JOUR:!

i/AL OF APPLIED MECHANICS

The result.. are pictorial in nature as shown in Figs. 7 to 9. Of

. particular importance are the stresses at the ends of the beam,
an effect unimportant for the long beams treated previously in
Analysis of Stresses and Strains Near the
this section. The experimental procedure was the same as used
for the long-beam study with the exception that no temperature
readings were obtained.
End of a Crack Traversing a Plate
Fig. 7 illustrates the time-dependent nature of the thermal
stresses as well as the fairly severe end effect.. for a beam of L/d = BY G. R. IRWIN,' WASHINGTON, D. C.
3.25. At the free ends, high shearing stresses can be observed
which extend into tlie mm •for a distance approximately equal A sub~ntial fraction .;f the IDJ';t;,,.;es-asooclated with energy, for example, from movement of.the forces applying ten-
to the depth of the beam. Such behavior would be anticipated· crack enension might be eliminated if the description of sion to the material. For convenience this is referred to here
on the basis of the Saint Venant effect. fracture ezperiments could include some reasonable esti- as the fixed-grip strain-energy release rate. Since the strain-
Figs. 8 and 9 represent element.. with beam dimensions of energy disappearance rate at any moment depends on the load
mate of the stress conditions near the leading edge of a
Lid = 1 and L/d = 0.308, respectively. It is interesting to note crack particularly at points of onset of rapid fracture and magnitudes rather than on movement of the points of load ap-
that the thermal-shock pattern at the upper edge and free end at points of fracture arrest. It is pointed out that for plication, use of the fixed-grip strain-energy release-rate concept
regions in Figs. 7 to 9 is almost identical for short times. It is aomewhat brittle tensile fractures in situations such that is not limited to fixed-grip experiments.
only after relatively long times that the fringe pattern becomes a generalized plane-stress or a plane-strain analysis is It is the purpose of this paper to describe the relation of these
characteristically different for the various L/d--ratio element... appropriate, the influence of the test configuration, loads, terms to the elastic stresses and strains near the leading edge of
and crack length upon the stresses near an end of the a somewhat brittle crack. For purposes of this paper "somewhat
CoNCLUSIONS
crack may be expressed in terms of two parameters. One brittle" means that a region of large plastic deformations may
On the basis of this exploratory study, it appears that the of these is an adjustable uuiform stress parallel to the exist close to the crack but does not extend away from the crack
pbototbermoelastic technique bas considerable promise as a direction of a crack ext-ension. It is shown that the other by more than a small fraction of the crack length.
quantitative tool for verifying thermal-stress analyses. In parameter, called the stress inteosity factor, is propor..
00
Previous investigations (3-7) have established a viewpoint with
addition, the ability to observe the time-dependent behavior of tional to the square root of the force tending to cause respect to the mecbanies of fracturing which may be summarized
complete thermal-stress fields places phototbermoelasticity in a crack extension. Both factors have a clear interpretation in part as follows:
unique position in the experimental thermal-stress-analysis field. and field of usefulness in investigations of brittle-fraclure The fixed-grip strain-energy release rate bas the same role as an
More specifically, the following are the conclusions of this in- mechanics. infiu!lllce controlling time rate of crack extension as the longitudi-
vestigation: nal load bas in eontrolling time rate of plastic extension of a tensile
INTRODUCTION
(a) The optical and physical properties pertinent to the bar. In the latter case the force per unit area tending to cause
analysis of thermal stresses have been obtained for Paraplex P-43 plastic extension is the longitudinal strees. In the former case a
over a temperature range from 70 to -40 F.
(b) For the disk and long-beam models which are representa-
tive of interference and thermal-gradient type of thermal-stre88
D URlNGandsubsequent to the recent World War, investi-
gations of fracturing have shared in the general growth
of applied-mechanics research. Among the fracture fail-
ures responsible for interest in this field were those of welded
motivating force per unit thickness can be defined quite generally
in terms of the rate of eonversion of strain energy to thennal
