Three-dimensional elastic crack tip interactions with shear transformation strains

THREE-DIMENSIONAL ELASTIC CRACK INTERACTIONS WITH SHEAR TRANSFORMATION STRAINS B.

L.

and

KARIH.ALOO

School of Civil and Xl~n~ng Eneineertns.

HUANG

XIAOFAS

The University

TIP

ofS>dney.

N.S \V X06.

Australia

Abstract -Three-dimensional t’l;wt~c Intcractiwts between ;t half-plane cr~dc dnd sources of internal stress such as unconstrained shexr transformation strains are anulyscd using Bueckner’s recent “Hetght fun&m” method of three-dirncnslunal crack analycis. r\nalytical exprewons are giben for the stress Inkwily factors induced along the crack front by unconstrained shear tmnsformation strains for several 4mple shaps of the trunsformation domaIn centred at the origin. When the ccntrc of the tr;msformutwn domain IF not coincident with the origin of coordinates analytical c\prcswms arc too cumhcr~~w~c t<>he (1f much practical value. In such inst.lnccs. numerical rcsulls .Irc prcscrttcd for transliwm;~twn dom;~~ns In the shape of a sphere and an ohlatc spheroid. The ~nllucncc ~~fthe oricnt.ltl~m t)f the l.Ittcr I\ aIs0 studied.

I II the I;IF~ few years considcrablc 01’ transti)rni;~tion

toughening

at ;I sharp

crack

Hudiansky

VI t/l.. I%3

front.

region.

This

continuu1n.

Iargc but unlikely R~cntly, in mode sourcc

problem

of internal I.

II

and

III

by Gao

by Rice

(I9s5). and G;IO (19SS) shear

signilic:tnl

clYcc1 upon

plant

:~nalytical crack front

simple

shapes

climcnsiomtl when

paper which

functions”

results

;Irc too cumbcrsomc

;lrc prcscntctl

spheroid.

The

for

for

particles.

the work stress The

stress

Kz(:‘)

intensity

factors

ilnd

;I

was gencralid functions”

work

rcportcd

strains

well-known

is

factors

crack

“weight

transformation

for which

(L~tmbropoulos.

particlcs

also

by Rice (I%S),

factors

induced

of transformation

analysis

is basd

Analytical

have iI

on

In all other

value.

and Gao

along

strains

trcatmcnt

in the shiipc

of the latter

;I half-

for SW~X~I.

Bucckncr’s

thrcc-

is prncticablc

only

GISCS the iIniilytici\l

In such instances, of ;1 sphcrc

numerical

and iIn obl:itl:

is also stud&i.

PRELIhllS~\RIES

the region

at location

rcportcd

intensity

at the origin.

domains

crack in an infinite such (hilt

treatment

of the transforming

1987).

is ccntrcd

of the orientation

;I half-plilnc

intensity

a half-plant

that the pioneering

It is however

2. Xl:\TIII.XlATIC/\L Consider

Rice’s

bc nod

to bc of much practical

Lvith its tip along :-itXis

for

the three-dimcnsion;ll

shear components

transf~,rnlation

inllucncc

particles

toughening.

(f3ucckcr.

domain

is csscntially

of transformed

bctwrcn

to dilatational

complcmcnts

of the transforming

“weight

the transformation

cxprcssions

using

arc given

by unconstrained

to what

strains.

orientation

tr~~nsli~rmalio1~

caprcssions

approximation

in

transformed

;I few particles.

is irrclcvant.

shape and

the whole

the stress

It should

is rcstrictcd

pxrticlcs

strains,

In the prcscnt (1958)

(IWS)

by I%ucckncr (19S7).

the shape of transforming

phase transformations

out over

interactions

19S2;

was mvdcllcJ

expressions

such ;IS transform;ltion

publishd

that

elastic

Evans,

and

zone spans only

( 19X5) gave cxuact analytical stress

to clilatational

plnnc stress

ticld such as that

lhc problem

is ;tcIcquatc if the number

I clue to thrc~-cli1ncnsic,nal

to modes

only

W;IS “smc:lred”

to bc valid if the transformed

Rice

rcccntly

I9S6)

subjoctd

partiulcs

19%).

