Fatigue of Materials

Written by a leading researcher in the field, this revised and updated second edition
of a highly successful book provides an authoritative, comprehensive and unified
treatment of the mechanics and micromechanisms of fatigue in metals, nonmetals
and composites.
The author discusses the principles of cyclic deformation, crack initiation and
crack growth by fatigue, covering both microscopic and continuum aspects. The
book begins with discussions of cyclic deformation and fatigue crack initiation in
monocrystalline and polycrystalline ductile alloys as well as in brittle, semi-
crystalline and noncrystalline solids. Total-life and damage-tolerant approaches are
then introduced in metals, nonmetals and composites along with such advanced
topics as multiaxial fatigue, contact fatigue, variable amplitude fatigue, creep-
fatigue, and environmentally assisted fatigue. Emphasis is placed upon scientific
concepts and mechanisms and the basic concepts are extended to many practical
cases wherever possible. The book includes an extensive bibliography and a problem
set for each chapter, together with worked-out example problems and case studies.
The book will be an important reference for students, practicing engineers and
researchers studying fracture and fatigue in materials science and engineering,
mechanical, civil, nuclear and aerospace engineering, and biomechanics.
Fatigue of Materials
SECOND EDITION

S. SURESH
Massachusetts Institute of Technology

CAMBRIDGE
UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 2RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org
Information on this title: www.cambridge.org/9780521570466
© Subra Suresh 1998

This publication is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.

First published 1998

Reprinted 2001 (with corrections), 2003, 2004

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data

Suresh, S. (Subra)
Fatigue of materials / S. Suresh. - 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 0 521 57046 8 (hardcover). - ISBN 0 521 57847 7 (pbk.)
1. Materials - Fatigue. 2. Fracture mechanics. I. Title.
TA418.38.S87 1998
620.1'126-dc21 98-15181 CIP

ISBN-13 978-0-521-57046-6 hardback

ISBN-10 0-521-57046-8 hardback

ISBN-13 978-0-521-57847-9 paperback

ISBN-10 0-521-57847-7 paperback

Transferred to digital printing 2006

To
My wife Mary and our children,
Nina and Meera
 Contents

Preface to the second edition xvii

Preface to the first edition xix

1 Introduction and overview 1

1.1 Historical background and overview 1
1.1.1 Case study: Fatigue and the Comet airplane 8
1.2 Different approaches to fatigue 11
1.2.1 Total-life approaches 12
1.2.2 Defect-tolerant approach 13
1.2.3 A comparison of different approaches 14
1.2.4 'Safe-life' and 'fail-safe' concepts 14
1.2.5 Case study: Retirement for cause 15
1.3 The need for a mechanistic basis 17
1.4 C o n t i n u u m mechanics 18
1.4.1 Elements of linear elasticity 20
1.4.2 Stress invariants 21
1.4.3 Elements of plasticity 22
1.4.4 Elements of linear viscoelasticity 26
1.4.5 Viscoplasticity and viscous creep 28
1.5 Deformation of ductile single crystals 29
1.5.1 Resolved shear stress and shear strain 30
Exercises 33

PART ONE: CYCLIC DEFORMATION AND FATIGUE CRACK

INITIATION 37
2 Cyclic deformation in ductile single crystals 39
2.1 Cyclic strain hardening in single crystals 40
2.2 Cyclic saturation in single crystals 40
2.2.1 Monotonic versus cyclic plastic strains 45
2.3 Instabilities in cyclic hardening 45
2.3.1 Example problem: Identification of active slip systems 47
2.3.2 Formation of dislocation veins 49
2.3.3 Fundamental length scales for the vein structure 52
2.4 Deformation along persistent slip bands 52
2.5 Dislocation structure of PSBs 53
2.5.1 Composite model 57

vn
viii Contents

2.5.2 Example problem: Dislocation dipoles and cyclic

deformation 58
2.6 A constitutive model for the inelastic behavior of PSBs 60
2.6.1 General features 60
2.6.2 Hardening in the PSBs 61
2.6.3 Hardening at sites of PSB intersection with the free surface 61
2.6.4 Unloading and reloading 62
2.6.5 Vacancy generation 62
2.7 Formation of PSBs 63
2.7.1 Electron microscopy observations 63
2.7.2 Static or energetic models 65
2.7.3 Dynamic models of self-organized dislocation structures 68
2.8 Formation of labyrinth and cell structures 69
2.8.1 Example problem: Multiple slip 71
2.9 Effects of crystal orientation and multiple slip 72
2.10 Case study: A commercial F C C alloy crystal 74
2.11 Monotonic versus cyclic deformation in FCC crystals 78
2.12 Cyclic deformation in BCC single crystals 79
2.12.1 Shape changes in fatigued BCC crystals 80
2.13 Cyclic deformation in H C P single crystals 82
2.13.1 Basic characteristics of Ti single crystals 83
2.13.2 Cyclic deformation of Ti single crystals 83
Exercises 84

3 Cyclic deformation in polycrystalline ductile solids 86

3.1 Effects of grain boundaries and multiple slip 86
3.1.1 Monocrystalline versus polycrystalline FCC metals 87
3.1.2 Effects of texture 89
3.2 Cyclic deformation of F C C bicrystals 89
3.2.1 Example problem: Number of independent slip systems 91
3.3 Cyclic hardening and softening in polycrystals 91
3.4 Effects of alloying, cross slip and stacking fault energy 95
3.5 Effects of precipitation 97
3.6 The Bauschinger effect 97
3.6.1 Terminology 98
3.6.2 Mechanisms 99
3.7 Shakedown 101
3.8 Continuum models for uniaxial and multiaxial fatigue 102
3.8.1 Parallel sub-element model 104
3.8.2 Field of work hardening moduli 106
3.8.3 Two-surface models for cyclic plasticity 110
3.8.4 Other approaches 112
3.9 Cyclic creep or ratchetting 113
3.10 Metal-matrix composites subjected to thermal cycling 115
 IX

3.10.1 Thermoelastic deformation 115

3.10.2 Characteristic temperatures for thermal fatigue 117
3.10.3 Plastic strain accumulation during thermal cycling 119
3.10.4 Effects of matrix strain hardening 120
3.10.5 Example problem: Critical temperatures for thermal fatigue
in a metal-matrix composite 122
3.11 Layered composites subjected to thermal cycling 123
3.11.1 Thermoelastic deformation of a bilayer 124
3.11.2 Thin-film limit: the Stoney formula 127
3.11.3 Characteristic temperatures for thermal fatigue 128
Exercises 129

4 Fatigue crack initiation in ductile solids 132

4.1 Surface roughness and fatigue crack initiation 132
4.1.1 Earlier observations and viewpoints 133
4.1.2 Electron microscopy observations 134
4.2 Vacancy-dipole models 137
4.3 Crack initiation along PSBs 141
4.4 Role of surfaces in crack initiation 143
4.5 Computational models for crack initiation 143
4.5.1 Vacancy diffusion 143
4.5.2 Numerical simulations 145
4.5.3 Example problem: Effects of vacancies 146
4.6 Environmental effects on crack initiation 147
4.7 Kinematic irreversibility of cyclic slip 148
4.8 Crack initiation along grain and twin boundaries 149
4.9 Crack initiation in commercial alloys 152
4.9.1 Crack initiation near inclusions and pores 152
4.9.2 Micromechanical models 155
4.10 Environmental effects in commercial alloys 156
4.11 Crack initiation at stress concentrations 157
4.11.1 Crack initiation under far-field cyclic compression 158
Exercises 162

5 Cyclic deformation and crack initiation in brittle solids 165

5.1 Degrees of brittleness 166
5.2 Modes of cyclic deformation in brittle solids 167
5.3 Highly brittle solids 169
5.3.1 Mechanisms 169
5.3.2 Constitutive models 170
5.3.3 On possible effects of cyclic loading 175
5.3.4 Elevated temperature behavior 176
5.4 Semi-brittle solids 179
5.4.1 Crack nucleation by dislocation pile-up 179
 Contents

5.4.2 Example problem: Cottrell mechanism for sessile dislocation

formation 180
5.4.3 Cyclic deformation 182
5.5 Transformation-toughened ceramics 184
5.5.1 Phenomenology 185
5.5.2 Constitutive models 187
5.6 Fatigue crack initiation under far-field cyclic compression 191
5.6.1 Example problem: Crack initiation under far-field cyclic
compression 196
Exercises 197

6 Cyclic deformation and crack initiation in noncrystalline solids 200

6.1 Deformation features of semi-/noncrystalline solids 200
6.1.1 Basic deformation characteristics 200
6.1.2 Crazing and shear banding 201
6.1.3 Cyclic deformation: crystalline versus noncrystalline materials 203
6.2 Cyclic stress-strain response 205
6.2.1 Cyclic softening 205
6.2.2 Thermal effects 207
6.2.3 Example problem: Hysteretic heating 207
6.2.4 Experimental observations of temperature rise 209
6.2.5 Effects of failure modes 210
6.3 Fatigue crack initiation at stress concentrations 211
6.4 Case study: Compression fatigue in total knee replacements 213
Exercises 217

PART TWO: TOTAL-LIFE APPROACHES 219

7 Stress-life approach 221
7.1 The fatigue limit 222
7.2 M e a n stress effects on fatigue life 224
7.3 Cumulative damage 227
7.4 Effects of surface treatments 228
7.5 Statistical considerations 231
7.6 Practical applications 235
7.6.1 Example problem: Effects of surface treatments 235
7.6.2 Case study: HCF in aircraft turbine engines 236
7.7 Stress-life response of polymers 237
7.7.1 General characterization 237
7.7.2 Mechanisms 238
7.8 Fatigue of organic composites 239
7.8.1 Discontinuously reinforced composites 240
7.8.2 Continuous-fiber composites 240
7.9 Effects of stress concentrations 242
7.9.1 Fully reversed cyclic loading 242
 XI

7.9.2 Combined effects of notches and mean stresses 243

7.9.3 Nonpropagating tensile fatigue cracks 244
7.9.4 Example problem: Effects of notches 244
7.10 Multiaxial cyclic stresses 246
7.10.1 Proportional and nonproportional loading 246
7.10.2 Effective stresses in multiaxial fatigue loading 247
7.10.3 Stress-life approach for tension and torsion 248
7.10.4 The critical plane approach 250
Exercises 254

8 Strain-life approach 256

8.1 Strain-based approach to total life 256
8.1.1 Separation of low-cycle and high-cycle fatigue lives 256
8.1.2 Transition life 257
8.1.3 Example problem: Thermal fatigue life of a metal-matrix
composite 260
8.2 Local strain approach for notched members 262
8.2.1 Neuber analysis 263
8.3 Variable amplitude cyclic strains and cycle counting 265
8.3.1 Example problem: Cycle counting 265
8.4 Multiaxial fatigue 268
8.4.1 Measures of effective strain 268
8.4.2 Case study: Critical planes of failure 269
8.4.3 Different cracking patterns in multiaxial fatigue 271
8.4.4 Example problem: Critical planes of failure in multiaxial
loading 273
8.5 Out-of-phase loading 276
Exercises 278

PART THREE: DAMAGE-TOLERANT APPROACH 281

9 Fracture mechanics and its implications for fatigue 283
9.1 Griffith fracture theory 283
9.2 Energy release rate and crack driving force 285
9.3 Linear elastic fracture mechanics 288
9.3.1 Macroscopic modes of fracture 288
9.3.2 The plane problem 289
9.3.3 Conditions of A^-dominance 295
9.3.4 Fracture toughness 296
9.3.5 Characterization of fatigue crack growth 296
9.4 Equivalence of Q and K 297
9.4.1 Example problem: Q and K for the DCB specimen 298
9.4.2 Example problem: Stress intensity factor for a blister test 300
9.5 Plastic zone size in monotonic loading 302
9.5.1 The Irwin approximation 302
9.5.2 The Dugdale model 303
xii Contents

9.5.3 The Barenblatt model 304

9.6 Plastic zone size in cyclic loading 304
9.7 Elastic-plastic fracture mechanics 307
9.7.1 The/-integral 307
9.7.2 Hutchinson-Rice-Rosengren (HRR) singular fields 308
9.7.3 Crack tip opening displacement 309
9.7.4 Conditions of/-dominance 310
9.7.5 Example problem: Specimen size requirements 312
9.7.6 Characterization of fatigue crack growth 313
9.8 Two-parameter representation of crack-tip fields 316
9.8.1 Small-scale yielding 318
9.8.2 Large-scale yielding 318
9.9 Mixed-mode fracture mechanics 319
9.10 Combined mode I-mode II fracture in ductile solids 320
9.11 Crack deflection 322
9.11.1 Branched elastic cracks 324
9.11.2 Multiaxial fracture due to crack deflection 326
9.12 Case study: Damage-tolerant design of aircraft fuselage 327
Exercises 328

10 Fatigue crack growth in ductile solids 331

10.1 Characterization of crack growth 331
10.1.1 Fracture mechanics approach 332
10.1.2 Fatigue life calculations 334
10.2 Microscopic stages of fatigue crack growth 335
10.2.1 Stage I fatigue crack growth 335
10.2.2 Stage II crack growth and fatigue striations 335
10.2.3 Models for striation formation 337
10.2.4 Environmental effects on stage II fatigue 340
10.3 Different regimes of fatigue crack growth 341
10.4 Near-threshold fatigue crack growth 343
10.4.1 Models for fatigue thresholds 345
10.4.2 Effects of micro structural size scale 346
10.4.3 Effects of slip characteristics 347
10.4.4 Example problem: Issues of length scales 351
10.4.5 On the determination of fatigue thresholds 352
10.5 Intermediate region of crack growth 354
10.6 High growth rate regime 357
10.7 Case study: Fatigue failure of aircraft structures 358
10.8 Case study: Fatigue failure of total hip components 364
10.9 Combined mode I-mode II fatigue crack growth 368
10.9.1 Mixed-mode fatigue fracture envelopes 369
10.9.2 Path of the mixed-mode crack 370
10.9.3 Some general observations 372
10.10 Combined mode I-mode III fatigue crack growth 373
Contents xiii

10.10.1 Crack growth characteristics 374

10.10.2 Estimation of intrinsic growth resistance 378
Exercises 379

11 Fatigue crack growth in brittle solids 383

11.1 Some general effects of cyclic loading on crack growth 384
11.2 Characterization of crack growth in brittle solids 385
11.2.1 Crack growth under static loads 385
11.2.2 Crack growth under cyclic loads 386
11.3 Crack growth resistance and toughening of brittle solids 388
11.3.1 Example problem: Fracture resistance and stability of crack
growth 389
11.4 Cyclic damage zone ahead of tensile fatigue crack 392
11.5 Fatigue crack growth at low temperatures 393
11.6 Case study: Fatigue cracking in heart valve prostheses 396
11.7 Fatigue crack growth at elevated temperatures 399
11.7.1 Micromechanisms of deformation and damage due to
intergranular/interfacial glassy films 399
11.7.2 Crack growth characteristics at high temperatures 402
11.7.3 Role of viscous films and ligaments 403
Exercises 406

12 Fatigue crack growth in noncrystalline solids 408

12.1 Fatigue crack growth characteristics 408
12.2 Mechanisms of fatigue crack growth 411
12.2.1 Fatigue striations 411
12.2.2 Discontinuous growth bands 413
12.2.3 Combined effects of crazing and shear flow 417
12.2.4 Shear bands 419
12.2.5 Some general observations 420
12.2.6 Example problem: Fatigue crack growth in epoxy adhesive 422
12.3 Fatigue of metallic glasses 424
12.4 Case study: Fatigue fracture in rubber-toughened epoxy 426
Exercises 430

PART FOUR: ADVANCED TOPICS 433

13 Contact fatigue: sliding, rolling and fretting 435
13.1 Basic terminology and definitions 435
13.2 Mechanics of stationary contact under normal loading 439
13.2.1 Elastic indentation of a planar surface 440
13.2.2 Plastic deformation 442
13.2.3 Residual stresses during unloading 443
13.2.4 Example problem: Beneficial effects of surface compressive
stresses 444
xiv Contents

13.3 Mechanics of sliding contact fatigue 445

13.3.1 Sliding of a sphere on a planar surface 446
13.3.2 Partial slip and complete sliding of a cylinder on a planar
surface 447
13.3.3 Partial slip of a sphere on a planar surface 448
13.3.4 Cyclic variations in tangential force 449
13.4 Rolling contact fatigue 451
13.4.1 Hysteretic energy dissipation in rolling contact fatigue 452
13.4.2 Shakedown limits for rolling and sliding contact fatigue 453
13.5 Mechanisms of contact fatigue damage 457
13.5.1 Types of microscopic damage 457
13.5.2 Case study: Contact fatigue cracking in gears 457
13.6 Fretting fatigue 462
13.6.1 Definition and conditions of occurrence 462
13.6.2 Fretting fatigue damage 463
13.6.3 Palliatives to inhibit fretting fatigue 466
13.6.4 Example problem: Fracture mechanics methodology for
fretting fatigue fracture 469
13.7 Case study: Fretting fatigue in a turbogenerator rotor 474
13.7.1 Design details and geometry 474
13.7.2 Service loads and damage occurrence 474
Exercises 481

14 Retardation and transients in fatigue crack growth 483

14.1 Fatigue crack closure 484
14.2 Plasticity-induced crack closure 486
14.2.1 Mechanisms 486
14.2.2 Analytical models 490
14.2.3 Numerical models 493
14.2.4 Effects of load ratio on fatigue thresholds 494
14.3 Oxide-induced crack closure 496
14.3.1 Mechanism 496
14.3.2 Implications for environmental effects 497
14.4 Roughness-induced crack closure 500
14.4.1 Mechanism 500
14.4.2 Implications for micro structural effects on threshold
fatigue 501
14.5 Viscous fluid-induced crack closure 503
14.5.1 Mechanism 503
14.6 Phase transformation-induced crack closure 504
14.7 Some basic features of fatigue crack closure 505
14.8 Issues and difficulties in the quantification of crack closure 506
14.9 Fatigue crack deflection 507
14.9.1 Linear elastic analyses 508
14.9.2 Experimental observations 511
Contents xv

14.9.3 Example problem: Possible benefits of deflection 512

14.10 Additional retardation mechanisms 515
14.10.1 Crack-bridging and trapping in composite materials 515
14.10.2 On crack retardation in advanced metallic systems 518
14.11 Case study: Variable amplitude spectrum loads 519
14.12 Retardation following tensile overloads 520
14.12.1 Plasticity-induced crack closure 521
14.12.2 Crack tip blunting 522
14.12.3 Residual compressive stresses 523
14.12.4 Deflection or bifurcation of the crack 523
14.12.5 Near-threshold mechanisms 524
14.13 Transient effects following compressive overloads 526
14.13.1 Compressive overloads applied to notched materials 529
14.14 Load sequence effects 529
14.14.1 Block tensile load sequences 530
14.14.2 Tension-compression load sequences 533
14.15 Life prediction models 534
14.15.1 Yield zone models 534
14.15.2 Numerical models of crack closure 535
14.15.3 Engineering approaches 536
14.15.4 The characteristic approach 536
Exercises 537

75 Small fatigue cracks 541

15.1 Definitions of small cracks 543
15.2 Similitude 543
15.3 Microstructural aspects of small flaw growth 544
15.4 Threshold conditions for small flaws 545
15.4.1 Transition crack size 545
15.4.2 Critical size of cyclic plastic zone 547
15.4.3 Slip band models 548
15.5 Fracture mechanics for small cracks at notches 550
15.5.1 Threshold for crack nucleation 551
15.5.2 Example problem: Crack growth from notches 552
15.6 Continuum aspects of small flaw growth 554
15.6.1 Two-parameter characterization of short fatigue cracks 554
15.6.2 Near-tip plasticity 556
15.6.3 Notch-tip plasticity 556
15.7 Effects of physical smallness of fatigue flaws 559
15.7.1 Mechanical effects 559
15.7.2 Environmental effects 561
15.8 On the origins of 'short crack problem' 562
15.9 Case study: Small fatigue cracks in surface coatings 564
15.9.1 Theoretical background for cracks approaching interfaces
perpendicularly 564
xvi Contents

15.9.2 Application to fatigue at surface coatings 566

Exercises 568

16 Environmental interactions: corrosion-fatigue and creep-fatigue 570

16.1 Mechanisms of corrosion-fatigue 570
16.1.1 Hydrogenous gases 571
16.1.2 Aqueous media 572
16.1.3 Metal embrittlement 574
16.2 Nucleation of corrosion-fatigue cracks 574
16.2.1 Gaseous environments 575
16.2.2 Aqueous environments 575
16.3 Growth of corrosion-fatigue cracks 577
16.3.1 Types of corrosion-fatigue crack growth 579
16.3.2 Formation of brittle striations 581
16.3.3 Effects of mechanical variables 583
16.3.4 Models of corrosion-fatigue 585
16.4 Case study: Fatigue design of exhaust valves for cars 586
16.5 Fatigue at low temperatures 588
16.6 Damage and crack initiation at high temperatures 589
16.6.1 Micromechanisms of damage 590
16.6.2 Life prediction models 594
16.7 Fatigue crack growth at high temperatures 598
16.7.1 Fracture mechanics characterization 598
16.7.2 Characterization of creep-fatigue crack growth 601
16.7.3 Summary and some general observations 603
16.8 Case study: Creep-fatigue in steam-power generators 604
Exercises 608

Appendix 609
References 614
Author index 659
Subject index 669
 Preface to the second edition

The first edition of this book was written primarily as a research monograph
for the Solid State Science Series of Cambridge University Press. Since its first
publication, however, the book has found wide readership among students and
practicing engineers as well as researchers. In view of this audience base which
evolved to be much broader than what the book was originally intended for, it
was felt that now would be an appropriate time for the preparation of an updated
and revised second edition which includes newer material, example problems, case
studies and exercises. In order to have the greatestflexibilityin the incorporation of
these new items in the book, it was also decided to publish the second edition as a
'stand-alone' book of Cambridge University Press, rather than as a research mono-
graph of the Solid State Science Series.
In writing the second edition, I have adhered to the objectives which are stated in
the preface to the first edition. In order to structure the expanded scope coherently,
the book is organized in the following manner. The introduction to the subject of
fatigue, the overall scope of the book and background information on some of the
necessary fundamentals are provided in the first chapter. The book is then divided
into four parts. Cyclic deformation and fatigue crack nucleation in ductile, brittle
and semi-crystalline or noncrystalline solids are given extensive coverage in Part
One. This is followed by discussions of stress-based and strain-based approaches
to fatigue in Part Two. Principles of fracture mechanics and the characteristics of
fatigue crack growth in ductile, brittle and semi-crystalline or noncrystalline solids
are then taken up in separate chapters in Part Three. Part Four comprises advanced
topics where, in addition to separate chapters on crack growth retardation under
constant and variable amplitude fatigue, small cracks, and environmental interac-
tions, a new chapter on contact fatigue is included. In each chapter, updated material
and references, as well as case studies and worked-out and practice problems are
included.
Since the publication of the first edition, numerous students, research colleagues
and engineers from many countries have provided valuable feedback, constructive
criticisms and suggestions. The first edition was translated into the Chinese language
in 1993 under the sponsorship of the Chinese Academy of Sciences. Z. G. Wang and
his colleagues of the State Key Laboratory for Fracture and Fatigue in Shenyang
deserve special thanks for diligently going over the entire manuscript of the first
edition and offering many helpful suggestions. Every attempt has been made to
respond to these comments in the second edition, while trying to adhere to the
main objectives of writing this book. The first edition as well as drafts of the second

xvii
xviii Preface to the second edition

edition were also used by the author as text material for subjects taught for graduate
students at the Massachusetts Institute of Technology, Harvard University and
Brown University.
I am thankful to many individuals who supplied me with information and original
micrographs for the second edition. They include H. Azimi, U. Essmann, W.
Milligan, T. Nakamura, A. Pineau, R.O. Ritchie and D.F. Socie. A number of
colleagues, especially J. Dominguez, T.C. Lindley, F.A. McClintock, T. Nicholas
and L. Pruitt, kindly read drafts of some chapters and offered helpful suggestions.
During the preparation of this second edition, I have had the good fortune to
interact extensively with four colleagues, A.E. Giannakopoulos, L.P. Kubin, H.
Mughrabi and C.F. Shih, who read many sections of the book, and provided helpful
criticisms, suggestions on key references, solutions to some practice problems and
advice on improvement of presentation. I am most grateful to them for their scho-
larly feedback and strong interest in this book.
I wish to thank S. Capelin of the Cambridge University Press for his strong and
continued support of this book project and for giving me considerable flexibility in
the preparation of the manuscript, and to M. Patterson for her efficient copy-editing
of the manuscript. K. Greene, D. LaBonte, G. LaBonte, C.-T. Lin and L. Ward
deserve special mention for their cheerful help with the preparation of figures and
references.
My research work on fatigue has been supported over the years by the U.S.
Department of Energy, Office of Naval Research, Air Force Office of Scientific
Research, and National Science Foundation. This support is gratefully acknowl-
edged. I thank the Department of Materials Science and Engineering at the
Massachusetts Institute of Technology for giving me the flexibility and the time
for the preparation of this second edition through the award of the R.P. Simmons
Endowed Professorship. In addition, I thank the Swedish Research Council for
Engineering Sciences for awarding me the Swedish National Chair in Mechanical
Engineering for the period 1996-98 at the Royal Institute of Technology (Kungl
Tekniska Hogskolan, KTH), Stockholm, where a significant portion of the new
material for the second edition was written. I thank the colleagues in the
Departments of Solid Mechanics and Materials Science and Engineering at KTH
for their hospitality during my stay in Stockholm.
Finally, I express my deepest gratitude to my wife Mary and our daughters Nina
and Meera for all their patience and support during the time I spent writing this
book. Without their care, affection and tolerance, this project could not have been
completed.

S. Suresh
 Preface to the first edition

Fatigue of materials refers to the changes in properties resulting from the

application of cyclic loads. Research into the deformation and fracture of materials
by fatigue dates back to the nineteenth century. This branch of study has long been
concerned with engineering approaches to design against fatigue cracks initiation
and failure. However, along with the development of 'science of materials' and
'fracture mechanics' in recent decades, fatigue of materials has also emerged as a
major area of scientific and applied research which encompasses such diverse dis-
ciplines as materials science (including the science of metals, ceramics, polymers, and
composites), mechanical, civil and aerospace engineering, biomechanics, applied
physics and applied mathematics. With the increasing emphasis on advanced mate-
rials, the scope of fatigue research continues to broaden at a rapid pace.
This book is written with the purpose of presenting the principles of cyclic defor-
mation and fatigue fracture in materials. The main approach adopted here focuses
attention on scientific concepts and mechanisms. Since fatigue of materials is a topic
of utmost concern in many engineering applications, this book also includes discus-
sions on the extension of basic concepts to practical situations, wherever appropri-
ate. In writing this book, I have attempted to achieve the following objectives:

(i) To present an integrated treatment, in as quantitative terms as possible, of

the mechanics, physics and micromechanisms of cyclic deformation, crack
initiation and crack growth by fatigue,
(ii) To provide a unified scientific basis for understanding the fatigue behavior
of metals, nonmetals and composites,
(iii) To develop a balanced perspective of the various approaches to fatigue, with
a critical analysis of the significance and limitations of each approach.

The topics addressed in this book are developed to the extent that the presentation
is sufficiently self-explanatory. The scope of the book is spelled out in Chapter 1.
Some background information on the relevant topics is also provided in the first
chapter to set the scene for later developments. This book could serve as a state-of-
the-art reference guide to researchers interested in the fatigue behavior of materials
and as a text for a graduate course on fatigue. Sections of the book could also be
used for an introductory course on fatigue for practicing engineers. Senior under-
graduate and graduate students taking courses on mechanical behavior of materials
or fracture mechanics may find this monograph useful as a supplement to their
textbooks.

xix
xx Preface to the first edition

Since there exists a vast amount of published information on fatigue of materials

which spans a time period of well over a hundred years, it is not feasible to cover
all the topics and results on this subject in a single monograph. This situation is
further compounded by the fact that fatigue is a research area in which one
encounters considerable empiricism and conflicting viewpoints. Therefore, I had
to rely on my own judgement in the selection of topics and references in order to
balance the aforementioned objectives of writing this book. My interpretations of
this research area have been shaped over the past twelve years through collabora-
tions, discussions and correspondence with numerous colleagues in the United
States and abroad. Their contributions are cited throughout the text and in the
bibliography.
In organizing various sections of this book, I have resisted the temptation to
categorize materials systems in terms of their individual composition or microstruc-
ture, as commonly done in many books on materials. Instead, I have chosen to
present a unified treatment of fatigue in different broad classes of materials while,
at the same time, pinpointing the significant role of microstructure in influencing
cyclic deformation and fracture. The mechanistic and mechanics aspects of fatigue
are developed in the early chapters primarily in the context of ductile metals and
alloys. These concepts are then extended to brittle solids (such as ceramics and
ceramic composites) and to semi-crystalline and noncrystalline solids (such as poly-
mers and organic composites) in later chapters of the book.
I am thankful to many colleagues who supplied me with information and original
photographs. These individuals include R.W. Hertzberg, H. Kobayashi, T.C.
Lindley, T. Nicholas, T. Ogawa, CM. Rimnac, R.O. Ritchie and M.T. Takemori.
I am indebted to J.R. Brockenbrough, C. Laird, M. Miller, H. Mughrabi, P.
Neumann, R.M.N. Pelloux and A.K. Vasudevan for reviewing drafts of various
chapters and for offering helpful comments. A special note of gratitude is extended
to A.S. Argon who provided insightful suggestions on the presentation of funda-
mental concepts and to C.F. Shih who kindly read drafts of all the sections dealing
with fracture mechanics.
My research work on fatigue has been sponsored over the years by U.S.
Department of Energy, National Science Foundation and Office of Naval
Research. This support is very much appreciated. My colleagues and students in
the Solid Mechanics and Materials Science Groups in the Division of Engineering at
Brown University have greatly contributed to my research by providing an intellec-
tually stimulating and friendly environment.
This project was initiated with the encouragement and support of R.W. Cahn. His
enthusiasm and interest in this work provided a strong motivation throughout the
preparation of the manuscript. A special mention should also be made of R. Bentley,
S. Capelin and I. Pizzie of Cambridge University Press for their efficient editorial
work on the manuscript and for promptly responding to my queries and concerns. T.
Judd skillfully helped with the preparation of figures.
Contents xxi

I express my profound gratitude to my wife Mary for her support, devotion and
patience throughout this project. The company of our little daughters Nina and
Meera provided joyful breaks from long hours of concentrated work.
Finally, I wish to dedicate this book to my mother Lakshmi on the occasion of her
sixtieth birthday. Her countless sacrifices for the sake of my education are gratefully
acknowledged.

S. Suresh
 CHAPTER 1

Introduction and overview

The word fatigue originated from the Latin expression fatigare which
means 'to tire'. Although commonly associated with physical and mental weariness
in people, the word fatigue has also become a widely accepted terminology in engi-
neering vocabulary for the damage and failure of materials under cyclic loads. A
descriptive definition of fatigue is found in the report entitled General Principles for
Fatigue Testing of Metals which was published in 1964 by the International
Organization for Standardization in Geneva. In this report, fatigue is defined as a
term which 'applies to changes in properties which can occur in a metallic material
due to the repeated application of stresses or strains, although usually this term
applies specially to those changes which lead to cracking or failure'. This description
is also generally valid for the fatigue of nonmetallic materials.
Fatigue failures occur in many different forms. Mere fluctuations in externally
applied stresses or strains result in mechanical fatigue. Cyclic loads acting in associa-
tion with high temperatures cause creep-fatigue; when the temperature of the cycli-
cally loaded component also fluctuates, thermomechanical fatigue (i.e. a combination
of thermal and mechanical fatigue) is induced. Recurring loads imposed in the pre-
sence of a chemically aggressive or embrittling environment give rise to corrosion
fatigue. The repeated application of loads in conjunction with sliding and rolling
contact between materials produces sliding contact fatigue and rolling contact fati-
gue, respectively, while fretting fatigue occurs as a result of pulsating stresses along
with oscillatory relative motion and frictional sliding between surfaces. The majority
of failures in machinery and structural components can be attributed to one of the
above fatigue processes. Such failures generally take place under the influence of
cyclic loads whose peak values are considerably smaller than the 'safe' loads esti-
mated on the basis of static fracture analyses.f

1.1 Historical background and overview

Fatigue is a branch of study which encompasses many scientific disciplines
and which offers a rich variety of phenomena for fundamental and industrial research.
From published reports, research on the fatigue of materials can be traced back to the

Failing to recognize this fact was the primary cause of catastrophic accidents involving the first com-
mercial jet aircraft, the Comet. A case study of the Comet aircraft failures is provided in Section 1.1.1.
 Introduction and overview

first half of the nineteenth century. Since that time, scores of scientists and engineers
have made pioneering contributions to the understanding of fatigue in a wide variety
of metallic and nonmetallic, brittle and ductile, monolithic and composite, and natural
and synthetic materials. It is not feasible to present, in a few pages, a comprehensive
survey of the historical development of these contributions to fatigue. Nevertheless, in
an attempt to highlight the salient topics of fatigue and to set the scene for the scope of
this book, an overview of major advances and of key areas in fatigue is given in this
section. The chapters to follow provide detailed discussions of various fatigue phe-
nomena along with the relevant historical backgrounds wherever appropriate.
The expression fatigue has been in use for a very long time. In the days of long
distance travel aboard sailing vessels, the straining of masts due to the frequent
hoisting of sails was referred to as fatigue. The first study of metal fatigue is believed
to have been conducted around 1829 by the German mining engineer W.A.J. Albert
(see Albert, 1838). He performed repeated load proof tests on mine-hoist chains
made of iron. One end of the chain was loaded while the chain was supported on
a 360-cm (12-ft) disc. The chain links were repeatedly subjected to bending, at a rate
of 10 bends per minute up to 100 000 bends, by a crank coupling which oscillated the
disc through an arc.
Interest in the study of fatigue began to expand with the increasing use of ferrous
structures, particularly bridges in railway systems. The first detailed research effort
into metal fatigue was initiated in 1842 following the railway accident near Versailles
in France which resulted in the loss of human lives (The Times of London, May 11,
1842; for a comprehensive description of this accident, see Smith, 1990). The cause of
this accident was traced to fatigue failure originating in the locomotive front axle. As
early as 1843, W.J.M. Rankine, a British railway engineer who later became famous
for his contributions to mechanical engineering, recognized the distinctive character-
istics of fatigue fractures and noted the dangers of stress concentrations in machine
components. The Institution of Mechanical Engineers in Britain also began to
explore the so-called 'crystallization theory' of fatigue. It was postulated that the
weakening of materials leading to eventual failure by fatigue was caused by the
crystallization of the underlying micro structure. In 1849, the British Government
commissioned E.A. Hodgkinson to study the fatigue of wrought and cast iron used
in railway bridges. The report of this commission (Hodgkinson, 1849) described
alternating bending experiments on beams whose midpoints were repeatedly
deflected by a rotating cam. In this time period, research on fatigue fracture was
also documented in the work of Braithwaite (1854) who employed the term fatigue
exclusively to denote the cracking of metals under repeated loading. (Braithwaite
(1854), however, credits one Mr. Field for coining this term. Poncelet (1839) is also
generally given credit for introducing the term fatigue in connection with metal
failure, although it had been used earlier in the context of other phenomena.)
A. Wohler conducted systematic investigations of fatigue failure during the period
1852-1869 in Berlin, where he established an experiment station. He observed that
1.1 Historical background and overview

the strength of steel railway axles subjected to cyclic loads was appreciably lower
than their static strength. Wohler's studies involving bending, torsion and axial
loading included fatigue tests on full-scale railway axles for the Prussian Railway
Service and on a variety of structural components used in small machines. His work
(e.g., Wohler, 1860) also led to the characterization of fatigue behavior in terms of
stress amplitude-life (S-N) curves and to the concept of fatigue 'endurance limit'.
The rotating bending machine widely used today for cyclically stressing metals is
conceptually the same as the one designed by Wohler. Although his rotating bending
apparatus had a maximum speed of only 72 revolutions per minute, one of his
fatigue test specimens was subjected to 132 250000 stress cycles without producing
fracture.
Another well known fatigue researcher of this era was W. Fairbairn who per-
formed tests on riveted wrought iron girders for the British Board of Trade; in
some cases, as many as 3100000 load cycles were applied. On the basis of his
experiments, Fairbairn (1864) concluded that the wrought iron girders subjected
to cyclic stresses with a maximum of only one-third of the ultimate strength
would fail. In 1874, the German engineer H. Gerber began developing methods
for fatigue design; his contribution included the development of methods for fatigue
life calculations for different mean levels of cyclic stresses. Similar problems were
also addressed by Goodman (1899).
The notion that the elastic limit of metals in reversed loading can be different from
that observed in monotonic deformation was popularized by Bauschinger (1886).
His work essentially identified the occurrence of cyclic softening and cyclic strain
hardening. Bauschinger also confirmed many of the results reported earlier by
Wohler. By the end of the nineteenth century, some eighty papers on fatigue had
been published in such diverse application areas as railway rolling stock axles,
crankshafts, chains, wire ropes, and marine propeller shafts (see, for example, the
survey by Mann, 1958).
Interpretations of fatigue mechanisms based on the old crystallization theory were
laid to rest by the pioneering work of Ewing & Rosenhain (1900) and Ewing &
Humfrey (1903). These researchers investigated the fatigue of Swedish iron and
published optical micrographs of cyclic damage on the specimen surface. It was
convincingly shown that slip bands developed in many grains of the polycrystalline
material. These slip bands broadened with the progression of fatigue deformation
and led to the formation of cracks; catastrophic failure of the specimen was insti-
gated by the growth of a single dominant flaw. They showed that the slip bands
intersecting the polished surface caused slip steps in the form of elevations and
depressions which we now commonly refer to as 'extrusions' and 'intrusions', respec-
tively. (The micromechanisms of fatigue damage and crack nucleation in metals
form the topics of discussion in Chapters 2-4 of this book.)
In 1910, O.H. Basquin proposed empirical laws to characterize the S-N curves of
metals. He showed that a log-log plot of the stress versus the number of fatigue
 Introduction and overview

cycles resulted in a linear relationship over a large range of stress. Significant con-
tributions to the early understanding of cyclic hardening and softening in metals
were also made by Bairstow (1910). Using multiple-step cyclic tests and hysteresis
loop measurements, Bairstow presented results on the hysteresis of deformation and
on its relation to fatigue failure. In France, Boudouard (1911) conducted fatigue
experiments on steel bars which were subjected to vibrations by means of an electro-
magnetic apparatus similar to the one designed earlier by Guillet (1910). The effect
of heat treatments on the fatigue resistance of steels was the subject of Boudouard's
study. Other notable contributions of this time period included those of Smith
(1910), Bach (1913), Haigh (1915), Moore & Seeley (1915), Smith & Wedgwood
(1915), Ludwik (1919), Gough & Hanson (1923), Jenkin (1923), Masing (1926)
and Soderberg (1939). In 1926, a book entitled The Fatigue of Metals was published
by H.J. Gough in the United Kingdom. A year later, a book bearing the same title
was published by H.F. Moore and J.B. Kommers in the United States. By the 1920s
and 1930s, fatigue had evolved as a major field for scientific research. Investigations
in this time period also focused on corrosion fatigue of metals (Haigh, 1917;
McAdam, 1926; Gough, 1933), damage accumulation models for fatigue failure
(Palmgren, 1924; Miner, 1945), notch effects on monotonic and cyclic deformation
(e.g., Neuber, 1946), variable amplitude fatigue (Langer, 1937), and statistical the-
ories of the strength of materials (Weibull, 1939). A prolific researcher of this period
was Thum (e.g., Thum, 1939) who, along with many German colleagues, reported
experimental results on such topics as fatigue limits, stress concentration effects,
surface hardening, corrosion fatigue and residual stresses in numerous publications.
Gassner (1941) was another prominent German researcher whose studies of variable
amplitude fatigue found applications in the German aircraft industry. (A compre-
hensive survey of the contributions of German engineers and scientists to the field of
fatigue, particularly during the period 1920-1945, can be found in Schutz, 1996.)
Chapters 7 and 8 discuss the key features of these developments in the context of
total-life approaches.
The occurrence df fretting was first documented by Eden, Rose and Cunningham
(1911) who reported the formation of oxide debris between the steel grips and the
fatigue specimen that was contacted by the grips. Tomlinson (1927) performed the
first systematic experiments on fretting fatigue by inducing repeated small amplitude
rotational movement between two contacting surfaces, and introduced the term 'fret-
ting corrosion' to denote the oxidation due to this repeated contact. The deleterious
effects of fretting damage on the fatigue properties of metals, as reflected in the for-
mation of pits on the fretted surface and in the marked reduction in fatigue strength,
were reported by Warlow-Davies (1941) and McDowell (1953). Discussions of fatigue
failures arising from fretting as well as sliding and rolling are considered in Chapter 13.
The notion that plastic strains are responsible for cyclic damage was established by
Coffin (1954) and Manson (1954). Working independently on problems associated
with fatigue due to thermal and high stress amplitude loading, Coffin and Manson
1.1 Historical background and overview

proposed an empirical relationship between the number of load reversals to fatigue

failure and the plastic strain amplitude. This so-called Coffin-Manson relationship
(described in Chapter 8) has remained the most widely used approach for the strain-
based characterization of fatigue.
Although the fatigue of metals by the development of slip bands and the slow
growth of microscopic flaws was documented in the work of Ewing and Humfrey in
the early 1900s, the mathematical framework for the quantitative modeling of fatigue
failure was not available. The stress analyses of Inglis (1913) and the energy concepts
of Griffith (1921) provided the mathematical tools for quantitative treatments of
fracture in brittle solids. However, these ideas could not be directly employed to
characterize the fatigue failure of metallic materials. Progress in this direction came
with the pioneering studies of Irwin (1957) who showed that the amplitude of the
stress singularity ahead of a crack could be expressed in terms of the scalar quantity
known as the stress intensity factor, K. With the advent of this so-called linear elastic
fracture mechanics approach, attempts were made to characterize the growth of
fatigue cracks also in terms of the stress intensity factor. Paris, Gomez &
Anderson (1961) were the first to suggest that the increment of fatigue crack advance
per stress cycle, da/dN, could be related to the range of the stress intensity factor,
AK, during constant amplitude cyclic loading. Although their original paper on this
topic was not accepted for publication by the leading journals in this field, their
approach has since been widely adapted for characterizing the growth of fatigue
cracks under conditions of small-scale plastic deformation at the crack tip. The
major appeal of the linear elastic fracture mechanics approach is that the stress
intensity factor range, determined from remote loading conditions and from the
geometrical dimensions of the cracked component, uniquely characterizes the pro-
pagation of fatigue cracks; this method does not require a detailed knowledge of the
mechanisms of fatigue fracture (Paris, Gomez & Anderson, 1961; Paris & Erdogan,
1963). (A detailed review of the fracture mechanics concepts and of their applications
to fatigue is provided in Chapter 9.)
The effects of various mechanical, micro structural and environmental factors on
cyclic deformation as well as on crack initiation and growth in a vast spectrum of
engineering materials have been the topics of considerable research in the past four
decades. In this period, substantial progress has been made in the understanding of
cyclic deformation and crack initiation mechanisms in fatigue, thanks to advances in
optical and electron microscopy. Notable among these developments are studies
which identified key microscopic features of fatigue deformation and fracture.
Thompson, Wadsworth & Louat (1956) demonstrated that slip bands along which
deformation was concentrated in fatigued metals persistently reappeared at the same
locations during continued cycling even after some material was removed near the
surface; they termed these surface markings 'persistent slip bands'. Zappfe &
Worden (1951) documented observations of characteristic ripple markings on fatigue
fracture surfaces which are now known as fatigue striations. Correlations of the
 Introduction and overview

spacing between adjacent striations with the rate of fatigue crack growth, first pub-
lished by Forsyth & Ryder (1960), became central to the development of various
theories for fatigue crack growth (Chapters 10 and 12) and to the analysis of fatigue
failures in engineering structures. Studies in this area by many researchers have
provided valuable information on substructural and microstructural changes respon-
sible for the cyclic hardening and softening characteristics of materials (Chapters 2
and 3) and on the role of such mechanisms in influencing the nucleation (Chapters 4-
6) and growth (Chapters 10-12) of fatigue cracks.
With the application of fracture mechanics concepts to fatigue failure, increasingly
more attention was paid to the mechanisms of subcritical crack growth. Conceptual
and quantitative models were developed to rationalize the experimentally observed
fatigue crack growth resistance of engineering materials (e.g., Laird & Smith, 1962;
McClintock, 1963; Weertman, 1966; Laird, 1967; Rice, 1967; Neumann, 1969;
Pelloux, 1969). Concomitant with this research, there was expanding interest in
understanding the processes by which the stress intensity factor range could be
altered by the very history of crack advance. An important contribution in this
direction came from the experimental results of Elber (1970, 1971) who showed
that fatigue cracks could remain closed even when subjected to cyclic tensile loads.
This result also implied that the rate of fatigue crack growth might no longer be
determined by the nominal value of stress intensity factor range, AK, but rather by
an effective value of AK which accounted for the details of fracture surface contact
in the wake of the advancing fatigue crack tip. Elber's work focused on the influence
of prior plastic deformation on crack closure during fatigue fracture. Although
Elber's conclusion about the role of crack closure in influencing fatigue crack growth
has since remained controversial, it also became evident from the studies of a num-
ber of researchers in the 1970s and early 1980s that Elber's arguments of premature
contact between the crack faces (based on the effects of prior crack tip plasticity)
represented just one mechanism associated with the phenomenon of fatigue crack
closure. From a survey of published information and on the basis of new results
obtained from their own investigations, Ritchie, Suresh & Moss (1980), Suresh,
Zamiski & Ritchie (1981), and Suresh & Ritchie (1984a) categorized the basic fea-
tures and implications of various types of crack closure and coined the expression
'plasticity-induced crack closure' for Elber's mechanism of crack face contact due to
prior plastic deformation. Further contributions to crack closure may arise from
fracture surface oxidation, viscous environments trapped within the crack walls and
stress-induced phase transformations. In addition, periodic deflections in the path of
a fatigue crack can cause reductions in the effective driving force for fatigue fracture
(Suresh, 1983a, 1985a) by partially 'shielding' the crack tip from applied stresses. A
discussion of the historical development of these concepts is presented in Chapter 14.
A significant outcome of the investigations of different types of crack shielding
processes is the realization that the rate of fatigue crack growth is not only affected
by the instantaneous value of imposed AK, but also by prior loading history and
1.1 Historical background and overview

crack size. As the mechanics of fatigue fracture become dependent on the geome-
trical conditions, the similitude concept implicit in the nominal use of fracture
mechanics, namely the notion that cracked components of different dimensions
exhibit the same amount of crack growth when subjected to the same value of
AK, is no longer applicable. This breakdown of the similitude concept is further
accentuated by the experimental observations that small fatigue flaws (typically
smaller than several millimeters in length and amenable to be characterized in
terms of linear elastic fracture mechanics) often exhibit growth rates which are
significantly faster than those of longer flaws (typically tens of millimeters in length),
when subjected to identical values of far-field AK. Furthermore, fatigue flaws of
dimensions comparable to or smaller than the characteristic microstructural size
scale often exhibit rates of crack growth which diminish with an increase in crack
length. Such crack growth cannot be satisfactorily analyzed in terms of available
theories of fracture mechanics. This so-called 'short crack problem', apparently first
identified by Pearson (1975), most severely affects the development of design meth-
odology for large structural components on the basis of experimental data gathered
from smaller-sized laboratory test specimens. It is, therefore, not surprising to note
that a significant fraction of research effort since the late 1970s has been devoted to
the study of crack closure phenomena and crack size effects on the progression of
fatigue fracture. Associated with this research effort are attempts to develop char-
acterization methodology for the propagation of fatigue flaws in the presence of
large-scale plastic deformation and in the vicinity of stress concentrations (see
Chapters 4, 7, 8 and 15).
Although fatigue failure under fixed amplitudes of cyclic stresses generally forms
the basis for fundamental studies, service conditions in engineering applications
invariably involve the exposure of structural components to variable amplitude
spectrum loads, corrosive environments, low or elevated temperatures and multiaxial
stress states. The development of reliable life prediction models which are capable of
handling such complex service conditions is one of the toughest challenges in fatigue
research. Although major advances have been made in these areas, the application of
fatigue concepts to practical situations often involves semi-empirical approaches.
Available models for fatigue involving conditions of multiaxial stress conditions,
complex load spectra and detrimental environments are discussed in Chapters 3, 7,
8, 10 and 16.
The majority of fatigue research reported in the open literature pertains to metallic
materials. There has, however, been a surge in interest aimed at nonmetallic materials
and composites which offer the potential for mechanical, thermal and environmental
performance hitherto unobtainable in conventional metals. This growing interest has
also generated a corresponding increase in research into the fatigue behavior of
advanced ceramics (e.g., Suresh, 1990a,b; Roebben et ah, 1996), polymers (e.g.,
Hertzberg & Manson, 1980; Hertzberg, 1995), and their composites. While the exis-
tence of cyclic slip has traditionally been considered a necessary condition for the
 Introduction and overview

occurrence of fatigue failure in ductile solids, it is now recognized that mechanical

fatigue effects in nonmetallic materials can arise in some cases from the kinematic
irreversibility of microscopic deformation under cyclic loads even in the absence of
cyclic dislocation motion. Mechanisms which impart kinematic irreversibility of
deformation during fatigue can be as diverse as microcracking, stress-induced
phase transformations, dislocation plasticity, creep, interfacial sliding or 'craze' for-
mation. An extension of existing knowledge on the fatigue of metallic systems to these
advanced materials and the identification of new mechanistic phenomena associated
with the fatigue of nonmetals and metal-nonmetal composites form the basis for
many ongoing research efforts. Chapters 3, 5, 6, 11, and 12 in this book provide a
detailed description of the fatigue characteristics of a wide variety of brittle and
noncrystalline materials, including many nonmetallic composites and layered solids.

Ll.l Case Study: Fatigue and the Comet airplane

The dramatic effect of subcritical crack growth by fatigue on the mechanical
integrity of aircraft structures was clearly brought to light by the series of crashes
involving the first commercial jet aircraft, the Comet. This airplane was manufactured by
the de Havilland Aircraft Company, England, and was designed to capture the rapidly
growing long-distance air travel business, spurred by the economic recovery of Great
Britain and Continental Europe after World War II. The fatigue failures of the Comet
cabin structures, which led to several accidents in the 1950s, also obstructed the
prominent role played by the British in the commercial jet aircraft industry. It is widely
believed that the fatigue problem of the Comet may have served as a catalyst in the
eventual emergence of the rival Boeing Aircraft Company in the United States as a world
leader in commercial aviation.
The use of the jet engine, pioneered by Sir Frank Whittle in Great Britain, for propelling
the commercial jet aircraft was still an untested proposition at the time of design of the
Comet. The fuel consumption rate of a jet engine was more than twice that of a piston
engine. In order to limit the fuel consumption rate of a jet aircraft to a level no higher than
that of one propelled by a piston engine, the plane had to travel twice as fast.| This meant
flying in the rarefied atmosphere of high altitude, typically around 12000 m (or 40000 ft),
which was more than double the altitude at which the World War II vintage aircraft flew.
The speed of sound is approximately 1200 km/h (or 760 miles per hour) at sea level; at an
altitude of 12000 m, it drops to about 1060 km/h. At this high altitude, termed the lower
stratosphere, the coldest air with a temperature of approximately -56 °C exists, and clouds

There are several basic requirements for the optimum performance of a passenger jet aircraft. Firstly, it
must travel as fast as possible for optimizing such factors as fuel efficiency, number of flights per unit
time period, and return on capital expenses. Secondly, it must fly below the speed of sound in order to
avoid a precipitous rise in the specific energy consumption, which is related to the ratio of thrust to
weight or to the ratio of drag to lift. Thirdly, the colder the air through which the aircraft flies, the
greater the efficiency of the jet engine. Fourthly, an aircraft should notflyat an altitude higher than what
is necessary because flying through rarefied atmosphere requires oversized wings. For passenger jet
aircraft, the first two requirements suggest a maximum cruising speed typically in the neighborhood
of 90% of the speed of sound or Mach 0.9, and the last two requirements suggest an optimum altitude of
10000-12000 m (Tennekes, 1996).
1.1 Historical background and overview

and thunderstorms are rare so that meteorological conditions do not impede flight
schedules during cruising. The colder outside air at such altitudes also enhances the
efficiency of the jet engines as the difference between the intake temperature and the
combustion temperature is raised.
A particularly important issue for high altitude flights was the design of the cabin
wherein the temperature and pressure had to be at near-ground levels for the comfort of
the passengers and the crew. The aircraft fuselage would have to be repeatedly stressed
from no pressure differential between the inside and the outside whilst on the ground to a
large pressure difference between the inside passenger cabin and the rarefied atmosphere
outside during cruising. The fuselage, therefore, had to be capable of withstanding high
stresses arising from cabin pressurization during such high altitude flights in thin air. It
would turn out that the fatigue stress cycles induced on the metal skin of the fuselage by
the repeated pressurization and depressurization of the cabin during each flight
contributed to the catastrophic fracture in several Comet airplanes (e.g., Dempster, 1959;
Petroski, 1996).
On the first anniversary of commercial jet aircraft operation, May 2, 1953, a de
Havilland Comet airplane disintegrated in mid-air soon after take-off from the airport in
Calcutta, India. The crash occurred during a heavy tropical thunderstorm. The official
organization investigating the crash concluded that the accident was the result of some
form of structural fracture, possibly arising from higher forces imposed on the airframe
by the stormy weather, or from the overcompensation by the cockpit crew in trying to
control the plane in response to such forces. Consequently, the design of the aircraft
structure was not viewed as a cause for concern.
On January 10, 1954, another Comet aircraft exploded at an altitude of 8230 m
(27 000 ft) in the vicinity of Elba Island in the Mediterranean Sea, after taking off from
Rome in good weather. Once again, no flaws in design were identified, and the aircraft
was placed back in service only weeks after this second crash.
The third accident took place soon afterwards on April 8, 1954, when a Comet
exploded in mid-air upon departure from Rome, after a brief stopover during a flight
between London and Cairo. The wreckage from the crash fell in deep sea water and could
not be recovered. This led investigators from the Royal Aircraft Establishment (RAE),
Farnborough, England, to renew efforts to recover pieces from the second crash over
Elba. Evidence began to emerge indicating that the tail section was intact from the Elba
crash, and that the pressurized cabin section had torn apart before fire broke out.
In order to probe into the origin of cabin explosion, RAE engineers retired a Comet
airplane from service and subjected its cabin to alternate pressurization and
depressurization, to about 57kPa (8.25 psi) over atmospheric pressure, by repeatedly
pumping water into it and then removing it. During such simulated cabin pressurization,
the wings of the aircraft were also stressed by hydraulic jacks to mimic wing loading
during typical flight conditions. After about 3000 pressurization cycles, a fatigue crack
originating in a corner of a cabin window advanced until the metal skin was pierced
through. Figure 1.1 schematically shows the location of cracks in a failed Comet airplane.
The Comet, being the first commercial jetliner, was designed and built at a time when
the role of fatigue in deteriorating the mechanical integrity of airframe components was
not appreciated, and when subcritical fatigue crack growth had not evolved into a topic
10 Introduction and overview

crack along top center line of fuselage

front fuselage separated at front spar

attachments in downward direction
rear fuselage
and tail unit
separated at
rear spar
failure probably downwards
symmetrical with starboard
wing failure
main failure
between ribs
12 and 13

frame 26
peeling off failure

secondary cracking by bending

of center portion over outer portion

direction of propagation
of main cracks

signs of fatigue
in skin at this corner
reinforcing
plates

peeling off failures

skin pulled over rivets on window frame peeling off failures

Fig. 1.1. Schematic diagram illustrating the location of fatigue cracks in a failed Comet
airplane. (After Petroski, 1996.)

of extensive research. It was assumed that the possibility of one fatigue cycle per flight,
due to cabin pressurization upon take-off and depressurization during landing, would not
be significant enough to advance any flaws in the fuselage to catastrophic proportions.
The cabin walls were designed to contain a pressure of 138 kPa (20psi), two and a half
times the service requirements. As an added demonstration of safety, the passenger cabin
of each Comet was pressurized once to 114 kPa (16.5 psi) in a proof test, before the plane
was placed in service. The investigative report of the Court of Inquiry into the Comet
failures noted that the de Havilland designers believed '... that a cabin (which) would
1.2 Different approaches to fatigue 11

survive undamaged a test to double its working pressure ... would not fail in service under
the action of fatigue...'. This notion was proven erroneous, at a significant cost to de
Havilland and to the British commerical aircraft industry.
The RAE tests revealed that the cabin failures in the first three Comet accidents were
due to fatigue cracking which was aided by stress elevation at the rivet holes located near
the window openings of the passenger cabin. In subsequent designs of the new Comet 4
models, which facilitated trans-Atlantic commercial jet travel for the first time, the
window sections were replaced with a new reinforced panel which had much greater
resistance to fatigue failure.

No aircraft has contributed more to safety in the jet age than the Comet. The lessons it
taught the world of aeronautics live in every jet airliner flying today.
D.D. Dempster, 1959, in The Tale of the Comet

1.2 Different approaches to fatigue

There are different stages of fatigue damage in an engineering component
where defects may nucleate in an initially undamaged section and propagate in a
stable manner until catastrophic fracture ensues. For this most general situation,
the progression of fatigue damage can be broadly classified into the following
stages:
(1) Substructural and microstructural changes which cause nucleation of per-
manent damage.
(2) The creation of microscopic cracks.
(3) The growth and coalescence of microscopic flaws to form 'dominant' cracks,
which may eventually lead to catastrophic failure. (From a practical stand-
point, this stage of fatigue generally constitutes the demarkation between
crack initiation and propagation.)
(4) Stable propagation of the dominant macrocrack.
(5) Structural instability or complete fracture.
The conditions for the nucleation of microdefects and the rate of advance of the
dominant fatigue crack are strongly influenced by a wide range of mechanical,
microstructural and environmental factors. The principal differences among differ-
ent design philosophies often rest on how the crack initiation and the crack propa-
gation stages of fatigue are quantitatively treated.
It is important to note here that a major obstacle to the development of life
prediction models for fatigue lies in the choice of a definition for crack initiation.
Materials scientists concerned with the microscopic mechanisms of fatigue are
likely to regard the nucleation of micrometer-size flaws along slip bands and
grain boundaries, and the roughening of fatigued surfaces as the crack inception
stage of fatigue failure. A practicing engineer, on the other hand, tends to relate
12 Introduction and overview

the limit of resolution of the (nondestructive) crack detection equipment (typically

a fraction of a millimeter) with the nucleation of a fatigue crack and with the
initial crack size used for design. Scattered within the limits of this broad range
of choices, there lies a variety of definitions for crack nucleation which are
specific to certain classes of fatigue-critical engineering applications. The total
fatigue life is denned as the sum of the number of cycles to initiate a fatigue
crack and the number of cycles to propagate it subcritically to some final crack
size. In light of the foregoing discussion on what constitutes crack initiation,
making a clear demarkation between crack initiation and crack propagation
can become a critical task.

1.2.1 Total-life approaches

Classical approaches to fatigue design involve the characterization of
total fatigue life to failure in terms of the cyclic stress range (the S-N curve
approach) or the (plastic or total) strain range. In these methods, the number of
stress or strain cycles necessary to induce fatigue failure in initially uncracked
(and nominally smooth-surfaced) laboratory specimens is estimated under con-
trolled amplitudes of cyclic stresses or strains. The resulting fatigue life incorpo-
rates the number of fatigue cycles to initiate a dominant crack (which can be as
high as some 90% of the total fatigue life) and to propagate this dominant flaw
until catastrophic failure occurs. Various techniques are available to account for
the effects of mean stress, stress concentrations, environments, multiaxial stresses
and variable amplitude stress fluctuations in the prediction of total fatigue life
using the classical approaches (see Chapters 7, 8 and 14). Since the crack initia-
tion life constitutes a major component of the total fatigue life in smooth speci-
mens, the classical stress-based and strain-based methods represent, in many
cases, design against fatigue crack initiation. Under high-cycle, low stress fatigue
situations, the material deforms primarily elastically; the failure time or the
number of cycles to failure under such high-cycle fatigue has traditionally been
characterized in terms of the stress range. However, the stresses associated with
low-cycle fatigue are generally high enough to cause appreciable plastic deforma-
tion prior to failure. Under these circumstances, the fatigue life is characterized
in terms of the strain range. An example of a situation, where the classical
(short-life) strain-based approach (also referred to as the low-cycle fatigue
approach) has found much appeal, involves the prediction of fatigue life for
the initiation and early growth of a crack within the strain field associated
with the fully plastic region ahead of a stress concentration (see Fig. 1.2). The
low-cycle approach to fatigue design has found particularly widespread use in
ground-vehicle industries.
1.2 Different approaches to fatigue 13

crack initiation
and early growth low-cycle
fatigue test
specimen

fatigue crack
growth test
specimen

plastic
zone

Fig. 1.2. Schematic diagram illustrating the various stages of fatigue in an engineering
component and the approaches used to estimate the fatigue life. (After Coffin, 1979.)

1.2.2 Defect-tolerant approach

The fracture mechanics approach to fatigue design, on the other hand,
invokes a 'defect-tolerant' philosophy. The basic premise here is that all engineering
components are inherently flawed. The size of a pre-existing flaw is generally deter-
mined from nondestructive flaw detection techniques (such as visual, dye-penetrant
or X-ray techniques or the ultrasonic, magnetic or acoustic emission methods). If no
flaw is found in the component, proof tests are conducted whereby a structure, such
as a pressure vessel, is subjected to a simulation test a priori at a stress level slightly
higher than the service stress. If no cracks are detected by the nondestructive test
method and if catastrophic failure does not occur during the proof test, the largest
(undetected) initial crack size is estimated from the resolution of the flaw detection
technique. The useful fatigue life is then defined as the number of fatigue cycles or
time to propagate the dominant crack from this initial size to some critical dimen-
sion. The choice of the critical size for the fatigue crack may be based on the fracture
toughness of the material, the limit load for the particular structural part, the allow-
able strain or the permissible change in the compliance of the component. The
prediction of crack propagation life using the defect-tolerant approach involves
empirical crack growth laws based on fracture mechanics. In terms of the require-
ments of linear elastic fracture mechanics, the defect-tolerant method is applicable
under conditions of small-scale yielding (i.e. away from the plastic strain field of any
stress concentrators), where the crack tip plastic zone is small compared to the
characteristic dimensions of the cracked component (including the crack size) and
where predominantly elastic loading conditions prevail. Various methods are avail-
able to incorporate the effects of mean stresses, stress concentrations, environments,
variable amplitude loading spectra and multiaxial stresses in the estimation of useful
14 Introduction and overview

crack growth life (see Chapters 7, 8, 10 and 14-16). This intrinsically conservative
approach to fatigue has been widely used in fatigue-critical applications where cat-
astrophic failures will result in the loss of human lives; examples include the aero-
space and nuclear industries.

1.2.3 A comparison of different approaches

The different approaches to fatigue also provide apparently different guide-
lines for the design of microstructural variables for optimum fatigue resistance.
These differences are merely a consequence of the varying degrees to which the
role of crack initiation and crack growth are incorporated in the calculation of useful
fatigue life. For example, in many structural alloys the resistance to the growth of
long fatigue cracks generally increases with an increase in grain size (or a decrease in
yield strength) at low AK values where a significant portion of subcritical crack
growth life is expended. On the other hand, the total fatigue life estimated on the
basis of stress-life plots generally exhibits the opposite trend; higher strength materi-
als and finer grained micro structures usually lead to a longer fatigue life. The appar-
ent contradiction between the two approaches can be reconciled by noting that the
former approach to fatigue deals primarily with the resistance to fatigue crack
growth, while the latter approach based on nominally defect-free laboratory speci-
mens focuses mainly on the resistance to fatigue crack initiation. The choice of a
particular microstructural condition for improved fatigue life is then predicated
upon the design philosophy for a specific application. Optimization of microstruc-
tural characteristics for improved resistance to both crack initiation and crack
growth would require a trade-off between the recommendations of the two
approaches.

1.2.4 'Safe-life9 and fail-safe' concepts

The safe-life and fail-safe design approaches were developed by aerospace
engineers. In the safe-life approach to fatigue design, the typical cyclic load spectra,
which are imposed on a structural component in service, are first determined. On the
basis of this information, the components are analyzed or tested in the laboratory
under load conditions which are typical of service spectra, and a useful fatigue life is
estimated for the component. The estimated fatigue life, suitably modified with a
factor of safety (or an ignorance factor), then provides a prediction of 'safe life' for
the component. At the end of the expected safe operation life, the component is
automatically retired from service, even if no failure has occurred during service (and
the component has considerable residual fatigue life). Although an estimate of life
may be obtained from practical tests on the actual component, the safe-life method is
intrinsically theoretical in nature. This procedure invariably has to account for
1.2 Different approaches to fatigue 15

several unknowns, such as unexpected changes in load conditions, errors in the

estimates of typical service load spectra, scatter in the test results, variation in
properties among different batches of the same material, existence of initial defects
in the production process, corrosion of the parts used in the component, and human
errors in the operation of the component. By selecting a large margin of safety, a safe
operating life can be guaranteed, although such a conservative approach may not be
desirable from the viewpoints of economy and performance. On the other hand, if
fatigue cracks are nucleated in the component during service, the component may
well fail catastrophically. As noted by Gurney (1968), the safe-life approach depends
on achieving a specified life without the development of a fatigue crack so that the
emphasis is on the prevention of crack initiation. The fail-safe concept, by contrast,
is based on the argument that, even if an individual member of a large structure fails,
there should be sufficient structural integrity in the remaining parts to enable the
structure to operate safely until the crack is detected. Components which have multi-
ple load paths are generally fail-safe because of structural redundancy. In addition,
the structure may contain crack arresters to prevent undesirable levels of crack
growth. (The case study in Section 9.12 provides an example of this concept.) The
fail-safe approach mandates periodic inspection along with the requirement that the
crack detection techniques be capable of identifying flaws to enable prompt repairs
or replacements. Whatever philosophy is employed in design, it is often preferable
(and even required in some safety-critical situations, e.g., aircraft and nuclear indus-
tries) that the critical components of a structure be inspected periodically. This step
eliminates dangerous consequences arising from false estimates and errors in the
design stage, especially with the safe-life approach.

1.2.5 Case study: Retirement for cause

The design and maintenance of gas turbine engine components have traditionally
involved estimates of fatigue life on the basis of low-cycle fatigue concepts. In this
approach, entire classes of components are retired from service when some pre-
determined design life is reached. For typical rotor components, such as discs which have
been serviced and maintained by the United States Air Force in the past several decades,
this approach meant that 1000 discs could be retired from service when, statistically, only
one among them had developed a small fatigue crack (typically 0.75 mm or less in depth).
In other words, 99.9% of the discs, which had considerable residual fatigue life, were
retired prematurely.
The development of a new fatigue control methodology for gas turbine engine
components provides an example of the practical use of thefoil-safeconcept. In 1985, the
United States Air Force began implementing the so-called 'retirement for cause' (RFC)
component life management methodology for its existing F100 aircraft engines,
principally for cost savings, in lieu of the traditional approaches based on pre-determined
life (Harris, 1987). In this program, the retirement of a gas turbine engine component
from service occurs when the unique fatigue life of that particular component (as opposed
16 Introduction and overview

to the predicted life for the entire population of the same component) is considered to be
utilized. The individual component is retired from service when there is a specific reason
or cause for removal from service, such as the existence of aflawof a certain (maximum)
allowable or detectable size. This system replaced classical low-cycle fatigue approaches
where an entire population of components of a certain type were retired, regardless of the
condition, when a pre-determined time or number of cycles was expended.
Analyses of crack growth and damage tolerance were carried out to identify which, if
any, components were candidates for RFC. Subsequently, a list of components and
combinations of inspection sizes and intervals were identified.! The F100 engine, for
which the RFC program has initially been implemented, is currently in service in the
twin-engine F-15 military aircraft built by the McDonnell Douglas Corporation and in
the single-engine F-16 fighter aircraft built by the General Dynamics Corporation. There
are over 3200 such engines in the operational inventory of the US Air Force. A total of
twenty-three components used in such parts as fan, compressor and low pressure turbine
rotors are being managed under this philosophy. The F100 engine overhaul manuals have
been revised such that the RFC procedure replaces the classical time to retirement
guidelines.
There were several developments for damage tolerance analysis and maintenance
which evolved from the RFC program. One such development is the refinement of a
nondestructive inspection method, based on eddy current monitoring, which is now
commonly used under the RFC program management. Another outcome of this program
is a cryogenic spin pit test for some titanium alloy discs. In this procedure, the component
is spun at a low temperature where the fracture resistance drops. If the disc does not burst
during the spin and safe operation is guaranteed above a certain 'inspection' size, the disc
is placed back in service until the next inspection. It was also demonstrated that if cracks
below the inspection size were to be present, the cryogenic spin pit test would not further
extend these cracks or cause the damage zone dimensions at the crack tip to be altered.
Whereas RFC was implemented on an F100 engine which was already designed and built,
improvements to damage-tolerant design were implemented by another effort termed the
ENgine Structural Integrity Program (ENSIP). Under ENSIP (Nicholas, Laflen, &
VanStone, 1986; Cowles, 1988), all critical structural components in an engine had to be
designed such that they could be inspected and, based on the inspection flaw size, could
be flown safely until the next inspection. Thus, all components became, by definition,
Candidates for RFC. Since the inception of such damage tolerance procedures in the
design Stage, failure incidents due to low-cycle fatigue have been essentially eliminated
from the US Air Force jet engine inventory.
The economic implications of such changes in failure control philosophy are also
substantial. Initial estimates by the US Air Force reveal that, over the time period 1986-
2005, the (life cycle) cost savings realized from the implementation of the RFC
methodology to the F100 engines alone will amount to nearly $1 billion. Additional
savings, amounting to as much as $655 million, are projected over this time period as a
consequence of reductions in labor and fuel costs arising from the extension of maintenance
intervals for the upgraded F100 core engines.

All of the components which were analyzed did not meet the criteria for RFC; the ones which failed the
criteria are still being retired after their design life, predicated upon low-cycle fatigue estimates, is
expended.
1.3 The need for a mechanistic basis 17

Another example of the fail-safe approach is the so-called leak-before-break criterion,

which was first proposed by Irwin (1964). This methodology is widely used in the
structural design of pressure vessels and pipes. The leak-before-break criterion is
developed as a means of ensuring that a pressure vessel or a pipe has the necessary
structural integrity and service use even if a surface crack propagates through the thickness
of the pressure vessel or pipe wall. Thus, the vessel or pipe would first 'leak' before any
catastrophic fracture occurs so that the fatigue flaw could be easily detected and repaired.
The implementation of this criterion would, therefore, require that the critical crack size at
the design stress level of the material be greater than the wall thickness of the vessel.
Examples of fracture mechanics analyses used in conjunction with the leak-before-break
criterion can be found in any book on the subject (e.g., Barsom & Rolfe, 1987).

1.3 The need for a mechanistic basis

Fatigue of materials is a branch of study which provides a broad variety of
complex mechanistic processes for scientific investigation. The size scales of observa-
tion that are of interest in this research area range from submicrostructural (even
atomistic) levels to dimensions of structural components spanning tens or hundreds
of meters. Implications of fatigue failure encompass many aspects of our lives. The
practical significance of fatigue has even generated fictional stories and motion
pictures whose themes have centered around fatigue failures (e.g., the novel No
Highway by Nevil Shute, 1948, and its motion picture adaptation entitled No
Highway to Heaven). The consequences of fatigue failure become most apparent
when stories of disasters, such as aircraft accidents involving the loss of human
lives, are publicized.!
Although considerations of fatigue failure are intimately tied to the practical
aspects of structural integrity in engineering components, the mechanistic and
scientific basis for the study of fatigue cannot be ignored because of the following
reasons:

(1) The size scale over which permanent damage occurs at the tip of a fatigue
crack is generally comparable to the characteristic micro structural dimen-
sion of the material, even if the component dimensions and the crack size are
orders of magnitude larger than the scale of the microstructure.
(2) The total life and fracture mechanics approaches provide methods for char-
acterizing the resistance of the material to crack initiation and growth under
cyclic loads. However, these concepts alone cannot offer a quantitative
description of the intrinsic resistance of the material to fatigue. This infor-
mation can be obtained only if there exists a thorough understanding of the

' The 1985 crash of a Japan Airlines Boeing 747 due to a catastrophic fatigue failure of the rear pressure
bulkhead and the resulting loss of 520 human lives is a case in point. A failure analysis of this accident is
presented in Chapter 10.
18 Introduction and overview

micromechanisms of failure. There is ample evidence from published work

on a wide variety of materials that subtle changes in the microstructure (and
the environment) can lead to drastic alterations in the extent of cyclic
damage and failure life. Thus, optimizing the microstructural characteristics
of a material for improving the fatigue resistance inevitably requires a scien-
tific knowledge of failure mechanisms.
(3) A significant portion of the fatigue crack growth life is spent at low AK
levels where the maximum crack tip opening displacement during a loading
cycle is typically smaller than a micrometer for most structural components.
Since this dimension is smaller than the characteristic microstructural size
scale in most materials, microstructural effects can markedly affect the
resistance to fracture even if the crack size is significantly larger than the
microstructural dimension.
(4) Even when a structural component is designed conservatively, fatigue failure
may occur because of unexpected changes in service conditions. Tost-mor-
tem' analyses of fatigue failures often involve tracing the origin of fatigue
failure via microscopic features present on the fracture surfaces, such as
'clam shell' markings and striations. These features can provide valuable
information about the location where fracture initiated as well as about the
magnitude of loads imposed upon the failed component. (See the various
case studies in this book for an illustration of this point.) A fundamental
knowledge of the link between the characteristic features observed on the
fatigue fracture surfaces, the microscopic mechanisms of failure and the
macroscopic rates of crack advance is vital to the success of such post-
mortem analyses.

1.4 Continuum mechanics

Continuum descriptions of cyclic deformation and fatigue failure require
an understanding of elasticity and plasticity theories. In this section, we present a
brief review of the theories of linear elasticity and plasticity which provide the
foundation for the derivations and discussions presented in various chapters of
this book. The contents of this section are intentionally brief and confined only
to those topics that are pertinent to the scope of this book. More elaborate details
and derivations on these topics can be found in standard textbooks (e.g., Hill,
1950; Malvern, 1969).
Consider the infinitesimal element of volume in a stressed solid, shown in Fig. 1.3.
The stress at a point is the local area intensity of the force which is transmitted
between adjacent parts of the solid through an imaginary surface that divides the
solid, such as the faces of the cubic element in Fig. 1.3. The stress can be represented
by its scalar components, o^-, normal and tangential to the surface. For the cartesian
1.4 Continuum mechanics 19

Fig. 1.3. Definition of stress components on an infinitesimal volume element.

coordinate system shown in Fig. 1.3, the stress components at a point are contained
in the matrix

(1.1)
°yy
°zz,
yz

where i,j = 1, 2, 3, and ay is the ith component of the force per unit area on a plane
whose outward normal points toward the positive Xj direction; X\ = x, x2 = y, and
x3 = z. Each infinitesimal volume element in the body must be in mechanical equili-
brium. Since there can be no net torque on the element, o^ = a^. Similarly, no net
force can act on the element, so that
i = 1,2,3, (1.2)
dx2
where bt is the ith component of the body force per unit volume. In the absence of
body forces, Eq. 1.2 can be expressed as
— = 0, (1.3)

where summation over j is implied. Equations 1.2 and 1.3 are known as the equili-
brium equations.
Under the influence of applied forces, let ut be the components of displacement at
a point in the body. The components of infinitesimal strain are defined as

(1.4)
2
When i ^j in this equation, the shear strains are obtained. However, it is impor-
tant to note that, for / ^y, Eq. 1.4 provides only one-half of the shear strains that are
commonly denned in engineering where y^ = 2eiJ. In Eq. 1.4, six components of
small strain, e^, have been expressed in terms of three components of the displace-
20 Introduction and overview

ment. This implies that the strains must be interrelated. For the strain components in
the x\-x2 plane,
du\ du2 1 /3wi du2
dXi dx2 2 \dx2 dxi
From Eqs. 1.5, one finds that

—-y-2-— h—^- = 0, (1.6)

for deformation on the xx-x2 plane. Equation 1.6 is one of the so-called compatibility
equations.
A state of plane stress or plane strain is characterized by the conditions
9( )/3JC 3 = 0 and a 13 = a23 = 0, with xx and x2 taken as independent variables.
For plane stress, the stress-strain relationships (to be discussed in the next subsec-
tion) are used along with the additional condition that <r33 = 0. Similarly, for plane
strain, the stress-strain relationships are applied with e33 = 0. An anti-plane state is
characterized by the conditions that on = G22 = a 33 = a 12 = 0.
Standard procedures of coordinate transformation (see, for example, Malvern,
1969) are used to derive the equilibrium and compatibility equations and strain-
displacement relationships for the cylindrical coordinate system from the results
discussed above for the cartesian reference system, Eqs. 1.3 and 1.6. These results
for the cylindrical coordinate reference are presented in Section 9.3.2.
The equilibrium conditions are fulfilled automatically if the stresses are expressed
in terms of the so-called Airy stress function x, which is defined by the relationships

3x2 dxf ox i ox2

in the cartesian coordinate system. Similarly, the compatibility condition, Eq. 1.6,
when expressed in terms of the Airy stress function, becomes

Similar expressions for the equilibrium and compatibility conditions in terms of

polar coordinates are given in Chapter 8. A complete solution is obtained by satisfy-
ing Eq. 1.8 and the prescribed boundary conditions.

1.4.1 Elements of linear elasticity

When the material undergoes only elastic deformation, the stresses and
strains are related by Hooke's law,
/ 1 r\\
C iikl^kh v-*-*^/

where Cijki are the elastic constants which, for isotropic material response, are
Cijki = k&ijhi + G(8ik8ji + 8u8jk). (1.10)
1.4 Continuum mechanics 21

X is known as the Lame constant and G is the shear modulus. 8{j is the Kronecker
delta with the property that 8tj = 0 for / / y and that 8tj — 1 for / =j. The isotropic
elastic constants defined by Young's modulus E and Poisson's ratio v are related to
G and k by the expressions

Young's modulus E is the ratio of the axial stress to the axial strain, whereas
Poisson's ratio v represents the ratio of transverse contraction to (axial) elongation
in simple tension. The strains can be related to the stresses in terms of the elastic
constants,

[ K + >L
1
r M

fe ( + <T33)],
[ K + ) ]
When the volume element shown in Fig. 1.3 deforms reversibly by an infinitesimal
strain increment de^, the stresses do work on the element by the amount
Aw = (Jijdtij = Cijkl€kid€ij. (1.13)

Under conditions of reversible and isothermal elastic deformation, the differential

work per unit volume Aw in Eq. 1.13 is also equal to the change in the Helmholtz free
energy AF, which is an exact differential. For the conditions stated, the work of
deformation is a single-valued function in strain space, such that

w = o cijki€ij€ki, (or) aij = —-, (1.14)

Z O€jj

which is useful for later discussion.

