Mixed boundary-value problems of two-dimensional anisotropic thermoelasticity with elliptic boundaries

International Journal of Solids and Structures 38 (2001) 5975±5994

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Mixed boundary-value problems of two-dimensional anisotropic thermoelasticity with elliptic boundaries Ching-Kong Chao *, B. Gao Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan, ROC Received 15 March 2000

Abstract The two-dimensional mixed boundary-value problems for an anisotropic thermoelastic body containing an elliptic hole boundary are considered in this paper. By using the formalism of Stroh [Phil. Mag. 7, 625±646], the approach of analytic function continuation and the technique of conformal mapping, a uni®ed analytical solution for elliptic hole boundaries and for general anisotropic thermoelastic media is provided. As an application, two typical examples associated with mixed boundary-value problems are solved completely. One is an indentation problem over an elliptic hole boundary, the other is a partially reinforced elliptic hole under a remote uniform heat ¯ow. Both the contact stress under the rigid stamp and the bonded stress along the reinforced segment are studied in detail and shown in graphic form. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Analytic function continuation; Conformal mapping; Elliptic hole boundary

1. Introduction In formulating the mixed boundary-value problems in continuum mechanics, the boundary conditions usually consist of two types. One is the potential type such as temperature, velocity potential, electrostatic potential and displacement; the other is the ¯ux type such as heat ¯ux, velocity, electrostatic charge and stress. Accordingly, for elasticity problems in solid mechanics, one may classify the displacements and the stresses as respectively the potential and ¯ux type quantities. The physical of the problem requires the potential quantities be bounded but allows the ¯ux quantities to be unbounded which may be named as ¯ux singularity. If the type of boundary conditions changes at a discontinuous point of the boundary, the resulting strength of singularity at this point will be signi®cantly enhanced due to the presence of both geometric and ¯ux singularities. This is the reason why the mixed boundary-value problems are more dicult to solve than the ordinary boundary-value problems such as the traction- and displacementboundary-value problems.

*

Corresponding author. E-mail address: [email protected] (C.-K. Chao).

0020-7683/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 6 8 3 ( 0 0 ) 0 0 4 0 3 - 0

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Mixed boundary-value problems in two-dimensional elasticity such as punch problems and interface crack problems have been solved and collected by Muskhelishvili (1953) and England (1971) for isotropic materials. In solving mixed boundary-value problems the analytical function approach is found to be a simpler and powerful procedure. As to anisotropic materials, the interface crack problems were investigated by Ting (1986), Qu and Bassani (1989), Suo (1990), and Gao et al. (1992); and the punch problems for an elastic half-plane were solved by Fan and Keer (1994) and Fan and Hwu (1996). Among them, the Stroh formalism (Stroh, 1958) and the analytic function approach have been utilized and found to be the most appropriate for solving mixed boundary-value problem of two-dimensional anisotropic elasticity. Through these methods, many exact closed-form solutions have been obtained for the elasticity problems with simpler geometry such as straight boundaries. Most of important practical applications of elasticity theory are concerned with solids which have curvilinear boundaries. However, due to the mathematical infeasibility, the corresponding problems associated with curvilinear boundaries have received less attention in the literature. Recently, a general solution for the problems of rigid stamp indentation on a curvilinear hole boundary of an anisotropic elastic body was given by Fan and Hwu (1998). To solve the problems with curvilinear boundaries, a one-to-one mapping needs to transform an awkwardly shaped region to a simple one. However, there are certain cases that a one-to-one mapping function cannot be found. For example, there is no exact closed-form solution for the generally anisotropic plate containing a polygonal hole for which the mapping function is not single-valued. This is due to the diculty of ®nding the mapping functions so that the three image points associated with three di€erent eigenvalues for anisotropic elasticity on the unit circle in the transformed domain are always coincident. The only exception occurs when the hole boundary is an ellipse in which the mapping functions can be given explicitly so that any point on the hole boundary under three di€erent mappings is mapped onto a single point on the unit circle in the transformed domain. In this paper, we like to deal with the mixed boundary-value problems of two-dimensional anisotropic thermoelasticity with elliptic boundaries. Based on the Stroh formalism (Stroh, 1958), the method of analytical continuation, and the technique of conformal mapping, a uni®ed analytical solution for elliptic hole boundaries and for general anisotropic thermoelastic media is provided. A typical example of an elliptic hole boundary indented by a rigid stamp is solved in detail where the exact closed-form solutions are obtained and the correctness of the present results is checked analytically by their reduced forms such as those for the corresponding isothermal problems (Fan and Hwu, 1998) and those for the stress boundary value problems (Chao and Shen, 1998). The problem with a partially reinforced elliptic hole under a remote uniform heat ¯ow is also solved completely and the correctness of the results is checked analytically as compared to the results for the stress boundary value problems (Chao and Shen, 1998). 2. Basic equations for plane thermoelasticity With respect to a ®xed rectangular coordinate system xi , i ˆ 1, 2, 3, let ui , rij , eij , T , hi be, respectively, the displacement, stress, strain, temperature, and heat ¯ux. For the uncoupled steady-state thermoelastic problems, the heat conduction, energy equation, strain displacement relation, constitutive law, and the equations of equilibrium consist of (Nowacki, 1962) hi ˆ hi;i ˆ

kij T;j kij T;ij ˆ 0

eij ˆ 12…ui;j ‡ uj;i †

…2:1† …2:2† …2:3†

C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

rij ˆ Cijks uk;s

bij T

rij;j ˆ Cijks uk;sj

5977

…2:4†

bij T;j ˆ 0

…2:5†

in which repeated indices imply summation, a comma stands for partial di€erentiation, and Cijks , kij , bij are the elastic constants, heat conduction coecients, and thermal moduli, respectively. For two-dimensional problems, the temperature T and the displacement ui are independent of x3 . The general solution to Eq. (2.2), which is similar in form to that for antiplane deformation in anisotropic materials, can be expressed as T ˆ 2 Re‰g0 …zs †Š;

zs ˆ x1 ‡ sx2

…2:6†

where g is an arbitrary function and the prime stands for di€erentiation with respect to its argument; Re denotes the real part of a complex function; the heat eigenvalue s is the solution of k22 s2 ‡ …k12 ‡ k21 †s ‡ k11 ˆ 0

