Determination of stress intensity factors and boundary element analysis for interface cracks in dissimilar anisotropic materials

,!%q&eering Fracmre Meelrcrnics Vol. 43, No. 4, pp. 603-614, 1992 Printed in Great Britain.

@X3-7944j92 $5.00 + 0.00 0 1992 F’ergamon Press Ltd.

DETERMINATION OF STRESS INTENSITY FACTORS AND BOUNDARY ELEMENT ANALYSIS FOR INTERFACE CRACKS IN DISSIMILAR AMSOTROPIC MATERIALS SANG BONG CHO and KAB RAE LEE Department of Machine Design, Kyungnam University, Masan 630-701, Korea YONG SHICK CHOY Department of Mechanical Engineering, Sungkyunkwan University, Suwon, Korea RYOJI YUUKI Institute of Industrial Science, University of Tokyo, 7-22-1, Roppongi, Minato-ku, Tokyo 106, Japan A~---After determining the eigenvectors and clarifying the stress intensity factors which are used in SUO’Srepresentation of the stress and displacement fieids for an interface crack in anisotropic dissimilar materials, an extrapolation method is proposed to determine the stress intensity factors of an interface crack in anisotropic dissimilar materials by numerical methods such as FEM or BEM. In accordance with the proposed method, BEM results of the stress intensity factors for the interface crack in various material combinations are shown and discussed.

INTRODUCTION such as ceramics or composite materials are being developed for the purpose of strengthening and lightening the structure. Also for this purpose, dissimilar or bonded materials such as ceramic-metal, polymer-composite and composit+metal have become much used in wide engineering fields. With the development of adhesive techniques, the stress analysis of bonded structures becomes important. Particularly, the problems of an interface or interface crack in dissimilar anisotropic materials which induces fracture are important from the viewpoint of strength. Because the interface in dissimilar anisotropic materials has singularity of stress, the crack originates from the interface. Hence, the fracture of dissimilar materials is induced mainly from the interface or interface crack. Williams [l], Erdogan [2], England [3] and Sih and Rice [4] analysed the interface crack problem shown in Fig. la, i.e. isotropic-isotropic materials. They showed that oscillation of stresses or overlapping of crack surfaces near the interface crack tip occurs when the boundary condition of crack surfaces is taken as traction free. Because these phenomena are unrealistic, modified interface crack models have been proposed by Comninou [5], Dundurs [6], Atkinson [7] and Mak et al. [8]. However, these models are, as yet, imperfect. On the other hand, because of the development of composite or other new materials, the problems of dissimilar anisotropic materials as shown in Fig. la-f should be considered. Some studies on dissimilar anisotropic materials were reported in refs [&lo]. Suo [12] discussed a definition of stress intensity factors for an interface crack and proposed the results that three uncoupled modes can be defined by conventional methods, such as the complex function. However, few numerical results of an interface crack in anisotropic bonded materials have been reported. Therefore, after determining the eigenvectors and clarifying the stress intensity factors which are used in Suo’s expression of the stresses and displacement fields for an interface crack in anisotropic dissimilar materials, we propose an extrapolation method to determine the stress intensity factors of an interface crack in anisotropic dissimilar materials by a numerical method such as BEM or FEM. We will discuss the proposed method through consideration and comparision with other results. NEW MATBRIALS

603

SANG

fd)

(f)

(e)

Fig. 1. Material types of interface cracks in dissimilar materials.

Fig, 2. Region near the interface crack tip in anisotropic dissimilar materials.

THE STRESSES AND D~PLACE~~ FOR A GENERALLY ANISOTROPIC MATElRIAL

The constitutive law connecting the stresses a# and strains tdl for a generally anisotropic material can be written in one of the following forms [12]: 6 =

j$

%jbjs

“1 =is

c#Lj

fll

where C%l= [e11,L~tL33,2~23,2L31,2~121~ (5,)

-~51t,~~,533,~23r531t5,2~T.

(2)

Here and throughout the paper, the superscript T denotes the transpose. The six-by-six matrices aidand c4 (uV= cg *) are conventional compliance and stiffness matrices. For a two-dimensional problem, i.e. with geometry and external loading invariant in the direction normal to the x, y-plane, the elastic field can be represented in terms of three functions &(q), &(z2) and &Jtj(zj),each of which is holomorphic in its argument, xi = x + S+y,Here S, are three distinct complex numbers with a positive imaginary part, which can be solved as roots of sixth-order ~lyno~~s to be listed shortly. With these holomorphic functions, the representations for displacements u, and stresses by are uj = 2Re[i$ a,h&f] Oli =

-2Re

i LgSjG(Zj) [

Du=2Re[~~~~~~~~~,]

(3)

1

fi=l,2,3)

(4) @f

and the complex variable zj is .q=x+Sy

(j=1,2,3).

