An alternative integral equation approach applied to kinked cracks in finite plane bodies

COMPUTER METHODS NORTH-HOLLAND

IN APPLIED

MECHANICS

AND ENGINEERING

84 (1990) 211-226

AN ALTERNATIVE INTEGRAL EQUATION APPROACH APPLIED TO KINKED CRACKS IN FINITE PLANE BODIES Nengquan LIU, Nicholas J. ALTIERO and Ukhwan SUR* Department of Metallurgy, Mechanics and Materials Science, Michigan State University, E. Lansing, MI 48824-1226, U.S.A.

Received 16 November 1989 Revised manuscript received 12 March 1990

In a previous paper an integral equation representation of cracks was presented which differs from the well-known ‘dislocation-layer’ representation. In this new representation, the equations are written in terms of the displacement-discontinuity across the crack surfaces. It was shown that the new technique is well-suited to the treatment of kinked cracks. In the present paper, this integral-equation representation is coupled to the direct boundary-element method for the treatment of finite bodies containing kinked cracks. The resulting approach is shown to be simple, yet very accurate.

1. Introduction

Recent attention has been focused on the development of integral-equation representations of cracks which can be coupled to the well-known boundary-integral equations for the treatment of cracks in finite bodies. In [2,3], a crack integral-equation representation is written in terms of the crack surface tractions. The tmknowns in this representation are the dislocation densities along the crack line. While this formulation is shown to be quite effective for curved cracks, the equations are shown to be invalid when the crack contains a kink, In [4-61, a crack integral-equation representation is written in terms of the resultant forces along the crack line. It is shown that, unlike the previous formulation, this one can handle kinked cracks. However, the unknowns in this representation are still the dislocation densities. Since these densities are singular at the crack tips and weakly-singular at kinks, a rather cumbersone numerical treatment is required. The crack integral-equation representation presented in [l] contains, as unknowns, the displacement discontinuities along the crack line. Since these are zero at crack tips and continuous at kinks, the numerical treatment of these equations need be no more complicated than the treatment of the boundary-integral equations themselves. In this paper, the crack integral-equation representation developed in [l] is coupled to the direct boundary-integral equation method and applied to finite plane bodies containing kinked cracks. The numerical treatment is quite straightforward, yet the results are shown to be extremely accurate. * Now

at Korea

00457825/90/$03.50

Advanced Energy Research Institute, Daedukdanji,

Chung-Nam, Korea.

@ 1990 - Elsevier Science Publishers BY. (North-Holland)

212

N. Liu et al., Kinked cracks in finite plane bodies

2. Theoretical development Consider an infinite isotropic elastic plane in which there is a point, i , at which some 'source' of stress is located and a point, x, at which the stresses are to be computeO~ At each of these points, we will be referring to internal 'surfaces', as shown in Fig. 1, described by unit normals ~ and n, respectively, and we will employ the following influence functions: ( u R ) i j ( x , ~ ) = t h e displacement in the i direction at x due to a unit force applied in the j direction at i in the infinite plane, (uc)ij(x, ~) = the displacement in the i direction at x due to a unit displacement discontinuity applied in the j direction at £ in the infinite plane, (TrR)ij(x, ~ ) = the value of ~ri at x due to a unit force applied in the ] direction at i in the infinite plane, (~rc)~y(x, i ) = the value of ~-~ at x due to a unit displacement discontinuity applied in the / direction at ~ in the infinite plane, where ~'i is a stress function defined such that the stress components are •¢rl ~" ffi Oxe '

cr2~ ffi

t~'2 Oxl '

0~1 ffi t~Tr2 cr,~ffi--Ox-~ Ox2 "

(2.1)

The influence functions are given in Appendix A. A plane elastic region D, with external boundary Fb, containing an internal piecewise smooth crack line F~, as shown in Fig. 2, is loaded by prescribed tractions tj on some part of the external boundary and prescribed displacements uy on the remainder of the external boundary. Then the direct boundary-integral equations and the integral equations developed g|

,.,.""'"T

Xi

Fig. 1. Source point, i, and field point, x, in the infinite plane.

213

N. Liu et al., Kinked cracks in finite plane bodies x 2

X

n

Fig. 2. Plane elastic region containing a crack. in [1] can be c o u p l e d as follows:

+ fro (uc),,(x, i),~u,(i) as(~), x on

(2.2)

Fb,

~,(~) = ~ (~s),j(~, z)tj(~) d~(i) - ~ (~c),j(~, i)~j(i) ds(i)

+ fro (,~c),~(x, i)Auj(i)ds(i), x on

(2.3)

F~, +

where i = 1, 2, j = 1, 2, summation on repeated indices iis implied, and Auj = U j - u j are the relative crack surface displacements.