…2:7†

Here, the assumption of positive de®niteness of the heat conduction coecients has been adopted such that the heat eigenvalue s cannot be a real number in Eq. (2.7). Once the temperature function g0 …zs † is determined, the general expressions for the displacement and stress functions can be expressed as (Ting, 1996) ui ˆ 2 RefAia fa …za † ‡ ci g…zs †g

…2:8†

/i ˆ 2 RefBia fa …za † ‡ di g…zs †g

…2:9†

za ˆ x1 ‡ pa x2 ; A ˆ fa1 ; a2 ; a3 g;

a ˆ 1; 2; 3

…2:10†

B ˆ fb1 ; b2 ; b3 g

…2:11†

Hereafter, bold symbols represent vectors or matrices. f is an arbitrary function, the elastic eigenvectors fa; bg and the elastic eigenvalue p can be determined from the following eigenvalue problem     N1 N2 a Nn ˆ pn; N ˆ …2:12† T ; n ˆ N3 N1 b where N1 ˆ T 1 RT ; N2 ˆ T Qik ˆ Ci1k1 ; Rik ˆ Ci1k2 ;

1

ˆ NT2 ; N3 ˆ RT 1 RT Tik ˆ Ci2k2 ;

Q ˆ NT3

the superscript T stands for the transpose of a matrix; while the vectors fc; dg satisfy the following equation      0 N2 b1 c Ng ˆ sg c; c ˆ …2:13† ; gˆ 1 NT1 b2 d where b1 ˆ fbi1 g, b2 ˆ fbi2 g and I is the identity matrix. The vector / given by Eq. (2.9) is the stress function which is related to the surface traction t by tˆ

o/ os

…2:14†

where s is the arc length measured along the curved boundary and the material is located on the right-hand side along the increasing path. Note that the general solutions in Eqs. (2.8) and (2.9) are for a state of plane strain and valid only when p1 6ˆ p2 6ˆ p3 6ˆ s. However, for the special case in which the heat eigenvalue s

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becomes equal to a single or double elasticity eigenvalue, pa , the solutions can still be derived by making use of the following identities S ˆ i…2ABT

I†;

H ˆ 2iAAT ;

Lˆ

2iBBT ;

c~ ˆ

Lc ‡ ST d

id;

…2:15†

where the real matrices S, H, L and vector c~ can be determined from c and b1 , b2 (Barnett and Lothe, 1973; Hwu, 1990). 3. Derivations of the general solution for curvilinear hole boundaries Consider the thermoelastic problem of an anisotropic body containing an elliptic hole in which the displacement and the temperature are speci®ed over a region L of the hole boundary while the remainder of the hole boundary L0 is assumed to be traction-free and insulated from heat ¯ow (see Fig. 1). These boundary conditions characterize the mixed boundary-value problems which would produce stress singularity at the tips of the region L as stated in the ®rst section. In order to solve the problem with an awkwardly shaped region, a transformation za ˆ ma …fa † is introduced in the present problem which maps the points of a region Sf with a circular hole boundary in the f-plane onto the points of a region Sz with a curvilinear hole boundary in the z-plane. With this mapping function, all the solutions given in the last section are expressed in terms of the complex variable f instead of z. For the later use of derivation, a compact matrix form solution which satis®es all the basic equations given in Eqs. (2.1)±(2.5) is written as T ˆ 2 Re‰g0 …f†Š Z Q ˆ …h1 dx2 h2 dx1 † ˆ 2 Re‰ikg0 …f†Š

…3:1†

u ˆ 2 Re‰Af…f† ‡ cg…f†Š

…3:3†

/ ˆ 2 Re‰Bf…f† ‡ dg…f†Š

…3:4†

…3:2†

with T

f…f† ˆ ‰f1 …f†; f2 …f†; f3 …f†Š where k ˆ …k11 k22

2 k12 †

1=2

and the argument has the generic form f ˆ x1 ‡ p …or s†x2 .

Fig. 1. An elliptic hole with mixed boundary conditions.

…3:5†

C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

5979

Once the solution of f…f† …or g…f†† is obtained for a boundary value problem, a replacement of f1 ; f2 ; f3 …or fs † should be made for each component function to calculate ®eld quantities (Suo, 1990). On transforming to the region jfj P 1 of the f-plane, the boundary conditions can then be expressed as T;n …r† ˆ 0;

u;n ˆ ^ u0 …r† r 2 L

…3:6†

hm ˆ tm ˆ 0;

r 62 L

…3:7†

where r ˆ eiw denotes the point on the unit circle of the f-plane; L is the union of n arcs Lk ˆ …ak ; bk †, k ˆ 1; 2; . . . ; n, ^ u0 …r† is the given function of the displacement gradient along the tangent direction n; hm and tm are the heat ¯ux and the traction function, respectively along the normal direction m; nT ˆ … cos h; sin h; 0†, mT ˆ … sin h; cosh; 0†; and h is the angle measured counterclockwise between the tangent vector n and the positive x1 -axis (Fig. 1). 3.1. The thermal ®eld We ®rst consider the thermal problem with the unknown function g…f† which satis®es the boundary conditions (3.6) and (3.7). To solve this function the temperature gradient hm is expressed in terms of g00 …f† as i oQ o h 0 ˆ ikg …f† ikg0 …f† hm ˆ on on     d  0  of ow ozs ox1 ozs ox2 d h 0 i of ow ozs ox1 ozs ox2 ikg …f† ˆ ikg …f† ‡ ‡ ; f ! r‡ df ow ozs ox1 on ox2 on ow ozs ox1 on ox2 on df …3:8† where ox1 ˆ cosh; on ox2 ˆ qow

sin h;

ox2 ˆ sin h; on

of ˆ if; ow

ozs ˆ 1; ox1

ow 1 ; ˆ ozs q… cosh ‡ s sin h†

ozs ox1 ˆ ˆ s; ox2 qow s ox1 2 ox2 2 qˆ ‡ ow ow

cosh …3:9†

On substituting Eq. (3.9) into Eq. (3.8) we ®nd hm ˆ or

kf 00 k g …f† ‡ g00 …f†; q qf

f ! r‡

  kr 00 ‡ k 00 1 hm ˆ g …r † ‡ g ; q qr r

…3:10†

…3:11†

where the superscript ‡ …or † denotes the boundary point is approached from the region S ‡ …or S †. Using the properties of holomorphic functions and applying the method of analytical continuation (England, 1971), we may introduce h0 …f† in the form 8 < kfq g00 …f†; f 2 S ‡   …3:12† h0 …f† ˆ : k g00 1 ; f 2 S qf f With the de®nition of Eq. (3.12), Eq. (3.11) can be rewritten as