(6)

On the basis of the two Airy-type stress functions, Lekhnitskii found that the Sj values satisfy the sixth-order characteristic equation [13]: ~~(S~~~(S) - u3m2

= 0

where MS) =i+$-

2cz4$+ a,

l&q = a,, S” - 2&s l,(S)

= a,3S3 -

{ai4 +

+ @a,, + u&s2 - 26&s -+a,

u&S + (6~~+ a&S - a,.

(7)

stressintensity factors and boundary element analysis for intcxfaa cracks

605

If one assumes that the roots are distinct, the six roots form three complex conjugate pairs, from which three S, values with positive imaginary parts can be selected. The elements of the matrixes A and L which are non-singular are given by

(9) and A Ii? =

%

s:+

ai2 -

a16&+ ~h(%%

-

&4)

A~h=a2,Sh+a22/Sh-a~+~*(~-a,,lS,) A3h

=

a41 sh

+

a42/sh

-

%

+

-

a,J3)

%t%

-

A13

=

rt3@,,

S:

+

=I,

&

=

rl3@21

S3

+

dS3

-

~126) +

a23 -

au/S3

A33

=

13@41

S3

+

ar21S3

-

G)

a43 -

aetlS3

=

1,2),

+

ad3

+

dsh) -

(h = !,2) 414

w-9

where tfh

z-e

(h 2

q3=

-$

h

4

(11)

3

THE STRESSES AND DISPLACIBIENT FIKLDS FOR AN INTERFACE CRACK IN DISSIMILAR ANIsOTRoPxc MAlFRL4Ls Considering a semi-intlnite and traction-free crack lying along the interface between two homogeneous anisotropic blocks with material I above and material II below (see Fig. 2), the stresses and displacement fields near the interface crack tip were obtained by Suo [12]. Here, the stresses and relative displacements near the interface crack tip are represented by r/Z instead of t using the stress intensity factors K = K, It X2 and & as in eqs (12) and (13). 1.9 1.8_ 1.1-

t Region 1

B $1.6&I 1.413 12 -

Region P

I

I

fl

I(b)

Fig. 3. (a) A central interface crack in a fhite dissimilar plate under uniform tenstion. (b) Model of BEM analysis.

1.1-

10L

,

,

0.p

0.2

83

,

,

0.4 0.5 -2dw

,

,

,

0.6

0.7

0.8

Fig. 4. Variation of nondimensional stress intensity factors of an interface crack against2u/w.

SAN0 BONG CHO er ai.

606

Since ifeqs (12) and (13) contain the term rk = e”‘“: the ambiguity of the dependence on the measuring unit of the crack length for values of the stress intensity factors occurs as in Sun and Jih [14]. The stresses a(r) = [aZ,, tra f* in the interface a distance T ahead of the crack tip are

and the relative displacements 6(r) = [a, ) 6, f a distance behind the crack tip are cash ~lt -KV(rfi)” + 2h KV(rIl)-” +2&V,

s

1

_

1

(13)

where dj(r)

=

Uj(T*

7t)

-

Uj(r,

--71)

(j

=

X,

Y)

(14)

and i is the arbitrary length which equals the total crack length in this paper, W and W, are eigenvectors and c is a bimaterial constant number, all to be determined from the following eigenvalue problem [ 15]: fi-‘V = e2”H-‘V,

(15)

and V and V, are the eigenvectors which are normalized by (H-‘+R-‘)V=[-j,*,*]‘=;W

(151

(H-‘+R-‘)Vo=[*,*,lJT=WO

(17)

where * signifies numbers determined by the eigenvalue problem of eq. (15), and two auxiliary vectors W and W, are intraduoed for convenience. For a tw~~e~ional problem of in-plane deformation, V, and W, become zero. IIere and throughout the paper, an overbar denotes complex ~njuga~on. Here, a positive definite Hermitian matrix H in eq. (15) involving bimaterial elastic constants is defined as

H = B, + B,,

(18)

where the matrix B is a positive definite Hermitian matrix as shown in eq. (19): B = iAL-‘.