1.4.2 Stress invariants

For any general three-dimensional stress state given by Eq. 1.1, orie can find
via coordinate tranformation three normal stresses, known as principal stresses,
which act on orthogonal planes that are free of shear stresses. The principal stresses,
orl5 a2 a n ( i ^3? a r e t n e roots of the cubic equation,
A.3 - IXX2 - I2k - h = 0 , (1.15)
where

73 =
The coefficients Ix, I2 and 73 are independent of the orientation of the coordinate
system chosen to describe the stress components. These coefficients are termed stress
22 Introduction and overview

invariants because the principal stresses are physical quantities at the point in the
solid under consideration. Any combination of these stress invariants also results in
an entity which is an invariant.
The normal mean stress or the hydrostatic stress, aH, is defined as

^H =5(^11 +^22 + ^33) = ^ ^ = " ^ . (1-17)

where the subscript k = 1, 2, 3. The hydrostatic stress aH causes a change only in
volume (and not in shape) in an isotropic continuum. From Eqs. 1.11, 1.12 and 1.17,
the compressibility g, the ratio of the negative of the dilatation e = en + €22 + £33 to
the pressure p = — aH, can be written as
<u8)
*—i-5rb-5-
where B is the bulk modulus.
The deviatoric components of stress are defined as
(1.19)

These deviatoric stress components, unlike the hydrostatic stress, bring about a
change of shape in the body and influence the plastic deformation. Analogous to
the stress invariants 71? I2 and 73, a new set of scalar invariants, / 1? J2 and J$, based on
the principal components of the deviatoric stress tensor, s\, s2 and s$, can be defined:
Jx = ou - Ix = 0,

J2 = 2S(JSiJ =
2^1 +
^2 ^ S
^ '

J
2 = 3 isijsjkSki) = S\ ^2^3 • (1-20)

1.4.3 Elements of plasticity

When a ductile metallic material is loaded beyond the elastic limit, it under-
goes permanent plastic deformation. Plastic flow in metals can be assumed incom-
pressible. The total strain imposed on an elastic-plastic solid €tJ can be written as the
sum of the elastic and plastic strains e| and e?, respectively. The elastic strains are
related to the stresses by Hooke's law, Eq. 1.9. The assumption of incompressibility
of the material during plastic deformation leads to the condition that e\\ = 0.
The plastic behavior under complex multiaxial loading conditions is described by
invoking constitutive laws which relate either the total strain to current stress state
(so-called deformation theories) or the increment of plastic deformation to the stress
and strain increment for a given state of the material (so-called flow or incremental
theories). Any general theory of plasticity has the following major components:
(1) A yield condition, which specifies the onset of plastic deformation for dif-
ferent combinations of applied stresses.
1.4 Continuum mechanics 23

(2) A hardening rule, which prescribes the work hardening of the material and
the change in yield condition with the progression of plastic deformation.
(3) A flow rule, which relates the increments of plastic deformation or the
components of plastic rate-of-deformation to the stress components.
Plastic deformation is determined by a yield condition which is a function/(cr^) of
the current stress state. In most cases, the associative flow rule is used which assumes
that the plastic strain increments are proportional to a function/(o^-) which depends
on the current plastic state of the material. All rate-independent plasticity theories
postulate that the material response is elastic for fiery) < 0- Elastic unloading occurs
from a plastic state when fiery) = 0 and (df /'dcry)dcry < 0, where day is the stress
increment. Plastic deformation occurs when fiery) = 0 and idf/der^doy > 0. If the
material deformation is isotropic, the yield function is an isotropic function of stress
such that/((jzy) = / ( / ! , 72, 73) =/(o r i, er2, a3)- Since the deformation of metallic mate-
rials is insensitive to moderate levels of hydrostatic stress <JH, the yield function
depends only on the deviatoric stress Sy. If the yield response of the material is the
same in tension and compression, i.e. if the material does not exhibit the Bauschinger
effect (Chapter 3),f(Sy) =f(—Sy). T h e n / is an even function of/ 3 .
The von Mises and Tresca yield conditions are the most widely used flow criteria
for metals. The von Mises condition states that
f = j2-k2= \sySij - k2 = 0, (1.21)

or, in terms of principal stresses,

2
\ ? ^i) 2 ] - k2 = 0. (1.22)
k is a constant for an elastic-perfectly plastic solid and it is influenced by the prior
strain history in a work-hardening material. Since J2 is a measure of the distortional
energy for an isotropic material, implied in the von Mises yield condition is the
assumption that plastic flow occurs when the distortional energy reaches a critical
value. For uniaxial tension, ox = cry, a2 = cr3 = 0, and k = a y / \ / 3 , where ay is the
tensile yield stress. For pure shear, ax = —cr3 = r y , a2 = 0, and k = r y, where ry is the
shear yield stress.
The Tresca yield condition states that the material will yield when the maximum
shear stress reaches a critical value:
\\ox-a,\=k, (1.23)
where the principal stresses are arranged in the order ox > a2 > cr3. For uniaxial
tension, the Tresca condition predicts that k = a y /2.
The yield condition fiery) = 0 for an isotropic material is represented in three-
dimensional stress space with the principal stresses a1? o2 and cr3 as the coordinate
axes. In this principal stress space, Fig. 1.4(tf), the shape of the yield surface repre-
sentative of the von Mises yield condition, Eq. 1.22, is a right circular cylinder, while
the Tresca yield condition, Eq. 1.23, is represented by the surface of a regular
hexagonal prism. The surfaces are parallel to the hydrostatic stress line OG, which
24 Introduction and overview

(b)

Fig. 1.4. (a) The yield surface drawn in three-dimensional principal stress space. The von Mises
and Tresca conditions for yield are represented by the right circular cylinder and the inscribed
hexagonal prism, respectively, (b) von Mises ellipse and Tresca hexagon for a state of biaxial
stress.

denotes the condition <j\ = a2 = cr^.'f The surfaces of the von Mises cylinder or the
Tresca hexagonal prism are perpendicular to the deviatoric plane or IT plane (such as
the plane ABCDEF in Fig. \A(a)), which describes the condition ox + <J2 + <r3 = 0.
In a biaxial stress state represented by the principal stress coordinates ox and cr2, the
von Mises yield condition is represented by an ellipse and the Tresca condition is
shown by the inscribed hexagon, Fig. 1.4(6). Since net work has to be expended on
the body during plastic deformation, the rate of energy dissipation is nonnegative,
such that
> 0. (1.24)
Geometrically, this condition implies that the yield surface must be convex. Thus the
yield condition requires the stress point to be on the yield surface and to be directed
outward from the surface. At a smooth point on the yield surface, Eq. 1.24 implies
that the incremental plastic strain vector de?- must be normal to the yield surface, and
that the vector denoting the incremental change in stress day must have an acute
angle with the strain vector. At a corner on the Tresca yield surface, this criterion
must be applied separately to the two surfaces intersecting the corner.
This normality of the plastic strain increment to the yield surface is reflected by the
flow rule,

del = dX^-, (1.25)

where dk is a positive scalar that can be taken to denote an effective stretching. By

analogy with Eq. 1.14,/ can be regarded as a plastic potential.

' This is so because if the stress state characterized by (au <J2, cr3) lies on the yield surface, so does
fa + crH, <r2 + <rH, <T3 + <7H), where <xH is any value of the hydrostatic stress defined in Eq. 1.17.
1.4 Continuum mechanics 25

Given the representation of the flow criterion in terms of the yield surface, it is
important to know how the yield surface changes during plastic deformation. The
theory of perfectly (or ideally) plastic solids assumes that the yield function is unaf-
fected by plastic deformation. If the material exhibits increasing resistance to plastic
deformation with plastic straining, the simplest approach to handle such strain hard-
ening is to invoke the so-called isotropic hardening model. During isotropic hard-
ening, the yield surface expands uniformly, but it has a fixed shape and its center
remains fixed in stress space. The dependence of the size of the yield surface on
deformation can be determined by developing a universal stress-strain relationship

(1.26)

which relates two scalar quantities through the function h: the effective stress ae
(which measures the size of the yield surface) and the effective plastic strain incre-
ment dep. If one uses the von Mises criterion for yield,

(1.27)
The numerical constant on the right hand side of this equation is chosen such that
ae = \an\ in uniaxial loading. The effective plastic strain increment is defined as
(1.28)
where the numerical factor is chosen such that in a state of uniaxial stress crn,
dep = devn = -2de p 2 = -2de p 3 .
An alternative statement of the isotropic hardening rule is obtained from the
argument that the size of the yield surface is a function F only of the total plastic
work, such that

= F(wp), wv = crjjdel., (1.29)

where the integration is carried out over the actual strain path. This condition
provides results equivalent to Eq. 1.26.
The use of isotropic hardening for tension-compression cyclic deformation in
many metallic materials does not rationalize the differences in elastic limit commonly
found between forward and reverse loading. In an attempt to account for this so-
called Bauschinger effect, an alternative hardening rule, known as kinematic hard-
ening, has been proposed. In the classical models of kinematic hardening, the yield
surface does not change its shape and size, but simply translates in stress space in the
direction of its normal. The application of isotropic hardening and various kine-
matic-type hardening rules to cyclic deformation is described in Chapter 3.
The discussions up to this point have focused on the incremental ox flow theories
of plasticity. There is also a different approach, known as the deformation or total
strain theory, which is adopted in plasticity problems mainly in view of its mathe-
matical simplicity. The deformation theory, where the total strain ezy is taken to be a
26 Introduction and overview

function of the current stress, is merely a nonlinear elasticity theory. It is assumed

here that
€* = vSij, (1.30)
where v is a positive scalar function during loading and a negative scalar function
during unloading. From Eq. 1.30, it is readily seen that
v =\^. (1.31)
ae takes the value given in Eq. 1.27 and ep = J(2/3)e?-e?-. This can be used in con-
junction with a universal stress-strain relationship of the form given in Eq. 1.26.
Under proportional loading (i.e. for loading in which the components of the stress
tensor vary in constant proportion), the incremental and deformation theories of
plasticity exhibit agreement, while substantial differences are encountered between
the two approaches for severely nonproportional loading.

1.4.4 Elements of linear viscoelasticity

Deformation which exhibits such features as stress-strain hysteresis, stress
relaxation, creep, or dynamic response to stresses which fluctuate sinusoidally with
time represents a material behavior which is both elastic and viscous. Such a defor-
mation is known as viscoelastic. Examples of viscoelastic behavior are commonly
found in high polymers.
Continuum models characterizing the viscoelastic constitutive behavior commonly
involve series and parallel arrangements of springs and dashpots. Figures l.5(a)-(c)
show three simple examples of such models where the viscoelastic solid is subjected
to a force F. The Maxwell element, Fig. 1.5(a), comprises a series arrangement of a
linear spring, which produces an instantaneous displacement u in response to F, and
a dashpot, whose velocity u is instantaneously proportional to F. The Kelvin-Voigt
element is a linear spring and a dashpot in parallel, Fig. l.5(b). The spring constant
and the coefficient of viscosity of the dashpot are denoted as /C and rj, respectively.
These idealized models qualitatively capture the relaxation, decay and creep phe-
nomena, although their quantitative predictions can deviate considerably from the
behavior of real viscoelastic materials. Further refinements to these models can be
made by incorporating additional elements. Figure 1.5(c) shows an example of a so-
called standard linear solid where an extra spring is placed in parallel with a Maxwell
element.
The applied force F and the displacement u at the point of application of the force
are related in the following way.

u=^ + -, F(0) = /C«(0), for the Maxwell model, (1.32)

F = Ku + w, w(0) = 0, for the Kelvin-Voigt model, (1.33)

Here, (t) after a variable denotes the value of the variable at time t, and (') denotes
the rate of change with t.
1.4 Continuum mechanics 27

Maxwell model ICelvin-Voigt model Standard linear model

rj V K
\ \|
\ V K \ fT
\ —[T WSAr—i+ F \ LL
\ \ K *F \ V 1———^te F
\ \ \ AAAAf J
\ \ \

Creep response

u u u

r
t t t
F\ F F
1

-11 i t
0
i i t
1 t

Relaxation response

F F F

K t t t
u
1
L
u

u0 fL Tor
u

(a) (b)
Fig. 1.5. Schematic arrangement (top row), creep response (middle row) and relaxation response
(bottom row) of (a) the Maxwell element, (b) the Kelvin-Voigt element, and (c) a standard
linear model. The creep response includes loading and unloading.

The Maxwell element thus characterizes steady creep under a constant load FQ
after an initial glassy response with the displacement Fo/K, in the following manner:

M(f) = ^ +^ . (1.34)

If a displacement u(t) is suddently applied such that u{i) = 0 when / < 0 and
w(/) — u0 when ^ > 0, stress relaxation characterized by the Maxwell solid takes
the form:
F{t) = UQ. (1.35)
The ratio r = rj/JC denotes the relaxation time which is the rate of decay of the force
according to the exponential law in Eq. 1.35 after the sudden imposition of the
28 Introduction and overview

displacement produces an instantaneous reaction by the linear spring. The relaxation

time t is the time required for the force to relax to l/e times its initial value JCUQ.
The Kelvin-Voigt element does not exhibit any instantaneous elastic response or
glassy response because the linear spring and the dashpot are in parallel. If a force Fo
is imposed on the element at ^ = 0 and held fixed with the passage of time, the
solution to Eq. 1.33 for u(t) gives the retarded elasticity response:

u(t) = ^(l-e-t/Tl (1.36)

where the retardation time r is the time required for {(F0/JC) — u} to be reduced by a
factor of l/e. Equation 1.36 thus indicates that the equilibrium displacement is
approached only asymptotically.
In the standard linear model shown in Fig. 1.5(c),
ER(u + x¥u) = F + r u F, ER • r F • u(0) = ruF(0), (1.37)
where r F is a constant which denotes the time of relaxation of u under constant F,
and ru is the time of relaxation of F under constant u. With different values of the
spring constants and viscosity, different values of ER, r F and ru result, and thus the
model may be described by a constitutive equation of the form: F -\-f\F = uxu + u2u,
where/j and ut are constants. This equation typically characterizes in a qualitative
way the deformation of a cross-linked polymer.
A generalized Kelvin-Voigt model, which comprises a number of Kelvin-Voigt
elements in series, or a generalized Maxwell model, which comprises a number of
Maxwell elements in parallel, are commonly used to 'fit' the creep response of metals
and polymers with greater degrees of precision (see, for example, Malvern, 1969). An
example of the transient viscoelastic response of polymers subjected to cyclic varia-
tions in applied stress is presented in Chapter 6.

1.4.5 Viscoplasticity and viscous creep

A fluid exhibiting Newtonian viscous behavior develops a straining rate
which is proportional to the applied shear stress, from the very onset of the applica-
tion of the stress. On the other hand, a viscoplastic (or Bingham-plastic) material
resists straining until a critical value of the shear stress is reached, beyond which the
rate of plastic straining is in proportion to the stress. Sour dough, clay, and paste are
materials which provide examples of such viscoplastic response.
Consider the case of simple shear with oxy = ayx = r and exy = ecyx = ec; all other
components of stress cr^ and creep strain rate eQy vanish. In this case of the so-called
Bingham material (Bingham, 1922), the onset of straining occurs when the absolute
value of r exceeds a critical value k, such that

= 0, if/„ < 0; 2 ^ c = / y r , if/„ > 0; /„ = 1- — . (1.38)

1.5 Deformation of ductile single crystals 29

Here, r\ is the coefficient of viscosity and/ y is a yield function. Assuming incompres-

sibility, Hohenemser and Prager (1932) generalized Bingham's model to the general
state of stress where

^ = 0, if/y < 0; 2n§ =fySij, iffy > 0; / y = ( l - -^j. (1.39)

Recall that stj is the component of the deviatoric stress defined in Eq. 1.19, and that
J2 is the second invariant of the deviatoric stress, Eq. 1.20. Further extensions of this
approach, including the assumption of compressibility, have been introduced in the
literature.
Most commonly, tensile tests are carried out at different strain rates and tempera-
tures to characterize the viscoplastic creep response of materials. The results of such
tests are commonly analyzed using empirical approximations of the form:
a = o,(ec)m{tC)\ (1.40)
c c
where a is the uniaxial tensile stress, e is the viscous part of the true strain, e is the
viscous strain rate, m is the strain-rate sensitivity parameter, n is the strain hardening
exponent, and a0 is a material parameter. In steady-state creep, where the strain rate
remains roughly constant, the constitutive response is given by the so-called Norton-
Odqvist law:

4=
3
2
where crQ is the von Mises effective stress, Eq. 1.27, ay is the yield strength, ey is the
yield strain rate and nc is the power-law creep exponent.
Since creep deformation is a thermally activated process, the mechanistic origins
of the evolution of strain rate are tied to an Arrhenius-type equation,
€c = Ae-(Q/RT)^ ( 1 4 2 )

where Q is the activation energy, T is the absolute temperature and R is the universal
gas constant. Note that Eq. 1.42 essentially denotes the temperature-dependence of
viscosity t] which is implicit in Eq. 1.38.

1.5 Deformation of ductile single crystals

The most comprehensive mechanistic details of cyclic deformation and fati-
gue fracture have been obtained on ductile single crystals. In an attempt to develop
some background information for the discussions of fatigue in metallic crystals (to
be presented in the next two chapters), this section focuses on the terminology and
methods used to describe the deformation of ductile single crystals. Further details
on this subject can be found in any introductory text on physical or mechanical
metallurgy.
30 Introduction and overview

1.5.1 Resolved shear stress and shear strain

The anisotropic deformation of single crystals is characterized in terms of
the resolved shear stress, r R , and resolved shear strain, y R , acting on a specific slip
plane along a specific slip direction. Figure 1.6 shows a cylindrical specimen of a
single crystal of cross-sectional area A, which is subjected to a uniaxial tensile or
compressive load P. Let the slip plane normal and the slip direction be oriented at an
angle 0O a n d ^o> respectively, with respect to the loading axis, n and b are the unit
vectors normal to the slip plane and along the slip direction, respectively. Xo ( =
90° — 0O) is t n e complementary of 0O.
The load P can be resolved along b to give a shear force, Pcos Ao, acting in the slip
direction. The corresponding initial shear stress, rRo, resolved on the slip plane is the
shear force, PcosA 0 , divided by the area of the slip plane, ,4/cos0 o , so that

TR = - C O S 0o = CTCOS 0 O (1.43)

The condition for the onset of plastic deformation is given by the so-called Schmid
law which states that a crystalline solidflows plastically when the resolved shear stress,
TRQ, acting along the slip direction in the slip plane reaches a critical value, rc:
a cos 0O cos Xo = a s i n x o c o s ^ o — Mo = rc. (1-44)
M is known as the Schmid factor, which has a maximum value of 0.5 corresponding
to the orientation 0O = Xo = 45°.

Fig. 1.6. A schematic diagram of a single crystal showing the orientations of the slip plane and
the slip direction.
7.5 Deformation of ductile single crystals 31

Consider the geometrical changes in the slip system as the initially cylindrical
crystal deforms plastically (Fig. 1.7). A reference gage length vector t0 along the
axis of the cylinder prior to deformation changes in both magnitude and direction
with the progression of slip. Let I be the instantaneous gage length vector at any
point during plastic deformation. It can be shown by simple geometrical arguments
that
« = «o + K R (Vn)b, (1.45)
where • denotes a dot product. In order to express I in terms of l0, consider
I • I = l0 • l0 + yll0 • n)2 (b b) + 2yRl0 . n) V b). (1.46)
Equivalently,
I2 = l02 (1 + 2yR cos 0o cos k0 + y£ cos2 0O). (1-47)
If yR is expressed in terms of the instantaneous and original gage lengths, I and £$,
respectively, and the initial orientation of the slip plane and slip direction 0O and k0,
we find that

YR
= (1.48)

In an actual tensile test on a single crystal, the ends of the crystal are gripped to the
testing machine and the gage length vectors £0 and I are confined to remain parallel
to the initial longitudinal axis of the cylinder (i.e. along the loading axis).
Consequently, the single slip displacement schematically illustrated in Fig. 1.7 is
tantamount to the rotation of both the slip plane and the slip direction toward the
tensile axis as deformation proceeds in a real experiment. The initial orientation
angles 0O and k0 will decrease with increasing plastic deformation.
The Schmid factor also relates the increment of shear strain dy& on a slip system
to the increment in longitudinal strain de. For instantaneous orientation angles 0
and k and small strain increments,
1
^ = H- = _L = (1 49)
de tR M cos 0 cos k'
When 0 = k = 45°, M = 0.5, rR = 0.5a.
The instantaneous value of resolved shear stress, rR, may be obtained by noting
that the glide plane, with the cross-sectional area ^4/cos0o, remains undistorted by
slip. Since the longitudinal axis at any point during slip is inclined at an angle k to
the slip direction,
p
xR = —cos 0o cos k = a cos 0o cos A. (1.50)
A
From Eq. 1.45, we note that
cos A = l-b/i= yj\ - ( £ 0 s i n V ^ ) 2 . (1.51)
Combining Eqs. 1.50 and 1.51, the instantaneous value of resolved shear stress can
be written as
32 Introduction and overview

Fig. 1.7. A schematic diagram of change in the gage length vector from £0 to £ due to slip.

r R = - cos0o V 1 - < (1.52)

The measurement of the applied force P and the instantaneous length of the
crystal I during a uniaxial test producing single slip in a single crystal can thus be
converted into a plot of the resolved shear stress rR and resolved shear strain yR
using Eqs. 1.52 and 1.48, respectively. It is seen from Eq. 1.52 that rR increases with
the elongation of the crystal (for afixedvalue of the applied force P). This phenom-
enon is known as 'geometrical softening'. It is important to note that, while the slip
system rotates toward the tensile axis in a monotonically loaded crystal, fully
reversed fatigue loads do not cause any orientation change.
Figure 1.8 shows a typical stress-strain curve at room temperature for an FCC
single crystal, oriented initially for single slip and subjected to uniaxial tension. Here
the variation of the resolved shear stress rR with the resolved shear strain yR exhibits
three distinct stages. Stage I commences at the critical stress r0 after an initial elastic
deformation. This region of 'easy glide' is characterized by primary slip and by
straight and uniformly spaced slip lines. As the crystal deforms and the slip systems
rotate with respect to the loading axis, secondary slip is activated. This onset of
secondary slip and the attendant decrease in mean slip distance generally marks the
beginning of stage II where the crystal exhibits a significant increase in the rate of
work hardening. With the progression of plastic deformation in stage II, the increase
in dislocation density and the propensity for cross slip promote the formation of
Exercises 33

III

Fig. 1.8. A typical stress-strain curve for an FCC single crystal exhibiting three distinct stages of
deformation.

dislocation cell structures. With an increase in applied stress (and/or temperature),

the dislocations become increasingly more capable of circumventing (by cross slip)
barriers which were generated during the high-hardening stage II deformation. The
onset of cross slip in stage III is accompanied by a reduction in work hardening.
Materials, such as aluminum, with a high stacking fault energy (i.e. with a high
propensity for cross slip), exhibit pronounced stage III deformation even at low
applied stresses. A detailed discussion of the mechanisms of monotonic deformation
in ductile single crystals, for different conditions of crystallographic orientation, test
temperature and strain rate, can be found in any textbook on mechanical metallurgy.
Several example problems illustrating the methods to identify active slip systems in
cubic crystals are presented in the next chapter.

Exercises
1.1 The design of the twin-engine Boeing 777 aircraft was launched in the late
1980s with the objective of capturing the long-distance air travel market.
The aircraft was designed to fill a gap between the four-engine 747 with a
passenger capacity of around 400 and the twin-engine 767 with a passenger
capacity of approximately 200. During the development of the 777, Boeing
had to face competition for the 300-350 passenger 777 aircraft from the
newer generation of planes introduced by rival manufacturers, i.e. the 323-
seat MD-11 from McDonnell-Douglas which replaced the DC-10 with more
efficient engines, and the four-engine A-340 and the twin-engine A-330 from
Airbus. The airlines which worked with Boeing in the development of the
34 Introduction and overview

111 forged an agreement early on in the design that its fuselage would be
much wider than that of the MD-11, A-330 or A-340.
(a) Speculate about the implications of the larger fuselage on the strength,
weight and power requirements.
(b) Speculate about the implications of the larger fuselage on damage tol-
erance considerations and design against fatigue fracture.
1.2 A circular shaft of length / and radius a is twisted at the ends by a torque T,
which results in an angle of twist a per unit length of the shaft. The shear
modulus of the material is G.
(a) Find the magnitude of the nonzero components of stress in the shaft.
(b) Derive expressions for the strain energy density and the total strain
energy.
1.3 For an isotropic elastic solid, derive expressions for the strain energy density
in terms of the components of stress or strain tensors for (a) plane strain and
(b) plane stress.
1.4 For a homogeneous and isotropic elastic medium, show that the principal
axes of the stress and strain tensors coincide.
1.5 The octahedral plane is defined as the plane which makes equal angles with
the principal stress directions.
(a) Show that the octahedral shear stress, which is the shear stress on the
octahedral plane, is

OW = \ V Ol - °lf + 02 - V3

where <J1? o2 and a3 are the principal stresses.

(b) Show that (3/2)c^ct = J2, where J2 is the second invariant of the devia-
toric stress tensor defined in Eq. 1.20.
(c) Describe the von Mises yield condition, Eq. 1.22, in terms of the octa-
hedral shear stress.
1.6 Equations which relate the plastic strain increments to the stress deviator
components are known as the Prandtl-Reuss equations (i.e. de? = s^dk
where stj- and dX are defined in Eqs. 1.19 and 1.25, respectively).
(a) Show that when/(azy) = J2, the plastic potential equations 1.25 become
the Prandtl-Reuss equations. / is the plastic potential function defined
in Eq. 1.25 and J2 is the second invariant of the deviatoric stress tensor
defined in Eq. 1.20.
(b) Show that the Prandtl-Reuss equations correspond to incompressible
plastic deformation.
1.7 Identify the slip systems with the same Schmid factor in a face-centered
cubic crystal where the stress axis for push-pull fatigue loading is along
the [110] direction.
Exercises 35

1.8 A cubic crystal contains a screw dislocation. The Burgers vector of the
dislocation is parallel to [001]. A crack propagates rapidly in the crystal
along the (001) plane in the [110] direction. Sketch the geometry and orien-
tation of the steps formed on the fracture surface.
Part one

CYCLIC DEFORMATION
AND FATIGUE CRACK
INITIATION
 CHAPTER 2

Cyclic deformation in ductile single crystals

Studies which link the origin of fatigue damage to microscopic deformation

processes date back to the work of Ewing & Rosenhain (1900) and Ewing &
Humfrey (1903) who reported cracking along traces of active slip planes in fatigued
iron. With the invention of electron microscopes, considerable progress has been
made in developing a detailed understanding of substructural and microstructural
changes induced by cyclic straining. Research work in the past several decades has
clearly established the existence of a rich variety of fundamental mechanisms that are
specific to cyclic loading conditions. A thorough knowledge of these phenomena is
essential for microstructural design for fatigue resistance in engineering materials.
The most conclusive results of deformation mechanisms in fatigue have been
obtained on high purity materials, in particular single crystals of face-centered
cubic (FCC) metals.f In commercial materials, on the other hand, microstructural
complexities often preclude clear identification and quantitative treatment of fatigue
mechanisms. Cyclic deformation mechanisms in commercial materials are also
strongly influenced by processing methods and impurity content.
In this chapter, attention is focused on the mechanisms and micromechanics of
deformation in single-crystalline metals and alloys with FCC crystal structures. Also
included are brief descriptions of known cyclic-deformation characteristics of body-
centered cubic (BCC) and hexagonal close-packed (HCP) crystals. Cyclic deforma-
tion in some rock salt crystals is addressed in Chapter 5. This discussion is followed
by a treatment of continuum formulations for uniaxial and multiaxial fatigue defor-
mation in polycrystalline ductile solids in Chapter 3. Fatigue crack nucleation in
monocrystalline and polycrystalline metals and alloys is considered in Chapter 4.
Cyclic deformation and crack initiation in brittle solids, such as ceramics and cera-
mic composites, and in semi-crystalline and noncrystalline solids, such as polymers
and organic composites, are considered in Chapters 5 and 6, respectively. Cyclic
damage has also been extensively investigated using stress-life or strain-life
approaches where the deformation, crack initiation and crack growth events are
all incorporated within the context of phenomenological continuum descriptions;
these methods are examined in Chapters 7 and 8.

' Engineering alloys are used occasionally in monocrystalline form in some fatigue-prone structures. An
example is a gas turbine engine for aircraft in which the turbine blades are made of nickel-base super-
alloy single crystals. See Section 2.10 for the cyclic deformation mechanisms in this alloy system.

39
40 Cyclic deformation in ductile single crystals

2.1 Cyclic strain hardening in single crystals

When well-annealed FCC single crystals, suitably oriented for single slip, are
subjected to cyclic strains under fully reversed loading, rapid hardening (that is, a
rapid increase in flow stress with increasing number of cycles) is noticed even in the
initial few cycles. With continued cyclic straining, the rate of hardening progressively
diminishes and a quasi-steady state of deformation, known as 'saturation', is
reached. Once saturation occurs, the variation of the resolved shear stress r R with
the resolved shear strain yR is not altered by further cycling and the stress-strain
hysteresis loop develops a stable configuration. (See Section 1.5.1 for definitions of
the resolved shear stress and the resolved shear strain, and Section 2.3.1 for an
example problem on the identification of active slip systems.)
It is well established that plastic strains are necessary for the inducement of fatigue
fracture in ductile single crystals. Under typical cyclic loading conditions involving
low values of imposed strains, the resolved plastic shear strain amplitude yp\ is only a
fraction of the total strain amplitude yt because of reversible slip, and it decreases
with cyclic work hardening. Consequently, fatigue tests conducted with a fixed total
strain amplitude do not provide a clear description of the progression of cyclic
deformation. Instead, the most common method of performing strain-controlled
tests involves fixed amplitudes of plastic strains. Figure 2.1 shows typical hysteresis
loops associated with initial hardening and the attainment of a stable hysteresis loop
upon saturation after Ns cycles. The maximum value of the resolved shear stress at
saturation is denoted as r s at the total applied shear strain of y ts .
A nominal measure of damage accumulation under cyclic loads is the cumulative
plastic strain which is defined as

U) (2.1)

where yvXi is the resolved plastic shear strain amplitude in the zth cycle and N is the
total number of cycles. For fully-reversed straining under a fixed plastic strain
amplitude y pl , T = 4yp\N. It is important to note that the parameter F is only an
approximate measure of global fatigue damage and that it does not accurately
capture the extent of permanent damage at low plastic strain amplitudes where a
significant fraction of dislocation motion can be reversible.

2.2 Cyclic saturation in single crystals

Experiments of fully reversed fatigue under fixed amplitudes of resolved
plastic shear strain, yph point to the existence of a saturation stress after initial
(rapid) cyclic work hardening. A plot of the peak resolved shear stress at saturation,
r s , as a function of ypX provides the 'cyclic stress-strain curve' for a single crystal,
which was first experimentally measured by Mughrabi (1978). Figure 2.2 schemati-
2.2 Cyclic saturation in single crystals 41

- compression —
• \ 600 S

yC J200

fit- ^

^ - "
1100

I50
H30 J
ten sion

,
-J20 0

-y^f^ r if
0
1/ ,

1
compression -m

15f
20 f- • " '
30 / •—• '

50 / " -"" *~~

100
1 ^ ^ ^ ^
200 f ' ^ ^
600 L - - ^ ^ ^ ^
» tension
1

resolved shear strain, yn

Fig. 2.1. Schematic illustration of typical variations in resolved shear stress, r R , as a function of
the resolved shear strain, y R , for an FCC single crystal oriented for single slip and fatigued at a
fixed value of resolved plastic shear strain amplitude = ypl. T R increases rapidly during the
initial fatigue cycles. After 7VS cycles, the hysteresis loops saturate with a peak tensile stress of rs
at a total applied shear strain of yts.

cally shows such a curve for FCC single crystals oriented for single slip. Actual
stress-strain data obtained for a number of FCC metals fatigued at different tem-
peratures are listed in Table 2.1.
There are three regions, marked A, B and C, which exhibit distinctly different
strain hardening characteristics in Fig. 2.2. At low values of plastic strain ampli-
tude (ypl < YPIAB)> denoted region A, work hardening occurs during cyclic loading.
(This hardening behavior is measured in terms of the saturation stress and is
different from the rapid cyclic hardening prior to saturation, at a fixed value of
yp\, discussed in Section 2.1.) Region A is followed by a 'plateau' in the stress-
strain curve (region B). This latter regime, where the saturation stress, r*, is inde-
pendent of the plastic strain, extends until YPIBC- A further increase in ypl results in
an increase in r*, stage C.
In region A, work hardening displayed by the cyclic stress-strain curve is almost
entirely due to the accumulation of primary dislocations. Microscopically, saturation
of the hysteresis loops corresponds to a state where a dynamic equilibrium is
achieved between bundles of edge dislocations and the surrounding matrix plied
42 Cyclic deformation in ductile single crystals

** 0 vol. % PSB 100 vol. % PSB

I
veins, persistent dislocation cells
bundles slip bands
or loop (PSBs)
patches

labyrinth
structures

threshold value
for crack formation
r pi, AB

Fig. 2.2. (a) Hysteresis loops with the resolved shear stress at saturation, r R , plotted against the
resolved plastic shear strain, ypl. The stress-strain curve is drawn through the tips of stable
hysteresis loops, (b) A schematic diagram showing different regimes of the saturation stress-
strain curve.

by screw dislocations. Under these conditions, fine slip markings are observed on the
free surfaces; the specimen could withstand an infinite number of fatigue cycles
because the cyclic plastic strain does not cause progressively degenerating damage.
One of the most visible features of cyclic saturation is the localization of slip along
bands. This process is nucleated at strain amplitudes corresponding to the beginning
of region B in the cyclic stress-strain curve (Fig. 2.2) and is intensified as the applied
plastic strain is increased. Early observations, e.g., Ewing & Humfrey (1903) and
Gough (1933), showed that fatigue failure initiated as a fine crack along those bands
where slip was particularly intense. These slip lines were termed 'persistent slip
2.2 Cyclic saturation in single crystals 43

Table 2.1. Cyclic stress-strain characteristics of FCC single crystals.

Material YplAB YplBC < (MPa) Reference

4 3
Cu (523 K) 1.0 x io- 1.0 x io- 14.0 Lisiecki & Weertman (1990)
Cu(295K) 6.0 x io- 5 7.5 x io- 3 27 .5 Mughrabi (1978)
Cu(77.4K) — 8.0 x io- 3 48 .0 Basinski, Korbel & Basinski
(1980)
Cu(4.2K) — — 73 .0 Basinski, Korbel & Basinski
(1980)

Cu-2.0 Al (at.%) 1.0 x 10~4 3.0 x 10"3 33.0 Wilhelm & Everwin (1980)
(295 K)
Cu-5.0 Al (at.%) — — 32.0 Woods (1973)
(295 K)
Cu-16.0 Al (at.%) — — 20.0-25.0 Li & Laird (1994)
(295 K)
Cu-2.0 Co (at.%) 3.0 x 10"4 5.0 x 10"3 27.5 Wilhelm & Everwin (1980)

(295 K)
1.0 xlO" 4 7.5xlO~ 3 52.0 Mughrabi, Ackermann, & Herz
Ni(295K) (1979)
1.0 xlO" 4 8.0 x IO-3 50.0 Bretschneider, Holste & Tippelt
Ni(293K) (1997)
7.5 x 10~5 5.0 x IO-3 20.5 Bretschneider, Holste & Tippelt
Ni(600K) (1997)
— — 12.0-16.0 Bretschneider, Holste & tippelt
Ni(750K) (1997)

Ag(295K) 6.0 x IO-5 7.5 x 10"3 17.5 Mughrabi, Ackermann & Herz
(1979)
Al-1.6 Cu (at.%) 1.5 x 10~5 1.5 x 10~3 95.0
Lee & Laird (1983)
(295 K)
Fe-llNi-16Cr-2Mo — — 59.0
Yan et al (1986)
(wt%)(295K)
Fe-19Ni-llCr-2Mo — — 59.0 Kaneko, Morita & Hashimoto
(wt%)(295K) (1997)

bands' (PSBs) by Thompson, Wadsworth & Louat (1956) who found that in Cu and
Ni, the bands persistently reappeared at the same sites during continued cycling even
after a thin layer of the surface containing these bands was removed by electropol-
ishing. Numerous studies, including Laufer & Roberts (1966), Lukas, Klesnil &
Krejci (1968), Watt, Embury & Ham (1968), and Woods (1973), have conclusively
44 Cyclic deformation in ductile single crystals

demonstrated that the PSBs form through the bulk of the single crystals, and that the
bands of coarse slip merely mark their egress at the specimen surfaces.
Static deformation experiments after fatigue loading (Broom & Ham, 1959) and
microhardness measurements on fatigue-induced slip bands (Helgeland, 1965) reveal
that the PSBs are much softer than the matrix. These results imply that, during
saturation in the plateau region of the cyclic stress-strain curve, essentially the entire
deformation is carried by the PSBs. In fact, the very formation of the PSBs appears
to be closely related to the occurrence of the plateau. These slip bands first appear at
Kpi ~ Yv\,AB> a n d their volume fraction,/, increases linearly to nearly 100% as ypl is
raised to a value of yPisc- For Cu, Ni and Ag, fatigued at room temperature, the
ratio of the threshold saturation stress for PSB formation (rPSB ^ r*) to the shear
modulus (G) is approximately the same (~ 6.5 x 10~4), Mughrabi, Ackermann &
Herz (1979). Figure 2.3 shows the markings exhibited by PSBs on the surface of a Cu
single crystal subjected to 15 000 cycles of fatigue at ypl values of 1.5 x 10~3 and

Fig. 2.3. PSB markings on the surface of a Cu single crystal subjected to 15 000 cycles of fully
reversed fatigue loads at two different values of resolved plastic shear strain amplitude, ypl. (a)
ypl = 1.5 x 10~3. (b) yp] = 4.5 x 10~3. Width of the specimen = 4.7 mm. (From Winter, 1974.
Copyright Taylor & Francis, Ltd. Reprinted with permission.)
2.3 Instabilities in cyclic hardening 45

4.5 x 10 3. Note the increase in the volume of the crystal covered by the PSBs due to
the increase in ypl.

2.2.1 Monotonic versus cyclic plastic strains

An approximate, but direct, comparison of strain hardening under mono-
tonic and cyclic loading can be made if the cumulative plastic shear strain amplitude
in a fatigue test is taken to be analogous to the resolved shear strain in a monotonic
tensile test. Figure 2.4 shows such a comparison for Cu. It becomes evident here that
strain hardening under monotonic tension occurs much faster than that under cyclic
loading. Whereas cumulative plastic strains T ~ 10 are necessary to achieve the
saturation stress, r* = 28 MPa (at ypl = 4.5 x 10~3), a shear strain of only 0.3 pro-
duces the same stress level in a monotonic load test. This difference becomes even
more substantial at lower values of ypl.

2.3 Instabilities in cyclic hardening

Cyclic strain hardening under a constant value of plastic strain amplitude
results in a monotonically rising stress-strain curve (Figs. 2.1 and 2.4). However, if
the single crystal is subjected to an increasing stress amplitude (ramp loading), the
resulting strain amplitude exhibits a sequence of maxima and minima (Fig. 2.5),
which were termed strain bursts by Neumann (1968). Although the absolute values
of stress amplitudes at which strain bursts take place are not well defined, their

cyclic
deformation

0 2 4 6 8
cumulative resolved shear strain

Fig. 2.4. Resolved shear stress plotted against cumulative resolved shear strain for copper single
crystals subjected to tension and fatigue. (After Wilkens, Herz & Mughrabi, 1980.)
46 Cyclic deformation in ductile single crystals

o
X

.S 4

10 20 30 40 50
resolved shear stress (MPa)

Fig. 2.5. Strain bursts observed in a copper single crystal subjected to increasing stress
amplitude at 7.1 kPa/cycle at 90 K. (After Neumann, 1968.)

relative occurrence seems to be perfectly periodic in that they appear whenever the
stress amplitude is raised by at least 11.5% within about fifty fully reversed fatigue
cycles. These strain bursts have been observed in single crystals of Cu, Ag, Mg and
Zn (Neumann, 1968) and of Cu-Al alloys (Desvaux, 1970).
The effects of loading mode on the possible occurrence of strain bursts can be
rationalized from the arguments presented by Neumann (1974, 1983). If the crystal is
subjected to sufficient fatigue cycles (at a given stress amplitude), hardening due to
the mutual trapping of dislocations progressively reduces ypl. This effect continues
until the mean free path for dislocation motion is smaller than their mean spacing.
Consequently, the probability of close encounters among dislocations decreases. If
the stress amplitude is slowly raised, the disintegration of dipoles leads to an ava-
lanche of free dislocations. Macroscopically, this process is manifested in the form of
a strain burst. At the higher stress level, the released dislocations are trapped again
during subsequent deformation.
This discontinuous hardening trend is also known to occur locally during the very
early hardening stage of a constant plastic strain amplitude fatigue test. Neumann
suggests that whether such local strain bursts can be seen macroscopically depends
strongly upon their interactions throughout the gage length of the specimen. Under
increasing stress amplitude conditions, experiments appear to show the occurrence of
strain bursts (coherently) through the entire gage length. However, coherent strain
bursts are not compatible with constant plastic strain control. Therefore, the dis-
continuous hardening behavior adds up to asynchronously or slowly varying macro-
scopic plastic strain amplitude in a strain-controlled test. If the fatigue specimens
which exhibit strain bursts are subjected to monotonic tensile deformation, they
undergo easy glide of zero strain hardening where coarse slip steps are formed
(Broom & Ham, 1959; Neumann, 1968).
2.3 Instabilities in cyclic hardening 47

23.1 Example problem: Identification of active slip systems

The discussions presented in the preceding sections and those to follow in
this and subsequent chapters in the context of cyclic deformation in ductile
single crystals require a quantitative understanding of the nature of microscopic
slip. Here, the procedures used to identify active slip systems are illustrated with
the aid of a worked example.
Problem:
Consider a face-centered cubic crystal that is subjected to tension loading.
Initially, the crystal with a circular cylindrical geometry and cross-sectional area
A, has its tensile axis oriented along [T23].
(i) Identify the slip system which is activated initially,
(ii) As plastic deformation occurs, show how the relative orientation between
the stress axis and the glide direction changes,
(iii) Calculate the applied tensile strain at which duplex slip involving the
primary and the conjugate slip systems is activated.
Solution:
(i) According to the Schmid law, the slip system which operates first is the
one for which the resolved shear stress, defined in Eq. 1.44, is a maximum.
The product cos 0O c o s ^o f ° r each of the twelve slip systems in an FCC
crystal is estimated from the 'cosine formula'. This formula states that the
cosine of the angle, 0, between the normals, (h\kili) and (h2k2l2), to two
planes in a cubic crystal is:f
, hxh2 + kfa + hh
cos 0 = — —. =. (2.2)

For example, cosine of the angle 0 between the normal to the plane (111)
and the direction [T23] is: (-1 + 2 + 3)/(Vl4>/3) = 4/(^14^3).
Similarly, cosine of the angle X between the slip direction [HO] and the
loading direction [T23] is: (—1 + 2 + 0 ) / ( A / I 4 > / 2 ) = - 3 / ( V l 4 v ^ ) .
For the FCC crystals which glide along the {lll}(110> slip system, the
three [UVW] ((110)) slip directions located on each of the four (hkl)
({111}) planes can be identified by recourse to the Weiss zone law which
states:
hU + kV + lW = 0. (2.3)
With the above information, the cosine of the angle between the slip plane
normal and the tensile loading axis, cos 0, and the cosine of the angle

More generally, if r^ and n 2 are the unit vectors normal to the two planes, then r^ • n 2 =
| n.! | - |n21 • cos</>. See Section 2.8.1 for further discussion.
48 Cyclic deformation in ductile single crystals

Table 2.2. Crystallographic geometric relationships for the FCC crystal with its
tensile axis initially along [123].

Slip (Plane)
system [Direction] Notation COS0 cos X cos 0 cos A
-3 -12
Primary A2
14V6
16
A3
1
A6
VT4V2 14V6

-3 -6
Conjugate B2
VT4V3 14V6
2 4
B4
10
B5
14V6

Cross-slip Cl 0

C3 0

C5 0
VT4V2
1
Critical (IiiXiio] Dl
14V6
12
D4

D6
14V6

between the glide direction on the slip plane and the tensile loading axis,
cos A, are determined for each of the twelve possible slip systems, as shown
in Table 2.2. For a given tensile load P and crystal cross-sectional area, A,
the last column of this table gives the initial Schmid factors, M, for the
different slip systems, and the maximum value of | cos0 • cos X\ identifies
the slip system with the highest resolved shear stress, as shown in Section
1.5.1. The last column of Table 2.2 reveals that for the given orientation of
the crystal, single slip occurs initially along the A3 slip system,
(ii) As slip deformation occurs along the A3 slip system, the axis of the
cylindrical crystal begins to rotate. Consequently, the stress axis
moves towards the [TOl] glide direction. This process is visualized most
2.3 Instabilities in cyclic hardening 49

010

Fig. 2.6. A stereographic projection showing the rotation of the loading axis, initially at [123],
towards the primary glide direction [TOl] until [112]. This rotation is marked by the arrow.

conveniently by examining the [001] stereographic projection, which is

shown in Fig. 2.6. (Basic descriptions of stereographic projection can be
found in an introductory text on mechanical metallurgy.) After this rota-
tion, when [Tl2] is reached, the stress axis lies between the zones [001] and
[Til], and both the primary and conjugate slip systems are equally
stressed.
(iii) The applied tensile strain, <?app, at which duplex slip involving the primary
and conjugate systems is activated is readily calculated with reference to
Fig. 1.7:

(2.4)
to £Q sin A
In the present case,

0 = cosZ{(T23)[T01]} = = 40.89°. (2.5)

Likewise,

= cosZ{(Tl2)[T01]} = >X = 30°. (2.6)

V6V2"
Combining Eqs. 2.4-2.6, one finds that e app = 0.31.

2.3.2 Formation of dislocation veins

Transmission electron microscopy (TEM) of cyclic strain hardening in Cu
single crystals, oriented for single slip and subjected to equal tension-compression
cyclic straining, show the following general trends.
50 Cyclic deformation in ductile single crystals

The initial few cycles of alternating strain produce dislocations that accu-
mulate on the primary glide plane (e.g., Basinski, Basinski & Howie, 1969;
Hancock & Grosskreutz, 1969).
Fully reversed cyclic loading produces approximately equal numbers of
positive and negative edge dislocations. One may envision the possibility
of frequent encounters between dislocations of opposite sign, which attract
one another. When such encounters occur over small distances, the strong
attraction between dislocations of opposite sign will trap the dislocations,
thereby creating a dislocation dipole. Only edge dislocations of opposite
sign are likely to form such dipoles because positive and negative screw
dislocations can easily cross slip and mutually annihilate (provided that
the stacking fault energy is sufficiently high). The process of mutual trapping
of edge dislocations continues until the entire dislocation arrangement
within the veins is composed of edge dislocation dipoles.
The generation of primary dislocations is a consequence of the geometrical
condition that, during fully reversed fatigue straining, there is no rotation of
the slip system with respect to the loading axis. Therefore, the primary slip
system remains the most highly stressed. Microscopy results by many
researchers, which document the absence of any lattice rotation between
adjacent channels, suggest that the average Burgers vector within the
veins is close to zero because of equal numbers of positive and negative
edge dislocations (e.g., Mughrabi, 1980). Thus, the veins do not produce
long-range internal stresses. One of the most notable distinctions between
monotonic and cyclic hardening of FCC single crystals at low amplitudes of
imposed strains is this absence of long-range internal stresses under fatigue
deformation. (The stresses due to the edge dislocation dipoles in the veins
are significantly more short-ranged (oc 1/r2) than those arising from the pile-
up of dislocations (oc l/^/r), where r is the radial distance from the core of
dislocation; see, for example, Neumann, 1983.) The absence of long-range
internal stresses has also been identified by X-ray measurements (Hartman
& Macherauch, 1963; Wilkens, Herz & Mughrabi, 1980).
With continued cycling, accumulation of dislocations occurs predominantly
in the form of mutually trapped primary edge dislocation dipoles.
These networks of edge dislocation dipoles are commonly referred to as
veins, bundles, or loop patches. The veins have an elongated shape; their
long axis is parallel to the primary dislocation lines and their cross sec-
tion normal to the long axis is equi-axed. The veins are separated by
channels, which are relatively dislocation-free and are of a size compar-
able to that of the veins. The width of the veins is about 1.5 (im in
copper fatigued at 20 °C. A reduction in temperature promotes a finer
dislocation structure.
The dislocation density within the vein is of the order of 1015 m~2 which
corresponds to a mean dislocation spacing of 30 nm. The dislocation density
2.3 Instabilities in cyclic hardening 51

within the channels is three orders of magnitude smaller, which implies that
the average dislocation spacing in the channels is comparable to the width of
the channels.
• The veins contribute to rapid hardening in the early stage of fatigue by
partially impeding dislocation motion on the primary slip system.
Increasing the number of cycles leads to an increase in both the dislocation
density within the veins and the number of veins per unit volume. This
results in an enlargement of the network of interconnected veins packed
tightly with primary edge dislocations which occupy up to 50% of the
volume of the material. In the dislocation-poor regions between the veins,
screw dislocations ply back and forth during cyclic straining.

Figure 2.7 is an example of the vein structure in a copper single crystal, oriented
for single slip under ypl control. This micrograph shows the view of the primary slip
plane which, in the notation used by Basinski, Korbel & Basinski (1980), is (111); the
primary slip vector is along [101] (which is normal to the vein structure). Primary
screw dislocations can be seen in the channels dividing the veins.

Fig. 2.7. A transmission electron micrograph of matrix vein structure in a section parallel to the
primary glide plane of a single crystal of Cu fatigued to saturation at 77.4 K. (From Basinski,
Korbel & Basinski, 1980. Copyright Pergamon Press pic. Reprinted with permission.)
52 Cyclic deformation in ductile single crystals

2.3.3 Fundamental length scales for the vein structure

The characteristic dimensions associated with the dislocation structure of
the veins can be rationalized in terms of the physical mechanisms, Neumann (1983).
The most fundamental characteristic length associated with the vein structure, i.e.
the mean spacing of the edge dislocations in the veins, can be related to the trapping
distance,rftrap,of the edge dislocations at the saturation stress, rs:
Gb
d -
"trap
where G is the shear modulus, b is the magnitude of the Burgers vector and v is
Poisson's ratio. Taking Gb = 11 Jm~2, v = 0.33 and rs = 28MPa (the maximum
value of the saturated peak stress for the vein structure) for Cu at room temperature,
one obtains dtmp < 24 nm. This compares reasonably well with the mean dislocation
spacing of 30 nm estimated from the measurement of dislocation densities.
For cyclic hardening to take place, there is a need for significant dislocation multi-
plication. In a high purity single crystal, this is only possible by the bowing out of
dislocations by a Frank-Read mechanism. In this process, the channel areas adja-
cent to the veins with a diameter equal to the Orowan-Frank-Read length

*.~l.55f (2.8,
are required. The channel diameter given in Eq. 2.8 represents the maximum distance
over which two ends of a semi-circular dislocation arc can be separated under a
resolved shear stress rs. Substituting the appropriate values for Cu, it is found that
dor & 0.6 um. This is of the order of the experimentally measured channel width of
1.5 um which separates adjacent veins in fatigued Cu.

2.4 Deformation along persistent slip bands

Winter (1974) developed a two-phase model to illustrate the mechanisms by
which plastic strain is carried by the PSBs. At equilibrium, where the stress ampli-
tude, the hysteresis loop and the volume fraction of the crystal occupied by the PSBs
are invariant during saturation (for fixed yp\), let the plastic strain amplitudes in the
PSBs and the matrix be }/PSB and ym, respectively. Noting that/ is proportional to
yph the average plastic strain amplitude may be expressed in terms of the law of
mixtures,
YPi =//PSB + (l-/)ym. (2.9)
This linear relationship suggests that yPSB and ym are constants, which are indepen-
dent of ypl. Winter argued that irrespective of the value of yph the softer phase
comprising the PSBs is fatigued at a strain amplitude of yPSB and the matrix (the
harder phase) at ym. The crystal deforms at the appropriate ypl by adjusting the
2.5 Dislocation structure of PSBs 53

relative amounts of the two phases (i.e. by changing/) within the plateau (region B)
of the saturation cyclic stress-strain curve. The parameters, ym and yPSB, in Eq. 2.9
correspond to y^AB and YPIBC> respectively, in Fig. 2.2 and Table 2.1. This line of
reasoning, albeit simplistic, provides an appealing rationale for the similarities in the
conditions that govern the formation of PSBs in single crystals of a variety of metals
and alloys.
Winter's two-phase model also has limitations (e.g., Brown, 1977). Firstly, the
model implies reversibility, contrary to the reality of a fatigue experiment where a
decrease in ypl does not lead to the disappearance of the PSBs. Secondly, the model is
not applicable for materials where fatigue loads alter the microstructure; for exam-
ple, precipitation-hardened alloys (in which cyclic straining can shear the strength-
ening precipitates) or work-hardened materials (in which fatigue softening occurs).
Thirdly, the two-phase model is not applicable to cyclic deformation involving dis-
location climb, i.e. above about half the melting point.
As the PSBs penetrate through the bulk of the crystal, the strain carried by them is
macroscopically reversible in that the local strain at the maximum value of the
applied tensile stress is identical to that at the compressive maximum (Finney &
Laird, 1975). On a polished surface, slip steps visible to the naked eye disappear
and reappear every quarter cycle. Although the surface slip steps form in proportion
to ypl, the displacements within the band are not fully reversed. This leads to the
formation of slip offsets and a rough topography within the bands which appears to
be the precursor to crack nucleation (see Chapter 4). There is considerable variation
in the thickness and distribution of slip, with the coarse PSBs consisting of narrower
slip bands known as micro-PSBs.
It is generally recognized that the PSBs are a pre-requirement for the formation of
fatigue cracks in pure crystals. Therefore, the observation of threshold stress (or
strain) for the formation of PSBs automatically implies the existence of a fatigue
(stress or strain) limit below which no cyclic failure occurs (Laird, 1976; Mughrabi,
1978).

2.5 Dislocation structure of PSBs

A PSB is composed of a large number of slip planes (~ 5000 in Cu at 20 °C)
which form a flat lamellar structure and span the entire cross section of the single
crystal. A periodic array of dislocation ladders or walls further divides the PSB
lamellae into channels, Fig. 2.8. Note that this figure is an oversimplification insofar
as the edge dislocations are not infinitely long, but are elongated dislocation dipole
loops (debris). The PSB walls, predominantly made up of edge dislocations, are
normal to the primary Burgers vector, b. The plastic deformation in the channels
in-between the walls occurs mainly by the glide of screw dislocations.
54 Cyclic deformation in ductile single crystals

gliding
screw bowing-out
, segment edge segment

;liding
screw segment

tens of micrometers

(c)

Fig. 2.8. Dislocation arrangements in FCC metals, (a) Vein structure in the matrix, (b) An
enlarged view of the dipolar walls and dislocation debris within a PSB. (c) A three-dimensional
sketch of PSB geometry formed in Cu at 20 °C. (After Mughrabi, 1980, and Murakami, Mura &
Kobayashi, 1988.)

The dislocation structure found within the PSBs is considerably different from
that of the matrix. The matrix contains, about 50% by volume, vein-like structures
consisting of dense arrays of edge dislocations. On the other hand, the PSB structure
is generated due to the mutual blocking of glide dislocations and the formation of
parallel wall (ladder) structures which occupy about 10%, by volume, of the PSBs
(Laufer & Roberts, 1966; Woods, 1973; Mughrabi, 1980). The walls are 0.03-
0.25 jim in thickness with a spacing of about 1.3 urn and they consist of dipoles.
The dislocation densities in the matrix veins and PSB walls are 1011—1012 cm"2 and
are two to three orders of magnitude greater than those in the in-between channels
containing screw dislocations. Weak-beam TEM studies of copper crystals reveal the
presence of about two-thirds vacancy dipoles and one-third interstitial dipoles of
primary edge dislocations (Antonopoulos, Brown & Winter, 1976; Antonopulos &
Winter, 1976). A schematic of the dislocation arrangements in the veins and in the
PSBs is given in Fig. 2.8.
Figure 2.9 is an arrangement of several electron micrographs showing the three-
dimensional geometry of the matrix veins and PSB structures in Cu. The (121) plane
clearly shows the ladder-like arrangement of the walls in the PSB, which are oriented
perpendicularly to the primary Burgers vector b. Figure 2.10 is a TEM micrograph
of a (121) section of a Cu crystal which shows the ladder structure in the PSB and the
matrix vein structure consisting of dislocation bundles.
2.5 Dislocation structure of PSBs 55

Fig. 2.9. A three-dimensional view, constructed from several TEM images, of the matrix vein
and PSB dislocations in Cu cycled to saturation at 20 °C at ypl = 1.5 x 10~3. The specimen
loading axis, indicated by the dashed line, is almost on the (121) plane and it makes an angle of
47° with the primary Burgers vector b. (From Mughrabi, Ackermann & Herz, 1979. Copyright
American Society for Testing and Materials. Reprinted with permission.)

The matrix veins accommodate plastic strains only of the order of 10 4, and hence
they undergo only microyielding. On the other hand, PSBs support high plastic shear
strains of the order of 0.01 and undergo macroyielding. This can involve dislocation
multiplication by the bowing-out of the edge dislocations (from the walls) and by
their transport along the channels; the screw dislocations in the channels may also
draw the edge dislocations out of the walls (see the schematic in Fig. 2.8(/?)). Figure
2.11 is an electron micrograph of a section parallel to the primary glide plane, (111),
of monocrystalline Cu at 20 °C. The primary edge dislocations bowing out of the
walls are also evident. The same sense of curvature is exhibited by dislocations of the
same sign.
Two different mechanisms have been proposed to rationalize the quasi-steady
state deformation of the saturation stress-strain curve. Deformation within the
matrix veins is believed to be accomplished by the back and forth ('flip-flop') motion
of dislocation loops that are produced by jogs during cross slip of screw dislocations
under cyclic straining (Feltner, 1965; Finney & Laird, 1975). It has been suggested by
Grosskreutz & Mughrabi (1975) that this flip-flop mechanism can accommodate
plastic strains of the order of 10~4, and hence accounts for the saturation behavior
56 Cyclic deformation in ductile single crystals

PSB

Fig. 2.10. TEM image of dislocation structures in a Cu crystal fatigued at ypl = 10 3 at room
temperature. A view of the (121) section revealing the matrix veins (M), PSB walls and screw
dislocations in the channels between the walls. (From Mughrabi, Ackermann & Herz, 1979.
Copyright American Society for Testing and Materials. Reprinted with permission.)

Fig. 2.11. TEM of a section parallel to the primary slip plane of single crystal Cu fatigued to
saturation at ypl = 5 x 10~ . Fast neutron irradiation was employed to pin the dislocations at
the peak tensile stress of the fatigue cycle. (From Mughrabi, Ackermann and Herz, 1979.
Copyright The American Society for Testing and Materials. Reprinted with permission.)
2.5 Dislocation structure of PSBs 57

in the veins either during rapid hardening or during saturation in region A. On the
other hand, a dynamic equilibrium between dislocation multiplication and annihila-
tion is considered responsible for saturation within the PSB in the plateau regime of
the cyclic stress-strain curve. Consequently, the local densities of edge and screw
dislocations are kept constant (Essmann & Mughrabi, 1979). Dislocation multiplica-
tion occurs by the bowing out of edge dislocations between the walls, whilst anni-
hilation occurs by climb of edge dislocations of opposite sign on glide planes 1.6 nm
apart in the wall structure of PSBs. Furthermore, the annihilation of screw disloca-
tions (on glide planes 50 nm apart) at room temperature is also believed to occur in
the dislocation-poor channels of the PSBs.

2.5.1 Composite model

Mughrabi (1981, 1983) proposed a composite model for fatigued crystals
containing dislocation walls and cell structures. His analysis examines the develop-
ment of long-range internal stresses as a natural consequence of deformation com-
patibility among the substructures. As large differences exist between the local
dislocation densities in the walls and channels of the PSBs, let rw and rc denote
the local flow stresses in the walls and channels, respectively. Under an imposed
far-field stress, the deformation of the composite, consisting of walls and channels,
occurs in the following sequence: (i) elastic straining of both the walls and channels,
(ii) elastic straining of the walls, plastic yielding of the in-between channels, and (iii)
plastic yielding of the channels and walls. Processes (ii) and (iii) denote plastic micro-
yielding and macro-yielding, respectively. The composite yields plastically under the
action of an applied stress r:
(2
*=/wT w +/cTc, *10)
where/ w and/ c are the area fractions of the walls and channels, respectively, in the
glide plane. If it is assumed, for strain continuity at the interface between walls and
channels, that the total strain yT is the same in the walls and channels,
YT = yPl,w + Xel,w = /pic + /el,c» ( 2 -11)
where yp\ w and yQ\ w are the plastic and elastic shear strains, respectively, in the walls,
and yp\c and y elc are the corresponding strains in the channels. From Eqs. 2.10 and
2.11 and the condition t h a t / c + / w = 1,
^w = T + / C ( T W - TC) = r + Gfc(yplc - ypl>w), (2.12)
where G is the shear modulus, and ypl c - ypl w = yel w - yel)C. Similarly,
^c = T - / W ( T W - TC) = T - G/w(Xpi,c - Kpi,w)- (2.13)
An outcome of Eqs. 2.12 and 2.13 is that at the macroscopic yield stress r, the local
stress in the walls is larger than r by
Ar w = Gfc(yplc - ypl,w), (2.14)
and the local stress in the channels is smaller than r by
Ar c = -G/W(ypi>c - y p l w ). (2.15)
58 Cyclic deformation in ductile single crystals

Equations 2.14 and 2.15 together must satisfy the condition that
/ c Ar c +/ w Ar w = 0. (2.16)
The local stresses, Eqs. 2.14 and 2.15, are long-range internal stresses that develop
due to different amounts of deformation in the walls and channels which have
different dislocation distributions. Thus, a crystal with a heterogeneous dislocation
distribution exhibits local deformation similar to that of a composite. This effect has
important implications for a wide variety of fatigue phenomena, such as cell forma-
tion (Sections 2.8 and 2.9), the Bauschinger effect (Chapter 3), cyclic slip irreversi-
bility and fatigue crack nucleation (Chapter 4).
The composite model discussed above rationalizes, in a simple manner, the
accommodation of plastic strains in the plateau regime of the cyclic stress-strain
curve. However, the development of quantitative criteria for the inception of a
fatigue crack requires detailed constitutive formulations of the inelastic deforma-
tion in the PSBs.

2.5.2 Example problem: Dislocation dipoles and cyclic deformation

Problem:
Consider an elongated edge dislocation dipole loop, which is shown sche-
matically in Fig. 2.\2{d). The length of the dipole loop is /d and the separation
distance of the dipole dislocations from glide plane to glide plane is s^.
(i) Examine the condition for the stability of the dipole loop under the
influence of an applied shear stress, by identifying the ratio, ld/sd, for
which the dislocation bowing stress, rbow, and the dipole passing stress,
r
diPoie> are just equal.
(ii) Now consider a fatigued Cu crystal whose dislocation arrangement con-
sists essentially of uniformly distributed edge dislocation dipoles. The
length of the dipoles, ld = 0.5 |im, and the glide plane separation,
sd = 20 nm. Assume that the dipole density Pdipoie = 1020m~3, and
that the cyclic plastic strains are accommodated exclusively by the dipole

I2sd
superjog
(retained from
cross slip)

(a) (b)
Fig. 2.12. (a) Schematic illustration of the dislocation dipole. (b) Glide path for dipole flip-
flop.
2.5 Dislocation structure of PSBs 59

flip-flop mechanism. Find the cyclic stress amplitude which would be

required for the operation of the flip-flop mechanism,
(iii) What would be the maximum shear strain amplitude?
Solution:
(i) The dipole passing stressf can be estimated from Eq. 2.7, from which it is
seen that
Gb
(2.17)

The dislocation bowing stress is

r
bow ^ ~T- (2.18)

Equating these two stresses, we find the condition for the stability of the
dipole loop to be

ld « 8TC(1 - v)sd, or ^ 1 5 . (2.19)

^d

(ii) Substituting G = 42 GPa, b = 0.25 nm, v = 0.33, and sd = 20 nm

(which are typical values for Cu) in Eq. 2.17, we find that the amplitude
of the stress necessary for the flip-flop mechanism is

(2 20)
^ -
This is of the order of the stresses seen for Cu at room temperature,
(iii) The total dislocation density is
P = Pdipole x k = !02° x 0.5 x 10~6 m" 2 = 0.5 x 1014 m" 2 . (2.21)
The average glide path, schematically sketched in Fig. 2.12(6), is (V2 • sd).
Since there are two dislocations in the dipole, the average glide path per
dislocation is,

Lgl = X- • V2 • sd = 0.7 x 20 = 14 nm. (2.22)

The shear strain amplitude is:

Yv\ = ±p • b - Lgl
= ±0.5 x 1014 (m~2) x 0.25 x 10~9 (m) x 14 x 10~9 (m)
= ±1.75x 10" 4 . (2.23)

' Strictly speaking, the dipole passing and dislocation bowing processes should be treated as functions of
dislocation displacement.
60 Cyclic deformation in ductile single crystals

2.6 A constitutive model for the inelastic behavior of PSBs

In this Section, we examine a constitutive model for PSBs in FCC single
crystals which is predicated upon the work of Repetto & Ortiz (1997), where a
complete list of references to prior studies on relevant crystal elasticity and plasticity
theories can be found. The formulation discussed here is amenable for implementa-
tion in finite-element simulations of fatigue crack initiation; some examples of such
simulations will be presented in Chapter 4.

2.6.1 General features

As a starting point to the analysis, assume that the plastic strains are carried
entirely within the PSBs, primarily by single glide, with the surrounding matrix taken
to be elastic. At the free surfaces where the PSBs exit the crystal, however, consid-
erations of multiple slip are necessary. Since transient cyclic deformation prior to
cyclic saturation constitutes only a small portion of the fatigue life, attention is
focused on deformation occurring after saturation. Consequently, the dislocation
densities in all the slip systems are assumed to remain constant.
The total deformation in the FCC crystal is envisioned as being composed of three
mechanisms which allow a multiplicative decomposition of the total deformation
gradient F (Lee, 1969):
F = FeFpFv. (2.24)
Here, F is the elastic part accounting for lattice distortion including rotation,! F p is
e

the plastic part accounting for the cumulative effect of slip in the PSBs (which leaves
the lattice undistorted as well as unrotated), and Fv is a factor representing the effects
of vacancy generation.
Let TR be the resolved shear stress acting in the slip system a along the direction s"
(i.e. a (110) direction) on a slip plane (i.e. a {111} plane) whose normal is nf\ If r is
the Kirchhoff stress tensor,
a
r ^ s V ; sa = ¥er and ma = F ^ V * , (2.25)
where the superscript T refers to the transpose of the tensor. The flow rule derived
from the kinematics of slip is written as (Rice, 1971):
ya r ® m01, (2.26)

where ya is the shear strain rate on the slip system a and ® denotes a vector dyadic
product. This constitutive formulation is completed by writing an equation for the
evolution of ya on the basis of the type of loading (e.g., fully reversed and yp\
controlled) and the material behavior (e.g., rate-independent plastic response).

t Details for extracting F e for FCC crystals can be found in Teodosiu (1982).
2.6 A constitutive model for the inelastic behavior of PSBs 61

2.6.2 Hardening in the PSBs

As the resolved shear stress TR causes dislocations to move within the PSB
along the primary slip system a, it is conceivable that this motion is restricted by the
mutual trapping of edge dislocations of opposite sign. The plastic hardening due to
this trapping process leads to a shear strain evolution which depends on r R as
follows:f

= bpaw\ 1 1 , (2.27)
|lP(4) j
where b is the magnitude of the Burgers vector, pa is the dislocation density in the
primary slip system, and w is the distance of separation between the walls in the PSB.
The term [1 — P(TR)] denotes the subfraction of destabilized dislocations that are
trapped by the nearest wall, and 1/[1 — P(TR)] signifies the average number of jumps
taken by a dislocation, from one dislocation wall to a neighboring wall in the PSB,
before being trapped in a stable position. P(ra) can be determined directly from
experimentally measured stable hysteresis loops of TR versus ypl, similar to the
ones shown in Fig. 2.2(a).

2.6.3 Hardening at sites of PSB intersection with the free surface

While considerations of single slip are appropriate for describing the defor-
mation within the ladder structure of a PSB, modeling of multiple slip is required at
the site of egress of a PSB at the free surface of the crystal. A possible mechanism for
hardening here is postulated on the basis of obstacles to the motion of primary
dislocations by forest dislocations. When the obstacles are randomly distributed in
the slip plane, the underlying kinetic equation can be analytically stated (Cuitino &
Ortiz, 1992) in the form of a hardening rule:

(2.28)

Here haa is the self-hardening modulus, hac = xaclyac is the characteristic hardening
modulus, xac = CGb^/nna is the characteristic flow stress (with G being the shear
modulus and C, a nondimensional constant), and y" = {bpa)/(2^/ff) is a character-
istic glide strain. The density of obstacles created by forest dislocations, na, takes the
form
(2.29)

where f3 denotes all relevant slip systems other than the primary system a. The
interaction coefficients a0^, which depend on the nature of the dislocation junctions

' Equation 2.27 is an oversimplification in that it does not account for the link between the hardening
process and the motion of screw dislocations in the channels or the increasing densities of loops and
dislocation debris.
62 Cyclic deformation in ductile single crystals

or jogs formed by the intersection of primary dislocations with forest dislocations,

have been exprimentally determined for the 12 slip systems of FCC crystals
(Franciosi & Zaoui, 1982). In the multi-slip regions of the PSB, the trapping mechan-
ism in the primary system and the forest hardening mechanism should both be
considered.

2.6.4 Unloading and reloading

The discussion up to this point deals with monotonic loading. Upon
repeated reversals of the load, the kinetic equation (Ortiz & Popov, 1982) for unload-
ing and reloading can be formulated as:
max
P(x0) = r f , , (2.30)
where rmax is the maximum resolved shear stress attained during loading and r0 is the
critical resolved shear stress for the onset of reversed slip. The function P carries the
same meaning as in Eq. 2.27. Implicit in this derivation is the notion that r0 -> 0
when the resolved shear strain ypl significantly exceeds a characteristic value, ypl c.
Thereafter, the reversed slip gradually facilitates the dissolution of the dislocation/
obstacle arrangement. Any subsequent reloading returns the deformation to con-
form to the virgin stress-strain response. During reloading, elastic deformation pre-
vails until rR -> r0. Thus, the model ostensibly accounts for the Bauschinger effect
(see Chapter 3), although its predictions have not been rigorously checked with
experimental observations.

2.6.5 Vacancy generation

Once cyclic saturation is attained in the plateau regime of the cyclic stress-
strain curve, the density of dislocations within the PSBs is maintained at an approxi-
mately constant value (~ 6 x 1015 m~2) by the establishment of a dynamic equili-
brium between dislocation multiplication and annihilation. It has been pointed out
(Mughrabi, Ackermann and Herz, 1979) that the bowing of edge dislocation seg-
ments out of the walls, at radii greater than the critical Frank-Read radius, is the
most probable mechanism for dislocation multiplication (see Fig. 2.11). Dislocation
annihilation is facilitated when two edge dislocation segments of opposite sign get
closer than the critical separation distance, ye & 1.6 nm (Essmann & Mughrabi,
1979).f Figure 2.13 is a schematic of an elongated dislocation loop in a PSB in
which the pure edge segments (which are parallel to the walls) are at the critical
separation yQ. The length of the loop L may be identified with the width of the
specimen. The reduction of the spacing y and the eventual annihilation of the

* Note that the above argument does not take into account the process of annihilation of screw disloca-
tions in the channels. In addition, care should be exercised in analyzing the annihilation process since it
may involve dislocation climb or merely a mechanical collapse due to very high local interaction stresses.
2.7 Formation of PSBs 63

Fig. 2.13. A schematic showing the annihilation of dislocation loops and the attendant
production of vacancies.

edge segments necessarily involves dislocation climb until the area of the loop Ly
vanishes by generating point defects (which, as seen earlier, are predominantly
vacancies). Vacancy generation also has the concomitant effect of promoting a
steady elongation of the PSB along the nominal slip plane, which gives rise to surface
protrusions. The rate of deformation induced by dislocation climb due to the anni-
hilation of edge dislocations is found (Repetto & Ortiz, 1997) to be
F V " 1 = cv s" OS" = (\paylya) S" ® S". (2.31)
The rate of elongation of the PSB is thus established via the vacancy generation rate
(cY) and the known slip rate. The implications of this PSB elongation process, along
with that of vacancy diffusion, to fatigue crack initiation will be considered in
Chapter 4.

2.7 Formation of PSBs

At the beginning of stage B in the saturation cyclic stress-strain curve,
structural changes must take place within the matrix to accommodate high values
of plastic strains because the dislocation veins in the matrix cannot accommodate
strains in excess of approximately 10~4. The ensuing formation of PSBs has been the
subject of a number of electron microscopy studies (e.g., Mughrabi, Ackermann &
Herz, 1979; Mecke, Blochwitz & Kremling, 1982; Jin, 1989; Holzwarth & Essmann,
1993).

2 J.I Electron microscopy observations

Holzwarth & Essmann (1993) presented a study of the mechanism by which
the matrix vein structure is transformed into the wall structure of a PSB. They
started with a saturated matrix vein structure in a Cu single crystal at ypl = 10~4,
64 Cyclic deformation in ductile single crystals

using a fully-reversed plastic-strain control test, until saturation to a cumulative

plastic strain of F = 15 at rs = 28 MPa at 300 K. The crystal was then subjected
to a sudden increase in ypl to 4 x 10~4 which caused an instant jump inflowstress to
33 MPa and which initiated the formation of PSBs. The number of fully-reversed
strain cycles at the higher ypl was varied from specimen to specimen in order to
gather information on the consecutive stages of PSB evolution. Continued cycling at
the higher ypl caused a rapid decline in theflowstress which dropped from 33 MPa to
a steady-state value of 28 MPa within 1000 cycles.
These experiments reveal that the transformation from the matrix vein structure to
the PSB wall structure most likely commences at the centers of the veins wherein
exist small areas that are dislocation-poor. These soft areas are surrounded by a
harder shell of higher dislocation density, wherein develop the first dislocation walls.
In the plateau regime, the walls shift at a rate of l-2nm/cycle, and this shift plays an
important role in establishing the typical ladder pattern in the PSBs.
Figure 2.14 shows the dislocation arrangements in a Cu crystal subjected uni-
formly to fully reversed plastic straining at yp\ = 4 x 10~4 (without a change in
ypi from a prior lower value). Here, an inhomogeneous matrix vein structure is

Fig. 2.14. The evolution of a PSB wall structure in the dislocation-poor region of the matrix
veins (marked by arrows), g = (111). yp\ = 4 x 1CT4. (From Holzwarth & Essmann, 1993.
Copyright Springer-Verlag. Reprinted with permission.)
2.7 Formation of PSBs 65

evident, with dislocation-poor interior regions in the veins. Consider the PSB in
this figure which cuts through a row of veins. From the geometry of the nascent
wall structure and the surrounding vein structure, it is noted that the walls
originate from the vein shells and that they have to move very little to establish
their spacing during PSB evolution. The destruction of veins is seen to begin
preferentially in the dislocation-poor regions where each vein forms two walls.
The scenario emerging from Fig. 2.14 is essentially the same as that put forth for
Ni by Mecke, Blochwitz & Kremling (1982), and for Cu by Jin (1989).

2.7.2 Static or energetic models

A quantitative description of the formation of dislocation structures within
the veins and PSBs can potentially be obtained from calculations of the equilibrium
positions of finite populations of dislocations. Although theoretical models of the
evolution of PSBs are in their infancy, the few results available to date appear to
provide some fundamental justification for the experimentally observed geometrical
arrangement of dislocations in fatigued crystals. Detailed numerical simulations are
necessary to establish a link between the conditions for the stability of dislocation
structures and the geometries of dislocation arrangements that develop during fati-
gue. The basis for the earlier models (such as the low-energy dislocation structures
(LEDS) model of Kuhlmann-Wilsdorf, 1979) was the Taylor-Nabarro lattice
(Taylor, 1934; Nabarro, 1952). This lattice consists of regular arrangements (of
parallel straight edge dislocations) extending to infinity. The infinite Taylor-
Nabarro lattices cannot explain the existence of dislocation-free channels between
the veins and the sharp transition in dislocation density from the veins to the in-
between channels.
It has been shown by Neumann (1983, 1986) that finite sections of Taylor-
Nabarro lattices of certain shapes are stable. Neumann has considered a regular
planar array of positive and negative edge dislocations (say, on the x-y plane)
with the dislocation lines extending along the z direction. The Burgers vector of a
dislocation is then given by b = (fs6, 0, 0), fs = ±1, b > 0. The position of the dis-
location, with respect to the origin of the coordinate system, is described by the
position vector r; = r(Xj, yj), j = 1,..., n, such that the slip plane of theyth disloca-
tion is y = yj. When a stress ra is applied, the force per unit length on the ith
dislocation is given by

(2.32)

(2.33)
66 Cyclic deformation in ductile single crystals

For mechanical equilibrium, all dislocation structures must be relaxed, i.e. the net
stress on a dislocation line (due to the applied stress and due to all the other dis-
locations) must be zero. This can be accomplished by shifting the dislocations within
their slip planes (i.e. with yjfixed)into appropriate positions Xj. Mathematically, this
process reduces to equating the right hand side of Eq. 2.32 to zero, which results in n
nonlinear equations for the n unknowns Xj. In this way, the stability of veins can be
modeled by using various shapes of the finite Taylor-Nabarro lattices as the starting
configurations.
Figure 2.\5{a) shows such a relaxed configuration of a diamond-shaped section of
Taylor-Nabarro lattice in which all the dislocations are at their equilibrium positions
at zero applied stress. It is found that, for this section, the application of a stress
results in the emergence of dipolar walls of dislocations from the polarization of the
initial configuration, Fig. 2.\5{b). Figure 2.15(c) shows the equilibrium configuration
of a wall structure of dislocations. Here the ratio of wall spacing to wall height is of
the order of unity, in concurrence with the experimental observations of ladder

W
1T1T1T1
1T1T1T1T1T1
ITlTlTlTlTITiTl
ITiTlTATiTlTlTlTiTl
1T1T1T1T1T1T1T1T1T1T1T1
1T1T1T1T1T1TIT1T1T1T1T1T1T1
ITITITITITITITITITITITITITITITI
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T1T1T1T1T1T1T1T1T1T1T1T1T1T1T1T1T
T1T1T1T1T1T1T1T1T1T1T1T1T1T1T
T1T1T1T1T1T1T1T1T1T1T1T1T
T1T1T1T1T1T1T1T1T1T1T
r1T1T1T1T
LT1T
T1T1T1T1T
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(b) XT 1
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TIT IT IT I T IT IT IT IT IT IT IT IT IT IT IT IT
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T IT IT IT IT IT IT IT IT IT IT
T IT IT IT IT IT IT IT IT
T IT IT IT IT IT IT
T IT IT IT IT
T IT IT
T

(c) IT XT XT XT XT
XT XT XT XT JET

XT -I P XT XT XT
XT XI XT XT

Fig. 2.15. (a) and (b) show equilibrium states of finite Taylor-Nabarro lattices at r a = 0 and
r a = 0.425Tmax> respectively, (c) is the computed equilibrium structure of dipole walls at
Dy = bd/lp. Figure (a) is an example of a 9 x 9 quadrupole structure. Figure (c) is an example
of walls (parallel columns) each of which consists of four dipoles. (From Neumann, 1986.
Copyright Elsevier Sequoia S.A. Reprinted with permission.)
2.7 Formation of PSBs 67

structures shown in Fig. 2.10. For the arrangements seen in Fig. 2.15(c), the dipole
strength is bd/lp, where d is defined in Fig. 2.15(c) and the inset in Fig. 2.16. (It
should be noted that for an isolated dipole, the dipole strength is defined by the
relationship Dy = b(y+ — y~), where y+ is the y coordinate of the positive dislocation
and y~ is the y coordinate of the negative dislocation in the dipole, and b is the
magnitude of the Burgers vector.)
When the applied stress ra reaches a certain critical value, r max , the initial disloca-
tion configuration (made up of multipoles) begins to decompose into dipolar walls.
Figure 2.16 shows the variation of the decomposition stress, r max , calculated from
the foregoing numerical simulation, as a function of the total number of dislocations
used in the model. In this figure, r max is normalized by the decomposition stress, t max ,
of the most narrow elementary dipole of the configuration, with the separation
distance between the dislocations in the dipole being d in a direction normal to
the slip plane. r^ ax (oc b/4d) is the passing stress of two dislocations (say, the zth
and &th dislocations which have opposite signs, with yt — yk = d). The veins in this

2.0

compressed
walls of 2Nd dipoles

1.5 pulled apart

o
ex 1.0
o diamonds of 1 x Nd quadrupoles
8

0.0
0 2 4 6
size of dislocation configuration, NA

Fig. 2.16. Normalized decomposition stress, T = rmax/Tmax, v s the size of dislocation

configuration for diamond-shaped Taylor-Nabarro lattice and for the wall structure. The inset
shows an elementary quadrupole for which h = Id. The matrix vein structure is represented
here by assemblies of elementary quadrupoles, as in Fig. 2.15(a). The PSB wall structure is
modeled by columns of dipole walls, as in Fig. 2.15(c). (After Neumann, 1986.)
68 Cyclic deformation in ductile single crystals

figure are composed of elementary quadrupoles (see inset in Fig. 2.16) which make
up a diamond-shaped Taylor lattice.
Figure 2.16 indicates that the wall structure is more stable than the vein structure
(at comparable values of r max ). This result provides a rationale for the experimental
observation that, at saturation, the decomposing veins cannot rearrange into a
different vein structure with smaller d values, but transform into a ladder structure.
This numerical model, albeit limited and very simplistic in its consideration of the
arrangements of dislocations in fatigued crystals, provides encouraging possibilities
for developing a physical basis for fatigue deformation.