N. Liu et al., Kinked cracks in finite plane bodies

214 3. N u m e r i c a l treatment

A simple numerical treatment of (2.2) and (2.3) is presented here in which the external boundary is approximated by M b straight elements and the crack by M¢ straight elements, as shown in Fig. 3. Equations (2.2) and (2.3) can then be written as Mb

cij(x)uj(x) = ~ fm [(uR).(X, i ) t j ( £ ) - (UC)ij(x, i)UjO/)] ds(£) m=l Mb+Mc

+

Y" fm(uc)o(x' i )

Au/(.f') ds(.~),

x on Fb ,

m=Mb+ 1

X2

crack element m

III

boundary elment m

m-i

Mb+l

M b

1

X!

Fig. 3. Discretized plane region containing a discretized crack.

(3.1)

N. Liu et al., Kinked cracks in finite plane bodies

215

Mb

~,(x) = ~ fm [(~R)o(x' i)tj(i) -

(~c)o(x,

i)u,(e)l ds(.~)

m=l Mb+ Mc

+

~" fm(Trc)ij(x' £) A u j ( i ) d s ( i ) ,

x on F c .

(3.2)

m=Mb+l

The displacements, tractions and displacement discontinuities on each element can be linearly approximated by u~(i) = N,(~:)u~ m-') + N2(~)u~ m) ,

i on element m of F b ,

(3.3)

ti(.~ ) = N,(~:)t~ 2m-t) + N2(g)t~ 2m) ,

i on element m of F b ,

(3.4)

Aui(i)=N,(~)Aa~")+ N2(~:) A u ~ +') , .~ on element m of C,

t~2m)

where", j( m ) , t(2m+l) , _j are values at the external boundary nodal point m (m = 1 , . . . , Au~m) are values at the crack nodal point m (m = g b + 2 , . . . , M b + M r ) , and

N1(~)=(1-~)/2,

N2(~)=(1+~)/2,

-1~<~:<~1.

(3.5) Mb),

(3.6)

Furthermore (m-l) + N 2 ( ~ ) x (m) ,

i-

{Nl(~)x

i on element m of F b ,

Nt(~)x(m) + N2(~)x(m+,) , .~ on element m of F¢,

ds(,~) = { [(s{bm)- s ~ " - ' ) ) / 2 ] d~: = [As~"/2] d ~ , [(s~''+')

"

"(m))/2] d~ = [Asm/2] d~ d)¢

9

.~ on element m of F , , .~ on element m of F~,

(3.7)

(3.8)

where As~' is the length of the external boundary element m and As m is the length of the crack element m. Inserting (3.3)-(3.8) into (3.1) and (3.2), we obtain

_(.).Uj(.) t,-ij

•

~ ASb/4

m=l

If.

+ Z ASb/4

!

~)(uR)ij(x ("), ~) d~] t(2m1)-I

(1-

(1 +

~)(uR),j(x ('), ~)d~: t(2m) _j

m=l

(m-l)

ASh~4

( 1 - ~)(UC),~(X~), ~) d~ uj

ASb/4

(m) (1 + ~)(UC)ij(X ("), ~)d~:]"Uj

m=!

Z m=l

[fm

N. Liu et al., Kinked cracks in finite plane bodies

216

"~+'°

+ Z

[L

AS'~/4

(1- ~)(uc)o(x ("), ~) d6] at,~ )

m=Mb+2

Mb+Mo-'

[L

As"~'I4 (1 + ~)(uc)q(x¢"),~:) d~]

+ 2

AUT+I)

(3.9)

m=Mb+l

-(")

.1-8-i

[L

( 1 - ~)('a'R)q(x (n), ~) d~] t(2m-1)_i

Z AS~/4

---

m=l

+

ZMbASb/4if. (x + 6)(#R).j(x¢"',~)d~]t~~')

m--1 Mb m=l

( 1 - ~)(#c)o(x 00, ~) d ~ ] u ~ ASb/4[L

2 AS'~/4[L (x + 6)(#c),~(x ¢"), ~)d6]-.j ¢') m=l

M b + Me

ASm/4[fm(1--~:)(Trc),(X 00, ,)d~:] Au:) m=Mb+2

+

~,

As~'/4

(1 +

~)(~'c)o(x ~"), ~)d~2

Au~ *t) ,

(3.10)

m = Mb + 1

( " ) = u/(xC")), where x ¢'') is the location of the external boundary nodal point n where u/ (n -- 1 , . . . , Mb), and 11"I") - ~'~(xC")), where x ('° is the location of crack element midpoint n (n-- M b + 1 , . . . , M b + Me). Equations (3.9) and (3.10) can be written in the following matrix form: [UCl{u} - [Ql{au} = [ U R I { 0 ,