(19)

The eigenvalue e*” is positive and thus c is real; if (6, V) is an eigenpair so is (-6, V) and, ~n~uently, since there are only three eigenvalues, E = 0 is an eigenvalue and the associated eigenvector can be chosen to be real. ~~T~ATI~N OF GENERALLY

OF EIGENVECTORS V AND W FOR AN INTERFACE CRACK ANISOTROPIC MATEIUALS OF IN-PLANE DEFORMATION

For a generaMy anisotropic material, the field quantities can be expressed by two complex functions & (zt ) and &2(z2), obtained by discarding &(z3) in eqs (31, (4) and (5). The matrices A, Table 1. Non-dimensional stress intensity factor F, for @I, = (EL :::

I .0577 1.0246

LO534 1 XI225

0.4 0.5 0.6 0.7 0.8

1.1094 1.1867 1.3033 1.4882 1.8160

1.1084 1.1870 1.3013 i A834 1.8311

Stress intensity factors and boundary element analysis for interface cracks

607

L, B and H am now two-by-two. The elements for A and L can be specialized from eq. (9) with ql = q2 = 0, while B is given by [12]: B = I’AL-’

=I,

=

Im(S,+ S2) -

@J,u,,

u,2)

where S, and S, are the roots of a characteristic 2u,J3

U,,S4-

-iv,

S2u,, -

u,2)

Im(S;, + S;,)

-a,

1

equation such as

+ (2u,2 + u&F

-

2u,,s

+ a,

= 0

(21)

and where S,=u,+i(,,

S,=u,+i[,,

S4=S2.

S3=S,,

(22)

Now, one can form the matrix H in eq. (15) for two anisotropic materials:

HI2-

41

H=B,+ii,,=

H,2 + G/W,,

ik/(Hll&)

&I

H22

1

(23)

where 4,

H 22=

[

42

= k,

t, -( a:+
+

(24)

65 + t2111+ [a,, (6 + tzll,,

r

>I

2

a:+r:

(25)

1

H,2=b,,(~,52+~2t,11,+b,,@,t2+~2t,% rJ(H,,H,)=[a,,(ol,a2-r,r,)

- 4211-

(26) [4,@,~2

- r, C2) - allI,,.

(27)

By substitution of eq. (23) into eq. (15), the bimaterial constant 6, solved from the eigenvalue problem of eq. (15) for two anisotropic materials, is J-l,L=2n

1-B 1+8

W-0

where

,h,,&)

(29)

’ =(/(,I,-Ht)* Table 2. Non-dimensional stress intensity factors 4 for (E),/(E),, F, = K,/u,,/(m); values in parentheses: K,/K,] m/m,

= 3.0

(Jw(~)”

= 10.0

This study

Yuuki [lq

0.2

1.018 (-0.099)

1.017 (-0.105)

(-kg

(-kg

0.3

1.045 (-0.097)

1.050 (-0.102)

1.028 (-0.161)

1.032 (-0.168)

0.4

(-EZ)

1.101 (-0.100)

1.078 (-0.156)

1.079 (-0.163)

0.5

1.175 (-0.093)

1.176 (-0.099)

1.150 (-0.156)

1.149 (-0.161)

0.6

1.285 (-0.093)

1.289 (-0.096)

1.251 (-0.157)

1.254 (-0.161)

0.7

1.459 (-0.098)

1.476 (-0.102)

1.412 (-0.160)

1.419 (-0.164)

0.8

1.771 (-0.106)

1.783 (-0.109)

1.704 (-0.169)

1.710 (-0.175)

h/W

This study

[upper values:

Yuuki (161

SANG BONG CHO et al.

608

For the interface crack problem of in-plane deformation of two generally anisotropic materials, we obtain the auxiliary eigenvector, W, of eq. (16) as follows:

w=

i&H, HzzI

H:,)IIH,,] - i(H,JH&]

(30)

= [ W, I:WJ*

By substituting eq. (30) into eq. (16), the eigenvector V is obtained as

(31)

where d=

{H,,H,(l

- C”)-

H:z)/&/tH,&z- H:z).