the mechanism

stress

(McMccking

; 1,a~iibropoulos.

two-di~nunsion~~l

thrc~-clirncnsio~i;~l

by an clcvatcd

of thcsc sturlics

IW.1

crack system

the c0’cct of transformed

;L tliscrctc,

triggcrcd

the majority

; l~utliansky,

as ;I two-tlilncnsion;II which

In

advan~c~ have hccn made in undcrstantling of’ccr;iniics

elastic solid.

The

s < 0 is cruckcd.

-_’ along sv I

the crack

crack lies on the pli~nc~~ = 0 Rice (19Sj) front

shoivcd

(2 = I. II.

Ill

that refer

the to

5s’

8.

tensile.

in-phne

shear

and

L.

KARIHALW

and

X.

Ht.itx<;

shear modes) due to a distribution

anti-plane

of transformation

strains E,:, over a volume C’ may be written as

where jr is the shear modulus

of the solid and L’L,,is the fundamental stress field in mode

r defined by

In eqn (2) !I~ 3 (I!,. /1,,, I!,,,), with components If,,. are weight functions for a half-plane cracked body. ci, is the Kronecker delta, Latin indices range over _v,Y, z and when preceded by :I comma

denote differentiation

with respect to that coordinate,

and summation

is

implied by repeated indices. Expressions for CT:,,,may be found in the work by Gao (1985). An explicit solution for II, wits given by Rice (198.5). Recently. Bueckner (19S7f derived !iI for all modes in terms of a Papkovitch-Ncuber

potential function G(.r,_r. z).

whcrc

and

if and (i, roprcscnt polar co-ordinates f’

in the (s, -_) plane such that the crack faces C“ itnd

can bc t~istin~tiishc~l by & = IT, -x,

respectiv4y.

In terms of C(.V, J*.z - :‘), the s, y. : components of Il, are

If,, = -(I

-Zv)G,,

-yG,,,

hf$.= 2( I - v)G., -_lG,,, h,, = - ( I - “t*)G,, -,rG,:,. ;

whcrc

(2 - v)y,, = -G,, (“--v)/t,, (Z-

f 2( i - v)ll,,

+ Ziff,,

= -i(G,,+ZN,,)+~(i

~~)$r, = -G,,-iG,,-?(I

-v)N,: -v)ff,;;

(6)

(7- v)g,,, = ;(I -v)(G.,.tZf~f.,)--2H.: (2 - V)fI,,f = -(l

-v)G,,.+2ff.,+Zi(l

c - ~)$I,, = i(l -v)fG,,+iG,;-?ff.,.,

--P)H.~

(7)

Three-dlmenslonal

elasttc crack tip interxtions

with shear transformation

593

strains

and

H = _vG., --XC.,. For pure dilatational

transformation

(8)

strains

eqn ( I) is considerably simplified :

Kf(:‘) = where superscript

F s ‘ c-;,or(.r._v.z) dx d!

dz.

D denotes dilatation. The problem of computing k’:(Y)

completely solved by Gao (1988).

analytically

was

In this case U:, are related only to G.,,. and Il/s,V which

arc readily calculated.

C:‘:, = Z( I +v) cos (fb/7)

I;” = -31 II

[

l-8

gi

+\‘)I//,,,,- (/I = II .

sin’ ((b/2)

I’,cl,, (11)

III)

whcrc

In the prcscnt paper WC complete the solution

for unconstrained

shear transformation

strains with I::, = c,!,, (n # IPI), for which eqn (I) reduces to

whcrc WC have used the superscript intensity

S to identify the pure shear contribution

factors. If h”:’dcnotcs the mode z stress intensity

by external loading, then under any arbitrary cxtcrnal loading the stress intensity

(internal)

to the stress

factor induced at the crack front transformation

strain

field and

factors may be obtained by superposition

A-‘,(:‘) =

(14)

K:‘(Y) + K;(i) + K;(Y).