energy during crack extension. This generalized force is the rate
fields, respectively, good correlation was obt..ined between the ■hips, gas-transmission lines, large oil-storage tanks, and pre&- of decrease of the fixed-grip strain energy with crack extension on
observed and theoretically determined fringe distributions. surized cabin planes. The propagation of a brittle crack across a unit-thickness basis. Also this energy rate can be regarded as
PaoTOTBERMOELASTic 8Tu»Y (Dux: F111:m) oF A Bll.lll
WITH L/d = 0.3 (c) For short beams, severe end effect.. were observed which one or more plates in which the average tensile stress was thought composed of two terms: (a) The strain-energy 1088 rate associated
(Dark apecb are due to dry ice,) extend for a distance approximately equal to the beam depth. to be safely below the yield strength is a prominent feature of with nonrecoverable displacement.. of the points of load applica-
During the initial stages of sudden temperature application, the these examples. tion (assumed zero in this discussion); and (b) the strain-energy
tory study was conducted of thermal stresses in short thermal-stress field at the upper edge and fre<rend regions As a result of these investigations there was a revival of in- loss rate associated with extension of the fracture accompanied
,f various L/d--ratios expo""d t-0 dry ice on one surface. appears to be independent of the beam dimensions. terest in the Griffith theory of fraoture strength (1). 1 It was only by plastic strains local to the crack surfaces. The second of
pointed out independently by Orowan (2) and by the author (3) these two terms, herein called g, appears to be the force compo-
that a modified Griffith theory is helpful in understanding the nent most directly related to crack extension and the one with the
development of a rapid fracture which is sustained with energy most practical usefulness.
from the surrounding stress field. Expositions of this idea have ·Determination of characteristic values of g for onset or arrest
been given (3, 4, 5) using such terms as fracture work rate and of rapid fracturing and the applications of such measurement.. to
strain-energy release rate. "fail-safe" design procedures have been discussed elsewhere ( 4, 5,
The basic idea of the modified Griffith theory is that, at onset 81 9). It will be shown here that the tensile stresses near the
of unstable fast fracturing, one can equate the fracture work per crack tip and normal to the plane of the crack are determined by
unit crack extension to the rate of disappearance of strain energy the force tendency g. The discussion is arranged so as to de-
from the surrounding elastically strained material. The term, velop relationships useful in the analysis of fracture experiments
disappearance of strain energy, refers to the loBB of strain energy whether the purpose of the work is to determine characteristic
which would occur if the system were isolated from receiving S-values or simply to determine the stress field near the leading
edge of the crack.
1 Superintendent, Mechanics Division, U.S. Naval Research Labo- The material of this paper is, at one point, related to Sneddon'•
ratory. analysis of streases near an embedded crack having the shape of a
t Numbers in parentheses ref~r to the Bibliography at the end of the
paper. flat circular disk ( 10). Otherwise, for simplicity and bearing in
Presented at the Applied Mechanics Division Summer Con~ mind the service fracture failures referred to in the foregoing,
ferenoe, Berkeley, Calif., June 13-15, 1957, of THB AlocRICAN So- discussion is restricted to a straight crack in a plate. It is as-
OIBTT 01' MIICHANICAL ElfGIN'llEBS.
Discussion of this paper should be addressed to the Secretary,
sumed the plate thickness is small enough compe.red to the crack
ASME, 29 West 39th Street, New York, N. Y., and will be accepted length so that generalized plane stress constitutes a useful two-
until Oct0ber 10, 1957, for publication at a later date. DisCU88ion dimensional viewpoint. In addition it is assumed the crack is
received, after the closing date will be returned. moving, as brittle cracks generally do move, along a path normal
N OT11: Statements and opinions advanced in papers are to be to the direction of greatest tension, so that the component of shear
understood as individual expressions of their authors and not those
of the Society. ManuscriJ)t received by ABME Applied Mechanics stress resolved on the line of expected extension of the crack is
Division, February 19, 1956. Paper No. 57-APM-22. zero.
361
 362 JOURNAL OF APPLIED MECHANICS SEPTEMIIER, 1957 363