(32)

EXTRAPOLATION METHOD TO DETERMINE THE STRESS INTENSITV FACTORS OF AN ~RFA~ CRACK IN DISSIMILAR ANISOTROPIC MATERIALS Here, we propose a method to determine the stress intensity factors for the defined by eqs (12) and (13) by means of extrapolating the numerical results displacements at points apart from the crack tip. Using eq. (12), we obtain the stress lu, for the interface crack in dissimilar anisotropic materials by the numerical results the crack tip as follows:

interface crack of stresses or intensity factor of stresses near

(33)

and the ratio &/lu, can be obtained by the following method: (34) where A=sin&

B=cosfJ

C=(W,cosB+

D=(W2cos8-

W,sin8),

W,sin8),

e=cln(r//) (35)

with the values W, and W, defined by eq. (30). It is necessary to evaluate the values 4 and K2 separately in order to characterize the behaviour of the interface crack and we can separate them from eqs (33) and (34). Table 3. Non~~e~on~ stress intensity factors of a central interface crack in a finite orthotropic dissimilar plate for 2afW = 0.4 lplane stress, (v& = (v& = 0.3; upper values: F, = ~/u~~(~a); v&es in pimnthescs: &/KI ; W.4 X = 0.5, (4 )I= 6%h = 100N/mm7

&/EI It, 0.5

0.45 0.4 0.3 0.1

p=l

p -0.49

1.300 1.3170 (-0.00566)

1.316 1.334 (-0.0086)

1.337 (-0.01207) 1.392 (-0.02813)

1.3556 (-0.01819) 1.415 (-0.04142)

1.331 1.349 (-0.009340) 1.371 (-0.01985)

I.697 (-0.01407)

1.746 (-0.124)

p I= 3.32

(-0.04579) 1.772 (-0.143)

Stress

intensity factors and

boundary elunent anaiysis for interface cracks

609

A similar formulation can also be deduced from the relative displacements of eq. (13). From eq. (13), 6, and 15,are obtained by &=d,K,+t,K, S,=d~K,+t&

(36)

where dk=Gk(cosfl +k tk =

sinQ+P,(sin8 -26 cos8) (k = 1 2) G,(2c cos 8 - sin 0) + P,(cos 8 + 2 sin 0) ’ *

(37)

The values G&and Fk in eq. (37) are defined by eq. (31). Therefore, the displ~ent extrapolation method is expressed as follows:

and K2 K=lim I

r-o

d2 -

4

t, (SJS,)

@J&) -

t2

*

From eqs (33), (34), (38) and (39) of the extrapolation method for dissimilar anisotropic materials, it is co&med that when anisotropic materials become orthotropic materials, Le. Hi2 = 0, the representations of the extrapolation method for orthotropic materials are consistent with Suo’s results 1121,and if anisotropic materials become isotropic, then H,, = H,, Hi2 = 0 in eqs (33) and (34), and the results for isotropic materials are consistent with Yuuki and Cho’s representations [16]. This means that Suo’s definition of stress intensity factors is consistent with the definition for an interfacial crack between two isotropic materials. These extrapolation methods for dissimilar anisotropic materials use the numerical results of stresses or displacements at points apart from the crack tip; therefore both numerical errors and the oscillation singularity problem can be avoided. It is expected that the proposed method can be useful for BEM analysis as well as FEM analysis without any special modifications. We consider the asymptotic problem. Consider a crack lying along the interface between two homogeneous anisotropic materials. If these two homogeneous anisotropic materials become an identical homogeneous anisotropic material, that is, c = 0 in eq. (15), it is not consistent with stress intensity factors for an interface crack and a homogeneous anisotropic material. If we define the

-ld-

‘h’

Fig. 5. Stress distributions at the interface of a finite dissimilar plate without a crack subjacted to uniform tension for /#II= 0”.

-l.OL

I . . 4

-‘-1.0

Fig. 6. Stress distributions at the intcrfacz of a linitc dissimilar plate without a crack subjected to uniform tension for f/J,= 30”.

Fig. 7. Stress distributions at the interface of a finite dissimilar plate without a crack subjected to uniform.tension for & = 45”.

Fig. 8. Stress distributions at the interface of a finite dissimilar plate without Bcrack subjected to uniform tension for d#$= 60”.

stress intensity factors for a crack in a homogeneous anisotropic material as 4 and I& for the in-plane problem, the following relations can be obtained from eqs (12) and (30):

K,, =K*.

(41)

When anisotropic materials become orthotropic materials, H,2 = 0, then the above relations become:

and when two materials become an identical isotropic material, H,,=ii Hu in the above equation.

Fig. 9. Stress ~~~~

at the iuterfhx of a Suite

Stress intensity factors and boundary element analysis for interface cracks

I

611

1: Composite

t=t

I: lsotropics

I

B-i -li I

a5

A

a

II

w

=

63 P

’t a2

(Ez/E$~=0.8

(Ez/E\I)*=0’5 \

\

a1 -

lE2hh=Q3 \

(E2/E1)1=O.S

\A

,

\ I

a0

- a2

Fig. 11. Variation

I

a3

I

I

06

0.7

I

0.4 as -2a/W

of K,/K, against h/W (4, = 09.