2.7.3 Dynamic models of self-organized dislocation structures

Attempts have also been made to develop analytical models of continuous
concentration fields of dislocations in fatigued crystals (Walgraef & Aifantis, 1985;
Aifantis, 1987). In this approach, the dislocations are represented only by a density,
p(x, t), which is a function of space and time. The to-and-fro motion of dislocations
under a cyclic stress is modeled in one dimension as a diffusion phenomenon with a
flux term D(d2p/dx2), where D is an effective diffusion coefficient and p is the density
of dislocations. Two populations of dislocations are considered: immobile disloca-
tions of density px (to represent those within the PSB walls) and free dislocations of
density p m (to represent those within the in-between channels). Walgraef and
Aifantis arrive at the following set of coupled differential equations for densities
of trapped and free dislocations:

dt dx
^ =D m ^ + Ppl (2.34)
at dxz
In these equations, the parameter P is related to the maximum stress or plastic strain
rate, x is the dislocation glide direction, and c measures the pinning rate of freed
dislocations by immobile dislocations. The function g(p{) is a dislocation generation/
loss function whose initial value is zero and whose derivate with respect to time is
positive at low dislocation densities (i.e. when dipole density increases) and becomes
negative at high densities (i.e. when increased dipole destruction occurs). The sub-
scripts T and 'm' denote quantities corresponding to the immobile and mobile
dislocations, respectively. (The reader may consult the original references for further
details.)
Solution of Eqs. 2.34 predicts instabilities in the form of oscillations in time and
spatial patterning. These two instabilities have been related to the occurrence of
strain bursts and the formation of PSBs, respectively. However, this analysis relies
on rather global assumptions on the diffusion of dislocations and ignores the specific
dislocation geometries within dipolar PSB walls and channels.
2.8 Formation of labyrinth and cell structures 69

Differt & Essmann (1993) have proposed a dynamic model for edge dislocation
walls within a reaction-transport modeling framework. In this work, the reaction
terms are consistent with the experimentally observed properties of fatigue-
induced dislocation structures (which overcome some of the limitations of the
Walgraef & Aifantis model). Two important length scales are introduced in the
reaction terms: (i) the critical annihilation distance of a dipole under the influence
of an applied stress and (ii) the critical distance for the spontaneous annihilation
of closely spaced dipoles. The analysis shows how the walls move. Edge disloca-
tions are deposited on the walls by the moving screws. If the fluxes on both sides
of the wall are not balanced, the wall is destructed on one side and reconstructed
on the other. The wall moves until the fluxes get balanced. This mechanism
rationalizes why a freshly-formed PSB is less periodic and imperfect than a
mature one.
In summary, the dislocation arrangements in fatigue can be broadly classified
into two basic groups (e.g., Glazov, Llanes & Laird, 1995) : (i) structures in which
equilibrium is maintained, and (ii) nonequilibrium self-organized dislocation struc-
tures. The former group includes the low-energy dislocation structure (LEDS)
discussed earlier in this section and includes the Taylor-lattice calculations. The
latter includes models of the type postulated by Walgraef & Aifantis. The non-
equilibrium structures have been shown to provide a rationale for the instigation of
fatigue instabilities such as the formation of ladder structures in PSBs and strain
bursts.
We close this section by noting that several approaches have emerged to ratio-
nalize the patterning seen in fatigue by recourse to computer simulations.
Hesselbarth & Steck (1992) have confirmed the Neumann model for equilibrium
structures through two-dimensional cellular automata simulations of stress-induced
patterning from an initial random configuration of edge dislocations. Devincre &
Pontikis (1993) have studied the periodicity of dipolar walls as a function of
applied stress and edge dislocation density, while full three-dimensional simulations
of dislocation structures under cyclic stressing have been undertaken by Devincre
& Kubin (1997).

2.8 Formation of labyrinth and cell structures

Electron microscopy of fatigued Cu crystals (oriented for single slip) pro-
vide further insights into the dislocation structures formed in the range of yp\
& 10~4-10~2 (Ackermann, Kubin, Lepinoux & Mughrabi, 1984). For ypl < 10"3,
the slip band structure is characterized by the hard matrix comprising veins and
the softer PSBs with the wall structure. An increased contribution of secondary slip
(which becomes more pronounced at higher ypl) and a gradual evolution of 'laby-
70 Cyclic deformation in ductile single crystals

2 um

Fig. 2.17. A view of the (010) section of a Cu crystal that was cycled to saturation at
ypl = 5 x 10~3, revealing the labyrinth structure. ((From Ackermann et al., 1984. Copyright
Pergamon Press pic. Reprinted with permission.)

rinth' and 'cell' structures are noticed for ypl > 2 x 10 3 . Figure 2.17 is a TEM
image of a (010) section of the Cu crystal that was subjected to fatigue at
ypl = 5 x 10~3. Note the formation of a labyrinth structure with its walls oriented
parallel to the (100) direction with a mean spacing of about 0.75 |im. The labyrinth
consists of two sets of orthogonal Burgers vectors; bx and b 2 denote the primary
and conjugate Burgers vectors, respectively. Labyrinth walls are also known to be
produced during cyclic deformation of Cu-Ni alloys (Charsley & Kuhlmann-
Wilsdorf, 1981), of ionic crystals (Majumdar & Burns, 1982) and of BCC metals
(e.g., Mori, Tokuwame & Miyazaki, 1979). The occurrence of walls of labyrinth
structures has been rationalized on the basis of a geometrical argument by
Dickson, Boutin & L'Esperance (1986), who considered a three-dimensional stack-
ing of twin dislocation loops of rectangular shape. The directions in which the
stacking of the loops was geometrically most favorable was shown to be consistent
with the crystallographic orientations of labyrinth structures actually observed in a
variety of metal crystals.
Ackermann et al. (1984) have suggested the following changes at the higher ypl
end of the saturation stress-strain curve (Fig. 2.2) for FCC crystals: matrix phase
with labyrinth structure —> PSBs and labyrinth structure —> cell structure.
Secondary slip (prevalent in region C) originates at the PSB-matrix interface
and spreads in the form of an expanding cell structure which fills the PSBs. The
transformation of all the PSBs into a cell structure appears to occur after 106
cycles. This marks the beginning of secondary hardening in region C. The above
structural changes reported for Cu have also been found in Ni (Mecke, Blochwitz
& Kremling, 1982). Figure 2.18 shows an example of a cell structure formed in Cu
in regime C.
2.8 Formation of labyrinth and cell structures 71

Fig. 2.18. A view of the (121) section of a Cu crystal that was cycled to saturation at ypl
1.45 x 10~ , revealing cell structures formed by multiple slip in regime C. The primary
Burgers vector is along b. (Photo courtesy of H. Mughrabi. Reprinted with permission.)

2.8.1 Example problem: Multiple slip

Problem:
(i) A single crystal of an FCC metal, with a (001) orientation along the stress
axis, is deformed until the onset of plasticity. How many equivalent slip
systems are activated? What is the Schmid factor for these slip systems? Is
the deformation accompanied by a change of axial orientation?
(ii) Answer the above questions for a BCC single crystal with a (111) orienta-
tion which deforms on slip systems of the type {TO 1} < 111).
Solution:
(i) A repeat of the calculations, similar to those listed in Table 2.2, readily
reveals that there are eight slip systems for this case with
M = |cos0-cosA| = l/(V3\/2) = 0.408. In other words, the slip sys-
tems A2, A3, B2, B4, Cl, C3, Dl and D4 have the same Schmid factor,
M. These eight slip systems comprise four {111} slip planes, each contain-
ing two (110) directions inclined equivalently to the stress axis [001]. The
resolved shear stress on the other four slip systems (A6, B5, C5 and D6) is
zero for this initial orientation because the four directions in the {111}
plane are perpendicular to the stress axis. Therefore, these slip systems are
not activated.!
The Schmid factor, M, can also be computed in a more concise manner,
for the slip system (11 l)[T01] for example, by ascribing the following

' I n a real experiment, however, errors in alignment and orientation may lead to the activation of slip
systems which differ from this predicted behavior. See Section 3.1 for a further discussion of this point.
72 Cyclic deformation in ductile single crystals

values: slip plane normal, n = [111], slip direction b = [101], and stress
axis, s = [001], such that

In view of the symmetry of the slip systems with respect to the stress axis,
no orientation change occurs.
(ii) For the BCC crystal, proceed similarly to the previous case. In this case,
there are six equivalent {110} (111) slip systems comprising three equiva-
lent (inclined) {110} planes, each containing two equivalent (111) slip
directions. The Schmid factor is also computed in a similar fashion for
these slip systems. For example, for the (110)[lll] slip system, assign:
n = [110], b = [111], and s = [111]. Substituting these values in Eq.
2.35 yields that M = 0.27. As in the previous case, in view of the sym-
metry of the slip systems with respect to the stress axis, no orientation
change occurs.

2.9 Effects of crystal orientation and multiple slip

As shown in Section 2.2, the cyclic deformation of FCC single crystals
oriented for single slip exhibits two prominent features: (i) the existence of a plateau
region, and (ii) the formation of persistent slip bands with their characteristic wall
structures. An extension of the fatigue mechanisms seen in monocrystalline FCC
metals to polycrystalline metals and alloys, however, requires a knowledge of cyclic
deformation in different orientations, including those involving multiple slip.
Table 2.3 summarizes the results from a number of experimental studies of the
effects of duplex and multiple slip on cyclic deformation. It is evident that the
possibility of occurrence of a plateau in the cyclic stress-strain curve, as well as
the extent of the plateau region are strongly influenced by the crystallographic
direction of the FCC crystal along which the fatigue loading is imposed. (See the
worked example in Section 2.3.1 for the identification of single and multiple slip
systems for a given crystal orientation.)
Gong, Wang & Wang (1997) have studied the effect of [001] multiple slip on the
cyclic stress-strain curve in Cu crystals. They found an absence of a plateau regime,
and in the strain range ypl = l.Ox 10~4—3.0 x 10~3, no PSBs were found to occur.
Fatigued [001] crystals comprise principally labyrinth structures of primary and cri-
tical dislocations, which can more easily accommodate multiple slip and cross slip than
the ladder structures. The labyrinths consist of two types of mutually orthogonal wall
arrangements of dislocations: one is always parallel to (001) (i.e. normal to the stress
axis), while the other is along (100) or (210) or (120). Figure 2.19 shows an example of
2.9 Effects of crystal orientation and multiple slip 73

Table 2.3. Effects of crystallographic orientation and multiple slip on the cyclic stress-
strain response of FCC single crystals at 20 °C.

Loading
Metal axis Observation Reference
Cu [001] Higher hardening rate than Kemsley & Pater son (1960)
[111] in single slip
[111] No plateau Lepisto & Kettunen (1986)
[112] Higher cyclic hardening rate Jin & Winter (1984) and Jin
[012] in multi-slip than in duplex (1983)
[122] slip
[001]
[034] Pseudo plateau than in single Gong et al (1995)
slip
[117] No plateau Gong et al (1995)
[001] No plateau, no PSBs Gong, Wang & Wang (1997)
[Oil] A clear plateau over Li et al (1998)
ypl = 1.1 x 1 0 " 4 - 7 . 2 x 10" 3 ;
small, irreversible rotation of
slip system during symmetric
tension-compression loading
which causes deformation
bands to form
Ni [001] Pseudo plateau in regime B Mecke & Blockwitz (1982)
[111]

such a structure. The interactions among different slip systems in the labyrinth and the
attendant formation of Lomer-Cottrell locks causes a much higher cyclic hardening
rate in multi-slip orientations than in single glide. The labyrinths accommodate dif-
ferent imposed plastic strains by appropriately adjusting their channel widths: an
increase in yp\ is accommodated by a reduction in the channel width of the labyrinth.
Another noteworthy feature of multiple slip during cyclic deformation is the
apparent improvement in the fatigue limit. In the [001] specimens, the fatigue
limit, denned as the critical value of ypl below which (nonpersistent) slip bands do
not form, is approximately 1.7 x 10~4, compared to 6.0 x 10~5 for single slip fatigue
of Cu. Further implications of such multiple slip for fatigue deformation and for the
cyclic response in polycrystals will be discussed in the next chapter.
74 Cyclic deformation in ductile single crystals

b=(a/2)[101]

Fig. 2.19. A transmission electron micrograph of the labyrinth structure formed in a single
crystal of Cu fatigued in equal tension-compression along the [001] stress axis at ypl = 1.8 x
10~3 at room temperature. The TEM foil is parallel to (T20). The symbols 'p' (for primary)
and 'c' (for critical) denote the projections of [101] and [101] screw dislocation segments on
the (120) plane along the [425] and [425] directions, respectively. (From Wang, Gong &
Wang, 1997. Copyright Elsevier Ltd. Reprinted with permission.)

2.10 Case study: A commercial FCC alloy crystal

Face-centered cubic Ni-base alloys find important technological applications
wherein high temperatures (up to about 1100 °C) are encountered. In some of these
components where essentially uniaxial stresses exist, as in the turbine blades of aircraft
jet engines, it is commonly known that the elimination of transverse grain boundaries
promotes substantial improvements in creep and fatigue lives. As a result,
directionally solidified and single crystalline superalloys are widely used as materials
for turbine blades in gas turbine engines.f
Nickel-base superalloys typically contain a solid solution of 5-20 wt% Cr for
enhancing oxidation resistance. In addition, Al and Ti (1-5 wt%) are added to increase
strength. Precipitation hardening results in the formation of an ordered Ni3(Al,Ti) y'
phase which has the Ll 2 structure. The volume fraction of the / phase can be varied over
the range 20-70% by controlling the Al and Ti concentration, while its size and shape can
be closely controlled by thermomechanical treatments. The excellent strength of Ni-base
superalloys at elevated temperatures is attributable to the presence of / precipitates. In
addition, elements such as Fe, Co, Mo, Nb, W and Ta, are added to monocrystalline
Ni-base alloys to solid-solution-strengthen the y and y phases.
In this section, we consider aspects of the microscopic cyclic deformation of a Ni-base
superalloy ((001) orientation for loading). The concentration, in wt%, of this alloy,
commercially known as PW (Pratt & Whitney) 1480, is Al (4.8%), Ti (1.3%), Ta (11.9%),

Single crystal Ni-base superalloys are also used as blades in the turbopump of the Space Shuttle main
engine.
2.10 Case study: A commercial FCC alloy crystal 75

Fig. 2.20. TEM micrograph of the initial microstructure showing the y/y precipitate
structure, g = {200}. (From Milligan & Antolovich, 1987. Copyright Metallurgical
Transactions. Reprinted with permission.)

Cr (10.4%), Co (5.3%) and W (4.1%), with the balance being Ni; the C content is 42
ppm. Figure 2.20 is the initial TEM micrograph showing the y/y precipitate structure of
this alloy prior to mechanical testing. This example also illustrates the evolution of
dislocation networks and stacking faults as a result of uniaxial cyclic loading and of
the interactions between dislocations and precipitates at different temperatures. The
results summarized here are from the work of Milligan & Antolovich (1987, 1991).
Distinct changes occur in the monotonic and fully-reversed cyclic deformation of the
PW 1480 alloy crystal in different temperature regimes. Orientation dependence of the
critical resolved shear stress in the family of {111} (110) slip systems, tension-compression
asymmetry as well as anisotropic strain hardening result from a highly anisotropic
octahedral slip, at temperatures typically below 760 °C. This trend has been attributed to
the ease of cube cross-slip and to a multitude of active slip systems. Above about 800 °C,
however, deformation is mostly isotropic, with the mechanism being largely governed by
the by-pass of y' particles by the dislocations which causes pronounced wavy slip. This
behavior is strain-rate sensitive, with increasing strain rate extending slip planarity and
anisotropic deformation of higher temperatures.
Figure 2.2\(a) shows a typical dislocation structure of PW1480 subjected to 6 cycles of
fully reversed strain-controlled fatigue. Dislocations and stacking faults with the y'
precipitates on the primary {111} slip planes are seen. The dislocation pairs (marked by
arrows) were determined to be of primary a/2(l 10) Burgers vector. Figure 2.2\{b) is from
the same area, with the primary slip plane viewed edge on. It is seen here that (i) the
dislocation pairs in the matrix are ostensibly in the same slip bands that contain the
faulted dislocation loops within the precipitates, and that (ii) the slip deformation is
highly planar. These micrographs also imply that the deformation mechanism involves
the shearing of the / precipitates by dislocations with the primary Burgers vector.
76 Cyclic deformation in ductile single crystals

R#

Fig. 2.21. Dislocation substructure in the superalloy crystal after six fully reversed cycles at
20 °C with ypl = 1.1 x 1013, and ypl = 8.7 x 103 s" 1 . (a) A bright-field TEM micrograph
showing stacking faults and faulted loops in the precipitate, and matrix pairs (arrows)
between the precipitates, b = {200}. (b) A bright-field micrograph of the same area looking
edge on. The planar nature of the structure is evident, g = {111}. (From Milligan &
Antolovich, 1991. Copyright Metallurgical Transactions. Reprinted with permission.)
 0.2 \im

Fig. 2.22. / precipitates in the PW1480 Ni-base superalloy sheared by {lll}a/2(110>

dislocations after a plastic strain of 0.08% at 705 °C. (a) A bright-field TEM micrograph.
g = {Til}, (b) A dark-field micrograph using the (030) superlattice spot, b denotes the
Burgers vector of the dislocation which is oriented at 45°. Dislocations are invisible here
because the g • b = 0 criterion is satisfied. (From Milligan & Antolovich, 1987. Copyright
Metallurgical Transactions. Reprinted with permission.)

Fig. 2.23. Dislocations by-passing the y precipitates after 765 fatigue cycles at
ypl = 2.3 x 10~3 and ypl = 8.7 x 10~3 s"1 at 1093 °C. g = {200}. (Photograph courtesy of
W.W. Milligan. Reproduced with permission.)

77
78 Cyclic deformation in ductile single crystals

Faulted dislocation loops are left behind by this shearing process as deformation debris.
A more direct evidence for such shearing is available in Figs. 2.22{d) and (b) which show
the shearing of a / precipitate by {111} a/2 (110) dislocations after a plastic strain of
0.08% at 705 °C.
Figure 2.23 shows an example of the dislocation by-passing of / precipitates in the
same alloy subjected to 765 fatigue cycles. This process essentially homogenizes the
macroscopic fatigue deformation response which becomes isotropic in tension and com-
pression.

2.11 Monotonic versus cyclic deformation in FCC crystals

There are some similarities between the dislocation structures discussed
above for cyclic deformation and those known to develop during uniaxial, mono-
tonic tension in FCC single crystals. At very low ypl (< 5 x 1CT4), the dislocation
configurations first generated during rapid cyclic hardening correlate well with the
substructures found during Stage I deformation of FCC monocrystals in monotonic
tension, with the exception that the matrix 'veins' seen in fatigue are akin to the cell
structures in Stage II of monotonic tension. At higher strain amplitudes, the disloca-
tion structures in the cyclic work hardening stage are similar to the unidirectional
Stage I configurations only during the first few cycles. The similarity then disappears
with the progressive formation of persistent slip bands. At high plastic strain ampli-
tudes, corresponding to regime C of the cyclic stress-strain curve, the formation of
cell structures and the associated rapid hardening during the early fatigue cycles finds
an analogy in Stage II deformation of FCC single crystals in unidirectional tension.
The formation of PSBs at the onset of saturation is somewhat analogous to coarse
slip band development during Stage III deformation of FCC single crystals in mono-
tonic tension.
There are also some significant differences between the dislocation structures
developed during monotonic and cyclic deformation:

• The density of dislocations produced during cyclic loading is significantly

higher than that generated, at comparable stresses, during monotonic
tension.
• During monotonic tensile deformation of a single crystal, both the slip
plane and the slip direction rotate toward the tensile axis (see Section
1.5). However, there is no such orientation change during fully reversed
cyclic loading of the crystal. This results in a preponderance of primary
dislocations and the absence of long-range internal stresses during cyclic
hardening.
• The evolution of persistent slip bands with their wall structure of edge
dislocations is specific to cyclic deformation.
2.12 Cyclic deformation in BCC single crystals 79

extrusion
intrusion

(a)

Fig. 2.24. (a) A series of steps resembling a staircase pattern produced by monotonic plastic
strain, (b) A rough surface consisting of hills and valleys produced by cyclic plastic strain.

A striking feature of fatigue deformation is the establishment of a saturated

state where the peak resolved shear stress is independent of the plastic shear
strain amplitude.
Because of the short-range interactions among dislocations during to-and-
fro motion in fatigue, there develops a high density of point defect clusters
(~ 1015 cm"3). The resultant friction stress can be as high as 25% of the
peak shear stress at 20 °C.
The flow stress of FCC crystals exhibits a stronger dependence on tempera-
ture and strain rate in fatigue than in tension.
One of the most visible distinctions between monotonic and cyclic deforma-
tion is in the development of surface roughness. Whereas monotonic loading
leads to the formation of surface slip steps which resemble a staircase geo-
metry, cyclic deformation produces sharp peaks and valleys (known as
'extrusions' and 'intrusions', respectively) at sites where the persistent slip
bands emerge at the specimen surface (Fig. 2.24). (Recall that the cumula-
tive plastic strains in fatigue are significantly greater than those in mono-
tonic tension, Fig. 2.4.) Details of the formation of extrusions and intrusions
are discussed in Chapter 4.

2.12 Cyclic deformation in BCC single crystals

Pure BCC crystals such as a-Fe, Mo and Nb, when oriented for single slip
are known to behave significantly differently from the FCC crystals. Generally, the
core of the screw dislocation in BCC metals does not dissociate, and the particular
nature of the screw dislocation core structure in BCC induces very high lattice
friction (so-called Peierls stress). Effects such as strain-rate sensitivity, strong tem-
perature dependence of cyclic deformation, relative mobility of edge and screw dis-
locations, as well as asymmetry of slip between tension and compression are a
consequence of the special role of screw dislocations in the BCC metals.
Mughrabi, Herz & Stark (1976), Mughrabi & Wuthrich (1976) and Mughrabi,
Ackermann & Herz (1979) report different regimes in the variation of mean
80 Cyclic deformation in ductile single crystals

saturation axial stress as a function of the axial plastic strain range during cyclic
deformation of a-Fe single crystals at 295 K. At low plastic strain amplitudes
(< 10~3), essentially no hardening occurs and the cyclic strain is a manifestation
of the motion of edge dislocations only. However, at higher strain amplitudes,
deformation proceeds by the large-scale motion of edge and screw dislocations
and culminates in the formation of a cell structure; pronounced cyclic hardening
as well as changes in the shape of the crystal are observed due to the asymmetric
slip of screw dislocations in tension and compression. These distinctions between
low and high strain fatigue are peculiar to BCC crystals. Although no PSBs have
been identified in either regime of plastic strain amplitudes, ill-defined bands of
slip, which could lead to crack nucleation, have been noticed. In agreement with
these findings, TEM investigations of dislocation structures ahead of fatigue
cracks (described in Chapter 4) have identified the existence of PSBs in polycrys-
talline Cu but not in pure a-Fe.
The following differences between FCC and BCC crystals point to some causes
for the distinctions in their fatigue response: (i) At 295 K and at low plastic strain
amplitudes, thermally-activated glide of screw dislocations as well as dislocation
multiplication are strongly suppressed in BCC a-Fe. (ii) Whereas FCC metals are
only weakly strain rate-sensitive, the flow stress of BCC metals is strongly depen-
dent upon the strain rate. For this reason, the cyclic stress-strain curves for BCC
crystals should be obtained at constant values of imposed strain rates. It is
generally seen that, as a consequence of dynamic strain-ageing,! high tempera-
tures, very low strain rates and the addition of impurity atoms (such as carbon,
nitrogen and oxygen) to the BCC metal promote cyclic damage that is more
similar to that found in FCC metals. Mughrabi, Ackermann & Herz (1979)
report that the addition of 30 weight ppm carbon to a-Fe single crystals leads
to cyclic stress-strain curves closer to those measured for FCC metals and slip
bands analogous to the PSBs. PSBs are also known to form in both the surface
and interior grains of polycrystalline low carbon steels (Pohl, Mayr &
Macherauch, 1980). These results show that any comparison of dislocation struc-
tures and cyclic slip characteristics reported by different investigators must be
made with caution because even a small impurity content can lead to marked
variations in fatigue micromechanisms.

2.12.1 Shape changes in fatigued BCC crystals

If slip occurs on different planes during tension and compression portions of
fatigue, a crystal must undergo shape changes due to this slip asymmetry. This effect,
observed in BCC crystals (Nine, 1973; Neumann, 1975; Mughrabi & Wuthrich, 1976;

' Dynamic strain-ageing refers to the phenomenon whereby certain materials (e.g., low carbon steels)
generally exhibit an increase in yield and fatigue strengths over certain temperature ranges as a result of
the interaction between dislocations and solute atoms (e.g., carbon and nitrogen). Details of the mechan-
isms of dynamic strain-ageing can be found in any textbook on mechanical metallurgy.
2.72 Cyclic deformation in BCC single crystals 81

Guiu & Anglada, 1980), effectively transforms an initially circular cross section of a
cylindrical crystal into an ellipse. Neumann showed that the shape change produced
by cyclic straining can be correlated with slip irreversibility, which is an important
factor for crack nucleation. Here we follow Guiu and Anglada for the derivation of
slip irreversibility due to shape changes in BCC crystals.
Consider a BCC crystal in which it is assumed that slip occurs on several different
planes in tension and compression in one direction defined by the unit vector b. Since
all these planes are parallel to b, they can be represented in terms of two basic
reference planes of the same zone with unit normal vectors i^ and n2, which can
be arbitrarily chosen. The glide strain in tension can be represented by the total glide
strain a l t in plane nx and by the total glide strain a2t in plane n2. Similarly, the total
glide strains in compression can be represented by a l c and a2c on planes nx and n2,
respectively.
In BCC crystals, any macroscopic slip plane can be visualized as being composed
of microscopic slip steps on planes of the {110} and {112} types. Assume that the
crystal is subjected to the same magnitude of plastic strain ep in both compression
and tension; the length of the crystal can then be taken to be unchanged. If the slip
direction for both cases is b, it can be shown that the net displacement after N
fatigue cycles of a point in the crystal is

lx(n 1 xn 2 )
Au(r) = 2Nyx (r.n)b,
n9
Here, r is the vector that locates the point under consideration with respect to the
origin of the coordinate system (Fig. 2.25), y\(= a l t — a l c ) is the net (irreversible)
shear strain on plane n l9 1 is the unit vector along the cylindrical crystal axis (i.e.
deformation axis), n is a vector which is perpendicular to both b and 1,
y = l^nj + y 2 n 2 | and y2 = a2t — a2c. The above result is most directly applicable
to BCC crystals which undergo shape changes as a consequence of glide along
(111). For such cases, a more convenient representation of Eqs. 2.36 is in terms of
the angles §, f, 0 t and 0 2 ; § is the angle between the tensile axis and the slip direction,
and 0 1? 02 a n < i f a r e the angles made by the reference planes n1? n2 and the maximum
resolved shear stress plane, respectively, with the plane of the same zone whose axis
is the slip direction. One finds that

(rn)b; ^
sinf 2cos§

and / g is the fraction of glide strain in plane i^ in tension which is not reversed in
compression, and e tot = 47Vep. If the initial diameter of the single crystal is do, it will
undergo a shape change into an ellipse which has major and minor diameters of d\
and J 2 , respectively, after N fatigue cycles. The following relationship is also
satisfied:
82 Cyclic deformation in ductile single crystals

Fig. 2.25. Nomenclature for the determination of shape changes induced by tension-
compression slip asymmetry during the cyclic straining of BCC single crystals. See text for
details. (After Guiu & Anglada, 1980.)

(2.38)

The slip irreversibility, Eqs. 2.36 and 2.37, is known to play an important role in
the nucleation of fatigue cracks in BCC crystals. Furthermore, microcrack nuclea-
tion at the boundaries of surface grains in polycrystalline a-Fe has also been
linked to the shape changes produced by the asymmetry of slip. Further discus-
sions of the effects of slip irreversibility on crack initiation will be provided in
Chapter 4.

2.13 Cyclic deformation in HCP single crystals

Hexagonal metals, such as Ti, Mg, Co, Zr and Be, are used in many engi-
neering applications where deformation and cracking under cyclic loads are topics of
interest. Compared to the volume of research done on the fatigue of FCC and BCC
metals, less information is available on the cyclic deformation response of hexagonal
closed-packed (HCP) metal crystals (see, for example, the earlier studies of Partridge
(1969), Stevenson & Vander Sande (1974) and Kwadjo & Brown (1978) on magne-
sium crystals).
Among HCP crystals, the most comprehensive experimental studies of cyclic
deformation have been carried out on Ti crystals. In this section, we review some
2.13 Cyclic deformation in HCP single crystals 83

results for Ti in order to compare and contrast its cyclic behavior with that of ductile
FCC metals and to identify micromechanisms which are specific to hexagonal
metals.

2.13.1 Basic characteristics of Ti single crystals

Titanium single crystals exhibit the following prominent features:
• Upon cooling, the BCC(/f) -> HCP(a) transformation occurs in Ti at
approximately 880 °C.
• As with BCC crystals, the deformation characteristics of Ti crystals are
strongly influenced by impurity and interstitial content.
• The stacking fault energy (SFE) of Ti is about 0.15 Jm~2 (as compared to
0.04 Jm~2 for Cu). Although the possibility of formation of stacking faults
in Ti has been questioned in view of its high SFE (which is expected to
promote cell formation), microscopy work on both monotonically and cycli-
cally loaded Ti crystals has shown the presence of stacking faults.
• In transition HCP metals such as Ti, the choice of the operative slip system
is strongly influenced by the electronic properties. The core of a screw dis-
location in Ti has a three-dimensional structure that preferentially moves in
the prismatic plane with a high Peierls force. Such an HCP crystal then
behaves like a BCC metal under monotonic loading, while the divalent
HCP metals, such as Be, Mg, Zn and Cd which have basal slip, behave
more like FCC metals.
• In HCP metals, deformation twinning is a dominant mode of plastic defor-
mation. Ti crystals deform by {1124}, {1122}, and {10Tl} twins during com-
pression along the oaxis, and by {10l2}, {1123}, and {1121} twins during
tensile loading along the oaxis.
We now examine the effects of crystal orientation and equal tension-compression
cyclic straining on substructure evolution in a high purity Ti single crystal (with
oxygen and carbon concentrations of 116 and 22 ppm, respectively, and total
other impurity content of 45 ppm). This discussion is derived from the work of
Gu et al (1994).

2.13.2 Cyclic deformation of Ti single crystals

The inset on the right side of Fig. 2.26 shows the unit triangle for the (0001)
stereographic projection for Ti and the orientations of three a-Ti single crystals,
marked A, B and C in the triangle, which were used for the fatigue tests. Specimen
A was oriented on the (0001)—(1010) border which is subject to duplex slip. Specimen
84 Cyclic deformation in ductile single crystals

400

1120

300

200

1010
100

0.4 0.8 1.2

Ae(%)
Fig. 2.26. Cyclic stress-strain curves for the three specimens of a-Ti single crystals. The inset
shows the orientations of the specimens A, B and C in the unit triangle. (From Gu et al., 1994.)

B was oriented near the (0001) corner, while specimen C was located near the middle
of the triangle. The saturated cyclic stress-strain curves (CSSC) for the three speci-
mens are plotted in Fig. 2.26. Orientations A and B represent the upper and lower
bounds, respectively, among the three cases considered here, while specimen C exhi-
bits an in-between response. Specimen B has a 'plateau-like' regime, similar to that of
FCC crystals. The dislocation substructures in each of the three fatigue specimens
were examined by TEM after the completion of the fatigue tests.
Gu et al. (1994) offer the following line of reasoning in an attempt to rationalize
the effect of crystal orientation on cyclic deformation in the a-Ti single crystals. (1)
Typically, the cyclic deformation is strongly influenced by the propensity for twin
formation. An increase in the occurrence of cyclic twins (in the sequence
B -> C -> A, among the three specimens considered here) causes a marked increase
in the cyclic hardening rate. (Under cyclic loading, {10T2}, {1121}, {1122} and {1123}
twins have been observed in Ti single and polycrystals.) (2) At fixed applied strain
amplitudes, orientations (such as Specimen C) which promote single slip and cross
slip give rise to planar dislocation dipole arrays and dislocation loops (similar to
fatigued FCC crystals), whereas cell structures are found in the specimens oriented
for duplex and multiple slip.

Exercises
2.1 Why does cyclic slip remain confined to the primary slip plane during the
formation of vein structure or the PSB structure?
2.2 Why are the vein structures and PSB structures in FCC crystals composed
mainly of edge dislocations?
Exercises 85

2.3 What are the characteristic dimensions associated with the geometry of the
vein structure and what is the physical basis for such dislocation configura-
tions? What are the effects of changes in test temperature (either increase or
decrease) on the geometry of dislocation arrangements within the vein struc-
ture and within the channels separating the veins?
2.4 Discuss the effects of stacking fault energy on the deformation of a ductile
solid in monotonic tension and in tension-compression fatigue.
2.5 A long crystal is bent to a semicircular shape with a radius of 20 cm. The
crystal has a square cross section (2 cm x 2 cm).
(a) If it is assumed that all bending is accommodated by the generation of
edge dislocations, calculate the total number of dislocations.
(b) If the magnitude of the Burgers vector of the edge dislocations is
0.32 nm, calculate the dislocation density.
2.6 Discuss the mechanisms responsible for dynamic strain-ageing.
2.7 There is a similarity between the hexagonal close-packed (HCP) and body-
centered cubic (BCC) crystal structures.
(a) Show which plane in the BCC structure is similar to the basal plane in
the HCP structure.
(b) Draw the arrangement of atoms in this plane and determine the stack-
ing arrangement normal to this plane in the BCC structure. Is the
stacking the same as in the HCP structure?
 CHAPTER 3

Cyclic deformation in polycrystalline

ductile solids

Following the discussion of cyclic deformation in ductile single crystals, we

now direct attention to polycrystalline metals and alloys. Firstly, experimental obser-
vations are discussed to show how the cyclic deformation of single crystals oriented
for multiple slip can be correlated with that of polycrystalline aggregates.
Continuum characterization of cyclic hardening and softening of polycrystals is
addressed, along with the effects of slip characteristics, alloying and precipitation.
The Bauschinger effect is then introduced, wherein both microscopic and continuum
viewpoints are presented to rationalize the dissimilar yield responses seen in tension
and compression during load reversals. Continuum models for uniaxial and multi-
axial cyclic deformation, which are of interest for engineering structures, are the
focus of subsequent sections. Lower bound and upper bound theorems for elastic
shakedown under cyclic loading are presented, and the origins of cyclic creep and
ratchetting are then examined. The chapter concludes with the derivation of critical
temperatures which signify distinct transitions in the cyclic deformation character-
istics of particle-reinforced metal-matrix composites and layered materials. While
this chapter confines attention to ductile crystalline solids, the cyclic deformation
characteristics of brittle solids as well as semi-crystalline and noncrystalline solids are
taken up in Chapters 5 and 6.

3.1 Effects of grain boundaries and multiple slip

The mechanisms of cyclic damage observed in ductile single crystals are also
known to be generally applicable to the deformation of near-surface grains in poly-
crystalline metals of high purity. However, the presence of precipitates, impurities,
inclusions and grain boundaries in commercial materials gives rise to fatigue defor-
mation characteristics which may significantly deviate from those discussed for
monocrystalline solids. It is pointless to include here the vast amount of empirical
information gathered to date on the cyclic stress-strain behavior of a wide spectrum
of engineering alloys. However, in an attempt to develop an overall perspective on
fatigue damage in many broad classes of materials, we present representative experi-
mental data and mechanistic interpretations on the effects of grain boundaries,
alloying and precipitation. Also included in these descriptions are the phenomeno-
logical constitutive laws for cyclic hardening and softening of metals and alloys.
Studies of polycrystalline metals have established that PSBs can form within the
bulk of the material. Winter, Pedersen & Rasmussen (1981) investigated fatigue

86
3.1 Effects of grain boundaries and multiple slip 87

deformation within the bulk of poly crystalline Cu, of 100-300 urn grain size, by
examining TEM foils taken at various depths below the free surface. At strain
amplitudes of about 10~4, PSBs confined to a single slip system and of a wall
structure similar to that of single crystals were identified in the interior of the poly-
crystalline fatigue specimen. Pohl, Mayr & Macherauch (1980) have also observed
PSBs in the interior sections of fatigued polycrystalline low carbon steel. Although
PSBs can traverse through low angle boundaries, large angle grain boundaries are
impervious to them. When the strain amplitude is raised to values beyond 10~3,
labyrinth and cell structures are observed in polycrystalline copper, as in the case
of single crystals (Chapter 2).
Experimental studies of polycrystalline metals have also identified the existence of
regimes in the cyclic stress-strain curve where PSB formation has a significant effect
(Lukas & Kunz, 1985). Specifically, the saturation stress-strain curves for coarse-
grained Cu exhibit a region of low cyclic strain hardening (somewhat analogous to
the plateau for the single crystal); the occurrence of this region corresponds to a
continuous increase in the volume of the material occupied by the PSBs, which carry
a higher plastic strain than the matrix.

3.1.1 Monocrystalline versus polycrystalline FCC metals

It was shown in Chapter 2 that single crystals of FCC metals, oriented for
single slip, exhibit a plateau in the cyclic stress-strain curve which spans a wide range
of resolved plastic shear strain amplitude, ypi, Fig. 2.2. In this plateau regime, with a
fixed value of the saturated peak resolved shear stress TS = r*, deformation localizes
along persistent slip bands whose volume fraction increases with increasing ]/pl in
order to accommodate cyclic deformation at a fixed value of r*. In the FCC crystals
oriented for multiple slip, however, well-defined PSBs are not commonly found; even
crystals oriented for duplex slip show only a very weak plateau (Table 2.1). Instead,
dislocation arrangements which promote multiple slip and cross slip, such as the
labyrinth structure in the [001] Cu crystals, are observed.
Studies of monocrystalline Cu oriented for multiple slip provide insights into the
cyclic response of poly crystals. As shown later in Section 3.3, the cyclic stress-strain
curve for many fine-grained FCC metals can be approximated, in a limited regime of
plastic strain amplitudes, by a power-law equation of the type:

where Aa/2 and Aepl/2 are the applied uniaxial stress and plastic-strain amplitudes,
respectively, k is an experimentally determined material constant, and n{ is the cyclic
strain hardening exponent. The equation can be recast into a form amenable for use
with single crystals by employing the Taylor factor, MT (= 3.06 for randomly-
textured polycrystalline FCC metals):
 Cyclic deformation in poly crystalline ductile solids

For fine-grained polycrystalline Cu, k' = 146 MPa and «f = 0.205 (Lukas & Kunz,
1985).
Figure 3.1 shows a comparison of the cyclic stress-strain curves for monocrystal-
line Cu, oriented for single slip and multiple slip, and polycrystalline Cu. The solid
line is the CSS curve for Cu single crystals oriented for single slip which shows a
plateau at a saturation stress of 28 MPa in regime B. The symbols are from the
experiments of Gong, Wang & Wang (1997) for [001] Cu single crystals which
have eight symmetric slip systems.! Note the absence of the plateau in the Cu multi-
slip crystal. The dashed line in Fig. 3.1 is the power-law function, Eq. 3.2, for Cu
polycrystals, corrected for the Taylor factor. It can be inferred from Fig. 3.1 that the
absence of a plateau in the polycrystalline FCC metals is similar to that in the
multiple-slip Cu single crystal. %
As a general trend, two factors clearly distinguish polycrystalline FCC metals
from single crystals oriented for single slip:

(1) Grains in a polycrystalline ensemble have a variety of slip orientations.

(2) The incompatibility of elastic and plastic deformation between adjacent
grains in a polycrystalline ensemble promotes local multiaxial loading and
predominantly multiple slip.

Forfine-grainedpolycrystalline FCC metals, these two factors lead to multiple slip

deformation which resembles the cyclic response of single crystals oriented for multi-
ple slip. For coarse-grained FCC metals, however, these two factors may not dom-
inate cyclic response since, statistically, the majority of the grains may not have their
orientations along the edges of the standard triangle of the stereographic projection.
Many of the crystal orientations here may fall within the triangle. Consequently,
coarse-grained polycrystalline FCC metals may exhibit cyclic deformation which
nearly resembles that of single-slip oriented monocrystals, with a low strain hard-
ening or even a mild plateau (e.g., Polak, Klesnil & Lukas, 1974).

' See the worked example in Section 2.8 for the identification of the symmetric slip system for the [001]
orientation. Although all eight systems would, in principle, be expected to have the same Schmid factor,
small errors during the alignment, machining or mounting of the fatigue specimen can lead to a pre-
ferentially higher Schmid factor in one of the eight systems. On the basis of their slip trace observations,
Gong et al. (1997) identify the B4 system to operate first as the primary system for which the resolved
shear stress and plastic shear strain are computed.
+ It is also of interest to note here that the work of Mughrabi & Wang (1981) showed an excellent
agreement between the cyclic deformation results at low plastic strain amplitudes (regime A) for Cu
single crystals (single slip orientation, where the effects of incompatibility stresses were ignored) and
polycrystals by employing the Sachs orientation factor. Above regime A, they found that the effective
orientation factor was larger than the Sachs factor, but smaller than the Taylor factor.
3.2 Cyclic deformation of FCC bicrystals 89

60 p -

40

20

10" 10" 10~2

Fig. 3.1. A comparison of the cyclic stress-strain (CSS) curves of Cu single crystals (—) and
polycrystals ( ) at 20 °C. The dashed line is a plot of the stress-strain response of Cu
polycrystals (from Lukas & Kunz, 1985), corrected by the Taylor factor, M T . The symbols (•)
denote the CSS response of Cu single crystals oriented for multiple slip with the loading axis
along [001]. (After Gong, Wang & Wang, 1997.)

3.1.2 Effects of texture

Differences in slip characteristics (i.e. single slip versus multiple slip defor-
mation) within the individual grains of a polycrystalline FCC metal are also known
to promote strong texture effects during cyclic loading (e.g., Mughrabi & Wang,
1981). The experiments of Llanes et ah (1993) and Peralta et ah (1995) on coarse-
grained polycrystalline Cu (grain size ~ 700 |im) reveal that a cyclically stronger
(lll)-(lOO) texture coaxial with the loading axis promotes a high cyclic hardening,
whereas a 'random' texture of nearly the same grain size results in a much softer CSS
curve as a result of dominant single slip. A comparison of the texture effects for the
coarse-grained and finer-grained Cu (~ 700 jim and 100 Jim grain size, respectively)
showed that the effects of crystallographic texture were much stronger than the
effects of grain size on the CSS curves.

3.2 Cyclic deformation of FCC bicrystals

When a bicrystal, whose component crystals are elastically and plastically
anisotropic, is subjected to cyclic loads, stresses evolve at the grain boundary in such
90 Cyclic deformation in poly crystalline ductile solids

a way that deformation is compatible between the grains. Crystallographic orienta-

tions of the two grains strongly influence the compatibility stresses near the bound-
ary which, in turn, may markedly alter the slip characteristics in the vicinity of the
boundary and hence influence the propensity for crack formation.
Here we briefly examine the experimental results of Hu & Wang (1997) on the
cyclic stress-strain response of a [345]/[Tl7] Cu bicrystal fatigued at 20 °C at
ypi = 2.8 xlO~ 4 — 6.45 x 10~3. They show that the dislocation structures of the
fatigued bicrystal are similar to those of multiple-slip-oriented single crystals (see
Section 2.9) and coarse-grained Cu polycrystals, especially at ypl > 9 x 10~4 where
predominantly cell structures evolve. Hu and Wang compute the resolved shear
stress rs and the resolved plastic shear strain amplitude ypl for the bicrystal in the
following manner:
+T 2 26.pi
= <JSM\, (3.3)

where the subscripts 1 and 2 denote the [345] (single-slip orientation) and [117]
(double-slip orientation) components, respectively, Mx and M2 are the Schmid fac-
tors for the primary slip systems BA of the two crystals, as and rs are the saturation
values of the axial stress applied to the crystal and the resolved shear stress, respec-
tively, and 6pi is the applied axial plastic strain amplitude. Figure 3.2 is a plot of rs vs
ypl for the [345]/[Tl7] Cu bicrystal for symmetric tension-compression loads with
plastic strain control. Note the absence of a plateau in the CSS curve. Also plotted
here are the stress-strain data for a coarse-grained Cu polycrystal with a grain size of

45

***
I 35 ,6°
O bicrystal
25 A polycrystal

_j I
15
1(T 10"

Fig. 3.2. Cyclic stress-strain curves of [345]/[117] Cu bicrystal (after Hu & Wang, 1997) and
coarse-grained Cu polycrystal (after Lukas & Kunz, 1985) appropriately modified by the Taylor
factor.
3.3 Cyclic hardening and softening in polycrystals 91

1.2 mm. For the Cu polycrystal, as and epl were converted into rs and yp\ using the
Taylor factor, MT = 3.06 in the following manner: rs = <rs/MT and yp\ = MTepl.
The CSS curve of the bicrystal overlaps with that of the coarse-grained polycrys-
tal, when modified appropriately by the Taylor factor, Fig. 3.2. (Recall that a similar
overlap was demonstrated in Fig. 3.1 between a Cu single crystal oriented for multi-
slip and a Cu polycrystal.) The foregoing results illustrate that under some condi-
tions, it is possible to correlate the cyclic deformation response of polycrystals with
the corresponding behavior seen in single crystals and bicrystals.

3.2.1 Example problem: Number of independent slip systems

Problem:
von Mises recognized that, in a deformed polycrystal, arbitrary shape
changes of a crystallite are possible only if at least five independent slip systems
operate. Provide a simple physical reasoning to justify this requirement.
Solution:
An arbitrary shape change is brought about by translations and rotations.
There are six possible degrees of freedom, in general: three for translation and
three for rotation. During plastic deformation of crystalline materials, however,
the volume remains constant because deformation occurs by shear in preferen-
tial slip systems. This leaves five independent components for the strain tensor.
Since shape changes must be accommodated by shear on the slip systems, arbi-
trary shape changes are possible only if at least five independent slip systems
operate.

3.3 Cyclic hardening and softening in polycrystals

The uniaxial deformation of engineering alloys subjected to cyclic loads is
usually characterized by the cyclic stress-strain (CSS) curve, similar to that discussed
for single crystals. The transient phenomena typically associated with cyclic defor-
mation are schematically illustrated in Fig. 3.3. In the case of constant amplitude,
fully-reversed stress control, Fig. 3.3(a), cyclic hardening or softening of the material
is reflected by a reduction or an increase, respectively, in the axial strain amplitude.
Similarly, under constant amplitude, strain-controlled fatigue loading, cyclic hard-
ening or softening of the material causes an increase or decrease, respectively, in the
axial stress amplitude, Fig. 3.3(6).
In both stress-controlled and strain-controlled fatigue, the respective strain ampli-
tude and stress amplitude reach a stable saturation value after an initial 'shakedown'
period. This saturation state gives rise to stable hysteresis loops. During 'shake-
down', there is a continual change in dislocation substructure until a stable config-
92 Cyclic deformation in poly crystalline ductile solids

A A A
vVVvf V- v -y -

(a) stress-controlled strain response for strain response for

loading cyclic hardening cyclic softening

y_v

(b) strain-controlled stress response for stress response for

loading cyclic hardening cyclic softening

Fig. 3.3. Phenomena associated with transient effects in fatigue, a, e and / denote stress, strain
and time, respectively.

uration representative of the saturated state is reached. Beyond this point, the hys-
teresis loop remains essentially the same cycle after cycle over the life of the test
specimen. The parameters used to describe the salient features of cyclic hysteresis
loops are defined in Fig. 3A(a). The locus of the tips of stable hysteresis loops
provides the cyclic stress-strain curve, Fig. 3.4(6).
Stress- and strain-controlled fatigue represent extremes of fully unconstrained and
fully constrained loading conditions. In real engineering components, there is usually
some structural constraint of the material at fatigue-critical sites. It thus seems
appropriate to characterize fatigue response of engineering materials on the basis
of data obtained under strain-controlled fatigue rather than cyclic stress-controlled
conditions.
Strain-controlled tests have gained increasing use in the determination of CSS
curves for engineering alloys. Three commonly used strain-controlled test methods
are indicated in Fig. 3.4(c). In the constant amplitude test, the specimen is cycled
within a constant plastic strain limit (until failure) to obtain a single stable
hysteresis loop. Multiple test specimens are needed to determine the entire CSS
curve using this method. In the multiple step method, a specimen is cycled
between constant plastic strain limits until a saturation loop results. Then the
plastic strain limits are incremented until another stable hysteresis loop is
obtained. This process is continued until the entire CSS curve is measured
from a single test specimen. In the incremental step method, the specimen is
subjected repeatedly to a strain pattern comprising linearly increasing and
decreasing amplitudes, from zero to a certain maximum total strain. The resulting
stable hysteresis loop provides the CSS plot. In some alloys exhibiting planar slip
deformation, the incremental step method provides a CSS response which is
different from the other direct methods because of the variations in the develop-
ment of dislocation structures.
3.3 Cyclic hardening and softening in polycrystals 93

cyclic stress-
strain curve'

I- 0
Ao-

constant plastic
strain limit
A A A A
V V V V V

multiple step,
increasing plastic

AY AA
VAV
strain limit after
saturation at each
step

incremental step,
A A repeated patterns
consisting of linearly
increasing and
decreasing strain limits
v
Fig. 3.4. (a) A schematic of a stable hysteresis loop and the nomenclature. Aee, Aep and Ae
denote the elastic, plastic and total strain range, respectively, (b) Cyclic stress-strain curve
drawn through the tips of stable hysteresis loops, (c) Procedures for obtaining cyclic stress-
strain curves.

The monotonic stress-strain behavior of ductile solids under uniaxial tension is

generally represented by a constitutive law, such as the Ramberg-Osgood relation-
ship:

(3.4)

where E is Young's modulus, A is a constant commonly referred to as the monotonic

strength coefficient, e is the uniaxial strain, o is the uniaxial stress, and nm is the
94 Cyclic deformation in poly crystalline ductile solids

Table 3.1. Monotonic and cyclic stress-strain characteristics of some common

engineering alloys.

Alloy Condition (MPa)/(MPa) nm/nf

Copper-base
OFHC Annealed 20/140 0.40/0.24
Brass 365 As-rolled 172/248 0.13/0.21
Cu-Be 172 As-drawn 641/490 0.02/0.15

Aluminum-base
2024 T4 303/448 0.20/0.09
6061 T651 290/296 0.04/0.10
7075 T6 469/517 0.11/0.10

Iron-base
SAE 1015 Normalized 225/240 0.26/0.22
Ferrovac E Annealed 48/159 0.36/0.19
SAE 1045 Q+T 1365/825 0.08/0.15
AISI 4340 Q+T 1172/814 0.07/0.15
Mar M-300 Annealed 952/800 0.03/0.08

Gy and a'y refer to monotonic and cyclic yield strengths, respectively.

Q and T refer to quenched and tempered conditions, respectively.
Source: Landgraf (1978) and Hertzberg (1995).

strain hardening exponent. The typical range of nm for alloys is 0-0.5. In the applied
mechanics community, the strain hardening exponent is often denoted by n = l/nm.
Here n = 1 for a linear elastic material and n = oo for an elastic-perfectly plastic
solid.
In an analogous fashion, the cyclic stress-strain response is characterized by the
relationship

Ae Aa
(3.5)
~2E 1A

where A' is the cyclic strength coefficient and nf is the cyclic strain hardening expo-
nent. For most metals, nf varies between 0.1 and 0.2 despite vast differences in their
cyclic hardening and softening characteristics. Table 3.1 provides a list of the strain
hardening characteristics in some common engineering alloys. As a general rule-of-
thumb, well-annealed, polycrystalline metals of high purity exhibit cyclic hardening
due to dislocation multiplication, as evidenced by an increase in the stress amplitude
3.4 Effects of alloying, cross slip and stacking fault energy 95

with fatigue cycles (at a fixed strain amplitude); work-hardened materials undergo
strain softening under cyclic loading. The rearrangement of prestrain-induced dis-
location networks due to fatigue causes cyclic softening.

3.4 Effects of alloying, cross slip and stacking fault energy

A clear demonstration of the relationship among the slip characteristics of
the material, the development of dislocation structures, and the overall cyclic stress-
strain behavior is found in the work of Feltner & Laird (1967a,b). They conducted
an investigation of the effects of prior loading history, slip planarity, plastic strain
amplitude and temperature on cyclic hardening and softening of polycrystalline Cu,
Al and Cu-7.5 wt% Al. They illustrated that pure FCC metals harden or soften
under fatigue loads to a characteristic steady-state saturation flow stress after an
initial transient period. For wavy slip materials, with a high stacking fault energy
and a strong propensity for cross slip, the steady-state cyclic flow stress is indepen-
dent of load history and of the degree of prior cold work at large plastic strain
amplitudes (see Fig. 3.5#); here only the plastic strain amplitude and temperature
influence theflowcharacteristics in fatigue.f Pure FCC metals fatigued at high strain
amplitudes exhibit a cell structure at steady state, regardless of the dislocation
structure which existed before cyclic straining. The cell size, which is independent
of the history of the material, increases with decreasing plastic strain amplitude and
increasing temperature.
In contrast to this behavior, when a planar slip mode is introduced by the
addition of 7.5 wt% Al in Cu (which reduces the stacking fault energy), the
cyclic stress-strain curve becomes sensitive to both prior deformation and thermal
history, Fig. 3.5(6). Inhibition of cross slip in this case precludes the attainment
of a common steady-state dislocation structure in fatigue from an initially hard
(cold-worked) or soft (annealed) condition. The low stacking fault energy Cu-7.5
Al alloy does not develop a cell structure during cyclic saturation, but exhibits
dissociated dislocations arranged in planar bands. Furthermore, the saturation
dislocation structures in the cold-worked conditions of the alloy were significantly
different, with the latter micro structure consisting of fewer bands and a greater
density of tangled interband dislocations that are apparently inherited from prior
cold work.
Experimental studies of substructure developments during fatigue in different
wavy slip metals have documented a one-to-one correspondence between the satura-
tion stress and the deformation-induced cell size. Based on their own investigations

' It is now generally recognized that the ease of cross slip plays a decisive role in determining the slip
mode. Stacking fault energy, however, is not the only parameter influencing cross slip. For example,
Gerold & Karnthaler (1989) demonstrate that short range order can also strongly influence cross slip.
96 Cyclic deformation in poly crystalline ductile solids

monotonic cw

monotonic cyclic cw
cold-worked (cw) |3
cyclic annealed
—
cyclic

— \ — ^ —
s* monoton r monotonic
a nnealed annealed
plastic strain plastic strain
(a) (b)
Fig. 3.5. (a) Schematic showing that the cyclic stress-strain curve is independent of prior
deformation history in wavy slip materials, (b) History-dependent behavior for planar slip
materials. (After Feltner & Laird, 1967a.)

and from a literature review, Plumtree & Pawlus (1988) developed the following
empirical relationship for the dependence of saturation stress on cell size:

(3.6)

In Eq. 3.6, crs is the saturation stress, a0 is the back stress, E is Young's modulus, b is
the magnitude of the Burgers vector, Xs is the linear intercept cell size and B is a
material constant which increases in proportion to the stacking fault energy (SFE).
The works of Pratt (1967), Feltner & Laird (1967b), Abdel-Raouf & Plumtree (1971),
Saga, Hayashi & Nishio (1977) and Kayali & Plumtree (1982) collectively show that
the constant B is 7.8 for Al (SFE = 200 x 10~3 Jm" 2 ) and 3.6 for Cu (SFE = 40 x
10~3 Jm~2). Equation 3.6 does not appear to be a function of the type of deforma-
tion in that it describes the steady-state flow stress-cell size data for both monotonic
and cyclic loading conditions.
For aluminum fatigued at a total strain range of 1 % at room temperature, the
cell size and the misorientation between neighboring cells remains constant from
the onset of saturation until final failure despite noticeable changes in the cell
morphology. One may then infer that the dislocation density remains relatively
constant upon the attainment of saturation. A survey of relevant literature shows
that the operative mechanism during cyclic saturation is the irreversible bowing of
dislocation segments from the cell walls. After being emitted from the cell walls,
the dislocation segments traverse the cell and enter the adjacent walls where they
are annihilated by dynamic recovery. This equilibrium between work hardening
(dislocation generation) and dynamic recovery (dislocation annihilation) is
believed to result in a constant maximum stress and dislocation density during
the saturation stage.
3.6 The Bauschinger effect 97

3.5 Effects of precipitation

Cyclic hardening and softening in precipitation-hardened alloys is promoted
by a mechanism in which the precipitate geometry and distribution as well as dis-
location-particle interactions can be altered by cyclic straining. Existing experimen-
tal evidence reveals that initial cyclic hardening occurs in these materials due to an
increase in dislocation density and due to dislocation-precipitate interactions (e.g.,
Brett & Doherty, 1978). Subsequently, cyclic softening is highly favored if the pre-
cipitates in the age-hardened alloy are easily sheared by dislocations, i.e. if they are
fine, closely spaced and coherent with the matrix. A particularly interesting feature
of deformation in these alloys is that the formation of PSBs causes cyclic softening
without reaching a saturation stage. Various mechanisms have been proposed to
rationalize cyclic softening in precipitation-hardened alloys:
(a) Reversion or re-solution by which the metastable strengthening precipitates
completely dissolve in the matrix after being cut by dislocations to a size smaller
than the critical size for particle nucleation (Broom, Mazza & Whittaker, 1957).
(b) Disordering of ordered precipitates due to the motion of single dislocations
through them which leads to a loss of strength (Calabrese & Laird, 1974).
(c) Micro structural inhomogeneities (due to pre-straining or quenching prior to
ageing) in the form of slip bands, that are depleted of strenghening metastable
precipitates and are replenished with stable particles (Laird & Thomas, 1967).
(d) Over-ageing, which leads to the substitution of metastable precipitates in the
matrix by coarsely distributed, stable ones (Broom, Mazza & Whittaker, 1957).
(e) Dissolution due to Ostwald ripening of unsheared precipitates, adjacent to the
sheared ones on the slip lines at the expense of the smaller precipitates in the slip
band (Sargent & Purdy, 1974). (Ostwald ripening refers to the process by which the
interfacial free energy between the precipitate and the matrix causes precipitate
coarsening upon further annealing, even after precipitation from supersaturated
solid solution is complete. The larger particles grow at the expense of the smaller
ones, thereby releasing excess surface energy and causing the microstructure to
coarsen; see, for example, Martin, 1980.) The difference between this and the rever-
sion mechanism is that the critical radius for Ostwald ripening is much larger than
the value for precipitate growth from the supersaturated solid solution.

3.6 The Bauschinger effect

The Bauschinger effect refers to the experimental result that, after a certain
amount of forward plastic deformation in tension or compression, the material
yields at a lower stress when the direction of loading is reversed than for continued
forward deformation (Bauschinger, 1886). A knowledge of the mechanisms under-
lying the Bauschinger effect is essential for the development of the constitutive
models for complex cyclic deformation, for a fundamental understanding of the
98 Cyclic deformation in poly crystalline ductile solids

work hardening phenomena, and for rationalizing such fatigue effects as mean stress
relaxation and cyclic creep (Chapter 8). For example, many commercial aluminum
alloys containing nonshearable strengthening precipitates (such as the peak-aged and
over-aged 7075 alloys used in aircraft applications) are stretched prior to temper
treatments to relieve thermal residual stresses. Since many of these alloys are known
to exhibit Bauschinger effects, low flow stresses may result under service conditions if
the material is loaded in a direction opposite to the stretching direction. It is also
known that the Bauschinger effect in precipitation-hardened commercial alloys can
persist even after the cyclic hysteresis loops are stabilized. On a more fundamental
level, the Bauschinger effect can be used to identify the contributions to strain
hardening from different kinds of dislocation mechanisms. The study of the
Bauschinger effect is, therefore, commonly regarded as a 'litmus test' for the validity
of strengthening theories in the sense that any complete hardening theory must be
capable of quantitatively accounting for the Bauschinger effect.

3.6.1 Terminology
One of the common methods of quantifying the Bauschinger effect involves
the definition of the reverse strain. This reverse strain is the magnitude of additional
strain after load reversal which makes the reverse yield stress equal in magnitude to
the maximum flow stress attained in the forward deformation. However, a realiza-
tion of the differences between forward and reverse flow stresses at any strain value
can be achieved by recourse to the construction schematically depicted in Fig. 3.6. In
Fig. 3.6(tf), ABC represents the forward deformation in uniaxial tension, with C
being the point of unloading. CD is the tensile unloading segment and DEF is the
reverse (compression) loading segment. The magnitudes of the stress and the accu-
mulated strain are replotted in Fig. 3.6(b) irrespective of the direction of loading.

|e| (accumulated)

(a) (b)
Fig. 3.6. (a) Schematic of the stress-strain curve for fully reversed loading, (b) Only the
magnitudes of the stress and the accumulated strain are replotted to illustrate the Bauschinger
effect.
3.6 The Bauschinger effect 99

Here, ABCD represents the tensile segment and DEF is the compression loading
portion. If the forward hardening segment is extrapolated (until point C'), the
instantaneous difference between the flow stress levels Aab in the (nearly parallel)
segments CC' and EF provides an indication of the Bauschinger effect in terms of
stress and characterizes the extent of 'permanent softening'. Similarly, the difference
in strain values Aeb between the forward and reversed deformation at the maximum
forward tensile stress amplitude (point C) characterizes the Bauschinger strain.

3.6.2 Mechanisms
The origins of the Bauschinger effect are related to the changes in disloca-
tion substructure induced by reversed loading and in the changes in the internal
stress systems. In polycrystalline metals where dislocation walls and subgrain bound-
aries form during forward straining, the dissolution of cell walls or sub-boundaries
upon stress reversals is considered a contributing factor to the Bauschinger effect
(Hasegawa, Yakou & Kocks, 1986). Furthermore, long-range internal stresses
induced by strain incompatibility between the PSB walls and channels can lead to
easier reverse flow in materials which form well-defined PSBs.
For particle-hardened alloys, the mechanistic basis for the Bauschinger effect is
often provided in terms of the interaction of dislocations with the strengthening
particles (Orowan, 1959). Age-hardened alloys can be broadly classified into two
groups: (i) those containing precipitates which are coherent with the matrix and
which can be sheared by dislocations, and (ii) those containing larger semi-coherent
or incoherent particles which are not penetrable by dislocations. The two types of
particles can give rise to very different hardening response in monotonic and cyclic
loading. Furthermore, the Bauschinger effects seen in the two cases are also dis-
tinctly different.
When the matrix contains permeable particles, mobile dislocations cut through
these barriers with essentially no dislocation pile-up. If the material is subjected to a
compressive stress following a tensile stress, the impingement of the shearable bar-
riers with dislocations once again leads to a reverse flow stress comparable in mag-
nitude to that seen in forward deformation due to the paucity of dislocation pile-up.
Consequently, there is little contribution to the flow stress from internal stresses
(Wilson, 1965; Stoltz & Pelloux, 1976).
In the case of alloys with coarse, incoherent and impermeable particles, the mono-
tonic stress-strain curve exhibits a large hardening rate. For these alloys, a simple
model to rationalize the Bauschinger effect is often formulated in terms of the
following three factors which influence the forward yield stress: (i) a contribution,
a0, to strength from solid solution hardening and the stress to bow out the first set of
dislocations past the obstacles, (ii) forest hardening, ad, due to the interactions of
mobile dislocations with forest dislocations, and (hi) a mean internal or back stress,
100 Cyclic deformation in poly crystalline ductile solids

<7M>which is exerted on the matrix by the particles. The total flow stress during
forward deformation is
orf = cr0 + crd + a^. (3-7)
At reverse strains of the order of the forward strain, the deformation must be cap-
able of overcoming cr0 and ad. However, ^ now assists reverse deformation, rather
than opposing it and hence the total reverse flow stress becomes
a r = cro-\-crd -OM> (3.8)
From Eqs. 3.7 and 3.8, Aab in Fig. 3.6(7?) becomes
Aa b = <r f -cr r = 2a^. (3.9)
The back stress which arises from the internal stress system in the material is very
localized in the matrix, and it is believed to be a consequence of the inhomogeneity of
plastic deformation on a microscopic scale. The term Aab is sometimes referred to as
'permanent softening'.
A large Bauschinger effect has been documented in Al-4Cu alloys containing
nonshearable 6' (CuAl2) precipitates, Al-Cu-Mg alloys containing S' (Al2CuMg)
precipitates, and Al-Cu-Zn-Mg alloys containing rj (MgZn2) precipitates (Wilson,
1965; Abel & Ham, 1966; Stoltz & Pelloux, 1976; Wilson & Bate, 1986). Using X-ray
diffraction measurements of directional lattice strains in two-phase cubic alloys,
Wilson (1965) demonstrated that reverse loading destroys the internal stresses, redu-
cing them to zero at some strain value. At this point, the stress difference between the
forward extrapolated and the knee portion of the reverse stress-strain curves, Fig.
3.6(6), was equal to 1.9OM. Here the mean stress in the matrix, a^, was estimated
using X-rays after unidirectional plastic deformation. Note that this experimental
correlation is quantitatively consistent with the prediction of Eq. 3.8.
Some of the most comprehensive studies of the Bauschinger effect have been
conducted on dispersion-strengthened metals. In their study of plastic deformation
in Cu-SiO 2 , Brown & Stobbs (1971a,b) and Atkinson, Brown & Stobbs (1974)
related the internal stresses to the formation of dislocation shear (Orowan) loops
around hard particles on many slip planes. In the unrelaxed state typical of low
temperatures and low strains, there are two contributions to strengthening: (i) A
homogeneous mean stress in the matrix is produced by the Orowan loops left around
the particles. This stress state is amenable to a transformation strain analysis because
the Orowan loops may be regarded equivalent to the transformation strains in
Eshelby's classical inclusion problem (Eshelby, 1957). This mean internal stress,
<7M, in the matrix opposes continued forward deformation during tensile loading,
(ii) The Orowan loops also give rise to an additional inhomogeneous stress locally at
the particle because the presence of the loops around the particles repels successive
dislocations. This increase in internal stress over and above the initial Orowan stress
has been termed the 'source-shortening stress' by Atkinson, Brown & Stobbs (1974).
The occurrence of a Bauschinger effect upon load reversal is then viewed as a con-
sequence of plastic relaxation which can arise due to (a) the removal of Orowan
3.7 Shakedown 101

loops by the formation of secondary dislocations and prismatic loops of primary

Burgers vectors (Atkinson, Brown & Stobbs 1974), or (b) the shrinkage of Orowan
loops by climb via pipe diffusion (Gould, Hirsch & Humphreys, 1974). These
mechanisms have also been extended to rationalize the Bauschinger effect in
metal-matrix composites reinforced with hard particles and fibers (e.g., Brown &
Clarke, 1977; Lilholt, 1977).

3.7 Shakedown
Ductile metals are often subjected to cyclic loads in such a manner that the
early fatigue cycles lead to the build-up of residual stresses. Repeated contact loading
in ball bearings and railway rails, for example, commonly engenders plastic yielding.
A consequence of this plastic flow is the generation of residual stresses, which can
possibly be of such a magnitude that a steady-state is attained after some load
reversals wherein a closed cycle of entirely elastic reversed deformation is promoted.
For this situation, there is no net accumulation of plastic strain in subsequent cycles
and the system is said to have undergone shakedown. In other words, the residual
stresses generated during the early load reversals alleviate the applied loads by
inhibiting further plastic deformation in subsequent cycles, and induce a state of
reversible elastic response. Note that during shakedown, the maximum applied load
alone is of a magnitude which violates the yield condition.
The limiting value of the applied load below which no continued accumulation of
plastic strain is possible during cyclic loading is known as the shakedown limit. If the
shakedown limit is exceeded, plastic strains continue to accumulate during each cycle
and this phenomenon is commonly known as ratchetting, cyclic creep or incremental
collapse.
The conditions governing the occurrence of shakedown can be formulated in
terms of the so-called shakedown theorems, for elastic-perfectly plastic solids:

• Statical, lower-bound theorem (Melan, 1938): Shakedown occurs whenever a

system of residual stresses, that satisfies equilibrium requirements, acts in
conjunction with applied loads in such a manner that the yield criterion is
not violated.
• Kinematical, upper-bound theorem (Koiter, 1956): If any kinematically
admissible plastic strain cycle could be identified in which the internal plastic
work is exceeded by the work done by the applied loads, then shakedown
will not take place. Under these conditions, cyclic plasticity and incremental
collapse are possible.

Elastic shakedown occurs when the development of residual stresses results in a

steady state in which cyclic deformation is purely elastic. Plastic shakedown, on the
other hand, refers to a state of deformation where a closed cycle of alternating
102 Cyclic deformation in poly crystalline ductile solids

plasticity occurs without any accumulation of plastic strains, ratchetting or incre-

mental collapse. Examples of both types of shakedown are discussed in Chapter 13
in the context of rolling contact fatigue. The shakedown limits for the thermal
fatigue of metal-matrix composites and for different types of contact fatigue are
considered in Section 3.10.2 and Chapter 13.

3.8 Continuum models for uniaxial and multiaxial fatigue

The two strain hardening rules which are widely used in metal plasticity are
those involving isotropic hardening and kinematic hardening. If the yield surface
(which represents the yield condition in stress space) expands uniformly during
plastic flow with no shape change or translation, isotropic hardening occurs. On
the other hand, kinematic hardening takes place when the yield surface does not
change its size or shape, but simply translates in stress space in the direction of the
outward normal. These hardening models can be written in terms of the functions F
and h:
F(aiJ-^j)-h(Xh) = 0, (3.10)
where o^ are the components of the symmetric stress tensor, ^ denote the translation
of the initial yield surface, Xh is a scalar function of the plastic strain and h(Xh) is a
function quantifying the expansion of the yield surface. When £y = 0, isotropic
hardening occurs and when h(Xh) is a constant, kinematic hardening is induced.
The hysteresis loop for an isotropically hardening solid is schematically sketched
in Fig. 3.1 (a). Here OAB represents the tensile loading segment with strain hard-
ening. When the load is reversed, yielding in compression commences at C, with the
stress at C having the same magnitude as at B. Continued compression loading
causes further hardening. If the load is reversed at Z>, subsequent tensile yielding
occurs at E when the stress reaches the same magnitude as at D. In descriptions of
isotropic hardening, the relationship between the increase in flow strength and
equivalent strain is independent of stress state. The Bauschinger effect is non-existent
for an ideally isotropically hardening material.|
The solid lines in Fig. 3.7(6) schematically represent the uniaxial stress-strain
curve for an elastic-plastic solid in which compressive yielding begins at point C
as a consequence of the Bauschinger effect. The dashed lines in this figure denote the
constitutive response derived from a kinematic hardening model (e.g., Prager, 1956;
Ziegler, 1959). If a linear hardening assumption is used (i.e. if ^ in Eq. 3.10 is
determined from the law d^y = cde?, where defj are the increments of the plastic
strain and c is a constant), the plastic portion of tensile deformation in Fig. 3.7(6) is

' If the ductile solid is reinforced with a brittle phase, the overall Bauschinger effect for the composite can
be rationalized even when the plastic deformation of the matrix is described by an isotropic hardening rule.
The Bauschinger effect in this case arises from the constrained flow of the matrix material between the
brittle particles (e.g., Llorca, Needleman & Suresh, 1990).
3.8 Continuum models for uniaxial and multiaxial fatigue 103

C
D
D

(a) (b)

Fig. 3.7. (a) Shapes of hysteresis loops associated with (a) isotropic hardening and (b) kinematic
hardening.

approximated by the straight line A'B''. Upon load reversal at B', the behavior
follows elastic unloading line BC\ Subsequent plastic compression is represented
by the dashed line CD' which is parallel to A'Bf. If c is assumed to be a function of
the stress invariants, the slope of the stress-strain curve at C will depend on the value
of the stress and hence on the location of C. Note that the kinematic hardening
model does predict the existence of the Bauschinger effect. This theory has been
implemented into many finite element codes.
Consider now the case of tension-compression loading over many cycles. The
kinematic hardening rule predicts that a steady state of alternating plastic flow
will set in after the first cycle of loading. On the other hand, the isotropic hardening
model suggests that the specimen will 'shake down' to an elastic state. However, as
seen in Figs. 3.3 and 3.4, a steady state of plastic strain amplitude is reached only
after an initial transient behavior. Thus, it is noted that simple isotropic and kine-
matic hardening rules may not provide an adequate description of plasticity, when a
material is subjected to periodic unloading and reloading along a different stress
path (e.g., Mroz, 1967).
In this section, a brief summary of some prominent models for cyclic plasticity is
presented, and their strengths and limitations are pinpointed. More comprehensive
reviews of these formulations are available in the literature (Drucker & Palgen, 1981;
Dafalias, 1984; Chaboche, 1986). Any complete model for cyclic plasticity must be
capable of rationalizing the following fatigue characteristics:

(1) the Bauschinger effect,

(2) elastic shakedown,
104 Cyclic deformation in poly crystalline ductile solids

(3) cyclic hardening or softening prior to the attainment of a stable hysteresis

loop,
(4) cyclic creep, ratchetting, or incremental collapse, whereby a mean plastic
strain accumulates in each cycle in the direction of the mean stress in
stress-controlled fatigue with a nonzero mean stress (see Section 3.9),
(5) mean stress relaxation, whereby the mean stress in a fatigue specimen sub-
jected to strain-controlled fatigue (with a nonzero mean stress) tends toward
zero with the progression of cyclic deformation (see Chapter 8),
(6) the characterization of small strain, multiaxial cyclic deformation under
proportional tension-torsion loading on the basis of equivalent strain ampli-
tudes which provide a cyclic deformation response which is equivalent to
that seen in uniaxial loading,
(7) the largest degree of cyclic strain hardening in nonproportional (out-of-
phase) tension-torsion, with a phase difference between tension and torsion
cycles of 90 degrees.

Many models for cyclic plasticity have been proposed over the years which ratio-
nalize the foregoing cyclic phenomena with varying degrees of success. In addition to
the Prager-Ziegler kinematic hardening model described earlier in this section, the
available approaches to modeling cyclic plasticity can be classified into the following
groups: (i) the parallel sub-element model, (ii) field of work hardening moduli,
(iii) two-surface models, and (iv) other developments involving combined
isotropic-kinematic hardening models or internal variable concepts. The following
sections describe the features of each of these approaches.

3.8.1 Parallel sub-element model

In the sub-element model of Masing (1926), Fig. 3.8(0), the solid is viewed as
being composed of a set of n elastic-perfectly plastic elements of the same dimen-
sions, but with different yield points. These elements are arranged in parallel and are
symmetric in tension and compression. Consider the following sequence of events:

(1) When the material is loaded in tension, the weakest element yields first at a
stress ox .
(2) If loading is continued further, the second weakest element yields at a tensile
stress a2.
(3) The total load on the material is a2- A (where A is the total cross-sectional
area), whereas the total load on element 1 is ct\Ai, where A\ is the area of
element 1. Upon unloading, the residual stress on element 1 equals
—(<j2 — or\) while the residual stress on each of the other (n — I) elements
(which are all elastic at this point) is (a2 — cr{)/(n — 1).
(4) If forward loading were to be continued beyond a2 to a far-field tensile
stress, a3, the residual stress on element 1 would become —(a3 — ax) and
3.8 Continuum models for uniaxial and multiaxial fatigue 105

(a)

Fig. 3.8. (a) Masing's parallel sub-element model, (b) Three types of kinematic hardening
behavior which can be extracted from the sub-element model. (After Asaro, 1975.)

on element 2, —(<J3 — a 2 ) and so forth. For this case, reverse plastic straining
would begin at —aR where |a R | = ox + a2 — cr3.
(5) If the forward tensile loading curve has the form a = / ( e ) , then the reverse
loading curve takes the form o = 2/rte).
On the basis of the Masing sub-element approach, Asaro (1975) presented micro-
mechanical arguments for three types of kinematic hardening. The premise here is
that, at fixed values of strain rate e and temperature T, the applied stress required to
achieve a certain mechanical state is a function of the internal structure such that
a = o(ot\, a2, a 3 , . . . ) . (3.11)
It is assumed that the internal structures form in the indicated sequence Qf 1 ,a 2 ,a 3
during forward loading and cause the yield surface to translate by the increment
d£y(c*i, of2, of 3 ,...). The order in which the recovery of individual events at occurs
during reverse loading determines the type of kinematic hardening.
If ai,«2» a3> e^c- (which are the microstructural variables determining the flow
strength) are viewed as being the plastic strains in the various sub-elements, then
one has the situation where the recovery has the sequence 2a l5 2a 2 , 2a 3 , etc. In other
words, element 1 deforms plastically twice as much in compression before element 2
yields as it did originally in tension. It is also implied here that (3cr/da1) is not a
function of a 2 , a 3 , etc. and likewise for (9<r/da2), (3or/da3), etc. This sequence, when
represented by a stress-strain plot, results in the type I behavior schematically shown
as loop OABCDEGHJB in Fig. 3.8(&). The type I kinematic hardening model ratio-
nalizes the Bauschinger effect in a number of alloys with shearable precipitates. In
these materials, dislocation cells are envisioned as the elastic-plastic elements which
generate the internal stress systems. The type I sub-layer model has also been found
106 Cyclic deformation in poly crystalline ductile solids

to provide a reasonable description of the Bauschinger effect in spherodized plain

carbon steels (Wilson & Bate, 1986).
If the structural operations occurring during the forward loading are reversed
upon unloading in exactly the same sequence, hardening of type II, loop
OABCDD'F'FGHH'A in Fig. 3.8(6), is obtained. The resulting loops contain inflec-
tion points which are symmetric about the origin. Reversible twinning could be
characterized by this scheme if the initial twins are not relaxed by the accompanying
plastic flow.
The third type of hardening arises from a recovery sequence a3,a2,a\, which is
opposite to that in the previous case. This behavior, path OABCDF"FGHH"A in
Fig. 3.8(6), appears to be represented by reversible collapse of linear dislocation pile-
ups and by the shape memory effect in Cu-Zn alloys (Pops, 1970). The type III
scheme manifests as reversible recovery on a microscale causing a nonlinear elastic
response.
The Bauschinger effect in aluminum alloys containing nonshearable precipitates,
dispersion-hardened copper crystals, and oxide-dispersed superalloys can be ratio-
nalized on the basis of type III kinematic hardening behavior. Typical characteristics
of materials which exhibit type II or type III behavior involve: (i) a large fraction of
elastic second phase, (ii) local dislocation reversibility, and (iii) stabilization of the
dislocation array around the precipitates against plastic relaxation. Extensions of the
Masing-type multicomponent models to thermo-mechanical fatigue have also been
explored by Maier & Christ (1996).

3.8.2 Field of work hardening moduli

Mroz (1967, 1969) introduced the concept of a 'field of work hardening
moduli' to generalize the known rules of kinematic hardening to cyclic loading.
This field involves a configuration of surfaces of constant work hardening moduli
in stress space. A similar concept was also proposed independently by Iwan (1967),
who rationalized the Bauschinger effect and multiaxial cyclic deformation using a
collection of yield surfaces. Although each one of the yield surfaces in such a model
obeys a linear work hardening rule, the combined effect of the nested yield surfaces
gives rise to a nonlinear work hardening law. In this section, we discuss in some
detail the anisotropic hardening model due to Mroz.
As a starting point to this discussion, it is instructive to define the linear and
nonlinear moduli in uniaxial deformation. The total strain increment de is composed
of the elastic part dee and the plastic part dep:
de = dee + dep. (3.12)
e Q p p
If da is the stress increment, de = da/Eu de = da/E , and de = da/E , where Eu
Ep, and EQ are the tangent, plastic and elastic moduli, respectively. From Eq. 3.12, it
is readily seen that
3.8 Continuum models for uniaxial and multiaxial fatigue 107

During elastic deformation, Et = Ee. As plastic deformation is initiated, there is a

change in Ep from an initial value of infinity (in the vicinity of the elastic limit) to
some finite value. The tangent modulus, Eu can be used as a direct measure of the
plastic modulus, Ep, during plastic flow because Ee is constant.
Consider a fatigue specimen of an initially isotropic material which is first sub-
jected to tensile loading, as indicated by the segment OABCDE in Fig. 3.9(a). Once
plastic deformation begins at point A, the tangent modulus decreases. Let the stress-
strain curve be approximated by n linear segments of constant tangent moduli
E{, E2,..., En, Fig. 3.9(a). In stress space, this approximation is represented by n
hypersurfaces So, Si,..., Sn9 where So is the initial yield surface and Si, S2,..., Sn
are the regions of constant work-hardening.
For the initially isotropic material, the yield surfaces So, Si,..., Snare similar and
concentric, and centered on the origin O. In two-dimensional stress space, these
surfaces form a family of circles, Fig. 3.9(b). Consider proportional loading, where
different components of the stress tensor increase in constant proportion to one
another. Assume first that all yield surfaces translate in stress space without changing
their shape or orientation. When the applied stress reaches the elastic limit, point A
in Fig. 3.9(<z), the initial yield surface So starts to move along the a2 axis until it
touches the circle Si at B. During this period, all the other surfaces remain fixed.
Between A and B, the plastic strain is defined by the tangent modulus E\. Similarly,
as the stress is increased from B to C, the surfaces SQ and Si translate together until
the point C is reached where *S0 and Si contact the surface S2 which until now had
been stationary. The tangent modulus between B and C has a constant value E2.
Beyond C, the surfaces SQ, SI and #2 translate together as the stress is raised until
point D is reached. Figure 3.9(7?) shows the situation where the applied stress corre-
sponds to the point E.
Now let the specimen be unloaded from the stress point E and reversed into
compression. When point G is reached, reverse plastic flow occurs; the surface So
starts to move downward until it touches surface S{. This corresponds to the stress
point H. The difference between stresses at H and G is twice the difference between
stresses B and A. Similarly, loading between the stress points HI and / / corresponds
to the previously described steps BC and CD. Note that the curve of inverse loading
is obtained by symmetry with respect to the curve OABCDE. The curve of reverse
loading EGHIJK is uniquely defined by the primary loading curve OABCDE. If
a = / ( e ) is the equation of the primary curve OABCDE and if a new coordiante
system (o\?) is chosen with its origin at E and rotated 180°, the curve EGHIJK is
defined by
a = 2/(1/2^). (3.14)
This result is identical to that predicted by the Masing model. Equation 3.14 also
indicates that aK = 1oE\ therefore, oK = —oE. From K, subsequent loading KNE is
108 Cyclic deformation in poly crystalline ductile solids

Fig. 3.9. (a) Representation of the stress-strain curve by regions of constant tangent moduli, (b)
Representation in stress space. (After Mroz, 1967.)

identical to that of EHK. Upon reaching point E, deformation follows a steady cycle
EHKNE.
Mroz also extended the above model to situations involving nonproportional
multiaxial loading. As an example, consider the case where the specimen is unloaded
partially to a stress point Q inside So from the stress point E, Fig. 3.9(6), whereupon
the stress point moves along QR contacting the surface So at T. The surface So must
move such that the point T makes contact with the surface S^ without intersecting it.
(Therefore, translation will not occur along the normal at T, i.e., along the extra-
polation of O0T.) Thus T will move to a point U on S}. As the stress point moves
along 77?, the surface So will touch Sx at W and subsequently they move together to
touch S2 at Z and so on. When the stress path meets the surface S4, the work
hardening modulus on the loading path becomes equal to that at E.
Mroz's geometrical model can be mathematically stated using a typical situation
sketched in Fig. 3.10. Si and Sl+{ are two yield surfaces whose centers are at O{ and
3.8 Continuum models for uniaxial and multiaxial fatigue 109

Fig. 3.10. Two yield surfaces Si and *S/+1; the translation of surface 5/ along PR is defined by its
relative position with respect to S[+l. (After Mroz, 1967.)

