. This is explained further in the earlier paper
(footnote 2) (and its appendices).
254 ELASTIC-PLASTIC FRACTURE

N o w i as a factor in the foregoing formula for Tappiied appears due to elastic

compliance of the test specimen with no crack present. Adding compliance to
the test arrangement by using in addition a spring bar is equivalent to adding
compliance to the uncracked test specimen. Therefore, analysis leads to an
equivalent (increased) length for the test bar, -Lequiv, by the relationship

OSB
-»-'equiv J-t 1+ (2)

where bse and bra are the elastic deflections of the spring bar and test bar
(with no crack) are under the same load. The equivalent length, iequiv,
should then be used to replace X in the instability formula. From compliance
calibrations, this could be determined for any particular test. The com-
pliances are given in Table 1 and it is noted that the compliance of the testing
machine and fixtures will have a negligible effect on results (when compared
with the much smaller spring constants of both the test specimen and spring
bars).
In addition, dc could be analyzed for any point on a load displacement
record by subtracting elastic displacements for the uncracked test bar and
spring bar from compliance calibration information and thus obtaining the
displacement due to the crack alone, 5crack. This includes both the elastic and
plastic deformations of the uncracked ligament sections. Then 6c is ob-
tained directly from geometry

44
(3)
L

Thus all factors in Tappiied can be obtained directly from specimen dimen-
sions, compliance calibrations, and a given point on the load displacement
record.
TABLE 1—Instability test component stiffnesses.

Spring
Component Size, in. Constant, lb/in.

A. test machine and Hxtures 320 000

B. test machine fixtures and spring bar LsB= 8 30 047
LSB = 11 13 035
LsB = 12 10 000
LsB = 14 6 472
LsB = 15 H 4 638
C. test bars (Z = 8 in.) no crack 100 000
a/W = 0.4 90 000
a/W = 0.5 50 397
a/W = 0.6 36 570
a/W = 0.7 19 217
 PARIS ET AL ON TEARING INSTABILITY THEORY 255

On the other hand, the so-called "tearing modulus," T^M, can be

evaluated in a quite independent manner. Indeed, Tmat depends only on
material properties and thus should be the same throughout the testing pro-
gram (except for a small effect of temperature). It depends only on the tear-
ing slope, dJ/da, of the J-integral R-curve (a plot of J versus crack length
change, Aa), the elastic modulus, E, and the flow stress, ao, of the material.
(For purposes herein, the flow stress was taken as the average of the yield and
ultimate strengths of the material.) Also, though Tmat therefore could be
measured separately, it could also be evaluated from each test in the follow-
ing manner.
During any test, / can be evaluated from the usual Rice et al pure bending
analysis." It is

/ =-—- X area (4)

bB
where the "area" is the area under the load versus displacement (due to the
crack) record up to any point at which / is desired. Changes in /, that is, in
A/, can be computed* from differences in results or
2
A/=—— X A(area) (5)
oB
On the other hand, changes in crack length are less easy to evaluate directly.
However, if test proportions are selected so that limit load is reached prior to
beginning of crack extension (that is, the Ju point), then crack extension can
be evaluated from the reduction in limit load due to crack growth. The
analysis is as follows. The limit load is
4ML 4
PL = —r' = —Ab^Bao (6)

where A is a constant. Differentiating for crack extensions (da = —db) gives

dPL = yABaoi-lbda) (7)

Dividing Eq 7 by Eq 6 and rearranging leads to

b APi

^Rice, J. R., Paris, P. C. and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing, ASTM STP 536, American Society for Teslii^ and Materials, 1973, pp.
231-245.
*The method here neglects correction terms, c(Mi$istent with methods in the literature (see
Paris et al (footnote 2), Appendix I).
256 ELASTIC-PLASTIC FRACTURE

Thus, upon reaching limit load, Aa can be evaluated from one point to the
next simply from the ligament size, b, the change in limit load, APL, and the
limit load itself, PL .
Hence, Jmat can be determined, by the methods described in the foregoing,
as

AJ E
r„a, = — 7 (9)
Aa ffo^
This method was used in the present testing program, that is, based on Eqs
5, 8, 9, and their implied assumptions. Now in Eqs 5 and 8 the remaining
ligament, b, appears in such a manner that in Jmat, Eq 9, upon substitu-
tion of Eqs 5 and 8, it is squared. In all computations, the original ligament
size was used, which implies that the analysis is limited to small crack exten-
sions compared with the ligament size.
Moreover, the/-integral method itself becomes suspect if substantial crack
extension occurs, so that restriction to small crack extension was already im-
plied.

Results of the Testing Program

The testing program is enumerated on Table 2. Test specimens were
machined from a single piece of material which was originally cut from a
rotor forging as a blank for a 4-T compact specimen of ASTM-A471 steel
(that is, NiCrMoV from Westinghouse Research as mentioned earlier). The
cracking plane and direction were as intended in the 4-T blank so that results
can be compared [see Paris et al (footnote 2) and Logsdon (footnote 3)].
Specimens are numbered according to a positive location system. First,
Slabs A, B, and C were removed from the blanks, starting with the side op-
posite to the notch location in the 4-T blank. They were then sliced into
Specimens 1 through 6 for each slab, dividing the thickness dimension of the
4-T blank. Therefore, each test specimen is numbered with a letter and
number identifying its location in the 4-T blank. Although care was taken in
identifying with location, no effect of location was necessarily anticipated,
nor was any found in test results to be described further here.
The test bars were machined with notches 0.2 in. deep and fatigue
precracked to give crack depths, a, from 0.313 to 0.713 in. (the same as a/W
values since W = 1 in.). Precracking loads were applied in the same testing
fixtures but with the spring bar removed to give tension-tension loading of
the notched side with maximum fatigue loads less than half the maximum
(limit) load in each stability test.
Test temperatures were chosen to be 130 and 230''C, so that each is at least
100 °C above the transition temperature of the material to avoid cleavage and
 PARIS ET AL ON TEARING INSTABILITY THEORY 257

^•1 oo ooooooooooooo
a, '. S' o '. - o o o o i o o p o o d o o o

If
^o oo I/) in r-. ^^ TT M <N *£) (N ^o 1^ ^^ I/)
»M^HOomoNro(Ni/)io^o>Qr^

m K

O.IO i/>oor^o>Oi/>'no^ooio c — c
^ e i^ — — > oo
r»^ ^ -tr i!?

ill » ? ^«^
I
U
00
O O —_ vO O -H CS r « vo O O O —I
• ^ •»!• t--' - H ' TT f S 00 r j —I ^ ' <> vO <N
f*5 r^ f S I/) f ^ TT ^ ir> m 1^ r^ -^ IP
<

o o o o ; OOOOOO
I
00 oo od GO

o o o o - o o o o o o o o o o o o o
f2 S.° fO f^ fO r*^ • r^ ro ro ro f*i f i f^ r*) f^ r^ **) fO ro
e

o o o o ' O O O O O O O O O O C5 d O
258 ELASTIC-PLASTIC FRACTURE

thereby examine upper-shelf tearing instability behavior as intended here.

Occasionally at 130 °C some minor pop-in was noted, which was entirely ab-
sent at 230 °C. However, tearing instability results of these tests show no
substantial difference for the two temperatures, as may be noted later.
Table 2 gives the length of spring used in each test and the equivalent span
of the test specimen, as determined from Eq 2 and compliance calibration in-
formation in Table 1.
The equivalent span and other test information from load displacement
records were then used to determine independently Tmat and Tappiied as
described in the preceding section. Thus the columns for Tmat and Tappued in
Table 2 may be compared to determine according to the theory if unstable
behavior is expected, that is, if Tmat ^ Tappued. These results can be com-
pared with the column of Table 2 marked "stability," where the actual
physical state of stability observed in each test is listed. Such a comparison is
shown in Fig. 2. For each test a point is plotted from its Jmat versus Tappiied
values. Lying on one side of the 45-deg theory line is a theoretical prediction
of stable or unstable behavior as noted. On the other hand, actual physical
behavior is noted for each point as solid points, stable and open points,

• - ZSO-C stable

a — 230" C unstable

• - ISO-C stable

O— 130°C unstoble

NOTE: half shaded stable

symbol indicates
morginal beliavior

40-

30-

applied

FIG. 2—Stability test results.

 PARIS ET AL ON TEARING INSTABILITY THEORY 259

unstable (and one-half shaded points as marginal). It is noted that the

agreement between physical behavior and theory is very good! Only two
points exhibiting adjudged unstable physical behavior lie on the stable side,
but very close to the theory line. Moreover, the data in Fig. 2 show that Tmu
is reasonably constant for a wide variety of variables (temperature and also
a/W, iequiv, etc. as the effective Tappucd). Thus this is felt to be a strong
verification of the theory for a single type of specimen—the bend test
specimen.

Farther Discussion of Load Displacement Records and Physical Instability

Physical instability behavior can be clarified and better understood by a
more specific discussion of load-displacement records from the testing pro-
gram. Furthermore, it is of special interest to make these observations for a
sequence of tests, where temperature and test specimen dimensions in-
cluding a/W are held constant and only the elastic compliance of the testing
system is varied by changing only the spring bar length. Referring to Table 2
for a set of tests, where the temperature is 130 °C and a/Wis about 0.5 (that
is, 0.510 to 0.518), it is seen that Tests A-2 {LSB = 8.0), B-2 (LSB = 11.0),
B-4 (LSB = 12.0), B-5 (LSB = 13.0), and B-3 (LSB = 14.0) comprise such a set
of tests. Figs. 3-7 show load displacement records from these particular
tests in the same respective order.
Now in each of these tests, tearing starts at the beginning of the descending
load portion where the record departs from the maximum load level in the
test. The beginning of tearing is denoted also as the /ic point, and values for
comparison are listed in Table 2 (it is noted that they are reasonably con-
stant). Tearing stability or instability then depends on the character of the
descending part of the load displacement records beyond this point of begin-
ning of tearing.
As noted from Figs. 3-7, as elastic compliance, that is, spring bar length,
is added, the descending portion of the records becomes steeper. Since the
testing system is in displacement control as measured by a linear variable dif-
ferential transformer (LVDT) in the hydraulic ram, when sufficient elastic
compliance is added to cause the descending load displacement record to be
vertical, unstable behavior ensues. Noting the sequence of behaviors with
added compliance in Figs. 3-7, it is judged that Figs. 6 and 7 show substan-
tially unstable behavior and that Fig. 5 shows substantially stable behavior
(though a short segment of vertical record occurs just after maximum load).
Moreover, when observing the test, unstable behavior as just described is
occasioned by a sudden increase in crack size and deformation of the test
specimen, whereas with stable behavior no sudden deformations occur. It is
by these observations and analysis of test records, such as Figs. 3-7, that the
"physical instability" was judged and recorded in the column so entitled in
Table 2.
260 ELASTIC-PLASTIC FRACTURE

2000

a.

1000

0.21
6 (In.)

FIG. 3—Load-displacement record for Test A-2, with LSB = {S in.), resulting in stable
behavior. (Displacement zero-point displaced slightly here and on Figs. 4- 7.)

It is relevant to observe here and to emphasize that as described in the

foregoing and in the more general analysis (footnote 2), tearing instability is
a system behavior involving not only the local fracture characteristics of a test
specimen (or structural component) but also the elastic compliance of the
overall test specimen and loading system. Perhaps cleavage fracture insta-
bility is a local material instability phenomenon, but it is clear here that tear-
ing instability is a general system type of instability. Moreover, this testing
program, where, in Table 2 and Fig. 2, stability is judged by theory, compar-
ing Tmat and Jappned, and judged separately by physical instability behavior,
clearly verifies the general approach taken by the theory of tearing instability
(footnote 2) and especially that tearing instability is not treatable as a local
phenomenon (in the Ku local critical field intensity sense).
 PARIS ET AL ON TEARING INSTABILITY THEORY 261

2000 -

1000-

0.30 0.36

FIG. 4—Load-displacement record for Test B-2, LSB == 11 in-, stable.

Some Additional Comments on Testing Results

In the preceding discussion, verification of the concepts of the tearing in-
stabiUty theory was emphasized as a system phenomenon, where transition
from stable to unstable behavior is caused by varying the system compliance,
that is, the spring bar length. In addition, other transitions caused by varying
local test specimen dimensions are observed from the data in the test pro-
gram.
For example, with other conditions identical, a switch from unstable
behavior is observed in Tests C-6 and B-5 due to a change in a/Witom 0.706
to 0.510, respectively; see Table 2. Since b = W — a, the implied effect is
noted in Eq 1 as an increase in b, which increases Tappiied, causing the transi-
tion to instability (even though total compliance increases).
Two other pairs of specimens, B-4 and A-6, and also C-3 and B-1, each ex-
hibits a switch from stable to unstable behavior due to a change in a/Witom
262 ELASTIC-PLASTIC FRACTURE

2000

1000 -

FIG. 5—Load-displacement record for Test B-4, LSB = 12 in., stable.

about 0.5 to 0.4. Both of these pairs were tested with spring bar lengths of 12
in., but one pair was at 130°C and the other at 230°C. Thus it was shown
that the switching of behavior from stable to unstable was not appreciably af-
fected by this temperature change, even though switching was induced by a
relatively small change in a/W. Again, these results were expected from the
theory, Eq 1, and the fact that temperature changes on the upper shelf only
weakly affect /-integial R-curve tearing behavior (footnote 3) (specifically,
dJ/da), flow stress, ao, and modulus, E. Therefore the results in Table 2
verify the theory even more strongly than the conclusions drawn in the
previous section herein.

Additional Experimentation whicli Seems Relevant for Future Worli

Although verification of the theory seems strong here, it would be interest-
ing to go beyond the limited scope of this initial testing program to ex-
plore effects of additional variables such as the following.
 PARIS ET AL ON TEARING INSTABILITY THEORY 263

2000

1000

FIG. (>—Load-displacement recordfor Test B-5. LSB = 13 in., unstable.

1. Test specimen thickness, that is, thicker specimens to verify that plane
strain was in fact fully achieved and thinner to explore the effects of plane
stress [especially in relation to J-integral plane-strain size criteria, for exam-
ple, size > (25 or 50)//ffo]•
2. Other specimen size effects such as proportionately scaling up dimen-
sions toward linear-elastic fracture mechanics behavior (that is, toward
large-scale structural component behavior).
3. A wider range of temperature variation to include large changes in the
upper shelf and their effects, as well as including temperatures down into the
transition range to observe effects of partial and greater amounts of cleavage
behavior.
4. Exploring the effects of material changes both through heat treatment
and other material processing (such as perhaps including Charpy upper-shelf
level changes as occur for irradiation damage, etc.), as well as other types of
materials which are vastly different, such as aluminum alloys (where cleavage
is nonexistent).
264 ELASTIC-PLASTIC FRACTURE

2000 -

1000

0.06 O.tZ 0.18 0.24 0.30 0.36 0.42

S (in.)

FIG. 7—Load-displacement record for Test B-3, LSB = 14 in., unstable.

5. Testing of other specimen configurations, such as tension-type

specimens instead of bending specimens, to assure relevance of results be-
tween different types of configurations (from laboratory specimens to struc-
tural components).
Therefore, though this testing program has produced very positive results,
much is left to be explored.

Conclusions
1. Tearing instability in three-point bend tests was shown to occur under
fully plastic plane-strain conditions.
2. The tests demonstrated the systems aspects of tearing instability by
transition from stable to unstable behavior through changes in loading
system compliance.
3. The tests demonstrated the effect of local cracked section geometry on
tearing instability by transition from stable to unstable behavior through
changes in a/W (that is, remaining ligament size effect).
4. A temperature change of 100°C (within upper-shelf Charpy behavior
 PARIS ET AL ON TEARING INSTABILITY THEORY 265

range) was shown to not affect tearing instability behavior appreciably for the
material tested, ASTM-A471 (NiCrMoV) rotor steel.
5. All test behavior patterns observed tended to support the theory of "in-
stability of the tearing mode of elastic plastic crack growth" as developed in
the earlier work (footnote 2) and its approach to the phenomenon.
6. The testing program described herein was intentionally limited in scope
in order to develop rapid results and thus has left many aspects of tearing in-
stability to be explored.

Acknowledgments
The support of this testing program at Washington University's Materials
Research Laboratory by the U.S. Nuclear Regulatory Commission^NRC) is
gratefully acknowledged. The interest and encouragement of NRC person-
nel, especially the late Mr. E. K. Lynn, and W. Hazelton, and R. Gamble,
was a prime factor in this work. Moreover, the timely provision of the
material tested by the Westinghouse Research Fracture Mechanics Group
under E. T. Wessel aided in an essential way to proper test planning without
time-consuming pretesting of material. The assistance of N. Nguyen in per-
forming the test is also acknowledged with thanks. The program was also
aided by the consulting assistance of Professors J. W. Hutchinson and J. R.
Rice.
/. D. Landes,^ H. Walker,^ and G. A. Clarke^

Evaluation of Estimation Procedures

Used In J-lntegral Testing

REFERENCE: Landes, J. D., Walker, H., and Clarke, G. A., "Evaluation of Estima-
tlOB Piocedoies Used in J-Integrai Testing," Elastic-Plastic Fracture, ASTM STP 668, J.
D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and
Materials, 1979, pp. 266-287.

ABSTRACT: Estimation techniques for the calculation of/ have enabled the develop-
ment of simpler data reduction methods for multiple specimen J-integral tests and also
prompted the development of single-specimen tests. This report describes an experimen-
tal program conducted to evaluate the accuracy of these estimation techniques. Com-
parisons between the values of J as calculated by the energy rate definition and those
calculated by the estimation techniques for compact toughness, three-point bend, and
center-cracked tension specimens are made.

KEY WORDS: elastic, methods, estimates, plastic, techniques, experiments, fracture,

crack propagation

The J-integral as proposed by Rice [1]^ is becoming widely used as a

parameter to characterize the fracture toughness of metals when the amount
of plasticity in the specimen or structure excludes the use of linear elastic
fracture mechanics parameters [2,3]. With the development of the fracture
toughness methodology based on /, various methods for experimentally
determining / from a load versus load point displacement test record have
been proposed [2-8], The goal of these proposed methods is to simplify the
experimental procedure and the data analysis in a/ic fracture toughness test.
Some of these methods were formally exact representations of the / energy
line integral value and others were approximations which were exact only for
limiting cases.
The first method proposed was the energy rate interpretation of the / line
integral. This method proposed by Begley and Landes [2] is formally exact
when the material behavior conforms to a deformation plasticity description.

' Fellow engineer, associate engineer, and senior engineer, respectively, Westinghouse Electric
Corporation, Research and Development Center, Pittsburgh, Pa. 15235.
^The italic numbers in brackets refer to the list of references appended to this paper.

266

Copyright 1979 b y A S T M International www.astm.org

 LANDES ET AL ON JINTEGRAL TESTING 267

The energy rate interpretation of J is expressed by [4]

(1)
B da

where
U — energy under the load-displacement curve,
a = crack length,
V = displacement of the applied force, and
B = specimen thickness.
The experimental approach for using Eq 1 to determine / is shown
schematically in Fig. 1. Load-displacement curves were generated for iden-
tical specimens with varying crack lengths and a procedure was followed to
evaluate the change in energy at a fixed displacement with change in crack
length, Fig. 1.
This method for determining / had some major disadvantages, the main
one being that several specimens, 5 to 10, had to be tested to simply develop
the calibration of / versus displacement. These specimens were not neces-
sarily sufficient to provide any information about the fracture toughness of
the material. A major step in the development of the test method occurred

Load

6
Q
Load

V, > V . > V,

FIG. 1—Energy rate determination of I.

268 ELASTIC-PLASTIC FRACTURE

when the beginning portion of the crack growth resistance curve was used to
determine the/ic fracture toughness [9]. This development was aided by the
work of Rice et al [6] which proposed methods for estimating/ as a function
of displacement for a single specimen. This work was originally based on the
analysis of a deeply cracked bend bar where the value of J is a function of the
work done on the cracked body

2Uc

where Uc is the work done in loading due to the introduction of a crack and b
is the uncracked ligament. Additional estimation formulas were developed
for other specimen geometries [6].
These estimation formulas for determining / from a single specimen were
instrumental in the development of the current procedures for measuring/ic
where a crack growth resistance curve generated either from several speci-
mens or a single specimen is used to determine Ju at the point of the in-
itiation of crack growth. Along with the development of the test method was
the development of some controversy about the exact form and the use of these
estimation formulas. Questions concerning the details of the use of expres-
sions like the one in Eq 2 were asked such as: How deep must the crack be in
a deeply cracked bend bar? Should UT, total energy, be used rather than
Uc? Should some modification be made to account for the small tension
component in a compact toughness specimen? Over what ranges of crack
length and to what accuracy do these estimations work in an actual ex-
perimental evaluation? In an attempt to resolve some of the questions, alter-
native methods for approximating/have been suggested [7,8,10].
The work reported in this paper was undertaken to answer these questions
from an experimental basis. Three specimen types that are most frequently
used for a J^ test were evaluated: the compact toughness specimen, the
three-point bend bar, and the center-cracked tension specimen. Calibration
curves were generated for each of these specimen types over a range of crack
lengths. Specimens were tested in monotonic loading with a small radius
blunt notch so that no crack extension could occur and a deformation
plasticity description of the flow behavior could be closely approximated.
The standard measure of/ was taken as the energy rate formulation ex-
pressed by Eq 1 which is formally correct and deviates from the energy line
integral definition of / only to the degree that the material flow properties
vary from specimen to specimen. The various estimation formulations of/ as
presented by Rice et al [6] were evaluated relative to the energy rate formula-
tion. Several of the alternative methods suggested in answer to some of the
questions posed in the foregoing were also evaluated. The results cover a wide
range of specimen conditions and provide an answer from an experimental
 LANDES ET AL ON J-INTEGRAL TESTING 269

basis to the question of how accurate are these estimation procedures for
determining /ic.

Experimental Procedure
The material used in this work was a high-strength HY130 steel whose prop-
erties are given in Table 1. Three specimen types were used for the in-
vestigation: the compact toughness specimen, the three-point bend bar, and
the center-cracked tension specimen. For each specimen configuration, 10
specimens were machined to the general dimensions shown in Fig. 2 with
various crack length to width ratios. The specimens were machined with a
blunt notch of radius 1.02 mm (0.04 in.). This was done to eliminate any
crack growth during the test. In order to determine that crack extension did
not take place, the specimens were heat tinted at the conclusion of each test
andfinallybroken open in liquid nitrogen. The specimens were then checked
for stable crack growth by examining the fracture surface for oxidation. At
no time during these experiments did any stable crack growth take place.
The range of crack length to width ratios tested was a/vv = 0.4 to a/vi =
0.85 for the three-point bend and compact toughness specimens, and
2a/>v = 0.4 to 2a/w = 0.85 for the center-cracked tension specimens. Load
versus load point displacement records for each specimen were taken to
prescribed displacement values. The area was then measured (by use of a
planimeter) in increments of .51 mm (0.02 in.) of displacement. A plot of the
area under the load displacement curve versus total crack length was made at
various displacement values. Calculations of the J-integral were then made
by the more exact energy rate definition of/ \4\. Estimations of the values of/
were made by methods described in the following section.

TABLE 1—Mechanical and chemical properties ofHY130 steel used in investigation.

Mechanical Properties

0.2% yield strength 974.9 MN/m^ (141.4 ksi)

Ultimate strength 1041.1 MN/m^ (151.0 ksi)
Elongation 20.0% . . .
Reduction in area 66.5% . . .
Room temperature Charpy energy 107.4 J (79ft• lb)

Chemical Composition, weight %

C Mn P S Si Ni Cr Mo V Al

0.12 0.79 0.004 0.005 0.35 4.% 0.57 0.41 0.057 0.059
270 ELASTIC-PLASTIC FRACTURE

114 r
1.02rv 50 8
>
84
42
38.10 I r \ p v _ t
h-'--^ ^42-
-194-
389
Center Cracked Tension

-1.02r

50.8
1.6^

117 J^ „o.'^
234-
Three Point Bend

-63.5-
-50.8—

12.7 d

61.0
M, 12.7
y 14.0
1.02r -*-
3.18d

All Dimensions
Compact Toughness in mm

FIG. 2—Blunt notch toughness specimens, 22.86 mm (0.90 in.) thick.

 LANDES ET AL ON J-INTEGRAL TESTING 271

Analysis
The estimation procedures for each specimen type are presented in this
section. In the interest of clarity, the analysis is presented in three separate
subsections, covering each specimen type.

Compact Toughness Specimen

Until recently, the estimation procedure in most common use was the one
developed by Rice at al [6]

where Uc is the total energy, UT, in the specimen minus the energy, Unc, that
would normally exist in the specimen if the specimen did not have a crack. In
practice, the energy due to the no-crack situation is negligible in a compact
specimen. Therefore the term Uc in Eq 3 can be replaced by the term UT,
which can simply be calculated from the area under the load versus load
point displacement curve. The terms B and £> in Eq 3 are the specimen
thickness and the remaining ligament, respectively. Equation 3 can be
rewritten into the form most often used to estimate the value of J for bend-
type specimens

24
'"If "'
where A is the area under the load versus load point displacement curve, as
shown in Fig. 3. In the development of this equation. Rice assumed that the
specimen was in pure bending or, at least, that the contribution due to ten-
sion was negligible. However, an analysis by Merkle and Corten [7] showed
that the tensile contribution could indeed cause a significant error in the
value of/ as estimated by Eq 4. The amount of correction to Eq 4 necessary
to account for the tension component is not only a function of the crack
length to width ratio but it is also a function of the total load and displace-
ment value. The proposed J-integral estimation equation by Merkle and Cor-
ten has

^2 a(l-2a- a') ['"' J P

(5)
272 ELASTIC-PLASTIC FRACTURE

Compact
And Bend
Bar Specimens

Load Point Displacement

2A
J = " - ^ ' ' hB(w-2a.

Load Point Displacement

FIG. 3—Description of the graphical evaluation of J from load versus load point displace-
ment records.

where

a = 2V(a/A)2 + (a/b) + V2 - 2(a/b + V2) (6)

Gi is the elastic strain energy release rate per unit crack extension and Vp is
the plastic displacement value.
Merkle and Corten have shown that, for a/w > 0.5, Eq 5 can be replaced
by one that contains total displacements [7] leading to the more readily
usable form [11]

a(l -2a- a^)\2A a{l-2a-

a^)2Pv
^ ~ V(l + a 2 ) (1 +a^y ) Bb (1 + a^y
Bb
(7)
A further simplification can be made if it is assumed that the complementary
energy is much smaller in magnitude than the total energy, resulting in the
following expression

I + a\2A
/ = (8)
1 + aV Bb
A comparison of these three forms of the Merkle-Corten equations can be
seen in Table 2. An additional method of estimating /, as proposed by Mc-
 LANDES ET AL ON J-INTEGRAL TESTING 273

Cabe [12] was evaluated. This method uses the secant offset technique to
calculate an effective crack length. This effective crack length along with the
effective modulus derived from the loading line is used to calculate an elastic-
plastic strain energy release rate Gi from

G, = -5r « / . (9)
E
where k is the effective stress intensity and E the effective modulus.

Bend Bar Specimens

A number of authors [8,10] have reported that problems exist when using
Eq 4 for bend bar specimens.
Srawley [10] shows that there is a considerable difference when using the
total energy as compared with the energy due to the presence of a crack. To
further illustrate this point, the estimation Eqs 3 and 4 were compared with
the strain energy release rate Gi in the elastic regime. In order to
demonstrate this comparison, the constant 2 in Eq 3 is replaced by the
variable /3. For a one-to-one comparison between / and G in the elastic
region, /3 will be 2.0.
The values of /S for three-point bend specimens with a span to width ratio
of 4 and for crack length to width ratios between 0.6 and 0.9 are listed in
Table 3 and are also plotted in Fig. 4, along with the curves originally
presented by Srawley [10]. These results show that by using the total energy,
UT, the value of /3 varies between 2.002 and 2.035, whereas by using the
energy due to the presence of a crack, Uc, the value of /3 varies between 2.03
and 2.638 for a/w's between 0.6 and 0.9.

Center-Cracked Tension Specimens

The/versus Aa R-curve generated from center-cracked tension specimens
are, in many cases different from those generated by compact specimens and
bend bar specimens. While the Ju point appears to be exactly the same, the
slope of the / versus Aa R-curve is greater for the center-cracked tension
specimen than for the bend-type specimens. One of the initial concerns was
the possibility that the value of / for large-scale plasticity was not being
estimated correctly by the equation developed by Rice et al [6]. This method
for estimating/ used the sum of the linear elastic energy release rate, and an
estimate of the value of/ in the plastic regime. The equation for the estima-
tion of/ for center-cracked tension specimens has the form

,= (ir|«i+^d^ (10)
E B{w — 2a)
274 ELASTIC-PLASTIC FRACTURE

TABLE 2—Comparison of} estimation techniques and J as calculated by energy rate definition
for compact toughness specimens.

Inches of Deflection"

a/w 0.020 0.040 0.060 0.080 0.100 0.120 0.140

0.4 523 1595 2996

366 1351 2680
423 1562 3099
503 1791 3432
507 1736 3287

0.45 3% 1401 2680

Jl 314 1202 2407
J3 360 1376 2755 ....
JA 372 1511 2966
Js 362 1342 2647

0.5 Ji 322 1219 2375 5045

Jl 277 1060 2143 3357
Ji 313 1200 2427 3801
J4 350 1310 25% 4000
Js 308 1256 2462 3728

0.55 Jl 274 1050 2082 3216 4430

Jl 242 925 1866 2972 4091
Ji 271 1035 2089 3327 4580
/4 292 1106 2203 3457 4725
/5 296 1081 2132 3299 4523

0.6 Jl 235 892 1800 2823 3858

Jl 209 799 1637 2590 3569
Ji 231 884 1811 2865 3948
JA 245 936 1891 2959 4052
Js 247 919 1829 2879 3957

0.65 Jl 200 748 1528 2420 3302 4331 5216

Jl 177 678 1395 2210 3055 3928 4800
Ji 193 741 1524 2415 3338 4292 5245
JA 205 777 1584 2490 3416 4375 5330
Js 216 802 1604 2544 3420 4467 5669

0.7 Jl 163 615 1270 2025 2762 3571 4373

Jl 146 565 1162 1823 2555 3267 4002
Ji 157 609 1254 1967 2757 3526 4319
JA 152 585 1201 1898 2635 3360 4125
Js 166 628 1279 2022 2828 3718 4613

0.75 Jl 127 492 1023 1624 2239

Jl 117 457 933 1477 2064
Ji 126 487 994 1574 2199
JA 131 508 1030 1619 2248
Js 132 502 1009 1598 2261
 LANDES ET AL ON J-INTEGRAL TESTING 275

TABLE 2—Continued.

Inches of Deflection"

a/w 0.020 0.040 0.060 0.080 0.100 0.120 0.140

0.8 Ji 90 381 790 1228 1733 2175 2642

J2 92 353 717 1135 1590 2038 2492
/.I 97 371 754 1194 1673 2144 2622
/4 103 389 784 1231 1714 2188 2667
/s 102 383 780 1234 1731 2275 2865

0.85 Ji 57 278 569 840 1250 1549 1810

J? 66 252 505 828 1127 1481 1814
7.1 67 254 509 834 1135 1491 1822
JA 69 262 524 860 1170 1538 1884
Js 70 266 542 871 1271 1604 1996

1 2A _/l+a\24
/i —— / 2 -
B da Bb '~\l+ay Bb

FIR
/ 4 == Gi +
2 (1 + a)' V
b(l + a^ -' J 0 P/BdVp+Y"
2 (1 - 2a
(1 +
-«2) [
©
Js
E

"1 in. = 25.4 mm.

where A is the area between the load versus load displacement curve and the
secant offset line to the displacement of interest. This area is shown in Fig. 3.

Results
The load versus load point displacement curves generated for the compact
toughness, bend bar, and center-cracked tension specimens are shown in
Figs. 5 through 7, respectively. After calculating the area under each curve at
various displacement values, the energy values for each area are plotted
against its corresponding displacement value. The value of J as defined by
the energy rate definition is then calculated by the technique as shown in Fig.
1. The/versus displacement curves for the compact toughness specimens are
shown as solid lines in Fig. 8 at various a/w ratios. The estimated values of/
determined by Eqs 4 and 8 are also shown in Fig. 8. It can be seen that the
estimated values from Eq 8 are in extremely good agreement with the more
exact definition of/, whereas Eq 4 considerably underestimates the value of/
byEql.
276 ELASTIC-PLASTIC FRACTURE

TABLE 3—Values of 0 for three-point bend specimen J estima-

tion equation, using total energy. Vj, and energy due to presence
of crack, Uc, only.

a/w j3 (Using f/c) /3 (Using i/r)

0.6 2.638 2.035

0.7 2.315 2.019
0.8 2.129 2.008
0.9 2.030 2.002

) (For Displacement "Due to Crack")

'Energy Due to Crack

II

fi (For Total Displacement)

• Total Energy

.6 .8 1.0
a/w

FIG. 4—Values of the nondimensional coefficient 0 as used in the form J — ^U/Bb within
the elastic range for a three-point bend specimen with S/W = 4. (The solid lines are from
RefXO and the points calculated from elastic compliances.)

The initial7estimation equation by Rice et al [6], the Merkle-Corten equa-

tion [7\, the proposed variation on the Merkle-Corten equation, Eq 8, and
the secant offset method by McCabe are all compared in Fig. 9 in a non-
dimensional form with the value of J as calculated by the energy rate def-
inition. It can be seen that Eq 8 most closely approximates the value of J
derived by Eq 1. It is quite fortuitous that Eq 8 is the simplest of the methods
used to account for the tension component in the area estimation procedures.
The values of / by the various estimation methods are compiled and
presented in Table 2 along with / as calculated by Eq 1.
As discussed earlier, the total energy, UT, appears to be more appropriate
for the estimation of/ for three-point bend specimens. A comparison of the
estimated value of/ by Eq 4 and the energy rate definition of/ is shown in
Fig. 10 with the more exact value of/ being represented by the solid lines. It
can be seen that the estimated value of/ is extremely close to the value of/ as
 LANDES ET AL ON J-INTEGRAL TESTING 277

Displacement, mm
0 1.0 2.0 3.0 4.0 5.0
1 1 1 1 1

HY-130 Steel
180 a _ 40
297"K (75"FI
W .4
IT-CT Specimens
160 -
-
- 35
.45

140

^,.5
120 -

100 - _^.55

20
80 - i .60

-
60 -1 _ _ 65

70 -
40 -///
.75
- 5
20 80
. .85
0 1 1 1 1 1
0.04 0.08 0.120 0.160 0.200
Displacement, in

FIG. 5—Load versus load point displacement curves for compact specimens at various a/W
ratios.

calculated by Eq 1 throughout the range oia/w ratios between 0.4 and 0.8.
Slight differences should be expected due to the possibility of inaccuracies in
curve fitting the energy versus crack length values used to calculate the value
of dU/da. The values of/ by the estimation Eq 4 and the energy rate defi-
nition of/ are presented in Table 4.
The comparison between the value of / by the energy rate definition for
center-cracked tension specimens and the estimation Eq 10 is shown in Fig.
11. It can be seen that at larger values of 2a/w the estimated value of/ is
somewhat lower than the value of/ as calculated by Eq 1. This may well be
due to the inaccuracy of curve fitting the values of the energy between the
load displacement curve and the offset secant curve, versus crack length at
larger values of crack length.
The slope of the energy versus crack length at large crack lengths is much
higher than at smaller crack lengths. Even small inaccuracies in curve fitting
278 ELASTIC-PLASTIC FRACTURE

Displacement, mm

3.0 il.O

HY-130 Steel
297''K(75°FI
3 Point-Bend Bars
S/W = 4

- 12

- 4

0 O.M 0.08 0.120 0.160 0.200 0.240

Displacement, in

FIG. 6—Load versus load point displacement curves for three-point bend specimens with
S/W = 4 at various a/W ratios.
 LANDES ET AL ON JINTEGRAL TESTING 279

Displacement, mm
0.25 0.5 0.75 1.0 1.25 1.5

HY-130 Steel
297°K(75°FI
Center Cracked
800
Tension Specimen

700 -

600

S 500

400 -

300

200

100

0 0.01 0.020 0.03 0.04 0.05 0.06

Displacement, In

FIG. 7—Load versus load displacement points for center-cracked tension specimens at various
2ii/yf ratios.
280 ELASTIC-PLASTIC FRACTURE

Displacement, mm

0.5 1.0 1.5 2.0 2.5 3.0 3.5

• y T T T

Symbol a/W ratio

o • .4 HY-130 Steel
1000 .5 297°K(75°F)
O • .6
IT-CT Specimen
0 • .7
Si.Xiu

.06 .08 .10

Displacement, in

FIG. 8—I versus displacement curves showing the comparison of the energy rate definition
of J (solid lines) and the estimated values of i by Eq 4 (solid points) and Eq 8 (open points)
for compact toughness specimens.
 LANDES ET AL ON J-INTEGRAL TESTING 281

Symbol Equation No. Method

4 Area Estimation
5 Merkle Corten
8 Modified Merkle-Coilen
9 Secant Offset

,-fbcMjj Displacement =
•••^^•-* 2.5 mm (0.10 in)

HY -130 Steel
0.8
297''K iTi'f)
IT - CT Specimen

II

. Displacement =
1.5 mm (0.60 in)
0.

0.

. Displacement =
0.5 mm (0.02 in)

.45 .50 .55 .60 .65 .70 .75

a/W Ratios

FIG. 9—Showing the comparison of the various estimates of J with the energy rate definition
of J at given displacement values.
282 ELASTIC-PLASTIC FRACTURE

Displacement, mm
1.0 2.0 3.0 4.0 5.0
—1—
-r
Symbol a/W ratios
.4
.5
.6

7000
120O
HY-130 Steel

MOO
1000 3 point Bend Bars
s/w=4
5000

800

4000

3000

2000

-1000

0.80 0.120 0.160 0.200

Displacement, in

FIG. 10—J versa* displacement curves showing comparison of the energy rate definition of
J (solid lines) and the estimated value of J from Eq 4. for three-point bend bar specimens.

at these higher values of crack lengths can have substantial effect on the
value of dU/da. With this in mind, the comparison of the estimated value of
/ with the energy rate definition ofJ appears to be quite good. The estimated
values of/ can be seen in tabular form in Table 5 along with the more exact
value of/ as calculated by Eq 1.

Discussion
The results from these studies answer most of the questions posed relative
to / estimation methods and illustrate, to within experimental limitations,
how well these approximations work. Each specimen will be discussed
separately.
The compact toughness specimen involved the greatest number of methods
for developing estimation of /. From these results it is clear that a simple
 LANDES ET AL ON J-INTEGRAL TESTING 283

s .
.
. t^ro
. r4o
h-r-
t^5
^ Q
^o««
f^lQO q O O
t-^M ' H ^
i/>»o ^ " ^
00»H ^ O N
r o i ^ i/5oo
r o f o <Nr»i

. . OOQ 0»0 r^^ ^ O fOCO (NfO

. . ^ ^ (NO ^fN ^ ^ 3-S ^U^
*2 ^**^ *Q^ « o o CT>r^ < N - ^
3
vO io»/> ^ ^ r o f O M f O <N<N

I
s .
,
.
, OOi-i fSOO
' *-iuo c o v o
, -^rt ^i/)
rOi/> f O f ^ ^ ' O
l o r ^ ro»/> ( N ^
c^c^ *Nro t/ir^
Oi/>
foa^
ooo
^ lOirt ^ • ^ r o r o r o m fN<N »Mrsi

r^oo 0000 ^ooo r-ro coo ^ P O N ^OQO

^in -^^ f^Q i/5fO '-•rs o<N rno
«, ^1-H
i/)io
r^io
'^'^
QOO
^ m
m m
m m
r^oo
r^rs
^ m i/>r-.
<Nr4 rt*-*

I ?i §2
r^ >« ... -_
Q ^O <<T 00
m ^ 00
- _ _ <NrN \ooo O ' ^
^ ^ ^ •«*• m i (N<N r4<N ^ ^ T H ^

S§
s 2g; 2^^ ^1/) m * ' ! CT^i-H '^r-^ oor^

^ -^ m m §111 »do
-5^
(N(N
<Nt^ T r r - • ^ r *
(NO r-f-
(N(N—<^^»H
(N-^
rnr-.
i^o
^

•* ro vO -H t-~ o ON CT\ • » r ~ 00 a^ ^p m a^ m
- ^ fO I— irt ro 00 <N 0 0 (N ^ ^ r-
n <N 0 0 <0 <S O (N (N
<*5 <*) O o fN <N KiS <S I N
ro n

I aiii
f S ON
g (N ON <M
S!
O r o vO O
Oi ON - N «
ON lO
<N 1/)

f NO 1^ ,1^ ^OO 5 0 0^0^ 0 1 ^ T-^iO

ii if S m o Q ( N "Ot^ m t ^
'Oi/)i7>'^mm(N(N
(^•H
(N

2 S SK . _ S t^
NO
^ ON ^O r l
NO n (N o gl SI
CS <N (N <N d (N

E
g

i !? !« !2
I 0
Ul
0 d
NO
0 d d
00
d d
284 ELASTIC-PLASTIC FRACTURE

Displacement, mm

0.25 0.50 0.75 1.0 1.25 1.50

- 7000

6000

5000

4000

- 3000

2000

1000

0.020 0.03 0.04

Displacement, in

FIG. 11—J versus displacement curves showing the comparison between the energy rate
definition of} {solid lines) and the estimated value of} from Eq 10for the center-cracked tension
specimen.

bending solution as expressed by Eq 3 provides the least accurate of the ap-

proximations. To determine / more accurately, some modification must be
made to account for tension. The modification proposed by Merkle and
Corten [7] provides a better approximation; however, a simplification of the
Merkle-Corten approach in Eq 8 provides the best approximation to / . The
procedure suggested by McCabe also provides a reasonable estimation of/.
The estimation formula for the bend bar in Eq 2 works best only for very
deeply cracked specimens. For specimens cracked in the range normally
tested, 0.5 < a/v> < 0.75, the total energy of loading should be used rather
than simply the energy due to a crack. This modification has been
demonstrated to work only for a span to specimen width ratio, S/W, of 4.
The estimation formula for the center-cracked tension specimen is also ac-
curate. Any difference that occurs in the development of the crack growth
resistance curve between this specimen and bend-type specimens comes from
sources other than this formula.
 LANDES ET AL ON J-INTEGRAL TESTING 285

g a* op So <N
11 ii
\0 CX) • ^ a\ CD
s
o
ills 00 ^ I a^ ro 00

1 s ss
)*1 f5 ° (*!
O <N
•^ ro
<N CM rt CO
ON ^
TT - ^
o -^
• ^ - ^

I o
i*H ^ o
13- r---(
opo
000
•^'-'
^t^
mr^
^ ^
^ot^
i-i,-(
1^1
v^{
o

i O f n O**? o n
r^<N ^ < N ' S o
n - w -<N(N
(N(N ^ f N irS(N
T)^
t-<f^
^fN
^ lO
fS<N
QOO^ l O - ^ C * ^
r*5i/> lOfO 0 < / )
r(N(N
^ i ^ oor--
fS<N Q O
rOfO
^ O
m
^"^
i/)0
T '«r(N
- H C<N «r> po

r-i t^ ^ -^ -rt 00 (Nt^ O r S fSi/l ^ F - (N-^ <NJ^ n*-!

g (NfN
QOlO
•Mrt
(^r-H
QOi/)
^ ^
m o
a^O
-H^
o i > oo(N r^a>
O r ^ O*-" T-IO^
(NrH (NfN (NT-^
Qoa^
(NrO
(N(N
(Nt^
Tfi-*
CNlfM
oio
S ^
(NfS
uo*^
OOS
(NfS

f o
fO'^
r^^
Or^
^ r ^ (NOO r o r o
!-*ON i ^ f s r*)oo
^ t ^ T-iOv ( N O
(^>^ 'O^tf
oi/> ^ < s
rO»-< (^<N
t ^ ^ r^io
O ^ i/)^
lOiT) ^Cr^
a^ y~i r*^^
^ ^ ^ON
00»d ' - ' 0 0
o ^H '-H —1 ^ ^ H T - < T H i - H ^ i - ( i - ( i - ( » - l t - l ^ f S ^

I OS ^ vC 00 fO a^ 00
S O 1^ lO ro <*) i-t

•s.

00
o o d
286 ELASTIC-PLASTIC FRACTURE

These results provide a sound basis for the continued use of J estimation
formulas for experimental evaluation of Ji^. However, they are only valid
when a deformation model of the plasticity behavior of the material is ap-
proximated by the test. Additional work should concentrate on developing
methods for experimentally approximating/for such cases as large amounts
of crack growth or periodic unloading during the test.

Conclosions
The following conclusions can be made from the comparisons of the
estimated values of J and the values of J as determined from Eq 1.
1. It is necessary to account for the tension component when estimating
the value of J for compact specimens.
2. The area estimation technique for compact specimens which approx-
imates the energy rate definition of J most accurately in this investigation was
a variation of the Merkle-Corten technique given by Eq 8.
3. The total energy, UT, should be used when estimating the value of/for
three-point bend specimens.
4. The value of 0 should be equal to 2 for three-point bend specimens
when the span to width ratio is 4.
5. The estimation values of/for center-cracked panel specimens appear
to closely approximate the values calculated by the energy rate definition of/.

Acknowledgments
This study was undertaken as a result of questions raised after examining
the results of the Cooperative Test Program by members of the ASTM
E24.01.09 Task Group on elastic-plastic fracture. The material used in this
study was the same material as used in the Cooperative Test Program, which
was generously supplied by the United States Steel Corp. Acknowledgment is
also made of the care taken in the testing portion of this program by P. J.
Barsotti, F. X. Gradich, and R. B. Hewlett of the Structural Behavior of
Materials Department of Westinghouse R&D Center. The work of W. H.
Pryle and Donna Gongaware, of the same department, is also appreciated
for the design and procurement of the specimens and the manuscript typing,
respectively.

References
[/] Rice, } . R., Journal of Applied Mechanics, Transactions, American Society of Mechanical
Engineers, June 1968, pp. 379-386.
[2] Begley, J. A. and Landes, J. D. in Fracture Toughness, ASTM STP 514, American Society
for Testing and Materials, 1972, pp. 1-20.
[3] Landes, J. D. and Begley, J. A. in Developments in Fracture Mechanics Test Methods
Standardization, ASTM STP 632, W. R. Brown, Jr. and J. G. Kaufman, Eds., American
Society for Testing and Materials, 1977, pp. 57-81.
 LANDES ET AL ON J-INTEGRAL TESTING 287

[4] Rice, J. R. in Fracture, Vol. 2, H. Liebowitz, Ed., Academic Press, New York, 1%8, pp.
191-311.
[5] Bucci, R. I., Paris, P. C, Landes, J. D. and Rice, I. R. in Fracture Toughness, ASTMSTP
514, American Society for Testing and Materials, 1972, pp. 40-69.
[6] Rice, I. A., Paris, P. C. and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing, ASTM STP 536, American Society for Testing and Materials, 1973,
pp. 231-245.
[7] Merkle, J. 0. and Corten, H. T., "A J Integral Analysis for the Compact Specimen, Con-
sidering Axial Force as Well as Bending Effects," ASME Paper No. 74-PVP-33, American
Society of Mechanical Engineers, 1974.
[8] Sumpter, J. D. G. and Turner, C. E. in Cracks and Fracture, ASTM STP 601, American
Society for Testing and Materials, 1976, pp. 3-18.
[9] Landes, J. D. and Begley, J. A. in Fracture Analysis. ASTM STP 560, American Society
for Testing and Materials, 1973, pp. 170-186.
[10] Srawley, J. E., IntemationalJoumal of Fracture, Vol. 12, No. 3, 1976, pp. 470-474.
[//] Embley, G. T., Knolls Atomic Power Laboratory, private communication, 1976.
[12\ McCabe, D. E. and Larides, J. D., this publication, pp. 288-305.
D. E. McCabe^ and J. D. Landes^

An Evaluation of Elastic-Plastic
Methods Applied to Crack Growth
Resistance Measurements

REFERENCE: McCabe, D. E. and Landes, J. D., "An Evalaation of Elastic-Plastic

Mettiods Applied to Cracli Growtii Resistance Measniements," Elastic-Plastic
Fracture, ASTM STP 668, I. D. Landes, J. A. Begley, and G. A. Clarke, Eds.,
American Society for Testing and Materials, 1979, pp. 288-306.

ABSTRACT: Information from tests on blunt notched specimens for J-integral

calibration by conventional / = —1/B dU/da analysis is used to evaluate the sig-
nificance of plastic zone adjustment to physical crack length in crack growth resistance,
KR, calculations. Secants are drawn to load versus displacement test records to
determine plastic zone adjusted crack lengths. Tests on three specimen geometries
[compact specimen (CS), single-edge notched bend (SENB), and center-notched
tension (CNT)] and on two materials (HY130 steel and 2024 aluminum) have shown
that this procedure develops KR values that are equivalent to J. This demonstration
opens possibilities that J can now be applied to cases where there is subcritical crack
growth such as in R-curve work, Kiscc, and possibly in creep cracking studies. Also,
this provides a simplified method for computing / experimentally on complex geom-
etries for which elastic Ki solutions are available.
Alternative J computational procedures are compared. These include / by a Ram-
berg-Osgood work-hardening law fit to load-displacement records, and J by area
approximation methods. The Ramberg-Osgood modeling appeared to work reasonably
well in tests on compact specimens but was found to be unreliable on SENB and CNT
specimens and therefore is not recommended. With no stable crack propagation, the
area approximation procedures for / determination produce reasonably accurate
estimates of 7 as might have been anticipated from past experience. Tests on the
compact specimen geometry required a Merkle-Corten correction procedure which
worked well on large crack aspect ratios, where a/w > 0.5, but tended to overcorrect
for short cracks, giving nonconservative results.

KEY WORDS; fracture (materials), J-integral, elastic, compliance, cracks, toughness,

deformation, plastic, crack propagation

The use of fracture mechanics on structural materials is presently a

reasonably well-established and practical practice, so long as the use is

'Senior engineer and fellow engineer, respectively. Structural Behavior of Materials,

Westinghouse R&D Center, Pittsburgh, Pa. 15235.

288

Copyright 1979 b y A S T M International www.astm.org

 MCCABE AND LANDES ON RESISTANCE MEASUREMENTS 289

restricted to the lower shelf or lower transition temperature range [1,2].^

Because of the difficulty in applying fracture mechanics to upper shelf
toughness evaluation, where there is extensive plasticity and slow-stable
crack growth to contend with, the problem has been handled in an interim
manner by defining fracture toughness in terms of conservative values.
Presently the toughness is defined using either Ju or crack-opening dis-
placement (COD) procedures where attention is concentrated on the onset
of slow-stable crack growth [3,4]. The conservatism contained within such
an approach is satisfying from a safety standpoint, but this more often
than not results in critical flaw size and stress level predictions that are not
indicative of the true (upper shelf) in-service performance. The true load-
carrying capability of flawed structures can be considerably underesti-
mated, depending upon geometry and the crack growth resistance capa-
bility of the materials used. Overdesign in materials selection can be as
serious an engineering problem as underdesign. Inexpensive alloys may be
ruled out for applications in which they would ordinarily be perfectly
suitable. Thus the refinement of our upper shelf instability prediction
capability can prove to be a very worthwhile objective.
The tool by which upper shelf instability predictions can be made is
available in the form of R-curve analysis. However, elastic-plastic methods
have not been developed for handling slow-stable crack growth and a
means for extending these methods is needed. In addition, instability
criteria under large-scale plasticity conditions are presently under review.
In all cases, R-curve development is accepted as an expression of funda-
mental material behavior, and modeling for instability prediction is pres-
ently subject to interpretation. One approach would be to incorporate all
elastic-plastic effects in the R-curve and then use the conventional match-
up procedure with elastic crack drive curves to predict instability [5]. A
second possibility is to consider a fresh approach through stable tear
modeling as suggested by Paris et al [6]. All considerations, however,
depend upon having a valid elastic-plastic R-curve to work with.
Several investigators have suggested that Ki calculated with a plastic
zone correction to crack size (effective crack size) produces KR values that
are equivalent to / [7,5]. This has been proposed for limited cases where
the plastic zones are relatively small in comparison to the overall size of a
surrounding elastic stress field. The experimental approach, using com-
pliance for determining this "effective" or plastic-zone corrected crack size
has been well established and demonstrated to be satisfactory for tests on
ultra-high-strength sheet materials. Here, the plastic zone is usually small
in comparison to overall crack size and any errors involved, either in
principle or in practice, tend to be minor. Therefore, it is not entirely clear
to some that the equivalency between / and KR has been adequately dem-

^The italic numbers in brackets refer to the list of references appended to this paper.
290 ELASTIC-PLASTIC FRACTURE

onstrated. The crux of the contention is that Gi or Ki become inaccurate

field parameters under any appreciable crack tip plasticity conditions and
that only / is a supportable computational approach. It is with this argu-
ment in mind that the present project was undertaken, namely, to explore
the significance of effective crack size when used in KR calculations. If
proved viable, this approach can then be justified for use under elastic-
plastic conditions where slow-stable crack growth intercedes at some inter-:
mediate point in the crack growth resistance development for the material.
The principal technique proposed in this investigation is to calculate KR
using compliance determined effective crack sizes in the linear-elastic
expressions for ^"1, and to compare these values with / obtained by con-
ventional means. For displacement levels where there is no slow-stable
crack growth, these comparisons give a one-to-one evaluation of the extent
of plasticity that can be handled by plastic zone corrected KR. The com-
parison is made for three specimen types: compact specimen (CS), single-
edge notched bend (SENB), and center notched tension (CNT). Much of
the raw data for this analysis were obtained from a companion project
entitled "Evaluation of Estimation Procedures used in J-Integral Testing"
[9], where the principal objective was to evaluate area approximation
methods for determining /. An additional study of value included in this
investigation is to compute / by other available or alternative means so that
we can compare and evaluate the reliability of these methods as well.

Compatational Methods
A number of computational methods which are in varied stages of
development and acceptance are available for calculating the J-integral.
This section is devoted to a somewhat simplified presentation of the tech-
niques tried. Most specimens were prepared with instrumentation designed
to provide the full range of data needed to employ the various compu-
tational approaches.

J-Integral by KR
Again, the principal point of interest is to test the validity of compliance-
determined effective crack length as an elastic-plastic methodology. The
procedure for effective crack size determination is outlined in Fig. 1, and
the crack size so determined is used in the K\ expression in place of the
actual crack size. The J-integral is estimated using the following expression

J = (XKR^/E

where
a = a constraint factor varying between 1 (plane stress) and 0.9 (plane
strain).
 MCCABE AND LANDES ON RESISTANCE MEASUREMENTS 291

KR = crack growth resistance using effective crack size, and

E = effective elastic modulus.
The effective elastic modulus is determined from the initial linear slope of
the test record. With initial crack size and theoretical compUance known,
the apparent elastic modulus behavior of the specimen can be determined.
This modulus will inherently contain crack tip constraint effects, thereby
justifying the use of a = 1 in conversion to / throughout the experiment.

J-Integral, Begley-Landes Method (B-L)

This experimental procedure for the determination of J-integral involves
the use of blunt notched specimens of identical geometry but with varying
initial crack sizes. The testing sequence, first outlined by Begley and
Landes [10], is used where the procedure is aimed at directly satisfying the
working definition/ = —\/B dU/da (see Fig. 2). Since this is the most
direct method for determining JR, it is regarded herein as the benchmark
of comparison for other computational procedures. Total energies up to
fixed displacement levels are determined from areas under load-displace-
ment records. This energy, normalized for material thickness, U/B, is
plotted against crack size. For a selected crack size, the family of slopes
plotted against displacement level constitutes the J-integral calibration
curve for that crack size. Calibration curves can be generated for any
chosen crack size within the range of a given family of U/B versus a curves.
This procedure is best performed using blunt notched specimens where
slow-stable crack extension is suppressed.

Elastic Compliance
Test Record
Calibration Curve

VLL ao fe
Load Line Displacement w w
Crack Aspect Ratio
KR = f ( P , aeff)
JR=KR2/E

FIG. 1—Illustration of secant technique for determining effective crack size.

292 ELASTIC-PLASTIC FRACTURE

Vw=L«'=0,50

6 Oispl.

Crack Length

FIG. 2—Calculation of] calibration curve by —l/B dV/da.

J-Integral—Area Approximation
An alternative method for determining /R is the area approximation
procedure, as suggested by Rice [11], which is more commonly used but is
perhaps subject to acceptably small errors (see Fig. 3). Here all necessary
information is obtained from the test record of one specimen. The approxi-
mation is to determine the energy input into the specimen, U, from the
area under the load-displacement record. This can be converted to JR by

JR = 2/Bb Mde = 2U/Bb

where
M = bending moment,
$ = bend angle,
B = specimen thickness, and
b = original uncracked ligament size.
 M C C A B E A N D LANDES O N RESISTANCE M E A S U R E M E N T S 293

_ 2 f^
— • •* " B b JQ Mde =2A/Bb
A = u i + Ug
B= Material Thickness
b = w - a - Uncrarted Ligament
Merl<le-Corten: Compact Specimens
J = J (elastic) + J (plastic)
2
J (elastic) = K o ' ^

J (plastic)=/,rj i f ' Pd(Ap)-./, rjS.'vdP

Where: u , = J AotiP
2 •'o P

Pd(A„)

- a(l-2a-n^>

Load Line Displ., \ i

1/2
a=[,^,l2,f).2] (f .1)
FIG. 3—J-integral determination by area methods.

The development is based upon the assumption that the specimen is in

pure bending and therefore can be regarded as being a good approxima-
tion only for deeply notched bend specimens. This same expression has
been used on compact specimens, but an unaccounted-for superimposed
tension component tends to make the pure bend model conservative,
especially so for crack aspect ratios, a/w, less than 0.5. Recently, Merkle
and Corten [12] have suggested a modified computational procedure for
the compact specimen (CS), which is also outlined in Fig. 3. Here energy
distributions are broken down into elastic and plastic contributions. The
plastic contribution is shown as a function of crack aspect ratio, a/w,
which accounts for superimposed tension, and typically JR is increased on
the order of 10 to 20 percent over values predicted by the pure bend
expression. This development can be simplified considerably from that
shown by combining terms such that / (elastic) is approximated from
other measurements (P and Vu) and with the Ti and r2 terms reduced to
tabular form.
For center-cracked panels. Rice et al [11] have set up an area under the
curve method of approximately the same form as the Merkle-Corten treat-
ment for compact specimens. Elastic contribution to / is treated inde-
pendently, and the / (plastic) area under the load-displacement record
corresponds to that between the load-displacement trace and a secant
drawn to the test record. To use / (plastic) = lA/Bb, where A is the area
294 ELASTIC-PLASTIC FRACTURE

just described and B the material thickness, the b dimension corresponds

to the sum of the two ligaments on either side of the central crack.

J-Integral—Ramberg-Osgood {R-O)
Another elastic-plastic approach available is to estimate the J-integral by
characterizing load-displacement records using the Ramberg-Osgood work-
hardening law

V = iV/P)oP + KiV/PJo-P"

where
V = load-line displacement,
P = applied load,
{V/P)o = initial load-line linear elastic compliance slope,
K = work-hardening coefficient, and
n = work-hardening exponent.
Figure 4 shows the development for calculation of J from the foregoing
expression. This development is basically similar to the more rigorous
Begley-Landes approach in that an attempt is made to define / in terms of
— l/B dU/da. The hazard present in the R-O development, however, is
that the work-hardening constants K and n are determined from one test
record and these may not necessarily correctly define the trend in load-
displacement records for changing initial crack size.

Uc = i VdP (Complementary Energy)

V = (V/P)o P+ K ( V / P ) " p "

da
p oa 0 n+1 oa 0

jR=p2/2f3(V/P)^ ^l + 2 l < i L , v / P ) " - l p " - l

n +1 0
G =P^/2 A IV/P)„=K ^/E
Od 0 0

JR = V E l+2Jin ,V/P,"-lp"-l
n+1 0

Load Line Displ. V

FIG. 4—Computation of } from Ramberg-Osgood work-hardening expression.

 MCCABE AND LANDES ON RESISTANCE MEASUREMENTS 295

It may be recognized that the expression for JR is essentially the same as

that developed by Eftis and Liebowitz for plasticity corrected G where the
term in brackets corresponds to C [13]. In experiments by the subject
investigators, on specimens having different initial flaw sizes, values of n
were shown to be variable, but this was not considered to be objectionable.
From the / viewpoint, this represents a significant oversight.

Experimental Program
The specimens used were blunt notched so that slow stable crack growth
was suppressed and all nonlinear effects observed on test records were due
to developing plasticity. J-integral calibration curves were developed ac-
cording to the Begley-Landes (B-L) procedure on three specimen geom-
etries of 23-mm-thick (0.9 in.) HY130 steel, [IT compact, 5.08-cm-wide
(2 in.) CNT, and 5.08-cm-wide (2 in.) by 20.32-cm-long (8 in.) three-point
bend specimens]. Variability of material was provided by an aluminum
alloy, 2024-T3 of 6.3-mm (0.25 in.) thickness, tested in the CS con-
figuration. Specimen dimensions are reported in Table 1. The crack
aspect ratios denoted in the next to last column in Table 1 are for test
records that were analyzed to compare the various / procedures. A signifi-
cantly larger population of specimens was used to develop the benchmark
values of/ by the B-L method [10].

Results and Discussion

The results of/ calculations on the blunt notched IT compact specimens
of HY130 and 2T compact specimens of 2024 A1-T3 are shown in Figs. 5
through 9. Five computational methods for determining / are compared.
The lines designated real / are best-fit curves through the open-square data
points which represent the B-L method and are regarded here as the
benchmark values. The lower fit curves represent elastic computation for
G (or elastic J) which is uncorrected for plasticity. Values of nominal stress
at the crack tip, corresponding to the various levels of displacement where
/-values were calculated, are shown in parentheses. Limit load correspond-
ing to a nominal crack tip stress level of 1.62 times material ultimate
strength [14] is indicated by vertical arrows.
The general observation that can be made here is that all methods
predicted / with reasonable accuracy, even with extreme plasticity ad-
justments ranging up to 100 percent of elastic values.
The / calculations derived from the Ramberg-Osgood work hardening
law tended toward slightly higher values of/. In order to fit load-displace-
ment records for different initial crack aspect ratios, the work-hardening
constants had to be varied appreciably as Table 2 shows. This suggests that
K and n have no significance with regard to the material flow properties.
296 ELASTIC-PLASTIC FRACTURE

1
I o o o o

•s
z

d § •&

I i/f d o d
o - ,r -T a, "
•<t o '-' • * 8§
d d CU so .5 e
o
U
S2

fs (s (s > 2
I
•c

i
u

II
all V5

u
l-l

0^ OvCT^<S il
i dddd •g ..

11 e
p

n II 3 II II

1
II H Z -C -S J<
un Z U tS - ( -H

XSS(
 ^ylCCAB'= AND LANDES ON RESISTANCE MEASUREMENTS 297

HY130
Compact Specimen IT

3000

fi 2000

1000 V Ramberg-Osgood
0 J by KR, Compliance

• J b y - i f^lReal)
Boa
* Jby2A/Bb
^ J by Merlile Corten
'(1291, M-C Corn = 1.35
'"nom'
.02 .04 .06 .08 .10
Load Line Displ., V ^ L " inches

FIG. 5—J-integral calibration curve for blunt-notched IT compact specimen: a/w = 0.4.

and, because of this, it is suggested that the R-O expression has no funda-
mental significance in the context of being a viable index of fracture
toughness. The expression appears only to be a convenient method for
curve fitting test records.
Values of/predicted from compliance-determined KR, according to Fig.
1, are shown as open-circle data points. Again, the elastic moduli, E, used
in conversion to / by KR^/E, are the "effective" values indicated by the
initial elastic slope of the test records. This is done because all compu-
tational procedures for / used herein are dependent upon initial slope, and
the use of an effective modulus is the best way to compare the compu-
tational methods on an equal basis.
These calculations of / from compliance-adjusted KR tended to be the
most consistent in comparison with the benchmark / curve. This not only
tends to support the suggestion that plastic zone correction to K\ is equiva-
lent to the computation of/, for small plastic zones embedded in dominant
elastic stress fields, but the present data have carried the suggestion well
beyond this limitation into large-scale plasticity. Therefore, these claims
now appear to be supportable at extensive plastic strain levels on the basis
of experimental evidence.
298 ELASTIC-PLASTIC FRACTURE

3000
HY130 Blunf
Compact Specimen - IT Real J
igh = 0.5
B = 0.9"

V Ramberg-Osgood
2000 o J by KR, Compliance

£ n j b y - i ~ (Real)
5 AJby2A/Bb
A J by Merkle-Corten
M-C Corr. s 1.20

1000

.02 .03 .04 .05 .06

Load Line Displ., V ^ L - inches

FIG. 6—J-integral calibration curve for blunt-notched 1T compact specimen; a/w = 0.5.

3000
HY130 Blunt
Compact Specimen IT
ao/w = 0.6
B = 0.9"

^ Ramberg-Osgood
o J by K p , Compliance
2000
° J b y - - 5 1 ^ (Real)
D Oa
A Jby2A/Bb
^ J by IVIerkle- Corten
M-C Corr. s 1- W

1000

.02 .03 .04 .05 .06

Load Line Displ. VLL-incties

FIG. 7—J-integral calibration curvefor blunt-notched IT compact specimen; a/w = 0.6.

 MCCABE AND LANDES ON RESISTANCE MEASUREMENTS 299

2024-T3
0
Compact Specimen 2T
ao/w = 0.451
B=0.25"

V Ramberq - Osgood
1500 o J by Kp, Compliance
Real J, /
o Jby-||iL,Real.
* J by 2A/Bb (Rice)
A J by Merkle-Corten
M-CCorr.= 1.28
c
s/
£ 1000 a A
c

~^
v
/ A'^
a/ ^ /'^nom'ksi
/^3.7)
'/
§; V(5i.o)
/M47.8)
500
a/(43.7)

§^38.5)

^(32)

•^—r 1 i 1 1 1 1— 1 1 J 1 1 1 i

0 .01 .02 .03 .04 ,05.06 .07.08 .09 .10 .11 .12.13
Load Line Displacement, VLL-inches

FIG. S—J-integral calibration curve for blunt-notched 2T compact specimen; a/w = 0.451.

The determination of / by area approximation methods was first calcu-

lated in terms of the Rice approximation [11], denoted by the closed
triangles. Although the Merkle-Corten method [12] is handled as a sepa-
rate and distinct calculation, according to Fig. 3, it can also be treated as a
correction procedure to the Rice pure bend approximation method. Here
we find that correction magnitudes tended to be constant over a range of
displacement levels for a given initial crack size. The magnitude of the
correction is dominated by initial crack aspect ratio, and Figs. 5-7 show
the correction to be nominally 1.14 for ao/w = 0.6, 1.20 for ao/w = 0.5,
and 1.35 for Oo/w = 0.4. From Figs. 5-9, it is concluded, therefore, that
the Rice approximation procedure will always yield conservative estimates
of / in compact specimens and that the M-C method improves the / es-
timate to compare favorably with real /. It is noted, however, that for short
crack aspect ratios, the M-C correction tends to overcorrect ai^d J estimates
will be on the nonconservative side.
Figures 10 and 11 compare / determinations in the center notched
tension specimens, CNT, listed in Table 1, for initial crack aspect ratios.
300 ELASTIC-PLASTIC FRACTURE

1500 r

1000

.04 .06 .08 .10

Load Line Displ, V|_L-inches

FIG. ')^-integral calibration curve for blunt-notched 2T compact specimen; a/w = 0.589.

2 Oo/yv, of 0.40 and 0.45. Again real / is represented by the curve which is
a best fit to the B-L method, open-square data points. The lower curves are
again the elastic values of G, and in these cases are almost an order of
magnitude less than real / . Levels of net section stress are shown and theo-
retical limit load levels are indicated by vertical arrows.
The J-integral derived from the Ramberg-Osgood fit did not work satis-
factorily in the CNT cases. The load-displacement records were almost
elastic-perfectly plastic in nature and, because of this, n-values were of the
order of 30 and K was of the order of 10^. These values are highly un-
realistic for the modeling of plasticity effects, and this evidently proved to
be the principal cause of the breakdown of/ prediction by R-O.
A good comparison between JR from KR and real / is shown for stress
levels well beyond limit load. This comparison tended to break down at

TABLE 2—Ramberg-Osgood work-hardening constants.

Material flo/w

HY130 0.4 108.4 2.866

HY130 0.5 187.9 X lO"* 5.751
HY130 0.6 805.4 X 10^ 6.376
2024-T3 0.451 13L8 3.075
2024-T3 0.589 260.6 X 10^ 5.019
 MCCABE AND LANDES ON RESISTANCE MEASUREMENTS 301

HY130
Center Notched Panel o
4500 2ao/w = 0.4 A

B = 0.9"
4000
V Ramberg - Osgood
/ i
° J Ijy K R , Coinpliance
3500 - a j b y - l i ^ , R e a l )
/Real J

3000
CNJ

c r
T 2500
• J t>y Kp, & Isida j .^
c V
V
2000 0 1 V
o/ V

1500 f V

Limit I
1^ '
1000 K.^/E
Load m
V V
500 / > ^ ( 1 5 5 ) O 5 7 ) ( 1 6 0 ) (Opg,)

^^r —1 1 1 1 1 1
0 .01 .02 .03 .04 .05
Displ. on 4.25" Span, 2V-incties

FIG. lO^J-integral calibration curve for blunt-notched center-cracked panels; 5.08 cm

(2 in.) wide, 2a/w = 0.4.

large displacements, however. Examination of the data output for these

calculations and for specimens with other initial crack aspect ratios, not
shown, indicated that this deviation developed when 2a (effective) became
greater than 80 percent of the total cross-sectional area. Further, it was
noted that the rapid increase in KR was apparently not aggravated by
compliance-indicated crack size nor by gross stress level. The sharp devia-
tion appears to be due to the secant expression, (sec ra/ynY''-, adjustment
to K\ for finite specimen width. It was interesting to note that if the poly-
nomial approximation to the Isidas' expression for the CNT configuration
is used (not supposed to be valid beyond la/w = 0.7), the fit to real / is
conservative for crack aspect ratios greater than 0.8. These data are in-
dicated by closed circles in Figs. 10 and 11. The modified area approxi-
mation method suggested by Rice [i/] for the CNT configuration, indicated
by open triangles, gave satisfactory values at all levels of displacement,
accurate to within 15 percent of real / \10\. This indicates that even at
displacements well beyond theoretical limit load, comparable estimates of
the fracture resistance of the material are possible using / . The calculation
of KR from a-effective, on the other hand, becomes extremely sensitive
302 ELASTIC-PLASTIC FRACTURE

HY130
Center Notched Panel
4000 2ap^,
w

3500 ^ Ramberg - Osgood

o J by K|^, Compliance

3000
• Jby-llf(Real)
A ,.. 0 2A
~ 2500 ^ J b y ^ + Bb
• J by K|; & Islda

- 2000

1500

1000

500
Limit Load

0 .01 .02 .03 .04 .05

Displ. on 4.25" Span, 2V-inches

FIG. 11—J-integral calibration curve for blunt-notched center-cracked panels; w = 5.08 cm

(2 in.) wide, 2a/w = 0.45.

o
HY130 Blunt
SENB 2"W
6000 ao/w = 0.4
B = 0.9"

5000 . V Ramberg - Osgood

o J by KR, Compliance 'Real J .

~ 4000 - °J'V-B|J"'«=I' °/ 7
^ Jby 2A/Bb

- 3000
V y

2000
.Ko/E
(287) '"nom'
1000

_ — O * ^ 1. ^ J , . . 1 i I 1 1 ' 1 1 1 1

0 .01 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 .07 .08.09 .10 .11.12 .13

Crosshead Displacement, A-inches

FIG. 12—J-integral calibration curve for blunt-notched SENB specimen; w = 5.08 cm

(2 m.), a/w = 0.4.
 MCCABE AND LANDES ON RESISTANCE MEASUREMENTS 303

~ o

HY130 Blunt
4000 - SENB 2" W
ao/w = 0.5
B = 0.9
3500 Real j /

3000 -

< 2500
.a

-
•- 2000
A
1500
/A
^K^/E

1000
V Ramberg Osgood
o J by Kp, Compliance
500 o J b y - i 7 ^ (Real)

r—^^ ' ' ' -J 1—I-

0 .01 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 7 . 0 8 . 0 9 . 1 0 . 1 1 . 1 2 .13
Crosshead Displacement, A-Inches

FIG. 13—J-integral calibration curve for blunt-notched SENB specimen; w = 5.08 cm

(2in.), i/v) = 0.5.

under such conditions and therefore tends to be unreliable, especially when

crack aspect ratios are greater than 0.8.
Figures 12 and 13 compare / calibration developed for blunt notched
SENB specimens. Two example crack aspect ratios of 0.4 and 0.5 are
shown. In these cases there was some difficulty in developing good Ram-
berg-Osgood fits to the test records, and the deviation in predicted /
perhaps reflects this problem. Again, J calculated from plastic zone ad-
justed KR values corresponds well with real / over the full range of dis-
placements. Since these specimens have negligible tensile component, the
Rice pure bend area approximation method compared quite favorably with
real / .

Sammaiy of Computational Methods

Tests on blunt notched specimens for three specimen geometries and for
two materials were helpful in exploring the strengths and weaknesses of
alternative J-integral computational methods. The principal categories are
(1) the Ramberg-Osgood modeling with work-hardening constants from
which / can be estimated, (2) area approximation methods, and (3) /
obtained from compliance-corrected KR values.
304 ELASTIC-PLASTIC FRACTURE

The Ramberg-Osgood work-hardening law can be used to fit load-

displacement records and the determination of / using K and n works
reasonably well so long as the test record does not approach elastic-perfectly
plastic behavior as had developed in the present CNT tests. However, each
variation in initial crack size required the development of unique work-
hardening constants. Because of this, there can be no solid rationale
developed to justify the use of this approach. In order to make the R-O
method work more effectively, it would be necessary to find the best fit of K
and n for a family of test curves generated from specimens of varied initial
crack sizes. For all the difficulty that this would involve, it would be more
expedient to determine J directly using the Begley-Landes procedure.
Area approximation methods of computing / proved to develop good
estimates of real / . In the case of three-point bend and centrally notched
specimens, the expressions suggested by Rice et al [11] worked satis-
factorily. For compact specimens, the Rice expression, developed for pure
bending, was substantially improved through tensile component adjust-
ments suggested by Merkle and Corten [12]. These M-C corrections tended
to be fixed over a range of displacements for a given initial crack aspect
ratio. In the present work, the adjustment varied between 35 and 14
percent for initial crack aspect ratios, a<,/w, varying from 0.4 to 0.6,
respectively. It was generally observed here that the best comparisons to
real / were obtained with the larger initial crack aspect ratios. For short
a/w, the corrected values tended to overestimate / .
The compliance technique of drawing secants to test records for pre-
dicting effective crack size, and the substitution of these values in elastic
stress intensity expressions for crack length, develops KR values that are
essentially equivalent to / . The procedure was tested on two materials and
on three specimen geometries. The comparison to real / was generally the
best for all computational procedures tried. The only exception to this was
for center-notched panels where a-effective was extended to more than 80
percent of the section width and applied stress was well beyond calculated
limit load. The compliance concept was tested over a considerable varia-
bility in test record shape, a comparison of which is shown in Fig. 14. The
point at which the CNT values of/ from KR started to deviate significantly
from real / is indicated by the vertical arrow.

Conclusioiis
1. Tests on blunt notched compact specimens, center-notched panels, and
single-edge notched bend specimens were used to evaluate KR (calculated
from compliance-indicated crack size) as an indicator of elastic-plastic
toughness. Comparisons were made between these results and / determined
directly using the Begley-Landes procedure. It was demonstrated that there
 MCCABE AND LANDES ON RESISTANCE MEASUREMENTS 305

SENB

HY130
Load Displacement Records
CS -VI = 2" ao/w = 0.4
SENB- w = 2 " ao/w = 0.4
CNT - w = 2 " 2ao/w = 0.4

.01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12
Displacement-inches

FIG. 14—Load-displacement records for three specimen types.

is equivalency between KR and J and that the equivalence is retained up to

theoretical limit load on all specimen geometries. Although this demonstra-
tion was made in the absence of subcritical crack growth, its main utility
would be to handle cases where subcritical crack growth develops such as
R-curves, iSTiscc, and possibly in creep studies. Also the concept could be
used to analyze for / experimentally and diminish the need for J analysis of
geometries where / solutions do not presently exist.
2. Area approximation methods for determining / were tested and were
confirmed as being reasonably accurate estimates of real / . For compact
specimens, the Merkle-Corten computational procedure was applied and
was found to yield better estimates of real / than the pure bend model. For
initial crack aspect ratios, a^/w less than 0.5, the M-C method tended to
slightly overcorrect, resulting in nonconservative estimates of/.
3. The Ramberg-Osgood work-hardening expression was applied to the
calculation of / and was found to be not completely reliable as a compu-
tational procedure. The technique appeared to work reasonably well on
compact specimens, but ran into difficulty in CNT and SENB tests. The
probable reason for this is that the R-O work-hardening constants, K and
n, in this type of application are values of convenience, lacking funda-
mental significance with regard to material properties.
306 ELASTIC-PLASTIC FRACTURE

References

[1] Barsom, J. M. and Rolfe, S. T., Journal of Engineering Fracture Mechanics, Vol. 2,
1971, p. 341.
[21 Shoemaker, A. K. and Rolfe, S. T., Engineering Fracture Mechanics, Vol. 2, 1971,
pp. 319-339.
[5/ J. D. Landes and J. A. Begley in Fracture Toughness, ASTM STP 514, American Society
for Testing and Materials, 1972, pp. 24-39.
14] British Standards D.D. 19, "Methods for Crack Opening Displacement (COD) Testing,"
1972.
[5] Fracture Toughness Evaluation by R-Curve Methods. ASTM STP 527, American Society
for Testing and Materials, 1973.
[6] Paris, P. C, Tada, H., Zahoor, A., Ernst, H., this publication, pp. 5-36.
[7] Irwin, G. R. and Paris, P. C , "Elastic-Plastic Crack Tip Characterization in Relation to
R-Curves," Plenary Paper for ICF-4, Fourth International Conference on Fracture,
Waterloo, Ont., Canada, June 1977.
IS] Turner, C. E. and Sumpter, J. D. G., IntemationalJoumal of Fracture Mechanics, Vol.
12, No. 6, Dec. 1976.
[9] Landes, J. D., Walker, H., and Clarke, G. A., this publication, pp. 266-287.
110] Begley, J. A. and Landes, J. D. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 1-20.
[77] Rice, J. R., Paris, P. C, and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing, ASTM STP 536, American Society for Testing and Materials, 1973,
pp. 231-245.
[12] Merkle, J. G. and Corten, H. T., Transactions, American Society of Mechanical
Engineers, Journal of Pressure Vessel Technology, Vol. %, No. 4, Nov. 1974, pp.
286-292.
[13] Liebowitz, H. and Eftis, J., Engineering Fracture Mechanics, Vol. 3, No. 3, Oct. 1971,
p. 267.
[14] Newman, J. C. Jr. in Properties Related to Fracture Toughness, ASTM STP 605,
American Society for Testing and Materials, 1976, pp. 104-123.
M. G. Dawes'

Elastic-Plastic Fracture Toughness

Based on the COD and J-Contour
Integral Concepts

REFERENCE: Dawes, M. G., "Elastic-Plastic Fiactoie Tonglmess Based on the COD

and /-Contoar Inleipral Concepts," Elastic-Plastic Fracture, ASTM STP 668, J. D.
Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and
Materials, 1979, pp. 307-333.

ABSTRACT: The paper reviews the definition, fracture characterizing roles, and
measurement of critical COD and /-values. It is proposed that COD should be de-
fined as the opening displacement at the original crack tip position. This definition
avoids much of the ambiguity of previous definitions based on the crack tip profile
and the elastic-plastic interface. Attention is drawn to a fundamental problem which
limits the general application of the /-contour integral concept to elastic-plastic de-
scriptions of the crack tip environment when cracks occur in overmatching yield
strength weld regions. A comparison of recent three-point single-edge notch bend
(SENS) testing techniques, based on the standard instrumentation used in A^ic tests,
shows there is a close mathematical link between the estimated values of COD and / .
Experimental data, obtained over a wide range of temperatures, are used to demon-
strate how the critical values of COD and J for unstable fracture are affected by varia-
tions in specimen geometry. Also, it is shown that measurements of/ic may lead to
overestimates of A'lc in materials having yield strengths less than approximately 700
N/mm^.

KEY WORDS: mechanical properties, fracture tests, crack initiation, toughness,

crack opening displacement, /-integral, elastic-plastic cracking (fracturing), fracture
properties, structural steels, crack propagation

Nomenclature
a Half length of through-thickness crack, or depth of surface crack
B Section and specimen thickness
dui Work term for/
£" £" for plane stress or £•/(! — p^) for plane strain
e Strain
«(, Strain tensor
' Principal research engineer, The Welding Institute, Cambridge, England.

307

Copyright 1979 b y A S T M International www.astm.org

308 ELASTIC-PLASTIC FRACTURE

G Crack extension force

J /-contour integral
K Elastic stress intensity factor
m Plastic stress intensification factor
P Load
PL Limit load
pe Potential energy
Q Load point displacement
9P Plastic component of q
rp Rotational factor after net section yielding in bend test
Ti Work term for /
U Work done
Ue Elastic strain energy
Up Work done in plastic deformation
V Clip gage notch mouth opening in bend test
Vp Plastic component of V
w Width of single-edge notched bend specimen or half width of cen-
ter-cracked tension specimen
w Strain energy density = i^ aydey
z Knife edge thickness
r6 Path contour for/
Crack tip opening displacement
•' 1 P Constants for elastic and plastic work terms, respectively
V Poisson's ratio
ffflow (ffr + au)/2
oy Stress tensor
Ou Tensile strength
<JY Uniaxial yield strength

The requirement for a yielding fracture mechanics (YFM) approach in

design is exemplified by elastically stressed structures that contain pre-
cracked regions in which the in-plane and antiplane dimensions, com-
bined with local stresses, result in unstable quasi-brittle fracture extension,
that is, the initiation of unstable fracture after the attainment of sufficient
plasticity in the crack tip region to invalidate a description of resistance to
crack extension in terms of the plane-strain stress intensity factor, K\e.
Attempts to match the crack tip region plasticity and constraint in these
situations commonly involve post net section yielding fracture behavior in
small laboratory specimens. This paper is concerned with two parameters
which attempt to characterize fracture toughness under such conditions.
These are the critical crack tip opening displacement (COD) [1],^ and a
critical value based on the/-contour integral [2].

^ The italic numbers in brackets refer to the list of references appended to this paper.
 DAWES ON COD AND J-CONTOUR INTEGRAL CONCEPTS 309

The following sections review the definition, fracture characterizing roles,

and measurement of critical COD and 7-values. The results of recent ex-
periments are also presented. Emphasis is given to three-point single-edge
notch bend (SENB) tests and specimen geometries that may be used to
assess the fracture toughness associated with both deeply buried cracks
and shallow cracks in weld metals and weld heat-affected zones (HAZs).

Crack Tip Opening Displacements

Approximately 17 years have passed since Wells [/] suggested COD as a
parameter which might be used to describe the capacity of material near
a crack tip to deform before crack extension. Numerous encouraging inves-
tigations in different countries have followed this suggestion, and in the
United Kingdom a simple COD design curve approach [3-7] has evolved
which has found widespread application to welded structural steels [8,9].
Nevertheless, doubts are still expressed regarding both the physical occur-
rence and the definition of COD. These aspects have been the subject of an
extensive review [10], which led to the following observations.

Physical Evidence of COD

The literature shows that the profiles of deformed cracks are dominated
by such factors as the orientation of the notch relative to the microstruc-
ture of the material, the type of microstructure, the grain size, and non-
metallic inclusions. Figure 1 illustrates some typical crack tip profiles [11],
starting with an undeformed 0.15-mm-wide notch. Fig. la, and successive
opening displacements in Fig. Ib-d. The extension ahead of the original
crack tip, indicated by x in Fig. Ic, is approximately equal to the region
which is generally referred to as the 'stretch' zone. This has been correlated
with both Kic values [12] and COD values [13-15].
Figure le and / show examples of COD [16, / 7] on a scale an order of
magnitude smaller than the examples referred to in the foregoing. At this
scale the crack tip profiles are dominated by microstructural features such
as grain boundaries and nonmetallic inclusions.

Analytical Models of COD

In the Wells small-scale yielding model [/] and the Dugdale strip yield-
ing-based models [3,18,19], there is an implicit assumption that the COD
occurs at both the original crack tip and the elastic-plastic boundary.
However, nothing is implied regarding radial displacements ahead of the
crack tip. It is perhaps for the latter reason rather than from definite ex-
perimental evidence that researchers became preoccupied with the search
for a square 'nose' in the crack tip profile, ostensibly located at the crack
tip.
310 ELASTIC-PLASTIC FRACTURE

a)

Incipient tear

e)

FIG. i^Examples of COD. (a) to (d): 0.15-mm-wide sawcut y.33 [11); (e) fatigue crack
X 330 [16]; and (f) ductile tear X 158 [17].

By considering shear strains in relation to a Prandtl field, Rice and

Johnson [20] predicted that the crack tip would be deformed by radial
displacements, such that for both small-scale and large-scale yielding the
displacement tangential to the original crack tip would be approximately
double the displacement ahead of this position. A similar amount of
stretching of the deformed crack ahead of the original crack tip position is
predicted by Pelloux's [21] alternating shear model. Thus, both Rice and
Johnson's [20] and Pelloux's [21] models show approximate agreement
with the physical evidence of deformed crack tip profiles.

Numerical Estimates of COD

Because of the complexity of analytical solutions for elastic-plastic mate-
rials, finite-element analyses have been used extensively in investigations of
 DAWES ON COD AND J-CONTOUR INTEGRAL CONCEPTS 311

crack tip profiles. However, the early investigations were preoccupied with
the search for a clearly defined nose in the crack tip profile. Unfortunately,
for tension situations and work-hardening materials the computed crack
tip profiles are generally more rounded. In these circumstances, therefore,
there was a problem of defining a near-tip COD. In response to work by
Srawley, Swedlow, and Roberts [22], Wells and Burdekin [23] suggested
that the COD should be redefined as the displacement at the elastic-plastic
boundary. While this definition is reasonable for small-scale yielding con-
ditions, it is not acceptable for more extensive yielding in materials that
work harden, since in these cases the elastic-plastic boundary may move
back a significant distance along the flanks of the crack. When this hap-
pens, the COD at the elastic-plastic boundary is dependent on crack
length, and the COD is not, therefore, a one-parameter description of the
near crack tip environment. Boyle [24] has suggested a number of alterna-
tive methods of defining COD from a rounded crack profile. Although
these definitions may be justified in relation to constant strain triangle
finite-element analyses, which have a single fixed node at the crack tip,
they appear to be unnecessary for those analyses [25,26] which use special
element designs to model the crack tip deformations more closely. Accord-
ing to Rice [26], it is essential to use sophisticated crack tip elements to
obtain satisfactory modeling of such deformations.
Fortunately, from the viewpoint of crack tip deformations, the foregoing
and other limitations of present finite-element analyses [27] are less impor-
tant in bending situations. This is because the notch flanks tend to remain
straight during bending, and the intersections of the tangents to the notch
flanks and crack tip give a COD which is negligibly smaller than the COD
a small distance behind the crack tip, that is, 8 in Fig. Ic.

Definition of COD
The foregoing considerations suggest that the Mode I COD can be de-
fined as the displacement at the original crack tip position, namely, the tip
of the fatigue precrack in a COD test specimen or a natural crack in a
structure. This definition recognizes the formation of a stretch zone ahead
of the original crack tip and avoids most of the problems associated with
earlier definitions based on the deformed crack tip profile and the elastic-
plastic boundary. Also, by defining the original crack tip as the reference
position, consistency is maintained with both experimental measurements
of COD and the early analytical models [1,3,18-21].

The /-Contour Integral Concept

Since rigorous analytical stress analysis solutions for cracked bodies in
'real' (elastic-plastic) materials have proved too difficult, analysts have
312 ELASTIC-PLASTIC FRACTURE

gained an insight into real material behavior by assuming materials which

have linear and nonlinear elastic behaviors. These studies have led several
workers [2,28,29] to derive path-independent line integrals which may be
used to obtain an approximate description of the crack tip environment
prior to fracture. The first of these integrals to gain prominence in engi-
neering fracture studies was that due to Rice [21. He defined the path inde-
pendent /-contour integral as

/ = \ Ldy - r, - ^ ds) (1)

which, for both linear elastic and nonlinear elastic material, was shown to
be equal to the potential energy release rate per unit thickness, that is

dP'

Furthermore, when limited to linear elastic behavior and small-scale

yielding, Eq 2 reduces to

J=G = — (3)

The important implication of path independence is that measurements of

/ remote from the crack tip can be used to describe conditions near the
crack tip. Also, if there is a singularity of stress or strain near the crack
tip, a critical value of/ can be used as a fracture characterizing parameter.
For incremental plasticity, which is the behavior more appropriate to
real materials, path independence has not been proved analytically. How-
ever, elastic-plasticfinite-elementcomputations have shown / to be virtually
path independent except for contours very close to the crack tip [24,30,31],
which is the region wherefinite-elementanalyses are least accurate.
Unfortunately, as demonstrated by Sumpter and Turner [32], when ap-
plied to a real material, the physical meaning of / as an energy release
rate, Eq 2, is lost, since in either deformation or incremental plasticity
the energy term is no longer potentially available for propagating a crack.
This is because a proportion of the energy has been dissipated in plastic
deformation. For an elastic-plastic material, therefore, Eqs 1 and 2 must
be interpreted as

1 dU
 DAWES ON COD AND J-CONTOUR INTEGRAL CONCEPTS 313

which represents the rate with respect to crack length of elastic and plastic
work done. In YFM, therefore, the use of Eq 4 relies not on energy bal-
ance arguments, but instead on path independence and the degree to
which the value of / is related to a singularity of stress or strain in the
crack tip region.
In the absence of complete solutions for cracks in elastic-plastic mate-
rials, present assessments of / as a fracture characterizing parameter de-
pend almost entirely on experimental fracture studies.

Relationships Between COD and /

Wells's [1] original small-scale yielding estimate of COD [8] was in the
form

may mayE

It follows from Eqs 3 and 5, therefore, that

/ = mar^ (6)

This relationship has been investigated for both small- and large-scale
yielding conditions. A review [10] of available analytical, numerical, and
experimental investigations shows that the factor m is generally between
approximately 1.0 and 2.0. However, discrepancies between the theoretical
and experimental values of m raise a number of questions regarding
definitions, methods of measurement, and also the relevance of discrete
values of ar. These questions will be returned to at a number of points in
the following sections.

COD and / as Fracture Characterizing Parameters

Since values of b are directly proportional to values of / (Eq 6), it is
helpful to discuss the critical values of these parameters together. How-
ever, it is first necessary to distinguish between the critical b/J values for
the onset of crack extension by unstable fracture and the critical values
for the onset of stable crack growth. As mentioned in the introduction to
this paper, in the context of YFM, unstable fracture refers to a quasi-
brittle fracture, which in the common structural steels is usually asso-
ciated with a significant proportion of cleavage crack growth. On the other
hand, stable crack growth is usually associated with a coalescence of the
voids that form around inclusions ahead of the crack tip. Unstable frac-
ture in the ductile/brittle transition temperature range may occur either
before or after the onset of stable crack growth.
314 ELASTIC-PLASTIC FRACTURE

In subsequent discussions, the critical b/J values for unstable and stable
crack extension will be symbolized by bc/Jc and bi/Ji, respectively. Further-
more, bc/Jc will be used exclusively to indicate fracture with a rising load.

Unstable Fracture: b^andJ^

There is much evidence in the literature to show that resistance to quasi-
brittle fracture may be affected in a given material by temperature, strain
rate, and section thickness. Similarly, it is well known that resistance to
fracture decreases with increasing notch acuity. Several major investiga-
tions have demonstrated that the aforementioned factors have a similar
effect on be values [3,11,33-36] and Jc values [37,38].
Recent investigations have also indicated that be values are dependent on
loading type and the in-plane dimensions of specimens [33,34,39-42]. This
dependency is such that for tension plates containing through-thickness
cracks and geometries having a/W < «0.5, the minimum be values occur
when a is approximately one half ofB. For a <f: B, be may be an order of
magnitude higher than the minimum value, whereas for a » 5 the be
values are generally no more than approximately double the minimum
value. From the viewpoint of simple laboratory tests for be, it is significant
that the minimum be values for through-thickness cracks in tension plates
are slightly underestimated by the preferred three-point SENB geometries
for Kic testing, that is, the ASTM Test for Plane-Strain Fracture Tough-
ness of Metallic Materials (E 399-74) and British Standard Methods for
Plane Straui Fracture Toughness (Kic) Testing (BS 5447-1977). The ex-
perimental evidence also indicates that full section thickness three-point
SENB specimens will underestimate the be values for surface notches in
tension plates provided that the bend specimen crack length and width
match the crack depth and section thickness for the tension plate. Ideally,
however, the laboratory specimen should be designed to match the plastic
constraint in the structural part of interest. This aspect of 6c testing, and
also Je testing, is especially relevant to the design of weld HAZ [43] and
weld metal [44] fracture toughness specimens.

Initiation of Stable Crack Growth: b\and J,

Before fracture mechanics approaches were used to assess resistance to
brittle fracture in structural steels, little, if any, consideration was taken
of small amounts of stable crack growth before unstable fracture. In fact,
the presence of a ductile 'thumbnail' on the fracture surface at the crack
tip was often interpreted as an indication of ductility. However, the defini-
tion of the critical event as that corresponding to the first detectable ex-
tension of a crack has focused attention on 6, and /, values when these
occur before unstable fracture. While there is some evidence to show that it
 DAWES ON COD AND J-GONTOUR INTEGRAL CONCEPTS 315

can be very conservative to use values of 6, and 7, in design [9], this as-
pect remains controversial and awaits the further development of theoreti-
cal R-curve relationships between, for example, bend test values of 6i/Ji,
6c/Jc, and unstable fracture in structural situations.
The fact remains, however, that the 6, and /, values come nearest to
being 'material properties.* For example, many studies of 6, [13,15,45-51]
and /, [50,52-56] have shown that under sufficiently 'plane-strain' condi-
tions these values are independent of geometry and loading type. For
steels, at least, a sufficient degree of plane strain for a constant /, is
generally ensured when [54]

a,BaindW- a >25 — ^ (7)

CTflow

Estimates ofKicfrom Jc or Ji
In the United States much interest has been expressed regarding the
prediction of Kic from critical /-values which have been obtained from
considerably smaller specimens than those required by the ASTM E 399-74
and BS 5447. For example, when the requirements of Eq 7 are met, the
values of /, or Jc prior to stable crack growth are termed Ju values, and
are used to predict i^ic from Eq 3, for example

Ki, = iJu£'r' (8)

A similar approach to the foregoing was used by Robinson and Tetelman

[49] to estimate A'lc values from critical COD values. These approaches
seem reasonable when the same micromodes of fracture initiation can be
guaranteed in both the small specimen and the much larger valid A^ic
ASTM E 399-74 specimen. In fact, under the latter conditions, a //-value
that meets the requirements of Eq 7 may underestimate a valid Ki^ when
this is based on up to 2 percent crack growth, which is permitted by
ASTM E 399-74.
Unfortunately, for those conditions when a standard Kic test gives a
valid result following unstable fracture with no prior stable crack growth,
values of/ic from small generally yielded specimens may overestimate Ku
via Eq 8, depending on the ductile/brittle transition temperature behavior
for the materials and designs of specimen. In Fig. 2, for example, at tem-
peratures below the ASTM E 399-74 plane strain fracture initiation mode
change (that is, from cleavage to microvoid coalescence), a /ic-value for
a specimen of thickness B might considerably overestimate the Jc value for
an ASTM E 399-74 valid Ku test specimen at the same temperature. Sev-
eral investigations [10,57,58] have drawn attention to this behavior, which
is featured in the experimental results. However, before describing these
316 ELASTIC-PLASTIC FRACTURE

Temperal ure

FIG. 2—Schematic diagram of the relationships between Jc, Ji section thickness (B), and
temperature.

and other recent experimental studies, it is helpful to describe the test

methods.

Three-Point SENB Tests

The emphasis on bend tests stems from the experience that these allow
the maximum economy in weld zone material, machining, and fatigue
cracking costs, and also the maximum versatility in specimen design and
testing equipment. The versatility in specimen design is considered es-
pecially important in regard to YFM assessments of cracks in discrete re-
gions of welds [43,44], as mentioned earlier.
Unless stated otherwise, the tests refer to full section thickness (B)
square specimens, or B X 2B section specimens complying with ASTM
E 399-74 and BS 5447.
 DAWES ON COD AND J-CONTOUR INTEGRAL CONCEPTS 317

COD Tests
Since standard K^ tests involve measurements of notch mouth opening
displacement, correlations between this displacement and COD offered the
prospect of a unified test method which could be interpreted in terms of
COD or/(Tic depending on the toughness of the material being tested. Con-
sequently this approach has been pursued by many investigators and has
been justified to a large extent by experiments [11,33,39,49], finite-ele-
ment analyses [33,59], and theoretical considerations [59,60]. The latter
investigations generally support the British Standards Institution (BSI)
Draft for Development, DD19, which was the basis for the COD tests
described later. However, it should be noted that a draft BSI standard
for COD testing is in the final stages of preparation. This new document
specifies that the COD should be calculated using the following relation-
ship [10]

^=17W+ 7 , . (9)
W-

which gives similar estimates to the former more complex relationships in

DD19, due to Wells [60].
In Eq 9 the value of AT is calculated from the load at the point of interest
on the load versus notch mouth opening record.

J-Tests
The estimation procedures in these tests were based on the areas under
single load versus displacement diagrams that were either directly or in-
directly related to the work done.
A general consideration of different specimen geometries led Sumpter
[31] to suggest the following relationship which can be used for any geom-
etry for which the elastic compliance and limit load are known

J= ^' ^' 4. ^p Up _ A i + ^?p Up .jQx

B{W - a) B(W - a) E' B(W -a) ^ '
Using this relationship, the total energy under the experimental load ver-
sus point displacement record is divided into elastic and plastic components
as shown in Fig. 3. The value of r)e is constant for a particular geometry
and loading type, and can be easily derived from the elastic compliance
and stress intensity factor [62]. The second term in Eq 10, and therefore,
ijp, may be obtained by treating plastic work done, Up, as that correspond-
318 ELASTIC-PLASTIC FRACTURE

p
2

P,
f Equal
1 areas
P
f "^
/ \ ^ \

FIG. 3—Schematic load versus load point displacement diagram.

ing to the deformation of a rigid plastic body, that is, Up = qp PL, where
qp is the plastic component of load point displacement and PL is the limit
load. Hence, from Eq 4

Jk. U. X
djPLqp)
BiW - a) B da

For three-point SENB specimens having a/W > 0.15, it can be shown
that rip = 2.0. Also, when these specimens are tested over a span of 4W,
and have 0.45 < a/W < 0.65, Eq 10 reduces to

2(Ue + Up)
PI _ 21/
/ =
BiyV - a) B(W - a)

which is the deep notch form used in the ASTM studies [54],
There are two problems concerning the measurement of load point dis-
placements in SENB tests. The first is the difficulty of separating the true
load point displacement of the test specimen from the displacements under
the loading points and in the testing system [49]. An investigation of these
displacements led the author [10] to develop the equipment shown in Fig.
4. With this equipment the vertical displacement of the notch mouth is
measured relative to the top surface of a 'comparison' bar. The bar rests
on pins which are attached to the specimen at the ends of the loading
span. The initial contact points between the comparison bar and the pins
are on the neutral axis of the specimen. It was shown [10] that the vertical
displacement of the notch mouth represents q to an accuracy of better
than ± 2 percent, provided that the total angle of bend is less than 8 deg,
which is approximately the maximum value of interest in fracture initia-
tion tests.
 DAWES ON COD AND J-GONTOUR INTEGRAL CONGEPTS 319

FIG. 4—Equipment for simultaneous measurements of q, using a linear transducer, and

V, using linear clip gage.