(3.11)

[eCl(u} - [Xl{Au} = [Pal{t} - { ~-},

(3.12)

where [UC] is 2M b × 2Mb, [Q] is 2M b x 2(M c - 1), [X] is 2M~ × 2(M~ - 1) and [PR] is 2M~ × 4M b. Thus

[t. l

-to1

[rJtec]-[rJ[Xl

[URI is 2Mb x 4Mb, [PCI is 2Me X2Mb,

[{.}__[ t..l t01 Au trlteRl tIl]{ }

(3.13)

where, as in [1], we have defined a nodal force matrix on the crack by {T} = [F2]I'n'}, where

(3.14)

N. Liu et al., Kinked cracks in finite plane bodies

I OI I I tr ]=

OI 0 0

0

o

o - ! o o o

0

0

0

1) 0 - I

217

0 01 (3.15) i

and I represents a 2 x 2 identity matrix.

It is well-known that one can obtain the diagonal 2 x 2 blocks of [UC] from rigid-body considerations. If we apply a rigid-body displacement, i.e., = U1

4

=

_.

4" ~_.

(31.._

..

"~-

u~Mb)

(3.16)

then the body is stress-free. Thus {t} = { 0 } ,

{ a u } = {0}

and (3.13) reduces to [UC]{u

I 1/2 u I u 2 . . . u

I u2} t =

{0}

(3.17)

,

so that

Mb U C(2i-1)(2i-1) ~- - E uc(2i-l)(21-1) ,

Mb u c(2i-1)(2i)-.~ - - E UC(2i-l)(2J)

j=1

i=~

j#i

j~i

Mb

(3.18)

Mb

UC(21)(2i-1) =. -- E UC(2i)(2]-l)

,

U C12i)(2i) -~ - E

/=I

UC(2i)(2/).

/=l

Once (3.13) has been constructed, we must impose four conditions at each nodal point m on Fb involving the boundary values u(m) 1

'

t~2m) '

t(2m+') "I

~

U

~m)

~

t~2m)

~

t(2m+') "2

'

and rearrange (3.13) accordingly to obtain

(3.19)

[A]{Z}={F} ,

where {Z} contains the unknown boundary values on Fb and the unknown matrix {Au}. Once we have obtained Au i at each crack nodal point, the displacement discontinuities normal and tangential to the crack surfaces at that point are A u . = A u l n I ~" A u 2 n 2 ,

Aut = Au2nl - A u l n 2 ,

and the nondimensional stress intensity factors can be determined from

(3.20)

218

N. Liu et al., Kinked cracks in finite plane bodies

K~l,=o= V"~8eG(1 + v) A u n ( e ) / o ' V " qra K.Is=. = V'~/8eG(1 + v) Aut(e)/o'v"rra

,

,

(3.21) K.l.=t = V ' ~ 8 e G ( 1 + v) A u . ( l - e ) / ~

,

+ v) Aut(l - e)/¢rv"~d,

g.l~=, = ~ e G ( 1

where e--->0, G, v are the shear modulus and Possion's ratio, respectively, and I is the length of the crack. It should be noted that, for a problem involving a traction free crack, { T} = {0} and (3.13) reduces to

[UC] -[Q] u ]{t} [F I[PCl - [r2l[Xl]{Au} = [ [rd[en] [Un] •

(3.22)

4. Results

Here we employ the coupled model to find numerical solutions for some finite domain problems. 4.1. Straight crack

In all straight crack problems, the crack is modeled by 12 elements and the external boundary is modeled by 40 elements. The crack discretization is shown in Fig. 4. The stress intensity factors, normalized with respect to o-V"~, as shown in (3.21), are calculated and are compared to [7]. (a) Straight central crack in a rectangular plate subjected ¢:~ uniform uniaxial tensile stress. A rectangular plate of height 2h and width 2b contains a central straight crack of length 2a. A uniform uniaxial stress, o', perpendicular to the crack direction, acts over the ends of the plate as shown in Fig. 5. In Table 1, the stress intensity factors are given for various ratios of a/b and h/b. (b) Straight central slant crack in a rectangular plate subjected to uniform uniaxial tensile stress. A rectanguhr plate of height 2.5b and width 2b, containing a crack of length 2a, is subiected to a uniform uniaxial stress o" at the ends. The crack is located centrally at an angle

][ _,Olaeach [ ~ -[-

I=

ASa each

::[ ,Olaeach I

2a

£ ffi .01a Fig. 4. Discretization for a straight crack.