The calculation of stress intensity factors due to unconstrained shear transformation strains prcscnts several diflicultics because the derivatives of G(.r,y,:-_=‘) with respect to .V and 2 such ns G.,,, G.,,: and therefore the functions U:, (i,j # J) involve real and/or imaginary parts of log-like complex functions.

General expressions

for G.,: and G,,,, are

given in Appendix A. Formally. the expressions (I 3) may be reduced to the following non-dimensional

form.

594

8. L. &RIHALOO

K:‘(=‘)= q (,

+

{d+q (I-cos

v)

and X. HLASG

’

COG~)+4+cos

4-2

d-2

x

c&

4)

1 , [ 1 +Er C,y(:-=') .r c$-COS'~) RJ(I-C“S~)+f$(cos 1

+3?$cos~(I-cos:f#J)

(l+cor9)_S~~(I_cos’~)

-cI_~fi(f-s

-&(I-2v)G.,,

11

3

G(f) = Tr+T

,.

-

C”cl ET ri JpRz(‘_v)

(2-3v)(-

I +cos 4-2

dV

cosz (b,

[

. +4;+v+(4-7v)

cos 9-2(2--3v)

C’,,/J(Z- I’)

+I:,:4(yq,:~~>

+ ;;

+Fr

‘_r

(-4+

[

? 4$(4-

l?v-(28+4v)

(I -v)(I -11’) (2 -v)

cos2 4)

13v+8v~+v(3--4v)

cos I$-(16-32~)

cos 0)

~0s~ 4)

(17)

Three-dimensional elastic crack tip interactions with shear transformation strains

595

where p = lm J[?(.~+iy)] To arrive at the non-dimensional duced the non-dimensional

= J(Zp)

form of stress intensity

(18)

factors (15)-( 17). we first intro-

variables identified by an asterisk

y* = _t,ia. z* = z/a.

s* = x/a,

sin (f#12).

and then for brevity omitted the asterisk. Calculation of Kg for any arbitrary

K;~+‘(z*‘)=

C’ involves

KS(z’)/(E,Ia)

complicated integrands

such as

Re (G.::,), Im (G.,,,) and Im (G,:,) (Appendix A) which require numerical integration. ever. the last term in the expression for KS(Y)

Likewise. the last term in the expressions for A’;(:‘)

and k-i,(/)

F,T is a function of y alone or F:~ is an even function

will vanish provided either

of _Vand the region V is symmetric

with respect to J’ (sue Appendix B). The expressions are further strained shear transformation

How-

will vanish if E_:~is a function of y alone.

simplified

if the uncon-

strains are functions of>valone and the region Vis symmetric

with rcspcct to J’. In this case, besides the last term involving G(s.J.

z). the terms containing

the variable /J = ,,/I!/J sin (4/z) also vanish.

3. SIMPLE As mention4

above. for an arbitrary

must be evaluated numerically. integrals as inlinite G(.r.r~.:).

1-KANSI~OKM/\TION

DOMAINS

transformation

domain I/ the integrals (l5)-(

series, providod there is no contribution

In this section we will

spheroidal region (Fig.

17)

In scvcral instances however it is possible to express the first

demonstrate

from

this “analytical”

the terms involving procedure on the

I)

(19) and then obtain

complete solutions

for

the spheroidal

(X” = c’,, = 2” = 0). For the spheroidal region (l9),

region centred at the origin

and indeed for an arbitrary

ellipsoidal

region to be considered numerically in the next section, we introduce ellipsoidal coordinates r, 0. (6 s = r cos 0 COS4 J* = It r cos 0 sin 4 (1 c = = -r sin U

(20)

such that in gcncral the coordinates of the centroid of region V, _K,).,v,,. zu transform

into

a

rO. U,,, (Jo. WC have used an overbar to distinguish the ellipsoidal coordinate 4 from the cylindrical coordinate 4 appearing in integrals (I 5)-( 17) and elsewhere. 6 is related to 4 through

cos (b = cos 6

co? qS+ $

I( For a spheroidal region, 4 = 4.