The second problem in measurements of q concerns SENB tests on

shallow-notched specimens, especially when the notches are located in weld
zones. As shown schematically in Fig. 5, situations can arise where plastic
hinges form in the base metal adjacent to notches in an overmatching
yield strength weld zone. In these instances the load point displacement
and, therefore, the estimated / , will not represent the effective value of /
in the notched region, which may involve a relatively small component of
work done in plastic deformation. Herein lies a fundamental difficulty in
applying the path independent /-contour integral concept to elastic-plastic
deformations in varying yield strength regions of welded structures. From
the viewpoint of a laboratory test, however, the effective value of J can be
estimated by considering contours around the crack, but within the welded
region, for example, by estimating / from measurements of the crack
mouth opening displacement. This can be done using the following re-
lationships (compare Eq 10) proposed by Sumpter and Turner [61]

WVp
(11)
E' BiW - a) a+ z + rpiW - a)
320 ELASTIC-PLASTIC FRACTURE

where rp = 0.4 for SENB specimens having a/W > 0.45, and 0.45 for
a/W < 0.45. Equation 11 has the added advantage that it can be used
with the standard instrumentation for ATic tests.
The only difficulty with Eq 11 concerns the definition of the limit load,
PL , on the load versus clip gage displacement record. For the experimental
work which follows, therefore, Eq 11 was modified to give

A-^ P1+P2 WVn

(12)
E' B{W - a) a+ z + rp(W - a)

where Pi and P2 are determined as shown in Fig. 3.

Finally it is worth noting the similarity between estimates of 5 and /
using Eqs 9 and 11, respectively. For instance, taking Pi = [1.5 orBiW —
ay]/4W it can be shown that the equations predict/ < 2ayS, that is, m <
2.0 (Eq 6). Also, it may be observed that the substitution of (Pi + P2)/2
for Pl (Eq 12) will generally result in larger post net section yield values
of m in materials having low ratios of yield to tensile strength.

Experimental Studies

Materials
The Ducol W 30 Grade B and BS 4360 Grade 50C steels used had the
chemical analyses and basic material properties summarized in Table 1.
Both materials were supplied in the form of 25-mm-thick normalized plate.

a1

€^
Weld metal yield
strength assumed
constant

Matc/iins ys

Undermatching YS plate

FIG. 5—Schematic load versus load point displacement behavior in weldment specimens
containing shallow notches.
 DAWES ON COD AND J-CONTOUR INTEGRAL CONCEPTS 321

- T — I • , -. ,- , 1 1

•20
Numbers refer to testing temperature,'C
(60-

UO - ' • " ' • /

-to*

120--
/ • -
r -40

100 -
y^ *-20
-so m*-6o
g SO -
- 4 0 * « -15

I 60
- 7 0 , ^ -90 "
/ -80
40 _
..»

20 -
/ -
- -100^*-70
'-looMf'-ao
/-so 1 1
/ 1 1 1 1 1
20 40 60 ao 100 120 HO
Critical J from equation 110),Nmm''

FIG. 6—Comparison of J-estimates from Eqs 12 and 10: Ducol W30 steel plate.

Comparison of Eqs 10 and 12

Although Eq 12 provides the most direct and simple method for ob-
taining /-values from a P versus V record, some doubt must surround the
use of a constant rotational factor, Vp, for different materials [33]. It was
of interest, therefore, to use the equipment shown in Fig. 4 to obtain a
comparison of the /-estimates from Eqs 10 and 12, that is, from simul-
taneous measurements of q and V. This was done using a series of full
section thickness (25 mm) Ducol W 30 SENB specimens having a/W =
0.5, W/B = 2.0, and S/W = 4. As shown in Fig. 6, there was excellent
agreement between the values of/,, estimated from Eqs 10 and 12 for tem-
peratures ranging from —100 to — 10°C. It may be noted that all the test
results in Fig. 6 refer to unstable fractures without prior stable crack
growth.
322 ELASTIC-PLASTIC FRACTURE

While the foregoing results are encouraging, it may be necessary to

check the relationships between q and V in different materials and dif-
ferent SENB geometries. For example, in the case of shallow-notched,
overmatching-yield-strength weld metal specimens (Fig. 5), it would be
helpful to check the relationships using plain material or all-welded metal
specimens having similar geometries to the ones of interest, for example,
by making simultaneous measurements of the plastic components of q and
V using the equipment in Fig. 4.

Measurements of Critical COD and J-Values

These tests were carried out to compare the effects of geometry and
temperature on critical values of COD and / for both the onset of stable
crack growth and unstable cleavage fracture.
Full thickness (25 mm) three-point SENB specimens were extracted
from the BS 4360 Grade 50C steel plate (Table 1) and prepared with
through-thickness notches. Tests were then carried out using the ASTM
E 399-74 instrumentation and the results were interpreted using the Wells
[60] DD19 COD relationship (which gives values similar to Eq 9) and the
slightly modified Sumpter and Turner [61] /-relationship, Eq 12.
Figures 7 and 8 show that the ductile/brittle transition temperature
ranges were generally raised by increasing the a/W ratios from 0.2 to 0.5,
and increasing the notch acuity from a 0.075-mm-radius notch to a fatigue
crack. This behavior was true for unstable quasi-cleavage fractures occur-
ring both before and after the onset of stable crack growth. The approxi-
mate lengths of stable crack growth in the fatigue precracked specimens
are shown in Fig. 9a and b, which are in the form of 'R curves'.
Figure 10 illustrates the near linear relationships between COD and /
over a wide range of temperatures. The data in Fig. 10 were combined
with more detailed tensile strength data (Fig. 11) to give the values of m
(Eq 6) in Fig. 12. This shows that, generally, m < 2.0 over a wide range
of temperatures, as predicted earlier.
Since there is a close mathematical link between the present COD and
/ estimation procedures, the 8c and Jc must be equally useful as fracture
characterizing parameters, at least for the three-point SENB specimen
geometries examined. This conclusion was confirmed by a variance ratio
analysis, which showed that the nondimensional standard deviations of the
total 8c and total Jc values for each geometry in Fig. 7 were equally in-
sensitive to geometry.

Experimental Estimates of Ku from he

Figure 13 shows the results of a partially completed program of work
on a BS 4360 Grade 50D steel. This material has a chemical analysis and
DAWES ON COD AND J-GONTOUR INTEGRAL CONCEPTS 323

§§

O TT
o o

o o
V
I
I ^1 Q!i

t 2§
f*^ ro fO f l '

t Ul
§1
U

a:;
J
I v^ -^ o a> \o
SS a^ ^ Q <N fS
I/) I/) -S t/> i/>

I
I n 00
o o
o o 00 ' ^ i-H rs ON
^ in 1/5 ^ lO
a
BQ
2Z
•^ ' ^ T!- f^ fO

is

•3
•B
§ U
o ea o
1
85
:3 V)
8
QOQ Q
324 ELASTIC-PLASTIC FRACTURE

u a 1 1 ' 1 1 1 - I- -1 .1
' ' ' '
a w ^
UV
A -~~0-2, --- = 10
" W B -
L Fatigue cracked specimens
0 e _ • ^.0.5. f-10 B = 25mm
A"-
A
0-5
a w / -
O - - « 0 5 , ~ -2 0 J /
om
BSi,3eO Grade 50C
Oi •m

§ /
03 •v^ -

02 -
ii AYC\\ )i;\ -
/ -x 0

0 1

0 A- , • 2 HII4 c« ' 1 ^ - * o, i I 1 1
-?5o -WO -;jo -i20 -m ~mo -90 -BO .-70 -SO -SO -W
a) Temperature.'C

I' T 1 1 -I r 1 1 '• T • 1 1
700
—r

g SOO - A
/
c 500 A/ ;
Oj

1 *oo
i»o^ / /
/ • / . •

,^ 300
1 _
s6
*^ oAa ,,
200 - ,v , , y^, '-w^os
^<^\\\\y\' -
A ^ / ^ ' ^
-," /OO

Oil » l« A? ic^ •94

-?50 -WO -130 -120 -110 -100 -90 -80
4 —? --0
o, , -70
, -60
, -50
, -40
1
b) Temperature ,' C

FIG. 7—Critical COD and J versus temperature for fatigue cracked specimens in BS 4360
Grade 50C steel.
 DAWES ON COD AND J-CONTOUR INTEGRAL CONCEPTS 325

0 8 T i l l 1 1 1 1 i ~T ^
Am m
1 A
07
^-i-] 1 0 075mm rad!us
machined notch
-
• F--f- /o
0 6 8 -- 25 mm •
o
0 5 W B / A
BSi360 Grade 50C •
0 i o» y / o •
•
o
03
- 05 ^ - 2 0
w e
02 -.^ O

^^^ o ^ ^ o
0 1

OAi « o , o.^-^ 1 1 1 1 1 1
-ISO -HO -130 -120 -no -700 -90 -80 -70 -60 -50 -40
a) Temperature ,'C

I 1 1 1 I 1 •I 1 1 I 1 1
700 o -

soo • -
:3
A
i 500 o -
1 ^,00
A
•
•
-
2 «
300 - • -
1 1 o
no
• A
1 200 • I o
-
A o
-V 100 8 o
o
0, \» 1 O 1 1 1 1 1 1 1 1
-150 -HO -130 -120 -110 -100 -90 -70 -60 -50 -UO
bl Temperature, 'C

FIG. 8—Critical COD and J versus temperature for machined notch specimens in BS 4360
Grade 50C steel.
326 ELASTIC-PLASTIC FRACTURE

•a
&

i
I

l-lMWf^'SaniDA r IDOftlJJ

•9

UJUJ '000 IDOIUJQ

 DAWES ON COD AND J-CONTOUR INTEGRAL CONCEPTS 327

1 1 I - T —r 1— »—1 r r
^
4
0 o
4
4
'••••
- < ^ -s ^
•Si! 04* ,
- 2g - I
6 ^c 6 0
-
IC « •"
5^7 ^„ _
•a
6 6 <o c
1 1 1 1 1 1 1 "U §
( sil^M ) " « " • 000

I
1 1 r 1 1 1 1 1 1
I
13
^ 5
Qj

•g
S 4 § ^ 2
" 6
S6 o 0 ? a
5i A

" 9
A

o
V -
o
1 Si
- k i d iltti _
<>j «o « " E
o o 1. to U'* E
o *
ojk o|i
^ -
•4 • o "
1 1 1 1 1. 1 1 , >

(SIHM) «"" '£703

328 ELASTIC-PLASTIC FRACTURE

I ~ -\ \ 1 r
700

S50

600

550
6
JO

t 500

450

1,00

350

300
-120 -WO -80 -60 -*0 -20 X
Tempemtun,'C

FIG. 11—Influence of temperature on the tensile properties ofBS 4360 Grade 50C steel.

mechanical properties similar to those summarized for the Grade 50C

material in Table 1. In these tests the small (10 mm) specimens were
extracted from near the center of the 100-mm-thick plates.
Figure 13 shows the valid J\c results (according to Eq 7) obtained for
specimens in each thickness. It can be seen from Fig. 13 that valid Ju
values [54] can give large overestimates of valid K\o values when applied
to materials which show significant shifts in the ductile/brittle transition
temperature range with variations in section thickness or in-plane dimen-
sions.

Conclusions
1. Since a deformed crack tip stretches beyond the original crack tip
position, it is proposed that the crack tip COD should be defined as the
displacement at the original crack tip position. This definition avoids much
 DAWES ON COD AND J-CONTOUR INTEGRAL CONCEPTS 329

1 • r T 1 1 1 1

a = 10
W .02.^
a ».0 5,-g- • 10 > Fatigue cracked
• W 8 = 25mm
o . 0 . 3 , f - = 20
W
25
BS i3S0 50 C
Jc ^m

•
A
20 •
• •
• • O
O 1 A
O
A O
A
§
;5 -

10 1 1 1 1 ' 1
-100 -90 -80 -70 -60 -SO -*0
Tempemture.'C

FIG. 12—ra versus temperature, BS 4360 Grade 50C steel.

of the ambiguity of previous COD definitions, which were based on the

crack tip profile and the elastic-plastic interface on the flanks of the crack.
2. When cracks are sited within weld regions, significant local variations
in yield strength cause a fundamental problem in the application of the J-
contour integral concept to an elastic-plastic material. For example, when
cracks exist in an overmatching yield strength region, estimates of J de-
rived from measurements on contours outside the weld region may grossly
overestimate the plastic work component of J within the weld region.
3. Full section thickness three-point SENB tests carried out at the tem-
peratures, strain rates, and section thicknesses of interest that give in-
valid ^ic results may be interpreted in terms of critical COD and /-values.
It can be shown that there is a close mathematical link between the COD
and /-values obtained with these interpretations. Also, when using the lat-
ter interpretations the experimental results indicated that COD and / are
equally useful as fracture characterizing parameters.
4. The critical values of COD and / for unstable cleavage fracture in
common structural steels are affected by section thickness and in-plane
geometry. It is important, therefore, that laboratory tests be designed to
match or overmatch the plastic constraint in the structural situations of
interest.
5. Values of/ic from small laboratory tests may give large overestimates
330 ELASTIC-PLASTIC FRACTURE

I
BS*3e0 trade SOD Steel
O 100mm
SeNB, B»2B
* 10 mm

A Valid K,

/Oactile brittle transitioit

eooo temperature behaviour
I
7000

';^ 6000-
J .
* /
!!^ 5000
6 /
iOOO /
/
3000
/ V
2000

Mid K/c
mo

-200 -150 -100 -50

Temperature 'C

FIG. 13—Estimates ofKicfrom he (valid according to Eg 7): BS 4360 Grade SOD steel.

of valid Ku values in common structural steels having yield strengths less

than approximately 700 N/mm^.

Acknowledgments
The help and encouragement of the author's colleagues is acknowledged,
and especial thanks are given to Mr. B. A. Wakefield and the staff of The
Welding Institute brittle fracture laboratory. The author also appreciates
the generosity of his colleague Dr. H. G. Pisarski for making available
the preliminary test results in Fig. 13.
 DAWES ON COD AND J-CONTOUR INTEGRAL CONCEPTS 331

References
[1] Wells, A. A. in Proceedings, Crack Propagation Symposium, Cranfield, England, Vol.
1, Paper B4,1961.
[2] Rice, J. R., Journal of Applied Mechanics, June 1968, p. 379.
[3] Burdekin, F. M. and Stone, D. E. W. Journal of Strain Analysis, Vol. I, No. 2, 1966,
p. 144.
[4] Harrison, I. D., Burdekin, F. M., and Young, I. G., "A Proposed Acceptance Stan-
dard for Weld Defects Based upon Suitability for Service," 2nd Conference on the
Significance of Defects in Welded Structures, The Welding Institute, London, England
1968.
[5] Burdekin, F. M. and Dawes, M. G. in Proceedings, Institution of Mechanical Engineers
Conference, London, England, May 1971, pp. 28-37.
[6] Dawes, M. G., Welding Journal Research Supplement, Vol. 53, 1974, p. 369s.
[7] Dawes, M. G., and Kamath, M. S. in Proceedings, Conference on the Tolerance of
Flaws in Pressurised Components, Institution of Mechanical Engineers, London, England
May 16-18, 1978, pp. 27-42.
[51 Draft British Standards Rules for the Derivation of Acceptance Levels for Defects in
Fusion Welded Joints, BSI WEE/37, Document 75/77081 D.C., British Standards Insti-
tute, Feb. 1976.
[9] Harrison, J. D., Dawes, M. G., Archer, G. L., and Kamath, M. S., this publication,
pp. 606-631.
[10] Dawes, M. G., "The Application of Fracture Mechanics to Brittle Fracture in Steel
Weld Metals," Ph.D. thesis (Council for National Academic Awards), The Welding In-
stitute, London, England, Dec. 1976.
[11] Nichols, R. W., Burdekin, F. M., Cowan, A., Elliott, D., and Ingham, T. in Proceed-
ings, Conference on Practical Fracture Mechanics for Structural Steel, Risley, England
April 1969, UKAEA/Chapman and Hall, Risley, 1969.
[12] Wessel, E. T. in Practical Fracture Mechanics for Structural Steels. UKAEA/Chapman
and Hall, Risley, England, 1969, p. HI.
[13] Green, G., Smith, R. F., and Knott, J. F. in Proceedings, Conference on Mechanics
and Mechanisms of Crack Growth, Churchill College, Cambridge, England, Paper 5,
April 1973.
[14] Broek, D. in Proceedings, Third International Conference on Fracture, Munich, Ger-
many, April 1973, Vol. 4, p. Ill 422.
[15] Fields, B. A. and Miller, K. J., "A Study of COD and Crack Initiation by a Replication
Technique," Cambridge University Engineering Department Report CUED/C-Mat/TR
17, Cambridge, England, Oct. 1974.
[16] De Morton, M. E. in Proceedings, WI/ASM Conference on Dynamic Fracture Tough-
ness, The Welding Institute, London, England, July 1976.
[17] Smith, R. F. in Proceedings, Institution of Metallurgists Spring Meeting, Newcastle,
England, Paper 4, March 1973, p. 28.
[18] Goodier, J. N. and Field, F. A. in Fracture of Solids, Drucker and Oilman, Eds., Inter-
science, New York, 1963, p. 103.
[19] Hahn, G. T. and Rosenfield, A. R., Acta MetaUurgica, Vol. 13, No. 3, 1965, p. 293.
[20] Rice, J. R. and Johnson, M. A. in Inelastic Behavior of Solids, McGraw-Hill, New
York, 1970, p. 641.
[21] Pelloux, R. M. ^., Engineering Fracture Mechanics, Vol. 1, No. 4, 1970, p. 697.
[22] Srawley, J. E., Swedlow, J., and Roberts, E., International Journal of Fracture Me-
chanics, Vol. 6, 1970, p. 441.
[23] Wells, A. A. and Burdekin, F. M., International Journal of Fracture Mechanics, Vol.
7, 1971, p. 242.
[24] Boyle, E. F., "The Calculation of Elastic and Plastic Crack Extension Forces," Ph.D.
thesis. Queen's University, Belfast, U.K., 1972.
[25] Levy, N., Marcel, P. V., Ostergren, W. J., and Rice, J. R., International Journal of
Fracture Mechanics, Vol. 7, No. 2, 1971, p. 143.
[26] Rice, J. R., International Jounml of Fracture Mechanics, Vol. 9, 1973, p. 313.
332 ELASTIC-PLASTIC FRACTURE

[27] Turner, C. E. in Proceedings, Conference on the Mechanics and Physics of Fracture,

Churchill College, Cambridge, England, Paper 2, Jan. 1975, pp. 2-1-2-16.
[28] Eshelby, J. D. in Solid State Physics, Vol. 3, Academic Press, New York, 1956, p. 79.
[291 Cherepanov, G. P., InternationalJoumal of Solids and Structures, Vol. 4, 1968, p. 811.
[30] Hayes, D. J., "Some Applications of Elastic-Plastic Analysis to Fracture Mechanics,"
Ph.D. thesis. University of London, London, England, 1970.
[31] Sumpter, J. D. G., "Elastic Plastic Fracture Analysis and Design Using the Finite
Element Method," Ph.D. thesis. University of London, London, England, Dec. 1973.
[32] Sumpter, J. D. G., and Turner, C. E., International Journal of Fracture Mechanics,
Vol. 9, No. 3, 1973, p. 320.
[33] Kanazavfa, T., Machida, S., Hagiwara, Y., and Kobayashi, J., "Study on Evaluation of
Fracture Toughness of Structural Steels Using COD Bend Test," International Institute
of Welding, Document X-702-73, June 1973.
[34] Serensen, S. V., Girenko, V. S., Deinega, V: A., and Kir'yan, V. I., Automatic
Welding, Vol. 28, No. 2, 1975, p. 1.
[35] Burdekin, F. M., Dawes, M. G., Archer, G. L., Bonomo, F., and Egan, G. R., British
Welding Journal, Vol. 15, 1968, p. 590.
[36] Priest, A. H. and May, M. J., "Strain Rate Transitions in Structural Steels," British
Iron and Steel Research Association Report MG/C/95/69, 1969.
[37] Egan, G. R., "Application of Yielding Fracture Mechanics to Design of Welded
Structures," Ph.D. thesis, London University, London, England, 1972.
[38] Kanazawa, T., Machida, S., Onozuka, M., and Kaneda, S. A., "Preliminary Study on
the J-Integral Fracture Criterion," International Institute of Welding, Document X-
779-75, May 1975.
[39] Ingham, T., Egan, G. R., Elliott, D., and Harrison, T. C. m Proceedings. Institution of
Mechanical Engineers Conference on Practical Application of Fracture Mechanics to
Pressure Vessel Technology, London, England, Paper C54/71, May 1971.
[40] Knott, J. F., Materials Science and Engineering, Vol. 7, No. 1, 1971, p. 1.
[41] Veerman, C. C. and Muller, T., "The Effect of Notch Depth and Pre-Strain on the COD
at Fracture," 2nd Progress Report of the European Community Research Program
6210-55/0/50, Nov. 1972.
[42] Kanazawa, T., Machida, S., Momota, S., and Hagiwara, Y. A. in Proceedings, 2nd
International Conference on Fracture, Brighton, England, Paper 1, 1969, p. 1.
[43] Dolby, R. E., and Archer, G. L. in Proceedings, Institution of Mechanical Engineers
Conference, London, May 1971, pp. 190-199.
[44] Dawes, M. G., Welding Journal, Vol. 55, No. 12, Dec. 1976, p. 1052.
[45] Harrison, T. C. and Feamehough, G. D., International Journal of Fracture Mechanics,
Vol. 5, 1969, p. 348.
[46] Fearnehough, G. D., Lees, G. M., Lowes, J. M., and Weiner, R. T. in Proceedings,
Institution of Mechanical Engineers Conference on Practical Application of Fracture
Mechanics to Pressure Vessel Technology, London, England, Paper C33/71, May 1971.
[47] Smith, R. F. and Knott, J. F. in Proceedings, Institution of Mechanical Engineers Con-
ference on Practical Application of Fracture Mechanics to Pressure Vessel Technology,
London, England, Paper C9/71, May 1971.
[48] Chipperfield, C. G., Knott, J. F., and Smith, R. F. in Proceedings, 3rd International
Conference on Fracture, Munich, Germany, Vol. 2, April 1973, pp. 1-233.
[49] Robinson, J. N. and Tetelman, A. S. in Fracture Toughness and Slow-Stable Cracking,
ASTM STP 559, American Society for Testing and Materials, 1974, pp. 139-158.
[50] Robinson, J. N., International Journal of Fracture Mechanics, Vol. 12, 1976, p. 723.
[51] Green, G. and Knott, J. F. in Proceedings, ASME Conference on Micromechanical
Modelling of Flow and Fracture, Troy, New York, American Society of Mechanical
Engineers, June 1975.
[52] Begley, J. A. and Landes, J. D. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 1-20.
[53] Landes, J. D., and Begley, J. A. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 24-39.
[54] Landes, J. D. and Begley, J. A. in Fracture Analysis, ASTM STP 560, American Society
for Testing and Materials, 1974, pp. 170-186.
 DAWES ON COD AND J-CONTOUR INTEGRAL CONCEPTS 333

[55] Clarke, G. A., Andrews, W. R., Paris, P. C , and Schmidt, D. W. in Mechanics of

Crack Growth, ASTM STP 590, American Society for Testing and Materials, 1976, pp.
27-42.
[56] Garwood, S. J., "The Measurement of COD and J at the Initiation of Crack Growth
and Their Interpretation," Imperial College, London, England, July 1976.
[571 Sumpter, J. D. G., Metal Science, Oct. 1976, p. 354.
[58] Milne, I., "The Ductile-Brittle Transition and Fracture Toughness of Ferritic Steels,"
to be published.
[591 Hayes, D. J. and Turner, C. E., IntemationalJoumal of Fracture Mechanics, Vol. 10,
No. 1, March 1974, p. 17.
[601 Wells, A. A. in Proceedings, Canadian Congress of Applied Mechanics, Calgary, Alta.,
Canada, 1971, pp. 59-77.
[61] Sumpter, J. D. G. and Turner, C. E. in Cracks and Fracture, ASTM STP 601, Ameri-
can Society for Testing and Materials, 1976, pp. 3-18.
[62] Turner, C. E., Materials Science and Engineering, Vol. 11, 1973, p. 275.
/ . Royer,^J. M. Tissot,^ A. Pelissier-Tanon,^ P. LePoac,^
and D. Miannay^

J-lntegral Determinations and

Analyses for Small Test Specimens
and Their Usefulness for Estimating
Fracture Toughness

REFERENCE: Royer, J., Tissot, J. M., Pelissier-Tanon, A., Le Poac, P., and Miannay,
D., "J-Integnl Detenninatioiu and Analyses for Small Test Specimens and Their
Usefulness for Estimating Fractnte Tongiiness," Elastic-Plastic Fracture, ASTM STP
668, J. D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing
and Materials, 1979, pp. 334-357.

ABSTRACT: General estimation procedures for the J-integral determination are review-
ed for the three-point bend and the compact tension specimens. Tests were made using
10 CD 9-10 steel. Experimental results are presented and are in partial agreement with
analytical results. The errors due to simplified analysis and experimental procedure are
explored. Toughness, as analyzed by the resistance curve technique, is shown to be size
dependent for bending and not for tension. Disagreement between the two loading
modes, if not fortuitous and due to the steel, suggests that simple strain and stress
analyses are not sufficient and that the loading procedure and the T-effect must be taken
into account.

KEY WORDS: crack propagation, /-contour integral, fracture tests, fracture proper-
ties, steels, elastic-plastic fracture, fracture initiation

Nomenclatnie
/ Energy line integral
V Pseudo strain energy release rate
G Elastic strain energy release rate
W Strain energy density

' Professor of mechanics and assistant professor, respectively, Ecole Nationale Superieure de
M£canique, 1, rue de la Nde, 44072 Nantes Cidex, France.
^Research consultant, Framatome, 77-81, rue du Mans, 92400 Courbevoie, France.
^Materials engineer and head. Fracture Mechanics Group, respectively. Commissariat k
I'Enetgie Atomique, service Metallurgie, B. P. No. 511, 75752 Paris-C61ex, France.

334

Copyright 1979 b y A S T M International www.astm.org

 ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 335

U Strain energy or work done in loading a specimen

U' Complementary strain energy or work
K\ Opening mode stress intensity factor
V Calibration factor
P Applied load
F Force per unit thickness
V Displacement of the applied load
C Specimen compliance
a Crack length
w, B Specimen width, specimen thickness
S Span in bending
p Notch acuity
Oy Uniaxial yield stress
au Ultimate tensile strength
E Young's modulus
V Poisson's ratio
Subscripts
e Value for the elastic behavior
p Value for the plastic behavior
L Value for the fully plastic limit
£ Value for plane strain
a Value for plane stress
To evaluate the toughness of low- or medium-strength metals with
specimens not satisfying size requirements of the ASTM Test for Plane-
Strain Fracture Toughness of Metallic Materials (E 399-74T) for linear
elastic behavior, three concepts are proposed for the elastoplastic range: the
J-integral, the crack opening displacement, (COD), and the equivalent-
energy concepts. These lead to different experimental and estimation pro-
cedures to obtain the adequate parameter and to determine its critical value
representing the toughness.
Unfortunately the results obtained by different investigators are not in
complete agreement. The controversy centers on the estimation procedure
and the possible relationships between the three concepts.
The first part of this paper deals with the calculation or estimation of/
from load displacement data of precracked specimens from experimental
and theoretical points of view and the second part with the influence of
specimen size, geometry, and mode of loading on the value of toughness.

Materials and Experimental Procednie

One 110-mm-thick rolled plate of 10 CD 9-10 steel with two heat
treatments was tested. Chemical composition, heat treatments, and tension
test results are given in Table 1.
336 ELASTIC-PLASTIC FRACTURE

TABLE 1—Properties ofmateriah tested parallel to the rolling direction.

Chemical Composition, percent by weight

C P S Si Mn Cr Mo Al Cu Sn

0.125 0.012 0.017 0.260 0.425 2.35 0.99 0.012 0.14 0.022

Heat Treatment

Material No.

1 normalized 940°C, 1 h, water quenched; tempered 700°C, 1 h

2 normalized 940°C, 1 h, water quenched; tempered 700°C, 1 h; stress re-
lieved 675 °C, 4 h

Room Temperature Tensile Properties (e" = 1.11 10 ~''s "')

Yield
Strength Tensile
0.2% Offset, Strength, Parameters (a = ao + Kep")
MPa
Material No. ay 0.2 au % CO AT n

1 520 650 22.6 502.9 51.4 0.707

2 500 625 23.8 483.4 48.9 0.704

Young's modulus: E = 206 000 MPa.

Poisson's ratio: v = 0.3.

Two specimen geometries were machined at midthickness in the

longitudinal (LT) direction with respect to rolling: the three-point bend
(TPB) and the compact tension (CT) specimens with dimensions as given in
Fig. 1. Specimens were precracked at different crack lengths, a, by fatigue
according to the ASTM specifications or drilled with a hole of different
radius, p, at the notch end. For short cracks in bend specimens, precracking
was done before machining. Crack length was measured at seven equally dis-
tant points, excluding the specimen faces, on the crack surface.
Specimens were tested with a constant crosshead velocity of 0.005
mm/s~'. Displacement of the load point, v, was measured with a clip gage
located on two knife edges, one fixed at the end of the upper roll and the
other fixed on a sliding bar supported by the ends of the lower rolls for bend-
ing and between the two center knife edges localized on the loading line in
the CT specimen. Load, P, was measured with the load cell. Electrical
monitoring was also performed by the d-c drop potential method with the
configuration as shown in Fig. 1 [/].''

^The italic numbers in brackets refer to the list of references appended to this paper.
ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 337

input
current

B w 0 c d e f g J
10 20 5 2 3 4 1.15 2 5.5
20 40 10 2 3 4 1.15 2 11
40 80 20 2 10 12 3.6 2 22
80 160 40 2 10 12 3.6 4 44

input current
^-^—n
A

output voltage

J
00

B w S
5 10 40
10 10 40
20 10 40
10 20 80
20 20 80
20 40 160
40 40 160

FIG. 1—CTand TPB specimens.

338 ELASTIC-PLASTIC FRACTURE

Theoretical Background
/ is defined for two-dimensional problems as the line integral

j=[(^,,-r^^)
where
W = strain energy density,
T = traction vector on path T, and
M = displacement vector.
Its properties are largely reviewed in the literature [2-7] and will not be
reported here. For an elastic material, / is representative of the potential
energy variation with respect to crack length. Begley and Landes [4] pro-
posed to extend this interpretation in the nonlinear range and /, or V as
denoted here, is given by

--(f). = I(-f).r
or

-(fi=i:(i7i-
with U the work of the applied force F per unit length of the crack front.
Thus V can be evaluated experimentally from the load displacement curves
for identical specimens of differing crack lengths. This method has been used
here at constant value of displacement.
The displacement v may be separated into two parts

^ *'no crack '" ''crack > O r V Velastic l Vp|„,jj

It follows that

' ~ ''crack ^^^ * ^ 'elastic i" ''^plastic ^ G "T V pi„,jj (2)

G being the elastic strain energy release rate given in different equivalent
forms

with E' = £7(1 — v^) in plane strain (subscript e) and £" = £ in plane
 ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 339

stress (subscript a), with /C, the stress intensity factor = (F/yfw) Y (a/w), w
being the width of the specimen.

2. G = i F ^ f (3)

with C the compliance v/F, generally tabulated as Ev/F or calculated as

cm + C(a) = cm + p7 1 ^ da or C = C(0)
E' Jo F^
(4)

e\i
3 G = - (^^') = - ^ ^ =
\ da /„ da
(5)
, C f. a\ U. ^ U,
'^ C y w J {w - a) ''• iw - a)

the prime denoting differentiation with respect to a/w and with Ue the elastic
strain energy.

. ^ _ 2Y^ f. a\ Ue _ Ue ,..
^- ^-WcV~^) -GT^^) - "' -(^r^~a) (^^
The functions and values used in this paper are given in the Appendix and
tabulated in Tables 2-4 with v = 0.3.
Analyses have been developed for calculating the value of V from a single
load displacement curve. They are based on three assumptions:
1. The actual load displacement curves are approximated by the two
limiting cases, purely elastic and rigid plastic behaviors [5,6]. That is

V =
_- /3(t/,M
^ '"•
+ t/.)\
"I"
_= +
,c:u.
\ 3a /v C v>

+,F'LU^_
-j^
Fi w — ije
Ue , Up
' w — a + ripw —"^—
a
^^^

with UeM the maximum elastic energy when v = vi = CFi, PL = BFL being
the limit load, and with Up = Fi(v — vi), the plastic energy, rjp values are
given in Tables 4 and 5.
340 ELASTIC-PLASTIC FRACTURE

? 5q

06 O

>£> OO O

in
•9 <»5 m «
I--' OO <S
i/> i/5 ^

= S

18 •<t- •«• i n
. - ' -H u i

I
^ f^ 'T 'H
Ov O ^
^0

235 r~ O CO
•*• T'
S'ON rj
U
>-)
< vO ON VO
vO ON
-; -; (N IN

Ill5=«
ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 341

1^
'2 ^c 00 i/l O •* lO
in 1^ o^
>n 00 (s
<s m <S (N r^

00 fO
>0 fN (N i^ O ^ 00
\0 O^ r^
o ^ O y-* m

fN ^ fN fS
^O "^ -^
fS ^ -Tj-
in i/i
\n \o ^

T^ 00
5 ON ^ n 00
00 ^ O "^ 00
po (*) f*^

«|S
«0
o n 00 5
in o I-- 00 so n so

1 lO 1^ O (N 00 <N r»)
-< <N <S

00 vD 00 ^C
0^ CM •<r •» o
r- 00 <N •* <s •» in

in 00 00 m
fo m 1^ 00

<

II III .
342 ELASTIC-PLASTIC FRACTURE

(N (N (N fS r4

o o

(N rn *-;
rsiri <N <N rj <N<NfN

S;S 8 8 8 2^2
^<N <N (N rJ (N(Nr4

8S
^ ^ r-i fN fS :^,-,--*

»-H^ fS Ol (N ^ ^ ^

g
T ^. '^.
«-H —i r-i (N fs w ^* ^*

T3
S
^ ^ ^* ^ fsj

"t?.

S 8
O O ^ -H fS O O O

S
<

,4*

Sfoss
2 § g * § 2 gh; i g^oo
U U U U CU
 ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 343

yri i n

M <N

^ ^O r o O
f S (N fN

• ^ 00 I— • *
"2 • ^ 00
• ^ (N
(N <N ( N (N (N rj O (N (N <N

00 r^ 00 0^ r-
•n • * in <N ^
<N (S ri ri (N (S d ( N (N <N

in 00 o
«S CN r-i ( N
^.
r>i <s
(N O
(N d r^ ( N ( N

1 s
r^
<N I N
n n
^
ro <N
^<N (N
d O
S8S
r>i r j

«> CN
1; S g in ^ ON
o a;
rr> o ro in (N ri d rr, f^ ( N
•a
s
oo Oi 0^
1^ • ^ in in ?
n • ^ rr ri (N

"S.
O 00
^
I
in
m
n
<:
H

^«ll
++
-r
I I I II
^ (li m Qj ii> v -^1
O O P^
" ^ II II II
Ou'g O. }) 1 CI. Q.
o* cr
u u u OS £,
344 ELASTIC-PLASTIC FRACTURE

2. The plastic displacement due to the crack is a function of the ratio of

the applied load to the fully plastic limit load [7], from which

Ue Up Up'
' w —a "^ w — a ' w —a

with Up' the plastic complementary work of the applied load, ric is null for
pure bending and its low values are given in Table 5 for the CT specimen.
3. The initial elastic behavior of the cracked body may be corrected for
plasticity by considering an equivalent elastic or effective crack size up to the
limit load [5]. This leads to an evaluation of the V versus v relationship. Thus
the first two assumptions give relations between V and areas easily measured
on the records. Such relations moreover, are more practical when rje = r/p or
when one contribution is greater than the others. Our first objective has been
to verify experimentally the adequacy of such an approach.
The J-integral has been advocated to represent strain and stressfieldsnear
the crack tip in the plastic range [2-4], so that the condition for onset of
growth from a precrack can be phrased in terms of/. As shown by Larsson
and Carlsson [9], however, in the small scale yielding regime, plastic zone
sizes are different for the same elastic limit condition, suggesting that
damage will vary with geometry. This phenomenon is probably attributable
to the effects of in-plane biaxiality as suggested by Rice [10], biaxiality being
negative in tension testing and positive in bend testing, but not influencing
greatly the J-value. For stably growing cracks, no similar characterizing
parameter has yet been identified. Begley and Landes [//], however, pro-
posed to determine a critical value, 7,^, from a resistance curve. Such a
technique is studied in the second part of this paper.

Results for the Estimation Procedure

For the two geometries, analysis has been made with the same procedure
as outlined in Fig. 2. For a geometry and size, areas under load displacement
record at given displacement v less than the displacement corresponding to
any encountered maximum load have been plotted as the normalized term
U/Bw^ versus a/w. The following function was fitted to the data by the
method of least squares so that the deviation E'(f/ — U^a)y/(Uy is
minimum, U^^j being the adjusted value

U ^ ^_ aN^
ao + 2ao —
w
Bw^ V w
(9)
+ a2 {^) + a3 (-^1 + «4 1-^ '
 ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 345

v,/w

(V/w)o°^
U/Bw^MPa ° U/Bw2,MPa^
TPB specimen CT specimen
FIG. 2—Schematic of the analysis.

This polynomial gives values of Vand U null for a = w and Vnull for a = 0,
which is in agreement with the physical significance of these parameters. The
degree of this function has been chosen over the range up to 12 to give the
minimum deviation. Then this function is derived with respect to a/vv to give
V/w versus a/w at given displacement. Finally V/w is plotted versus v/w or
(U/Bw^)^ij and the best fit to the data is found.
346 ELASTIC-PLASTIC FRACTURE

Three-Point Bend Specimen

Seventeen (5 X 10 X 40), twenty (20 X 10 X 40), and five (20 X 20 X 80)
precracked specimens of the first material with diff^erent relative crack
lengths over the range 0 to 0.77, 0 to 0.79, and 0 to 0.75, respectively, were
loaded up to a relative displacement v/w — 0.128, 0.128, and 0.104, respec-
tively. The observed elastic compliance for v/w < 0.010 is reported in Table
2. For plain bar, the actual compliance is greater than the theoretical, em-
phasizing perhaps the indentation effects of rolls or revealing the inaccuracy
of the equation; the crack contribution is relatively independent of size and
in good agreement with theoretical values. Corresponding values of r/<,
calculated with Eq 6 with E' = E are reported in Table 4 and are slightly
greater than the generally accepted value of 2 for a/w > 0.5.
Due to the appearance of the curves of Fig. 2, the method outlined in the
introduction leads to a decomposition of the V/w versus U/Bw ^ relationship
into three ranges. The first one is defined between U/Bw ^ — 0 and U/Bw ^
= {U/Bw^)I by an initial slope at the origin, giving the presumed value rie
and a final slope at {V/w)i, {U/Bw^)\, giving the presumed value Tjpj
{V/w)i, (U/Bw^)i, and r/p are the values obtained from the determination
for the second range. The polynomial of order three in U/Bw^ fulfilling these
requirements is given as

V_ Ve U
w + w
w

U
\Bw'J, (r/; .+ Irje)
Bw^
(10)
—
(' wj iWvi J
u
Vfiwv, iv . + VP)
\ Bw^
- 2
+ 1 a ;.
w [UH-VJ
The second one is linear between the limits {U/Bw^)i and {U/Bw'^)i such
that

(11)
w / a\ Bw^ \W/o
w
with (V/w)o being the intercept with the V/w axis.
 ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 347

Beyond this, no relationship is proposed and we just report the maximum

values (V/vv) M and (U/Bw ^) M attained.
Values of the variables with a fit within 2 percent are given in Table 6. We
note that the linear relationship is sufficient for a/w > 0.6, coefficients are
relatively independent of geometry, and the mean value is 1.86. This value is
lower than the values given in Table 4. However, we have seen in a similar
treatment that two is obtained if the parameter V is plotted versus C/crack in-
stead of U.
From a practical point of view, we note that the V/w versus a/w relation-
ship at given displacement shows a maximum localized near a/w = 0.3 and
going toward increasing a/w with decreasing v/w. Therefore, when several
specimens are used to determine toughness, it seems judicious to choose a/w
near this range, if some scatter at precracking is to be assumed, because V/w
varies slowly with a/w. Moreover, our results show that an efficient and easy
initial calibration may be obtained with as few specimens as five, an un-
cracked specimen included, which is proved by the (20 X 20 X 80) specimen
testing for which the results are in the scatterband.^
Some limitation may subsist after this study, however, because slow crack
growth always occurred during testing.

Compact Tension Specimen

To alleviate this last restriction, testing was made with single size
specimens of the second material, CT 20 mm, but with three notch radii: p =
0 obtained by fatigue precracking, p = 0.5 mm, and p = 2 mm. Several
relative crack lengths were investigated: 13 ranging from 0.255 to 0.853 for p
= 0, 12 ranging from 0.250 to 0.803 for p = 0.5 mm, and 12 ranging from
0.250 to 0.801 for p = 2 mm. Slow crack growth was observed only for p = 0.
Plain specimens were not considered due to a lack of definition. In this kind
of test, indentation by the pins has no direct effect on measured displace-
ment, but for short cracks, plasticflowoccurring through pin holes, as noted
in the Appendix, obliged us to reject our tests with lower crack length than
mentioned in the foregoing. Experimental compliances are given in Table 3
and the corresponding ij^ in Table 5; compliance for v/w < 0.003 increases
with increasing radius in agreement with theory [/]; and" when testing was
stopped for the first limit load encountered for p = 0, that is, for v/w <
0.03, limit loads for other radii were not yet observed and must increase with
increasing radius. Better agreement with theory is noted for the medium
radius; the improvement on p = 0 may be due to a straight crack front.
In view of the appearance of the curves in Fig. 2, two analytical treatments
were tried. The first one, called the "polynomial function," is the same as ap-

* Actually, four specimens are sufficient to determine the four coefficients of the polynomial
but with five specimens deviation is tested.
348 ELASTIC-PLASTIC FRACTURE

\ 5«

oooor-^o^<s-"0 o^odr-^^ior^(N*HO

s 05
o vo o;
t^ r j (N 00 vp
o oq I/) ^ 00
( S - H - J - H O O O O O niNCM — O O O O O

e
.o 1^ o o m <N so 1^
o o o o o o o o o o o o o o o o o o
I I I I I I I I I I I I I I I

(Nr^'-'O^Qroooo ^/)^^f5l/)^ma^oo*^ ^ ^ Q ^ f ^ ^ r ^ o ^
^^t^9^C^000DQ0O^^ Ol/)00^v^9^00000^ Ol/^00^O^00a00000

>

•2> 0 0 « - H — c r t — . « « O —•

^(Nro-^iA)Ol^oO(^ r-^(Nr^'^t/)^l^0O(ys
alS
DO odoocJoooo ooooooooo ooooooooo

X X X
o o o
X X
IT) o
 ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 349

plied to bend testing; the second, called "the exponential function," was in-
tended to cover a larger range, and the function is given by

Jlp
1 - ^
w
(12)
U (V\) U
Bw ^ \w /o\ exp — r Bw
with r taking the value

np L
r =
1 -

to provide a good fit; the curve goes through the experimental point
{U/Bw^)u{V/w)i.
Values of the variables with a fit within 2 percent are given in Table 7. We
see that the linear relationship is sufficient for a/w > 0.7 for p = 0 and for
a/w > 0.6 for the two other radii, and that T;^ = rjp is relatively radius in-
dependent with a decreasing value with increasing crack length, but this
value is lower than the theoretical ones reported in Table 5. Below this limit,
some decreasing trend is observed which is in disagreement with theory. No
clear explanation has been found, though limited backward plasticity may
be the reason. Moreover, we may conclude that slow crack growth does not
affect these results very much in the interpolation range.
From a practical point of view, we note that the V/w versus a/w relation-
ship at a given displacement as obtained in our treatment shows a maximum
at about a/w = 0.4 which is displaced by v/w in the reverse direction of
bending. Thus more accuracy is to be assumed when testing over this range.
Moreover, we have observed that the Merkle and Corten treatment [8] leads
to calibration curves displaced toward higher V/w values.

Results for the Fracture Criterion

The previous calibration results have been used to obtain the crack growth
resistance curves with some extension to other specimen sizes. The double
bend technique [11] has been applied with heat tinting or with brittle frac-
ture at — 196°C to measure the ductile crack extension Aa. Partial results
are shown in Figs. 3 and 4, with the extreme point corresponding to the
maximum load. No curve could be easily fit to data points.
For the effect of specimen size, we see that what is partially attributed to
the stretch zone development is at the left of the V = (a^ -f a„) Aa straight
350 ELASTIC-PLASTIC FRACTURE

lOvOv^^O-^r^tNO • ^ s O ' ^ v O T r n ^ O TtvOt^\OlOfn-HO

n ^ r - ( N 0 Q r ^ O i / ) a^Troo^ol^^a^l/) l^rN^D^/)^^loo^lA)
fOi^o-^roq-^iNO r^ji-^o-^roottwo r^oo^-^roo-^—"o

^Hr^t---fNooi^oirt (yNTt-QO(T)r^\oo^i/> (N<Nro*/)i^i/><?^io

0 0 0 - ^ O 6 T ) - ( N O (NI~-O^OO'^'-<O «OO^'<J;<IO^«O
(Nf^rj^oooo rofNr^^oooo r^(Nr>i^oooo

00 t-- ^ t ^ •» O i n r^
TT ^ 00 - ^ >i ^ viS -^
•-H O O O O O O O ^ ' —' O O O O O O •rt -H o o e> o o o

J - H O O O O O O O ^ O O O O O O O - H O O O O O O O

^£> fO 00
O—<»H<N<N<N<N-^ O-H-^fNININtN-^ O ^* ^ *N <N r4 fN --'

*; (N<N<Nr>lfN(N(N—H rNl(N(NrN(N(NfNI»^ rsl(N<N<NfNfS(Nw

u fS^^•^l/>^£>^-«o^ «Nro^»/>^or-;00a> iNco'^io^r^ooO;

« S
n o o o o o o o o o o o o o o o o o o o o o o o o

rH^«HfNfS(N(N(N » ^ ^ ^ ( N < N ( N ( N ( N ^•-H^iNtNtNCNrS

J
TO ' ^

1/5 S "
O 2 c
^ o
>. o,
(/5
"3 ><
 ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 351

CM
E X O
Specimen size a/w
X 5x10 x 4 0
• 10x10 x 4 0 .7
• 20x10 x 4 0
A 10x20X 80
D 20x20X 8 0
• 20x40x160
O 40x40x160
1 ,
2 .4 .6
Crack extension Aa.mtn
a) Effect of specimen size'

1/

•*•/
w .2i
E
Specimen size a/w
• .3
> / 20x20x80 o .5
.1.. A .7

ol
.2 .4 .6 .8
Cracit extension Aa mm

b)Effect of crack size

FIG. 3—Resistance curves o/V versus An for TPB specimen and Material 1.

line. No true initial extension can be pointed out, because very early, all
along the fatigue crack front, several small ductile tunnels appear as observ-
ed by microfractography, and because when the deviation point from the se-
cond straight line on the drop potential versus displacement record occurs in
the CT specimens, propagation has taken place all along the crack length.
However, when fibrous fracture concerns all the front, some trends can be
deduced in spite of the scatter of the data points: for the CT specimen the
resistance is independent of size; for the TPB specimen, for similar
geometries, resistance and resistance gradient decrease with increasing size;
352 ELASTIC-PLASTIC FRACTURE

/
*/
• ^ /

»'/
*/

.2.. /
/ A A
/ A HO

S A f'c Specimen size a/w

> • 20x40x160
ACTIO .7
OCT 20
ACT 40
X CT80
•/
+
.V .4 .6 ^8
Crack extension Aa.mm

a) Effect of specimen size

.2..

s Specimen size a/w

• .4
> 1-. CT20 O .55
A .7

/
I
.2 .4 .6 .8 1
Crack extension Aa mm

b) Effect of crack s i z e
FIG. 4—Resistance curves ofV versus Aafor CT specimen and Material 2.

for a geometry where thickness is only varying, resistance and resistance gra-
dient increase with increasing size. No explanation is apparent. However,
these observations may be put together with the extension mode: in all
geometries, onset of crack growth occurs at midthickness; then, in CT
specimens this initial "fibrous thumbnail" develops to spread uniformly all
along the fatigue front; in TPB specimens, this thumbnail develops forward
by tunneling when thickness is low and stops and two other thumbnails
 ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 353

develop on both sides when thickness is high. The two extreme modes may be
relevant to plane-stress and plane-strain conditions.
For the effect of initial crack length, nothing is noted.
For the effect of geometry, according to our limited data and with the use
of 7; = 1.86 and 2.2 in the relation V = riU/B (w - a) for a/w = 0.7, for
the two geometries, nothing is noted.
From a practical point of view, in our state of knowledge, it seems very dif-
ficult to define a critical value to characterize toughness, though it should be
possible to define a critical Aa value for which all values would be the same.
However, crack growth resistance may represent toughness. In this respect,
CT specimens appear more attractive, but their behavior may be fortuitous
here and due to the material under investigation. But it seems reasonable to
consider toughness as geometry and size dependent. Increasing size would
allow resistance over a larger range to be obtained.

Conclusion
V/w versus U/Bw ^ relationships have been established for the TPB and
CT specimens. Due to normalizing, they are to cover all specimen and crack
sizes. From them it is shown, with a limitation due to the uniqueness of the
material under study, that toughness is independent of crack length and
loading mode, but depends on size.

Acknowledgment
This investigation was made possible by a research grant from the Delega-
tion Generale a la Recherche Scientifique et Technique. The support of our
respective laboratories is gratefully acknowledged.

APPENDIX
Three-Point Bend Specimen
Elastic Behavior
. Srawley 112]:

Ki = B-Jw

(13)
0 1^ «W-,.^ ,„, a ,-,-./a'^
- - 1.99 - — 1 - ~ 2.15 - 3.93 — + 2.7
w/ \w
2(1 - ^ 2 ^ 1 - "
w/ \ w
354 ELASTIC-PLASTIC FRACTURE

for 0 < a/w s 1

Elastic beam theory:

C(0)
4E w^ -f(F"-) (14)

Bucci et al [5]:

C(0) = ^ ^ 1.04 + 3.28 ( — ) (1 + c) (15)

Srawley [/2] (from):

25^
C(a) = - 19.37 — + 8.72 (—) - 6.10 {—
£"tv2 w \w I Xw,

+ 2.98 l - ^ j - 0.82 ( - ^ j + 13.54 In ( l + 2 - ^

(16)
a a
- 2.26 In 1 10.39 - 0.57
1 + 2 1 -

^ 2 - ^
w \ w
+ 0.49
w
Fully Plastic Behavior
In plane strain:
Pu = ; 8 / e r / y (w - a)^ (17)

where
/3 = 1 for a Tresca material,
2/V3 — for a Von Mises material,
Of = uniaxial tensile flow stress = {oy + au)/2,
Oy — uniaxial yield stress, and
ff„ = ultimate tensile strength.
From Ewing [13]:

f = 1.261 for — > 0.290 (Charpy notch) 0.296 (sharp notch)

/ = 1 for — = 0

From Chell and Spink [14]:

f = 1.261 - 2.72 10.31 - —) for 0 < — < 0.31

 ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 355

with the assumption of/ being a continuous and differentiable function between the
Ewing' limits from which V is continuous—from adjustment to Knott's results [75] in
w/(w — a) for Charpy notch ,

/ = 1.26 - 1.56 1.408 for 0 < — < 0.29 (18)

^w — a

and with the other condition of V = 0 for a/w = 0

/ = 1.261 - 3.8314 (0.261 - a/w)^ (19)

In plane stress:
20 B , „
(20)

for a/w > 0.02 for a Von Mises material and a/w > 0 for a Tresca material.

Compact Tension Specimen

Elastic Behavior
Srawley [12]:

K, =
0.886 + 4.64 — -- 13.32
ii)'-- 5.6 (t)1
fe)' + ''•''
B-Jw 3/2
1 - ^
(21)
for 0.2 < a/w < 1
Adams and Munro [16]:
2.2
(22)
Srawley [12] (from):
cm = ^

C(a) = 119.11 — + 75.04 i — f - 92.83 (—]

w \w/ \wi

- 69.00 l-^j + 27.01 IAJ

fey + 32.76 K
(23)
- 15.62 7.84 l - ^ j + 134.84 In (1 - -^

a
w \ w w
+ 15.82 - 9.64
1 -
356 ELASTIC-PLASTIC FRACTURE

with the restriction of Y (a/w) not defined for (a/w) < 0.2.
Newman [17]: tabulated values of EC.

Fully Plastic Behavior

Ewing and Richards [18] (lower-bound solution):

PL = ffofwB 2.7 + 4.59 ( — 1 + 1.7 (24)

w

PL - a/wB (1 + /3) 1 + 1 + ^ - (25)

w

9
M
L^NV-
CO
X^3 -2
9 y^L 4 •
CM

a
m

1. FL= .325 w'k

2 . FL^ .3w^k
Partial enlarged view of solution 3.
3. FL= .18w'k
4 . EWING and RICHARDS [17]
(k = shear yield stress)

FIG. 5—Schematic of slip line fields in the CT specimen.

For the geometry under consideration, these solutions are valid above a limit which is
given for plane strain by the upper-bound theorem as a/w ~ 0.27, or with more re-
fined slip line fields as shown in Fig. 5 as a/w - 0.25. Therefore, due to the strain
hardening of the material, flow may take place through the pinhole for values of a/w
less than, say, 0.35.

References
[/] Ritchie, R. O., Garrett, G. G., and Knott, J. F., International Journal of Fracture
Mechanics, Vol. 7, 1971, pp. 462-467.
[2] Rice, J. R. in Fracture, H. Liebowitz, Ed., Vol. 2, Academic Press, New York, 1968, pp.
191-311.
[3] McClintock, F. A. in Fracture, H. Liebowitz, Ed., Vol. 3, Academic Press, New York,
1971, pp. 47-225.
[4\ Begley, J. A. and Landes, J. D. mFracture Toughness, ASTMSTP514, American Society
for Testing and Materials, 1972, pp. 1-23 and pp. 24-39.
[5] Bucci, R. J., Paris, P. C, Landes, J. D. and Rice, J. R. in Fracture Toughness, ASTM
STP 514, American Society for Testing and Materials, 1972, pp. 40-69.
[6] Rice, I. R., Paris, P. C, and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing, ASTM STP 536, American Society for Testing and Materials, 1973,
pp. 231-245.
 ROYER ET AL ON ESTIMATING FRACTURE TOUGHNESS 357

[7] Miannay, D. and Pelissier-Tanon, A., M^canique Materiaux Electricite, No. 328-329,
1977, pp. 29-40.
\8] Merkle, J. G. and Corten, H. T., Journal of Pressure Vessel Technology, No. 11, 1974, pp.
286-292.
[9] Larsson, S. G. and Carlsson, A. F., Journal of the Mechanics and Physics of Solids, Vol.
21, 1973, pp. 263-277.
[10] Rice, J. R., Journal of the Mechanics and Physics of Solids, Vol. 22, No. 1, 1974, pp.
17-26.
[//] Landes, J. D. andBegley, J, A. \n Fracture Analysis, ASTMSTP560. American Society for
Testing and Materials, 1974, pp. 170-186.
[12] Srawley, J. E., International Journal of Fracture Mechanics, Vol. 12, No. 3, 1976, pp.
475-476.
[13] Ewing, D. J. F., Journal of the Mechanics and Physics of Solids, Vol. 16, 1968, pp.
205-213.
[14] Chell, G. G. and Spink, G. M., Engineering Fracture Mechanics, Vol. 9, 1977, pp.
101-121.
[15] Knott, J. F. in Fracture 1969, Proceedings of the Second International Conference on Frac-
ture, Chapman and Hall Ltd., London, 1969, pp. 205-218.
[16] Adams, N. J. I. and Munro, H. G., Engineering Fracture Mechanics, Vol. 16, 1974, pp.
119-132.
[17] Newman, J. C , Jr. in Fracture Analysis, ASTM STP 560, American Society for Testing
and Materials, 1974, pp. 105-121.
[18] Ewing, D. J. F. and Richards, C. E., Journal of the Mechanics and Physics of Solids, Vol.
22, 1974, pp. 27-36.
/. Milne' and G. G. ChelP

Effect of Size on the J Fracture

Criterion

REFERENCE: Milne, I. and Chell, G. G., "Effect of Size on the /Fractnre Criterion,"
Elastic-Plastic Fracture, ASTM STP 668. J. D. Landes, J. A. Begley, and G. A. Clarke.
Eds., American Society for Testing and Materials, 1979, pp. 358-377.

ABSTRACT: Experimental evidence showing the size dependence of Ju in ferritic

steels is presented. This behavior is described in terms of a relatively simple mechanistic
model for cleavage fracture. It is shown that the mechanisms of cleavage fracture and
ductile slow crack growth are always in competition and this leads to the behavior
frequently encountered in fracture tests where initially ductile crack extension leads
to fast brittle failure. The size eifect is also discussed in terms of a shift in the ductile
brittle transition temperature. The implications on failure assessment procedures are
mentioned.

KEY WORDS: fracture criterion, size effect, cleavage failure, slow crack growth,
/-integral, fracture toughness, ferritic steels, ductile brittle transition (toughness),
failure assessment, elastic-plastic, crack propagation

Although in the small-scale yielding regime failure can be characterized

a one-parameter criterion, such as the fracture toughness Kic, it is not
clear if this is applicable after appreciable yielding. Experimental evidence
based upon evaluation of the 7-integral at failure suggests that under some
circumstances this may be the case [1].^ Alternatively, there is an increasing
amount of evidence demonstrating situations in which this is not so, and a
geometry effect is apparent. In this paper we briefly review the latter evidence
and investigate some of its implications on fracture toughness testing and
service assessments, confining ourselves to ferritic steels. Existing concepts
and models of fracture are used to provide a description of fracture behavior
consistent with the observed size dependence of the / failure criterion. The
model predictions are related to fractographic features and the occurrence
of slow crack growth prior to fast failure.

'Research officer and Fracture Mechanics Project leader, respectively. Materials Division,
Central Electricity Research Laboratories, Kelvin Avenue, Leatherhead, Surrey, U.K.
^The italic numbers in brackets refer to the list of references appended to this paper.

358

Copyright' 1979 b y AS FM International www.astm.org

 MILNE AND CHELL ON J FRACTURE CRITERION 359

Failure Criterion
The failure criterion is based on the attainment of a critical value, /k,
of the parameter/ [1,2]. Consistent with the usual experimental method of
determining/ [2], we interpret7ic as characterizing the maxima in the total
energy. This interpretation is similar to that proposed by Griffith for brittle
failure. Jic therefore represents a critical force which is exerted on the crack
and plastic zone at failure. It should be noted that the energy released
by the propagation of the crack an increment Aa is not — JAa [1,3].
The problem of relating this macroscopic parameter / to the metallurgical
mechanisms of failure is still unresolved. The assumption that/characterizes
the crack tip environment [/], although supported by some theoretical
evidence [4], is still not proven in the case of real materials. Thus while it
may have credence for monotonically loaded cracked bodies at constant
temperature, it is certainly not true in general [5].
To aid comparison with fracture toughness, we define a plastic stress
intensity factor, Kp, which is equal to a critical value Ku at failure and is
related to / through the equation

where E is Young's modulus and v Poisson's ratio. Putting / = Jw in Eq 1

provides the necessary relationship between Ku and / k . In circumstances
where plastic deformation is negligible, Kp is, of course, equal to the stress
intensity factor calculated linear elastically. It thus follows that, provided
/ic is a valid fracture criterion, Ku is numerically equal to ^ic since Ju =
(1 - v^)KuyE.

Experimental Evidence Showing a Geometry Dependence of J/c

Much of the work in elastic-plastic fracture mechanics has been aimed
at establishing the equivalence of Ku and Ku. This has been successful
within limits and the current argument tends to center around the definition
of these limits [2]. This tends to beg the question, however, since there
obviously is c geometry dependence to Ju. Moreover, size limits cannot be
imposed on real structures. In particular, where tests have been performed
without regard to geometry limitations, large variations in Ju, and hence
Ku, have been observed as a function of crack size a, a/w (where w is the
specimen width), and specimen size. Table 1 lists much of the recent work
performed in this area. Clearly the effect is not confined to one type of
steel or one specimen geometry. This effect is also dramatically illustrated
in Fig. 1, where the results for the single-edge notched tension (SENT)
tests listed in Table 1 are plotted in terms of Ku/Ku against the crack
360 ELASTIC-PLASTIC FRACTURE

TABLE 1—Published vmrk where a size dependence of toughness in ferritic steels has been
noted.

Reference Material Testing Geometry

[6] maraging steel three-point bend

[7] high-temperature bolting steel three-point bend
and a rotor steel
[«] pressure vessel steel three-point bend
[9] high-temperature bolting steel single edge notched tension
[10] high-temperature bolting steel single edge notched tension
and rotor steel
[11] A533B pressure vessel steel compact tension

length a, normalized for convenience by K\^/a^, ff„ being the ultimate

tensile strength. The values oiKw were obtained using a load displacement
curve fitting technique which enables/to be determined for a given specimen
using only the recorded data for that specimen \lO,30\. The increase in K\i

3.0 r

25J VALIDITY LIMIT

VALIDITY LIMIT

2.5 ^

FIG. 1—Effect of size on Kij/or single-edge notched tension specimens.

 MILNE AND CHELL ON J FRACTURE CRITERION 361

occurs not only as a/w decreases, but also as the specimen size decreases
although this is not apparent from the figure.
It is interesting to note that if

a = 257,c/ffy( = 25(l - v^Wu^/Eay)

the minimum value required for a valid Jic i,K\c) value [J2], then the curve

aOuVKyi' = 0.1 {K^^/KxiY

in Fig. 1 (dotted line) is the validity limit below which size effects should
become apparent. The data in Fig. 1 thus meet the proposed validity re-
quirement for a /ic analysis, indicating that this proposal is inadequate.
A limit of 150 J\JOY, full line in Fig. 1, would be more satisfactory \10\.

Fractography
The behavior illustrated in Fig. 1 could be attributed to several causes:
slow crack growth prior to fast fracture, a change in the mode of failure,
or the inability of the analytical technique to correctly predict the failure
criterion. This latter point can be countered by the similarity of the pre-
dictions using different methods of / analyses [/(?] and by the fact that in
at least two instances \6,S\ the failure criterion calculated without any
plasticity correction still exceeded K\^. Thus we are left with having to
explain the phenomenon in terms of mechanistic or fractographic behavior.
To investigate these features, the fracture surfaces of four of the steels
listed in Table 1 were examined in detail using a high-resolution Camebax
scanning electron microscope. The four steels, whose composition and
mechanical properties are listed in Table 2, were
(A) BS 1501 271 A: A fine grained, pearlitic pressure vessel steel which
had been tested in three-point bend at 130°C \8\. The fracture toughness
of this steel had previously been measured at between 56 and 69 MNm~^^^
at —30°C, yet small specimens produced/if u values as high as 270 MNm^^^^.
(B) A medium-strength quenched-and-tempered bainitic steel which
had been tested in three-point bend between — 70°C and room temperature
\12\. This steel suffered from bands of inclusions; mainly of manganese
sulphide and titanium nitride, oriented in the rolling plane. Loss of linearity
in the load displacement curves, where this occurred, was primarily due to
the slow stable crack growth. Since the initiation point was not known,
K\i was not determined.
(C) A quenched-and-tempered bainitic rotor forging steel which had
been tested using SENT specimens at room temperature \10,13\. Results
shown in Fig. 1 are for this steel.
(D) A high-temperature bolting steel heat treated to give a bainitic
362 ELASTIC-PLASTIC FRACTURE

3 8

1/1
o O S
d d d

s q
d d d

00
q q q
d d d

00
00 § in

00
•a
c d d

I U o
d

i2

o
>= z

^ i§|l|tfl
I
 MILNE AND CHELL ON J FRACTURE CRITERION 363

Structure [10]. Tests were performed on SENT specimens and the results
are also shown in Fig. 1.
In each case the area studied was confined to that region immediately
ahead of the fatigued starter crack.
Despite the differences in the four steels studied, the fracture surfaces
exhibited similar features which could be categorized in the following way.
1. Fully brittle, as in Fig. 2. Here immediately adjoining the stretch
zone the fracture surface was made up almost entirely of cleavage facets,
often with microcracks and sharp stepped features associated with it. There
was no evidence that changes in orientation from one cleavage facet to
another involved any substantial amount of ductile fracture. These frac-
tures were associated with the lower temperatures of Steel B and the lower
K^ values in the other steels {Ku/Ku < 1.3).
2. Brittle, but with isolated ductile regions, as in Fig. 3. Here, although
the fracture surface was predominantly brittle, as in Fig. 2, regions of
ductile fracture, generally no larger than a grain, occurred randomly dis-
tributed along the tip of the stretch zone. The dimpled features of these
areas were always much finer than the main features associated with ductile
slow crack growth (compare Figs. 3 and 4), were similar to that observed
in the shear lip regions, Fig. 5, but were never associated with inclusions.
It should be emphasized that the ductile areas were always isolated from
each other and should not be confused with ductile slow crack growth.
They were, however, present in those specimens which exhibited high
Kit values {Ku/Ku > 1.3) and in particular although not exclusively
where the crack lengths were short. There were less of these regions of
ductile fracture in areas of the fracture surface remote from the fatigue
starter crack.
3. Ductile slow crack growth as in Fig. 4. These regions were observed
only for the shorter cracked specimens of the smallest size of Steels A and
D, and for the higher temperatures of Steel B. In this latter instance some
tests failed entirely in the slow ductile mode, while others started in the
ductile mode but changed eventually to fast brittle fracture. Often in these
cases there was a sharp transition between the ductile and the brittle re-
gions, yet some brittle fracture could be observed well within the slow
crack growth areas and some ductile dimpling within the •predominantly
brittle region. TTie ductile areas contained features on two different scales:
(1) voids 15 to 25 fim in size which generally contained inclusions or other
nonmetallic particles and (2)finedimples less than ~ 2 nm in size.
The dimpled regions tended to link one void with another. The individual
dimples were comparable in scale to similar features observed in the duc-
tile r ^ o n s in 2 of the foregoing, and also to the ductile dimples observed
in the shear lip regions (Fig. 5).
The voids, on the other hand, apart from containing inclusions, had a
similar appearance to the stretch zones developed in all of the specimens.
364 ELASTIC-PLASTIC FRACTURE

.,[ T'f^ -C

>t,

I
e

i
 MILNE AND CHELL ON J FRACTURE CRITERION 365

these being characteristic of stretch zones typically found in steels of this

nature [14]. This similarity between the stretch zones and the voids sug-
gests that the voids are really a quasi-continuous extension of the stretch
zone.
Ductile tearing of this nature, that is, void growth generally around in-
clusions linked by bridges of ductile dimples, can occur either at plastic
collapse or during slow stable crack growth. The latter is crack tip con-
trolled rather than controlled by the dimensions of the uncracked liga-
ment, and as such, although the microprocesses appear to be the same,
the mechanical description of failure is different from plastic collapse. As
mentioned previously, this slow crack growth is not just a special case of
ductile fracture but is a mode of propagation in its own right [12].
This should be compared with the general observation that, where the
fracture process was fast, a high measure of fine ductile dimple fracture
was observed on the surfaces of specimens where Ku was much higher than
Kic; for example, compare Fig. 3a and 2b.

Descriptions of Fracture Beliavior Based upon Simple Mechanistic Models

Cleavage Fracture
Recently a model of cleavage fracture from sharp cracks has been pro-
posed based upon the postulate that fracture will occur when the stress
normal to the crack plane exceeds the cleavage stress over a characteristic
distance ahead of the crack tip [15]. This distance is associated with some
microstructural feature such as the grain size or carbide spacing. The
model is appealing since it explains an apparent anomaly in the compara-
tive magnitudes of Charpy energies and fracture toughness values of two
steels [16], as well as the increase in toughness with temperature due to
changes in yield stress [15,31].
In applying this model to the size effect described in the foregoing and to
relate / to the failure mechanism, we assume that the stress field ahead
of the crack can be characterized by /, even in the large-scale yielding
regime. Hence, contrary to general belief, when the failure mechanism
is taken into account, a size dependence of / is qualitatively predictable.
If ff represents a flow stress and Xo the characteristic distance over which
the normal stress OYY must exceed the cleavage fracture stress a/, then
the point of failure, F, for a given specimen and material, is shown schemati-
cally in Fig. 6 for plane-strain conditions. The quantity X, a measure of
the distance ahead of the crack, has been normalized by EJ/ a^. In a
smaller specimen of identical material loaded to the same value of /, the
stress field directly ahead of the crack should be the same as before. How-
ever, let us assume that in the smaller specimen failure occurs after large-
scale plasticity such that some through-thickness deformation occurs with
366 ELASTIC-PLASTIC FRACTURE

\ ViCMiiiiilK >e » a

ut rvj
 MILNE AND CHELL ON J FRACTURE CRITERION 367

£S i ^ i l
368 ELASTIC-PLASTIC FRACTURE
 MILNE AND CHELL ON J FRACTURE CRITERION 369

50>tm

FIG. 5—Ductile features of a shear lip.

a resulting loss of stress triaxiality. The new stress field, although still
characterized by the same value of 7, will now fall below the previous level
near the crack tip, and thus, when / = /ic, failure will not occur because
the critical stress condition for cleavage is not satisfied (see Fig. 6). To
satisfy this condition, extra load must be added so that / at failure be-
comes greater than Ji^. This results in a value of Ku which exceeds Kic.
Since the effect is likely to be most pronounced in failures occurring after
general yielding, the extra load needed to fracture the specimen will re-
sult in an even greater loss of stress elevation. Thus in this regime the
conditions necessary to attain cleavage are in direct competition with the
consequences of trying to attain it. At some stage, cleavage will not be at-
tainable and another mode of failure will take over.
After general yielding there is some loss of constraint. A measure of
the plastic constraint, R, in the specimen is given by the ratio of plane-
strain to plane-stress collapse loads. If the observed increase in Ku is a
consequence of loss of stress elevation, then it should be less pronounced
in the more highly constrained geometries. The predictions of slip-line field
theory for both three-point bend [18] and SENT geometries [19] indicate
increasing values of/? with increasing ratio a/w. (In the case of SENT this
370 ELASTIC-PLASTIC FRACTURE

STRESS RESULTING FROM LOSS

TRIAXIALITY (SMALL SPECIMEN)

001 002

EJ

FIG. 6—Schematic representation of the stress profiles ahead of cracks in large and small
specimens with the same J-value.

peaks at about a/w = 0.5 and decreases again.) This is in agreement with
the observed increase in Ku as crack length a is decreased in a specimen
of given size.
There are effects due to crack tip blunting in addition to the complica-
tions inherent in trying to achieve the critical stress over the critical dis-
tance where plasticity is large. Blunting creates a localized area of plane
stress ahead of the crack and a resulting intense strain region [77] which
will amplify the effects discussed in the foregoing. Indeed the ductile ap-
pearance of the stretch zone is a direct surface manifestation of this strain-
ing. Furthermore, if p is the root radius of a notch, the maximum stress
elevation ahead of it will depend on a/p and the effect of blunting will be
most pronounced for small crack sizes. Since the increasing values of
Ku are consistent with increasing stretch zone size and, in general, since
p will be a function of the stretch zone size, the effect of the blunting is
to make the attainment of the critical stress more difficult.
The increase in the amount of ductility on the fracture surfaces of speci-
mens with increasing ATij values clearly reflects the increasing amount of
strain occurring ahead of the crack as triaxiality is lost. The rise in Ku
values as size is decreased (see Fig. 1) is also consistent with the sudden
 MILNE AND CHELL ON J FRACTURE CRITERION 371

increased sensitivity of J to the applied load after general yielding is reached

(Fig. 7a). After general yielding, the macroscopic deformation of the speci-
men becomes progressively more consistent with plane-stress rather than
plane-strain conditions [20] and large displacements occur (Fig. 7b). In
this region small changes in load can result in large changes in both the
value of/ and displacements which are consistent with the sudden increase
in Ku.

Ductile Slow Crack Growth ^

As previously mentioned, at some stage the critical stress criterion can-
not be satisfied because the "maximum achievable stress levels are essentially
limited, even with continuous strain hardening" [17]. This is "suggestive
of abrupt toughness transitions," although, unlike Ref 17, we refer to
changes in the value of the failure parameter at constant temperature, and
not to changes due to increases in temperature or loading rate. Eventually
a transition in failure mode occurs, cleavage fracture cannot be initiated,
and the crack advances instead in a stable manner.
The initiation of this slow crack growth arises due to the formation of
voids ahead of the crack tip, and their subsequent linking by local necking
of the remaining material [17]. Thus the crack can be thought to advance
in a series of jumps [26] as each void becomes linked with its neighbor.

PLANE STRAIN

DISPLACEMENT

FIG. 7—Schematic representation showing sudden increase in J and displacement as a

function of load after general yielding.

To the authors' knowledge there is no evidence of slow crack growth by brittle cleavage,
although a quasi-static fast fracture mode is possible in specimens where the stress intensity
factor falls with increasing crack length, or where crack front geometry changes result in a
lower stress intensity.
372 ELASTIC-PLASTIC FRACTURE

The crack tip displacement required to do this is inversely related to the

spacing of the metallurgical features, inclusions, precipitates, and grain
boundary triple points, around which the voids nucleate. Thus the micro-
structure will have an important influence over the incidence of slow crack
growth, and we would expect steels containing a high density of inclusions,
or spheroidals carbides, to more readily exhibit slow crack growth prior
to fast fracture than those steels without any obvious nuclei where holes
can nucleate.
For voids to initiate and grow around inclusions, there must be a large
amount of plastic strain resulting from loss of stress triaxiality. Never-
theless, voids will not grow without a certain amount of stress elevation
[17,23]. This is borne out by observations on the fracture surfaces of speci-
mens where slow crack growth occurs. The ductile stable crack is held up
at the surfaces, where triaxiality is a minimum, and advances most rapidly
in the center of the specimen in a thumbnail geometry. In the surface
regions of the specimen where there is no stress elevation, ductile shear lips
develop without the presence of large voids around inclusions. These shear
lips are similar to those developed on surfaces which have failed by fast
fracture, and contain fine ductile dimples. Moreover, specimens which have
been side grooved to artificially increase constraint [24], and which fail
by slow crack growth, do so with a uniformly straight crack front rather
than a thumbnail. These considerations lead to a mechanism for slow crack
growth based upon the postulate that ductile crack propagation will occur
when a critical plastic strain (which will depend on the state of stress) is
exceeded over a characteristic distance associated with the microstructure
(for example, inclusion spacing) [17,21,22].
Frequently after some amount of slow crack growth the crack propagates
in a brittle manner. This can result from several causes, not least the hetero-
geneity of the material. Regardless of this, as the geometry of the crack
front becomes more convex, through-thickness deformation diminishes
and triaxiality increases local to the advancing crack tip. It has also been
suggested that the stress level is raised further by crack tip sharpening
as the crack propagates [25]. The overall effect is to make cleavage more
favorable so that a change from ductile slow crack growth to brittle fast
fracture can occur.
In summary, both slow crack growth and cleavage are favored by a large
degree of triaxiality, but slow crack growth also needs large plastic strains
to induce void growth. For slow crack growth to be preferred, these con-
ditions must be satisfied before the stress ahead of the crack can be elevated
above a/ over the critical distance for cleavage [31]. The mode of failure
which predominates depends upon microstructural features (for example,
carbide and inclusion spacings) as well as mechanical effects such as loss
of through-thickness constraint and crack tip blunting. In a sense, there-
fore, the fast cleavage and slow ductile modes of crack propagation are
 MILNE AND CHELL ON J FRACTURE CRITERION 373

always in competition, even at temperatures below the ductile brittle transi-

tion.

Geometry Dependence of Ku and the Ductile Brittle Transition

We assume that the plane-strain fracture toughness, Kic, is, as postulated,
a genuine material parameter which is measurable only on large enough
(valid) specimens. Brittle cleavage fracture is then expected in the lower-
shelf region; fast ductile fracture, which is distinct from slow crack growth
and has not so far been discussed here, is expected in the upp^r-shelf re-
gion. These two regions are linked by the transition region, which is a
mixed fracture mode.
The observed increases in Ku as a function of decreasing a, a/w or
specimen size, and associated with extra ductility in the fracture surface,
have an analogy in the effects of temperature on toughness in the ductile
brittle transition region. This transition, in body centered cubic alloys,
and the associated notch sensitivity was first explained by Orowan [27]. It
is a consequence of the temperature dependence of the yield or flow stress
in these materials, and the temperature independence of the cleavage stress.
Increasing geometric constaint, by notching a specimen or by increasing its
size, raises the stress triaxiality and increases the ductile brittle transition
temperature, creating the notch or size sensitivity. There is no apparent
reason why these concepts should not extend to specimens containing
sharp cracks. Indeed it is of interest to note that in a fracture analysis
of invalid compact tension test data on A533B steel, it was concluded that
there were indications of a size effect on the ductile brittle transition tem-
perature [11].
Using models of cleavage failure it should be possible to quantify the
effects of size, at a given temperature, T, in terms of a shift in the ductile
brittle transition temperature. For example, the model proposed in Ref 15
allows changes in toughness resulting from variations in yield stress and
stress elevation to be taken into account. If the stress field ahead of a
crack is known, and the effects of crack tip blunting are suitably simulated,
then the model enables the effects of size to be expressed in terms of a
change in yield stress. This can be directly related to a temperature shift
through the yield stress-temperature curve.
Thus continuum analyses based upon only macroscopic variables (for
example / ) cannot predict Ku from an invalid test if the size, geometry,
and material of the specimen is such that Ku is greater than Kjc at the
testing temperature. For A^^ to be obtained from such a test, the mechanism
for failure must be defined. Indeed the attainment of a critical value of/,
although macroscopically necessary, is not, on the microscale, always a suf-
ficient condition to initiate fast fracture. Hence, if Ku (T) represents the
temperature dependence of the toughness, we can write
374 ELASTIC-PLASTIC FRACTURE

Kv{T) = K,.{T + AT)

where AT is the shift in the transition curve resulting from size effects.
In the transition region the difference between Kv and ^ic depends
very much upon the magnitude of this shift in the transition curve. For a
given shift, a steel with a sharp transition will show a greater effect than
one with a gradual transition. The lower end of the transition is always
gradual, so tests performed in this region may exhibit only a small in-
crease in Kv over Kh, which may be contained within the experimental
scatter band.
The foregoing description has excluded the possibility of slow crack
growth, which complicates the problem even further and makes a general
description difficult. It also leads to questions concerning the relationship
of failure parameters determined at the initiation of slow crack growth
to the same failure parameters calculated at the onset of fast fracture.
These questions have not, as yet, been satisfactorily resolved.

Failnre Assessments
Although much of the previous discussion has revolved around the micro-
processes of fracture, these have to be represented in some mechanical
way (that is, via macroscopic variables) before they can be used in an assess-
ment. Figure 8 represents how the order of events (from Path 1 to Path 5)
leading to failure of a cracked body can change as the triaxiality ahead
of the crack is reduced. This also shows that, excluding fast ductile failure,
there are only two mechanical descriptions of failure, brittle fracture and
plastic collapse [28], The question to be answered in a failure assessment
is which of the alternative paths to failure will be followed by the structure,
and how can some measure of control be introduced into each of these
paths? Once the path to failure has been established, the difficult task of
obtaining relevant materials parameters must then be faced.
Current assessments are based upon initiation data. Thus if the two failure
limits can be reconciled, and there are procedural manuals now available
for doing this [29], the problem can be handled in principle. However, for
tough materials, the size limitations of test specimens will cause them to
fail on any of the paths from 2 to 5. The previous discussion and the data
used therein indicate that there is a risk that specimens failing along Path
2 may produce Kv in excess of Ku, especially in the ductile brittle transi-
tion region. This can lead to an overestimate of the defect tolerance of a
structure. Moreover, for those specimens which follow Path 3, slow crack
growth initiation occurs before the specimen reaches its full load bearing
capacity. Here it is generally thought that there is a likelihood of under-
estimating the defect tolerance of the structure. This is not always the case
since Ku is definable as the minimum possible value for Ku at the relevant
 MILNE AND CHELL ON J FRACTURE CRITERION 375

CRACKED
BODY.
INCREASING
LOAD

LINEAR
ELASTIC
DISPLACEMENT

ELASTIC-
1" PLASTIC
EFFECTS

INCREASING
TRIAXIALITY
BRITTLE
FRACTURE

FIG. 8—Different failure paths which a cracked body can follow.

temperature. It follows therefore that the ductile initiation value for Ku

can also exceed Kk- This argument is equally applicable to Path 4, of
course, since the only difference between Paths 3 and 4 is that sufficient
triaxiality for cleavage cannot be generated in the latter instance. Thus
a simple mechanical representation of the failure conditions of a body is
not sufficient to accurately predict the load-carrying capacity of a struc-
ture where failure is likely to be by any route other than Paths 1 and 5.
There are two things needed to cover Routes 2 to 4 in a way that avoids
undue pessimism.
1. It is necessary to accurately assess the triaxiality local to the crack
tip, taking into account the changing shape of a growing crack and the
location of the crack front.
2. A way must be discovered to relate how this triaxiality determines
which of the microprocesses, cleavage or void growth, is favored.
If the triaxiality in a structure cannot be developed to the level where
cleavage is initiated, assessment can be based upon the amount of growth
376 ELASTIC-PLASTIC FRACTURE

necessary to reach plastic collapse under the applied loads. Alternatively,

it can be based upon the geometric changes required of the crack to in-
voke the triaxiality necessary for cleavage. Clearly, despite the current
activity in helping to understand the mechanics of slow crack growth, we
are a long way from developing the techniques for solving this problem.
In the meantime the problems discussed in the foregoing are best handled
by designing to initiation. In a slow crack growth situation, this is highly
pessimistic since it does not use the full load bearing capacity of the struc-
ture. However, it should be recognized that there is a chance that the
initiation data obtained from small specimens could exceed Kic.

Conclusions
1. There is a distinct possibility for ferritic steels that fracture toughness
values obtained by elastic-plastic analyses of invalid-sized specimens can
exceed Ku.
2. This behavior can be qualitatively described using relatively simple
mechanistic models of fracture, and results from a loss of stress triaxiality
ahead of the crack due to loss of through thickness constraint and crack
tip blunting.
3. This size effect can be represented in terms of a shift in the ductile
brittle transition temperature.
4. In general the attainment of a critical value of J, although a macro-
scopic requirement for fast fracture, is not necessarily a sufficient one on
the microscopic level.

Acknowledgments
The authors wish to thank Drs. V. Vitek and I. L. Mogford for their
comments on the manuscript.
This work was performed at the Central Electricity Research Laboratories
and is published by permission of the Central Electricity Generating Board.

References
[/] Landes, J. D. and Begley, J. A. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 24-39.
[2] Begley, J. A. and Landes, J. D. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 1-23.
[3] Vitek, V. and Chell, G. G., Materials Science and Engineering, Vol. 27, 1977, p. 209.
[4] McClintock, F. A., in Fracture, Vol. 3, H. Liebowitz, Ed., Academic Press, New York,
1971, p. 47.
[5] Chell, G. G. and Vitek, V., International Journal of Fracture Mechanics, Vol. 13, 1977,
p. 882.
[6\ Brown, W. F. and Srawley, J. E., Plane Strain Crack Toughness Testing, ASTM STP
410, American Society for Testing and Materials, 1966, p. 16.
[7] Chell, G. G. and Spink, G. M., Engineering Fracture Mechanics, Vol. 9, 1977, p. 101.
 MILNE AND CHELL ON J FRACTURE CRITERION 377

[8] Milne, I. and Worthington, P. J., Materials Science and Engineering, Vol. 26, 1976, p.
585.
[9] Chell, G. G. and Davidson, A., Materials Science and Engineering, Vol. 24, 1976, p.
45.
[10] Chell, G. G. and Gates, R. S., International Journal of Fracture Mechanics, Vol. 14,
1978, p. 233.
[//] Sumpter, J. D. G., Metal Science, Vol. 10, 1976, p. 354.
[12] Milne, I., Materials Science and Engineering, Vol. 30, 1977, p. 241.
[13] Batte, D. A., Blackburn, W. S., Elsender, A., Hellen, T. K., Jackson, A. D., and
Poynton, W. A., to be published.
[14] Elliott, D., "The Practical Implications of Fracture Mechanisms," Spring Meeting,
Institute of Metallurgists, Newcastle-upon-Tyne, U.K., 1973, p. 21.
[15] Ritchie, R. C , Knott, J. F., and Rice, J. R., Journal of the Mechanics and Physics of
Solids, Vol. 21, 1973, p. 395.
[16] Ritchie, R. O., Francis, B., and Server, W. L., Metallurgical Transactions, Series A,
Vol. 7A, 1976, p. 831.
[17] Rice, J. R. and Johnson, M. A. in Inelastic Behavior of Solids, M. F. Kanninen et al,
Eds., McGraw-Hill, New York, 1970, p. 641.
[18] Ewing, D. J. F., Ph.D. thesis, Cambridge University, Cambridge, England, 1969.
[19] Ewing, D. J. F. and Richards, C. E., Journal of the Mechanics and Physics of Solids,
Vol. 22, 1974, p. 27.
[20] Andersson, H., Journal of the Mechanics andPhysics of Solids, Vol. 20, 1972, p. 33.
[21] McClintock, F. A., International Journal ofFracture Mechanics, Vol. 4, 1968, p. 101.
[22] MacKenzie, A. C , Hancock, J. W., and Brown, D. K., Engineering Fracture Mechanics,
Vol. 9, 1977, p. 167.
[23] McClintock, F. A., Journal of Mechanics. Vol. 35, 1968, p. 363.
[24] Green, G. and Knott, J. F,, Metals Technology, Vol. 2, 1975, p. 422.
[25] Hancock, J. W. and Cowling, M. J. in Fracture, 1977, 4th International Conference on
Fracture, D. M. R. Taplin, Ed., University of Waterloo, Waterloo, Ont., Canada, Vol.
2,1977.
[26] Clayton, J. Q. and Knott, J. F., Metal Science. Vol. 10, 1976, p. 63.
[27] Orowan, E., Reports of Progress in Physics. Vol. 12, 1948, p. 185.
[28] Dowling, A. R. and Townley, C. H. A., International Journal of Pressure Vessel Piping,
Vol. 3, 1975, p. 77.
[29] Harrison, R. P., Loosemore, K., and Milne, I., Report No. R/H/6-Revision 1, Central
Electricity Generating Board, 1977.
[30] Chell, G. G. and Milne, I., Materials Science and Engineering, Vol. 22, 1976, p. 249.
[31] Rawal, S. P. and Gurland, J., Metallurgical Transactions, Series A, Vol. 8A, 1977, p.
691.
[32] Landes, J. D. and Begley, J. A. in Fracture Analysis. ASTM STP 560. American Society
for Testing and Materials, 1973, pp. 170-186.
C. Berger,' H. P. Keller,^ and D. Munz^

Determination of Fracture Toughness

with Linear-Elastic and
Elastic-Plastic Methods

REFERENCE: Berger, C , Keller, H. P., and Munz, D., "DctenninafliMi of Ffactora

TooiiJiiKn with LineaivEhrtic and Elastic-PUstic MeAods," Elastic-Plastic Fracture.
ASTMSTP 668, J. D. Landes, J. A. Begley, and G. A. Qarke, Eds., American Society
for Testing and Matoials, 1979, pp. 378-405.

ABSTRACT: Fractute toughness was determined for two nickel-chromium-molybdenum

steds with linear-elastic and elastic-plastic methods. From the evaluation of linear-
elastic/ideal plastic load-diq>lacement curves and from the experimental results it
follows that the relation of Merkle and Corten should be applied for J-integral determi-
nation for compact specimens. The extrapolation method for the determination of a
critical /-value implies some problems. As an alternative it is proposed to determine /
at a fixed distance from the blunting line. A comparison between stress intensity
factors calculated from / and with linear-elastic methods shows that linear-elastic
fracture mechanics can be applied to much smaller specimens than given by the ASTM
Test for Bane-Strain Fracture Toughness of Metallic Materials (E 399-74). The
equivalent-energy method and measurement of crack tip opening displacement with
different clip gages at different distances from the crack tip yield stress intensity factors
in agreement with the J-integral method.

KEY WORDS: crack propagation, fracture tests, steel, I-integral, linear-elastic

fracture mechanics, crack opening displacement

Nmn^idatiiie
a Crack length
Aa„ax Crack extension at maximum load
Aost Crack extension due to crack blunting
b Ligament width
B Specimen thickness
COD Crack tip opening displacement
CODio Plane-strain crack tip opening displacement at the onset of
crack extension
' Research engineer, Kraftwerk-Union AG, Muelheim, Germany.
^Research engineer and division head, respectively, Deutsche Forschungs- und Versuch-
sanstalt far Luft- und Raumfahrt, Colc^ne, Germany.

378

Copyright 1979 b y A S T M International www.astm.org

 BERGER ET AL ON FRACTURE TOUGHNESS 379

E Young's modulus
FL Limit load for ideal plastic behavior
G Strain energy release rate
G J-integral calculated according t o Eftis and Liebowitz
J J-integral according t o Eq 2
/lo Plane-strain J-integral at the onset of crack extension
J\, Ji, J3,
JA, JAU JA2 J-integral according to approximate Eqs 3-8
K Stress intensity factor calculated with Eq 10
K* Stress intensity factor calculated with Eq 10 and plasticity
correction
Kio Plane-strain stress intensity factor at the onset of crack extension
Kj Stress intensity factor calculated from / with Eq 32
KQ Stress intensity factor determined with 5 percent secant method
Kcqu Stress intensity factor calculated according t o equivalent
energy method
U Deformation energy
Uo Deformation energy at onset o f crack extension
Unoa Deformation energy of a specimen without a crack
Ua Elastic component of deformation energy
W Specimen width
Y Function of a/ W; see Eq 10
a Size factor; see E q 31
j8 Size factor for ligament width; see Eq 33
7 Function of a/ W; see E q 12
5 Load point displacement
8„ Load point displacement due to the crack
6nocr Load point displacement of a specimen without a crack
Oy Yield strength
fffi Mean flow stress
p Poisson constant (0.33)
ij/ Function of a/ W; see Eq 9

For the determination of fracture toughness Ku with subsized specimens,

elastic-plastic evaluation methods are used. The most important methods
are known under the terms J-integral, method of Eftis and Liebowitz,
crack tip opening displacement (COD), and equivalent energy. For all
methods a critical load of the load-displacement curve for the evaluation
of Kic has to be determined. For specimens with large plastic zone sizes,
the secant method of the ASTM Test for Plane-Strain Fracture Toughness
of Metallic Materials (E 399-74) for linear-elastic fracture mechanics
evaluation cannot be applied directly. Also, the maximum load, sometimes
used during the development of elastic-plastic fracture mechanics, is not
suitable, because very often crack extension begins before the maximum
380 ELASTIC-PLASTIC FRACTURE

load is reached and then the maximum load depends on the specimen
geometry in a complicated manner. Therefore, in all elastic-plastic evalua-
tion methods the attempt is now made to use the load JFO at the onset of
crack extension. Determination of this load, however, is the crucial point
in all elastic-plastic methods.
There are two prerequisites for an elastic-plastic procedure for the
determination of plain-strain fracture toughness. The critical value, for
example, /lo or CODio, has to be independent of specimen size in a broader
range than Kio. Furthermore, there must exist an unequivocal relation
between the elastic-plastic parameter, for example, / or COD, and the
stress intensity factor K in the linear-elastic region.
The most promising method seems to be the J-integral procedure. After
the first experimental investigation by Begley and Landes [1,2Y with the
pseudo-compliance method, requiring specimens with different crack
Jengths, approximate methods for the determination of / from one load-
displacement curve were sought.
In this paper results are presented for two alloy steels. A comparison
between the different evaluation procedures is made; the problem of
determining the onset of crack extension is discussed, and the effect of
specimen size on the critical values is shown.

The J-integral
The J-integral evaluation is based on the relation

7 = - - ^ ^ (1)
•' B da ^^'
Begley and Landes [1,2] used specimens of different crack length and
calculated / with the equation

, _ J_ U{a + Aa) - U{a) ,.,

'' B Aa ^^^

In Eq 1, dU is the change of the deformation energy during crack propaga-

tion da, whereas in Eq 2 the difference of the deformation energy of two
specimens with crack length a and a + Aa is determined. By finite-element
calculations it was shown that / , according to Eq 2, leads to higher values
than according to Eq 1 [J]. The pseudo-compliance method of Begley and
Landes requires a large number of specimens. Therefore, different ap-
proximate equations were developed to determine / from one load-displace-
ment curve. For compact specimens these equations are as follows.

•'The italic numbers in brackets refer to the list of references appended to this paper.
 BERGER ET AL ON FRACTURE TOUGHNESS 381

Rice, Paris, and Merkle [4]:

/i = ; ^ ( f / - i / n o c r ) (3)

Landes and Begley [5]:

J2 = j^U (4)

Kanazawaetal[6]:

^=
.i l-±\u+(l-V)FS-lu^
b W) \W b) W
(5)

Merkle and Corten [7]:

/4 = G + /pi = G + - ^ [Di JFrfSp, + D2 jSpirfFl (6)

= G + - ^ ((D, - D{)U + DiFh - (Z), + Di)UA (6a)

Merkle and Corten simplified [7]:

/4i = ^ lDiiFd6„ + D2jScrdFl (7)

= j ^ [(£>i - Z)2)t/ + DiFd - (£>, + Z)2)f/„<Kr] (7a)

Merkle and Corten simplified, replacing the displacement due to the crack
by the total displacement 6:

Jn = j^[Di\Fd8 + DilddF] (8)

= ^[(Di-D2)U + D2F6] (8a)

In these equations there is

li = (9)
382 ELASTIC-PLASTIC FRACTURE

where Y(a/W) is given by the linear-elastic relation

V2[l + (a/W)^V''^ - (1 + a/W)

y= nr^Ti^F (12)

By means of idealized load-displacement curves it is possible to calculate

/ with the different approximate equations and to compare the results with
/from Eq 1.
1. For linear-elastic behavior there is

±^hL+Jj^=?^L^^^Y^da/W + A) (13)

F^ (1 - v^)
U = —^-——- (A + \Y^ da/W) (14)
KB

jn{\— ^2)
t/nocr = ^^„ . X A (15)
EB

1 dU F^ (1 - v^)
•"-BIT' EB^W *" ««
For calculation of the different /, according to Eqs 3-8 and of/ accord-
ing to Eq 16, Y was calculated using the equation of Srawley {8\. For the
total displacement, values of Gross [9], given in Table 1, were used. These
values were calculated assuming a more realistic load distribution than
earlier calculations by Roberts [10]. The displacement of a specimen
without a crack is dependent on the load distribution and on the gage
length of the extensometer. In this investigation, A in Eq 13 was obtained
by comparing the values of Gross for the total displacement and JY"^
da/W, using the equation of Srawley [8\. An average value of ^4 = 1.92
was obtained (see Table 1). In Fig. 1 the ratios / / / are plotted against
a/ W. These ratios are independent of the materials properties. The calcu-
lations of Kanazawa et al (/;) and of Merkle and Corten (/4) lead to
correct /-values. The simplified equation of Merkle and Corten, based on
the total displacement (/42), leads to correct values for a/W = 0.6, whereas
h = 2U/Bb is about 17 percent below the correct value.
 BERGER ET AL ON FRACTURE TOUGHNESS 383

2. For linear-elastic/ideal plastic behavior, a linear F — S curve is

assumed until a limit load FL is reached, which is given according to
Merkle and Corten [7] by

FL = a,B(W-a)y (17)

where y is given by Eq 12.

The displacement where FL is reached is

dL = ^ (W-a)y(lY^da/W + A) (18)

Calculation of J, Jt, Ji, /,, /,, and Jn leads to

6 - y S i - ^ ff, m i - a/W) yA
h/J = (19)
A(6 -6i)+ -— ay W{1 - a/WYyY^

6- y6i
h/J = (20)
A(« - 61) + 2 ^ , a^ W'd - a/Wy y Y^

8- 6L+-^ ayWil - a/WYyY^

h /J = ^ ~ (21)
2D, (5 - 6i) + - ^ a, W{\ - a/W^yY^

h/J = 1 (22)

J.yj = ^-^iD.-D.V2D. ^^3^

5 - & + 2FF, ''^ ^^* ~ a/W)^Y^y

For 6 — 00, / , 2 / / approaches 1, whereas/i// and/a// approach 1/Di and

/a// approaches l/2Dt. In Fig. 2 the different ratios are plotted against
/for a/W = 0.6, o^, = 500 N/mm^,i: = 2 X 10* N/mm^, .» = 0.33, W =
40 mm, andv4 = 1.23.
From these calculations the following conclusions can be drawn:
1. J2 = 2U/Bb underestimates/also for fl/yy = 0.6.
2. The equation of Kanazawa et al, which is correct for elastic behavior,
underestimates/ in the elastic-plastic region.
384 ELASTIC-PLASTIC FRACTURE

TABLE 1—Displacements for compact specimens; see Eq 13.

a/W A+ lY^da/W lY^da/W'' A

0.3 6.20 4.38 1.82

0.4 10.49 8.51 1.98
0.5 17.57 15.58 1.99
0.6 30.75 28.85 1.90
0.7 60.49 58.60 1.89

"Gross [9].
''rfromSrawley[«].

3. The equation of Merkle and Corten is a good approximation of J. For

a/W = 0.6 the simplified equation leads to correct /-values.

The Method of Efds and Liebowitz

Eftis and Liebowitz [11,12] discussed fracture behavior for nonlinear
load-displacement curves. On the basis of energy considerations they
introduced a fracture criterion G for stable crack propagation. Using Eq 1 or
2 for J-integral definition, <5 is then identical with / for the onset of crack
extension.

1.0
k /J. J 3/J

.9
y^^uil J^/J4l u ^^^
.8

.7 / y ^ 2 nx

.6 y\l^

.5

.3 .U .5 .6 .7
a/W
FIG. 1—Ixllfor linear-elastic behavior for compact specimens.
 BERGER ET AL ON FRACTURE TOUGHNESS 385

\ J42/J
\ 1
\ - /1 -

J|/J

yJa/J

20 40 60 80 100
J, N/mm
FIG. 2—J;/J for linear-elastic/ideal plastic behavior for compact specimens.

Besides the more fundamental considerations, Eftis and Liebowitz pro-

posed an evaluation procedure for (5 or /, respectively. In this procedure
the load-displacement curve is described by

S=F/M + k{F/M)'- (24)

with the two parameters n and k, leading to

0=J=G Ink /FV

(25)
1+
n + 1 VM

n and k can be determined from two points, (Si, Fi) and (&, F2), on the
load-displacement curve according to

62 - F2/M
Ig 61 - Fi/M
(26)
\gF2/Fi

M\"
k = {di- Fi/M) (27)

The validity of Eq 24 has to be proved. The crack length dependence of

the nonlinear component of S in this equation is given by the term (F/M)",
where M is the slope of the linear part of the load-displacement curve.
There is some arbitrariness in the choice of (F/M)". It can be shown
386 ELASTIC-PLASTIC FRACTURE

that different results for / are obtained, if the nonlinear component of 6

in Eq 24 is replaced for instance by * X jp" X Af ~^.