~.[

N. Liu et al., Kinked cracks in finite plane bodies

1T11*21T

219

x2

T

TI

l

ff

T 2a

D,,I xI

2.5 b

llllil

I L I I

I.

"l

2b

o

2b

I

Fig. 5. Geometry and loading for a straight central crack in a finite plate.

Fig. 6. Geometry and loading for a straight central slant crack in a finite plate.

Table 1 Stress intensity factors for a straight central crack in a finite plate

Table 2 Stress intensity factors for a straight central slant crack in a finite plate

K~ (h/b ffi 1.0)

K~ (h/b ffi 0.4)

~, = 22.5

K!

Kn

a/b

Present

Ref. [7]

Present

Ref. [7]

a/b

Present

Ref. [7]

Present

Ref. [7]

0.2 0.3 0.4 0.5 0.6 0.7

1.07 1.12 1.21 1.31 1.47 1.67

1.07 1.12 1.21 1.32 1.47 1.67

1.25 1.51 1.83 2.24 2.80 3.66

1.25 1.52 1.84 2.24 2.80 3.66

0.1 0.2 0.3 0.4 0.5 0.6

0.148 0.154 0.164 0.180 0.187 0.200

0.148 0.160 0.164 0.180 0.188 0.200

0.356 0.367 0.386 0.413 0.425 0.439

0.358 0.366 0.367 0.390 0.404 0.416

3' to the direction of cr as shown in Fig. 6. In Tables 2-4, the stress intensity factors are given for various ratios of a/b and various angles 3'. 4.2. Kinked crack In the kinked crack problems, both stress intensity factors and relative crack suface displacements are reported. The material properties selected E, v, are given in the following descriptions.

N. Liu et al., Kinked cracks in finite plane bodies

220

Table 4 Stress intensity factors for a straight central slant crack ir a finite plate

Table 3 Stress intensity factors for a straight central slant crack in a finite plate "y = 45

Kl

Kn

~ - 67.5

KI

Kn

a/b

Present

Ref. [7]

Present

Ref. [7]

a/b

Present

Ref. [7]

Present

Ref. [7]

0.1 0.2 0.3 0.4 0.5 0.6

0.500 0.513 0.534 0.550 0.610 0.616

0.500 0.517 0.538 0.550 0.618 0.606

0.500 0.506 0.518 0.522 0.535 0.551

0.500 0.502 0.510 0.522 0.538 0.551

0.1 0.2 0.3 0.4 0.5 0.6

0.861 0.868 0.900 0.959 1.002 1.085

0.868 0.870 0.900 0.958 1.003 1.118

0.355 0.354 0.359 0.374 0.377 0.388

0.351 0.351 0.356 0.374 0.369 0.380

(a) Symmetric V-shaped crack in a square plate. This example involves a symmetric V-shaped crack in a square plate subjected to pure shear stress as shown in Fig. 7. Each straight crack segment is modeled by 22 elements of equal length and the external boundary is modeled by 72 elements. In Figs. 8 and 9, the relative crack surface displacements are plotted. E = 205, v = 0.3. (b) Non-symmetric V-shaped crack in a square plate. This example involves a non-symmetric V-shaped crack in a square plate subjected to uniform uniaxial tensile stress as shown in Fig. 10. Each straight crack segment is modeled by 12 elements of equal length and the external boundary is modeled by 72 elements. In Figs. 11 and 12, the relative crack surface displacement are plotted. E - 200. v - 0.3. (c) Z-shaped crack in a rectangular plate. The first example involves a Z-shaped crack in rectangular plate subjected to a uniform uniaxial tensile stress at one end and a sliding support on the opposite end as shown in Fig. 13 ga

t t 36

(3

I_l_

36

Fig. 7. Geometry and loading for a symmetric V-shaped crack in a finite plate.