I2 .

sin’ 6

)

(21)

596

B. L. KAR~HALOO

and X.

HL-ASG

la)

/

The lower (I) ;lntI upper (14) limits ofintcgr:rtion Gll~UlilttXl

for an ;irhitrary cllipsoidai region drc

;lS fOllOWS.

(31, where

and zcis the an&

bctwccn radius vectors r,, und r such that cos

SL =

cos 0 cos (J;- (6,)) + sin U sin O,,.

(24)

For p. 2 I (whcrc p. = [(XX) + (r~_~,,/h)‘]“‘. not to be confused with cylindrical coordinate p = (s’+_v’)’ ’ appearing in integrals (I?+( 17) and elscwhcrc),

where

Three-dimmsional elastic crack tip interactions with shear transformation strains

597

Y

I ----

4Yl x*ly)

\ \ !

I x

I’

/’

IJ

x’*y’,,’

(26)

y I, -0 -

(27)

For p,, c I, -x

< $ < n. Finally.

the limits

on 0 UC’ calculated

from

the solution

to the

incqualily I -rX WC note first

sin’

2 >/ 0.

(33)

that

o,,=

0

0;

Z” =

n/2 ;

PO

tan-’

otherwise.

=o

(29)

598

8. L. KARIHALCKIand X. The

lower and upper limits

on 0 following

HCASG

from the solution

of (28) are given in

Table I. In Table I

zz = tan-’

r; - I -cos’

+fJ,, -._I_

1 r3 sin 20,, cos (&--61;,,)

1

For the special region described by

F,, = 0.thereby

considerably

simplifying

(30)

(19). cz= h.d;,, = 0 and the limits

borne in mind that when I.Y~~Id 0 and Ij.,,l -C h the integrals mathematical sense. For

[.r,,J ,< 0 the integrals

can

[t should however be

have n meaning only in the

be interpreted

/_I*~~/ 2 h. For the region (1% and constant transformation simplify

~6 = (b, such that I),, = .x1,.

of integration.

strains

physically the integrals

only when

(IS)-( 17)

to read

where (rn # n, N = I, If, III)

and Y!” between C/.P:tnd between cos O/sin 0 muns in (34) togcthcr with the corresponding co&icicnt

that cithcr cos 0 or sin 0 term appears n;,/L~‘,t,,,]or o;,[V,:,,,].

Three-dimensional In (U).k

=O.

l.Zand

clastlc crack tip interactions

i=O.

with shear transformation

strains

599

l.2.3and

(35)

The only non-zero coefFicirnts a;,[C’t]

and L~,[LLI are (36)