The Equivalent-Energy Concept

The equivalent-energy concept, developed by Witt and Mager [13,14],
is an essential empirical method for the determination of Ku with small
specimens. Witt and Mager used the maximum load for the ific-determi-
nation. Later on, however, the load Fo at the onset of crack extension was
used [15,16]. The stress intensity factor at the onset of crack extension
Kcqa is calculated by

K^, = -^^=M^Y(a/W) (28)

BjwTh

where
Fi = arbitrary load within the linear region of the F-5-cuTve,
U\ = corresponding deformation energy, and
l/o = deformation energy at JFO.
Begley and Landes [17] have shown that the equivalent energy concept
and the J-integral concept do not give identical results. For linear-elastic/
ideal plastic material behavior the two methods can be compared. The
comparison is made in the form of the ratio p = J/{K^^a/E'). This ratio
is equal to one if both methods coincide. For 6 < 6L, that is, in the elastic
region, there i% p = 1. For 5 > 8i, p deviates from 1 and reaches a
boundary value for 6 — oo. For compact specimens this boundary value is
given by

- T/(tr2 /^'^ - (U + (a/Wy]^^^ + y/2] {\Y'da/W + A)

In Fig. 3, />» is plotted against a/W. It can be seen that the equivalent-
energy method leads to higher values than the J-integral method. For
a/W = 0.6 the difference is about 10 percent.

Crack Tip Opening Displacement

In most experimental investigations an attempt is made to calculate
COD from crack mouth displacement. Different relations between COD
and crack mouth displacement have been published, leading to different
COD-values [15,18-21]. Therefore it is useful to measure COD at different
distances from the crack tip and to find COD by extrapolation to the
crack tip. By displacement measurements and finite-element calculations
 BERGER ET AL ON FRACTURE TOUGHNESS 387

I.U

.9

^ .8

—>
.5

.3 .U .5 .6. .7
a/W
FIG. 3—J/(K^e,«/£") versus a/Vffor linear-elastic/ideal plastic behavior (6 — oo).

it could be shown that for compact specimens there exists a linear relation
between COD and distance from the crack tip only up to the region of
the pinholes [22].
The relation between COD and K is given by

K^ (1 - v^)
COD = C (30)
E X ay

The constant C is not exactly known. Finite-element calculations resulted

in C = 0.5 for three-point bend specimens [23,24]. Experimental investiga-
tions have shown that C depends on the material. Robinson and Tetelman
[25] found C = 1 for different materials. Later on, different values of C
between 0.38 and 1 were determined [26-28].
Thus there are two uncertainties in the determination of A'lc from crack
opening displacement: first, the determination of COD, and second, the
relation between COD and K.

Experimental Procedure
Two nickel-chromium-molybdenum steels were investigated. The chemical
composition, heat treatment, and mechanical properties are given in
Table 2. Steel 1 was available as a turbine disk (outer diameter 2925 mm,
inner diameter 885 mm, thickness 670 mm). Steel 2 was supplied as bars
of dimensions 450 by 250 by 100 mm. From both materials compact
specimens of different sizes, given in Tables 3 and 4, were machined. The
larger specimens had a W/B ratio of about two. The specimens with B —
388 ELASTIC-PLASTIC FRACTURE

•Q

> g .
o 1 •
(fa

t^ o
z ^ <s
r^ ^* ? :
SV
Q 1 •
Z
o in m
S m 't
o o
^ §1
1 u
in 00 Z
*-
a
.a«
«
1
•5
e
i« «
a
1 8 : <

r
o t4M
o> o
s:
{
.§
"a
e
o

1E
<rt Is
d d 1
H
1
U !
•«
9
•o

E
6 a. S o
d d
1
S 1
a
o
op ;

1 1
i1
o
B

1
e >0 -H

d d

.s M

. E
f o oo 1 S ^
9< r~-
S
2rt
55 z iM
d d
a s.
ea a"
<
H ^"•o (S 3 *i
c;
.E «
oo <s
II 1
u
d d
II z
e
iS §
^n'U,
1
1
V
?
1
s
i12
11
g
0 %
S
II 11 11 ZB<
VI I/]
BERGER ET AL ON FRACTURE TOUGHNESS 389

'O

S2
o m<5 in^
(N « ^

o o n
« lis m
(N — —

(N « rH

'O S;

O t ^ O^CT^OO

^1 <N i n

§m
0 IT) ^
( s •-<
I
390 ELASTIC-PLASTIC FRACTURE

Tj- TT Qo o m
Cn OV ^ ^-H ^
- - -H -H ( N (N

•o
"5>

1 00 I^ <^ 0< 1^
« rt rt rt (N

^O O 0 ^ (N TH
t^ r^ so o^ i^

I r* (N a^ Os 00
^ so ^C OO I/)
« -H - « rt (N

a
5

ttJ
< ^ 00 00 00 0 ^
1/7 ^ TT ^ fn

^i
 BERGER ET AL ON FRACTURE TOUGHNESS 391

5 mm and 5 = 9 mm had a width of 50 mm. The specimens were pre-

cracked by fatigue up to a crack length of a/W = 0.6. The specimens
were loaded to different load levels and the load-load point displacement
curves recorded. For some specimens with B = 50 mm and B = 100 mm,
crack opening displacement was measured with four clip gages at different
distances from the crack tip. For specimens with 5 = 25 mm, displace-
ment could be measured only with three clip gages. The different displace-
ments measured were extrapolated to the crack tip for the determination
of COD as shown in Fig. 4. Crack extensions were measured with a
scanning electron microscope (SEM) on the fracture surface after unloading
and fatigue cracking of the specimens. At least 10 measurements were
made at equal distances along the crack front and an average crack ex-
tension was calculated. The stretched zone width was included in the crack
extension. For some specimens the width of the stretched zone was measured
also at different points along the crack front.
Some results for Steel 2 have already been published [29]. In this paper,
some additional results are presented.

Results and Discussion

Determination of critical J-values
At first for all specimens, JA (Merkle/Corten relation) was calculated
and plotted against crack extension.

2.5
F=2UKN V

UJ
2 2.0-
liJ
U o
< F=178KN.,,^
_i
^ 1 0

Q
o
o F=124KN-.^^^_^
?1.0
z O %v
LU ^o
a.
o N. crack tip
^ as load line
/ ^ v ^ \ l
<
on
o
25 50 75 100
DISTANCE FROM THE SPECIMEN SURFACE,mm

FIG. 4—Crack opening displacement for different loads versus distance from the specimen
surface for a specimen with B = 50 mm of Steel 1.
392 ELASTIC-PLASTIC FRACTURE

Results for Steel 1 are shown in Figs. 5 and 6. In Fig. 5 the crack
extensions at maximum load Aam>x are marked for the smaller specimens.
It can be seen that some points are included in the figures where the
specimens are loaded beyond maximum load. Up to a crack extension of
about 0.3 mm, / increases considerably. At larger crack extensions the
slope of the /-Aa-curve is smaller. From Fig. 6 it can be seen that the
/-Aa-curves intersect the blunting line / = 2an X Aa at a crack extension
between 40 and 50 /^m. It was assumed [5] that the crack extension up
to Aa = //2fffi is due to the blunting of the crack tip, which can be seen
on the fracture surface as a stretched zone Aost. This assumption could
not be confirmed by fracture surface observations in the SEM. For all
specimen sizes, the stretched zone was measured at different points along
the crack front. An example of the stretched zone between fatigue crack
and static fracture is shown in Fig. 7a. The average values of Aost are
plotted against specimen thickness in Fig. 8. For the larger specimens a
stretched zone width of about 28 nm was found, and for the smaller
specimens a lower value of about 20 /im was measured. These stretched-
zone values are lower than the values determined by the intersection of
the J-Aa-curve with the blunting line.

1,00

e
e
300

200

1.0 1.5 20
CRACK EXTENSION Aa,mnn
FIG. 5—J-Aa-curve/or Steel 1.
 BERGER ET AL ON PRACTURE TOUGHNESS 393

/u2(}^•^a
OA
200 A
E • A
m O • • A
E m A
• •
^ o
• • A
/ ^
A
100 o/
• B.nim Wmm
A 0 100 200
/// • 50 100
• 25 50
A U 28
A 5 50
1
^'^st 0,1 0,2 0.3
CRACK EXTENSION Aarnm
FIG. 6—Initial part of J-Aa-curve for Steel 1.

For Steel 2, crack extensions were measured only up to about 0.3 mm.
As an example, a/4-Aa-curve for specimens with 5 = 9 mm, W = SQ mm
is shown in Fig. 9. Two straight lines were drawn through the points.
Originally it was assumed that crack extension begins at the intersection
of the two straight lines [29]. Detailed investigations with the SEM, how-
ever, have shown that crack extension occurs also below the intersection
point. In Fig. lb the extension of the stretched zone is shown. In Fig. 8
it can be seen that almost the same values Aost were observed as for Steel
1. Again a decrease with decreasing thickness occurred.
From these results a generalized /-Aa-curve can be drawn (see Fig. 10).
It is supposed that crack blunting begins if the maximum load during
fatigue precracking is exceeded. The corresponding/is called//„»«. Between
Jfmtx and /o crack blunting occurs, leading to a stretched zone Acst on the
fracture surface. For the investigated steels, / increases considerably at the
beginning of stable crack extension. Then there exists a transition region
or—as shown in Fig. 9—a kneepoint. The intersection of the "blunting
line" / = 2ffn Aa with the /-Aa-curve can occur in the steep region (at
Point 2 in Fig. 10a for Steel 1) or in the flat region (at Point 2 in Fig. lOA
for Steel 2).
During the development of the /-integral method it was suggested that
/o at the onset of stable crack extension should be determined to predict
J^ic for large structures. For materials with a steeply rising crack growth
resistance curve or with a large amount of crack blunting before the onset
of crack extension, an exact determination of/„ can be very difficult. It is
394 ELASTIC-PLASTIC FRACTURE

o
o

I
i
I

&
.o

o
m
 BERGER ET AL ON FRACTURE TOUGHNESS 395

I/O
ts
o
< A
a
g30
A
Nl
o
(
Q
UJ

IXl 2-^°
a:
O)

10 0 steel 1
A steel 2

20 AO 60 80 100
THICKNESS B.mm
FIG. 8—Stretched zone versus specimen thickness.

300
A:zlQ^y AQ

.0'''''^

•'Cr^
E ^^n-7^
e 200
c
Y /

100

V
0 0.1 0.2 0.3

CRACK EXTENSION Aa,mm

FIG. 9—J-£i3i-curvefor specimens with B = 9 mm o/irf W = 50 mm of Steel 2.

396 ELASTIC-PLASTIC FRACTURE

•9

"5

o
I
o
O "^
 BERGER ET AL ON FRACTURE TOUGHNESS 397

also a subject of discussion whether it is desirable to determine Jo for a

steeply rising crack growth resistance curve or, better, a /-value after some
crack extension. In linear-elastic fracture mechanics, a "/sTic" is determined
also after some crack extension. The problem is to find a procedure which
is simple and leads to unequivocal values independent of specimen size.
The extrapolation method proposed by Landes and Begley [17] is an
attempt to find such a /-value. With regard to the test results for the two
steels, this method has some critical points:
1. For small specimens the maximum load can occur at small crack
extensions. Then the extrapolation method can be applied only if measuring
points beyond maximum load are included. For instance, for specimens
with 5 = 14 mm of Steel 1, the maximum load occurred at Aa = 275 ^m.
Therefore the range of Aa is too small to determine a straight line (see Fig.
11). It is possible to calculate J-integral for specimens loaded beyond
maximum load. It has to be proved, however, if the measured /-value is
affected by the unloading of the whole specimen.
2. If the transition region from the steep to the flat region of the /-Aa-
curve occurs beyond the blunting line, then the intersection with the
blunting line can depend strongly on the number of points below the
transition region.
For the foregoing reasons, some modifications of this evaluation proce-
dure should be considered. One possibility is to define the critical / at a
fixed deviation from the blunting line—for instance, at the intersection of
the /-Aa-curve with the straight line parallel to the blunting line at a
distance of 0.1 mm (see Fig. 10).

300

AA

AOmax
(B=5mm)
AO
max
(B=Umm)

0.3 ou 0.5
CRACK EXTENSION Aamm
FIG. 11—J-Aa-curve for specimens with B = 5 mm, W = 50 mm and B = 14 mm.
Vf = 28 mm for Steel 1.
398 ELASTIC-PLASTIC FRACTURE

The possible critical /-values, mentioned in the foregoing, are marked in

Fig. 10:
1. at Afl = Aost (Point 1),
2. at the intersection of the/-Aa-curve with the blunting line (Point 2),
3. at the intersection of the /-Aa-curve with a parallel line at a distance
Aoo to the blunting line (Point 3),
4. at the intersection of the blunting line with a straight line through the
points, neglecting all points within the steep part of the /-Aa-curve (Point
4), and
5. at the kneepoint of the /-Aa-curve (Point 5).

Effect of Specimen Size on the Critical i-Values

The effect of specimen size on the /-Aa-curve can be seen from Fig. 6.
For crack extensions up to 0.3 mm there is no effect of the specimen size
for the proportionally sized specimens. The points for the specimens with
5 = 5 mm, W = 50 mm are at the lower bound of the scatter band. In
the range between 0.3 and 0.5 mm the points for the specimens with B =
5 mm and B = 100 mm are above the points of the specimens with B =
25 mm and JB = 50 mm (see Fig. 5).
Some of the different /-values, mentioned before, are plotted against
specimen thickness for Steel 1 in Fig. 12. The values according to 1, 2, and
3 of the foregoing are almost independent of thickness.
The extrapolation method leads to/-values which increase with increasing
thickness.
The results for Steel 2 can be seen from Fig. 13. / at Aa = Aast and at
the kneepoint of the /-A a-curves is almost independent of thickness for
B > 9 mm. / at the intersection with the blunting line, however, is much
higher for specimens with B = 100 mm than for the smaller specimens.
At small thickness in the range of J? = 5 mm, all/-values increase.
From these results it can be seen that, at small crack extension, / is
independent of specimen size above a critical thickness. It was assumed
that / increases below a critical thickness given by

B = aX — (31)
ay

with a between 25 and 50 [5,30]. For Steel 1, the /-values tend more to a
decrease than to an increase. Possibly the smallest specimen investigated
had a thickness larger than that given by Eq 31. From / = 150 N/mm
(intersection of/-Aa-curve with the parallel to the blunting line) a minimum
thickness ofB = 4.4 mm is calculated with a = 25. From/ = 84 N/mm^
(onset of crack extension) a minimum thickness ofB = 2.5 mm is calculated.
For Steel 2 for/ = 162 N/mm (kneepoint of the/-Aa-curve), the minimum
 BERGER ET AL ON FRACTURE TOUGHNESS 399

200

E
£

100

o extrapolation
• at intersection with 2Gti(Aa-0.l)
a at intersection with J = 20,|Aa
^ at Aa = Aast
20 40 60 80 100
THICKNESS B. mm
FIG. 12—Effect of specimen size on different i-values for Steel I.

thickness is B = 8.2 mm; for / = 100 N/mm (onset of crack extension),

B = 5 mm.
From these calculations it can be concluded that Eq 31 with a = 25 is
a good approximation to the minimum specimen thickness.

Comparison Between the Different Evaluation Procedures for J

In this investigation all /-values were determined from load-displacement
curves of specimens with one crack length. Therefore, /-values according to
Eq 2 could not be determined.
The different /-values for both steels are given in Tables 3 and 4. From
these tables the following conclusions can be drawn:
1. /z = 2U/Bb is smaller than/4 (correct Merkle/Corten equation).
2. The simplified Merkle/Corten equation, based on the total displace-
ment (/«), leads to slightly higher values than the correct Merkle/Corten
equation.
3. The equation of Kanazawa (/j) is smaller than/4, especially for small
specimens.
4. The evaluation procedure of Eftis and Liebowitz agrees with JA within
10 percent.
These results generally are in agreement with the prediction from the
idealized load-displacement curves made in the foregoing. Consequently,
400 ELASTIC-PLASTIC FRACTURE

300

£
£
200

100

a at intersection with J=20f[-Aa

• at kneepoint of J-Ao-curve
A a t AOr^^Ost

20 ^0 60 80 100
THICKNESS B,mm
FIG. 13—Effect of specimen size on different J-values for Steel 2.

the Merkle/Corten relation for J-integral determination for compact

specimens is recommended.

Comparison of J-Integral with Linear-Elastic Determined Stress Intensity

Factor
For large specimens, Kj calculated from/with

JE
Kh = (1 - v^) (32)

should be identical to K calculated by the linear elastic Eq 10. For smaller

specimens there is K < Kj, but still K* = Kj, where K* is calculated by
the linear elastic equation, but where crack length a is replaced by a +
(l/6ir) {K/oyY. For even smaller specimens there is also K* < Kj. The
minimum specimen size, for which K* = Kj, is given by the ligament
width criterion
 BERGER ET AL ON FRACTURE TOUGHNESS 401

W-a = 01—^' (33)

.ffv

For Steel 1, different /iT-values are plotted against specimen thickness for
the proportionally sized specimens in Fig. 14: KQ ( A S T M secant method),
K, K*, and KM (from J4) for two critical loads, corresponding to Aa =
Aost and to the /-value determined with the extrapolation method. From
Fig. 14 and the values listed in Table 5, the following conclusions are
derived.
1. Comparing KQ with K at Aa = Aost shows that for the specimens
with B = 50 mm and B = 100 mm, crack extension begins below KQ, and
for specimens with 5 = 14 mm and B = 25 mm, above KQ.
2. For the onset of crack extension (Aa = Aost), K* and Kj4 are identical
for all specimen sizes. Therefore the critical W — a of Eq 33 is smaller
than 11.2 mm (for specimens with B = 14, W = 2S mm, and a/W =
0.6), leading to /3 < 0.40.
3. For the critical point, determined with the extrapolation method,
Kj4 and K* agree also for the smallest specimens, for which the extrapola-
tion method could be applied (B = 25 mm, VT = 50 mm, and a/W =

ouu

II

200
8
8 0

•
0
e
*
c
0
i
« 0
If
a 0
100 <II Q.
•^KQ 0
t_

0 X
<1 0)

Kji 0 0

K* • •
K * e

20 UO 60 80 100
THICKNESS B, m m
FIG. 14—Effect of specimen size on different K-valuesfor Steel 1.
402 ELASTIC-PLASTIC FRACTURE

•9- QD O^
<N 00 vO
(N -H «

-N 00 t-
§. -H f. f-
<S —< "N

a
00 ^ O
.-< 00 r~
<S -H «

r- 5 rt
»- 00 I^
pa — rN

•o r^ ui o -"T
>«• <N CN <N «

?!

^ <N — «

^1 f*4 ^^

« . • ! 8SJQ2;'"
 BERGER ET AL ON FRACTURE TOUGHNESS 403

0.6). The minimum ligament width therefore is smaller than 20 mm,

leading to/3 < 0.41.
For Steel 2, /3 = 0.4 was found in an earlier investigation [29].
From these results it can be concluded that linear elastic fracture
mechanics can be applied to much smaller specimens than given by ASTM
Method E 399-74 if the plasticity correction is used for the AT-calculation.

Other Elastic-Plastic Methods for K/c Determination

The A^-values determined from COD for Steel 1 are given in Table 5.
KcoD was calculated from COD by means of Eq 30 with C = 1. There is
an excellent agreement between Kj4 and KCOD for Steel 1. Therefore it can
be concluded that, for the investigated steel, C = 1.
The stress intensity factors Ktqa determined with the equivalent-energy
method are given in Tables 4 and 5. In accordance with the predictions
from the idealized load-displacement curves, there is very good agreement
between X ' ^.ndATequ.

Conclasions
From the evaluation of idealized load-displacement curves, especially
linear-elastic/ideal-plastic behavior, and from experiments on two nickel-
chromium-molybdenum steels, the following conclusions can be drawn for
compact specimens.
1. From the different equations for J-integral determination from one
load-displacement curve, the relation of Merkle and Corten even in its
simplified form yields the best results.
2. Crack extension begins below the /-value determined with the extrap-
olation method. This method can be applied only for small specimens, if
data points beyond maximum load are included. As an alternative it is
proposed to determine / at a fixed distance from the blunting line.
3. A comparison between stress intensity factors calculated from / and
linear-elastic including the plasticity correction shows that linear elastic
fracture mechanics can be applied to much smaller specimens than given
by ASTM Method E 399-74.
4. The equivalent-energy method agrees fairly well with the J-integral
method, if the evaluation is made at the same load.
5. Crack tip opening displacement can be determined using different
clip gages at different distances from the crack tip. A"COD calculated from
COD with C = 1 in Eq 30 agrees with Kj calculated from / .

Acknowledgment
We wish to thank J. Eschweiler and F. Vahle for their help during the
404 ELASTIC-PLASTIC FRACTURE

performance of the tests. The financial support of the Deutsche For-

schungsgemeinschaft is gratefully acknowledged.

References
[1] Begley, I. A. and Landes, J. D. in Fracture Toughness, ASTM STP 514. American
Society for Testing and Materials, 1972, pp. 1-20.
[2] Landes, J. D. and Begley, J. A. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, pp. 24-39.
[3] Boyle, E. F., "The Calculation of Elastic and Plastic Crack Extension Forces," Ph.D.
Thesis, Queen's University, Belfast, U.K., 1972.
[4] Rice, J. R., Paris, P. C , and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing, ASTM STP 536, American Society for Testing and Materials, 1973,
pp. 231-245.
[5] Landes, J. D. and Begley, J. A. in Fracture Analysis, ASTM STP 560, American
Society for Testing and Materials, 1974, pp. 170-186.
[6\ Kanazawa, T., Machida, D., Onozuka, M., and Kaned, S., "A Preliminary Study on
the J-Integral Fracture Criterion," Report No. IIW-779-75, University of Tokyo, Tokyo,
Japan, 1975.
[7] Merkle, J. R. and Corten, H. T., Journal of Pressure Vessel Technology, Transactions,
American Society of Mechanical Engineers, Vol. %, 1974, pp. 286-292.
[8] Srawley, I. B., Engineering Fracture Mechanics, Vol. 12, 1976, pp. 475-476.
[9] Gross, B., unpublished results.
[10] Roberts, E., Materials Research & Standards, Vol. 9, 1969, p. 27.
[//] Liebowitz, H. and Eftis, J., Engineering Fracture Mechanics, Vol. 3, 1971, pp 267-281.
[12] Eftis, J., Jones, D. L., and Liebowitz, H., Engineering Fracture Mechanics, Vol. 7,
1975, pp. 491-503.
[13] Witt, F. J. and Mager, T. R., "A Procedure for Determining Bounding Values on
Fracture Toughness Ku at any Temperature, Report ORNL-TM 3894, Oak Ridge
National Laboratory, 1972.
[14] Witt, F. J. and Mager, T. R., Nuclear Engineering and Design, Vol. 17, 1971, pp. 91-
102.
[15] Robinson, J. N. and Tetelman, A. S., "Comparison of Various Methods of Measuring
Kic on Small Precracked Bend Specimens that Fracture After General Yield," Technical
Report No. 13, School of Engineering and Applied Science, University of California, Los
Angeles, Calif.
[16] Schieferstein, U., Berger, C , Czeschik, H., and Wiemann, W. in Berichtsband der 8.
Sitzung des Arbeitskreises BruchvorgSnge, Deutscher Verband flir Materialprufung,
1976, pp. 50-57.
[17] Begley, J. A. and Landes, J. D. in Progress in Flaw Growth and Fracture Toughness
Testing, ASTM STP 536, American Society for Testing and Materials, 1973, pp. 246-
263.
[18] "Methods for Crack Opening Displacement (COD) Testing," Draft for Development 19,
British Standards Institution, 1972.
[19] Barr, R. R., Elliott, D., Terry, P., and Walker, E. T., Journal of the Welding Institute.
Vol. 7,1975, pp. 604-610.
[20] HoUstein, T., Blauel, J. G. and Urich, B., "Zur Beurteilung von Rissen bei Elasto-
Plastischem Werkstoffverhalten," Report of Institut fflr FestkOrpermechanik der
Fraunhofer-Gesellschaft, Freiburg, Germany, 1976.
[21] Schmidtmann, E., Ruf, P. and Theissen, A., Materialprufung, Vol. 16, 1974, pp. 343-
348.
[22] Berger, C. and Friedel, H., unpublished results.
[23] Levy, N., Marcal, P. V., Ostergren, W. J., and Rice, J. R., International Journal of
Fracture Mechanics, Vol. 7, 1971, pp. 143-150.
[24] Hayes, D. J. and Turner, C. E., International Journal of Fracture Mechanics, Vol. 10,
1974, pp. 17-32.
 BERGER ET AL ON FRACTURE TOUGHNESS 405

[25] Robinson, J. N. and Tetelman, A. S. in Fracture Toughness and Slow-Stable Cracking,

ASTM STP 559. American Society for Testing and Materials, 1974, pp. 139-158.
[26] Robinson, J. N., IntemationalJournal of Fracture Mechanics, Vol. 12, 1976, pp. 723-
737.
[27] HoUstein, T. and Blauel, J. G., International Journal of Fracture Mechanics, Vol. 13,
1977, pp. 385-390.
[28] Chipperfield, G. G., International Journal of Fracture Mechanics. Vol. 12, 1976, pp.
873-886.
[29] Keller, H. P. and Munz, D. in Flaw Growth and Fracture. ASTM STP 631. American
Society for Testing and Materials, 1977, pp. 217-231.
[30] Paris, P. C. in Fracture Toughness, ASTM STP 514, American Society for Testing and
Materials, 1973, pp. 21-22.
D. Munz^

Minimum Specimen Size for the

Application of Linear-Elastic
Fracture Mechanics

REFERENCE: Munz, D., "Minimum Specimen Size for tiie Application of Linear-
Elastic Fiactme Meclianics," Elastic-Plastic Fracture. ASTM STP 668. J. D. Landes,
J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and Materials,
1979, pp. 406-425.

ABSTRACT: The minimum thickness and the minimum ligf lent width for the
determination of plane-strain fracture toughness with linear-ela^dc methods can be
considerably smaller than given by the ASTM Test for Plane-Strain Fracture Toughness
of Metallic Materials (E 399-74). For the ligament width, the factor 2.5 in the size
requirement equation can be replaced at least by 1, possibly by 0.4. The size dependence
of KQ determined with the 5 percent secant method is due to plasticity at the crack
tip and to the existence of a rising plane-strain crack growth resistance curve. With a
variable secant, adjusted to the specimen width, it is possible to determine size-indepen-
dent fracture toughness values.

KEY WORDS: fracture properties, crack propagation, toughness, tests, aluminium

alloys

Nomenclatofe
a Crack length
a* Effective crack length
Aa Crack extension
AOKQ Crack extension at KQ
Aac Crack extension at Ku (onset of unstable crack extension)
B Specimen thickness
Be Minimum thickness according to ASTM method E 399-74
.ffpi.st. Minimum thickness of a specimen with plane-strain region in
the center
Bis. Minimum thickness of a proportional-sized specimen, for
which linear-elastic fracture mechanics can be applied

'Division head, Deutsche Forschungs- und Versuchsanstalt fur Luft- and Raumfahrt,
Cologne, FR Germany.

406

Copyright 1979 b y AS FM International www.astm.org

 MUNZ ON SPECIMEN SIZE 407

C Constant in ifepi = Ca, 4W

COD Crack tip opening displacement
E Young's modulus
F„ Load at the onset of crack extension
G Strain energy release rate
G* Strain energy release rate calculated with a*
JioPlane-strain J-integral at the onset of crack extension
K Stress intensity factor
K* Stress intensity factor calculated with a*
Ko Stress intensity factor at the onset of crack extension
Kio Stress intensity factor at the onset of crack extension under
plane strain
KQ Stress intensity factor at the 5 percent secant intersection
AQpl KQ, if AQ > KQ
Ko.i Stress intensity factor at 0.1-mm crack extension
K, Stress intensity factor, determined with a variable secant
according to Eq 20
nto Slope of the linear part of the i^-v-curve
m Slope of the secant
rpi Radius of plastic zone
V Crack mouth displacement
Vei Elastic component of v
Vpi Plastic component of v
Vcr Component of v due to crack extension
AV = Vpi + Vcr
W Specimen width
Wo Specimen width for which Ko = KQ
(W — a)us, Minimum ligament width for which linear-elastic fracture
mechanics can be applied
{W — a)c Minimum ligament width according to ASTM Method E
399-74
a Constant in minimum thickness relation, Eq 7
/3i Constant in minimum thickness relation, Eq 3
02 Constant in minimum ligament width relation, Eq 4
Qy Yield strength
V Poisson's ratio
w Plastic zone size

According to ASTM Method E 399-74, the size requirements for the

determination of plane-strain fracture toughness Ku are given by

«'="(!f)
B >B, = 2.S(—^) (1)
408 ELASTIC-PLASTIC FRACTURE

(W-a)>{W- ah = 2.5 (-fj (2)

It is assumed that the same size factor of j3 = 2.5 for thickness B and
ligament width W — a has to be used. There are, however, different require-
ments for thickness and width. The critical thickness—called 5pi.s,.—is given
by the requirement of a sufficient amount of plane strain along the crack
front in the center of the specimen. From J-integral investigations it is
known that the factor 2.5 in Eq 1 can be reduced considerably [1,2].^ The
critical ligament width—called (W — a)LE—is given by the requirement of
a sufficiently small plastic zone size to. Munz et al [2,3] have shown also
that the factor 2.5 in Eq 2 can be reduced. For these reasons the size
requirements are written in the following form:

B > 5pu,. = /3i f ^ V (3)

a

(W~a)>(W- a),E = 02 (^^y (4)

If these requirements are fulfilled it should be possible to determine a

size-independent fracture toughness Ku. However, if the 5 percent secant
method of the ASTM Test for Plane-Strain Fracture Toughness of Metallic
Materials (E 399-74) is used, Kic can depend on the ligament width, also
for W — a > {W — a)LE. There are two reasons for the effect of ligament
width on Kid the existence of a rising plane-strain crack growth resistance
curve and the plastic deformation at the crack tip. After discussing the size
limits for plane strain and for linear elastic fracture mechanics, it will be
shown in this paper that size-independent "/iric"-values can be determined
using a variable secant, adjusted to the specimen size.

Stable and Unstable Crack Extension

There are two possible defmitions of fracture toughness Ku. It can be
defined as the critical stress intensity factor at the onset of unstable crack
extension under plane-strain conditions. In ASTM Method E 399-74, "Ki^
is based on the lowest load at which significant measurable extension of the
crack occurs," that is, at the onset of stable crack extension. During the
development of the standard method for Kic determination, it was assumed
that for thick enough specimens, failing under plane-strain conditions, no
stable crack growth but immediate unstable crack extension occurs. From
theoretical [4] and experimental [3,5-7\ investigations, however, it is known
^The italic numbers in brackets refer to the list of references appended to this paper.
 MUNZ ON SPECIMEN SIZE 409

that under plane-strain conditions stable crack growth can also occur. In
this case, plane-strain stable crack growth can be characterized by a rising
Ki, Aa-curve (see Fig. 1). Kia at the onset of stable crack extension is
independent of specimen size, if Eqs 3 and 4 are fulfilled. K^ at the onset
of unstable crack extension depends on the crack length or specimen width,
respectively [8]. With the 5 percent secant method a "fracture toughness"
KQ at a crack extension of 2 percent or less is determined. Therefore KQ
increases with increasing specimen width or crack length, respectively. It is
a point of discussion which value of K along the Ki, Aa-curve should be
used to characterize the fracture behavior. Should it be K\o or a value near
Kio, or should the characterization be done with Kw or another /f-value in
the upper range of the K, Aa-curve?

Minimum Thicloness for PIaii<> Strain

No exact solution exists of the three-dimensional stress distribution
within the plastic zone in the near crack tip region. Of special interest is
the stress Oz in the direction of the crack front and the minimum specimen
thickness B^,a. for which a plane-strain region exists in the center of the
specimen. From experimental investigations, some approximate values of
BfiM. or /3i in Eq 1, respectively, can be determined.
Vosikowsky [9] found in steels that the thickness constraint at the crack
tip begins to collapse when the size of the plastic zone at the surface (plane
stress) is equal to the specimen thickness. With

Wplane stress — I (5)

AQKQ AQC Aa

FIG. i—Plane-strain Ki-A&-curve.

410 ELASTIC-PLASTIC FRACTURE

ffi = 0.32 is calculated. Robinson and Tetelman [10] measured the trans-
versal strain parallel to a notch and found

^. = 25 COD = ^ ^ ^ " " ' ^ " ^ (6)

For an aluminium alloy with Oy = 500 N/mm^ there is 0i = 0.18; for a

steel with Oy = 1000 N/mm^ and £ = 2 X 10^ N/mm^ there is |8i = 0.11.
The minimum thickness for plane strain can also be found indirectly by
measuring the stress intensity factor at the onset of crack extension Ko with
specimens of different thickness but constant width. It can be assumed that
Ko deviates from Kio for B < Bpia.- It is also possible to find fipist. from
the thickness-dependence of the J-integral at the onset of crack extension
Jo. Jo should deviate from the /lo also at 5pi.st.. The minimum thickness in
J-integral investigations for the determination of/lo is given in the form

5p,.s.. - a X — - a — (7)

or

Pi — = (8)
E
a-values of 25 and 50 were proposed [1,11]. For a = 25, Eq 8 is identical
toEq6.
From all these considerations it can be concluded that the minimum
specimen thickness for the determination of plane-strain fracture toughness
can be given by Eq 8 with a = 50 or less.
A comparison for the minimum thickness between Be according to Eq 1
and Bpi.st. according to Eq 8 with a = 25 and a = 50 is given in Table 1
for some materials. It can be seen that the minimum thickness of ASTM
Method E 399-74 can be reduced considerably. (The ratio BuE/Bpi.a. in
Table 1 is explained later on.)

TABLE 1—Comparison of minimum thickness Be, Bp/.j<. and BLE-

Bc/Bp\.st BLE/Bpi. St.

E. ay.
Material N/mm^ N/mm^ a = 25 a = 50 a — 25 a( = 50

Al-alloy 7 X 10" 400 20 10 3.1 1.6

Steel 2 X 10* 400 56 28 9.0 4.5
800 28 14 4.5 2.2
Ti-alloys 1,2 X 10* 1000 13 7 2.1 1.1
 MUNZ ON SPECIMEN SIZE 411

Minimom Ligament Width

The minimum ligament width for the application of linear elastic frac-
ture mechanics in terms of the factor 0i in Eq 4 can be obtained by
comparison of J-integral and strain energy release rate G. Such a com-
parison can be made by finite-element calculations. The results can be
plotted in the form of/ZG-ff/a^-curves (Fig. 2). At a critical G, the/-curve
deviates from the G-curve. The deviation occurs at considerably higher
values, if strain energy release rate is calculated using Irwin's plasticity
correction with an effective crack length

a* = a + r p , = a + ^ ( ^ ) ' (9)

Stress intensity factor and strain energy release rate calculated with a* are
designated K* and G*. For sufficiently large specimens or sufficiently low
loads, K*=K and G* = G.
For three-point bend specimens with a/W = 0.5, the calculations of
Hayes [12] for a non-work-hardening material showed that G* was in
agreement with/up to F/BWoy = 0.10, leading to 182 = 0.45. MarkstrOm
and Carlsson [13] calculated / and G for compact specimens for linear
elastic/ideal plastic behavior. From these results, 182 = 0.42 is obtained.
Another method to find out the minimum ligament width is the determi-
nation of the onset of crack extension for specimens with different ligament
width. In Fig. 3, Ka calculated from iv> and crack length a and K*o calcu-
lated from Fo and a* are plotted schematically against W — a. Below a
critical ligament width (W — a)LE K*o is lower than Kio = K*io.
Munz and coworkers [2,3,6,14] have determined K*e for some materials
with specimens of different size. In Table 2 the minimum ligament width
and 182 are given. For some materials, only an upper limit for {W — OKE
can be given, because it was possible to determine Kio also with the smallest

J/G
G7G

0/Gy
FIG. 2—I/G and J/G* versus a/a,.
 412 ELASTIC-PLASTIC FRACTURE

^ — 1 r_-—" 1

Ko>

Ko
^10

(W-QILE W-a
FIG. 3—Stress intensity factor Ko and K*o at the onset of crack extension versus ligament
width.

specimen. From Table 2 it can be seen that in any case 182 = 2.5 can be
replaced by /3 = 1.0. From the results for the steels, 182 = 0.4 is suggested.
For proportionally sized specimens with W/B = 2, the minimum
specimen size for the determination of plane-strain fracture toughness is
given by the minimum ligament width (W — a)LE. For even smaller speci-
mens, elastic-plastic methods, such as J-integral, have to be applied. The
lower size limit for these methods is given by the minimum thickness fipi.st..
The range of specimen size, where only elastic-plastic methods can be
applied for fracture toughness determination in terms of thickness, is
given by the ratio BIE/BP\M.. For proportionally sized specimens with
W/B = 2 and a/W = 0.5, BLE = {W - ahs and

Bis./Bp\,sx. — ^
Pi

For some materials this ratio is given in Table 1 for Q2 = 0.4 and 0i

TABLE 2—Minimum ligament width (W — a)u:for the determination offracture toughness

for different materials.

By, Kio, (W-a)c, (W-a)LE.

Material N/mm^ MN m-^'2 mm mm 02 Ref

Ti-6A1-4V 910 48 7.0 < 3 <1.1 [61

7475-T7351 426 40 22.0 < 6.25 <0.7 [51
7475, annealed 326 44 45.5 <20 <1.1 \2]
NiCrMo Steel I 497 188 358 56 0.39 [21
NiCrMo Steel II 850 141" 70 <11.2 <0.41 1/41
850 188* 121 <20 <0.41 [14]

" Onset of crack extension.

''Determined according to ASTM Task Group Elastic Plastic Fracture.
 MUNZ ON SPECIMEN SIZE 413

according to Eq 8 with a = 25 and a = 50. From this table it can be

seen that there is only a small range of specimen size where elastic-plastic
methods have to be applied for the determination of plane-strain fracture
toughness.

Plastk; CeaqpeneBt «f COD

The plastic deformation at the crack tip leads to a deviation from the
linear-elastic load-displacement curve. For small specimens the deviations
until the 5 percent secant intersection at KQ can be due only to the plastic
deformation. The plastic component of crack opening displacement Vpi
can be calculated by finite-element methods. A simple estimation was
made by Irwin, assuming linear-elasticity with the crack tip in the center of
the plastic zone. The elastic component of COD is given by

v.. = -^fia/W) (10)

According to Irwin the plastic component then is given by

"•"* da ^ ' ' ' - EBW da/W ^ 6,r \aj ^ ^

and

^ = ^' (^X (12)

For three-point bend specimens with a/W = 0.5, f'/f = 5 and therefore

vp, ^ 0.265 /K\^

(13)

For the 5 percent secant method, Av/va = 0.0526. If the onset of crack
extension occurs at Kjo > KQ, then at KQ the total deviation from the
linearis v-curve Av is identical to Vpi and

KQ = KQf, = CjWXay (14)

with C = 0.45.
Finite-element calculations lead to somewhat different results [15-17].
The C-values obtained are listed in Table 3. An average value of 0.72 was
found.
414 ELASTIC-PLASTIC FRACTURE

TABLE 3—C in KQPI = CVWffy, calculated by finite-element method.

c Work Hardening Ref

0.73 low Brown and Srawley \22\

0.67 low Markstrom [23]
0.78" high MarkstrOm [23]
0.69* no Hayes and Turner [24]

" Extrapolated value.

* From Fig. 7of Ref24

C can also be obtained from experimental investigations from measured

KQf\ and a^. Results for aluminum alloys and a low alloy steel are given in
Table 4. With two exceptions, the values are between 0.50 and 0.60. From
all results, an average value of C = 0.55 is obtained. This experimental
value is between the value from finite-element calculations and the estima-
tion of Irwin. From results of Griffis and Yoder [18] for an aluminum
alloy, C = 0.50 can be obtained. It is not clear why the experimental C-
values are below the calculated ones.

Specimen Size Effect on Kg

Within the application range of linear-elastic fracture mechanics, KQ,
determined with the secant method, can be dependent on specimen size.
In different investigations it was found that KQ is almost independent of
thickness but can increase considerably with increasing width [3,6,19].
Two effects can be responsible for the increase of KQ with increasing
width: the plastic deformation at the crack tip and the rising crack growth
resistance curve. The increase of KQ with increasing width is shown
schematically in Fig. 4. The maximum possible KQ is KQPU if Ko > KQ.
For small specimens, KQ — KQPU At a critical width Wo, crack extension
starts at KQ. For a level K, Aa-curve there is no further increase of KQ.
For a rising K, Aa-curve, KQ increases according to slope of the K, Aa-
curve. The stable crack extension AUKQ at KQ increases with increasing
width, beginning at Wa.
Figure 5 shows results obtained with three-point bend specimens of the
aluminum alloy 7475-T7351. KQ is plotted against width and a considerable
increase can be seen. With the electrical potential method for the onset of
crack extension, A'lo = 40 MNm"^^^ was found. In Fig. 5 also, the -^gpi-
VF-relation according to Eq 14 with C = 0.55 is plotted. For the specimens
with W = 12.5 mm and W = 25 mm, crack extension occurs at Ka > KQ
and therefore KQ = KQ^I. The increase of KQ from W = 50 mm to W^ —
100 mm is due to the rising K, Ac-curve.
With the electrical potential method it was also possible to determine
K*o.i at a crack extension of 0.1 mm. The calculation was done with the
 MUN2 ON SPECIMEN SfZE 415

s ss "2 ^2
<3 i/i
iii »o »o •* ift lO lO lO oi
o d ddSSS o o o d d d odd o o o o o o

o^ r- <N to m p lo (> q "~! 1

I/) C\ 0^ fO «-! »0
«!Q Q r? "^
a SR (N OO »0 • * ^O 1/5
I
"1 "1
(N M (N fS fS iQ2' «> 2C3

^•1 aas acj fS fN r» fN (N ? 20 ) lO fN l o m

I
^
.1 f) \0 rt f> ro
(S
•
n g rS o
^ ^ ^ ^ ^ s ?§

II 'O * i '^ '•O **) "7 "1

d d d d d d d

I
•<»•

m
ii •J « J t« H
H J t/j H irt
H
i-j
fr^T"
•JH

16 11 i I"

ll ii fig
•a
•c
figs
 416 ELASTIC-PLASTIC FRACTURE

K.AQ

FIG. 4—KQ (5 percent secant) and crack extension at KQ versus specimen width for a
level (a) and a rising (b) K-Aa-carve.

60

40 A/*^

•/ LU
O QC
< UJ
z li- \-
•
a. -2
20 J B, mm
1 50
25
> 12.5

20 40 60 80 100
SPECIMEN WIDTH W. mm

FIG. 5—Effect of specimen width on Kofor three-point bend specimens of aluminium alloy
7475-T7351.
 MUNZ ON SPECIMEN SIZE 417

plasticity correction. From Fig. 6 it can be seen that K*o.i is independent

of width. Thus it is possible to apply linear-elastic fracture mechanics also
to the smallest specimens with W = 12.5 mm, showing again that the
increase of KQ with increasing W is not due to failing of linear-elastic
fracture mechanics.

Variable Secant for the Determination of Size-Independent /T-Values

The introduction of the 5 percent secant method was an important step
during the development of a standard procedure for fracture toughness
determination. This method has two disadvantages. A specimen size effect
occurs for some materials, and for small specimens a K-\a\ue is determined
which is lower than Ko at the onset of stable crack extension. With a
variable secant with a slope adjusted to the specimen size, it is possible to
determine size-independent A'-values.
To determine the relation between the slope of the secant and the crack
extension, it is useful to decompose the crack opening displacement v into
the three components

V = Vei + Vpi + Vcr (15)

60

2 a
A T
• • ^
40 m

Ul
o cc
< Ul
U. 1-
Q:
-1
2
1J _
• O CO tJ B, mm
o 50
* 25
• o 12,5

20 40 60 80 100
SPECIMEN WIDTH W, mm

FIG. 6—Stress intensity factor K*o./ at 0.1-mm crack extension versus specimen width
for the aluminium alloy 7475-T73S1.
418 ELASTIC-PLASTIC FRACTURE

Vei is given by Eq 10. With regard to Eqs 12 and 14 between Vpi/vd and
{K/oyY/ W, a linear relation can be assumed

, _ . ^ (K\ _ 0.0526 1 / A - y ,.,,

vp,/ve, - A ^[^^^ - — —i^-) (16)

ForC = 0.55, A =0.174.

The component Vcr, due to crack extension, is given by

and

Vel / W

For three-point bend specimens with a/W = 0.5, f'/f = 5 and

Av vp, + Vcr 0.174 /KV _^ ^ ^ Aa
— = ^ = -Tjr- — + 5 X—rr (19)
Vei Vei w \ay/ yy
If specimens with different width but identical K, Aa-curves are tested, it
can be seen from Eq 19 that the secant intersects the F, v-curve at the
same K or Aa, respectively, for Av/vei X W = const.
Size-independent K-\a.lues then can be obtained according to the follow-
ing procedure. For tests with different specimen sizes a specimen with a
width W = Ws, to which the 5 percent secant is applied {Av/va = 0.0526),
is used as a reference. Then for a specimen with an arbitrary width W there
has to be
Av Ws
— = 0.0526 —, (20)
Vel W

Between Av/va and the change of the slope of the secant Am the relation
holds
mo- m _Am_ I
mo mo Vei/Av + 1
leading for the secant to
Am 1 (22)
m„ l + l9W/Ws

where mo is the slope of the linear part of the F, v-curve.

 MUNZ ON SPECIMEN SIZE 419

For the three-point bend tests of the aluminum alloy 7475-T7351, for
which the Ka-W-relation was shown in Fig. 5, /iT-values called Ks were
determined with the variable secant. As a reference width Ws = 50 mm
was used. In Fig. 7, Ks is plotted against width.
Comparing Figs. 5 and 7, it is shown that there is a much smaller
increase of Ks than of KQ with increasing width. This small increase dis-
appears, if the original crack length a is replaced by a corrected crack
length a + 0.5 mm, leading to K*s (see Fig. 8).
Similar results were obtained for the titanium alloy Ti-6A1-4V. As can
be seen from Fig. 9 also, a considerable increase of KQ with width was
observed for this alloy [6]. K*s, however, obtained for a reference width
of 40 mm, is independent of width (Fig. 10).

Fracture Toughness Determination witii Sabsized Specimens

For a rising plane-strain K, Aa-curve, the question arises which value
along the curve should be used as a fracture toughness value. This problem
shall not be discussed here. With the variable secant it is possible to
determine a /iT-value at small or larger crack extension, depending on the
choice of the reference width.
For small specimens, the slope of the secant has to be low enough to

60

o I

IP *
•
AO •
o

Hi
O Q
< li]
cr. 2
20 to (.} B, mm
cI 50
* 25
• c) 12.5

0 20 ^0 60 80 100
SPECIMEN WIDTH W. mm
FIG. 7—Effect of specimen width on Ksfor the aluminium alloy 7475-T73S1.
420 ELASTIC-PLASTIC FRACTURE

60

40

E o ai
z
if z
20 CE
o D, mm
a 50
25
o 12,5

20 40 60 100
SPECIMEN WIDTH W. mm

FIG. $—Effect of specimen width on K*s/or the aluminium alloy 7475-T7351.

make sure that the secant intersects the F, v-curve at K, > Kio. The
necessary slope can be obtained by means of Eq 19 with Aa = 0, leading to

Av 0.174 /KV
(23)
v., ^ W a, J

Taking into account the scatter of C in Eq 14, use of

Av 0.2 /K V
(24)
v., ^ WK'^y

is proposed. In addition to the already mentioned results, for some aluminum

alloys, Ku obtained from small three-point bend tests with .8 = 12 mm,
W = 12 mm and a span of 48 mm was c(»npared with Kic from compact
specimens with B = 25 mm, W = 50 mm. For the compact specimens the
5 percent secant (Av/vd = 0.0526) was used; for the bend specimens
according to Eq 20, Av/va — 0.22 or Am/nto = 0.18. The results are
given in Table 5. In Fig. 11, K*ic for the compact specimens is plotted
against K*s for the three-point bend specimens. Nearly all measuring
points are along a line for which K*ic is 5 percent below K*,. This small
deviation may be due to the effect of specimen type. In some investigations
 MUNZ ON SPECIMEN SIZE 421

100

O
eo O'x
O
11

' <)
60 I 1
z
8
40 a
B, mm
X 78
o 39
20 " 20
0 10
-» 5
0 2

20 40 60 80
SPECIMEN WIDTH W, mm
FIG. 9—Effect of specimen width on KQ for three-point bend specimens of the titanium
alloy Ti-6Al-4V.

comparing compact and three-point bend specimens, such an effect was

observed [20,21]. For some tests the difference was larger than 5 percent.

Conclusions

1. The minimum specimen thickness for the determination of plane-

strain fracture toughness, which is due to the requirement for a plane-
strain region in the center of the specimen, is much smaller than given in
ASTM Method E 399-74. The factor 2.5 of the thickness requirement can
be replaced by aoyil — v^)/E, with a between 25 and 50.
2. The minimum ligament width for the application of linear-elastic
fracture mechanics is also much smaller than given by ASTM Method E
399-74. If K is calculated with Irwin's plasticity correction, the factor 2.5
can be replaced at least by 1, possibly even by 0.4.
3. For proportionally sized specimens with W/B = 2, there is only a
small range of specimen size where elastic-plastic methods have to be
applied for the determination of plane-strain fracture toughness.
4. For specimens for which linear elastic fracture mechanics can be
applied, size-dependent /sTic-values can be obtained if the 5 percent secant
422 ELASTIC-PLASTIC FRACTURE

80
1 O
8 ^ i;] a
X
>
60

I 40
B, mm
X 78
O 39
20 ^ ' 20
a 10
'^ 5
o 2

20 40 60 80
SPECIMEN WIDTH W, mm

FIG. 10—Effect of specimen width on K*s for the titanium alloy Ti-6Al-4V.

TABLE 5—Comparison of stress intensity factors (in MNm ^'^) of3-point bend and compact
specimens of different size.

Compact, IV = 50 mm
3-Point Bend,

Material

Al-alloy
Specimen
Orientation

T-L
VK = 12 mm

Ks

35.7
K*,

42.5
Kic

38.7
K*ic

40.3
-m 22.7
7475- L-T 39.2 46.5 43.5 45.3 27.0
T7351 ST 32.1 38.0 34.5 35.9 19.8

Al-alloy L-T 39.8 45.8 48.9 50.5 37.0

7075(1) T-L 38.0 44.2 40.6 41.9 24.8

Al-alloy L-T 33.5 38.6 31.8 32.8 13.8

7050 T-L 36.5 42.3 33.8 34.9 16.6
ST 24.6 28.5 26.5 27.4 H.2

Al-alloy L-T 21.3 33.1 36.0 37.2 25.6

2024 (I) T-L 21.3 33.5 34.6 35.7 31.1
S-T 19.6 28.4 26.6 27.5 21.6

Al-alloy L-T 29.1 34.0 32.3 33.4 13.3

7075(11) T-L 27.4 31.8 28.9 29.9 11.6
S-T 21.6 24.8 23.1 23.8 8.6
 MUNZ ON SPECIMEN SIZE 423

50

• T 7050
• e 7475
• a 2024 (I)
• • 7075 (H)
A 7075 (I)

e
2:

AO 50
K* (3-point bend), MNm-3/2

FIG. 11—K*/c/or compact specimens with Vf = 50 mm versus K*s/or three-point bend

specimens with Vl — 12 mm.

method is applied. The size effect is due to the plastic deformation at the
crack tip and to the existence of a rising plane-strain crack growth resis-
tance curve.
5. With a variable secant, adjusted to the specimen width, it is possible
to determine size-independent fracture toughness values.

Acknowledgments:
The author thanks J. Eschweiler for performing the tests thoroughly.
The financial support of the Deutsche Forschungsgemeinschaft is gratefully
acknowledged.

APPENDIX
Materiab and Experimental Procednie
The experimental results were obtained for different aluminum alloys, a titanium
alloy, and two steels:
424 ELASTIC-PUSTIC FRACTURE

1. Aluminum alloy 7475-T7351, plate of thickness 63 mm, fracture toughness

and jrield strength dependent on the location of the specimens; therefore, specimens
from the center of the plate were distinguished from specimens from the surface.
2. Aluminum alloy 7475-T7351, plate of thickness 63 mm; specimens were an-
nealed for 15 h at 180°C.
3. Aluminum alloy 2024-T351 (I), plate of thickness 130 mm.
4. Aluminum alloy 2024 (II), plate of thickness 10 mm.
5. Aluminum alloy 7050-T73651, plate of thickness 100 mm.
6. Aluminum alloy 7075-T7351, plate of thickness 32 mm.
7. Aluminum alloy 7075 (II) (treatment not specified), plate of thickness 63 mm.
8. Titanium alloy Ti-6A1-4V, plate of thickness 82 mm.
9. Nickel-Chromium-Molybdenum Steel I with the composition 0.32C, 4.2Ni,
1.68Cr, 0.43MO, 0.41Mn, 0.28Si, 0.016P, 0.014S, bars of cross section 100 by 250
mm in normalized condition.
10. Nickel-Chromium-Molybdenum Steel II with the composition 0.28C, 3.47Ni,
1.55Cr, 0.35MO, 0.26Mn, 0.09V, 0.19Si, 0.006P, 0.007S, 0.005Sn, 12 h 850''C,
water-quenched, 29 h 600°C; specimens were cut from a turbine disk.
Compact and three-point bend specimens of different size were cut from the plate
and precracked in fatigue. Some of the compact specimens also were used for J-
integral evaluation. For these specimens, instead of crack mouth displacement, the
displacement at the load line was measured.

References
[/] Landes, J. D. and Begley, J. A. in Fracture Analysis, ASTM STP 560, American Society
for Testing and Materials, 1974, pp. 170-186.
[2] Keller, H. P. and Munz, D. in Flaw Growth and Fracture, ASTM STP 631, American
Society for Testing and Materials, 1977, pp. 217-231.
[3] Munz, D., "Fracture Toughness Determination of the Aluminium Alloy 747S-T7351
with Different Specimen Sizes," Deutsche Luft- und Raumfahrt, Forschungsbericht
DLR-FB 77-04, 1977.
[4] Rice, J. R. in Mechanics and Mechanisms of Crack Growth, Proceedings, British Steel
Corp., Cambridge, U.K., 1973, pp. 14-36.
[5] Robinson, J. N., "The Critical Crack-Tip Opening Displacement and Microscopic and
Macroscopic Fracture Criteria for Metals," Ph.D. thesis. University of California, Los
Angeles, Calif., 1973.
[6] Munz, D., Galda, K. H., and Link, F. in Mechanics of Crack Growth, ASTM STP 590,
American Society for Testing and Materials, 1976, pp. 219-234.
[7] Green, G. and Knott, J. F. Journal of the Mechanics and Physics of Solids, Vol. 23,
1975, pp. 167-183.
[8] Srawley, J. E. and Brown, W. F. in Fracture Toughness Testing and its Application,
ASTM STP 381, American Society for Testing and Materials, 1965, pp. 133-198.
[9] Vosikowsky, O., International Journal of Fracture Mechanics, Vol. 10, 1974, pp. 141-
157.
[10] Robinson, J. N. and Tetelman, A. S., International Journal of Fracture Mechanics,
Vol. 11, 1975, pp. 453-468.
(//] Paris, P. in Fracture Toughness, ASTM STP 514, American Society for Testing and
Materials, 1972, pp. 21-22.
[12] Hayes, D. J., "Some Applications of Elastic-Plastic Analysis to Fracture Mechanics,"
Ph.D. thesis. University of London, London, U.K., 1970.
[13] MarkstrOm, K. M. and Carisson, A. J., "FEM-Solutions of Elastic-Plastic Crack
Problems—Influence of Element Size and Specimen Geometry," Publication No. 197,
Hallfastnetslara, KTH, Stockholm, Sweden, 1973.
[14] Berger, C, Keller, H. P., and Munz, D., this publication, pp. 378-405.
 MUNZ ON SPECIMEN SIZE 425

[15] Brown, W. F. and Srawley, J. E. in Review of Developments in Plane Strain Fracture

Toughness Testing, ASTM STP 463, American Society for Testing and Materials, 1970,
pp. 216-248.
[16] MarkstrOm, K. H., Engineering Fracture Mechanics. Vol. 4, 1972, pp. 593-603.
[17] Hayes, D. J. and Turner, C. E., InternationalJoumal of Fracture Mechanics, Vol. 10,
1974, pp. 17-28.
[18] Griffis, C. A. and Yoder, G. R., Transactions, ASME, Journal of Engineering Materials
Technology, American Society of Mechanical Engineers, Vol. 98, 1976, pp. 152-158.
[19] Kaufman, J. G. and Nelson, F. G. in Fracture Toughness and Slow-Stable Crack
Growth, ASTM STP 559, American Society for Testing and Materials, 1974, pp. 74-85.
[20] Hall, L. R. in Fracture Toughness Testing at Cryogenic Temperatures, ASTM STP 496,
American Society for Testing and Materials, 1971, pp. 40-60.
[21] Munz, D., unpublished results.
[22] Brown, W. F., Jr. and Srawley, J. E. in Review of Developments in Plane Strain
Fracture Toughness Testing, ASTM STP 463, American Society for Testing and
Materials, 1970, pp. 216-248.
[23] MarkstrOm, K. M., Engineering Fracture Mechanics, Vol. 4, 1972, pp. 593-603.
[24] Hayes, D. J. and Turner, C. E., International Journal of Fracture Mechanics, Vol. 10,
1974, pp. 17-32.
W. R. Andrews' and C. F. Shih'

Thickness and Side-Groove Effects

on J- and 6-Resistance Curves for
A533-B Steel at 9 3 X

REFERENCE: Andrews, W. R. and Shih, C. F., "TUcknen and SMe-GrooTC Effecto

an J- and 5-Registance Curves for A533-B Steel at 93 °C," Elastic-Plastic Fracture. ASTM
STP 668, J. D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for
Testing and Materials, 1979, pp. 426-450.

ABSTRACT: A test program was conducted to determine the effects of specimen

thickness variations, side grooves, and crack length variations on the deformation and
ductile fracture of A533-B, Cl-1 steel at 93 °C.
The crack extensions were estimated using the correlation between elastic compliance
and crack length. Crack extensions were also estimated using a correlation among crack-
opening displacements, load line displacements (5 — Vi), and crack length. The in-
ferred estimates of crack extension were supp'emented by some measurements on heat-
tinted fracture surfaces. The results suggest that the observation of thickness or side-
groove effects on crack-extension resistance curves is dependent on the method of
measuring crack extension.
The compliance correlation method was less sensitive to crack extension and showed a
classical thickness effect: increased crack growth resistance with decreasing thickness,
and decreased resistance with the use of side grooves. The 8 — VL correlation method
was more sensitive to crack extension and showed no effect of thickness or of side grooves
on crack growth resistance. The presence of side grooves promoted flat fracture and sup-
pressed shear lips. Specimens without side grooves developed large shear lips.

KEY WORDS: ductile fracture, testing, crack extension, side grooves, compact
specimen, thickness effects, J-integral, crack-opening displacement, crack propagation

Ductile fracture of low-alloy steel initiating from crack-like defects has

been studied in the past, and a number of criteria for predicting initiation
and continued stable tearing have been advanced. None of the criteria ad-
vanced have found the acceptance that /L ic has found for brittle fracture
ASTM Test for Plane-Strain Fracture Toughness of Metallic Materials (E
399-74). The evaluation of these criteria has been made difficult by the frac-

' Engineer-Mechanics of Materials, Materials and Processes Laboratory and research

engineer, Corporate Research and Development, respectively. General Electric Company,
Schenectady, New York 12345.

426

Copyright 1979 b y A S T M International www.astm.org

 ANDREWS AND SHIH ON A533-B STEEL AT 93°C 427

ture process and the difficulty of observing the process through the thickness
of the specimen.
A review of the available data, primarily those from the Heavy Section
Steel Technology (HSST) Program [1],^ revealed that both flat fracture and
shear lip formation play important roles in the fracture of A533-B steel.
Compact specimens as large as 508 mm wide by 254 mm thick produced
shear lips on nearly 60 percent of the specimen thickness. In these
specimens, the fracture surfaces indicate the sequence of events in the pro-
cess of ductile tearing. The initiation of fracture occurred at the center of the
specimen thickness and proceeded by flat, ductile tearing to form a
characteristic thumbnail-shaped crack front. At some critical depth of the
crack front, the side ligaments began tearing to form shear lips adjacent to
the surface. The width of the shear lips increased as the crack progressed un-
til 60 percent of the specimen thickness fractured by 45-deg shear. This pro-
cess was observed in practically all specimen sizes.
Thus, the phenomenon of ductile fracture takes on a complex, three-
dimensional aspect not found in brittle fracture. The flat fracture near the
center thickness develops under nearly plane-strain constraint, whereas the
shear lips near the surfaces develop under plane-stress deformation.
The problem is to select fracture criteria which are independent of
specimen geometry and size. The objective of a larger program [2,3], of
which these tests are a part, is to evaluate plastic fracture criteria beyond
small-scale plasticity using results of compact specimen, center-cracked
plate specimen, and double-edge notched plate specimen tests. Finite-
element calculations based on these specimen geometries were carried out to
provide detailed computations of several potential fracture parameters for in-
itiation, stable growth, and instability [4], The preliminary observations in-
dicate that a fully three-dimensional analytical model is needed to simulate a
standard compact specimen when the mode of failure is nonplanar. To avoid
the expense of 3-D modeling, incorporating criteria for flat and for shear
fracture, the use of side grooves in compact specimens was tried with the ob-
jective of simplifying the fracture process to approximate a 2-D, plane-strain
model. To accomplish this, the side grooves were expected to suppress to a
minimum the formation of shear-lips and to produce a flat-fracture crack
which has a straight leading edge through the thickness.
This program investigated the effects of specimen thickness (B), side-
groove depth, and initial crack depth (ao) on compact specimen tests. Two
methods for estimating the crack extension (Aa) were used. One method, us-
ing the crack-length correlation with elastic compliance [5], measures the
average crack depth, whereas the second, using the correlation among crack-
tip opening displacement, load-line displacement (6 — VL), and crack
length, measures the crack extension near the center of the specimen. These

^The italic numbers in brackets refer to the list of references appended to tliis paper.
428 ELASTIC-PLASTIC FRACTURE

tests showed that the crack growth resistance was affected by specimen
thickness and side grooves when estimated with the elastic compUance cor-
relation, but these effects were negligible with the d — Vi correlation.

Material
The composition of the test materials is given in Table 1, the mechanical
properties in Table 2, and the Charpy V-notch impact properties in Table 3.
Test Material 1 was from the same heat and heat treatment as was used in
the Welding Research Council (WRC) survey on mechanical properties of
A533-B steel [6]. The source material was nozzle dropouts from 165-mm-
thick (6VJ in.) plate. Test Material 2 was A533-B plate designated in this
program as EPRI-Ol-GE-02. This plate was rolled and quenched and
tempered as a 203-mm-thick (8 in.) plate.

Test Specimens
The specimen geometry was based on the standard compact specimen
(ASTM E399-74) with modifications to permit measurement of the load-line
deflection (VL) and the opening displacement near the crack tip (V;v), Fig. 1,
using a linear variable differential transformer (LVDT) and an extension rod
across the crack. The varied test specimen dimensions and precrack lengths

TABLE 1—Ladle analysis of test material, weight percent.

Heat No. Mn Cu Si Ni Cr Mo Al

A0999-1 0.22 1.32 0.010 0.014 0.14 0.19 0.60 .09 0.50 0.026

B0256, 0.20 1.22 0.011 0.005 0.15 0.66 0.55

0.19 1.22 0.009 0.016 0.15 0.65 0.54

TABLE 1—Longitudinal tension test results at 93°C.

Material 1 Material 2
A0999-1 B0256
Identification 30319 30321 30322 30XXX T-Specimen

Tensile Strength, MPa 555 555 552 542 574

Proportional Limit, MPa 368 381 400 423 426
0.02 % yield strength, MPa 409 421 423 439 441
0.2 % yield strength, MPa 421 430 425 436 443
Percent Elongation, 4.57 cm 26.1 26.7 24.4 26.1 26.7
Percent Reduction of Area 70.0 71.8 65.7 72.3 71.6
Fracture stress 1150 1238 1085 1091 1262
 ANDREWS AND SHIH ON A533-B STEEL AT 93°C 429

TABLE 3—Charpy impact test results (L-T orientation).

Material 1
A0999-1

Identification 30319 30321 30322 30XXX B0256

40.77T.T., °C -39 -40 -40 -46 < -32

6 7 . 8 / T . T . , °C -26 -29 -19 -17 < -32
50 percent FATT, °C - 4 -18 -18 -12 - 7
0.89mmL.E.T.T., °C -29 -23 -32 -32 < -32
Energy at 93 °C, J 154 165 190 171 190

NOTES:
L-T = Longitudinal-transverse (see ASTM E399-74, Fig. 9).
T.T. = Transition temperature.
FATT = Fracture appearance transition temperature.
L.E.T.T. = Lateral expansion transition temperature.

are given in Table 4. Both the nine-point average crack length and center
thickness crack length are listed. In subsequent discussion of results, the
elastic compliance is referred to the nine-point average crack length and the
limit load with the center-thickness, or maximum depth, of the precrack.

Test Procedure
The specimens were loaded in an Instron 1330-kN (300 kip), four-column
load frame, using closed-loop position control of the loading ram. The
testing machine and grips had a compliance of 2.9 X 10"'' mm/N (5.1 X
10"' in./lb). The ram opened the load points of the specimen at about 7.6
mm/h (0.3 in./h). The load-line displacement versus load (P) was recorded
on an A^-y plotter. The load-line displacement, the load, the near-tip crack
opening displacement, and the ram position were recorded on digital
magnetic tape using a Vidar data logger. The transducer signals were
scanned sequentially every 3 s at a scan rate of about 30 channels per second.
The data were recorded to the nearest millivolt using a ± 10 V standard
range. The magnetic tape was placed on file on a large, general-purpose
computer and the values of 6, Aa, and/i (J-integral) were computed using a
FORTRAN computer program.
The crack-tip opening displacement was estimated from the measure-
ments of VL and V^, assuming the arms of the compact specimens to be
rigid (no bending), by extrapolating the two measurements to the center-
thickness crack depth.
Two techniques were used for estimating Aa. One method is the unloading
compliance method [5] in which the elastic compliance is used to estimate Aa
through the use of combined analytical [7] and experimental [8] correlations.
The compliances were estimated using a linear least-squares best fit to the P
430 ELASTIC-PLASTIC FRACTURE

NOTES:

® DIMENSIONS IN MILLIMETERS
WITH INCH D I M I L N S I O N S IN
PARENTHESES
® DIMENSION J GIVEN IN TABLE 4

0.075 ( 0 0 0 3 ) MAX A DIAMETER.

NOTCH HOOT RADIUS

PART S
1 2 5 4 11.001
SIZE A B C D E f G H J K
2 12.7 (O50I
4T
MM SO. 8 0 lOI.e 203.20 2 5 4 . 0 2 4 4 . 0 0 122X10 112.00 6 6 . 0 0
INCHES 2 . 0 0 0 4 . 0 0 aaoo 10 19.600 4 . 8 0 0 4 . 4 0 0 2 . 6 0 0
4 10.2 3 6.4 (0.25)
® 0.40 DETAIL 8 - 8

FIG. la—Compact fracture specimen: basic design.

HOLE TO BE CENTERED ON
1/2-20 UNF-2B THICKNESS OF SPECIMEN

DIMENSIONS IN MILLIMETERS
WITH INCH DIMENSIONS IN PARENTHESES

© MAKE FROM 4T COMPACT SPECIMEN

FIG. lb—Compact fracture specimen: hole for LVDT displacement measurement V N .

 ANDREWS AND SHIH ON A533-B STEEL AT 93°C 431

TABLE 4 —Test specimen geometry and material properties.

Crack Length,
mm
Nominal B, Specimen Test
Thickness mm Thickness, in. Identification Material 9-Point Avg Center

1 25.4 1.00 30319-3 101.6 113.5

2.5 50.8 2.50 30319-1 115.3 117.3
2.5 50.8 2.50 30319-4 134.6 136.1
4 101.6 4.00 30322-1 122.7 129.3
4 101.6 4.00 30321-1 141.0 146.6
4 101.6 4.00(25% SO" 30XXX-1 128.3 131.3
4 101.6 4.00(25% SG) 30XXX-2 143.5 145.5
4 101.6 4.00(50%SG) 30322-2 127.5 128.3
4 101.6 4.00(50%SG) 30321-2 141.0 141.5
4 101.6 4.00(25%SG) T52'' 2 117.2 117.2
4 101.6 4.00(12.5%SG) T-71 2 115.8 115.8
4 101.6 4.00(12.5%SG) T-32 2 125.0 125.0
4 101.6 4.00(12.5%SG) T-21* 2 134.0 134.0
4 101.6 4.00(12.5%SG) T-31 2 136.0 136.0
4 101.6 4.00(12.S%SG) T-62 2 144.5 144.5
4 101.6 4.00(12.5%SG) T-22* 2 145.8 145.8
4 101.6 4.00(25%SG) T-51 2 149.5 149.5
4 101.6 4.00(25%SG) T-61 2 162.7 162.7
4 101.6 4.00(25%SG) T-41'' 2 169.8 169.8

NOTES:
Modulus of elasticity = 200 GPa.
Flow stress. CTK = 490 MPa.
«SG = side grooved.
''Specimen was heat-tinted following test.

versus VL data, obtained on unloading, ignoring the first one to four data
points at each unloading. The calculated crack extension values were cor-
rected for the error in compliance measurement due to the finite deflection of
the load line.
The second method is based on the unique relationship between 8 and VL
which holds provided no crack extension occurs. When crack extension oc-
curs, the measured excess of 8 over the unique value calculated for the
original crack length is used to estimate the extent of the crack extension.
The derivation of this method is shown in the Appendix. This latter method
for estimating crack extension will be referred to as the S — Vi method. Four
were terminated with a heat-tinting operation which provided calibration
points for the d — Vi estimate of Aa. (The heat tinting was performed by
heating the cracked specimen to about 260°C (500°F) for 4 h, cooling to
room temperature, and breaking.) The heat tint points were in good agree-
ment with compliance estimates of Aa, but only the T-41 result is within the
range reported here.
The values of/i were calculated using the Merkle-Corten relationship [9]
in the form
432 ELASTIC-PLASTIC FRACTURE

T~ 2/1 I 2PVt ...

where

a, = 1.222468 - 0.637295 (ao/W)

(2)
+ 0.614937 (ao/W)^ - 0.200797 (ao/Wy

02 = -0.006771 + 0.595163 (ao/W^)

(3)
- 0.940241 (ao/W)^ = 0.353779 {ao/Wy

where
A = area under the load deflection curve,
BN — net thickness,
W = specimen width,
ao = initial fatigue-crack depth (nine-point average), and
P = maximum load reached at or prior to the measurement point.
This expression for Ji was found to be in excellent agreement with J\
evaluated using finite-element computations along a contour remote from
the crack tip for both stationary and growing cracks [5].
A method for obtaining silicone rubber replicas of the crack tip was applied
to the specimens not heat tinted. The procedures used are detailed else-
where [70].

Results and Discussion

The results are presented in two parts. The first part discusses results prior
to crack extension and the second focuses on crack initiation and growth
results.

Deformation
The deformation of the specimens in terms of load-line deflection versus
load is summarized in Fig. 2 using normalized axes. A discussion of the nor-
malizing parameters is found in the Appendix. In Fig. 2, the effect of side
grooves on the elastic compliance is made evident by the different slopes in
the linear, rising load portion of the curves. The effects of side grooves on the
elastic compliance of compact specimens are evaluated in greater detail in
Ref 6. The differing limit loads for large plastic deformations are consistent
with the transition from plane-strain to plane-stress plastic deformation as
the thickness is reduced. The plane-strain and plane-stress limit loads based
on slip-line field solutions [4,11] are indicated in Fig. 2. The transition in
limit loads from plane-strain to plane-stress levels was observed only when
 ANDREWS AND SHIH ON A533-B STEEL AT 93°C 433

4.0-

LIMIT LOADS;

J<o *•
)K5 O-X-O

2.0 3.0
EVL
ffy W ( 2 + o / W ) 2

FIG 2—Summary of load versus load-line displacement curves, compact specimens, a/W >
0.55.

the center-thickness crack length was used to calculate notch stress and limit
stress. This distinction in crack-length measure was made necessary by the
curvature of the crack front in the non-side-grooved specimens (See Table 4).
The normalized values of/i with increasing deflection are shown in Fig. 3
for varying specimen thickness. The figure gives evidence that any unique
relationship between / | and 6 deteriorates when there is significant cross slip
(out-of-plane slip). This observation is consistent with the transition from
plane-strain to plane-stress constraint, and is a direct consequence of the
variation of limit loads seen in Fig. 2.
The relationships between J\ and 5 also span a range for varying specimen
thicknesses (Fig. 4), and fall in the same order as observed in Fig. 3. It was
found that on normalized coordinates for 6 versus VL , a unique relationship
exists prior to crack initiation. This unique relationship is developed in the
Appendix and forms the basis for a sensitive method for estimating crack ex-
tension at the specimen mid-thickness.

Cracking
Fracture Appearance—The fracture surfaces of the various specimens are
contrasted in Fig. 5. In every case, the use of side grooves, 12 Vi percent of the
gross thickness or deeper, promoted flat fracture. Some obvious lateral con-
434 ELASTIC-PLASTIC FRACTURE

14.0

PLANE STRESS
LIMIT LINE

3.0

ITyW ( 2 * 0 / W ) ^

FIG. 3'-Summary o/J/ versus load-line displacement curves, compact specimens, a/W >
0.55.

traction of the sides occurred in those specimens side grooved to 12 Vi per-

cent. To be perfectly clear, percent side groove depth is calculated

B - Bj,\
percent SG = 100 (4)

where Bf, is the net specimen thickness.

When side grooves were not employed, shear lips formed which were of
constant absolute size for the several thicknesses. The shear lips formed an
increasing proportion of the fracture surface as the thickness decreased and
also as distance increased from the precrack-tip in the direction of crack prop-
agation. This latter relation held until the crack approached the back sur-
face to a distance of about three-quarters to half the specimen thickness, at
which point the proportion to total thickness of shear lip became constant at
about 60 percent. The thinnest specimen (B/W = 0.125) developed a full
shear fracture after a short transition from the originally flat fatigue
precrack. These results are consistent with those of Merkle [/] discussed in
the introduction.
Crack Extension Estimates Using Compliance—The /-resistance curves
developed for Test Material 1 using the unloading compliance estimates of
crack extension are shown in Fig. 6. The test results show the effects of side
 ANDREWS AND SHIH ON A533-B STEEL AT 93»C 435

inches
0.01 0.02 0.03 0.04
—I 1— 5000
—r— 1—

800
4T, 25%

4000

600

3000
E
z

400-

2000

200
1000

0.2 0.6
8, mm

FIG. 4—Relationships between J; and S, compact specimens, A533 Grade B, Class 1, steel;

grooves and thickness. It is seen that the values of/i for crack initiation, Jc,
are dependent upon specimen thickness and on side grooving. The values of
/c increased as the thickness was reduced from the 102-mm (4 in.) side-
grooved specimens (largest effective thickness) to 102 mm (4 in.), to 63.5 mm
(2.5 in.) and then decreased for the 25.4-mm (1.0 in.) thickness. These
results suggest that /c is dependent on the degree of plane-strain constraint
achieved in the fracture process zone.
The slopes of the /i-resistance curves, Fig. 6, increased with decreasing
thickness, reflecting the formation of large shear lips in specimens without
side grooves. The increased slope with decreased thickness correlated with
the ratio of the estimated plastic zone size to the thickness

JE (5)
^ c = -

where ay is the average of the ultimate and 0.2 percent yield strengths. This
relationship is shown in Fig. 7. A similar, nearly linear, relationship was
found by Lake [12] for an aluminum alloy.
 436 ELASTIC-PLASTIC FRACTURE

THICKNESS
(mm)
0

6.35(12.5%)

12.7(25%)

25.4 (50%)

SIDE-GROOVE DEPTH
mm (%)

FIG. 5—Fracture surfaces of 4T compact specimens, A533 Grade B, Class 1 steel tested at
 ANDREWS AND SHIH ON A533-B STEEL AT 93'C 437

7000

-6000
1000

5000

800

4000

600

-3000

400

2000

200-
- 1000

FIG. 6—Effects of thickness and side grooves on resistance to crack growth, estimated using
compliance correlation, 4T compact specimens. A533-B CI steel (Material I) tested at 93°C.

The 7i-resistance curves developed for Material 2 using unloading com-

pliance estimates of Aa are shown in Fig. 8. These tests indicate that initial
crack length (a/w > 0.55) has no effect on the resistance curves.
Crack Extension Estimates Using 8 — VL—The results of the thickness
and side-groove effect studies (Material 1) when evaluated using the 6 — VL
estimates of Aa are shown in Fig. 9. Note that there is no effect of thickness
on Jc, 8c, or on the slopes of the resistance curves. The side-mounted gage
mentioned in the legends to Fig. 9 was mounted across the crack tip on the
side of the specimen, thus representing a slightly different measurement than
was made for the other specimens. Comparison of the results in Fig. 9a with
those in Fig. 6 indicates that the 8 — VL method was more sensitive to crack
initiation; it seems to be representative of the crack extension at the center of
the specimen. The compliance technique, in contrast, represents a through-
438 ELASTIC-PLASTIC FRACTURE

350-
50

300 SPECIMEN THICKNESS, in,

4.0 2.5 'o
4 0 .

a.
"i
2 2S0 35 S
d

30 I
< 200
E
<
150 THICK
O-" S I D E - GROOVED
J SPECIMENS
SPECI

100
8 10 l< 18 20
PQ = J(E /Bo

FIG. 7—Correlation of slopes of i[-resistance curves, crack growth estimated using com-
pliance correlation. 4T compact specimens, A533-B Cl-1 steel (Material 1) tested at 93 °C.

thickness average crack extension. A further reduction in the data scatter is

noted when the data are presented as a 5-resistance curve (Fig. 9b).
Further evidence that supports the validity of the d — VL estimates of Aa
was found in silicone rubber crack-tip replicas [10], The replicas, obtained
for the 15 specimens not heat tinted (Table 4) were sectioned and observed in
profile, and gave 8 values which agreed with those inferred from the two
displacement measurements. (See Test Procedure section and Fig. 1.) Fur-
thermore, the critical values of 6 for crack initiation were in agreement be-
tween estimates from the silicone rubber profile (0.2 to 0.7 mm) and
estimates using the 8 — Vi technique (0.3 to 0.5 mm).
The results of evaluating tests of Material 2 side-grooved specimens with
varied crack lengths using the 8 — Vi method are shown in Fig. 10a. When
compared with Fig. 8, the range of/c estimates is reduced and the scatter in
the crack growth resistance values of/i is reduced.
Further reduction of scatter was again noted for these when presented as a
6-resistance curve (Fig. 106).
The 6 — VI method for estimating crack extension resulted in Ji- and
5-resistance curves which are independent of thickness and of the presence of
side grooves. The method provided a measure of mid-specimen crack exten-
sion. This observation is important because the measured limit loads cor-
related with mid-specimen crack length and not with the nine-point average
 ANDREWS AND SHIH ON A533-B STEEL AT 93°C 439

01 02 03
T —r- - • ^

0
A

D O
0 "
A
SPECIMEN
a <3| ° T-52 a
^ 0 A T-71 a
cP T-32 0
T-21 0
T-SI A
T-62 •
T-22 0
T-51
TSI
a0
HEAT TINT X
(T-41)

mm
CRACK EXTENSION

FIG. 8—Resistance to crack growth estimated using compliance correlation, side-grooved 4T

compact specimens, variable crack length, A533-B Cl-1 steel (Material 2) tested at 93°C.

crack length when the crack fronts were not straight. Like the curves
developed using compliance estimates of Aa, the/i- and 5-resistance curves
were independent of initial crack length. The 5 — Vi method resulted in a
greater sensitivity to short crack extensions.

Conclusions
1. The load-deflection curves in compact specimens showed variations of
limit loads between plane-strain and plane-stress limits as the specimen
thickness was reduced from B/W = 0.5 to B/W = 0.125. A similar and con-
sistent variation was found for the J-integral deflection curves. 5 deflection
curves were independent of thickness.
2. Side grooves ranging from 12*72 percent of gross thickness and deeper
successfully suppressed shear-lip formation in A533-B steel at 93 °C.
440 ELASTIC-PLASTIC FRACTURE

SPECIMEN.

o 30319 - 3 lin.

D 30321 - 2 50%S.G.

0 30322- 2 5 0 % S.G

A 30322 - 1 4 in. (SIDE MOUNTED GAGE)

A 30321 - 1 4 In,
_1
a 30319 - 1 2 ^ in
Q 30319 - 4 2^ In
0 30XXX - 2 25% SG

0 30XXX- 1 2 6 % SG,

A Q
O

A r^
<« o
C

mm
CRACK EXTENSION

(a)
 ANDREWS AND SHIH ON A533-B STEEL AT 93°C 441

O OS f

SPECIMEN
30319 - 3 I in.
30321 - 2 5 0 % S.G.
30322 - 2 S 0 % S.G.
_ 0.03
30322- I 4 in.
30321 - I 4 in.
30319 - I 2''2in. .
30SI9 - 4 2l^in.
30XXX-2 25%S.6.
30XXX- I 25% SG..
* SIDE MOUNTED GAGE

—I 0
4 6 10
CRACK EXTENSION, mm

(b)

FIG. 9—(a) J /-resistance and (b) &-resistance to crack growth estimated using 6 — VL tech-
nique, 4T compact specimens, variable thickness and side grooved, A533-B, Cl-1 steel
(Material 1), tested at 93°C.
442 ELASTIC-PLASTIC FRACTURE

0.1 Q3

-r

a
o
o

a 0 SPECIMEN

T-52 ^
T-71 D
T-32 O
T-21 0
T-31
T-22
^
0
T-51 A
T-61 0
HEAT TINT X
(T-41 )

CRACK EXTENSION

(a)
 ANDREWS AND SHIH ON A533-B STEEL AT 93-0 443

Q2 03
—r- —!—

o 0

SPECIMEN
&.
•M2 o
T-71
T-M
•
O
T-Zf 0
T-JI ii
T-22 o
T-«l a
T-61 0
HEAT TINT X
<=^<?_. (T-»l)

r-}sc
4 6
CRACK EXTENSION,mm

FIG. 10—(a) J /-resistance and (b) 8-resistance to crack growth estimated using 5 — VL tech-
nique, side grooved, 4T compact specimens, variable crack length, A533-B Cl-1 (Material 2)
steel tested at 93°C.
444 ELASTIC-PLASTIC FRACTURE

0020 NO CRACK EXTENSION

D COD ADINA FLOW THEORY

X CRACK LINE EXTRAPOLATION ^
ADINA FINITE DEFORMATION
T-52 FLOW THEORY
A T-52 EXPERIMENTAL
0.015
.O T-41 EXPERIMENTAL
NO GROWTH REFERENCE
FOR EXPERIMENTAL DATA
yk
^

/^X
0.010

0,005
o/^Z^

1 \
001 OOZ .005 J006
0 m% .004

W(2rto/W)'

FIG. 11—Relationship between crack-tip opening displacement (5) and load-line displace-
ment {VOfor compact specimens, a/W > 0.55, prior to crack growth (A = crack length).

3. Shear lip dimensions in non-side-grooved (smooth) specimens were

relatively independent of specimen thickness.
4. The observation of the effects of specimen thickness and of side-groove
depth in the specimen is dependent on the method of estimating the crack ex-
tension. The methods used are the elastic compliance technique and the 5 —
VL technique, which used the correlation among crack-tip opening displace-
ment (6), load line displacement (VL), and crack extension.
5. Crack extensions estimated using the 8 — VL technique measure mid-
thickness crack extensions and result in /i- and S-resistancje curves which are
independent of thickness in the range studied, are unaffected by the use of
side grooves, and are independent of crack length when the crack length to
width ratio, a/W, is greater than 0.55.

Acknowledgments
The authors are grateful to D. J. Tinklepaugh and D. F. St. Lawrence for
laboratory testing and to S. Yukawa and D. F. Mowbray for technical con-
sultation.
ANDREWS AND SHIH ON A533-B STEEL AT 93°C 445

NO CRACK GROWTH

•FATIGUE CRACK TIP

CRACK GROWTH

V|_ = CONSTANT

FIG. 12—Schematic crack-tip geometry.

 446 ELASTIC-PLASTIC FRACTURE

0.1 03
—I- —r-

5 «

SPECIMEW T-52

^_ Nf|X)W-o<,
»

o la
X T-41
(HEAT TINT)

mm
CRACK EXTENSION

FIG. 13—Sensitivity of crack-growth estimates to variations in y (Eg 26), side-grooved 4T

compact specimen, T-52; A533-B CI-J steel tested at 93 °C.
 ANDREWS AND SHIH ON A533-B STEEL AT 93'C 447

APPENDIX
Derivation of Basis for Crack Extension Calcalation From &— Vi Measurements
Srawley and Gross [13] observed for compact specimens that the dimensionless
coefficient

P "" (2+a/W) - 1-312-Q (6)

is approximately constant for large a/W (that is, a/W > 0.55). This observation may
be coupled with relations for elastic fracture mechanics
Ki^ = GiE' =JiE' (7)
where JE" = E — (plane stress) a n d £ ' = £7(1 — v^) — (plane strain) and
lA

where A is the area under the load-displacement curve. For elastic loading

A^\PdVL=\pVL (9)

and
K\ = stress intensity factor,
G\ = elastic strain-energy release rate,
/i = J-integral,
a — crack length,
B — specimen thickness,
P = applied load,
W = specimen width,
WL = load line displacement,
E = Young's modulus,
V = Poisson's ratio, and
Q = a constant.
Substituting Eq 8 into Eq 7, we get
1AE
^•' = fi(MFZ^^/^''/'^) "°^
The factor
f(a/W) i 3/(2 + a/W)
in Eqs 8 and 10 is an approximate correction for the tensile loading and for elastic
conditions. J\ so calculated differs from that using the Merkle-Corten correction [9] by
3 to 4 percent for a/Yf > 0.5.
448 ELASTIC-PLASTIC FRACTURE

Substituting Eq 9 into Eq 10 and this in turn into Eq 6 yields

Introducing the notch stress ratio

g ^ 2P(2 + a/W) ^ 6 VLE

ay OYWBiX - a/W)'^ Q^ arWd + a/Wy ^ '

Equation 12 suggests plotting a load versus load-line deflection curve on coor-

dinates

- / _ 2P(2 + a/W)

and

VLE
(14)
aYW(2 + a/W)^

which has an elastic slope of b/Q^ or 3.49.

Considering fully plastic deformation of the compact specimen, the limit load [10]
for plane-strain constraint is

•^"ffl = 2.52 (15)

ay
The curve resulting from the combined elastic and plastic deformation, Fig. 1, is in-
dependent of crack length for all a/W > 0.55 and for plane-strain constraint.
A second plot in which J\ is normalized may be developed by integrating the nor-
malized load-deflection plot. Without going through the step-by-step procedure, the
result is

WayHi - a/W) ~ ^[ayW(2 + a/W)^) ^^^^

Again, similar to the normalized load-deflection curves, the functional relationship is

independent of crack length for a/W > 0.55 as long as no crack growth occurs. See
Fig. 2.
In the figure, J\ is a unique function for a given level of constraint at the crack tip,
but different functions exist for differing levels of constraint. The relationship for
crack-tip opening displacement (6) can be examined by noting

/i = a X 5 X ffj- (17)

where a is a proportionality constant, and is dependent on the degree of constraint.

Substituting Eq 17 into Eq 16

Way{\-a/W) •" {ayW(2 + a/W)^

 ANDREWS AND SHIH ON A533-B STEEL AT 93°C 449

The empirical data available from the dual-gage estimates of 5 and from finite-
element calculations show that the relationship given in Eq 18 is unique and indepen-
dent of a/W and of the degree of constraint as long as no crack extension occurs
(Fig. 11).
Simplifying Eq 18 and rearranging

(1 - a/W)

Assuming this relationship is unique for a/W > 0.55 and/( VL) is independent of
a/W, Eq 19 can be used to estimate crack extension if VL and 6 are known in-
dependently. Taking the derivative of Eq 19

3(6*) (a/Vf — 4)
a(a/VF) •'^ '•' (2 + a/VK)3

where 5* is measured at the current crack tip (Fig. 12). In Fig. 12

A6* = 6(2) - 6(1) (21)

6 is measured at the original crack tip, thus a correction is needed to relate the virtual
value of 6, 6(2), to the measured value of 6. Referring to Fig. 12, similar triangles give
the relationship

b_ _ 6(2)
(22)
R R- Aa

Assuming
R = 7(5) (23)

gives

6 = 6(2) + - (24)
7
Solving Eq 21 for 6(2) and substituting the result, with A(6*), into Eq 24 gives

6 - 6 ( , ) + A a ( ^ - ^ X ^ 2 + ao/VV)3 + ^ j (25)

Rearranging and solving for Aa

1 , Aao/W-4) ^^°'
y •^^'''{l + ao/W)

where

f(x) =f(Vi)/W(2 + ao/W)^

(27)
= 8a)/{W-ao)
450 ELASTIC-PLASTIC FRACTURE

If the value or values of 7 are determined, and if the function/(j:) is established,

the change in crack length, Aa, may be calculated using Eq 26.
The function/(jc) was established in part experimentally and in part analytically.
Experimentally, the specimen with the deepest crack, T-41, had the largest load-line
displacement and the largest value of x (Eq 18) prior to cracking initiation. The 8 —
VL relationship for this specimen was used to cracking initiation. Beyond initiation
the relationship was established with the help of finite element calculations for
Specimens T-61 and T-52. The calculations for these two specimens established that
the/(jc) versus JC curve (Eq 18) is unique and independent of a / W a s long as crack ex-
tension does not occur.
The values of 7 which gave the best results are given by the relationship

{N)Xf(x)X(W-ao)
y= == (28)

where A^ is a dimensionless constant.

The sensitivity of the slope of the /i-resistance curve is indicated in Fig. 13. The
results of using two values of iV for calculating Aa for Specimen T-52 are shown. For
reference, a single heat-tint-derived crack extension value is plotted. Note that varying
N changes the slope of the crack extension curve, but does not change the criticalJi at
crack initiation.

References
[/] Merkle, J. G., Oak Ridge National Laboratory, private communication.
[2] Wilkinson, J. P. D., Hahn, G. T., and Smith, R. E., "A Program to Study Methods of
Plastic Fracture," Proceedings. American Society for Metals/American Society for Non-
Destructive Testing. Fourth Annual Forum on Prevention of Failure Through Non-
Destructive Inspection, Tarpon Springs, Florida, June 15, 1976.
[3] Wilkinson, J. P. D., Hahn, G. T., and Smith, R. E., "Methodology for Plastic Frac-
ture—A Progress Report," Proceedings, Fourth International Conference on Structural
Mechanics in Reactor Technology, San Francisco, California, Aug. 1977.
(•*] Shih, C. F., deLorenzi, H. G., and Andrews, W. R., this publication, pp. 65-120.
[5] Clarke, G. A., Andrews, W. R., Paris, P. C, and Schmidt, D. W. in Mechanics of Crack
Growth. ASTM STP 590, American Society for Testing and Materials, 1976, pp. 27-42.
[6] Hodge, J. H., "Properties of Heavy Section Nuclear Reactor Steel," Welding Research
Council Bulletin No. 217.
[7] Jewett, R. P., Closed Loop Magazine, MTS Corp, Vol. 4, No. 3, Summer 1974.
[8] Shih, C. F., deLorenzi, H. G., and Andrews, W. R., InternationalJournal of Fracture
Mechanics, Vol. 13, 1977, pp. 544-548.
[9] Merkle, J. G., Corten, H. T., Transactions KSME, Journal of Pressure Vessel Technology,
Nov. 1974, pp. 286-292.
[10] Shih, C. F., deLorenzi, H. G., Yukawa, S., Andrews, W. R., van Stone, R. H., and
Wilkinson, J. P. D., "Methodology for Plastic Fracture," Contract RP-601-2, Third
Quarterly Report, 1 Nov. 1976 to 31 Jan. 1977 for Electric Power Research Institute, Palo
Alto, Calif., 16 March 1977.
[//] Green, A. P. and Hundy, B. B., Journal of the Mechanics and Physics of Solids, Vol. 4,
1956, pp. 128-144.
[12] Lake, R. L. in Mechanics of Crack Growth, ASTM STP 590, American Society for Testing
and Materials, 1976, pp. 208-218.
[J3] Srawley, J. E. and Gross, B., Compendium, Engineering Fracture Mechanics, Vol. 4,
1972, pp. 587-589.
/. A. Joyce^ and J. P. Gudas^

Computer Injeractive Jic Testing of

Navy Alloys

REFERENCE: Joyce, J. A. and Gudas, J. P., "Computer InteractiTeyic Testing of Navy

AUoys," Elastic-Plastic Fracture. ASTMSTP668. J. D. Landes, I. A. Begley, and G. A.
Clarke, Eds., American Society for Testing and Materials, 1979, pp. 451-468.

ABSTRACT! A computer interactive unloading compliance single-specimen Jic test pro-

cedure has been developed. This procedure utilizes an on-line minicomputer to analyze
digitized load-displacement data during testing. Unique values of Ji and crack length are
determined from compliance measurement on short unloadings along the load displace-
ment record. The test procedure is presented in detail and analysis procedures are
discussed.
Three tasks which demonstrate the validity and utility of the computer interactive test
method are discussed. Results for single-specimen and multiple-specimen tests are
presented for HY 130, lONi steel, 17-4PH steel, Ti-7Al-2Cb-lTa, and Ti-6A1-4V which
show close correspondence between the two methods. Tests on 17-4PH steel compact ten-
sion specimens with various thicknesses and crack lengths are summarized and dimen-
sional effects on/ic and the/-Aa resistance curve slopes are discussed. Finally, tests on
HY 130 specimens with various notch root radii demonstrate effects of notch acuity on
Jic-

KEY WORDS: elastic-plastic fracture, toughness testing, single-specimen tests,

multiple-specimen tests, computer interactive testing, high-strength steels, titanium
alloys, specimen geometry effects, notch acuity, unloading compliance test method,
crack propagation

The objective of this effort was to develop a single-specimen test procedure

to evaluate the elastic-plastic fracture toughness parameter, Ju, with the re-
quirement that the method be practical for use in conventional materials
testing laboratories. Single-specimen Ju testing is potentially superior to
multiple-specimen procedures for several reasons. In the first place, single-
specimen tests define the /i versus crack growth resistance curve more
thoroughly. The determination of a unique/ic data point from each specimen
allows for analysis of material variability and facilitates testing over a range
of temperatures and various environments.
'Assistant professor of mechanical engineering, U.S. Naval Academy, Annapolis, Md. 21402.
^Head, Fatigue and Fracture Branch, David W. Taylor Naval Ship Research and Develop-
ment Center, Annapolis Laboratory, Annapolis, Md. 21402.

451

Copyright 1979 b y AS FM International www.astm.org

452 ELASTIC-PLASTIC FRACTURE

Single-specimen J\c evaluation methods have been developed by Clarke et

al [7p utilizing complicated electronic signal amplification which involves
considerable data evaluation and calculation after the test is completed. The
method described herein involves the analysis of conventional load-
displacement signals through the use of readily available electronic in-
struments and a minicomputer. The method involves digitizing analog load-
displacement data, and on-line, real-time computer interactive determina-
tion of crack length and/i. The immediate calculation of crack length allows
the test engineer to properly space unloadings, vary test machine speed, or
vary unloading length during the test to obtain optimum results from each
specimen. Further, the method provides for storage of both digitized load-
displacement data and /i versus crack extension data for future retrieval and
analysis.

/ic Test Methods

The application of/i as an elastic^plastic fracture toughness criterion is
based on the nonlinear elastic singularity solution of Hutchinson [2] and Rice
and Rosengren [3] which gives the stress, strain, and displacement com-
ponents near a sharp crack in an incompressible deformation theory plastic
material with a uniaxial stress-strain relation

(7 = adepr (1)

as

' / l ^ N/(N+1)
-i =
) sm
' / i ^,1/(^+1)
) ^m (2)

' h ',N/(N+l)

where Syid), ey{d), uaid), and I„ are functions of ^ andiV, and r and d defme a
polar coordinate system about the crack tip. Only/i in Eq 2 depends on the
applied boundary conditions or specimen geometry and thus/i sets the inten-
sity of the stress, strain, and displacement singularity in a manner completely
analogous to the stress factor Ki in the Williams [4] and Irwin [5] elastic solu-
tion. Begley and Landes [6] further proposed that a critical value of/i exists
(which is a material property) such that crack extension occurs when/i > Jic
^The italic numbers in brackets refer to the list of references appended to this paper.
 JOYCE AND GUDAS ON NAVY ALLOYS 453

For the special case that A/^ = 1 in Eq 1, Eq 2 reduces to the Irwin-Williams

elastic solution, and for this case Rice [7] has demonstrated that

J, = 'K^K.' (3)

where
E = material modulus of elasticity,
V = Poisson's ratio, and
Ki = linear elastic stress intensity factor.
Rice et al [7] has shown that for specimens in which only one length dimen-
sion is present, Ji can be evaluated approximately by

24

where
A = area under the load-load point displacement curve,
b = uncracked ligament, and
B = specimen thickness.
To obtain/ic, /i must be evaluated at the point on the load-displacement
curve where crack extension initiates. To determine this point of crack initia-
tion, Landes and Begley [8] proposed a multispecimen test procedure which
includes the following steps:
1. For each material, temperature, environment, etc., at least four in-
dividual specimens are loaded to different crack opening displacement
(COD) values and a load-displacement plot is developed for each specimen.
2. Crack extension is then marked by heW tinting or fatigue cracking.
3. Specimens are then pulled apart and crack extension is measured over
nine evenly spaced points, including the centerline of the specimen and ex-
cluding the points at each surface.
4. /i-values are then calculated from the load-displacement record using
Eq4.
5. For each material, a plot of / versus crack extension, Aa, is con-
structed. A least-squares straight line is fit to all data. The critical/ic value is
obtained at the intersection of the foregoing line and the crack opening
stretch line defined by the relationship

J = 2afl„„ Aa (5)

where anow is the average of the material yield strength and ultimate tensile
strength, and Aa is the crack extension.
454 ELASTIC-PLASTIC FRACTURE

6. Test validity is determined by the following relationship

B > ^'ys (6)

where the value jS has been suggested as 50 by Paris [10] or 25 to 40 by

Landes and Begley [9].
Alternative methods to the multiple-specimen method just outlined have
obtained Jw from a test on a single specimen. For this type of test, a pro-
cedure is required which gives an accurate measure of crack extension while
the specimen is under load so that the /i at initial crack extension can be
determined. Clarke et al [/] has used an unloading compliance method to
evaluate crack extension. Carlsson and Markstrom [//] have used electrical
impedance and eddy current techniques for this purpose. The description of
the computer interactive unloading compliance test method developed in this
effort is presented in the next section.