O/+1, respectively. These centers are defined by the position vectors $$ and ^ / + 1 \
respectively, from the origin O. Sj and 5/+i, respectively, are defined by the following
equations:
Pii - 4°
4°) - (otfV1 = 0, (^ - 4/+1)) - {4 =0, (3.15)
l
where/ is a homogeneous function of order nx of its arguments and a 0 and <TQ/+1) are
constants. If P lies on the surface Sh the instantaneous translation of S[ will occur
along PR, where R is a point on S /+1 corresponding to the same direction of outward
normal. The position of R is determined by drawing from O/+1 a vector Oi+\R
parallel to OjP. From the homogeneity of/ in Eq. 3.15 and by denoting the stresses
at P and 7? by off and oV+l\ respectively, one obtains

The translation of St is given by

(3.17)

The parameter d^ is obtained from the continuity condition that the stress point
remain on the yield surface,

l ; r - = 0. d<7 = (3.18)

Mroz's model can be extended to situations where, in addition to translation, the

yield surfaces expand or contract. An important difference between the Mrbz model
and the Prager-Ziegler model is in the direction in which the yield surface translates.
110 Cyclic deformation in poly crystalline ductile solids

The Mroz formulation captures many of the salient features of uniaxial fatigue
deformation described earlier in this section. However, it does poorly for asymmetric
cyclic deformation (White, Bronkhorst & Anand, 1990).

3.8.3 Two-surface models for cyclic plasticity

Dafalias & Popov (1975) and Krieg (1975) independently proposed conti-
nuum models for nonlinearly hardening ductile solids subjected to random uniaxial
and multiaxial cyclic loads. Their approach introduces the concept of a bounding
surface in stress space which always encloses the loading surface. The two-surface
model provides a practical simplification of the aforementioned multiple loading
surface approach of Mroz by analytically prescribing the variation of the plastic
modulus.
Consider a ductile solid which is subjected to a fully reversed stress cycle; the
stress-strain curve for this loading history is shown in Fig. 3.11. The material
deforms elastically along OA; path ABC denotes plastic deformation. At point C,
the material is elastically unloaded along CD. This is followed by plastic loading in
compression along DEF. FG denotes the path for elastic unloading at F and GHBC
represents subsequent plastic reloading. Beyond point B, the line HXr coincides with
the prior loading line BC. Corresponding to each load reversal, three regions, which
characterize the manner in which the plastic modulus Ep changes, can be identified.
Consider, for example, the loading path starting at F. The first part FG represents
elastic deformation. The second part GH represents the plastic portion in which the

Fig. 3.11. Schematic illustration of the line bounds in stress-strain space. (After Dafalias «
Popov, 1975.)
3.8 Continuum models for uniaxial and multiaxial fatigue 111

value of Ep changes from point G to point H. The third part HX' also represents
plastic behavior; however, Ep has a constant value in the region. This third part lies
on lines such as XX' or YY' which provide bounds.
Dafalias & Popov postulated that the instantaneous value of the plastic modulus
Ev is determined by (i) the relative position of the current plastic loading point with
respect to the bounding line (<5ys in Fig. 3.11) and (ii) the amount of plastic work,
wp = Jcrde p , accumulated during plastic deformation prior to the elastic deforma-
tion preceding the current plastic state. The value of 8ys at the initial yield point is <5in.
All loading states lying on a line drawn parallel to XX' in Fig. 3.11 have the same
values of 8ys and Ep.
The generalization of the above model to multiaxial loading conditions in stress
space is schematically illustrated in Fig. 3.12. This figure shows two circles, where the
inner circle with its center at k represents the loading surface and the outer circle with
its center at r represents the bounding surface. Let point a on the loading surface
represent the current plastic state of the material. Given point a, there is a corre-
sponding point b on the bounding surface, and the distance between a and b, denoted
as 8ys, determines the value of the generalized plastic modulus at point a for the
multiaxial case. Point b can be defined in a number of ways: (i) if the two surfaces are
congruent, b can be taken as the point determined from the condition of congruency
with respect to a; (ii) b can be obtained by the intersection of the normal to the
loading surface at a with the bounding surface; or (iii) b can be taken as the point of
intersection of the line drawn through ka with the bounding surface, as shown in Fig.
3.12. If ay are the coordinates at a and a* are the coordinates at b, the distance, <Sys,
can be denned as

= y/(pfj - Oij) (or*. - Otj). (3.19)

Let K be the generalized plastic modulus for the multiaxial case. Assuming the
associated flow rule, Eq. (1.25), and that the plastic strain increment dep is proper-

Fig. 3.12. Schematic illustration of the loading and bounding surfaces in stress space and of the
motion of these surfaces during multiaxial loading. (After Dafalias & Popov, 1975.)
112 Cyclic deformation in poly crystalline ductile solids

tional to the projection da of the stress increment da^ on the unit normal co to the
loading surface,

del-= ^ w t j . (3.20)

In Fig. 3.12, c is the point on the bounding surface where the outward normal to it
lies on the same direction as the normal to the loading surface. If m is the unit vector
along the line that joins a and c, the translation of the bounding surface along m
fulfills the condition that the point of contact is also the current stress state.
Dafalias & Popov formulated various conditions for the deformation and transla-
tion of the loading surface as well as of the bounding surface according to different
hardening rules, particularly with reference to well known kinematic hardening rules.
In general, the inner yield surface is taken to follow kinematic hardening rules while
the outer yield surface follows isotropic hardening. The use of the two-surface model
to rationalize cyclic creep is illustrated in Section 3.9.
Modifications of the two-surface models of cyclic plasticity have been examined by
Moosbrugger & McDowell (1989) within the context of various isotropic and kine-
matic hardening formulations. These workers also show that the two-surface models
for rate-independent plasticity offer superior correlation with experimental data on
nonproportional cyclic deformation in stainless steels.

3.8.4 Other approaches

Bodner & Partom (1975) introduced an internal state variable approach to
characterize the resistance of the material to plastic flow. The internal variable,
referred to as the 'hardness', was taken to be a function of the plastic work. This
approach was extended by Bodner, Partom & Partom (1979) to model uniaxial cyclic
loading of elastic-viscoplastic materials. The evanescent hardening model is a phe-
nomenological approach analogous to the Prager-Ziegler hardening rule, with the
exception that the direction and amount of displacement of the yield surface are
modified by the back stress (e.g., Chaboche, 1986). The endochronic or internal time
approach uses an internal time variable as a measure of the history of deformation
(e.g., Valanis, 1980).
White, Bronkhorst & Anand (1990) have formulated a phenomenological rate-
independent plasticity model with combined isotropic-kinematic hardening. Here,
the isotropic component of the resistance to plastic flow is shown to soften initially
during a load reversal. This softening is taken to represent a dynamic recovery of the
underlying dislocation substructure. The model captures the basic characteristics of
small strain cyclic deformation in uniaxial loading, and some aspects of nonpropor-
tional strain cycling in tension-torsion. However, the predictions deviate signifi-
cantly from experiments for large strain torsion.
3.9 Cyclic creep or ratchetting 113

3.9 Cyclic creep or ratchetting

As alluded to in the preceding section, when a fixed amplitude of cyclic
stresses is imposed on a material, a phenomenon known as cyclic creep or ratchetting
takes place if the plastic deformation during the loading portion is not opposed by
an equal amount of yielding in the reverse loading direction. Figure 3.13(a) shows an
example of this process for a fatigue softening material subjected to cyclic loads with
a tensile mean stress, where cyclic creep occurs in the direction of increasing tensile
strains. Here the accumulation of damage is accelerated by two processes: (i) an
increase in cyclic plastic strain from cycle to cycle as a result of cyclic softening,
and (ii) the displacement of the mean strain to higher tensile strain levels. Similarly, a
compressive mean stress may enhance the conditions for buckling, as shown in Fig.
3.13(&). Cyclic creep can also be found, even under equal tension-compression load-
ing, in materials in which pronounced yield anisotropy exists between tension and
compression. Examples include cast iron, maraging steel and most composites.
From a continuum viewpoint, cyclic creep may be rationalized by recourse to the
two-surface models of cyclic plasticity (Section 3.8.3). Consider, for example, the
case of cyclic creep under a tensile mean stress, Fig. 3.14. Assume that cyclic loading
is imposed between the uniaxial stresses, a2 > 0 and ox < 0, with (a2 — | ax |)/2 > 0.
In Fig. 3.14, Au A2, A3, A4, etc., denote points at which load reversal occurs. kx is
the point in stress space where yielding initiates prior to approaching the bound ax in
compression. Similarly, k2 is the corresponding point for the onset of yield prior to
reaching the bound cr2 in tension. The value of the plastic modulus Ep changes
between kx and ox and between k2 and o2, according to the distance 8ys of the
instantaneous stress point from the bounding lines YY' or XX'\ respectively. As
noted in Section 3.8.3, the smaller the value of <5ys (i.e. the closer the stress point is
to the bounding line), the smaller is the value of Ev.
Assume that at the onset of plastic deformation, the distance between the lines
a = k2 and XX' is smaller than the distance between the lines a = kx and YY'. In this
case, the values of the distance <5ys for tensile deformation are smaller than the
corresponding values for compressive deformation. On the other hand, the plastic
work done before each load reversal, Wv = Jcrde p , is approximately equal.

{a) (b)

Fig. 3.13. Cyclic creep under (a) tensile and (b) compressive mean stress.
114 Cyclic deformation in poly crystalline ductile solids

Fig. 3.14. A rationale for the occurrence of cyclic creep in terms of the bounding surface
concept. (After Dafalias & Popov, 1975.)

Therefore, from Section 3.8.3, the plastic modulus is smaller for the stress states lying
on the loading paths k2o2 than for the ones on kxox. The cyclic loops, therefore, are
not closed and they progressively shift to the right of the strain axis as shown in Fig.
3.14. This argument thus provides one rationale for the occurrence of cyclic creep.
With continued shift to increasing mean strains, the distance between a = k\ and
XX' and that between a = k2 and YY' tend to become equal. At this point, the
plastic moduli also become equal and the loops become stabilized.
Cyclic creep or ratchetting has major implications for fatigue in a variety of
engineering applications. For example, in rolling or sliding contact fatigue, combi-
nations of high normal and shear tractions are imposed on a thin layer of material
near the contacting surfaces. Within this layer, which is 10-50 jim thick in gears and
bearings and about 2 mm thick in railway track, shear strains as high as 100% are
accumulated with repeated applications of the load. Under conditions of free rolling
or low traction (where the traction coefficient, i.e. the ratio of the shear to the normal
traction, /xt < 0.25), an unsymmetrical cycle of shear is imposed on the highly
deforming region which is confined beneath the surface. For high traction
(/xt > 0.25), on the other hand, the critically stressed material lies at the surface.
For this latter case, the material is subjected to a nonproportional load cycle which is
made up of tension, followed by shear, followed by compression. An example of the
application of cyclic creep models to contact fatigue problems is found in the work of
Bower & Johnson (1989), who employed phenomenological nonlinear kinematic
hardening models for analyzing the effects of strain hardening on cyclic creep con-
tact fatigue. Further discussions of cyclic creep and ratchetting can be found in
Section 13.4.4.
3.10 Metal-matrix composites subjected to thermal cycling 115

3.10 Metal-matrix composites subjected to thermal cycling

The deformation mechanisms of precipitation-hardened and particle-rein-
forced alloys under cyclic mechanical loads were discussed in the earlier sections of
this chapter. We now direct attention at continuum analyses of cyclic plastic defor-
mation in metal-matrix composites. The problem considered here pertains to situa-
tions where the reinforced metal is subjected to cyclic variations in temperature. The
stresses and plastic strains in the composite arise as a result of thermal expansion/
contraction mismatch between the matrix and the reinforcement. The thermal
cycling conditions to be analyzed here are such that plastic flow in the matrix
dominates over any creep or thermal shock effects. We seek to derive closed-form
solutions for different characteristic temperatures which signify the onset and spread
of cyclic plasticity in the metallic matrix. As shown later in Chapter 8, these results
for cyclic plasticity are useful for estimating the fatigue lives of reinforced metals, by
invoking different damage and failure criteria. The results presented here are from
Olsson, Giannakopoulos & Suresh (1995).

3.10.1 Thermoelastic deformation

Consider a ductile alloy which is reinforced uniformly with spherical parti-
cles of a brittle material, e.g., a ceramic. Figure 3A5(a) shows a spherical volume
element of the composite, where the outer hollow sphere is the matrix alloy (phase 1)
and the concentric inner sphere is the ceramic reinforcement (phase 2). Each phase is
assumed to be homogeneous and isotropic, with the interface between them
mechanically well-bonded. For the volume element with spherical symmetry, the
field quantities are functions only of the radial position r, the material properties,
and the concentration of the brittle phase. The radial and tangential directions are
identified with subscripts r and 0, respectively. The inner region occupied by the
ceramic particle is 0 < r < rh and the outer region occupied by the metal is

IA71

D F Time

(b)

Fig. 3.15. (a) Geometry of the representative volume element for the thermal cycling analysis for
the metal-matrix composite, (b) Thermal loading history.
116 Cyclic deformation in poly crystalline ductile solids

r{ < r < ro. T h e v o l u m e fraction of the ceramic particle is, / p = (rj/r o ) 3 , while t h a t
occupied by the metal is (1 -fv). The concentration of the particles,/p, can be dilute
or nondilute with only the proviso that the particles be well dispersed within the
matrix without forming an interpenetrating composite. For the small strain analysis
considered here, the radial and tangential strains are related to the radial displace-
ment u by

er=fr, ^ , «,=£<«,). (3-21)

The last of these three equations provides the condition for compatible strains. The
equation for equilibrium in the radial direction is:

^ + ?(<xr-or,) = 0. (3.22)

If both the matrix and the particle initially deform elastically,t the stress-strain
relations are written as

J'^^ (3.23)
where E is Young's modulus, v is Poisson's ratio, and a is the thermal expansion
coefficient, with the subscripts 1 and 2 denoting the matrix and the particle, respec-
tively.
Consider a uniform change in temperature, AT, from an initial stress-free tem-
perature. For an increase or a decrease in temperature, A 7 > 0 o r A T < 0 , respec-
tively. For this pure thermal loading case, where the composite is free of any external
kinematical constraint, the appropriate outer boundary condition for the volume
element in Fig. 3A5(a) is that ar = 0. The stresses in the spherical particle are then
readily determined from straightforward thermoelastic analysis to be
2El (a2-ax)(l-fp)AT
v P cr p p 5 (3.24)
3(1-

and the stresses in the matrix are

In Eq. 3.24, the elastic mismatch parameter Mel is defined as

Ex [1-2V! \-2v2\
(326)
N \
For a metallic matrix with a ceramic reinforcement (c^ > a2) and for a decrease in
temperature, AT < 0, the interface pressure ar2 — —p < 0.
' The thermomechanical properties of the two phases are assumed to be temperature-independent over
the range of thermal excursions considered here. The temperature-dependence, however, can easily be
incorporated within the context of this analysis by recourse to numerical solutions.
3.10 Metal-matrix composites subjected to thermal cycling 117

3.10.2 Characteristic temperatures for thermal fatigue

If the change in temperature AT" from the initial stress-free temperature
becomes sufficiently large, plastic yielding occurs in the ductile matrix. Since the
effective stress in the matrix is

^e = K l - <*r\ I = 2 ( 1 lf\ \P\ ( ~ ) ' (3'27)

the maximum effective stress develops at the interface which is the site for the onset
of plastic flow. The temperature change necessary to initiate plastic flow at the
interface, A7"1? is found by setting (crQ)l = ay at r = ri? where ay is the yield strength
of the matrix:

(3.28)

The plastic zone which commences at the interface spreads outwards.

The spread of the plastic zone is analyzed conveniently by assuming that the
matrix is an elastic-perfectly plastic solid. Within the volume element shown in
Fig. 3.15(a), three deformation regions can be identified: (i) the volume occupied
by the reinforcement particle within which only elastic deformation occurs, (ii) an
elastic-plastic region, r-x < r < rp, within which (crQ){ = cry, and (iii) an elastic outer
region, rp < r < ro, with a stress-free boundary at r = ro. Continuity of radial stress
and displacement across the boundaries between these regions and the condition that
{aQ)x = ay, lead to
\(a2-al)ATl\ =
or (1 - Vl)
21ogp +- 1 - Mi (3.29)

This equation provides a direct connection between the size of the plastic zone rp and
the magnitude of thermal strain (a2 — a{)ATi, and reduces to Eq. 3.28 when rp = r{.
As the plastic zone continues to spread under a monotonic temperature change
| AT|, another critical value of | AT|, which we shall term | A7"31, is reached at which
the entire matrix becomes plastic. The magnitude of AT3 is determined by setting
rp = ro in Eq. 3.29:
- v O f l 2M eli 1
(3.30)
3
-«ill/P J
Now consider temperature cycles, 0 < |A7"| < ATa, commencing at the stress-free
temperature,! Fig. 3.15(&). The first unloading in the thermal cycle corresponds to
the excursion from point A to point B.
If the yield strength of the matrix is the same in tension and compression, it can be
readily shown that there exists a characteristic temperature change, AT2 = 2AT1?

' Equations 3.28 and 3.30 pertain to thefirstthermal loading ramp from | AT| = 0 to \AT\ = ATa (point
A in Fig. 3.15(6)). In general, AT and ATa can be positive or negative. For simplicity, we consider
positive values of the characteristic temperatures in subsequent discussion.
118 Cyclic deformation in poly crystalline ductile solids

below which a zone of reverse plasticity does not occur. In other words, if
|A7a| < AT2, there is no accumulation of plastic strains during thermal cycling.
Therefore, A T2 represents a shakedown limit for the thermal fatigue of a metal-
matrix composite. For \AT.d\ < AT2, only elastic conditions prevail after the first
unloading (i.e. after point B).
If, on the other hand, |ATa| > AT2, a zone of reversed plastic flow of radius rc
develops at the particle-matrix interface. Repeated thermal cycling causes rc to
expand outward. Referring to Fig. 3.15(6), if the temperature is first increased
(decreased) from some initial stress-free temperature (point O) to a maximum (mini-
mum) value at point A, and then decreased (increased) to the current temperature
A T, the size of the reversed or cyclic plastic zone rc is given by
\(a2-al)(AT!i-AT)\ =

2iog
ftH 6
2<7y(l —^ i ) [ i - c i n - i " i i _ _ i ' c i , - i i / - c \i

The right hand sides of Eqs. 3.29 and 3.31 differ only by a factor of 2 and by a
( 3 31)

change from rp to rc. A remarkable feature of Eqs. 3.28-3.31 is that both the
monotonic and cyclic plastic zones initiate at the particle-matix interface and spread
monotonically outwards during cyclic variations in temperature.
As the cyclic plastic zone spreads outwards, there exists another critical tempera-
ture at which the entire matrix begins to undergo reversed plastic yielding. This
characteristic temperature AT4 is obtained by setting rc = r0 in Eq. 3.31, from
which it is readily seen that AT4 = 2AT3. Figure 3.16 schematically shows the
spread of rp and rc for different values of A Ta with respect to the various character-
istic temperatures.
For the uniform spatial distribution of spherical particles, the conditions that
\AT{\ < \AT2\ and that lAT^I < |Ar 3 | always hold. However, the condition that
IA 721 < IA 7^ | need not always be true, i.e. the first monotonic temperature change
from O to A in Fig. 3.15(6) may cause complete yielding of the matrix with no cyclic
plastic zones induced upon subsequent unloading (from A to B). From Eqs. 3.28 and
3.29 and from the condition for shakedown (AT2 = 2 A 7^), it can be shown that
there exists a critical concentration of the reinforcement, / p = /p*, at which
A 7^ = A 7*3, which obeys the equation:

1 _ 2/p* + ^Sl/ p *{log(^) + 2(1 -/p*)} = 0. (3.32)

The value of/p is unique for afixedcombination of the matrix and reinforcement,
and is bounded by 0.203</p <0.586. If/p > / p , AT2 > AT3, and the composite
matrix undergoes no plastic strain accumulation during thermal cycling, although
the matrix is fully yielded during the first thermal excursion from the initial stress-
free state.
3.10 Metal-matrix composites subjected to thermal cycling 119

\ATa\ < A77! ATX < |Ar a | < AT3 AT3 < \ATt

(b)

\ATa\<AT2 AT2<\ATa\<AT4 AT4<\ATa\

(c)

AT3 < \ATa\ < AT2 A7\ < \ATa\ < AT4 AT4 < \ATa\

Q elastic response
£ 2 plastic zone (monotonic)
8 2 reversed plastic zone (cyclic)

Fig. 3.16. The evolution of monotonic and cyclic plastic zones at different temperature
amplitudes, (a) Monotonic thermal loading corresponding to point A in Fig. 3.15(7?).
(b) Temperature cycling for AT2 < AT3 corresponding to points B, C, D,... in Fig. 3.15(6).
(c) Temperature cycling for AT2 > AT3 corresponding to points B, C, D,... in Fig. 3.15(6).

3.10.3 Plastic strain accumulation during thermal cycling

Consider the situation where an elastic-ideally plastic metal reinforced with
ceramic particles is thermally cycled in such a way that \AT2\ < \ATa\ < |A7"3|,
which is a possible scenario for/ p </p*. The general expressions for the tangential
and radial components of the plastic strain in the matrix are

(3.33)

where S = sign{(a2 — oi\)AT} = {(a2 — ai)AT}/\(a2 — a\)AT\ is either +1 o r - 1 ,

depending on whether the plastic flow is forward or reversed. The constant C\ is
obtained from the continuity conditions to be

for |Ar a | < |Ar 3 |. (3.34)

Since the radial stress and the effective stress (and hence the tangential stress) are
continuous across the plastic zone boundary, the plastic strains are also continuous.
120 Cyclic deformation in poly crystalline ductile solids

From Eq. 3.33, the effective plastic strain is found which, for the present geometry,
reduces to
1
g (3.35)
because of proportional plastic strains. At the initial stress-free temperature (point
O), 4 = 0.
Beginning with the first thermal excursion O-A, the accumulated plastic strain is
found by substituting Eq. 3.34 in Eq. 3.35 and integrating Eq. 3.35:

,,<,<,„. ,3.36)

Unloading from A to B in Fig. 3.15(&) causes a reversed yield zone of radius rc to

develop. Inside this zone, <^Jf(r) differs from that given in Eq. 3.36. Upon changing
the temperature from B to C, theflowcondition is satisfied up to rp, but in the region
rc < r < rp, the plastic strains remain unaltered. Thus, for |AT2| < |ATa| < |Ar 3 |,
and constant-amplitude temperature cycling, 0 < \AT\ < |ATa|, the plastic strains
change only inside the cyclic plastic zone. The radius of this zone, rc, is found from
Eq. 3.31 by setting AT = 0 (fully unloaded state). Using Eqs. 3.34 and the con-
tinuity of plastic strains, the accumulated effective plastic strain after NT tempera-
ture reversals is found as

If IAT3I < | A r a | < |AT4|,

CPI,NT _ 2ay(l - vQ [/ro\3/ | / p ^i|o? 2 - ttil|Ara - Ar 3 [\ _ 1
^y(l - Vi) / J

The reversed flow zone size rc in Eq. 3.38 is given by Eq. 3.31. The first term on the
right hand side of Eq. 3.38 gives e^ T during the first temperature excursion from O
to A, and the second term gives the accumulated value in the next (NT — 1) tem-
perature reversals. It should be noted that the rate of plastic strain accumulation in
Eqs. 3.37 and 3.38 is the same, i.e. the second terms on the right hand side of these
two equations are identical. Thus, the difference in the magnitude of the accumu-
lated plastic strain is only that due to the difference developed in the first tempera-
ture excursion.

3.10.4 Effects of matrix strain hardening

The preceding results were restricted to metallic matrices which do not show
any strain hardening. If the matrix undergoes cyclic strain hardening, the following
modifications to the results are anticipated.
3.10 Metal-matrix composites subjected to thermal cycling 121

(1) For cycling between fixed temperature limits, the four characteristic tem-
peratures, ATh / = 1 , . . . , 4 , are constants for an elastic-ideally plastic
matrix. For a strain-hardening matrix, however, these temperatures change
with the number of thermal cycles; they can still be defined based on the
plastic strain distribution.
(2) The accumulation of plastic strain near the particle-matrix interface
increases the flow strength of the hardening matrix. Further accumulation
of plasticity is thereby suppressed.
(3) The result of the preceding section that plastic strain accumulation occurs
only within the cyclic plastic zone, still applies to the hardening matrix. The
rate of accumulation, however, is affected by the hardening rate.

Consider a matrix alloy with isotropic, linear strain hardening //, which
captures the cyclic hardening characteristics of the matrix. The matrix yield
strength ay is now modified to include hardening in the following way:
cry +Heller). Let H = 27/(1 - vx)/Ex. It can be readily shown (Olsson,
Giannakopoulos & Suresh, 1995) that the plastic strain increment during each
load reversal is explicitly given by

For H = 0 (no hardening), the results of the preceding section are recovered. An
examination of Eq. 3.40 reveals that Aej^ 7 -> 0 as NT -> oo. Therefore, unlike the
case of an ideally plastic matrix for which the plastic strain accumulation during
thermal fatigue is unbounded, the accumulated plastic strain for a strain-hardening
matrix asymptotically approaches a limiting value at large numbers of thermal cycles
in the following manner:

(^)3|[(^)3] (3-40)
where Ae^ 1 is the plastic strain accumulated after the first thermal excursion. The
limiting strain in Eq. 3.40 has a maximum value at the particle-matrix interface. If
the ductility of the matrix alloy exceeds this maximum value, an endurance limit for
thermal fatigue is expected. This temperature endurance limit is higher than A T2
because of strain hardening.
All of the foregoing analytical results pertain to situations where Eu v1? a1? a2,
ay and H are temperature-independent. The same approach can be adapted for
analyzing thermal fatigue by including the temperature-dependence of these prop-
erties by recourse to numerical simulations (see, for example, Olsson et al.,
1995).
122 Cyclic deformation in poly crystalline ductile solids

3.10.5 Example problem: Critical temperatures for thermal fatigue

in a metal-matrix composite
Problem:
A 2024-T6 aluminum alloy, reinforced with 20 volume % of spherical SiC
particles, is thermomechanically processed at 250 °C. At this temperature, the
composite is almost free of internal stresses and its reinforcement is well-bonded
to the matrix. The isotropic properties of the matrix are: ax = 22 x 10~6 °C~1,
Ex = 72 GPa, vx = 0.33, ay = 393 MPa, and melting temperature,
?meit = 660 °C. (All the symbols are defined in Section 3.10.1.) The matrix
exhibits essentially no strain hardening. The isotropic properties of the SiC
particles are: a2 = 4.7 x 10"6 °C~1, E2 = 450 GPa, v2 = 0.17, and
Tmelt = 660 °C.
(i) If the reinforced alloy is uniformly cooled to room temperature (25 °C),
what is the temperature at which plastic yielding begins during cooling
from the processing temperature? (In order to obtain analytical results,
assume that the mechanical properties of both phases of the composite
are independent of temperature for the range of temperatures and heat-
ing/cooling rates considered.)
(ii) What is the volume fraction of the matrix which has undergone plastic
yielding upon cooling?
(iii) If the composite is thermally cycled between room temperature and the
processing temperature, compute the plastic strain accumulation per
cycle.
(iv) Now suppose that the yield strength of the matrix decreases linearly with
increasing temperature, and that the Poisson's ratios, Young's moduli
and thermal expansion coefficients for both phases remain temperature-
independent, for the range of temperatures considered. Explain briefly
how the temperature dependence of yield strength of the matrix would
affect (a) the critical temperatures A 7^ and A 7^ and (b) the monotonic
plastic zone radius rp.
Solution:
It is given that:/p - (rjrof = 0.2, and r proc = 250 °C, r room = 25 °C,
and |AT| = r proc - r room = 225 °C. Since \AT\ is less than one half of the
melting temperature, it may be assumed that creep effects are less dominant
compared to plasticity effects (i.e. when long hold periods at temperatures are
not involved). Denoting the properties of the matrix by the subscript T and
those of the particle by the subscript '2' and substituting the appropriate
values in Eq. 3.26, it is found that the elastic mismatch parameter
Mel = 0.35.
3.11 Layered composites subjected to thermal cycling 123

(i) Substituting the numerical values of the various terms in Eq. 3.28, we
find that the temperature change, from the initial stress-free temperature
(250 °C) at which plastic yielding begins is: | AT^ | = 172 °C, i.e. at 250 -
172 - 78 °C.
(ii) To find the plastic zone radius rp for a temperature change of
\AT\ = 225 °C, replace A ^ in Eq. 10.2 by \AT\. Substituting the appro-
priate numerical values for |A!T| and the various material parameters, it
is seen that
/r\3 /r\ 3 /r\3
1.0467M -0.2333 log -^1 =1.0644, or U « 1.025. (3.41)
\ri/ Vi/ VJ
The volume fraction of the matrix which has undergone plastic yielding
upon cooling from the processing temperature to room temperature is:
J J /• f A. \ 3 1
-1 . (3.42)

Substituting the value of rp from Eq. (2) into this equation, it is seen that
Kpl « 0.63%.
(iii) We note that the characteristic temperature change below which a zone
of reversed plastic flow does not develop in the matrix is:
|AT 2 | = 2|A7^!| = 344 °C. Since the temperature range for thermal
cycling |AT| = 225 °C < |AT 2 |, it is apparent that repeated thermal
cycling between room temperature and the processing temperature will
not lead to any continued accumulation of plastic strains in the matrix,
(iv) (a) As the yield strength decreases with increasing temperature, both A 7^
and A T3 will be lowered compared to the case where the yield strength is
independent of temperature, (b) The monotonic plastic zone size, rp,
increases because of the lower yield strength at higher temperatures.
There is a corresponding increase in the volume of the matrix material
which has undergone plastic yielding.

3.11 Layered composites subjected to thermal cycling

Layered materials represent a broad class of composites which find diverse
applications in mechanical, thermal, electronic, magnetic, ferroelectric, optical and
biomechanical components or devices. They also facilitate maintenance of clearance/
tolerance or enhance appearance and biocompatibility. In many applications, fluc-
tuations in externally imposed stresses or pulsations in internal stresses arising from
such factors as the thermal expansion mismatch between the layers during repeated
temperature excursions can lead to cyclic deformation and fracture. Examples of
layered structures in which thermal cycling or thermo-mechanical fatigue is a topic
124 Cyclic deformation in poly crystalline ductile solids

of interest include: (i) laminated composites comprising organic or inorganic

matrices and discontinuous or continuous reinforcements, (ii) thermal-barrier or
wear-resistant coatings comprising a ceramic outer coating and an in-between metal-
lic bond coat on a metallic substrate, and (iii) semiconductor devices with patterned
metallic conduction lines, passivated by a glassy layer or a ceramic or a polymer, on
the Si single-crystal substrates.
In this section, we consider the cyclic deformation response of layered structures
subjected to fluctuations in temperature. This section begins with a basic treatment
of the thermoelastic deformation of a bilayer by recourse to classical beam/plate
theories. This is followed by the derivation of stresses in thin films on substrates
which are subjected to thermal excursions. Attention is then directed at some critical
temperatures for the occurrence of distinct transitions in the cyclic deformation
response of a ductile layer in a metal-ceramic bilayer system which undergoes cyclic
variations in temperature, similar to the analyses presented in the preceding section
on metal-matrix composites.

3.11.1 Thermoelastic deformation of a bilayer

Consider a general bilayer, schematically shown in Fig. 3.17(a), which is
subjected to uniform temperature changes at all times. The uniform thicknesses of
layers 1 and 2 are hx and h2, respectively. For a start, the interface between the two
layers is assumed to be mechanically well bonded, and the two layers are taken to
exhibit in-plane elastic and thermal isotropy with Young's moduli, Ex and E2,
Poisson's ratios, v{ and v2, and coefficients of thermal expansion, ax and a2. The
in-plane shape of the bilayer is a rectangle with length Lx and width Ly.
Let the bilayer plate be flat and devoid of any internal stresses at some initial
reference temperature JQ, which could be identified with the processing, diffusion-
bonding or softening temperature. Let the temperature of the bilayer be now chan-
ged uniformly by AT to some instantaneous value T (with T — To = AT > 0 denot-
ing heating and AT < 0 denoting cooling) in such a way that the entire plate is
always at the same temperature T. The thermal expansion or contraction mismatch
between the layers causes gradients in stresses to develop through the thickness of
each layer. As a result, the initially flat plate bends with a curvature /cR (which is the
inverse of the radius of bending). If Lx > h\ + h2 and Ly ^> hx + h2, a state of
equi-biaxial stresses exists in the plate such that oxx = oyy and exx = eyy (i.e. the
curved plate has the same state of stress as the skin of a balloon which is internally
pressurized), t

' Near the free edges of the bilayer where the interface intersects the free surfaces, the stresses have to be
modified locally so as to satisfy the stress-free boundary conditions at the free surfaces of the uncon-
strained bilayer plate. Consequently, there develops a three-dimensional stress state which comprises
both in-plane and out-of-plane normal and shear stresses. The size of this 'edge zone', measured from
each edge, is of the order of the total layer thickness. The analysis presented in this section is valid only
away from this edge-zone.
3.11 Layered composites subjected to thermal cycling 125

layer 1

layer 2

(a)

|AJC|

onset of
inelastic deformation ideally plastic
\

elastic ^^ If
' ^ elastic

residual
curvature
X 1
unloading

onset of reverse
inelastic deformation

1 _ i\AT\
(b) AT3

Fig. 3.17. (a) Geometry of the bilayer and the associated nomenclature, (b) Schematic
representation of the variation of curvature with temperature for a metal-ceramic bilayer where
the metallic layer is elastic-perfectly plastic. All material properties are assumed to be
independent of temperature.

The thermoelastic strain in the bilayer, in regions away from the free edges, con-
sists of two parts: (i) an in-plane normal strain, e0, which arises from a uniform
stretch or contraction, and (ii) the strain due to bending, KRZ, where z is the thickness
coordinate:
€xx(z) = eyy{z) = e(z) = eo + KRZ. (3.43)

The only nonzero components of stress in the two layers are

Ex(z)
yyA l
\-vx{z)
-Mz)-[a2(z)AT]}. (3.44)
- v2(z>

The subscripts T and '2' refer to quantities associated with layers 1 and 2, respec-
tively, in Fig. 3.11 (a). Static equilibrium dictates that the net force and the net
126 Cyclic deformation in poly crystalline ductile solids

moment arising from the stresses in Eq. 3.44 should be offset by any externally
imposed force Fap and moment, M ap . For the unconstrained bilayer plate subjected
only to a uniform temperature excursion, these force and moment equilibrium con-
ditions give:

o2(z)&z + ox(z)dz = 0, and

J-h2 Jo
f a2z(z)dz + f l axz(z)dz = 0. (3.45)
l-h2 Jo
Combining Eqs. 3.43-3.45, and solving for €0 and /cR yields
AT
H C
+C + C(4h2 (3-46)

and

/cR = —6h x h 2 (hi + h2), (3.47)

C4
where
2 A 2 A

C\ = E\hi, C2 = E2h2, C3 = E\E2h\h2,

C4 = d + C2 + [C3(4A? + 6hxh2 + 4A|)],
i)! =ElE2(al -a2)AT,
Ei = ——, and E2 = -^—. (3.48)
1 — Vi 1 — v2
Substituting Eqs. 3.46-3.48 into Eq. 3.44, it is readily shown that the stresses at the
outer surfaces of the two layers and at the interfaces are:

(3-49)

Note that the stresses vary linearly with z within each layer to a maximum value at
the interface, and that there occurs a sharp jump in the magnitude of the stress at the
interface between the two layers. While the foregoing analysis pertains to a bilayer,
the same approach can be used to analyze the thermoelastic deformation of a multi-
layer comprising any number of layers. In such a case, the equilibrium equations, Eq.
3.45, can be solved numerically using a personal computer where additional effects
3.11 Layered composites subjected to thermal cycling 127

involving temperature-dependence of material properties in the multilayer, plastic

yielding and strain hardening in the ductile layers, in-plane anisotropy of elastic
properties, continuous gradients in composition and properties through one or
more layers, steady-state creep, as well as heat conduction arising from nonuniform
temperature variations through the thicknesses of the layers can all be incorporated
(e.g., Finot & Suresh, 1994).

3.11.2 Thin-film limit: the Stoney formula

In many practical applications, thin coatings are deposited on much thicker
substrates. Examples include ceramic coatings on metal alloys for protection against
elevated temperatures, environmental attack or contact fatigue damage. A very
useful result, commonly known as the Stoney formula (after Stoney, 1909), for the
estimation of the uniform stress in a thin film on a thick substrate can be derived
from the above bilayer result.
Consider the thin-film limit of the bilayer shown in Fig. 3.17(a), where h\ <^C h2.
With this geometric condition, Eq. 3.47 becomes

KR = 6=J--j(«i -a2)AT, hx<^h2. (3.50)

In addition, Eqs. 3.48 and 3.49 give, for the thin-film limit,
O\\z=zhx ^ —Eiipt\ —a2)AT, and cr\\z=o ^ —Ei{ot\ —a2)AT. (3.51)
In other words, the equi-biaxial stress in the thin film is essentially uniform. It is also
readily seen from Eq. 3.49 that, for hx <^ h2, the magnitude of the stresses in the thick
substrate is very small.
Combining Eqs. 3.47 and 3.51, and denoting the various parameters associated
with the film and the substrate with the subscripts 'film' and 'sub', respectively,

(3.52)

where Rc 0 is the initial radius of curvature of the thin-film/substrate system at some

reference temperature and Rc is its radius of curvature after some temperature
change AT". In other words, if the radius of bending before and after a temperature
change can be measured experimentally, the stress in the film is computed easily
using Eq. 3.52.f Note that once the curvature change is known for a thermal excur-
sion, the film stress is determined with the knowledge only of the film and substrate
thicknesses and of the elastic properties of the substrate. In other words, the elastic
properties of the film, which are often very difficult to obtain, are not needed. The
minus sign for the film stress in Eq. 3.52 simply implies that if the bilayer is convex

' In view of its simplicity, the Stoney equation, Eq. 3.52, is commonly used to determine the thin film
stresses in the microelectronics and structural coatings industries where scanning laser interferometry
methods are widely used to determine the radius of curvature of the thin-film/substrate system before
and after thermal excursions.
128 Cyclic deformation in poly crystalline ductile solids

shaped on the thin-film side (i.e. positive /cR), then the stress in the film should be
compressive. From Eq. 3.50, we see that such a situation arises when the coefficient
of thermal expansion of thefilmis higher than that of the substrate and the bilayer is
subjected to heating from a stress-free initial reference temperature.

3.11.3 Characteristic temperatures for thermal fatigue

Four different characteristic temperatures, which signify different distinct
transitions in the evolution of cyclic deformation, were defined in Section 3.10.2 for
the thermal cycling of a metal reinforced with brittle spherical particles. Four such
characteristic temperatures for the thermal fatigue can also be analytically derived
for the bilayer in Fig. 3Al(a) (for any arbitrary combination of layer thicknesses hi
and h2) where one or both layers is a metal (Suresh, Giannakopoulos & Olsson,
1994).
Let layer 1 in Fig. 3.17(a) be a metal and layer 2 be a brittle solid, such as a
ceramic. With this choice, we see that in general, (ax — a2) > 0. Let the plastic
deformation of the metallic layer be characterized by an elastic-perfectly plastic
deformation behavior, with a yield strength, ay{ (which is the same in magnitude
but opposite in sign for tension and compression). Layer 2 remains a linear elastic
solid throughout any changes in temperature. Let the bilayer be subjected to uniform
temperature changes which fluctuate with a constant amplitude ATa.
Starting with the stress-free temperature To, consider the first uniform change in
temperature |AT|. Initially both layers deform elastically. When |AT| reaches a
critical value |A7\|, the metallic layer begins to yield plastically at the interface.
Setting the stress at the interface in layer 1 to equal ayl for this critical condition,
it is found that

(3 53)
-
where the various parameters are defined in Eq. 3.48.
If now the temperature continues to change monotonically beyond lAT^I, the
plastic zone which initiates at the interface spreads outwards to the free surface of
the metallic layer. After a further temperature change, there occurs a critical condi-
tion, \AT\ = \AT3\ that the entire metallic layer becomes fully plastic, where

(3.54)

If the metallic layer does not strain harden, this limit temperature for full yielding
also signifies the condition for a limiting curvature, /cR L. That is, any further change
in temperature beyond lAT^I does not cause any change in the curvature of the
bilayer. This limiting curvature is
Exercises 129

h T , , , , , , (3-55)
"2^2

If the applied temperature amplitude for thermal cycling | A Ta | is always smaller

than |AT7!|, plastic deformation will not occur in the metallic layer. If
| AT"! | < \ATa\ < | AT3\, the metallic layer will not undergo complete plastic yielding.
Figure 3.17(Z?) schematically shows the evolution of curvature as a function of tem-
perature for a metal-ceramic bilayer where the metallic layer exhibits elastic-per-
fectly plastic response.
For fluctuations in temperature, there are two additional limiting possibilities for
the evolution of cyclic plastic zones in the metallic layer: (i) there occurs no reversed
plastic yielding in layer 1, or (ii) the limiting curvature (KR = KR L ) which develops at
| ATJ = I Ar 3 | is exactly reversed during each temperature reversal. Consider case (i)
for which a characteristic temperature, | A 7^21 = 21A ^ |, exists. When |Ar a | < |Ar 2 |,
no reversed plastic zone develops and thermal cycling does not lead to the accumu-
lation of plastic strains in the metallic layer. It can be shown (e.g., Suresh,
Giannakopoulos & Olsson, 1994) that the highest characteristic temperature corre-
sponding to the complete reversal of the limiting curvature from KR = KRL to
KR = — K R L, case (ii), is: | A7"4| < 21AT^ |. For a general bilayer with any general
combination of hx and /z2, it is typically seen that |AT4| > |Ar 3 | > |AT2| > | A ^ |.
For the thin-film limit,

ATX = AT3 =W7^1 -, h « h2,

Ex(ax -a2)
AT2 = AT4 = 2AT3, hx<^h2. (3.56)

The effects of temperature-dependence of material properties, strain hardening in

the metallic layer, and free edges (where singular fields and multiaxial stress
states develop) on the evolution of monotonic and cyclic plastic deformation
in the metal-ceramic bilayers are addressed in Suresh, Giannakopoulos & Olsson
(1994).

Exercises
3.1 It was shown in Chapters 1 and 2 that the resolved shear stress in a single
slip system is related to the applied stress by the Schmid factor, M. For a
poly crystal, a = M T r, where M T is commonly referred to as the Taylor
factor. The value of M T is computed such that the continuity of slip at
the grain boundary satisfies the requirement for five independent slip sys-
tems to operate in each grain. The values of MT are computed by invoking
the principle of virtual work. Convince yourself, by consulting appropriate
130 Cyclic deformation in poly crystalline ductile solids

references and standard textbooks on mechanical metallurgy, that for ran-

dom orientations of grains,
(a) For FCC polycrystals, MT = 3.06.
(b) For BCC polycrystals, MT = 2.00.
(c) Taylor-type calculations are not possible for HCP polycrystals where
twinning and nonbasal slip influence deformation.
3.2 The Taylor factor MT is based on the assumption of multiple slip. Sachs
formulated an alternative orientation factor M s (= 2.24 for FCC polycrys-
tals), assuming that individual grains deform in single slip. (Consult stan-
dard textbooks on mechanical metallurgy where appropriate references can
be found.)
(a) What are the limitations of this assumption in terms of plastic
strain?
(b) It has been observed that the orientation factor in general is closer to the
Sachs factor for moderately deformed crystals, but then assumes larger
values approaching MT at higher imposed strains. Why?
3.3 Silver solder joints can withstand a much higher normal stress than a tensile
specimen made entirely of the solder material. Why?
3.4 Discuss possible effects of reinforcement volume fraction and shape (i.e.
spheres, particles with sharp corners, and whiskers) on the development
of the Bauschinger effect in metal-matrix composites.
3.5 Two engineering materials are subjected to fully-reversed, strain-controlled,
cyclic deformation at room temperature. The first, a substitutional solid
soution alloy, exhibits an isotropic hardening response. The second is a
composite whose matrix is identical to the first alloy; spherical ceramic
particles, 25% by volume, are uniformly dispersed in this matrix.
(a) Speculate on the dislocation-particle interactions in the second material
during cyclic plastic deformation.
(b) Schematically sketch the shape of the cyclic stress-strain curve during
repeated tension-compression loading.
(c) Discuss the type of hardening rule you would use to model the cyclic
constitutive behavior of the composite.
3.6 Using the bounding surface concept and a rationale similar to that used in
connection with Fig. 3.14, provide an explanation for the occurrence of
cyclic creep in the direction of increasing compressive strain.
3.7 Discuss the cyclic deformation of a material in terms of the two-element
Masing model, assuming elastic-perfectly plastic response.
(a) Construct the hysteresis loops for a small amplitude at which only the
softer elements yield plastically and for a larger amplitude at which both
elements yield plastically.
Exercises 131

(b) Show that the so-called 'permanent softening' is expected only in one of
the two cases and that the amount of permanent softening is equal to
twice the internal back stress in the soft elements.
(c) Considering both cases, can permanent softening be considered an
unambiguous measure of the back stress?
(d) Describe the occurrence of the Bauschinger effect in terms of deforma-
tion-induced internal stresses and the yielding of the softer elements
upon load reversal.
3.8 Discuss the effects of cyclic strain hardening on the shakedown limit derived
in Section 3.10.2 for the metal-matrix composite subjected to thermal cycling.
3.9 An aluminum film, 1 urn in thickness, is deposited onto a silicon substrate
which is 500 fim thick and 100 mm in diameter. The isotropic properties of
Al and Si are: EAl = 66 GPa, vA1 = 0.33, aM = 23 xlO'6 °C~\ Esi = 130
GPa, vsi = 0.28, asi = 3xl0~ 6 °C~1. At some reference stress-free tem-
perature, this thin-film/substrate system is flat (i.e. zero curvature). This
bilayer is now uniformly cooled by 50 °C.
(a) What is the average stress in the aluminum film?
(b) Is it tensile or compressive?
(c) Describe the direction in which the bilayer bends during the above
thermal excursion.
(d) If the yield strength of the thin aluminum film is 140 MPa, what is the
temperature change needed to cause plastic yielding in the aluminum
film?
(e) If the thin-film/substrate system is thermally cycled between the stress-
free initial temperature and some high temperature TcyciQ9 what is the
minimum value of r cycle needed to induce fully reversed plastic flow in
the entire aluminum film?
3.10 A 50 mm x 50 mm bilayer plate is made by diffusion bonding a plate of pure
Ni, 3 mm in thickness, to a plate of pure A12O3, 2 mm in thickness, at
827 °C. The isotropic properties of Ni are: Em = 214 GPa, vNi = 0.31,
and a Ni = 17.8 xlO" 6 °C" 1 at 827 °C and 13.4 xlO" 6 °C" 1 at 20 °C. The
yield strength of Ni as a function of temperature, <7yNj(r), are: 148 MPa
(20 °C), 140 MPa (227 °C), 115 MPa (427 °C), 69 MPa (627 °C) and 45 MPa
(827 °C). The isotropic properties of the polycrystalline A12O3 plate are:
£AI 2 O 3 = 380 GPa, vAl2o3 = 0.25, and aAl2o3 = 9.4 x 10~6 °C~l at
827 °C and 5.4 x 10"6 °C~l at 20 °C. You may ignore the strain hardening
characteristics of Ni for the purpose of this problem.
(a) If the bilayer is uniformly cooled to room temperature (20 °C), what is
the temperature at which plastic yielding begins during cooling from the
bonding temperature?
(b) Is it possible to cause yielding of the entire Ni layer before reaching the
room temperature?
 CHAPTER 4

Fatigue crack initiation in ductile solids

The initiation of fatigue cracks is an event whose very definition is strongly

linked to the size scale of observation. For example, materials scientists are likely to
consider the nucleation of flaws along persistent slip bands as the initiation stage of
fatigue failure, whilst a mechanical engineer may associate the resolution of crack
detection with the threshold for crack nucleation. Between this wide range of view-
points lies a variety of failure mechanisms that are affiliated with the inception of
microscopic flaws at grain boundaries, twin boundaries, inclusions, micro structural
and compositional inhomogeneities, as well as microscopic and macroscopic stress
concentrations. The differences in the approaches to fatigue crack initiation consti-
tute the fundamental distinction between the fatigue design philosophies currently
practiced in industry. From a scientific standpoint, developing a quantitative under-
standing of crack initiation processes must be regarded as one of the most important
tasks.
In this chapter, attention is first directed at the mechanisms of fatigue crack
initiation in nominally defect-free (unnotched) pure metals and alloys, and commer-
cial materials. Models for fatigue crack initiation are described and their significance
and limitations are pinpointed. Also addressed are the mechanisms by which fatigue
cracks initiate ahead of stress concentrations under fully compressive cyclic loads.
Continuum aspects of crack initiation based on stress-life and strain-life approaches
are addressed in Chapters 7 and 8, respectively, where the initiation of fatigue cracks
at stress concentrations under tension and tension-compression fatigue are treated.
Possible ways in which the seemingly conflicting viewpoints based on crack initiation
and crack growth can be brought together are described in Chapter 15. Fatigue crack
initiation in brittle solids and noncrystalline solids are considered in Chapters 5 and
6, respectively.

4.1 Surface roughness and fatigue crack initiation

The origin of fatigue cracks in metals and alloys of high purity is often
rationalized by mechanisms of the type first proposed by Wood (1958). The basic
premise of Wood's postulate is that repeated cyclic straining of the material leads to
different amounts of net slip on different glide planes. The irreversibility of shear
displacements along the slip bands then results in the 'roughening' of the surface of
the material. This roughening is manifested as microscopic 'hills' and 'valleys' at sites

132
4.1 Surface roughness and fatigue crack initiation 133

where slip bands emerge at the free surface. The valleys so generated function as
micronotches and the effect of stress concentration at the root of the valleys
promotes additional slip and fatigue crack nucleation.

4.1.1 Earlier observations and viewpoints

The first documentation of slip-induced surface roughening during fatigue
was by Forsyth (1953). He reported that, in a solution-treated Al-4 wt % Cu alloy,
thin ribbons of the metal (0.1 um thick and 10|im long) 'extruded' at the specimen
surface from the persistent slip bands. Similar extrusions were also seen in single
crystals and polycrystals of silver chloride (Forsyth, 1957). The valleys and hills
formed on the fatigued surface are commonly referred to as 'intrusions' and 'extru-
sions', respectively. The formation of an intrusion-extrusion pair during fatigue was
identified by Forsyth & Stubbington (1955) in Al-4.5 wt% Cu and by Cottrell &
Hull (1957) in Cu. Figures 4A(a) and (b) show a pair of intrusions and extrusions,
which were formed within 1 % of the expected fatigue life, along slip bands in Cu
cyclically strained at —183 °C. Both micrographs were taken from the surface of the
same fatigue specimen.
Wood's hypothesis on the creation of surface roughness due to the to-and-fro
motion of slip bands does not explain why the intrusions progressively deepen. In
an attempt to rationalize the development of net slip offsets, Mott (1958) proposed a
qualitative model. His suggestion was that screw dislocations, moving along differ-
ent paths in the slip bands of a crystal during forward and reverse glide, repeat their
paths by cross slip. The screw dislocations complete a circuit during a fatigue cycle;
the volume encompassed by the circuit is then translated parallel to the dislocation
by a distance equal to its Burgers vector. This displacement manifests itself in the

Fig. 4.1. (a) Intrusions and (b) extrusions along slip bands in polycrystalline Cu fatigued at
-183 °C. (From Cottrell & Hull, 1957. Copyright The Royal Society, London. Reprinted with
permission.)
134 Fatigue crack initiation in ductile solids

form of an extrusion at the specimen surface. Although Mott's proposal formed the
basis for a number of subsequent models for surface roughening, its basic feature
that screw dislocations travel in a closed circuit has not been convincingly substan-
tiated by experiments. Kennedy (1963) emphasized the need for a gating mechanism
which would modify the forward-reverse oscillations of screw dislocations into irre-
versible displacements. The formation of obstacles to dislocation motion, such as
creation of jogs as a consequence of edge-screw intersections and the intersection of
two screw dislocations with a third dislocation at a node in a free surface, have all
been suggested as possible gating mechanisms which would provide net irreversible
slip during fatigue (see Lin & Lin, 1979 for a critical review).
The first quantitative statistical model for random slip leading to the formation of
hills and valleys on fatigued surfaces was published by May (1960a,b) who adapted a
variation of Mott's cross slip mechanism. The assumption here is that the reverse
glide of dislocations is shifted from the forward path in a random manner. If the
amount of this shift is comparable to the width of the slip band, this is tantamount to
stating that random distribution of slip in each half-cycle is independent of the
distribution in the previous cycles. Thus, hills and valleys are formed on the surface,
with subsequent slip concentrating in the valleys in proportion to their depths. May
assumed that if f(z, N) is the fraction of the valleys (of width w) with a depth
between z and z + dz after TV cycles of fatigue, /(z, N) obeyed a diffusion equation

where k and fi are factors of the order of unity, b is the magnitude of the Burgers
vector, and y is the plastic strain in the slip band which is equivalent to yPSB in Eq.
2.9. Solving Eq. 4.1, May showed that

= Fexp - \ \ ^ T w j - V ^ \ IbyN , (4.2)

where F is a slowly varying function of z and N. For small values of N, the exponential
factor is very small and the deepening of the valleys would not be significant. The main
criticism of this statistical model is that it does not contain a sufficient number of
physical variables which incorporate the random slip process in a realistic form.

4.1.2 Electron microscopy observations

A quantitative measurement of the height of slip steps formed during the
fatigue of Cu single crystals has been conducted by Finney & Laird (1975) and Laird,
Finney & Kuhlmann-Wilsdorf (1981). Using interferometric measurements, they
showed that surface slip steps form in proportion to the applied plastic strain.
However, individual steps were found to undergo irreversible slip. The irreversibility
of slip steps was noticed even in the inert environment of dry nitrogen. Later studies
by Cheng & Laird (1981) revealed that crack nucleation occurred preferentially at
the site of the PSBs with the highest slip offset and the largest strain localization. For
4.1 Surface roughness and fatigue crack initiation 135

Cu single crystals, ypl and the slip offset (denoted by ndb, where nd is the number of
dislocations and b is the modulus of Burgers vector) were related by the expression,
y pl oc (ndb)m*. (4.3)
For Cu single crystals subjected to a plastic strain amplitude in the range of 10~3 to
10"2, mp ^ 0.78 and ndb = 0.3-3 urn.
A technique, known as the taper-sectioning method, introduced by Wood (1958),
has allowed high-resolution imaging of surface morphology in fatigued crystals. In
this method, the specimen is sectioned along a plane which is oriented at a small
angle a (i.e. a few degrees) to the specimen surface. The profile of the surface, as
observed on the sectioned plane, is magnified by a factor of (1/ sin a), over and above
the magnification obtained by other visual methods such as scanning electron micro-
scopy, optical microscopy and optical interferometry. Hunsche & Neumann (1986)
have refined this technique to obtain sections with sharp edges normal to the speci-
men surface so that the surface features can be recorded to a resolution of 20 nm in
the scanning electron microscope. Basinski & Basinski (1984), Hunsche & Neumann
(1986), and Ma & Laird (1989a,b) have utilized this method to examine the details of
surface roughening in fatigued Cu. Their results collectively indicate the following
general trends:
(1) The surface of the fatigued crystal is covered with PSB extrusions, intrusions
and protrusions. A protrusion is a surface uplift (a large extrusion), many
micrometers in height, where a macro-PSB, tens of micrometers wide and
containing tens of matrix and/or PSB lamellae, emerges at the free surface.
A protrusion may contain several intrusions and extrusions. Figure 4.2 is a
scanning electron micrograph obtained using the sectioning method which

Fig. 4.2. Protrusions with extrusions and intrusions on the surface of a Cu crystal fatigued at
room temperature for 120000 cycles at ypl = 0.002. (From Ma & Laird, 1989a. Copyright
Pergamon Press pic. Reprinted with permission.)
136 Fatigue crack initiation in ductile solids

shows PSB protrusions with superimposed extrusions and intrusions in a Cu

crystal fatigued for 120000 cycles at a constant resolved plastic shear strain
amplitude of 0.002.
(2) The extrusions, with a triangular cross section (base width ^ 1 — 2 jim and
height ^ 2 — 3 jim), grow at rates of 1 — 10 nm (cycle)"1, whereas the growth
rate of protrusions is an order of magnitude smaller.
(3) The lamallae within the protrusions undergo constrained slip in the direc-
tion of the primary Burgers vector. Concomitantly with the formation of
protrusions, there develops a population of negative protrusions or
encroachments.
(4) The protrusion height increases in proportion to the width of the macro-
PSBs.
(5) Optical interferometry indicates that a greater than average local strain
develops at the PSB-matrix interface.

The contour of a PSB profile, imaged directly by focusing on the specimen edge, is
provided in Fig. 4.3. This figure shows a protrusion on the side surface of a Cu
specimen fatigued at 77 K for 35 000 cycles at ypl = 0.002. The loading axis is along
the vertical direction and b is the direction of the primary Burgers vector. A similar
extrusion/protrusion is found on the opposite side of the crystal (not shown in Fig.
4.3).
Numerical simulations of the random slip processes have also been attempted to
quantify the extent of surface protrusions produced by fatigue. Differt, Essmann &

Fig. 4.3. The contour of a PSB profile created in a Cu crystal. (From Differt, Essmann &
Mughrabi, 1986. Copyright Taylor & Francis, Ltd. Reprinted with permission.)
4.2 Vacancy-dipole models 137

Mughrabi (1986) considered cyclic slip of a stack of planes by a random distribution

of irreversible microscopic displacements. The mechanism for the development of
slip irreversibility was taken to be the dynamic equilibrium between dislocation
multiplication (at a Frank-Read source) and annihilation within the PSBs (see
Section 4.2), which produces microscopic extrusions and depressions on the speci-
men surface. For an individual PSB, the mean width (peak to valley) of the rough-
ness profile was predicted to be
w = 2F^NbyFSBpPSBhp, (4.4)
where F is a nondimensional factor, b is the magnitude of the Burgers vector for the
dislocations within the PSB, N is the number of fatigue cycles, /?PSB is the ratio of the
irreversible strain to the total strain within the PSB, and hp is the thickness of the
PSB. For Cu single crystals, taking b = 0.256 nm, hp = 1000 nm (for a PSB lamella
composed of 5000 atomic planes with a mean spacing of 0.2 nm), /?PSB ^ 0.3,
KPSB = 0.0075 (from Section 2.4) and F ~ 1, the mean height of the extrusion at
TV = 106 strain cycles is found to be ~ 0.76 |im. This value is of the order of the slip
step height observed in experiments. It should be pointed out that this model, where
the quantitative estimates for the parameters for slip within a PSB are only rough
estimates, cannot account for the interaction among closely spaced PSBs or for the
formation of intrusions below the average level of the surface.

4.2 Vacancy-dipole models

The nucleation of PSBs at the beginning stages of cyclic saturation is also
accompanied by the formation of extrusions. This surface roughening appears to be
instigated when the average dislocation distance in the fatigued matrix approaches
the annihilation distance for dislocations. Transmission electron microscopy studies
reveal that there is a critical spacing between dislocations, below which their anni-
hilation is favored. Consider two screw dislocations of opposite signs (left hand
screw SLH and right hand screw SRH), gliding in the PSB channels (see Fig.
4A(a)). When the distance between this pair of dislocations becomes less than a
critical spacing, ys « 50 nm at room temperature, they annihilate each other by
cross slip (Mughrabi, Ackermann & Herz, 1979; Essmann, Gosele & Mughrabi,
1981). Similarly, a dipole consisting of edge dislocations of opposite sign will anni-
hilate to form a vacancy if the spacing of the edge dislocations, j e , becomes smaller
than about 1.6 nm, as shown in Fig. 4A(a). (Note that the edge dislocation pair in
Fig. 4A(a) is a vacancy dipole. An interstitial dipole is one where the signs of the
edge dislocations on the two parallel planes, 1 and 2, are opposite to those shown in
this figure.) It has been found in TEM that the majority of the dislocation dipoles
that are observed after fatigue are of vacancy-type (Antonopoulos, Brown & Winter,
1976). This result also finds direct experimental support in the electrical conductivity
measurements by Johnson & Johnson (1965) and Polak (1970), which show that
138 Fatigue crack initiation in ductile solids

7
5RH
:i
1-
2- nzi
(a)

S ^ interface
dislocation

Fig. 4.4. (a) The critical annihilation distance for screw and edge dislocations, (b) Mechanism of
extrusion formation by combined glide and dislocation annihilation, (c) Irreversible slip in the
PSB creating effective interfacial dislocations which put the slip band in a state of compression.
(d) The combined effects of applied stresses and internal stresses. Bigger arrows indicate
repulsive forces on interfacial dislocations and smaller arrows denote forces caused by the
applied load. (After Essmann, Gosele & Mughrabi, 1981.)

vacancies, of concentration ~ 10 4, are generated during fatigue of metals. Such

vacancy generation has also been clearly demonstrated by Argon & Godrick
(1969) in LiF crystals fatigued at elevated temperature (see Chapter 5). It therefore
appears reasonable to suppose that individual or clusters of vacancies produced by
cyclic slip are responsible for the swelling of the material which produces protrusions
and extrusions in fatigue, as in Fig. 4.3.
Essmann, Gosele & Mughrabi (1981) developed a model for surface roughening
and crack nucleation on the basis of the hypothesis that the annihilation of the
dislocations within the slip bands is the origin of slip irreversibility. Without anni-
4.2 Vacancy-dipole models 139

hilation, the to-and-fro motion of dislocations within the walls and channels of the
PSB would be reversible and no permanent changes in surface topography would
result. Essmann et al. propose the following sequence of events:
(a) Dislocations that are generated at dislocation sources (e.g., point S in Fig.
4.4(6)) are terminated by mutual annihilation before the reversal of strain.
(b) The annihilation of vacancy-type dipoles, shown in Fig. 4.4(7?), is the dominant
point defect generation process. Dislocations moving during the tensile portion of
the fatigue cycle are denoted by solid symbols and those moving during the com-
pression portion by open symbols. During tensile loading, slip is transmitted across
the specimen by the sequence of microscopic processes extending from A to A'.
(c) At locations where the edge dislocations are annihilated (e.g., the PSB walls in
Cu), the plane on which slip is dominant is changed because of the annihilation
process. Therefore, the effective slip plane A-A' is not parallel to the primary
Burgers vector b, but is slightly inclined to b. Slip steps are created at surface loca-
tions A and A' during the tensile portion of fatigue.
(d) On reversing the strain into compression, slip steps are formed at B and Bf by a
similar process. The steps A-B and A'-Bf thus constitute an extrusion. If interstitial-
type dipoles, rather than vacancy-type dipoles, are considered, then intrusions,
rather than extrusions, form by a process analogous to that described in Fig. 4.4(6).
(e) The extrusion in Fig. 4A(b) ceases to grow when the concentration of vacancies
formed by edge dislocation annihilation attains a saturation value, [Cv]sat
(%3x 10~4 for Cu), within the slip band. At saturation, the effective slip plane no
longer deviates from b.
if) While Fig. 4.4(6) illustrates the formation of extrusions by two microscopic slip
processes, Fig. 4.4(c) schematically shows this phenomenon for the situation invol-
ving the superposition of multiple slip processes. The path X-Y denotes the zig-zag
glide of a combined slip process aided by annihilation, just as in Fig. 4.4(6). For
clarity, the other paths are merely represented by straight lines. These lines join edge
dislocations which have either survived the annihilation process and arrived at the
free surface or have been deposited at the PSB-matrix interface. These latter inter-
face dislocations have the same sign and give rise to internal stresses. Thus, the net
result of the irreversible slip process after one cycle is a row of edge dislocations at
the PSB-matrix interface. The extra half planes of atoms of these edge dislocations
face into the PSB.
(g) If all the interface dislocations emerge at the free surface, the resulting surface
roughness of the PSB lamellae in the direction of the active slip vector is given by
e = [Cv]sar4>/cos0, (4.5)
where d0 is the diameter of the crystal and 0 is the angle between b and the specimen
surface. If all the dislocations remain at the PSB-matrix interface, the mean separa-
tion S[ of the interface edge dislocations is
(4.6)
140 Fatigue crack initiation in ductile solids

Taking typical values for b and [Cv]sat f° r Cu, one obtains s-x ^ 1 urn.
(h) The arrangement of interface dislocations in Fig. 4A(c) leads to an elastic
compressive stress within the PSB acting along b and to a tensile stress in the matrix
adjoining the PSB. Essmann et al. estimate that this compressive stress in the PSBs is
of the order of 2MPa for Cu.
(f) The combined effect of the applied stress and the internal stress produced by
the interface dislocations is shown in Fig. 4A(d). The bigger arrows denote the
internal stress arising from the mutual repulsions of the interface dislocations. The
smaller arrows refer to the stress resolved in the direction of the PSB due to the far-
field axial load, which reverses sign during every half fatigue cycle. Thus, A and A'
serve as stress concentration points in tensile loading where the internal stress and
the applied stress combine to produce high local stresses. B and Bf are stress con-
centrating sites during compression where the two stresses oppose each other.
Starting with a set of initial assumptions similar to those of Essmann et al.,
Antonopoulos, Brown & Winter (1976) developed a model of dislocations at PSB-
matrix interfaces by considering a continued increase in the density of vacancy
dipoles during fatigue. Their approach is different from the foregoing model of
Fig. 4.4 in that Antonopoulos et al. did not consider the annihilation of vacancy
dipoles (which are continually replenished) to form vacancies. They concluded that
the material in the PSB isfiber-loadedin tension parallel to b after the attainment of
cyclic saturation. This prediction is contradictory to that of Essmann et al. A pos-
sible clue to this apparent contradiction can be obtained by noting that both models
assume the sign of the internal stress to remain unchanged during fully reversed
loading. Brown & Ogin (1985) have pointed out that if the effective slip plane
does not switch (from A—A to B—B! in Fig. 4.4) as the applied load is changed
from tension to compression, then the internal stress also changes sign. Despite
their differences, both models clearly pinpoint the significant role of vacancy dipoles
and interfacial dislocations in promoting surface roughness. Both groups of authors
also propose that cracks initiate at the surface steps created at the PSB-matrix
interface. This prediction is consistent with a variety of experimental observations,
to be described in Section 4.3.
Micromechanical models have also been proposed to describe the formation of
intrusions and extrusions on the surfaces of metals due to glide on parallel planes
(Lin & Ito, 1969; Lin & Lin, 1979; Tanaka & Mura, 1981). A feature common to
these analyses is the assumption that the forward and reverse slip displacements
during a fatigue cycle are accommodated within two closely-located, parallel layers,
i.e. the most favorably oriented slip planes. Such an assumption apparently finds its
basis from the experiments of Forsyth (1953) and Charsley & Thompson (1963), who
found that the slip plane accommodating plastic deformation during the forward
(tensile) loading and the one during reversed (compressive) loading are closely
spaced, but distinct from each other. Both slip planes are still part of the same
slip band.
4.3 Crack initiation along PSBs 141

If the dislocations piled up on a slip plane (layer I) under the maximum

(positive) shear stress glide in the opposite direction along the same slip band
upon load reversal, there is no net dislocation buildup. In order to incorporate
slip irreversibility in the model, it is postulated that the irreversibility of disloca-
tion motion in the two adjoining layers arises from the different levels of back
stress during slip in the forward direction on layer I and in the reversed direction
on layer II (Tanaka & Mura, 1981). The first tensile loading causes dislocation
pile-up on layer I. The positive back stress (which opposes the stress causing
dislocation motion) due to positive dislocations on layer I facilitates the pile-up
of negative dislocations on layer II during reversed loading. The back stress due
to dislocations on layer II helps further pile-up of dislocations on layer I during
the following forward cycle. This process leads to dislocation pile-up with increas-
ing number of fatigue cycles.
This model has been modified by Venkataraman et al. (1990) to represent the
matrix-PSB interfaces as an array of vacancy dipoles. The analysis shows that
there exists a critical number of fatigue cycles above which the accumulation of
interface dislocations becomes energetically unfavorable as a result of an
increase in stored elastic strain energy. Crack initiation occurs in the PSB at
this point.

4.3 Crack initiation along PSBs

The interface between the PSB and matrix is a plane of discontinuity across
which there are abrupt gradients in the density and distribution of dislocations. One
may then expect these interfaces to serve as preferential sites for fatigue crack
nucleation. As noted earlier, this was also the inference derived from the model of
Essmann, Gosele & Mughrabi (1981). Direct experimental evidence of crack initia-
tion at the interface has also been obtained by Hunsche & Neumann (1986) and Ma
& Laird (1989a,b).
Figure 4.5 shows a fatigue crack nucleated along the leading edge of a macro-PSB
protrusion in Cu fatigued at a constant plastic strain amplitude of 0.002.
Concomitant interferometric observations of this test specimen revealed that the
strains within the PSB are highly inhomogeneous and localized at the PSB-matrix
interface. These results imply that fatigue crack initiation is strongly biased by the
roughening of the surface. The population of these fatigue cracks increases linearly
with the number of cycles and the applied strains. Furthermore, the statistical dis-
tribution of the crack sizes corresponds to the distribution of localized strains mea-
sured at the PSBs using white light interferometry.
Direct evidence of crack initiation and early crack growth along PSBs has been
obtained in fatigued, polycrystalline Cu by Katagiri et al. (1977). With high voltage
TEM observations of dislocation arrangements ahead of pre-cracks, Katagiri et al.
142 Fatigue crack initiation in ductile solids

Fig. 4.5. Fatigue crack initiation (denoted by an arrow) at a PSB-matrix interface in a Cu

crystal fatigued for 60000 cycles at ypl = 0.002 at 20 °C. (From Ma & Laird, 1989b. Copyright
Pergamon Press pic. Reprinted with permission.)

clearly established that, in materials that form PSBs, crack nucleation and early
crack growth occur in the PSB. Figure 4.6 is a TEM image showing two PSBs (of
length ~ 100 jam, which is comparable to the grain size) in a Cu specimen where a
surface layer, approximately 2 jam thick, containing intrusions has been removed by
electropolishing. A nascent crack is seen within one of the PSBs and the presence of
the crack does not appear to modify their dislocation structure. An optical micro-
graph of the crack is also shown in the inset. A similar TEM image of the Cu
specimen obtained by Katagiri et al. also revealed cracks nucleating along the
PSB-matrix interface at the root of a surface intrusion.

20 jim

electron
diffraction pattern

Fig. 4.6. A nascent fatigue crack along the ladder structure of a PSB in fatigued, polycrystalline
copper. Inset at left is an optical micrograph showing the location of the crack with respect to
the free surface. (From Katagiri et al., 1977. Copyright Metallurgical Transactions. Reprinted
with permission.)
4.5 Computational models for crack initiation 143

4.4 Role of surfaces in crack initiation

As discussed in Chapter 2, it is now well accepted that a single PSB can
extend throughout the cross section of a ductile monocrystal and that, in poly-
crystalline metals, slip bands form even within interior grains. It is, therefore, of
interest to investigate whether a rough surface topography created by the pre-
sence of intrusions alone is responsible for crack nucleation or whether internal
dislocation structures can play an equally significant role in determining the
nucleation events and the overall fatigue life. In polycrystals (see Chapter 3),
PSBs formed within interior grains produce slip that is confined to the individual
grain. The large transfer of material causing the creation of rough topography is
possible at surface grains, but not in the interior ones because of constraint from
the surrounding matrix.
Experiments dating back to the work of Thompson, Wadsworth & Louat (1956)
have suggested that removing the intrusions and extrusions by electropolishing the
specimen surface increased fatigue life. More conclusive evidence for the argument
that the surface geometry determines fatigue life is found in the study of Basinski,
Pascual & Basinski (1983). These researchers demonstrated that, even in single crys-
tals where coarse PSBs traverse through the bulk, elimination of surface roughness
by electropolishing leads to a drastic enhancement in total fatigue life. This rejuve-
nation takes place at both low and high plastic strain amplitudes, i.e. at both the
beginning and end of regime B in Fig. 2.2.

4.5 Computational models for crack initiation

Finite element simulations of fatigue crack initiation have been carried out
by recourse to the constitutive model for the inelastic deformation of PSBs, which
was discussed in Section 2.6. The roughening of the surface of the crystal due to
vacancy diffusion and the ensuing crack initiation process, as revealed by computa-
tional simulations based on these models, are considered here.

4.5.1 Vacancy diffusion

The concentration of vacancies produced within the PSBs is markedly larger
than that in the vicinity of the free surface. The annihilation of edge dislocation
pairs, schematically sketched in Fig. 2.13, also causes the vacancy concentration in
the PSB to surpass that in the matrix. The attendant steep gradient in vacancy
concentration provides a driving force for the diffusion of vacancies within the
PSBs in the direction of the free surface. This mechanism occurs by lattice diffusion
in the crystal matrix as well as by pipe diffusion along the dislocation lines. Since
high-cycle fatigue necessarily involves a sufficiently long period of time for the
nucleation of a critical fatigue flaw in initially smooth-surfaced high-purity single
144 Fatigue crack initiation in ductile solids

crystals, one would expect sufficient time to be available for such diffusive processes,
even at room temperature.
The evolution of vacancy concentration can then be estimated on the basis of the
following considerations and approximations (Repetto & Ortiz, 1997). (1) The stress
concentrations arising at the free surface by the roughening induced by the egress of
the PSB produces strain energy density gradients, which in turn promote stress-
assisted diffusion. (2) The vacancy concentration at the free surface remains at its
equilibrium value given by

ceq = exp- A C '/* r , (4.7)

where AGV is the incremental change in free energy per vacancy, k is Boltzman's
constant, and T is the absolute temperature. (3) An effective diffusion coefficient for
vacancy mobility by both pipe and lattice diffusion is determined from a rule-of-
mixture type approximation based on the dislocation density. (4) Since the screw
dislocations in the channels between the walls are parallel to the direction of vacancy
flux to the free surface, they are postulated to contribute more strongly to vacancy
diffusion than edge dislocations.
With the above line of reasoning, the flux of vacancies from the PSBs to the free
surface is expected to follow the diffusion equation:

^ = Vo • j[AaJ + (b2PsDpipes ® s)](v oCv + A ; A w)} + \pay2eya, (4.8)

where Z)lat and Dp[pQ are the lattice and pipe diffusion coefficients, respectively, I is
the identity tensor, b is the magnitude of the Burgers vector, ps is the density of screw
dislocations in the channels, s is the direction of the Burgers vector in the unde-
formed configuration of the crystal, R is the universal gas constant, and W is the
elastic strain energy density. The last term on the right hand side of Eq. 4.8 is the
vacancy generation rate defined in Eq. 2.31. The symbol Vo is the material gradient
operator defined such that (VQ/) 7 = 3//3X 7 , where (X{, X2, X3) refer to the material
reference frame defined with respect to the undeformed configuration of the crystal.
The term within the square brackets in Eq. 4.8 equals - J v , where J v is the vacancy
flux through the crystal.
The net outward flow of vacancies causes the surface of the crystal to move
inward, thereby forming an intrusion. The outward velocity of the vacancies,
relative to the undeformed configuration of the crystal, is Vv = J v /c v . If one
considers a small area on the undeformed surface of the crystal with a unit
outward normal N, the inward velocity of the surface region due to the egress
is VN = J v - N.f

' It is worth noting that the possibly significant process of annihilation of screw dislocations in the
channels is not taken into account in this formulation.
4.5 Computational models for crack initiation 145

4.5.2 Numerical simulations

The constitutive model discussed in Section 2.6 and the fully-coupled
vacancy diffusion model discussed in Section 4.5.1 have been incorporated into a
finite-element program by Repetto & Ortiz (1997) with the aim to simulate surface
roughening in a Cu single crystal. Figure 4.7 shows the deformed mesh from the
finite-element simulation where a large surface protrusion is clearly visible. A
comparison of Fig. 4.7 with Fig. 4.3 shows that experimental trends of protrusion
formation are captured by the numerical model. The shape of the protrusion is
determined by a competition between two mechanisms: lengthening of the PSB
which causes material to be pushed out of the matrix, and vacancy flux which
causes the surface to recede. The former process dominates in the center of the
PSB which facilitates the formation of a protrusion. It is also seen that PSB
formation, multiple slip at corners and vacancy flux cause grooves to form at
both the corners of the protrusion. These grooves serve as critical sites for the
nucleation of fatigue cracks.

|u,m

376.5 -

376.0 -

375.5

375.0 -

459.0 459.5 460.0 460.5 461.0 461.5

Fig. 4.7. Surface roughening in a [125] Cu single crystal oriented for single slip under symmetric
tension-compression loading with ypl = 6x 10~3. The figure shows the deformed mesh from the
finite-element simulation after about 65 000 cycles. (After Repetto & Ortiz, 1997.)
146 Fatigue crack initiation in ductile solids

4.5.3 Example problem: Effects of vacancies

Problem:
Repeated cyclic loading on an FCC crystal can produce a high density of
vacancies. Show that if a cubic crystal is filled randomly with vacancies to a

AL Aa

where AL/L is the fractional change of the external edge dimension of the
crystal due to the introduction of the defects, and Aa/a is the fractional change
of the lattice parameter (as measured by X-ray diffraction) due to the defects.!
Solution:
In order to arrive at the desired result, carry out the operation of intro-
ducing vacancies in the crystal in the following steps:
Step 1:
Create the vacancies inside the crystal by taking atoms from the interior
and placing them on various free surfaces of the crystal. Apply, however, a set of
body forces around the vacant sites so that no strains develop due to atomic
relaxation. If Lt (i = 1, 2, 3) are the linear dimensions of the crystal, and if ALt
denote the corresponding changes in linear dimensions, the total fractional
volume change is
x 2
- V — _|_ _|_ J (A Q\

But,

f,^M _,M -x.. (4,0)

L IV Jstepl [ L Jstepi
In addition, note that (Aa/a) = 0 for step 1 since no atomic relaxation is
permitted.
Step 2:
Now release the body forces. During the subsequent relaxation, the crys-
tal shrinks homogeneously because the vacancies are distributed at random. In
this process, the fractional decrease in the lattice parameter (i.e. the fractional
decrease in interplanar spacings) must equal the fractional decrease in the linear
dimensions of the crystal. Therefore,

a
^ Jstep2 L ^ Jstep2 L Jstep2

' The author thanks Professor R.W. Balluffi of the Massachusetts Institute of Technology for bringing
this problem to his attention.
4.6 Environmental effects on crack initiation 147

Combining Eqs. 4. 10 and 4.11, it is seen that

jAfli
-
Jstepl
+
if] - 3 —
L ^ Jstep2 L a
Jstep2
+xv
(4.12)

However,
3
Jstepl [irl = = 3 ^ 1 = 3[-1.
L ^ Jstep2
L ^ J total

La Jstepl •4-1 -•4-1 = 3 (4.13)

L a JsteP2 L « J total
Using Eqs. 4.10-4 13, it is seen that
Aa\
U

4.6 Environmental effects on crack initiation

Surface roughening and fatigue crack initiation can occur in pure materials
in vacuo and at temperatures down to 4.2 K (McCammon & Rosenberg, 1957).
However, the fraction of fatigue life at which crack nucleation occurs can be sig-
nificantly affected by the test environment. There is a wealth of experimental evi-
dence indicating that the environment plays an important role in dictating the extent
of slip irreversibility and fatigue life. For example, it was demonstrated a long time
ago by Gough & Sopwith (1932) and by Thompson, Wadsworth & Louat (1956) that
fatigue life is markedly improved in vacuo or in dry, oxygen-free media as compared
to moist laboratory air.
Consider the case of fatigue in pure metals in vacuo or in an inert environment.
Single slip during the tensile loading cycle produces slip steps at the surface. The
extent of surface slip offset can be diminished by reverse slip during unloading or
subsequent compression loading in fully reversed fatigue. In inert environments,
surface roughening during fatigue occurs primarily by a random process (see
Section 4.1.1). On the other hand, when slip steps form during the tensile portion
of a fatigue cycle in laboratory air or in a chemically aggressive medium, the
chemisorption of the embrittling species (such as oxygen or hydrogen) or the
formation of an oxide layer on the freshly formed slip step makes reverse slip
difficult on the same slip plane upon load reversal. In the embrittling medium,
this process (schematically shown in Fig. 4.8) can provide a mechanism of
enhanced surface roughening as well as easier transport of the embrittling species
to the bulk of the material preferentially along the persistent slip bands, thereby
facilitating crack nucleation.
148 Fatigue crack initiation in ductile solids

'//////
tension '////MY

compression
V///////////

V
second tension WZW/'
incipient crack

Fig. 4.8. A model for fatigue crack nucleation near a free surface by the synergistic effect of
single slip and environmental interactions. (After Thompson, Wadsworth & Louat, 1956, and
Neumann, 1983.)