a;,[C”l-‘,-1: rr:,,, = -(7-3~). UC; ” = - 41,. tr;, = 32~.

~~~S~i:‘r’ I:

(I;;, = (2-3~). nc,, = 4(4-7v).

ofsin

=

-S(3--3jv)


(1’;, = 32 - 4,.,

The next step in the evaluation ofintcgrals in (3-I) in powers of sin 0 and cos 0. This multiple sums involving powcrs

tr’;:

ai: = 64(1-v),

11’;,) = - 32.

tr;;,, = 1+ I’.

air,: = -2(3-3v)

(3 I)-(33)

(I’&, = 37,.

(37)

([i, = _ 33.

(40)

is to expand rU. r,and h, appearing

allows us to cxprcss the inkgrills

(3l)-(33)

as

0 and cos 0, but WC find that the resulting cxprcssions

for K,:(O) arc too cunibcrsomc to bc of :lny practical USC thorcby negating any advantage this proccdurc may have over the direct numerical integration

of (IS)-(

17) that will be

prcscntcd bslow. tlowcvcr. for a spheroidal region (a = h) centrcd at the origin, the integrals (31) (33) can in fact bc cvaluatcd as ;L single intinite expansion in powcrs

of sin

sum using the procedure involving

t) and cos 0.

In this special GISC, it can bc shown that the only non-zero components of the funclamcntal fields U,:,, ;trt: U’,‘,. and Ul.!. Therefore.

for constant transformation

strains, we

have

K; = 0

(41)

(42)

2

k’ihuv = (I+:

s

y U:,‘j

dV

K;(O) = 0

16

+ IS(I+V)(VI+I)R

-g(ll-9vJ(r;)_B(;.~z+;)}

(43)

(44)

(45)

600

B.

i.. KARIHALOC)

,inJ X. HL.AX

(46)

K’:(O)= 0

(47)

In (44). (5’)

The :tnalytic:~l expressions (SO)-(Q) gcncral numerical integration

wcrc :I!SO useful for chcckins

4. NIJhiERICAL

For an arbitrary

the accuracy of the

procedure which is dcscrihed in the next section.

ellipsoidul domain

RESULTS

AND

DISCUSSION

1’ ccntrod at .x,,, .I’,,. I,,

and constant transformation strains E,:_, the integrals (I 5)-( 17) wet-c cvatuatcd numerically using the Gaussian method. To enhance the rate of convcrgcncc of the singular integrals as R -0. ellipsoidal coordinates (20) were chosen which alIo\scd formal intc~r~tion of (I 5)-( 17) with respect to the coordinate r such that the inkgrands took the form

Three-d~mrnsional

elastic crxk

tip interactions

with shear transformation

stems

601

where the upper and lower limits of integration on r are given by (23). The ranges of integration with respect to 8 and c,$given by expressions (25)-(29) were each divided into 2’ = 8 intervals. with each interval being represented by six Gauss points. The accuracy of the numerical integration scheme was checked in two ways. First. by comparing with the analytical results (Xl)-(52). ft was found that the numerical results differed from these by less than +O.Ol. Secondly. by increasing the number of intervals to 2’ = 16. It was found that the results hardly differed from those obtained with only 2’ = 8 intervals. The numerical integration scheme was used for an exploration of the problem at hand for a general eIli~soidal domain V. In particular, attention was focussed on the spherical domain (n = h = c) and the ablate spheroidal domain with a = b and a/c = 5. The choice of these two shapes was dictated by the fact that zirconia particles are spherical when found in an alumina matrix or DSthin ablate spheroids when found in partially stabilized zirconin. In an extensive numerical exploration of complex and singular integrals it is essential to have at least some independent checks on the qualitative, if not the quantitative, accuracy of the results. This was done by a close examin~ltion of the properties of C& appearing in integrals ( 15).-( 17). It was found. for instance, that CC,, possess several symmetry properties with respect to J*and : which intlucncc the final result (i.e. K.:). The behaviour of ZC,, with respect to ~8and z for various ~1. II and a is evident from Table 2. Moreover, it was found that A-:(-y)