REPBllSBNTATIVE 8Tuss F'JELDs Assocu.TIJD WJTB CB.A.cu

: of plane atrain. The preceding comment becomes intuitively ob-
A paper by Weotergaard (11) g&ve a convenient eemi-inverae
method for oolving a -i.ain olasa of p!ane-ctrain or plane-auess
RECIMAASLE
CRACK OPEMING
I:: ,
vious when one conaiders thati in Sneddon's example, all particle
displaoomentz lie in planeo which contain the axis of oymmetry.
problems. Let :Z, Z, and Z' repreaent successive derivatives with '· ' These p1an.. would approximate to a set of parallel plan.. within
reepeot to• of a function i!'(,), where• is (z + iy). Assume that
the Airy stress function may be represented by
i t p
any region whOBe dimensions are very small compared to distance
from the region to the a.xis of symmetry.
F - Rei!'+ y lm:Z ................. [1] FoBCE TENDING TO CAUSE CRACK. Ex.TBNSION

then -~!~~- As the crack extends, an energy transfer from mechanical or

strain energy into other form& occurs in the vicinity of the crack.
"• -

u, -
~
l>'F

l>'F
a.• -
= ReZ

ReZ
- ylmZ' ... ....•..•..• [2]

+ ylmZ'.. . ........ (3]

'
. i, - - - -.o ~-·-·-'
I
I
I
'
l
1-
:
:P a - - - - - .l
I
b_.
:
'
I
I
I
.,
STRAIN
GAUGE
The prooess is such that tran&fer of strain energy to heat dorni-
·- · nateL · -·
S is the magnitude of this energy excha.nge associated with unit
extension of the crack and may be regarded a, the force tending to
-

cauee crack extension. Thia may be aeen as follows:

FIG. 1 OnmNG OJ' A CRACE BT WBDG:ll F0RC38 RIILA.TION o:r r AND f TO I/ AND (.1: - a) AND ExAKPLJ:8 or The linear eluticity relation, resulting from Equations (5]
l>'F
T., = - <>rl>y- -yReZ'... . ... [4] L0c.ATJOM8 l'OB 8'nu.IN GA.GM through [11] oorreopond to a parabolic shape' for the oraok open•
In all of these problems a uniform compreslion, - 11.. , may be ing near the crack tip. In Fig. 3 the origin of z, v-oo-ordinates baa
By choices of the function Z(z), Weatergaard showed 110lution11 for added to the value of u,,, given by Equation [2i Sinoe linearized end of the crack serve to determine the "crack-tip stress distribu- been shifted so that the crack opening, shown by the dashed line,
stress distribution as influenced by bearing pressures or cracks in elasticity relations are assumed to apply, one may obtain the Z- tion.''
a variety of situationa. The class of problems which can be solved function for combined tension and wedge action by adding the Consider for all of the five examples the substitution of varia-
in this way is limited to those such that T S7I is zero along the ~xis. appropriate Z-function for tension to the appropriate Z-funetion blea
In particular, if a large plate cont.a.ins a single crack on the :t- for a pair of wedge forces.
a.xis whose length is amall compared to the plate dimensions or a AB an extending era.ck moves aeroee a plate of finite width the
z•a+ri1
oolinea.r series of such cracks, and if the applied loads are Neb crack may attain sufficient length so that the tensile foroes acting where
that T,, 11 is ero along the :r--axis, then the stre81!!1 distribution is to ca.use crack extension a.re not sufficiently accurate when ob--
readily constructed with the aid of Westergaa.rd's semi-inverse tained using infinite plate relations such as those of Equations r' - (z - a)• + y• and tan 8 - y/(z - a)
prooedures. (5] and (7 ]. The major adjustment required is that of the total aa shown in Fig. 2.
Two ex&m.plea of such problems were given by Westergaard load across the z-axis from the end of the crack to the side of the
(11) as follow,: plate. A convenient way to make this adjustment, if the era.ck is
If one aanµne1 quantities such as r/a a.nd r/(a - b) ma.y be
negl~ in comparison to unity, one finds in each case \ • '.,x-
1 A central straight crack of length 2a along the :i:-axis in an centered, is to use expreseions for Z such as in examples 2 and 5.
infinite plate with a biaxial field of tension u at large distances The side boundaries of the plate would then be taken to occur at
from the era.ck % ""' -l/2 and% _, +l/2. In the stress distributions resulting
from examples 2 and 5 the shearing stress -r-. is zero along the aide
"• -(~r~(:~ (1 + sin{ sin~) . . . (10]
TENSION f
~--;