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hm ˆ h0 …r‡ †

h0 …r †

…3:13†

In a similar way, the temperature gradient in the direction of n can be obtained as dT i ˆ ‰h0 …r‡ † ‡ h0 …r †Š dn k

…3:14†

Substituting the boundary conditions (3.6) and (3.7) into Eqs. (3.13) and (3.14), we obtain the following Hilbert problem h0 …r‡ † ‡ h0 …r † ˆ 0;

r2L

…3:15†

h0 …r‡ †

r 62 L

…3:16†

h0 …r † ˆ 0;

The solution can be easily found as h0 …f† ˆ X0 …f†p0 …f† where p0 …f† is a polynomial function and X0 …f† is the basic Plemelj function de®ned as n Y 1=2 1=2 …f aj † …f bj † X0 …f† ˆ

…3:17†

…3:18†

jˆ1

The determination of p0 …f† may be found from the behavior of the temperature function at in®nity and origin which has the form for large j zj as (Chao and Shen, 1998) h0 2 z ‡ r0 zs log zs ‡ l0 log zs ‡ O…1† 2 s where h0 is related to the uniform heat ¯ux applied at in®nity de®ned as (Chao and Shen, 1998) g…z† ˆ

h0 ˆ

q0 … cosc0 ‡ s sin c0 † ik…s s†

…3:19†

…3:20†

with q0 being the magnitude of heat ¯ux and c0 the angle measured from the direction of heat ¯ux with respect to the positive x1 -axis; the constant r0 in Eq. (3.19) can be found from the condition of the resultant ^ applied at the entire body as well as the condition that the temperature ®eld is required to be heat ¯ow Q single-valued. These conditions are expressed as ^ ‰Q…f†Šc ˆ 2 Re‰ikg0 …f†Šc ˆ Q;

…3:21†

‰T …f†Šc ˆ 2 Re‰g0 …f†Šc ˆ 0

…3:22†

where ‰g…f†Šc denotes the jump of a complex function g…f† when enclosing any contour c. Substituting Eq. (3.19) into Eqs. (3.21) and (3.22) and solving for r0 , we have r0 ˆ

^ Q 4pk

…3:23†

The remaining unknown constant l0 in Eq. (3.19) will be determined once the geometry of a curvilinear hole is given. With the transformation zs ˆ ms …f† and the de®nition of Eq. (3.12), the in®nity and origin conditions for h0 …f† can be found by using Eq. (3.19) as " #   kf 00 kf r0 l0 1 0 h0 ‡ h0 …f† ˆ g …f† ˆ ; j fj ! 1 …3:24† ms …f† ‡ O q q f ms …f† …ms …f††2

C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

and

"   k 00 1 k r0 g h0 ‡ h0 …f† ˆ ˆ qf qf ms …1=f† f

#

l0 …ms …1=f††

2

m0s

  1 ‡ O…f†; f

jfj ! 0

5981

…3:25†

3.2. The stress ®eld Having the temperature function, we now intend to ®nd the stress function f…f† in Eqs. (3.3) and (3.4) which satis®es the boundary conditions (3.6) and (3.7). In a way similar to the previous approach, the surface traction tm along the hole boundary can be expressed as i o/ o h ˆ Bf…f† ‡ B f…f† ‡ dg…f† ‡ d g…f† tm ˆ on on i if i h 0 ˆ ‰Bf 0 …f† ‡ dg0 …f†Š ‡ B f …f† ‡ d g0 …f† ; f ! r‡ …3:26† q fq where we have used the relations (3.9) by the replacement of zs and s with za and pa , respectively. Eq. (3.26) can then be rewritten as "    # i ir h 0 ‡ i 1 1 0 0 ‡ 0 Bf …r † ‡ dg …r † ‡ ‡ dg …3:27† tm ˆ Bf q qr r r Now we introduce a new holomorphic function as ( 0 ‡ 0 f‰Bf h …f† ‡ dg …f†Š; i f2S 0 H …f† ˆ 1 Bf 0 …1=f† ‡ dg0 …1=f† ; f 2 S f

…3:28†

With the de®nition of Eq. (3.28), Eq. (3.26) can be replaced by H0 …r‡ †

H0 …r † ˆ iqtm

…3:29†

Similarly, the displacement gradients along the tangent direction n of the hole boundary can be obtained as   1 …3:30† H0 …r‡ † ‡ MM H0 …r † 2M Im …c AB 1 d†rg0 …r‡ † ˆ qMu;n where M ˆ iBA 1 is the impedance matrix; Im denotes the imaginary part of a complex function. Using the boundary conditions (3.6) and (3.7), we obtain the following Hilbert problem from Eqs. (3.29) and (3.30) as n  o 1 H0 …r‡ † ‡ MM H0 …r † ˆ M q^ u0 ‡ 2 Im …c AB 1 d†rg0 …r‡ † ; r 2 L …3:31† H0 …r‡ †

H0 …r † ˆ 0;

r 62 L

The general solution to this Hilbert problem is Z n  1 1 1 X…f† ‰X‡ …t†Š M q^ u0 ‡ 2 Im …c H0 …f† ˆ 2pi f L t

…3:32†

AB 1 d†tg0 …t†

o

dt ‡ X…f†p…f†

…3:33†

where p…f† is a polynomial with an appropriate singularity at in®nity and origin; X…f† is the basic Plemelj function de®ned as X…f† ˆ KC…f† where

…3:34†

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** K ˆ ‰k1 ; k2 ; k3 Š;