4.7 Kinematic irreversibility of cyclic slip

The preceding discussions focused on the irreversibility of displacements
during the forward and reverse slip of dislocations. The various mechanisms of cyclic
slip irreversibility can be summarized as follows:

(1) Cross slip of screw dislocations and different paths for their forward and
reverse glide during a complete fatigue cycle.
(2) The extrusions, with a triangular cross section (base width ^ 1 — 2 um and
height ~ 2 — 3 |im), grow at rates of 1 — 10 nm (cycle)"1, whereas the growth
rate of protrusions is an order of magnitude smaller.
(3) Random distribution of slip (independent of prior history) with the progres-
sive, preferential deepening of valleys at surfaces.
(4) Dislocation-dislocation interactions leading to the formation of nodes, jogs
or dislocation locks which impede motion during part of a fatigue cycle.
(5) Production of point defects during saturation, due to the dynamic equili-
brium between dislocation generation and annihilation.
(6) Irreversibility due to shape changes (see Section 2.12.1) as well as differences
in dislocation back stress due to slip on different glide planes during the
tension and compression portions of fatigue.
(7) Reduction in slip displacement during unloading due to the adsorption of an
embrittling species or due to the oxidation of slip steps, and the attendant
creation of net slip irreversibility.
4.8 Crack initiation along grain and twin boundaries 149

In brittle solids and semi- or noncrystalline materials, kinematic irreversibility of

cyclic deformation can come about even in the complete absence of dislocation
motion. The underlying mechanisms are examined in Chapters 5 and 6.

4.8 Crack initiation along grain and twin boundaries

The nucleation of fatigue cracks at grain boundaries occurs under the influ-
ence of embrittling environments (which preferentially attack grain boundaries and
the particles, if any, on them) and elevated temperatures (at which grain boundary
cavitation and sliding are promoted). Intergranular failure is also commonly
observed in brittle solids due, at least in part, to the residual stresses induced by
thermal contraction mismatch between adjacent grains or to the presence of grain
boundary glass phases. The occurrence of grain boundary fatigue crack nucleation in
a ductile solid, in the absence of grain boundary particles, creep deformation or
environmental influences, is relatively less common. There have been some docu-
mented cases of purely mechanical fatigue failure along grain facets (e.g., Porter &
Levy, 1960; Kim & Laird, 1978; Figueroa & Laird, 1983; Watanabe, 1985).
Using optical interferometric measurements of slip step heights at grain facets in
fatigued copper, Kim & Laird (1978) noted that fatigue cracks may nucleate at grain
boundaries if: (i) the grain boundaries separate highly misoriented grains, (ii) the
active slip system of at least one of the grains is directed at the intersection of the
boundary with the specimen surface, and (iii) the traces of the high angle grain
boundaries in the free surface make a large angle (30-90°) with the tensile stress axis.
In general, grain boundary cracking may arise from one of two mechanisms dur-
ing cyclic loading:
(1) At low to intermediate plastic strain amplitude, the impingement of PSBs at
grain boundaries causes cracking (e.g., Figueroa & Laird, 1983; Mughrabi,
et ai, 1983). Figure 4.9(a) shows an example of this crack initiation mechan-
ism after 7000 cycles at a plastic strain range of ±5 x 10~4 in copper at sites
where primary slip bands intersect a grain boundary.
(2) At high plastic strain amplitudes, grain boundary cracking occurs as a con-
sequence of surface steps formed at the boundary. Figure 4.9(b) is an exam-
ple of this process. It shows optical interferograms of a slip-step of height
0.9 |im formed in polycrystalline Cu after 60 cycles of fully reversed fatigue
loading at a plastic strain of ±7.6 x 10~3. Fringes shifted from left to right
indicate depressions.
In BCC metals such as commercially pure iron, intergranular crack nucleation has
also been observed under reversed bending and push-pull axial loading over the
cyclic frequency range 0.01-1000 Hz (Guiu, Dubniak & Edward, 1982). It was
mentioned earlier (Section 2.12) that the asymmetry of slip associated with the
glide of screw dislocations in tension and compression can induce shape changes
150 Fatigue crack initiation in ductile solids

20 nm |

Fig. 4.9. (a) Nucleation of flaws (denoted by arrows) along a grain boundary. (From Figueroa
& Laird, 1983. Copyright Elsevier Sequoia, S.A. Reprinted with permission.) (b) White light
interferograms showing slip-step formation at grain boundary in fatigued Cu. (From Kim &
Laird, 1978. Copyright Pergamon Press pic. Reprinted with permission.) The dark diagonal
lines parallel to the arrow are fiducial markers whose separation is 100 urn.
4.8 Crack initiation along grain and twin boundaries 151

in BCC single crystals. The surface roughness created by similar shape changes in the
near-surface grains of polycrystalline BCC metals, such as a-iron, can cause inter-
granular crack nucleation (Mughrabi, Herz & Stark, 1981).
The process of slip involves a simple translation of atoms across a glide plane such
that a rigid block of solid on one side of the slip plane moves with respect to the
other in the direction of slip. Slip occurs by translations in whole multiples of the
Burgers vector, so that the relative crystallographic orientation of different regions in
a slipped material remains the same. On the other hand, a twin boundary is a surface
where the atom positions in the twin on one side of the boundary are a mirror image
of those in the untwinned matrix material on the other side of the boundary.
Therefore, one observes a shape change in a twinned body.
Although twin boundaries are grain interfaces with the lowest energy, their role in
crack nucleation has long been known (e.g., Thompson, Wadsworth & Louat, 1956;
Boettner, McEvily & Liu, 1964; Neumann & Tonnessen, 1988). In FCC metals, twin
boundaries are parallel to the slip planes so that the PSBs can fit into the region of
high local stresses at the boundary. Therefore, the geometric relationship between
the boundary and the slip plane may provide possible clues to the role of twins in
fatigue crack nucleation. An intriguing aspect of fatigue crack nucleation at twin
boundaries is that, in a stack of lamellar twins, there is a propensity for slip bands
and cracks to form only at every other twin boundary (Boettner, McEvily & Liu,
1964). This trend has also been studied in greater detail in the context of annealing
twins formed in polycrystalline Cu, Ni and austenitic stainless steel (Neumann &
Tonnessen, 1988). Figure 4.10 shows an example of fatigue crack nucleation at every
other twin boundary in Cu fatigued at room temperature.

30 nm

Fig. 4.10. Nucleation of fatigue cracks along every other twin boundary (indicated by arrows) in
polycrystalline Cu fatigued at room temperature. (From Neumann & Tonnessen, 1988.
Reprinted with permission from P. Neumann.)
152 Fatigue crack initiation in ductile solids

Using the taper sectioning technique (discussed in Section 4.1.2) and electron
channeling method in the SEM, Neumann & Tonnessen have detected the orienta-
tions of grains and the nucleation of microcracks at twin boundaries in FCC metals.
They found that, at low imposed stress amplitudes, PSB formation within the grains
was suppressed. However, PSBs were found exclusively parallel to and coincident
with twin interfaces. Careful grain orientation measurements revealed that, even
when slip activity ceases within the interior of grains, slip bands are activated at
some twin interfaces as a consequence of local stress concentrations.
Neumann & Tonnessen have rationalized the observations of fatigue crack for-
mation at every other twin boundary utilizing a mechanism which relies on the
elastic anisotropy of the material containing the twins. For example, the shear
modulus of Cu varies with direction by as much as a factor of 3.2. In order to ensure
strain compatibility at twin boundaries in the elastically anisotropic material, inter-
nal stresses must be generated in the vicinity of the twin boundaries. Consider a stack
of lamellar twins, where the crystallographic orientation of the lamellae changes
back and forth from that of the matrix to the twin to the matrix as one traverses
across the boundaries. With the change in orientation, there is also a change in the
direction of internal stresses. The internal stresses act in concert with the resolved
stresses from the applied loads at every other twin boundary. When the resultant
stress is of sufficiently high magnitude, a PSB is formed near the twin-matrix inter-
face and eventually develops into a fatigue crack. At the alternate boundaries, the
internal stresses oppose the resolved stresses so that slip is obstructed. Neumann &
Tonnessen used this approach to predict the internal stressfieldin the vicinity of twin
boundaries using elasticity theory. Their simulation, in conjunction with electron
channeling measurements of orientation of grains, correctly predict the twin bound-
aries at which fatigue cracks are likely to nucleate.

4.9 Crack initiation in commercial alloys

In engineering components made of commercial materials, the principal
sites of heterogeneous fatigue crack nucleation include voids, slag or gas entrap-
ments, inclusions, dents, scratches, forging laps and folds, macroscopic stress con-
centrations, as well as regions of microstructural and chemical nonuniformity. While
surface grains are the most likely locations for crack initiation in metals and alloys of
high purity, the formation of fatigue cracks is feasible at both near-surface and
interior locations in commercial alloys.

4.9.1 Crack initiation near inclusions and pores

The fatigue lifetime and the maximum fatigue strength of commercial alloys
are decreased by the presence of inclusions and pores (generally classified as defects).
4.9 Crack initiation in commercial alloys 153

The mechanisms of fatigue crack initiation at defects depend upon a number of

mechanical, micro structural and environmental factors. These factors involve the
slip characteristics of the matrix, the relative strength values of the matrix and the
defect, the strength of the matrix-inclusion interface and the relative susceptibility of
the matrix and the inclusion to corrosion in the fatigue environment (e.g.,
Cummings, Stulen & Schulte, 1958; Bowles & Schijve, 1973).
The effect of inclusions on fatigue crack initiation is often specific to the alloy
system. Here we consider three examples.
(a) In high strength steels containing MnS particles, the initial stage of fatigue
damage is the debonding of the inclusion from the matrix. This occurs by a quasi-
static mode of failure in that the interfacial separation is induced during the very first
tensile loading at far-field stress levels close to the threshold stress range for infinite
fatigue life. Figure 4.11 shows an example of a partial debonding of an MnO-SiO2-
A12O3 inclusion from the matrix of a 4340 steel and the nucleation of a fatigue crack
normal to far-field tension.
(b) In aluminum alloys, constituent particles such as the S-phase (Al2CuMg) and
/3-phase (Al7Cu2Fe), typically 1-10 |im in diameter, provide sites for fatigue crack
nucleation. The type of cracking, however, is a function of microstructure and test
conditions, (i) In 2024-T4 aluminum alloy, Grosskreutz & Shaw (1969) noticed the
debonding of the particle-matrix interface after cyclic damage within the matrix
over a large number of fatigue cycles, (ii) Another type of crack nucleation in the
2024-T4 alloys involves cracking along slip bands emanating from or terminating
at the inclusions. Figure 4.12 presents the experimentally determined relative

Fig. 4.11. Scanning electron micrograph showing the nucleation of a fatigue crack normal to the
tensile axis (vertical direction) at the site of an MnO-SiO 2 -Al 2 O3 inclusion which is partially
debonded from the 4340 steel matrix denoted M. (From Lankford & Kusenberger, 1973.
Copyright Metallurgical Transactions. Reprinted with permission.)
154 Fatigue crack initiation in ductile solids

• Al 2 CuMg (S phase)
X Al 7 Cu 2 Fe(/3 phase)

£ 2

5 10 15
particle thickness (|im)

Fig. 4.12. Relative probability of crack initiation versus the constituent particle thickness
normal to the stress axis for S and f$ inclusions in 2024-T4 aluminum alloy. (After Kung & Fine,
1979.)

probability for the initiation of fatigue cracks at two different types of constituents,
S-phase and /3-phase particles, in a commercial 2024-T4 aluminum alloy as a
function of the particle thickness, measured in the direction normal to the stress
axis. Here, the number of particles in the vicinity of which matrix cracks were
initiated was determined as a function of the particle size. This number was divided
by the particle size distribution to give the relative fatigue crack initiation prob-
ability curves (Kung & Fine, 1979). Note the precipitous increase in crack nuclea-
tion probability with an increase in the size of the inclusion and the scatter
associated with the measurements due to the variability in the size and distribution
of the inclusions.
(c) In high strength nickel-base superalloys, crack initiation has been identified
with the existence of large defects, either pores or nonmetallic inclusions (e.g.,
Hyzak & Bernstein, 1982). At room temperature, crystallographic cracking at or
near the surface is initiated at the sites of the defects at both low and high strain
ranges. At an elevated temperature of 760 °C, low strain range fatigue results in
crack nucleation at the interior of the specimen. Figure 4.13 shows an example of
this process in a complex nickel-base superalloy (made by powder metallurgy
methods conforming to a commercial designation AF-115 and consisting princi-
pally of Ni-Cr-Co-W-Ti-Al-Mo-Hf-Cb-C) where subsurface crack initiation
occurs at an HfC>2 inclusion. At high strain range values, however, near-surface
crack nucleation is dominant.
4.9 Crack initiation in commercial alloys 155

Fig. 4.13. Subsurface fatigue crack initiation at an HfO2 inclusion in an AF-115 nickel-base
superalloy at 760 °C. (From Hyzak & Bernstein, 1982. Copyright Metallurgical Transactions.
Reprinted with permission.)

4.9.2 Micromechanical models

Several analytical models which estimate the number of fatigue cycles
required for crack nucleation in the vicinity of defects have been developed over
the years. These models, which have been compared with experimental results with
varying degrees of success, can be broadly categorized into the following groups:
{a) Analyses wherein the voids or debonded inclusions are regarded as notches and
the fatigue limit is estimated from the reduction in fatigue strength due to these
defects (e.g., Tanaka & Mura, 1982). Notch effects on crack initiation are addressed
in Chapters 7 and 8.
(b) Analyses wherein the pile-up of dislocations at the inclusion is considered
responsible for cracking of the inclusion or interfacial debonding with the attendant
inception of a fatigue flaw within the matrix.
(c) Formulations wherein the threshold stress intensity factor range is related to
the square root of the (statistical average) projected area of the defects on a plane
normal to the maximum tensile stress (e.g., Murakami & Endo, 1986).
In a set of models developed for fatigue crack nucleation from intermetallic par-
ticles, Chang, Morris & Buck (1979) and Morris & James (1980) considered a dis-
location pile-up process which views crack initiation as being composed of two
successive events: cracking inside the brittle particle and the onset of crack advance
from the brittle particle into the ductile matrix. The first stage of particle cracking
was assumed to occur when a critical elastic strain energy was reached inside the
156 Fatigue crack initiation in ductile solids

particle near the site of dislocation pile-up. The second stage, i.e. crack advance
inside the matrix, was assumed to occur when the total energy of the system attained
a minimum. The total energy comprises four terms: the elastic strain energy from the
stress field of the piled-up dislocation array, the effective surface energy needed for
crack advance, the work done by the applied stress in opening the crack, and the
elastic strain energy of the crack under the applied stress field.
Tanaka & Mura (1982) presented an extension of the parallel layer model for
surface roughening and slip band fracture to include crack initiation from interme-
tallic particles in high strength steels and aluminum alloys. The initiation of the crack
was assumed to be determined by the energy criterion that the fatigue flaw initiates
when the self strain energy of dislocation dipoles accumulated at the inclusion
reaches a critical value. Tanaka & Mura considered three different processes of
fatigue crack initiation: a slip band crack emanating from a debonded inclusion, a
slip band crack initiating from an uncracked inclusion and inclusion cracking due to
the impingement of slip bands. For the last mechanism, which is believed to be
representative of inclusion cracking in aluminum alloys, the solution was obtained
using Eshelby's equivalent inclusion method (Eshelby, 1957).

4.10 Environmental effects in commercial alloys

If a cyclically loaded engineering component is exposed to a chemically
aggressive medium during service, preferential attack of the environment at select
locations on the material surface may provide nucleating sites for fatigue cracks (Fig.
4.14). These sites are generally corrosion pits which form at surface locations where:

(1) Slip steps or intrusions are created at the surface.

(2) Grain boundaries, either embrittled by temper treatments (as in alloy steels)
or surrounded by precipitate-free zones (as in age-hardened alloys) intersect
the surface.
(3) The protective oxide layer on the surface is partially broken, exposing the
underlying fresh metal to preferential chemical attack.
(4) Inclusions, such as MnS particles in steels, debond from the surrounding
matrix at near-surface locations.
(5) One of the constituent phases in a multiphase alloy is preferentially cor-
roded.

Corrosion pits are typically smaller than a millimeter in depth and serve as micro-
notches which locally elevate the stress level. Furthermore, the pH level of the
corrosive medium inside the pit can be more acidic than that in the bulk, causing
possible acceleration in the rate of fatigue crack growth. Experimental results
obtained for Ni-Cr-Mo-V steels, austenitic stainless steels and aluminum alloys
4.11 Crack initiation at stress concentrations 157

Fig. 4.14. A fatigue crack initiated at corrosion pits in stress relief groove in a low pressure
turbine rotor made of Ni-Cr-Mo-V steel. (From Lindley, 1982. Reprinted with permission
from T.C. Lindley.)

have established that the formation of corrosion pits on the initially smooth surfaces
of the fatigue specimen results in a significant reduction in the fatigue strength.
Figure 4.14 shows an example of fatigue cracks initiated at corrosion pits in stress
relief groove in a low pressure turbine rotor made of a Ni-Cr-Mo-V steel. The rotor
consists of stepped shafts with shrunk-on discs which carry the turbine blades. A
combination of the steady mean stress due to the shrink fit and the pulsating stresses
caused by the self-weight bending resulted in the initiation of fatigue cracks at
corrosion pits. In the case of the turbine component shown in Fig. 4.14, the fatigue
cracks led to the complete failure of the low pressure turbine shaft.

4.11 Crack initiation at stress concentrations

The initiation of fatigue cracks at stress concentrations is a topic of con-
siderable interest in a wide variety of engineering applications. The issue of crack
initiation at notches and other stress raisers subjected to cyclic tension or tension-
compression fatigue has been investigated extensively in the context of stress-life
approach, strain-life approach and fracture mechanics approach. These approaches
will be addressed in Chapters 7-9.
Here we consider the processes by which fatigue cracks initiate ahead of stress
concentration under fully compressive cyclic loads, primarily because the mechan-
istic origins of crack inception in this case can be treated without recourse to the total
158 Fatigue crack initiation in ductile solids

life approaches or defect-tolerant approach. Fatigue crack initiation in ceramics and

polymers subjected to cyclic compression loading is examined in Chapters 5 and 6,
respectively.

4.11.1 Crack initiation under far-field cyclic compression

The analysis of fatigue cracking is generally based on the premise that fully
compressive cyclic loads or compressive stresses superimposed periodically or ran-
domly upon tensile cyclic loading patterns do not significantly modify the inception
and growth of fatigue cracks. This notion arises from the tacit assumption that
fatigue cracks remain closed during compression loading or even below a certain
tensile load known as the crack closure load (see Chapter 14). There is, however, a
wealth of information available in the literature which has unequivocally established
that far-field compressive stresses can have a marked effect on both the initiation and
propagation of fatigue cracks in both brittle and ductile solids. A neglect of the
influence of compression cycles in fatigue can lead to a nonconservative estimate
of the useful fatigue life.
The initiation and growth of fatigue cracks under the influence of imposed cyclic
compressive stresses are problems of significant practical importance in a wide vari-
ety of engineering applications. Examples of metallic structural components periodi-
cally subjected to compressive stress cycles include: landing gear and wing-root
sections of aircraft,! coil springs, shellings and fillets joining the web and head of
rails in railroad applications, near-surface regions of shot-peened materials, weld-
ments containing residual compressive stresses, and components of deep-diving sub-
mersibles. The occurrence of this phenomenon can easily be tested in cast iron by
subjecting it to a few cycles of zero-compression-zero loading. Small fatigue cracks,
initiating around the defects and cavities in this material, are readily visible after
such compression fatigue.$
It has long been recognized (e.g., Gerber & Fuchs, 1968) that the application of
uniaxial cyclic compressive loads to notched plates of metallic materials causes the
nucleation and growth of fatigue cracks along the plane of the notch, in a direction
normal to the far-field compressive stress. The cracks propagate at a progressively
slower rate until complete crack arrest takes place at a fatigue crack length, a* (Fig.
4.15). Figures 4.16(a) and (b) are examples of cracks initiated ahead of stress con-
centrations in an intermetallic and a metal-matrix composite, respectively, under
fully compressive cyclic loads. Although this phenomenon of noncatastrophic
crack growth from notches under cyclic compression is apparently similar to the

' The compression fatigue failure of upper spar cap on the wing of an F-15 military aircraft is a typical
example (see Rich, Pinckert & Christian, 1986).
+ A case study of fatigue cracking under cyclic compression in a total hip femoral component is presented
in Chapter 10.
4.11 Crack initiation at stress concentrations 159

-time

uIT
n min

,-<cj
(a)

number of compression cycles, N

(b)

Fig. 4.15. Schematic showing (a) the loading of a notched specimen in cyclic compression and
(b) typical variation of crack length, measured from the notch tip, as a function of the number
of compression cycles.

behavior of 'nonpropagating' cracks nucleated under cyclic tension (Chapter 7), the
mechanisms underlying the two processes are different.
Observations of controlled crack initiation in cyclic compression of notched metal-
lic materials were first reported in the 1960s. Hubbard (1969) reported a fracture
mechanics-based study of crack growth in a center-notched plate of 7075-T6 alumi-
num alloy. He found that fatigue cracks grew from the tip of the notch over distances
of several millimeters. Similar observations have been made subsequently in a variety
160 Fatigue crack initiation in ductile solids

50 nm

Fig. 4.16. Examples of mode I fatigue cracks initiated at stress concentrations under far-field
cyclic compression :(a) Ti-48 Al intermetallic with a predominantly y-phase micro structure. (6)
Al-3.5 Cu alloy reinforced with 20 volume % of SiC particles. The cyclic compression loading
axis is vertical in both cases. (Photographs courtesy of P.B. Aswath and Y. Sugimura,
respectively.)

of ferrous and nonferrous alloys (Saal, 1971; Reid, Williams & Hermann, 1979;
Suresh, 1985b; Pippan, 1987).
The mechanism by which a fatigue crack initiates and advances in a direction
normal to the imposed compression axis is dictated by the development of a cyclic
plastic zone ahead of the notch tip upon unloading from the far-field compressive
stress. In zero-tension fatigue, there develops a region of reversed flow ahead of a
tensile crack within which residual stresses comparable in magnitude to the flow
4.11 Crack initiation at stress concentrations 161

stress in compression exist.f If one considers the case of a 'sharp' nonclosing notch
which is subjected to a zero-compression-zero fatigue cycle, it is seen that the reverse
flow induced within the monotonic plastic zone ahead of the notch tip upon unload-
ing from the maximum compressive stress generates a zone of residual tensile stresses
at the notch tip (Fig. 4.17). Full field finite element simulations of the generation of
residual tensile stresses in notched plates of ferrous alloys have been reported by
Holm, Blom & Suresh (1986). Quantitative and in-situ measurements of the evolu-
tion of residual tensile stresses ahead of stress concentrations subjected to cyclic
compression loading have been reported by Pruitt & Suresh (1993); see Chapter 6.
Residual stresses are induced ahead of the notch tip during unloading from the
far-field compressive stress because there is no contact (closure) in the wake of the
notch tip. (If a long, sharp fatigue crack, rather than a notch, is subjected to cyclic
compression, such a residual tensile field may not be generated because of the com-
plete closure of the crack during far-field compression loading and unloading.) Once
a fatigue crack emanates from the notch tip, the faces of the crack tend to remain
partially or fully closed during some portion of the loading cycle. Experimental

load

VV\ time

load

time
0

W
Fig. 4.17. (a) A schematic of a zone of residual compression ahead of a sharp notch (with a
small included angle at the notch tip) subjected to cyclic tension in an elastic-perfectly plastic
solid. rc is the cyclic plastic zone defined in Eq. 9.74. (b) A zone of residual tension for the
nonclosing notch subjected to cyclic compression.

' As shown in Section 9.6, the size of the cyclic plastic zone ahead of a stationary fatigue crack in an
elastic-perfectly plastic solid subjected to zero-tension cyclic loading in plane stress is about one-quarter
the size of the monotonic plastic zone.
162 Fatigue crack initiation in ductile solids

measurements of crack closure (Suresh, Christman & Bull, 1986) reveal that as the
length of the fatigue crack increases, the fraction of the loading cycle during which
the crack remains open progressively diminishes. This increasing closure causes the
crack to arrest completely after growth over a distance a*, as shown in Fig. 4.15. The
total distance of crack growth ahead of the notch is a complex function of such
variables as the size of the residual tensile zone created during the first cycle, the
stress state, load range, notch tip geometry and the microscopic roughness of the
fatigue crack faces. The distance a* is dependent upon the rate of exhaustion of the
residual tensile zone and the rate of increase in closure stress with increasing fatigue
crack length.
As crack advance under far-field cyclic compression is governed by local tensile
stresses, the influence of micro structure on crack growth behavior is found to be
similar to that seen under far-field cyclic tension. However, the mechanisms of
fracture surface contact and abrasion are different for the two cases.
Since the size and shape of the monotonic and cyclic plastic zones and the extent
of crack closure are strongly influenced by whether plane stress or plane strain
conditions prevail (see Fig. 9.15 and Chapter 14), the stress state is expected to
have a marked effect on the characteristics of crack initiation and growth in notched
components subjected to cyclic compression. Plane stress conditions, where the
monotonic and reversed plastic zone sizes directly ahead of the notch tip are
about three times greater than those for plane strain, Eq. 9.77, a substantially faster
rate of crack initiation and a greater crack growth distance a* are promoted in plane
stress than in plane strain. Holm, Blom & Suresh (1986) conducted a numerical
simulation of the effects of stress state on crack growth in edge-notched plates of
a bainitic steel under cyclic compression using elastic-plastic finite element calcula-
tions with an isotropic hardening model. Geometrical aspects of crack propagation
were modeled by releasing the crack tip node at the peak of each compression cycle
(amax), by changing the boundary conditions, and by solving the contact problem to
determine the stress level of the compression cycle (acl) at which the freshly formed
crack faces would first contact during the loading portion. Their results reveal that
the crack remains open during a larger fraction of the compression cycle in plane
stress than in plane strain.

Exercises
4.1 The roughening of the surface of a material by the formation of slip steps
plays an important role in the nucleation of fatigue cracks. Consider a cube
which is made of an FCC single crystal. One of the corners of the cube is
located at the origin of the Cartesian coordinate system (with axes x, y and
z, and unit vectors along the axes i, j and k, respectively). A closed disloca-
tion loop lies on the plane x — a/2, where a is the edge length of the cube.
Exercises 163

The loop consists of dislocation segments of pure edge, pure screw and
mixed edge-screw characters. If the loop expands in its own plane under
the influence of an applied stress, slip steps will be formed on some faces of
the crystal when all segments of the loop intersect the faces of the crystal.
Indicate the faces of the cube on which the slip steps will form in the
following cases:
(a) The Burgers vector of the dislocation loop is b = b\.
(b) The Burgers vector of the dislocation loop is b = bk.
4.2 An edge dislocation (±x) is located on the plane x = x\\ x, y and z are the
axes of the Cartesian coordinate system and i, j and k are the corresponding
unit vectors along these coordinate axes. The Burgers vector of J_i is bi = b\
and the dislocation line is parallel to k. A second edge dislocation line J_2>
also oriented parallel to k, is at the origin. Calculate the glide and climb
forces J_! would experience due to the presence of ±2> m terms of xx and 0
(i.e. the angle between the x-axis and the line on the x-y plane connecting the
two dislocations) for the case where the Burgers vector of J_2 *s ^2 = bi.
4.3 A crystal contains a single edge dislocation (Burgers vector, b = bi and
dislocation line vector parallel to k, where i, j and k are the unit vectors
along the Cartesian coordinate axes, x, y and z, respectively).
(a) The crystal is subjected to a tensile stress at = axx. Calculate the force
on the dislocation. Is it a glide or a climb force?
(b) The stress is now reversed into compression with the compressive stress
<TC = — 2oxx. Recalculate the magnitude and direction of the force on the
dislocation.
(c) Find the magnitude and direction(s) of the maximum force on a screw
dislocation (oriented parallel to k with a Burgers vector, b = b k) in a
crystal subjected to a shear stress, oxz.
4.4 Consider two parallel screw dislocations of the same sign. Obtain an expres-
sion for the force on the dislocations as a function of their relative positions.
Comment on the stability of this arrangement.
4.5 Describe the process of point defect production by the annihilation of two
edge dislocations of opposite signs which are separated by two atomic
planes. The extra planes of atoms in both the dislocations are on the outside
of the slip planes for the dislocations. Use schematic diagrams to show the
orientations, glide directions, and atomic arrangements before and after
annihilation. Are the point defects vacancies or interstitials?
4.6 Consider two parallel edge dislocations of the same sign. Let 0 be the angle
between the line connecting the two dislocations and the direction of their
Burgers vectors.
(a) Calculate the variation of the glide force and the climb force for the two
dislocations as a function of the relative positions of the dislocations
(i.e. as a function of 0).
164 Fatigue crack initiation in ductile solids

(b) Which of the following configurations is more stable and why?

Configuration A: The two dislocations align themselves one below the
other along a direction normal to their slip planes. Configuration B: The
two dislocations align themselves one beside the other along a direction
normal to their slip planes.
(c) Give examples of at least two practical situations where the more stable
configuration in part (b) is observed.
(d) Repeat part (a) for the situation where the two parallel edge dislocations
are of opposite sign.
4.7 A dislocation dipole contains two coplanar edge dislocations of the same
magnitude of Burgers vector but opposite signs. The dislocation lines are
separated by a distance 8.
(a) Calculate the shear stress due to the dipole on a parallel plane a distance
k away from the plane of the dipole, for k > 8.
(b) Determine the positions of equilibrium of a positive edge dislocation on
the y = k plane.
 CHAPTER 5

Cyclic deformation and crack initiation in

brittle solids

In this chapter, we examine the mechanics and mechanisms of cyclic damage

and crack nucleation in a wide range of brittle materials, including ceramics, glasses
and ionic crystals. The fatigue behavior of brittle polymers is considered in the next
chapter. Although the discussion of cyclic deformation and fatigue crack initiation
for ductile materials was provided earlier in separate chapters, the corresponding
descriptions for brittle materials warrant a single, unified presentation because of the
nebulous demarkation between deformation mechanisms and flaw nucleation. For
example, crack nucleation along grain boundaries can be regarded as the initial stage
of the cyclic deformation process in some brittle materials. Fatigue crack growth in
brittle ceramics and polymers are considered in Chapters 11 and 12, respectively.
It is also pertinent at this juncture to clarify the terminology used in the descrip-
tion of cyclic failure of brittle solids. In the metallurgy, polymer science and mechan-
ical engineering communities, the word fatigue is a well accepted term for describing
the deformation and failure of materials under cyclic loading conditions. However,
in the ceramics literature, the expression static fatigue refers to stable cracking under
sustained loads in the presence of an embrittling environment (which is commonly
known as stress corrosion cracking in the metallurgy and engineering literature). The
expression cyclic fatigue is used in the ceramics community to describe cyclic defor-
mation and fracture. In keeping with the well-established universal conventions, and
in an attempt to avoid confusion, we use the term fatigue in this book to denote
deterioration and fracture of both metals and nonmetals due only to cyclic loads.
Earlier discussions of fatigue in metallic materials (Chapters 2-4) underscored the
role of kinematically irreversible cyclic slip in promoting permanent damage and
crack nucleation. Since cyclic stress effects in ductile metals are intimately related to
the to-and-fro motion of dislocations, it has traditionally been assumed that disloca-
tion motion (slip) is a necessary condition for fatigue. A broader examination of the
requirements for cyclic fracture in both ductile and brittle solids clearly indicates that
kinematically irreversible microscopic deformation can arise not only from cyclic
slip, but also from microcracking, martensitic transformation, interfacial sliding or
creep. The processes which impart kinematic irreversibility to microscopic deforma-
tion during the fatigue of brittle materials include:

(1) Frictional sliding of the mating faces of microcracks that are nucleated at
grain boundaries (in single phase systems), at interphase regions (in multi-
phase systems), and along the interfaces between the matrix and the rein-
forcement (in brittle composites) under the influence of the applied loads.

165
166 Cyclic deformation and crack initiation in brittle solids

(2) Progressive wear and breakage, under repeated cyclic loading, of bridging
ligaments which connect the faces of microcracks and long flaws in brittle
solids at low and elevated temperatures.
(3) Wedging of the mating surfaces of microscopic and macroscopic flaws by
debris particles which are formed as a consequence of repeated contact
between the crack faces, especially under fully compressive or tensile-com-
pressive cyclic stresses.
(4) Microcracking due to the release of thermal residual stresses at grain bound-
aries and interfaces, which gives rise to a permanent transformation strain.
(5) The inelastic strain arising from shear or dilatational transformations such
as mechanical twins or martensitic lamellae.
(6) The viscousflowof glassy phases that are introduced during processing and/
or formed as a result of environmental interactions at elevated temperature,
and the associated interfacial cavitation in ceramics and ceramic composites
during high temperature fatigue. The strain-rate dependence of viscous
deformation causes the fatigue response of the brittle solid with glassy
films to be both time-dependent and cycle-dependent.

5.1 Degrees of brittleness

Before embarking on a discussion of the mechanisms of cyclic damage in
brittle solids, it is essential to examine as to what constitutes brittle behavior. In
silicate glass, which is a good example of a highly brittle solid, the strength of the
atomic bonds primarily determines the resistance to fracture. Other examples of very
brittle materials include crystals of diamond structure, quartz and sapphire, and
many ceramics at room temperature (Table 5.1). As noted by Lawn & Wilshaw
(1975), flaws in highly brittle solids are characterized by three features: (i) the
flaws need not be large in size to affect the strength significantly, (ii) the flaws are
induced mainly on the surface of the material, as a result of such mechanisms as
contact damage, although flaws in the bulk may also be introduced during proces-
sing (see next section), and (iii) theflawsexhibit wide variations in size, location and
orientation, unless they are intentionally introduced by a controlled process. Design
of brittle materials on the basis of Weibull statistics (Weibull, 1939) separates flaw
populations into surface and volume flaws.
Nonbrittle or ductile solids, such as FCC metals, in which (dislocation) plasticity
plays the dominant role in controlling cyclic deformation and fracture, are at the
other end of the spectrum in this classification of brittleness. In-between these two
distinct groups of brittle and nonbrittle solids lies a category of materials called semi-
brittle solids (Table 5.1). In this class of materials, limited plasticflowoccurs prior to
the growth of a brittle crack. Consequently, the component of the resolved shear
stress on the active slip plane becomes as important a factor in deformation and
5.2 Modes of cyclic deformation in brittle solids 167

Table 5.1. Classification of the degree of brittleness according to crack initiation

mechanisms and examples of material types which fit this clasification at room
temperature.

Classification Main factors Material

Highly brittle Bond rupture Diamond structure, zinc
blende structure, silicates,
alumina, mica, B, W,
carbides, nitrides

Semi-brittle Bond rupture, Sodium chloride structure,

dislocation other ionic crystals, HCP
mobility metals, most BCC metals,
glassy polymers

Nonbrittle Dislocation FCC metals, nonglassy

or ductile mobility polymers, silver halides,
some BCC metals

Source: Lawn & Wilshaw, 1975.

fracture as the component of the normal tensile stress on the ensuing crack plane.
This inference is also supported by experimental observations which show that
cracks in semi-brittle solids are nucleated as easily in compression as in tension.
(This behavior is distinctly different from that of brittle solids, which are many
times stronger in compression than in tension.) Semi-brittle solids generally do not
contain a sufficient number of slip systems to accommodate plastic strains. Recall
from Chapter 3 that five independent slip systems are needed to ensure strain com-
patibility in a polycrystalline solid. Therefore, the initiation of a brittle crack may be
the principal factor responsible for relieving the strains accommodated by a limited
amount of slip in these materials. Damage evolution by craze formation is another
mode of semi-brittle deformation in polymers, which is discussed in the next chapter.
Increases in temperature tend to reduce the degree of brittleness in a material.
While the classification in Table 5.1 is primarily intended for quasi-static loading
conditions, it should be noted that strain rate and temperature strongly affect the
degree of brittleness in many materials.

5.2 Modes of cyclic deformation in brittle solids

In ductile metals and alloys, the occurrence of cyclic deformation and fati-
gue fracture is clearly identified with to-and-fro dislocation motion either in the bulk
168 Cyclic deformation and crack initiation in brittle solids

or in the immediate vicinity of the crack tip. In brittle solids, however, it is generally
not so straightforward to identify a clear cyclic effect. This difficulty arises because
fluctuations in applied loads can lead to a large influence on the cyclic stress-strain
curve and in the crack initiation/propagation life (compared to static loads of the
same peak value) even when there are no discernable differences in the deformation
and damage mechanisms between the static and cyclic loading cases. In this chapter,
and in Chapters 6, 11 and 12, we demonstrate the following effects of cyclic loading
on deformation and cracking in brittle solids.
(1) Cyclic loading resulting in an increased hysteresis or gradual shift in the
stress-strain curve as a result of progressive damage evolution involving
microcracking, crazing, cavitation or phase transformations at both low
and high temperatures.
(2) Cyclic response of semi-brittle ionic crystals at elevated temperatures
wherein the dislocation structures are apparently similar to those of fatigued
ductile FCC crystals at room temperature.
(3) Elevated temperature cyclic response of ceramic composites in which the
cyclic deformation characteristics at the tip of a crack are distinctly different
from those immediately ahead of a statically loaded crack.
In addition, it is shown in this chapter and in Chapters 6, 11 and 12 that an
identifiable fatigue effect, as seen by an enhanced or reduced time to failure or
crack propagation rate compared to a static load of the same peak value or mean
value, can occur in many brittle solids at room and elevated temperature, even when
the mechanisms of deformation and damage are the same under both static and cyclic
loads. The following processes which lead to such fatigue effect are considered in this
connection.
(1) The formation of crazes during cyclic loading results in cyclic softening in
brittle polymers (next chapter).
(2) The inducement of a permanent deformation (such as microcracking or
crazing) within the cyclic damage zone ahead of the notch in a brittle cera-
mic or polymer at room temperature results in a mode of crack initiation
and growth under cyclic loads which is distinctly different from that
observed under monotonic compression loads (see Chapters 6, 11 and 12).
(3) The progressive breakdown of bridging ligaments in the wake of a fatigue
crack results in a much higher crack velocity at room temperature under
cyclic loads than under sustained loads of the same peak stress intensity
(Chapter 11).
(4) The bridging of the wake of a fatigue crack by glassy ligaments (formed
from the viscous flow of the glassyfilmsleft from the processing additives or
formed in-situ as a result of environmental interactions) leads to a much
lower crack velocity at high temperature under cyclic loads than under
sustained loads of the same peak stress intensity (Chapter 11).
53 Highly brittle solids 169

5.3 Highly brittle solids

In highly brittle solids with strong covalent or ionic bonding and very little
mobility of point defects and dislocations, no distinct differences are known to exist
between static and cyclic loading conditions as far as the mechanisms of microscopic
deformation or microcrack nuleation are concerned. However, once microscopic
flaws are induced in a 'static mode', frictional sliding of the faces of the cracks
can impart an apparent plasticity to the deformation of the material, most notably
under compression loading conditions. Consequently, one may envision the possi-
bility of observing differences in the manner and rate of growth of the internal flaws
under mono tonic and cyclic loading conditions.

5.5./ Mechanisms
There exist several possibilities for introducing crack nuclei during the fab-
rication and service of a brittle solid:
(a) Brittle solids contain a population of small microscopic flaws, commonly
known as Griffith flaws. The presence of these flaws can cause marked reductions
in fracture strength. On the free surface, the Griffith flaws can be initiated due to the
impingement of hard dust particles, such as quartz, which are prevalent in the atmo-
sphere. Within the bulk, defects such as pores, inclusions or gas bubble entrapments
are likely to develop in a commercially-processed material. These internal defects
serve as potential sites for the nucleation of a dominant crack.
(b) The free surface is almost invariably 'rough' on an atomic level. Surfaces
typically consist of steps, grooves, ridges, pits, etc., as a result of crystal growth,
dissolution, cleavage or ion bombardment, even when the surface preparation tech-
niques and the specimen surfaces are 'clean'. These atomically rough surface features
can serve as local stress raisers in brittle solids where atomic bond rupture is the
principal mode of failure.
(c) In materials such as ceramics, rocks, cement mortar and concrete, distributed
microcracking in the bulk is known to occur along grain facets and/or interfaces. In
noncubic, single phase brittle solids and brittle composites, residual stresses gener-
ated at grain boundary facets and interfaces give rise to microcracking during cool-
ing from the processing temperature as a result of thermal contraction mismatch
between adjacent grains or phases. Furthermore, the residual stresses may aid in the
nucleation of intergranular flaws under the influence of an external stress. In trans-
formation-toughened ceramics (to be discussed in Section 5.5), microcracking occurs
in conjunction with stress-induced martensitic transformations.
(d) Brittle materials, when exposed to certain embrittling environments, can suffer
strength degradation and increased susceptibility to flaw nucleation. For example,
the large sodium ion in the glass network is replaced by a smaller species, such as H +
in an acid solution or Li+ in a molten salt. Furthermore, local devitrification of glass
can provide a preferential site for crack nucleation.
170 Cyclic deformation and crack initiation in brittle solids

Although it is experimentally difficult to identify and quantify the extent of stable

microcracking in smooth specimens of brittle solids at room temperature, especially
under tensile loading conditions, observations of confined microcracking damage
ahead of stress concentrations and cracks have been reported for a wide variety of
brittle solids. Figure 5A(a) shows a zone of grain boundary microcracks formed
ahead of a notch in a single edge-notched specimen of A12O3 (average grain size
« 18um) subjected to compression loading. Figure 5.\{b) documents a zone of
microcracks, as observed in a TEM foil taken 0.5 (im below the tensile fracture
surface, in a transforming ceramic which contains an A12O3 matrix with 23% meta-
stable tetragonal ZrO2. In this figure, microcracks were nucleated around the ZrO2
particles following their transformation from a tetragonal to a monoclinic phase (see
Section 5.5 for details of phase transformations).

5.3.2 Constitutive models

In a linear elastic solid containing a population of open microcracks, the
flaws are generally characterized by a nondimensional density /?, which depends on
their geometry and spatial distribution:

(5.1)

where 7VC is the number of microcracks per unit volume, S is the area of a microcrack
and P is its perimeter. The symbol { } denotes a volume average.

Fig. 5.1. (a) A zone of grain boundary microcracks formed ahead of a single edge-notch in an
A12O3 subjected to uniaxial compressive stresses in a direction normal to the plane of the notch.
(From Suresh & Brockenbrough, 1988. Copyright Pergamon Press pic. Reprinted with
permission.) (b) Microcracks ('me'), denoted by arrows, as seen in a TEM foil taken 0.5 Jim
from the tensile fracture surface in ZrO 2 -toughened A12O3. (From Ruhle, Clausen & Heuer,
1986. Copyright American Ceramic Society. Reprinted with permission.)
5.3 Highly brittle solids 171

The change in elastic moduli as a function of the microcrack density, geometry

and orientation has been investigated. Some representative results for a select num-
ber of crack systems are given in Fig. 5.2 for an initially isotropic material with an
initial Poisson's ratio, VQ = 0.25. The indicated moduli E/EQ are measured in a
direction normal to the crack planes and the loading is such that all microcracks
are open. Figure 5.2 compares the results for various crack systems in terms of the
crack density /3, as denned in Eq. 5.1. For a random distribution of equal-sized
penny-shaped cracks, p = Nca3, where a is the crack radius (Budiansky &
O'Connell, 1976).
Generalized constitutive models for multiaxial loading conditions are well devel-
oped for elastic-plastic solids (see Chapter 3). A number of formulations, analogous
to the deformation and incremental theories of plasticity (Section 1.4), have evolved
from both experimental and theoretical attempts to characterize the constitutive
response of brittle solids, such as concrete and rocks, in monotonic compression
(e.g., McClintock & Walsh, 1962; Nemat-Nasser & Horii, 1982).f

three-dimensional slits

random pennies

two-dimensional slits

Fig. 5.2. Variation of the elastic modulus E for a microcracked solid, normalized by its modulus
Eo when no microcracks exist, is plotted as a function of the microcrack density (3 for the
indicated crack systems. (After Laws & Brockenbrough, 1987.)

t The preponderance of research effort directed at monotonic compression deformation is a consequence

of the fact that brittle solids are many times stronger in compression than in tension and that they are
most commonly used in compression-dominated applications.
772 Cyclic deformation and crack initiation in brittle solids

The development of constitutive models for a microcracking solid in monotonic

and cyclic deformation requires the assumption of a microcrack nucleation criterion.
The nucleation of microcracks is dictated by such complex factors as the composi-
tion of the matrix and grain boundary phases, the occurrence of any stress-induced
phase changes in the microstructure, characteristic microstructural dimensions such
as grain size, prior processing history (and the attendant thermal residual stresses
generated upon cooling from the processing temperature), stress state and test tem-
perature.
A characteristic stress-strain curve commonly used for the constitutive behavior
of a microcracking ceramic in one cycle of zero-tension-zero loading is schemati-
cally shown in Fig. 5.3. Below a certain threshold stress a0, the material deforms as if
it were an uncracked solid with Young's modulus EQ and Poisson's ratio v0- Beyond
(To, microcracking is assumed to increase continuously until a certain saturation
stress as is reached. A completely saturated state of microcracking, independent of
the applied stress, is assumed based on the notion that the sites for microcrack
nucleation become exhausted above some applied stress level, <rs, when the local
(thermal) residual stresses dictate microcracking (e.g., Hutchinson, 1987). Above
<rs, a reduced elastic modulus Es governs elastic deformation. Unloading from ten-
sion gives rise to a permanent strain eT, Fig. 5.3. This strain is viewed as a transfor-
mation strain arising from microcracking due to the release of local residual stresses.
Criteria have been postulated for the evolution of microcrack density /3 as a
function of the applied stress (between the threshold <r0 and the saturation as).
Microcrack nucleation at a critical value of principal tensile stress (e.g., Ortiz &

/ 1

Fig. 5.3. Stress-strain curve for a microcracking material subjected to a zero-tension-zero load
cycle. The strain axis is enlarged for clarity.
53 Highly brittle solids 173

Giannakopoulos, 1990) and at a critical mean stress (e.g., Hutchinson, 1987) have
been considered. Other constitutive models for isotropic microcracking have also
been proposed where microcrack nucleation between the threshold and saturation
stages is considered to be governed by an effective stress a. For example,
Charalambides & McMeeking (1987) use the following small strain, nonlinear elastic
constitutive relation to characterize the deformation of a microcracking solid:
EetJ = [h(d) + v]^ - vakk8tj, (5.2)
where atj and etj are components of the stress and strain tensors, respectively, v is
Poisson's ratio, 8^ is Kronecker's delta (defined in Section 1.4),

> (5-3)

for a random distribution of penny-shaped microcracks, and

£ = y/<*ij°ij- (5.4)

With respect to the different regimes of the loading curve in Fig. 5.3, Charalambides
& McMeeking assumed that
P=0 for a < a0,
ft = k{a — a 0 ) for <70 < a < as,
/3 = ps = k(as-G) for a>as, (5.5)
where k is a factor which governs the rate of microcracking with stress.
In an independent study, Brockenbrough & Suresh (1987) used a similar assump-
tion of penny-shaped flaws to develop a constitutive model for their numerical
simulation of compression fatigue crack nucleation in highly brittle solids. Using
in situ video photography and acoustic emission measurements, they measured the
threshold stress <x0 f ° r microcrack nucleation in edge-notched specimens of mono-
lithic alumina loaded in uniaxial cyclic compression (Fig. 5.\a shows notch tip
microcracking from one such test). Microcrack nucleation was assumed to follow
the relationship:
- - 1 ) for |<x|>|a o |, £ < & ,

£=0 for \a\ < |a o |, (5.6)

where A is a factor which, analogous to k in Eq. 5.5, controls the rate of micro-
cracking with stress, n is an exponent of order unity and a is as defined in Eq. 5.4.
fi = ps for a > crs. Unlike the earlier approaches, the assumption of a saturation in
microcrack density is not essential for the numerical procedure employed for mod-
eling crack initiation in cyclic compression. Figure 5.4 illustrates the stress-strain
curve (path A) for compression loading.
Let the total density of open microcracks which exist at the maximum far-field
compressive stress of the fatigue cycle be /3 max . Out of this total population, let a
certain fraction of microcracks, of density /3U, close fully upon complete unloading
174 Cyclic deformation and crack initiation in brittle solids

Fig. 5.4. Constitutive behavior of a microcracking brittle solid in cyclic compression. The strain
axis is enlarged for clarity.

from the far-field compressive stress. The development of permanent strains after
one compression cycle can then be quantified by an unloading parameter,

X— 1 — (5.7)
Pmax

The unloading path B in Fig. 5.4 corresponds to the idealized situation where all the
microcracks, of density /3 max (= /3S if saturation occurs) existing at the maximum far-
field compressive stress, gradually close upon unloading. In this case, fiu = j#max?
X = 0 and no permanent strains exist in the fully unloaded state. Path D represents
the other extreme case where all the microcracks nucleated during compression
loading are blocked by the presence of debris within them or they are locked in
friction. Here, X = 1 and unloading occurs with the same slope as the initial loading
portion of curve A. If the frictional sliding taking place between the faces of the
microcracks during compression loading is partially reversed upon unloading,
0 < Pu < Anax> 0 < A. < 1, and unloading occurs along path C. Thus the linear
unloading parameter X conveniently characterizes permanent strains representing
the entire range of linear unloading paths. The elastic secant moduli during unload-
ing are given by
Eu = E0X + Em(\ - XI vu = v0X + v m (l - X). (5.8)
Em and vm are the values of Young's modulus and Poisson's ratio, respectively, at
the maximum far-field compressive stress.
5.3 Highly brittle solids 175

Using a finite element analysis, Brockenbrough & Suresh (1987) showed that when
the unloading path leads to permanent strains, i.e. for k > 0, the resulting residual
stresses that arise in the matrix material in the vicinity of a stress concentration are
distinctly different from those induced under monotonic loading conditions. This
effect of permanent strains causes a mode of failure which is unique to cyclic loading
conditions. More significantly, the results imply that kinematic irreversibility of
microscopic deformation (in this case, the differences in the opening and closing
of microcracks) occurring during fatigue in a brittle solid is qualitatively similar to
the development of stress-strain hysteresis due to slip irreversibility in metal fatigue.
Note the similarity of the above formulation for the compression fatigue of brittle
solids to the process of compression fatigue in metals discussed in Section 4.11. A
similar conceptual framework was used by Lawn et al. (1994) to rationalize the
progressive evolution of microcracking during cyclic indentation of brittle ceramics.
A constitutive model similar to that shown in Fig. 5.4 has also been used to predict
the size and shape of the cyclic damage zones developing ahead of tensile fatigue
cracks in microcracking ceramic materials, Suresh & Brockenbrough (1990). These
cyclic damage zones are taken up for discussion in Chapter 11.

5.3.3 On possible effects of cyclic loading

Cyclic load experiments conducted on smooth-surfaced specimens of brittle
solids generally exhibit considerable scatter in number of cycles to failure or time to
rupture. This difficulty arises as a consequence of the variability associated with
initial flaw populations which, as discussed earlier, depend strongly on such factors
as processing methods and specimen surface preparation techniques. In situations
where a small artificial surface flaw is introduced to expedite failure, the results often
are a strong function of the initial flaw size.
Available experimental results generally point to some effects of cyclic loading on
time to rupture vis-d-vis that under static loads on account of the following possible
mechanisms:!
(1) When environmental effects primarily dictate the rupture behavior, the life-
time is controlled by time-dependent damage and failure mechanisms (see
Chapter 16) and cyclic frequency or waveform has no effect the time to
rupture (see Fig. 5.5).
(2) When purely mechanical cyclic loading leads to a progressive evolution of
damage as, for example, in the exacerbation of intergranular microcracking
during repeated cyclic loading (especially in the compression portion of
tension-compression loading), a detrimental effect of cyclic loading can
result. Under such circumstances, higher cyclic frequencies are generally
more damaging.

' Usually, for comparison of times to rupture under static and cyclic loads, the maximum nominal stress
corresponding to the fatigue cycle is taken to be the same as that of the static stress for smooth specimens.
176 Cyclic deformation and crack initiation in brittle solids

extrinsic factors
promoting a
beneficial effect
of cyclic loading ^ ^ — —
*» "^
) ruptiire (1<3g scale)

time-dependent •
rupture
^ ^ no fatigue effect

controlled by ~^»^ i
(ii
environment synergistic * i ^ ^
IU11

effects of . ^ ^
+•"
environment 1 ^^ ^
and fatigue | cycle-dependent failure
causing detrimental effect of fatigue
cyclic frequency (log scale)

Fig. 5.5. A schematic illustration of possible beneficial or detrimental effects of cyclic loading on
the time to rupture in brittle solids.

(3) When extrinsic factors, such as deflection of microscopic flaws, wedging of

the faces of microcracks by debris particles which are produced as a result of
repeated contact or the closure of microcrack faces, prominently influence
cyclic deformation behavior, an apparently beneficial effect of cyclic load-
ing, vis-a-vis static loading, is feasible, as shown in Fig. 5.5.

5.3.4 Elevated temperature behavior

An elevation in temperature tends to lower the degree of brittleness in most
materials. In some ceramics, several micromechanisms concurrently play an active
role in dictating the overall creep deformation. These mechanisms include: atomic
diffusional processes within the bulk of the grains and along grain boundaries and
interfaces, grain boundary sliding, and even dislocation plasticity. In some other
systems, such as many aluminum oxides subjected to certain combinations of tem-
peratures and strain rates, apparent ductility at elevated temperatures may simply
arise from the growth of microcracks along the facets of elastically deforming grains.
The growth of intergranular flaws may be caused by local diffusion processes or by
viscous flow of any amorphous films deposited along grain facets from the additives
or impurities introduced during fabrication.
This latter mechanism of interfacial microcracking by viscous flow of an amor-
phous phase in an otherwise elastic medium can be illustrated with the example of a
SiC whisker-reinforced A12O3 matrix composite. It is known that SiC, when exposed
to oxygen-containing media at temperatures typically in excess of 1250 °C, oxidizes
5.3 Highly brittle solids 177

to form an amorphous SiO2 glass phase. The viscous flow of this phase at the
elevated temperature leads to the nucleation of cavities and microcracks along the
interfaces between the alumina matrix and the SiC particles. Figure 5.6(a) shows an
example of interfacial cavities formed at every corner of a SiC particle in an A12O3-
33 volume% SiC whisker composite fatigued in 1400 °C air. The presence of the
amorphous interfacial film and the kinematically irreversible cyclic displacements
associated with the opening and closing of the interfacial microcracks promote
tensile fatigue deformation mechanisms which can be different, in some cases,
from those seen under sustained tensile loads. The reinforcement phase also breaks
under cyclic loading conditions; Fig. 5.6(b) is a micrograph of a SiC whisker broken
by cyclic tensile loads in 1400 °C air. In this figure, the meniscus of the silica glass
phase can be seen within the broken whisker.
Environmental effects at high temperatures can influence deformation and frac-
ture in brittle solids subjected to cyclic loads in a manner which is different from that
observed under static stresses. If the material does not contain any macroscopic
cracks or stress concentrations, the fatigue mechanisms described above are confined
to the near-surface region, where oxygen is available. If through-thickness stress
concentrations are present in the material, the transport of the environment to the
tip of the defect causes a damage zone to develop under the influence of an applied
stress. In the example shown in Fig. 5.6, the oxidation of SiC whiskers giving rise to
interfacial microcracking is essentially the same for monotonic and cyclic loading
conditions. However, mechanisms associated with microcrack opening and closure,
bridging of the flaws by reinforcement particles and the breaking of debonded whis-
kers are affected in different ways depending upon whether the composite is sub-
jected to monotonic or cyclic loads.
Direct tensile fatigue tests at 1000-1200 °C on hot-pressed Si3N4 unidirectionally
reinforced (along the tensile axis) with 30 vol.% SiC (SCS-6) fibers also reveal a
gradual reduction in the elastic modulus with the progression of fatigue damage
(Holmes, 1991). No such changes in compliance were observed in specimens which
are subjected to low stress cycles below the endurance limit (at 2 x 106 cycles).
Fatigue-induced changes in elastic properties were observed at maximum stress levels
that were above the monotonic proportional limit, with both mechanical fatigue and
creep influencing progressive damage. This regime of fatigue is characterized by
decreasing stiffness, increasing stress-strain hysteresis, and strain ratchetting (cyclic
creep). However, when the maximum stress level in fatigue is below the monotonic
proportional limit, only creep deformation occurs. This creep regime of loading is
characterized by strain ratchetting, but no change in elastic properties. Holmes has
also shown that the fatigue life deteriorates markedly with decreasing tensile load
ratio R. Figure 5.7 contains experimental data on the changes in the cyclic stress-
strain hysteresis loops with increasing number of fatigue cycles. Note the occurrence
of cyclic creep (as in metals, Section 3.9) and of increase in specimen compliance due
to repeated stress cycles.
178 Cyclic deformation and crack initiation in brittle solids

0.2 n

Fig. 5.6. Microscopic deformation mechanisms ahead of a crack tip in an Al 2 O 3 -33 volume%
SiC whisker composite fatigued at a stress ratio, R = crmin/crmax = 0.15 and a load frequency of
0.15 Hz (sinusoidal waveform) in 1400 °C air. (a) TEM photograph showing nucleation of
cavities at the interfaces between SiC whiskers and matrix A12O3 grains, (b) A SiC whisker
broken under cyclic loads. The meniscus of the amorphous glass phase can be seen within the
broken whisker. (From Han & Suresh, 1989. Copyright American Ceramic Society. Reprinted
with permission.)
5.4 Semi-brittle solids 179

300

200

100

0.1 0.2
strain (%)

Fig. 5.7. Uniaxial tensile cyclic stress-strain behavior of a Si 3 N 4 -SiC fiber composite at
maximum stress levels above the proportional limit (196 MPa). The number of stress cycles is
indicated along with each hysteresis loop. Test temperature, T = 1200 °C, laboratory air
environment, R = 0.1, and vc = 10 Hz. Number of cycles to failure, Nf — 6.5 x 104. (After
Holmes, 1991.)

5.4 Semi-brittle solids

Among nonmetallic crystalline materials conforming to the definition of
semi-brittle behavior, the fatigue deformation characteristics of ionic crystals have
been the subject of much investigation. Experimental studies of monotonic deforma-
tion in rock-salt-type materials have shown that the pile-up of dislocations at obsta-
cles or at other dislocations can lead to the nucleation of cracks. It is also known
that, under some restricted conditions of temperature and strain rate, cyclic loading
can produce slip bands and cellular structures in rock salt crystals similar to those
observed in FCC metals.

5.4.1 Crack nucleation by dislocation pile-up

When the resolved shear stress exceeds a critical value on a favorably
oriented low index plane of a semi-brittle crystal, such as MgO, dislocation sources
of the Frank-Read type are activated. If the glide of the dislocation loops, moving
outwardly from the source, is impeded by an obstacle such as a grain boundary or an
inclusion, a pile-up of the dislocations occurs at the barrier. Since the piled-up
coplanar dislocations are of the same sign, the mutually repulsive forces between
them cause a concentration of stress at the obstacle. One may then postulate (Zener,
1948) that a brittle crack nucleates at the obstacle if the energy associated with the
pile-up of dislocations is sufficient to compensate for the surface energy term ys
associated with the creation of new crack surfaces. This scenario is the basis of
180 Cyclic deformation and crack initiation in brittle solids

the well known 'Petch' relationship (Petch, 1953) for the grain size-dependence of
yield strength:

o fe (59)

where rxy is the resolved shear stress on the glide plane, r®v is the friction stress
(which is to be overcome before the dislocations glide on the slip plane), G is the
shear modulus and d% is the grain size of the material. Equation 5.9 is derived by
assuming that the Frank-Read source is located at the center of the grain.
Evidence of dislocation pile-up at grain boundaries and of the attendant nuclea-
tion of a crack at the boundary is available for a number of semi-brittle solids. The
birefringence of transmitted polarized light in Fig. 5.8(a) shows stress concentra-
tions, generated at the tip of slip bands obstructed by a grain boundary, in a bicrystal
of MgO. Transgranular crack formation at the site of slip band obstruction at the
grain boundary is evident in the micrograph of the etched MgO bicrystal, Fig. 5.8(6).
Figure 5.9 illustrates some mechanisms by which the pile-up of dislocations can
nucleate a brittle crack. The nucleation of a microcrack or a wedge-shaped cavity by
the pile-up of dislocations at an obstacle such as a grain boundary (GB) is shown in
Fig. 5.9(a). The stress gradients ahead of a dislocation pile-up are similar to those
found in front of a shear (mode II) crack. If the interface in Fig. 5.9(a) is weak,
preferential cracking occurs along the interface.
The crack nucleation mechanism shown in Fig. 5.9(6), which was first proposed by
Cottrell (1958), is a process in which two intersecting slip planes provide the nucleus
for a crack, even in the absence of a pre-existing obstacle to slip. This process has
also been suggested as a mechanism for the initiation of subsurface fatigue cracks
along {001} cleavage planes of /3 phase in Ti alloys composed of a-fi duplex micro-
structures (Ruppen et al., 1979).

5.4.2 Example problem: Cottrell mechanism for sessile dislocation

formation
Problem:
Consider the dislocation reaction involving a [111] dislocation on the
(101) slip plane and the [111] dislocation on the (101) plane in iron. This process
is similar to that schematically sketched in Fig. 5.9(b).
(i) Write down the dislocation reaction and find the product dislocation,
(ii) Show that this dislocation reaction is energetically feasible,
(iii) Show that the product of this dislocation reaction is a sessile dislocation.
Solution:
(i) The Burgers vector of the product dislocation is the vectorial sum of the
Burgers vectors of the dislocations on the two intersecting planes:
5.4 Semi-brittle solids 181

Fig. 5.8. {a) Stress concentrations revealed by the birefringence of transmitted polarized light in
an MgO bicrystal at a location where a grain boundary obstructs slip. (From Ku & Johnston,
1964. Copyright Taylor & Francis, Ltd. Reprinted with permission.) (b) Transgranular cracks
formed at sites where slip is impeded by the grain boundary in the MgO bicrystal. (From
Johnston, Stokes & Li, 1962. Copyright Taylor & Francis, Ltd. Reprinted with permission.)
182 Cyclic deformation and crack initiation in brittle solids

(a)

Fig. 5.9. Mechanisms for the nucleation of cracks by (a) dislocation pile-up at a grain boundary
(GB) and (b) dislocation reactions.

a 4001]. (5.10)
J (ioi) + 2 ^
Thus, the Burgers vector of the product dislocation is a[001].
(ii) Note that the energy of a dislocation is proportional to the square of the
magnitude of the Burgers vector. The sum of the energies of the two
reacting dislocations is proportional to 3a /2. The energy of the product
dislocation is proportional to a . Since there occurs a net reduction in
energy, this reaction is energetically favorable.
(iii) The product dislocation is of pure edge character, whose extra plane of
atoms lies parallel to the (001) cleavage plane of BCC a-iron. Since this
dislocation is not favorably oriented for slip (i.e. {110} plane and (111)
direction), it cannot glide. It is a sessile dislocation and forms an obstacle
to other dislocations gliding down the (101) and (TOl) planes.

5.4.3 Cyclic deformation

Under appropriate cyclic loading conditions, semi-brittle solids can exhibit
characteristics of fatigue damage which are apparently similar to those found in
ductile metal crystals. In his work on surface roughening, Forsyth (1957) demon-
strated that intrusions and extrusions form on the surfaces of cyclically strained
AgCl crystals in much the same way as the development of surface roughness in
FCC metals and alloys (Chapter 4). Figure 5.10 is a micrograph, prepared under
transmitted light, showing surface extrusions and crevices in AgCl. Forsyth found
that the crevices eventually developed into fatigue cracks.
Conditions for the possible development of cell structures in BCC and HCP metals
were examined in Sections 2.12 and 2.13, respectively. There has long been a search
for similar ductile modes of fatigue deformation in semi-brittle crystal structures.
Earlier investigations (e.g., McEvily & Machlin, 1961; Subramanium & Washburn,
1963; Argon & Godrick, 1969) of reversed fatigue in a rotating bend configuration in
NaCl, LiF and MgO at room temperature did not reveal any substructural devel-
5.4 Semi-brittle solids 183

Fig. 5.10. Intrusions and extrusions formed on the surface of fatigued AgCl crystal. (From
Forsyth, 1957. Copyright The Royal Society, London. Reprinted with permission.)

opments comparable to those found in cyclically strained FCC metals. It has, how-
ever, been shown by Argon & Godrick (1969) that the elevated temperature fatigue
deformation of LiF is similar to the room temperature cyclic deformation character-
istics of ductile FCC metals. They found that above 673 K (which is 59% of the
absolute melting temperature), cross slip and dislocation climb were favored. Cyclic
loading in this temperature regime produced pores throughout the highly strained
volume of the crystal. Continued cyclic straining resulted in a gradual change in the
density of the crystal as a consequence of cavitation. Argon and Godrick measured a
fractional density change of 3 x 10~8 per cycle in LiF crystals fatigued at 783 K at a
strain amplitude of 2.5 x 10~3. Similar pore development was also observed in AgCl
crystals which were fatigued above 423 K (58% of the absolute melting temperature).
Majumdar & Burns (1981) also employed direct push-pull fatigue loading in
smooth specimens of LiF crystals at elevated temperatures and showed that disloca-
tion banding occurred within subgrains. Microcracking also appeared to take place
along the (110) directions at low strain amplitudes and at large numbers of fatigue
cycles. Associated with the bands are alternate regions of high and low dislocation
density which seem to be sites where dynamic recovery occurs. As the temperature is
raised, the behavior of the LiF crystal appears to resemble more closely that of FCC
metals. A TSB-like' slip pattern emerges in the fatigued crystal at a temperature of
573 K, and at low strain rates and strain amplitudes. Ladder-like dislocation struc-
tures appear to exist within the PSBs, Fig. 5.11. The arrangement and the spacing of
dislocations within the ladders of PSBs, however, are different from those of ductile
crystals. The fatigued crystal also exhibits a plateau where the saturation value of the
shear stress is independent of the plastic strain amplitude as in FCC crystals (Fig.
2.2). Further increases in test temperature and strain amplitude cause a cellular
structure to form, similar to the trend seen in regime C of Fig. 2.2.
184 Cyclic deformation and crack initiation in brittle solids

Fig. 5.11. PSB-like dislocation structure in LiF single crystal fatigued at 573 K. A6 p /2 = 0.5%,
and € — 10~3 s" 1 . Note that the rungs of the PSB ladder structure are bent, as, for example, at
location A. The curved dark lines running across the micrographs are cleavage steps and their
positions oscillate with the PSB structure, probably as a result of local stresses. (From
Majumdar & Burns, 1982. Copyright Pergamon Press pic. Reprinted with permission.)

Majumdar & Burns (1987) also conducted fully reversed fatigue tests on MgO
single crystals at 743 K. They found that dense bundles of dislocations developed
as a consequence of reversed straining, similar to the vein structure evolving from
the early stages of fatigue in FCC metals (Fig. 2.2). These bundles were aligned
normal to the Burgers vector. Bowed out screw dislocations were observed
between the edge dislocation bundles suggesting that the screws were largely
mobile.
In summary, one of the principal contributing factors for the pronounced brittle-
ness of ionic crystals at low temperatures is the limited possibility for cross slip.
However, when certain combinations of temperature and strain rate favor cross slip,
it is not surprising to observe the aforementioned similarities between their fatigue
deformation characteristics and those of ductile FCC metals.

5.5 Transformation-toughened ceramics

Transformation-toughened ceramics constitute a special class of brittle
solids in which 'plasticity' can be introduced by means of phase changes under the
influence of an applied stress. The phase change imparts a propensity for nonlinear
deformation and for stable fracture in monotonic and cyclic loading. Like the TRIP
steels which exhibit TRansformation-Induced Plasticity, transforming ceramics offer
5.5 Transformation-toughened ceramics 185

the possibility of optimizing strength and ductility by the proper dispersion of a

metastable phase in a stable matrix.

5.5.1 Phenomenology
Tetragonal (/) to monoclinic (m) phase changes occur as a martensitic trans-
formation in ceramics which contain metastable tetragonal ZrO2. This tetragonal
phase may be present in a stable cubic matrix phase in the form of a precipitate (as in
partially stabilized zirconia, PSZ), or a dispersoid (as in ZrO2-toughened alumina,
ZTA), or may be formed as the fine matrix phase in the (nearly) 100% t-ZrO2
polycrystals, TZP. Since the seminal paper of Garvie, Hannick & Pascoe (1975)
on this topic, it has been acknowledged that the dilatational and shear strains
accompanying the / to m transformation can account for the remarkable toughening
properties of ZrO2-containing ceramics at low temperatures (typically below 700 °C).
Among the various transforming ceramic micro structures, a large body of
research has centered around the monotonic and cyclic deformation and fracture
characteristics of ZrO 2 , partially stabilized with MgO (commonly referred to as Mg-
PSZ). Figure 5A2(a) is a micro structure of a peak-aged (maximum strength) Mg-
PSZ containing 9 mol.% MgO. This material is composed of cubic phase zirconia
grains, of 50jim average diameter, with the MgO in solid solution. Lens-shaped
tetragonal precipitates, which are 300 nm long and metastable at room temperature,
populate the interior of the grains. The tetragonal oaxis of the precipitates is parallel
to their smallest dimension. The precipitates are oriented within the cubic phase
grains in such a way that their oaxis is parallel to one of the three cubic axes.
Experimental studies by Chen & Reyes Morel (1986) and others reveal that the
mechanism by which microscopic strains induced by martensitic transformation are
converted to macroscopic plastic strains is via shear localization. Figure 5.\2(a)
shows an interior section of the material, deformed under a hydrostatic compressive
stress of 200 MPa, showing shear bands within the grains. It is seen that the shear
bands, which span the entire grain, have different orientations in different grains; the
bands within individual grains appear to be parallel. In order for the transformation
of individual tetragonal particles to cause a macroscopic shear strain via shear
banding, it appears necessary that the particle transformation be correlated, as
shown in Fig. 5A2(b).
Shear bands in 'transformation plasticity' can thus be deemed functionally equiva-
lent to the slip bands in 'dislocation plasticity' of polycrystalline metals. Although
the shear characteristics of the two phenomena are similar in this respect, there are
also some major differences between them. While dislocation plasticity is volume-
preserving, transformation plasticity induces both microscopic and macroscopic
dilatational strains. For unconstrained t -> m transformation in Mg-PSZ, the max-
imum amounts of volumetric and shear strains are 0.04 and 0.16, respectively. When
embedded in an elastic matrix, correlated transformation of the particles leads to the
186 Cyclic deformation and crack initiation in brittle solids

(b)

Fig. 5.12. (a) Transgranular shear band formation due to martensitic transformation. (From
Chen & Reyes Morel, 1986. Copyright American Ceramic Society. Reprinted with permission.)
(b) Correlated transformation of particles leading to the formation of a shear band.
5.5 Transformation-toughened ceramics 187

nucleation of shear bands, whose orientation varies from grain to grain. Thus, the
average shear strain over the particle for constrained transformation is less than 0.16,
although the dilatant transformation strain is still 0.04 for the particle. It is also
known that martensitic transformation can lead to microcracking as a consequence
of the intersection of shear bands and grain boundaries, as well as by the decohesion
of the transformed particle from the surrounding matrix. The opening of a popula-
tion of microcracks also produces macroscopic dilatational strains.
Permanent phase changes cause stress-strain hysteresis in transforming ceramics
upon loading and unloading, similar to the behavior found in metallic materials.
Figure 5.13 is a stress-strain diagram of the peak-aged Mg-PSZ subjected to one
cycle of uniaxial tensile loading. Beyond a tensile stress of about 275 MPa, fully
irreversible phase transformation takes place; the resulting constitutive response
becomes highly nonlinear. Elastic unloading is observed in the nonlinear regime;
reloading occurs with an elastic modulus which is identical to that of the initial
elastic regime. These results indicate how nonlinear effects associated with marten-
sitic phase transformations can provide a mechanism for stable fatigue damage to
occur in nominally brittle solids.

5.5.2 Constitutive models

Constitutive models for general multiaxial loading conditions have been
proposed to quantify the monotonic and cyclic deformation response of transform-
ing ceramics in uniaxial loading. Initial developments in this direction have come
from attempts aimed at estimating the extent of toughening due to phase changes
ahead of a crack tip (e.g., McMeeking & Evans, 1982; Budiansky, Hutchinson &

400

200

1 2
strain (x 103)

Fig. 5.13. Uniaxial tensile loading and unloading behavior of peak-strength Mg-PSZ. (After
Marshall, 1986.)
188 Cyclic deformation and crack initiation in brittle solids

Lambropoulos, 1983). These models, which are described in some detail below,
neglect the shear component associated with the transformation and invoke the
assumption that the dilatant transformation occurs at a critical mean stress am.
In the analysis of Budiansky et al., the constitutive behavior is formulated for a
linear elastic matrix with embedded metastable particles which undergo irreversible
inelastic volume expansion. Assume that the matrix material deforms linearly under
both hydrostatic tension and compression with bulk modulus B, according to
1
c>m = —<yjrk = (5.11)

where otj and etJ are stress and strain tensors, respectively, and ekk is the total
dilatation.
When the mean stress due to a monotonically increasing load is less than a critical
value (7^, the particles satisfy Eq. 5.11 with the same bulk modulus B, as shown in
Fig. 5.14. However, once am > o^, the incremental response of the particle is gov-
erned by Bf where
&m = B'ekk. (5.12)
The inelastic or transformed portion of the dilatation 0p is the difference between the
total and elastic dilatation:

c/p — €kk . ^0.1J)

Upon complete transformation, the incremental response is again governed by the

bulk modulus B at large values of ekk, as per the relation
&m = Bekk. (5.14)

crl

particles matrix composite

(a) (b) (c)
Fig. 5.14. Stress-strain response due to purely dilatant transformation of particles in the elastic
matrix of the two-phase composite. The shear behavior is linear and the shear modulus is the
same for both the particle and the matrix. (After Budiansky, Hutchinson & Lambropoulos,
1983.)
5.5 Transformation-toughened ceramics 189

In this model, the particle and the matrix are assumed to exhibit the same shear
behavior; the shear modulus of the composite is G at all strains. The incremental
response of the composite for am > a^ is
a = + (515)
= Bekk

where fp is the volume fraction of the toughening particles. In the intermediate

segment of Fig. 5.14, the dilatation of the transformed composite is

with 0 =fp0p. The maximum dilatation upon complete transformation is 0T =fp0p.

If Bf < —4G/3 in Eq. 5.15, a particle in an infinite elastic matrix with bulk and shear
moduli B and G, respectively, transforms completely to 0p as soon as the critical
mean stress is imposed. Even if such a transformation occurs at a critical mean stress,
a distribution of critical mean stresses may exist in a composite consisting of a
distribution of particle sizes. In this case, the incremental bulk modulus may never
drop below 4G/3. If the particle distribution is sufficiently wide, supercritical trans-
formation, i.e. B < —4G/3, may never happen. Budiansky et at. term B = —4G/3 as
the condition denoting a critically transforming composite and B > —4G/3 as the
condition denoting a subcritical transformation. This latter case, with G > 0, is the
condition for a real longitudinal wave speed.
Budiansky et al. use an incremental formulation, similar to that employed for an
elastic-plastic solid (Section 1.4.3), where loading occurs on the transforming branch
of the stress-strain curve if €kk > 0, and

&m = B k k k and 0 = 1 1 - -Ukk. (5.17)

Unloading occurs if ekk < 0 and

°m = Bekk and 0 = 0. (5.18)
T
Upon unloading to zero stress, the permanent dilatation is 6. If 0 < 0 , only partial
(subcritical) transformation is seen. For this purely dilatant transformation, the
stress-strain relations in three dimensions are

€tj = ^Stj +^mSy +\&&ip °ij = 2GeU + B(€kk ~ 0)SiJ9 (5.19)

where Sy are the components of the stress deviator (Section 1.4.2) and
etj (= €y — [5y€^]/3) are the components of the strain deviator, respectively.
In an attempt to address the issue of combined dilatational and shear effects on
the multiaxial constitutive response, an experimentally based formulation has been
suggested by Chen & Reyes Morel (1986). They carried out unconstrained and
constrained compression experiments on cylinders of Mg-PSZ at room temperature.
Their experimental results, showing the variation of axial plastic compression strain
(denoted -A), radial strain (denoted R) and volumetric strain (denoted V) as a
function of the differential axial compressive stress D for a superimposed pressure
190 Cyclic deformation and crack initiation in brittle solids

P = 200 MPa, are plotted in Fig. 5.15. A, R and V scale with the applied differential
stress in the ratio of —2 : 3 : 4.
The yield condition for transformation plasticity, which can be derived from the
multiaxial compression experiments, is

(5.20)

where a e is the effective stress, Eq. 1.27, am is the mean pressure, and a* and a^ are
measures of hardness.
The experiments of Chen & Reyes Morel also suggest the following yield criterion
consistent with Eq. 5.20:
Yc= 70c + aci>, (5.21)
c
where YQ and Y are the compressive yield stress values at pressures of 0 and P,
respectively, and ac is a constant, which is about two for a wide range of strain
values, except for very small and very large strains. Experiments on Mg-PSZ suggest
that a* = 3Fo/5 and a^ = O.57o- The numerical value ac = 2, along with the
assumption of normality flow, produces a ratio A : R : V of — 2 : 3 : 4, which is
consistent with the results of Fig. 5.15. Furthermore, since this formulation is phe-
nomenological, it also accounts for the effects of microcracking (induced by trans-
formation) on deformation.

0.018

0.016

0.014

0.012

1 0.010

I 0.008
"3,

0.006

0.004

0.002

1000 1200 1400 1600 1800 2000 2200 2400

|S| (MPa)

Fig. 5.15. Experimentally determined variation of axial (^4), radial (R) and volumetric (V)
strains plotted as a function of the differential axial compressive stress E for Mg-PSZ
(maximum strength condition) under an imposed hydrostatic pressure, P = 200 MPa and a
strain rate, 6 = 1 0 . (After Chen & Reyes Morel, 1986.)
5.6 Fatigue crack initiation under far-field cyclic compression 191

This experimentally based model has been used to rationalize observations of

tension and compression yield anisotropy in Mg-PSZ. Suresh & Brockenbrough
(1988) have also implemented this constitutive formulation, in conjuction with the
incremental (flow) theory of plasticity, to model the development of transformation
zones and residual stresses ahead of stress concentrations in Mg-PSZ subjected to
cyclic compression loads. For compression fatigue, the model (incorporating elastic
unloading and reloading behavior in fatigue) provided accurate predictions of the
direction of crack nucleation as well as the maximum distance of fracture.

5.6 Fatigue crack initiation under far-field cyclic compression

The possibility that fatigue crack initiation and stable crack growth, attri-
butable solely to cyclic variations in applied loads, can occur at room temperature
(even in the absence of an embrittling environment) in single phase ceramics, trans-
formation-toughened ceramics, and ceramic composites was first demonstrated for
cyclic compression loading of notched plates (Ewart & Suresh, 1986, 1987). This
fatigue crack growth phenomenon is a true mechanical fatigue effect on account of
the following experimental observations: (i) Fatigue crack initiation and growth
occur perpendicularly to the compression axis in brittle solids subjected only to cyclic
loads. Monotonic compressive stresses promote a splitting mode of failure parallel
to the stress axis, (ii) There is a gradual increase in crack length with an increase in
the number of compression cycles, (iii) Although an embrittling environment can
modify the rate of crack growth, fatigue fracture in cyclic compression is not a
consequence of environmental effects in that it can take place even in vacuo. (iv)
The rate of crack growth is strongly influenced by such mechanical fatigue variables
as mean stress, stress range, and stress state, (v) The overall crack growth character-
istics of fatigue crack initiation and growth at a macroscopic level are qualitatively
similar to those observed in ductile metals and alloys (see Section 4.11) and semi- or
noncrystalline polymers and polymeric composites (see next chapter).
Brittle solids such as ceramics, rocks or concrete are known to fracture in mono-
tonic compression in one of several mechanistically dissimilar modes (McClintock &
Walsh, 1962; Nemat-Nasser & Horii, 1982):

(1) The frictional sliding of pre-existing flaws produces opening displacements

at the crack tip which are sufficient to initiate a locally tensile mode of
failure. Consequently, in unconstrained axial or radial compression, cracks
extend across planes of local maximum principal tension and curve in a
direction parallel to the maximum principal axis of compression. This pro-
cess causes the material to 'split' parallel to the compression axis.
(2) The coalescence of pre-existing microcracks or weak interfaces results in a
shear failure under small confining pressures.
192 Cyclic deformation and crack initiation in brittle solids

(3) Large confining pressures promote a relatively more homogeneous, pseudo-

ductile deformation by limiting the coalescence of micro-defects.