FOCAL

,

,

,

,

,

0.5

06

07

OE

I

08

at point A

0.3

0.4 -2a/W

Fig. 12. Variation of nondimensional stress intensity factors of an interface crack at point A against Za/W (fp, = 300).

RESULTS AND DIS~SSIO~

We analyse the interface crack problem in a finite plate of dissimilar anisotropic materials as shown in Fig. 3 under uniform tension by the BEM [17j and calculate the stress intensity factors Ki and KJK, by the proposed displacement extrapolation method of eqs (38) and (39). The non-dimensional stress intensity factor Fi defined as follows: jr;;.= J(G

+ G)/a~J(xa)

= Ki/g~J(xa)

(44

is used in the numerical results throughout the paper. A finite plate with a central interface crack in dissimilar anisotropic materials is subdivided into two regions of 80-90 elements for each region as shown in Fig. 3 and analysed under the plane stress condition, I

Pt

I: Composite

I: Canmite II: Is&pics H/W=10 p = 1.0 @I=300 (Ehi = (Ed1

(E&&h =0.3

S: -2a/w Fig. 13. Variation

of KJK, agninst 2a/W at point A (4, = 3w.

.* a2

a3

L

I

I a0

a4 -2a/w

as

a6

67

ad

Fig. 14. Variation of nondimensional stress intensity fattars F, of an interface crack at point A against Zu/W (f#, = 45”).

SANG BONG CHO et al.

-2alW

Fig. 15. Variation of KJK, against 2u/W at point A (lp, = 45”).

-2alW Fig. 16. Variation of nondimensional stress intensity factars F, of an intexface crack at point A against Y&z/W GA = @"I.

We analyse a central interface crack problem as shown in Fig. 3 in a finite plate of dis@milar isotropic materials and Fig. 4 shows the non-dimensional stress intensity factors, F,, obtained by the displacement extrapolation method against 2ujW for various values of (E),/@),, and (v), = (v)z,= 0.3. Table 1 shows the values of & for a crack in homogeneous material and a comparison of BEM results with Isida’s results [18]. Table 2 shows the values of Ft against the values of (B),/(E),, and 2u/ W, and compares these results with Yuuki and Cho’s results [lq. Because our results are nearly consistent with other results, our proposed extrapolation method is valid.

t

tat

I: Composite P:Isotmpics H/W t 1.0 P=l.O

0.5 -

01= 600 i Eh = Ed1

a0 Fig.

17.

, -- 0.2

0.3

0.4 -2a/W

0.5

Variation of K& qaimt (#! = ao”).

0.6

I 0.7

I 0.8 -2a/W

Za/W at point A

Fig. 18. V&@ of non-&mensionaI stress intmsity fao tars F, of an interface crack at point A against 2u/W (6, = 90”).

stress intensity factors and boundary

elementauaiysis

for interface cracks

613

Next we analyse a centrai interface crack in a finite orthotrupic (region &orthotropic (region II) plate subjected to uniform tension, as shown in Fig. 3. Table 3 shows the results of the case where (EJE,), is fixed at 0.5 but (l&/E,),, is variable. From Table 3, it can be seen that the values of Fi for the dissimilar plate approach smoothly the value for the dissimilar orthotropic plate [(EJE,), = 0.51. In this case, we can calculate Ki and Xi, using eqs (42) and (43). The values of p in Table 3 define the relation of material constants as follows in refs [12] and [19]:

-2v,,

1

where E, = E,, and Ez = E, are the Young’s moduli of the flbre direction and the direction normal to the fibre respectively. v12is Poisson’s ratio and G12is the shear modulus. From the above results, it can be considered that our proposed extrapolation method and BEM results are valid. Central interface crack in a finite plate of d&similarkotropic and anhotropic materials