Computer Interactive J^ Test Method

The schematic of the computer interactive test arrangement is shown in
Fig. 1. In this setup, load cell and clip gage signals are conventionally
amplified and fed to a scanner which is interfaced to a digital voltmeter
through an IEEE Standard 488-1975 interface. The digitized data are then
made available to a microprocessor with magnetic tape cartridge and
cathode ray tube (CRT) graphics capability. The test arrangements at the
U.S. Naval Academy and the David W. Taylor Naval Ship Research and
Development Center (DTNSRDC) also employ a peripheral interactive
graphics plotter interfaced with the computer. The test results reported
herein utilized an Instron Model TTD universal test machine and a Tinius-
Olsen test machine, both of which are screw-type displacement-controlled
devices. The computer used for this testing was a Tektronix Model 4051.
A single-specimen Jic test is conducted with this apparatus using the
following steps:
1. Load and COD transducers are calibrated using the test machine elec-
tronics and a precision micrometer, respectively. This step yields the slope
and intercept of a straight-line calibration curve for each data channel.
These values are stored in a magnetic tape file for use by the/ic test program.
2. The clip gage is attached to razorblade knife edges mounted on the load
line and the specimen is inserted in the test machine.
3. Initial crack length is determined by loading the specimen between 10
; and 50 percent of the expected maximum test load. Between 30 and 50 load-
displacement data pairs are gathered and the computer estimates the initial
crack length (a/w) using the relation from Saxena and Hudak [12]
 JOYCE AND GUDAS ON NAVY ALLOYS 455

— = 1.000196 4.06319f/x + 11.242t/x2

-706.043f/:c3 + 464.335f/x'' - 650.677f/x5 (7)

where

Ux = 1/2 and E' =

BE'b (1 - v^)
+1

where
E and i* = specimen elastic modulus and Poisson ratio, respectively,
B = specimen thickness, and
b/P = load-line compHance.
The specimen compliance, the least-squares correlation, and the crack
length estimate are determined and displayed after each unloading and
loading on the computer CRT screen. A least-squares correlation of the
unloading data to a straight line of 0.9999 or greater and crack length
estimates varying by ±0.05 mm are generally obtainable and are required
before continuing the test. With the initial crack length estimate completed,
the specimen is returned to zero load and the test is begun by starting the test
machine and computer data acquisition simultaneously.

ROM a RAM
Test Moehine Control Memory

Magnetic Tape
for Data Storage

A/D

-<c
J^
I Digitizing
Module
[US
Connputer/Terminal
77m

Aa
Interactive Groptiics
Plotters

FIG. 1—Schematic of computer interactive J/c test apparatus.

456 ELASTIC-PLASTIC FRACTURE

4. Load-displacement data pairs obtained by the computer are plotted on

line by the interactive graphics plotter, producing a typical plot as shown in
Fig. 2. The area under the load-displacement curve is obtained from each
data pair using a trapezoidal quadrature. Unloadings of approximately 10
percent are initiated and spaced at the discretion of the operator. For each
unloading, involving 25 to 45 load-displacement pairs, the computer
develops the best-fit straight-line slope, which is substituted into Eq 7 to give
a crack length estimate. At each unloading, estimated crack length, change
in crack length, /i, and correlation of the regression analysis are determined
and printed on the CRT screen. When the desired maximum crack extension
is obtained, the test is terminated and the data file is closed.
5. Finally, an additional program is introduced which operates on the/i-
Aa data file to obtain/u. This program fits a least-squares straight line to all
data with Aa > J/2anow + 0.05 mm, then obtains/k by solving for the in-
tersection of the foregoing line and the crack opening stretch line.
A critical requirement for a successful test of this type is that an accurate
estimate of crack length be made from the data obtained on each unloading.
A typical unloading is shown greatly magnified in Fig. 3. The present pro-
gram eliminates the upper 2 percent of the unloading curve based on max-
imum load and then uses all other points, both unloading and loading, to ob-
tain a slope or compliance estimate. A slight delay in data gathering for slope
calculation is then experienced before data acquisition is resumed. Figure 3

8.5 t I.S 2
COD MM.

FIG. 2—Plot of single-specimen load-displacement data for HY130 steel.

 JOYCE AND GUDAS ON NAVY ALLOYS 457

33

32

31 .

1.22 1.24 1.26 1.26 1.3 1.32

COD MM.

FIG. 3—Plot of single-specimen unloading detail.

shows a slight hysteresis between loading and unloading, but nearly identical
slopes are obtained. A correlation greater than 0.995 is typical for unloadings
involving 25 to 40 data pairs.

Applications of the Computer Interactive Ju Test Method

The computer interactive Ju test procedure has been employed in several
programs at the U.S. Naval Academy and DTNSRDC. The results of three
separate efforts are discussed herein. These include:
1. Evaluation of/ic of five Navy steels and titanium alloys and comparison
with multiple-specimen data.
2. Determination of specimen size limitations on thickness and remaining
ligament for 17-4PH steel.
3. Evaluation of effects of notch acuity on /ic measurement of HY 130
steel.

J/c Testing ofNayy Steels and High-Strength Titanium Alloys

Computer interactive single-specimen tests and conventional multiple-
specimen tests were performed on three high-strength steels and two high-
strength titanium alloys. The objective of these tests was to assess the ability
of the computer interactive test procedure to produce 7ic and/i-Aa test data
which correlated with that produced with the multiple-specimen method.
The chemical composition of the test materials included in this effort is
458 ELASTIC-PLASTIC FRACTURE

described in Table 1 and the mechanical properties are presented in Table 2.

The lONi steel was provided in the form of 38-mm plate and the Ti-7-2-1
alloy was provided in the form of 102-mm plate. The other materials were in
the form of 25-mm plate. Modified compact specimens (ITCT) were used in
these tests. In all cases, the crack plane was produced in the T-L orientation.
All specimens were fatigue cracked as per the ASTM Test for Plane-Strain
Fracture Toughness of Metallic Materials (E 399-74) to a total notch depth of
38 mm (a/w = 0.75). During/ic testing, maximum crosshead speed was 0.25
mm/min. The multiple-specimen test procedure was that by Landes and
Begley [9] described earlier, and the computer interactive test procedure
described in the preceding section was employed. Specific/k-values for these
single- and multiple-specimen tests are reported in Table 3. The /i versus
crack extension resistance curves for HY-130, lONi steel, 17-4PH steel,
Ti-7-2-1, and Ti-6-4 are presented in Figs. 4-8, respectively. Each figure
shows the blunting line and least-squares linear regression fit of individual
data points.
Analysis of multiple- and single-specimen test data shows that both test
methods produce equivalent /ic-values. The data in Table 3 show that 7ic
calculated from single-specimen tests tends to run higher than Ju from
multiple-specimen tests. With the exception of Ti-7-2-1, the single-specimen
method consistently produces high/i-Aa resistance curve slopes. This dif-
ference results from the fact that considerable crack front curvature upon
crack extension existed with all materials except Ti-7-2-1. When cracks tun-
nel, the compliance calculated crack length is smaller than the average of
nine measured points across the specimen. Final nine-point crack length
measurements for the single specimen are shown in Figs. 4-8 and suggest
equivalence between single- and multiple-specimen tests when an identical
crack length measurement technique is used.

Specimen Size Limitation Analysis

This effort involved using a basic IT compact specimen configuration with
a range of thicknesses and fatigue crack lengths to determine effects of these
changes onJu and the/i-Aa resistance curve slope. The 17-4PH steel (a,, =
895 MPa) used in the test method correlation study was used in this task, but
crack planes were placed in the L-T orientation. Tests were carried out at
ambient temperatures using the Instron TTI> test machine. The computer
interactive test procedure described earlier was employed for all tests and in
this case the correction for axial force on the ITCT specimen was included in
the/ic calculation [13].
Figure 9 shows the Ji-Aa resistance curves for three specimens with
thicknesses of 25.4, 9.5, and 5.0 mm, respectively. For these tests, the initial
crack length was identical at a/w = 0.72. /k-values for these three
specimens are presented in Table 4. /^-values for specimens with 25.4 and
 JOYCE AND GUDAS ON NAVY ALLOYS 459

00
H8 8
o d
in
Ui
— <N
< O"=

o o
s

a ;? K 88

I I' 5 lO lo
•*dd
o

1-1 - ^ l/> §;
Udd

tu :8;
o
<
-H 00 1^
o
o o d
1^
d
t2
ft-SS 8 s d d
d d d d
in <N
Itsg s z 8o
dd

1}
O -H
U o o

o
(N < TT
r^ Bu vi
!i!
460 ELASTIC-PLASTIC FRACTURE

s
Of

§i
-H 0^ ON —c O -H

u^

s2
•c B
5<«

19
•a 0 .

"1?. -O 00

si

2 3 S -s
<U 13

>-) I?
oa
<
" O -• (J O ,-
U o
or)
s
o ^
U
o
i/>
m
^
oo
o
^
lo
o So"'
o
o If.
W wo
D
Z

0.
feo tJi4
I111
£ - b'i^
 JOYCE AND GUDAS ON NAVY ALLOYS 461

TABLE 3—Summary ofJu test results for Navy steels and high-strength titanium alloys.

•^Ic, kPam Equivalent A"ic, mPa-mVj

Single Multi- Single Multi-

Material Specimen specimen Specimen specimen

HY130 208 186 213 202

10 Ni steel 138 118 174 161
17-4PH steel 112 106 157 153
Ti-7Al-2Cb-lTa 73 71 97 %
Ti-6A1-4V 48 39 78 71

9.5 mm thickness agree within 1 percent while the specimen which was 5.0
mm thick produced a /ic-value which was nearly twice that of the thicker
specimens. The analyses included in Table 4 based on the Paris [10]
thickness requirements for a valid 7ic test show that the thickest specimen is
definitely valid, the intermediate thickness is questionable, and the thinnest
specimen is not valid. This shows excellent agreement with these single-
specimen /ic test results. Figure 9 also shows that the Ji-Aa points at large
crack extension fall below the extrapolated straight-line fit to data for Aa <
1.1 mm. Therefore the /ic-values reported in Table 4 were calculated ex-

see

45e

4ee

358.

3ee

K 258.
p
A
» 208.
n
158.
O MULTIPLE SPECIMEN
lee.
0 SINGLE SPECIMEN

58. • SINGLE SPECIMEN. MEAS. A A MAX

-e.2s e e.2s e.s e.7E i 1.2s i.s I.7E 2 2.25 2.5

DELTA A MM
FIG. 4—Plot of J versus crack extension data for HY130 steel.
462 ELASTIC-PLASTIC FRACTURE

2sa

286.

O MULTIPLE SPeCIHEN
O S W e U SPECIHEN
•siNeLE sPEcmEN. rcAS AA MAX

e.2s e.E e.TE i I.2S I.E 1.76 2 Z.ZS 2.E 2.76

DELTA A m
FIG. 5—Plot of J versus crack extension data for lONi steel.

358

2Se

288.
K
P
A IS

188
O MULTIPLE SPECIHEN
0 SIN6LE SPECIHEN
• SIN6LE SPECIHEN. HEAS AA HAX

el
-8.2S B a.2S 8.5 8.7S I t.25 t.5 1.76 2 2.25 2.5 2.75
DELTA A HH

FIG. 6—Plot ofl versus crack extension data for 17-4PH steel.
 JOYCE AND GUDAS ON NAVY ALLOYS 463

!«,.

I«. 0 •

128.

180.

ee.
O MULTIPLE SPECIMEN
0 SCNGLE SPECWEN
28. • SINSLE SPEC. MEAS. AA MAX

.25 8 e.2S 8.5 8.75 I 1.25 I.5

DELTA A MM

FIG. 7—Plot of i versus crack extension data for Ti-7Al-2Cb-lTa.

ISflL

les.

8t.

O MULTIPLE SPECDBi
0 SOKLE SPECIHEN
28.

-e.25 a 0.ZS 0.5 e.75 t 1.2 I.S 1.76

DELTA A HH

FIG. &—Plot ofi versus crack extension data for Ti-6Al-4V.

464 ELASTIC-PLASTIC FRACTURE

-a.i a.i 0.3 e.s 9.7 8.9 I.I 1.9

DELTA A MM.

FIG. 9—Plot of J versus crack extension data for 17-4PH steel as related to specimen
thickness.

eluding data pairs with Aa > 1.1 mm. Data on specimens of various thick-
ness have shown an increase in the initial Ji-Aa curve slope with decreas-
ing thickness, but subsequent decrease in slope. This is shown clearly with
the 5.0-mm specimen and suggests that insufficient thickness can result in
nonconservative/ic measurements for the material investigated.
The second part of this effort involved developing Ji-Aa data with ITCT
specimens with crack lengths in the range 0.72 to 0.92 a/w. Figure 10 shows
the /i-Aa resistance curves for the three test specimens and /rvalues are
reported in Table 4. /ic-values for the specimens with a/w = 0.72 and 0.80
agree within 3 percent while the resistance curve slopes agree within 10 per-

TABLE 4—Summary ofJic test results for 17-4PH steels with various thickness specimens and
various crack lengths.

Thickness, Crack dJ\/da,

Specimen mm Length, a/w / i c , kPam 25 to 507ic/5ys, mm kPa-ra/mm

1 25.4 0.72 262 7.3 to 14.6 379.5

2 9.5 0.72 264 7.4 to 14.7 551.3
3 5.0 0.72 473 13.1 to 26.4 388.7
4 25.4 0.80 254 7.1 to 14.1 345.0
5 25.4 0.92 300 8.5 to 17.0 745.2
 JOYCE AND GUDAS ON NAVY ALLOYS 465

laee,

FIG. 10—Plot of J versus crack extension data for 17-4PH steel as related to specimen liga-
ment length.

cent. The most deeply cracked specimen {a/w = 0.92), however, gives a
slightly higher value of Ju and a much steeper resistance curve slope, in-
dicating that the error in J\c introduced by subsize ligament effects is non-
conservative for the material investigated.

Crack Tip Acuity Effects

The / i of HY 130 plate was evaluated to determine the effects of notch tip
acuity on the Jan measurement. Two specimens were produced conven-
tionally, and four specimens were produced with machined notches to depths
in the range 0.7 < a/yt < 0.76 and root radii of 0.051 and 0.076 mm. The
two regularly machined specimens were fatigue cracked to depths of 0.74 and
0.79 a/w according to ASTM E 399-74 criteria. All tests were carried out at
ambient temperature utilizing the Tinius-Olsen test machine. The single-
specimen procedure described earlier was followed and the correction for ax-
ial force on the ITCT specimen was included in the/ic calculation [13]. The
JI versus crack extension curves for these tests are presented in Fig. 11 and
results are summarized in Table 5. It can be seen that crack tip geometry
substantially affects the Ju measurement. The lowest values were produced
with the fatigue crack specimens. Among the specimens with machined
notches, the apparent/ic-values related to the 0.051-mm root radii were below
those related to the 0.076-mm root radii. This result is consistent with that
466 ELASTIC-PLASTIC FRACTURE

eea

sse.

sea.

450.

480.

3S0.

300.

260.

200.

1E0.
D O 0.08 m RADIUS
108. A O 0.05 HH RADIUS

50. Q o FATIGUE CRACKED

-0.2 0.2 0.4 0.6 0.8 1.2 1.4 1.6

DELTA A MM

FIG. 11- -Plot of J versus crack extension data for HY130 steel as related to specimen crack
root radius.

obtained by Begley and Logsdon [14] for A471 nickel-chromium-molybde-

num-vanadium rotor steel.
It should also be pointed out that the/ic-values determined from fatigue
precracked specimens are substantially lower than those reported for HY 130
in the test method correlation task. The HY 130 plate used for this notch
acuity study was not traceable to a particular producer or vintage, precluding
the use of these Ju data to describe the HY 130 system. On the other hand,
these data point up the sensitivity of the Jic measurement and the computer
interactive test procedure in evaluating fracture toughness from the simple
quality-assurance standpoint.

TABLE 5—Summary ofJic test results for HY 130 steel with various notch root radii.

Specimen Notch radius, mm /ic , kPam dJi/da, kPa-m/mm

FLF-2 fatigue cracked 128 186.2

FLF-4 fatigue cracked 103 182.5
FLF-5 0.08 446 123.5
FLF-6 0.08 508 99.5
FLF-7 0.05 407 104.9
FLF-8 0.05 381 119.7
 JOYCE AND GUDAS ON NAVY ALLOYS 467

Conclusions
The computer interactive unloading compliance /ic test method has been
shown to produce equivalent /ic-values for the steels and titanium alloys
tested when compared with multiple-specimen data. These single-specimen
tests show high Ji-Aa resistance curve slopes when crack tunneling occurs
because effective crack length is shorter than that calculated from nine
measurements across the thickness.
The computer interactive test method is seen te possess several advantages
in comparison with the multiple specimen method. In the first place, the
computer interactive method produces more complete and consistent /i-Aa
resistance curve data. The immediate calculation of/ic and crack extension
after each unloading gives the test engineer the capability to space
unloadings evenly, to repeat a particular unloading, to change machine
speed, etc., so as to obtain optimum results from each specimen. The test
method allows for evaluation of material variability and is adaptable for
testing at different temperatures and in various environments. Magnetic tape
storage of digitized load-displacement data allows for future reanalysis as the
/ic fracture criterion develops. Finally, the fact that a unique / k test result is
produced from each test suggests that Jic testing can be carried out on a
routine basis as is K^ testing. These advantages are enhanced by the fact
that the test method described herein is readily adaptable to the new genera-
tion of computer interactive test machines now being made available.
The computer interactive unloading compliance method was successfully
utilized to evaluate effects of specimen thickness, remaining ligament, and
notch acuity on/u and the shape of the7i-Aa resistance curve.

Acknowledgment
The authors acknowledge the Naval Sea Systems Command (NAVSEA
03522), the National Science Foundation (Contract ENG76-09623), and the
Structures Department of DTNSRDC for supporting various aspects of this
research.

References
[/] Clarke, G. A., Andrews, W. K., Paris, P. C , and Schmidt, D. W. mMechanics of Crack
Growth, ASTM STP 590, American Society for Testing and Materials, 1976, pp. 27-42.
[2] Hutchinson, J. Vf., Journal of Mechanics and Physics of Solids, Vol. 16, 1968, pp. 13-31.
[3] Rice, J. R. and Rosengren, G. E., Journal of the Mechanics and Physics of Solids, Vol. 16,
1968, pp. 1-12.
[4] Williams, M. L., Journal of Applied Mechanics, Vol. 24, 1957, pp. 109-114.
[5] Irwin, G. R., Journal of Applied Mechanics, Vol. 24, pp. 361-364.
[6] Begley, J. A. and Landes, J. D. in Fracture Mechanics. ASTM STP 514, American Society
for Testing and Materials, 1972, pp. 1-20.
[7] Rice, J. R., Journal of Applied Mechanics, Vol. 35, 1%8, pp. 379-386.
468 ELASTIC-PLASTIC FRACTURE

18] Rice, J. R., Paris, P. C, and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing. ASTM STP 536, American Society for Testing and Materials, 1973,
pp. 231-245.
19] Landes, J. D. and Bee|ey, J. A. in Fracture Analysis, ASTM STP 560, American Society
for Testing and Materials. 1973, pp. 170-186.
[10] Paris, P. C , Discussion to J. A. Begley and J. D. Landes in Fracture Mechanics. ASTM
STP 514, American Society for Testing and Materials, 1972, pp. 21-22.
[11] Carlsson, A. J. and MarkstrOm, K. M. m Proceedings, Fourth International Conference on
Fracture, Waterloo, Ont., Canada, 1977, pp. 683-691.
[12] Saxena, A. and Hudak, S. J., Jr., "Review and Extension of Compliance Information for
Common Crack Growth Specimens," Westinghouse Scientific Paper 77-9E7-AFCGR-P1,
Pittsburgh, Pa., 1977.
[13] Merkle, J. G. and Corten, H. T., Journal of Pressure Vessel Technology, Transactions,
American Society of Mechanical Engineers, Vol. 96, Nov. 1974, pp. 286-292.
[14] Logsdon, W. A. and Begley, J. A., Engineering Fracture Mechanics, Vol. 6, 1977, pp.
461-470.
A. D. Wilson^

Characterization of Plate Steel

Quality Using Various Toughness
Measurement Techniques

REFERENCE: Wilson, A. D., "Characterization of Plate §teel Quality Using Various

Toughness Measurement Techniques," Elastic-Plastic Fracture, ASTM STP 668. J. D.
Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and
Materials, 1979, pp. 469-492.

ABSTRACT: The fracture toughness properties of three steels (A516-70, A533B Class 1,
and HY-130) are determined in two steel-quality levels (conventional and calcium
treated). In addition, a conventional quality A543B Class 1 steel is similarly examined
at two locations (quarterline and centerline). Both investigations involved comparing
steels with differing inclusion structures. The primary effort was to establish the upper-
shelf toughness differences found using the Charpy V-notch impact and dynamic tear
tests compared with those found using J-integral determinations ofJu and Kic- The/ic
determinations appeared to be more sensitive to changes in inclusion structure than
either the Charpy V-notch or dynamic tear tests. This was established by comparing
the toughness between quality levels and by measuring the anisotropy of toughness
within a steel. Comparisons are made with the Charpy V-notch impact-Zfic upper-shelf
correlation values. Comments concerning the suggested graphical method for 7ic deter-
mination are also given.

KEY WORDS: crack propagation, fractures (materials), inclusions, steels, plastic

properties

The quality of steel plate can be significantly affected by nonmetallic

inclusions. The presence of inclusions, such as sulfides and oxides, primarily
affects the ductile behavior of the steel. Therefore when the quality of a
particular grade of steel is improved by inclusion reduction or modification,
the benefits are commonly assessed by conventional testing determinations
such as the tensile percent reduction of area or Charpy V-notch (CVN)
upper-shelf energy [1-3].^ In addition, the dynamic tear (DT) test upper-
shelf energy can be used to quantify quality enhancement [2,3]. While these
measurements give relative indications of improvement, the results cannot
'Research engineer, Lukens Steel Company, Coatesville, Pa. 19320.
^The italic numbers in brackets refer to the list of references appended to this paper.

469

Copyright 1979 b y AS FM International www.astm.org

470 ELASTIC-PLASTIC FRACTURE

be used directly in design. However, through fracture mechanics by de-

termining fatigue crack propagation and fracture toughness properties,
these benefits can be directly related to design. Previous work [4] has re-
ported the influence of inclusions on the fatigue crack propagation properties
of steels. In this study the effect on the plain-strain fracture toughness,
Kic, will be established.
There is a concern for obtaining the Kic properties of improved quality
steels for a number of reasons. Existing structural designs may already be
based on Kic determinations made on conventional quality steels. Thus, if
it is possible that plain-strain conditions may exist in a structure, any design
modification to accommodate the improved steel quality would require the
actual improved Kic properties. In addition, it is of interest to determine if
existing ^ic correlations with CVN properties are still applicable and to
establish whether tensile ductility, CVN, and DT testing provide reliable
indications of the enhancement in Ku of improved quality steels. However,
to obtain upper-shelf/iTic values, to the ASTM Test for Plane-Strain Fracture
Toughness of Metallic Materials (E 399-74) standards, for common struc-
tural steels would require extremely large and expensive test specimens.
The J-integral testing approach provides a test method for obtaining Ku
values using reasonable-sized specimens.
The conception [5] and development [6-10] of the J-integral as a fracture
criterion and testing technique has been a significant contribution to
metallurgists interested in obtaining fracture toughness properties of ductile
materials. This is because A'lc values can be determined from the following
relationship

Ju = ^-^Ku' (1)

where Ju is the critical value of J at the point of crack extension and v and
E are Poisson's ratio and the modulus of elasticity, respectively. In steels,
for example, J-integral testing has been used to establish the fracture tough-
ness in the transition range and on the upper shelf [11-13] of a number of
alloys. Also emphasizing the interest in the J-integral test technique has
been the formation of an ASTM task group, originally E24.01.09 and at
present E24.08.04, which has developed guidelines for Ju determinations
[14].
In this investigation, a carbon steel (A516-70) and two alloy steels (A533B
Class 1 and HY-130) are characterized in two quality levels (conventional
and calcium treated). In addition, a conventional quality A543B Class 1
steel is examined at two locations (quarterline and centerline). In both
programs the tensile ductility, CVN, and DT upper-shelf energies and Ju
on the upper shelf are established. The primary intent of these evaluations
is to determine how the conventional test techniques, particularly the CVN
and DT, rate material quality compared with the Ju measurements.
 WILSON ON PLATE STEEL QUALITY 471

Experimental Details

Materials
Four steels of a wide range of strength levels were studied. The A516-70
carbon steel has a minimum 0.2 percent offset yield strength (0.2YS) of
262 MPa (38 ksi), the A533B Class 1 low-alloy steel a minimum 0.2YS of
345 MPa (50 ksi), the HY-130 alloy steel a minimum 0.2YS of 896 MPa
(130 ksi), and the A543B Class 1 alloy steel a minimum 0.2YS of 552 MPa
(80 ksi). The concern for determining the Ku at the upper shelf for these
four steels is due to their primary operating temperature in a number of
applications being on the upper shelf. To obtain plain-strain conditions
in each of these steels with a Ku of 165 MPaVm (150 ksiVin.) and an average
0.2YS would require a thickness of about 559 mm (22 in.), 305 mm (12 in.),
76 mm (3 in.), and 178 mm (7 in.), respectively, for the A516, A533B,
HY-130, and A543 steels. These thicknesses of material have been produced
for these steels and thus the concern for K^ values exists.
The A516, A533B, and HY-130 steels were characterized in two quality
levels, namely, steel made by conventional steelmaking practices (CON)
and by a calcium treatment (CaT). The mechanical properties and non-
metallic inclusions resulting from these two practices have been reported in
detail [1-4]. Briefly, for aluminum-killed, fine-grained steels the CON
steels have higher sulfur levels, lower toughness and ductility properties,
and anisotropy of these properties due to the presence of two kinds of
inclusions. Type II manganese sulfide and galaxies of alumina inclusions
lead to this behavior. Figure 1 shows these manganese sulfide inclusions
in the CON A533B steel and indicates their elongated and pancaked nature
due to their plastic behavior at hot rolling temperatures. The alumina
galaxies shown in Fig. 2 for the same steel do not individually deform, but
as a group the galaxies are rotated and aligned in a planar fashion due to
rolling. Both of these kinds of inclusions lead to the lower level of properties
and anisotropy in CON steels.
CaT prevents the formation of both of the foregoing kinds of inclusions
by both desulfurization and inclusion shape control. The remaining in-
clusions in these steels, as shown in Fig. 3 for CaT A533B, are duplex-
round compact inclusions. The calcium modification of these inclusions
makes them harder at hot-rolling temperatures and thus they do not
elongate. CaT steels therefore tend to have better toughness and ductility
properties with improved isotropy of these properties, as well as lower sulfur
levels.
The chemistries of the steels examined in this part of the study are given
in Table 1. In addition to conventional steelmaking techniques, the CON
HY-130 material was treated with a ladle flux practice to obtain the rather
low sulfur level indicated. However, there is no inclusion shape control in
this practice [1,2] and thus there is still anisotropy present.
472 ELASTIC-PLASTIC FRACTURE

FIG. 1—Composite of tight photomicrographs from CON A533B steel indicating morphology
of largest Type II manganese sulfide inclusions.

The A543 steel studied was of CON quality level. The purpose of this
part of the program was to compare the properties of this steel at the
quarterline (QL) (quarter thickness) and centerline (CL) (center thickness)
locations of the plate. Because of the solidification behavior of large steel
ingots, there normally are larger inclusions in both size and number at the
CL. This leads to poorer upper-shelf energies and tensile ductilities. The
chemistry of this plate is given in Table 1.
The A516 and HY-130 plate steels, which were nominally 51 mm (2 in.)
in thickness, were tested at the centerline of the plates. Tension testing was
performed in the longitudinal (L) and transverse (T) orientations. CVN,
DT, and Jic testing was performed in the longitudinal (LT) and transverse
(TL) orientations. The thicker A533B and A543 plates were tested in all
three testing orientations, namely, L, T and through-thickness (S) tensiles
and LT, TL and through-thickness (SL) CVN, DT, and/ic tests. The A533B
CON steel was tested at the QL, while the CaT was tested at the CL. As
mentioned previously, the A543 tests were performed at the QL and CL.
 WILSON ON PLATE STEEL QUALITY 473

FIG. 2—Composite of light photomicrographs from CON A533B steel indicating morphology
of largest galaxies of alumina inclusions.

Testing Techniques
The tension, CVN, and DT tests were all performed according to the
applicable ASTM specifications, namely: Tension Testing of Metallic
Materials (E 8-69); Notched Bar Impact Testing of Metallic Materials
(E 23-72); and Test for Dynamic Tear Energy of Metallic Materials (E 604-
77). The tensile properties were obtained at room temperature (RT) using
two 6.4-mm-diameter (0.252 in.) specimens. The full transition curve was
obtained in both CVN and DT testing and the respective upper-shelf
energies were obtained by averaging 3 to 5 CVN and 2 to 3 DT results
which had 100 percent ductile fracture appearance. The CVN tests used
the conventional 10-mm-square (0.394 in.) specimen and the DT tests used
the 16-mm-thick (5.8 in.) specimen. The SL-oriented DT tests for the
A533B CON material at the QL were peirformed on specimens wdth welded-
on extensions. No SL-oriented DT tests were performed at the QL of the
A543 material.
474 ELASTIC-PLASTIC FRACTURE

FIG. 3—Composite of tight photomicrographs from CaT A533B steel showing calcium-
modified, shape-controlled inclusions.

The J-integral tests in this study were performed using the single-speci-
men compliance-unload technique developed by Clarke et al [15]. The
single-specimen J-integral (SSJ) technique allows obtaining a full / versus
crack extension (Aa) resistance (R)-curve, having about 15 to 25 points,
using a single specimen. Two specimens were tested for each material at a
temperature expected to give upper-shelf behavior. The A543 and HY-130
steels were tested at RT and the A516 and A533B steels were tested at
+93°C (+200°F) to assure fully ductile conditions. Those temperatures
also coincided with the upper shelves of the CVN and DT tests. The ele-
vated temperature was obtained by wrapping the specimen with resistance
heating tapes and insulating with glass wool. The specimen used was the
compact design, deeply notched, with provision made for load-line dis-
placement measurements on the specimen. Specimens 25 mm (1 in.) thick
were tested in all cases except for the CON A533B and CON HY-130
steels, where 16-mm-thick (5/8 in.) specimens were tested because of
limited material availability. The precrack lengths were controlled so that
the a/w for most tests was nominally 0.75, where a is the crack length to
 WILSON ON PLATE STEEL QUALITY 475

o o
o d d d
o —•
§ 1
d d
^=
d d
H

I/) •£>
o q ^?

o O -^

<:> S do d

« (N ro 00 1/5

d d d d d d d

38
o d
i
d d
o 88
d d
o
d
B.
^ q ?
^— " '^ d d d

u
.J
(S O
q q §i 88 o
BO d d d d d d d

in m oo o (N 00 i^
rs IN + ^
+1
•a "
S. II

»J O
^ E ^O O 1/5 I/)
c o

>^ C op
• ^

"rt (U
3 > z
a -] O a O 'ca O c« o 11 E
O O U O u
Q
-W
C
u ++
ffi 1
V
b ++ ++ +
H "ea S
z z
ill
S
'2
z u
lui
-S I '2
••-•
en <
476 ELASTIC-PLASTIC FRACTURE

the centerline of loading and w is the specimen width. The a/w for the
CaT A533B tests was 0.60. The loading rate during the tests was 0.953
mm/min (0.0375 in./min).
The Jic determinations were made from the / versus Aa R-curves follow-
ing the guidelines of Clarke [14]. Not all specimens met the suggested
specimen size requirement of this procedure. The / values used in the
R-curves were corrected to account for the tension component in the com-
pact specimen using the Merkle-Corten correction [16]. In addition, a
correction to account for the rotation of the compact specimen during
testing was developed by Donald [17]. The rotation correction of the Aa
values ranged from less than 1 percent for the HY-130 tests to as much as
25 percent for the A516 CaT tests.

Results
The results of the conventional mechanical property tests, tension, CVN,
and DT, are presented in Table 2. The items to particularly note are the
tensile percent reduction of area (RA) and the CVN and DT upper-shelf
energies (CVN USE and DT USE). The improvement in the level of duc-
tility arid toughness in the A516, A533B, and HY-130 for the CaT quality
is readily apparent. In the A543 steel the QL toughness and ductility
levels are better than those at the CL.
Tension tests at +93°C (+200°F) for the A533B CaT steel indicated
that the 0.2YS and ultimate tensile strength (UTS) values are about 21
MPa (3.1 ksi) and 36 MPa (5.2 ksi) lower, respectively, than at RT.
For A516, the 0.2YS and UTS are reduced by 18 MPa (2.6 ksi) and 35
MPa (5.1 ksi), respectively, at +93°C (-l-200°F) versus RT. These modifi-
cations were made to the tensile strengths used in the later/ic analyses.
Although the tensile ductility, particularly the percent RA, can be a
good measure of steel quality, as shown in Table 2, it is not commonly
considered a measure of toughness. The CVN USE and DT USE are
measures of notch toughness and thus are often related to fracture tough-
ness values. Therefore, the percent RA values will not be used in the
comparisons developed later in the paper, while the CVN USE and DT
USE will be used extensively.
The /-integral results are displayed in Figs. 4-7 in "the form of the /
versus Aa R-curves for the A516, A533B, HY-130, and A543 steels,
respectively. An average "blunting line" for all of the data of the particular
steel grade is also given for reference purposes. Actual material tensile
properties were used in each actual Ju determination. The "blunting line"
is determined from
J = 2FSAa (2)
where FS is the flow stress [FS = (0.2YS + UTS)/2]. It can be roughly
 WILSON ON PLATE STEEL QUALITY 477

lo 1/) O IQ O in o
(N 00 1/5 in cF
o o TT O 00 00 ^

til —< 00
o Q m
^ O r^
^ vO r ^
.J

ir

as <

00 t o r -
l-J
rn f^ (N

« c

I B .J
o
<
o o
0 0 >£>
u
. in
o^ —
00
>o X
H fJ J
J H (/) Sr^ h ri ^ X £
J H J H i^ J H J H 5?

« ii
ft. 3
o
S
V U v
^ 2 c 2
u "y
ai S .
0 *- w»
•5 V « c
3 ^ (-• C "C
'^ i >
t'l
00-S
3 !n «
.2
u
(L> 3 s C 0 ii
60 0 ca fc- c w 3 >c
H H O 0 0
a * TS .^ ^ o« (J
^
478 ELASTIC-PLASTIC FRACTURE

mm
0 .5 10 1.5 2.0 2S
— I 1 1 1 1 1 1 I 1

.10

FIG. 4—J versus An R-curves for A516 steels. Line indicates average "blunting line" for
these steels; for reference purposes. Eg 2. (KN/m = lb/in. X O.I 751).

noted by examining Figs. 4-6 that for each steel grade the R-curve is
shifted to higher / levels, indicating more toughness for the CaT steels.
This is particularly indicated in the TL and SL orientation. Figure 7 shows
that the R-curves for the QL A543 steel indicate tougher behavior than at
the CL in all testing orientations.
In this investigation the critical value of/ is determined by two methods,
namely, that Ju determined using the graphical analysis technique (/IC)G,
[14], and that determined at the point of first load drop from the load-
displacement curve, (/IC)FLD [15]. In the graphical-analysis method the
C^IC)G was taken at the intersection of the "blunting line" and the visually
determined best-fit line through points having the required amount of
crack extension according to Ref 14. The (/IJFLD was established at the
point of maximum load, since all of the load-displacement curves were
smooth with no discontinuities. These determinations are given in Table 3.
Those (/I<:)G values that meet the suggested validity requirements [14],
 WtLSON ONf PLATE STEEL QUALITY 479

mm
10
-1— 1
U
1 r-
2J0 25

A533B

8000 lT(CoT|

.TL(CoT)
6000
c

4000 Sl(CaT)

"UCON)
2000
, . " . >SL(CON)

.08 .10

FIG. 5—J versus Aa R-curves for A533B steels. Line indicates average "blunting line" for
these steels; for reference purposes. Eg 2. (KN/m = lb/in. X 0.1751).

including that for specimen size, are noted. The specimen size requirement
demands that

B,b>: 25J/FS (3)

where B is the specimen thickness and b the initial remaining ligament

length of the specimen. Those (/IJFLD points which would also meet this
specimen size requirement are also noted in Table 3. All of the specimen
tests which did not meet the validity or specimen size requirements failed
due to the ligament length, b, being undersized.
The applicable A'lc values were calculated for each of the/ic determinations,
using Eq 1 with a value for v of 0.3 and a value for E of 207 000 MPa
(30 X 10* psi), and are listed in Table 4. Also, the Rolfe-Novak-Barsom
(RNB) upper-shelf/iTic-CVN correlation [18,19] was used to determine an
480 ELASTIC-PLASTIC FRACTURE

mm
0 -5 1.0 i.s 2,0 2S
1 1 1 1 1 \^—I ^—I Sr

4000 HY-130 IT(CaT)

IT (CON)
3000
. Tl(CaT)
c

2000

, 'TLICON)

1000

_ ( — — ( ( 1 1 1 1 —

.04 .06 .08 .10

Aa,in
FIG. 6—J versus Aa R-curves for HY-130 steels. Line indicates average "blunting line" for
these steels: for reference purposes, Eq 2. (KN/m = lb/in. X 0.1751).

additional A^ic value using the CVN USE values of Table 2. This correlation
is

/ Ku V ./CVN USE „_A ,.,

where 0.2YS is in ksi, CVN USE in ft-lb, and Kic in ksi y/m.

Discnssion

Rating Steel Toughness

In order to determine how each of the toughness measurement tech-
niques rates the quality of the steels, a normalizing technique was used.
 WILSON ON PLATE STEEL QUALITY 481

mm

A543
. . 'lT(Ql)

3000
f »

* t

.Tl(Ql)
. "• 'IT(CI)
2000 / ' • .' c •
e / •• .'"»", .-SlIQl)
/ . < » » 0 °
.• < • . * . "
/ ' ' ' -f ° ° . 'TliCl)

•Z" • . •
U «""'' • .
1000
i .<". • . .. •• . •
:ifV - ". - si(ci)
'S' u •" •"
af•
Lk> JJ -•' "

0 J• • 1 ( 1 1-..—•• ..». t t . *•-•.-••,

.02 .04 .06 .08 .10

Aa,in
FIG. 7—J versus Aa R-curves for A543 steels. Line indicates average "blunting line" for
these steels: for reference purposes. Eq2. (KN/m = lb/in. X 0.1751).

The normalizing method used determined ratios of toughness levels for

each test technique and then compared these ratios between techniques.
Two types of ratios were calculated. The first were anisotropy ratios. These
quantify the amount of anisotropy of a particular property in a steel by
ratioing the property levels in each testing orientation. This is done by
dividing the TL or SL value of a property by the LT value. Thus a steel
with a great deal of anisotropy such as the CON A533B material would
have anisotropy ratios TL/LT and SL/LT of 0.55 and 0.32, respectively,
for CVN USE, while the CaT A533B steel ratios would be 0.81 and 0.66,
respectively, for the same property. This indicates the more isotropic
nature of the CaT steel.
The second type of ratios determined were quality ratios. These quantify
the amount of improvement of a particular property by rating the property
in a particular orientation of the steel in the poorer quality to that in the
482 ELASTIC-PLASTIC FRACTURE

" ^ i/> i/> n f*i '-H 00 go -tr

-,6 r-- ro O O '-I m r-- (N
I 00 Tf ^O irj <T) (N o "^ r--
f <S TJ- fO (N --H ^
I -H o --^ Tj- r- r-
I
I
I
••—t
r--
<N
00
i-H
-^
CO
TT
m
0^
fs
(N
(^
rj
^H
lO
—*
^
1^^
' ^O O oC CN 00 i/i" ^O (N 0 0
I i-H r- ro •-' fs ^o ^H s ^ TT
1 o r^ 00 r-- n (N
» r o - ^ CD <N -PH ^
o ^- ^

" (No ^to - lo mo o

^ w r - CO
•^ O ^
00 r ^ 0^
f S ^ TT ^ <N ^ ^ t~- ^
(N ->£) TT »N ^ ^
(N <N 00
_ _ ' 1 0 t~- f s i/i o^ t^ ^r
^-1 \0 r^ O^ r^ ^->
f S UO TT ^ - H rt
o !-- r--

8 o" o o
T t TT 0 0 O (N O
o" o 00'
^1?
o r o I/) Tj- (N --< w
(N \ 0 TT <N ^-< '-H

o.
o
• «

ti
a ^ 00 <N r - lo uo 00 r ^ 00 r o
^=i i/> 0 ro 0 -.o r^ ro
c f^ 0^ J~- ^ I/) 0^ vO 00 10 (T)
^ ~ C^ ^.^ --^ -.^ s.^ s_^ - '
^^
C^ , 0 ,-H lo ^ a^ r-~ 0 CO ro r-
o^-^<N^-^
*=;
•w
0
00
"O
00
ro
r~-
'^
00
0
1/)
0
0^
O^
i/l
f*^ 00 0^
00 i/1 n
c JJ

1 1 r ^ ' ^ 0 * • ^ ' r o ' 0^ r s '

;-* ^ o r-- TT (N 0 0 0 0
r - ^ r-- ^ \ 0 06} v ^
'"' ^^ , .-' C" a '
»H <N r o
5 0 ^ 10
00 ^ 0 r o

•l f
& ^
^ E
r n f*^ 00 fO lO ' O 00 ^£> ^ *N
^ ON ^ <N r - - ^ CM 10 r-- <^
( S ' O ^ O 10 O lO r-- uo f*>
I O " ^ O O I/) (N ' t
^ w t-H 0 0 <N t ^ 00 00 (N ^
Q
f<j vO t-- O lO ^ Tf b •
I t-. vo rri
iTi o irf 10 i/i" o r4" o !«
U ^ r-- rN o r j (Ni r^ -t-» ' —
I-)
o •"• ir> lO ^ ^ »-i i/l oo'o'X c -1
DO ^ '0 °
< <N <S S
(^ U-) m

OS S |

. ? £ II
H H H fJ J H hJ H J J c •= -
J H iJ H c/i ^ H J H V5

o S 1;
XI C P
0. ja t»

flitIsAl
°< E A
M 0 ^
»•
O4 V>
£ o
(A

^o
^
ea
ro
m
^
m

^
^llim
 WILSON ON PLATE STEEL QUALITY 483

r^ u*) <N m o n fs
00 r- r^ O 00 •<r (N
t-H TH fS fS ---I (N (N
z
OS

z r4 >o O (S 00 -< >o

o ro O
^H TH
OS Tj- o
T-4 TH ^H
lO 0^
(N —i

t^ o IQ rt • ^
<N O Si?! o o 0 0 SO l/>

-H ^ l/>
r-~ ro r-- O OS
r o r o <N
oC-^'
r-T v^ o^ o o
^ 1/5 CT^ <N (N
r o fO (N

l-l
H
E
•x-x'x
rj- 00 ^ O 0^
TT r o (N (N w OS
r oCT^lO ^ »/)
> 1-^ fo
O so
n t - I/) O 0^
TT r o <N <N ^

.K lO - ^ n
s o 00 r ~
•<r r n rsi

Z oe etc ac
o ^H ro so i/> 1/5
o •^ t^ ro t^ -^ 9
CJ W W w S Q
• ^ 00 OS a> o^ 00 o
•* so
I ro
Csl
sO
^s©
(N
T^
r^ •^
^^ ^o
00 00 (S r^
•^ i^ TT r- lo
Cs) W rt -H rt
ao •a
< e •o
l-l o
U
g s» ,
s ^ ^ - X o «*-o «.
P ^ 0 0 t-- UO
0^ o 6 r ^
(N ^ ^
l O (D O a 2;.s X
£OS• 00
3 : 0r o t*- "Ol
<N - - -H o OS iTi^
Os" OO' •^'' Z
00 r~ - *
o <N ^ -H
3 5"

If c 3,1=
u V <it

Sf^ J H i« Sf^ J (J wJ
H H
II C r- C o

jb S 5 . 0
00 5, "> -a

so
n
ro H> S o
lO >o
< < ^ ^
484 ELASTIC-PLASTIC FRACTURE

better-quality condition. Therefore, for the A516, A533B, and HY-130

steels, the ratios compare CON and CaT toughness levels, CON/CaT. In
the A543 steel, this ratio would be CL/QL. Thus, again using the A533B
steel CVN USE for comparison, the ratios would be 0.72, 0.49, and 0.35,
respectively, for the LT, TL, and SL orientations. This shows that the
largest improvement in CVN USE is obtained by the CaT steel in the SL
orientation.
In comparing the ratios for each testing technique it was soon found that
the CVN USE and DT USE ratios compared well with each other and were
closer to the J^ ratios than the Kic ratios. This is plausible since these
three parameters are all energy related. Therefore the two Ju determination
ratios, (/IC)G and (/IC)FLD, were compared with each of the other toughness
ratios as shown in Fig. 8. The ratios for the Ki^ determinations from each
Ju analysis (Eq 1) were compared with the ratios of K^ values calculated
using the RNB correlation (Eq 4) and are also shown in Fig. 8. All six of
these graphs also have 45-deg lines for reference purposes.
Generally it can be concluded from these graphs that the ratios obtained
from conventional toughness tests are not the same as those obtained by
Jic testing. More specifically, the J,c test is shown to be more sensitive to
changes in inclusion structure than either CVN or DT tests. This is indi-
cated by the general position of the data points to the right of the 45-deg
perfect agreement lines in all six plots in Fig. 8. This is shown particularly
for the invalid Ju data points and to a lesser extent for the valid data. For
example, in Fig. 8a it is generally shown that when the (Jic)c ratio is 0.4
the CVN USE ratio is 0.6, demonstrating that the (/IJG tends to show more
anisotropy in a steel and larger differences between quality levels of a steel
than the CVN USE. What this means specifically to these comparisons
is that for three of the steels the Ju determinations demonstrate that CaT
results in more improvement in steel quality than would be identified by
the CVN or DT tests. For the A543 steel the Ju determinations indicate a
larger difference between CL and QL quality than shown by the conven-
tional tests.
These differences result even though the appearance of the fracture
mode in each of the test specimens is the same. The appearance of the
manganese sulfide Type II inclusion formations on an SL SSJ fracture of
the CON A533B steel is shown in Fig. 9 and the calcium modified in-
clusions on a SL SSJ fracture of the CaT A533B steel is shown in Fig. 10.
Similar fracture appearance has also been noted on tensile and CVN
fractures [2] and also on DT fractures.
There are two reasons why the Ju data are more sensitive to inclusion
effects. Most evident is that a sharp crack starter is used in the Ju test
compared with the machined notch in the CVN and the pressed notch in
the DT tests. This may therefore indicate a different interaction between
the crack or notch and the inclusions in the steel. An additional reason for
 WILSON ON PLATE STEEL QUALITY 485

1.0- IJO-

.8 .8
»•%••
.6
^%
-P W

.2 • ' ^ 0 2 / O O

I _L_ L 0._
°0 .2 .4 .6 J IJO .2 .4 .6 .8 10
(a) CVN USE (b.) CVN USE

o ASM U)-
U)- o A533B A
A HY-130 A '
.8 D A543
fi .8
.'O A •/
49 J* 0 ^
.4 < : ^ » ::t.4

.2 'o°«o .2
/ o o

0 _L J L 0, J L _L J L
.2 .4 6 .8 10 0 .2 .4 .6 .8 10
c. DT USE (d.j DT USE

t .^A
IJOI- 10

.8 . # ' - 8
^SA
*i\t
w
.6 1*
^
/ "o
/ G O / oo
i^.4 L.4 — /
/

/
.2 — /
.2
/
_L J I L n/ 1 1 1 1 1
00 .2 .4 .6 .8 U) 0 .2 .4 .6 .8 10
(f.) (K,c)^RNB

FIG. 8—Comparison ofanisotropy and quality ratios determined from the various toughness
measurements. Lines indicate equality between the ratios. Solid points are valid J/c results.
486 ELASTIC-PLASTIC FRACTURE

FIG. 9—Scanning electron microscope fractograph of Z-integral specimen in SL orientation

of CON A533B steel showing manganese sulfide Type 11 inclusions.

the differences just mentioned is that the CVN and DT tests are conducted
at impact or high loading rates, while the Ju tests are performed at a static
or low loading rate. This may be the controlling influence since the data
points for the lower-strength steels (A516 and A533B), which are more
loading rate sensitive, are more prominently shifted to the right in the
graphs in Fig. 8. This may indicate that the rate sensitivity of the steel
is the reason for the different effect of inclusion structure for the /ic data.
This could only be checked, however, by performing Ju tests at impact
loading rates.

Comparing Jh Determination Methods

In this paper two methods have been used to determine /ic, namely, the
graphical technique and the first load drop method. Figure 11 presents a
plot comparing the Kic values obtained from the respective 7ic determina-
tions for each of these methods. Also given on the graph are the 45-deg ref-
erence line and ± 10 percent lines surrounding this line. This compilation
reveals that, generally speaking, using ± 10 percent to account for possible
experimental and analytical errors, the (A'IC)G is equal to the (KIC)FLD when
 WILSON ON PLATE STEEL QUALITY 487

FIG. 10—Scanning electron microscope fractograph of i-integral specimen in SL orientation

of CaT A533B steel showing calcium-modified, shape-controlled inclusions.

the /ic data being used are valid. When the /ic data are invalid, that is, not
meeting specimen size requirements, there is more of a deviation from this
relationship. If this relationship—indicating that the two /ic determination
methods give identical results for valid data—holds for a number of other
steels or metals, the J-integral testing and analysis procedure would be
simplified significantly. This is because the FLD number can be obtained
from a single specimen without the additional instrumentation required by
the SSJ technique. The results given in Fig. 11 therefore suggest that other
materials should be examined for the presence of this correlation.
If the foregoing relationship exists, it would also appear to allow com-
ment on the specimen size requirement for /ic determinations [14]. In
particular, the two CON HY-130 LT orientation Ju values and the two
CON A533B SL orientation results would appear to be close to validity
judging by the fact that their graphical and FLD Ku values are close or
within the scatterband of Fig. 11. Also, on the other hand, the valid A516
data points falling outside the scatterband suggests that possibly these
points should not be considered valid. These observations are examined
next.
488 ELASTIC-PLASTIC FRACTURE

MPtt/m

FLO

FIG. 11—Comparison o/K/c values calculated for various J/c analytical methods, graphical
and first load drop. Dashed line represents equality with scatterband around it indicating
± \0 percent. Solid points are valid he results.

Comments on he Analysis Guidelines

The graphical Jic analysis method used here is that suggested by the
guidelines of Clarke [14]. The specimen size requirement in these guide-
lines is covered by Eq 3. If the factor of 25 in this equation had been 18,
the two HY-130 and two A533B values mentioned in the preceding section
would be considered valid. However, this would only accentuate the prob-
lem that appears to be present for the two valid A516 CON values. This
suggests that possibly more weight should be given to the strength level of
the steel in Eq 3. This could be done by making the equation similar to
the form of the ASTM E 399-74 specimen size requirement, for example

JE
B,b s
X {Fsy (5)

where X is a number like 15. This equation would invalidate the questioned
A516 results while allowing the two HY-130 values to be considered valid;
however, the two A533B values would also be invalid.
 WILSON ON PLATE STEEL QUALITY 489

By way of observation, it was also found that generally the points on the
/ versus Aa R-curve at low Aa values did not fall on the "blunting line."
These points tended at the start to be at zero crack extension and then to
come above the blunting line. This has been reported previously also by
Clarke et al using the SSJ technique [15]. The lack of correspondence be-
tween the data points and the blunting line also appeared to be independent
of whether the test turned out to give a valid or invalid Ju.
An additional remark that can be made on the graphical Ju procedure is
that the points closest to the blunting line should be given more weight in
the determination of the line to extrapolate back to the blunting line to
obtain the /ic value. This is shown in Fig. 12 for a CON A533B SL-oriented
specimen. If an average line using all of the points were used, a /ic of
4.85 N/m (850 lb/in.) would have been determined. However, if only the
nearest points to the blunting line were considered, the Ju of 3.54 N/m
(620 lb/in.) would be found. The latter result is also closer to the FLD
value.

Comment on CVN Upper-Shelf Correlation

The RNB CVN upper-shelf correlation (Eq 4) was found to be initially
applicable to steels with 0.2YS greater than 758 MPa (110 ksi) [18,19]. It

FIG. 12—J versus Aa R-curve for CON A533B steel in SL orientation. If all data points
are used, a J/c at Point " a " of 4.85 N/m (850 Ib-in.) is obtained. If only points at lower Aa's
are used, a he at Point "b" of 3.54 N/m (620 Ib-in.) is determined.
490 ELASTIC-PLASTIC FRACTURE

has since been found to be useful for rotor forging steel CVN-iiric correla-
tions with 0.2YS values down to 552 MPa (80 ksi) [20]. It also has been
found useful for A533B steels with 0.2YS values as low as 414 MPa (60 ksi)
[21]. In addition, Paris has commented that the J-integral concept has
made this empirical CVN-ZiTic correlation appear more reasonable from a
technical standpoint [22]. Figure 13 shows plots of/sTic determined by the
Jic methods plotted versus the Ku obtained from the RNB correlation. It
can be immediately noted that almost all of the valid Ku from /ic values
are lower than those predicted by the correlation. This is explainable since
the .7ic determination is made at the point of crack initiation, while the
Kic is determined after an allowable amount of crack extension (5 percent
secant offset). Thus the /ic-determined values should be lower. The invalid
Kic from /ic points, on the other hand, tend to be above the line. This is
most likely a result of the large amounts of plasticity involved in these
fractures. In addition, the points above the line tend to be from the lower-
strength steels, A533B and A516, which would be expected to have a sig-
nificant effect of loading rate on upper-shelf toughness. Therefore, the
RNB correlation would not be expected to hold up, because the Kk test is
performed at a slow loading rate and the CVN at a fast loading rate.

Conclasions
1. The /ic values appear to be more sensitive to changes in inclusion
structure than either the CVN or DT tests. Therefore Ju tends to indicate
MPaVm MPa/nT
soo too 200 100 200
i 1 3UU 1
OA5)6
O OAS33B
AHY-130 400

400 DA543
400
0 0

0
^ o 300
300 oo
•
// o
O
Q
-.J
0
UL
/
^ / U o O /' %
200
'200 ^ ° ''1 \
O ' • ^
—200 f- /I*

100 m 100 f K>0

1 i 1 r
1(50 200 300 100 200 3019
1^ (•^IJRNB .k""^
" RNB

FIG. 13—Comparison ofKic values obtained from he determinations and from CVN upper-
shelf correlation. Dashed lines indicate equality. Solid points are valid he results.
 WILSON ON PLATE STEEL QUALITY 491

that there is more improvement by calcium treatment than would be indi-

cated by these conventional tests.
2. The Jic determinations obtained by both the graphical method and
first load drop technique were found to be the same for valid /ic tests.
3. It is suggested that the / u specimen size requirement equation be
modified to use a factor of 18 rather than 25 or use a new requirement
which places more emphasis on the strength level of the steel, or use both.
4. The Kic determined by the Rolfe-Novak-Barsom CVN-ZTic upper-shelf
correlation was found to be conservative when compared with the Ku ob-
tained from Jic, for invalid data. For valid data, the Ku from Jic gives a
lower result than that predicted by the CVN correlation.

Acknowledgments
The author would like to acknowledge especially the contributions of
J. Keith Donald, vice president, and David W. Schmidt, staff engineer, of
the Del Research Division of the Philadelphia Suburban Corp. for per-
forming the single-specimen J-integral tests. In addition, their general
comments, suggestions, and assistance during this research program are
greatly appreciated.
The guidance and comments provided by John A. Gulya during the
experimental and writing aspects of this research are also appreciatively
acknowledged.

Disclaimer
It is understood that the material in this paper is intended for general
information only and should not be used in relation to any specific appli-
cation without independent examination and verification of its applicability
and suitability by professionally qualified personnel. Those making use
thereof or relying thereon assume all risk and liability arising from such
use or reliance.

Rrferences
[/] Wilson, A. D., "The Interaction of Advanced Steelmaking Techniques, Inclusions,
Toughness and Ductility in A533B Steels," American Society for Metals, Technical
Report System No. 76-02, 1976.
[2] Wilson, A. D., "The Effect of Advanced Steelmaking Techniques on the Inclusions
and Mechanical Properties of Plate Steels," presented at American Institute of Mining
Metallurgical and Petroleum Engineers Annual Meeting, Atlanta, Ga., March 1977, to
be published.
[3] Wilson, A. D., "The Influence of Thickness and Rolling Ratio on the Inclusion Be-
havior in Plate Steels," presented at American Society for Metals Materials Conference,
Chicago, 111., Oct. 1977, to be published in Metallography, International Metaliographic
Society.
492 ELASTIC-PLASTIC FRACTURE

[4] Wilson, A. D., Journal of Pressure Vessel Technology, Transactions, American Society
of Mechanical Engineers, Vol. 99, Series J, No. 3, Aug. 1977, pp. 459-469.
[5] Rice, J. R., Journal of Applied Mechanics, Transactions, American Society of Mechanical
Engineers, Vol. 35, Series E, June 1968, pp. 379-386.
[6] Begley, J. A. and Landes, J. D. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 1-23.
[7] Landes, J. D. and Begley, J. A. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 24-39.
[8] Landes, J. D. and Begley, J. A. in Fracture Analysis, ASTM STP 560, American Society
for Testing and Materials, 1974, pp. 170-186.
[9] Bucci, R. J., Paris, P. C , Landes, J. D., and Rice, J. D. in Fracture Toughness, ASTM
STP 514, American Society for Testing and Materials, 1972, pp. 40-69.
[10] Rice, J. R., Paris, P. C , and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing. ASTM STP 536, American Society for Testing and Materials, 1973,
pp. 231-245.
[//] Logsdon, W. A. in Mechanics of Crack Growth, ASTM STP 590, American Society for
Testing and Materials, 1976, pp. 43-60.
[12] Marandet, B. and Sanz, G., "Characterization of the Fracture Toughness of Steels by
the Measurement with a Single Specimen o f / u and the Parameter /Ced," presented at
the Tenth National Symposium on Fracture Mechanics, American Society for Testing
and Materials, 23-25 Aug. 1976.
[13] Logsdon, W. A. and Begley, J. A., Engineering Fracture Mechanics, Vol. 9, 1977,
pp. 461-470.
[14] Clarke, G. A., "Recommended Procedure for J\c Determination," presented at the
ASTM E24.01.09 Task Group Meeting, Norfolk, Va., American Society for Testing and
Materials, March 1977.
[15] Clarke, G. A., Andrews, W. R., Paris, P. C , and Schmidt, D. W. in Mechanics of
Crack Growth, ASTM STP 590, American Society for Testing and Materials, 1976,
pp. 27-42.
[16] Merkle, J. G. and Corten, H. T., Journal of Pressure Vessel Technology, Transactions,
American Society of Mechanical Engineers, Vol. 96, Series J, No. 4, Nov. 1974, pp.
286-292.
[17] Donald, J. K., "Rotational Effects on Compact Specimens," presented at the ASTM
E24.01.09 Task Group Meeting, Norfolk, Va., American Society for Testing and Ma-
terials, March 1977; available from Del Research Division, Philadelphia Suburban Corp.,
, 427 Main St., Hellertown, Pa. 18055.
[18] Barsom, J. M. and Rolfe, S. T. in Impact Testing of Metals, ASTM STP 466, American
Society for Testing and Materials, 1970, pp. 281-302.
[19] Rolfe, S. T. and Novak, S. R. in Review of Developments in Plain Strain Fracture-
Toughness Testing, ASTM STP 463, American Society for Testing and Materials, 1970,
pp. 124-159.
[20] Begley, J. A. and Logsdon, W. A., "Correlation of Fracture Toughness and Charpy
Properties for Rotor Steels," Westinghouse Research Laboratories Scientific Paper
71-1E7-MSLRF-P1, Pittsburgh, Pa., July 1971.
[21] Sailors, R. H. and Corten, H. T. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 164-191.
[22] Paris, P. C , written discussion to Ref 6, pp. 20-21.
W. L. Server^

Static and Dynamic Fibrous

Initiation Toughness Results for
Nine Pressure Vessel Materials

REFERENCE: Server, W. L., "Static and Djrnaintc Fibrous Initiation Tonghnen

Resnlts for Nine PrcMore Vesiel Materials," Elastic-Plastic Fracture. ASTM STP
668, J. D. Landes, J. A. Begley, and G, A. Clarke, Eds., American Society for Testing
and Materials, 1979, pp. 493-514.

ABSTRACT: The upper-shelf toughness regime corresponds to the normal operating

temperature for ferritic nuclear pressure vessels. However, actual fibrous initiation
toughness data have not been available for a wide variety of heats of steels and weld-
ments. In fact, much of the data available have assumed that fracture initiation oc-
curred at maximum load (equivalent energy approach), which is not the case for most
materials tested on the upper shelf. Multispecimen (25.4 mm thickness compact or
bend) initiation tests were, therefore, performed in the upper shelf temperature region
to determine Jic for five base metal (two A533B-1, one A508-2, and two A302B) and
four weld metal (two manual arc and two submerged arc) heats of nuclear ferritic
steel. All of the heats were investigated under quasi-static loading (loading time ~ 100
s) and under either closed-loop hydraulic loading (loading time ~ 100 ms) or con-
trolled impact loading (loading time ~ 1 ms). For all materials investigated, crack
initiation occurred prior to maximum load, and initiation toughness values {Ksc)
were up to 50 percent less than the equivalent energy (maximum load) toughness
values. The initiation toughness results at 177°C also increased with decreasing loading
time for all but one heat tested. HSST Plate 02 (A533B-1 steel) was tested at 71 °C
using all three loading rates; there were little differences observed for the Plate 02
tests due to loading rate, although the intermediate loading rate results were slightly
lower than the rest. The data were analyzed using a simple statistical approach to
obtain approximate confidence limits for the Jic values obtained. A comparison of
Jic values obtained using three-point and nine-point fibrous crack length averages
across the specimen thickness was made. The nine-point average generally gave higher
values of /ic, but the variance about the regression line was not consistently lower
than for the three-point average.

KEY WORDS: fractures (materials), mechanical properties, test, /-contour integral,

pressure vessels, steels, statistics, initiation toughness, crack propagation

Fracture-safe design analyses are generally based upon a critical frac-

ture parameter which is measurable in the laboratory and can be assumed
• Vice president. Fracture Control Corp. Goleta, Calif. 93017.

493

Copyright® 1979 b y A S T M International www.astm.org

494 ELASTIC-PLASTIC FRACTURE

to be equivalent for both the structure and the laboratory specimen. The
need for convenient, small specimens and relatively easy laboratory test
procedures is obvious. In particular, linear elastic fracture mechanics
(LEFM) and its critical fracture criterion, Ku, are widely accepted for
fracture-safe design purposes. However, the laboratory method used to
measure Ku per the ASTM Test Method for Plane-Strain Fracture Tough-
ness of Metallic Materials (E 399-74) requires stringent limitations on
minimum specimen size requirements. These size requirements may not
be overly restrictive for high-strength materials, but for lower-strength
steels the size requirements demand specimens which are often larger than
the full section size of the structure. Therefore, the J-integral concept has
been proposed in the United States [1]^ to provide an extension of LEFM
for large-scale plastic behavior, both in the laboratory test and in the struc-
ture itself.
The J-integral is basically a two-dimensional, path-independent line
integral applicable to both elastic and elastic-plastic response when coupled
with a plastic deformation theory. The J-integral can also be interpreted
in terms of a potential energy difference per unit thickness (3 U) between
two identically loaded bodies having infmitesimally differing crack length
(da), that is

da
& = constant (1)

Rice et al [3] have developed a simple, approximate expression for / when

deeply cracked specimens are loaded in bending

2\' Pd8
_ J0
(2)
Bb

where the integral term is the area under the load-displacement (P — 8)

curve to some deflection 6i, fi is the specimen thickness, and b is the speci-
men ligament depth. The value of 7 at fracture initiation is defined as the
critical fracture parameter, /ic. The original derivation of Eq 2 for three-
point bend specimens referred only to the portion of the load-displacement
curve due to a crack; therefore, the uncracked body energy should be sub-
tracted out. However, recent analytical and empirical results indicate that
Eq 2 is more accurate when the uncracked body energy is included [4,5].
Merkle and Corten [6] have also presented a correction for compact speci-
men testing which accounts for the tension component. Again, the inclu-
sion of the compact specimen correction has not been shown to be empiri-
^The italic numbers in brackets refer to the list of references appended to this paper.
 SERVER ON FIBROUS INITIATION TOUGHNESS 495

cally necessary. In the work that follows, the calculation of / is based on

Eq 2 using the complete load-displacement curve; the effects of including
the corrections are discussed at the end of the paper.

Experimental Procedure

Materials and Test Specimens

The nuclear pressure vessel materials used in this investigation are listed
in Table 1 along with their relevant mechanical properties [7-12]. The
materials were chosen based upon their relatively low level, within a material
group, of equivalent energy (maximum load) fracture toughness as determined
from instrumented precracked Charpy tests [13]. In particular, the heats
EN (A302B steel) and BAS (submerged arc weld metal, Linde 80 flux) are
very low upper-shelf materials with high copper levels (0.20 and 0.33 weight
percent, respectively). The low initial shelves plus the high copper levels
make these heats highly suspect after in-service neutron irradiation damage.
Compact and bend test specimens were machined at quarter thickness
from plate and forging materials and not within 12.7 mm of the surfaces
for weld metal materials. All specimens were notched and fatigue pre-
cracked to a crack depth to specimen width ratio (a/w) of between 0.5
and 0.6 with the crack running in the rolling, forging, or welding direction
(TL orientation; see ASTM E 399-74). The compact specimens were machined
so that the clip gage could be mounted at the specimen load-line.

Testing and Analysis Procedure

The general procedure for t h e / initiation testing followed the early guide-
lines proposed by the ASTM Task Group on Elastic-Plastic Fracture Criteria
[14]. The first eight heats of steel listed in Table 1 were tested under quasi-
static loading ( — 100 s loading to Ju) using 25.4-mm-thickness compact
specimens at 177°C. These same steels were then tested either at 177°C
as dynamic 25.4-mm-thickness compact specimens ( — 100 ms loading to
Jic', heats CJ, EBB, and BAS) or dynamic 25.4-mm-thickness bend speci-
mens (~ 1 ms loading to Ju, heats EN, NA, EG, and EK). The HSST
Plate 02 material was tested at room temperature under quasi-static loading
and at 71 °C under all three rates of loading using 25.4-mm-thickness
specimens.
Compact specimens were loaded to different displacement values in a
closed-loop MTS test machine (89-kN capacity) in displacement (ram)
control for both static and dynamic tests. For the dynamic tests, no over-
shoot in load or deflection was allowed; therefore, a change in rate during
the last few milliseconds of loading occurred. The real effect of this de-
crease in loading rate is not known, but there is definitely a difference
in the test results as compared with static loading [15].
496 ELASTIC-PLASTIC FRACTURE

Hi
g g g

ills.!
^ rt ^ "O (N
^ i/> 1/^
(N rl *^

III- « s

I I I I
s
;-^ t^ ^
I I I I I

hy
E3 s
in >A I/)

ill •*
vp
<s
^
rJ
O S vO

Si
00

1
lo

§
t-^

!? 5?
^ r^

? I

l l IP in
2
III i l l Pijiii ill
0\
K
CQ

ca
,_,
=0

m
—
S g
z us
s
UA
NT
OS

(A
•<
« SB
i«* II
m
 SERVER ON FIBROUS INITIATION TOUGHNESS 497

The dynamic bend initiation technique developed for a drop tower im-
pact test is shown in Fig. 1. Hardened-steel deflection stops were used to
stop the falling tup at differing amounts of deflection. When the tup strikes
the deflection blocks, a sudden increase in the load signal occurs, marking
the stopping event. A typical load signal is shown in Fig. 2. It should be
noted that a few millimeters of deflection can still occur due to the elastic
brinelling in the stop blocks and the tup. However, the values of J were
calculated from the load-time trace when the tup hit the deflection stop;
these resulting / values are therefore slightly conservative (low). The drop
tower mass for the 25.4-mm-thickness specimens was 961 kg, and the im-
pact velocity was 1.41 m/s. This impact velocity meets the current re-
quirements for reliable instrumented impact testing (see Ref 13 and 16-20
for review of these requirements).
After loading, each specimen was heat-tinted at 288°C for 15 to 30 min.

FIG. 1—Schematic diagram of the drop tower stop-block arrangement.

EIB-T14
7rc

HIT
DEFLECTION
STOPS

FIG. 2—Impact test record for HSST Plate 02 specimen tested at 71 °C.
498 ELASTIC-PLASTIC FRACTURE

Specimens were then broken apart at — 70°C to reveal the amount of

fibrous crack growth. The amount of crack growth was taken to include
all extension from the end of the fatigue crack to the end of the heat-
tinted marking (thus including the stretch zone). J values were then cal-
culated using Eq 2. For the impact bend tests, no direct measure of dis-
placement was possible, and the value of the load-displacement integral
was determined from the load-time trace as follows:

Vo Pdt
Wx = Vo\ Pdt 1 ^ (3)
' 0 4£'o

where
Wi = velocity-corrected energy value representing the total specimen
plus machine energy consumed up to time t\ (when the stop block
is hit),
Va = initial impact velocity, and
EQ = total energy available (1/2 MVo^).
The value of W\ is then corrected for extraneous compliance contributions
[21,22], giving a value of energy (£"1) to be used in Eq 2

£ , = W , _ ^'^roofer Vo^ar'
(4)
2 'ar 8E0 EB

The time and load at general yield {tar and Par) are used to determine the
total system compliance, and the known nondimensional specimen compli-
ance {CND) from finite element and boundary collocation studies [23] is
then used to correct for the elastic extraneous energy contribution.
Once the crack extensions (Aa) have been measured using either a three-
point or a nine-point average for each specimen tested and the / values
calculated, a plot of / versus Aa is constructed. The straight line repre-
senting crack blunting is assumed to be known with certainty and is drawn
with a slope equal to 2ff/ (see Fig. 3), where Of is the flow stress indicative
of the specimen testing temperature and loading rate (equal to the average
of the yield and ultimate stresses) [14]. Values of a/ indicative of impact
loading were estimated from instrumented standard Charpy V-notch re-
sults [7-13], which were analyzed using extrapolated slipline field solutions
which include the indentor [24]

Of = (0.0467 mm-2) x ( ^ ^ ^ ^ ^ - ^ ) (5)

where PGY and PM are the general yield and maximum loads from the
 SERVER ON FIBROUS INITIATION TOUGHNESS 499

1200

Of'see MPa

800 -'
>S=243MPa

= 333 ±90 kJ/m'

200

Aaj p,, mm

FIG. 3—Regression line and 95 percent confidence limits (based on J/J X S^)/or dynamic
bend results of heat EG at I77°C.

instrumented Charpy trace. The values of a/ for the dynamic compact

tests were estimated by interpolating the static tensile results and the im-
pact Charpy results on the basis of loading time [13]

Of — 0.6 (a/, impact) + 0.4 (a/, static) (6)

The values of Of obtained at the appropriate test temperatures and loading

rates are listed in Tables 2 and 3. The best line fit of the experimental
/ versus Aa values (not on the blunting line) which intersects with the 2af
blunting line describes the point of fibrous initiation (/k). Typical plots
for static and dynamic tests are shown in Figs. 3-5. The best line fit and
confidence limits about the values of/ic were investigated by performing a
statistical analysis of the data obtained. Also, the effects of using a three-
point and a nine-point average for Aa were investigated. These analyses
are presented in the next section.

Results and Discussion

Analysis of the Data

The /ic results obtained from three-point average Aa values were re-
ported by Server et al [13] and are listed in Tables 2 and 3. The amount
500 ELASTIC-PLASTIC FRACTURE

ao «
^^ I/)

3^ ^ ^
ro ^ •<»• r o

3^ ^ ^
00 l-~ f - O rt O <S 0 0 l O CT^

3^
3-
Tf 00 <S 5 >0 t-

o^ Wi vAOoo^ CT^r^vooo i^PSQO^

^^ (S «

3^ ,^ ^ ^
oo^ oooo^r> r ^ ^ , ^ - ---
E' fOl/J O O O Q O •^fOO^O 1-4Qprn9^

<a« f l S l § l S5«S
00
00
01
Q
1/5 i
^ O vo S ^ \0 t^ k

^^1 T t ^ t ^ t "^ ' tC

00 «»1
3 g
1=
u flu

I |8|8
Pr P
r2
1/) i %

n
n
CQ § z
 SERVER ON FIBROUS INITIATION TOUGHNESS 501

S8 S
S

OS Tf h- 0^
l£
- ^ =5

S^ r ^ CO
oo o _
g2 Si

rs 00 t^ r^
00 -H 1^ O
s§

l-fS
•» <o
I-- PO
00
u-> 00
Ov ^o r^ 00
o» I '"' •—'
r n i/l
S5!ii

^ fO

o o o <s lO >r> 00 m 0^ l o in io 00 •M
m rt
rt <N
g g
ro vo
n • *
(S m
l / > lO -H 0^
( S m (S (N
00
<N
^
v H 1-H 1 <N
rg

^
i k
^
m
1^
00
lO
o
>o
TT
a>
(N
m
00
r^
m
S
1 «
00
0^
•<r
in
vO
m 11 sila u u

§ 8-=
•c
___^ ^_^ ^_^ O ^fO_^ ^_^ ^^ 2
00 l~-
S 00 | g 8 U1 S
<N (S m m (S s
m N <S <s t ^^
00 o
•<j-
M
ID . =
OS O
1 t t t t ^
T
1
1 &I
1
Ov
t- t-
o>
00
• * 00
r5
Tf
<N
00
ON
vO
U1
J=
•*-*
^i" 5 •*>
I•R! lo u
<N i i
^ H
(N r^ ^<N CM <N ^H
J3 •d
i:5 .5^ III
<O
• < U u
^1:
.a '^
«<-'
t;
i| ta U
«
lO
1£ u GO U lO
tsw
1(4 v>
g.U
U (rt
1S^
•K
1 (/I
(4 U
£
•o
11
>2
S i
1tilil
1 i 1
1 1a a
XI

1
<s
m
<
2
< •a
IS
1 1< in
<: 1i
2 2 I/)
< 1 1 °2ii
S ^
.1 s
z o "7 Z B- </l
1^ y ^^ < Q -O 1 < U U
(J 00
u n
502 ELASTIC-PLASTIC FRACTURE

J'ali

aai^
i\i^

I 00 -^ 5 \ o r-^ o^ 00
• •<*• <*! o »o '-I »/) m

ft.

1 i
a u
o
*l-l

u "2
a •S
B
* * M

u
8
1
.S 8 c
a S.J
1 » Q*
^or^fooo(^o^-<^ h a> M
vOOf^OOOOfslO'-' £ ^
^3 c S3
I 1 •d v
I "ti *N
"S
1•s Ic i5 c
Co . M ^ M

fc'
11
WJ + j
•g
,o CO s
s C»! - , oE(2 S
ifl 5cd fc-
«M v u

fi
• =

<\n <v> ou o<: >

Q.
.a
ts
.a
s
1% II£ * 3 n
•o a
•? ti <4
ts t; •o •o . ••oii a

in
(S
V

^
1^
V
1

1
1e
f o s II
B U o
O
S

?P T z; cu V)
 SERVER ON FIBROUS INITIATION TOUGHNESS 503

Of = 498 MPa

FIG. 4—Regression line and 95 percent confidence limits (based on log deviate of J) for
static compact results for heat BAS at 177°C.

of crack extension was later remeasured to provide nine-point average

values for comparison with the three-point average results. When the nine-
point values were being averaged, it became obvious that some of the
"equivalent" three-point values were different than those obtained earlier.
Investigation of the measurement procedures revealed that the accuracy
of location through the thickness of the earlier three-point values was
~0.15 mm. The later nine-point measurements were more accurate (<0.05
mm) with respect to thickness location. Since slightly different points of
measurement were used for measuring the bowed crack fronts, some in-
dividual results differ markedly. However, the averages of the three-points
in each case were usually within 0.10 mm.
The 7ic values obtained using a least-squares linear fit to the "new"
three-point and nine-point averages for Aa are listed in Tables 2 and 3.
The first value is the three-point average value and the second (in parentheses)
is the nine-point average. The slope of the least-squares fit (13), the num-
ber of degrees of freedom (<^) used for the fit (number of points minus 2),
and the overall mean variance (S^) were computed and are also listed in
the tables. It is obvious that a best fit requires many data points and low
variance about the line. The comparison of the two three-point average
/ic values is further obscured by the manual curve fits (some of which had
curvature) used to obtain the original values in Ref 13. Due to these dif-
ficulties, no conclusions will be drawn with regard to the three-point average
504 ELASTIC-PLASTIC FRACTURE

000 •
o,= 581 MPa
900 •

800 •

700 • . • ^^^^^ . • *
J = 2af Aa
600 -

• ^ ^"^'^^^ < \ - -
^= 334 MPa
- • • '

600 -

400 -

300 ' / B

/ • J|j = 304 ±84 kJ/m^

200 -

100 -

n - 1 1 1 '»- 1 •' »• — 1 1
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Aag p,, mm

FIG. 5—Regression line and 95 percent confidence limits {based on Aa/Aa X S^) for
dynamic compact results for heat BBB at IIT'C.

comparisons; the approach of a linear regression fit is a more consistent

analysis than the manual curve fits.
A statistical analysis of the least-squares linear fit allows approximate
confidence limits to be placed upon the intersection point of the two straight
lines. If the variance of the data does not vary over the linear fit, the 95
percent confidence limits at /k can be computed as \2S\

(7)

where
n = number of points used to fit the linear line,
t = value of the ^-distribution for n — 2 degrees of freedom,
Aa = average value of Aa from the data,
Attc = value of Aa at Jic, and
5 = estimated value of the standard deviation taken as the square
root of the variance about the line (5^).
If the overall variance about the line was used, very large confidence limits
developed for most cases. These large values are due to the few number of
 SERVER ON FIBROUS INITIATION TOUGHNESS 505

data points available for the fit and possibly due to a change in variance
at higher levels of/ and Aa.
The proper approach when the variance is not constant is to determine
the variance distribution, to weight each individual data point by the inverse
of the associated variance, and to then determine the regression parameters
by minimizing the weighted sum of squares [25], This approach would
give different values for the regression parameters and would allow a
weighted variance to be used in Eq 7. Due to the limited data, this approach
was not possible. Instead, the overall variances were adjusted (weighted)
without altering the regression parameters. Four schemes were used to
weight the confidence limits to more realistic values:
1. ^ssume thevariance is a linear function of / departing from the
mean/[52 X ( / / / ) ] .
2. Assume the variance is a linear function of Aa departing from the mean
Aa[S^ X (Aa/Aa)].
3. Use a log deviate of J for computing the variance.
4. Use a log deviate of Aa for computing the variance. _
The corrections of variance using linear functions related to / and Aa
are straightforward and the results for two series over the entire data range
are shown in Figs. 3 and 5, respectively. The log deviate approach for /
weights the smaller / data by use of logarithms

l n J = l n ( a +/3Aa) (8)

where a and /3 are the linear regression parameters. The variance is

V* = ^lnJ-In jy ^^j
in- 1)

and the 95 percent confidence limits at Ju become

Ju{exp[t(V*r']] (10)

Ju{exp[tiV*y']]-'

Results for the log deviate of / approach are shown in Fig. 4 over the
entire data range. The log deviate approach for Aa requires that the original
regression be computed as Aa on / ; the variance and limits therefore re-
flect ranges in Aa rather than / . These Aa ranges are then converted to /
limits by using the slope of the regression line. A similar form of limits
as shown in Fig. 4 is obtained for the log deviate of Aa. Tables 2 and 3
list ± 95 percent confidence limits at /ic based on all four methods.
It is important to note that the confidence limits developed assume
that the blunting line slope of 2<T/ is known with absolute certainty. If this
506 ELASTIC-PLASTIC FRACTURE

line has some uncertainty (as it may have), the confidence limits would
be inflated.

Results for Eight Heats Tested at 177°C

The results in Table 2 indicate that the nine-point /ic values are generally
greater than the three-point values, even though the slopes of the lines (/3)
are much steeper. These higher values can be interpreted as indicating
more plane-stress behavior, since the crack average includes surface mea-
surements, while the three-point values may be indicative of more plane-
strain behavior. The variances about the lines are not consistently lower
for the nine-point average. It is evident that the two averaging techniques
can result in different values of/ic. No one value for the 95 percent confidence
limits at Ju gives consistently high or low values; the value to choose is
rather arbitrary, but eliminating the highest and lowest values and taking
an average of the two intermediate limits allows a qualitative evaluation
of the ranges that Ju can be expected to have. This approach indicates
that the deviation of/ic is generally lower for the three-point average values
(see the arrows in Table 2 designating the Ju values with the lowest limits).
For all but one heat (BKM) the dynamic Ju values are higher than the
static values, and the slope of the lines (/3) increases with increasing loading
rate. Also indicated in Table 2 are the values of J at maximum load (/max).
In many cases, only one test was carried beyond maximum load; the /max
result for materials which have more than one test beyond maximum load
is computed as an average value. It is obvious that the maximum-load
values are significantly higher than the initiation values.
Values of Ju can be converted to equivalent stress intensity values (Ku)
by the equation

EJu
Ku — ) (11)
1

where v is Poisson's ratio. All 7ic values in Tables 2 and 3 meet the cur-
rent specimen size criterion for validity [14].

25/ic
a,b,B>
Of (12)
As indicated in Table 2, however, not all test points (usually the largest
deflection results) meet the size criterion of Eq 12 for individual / results.
It should be noted that Ku can be equivalent to large specimen linear
elastic Ku values; however, the measurement point for Ju is not always the
same point as for Ku [26], and statistical size effects can sometimes be-
come important [13].
 SERVER ON FIBROUS INITIATION TOUGHNESS 507

Jic values were chosen for either three-point or nine-point average crack
advance based upon the lowest values of the average 95 percent confidence
limits as discussed previously. These values were converted to equivalent
A'jc values using Eq 11. The Kjc values are shown in Fig. 6 as a function
of the Charpy V-notch upper-shelf energy level. Also shown superimposed
on the graph are lines obtained for different yield stresses for the Rolfe-
Novak upper-shelf correlation which was based on higher-yield-strength
steels (oy > 690 MPa) [27\

K\c
= 5 — - 0.05 (13)

where Ku is in units of kips per square inch by square root inch, the yield
stress {oy) is in units of kips per square inch, and the Charpy V-notch energy
(C,) is in units of foot pound force. It is interesting to note that there is
good agreement between the data and Eq 13 for the static fracture results
at a yield stress level of 400 MPa; the yield stress at 177°C for all the
materials is near 400 MPa. The dynamic results agree favorably with the

280 r-
(jy = 600 M P a / ^
500 MPa

-
^ / ^ y = 400

200 - 1
BKM
/ ^ 5^1 cj

S! 160 - ©A
I
E
^ 1
z • EG
s 1
1
5? 120 - BAS

// 1 MA
' EN
80 - Of
LOADING
SYMBOL TEST TYPE TIME, s

_ • STATIC COMPACT ~102

A DYNAMIC COMPACT -10-1

e DYNAMIC BEND -10-3

1 1 1 1 1 1
40 80 120 160 200 240

CHARPY V-NOTCH UPPER SHELF ENERGY, J

FIG. 6—Initiation toughness results at 177°C compared with the Rolfe-Novak correlation.
508 ELASTIC-PLASTIC FRACTURE

other yield stress lines drawn in Fig. 6; again, these yield stress levels are
indicative of yield stresses at 177°C (using an approach similar to Eq 5
and 6 for yield only). Therefore, it appears that at 177°C the Rolfe-Novak
correlation can be used to estimate levels of fracture toughness based upon
Charpy V-notch energy and the yield stress. However, several of the data
points fall below these lines, especially at the lower Charpy V-notch energy
levels. Also, application of this approach to other temperatures on the
upper shelf may be misleading. For example. Fig. 7 shows that static upper-
shelf toughness results obtained from another program using heat CJ [28]
and the results obtained here. Also, cleavage-initiated fracture results
[7,11,13] are shown as a function of temperature. It appears that the fibrous
fracture toughness shelf reaches a peak at the fracture mode transition
and then decreases with increasing temperature, while the Charpy V-notch
impact energy is relatively flat and fixed over this same temperature range
(as is the yield stress). This trend in fracture toughness on the upper"shelf
has been observed elsewhere; for example, see Ref 26. Therefore, the good
agreement of the data with the Rolfe-Novak correlation is perhaps fortuitous,
although there is a basic trend indicative of the correlation.
The Jmax values in Table 2 can also be converted to stress-intensity values
using Eq 11. The values for Km„ have been shown to be the same as equiva-
lent energy toughness values obtained using energy to maximum load
[29]. Since there was a large disparity between Ju and /max, the stress-
intensity factors will also vary according to the square root. Equivalent
energy fracture toughness based upon maximum load produces a highly
optimistic measure of upper-shelf fracture toughness.

•
260 -
• A
200 •

f
6e
_
vi 150
A O PRECRACKED CHARPY
D D Y N A M I C I T COMPACT

^ 100 e
A
^
STATIC I T COMPACT
D Y N A M I C 4T COMPACT

A OPEN POINTS ARE LINEAR ELASTIC

CLEAVAGE INITIATION
50 o O o HALE FILLED POINTS ARE ELASTIC-PLASTIC
CLEAVAGE INITIATION
NDTT ' " ' • ' N O T FILLED POINTS ARE ELASTIC PLASTIC
FIBROUS INITIATION

0 1
-100
t . 1 1
200
TEMPERATURE, °C

FIG. 7—Initiation toughness results for heat CJ.

 SERVER ON FIBROUS INITIATION TOUGHNESS 509

HSST Plate 02 Results

The results for HSST Plate 02 were analyzed in the same manner as
the eight heats tested at 177°C (see Table 3). The following observations
were the same as for the other eight materials:
1. The nine-point average/ic values are larger than the three-point average
values.
2. There is no consistent trend for the variance about the lines between
the nine- and three-point crack averages.
3. Dynamic loading increases the slope of the regression line.
4. The/max values are substantially higher than the Ju values.
Using the same approach as before for choosing /ic (lowest average con-
fidence limits) and converting to Kic values (Eq 11) gives the results shown
in Fig. 8. The 25°C static result is at the low end of the temperature range
where fibrous initiation occurs. The 71 °C static result increased by ~25
percent over the 25°C result. This increase is also consistent with the re-
sults shown in Fig. 7. It is likely that tests at 177°C would give a Jic value

300

200

150

100
i
s
<
z
>-
o

_L
-2 -1 0
10910 (LOADING TIME, s)

FIG. S—HSST Plate 02 results at 71 °C.

510 ELASTIC-PLASTIC FRACTURE

less than the 71 °C result and probably close to the room-temperature value.
The effect of loading rate at 71 °C is somewhat puzzling. The impact bend
toughness is only slightly larger than the static toughness, but the inter-
mediate-rate dynamic compact toughness is lower than either the static
or impact results.

Uncracked Body Energy and Compact Tension Component Corrections

The elastic uncracked body energy correction for the three-point bend
test (including shear) can be calculated as

lOPi^
U no crack _, _ (14)
Jbts
The energy correction for the materials studied would be less than 10
kJ/m^ (20 kJ/m^ in terms of Ju). For heats EN and NA this correction
would reduce the impact Jic values by —20 percent, whereas for the other
bend results the correction would be a decrease of less than 7 percent.
The tension component correction for the compact specimen [6\ results
in a revision of Eq 2

where 71 and 72 are variables dependent upon the a/w ratio. For the ma-
terials studied here (with a/w = 0.52), the correction for Ju would be an
approximate 20 percent increase.
The increase due to the compact specimen correction and the decrease
due to the bend correction are shown by arrows in Fig. 9. Only in the
cases of heats EN and NA is there a notable change in results—the re-
sults for the A302B steels show very little loading rate effect when the off-
setting corrections are made. There is divided opinion among the technical
community concerning these corrections (as stated earlier), and it is not
evident from this work that the corrections are needed. Note, however,
that the corrections were made for the already determined initiation values,
not for the total original raw data; it is quite possible that the regression
slopes, the variance about the lines, and possibly the /ic values would have
changed had the corrections been made to the individual data points.

Summary
This paper has investigated the fibrous initiation toughness results for
nine nuclear pressure vessel materials, in addition to the variation that
 SERVER ON FIBROUS INITIATION TOUGHNESS 511

360 -

II
_i o
o

50

— CD —

< < 2<

EG NA EK BKM CJ BAS

FIG. 9—Effect of correcting the J/c values for elastic uncracked beam energy and tension
component forces.

these values may have. From the results presented, the following conclu-
sions and observations can be made:
1. An experimental technique has been developed to measure /k for
fibrous initiation under impact three-point bend loading.
2. Dynamic loading increases the slope of the regression line (R-curve),
and, in almost all cases, the Ju results for dynamic loading are higher
than static values.
3. Maximum load values of J are significantly higher than the initiation
J\e results.
4. There appears to be a functional relationship between initiation tough-
ness and Charpy V-notch energy on the upper shelf. However, there also
512 ELASTIC-PLASTIC FRACTURE

appears to be a drop-off in toughness (once the maximum has been reached)

while the Charpy level remains relatively unchanged over the same tem-
perature range.
5. Three heats of materials had initiation stress intensity toughness
results less than 150 MN-m~^^^ under static loading. These heats were two
A302B heats (EN and NA) and a submerged arc weld metal (heat BAS).
Not only do these materials have low upper-shelf toughness, but two heats
(EN and BAS) also have high copper levels, making them markedly sus-
ceptible to in-service neutron irradiation damage.
Summarizing, the areas of concern which should be resolved in the
future are:
1. Whether three-point or nine-point average crack fronts (Aa) should
be used for determining Jic. These two approaches appear to give different
results, with the nine-point average giving higher results. However, the
95 percent confidence limits are generally lower for the three-point average
regression line.
2. The certainty in knowing that the 2af crack blunting line is generally
correct. It seems best to use a a/ indicative of the loading rate and test
temperature.
3. The type of confidence limits to be used to determine the confidence
of the straight-line curve fit with regard to magnitude (a) and slope (/S). In
particular, the distribution of the variance about the curve should be
determined.
4. The necessity for correcting three-point bend data for elastic un-
cracked body energy and the compact data for tension component forces.

Acknowledgments
Support under Electric Power Research Institute Research Project RP
696-1 is gratefully acknowledged. The author is indebted to Dr. R. A.
Wullaert and Dr. W. Oldfield for their discussions and reviews of the
manuscript. Special thanks also goes to J. W. Sheckherd for his perfor-
mance of the testing.

References
[/] Begley, J. A. and Landes, J. D. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 1-20.
{2] Rice, J. R. in Fracture, H. Liebowitz, Ed., Vol. II, Academic Press, New York, 1968,
pp. 191-311.
[3] Rice, J. R., Paris, P. C , and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing, ASTM STP 536, American Society for Testing and Materials, 1973,
pp. 231-245.
[4] Sumpter, J. D. G. and Turner, C. E. in Cracks and Fracture, ASTM STP 601, American
Society for Testing and Materials, 1976, pp. 3-18.
[5] Robinson, J. N., "An Experimental Investigations of the Effect of Specimen Type on
 SERVER ON FIBROUS INITIATION TOUGHNESS 513

the Crack Tip Opening Displacement and J-Integral Fracture Criteria," International
Journal of Fracture, Vol. 12, No. 5, 1976, pp. IIZ-I^I.
[6] Merkle, J. G. and Corten, H. T., "A J Integral Analysis for the Compact Specimen,
Considering Axial Forces as Well as Bending Effects," Journal of Pressure Vessel Techno-
logy, Nov. 1974, pp. 286-292.
[7] Wullaert, R. A., Oldfield, W., and Server, W. L. "Fracture Toughness Data for Fer-
ritic Nuclear Pressure Vessel Materials; Task A," Final Report of Electric Power Re-
search Institute on Research Project RP 232-1, EPRI NP-121, Electric Power Research
Institute, April 1976.
18] Server, W. L., Sheckherd, I. W., and Wullaert, R. A., "Fracture Toughness Data for
Ferritic Nuclear Pressure Vessel Materials; Task B—Laboratory Testing, Final Report,"
EPRI NP-119, Electric Power Research Institute, April 1976.
[9] Van Der Sluys, W. A., Seeley, R. R., and Schwabe, I. E., "Determining Fracture
Properties of Reactor Vessel and Forging Materials, Weldments, and Bolting Materials,
Final Report," EPRI NP-122, Electric Power Research Institute, July 1976.
[10] Loss, F. J., Ed., "Structural Integrity of Water Reactor Pressure Boundary Components,"
NRL Report 8006, NRL NUREG 1, Naval Research Laboratory, Aug. 1976.
[7/] Marston, T. U., Borden, M. P., Fox, I. H., and Reardon, L. D., "Fracture Toughness
of Ferritic Materials in Light Water Nuclear Reactor Vessels, Final Report," EPRI
232-2, Electric Power Research Institute, Dec. 1975.
[12] Oldfield, W., Wullaert, R. A., Server, W. L., and Wilshaw, T. R., "Fracture Tough-
ness Data for Ferritic Nuclear Pressure Vessel Materials; Task A—Program Office,
Control Material Round Robin Program," Effects Technology, Inc. Report TR 75-34R,
July 1975.
[13] Server, W. L., Oldfield, W., and Wullaert, R. A., "Experimental and Statistical Re-
quirements for Developing a Well-Defmed Km Curve," EPRI NP-372, Electric Power
Research Institute, May 1977.
[14] ASTM Task Group E24.01.09 on Elastic-Plastic Fracture Criteria; Chairman, J. A.
Begley and J. D. Landes.
[15] Logsdon, W. A. and Begley, J. A. in Haw Growth and Fracture, ASTM STP 631, American
Society for Testing and Materials, 1977, pp. 477-492.
[16] Ireland, D. R., Server, W. L., and Wullaert, R. A., "Procedures for Testing and Data
Analysis," Effects Technology, Inc. TR 75-43, Oct. 1975.
[17] Server, W. L., Wullaert, R. A., and Sheckherd, J. W., "Verification of the EPRI Dynamic
Fracture Toughness Testing Procedures," Effects Technology, Inc. TR 75-42, Oct. 1975.
[18] Server, W. L., Wullaert, R. A., and Sheckherd, J. W., in Flaw Growth and Fracture,
ASTM STP 631, American Society for Testing and Materials, 1977, pp. 446-461.
[19] Server, W. L., "Impact Three-Point Bend Testing for Notched and Precracked Speci-
mens,"/o«ma/o/resting and Evaluation, Vol. 6, No. 1, 1978, pp. 29-34.
[20] Oldfield, W., Server, W. L., Odette, G. R., and Wullaert, R. A., "Analysis of Radia-
tion Embrittlement Reference Toughness Curves," Fracture Control Corp. FCC 77-1,
Semi-Annual Progress Report No. 1 to the Electric Power Research Institute on Re-
search Project RP 886-1, March 1977.
[21] Server, W. L., and Ireland, D. R. in Instrumented Impact Testing, ASTM STP 563,
American Society for Testing and Materials, 1974, pp. 74-91.
[22] Server, W. L., Ireland, D. R., and Wullaert, R. A., "Strength and Toughness Evalua-
tions from an Instrumented Impact Test," Effects Technology, Inc. TR 74-29R, Nov.
1974.
[23] Saxton, H. J., Jones, A. T., West, A. J. and Mamaros, T. C. in Instrumental Impact
Testing, ASTM STP 563, American Society for Testing and Materials, 1974, pp. 30-49.
[24] Server, W. L., "General Yielding of Charpy V-Notch and Precracked Charpy Specimens,"
Journal of Engineering Materials and Technology, Vol. 100, 1978, pp. 183-188.
[25] Davies, O. L. and Goldsmith, P. L., Eds., Statistical Methods in Research and Pro-
duction, Hafner, New York, 1972, p. 195.
[26] Landes, J. D. and Begley, J. A., "Recent Developments in 7ic Testing," Westinghouse
Scientific Paper 76-1E7-JINTF-P3, May 1976.
[27] Rolfe, S. T. and Novak, S. R. in Review of Developments in Plane-Strain Fracture Tough-
ness Testing, ASTM STP 463, American Society for Testing and Materials, 1970, pp.
124-159.
514 ELASTIC-PLASTIC FRACTURE

[25] Bbrden, M. P. and Reardon, L. D., "Sub-Critical Crack Growth in Ferritic Materials
for Light Water Nuclear Reactor Vessels," EPRI NP-304, Electric Power Research
Institute, Aug. 1976.
129] Merkle, J. G. in Progress in Flaw Growth and Fracture Toughness Testing, ASTM
STP 536. American Society for Testing and Materials, 1973, pp. 264-280.
W. A. Logsdon^

Dynamic Fracture Toughness of

ASME SA508 Class 2a Base and
Heat-Affected-Zone Material

REFERENCE: Logsdon, W. A., "Dynamic Fracture Toughness of ASME SA508 Class

2a Base and Heat-Affected-Zone Material," Elastic-Plastic Fracture. ASTM STP 668,
J. D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and
Materials, 1979, pp. 515-536.

ABSTRACT: The American Society of Mechanical Engineers (ASME) Boiler and

Pressure Vessel Code requires that dynamic fracture toughness data be developed for
materials with specified minimum yield strengths greater than 345 MPa (50 ksi) to
provide verification and utilization of the ASME specified minimum reference toughness
KiR curve. In order to qualify ASME SA508 CI 2a pressure vessel steel [minimum yield
strength equals 450 MPa (65 ksi)] per this requirement, dynamic fracture toughness
tests were performed on three heats of base and heat-affected-zone (HAZ) material
from both automatic and manual submerged-arc weldments. Linear elastic Ku results
were obtained at low temperatures while J-integral techniques were utilized to evaluate
dynamic fracture toughness over the transition and upper shelf temperature ranges.
Loading rates in terms of/f were on the order of 2.2 to 4.4 X 10^ MPaVm/s (2 to 4 X
10^ ksivin. /s). Tensile, Charpy impact, and drop weight nil ductility transition (NDT)
tests were also performed. All dynamic fracture toughness values of SA508 CI 2a base
and HAZ material exceeded the ASME specified minimum reference toughness KIR
curve. Upper shelf temperature resistance curves obtained by the standard multiple-
specimen / test technique, where each specimen was loaded dynamically to a specific
displacement, and resistance curves obtained via specimens loaded dynamically to
failure, yielded essentially identical/id values.

KEY WORDS; steel—A508, tensile, Charpy, dynamic, fracture, toughness, weldments,

heat-affected-zone, crack propagation

The fail-safe performance of pressure-retaining vessels involved in nuclear

applications (pressurized water reactors, etc.) can depend greatly on the
ability of various structural materials to sustain high stress/strain in the
presence of flaws. Pressure-retaining materials for vessels utilized in nuclear
applications must comply with minimum dynamic fracture toughness stan-

' Senior engineer. Structural Behavior of Materials, Westinghouse R&D Center, Pittsburgh,
Pa. 15235.