N. Liu et ai., Kinked cracks in finite plane bodies 0.40-

"+-'--r-'r--T--~'-

*

'= - I " - r '

I

+' - r ' - V - w - ' - - +

l i i II II

l ul Iii

Hll I

H

II

i'

H

H

I

I

11

H

H

Ul

m

0.20-

"

II

m H

+

HH

HH I

,

mllll

H

0.30.

'

221

Ill

H

H

II

H

I

H H

0.10H

0.00

H

'

+

'

I

2

'

'

'

I

-8

'

'

'

I

-4

'

'

"

I

0

'

'

'

I

4

'

8

Fig. 8. Relative crack surface displacement &u, for body containing symmetric V-shaped crack and subjected to pure shear.

0.050-

,

,'

r

[

,

,

,

1

,

,

T

I

r

-,-

,

"l'-v

-r-r--T--~

H lit I Ill H lil |

I H

H

m

H

0.030

•

H

,,

H

= II

I

H II

0.010"

m m

-0.010" | I

I" I

-0.030"

m m

iii

= • iiiii I

|11 mmli mmmUl 1 i l

-0.050

' -' 2

'

'

I -8

'

'

+

, -4

'

'

'

t 0

'

'

'

, 4

'

"

'

, 8

'

'

' 12

Fig. 9. Relative crack surface displacement &u2 for body containing symmetric V-shaped crack and subjected to

pure shear•

and the second example involves uniaxial tensile stress as shown in Fig. 16. The middle straight crack segment is modeled by 20 elements of equal length and each of the two other segments are modeled by 5 elements, as shown in Fig. 17. The external boundary is modeled by 90 elements. In Figs. 14 and 15, the relative crack surface displacements are plotted. In Tables 5 and 6, the stress intensity factors are reported and compared to [3, 8]. E - 205, p =0.3.

N. Liu et al., Kinked cracks in finite plane bodies

222

X2

T

I I

60 °

60 °

xj

36

I)

I I I I

l-

Fig. 10. Geometry and loading for a non-symmetric V-shaped crack in a finite plate.

0,200----,----,'---r'-~

.... ,'--r-~,

.... ,

!

",

.... ,'-'

I

",

"",""-T"

,---r----i

IN

m llllll

0,160, HI|HI iD a

0,120, II

II m m m '

[]

'

~" .

| an mlmnnmmmmm| It IN

m

m m m

i

IN

0.080'

m II m

0.040'

0.000

,,

•

Fig. 11. Relative crack surface displacement ~u. for body containing non-symmetric V-shaped crack and subjected to uniform uniaxial tensile stress.

5. Conclusions The integral-equation representation developed in [1] has been coupled to the well-known direct boundary-integral equation method and finite plane bodies containing kinked cracks have been solved using this coupled technique. The results are very close to results obtained using other methods but the present approach can be applied to arbitrary regions containing arbitrary cracks in a simple, straightforward manner.

N. Liu et al,, Kinked cracks in finite plane bodies 0.120-

--

1----+r'---I

0.0800.040-

---I--

"-'P---'-I

"+-

"-t-

,

t

'

'

i

'

'

I

223

I

,

•"mln1"1"M1n'=IEIEI E m

0.000"-

-0.040 m H

-0.080

,,. i N

-0.120" -0.160

m

m

"t • ,,, =., ,,,, ...., . I " " '

'

~3

i

,

-6

/ 0

,,

,

I 3

,

,

,

,

6

,

+

,

9

2

Fig. 12. R e l a t i v e crack surface d i s p l a c m e n t Au, for b o d y containing n o n - s y m m e t r i c V-shaped crack and subjected to u n i f o r m uniaxial tensile stress.

~X2

I I

T ii o 2a = ,5

xi 25

2a =

.5

(

20

=1

Fig. 13. G e o m e t r y a n d loading for a Z - s h a p e d crack in a finite plate, case 1.

N. Liu et al., Kinked cracks in finite plane bodies

224

0.0090-

,

,

'

!

--'

v

,

'

I

~

~

m

0.0060-

---

II

mm

0.0030-

m

m,

m

0.0000-

• n

II m

m m

m

-0.0030-

mm

toNI m

-0.oo60.

m

'

'

m

llll i

m

Iiii n

'

6

'

-'3

-6

'

*

~

'

6

Fig. 14. Relative crack surface displacement Au x for body containing Z-shaped crack and subjected to uniform uniaxial tensile stress on one end and a sliding support on the opposite end.