Figures 5-9 show the distribution of the stresses at the interface without a crack subjected to a uniform tension stress when the fibre angles, & in region I, are 0”, 30”, 45”, 60” and 90”, and p [eq. (4511is 1. When +i = 0”, bv is nearly uniform and l;cyis nearly zero at the interface. However, when +i = 30”, 45” and 60”, the value of ov at the centrai part of the interface increases with decreasing (EJE,),, but the value of ou at the nearby traction-free edge decreases with decreasing (EJE, jr. However, when Ibt= 90”, the ~s~bu~on of stresses is equal to the case of #i = 0”. Because these phenomena contrast somewhat with the case of i~tropi~-isotropic dissimilar plate [16], these results can be assumed from the relations of the material constant for adhesive materials in each region, The accurate reason for these contrasting results must be examined in the future. Figures 10-19 show the results of non-dimensional stress intensity factors, F, and J&/k;, for a central interface crack subjected to a uniform tension stress when the fibre directions, 4, in region I, are 0”, 30”, 45”, 60” and 90”, and isotropic material is in region II against the various values U%/& 11s #1 and 24 K I:Cmposite I:Isotqiis

H/W = 1.0 p =I.0 2a/W=O.4

. I:Composite P:lsatrupics H/W =I.0 P=l.O ch =900 (0s =(E1h

= to-

(Ez/E1fr=l.O

-:

t

;

_

:

fE24%=03

Q

:

~Et/Elfr=aS

d : (Ez/E1)1=0.3

cm-

Ql t\

0

8

0.0I..‘. a2

\

1

I

0.5 Ml a7 ae 0.4 -2a/W Fig. 19. Variation of X,/K1 against h/W at point A C#,= Qw. 0.3

Fig. 20. Variation of ~on-~me~io~i stress intensity factars F, of an interface crack against CpIand Za/Ur = 0.4.

SANG BONG CHO et al.

614

When r#~,= 0”, 30” and 45”, the values of Fi increase with decreasing (E2/E,)i, but the range of variation of Ft decreases with increasing Cp,at a certain crack length. Also, when r#+= 60” and 90”, the values of 4 decrease with decreasing (E,/E,), in contrast with the case of Cp,= 0”, 30” and 45”. It is assumed that these contrasting results can occur by the definition of stress intensity factors in eqs (33) and (34) containing the term of adhesive material constants (E, , E2). When 4i = 0” and 90”, the values and the range of variation of K,/K, are smallest; on the other hand, when 4, approaches 45”, the values and the range of variation of lu,/K, are larger. Figure 20 shows the results of 15;against the fibre angle Cprand 2a/w = 0.4. When f&/E,), approaches 1, the values of Fi approach the Fi of (E,/E, ), = 1 in Fig. 20, and the reverse phenomenon of & against the values of &/E,)t occurs at the nearby (pi = 60”. CONCLUSIONS The following conclusions are obtained. (1) The eigenvectors which are used in Suo’s representation of stresses and displacements for an interfacial crack in dissimilar anisotropic materials were obtained, and the extrapolation equations for numerical methods such as FEM and BEM were proposed in this paper. (2) It was confirmed that our proposed extrapolation method can be applied to various interface crack problems in generally anisotopic dissimilar materials. REFERENCES M. L. Williams, The stress around a fault or crack in dissimilar media. Bull. seismof. Sot. Am. 49, 199-204 (1959). F. Erdogan, Stress distribution in bonded dissimilar materials with crack. J. uppf. Mech. 87, 403410 (1965). A. H. England, A crack between dissimilar media. J. uppl. Mech. 32, 400402 (1965). G. C. Sih and J. R, Rice, The bending of plate of dissimilar materials with cracks. J. appl. Mech. 86,477-482 (1964). M. Comninou, The interface crack. J. appl. Mech. 44, 631-636 (1977). J. Dundurs and A. K. Gautesen, An oppo~unisti~ analysis of the interface crack. Znt. J. Fracture 36, 151-159 (1988). C. Atkinson, The interface crack with contact zone (an analytical treatment). lnt. J. Fracture 36, 161-177 (1982). A. F. Mak, L. M. Keer, S. H. Chen and J. L. Lewis, A no-slip interface crack. J. appl. Mech. 47, 347-350 (1980). M. Imanaka, W. Kishimoto, K. Okita, N. Nakayama and H. Nagai, Fatigue life estimation of adhesive shaft joints. Int. J. Fracture 41, 223-228 (1989).

T. C. Ting, Explicit solution and invariance of the singularities at an interface crack in anisotropic composites. ht. J. Solids Structures 22, 965-983 (1986). S. Wang and I. Choi, The interface crack between dissimilar anisotropic

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M. Vable and D. L. Sikarskie, Stress analysis in plane orthotropic material by the boundary element method. Znt. J. Solids Structures 24, l-l 1 (1988). (Received 16 September 1991)