515

Copyright 1979 b y AS FM International www.astm.org

516 ELASTIC-PLASTIC FRACTURE

dards as set forth in Sections III and XI of the American Society of Mechani-
cal Engineers (ASME) Boiler and Pressure Vessel Code [1].^ In brief, for a
particular selected material, the dynamic fracture toughness [which has been
temperature corrected based on drop weight nil-ductility transition (NDT)
tests and Charpy impact tests] [1.2] must lie above an ASME specified
minimum reference toughness Km curve. This Km concept is based on lower-
bound dynamic fracture toughness and crack arrest data generated on
ASTM A533 Gr B CI 1 and ASTM A508 CI 2 pressure vessel steels and can
be considered as a conservative representation of the dynamic fracture
toughness of those pressure vessel materials with specified minimum yield
strengths up to 345 MPa (50 ksi).
The present state of the art in nuclear pressure vessel technology calls for
higher-strength materials such as ASME SA533 Gr A CI 2 or ASME SA508
CI 2a [minimum yield strengths equal 485 MPa (70 ksi) and 450 MPa (65
ksi), respectively]. The ASME Boiler and Pressure Vessel Code permits the
use of higher-strength materials [greater than 345 MPa (50 ksi) minimum
specified yield strength] for pressure vessels; however, Appendix G of the
Code requires that dynamic fracture toughness data need be developed to
enable verification and use of the ASME specified minimum reference
toughness Km curve relative to these new materials.
To develop this data base relative to ASME SA508 CI 2a pressure vessel
steel, dynamic fracture toughness tests were performed on three heats of
base and heat-affected-zone (HAZ) material. Linear elastic Ku results were
obtained at low temperatures while J-integral techniques were utilized to
evaluate dynamic toughness over the transition and upper shelf temperature
ranges. Support tests (tensile, Charpy impact, and drop weight NDT) were
performed to permit a comparison of toughness results with the ASME
specified minimum reference toughness Km curve.

Material, Mechanical Properties and Weld Parameters

ASME SA508 CI 2a is a quenched-and-tempered vacuum-treated carbon
and alloy steel typically utilized in forgings for nuclear pressure vessel ap-
plications such as vessel closures, shells, flanges, tube sheets, rings, heads,
and similar components. Chemical compositions and heat treatments of
SA508 CI 2a base and HAZ material are outlined in Table 1. Parameters
describing the automatic and manual submerged-arc weldments are pre-
sented in Table 2. Throughout this paper the weldments are identified as
follows:
TO-material/TO-weld wire

Each weld was post-weld stress-relieved at 607°C (1125°F) for 3 to 3.5 h.

^The italic numbers in brackets refer to the list of references appended to this paper.
 LOGSDON ON DYNAMIC FRACTURE TOUGHNESS 517

o— •o
O O O O O O O O
o o o o o o o o
V V V

o
O O O O O O O O

sC
o o o o o o o o
s
_a^OtOf^'Nnu^ro

I o o o o - ^ « - i o

3 3
tr
u
o o o o o o o o V l>4 c
J<!
U
S
2
u s
<yvoO(Nr^oooi~~ c •a
B
u
o o o o o o o o
•a
s
o o o o o o o o
efl

>0
n
xT s
"a.
j:
lo
2 •o
I ^ O O ^ O O i O i O x:
8 ^o ^ ^ (S fS *- ^
o o o o o o_
2

I
O O O O O O O O
o o
O f
sO s
ul TT .

so ^ "-O sD 0^ (N '
O O O O O ^ ^ f S
S B
2
•o
O O O O O O O O
i-l
CO

< O 1>

illiiifi
il
i!
«^
(4 c4 <4 c4 u 0) a> u
CO "^
00 00
§1

S lO
00
r^ O^ f^ ^ CTv 0^
00 00 O O O O
518 ELASTIC-PLASTIC FRACTURE

^ •*-• «-i.
P^ cu cu c4
8§ .5 c •- 5 o:: a: CEi o o f*^ 1-1
o o ^ 00
(M • - vO S
u
QQo Qo O S'
lO <^
\ \ \ •< O i« O •« o -a ^
t-i
^ ^ ^-^ ^-^ 0) (N t-~ 00
E B £ o- r>» ro r j lO U^ "^ 0*^
o o w ^ o o o r~- (N ^T' "o
O - ^ 00 "^ ,—1 ^.^ tU
TT ^ TT 00 m o o
fO
^ (ro
N ^O
^
o&^« Cb
o o d o ° o 2
fO - ^ 0^ <^ n TJ- OV o -o _ y
^ <N -^^
s;s§u §;gg t*
ro "^ "^ a> r-i •'J- - ^
O c
• ^ ^ r-- C9

5
666 ^ -.666 *^
H H H H • f ^ t ' H (N • • r-- i-i ^ 6
5

cit

a. E 2 "S
fcM
en
• \
3
u c x: u
O • » TT - ^ Q fN r ^ r ^ u.
b u. o
:a
•^
rt UO
^ ^
t^
\ .
<?^ 0 O I/) cn
ft. E„ OO 00 00 c •o OI/)
O (N
O ^ o.
I lO
2 ***•
m
"^
f*^
i/> |8S c rvi
ca
lo ^
.a
s
1
3
u E s u ou
o o o o
2 J H <N O o (N o r-
« vO o
< f^ fO • a —* ( N ^

£ 3 s-i
i
«-s
.III
iiJ til
 LOGSDON ON DYNAMIC FRACTURE TOUGHNESS 519

Tensile requirements for SA508 CI 2a call for a minimum yield strength of

450 MPa (65 ksi), a range in ultimate strength of 620 to 795 MPa (90 to 115
ksi), and minimum total elongation and area reductions of 16 and 35 per-
cent, respectively. Table 3 and Figs. 1 and 2 summarize the tensile properties
of SA508 CI 2a base plate and HAZ material. The HAZ material from weld-
ment 5389/4109 displayed an ultimate strength slightly above the specified
maximum. Compared with the base metal, the HAZ material typically
demonstrated moderately superior strengths and elongations and inferior
area reductions.
Dynamic fracture toughness data are typically plotted versus T RTNDT
for comparison with the ASME specified minimum reference toughness
/TiR curve, where RTNDT is defined as a reference temperature. The method
for establishing a reference temperature is outlined in detail in Section III,
Division I and Subsection NB-2331 of the ASME Boiler and Pressure Vessel
Code [/]. Table 3 summarizes the drop weight NDT temperature and Charpy
V-notch impact properties necessary to determine the reference temperatures
relative to SA508 CI 2a base and HAZ material. Charpy impact properties
are also illustrated in Figs. 3 and 4. Drop weight NDT temperatures and
Charpy V-notch impact properties of SA508 CI 2a HAZ material were su-
perior to those of the base material. In addition, SA508 CI 2a HAZ material
reference temperatures were defined by the drop weight NDT temperatures
whereas base metal reference temperatures were defined by Charpy V-notch
impact properties (in two cases by the energy absorbed and in one case by
lateral expansion). These results are in direct contrast to those previously
developed for SA533 Gr A CI 2 base plate and weldments [J].

Experimental Procedures
All dynamic fracture toughness tests were performed on 2.5 cm-thick
(1.0 in.) precracked compact toughness (CT) specimens with the exception
of two base metal tests (TO-4584). The smaller CT specimens were tested on
a servohydraulic MTS machine with load frame and load cell capacities of
22 680 kg (50 kips) and 9072 kg (20 kips), respectively. Dynamic capability
was realized by employing a 341 litres/min (90 gpm) MTS Teem valve (two
stage with feedback). Loading rates in terms of A^ were on the order of 2.2 to
4.4 X lO"* MPaVm/s (2 to 4 X lO"* ksiVm./s). Load versus time, displace-
ment versus time, and load versus displacement traces were recorded for
each test. The larger CT specimens were tested in a facility previously de-
scribed by Shabbits [4].
Some specimens tested at low temperatures were linear elastic and simi-
lar to those described by previous investigators [4-6\. The majority of test
specimens, however, were in the elastic-plastic regime where J-integral test
techniques applied [7-9]. Dynamic instrumented precracked Charpy tests
have been previously employed to obtain dynamic fracture toughness values
520 ELASTIC-PLASTIC FRACTURE

TABLE 3—Mechanical, drop weight, and impact

Nil Ductility SOftlb*

Transition Energy
Temperature Temperature

TO-Number Base or HAZ° op °C op OQ

4584 Base 0 -18 145 63

5387 Base 20 - 7 115 46
5389 Base 50 10 130 54
4585/4109 HAZ (automatic) -30 -34 15 - 9
4585/3993, 4004 and 4009 HAZ (manual) -10 -23 50 10
5387/4109 HAZ (automatic) -20 -29 15 - 9
5389/4109 HAZ (automatic) -10 -23 -5 -21

ASTM requirements Charpy impact at 21 °C (70°F) = 48 J (35 ft lb) (mini-

mum average value of three specimens)

NOTE— RT = reference temperature; NDT = nil ductility transition temperature.

"Heat-affected zone.
*lftlb = 1.356J.

at upper shelf temperatures. Because crack growth initiation often occurs

prior to the maximum load point and because the actual initiation point
cannot be ascertained, dynamic instrumented precracked Charpy tests
typically overestimate a material's dynamic fracture toughness at transition
and upper shelf temperatures and as such were not included in this study
[3.10].
The dynamic test techniques employed in this investigation can be divided
into two categories: (1) load-to-failure and (2) dynamic resistance curve.
These test techniques are described and illustrated in Ref 3 and will be
briefly reviewed herein.

Load-to-Failure
All ASME SA508 CI 2a specimens tested at temperatures below that
where upper shelf fracture toughness behavior was first experienced were
loaded dynamically to failure and sustained cleavage-controlled fractures.
The onset of crack extension was abrupt and unambiguous. There was no
stable growth. A sudden drop in the load deflection curve occurred at the
fracture point. Inertial loading effects were negligible at the testing speed
utilized. At low temperatures the load versus displacement records were
linear and the fracture toughness was calculated directly from the failure
load as outlined in the ASTM Test for Plane-Strain Fracture Toughness of
Metallic Materials (E 399-74), although in some cases the specified size
criterion was not met by the 2.5 cm-thick (1.0 in.) CT specimens.
 LOGSDON ON DYNAMIC FRACTURE TOUGHNESS 521

properties ofSASOS CI 2a pressure vessel steel.

35-miI Lateral Mechanical Properties (room temperature)

Expansion
Temperature RTNDT Cys Out
Elonga- Reduction
op °C op °C ksi MPa ksi MPa tion, % in Area, %

160 71 100 38 88.3 608.8 105.5 727.4 21.7 62.4

90 32 55 13 83.2 573.6 99.3 684.6 20.5 57.0
105 41 70 21 89.1 614.3 105.9 730.2 20.4 59.2
20 -7 -30 -34 93.0 641.2 111.6 769.5 25.7 49.1
40 4 -10 -23 93.9 647.4 105.2 725.3 21.8 38.9
35 2 -20 -29 97.2 670.2 112.2 773.6 33.6 47.5
20 -7 -10 -23 101.9 702.6 117.8 812.2 35.0 55.0

65 450 90 620 16 35
to to
115 795

At transition temperatures, nonlinear load versus displacement records

were observed although the specimen fractures were cleavage controlled.
Fast fracture occurred at maximum load. For these tests, / was calculated
from the estimation method outlined by Rice et al [//]. Corresponding
Kii values were calculated from the relationship between elastic-plastic
and linear elastic fracture mechanics parameters [9]. The criterion for
determining if a fracture was cleavage initiated consisted of evaluating as
follows the amount of stretching (blunting) experienced by the specimen

_ 0.55 JM
Aa <
Of

where

Aa = average amount of stretching (blunting),

JM — J calculated at the maximum load point, and
Of = flow stress midway between the material's yield and ultimate
stresses.
For ferritic steels such as SA508 CI 2a, compliance with the foregoing
requirement indicates cleavage initiation; if Aa is larger, the mode of
fracture initiation is fibrous. Thus in dynamic fracture toughness testing
it is not uncommon to obtain nonlinear load versus displacement records
522 ELASTIC-PLASTIC FRACTURE

Temperature, "C
0

Symbol TO-Number
4584
5387
A 538?

Closed (>ls.= Ultimate Strength

Open Pis. = 0 . 2 * Yield Slrenglh

-100 -50 0 50 100

so 1 I 1
'
70 -

60 -.

!
50
s
4 Symbol TO-Number

i
4584
40 5387 -
5389

iO Closed Pis- Reduction n Area >

Open Pts.= Elongation )
20 ^^i^=^^:^:^^-'^=^:^

10 1 1 1 1 1 ' 1
0 50
Temperature, ^

FIG. 1—Tensile properties of SA508 CI 2a base material.

 LOGSDON ON DYNAMIC FRACTURE TOUGHNESS 523

Temperature, *C
0

Symbol TO-Material/ Automatic

TO-Weld Wire or Manual
0 4585/4109 Automatic
f3993
0 4585/ 4004 Manual
[4009
D 5387/4109 Automatic
A 5389/4109 Automatic

^- 110

Closed Pls.= Ultimate Strength

Open Pts.= 0.2% Yield Strength

Symbol TO-Material/ Automatic

TO-Weld Wire or Manual
0 4585/4109 Automatic
[3993
O 4585/ ] 4004 Manual
[4009
a 5387/4109 Automatic
A 5389/4109 Automatic

£. 50

Closed Pts.= Reduction in Area

Open Pts.= Elongation
_] 110
-150 -100 0 50 100 150
Temperature, *T

FIG. 2—Tensile properties ofSASOS CI 2a heat-affected zone material.

524 ELASTIC-PLASTIC FRACTURE

ftfflplfJtur*. "C
» 100 ISO ?00 ?S0
n I

IM- Entrgy Miiortwd

tmk FrKturt
ItO U l t r t l Eipinslon

5 100

S 100

15 3

TBdiperiture, *C
50 100 150 ZOO
—T 1 ^ 1 —

Energy A b i o r M
BriHIe FrMure

I 100

LO

Lateral Expinsian

^ SO

IS

FIG. 3—Charpy V-notch impact properties ofSASOS CI 2a base material.

 LOGSDON ON DYNAMIC FRACTURE TOUGHNESS 525

and cleavage-controlled fracture initiation without some form of stable

fibrous crack growth (ductile tearing). Although rather infrequent, this
fracture behavior can also occur under quasi-static loading rates—generally
at lower temperatures and over a smaller temperature range for a given
material.
At initial upper shelf temperatures, specimens loaded dynamically to
failure experienced fractures which displayed a zone of ductile tearing
followed by cleavage rupture. The point of fibrous crack initiation was not
apparent from the load-displacement records, which often exhibited some
load drop prior to fracture. Calculating a fracture toughness based on
maximum load is clearly not related to the point of crack growth initiation.
Crack growth may in fact occur prior to or after the maximum load. There-
fore, it was not possible to obtain a dynamic fracture toughness value from
a single specimen loaded-to-fallure at upper shelf temperatures.
A schematic of this combined fracture behavior experienced by speci-
mens loaded dynamically to failure at upper shelf temperatures is illustrated
in Fig. 5. This schematic clearly illustrates the interaction of the two basic
fracture processes. The only modification to this schematic as a result of
dynamic loading is that the crosshatched zone of ductile tearing followed
by cleavage rupture would span a larger temperature range. All of the
tests loaded to failure at upper shelf temperatures for SA508 CI 2a base or
HAZ material [maximum temperature equaled 66°C (150°F)] displayed a
region of ductile tearing followed by cleavage rupture. Increasing the maxi-
mum test temperature approximately 27 °C (50 °F) would have resulted in
totally fibrous, ductile fractures. The purpose of applying the previously
stated requirement for cleavage initiation would guarantee that a particular
dynamic fracture toughness test result occurred prior to the crosshatched
zone of ductile tearing followed by cleavage rupture.

Dynamic Resistance Curve

To obtain clearly defined dynamic fracture toughness values at upper
shelf temperatures, it was necessary to employ a resistance curve test tech-
nique identical to that set forth by Landes and Begley in Ref 9 for quasi-
static fracture toughness testing. This technique is applicable to the ductile
tearing upper shelf fracture regime where the onset of crack growth cannot
be ascertained from the appearance of the load-deflection record. Compact
toughness specimens were dynamically loaded to a specific displacement
(not to failure), unloaded, heat tinted, and broken open to reveal the
amount of stable crack growth.

Results and Discussion

The dynamic fracture toughness values generated on ASME SA508 CI 2a
526 ELASTIC-PLASTIC FRACTURE

ijui 'uoisueflxjifjairi

§ 8 3 3 3 f3

S 3 ^ § 8 a 3 9 S
f twqjosqv X£jMi]

ujui uoKutdx] t r j s i n

u| 'uoisutdx] | f j » i r i

s s £ 3
& o

- -

fli 1 ° t
;

i:
C Z 0

^
° - :>^ ^ .
s o
^
hi § 8
/

S
'
3 3
^

f 'p.in'fSfjv ^fij.'ui
 LOGSDON ON DYNAMIC FRACTURE TOUGHNESS 527

uitu 'utttuedi] itaig}

u| 'U0|surdit3 rtMin
2 S
1
«
1
% 1
&
1

g- <
/
g •
1 D /•

V
8

111
• 1 ! I 0 o o\ ^
•5~r"<

y^
R
/ 0

xv
1
• <
/ \\
\>
3
1
i1 1
1 1

9 § 8 a 3 9 !5

1
S 3
I
§
\
§
i
8
r 'p9qj«(iv X6iiu3
1
3 3
1
9
1
!
1
3
1

uJuj 'uofsuedv] \ti»yf)

.1

f
S

3 R 8 3 3 3 R
iu»j«l U n p t j j siiiijg pur qi u cxqjcsqv ^6'*']

a 3 3 R 8 a 3 3 R
528 ELASTIC-PLASTIC FRACTURE

*" Cleavage Toughness

Ductile Tear
Initiation &- — - a — — p—
* ^ ^ Tougtiness

^ Zone of Ductile Tear

Followed by Cleavage Rupture

2
° Cleavage fracture J . - K

57 Some Stable Crack Growth Followed

by Cleavage Jj^ i Kj^^

D K. on Upper Stielf (hypothesized)

• K, Predicted from J

Test Temp.
(Dashed Line Means That the Extension Has
Not Been Adequately Demonstrated)

FIG. 5—Schematic ofKic transition temperature curve.

base and HAZ material are plotted versus T — RTNDT for comparison with
the ASME specified minimum reference toughness Km curve in Figs. 6
and 7, respectively, and versus temperature in Fig. 8. Single upper shelf
dynamic fracture toughness values generated via the dynamic resistance
curve test technique on SA508 CI 2a base metal are also included in Figs.
6 and 8. In all cases the dynamic fracture toughness of SA508 CI 2a base
and HAZ material exceeded the ASME specified minimum reference
toughness Km curve. Gillespie and Pense previously developed quasi-static
fracture toughness data on SA508 CI 2a which also fell above the Km curve
[12]. Therefore, this 450 MPa (65 ksi) minimum yield strength material is
acceptable for nuclear pressure vessel structural applications from a dynamic
fracture toughness standpoint.
The dynamic fracture toughness, drop weight NDT temperatures, and
Charpy V-notch impact properties of SA508 CI 2a HAZ material were
superior to those of the base material. The fracture toughness behavior
demonstrated by SA508 CI 2a was quite unlike that previously reported
for SA533 Gr A CI 2, where at any given temperature the average base
plate dynamic fracture toughness surpassed that of the weldments by
approximately 30 percent [3]. Recall that the bases for defining reference
temperatures relative to SA508 CI 2a and SA533 Gr A CI 2 pressure vessel
steels (whether drop weight NDT temperatures or Charpy V-notch impact
properties) were also in direct contrast. This toughness superiority displayed
by SA508 CI 2a HAZ material was not manifested as increased conservatism
when the fracture toughness values were compared with the ASME specified
minimum reference toughness Km curve. The superior drop weight NDT
temperatures, Charpy impact properties, and resulting reference temper-
 LOGSDON ON DYNAMIC FRACTURE TOUGHNESS 529

T - RT^rjj, Temperature, *C

-25 0

Specimen
Symbol TO-Number
Size
0 4584 2TCT
a 5387 ITCT
A 5389 ITCT
1^ 200
Open pts. = l o a d - t o - F a i l u r e Test Technique
200 s
Closed Pts. = Dynamic Resistance Curve Test Technique

f. 150

~ ASME Specified M i n i m u m
Reference Toughness K.^

-200 -150 -100 -50 0 50 100 150 200

T - ^T^Qj. Temperature, *T

FIG. 6—Fracture toughness versus T — RT^DTfor SA508 CI 2a base material.

atures displayed by the HAZ material actually penalized the HAZ dynamic
fracture toughness values by shifting them such that the HAZ and base
metal values both demonstrated the same degree of conservatism relative
to the ASME specified minimum reference toughness Km curve.

T - R T j j . ^ , Temperature, **€

-25

TO - Material/ Automatic
Symbol
TO - Weld Wire or Manual

o 4585/4109 Automatic

• ,[3993
4585/^4004 Manual
14009
200- • 5387/4109 Automatic

5389/4109 Automatic 200 i

ITCT Specimens
150

- A S M E Specified M i n i m u m
Reference Tougtiness K,™
Curve ^^

-50 0 50
T - R T ^ - _ , Temperature, T

FIG. 7—Fracture toughness versus T ~ R T^DTfor SA508 CI 2a heat-affected zone material.

530 ELASTIC-PLASTIC FRACTURE

Temperature, "C
0

TO- Base or Specimen

Symtwl
Number HAZ , Size

45M Base ZICI

5387 Base ITCT
5389 Base ITCT

• 4585/4109
(3993
HAZ ' A u l o l ITCT

» 4585« 4004
14009
HAZ'ManuaM ITCT

• 5387/4109 HAZ ' A u t o l ITCT

A 5389/4109 HAZ l A u l o l ITCT

?5 50
Temperature, 'T

FIG. 8—Dynamic fracture toughness ofSA508 CI 2a base and heat-affected zone material.

Concerning SA508 CI 2a base metal, superior dynamic fracture tough-

ness was demonstrated by TO-5387, which also produced the lowest yield
strength, ultimate strength, and reference temperature. TO-5389 displayed
the lowest dynamic fracture toughness and ductility (see Fig. 1) plus the
highest drop weight NDT temperature.
The dynamic fracture toughness of SA508 CI 2a HAZ material manu-
factured utilizing automatic submerged-arc welding (4585/4109) sub-
stantially exceeded that of the corresponding manual weldment. The HAZ
of this manual weldment also demonstrated the poorest ductility and
Charpy impact properties. As was the case with tbe base metal, the HAZ
material from weldment 5389/4109 displayed the lowest dynamic fracture
toughness of the automatic submerged-arc weldments.
The SA508 CI 2a HAZ dynamic fracture toughness data exhibited rela-
tively large scatter. Some CT specimens demonstrated step-type crack fronts
as the fracture plane, which normally remained in the HAZ material,
searched out the path of least resistance. This typically produced higher
dynamic fracture toughness values than when the fracture plane was identi-
cal with that of the fatigue precrack.
Resistance curves relative to the single upper shelf dynamic fracture
toughness values generated at de^C (150°F) on SA508 CI 2a base metal
(rO-5387 and TO-5389) are illustrated in Fig. 9. Based on Madison and
Irwin's equation for estimating dynamic yield strength as a function of
temperature and test speed [6,13], the dynamic yield strengths of SA508 CI
 LOGSDON ON DYNAMIC FRACTURE TOUGHNESS 531

Crack Growth cm
0 .025 .05 .075 .10 .125
1600 1 1 1 1 1

1400 -
^^•'"""^ ^^.-^''^
1200 - D ^^--'''''^ -
1000 '^^^^ A ^^."""''''^

800 -""""^ A
fU Specimen
Symbol TO-Number Size

600 D 5387 ITCT

A 5389 ITCT

400 - / remperature = 150°F {66°CI

200

1 1 1 1 1 1 1 1 1
.02 .025 .03
Crack Growth, in

FIG. 9—Standard J resistance curves for SA508 CI 2a base material at a temperature of

66°CU50°F).

2a base material (TO-5387 and TO-5389) increased by an average of only

4.9 percent over the static yield strength values. Therefore, the blunting
lines in Figs. 9 and 10 were determined utilizing quasi-static yield and
ultimate strength values. Note the slopes (dJ/da) of these two resistance
curves (determined via least-squares linear regression) are nearly identical.
Further support relative to the dynamic resistance curve test technique
is demonstrated through Fig. 10, which illustrates modified resistance
curves developed on specimens loaded dynamically to failure over the upper
shelf temperature range of 24 to 66°C (75 to 150°F). The true upper shelf
dynamic fracture toughness of SA508 CI 2a should be nearly constant over
this small temperature range when determined via specimens which follow
the ductile tear initiation toughness curve of Fig. 5 (that is, via the standard
resistance curve test technique where specimens are loaded to specific dis-
placements). Since ductile tearing occurred in each of the tests included in
Fig. 10, the point on the load-deflection curves where fibrous crack growth
first initiated was not obvious, and calculating individual /m values was
impossible. As previously mentioned, each of the fracture surfaces displayed
a region of fibrous, ductile tearing immediately adjacent to the precrack
followed by an area of cleavage fracture. For comparison. Fig. 11 illustrates
representative fracture surfaces from a series of test specimens loaded to
specific displacements (heat tinted, TO-5387) and a series loaded dy-
namically to failure (TO-5389). Measuring the fibrous, ductile tearing
type crack growth on the dynamically failed specimens (three-point average)
and plotting it versus / (calculated based on the total area under the load-
532 ELASTIC-PLASTIC FRACTURE

Crack Growth (fibrous before cfeavag 1, cm

0 .05 .10 .15 .20 .25 .30
2200 [ 1 1 1 1 1
^(1501

2000 " -
(1501 ^ ^
1800
~
- .30

1600

1400
-
(125) ^ ^ ^ ^

1200

Specimen
-
/n('5)^;Sn25l Symbol TO-Number
1000 - / (751 n , ^
Size
• 5387 ITCT
/ ^/liooi / • A 5389 ITCT
800 - riiooip^ Brackets, ( ) = Test Temperature i n
Degrees fatirentie t
600

400 1 1 1 1 1 ! 1 1 1 1 1 1 1
.04 .05 .06 .07 .08 .09
Crack Growtti (fibrous before cleavage), in

FIG. 10—Modified J resistance curves for SA508 CI 2a base material.

deflection curve to abrupt failure) resulted in the modified resistance curves

pictured in Fig. 10. A nine-point average measure of fibrous crack growth
was impractical due to interaction of the shear lip formation with the crack
extension adjacent to the specimen precrack. Note again that there is little
heat-to-heat variation in resistance curve slope.
A direct comparison of the standard and modified resistance curves is
illustrated in Fig. 12. When the identical average measure of fibrous crack
extension is employed (three-point average), the standard and modified
resistance curves are essentially identical in terms of both critical J (JM)
and slope (dJ/dd). Dynamic fracture toughness values derived from both
the standard and modified resistance curves are compared in Table 4.
Fracture toughness values obtained from these totally independent re-
sistance curves are surprisingly similar. Obviously, no deceleration occurred
for the SA508 CI 2a tests loaded dynamically to failure, where the extent
of ductile growth was fortunately marked by a change in fracture mode.
The similarity of test results supports the contention that deceleration also
did not unduly affect dynamic fracture toughness values in dynamically
interrupted tests.

Conclusions
1. All dynamic fracture toughness values of ASME SA508 CI 2a base and
HAZ material exceeded the ASME specified minimum reference toughness
 LOGSDON ON DYNAMIC FRACTURE TOUGHNESS 533

TO-5%7
Specimens Loaded
Dynamically to
Specific
Displacements

TO-5389
Specimens Loaded
Dynamically to
Failure

-1*1 "'^S^r* -• ."V^. •

FIG. 11—Fracture surfaces from a series of specimens loaded dynamically to specific dis-
placements (TO-5387) and a series loaded dynamically to failure (TO-5389).

KIR curve. Therefore, this 450 MPa (65 Icsi) minimum yield strength ma-
terial is acceptable for nuclear pressure vessel structural applications from
a dynamic fracture toughness standpoint.
2. The dynamic fracture toughness, ductility, and Charpy impact properties
of SA508 CI 2a HAZ material manufactured utilizing automatic submerged-
arc welding substantially exceeded those of the corresponding manual
weldment.
3. Upper shelf temperature resistance curves obtained by the standard
multiple-specimen test technique (dynamically load each specimen to a
534 ELASTIC-PLASTIC FRACTURE

^ §

a.

=2

I I

I ^

•=,

I t
3

.J o
<
I O irt
1/1 t--
O O
00 00

3
Z
S5,?<^Sr^<?
6
 LOGSDON ON DYNAMIC FRACTURE TOUGHNESS 535

Crack Growth, cm
.05 .10 .15 .20 .25 .30
2200 I 1 1 1 1 1

2000 - .35
,'''
1800 - ^ .30
1600
Heat 5387 ^' ,•''
1400 .Z5~g

1200
_
.20 ^-
y 5 ^ . , ^ „ Heat 5389 Cracli Growth
1000 Measurements
" 1 y^'''^ ^J^ \ J
\AY' .15

x^
Per Speci men
800
,^ . i
— o— standard Resistance
Standard Resistance Curves|
CurveSj
9
3
600
i Modified Resistance Cu rves 3 .10
400 1 1 1 i' 1 1 1 1 1 I I
0 .01 .02 .03 .04 .05 .06 .07 ,08 .09 .10 .11 .12 13
Crack Growth, i n .

FIG. 12—Standard and modified J resistance curves for SA508 CI 2a base material.

Specific displacement and heat tint to mark the degree of stable crack
growth) and resistance curves obtained via specimens loaded dynamically
to failure (where a region of fibrous, ductile tearing adjacent to the pre-
crack was observable due to a change in fracture mode) were essentially
identical in terms of both critical / {Ju) and slope (dJ/da). Therefore, decel-
eration did not unduly affect dynamic fracture toughness values in dy-
namically interrupted tests.

References
[/] ASME Boiler and Pressure Vessel Code, American Society of Mechanical Engineers,
New York, 1974.
[2] PVRC Recommendations on Toughness Requirements for Ferritic Materials, Appendix 1,
Derivation of A'IR Curve, WRC Bulletin 175, Welding Research Council, Aug. 1972.
[3] Logsdon, W. A. and Begley, J. A. in Flaw Growth and Fracture, ASTM STP 631, Ameri-
can Society for Testing and Materials, 1977, pp. 477-492.
[4] Shabbits, W. C , "Dynamic Fracture Toughness Properties of Heavy Section A533
Gr B C! 1 Steel Plate," Technical Report No. 13, Heavy Section Steel Technology Pro-
gram, Dec. 1970.
[5] Bush, A. J. in Impact Testing of Metals. ASTM STP 466. American Society for Testing
and Materials, 1970, pp. 259-280.
[6] Paris, P. C , Bucci, R. J., and Loushin, L. L. in Fracture Toughness and Slow-Stable
Cracking, ASTM STP 559, American Society for Testing and Materials, 1974, pp. 86-98.
[7] Begley, J. A. and Landes, I. D. m Fracture Toughness. ASTM STP 514. American Society
for Testing and Materials, 1972, pp. 1-23.
[8] Landes, J. D. and Begley, J. A. in Fracture Toughness, ASTM STP 514, American
Society for Testing and Materials, 1972, pp. 24-39.
[91 Landes, J. D. and Begley, J. A. \n Fracture Analysis, ASTM STP 560, American Society
for Testing and Materials, 1974, pp. 170-186.
[10] Stahlkopf, K. E., Smith, R. E., Server, W. L., and Wullaert, R. A. in Cracks and
Fracture, ASTM STP 601, American Society for Testing and Materials, 1976, pp. 291-
311.
536 ELASTIC-PLASTIC FRACTURE

[11] Rice, J. R., Paris, P. C , and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing, ASTM STP 536, American Society for Testing and Materials, 1973,
pp. 231-245.
[12] Gillespie, E. H. and Pense, A. W., "The Fracture Toughness of High Strength Nuclear
Reactor Materials," Department of Metallurgy and Materials Science, Lehigh Uni-
versity, Bethlehem, Pa., March 26, 1976.
[13] Madison, R. B. and Irwin, G. R., Journal of the Structural Division, Proceedings of the
American Society of Civil Engineers, Sept. 1971, pp. 2229-2242.
R. L. Tobler' and R. P. Reed'

Tensile and Fracture Behavior of a

Nitrogen-Strengthened,
Chromium-Nickel-Manganese
Stainless Steel at Cryogenic
Temperatures*

REFERENCE: Tobler, R. L. and Reed, R. P., "Tensile and Fracture Behavior of a Ni-
trogen-Strengthened, Clironiiam-Niclcei-Manganese Stainiess Steel at Cryogenic Tem-
peratures," Haiftc-Ptorfc Frarture, ASTMSTP668, J. D. Landes, J. A. Begley, and G.
A. Clarke, Eds., American Society for Testing and Materials, 1979, pp. 537-552.

ABSTRACT: J-integral fracture and conventional tensile properties are reported for an
electroslag remelted Fe-21Cr-6Ni-9Mn austenitic stainless steel that contains 0.28 per-
cent nitrogen as an interstitial strengthening element. Results at room (295 K), liquid-
nitrogen (76 K), and liquid-helium (4 K) temperatures demonstrated that the yield
strength and fracture toughness of this alloy are inversely related and strongly
temperature dependent. Over the investigated temperature range, the yield strength
tripled to 1.24 GPa (180 ksi) at 4 K. The fracture toughness, as measured using 3.8-cm-
thick (1.5 in.) compact specimens, decreased considerably between 295 and 4 K. During
plastic deformation at 295 K the alloy undergoes slight martensitic transformation, but
at 76 and 4 K it transforms extensively to martensites. The amount of body-centered
cubic (bcc) martensite formed during tension tests was measured as a function of elonga-
tion.

KEY WORDS: cryogenics, fracture, low-temperature tests, martensitic transformations,

mechanical properties, stainless steel alloys, crack propagation

Recently, austenitic stainless steel strengths have been increased con-

siderably by the substitution of nitrogen and manganese for nickel. In addi-
tion to providing interstitial and solid solution strengthening, these elements
serve to increase austenite stability with respect to martensitic transforma-
tions. Compared with nickel, these elements are more abundant and less ex-
pensive. The alloy studied in this report, Fe-21Cr-6Ni-9Mn-0.3N (21-6-9),
•National Bureau of Standards contribution, not subject to copyright.
' Metallurgist and section chief, respectively. Cryogenics Division, Institute for Basic Stan-
dards, National Bureau of Standards, Boulder, Colo. 80302.

537

Copyright 1979 b y A S T M International www.astm.org

538 ELASTIC-PLASTIC FRACTURE

has a room temperature yield strength nearly twice that of AISI 304.
Available tensile and impact data [1-4]^ suggest that the 21-6-9 alloy retains
good toughness at low temperatures, leading to consideration of its use for
applications benefiting from high strength and toughness.
Accordingly, 21-6-9 is currently being considered for such critical com-
ponents as the coil form for the prototype controlled thermonuclear reaction
superconducting magnets and the torque tube for rotating superconducting
machinery. To insure satisfactory service life and to compare with other can-
didate materials, it is necessary to evaluate the fracture resistance of the
alloy. This study presents the first fracture toughness data for this alloy.

Material
The electroslag remelted 21-6-9 austenitic stainless steel plate was pro-
cessed and donated by Lawrence Livermore Laboratories, Livermore, Calif.
The chemical composition (in weight percent) of this heat is 19.75Cr-7.16Ni-
9.46Mn-0.019C-0.15Si-0.004P-0.003S-0.28N. This steel was soaked at 1366
K for 4 h, then cross-rolled from 30.5 by 30.5 by 10-cm (12.2 by 12.2 by 4-in.)
slabs to 50 by 50 by 3.6-cm (20 by 20 by 1.44-in.) plate. Rolling was com-
pleted in 12 steps, using five 90-deg rotations. The final plate temperature
after this hot rolling was 1089 K. Each plate was then annealed at 1283 K for
1 Vi h and air cooled, followed by an anneal at 1366 K for 1V2 h and a water
quench. The resultant hardness was Rockwell B92 and the average grain
diameter was 0.16 mm (0.0064 in.).

Procedure

Tensile
Tension specimens were machined following the ASTM Standard Methods
of Tension Testing of Metallic Materials (E 8-69). Thfe reduced section diam-
eter was 0.5 cm (0.1 in.) and gage length was 2.54 cm (1 in.). The tension
axis was oriented transverse to the final rolling direction. Tests were per-
formed at a crosshead rate of 8.3 X 10"" cm/s, using a 44.5-kN (10 000 lb)
screw-driven machine that was equipped with the cryostat assembly designed
by Reed [5]. The tests at 295 K were conducted in laboratory air, whereas
tests at 76 and 4 K used liquid nitrogen and liquid helium environments,
respectively. Load was monitored with a commercial load cell while specimen
strain was measured with a clip-on, double-beam, strain-gage extensometer.
Yield strength was determined as the stress at 0.2 percent offset plastic
strain.

^The italic numbers in brackets refer to the list of references appended to this paper.
 TOBLER AND REED ON TENSILE AND FRACTURE BEHAVIOR 539

Magnetic
To detect the amount of ferromagnetic, body-centered cubic (bcc) marten-
sitic phase in the paramagnetic, face-centered cubic (fee) austenitic matrix, a
simple bar-magnet torsion balance was used [6], Previous measurements on
iron-chromium-nickel (Fe-Cr-Ni) austenitic steels established a correlation
between the force required to detach the magnet from the specimen and the
percent bcc martensite [6]. The same correlation was used for this study to
estimate the amount of bcc martensite in the iron-chromium-nickel-
manganese (Fe-Cr-Ni-Mn) alloy.

Fracture
The J-integral specimens were 3.78-cm-thick (1.488 in.) compact
specimens of a geometry described in the ASTM Test for Plane-Strain Frac-
ture Toughness of Metallic Materials (E 399-74). The specimen width, W,
and width-to-thickness ratio, W/B, were 7.6 cm (3.0 in.) and 2.0, respec-
tively. Other dimensions are shown in Fig. 1. The notch, machined parallel
to the final rolling direction of the plate, was modified to enable clipgage
attachment in the loadline.
The J-integral specimens were precracked at their test temperatures, using
a 100-kN (22 480 lb) fatigue testing machine and cryostat [7]. All fatigue
operations were conducted using load control and a sinusoidal load cycle at
20 Hz. Maximum fatigue precracking loads (P/) were well below the max-
imum load of/tests (Pma), as indicated in Table 1. The maximum stress in-

1.9ein D i a .

FIG. 1—Compact specimen for fracture testing of Fe-21Cr-6Ni-9Mn alloy (1 cm = 0.4 in.).
540 ELASTIC-PLASTIC FRACTURE

TABLE 1—Precracking parameters for i-integral test specimens.

Test Pf/Pmn Kf. Relative Crack

Temperature, K X 100, % MPam'^' Length, a / W ae/a

295 40 to 45 48 to 54 0.638 0.90 to 0.91

76 22 to 27 52 to 63 0.640 0.87 to 0.90
4 30 to 35 52 to 63 0.64 to 0.795 0.88 to 0.89

tensities during precracking {Kf), thefinalrelative crack lengths (a/ W), and
the edge-crack-to-average-crack-length ratios (a^/a) at each temperature are
also listed in Table 1. After precracking, the specimens were transferred to a
267-kN (60 000 lb) hydraulic tension machine for fracture testing. Thus, the
76 and 4 K fracture specimens were warmed to room temperature between
precracking and / testing at 76 and 4 K. This was necessary since the load
limitations of the 100-kN (22 480 lb) fatigue machine precluded loading this
alloy to fracture at low temperatures.
The J-integral tests followed a resistance curve technique similar to that
described originally by Landes and Begley [8]. A series of nearly identical
specimens was tested at each temperature. Each specimen was loaded to pro-
duce a given amount of crack extension. The specimens were then unloaded
and heat tinted or fatigued a second time to mark the amount of crack exten-
sion associated with a particular value of/. The oxidized zone of crack exten-
sion (including blunting, plus material separation) could be identified and
measured after fracturing the specimen into halves.
Using the approximation for deeply cracked compact specimens [9]

J = 2A/B(W-a) (1)

the value of J for each test was calculated from the total area, A, under the
load-versus-deflection record. The values of J obtained at each temperature
were plotted versus crack extension, Aa, which was measured at five loca-
tions equidistant across the specimen thickness, and averaged.
The critical value of the J integral, Jic, defined as the / value at the initia-
tion of crack extension, was obtained by extrapolating a reasonable fit of the
J-Aa curve to the point of actual material separation. An estimation of the
plane-strain fracture toughness parameter, denoted KidJ), was made
using [8]

K,,V)^- ;(/.c) (2)

where E is Young's modulus and v is Poisson's ratio. At room temperature,

E = 195 GPa (28 306 ksi) and i> = 0.287; at 76 and 4 K, £• = 203 GPa
 TOBLER AND REED ON TENSILE AND FRACTURE BEHAVIOR 541

(29 467 ksi) and v - 0.278, according to Ledbetter's measurements by an

acoustic technique [10],

Results and Discussion

Tensile
The yield and tensile strengths, elongation, and reduction of area were ob-
tained for the 21-6-9 alloy at 295, 76, and 4 K. These data are summarized in
Table 2. The results from this study are combined in Figs. 2-4 with the un-
published results of Landon [/] for the same heat, also hot rolled and an-
nealed, and with the results of Scardigno [2], Malin [3], and Masteller [4] for
annealed bar stock. The spread of the Malin data represents results from
both the longitudinal and transverse specimen orientations. Agreement is
very good, except that the ultimate-strength data of Masteller are consis-
tently higher than the average of the other data.
Typical stress-strain curves at each temperature are presented in Fig. 5.
The pronounced discontinuous yield behavior at 4 K probably is associated
with adiabatic specimen heating of the type described by Basinski [//]. Note
that at 4 K the materials' specific heat is very low so that plastic deformation
may cause significant heat evolution. Significant local heating is indicated,
as theflowstress drops to stress levels less than sustained at 76 K. These load
drops should not be attributed to martensitic phase transformations, for
three reasons: (1) More extensive transformation was detected in this alloy at
76 K than at 4 K (see later discussion) and no discontinuities in the stress-
strain mode at 76 K were observed; (2) load drops have been observed in both

TABLE 2—Tensile properties ofFe-2ICr-6Ni-9Mn alloy.

Yield Strength, Elongation,

0.2% Offset, Tensile Strength, 2.5-cm Gage Reduction of
Temperature, K GPa GPa Length, % Area, %

295 K 0.350 0.696 61 79

0.357 0.705 61 78
average 0.353(51 ksi") 0.701 (102 ksi) 61 78

76 K 0.913 1.462 42 32
0.886 1.485 43 41
average 0.899 (130 ksi) 1.474 (214 ksi) 43 37

4K 1.258 1.633 16 40
1.224 1.634 NA* NA
average 1.241 (180 ksi) 1.634 (237 ksi) 16 40

"1 ksi = 6.894 X 10~3GPa.

*NA = inot available.
542 ELASTIC-PLASTIC FRACTURE

2.0
V Tensile O This Study
Yield A Scardigno (1974)
V
1.6 D London ( 1 9 7 5 )
V Mosteller ( 1 9 7 0 )
o
a.
O 1.2
4 ^^\ I Molin (1970)

0.8 -

-
0.4

I I I . 1 1 1 1 1 1 1
SO 100 150 200 250 300
TEMPERATURE, K

FIG. 2—Summary of tensile and yield strength data as a function of temperature for the Fe-
21Cr-6Ni-9Mn alloy (1 GPa = 145.16 ksi).

V Mosteller ( 1 9 7 0 )
0 This Study
A Scardigno ( 1 9 7 4 )
• London (1975)
1 Molin (1970)

100 150 200 300

TEMPERATURE, K

FIG. 3—Summary of tensile elongation as a function of temperature for the Fe-21Cr-6Ni-

9Mn alloy.

metastable (for example, AISI 304) and stable (for example, AISI 310)
austenitic stainless steels at 4 K and no distinction is apparent between the
two alloy groups [12]; and (3) in austenitic steels the amplitude and fre-
quency of the load drops at 4 K are a function of the strain rate [12] which
would be expected if local heating were responsible.
Another indication of significant local heating is the rise of the reduction
of area to values higher than obtained during 76 K tests. Specimens tested at
 TOBLER AND REED ON TENSILE AND FRACTURE BEHAVIOR 543

100 150 200 250 300

TEMPERATURE, K

FIG. 4—Summary of tensile reduction of area as a function of temperature for the Fe-21Cr-
6Ni-9Mn alloy.

FIG. 5—Stress-strain curves for the Fe-21Cr-6Ni-9Mn alloy at 295, 76. and 4 K (1 GPa
145.16 ksi).
544 ELASTIC-PLASTIC FRACTURE

4 K developed very local areas of increased plastic deformation, which

resulted in sizable specimen necking prior to fracture. From Figs. 3 and 4, it
is clear that the 21-6-9 alloy shows a significant decrease of ductility below
195 K, and tensile elongation decreases progressively between 195 and 4 K.
A primary advantage offered by this alloy is its high yield strength com-
pared to other austenitic alloys. At room temperature the yield strength of
the 21-6-9 alloy is about 0.38 GPa (55 ksi), compared with AISI 300 series
(Fe-Cr-Ni) steel values of 0.21 to 0.25 GPa (30 to 35 ksi). The yield strength
of the 21-6-9 steel approximately triples to a value of 1.24 GPa (180 ksi) as
the temperature is decreased to 4 K. Similarly, the Fe-Cr-Ni austenitic alloys
achieve values about double or triple their room temperature values of 0.42
to 0.76 GPa (60 to 110 ksi) at 4 K. Therefore, the strength advantage offered
by the 21-6-9 alloy is greatest at low temperatures.

Fracture
The load-versus-load-line deflection curves for compact specimens at 295,
76, and 4 K are shown in Fig. 6. The curves at 295 K extended to larger
deflections than indicated on the axis of the diagram. The fracture test data
are tabulated in Table 3. At no temperature could valid/fic data be measured
according to ASTM E 399-74. The 5 percent secant offset data are denoted
KQ because the thickness and crack front curvature criteria were not satis-
fied. Using5 > 2.5 (KQ/oyY, a specimen thickness of 4.2 cm (1.7 in.) at 4 K
is required, slightly larger than the 3.8-cm (1.5 in.) thickness tested. The
crack front curvatures shown in Fig. 7 are also excessive. The surface crack
lengths are 88 to 89 percent of the average of internal crack lengths, whereas
90 percent is specified in ASTM E 399-74 as the minimum deviation.
The /-versus-Aa results at room temperature are plotted in Fig. 8. Ductile
tearing (slow, stable cracking) occurred at this temperature, and large ap-
parent crack extensions were observed due to crack-tip deformation. Only in
two specimens at the highest values of Aa was actual material separation
noted. These two values fall on the same trend line as the specimen data that
did not exhibit material separation. Furthermore, the recommended blunt-
ing line, / = 2Aaa/, does not match the experimental trend. Therefore the
response of this extremely ductile material to J-integral tests at room tem-
perature is inconclusive, with no well-defined /ic measurement point observ-
able.
The room temperature behavior may result from failure to meet the / test
specimen size criterion. According to the tentative criterion suggested by
Landes and Begley [8], the specimen thickness for valid/ic measurements
should satisfy the relationship

B > a{J/af) (3)

where a is 25 and Of is the average of the yield and tensile strengths. In the
 TOBLER AND REED ON TENSILE AND FRACTURE BEHAVIOR 545

200
1 1 1 1

— ^76K _
150 —

/ ^ ^ \ 4 K
100 —
O
<
O
/ 295K
' ^
50 —

Compact Specimens, B = 3.8cm

a/W S 0.64

1 1 1 1
.1 .2 .3 .4 .5

LOADLINE DEFLECTION, cm

FIG. 6—Typical load-deflection curves for compact specimens at 295, 76. and 4 K for an-
nealed Fe-21Cr-6Ni-9Mn alloy (1 kN = 224.8 lb: 1 cm = 0.4 in.).

tests at 295 K, we tentatively estimate the critical / values to be in the range

925 to 1350 kJ-m-2 (5285 to 7714 in.-lb-in.-2). Using the flow stress value
of 0.527 GPa (76.5 ksi), the J-integral results at room temperature are invaUd
for the specimen thickness tested here. A specimen thickness of 6.3 cm (2.5
in.) may be needed to insure valid data, according to Eq 3.
The /-resistance curve at 76 K is also shown in Fig. 8. The data fit a
regular trend, with the exception of the point representing the largest ob-
served crack extension (not shown). The curve drawn through the remaining
data indicates that crack extension initiates at a/ic value of about 340 kJ • m"^
(1943 in. -Ib-in."^). The corresponding value oiKiJi.!), estimated using Eq 2,
is 275 MPa-m''2 (250 ksi-in.'^^).
At 4 K, the alloy approached linear-elastic behavior, but the results of the
first three tests failed to satisfy the ASTM E 399-74 validity criteria for direct
Ku measurements. Consequently, eight additional / tests were conducted
and these results are included in Fig. 8. The /-Aa curve at 4 K is nearly hori-
zontal, indicating a / k value of about 150 kJ-m"^ (857 in.-lb-in.~^); the
546 ELASTIC-PLASTIC FRACTURE

TABLE 3—Fracture results for 3.8-cm-thick U.5 in.) compact specimens of Fe-21Cr-6Ni-9Mn
alloy.

Temperature, KQ" Aa.

K a/W MPam'^2 kJm"2 cm kJm"2

295 0.638
0.636
58
61
177
744
0.013*
0.051*
]
0.640 55 905 0.069* L between 905
0.635 63 1355 0.097'' [ and 1355
0.642 50 1423 0.112''
J
76 0.612 134 261 0.0 ^
0.634 153 413 0.028
0.640 131 499 0.053
0.637 137 674 - 340
0.079
0.645 130 788 0.091
0.643 130 698 0.198^

4 0.645 164 NA NA ">

0.648 162 NA NA
0.643 159 NA NA
0.670 NA 100 0.0
0.655 167 147 0.020
0.670 158 149 0.080 >. 150''
0.656 160 162 0.076
0.750 NA 89 0.0
0.725 NA 191 0.0313
0.725 NA 274 0.105
0.755 NA 141 0.033 ^

"Calculated from ASTM E 399-74.

* Apparent crack extension due to crack-tip deformation only.
''Crack extension due to deformation and material separation.
''Incorrect/ic values were reported for this alloy in Ref 13, due to transcribing errors.
NOTES: 1 in-lbin."^ = 0.175 kj-m"^; 1 in. = 2.54 cm; 1 ksiin.'^^ = 1.099 MPam'^^;
NA = not available.

KiJiJ) estimate from Eq 2 is 182 MPa-m'^^ (165 ksi-te.'^^). Data comparison

with other alloys is made in Fig. 9, which indicates tnat the 21-6-9 alloy offers
relatively high toughness for its strength level. Therefore it is attractive for
some applications at temperatures as low as 4 K.

Phase Transformations
After tension tests at 76 and 4 K, the deformed specimens were magnetic.
Therefore, these specimens were measured, using bar-magnet torsion
balance equipment [5], to correlate magnetic attraction with specimen
reduction of area. The magnetic readings were converted to percent bcc
martensite and the reduction of area converted to elongation, assuming con-
stant volume. These data are plotted in Fig. 10. Typical microstructures are
shown in Fig. 11.
TOBLER AND REED ON TENSILE AND FRACTURE BEHAVIOR 547

7^
*
f
1

CSI
548 ELASTIC-PLASTIC FRACTURE

1600

1400

1200

1000 —

Aa, cm
FIG. 8—The i-integral as a function of crack extension at 29S, 76, and 4 Kfor annealed alloy
Fe-21Cr-6Ni-9Mn (1 in. Win. ~^ = 0.175 kj-m~^; 1 cm = 0.4 in.).

Although not positively identified, it is probable that hexagonal close-

packed (hep) martensite also formed in the 21-6-9 alloy during low-tempera-
ture deformation. The microphotographs after tensile deformation at 4 K
show transformed regions which are parallel to the {111} slip band traces.
These appear identical to the hep areas identified in earlier research on AISI.
304, an Fe-Cr-Ni alloy [6.12].
The amount of bcc martensite formed is large and only slightly less than
that which is formed in AISI 304 at the same temperatures [6,12]. Perme-
ability values of the order of 10 were measured in heavily deformed specimen
portions at 76 K, but it is difficult to identify bcc martensite in the Fig. 11
photomicrographs. Normally, in austenitic stainless steels the bcc marten-
sitic product has an acicular, plate-like morphology with the habit plane of
the plate not {111}. Examination of specimen microstructures, typified by
Fig. 11, indicate that only at {111} band intersections are the distinctive
plate-like microstructures observed.
 TOBLER AND REED ON TENSILE AND FRACTURE BEHAVIOR 549

lU" I I 1 1 1 1 1 11 1 1 I I I 1

i T = 295. 4K
-

0 • Ti Alloys
-
A • Al Alloys
7 • INC Alloys
D
D \ a • Steels
V N
D D
CO
CO \ ^ 2 1 6 9 alloy, T=4K
D
io2 - -
D
O
- • ^Optimum Properties -
_ • v^ _
- V °\ -
- ^ A ^ -
< A
•
0

• • \
-1
10'
-
1 1 1 1 1 1 1I 1 i\ 1 1 1 1 "
10'^ 10-2

STRENGTH LEVEL,<ry/E

FIG. 9—Mechanical properties of 21-6-9 alloy at 4 K. as compared with data for other alloys
from Ref 13 (/ in. -Win. '^ ^0.175 kJ-m'^).

There is clear evidence that the amount of the transformation is sup-

pressed, as a function of either stress or strain, as temperature is lowered
from 76 to 4 K. This is similar to the Fe-Cr-Ni (AISI 304) alloy martensitic
transformation behavior [6,12], where formation of the hep martensitic
phase was suppressed at temperatures between 20 and 4 K. Apparently, in
the complicated energy balance affecting martensitic transformation for
these alloy systems at low temperatures, the increase of flow stress and the
decrease of dislocation mobility more than offset the gradually increasing
free energy difference between the structures.
It is not clear that martensitic transformations are deleterious to material
application. Normally, the stress levels used in service are less than the yield
strength, and no martensitic transformations should occur. The complexities
and concern usually are discussed when one considers welds and weld tech-
niques. Chemical segregation and stress concentrations are then more likely,
rendering particular sections less stable and, locally, stressed above the yield
strength. In these situations martensitic products will form.
AISI 304 behaves in a similar manner; it is stable on cooling to low tem-
peratures but transforms to hep and bcc martensitic products during plastic
deformation. But, unlike 21-6-9 alloy, the fracture toughness of annealed
550 ELASTIC-PLASTIC FRACTURE

20 40 60

ELONGATION, percent
FIG. 10—Estimated percent hcc martensite that forms during tension tests as a function of
tensile elongation.

AISI 304 remains extremely high at 4 K [14], implying that martensitic

transformations are not necessarily detrimental to fracture toughness. This is
less certain in the case of the Fe-Cr-Ni-Mn-N alloy, however, where the
toughness rapidly decreases between 76 and 4 K. For appropriate safety of
operation at 4 K, additional research is necessary to understand the effect of
martensitic transformations on the fracture toughness of stainless steels.

Conclusions
1. The fracture toughness of the 21-6-9 austenitic stainless steel exhibits
an adverse temperature dependence between 295 and 4 K, but retains a
respectable Jic toughness of 150 kJ-m"^ (857 in.-lb-in.""2) at 4 K. Linear-
elastic behavior was approached at 4 K.
2. The yield strength of the 21-6-9 alloy is also strongly temperature
 TOBLER AND REED ON TENSILE AND FRACTURE BEHAVIOR 551

(a) (b)
FIG. 11—Microstructures of alloy 21-6-9 after deformation at 4 K. Bands lie on \111\
austenitic planes and probably represent hep and bcc martensite: (a) X 440, (b) X 660.

dependent, tripling between room temperature and 4 K, and reaching a

value of 1.24 GPa (180 ksi) at 4 K.
3. During plastic deformation at 76 and 4 K, bcc martensite was identi-
fied in increasing amounts as a function of strain. Suppression at 4 K, com-
pared with 76 K, of the amount of bcc martensite was found.

Acknowledgments
The authors thank D. P. Landon, Lawrence Livermore Laboratories, for
supplying the test material. Dr. R. P. Mikesell conducted the tension tests,
R. L. Durcholz contributed technical assistance to tension, fracture, and
metallographic preparation, and Dr. M. B. Kasen provided the photomicro-
graphs.

References
[/) Landon, P. R., Unpublished data, Lawrence Livermore Laboratories, Livermore, Calif.,
1975.
[2] Scardigno, P. F., M.Sc. degree thesis, Naval Postgraduate School, Monterey, Calif.,
AD/A-004555, 1974.
[3\ Malin, C. O., NASA SP-5921(01), Technology Utilization Office, National Aeronautics
and Space Administration, Washington, D.C., 1970.
552 ELASTIC-PLASTIC FRACTURE

[4] Masteller, R. D., NASA CR-72638(N70-27114), Martin Marietta Corp., Denver, Colo.,
1970.
[5] Reed, R. P. in Advances in Cryogenic Engineering, Vol. 7, K. D. Timmerhaus, Ed.,
Plenum Press, New York, 1962, p. 448.
[6] Reed, R. P. and Guntner, C. J., Transactions, American Institute of Mining Engineers,
Vol. 230, 1964, p. 1713.
[7] Fowlkes, C. W. andXobler, R. L., Engineering Fracture Mechanics, Vol. 8, 1976, p. 487.
[8] Landes, J. D. and Begley, J. A. in Fracture Analysis, ASTM STP 560, American Society
for Testing and Materials, 1974, pp. 170-186.
[9] Rice, I. R., Paris, P. C , and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing, ASTM STP 536, 1973, pp. 231-245.
[10] Ledbetter, H. M., Materials Science and Engineering, Vol. 29, 1977, p. 255.
[//] Basinski, Z. S., Proceedings of the Royal Society, London, England, Vol. A240, 1957,
p. 229.
[12] Guntner, C. J. and Reed, R. P., Transactions, American Society for Metals, Vol. 55,1962,
p. 399.
[13] Tobler, R. L. in Fracture 1977, D. M. R. Taplin et al, Eds., University of Waterloo Press,
Waterloo, Ont., Canada, 1977, p. 839.
[14] Reed, R. P., Clark, A. F., and van Reuth, E. C , Eds., Materials Research for Super-
conducting Machinery III, AD-A012365/3WM, National Technical Information Service,
Springfield, Va., 1975.
W. H. Bamford' and A. J. Bush'

Fracture Behavior of Stainless Steel

REFERENCE: Bamford, W. H. and* Bush, A. J., "Fracture Behavior of Stainless

Steel," Elastic-Plastic Fracture, ASTM STP 668, J. D. Landes, I. A. Begley, and G.
A. Clarke, Eds., American Society for Testing and Materials, 1979, pp. 553-577.

ABSTRACT: An experimental program has been carried out to characterize the

fracture properties of austenitic stainless steel piping and plate material. Characteriza-
tion was in terms of the J-integral, and several specimen types were tested, including
compact specimens, center-cracked panels, and three-point bend specimens.
Several methods of monitoring crack extension were used in the program, including
unloading compliance, electrical potential, and acoustic emission in addition to the
multiple-specimen heat tinting method used for baseline data. These methods are
compared and evaluated in detail.
In addition to determining Jic values for the material, Paris's proposed tearing
modulus is evaluated, and various proposed specimen size requirements are discussed.

KEY WORDS: fracture properties, stainless steel, J-integral, toughness, piping,

tearing modulus, compliance, crack propagation.

The fracture behavior of reactor coolant piping is an important considera-

tion in assessing the integrity of a nuclear reactor system. The piping of
interest here is very large—73.66 to 83.82 cm (29 to 33 in.) in diameter and
5.08 to 7.62 cm (2 to 3 in.) in thickness. The piping carries an internal
pressure of 15.52 MPa (2250 psi) and is subject to various thermal and
bending loadings as well.
This reactor coolant piping is manufactured of stainless steel—either
forged or centrifugally cast. Because of its extensive ductility, quantitative
characterization of the fracture properties of stainless steel has not been
possible until recently, with the development of the J-integral. A test pro-
gram was carried out to characterize the Jic properties of two types of stain-
less steel piping material. Three different specimen types were tested, and
data were prepared in accordance with recommended ASTM procedures
[1].^ In addition, several methods of monitoring crack extension were
evaluated, and a brief discussion is provided on size requirements. Ahhough
' Senior engineer, Westinghouse Nuclear Energy Systems, and senior engineer, Westinghouse
R&D Laboratories, respectively, Pittsburgh, Pa.
^The italic numbers in brackets refer to the list of references appended to this paper.

553

Copyright' 1979 b y A S T M International www.astm.org

554 ELASTIC-PLASTIC FRACTURE

the majority of the data were obtained for piping materials, additional tests
were performed on 304 stainless steel plate material, which showed equiva-
lent results for/ic and leads to the conclusion that the results of this program
apply to 304 and 316 stainless steel in general.
Because of its extensive ductility, stainless steel is a particularly good
material for evaluating the adequacy of the proposed /ic testing methods.
Comments are made on test methods, data presentation, and validity
criteria.

Experimental Program

Materials and Specimens

Three materials were tested in the program, two types of stainless steel
reactor coolant piping and one heat of 304 stainless steel plate.^ The majority
of the specimens tested were machined from production heats of reactor
coolant piping material, one of forged 304 and the other of centrifugally
cast 316 stainless steel. The chemistry and heat treatment of the materials
are given in Tables 1 and 2.
Three specimen types were tested, as shown in Fig. 1. These were com-
pact specimens, three-point bend specimens, and center-cracked panels.
Most of the specimens were 5.08 cm (2 in.) thick, with only a few of the
compact specimens 2.54 cm (1 in.) in thickness. The tests were conducted
at room temperature and 316°C (600°F). A summary of the combinations
of materials, geometries, and test conditions is provided in Table 3.
Compact specimens were machined with cracks oriented in both the
axial and circumferential directions for both the piping materials, and for
the forged piping 2.54-cm (1.0 in.) compact specimens were oriented with
the crack propagating in the through-thickness direction. No directional
affects were observed for the cast piping material, so through-thickness
tests were not done. Because of size limitations off the actual piping, the
three-point bend specimens and center-cracked panels were all machined
with the crack propagating circumferentially. The specimen orientations
are shown in Fig. 1.
All the specimens were precracked in air at room temperature prior to
testing, following the guidelines of ASTM Test for Plane-Strain Fracture
Toughness of Metallic Materials (E399-74). The precracking was done with
a sinusoidal tension-tension loading in all cases except one. To minimize
the crack front curvature of the precrack in one of the bend bars, it was
precracked in compression. The crack front produced was much straighter,
but since the technique appeared to influence the results (as seen in Fig.
14), it was discontinued.

^This plate was supplied by L. A. James of Hanford Engineering Development Laboratory.

 BAMFORD AND BUSH ON STAINLESS STEEL 555

00
in
d d

•5

3 £

I o s
ON
O
d d d

o
o
d d
1-1
a
<
00

d d

E
9 (J
2 _ B <V •o
o

-II rs so":

< 4=
U
M

<
U tt. m
556 ELASTIC-PLASTIC FRACTURE

I 2
en c

.0 . n
006^
§

! | '

5.

I iP
o a>

I h^ 05 n

CM

U
.J

u ^ u

sa
:'?i?2-S?
ill
'.O'^ u
s„
2!8 z "3.

l5 sQ £ -H OS

(b
u <
 BAMFORD AND BUSH ON STAINLESS STEEL 557

-Axial Specimen

^
1 [ 1
3-Polnt Bend Bar 0.5,0.6
W
FIG. 1—Specimen geometries and orientation (1 in. = 2.54 cm).

Test Apparatus and Procedure

Because of the high ductUity of stainless steel in the temperature range
of interest, the fracture properties are best discussed in terms of an elastic-
plastic parameter, and for this study the J-integral was chosen. Efforts
were made to determine Ju'in a manner consistent with the recommended
procedure under development presently by ASTM [/]. This procedure in-
volves the determination of Ju for the material and condition of interest
558 ELASTIC-PLASTIC FRACTURE

S
0^ O Ov r-» 00 fO
^
•^
O
o
^
I/)
fO
r^
O
fo
0^
<N
8
•^ ^ (N fs r4 TT
A

[ _ f - , 0 0 0 ( _ U H H ° ° °

a.
s Si

1 1 i 11 I i
•2 is •o
•5
Sc g

I f 1 •y I
e
1^ (n •s
k,
5t
-k*
^
Q ro (N ro ^H (N rrt
H a
•t.u
•^
1
XUJ
J
w
e
_o
<• 13
H e ill
•a^-g
•if
illllllll
u
•c
o

i!tl
•SSaJI

Mill
A e
.s ji
V
'H &
s
u 13
I fill
-s 'og
•o
V S
Se Q.
fcri

£
M

a
3 •s
vO
^
(/5
(A
nil
s
 BAM FORD AND BUSH ON STAINLESS STEEL 559

by plotting of / versus subcritical crack extension. The value of / ^ was

determined from this plot at the point corresponding to zero apparent crack
extension. Producing such a plot of/versus crack extension can be accom-
plished by testing multiple specimens to produce different amounts of
stable crack growth, and heat tinting the specimens to mark the crack.
It is also possible to produce such a plot from a single specimen, provided
a reliable method can be found for determining crack length without
breaking the specimen.
All specimens were tested in electrohydraulic test machines, with both
load and load line displacement recorded during the test. In addition, the
experimental apparatus included several methods for crack length deter-
mination, including acoustic emission, electrical potential, and elastic
compliance. The electric potential and compliance methods were used with
the three-point bend tests, and the acoustic emission and electric potential
methods were used in conjunction with the center-cracked panel tests. Al-
though these methods were investigated, the primary method used for ob-
taining data was the multiple-specimen technique, where a series of speci-
mens was tested to different amounts of crack extension, and the specimens
were then heat tinted and broken apart so the stable crack extension could
be measured.
Heat Tinting—Since the heat tinting method has long been successful,
it was used to obtain baseline data. In the heat tinting method, multiple
specimens are used in which specimens are (1) loaded to some predeter-
mined displacement to obtain an estimated crack length Aa, (2) unloaded,
(3) heat tinted to mark the crack advance, and finally (4) cooled and broken
apart by further loading in order to expose the heat tinted surface for the
actual measurement of Aa. In the 24°C (75°F) series of tests, heat tinting
was done by placing the specimens in a furnace at 316 °C (600 °F) for a
minimum of 4 h, and for the 316°C (600°F) series the heat tinting was
done at 427°C (800°F) for a minimum of 4 h.
Loads were measured with a combination load cell-loading tool for the
bend tests, while the center-cracked panel and compact specimen tests
employed the testing machine load cell outputs directly. Displacements
were measured at the centerline of the loading tool to eliminate bending
effects for the bend tests, and by a clip gage mounted across the center of
the crack for the center cracked panels. The compact specimens employed
a clip gage mounted over the specimen front to read load line displacement.
Electrical Potential—To explore other ways of determining crack initia-
tion and advance in the stainless steel material, the electrical potential
method was also tried. This method is shown schematically in Fig. 2 [/].
In this technique a constant current is applied to the specimen during
loading. As the crack advances, the resistance of the specimen increases
and the change is measured as an increase in the electrical potential. In
the present tests both load and displacement versus electrical potential
560 ELASTIC-PLASTIC FRACTURE

DISPLACEMENT

FIG. 2—Electrical potential method for measuring crack advance.

were recorded simultaneously and a record of the curves obtained is shown

in Figs. 3£> and 3c. The region for the start of crack advance as determined
from the heat tinting tests already described is shown on the curves. As
Figs. 3i and 3c show, there is no clearly defined correlation between the
electrical potential curve slope changes and crack initiation for these mate-
rials. Use of the electric potential method for this test would have resulted
in an implied J^ value much lower than the true value, as shown in Fig. 3c.
Similar results were obtained when the technique was applied to the
center-crack panel test. In this case the implementation was somewhat
more difficult, because the loading pins were electrically insulated from
the specimen. A plot of electrical potential versus load for Specimen SW-35
is provided in Fig. 4. This specimen was tested to failure, and again the
electrical potential output underpredicted the onset of crack extension by
a considerable amount, as shown in the figure.
The shortcomings of the electrical potential method in predicting the
onset of crack extension for these materials were not altogether unexpected,
because the extensive plasticity developed in the specimens alters the resis-
tivity of the material, and this probably led to the premature signal.
Acoustic Emission—The acoustic emission method was used to attempt
to determine the onset of crack extension for one center-cracked panel,
Specimen SW-35. The method relies on the fact that crack growth results
in the release of energy, some of which is in the acoustic frequency range.
High ultrasonic frequencies are generally measured, to minimize inter-
ference from rubbing and other mechanical sources of noise. In spite of
this, mechanical interference is a significant problem.
The test setup is shown schematically in Fig. 5. An acoustic emission
sensor was mounted on the face of the center-cracked panel above the
 BAMFORD AND BUSH ON STAINLESS STEEL 561

FIG. 3—Electrical potential test results for three-point bend specimen Cl-68 (1 in. = 2.54
cm; 1lb = 0.4536 kg; 1 lb/in. 2 = 6.895 kPa).
562 ELASTIC-PLASTIC FRACTURE

Load (KIPS)

250 \.
/l
200 / 1
-« RANGE
150 1 Of J IC
,,
~
100

50

1 1 1 1 1 1 1 1 1
SO 100 150 200 250 300 350 400
Electrical Potential (Millivolts)

FIG. 4—Electrical potential test results for center-cracked panel specimen SW-35 (1 kip
4448 N)

crack tip at one end, held in place with a spring loading and acoustically
coupled to the specimen with conductive grease. Output from the sensor
was amplified and then fed through a rate meter, which is actually an
averaging device. The rate meter averages the pulses over discrete periods
of time so that they can be mechanically recorded; the pulses actually
occur over such short periods that the recording response is not fast enough
to pick them up.
The results of the test are also presented in Fig. 5, where the acoustic
count rate is displayed as a function of load. The figure shows a large
increase in count rate at about 1.0 MN (225 000 lb). This value is somewhat
below the true Ju for the material as measured with the multiple-specimen
tests (shown in Fig. 9). While this result is somewhat disappointing, it was
not altogether unexpected. Stainless steel is a parti^larly poor material for
acoustic emission, because it is not only a low emitter but also a poor trans-
mitter of acoustic noise. Another important factor which undoubtedly
influenced the results is the extensive plasticity developed in the specimen,
which also produces acoustic emission.
Elastic Compliance—The elastic compliance method is illustrated sche-
matically in Fig. 6 [2]. During the loading cycle, load drops of approxi-
mately 10 percent are made at various intervals. The changes in the slope
of the linear portion of the load-displacement curve during the load drop
should be a measure of any changes in crack length. Because the slope
change could not be measured with sufficient precision on the general or
conventional load-displacement curve, as shown in Fig. 6a, a second curve
shown in Fig. 6b was recorded simultaneously with greatly amplified scales.
To facilitate the amplification in the present test series, most of the elastic
 BAMFORD AND BUSH ON STAINLESS STEEL 563

Acoustic Emiuion 80 Db
Sensor on
Spaclmen Amplifier

Discriminator

X-Y Racorders Rate Meter

Schematic of Test Apparatus

Acoustic Emission (Counts Per Second)

100,000

10,000 —

1000 -

100 —

0 t-
_l_
100 ISO 200 2S0 300
Load (KIPS)

FIG. 5—Acoustic emission results for center cracked panel specimen SW-35 (I kip = 4448 N).

contribution was electronically subtracted from the curve using a special

instrumentation package developed for this purpose.
Because of the precision required in the amplified curve, hystersis in
the output of the loading and displacement measuring system must be kept
to a minimum. Therefore, to measure load, rather than use the testing
machine load indicator, a combination load cell-loading tool was designed
and used for the bend tests. To measure displacements, various methods
were tried. A three-point beam system, developed for making bend bar
compliance measurements, was first tried. In this system, three strain-
564 ELASTIC-PLASTIC FRACTURE

^ .
•
1

, 1
^ "

-} 3
T^.
jk ;,
• ^ " ^ ~ - ^

I~~"^L. _ _.
r
«i
rU i • »
!

V d

ft r —j
3

^
^
i ^
• -

- - . . . ^

- •
•T • ,:
peffl
BAMFORD AND BUSH ON STAINLESS STEEL 565

Is

IS
u 0(1
-S "^
Q
•y •a
|p
s
"3
1^ • ^ ^

^b •S
.c (N

IS
IS

1!
o

peoT
566 ELASTIC-PLASTIC FRACTURE

gaged cantilever beams are used—one beam contacts the specimen at the
centerline of loading point and the other two are placed over the support
points. The strain gages on the beams were wired in a Wheatstone bridge
configuration so that only the vertical displacement of the beam relative
to the support points was recorded. Other methods used to determine
displacements were strain gages mounted both above the crack tip and
near the center loading point on the compression surface of the bend speci-
mens. Also, a clip gage was placed across the crack mouth opening.
A typical set of load displacement curves obtained using the mouth
opening clip gage is shown in Fig. 66. The curves shown are considerably
reduced in scale for presentation, but are representative of the type of
traces obtained for all the four measurement systems. The strain gage on
the side of the specimen was the only one to show any increase in hysteresis.
All of the systems showed an initial decrease in displacement or strain
over the first few load drop cycles, indicating a pseudo-decrease in crack
length, except for the case where a strain gage was placed near the top of
the crack.
To investigate whether or not the decrease in strain or displacement shown
in the compliance curve may have been caused by support conditions, a
high-pressure lubricant was applied to the ASTM E 399-74 recommended
rollers [3] used to support the specimen. Since electrical potential measure-
ments were also scheduled, insulators at the supports would be required.
Therefore, while investigating support conditions, Micarta plates along
with the lubricant were used, during some of the tests. When using the
lubricant, the rollers, instead of rolling, slid to the back of the support
block. Rather than have the rollers slip during the early part of the test
and affect the curves, the rollers were placed against the back support at
the beginning of the test. To prevent brinelling of the specimen, hardened
steel plates were placed between the rollers and the specimen. Regardless
of the support conditions, the initial decrease in compliance occurred.
For large displacements, the curves having the least hystersis and smoothest
appearance were obtained using the lubricant; therefore, lubricant was
used for all of the tests reported here.
Application of the compliance method to the bend tests was singularly
unsuccessful, and an adequate explanation for this behavior was not ob-
tained. It appears to be related to the extremely large displacements for
the bend bars combined with extensive plasticity present in the tests. Further
complications arose from the fact that the data points follow the blunting
line very closely, with no sharp deviation at all, as is discussed later.
Recent tests with three-point bend specimens of a high-strength steel
have verified the adequacy of the compliance method used in the present
tests. Results showed excellent agreement with multiple-specimen tests,
and tend to support the contention that the extensive plasticity and large
displacements present in these tests led to the lack of success.
 BAMFORD AND BUSH ON STAINLESS STEEL 567

The compliance method was also applied to a 5.08-cm-thick (2.0 in.)

compact specimen, with considerably more success. The same techniques
were used as previously explained, and the results are shown in Fig. 7. In
this case, no pseudo-decrease in crack length was obtained, and the results
of the test agreed very well with the multiple-specimen test results, as
summarized in Fig. 8.
J (In. Lb/ln.^)
10000

0.0 0.02 0.040.060.080.100.12 0.14 0.16 0.18

Delta A (In.)
FIG. 7—Compliance method results for compact specimen SW-3 (1 in. = 2.54 cm: 1
in. lb/in. ^ = 0.0001751 MJ/m').

i (ln.-Lb./ln?)

Blunting Line
(Room Temp.) Blunting Line
(eocF)
18000 - \ 'A
16000
14000
12000
10000
8000
6000
4000
LEGEND:
2000 I Circumferential Orientation. RT
O Circumferential Orientation, 600<*P
0 A Axial Orientation, RT
a Axial Orientation, eOO^F

0.10 0.20 0.30 0.40 0.50

Aa (Inches)
FIG. 8—/ic determination, 316 cast stainless steel, compact specimens (1 in. — 2.54 cm:
1 Ib/in.^ = 0.0001751 MJ/m^).
568 ELASTIC-PLASTIC FRACTURE

Results
The data were analyzed by plotting the J-integral values obtained as a
function of crack extension for each material type and specimen type.
The value of / for the compact specimens was calculated from the ex-
pression proposed by Merkle and Corten [4] which is presently recom-
mended by ASTM [1]

2A 2P8 .^^

where
a 1,0 2 = coefficients developed by Merkle and Corten [4] to account for
the tension component in the compact specimen; the values of
a I and a 2 are functions of the crack depth of the specimen,
A = area under the load versus load point displacement curve,
B = thickness of the specimen,
b = remaining ligament of the specimen,
P = final load value, and
8 = final load point displacement.

For the three-point bend specimens the J-integral was calculated from
the expression originally developed by Bucci, et ai [5]. Using the same
symbols as Eq 1, the expression is given by

The expression for / for the center-cracked panel specimens was based
on the estimation method proposed by Rice et al [6]. The expression results
from the summation of the linear elastic strain energy release rate G added
to the plastic portion of the loading, and is

A*
J=G + - (3)

where G is the linear elastic strain energy release rate, and the value of
A* is the area under the load displacement curve between that curve and a
straight line drawn from the origin to the point of interest. The remaining
symbols are the same as defined in Eq 1.
The cracks tended to lead somewhat in the center of the specimen, and
so measurement of crack advance was accomplished by two methods, an
area averaging method, where the area of advance was actually measured
and divided by the specimen width, and a nine-point averaging method.
 BAM FORD AND BUSH ON STAINLESS STEEL 569

These two methods gave very consistent results, so the nine-point averaging
was used. The Ju value was determined by a least-squares best fit of the
data to a straight line and analytical determination of its intersection with
the so-called "blunting" line, given by

/ = 2 ffo Aa (4)

where
Aa = crack extension,
ao = flow stress = Vi {ay + Ou),
<jy = 0.2 percent offset yield stress, and
ff„ = ultimate strength.
Test results showed that the stainless steels investigated are extremely
tough, and consistent in their properties. Results for compact specimen
tests of the cast 316 stainless steel, Fig. 8, show that there is no effect of
orientation on the results, although both the slope of the / versus Aa curve
and the/ic value are somewhat temperature dependent. The experimentally
determined Jic values for all the steels tested are summarized in Table 3.
Center-cracked panel tests conducted at room temperature are summarized
in Fig. 9 and show remarkable consistency with both the slope of the curve
and Jic value obtained from the compact specimens, as shown in Tables 3
and 4.

J (ln.-Lb./ln?)

24000
/*

20000

16000 Blunting /
\ Line /

12000
\ y^

8000

4000 /•

0
1 1 1 1 1
0.1 0.2 0.3 0.4 0.5
Aa (Inciies)

FIG. 9—1 ic determination, 316 cast stainless steel, center-cracked panels (1 in. = 2.54 cm;
1 in. lb/in. ^= 0.0001751 MJ/m^).
570 ELASTIC-PLASTIC FRACTURE

UO ^ 'H fO t^ n ^ r o CT^ ^ r^
00 t ^ f») (~- 3 : oo
- H • ^ 00

^ *^. '^. "^uS '^

( N t-^ s d ON
O I-- O ;
O ^ ro
<N
^
TH *0 '—I r o ••—»

-H r o ^ O lO 1/1 —I v o

S• ^' O1\0^ ^ 1>- 00

ro -^ 00
f S lO r o
3 <N
n (N ^o

Si
il sii
s

I/) 0^ fS
A I

t
:S 00 5 to ^^
00 O « oo
S -H -H o d

o o o o o O I-; o o i^ o
nj- <N <N P4 <N -<" ri " r-i (N -H ^

^ vO >0 ^
i-i H H 2 S iS
Oi Oi r^ m rr>
i""^
Oi Oi

t/5
mil ik! I
M
s li 11
 BAMFORD AND BUSH ON STAINLESS STEEL 571

Tests of the forged 304 stainless piping material again showed a temper-
ature dependence of the data, although less pronounced than that of the
cast piping, as seen in Fig. 10. The orientation of the specimens again
appeared to have little effect on the results, although some scatter is evident.
To determine further whether orientation was important, radially oriented
2.54-cm-thick (1.0 in.) compact specimens were machined, and test results
are shown in Fig. 11. It can be seen that there is a very slight effect, in
that /ic at 316°C (600°F) is somewhat lower for the radial direction while
the slope of the / versus Aa curve is somewhat higher. However, a definite
conclusion as to orientation effect cannot be reached because of the scatter
in the data.
Compact specimens were also machined from 304 stainless plate material,
and these 2.54-cm-thick (1.0 in.) specimens were tested at 316°C (600°F).
Results are shown in Fig. 12, and indicate that the/u value for this material
is somewhat lower than for the piping steels tested at the same temperature,
although the slope of the / versus Aa curve is slightly higher. Also, much
less scatter is evident in these data.
Considerable difficulty was encountered in interpreting the data ob-
tained from the three-point bend specimens, as shown in Fig. 13 and 14.
The data display less scatter than the compact specimens, and have the
same trends, in that the slope of the / versus Aa line decreases with tem-
perature. Unlike the compact specimens, however, the slope of the / versus
Aa line is nearly equal to that of the blunting line for both materials at

J (In.-Lb./ln?)

0.10 0.20 0.30 0.40 0.50

Aa (Inches)
FIG. 10—J Ic determination, 304 forged stainless steel, compact specimens (1 in. = 2.54 cm;
1 in.lb/in.^ = 0.0001751 MJ/m^).
572 ELASTIC-PLASTIC FRACTURE

J (In.-Lb./ln?)

22000 LEGEND:
• IT Compact Specimens
20000

18000 Blunting /
Line v /
16000

14000 -
12000 / ^/
10000 -
8000 m/
6000
•
4000
2000
fm
0
1 1 1 1
0 0.10 0.20 0.30 0.40 0.50
Aa (Indies)

FIG. 11—I ic determination, 304 forged stainless steel, radial orientation, compact speci-
mens (1 in. = 2.54cm: 1 in. Ib/in.^ = 0.0001751 MJ/m^).

J (In.-Lb./ln?)

Blunting y'
r^Line IT

1 /
6000

4000
1' /
2000 /
-
0 J1 1 1 1 1
0.1 0.2 0.3 0.4 0.5

Aa (Inches)

FIG. 12—] ic determination, 304 stainless steel plate, compact specimens (1 in. — 2.54 cm;
1 in. lb/in.^ = 0.0001751 MJ/m^).

room temperature. This is a remarkable result, because it indicates that

the crack has little or no influence on the failure of these specimens, that
instead the specimen simply tears apart. This would be understandable if
similar behavior were observed for the other specimen types, because it
is well known that stainless steel is not particularly notch sensitive. But
this implies that the fracture behavior of this material is somewhat geometry
dependent, at least when portrayed in the manner presently recommended
[/]. An alternative explanation is that the present blunting line concept
 BAMFORD AND BUSH ON STAINLESS STEEL 573

J (ln.-Lb./ln.>)

44000 ^ Blunting Lin*

(RooniTaflip.)
36000 —
\/
320(K) TS'F TasU
28000 —
24000
'"fir As = 0.89
479S0
Blunting
20000 _Llne
(600»F)i
16000
—• \ y ,600°F Tests
12000
8000
4000
0
" • • 1 1 1 1 1
0.20 0.40 0.60
Aa (InchM)
FIG. 13—J/c determination, 316 cast stainless steel, three-point bend specimens (I in. =
2.54 cm; 1 in. lb/in.^ = 0.0001751 MJ/m^).

J (In.-Lb./ln?)

_ Blunting Line
28000 "" (Room Temp.)

24000
xfj
/Blunting /
20000 Line /

16000 ^
II
m 1^
600°Fl/

12000 ^ J 1 J
Legend:
8000 • RT.
- Ill / • 600°F
4000 " ^f^ X HT—Preorackedin
Compression
0 -^ 1 1 1 1 1
0 0.10 0.20 0.30 0.40 0.50 0.60
A a (Inches)

FIG. 14—I IQ determination, 304 forged stainless steel, three-point bend specimens (1 in.
2.54cm; 1 in. Ib/in.^ = 0.0001751 MJ/m').
574 ELASTIC-PLASTIC FRACTURE

needs to be modified to cover the full range of material fracture charac-

terization, and to include specimen geometry effects.
Results of the bend specimen tests are consistent for the two materials
and where portrayed in the recommended manner produce / ^ values which
are much different than results from compact specimens and center-cracked
panels in all but one case. The bend specimen tests were very carefully
done, and no anomalies could be found in the testing procedures, so it
appears that the data presentation methods may need improvement for
this type of test with very ductile materials.
As a sidelight to this investigation, Paris's proposed tearing modulus
[7] was evaluated. The tearing modulus is defined as

E dJ
a^ da

where
E = Young's modulus,
ffo = flow stress, and
dJ/da = slope of the J versus Aa curve.
Results of this calculation are given in Table 4, and show that the tearing
modulus is quite large for this material. However, it is certainly not in-
dependent of specimen geometry, as has been proposed. This finding
agrees with conclusions reached by several other investigators for other
materials.

Conunents on Size Requirements

The proposed size requirements of ASTM [i] apply to both the specimen
thickness, B, and the remaining ligament, b, and are

B,h> 751/ao

whereffo—flowstress = ^lia, + ff„).

All the specimens tested meet this requirement at Ju except the 2.54-cm
(1.0 in.) radially oriented compact specimens, as shown in Table 4. This
does not mean to imply, however, that the tests all meet the proposed size
requirements, which apply to individual specimens. As may be seen in
Table 4, the specimens are close to the limit of the requirements even at
/ic, so many of the specimens do violate the criteria. As seen in Fig. 8
through 14, there is no apparent change in the fracture behavior once
the proposed requirement is violated. The data remain on the same straight
line, and may even display less scatter at longer crack lengths. This implies
 BAMFORD AND BUSH ON STAINLESS STEEL 575

that the size requirement may be too restrictive for very ductile materials.
Further, the very high strain hardening of these austenitic stainless steels
implies an enhancement of the dominance of the crack tip singular field,
and thus a lessening of the size requirement.
Several authors have recently proposed other criteria for applicability of
the J-integral to characterization of elastic-plastic fracture. Even though
the limits of these criteria are not yet well developed, it is of interest to
calculate the parameters involved.
Hutchinson and Paris [8] proposed that one important requirement
would be

b dJ

where the symbols have been previously defined, to ensure that propor-
tional loading takes place in the specimen. This parameter has been eval-
uated at / = /ic, and results are summarized in Table 4, showing that the
parameter ranges from 13 to greater than 90 for the tests reported here.
An interesting point is that the highest values of this parameter were ob-
tained for the three-point bend specimens. All the data obtained for the
stainless steels tested appear to meet this criterion.
McMeeking and Parks [9] have also proposed a criterion for / dominance
of the crack tip field for a specimen, which will be called Q in this work

-, bao

where the symbols have been previously defined.

This parameter is also tabulated in Table 4. McMeeking and Parks
claim that the parameter Q should be much greater than 200 for a center-
cracked panel, but for bend type specimens the value of Q need not be
nearly as high, although they make no quantitative recommendation. Table
4 shows clearly that the center-cracked panel does not meet their proposed
value, but, since no recommendations were made for other specimens,
the numbers are provided for information only.

Conclusions
1. Three specimen types were tested at two different temperatures, room
temperature and 316 °C (600°F). Of these, the most efficient specimen was
found to be the compact specimen, although good agreement was obtained
between compact and center-cracked panel specimens. The three-point
bend specimens gave results which were inconsistent and difficult to inter-
576 ELASTIC-PLASTIC FRACTURE

pret, and thus should be avoided for characterizing very ductile materials
according to the presently recommended practice. Note that these speci-
mens have been found to be quite adequate for characterizing materials
which do not harden extensively.
2. The only suitable methods for obtaining / versus crack extension in-
formation on very ductile materials were found to be the unloading com-
pliance method and the multiple-specimen heat tinting technique.
3. The presently recommended procedures for data interpretation pro-
duce consistent results for compact specimens and center-cracked panels,
but may need to be improved for three-point bend specimen results for
ductile materials. The proposed validity criteria appear to be too restrictive
for ductile materials, a conclusion which is supported by the consistent
specimen behavior before and after violating the proposed requirement.
Further evidence is provided by consideration of the validity criterion re-
cently proposed by Hutchinson and Paris [8], which the specimens clearly
meet.
4. Results of the tests show that the three materials were all very tough
at both room temperature and 316°C (600°F), with / k equal to about 0.79
MJ/m2(4500 in. lb/in.2) at room temperature, and ranging from 0.26 to
0.40 MJ/m2 (1500 to 2500 in.-lb/in.^) at the higher temperatures. It is also
important to note that Jic is a very conservative measure of the fracture
resistance of this material, since considerable stable crack growth occurs
prior to fracture. In one specimen, for example, a value of/ = 8.40 MJ/m^
(48 000 in.-lb/in. ^) was sustained without failure.

Acknowledgment

The authors wish to express their appreciation for the helpful advice
received from Jim Begley, John Landes, and Garth Clarke during the testing.
Also thanks are due to Lou Ceschini, who performed the center-cracked
panel tests, and to Andy Manhart, who assisted with the electrical potential
and acoustic emission measurements.

References
[/) "Recommended Practice for the Determination of Ju" as detailed in correspondence
from G. A. Clarke to ASTM Task Group E24.01.09 dated 10 March 1977.
[2] Clarke, G. A., Andrews, W. R., Paris, P. C , and Schmidt, D. W. in Mechanics of
Crack Growth, ASTM STP 590, American Society for Testing and Materials 1976, pp.
27-42.
[3] ASTM Book of Standards, Part 10, American Society for Testing and Materials, 1976,
pp. 471-490.
[4] Merkle, I. G. and Corten, H. T., Transactions, American Society of Mechanical Engineers,
Journal of Pressure Vessel Technology, Series I, Vol. 96, No. 4, Nov. 1974, pp. 286-292.
[5] Bucci, R. J., Paris, P. C , Landes, J. D., and Rice, J. C. in Fracture Toughness, ASTM
STP 514, American Society for Testing and Materials, 1972, pp. 40-69.
 BAMFORD AND BUSH ON STAINLESS STEEL 577

[6] Rice, J .R., Paris, P. C , and Merkle, J. C , in Progress in Flaw Growth and Fracture
Toughness Testing. American Society for Testing and Materials, ASTM STP 536. 1973,
pp. 231-245.
[7] Paris, P. C , Tada, H., Zahoor, A., and Emst, H., this publication, pp. 5-36.
[5] Hutchinson, J. W. and Paris, P. C , this publication, pp. 37-64.
[9] McMeeking, R. M. and Parks, D. M., this publication, pp. 175-194.
Applications of Elastic-Plastic
Methodology
G. G. Chel?

A Procedure for Incorporating

Thermal and Residual Stresses into
the Concept of a Failure
Assessment Diagram

REFERENCE: Chell, G. G., "A Procedore for Incorporating Thermal and Residual
Stresses into the Concept of a Failure Assessment Diagram," Elastic-Plastic Fracture,
ASTM STP 668, J. D. Landes, J. A. Begley, and G. A. Clarke, Eds., American
Society for Testing and Materials, 1979, pp. 581-605.

ABSTRACT: The Failure Assessment Curve proposed by Harrison, Loosemore, and

Milne has been rederived and interpreted in terms of an equivalent J-integral analysis.
Comparison with computed values of/ indicates that the curve is a good approximation
to a lower-bound failure criterion for mechanical loading. The J-integral interpretation
enables thermal, residual, and secondary stresses to be included within the concepts
of the Failure Assessment Diagram. A procedure is introduced which transforms points
on a failure diagram obtained, for instance, from a J-integral analysis, into approximately
equivalent points on the Failure Assessment Diagram. This procedure is particularly
useful for failure assessments involving thermal, residual, or secondary stresses where
a plastic collapse parameter is not definable. The procedure assumes that a plastic
stress intensity factor can be estimated. A method of assessing the severity of a
mechanical load superposed on an initial constant load is also presented. Examples
showing the applications and advantages of the technique are given.

KEY WORDS: failure criterion, assessment curve, J-integral analysis, thermal

stresses, residual stresses, secondary stresses (fracture), assessment diagram, fracture
(materials), elastic-plastic, post-yield, crack propagation

Nomenclature
L Generalized load
La Applied load
Xi Plastic collapse load
Lt Load at fracture

'Fracture Mechanics Project leader. Materials Division, Central Electricity Research

Laboratories, Kelvin Avenue, Leatherhead, Surrey, U. K.

581

Copyright 1979 b y AS FM International www.astm.org

582 ELASTIC-PLASTIC FRACTURE

LK Fracture load determined using linear elastic fracture mechanics

(LEFM)
Xi Initial loading
a Generalized stress = AL/Bw
Oa Applied stress
ffi Plastic collapse stress
Of Fracture stress
a/ Relaxed stress determined elastic-plastically
CT/ Relaxed stress determined linear elastically
a Flow stress
Y Geometric function
a Crack size
a' Effective crack length determined using Irwin's first-order plasticity
correction
K\ Stress intensity factor
Kp Plastic stress intensity factor = ^fW'Jor K\ determined using effec-
tive crack length a'
Kc Fracture toughness
a Plastic constraint factor
/ J-integral
J\ J determined linear elastically, = K-^/E'
E Young's modulus
V Poisson's ratio
Kr Ki/K^, a failure assessment coordinate
K/ Value of Kr at failure
K/Q Value of K/ lying on the Failure Assessment Curve immediately
above the failure assessment point Q
Kr' Value of ^ , due to initial loading
Sr (To/ffi, a failure assessment coordinate
5/ Value of Sr at failure
5/2 Value of 5, corresponding to the point Q
Sr' Value of 5r due to initial loading
d Displacement at loading point
da Component of displacement due to uncracked body
B Thickness of component
W Width of component
A Geometric constant

Recently a procedure for assessing the integrity of cracked components

in the linear elastic and post-yield fracture regimes has been proposed [/].^
To simplify elastic-plastic fracture analyses a Failure Assessment Diagram
was introduced. This enables the integrity of cracked plant to be ascertained
^The italic numbers in brackets refer to the list of references appended to this paper.
 CHELL ON A FAILURE ASSESSMENT DIAGRAM 583

through two separate calculations based on the two extremes of fracture

behavior, namely, linear elastic and fully plastic. These two calculations
provide a point on the Failure Assessment Diagram, and the relative
position of this point to a Failure Assessment Curve determines the integrity
of the structure. If the point falls below the Failure Assessment Curve, the
structure is safe; if on or above it, failure is predicted. The Failure Assess-
ment Curve interpolates between the critical conditions necessary for
fracture in the two extremes and is based on the fracture equation given
in Ref 2 as modified and generalized in Ref J.
The Failure Assessment Diagram provides a valuable contribution to a
very complex problem. It not only reduces difficult concepts to an easily
comprehensible pictorial representation, but also bypasses the need to
perform detailed elastic-plastic calculations. Furthermore, progress has
been made in validating the application of the Failure Assessment Diagram
for real problems. Experimental data are available which demonstrate that
the Failure Assessment Curve is a lower bound, and hence safe [/].
Although it is recommended that the Failure Assessment Diagram be
used together with lower-bound data and conventional engineering safety
factors, and therefore the detailed form of the Failure Assessment Curve
is not important, it is still valuable to know how the Failure Assessment
Curve compares with other possible curves based, for example, on a J-
integral analysis. Furthermore, at the present time it is not at all clear
how failure assessments which involve thermal or residual loadings are to
be included on the Failure Assessment Diagram. These loadings do not
contribute to the plastic collapse load as formally defined in plastic limit
load theory, but may contribute strongly to fracture in the post-yield
regime. It is the purpose of this paper to go some way toward answering
these questions.

Analytical Basis of the Failure Assessment Diagram

The interpolation fracture formula used [3] is

X, = X . ^ c o s - [ e x p ( - ^ ) ] (1)

where
Lf = generalized fracture load, for example, pressure,
Li = value of the generalized load corresponding to plastic collapse, and
Lk = fracture load determined from linear elastic fracture mechanics
(LEFM).
Both Xi and X* depend on crack length. Equation 1 can be written in
terms of a generalized stress, a (the load L divided by a geometric constant
584 ELASTIC-PLASTIC FRACTURE

of the dimensions of length squared) and to show the explicit dependence

on the fracture toughness, Kc, as

a,= a . ^ c o s - [ e x p ( - g ^ ^ ) ] (2)

where a is the crack length, y is a function dependent on flaw shape and

size, structural geometry, and loading system, such that for an applied
stress (7a, the stress intensity factor, Ki, is given by

Kl = Oa^Y (3)

The suffixes/and 1 have the same meaning as for loads £ . Equation 2 can
be rearranged to read

In defining a plastic stress intensity factor, Kp, for an applied stress

a a, as

K\ = —aY'aj^
—-aJ^ar In sec ( :-—
r^) (5)
TT^ \2ai

it can be seen from Eq 4 that at fracture, when Oa = a/

Kp incorporates the effects of plasticity. It can also be related to the J-

integral [4] through the equation

Kp = yfETJ (6)

where £ " = E, Young's modulus for plane stress, or E' = E/{1 — t^),
where v is Poisson's ratio, for plane strain. Hence, when fracture is governed
by crack tip events, that is, characterized by Kc, the fracture Eq 1 can
be interpreted as a post-yield fracture expression based upon an approximate
functional form for the J-integral, given by Eqs 6 and 5 [5]. In this context
Eqs 5 and 6 were proposed and successfully employed in obtaining valid
fracture toughness values from invalid specimen test data [6].
Dividing both sides of Eq 4 by A^i^, using Eq 3, and inverting the result
gives
 CHELL ON A FAILURE ASSESSMENT DIAGRAM 585

8ffl^ , / TTffaX
Ki/Kp = In sec I (7)
V2a,
With Kp = /sTc, Oa = Of, and K\ evaluated for a/, Eq 8 represents the
Failure Assessment Curve [1], that is

8ai^ , (•KoA
Ki/K, = ——: In sec -— (8)
TT^ff/^ \2ai /

It can be seen that Eq 7 has the same functional relationship between

K\/Kp and aa/a\ as that given by the Failure Assessment Curve for the
dependence of Ki/K^ on a//ffi. Hence a failure curve may be obtained
from any elastic-plastic theory which relates K\/Kp or, equivalently, (J\/J) '^^
to a„/ffi, where /i is the linear elastic value Ki^/E'. This forms the basis
of the present paper.

The Failure Assessment Diagram

This is constructed with respect to the two axes K and S where the
coordinates of points are [1]

Kr ^^ Kl/Kc

Sr = a„/<Jl

The distance of each point {Kr, Sr) from the origin is linearly proportional
to the load characterizing parameter a, as can be seen from the definition
of Kr and Sr. Equation 8 is a curve on the diagram corresponding to the
loci of points (/iT/, 5/) which coincide with the onset of failure (Fig. 1).
Therefore any loading or crack size which produces a point under the curve
is safe; conversely, failure will occur if it is on or outside. If {Kr, Sr) is
below the curve, then failure can occur either by increasing the load or the
crack size. In the case of the former, the fracture load for a given crack
length is easily determined using the property that the point {Kr, Sr) is
linearly proportional in a to its distance from the origin. In the case of the
latter, the path traversed by the point {Kr, Sr) as it moves toward the
Failure Assessment Curve will be called the path to failure. Such a path
AB is shown in Fig. 1 as the crack length increases from a\ to ag. Once a
path to failure has been calculated for a given loading, other paths to
failure, or parts of them, for other loadings can be easily calculated using
the same linear proportionality as before, and for each point on the path
determining the corresponding point for the new loading. Such a path
A 'B' corresponding to half the loading that generates the path AB is
shown in Fig. 1.
586 ELASTIC-PLASTIC FRACTURE

PATH TO F A I L U R E
FOR STRESS o AS CRACK
LENGTH VARIES FROM
a, TO ag

PATH TO FAILURE
FOR STRESS 0 So
AS CRACK LENGTH
Q VARIES FROM ai TO 33

FIG. 1—Failure Assessment Diagram.

Examples of Failure Carves

Mechanical Loading
The post-yield solutions in Refs 2 and 3 are based on the Bilby, Cottrell,
and Swinden [7]-Dugdale [8] (BCS-D) model of a yielded crack in an
infinite body subject to uniform applied stress. Hence in this case, Eq 8,
with ffi identified as the ultimate tensile strength/-represents the Failure
Assessment Curve. An analytical solution for the penny-shaped crack in
tension is available [9] and can be expressed in terms of Kp [10] so that the
equation of the failure curve can be written down directly as

A-./ySTp = {a./ai) {2[1 - (1 - a^Va,^)'^^]) (9)

This curve is shown in Fig. 2a.