C…f† ˆ

n Y

++ …f

aj †

…1‡da †

…f

bj †

da

…3:35†

jˆ1

The angular brackets hh ii stands for the diagonal matrix in which each component is varied according to the Greek index a; da and ka , a ˆ 1; 2; 3 are the eigenvalues and eigenvectors of (Fan and Hwu, 1996)  1  M ‡ e2pid M 1 k ˆ 0 …3:36† In order to ®nd the arbitrary polynomial vector p…f†, we consider the complex function vector f…z† which has the form for large jfj as f…z† ˆ hhza iiq ‡ hh log za iir ‡ hhH …za †ii ‡ O…1†

…3:37†

1 where q is related to the stresses r1 ij and strains eij at in®nity as (Eshelby et al., 1953; Ting, 1988) T 1 q ˆ AT t1 2 ‡ B e1

with

8 19 < r12 = r1 ; t1 2 ˆ : 22 ; r1 32

…3:38† 8 1 9 < e11 = e1 e1 1 ˆ : 121 ; 2e13

…3:39†

The complex constant r is related to the resultant force p^ applied on the entire body and the complex function H …za † will be determined to counterbalance the singular term r0 zs log zs of the temperature function given in Eq. (3.19). For convenience of the calculation, the complex function H …za † is chosen as the form h…f† log f and using the transformation za ˆ ma …f†, the stress function f…f† in Eq. (3.37) becomes f…f† ˆ hhma …f†iiq ‡ hh log …ma …f††iir ‡ hh log fiih…f† ‡ O…1†

…3:40†

In order to determine the unknown constant r and unknown function h…f† in Eq. (3.40), the condition of single-valued displacements and the condition of the resultant force ^p applied over the entire body must be satis®ed to yield h i Af…f† ‡ A f…f† ‡ cg…f† ‡ cg…f† ˆ 0 …3:41† c

h

Bf…f† ‡ B f…f† ‡ dg…f† ‡ d g…f†

i c

ˆ^ p

…3:42†

By direct substitution of Eqs. (3.19) and (3.40) into Eqs. (3.41) and (3.42), we may immediately have the following equations Ar

Ar ‡ cl0

Br

Br ‡ dl0

cl0 ˆ 0

Ah…f†

^ p 2pi A h…f† ‡ cr0 ms …f†

c r0 ms …f† ˆ 0

Bh…f†

B h…f† ‡ dr0 ms …f†

dr0 ms …f† ˆ 0

d l0 ˆ

…3:43†

Solving for r and h…f† in Eq. (3.43), we obtain rˆ and

1 T A ^ p 2pi

AT dl0

d l0




BT cl0

cl0




…3:44†

C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

h…f† ˆ







BT c ‡ AT d r0 ms …f† ‡ BT c ‡ AT d r0 ms …f†

5983

…3:45†

With the de®nition of Eq. (3.28), the in®nity and origin conditions for H0 …f† can be obtained by using Eqs. (3.19) and (3.40) as     0 

0  fma …f† 0 0 H …f† ˆ B fma …f† q ‡ r ‡ hhf log fiih …f† ‡ h…f† ‡ d h0 ms …f† ‡ r0 log …ms …f†† ma …f†    l0 1 ‡ r0 ‡ ; j fj ! 1 …3:46† fm0s …f† ‡ O f ms …f† and

9 8 * * 0  1  ++      = < 1  1  ma f log f 1 1 H0 …f† ˆ B m0 q‡ r h0 ‡h : f a f f f f ; fma …1=f†          1 1 l0 1 0 1 ‡ d h0 m s ms ‡ r0 log ms ‡ r0 ‡ ‡ O…f†; jfj ! 0 f f f ms …1=f† f

…3:47†

Thus far we have completed the derivations of the general solution; the only unknown functions p0 …f† in Eq. (3.17) and p…f† in Eq. (3.33) will be obtained with the help of Eqs. (3.24)±(3.47).

4. Rigid stamp indentation on an elliptic hole boundary Although the solutions presented in the last section are derived for any curvilinear boundary, the exact closed-form solutions can only be obtained when the transformation functions are single-valued. A typical geometry of curvilinear boundaries is the one with elliptic boundaries whose transformation function  zs ˆ ms …fs † ˆ 12 …a ibs†fs ‡ …a ‡ ibs†fs 1 …4:1† or

 za ˆ ma …fa † ˆ 12 …a

ibpa †fa ‡ …a ‡ ibpa †fa 1



…4:2†

is found to be single-valued when the points outside the elliptic hole with the major length 2a and the minor length 2b of the z-plane are designated to map onto the points outside the unit circle of the f-plane. Suppose ^ approached from the negative x1 -axis and is loaded by a that the hole is subjected to a resultant heat ¯ow Q rigid stamp with a resultant force ^ p along the segment between …a cos u; b sin u† and …a cos u; b sin u† which is mapped onto an arc L ˆ …e iu ; eiu † in the f-plane. With this speci®cation, the general solution given in Eq. (3.17) becomes h0 …f† ˆ X0 …f†p0 …f† where X0 …f† ˆ f

e

iu




…4:3† 1=2

f

eiu




1=2

…4:4†

and the polynomial p0 …f† is determined from the in®nity and origin conditions of the temperature function. By direct substitution of Eq. (4.1) into Eqs. (3.24) and (3.25) and knowing that h0 ˆ 0 in the present problem, we have   k 1 h0 …f† ˆ r0 ‡ O ; j fj ! 1 …4:5† q f

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and k r0 ‡ O…f†; q

h0 …f† ˆ

j fj ! 0

…4:6†

where r0 is given by Eq. (3.23). Moreover, X0 …f† given in Eq. (4.4) has the form for large jfj as   1 cosu 1 X0 …f† ˆ ‡ 2 ‡ O 3 f f f and for small jfj as X0 …f† ˆ