z

-A’:(y);

-0

-

-

A’;( -:)

=

-K.f(:);

,)‘,,

=

K;v,(-y)

=

K,\;(y):

=o= 0.

-0 0

(56)

Finally, it was found that the donlin~lnt contributions to h’f, k’i and A$, came from i&._ Utt and Uvj. respectively. The remaining U:,, made but littlecontribution to k’z. Therefore, hlf;. h”.;. h?;, exhibited essentially the respective symmetry bchaviour of U’,,, Uit and Uif,: (Table 2). This observation was useful not only for performing a qualitative check on the numerical results, but also in simplifying the graphical presentation of the latter, in that the stress intensity factors could be plotted to within a factor of the strain t& corresponding to the doi~~in~lntUk. Thus in the figures to follow, the scale for KS should actually be read as A’:‘/er,..In dimensioned quantities this scale represents (KfE,ja)/e&. Similarly, the scale for h”;;Iin dimensioned yuantitics should read (K~&/u)/E~~, and for Ki, should read (K;,f:‘Ju)/i:~:. It is now possible to comment on the typical behaviour of AL::’ as demonstrated

by the numerical integration procedure (V = 0.3). This behuviour was found to be qualitatively totally consistent with the above observations. Typically, IKrc’pI for ujc = I was found to be consistently larger than that for u/c = 5 (Fig. 3). However, the variation of IKK:‘Iwith I=,11> 0 was similar for both shapes; IA’ij decreased with increasing Iz,,l 2 0, initially rather slowly but then more rapidly.

Trrhlc 2. Behaviour mm

of Liz

a = II

a =

III

602

B. L. KARIHALOO

and X.

HLXSG

The shape of region V has ;I far more pronounced influence upon lK;l IKtl.

lK;il

than upon

for (I/C*= 5 is not only much smaller than for a/c = I but it diminishes

rapidly

with increasing /z,,/ > 0. For q’~’ = 1, fK;l seems to achieve (Fig. 4) the maximum value at or near Izol = i/2 and then dccrcases with increasing IzoJ > l/2. As far as Ilrlt,l

is concerned, the shape and location of V seems to have an cfTcct on it

similar to that on IK;‘l. Thus. 5). However.

IK:,l

for (I/C = 1 is consistently

larger than for N/L’ = 5 (Fig.

for both shapes it decreases with increasing I:J

decrease is ditrcrcnt.

For

a/c = 5, it decreases rapidly with

> 0. although the rate of increasing izi,l > 0, but for

U/C = I it decreases rather slowly in the beginning. Up to now, we wcrt’ only interested in the shape of the tansformation found in particular that spherical particles give :L larger IK:I

domain. WC

than do ablate spheroids with

long axis normal to the crack front (i.e. parallel to .r-direction). It is known howcvcr (Lambropoulos, 1956). that the orientation of the long axis of obltite spheroids r&five to the truck plane has I\ significant inffucnce on the t’xtcnt of change in lu, due to dil~lt~tion~l

tr~tnsform~ltion

cxaminc the inlluencc of orientation due to shear transformations

strains.

It is therefore of some interest to

of oblatr: spheroids on the stress intensity

strain. WC consider an additional orientation.

factors li.z

nomoly when

the long axis is par:~lIcI to the crack front (i.e. parallel to z-direction). An exnmplc of this orientation of oblatc spheroids which corresponds in size to the previously considcrcd ori~nt;~tion (long axis parallel to x-direction)

is simulated by choosing u = h, G/C= 0.2.