Sy •• ,
"
Z(z) - (I - (a/z)•]'I• ............... [5} boundaries. However, the side boundaries are represented as
posaessing & distribution of z-direction loads which should be
and ,-o
8
2 A series of equally spaced straight cracks of length 2a, on absent. Depending upon the objectives of the stl'88I a.nalysie this
the z-axis in an infinite plate with biaxial stress u, as before, and
with the distance between the crack centers1 l
defect may be outweighed in importa.nce by the convenience of
having an approximate solution of the problem in compact form.
"• - (~r ;(:) (I - sin¾ sin~) - ,,_ ... [11] FIG. 3 LINZ.t.B-ELA.BTIO-Tmo&T CRACK. OPJENJNG8 AND $T1t£88ES
Nil.AB EJfD oJ' .a. CJU.ox
Suppose, next, that the situation to be studied HI a crack extend-
ing acr088 a finite-width pla,,te from one of the plate side bound- where Eis Young'• modulue. g is independent of r a.nd of (J and
Z(z) - [1 - (em,,.-a/Z)']w .... . [6]
aries. Let the intersection of the crack with the side boundary be
the origin of co-ordinates and let the line of crack extension be the
will be disoueiied in following aeotione of this paper.
For a crack traversing & plate, the thickness of which is con-
extende to z - a. It is aoeumed a is very small compared to the
length of the crack. If v-direction tensions given by
SID 1rz/l
positive portion of the z-axis, the end of the crack being at .i: ... a. eiderably smaller than the crack length, a generalized plane--etrees ~)•;. I
viewpoint ia appropriate and v. ie zero. However, for comparison B,(p) - p ( .. V(2") .. (12]
Three additional examples obtainable with the semi-inverae It will be aseumed that weight. or blocks have beeo s e t ~ the
procedure suggested by Westergaard are as followz: side boundaries so as to prevent or greatly reduce the tendency of with result. obtained by Sneddon (10) one may cormider for the
3 Single crack along the z..axis extending from -a to a with these boundaries to move in the negative z-direction &a the crack moment the set of three extensional stresses which would pertain are exerted on the edges of the Ol'aOk from z - 0 to z - a, a.nd p
a wedge action applied to produce a pair of "splitting forces" of extends. In this event Z-function& similar to thOBe of examples to a p!ane-ctrain analyais. Sneddon studied tho etreoe distribution is increased frot11 sero to 1, the craok is cloeed up eo that the cr&ek
magnitude P located at z - b ( ... Fig. I) 2 and 5 may again be employed as a convenient means for obtain- predicted by linear elastic theory in the vicinity of a ''penny- opening appears to end a.t the origin as shown by the full line.
ing a compact approximation to the streaa distribution. In thia shaped'' crack embedded in a much larger eolid material and 1Ub- The factor p may be regarded aa a proportional loading perame-
Pa [1 - (b/a)•J½ situation the side bound&ries of the plate would be a88Umed to be jeoted to tension perpendicular to the plane of the crack. For the ter. To the l&ID.e approximation as Equation [10], the crack
Z(,) - ,r(z - b)z I - (a/z)' · · · · · · · · · · l7 J at% - 0 and at,z _, l/2. exteneional st"resees in the· close neighborhood of the crack outer opening from z - 0 to z _,· a at a.ny time during the closure opera-
4 The situation of example 3 with an additional pair of forces In any of the foregoing examples the only streas acting at the boundary, Sneddon gave expreesions identical to Equations [IO} tion is given by
edges of the crack is the optional added stress in the %-direction and [11] with reprd to the functional relationship of "• and "•
of magnitude P at z = -b
2Pa [I -
(b/a)']'I•
- er••· An uncertainty as to proper choice of ,r•• exists for the
example discussed previously of a erack extending: from one side
to r and 8. A third extensional etreoe directed parallel to the
outer boundary of the penny-shaped crack was given by Sneddon
v(p) - (I - p) B =: /'V
2(E")' [2(a - z)]. [13]

Z(,) - .-(•• - b') I - (a/z)' . . ....... [SJ of & finite-width plate. In addition, if the crack moves ~pidly, with the remark that no oounterpart to this third extensional
Since the degree of closure is a linear function of s. the work done
determination of the stress distribution away from the crack will 8tre88 existed in a two-dimensional analys.is of streases near a
by the closing forcea as p is varied from sero to 1 is given by
5 Example 3 repeated e.long the z..axis at intervals l, and with require a dynamic-stress analysis. crack. However, the remark appliea only to the two--dimensional
the wedge action centered so that bis zero
8T.u.as ENVIRONMENT o:r THE ENn oF TBB Ca.A.ex
analyais a8IIIIJlling generaliled plane etreoe. For the two--dimen•
eional analysis aamming plane strain the third extensional etreoe,
which ii Poisson's rat.io timea the sum of er-• a.nd "•' as in Equations
f B,(l)v(O)d,: -! f("' ~ "')"' d,; - ag ... (14]

However, the streu distribution near the end of the crack can be
[IO] and [11], is the counterpart to Sneddon', third extensional Thus a9 ia the "fixed grip'' loss of energy from the strain--energy
expreeaed (a) in<rependently of un-i.aintiea ;,r both m,gnitude of
streN oomponent. Tht11 for any small region around the outer field u the crack extende by the amount a and the generalized
applied loads and of the dynamic unloading influenceo, and (b) in
such a way that records from BeV"eral main gages pl&eed near the
boundary of Sneddon', penny..haped oraok, the etreoeee, otrains, force interpretation of9 is apparent.
and diaplacements correspond to a situation which ii locally one For mathematical Bimplicity the foregoina: calculation was