0.080-

"-'--',---~

|

'

'

I

i

i m m l m m m m m

mm

m

m u

0,060"

"

m

IN

llml

=

m

0,040"

m m

m

m

U

mml

m

0.020"

m

0.000

, -6

, -'3

'

'

6

'

~

"

'--6

Fig. 15. Relative crack surface displacement Au= for body containing Z-shaped crack and subjected to uniform uniaxial tensile stress on one end and a sliding support on the opposite end.

Table 5 Stress intensity factors for the Z-shaped crack of Fig. 13 in a finite plate

Present Ref. [3] Ref. [8]

Kl.

Kib

K..

Kub

4,516 4,502 4,555

0,321 0,325 0,335

4.410 4,397

0.363 0,405

Table 6 Stress intensity factors for the Zshaped crack of Fig. 16 in a finite plate

Present Ref. [3]

Kl

K.

4.592 4.600

0.382 0.410

N. Liu et al., Kinked cracks in finite plane bodies

225

g2

~

II

I I

B

¢J

2a = .5

25

.02a each

-41

.97a each

.97a each

oo, "<

2a = .5

Nil ~x

NffeN

lltll Ii

20

-I

g = .02 a

Fig. 16. Geometry and loading for a Z-shaped crack in a finite plate, case 2.

Fig. 17. Discretization for a Z-shaped crack.

Appendix A. Influence functions The influence functions for plane stress can be written as

(uR)# = [ - ( 3 - v)Sij log p + (1 + v)q,qjl/S~rG ;

(A.I)

(uc),~ = [2(1 + v)(~,q~ - n2q~) + (1 - V)~lq , + (3 + v)~2qel/4~rp , (uc)12 = t2(1 + v ) ( - a 2 q ~ - nlq2) - 3 + (1 + 3v)tieq ~ + (3 + v)~q2]/4~rp, -

(A.2)

3

(uc)21 = [2(1 + v ) ( - n 2 q 1 - ~ q ~ ) + (3 + v)a2q~ + (1 + 3v)~q2]/4~rp , (uc), 2

[ - 2 ( 1 + v)(nlqi

(~rR)~ - [ - 2 ~

-

a2q~) + (1 - v)a2q2 + (3 + v)a,q~]/a~rp,

- (1 + v)qlq2]/a~r,

( ~ ' R ) n = [(1 - v)log p + (1 + v)q~]/a~r, (~'R)21 = [ - ( 1 - ~,)log p - (1 + v)q~]/4"tr, (~'R)22 = [ - 2 0 + (1 + v)q~q2]/acr ;

(A.3)

226

N. Liu et al., Kinked cracks in finite plane bodies

G( I + v) 2~rp

- 3I + 2~lq ~_ ri2ql - 3~iq2] [2n2q

(71"C)12~_ (,./FC)21 G(I + u) [2ti1q31_ 2t72qa2- t~lqI + r~2q2] " 2"n'p ~-

(A.4)

(Irc).,., + I:) "-= G(I2xrp [-2r~q3~- 2r~'q2 + 3r~2q' + n,q2], where p " - [ ( X 1 -- ,~1) 2 -I" (X 2 -- .,~'2)2] 112 ,

ql =

X l -- "~1 ,

P

q2 =

X2 -- "~2

O= arctan[ q~r~ - q~rT._._32] qlr71 + q2v~2

P

(A.5)

"

(A.6) (A.7)

and G, v are the shear modulus and Poisson's ratio, respectively. For plane strain, u must be replaced by u/(1 + v).

Acknowledgment

The support of this work by the State of Michigan through its Research Excellence Fund is most gratefully acknowledged.

References U. Sur and N,J, Altiero, Internat. J. Fracture 38 (1988) 24-41. C.F. Sheng, J. Appl, Mech. 54 (1987) 105-109, W.L. Zang and P. Oudmundson, lnternat, J. Fracture 38 (1988) 275-294. Y.K. Chueng and Y.Z. Chen, Engrg, Fracture Mech. 28 (1987) 31-41, Y.K. Chueng and Y.Z. Chen, Theoret, Appl, Fracture Mech, 7 (1987) 177-185, W.L. Zang and P. Oudmondson, in: K, Salanm et al., eds,, Advances iq Fracture Research, Vol. 3 (Pergamon, Oxford, 1989) 2127-2134. [7] D,P. Rooke and D,J, Cartwright, Compendium of Stress IntensityFactors, Her Majesty's Stationery Office, London, 1975, [8] K,Y. Lee, D.S. Won and H,J, Choi, Engrg. Fracture Mech, 27 (1987) 75-82.

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