Under cyclic compression loading conditions, however, brittle solids with stress
concentrations exhibit a completely different mode of crack initiation which is
macroscopically similar to that found in metallic materials (Section 4.11). When
notched plates of brittle solids are subjected to some combinations of (cyclic com-
pression) stress amplitude and mean stress, confined microcracking occurs at the tip
of the notch (see Fig. 5Aa). If even a fraction of this microcracking deformation at
the notch tip is permanent (i.e. when microcracks do not open and close the same
way during compression loading and unloading), residual tensile stresses are created
within the microcrack zone upon unloading from the maximum far-field compres-
sion stress. Similarly, permanent strains associated with stress-induced phase
changes, frictional sliding, or creep can also cause residual tensile stresses to develop
ahead of stress concentrations under far-field cyclic compression. Examples of fati-
gue crack growth in cyclic compression ahead of stress concentrations in brittle
solids are shown in Figs. 5A6(a) and (b).
Brockenbrough & Suresh (1987) conducted a finite element simulation of near-
tip fields in notched plates of a polycrystalline aluminum oxide subjected to
uniaxial cyclic compression. They used a constitutive model for microcracking
which was described earlier in Fig. 5.4 and in Eqs. 5.4-5.8. Their predictions
of the variation of residual stresses normal to the plane of the notch, ayy
(which is the stress ayy perpendicular to the plane of the notch, normalized by
the product of the maximum far-field compressive stress a°° and the elastic stress
concentration factor for the notch tip Kt), are plotted in Fig. 5.17 as a function
of the distance directly ahead of the notch tip x, normalized by the notch tip
radius p. These computations were made using the following values for the vari-
ables in Eq. 5.6: cro/(cr°°Kt) = 0.0076, A = 0.004, n = 1, and & = 0.4 for penny-
shaped microcracks. When the microcracking deformation ahead of the notch tip
leaves no permanent strains, i.e. when X = 0, (see Eq. 5.7 and Fig. 5.4), essentially
no residual stresses are induced ahead of the notch tip after one compression
cycle. On the other hand, when A > 0, a region of large residual tension is created
at the notch tip after the far-field compressive stress is removed. Note that
residual stresses are self-equilibrating. A zone of residual tension in the immediate
vicinity of the notch tip is accompanied by a zone of residual compression away
from the notch tip, Fig. 5.17.
The effect of this residual stress zone is that residual tensile stresses easily exceed
the tensile strength of the brittle solid over a distance of the order of the notch root
radius. This implies that a mode I crack will develop over this distance upon unload-
ing from the far-field compression stress, an inference which is supported by experi-
mental observations. Once a fatigue crack initiates from the notch tip, its rate of
growth during subsequent compression cycles is dictated by such factors as the
exhaustion of the residual stress zone created during the first cycle, formation of
5.6 Fatigue crack initiation under far-field cyclic compression 193

r . '"

(b)

Fig. 5.16. Examples of mode I fatigue cracks initiated at stress concentrations under far-field
cyclic compression: (a) Polycrystalline A12O3. (b) Cement mortar. The compression axis is
vertical in both cases. (From Ewart & Suresh (1986) and Suresh, Tschegg & Brockenbrough
(1989), Reprinted with permission.)

debris particles of the brittle solid due to repeated contact between the crack faces,
the generation of a residual stress field in subsequent cycles, and the development of
closure due to an increase in crack length. Experiments by Ewart & Suresh (1987) on
polycrystalline alumina show that the high frequency contact between the crack faces
generates debris particles of the ceramic, typically of the order of 1 jim in size, within
the crack. The presence of the debris particles promotes a wedging effect. Periodic
removal of the debris particles by ultrasonic cleaning of the crack leads to a sig-
nificant increase in the total distance of crack growth (Fig. 5.18). In ceramics with
grain sizes in the range 1-30 jim, crack growth over a distance of the order of 1 mm
194 Cyclic deformation and crack initiation in brittle solids

U.JU

0.45 -

0.40 -

0.35

0.30

0.25 Ir— A = 1.00

U A= 0.75
0.20

•\i~ A
= 0.50
0.15

0.10 \ y\ A = 0.25

0.05 A
^0.00
0.00
^ — = 1 ^ 0.50
=^-0.75
-0.05

-0.10 i i t i i i

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

x
Fig. 5.17. Finite element predictions of the normalized stress ayy as a function of the normalized
distance x/p directly ahead of the notch tip for different unloading paths, A.. (After
Brockenbrough & Suresh, 1987.)

0 8 16 24 32
number of compression cycles, N (x 10~4)

Fig. 5.18. Variation of fatigue crack length a, measured from the notch tip, as a function of the
number of compression cycles in a-alumina of grain size = 1 8 urn (curve A). Curve B shows the
increase in crack growth rates as a consequence of reducing crack closure by the removal of
debris particles from the crack after every 5000 compression cycles using ultrasonic cleaning.
(After Ewart & Suresh, 1986.)
5.6 Fatigue crack initiation under far-field cyclic compression 195

has been observed under fully compressive far-field cyclic loads. The fatigue cracks
arrest after growth over this distance because of the development of crack closure in
cyclic compression and the exhaustion of the residual tensile field.
At the maximum far-field compressive stress, there exists a state of compression
immediately ahead of the notch tip. Complete unloading, however, gives rise to a zone
of residual tension at the notch tip. Numerical simulations by Brockenbrough &
Suresh (1987) reveal that residual tensile stresses are generated only after unloading
occurs below a certain critical value, crcr, of compressive stress, i.e. when
<rcr < <7°° < amax, Fig. 5.17. This result implies that the mean stress of the compression
fatigue cycle will have an important effect on the magnitude and extent of the residual
stressfield.As the mean stress is pushed far below the zero level, the extent of residual
tension will decrease and, consequently, crack initiation from the notch tip will become
more difficult. Such predictions have been confirmed independently by experimental
studies of the effects of cyclic compressive mean stress on crack initiation in /3-alumina
(James, Tait & Mech, 1991) and in brittle polymers (Pruitt & Suresh, 1993).
An important feature of the phenomenon of crack initiation from notches under
cyclic compression is that both monolithic and composite (brittle or ductile) solids
with vastly different microscopic deformation modes exhibit a macroscopically simi-
lar mode I fatigue crack growth behavior. The principal reason for this universal
trend is that residual tensile stresses are induced within the notch tip damage zone
during cyclic compression as long as permanent deformation occurs ahead of the
notch tip; such permanent deformation can be a consequence of dislocation plasti-
city, phase transformation, microcracking, interfacial sliding or creep. Since the zone
of residual tension is embedded within a compressive residual stress field, crack
initiation and growth in cyclic compression is intrinsically stable even in brittle
solids. Results of numerical analysis of the residual stress fields ahead of notches
subjected to cyclic compression have been reported by Suresh & Brockenbrough
(1988) and Suresh (1990a) for single phase ceramics, transforming ceramics, creeping
solids, and cement mortar.
Experimental studies of crack initiation in cyclic compression in ceramic compo-
sites have pointed out the potentially deleterious effects that fatigue loads can have on
the service life of a structural component (see, for example, Suresh, 1990a). Consider,
for example, the case of hot-pressed Si3N4 which is reinforced with SiC whiskers. It is
known that the addition of 20-30 volume% SiC to the Si3N4 matrix can lead to an
increase in fracture toughness by more than a factor of two over that of the unrein-
forced matrix material. However, when the composite contains stress concentrations,
the application of cyclic compressive loads causes fatigue cracks to initiate more easily
in the composite than in monolithic Si3N4. This effect can be rationalized by noting
that the stress-strain curve for the composite is more nonlinear than that for the
matrix material. Unloading from a far-field compressive stress can promote a higher
degree of permanent deformation which, in turn, may lead to larger residual tensile
stresses ahead of the stress concentration (see Fig. 5.17).
196 Cyclic deformation and crack initiation in brittle solids

Crack initiation ahead of stress concentrations subjected to cyclic compression

also offers a capability to introduce controlled fatigue pre-cracks in brittle materials
prior to the determination of fracture toughness, creep crack growth, or fatigue
crack growth in tension. An advantage of this technique is that fatigue cracks can
be introduced in brittle solids such as ceramics and ceramic composites and in ductile
metals using similar cyclic compression test conditions and specimen geometries.
Furthermore, the cyclic compression pre-cracking method is the only known tech-
nique for introducing fatigue pre-cracks in circumferentially notched cylindrical rods
of brittle solids (which are used for quasi-static and dynamic mode I and mode III
fracture tests).

5.6.1 Example problem: Crack initiation under far-field cyclic

compression
Problem:
A polycrystalline ceramic plate containing a through-thickness edge notch
is subjected to uniaxial cyclic compression loading at room temperature. The
specimen is placed between two perfectly parallel surfaces and the cyclic com-
pressive loads are applied in a direction normal to the plane of the (nonclosing)
notch. The peak load of the fatigue cycle is amax and the lowest load is a min . (For
fully compressive cyclic loading, where a max and amin are both negative,
^min < < W , l<wl < |tfmin| and R = |or min |/|a max | > 0.) Assuming that grain
boundary microcracking due to the applied loads is the primary source of
inelastic deformation, answer the following questions.

(i) Four experiments are conducted where amin is held fixed and constant-
amplitude compressive stress cycles with the following R ratios are used:
(a) 2, (b) 10, (c) 30, and (d) oo (which corresponds to zero-compression
cyclic loading with a max = 0). How would you expect the initial rate of
crack growth and the maximum distance of crack growth under cyclic
compression (before complete crack arrest) to vary as a function of R for
these four cases?
(ii) Three additional experiments are conducted where <7max is held fixed and
the following R ratios are used: (a) 2, (b) 10, and (c) 30. How would you
expect the initial rate of crack growth and the maximum distance of crack
growth under cyclic compression (before complete crack arrest) to vary as
a function of R for these three cases?

Solution:
On the basis of the information provided, assume that the trends pre-
dicted by the simulations in Fig. 5.17 apply to the polycrystalline ceramic con-
sidered here.
Exercises 197

(i) It is evident from the preceding section and the given information that,
for fixed a min , the extent of permanent damage developed ahead of the
notch during compression loading is the same in all four cases. The extent
of residual tensile stresses generated upon unloading then is dependent on
the <rmax. That is, the closer is the value of amax to zero far-field stress (and
hence the higher the R ratio), the larger would be the tensile stress gen-
erated at the notch tip. Therefore, the maximum tensile residual stress
field (or a distance ahead of the notch tip over which a tensile stress of
some level is developed) would be expected to increase with an increase in
the applied R ratio. Consequently, the initial rate of crack growth and the
maximum distance of crack growth under cyclic compression (before
complete crack arrest) would both increase with an increase in R ratio.
(ii) For a fixed value of applied a max , amin increases with increasing R. Larger
the value of R, greater would be the amount of permanent deformation
(in this case, microcracking) at the notch-tip. As a simple approximation,
one would then expect X to be higher for higher values of R. From Fig.
5.17, it could then be argued again that the extent of tensile residual stress
field, the initial rate of crack growth and the maximum distance of crack
growth under cyclic compression (before complete crack arrest) would all
increase with an increase in R ratio.

Exercises
5.1 Two dislocations in BCC a-iron with Burgers vectors a o /2 [111] and
ao/2 [111] (where a0 is the lattice dimension) glide along two intersecting
{110}-type slip planes meet.
(a) Write down the dislocation reaction and show that it is energetically
favorable.
(b) Discuss the orientation of a possible cleavage crack which could be
triggered by this Cottrell mechanism.
5.2 A cubic crystal contains a mixed dislocation (i.e. a dislocation with com-
bined edge and screw components) with a Burgers vector parallel to [001].
What would be the orientation and geometry of the steps on the cleavage
fracture surface for a crack propagating along the cleavage plane (001)?
Discuss various possibilities for the crack-dislocation intersection.
5.3 Consider a brittle elastic solid which is subjected to cyclic loading. The
material is first loaded in uniaxial tension. At a threshold tensile stress a T ,
microcracks initiate in the material; a further increase in tensile stress to a
peak value of ^Tmax results in a progressive increase in both the number of
microcracks per unit volume and the average size of the microcracks, and in
198 Cyclic deformation and crack initiation in brittle solids

the release of the residual thermal strain. The specimen is unloaded at crTmax.
During unloading, the microcracks remain open until a far-field tensile
stress of a Tcont . Below orTcont, lowering the far-field stress to zero and sub-
sequent load reversal into compression results in a gradual recovery of
stiffness until all the microcrack surfaces are closed (where the far-field
stress is trCclose). Further increases in compressive stress to a peak value of
a Cmax are borne by the fully dense elastic solid. Unloading from the far-field
compressive stress results in the elastic recovery of a fully dense solid until
the microcracks reopen at a far-field compressive stress of a Copen , with
I^Copenl > I^Cclosel-
(a) Plot the variation of the far-field stress against the average inelastic
strain (i.e. total strain minus the elastic strain for the fully dense
solid) for one full fatigue cycle involving zero-tension-zero-compres-
sion-zero loading.
(b) How is the stress-strain curve in (a) affected by the presence of a popu-
lation of pre-existing microcracks prior to fatigue loading.
(c) Suppose that, in (a), frictional sliding occurs between the faces of the
microcracks (as, for example, along grain boundaries or shear bands)
upon unloading from the far-field compressive stress. Schematically
sketch the stress-strain curve by incorporating the sliding mechanism.
5.4 Cyclic indentation of a brittle solid exhibits some features which are quali-
tatively similar to those discussed in connection with the cyclic compression
loading of a notched brittle solid. Discuss the similarities and differences
between the initiation of a fatigue crack in a notched brittle solid subjected
to cyclic compression loading and the cracking that occurs beneath the tip of
an indenter repeatedly pressing against the surface of a brittle solid.
5.5 The mechanism for the nucleation of a crack by the pile-up of dislocations,
Fig. 5.9(#), at an obstacle (such as a grain boundary) was proposed by A.N.
Stroh (Proceedings of the Royal Society, London A223, p. 404, 1954, and
A232, p. 548, 1955). Since the dislocation pile-up leads to stress concentra-
tion in much the same way as a mode II crack, the near tip stress fields ahead
of a mode II crack, Eqs. 9.47, can also be used to model the fields ahead of
piled-up dislocations. Stroh postulated that cleavage fracture will nucleate
at an obstacle when the maximum local tensile stress, aee reaches a critical
value.
(a) Using Eq. 9.47, show that aee is a maximum at an angle of-70.5° from
the plane of dislocation pile-up.
(b) Following a procedure similar to that used in the previous problem on
the Petch relationship, show that the cleavage fracture stress takes the
form:
aF = ai+kFd~l/2,
Exercises 199

where ax is the friction stress associated with lattice resistance, k¥ is a

material constant, and dg is the mean spacing of the obstacles.
5.6 List three reasons why the crack growth characteristics of brittle ceramics
and ceramic composites from stress concentrations subjected to cyclic com-
pression are qualitatively similar to those seen in notched metallic plates.
5.7 A silicon nitride ceramic matrix is reinforced with 10, 20 and 30 volume % of
SiC short fibers. Uniaxial tensile tests conducted on the unreinforced as well
as the reinforced ceramics show that the stress-strain curve exhibits greater
nonlinearity with an increase in the concentration of SiC. An increase in SiC
content also leads to an apparently higher overall fracture toughness.
Discuss possible effects of SiC reinforcement and the effects of SiC concen-
tration on the propensity of notched silicon nitride matrix ceramic to sub-
critical crack growth under cyclic compression.
 CHAPTER 6

Cyclic deformation and crack initiation in

noncrystalline solids

The cyclic deformation and fatigue crack initiation characteristics of semi-

crystalline and noncrystalline solids are the subjects of discussion in this chapter.
Attention is devoted to a consideration of the response of polymers and organic
composites. The principal objective of this chapter will be to document the pro-
minent mechanisms of fatigue in amorphous materials and, in particular, to pre-
sent a detailed description of the similarities and differences between the fatigue
characteristics of crystalline and noncrystalline materials. As in the case of brittle
ceramics and composites, the discussion of cyclic deformation and crack initiation
is presented here in a single chapter because in many instances, microscopic crack-
ing processes (such as crazing) constitute prominent mechanisms which influence
fatigue response. Also addressed in this chapter is the topic of fatigue crack initia-
tion at stress concentrations in polymers. Stress-based and strain-based approaches
to fatigue life of polymers are considered in Chapters 7 and 8, respectively. Fatigue
crack growth in semi-/noncrystalline solids is taken up in Chapter 12.

6.1 Deformation features of semi-/noncrystalline solids

6.1.1 Basic deformation characteristics
A noncrystalline material can be a metallic glass (e.g., Pd-20 at.% Si alloy),
an inorganic glass (e.g., silicate glass), or an organic glass (i.e. a polymer).
Amorphous materials exhibit a variety of permanent deformation modes depending
on their basic structural unit. This fundamental unit is a single atom for a metallic
glass whereas, for an inorganic glass, the building block is a SiO4 tetrahedron. A
long-chain molecule built upon a tetravalent carbon atom is the basic structural unit
for an organic glass. The deformation of an amorphous solid can take place homo-
geneously through the bulk of the material or heterogeneously along shear bands or
crazes. On a macroscopic scale, the deformation of an amorphous solid can be as
varied as linear elastic, plastic or Newtonian viscous depending on the temperature,
applied stress and strain rate. With the increasing technological importance of mate-
rials such as plastics in structural applications, the fatigue of amorphous solids has
also become a topic of major practical interest.
The formation of long-chain molecules is the main feature of the structural con-
stitution of polymers that differs from the basic building block of atomic solids (such
as ceramics and metals) or space networks (such as oxide glasses). The obstacles to

200
6.1 Deformation features of semi-jnoncrystalline solids 201

complete crystallization in polymeric solids arise from the very existence of the long-
chain molecule and of the chain branches and side groups. Consequently, the struc-
ture of a polymer can be fully amorphous or of a semi-crystalline arrangement in
which folded molecular chains are formed amid an amorphous phase.
With the exception of some modes of deformation such as craze formation or the
rotation of molecular chains, the mechanisms of deformation and failure in poly-
meric solids exhibit many similarities to those discussed earlier in this book for
metals and ceramics. Under cyclic loading, polymers display deformation modes
(such as stress-strain hysteresis and cyclic softening) and subcritical crack growth
analogous to atomic solids. It was shown earlier in the context of fatigue in metals
and alloys (Chapters 2-4) and in ceramic materials (the preceding Chapter) that
kinematic irreversibility of microscopic deformation is an important reason for the
onset of true mechanical fatigue effects. In polymeric solids, such kinematically
irreversible deformation can be manifested as crazing, shear band formation, rota-
tion or other changes in the orientation of molecular chains, or a combination of
these mechanisms.

6.1.2 Crazing and shear banding

Homogeneous deformation in glassy polymers is effected by the displace-
ment of segmented molecular chains and by the gradual orientation of the long axis
of the initially unaligned molecules with the tensile loading axis. Crazing and shear
flow are the two most common modes of deformation during the fatigue of poly-
mers. Crazing generally has the connotation of brittle failure, whereas deformation
by shear banding represents a more ductile process. The relative dominance of these
two processes during fatigue is dictated by many mutually competitive mechanisms
which depend on the molecular structure, specimen geometry, processing methods,
test temperature, loading rate, stress state, fiber reinforcements, and the degree of
plasticization.
Crazes occur in glassy polymers, such as polystyrene (PS), polymethylmethacry-
late (PMMA) and polysulfone (PSF) and in some semi-crystalline ones such as
polyethylene (PE), polypropylene (PP), polyethyleneteraphthalate (PETP), and
polyoxymethylene (POM) (see, e.g., Kramer (1983) and Kramer & Berger
(1990)). When these polymeric solids are subjected to tensile stresses at low tem-
peratures, fine crack-like features appear which apparently resemble surface flaws
in ceramics. There is a continuity of material across a craze, whereas the faces of a
Griffith crack in a brittle solid are fully separated. Crazes are always oriented
perpendicularly to the maximum principal tensile stress, while the Griffith flaws
in a brittle solid with an equi-axed grain structure essentially have a random
distribution. However, crazes can be regarded as the polymer analogue of dilatant
transformations arranged perpendicularly to the maximum principal tensile stress.
202 Cyclic deformation and crack initiation in noncrystalline solids

In this sense, crazes lead to inelastic strains in much the same way as the shear
and/or dilatant transformations in metals and ceramics (such as mechanical twins
or martensitic lamallae). Furthermore, purposely promoting the formation of
crazes at lower stress levels in an attempt to develop appreciable dilatational
plasticity is viewed as a possible means of toughening polymers (e.g., Argon,
1989). This approach is conceptually similar to the mechanisms of microcrack or
transformation toughening in brittle solids (see Chapter 5). Cyclic deformation and
the subcritical advance of fatigue fracture in many polymers are dictated by the
nucleation, growth and breakdown of crazes.
A craze contains fibrils of highly oriented molecules (craze matter) separated by
porous regions. The density of the craze matter is only 40 to 60% of the matrix
density and the molecular fibrils are oriented along the direction of the maximum
tensile stress. Figures 6.1 (a) and (b) are electron micrographs of a newly formed craze

Fig. 6.1. (a) A newly formed craze in a thin slice of PS. Note the orientation of the craze normal
to the tensile stress axis (indicated by the arrow) and of the fibrils parallel to the tensile stress
axis, (b) An electron micrograph of the central section of another craze. (From Beahan, Bevis &
Hull, 1971. Copyright Taylor & Francis, Ltd. Reprinted with permission.)
6.1 Deformation features of semi-jnoncrystalline solids 203

in a thin slice of PS which was loaded on a mini-tensile straining device. Note that
the craze is oriented normal to the uniaxial tensile stress and that the fibrils within
the craze are aligned with the tensile axis. (The craze geometry is discussed in further
detail in Chapter 12.) It is generally known that the criterion for the nucleation of a
craze in a three-dimensional stress state is of the form:
(6.1)

where <rmax and amin are maximum and minimum principal stresses, respectively,
A(T) and £l\(T) are material constants which depend on the temperature, and aH
is the hydrostatic stress (Section 1.4). Even if the overall hydrostatic stress is nega-
tive, a craze can advance if there exists one tensile stress component (with the craze
extending normal to the tensile stress direction). The nucleation of crazes in crazable
polymers has been investigated in detail, and the criteria for their formation have
been developed in terms of the tensile hydrostatic stress component and the devia-
toric stress component of the imposed stress state (e.g., Argon & Hannoosh, 1977;
Argon, 1989).
The growth of crazes is also a topic of considerable interest because cracking in
many polymers is preceded by the formation of a craze. In this sense, the craze zone
ahead of a crack in a polymer is somewhat analogous to the plastic zone in front of a
crack in a ductile metallic material. Under an imposed stress, the craze extends by
drawing more polymer from its surface into the fibril while the fibrils themselves
deform by creep, Kramer & Hart (1984); see Section 12.2 for further details.
Shear localization is also a prominent feature of deformation and failure in many
polymers. At stress levels lower than the tensile strength of the glassy polymer,
'plastic' deformation can be initiated by the formation of shear bands. In polymers
susceptible to shear banding, the onset of 'yielding' is associated with the inception
of shear bands. Figure 6.2 shows shear bands in plastically stretched PETP. Shear
bands are always oriented along the direction of the maximum shear stress.

6.1.3 Cyclic deformation: crystalline versus noncrystalline materials

There are many apparent differences in the cyclic deformation characteris-
tics of metals and polymers at room temperature. These distinctions stem from the
differences in their homologous temperature (in relation to the test temperature),
thermal diffusivity and structural constitution.
(1) Whether a metallic material undergoes cyclic hardening or softening
depends on such factors as the degree of prior cold work, composition,
and heat treatment. Polymeric materials, on the other hand, exhibit only
cyclic softening behavior. Drastic changes in composition, molecular struc-
ture, temperature, and strain rate alter only the degree to which cyclic soft-
ening occurs in polymers. Even cyclic stability is rarely observed; it may
occur only under conditions of small strain amplitudes.
204 Cyclic deformation and crack initiation in noncrystalline solids

Fig. 6.2. Shear bands (sb) and crazes (c) in plastically stretched PETP. (From Argon, 1980.
Copyright Academic Press. Reprinted with permission.)

(2) The second major distinction between fatigue of metals and polymers is the
degree to which the loading rate influences the stress-strain characteristics
and failure modes. For many polymeric materials, even room temperature is
a significant fraction of the homologous temperature. Furthermore, hystere-
tic damping effects are substantial in thermoplastics as a result of their
strong nonlinear viscous behavior, which is not the case for metals. These
factors, coupled with the poor thermal diffusivity of polymers, can result in
marked increases in temperature during cyclic deformation at sufficiently
high strain rates. The attendant thermal softening becomes an important
consideration and may even dominate over any intrinsic mechanical fatigue
effects. These interactions between adiabatic thermal effects and mechanical
fatigue are complex. For certain combinations of stress amplitudes and
strain rates, thermal effects exacerbate the fatigue failure process while, in
other cases, the combination of the thermal and mechanical effects is not
deleterious to fatigue life. The imposition of low strain rates to circumvent
6.2 Cyclic stress-strain response 205

thermal softening effects may promote other time-dependent deformation

processes such as creep.
(3) Repeated dislocation glide along crystallographic slip systems is the domi-
nant mode of fatigue deformation in metallic materials. However, there
exists a variety of microscopic deformation modes in polymeric materials.
The underlying mechanisms may include homogeneous deformation invol-
ving normal and shear yielding, disentanglement, reorientation and slip of
chain segments, crystallization, or heterogeneous deformation arising from
shear banding and crazing.

6.2 Cyclic stress-strain response

Comprehensive studies of the cyclic stress-strain characteristics of poly-
meric materials have been reported by Rabinowitz & Beardmore (1974) and
Beardmore & Rabinowitz (1975). These investigators have shown that cyclic soft-
ening is the general fatigue phenomenon in ductile polymers, irrespective of the
underlying molecular structure, although cyclic softening becomes more pronounced
with increasing ductility. Amorphous and semi-crystalline polymers as well as poly-
mer-matrix composites exhibit cyclic softening. Changes in the extent of crystallinity
mainly affect the degree and rate of cyclic softening. For example, when subjected to
low strain amplitudes, amorphous polymers exhibit an incubation period prior to
strain-softening; this incubation period diminishes with increasing values of imposed
strain amplitudes. The cyclic stress-strain curves of homopolymers develop stable
hysteresis loops. However, polymer-matrix composites undergo cyclic softening
throughout the fatigue life without ever attaining a saturated state. This behavior
may result from the gradual development of microscopic damage in the matrix, in
the reinforcement or along the matrix-reinforcement interface.

6.2.1 Cyclic softening

Figure 63{a) shows the cyclic softening behavior of polycarbonate (PC)
subjected to strain-controlled fatigue at 298 K. While the stress-strain hysteresis
loops are generally symmetric about the origin in metallic materials, the initial
hysteresis loop (in the first full cycle of loading) for polycarbonate exhibits a pro-
peller-like shape at room temperature as a consequence of inelastic deformation. By
repeating this strain-controlled fatigue experiment at several different strain ampli-
tude levels, the cyclic stress-strain curve shown in Fig. 63{b) is obtained (which is the
same as the strain-controlled test method described in Chapter 3 for metals). In the
low strain regime, the monotonic and cyclic tension responses are identical. The
amount of cyclic softening increases with increasing strain values. A compressive
206 Cyclic deformation and crack initiation in noncrystalline solids

25 MPa i-

12
strain (%)
(b)

Fig. 6.3. Cyclic softening in PC subjected to strain-controlled fatigue at 298 K. (a) Change in the
size and orientation of the hysteresis loops with the progression of fatigue deformation, (b)
Stress-strain response in monotonic tension and fatigue. (From Rabinowitz & Beardmore,
1974. Copyright Chapman & Hall. Reprinted with permission.)

stress-strain curve in monotonic and cyclic loading can also be obtained in a similar
fashion. The origin of the differences between monotonic and cyclic response is
linked to the molecular rearrangements. During cyclic deformation, the strains are
accommodated by molecular rearrangements at the microscopic level, whereas
monotonic loading promotes more macroscopic permanent deformation associated
with molecular rearrangements.
6.2 Cyclic stress-strain response 207

6.2.2 Thermal effects

As noted earlier, the local temperature rise associated with hysteretic heating
in fatigue-loaded polymers can result in thermal softening even in room temperature
fatigue for certain combinations of strain rates and cyclic stresses/strains. The typical
differences between the thermal and mechanical fatigue effects are illustrated in the
photographs of failed specimens of PMMA shown in Fig. 6.4.

6.2.3 Example problem: Hysteretic heating

Problem:
The temperature increase in the polymer is precipitated by the accumula-
tion of hysteretic heat during each fatigue cycle. Consider a polymer subjected to
a sinusoidal variation of cyclic stress,
a = a0 sin cot, (6.2)
where a is the stress at time t, a0 is the peak value of the stress cycle, and
co = 2nvc is the angular frequency, vc being the number of stress cycles per
unit time. If the viscoelastic behavior of the polymer is linear (see Section
1.4.4), the strain will also fluctuate sinusoidally, but it will be out of phase
with the stress by a phase angle 8, which is a function of co. The peak values
of the stress and strain, <r0 and e0, respectively, are related by the complex
modulus E*, where

Fig. 6.4. Examples of failure due to cyclic thermal softening (top) and mechanical fatigue
(bottom) in PMMA. The total length of the test specimen is 110 mm. (From Constable,
Williams & Burns, 1970. Copyright Council of the Institution of Mechanical Engineers.
Reprinted with permission.)
208 Cyclic deformation and crack initiation in noncrystalline solids

f
Lf, \E*\ = ^ = J(Ef + {Ef. (6.3)
6
£ is known as the storage modulus and it denotes the ratio of the stress in phase
with the strain to the strain. The imaginary number i = V—T. E is the loss
modulus and it is the ratio of the stress which is 90° out of phase with the strain
to the strain. The linear viscous deformation due to sinusoidal fatigue loading
can also be characterized in terms of the complex compliance,

D*=± = iy-iD", (6.4)

where, analogous to the storage modulus and loss modulus, D' and D" denote
the storage compliance and loss compliance, respectively. The entity
tan 8 = E /E = D"ID' is known as the loss tangent.
(i) Find the rate of hysteretic energy dissipated per fatigue cycle.
(ii) If H represents the heat transfer coefficient for loss of heat from the
specimen surface to the surroundings, and T and To are the instantaneous
specimen temperature and the ambient temperature, respectively, derive
an expression for the rate of temperature increase assuming adiabatic
heating conditions.
Solution:
(i) The rate of hysteretic energy dissipated as heat, Q, per unit volume of the
material during fatigue loading is given by

where k is the strain rate and the superscripts ' and " indicate the real and
imaginary components, respectively (Constable, Williams & Burns, 1970).
Combining Eqs. 6.2 and 6.5 with the result that o1 = a/2(l + tan2 6), one
obtains the energy loss per stress cycle,
tan
' (6.6)
E l+tan2<5
The average energy dissipation rate per unit volume is
Q= co_ = nvA tana = ^ (6J)
* ^ In E l+tan 2 5 ° V

For cyclic loading with a zero mean stress, the peak stress value of the
fatigue cycle is the same as the stress amplitude Aa.
(ii) If adiabatic heating conditions prevail (i.e. heating in which all of the heat
generated within the polymer is manifested as a temperature rise and
none is lost to the surroundings), the time rate of change of temperature
dT/dt is given by
6.2 Cyclic stress-strain response 209

dT nvcD/fa0
(6.8)
~d7
where p is the mass density and cp is the specific heat. In reality, however,
some heat is lost to the surroundings. Equation 6.8 may be modified to
account for heat loss:
dT nvcD"crQ HA
(T-To). (6.9)
~d7 pcp pcp V

Here A and V are the surface area and volume of the fatigue test speci-
men, respectively. Note that the loss compliance D" depends strongly on
both tempeature and strain rate. As the specimen temperature increases
and approaches a critical softening temperature, the specimen becomes
too soft to support the load and suffers catastrophic fracture.

6.2.4 Experimental observations of temperature rise

The experimentally determined rise in temperature T for polytetrafluoro-
ethylene (PTFE) subjected to fatigue at a frequency of 30 Hz in the room tempera-
ture environment is plotted in Fig. 6.5 as a function of the number of fatigue cycles
for several different values of the imposed stress range Aa. The endurance limit A<re
of this polymer (i.e. the stress range below which fatigue failure does not occur for at
least 106 stress cycles) is 6.5 MPa for the conditions of the experiment. Figure 6.5
reveals that, when A a > Aae, a rapid increase in temperature occurs with increasing

210 ACT (MPa)

A 10.3
£9.0
150 ^ D
C8.3
c 0 7.6
B * ¥
¥ I 1 £6.9
A 1 1 1 F6.3
90

1JJJ J 1 1
F

i
103 104 105 106 107
number of cycles to failure, Nt

Fig. 6.5. Effect of the applied stress range ACT on temperature rise in PTFE subjected to stress-
controlled fatigue. The symbol x denotes failure of the specimen. (After Riddell, Koo &
O'Toole, 1966.)
210 Cyclic deformation and crack initiation in noncrystalline solids

number of cycles. The rate of initial elevation in temperature is higher for the larger
values of Aa. However, when Aa < Aae (i.e. for curve F), the temperature increase
is not sufficient to cause thermal failure. Consequently, the temperature stabilizes
after prolonged fatigue loading, and the specimen is essentially capable of sustaining
an infinite number of fatigue cycles.
Koo, Riddell & O'Toole (1967) performed fatigue experiments in an attempt to
quantify the effect of thermal softening on fatigue life in three fluoropolymers:
PTFE, polychlorotrifluoroethylene (PCTFE), and polyvinylidenefluoride (PVF2).
They derived estimates of the loss compliance D" during fatigue on the basis of
dynamic modulus measurements using a torsional pendulum. It was found that
hysteretic heating caused a significant increase in loss compliance (Fig. 6.6). With
an elevation in material temperature, the specimen became too soft to fail by purely
mechanical fatigue, but instead suffered a loss in fatigue strength by thermal soft-
ening. This damage was at least partially recoverable by annealing the specimen in
the early and intermediate stages of temperature rise (see Fig. 6.5). The most sig-
nificant effect of temperature rise on fatigue damage was observed in the stress cycles
just prior to final failure.

6.2.5 Effects offailure modes

In addition to the aforementioned effects of homogeneous deformation on
cyclic response, the microscopic failure modes responsible for inhomogeneous defor-
mation can also strongly influence the cyclic stress-strain behavior. Figure 6.7 illus-
trates the cyclic deformation characteristics of PC at 77 K. At this temperature, PC is
susceptible to craze formation. Since crazes form only under local tensile stresses,
cyclic softening is observed only in the tensile portion of fatigue, and the hysteresis

2.0 x l O 1 1

2 1.5 xlO"11
x

£ 1 x 10~u

Q 12
5x10
I I 1
20 60 100 140

Fig. 6.6. Change in loss compliance D" due to temperature rise in PTFE subjected to stress-
controlled fatigue at room temperature at a cyclic frequency of 30 Hz. (After Koo, Riddell &
O'Toole, 1967.)
6.3 Fatigue crack initiation at stress concentrations 211

lOOMPar-

Fig. 6.7. Anomalous fatigue deformation of PC at 77 K where craze formation leads to cyclic
softening only in the tensile portion of cyclic loading. (From Rabinowitz & Beardmore, 1974.
Copyright Chapman & Hall. Reprinted with permission.)

loop remains stable in the compression portion. This type of anomalous softening is
also seen in the stress-strain response of polymers when the deformation mechanism
involves the stable growth of microscopic cracks under fatigue loading (Beardmore
& Rabinowitz, 1975).
The overall fatigue response of a polymer is dictated by a combination of factors
involving the molecular structure, deformation modes, and cyclic loading conditions
(Hertzberg & Manson, 1980, 1986), which include:
(1) polymer composition, molecular weight and distribution, and thermody-
namic state,
(2) structural and morphological changes induced by the mechanical loads and
the environment, such as bond breakage, molecular alignment and disen-
tanglement, or crystallization,
(3) the type of deformation, such as elastic, linear viscoelastic or nonlinear
viscoelastic response,
(4) the mode of microscopic failure, such as crazing or shear banding,
(5) thermal softening, and
(6) the time-scale of the experiment vis-a-vis the kinetic rate of the processes
causing structural changes.

6.3 Fatigue crack initiation at stress concentrations

It was shown in Sections 4.11 and 5.6 that the application of uniaxial cyclic
compressive loads to notched plates of metals and ceramics, respectively, resulted in
212 Cyclic deformation and crack initiation in noncrystalline solids

the nucleation and growth of fatigue cracks along the plane of the notch. A similar
phenomenon was reported for polymeric materials by Pruitt & Suresh (1993).
Figures 6.8(#) and (b) show examples of fatigue crack growth normal to the far-
field cyclic compression axis in notched specimens of an untoughened polystyrene
and a high-impact polystyrene comprising 7.5 wt% butadiene rubber in the form of
gel particles with an average diameter of 1-2 urn. (The rubber particles are added to
the polystyrene to enhance its toughness through increased craze formation.)
Inelastic deformation at the tip of notch, arising from such irreversible deforma-
tion processes as shear localization and rotation of molecular chains, generates a
zone of residual tensile stresses upon unloading from the first compression cycle. If
the magnitude of such compressive stresses exceeds the craze strength, crazes
oriented parallel to the plane of the notch and normal to the far-field compression

{<*)

0.1 mm

0.1 mm

Fig. 6.8. Fatigue crack initiation and growth normal to the cyclic compression axis in (a) an
untoughened PS (weight-average molecular weight, M w = 300 000 and polydispersity = 2.4),
and (b) a high-impact polystyrene comprising 7.5 wt% butadiene rubber in the form of gel
particles with an average diameter of 1-2 jam (M w = 240000 and polydispersity = 2.8). The
compression axis is vertical. (From Pruitt & Suresh, 1993. Copyright Taylor & Francis, Ltd.
Reprinted with permission.)
6.4 Case study: Compression fatigue in total knee replacements 213

axis (i.e. normal to the local maximum tensile stress) are induced ahead of the notch-
tip. This provides a strong kinematically irreversible damage mechanism for the
generation of residual tensile stresses in the subsequent cycles and for the advance
of the crack.
The white region immediately ahead of the notch in Fig. 6.8(Z?) is approximately
indicative of the region in which crazing occurs during cyclic compression. Figure
6.9(a) shows a transmission electron micrograph of a typical craze observed within
this region during the cyclic compression loading of the toughened polystyrene.
Figure 6.9(6) shows the craze penetrating through the rubber particle and the
matrix. The features of the craze formed under imposed cyclic compression are
the same as those produced during monotonic or cyclic tension (Pruitt & Suresh,
1993).
In-situ photoelastic and laser interferometric measurements have also been carried
out in notched plates of a photoelastic resin to quantify the evolution of residual
tensile stresses ahead of the stress concentration upon unloading from the far-field
compression axis. Figure 6.10 shows the evolution of an increasing tensile residual
stress field at the notch tip upon unloading from a maximum applied compressive
stress of-16.5 MPa. Plotted in Fig. 6.10 are the contours of constant oyy (i.e. stress
normal to the plane of the notch) at different stages of unloading. At an applied
compressive stress of-2.76 MPa, tensile stresses with a magnitude in excess of +4.96
MPa span a distance of 0.07 mm ahead of the notch tip. Upon unloading to -0.92
MPa, the stresses exceed +7.6 MPa over a distance of 0.11 mm. Further unloading
to -0.55 MPa causes the residual tensile stresses to exceed + 9.1 MPa over a distance
of nearly 0.1 mm. Given that the tensile strength of the brittle photoelastic resin is
only 6.7 MPa, the stress measurements shown in Fig. 6.10 provide a justification for
the nucleation of a fatigue crack ahead of the notch under cyclic compression load-
ing. The overall initiation and growth characteristics of compression fatigue cracks
in polymers are qualitatively similar to those of metals and ceramics (see Sections
4.11 and 5.6).

6.4 Case study: Compression fatigue in total knee replacements

Compression-dominated fatigue is a common occurrence in orthopedic polymer
inserts used for total knee replacements. Figure 6.11 schematically shows the geometry of
interest and the associated nomenclature for a total knee replacement. Ultrahigh
molecular weight polyethylene (UHMWPE) has been successfully used as a material for
total knee and hip orthoplasty. Quantitative analyses of the polymer damage in the total
knee replacements have shown that the degree of degradation is linked to compression-
dominated fatigue processes which stem from cyclic contact between the metal and the
UHMWPE components of the artificial joints (Wright, Burstein & Bartel, 1985).
214 Cyclic deformation and crack initiation in noncrystalline solids

Fig. 6.9. (a) A transmission electron micrograph of a typical craze produced ahead of the notch
during the cyclic compression loading of the toughened polystyrene, (b) shows the craze
penetrating through the rubber particle and the matrix. The far-field compression axis is
approximately perpendicular to the craze in both figures. (From Pruitt & Suresh, 1993.
Copyright Taylor & Francis, Ltd. Reprinted with permission.)

As the articulating surfaces of the knee joint move during flexion, the polymer
component is subjected to complex stress distributions within and at the surface of the
UHMWPE insert. In a total knee replacement, the polymer insert is compressed by the
metal component and results in compressive stresses perpendicular to the articulating
surface (Bartel, Bicknell & Wright, 1986). The delamination and pitting of the tibial insert
6.4 Case study: Compression fatigue in total knee replacements 215

A + 1.9 MPa
A —- + 0.6 MPa B + 3.4 MPa
B —- -f LBMPa C +5.5 MPo
C — — + 3.2MPO D +7.6 MPo
0 - — • 4.9 MPo

(a) (*)

A —
+ 6.7 MPq
+ 2 9 MPo
L, 0.3mm

B -- + 5.4 MPo
C — + 7.4 MPo
0 — + 9.1 MPQ

Fig. 6.10. In-situ photoelastic and laser interferometric measurements of the evolution of tensile
residual stresses in a photoelastic resin during different stages of unloading from a maximum
far-field compressive stress of-16.5 MPa. Far-field compressive stress magnitudes: (a) -2.75
MPa, (b) -0.92 MPa, and (c) -0.55 MPa. (After Pruitt & Suresh, 1993.)

also produces polymer debris which is known to cause osteolysis, infection and loosening
of the implants (Mirra, Marder & Amstuz, 1982).
Bartel, Bicknell & Wright (1986) have used a three-dimensionalfinite-elementmodel
(FEM) to analyze the stresses in condylar-type knee replacements, Fig. 6.12. This
model comprises a metal-backed UHMWPE with a uniform concave surface and the
arrangement is loaded by a metal component with a convex surface. The contact
surfaces are defined by two radii of curvature as the knee prosthesis has distinct radii
for extension and flexion. For the analysis, the contact forces were chosen to simulate
in-vivo conditions with femoral-tibial contact forces ranging from 4.3-4.9 times the
216 Cyclic deformation and crack initiation in noncrystalline solids

Fig. 6.11. A schematic representation of the geometry of interest in total knee replacement.

body weight of a typical patient. For the total knee replacement, it was found that the
maximum principal stress occurred at the surface of the UHMWPE and along the
tangent to the articulating surface. The greatest magnitudes of stresses were found to be
compressive and located at the center of contact. Further, the stresses were found to be
cyclic in nature through the natural action of flexion and extension. The model
demonstrated that the maximum principal stress at a point near the surface of a total
condylar type tibial knee component can range from 10 MPa of tension to more than
20 MPa of compression as the contact area sweeps across the surface in the action of knee
flexion, Fig. 6.12. From this work, it was established that the primary damage mechan-
isms in the total knee replacement are driven by compressive or compression-dominated
cyclic loading.
In an attempt to simulate the potentially deleterious effects of cyclic compressive
loads in nucleating fatigue cracks at corners and stress concentrations in knee
prosthesis, Pruitt & Suresh (1993) and Pruitt et al. (1995) carried out systematic
experiments of compression fatigue cracking at notches in sterilized and unsterilized
UHMWPE. On the basis of these laboratory experiments and from the information
available from knee replacement components, it was concluded that the inception and
growth of sharp fatigue cracks which have initiated under cyclic compressive loading
can be further extended by subsequent action of cyclic tensile stresses to such a critical
crack length that pitting, delamination and fatigue failure can occur at the UHMWPE
surface.
Exercises 217

articulating
surfaces

region
analyzed by
finite elements

{a)

A.I mm
anterior

10.9 mm

- edge of
contact area

Fig. 6.12. (a) A schematic of the condylar-type tibial insert made of UHMWPE. The finite
element mesh is also superimposed in this figure to illustrate the geometry analyzed, (b)
Magnitudes of the maximum principal stress (in units of MPa) at the surface of the knee
replacement during extension, (c) Magnitudes of the maximum principal stress (in units
of MPa) at the surface of the knee replacement during flexion. (After Bartel, Bicknell &
Wright, 1986.)

Exercises
6.1 A mechanical test is conducted on three different materials. In this experi-
ment, a constant tensile stress is applied instantly to a specimen. The first
material, a rubber, deforms rapidly during the application of the stress, with
218 Cyclic deformation and crack initiation in noncrystalline solids

the extent of deformation becoming progressively less until it reaches an

equilibrium gage length. The second, a thermosetting material, reacts to the
applied stress by deforming almost instantaneously to an equilibrium exten-
sion. The third, a thermoplastic material, initially deforms to substantial
extension in response to the stress and then deforms further with the exten-
sion varying linearly with time.
(a) Describe the behavior of each material with an appropriate combina-
tion of springs and dashpots.
(b) Describe the behavior of each material following the removal of the
tensile stress.
6.2 Discuss the effects of (a) quenching rate of molten polymer and (b) plasti-
cization on the dynamic mechanical properties (such as loss modulus). Also
discuss the implications of such effects to fatigue deformation.
6.3 Two organic composites are made with the same matrix material, same size
and concentration of unidirectional fiber reinforcements, and same interfa-
cial properties between the fibers and the matrix. The spatial distribution of
the fibers in the matrix, however, is different for the two cases. If crazing
occurs in the polymeric matrix of the two composites, would you expect the
monotonic and cyclic stress-strain response to be different for the two
materials? Explain.
6.4 If crazing is the primary mode of damage in a polymer during cyclic defor-
mation, would you expect cyclic softening to occur in both the tensile and
compressive portions of a fully-reversed stress cycle? Explain.
6.5 If shear banding is the primary mode of damage in a polymer during cyclic
deformation, would you expect cyclic softening to occur in both the tensile
and compressive portions of a fully-reversed stress cycle? Explain.
6.6 It was shown in Section 6.3 that mode I fatigue crack growth occurs in
notched specimens of polymers when subjected to fully compressive far-
field cyclic loads.
(a) What are the mechanisms responsible for this effect?
(b) Discuss the possible effects of compressive stress amplitude and load
ratio on the characteristics of crack initiation and growth from notches
in polymers subjected to fully compressive cyclic loads.
(c) Can a fatigue crack be induced along the plane of a notch in a unidir-
ectionally reinforced graphite-epoxy composite subjected to uniaxial
cyclic compression, with the axis of the graphite fiber parallel to the
compression axis and with the plane of the notch perpendicular to the
compression axis?
6.7 Repeat the example problem in Section 5.6.1 for the case of a notched plate
of a polymer which is subjected to applied cyclic compression loading nor-
mal to the plane of the notch.
Part two

TOTAL-LIFE APPROACHES
 CHAPTER 7

Stress-life approach

The preceding chapters were concerned with the evolution of permanent

damage under cyclic deformation and with the attendant nucleation of a fatigue
crack. While these discussions pertain to micromechanical processes, phenomenolo-
gical continuum approaches are widely used to characterize the total fatigue life as a
function of such variables as the applied stress range, strain range, mean stress and
environment. These stress- or strain-based methodologies, to be examined in Part
Two, embody the damage evolution, crack nucleation and crack growth stages of
fatigue into a single, experimentally characterizable continuum formulation. In these
approaches, the fatigue life of a component is defined as the total number of cycles
or time to induce fatigue damage and to initiate a dominant fatigue flaw which is
propagated to final failure. The philosophy underlying the cyclic stress-based and
strain-based approaches is distinctly different from that of defect-tolerant methods
to be considered in Part Three, where the fatigue life is taken to be only that during
which a pre-existing fatigue flaw of some initial size is propagated to a critical size.
The stress-life approach to fatigue was first introduced in the 1860s by Wohler.
Out of this work evolved the concept of an 'endurance limit', which characterizes the
applied stress amplitude below which a (nominally defect-free) material is expected
to have an infinite fatigue life. This empirical method has found widespread use in
fatigue analysis, mostly in applications where low-amplitude cyclic stresses induce
primarily elastic deformation in a component which is designed for long life, i.e. in
the so-called high-cycle fatigue (HCF) applications.! When considerable plastic
deformation occurs during cyclic loading as, for example, a consequence of high
stress amplitudes or stress concentrations, the fatigue life is markedly shortened.
Here, fatigue design inevitably calls for the so-called low-cycle fatigue (LCF)
approach. Realizing the important role of plastic strains in inducing permanent
fatigue damage, Coffin (1954) and Manson (1954) independently proposed a plastic
strain-based continuum characterization of LCF.
This chapter deals with the stress-life approach to fatigue where the effects of
stress concentrations, mean stresses, surface modifications, variable amplitude cyclic
loads, and multiaxial loads are also discussed. Total-life characterization of fatigue
in nonmetallic materials is also addressed wherever appropriate. The stress-based
and strain-based approaches have found widespread application, most notably in the
design of many fatigue-critical components for automobiles and other surface vehi-
cles. Strain-based approaches to total fatigue life are considered in the next chapter.

' A case study of the HCF fatigue problem in aircraft gas turbine engines is presented in Section 7.6.

221
222 Stress-life approach

7.1 The fatigue limit

Methods for characterizing the fatigue life in terms of nominal stress ampli-
tudes using experimental data obtained from rotating bend tests on smooth speci-
mens emerged from the work of Wohler (1860) on fatigue of alloys used for railroad
axles. In this approach, smooth (unnotched) test specimens are typically machined to
provide a waisted (hour-glass) cylindrical gage length, which is fatigue-tested in
plane bending, rotating bending, uniaxial compression-tension (push-pull) or ten-
sion-tension cyclic loading. Test methods for determining the stress-life response are
spelled out in detail in ASTM Standards E466-E468 (American Society for Testing
and Materials, Philadelphia).
From such an experiment, the stress amplitude aa for fully reversed loading (equal
to one-half of the stress range from the maximum tension to maximum compres-
sion), is plotted against the number of fatigue cycles to failure, 7Vf, Fig. 7.1. The solid
line illustrates the stress-life plot (also known as the S-N curve) observed for mild
steels and other materials which harden by strain-ageing. Under constant amplitude
loading conditions, these alloys exhibit a plateau in the stress-life plot typically
beyond about 106 fatigue cycles. Below this plateau level, the specimen may be cycled
indefinitely without causing failure. This stress amplitude is known as the fatigue
limit or endurance limit, crQ. The value of ae is 35% to 50% of the tensile strength aTS
for most steels and copper alloys. The intercept of the stress-life curve with the
ordinate is crTS at one-quarter of the first fatigue cycle. (For single crystals of FCC
metals, the steady state value of the peak saturation stress r*, Fig. 2.2, may be
regarded as an endurance limit.)

103 104 105 10° 10'

cycles to failure, JVf

Fig. 7.1. Typical S-N diagram showing the variation of the stress amplitude for fully reversed
fatigue loading of nominally smooth specimens as a function of the number of cycles to failure
for ferrous and nonferrous alloys.
7.1 The fatigue limit 223

Table 7.1. Cyclic endurance limit of some common engineering alloys.

Material Condition aTS (MPa) (Jy (MPa) ae (MPa)

a
AI alloys
2024 T3 483 345 138
6061 T6 310 276 97
Steelsb
1015 Annealed 455 275 240
1015 60% CW 710 605 350
1040 Annealed 670 405 345
4340 Annealed 745 475 340
4340 Q&T''* (204 °C) 1950 1640 480
4340 Q & T ' ' (538 °C) 1260 1170 670
HY140 Q&T^ (538 °C) 1030 980 480

a
Endurance limit based on 5 x 108 cycles. Source: Aluminum Standards and Data, The
Aluminum Association, New York, 1976.
b
Endurance limit based on 107 cycles. Source: Structural Alloys Handbook, Mechanical
Properties Data Center, Traverse City, Michigan, 1977.
'Refers to quenched and tempered condition; the data within parentheses refer to tempering
temperature.

Many high strength steels, aluminum alloys and other materials do not generally
exhibit a fatigue limit (see dashed line in Fig. 7.1). For these materials, cra (or ACT)
continues to decrease with increasing number of cycles. An endurance limit for such
cases is defined as the stress amplitude which the specimen can support for at least
107 fatigue cycles. Table 7.1 lists the fatigue endurance limits for a variety of engi-
neering alloys along with crTS and the monotonic yield strength, cry.
If Fig. 7.1 is redrawn on a log-log scale, with the (true) stress amplitude plotted as
a function of the number of cycles or load reversals! to failure, a linear relationship
is commonly observed. The resulting expression relating the stress amplitude,
o-a = Acr/2, in a fully-reversed, constant-amplitude fatigue test to the number of
load reversals to failure, 27Vf, is (Basquin, 1910)

-— = aa = crf(2Nf) , (7.1)

where o\ is the fatigue strength coefficient (which, to a good approximation, equals

the true fracture strength a f , corrected for necking, in a monotonic tension test for
most metals) and b is known as the fatigue strength exponent or Basquin exponent

' A constant amplitude cycle is composed of two load reversals. As shown later in this chapter and the
next one, the use of the number of load reversals, instead of the number of fatigue cycles, is helpful in
analyzing variable amplitude fatigue.
224 Stress-life approach

crack initiation

Nf (log scale)

Fig. 7.2. Contributions of crack initiation and crack propagation processes to total fatigue life
in a nominally smooth specimen.

which, for most metals, is in the range of —0.05 to —0.12. Typical values of o\ for
many engineering alloys are tabulated in the next chapter.
The S-N curve schematically shown in Fig. 7.1 strictly pertains to the total
fatigue life of a nominally smooth-surfaced, 'defect-free' material. Here total life
implies the number of cycles to initiate fatigue cracks in the smooth specimen plus
the number of cycles to propagate the dominant fatigue crack to final failure. This
two-stage process involving initiation and propagation is represented in the S-N
curve shown in Fig. 7.2. The fraction of the fatigue life which is expended in nucle-
ating a dominant fatigue crack of engineering size (typically a fraction of a mm) may
vary from essentially 0%, for specimens containing severe stress concentrations,
rough surfaces or other surface defects, to as high as 80% in very carefully prepared,
nominally defect-free, smooth specimens of high purity materials.

7.2 Mean stress effects on fatigue life

The aforementioned empirical descriptions of fatigue life pertain to fully
reversed fatigue loads where the mean stress of the fatigue cycle om is zero.
However, fully reversed stress cycles with a zero mean stress are not always repre-
sentative of many applications. The mean level of the imposed fatigue cycle is known
to play an important role in influencing the fatigue behavior of engineering materi-
als. Figure 7.3 schematically shows a fatigue cycle of sinusoidal waveform with a
nonzero mean stress. In this case, the stress range, the stress amplitude and the mean
stress, respectively, are defined as
7.2 Mean stress effects on fatigue life 225

Fig. 7.3. Nomenclature for stress parameters which affect fatigue life. The variation of stress a
with time t is shown.

Aor = -Or (7.2)

The mean stress is also characterized in terms of the load ratio, R = <rr^n/crmax. With
this definition, R = — 1 for fully reversed loading, R = 0 for zero-tension fatigue, and
R = 1 for a static load.
When the stress amplitude from a uniaxial fatigue test is plotted as a function of
the number of cycles to failure, the resultant S-N curve is generally a strong function
of the applied mean stress level. Figure 1.4(a) shows the typical S-N plots for
metallic materials as a function of four different mean stress levels, crml, am2, a m3
and <rm4. One observes a decreasing fatigue life with increasing mean stress value.
Mean stress effects in fatigue can also be represented in terms of constant-life
diagrams, as shown in Fig. lA(b). Here, different combinations of the stress ampli-
tude and mean stress providing a constant fatigue life are plotted. Most well known
among these models are those due to Gerber (1874), Goodman (1899),f and
Soderberg (1939). The life plots, represented in Fig. 7.4(Z>), are described by the
following expressions:

Soderberg relation : <ra = ora|a =0| 1 —}, (7.3)

Modified Goodman relation : aa = a a | a =0| 1 — |, (7.4)

I °TS J

Gerber relation : (7.5)

where a a is the stress amplitude denoting the fatigue strength for a nonzero mean
stress, cra\am=0 is the stress amplitude (for a fixed life) for fully-reversed loading
(am = 0 and R = — 1), and <ry and <rTS are the yield strength and tensile strength of
the material, respectively.

' The modified Goodman equation, Eq. 7.4, is generally considered to be a modification of methods
originally proposed by a number of different engineers.
226 Stress-life approach

(b)
Fig. 7.4. (a) Typical stress amplitude-life plots for different mean stress values, (b) Constant life
curves for fatigue loading with a nonzero mean stress.

As a general rule-of-thumb, the following observations can be made about the

foregoing models for the effects of mean stress on fatigue life.

(1) Equation 7.3 provides a conservative estimate of fatigue life for most engi-
neering alloys.
(2) Equation 7.4 matches experimental observations quite closely for brittle
metals, but is conservative for ductile alloys. For compressive mean stresses,
however, it is generally nonconservative. To circumvent this problem, one
may assume that compressive mean stresses provide no beneficial effect on
fatigue life.
(3) Equation 7.5 is generally good for ductile alloys for tensile mean stresses. It
clearly does not distinguish, however, between the differences in fatigue life
due to tensile and compressive mean stresses.

The constant life diagram for different mean stress levels, also commonly referred
to as the Haigh diagram (Haigh, 1915, 1917), is schematically represented as shown
in Fig. 7.5. In this figure, the maximum and minimum stresses of the fatigue cycle,
both normalized by the tensile strength, are plotted. The dashed lines denote experi-
mentally determined values of combinations of maximum and minimum stress levels
(representing different mean stresses) which represent constant fatigue lives for the
indicated number of cycles. This figure affords a convenient graphical representation
of the effects of mean stress on S-N fatigue response, although considerable experi-
mental effort is needed to determine empirically the information needed for this plot.
Similar diagrams are also developed for notched members where the net-section
stresses are used.
While the Basquin relation, given by Eq. 7.1, is valid only for zero mean stress,
Morrow (1968) has presented a modification of the Basquin relation which accounts
for mean stress effects (for any am) in the following form:
f (7.6)
7.3 Cumulative damage 227

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2

Fig. 7.5. A schematic representation of the Haigh diagram showing constant life curves for
different mean stress levels in terms of the maximum and minimum stresses of the fatigue cycle.

The number of cycles to fatigue failure for any nonzero mean stress, iVf, can then be
written as
\/b
Nf = 1 - (7.7)

where TVf 1^ = 0 *s the number of cycles to failure for zero mean stress.

7.3 Cumulative damage

The principles of stress-based characterization of total fatigue life are only
relevant for constant amplitude fatigue loading. In reality, however, engineering
components are invariably subjected to varying cyclic stress amplitudes, mean stres-
ses and loading frequencies.
A simple criterion for predicting the extent of fatigue damage induced by a parti-
cular block of constant amplitude cyclic stresses, in a loading sequence consisting of
various blocks of different stress amplitudes, is provided by the so-called Palmgren-
Miner cumulative damage rule (Palmgren, 1924; Miner, 1945). Implicit in this linear
damage rule are the assumptions that:
(1) The number of stress cycles imposed on a component, expressed as a per-
centage of the total number of stress cycles of the same amplitude necessary
to cause failure, gives the fraction of damage.
(2) The order in which the stress blocks of different amplitudes are imposed
does not affect the fatigue life.
228 Stress-life approach

(3) Failure occurs when the linear sum of the damage from each load level
reaches a critical value.
If nt is the number of cycles corresponding to the /th block of constant stress
amplitude aai in a sequence of m blocks, and if Nfi is the number of cycles to failure
at aai, then the Palmgren-Miner damage rule states that failure would occur when

It will be shown in later chapters that damage accumulation and failure under
variable amplitude loading conditions are dictated by several concurrent mechan-
isms and that the linear damage rule may lead to erroneous predictions of variable
amplitude fatigue behavior in many situations. For example, the Palmgren-Miner
damage rule predicts a greater degree of fatigue damage due to a higher amplitude of
cyclic stresses. However, it is well established that tensile overloads applied to
notched and cracked metallic materials reduce the rate of fatigue crack growth
and that the application of compressive overloads generally has the opposite trend
(see Chapter 14).
Even for smooth specimens, the linear damage rule may provide incorrect results
because of its omission of load sequence effects. Consider a smooth fatigue specimen
which is subjected to nx and n2 cycles of two different amplitudes of cyclic stresses,
a al and a a2 , respectively, which have the same R ratio. As shown in Fig. 7.6,
a
ai > aa2- Let the fatigue life (number of cycles to failure) at cr^ and a a2 be Nn
and JVf2, respectively.
Case 1:
First consider the loading sequence in which cra2 is applied after oral. The
extent of damage induced by this lower stress level may be in excess of the damage
rule prediction of n2/Nf2, if the preceding application of <ral for n\ cycles had
nucleated cracks or significantly contributed to the number of cycles necessary at
the lower stress level to nucleate cracks. Consequently, one may observe for this case
that ]T ni/Nfi < 1. In this case, damage may occur even if cra2 is below the endurance
limit.
Case 2:
Now consider the reverse situation where the application of a a2 precedes
that of <jal. If the material strain-ages, the application of aa2 prior to <xal may
enhance the fatigue limit even if a a2 is lower than the fatigue limit. This phenomenon
is known as coaxing. For this case, J2

7.4 Effects of surface treatments

The free surface of a component is a common site for the nucleation of a
fatigue crack. Therefore, the manner in which the surface is prepared during man-
7.4 Effects of surface treatments 229

Case 1 Case 2

n2 cycles

(a) (b)

Fig. 7.6. Block loading sequence for (a) case 1 and (b) case 2. (c) Fractional life expended as
estimated by the Palmgren-Miner rule.

ufacturing of the component has a decisive role in dictating the initiation life for
fatigue cracks.
There exists a variety of surface treatments, such as carburizing, nitriding, flame
hardening, induction hardening and shot-peening, which are designed to impart high
strength, wear resistance or corrosion resistance locally in the near-surface regions of
the material. Furthermore, common machining operations such as grinding, polish-
ing and milling cause different degrees of surface roughness to develop. The valleys
on the rough surface serve as stress concentrations, which, in turn, induce different
levels of resistance to fatigue crack nucleation (see Chapters 4 and 15 for further
discussions on this issue).
In addition to the roughness of the surfaces, the residual stresses that are induced
by the surface treatments have an important effect on the fatigue life. Residual
stresses are generated in a component as a consequence of thermal, chemical or
mechanical treatments:

(1) Mechanical working, which causes nonuniform plastic deformation, may be

used to provide a favorable residual stress distribution. Shot-peening of
surfaces, surface rolling offillets,and auto-frettage of gun tubes or pressure
230 Stress-life approach

vessels are some examples. Deleterious residual stress patterns may also
arise from mechanical working, as, for example, in the undesirable devel-
opment of tensile stresses due to cold straightening (e.g., Osgood, 1982).
(2) Local volume changes associated with precipitation, chemical reactions, or
phase transformations induce residual stresses in a component. Case hard-
ening of a surface by nitriding produces compressive stresses in the diffusion
region.
(3) Each fabrication technique, such as grinding, milling, polishing, rolling, and
welding, produces its own characteristic residual stress patterns.
(4) Even in the absence of phase changes, nonuniform thermal expansion or
contraction produces residual stresses. An example of the development of
thermal residual stresses can be found in processes involving rapid quench-
ing and in thermal fatigue.

Residual stresses arising from fabrication or surface and heat treatments, when
superimposed with the applied fatigue loads, alter the mean level of the fatigue cycle
and the fatigue life for crack nucleation. In general, residual stresses affect the fatigue
behavior of materials in the same way as the static mechanical stresses superimposed
on a cyclic stress amplitude. Therefore, residual stresses are favorable, if compressive,
and detrimental, if tensile; this is especially true for high strength materials. The ben-
eficial effect of residual stresses introduced by surface treatments becomes less signifi-
cant at larger applied stresses (at lower fatigue lives) because a large amplitude of the
pulsating stress easily 'relaxes' the residual stress, particularly in softer materials.
Consider, for example, the case of shot-peening, which is widely used to improve
the fatigue life of many engineering structural parts. Examples of shot-peened com-
ponents include chassis, valve springs, gears and shafts for automobiles, and exhaust
stack for aircraft engines. In the shot-peening process, a stream of small, hard
spheres (typically 0.1 to 1 mm in diameter) is shot at a surface which is to be treated.
Depending on the diameter of the shots, the velocity of their impingement on the
surface and the duration of the process, the maximum (long range) compressive
residual stress generated by the localized plastic deformation of the surface layer
can reach about one-half the yield strength of the material. The compressive residual
stress zone spans a depth of about one-quarter to one-half the diameter of the shots.
As the shot-peened surface layer has a compressive mean stress, it acts to enhance
significantly the total fatigue life by reducing the damaging effect of the tensile
portion of fully reversed cyclic loads. Figure 7.7 shows an example of the beneficial
effect of shot-peening on the endurance limit of steels with different levels of surface
finish. A worked example quantifying the benefits of shot-peening in contact fatigue
applications is presented in Chapter 13, where the combined effects of surface rough-
ness, peening and lubrication on S-N fatigue behavior and the endurance limit are
also examined.
Caution should be exercised in designing the parameters for shot-peening. If shot-
peening is done for a longer time span than necessary, it may induce cracks along slip
7.5 Statistical considerations 231

1000
. as-fabricated
* shot-peened after fabrication

750

polished or ground

500

250

2000

Fig. 7.7. Beneficial effect of shot-peening on the fatigue life of steels of different strength levels.
The endurance limit cre (defined at 2 x 106 stress cycles) is plotted against the tensile strength,
flrjs. (After Mann, 1967.)

bands in such materials as Ni-base superalloys. In addition, the surface roughness

induced by shot-peening can also, in some cases, make it easier for fatigue cracks to
initiate, thereby countering the beneficial effects of compressive residual stresses.
Laser shock peening is another surface treatment method where a high intensity,
short pulse duration laser beam is impacted on a surface to induce reversed plastic
flow and compressive residual stresses. Here, residual compressive stresses extend
deeper into the material (typically up to 2.5 mm) than in shot-peening. As an exam-
ple, consider the turbine engine for aircraft in which the leading edge of the fan blade
is laser shock peened to improve the resistance to fatigue cracking in the presence of
foreign object damage (see Section 7.6).

7.5 Statistical considerations

There are a number of sources of uncertainty in the analysis of fatigue
results, in general, and in the use of the stress-life approach. These arise from:
(1) Uncertainties and/or errors in the estimation of material properties which
include microstructural variability from one specimen or batch to another as
232 Stress-life approach

well as experimental errors in the measurement of properties in the same

batch of materials.
(2) Uncertainties in the modeling of applied stresses, for a given service condi-
tion and environment. This variability stems from two sources: (i) the varia-
bility in stress amplitudes during a known service cycle as a consequence of
such factors as vibrations, and (ii) lack of knowledge about the exact dis-
tribution of stress cycles which occur over the design.
(3) Uncertainties in the a priori estimation of the 'environment', and in the
ensuing variation in loading intensity. For example, in wind-turbine and
high-speed transportation applications, the 'environment' and 'loading
intensity' refer to the wind speed. In the fatigue analysis of automobiles,
they generally refer to the 'driver profile' or the 'road condition'. In offshore
structural design, they refer to the 'sea state', e.g., Veers, 1996.
(4) Uncertainties in modeling, predictions and life estimates for fatigue pro-
cesses.
Such uncertainties are analyzed using known statistical approaches to derive the
level of reliability or probability of failure. A detailed review of such approaches to
fatigue can be found in the following references: ASTM STP 744 (The American
Society for Testing and Materials, Philadelphia), Tangjitham & Landgraf (1993),
Sobczyk & Spencer (1992), Wirsching (1995), and Veers (1996). Here we present
some basic concepts commonly adopted for the statistical characterization of fatigue
data.
A parameter which is widely used to describe the uncertainty is the coefficient of
variation (CVar) which is defined as:
i Ns * Ns

CVar = - ^ ; ™x,rv = TT X ! X™i> °£rv = TT

Here, Ns denotes the number of samples of a random variable xrv (such as the stress
amplitude in an S-N fatigue test), and mx rv and ax rv denote the mean and the
standard deviation, respectively.
A distribution in the value of the random variable is usually characterized in terms
of normal distribution, log-normal distribution, or Weibull distribution. For exam-
ple, the probability density function for normal distribution is defined as

(7.10)

where the mean mx rv and the standard deviation ax rv are as defined previously.
Usually, the fatigue strength or the endurance limit values listed from experiments
represent the arithmetic mean derived from multiple experiments. In brittle solids,
such as ceramics and polymers, and in metallic alloys with considerable microstruc-
tural variability arising from processing, the extent of scatter in fatigue data may be
large as a result of a large scatter in microscopic flaw size distribution. Consequently,
7.5 Statistical considerations 233

different sets of experiments conducted on the same material may not give the same
arithmetic mean of the critical strength parameter. To address this issue, Weibull
(1939) proposed the concept of a probability of failure, P, at a given failure strength,
crf normalized by an average value of a critical stress <rcr ave (which may be identified
with the average value of tensile fracture strength cru for a brittle solid or with the
fatigue endurance limit, a e for a metallic alloy). At low values of <xf, P -> 0, and at
very high values of a f , P -> 1. Weibull defined the failure probability as follows:
-( — I ' (7-H)
Wo/ J
where raw is known as the Weibull modulus, and af 0 is a reference strength. Note
that P in Eq. 7.11 represents the fraction of the total number Ns of identical test
specimens in a batch for which the failure strength falls below <rf. Similarly, Ps, the
probability of survival, represents the fraction of the total number Ns of identical test
specimens in a batch for which the failure strength exceeds <jf. When crf/<7f 0 = 1,
P = 1 — e x p ( - l ) = 0.633. In other words, the reference stress Of0 represents the
stress level for which the cumulative probability of failure of all specimens in the
same batch at a stress level af or lower, is 63.3%. Equation 7.11 can be rearranged to
give

rLp = «p{(—) (or) l n

rW-) • < 7 - 12 )
i- p [ Wo/ J i-p VW
Taking the logarithm of both sides once again,
ln[ln—!—1 = mwln (-*-). (7.13)
L l ~ FJ Wo/
It is readily apparent that a plot of the double-logarithm of 1/(1 — P) (ordinate)
against the logarithm of <rf (abscissa), based on experiments conducted on a number
of identical specimens, yields a straight line whose slope is raw and whose intercept
with the ordinate is —rawln<7f 0 . Such a plot is referred to as the Weibull diagram.
Figure 7.8(tf) schematically shows the probability of failure, Eq. 7.11, and Fig. 7.8(Z?)
illustrates the Weibull diagram, Eq. 7.13. In practice, the Weibull modulus is com-
monly determined using the following steps. (1) A data sample of 7VS identical speci-
mens from the same batch of materials is tested to failure under static and/or cyclic
loading conditions, and the appropriate strength value af for each specimen is mea-
sured. (2) The values of af are ordered from the smallest to the largest. The smallest
strength value is ascribed an index value / of 1, the next higher value an index value
of 2, and so forth. The highest strength is then assigned an index of 7VS. (3) Calculate
p = i/(Ns + 1), for 1 < / < Ns. (4) Calculate ln{ln[l/(l - P)]} and plot it against
lna f . (5) The slope of this plot gives the Weibull modulus raw.
The overall objective of incorporating probabilistic analyses into fatigue design is
to ensure that a low probability exists for a combination of higher than average
cyclic stress amplitude and a lower than average fatigue endurance limit (or stress
234 Stress-life approach

1.0

0.0
(a)

-wwln(af0)

(b) ln(af)

Fig. 7.8. (a) A Weibull plot of failure probability against the normalized failure strength, (b) The
Weibull diagram, based on Eq. 7.13, whose slope is the Weibull modulus, raw.

amplitude at a fixed life) to cause failure. In practical design involving the stress-
based approach to total fatigue life, however, an endurance limit is first established
on the basis of experiments conducted on carefully prepared smooth test speci-
mens. This limit is then lowered by applying modifying factors (commonly known
as the 'knock-down factors') to account for such effects as surface finish, size
effects and constraints, temperature, stress corrosion, fretting, and numerous
unknown effects. The damaging effects of repeated contact and corrosion at sur-
faces (arising, for example, from fretting fatigue), and the mitigation, at least in
part, of such deleterious effects by recourse to shot-peening and lubrication are
addressed in Chapter 13.
7.6 Practical applications 235

7.6 Practical applications

7.6.1 Example problem: Effects of surface treatments

Problem:
A high strength steel is to be used as a material for leaf springs in a
ground transportation vehicle. During service, cyclic loading is expected to
result in a load ratio, R = 0 (i.e. zero-tension cyclic loading). The 'as-heat-
treated' condition of this alloy has an endurance limit, <re, of 400 MPa which
was determined from laboratory tests conducted under fully reversed cyclic
loading conditions. The tensile strength of the steel is 1500 MPa. The surface
finish of the final product is expected to have an average roughness which, from
prior experience in fatigue design, is known to 'knock down' the endurance limit
by 40% from that measured in 'smooth' laboratory specimens. Shot-peening of
the product prior to the assembly of the vehicle is known to introduce a max-
imum surface compressive stress of-550 MPa. Determine the maximum stress
amplitude which the surface of the leaf spring, which has to be designed for an
infinite fatigue life, can sustain.

Solution:
The endurance limit of the product with a rough surface finish is 40%
lower than the value measured in the laboratory. The design endurance limit
then is: 400 x 0.6 = 240 MPa. This value pertains to fully reversed cyclic stres-
sing, i.e. for R = -1 or om = 0.
The applied loading involves R = 0 and a stress amplitude, a a . From Eq. 7.2,
it is seen that a a = a m = Aa/2 for R = 0. The compressive surface stress arising
from shot-peening lowers the mean stress in surface to a new value: am = (am -
550) = (a a - 550) MPa. Assuming that the modified Goodman approximation,
Eq. 7.4, provides a reasonably accurate measure of the high-cycle fatigue life
under nonzero mean stress, we see that

? + ^=L (7 14)
'
Substituting the appropriate values,

—-—I— =1 (1 15^
v
240 1500 ' ' J
This equation gives: <ra = 283 MPa. Thus, the leaf spring could be subjected
to an applied stress amplitude of up to 283 MPa for infinite life. (Note that the
possible beneficial effect of shot-peening in enhancing the fatigue endurance
limit has not been taken into account in the numerical calculations. Thus, the
stress amplitude estimate is likely to be very conservative.)
236 Stress-life approach

7.6.2 Case study: HCF in aircraft turbine engines

Fatigue cracking under low-amplitude, high-cycle loading is the dominant failure
process in a number of engineering applications. In this section, which is largely
predicated upon a review by Cowles (1996), we examine the origins and effects of high-
cycle fatigue in advanced gas turbine jet engines used in military aircraft.
The principal cause of failure of components, from which a gas turbine jet engine of a
modern military aircraft is made, is high-cycle fatigue. As illustrated in Fig. 7.9(a), fatigue
failure accounts for 49% of all component damage in jet engines. High-cycle fatigue
(HCF) is responsible for nearly half of all these failures, whilst low-cycle fatigue (LCF)
and all other modes of fatigue lead to the remainder of fatigue failures in roughly equal
proportions. Failure by HCF affects a variety of engine components, as shown in Fig.
7.90).
The origin of HCF in the gas turbine aircraft engine can be attributed to one or more
of the following causes.
(1) Mechanical vibration arising from rotor imbalance (which affects plumbing,
nonrotating structures and external members) and rub (which affects blade tips
and gas path seals).
(2) Aerodynamic excitation occurring in upstream vanes, downstream struts and
blades, whereby engine excitation frequencies and component response
frequencies corresponding to different modes of vibration may overlap.
(3) Aeromechanical instability, primarily in blades, accompanying aerofoil flutter.
(4) Acoustic fatigue of sheet metal components in the combustor, nozzle and
augmentor.
The above sources of low-amplitude, HCF are augmented by the following damage
processes which create microscopic notches and other sites at which fatigue cracks can
nucleate and advance subcritically to catastrophic proportions.

1% thermal cases & housings

(a) (b)

Fig. 7.9. (a) Distribution of different failure modes in jet engines, (b) Susceptibility of
different components to HCF problems. (After Cowles, 1996.)
7.7 Stress-life response of polymers 237

(1) Foreign object damage (FOD), which usually occurs in compressor and fan
blades: FOD can induce micronotches, tears, dents and gouges that may vary in
dimensions from tens of micrometers to tens of millimeters, depending on the
size, nature and severity of impact of the foreign object. Sources of FOD are as
diverse as sand particles and birds. As noted earlier, the leading edge of the fan
blade, which is susceptible to FOD, is laser shock peened for improved fatigue
resistance.
(2) Domestic object damage (DOD), which arises from a dislodged debris or
component from another location of the engine.
(3) Fretting fatigue, which occurs at blade and disc attachment surfaces (dove-tail or
fir-tree section), bolt flanges, and shrink-fit areas. See Chapter 13 for terminology
definitions and detailed discussions pertaining to fretting fatigue.
(4) Galling, which occurs in the same regions as fretting, except that it involves greater
displacements due to major engine throttle and speed changes.
(5) HCF-LCF interactions, where HCF is exacerbated by LCF as, for example, when
creep and thermo-mechanical fatigue in hot sections (such as turbine blades) cause
further reductions in fatigue life, over and above that due to vibrations.

Current methods to assess the HCF life of critical components in aircraft gas
turbine engines entail the following general steps, (i) Appropriate stress analysis
(largely based on the finite-element method) are performed to determine the mean
stress level, (ii) Structural dynamics simulations are carried out to determine resonant
frequencies and excitation modes, (iii) The design of the component is then carried
out in such a way that (a) it meets the criteria for safe life for HCF with the
appropriate mean stress level (using the modified Goodman diagram), and (b) no
resonance-related problems arise. The parameters that serve as input to design are
gathered from specimen testing and component testing, and the stress-life approach is
empirically modified to allow for reductions in life due to FOD, DOD, fretting and
galling.

7.7 Stress-life response of polymers

7.7.1 General characterization
The characterization of the fatigue life of polymeric materials on the basis of
stress amplitudes is done in much the same way as for metals. Figure 7.10 is a typical
Wohler (S-N) curve for a polymer, where three distinct regions are seen in
the variation of the stress amplitude cra as a function of the number of cycles to
failure Nf.
The existence of region I and the slope of the A<r-Nf plot in this region
depend on whether crazes form at the high values of Acr, and on whether the
crazes cause microscopic cracks to nucleate. PS and PMMA, which are prone to
craze formation, exhibit a well defined region I where the fatigue life depends
strongly on Aa. If the maximum tensile stress in the very first cycle is not
238 Stress-life approach

Fig. 7.10. Schematic representation of the typical variation of stress amplitude <xa with the
number of cycles to failure 7Vf for polymeric materials.

sufficiently large to form crazes, a distinct region I may not exist and the slope
of the Aa-Nf curve at the high Aa end will be a mere extrapolation of that in
region II.
The dependence of total fatigue life on the stress amplitude in region II is a
manifestation of the enhanced role of microscopic crack growth on fatigue fracture.
At the higher aa end of region II, slow growth of crazes and their transformation into
cracks are dominant mechanisms of failure. A slope of 14MPa per decade of iVf
seems to be characteristic of region II in a variety of polymeric materials fatigued at
room temperature. As the test temperature is raised, there is a competition between
shear banding and crazing in this region, as in the case of tensile deformation.
The high-cycle fatigue region represented in region III essentially forms the
endurance limit for the polymer. The fatigue life here is controlled by the incuba-
tion time for the nucleation of microscopic flaws. The relative dominance of
nucleation and growth of crazes and cracks constitutes the main distinction
between regions II and III.