1

f cosu ‡ O f2




…4:7†

…4:8†

Using the properties given in Eqs. (4.5)±(4.8), Eq. (4.3) yields to ®nd the polynomial p0 …f† as p0 …f† ˆ

kr0 …1 ‡ f† q

…4:9†

Substituting Eq. (4.9) into Eq. (4.3) and applying Eq. (3.12), we ®nd h0 …f† ˆ

kf 00 kr0 …1 ‡ f† g …f† ˆ p q q …f e iu †…f eiu †

…4:10†

and the temperature function g0 …f† can be determined by integrating Eq. (4.10) with respect to f as      f cosu 1 cos u g0 …f† ˆ r0 sinh 1 sinh 1 …4:11† sin u f sin u ^0 …r† ˆ 0 which implies the pro®le of the stamp is compatible with In the present problem we assume u that of the hole boundary. With this condition and the obtained temperature function (4.11), the general solution given in Eq. (3.33) becomes     Z r0 1 cos u 1 1 t 0 1 ‡ ‰X …t†Š M Im …c AB d†t sinh H …f† ˆ X…f† f sin u pi L t     1 cosu a2s sinh 1 a1s dt ‡ X…f†p…f† …4:12† t sin u t2 where a1s ˆ …a

ibs†=2, a2s ˆ …a ‡ ibs†=2, and X…f† is de®ned as DD EE X…f† ˆ KC…f† ˆ K …f e iu † …1=2† iea …f eiu † …1=2†‡iea ; iea ˆ 12 ‡ da

…4:13†

To obtain the polynomial vector p…f†, the properties of the stress function H0 …f† at in®nity and origin must be used. By direct substitution of Eq. (4.1) into Eqs. (3.46) and (3.47), we ®nd    f 1 0 H …f† ˆ B r ‡ hha ibpa iiq ‡ O ; jfj ! 1 …4:14† 2 f and

  1 H0 …f† ˆ B r ‡ hha ‡ ibpa iiq ‡ O…f† ; 2f

jfj ! 0

…4:15†

Notice that the unwelcomed singular term f log f, which has been cancelled out by the proper choice of the function h…f† de®ned in Eq. (3.45), does not appear in Eqs. (4.14) and (4.15).

C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

In addition, C…f† given in Eq. (4.13) can be expanded for large jfj as   1 G1 G2 G3 1 C…f† ˆ ‡ 2 ‡ 3 ‡ 4 ‡ O 5 f f f f f and for small jfj as

C…f† ˆ e

2uea



5985

…4:16†

1 ‡ G4 f ‡ G5 f2 ‡ G6 f3 ‡ O…f4 †

…4:17†

where G1 ˆ hh cosu ‡ 2ea sin uii; G2 ˆ

1 …1 ‡ 4e2a †…1 4

G3 ˆ

  1 3

G4 ˆ hh cosu G5 ˆ G6 ˆ

1 4

 cos2u† ‡ cos 2u ‡ 2ea sin 2u ;

  23 15 ‡ e2a ea sin 3u ‡ 4 8

    9 2 3 3 2 ea cos3u ‡ ‡ ea cos u 2 8 2



  1 ; ‡ e2a /a sin u 4

2ea sin uii;


1 ‡ 4e2a …1

  1 23 3 4

cos2u† ‡ cos 2u

  15 e2a ea sin 3u ‡ 8

 2ea sin 2u ;        9 2 3 3 2 1 ea cos3u ‡ ‡ ea cosu ‡ ‡ e2a ea sin u 2 8 2 4 …4:18†

Using the properties given in Eqs. (4.14)±(4.17), Eq. (4.12) yields to determine the polynomial vector p…f† as p…f† ˆ e2 f2 ‡ e1 f ‡ e0 ‡ e 1 f

1

…4:19†

where the constant coecients e 1 ; e0 ; e1 ; e2 satisfy the following relation (see Appendix A) 2Ke2 ˆ Bhha

ibpa iiq;







 Ke1 ‡ K 12 iea eiu ‡ 12 ‡ iea e iu e2 ˆ Br; K e 2uea e 

2ue 



 a e0 ‡ e 2uea 12 ‡ iea eiu ‡ 12 iea e iu e 1 ˆ K e

1

ˆ

1 Bhha 2

‡ ibpa iiq;

Br

…4:20†

Solving Eq. (4.20) for the four unknowns e 1 ; e0 ; e1 ; e2 , the problem is completely solved. Since the solutions obtained in this paper are new and no other analytical solution is available in the literature, the correctness of the present results can only be checked analytically by their reduced forms such as those for the corresponding isothermal problems and those for stress boundary value problems. We ®rst consider a special case that an anisotropic body containing an elliptic hole is subjected to a resultant force ^p applied on the rigid stamp. In this case q ˆ 0 and the unknown constant coecients given in Eq. (4.20) become e

1

ˆ 0;

e0 ˆ

hhe2uea iiK 1 Br;

e1 ˆ K 1 Br;

e2 ˆ 0

…4:21†

T

p as found from Eq. (3.44). where r ˆ …1=2pi†A ^ By combining Eqs. (3.28), (4.12), (4.19) and (4.21), the ®nal expression of f 0 …f† is obtained as    2uee  1 e T B 1 KC…f† K 1 BAT ‡ f 0 …f† ˆ K 1 B A ^p 2pi f which is identical to that provided by Fan and Hwu (1998).

…4:22†

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C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

Next we consider an anisotropic body containing an elliptic hole which is subjected to a remote uniform heat ¯ow with an angle c0 measured from the positive x1 -axis. If the whole boundary of the elliptic hole is traction-free and insulated, the displacement-prescribed and temperature-prescribed segment L will vanish, and the present mixed boundary-value problem reduces to a stress boundary-value problem. In such a problem, the Plemelj functions X0 …f† in Eq. (3.18) becomes a unity and the solution will be reduced to a polynomial form h0 …f† ˆ p0 …f†

…4:23†

where p0 …f† can be determined from the in®nity and origin conditions of the temperature function. Substituting Eq. (4.1) into Eqs. (3.24) and (3.25) with knowing that r0 ˆ 0, we have " #     kf l0 1 k 1 0 h0 a h0 …f† ˆ …f† ‡ O h f ‡ O ˆ …4:24† m 1s 0 s 2 q f q f …ms …f†† for large jfj and

" k h0 …f† ˆ h0 qf

#

l0 2

…ms …1=f††

m0s

  1 k 1 a1s h0 ‡ O…f† ‡ O…f† ˆ f q f

…4:25†

for small jfj. By combining Eqs. (4.24) and (4.25), we may obtain the following result h0 …f† ˆ

kf 00 k g …f† ˆ a1s h0 f q q

a1s h0 f

1



…4:26†

where a temperature function g…f† can then be determined by integrating Eq. (4.26) twice with respect to f and zs as
g…f† ˆ 12h0 a21s f2 ‡ a1s a1s h0 a1s a2s h0 log f ‡ 12h0 a1s a2s f 2 …4:27† and comparing Eq. (4.27) with Eq. (3.19), the constant l0 can be found as l0 ˆ a1s a1s h0

a1s a2s h0

…4:28†

Since the entire boundary of the elliptic hole is traction-free, the solution given in Eq. (3.33) reduces to H0 …f† ˆ p…f†