There are several computed / against load curves in the literature. These
have been used, together with the elastic solutions, to obtain failure curves
for the following geometries: three-point bend with span-to-width ratios of
4 [11] and 8 [12], center-cracked plate in tension [13], and a crack emanating
from a hole in a plate in tension [13]. All the computations were in plane
strain and the results are shown in Fig. 2a. Plane-stress solutions for a
 CHELL ON A FAILURE ASSESSMENT DIAGRAM 587

<?•"Ln
p-1

o
/= // ^' \
/ \
/ /^r * 1-
^_
// / /• /
Z
I- ?
UJ
s:
11
l/t
UJ
vn UJ
—
<>
/ / ^ cc
3
' 1/
1
-U J3
D1C l-l 3 9.
1 fjo° U.
If ^s g>
s*
1« 1 Q
•s 1^

— • « •a,
•t! K
i •S
•V.
"a 3
V • >
£ o
1 1 1 1 e «o

11
o o
u S

i9
•a
e ^
^ ^
^
o. to
1
U
g
•s
s
o o a 2 Q^
X s r S •a
J! •^
<z z u1 •S
< i-cc 0<. a.<
_l
.§ e
c a
•e, S
<
ec o u z
LU z m UJ S 0
z u
<
C£
o
111 u^ 1- m CD «^
1- 0.
cc
<
z
< z • «

UJ u T. o o 13
z Ui UJ
(/> K UJ Q. a. S .3
Ul
-< >
z- 1- u
^ UJ •§
1
Q. z z
< Za.
Ul UJ OC
UI
a.
X
SO
1Ih

Q. U U I - I- ^1
a
!I u •e
i«.
u o
h
li :5 »
1^•«
<s • «
-•«
< 3
U. ( J d Pi
»-4
[fa
^
»51
a.
588 ELASTIC-PLASTIC FRACTURE

single-edge cracked plate in tension,^ three-point bend with span-to-width

ratio of 8, and a center-cracked plate are shown in Fig. 2b.
From Fig. 2 it is clear that using a / analysis does not produce a universal
failure curve. Associated with each structure is a set of failure curves
depending on crack shape, size, and loading. However, the Failure Assess-
ment Curve does provide a good approximation to a lower-bound curve.
Furthermore, the differences between the curves are insignificant compared
with the uncertainties inherent in an assessment of any real structure.

Fixed-Grip Loading
Using Eqs 6 and 5 to represent /, the effects of fixed-grip loading on
Kp have been determined [5]. The result is given by Eq 5, where the symbols
have the same meaning but now oi depends on the fixed displacement d and
crack length. This dependence is obtained by solving the following equation
for Oa

where
da = displacement due to the uncracked body subject to the effective,
relaxed load La,
B = thickness of the body, and
A = a geometric constant such that o<, = La/A and/ = Kp^/E'.
The elastic solution is obtained by writing/ = Ki^/E'.
The dependence of oi on d and a obtained linear elastically differs from
that obtained elastic-plastically. The failure curve is thus represented by
the equation

8 /(Ti
Ki/Kp = r2
In sec (11)
ffa' 2(7,

where superscripts e and p signify the values of Oa which satisfy Eq 10 in

the elastic and elastic-plastic cases, respectively. The results of evaluating
Eq 11 subject to Eq 10 are shown in Fig. 3 for center-cracked plate and
three-point bend geometries. For these, ai is given by

ffi = a (1 — a/w)

^Neale, B. K., private communication.

 CHELL ON A FAILURE ASSESSMENT DIAGRAM 589

and

(7i = 2a (1 — a/w)^

respectively, where a is a flow stress. Further, Sr = Oa'/ai to maintain

the linear relationship between loading (either stress a a' or displacement d
in this case) and the distance of the point (Kr, Sr) from the origin. The
center-cracked plate results are given for two normalized gage lengths L*,
that is, separation of loading pins divided by half the plate width w.
From Fig. 3 it is clear that now the failure curves not only depend
strongly on the specimen geometry and crack length, but also on the gage
length over which the displacements are imposed. Two things are immediately
striking about Fig. 3. The first is that Sr can exceed 1 without failure
occurring. This is due to the definition of Sr in terms of the elastically de-
termined relaxed stress a / . The presence of plasticity further relaxes the
induced stress, so that althoughCT/may exceed oi, a/ cannot. The
second is that K\/Kp can exceed 1. This can happen when the displace-
ment loading is such that K\ decreases with increasing crack length. Since,
in a sense, plasticity increases the effective crack length (for example.

FAILURE CENTRE CRACKED PLATE

CURVES THREE POINT BEND, SPAN WIDTH = 4
0 5= A

I 0 _ 0 25

0 5, L* = 4
08

0 25. L* = 4

06 — 0 5, L' = 16

0 4

FAILURE ASSESSMENT
CURVE
0 2

02 0 4 0 6 08 1.0 1,2
S

FIG. 3—Failure curves for fixed-grip loading.

590 ELASTIC-PLASTIC FRACTURE

Irwin's first-order plasticity correction procedure involves adding a term

proportional to Ki^ to the original crack length, and treating the new
crack linear elastically), Kp will be less than Ki if the latter decreases with
crack length. This is the case for three-point bend [75] and explains
the apparent anomalous behavior.
The foregoing examples demonstrate that cracked structures subject to
secondary loading can be described in terms of the failure curve concept,
but that in some cases the Failure Assessment Curve may be far too
conservative as a lower bound.

Approximate Transformation of Failure Assessment Points

To avoid the risk of overconservatism in, for example, secondary loading,
and to take into account thermal or residual loading, it could be argued
that every structure should be looked at independently with reference to
its own failure curve. This is unsatisfactory since it would involve the
generation of a multiplicity of failure curves to cover every situation. This
can be avoided, however, if failure assessments can be referred to a single
curve. Thus it is useful to produce a means of transforming failure assess-
ment points from a given failure diagram to corresponding points on the
Failure Assessment Diagram. The transformation should not depend on
the failure coordinate S,; this parameter is not defined for self-equilibrated
thermal and residual loadings since thdse do not affect the plastic collapse
load.
Let all quantities that refer to the actual failure diagram be denoted by
primes. A typical diagram containing a failure curve A 'B' and an assess-
ment point P' with coordinates {Kr', 5 / ) is shown in Fig. 4a. If the
applied loading is increased, failure will occur at the point P/ with co-
ordinates {K/, S/'). A simple graphical procedure for locating an equiv-
alent point P and failure point Pf on the Failure Assessment Diagram that
avoids specifying a value for Sr' is easily constructed by putting Kr —
Kr' and Kr^ = Krf, drawing in the line Oi^, and hence finding the co-
ordinates Sr and Srf as shown in Fig. 4A, where superscript/ signifies that
Kr and Sr are evaluated at the failure load. This procedure is satisfactory
provided the failure curve A'B' is known, but this requires a comprehen-
sive elastic-plastic fracture analysis for its determination. Furthermore,
as just mentioned, if the applied loading is thermal or residual, the failure
curve cannot be constructed at all because S', is not known. These two
disadvantages limit the scope of this simple graphical technique. However,
an approximate means of obtaining an equivalent failure assessment point
to P ' on the Failure Assessment Diagram, which does not involve either a
detailed knowledge of the failure curve A'B' or Sr', is possible, provided
that a pessimistic evaluation of the elastic-plastic stress intensity factor Kp
is performed at the load level corresponding to P'. This pessimism is
 CHELL ON A FAILURE ASSESSMENT DIAGRAM 591

<
DC

< S

LU
i

ec a

-^i,

UJ lU

«:>
r
1
= (C
-1 3
<
oc ^
-a
<<J u
<
o ^
\
LXJ
K
"S
" .
M
3
-J
— O•O <
U. 's^^
UJ
V) |S
\ ^ z
N •V UJ as
-_ / N
II
UJ
a/ N"^ u.
'o UJ

/ "i
in
1*V j_ >v
"^
r^
^^ •i', S?
^-5
•»»/» "V — o
/ o
/ *^
/ I 1 1 1 5

on
592 ELASTIC-PLASTIC FRACTURE

compatible with normal engineering safety practice, as well as the safety

procedures recommended when using the Failure Assessment Diagram [1].
Since, as can be seen from Eq 7, a failure curve is obtained by plotting
Ki/Kp as a function of applied load, it is clear that the point Q/' which
lies on the failure curve A 'B' immediately above the point P' (Fig. 4a)
has coordinates {K/-^', 5/0 ) where S/-^' = S/ and K/^' ^ Ki/Kp. The
equivalent point to Qf' on the Failure Assessment Diagram is Qf with
coordinates (K/<^',S/0') (Fig. 4b), where S/o can be obtained graphically.
Hence an approximately equivalent point to P ' on the Failure Assessment
Diagram is given by the point R with coordinates (Kr', S/-^) and this will
be a pessimistic point provided that QRf/OR < OP/'/OP, where Rf is the
point where the line OR cuts the Failure Assessment Curve.
This approximate procedure will, of course, be exact if the actual failure
curve is coincident with the Failure Assessment Curve. The accuracy of
the approximations and the degree of pessimism therefore depend on the
assumption that the Failure Assessment Curve provides a realistic lower-
bound failure criterion. The results just given indicate that for mechanical
and secondary loading this is the case. The procedure will not work if
Ki/Kp > 1, a situation which may occur if Ki decreases with increasing
crack length. In these cases a pessimistic assessment can be made using
linear elastic fracture mechanics and putting S/o = 0, so that failure is
predicted when Kr= 1.
Even when thermal and residual stresses are self-equilibrated within the
substructure subject to a failure analysis, they can still contribute sub-
stantially to the J-integral. Within the context of the BCS-D model of a
crack with plastic yielding, thermal stresses can result in a value of Kp
which greatly exceeds K\ [5,18], This conclusion has recently been verified
by elastic-plastic finite-element calculations performed on a center-cracked
sheet subjected to thermal loading [17]. Hence thermal and residual stresses
can contribute to elastic-plastic failures by reducing the tolerance of the
structure to mechanical loading. Since the failure parameter Sr is a measure
of elastic-plastic effects (zero for purely elastic, one for purely plastic), its
definition must be mbdified to incorporate thermal and residual stresses.
This can be accomplished using the transformation technique developed in
the foregoing in order to obtain an apparent value of Sr, and hence an
equivalent failure assessment point, on the Failure Assessment Diagram.
The graphical procedure outlined for determining the point R is straight-
forward. However, it is computationally convenient to have an analytical
method for obtaining the coordinates of the approximate assessment point
R. The Failure Assessment Curve is described by the relationship

Krf = F{Srn (13)

where F{Sr) is the function on the right-hand side of Eq 7 or 8. For a

 CHELL ON A FAILURE ASSESSMENT DIAGRAM 593

given value of K/, Sr^ can be obtained as the solution of Eq 13. A solution
which is 96 percent accurate is given by

-24 1 >
5/ = 1 — exp {K/Y - 1 (14)

Thus, putting K/ = Kr^-^' in Eq 14 gives the coordinate Sr^-^ and hence

the coordinates of the point R as (Kr', S/-^).

Example of Transformation Procedure: Secondary Loading

Secondary stresses may arise, for example, when one part of a structure
is heated or cooled with respect to another part and, as a consequence,
impose boundary displacement loading on the other part. An example of
the use of the transformation procedure for such a case is given in the
following.
Consider a center-cracked plate subject to fixed displacement d. The
relevant details concerning the values of the initial stress, fracture toughness,
etc. are shown in Fig. 5. The plate contains a sharp defect of length-to-
width ratio (a/vv) of 0.2. Two things are required: first, the path to failure,
so that the critical defect size can be determined under the given displace-
ment loading, and second, the maximum displacement that can be imposed
before the initial defect will result in failure.
In the first part of the calculation, K\/Kp is evaluated for a/w = 0.2,
0.3, 0.4, and 0.5 using Eqs 10 and 11. The transformed points given by
Kr' (= Ki/Kc) and S/o (obtained from Eq 14 with K/ = Ki/Kp)are then
plotted on the Failure Assessment Diagram (Fig. 5a). It is clear that
failure occurs at the point Pi corresponding to a/w = 0.5 on the path to
failure. The second part is easily calculated. The distance of the transformed
point, corresponding to a/w = 0.2, from the origin is measured on the
Failure Assessment Diagram. The ratio of this distance divided by the
length of the line to the Failure Assessment Curve which passes through
the origin and the transformed point (OP2) is determined. The original
displacement of 0.3 mm multiplied by the inverse of this ratio gives the
maximum allowable displacement to be 0.4 mm. The points on the diagram
corresponding to displacements of 0.375 and 0.45 mm are also shown in
Fig. 5a for comparison.
The path to failure and the effect of increasing displacement are shown
in Fig. 5b with respect to the reference failure diagram. Also shown are
the failure curves corresponding to a/w = 0.5 and 0.2. The failure points
Pi' and Pi' are equivalent to the two points Pi and P2 in Fig. 5a. It can
be seen that in this case the approximate transformation procedures
predict a displacement at failure in good agreement with the value of
about 0.4 mm obtained using the reference failure diagram.
594 ELASTIC-PLASTIC FRACTURE

\
\ ; / E —1
/ in
^1
E
Q. J^k/ E —
^S^
z ^ \ o
/ / *^ t3 II
•5-aa

Z * - */» I/
/''^o
\E
o 1
ai
?\ \
o
— cs ^

II
O m ^ 1 ° z 3 \
11 1/1"^
"* 1
1 UJ x: < \
U.
1 > u \
V O tt<g
U 1- 1-
\

i
X
oe * "^ \
3 1- I
- • n: • -
D. \
<S9 \
U. U- £
\
1 1 1 1 \

I!
1 • ^

^
a
•^
p
1-
•3
s £,
:j
u
u a.
^s c • «

tt aa
^ u
CI, 0 cC
"Ct /—. o
X

— _;
= ill
O ^ ftj

M
 CHELL ON A FAILURE ASSESSMENT DIAGRAM 595

Initial Loads
A situation which may often occur in practice is when a load L is added
to a structure which already experiences a constant initial load X, which is
independent of Z. Here an assessment is required with respect to the load
L. The initial load could arise, for example, from residual stresses in a
weld that has not been stress relieved. If Li were applied mechanically,
then the plastic colli^pse load, Li, would depend on X,. If the initial loading
were by self-equilibrated thermal or residual stresses, then the plastic
collapse load would be independent of X,. In both cases, however, X, would
produce a failure assessment point on the Failure Assessment Diagram.
The coordinates of this point {K,\ S,') for mechanical loads can be obtained
following the procedures advocated in Ref 1. For thermal or residual
stresses they can be determined using the transformation procedure after
evaluation of the ratio Ki/Kp. Such an assessment point P is shown in
Fig. 6a. Any additional loading will cause the assessment point P to move
due to an increase in Kr plus a further shift in Sr. It is therefore proposed
that relative to the added load X, the initial loading has the effect of
moving the origin from 0 to the point P while still retaining the same
Failure Assessment Curve centered on the origin at 0, the differences being
that the /T-axis records a nonzero value when X = 0 due to the loading
Li, and the new S'-axis, which still goes from 0 to 1, is contracted into the
reduced length 1 — S; [PB" in Fig. db, where & B" corresponds to the
point {Kr\ 1)] relative to the axis OB. The new Failure Assessment Curve
relative to the new coordinate system is A'B', and the assessment point
P' for the load X is linearly proportional to its distance PP' from the new
origin at P. The logic of this construction can be seen when the load X,
is of the same type as the added load X. In this case it can be shown that
the point P' with coordinates {Kr', Sr') determined with respect to the
origin at P is, for assessment purposes, equivalent to the point P' with
coordinates (Kr, Sr) determined with respect to the origin 0 on the Failure
Assessment Diagram.

Examples of Initial Loading

Thermal and Mechanical Loading

Recently the effects of thermal loading on a center-cracked plate have
been calculated using an elastic-plastic finite-element program [17]. The
plate, of width 2w, contained a crack of length a/w = 0.26 and was
subjected to a uniform applied tensile stress Oa superposed on the thermally
induced stress a{x)

^ = 0.6 - l.Six/w)'
a
596 ELASTIC-PLASTIC FRACTURE

I
I

I
P
I!
•I
 CHELL ON A FAILURE ASSESSMENT DIAGRAM 597

where a equals the yield stress and the center of the plate is the origin of
coordinates. Since both /i and / were determined as a function of Oa, a
failure curve, shown in Fig. 7a as a dashed line, can be drawn where Sr is
defined in terms of the mechanical load. Also shown, for reference, is the
failure curve corresponding to the case of mechanical loading only (dotted
line) as well as the Failure Assessment Curve (full line).
Following the procedures outlined in the previous section, all points on
the failure curve AB can be transformed to points lying on a failure curve
A'B" on a new failure diagram with origin P and axes K' and S' (Fig.
7b). The coordinates of P with respect to the origin at 0 are {K/, S/) where
the value of K/ will depend on the value of Kc, but the value of 5r' will be
given by the transformation procedure applied to the ratio K\/Kp when
<7o = 0. In the present case the ratio is 0.98, so that the value of Sr'
obtained either graphically or using Eq 14, with/sT/ = 0.98, is approximately
0.3. Using this value, the transferred failure curve, A 'B", was constructed
and, as can be seen in Fig. lb, it is very similar to the part of the Failure
Assessment Curve A 'B' which traverses the new coordinate system.
To illustrate the failure assessment procedure, consider the following
example, which is to find the applied stress at failure for the given initial
thermal loading and crack length. The material properties and plate
dimensions are given in Fig. 7. When ff„ = Q, Ki = 40.3 and hence the
initial assessment point, P, forming the origin of the new axes, has the
coordinates {K,' = 0.403, S,' = 0.3) since, from the foregoing, Kr^ =
KxIKp = 0.98. When <Ja = <JX= 440 M N m - ^ Kx = 105.5 MNm"^^^ and
the assessment point coordinates with respect to the new axes K' and S'
are (K/ = 1.055, S/ = 1). This is shown as the point P ' in Fig. lb. The
approximate procedure based upon the Failure Assessment Curve A 'B'
gives the failure point as Pj, and hence the failure stress, Oa^, is {PPf/PP')
X 440 MNm"^ = 292 MNm"^. The equivalent assessment points on the
reference failure diagram (Fig. la) are also marked P, Pj', and P ' , and
the failure stress is calculated to be 277 MNm"^, in good agreement with
the approximate answer. The value of 277 MNm"^ could have been
obtained, of course, from Fig. lb as {PPf/PP') X 440 MNm-^.
The predictions of LEFM are obtained by taking the failure curve to be
the line Kr = \. Thus from Fig. 7 the failure stress is {PP'VPP') X 440
MNm"^ = 402 MNm~^, considerably in excess of the value obtained from
the reference curve.

Residual and Mechanical Loading

The same procedure can be followed here as for the thermal loading.
Consider the initial residual stress

- ^ = 0.8 + 1.0667
w
598 ELASTIC-PLASTIC FRACTURE

Bo I;
^"^
^ «.
oa k.as
~
— a
e o

o
<
ec 1^
•s-s
c^ a
o<
Ui °-s
•§ 5
tc u •«
3
« -e
-1
<
u.
H«) =
a
HI
O
u.
*i
S =3

z
<
ec
h-
?
'^ is

.a
fa
%, ^
" ? -i
55 ») 5,
.. S .e
«; w -5
^ 2T3
•1 S3 ^
5!X S

K^
11^X
•J«1

<
T. l|
"" p ^

< lis
UJ
.r^l
tc
3
-1
Ia s ?«
=§ 5-g
<
U.
•~ «) Si
UJ
Sft-S
u
z !
ai ~- ? =
a: •B ^ = 3
UJ
u.
UJ •.=| ts ~|
oe
n
ill
•^ u a h4t
o.-a je
iii
« 'S S
f^^T3 p
(- o .a
2C'-S§ lg
s1
 CHELL ON A FAILURE ASSESSMENT DIAGRAM 599

which is symmetric about the center of a plate of width 2w and the additional
mechanical tensile stress a„. An elastic-plastic solution to this problem can
be obtained using the strip yielding model solutions for plates of finite
width [18]. These model solutions will give, in general, pessimistic values
of Kp as a function of a a for the center cracks of varying sizes, and hence
enable pessimistic reference failure curves to be determined. These are
shown in Fig. 8 for a/w = 0.05, 0.2, and 0.5, together with the material
constants and plate dimensions. Using the procedures outlined in the fore-
going, the three failure curves can be transformed into equivalent curves
on the Failure Assessment Diagram.
Figure 8b shows these curves superposed on part of the Failure Assess-
ment Diagram where the S axis has, for convenience, been magnified by
two. Each of the curves has an associated origin corresponding to the
coordinate S/, which has the values 0.885, 0.81, and 0 for the crack sizes
a/w = 0.05, 0.2, and 0.5, respectively. The K/ coordinates are 0.260,
0.547, and 0.792, corresponding to Ki values of 26.0 MNm-^''^ 54.7
MNm"^'2^ and 79.2 MNm"'''^ when Oa = 0. Relative to the transformed
axes {K' and S' for a/w = 0.05 and K" and S" for a/w = 0.2), the
failure assessment points, P' and Q', evaluated for the plastic collapse
stresses, 475 M N m - ^ 400 MNm-^, are (A"/ = 0.558, Sr' = 1) and
{Kr" = 1.06, Sr" = 1).
The equivalent points on the reference failure diagram are shown in
Fig. 8a. The approximate transformation procedure then gives the follow-
ing failure stresses: for a/w = 0.05, a / = (PPj/PP') X 475 MNm-^ =
405 MNm~^ compared with the reference value of {PPf/PP') X 475 =
266 MNm-2; for a/w = 0.2, a/ = {.QQf/QQ') X 400 = 159 MNm.-^
compared with the reference value oi{QQf/QQ.') X 400 = 150 MNm-2.
In this case the failure stresses are higher than the reference failure stresses
and to a large extent this is probably due to the pessimistic failure curves
resulting from an analysis based upon strip yielding model solutions. The
failure stresses obtained using linear elasticity are 1191 MNm~^ for a/w =
0.05 and 353 MNm-^ for a/w = 0.2.
In Fig. 9 is shown a series of paths to failures which were determined
using the transformation procedure and the results from the residual
stress example. The three paths (dashed lines) correspond to applied
stresses, ff„, of 0, 100, and 200 MNm~^ From these paths to failure it can
be seen that when a a = 100 MNm~^, failure is predicted for a crack
length a/w = 0.34, and when a a = 200 MNm~^, the critical defect size
is a/w = 0.16, assuming as before that Kc = 100 MNm"^^^. The results
obtained from the strip yielding model are a/w = 0.33 and 0.11, respectively.
Again, as in the case of failure loads, the shape of the failure curve for
small crack sizes means that the approximate transformation procedure
is optimistic. The critical defect sizes obtained from a linear elastic analysis
are approximately a/w = 0.45 and 0.31 for the two stress levels, showing
600 ELASTIC-PLASTIC FRACTURE

9lS
/'
(/I
o3
£ •o

/ ScT// »;
O <
oe 1
%/\ u h.
/ ^
a£ < Q
/ / o ^ o S
f UJ s
\ o «
3
•a
d _J
<
1
3C o lu
)/) u. e-
u
o §•
hA II
o UJ
•^
3 r- £ » i
(/I O ec *> r
UJ o o Si a
UJ u.
K t-
UJ tso
ee 1/1
-1 3
< 1-
o
-J
1-
W
<
ee
3
11
5
—
4C :b
a ^
••«
o ,«
o**-
». H)
•^•s
Is
«l •>
^S'g
a -,
Vi ^t
o ^^
Q V
z ^•s
/ <
/ /
14
^ / \ / < e ai
Q s:
o •-
t>
S
•-•
/ / \ / UJ
cc
k
«i
•«
u

// , / W
3

<
u.
P•"
// \ / oiTll "P
In
1*
UJ

11 \ / A u
z
Ui
I / \ UJ
U.
14
A
"B 1
/ \ UI •S i
cc ^2 "^
>
.& a
- t>

1
/
1 \
1 /
/ \
'ifl
^1
^rg

/I i\ / V 1 V 1 !
O. ri

o
1
 CHELL ON A FAILURE ASSESSMENT DIAGRAM 601

FIG. 9—Construction of paths to failure {dashed lines) for three different stress levels
superposed on an initial residual stress.

that the transformation procedure is again much superior to this approxi-

mation.
It is interesting to note that because the origins, to which assessments
are referred, are dependent on the defect size, the paths to failure are no
longer similar as the applied load is increased. This is in contrast to the
case when only a single mechanical load is applied (see Fig. 1).

Discussion
From the two examples it can be seen that the approximate transforma-
tion procedure based upon the Failure Assessment Curve can, in some
circumstances, produce an optimistic value for the failure load. This is
particularly true if the initial loading is so severe that the ratio K\IKp
evaluated without additional mechanical loading is less than about 0.8 and
if the actual failure assessment curve differs considerably in shape from the
Failure Assessment Curve (see Fig. 2a; a/w = 0.05). However, the technique
602 ELASTIC-PLASTIC FRACTURE

still produces answers which are considerably better than those given using
linear elastic fracture mechanics LEFM.
One of the advantages and simplifying features of the Failure Assessment
Diagram concept is that only the elastic and fully plastic solutions are
necessary for a failure analysis. In order to maintain that principle, methods
of specifying Kp based solely on elastic analyses are required, and the
obvious method would be to adopt Irwin's first-order plasticity correction
as a means of incorporating elastic-plastic effects. Within this approxima-
tion the value oiKp for a crack length a is given as

Kp = KM')

where a' is an effective crack length given by

and a is a constant which incorporates the constraint on plastic deforma-

tion. For initial loading by self-equilibrated thermal or residual stresses, it
is unlikely that the degree of plastic deformation prior to the addition of a
mechanical load will be so great that Irwin's first-order correction, along
with a pessimistic choice for a, will not be adequate enough to give an
upper bound to Kp. This estimate would then produce an upper-bound
value for Sr'. Suggested values for a are, perhaps, the plane-stress value
of ir/16 to be used for plane-strain problems, and double this value, TT/S,
to be used for plane stress or where loss of plastic constraint is possible.
In the residual stress example, remembering that the strip yield model
simulates plane stress, this choice would have produced a failure stress of
approximately 325 MNm~^ compared with the reference value of 266
MNm~^, and the previously calculated approximate value of 405 MNm~^.
The foregoing problem is encountered only when the initial loading is
thermal or residual. If it is mechanical, then the Failure Assessment
Curve provides a realistic lower-bound failure criterion. Furthermore, the
shift in the position of the origin for initial mechanical loading is determined
using the Failure Assessment Curve procedures [1] and hence will already
have built-in safety factors.
In tough materials, particularly those which fail by ductile mechanisms,
stable crack growth may precede an instability leading to fast fracture. In
these circumstances it is recommended that the toughness value at the
initiation of crack growth be used in the evaluation of the assessment
parameter K, [/]. The problem of predicting the instability point is a very
difficult one and depends not only on material properties but also on the
loading system and the geometry of the component. In the case of ductile
 CHELL ON A FAILURE ASSESSMENT DIAGRAM 603

failures the problem has recently been discussed [19]. In terms of the Failure
Assessment Diagram the effects of system loading conditions are reflected
in the ratio K\/Kp (compare the earlier section on fixed-grip loading) and
hence the approximate transformation procedure adequately caters for
these as regards an assessment based upon initiation.
Some of the advantages of basing failure assessments on the Failure
Assessment Diagram rather than on an explicit elastic-plastic failure
analysis have already been mentioned. Besides the simplicity of a dia-
grammatical representation, and the ease with which changes in the
applied loading can be determined, the Failure Assessment Diagram also
allows the effects of variations in fracture toughness and flow stress to be
studied. For example, the effect of dividing the toughness (or flow stress)
by two is to double the value of Kr (or Sr) and move the failure assessment
point accordingly.
If initial loading is present, the same procedures apply for variations in
toughness, but the effects of changes in flow stress on the apparent value
of Sr due to the initial load are more complicated, but still relatively
easily calculated. By applying the transformation procedure to initial
loading, most of the advantages of the Failure Assessment Diagram can be
recovered with respect to additional mechanical loading.

Summary of Failure Assessment Procedure

A Single Mechanical Load

The procedure in this case for obtaining a failure assessment point is
outlined in Ref / . To allow for crack growth due to fatigue, for example, it
is useful to draw a path to failure on the Failure Assessment Diagram for
a particular load level. Other paths to failure are then easily constructed
(see Fig. 1) and hence useful information extracted, such as the variation
of critical defect size with applied load. For each crack length, only two
calculations are necessary, a linear elastic determination of ^ i to obtain
Kr, and a Umit analysis for ai to obtain Sr. If the assessment point is
above or on the Failure Assessment Curve, failure is predicted; if below,
the structure is safe at the given load.

A Mechanical Load Superposed on a Constant Initial Mechanical Load

The same procedure as in the foregoing is followed to obtain a failure
assessment point (Kr, Sr) for the constant initial load. This point then
forms the origin of a new failure diagram where the K-a.xis goes from Kr
to 1 and the 5-axis goes from 0 to 1 and is contracted into the length
1 — Sr. With respect to these new axes, the failure assessment procedure
604 ELASTIC-PLASTIC FRACTURE

is then repeated for the additional loading to obtain an assessment point

(Kr', Sr'). The magnitude of the additional load is directly proportional
to the distance of the point {Kr', Sr') from the new origin.

A Mechanical Load Superposed on a Constant, Initial Thermal, or

Residual Stress
The ratio Kr^ = K\/Kp due to the initial loading must first be determined.
This can be accomplished using an elastic analysis and utilizing Irwin's
first-order plasticity correction factor made pessimistic by a suitable choice
of a in Eq 15. Then, either graphically or using Eq 14, the apparent value
of Sr corresponding to Kr^ can be found. The point {Kr, Sr), where K, is
obtained from the elastic stress intensity factor due to the initial loading,
now forms the origin of a new failure diagram based on the old /iT-axis
and a reduced 5'-axis, as in the case of initial mechanical loads described
in the foregoing. The assessment point due to any additional mechanical
load with respect to the new axes is then found as before.

Conclusions
1. The Failure Assessment Curve provides a realistic lower-bound failure
criterion for most mechanical loading situations.
2. The curve can be interpreted as being equivalent to an elastic-plastic
analysis based upon an approximate functional form for the J-integral.
3. Using this interpretation, failure curves can be constructed from any
elastic-plastic analysis which relates the ratio of elastic to plastic stress
intensity factors to the applied stress divided by the collapse stress.
4. This interpretation also allows thermal, residual, and secondary
stresses to be included within the framework of a failure diagram.
5. In the presence of initial loading, either by mechanical loads or by
thermal and residual stresses, a failure assessment point can be found
which forms the origin of coordinates with respect to a new failure diagram
based on part of the Failure Assessment Curve. The assessment of any
additional loading must be made with respect to a set of axes centered at
this new origin of coordinates.
6. This procedure allows thermal and residual stresses to be incorporated
within the concept of the Failure Assessment Diagram.

Acknoviledgment
The author would like to thank Dr. I. Milne for helpful discussions and
Dr. I. L. Mogford for his comments on the manuscript.
This work was performed at the Central Electricity Research Laboratories
and is published by permission of the Central Electricity Generating Board.
 CHELL ON A FAILURE ASSESSMENT DIAGRAM 605

References

[/1 Harrison, R. P., Loosemore, K., and Milne, I., "Assessment of the Integrity of Structures
Containing Defects," CEGB Report No. R/H/R6, Central Electricity Generating Board,
U.K., 1976.
[2] Heald, P. T., Spink, G. M., and Worthington, P. J., Materials Science and Engineering,
Vol. 10,1972, p. 129.
[3] Dowling, A. R. and Townley, C. H. A., International Journal of Pressure Vessels and
Piping, Vol. 3, 1975, p. 77.
[4] Rice, I. R. in Mathematical analysis in the mechanics offracture. Vol. 2 (H. Liebowitz,
Ed.), Academic Press, New York, 1968, p. 191.
[5] Chell, G. G. and Ewing, D. J. F., International Journal of Fracture Mechanics. Vol. 13,
1977, p. 467.
[6] Chell, G. G. and Milne, I., Materials Science and Engineering, Vol. 22, 1976, p. 249.
[7] Bilby, B. A., Cottrell, A. H., and Swinden, K. H. in Proceedings. Royal Society, Vol.
A272, 1963, p. 304.
\8\ Dugdale, D. S., Journal of the Mechanics and Physics of Solids, Vol. 8, 1960, p. 100.
[9] Keer, L. M. and Mura, I. in Proceedings. 1st International Conference on Fracture,
T. Yokobori, T. Kawasaki, and J. L. Swedlow Eds., Published by Japanese Society for
Strength and Fracture of Materials, Tokyo, Vol. 1, 1965, p. 99.
[10] Chell, G. G., Engineering Fracture Mechanics. Vol. 9, 1977, p. 55.
[//] Hayes, D. I. and Turner, C. E., International Journal of Fracture, Vol. 10, 1974 p. 17.
[12] Sumpter, i. D. G. and Turner, C. E., work reported by P. Chuahan in General Electric
Co. Report No. W/QM/1974-14, 1974.
[13] Sumpter, J. D. G., "Elastic-Plastic Fracture Analysis and Design Using the Finite
Element Method," Ph.D. thesis. Imperial College, London, U.K., 1973.
[14] Andersson H., Journal of the Mechanics and Physics of Solids. Vol. 20, 1972, p. 33.
[15] Chell, G. G. and Harrison, R. P., Engineering Fracture Mechanics, Vol. 7, 1975,
p. 193.
[16] Chell, G. G., International Journal of Pressure Vessels and Piping, Vol. 5, 1977,
p. 123.
[17] Ainsworth, R. A., Neale, B. K., and Price, R. H. in Proceedings, Conference on the
Tolerance of Flaws in Pressurized Components, Institution of Mechanical Engineers,
London, U.K., 16-18 May 1978.
[18] Chell, G. G., International Journal of Fracture Mechanics, Vol. 12, 1976, p. 135.
[19\ Paris, P. C , Tada, H., Zahoor, A., and Ernst, H., "A Treatment of the Subject of
Tearing Instability," USNRC Report NUREG-0311, National Research Council,
Aug. 1977.
/. D. Harrison,^ M. G. Dawes,^ G. L. Archer,^
and M. S. Kamath^

The COD Approach and Its

Application to Welded Structures

REFERENCE: Harrison, J. D., Dawes, M. G., Archer, G. L., and Kamath, M. S.,
"The COD Approach and Its Application to Welded Structaics," Elastic-Plastic
Fracture. ASTM STP 668, J. D. Landes, J. A. Begley, and G. A Clarke, Eds.,
American Society for Testing and Materials, 1979, pp. 606-631.

ABSTRACT: The crack opening displacement (COD) approach has, since its inception
as a fracture initiation parameter in yielding fracture mechanics, gained increasing
acceptance both as a viable research tool and an engineering design concept. In the
United Kingdom, The Welding Institute has pioneered the application of COD in the
structural fabrication industry largely through the development of the COD design
curve. This paper is a representation of the philosophy underlying design curve
applications and illustrates the practical significance of COD by drawing on case
studies from various welded structures.
Following a brief appraisal of the origins of the design curve, the paper outlines
procedures for the use of COD in design, that is, either as a basis for material selection
or in setting up acceptance levels for weld defects. The reliability of a small-scale
test prediction from the design curve has been investigated on a statistical basis from a
survey of more than 70 wide-plate tension test results in which the material had also
been categorized by COD. Specific practical examples are then discussed covering the
various types of application, material selection defect assessment, and failure investiga-
tion.
Structures included in these examples are offshore rigs, oil and gas pipelines,
pressure vessels, etc., with special emphasis on the manner in which small-scale COD
test results are translated to the structural situation.
Finally, the paper includes information on the considerable range of structures to
which the concept has been applied during the pastfiveyears.

KEY WORDS: mechanical properties, fracture test, crack initiation, toughness, crack
opening displacement J-integral, elastic-plastic cracking (fracturing), fracture properties,
structural steels, design, crack propagation

'Head of Engineering Research, principal research engineers, and research engineer,

respectively. The Welding Institute, Abington Hall, Abington, Cambridge, England.

606

Copyright 1979 b y A S T M International www.astm.org

 HARRISON ET AL ON COD APPROACH 607

Nomenclatare

a
Depth of surface crack or half height of buried crack
OCT
Critical value of a for unstable fracture
flmax
M a x i m u m allowable value of a
a
Half length of through-thickness rectilinear crack
OCT
Critical value of a for unstable fracture
flraax
M a x i m u m allowable value of a
BSection or specimen thickness
c
Half length of buried or surface crack
EYoung's modulus
e
Strain
cy
Yield strain = OY/E
G\Mode I crack extension force
/
The /-contour integral
K\Mode I stress intensity factor
Ku Critical plane-strain stress intensity factor
LPlastic constraint factor
mPlastic stress intensification factor
Ma,M^ Correction factors for buried cracks at the edge of the crack
nearest to a free surface due to that free surface and due to the
remote free surface
M Correction factor for buried cracks = Mo X M^
Mt Finite thickness correction factor for surface cracks
Ms Free-surface correction factor for surface cracks
p Depth below surface of buried defects
R Radius of holes
T Wall thickness of cylindrical vessels
max
max
W Half width of CNT a n d D E N T specimens
6 Crack tip opening displacement (COD)
be Critical value of 6
bm b at first attainment of m a x i m u m load plateau in bend test
T Constant = 3.142
a Applied stress
ax Effective stress = ( a X SCF) + OR
OY Uniaxial yield stress
* Nondimensional C O D = bE/lTtava
#2 Complete elliptic integral of second kind.
SCF Elastic stress concentration or, where localized uncontained
yielding occurs, the strain concentration factor
608 ELASTIC-PLASTIC FRACTURE

The theoretical and experimental basis for crack opening displacement

(COD) as a fracture characterizing parameter in yielding fracture mechanics
is described by Dawes [1]^ elsewhere in this publication. In order to place
the application of the COD approach in context it is convenient to think
in terms of the brittle-to-ductile transitional behavior encountered with
rising temperature in structural steels. If single-edge notched bend specimens
of thickness B equal to that of the structure, of width W = 2B, and with
notch depth a = B (the preferred COD test specimen geometry), are
tested over a range of temperatures, the following behavior may be expected.
At low temperature, failure will occur under elastic conditions, the test
will give a valid value of K\o to the ASTM Test for Plane-Strain Fracture
Toughness of Metallic Materials (E 399-74), and the result may be applied
in structural analysis using linear elastic fracture mechanics (LEFM). With
increasing temperature, the toughness will rise until the A'lc measurement
capacity of the specimen (the greatest possible capacity for the given
thickness) is exceeded and failure will occur only after significant yielding.
COD will then be determined from the test record and the result may be
applied using yielding fracture mechanics as explained later.
With further increases in temperature, the material will behave in a
fully ductile manner so that the specimen does not fracture but fails by a
simple plastic instability. In such cases, structural failure will also be by
plastic instability and this will be assessed by limit load analysis.
Proposals for a weld defect assessment method based on a continuous
approach covering these three regimes are currently in an advanced stage
of drafting by a British Standards Committee. The approach may be
summarized as follows:

Analysis
Specimen Behavior Structural Behavior Method
Kio elastic LEFM
COD contained yielding YFM
fully plastic plastic instability limit load

The current paper deals only with the proposed method of application of
yielding fracture mechanics. Because of the greatly increased complexity of
rigorous elastic-plastic analyses compared to LEFM, the approach is
simplified and employs a 'design curve' which is semi-empirical. This
curve is considered to be conservative and makes possible swift assessments
in practical situations, but more accurate analyses of specific problems are,
of course, possible.

^The italic numbers in brackets refer to the list of references appended to this paper.
 HARRISON ET AL ON COD APPROACH 609

Derivation of the Design Curve

The evolution of the COD design curve has been described in detail by
Dawes and Kamath [2]. The basis was established by Burdekin and Stone
[3], who studied the extension of the Dugdale strip yield model into the
general yielding regime.
The design curve takes the form of a relationship between the nondimen-
sional COD, $, and the ratio of applied strain to yield strain, e/er; * is
defined as 6c/2ireyamax- The applied strain, e, is taken as the local strain
which would exist in the vicinity of the crack, were the crack itself not
present. The significance of Omax should be stressed. The design curve has
always been intended, as the name implies, to be one which can be used
directly in design. Its purpose is to give conservative predictions of the size
of defect which can be allowed to remain in a structure without repair
and it is not intended to predict criticality. Thus, Omax should be smaller
than the critical defect size act.
The original curve of Burdekin and Stone was changed to take account
of later experimental findings, first by Harrison et al [4], then by Burdekin
and Dawes [5], and was finally set out in its current form by Dawes [6],
This is given by

* = (—V for — < 0.5 (la)

* = (—) -0-25 for — > 0.5 (Ife)

\ey/ ey

It can be shown that, for small-scale yielding, 6 is related to d by

Gi — mayd
where the plastic stress intensification factor m is equal to 1 for plane stress.
Hence
Gi ^ ^ ^ g^Tra
<JY OYE OYE

or

ZTreya 2 \ffr/ 2 \er/

Thus Eq la has a factor of safety of 2 on defect size based on the plane

stress equivalence between 6, K, and G at low stresses.
610 ELASTIC-PLASTIC FRACTURE

Method of Application
For situations where the effective ratio of defect size to plate width a / W
is less than about 0.1 and where the nominal design stress a is less than the
yield stress of the base material, Dawes [6] proposed that Eq 1 could be
rewritten as follows in terms of stress

amax = — T for — ^ "-5 (3a)

_ bcE (7i
flmax = —. ... , for — > 0.5 (3b)
2ir{<Ti — O.zSffy) ay

where a\ is the total pseudo-elastic stress in the vicinity of the defect.

While a\ may be above yield, the structure itself may still behave in a pre-
dominantly elastic manner. This is because the yielding in the zone under
consideration is contained by the surrounding elastic material. In welded
structures, contained yield occurs as a result of residual stresses which
may themselves be equal to the yield stress and may be additive to the
applied stress. Contained yielding also occurs at stress concentrations
where pseudo-elastic stresses may be well above yield.
For general applications, the values of ai as given in Table 1 were sug-
gested.

Part-Through Surface and Buried Defects

The design curve was originally formulated on the basis of through-
thickness defects. Dawes [6] suggested that part-through cracks could be
dealt with by assuming that, for contained yielding situations, the param-
eters governing flaw shape effects would be the same as those under

TABLE 1—Total pseudo-elastic stress values.

Crack Location Weld Condition ai

Remote from stress stress relieved" a

concentrations as welded a -\- ay''
Adjacent to stress stress relieved" SCF*^ X a
concentrations as welded (SCF X a) •¥ ay''

" Here it is assumed that post weld heat treatment (PWHT) has eliminated all the residual
stresses. Often this will not be so and it is prudent to make some allowance for the residual
stress remaining after PWHT.
'' It has been assumed that residual stresses of yield point magnitude will exist in as-welded
structures. While this is true for stresses along the weld, transverse residual stresses can
often be assumed to be lower than yield in specific cases.
"^ Strain concentration factor.
 HARRISON ET AL ON GOD APPROACH 611

linear elastic conditions. It was realized that this approach could not be
justified rigorously, but it seems unlikely that elastic-plastic solutions for
the part-through crack will be available for some time to come. The follow-
ing LEFM expression was used to describe a semi-elliptic surface crack

^i = 1 (4)

For the equivalent through thickness crack of length 2a

^i = a'flta

Thus

a a fMtMs \ ^
(5)
B 5 V *2

Values of

M,Ms\_ /a_ a_
$2 / \fi , 2C

were taken from a survey by Maddox [7] and a/B is plotted against a/B
in Fig. 1.
With the exception of deep surface cracks, Fig. 1 agrees closely with
formulas proposed more recently by Newman [8].
For buried elliptical cracks, the equivalent equation to (5) is

M was calculated as the product of the magnification factors M^ and

Mo applicable to the stress intensity factor at the end of the minor axis
which approaches nearest to a free surface. Mo is the magnification factor
at that point due to the presence of the near surface and M^ is that due to
the presence of the more remote free surface. M, and Mo were taken from
the work of Shah and Kobayashi [9]. For a/c = 0, M was derived from
Feddersen's relationship [10]

M = (^sec
(sec -—- j
612 ELASTIC-PLASTIC FRACTURE

ic 1 05 0-4 03' 02 01
a
2c

2c

J
1" •
'v^ J TT
01
] •

001 i
0 01 10

FIG. 1—Relationships between surface crack dimensions and equivalent through-thickness

crack dimension si.

Figure 2 is a plot of a/ip + a) against a/[2(p + a)] or a/B for buried

defects.

Recent Experimental Justification for the Design Curve

The implementation of the COD approach through the simple design
curve, which takes into consideration the effects of residual stresses and
geometric stress concentrations, has found wide application to welded
structures. However, because of its semi-empirical origins and inherent
simplicity, the design curve, as already stated, predicts maximum allowable
crack sizes and not critical crack sizes, with a margin of saftey only vaguely
estimated as being approximately 2.0. It was decided, therefore, to carry
out an assessment of the COD design curve by making a comparison of
the allowable crack sizes predicted by the small-scale COD tests and the
critical crack sizes at fracture in wide-plate tests [11]. From a survey of the
published literature and work carried out at The Welding Institute, a
total of 73 sets of small- and large-scale tests was compiled. The results
were then analyzed on a statistical basis. The main steps and observations
from these analyses are summarized in the following.
Initially, the test data were processed, as shown in Fig. 3, to give safety
 HARRISON ET AL ON COD APPROACH 613

FIG. 2—Relationships between buried crack dimensions and equivalent through-thickness

dimension a.

factors I and s for through-thickness and surface cracks, respectively. A

comparison between the predicted allowable and the critical crack sizes
obtained is shown in Fig. 4 with some of the important probability levels
indicated. This shows that, on average, the design curve has a built-in
factor of safety of approximately 2.5, and the maximum allowable size
derived from the curve implies a 95.4 percent probability of survival with
respect to the wide-plate test. However, when the scatter in results is taken
into consideration, there appears to be little scope for modifying the shape
of the design curve.
An examination of the variables in the wide-plate tests drew attention to
the influence of residual stresses on brittle fracture. For situations where
it was reasonable to assume no residual stresses, for example, plain plate
and some stress-relieved weldments only. Fig. 5a shows that critical values
of # were generally well below the design curve. However, when the results
for as-welded plates are added to the plot and if residual stresses are still
assumed to be zero (Fig. 5b), it can be seen that a significant proportion of
the as-welded plate specimen results fall to the left of the design curve. In
other words, the failure stress assumed was lower than that to be expected
from the design curve for the known value of $. The as-welded wide-plate
results fall within the same general scatter band as those for plates assumed
to be free from residual stress and were thus safely predicted by the design
614 ELASTIC-PLASTIC FRACTURE

Wide Plato Ilaterial

test xat-
Thickness
Temperature
Notch location
Orientation

1
Obtain:
1
Dbtain:
Fracture stres
Constants: E, y 5
or strain (if c
failure after Usually minimum
net section 3 tests
yielding)

-£>-

Calculate:
a using
max "
equations (3)

L
Convert a to equivalent a
(through max ^ max
crit
thickness) for surface crack in wide plate test
using Fig. 1
(surface)

FIG. 3—Method of processing COD and wide-plate test data.

 HARRISON ET AL ON GOD APPROACH 615

-1 1 1 1 \ 1 1 \ r
52

49

*f-

36

32

28

2i

S 20

O .^ Plainpiate tests
• /^Weldments with negtigi ble residual stresses assumed
• ^Weldments with residual stresses
Circles Through-thicltness notch
Triangles Surbcenotch

I -1- -J_ _J_ _L

4 a 12 16 20 24 28 32
^max. or "max (mm)

FIG. 4—Comparison of critical and maximum allowable crack sizes showing probability
levels and safety ratios.
616 ELASTIC-PLASTIC FRACTURE

10 1
/ O
0 ^ Plain plate tests
• A. Weldments with negl igible
residual stresses assumed 4 -ra 5
• ^ Weldments with residual
stresses / 0
Circles Through -thickness notch / A ^16 S
Triangles Surface notch
— 16 7
0 A — 12 5

u -^10-3
Design /
^ A
Curve /
10 7 ° -
o /
/ *

b / 09

/ o
I'M

/ °
/ o
01

1
0> 1-0
"/ or ";
6)' (Ty-
(a) plain plate and weldments with negligible residual stress
effects leg: stress relieved welds)

FIG. 5—Design curve relationships between nondimensional COD and applied strain and
stress (normalized), with experimental COD/wide-plate test results.
 HARRISON ET AL ON GOD APPROACH 617

Design Curve

10-

i-en

01

001
-±. or3-
ey try
(b)All wetdmenU, but residual stress neglected m design
curve calculation.lncludes weUments from figuie S(a)

FIG. S—Continued.
618 ELASTIC-PLASTIC FRACTURE

^en

0 01
01 10
—'-or—'-
BY try
(c)Weldmentsy:e!c! magnitude residual stress Included

FIG. 5—Continued.
 HARRISON ET AL ON COD APPROACH 619

curve only when full-yield residual stresses were assumed to be active

(Fig. 5c).
The assessment also showed that when residual stresses were present
there were no significant differences in the average factors of safety for
through-thickness and surface cracks. In the absence of residual stresses,
however, the results suggested that the factors of safety were slightly
higher in the case of surface cracks.

Numerical Assessments of the Design Curve

Sumpter [12] and Sumpter and Turner [13] report the results of elastic-
plastic finite-element analyses. Some of these were for an elastic perfectly
plastic material, but some assumed a work-hardening law approximately
equal to that for the ASTM Specifications for Pressure Vessel Plates, Alloy
Steel, Quenched and Tempered, Manganese-Molybdenum and Manganese-
Molybdenum-Nickel (A 533B-74). The latter were compared by Sumpter
with the COD design curve. The following geometries were studied.

1. Edge cracked plate a/W = 0.1,

2. Crack at the edge of a hole of radius I?, a/R = 0.05, 0.1, and 1.0, and
3. Radial crack at the bore of a pressurized cylinder of thickness T, a/T
= 0.03.

The results are plotted in Fig. 6. For the edge-cracked plate there was close
agreement between the finite-element analysis results and the design curve.
For cracks at the edge of a hole, the design curve was shown to be con-
servative, but not excessively so for ratios of crack length to hole radius, a/R,
of 0.05 and 0.10. As stated in the section dealing with the application of the
design curve, the recommended procedure is to calculate the value of ai as
SCF X a. Sumpter shows that this procedure becomes very pessimistic for
the unusual case of very long cracks at the edge of a hole with a/R = 1.0.
However, as Burdekin and Dawes originally suggested, it is more realistic
for a/R > 0.2 to assume that the crack is one of total length a + 2/? in a
stressfieldequal to the membrane stress. If this procedure is adopted for the
results for a/R = 1.0, the plot of * against e/er comes closer to that for a/R
= 0.1, but remains very conservative. The mean factor of safety on $ be-
tween the results for a/R = 0.05 and 0.1 and the design curve for a given
value ofe/ey is 2.0.
For the radially cracked cylinder, a comparison was made with the design
curve for one ratio of crack depth to cylinder wall thickness, a/T = 0.03.
The design curve was again found to be conservative with factors of safety on
* of 4.5 at e/er = 0.6, 2.5 at e/er = 1-0, and 1.2 at e/er = 1-6.
620 ELASTIC-PLASTIC FRACTURE

-a

"a,

r
d

•-en
 HARRISON ET AL ON COD APPROACH 621

The / Design Carre

Begley et al [14] have proposed a / design curve which is in essence very
similar to the COD design curve. The similarity between the two approaches
has been discussed by Merkle [15].
The design curve of Ref 14 takes the form

Eiraey^^ = ( yerj
7:] fo"- iey^ l - O (7a)
and
T r ^ = —-1 for — >1.0 ab)
Eiraey^ ey ey
It will be seen that these are similar in character to Eq 1. It is generally stated
that J = m ay 8, where w is a plastic stress intensification factor which
ranges from about 1.0 to 2.0. Substituting f o r / and assuming m = 1.0,
Eqs 7a and 7b reduce to

1 / eV e
* = —(— for — < 1.0 (8a)
2 \ey/ ey

and

* = — - 0.5 for — > 1.0 (8i)

ey ey

For m = 2.0 they reduce to

# = — — for — < 1.0 (9a)

4 \ey/ ey
and

$ = ^(— - 0.25\ for — > 1.0 (%)

2 \ey J ey
These are plotted for comparison with the design curve in Fig. 7.
It should be borne in mind that, while the COD design curve is intended
to be conservative and is empirically based on the results of wide-plate
tests on specimens where it may be assumed that m varied, the curve of
Begley et al is intended to predict critical conditions. Viewed in this light,
it is felt that the two approaches are in reasonable agreement.
622 ELASTIC-PLASTIC FRACTURE

1 1 1

-
\
\
\ \
\ \ \
\ \ e
o
\\ \\ \
\ \ \
_ \ \
\ \ \
\ \ \
\ \
\ \ o
\ \ b^ \ •;?
\
\
\
\ ^
\
\
V .1
\ \
<ii
\ \ \ §
\ \ \
Q: Q,
§

v'^
Q
3

•5)
" \ \ \
I
\ \ \
1
1 s
1
1

1 1 ^
H-e-i 1
 HARRISON ET AL ON COD APPROACH 623

Experience in tlie Practical Application of the COD Design Curve

The design curve approach has been applied in recent years to a great
range of structures in a variety of contexts.
The applications may be broadly classified into four major groups:
1. Material selection (design stress and defect levels predetermined).
2. Acceptance levels for weld defects (decisions regarding known defects
or fixing defect acceptance standards where material and design stress are
predetermined).
3. Fixing allowable stress (material and inspection level predetermined).
4. Failure analysis.
Lists of some of the applications with which the authors have been
concerned over the past five years are given in Tables 2-6. These are by
no means exhaustive. Some specific examples are discussed in greater
detail in the following.

Material Selection
If the design stress is known and a size of defect which might escape
detection by nondestructure testing (NDT) is assumed, the design curve
may be used to determine the required level of toughness. Table 2 lists
welding consumable manufacturers whose products have been tested by
The Welding Institute in order to establish whether toughness levels fixed
in this way have been achieved. Table 3 lists a range of structures where
this approach has been used. Two specific examples are given in the
following.
Offshore Production Platforms—The production platforms for British
Petroleum's (BP) Forties Field in the North Sea involved a quantum jump
in size over the great majority of similar structures. Because of the greater
depths in the North Sea and the severity of the environment, structural
steels of higher strength (320 N/mm^ yield, 500 N/mm^ tensile strength)
and increased thickness (60 to 100 mm) were employed. Would such
structures be safe if the welds in stress concentration regions at the massive
intersections (nodes) remained in the as-welded conditions, or should they
be post-weld heat treated (PWHT) as would be mandatory for pressure
vessels built of similar materials? It was assumed that surface and buried
defects up to 12.5 and 25 mm deep, respectively, might escape detection

TABLE 2—Welding consumable manufacturers for whom COD tests have been carried out
with the objective of meeting specific requirements.

Arcos BOC Murex Esab

Big 3 Lincoln UK GKN Lincoln Kobe Steel
Lincoln Electric West Falische Union Varios Fabrieken
Metrode Oeriikon Phillips
624 ELASTIC-PLASTIC FRACTURE

TABLE 3—Cases where the COD design curve has been used as a basis for material selection.

Class of Structure Company Project

Pipelines Aramco LNG" pipeline

Aramco spiral weld gas pipeline
Bechtel International offshore gas pipeline
HP crude-oil pipeline
HP automatic MIG'' girth welds for
Ninian and Forties field lines
Brown and Root offshore flare line
Brown and Root weld overlay on offshore riser pipe
Canadian Artie Gas natural gas pipeline
Conoco offshore gas pipeline
Hoesch pipeline steel
Gasunie thick-walled pipeline
Shell oil pipeline girth welds
Offshore structures BP selection of weld procedures and
decision regarding post-weld heat
treatment
Conoco crane pedestals
Highland Fabricators production platform
Laing offshore production platform
McDermott production platform
Phillips Petroleum general specification for offshore
structures
Redpath Dorman Long production platform
Shell general specification for production
platforms
Shell jack-up rig
Pressure vessels and Air Products ctyogenic vessels (9%Ni steels)
boilers Clarke Chapman boiler drum (decision regarding post-
weld heat treatment)
Intemation Combustion boiler drum
Nuclear Central Electricity stainless steel weld for AGR*^ boiler
Generating Board
Nuclear Power Company steam drum SGHWR''
The Nuclear Power Group circulator outlet gas duct/liner
weld in AGR
Miscellaneous AUis Chalmers general purposes
Aramco plates for low-temperature service
Ove Arup cast steel weldraents for building
frame
Capper Neil LPG' storage tanks
Capper Neil oil storage tanks
Central Electricity penstocks for pumped storage scheme .
Generating Board
Chicago Bridge & Iron LNG storage tanks
High Duty Alloys crash barriers
High Duty Alloys repair procedures for 12 OOO-t
extrusion press
Johns & Waygood high-rise building
Kockums high heat input welding for ships
Lindsey Oil Refineries oil storage tanks
Uddeholms 9%Ni steel for LNG tanks
Whessoe S-m low-speed wind tunnel
Whessoe oil storage tanks

"Liquefied natural gas. ''steam-generated hot-water reactor.

^ Metal inert gas. 'Liquefied petroleum gas.
"^ Advance gas-cooled reactor.
 HARRISON ET AL ON COD APPROACH 625

TABLE 4—Cases where the COD design curve has been used to fix acceptance levels for
weld defects and inspection sensitivity.

Company Project

Clarke Chapman electroslag welds in boiler drum

Central Electricity Generating Board oil storage tanks
Elliott compressor rotor
Gasunie gas pipelines
Shell oil pipelines
Shell offshore pressure vessels

TABLE 5—Assessment of weld defects.

Class Company Project

Pipelines BP Aleyaska pipeline

BP refinery pipes
BP ethelyne pipeline
Danish Weld Inst spiral-welded oil products line
Gasunie gas pipelines
Metallurgical Consultants spiral-welded pipe
Unit Inspection spiral-welded pipe
Offshore Conoco (rffshore crane pedestals
structures etc. Occidental deck modules
Shell flash welds in anchor chain links
Pressure vessels and British Gas gas bullets
boilers etc. Clarke Chapman steam and mud drums
ICI hydro desulphurizer
Richard Ross nozzle welds
Whessoe pressure vessels
Miscellaneous Colocotronis supertanker
Dorman Long (South Africa) pumped storage scheme penstocks
Cleveland Potash mine shaft lining

TABLE 6—Failure investigations.

Company Structure Type

Bechtel International pile for offshore platform

BP oil tankers
Conoco offshore gas pipeline
Conoco rolled beams for deck module
Noble Denton leg for offshore structure
Shell pile for offshore platform
South of Scotland Electricity Board steam riser pipe
South of Scotland Electricity Board feedwater header forging
626 ELASTIC-PLASTIC FRACTURE

by NDT in these complex structures. Photoelastic analysis of a typical node

using the frozen stress method indicated a maximum SCF of 8, and the
nominal design stress was 75 N/mm^ or approximately Vi ay. The required
COD value determined from the design curve is given by Eq 3b as

dc — — = — ((Ti — 0.25 (JY)

E

Hence, if the nodes are as-welded

(71 = (SCF X a) + OR = (8 X V4 ay) + ay = 3ay

and substituting for ay and a max gives

8c = 0.37 mm

COD tests were carried out at the design temperature of — 10°C on the
parent steel, BS 4360 Grade SOD, which gave a minimum 5„ = 0.49 mm,
but the best weld metal, out of the total of 17 tested, gave only 8c =
0.12 mm.
It was clear that it would be necessary to heat treat the nodes in order to
obtain the required defect tolerance. It after PWHT, OR is assumed to be
zero, a I becomes 8 X V* ay = lay. This gives a required COD of

8c = 0.24 mm

Not only did PWHT lower the required COD, but it significantly increased
the toughness of the weld metal. Five weld metals were found giving
satisfactory toughness and one had a minimum value of 6m = 0.49 mm.
This study (which has been described more fully elsewhere [16]) showed
quite clearly that it was necessary to heat treat the node regions to ensure
safety of the complete structures. This decision had a marked effect on the
design philosophy adopted.
Specification of Toughness of Girths Welds in Large-Diameter Pipeline
for Service in Arctic Regions—Normally girth welds are not highly stressed
in the longitudinal direction and hence fracture risks are very low. When
the line goes through areas liable to subsidence or earthquakes or both,
however, high longitudinal bending stresses can develop. In this particular
case, strains up to 0.5 percent due to the aforementioned causes were
envisaged, which meant that weldments with good fracture resistance were
needed, particularly in view of the associated low ambient temperatures
(down to — 40°C). Semiautomatic and manual welding processes were
considered in conjunction with pipe to API-5LX70 and 19-mm wall thick-
ness. The fracture resistance of the various regions of the heat-affected
 HARRISON ET AL ON COD APPROACH 627

zones (HAZs) and weld metals was thoroughly examined by the COD test at
the minimum service and other selected temperatures on specimens taken
from welds made under field conditions. These tests demonstrated clearly
that the fracture toughness of the HAZs was adequate at all temperatures,
but that there could be difficulties in the weld metals, both manual and
semiautomatic.
In order to judge the validity of the maximum allowable flaw sizes
predicted from the COD results, a series of full-size bend tests was carried
out in a specially made rig. The girth welds in the pipe lengths contained
crack-like defects of suitable size in the center of the weld deposits, which
were made under typical field conditions. The weld area was placed at the
position of maximum bending moment in the rig and cooled to the required
temperature. Load was slowly applied up to failure and the maximum
applied strain measured by electric-resistance strain gages. After failure,
the depth of the actual defect Ocr was measured and compared with the
predicted maximum allowable depth amax. In the calculations it was
assumed that the peak tensile residual stress level transverse to the girth
welds would be between 0.5 and 0.75 er- Using these values of residual
stress, the minimum COD values from the small-scale tests, the measured
applied strain values, and Eqs 3b and 5, ratios of a^/a^^ between 2 and 3
were obtained, which is consistent with the normal experience from wide-
plate tests as indicated in Fig. 4.

Acceptance Levels for Weld Defects

Table 4 lists a number of cases where the design curve has been used to
fix NDT requirements at an early stage in design or construction. Table 5
lists cases where the concept has been used to assess the significance of
known defects where repair was felt to be undesirable for reasons of cost,
delivery, or because of the possibility of introducing more harmful defects
in the course of the repair.
Defects in the Alyeska Pipeline—As a result of disclosures to the press of
falsification of radiographs of the girth welds in the Alyeska crude oil
pipeline, all the X-ray films for the part of the line completed at that stage
were reexamined. This audit indicated that there were defects larger than
those permitted by the construction code, API-1104, in some 2955 of the
30 000 welds audited. The defect acceptance levels in API-1104 are set in
order to maintain a certain level of workmanship and bear no relationship
to the performance of pipelines in service. Nevertheless, the code had
been adopted by the Department of Transportation (EMDT) as a basis for
licensing the pipeline. The pipeline company through BP asked The Weld-
ing Institute to help to develop a submission to DOT for waivers to the
code. This case was based on the design curve and, in terms of the numbers
of specimens tested (some 450), represented the largest single case of its
application.
628 ELASTIC-PLASTIC FRACTURE

The effects of weld procedure (three different procedures), position

around pipe circumference, and notch orientation were studied. Nine notch
positions and orientations with respect to the weld metal and HAZ were
investigated. Tests were carried out at 0, —12, —18, —29, and —40°C. In
fact — 12°C was chosen as the design basis to allow for the possibility of
cold pressurization during start-up. Because of the considerable variety of
microstructures sampled, there was wide scatter in the COD values, but a
lower bound of 0.1 mm was used. At —40°C, one specimen gave a result
as low as 0.025 mm. Although these COD values are relatively low, no
specimen gave a result which could possibly be interpreted as a valid Ku
value for the specimen thickness of 12.7 mm (0.5 in.). The stresses con-
sidered in the analysis included pressure, thermal, bending due to expansion
and self-weight, earthquake, and residual. It was found as a result of the
analysis that none of the defects required repair.
The National Bureau of Standards carried out an independent assessment
[17]. In Ref 77 an analysis is reported by Begley, McHenry, and Read
based on a different interpretation of the COD test. This suggested that
the Alyeska proposals were conservative by factor of about 1.2 to 2.0.
However, it is believed that the Begley, McHenry, and Read approach is
aimed at predicting critical conditions while the design curve already
incorporates a factor of safety of 2.5 on the best estimate for criticality.
Thus the two approaches seem to be in good agreement.
The DOT accepted in principle the case for waivers, but asked for a
further safety margin of 2 to allow for problems of X-ray interpretation.
On this basis a small number of repairs were required.
In fact, many of the defects were repaired because the negotiations of
the case were time-consuming and because the pipeline company could not
afford to waste this amount of time when the proposal might have been
rejected. The important point, however, is that a major U.S. Government
authority, that is, DOT, accepted the principle of using a yielding fracture
mechanics analysis to derive defect acceptance levels in a large pipeline
project. Furthermore, some defects in a section of pipe crossing the Koyukuk
River, which would have cost about $5 million to repair, were accepted.

Fixing Allowable Stresses

The use of fracture mechanics to fix design stresses is less common, but
one case involving liquefied natural gas (LNG) storage tanks can be cited.
A program of COD and wide-plate tests was carried out on weldments
in 9 percent nickel steel, 18 mm thick, mostly at — 164°C, which was the
minimum design temperature. The main objective of the work was to see
if the design stress level could be safely raised from the API-620Q value
of 196.5 N/mm^ to 290 N/mm^. Two plate materials, A553 and A353,
were examined and two suitable weld metals. The COD tests showed that
 HARRISON ET AL ON GOD APPROACH 629

all the materials were fully ductile at —164°C. Values of a max between 7
and 19 mm were obtained for a design of 290 N/mm^ using COD values at
maximum load in conjunction with equations 3b and 5 and making
conservative assumptions about effects of distortion and residual stresses.
Through-thickness defects 20 to 25 mm long were incorporated in various
weld regions in wide-plate specimens and tested at — 164°C. The fact that
all the plates failed at stresses above the yield stress of the weld metal
indicated that the approach was very conservative. The work was carried
out about five years ago and, in view of the complete ductility of the
materials involved, it is now thought that a limit load approach, as sug-
gested in the introduction, would be more appropriate and less conserva-
tive. Nevertheless, it was clearly demonstrated that an increase in design
stress level to 290 N/mm^ was reasonable in terms of tolerance to severe
fabrication defects. This resulted in a significant reduction in the cost of
the LNG tanks.

Failure Investigation
A number of instances where structural failure conditions have been
compared with predictions from the design curve are listed in Table 6.
These represent some of the most interesting applications of the design
curve, but space permits only one example to be discussed in more detail.
The example chosen is the Cockenzie boiler drum. This has already been
discussed sometime ago by Burdekin and Dawes [5] and by Ham [19],
but it may now be reassessed in the light of the revised design curve and of
the defect shape corrections in Fig. 1.
The failure [19] during hydrostatic test at nominal stress level of 0.55
OY occurred from a large semi-elliptical surface defect 81 mm deep by
325 mm long at the edge of an attachment. The material, which was a
low-alloy structural steel, had a thickness of 141 mm, a yield stress of
376 N/mm^, and minimum COD value of 0.25 mm in the stress-relieved
condition.
Substitution of the foregoing values into the design curve gives amax =
74 mm that is, Omax/fl = 0.52. From Fig. 1, am^/B = 0.49, giving a
maximum depth of 69 mm for a surface defect with aspect ratio 0.25.
The factor of safety in this case is 1.17. This is smaller than usual, but it
could be influenced by two factors which were ignored in the analysis,
both of which would tend to increase it. First, residual stress was assumed
to be zero. It is probable, however, that some residual stress will have
remained since the material was thick and the geometry was complex.
Second, the defect was close to a nozzle and the local stress may have
been elevated because of this. The details available are not sufficient for
either of these possible effects to be assessed; however, use of the design
curve still gives a reasonable explanation of the failure.
630 ELASTIC-PLASTIC FRACTURE

Conclasions
The COD test is a useful method of studying the fracture toughness of
materials in the transition region between linear elastic behavior, where
Kic should be used, and fully ductile behavior, where a limit load approach
is appropriate. It was concluded from the statistical analysis of 73 sets of
tests, where predictions from the design curve were compared with the
results of large-scale tests, that the average inherent safety factor is ap-
proximately 2.5. The analysis also revealed a 95 percent probability of the
predicted allowable crack size being smaller than the critical crack size. It
was shown that the approach described is comparable with design curves
derived from finite-element analyses and /-analyses. It was concluded from
the several practical examples described that the design curve can be
successfully used in at least three different ways: (1) selection of materials
during initial design stage, (2) specification of maximum allowable flaw
sizes at design or after fabrication to establish the necessity for repairs, and
(3) failure analysis.

References
[/] Dawes, M. G., this publication, pp. 306-333.
[2] Dawes, M. G. and Kamath, M. S., "The Crack Opening Displacement (COD) Design
Curve Approach to Crack Tolerance," Conference on the Significance of Flaws in
Pressurised Components, Institution of Mechanical Engineers, London, England, May
1978.
[3] Burdekin, F. M. and Stone, D. E. W., Journal of Strain Analysis. Vol. 1, No. 2, 1966,
p. 194.
[4] Harrison, J. D., Burdekin, F. M., and Young J. G., "A I>roposed Acceptance Standard
for Weld Defects Based Upon Suitability for Service," 2nd Conference on the Significance
of Defects in Welded Structures, The Welding Institute, London, England, 1968.
[5] Burdekin, F. M. and Dawes, M. G., "Practical Use of Linear Elastic and Yielding
Fracture Mechanics with Particular Reference to Pressure Vessels," Conference on
Application of Fracture Mechanics to Pressure Vessel Technology, Institution of
Mechanical Engineers, London, England, May 1971.
[6] Dawes, M. G., Welding Journal Research Supplement, Vol. 53, 1974, p. 369s.
[7] Maddox, S. I., International Journal of Fracture Mechanics, Vol. 11, No. 2, April 1975,
pp. 221-243.
[8] Newman, 1. C. in Part-Through Cracks Life I'rediction, American Society for Testing and
Materials, 1979.
[9] Shah, R. C. and Kobayashi, A. S., International Journal of Fracture Mechanics, Vol. 9,
No. 2, 1973, p. 133.
[10] Feddersen, C. E. in Discussion to Plane Strain Fracture Toughness Testing of High-
Strength Metallic Materials, ASTM STP 410, 1967, p. 77.
[11] Kamath, M. S., "The COD Design Curve: An Assessment of Validity using Wide Plate
Tests," The Welding Institute Members Report 71/E, 1978, to be published.
[12] Sumpter, J. D. G., "Elastic-Plastic Fracture Analysis and Design Using the Finite
Element Method," Ph.D. thesis. University of London, Dec. 1973.
[13] Sumpter, J. D. G. and Turner, C. E., "Fracture Analysis in Areas of High Nominal
Strain," 2nd International Conference on Pressure Vessel Technology, San Antonio,
Tex., Oct. 1973.
[14] Begley, J. A., Landes, J. D. and Wilson, W. K. in Fracture Anafysis, ASTM STP 560,
American Society for Testing and Materials, 1974, pp. 155-169.
 HARRISON ET AL ON COD APPROACH 631

[/5] Merkle, I. G., International Journal of Pressure Vessels and Piping, Vol. 4, No. 3,
July 1976, pp. 197-206.
[16] Harrison, J. D. in Performance of Offshore Structures. Series 3, No. 7, Publication of
the Institution of Metallurgists, London, England, 1977.
[17] Berger, H. and Smith, J. H., Eds., "Consideration of Fracture Mechanics Analysis and
Defect Dimension Measurement Assessment for the Trans-Alaska Oil Pipeline Girth
Welds," National Technical Information Service Report PB-260-400, Oct. 1976.
[18] Harrison, J. D. and Carter, W. P., "The Use of 9%Ni Steel for LNG Application,"
Proceedings, Conference on Welding Low Temperature Containment Plant, The Weld-
ing Institute, London, England, Nov. 1973.
[19] Ham, W. M., Discussion to Conference on Practical Application of Fracture Mechanics
to Pressure Vessel Technology, Institution of Mechanical Engineers, London, England,
May 1971.
H. I. McHenry,' D. T. Read,' and J. A. Begley'

Fracture Mechanics Analysis of

Pipeline Girthwelds*

REFERENCE; McHenry, H. I., Read, D. T., and Begley, J. A., Fracture Mechanics
Analysis of Pipeline Girthwelds," Elastic-Plastic Fracture, ASTM STP 668, J. D.
Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and
Materials, 1979, pp. 632-642.

ABSTRACT: Size limits for surface flaws in pipeline girthwelds are calculated on the
basis of fracture mechanics analysis. Parameters for the analysis were selected from
data on a 1.22-m-diameter (48 in.), 12-mm-thick (0.46 in.) pipe welded by the shielded
metal-arc process. The minimum fracture toughness of the welds as determined by the
crack opening displacement (COD) method was 0.1 and 0.18 mm (0.004 and 0.007 in.),
depending on the flaw location. The yield strength of the welds was 413 MPa (60 ksi).
Because the toughness to yield strength ratio was high, elastic-plastic fracture mechan-
ics analysis methods were required to determine critical flaw sizes. Four approaches
were employed: (1) a critical COD method based on the ligament-closure-force model
of Irwin; (2) the COD procedure of the Draft British Standard Rules for Derivation of
Acceptance Levels for Defects in Fusion Welded Joints; (3) a plastic instability method
based on a critical net ligament strain developed by Irwin; and (4) a semi-empirical
method that uses plastic instability as the fracture criterion developed by Kiefner on
the basis of full-scale pipe rupture tests. Allowable flaw sizes determined by the Draft
British Standard method are compared with the critical flaw sizes calculated using
critical-COD and plastic instability as the respective fracture criteria. The results for
both axial- and circumferential-aligned flaws vary significantly depending on the
analysis model chosen. Thus, experimental work is needed to verify which model most
accurately predicts girthweld behavior.

KEY WORDS; carbon-manganese steel, fracture mechanics, fracture toughness, pipe-

line, radiographic inspection, weld flaws, weldments, crack propogation

During the summer of 1976, the National Bureau of Standards (NBS)

assisted the Office of Pipeline Safety Operations (OPSO) of the Depart-
ment of Transportation (DOT) in the evaluation of fracture mechanics as
a method of calculating allowable flaw sizes in pipeline girthwelds. The

•Contribution of the National Bureau of Standards; not subject to copyright.

'Metallurgist and physicist, respectively. National Bureau of Standards, Boulder, Colo.
80302.
^Associate professor, Ohio State University, Columbus, Ohio.

632

Copyright' 1979 b y AS FM International www.astm.org

 MCHENRY ET AL ON PIPELINE GIRTHWELDS 633

approach taken by NBS was to calculate allowable flaw sizes in a specific

pipeline in accordance with guidelines established by OPSO [1].^ The
material property data, the pipeline stresses, and the analysis approaches
used are described in this paper. The overall NBS program has been de-
scribed in a report to the DOT [2].

Pipeline Weldment Properties

The pipeline evaluated was built from 1.22-m-diameter (48 in.), API
5LX-65 steel pipe having wall thicknesses of either 12 or 14 mm (0.46 or
0.56 in.). It was welded by the manual shielded metal-arc process using
AWS E7010G and E8010G electrodes. The mechanical properties of sev-
eral pipeline segments containing girthwelds were evaluated by NBS [2],
the Welding Institute [3], and Cranfield Institute of Technology [4]. The
test results were used as the basis for selecting conservative values for use
in the analysis as summarized in the following.
The tensile properties of interest in the analysis were the yield strength
(o>), the flow strength (a), and Young's modulus (E). The yield strength
values used were 448 MPa (65 ksi) for the base metal and 413 MPa (60 ksi)
for the weld metal. The flow strength value was taken as the yield strength
plus 68.9 MPa (10 ksi) as recommended by Kiefner et al [5] on the basis of
extensive pipeline studies. Young's modulus was 208 GPa (30.2 X 10^ ksi)
for both the base metal and the weld metal.
The fracture toughness of the pipeline weldments was determined for
notch locations in the weld metal, the heat-affected zone (HAZ), and the
base metal at temperatures ranging from — 40 to 0°C (—40 to 32 °F). The
tests were conducted using the crack opening displacement (COD) method
by the British Welding Institute [3] and by Cranfield Institute of Tech-
nology [4]. The COD fracture-toughness values used to establish allowable-
flaw-size curves were 0.1 mm (0.004 in.) for nonplanar weld defects, which
tend to be randomly located in the weld, and 0.18 mm (0.007) for planar
defects, which tend to be on the inside surface. These values were the min-
imum values obtained for the applicable notch orientations at — 12°C
( + 10°F), 10 deg C (18 deg F) below the minimum exposure temperature
possible during service.

Pipeline Operating Stresses

The maximum credible stresses in the appropriate orientations were used
in the critical flaw size calculations. Girthweld flaws are typically oriented
circumferentially, and consequently the axial stresses are used in the anal-
ysis. Arc burns are typically spots or axially aligned drags, and flaw growth
would be caused by the hoop stresses.
^The italic numbers in brackets refer to the list of references appended to this paper.
634 ELASTIC-PLASTIC FRACTURE

Axial stresses in the pipeline during service are caused by the internal
pressure, thermal expansion, and earthquake loadings. These stresses are
superimposed on a pipe bending stress due to soil settlement. The maxi-
mum credible stress in the axial direction is 398 MPa (57.7 ksi) caused by
the hypothetical condition of extended winter shutdown, followed by full
pressurization. The maximum credible stress includes a bending stress
caused by 15 cm (6 in.) of soil settlement in 30 m (100 ft) of pipe length.
The maximum axial stress of 398 MPa (57.7 ksi) was used for critical flaw
size calculations for weld defects.
Hoop stresses in the pipeline are caused exclusively by internal pressure.
Maximum hoop stresses are 72 percent of Oy, 322 MPa (46.8 ksi), during
normal operation; 80 percent of Oy, 358 MPa (52.0 ksi), during surges;
and 95 percent of a,, 425 MPa (61.8 ksi), during hydrotest. The maximum
credible stress during pipeline operation, that is, the stress of 358 MPa
(52.0 ksi) due to a pressure surge, was used for critical crack size calcula-
tions for arc bums.

Analysis Methods
Critical flaw sizes were calculated using four distinct fracture-mechanics
analysis methods and the appropriate maximum-credible-stress and ma-
terial-property information. The fracture-mechanics models were (1) the
critical-COD method, (2) the Draft British Standard method, (3) the
plastic-instability method, and (4) the semi-empirical method. Each of
these methods is described in the following. In each method, weld flaws
are assumed to be equivalent to surface cracks equal in size to the weld
defect.

Critical-COD Method
This model is based on the critical (COD) concept. Crack extension that
could cause leakage occurs when the COD value at the crack tip (desig-
nated 6) exceeds a critical value: the COD fracture toughness. 5 is calcu-
lated using a ligament-closure-force model developed by Irwin [6] and
based on plasticity-corrected linear-elastic theory. In this approach, the
surface crack is modeled as a through-thickness crack in a wide plate
coupled with closure forces due to the ligament. The opening of a through
crack of length, /, in a plate under a remote tensile stress, a, is given by.

d = 2la/E (1)

For a surface crack, the opening of Eq 1 is reduced by the remaining

ligament. The effect of ligament depth can be estimated by a closing force
 MCHENRY ET AL ON PIPELINE GIRTHWELDS 635

distributed over the face of the crack. Assuming the ligament is yielded,
the total closing force, F^, is
Fc = lit - a) ff (2)
where
a = crack depth,
( = pipe thickness, and
ff = flow strength.
Distributing this closing force over the crack-face area, It, gives a closing
stress,ffc,on the equivalent through crack of

ffc = (1 - a/t) CT (3)

This closing stress opposes the remote stress, a. The resultant opening of
the surface crack is then

6 = 2/ (ff - ffc) /E (4)

To account for the additional crack opening due to crack tip plasticity,
the effective crack length, which includes Irwin's [7] plasticity correction,
Ty, is used in place of /. The resulting expression when Eq 3 is substituted
into Eq 4 becomes

2(1 + 2ry)
6 = a-il-f)a (5)

where r, = (l/2x) (K/ay)^ and K = a ^vl/l = stress intensity factor. The

residual stresses can be accounted for by assuming they are of the self-
equilibrating type resulting from weld shrinkage. In this case, one can
assume that yield point stresses act over a distance comparable to the weld
size or pipe thickness. If so, a displacement. A, of A = Oy t/E will relieve
the residual stress. An approximation of the contribution of such a residual
stress to 6 is simply to add A to the applied 8, or equivalently to subtract A
from 6<.. For Oy = 414 MPa (60 ksi), E = 208 GPa (30.2 X 10^ ksi), and
t = 13 mm (0.51 in.), A = 0.025 mm (0.001 in.). Thus, when using Eq 5,
failure occurs when the sum of 6 plus A exceeds the fracture toughness, 6c.
The same value of A was used to account for the residual stresses in 12- and
14-mm-thick (0.46 and 0.56 in.) pipe.

Draft British Standard Method

This procedure is described in the Draft British Standard Rules for
636 ELASTIC-PLASTIC FRACTURE

Derivation of Acceptance Levels for Defects in Fusion Welded Joints [8].

The model is based on an "allowable COD" concept; crack extension will
not occur if the flaw size is limited by the relationship

2ira
a + Or
8c = 0.25 (14)

where a is an allowable-flaw-size parameter based on a conservative inter-

pretation of wide-plate test results. The residual stress, Or, is assumed to be
equal to the yield stress, o>. The relationship between a and the flaw di-
mensions, a and /, and the pipe thickness, t, is given in Fig. 1. This figure
relates the surface flaw dimensions, a and /, to the half-length, a, of an
equivalent through-thickness flaw for the case of flat plates. In this study,
curvature effects were neglected and Fig. 1 was used directly for all cir-
cumferential flaws and for axial flaws up to 8 cm (3.2 in.) in length.

Plastic-Instability Method
This model applies to circumferential flaws and was developed by Irwin
[9] on the basis of investigations of net ligament fractures from part-through

0.8

0.5

*. 0.1

0.05

I I I I ] I I I I J I I I I I L
0.01
0.01 0.1 1.0
a/t

FIG. 1—Draft British Standard relationship between actual flaw dimensions and the
parameter a for surface flaws.
 MCHENRY ET AL ON PIPELINE GIRTHWELDS 637

cracks in flat plates of X-65 line pipe and estimated corrections for bulging
effects in pressurized cylinders. Plastic instability leading to rupture occurs
when the net ligament strain, €„, reaches a critical value, fc

60 - (a - y
e„ = ^^ (15)

where So — COD at the mid-thickness and 0/2 is the rotation of the crack
surface due to bulging. The failure condition selected on the basis of the
flat-plate tests [70] was tc — 0.18. Details regarding the evaluation of 60
and d on the basis of shell theory for the specific geometry, yield strength,
and applied stresses applicable to the pipeline are given by Irwin [9].

Semi-Empirical Method
This model applies to axial flaws and was developed by Kiefner et al [5]
on the basis of full-scale pipe rupture tests. Plastic instability leading to
rupture occurs when the applied stress reaches a critical value related to
the flaw size, material flow strength, a, and pipe dimensions.

1 — a/tM

,, /, , 0.628/^ 0.0034/^ \'72

where D is the pipe diameter.

Results and Discussion

Critical and allowable (per the Draft British Standard) flaw sizes were
calculated using the applicable fracture-mechanics models, material-
property data, and pipeline operating stresses for circumferential and axial
flaws. The results are plotted in figures as critical-flaw-size curves with
flaw depth as the 3;-axis and defect length as the x-axis.
Since all flaws are considered surface cracks, the principal differences in
the three types are orientation and location. Flaw orientation determines
whether the applicable stresses are axial or hoop. Flaw location is used to
establish the applicable minimum fracture-toughness, 0,1 mm (0.004 in.)
for randomly located flaws and 0.18 mm (0.007 in.) for surface flaws.
Calculated sizes of circumferential flaws are plotted in Fig. 2 for each of
the applicable analysis methods. Allowable flaw sizes determined by the
638 ELASTIC-PLASTIC FRACTURE

CRACK LENGTH (in)

6 8 10
1.25

1.0

0.75

K 0.5

0.25

15 20 25
CRACK LENGTH (cm|

FIG. 2—Comparison of allowable circumferential flaw sizes determined by the Draft British
Standard method with critical circumferential flaw sizes determined using critical-COD and
plastic instability as the respective fracture criteria.

Draft British Standard method are compared with two critical flaw size
curves determined using critical COD and plastic instability as the respective
fracture criteria. For the critical-COD and the Draft British Standard
methods, a toughness value of 0.1 mm (0.004 in.) was used. This is the
minimum toughness measured [3,4] for through-thickness notches in the
weldment and is applicable to randomly located flaws such as porosity and
slag inclusions. For the plastic instability method, a critical ligament strain
of 0.18 mm (0.007 in.) was used as the failure criterion and all flaws were
located on the exterior surface, the worst-case location when bulging is
considered.
Results of flaw size calculations using higher weld toughness are shown
in Fig. 3, where a critical-COD value of 0.18 mm (0.007 in.) was used.
This is the minimum toughne;>s measured [3,4] for surface notches in the
weldment and is considered applicable to surface defects such as lack of
penetration and lack of root fusion.
In Fig. 4, calculated sizes of axial flaws are plotted for each of the appli-
cable analysis methods. Here, as in Fig. 2, allowable flaw sizes determined
by the Draft British Standard method are compared with two critical-flaw-
size curves using critical COD and plastic instability (the semi-empirical
curve) as the respective fracture criteria. For the critical-COD and Draft
British Standard methods, a toughness value of 0.18 mm (0.007 in.) was
used. The surface notch toughness [0.18 mm (0.007 in.)] was used because
the only axial-aligned flaws considered were arc bums on the surface. For
analysis purposes, arc burns were considered surface cracks of length equal
to the arc bum length and depth equal to the depth estimated from a
 MCHENRY ET AL ON PIPELINE GIRTHWELDS 639

CRACK LENGTH (in)

6 8 10 14 16
0.5

«c = 0.18mm (0.007in)
0.4

0.3:

0.2 i

0.25 Draft British Standard 0.1

10 15 20 25 30 35 40
CRACK LENGTH (cm)

FIG. 3—Comparison of allowable circumferential flaw sizes determined by the Draft

British Standard method with critical circumferential flaw sizes determined by the critical-COD
method for the case 5c = 0.18 mm (0.007 in.).

CRACK LENGTH (in)

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
1.25 - I I I 1 1 1 1 0.5

s,^^__^ X- Semi-Empirical

1.0 0.4

Critical C O D ^ ^ — - . , , _ _ _ _ ^
0.75 - "0.3

• \ ^ ^ Draft British Standard

0.5 - -0.2 y
o-ys = 413MPa (601<si)

5= = 482MPa(70ksi)
0.25 0.1
8c = 0.18mm (0.007in)

1 1 1 1 1 1 1 1
0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
CRACK LENGTH, (cm)

FIG. 4—Comparison of allowable axial flaw sizes determined by the Draft British Standard
method with critical axial flaw sizes determined using critical-COD and plastic instability
{the semi-empirical curve) as the respective failure criteria.
640 ELASTIC-PLASTIC FRACTURE

metallographic correlation discussed elsewhere [2], The semi-empirical re-

sults were obtained using a flow stress value of 482 MPa (70 ksi) to calcu-
late plastic collapse. Notice that the crack length axis in Fig. 4 extends to
8 cm (3.2 in.) instead of 40 cm (16 in.) as used in Fig. 2 and 3. The shorter
axis was used because the short arc-burn lengths were of principal interest.
The smallest flaw sizes were calculated using the Draft British Standard
method. This method is intended to calculate allowable flaw sizes (flaw
sizes regarded as safe) and contains conservative assumptions and safety
factors absent in the other models. The principal conservative elements in
this method include the relationship between toughness, stress level, and
crack size; the treatment of crack aspect (depth to length) ratio; and the
assumed residual-stress level. The relationship between toughness, stress
level, and crack size is based on a lower-bound interpretation of wide-plate
test results; failure does not occur under the conditions used (in this investi-
gation) to calculate "critical" flaw size. Linear-elastic theory is used to
calculateflaw-shapeeffects; beneficial effects of stress redistribution due to
plastic strain in the ligament are neglected. Under linear-elastic conditions,
the stress intensity at the leading edge of a surface crack rapidly increases
in severity between aspect ratios of a// = 0.5 and 0.1. Thus, the most severe
conditions are attributed to relatively short cracks, and the Draft British
Standard curves characteristically become asymptotic at crack lengths less
than 50 mm (2 in.) Residual stresses are assumed to equal the yield strength
of the weld and this stress system is added to the applied stress. This con-
servatism is partly offset by the empirical nature of the stress/flaw-size/
toughness relationship.
The largest flaw sizes were calculated by Irwin's plastic-instability model
for weld flaws and by Kiefner's semi-empirical model for arc burns. The
failure criterion for both these models is plastic collapse of the ligament.
The credibility of both models is enhanced by their relationship to large-
scale test results on pipeline steels. Irwin's results apply to flaws at the
exterior surface, the most severe location when bulging is considered.
Buried or internal flaws, which are more common, can be approximately
30 percent longer before reaching critical size. Further calculations by
Irwin and Albrecht [9] show that using a critical strain of 0.12 instead of
0.18 reduces the allowable length by less than 25 percent and would not
significantly affect the relative position of the curve.
Flaw sizes calculated using the critical-COD model fall between those of
the Draft British Standard and the plastic-instability models. The results
are higher than the Draft British Standard results because failure occurs at
a critical flaw size instead of at an allowable flaw size. The results are
lower than those obtained by the plastic-instability models because the
critical-COD level usually is reached at stable values of ligament strain.
Care should be taken in using the critical-COD model because the dif-
 MCHENRY ET AL ON PIPELINE GIRTHWELDS 641

ference between the flow strength of the material and the applied stress
strongly influences the position of the curves as shown in Fig. 5.

Conclusions and Recommendations

The critical flaw sizes calculated using fracture mechanics vary signifi-
cantly depending on the fracture criterion chosen, that is, critical crack size,
allowable crack size, or plastic instability. Thus, experimental work is
needed to verify which fracture criterion most accurately models girthweld
behavior. In addition, further analytical development is needed to improve
the models evaluated.
The critical-COD and the Draft British Standard methods yield similar
results for short deep flaws and for long shallow flaws. Thus, tests are

FLAW LENGTH, in
6 8 10 12
056

0.52

048
o-max = 398 GPa (57.7 ksi)
o-y = 413 GPa (60 ksi)
0.44
Sc = 0 18 mm (0 007 in)
t = 12 mm (0.462 in) - 040

0.36

552 GPa (80 ksi) 0 32 -

iT = 517 GPa (75 ksi) t—

? = 483 GPa (70 ksi) 0.28 I

T = 448 GPa (65 ksi| <
- 0.24 i

- 0 20

016

- 012
2 -

008

0.04

20 30 40
0
FLAW LENGTH, cm

FIG. 5—Effect of changes in the difference between the flow stress and the applied stress
on allowable flaw sizes calculated by the critical-COD method.
642 ELASTIC-PLASTIC FRACTURE

needed to determine flaw shape effects under conditions of ligament yield-

ing for the a/l range of 0.5 to 0.01.
The plastic-instability analyses indicate that relatively large flaws are
required for plastic collapse of the ligament. For welds with COD tough-
ness levels less than 0.2 mm (0.008 in.) crack extension is generally pre-
dicted to occur prior to plastic collapse. Experiments are needed to estab-
lish the governing conditions for each failure mode.

Acknowledgment
This work was sponsored by the U.S. Department of Transportation,
Office of Pipeline Safety. The authors wish to express appreciation to
Lance Heverly of OPSO, the project monitor; to Drs. Richard P. Reed
and Maurice B. Kasen of NBS, the task leaders; to Harold Berger of NBS,
the program manager, and to G. M. Wilkowski of Battelle, who critically
reviewed the manuscript.

References
[/] Office of Pipeline Safety Operations notice in the Federal Register, Aug. 13, 1976.
[2] Consideration of Fracture Mechanics Analysis and Defect Dimension Measurement
Assessment for the Trans-Alaska Oil Pipeline Girth Welds," H. Berger and J. H. Smith,
Eds., NBSIR 76-1154, National Bureau of Standards, Gaithersburg, Md., Oct. 1976.
[3] Harrison, J. D., "COD and Charpy V Notch Impact Tests on Three Pipeline Butt Welds
Made in 1975," Welding Institute Report LD 22062/5, July 1976.
[4\ Spurrier, J. and Hancock, P., "Crack Opening Displacement and Charpy Impact Test-
ing at Cranfield Institute of Technology for British Petroleum Trading Co., Ltd., "Cran-
field Institute of Technology, Cranfield, U.K., July 1976.
[5] Kiefner, J. F., Maxey, W. A., Eiber, R. J., and Duffy, A. F. in Fracture Toughness,
ASTM STP 536, American Society for Testing Materials, 1973, pp. 461-481.
[6] Irwin, G. R., "Fracture Mechanics Notes," Lehigh University, Bethlehem, Pa., 1969.
[7] "Fracture Testing of High Strength Sheet Materials," First Report of Special ASTM
Committee, ASTM Bulletin, American Society for Testing and Materials, Jan. 1960.
[8] Draft British Standard Rules for the Derivation of Acceptance Levels for Defects in
Fusion Welded Joints, British Standards Institution, London, U.K., Feb. 1976.
[9] Irwin, G. R. and Albrecht, P. in Consideration of Fracture Mechanics Analysis and
Defect Dimension Measurement Assessment for the Trans-Alaska Oil Pipeline Girth
Welds. H. Berger and J. H. Smith, Eds., NBSIR 76-1154, National Bureau of Standards,
Gaithersburg, Md., Vol. 2, Appendix D, Oct. 1976.
[10] Irwin, G. R., Krishna, G., and Yen, B. T., Fritz Engineering Laboratory Report 373.1,
Lehigh University, Bethlehem, Pa., March 1972.
L. A. Simpson^ and C. F. Clarke^

An Elastic-Plastic R-Curve
Description of Fracture in Zr-2.5Nb
Pressure Tube Alloy

REFERENCE: Simpson, L. A. and Clarke, C. P., "An Ehutic-Plastic R-Curve Descrip-

tion of Fnurtnie in Zr-2.5Nb Piessoie Tube ABoy," Elastic-Plastic Fracture, ASTM STP
668, J. D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing
and Materials, 1979, pp. 643-662.

ABSTRACT: An R-curve approach was investigated with the aim of establishing a means
of predicting critical crack lengths in Zr-2.5Nb pressure tubes using small fracture-
mechanics specimens. Because of the elastic-plastic nature of the fracture process and
limitations on the maximum specimen size, conventional R-curve methods were not ap-
plicable. The crack growth resistance was therefore expressed in terms of the crack open-
ing displacement (COD) and R-curves were plotted for several sizes of specimens and
crack lengths at 20 °C and at 300 °C. The effect of hydrogen on R-curve behavior at these
two temperatures was investigated as well.
Conventional clip-gage methods were not suitable for this work. Crack length was
determined from electrical resistance, and COD, at the actual crack front, was deter-
mined from photographs of the specimens taken during testing. Crack length and speci-
men size had little, if any, effect on the R-curve shape. A method for expressing crack
growth resistance in terms of the J-integral was also investigated and appears to be
consistent with the COD approach. The effects of hydrogen and temperature on R-
curve shape are consistent with their known effects on the mechanical behavior of Zr-
2.5Nb. Finally, predictions of critical crack length in pressure tubes obtained by match-
ing R-curves to crack driving force curves are consistent with published burst-testing data.

KEY WORDS: crack propagation, fracture, R-curves, metals, zirconium, pressure

tubes, potential drop

The CANDU^ nuclear reactor system uses cold-worked Zr-2.5Nb pressure

tubes as the primary coolant containment. At present, critical crack length
data are obtained by burst testing full-size tube sections [1]J The work
described here is part of a program to develop a framework for predicting

'Research officer and research technician, respectively, Materials Science Branch, Atomic
Energy of Canada Ltd., Whiteshell Nuclear Research Establishment, Pinawa, Man., Canada.
^The italic numbers in brackets refer to the list of references appended to this paper.
•^CANndsL Deuterium C/ranium.

643

Copyright' 1979 b y AS FM International www.astm.org

644 ELASTIC-PLASTIC FRACTURE

critical crack lengths in these tubes from small fracture-mechanics

specimens. The advantages of using small specimens are material conserva-
tion, experimental convenience, and a more suitable specimen geometry for
studying the micromechanisms of the fracture process.
The feasibility of using small specimens depends on the development of a
geometry-independent fracture criterion. The typical pressure tube, as used
in the Pickering reactor design, has a mean diameter of 10.7 cm and a wall
thickness of 4.1 mm. For the most serious type of defect, a through-thickness
crack lying in the axial direction, propagation will occur under near-plane
stress conditions so that a plane-strain linear elastic fracture-mechanics
(LEFM) approach is not feasible. Early attempts at establishing a fracture
criterion for zirconium alloys dealt with 5 max, the crack opening displace-
ment (COD) at instability or maximum load. Some limited success in pre-
dicting tube behavior was achieved with this criterion [2,3]; however, there
were also discrepancies which suggested that 6 max was geometry dependent
[4]. More recently, the first author [5] confirmed this geometry dependence
but also suggested that 6,, the COD at crack initiation, may be geometry in-
dependent. Subsequent to initiation, however, Zr-2.5Nb tolerates con-
siderable slow, stable crack growth under rising load and 6, is therefore un-
necessarily conservative as a fracture criterion. For the same reason, the use
of the J-integral at crack initiation is unsuitable.

R-Curve Methods
An R-curve, briefly, is a plot of the resistance to further crack extension in
a specimen undergoing slow, stable crack growth, against the extent of this
stable crack extension. It has been suggested [6] that the R-curve for a
material of fixed thickness is geometry independent. If this is so, the failure
condition for any geometry can be determined from the point of tangency of
the R-curve with the plot of crack driving force against crack length for that
geometry. These techniques and concepts are well documented in Ref 6 and
many other papers in the literature dealing with R-curves and will not be
repeated here.
The geometry independence of the R-curve is still a debatable concept and
should be established for a particular material. For example, work by Adams
[7] on two high-strength aluminum alloys suggests that the R-curve depends
on specimen configuration. Thus, one aim of this work is to assess the
geometry dependence of R-curves for Zr-2.5Nb.
Traditionally, the crack growth resistance, KR , has been calculated using
LEFM equations for the stress-intensity factor and the effective crack length
(corrected for plastic zone contribution) for a particular type of specimen.
The stress-intensity factor has significance only if the in-plane specimen
dimensions of crack length and ligament size are large compared with the
plastic zone size. The ASTM Recommended Practice for R-curve Determina-
 SIMPSON AND CLARKE ON PRESSURE TUBE ALLOY 645

tion (E 561-75T) states that the uncracked ligament size should exceed (4/x)
{Kmix/oyy where K^^ is the maximum K level in the test and Oy is the yield
stress. Thus very large specimens are required for calculations of KR by
LEFM equations from measurements on tough materials. Table 1 gives the
yield strengths of Zr-2.5Nb at 20 and 300°C (maximum reactor operating
temperature) and the minimum in-plane specimen dimensions calculated
from the foregoing criterion, assuming (conservatively) /iTmax = 100
MPa/m'^^
Test specimens must be cut from flattened pressure-tube material to ob-
tain the relevant microstructure and mechanical properties. The diameter of
the tubes limits the practical specimen size to about 60 mm although,
because of the large nonuniform deformations experienced in flattening large
specimens, sizes of the order 35 mm are preferred. Thus the LEFM approach
was not suitable for determining R-curves in this work.
Recently a number of investigations have considered the use of elastic-
plastic fracture parameters such as COD [8,9] and the J-integral [8-11] to
describe crack-growth resistance. For steel, McCabe [8] converted COD, 6,
to an effective KR via

KR = m{E X a„ X 6)1^2 (j)

where
E = Young's modulus, and
m = constant = 1.0.
The validity of Eq 1 should be verified for a particular material as various
derivations of Eq 1 give m values between 1 and 2.
While the J-integral is not well defined for situations in which the crack-tip
region is unloaded, attempts have been made to calculate it subsequent to
stable crack growth [8-11]. The usual assumption is made that the/value of
a specimen following some crack extension from a to Aa is the same as in a
nonlinear elastic specimen of initial crack length a + Aa loaded to the same
value of load or displacement or both with no crack extension. A valid/can
be calculated for the latter specimen, so the problem reduces to calculating
TABLE 1—Minimum in-plane specimen dimensions for LEFM calculation of
KR^

Temperature, C° oy, MPa

± -"»• m a .