7.7.2 Mechanisms
The mechanisms controlling the total fatigue life of polymers also vary with
many morphological, thermal, mechanical and environmental factors.
(1) When the cyclic loading involves high strain rates, the material is prone to
hysteretic heating and to thermal softening. Consequently, increasing the
test frequency (typically beyond 10 Hz for experiments conducted in the
laboratory environment) results in a reduction in fatigue life. In an attempt
to address these issues, the American Society for Testing and Materials
(Philadelphia) developed standard methods in 1971 for the fatigue testing
of polymers. These procedures are spelled out in Specification D-671-71.
The standard calls for the measurement of temperature at fatigue failure
unless it can be demonstrated that the heat rise is not significant.
Furthermore, when thermal softening controls fatigue fracture, this stan-
dard defines the fatigue failure life operationally as the number of loading
7.8 Fatigue of organic composites 239

cycles at a fixed stress amplitude which leads to an apparent reduction in

modulus of 30% from the commencement of the fatigue test.
(2) The increase in the temperature of the polymer during high frequency fati-
gue loading also depends on the dimensions of the specimen. In thinner
specimens, a greater fraction of the heat generated by damping is lost to
the environment. Therefore, thicker fatigue test specimens, which retain a
larger fraction of the temperature rise induced by hysteretic heating, exhibit
a lower fatigue endurance limit (Riddell, Koo & O'Toole, 1967). In an
analogous fashion, increases in test temperature (caused by external heating)
also cause reductions in fatigue life.
(3) Most polymeric solids exhibit a longer fatigue life in inert environments
than in chemically aggressive media. Such deleterious effects of aggressive
environments are caused by complex synergistic interactions between the
structure of the polymer and the surrounding medium. For example, the
absorption of a nonsolvent can reduce the glass-transition temperature and
modulus to the point that cavitation and crack initiation are promoted
(Gent, 1970). Some liquid media, such as water, change the state of a
polymer by processes involving plasticization or antiplasticization.
Similarly, a small amount of acetone can transform polycarbonate from
a glass to a crystalline solid. Specific combinations of material, environ-
ment and load can also improve the fatigue resistance (Manson &
Sperling, 1976).
(4) Although current understanding of the effects of molecular properties on
fatigue life is incomplete, it appears that the total fatigue life increases with
increasing molecular weight (Sauer, Foden & Morrow, 1977) and crystal-
linity (Riddell, Koo & O'Toole, 1967), and with decreasing level of cross-
linking (Sauer, 1978).
(5) As in the case of metallic materials, the total fatigue life derived from con-
stant stress-amplitude fatigue decreases with increasing mean stress.

7.8 Fatigue of organic composites

The reinforcement of polymeric matrices with particles, whiskers and con-
tinuous fibers provides many possibilities for improving the strength, stiffness and
resistance to creep and fatigue. The extent to which the improvements in the resis-
tance to fatigue are realized is, of course, dictated by the specific combinations of
matrix and reinforcement materials, processing methods, geometrical factors asso-
ciated with the arrangement of the reinforcement phase in the matrix, and interfacial
characteristics. Discontinuous and continuous fiber reinforcements of glass, boron,
carbon, and high modulus organic materials such as aromatic polyarides are
commonly used in the synthesis of polymeric composites. Polymers widely used
240 Stress-life approach

as matrices for organic composites include polyesters, epoxies, Nylon 66, PC,
polyphenylenesulphide, polyamideimide, PSF, PVC, and polyetherimide.

7.8.1 Discontinuously reinforced composites

The fatigue behavior of polymers reinforced with particulates and short
fibers has been extensively reviewed in Hertzberg & Manson (1980, 1986). In this
section, we present two examples to illustrate: (i) how the introduction of a strong
second phase can enhance the resistance of a polymer to high cycle fatigue, and (ii)
how such improvements are influenced by the specific choice of the reinforcement
phase.
The stress-life curves for tension fatigue in several injection-molded composites
with PSF matrices and with carbon or glass (short) fibers are presented in Fig.
7.11. Also included in this figure, for comparison purposes, are the stress-life
characteristics of the unreinforced matrix material with a similar processing his-
tory. It is clear that the polymeric composites offer a superior (total) fatigue life
than the unreinforced matrix, with the resistance to fatigue fracture increasing
with increasing concentration of the glass fibers. The superiority of the reinforce-
ment with carbon fibers to that of glass fibers is also evident. The high stiffness
of the carbon whiskers contributes to the fatigue life by reducing the cyclic
strains in the matrix at a given stress amplitude. Furthermore, the high thermal
conductivity of carbon moderates the temperature rise due to hysteretic heating.
It should, however, be noted that the beneficial effects of whisker reinforcements
shown in Fig. 7.11 may be markedly reduced if the fatigue tests are conducted at
higher cyclic frequencies where thermal softening begins to control the fatigue
life.

7.8.2 Continuous-fiber composites

Cyclic loading of organic composites with continuous and unidirectional
fibers can engender a more deleterious effect on service life than monotonic loading.
The residual strength, toughness, and elastic response are all adversely affected by
prolonged cyclic loading. This effect is clearly demonstrated in the stress-life data
plotted in Fig. 7.12 for an organic composite with an epoxy resin (Bakelite ERL
2256) matrix and 0.6 volume fraction of unidirectional E-glass fibers. When the
composite is subjected to tension fatigue at a stress ratio of 0.1 and a test frequency
of 19 Hz in the direction of fiber orientation, an endurance limit of about 686 MPa
results. This value is significantly lower than the static tensile strength of 1245 MPa.
Also shown in Fig. 7.12 are the results of fatigue tests conducted for different
orientations, </>, of the tensile axis with respect to the fiber direction. It is seen that
an increase in 0 leads to a monotonic reduction in the endurance limit, with the
7.8 Fatigue of organic composites 241

200
I, • PSF
V PSF+10% glass
O PSF+ 20% glass
A P S F + 40% glass
O PSF+ 40% carbon

Fig. 7.11. A stress-life plot showing the variation of maximum tensile stress <jmax as a function
of the number of fatigue cycles to failure 7Vf for injection-molded PSF which is reinforced
with different amounts of short glass fibers and with 40% short carbon fibers. The results
are based on room temperature fatigue tests conducted at R = 0.1, and vc = 5-20 Hz. The
small arrows denote that no failure occurred at the indicated number of cycles. (After Mandell
et ai, 1983.)

matrix playing an increasingly dominant role in the fatigue process. Similar char-
acterizations of fatigue failure have also been developed using strain-based
approaches (e.g., Dharan, 1975; Talreja, 1987).
Nondestructive test methods based on acoustic emission, radiography, and mod-
ulus and damping measurements of fiber composites subjected to fatigue loads show
fiber breakage, debonding, or matrix cracking fairly early in the fatigue process. The
growth of such microscopic degradations continues to result in the lowering of the
stress at which failure occurs with increasing numbers of fatigue cycles. In view of this
progressive degradation, the useful fatigue life of many continuous-fiber composites is
often defined as the number of cycles needed to cause a certain reduction in the elastic
modulus. It is also common to characterize the distributed (microscopic) fatigue
damage using Weibull statistics (see, for example, Talreja, 1987). At high cyclic fre-
quencies, hysteretic heating also becomes an important factor. Furthermore, anom-
alous improvements in fatigue life may result in some cases due to the realignment of
fibers, crack blunting, periodic overloads or frequency fluctuations.
242 Stress-life approach

1200

1000

800

\ 600

200

100
.^ = 60°

102 103 104 105 106 107 108

Fig. 7.12. Stress-life plots showing the variation of maximum tensile stress a max as a function
of the number of fatigue cycles to failure N? for an epoxy resin matrix reinforced with 60
volume % unidirectional glass fibers. 4> denotes the angle between the fiber direction and the
tensile axis. The tests were conducted at room temperature at R = 0.1 and vc = 19 Hz. (After
Hashin & Rotem, 1973.)

7.9 Effects of stress concentrations

The discussion up to this point has focused on nominally smooth-surfaced
solids. However, engineering structures invariably contain stress concentrations
which are the principal sites for the inception of fatigue flaws. The stress and defor-
mation fields in the immediate vicinity of the stress concentration have a strong
bearing on how the fatigue cracks nucleate and propagate. In this section, we exam-
ine continuum approaches based on stress-life concepts which deal with the issue of
fatigue failure ahead of stress concentrations.

7.9.1 Fully reversed cyclic loading

The theoretical elastic stress concentration factor Kt relates the local stress
ahead of the notch tip to far-field loading and is defined as the ratio of the maximum
local stress <rmax to the nominal stress S. Under fatigue loading conditions, the elastic
stress concentration factor is replaced by the so-called fatigue notch factor:
unnotched bar endurance limit
notched bar endurance limit
In general, fatigue experiments suggest that notches produce a less stress concentrat-
ing effect than predicted by theoretical elastic analysis such that Kf < Kt; Kf -> Kt
for large notch-root radii and for higher strength materials. The degree of agreement
between theoretical predictions of elastic stress concentration and actual effects is
often measured by the so-called notch sensitivity index which is defined as
7.9 Effects of stress concentrations 243

<717)
The parameter q varies from zero for no notch effect to unity for the full effect
predicted by the elasticity theory. Kt is a function only of the component geometry
and loading mode and is available in many handbooks (e.g., Peterson, 1959).
However, Kf is determined from experimental measurements or empirical, engineer-
ing 'rules-of-thumb'. An example of such a measure of Kf is the well known Peterson
equation for ferrous wrought alloys (Peterson, 1959):

where An is a constant whose value depends on the strength and ductility of the
material (An ~ 0.25 mm for annealed steels and An ~ 0.025 mm for steels of very
high strength) and p is the notch-root radius.
The stress-life approach is employed for high-cycle fatigue failures ahead of stress
concentrations by appropriately modifying the smooth specimen (unnotched) endur-
ance limit ae. This involves either dividing the stress range A a for all fatigue lives by
the factor Kf (which often leads to very conservative results) or merely reducing the
fatigue limit ae by a factor of Kf. This method is unsuitable for situations where
considerable plastic deformation occurs ahead of the stress concentration.

7.9.2 Combined effects of notches and mean stresses

Now we consider a notched member which is subjected to cyclic loading
with a nonzero mean stress in such a way that only elastic conditions prevail at all
times throughout the member. If S, Sm, and Sa are the instantaneous value, the mean
value, and the amplitude, respectively, of the nominal far-field cyclic stress imposed
on the notched member (Fig. 7.13), the local stress amplitude and the mean stress at
the tip of the notch can be computed from the fatigue notch factor Kf such that

where the peak value of K^S is always smaller than the yield strength of the material
cry in both tension and compression. In order to assess the effect of the mean stress
on fatigue life, the modified Goodman equation, Eq. 7.4, or some other approach
discussed in Section 7.2 may be employed. Using the modified Goodman diagram
for the notched member, we see that

max y
K{ \ aTS j
Figure 7.13 schematically shows the modified Goodman diagram for smooth and
notched specimens which are subjected to cyclic loading under a nonzero mean
stress. These specimens are subjected to an applied stress amplitude of *Sa and
KfSz, respectively, as indicated in the figure. Note that for the elastically deforming
notched member, the intercepts in both the ordinate and the abscissa are reduced by
a factor of Kf as compared to the unnotched fatigue specimen.
244 Stress-life approach

Kf

Fig. 7.13. The effect of mean stress on fatigue life as determined from the modified Goodman
diagram for notched and unnotched fatigue specimens.

7.9.3 Nonpropagating tensile fatigue cracks

An intriguing phenomenon associated with tensile fatigue cracks growing
ahead of stress concentrations involves the so-called nonpropagatingflaws.It was
first shown by Frost & Dugdale (1957) and Frost (1960) that fatigue cracks emanat-
ing from notches can arrest completely after growing some distance. A number of
subsequent studies have unequivocally documented the existence of nonpropagating
fatigue cracks ahead of stress concentrations in ductile solids (e.g., Smith & Miller,
1978; El Haddad et al 1980; Tanaka & Nakai, 1983). Experimental measurements
based on total life have shown that the nonpropagation or arrest of fatigue flaws
occurs only ahead of sharp notches, above a certain critical value of Kt. Figure 7.14,
where the fatigue limit is plotted as a function of Ku schematically illustrates the
conditions for the occurrence of nonpropagating fatigue flaws. Here, the threshold
stress for crack initiation, expressed as the unnotched fatigue limit ae divided by K{
or Ku is lower than the stress required to cause catastrophic fracture of nonpropa-
gating cracks above the critical Kt. A discussion of the possible causes for the arrest
of fatigue cracks ahead of stress concentrations is provided in Chapter 15 in con-
nection with the topic of short fatigue cracks.

7.9.4 Example problem: Effects of notches

Problem:
A circumferentially notched cylindrical rod of a high-strength, low alloy
steel is subjected to repeated bending at a nominal stress amplitude,
7.9 Effects of stress concentrations 245

complete failure
I
A\T\\\\T\\\
nonpropagating cracks
no cracks

— Aae/Kt

Fig. 7.14. Threshold stress range (fatigue limit) for crack initiation at a notch tip, characterized
by the unnotched fatigue limit Acre divided by Kf or Kt and plotted as a function of Kt.
(After Frost, Marsh & Pook, 1974.)

Sa = 200 MPa, and a mean stress, Sm = 250 MPa. For the notch geometry of
the rod, the elastic stress concentration factor, Kt9 and the notch sensitivity
index, q, are found from appropriate handbooks to be 2.3 and 0.92, respectively.
Fully reversed cyclic stress tests conducted on smooth (unnotched) laboratory
specimens give the following material parameters for the Basquin equation, Eq.
7.1: Of = 1000 MPa and b = -0.12. The endurance limit was estimated from
these tests to be 280 MPa. The yield and tensile strengths of the steel are
ay = 600 MPa and a TS = 1050 MPa. Estimate the high-cycle fatigue life of
the notched rod.
Solution:
From Eq. 7.17, we find that the fatigue notch factor, Kf = 1.2. For the
given loading conditions, i.e. Sm = 250 MPa, 5 a = 200 MPa, we thenfindan
equivalent value of fully reversed cyclic stress, SaLm=(b f ° r a n unnotched speci-
men using Eq. 7.20. For this purpose, we rewrite Eq. 7.20 as:
i - i

Substituting the appropriate numerical values, it is seen that

£al<rm=o = 336 MPa. (Note that the requirement in Eq. 7.20 that
Kf' Slmax < Oy is satisfied.) The equivalent value of SaLm=o = 336 MPa exceeds
the fatigue endurance limit of 280 MPa. Thus, the specimen is expected to have a
finite life. Using Eq. 7.1,
oi(2Nf)b= 1000(2TVf)-°-12 , =» 27Vf = 8855. (7.22)
Thus, the number of stress reversals to failure is 8855, or the number of cycles to
failure is 4427.
246 Stress-life approach

7.10 Multiaxial cyclic stresses

The discussions up to this point dealt with total fatigue life characterization
for loading on the basis of uniaxial cyclic stresses involving primarily elastic condi-
tions. There are, however, many practical applications where fatigue-critical compo-
nents undergo multiaxial cyclic loading. For example, the fuselage of an aircraft is
subjected to hoop and longitudinal cyclic stresses due to repeated pressurization and
depressurization. Similarly, pressure vessels, tubes and pipes are subjected to biaxial
stresses which arise from internal pressures. Transmission shafts in automobiles
experience combined shear stresses arising from torque which is superimposed on
axial stresses generated by bending.
In general, different modes of imposed loads (such as tension and torsion) may
occur either in a static mode (as, for example, due to a steady bending moment, mean
pressure or steady torque) or they may fluctuate in phase or out of phase. The overall
fatigue life under general multiaxial conditions is inevitably dictated by the complex
phase relations between the different modes of loading. In this section, we consider
stress-based approaches to total life under multiaxial loading conditions. The corre-
sponding discussions of multiaxial strain-based approaches are taken up in the next
chapter.

7.10.1 Proportional and nonproportional loading

Consider a fatigue specimen which is subjected to multiaxial cyclic loads.
Proportional loading is considered to occur if, during changes and fluctuations in the
imposed loads, the different components of the stress tensor vary in constant pro-
portion to one another. At a given reference point in the material, proportional
loading exists if the principal stresses at that point, a1? a2 and a3, vary in the
following manner:

^ = Xlt ^ = X2. (7.23)

Here X\ and X2 are scalar constants which may vary from point to point, but are
constants for a given material point in the solid.
As an example, an increase in the internal pressure of a close-ended, thin-walled
cylindrical tube causes the hoop stress <rhoop and the axial stress aaxial to vary in
proportion to each other so that at all times,tfhoop/^axiai= 2. If the thin-walled
cylinder is subjected to an axial tension P and a torque about the axis T, propor-
tional loading occurs only if P oc T. Any other mutual variation of P and T gives
rise to nonproportional loading. Further discussions of nonproportional and out-
of-phase loading are considered in the next chapter in the context of strain-life
fatigue.
7.10 Multiaxial cyclic stresses 247

7.10.2 Effective stresses in multiaxial fatigue loading

We begin with a consideration of fully reversed cyclic loading (zero mean
stress) where all cyclic loads are perfectly in phase or 180° out of phase with one
another. Let ax, oy and oz denote the three normal components of stress applied to a
fatigue specimen and let rxy, ryz and xzx denote the three shear stress components. Let
<71? <72, cr3 denote the principal stresses, with o\ > cr2 > cr3. The effective stress which
characterizes the deformation of the material is then assumed to be that given by the
von Mises criterion based on the octahedral shear stress,! Toct:

= "/5 r oct = -7=yJOl - <*lf + O2 - °3

= "W
(7.24)

Another measure of the intensity of the multiaxial stress state (and used in the Tresca
yield criterion which was discussed in Section 1.4.3) is the maximum shear stress,
r max , which acts on at most three mutually perpendicular sets of planes that intersect
the principal stress axes at 45°:
ox -
= max (7.25)

For fluctuating multiaxial stresses, Eq. 7.24 is written in terms of the amplitudes of
principal stresses aza (/ = 1, 2, 3) and the amplitude of the effective stress a a e as

^a,e = -7= V Ola ~ ^2a)2 + O2a ~ ^ a ) 2 + 03a ~ ^la)2- (7-26)

In the most elementary stress-life analysis for multiaxial fatigue involving a zero
mean stress, the Basquin exponent b and the fatigue strength coefficient a[ in Eq. 7.1,
which are determined from experiments conducted on smooth specimens subjected
to fully reversed uniaxial cyclic loads, are used in conjunction with the following
stress-life equation for multiaxial stresses to determine the fatigue life, JVf:
(7.27)

In some cases, a steady, nonfluctuating load in one mode (such as bending) may be
superimposed on cyclic stresses in another mode (such as cyclic torsion). Here, the
effects of mean stress on fatigue failure should be considered. If it is assumed that the
controlling mean stress is related to the steady value of the hydrostatic stress, an
effective value of this mean stress, a m , may be defined in terms of the mean values of
the principal stresses, crim (i = 1, 2, 3), as
+ °zm- (7.28)

' Recall from Chapter 1 that the octahedral shear stress is the resolved shear stress on the n-plane, which
is the plane oriented at equal angles to the three principal stress directions.
248 Stress-life approach

Alternatively, an effective mean stress may be defined based on the octahedral shear
stress as

<Ve = -J= V Olm ~ ^2mf + fern ~ ^m) 2 + <>3m ~ ^lm)2, (7-29)

The simplest way to invoke stress-based approaches to fatigue in the presence of

multiaxial stresses which involve a nonzero mean stress is to combine the effective
stress amplitude <ra e , determined from Eq. 7.26, and the effective mean stress as, for
example, determined from Eq. 7.29, with the modified Goodman equation to obtain

With this equation, the generalization of the modified Goodman equation to multi-
axial fatigue then involves an equivalent fully-reversed uniaxial cyclic stress. The
fatigue life under multiaxial cyclic stresses involving nonzero mean stresses is next
determined using Eq. 7.27.
One of the major drawbacks of such effective stress approaches is that the differ-
ing effects of axial tension and compression mean stresses in a multiaxial fatigue test
may not be accurately captured. In addition, the orientation of fatigue cracks with
respect to the loading axes is not quantitatively determined from such criteria.
Furthermore, a wide variety of experimental observations (to be discussed next)
reveal that the normal stress also plays a critical role in influencing fatigue lives in
multiaxial loading.

7.10.3 Stress-life approach for tension and torsion

For uniaxial tensile loading with a nonzero mean stress, the only nonzero
components of stress are a l a and a l m . For this case, as anticipated, Eq. 7.30 reduces
to Eq. 7.4. Now consider pure torsion loading only. The only nonzero components
of stress in this case are the amplitude and mean value of the shear stress, rxySi and
T
xy,m> respectively. From Eqs. 7.24, 7.26 and 7.28, it is readily seen that
cra,e = A/3TJCJ,ia and a m = 0. (7.31)
An interesting point to observe here is that am = 0 even if the mean shear stress for
torsional loading is nonzero. A variety of independent experimental studies (see, for
example, the work dating back to Sines, 1959) have shown that a superimposed mean
static torsion has no effect on the fatigue limit of metals subjected to cyclic torsion,
although a superimposed static tension or bending stress has a marked effect on the
fatigue life in normal cyclic loading.
The differing effects of torsion and tension loading on the microscopic and macro-
scopic modes of cracking and on the overall fatigue life are illustrated in Figs. 1 A5(a)
and (&), respectively, for an Inconel 718 alloy. Essentially throughout the fatigue life,
failure occurs on planes of maximum shear. This damage process is classified as
regime A in Fig. l.\5(a). In this regime, the cracks were confined to the planes of
7.10 Multiaxial cyclic stresses 249

regime B

10 10 10° 10'
Nf (cycles)

regime A regime B
1.0

0.8

0.6

0-4

0.2

104 105 10° 10'

f (cycles)
Fig. 7.15. The evolution of different regimes of fatigue failure during (a) torsion fatigue and (b)
tension fatigue of an Inconel 718 alloy (yield strength, ay = 1160MPa. (From Socie, 1993.
Copyright American Society for Testing and Materials, Philadelphia. Reproduced with
permission.)

maximum shear through the fatigue life. The fraction of life, N/Nf, expended in
initiating a dominant shear crack is less than 0.1; the remaining 90% of life is spent
in propagating this shear crack. For Nf > 106 cycles, the local mode of failure occurs
on planes oriented normal to the local principal tensile stress, with the microscopic
tensile cracks oriented at 45° to the shear cracks. This failure process is categorized
as regime B in Fig. 7A5(a). The fraction of life expended in initiating the dominant
crack rises to 0.2 at N/Nf ~ 107 cycles.
When the Inconel 718 alloy is subjected to axial tension fatigue, Fig. 7.15(6), a
different failure pattern emerges throughout the life. In both the low-cycle and high-
250 Stress-life approach

cycle fatigue failure regimes, the macroscopic crack plane is approximately normal to
the tensile loading axis. For JVf < 105 cycles, regime A, the local mode of micro-
scopic cracking is along planes of maximum shear stress. As discussed in detail in
Chapter 10, this microscopic mode of initial crack advance in tension fatigue is
commonly referred to as Stage I, where single-slip failure along planes of local
maximum shear induces a serrated or faceted fracture morphology. Such features
are clearly evident in the micrograph shown in Fig. l.\5(b). Within regime A, the
fraction of life expended in nucleating a dominant fatigue crack gradually rises from
approximately 0.1 at JVf ~ 103 cycles to about 0.4 at Nf ~ 105 cycles. For N{ > 106
cycles, regime B, a tensile mode of failure emerges under imposed tension fatigue,
with the fraction of total life expended in nucleating a dominant fatigue flaw (1 mm
in size) gradually rising to as high as 0.9 at JVf ~ 107 cycles. Such a mode of failure is
commonly referred to as Stage II, whose microscopic mechanisms are examined in
detail in Chapter 10.
The extent to which regimes A and B individually dominate the total fatigue life,
and the fraction of total life expended in creating a dominant flaw within each of
these regimes is a strong function of the composition and micro structure of the
material, and of the test environment. Locally tensile failure patterns can also be
induced in some alloys subjected to cyclic torsion, especially in the high-cycle fatigue
regime.

7.10.4 The critical plane approach

When the multiaxial fatigue cycles involve proportional loading, the most
critical plane on which damage and cracking occur (e.g., the plane on which the
tensile normal stress or the shear stress is a maximum) stays fixed relative to the
loading axis. Under such circumstances, the critical plane can be identified in a
relatively straightforward manner, as discussed later in this section. Such a situation
continues to hold even if the cyclic stresses are of variable amplitude. By contrast,
when the components of the principal stress tensor do not vary in constant propor-
tion to one another, the orientations of the critical planes in the specimen vary with
time and loading path. As a result, an instantaneous definition of these critical planes
derived from local stress analyses is needed. The critical plane approach to multiaxial
fatigue life estimation then involves the following steps: (1) identification of the
orientation of the plane which represents the critical site in the material for the
onset of damage based on some criterion (such as the maximum normal/shear
stress/strain criterion, or a combined normal stress/shear stress criterion) and (2)
estimation of the number of cycles to initiate a fatigue crack or the total number of
cycles to failure using experimental results on stabilized cyclic stress-strain curves
and fatigue lifes, preferentially from uniaxial stress-life data obtained under fully
reversed loading conditions (in conjunction with the modified Goodman diagram or
7.10 Multiaxial cyclic stresses 251

some other approach to account of the effect of mean stresses; see later discussion in
this chapter).
Figure 1 A6(a) schematically shows a fatigue specimen subjected to bending-tor-
sion cyclic loads. The reference coordinate axes, x, y and z, are marked on the figure.
Let the orientation of a critical plane be defined with reference to this original
coordinate system with the aid of two angles: the angle, y, between the axis of the
specimen (the x axis) and the line of intersection of the critical plane with the surface,
and the angle, cp, between the normal to the critical plane, which is along the z
direction in Fig. 7.16(&), and the z axis. A new set of coordinate axes, x\ y and
zr, is defined in Fig. 7.16(7?) to facilitate the visualization of the critical plane orienta-
tion. Using straightforward methods for coordinate transformation, the components
of the stress and strain tensor in the x\ y and z1 coordinate system are written as

cos^siny — sin^sinyX
cos cp cosy sin cp cosy I. (7.32)
—sincp — cos<p /
Then the important components of the stress acting on the potentially critical plane
are identified as: the normal stress (o^vX t n e (in-plane) shear stress (cry^) which
produces a mode II type crack opening, and the (out-of-plane) shear stress which
produces a mode III type crack opening (see Chapter 9 for further discussions of
mode II and mode III).

(a) (b)
Fig. 7.16. Schematic of a fatigue specimen subjected to bending-torsion cyclic loads (a) and the
definition of the critical plane and the associate nomenclature (b).
252 Stress-life approach

In general, the identification of the orientations of the critical planes for general
multiaxial loading with nonproportional stress fluctuations can become very cum-
bersome. However, as reviewed by Chu, Conley & Bonnen (1993), some simplifica-
tions can be extracted by noting that most fatigue cracking initiates at the surface.
Given the state of plane stress at the surface elements of the specimen, criteria based
on mode I (tensile failure) and mode III (torsional or out-of-plane shear failure) type
damage lead to cp = 90° as the critical orientation, whereas mode II type failure
based on in-plane shear stresses lead to <p = 45° as the critical orientation. With this
fixed orientation of cp, only variations in the angle y need to be calculated. If such
approximations cannot be applied to a given material or loading situation, detailed
computations based on the critical plane approach need to be undertaken.

7.10.4.1 Critical plane criteria invoking normal and shear stresses

On the basis of experimental observations of fatigue crack initiation pat-
terns in steels and aluminum alloys, Findley (1959) postulated a criterion for multi-
axial fatigue failure which is predicated upon the combined effects of normal stress
and maximum shear stress. In his model, fatigue failure is deemed to occur when

Here, a and f$ are material parameters determined experimentally, which are con-
stants for a given life. An increase in the maximum normal stress an max acting on the
plane of the critical alternating shear stress causes a corresponding reduction in the
permissible cyclic shear stress ra as per Eq. 7.33, for a fixed fatigue life. McDiarmid
has proposed several variations of the Findley criterion, the most recent of which
attempts to account for different types of cracking patterns observed during multi-
axial fatigue (McDiarmid, 1994).
The justification for the incorporation of a normal stress component, in the form
of either a maximum normal stress or a hydrostatic stress, in Eq. 7.33 and in other
critical plane criteria presented in subsequent sections and in the next chapter, stems
from the differing effects on crack growth from tensile and compressive mean
stresses.f From a mechanistic viewpoint, one possible rationale for such an effect
is that interlocking and closure of the microscopic irregularities of the crack faces
seen, for example, in the Stage I transgranular crack growth process seen in Fig.
7.15(6), can influence the transmission of shear tractions. The magnitude of tensile or
compressive normal stresses can consequently determine how effectively such con-
tact between crack faces and the attendant transmission of shear loads occurs. One
may then envision the possibility of a minimization or elimination of such contact,
and a resultant increase in the rate of advance of the crack, due to tensile mean
stresses. Such a situation is demonstrated in the case study on multiaxial fatigue
' Historically, criteria for multiaxial fatigue failure on the basis of both shear and normal stresses have
also been considered in the earlier works of Gough, Pollard & Clenshaw (1951) and Stulen & Cummings
(1954). A survey of historical developments in the area of multiaxial fatigue is given by Garud (1981).
7.10 Multiaxial cyclic stresses 253

presented in Chapter 8. A number of criteria have also been put forth to identify
critical planes of failure on the basis of different combinations of normal and shear
strains for proportional and nonproportional loading. These approaches are also
discussed in Chapter 8.

7.10.4.2 Criteria invoking effective stresses

In order to develop a criterion which quantitatively captures the effects of
superimposed normal loads on the observed fatigue limits, Crossland (1956) postu-
lated that
ax< A (7-34)
where / 2 a is the amplitude of the second invariant of the stress deviator defined in
Section 1.4, a H m = cfm/3 (see Eq. 7.28) is the mean value of the hydrostatic stress
during a multiaxial fatigue loading cycle, and crH max is the maximum value of the
hydrostatic stress during multiaxial loading; <7H,max — ^Hm + aH,a (where a^^ is the
amplitude of the hydrostatic stress during a multiaxial fatigue loading cycle), and a
and fi are material constants determined experimentally.
Sines (1959) modified the Crossland criterion in the following manner:
V-hl + a<rHtm<P' (7.35)
Let re _! be the endurance limit measured from a fully reversed cyclic torsion test
(where a H m = 0 and J2a = re _i) and let a B 0 be the endurance limit determined
from a zero-to-tension (R = 0) uniaxial or bending cyclic load test (where
^2a = ^B,O/V3 and crUm = crB0/3). Then, a = [(3Te>_i/orB)0) - V3] and 0 = re_{
in Eq. 7.35. The Sines criterion correctly captures the experimental observations that
(a) the fatigue limit in torsion is independent of a mean shear stress and that (b) the
endurance limit in bending varies linearly with a static normal stress. Following the
same line of reasoning, the material parameters in Eq. 7.34 are:
a = [(3r e ,-I/OB _ 0 — V3] and /? = re _1? where OB,-I *S t n e endurance limit mea-
sured under fully reversed bending.
A number of subsequent variations of the Crossland (1956) and Sines (1959)
criteria have been reported. For example, Dang Van (1973) has formulated a criter-
ion where a linear combination of the shear stress acting on the plane of maximum
shear stress and the instantaneous hydrostatic pressure, both of which are deter-
mined from local stress analyses, are used in an equation analogous to Eqs. 7.34 and
7.35. Such a criterion has also been suitably modified for damage and cracking
predictions under in-phase and out-of-phase loading in multiaxial fatigue as well
as contact fatigue. A variation of such an approach has also been suggested by
Papadopoulos et al. (1997) who use volume-averaged stress quantities, ostensibly
to simulate differing crystallographic orientations in a polycrystalline ensemble.
Experimental studies of proportional and nonproportional loading with fully
reversed cycles in a variety of high-strength metals have shown that the ratio of
the fatigue limit in torsion to that in normal/bend loading, re?_i/crB _x ~ 0.57-0.80
254 Stress-life approach

(Zenner, Heidenreich & Richter, 1985; Froustey & Lasserre, 1989). For a phase
difference of 0° between torsional and bending stress cycles, all of the foregoing
theories give essentially the same result. For 90° out-of-phase loading, larger differ-
ences are seen among the different criteria. A quantitative comparison of the pre-
dictions of the foregoing theories with experimental results for in-phase and out-of-
phase loading of high-strength metals is given in Papadopoulos et al. (1997).

Exercises
7.1 The S-N curve for an elastic material is characterized by the Basquin rela-
tionship, o^ — C • Nf, where C is a material constant, <ra is the stress ampli-
tude, Nf is the number of fully reversed stress cycles to failure and b is the
Basquin exponent approximately equal to -0.09. When the stress amplitude
is equal to the ultimate tensile strength of the material in this alloy, the
fatigue life is 1/4 cycle. If a specimen spends 70% of its life subject to
alternating stress levels equal to its fatigue endurance limit ae , 20% at
l.l<re and 10% at 1.2<re, estimate its fatigue life using the Palmgren-Miner
linear damage rule.
7.2 Explain why the modified Goodman diagram can be rewritten in terms of
the endurance limit, cre, as

where cre\am=o is the endurance limit for zero mean stress cyclic loading.
7.3 A circular cylindrical rod with a uniform cross-sectional area of 20 cm is
subjected to a mean axial force of 120 kN. The fatigue strength of the
material, a a = 0%, is 250 MPa after 106 cycles of fully reversed loading
and <rTS = 500 MPa. Using the different procedures discussed in this chap-
ter, estimate the allowable amplitude of force for which the shaft should be
designed to withstand at least one million fatigue cycles. State all your
assumptions clearly.
7.4 A rotating bending machine produces a pure bending moment uniformly
over the gage length of a fatigue specimen. Show that, in a bending specimen
rotating at an angular velocity &>, the cyclic stress will be of the form
Aa = A sin&tf, where A is the peak stress amplitude and t is the time.
7.5 Why is tempered glass more resistant to tensile fracture than ordinary glass?
7.6 A cylindrical shaft of circular cross section is subjected to a bending moment
of constant amplitude over its entire length. The deformation of the shaft
material can be approximated by an elastic-perfectly plastic constitutive
model and the yield strength in tension can be considered equal to that in
compression. If the outer fibers of the shaft yield plastically over a depth of
1/4 of the diameter during the application of the bending moment, qualita-
Exercises 255

tively discuss the development and distribution of mechanical residual stres-

ses along the diameter of the shaft when the bending moment is fully
removed.
7.7 Find the nonzero components of the principal values of the mean stress aim
(i = 1, 2, 3) and the nonzero components of the principal stress amplitudes
<j/a (7 = 1, 2, 3) for uniaxial cyclic loading with R = 0.2.
7.8 A thin-walled cylinder with closed ends is subjected to a steady twisting
moment T, which is oriented parallel to the axis of the cylinder. The internal
radius of the cylinder is rx and its wall thickness is t.
(a) If the cylinder is now subjected additionally to an internal pressure p
which fluctuates between zero and a maximum value, pm2LX, comment on
the variation in the principal stress directions during a complete cycle in
P-
(b) Under what conditions would you expect the principal stresses to be
nearly aligned with the axial and hoop directions?
 CHAPTER 8

Strain-life approach

The information derived from cyclic-stress-based continuum analysis mainly

pertains to elastic and unconstrained deformation. In many practical applications,
engineering components generally undergo a certain degree of structural constraint
and localized plastic flow, particularly at locations of stress concentrations. In
these situations, it is more appropriate to consider the strain-life approach to
fatigue.
In this chapter, strain-based approaches to total fatigue life are discussed for
smooth-surfaced and notched components subjected to constant amplitude and vari-
able amplitude cyclic loading of metals and nonmetals. Also examined are available
methods for fatigue life estimation for multiaxial cyclic loading.

8.1 Strain-based approach to total life

Coffin (1954) and Manson (1954), working independently on thermal fati-
gue problems, proposed a characterization of fatigue life based on the plastic strain
amplitude. They noted that when the logarithm of the plastic strain amplitude,
Aep/2, was plotted against the logarithm of the number of load reversals to failure,
27Vf, a linear relationship resulted for metallic materials, i.e.

^Wf(2tff)c, (8.1)

where ef is the fatigue ductility coefficient and c is the fatigue ductility exponent. In
general, ef is approximately equal to the true fracture ductility ef in monotonic
tension, and c is in the range of —0.5 to —0.7 for most metals. Typical values of
€f and c for a number of engineering alloys are listed in Table 8.1.

8.1.1 Separation of low-cycle and high-cycle fatigue lives

Since the total strain amplitude in a constant strain amplitude test, At/2,
can be written as the sum of elastic strain amplitude, Aee/2, and plastic strain
amplitude, Aep/2,

f =f + ^, (8.2)
256
8.1 Strain-based approach to total life 257

Table 8.1. Cyclic strain-life data for some engineering metals and alloys.

Material Condition ayi (MPa) a[ (MPa) b c

Al alloys*
1100 annealed 97 193 1.80 -0.106 -0.69
2014 T6 462 848 0.42 -0.106 -0.65
2024 T351 379 1103 0.22 -0.124 -0.59
5456 H311 234 724 0.46 -0.110 -0.67
7075 T6 469 1317 0.19 -0.126 -052
Steels*
1015 aircooled 228 827 0.95 -0.110 -0.64
4340 tempered 1172 1655 0.73 -0.076 -0.62
Ti alloys^
Ti-6A1-4V solution-treated + aged 1185 2030 0.841 -0.104 -0.69
Ni-base alloys*
Inconel X annealed 700 2255 1.16 -0.117 -0.75

t Source: Osgood, 1982

* Refers to the monotonic yield strength

the Coffin-Manson relationship, Eq. 8.1, provides a convenient engineering expres-

sion for characterizing the total fatigue life. Using the Basquin equation (Eq. 7.1)
and noting that
(8 3)
^=£=% -
where E is Young's modulus, it is found that

(8.4)
Combining Eqs. 8.1, 8.2 and 8.4, one obtains

(8.5)
The first and second terms on the right hand side of Eq. 8.5 are the elastic and plastic
components, respectively, of the total strain amplitude. Equation 8.5 forms the basis
for the strain-life approach to fatigue design and has found widespread application
in industrial practice. Table 8.1 provides a list of representative data for the stress-
life and strain-life characterization of some commonly used engineering alloys.

8.1.2 Transition life

The variations of the elastic, plastic and total strain amplitudes are plotted
in Fig. 8.1 as functions of the number of load reversals to failure, 27Vf, from Eqs. 8.4,
258 Strain-life approach

27Vf (log scale)

Fig. 8.1. The total strain amplitude versus life curve obtained from the superposition of the
elastic and plastic strain amplitude versus life curves.

8.1, and 8.5, respectively. In order to examine the implications of Fig. 8.1 for 'short'
and 'long' fatigue lives, it is useful to consider a transition life, which is defined as the
number of reversals to failure (27Vf ) t at which the elastic and plastic strain amplitudes
are equal. From Eqs. 8.1 and 8.4,

(2N{\= (-V) . (8.6)

At short fatigue lives, i.e. when 2Nf < (2JVf)t, plastic strain amplitude is more
dominant than the elastic strain amplitude and the fatigue life of the material is
controlled by ductility. At long fatigue lives, i.e. when 27Vf ^> (27Vf)t, the elastic
strain amplitude is more significant than the plastic strain amplitude and the
fatigue life is dictated by the rupture strength. Optimizing the overall fatigue
properties thus inevitably requires a judicious balance between strength and duc-
tility (e.g., Mitchell, 1978).
Mean stress effects have also been incorporated into the uniaxial strain-based
characterization of fatigue life in a simple manner (Morrow, 1968). Assuming that
a tensile mean stress reduces fatigue strength o\, such that (see Eq. 7.6):

the strain-life relationship, Eq. 8.5, can be rewritten (see Section 7.2) as

(8.8)

Cycle-dependent relaxation of the mean stress under strain-controlled fatigue load-

ing is a counterpart of the cyclic creep mechanism discussed earlier (Section 3.9) in
connection with stress-controlled fatigue. Consider a ductile solid which is sub-
jected to a fixed amplitude of cyclic strains, where the mean strain is tensile, Fig.
8.2(<z). If the material exhibits cyclic softening behavior, the flow stress at the
8.1 Strain-based approach to total life 259

A C

Fig. 8.2. Mean stress relaxation in a cyclically softening material subjected to strain-controlled
fatigue.

imposed strain level C is expected to be lower than that at A. With a tensile mean
strain level, the tendency for similar behavior in compression is not significant, and
consequently, the shape of the hysteresis loop for the portion C to D will be
roughly the same as that from A to B. This process results in a progressive
reduction in the mean stress with increasing strain cycling, as shown in Fig.
8.2(6). The rate of decrease in mean stress progressively diminishes as the mean
stress level approaches zero.
Mean stress relaxation can also occur in cyclically hardening materials (e.g.,
Sandor, 1972), although the mechanisms here are less obvious. Cyclic hardening
reduces the plastic strain range and increases the stress range for a fixed total strain
amplitude. With reference to Fig. 8.2(a), the material develops a higher flow stress at
C than at A. However, with a tensile mean stress, the material yields more in tension
than in compression. This preferential plastic straining alters the shape of the hyster-
esis loops in such a way that the stress at point D is lower than that at B, although C
is at a higher stress level than at A. The net result is that a relaxation of mean stress
occurs in a cyclically hardening material as well.
We conclude this section by noting that the low-cycle fatigue behavior of poly-
meric materials is also often characterized empirically using strain-based approaches.
Figure 8.3 shows the variation of total strain amplitude Ae (log scale) as a function
of the number of load reversals to failure 27Vf (log scale) for polycarbonate at 298 K.
Note the similarity of this curve with the corresponding one for metals presented in
Fig. 8.1.
260 Strain-life approach

0.2

0.1
0.08

0.06
Ae
0.04

0.02

0.01
0.008
10° 102 104 106
2N{

Fig. 8.3. Total strain amplitude Ae plotted against the number of load reversals to failure 2Nf
(log-log scale) for polycarbonate at 298 K. (After Beardmore and Rabinowitz, 1975.)

8.1.3 Example problem: Thermal fatigue life of a metal-matrix

composite
Problem:
Refer to the example problem of a metal-matrix composite (Section
3.10.5) which is subjected to thermal cycling. Now consider situations under
which the metallic matrix of the composite undergoes reversed plastic yielding
during thermal cycling.
Let the matrix, a 2024 aluminum alloy, be reinforced with 25 volume % of
spherical SiC particles. The composite is first cooled from a stress-free tempera-
ture of 400 °C to 25 °C. Subsequently, it is thermally cycled between these two
temperature limits. It was determined from experiments and microscopic obser-
vations that damage arising from plastic deformation was more significant for
thermal fatigue life than that from creep, for the particular testing conditions
employed.
(i) Compute the cyclic plastic zone size for one temperature reversal between
40 °C and 25 °C. Assume that the properties of the matrix alloy are the
same as those given in the example problem in Section 3.10.5. In order to
get analytical results, assume for a start that the mechanical properties of
both phases of the composite are independent of temperature for the
range of temperatures and heating/cooling rates employed.
(ii) Find the plastic strain accumulation per thermal cycle between 400 °C and
25 °C.
8.1 Strain-based approach to total life 261

(iii) If the fatigue ductility coefficient, e[ = 0.22, and the fatigue ductility
exponent, c = -0.59 (in Eq. 8.1), find the number of temperature rever-
sals to failure. Identify the location at which failure initiates on the basis
of Coffin-Manson criterion.
(iv) The results obtained in the above three parts are an oversimplification
in that they do not account for the temperature-dependence of material
properties. If the properties vary with temperature in the same manner
as in part (iv) of the example problem in Section 3.10.5, discuss the
effects of such temperature-dependence on your results in items (i)-(iii).
Solution:
It is given that:/ p = (r-Jrof = 0.25, and Tini = 400 °C, r room = 25 °C,
and the applied temperature amplitude (see Fig. 3.16(Z?)),
|AT a | = r ini — r room = 375 °C. It is noted that creep effects are less dominant
compared to plasticity effects. Denoting the properties of the matrix by the
subscript T and those of the particle by the subscript '2' and substituting the
appropriate values in Eq. 3.26, it is found that the elastic mismatch parameter
Mel = 0.35. Substituting the numerical values of the various terms in Eq. 3.28,
we find that the temperature change, from the initial stress-free temperature
(400°C) at which plastic yielding begins is: lAT^I = 174°C, i.e. at 400 -
174 = 226 °C. The temperature change beyond which any reversal in tempera-
ture induces a reversed plastic zone is \AT2\ =2\ATl\ = 348°C, i.e. at 400 -
174 = 52 °C. Since the first cooling is down to 25 °C, subsequent thermal
cycling would induce a reversed yield zone.
(i) Substituting the numerical values of geometrical and material parameters
into Eq. 3.31, and solving iteratively for the reversed plastic zone rc, we
see that that (rc/r{f « 1.06.
(ii) It can easily be shown, from the information provided in Section 3.10.3,
that the maximum plastic strains develop at the particle-matrix interface,
i.e. at r = rx. The second term on the right hand side of Eq. 3.37 can be
used to estimate the rate of plastic strain acumulation per cycle.
Substituting r = rx into this term, we note that the rate of plastic strain
accumulation per cycle (for the given thermal amplitude ATa), is

-1 (8.9)

Substituting the appropriate material properties and the result for rc from
part (i), it is seen that A6pl = 4.4 x 10"4.
(iii) The interface between the particle and the matrix develops the maximum
plastic strain which fully reverses during one complete thermal cycle.
From Eq. 8.1,
262 Strain-life approach

Substituting the values of e[ and c from the given information, and

noting that the plastic strain range during a thermal fatigue cycle is
Aepl = 4.4 x 10~4 and that the plastic strain amplitude is
Aepl/2 = 2.2 x 10~4, from part (ii), it is seen that 2Nf « 36000.
(iv) As discussed in part (iv) of Section 3.10.5, a decrease in yield strength of
the matrix with increasing temperature causes the monotonic plastic
zone to increase. However, the cyclic plastic zone size is not expected
to be affected to a significant extent because of the counteracting effect of
an increasing yield strength during cooling. Therefore, the cyclic plastic
zone size, the plastic strain accumulation per cycle and the total life
would be expected to be affected only in a moderate manner. Precise
estimates of the effects of changing properties during thermal cycling can
only be quantified through numerical solutions of the relevant equations.

8.2 Local strain approach for notched members

The local strain approach relates deformation occurring in the immediate
vicinity of a stress concentration to the remote stresses and strains using the con-
stitutive response determined from fatigue tests on simple laboratory specimens. The
analysis is divided into two steps:
(1) From a knowledge of the imposed loads on a notched component, the local
stress and strain histories at the tip of the notch must be known.
(2) The fatigue life that can be expected for the local stress and strain histories
must be determined.
For the first part, either simple analytical expressions or detailed finite element
simulations of the notch tip deformation (using constitutive laws and hardening rules
described in Section 3.3) are developed to relate the local stresses and strains to far-
field loading. Alternatively, the notch tip deformation is experimentally monitored
with the aid of strain gages or other displacement/strain measurement techniques.
This is perhaps the most challenging aspect of design against fatigue fracture ahead
of stress concentrations. For the second part, the damage accumulation from the
local stress and strain histories must also be estimated so that the safe fatigue life of
the component can be assessed based on the low cycle fatigue properties measured
on smooth laboratory test specimens. For design purposes, it is often more conve-
nient to relate notch tip fields to nominal (far-field) loading by simple engineering
approximations. This section describes some such commonly adapted methods.
8.2 Local strain approach for notched members 263

8.2.1 Neuber analysis

The stress and strain concentration factors are of the same value when only
elastic deformation occurs at the tip of the notch. However, once the material yields
at the notch tip, the stress and strain concentration factors take different values.
(Note that the strain concentration factor K€ is the ratio of the maximum local strain
to the nominal strain.) Under conditions of plastic deformation, the theoretical
elastic stress concentration factor is given approximately by the geometrical mean
of the stress and strain concentration factors, Ka and K€, respectively, as per the well
known Neuber's rule (Neuber, 1961):
Kt = ^ K € . (8.11)
The prediction of fatigue lives for notched members has found increasing use of local
strain approaches, which are modifications and applications of the Neuber's rule to
fatigue loading conditions. In these approaches, the following expressions for fatigue
notch factors are found to provide satisfactory predictions of the fatigue behavior in
notched members of a wide variety of steels (Topper, Sandor & Morrow, 1969;
Dowling, Brose & Wilson, 1977):
Kf = /KX (8.12)
for plastic deformation ahead of notches and

for elastic deformation. In Eq. 8.13, Aa00, Aa and Ae are the amplitudes of the fully
reversed nominal stress, notch tip stress and notch tip strain, respectively. For fixed
values of the imposed stress range Aa°°, Eq. 8.13 is an equation of a rectangular
hyperbola,
AaAe = (KfAa°°)2/E = constant. (8.14)
There is a family of curves, with different combinations of Aa and Ae, which satisfies
this equation. Kf can be uniquely determined by simultaneously solving Eq. 8.13
with the cyclic stress-strain constitutive equation, Eq. 3.5, for the material.
Multiplying both sides of Eq. 3.5 by Aa, one obtains

K,< • (8-15)

From Eqs. 8.14 and 8.15,

(Aa) 2 [Aor] 1/Wf
+ A a = . (8.16)
IE \2K'\ 2E
The factor '2' is introduced in the denominator on both sides of the equation because
the stress amplitude is one-half of the total range, Aa°°, for fully reversed loading.
Figure 8.4 is a schematic diagram of the approach outlined in Eqs. 8.11-8.16.
Since the stress-strain response for the region at the tip of the notch must coincide
264 Strain-life approach

cyclic o-e curve.

" hysteresis curve (Acr vs. Ae)

- AcrAe = constant

Fig. 8.4. Schematic illustration of the procedure used to determine the local stresses and strains
at a notch tip.

with the characteristic cyclic stress-strain behavior of the material, the local stress ax
and the local strain €{ at the notch tip corresponding to a far-field tensile stress o™ is
determined by the intersection of two curves: (i) the Neuber hyperbola represented
by the condition that ae = (K{a°°)2/E = constant, and (ii) the cyclic stress-strain
curve given by e = a/E + (a/Kf)l/rl{. Point Q in Fig. 8.4 represents the local stress-
strain coordinates at the notch tip corresponding to a far-field tensile stress crf°.
If the far-field stress is now reversed to a (compressive) value af°, the stress range
causing this reversal is Aa°° = of0 — a™. The corresponding local stress range and
strain range values for the notch tip are Aa and Ae, respectively. To determine these
values, the origin of the stress-strain coordinate system is now located at point Q.
The stress-strain hysteresis curve obtained from the cyclic deformation tests is now
used in conjunction with the Neuber rule, AaAe = (KfAa°°)2/E = constant, to
locate the stress-strain coordinates for the notch tip for a far-field stress value of
erf0. This is represented by Eq. 8.16 and by the point R in Fig. 8.4.
All subsequent reversals of loading employ Eqs. 8.14—8.16 to determine the local
fields at the notch tip. This local strain approach can easily be implemented in a
computer code.
In practical situations, particular attention has to be paid to the relationship
between accumulated damage under variable amplitude loading and that measurable
in a laboratory specimen under constant amplitude loading conditions. Cycle count-
ing techniques have been developed to reduce complex fatigue loading histories to a
series of discrete events so that cyclic damage could be properly accounted for. A
number of counting techniques have been proposed to accomplish these goals: these
include the so-called rainflow counting, range pair, level crossing and peak counting
methods.
8.3 Variable amplitude cyclic strains and cycle counting 265

The extent to which any of the aforementioned methods for local strain analysis
will provide successful predictions of fatigue life in a material depends on the frac-
tion of fatigue life expended to initiate small flaws ahead of the notch and on the
remaining fraction to propagate this flaw to failure. In addition to the local stress
and strain calculations, fracture mechanics-based analyses of the stress and deforma-
tion fields ahead of notches are of considerable interest in developing a rational
approach to this notch fatigue problem.

8.3 Variable amplitude cyclic strains and cycle counting

In variable amplitude fatigue, cycle counting methods are often employed to
reduce the random load history into a series of discrete events which can be analyzed
using the laboratory data obtained for constant amplitude fatigue loads. Cycle
counting methods are commonly used in many life prediction models in the context
of both local strain approach and defect-tolerant approach. A number of cycle
counting methods have been developed over the years, on the basis of 'trial-and-
error' approaches and 'educated guesswork'. Although empirical in their formula-
tion, such counting methods have been tested (with varying degrees of success) on a
wide variety fatigue-critical engineering components and have been found to provide
useful guidelines for fatigue design. A detailed description of the various cycle count-
ing techniques can be found in the reviews by Dowling (1972), Fuchs & Stephens
(1980), Bannantine, Comer & Handrock (1990) and Dowling (1993). In this section,
an example of the widely used rainflow counting procedure is presented to illustrate
how complex loading histories imposed on a component can be reduced to a series of
stress-strain hysteresis loops.

8.3.1 Example problem: Cycle counting

Problem:
Figure 8.5(a) shows an example of a strain-time loading sequence. The
corresponding stress-time history (derived, for example, from uniaxial fatigue
tests on smooth specimens) is shown in Fig. 8.5(Z?). (Note that a clear functional
relationship does not exist between the stress-time and strain-time plots because
of plastic deformation of the material. As an example, events 3-4 and 5-4' have
identical values of mean strain and strain amplitude. However, the correspond-
ing values of mean stress and stress amplitude are different.) Simplify the strain
history using the rainflow counting method, and suggest strategies for fatigue
life estimate for this variable amplitude cyclic straining.
Solution:
In order to simplify the strain history using the rainflow counting
method, it is more convenient to replot Fig. 8.5(a) with the time axis oriented
266 Strain-life approach

4 4' 2 4 4'

time

(a) (b)

2, 2' 4, 4'

Fig. 8.5. (a) A random strain-time history imposed on a ductile solid, (b) The corresponding
stress-time history, (c) Fig. (a) replotted with the time axis pointing downward, to illustrate
the rainflow technique, (d) Stress-strain hysteresis loops extracted from the rainflow counting
method. (After Landgraf & LaPointe, 1974.)

downward, as shown in Fig. 8.5(c). Imagine now that the lines connecting the
strain peaks are a series of 'pagoda roofs' and that rain is dripping down these
roofs (hence the name 'rainflow counting method'). Several rules are imposed on
rain dripping down the roofs so that equivalent hysteresis loops can be extracted
from the strain history (e.g., Dowling, 1972; Anzai & Endo, 1979). The follow-
ing rules are imposed to define the flow of rain on the roofs: (i) The strain
history is plotted such that the first and last peaks or valleys have the largest
magnitude of strain. This ordering eliminates counting half cycles, (ii) Rainflow
initiates at each peak (such as point 1) and is allowed to drip down continuously.
However, the flow of rain from a peak must stop whenever it drips down a point
which has a more positive strain value than the one from which it drips. For
example, rain dripping down peak 2 must stop opposite peak 4 because the
latter location has a more positive strain value than the former location.
Similarly, the flow must stop when it comes opposite a minimum more negative
than the minimum from which it is initiated. For example, flow from valley 5
must stop opposite valley 6 because the latter location is more negative than the
former, (iii) Rainflow must stop if it encounters rain from the roof above. For
example, during flow from points 3 to 4, rain dripping down from point 2 is
8.3 Variable amplitude cyclic strains and cycle counting 267

encountered and hence flow must stop at point 2'. Note that every part of the
strain-time history is counted once and only once.
Now apply the above rules to the strain history in Fig. 8.5(c). Rainflow begins
on the outside at the highest peak strain and follows the pagoda roof down to
the peak at 2. At this point, the flow drips down to location 2' and continues
down to point 4. From 4, rainflow continues down to location 4' (which has the
same strain magnitude as that at 4). From there, the flow path is along 4' to 6, 6
to 7, and 7 to V. The stress-strain path from this sequence of events corresponds
to the hysteresis loop defined by the circuit 1 —> 4, 4r —> l r , i.e. the outermost
loop in Fig. 8.5(d).
Three additional hysteresis loops can be defined from the rainflow analysis of
the remaining paths. These include paths 2 -> 3 -» 2\ 5 ->> 4' - • 5', and
6 —• 7 -> 6/ in Fig. 5(c). Note that these hysteresis loops are not symmetrical
about the origin of the strain axis.
If the random strain history shown in Fig. %.5(a) were to be repeated m
times, the rainflow method would characterize all of these random loading
events in terms of the four hysteresis loops shown in Fig. 8.5(d), with each
loop repeated m times. Constant amplitude, uniaxial fatigue data can now be
generated on smooth laboratory test specimens using these hysteresis loops
and the material constants b, c, cr'f and ef are determined experimentally.
Mean stress effects on fatigue life are accounted for by recourse to Eq.
8.8, for example, where Ae and a m appropriate for a particular strain cycle
are substituted along with the foregoing material properties to obtain the
number of cycles to failure, 7Vf, representative of that strain cycle. These
results could be used in conjunction with the Palmgren-Miner rule to sum
the fatigue damage. Specifically, this entails the summation of damage for
each strain cycle to obtain

(8-17)

where M ( = 4 in the example shown in Fig. 8.5) is the total number of key
events (in the form of strain cycles) identified from the rainflow counting
method and d is the accumulated fractional damage. Alternatively, the
cycle counting method can be used in conjunction with the life prediction
methods employing variable amplitude crack growth, which are discussed in
Chapter 14.
Computer algorithms for cycle counting are available in the ASTM Annual
Book of Standards section 3, vol. 03.01, 1986 (American Society for Testing and
Materials, Philadelphia). Some simple algorithms have also been published by
Downing & Socie (1982).
268 Strain-life approach

8.4 Multiaxial fatigue

Stress-based phenomenological approaches to total life under multiaxial
fatigue were discussed in the preceding chapter. We now consider multiaxial loading
under low-cycle fatigue conditions where combinations of different loading modes
induce failure which is at least partly dictated by the critical values of different
components of cyclic strains.

8.4.1 Measures of effective strain

In parallel with the definitions of effective stress quantities presented in the
context of multiaxial stress-life approach in the preceding chapter, it is instructive to
examine first the basic definitions of some effective strain quantities in the context of
low-cycle fatigue. Following the discussion in Section 7.10, the effective intensity of
the multiaxial stress/strain state is usually characterized in terms of the conjugate
scalar pairs roct and yoct (i.e. the octahedral shear stress and strain, respectively) in
the von Mises distortional energy theory or in terms of rmax and ymax (i.e. the
maximum shear stress and strain, respectively) in the Tresca yield theory. The effec-
tive strain in multiaxial loading is

where for the fully plastic state, the Poisson's ratio, v = 0.5 and for the elastic state,
v = 0.33 for most metals and alloys. In Eq. 8.18, e1? e2 and e3 are the three principal
strains, with ex > e2 > €3- The effective plastic strain based on the distortional energy
theory is written as

where the superscript 'p' denotes plastic strains. The corresponding effective strain
measures based on the maximum shear strain values are
6
_ /max _ 1 ~ 63 i p _ 2 p 2 , p px
a n a 6
^e - Y ^ - j + y » e - j /max ~ 3 V61 ~ €3)' ^.ZUj

The objective of the theories of multiaxial fatigue is to predict fatigue life under
complex loading conditions in terms of laboratory data of strain-life curves gathered
from uniaxial fatigue tests using simple criteria for failure. If the amplitude of the
maximum principal strain, A^/2, determines failure, Eq. 8.5 may be rewritten to
obtain

^ (2Nf) + 4(2Nfy. (8.21)

In terms of the von Mises criterion, the strain-based expression for multiaxial fatigue
life becomes
8.4 Multiaxial fatigue 269

where the effective strain is denned in Eq. 8.18. Similarly, with the Tresca criterion, it
is useful to note the correlation between axial strain and shear strain,

Combining Eqs. 8.22 and 8.23 and taking v = 0.3 for elastic deformation and v = 0.5
for plastic deformation, it is seen that
(8.24)

The main drawback of these effective strain measures is that they do not adequately
capture the effects of mean stress on multiaxial fatigue life. In an attempt to over-
come this limitation, Smith, Watson & Topper (1970) suggested a simple 'energy-
based approach' to account for mean stress effects. In their approach, multiplying
both sides of Eq. 8.5 by the maximum stress (crmax = crm + {Aa}/2) results in

f ^f2b b
c. (8.25)
This model is predicated on the premise that no fatigue damage occurs when
<jmax < 0, which can be at variance with experimental observations.
These rather strong limitations of the foregoing theories led to considerations of
other approaches, such as the critical plane approach to multiaxial fatigue failure
(Section 7.10), which is taken up in the following subsections specifically in the
context of low-cycle fatigue.

8.4.2 Case study: Critical planes of failure

Conditions governing the identification of critical planes for the onset of fatigue
damage and cracking were discussed in Section 7.10.4 by recourse to the stress-life
approach to high-cycle multiaxial fatigue. In this section, we present a particular case
study which illustrates the existence of critical planes of failure for strain-controlled,
constant-amplitude cycling of an Inconel 718 alloy, which can be predicted on the basis of
the maximum shear strain amplitude. In addition, this example, which is derived from the
works of Socie & Shield (1984) and Socie (1993), clearly illustrates the pronounced effect
of the normal stress on the rate of progression of damage and cracking on the critical
plane.
Consider the three multiaxial loading paths schematically illustrated in Figs. 8.6(a)-(c).
These figures signify three different experiments conducted with different combinations
of axial strains and torsional shear strains on cylindrical tubes of Inconel 718 alloy. The
multiaxial loading in these three experiments is designed such that they all have cyclic
proportional straining with the same maximum shear strain amplitudes. Each specimen
also experiences a static mean strain. The differences between the three experiments are
that the normal stresses and strains across the plane of the maximum shear strain are
270 Strain-life approach

{a) (b) (c)

shear
strain

-70°

Fig. 8.6. (a)—(c) Three different proportional loading paths for the multiaxial straining
experiments which have the same maximum shear strain amplitudes, but different normal
stresses and strains across the planes of maximum shear strain, (d) Mohr's circle construction
showing the orientations of the planes of maximum shear strain, (e) A schematic illustration
of the orientations of the two planes of maximum shear strain. (After Socie, 1993.)

different. Figure 8.6(d) is the Mohr's circle of strain for these cases, where it is seen that
two mutually perpendicular planes are subjected to the same maximum shear strain
amplitudes. The orientations of these planes are schematically sketched in Fig. 8.6(e).
Since the sign of the shear strain has no physical significance, these two planes would be
expected to undergo the same extent of damage if shear strain amplitude alone
determined the evolution of damage. Figures 8.7(a)-(c) show the orientations of the
cracks formed for the three different loading paths sketched in Figs. 8.6(a)-(c),
respectively. It is apparent that the cracking angles seen in Figs. S.l(a)-(c) closely match
the orientations of maximum shear strain planes predicted by the Mohr circle in Figs.
8.6(d) and (e). This also implies in this case that the maximum shear strain plane controls
the evolution of damage and cracking in the Inconel 718 alloy. The loading paths shown
in Figs. 8.6(a) and (c) give rise to crack planes oriented at -70° and + 20°, respectively, as
seen from the angles of cracks in Figs. SJ(a) and (c), respectively. The loading history
shown in Fig. 8.6(6) promotes a crack angle of -20°, as seen in Fig. 8.7(6), because its
loading direction is the reverse of that in the other two cases.
8.4 Multiaxial fatigue 271

am*

Fig. 8.7. (a)-(c) Observations of crack paths for the proportional straining histories shown in
Figs. 8.6.(a)-(c), respectively. (From Socie (1993). Copyright the American Society for
Testing and Materials. Reproduced with Permission.)

It is also apparent from Fig. 8.6(e), however, that the maximum normal strains on the
two critical planes with the highest shear strain amplitude are different. Although the
loading paths followed in the experiments entail proportional straining, the principal axes
of stress and strain are not coincident, and consequently, the maximum normal stresses
on the two planes of maximum shear strain amplitude are not the same. Observations of
subcritical crack growth on the different planes clearly reveal that the maximum shear
strain planes which have the highest tensile normal stress across them exhibit the highest
crack propagation rates and the lowest fatigue lives. Compressive normal stresses inhibit
the advance of cracks, while the tensile normal stresses facilitate crack growth.
The results discussed in this section thus clearly show how maximum shear strain
amplitudes govern the initiation of fatigue cracking and how mean stresses on the planes
of maximum shear influence fatigue crack growth and overall fatigue life. It is thus
evident that the critical plane theories for multiaxial fatigue should incorporate the effects
of both shear and normal components of multiaxial stresses/strains for life prediction.
This case study also serves to provide a mechanistic justification for the various stress-life
criteria described in Section 7.10 and for the additional criteria based on cyclic strains to
be presented in the following sections.

8.4.3 Different cracking patterns in multiaxial fatigue

Brown & Miller (1973) suggest two different patterns for crack growth at
surfaces of materials subjected to multiaxial fatigue loading on the basis of the
orientations of the planes of maximum shear strain amplitude with the free surface.
Consider the situation schematically sketched in Figs. 8.8(a) and (e), where a cubic
element of a material is subjected to multiaxial strain cycles. The free surface plane is
marked in these figures. Figures 8.8(6) and (c) show the planes of maximum shear
strain amplitude wherein a small, stage I crack initiates by single shear process at the
free surface and advances during subsequent cycling. In these figures, the shear stress
acts parallel to the free surface (i.e. the front surface of the cube) and there is no
shear stress acting normally to that surface. That is, the unit normal vectors to the
272 Strain-life approach

Case A

el>€2>€3

surface ^
plane
mmm
id)

CaseB
€l>e2>e:

surface
plane

if) (g) (h)

Fig. 8.8. Schematic illustrations of case A and case B fatigue cracking in multiaxial fatigue: (a)
and (e) show multiaxial strains, (b), (c), if) and (g) denote planes of maximum shear strain
amplitude and the planes and directions of stage I crack growth for case A and case B. (d) and
(h) show the planes and directions of stage II crack growth for case A and case B, respectively.

planes of maximum shear strain amplitude lie on the specimen surface plane. Under
these circumstances, the cracks advance more in a direction parallel to the surface
than normal to the surface, thereby increasing the aspect ratio of the crack. This
pattern of cracking has been termed 'case A' by Brown & Miller. When the case A
cracks become longer, i.e. when their critical dimensions span several grain dia-
meters, stage II crack growth occurs as a result of simultaneous or alternating slip
involving more than one slip system. At this time, the direction of crack advance and
the plane on which it occurs are as shown in Fig. 8.8(<f).
Now consider the possibility shown in Fig. 8.8(^) where the magnitudes of princi-
pal strains are such that the planes of maximum shear strain amplitude occur on the
planes schematically sketched in Figs. 8.8(/) and (g). Here, the stage I cracks initiate
at the surface and advance at 45° angles into the material, and this mode of cracking
has been termed 'case B\ (The fatigue crack growth process along persistent slip
bands (Chapter 4) for ductile single crystals represents case B cracks.) The direction
of stage II crack growth for case B is also from the free surface into the material, as
sketched in Fig. 8.8(A). Uniaxial tension fatigue leads to the same shear stress for
case A and case B and hence it can facilitate either mode of failure. Mixed tension-
torsion fatigue loading, however, invariably promotes case A cracks.
Brown & Miller (1973) postulate that the criterion for cracking for case A and case
B follows the general relationships:
A ymax ,
"=/a (8.26)
8.4 Multiaxial fatigue 273

where Aymax/2 and Aen/2 are maximum shear strain amplitude and normal strain
amplitude, respectively, during the low-cycle fatigue loading, and / a and / b are
different nonlinear functions of their arguments for case A and case B, respectively.
The examples shown in the preceding subsection on Inconel 718 and the work of
Fatemi & Socie (1988) demonstrate that the peak normal stress to the plane of
maximum shear strain amplitude influences the propagation of stage I cracks
under a variety of multiaxial loading conditions that induce case A and case B
cracking.

8.4.4 Example problem: Critical planes offailure in multiaxial

loading
Problem:
The cylindrical tube is subjected to different combinations of fluctuating
torsional moment (AT) and static axial load (P) as well as static internal pres-
sure (/?). On the basis of the discussions presented in Sections 7.10.3, 7.10.4, 8.4.2
and 8.4.3, suggest possible orientations of (a) planes of maximum shear strain
amplitude, (b) planes of maximum normal stress amplitude, (c) planes on which
stage I shear cracks may form and (d) planes on which tensile cracks may form,
and (e) the expected relative fatigue life, for the following four cases of loading.!
(i) Case 1: The thin-walled cylindrical tube which is subjected to cyclic tor-
sion loading only (i.e. AT is nonzero, P = 0 and p = 0).
(ii) Case 2: Repeat the above problem for the situation where a static axial
tensile load is imposed on the cylindrical tube which is subjected to cyclic
torsional loading (i.e. AT is nonzero, P > 0 and p = 0).
(iii) Case 3: Repeat the above problem for the situation where a static axial
compressive load is imposed on the cylindrical tube which is subjected to
cyclic torsional loading (i.e. AT is nonzero, P < 0 and p = 0).
(iv) Case 4: Repeat the above problem for the situation where a static axial
compressive load and a static internal pressure are imposed on the cylind-
rical tube which is subjected to cyclic torsional loading (i.e. AT is non-
zero, P < 0 and p is nonzero).
Solution:
For all the cases considered in this problem, we indicate orientations of
critical planes where the cyclic strains are marked by double-ended arrows and
static stresses are indicated by single-ended arrows. Possible critical planes are
denoted by dashed lines.
(i) Case 1: For pure torsional cyclic loading, the maximum shear strains
occur along the axial and hoop directions. These are also the planes

' This problem is adapted from a discussion in Socie (1993).

274 Strain-life approach

along which stage I, shear cracks (see Sections 7.10.3 and 7.10.4) are
likely to form. The planes on which the tensile normal strains are max-
imum are oriented at 45° to the axis of the tube. Tensile cracking would
be expected to occur on these planes. These results are schematically
sketched in Fig. 8.9.
(ii) Case 2: For torsional cyclic loading with superimposed axial tension,
there develops a superimposed tensile stress on only one of the two
shear planes (i.e. the added tensile stress acts only parallel to the axial
direction). Shear cracking due to stage I would then preferentially occur
only on this plane (see Fig. 8.10), and the overall fatigue life would be
expected to be smaller than that in part (i). Superimposed tension is not
expected to alter the stresses on the vertical shear plane. The superim-
posed static tension would, however, raise the stresses equally on both the
tension planes oriented at 45° to the tube axis. The resulting possibilities
for failure are schematically shown in Fig. 8.10. Thus, the addition of an
axial tensile load to a cyclically twisted cylindrical tube is expected to be
detrimental to fatigue life, irrespective of whether the failure occurs by a
shear mode or a tensile mode.
(iii) Case 3: For torsional loading with superimposed axial compression, one
of the two shear planes (i.e. the plane normal to the applied compression
axis) develops a normal compressive stress and hence is not expected to
develop any cracks. On the other hand, the other shear plane which is
parallel to the applied compression axis can easily develop a shear crack,
and hence the overall fatigue life may not be higher compared to the
shear failure for Case 1 or Case 2. The two 45° planes on which tensile
stresses develop would both exhibit a reduced propensity for cracking as

AT
|
"T
I
\

cyclic shear strain («•—•) cyclic tensile strain (*—*•)

cyclic torsion (AT)

stage I microcracks tensile damage
(shear damage)

Fig. 8.9. A schematic illustration of the evolution of cyclic shear strains, normal strains, shear
damage and tensile damage for cyclic torsional loading of the cylindrical tube, Case 1.
8.4 Multiaxial fatigue 275

f % /

\
/ \
/
cyclic shear strain (•*—») cyclic tensile strain (•*—•>)
static tensile stress (—•) static tensile stress (—••)

cyclic torsion (AT) stage I microcracks tensile damage

static tension (P) (shear damage)

Fig. 8.10. A schematic illustration of the evolution of cyclic shear strains, normal strains,
shear damage and tensile damage for cyclic torsional loading of the cylindrical tube with a
superimposed axial tensile load, Case 2.

a consequence of the applied compressive load. These possibilities are

schematically sketched in Fig. 8.11. One may thus conclude that while
the superimposed compressive load may offer a beneficial effect on fati-
gue life for materials which fail by tensile failure under cyclic torsion, it
may not have any effect on fatigue life for materials which undergo a
stage I shear failure.
(iv) Case 4: For torsional loading with a superimposed static compression as
well as a static internal pressure, a compressive normal stress is induced

AT

cyclic shear strain (•*—») cyclic tensile strain (*—•>)

static compressive stress (—•>)

cyclic torsion (AT) stage I microcracks tensile damage

static compression (-P)

Fig. 8.11. A schematic illustration of the evolution of cyclic shear strains, normal strains,
shear damage and tensile damage for cyclic torsional loading of the cylindrical tube with a
superimposed axial compressive load, Case 3.
276 Strain-life approach

AT

cyclic shear strain (*—••) cyclic tensile strain (•*—••)

static tensile stress
or compressive stress (—••)

cyclic torsion (AT) stage I microcracks tensile damage

static compression (-P) (shear damage)
static internal pressure (p)
Fig. 8.12. A schematic illustration of the evolution of cyclic shear strains, normal strains,
shear damage and tensile damage for cyclic torsional loading of the cylindrical tube with a
superimposed axial compressive load and a static internal pressure, Case 4.

on the horizontal shear plane due to axial compression, and a tensile

hoop stress develops on the vertical shear plane. Thus, shear failure,
should it occur, would preferentially take place on the vertical shear
plane (Fig. 8.12). Because of the superimposed tensile opening stress on
this plane, materials which exhibit a shear mode of failure would be
expected suffer a reduction in the fatigue life. On the two tension planes
oriented at 45°, the stresses induced by the axial load would offset those
stresses arising from the hoop stress; hence the overall tensile stress would
remain the same as in Case 1 on these two planes (Fig. 8.12). Thus the
superimposed static stresses would be expected to have a large influence
on the shear mode of failure while having no effect on the tensile mode of
failure. This situation is thus the opposite of that seen in Case 2.
The situations considered in this problem thus clearly show the need for
carefully examining the evolution of damage and cracking on all the
potential critical planes.

8.5 Out-of-phase loading

As a start to this discussion, the loading path for proportional and non-
proportional loading is schematically illustrated in Fig. 8.13. A cylindrical specimen
which is subjected to combined tension and torsion cyclic loads is shown in Fig.
8.13(#). The variations of axial strain e and shear strain y/V3 with time are illu-
strated in Fig. 8.13(6) for proportional loading. Here, the axial and shear strains vary
8.5 Out-of-phase loading 277

\J time

•3 +

r\
\j time

(a)
(b)

time proportional
loading path
normal strain

'S +
/
90° out-of-phase
\j time
loading path

(c) (d)
Fig. 8.13. (a) A cylindrical specimen subjected to combined axial and torsional fatigue loads, (b)
Strain-time history for proportional loading where the axial and torsional strains are in phase.
(c) Strain—time history for nonproportional loading where the axial and torsional strains are 90°
out of phase, (d) The loading path for in-phase and 90° out-of-phase loading in strain space, for
the strain histories considered here.

in constant proportion to each other and the phase angle between them is zero.
Multiaxial fatigue involving 90° out-of-phase variation between the axial and
shear strains is shown in Fig. 8.13(c). The loading path for proportional loading
corresponds to a straight line in strain space, Fig. 8.13(d), where the axial strain is
plotted against the shear strain. The 90° out-of-phase loading path is a circle in
normalized strain space.
There exists experimental evidence which appears to suggest that in-phase cyclic
straining is more damaging to fatigue life at low strain amplitudes, while out-of-
phase cyclic straining is more damaging at high strain amplitudes, when the ampli-
tudes of the applied tension (or bending) or torsion for proportional and nonpro-
portional loading are comparable. It should, however, be noted that to obtain the
same strain range, the applied normal and shear strains for nonproportional loading
must be increased compared to those for proportional loading. This is because for
27S Strain-life approach

comparable applied strains, the maximum strains are smaller for nonproportional
loading than for proportional loading. When the maximum strains are held compar-
able for the two cases, it is almost always seen that nonproportional loading is at
least as damaging, and generally more damaging, than proportional loading.
Available approaches to handle nonproportional fatigue loading can be broadly
classified into two groups. The first group regards effective values of cyclic stress or
strain without regard to their variations along specific planes or crack growth direc-
tions. For example, Taira, Inoue & Yoshida (1968) integrated the octahedral shear
strains throughout a cycle, in order to circumvent the problem of changes in the
direction of stresses. The second group of methods used to characterize nonpropor-
tional loading considers the conditions on a critical plane, which were discussed in
detail in the preceding subsections.
It is of interest to consider here the multiaxial fatigue experiments of Lamba &
Sidebottom (1978) on tubular specimens of OFHC copper. Their results show that a
plot of the maximum Mises equivalent plastic stress range with the maximum plastic
strain range provides a unique (stable) hysteresis loop for symmetric strain-con-
trolled loading in tension-compression, cyclic torsion or in-phase tension-torsion.
Here the cyclic deformation is independent of the loading direction as long as the
loading direction remains unaltered during the fatigue test (i.e. for proportional
loading).
A completely different picture emerges when the tension and torsion cyclic loads
are 90° out of phase with each other. For a fixed value of maximum plastic strain
range, the out-of-phase fatigue test results in a 40% higher stress level (in the stabi-
lized hysteresis loop) than the proportional or in-phase multiaxial test. Similar trends
have also been seen by Kanazawa, Miller & Brown (1979). This additional hardening
occurs for out-of-phase loading as well as for any changes in cycling direction. These
differences do not depend on whether the fatigue specimen is initially subjected to
uniaxial loading or not. However, this additional hardening obtained in out-of-phase
cyclic loading can be erased if uniaxial fatigue having the same maximum strain
range is imposed after the multiaxial test.

Exercises
A metallic material is shot-peened \n an attempt to improve its fatigue life.
The shot-peening process results in a compressive residual stress of 250 MPa
at the surface of the material. The material has the following monotonic and
fatigue characteristics in the as-fabricated condition (before shot-peening):
E = 210 GPa, ^ = 1000 MPa, o{ = 1100 MPa, 4 = 1.0, n{ = 0.13,* =
-0.08, and c = -0.63. (See the discussions related to Eqs. 3.5, 7.1 and 8.5 for
the definition of these variables.) On a log-log plot, show the variation of the
total strain amplitude Ae/2 with the number of reversals to failure 27Vf for the
material in both the as-fabricated and shot-peened conditions.
Exercises 279

8.2 Consider a double-edge-notched plate of the following dimensions: width

W = 2.54 cm, height H = 3.05 cm, thickness B = 0.25 cm, length of each
notch a0 = 0.254 cm and the net-section, theoretical stress concentration
factor at the notch-tip Kt = 2.4. The monotonic and fatigue properties of
the engineering alloy are as follows: E = 210 GPa, Ar = 1050 MPa,
n{ = 0.1, a\ = 1150 MPa, e'f = 1.15, b = -0.075, and c = -0.7. (These
variables are defined in connection with Eqs. 3.5 and 8.5.) The plate is
subjected to a fully reversed (R = -1) tensile load of maximum value
equal to 44.5 kN. Assuming that the plastic strains developed in the plate
are small, determine the total fatigue life of the notched plate using the
Neuber analysis.
8.3 In the example problem in Section 8.1.3, discuss the possible effects of cyclic
strain hardening on the thermal fatigue life.
8.4 If the compressive axial load in Case 4 of the example problem in Section
8.4.4 were to be replaced by a tensile axial load, suggest possible orientations
of (a) planes of maximum shear strain amplitude, (b) planes of maximum
normal stress amplitude, (c) planes on which stage I shear cracks may form
and (d) planes on which tensile cracks may form, and (e) the expected
relative fatigue life.
Part three

DAMAGE-TOLERANT
APPROACH
 CHAPTER 9

Fracture mechanics audits implications for

fatigue

One of the most successful applications of the theory of fracture mechanics

is in the characterization of fatigue crack propagation. An analysis of fatigue flaw
growth based on fracture mechanics inevitably requires a thorough understanding of
the assumptions, significance and limitations underlying the development of various
crack tip parameters. An important part of such a study of fracture mechanics is the
identification of the regions of dominance of the leading terms of asymptotic crack
tip singular fields. The appropriate conditions for the dominance of critical fracture
parameters are obtained from a knowledge of the accuracy of asymptotic continuum
solutions and from the mechanistic understanding of microscopic deformation at the
fatigue crack tip.
In this chapter, we present a focused discussion of the theories of linear elastic and
nonlinear fracture mechanics that are relevant to applications in fatigue. Details of
the mechanisms of fatigue crack propagation are examined in the following chapters.

9.1 Griffith fracture theory

Modern theories of fracture find their origin in the pioneering work of
Griffith (1921) who formulated criteria for the unstable extension of a crack in a
brittle solid in terms of a balance between changes in mechanical and surface ener-
gies. Consider a through-thickness crack of length 2a located at the center of a large
brittle plate of uniform thickness B, which is subjected to a constant far-field tensile
stress a (Fig. 9.1). Griffith postulated that, for unit crack extension to occur under
the influence of the applied stress, the decrease in potential energy of the system, by
virtue of the displacement of the outer boundaries and the change in the stored
elastic energy, must equal the increase in surface energy due to crack extension.
Using the stress analysis of Inglis (1913) for an elliptical hole in an infinite elastic
plate, Griffith deduced the net change in potential energy of the large plate, shown in
Fig. 9.1, to be

WV = -?*£». (9.1)
where, for plane strain and plane stress, respectively,

E' = -^-~ and E' = E. (9.2)

1 vz

283
284 Fracture mechanics and its implications for fatigue

8a—*\

mi in
Fig. 9.1. A large plate of an elastic material containing a crack of length 2a.