…4:29†

Substituting Eqs. (4.1) and (4.2) into Eqs. (3.46) and (3.47) with knowing that q ˆ 0 and h…f† ˆ 0, the properties of H0 …f† at in®nity and origin, respectively are found as H0 …f† ˆ Br ‡ dl0 ‡ O…f 1 †;

jfj ! 1

…4:30†

and H0 …f† ˆ Br ‡ d l0 ‡ O…f†;

j fj ! 0

…4:31†

where r ˆ AT …dl0 d l0 † BT …cl0 cl0 †. Notice that during the derivation of Eqs. (4.30) and (4.31) the complex functions dh0 ms …f† and d h0 ms …1=f† associated with the solutions of the homogeneous problem due to a uniform heat ¯ow, which will not produce stress, have been subtracted from Eqs. (3.46) and (3.47), respectively. By combining Eqs. (4.29)±(4.31), the ®nal expression for H0 …f† can be obtained as H0 …f† ˆ Br ‡ dl0 where Br ‡ dl0 is a real vector. Moreover, the temperature function can be obtained as

…4:32†

C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

g…f† ˆ a1s a1s h0




a1s a2s h0 log f ‡ 12 h0 a1s




h0 a2s a2s f

2

5987

…4:33†

The solution of f…f† can then be found by applying the ®rst term of Eqs. (3.28) and (4.33) to Eq. (4.32) and integrating the results with respect to f as
f…f† ˆ 12B 1 d h0 a2s h0 a1s a2s f 2 ‡ r log f …4:34† which is proved to agree with the one obtained earlier by Chao and Shen (1998). 5. A partially reinforced elliptic hole under remote heat ¯ow In this section, we consider a partially reinforced elliptic hole embedded in an in®nite anisotropic medium under a remote uniform heat ¯ow (see Fig. 2). Suppose that the hole is subjected to a uniform heat ¯ow q0 with an angle c0 measured from the x1 -axis and is isolated on the whole boundary including the reinforced segment between …a cos u; b sin u† and …a cosu; b sin u† which is mapped onto arc L ˆ …e iu ; eiu † in the f-plane. We know that the reinforced inclusion produces a rotation under uniform heat ¯ow, therefore the displacement vector of the reinforced inclusion can be found as (Ting, 1996)   ^ ie iu y …5:1† u ˆ e ‰a cosum…0† b sin un…0†Š ˆ e Re where y ˆ bn…0† ‡ iam…0† and e is a rotation angle. The displacement gradient along the tangent direction n of the hole boundary can be obtained as  o^ u e  y ˆ u;n ˆ …5:2† ‡ ry on 2q r and the temperature function g0 …f† can be determined by di€erentiating Eq. (4.33) with respect to f as

g0 …f† ˆ h0 a2s h0 a1s a2s f 3 h0 a2s h0 a1s a1s f 1 …5:3† With the results of Eqs. (5.2) and (5.3) and the properties of H0 at in®nity and origin given in Eqs. (4.30) and (4.31), the ®nal expression of the stress function H0 , similar to the previous case, can be obtained as

Fig. 2. A partially reinforced elliptic hole under a remote uniform heat ¯ow q0 .

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C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

 h   1  e  y 1 I ‡ MM M ‡ fy ‡ i c 2 f   
1 c A B d h0 a2s h0 a1s a2s f2

H0 …f† ˆ

AB 1 d




h0 a2s


h0 a1s a2s f 2 h0 a2s 
i h0 a1s a1s ‡ X…f†p…f† h0 a2s


 h0 a1s a1s

where X…f† is given in Eq. (4.13) and the polynomial vector p…f† is de®ned as p…f† ˆ e3 f3 ‡ e2 f2 ‡ e1 f ‡ e0 ‡ e 1 f

1

2

‡ e 2f

…5:5†

The constant coecients in Eq. (5.5) satisfy    1  1 1 e3 ˆ iK 1 I ‡ MM M c A B d h0 a2s e2 ˆ K

1

e1 ˆ K

1

  1 e 1 I ‡ MM M y 2 
h0 a1s a1s

e

2

ˆi

e

1

ˆ

e0 ˆ









e2uea e2uea e2uea  c







K

K

i

1

1

K 1

G 1 e3

  1 h 1 i I ‡ MM M c

…Br ‡ dl0 †

G 1 e2

I ‡ MM

 …Br ‡ dl0 † 1

AB d



h0 a2s


AB 1 d h0 a2s


h0 a1s a1s



c

1

AB d



h0 a2s

G2 e3

  1 1 I ‡ MM M c




h0 a1s a2s

1



1

M

e y 2


AB 1 d h0 a2s G4 e


h0 a1s a2s

2

  1 h
1 i I ‡ MM M c AB 1 d h0 a2s 
i h0 a1s a1s G4 e 1 G5 e 2


h0 a1s a1s …5:6†

where r ˆ AT …dl0 d l0 † BT …cl0 cl0 † as found from Eq. (3.44) by letting p^ ˆ 0. As for the rotation angle e , we impose the condition that the resultant moment about the x3 -axis due to the traction tm on the elliptic boundary vanishes, i.e., Z ‰a cosum…0† b sin un…0†Štm q du ˆ 0 …5:7† L

With the help of Eqs. (3.29) and (3.9), Eq. (5.7) can be replaced by Z h i i …r 2 y y† H0 …r‡ † H0 …r † dr ˆ 0 L 2