{- ay
mm
r
20 800 20
300 533 45

''A-max=100MPa/m'
646 ELASTIC-PLASTIC FRACTURE

the/values for the equivalent specimens for various crack lengths. Garwood
et al [10,11] have developed a convenient method for calculating / values
following stable crack extension in deeply cracked compact tension or bend
specimens from a single load (P)-load point deflection (dp) curve. For small
increments of crack growth, they derive

J -J ^-"'- 12 ^ " - ^ ' - (2)

J„ = 7 at «th point on P-8p curve

where
W = specimen width,
B = specimen thickness,
U„ = area under the P-8p curve up to a point, n, on the curve, and
a„ = crack length at point, n, on the P-6p curve.

j = 2^° (3)

where
U„ = area under the P-8p curve up to crack initiation (or any arbitrary
point prior to initiation), and
Jo = corresponding initial value of J.
With this equation, /„ can be calculated from a single load-load point
deflection curve provided crack extension is simultaneously monitored.
In this work the techniques for applying these R-curve methods to
pressure-tube material are developed and the effect of temperature and
hydrogen content on R-curve behavior are examined. An initial assessment is
also made of the ability of R-curves to predict pressure-tube failure.

Experimental

Specimen Preparation
Factors affecting the choice of specimen size were:
1. The need to test at 300°C in a furnace.
2. The need to cut specimens directly from pressure tubes to obtain the
relevant material condition.
3. The need to minimize deformation imparted to the specimens when
flattened.
The compact tension specimen (CTS), Fig. 1, was chosen for this study in
SIMPSON AND CLARKE ON PRESSURE TUBE ALLOY 647

^i

icz: ^

FIG. 1—Compact tension specimens used in this study.

648 ELASTIC-PLASTIC FRACTURE

three sizes specified by their width (W) dimension of 17, 34 and 68 mm. The
other dimensions are in the proportions recommended in the ASTM Test for
Plane-Strain Fracture Toughness of Metallic Materials (E 399-72), except for
thickness, which, after machining, was 3.75 to 3.80 mm for all specimens.
Most of the testing was done on 34-mm specimens with some 17- and 68-mm
specimens tested at 20°C to study geometry (size) effects. Two crack length
ranges were studied as well with a/W « 0.3 and 0.6. Fatigue precracks were
initiated in all specimens using maximum stress-intensity factors less than 20
MPa/mi^2.
Some specimens were gaseously hydrided at 400 °C to levels of 200 /xg/g to
study the effect of hydrogen on R-curve behavior. (The hydriding conditions
were chosen to have a minimal effect on structure. Hydrogen exists as zir-
conium hydride when present in excess of its solubility limit of ~ 1 fig/g at
20 °C and ~ 65 /ig/g at 300 °C [5,12] and under certain conditions is a factor
in causing embrittlement.)

Measurement of Crack Length

Most R-curve determinations in the past have used compliance
measurements to follow crack extension. The need to test at elevated
temperatures prevented the use of the clip gages necessary for this approach,
and our experience in using the d-c potential drop method to follow
hydrogen-induced subcritical crack growth [13] suggested that this technique
would be highly suitable, particularly as it gives a continuous reading of
crack extension. A constant current ( — 10 A for 34-mm specimens) was
passed through the specimen during the test via the screw-in copper leads
shown in Fig. 2. The potential drop across the crack was monitored using zir-
conium leads and a chart recorder with 100-/iV full-scale sensitivity.
The change in potential drop with crack extension is reported to be linear
in a/W for the CTS geometry [14] for 0.3 < a/W < 0.7. This was confirmed
for our 34-mm specimens by following a fatigue crack in which the fracture
surface was periodically marked (~ every 1 mm) by overloading. The results
from two specimens are shown in Fig. 3. By using the slope of this curve, the
amount of crack extension in any specimen was calculated from the change
in potential drop from the start of the test. This technique is capable of
detecting crack extension of less than 10 iim [13], which is more than ade-
quate for the present task.

COD Measurement
In most R-curve studies to date, where COD measurements were required,
they were determined from clip-gage readings at the crack mouth. These
calculations usually assumed that the specimen rotated about a fixed center
in the ligament [8,15], Preliminary testing on Zr-2.5Nb specimens indicated
 SIMPSON AND CLARKE ON PRESSURE TUBE ALLOY 649

CURRENT
LEAD

POTENTIA
LEADS S .

FIG. 2—Experimental arrangement for recording COD and crack extension.

that no suchfixedcenter existed [5], even when the reduction in ligament size
with crack extension was accounted for. Therefore, COD was measured
directly on each specimen by photographing pairs of microhardness indenta-
tions on opposite sides of the crack mouth as the specimen was loaded [15].
For each load level, crack mouth displacements were measured, plotted
against distance from the original crack tip as in Fig. 4, and a line was drawn
through to the apparent center of rotation on the abscissa. The intercept at
the original crack front was the COD at that point; however, our interest was
in the COD at the actual crack tip. This was found by marking the position
of the crack front, as determined by the potential drop data, on each line
(load level) in Fig. 4. Joining these points yielded a locus of the actual COD
during the test.

R-Curve Determination
Using the COD measurements just described and Eq 1, R-curve deter-
minations were carried out at 20 and 300 °C for specimens containing as-
received hydrogen (~ 10 ixg/g) and 200 /ig/g hydrogen. The 300°C tests were
done in a furnace containing a window to allow photographic recording of
650 ELASTIC-PLASTIC FRACTURE

FIG. 3—Fractional change in potential drop across compact tension specimen versus crack
extension.

COD. Specimens were loaded well past maximum load in all cases except for
hydrided material at 20 °C, where instability occurred shortly after maximum
load. The specimens were heat-tinted at 300 °C (if they had not already been
tested at that temperature) prior to final fracture to identify the region of
slow stable crack growth on the fracture surface. The amount of stable crack
growth was measured and used in conjunction with the total change in poten-
tial drop to check the calibration of Fig. 3.

J-Integral Determination
J-integral values were determined from plots of load versus load-point
displacement (the latter can easily be determined from plots similar to Fig. 4)
 SIMPSON AND CLARKE ON PRESSURE TUBE ALLOY 651

'"T ' / r •- { -:"»r

0.5

S a 4 Z 0 2 4 6
WtTANCE FROM FATICUi CRACK TIP (mm)

FIG. 4—Method for determining COD from displacement measurements on crack face.
Each numbered straight line represents a set of displacements for a given load level. The curved
line on the right indicates the magnitude and position of the COD at the actual crack tip.
Inset shows microhardness indentations usedfor crack face displacement measurements.

at each loading stage of the test. The areas under the P-dp curves were
measured with a planimeter. These determinations were confined to the
deeply cracked specimens since Eq 2 assumes predominately bending condi-
tions.

Results and Discussion

Stable Crack Development

Fracture surfaces of typical specimens after varying amounts of stable
crack growth are shown in Fig. 5. While these specimens all contained as-
received hydrogen levels (~ 10 /*g/g) and were tested at 20 °C, they are also
652 ELASTIC-PLASTIC FRACTURE

SPECIMEN THICKNESS = 3.75mm

FIG. 5—Typical morphology of slow stable cracking in '/,r-2.5Nb at various stages of de-
velopment. (F = fatigue crack surface: S = slow stable crack surface: end of stable crack
marked for clarity.)
 SIMPSON AND CLARKE ON PRESSURE TUBE ALLOY 653

typical of the tests at 300 °C on both nominal and hydrided material. The
first notable feature is that the fatigue precracks were not always straight but
were often restrained near one surface. This was observed in about 50 per-
cent of the specimens and is attributable to residual stresses created by the
tube-flattening treatment. Annealing the specimens prior to fatigue
precracking was ruled out because of the possibility of altering the cold-
worked structure of the material and the apparent insensitivity of R-curve
shape to the initial fatigue-crack configuration.
As shown in Fig. 5, stable crack development commenced with initiation
near the specimen midsection. The crack assumed a triangular shape as it
tunneled forward and, after propagating a distance roughly equal to the
specimen thickness, the crack front tended to straighten. This was accom-
panied by a transition from flat fracture to fracture on planes inclined about
45 deg to the original crack plane (slant fracture). When the ligament size
was reduced to about 5 mm, the fracture surface became flat again.
Stable crack growth continued well past maximum load in all tests, except
for the hydrided specimens at 20 °C, which failed abruptly soon after max-
imum load. In these latter specimens, tunneling was more pronounced and
development of slant fracture did not occur, although small shear lips
formed at the specimen surfaces. While it was tempting to associate the flat
fracture surface with plane-strain conditions, examination of the fracture
surface in Fig. 6 revealed that splitting occurred along the hydride platelets,
which were mostly oriented at right angles to the crack front and crack plane
(that is, platelet normals were in the specimen thickness direction [5]). This

[H] = 10Hg/g
FIG. 6—Effect of hydrogen on fracture morphology at 20°C. (F = fatigue crack surface;
S = slow stable crack surface.) Bar indicates 100 jim.
654 ELASTIC-PLASTIC FRACTURE

Splitting is a common effect in hydrided Zr-2.5Nb near room temperature

and causes the specimen to delaminate into a number of parallel thin
specimens. Microscopic examination reveals that fracture of the individual
lamina is by ductile tearing [5].

R-Curves Geometry Independence

It was mentioned earlier that the choice of w = 1 in Eq 1 was somewhat
arbitrary. A test of validity would be to compare KR calculated from Eq 1
with A'LEFM, the stress intensity factor calculated from the LEFM analysis for
the compact tension specimen, in the early stages of loading, where LEFM
analysis has validity. This is done for several specimens in Fig. 7 where /STLEFM
for a particular loading stage is plotted against KR calculated from Eq 1. To
extend the valid range of A^LEFM, it is calculated using the plastic zone-
corrected crack length (by adding the plastic zone size to the actual crack
length). While there is some scatter, the data in Fig. 7 are distributed
uniformly about the line representing m = 1. This suggests that m = 1 is an
appropriate choice to make KR , as calculated from Eq 1, compatible with an
LEFM calculation of the crack growth resistance.
The i?-curves calculated from Eq 1 are plotted in Figs. 8a to 8e. The 20°C
data for nominal hydrogen are divided between Figs. 8a and 8i>, representing
shallow and deeply cracked 34-mm specimens, respectively. Figure 8a also
includes results from two 68-mm specimens, and Fig. 8£> includes a typical

eo
1 1 1 r \ r
// ^' V-
70

60
X'
/ ' y ^ 1
50 / ^/ / ^ '^
/
5 40

30
O
/ / /
/>/ V ^
/ /
P'
</
SYMBOL SPEC.
"•

~
A 308
//.''
20 V^ o 296
X 307
^/'^
0 306
10 # 294
^/^ •
1 1 1 1 I 1 1 1

10 20 30 40 50 60 70 80
Kn(MPam>/,)

FIG. 7—Comparison of KR as calculated by LEFM methods with values calculated from

COD. Solidline: m = 1.0; dashed lines: m = 1.0± 20percent.
 SIMPSON AND CLARKE ON PRESSURE TUBE ALLOY 655

result from a 17-mm specimen. Figures 8a and 8b suggest a crack length

dependence of R-curve shape in the latter stages of its development, with the
shallow-cracked specimens attaining higher plateau levels (for 34-mm
specimens). When the shallow cracks were extended to depths typical of
those in the deeply cracked specimens, the KK values fell off toward the
plateau values of the latter. This, combined with the tendency of deep cracks
to revert to flat fracture, suggests that the crack may interact with the stress-
free back surface of the specimen when the ligament size approaches approx-
imately 10 mm. The two 68-mm specimens in Fig. 8a show the opposite crack
length effect (higher plateau for deep crack).
The 34-mm specimens showing high plateau levels in Fig. 8a developed
single, slant-fracture surfaces turned more than 45 deg from the original
crack plane. This presents some concern over the interpretation of the KR
values since the fracture contains a Mode III component and COD is
measured in the loading direction, not normal to the crack plane.
No clear effect of crack length was observed for either set of tests at 300 °C
(Figs. M and 8e); in fact, in Fig. 8d, the shallow-cracked specimen yields a
slightly lower R-curve than the others, that is, an opposite effect of crack
length to that suggested at 20°C (for 34-mm specimens) by Fig. 8a and 86.
The R-curves for hydrided material at 20°C were also insensitive to crack
length (Fig. 8c). Because of this, and because the differences reported for as-
received material at 20°C are significant only at the later stages of R-curve de-
velopment (where some question exists about the meaning of COD), we con-
clude that R-curves in Zr-2.5Nb may well be geometry (size) independent.
Certainly, further investigation is justified to resolve this question convincingly,
possibly using a completely different specimen geometry.

R-Curves—Effect of Temperature and Hydrogen

The R-curves at 300 °C for as-received and hydrided material are in-
distinguishable. In the as-received material, all the hydrogen would have
been in solution, whereas hydrides would have been present in the hydrided
material. Thus hydride has no effect at this temperature. At 20°C, hydride
has a definite embrittling effect, demonstrated by the R-curves in Fig. 8c,
which are much flatter and have a lower plateau value than the others. This
behavior is consistent with earlier work on the fracture properties of zir-
conium alloys [1,16], where hydrogen up to 400 ixg/g has a detrimental effect
only below about 150°C.
An interesting comparison can be made here with McCabe's results [8] for
a carbon-manganese steel, which suggest that the R-curve is temperature in-
dependent above the Charpy transition temperature. Hydrogen causes an up-
ward shift in the transition temperature in Zr-2.5Nb [IT] from just below
room temperature to about 150 °C for a concentration of 200 /ig/g. Thus all
tests except those at 20 °C with 200-/ig/g hydrogen were carried out in the
656 ELASTIC-PLASTIC FRACTURE

ISO

160
T = 20 C
H I = 10 , f / g
W = 34 mm

306 0.33
309 0.28
307 0.29
308 0.29
376 0.29
68 mm
375 O.60

20 30 40 50 60 70 80 90 100 110 12 0 (30 140 ISO

CRACK EXTENSION. M (mm|

I
i

_1_ -L. I J_ J_ _1_

10 20 30 40 50 60

STABLE CRACK EXTENSION. i» (mm)

 SIMPSON AND CLARKE ON PRESSURE TUBE ALLOY 657

I 1 1 1 1 1

160 -

140 - -s^ -
o
o
120 - J^)tS> t -
re 4-
T = 20°C
_/ a
I 100 ~/o
/ 0 „
H] = 200 Mg/g
W = 34 mm
-

80 o »/ -
) Y
60 v7 SYMBOL SPEC. a/W -
• 323 0.S3

40
7 A

0
326
322
0.32
0.53
327 0.33
D 301 0.S9
20

1 1 1 1 1
10 20 30 40 50 60 70

STABLE CRACK EXTENSION, Hi (mm)

20

_L. I I _L. 1
_!_
10 20 30 40 50 60

STABLE CRACK EXTENSION, Aa (mm)

FIG. 8—R-curves for Zr-2.SNb specimens determined from COD.

658 ELASTIC-PLASTIC FRACTURE

200 1 1 1 1 1

~^*
160 -

140 -

M
- J 9/ ~
n T = 300'C
H| = 200 ^g/g
-
1* 0 W = 34 mm
80
i A/
6 0 Jr SYMBOL SPEC. a/W -
P • 293 0.60
40! y + 325 0.33 -
o 299 0.59
20
•

1 1 1 1 1
10 20 30 40 50 60 70
STABLE CRACK EXTENSION, Aa (mm)

FIG. 8—Continued.

upper-shelf region and, as for the steel results, the R-curves were temperature
independent.

J-Integral Measurements
The J-integral was calculated as a function of crack extension, using Eq 2,
for all the deeply cracked 34-mm specimens. The exact physical significance
of/ as measured in this manner is not completely clear. The critical assump-
tion by Garwood et al [10] is that the difference between the energy under the
actual load-displacement curve and that for the hypothetical specimen
loaded to the same load and deflection is the energy taken up by crack
growth. They admit that no proof of this assumption exists. Also, the ac-
curacy of the J calculation will be dependent on minimizing the segments of
crack growth between calculation points to some as yet undefined optimum
value. Because of these uncertainties, the credibility of the J-integral results
will simply be discussed in terms of their self-consistency with the KR data
obtained by COD measurements.
The initial value of 7, /o, was not the initiation value as chosen by Garwood
and Turner [//]. Their analysis is equally valid if/o is chosen anywhere in the
linear portion of the load-displacement curve (that is, prior to initiation) and
 SIMPSON AND CLARKE ON PRESSURE TUBE ALLOYS 659

this procedure was followed here. Plots of /„ versus Aa yield the same
qualitative shapes as the KK plots in Figs. 8a to 8e. A general comparison
between the two methods is obtained by plotting/„/aj, against 6 in Fig. 9.
After some initial curvature, there is a generally good straight-line correla-
tion between these two parameters which can be described by the equation
/„ = lAoyib - do) (4)
where 6o = 0.032 mm. Thus, after some initial loading, /„ is linearly depen-
dent on 6. Calculations of/ based on simple yield models [18] suggest that
J = OyS (5)
In the early stages of loading, the primary contribution to/will be elastic [19]
and
/oc6 2oc62 (6)

the latter proportionality resulting from the method of COD measurement

used here (Fig. 4). This is consistent with the initial curvature in Fig. 9.
Thus the calculation of/ beyond crack initiation, using Eq 2, is in good
agreement with the foregoing relationships between / and COD. The factor
of 1.1 could easily be accounted for by work hardening, since Eq 5 is based
on perfectly plastic behavior. Further work is underway to substantiate the
use of//} as a fracture criterion, including the calculation of/ in a cracked

1 1 1 1 "•

SYMBOL SPEC. IH] T A

cg/l deg. C

04
+
o 300
294
10
10
20
20
^X°
/
299 200 300 +
& 293 20b 300
•
o 298 10 300
oZ

301 200 20
03
• 323 200 20
*
X 322 200 20
/ -
o /

+
1
>
02
A
/ t '
* / • = 1,1(6 -04)32)
• jm
*y/ff x •
4 » / 0
•
01
4
•
•
•
• 1
1
02 03 04 05
6 (mm)

FIG. 9—Relationship between 1 measurements and COD.

660 ELASTIC-PLASTIC FRACTURE

tube body as a function of crack length and pressure, using finite-element

analysis. JR curves will then be used for critical crack length prediction via a
tangency condition similar to that used for KR curves.

Critical Crack Length Prediction

The ultimate test oiKR as a fracture criterion is in its ability to predict the
failure condition of a structure, in this case the Zr-2.5Nb pressure tubes. To
determine the critical crack length at operating pressure, it is necessary to
place the R-curve such that it is just tangent to the crack driving force curve
as in Fig. 10. The crack driving force for an axially cracked tube is given by
[20]
8a ffv irMa 1/2
K = X In sec (7)

where
a = crack half-length,
a = hoop stress in pressure tubes, and
M = magnification factor due to tube curvature.
A "flow stress" is often used in place of the yield stress, Oy, in Eq 7, which
takes into account work-hardening. It is not used here because its selection is
somewhat arbitrary, and, because the material is cold-worked, it has a
negligible effect on the crack driving force curve over the range of crack
lengths of interest. M is given by

M = l + 1.255 -0.0135 (8)

Rt iRty
300 -T 1 1 r 1 1 r—

T = 300°C
533 MPa

200

100-

76 78 aO 82 84 86 88 90 92 94
2a (mm)

FIG. 10—Critical crack length prediction for pressure tubes at SOCC obtained by matching
upper (UB) and lower (LB) bounds ofR-curves with crack driving force curve.
 SIMPSON AND CLARKE ON PRESSURE TUBE ALLOY 661

where
R = tube radius, and
t = tube wall thickness.
In Fig. 10, the data for hydrided and as-received material at 300 °C are
combined and represented by R-curves corresponding to the upper and lower
bounds of the scatter. An operating hoop stress, in the tube, of 125 MPa is
assumed and the crack driving force (Eq 7) is plotted against axial crack
length, 2a, for a yield stress of 533 MPa. The lower-bound R-curve indicates
that a 75-mm crack will grow stably as the operating stress is applied until it
just reaches criticality at operating pressure and a length of 84 mm. Simi-
larly, the upper-bound curve predicts that an 84-mm crack would become
unstable if loaded to operating stress. Thus the data predict a range of
critical crack lengths between 75 and 84 mm. Ideally this should be com-
pared with burst-testing data on identical material. Because the pressure
tubes are extruded hot, and undergo some cooling during the process, the
strength can vary 20 percent over the length of the tube [21]. Yield strengths
were not provided with the burst-testing data [1,17], which were obtained
several years ago, and the choice of 533 MPa may be inappropriate. Most of
the burst data pertain to irradiated material which will also cause significant
changes in yield stress. In spite of these difficulties, the lower bound of the
burst-testing data [17] indicates a critical crack length of 70 to 75 mm at 125
MPa, in excellent agreement with the R-curve prediction. Certainly further
work is justified. The next logical step is to make direct comparisons by cut-
ting compact tension specimens from previous burst sections, eliminating the
effect of variations in material properties.

Sanunaiy and Conclusions

Because of the considerable amount of stable crack growth which
Zr-2.5Nb will tolerate, an R-curve approach seems to be the most ap-
propriate to use in the development of a fracture criterion. The geometry in-
dependence of the R-curve has not been unequivocally established in this
work. However, the small crack length dependence at 20°C for as-received
material is reversed at 300 °C (Fig. 8d) and both dependencies may therefore
reflect material variations rather than geometry effects. The calculations of
KR using COD are fully consistent in the LEFM limit. The expression of
crack-growth resistance in terms of JR is also compatible with the COD
measurements, which suggests that the /-integral approach may be an
equally appropriate way of deriving an R-curve. The potential-drop method
has proved to be an accurate and convenient means of following stable crack
extensions, especially at elevated temperatures where displacement-gage
methods are unsuitable.
Finally, the effect of temperature and hydrogen on the R-curves
themselves is fully consistent with the established effects of these parameters
662 ELASTIC-PLASTIC FRACTURE

on material properties. The application of an R-curve to the prediction of

critical crack length in a tube, while preliminary, shows promise and war-
rants further development.

References
[/]' Langford, W. J. and Mooder, L. E. J., Journal of Nuclear Materials, Vol. 39, 1971, pp.
292-302.
[2] Fearnehough, G. D. and Watkins, B., IntemationalJoumal of Fracture Mechanics, Vol.
4, 1968, pp. 233-243.
[3\ Henry, B., "La Prevision des Conditions Critiques de Rupture de Tubes de Pression en
Zr-2.5% Nb par le CritSre de IVlargissement critique de Fissure," Euratom Report EUR
5017f, Ispra, 1973.
[4] Pickles, B. W., Canadian Metallurgical Quarterly, Vol. 11, 1972, pp. 139-146.
[5] Simpson, L. A., "Initiation COD as a Fracture Criterion for Zr-2.5% Nb Pressure Tube
Alloy" in Fracture 1977, Vol. 3, D. M. R. Taplin, Ed., University of Waterloo Press,
Waterloo, Ont., Canada, 1977.
[6] McCabe, D. E. and Heyer, R. H. in Fracture Toughness Evaluation by R-Curve Methods,
ASTM STP 527, American Society for Testing and Materials, 1973, pp. 17-35.
[7] Adams, N. J. in Cracks and Fracture, ASTM STP 601, American Society for Testing and
Materials, 1976, pp. 330-345.
[5] McCabe, D. E. in Flaw Growth and Fracture, ASTM STP 631, American Society for
Testing and Materials, 1977, pp. 245-266.
[9] Tanaka, K. and Harrison, J. D. "An R-Curve Approach to COD and J for an Austenitic
Steel," British Welding Institute Report No. 7/1976/E, July 1976.
[10] Garwood, S. J., Robinson, J. N., and Turner, C. E., IntemationalJoumal of Fracture,
Vol. 11, 1975, pp. 528-530.
[//] Garwood, S. J. and Turner, C. E., "The Use of the J-Integral to Measure the Resistance of
Mild Steel to Slow Stable Crack Growth" in Fracture 1977. Vol. 3, D. M. R. Taplin, Ed.,
University of Waterloo Press, Waterloo, Ont., Canada, 1977.
[12] Kearns, J. J., Journal of Nuclear Materials, Vol. 27, 1968, pp. 64-72.
[13] Simpson, L. A. and Clarke, C. F. "The Application of the Potential Drop Technique to
Measurements of Sub Critical Crack Growth in Zr 2.5% Nb," Atomic Energy of Canada
Ltd., Report No. AECL 5815, 1977.
[14] Mclntyre, P. and Priest, A. H., "Measurement of Sub Critical Flaw Growth in Stress Cor-
rosion, Cyclic Loading and High Temperature Creep by the DC Electrical Resistance
Technique," Bisra Open Report MG/54/71, British Steel Corp., London, 1971.
[15] Ingham, T., Egan, G. R., Elliott, D., and Harrison, T. C. in Practical Applications of
Fracture Mechanics to Pressure Vessel Technology, R. W. Nichols, Ed., Institution of
Mechanical Engineers, London, 1971, pp. 200-208.
[16] Watkins, B., Cowan, A., Parry, G. W., and Pickles, B. W. in Applications-Related
Phenomena in Zirconium and Its Alloys, ASTM STP 458, American Society for Testing
and Materials, 1969, pp. 141-159.
[17] Ells, C. E. in Zirconium in Nuclear Applications, ASTM STP 551, American Society for
Testing and Materials, 1974, pp. 311-327.
[18] Rice, J. R. in Fracture, H. Liebowtiz, Ed., Academic Press, New York, 1%8, Chapter 3,
pp. 191-311.
[19] Knott, J. F., Fundamentals of Fracture Mechanics, Wiley, New York, 1973, pp. 170-171.
[20] Kiefner, J. F., Maxey, W. A., Eiber, R. J., and Dufiy, A. R. in Progress in Bow Growths
and Fracture Toughness Testing, ASTM STP 536, American Society for Testing and
Materials, 1973, pp. 461-481.
[21] Evans, W., Ross-Ross, P. A., LeSurf, J. E., and Thexton, H. E., "Metallurgical Properties
of Zirconium-Alloy Pressure Tubes and Their End Fittings for CANDU Reactors," Atomic
Energy of Canada Ltd., Report No. AECL-3982, Sept. 1971.
B. D. Macdonald^

Correlation of Structural Steel

Fractures Involving Massive
Plasticity

REFERENCE: Macdonald, B. D., "Correlation of Structural Steel Fractures luTolving

Massive Plasticity," Elastic-Plastic Fracture. ASTM STP 668. J. D. Landes, J. A.
Begley, and G. A. Clarke, Eds., American Society for Testing and Materials, 1979,
pp. 663-673.

ABSTRACT: A three-dimensional, elastic-plastic fracture strength correlation for A36

and HSLA structural steel connections containing discontinuities was determined.
The fracture specimens comprised beam-column connections in which one column
flange contained a mid-thickness plane of discontinuity. Beam loading or direct
tension applied normal to the column face imposed tensile load transfer around the
boundaries of the discontinuity. Fracture extension was mixed mode (crack opening
and edge sliding), and inclined toward the free surface on the web side of the column
flange containing the discontinuity. Successful correlation for these specimens was
accomplished with a plastic stress singularity strength model, if the discontinuity was
sufficiently large. The average singularity strengths at ultimate load were 64.6 MNm ~^'^
(58.7 ksi in.'''^) for HSLA steels, and 53.7 MNm"^'^ (48.8 ksi in.'^^j fo, ^ 3 ^ ^^^
The percent coefficient of variation was 6.4 percent for HSLA steels and 8.4 percent for
A36 steel.

KEY WORDS: fracture (materials), failure, cracking (fracturing), elastic theory,

plastic theory, tensile properties, stress-strain diagrams, bend tests, analyzing steels,
structural steels, crack propagation.

The problem that initially required this research was the need to determine
the residual strength of moment connections in which one column flange
contains a mid-thickness plane of discontinuity, or lamination, shown
cross-hatched in Fig. la. The ultimate aim of this research is to develop
a practical methodology for evaluating the residual strength of cracked
structural steel elements. Presently available evaluations are often grossly
conservative because they do not adequately account for one or more of
the following modes of behavior observed in cracked structural steel com-
ponents:
'Research engineer. Research Department, Bethlehem Steel Corp., Bethlehem, Pa. 18016.

663

Copyright' 1979 b y A S T M International www.astm.org

664 ELASTIC-PLASTIC FRACTURE

A. Moment Connection

/ / / / / / / / /
B. Pull Tab Type Specimen

C. Beam Loading Specimen

FIG. 1—Fracture specimens.

1. Structural steel elements do not generally operate in the plane-strain

regime, which is the cracked-structure response quantified by many cur-
rently accepted residual strength calculations.
2. Massive plasticity may accompany final fracture, and such plasticity
is not properly represented in existing residual strength models.
3. Strong three-dimensional effects cannot be adequately estimated with
the two-dimensional analysis common to current cracked-structure models.
By combining three existing concepts, a correlation was developed be-
tween residual strength tests on structural steel subassemblages which did
not behave in plane strain, exhibited massive plasticity, and were strongly
three-dimensional. The concepts used are:
1. Three-dimensional elastic-plastic finite-element stress analysis.
2. Strength of the plastic stress singularity in the neighborhood of a
material discontinuity.
3. Maximum tensile stress theory of fracture.
 MACDONALD ON MASSIVE PLASTICITY 665

Specimens and Tests

The fracture specimens comprised beam-column connections in which
one column flange contained a naturally occurring mid-thickness plane of
discontinuity, shown cross-hatched in the moment connection sketch of
Fig. la.
Twenty-five welded beam-column specimens were tested [1].^ Nine speci-
mens were three-plate welded column sections, 14 were rolled column sec-
tions with one flange removed and replaced by a plate containing a dis-
continuity, and two were as-rolled column sections without discontinuities.
Initial discontinuity widths were 0 to 120 mm (0 to 4.74 in.) in 203 to 406
mm (8 to 16 in.) flange widths. Flanges were A36 and the HSLA steels,
A572 GrSO and A588, 24 to 51 mm (Wif, to 2 in.) in thickness. Column
flange through-thickness loading was applied by beam tension flanges or
tension pull tabs 203 to 356 mm (8 to 14 in.) in width and 14 to 32 mm
{Vib to I'A in.) in thickness.
Twenty-two pull-tab-type specimens and three beam-loading specimens,
Fig. lb and Ic, respectively, were tested. Several specimens were subjected
to axial column load in addition to pull-tab loading, and several others
contained tension stiffeners recommended by the American Institute of
Steel Construction (AISC) [2]. Residual strength correlation was obtained
for specimens in which the initial discontinuity width was large enough to
precipitate failure by unstable extension of the discontinuity.

Crack-Tip Considerations
Hilton and Hutchinson [3] have presented the concept of using plastic
stress singularity strength, K, to predict plastic fracture instability in a
cracked Ramberg-Osgood material. They contend that, as in linear elastic
fracture mechanics, K would attain a critical value at the onset of fracture
instability. They also contend that if the dominant crack-tip singularity
were known, then K could be determined with the aid of finite-element
(FE) stress analysis.
Figure 2 identifies a crack-tip polar coordinate system originating at
the normal to the edge of the discontinuity. It lies in the mid-plane of the
pull tab or the beam tension flange, and the Z-direction is parallel to the
edge of the discontinuity. Hutchinson [4] derived the r^'''^ stress singularity
for a bilinear hardening material. Fig. 3, assuming all the material sur-
rounding the crack tip to yield. The r~^'^ stress singularity was assumed
to be valid for the multilinear hardening (MLH) material. Fig. 3, used in
the present study.

^The italic numbers in brackets refer to the list of references appended to this paper.
666 ELASTIC-PLASTIC FRACTURE

FIG. 2—Crack-tip coordinates.

Bilinear Hardening
Multilinear Hardening

FIG. 3—Stress-strain behavior.

Finite-Element Analysis

Three-Dimensional Finite-Element Model

Complicated specimen geometry and plasticity observed in the tests
indicated the need for elastic-plastic FE modeling. A two-dimensional
elastic-plastic FE model was tried but was not representative of observed
test behavior. That study used plane-strain boundary conditions on the
column and plane-stress conditions on the pull tab.
A three-dimensional elastic-plastic FE model was used to analyze the
 MACDONALD ON MASSIVE PLASTICITY 667

laminated column flange, the column web, and the beam tension flange or
the pull tab, Fig. 4. The column flange material outside the width of the
pull tab was found to be unstressed and was deleted from the model. Three
mutually perpendicular planes of symmetry divided the specimen through
the web center, web mid-thickness, and pull tab mid-thickness. Symmetry
boundary conditions were established wherever these planes touch the
model. The FE models (shaded area in Fig. 4) were analyzed using ANSYS,
a general-purpose large-scale computer program for the solution of struc-
tural and mechanical engineering problems. The model contained 341
elements and 670 nodes. The 293 nonsingular elements were 8-node iso-
parametric bricks and the 48 singular elements are described in the fol-
lowing. No lamination extension was allowed during loading of the FE
model since no slow stable crack growth was found in the post-failure
sectioned structures.
The FE model side elevation. Fig. 5, shows the four levels of elements
established along the length of the column; t is the specimen pull tab or
beam tension flange thickness. The cluster of lines in the flange shows the
elements at the tip of the planar discontinuity. Twenty-nine elements
comprised the pull tab. The FE models were loaded at the pull tab ex-
tremity to the nominal stress at failure determined from the fracture
tests. The elastic-plastic iterative solution was based on the initial stress
method. The largest plastic strain increment in the last iteration was 5

FIG. 4—Finite-element model.

668 ELASTIC-PLASTIC FRACTURE

Z
I

4t

2t

f/2
Column Web Flange
Ml 1 II
Pull Tab
!
FIG. 5—Side elevation.

percent of the elastic strain. The von Mises equivalent stress was used as a
measure of yielding.
The plan view of the FE model of the column is shown in Fig. 6. Each
level of elements along the length of the column contained 78 elastic-plastic
elements. Twelve singularity elements in each level, shown shaded in Fig.
6, surrounded the crack front. The radial extent of these elements was in
constant proportion, 0.0133, to the initial discontinuity width.
These wedge-shaped singularity elements have been adapted from
Tracey's [5] three-dimensional elastic element to include a five-point MLH
approximation. Fig. 3, to the engineering stress-strain autographic record
of the tensile test of each material used. The singularity elements were
used to determine the plastic stress singularity strength along the crack
front. The normal and shear stresses in the r — ^ plane varied as r~''^.
The Z stresses were nonsingular, and the Z shear stresses were insignifi-
cantly small in the singularity elements.

Results
The FE solution for the plastic yield zone at the mid-plane of the pull
tab is shown in Fig. 7 for one test. The yield zone intersected the free sur-
face of the column flange on the side toward the column web. This was
corroborated by the whitewash spalling observed during the same fracture
test, as shown in Fig. 8.
Figure 7 also shows that the region ahead of the discontinuity and slightly
toward the web side of the column flange behaved elastically. Observed
fracture instability was directed toward this elastic region, as was anticipated
by Sih [6]. Note that mixed-mode fracture (crack opening and edge sliding)
occurred despite the symmetry of the problem.
 MACDONALD ON MASSIVE PLASTICITY 669

FIG. 6—Column cross section.

D ELASTIC MATERIAL
I YIELDED MATERIAL

FIG. 7—Yield zone.

670 ELASTIC-PLASTIC FRACTURE

FIG. 8—Fracture test showing whitewash spalling.

Mixed-mode fracture response was taken into account by using the

maximum stress theory of fracture [7]. Therefore, it was assumed that
the plane of fracture extension was indicated by the vanishing of ar«. Let

a, = K(e)/(2%ry'^

where K{d) is dependent on the boundary conditions and orientation with

respect to the plane of discontinuity. It is assumed that at uhimate load

Kf = K(0) I „^=o = <'e I »^=o (2w)>^2

Thus, the plastic stress singularity strength is assumed to take on a

critical value, Kf, with incipient fracture. Stresses in the singularity ele-
ments, shown shaded in Fig. 6, were evaluated at the element centroid.
Hence, r equals the radial distance from the crack front to that element
centroid for purposes of the FE crack-front calculations.
The variation of ar« for the FE model of one fracture test is shown in
Fig. 9. The plane of expected fracture extension is indicated by (Tro = 0 at
about ^ = — 30 deg. The actual failure angle was about 6 = —45 deg for
 MACDONALD ON MASSIVE PLASTICITY 671

-45 -15
Q .degrees

(Iksi - 6.9 MNiti'* , I ksi in'4 = I.I MNm'%)

FIG. 9—Crack-tip behavior.

this test, as shown in Fig. 10. The variation of K{d) for this test is also
shown in Fig. 9. Along the plane Ore — 0 the plastic singularity strength,
K(d), was assumed to take on its critical value, Kf, in this case about 68.8
MNm^^^^ (62.5 ksi-in.'^^). The tendency for fracture extension rather than
material flow along this plane was also indicated by the von Mises equiva-
lent stress, aeq, exhibiting a relative minimum, where art = 0. Recall that
the predicted direction of fracture extension pointed toward the elastic
region nearest to the crack border, as was shown in Fig. 7. This proximity
of the elastic region was consistent with the local minimum in aeq.

COLUMN WEB COLUMN FLANGE

PULL TAB
FIG. 10—Fracture test cross section.
672 ELASTIC-PLASTIC FRACTURE

Fracture Model Results

Plastic singularity strength, Kf, at ultimate load plotted versus initial
flaw width, Fig. 11, shows the fracture correlation for the two types of
steel tested. The A36 data represent four different heats of steel, and the
HSLA data represent six different heats. The average plastic singularity
strengths at ultimate load were 64.4 MNm-^^^ (58.7 ksi-in.»^2) for HSLA
steels, and 53.2 MNm-^/z (43.4 ksi-in.'^^) for the A36 steel. The present
coefficient of variation was 6.4 percent for the HSLA steels, and 8.4 per-
cent for the A36 steel.

Conclusions and Suggestions for Future Research

The combination of (1) three-dimensional elastic-plastic FE stress analy-
sis, (2) the plastic stress singularity strength for a crack in an MLH ma-
terial, and (3) the maximum tensile stress theory of fracture yields a good
model of the fracture tests performed. This analysis is currently being
applied to beam and axially loaded fracture toughness specimens in order
to find a consistent interpretation of large-scale plasticity and through-

60-
..O.a-
50k --overage- ::fl:::4K
W
40

30 (symbol key shown below)

20

10 A36

70
60 -overage —-——
•—na-o
^- '^
—-D—•-"-
50
40 0 pull tab loading only
II II II w/tension stiffeners
•
30 0 II II II <r column loading
• beam loading
20|-
10 HSLA

0,

INITIAL FLAW WIDTH ,05 .inches

(Iksi-inl^- I.IMNin%,lin-25.4inm)

FIG. n—Strength model.

 MACDONALD ON MASSIVE PLASTICITY 673

thickness effects. Future work should include the strength analysis of

notch toughness specimens of other geometries by this method. Low-cycle
fatigue crack growth problems may also be amenable to this analysis when
large-scale plasticity is a primary influence. The ultimate objective of such
research is to develop a practical methodology for evaluating the strength
of structural steel elements when a crack is present.
Acknowledgment
This research was sponsored by the American Iron and Steel Institute
and monitored by the Task Force, Project 164. E. L. Meitzler and R. L.
Kieffer were instrumental in the execution of this work.

References
[/] Macdonald, B. D., "Effect of Laminations on Moment Connections," submitted for
publication in the American Society of Civil Engineers, Journal of the Structural Division.
[2] Manual of Steel Construction, 7th ed., American Institute of Steel Construction, New
York, 1970, pp. 5-40.
[3] Hilton, P. D, and Hutchinson, J. W., Engineering Fracture Mechanics, Vol. 3, 1971,
pp. 435-451.
[4] Hutchinson, J. W., Journal of the Mechanics and Physics of Solids, Vol. 16, 1968, pp.
13-31.
[5] Tracey, D. C , Nuclear Engineering and Design, Vol. 26, 1973, pp. 1-9.
[6] Sih, G. C. and Macdonald, B. D., Engineering Fracture Mechanics, Vol. 6, 1974, pp.
361-386.
[7] Sih, G. C. and Liebowitz, H., Eds., Fracture, Vol. 2, 1968, p. 94.
/. G. Merkle'

An Approximate Method of
Elastic-Plastic Fracture Analysis for
Nozzle Corner Cracks*

REFERENCE: Merkle, ] . G., "An Approximate Method of Elastic-Plastic Fracture

Analysis for Nozzle Comer Cracks," Elastic-Plastic Fracture. ASTM STP 668, J. D.
Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and
Materials, 1979, pp. 674-702.

ABSTRACT: Two intermediate test vessels with inside nozzle corner cracks have been
pressurized to failure at Oak Ridge National Laboratory (ORNL) by the Heavy Section
Steel Technology (HSST) Program. Vessel V-5 leaked without fracturing at 88°C
(190°F), and Vessel V-9 failed by fast fracture at 24°C (75°F) as expected. The
nozzle corner failure strains were 6.5 and 8.4 percent, both considerably greater than
pretest plane-strain estimates. The inside nozzle corner tangential strains were nega-
tive, implying transverse contraction along the crack front. Therefore, both vessels
were reanalyzed, considering the effects of partial transverse restraint by means of the
Irwin (3ic formula. In addition, it was found possible to accurately estimate the
nozzle comer pressure-strain curve by either of two semi-empirical equations, both of
which agree with the elastic and fully plastic behavior of the vessels. Calculations of
failure strain and fracture toughness corresponding to the measured final strain and
flaw size are made for both vessels, and the results agree well with the measured values.

KEY WORDS: fracture mechanics, fracture toughness, elastic-plastic analysis,

fracture strength, pressure vessels, nozzles, cracks, stress concentrations, crack
propagation

Nomenclature
Ai, A2 Terms from which the real root of Eq 17, a cubic equation, is
calculated, dimensionless
a Crack depth, cm (in.)
•Work done at Oak Ridge National Laboratory, operated by Union Carbide Corp. for the
Department of Energy; this work funded by U.S. Nuclear Regulatory Commission under
Interagency Agreements 40-551-75 and 40-552-75. By acceptance of this article, the publisher
or recipient acknowledges the U.S. Government's right to retain a nonexclusive, royalty-free
license in and to any copyright covering the article.
' Senior development specialist. Oak Ridge National Laboratory, Oak Ridge, Tenn. 37830.

674

Copyright 1979 b y AS FM International www.astm.org

 MERKLE ON NOZZLE CORNER CRACKS 675

B Plate thickness, c m (in.)

C Linear elastic fracture mechanics (LEFM) shape factor based on
local stress, dimensionless
Cn L E F M shape factor based on nominal stress, dimensionless
E Modulus of elasticity, M P a (ksi)
Es Strain-hardening tangent modulus, M P a (ksi)
Ki Mode I elastic crack-tip stress-intensity factor, MN • m ~'^^ (ksi 'Jin.)
Kic Plane-strain fracture toughness, MN-m"^'^(ksi Vim)
Kici Fracture toughness measured with a specimen of thickness d and
calculated from the test data by the equivalent energy procedure,
MN-m-3'2(ksiViir)
Kc Non-plane-strain fracture toughness, MN • m"3'^,(ksi -Jin.)
Kt Elastic stress concentration factor, dimensionless
Ke Inelastic strain concentration factor, dimensionless
K„ Inelastic stress concentration factor, dimensionless
M Initial slope of the pressure-strain curve, MPa (ksi)
p Pressure, MPa (ksi)
p*f Elastically calculated failure pressure, MPa (ksi)
Par Gross yield pressure, MPa (ksi)
Tc Nozzle corner radius of curvature, cm (in.)
r, ftside radius of vessel cylinder, cm (in.)
Tm Mid-thickness radius of vessfel cylinder, cm (in.)
r„, Inside radius of nozzle, cm (in.)
r„ Outside radius of vessel cylinder, cm (in.)
Tz Effective nozzle radius, cm (in.)
t Thickness of vessel cylinder, cm (in.)
jS Shell analysis parameter, dimensionless
j8ic Plane-strain plastic zone size parameter, dimensionless
/3c Non-plane-strain plastic zone size parameter, dimensionless
5c Calculated crack opening displacement, m m (in.)
e Notch root strain, dimensionless
X Applied strain, dimensionless
X<j Stress-strain parameter, dimensionless
X/ Failure strain, dimensionless
X/o Calculated failure strain for plane-strain conditions, dimensionless
Xj Strain at t h e onset of strain hardening, dimensionless
\Y Yield strain, dimensionless
V Poisson's ratio, dimensionless
p Notch root radius, cm (in.)
Oh Nominal hoop stress in vessel cylinder, MPa (ksi)
OY Yield stress, MPa (ksi)

The development of fracture mechanics methods of analysis has made it

possible to quantitatively examine a given structural design and material
676 ELASTIC-PLASTIC FRACTURE

selection to determine if there are sufficient margins between the specified

flaw sizes, material properties and loading conditions, and those that
could cause failure. In the case of a welded steel pressure vessel, two
types of situations involving flaws need to be considered in a fracture safety
analysis. The first is a flaw attempting to propagate out of an embrittled
region, wherever one might exist, and the second is the attempted unstable
extension of a flaw growing by fatigue in sound material. Precautions
against the first type of failure (the nonarrest of a propagating crack) are
based on defining the size and shape of a boundary surrounding the
embrittled region in sound material and treating this boundary as the size
of a crack that must arrest. This is the concept underlying the use of the
reference flaw size and the reference (crack arrest) fracture toughness in
nuclear pressure vessel design [1].^ Precautions against the second type of
failure (static initiation of a crack formed and growing by fatigue in sound
material) can be based on fracture mechanics analysis methods that use
the static initiation fracture toughness. Methods for considering, by
analysis, the possible stable growth of cracks under monotonically increas-
ing loads are now being developed [2-4], but the analysis to be discussed
here does not include this phenomenon explicitly. Instead, stable crack
growth will be treated approximately by using a maximum load fracture
toughness determined from a test specimen in which some stable crack
growth may have occurred before failure. Depending on the method of
analysis, the amount of stable crack growth that may occur in the structure
before failure may also be estimated, based on test data, and added to the
original crack size. The reasonableness of this approach will be evaluated
by comparing calculations with experimental data obtained from two
Heavy Section Steel Technology (HSST) Program intermediate pressure
vessel tests.

Statement of the Problem

The particular fracture prevention problem being considered here is that
of preventing the unstable extension of a crack formed and growing by
fatigue at the inside corner of a nozzle in a pressure vessel under vessel
internal pressure loading. It will be shown that a relatively simple method
of analysis can provide useful approximate results for this type of problem,
provided that two important features of the problem, both of which have
been observed experimentally, are considered. The first important feature
is the dependence of the inside nozzle corner pressure-strain curve on the
elastic stress concentration factor of the nozzle comer and on the gross
yield pressure of the vessel cylinder. The second important feature is the
apparent beneficial effect of transverse contraction at the inside nozzle

^The italic numbers in brackets refer to the list of references appended to this paper.
 MERKLE ON NOZZLE CORNER CRACKS 677

corner, under vessel internal pressure loading, on the toughness governing

the extension of an inside nozzle corner crack.
The type of crack being considered is assumed to lie in the plane con-
taining the axis of both the nozzle and the vessel (the longitudinal plane),
because the inside nozzle corner stress concentration factor for pressure
loading, here defined as the ratio of the peak nozzle corner stress to the
average cylinder hoop stress, is known to be a maximum in this plane. In
addition, cyclic pressure experiments have shown that fatigue cracks form
first at this location [5]. The problem is relevant to the fracture safety
analysis of nuclear pressure vessels because cracks formed by thermal
fatigue have occurred around the inside corners of Boiling Water Reactor
(BWR) feedwater nozzles [6]. Previous example calculations have also
shown that inside nozzle corner cracks of sufficient initial size can grow
appreciably by fatigue [7], thus increasing the importance of developing
fracture analysis methods for such flaws. Since local yielding is permitted
at nozzle corners by the ASME Code design rules, provided that rules
regarding low-cycle fatigue prevention can also be satisfied [8], it is clear
that satisfactory estimates of strength for vessels containing nozzle corner
flaws cannot be made with only linear elastic fracture mechanics (LEFM)
methods of analysis. Therefore, there is a need for elastic-plastic fracture
analysis methods, simple enough for code application, by which such
estimates of strength, in terms of load, can be made. The objective of this
paper is to demonstrate a means of calculating the conditions for stable
and unstable crack extension at the inside corner of a nozzle in a pressure
vessel, under internal pressure loading, using some experimentally based
approximations that appear to be both physically rational and reasonably
accurate.

Experimental Results and Implications

Two intermediate test vessels containing A508 Class 2 forged nozzles
with fatigue-sharpened inside nozzle corner cracks, designated Vessels V-5
and V-9, have been tested to failure by the HSST Program, which is
managed for the U.S. Nuclear Regulatory Commission by the Oak Ridge
National Laboratory (ORNL). The design of these vessels is shown in
Fig. 1, and a general view of two intermediate test vessels as delivered, one
of which contains a nozzle, is shown in Fig. 2. The data pertinent to the
analysis of Vessels V-5 and V-9, except for the fracture toughness properties
of the nozzle materials, are listed in Table 1. The tests were performed by
ORNL, and a detailed report on the testing procedures, analyses, and
experimental results is available [9].
Each vessel contained one fatigue-sharpened surface crack, approximately
3.05 cm (1.2 in.) deep, in the inside nozzle corner nearest to the vessel
head, as indicated in Fig. 1. Each flaw was prepared by first sawing a
678 ELASTIC-PLASTIC FRACTURE

39-in-OD SHELL

VERTICAL SECTION

FIG. 1—Design dimensions for intermediate test vessel with 22.86-cm-lD (9-in.) test nozzle
(1 in. = 2.54 cm).

FIG. 2—General view of two HSST Program intermediate test vessels, showing bolted-on
closure head used for all vessels and welded-in nozzle used for Vessels V-5 and V-9.
 MERKLE ON NOZZLE CORNER CRACKS 679

1|
-2J

^•_B1 ^^ .5 '
,.9 .S .5 o •
00 ro CQ a^ rrt .2 ° IT) O
O S
^ -^ -e;
OS. Eu i^E E
U
•5

lir ro ^ 00 <N
12
00
g

op a -=•
•o
si- •a -ss :32 .a
F:' •»:s
's ^ ?'£ ^ (N
o
•". in «
c c —
:3
liiiii SI i l
V)

>
•a i i ii i§ s
Sl^l^ •«• m 00
00 1^

m VO O
ii ?s
^2
£RS
S2
T3
O
C

«>

a
n
680 ELASTIC-PLASTIC FRACTURE

20-mm-deep (0.80-in.) slot across the nozzle corner; then welding a steel
boss over the opening of the slot; next applying cyclic hydraulic pressure
to the notch cavity through a hole drilled in the boss until ultrasonic
measurements made from the outside nozzle corner, in the notch plane,
indicated sufficient fatigue flaw growth; and finally removing the weld
boss by flame cutting and grinding. This difficult procedure required
cutting, welding, and grinding to be done by a worker inside the vessel,
a process requiring special equipment and safety precautions as described
in more detail in Ref 9. The pretest ultrasonic estimates of crack front
depth and shape for Vessels V-5 and V-9 were quite similar [9]. The
pretest ultrasonically estimated crack front configuration for Vessel V-9
is shown in Fig. 3. The inflections in the crack front shape are believed to
be due to the effects of the weld boss. Their effects on the test results,
which are believed to be minor, will be discussed later.
Static fracture toughness data for the nozzle material of Vessel V-5
were obtained before the test using precracked Charpy V-notch (PCCV)
[9] and a combination of 0.85T and 2.0T compact specimens [10]. Fracture
toughness values at maximum load were calculated for each specimen,
from its load-displacement diagram, by the equivalent-energy procedure
[//]. This calculation procedure was justified by the known substantial
agreement between J-integral and equivalent-energy toughness calculations

NOZZLE

FATIGUE CRACK FRONT

ESTIMATE

VESSE
ULTRASONIC DATA

SAW CUT NOTCH

FIG. 3—Pretest estimate of fatigue crack front position in the inside nozzle corner of
intermediate test vessel V-9, based on ultrasonic data.
 MERKLE ON NOZZLE CORNER CRACKS 681

for the same points on the load-displacement diagrams of notched beams

and compact specimens [12]. Nevertheless, the omission of stable crack
growth measurements from these data did lead to disadvantages in their
application. At 93 °C (200°F), the maximum load toughness values obtained
from the PCCV and 0.85T compact specimens ranged from 159 to 236
MN-m-^^2 (145 to 215 ksi VirT.). At the same temperature, the two 2.0T
specimens tested [9,10] gave toughness values of 245 and 265 MN-m"^^^
(223 and 241 ksi VitT). Considering the range of data for each specimen
size and the increase in both stable crack growth and static upper-shelf
toughness values at maximum load with increasing specimen size generally
observed in resistance curve testing, the latter value of 265 MN-m"^^'^
(241 ksi Vnr.) was selected as the toughness value to be used for analyzing
the flawed 15.2-cm-thick (6-in.) Vessel V-5 nozzle forging.
Static and dynamic fracture toughness data for the nozzle material of
Vessel V-9 were obtained before the test, using precracked Charpy V-notch
and a combination of 0.85T, 1.5T, and 2.0T compact specimens [9,10].
These data are plotted versus temperature in Fig. 4. A vessel test temperature
of 24°C (75°F), which is below the dynamic upper-shelf temperature, was
selected in order to produce a fast-running fracture as a test result. The
toughness data shown in Fig. 4 indicated that there was no consistent
effect of specimen size on the static fracture toughness of the Vessel V-9
nozzle material at 24°C (75°F). This is because (1) the 1.5T specimens
gave values near the middle of the static toughness range, (2) both greater
and lesser values were obtained from smaller specimens, and (3) the
minimum and maximum values were obtained from the 2T specimens.
Consequently, it was decided to make static initiation calculations for
three toughness values covering the full range of the values measured at
24°C (75°F): 159, 220, and 298 MN-m-^/^ (145, 200, and 271 ksi Vm".).
Because the steepest part of the dynamic fracture toughness transition
curve occurs at 24 °C (75 °F) and the range of dynamic values extends
from below to above the range of static values, both stable crack growth
and "popins" were considered possible [9].
The result of the test of Vessel V-5 at 88°C (190°F) was a leak without a
fracture, which occurred at a pressure of 183 MPa (26 600 lb/in. ^). The
position of the crack front was measured continuously during the test by
an ultrasonic sensor located on the outside surface of the nozzle directly
opposite the fatigue-sharpened crack front. Stable crack growth was first
detected at a pressure of 124 MPa (18 000 lb/in. ^), and above that pressure
the crack front continued to advance stably until it penetrated the outer
surface near the ultrasonic crystal [9]. The point of leakage was barely
visible and there was no visible distortion of the vessel. A closeup view of
the point of leakage in Vessel V-5 is shown in Fig. 5. The result of the
test of Vessel V-9 at 24 °C (75 °F) was a fast fracture as expected because
of the test temperature selected for that purpose. Ultrasonic data did
682 ELASTIC-PLASTIC FRACTURE

50 100
TEMPERATURE (°F>

FIG. 4^Static and dynamic Kud values for Vessel V-9 nozzle material [1 in. = 2.54 cm;
1 ksi -JE. = 1.0988 MNm -^^'•, °C = 5/9 ("F - 32)].
 MERKLE ON NOZZLE CORNER CRACKS 683

FIG. 5—Closeup view of leak point adjacent to ultrasonic base block on nozzle of Vessel
V-5 (arrow shows flaw penetration to surface).
684 ELASTIC-PLASTIC FRACTURE

FIG. 6—Closeup view offractured nozzle in Vessel V-9: test temperature was 24°C (75°F).
 MERKLE ON NOZZLE CORNER CRACKS 685

indicate that some stable crack growth occurred before failure, commencing
at 145 MPa (21 000 lb/in. 2) and totaling about 1.27 cm (0.5 in.) just
before failure at 185 MPa (26 900 lb/in. ^). A closeup view of the fractured
nozzle in Vessel V-9 is shown in Fig. 6.
The circumferential strain values measured on the outside surfaces of the
cylinders of Vessels V-5 and V-9, which are shown plotted in Fig. 7,
indicate that the cylinders of both vessels were fully yielded at failure. The
strains measured at the inside nozzle comers opposite the flaws, for
Vessels V-5 and V-9, are shown plotted in Fig. 8. The nozzle corner
strains at failure for Vessels V-5 and V-9 were 6.5 and 8.4 percent, respec-
tively. Both of these strains are remarkably large compared with the
maximum previously measured strain tolerance of the same material for a
4.75-cm-deep (1.87-in.) flaw in the cylindrical region of an intermediate
test vessel [13], which was 2 percent.
The flaw region of Vessel V-5 has not yet been sectioned for post-test

STRAIN (%)

FIG. 7—Pressure versus outside circumferential strain in vessel cylinder for intermediate
test vessels V-5 and V-9 (/ lb/in. ^ = 6895 Pa).
686 ELASTIC-PLASTIC FRACTURE

^^
—'"H

jO-

20,000
J?
//
A*
14
15,000
T1
JJ • GAGE 24. V - 6
j i
A GAGE 24, V - 9

10.000
f /^^"""^ ®

5000

FIG. 8—Pressure versus inside uncracked nozzle corner circumferential strain for inter-
mediate test vessels V-5 and V-9 (1 lb/in. ^ = 6895 Pa).

examination, but the fracture surfaces containing the original nozzle

corner flaw in Vessel V-9 have been separated, with the results shown in
Fig. 9. The original fatigue-sharpened crack in Vessel V-9 was very close
to the size and shape estimated by ultrasonics before the test [9] (see
Fig. 3). Furthermore, stable crack growth, the extent of which can be seen
in Fig. 9, did increase the average crack depth by about 1.27 cm (0.5 in.),
and it also eliminated the inflections in the crack front shape before failure.
Pre- and post-test estimates of the circumferential strains at failure at
the unflawed nozzle corners opposite the flaws in Vessels V-5 and V-9,
and the corresponding pressures, were made by ORNL and by others,
using several different methods of elastic-plastic fracture analysis, all of a
semi-empirical nature [9]. All of the direct estimates of the nozzle corner
failure strains were low, most by a wide margin. Of these estimates, those
that did not assume plane-strain toughness conditions were more accurate
than those that did [9]. Thus it was apparent that some aspect of nozzle
corner geometry was causing the strain tolerances for nozzle corner cracks
to be substantially greater than would be expected for the same size surface
cracks in the cylinder of a pressure vessel, where it is known that they
would be subjected to plane-strain conditions. In fact, the measured nozzle
 MERKLE ON NOZZLE CORNER CRACKS 687

FIG. 9—Closeup view offlaw in fractured nozzle of intermediate test Vessel V-9.

corner failure strains were closer to the previously measured failure strains
for surface-flawed uniaxial tension bars near and in the upper-shelf
temperature range [14]. Thus it was clear that the tendency of a plane-
strain analysis to underpredict nozzle corner failure strains by a wide
margin, for high toughness conditions, must be due to either an error in
the LEFM portion of the calculation or to the assumption of full transverse
restraint around the crack front. The possibility of large errors in the
LEFM portion of the pretest estimates was subsequently dismissed because
(1) calculations based on several different methods for estimating the
LEFM shape factor for nozzle corner cracks had given similar results [9];
(2) the method used by ORNL, based on Derby's epoxy model test data
[15], was confirmed by later photoelastic experiments [16]; and (3) the
difference between shape factor values estimated from Derby's data [15]
and those based on the solution for an edge crack extending from a hole in
a plate [/] was explained by Embly [9,17] as being due to the effects of
pressure in the crack, effects that are experimentally included in the
former solution but analytically neglected in the latter. For this reason, the
experimentally measured principal strains at the unflawed nozzle corners
opposite the flaws in both vessels were examined closely (see Tables 2 and
688 ELASTIC-PLASTIC FRACTURE

3). Both sets of strain readings indicated the occurrence of considerable

transverse contraction in the plane of the crack at the nozzle corner, thus
implying that full transverse restraint does not exist for nozzle corner
cracks at that location, under vessel internal pressure loading. This phe-
nomenon will be discussed further in the section on analysis.
The pretest estimates of failure pressure for Vessel V-5 were based on an
elastic-plastic nozzle corner pressure-strain curve calculated by the finite-
element method [9]. However, this curve proved to be inaccurate with
respect to the experimental data obtained for Vessel V-5, because it
underestimated the elastic stress concentration factor and overestimated
the pressures for given strains in the elastic-plastic range. Therefore, the
pretest calculations for Vessel V-9 were based on the experimentally
measured pressure-strain curve for Vessel V-5 shown in Fig. 8, and it was
recognized that improved methods for estimating elastic-plastic nozzle
corner pressure-strain curves would be required as part of any practical
method of fracture analysis for nozzle corner cracks.

Analysis
The objectives of the post-test analysis developments to be discussed in
this section were principally to develop an improved method for estimating
TABLE 2—Principal stress and elastic stress-concentration factor values at the inside un/lawed
nozzle corner of intermediate test vessel V-9, calculated from experimental strain data.

Stress, MPa
Pressure,
MPa (ksi) a\ 02 ai Remarks K,

6.9 (1.0) 63 0.4 - 6.9 elastic 4.05

13.8 (2.0) 134 3.5 -13.8 elastic 4.33
34.5 (5.0) 302 6.2 -34.5 elastic 3.89
55.2 (8.0) 419 - 9.9 -55.2 yield
68.9 (10.0) 405 -47.8 -68.9 yield
75.8(11.0) 399 -75.8 -75.8 yield (corner)

TABLE 3—Stress-concentration factor estimates for identical nozzles in an intermediate

test vessel and a reference calculational model of typical PWR" vessel design.

Reference
Intermediate Test Calculational
Term Vessel with Nozzle Model of PVV^R Vessel

Nozzle mean radius, r 19.05 cm (7.5 in.) 19.05 cm (7.5 in.)

Cylinder mean radius, r„ 41.91 cm (16.5 in.) 229.24 cm (90.25 in.)
Cylinder thickness, t 15.24 cm (6.0 in.) 21.59 cm (8.5 in.)
(3 0.484 0.174
K, 4.16 2.71

° PWR = Pressurized Water Reactor

 MERKLE ON NOZZLE CORNER CRACKS 689

elastic-plastic nozzle corner pressure-strain curves, and to find one or more

reasonable methods for considering the combined effects of nominal yield-
ing and partial transverse restraint conditions on the criteria governing
the extension of nozzle corner cracks. Two analysis procedures, differing
principally only in the relationship used between flaw size, toughness, and
failure strain in the elastic-plastic range, were developed. Both procedures
use the same estimates of the nozzle corner pressure-strain curve, and the
same LEFM relationship between vessel internal pressure and the crack-tip
stress intensity factor, for elastic conditions. Both procedures also use the
same equation for estimating the increase in fracture toughness due to less-
than-fuU transverse restraint. For elastic-plastic conditions, one procedure
uses LEFM to calculate the relation between failure strain, flaw size, and
toughness, and the other procedure uses the tangent modulus method
equations for bending. The latter method is an incremental application of
Neuber's equation for estimating inelastic stress and strain concentration
factors, and is described in detail in Ref 13. In this paper, the former
procedure (using LEFM based on strain) is used to estimate failure strains
for the given toughness values, using the initial flaw sizes, and the latter
procedure (using the tangent modulus method) is used to estimate tough-
ness values for the given failure strains, using the flaw sizes measured at
or near failure. The development of the analysis procedures, and the
calculated results for intermediate test vessels V-5 and V-9, are discussed
in the following. The calculated results are summarized in Table 4, which
also lists the numbers of the equations and figures used to obtain the
calculated results.

Pressure-Strain Curve Estimates

In principle, the nozzle corner pressure-strain curve should be bounded
by two tangents, the first representing the initial elastic behavior of the
nozzle at low pressures, and the second being the line of constant pressure
that defines the gross yield pressure of the nozzle region. By comparing
the measured nozzle corner pressure-strain curves for Vessels V-5 and V-9
shown in Fig. 8 with the measured pressure-strain curves for the vessel
cylinders remote from the nozzles shown in Fig. 7, it can be seen that the
gross yield pressures indicated by both figures are essentially the same.
This is consistent with the assumption that nozzle design by the area
replacement method specified by the ASME Code [8] serves to prevent the
gross yield pressure of a nozzle region from becoming less than that of the
cylinder into which the nozzle is inserted. Therefore, for estimating pur-
poses, the gross yield pressure of a nozzle region designed by the area
replacement method will be assumed to be identical to that of the cylinder
into which the nozzle is inserted.
Previous comparisons between theory and experiment have shown that
690 ELASTIC-PLASTIC FRACTURE

I.
00 .S
IIS
r-
hn$
^-*^or<
s 2 ^1 1 !j
s1 s1 s1 .? <^ is *?
ee B 3 06" sgSSS s ^B S
|2(2 B
^n ^ 1
—5 S ^ Sz S z Sz s
(^ "^ s^ 800
00 M
• ^ e
ON

M "* ro
'00. K S$
— CN <N srt 0^ -fN1 ^
i <N 0 -H 1-1 r - —.
S'. ^!^
-H 00 TT 0 <N 0 sq^s
(S ^ i n n
^

1 ^
S
m
ii ST

ill z
1 * <N 2 i / i 00 0
K
.^
0

0 <N n iO <N

11
!i 3 '^'
M ^g u^ III.
lilJ is mi s

•J li

ill Hill
n
< li 1
M: 8. -
 MERKLE ON NOZZLE CORNER CRACKS 691

the gross yield pressure of an intermediate test vessel cylinder can be closely
estimated by the equation

Per = 1.04 ay In (ryr,) (1)

where r„ and r, are the outer and the inner vessel cylinder radii, respectively.
In Eq 1, the factor 1.04 is an empirical factor based on both intermediate
test vessel and small-scale steel model test data, and the remainder of the
equation is based on the Tresca (maximum shear stress) yield criterion.
From Table 1, the room temperature yield stresses of Vessel V-5 and
Vessel V-9 cylinder materials were 500 and 475 MPa (72.5 and 68.9 ksi),
respectively. Therefore, assuming test temperature yield stresses of ay =
476 MPa (69 ksi) for both vessel cylinders, and using ro/r, = 1.44, Eq 1
gives Par = 182 MPa (26.4 ksi).
Although pretest estimates of the elastic stress concentration factor of
the nozzle corners in Vessels V-5 and V-9, based on both elastic finite-
element analysis [18] and epoxy model strain-gage data [15], were approxi-
mately 2.9, the experimental strain data obtained from both vessels indi-
cated a value close to 4. Apparently the finite-element mesh size used
analytically and the strain gages used experimentally on the epoxy models
were^jgot small enough relative to the other nozzle dimensions to determine
the true peak nozzle comer strain. The principal stresses calculated from
the measured principal strains at low pressures on the unflawed inside
nozzle comer of Vessel V-9 are listed in Table 2. These stresses were
calculated from Hooke's law before yielding, and with the aid of the
Tresca yield criterion after yielding [9]. Not only is the initial elastic stress
concentration factor close to 4, but the intermediate principal stress is
initially small and tends to become compressive, eventually equaling the
vessel internal pressure after local yielding occurs. In addition, the measured
values of the nozzle corner stress concentration factor for Vessels V-5 and
V-9 were found to be consistent with an analysis derived by Van Dyke
[19] for calculating the stresses around a circular hole in a cylindrical
shell. The value of the elastic stress concentration factor of the hole, at the
longitudinal plane, is given by Van Dyke's analysis as

K, = 2.5 + ^ |S2 (2)

where

^' = '-""W^
692 ELASTIC-PLASTIC FRACTURE

and where
r = hole radius,
r„ = cylinder midthickness radius, and
t = cylinder thickness.
Applying Eqs 2 and 3 to the nozzle design shown in Fig. 1, both for the
case of an intermediate test vessel cylinder and for a cylinder of typical
reactor vessel dimensions, gives the results shown in Table 3. The value of
Kt for the nozzle in an intermediate test vessel is 4.16, but the value of
Kt for the same nozzle inserted into a typical reactor vessel is only 2.71,
because of the influences of the cylinder mean radius and thickness, both
of which occur as factors in the denominator of Eq 3.
Having resolved both the estimates of the gross yield pressure and the
elastic stress concentration factor, two semi-empirical equations were
developed for estimating the elastic-plastic nozzle corner pressure-strain
curves of Vessels V-5 and V-9. The initial elastic slopes of these curves
were both determined by using the calculated elastic stress concentration
factor, and by assuming that the intermediate principal stress at the inside
nozzle corner was compressive and equal to the vessel internal pressure.
Thus the initial slope, M, of the nozzle corner pressure-strain curves was
calculated from
E
M = —— (4)
K, i^] + 2v

FoiE - 2068 MPapercent-i (300 ksi-percent''), K, = 4.16, and v -

0.3, Eq 4 gives M = 208 MPa-percent"* (30.12 ksi-percent"').
The first semi-empirical equation was based on the assumption that the
slope of the pressure-strain curve decreases linearly with increasing pressure,
and reaches zero at the gross yield pressure. The resulting equation is

p = Pay a - e-^^poY)) (5)

where X is the nozzle corner strain. For the intermediate test vessel nozzle
corners, substituting the values of par and M determined from Eqs 1 and 4
gives

p = 26.4(1 - c - ' " " ^ ) (6)

where p is in ksi and X is in percent. Equation 6 is shown plotted in Fig.

10, which demonstrates that it fits the data from Vessel V-5 with considerable
accuracy.
The second semi-empirical equation was based on plotting the measured
 MERKLE ON NOZZLE CORNER CRACKS 693

FIG. 10—Comparison of calculated and measured nozzle corner pressure-strain curves for
intermediate test vessels V-5 and V-9.

pressure divided by the measured strain versus the measured pressure, for
Vessel V-9, from which it was deduced that the two quantities plotted
could be approximately related by the equation of an ellipse, namely

(7)
\M\j [par J

Rearranging Eq 7 gives

PGY

'-m
(8)

Again, for the intermediate test vessel nozzle corners, substituting the
values of Par and M obtained from Eqs 1 and 4 gives

_ 26.4
(9)
8765\2
694 ELASTIC-PLASTIC FRACTURE

where p is in ksi and X is in percent. Equation 9 is shown plotted in Fig.

10, which demonstrates that it fits the data from Vessel V-9 with equal
accuracy. Thus it appears that either or both of the simple semi-empirical
expressions discussed in the foregoing can be used to obtain good estimates
of elastic-plastic nozzle corner pressure-strain curves for use in elastic-
plastic fracture strength calculations.

Fracture Analyses
Both methods of analysis developed make direct use of the LEFM
solution for the problem being analyzed. Thus, for the intermediate test
vessels with nozzle corner cracks, the experimental curve obtained by
Derby [75] for a series of small, thick-walled epoxy model vessels, which
were approximately geometrically similar to the intermediate test vessels,
was used. This curve, shown in Fig. 11, gives the nondimensional LEFM

1 1 1 1 1
V K. - LARGE EPOXY VESSELS
o \ 5L o ' >v o
.^^ d
To 0
- SMALL, THICK-WALLED
2 -
EPOXY VESSELS
—-^•—.^_

YUKAWA'S FLAT- PLATE

MODEL

• SMALL, THICK- WALLED EPOXY VESSELS

o LARGE EPOXY VESSELS
1 1 1 1 1 1
0.2 0.4 0.6 0.8 1.0 1.2 t.4

FIG. 11—Summary of experimental results obtained from ORNL nozzle corner crack
epoxy model fracture tests [15] and comparison with hole in flat plate approximation [1].

flaw shape factor based on the nominal cylinder hoop stress, which is
defined by

K,
Cn = (10)
Oh yv a

In Eq 10, Oh is the nominal cylinder hoop stress, defined by

o>.=P\J (11)
 MERKLE ON NOZZLE CORNER CRACKS 695

In Fig. 11, C„ is given as a function of a/r^, where r^ is the effective nozzle

radius defined by

rz = r„, + n (1 - 1/V2) (12)

where r„, and r^ are the inside nozzle radius and the inside nozzle corner
radius of curvature, respectively. For the intermediate test vessel nozzles,
from Fig. 1, r„, = 11.43 cm (4.5 in.) and tc = 3.81 cm (1.5 in.), so that
Eq 12 gives r^ = 12.55 cm (4.94 in.). For both methods of analysis, the
LEFM shape factor based on the peak nozzle corner stress is calculated
from

C= ^ (13)

where, for Vessels V-5 and V-9, Kt = 4.16 as determined previously.

The representation of the effects of partial transverse restraint on fracture
toughness is the same in both methods of analysis. The concept under-
lying this part of the calculations is that the nominal strain in the direction
tangent to the crack front, in the plane of the crack, is the primary agent
of transverse restraint \13\. When this strain is zero, plane-strain toughness
conditions prevail, but, when this strain is a contraction, the toughness is
elevated above the plane-strain toughness. If the transverse contraction
strain is approximately equal to or greater than that corresponding to
uniaxial tension, the toughness elevation can be estimated from Irwin's
empirical formula [20]

^ = vm:4^ (14)
Ale

For a through crack, /3ic is defined by [20]

(15)
B

where B is specimen thickness. However, for a part-through surface crack,

an alternate definition

^.c = ^ - ^ (16)
2a
696 ELASTIC-PLASTIC FRACTURE

is used here, in order for the denominator in the expression for /3ic to
retain its identity as twice the distance from the point of greatest transverse
restraint on the crack front to the nearest free surface, not including the
crack surface [13].
Whereas Eq 14 is convenient for estimating the toughness elevation due
to less than full transverse restraint when the plane-strain toughness is
known, a rearrangement of Eq 14 is necessary for determining the plane-
strain toughness when the known value of toughness is a non-plane-strain
value. This rearranged equation is

/S^ic + (5/7)/3,c - (5/7)/3c = 0 (17)

where

'KA' IK.

0. = ^ - ^ or ^ ^ (18)
B 2a

as appropriate. The solution to this equation is

^u^Ai'^i-Ai''^ (19)

where

Ai - Vw2-f 0.0135 + m (20)

Ai = Vw2 +0.0135 - m (21)

and

m = (5/14)|3c (22)

The toughness elevation is then determined from

^c _ iS
(23),
/i^ic "V /3 ic

The estimate of nozzle corner failure strains by the method of LEFM

based on strain begins with the combination of Eqs 10 and 11, rearranged
and symbolically changed to read

P*f = —7-^ (24)

c„ ( - 7 - 1 V ^
 MERKLE ON NOZZLE CORNER CRACKS 697

In Eq 24, />*/ is the elastically calculated failure pressure and Ku is the

plane-strain fracture toughness. The failure strain for plane-strain condi-
tions is calculated from

X/o = ^-^ (25)

M
The failure strain for non-plane-strain conditions is then calculated from

X/=(|^J\/„ (26)

where the ratio {KJKu) is obtained from Eq 14. The estimated failure
pressure is then calculated from Eq 9. Calculated results for Vessels
V-5 and V-9 are shown in Table 4. The three values of failure strain and
pressure listed for Vessel V-9 are those corresponding to the initial flaw
size and the three measured fracture toughness values listed in the upper
part of the table. For Vessel V-5, the calculated failure strain and failure
pressure, based on the initial flaw size, are only slightly conservative, and
the same is true of the strain and pressure corresponding to the maximum
fracture toughness value measured for Vessel V-9. Noting the large dif-
ferences between the plane-strain and the non-plane-strain estimates of
failure strain for both vessels, it is clear that considering the effects of
transverse restraint is essential to the accuracy of the analysis.
The calculations of the plane-strain fracture toughnesses corresponding
to the measured values of nozzle corner strain and flaw size by the tangent
modulus method were based on the directly measured flaw size at failure
for Vessel V-9 (see Fig. 9), and the last ultrasonically measured flaw size
in Vessel V-5 before the pressure began to decrease [9]. Note that the flaw
in Vessel V-5 was 8.4 cm (3.3 in.) deep at a pressure of 183 MPa (26.5 ksi),
and therefore underwent approximately 12.7 cm (5 in.) of stable crack
growth during the last 0.7-MPa (100 lb/in. ^) rise in pressure.
Because of the steep strain gradient in the nozzle corner region, the
tangent modulus equations for the case of bending [13] were used for
these toughness calculations. The derivation of these equations is given in
Appendix H of Ref 13. Briefly, this method of analysis is based on the
Neuber equation for inelastic stress and strain concentration factors

K,K, = K,^ (27)

written in incremental form and then rearranged so that the increment in

the notch ductility factor rfeVp, where e is notch root strain and p the notch
root radius, appears on the left-hand side of the equation and only measur-
able quantities appear on the right-hand side. For a trilinearized stress-
698 ELASTIC-PLASTIC FRACTURE

strain curve and the case of bending, with the applied strain in the strain-
hardening range, the notch ductility factor increments were calculated
from the equations given in the following [13]. For the elastic range

Ae^ = 2C^la4E/E, Xy (28)

For the transition range

AeV^ =4CV^>/£7£7(VXA^- \Y) (29)

For the strain-hardening range

AeV;^ = 2CV^ fVX/(v + x j - VuxTTX)

+ Xwln (30)
VX7 + VX, + Xrf _

where

Xrf — Xs (31)

and where
XK = yield strain,
Xj = strain at the onset of strain hardening,
X/ = applied or failure strain,
E = elastic modulus, and
Es = strain-hardening tangent modulus.
For both vessels, the value oi Es was taken as 20.7 MPa-percent"' (3.0
ksi-percent"'), and Xj was taken as 1.2 percent. The total values of effp
were calculated by adding the values obtained from Eqs 28, 29, and 30,
and the values of Kc/ar were then obtained from [13]

Ik 6/ V^ (32)
Or 20X»

The values of Ku/ar were then obtained by dividing the results of Eq 32

by the values of KJKu obtained from Eq 23. The resulting toughness
values for both vessels are listed in the lower part of Table 4. Both plane-
strain toughness values compare well with the measured values for the two
nozzle materials. The non-plane-strain toughness ratios, KJoy, may look
 MERKLE ON NOZZLE CORNER CRACKS 699

high, but the calculated crack-tip opening displacements listed at the

bottom of Table 2, as calculated from

are both very close to the crack mouth opening displacements measured at
the pressures used for the calculations [9]. Thus the necessity for consider-
ing partial transverse restraint effects for nozzle corner cracks under vessel
internal pressure loading is again indicated.

Discussion
The experimental data obtained from intermediate test-vessels V-5 and
V-9 revealed the need for improved accuracy in the representation of
several factors involved in the fracture analysis of nozzle comer cracks.
Although the LEFM relationship between vessel internal pressure and the
crack-tip stress intensity factor was considered to be satisfactory, the
finite-element method estimate of the nozzle corner pressure-strain curve
made before the test of Vessel V-5 was not considered satisfactory, in
e i t h e r ^ e elastic or the elastic-plastic ranges. Furthermore, the reasonable-
ness of a method for extending LEFM into the elastic-plastic range for
nozzle corner cracks required demonstration, and it was found that such a
demonstration would require the consideration of transverse restraint
effects on toughness as well as the effects of nominal yielding on crack-tip
behavior per se.
The latter requirement was made evident by the tendency of pretest
plane-strain analyses to underpredict nozzle corner flaw strain tolerances,
for pressure loading, and the contraction strains measured on the unflawed
inside nozzle corners of Vessels V-5 and V-9. Consequently, additional
approximate non-plane-strain analyses were performed for both vessels
with considerably improved results. These relatively simple calculations
were performed by two partially different methods, namely, LEFM based
on strain, and the tangent modulus method. In both methods of analysis,
C^ is a factor in the expression for the toughness corresponding to a
certain strain and flaw size, and the other factor is a function of strain,
uncracked geometry, and material properties. Two accurate analytical
approximations for the pressure-strain curve were developed, and these
approximations are useable in both methods of fracture analysis.
One difference between the two methods of analysis, as applied here,
was that stable crack growth was neglected in one of the analyses, but
was considered in the other. In estimating failure strains by the method of
LEFM based on strain, the original crack sizes were used. Nevertheless,
the results were slightly conservative. In calculating the toughnesses
700 ELASTIC-PLASTIC FRACTURE

corresponding to given nozzle corner strain levels by the tangent modulus

method, stable crack growth was considered, and the results compared well
with the measured toughness values. It follows that stable crack growth
should be considered when estimating failure strains by the latter method,
in order to avoid unconservative results.
In developing approximations of the type presented in this paper, it is
appropriate to recognize their limitations, and to anticipate possible
improved approaches to the problem. One of the main limitations inherent
in the analysis results presented in this paper is their dependence on
equivalent-energy maximum load toughness values not accompanied by the
corresponding stable crack growth values. It is possible that the large
values of toughness developed by the nozzle corner cracks in intermediate
test vessels V-5 and V-9 are the combined result of stable crack growth
and partial transverse restraint. However, without toughness data including
stable crack growth values, these effects cannot be separated. With such
data, the analysis methods developed in this paper could still be used to
calculate strength as a function of toughness and current flaw size.
Because the toughness elevation due to less-than-fuH transverse restraint
appears to increase with increasing toughness, it also follows that near
plane-strain conditions may exist for low toughness values. Thus the
benefits of decreased transverse restraint cannot be taken for granted, and
they should be better defined experimentally for low toughness conditions.
For example, such experiments could be conducted by testing vessels
containing nozzle corner flaws at or below their transition temperatures. By
combining the data thus obtained with existing data, it could then be
determined under what conditions and to what extent partial transverse
restraint and stable crack growth can be relied upon to increase toughness
values above those sufficient for plane-strain crack initiation, for nozzle
corner flaws.

Conclusions
An approximate method of elastic-plastic fracture analysis has been
developed for calculating the conditions governing the stable or unstable
extension of an inside nozzle corner crack in a pressure vessel, under
internal pressure loading. The approximations used in the analysis include
(1) an estimate of the inside nozzle corner elastic-plastic pressure-strain
curve, based on the elastic stress concentration factor of the nozzle corner
and the fully plastic pressure of the vessel cylinder; (2) an estimate of
the toughness elevation due to less-than-full transverse restraint, based on
the Irwin /3ic formula; and (3) one of two approximate elastic-plastic strain
versus toughness relations, the first being LEFM based on strain, and the
second being the tangent modulus method. The method of analysis is
developed with the aid of experimental data from two HSST Program
 MERKLE ON NOZZLE CORNER CRACKS 701

intermediate test vessels with inside nozzle corner cracks, both of which
developed high fracture toughness values, and example calculations are
made for both vessels. It is noted that the method of analysis could be
improved by using resistance curve toughness data instead of equivalent-
energy maximum load data, because the latter data do not permit an
estimate of stable crack growth as a function of applied load. It is also
noted that the effects of transverse restraint on toughness are expected to
decrease as toughness decreases, and therefore that additional experiments
on steel vessels under low toughness conditions, which have not yet been
conducted, are desirable for examining the accuracy of the method under
these conditions.

References
[/] PVRC Ad Hoc Group on Toughness Requirements, "PVRC Recommendations on
Toughness Requirements for Ferritic Materials," WRC Bulletin 175, Welding Research
Council, Aug. 1972.
[2] ASTM Task Group E24.01.09, "Recommended Procedure for Ji^ Determination,"
draft document dated 1 March 1977.
[3] Shih, C. F. et al, "Methodology for Plastic Fracture," Fourth Quarterly Progress
Report to Electric Power Research Institute, SRD-77-092, General Electric Company,
Schenectady, N. Y., 6 June 1977.
[4] Paris, P. C , Tada, H., Zahoor, A., and Ernst, H., this publication, pp. 5-36.
[5] Pickett, A. G. and Grigory, S. C , Transactions, American Society of Mechanical
Engineers, Journal of Basic Engineering, Vol. 89(C), Dec. 1967, pp. 858-870.
[6] Stahlkopf, K. E., Smith, R. E., and Marston, T. U., Nuclear Engineering and Design,
Vol. 46, No. 1, March 1978, pp. 65-79.
[7] Mager, T. R. et al, "The Effect of Low Frequencies on the Fatigue Crack Growth
Characteristics of A533, Grade B, Class 1 Plate in an Environment of High-Temperature
Primary Grade Nuclear Reactor Water," WCAP-8256,- Westinghouse Electric Corp.,
Pittsburgh, Pa., Dec. 1973.
[8] ASME Boiler and Pressure Vessel Code, Section III, Division I, Nuclear Power Plant
Components, 1974 edition.
[9] Merkle, J. G., Robinson, G. C , Holz, P. P., and Smith, J. E., "Test of 6-In.-Thick
Pressure Vessels. Series 4: Intermediate Test Vessels V-5 and V-9 With Inside Nozzle
Comer Cracks," ORNL/NUREG-7, Oak Ridge National Laboratory, Oak Ridge, Tenn.,
Aug. 1977.
[10] Mager, T. R., Yanichko, S. E., and Singer, L. R., "Fracture Toughness Characteriza-
tion of HSST Intermediate Pressure Vessel Material," WCAP-8456, Westinghouse
Electric Corp., Pittsburgh, Pa., Dec. 1974.
\11] Witt, F. J. and Mager, T. R., "A Procedure for Determining Bounding Values on
Fracture Toughness Ku at Any Temperature, ORNL-TM-3894, Oak Ridge National
Laboratory, Oak Ridge, Tenn., Oct. 1972.
[12] Merkle, J. G. and Corten, H. T., Transactions, American Society of Mechanical
Engineers, Journal of Pressure Vessel Technology, Vol. 96, Series J, No. 4, Nov. 1974,
pp. 286-292.
[13] Bryan, R. H. et al, "Test of 6-Inch-Thick Pressure Vessels. Series 2: Intermediate
Test Vessels V-3, V-4 and V-6," ORNL-5059, Oak Ridge National Laboratory, Oak
Ridge, Tenn., Nov. 1975.
[14] Grigory, S. C, Nuclear Engineering and Design. Vol. 17, No. 1, 1971, pp. 161-169,
[15] Derby, R. W., Experimental Mechanics, Vol. 12, No. 12, 1972, pp. 580-584.
[1.6] Smith, C. W., JoUes, M., and Peters, W. H., "Stress Intensities for Nozzle Cracks in
702 ELASTIC-PLASTIC FRACTURE

Reactor Vessels," VPI-E-76-25, Virginia Polytechnic Institute and State University,

Blacksburg, Va., Nov. 1976.
[77] Embly, G. T., "Stress Intensity Factors for Nozzle Corner Flaws," Knolls Atomic
Power Laboratory, Schenectady, N. Y., draft dated July 1974 (to be published).
[18] Krishnamurthy, N., in Proceedings, First International Conference on Structural
Mechanics in Reactor Technology, Paper G 2/7, Vol. 4, Berlin, Germany, 1971.
[19] Van Dyke, P., American Institute of Aeronautics and Astronautics journal. Vol. 3,
No. 9, 1965, pp. 1733-1742.
120] Irwin, G. R., Krafft, J. M., Paris, P. C , and Wells, A. A., Basic Aspects of Crack
Growth and Fracture, NRL Report .6598, U.S. Naval Research Laboratory, Washington,
D.C., 21 Nov. 1967.
M. M. Hammouda^ and K. J. Miller^

Elastic-Plastic Fracture Mechanics

Analyses of Notches

REFERENCE: Hamtnouda, M. M. and Miller, K. J., "Elastic-Plastic Fractoie

Meciumics Analyses of Notches," Elastic-Plastic Fracture, ASTM STP 668, J. D.
Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing and
Materials, 1979, pp. 703-719.

ABSTRACT: Fatigue failures invariably start at a surface notch. Crack initiation is

due to plasticity while crack propagation can continue in an elastically stressed material
due to the crack generating its own crack tip plasticity.
From elastic-plastic analyses, it is possible to predict the effect of notch plasticity
on the behavior of short propagating cracks. Elastic-plastic fmite-element analysis
solutions to cracks in notch fields indicate that a crack will initially propagate at a
decreasing rate until the crack can generate a crack tip plasticity that is greater than
the elastic threshold stress intensity condition.

KEY WORDS: notch root radius, notch depth, crack initiation, crack propagation,
nonpropagating crack, threshold stress intensity factor, plain fatigue limit, notch
fatigue limit, stress concentration factors, strength reduction factors, notch stress-
strain field, crack tip stress-strain field, bulk stress-strain field, elastic-plastic finite-
element analysis

Nomenclatiiie
D Notch depth
e Notch contribution to fatigue crack length
K Stress intensity factor
AK Stress intensity factor range
A/STxh Threshold stress intensity factor range
KT Theoretical elastic stress concentration factor
I Fatigue crack length
N Number of cycles
Nf Number of cycles to failure
dl/dN Fatigue crack growth rate
A Length of plastic shear ear at crack tip
a Stress
* Research fellow and professor, respectively, Faculty of Engineering, University of
Sheffield, Sheffield, U.K.

703

Copyright 1979 b y A S T M International www.astm.org

704 ELASTIC-PLASTIC FRACTURE

ffe Fatigue limit for plain specimens

ao Fatigue limit for a particular notch profile
Oy Yield stress
p Notch root radius
Fatigue studies may be divided into two categories. The first is termed
the high-strain, low-endurance regime. Here bulk plasticity occurs and the
cycles required for crack initiation are negligible, all lifetime being con-
cerned with propagating a crack to critical dimensions [1].^ The second
regime is that at low cyclic stresses and here plasticity is extremely localized.
In this case, crack initiation will dominate the lifetime of the specimen or
component [2].
In both regimes it is clear that the fatigue process is due to the material
suffering irreversible, that is, plastic, deformation. In the case of smooth
and flat surfaces, should the stress range level in the bulk material be very
low, but just above the fatigue limit, the plasticity will be restricted to
favorably oriented slip bands within one or two surface crystals and may
be defined as microplasticity. In these circumstances, once a crack is
initiated its growth will continue due to self-generated crack tip plasticity.
This latter form of plasticity can be characterized by invoking linear elastic
fracture mechanics (LEFM) analyses [3] that describe the elastic stress
intensification at the crack tip.
In most components a crack is initiated at a stress concentration such
as a notch, which generates bulk but localized plasticity. Such macro-
plasticity is grain size independent. The notch may be large or small, that
is, a keyway or machining scratch. The notch geometry and the bulk stress
field control the limits between the initiation and the propagation phases
of the crack. Thus in order to understand the fatigue behavior of com-
ponents it is necessary to study the interactive role of micro- and macro-
plasticity of defects in notches during the phases of crack initiation and
propagation.
The present paper examines the behavior of short cracks in notches that
are subjected to various levels of stress which induce differing degrees of
plasticity. Elastic-plastic finite-element analyses are used to predict a safe
bulk stress level below which notch plasticity will not cause fatigue failure
although cracks will be initiated.

Previous Work
The role of plasticity in fatigue crack growth is best illustrated by ref-
erence to cracks in biaxially stressed plates. Consider a plate in the xy-plane,
containing a through central crack whose normal is in the j-direction. Let
the plate be subjected to a positive Oy stress and also a Ox stress that has
^The italic numbers in brackets refer to the list of references appended to this paper.
 HAMMOUDA AND MILLER ON ANALYSES OF NOTCHES 705

magnitudes +ay, 0, and —Oy to equate with equibiaxial, uniaxial, and

shear loading conditions. It has been shown that for constant values of
Oy the crack tip elastic stress intensity factor Ki is unaffected by the magni-
tude and sense of the stress ax applied on the plane perpendicular to the
crack front [4], Crack growth rates are affected [5], however, because the
size, shape, and orientation of the crack tip plastic zone are dependent on
the state of biaxial stress [4], Knowing the size of the plastic shear ears
at the crack tip, it is possible to predict crack growth rates for various
degrees of biaxiality [6].
It follows that elastic solutions to notch problems need to be re-evaluated
and modified to account for the various roles of notch and crack tip plas-
ticity and also the state of stress biaxiality. For these reasons alone, theoret-
ical stress concentration factors and their derivatives, for example, Neuber/
Stowell equivalent strain concentration factors, are unsatisfactory since
they do not admit to the presence of a crack and hence do not differentiate
between initiation Stage I and Stage II propagation phases of the fatigue
process [7]. Neither do they define the extent and strength of the notch
stress-strain field and cannot predict size effects [8] or the phenomenon
of nonpropagating cracks [9].
To overcome these limitations, some recent analyses [10,11] based on
the physical processes of fatigue have been concerned with the propagation
of cracks within notch fields to determine the boundary conditions for
initation and nonpropagating cracks. In summary form, the conclusions
reached on a basis that a crack in a notch field can be equated to a crack
in a plain specimen when both cracks have the same velocity, that is the
same crack tip condition (not necessarily the same AK field), were as
follows:
1. The equivalent length of a fatigue crack of length / within the notch
field can be stated as

L^l + e (1)

where e increases from zero to the depth of the notch, D, as / increases

from zero to the boundary of the notch field. Beyond the edge of the notch
field, the equivalent crack length is simply defined as

L^l +D (2)

It follows that the notch field is that which extends from the notch root to
a point in the bulk of the material at which the effective crack length is
given by Eq 2.
706 ELASTIC-PLASTIC FRACTURE

2. The depth of the notch field in uniaxially stressed plates is approxi-

mately given by

0.13 sfD^ (3)

for a very wide range of notches of engineering importance. Here p is the

notch root radius. For very sharp notches, D » 0.13 VDp and the effec-
tive crack length can be approximated to D.
3. The equivalent length of a crack within the notch field is given by

1 + 7.69 I (4)
-V P

and the stress intensity factor of a crack within the notch field can be de-
fined as

K = 1 + 7.69 a y/rrl (5)

It follows that the term prior to aVx/ can be considered as a fatigue crack
concentration factor which may be equated to the strength reduction factor
Kf or theoretical stress concentration factor KT, although neither of these
terms allows for the presence of a crack. Outside the notch field

K = a V7r(/ + £>) (6)

4. For sharply notched plates, an initiated crack will not propagate if

the bulk stress level

ff <
yfD (7)

where M is associated with the geometric factor of the stress intensification

of cracks in bodies of different shape. For an edge nonpropagating crack
where / <: l^, the term M is equal to 0.5, that is, (1.12 Vx)-'.
These important conclusions are illustrated in Figs. 1 and 2. Equation (3)
successfully combines both the size and shape effect of notches long known
to affect the fatigue behavior of components and was derived for a very
wide range of notch profiles by considering the interaction between the
crack tip and notch elastic-stress fields. This present paper now considers
the role of notch plasticity in the development of very short cracks.
 HAMMOUDA AND MILLER ON ANALYSES OF NOTCHES 707

FICri-^AtotcA contribution to fatigue crack length from linear elastic fracture mechanics
analyses [10].

Ot

FAILURE

'^'niiin/i//7)////////u////
I NON-PROPAGATING

I SAFE
C^H-7-69^]
0-5

lO

FIG. 2—Fatigue regimes for notches with different elastic stress concentration factors [11].
708 ELASTIC-PLASTIC FRACTURE

Present Work
The problem of understanding the behavior of the very short fatigue
crack concerns the dominant role of plasticity in the very early crack growth
regime. A very short fatigue crack is almost impossible to monitor in ex-
perimental growth rate studies, and so a theoretical crack growth analysis
is required in order to assess the lifetime of the crack in this phase. Such
a lifetime can be infinite in the case of an initiated but eventually non-
propagating crack. During this phase, notch plastic zones are bigger than
the extent of crack tip plasticity and hence elasticity cannot describe crack
tip conditions. Consider Fig. 3 and a very short fatigue crack. The bulk
stress field controls the extent of the plastic zone, although it has minimal
effect on the extent of the notch field. In this analysis it will be assumed
that, immediately the crack is initiated, the plastic shear ears at the crack
tip [12] will extend to the initial elastic-plastic boundary for the maximum
tensile load applied. The length of the shear ears. A, during propagation
can be determined from an elastic-plasticfinite-elementanalysis for a crack

TrrTTTTTrrrrrjrr»*»i*»iiiit»****t* iTTmTl
FIG. 3—Crack tip and notch plastic fields.
 HAMMOUDA AND MILLER ON ANALYSES OF NOTCHES 709

of any length. Figure 4 shows the type of finite-element idealization used

in this study.
The mathematical procedure used in the elastic-plastic finite-element
analysis was identical to that reported by Yamada et al [13] which invokes
the Prandtl-Reuss equations for plastic flow in a von Mises material. The
length of the shear ears is equated to the maximum extension of the plastic
zone radiating from the crack tip.
Stress-strain analyses were carried out for plane stress conditions at two
notch profiles, one shallow and one sharp. The first had a depth of 16 mm
and a root radius of 16 mm. Analyses were for cracks of length 0, 0.035,
0.070, 0.105, 0.210, 0.430, 0.570, and 1.160 mm. The second notch had
a depth of 16 mm and a root radius of 2.25 mm with crack lengths of 0,
0.035, 0.070, and 0.35 mm. The material stress-strain behavior was as
follows: Young's modulus, 206 GN/m^; cyclic yield stress, 176 MN/m^
(that is, 0.6 of the monotonic yield stress, Ref 2); and work-hardening
modulus (slope of the stress-strain curve beyond yield) zero for the first
notch and 10330 MN/m^ for the second notch. Such behavior is typical of
yield and work-hardening behavior of the mild steel used both in the pres-
ent experimental program and by Obianyor and Miller [14] to study the
effect of stress overloads on threshold stress intensity values. The material
has the following composition:

0.14C-0.58Mn-0.16Si-0.008N remainder Fe

Figure 5 provides a schematic of the results of the finite-element anal-

yses. At the LEFM threshold limit, which is here determined experimentally
and is similar to that quoted in Ref 14, there is a known stress intensity
field. This field characterizes a small plastic zone at the crack tip, the ex-
tent of which can be calculated to determine A by assuming a von Mises
nonhardening material, plane stress conditions, and a yield condition equal
to twice the cyclic yield stress. The line designated "no crack" represents
the elastic-plastic boundary to which A will extend for just-initiated cracks.
It follows that short nonpropagating cracks of length I < h can develop
in the shaded area due to notch plasticity and that the length which a non-
propagating crack can attain is stress-level dependent. For a crack to
eventually stop, ^K-n must not be exceeded. Thus, according to LEFM,
the critical nonpropagating crack size can be expected to decrease as stress
range increases. However, Frost [15] has shown experimentally that non-
propagating crack length increases with increasing stress level. This con-
flict is due to the fact that early crack growth is due to plasticity and can-
not be described by LEFM parameters. As stress level increases, the ex-
tent of the plastic zone increases and nonpropagating cracks in these zones
can therefore be longer. Thus Point A represents the limiting stress level
below which initiation will not take place, while Point B is an upper-bound
710 ELASTIC-PLASTIC FRACTURE

t t t t t f t f t t t

168 mm

See Fig. 4b

I I 2 48mm (not to scale)

-105 mm -

FIG. 4a—Half plate, with circular edge notch: p = 16 mm, D = 16 mm.

FIG. 4b—Finite-element idealization of crack tip zone.