Here E is Young's modulus and v is Poisson's ratio. The surface energy of the crack
system in Fig. 9.1 is
Ws = 4aBYs, (9.3)
where ys is the free surface energy per unit surface area. The total system energy is
then given by
na2a2B A n
u =wP =T- + 4aBYs. (9.4)

Griffith noted that the critical condition for the onset of crack growth is:
2
dU dJ¥j> dWs
+ 2ys = 0, (9.5)
U^i UwH UvH IL

where A = 2aB is the crack area and dA denotes an incremental increase in the crack
area. Note that the total surface area of the two crack faces is 2A. The resulting
critical stress for fracture initiation is

orf = (9.6)
As the second derivative d2 U/da2 is negative, the above equilibrium condition, Eq.
9.6, gives rise to unstable crack propagation.
Griffith's idealized model for brittle fracture considers a sharp crack for which the
near-tip stresses exceed the cohesive strength of the material. In common engineering
materials, nonlinear deformation processes are induced near the crack tip under the
influence of the applied stress. Thus, although the Griffith concept laid the founda-
tion for the physics of fracture, its energy balance considerations cannot be directly
9.2 Energy release rate and crack driving force 285

applied to most engineering solids. Orowan (1952) extended Griffith's brittle fracture
concept to metals by simply supplementing the surface energy term in Eq. 9.6 with
plastic energy dissipation. The resultant expression for fracture initiation is

_ \2Ejy, + yv)
(9.7)
na
where yp is the plastic work per unit area of surface created. Note that yp is generally
much larger than ys.

9.2 Energy release rate and crack driving force

Consider an elastic plate of uniform thickness B, as shown in Figs. 9.2 and
9.3. The plate contains an edge crack of length a. Let the plate be rigidly fixed at the
upper end. The lower end of the plate is loaded with a force F and the displacement
of the load point is u. (One may, in a broader sense, regard the variables F and u as a
generalized force and a generalized displacement, respectively. F and u are work
conjugate variables. For example, when F is taken to be the torque, u is the corre-
sponding rotation.) The total mechanical potential energy of the cracked plate, WP,
is defined as
WF = O - W¥, (9.8)

u + du
displacement

(b)

Fig. 9.2. (a) An elastic plate containing an edge crack subjected to dead weight loading.
(b) Changes in the force-displacement curve and components of mechanical energy during
incremental crack growth.
286 Fracture mechanics and its implications for fatigue

displacement
(a) (b)
Fig. 9.3. (a) An elastic plate containing an edge crack subjected to displacement controlled
loading, (b) Changes in the force-displacement curve and components of mechanical energy
during incremental crack growth.

where O is the stored elastic strain energy and WF is the work done by the external
forces.
Irwin (1956) proposed an approach for the characterization of the driving force
for fracture in cracked elastic bodies, which is conceptually equivalent to that of the
Griffith model. Irwin introduced, for this purpose, the energy release rate Q which is
defined as

(9.9)
Consider the estimation of Q for the following two loading situations.
Case (1): Load-control or dead-weight loading
The cracked plate is subjected to afixedforce F by the application of a dead
weight as shown in Fig. 92{a). In this load-controlled case, the components of
mechanical energy (for a fixed crack length a) are written as
Fu
and W¥ = Fu, (9.10)

where T, in general, is the applied load (which equals a fixed value, F, for the
particular case of dead-weight loading). Combining Eqs. 9.8 and 9.10, it is seen that
Wp = - * = - y . (9.11)
Now consider an increase in crack length from a to a + 8a. This causes a correspond-
ing increase in displacement from u to u + 8u under the fixed force F, as shown in
Fig. 9.2(b). From Eqs. 9.9 and 9.11, the energy release rate for dead-weight loading is
written as
'du\
(9.12)
, IB \% F fixed
9.2 Energy release rate and crack driving force 287

As shown in Fig. 9.2(6), the advance of the crack by an increment 8a (with F fixed)
leads to a net increase in the stored strain energy by the amount

Z Z
F fixed
Case (2): Displacement-controlled loading
Now consider the situation shown in Fig. 9.3, where the displacement u is
controlled and the force F varies accordingly. When the crack advances by an
increment 8a under a fixed displacement w, the change in W? is zero, and hence
= <5O. From Eq. 9.9,
=
or <?=-- — ~y~ — ' (9*14)
u fixed •" L ° ^ J u fixed L ®a J M fixed
As shown in Fig. 9.3(6), the advance of the crack by an increment 8a (with u fixed)
leads to a net decrease in the stored strain energy by the amount

=-f. Z
(9.15,
u fixed
The compliance, C, of a cracked plate, which is the inverse of the stiffness, is
defined as
u
(9.16)
Combining Eq. 9.16 with Eqs. 9.12 and 9.14, it is seen that

&£
IB da
Equation 9.17 holds for both load control and displacement control, i.e. the energy
release rate Q is independent of the type of loading.! This result can also be ratio-
nalized by noting that the magnitudes of the change in stored energy under load
control (Eq. 9.13) and displacement control (Eq. 9.15) differ only by (8F • Su)/2,
which is a negligible quantity. For crack advance by an increment 8a with a given
F and w, therefore,

(9.18)
F fixed u fixed
It is noted that the definition of Q given in Eq. 9.9 is valid for both linear and
nonlinear elastic deformation of the body (see the Section 9.7.1). Q is a function of
the load (or displacement) and crack length, and is independent of the boundary
conditions (i.e. type of loading) for the cracked body. The Griffith criterion for
fracture initiation in an ideally brittle solid can be re-phrased in terms of Q such that

^ 2 y s . (9.19)

' This is to be expected since, as shown later, the critical value of the energy release rate is related to the
fracture toughness, which is a property of a material.
288 Fracture mechanics and its implications for fatigue

9.3 Linear elastic fracture mechanics

While the aforementioned theories regard fracture from an energy stand-
point, the critical conditions for the growth of flaws can be formulated in more
precise terms by means of linear elastic stress analyses. In this section, we provide
a detailed discussion of linear elastic fracture mechanics.

9.3.1 Macroscopic modes offracture

Before considering the variation of the stress and deformation fields in
cracked bodies subjected to external loads, it is appropriate to examine the different
modes of fracture. The crack surface displacements in the three basic modes of
separation are schematically shown in Fig. 9.4. Mode I is the tensile opening mode
in which the crack faces separate in a direction normal to the plane of the crack and
the corresponding displacements of the crack walls are symmetric with respect to the
x-z and the x-y planes. Mode II is the in-plane sliding mode in which the crack faces
are mutually sheared in a direction normal to the crack front. Here the displacements
of the crack walls are symmetric with respect to the x-y plane and anti-symmetric
with respect to the x-z plane. Mode III is the tearing or anti-plane shear mode in
which the crack faces are sheared parallel to the crack front. The displacements of
the crack walls in this case are anti-symmetric with respect to the x-y and x-z planes.
The crack face displacements in modes II and III find an analogy to the motion of
edge dislocations and screw dislocations, respectively.
Irwin (1957), using the analytical methods of Westergaard (1939), quantified the
near-tip fields for the linear elastic crack in terms of the stress intensity factor. Since
the methods of characterizing the propagation of long fatigue cracks under a vast
spectrum of microstructural, environmental and loading conditions are largely based
on linear elastic fracture mechanics, we present here a detailed derivation of near-tip
fields for mode I linear elastic fatigue cracks.

(a) (b) (c)

Fig. 9.4. The three basic modes of fracture, (a) Tensile opening (mode I), (b) In-plane sliding
(mode II). (c) Anti-plane shear (mode III).
9.3 Linear elastic fracture mechanics 289

9.3.2 The plane problem

Consider a semi-infinite crack in an infinite plate of an isotropic and homo-
geneous solid (Fig. 9.5) in an attempt to develop crack solutions for plane strain and
generalized plane stress. For the plane problem (i.e. for modes I and II), the in-plane
displacements ur and ue in the radial and angular coordinate directions, respectively,
of a crack tip element are functions only of the polar coordinates r and #, centered at
the crack tip. In the absence of body forces, the equilibrium equations (see Section
1.4) in polar coordinates are
forr \ <forl
dr r dO

(9.20)

where r and 0 are the polar coordinates shown in Fig. 9.5. The in-plane strain
components are related to the in-plane radial and angular displacements according
to
dur
€„ —•

ur 1 due
= +
7 7 Ho'
= /T\ H dur due _ M
(9.21)

The condition of strain compatibility (in polar coordinates) requires that

€00 2 few 1 d2€rg 1 der^ J_ de^ _ j _ 96^ _
2 2
(9.22)
dr r dr r drdO r dO r2 dO2
The components of stress and strain in the r-6 plane are related by Hooke's law
wherein, for plane stress (azz = 0),

Fig. 9.5. Coordinate system and stresses in the near-tip region of a crack in a plate.
290 Fracture mechanics and its implications for fatigue

Eerr = arr - vaoe,

E^ee = °ee — vam
2fier0 = \xyrQ = are, (9.23)
where \i is the shear modulus and, for plane strain (ezz = 0),
2/X6 rr = (1 - V)On - VCTM,

= (1 - v)aee - varr,
= or9. (9.24)
For the plane problem, the equations of equilibrium, Eqs. 9.20, are satisfied when the
stress components are expressed by the Airy stress function x through

The compatibility condition, Eq. 9.22, when expressed in terms of the Airy stress
function, satisfies the biharmonic equation,

V ^ Z ) = O, V > = £ + I A+ - L | 1 . (9.26)
The boundary conditions for the plane problem of the plate containing a traction-
free crack are
aee = ar0 = 0 for 0 = ±n. (9.27)
The choice of the Airy stress function for the present crack problem should be such
that x n a s a singularity at the crack tip and is single-valued. A possible form of /
which satisfies this requirement is
x = ftp(rj0) + q(r,0), (9.28)
where p and q are harmonic functions of r and 6 which satisfy the Laplace equations
V2p = 0 and V2q = 0.
Following the approach of Williams (1957), we consider solutions of separable
form for the Airy stress function, x — ^(r)©(#)? based on
p = Air* coskO + A2rx sinkO,
q = Blr{x+2)cos(k + 2)6 + B2r{x+2) sin (k + 2)0, (9.29)
which lead to
X = r(x+2) [Ax cos kO + Bx cos (k + 2)0]
+ r{M)[A2 sin kO + B2 sin (A + 2)0]. (9.30)
This equation consists of a symmetric part (the term within the first set of brackets
on the right of the equality sign) and an anti-symmetric part (the term within the
second set of brackets on the right of the equality sign). The symmetric part, which is
an even function of 0, provides the mode I solution for crack tip fields and the
antisymmetric part, which is an odd function of 0, provides the mode II solution.
Taking only the first term here to obtain the mode I fields,
9.3 Linear elastic fracture mechanics 291

d2v
-± z = (X + 2) (X + 1) rk[Ax cosXO + Bx cos (X + 2)6>],
dr
d (\ dX

= (X + 1) / [ ^ ! sinA.6> + (A + 2 ) ^ sin (X + 2)0]. (9.31)

Now applying the boundary conditions, Eq. 9.27, one obtains
(Ai + BX) COSAJT = 0,
[XAX + (A.+ 2 ) ^ ] sinA.7E = 0. (9.32)
The admissible cases are: (i) COSAJI = 0 and hence

X = ^ l . (9.33)
where Z is an integer including zero, and

B A (9 34)
'
or (ii) sin Xn = 0 and hence
A, = Z and ^ = -Ax. (9.35)
Since the governing equations 9.20-9.24 are linear, any linear combination of the
admissible solutions also provides a solution. Hence, from Eqs. 9.33-9.35,

*=f, (9.36)
where Z is a positive or negative integer, including zero. Although, from a purely
mathematical standpoint, there is no basis to reject any value of A, the solution can
be chosen to be of the lowest order singularity which is consistent with physical
arguments. From Eqs. 9.31, it is seen that otj ~ / and e(j ~ / . Therefore, the strain
energy density is given by

<t> = \°ijtij - r2\ (9.37)

The total strain energy within any annular area of inner and outer radii r0 and R,
respectively, centered at the crack tip, is
'2n rR i o2n oR

J - a^ij r dr dO ~ r(2A+1) dr 60. (9.38)

0 Jr0 ^ JO Jr0
Invoking the argument that <& should be bounded (i.e. O < oo) as r0 ->• 0, we see that
X > — 1. (X = — 1 gives the trivial solution that atj = 0.) Note that, since the displace-
ments Uf ~ r (A+1 \ the boundedness of displacements also requires that X > — 1. Thus,
the physically admissible values of X are
A = - ^ , 0, i , 1, ^ , 2 , . . . , (or) X= |, (9.39)
where Z is - 1 , 0, or a positive integer. Taking the most dominant singular term
represented by X = —1/2 (and Bx — Ax/3),
292 Fracture mechanics and its implications for fatigue

X = [cos^ + I c o s f ] + o ( r 2 ) + O 0 - 5 / 2 ) + "--,
(9.40)
The second term on the right hand side of Eq. 9.40, with an exponent of 0 for r, is a
nonsingular, but nonvanishing, term. The higher order terms, with exponents greater
than zero, vanish as r —• 0. Rewriting Ax =

TSix8Jx

(terms which vanish at crack tip), (9.41)

where there is no summation over x in the second term on the right hand side. K\ is
known as the stress intensity factor for mode I loading. 8y is the Kronecker delta
defined in Section 1.4.1. In terms of the in-plane stress components,

K5\M alyy(e)/
+ (terms which vanish at crack tip), (9.42)
where the first term is the leading singular term for linear elastic mode I crack
problems. The second term, generally referred to as the 'T term', contains the non-
singular stress GXX — T (Williams, 1957; Irwin, 1960; Larsson & Carlsson, 1973; Rice,
1974). For example, a brittle crack of length 2a lying on the x-z plane under
remotely uniform biaxial stresses a^x and a™, is subjected to
Kl = ay^y^Ka' and T = axxx-cr^r (9.43)
Although the leading singular term of the asymptotic solution, Eqs. 9.41-9.43, is
adequate for characterizing most linear elastic fatigue crack growth problems, the
omission of the T-stress can introduce significant errors in certain fatigue situations.
Examples of such situations include: (i) short fatigue cracks, (ii) cracks subjected to
mixed-mode loading where the in-plane shear stresses are substantially larger than
the tensile stresses, and (iii) small cracks inclined at a small angle to the far-field
tensile axis. Furthermore, different geometries of cracked specimens can influence
the near-tip yield behavior in different ways because of the differences in the T-stress
term. This effect and the attendant influence on fatigue crack closure are considered
in Chapter 14.
For the plane problem, the leading terms for mode I stress fields in cartesian
coordinates are
. 0 . 30
1 — s i n - sin —

. 0 . 30
cos- 1 + sin- sin — (9.44a)
. 0 30
sin- cos —
9.3 Linear elastic fracture mechanics 293

When written in cylindrical coordinates, the stressfieldsfor mode I have the follow-
ing leading terms:
1 + sin 2 -
0
cos- 2o
2 cos -
Ore .00
sin- cos-
= vx(prr

9z = 0- (9 Mb)
The corresponding displacements are
0 30
(2K- l)C0S--C0Sy

IE
(2K + 1) s i n - - s i n —

0 30
(2K — 1) cos - — cos —

IE n , fi . 30
(1+v) -(2K- 1) sin-+ sin —

Uz = (9.45)
"
For plane stress,
(3-v)
K = vx = 0 , v2 = v,
"(1+v)'
and, for plane strain,
K = (3- 4v), vi = v, v2 = 0. (9.46)
The term KY in Eqs. 9.44 and 9.45 is the mode I stress intensity factor which incor-
porates the boundary conditions of the cracked body and is a function of loading,
crack length and geometry. For plane problems, it is independent of the elastic
constants.
The near-tip fields for mode II can be derived in a similar fashion by applying the
boundary conditions, Eq. 9.27, to the antisymmetric part of the Airy stress function,
Eq. 9.30. The resulting asymptotic solutions for mode II are:
.Of 0 30
— sin- 12 + cos- cos —
. 0 0 . 30
sin- cos- sin — (9.47a)
2nr
0 (. . 0 . 30\
cos- I I - sin- s m y l

and, in cylindrical coordinates,

294 Fracture mechanics and its implications for fatigue

sin- I 1 — 3 sin2-J

— 3 s i n - cos -
\f2nr
cos- 1 — 3sm -
2 V 2
Gzz = VX{GXX + cryy) = VX((7rr

a
xz — ayz — arz ~ °Qz =
0. (9.476)
0 30
(l+v) I (2*:+ 3)sin- + sin —
0 30
—(1+v) |(2/c —3)cos- + cos—-

0 36"]
(l + v) i 0 i
36
ur (2/c—l)sin- + 3sin —
ue IE V2n 9 39~\
(l + v) -(2/c+l)cos- + 3cos

(9.48)

For mode III, it can be shown (e.g., Anderson, 1995) that

— sin-

^ «!

sin-

cos-j

xx yy ff 00 zz '

axy = are = 0 . (9.49)

ux = uy = ur = ue = 0. (9.50)
The above singular solutions for all three modes of fracture indicate that the
stresses and displacements, respectively, are of the form

(9.51)
9.3 Linear elastic fracture mechanics 295

where the subscript M refers to the modes of failure, I, II, and III. The appropriate
stress intensity factor for each mode is denned as

= lim <V2Jtrcryy
r-»0 0=0

= lim <
0=0

= l i m < V2nrcryz (9.52)

0=0)
Stress intensity factor calibrations for a wide range of specimen and crack
geometries, and loading conditions can be found in the Appendix.

9.3.3 Conditions of K-dominance

The stress intensity factors in Eqs. 9.51-9.52 are a measure of the intensity of
the near-tip fields under linear elastic conditions. The radial component term \flnr
and the angular component term 6^(0) in Eq. 9.51 depend only on the spatial
coordinates. These terms determine the distribution of the near-tip fields.
There exists an annular zone ahead of the crack tip, known as 'the region of K-
dominance', within which the stress intensity factor provides a unique measure of the
intensity of stress, strain or deformation. The outer radius of the annular zone is
determined by the radial distance at which the approximate, asymptotic singular
solutions Eqs. 9.44-9.50 deviate significantly (say, by more than 10%) from the
full elasticity solutions which include the higher order terms, e.g., the T-term and
the nonsingular terms in Eqs. 9.40-9.43. The higher order terms need to be included
beyond the annular zone because they begin to influence the overall accuracy of the
solutions. The foregoing near-tip solutions were determined on the basis of the
assumption of a sharp crack (of tip radius -> 0). However, the ^-concept holds
even when the crack is not sharp and when there is nonlinear deformation in a small
zone near the crack tip. In ductile solids, the material at the crack tip yields when the
near-tip stresses exceed the flow strength and the linear elastic solutions lose their
validity within this plastic zone. Even if the crack tip plastic deformation zone is very
small or nonexistent (as, for example, in the case of brittle ceramics), the foregoing
continuum solutions are not expected to hold within the near-tip region of intense
deformation (such as the zone of microcracking or phase transformation). Thus, the
inner radius of the region of ^-dominance is dictated by the size scale of microscopic
failure processes (generally referred to as the 'process zone'). The near-tip fields
within the plastic zone will be considered in Section 9.7.
The usefulness of the ^-fields to characterize the onset or continuation of crack
advance in materials that undergo inelastic deformation, such as plasticity, creep,
microcracking or phase transformations, is predicated on conformity to the so-called
'small-scale yielding' condition. This condition requires that the crack tip zone of
inelastic deformation, whatever its origin, be confined well inside the region of
296 Fracture mechanics and its implications for fatigue

^-dominance over which the asymptotic results, Eqs. 9.44-9.50, provide a reason-
able approximation to the full solution. As alluded to earlier, an understanding of
the conditions of ^-dominance is essential for the characterization of fatigue fracture
involving highly crystallographic crack growth, mixed-mode loading conditions or
small fatigue flaws. A detailed discussion of each of these cases will be taken up in
later chapters.

9.3.4 Fracture toughness

In linear elastic fracture mechanics, the initiation of crack advance under
monotonic, quasi-static loading conditions is characterized by the critical value of
the stress intensity factor, Kc. The value of Kc is a function of the mode of loading,
the chemical environment, the material microstructure, the test temperature, strain
rate, and the state of stress (i.e. plane stress or plane strain). The experimental test
specimens used for the determination of critical stress intensity factor must conform
to the requirements of small-scale yielding and other conditions of ^-dominance
which are spelled out in detail in the fracture test standard E-399 developed by the
American Society for Testing and Materials (Philadelphia) in 1974. The critical value
of the mode I stress intensity factor measured under plane strain conditions is
commonly referred to as the fracture toughness, Kic, of the material at the particular
test temperature. (Plane strain conditions are assumed to exist in the fracture tough-
ness test specimen when the thickness of the test specimen is at least about 25 times
the monotonic plastic zone size, which is denned in Section 9.5.) The corresponding
fracture toughness values in the sliding and tearing modes are designated as Kuc and
KiuC9 respectively.

9.3.5 Characterization of fatigue crack growth

Under cyclic loading conditions, the onset of crack growth from a pre-
existing flaw or defect can occur at (maximum) stress intensity values that are well
below the quasi-static fracture toughness. For conditions of small-scale yielding,
where the nonlinear zone at the crack tip is a mere perturbation in an otherwise
elastic material, Paris, Gomez & Anderson (1961) and Paris & Erdogan (1963)
postulated that the growth of a crack under cyclic loading should be governed by
the 'law',

^ = CAKm, (9.53)

where da/dN is the change in the length of the fatigue crack per load cycle (a is the
crack length and N is the number of fatigue cycles) and AK is the stress intensity
factor range denned as
= Kmax-Kmm. (9.54)
9.4 Equivalence of Q and K 297

KmzLx a n d ^min? respectively, are the maximum and minimum stress intensity factors
corresponding to the maximum load, P m a x (or maximum nominal stress, crmSiX) and
the minimum load, Pmin (or minimum nominal stress, crmin). Recall that
I m a x = YcrmSLxy/na and Kmin = Ycrmin^/na for a center-cracked plate containing a
crack of length 2a which is subjected to tensile fatigue with a far-field stress range,
ACT = crmax — <rmjn. Y is the finite size correction factor for the plate. The terms C and
m in Eq. 9.53 are empirical constants which are functions of the material properties
and microstructure, fatigue frequency, mean stress or load ratio, environment, load-
ing mode, stress state and test temperature. The empirical crack growth law, Eq.
9.53, due to Paris et al. is the most widely used form of characterizing fatigue crack
growth rates for a vast spectrum of materials and test conditions. Equation 9.53 also
represents one of the most useful applications of the theory of linear elastic fracture
mechanics. Further details of the fracture mechanics-based approach to characteriz-
ing fatigue will be considered in subsequent chapters.

9.4 Equivalence of Q and K

The stress intensity factor approach to fracture has a direct equivalence to
the energy approach. Consider the definition of energy release rate for the displace-
ment boundary value problem involving the fixed-grip case, Eq. 9.17. Here, the
change in strain energy (per unit thickness of the crackfront) as the length of the
crack is increased from a to a + 8a is given by

r +8a i
^{cfyyUy ~\~ O%yU% ~\~ O,yU ^ AX.

The factor 2 before the integral sign appears because of the displacement of the two
(9.55)

opposing crack surfaces, and the factor 1/2 after the integral sign is introduced
following the assumption of proportionality between the stresses and displacements.
Substituting the appropriate stresses atj- forr = x — a and 0 = 0 and displacements ut
for r = a + da — x and 6 = 0 from the previous section into Eq. 9.55, it is readily
seen, in the limit of 8a -> 0, that

= Gi + Gn + Gin. (9.56)
da
The energy release rate G and the stress intensity factors Kh Ku and Km in the three
modes of fracture are uniquely related. For the general three-dimensional case
involving plane strain and anti-plane strain loading,
2
(1 - V2)) , 2 , K2\ , (1+V) K2 rQ c?x
y —— ^ — \Ki +AII) H ^ — ^III K7-5/)
and, for plane stress,
G = ± {Kl + K%). (9.58)
298 Fracture mechanics and its implications for fatigue

Note that, when the crack advances in its own plane, i.e. for self-similar, coplanar
crack growth, the energy release rates for the different modes of fracture are simply
additive. This relationship also provides a means for developing mode-invariant
criteria for the onset of failure under multiaxial loading conditions.

9A.I Example problem: Q and K for the DCB specimen

Problem:
The double-cantilever beam (DCB) specimen is one of the widely used
fracture test specimens for both brittle and ductile solids.
(i) Using the concepts discussed in Section 9.2, calculate the energy release
rate for the DCB specimen subjected to a mode I load as shown in Fig.
9.6.
(ii) Using the result from (i) and Eqs. 9.57 and 9.58, determine the stress
intensity factor for the DCB specimen.
(hi) Discuss how stress intensity factor calibrations may be obtained experi-
mentally for the DCB specimen using measurements of compliance
changes.
Solution:
(i) As illustrated in Fig. 9.6(a), the out-of-plane end deflection u/2 of a
slender cantilever beam (a ^> 2h) of length a and height h subject to an
end load F is given by beam theory:

2 3£7' 12 '
From Eq. 9.17 and Fig. 9.6(c),

2B da ~ BEI ~ EB2h3 ' K

'
(ii) From Eqs. 9.57, 9.58 and 9.60, the mode I stress intensity factor for plane
stress is derived as
K\ UF2a2 a F
G K 2 V (961)
*
and, for plane strain, as
V a F
^

(iii) From Eq. 9.60, for plane strain,

9.4 Equivalence of Q and K 299

2h

±
Y

\IC{d)

displacement, u
(b)

da

Fig. 9.6. (a) The double cantilever beam specimen, (b) A plot of the load versus the
displacement showing the compliance, (c) Compliance change as a function of crack length.
300 Fracture mechanics and its implications for fatigue

where /cf(a) is the stress intensity factor per unit applied load and
c(a) = C(a)/B.
In order to calibrate the stress intensity factor for a DCB specimen, prepare a
number of specimens with different crack lengths, a. Use a clip gage to measure
the opening displacement u at the loading point, shown in Fig. 9.6(#), as a
function of F. For each specimen with a known crack length, plot F versus u
as in Fig. 9.6(c) and the slope C(a). Control the load such that no crack exten-
sion occurs during this experiment. From similar results on multiple DCB speci-
mens, plot c(a) versus a. Find kf(a) and Ki(a) using Eqs. 9.63 and 9.64.

9.4.2 Example problem: Stress intensity factor for a blister test

Problem:
The interface toughness of an adhesive is to be estimated on the basis of a
proposed method which entails the bonding of a thin elastic disk with the
adhesive on to a rigid substrate. Pressure is applied to the bonded side of the
disk through a tiny hole in the substrate, by pumping a fluid. This pressurization
results in the partial debonding of the disk and the formation of a 'bulge' or
'blister' in the disk. This method is hence referred to as the pressurized 'bulge
test' or 'blister test'.
Consider a two-dimensional version of this test which is shown in Fig. 9.7. A
thin beam is adhesively bonded to a rigid surface through which an incompres-
sible fluid is forced resulting in a blister of half-width /. Assuming that all
concepts of linear elastic fracture mechanics may be extended to this case,
apply the compliance method to determine the tensile 'crack-tip' stress intensity
factor in terms of the blister dimension / and the piston displacement q. Assume
that q = 0 when the beam is undeflected, and that all length dimensions normal
to the plane of the figure are unity. Let the thickness of the beam and its elastic
modulus be h and E, respectively, as shown in Fig. 9.7. You are given the
following information.
(i) The transverse deflection of an elastic double cantilever beam of length 2/
subjected to a uniform pressure p is

*y2.
(ii) A measure of compliance c(l) may be defined as q = c(l)Q, where Q is the
force acting on the piston per unit thickness as shown in the figure.
9.4 Equivalence of Q and K 301

Fig. 9.7. A schematic of the pressurized bulge or blister test and the associated nomenclature.

I w(x)dx = qH.

(iv) From the compliance interpretation of the energy release rate, Q, it is

(9.66)

known that

(9.67)
* - % * •

Solution:
The compliance of the system is defined as
q = c(l)Q. (9.68)
Observe that (i) the pressurized liquid is incompressible, (ii) the pressure under
the blister is equal to the pressure in the piston, and (iii) the pressure in the
piston is directly related to Q by

(9.69)
Substituting Eq. 9.65 in Eq. 9.66, we obtain
_ 15 EqHh3
P (9.70)
~T~~J~~'
Now substitute Eq. 9.69 into Eq. 9.70 to get
8 1 /5 ^
(9.71)
Arranging this equation into the form of Eq. 9.68, we note that
302 Fracture mechanics and its implications for fatigue

(9 72)
-
The compliance interpretation of Q is given by Eq. 9.67. Noting that there are
two crack tips at the blister in Fig. 9.7 and taking the result from Eq. 9.72,

But q = c(l)Q. With this result, Eq. 9.73 is rewritten as

For plane stress, we take this equation with the link between KY and Q to get

I 32^ f> V
Young's modulus E appears in this equation because KY has been written in
terms of the displacement q. Instead, if use is made of Eq. 9.71, the stress
intensity factor becomes
[2 I2
(976)

This equation gives us the desired result.

9.5 Plastic zone size in monotonic loading

9.5.1 The Irwin approximation
Estimates of the boundary of the plastic zone ahead of a crack in a ductile
solid can be derived by considering the crack tip zone within which the von Mises
equivalent stress (which is defined in Eq. (1.22) and calculated from the stress ana-
lysis of Section 9.3) exceeds the tensile flow stress ay (Irwin, 1960). The extent of the
plastic zone rp ahead of the crack tip (0 = 0) is proportional to the square of the
stress intensity factor. In mode I,

1 (K \
rnp = — (— 1 , for plane strain,
3TC \ayj
1 / jr \ 2
rp = - ( — ) , for plane stress. (9.77)
n\ayj
Precise analyses of the plastic zone size and shape in modes I and II in strain-
hardening solids are discussed in Section 9.10 for plane strain and plane stress.
Following similar arguments, the plastic zone size ahead of a mode III crack is
found to be
9.5 Plastic zone size in monotonic loading 303

(9.78)

where xy is the shear yield stress of the material.

9.5.2 The Dugdale model

The size of the yield zone ahead of a mode I crack in a thin plate of an
elastic-perfectly plastic solid (subject to plane stress deformation) was estimated by
Dugdale (1960). In the Dugdale model, the plastic region is envisioned as a narrow
strip (of near-zero height) which extends a distance rp ahead of the crack tip and is
loaded by the traction ayy = oy over the length rp (Fig. 9.8). If ayy were to be zero
over the whole length \x\ < a + rp, y = 0, a far-field tensile stress cr°° would produce
a positive stress intensity factor KY = a^^/nia + r p ) (in an infinitely large plate). If
the traction ayy = ay were to be applied simultaneously along the length of the strip
a < \x\ < a + r p , it would superimpose a negative stress intensity factor KY on KY
where

(9.79)

The requirement of bounded stresses at the point x = a-\- rp provides the condition
that Ki + Ki =0. Solving for r p , one finds that
r /ncr°°\
-^p = sec - — - 11. (9.80)
a \2ayJ
For cr°° <3C oy and hence for rp <^C a, this equation asymptotically leads to a plastic
zone size

t t t t

i i i i i
Fig. 9.8. A schematic representation of the Dugdale plastic zone model.
304 Fracture mechanics and its implications for fatigue

(9.81)

This asymptotically exact result due to Dugdale compares well with the Irwin
approximation, Eq. 9.77, for plane stress.
In the above model, one notes that an opening displacement 8 = 2v(a) at x = ±a
and y = 0 exists (which may be regarded as a consequence of necking ahead of the
crack). It can be shown that the crack tip opening displacement takes the form
sec
(^H (9-82)
or asymptotically, when a°° «(T y ,

(9.83)
CTyE

9.5.3 The Barenblatt model

The strip zone model due to Barenblatt (1962) provides an analogue for
ideally brittle materials of the Dugdale plastic yield zone analysis. For the perfectly
brittle solid, one may envision, in Fig. 9.8, that the crack face traction oyy = orth,
where ath is the theoretical bond rupture strength « E/10 (see, for example, Lawn,
1993). The critical condition for the fracture of the brittle solid may then be
expressed in terms of the critical size of the cohesive zone at the crack tip,
rp = rc0, or in terms of the critical crack opening displacement 8C = 2vc (Rice,
1968) such that

\ ^ = 2ys. (9.84)

9.6 Plastic zone size in cyclic loading

The existence of a reversed plastic zone ahead of a fatigue crack has long
been recognized (e.g., Paris, 1960; McClintock, 1963; Rice, 1967). The stresses within
the reverse yield zone at the fatigue crack tip can be estimated from the analysis of
Rice (1967). Consider a cracked plate of an elastic-perfectly plastic solid which is
subjected to a far-field tensile load P. The yielding of the material at the crack tip
under the influence of this load creates a monotonic plastic zone of dimension rp
given by Eq. 9.66 and shown in Fig. 9.9(a). For large plane strains and well-devel-
oped plane stress yielding, one may expect proportional plastic flow to hold, i.e. the
components of the plastic strain tensor are expected to remain in constant propor-
tion to one another at each point of the plastic region. If the load P is reduced by an
amount AP to a tensile load P — AP, reverse plastic flow is instigated. If the extent
of crack tip blunting is assumed to be negligible, the infinitely large stress concentra-
9.6 Plastic zone size in cyclic loading 305

(*,0)

-2ay
(a)

• \P-AP
, 0)
cyclic plastic zone
monotonic plastic zone

\ \\
Fig. 9.9. Schematic representation of the development of cyclic plastic zone upon unloading.
(After Rice, 1967.) (a) Monotonic plastic zone created by a far-field load P. (b) Stress
distribution due to the reduction of the load by AP which, when superimposed with (a), gives
the result in (c).

tion factor at the tip of the sharp crack leads to the formation of a reversedflowzone
which is embedded within the monotonic plastic zone.
For proportional loading, the changes in the near-tipfieldsdue to the reduction of
the load are given by the solution derived earlier for monotonic loading (Section 9.3)
with the exception that the loading parameter is replaced by the load range AP and
that the yield stress and strain are replaced by twice their values corresponding to the
load P. This modification is introduced to obtain the correct values of stresses in the
reversed flow zone after subtracting the changes due to the load reduction AP, Fig.
9.9(6). If closure of the crack faces is not encountered, the superposition of the near-
306 Fracture mechanics and its implications for fatigue

tip stresses in the fully loaded state, Fig. 9.9(<s), and in the partially unloaded state,
Fig. 9.9(&), leads to the stress distribution at the tip of the crack under the far-field
load P — AP, Fig. 9.9(c). Thus, for a crack which is only partially unloaded from a
far-field tensile load, there exists within the monotonic plastic zone a region of
reversed flow (termed the 'cyclic plastic zone') of size rc in which residual compres-
sive stresses are induced. For an elastic-perfectly plastic solid undergoing propor-
tional flow, the stress within the cyclic plastic zone is equal to the flow stress in
compression (—ay). The size of rc is derived by replacing KY by AKi and ay by —2ay
in Eq. 9.66 so that, for plane stress,
I/A*
(9.85)
n\2a
For zero-tension-zero loading, AKj = Ki and rc = r p /4. For materials which
cyclically harden or soften, ay in Eq. 9.85 should be replaced by the cyclic yield
strength, oy.
There are some interesting consequences of reversed plastic flow during unloading
from a far-field load:
(1) Even after the far-field load is fully removed, there exists a zone of residual
compressive stress ahead of the fatigue crack which has (previously) been
subjected to far-field cyclic tension. This residual stress zone can have
important implications for transient crack growth phenomena observed
under variable amplitude fatigue (see Chapter 14).
(2) Since residual stresses are self-equilibrating, the residual compressive
stresses at the crack tip must be offset by residual tensile stresses away
from the crack tip.
(3) If a nonclosing flaw (such as a sharp notch) is unloaded from a far-field
compressive load, reversed plastic flow at the notch-tip creates residual
tensile stresses. This zone of residual tension can induce stable mode I
fatigue crack growth in notched plates loaded in uniaxial cyclic compression
(see Chapter 4).
(4) A cyclic variation in the stress intensity factor A^ I ? from 0 to Kx, gives rise
to a cyclic crack tip opening displacement A<5t, which is one half of the total
opening displacement 8 under a monotonic stress intensity factor Ki9 Eq.
9.83:
\K2
A5t^—L. (9.86)
layh
(5) In the absence of crack closure, the value of rc and cyclic variations in
stresses, strains and displacements depend only on AP and are independent
of the maximum load P.
(6) When crack closure does not occur, the plastic superposition is valid up to
the point when rc = rp, that is, for complete load reversal involving equal
tension-compression fatigue.
9.7 Elastic-plastic fracture mechanics 307

Whereas the cyclic plastic zone size for a stationary fatigue crack in an elastic-
perfectly plastic solid is one-quarter the size of the monotonic plastic zone, a plane
stress analysis by Budiansky & Hutchinson (1978) shows that the reversed yield zone
for an extending fatigue crack is less than 10% of the Dugdale monotonic yield zone
size. This point is taken up for further discussion in Chapter 14. Analyses of cyclic
damage zones ahead of fatigue cracks in ceramics are discussed in Chapter 11.

9.7 Elastic-plastic fracture mechanics

The stress intensity factor K provides a unique characterization of the near-
tip fields under small-scale yielding conditions, while the corresponding loading
parameter for the characterization of monotonic, nonlinear fracture in rate-indepen-
dent materials is the /-integral proposed by Rice (1968). Although some of the
features of this integral were embedded in the energy concepts derived by Eshelby
(1956) and were discussed independently by Sanders (1960) and Cherepanov (1969),
the particular form of this line integral proposed by Rice (1968) has led to the
unifying theoretical basis for nonlinear fracture mechanics.

9.7.1 The J-integral

Consider a cracked body subjected to a monotonic load, Fig. 9.10.
Assuming that the tractions T are independent of crack size and that the crack

Fig. 9.10. A contour around a crack tip and the nomenclature used in the definition of the /
integral.
308 Fracture mechanics and its implications for fatigue

faces are traction-free, the line integral / along any contour T which encircles the
crack tip is given by
(987)
fA -
where u is the displacement vector, y is the distance along the direction normal to the
plane of the crack, s is the arc length along the contour, T is the traction vector and w
is the strain energy density of the material. The stresses atj are related to w by the
relation a^ — dw/dey. For a material which is characterized by linear or nonlinear
elastic behavior (i.e. by deformation plasticity), / is independent of the path T taken
to compute the integral.
Rice (1968) showed that / is the rate of change of potential energy (with respect to
crack advance) for a nonlinear elastic solid and that J reduces to the energy release
rate Q for a linear elastic material:

(9.88)
da
If F is regarded as the contour which just encircles the cohesive zone in the
Barenblatt model, it is found that / is equal to the energy release rate given in Eq.
9.84: / = Qc = 2ys. Furthermore, if the line integral is applied to the Dugdale model,
the following relationship between / and the crack tip opening displacement Su Eq.
9.83, is obtained:
/ = ay8t. (9.89)

9.7.2 Hutchinson-Rice-Rosengren (HRR) singular fields

Consider an elastic-power law plastic material whose uniaxial constitutive
response is characterized by the Ramberg-Osgood relationship, Eq. 3.4. Since the
elastic strains are negligible compared to the power law terms near the crack tip, Eq.
3.4 can be approximated by the pure power law, e/ey = a{o/oy)n, where ey = ay/E.
This relationship, when generalized to multiaxial stress states using the J2 deforma-
tion theory (see Section 1.4.3), can be written as

For nonlinear elastic solids undergoing monotonic, small-strain deformation,

Hutchinson (1968) and Rice & Rosengren (1968) showed that the strength of the
near-tip fields is the /-integral and that the stresses, strains and displacements exhibit
r-i/(*+i)9 r-n/{n+\) a n d ri/(/i+i) s m g u l a r i t y ? respectively. These so-called 'HRR fields'
are written as
9.7 Elastic-plastic fracture mechanics 309

, n),

aayeylnrj
J__\ rl/(n+D Qfr ny (991)
otayeylj
The universal functions 6^.(0, «)5 ^(0, w), and ut{0, n) in Eqs. 9.91 vary with the polar
angle 0, the strain hardening exponent n and the state of stress, i.e. plane stress or
plane strain. The factor /„ depends mildly on the strain hardening exponent n.
Since / is a measure of the intensity of crack-tip fields, the onset of crack advance
under quasi-static loads can be formulated on the basis of a critical value, J = Jc.
When conditions of/-dominance (see Section 9.7.4) are satisfied in the plane strain
test specimen, the measured critical value of plane strain fracture toughness is
denoted by / I c . Detailed test procedures for the experimental measurement of / I c
are spelled out in the test standard E-813 developed by the American Society for
Testing and Materials (Philadelphia) in 1981. Under linear elastic, plane strain con-
ditions,

/ i c = — (1 - v 2 ) . (9.92)

This relation is used to infer an equivalent Kic value from / I c measurements in high
toughness ductile solids in which valid Kic testing will require unreasonably large test
specimens.

9.7.3 Crack tip opening displacement

Expressions for the crack tip opening displacement 8U Eqs. 9.83 and 9.89,
were derived earlier using the Dugdale yield zone model. More precise estimates of 8t
can be obtained for strain-hardening materials from the solutions (Eqs. 9.91) for
crack face displacements. The definition of 8t is somewhat arbitrary because the
distance between the crack faces, 8 = uy(x, 0 + ) — uy(x, 0~), varies as (—x)1/(w+1) as
the crack tip is approached. A commonly used operational definition of 8t is based
on the distance between two points on the upper and lower crack faces where two
45° lines drawn from the deformed crack tip intercept the crack faces (Fig. 9.11).
With this definition,

St = dn^, (9.93)

where dn is a function of a, ey, and n. dn ranges in value from about 0.3 to 0.8 as n is
varied from 3 to 13 (Shih, 1981).
The crack tip opening displacement provides a measure of the size of the region at
the crack tip where finite strain deformation is dominant. The condition for the onset
of quasi-static fracture can also be stated as 8t = 8tc, where 8tc is a critical crack tip
310 Fracture mechanics and its implications for fatigue

Fig. 9.11. Definition of crack tip opening displacement, <5t.

opening displacement for the material under consideration. This approach to deter-
mining critical conditions for fracture initiation is sometimes appealing in that 8t
provides a physical length scale for fracture which is often needed for developing the
vital link between microscopic failure processes and macroscopic fracture toughness.
The magnitude of 8t varies continuously during fatigue due to the fluctuations in
load. The effective range of 8t in a given fatigue cycle is determined by the following
factors: (i) the extent of reversed flow ahead of the crack tip, as seen in the devel-
opment of Eq. 9.86, (ii) the roughness of the fracture surfaces (which is influenced by
the microscopic fatigue mechanisms and the microstructural size scale), and (iii) the
presence of any corrosion films or residual stretch of plastically deformed or trans-
formed material on the fracture surfaces which cause premature closure of the crack
even at a far-field tensile stress (see Chapter 14).
Noting the connection between / and K from Eqs. 9.57, 9.58 and 9.88, an expres-
sion for cyclic crack tip opening displacement can be derived. It is of the form given
in Eq. 9.86.

9.7.4 Conditions of J-dominance

The /-integral provides a unique measure of the strength of singularfieldsin
nonlinear fracture. However, some conditions must be ascertained before / can be
used to characterize fracture in real ductile materials. The following list of require-
ments for the use of/ is taken from a review article by Hutchinson (1983):

(1) The deformation theory of plasticity must be an adequate model of the

small-strain behavior of real elastic-plastic materials under the monotonic
loads being considered.
9.7 Elastic-plastic fracture mechanics 311

(2) The region in which finite strain effects dominate and the region in which
microscopic failure processes occur must each be contained well within the
region of the small-strain solution dominated by the singular fields, Eqs.
9.91.

The HRR fields exactly satisfy the first requirement for the use of deformation
theory of plasticity if proportional loading occurs everywhere. Then, the singularity
fields, Eqs. 9.91, based on the deformation theory assumptions are also solutions to
the corresponding J2 flow (incremental) theory equations (Section 1.4.3). Although
this requirement for proportional loading is not exactly fulfilled in general in an
elastic-power law plastic solid, the conditions encountered during the application
of monotonic, uniaxial loads to stationary cracks do provide a reasonable justifica-
tion for the use of deformation theory.
The aforementioned second requirement for the validity of / to characterize non-
linear fracture provides the physical basis for determining the inner radius, r0, of the
annular zone of /-dominance. Let R denote the outer boundary of the zone of J-
dominance which may be taken as the maximum distance ahead of the crack tip
within which the singularity solutions, Eqs. 9.91, match (say, within 10% error) the
full solutions (estimated using techniques such as the finite-element method) for the
particular specimen geometry under consideration. Finite-element, flow theory
model calculations of the crack-tip fields by McMeeking (1977) and McMeeking
& Parks (1979), which take into account crack tip geometry changes, reveal that
the finite strain effects are significant only over a distance of at most 38t. This result
provides a measure of r0. From a microstructural standpoint, r0 should be bigger
than the size of the process zone; for example, the grain size for transgranular
cleavage or intergranular fracture, and the mean spacing of void-nucleating particles
for ductile failure by void growth. Numerical simulations of near-tip fields for small-
scale yielding conditions in power law hardening materials show that the HRR
singular solutions hold over a distance of 20-25% of the size of the plastic zone
directly ahead of a mode I crack for essentially the entire range of strain hardening
exponents found in ductile alloys.
Under large-scale yielding, however, the size of the region of / dominance is
strongly dependent on specimen configuration. For these cases, where the entire
uncracked ligament may be fully engulfed in a plastic zone, the size of the region
of J dominance, R, is as small as 1 % of the length of uncracked ligament for a
center-cracked tension specimen or 7% of the length of the uncracked ligament for a
deeply cracked bend bar or a compact tension specimen (McMeeking & Parks,
1979). This is the reason why standardized test procedures for determining the
critical value of J in quasi-static loading, / Ic , require the use of deeply cracked
bend or compact tension specimens where the initial pre-crack length to the specimen
width ratio is at least 0.5 (see ASTM Standard E-813 for / Ic testing; American
Society for Testing Materials, Philadelphia). With this requirement and the result
that the zone of finite strains at the crack tip spans a distance of 38U it can be shown
312 Fracture mechanics and its implications for fatigue

that the minimum uncracked ligament size, b, needed to obtain a valid / Ic estimate is:
b = 25/ Ic /a y .
For /-controlled crack growth, Hutchinson & Paris (1979) have suggested that the
regime of elastic unloading and nonproportional loading should be confined to well
within the zone of /-dominance. In other words,

— » - and Aa«iv. (9.94)

da xv
The growth of cracks also causes changes in the near-tip fields as compared to
those predicted by the HRR solutions for stationary cracks. Asymptotic analyses of
near-tip stress fields during quasi-static crack growth in an elastic-perfectly plastic
solid reveal that the stresses are generally unchanged from the Prandtl slip line fields
representative of stationary crack tips (e.g., Rice, Drugan & Sham, 1980), except
behind the crack tip where differences of about 10% emerge as a result of a wedge of
elastic unloading for 112° <0< 162° (see Fig. 9.5 for the coordinate system).
However, the near-tip strain distribution for a growing crack has a logarithmic
singularity which is weaker than the \/r strain singularity for a stationary crack.

9.7.5 Example problem: Specimen size requirements

Problem:
A low strength steel is to be used in an application which requires appre-
ciable damage tolerance. It is known from prior testing that the alloy has a plane
strain fracture initiation toughness, Kic = 175 IMPa^m". The yield strength of
the steel is ay = 350 MPa.
(i) If the fatigue crack growth characteristics of this alloy in zero-tension
cyclic loading are to be experimentally measured over the entire range of
crack growth until final failure, determine the minimum specimen size
requirements for a compact (tension) specimen which conform to
ASTM Standard E-399. It is desired to have plane strain conditions dur-
ing the entire range of crack growth. A schematic of the geometry of the
compact specimen is given in Fig. A.2(a) in the Appendix. It may be
assumed that the final failure occurs in this case when the the maximum
stress intensity factor for the fatigue cycle approaches Kic.
(ii) For the same alloy and specimen geometry, determine the minimum speci-
men size requirements for a 'valid' / I c test which conforms to the speci-
fications of ASTM Standard E-813.
Solution:
For the steel, the values of Young's modulus and Poisson ratio are taken
as E = 210 GPa and v = 0.33.
9.7 Elastic-plastic fracture mechanics 313

(i) In order to carry out a 'valid' plane strain fracture initiation toughness
test, small-scale yielding should be ensured. For the compact specimen
shown in Fig. A.2(a), this implies that the length of the crack, a, the size
of the uncracked ligament, (W — a), and the specimen thickness, B,
should be larger than at least 25 times the plane strain plastic zone size
(Eq. 5.1), i.e.

a, (W-a),B>25x — (—) . (9.95)

3n\ay /
It is thus seen that a, (W -a), B> 66.3 cm (26.12 inches). This rather
severe requirement for specimen size makes testing very difficult,
(ii) For / I c testing which conforms to ASTM specifications, start with a
deeply cracked compact specimen whose initial crack length to width
ratio a/W > 0.5. The geometrical requirement in this is that the
uncracked ligament length be
(W-a)>25^. (9.96)

For small-scale yielding, / I c and Kic are related by Eq. 9.92 so that the
above equation can be rewritten as
K2
(W-a)>25—(\ -v2). (9.97)
E(Jy
Substitution of the appropriate numerical values gives the result that
(W — d) = 9.3 mm. For materials with high fracture toughness and
low yield strength values, the measurement of / I c thus provides an appeal-
ing test procedure for the estimation of fracture toughness.

9.7.6 Characterization offatigue crack growth

Although conditions of nonproportional loading and the occurrence of
elastic unloading would appear to violate the fundamental basis on which the appli-
cation of/-integral to fracture problems is predicated, Dowling & Begley (1976) and
Dowling (1977) have proposed a power law characterization of fatigue crack
advance under elastic-plastic conditions based on the cyclic /-integral, / c (also
referred to as A/), during a fatigue cycle, i.e. da/AN oc (/ c ) m , where m is an
exponent analogous to m in Eq. 9.53. This approach, despite its apparent short-
comings, provides suprisingly good characterization of the growth of short fatigue
cracks of length comparable to the near-tip plastic zone size and longer fatigue flaws
in nearly fully yielded specimens in some materials under certain cyclic loading
conditions.
For the cyclic loading of a specimen under displacement control, the /-integral is
usually determined in a straightforward manner by employing Eqs. 9.56 and 9.87
314 Fracture mechanics and its implications for fatigue

(Dowling & Begley, 1976). This procedure is schematically illustrated in Fig. 9.\2{a).
Here the rising part of the load-displacement hysteresis loops for two different crack
lengths ax and a2 are shifted to a common origin. The /-integral is then obtained
from the potential energy difference corresponding to the shaded area and from Eqs.
9.56 and 9.88.
F o r the cyclic loading of a specimen under load control, there is some ambiguity in
defining the proper limits of integration in the determination of the strain energy.

(a)

a, a0

(b)

(c)
Fig. 9.12. Determination of/-integral with stabilized cyclic hysteresis loops, (a) Hysteresis loops
for two different crack lengths in displacement-controlled fatigue and the translation of the
rising part of the stabilized hysteresis loop to a common origin, (b) Similar method for load-
controlled fatigue with the minimum load being employed as the reference point, (c)
Determination of / using a single specimen.
9.7 Elastic-plastic fracture mechanics 315

Sadananda & Shahinian (1979) have suggested use of the minimum load of the
fatigue cycle as a reference point for shifting the load-displacement curves, Fig.
9.12(&). Here the rising portions of the load-displacement curves are translated to
a common origin at the minimum load. The shaded area gives the potential energy
from which the /-integral for cyclic loading can be computed. Note that these
methods require the measurement of at least two load-displacement curves corre-
sponding to two different crack lengths. Approximate methods have been developed
in an attempt to overcome this limitation and to determine / from a single specimen.
Figure 9.\2{c) schematically illustrates this method. If Ae is the total area under the
load-displacement curve (shaded region), / for characterizing fatigue crack growth
(for an edge-cracked or compact tension specimen) is determined from

J =
Tb ^IAQ + a2P(8
™* ~ 5 ™ n) }' (9 98)
'

where B is the thickness of the specimen, b is the length of the uncracked ligament, P
is the load, <5max and 8min are the maximum and minimum displacements shown in
Fig. 9.12(c), and ax and a2 are correction coefficients which are functions of the
crack length and have been published by Merkle & Corten (1974).
The justification for the above methods apparently rests on the argument that the
phenomenological constitutive models for cyclic plasticity (Chapter 3) can be for-
mulated in terms of stable hysteresis loops and that, if such loops can be mathema-
tically translated to a common origin after each load reversal, the requirement for
the stress to be proportional to the current plastic strain can be effectively satisfied.
Although many researchers have employed the /-integral to characterize fatigue
fracture at room and elevated temperatures, it should be noted that the cyclic /
approach can seriously violate the basic assumptions leading to the development
of the /-integral. Severe nonproportional loading and the rapid advance of the
fatigue crack promote conditions where material descriptions based on the deforma-
tion theory of plasticity do not hold. At this point, experimental documentation of
a reasonably good characterization of fatigue crack growth under some elastic-
plastic conditions is the main justification that can be provided for the application
of/-integral to cyclic loading. See Chapter 15 for a further discussion of this topic.
An alternative approach for characterizing fatigue crack growth under elastic-
plastic conditions is often formulated in terms of crack opening displacements. Here,
the fatigue crack growth rate is envisioned as being proportional to the cyclic crack
tip opening displacement, A<5t, defined in Eq. 9.86 such that da/dN oc A<5t. This
type of analysis provides a size scale, A<5t, for comparisons with striation spacing,
residual crack wake stretch or fracture surface oxide thickness to correlate crack
growth and crack closure (see Chapters 10 and 14). Furthermore, the crack opening
displacement offers a convenient means for comparing the fatigue crack growth rates
in different modes of fracture on a common scale. It can be seen from Eq. 9.93 that
characterizations of crack advance based on / and on 8t are essentially equivalent for
316 Fracture mechanics and its implications for fatigue

proportional loading. The implications and limitations of the approaches for fatigue
crack growth under nonlinear fracture conditions are examined in later chapters.

9.8 Two-parameter representation of crack-tip fields

Consider the small-strain, linear elastic solution, Eqs. 9.41-9.43, where the
first two terms provide an adequate solution for the crack tip stress fields:

|L4(0) -^TSuSy. (9.99)

Under small-scale yielding, different levels of hydrostatic stresses induced at the

crack tip in different specimen geometries can be characterized in terms of the non-
dimensional parameter, T/ay, where oy is the yield strength (Bilby et aL, 1986;
Harlin & Willis, 1988; Betagon & Hancock, 1991; Parks, 1992). The use of KY and
T, as two parameters which fully characterize the near-tip fields, becomes increas-
ingly more invalid as the plastic zone size at the crack tip enlarges beyond the limits
of small-scale yielding.
It was ostensibly first noted by McClintock (1971) that the states of near-tip stress
fields in fully yielded cracked specimens require more than a single parameter, such
as the /-integral, to adequately capture the triaxial fields. For example, center-
cracked panels with large-scale yielding exhibit markedly reduced triaxial stresses
at the crack tip than fully-yielded, deeply-cracked compact and bend specimens. In
order to account for the role of differing states of triaxial stresses in influencing
crack-tip fields and to overcome the inadequacy of the K^-T two-parameter char-
acterization under large-scale yielding, O'Dowd & Shih (1991, 1992) have proposed
the so-called two-parameter J-Q theory. The salient features of this theory are
outlined next.
Within the context of the small-strain deformation plasticity theory and an elastic-
power law plastic material model, the stress fields at the crack tip take the form

a, = <TV I ) <TZ7(0, n) + higher order terms. (9.100)

3 y J
\otay€yInr)
The first term on the right hand side of Eq. 9.100 may easily be recognized as the
HRR singular stressfield,(cy)HRR, see Eq. 9.91. Higher order asymptotic analyses of
crack-tip fields under large-scale yielding have been carried out by Li & Wang
(1986), Sharma & Aravas (1991) and Xia, Wang & Shih (1993). These studies reveal
that, in the region ahead of the crack tip where \6\ < n/2 and J/cry < r < 5J/cry9 Eq.
9.100 can be approximated in the following manner:

where Q is a measure of the crack tip stress triaxiality, defined as

9.8 Two-parameter representation of crack-tip fields 317

(9.102)

That is, in this operational definition, Q represents the difference, normalized by the
yield strength cry, between the actual hoop stress at the crack tip and that given by the
HRR singular field at a fixed distance 2J/cry directly ahead of the crack tip. The
distance r = 2J/ay is chosen so as to lie just outside the blunting zone characterized
by the finite strains; under such conditions, Q is found to be essentially independent
of r.
Alternatively, the ^-dominated small-scale yielding field, (tf//)SSY;T=o> c a n a ^ so
serve as the reference solution. O'Dowd & Shih (1991, 1992) define Q as

at 0 = 0, r = —. (9.103)

In terms of the hydrostatic or mean stresses, Q can also be defined as

. „ 2/
r=- (9.104)

Equations 9.102 and 9.103 can be interpreted in the following manner. Positive
(negative) values of Q raise (reduce) the level of crack-tip hydrostatic stress from
that given by the HRR solutions or the small-scale yielding solution.
The variation of hoop stress, aee, as a function of the normalized distance ahead of
the crack tip, r/(J/cry), for plane strain and for E/ay = 500 and v = 0.3 are plotted
in Fig. 9.13(a) and (b) for small strain (HRR solution) and finite strain, respectively,
for different values of n and for 6 = 0. These pertain to the reference fields: T = 0
and 2 = 0.

6.0 r 6.0
\ \
•\ \ y
\ \ n=3
5.0 . \ \ r* » 5.0 \
s/*= 3
Y
/
\ \
4.0 -
\ ^5 ^v^- 4.0 - -. 5

10 " ^ ^ *
•'•• . 2 0 ' • s
3.0- 3.0-
n = 00 "/ / n = oo '" - "
s
i | i , . , 1 , i
2.0 2.0
0.0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 5.0
rl(Jlay) rl(Jlcfy)
(a) (b)
Fig. 9.13. The variation of hoop stress, ae9, as a function of the normalized distance ahead of the
crack tip, r/(J/cry), for plane strain and for E/ay = 500 and v — 0.3 for (a) small strain (HRR
solution) and (b) finite strain, respectively, for different values of n and 0 = 0 for the reference
fields: T = 0 and Q = 0. (After O'Dowd & Shih, 1991, 1992.)
318 Fracture mechanics and its implications for fatigue

O'Dowd & Shih have calculated Q and Qm for several finite width geometries and
find the difference between them to be always smaller than 0.1. Consequently, the
definition of Q given in Eq. 9.103 has been recommended as the basis upon which
crack-tip stress triaxiality is quantified.

9.8.1 Small-scale yielding

The two-parameter characterization of crack tip fields by Ki and T or by /
and Q have equivalence under small-scale yielding. Recall from Section 9.7.1 that
under plane strain and small-scale yielding,

/ = ^(l-v2). (9.105)
Also, from dimensional considerations, it can be shown that

( 9 - 106 )
r
ay oy
Making use of Eq. 9.103, we write

For a power-law hardening solid characterized by the two-term solutions in Eqs.

9.99 and 9.101, a unique relationship can be defined between Q and T in the follow-
ing form:
/ T \
(9.108)
for given E/ay and v. Figure 9.14 shows the variation of Q (based on the definition in
Eq. 9.103) with T/cry for different values of n. These plots are essentially the same for
both small strain and finite strain deformation at crack tip.

9.8.2 Large-scale yielding

Under large-scale yielding, the equivalence between K and / , and that
between T and Q breaks down. For this situation, O'Dowd & Shih (1991, 1992)
identify the following crack tipfieldsfor plane strain based on the two parameters /
and Q:

(1^9e9,n,v,Q). (9.109)
y \J/cry ay )
Within the J-Q annulus, the fields are adequately represented by the form of Eq.
9.107. Values of Q for a variety of specimen geometries and crack sizes (a/W) can be
found in O'Dowd & Shih (1991, 1992). Implications of the two-parameter approach
to fatigue crack growth characterization is discussed in Chapter 15.
9.9 Mixed-mode fracture mechanics 319

0.5 r—

0.0

-0.5

-1.0

-1.5

-2.0
I
-1.0 -0.5 0.0 0.5 1.0
Tidy
Fig. 9.14. The variation of Q with T/ay for different values of n (for E/ay = 500 and v = 0.3).
(After O'Dowd & Shih, 1991, 1992.)

9.9 Mixed-mode fracture mechanics

The stress analyses discussed thus far have centered around the development
of near-tip fields for stationary, mode I cracks. Fatigue cracks in structural materials,
however, are generally subjected to combined-mode loading conditions. In order to
develop proper design methodologies for combined-mode fatigue fracture, it is
necessary to gain insights into the nature of near-tip fields under conditions of
load mixture. Even when a component containing a fatigue crack is subjected to
purely tensile far-field loading, mixed-mode conditions may prevail ahead of the
crack tip if it is inclined at some arbitrary angle to the tensile axis, or microstructural
and environmental factors promote a nonplanar fatigue crack growth. In composite
materials consisting of two dissimilar components, a fatigue crack located at the
interface between the two materials is subjected to local mixed-mode loading even
when the far-field stresses are purely in mode I. The inducement of such mixed-mode
conditions at the tip of a fatigue crack can lead to pronounced changes in the
'effective driving force' for crack advance, the size of the crack tip plastic zone,
the rate of crack propagation, the extent of fatigue crack closure and the microscopic
mechanisms of damage.
320 Fracture mechanics and its implications for fatigue

Among mixed-mode fracture problems, complete solutions of the near-tip fields

are available for combined mode I-mode II cracks subjected to small-scale nonlinear
deformation due to power law plasticity (Shih, 1974) and transient power law creep
(Brockenbrough, Shih & Suresh, 1991). We first present here the elastic-plastic
solutions for mode I-mode II cracks in strain-hardening materials. This is followed
by a discussion of mixed-mode near-tip fields for deflected and branched cracks.

9.10 Combined mode I-mode II fracture in ductile solids

The linear elastic fracture mechanics solutions, Section 9.3, and the HRR singular
fields, Section 9.7.2, can be extended to provide the near-tip fields for mode I-mode
II crack problems. Under conditions of small-scale yielding, the far-field stress com-
ponents for a crack subjected remotely to tensile opening and sliding stress intensity
factors, KY and Kn, respectively, are given by

^ .aljiO) + Ku5$
5$(0)l (9.110)

where r and 0 are the polar coordinates centered at the crack tip (see Fig. 9.5) and
ojj(0) and ojj(0) are the dimensionless universal functions described in Eqs. 9.44 and
9.47. For small-scale yielding, the /-integral is related to mixed-mode stress intensity
factors by

where E' is defined in Eq. 9.2. The relative strengths of K^ and Kn can be char-
acterized in terms of an elastic mixity parameter, Me, which is defined as (Shih,
1974)
aee(r, 0 = 0) 2 K,
Afc=-tan"1 lim = -tan- l K . (9.H2)

In this characterization, Me = 0 for pure mode II, Me = 1 for pure mode I, and
0 < M e < 1 for different mixities of modes I and II.
The near-tip fields for the mixed-mode crack problem in an elastic-plastic solid
whose constitutive response is represented by the nonlinear elastic (deformation)
theory are analogous to the HRR fields for mode I described in Eq. 9.91 and are
of the form

<re = ayKu r~1/("+1) <re(0, M p ,n),

, it). (9.113)
9.10 Combined mode I-mode II fracture in ductile solids 321

The dimensionless functions <rzy, cre, e?, and ut depend only on the polar angle 0, the
strain hardening exponent n, and the near-tip plastic mixity parameter, M p , which is
defined similar to M e as
croo(r, 0 = 0)}
M^-tan"1 lim
n
2
-tan -1 (9.114)
71

where M p equals 0 for pure mode II, 1 for pure mode I, and 0 < M p < 1 for
different mixities of modes I and II. The strength of the singular fields given in
Eqs. 9.113 is the parameter K^. The superscript P denotes that it is a plastic stress
intensity factor and the subscript M refers to the mixed-mode condition. A definite
meaning to this parameter has been given (Shih, 1974) by setting the maximum value
of the ^-variation of the effective stress, <je = v/{(3/2),f^y}5 to unity where
stj = oij — (o'kk/^ij- The strength of the dominant mixed-mode singularity, K^,
can be related to the /-integral via Mp:

J= ——[Ki +Kii)=—=-ln[M ){KM) > (9.115)

where In(Mv) is a numerical constant which is a function of the strain hardening

exponent n and the plastic mixity factor M p . Thus, for combined-mode fracture, the
two parameters, Kp and M p , completely specify the near-tip fields for a given value
of ft, irrespective of whether small-scale yielding conditions prevail. A complete
description of the near-tip fields in terms of / and M e (or, equivalently, KY and
Ku) requires that the relationship between M e and M p be specified. By performing
a detailed finite element analysis of mixed-mode loading under small-scale yielding,
Shih (1974) has computed the relationship between M e and M p for values of ft =
1-99. While M e = M p for n = 1, M p > M e for n > 1, with the difference between
M p and M e increasing with increasing ft.
The (circumferential) 0-variations of the universal dimensionless functions <rzy, €y
and cre are shown in Fig. 9.15 in order of increasing asymmetry. One observes a
pronounced decrease in the hoop stress oe0 and in the hydrostatic stress, and a
noticeable change in the distribution of plastic strains as the mode of loading is
changed from pure tension (mode I) to pure sliding (mode II).
This effect of plasticity on the near-tip fields is more directly evident in Fig. 9.16,
where the elastic-plastic boundaries are shown as functions of n and M e for plane
stress and plane strain. At the same amplitude of far-field loading (that is, at a fixed
amplitude of / ) , the plastic zone size in mode II is up to five times bigger than that in
mode I. This result implies that any deviation of the fatigue crack from the nominal
mode I growth plane, as often found in crystallographic crack propagation, causes
not only a change in near-tip stress intensity factors due to the inducement of mixed-
mode conditions, but also leads to an increase in the size of the plastic zone ahead of
the crack tip. The consequences of this effect are especially important in the study of
322 Fracture mechanics and its implications for fatigue

-1.5
-180 90 180

Fig. 9.15. Circumferential variations of stresses and strains shown in order of increasing load
asymmetry for small-scale yielding and plane strain loading conditions. Strain hardening
exponent, n = 13. (From Shih, 1974. Copyright American Society for Testing and Materials.
Reprinted with permission.)

short fatigue cracks where plastic zone dimensions comparable in size to the crack
length cause uncertainties in characterization.

9.11 Crack deflection

Tensile fatigue cracks in both brittle and ductile solids can deviate signifi-
cantly from their normal mode I growth plane under the influence of far-field multi-
axial stresses, interaction of the crack tip with micro structural inhomogeneities such
as grain boundaries and interfaces, variable amplitude loading in the form of over-
loads, crystallographic separation, or the embrittling effect of an aggressive environ-
ment. As the local, near-tip stress intensity factor which is responsible for the
9.11 Crack deflection 323

ya;

-0.6

plane strain
Fig. 9.16. Contours of plastic zones shown for different values of the strain hardening exponent
n for mode I and mode II cracks subjected to plane stress and plane strain loading. (After Shih,
1973, 1974.)

advance of a deflected or branched crack (i.e. an 'effective driving force') can be

considerably smaller than that of a straight crack of the same (projected) length,
periodic changes in crack path lead to apparently beneficial (slower) fatigue crack
growth rates (for a fixed amplitude of far-field loading). The deflection of a tensile
crack from the nominal mode I plane induces mixed-mode near-tip conditions even if
the far-field loading is purely mode I. Under cyclic loading conditions, the increase in
the roughness of the fracture surface created by crack deflections can also cause an
increase in the far-field stress at which the crack faces begin to close upon unloading
from the maximum tensile stress. This enhanced closure level leads to further reduc-
tions in the near-tip 'driving force' for fatigue crack growth (see Chapter 14).
It should be recognized at the outset that micro structurally induced crack deflec-
tion in engineering materials is a problem for which it is very difficult to develop
accurate continuum solutions. The mixed-mode fracture solutions are useful for
deflected cracks only if the kink length greatly exceeds the size scale of the micro-
structural inhomogeneity responsible for crack deflection and the size scale of the
near-tip plastic zone. Furthermore, if the entire crackfront does not deflect uni-
formly, multiaxial conditions involving modes I, II and III would be expected to
develop. The discussions in this section are, therefore, intended to provide only a feel
for the changes in 'effective driving force' arising as a consequence of simple deflec-
tions in the path of a linear elastic fatigue crack.
324 Fracture mechanics and its implications for fatigue

9.11.1 Branched elastic cracks

Figure 9.\l{a) is a schematic representation of an idealized line crack con-
taining a kink of length b inclined at an angle a from the plane of the main crack of
length a. A symmetrically forked crack with an included angle 2a is shown in Fig.
9.17(6). For a pupative kink (b -> 0), the local mode I and mode II stress intensity
factors, k\ and k2, respectively, can be expressed in the form

Fig. 9.17. A schematic representation of (a) kinked and (b) forked crack geometries and the
associated nomenclature.
9.11 Crack deflection 325

kx = an{a)Kl+aX2(a)Kll,
k2 = a2i(a)Kl + a22(a)Ku, (9.116)
where KY and Ku denote the mode I and mode II stress intensity factors for the main
crack in the absence of the kink or fork. To a first order approximation in a, the
dimensionless factors for the infinitesimal kink are (Bilby, Cardew & Howard, 1977;
Cotterell & Rice, 1980)

( 3 cos - +
3/ . a 3a
-lsm- + —

«22(«) = 4 (COS - + 3 COS y I. (9.117)

Suresh & Shih (1986) have presented a summary of available stress intensity factor
solutions for kinked and forked cracks as functions of the deflection angle and the
length of the deflected part of the crack. The variations of the near-tip stress intensity
factors, kx and k2, for a line crack containing a kink of length b = 0.1 a and subjected
to a far-field tensile stress intensity factor Ki are plotted in Fig. 9.18(a) as a function
of the kink angle, a. Similar results for a symmetrically forked crack with a fork
length b = 0.1a are presented in Fig 9AS(b) as a function of the included fork angle,
2a (Kitagawa, Yuuki & Ohira, 1975; Lo, 1978). Note that k2 vanishes at 2a = 32° for
b/a = 0.1. A similar observation was made by Bilby, Cardew & Howard (1977) who
calculated the included angle 2a for k2 = 0 to be 36° for b/a = 0.025.

1.0
b
= 0.1
a
^ \
0.5 - -kJK,

0.0
; K

-0.5 -
. , i , . i ,
60 120 180
2a (degrees)
(b)
Fig. 9.18. Variation of normalized kx and k2 for b/a = 0.1 as a function of (a) kink angle a and
(b) fork angle 2a. (After Kitagawa, Yuuki & Ohira, 1975.)
326 Fracture mechanics and its implications for fatigue

The studies discussed above also show that for b/a > 0.5, kx and k2 are indepen-
dent of b/a. A similar trend is observed for symmetrically forked elastic cracks. This
is consistent with the known result that the kinked and forked crack solutions for
b/a > 0.5 approach those for a crack inclined at an angle f$ (radians) (/* = n/2 — a)
to a remote tensile stress intensity K^.
-£r = sin/* cos/?. (9.118)

9.11.2 Multiaxial fracture due to crack deflection

For an elastic crack containing an infinitesimal kink (b/a -> 0) subjected to
pure mode I far-field loading, the functions a\\ and a\2 in Eq. 9.116 can also be
obtained to a reasonable degree of accuracy from coordinate transformations
(Lawn, 1993). Here,
fci=*iSk, k2 = Kl5lr99 (9.119)
where o\e and o\Q are defined in Eq. 9.44.
The crack deflection processes discussed thus far pertain to the tilt configuration
represented by the rotation of the crack plane about an axis parallel to the crack
front, axis Oz in Fig. 19(a). Whereas this tilt deflection gives rise to near-tip mode I-
mode II displacements for far-field mode I loads, the twist deflection represented by
the rotation of the crack plane about an axis parallel to the initial direction of crack
advance, axis Ox in Fig. 9.19(£), induces mode I and mode III displacements at the
crack tip. For an infinitesimal twist angle 0,

z/ Sa

Fig. 9.19. Crack deflection leading to (a) tilting and (b) twisting of the crack front.
9.12 Case study: Damage-tolerant design of aircraft fuselage 327

^ k^K&t, (9.120)
where the functions a\^ and o\>^ can be found listed in Lawn (1993). Equations 9.120
provide only a crude result since the twisting of a crack plane, Fig. 9.19(Z>), causes the
formation of steps along the crack path; this may promote mode II displacements as
well. Furthermore, the near-tip stress intensity factors may deviate considerably
from the predictions of Eqs. 9.120 as a consequence of frictional sliding along the
deflected segments of the crack.
The elastic solutions for kx and k2 for kinked or forked cracks are strictly valid
only when the plastic zone size is smaller than the zone of dominance of the kx and k2
singular fields, which itself is a fraction of the kink or fork length b. From a knowl-
edge of (i) the elastic solutions for kx and k2 as a function of the deflected crack
geometry, (ii) the universal mixed-mode plastic near-tip fields, and (iii) the numeri-
cally determined relationship between Me and M p as a function of the strain hard-
ening exponent n for a given deflected crack geometry, Suresh & Shih (1986) have
determined the near-tip fields ahead of a kinked or a forked crack for plane strain
and small-scale yielding conditions. Their results show that the combined effects of
crack tip plasticity and crack deflection can promote a more beneficial crack growth
resistance than deflection alone.

9.12 Case study: Damage-tolerant design of aircraft fuselage

The fuselage of a passenger jet aircraft is one of the most complicated structures in
the aircraft. It is a stiffened, thin-walled cylinder. Its diameter to wall thickness ratio is
approximately 2000, somewhat analogous to a fully pressurized toy balloon. At cruising
altitude, the differential pressure between the passenger cabin and the outside is
approximately 55 kPa (8 psi), as discussed in Section 1.1.1. The cabin pressure causes a
tensile stress in the hoop direction. The bending induced by routine flight loads causes
tensile loads in the crown section (top part of the fuselage), shear loads on the sides of the
fuselage, and compressive loads in the bottom of the fuselage (keel) in the longitudinal
direction (e.g., Budiman, 1996).
Stiffeners are used in the following manner in order (i) to prevent bending, (ii) to avoid
instability due to bending and (iii) to maintain the circular shape of the fuselage.
(1) Stiffeners in the circumferential direction, termed the frames. The spacing between
neighboring frames in a medium-range passenger aircraft, such as the Boeing 737,
is 500 mm (20 in.).
(2) Stiffeners in the longitudinal direction, termed the longerons. The spacing between
neighboring longerons in a medium-range passenger aircraft, such as the Boeing
737, is 250 mm (10 in.).
Consider the following possibility for an accident. A turbine blade, dislodged from one
of the jet engines of the aircraft during the course of a flight, penetrates the skin of the
fuselage which is fully pressurized. The likely orientation for a through-thickness crack
formed by this impact would be along the hoop direction. The hoop stress of the
328 Fracture mechanics and its implications for fatigue

cylindrical fuselage with stiffeners, however, is nearly twice that of the longitudinal stress.
Consequently, the worst situation here would be for the crack to be oriented along the
axial direction. In order to arrest the advance of such a crack, tear straps or crack
arresters are commonly introduced under the frames, midway between two neighboring
frames, and under the longerons. The thickness of the tear straps is approximately the
same as that of the fuselage skin.
The spacing between adjacent tear straps provides the basis for the maximum crack
size in design. It means that, for the medium-range passenger transport aircraft, the
fuselage has to be designed with provisions for the existence of a longitudinal crack as
long as 500 mm.

Exercises
9.1 In the compliance calibration of an edge-cracked fracture toughness test-
piece of an aluminum alloy, it was observed that a load of 100 kN produced
a displacement between the loading pins of 0.3 mm when the crack length
was 24.5 mm, and 0.3025 mm when the crack length was 25.5 mm. The
fracture load of an identical testpiece containing a crack of length
25.0 mm was 158 kN. Calculate the critical value of the mechanical potential
energy release rate Q at fracture and hence the plane strain fracture tough-
ness Kic of the alloy. All testpieces were 25 mm thick. For the alloy, Young's
modulus, E = 70 GPa and Poisson's ratio, v = 0.3.
9.2 Starting with the unsymmetric part of the Airy stress function, Eq. 9.30,
where

and assuming that x is separable, i.e. x = R(r) • ©(#), derive the leading term
of the asymptotic singular solution for mode II,

following the procedure discussed in Section 9.3.2 for Mode I. Write com-
plete expressions for the different components of the stress field and com-
pare your results with Eq. 9.47.
9.3 Use the result from the previous problem to show that the expression for the
leading terms for the displacements is of the form

y
where /x is the shear modulus. Derive complete expressions for ux and uy and
compare your results with those given in Eq. 9.48.
9.4 A piston (89 mm in diameter) is designed to increase the internal pressure in
a cylinder from 0 to 55MPa. The cylinder (closed at the other end!) is
200 mm long with internal diameter = 90 mm, outer diameter = 110 mm,
Exercises 329

and made of 7075-T6511 aluminum alloy (extruded bar with yield

strength = 550 MPa, and Kic = 30 MPav^n)- On one occasion, a malfunc-
tion in the system caused an unanticipated failure and the cylinder burst.
Examination of the fracture surface revealed a metallurgical defect in the
form of an elliptical flaw 4.5 mm long at the inner wall, 1.5 mm deep, and
oriented normal to the hoop stress in the cylinder. Compute the magnitude
of the pressure at which failure took place. (For the purpose of this problem,
assume that the stress intensity factor for the elliptical flaw is
K = (1.12/^/Q)cr y/na, where a is the appropriate normal stress and Q is
a shape factor. The calibration for Q as a function of the aspect ratio of the
flaw is given in Section A.7.)
9.5 A steel used for an engineering application has a specified yield strength of
1000 MPa and a plane strain fracture toughness of 150MPa,>/in~.
(a) Calculate the minimum dimensions required to carry out a 'valid' plane
strain fracture toughness test.
(b) Estimate the weight of the single edge cracked bend specimen and of the
compact specimen of this steel which would have sufficient dimensions
to provide valid plane strain fracture toughness. (See the Appendix
section for K calibrations for different specimen geometries.) Assume
that a/W = 0.45.
(c) Estimate the load capacity of the mechanical testing machine you would
need to carry out the fracture test.
(d) If the available testing machine has a load capacity of 200 kN, do you
need to alter the dimensions of the specimens to obtain a valid Kicl If
so, what will you do?
9.6 The ASTM standard test method based on linear elastic fracture mechanics
(standard E-399, 1974) for plane strain fracture toughness of metallic mate-
rials requires that the characteristic specimen dimensions (such as the crack
size, the size of the uncracked ligament and specimen thickness) be greater
than 2.5 x (ATIc/ory)2, where Kic is the mode I fracture toughness in plane
strain and ay is the yield strength of the material. Similarly, the standard test
method for the elastic-plastic fracture toughness / I c (standard E-813, 1981)
requires that the size of the uncracked ligament and the specimen thickness
be greater than 25/ Ic /a y , where / I c is the critical value of the /-integral for
fracture initiation.
(a) Noting the relationship between / and K, Eqs. 9.57 and 9.8, derive an
expression for the ratio bK/bj for a linear elastic material, where bK is
the uncracked ligament length for a K\c test specimen and bj is that for a
/ I c test specimen. cry, E and v, respectively, are yield strength, Young's
modulus and Poisson's ratio for the material.
(b) Calculate the ratio bK/bj for a steel for which Kic = 250
ay = 350 MPa, E = 200 GPa and v = 0.33.
330 Fracture mechanics and its implications for fatigue

9.7 For plane strain and small-scale yielding conditions, the region of /-dom-
inance spans a distance R of up to 25% of the size of the monotonic plastic
zone at the crack tip. Assuming the requirement for /-dominance to be that
R > 38t (where 8t is the crack tip opening displacement defined in Section
9.7.3) and noting that 8t & 0.6//a y for a low hardening material, prove that
the requirement for /-dominance is always satisfied for a low hardening steel
with the following properties: E/cry = 500 and Poisson's ratio, v = 0.33.
9.8 Two engineering alloys, of the same overall yield strength (300 MPa) and
elastic properties, are being considered as candidate materials for a parti-
cular structural application. Material 1 has a grain size of lOjim, while the
grain size of Material 2 is 220 jim. Both alloys fail in an intergranular
fracture mode by the nucleation, growth and coalescence of voids at grain
boundaries. The structural component, for which the alloys are being con-
sidered, is subjected to a tensile stress of 85 MPa. It contains a through-
thickness single edge crack of length a = 25 mm. The thickness of the part
containing the crack is 30 mm and its width is 100 mm. (The equations
necessary to calculate the stress intensity factor for this problem can be
found in the Appendix.)
(a) Comment on the extent and validity of /-dominance for the two mate-
rials under consideration.
(b) An examination of the ASTM standard (E813) for elastic-plastic frac-
ture toughness testing of these two materials reveals the requirement
that the depth of the initial crack (i.e. the notch length plus the length of
the fatigue pre-crack) be at least one-half of the width of the specimen
(e.g., compact tension, three-point bend specimen, or four-point bend
specimen). Why?
9.9 Derive expressions for the near-tip mode I and mode II stress intensity
factors, kx and k2, respectively, for a center-cracked plate containing a
crack which is inclined at an angle fi to the far-field tensile load and sub-
jected to a far-field mode I stress intensity factor, 7^. Check your answers
with Eq. 9.118.