…5:8†

and substituting Eq. (5.4) into Eq. (5.8) and applying residue theory, we have a rotation angle as e ˆ

yT Khhe 2uea iiS1 yT KS2 S3 ‡ S4 ‡ S5

where the constant vectors in Eq. (5.9) are expressed as

…5:9†

C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

S1 ˆ Y1 S2 ˆ Y2

…G2

G21 †e3 ‡ G4 Y2

…G5

G24 †e 2

T

2uea

S3 ˆ y Khhe T

S4 ˆ y Khhe S5 ˆ

2uea

Y1 ˆ K

1

1

G6 †e

G31

2

G3 †e3

iiG1 Ry

…5:10†

G24 †Ry ‡

yT K…G2 2

G21 †Ry

yT KRy



  1 h 1 …Br ‡ dl0 † i I ‡ MM M c    io 1 c A B d h0 a2s h0 a1s

Y2 ˆ hh e2uea iiY1 RˆK

…2G1 G2

G34

iiG4 Ry

yT K…G5 2

and

‡ G1 Y1

…2G4 G5

I ‡ MM

1



1

5989


AB 1 d h0 a2s


h0 a1s a1s …5:11†

M

Now we have completed the solution for the problem with a partially reinforced elliptic hole under a remote uniform heat ¯ow. When the reinforced segment is assumed to vanish, the solution given in Eq. (5.4) can be reduced to the one given in Eq. (4.29) corresponding to an in®nite plate with an elliptic hole subjected to a uniform heat ¯ow at in®nity (Chao and Shen, 1998).

6. Illustrative examples In order to demonstrate the use of the present approach and to understand clearly the physical behavior of the mixed boundary-value problems, numerical examples associated with an elliptic hole boundary under indentation and a partially reinforced elliptic hole under a remote uniform heat ¯ow will be discussed in this section. 6.1. A rigid stamp on an elliptic hole boundary We ®rst consider that an in®nite body containing an elliptic hole with b=a ˆ 0:6 is subjected to a re^ approached from the negative x1 -axis and is loaded by a rigid stamp with a resultant sultant heat ¯ow Q force ^ p ˆ …^ p; 0; 0† along the segment u ˆ 30°. The material properties considered in the present study are chosen as E11 ˆ 144:8 Gpa;

E22 ˆ E33 ˆ 9:7 Gpa;

G12 ˆ G23 ˆ G13 ˆ 4:1 Gpa; a11 ˆ

0:3  10

6

1

K ;

m12 ˆ m23 ˆ m13 ˆ 0:3

k11 ˆ 4:62 W m

1

a22 ˆ a33 ˆ 28:1  10

K 1; 6

K

k22 ˆ k33 ˆ 0:72 W m 1 K

1

…6:1†

1

Based on the material properties listed above, the thermal eigenvalue s and elasticity eigenvalues pa can be determined from Eqs. (2.7) and (2.12), respectively. The matrices K and C…f† can also be found from Eqs. (3.15) and (3.36) as the stamp location is given. It is found that, based on the material properties given in Eq. (6.1), the eigenvalues da …a ˆ 1; 2; 3† corresponding to Eq. (3.36) are d1 ˆ 0:5 0:0863i, d2 ˆ 0:5 ‡ 0:0863i, d3 ˆ 0:5 which implies that the stresses would change sign near the ends of the contact portion. However, the oscillatory zone at such locations is extremely small for the case of a rigid stamp. By

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C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

using the transformation (4.1) and (4.2), the general solution of h0 …f† and H0 …f† can be found from Eqs. (4.10) and (4.12), respectively. Consequently, the function g…f† and f…f†, respectively can be determined from Eqs. (3.12) and (3.28), and the stress function /…f† is then obtained from Eq. (3.4). The contact stress rmm under the rigid stamp is related to the stress function /…f† by rmm ˆ mT …h†/;n

…6:2†

Notice that during the calculation of contact stress de®ned in Eq. (6.2), a replacement of fs ; f1 ; f2 ; f3 has been made for each function stated above. For the purpose of clearly expressing the e€ect of material properties, geometric con®guration and applied loading on the contact stress, the nondimensional parameter k de®ned as k ˆ

^ sin u a11 E11 Qb k11 p^

…6:3†

is used which must be properly chosen such that the condition of a negative (compressive) contact stress should be satis®ed. In general, k ranges from 1 to 1 which describes the indentation problem under both thermal and mechanical loading conditions. In the present case, the perfect contact is found to maintain throughout the punch face as k ranges from ±0.882 to 0.067. For k P 0:067, corresponding to a suciently large heat ¯ux into the in®nite body from the rigid punch, tensile contact stresses are predicted near the ends of the rigid stamp. For k 6 0.882, tensile contact stresses are predicted near the middle of the rigid stamp which will result in imperfect contact. The nondimensionalized contact stress ^ rmm =…p=2b sin u† with k ˆ 0:5 and 0 is shown in Figs. 3 and 4, respectively which indicates that the stress singularity is found near the ends of the rigid stamp. Notice that the oscillatory behavior near the ends of

Fig. 3. The nondimensionalized contact stress rmm =…^ p=2b sinu† along the elliptic hole indented by a rigid stamp with k ˆ

0:5.

C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

5991

Fig. 4. The nondimensionalized contact stress rmm =…^ p=2b sinu† along the elliptic hole indented by a rigid stamp with k ˆ 0.

the rigid stamp cannot be seen from these ®gures since the oscillatory zone is extremely small as discussed in the last paragraph. 6.2. A partially reinforced elliptic hole under remote heat ¯ow As a second example, we consider an in®nite body containing an elliptic hole with a partially reinforced segment u ˆ 30° under a remote uniform heat ¯ow (see Fig. 2). The material properties used in this example are listed in Eq. (6.1). With this speci®cation, the stress function /…f† can be obtained from Eq. (3.4) with the help of substituting Eqs. (5.3)±(5.6) into Eq. (3.28). Since the transformation function is singlevalued along the elliptic hole boundary, the exact solution of the radial stress rmm k11 =a22 q0 G12 a along the reinforced segment can be found by applying Eq. (6.2). The results shown in Fig. 5 (c0 ˆ 0°) and Fig. 6 (c0 ˆ 30°) reveal that the stress becomes unbounded at the ends of the reinforced portion. The oscillatory behavior at the ends of the reinforced portion cannot be seen from Fig. 5 or Fig. 6 since the oscillatory zone at such locations is extremely small.