 HAMMOUDA AND MILLER ON ANALYSES OF NOTCHES 711

^ CRACK LENGTHS
^-J i, < i 2 < i 3 < i 4 ^^-^ ^.'•<:P^
LU
M
^.^^^^^^'^^ ^ ^-^^^^*^^^~^
Co
UJ
^^^^^^^^\^^-
y^'^^^_.^^'^
z
O '^S-^^"^
hsj

o LEFM / ' ^ A
1- THRESHOLD X /
tO / / y ^^'v
<
/
/

h///
y X

p
^^t;

o
o
_l
'^ '^ I W

LOG % • „

FIG. 5—Schematic representation of the effect of increasing stress and crack length on
crack tip plasticity for a given notch profile.

stress-level solution for propagation to failure. At stress levels below Point

A an existing flaw of length greater than /3 is necessary for it to propagate
to failure according to LEFM. At stress levels above Point B, failure will
occur whether or not there preexists a flaw. Finally, Point C indicates that
the same size of plastic shear ear exists when / = 0 and I = h and that
when 0 < I < h the crack is decreasing in growth rate because A is de-
creasing.
Figures 6 and 7 present finite-element analyses for the shallow and sharp
notch profiles, respectively, and these can be compared with the schematic
of Fig. 5.
To summarize, it is best to refer to Fig. 8. If a crack has a length suf-
ficient to generate crack tip plasticity greater than that characterized by
A/JT-Th, then the crack growth can be interpreted by a conventional LEFM
analysis. To attain this critical crack length, it is necessary for crack growth
to be a consequence of notch-generated plasticity. Should the plasticity
not be of sufficient extent to develop this critical crack length, then a non-
propagating crack results. Figures 6 and 7 therefore define for a given
notch geometry the stress boundaries between initiation and propagation
of a crack based on a knowledge of the extent of crack tip plasticity A.
The critical crack length is thus seen to be a function of applied stress
level and notch profile.
It has been previously stated that it is difficult to experimentally monitor
short crack growth behavior. To establish the validity of the present theory,
however, it is possible to determine safe conditions for cracked specimens.
Thus Fig. 9 presents the results of tests on prior edge-cracked mild steel
specimens cycled in zero-tension. From these tests AK-n was determined
712 ELASTIC-PLASTIC FRACTURE

lOO

•+ OOOO
003S
E X 0 070
E
a 0 lOS
x" T 0-2IO
10 A 0'430
z
• 0 570
o 1 I60
<
UJ

< THRESHOLD— —
UJ
X oio •'
I/)

o
<

o-oiO-l 0-2 0-4 OS

BULK STRESS / f f „

FIG. 6—Elastic-plastic finite-element analysis results for cracks in a shallow-edge circular

notch: p = 16 mm, D — 16 mm.

as 5.35 MN/m~^^^, which agrees with published work on the same ma-
terial [14], A second series of tests on edge notched plates, similar to that
shown in Fig. 4a, determined the endurance limit, that is, that limit below
which cracks were initiated but not propagated through the notch field and
on to failure; see Fig. 10. From a knowledge of AK-n and do it is possible
to draw the limiting conditions for safety a la Kitagawa [16]; for example,
see Fig. 11. In this latter figure are the data points of the present theoret-
ical elastic-plasticfinite-elementanalyses which are in close agreement with
the experimentally determined boundary conditions, the former values
being derived from Fig. 6 for plastic zone sizes equal to that corresponding
to AKrh in a simple cracked member.
Thus the safe stress levels for cracks of different length in various notched
configurations can be determined from elastic-plasticfinite-elementanalyses
such as those shown in Figs. 6 and 7. Note that for very short crack lengths,
LEFM analyses should not be employed. This is because LEFM can char-
acterize the extent of plasticity only when (1) the plastic field is small in
comparison with the elastic stress intensification field and (2) the extent
of crack tip plasticity is physically meaningful. When cracks are less than
0.25 mm, the crack tip plastic zone size is measured in units of angstroms
and hence LEFM characterization of fracture processes is no longer ap-
plicable.
It now remains to modify the Smith analysis [10] to account for plasticity
effects on short crack growth rates. An assumption in the present approach
 HAMMOUDA AND MILLER ON ANALYSES OF NOTCHES 713

8
o ^
<£l
t^
'-' in
UJ
ct
1«
3
H
to ;^
^
^
_l
s
•3
3 a
m g
•3
S
^
i

e in O o 1
<n 1^ m
E
o O m
1
O o
o O ^
o
1
-J
4 o O
• + X
0)

U1
1^
_5
§
^^ ' HIONBT aV3 aV3HS DllSVld
714 EUSTIC-PLASTIC FRACTURE

TOTAL

\ 7

BULK CRACK TIP

PLASTICITY _ PLASTICITY
CONTROL CONTROL
(EPFMl (LEFM)

CRACK LENGTH
Corresponds to
AKth

FIG. 8—Plastic and elastic fracture mechanics characterization of fatigue crack growth.

o
V
^ lO

o 5
1
AKth

1
X. 1
lO^ lO* lO'
Cycles to Failure Nf

FIG. 9—Experimental data for determining the threshold stress intensity factor.
 HAMMOUDA AND MILLER ON ANALYSES OF NOTCHES 715

300

200-
3
'H.
E
<
ft

lOO
lO^ ID*
Cycles to Failure Nf

FIG. 10—Experimental data for determining the shallow-notch fatigue limit: p = 16 mm,
X> = 16 mm.

0-5

<
o-2-s;

O I

FIG. l\~Comparison of experimental and theoretical (elastic-plastic finite element) results

for the shallow notch.

was that A for short cracks will extend to the elastic-plastic boundary at
the root of the notch, and so, as the very short crack grows, the value of A
will initially decrease, producing a decrease in crack growth rate until the
crack is long enough to grow under LEFM control; see Fig. 8. Since e is
determined from the equivalence of crack velocity, which is a function of
crack tip plasticity conditions, this means that the notch contribution term
e of Eq 1 will initially decrease while the crack is in the notch plastic zone.
This effect can be determined from the crossover behavior of the curves
of Figs. 6 and 7. While this effect is small, Fig. 12 also shows that e is a
function of stress level since this controls the extent of plasticity and early
growth rate.
716 ELASTIC-PLASTIC FRACTURE

0-2

FIG. 12—Notch contribution to fatigue crack length from elastic-plastic fracture mechanics
analyses for different stress levels o/oy: (a) 0.45, (b) 0.35, (c) 0.31.

Discussion
Although the agreement between theory and experiment depicted in Fig.
11 is very good, continuum mechanics analyses, be they elastic or elastic-
plastic, can be in error when crack lengths and growth rates are of the
order of microstructural features such as grain size. The minimum finite-
element size used in this study is 0.035 mm, that is, comparable to the
grain size of mild steel. The AK-n plastic zone size is 0.13 mm. It follows
that while the present method cannot model the plastic behavior of a single
grain, the very small mesh size can give some assessment of continuum
plastic behavior around threshold conditions. It should be noted that the
program was used only to determine the elastic-plastic boundary.
The form of the base curve (/ = 0) in Fig. 5 is important. For a shallow
notch. Fig. 6, the stress level is critical to the extent that, should a crack
be initiated, then it will propagate to failure since threshold conditions are
immediately exceeded as indicated by the steepness of the base curve. This
condition is equivalent to Point X in Fig. 2. For a sharp notch, however
(see Fig. 7), a crack may be initiated at a low stress value but it will not
grow since the threshold is not exceeded. This is equivalent to Point Y in
Fig. 2. Should the stress level be increased slightly, then an initiated crack
may grow but still not propagate to failure unless the stress is increased to
a level indicated by Point Z in Fig. 2, which approximates to Point B in
Fig. 5. Thus, the fatigue failure of components from notches is seen to be
a function of the applied stress level a, the notch profile parameters p and
D, and the material property, AKn.
Another aspect of Figs. 6 and 7 that is important concerns the crossover
behavior of the "no crack" curve (J = 0). As an example, consider the
 HAMMOUDA AND MILLER ON ANALYSES OF NOTCHES 717

sharp notch, Fig. 7, and a stress level a equal to 0.2 Oy. As soon as a crack
is initiated, the plastic shear ear length decreases until the crack length is
approximately 0.04 mm. However, it attains its original length when /
equals 0.07 mm. It follows that the crack growth rate decreases and then
increases as depicted in Fig. 8. Because the crack growth rate decreases,
then the notch contribution factor e, based on an equivalence of crack
growth rates, should decrease faster than the increase in the fatigue crack
length I. The present studies show this to be the case, Fig. 12, for values
of / up to about 0.1 mm for the shallow notch subjected to stress levels
a > 0.3 Oy approximately. These studies also indicate as shown in Fig. 12
that even without a fatigue crack the notch has an equivalent crack length
e which is stress-level dependent. In the present case, initial values of e/D
are 0.36 and 0.64 for stress levels 0.35 Oy and 0.45 Oy, respectively, for the
shallow notch. Thus, early life fatigue can have exceedingly high crack
growth rates although the cracks are extremely small. In this regime, linear
elastic fracture mechanics does not apply. It is recommended that in sharp
notch situations designers use equivalent crack lengths derived from Eq 2;
that is, they should assume that e/D is unity.
It is interesting to compare present results with other recent work. Jerram
{17] concludes that the elastic stress intensity factor is not a suitable crite-
rion for predicting crack initiation and that the fatigue crack length used
in his paper to define initiation, namely, 10 ^m (0.0004 in.) was too large.
We would add that no elastic parameter is suitable; see Figs. 8 and 12.
Ohji et al [18\ studied, by finite-element methods, cracks emanating from
notches, but these long cracks were 0.25 mm in length and therefore well
beyond the notch stress field. They argued that nonpropagating cracks
are due to crack closure and can be assessed by effective values of stress
intensity factors or local strain range values or both. Thus they introduced
a criterion which stated that a crack will continue to propagate if the mag-
nitude of the strain range, at a certain characteristic distance ahead of the
crack tip, exceeds a critical value. Thus the work of Obianyor and Miller
[/4] is relevant since they show that as a crack grows into a rapidly decreased
stress field, due to the application of a prior overload, the threshold stress
intensity factor increases, thus increasing the possibility of a nonpropagating
crack. Now Kotani et al [19], like Jerram, examined stress-strain behavior
ahead of the crack tip by invoking the Neuber relationship, but once again
crack "initiation" lengths were such as to be propagating cracks and out-
side the notch stress field. Their work showed that specimens with stress
concentrations had "initiation" lives greater than plain specimens on a
basis of local stress range values. This was undoubtedly due to initially
decreasing crack propagation rates in the notched specimens with cracks
growing into much lower stress fields.
It appears that two classes of notch crack lengths have to be considered.
The first concerns the very short crack whose initiation and early growth
718 ELASTIC-PLASTIC FRACTURE

are not amenable to LEFM analyses, while the second type concerns the
longer but still small crack that is amenable to LEFM if the stress level is
high enough. Nevertheless, cyclic plasticity controls the birth and the early
growth of both types of crack. The present work indicates that for the
sharp notch a crack length of the order of 0.1 mm can develop at the notch
root, due to plasticity, and failure will still not occur. This length is of the
same order as the length of a crack necessary for the application of LEFM
analyses; see Fig. 11. On the other hand, shallow notches require higher
stress levels to develop the notch root plasticity that initiates cracks. Such
plasticity is well contained within the notch field; compare the 0.03-mm-
deep with the 2.08-mm-deep notch field. These short cracks are not amen-
able to LEFM analyses and the notch contribution factor e is stress level
dependent since this controls the extent of plasticity. These initiated cracks
will not cease propagation, however, because the threshold limit is easily
exceeded (see Fig. 6) due to the higher stress levels and the strain concen-
tration feature of the notch.
Finally, it should be noted that both classes of cracks will slow down as
they come close to the elastic-plastic strain boundary.
All the foregoing work has confined itself to a two-dimensional apprecia-
tion of fatigue crack initiation and growth. Work is now continuing on a
three-dimensional appreciation of crack growth of very short cracks at
notch roots.

Conclusions
1. Notch root plasticity controls the early stage propagation of fatigue
cracks in notches and in this regime LEFM analyses do not apply.
2. Elastic-plastic fracture mechanics can account for fatigue crack growth
below the elastic threshold stress intensity condition by considering the
interaction between crack tip and notch field plasticity.
3. Elastic-plastic fracture mechanics can account for decreasing crack
growth rates and the production of nonpropagating fatigue cracks.

Acknowledgments
The authors would like to thank British Gas for providing a research
scholarship to support M. M. Hammouda.

References
[/] Ham, R. K., in Proceedings, International Conference on Thermal and High Strain
Fatigue, Institution of Metallurgists, London, England, 1%7, pp. 55-79.
[2] Miller, K. J. and Zachariah, K. P., Journal of Strain Analysis. Vol. 12, No. 4, 1977,
pp.262-270.
 HAMMOUDA AND MILLER ON ANALYSES OF NOTCHES 719

[3] Irwin, G. R., Transactions ASME, Journal of Applied Mechanics, VoL 24, 1957, p. 361.
[4] Miller, K. J. and Kfouri, A. P., International Journal of Fracture, Vol. 10, No. 3, 1974,
pp.393-404.
[5] Hopper, C. D. and Miller, K. J., Journal of Strain Analysis, Vol. 12, No. 1, 1977, p. 23.
[6] Miller, K. J., in Proceedings, "Fatigue 1977" Conference, Cambridge, England, Metal
Science Journal, Vol. 11, Nos. 8 and 9, 1977, p. 432.
[7] Forsyth, P. J. E., iti Proceedings, Symposium on Crack Propagation, Cranfield, England,
1961, p. 76.
[8] Coyle, M. B. and Watson, S. J., in Proceedings, Institution of Mechanical Engineers,
Vol. 178, 1963, p. 147.
[9] Frost, N. E., Marsh, K. J., and Pook, L. P., Metal Fatigue, Oxford University Press,
Oxford, England, 1974, p. 173.
[10] Smith, R. A. and Miller, K. J., International Journal of Mechanical Sciences, Vol. 19,
1977, pp. 11-22.
[//] Smith, R. A. and Miller, K. J., International Journal of Mechanical Sciences, Vol. 20,
1978, pp. 201-206.
[12] Tomkins, B., Philosophical Magazine, Vol. 18, No. 1S5, 1968, p. 1041.
[13] Yamada, Y., Yoshimura, N., and Sakuri, T., International Journal of Mechanical
Sciences, Vol. 10, 1968, p. 343.
[14] Obianyor, D. F. and Miller, K. J., Journal of Strain Analysis, Vol. 13, No. 1, 1978,
pp. 52-58.
[15] Frost, N. E., The Engineer, Vol. 200, 1955, pp. 464 and 501.
[16] Kitagawa, H. and Takahashi, S., in Proceedings, Second International Conference on
Mechanical Behavior of Materials, Boston, Mass., 1976, p. 627.
[17] Jerram, K., "Fatigue Crack Initiation in Notched Mild Steel Specimens," Report 1972,
Central Electricity Generating Board, RD/B/N1994.
[18] Ohji, K., Ogura, K., and Ohkubo, Y., Engineering Fracture Mechanics, Vol. 7, 1975,
p. 457.
[19] Kotani, S., Koibuchi, K., and Kasai, K., "The Effect of Notches on Cyclic Stress-
Strain Behaviour and Fatigue Crack Initiation," Report of the Mechanical Engineering
Research Laboratory, Hitachi Laboratory, Tsuchiura, Japan.
W. R. Brose^ and N. E. Bowling^

Size Effects on the Fatigue Crack

Growth Rate of Type 304 Stainless
Steel

REFERENCE: Brose, W. R. and Dowling, N. E., "Size Effects on the Fatigne Crack
Growth Rate of Type 304 Stainless Steel," Elastic-Plastic Fracture, ASTM STP 668.
}. D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for Testing
and Materials, 1979, pp. 720-735.

ABSTRACT: Planar size effects on the fatigue crack growth rate of AISI Type 304
stainless steel characterized by linear-elastic fracture mechanics were experimentally
investigated. Constant-load amplitude tests were conducted on precracked compact
specimens ranging in width from 2.54 to 40.64 cm (1 to 16 in.). The da/dN versus AK
data are compared on the basis of several size criteria which are intended to limit
plasticity and thus enable linear-elastic analysis of the data. Also, the cyclic J-integral
method of testing and analysis was employed in the fatigue tests of several specimens
undergoing gross plasticity. The cyclic J crack growth rate data agree well with that
from the linear-elastic tests. It is argued that an appropriate size criterion for linear-
elastic tests must limit the size of the monotonic plastic zone and thus be based on
J^max, the maximum stress intensity. While the size criteria considered vary widely in
the amount of plasticity they allow, they provide comparable correlations of crack
growth rate. Thus the use of the most liberal criterion is justified.

KEY WORDS: 304 stainless steel, fatigue crack growth rate, size effects, size criteria,
plasticity, cyclic J-integral, crack propagation

In characterizing static fracture, the techniques of linear-elastic fracture

mechanics can be applied if the cracked body is predominantly elastic at
fracture. In a fracture toughness test, certain minimum specimen size
requirements assure this condition by restricting the size of the crack-tip
plastic zone with respect to that of the specimen. An analogous size re-
quirement is probably necessary for fatigue crack growth data character-
ized by linear-elastic fracture mechanics.
Size effects in the areas of fracture and fatigue are actually of two kinds.
The first concerns specimen or component thickness as it affects the
'Engineer and senior engineer, respectively, Structural Behavior of Materials Department,
Westinghouse R&D Center, Pittsburgh, Pa. 15235.

720

Copyright 1979 b y A S T M International www.astm.org

 BROSE AND DOWLING ON FATIGUE CRACK GROWTH 721

amount of transverse constraint and thus the state of stress at the crack
tip. The importance of thickness in the area of fracture is well docu-
mented. In fatigue, thickness effects reported in the literature are inconsis-
tent. Increasing thickness has been observed to increase, decrease, and
have no effect on growth rate [1,2].^ Also, thickness can be considered to
be a controlled variable in determining crack growth rate.
The other size effect involves what is known as planar size, generally
identified as specimen width, the dimension perpendicular to the thickness
direction and parallel to the crack plane. It is planar size which, for a
given material, determines the degree of plasticity at a given K level in a
cracked body. Size requirements for fatigue which limit plasticity and thus
enable linear elastic analysis of the data must be related to planar size.
This paper is concerned with planar size effects on the fatigue crack
growth rate of annealed AISI Type 304 stainless steel, a material widely
used in the nuclear industry. Planar size effects are of interest in this
material because of its relatively low monotonic yield strength. Relatively
large specimens may be needed to obtain fatigue crack growth data under
predominantly elastic conditions.
Test results are presented for specimens ranging a factor of 16 in size.
Several size criteria are examined in terms of the degree of plasticity which
they permit. Also examined is the effect of the observed plasticity on
fatigue crack growth rate and thus the ability of these size criteria to
produce size-independent correlations of growth rate. In addition to the
linear-elastic tests, an elastic-plastic experimental and analytical technique
is employed to generate fatigue crack growth data under conditions of
gross plasticity. The data obtained by this technique on small specimens
are compared with that from larger specimens under elastic conditions.

Plasticity and Size Criteria

Since planar size effects in linear-elastic fatigue crack growth are related
to the development of plasticity, it is instructive to examine the stress-
strain behavior near the tip of a growing fatigue crack in a ductile metal.
Figure 1 schematically illustrates this behavior for an idealized elastic-
perfectly plastic material. Two separate plastic zones are identified. Within
the monotonic plastic zone, increments of yielding occur at the maximum
point of each loading cycle while the cyclic behavior is still elastic. In the
cyclic plastic zone, incremental yielding continues to occur on each loading
cycle, and there is inelastic cyclic action as well. The size of these plastic
zones can be estimated by the Irwin plastic zone size equation as shown in
Fig. 1 and as explained by Paris [3]. For zero-to-tension loading, R = 0,

^The italic numbers in brackets refer to the list of references appended to this paper.
722 ELASTIC-PLASTIC FRACTURE
 BROSE AND DOWLING ON FATIGUE CRACK GROWTH 723

the cyclic plastic zone is only one-fourth the size of the monotonic plastic
zone. This difference in size increases as R increases.
Due to bending in the compact specimen, Fig. 2, the fully plastic con-
dition occurs when the region of material yielded in monotonic tension
spans about half the width of the uncracked ligament and meets with a
similar zone of material yielded in compression and extending from the
back of the specimen. The crack-tip cyclic plastic zone is thus only about
one-eighth the size of the remaining specimen ligament at fully plastic
yielding when ^ = 0.
Thus, as has been previously proposed [4], it appears reasonable to
establish a size criterion for linear-elastic fatigue crack growth testing
which would limit the amount of monotonic plasticity developed. One
approach is to limit plasticity as reflected in the specimen load-deflection
behavior which is directly measurable. A somewhat arbitrarily chosen form
of such a criterion requires

' plastic s V^ (1)

where
' plaslicj •'max •'max »
Vmax = maximum measured specimen deflection, and

a, crack length

H, half-height

B =thickness

ap, machined notch length

H/W=0.6 B/W=0.5 an/W=0.25

FIG. 2—Compact specimen geometry employed in linear-elastic tests.

724 ELASTIC-PLASTIC FRACTURE

Vmax" = corresponding deflection calculated on the basis of elastic

behavior [/, 4].
Equation 1 can be rewritten as

'max ^ ^ 'max (2)

The fatigue crack growth data presented later will be examined in light of
this criterion.
It is not always convenient to measure specimen deflection response, and
also a great deal of fatigue crack growth data exist without accompanying
deflection measurements. A size criterion based on calculated quantities is
thus desirable. Such a size criterion should be based on /imax, the max-
imum stress intensity in a loading cycle, rather than AK, the range of stress
intensity. One such criterion is being considered for adoption in a forth-
coming ASTM standard [4]. For compact specimens, it is required that

iW-a)^-{ - ^ (3)

where W — a is the uncracked ligament length (specimen width minus

crack length) and Oy is the 0.2 percent offset monotonic yield strength.
Data on a 1310-MPa (190 ksi) yield strength lONi steel appear to conform
well to this size criterion [4].
It can be argued that monotonic yield strength is not an adequate single-
parameter index of the degree of plasticity developed in a cracked body,
since it does not take into account the large difference in strain-hardening
capacity different metals can exhibit. Considering two metals with the
same static yield strength, the one with higher strain-hardening character-
istics will indeed produce less extensive plasticity in a cracked body at the
same stress intensity. A first-order correction for the effect of strain-
hardening would be to substitute a value of flow stress,ffnow,equal to the
average of the yield and ultimate strengths, for Oy in Eq 3. Both the Oy and
fffio» size criteria will be applied to the data presented later on 304 stainless
steel, a material which exhibits extensive strain hardening.
While the three size criteria presented in the foregoing are of primary
interest in this paper, two other conditions of plasticity are here described.
They represent possible upper and lower bounds in terms of plasticity
within which any reasonable size criterion would fall. So-called nominal
yielding occurs when the nominal stress or "P/A + Mc/F' stress reaches
the yield, On — Oy. The other bound is the calculated condition of fully
plastic limit load, P = PLL. Rice's solution for this condition for the
compact specimen, presented in Ref 5, is employed in this paper. Yield
strength rather than flow stress was used in the equation for the results
presented herein.
 BROSE AND DOWLING ON FATIGUE CRACK GROWTH 725

Experimental Procedure
The specimens tested in this study were machined from a 6.35-cm-thick
(2.5 in.) rolled plate of solution-annealed 304 stainless steel. Additional
test material information is given in Table 1. Specimens were taken from
the plate in the longitudinal-transverse (L-T) orientation, defined in the
ASTM Test for Plane-Strain Fracture Toughness of Metallic Materials
(E 399-74). In this orientation, the specimen crack plane is perpendicular
to the rolling direction of the plate.
The linear-elastic fatigue crack growth tests are now described. Compact
specimens of the geometry shown in Fig. 2 were employed. In this paper,
specimen size is identified by width. All other dimensions are as in Fig. 1
except that, as noted, the thickness sometimes differs. The largest speci-
men tested is shown in Fig. 3.
All testing was performed with servo-controlled, electrohydraulic test
machines. Constant-amplitude tension-tension load control was employed
with a ratio of minimum to maximum load, R, equal to 0.05, and fre-
quencies in the range of 10 Hz. At the very end of tests, the frequency was
decreased to facilitate crack length measurement. Deflections were mea-
sured 0.483 cm (0.190 in.) away from the specimen front face. In Fig. 2,
the specimen front face is to the left of the load line. Crack growth was
monitored visually using a calibrated traveling microscope in conjunction
with scribe lines placed on the specimen surface. Crack length versus cycles
data were reduced to crack growth rate versus stress intensity range using a
seven-point incremental polynomial fitting technique [6] and a stress-
intensity solution available in the literature [7\.
The cyclic / tests were performed in accordance with the experimental
and analytical techniques described by Bowling and Begley [8], Test
specimen geometry was that shown in Fig. 2 except that the notch was
modified to permit deflection measurement at the load line, and the
machined notch length was given by a„/W = 0.465. Specimen width was
5.08 cm (2 in.).
Cyclic / tests are conducted under deflection control to a sloping line.
TABLE 1—Test material information.

Description: AISI 304 stainless steel; Jessop Steel Co. heat 24348
Condition: hot-rolled, annealed and pickled
Geometry: plate, 6.35 by 60 by 60 cm (2.5 by 24 by 24 in.)
Chemistry: 0.058C-1.48Mn-0.035P-0.012S-0.38Si-8.90Ni-18.15Cr-0.44Mo-0.17Co-0.57Cu-Fe
remainder (values in weight %)
Tensile properties:"
Offset yield strength, MN/m^ (ksi) 269 (39)
Ultimate tensile strength, MN/m^ (ksi) 579 (84)
True fracture strength, MN/m^ (ksi) 1920 (279)
Reduction in area, % 82
Charpy impact energy," m-N (ft-lb) 320 (236)

"Longitudinal orientation with respect to rolling direction of plate.

726 ELASTIC-PLASTIC FRACTURE

FIG. 3—Large compact specimen andfatigue test apparatus.

This control condition is illustrated in Fig. 4 along with some of the load-
deflection loops obtained in one test. A special analog control circuit was
used to impose this condition in which neither load nor deflection ampli-
tude is constant. Note that as the test progresses and crack length in-
creases, the maximum load drops while the maximum deflection increases.
The minimum deflection is always zero. The amount of cyclic plasticity
 BROSE AND DOWLING ON FATIGUE CRACK GROWTH 727

304 SS
Spec. 2435B-10,W=2 in.

17 Cycles

FIG. 4—Load versus deflection loops during a cyclic i test under deflection control to a
sloping line.

gradually increases with crack length, producing gradually increasing

values of AA and crack growth rate. The cyclic / test technique is valuable
in that it provides extensive cyclic plasticity without the large mean de-
flections, or specimen ratchetting, that can occur at the end of a load
control test.
The method of calculation of A/for a given loading cycle is shown in Fig.
5. It is based on the Rice et al [9] approximation for /, and employs the
area under the load-deflection curve on the loading half-cycle. Only the
area above the macroscopic crack closure load, which was considered to be
at the point of inflection on the unloading half-cycle curve, was used.
Further details on / in fatigue are given in Refs 8,10, and / / .
728 ELASTIC-PLASTIC FRACTURE

p
D
1.

/ \ \ ~'' V \
AP
X '''••''' \ X \ e

0
C/\\ '^'/^ ''\\N E
6

B/\ ix^^\ \ \ \ \ \ \ \ \ F •

// Estimated Closure Point

A G
2 (Hatched Area)
AJ

FIG. 5—Operational definition of cyclic J.

Results and Discussion

The measured specimen plasticity during the fatigue crack growth tests
is now examined. Figure 6 compares the measured maximum specimen
deflection with that predicted by elastic behavior for the test on one speci-
men. There is a plastic component of deflection through most of the
specimen life, and the plastic deflection increases rapidly near the end of
the test. While not shown in the figure, the measured deflection range,
agreed closely with that predicted elastically for the
duration of the test. This indicates that while the size of the region yielded
monotonically becomes very large at the end of the test, the cyclic plastic
zone remains small. These measurements concur with behavior expected
from analytical considerations presented earlier.
The degree of plasticity permitted by each of the size criteria discussed
earlier is also shown in Fig. 6. The stress-intensity range at which each
criterion is violated is listed for each specimen in Table 2. Note that the
order in which the size criteria are violated is unchanged for different
specimen sizes. By definition, the plasticity permitted by the deflection
criterion corresponds to Vma, exceeding Vm^x by ^ factor of two. At the
criterion based on yield strength, this factor,is about 1.5, while for the
 BROSE A N D D O W L I N G O N FATIGUE CRACK G R O W T H 729

.090

304 SS
Spec. 2434D-9,W=2 in.
.080

2 W-a=4/7i(K^ax''<'y'
.070 - 3 Vmax=2V^ax
4 W-a=4/Ti(Kn,ax/<'f|ow'
P=P
LL
.060 -

1 in. =2. Mem

.050

= .040

.030

.020

.010 --

.100 .200 .300 .400 .500 .600 .700

a - i ^ , Crack Length Beyond Notch, inches

FIG. 6—Measured and elastically calculated maximum specimen deflection.

criterion based on flow stress, the factor is about four. The values of
these plastic deflection factors at the three size criteria were roughly in-
dependent of specimen size.
The degree to which the plasticity allowed by the various size criteria
affects fatigue crack growth rate is now examined. Figure 7 contains the
da/dN versus A/sT data obtained in the linear-elastic tests. The shaded
points differ from the unshaded points only in that the shaded points
violate the size criteria based on yield strength, Eq 3. Figures 8 and 9
contain the same data except that the shaded points violate the size criteria
based on deflections and on flow stress, respectively.
In each of these plots the data obtained from the cyclic / tests are also
730 ELASTIC-PLASTIC FRACTURE

,.-^ ^--^^vO

g?s§l

"c
'i•§ ,o
B
?"
•a •n
^ u
u
:^
«,
->
Q>

d"
••M

t3
-•^
Q
> •

v.
^
1^
<
1

^i
VI
03 lo lo ,^ in § . Al
v
VI I
S Al
VI ° is I
'^^S-S-^o
5;§2:2s
If!t|il
22?
rt <N n -n-
SPSS
 BROSE AND DOWLING ON FATIGUE CRACK GROWTH 731

1 1 1 1 1 111I 1 1 1
-
304SS o
8
- oo -
T (5>
o
-3 - "

- -

1 '°

-4 JliO /
10 -
5"°V
« • vi
-
tf<^/I—Factor of 3.8
5/ scatterband of James
-
v/
\r*
• A » V / Linear-Elastic Tests ,
/ D ' / o W=lin.
o W=2
-5 A W =4, B =0. 5
^ S /
V / • W=4 -
V W=16. B=2

Cyclic J Tests -
O W=2

l§ Shaded Points Violate:

-
/ o / W-ai4/Ti(Knax/0y)^

-6
lin. =2.54 cm ,,,
ll(si/in'.=l.lMn-m -
1 1 1 1 1 J 1 1 1 ; ~
10 100
AK, Stress Intensity Range, Icsi/IfT

FIG. 7—Comparison of data with size criterion based on yield strength.

shown. Values of AJ were converted by AK = VAJ-E. The elastic-plastic

data shown are actually a compilation of data from tests on four separate
specimens covering different but overlapping ranges of A/ and crack
growth rate.
Also shown on these plots are upper and lower scatterbands for a large
body of fatigue crack growth data on annealed 304 stainless steel compiled
by James [12]. These data represent 10 different specimen geometries
732 ELASTIC-PLASTIC FRACTURE

T
\ r ( M M 1 1

o
304SS
8
oo

-3 o
»o
o
o
-
1 »o -
• / o
/ */

-4 J <»/
— t'ol ~
'•CF/
- • ^ -
- • % i < ^ ' — Factor of 3.8 -
-A?/ scatterband of James

•5
- Linear-Elastic Tests -
OW=lin.
V /
o W=2
-5 AW =4, B = 0 . 5 -
jiy
V
- aw=4 -
- ^By vw=16, B=2 -
- Cyclic J Tests -
o w=2

Sfiaded Points Violate:

/ o /
V
" max i V' ^max

-6
lin.=2.54cm
-3/2
1 ksi /Trf. =1.1 Mn-m
1 1 1 1 , ,|. 1 ]

10 100
AK, Stress Intensity Range, ksi •{\n.

FIG. 8—Comparison of data with size criterion based on deflections.

and several heats of material. However, the specimens cover a narrow

range of size and are mostly relatively small.
Consider all the linear-elastic test data except perhaps the last three
points for each specimen at the high-growth-rate end. The spread in
fatigue crack growth increases only slightly with AK, from a factor of 2.5
at the low end to a factor of about three at the high end. Also, the data
agree quite well with the scatterband of James, which is based primarily on
relatively small specimens. As AK increases, the difference in the degree of
plasticity existing in specimens of different size increases. On this basis,
 BROSE AND DOWLING ON FATIGUE CRACK GROWTH 733

1 1 1 1 1 1 1 1 I I
-
o
304 5S
8
o
o
»<s>
%
„-3 oo —
o
o
9

7V
— gao 1 -
- •
- • ' W o ' / l — factor of 3.8
„/ scatterband of James
-
o La v / Linear-Elastic Tests
/ o W=l in.
- O/ Op
V / 0 W=2
D ' A VV=4, B=0.5
D W=4
—
-
1s V
v W=16, B=2

Cyclic J Tests
-

0<^„
o W=2

- ff Shaded Points Violate:

y <s / W-a>4/„(K^3,/o,|„„)2
/§
/ 0 /

-6 -_
l i n . =2.54 cm , "
Iksi 7irr = l.lMn-m "^ '
- 1 1 1 1 1 1 1 1 1 1

10 100
A K, Stress Intensity Range, l(si {\r\.

FIG. 9—Comparison of data with size criterion based onflow stress.

the experimental data indicate that the plasticity experienced in these tests
has little effect on crack growth rate. It is not surprising, then, that the
size criteria employed in Figs. 7, 8, and 9 provide comparable correlations
of crack growth rate.
The data do show, however, a consistent layering with specimen size. At
a given A^, crack growth rate increases as specimen size decreases. So it
is likely that plasticity does have some effect here on growth rate. Perhaps
the relative size of the cyclic plastic zone, which was always small in the
subject tests, has first-order control of the validity of the linear-elastic
734 ELASTIC-PLASTIC FRACTURE

analysis. A large degree of monotonic plasticity could have a secondary

effect, though, and be a factor in the layering observed.
Since the linear-elastic test specimens represent a factor of four in thick-
ness, a possible contribution to the spread in growth rate from this source
should be considered. However, two previous investigators found no sig-
nificant effect of thickness on growth rate in annealed 304 stainless steel.
James [13] compared growth rates from foil specimens 0.254 mm (0.010
in.) thick with those from specimens of conventional thickness which
correspond to the scatterbands already described. Shahiniam [2] tested
specimens ranging in width from 0.762 to 2.54 cm (0.3 to 1 in.).
Finally, it is noted that the data obtained in the cyclic / tests on the
5.08-cm-wide (2 in.) specimens compare very well with those from the
linear-elastic tests. This agreement is a positive reflection on the ability of
the cyclic / method to characterize crack growth rate data under conditions
of gross plasticity. Also, it is a further indication of the lack of a strong
size effect due to plasticity in the linear-elastic tests. This is because the
plasticity in the elastic-plastic tests is directly accounted for by the use of
/ , while it is not directly accounted for in the linear-elastic tests.
Regarding the linear-elastic data, duplicate tests, as well as data at
other /?'s and starting stress intensities and on other geometries, would of
course strengthen the conclusions reached here regarding 304 stainless
steel. Elevated-temperature data on specimens of different sizes should also
be obtained since this material is used widely in that situation. Of more
importance, though, is that data on other high strain hardening materials
be generated before conclusions regarding size effects on fatigue crack
growth of 304 stainless steel are considered to have general applicability.
An investigation of these variables is currently underway at Westinghouse.

Smninaiy and Conclusions

In order to investigate planar size effects on the fatigue crack growth
rate of annealed 304 stainless steel, compact specimens ranging a factor of
16 in width were tested using linear-elastic fracture mechanics techniques,
as were compact specimens employing the cyclic / elastic-plastic technique.
The following conclusions are offered.
1. Size criteria intended to limit plasticity during fatigue crack growth
should restrict the size of the monotonic plastic zone with respect to speci-
men size, and should thus be based on Kmax rather than on AK.
2. Monotonic plasticity did not appear to strongly affect crack growth
rate in the subject material. Thus the three different size criteria con-
sidered, while allowing widely different amounts of plasticity, provided
comparable correlations of crack growth rate.
3. Fatigue crack growth rates obtained under gross plasticity and char-
acterized by/agreed well with those from linear-elastic tests.
 BROSE AND DOWLING ON FATIGUE CRACK GROWTH 735

4. Data at other load ratios, on other geometries, and especially on

other strain-hardening materials, are needed before the observations on the
utility of the various size criteria can be considered to have general ap-
plicability.

Acknowledgments
The experimental work was conducted in the Mechanics of Materials
Laboratory under the direction of R. B. Hewlett. This laboratory is op-
erated by the Structural Behavior of Materials Department managed by
E. T. Wessel. A number of technicians participated in the experimental
work, and the care exercised is greatly appreciated. Mr. P. J. Barsotti is
especially thanked for preparing the oscillograph system for deflection
measurement. This work was sponsored by the Westinghouse Advanced
Reactor Division, Waltz Mill, Pa.

References
[/] Dowling, N. E. in Flaw Growth and Fracture, ASTM STP 631. American Society of
Testing and Materials, 1977, pp. 131-158.
12] Shahinian, P., Nuclear Technology. Vol. 30, Sept. 1976.
[3] Paris, P. C , Fatigue—An Interdisciplinary Approach. Syracuse University Press,
Syracuse, N.Y., 1964, pp. 107-127.
[4] Hudak, S. J., Jr., Saxena, A., Bucci, R. J., and Malcolm, R. C , "Development of
Standard Methods of Testing and Analyzing Fatigue Crack Growth Rate Data—Third
Semi-Annual Report," AFML Contract F33615-75-5064, Westinghouse Research
and Development Center, Pittsburgh, Pa., March 10, 1977.
[5] Hudak, S. J., Jr. and Bucci, R. J., "Development of Standard Methods of Testing
and Analyzing Fatigue Crack Growth Rate Data—First Semi-Annual Report," AFML
Contract F33615-75-5064, Westinghouse Research and Development Center, Pittsburgh,
Pa., Dec. 16, 1975.
[6] Qark, W. G., Jr., and Hudak, S. J., Jr., "The Analysis of Fatigue Crack Growth Rate
Data," Westinghouse Scientific Paper 75-9E7-AFCGR-PJ, Aug. 26, 1975, to be published
in Proceedings. 22nd Sagamore Army Materials Research Conference on Application of
Fracture Mechanics to Design, Sept. 1975.
[7] Saxena, A. and Hudak, S. J., Jr., "Review and Extension of Compliance Information for
Common Crack Growth Specimens," Westinghouse Scientific Paper 77-9E7-AFCGR-P1,
May 3, 1977.
[*] Dowling, N. E. and Begley, J. A. in Mechanics of Crack Growth, ASTM STP 590.
American Society of Testing and Materials, 1976, pp. 82-103.
[9] Rice, J. R., Paris, P. C. and Merkle, J. G. in Progress in Flaw Growth and Fracture
Toughness Testing. ASTM STP 536. American Society of Testing and Materials, 1973,
pp. 231-245.
[10] Dowling, N. E. in Cracks and Fracture. ASTM STP 601, American Society of Testing
and Materials, 1976, pp. 19-32.
[11] Dowling, N. E. in Cyclic Stress-Strain and Plastic Deformation Aspects of Fatigue Crack
Growth, ASTM STP 637, American Society for Testing and Materials, 1977, pp. 97-121.
[12] James, L. A. m Atomic Energy Review, Vol. 14, No. 1, 1976, pp. 37-86.
[13] James, L. A., "HEDL Magnetic Fusion Energy Programs Progress Report—January-
March 1977," Westinghouse Hanford Co., Energy Research and Development Adminis-
tration Contract EY-76-C-14-2170.
D. F. Mowbray^

Use of a Compact-Type Strip

Specimen for Fatigue Crack Growth
Rate Testing In tiie High-Rate
Regime

REFERENCE: Mowbray, D. F., "Use of a Compact-Type Strip Specimen for Fatigue

Cnicit Growtli Rate Testing in tlie High-Rate Regime," Elastic-Plastic Fracture, ASTM
STP 668, J. D. Landes, J. A. Begley, and G. A. Clarke, Eds., American Society for
Testing and Materials, 1979, pp. 736-752.

ABSTRACT: Fatigue crack growth in chromium-molybdenum-vanadium steel was

studied in the high-rate regime of 2.5 X 10~^ to 2.5 X 10~' mm/cycle with a compact-
type strip specimen. The specimen was found to give rise to constant growth rates over
large increments of crack length when cycled under simple load control. Use of the
crack opening load ranges and an approximate J-integral analysis showed that the
growth was occurring under essentially constant AJ.

KEY WORDS: fatigue crack growth, fracture mechanics, J-integral, crack propagation

Nomenclatiue

a Crack length
aeff Effective crack length, a + r^
B Specimen thickness (gross section)
B„ Strip thickness at minimum section
Ci, y Constants in Dowling-Begley crack growth relationship
e, p Subscripts indicating elastic and plastic parts
E, V Elastic constants
G Strain energy release rate
/ Path-independent integral
K Stress intensity factor
N Cycles

' Manager, Mechanics of Materials Unit, Materials and Processes Laboratory, General
Electric Company, Schenectady, New York 12345.

736

Copyright 1979 b y A S T M International www.astm.org

 MOWBRAY ON COMPACT-TYPE STRIP SPECIMEN 737

P Load
PL Limit load
ry Plastic zone size, {K/loyY
R Ratio of minimum to maximum load in a fatigue cycle
U Potential energy
W Specimen width (gross section)
Wa Strip width
W Effective specimen width, W^/Wij
6 Load point deflection
A Indicates the range of a variable
Oy Yield stress

Fatigue crack growth at high rates^ is of current interest because of its

role in the low-cycle fatigue damage process. This process consists mainly
of the propagation of small cracks, the growth of which occurs at high
rates due to the highly (plastically) strained surrounding material. It is
commonly reported that as much as 99 percent of the measured life in low-
cycle fatigue specimens involves propagation.
The foregoing observations have prompted a limited amount of work on
characterizing growth at high rates [1-5],^ as well as several models which
relate crack growth to low-cycle fatigue [6-9]. The models appear to demon-
strate a good correspondence between crack growth and low-cycle fatigue
relationships. It is suggested that their validity and usefulness can be
furthered by the acquisition and characterization of more crack growth
rate data.
The number of investigations of crack growth in the high-rate regime
has remained, for two prominent reasons, quite small. One reason is con-
nected with difficulties in testing with small-size test specimens and stan-
dard control procedures. Unstable crack advance commonly results as
such tests proceed into the high-growth-rate regime, thus precluding the
possibility of obtaining any useful data. The second reason centers on the
apparent absence of a suitable controlling variable with which to correlate
data between tests and specimen types. Most crack growth investigations
in recent years have been based on the stress intensity factor as the con-
trolling variable. The stress intensity factor is defined for linear elastic
material only, and loses meaning as a controlling variable when the crack-
tip plastic zone becomes a significant fraction of the crack length. This is
normally the case in the high-growth-rate regime with small-size laboratory
specimens. Hence, most investigations are terminated at rates less than
10~-'mm/cycle.

^The range of crack growth rates in the high-rate regime is considered in this paper as
10"''to 10 ~'mm/cycle.
•'The italic numbers in brackets refer to the list of references appended to this paper.
738 ELASTIC-PLASTIC FRACTURE

A nonlinear fracture mechanics approach was recently explored by

Dowling and Begley [3] with apparent success. They obtained crack growth
rate data for A533-B steel in the high-growth-rate regime with deep-notched
compact specimens. In order to keep the growth rates stable, they utilized
an analog computer to decrease the load-point deflection in a prescribed
fashion as the crack length increased. The test results were successfully
correlated with an operational definition of the J-integral, which considered
that only the loading during crack face opening results in damage. The
crack closure point during a cycle was detected by noting the point in the
load-deflection curve where the unloading slope changed curvature. The
load range corresponding to the crack closure point and maximum load
was used to compute A/.
Dowling has subsequently added confirmation to the correlation with
tests on deep-notched center-cracked specimens [4] and smooth bar low-
cycle fatigue specimens in which the growth of small surface cracks was
charted [5]. The results from the first two investigations are shown in Fig. 1.
In the high-growth-rate regime, they fit a power type relationship of the
form

da
dN = C, AP (1)

where da/dN is the crack growth rate and Ci and y are material constants.
Considering that the AJ — da/dN curve is independent of geometry, the
approach of Dowling and Begley should have general applicability. This
approach also reduces to, and extends, the linear elastic fracture mechanics
approach because of the relationship of / to K.* Data obtained from AJ-
testing overlap and extend to higher growth rates data obtained in the
nominally elastic range and correlated with AK. Figure 1 illustrates this
result for the A533B steel.
There is one prominent objection to, and one practical difficulty in,
applying the J-integral to fatigue crack growth. The objection centers about
the mathematical definition of / . In the strict mathematical sense, it is
valid only within the confines of deformation plasticity theory [10], which
excludes consideration of unloading. Dowling and Begley [3] approached
this objection on the basis that / may have more applicability than the
current mathematical definition indicates. Theirs is an operational defi-
nition of J implying that the stress and strain fields near the crack tip dur-
ing the loading half of a fatigue cycle are defined by / , despite the inter-
mittent unloading.
The difficulty with practical applications is in the determination of J — a
relationships. There are only a limited number of configurations for which
*J = K ^aE, where a = 1 for plane stress and 1 — c ^ for plane strain.
 MOWBRAY ON COMPACT-TYPE STRIP SPECIMEN 739

0.001 ooi oi
•2
10

10

10

10

g,0=
LINEAR PORTION
do/dN= C, a j '

10

g.to

ELASTIC-PLASTIC TESTS
A cc. W = r
CT, W = 2"
•
LINEAR ELASTIC TESTS
A CC, W = 1"
o CT, W = 2"
a CT, W • 2.5"
o CT, W 8 "

-8
10 - CC - CENTER CRACKED SPECIMENS
CT= COMPACT SPECIMENS
DATA FROM 2 0 TESTS

a_il. I I I
I 10 10 10 10
flj OR ( f i K l ^ / E (IN-LB/IN^)

FIG. 1—Correlation ofA533B steel crack growth rate data (from Ref4).

J is known or can be directly measured. Most deep-notch configurations

allow / to be deterpiined during test by measuring the area enclosed by the
load-displacement curve. Most other configurations require, at present, a
complex set of experiments or elastic-plastic analysis to determine the/ — a
relationship. When examining this approach, these difficulties must be
740 ELASTIC-PLASTIC FRACTURE

balanced against the fact that any approach involving nonlinear material
behavior will have similar difficulties.
The present work was undertaken to investigate further the possibility of
utilizing the /-integral to correlate crack growth rate data. A somewhat
different approach was taken in the testing. A specimen was sought which
would allow stable growth rates to be obtained while utilizing simple control
procedures. It was found that a compact-type strip specimen first used by
McHenry and Irwin [11] possessed these characteristics. A slightly modified
version of their specimen design gave rise to constant growth rates over
large increments of crack length when cycled under simple load control.
No simple J-integral solution could be evolved for this configuration, how-
ever, and it was necessary to employ estimation procedures to compute AJ.

Test Program

Material
The test material was chromium-molybdenum-vanadium (Cr-Mo-V)
steel forging. The chemical composition of the steel is given in Table 1.
Table 2 gives the standard tension test properties and cyclic stress-strain
curve properties. The latter were determined by means of the incremental
step test [12].

Specimen
The modified strip specimen is shown in Fig. 2. It is of the compact type,
with deep side grooves part-way across its width. The full-thickness section
remaining beyond the grooves provides the stiffness for crack growth rates
to remain stable under constant-loading cycling. The specimen differs from
that of McHenry and Irwin in that it is 25.4 mm wider and has machined-in
knife edges on the load line. The larger width gives the specimen a useful
crack length range of 50 mm. The knife edges allow for the measurement
of load-line displacement, as required in /-computations.
A stress intensity factor solution for the specimen was determined by the
TABLE 1—Chemical composition.

Composition, Weight %
c Mn P, max S, max Si

0.25 to 0.35 0.7 to 1.0 0.025 0.025 0.15 to 0.35

Ni, max Cr Mo V Fe

0.5 0.85 to 1.25 1.0 to 1.5 0.2 to 0.3 balance

 MOWBRAY ON COMPACT-TYPE STRIP SPECIMEN 741

TABLE 2—Mechanical properties.

Tension Test

0.2% yield strength, MN/m^ 676

9.02% yield strength, MN/m^ 634
Ultimate tensile strength, MN/m^ 820
Elongation in 50.8 mm, % 22
Reduction in area, % 66
Cyclic Stress-Strain"

Cyclic strain hardening exponent, (n ') 0.144

Cyclic strength coefficient (A"), MN/m^ 1207
0.2% offset stress, MN/m^ 503

VI
31.8 9.63

[1
-V
s
+
^
Bn = 3 . l 8

ALL DIMENSIONS IN M I L I M E T E R S

FIG. 2—Compact-type strip specimen.

experimental compliance method. A description of the experimental tech-

nique utilized is given in Ref 13. The following polynomial form describes
the result obtained (refer to Fig. 2)

K = 9.48 + 28.98
yfBBnW' W
+ 29.44
W (2)
742 ELASTIC-PLASTIC FRACTURE

where W is an effective width defined after McHenry and Irwin [//] as

It was noted by McHenry and Irwin that the /(T-solution for the strip
specimen could be expressed to within a few percent accuracy by

where C is a constant. This is also true for the present design, with C = 38.
The accuracy is ± 4 percent for a/W in the range of 0.1 to 0.4.

Test Procedure
The specimens were tested in a servo-hydraulic testing system under load
control. All testing was at room temperature. The cyclic frequency was
varied in each test from ~ 10 to 0.01 Hz, depending upon the current crack
growth rate. Crack lengths were monitored with a telescopic filar gage
having a 0.125-mm division scale in the field of view. Displacement across
the knife edges was measured with a clip gage. Loops of load versus dis-
placement were recorded periodically.
The program included crack growth rate tests on four specimens. Each
specimen was tested at a series of constant load ranges for R = 0.1.
Generally, four load ranges per specimen were attemped with each suc-
cessive range increased above the previous one. The cracks were propagated
~ 10 mm at each load range.

Test Results
Example crack growth data from two of the tests are plotted in Figs. 3
and 4 on linear coordinates. Each plot represents the results from one speci-
men in which four successively higher load ranges were imposed. (Note
that the data for each load range are defined by different scales on the
abscissa.)
Figures 3 and 4 indicate a unique crack growth rate response for the
specimen in that the data define linear curves or constant growth rates.
This response is apparently insensitive to the absolute load levels and crack
length range of ~40 mm. Different starting load levels were used for the
four specimens so that in general there were differing load-range/crack-
length combinations. The only departure from the linear trend appears
in the initial 1/2 mm of crack propagation at a new load level.
 MOWBRAY ON COMPACT-TYPE STRIP SPECIMEN 743

SCALE ©
AP=32000N
do/dN = 1.8x10''mm/CYCLE

SCALE ( D
&P- 2 8 0 0 0 N
do/dN=3.8Klcr^m/CYCLE

SCALE @
HP- 2 4 0 0 0 N
do/dN = 8-9KI0''mm/CYCLE

•TRANSITION FLAT-TO-SLANT

LE @
aP= 2 0 0 0 0 N
do/dN= I.SxIcr^m/CYCLE

80 100
200 250
1600 2000
8000 10000
CYCLES,N

FIG. 3—Crack growth data for specimen 4 (Cr-Mo-V steel).

Note that at one of the load ranges in each test the slope of the linear
line defined by the data does change slightly. This appears to always co-
incide with the transition to fully slant fracture. The growth rate increases
approximately 10 percent when this takes place.
Fatigue crack growth rates are summarized in Table 3. They were de-
termined by fitting linear curves to the crack length versus cycles data and
differentiating the analytic expression describing the fitted curve. Least-
squares regression analysis was used in the fitting. With the initial one or
two data points at each load range omitted, deviations from a straight-line
relationship were less than 2.0 percent.
744 ELASTIC-PLASTIC FRACTURE

SCALE O
flP= 34 0 0 0 N
do/dN= 2 2XI0"'mm/CYCLE

SCALE (3)
/iP= 2 4 0 0 0 N
d o / d N = 4 . 3 IO"'mm/CYCLE

18 -J_ _L.
© 0 20 40 60 80 100
® 0 200 400 600 800 1000
(D 0 800 1600 2400 3200 4000
CYCLES, N

F I G . 4—Crack growth data for specimen 5 {Cr-Mo-V steel)-

Analysis of Data
An analysis of the data was based on the previously discussed operational
definition of A/. In lieu of a simple/-solution for the strip specimen, values
of A/ were calculated using an estimation procedure developed by Bucci
et al [14]. These authors demonstrated the procedure quite accurately by
comparing calculated and experimental results for different specimen
geometries. Others [15,16] have subsequently shown favorable results with
the same approach.
 MOWBRAY ON COMPACT-TYPE STRIP SPECIMEN 745

TABLE 3—Summary of crack growth rate data.

Specimen Load a (Range), da/dN,

No. Range, N mm mm/i cycle Ay, MJ/m2 ^.K^/E, MJ/m2

1" 7100 18 to 23 6.1 X 10 ~5 0.0039 0.0039

2 10900 18 to 26 1.6 X 10 ~'' 0.0096 0.0096

11300 26 to 36 2.1 0.010 0.010
11800 36 to 46 4.8 0.014 0.014
12500 46 to 56 1.3 X 10"3 0.028 to 0.030 0.027 to 0.028
13100 56 to 66 1.6 0.038 to 0.040 0.033 to 0.035

3 17800 20 to 26 9.9 X 10"" 0.021 0.021

20000 26 to 31 2.0 X 10 ~3 0.032 to 0.033 0.027 to 0.029
22000 31 to 36 3.5 0.048 to 0.050 0.040 to 0.042
24000 36 to 41 8.1 0.080 to 0.086 0.066 to 0.071
26000 41 to 46 2.2 X 10-2 0.116 toO.119 0.094 to 0.097
30000 46 to 53 1.2 X 10-' 0.213 to 0.219 0.156 to 0.160
34000 53 to 58 3.0 0.319 to 0.332 0.221 to 0.230

4 20000 20 to 31 1.5 X 10-3 0.032 to 0.036 0.027 to 0.030

24000 31 to 41 8.9 0.078 to 0.084 0.064 to 0.070
28000 41 to 48 3.8 X 10-2 0.171 to 0.176 0.129 to 0.133
32000 48 to 58 1.8 X 1 0 - ' 0.269 to 0.279 0.186 to 0.193

5 24000 20 to 31 4.3 X 10-3 0.049 to 0.053 0.042 to 0.045

28000 31 to 41 1.8 X 10-2 0.108 toO.114 0.087 to 0.091
32000 41 to 48 1.2 X 1 0 - ' 0.198 to 0.205 0.147 to 0.153
34000 48 to 58 2.2 0.289 to 0.310 0.201 to 0.216

"Used for experimental compliance analysis.

The estimation procedure is based on the definition of / in terms of

potential energy, stated as

J= (5)
B\da )i

where U is the potential energy at load-point displacement & (or area under
the load/load-point displacement curve). / and U are partitioned into elastic
and plastic components, such that

/ . + Jp (6)

/. = (7)
B \ da /s

1 /dU„
/.- (8)
B \ da Ji
where the subscripts e and p designate elastic and plastic, respectively.
746 ELASTIC-PLASTIC FRACTURE

Je IS equivalent to the elastic strain energy release rate, G, and this in

turn to (K^/E) for plane stress. Bucci et al found it most accurate to com-
pute G as a function of the plastic zone corrected crack length; that is

K(aaty
Je — G(aeff) — (9)

UtB — a + r, (10)

where r^ is the plastic zone length for plane stress

1 / K
(11)
lit \ 2CTV

where o> is the material yield stress. In this application, the conventional
yield stress is multiplied by a factor of 2 to account for cyclic plastic ac-
tion [17\.
Computations of/«involve two steps. First, K and r, are computed based
on a, and then K is recomputed based on a as- Because of the cyclic prob-
lem, Oy was taken as the 0.2 percent offset stress on the cyclic stress-strain
curve.
Up is estimated as the product of the limit load, PL, and the plastic load-
line displacement, 6p, such that

1 diPLdp)
Jp=-
B aa dp / dPi
(12)
BW\da/W /i

Computations of Jp were made utilizing plastic displacements from the

recorded load-displacement loops, and a lower-bound limit-load solution
for a compact specimen. The latter, developed by Merkle and Corten [18],
is expressed as

PL = 1.26a„ BW (1 - a/W) a (13)

where the 1.26 multiplier has been added to account for plastic constraint
via the Green and Hundy [19] solution for bend specimens, and a is de-
fined by

2 a/W \^ a/W 2 a/W

a— 1 - a/W +4 -a/W +2 1 -a/W + 1 (14)
 MOWBRAY ON COMPACT-TYPE STRIP SPECIMEN 747

Differentiation of Eq 13 yields the result required for evaluation in Eq 12.

The derived expression was applied to the strip specimen assuming it had
constant cross-sectional dimensions defined by the equivalent quantities,
S/BB7 indWyWa.
Values of AJ were calculated for each load range in increments of crack
length of 1.25 mm. The crack opening load ranges were established by
locating the inflection points on the unloading line of the load-displacement
records. Figure 5 shows some typical records with the estimated inflection
points indicated by arrows. Location of the inflection points was rather
subjective with this specimen and material because of the small percentage
of plastic deflection which develops. (The heavy ligament at the back face
tends to prevent yielding on a gross scale.) The trend in behavior is clear,

5000 N

0 = 3l.8mm 33 34 2 35.6 36.8

SPECIMEN NQ,4 , flP = 3 2 0 0 0 N

FIG. 5—Example load-displacement records.

748 ELASTIC-PLASTIC FRACTURE

however, in that the crack closure load increases with increasing crack
length.
The values of A / calculated in the fashion described in the foregoing are
listed in Table 3. A range of values is given corresponding to each incre-
ment of crack length at which the applied load was maintained constant.
The range of AJ indicates how much variation was calculated for the
indicated crack length range. Examination shows that all values were within
± 4 percent of the median value. Hence, there appears a near constancy in
AJ, deriving from a balancing of the increasing crack length with a de-
creasing crack opening load range. Shown in the adjoining column of
Table 3 are the linear elastic fracture mechanics based values, AK^/E.
The difference between the AJ and AK^/E values indicates the extent of
the nonlinear correction. These differences vary from zero at growth ratei
of 2.5 X 10~^ mm/cycle to —40 percent at 2.5 X 10~' mm/cycle.

Discussion
From the testing point of view, the strip specimen provides an excellent
means for generating high growth rate data. It is not limited by any type
of instability, and can be utilized to obtain very high growth rates with
simple load control. Although rates as high as 3.0 X 10"' mm/cycle were
obtained in this investigation, higher rates could apparently have been
achieved without concern for ratchetting.
Of further significance with regard to the specimen performance is the
constant growth rate obtained under simple load control. This is a desirable
feature in any crack growth rate test because it (1) limits scatter, and (2)
means the parameters controlling the crack growth process are being kept
constant in the test. In this case, it appears that AJ for crack surface
opening was being maintained constant. This is stated with caution be-
cause AJ was not directly measurable and there was an element of sub-
jectivity in selecting the crack closure load points. Further analysis of the
specimen and tests on other materials are needed to confirm the potential
constant AJ feature. In any event, what has been observed lends support
for AJ as the controlling variable.
A plot of the test results is shown in Fig. 6 on logarithmic coordinates
of da/dN versus A/(or AK^/E in the nominally elastic range). The resulting
correlation is very good, showing no dependence on load-range crack-length
combination. Also, the scatter is minimal considering the subjective nature
of the load range interpretation, and the fact that the customary independent
variable {AK) is squared in this representation. Further evidence of the
generality of the correlation is shown in Fig. 7, where data obtained for the
same material with two other specimen types (single-edge notch and stan-
dard compact specimens) are plotted together with strip specimen data.
Most of the auxiliary data lie in the applicable range for linear elastic
 MOWBRAY ON COMPACT-TYPE STRIP SPECIMEN 749

I0"2 10"'
AJ OR a K ^ / E (MJ/m^)

FIG. 6—Correlation of strip specimen data for Cr-Mo-V steel.

fracture mechanics, and are correlated on the basis of AK^/E. However,

in the growth rate range of 2.5 X 10 -^ to 2.5 X 10 "^ mm/cycle, the
single-edge notch specimen was in the plastic range. Values of AJ were
determined by the same estimation procedure applied to the strip speci-
men.^
The combined set of data (Fig.7) covering the growth rate range 5.0
X 10"* to 5.0 X 10"^ mm/cycle appears to fit in a reasonably narrow
scatter band of two parallel lines similar to that observed by Dowling for

^Use of the SEN specimen to even higher growth rates was not pursued because of buckling
problems.
750 ELASTIC-PLASTIC FRACTURE

ELASTIC- PLASTIC
• STRIP, B =0.125
• SEN , e- 0.125
LINEAR ELASTIC
o STRIP, B 0.I2S
A IT CS B 1.0
0 IT CS B = 0.3

I 10

10"' I0"2
4K OR 4 K V E (MJ/rh' )

FIG. 7—Correlation of all available data for Cr-Mo-V steel.

A533-B steel (see Fig. 1). A least-squares fit of the data in this growth
rate range yields the following relationship

^=1.68(A7)'« (15)

For crack growth rates beyond 5.0 X 10 ~^ mm/cycle the data define an
upward swing toward unstable behavior. This is unlike that for A533-B
data (Fig. 1), which exhibit a straight-line relationship to rates of at least
2.5 X 10"' mm/cycle. The difference is reasonable, however, since the
Cr-Mo-V steel possesses considerably less toughness: 7ic ~0.01 MJ/m^
versus 0.175 for A533-B.
It is noted that the foregoing observations concerning AJ as the con-
trolling variable rest in part on the accuracy of the estimation procedure as
employed herein. The accuracy has not been checked very well in this
 MOWBRAY ON COMPACT-TYPE STRIP SPECIMEN 751

investigation. One partial check was obtained by showing that some auxil-
iary data in the elastic range blend fairly well with data in the high-growth-
rate regime. A more complete check could be provided by comparing with
data obtained on specimens in which / is directly measurable. If the esti-
mation procedure proves an accurate means for determining A/, it will
be of considerable practical value in applications involving high strain
loadings.
As a final item, it is suggested that the estimation procedure employed
for computing A/ could be used in a broader sense to correct existing data
which are expressed in terms of A^, but which are actually beyond the
range of validity of linear elastic fracture mechanics. That is, data from an
individual test are oftentimes characterized in terms of AK even though
part of or all of the test was conducted at net-section stress levels too high
for the stress intensity factor to remain valid as a crack-tip stress field
parameter. For these cases, the proposed estimation procedure could be
used to correct for the nonlinear material effect, thus extending the range
of validity of the test results to higher growth rates.

Summary
A fatigue crack growth study was carried out in the high-growth-rate
regime. A compact-type strip specimen was employed in the testing. This
specimen was found to have some excellent qualities for testing in the high-
rate regime. It exhibited extremely stable behavior and gave rise to con-
stant crack growth rates under simple load control. The data were analyzed
in terms of the /-integral by applying an approximate calculational pro-
cedure. The resulting correlation of data tend to support the Dowling and
Begley hypothesis that the crack growth rate is controlled by the range of
/ operative in opening the crack surfaces.

References
[/] Solomon, H. D., Journal of Materials, Vol. 7, 1972, p. 299.
[2] Gowda, C. V. B. and Topper, T. H., in Cyclic Stress-Strain Behavior-Analysis. Experi-
mentation and Fracture Predictions. ASTM STP 519. 1973, p. 170.
13] Dowling, N. E. and Begley, J. A. in Mechanics of Crack Growth, ASTM STP 590.
American Society for Testing and Materials, 1976, p. 83.
[4] Dowling, N. E. in Cracks and Fracture. ASTM STP 601. American Society for Testing
and Materials, 1976, p. 19.
[5] Dowling, N. E. in Cyclic Stress-Strain and Plastic Deformation Aspects of Fatigue
Crack Growth. ASTM STP 637. 1977, pp. 97-121.
[6] Mowbray, D. F. in Cracks and Fracture. ASTM STP 601. American Society for Testing
and Materials, 1976, p. 33.
[7] Tompkins, B., Philosophical Magazine. Vol. 18, 1968, p. 1041.
[8] Boettner, R. C , Laird, C , and McEvily, A. J., Transactions, Metallurgical Society
of the American Institute of Mining Engineers, Vol. 233, 1965, p. 379.
[9] McEvily, A. J., Beukelman, D., and Tanaka, K. in Proceedings. International Con-
ference on Mechanical Behavior of Materials, Japan, 1972, p. 269.
752 ELASTIC-PLASTIC FRACTURE

[10] Rice, J. R., Transactions, American Society of Mechanical Engineers, Journal of Applied
Mechanics, Vol. 35, June 1968, p. 379.
[11] McHenry, H. I. and Irwin, G. R., Journal of Materials, Vol. 7, 1972, p. 455.
[12] Landgraf, R. W., Morrow, J., and Endo, T., Journal of Materials, Vol. 4, 1969, p. 176.
[13] LeFort, P. and Mowbray, D. F., Journal of Testing and Evaluation, Vol. 6, No. 2,
March 1978.
[14] Bucci, R. I., et al in Fracture Toughness, Proceedings of the 1971 National Symposium
on Fracture Mechanics, Part II, ASTM STP 514, American Society for Testing and
Materials, 1972, p. 40.
[15] Shih, C. F. and Hutchinson, J. W., "Fully Plastic Solutions and Large Scale Yielding
Estimates for Plane Stress Crack Problems," Harvard University Report DEAP S-14,
1975; to appear in Transactions, American Society of Mechanical Engineers, Journal of
Applied Mechanics.
[16] Sumpter, J. D. G. and Turner, C. E. in Cracks and Fracture, ASTM STP 601. 1976,
p. 3.
[IT] Paris, P. C. in Fatigue—An Interdisciplinary Approach, Burke, Reed, and Weiss, Eds.,
Syracuse University Press, Syracuse N.Y., 1963, p. 107.
[18] Merkle, J. G. and Corten, H. T., Transactions, American Society of Mechanical
Engineers, Journal of Pressure Vessel Technology, Nov. 1974, p. 286.
[19] Green, A. P. and Hundy, B. B., Journal of the Mechanical and Physics of Solids, Vol. 4,
1956, p. 128.
Summaiiy
 STP668-EB/Jan. 1979

Summary

The papers in this publication can be divided into three major sections:
(1) the presentation and analytical evaluation of elastic-plastic fracture
criteria; (2) experimental evaluation, including both the toughness evaluation
of materials in the elastic-plastic regime and the evaluation of various
fracture criteria and characterizing parameters; and (3) application of
elastic-plastic methodology to the evaluation of structural components,
including the application to fatigue crack growth analysis.
These papers demonstrate that the elastic-plastic fracture field is in a
stage of rapid development. New approaches and parameters are emerging
and no single approach has been adopted by all of the workers in this
field. However, the field has reached a state of development where certain
trends can be identified. Ductile fracture characterization, which in the
past had largely been based on criteria taken at the point of initiation of
stable crack growth, has been extended so that stable crack growth and
ductile instability are analyzed. Fracture-characterizing parameters, which
are mainly divided into field parameters and local crack-tip parameters,
were once viewed as presenting opposing approaches but are now generally
regarded as having a common basis. Detailed aspects of developing elastic-
plastic fracture techniques, such as establishing limitations on the use of
criteria and on test specimen type and size, are now being actively pursued,
implying that a level of confidence in the underlying concepts has been
reached. Further evidence of this confidence is demonstrated by attempts
to apply the methodology to structural analysis and phenomena other than
fracture toughness.
Results from individual papers are summarized in the following sections.

Elastic-Plasdc Fracture Criteria and Analysis

The papers in this section are concerned with the development of criteria
and parameters to characterize elastic-plastic fracture. In many papers,
finite-element analysis is applied to determine how well these parameters
characterize the crack-tip stress and strain fields under conditions of
large-scale plasticity. Much of the emphasis in these papers is on stable
crack growth and instability characterizations of fracture.
Paris et al have proposed a method for characterizing fracture at the
point of ductile instability. Characterization of stable crack growth is based
on the J-integral where the slope of the / versus crack growth resistance

755

Copyright 1979 b y A S T M International www.astm.org

756 ELASTIC-PLASTIC FRACTURE

curve is given by a nondimensionalized parameter called the tearing

modulus, T. The instability condition is formulated in a manner similar to
the linear elastic fracture mechanics (LEFM) R-curve approach. When an
applied mechanical crack drive of a specimen or structure, labeled Tappued,
is equal to or greater than the material resistance to crack advance, labeled
-/ material (Tappiied ^ ^material), tearing instability ensues. The instability condi-
tion was formulated for a large number of geometries and loading con-
figurations in this paper. The result is a simple methodology for using
laboratory tests to evaluate the tearing instability condition for many types
of structures. Some initial experimental verification of this method is given
in a second paper by Paris et al in this "publication; however, much more
remains to be done. This proposed method suggests many areas for future
research both in analytical developments and in experimental verification
and material property evaluation.
A paper by Hutchinson and Paris provides some rationale for the
method proposed in the previous paper by taking a theoretical approach
to evaluate the use of the J-integral for characterizing stable crack growth.
A criteria for /-controlled crack growth is formulated by determining a
region of proportional loading ahead of an advancing crack. The criterion
is formulated in terms of a nondimensional size parameter, w, which is a
ratio of the size of the proportional loading region to the uncracked liga-
ment. The condition for/-controlled and growth is w 5S> 1.
Shih et al evaluated five parameters for characterizing stable crack
initiation and growth, using nine criteria for the evaluation. The two
chosen as the most viable were the J-integral and crack-tip opening dis-
placement, 5. Finite-element investigations show that both parameters
characterize the near-tip deformation. For stable crack growth, non-
dimensional parameters were developed similar to those of Paris et al and
labeled 7> for crack growth characterized by / and Ts for crack growth
characterized by a crack opening angle. A /-characterization of the crack
growth was determined to be valid up to a crack extension equal to 6
percent of the remaining ligament; however, crack opening angle remained
constant over a much larger range of crack extension, suggesting that
Ti would be preferred over Tj. Complete ductile fracture characterization
is given by a two-parameter approach, either /k and 7> or 6c and Tj, which
characterizes both the initiation and the growth of a stable crack. The
analysis suggested the use of side grooves for experimental evaluations to
ensure uniform flat fracture.
Kanninen et al used a finite-element approach to evaluate eight param-
eters for stable crack growth and instability. They also set nine require-
ments for choosing the appropriate parameter to characterize stable crack
growth. From the parameters evaluated, four were found to vary with
crack extension while four did not. The four that did not vary were: (1)
crack tip opening angle, (2) work involved in separating crack faces per
 SUMMARY 757

unit area of crack growth, (3) generalized energy release rate based on a
computational process zone, and (4) critical crack-tip force for stable
crack growth. These four parameters were judged to be more suitable for
stable crack growth and instability characterization. The concept of a
/-increasing R-curve was viewed as being fundamentally incorrect because
the crack-tip toughness does not increase with an advancing crack.
Sorensen used finite-element techniques to study plane-strain crack
advance under small-scale yielding conditions in both elastic-perfectly
plastic and power hardening materials. The stress distribution ahead of a
growing crack was found to be nearly the same as that ahead of a stationary
crack; however, strains are lower for the growing crack. When loads are
increased at fixed crack length, the increment in crack-tip opening is
uniquely related to the increment in / ; when an increment of crack advance
is taken at constant load, the incremental crack tip opening is related
logarithmically to / . When separation energy rates are calculated for large
crack growth steps, the use of 7 as a correlator is sensitive to strain harden-
ing properties and details of external loading.
McMeeking and Parks used finite-element techniques to study specimen
size limitations for /-based dominance of the crack-tip region. They
analyzed deeply-cracked center-notched tension and single-edge notched
bend specimens using both nonhardening and power loading laws where
deformation was taken from small-scale yielding to the fully plastic range.
The criterion used to judge the degree of dominance was the agreement
between stress and strain for the plastically blunted crack tip with those
for small-scale yielding. They found good agreement for the bend specimen
when all specimen dimensions were larger than 25 //ao, where Oo is the
tensile yield. This size limitation is equivalent to one proposed for / c
testing. The center-notched tension specimen, however, would require
specimen dimensions about eight times larger (200 J/a„), although loss of
dominance is gradual and this requirement is somewhat arbitrary.
Nakagaki et al studied stable crack growth in ductile materials using a
two-dimensional finite-element analysis. They looked at three parameters:
(1) the energy release to the crack tip per unit crack growth, using a global
energy balance; (2) the energy release to a finite near-tip "process zone"
per unit of crack growth; and (3) crack opening angle. Their work con-
firmed numerically an earlier observation by Rice that the crack-tip energy
release rate approaches zero as the increment of crack advance approaches
zero for perfectly plastic material. From these present results, they are not
ready to propose an instability criterion. However, they cannot base such a
criterion on the magnitudes of an energy release parameter since these
depend on the magnitudes of the growth step; therefore, a generalized
Griffith's approach cannot be used for ductile instability.
Miller and Kfouri presented results from a finite-element analysis of a
center-cracked plate under different biaxial stress states. Comparisons were
758 ELASTIC-PLASTIC FRACTURE

made of: (1) crack-tip plastic zone size, (2) crack-tip plastic strain intensity
and major principal stresses, (3) crack opening displacements, (4) J-
integral, and (5) crack separation energy rates. They found that, for
biaxial loading, brittle crack propagation can be best correlated with
plastic zone size. Crack-tip plastic strain intensity is more relevant to
initiation while crack opening displacement is more relevant to crack
propagation. Stable crack propagation was not uniquely related to J.
D'Escatha and Devaux used elastic-plastic finite-element computations
to evaluate a fracture model based on a three-stage approach—void
nucleation, void growth, and coalescence. The purpose of this model is
to predict the fracture properties of a material represented as the initiation
of cracking, stable crack growth, and maximum load. The problem in a
fracture model is to use two-dimensional analysis to predict fracture for a
more realistic three-dimensional crack problem. Various parameters used
to correlate stable crack growth were evaluated by this model, including
crack opening angle, J-integral, and crack-tip nodal force. The next step
will be an experimental evaluation of the present results.
The papers in this section were mainly concerned with the presentation
and analytical evaluation of ductile fracture criteria. A common theme is
that fracture evaluation should include more than simply the initiation of
stable crack growth; stable crack growth characterization and ductile
instability prediction must also be included. While there is no agreement
as to which parameter should be used, the types of parameters are mainly
field-type or crack-tip parameters. Field-type parameters such as the J-
integral have a lot of appeal and are shown to be useful for correlating
stable crack growth under a restricted set of conditions. A crack-tip
parameter such as crack opening angle has fewer restrictions and has more
general support for correlating stable crack growth. The results presented
here suggest many areas for future study. More analysis is needed to
determine the best single approach to ductile fracture characterization.
The approaches presented must be evaluated with critical experimental
studies. The optimum approach must lend itself to relatively simple
evaluation of material properties and must be easily applicable to the
evaluation of structural components. This approach may include one or a
combination of methods suggested here or may be one that is developed in
future studies of ductile fracture criteria.

Experimental Test Techniques and Fracture Toughness Data

The papers in this section deal with experimental evaluation of elastic-
plastic techniques and fracture toughness determination for several mate-
rials. A number of papers deal with various aspects of the analysis used
to determine the J-integral from the experimental load versus load-line
 SUMMARY 759

displacement records for various specimen types. A critical evaluation of

the present analysis techniques along with proposed new techniques for
elastic-plastic specimen analysis are presented in this section. Also included
are a number of papers describing the results of elastic-plastic fracture
toughness testing using both J-integral and crack opening displacement
(COD) techniques.
The paper by Paris et al outlined the test procedure and results used to
verify the tearing instability model described in the previous section. An
experimental technique with a variable-stiffness testing system was used
by Paris et al to vary the applied tearing modulus, Tappued. for each test.
The value of TappUed at the point of ductile instability was determined
by continuously increasing the value of -* applied until instability occurred.
This value of TappUed was then compared with the value of the material
tearing modulus, J- material * determined from the slope of the / versus crack
extension curve developed for the material of interest. The results of the
tests on single-edge notched bend specimens showed extremely good agree-
ment between the predicted value of instability and the actual experimentally
determined instability for the material tested. It was emphasized by the
authors that, as the applied tearing modulus is a function of the compliance
of the overall system, the ductile instability phenomenon is very much
dependent on the overall stiffness of the testing system or the structure
under consideration. Future research in this area was discussed by the
authors and consisted of testing a wider range of specimen types and
variable geometries of a given generic specimen type.
Landes et al evaluated the approximation techniques used to calculate
the value of J from the area under the load displacement curves for the
most commonly used test specimens. This was accomplished by testing
compact, three-point bend, and center-crack tension specimens each with
blunt notches of various lengths. The values of / determined from the
energy rate definition of the J-integral were compared with the various
area methods of approximating / to evaluate the accuracy of the various
approximation techniques. It was found that a correction factor for the
tension component in a compact specimen was necessary. A modified
Merkle-Corten correction factor was proposed for both simplicity and
accuracy when calculating the value of / for a compact specimen. The
three-point bend approximation was found to be accurate if the total
energy aplied to the specimen in the approximation formula is used. The
value of/ calculated from the approximation formula for the center-cracked
panel was also found to be quite accurate when compared with the value of
J calculated by the energy rate definition.
McCabe and Landes proposed the use of an effective crack length to
calculate the resistance to crack growth by the KR technique. It was found
that by using a secant method to calculate the effective crack length, the
760 ELASTIC-PLASTIC FRACTURE

value of J at any point on the load displacement curve could effectively be

calculated by using the relationship between K and J. A comparison of the
results from this technique with the values of/ calculated from the energy
rate definition of J and the value of J calculated from the Ramberg-Osgood
approximation of the load displacement curves was presented. The results
from the secant method showed that this technique is a very good approxi-
mation to the value of 7 calculated from the energy rate definition.
In the next two papers by Dawes and Royer et al, the effect of specimen
thickness on the critical value o f / w a s noted. Dawes presented data show-
ing that the critical values of both COD and / can be affected by section
thickness and that therefore care should be taken to match or overmatch
the plastic constraint in the test specimen to that of the structure. Dawes
also proposed that the crack-tip COD should be defined as the displace-
ment at the original crack-tip position. The data presented by Dawes show
that is it possible to overestimate the value of K^ when using results from
a/ic test on smaller specimens. While Royer et al also note a size effect for
the three-point bend specimen, none was found for the compact specimen.
It was pointed out that while the compact specimen showed no effect of
size on the critical value of/, this result may possibly be fortuitous due
to the type of material under investigation.
The paper by Milne and Chell discusses a proposed mechanism which
may account for the specimen size effect on /k found by them. For ferritic
steels, a shift in the transition temperature due to increased triaxiality of
larger specimens may well account for a size dependency on / c . It is
concluded by Milne and Chell that obtaining Kic from the small-specimen
/ic test can possibly lead to nonconservative values of A^ic. The mechanism
attributed to this phenonmenon is one of a loss of through-thickness con-
straint which may cause crack-tip blunting during the test.
Using an analysis which employed the assumption of elastic-perfectly
plastic behavior, Berger et al evaluated the various forms of the Merkle-
Corten formula for the correction factors for the tension component in the
compact specimen. A comparison was made of the equation which separates
the elastic and plastic portions of the load displacement curve with various
modified forms of the correction formula. It was found that the simplified
form of the Merkle-Corten equation, which utilizes the total displacement
as limits of integration, slightly overestimates the previously discussed
form. The authors also proposed that a fixed displacement value should be
used to determine the critical value of / rather than the intersection of
the/versus Aa line and the theoretical blunting line.
Munz suggested in his paper that linear-elastic toughness testing need
not be restricted to the size criteria defined in the ASTM Test for Plane-
Strain Fracture Toughness of Metallic Materials (E 399-74). It was noted
that the size dependence of KQ as determined by the 5 percent secant
offset method is due primarily to the crack-tip plasticity and the existence
 SUMMARY 761

of a rising plane-strain crack growth resistance curve. A proposed variable

secant method is presented which would allow specimens of up to six times
smaller than the present size criterion permits to be tested for KQ values.
Andrews and Shih presented a study on shear lip formation during
testing and the effects of side grooving specimens. They noted that the
shear lip dimensions found in the specimens were independent of the
specimen dimensions. However, side grooving the specimens to a depth of
I2V2 percent of the thickness completely suppressed shear lip formation.
The / versus A a crack growth resistance curve was shown to be affected
both by the thickness of the specimen and side grooving of the specimen.
By measuring the crack-tip opening displacement using a linear variable
differential transformer (LVDT) near the center of the specimen, Andrews
and Shih showed that a crack growth resistance curve could be developed
which is independent of specimen geometry and side grooving.
An interactive computerized J-integral test technique was described in
the paper by Joyce and Gudas. By using a data acquisition system along
with a computer, they showed excellent agreement between the values of
/ic' obtained from the unloading compliance technique and those obtained
from the heat tinting method. The advantages of an interactive data
reduction process occurring while the test is still in progress were discussed.
This technique also allows for future reanalysis of the data by storing the
data points on a magnetic tape system. The data from a test sequence on
the computerized test technique showed that a nonconservative error in
/ic could be obtained when using specimens with subsized remaining
ligaments or specimens with insufficient thickness.
In the paper by Wilson an evaluation of a number of toughness testing
methods to characterize various plate steels was made. The methods
evaluated were Charpy V-notch (CVN), dynamic tear (DT), and Ju. The
materials tested were A516, A533B, and HY130 manufactured by conven-
tional steel-making techniques and also by a calcium-treated technique. A
conventionally manufactured A543 material was also evaluated at the
centerline and quarter-point positions of a plate. It was found that the
/ic method of testing was far more sensitive to material quality than the
other methods. It is postulated that this sensitivity may well be due to the
acuity of the crack in the Ju specimen compared with the machined notch
and the pressed notch of the CVN and the DT specimens, respectively.
The results of these tests show that the Jjc tests indicate a significant
improvement in the toughness of the calcium-treated steels over the con-
ventionally manufactured steels.
Nine pressure vessel materials were evaluated using static and dynamic
initiation toughness results in the paper by Server. It was found from these
test results that a nine-point average of the crack front gave a higher value
of 7ic than a three-point average of the crack front. The dynamic test
values always gave greater slopes of the / versus A a crack growth resistance
762 ELASTIC-PLASTIC FRACTURE

curves and in many cases the dynamic values of J^ were higher than the
corresponding static values.
Logsdon presented the results of a dynamic fracture toughness test on
SA508 CI 2a material using elastic-plastic techniques. A temperature-
versus-toughness curve at testing rates up to 4.4 X 10" MPaVm/s was
developed using the Ku procedure at low temperatures and /w at higher
temperatures. The results of these tests show that this material is suitable
for nuclear applications. It was also shown that the necessary deceleration
of the /id multispecimen test, due to the speed of testing to prescribed dis-
placement values, had no effect on the results of the Ju test.
In the paper by Tobler and Reed a presentation of the techniques used
to test an electroslag remelt Fe-21Cr material at cryogenic temperatures
was made. The toughness values at 4, 76, and 295 K were found by using
/ic techniques. It was noted from the tension test results that, once plastic
deformation occurred, a slight martensitic transformation took place at
room temperature; at 76 and 4 K, however, an extensive martensitic
transformation took place. The toughness of this material was found to be
adversely affected as the temperature was reduced from 295 to 4 K while
the yield strength increased by a factor of 3.
The problems of testing high-ductility stainless steel were presented in a
paper by Bamford and Bush. Tests were conducted on 304 forged and 316
cast stainless steel at both room temperature and 316°C. The authors
pointed out that the present recommended size requirements for Jic may be
too restrictive as no change was noted in the slope of the crack growth
resistance curve when passing from the proposed valid region to the
nonvalid region. An acoustic emission system was also used in order to
detect the initiation of crack growth. While the acoustic emission test
showed large increases in count rate during the test, there was no obvious
means of detecting crack initiation. The extensive plasticity achieved
during the test also obscured the crack initiation point as defined by an
increase in the electric potential of an electric potential system used. The
unloading compliance technique was found to work favorably on the
compact specimen; however, difficulty was encountered when using the
three-point bend specimen.
The papers in this section were concerned mainly with the evaluation
of various elastic-plastic criteria using experimental methods. There were
basically two areas of investigation in this section: (1) the evaluation of the
actual criteria, and (2) the results of fracture toughness testing when using
a particular criterion. While a number of papers show an effect of size on
both COD and the J-integral, others do not. Various testing procedures
are used to show these size effects, creating a future need for a common
method of testing. This section also shows encouraging results in the
development of an instability criterion for ductile fracture. Future work
in these areas should of course be directed at both size effects on the
 SUMMARY 763

various elastic-plastic criteria and on the development of a test technique

which correctly describes initiation and stable crack growth resistance up
to and including ductile instability. The papers presented in this section
will aid future studies in elastic-plastic criteria and testing methods.

Applications of Elastic-Plastic Methodology

The use of elastic-plastic fracture methodology to analyze structural
components marks its emergence from the status of being mainly a re-
search technique to that of being a useful engineering tool. The papers in
this section include generalized methods for applying elastic-plastic fracture
methodology, specific applications to structural components, and the appli-
cation of elastic-plastic parameters to fatigue crack growth-rate correlation.
Chell discussed methods for using a Failure Assessment Curve to make
failure predictions for structures subjected to thermal, residual, or other
secondary stresses where a failure collapse parameter is not definable.
A procedure is introduced which transforms points on a failure diagram
from an elastic-plastic fracture analysis into approximate equivalent
points on the Failure Assessment Diagram. A method for assessing the
severity of a mechanical load superposed on an initial constant load is
also presented. The paper concludes that the Failure Assessment Curve
will provide a good lower-bound failure criterion for most mechanical
loading.
Harrison et al reviewed methods for applying a COD approach to the
analysis of welded structures. The COD test is particularly useful in
studying fracture toughness of materials in the transition between linear-
elastic and fully plastic behavior. Design curves are developed relating a
nondimensional COD to applied strain or stress. These curves are useful
for (1) selection of materials in design of structures, (2) specification of
maximum allowable flaw sizes, and (3) failure analyses. Many examples
are cited where the COD design curve has been used for these evaluations
on structures designed for real applications. Examples of structures
analyzed by COD methods include pipelines, offshore structures, pressure
vessels, and nuclear components.
McHenry et al used elastic-plastic fracture mechanics analysis methods
to determine size limits for surface flaws in pipeline girthwelds. Four
criteria were used: (1) a critical COD method based on a ligament-closure
force model, (2) the COD procedure based on the Draft British Standard,
(3) a plastic instability method based on critical net ligament strain, and
(4) a semi-empirical method based on full-scale pipe rupture tests. The
critical flaw sizes determined varied significantly, depending on the frac-
ture criterion chosen, and experimental work will be needed to determine
which method most accurately predicts girthweld fracture behavior.
764 ELASTIC-PLASTIC FRACTURE

Simpson and Clarke used a crack growth resistance (R-curve) approach

based on small fracture mechanics type specimens to determine critical
crack lengths in Zr-2.5Nb pressure tubes. The R-curves were based on
COD as the mechanical characterizing parameter. Their results showed
little specimen size effect on the R-curve shape. R-curves based on a
J-integral approach were shown to be consistent with the COD approach.
Effects of temperature and hydrogen on the R-curve shape were investigated.
Predictions of critical crack lengths in pressure tubes based on an R-curve
procedure gave results which were consistent with published burst testing
data.
Macdonald used a three-dimensional elastic-plastic fracture model to
correlate the fracture strength of two structural steels in the form of
beam-column connections. The model was based on the combination of
(1) a three-dimensional elastic-plastic finite-element stress analysis, (2) a
plastic stress singularity for a crack, and (3) the maximum tensile stress
theory of fracture. From these a plastic singularity strength parameter,
Kf, was developed. Cracking occurred by mixed mode (crack opening and
sliding). Experimental results correlated with Kf showed a relatively small
scatterband.
Merkle used approximate elastic-plastic fracture methods to analyze the
unstable failure condition for inside nozzle corner cracks in intermediate
test vessels. The method was applied to two vessels tested in the heavy
section steel technology (HSST) program (Vessels V-9 and V-5). Semi-
empirical methods were developed for estimating the nozzle corner pressure-
strain curve. Two approximate methods of fracture analysis were used: one
used an LEFM approach based on strain which did not consider stable
crack growth; the second used a. tangent modulus method which in-
corporated stable crack growth by using a maximum load fracture tough-
ness value. The beneficial effect of transverse contraction was included
in the analysis. Calculations of failure strain and fracture toughness agreed
well with measured values.
Hammouda and Miller used elastic-plastic finite-element analyses to
predict the effect of notch plasticity on the behavior of short cracks under
cyclic loading. This analysis was used to predict crack growth behavior in a
regime where LEFM methods do not apply. Consideration of the inter-
action between the crack tip and notch field plasticity can account for
fatigue crack growth where a linear elastic analysis would predict that the
fatigue threshold stress intensity factor is not exceeded. Crack propagation
from a notch initially proceeds at a decreasing rate and in some cases
cracks initiate but become nonpropagating.
Brose and Dowling studied the effect of planar specimen size on the
fatigue crack growth rate properties of 304 stainless steel on specimen
widths of 5.08 and 40.64 cm (2 and 16 in.). The objective was to evaluate
size criteria intended to limit crack growth testing to the linear elastic
 SUMMARY 765

regime and to evaluate the use of a cyclic value of J-integral, AJ, for
correlation of crack growth rate data on specimens undergoing gross
plasticity. The results show that crack growth rates correlated by AJ on
small specimens having gross plasticity are equivalent to results from large
specimens in the linear elastic regime, where the data are correlated by
AK. No significant size effects were observed.
Mowbray studied fatigue crack growth of chromium-molybdenum-
ranadium steel in the high-growth-rate regime where a cyclic J-integral
value, AJ, was used to correlate growth rate. A compact-type strip speci-
men was used which gave rise to constant crack growth rate under simple
load control at essentially constant AJ. These results supported previous
results by Dowling and Begley which showed that crack growth rate in
the high-growth-rate regime is controlled by AJ. An approximate analysis
was used to determine AJ from cyclic load range for the strip specimen.
The papers in this section consider methods for applying elastic-plastic
fracture techniques to the analysis of structures. The prominent technique
for using small-specimen results to analyze large structural components is
one based on crack opening displacement concepts. The COD was one of
the first proposed elastic-plastic fracture parameters and has gained some
degree of acceptance as an engineering tool. Other methods for application
of elastic-plastic techniques include the Failure Assessment Diagram,
R-curve techniques, plastic instability, and the plastic stress singularity.
Again, no single method of analysis is generally accepted; many areas for
future studies are identified by these papers.
A cyclic value of J-integral, AJ, is shown experimentally to correlate
fatigue crack growth rate in the high-growth-rate regime. This approach is
gaining more acceptance and has promise of becoming a useful tool for
analyzing fatigue crack growth under large-scale plasticity.

/. D. Landes
G. A. Clarke
Westinghouse Electric Corp. Research and
Development Center, Pittsburgh, Pa.;
coeditors.
 STP668-EB/Jan. 1979

Index
Closure
Accoustic emission, 541, 560 Load, 727, 738
Airy's stress function, 201 Stress, 12
Antibuckling guides, 131 Compact specimens, 27, 78, 79, 269,
Area estimation procedure, 271, 290, 347, 355
276, 286 Compliance calibration, 252
Complimentary work, 344
Computer interactive testing, 451
Crack driving force, 659
B Crack growth
Initiation, 49
Bauschinger effect, 200 Simulation, 71
Bend specimens Stable, 49,126,131
Deeply cracked, 38, 45
Unstable, 53, 226
Three point, 17, 49, 236, 269, 346,
Crack opening angle (COA), 71, 88,
353
98,115,116,124, 203
Biaxiality, 215
Crack opening displacement (COD),
Blunting line, 393, 489,544
88, 118, 195, 316, 328, 386,
Body centered cubic, 539
608
Boundary layer analysis, 186, 216,
COD design curve, 309, 623
219 Crack tip
British Standards Institute, 317, Acuity, 370, 465
608, 635 Force, 124
Brittle fracture, 363 Opening ration, 180
Crack velocity, 715
Creep studies, 305
Criteria
Failure, 67, 604
Center cracked panel, 71, 101, 108, Instability, 13, 27
269,290 Recoverable energy, 128
Center cracked strip, 9, 10, 59, 71 Tresca, 20,691
Charpy correlation, 490 Von Mises, 20,154, 668
Charpy energy, 508 Critical
Clevage Crack length, 659
Fracture, 365 Crack opening displacement, 634
Instability, 5, 15, 23, 260, 322 Energy release rate, 148
Rupture, 525 Thickness, 408

767

Copyright' 1979 b y A S T M International www.astm.org

768 ELASTIC-PLASTIC FRACTURE

Cyclic/, 725 Equivalent energy, 379, 386, 403

Cyclic plastic zone, 721 Equivalent length, 254, 705

D
Damage function, 231 Face centered cubic, 539
Deep surface flaw, 13 Failure assessment diagram, 582,597
Deformation theory of plasticity, 43, Failure curve, 586
61,80,94,112,115 Fatigue, 704
Double cantilever beams, 14 Fatigue crack growth, 722, 731, 742
Double edged cracked strip, 11, 23, Fatigue failure, 716
56 Finite element
Ductile-brittle transition, 332, 373 Constrant strain elements, 76,
Ductile fracture, 65, 230 125,155,165
Ductile tearing, 365 Elastic-plastic, 74, 123, 131, 199,
Dynamic 227
Compact tests, 499 Equations, 153
Fracture toughness, 515,532,681 Hybrid displacement model, 199
J-Integral tests, 506 Isoparametric elements, 76, 165,
Resistance curves, 41,525 216
Tear energy, 473 Mesh, 80, 97,157,165, 246
Yield strength, 530 Model, 74, 80, 202
Three dimensional, 664
Finite strain studies, 92
E First load drop, 478, 486
Eddy current, 454 Flowtheory, 62, 94
Effective Incremental, 43, 80
Crack size, 289 /2, 68, 70, 80,113
Elastic modulas, 291 Fracture parameter, 72,104,110
Elastic span, 251
Elastic compliance, 427, 444, 562,
741
Elastic-plastic deformation, 6,40 G, strain energy release rate, 28,
Elastic shortening, 19 204, 272,338
Electrical potential, 336, 415, 559, Gaussian integration, 201
648, 661 Geometry dependance, 359, 654
Elliptical surface flaw, 73, 230 Girth welds, 626, 633
Energy Grain boundaries, 309
Deformation, 380
Rate definition, 286,276
H
Separation rate, 70, 71
Epoxy model, 694 Heat tinting, 78,431,559
Equi-biaxial state, 44 Hydrostatic stress, 166
 INDEX 769

I N
Irradiation damage, 23, 263, 661 National Bureau of Standards, 628,
Incremental polynomial, 725 633
Instability, 5,13, 27, 66, 637 Neuber's equation, 689, 697, 705
Instrumented Charpy, 495 Nodal force, 171,197,218,248
Nodal release, 155,168
Nonmetallic inclusions, 309
Nonpropagating cracks, 709
Notch ductility factor, 697
/-controlled crack growth, 38, 42,
Notch plasticity, 706
43,113,186
Notch round bars, 18,58
/-dominance, 177,186
Nozzle comer, 676, 686
/-resistance curve, 5, 39, 66,464, 644
Nuclear pressure vessels, 123, 495,
516, 676
Nuclear reactor, 643, 677
K
K-field, 103
/Tic test, 105 O
Kirchhoff stress, 178 Offshore oil platform, 623

Large-scale yielding, 37 Part through cracks, 610,695

Least squares fit, 504, 750 Path independence, 93
Limit load, 18,344 Phase transformation, 546
Limit moment, 14,18, 48 Photo-elastic analysis, 626
Linear elastic fracture mechanics, 13 Plane strain, 7, 9, 55,128
Liquified natural gas tanks, 628 Plane stress, 51, 59, 67
Log deviate, 505 Plastic collapse load, 582,595
Linear variable displacement trans- Plastic constraint,'746
ducer (LVDT), 131 Plastic zone
Cyclic, 721
Monotonic, 721
M
Shape, 168
Margin of safety, 66 Size, 104, 215
Martensite transformation, 537, 549 Plasticity theory
Metallurgical mechanisms, 359 Deformation, 43, 61, 80, 94, 112
Microstructure, 372 Incremental flow, 43,80
Minimum ligament, 411 /2flow, 68, 70,80, 113
Minimum specimen thickness, 421 Prandtl-Reuss, 133, 154, 191, 709
Mixed mode fracture, 73 Post yield fracture, 582
Multiple specimen test, 566 Prandtl slip line, 167
770 ELASTIC-PLASTIC FRACTURE

Prandtl stress, 166 Specimen size requirements, 491

Precracked Charpy specimen, 680 Stable crack growth, 49, 126, 131
Pressure vessels, 123, 495, 516, 628, Steels
676 A508,561
Process zone, 70, 103,118, 127, 138, A516, 471
210, 435 A533,67,471,516
Proportional loading, 41 Austenitic stainless, 548,554
CrMoV, 740
Ferritic, 521
HY103, 471
Quasi-brittle fracture, 314 NiCrMo, 387
Rotor forging, 361
Stiffness matrix, 155
R
Strain energy density, 127
/?-values, 127 Strain energy release rate, 28, 204,
Ramberg-Osgood stress strain, 50,59 272,338
Rate sensitive materials, 23 Strain field, 137,151
Reactor coolant piping, 553 Strain hardening, 38, 49, 51, 60, 68,
Reference toughness curve, 553 105,172, 724
Residual stress, 592, 619, 629, 640, Strain hardening laws,
653 Isotropic, 165,178
Resistance curves, 5,39, 66,464, 644 Multilinear, 665
Rubber infiltration, 74, 102, 438 Power hardening, 176
Ramberg-Osgood, 207, 294
Stress
Maximum hoop, 634
Secant method, 409, 544 Prandtlfield, 107, 166
Secant offset, 273 Residual, 592, 619, 629, 640, 653
Semi-elliptical surface crack, 236 Secondary, 593
Separation energy rates, 172, 216, Thermal, 592
225 Stress intensity
Shearlip,3, 63,427, 434 Factors, 198, 717
Short crack lengths, 712 Magnification factors, 611
Side grooves, 77,101, 427, 434 Stretch zone, 309, 365, 391
Silicone rubber replicas, 438 Strip yield model, 31,309, 609
Single parameter characterization, Submerged arc weld, 512
67,358
Single specimen tests, 451
Singularity, 41
Size independence of /?-curves, 7 Tangent modulas, 689, 697
Slip line fields, 10,22,106 Tearing instability, 6, 14, 25, 251
Slip line solutions, 176 Tearing
Small-scale yielding, 37 Instability, 6.14. 25. 251
Specimen size effect, 398,414 Modulas, 8, 24, 118, 255, 574
 INDEX 771

Resistance, 20 Growth, 74, 176, 191, 231, 375

Stable, 6, 365 Nucleation, 74, 234
Temperature dependence of tough-
ness, 373
Tension component in bend speci- W
mens, 2, 71
Tension testing, 473 Work density, 71
Testing machine stiffness, 28
Through cracks, 236
Triazial
Stress, 247, 369 X-rays, 627
Tension, 176, 372, 375

U
Uncracked body energy, 510 Yielding
Unloading compliance, 78, 429,559, Large scale, 37
562 Small scale 37
Surface, 191

Variable secant method, 417

Void
Coalescence, 74,176, 234 Zirconium, 644