When the long axis is parallel to the crack front (i.e. parallel to z-direction) numerical computations for N = h, a/c = 0.2 show. as expected. that the variation of K:(O) is more pronounced with Izol than was the case for rr = h. n/c = 5. Thus the maximum value of Ik’,j increases from around 4 at :,, = 0 to about I4 at I:,J = 0.5 and further to about 75 at {zol = 1.0. The incrcasc is cvcn greater in IKnl. Its maximum value increases from about 0.6 at z* = 0 through about 22 at jzO/ = 0.5 to about 850 at /zll/ = I .O. The largest inorcasc is noticed in J&j. Its maximum value increases from about 8 at z. = 0 through about 1-I

Three-dimensional

-6 t

Fig. 4. V;lri:ltion

rlltstic crack tip interactions

2,=-l.

of A’:,(O) wth

with shear transformation

y,=-1, Y

vI, lilr

two lypical values of ,ru and z,, for :I spherical ((1 = h = c).

5

1

Fig. 5. Vsriation

603

strains

of F;;,(O) with .x0 for two typical values of _voand of tr!c ((I = h)

region

b0-l

8. L. iCutwroo

and

X.

H~~sti

at I:,,/ = 0.5 to ;t colossal IS50 at I:,,/ = 1.0. This behaviour is quite consistent with that expected from analytical considerations. Similar changes to K:(O) can be expected when the tong axis is normal to the crack plane (i.e. parallel to J,-direction). In conclusion, it should be added that further in-depth numerical exploration is in progress to pain a deeper insight into the extent of variation in K:(-_‘) due to changes in the position and orientation of 1’. In particular. attempts are being made to derive useful analytical expressions for K:(-_‘) in the limit as AI--. 0.

REFERENCES

(A2)

where

Three-dimensional

elastic crack tip interactions

G

with shear transformatloo

straios

605

I

Rez=Co i

i

Im g = C,

c

RewRe;j-ImwIm;s b Re w Im f

+ Im w Re

,

6

(AS)

In (A5)

such that

G(S..v.I-_,‘)

=

50 i

nnd

(A6)

(A7)

whcrc the *‘-”

sign is chosen when (-_- -_‘) < 0. t~‘mally. G.I,z is yivcn by

where

(A91

and

Rc:=~& c ImA=

c

I d’d’.:

(-_-,_‘)[32’-(:-;‘)‘I*

(AlO)

606

8. L. KARIKUOO

and X.

APPENDIX

HUNG

B

In order to prove that the last term in each of the three expressions ( I5)-( of 1 alone. consider the region I’ with the closed. smooth surface

J,

.r,(t.) :,(.r._v)

17) vanishes when E:< is 3 function

Q.V S.&z

s .x s .x:(1.) < : 5s 2~(.~..1.).

(BI)

the region &*consider the subregion &” consisting of the torus
(B’)

R&king

to Fig. 2 and assuming in the tirst instance z, (0.0) # 0 arid s,(O. Ot f 0, we have

The lir\t Integral in the right-h;lnd side of (B3) vanishes bccausc

ofcontinuity conditions

. J

cf;(,19C..l(.r..tf.2) d t’ = - 2c_~(fh:)[G(r:\‘(I -fl’), fIt:,z,(r:V’(I - fF), Or:)) -G( -r:J(I

-ff”).flL:If

-r:J(

I

-ctp,tk:))

-G(EJ(I -f~‘),il,:, :,(c,‘(l -flq,ij,:)) 4-a -I:vs(I -fI~).fk.:,( -E”‘(

%lcc 1~10.0) f 0. z,(O,O)

I

-f~~).flc))j,:.

(135)

# 0, it is clear from (BS) that

!Y

1:

c.!,(.v)G.,,(.r.,s,,-)dl. = 0.

(Dh)

I

(B7)

In ;Lsimilar mxnner. it may lx shown that

lirn

r-0

I.

c;‘,(_r)bG,,,,dr- = 0.

WC will now prove the rcsult5 (B6). (87) when -_z(O.O) = 0. In this cxsc of cvursc :,(O,O) # 0. From (B5) it is clear that .x - O(r) and y _ O(E). In view of the fact that S’ IS a qualretic 111;1t: _ O(c). Conscqucntly.

; - O(\‘E).

q -

O(,!c).

In

Jrt

>

-O(l)

surfacc it follows

(IW)

Threedimensional

elastic crack tip interactions

with shear transformation

strains

607

and

G----.

I (69)

O(\‘E)

In the right-hand

side of (BSJ. the terms

(I-f)‘).Bc.~:(~~‘(l--~).fJ~))c

G(~E\

(BIO)

cm be shown to be of O(v;e). Therefore. these terms vanish as E + 0. and we recover the result (86). Likewise. in deriving the result (87) we would encounter the terms J.G.,(~E\‘(I which may be rewntten

-O’)./)R.I:(~E\‘(~

-H-‘).&))E.

(BIO

as

(BI?)

In view of (BX) and (BU). the term (812) (87).

is of O(,‘C). Therefore

it will vanish as E -

Il. and we mover

the result