7. Conclusions A general solution for the mixed boundary-value problems of two-dimensional anisotropic thermoelasticity is obtained by employing the Stroh formalism, the method of analytical continuation, and the technique of mapping an elliptic curve to a unit circle. Through these general solutions, two typical examples are fully discussed. Because the transformation function used in our solutions is always singlevalued, we can obtain the exact solutions in the derivations. Notice that the solution derived in Sections 4

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C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

Fig. 5. The nondimensionalized radial stress rmm k11 =a22 q0 G12 a along the reinforced segment with c0 ˆ 0°.

Fig. 6. The nondimensionalized radial stress rmm k11 =a22 q0 G12 a along the reinforced segment with c0 ˆ 30°.

and 5 are valid for the entire full ®eld only when a replacement of fs ; f1 ; f2 ; f3 should be made for each component function to calculate the displacement and stress from Eqs. (2.8) and (2.9). This observation, which was pointed out by Suo (1990), makes the analytic function continuation possible. The mixed boundary-value problems associated with a point heat source can also be derived in a similar fashion by choosing the proper expression of H …za † in Eq. (3.37) such that the existence of single-valued displacements in the entire system is guaranteed.

C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

5993

Appendix A ^ can be expressed as The external applied force p Z Z Z h i i H0 …r‡ † H0 …r † 0 0 ‡ ^ pˆ H …r † H …r † dn ˆ dr tm dn ˆ r L L q L Z Z    1

C‡ …r† 1 1 K 1 ‡ e 2pea ˆ ‰C‡ …t†Š K 1 M2 Im …c AB 1 d†tg0 …t† dt dr 2pi r t r L L Z C‡ …r† C …r† p…r† dr ‡K r L

…A:1†

By changing the order of integration, the integral term in Eq. (A.1) can be rewritten as Z Z   C‡ …r† 1 dr dt …A:2† ‰C‡ …t†Š 2K 1 M Im …c AB 1 d†tg0 …t† r† L L r…t R which is in fact zero since the integral term L …C‡ …r†=r…t r†† dr is found to vanish. Thus, Eq. (A.1) can be reduced to Z Z Z H0 …r‡ † H0 …r † C‡ …r† C …r† ^ dr ˆ K p…r† dr …A:3† pˆ tm dn ˆ r r L L L The integral terms in Eq. (A.3) can be expressed as I

H0 …f† df ˆ f

c

ˆ and

I K

c

Z c1

Z

L

H0 …r‡ † dr ‡ r 0

‡

H …r † dr r

Z Z

c2

H0 …r † dr ‡ r 0

L

H …r † dr ‡ r

Z Z

c0

H0 …f† df ‡ f 0

c0

H …f† df ‡ f

Z Z

c1

H0 …f† df f

H0 …f† df ˆ 0 f c1

Z

Z Z C…r‡ †p…r† C…r †p…r† C…f†p…f† dr ‡ dr ‡ df r r f c c1 c0  Z1 C…f†p…f† ‡ df f c Z 1 ‡ Z Z C…r †p…r† C…r †p…r† C…f†p…f† ˆK dr dr ‡ df r r f L L c0  Z C…f†p…f† ‡ df ˆ 0 f c1

C…f†p…f† df ˆ K f

…A:4†

…A:5†

where the integration contour is shown in Fig. 7. On substitution of Eqs. (A.4) and (A.5) into Eq. (A.3) we have the following relations Z Z H0 …f† C…f†p…f† df ˆ K df …A:6† f f c0 c0 and Z c1

H0 …f† df ˆ K f

Z c1

C…f†p…f† df f

…A:7†

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C.-K. Chao, B. Gao / International Journal of Solids and Structures 38 (2001) 5975±5994

Fig. 7. The integration contour for Eqs. (A.4) and (A.5).

Once the conditions of H0 …f† and C…f† at in®nity and at the origin are given, the coecients in p…f† can be determined by solving Eqs. (A.6) and (A.7) in a straightforward manner.

References Barnett, D.M., Lothe, J., 1973. Synthesis of the sextic and the integral formalism for dislocation, Green's function and surface waves in anisotropic elastic solids. Phys. Norv. 7, 13±19. Chao, C.K., Shen, M.H., 1998. Thermal stresses in a generally anisotropic body with an elliptic inclusion subject to uniform heat ¯ow. J. Appl. Mech. 65, 51±58. England, A.H., 1971. Complex Variable Methods in Elasticity. Wiley Interscience, London. Eshelby, J.D., Read, W.T., Shockley, W., 1953. Anisotropic elasticity with applications to dislocational theory. Acta Met. 1, 251±259. Fan, C.W., Hwu, C., 1996. Punch problems for an anisotropic elastic half-plane. J. Appl. Mech. 63, 69±76. Fan, C.W., Hwu, C., 1998. Rigid stamp indentation on a curvilinear hole boundary of an anisotropic elastic body. J. Appl. Mech. 65, 389±397. Fan, H., Keer, L.M., 1994. Two-dimensional contact on an anisotropic elastic half-space. J. Appl. Mech. 61, 250±255. Gao, H., Addudi, H., Barnett, D.M., 1992. Interfacial crack-tip ®eld in anisotropic elastic solids. J. Mech. Phys. Solids 40, 393±416. Hwu, C., 1990. Thermal stresses in an anisotropic plate disturbed by an insulated elliptic hole or crack. J. Appl. Mech. 57, 916±922. Muskhelishvili, N.I.. Some Basic Problems of the Mathematical Theory of Elasticity, Noordho€, London. Nowacki, W., 1962. Thermoelasticity. Addison and Wesley, Reading. Qu, J., Bassani, J.L., 1989. Cracks on bimaterial and bicrystal interfaces. J. Mech. Phys. Solids 37, 417±433. Stroh, A.N., 1958. Dislocation and cracks in anisotropic elasticity. Phil. Mag. 7, 625±646. Suo, Z., 1990. Singularities, interfaces, and cracks in dissimilar anisotropic media. Proc. R. Soc. Lond. A 427, 331±358. Ting, T.C.T., 1986. Explicit solution and invariance of the singularities at an interface crack in anisotropic composite. Int. J. Solids Structures 22, 965±983. Ting, T.C.T., 1988. Some identities and the structures of Ni in the Stroh formalism of anisotropic elasticity. Q. J. Appl. Math. 46, 109± 120. Ting, T.C.T., 1996. Anisotropic Elasticity-Theory and Applications. Oxford Science, New York.