Green's function for an anisotropic film-substrate embedded with a screw dislocation

Engineering Analysis with Boundary Elements 29 (2005) 624–635 www.elsevier.com/locate/enganabound

Green’s function for an anisotropic film-substrate embedded with a screw dislocation H.Y. Wang, M.S. Wu* Division of Engineering Mechanics, School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, Singapore 639798 Received 22 September 2004; revised 6 January 2005; accepted 8 January 2005 Available online 8 April 2005

Abstract In this paper, the Green’s function for an elastically anisotropic dissimilar film-substrate embedded with a screw dislocation is obtained analytically in series form by the method of continuously distributed image dislocations. The film-substrate interface conditions are satisfied exactly a priori, and it remains to satisfy only the free surface condition. This leads to a Fredholm integral equation of the second kind with the image dislocation density as the unknown. The solution is expressed in the form of a Neumann series, whose terms are certain integral transforms of rational functions. The series can be used as Green’s function for a broad range of thin film problems, e.g., mode III crack, screw dislocation pileup, and edge stress concentration problems. Numerical studies of the Green’s function focus on a screw dislocation in various semi-infinite film-substrate systems. The solutions satisfy the free surface condition very well using only three terms of the series. The importance of the free surface, elastic anisotropy and material inhomogeneity is emphasized in the numerical investigations. q 2005 Elsevier Ltd. All rights reserved. Keywords: Green’s function; Screw dislocation; Film-substrate; Image method

1. Introduction Thin films are ubiquitous in contemporary technological applications. Defects such as dislocations and cracks play a critical role in the performance and reliability of thin-film devices. As such, the elastic stress field of a dislocation in a film-substrate is of central interest for the analysis and design of such devices. The early works focus on the problem of a screw dislocation in homogeneous and inhomogeneous isotropic materials, e.g., a half space ([1]), an infinite B–A–B tri-layer consisting of a strip A bounded by semi-infinite B layers ([2]), and a multilayer comprising alternate A and B layers ([3,4]). The effect of elastic anisotropy was considered by Barnett and Lothe [5], who obtained solutions for a mixed dislocation in an anisotropic half space. Subsequently, dislocations in anisotropic half spaces, bimaterials, trimaterials, strips and layered media were investigated, e.g., Suo [6], Ting [7], Alshits and * Corresponding author. Tel.: C65 6790 5545; fax: C65 67911859. E-mail address: [email protected] (M.S. Wu).

0955-7997/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2005.01.013

Kirchner [8,9], Choi and Earmme [10], and Wu et al. [11]. Alshits and Kirchner [8,9] obtained solutions for singular sources in a multilayer by means of the Fourier transform method. This requires the inversion of the transform. Choi and Earmme [10] studied the problem of a general dislocation in an anisotropic trimaterial by the method of analytic continuation and the Schwarz-Neumann alternating technique. The solution is expressed in terms of infinite series for the analytic functions from which the elastic field can be derived. Using this technique, Choi and Earmme [12] also obtained closed-form solutions for a screw dislocation in a homogeneous strip. In contrast, Wu et al. [11] used the image dislocation method to obtain exact solutions for the problem of an edge dislocation in an infinite anisotropic bimaterial. The solutions for screw and edge dislocations in a domain can also be used as Green’s functions for solving a broad range of problems, e.g., crack, dislocation pileup, dislocation wall, and edge stress concentration problems. A brief review of the literature shows the importance of Green’s functions for the boundary element method (BEM). For instance, Pan et al. [13] obtained the Green’s function for homogeneous, generally anisotropic solids, while Denda [14] developed the boundary element method using

H.Y. Wang, M.S. Wu / Engineering Analysis with Boundary Elements 29 (2005) 624–635

the Green’s functions for the line force and the dislocation dipole in dissimilar isotropic bimaterials. The use of such functions for a bimaterial in the BEM does not require boundary elements to model the interface. This leads to a gain in the accuracy of the numerical results and a reduction in the solution effort. In this paper, we use the image dislocation method to develop the Green’s function for a screw dislocation in an anisotropic film-substrate which has an interface and a free surface. The problem of an edge dislocation is considered elsewhere ([15]). The method does not require the inversion of transform, and the solutions for the image density and the stress field can be obtained formally in exact series form. The interface boundary conditions are satisfied a priori while the free surface condition is satisfied exactly if the entire series is considered. The terms of the series represent progressively weaker interacting effect between images on the interface and the free surface. Mathematically, these terms are certain H and I transforms of rational functions with well-defined properties. As a result, the series solution can be further manipulated if needed and can be used as a Green’s function. The organization of the paper is as follows. The mathematics needed for the formulation and solution is summarized in Section 2. The problem formulation and the solutions are presented in Section 3. Numerical results are explained in Section 4, and conclusions are given in Section 5.

The stress components of a mixed dislocation with screw and edge components in an anisotropic homogeneous solid can be written in the form of a rational function of first-order to second-order polynomials, as shown in Hirth and Lothe [16]. The rational function can be divided into two parts P and Q: Pðx K xr ; a; bÞ Z

x K xr C a ; ðx K xr C aÞ2 C b2

(2.1)

Qðx K xr ; a; bÞ Z

b ; ðx K xr C aÞ2 C b2

(2.2)

where aZA(yKyr), bZB(yKyr), x and y denote the rectangular coordinates of the point of interest, xr and yr the position of a certain dislocation source, and A and B some material parameters. It can also be shown that the limiting values of P and Q as a/0 and b/0 are given by: 1 ; x K xr

Define also the integral transforms H and I of the image dislocation density function r(x): ð 1 N PðxKxr ;a; bÞrðxr Þdxr ; (2.5) Hða; bÞ½rðxÞ Z p KN ð 1 N Qðx Kxr ; a; bÞrðxr Þdxr ; (2.6) Iða;bÞ½rðxÞ Z p KN where the kernels are the rational functions P and Q defined previously. The unknown r(x) is to be sought in the form of the rational functions P and Q and their iterated H and I transforms. Consequently, the properties of H and I play a key role in the formulation and solution of the problem by the image method. Specifically, the H and I transforms of P and Q yield P and Q with additive arguments: ð 1 N Hða;bÞ½PðxKxs ; a; bÞ Z PðxKxr ;a; bÞ p KN (2.7) Pðxr Kxs ; a;bÞdxr ZKQðxKxs ; aCa;jbj CjbjÞ; Hða; bÞ½QðxKxs ; a; bÞ Z sgnðbÞPðxKxs ; a Ca; jbj CjbjÞ; (2.8) Iða;bÞ½PðxKxs ; a; bÞ Z sgnðbÞPðxKxs ; aCa;jbj CjbjÞ; (2.9) Iða;bÞ½QðxKxs ; a;bÞ ZsgnðbÞsgnðbÞQðxKxs ; aCa;jbj CjbjÞ;

(2.10)

where all the results involve the addition of the absolute values of the arguments b and b. The H and I transforms also obey the following iterative rules:

2. Integral transforms with rational kernels: the H and I transforms

Lima;b/0 Pðx K xr ; a; bÞ Z Pðx K xr ; 0; 0Þ Z

625

(2.3)

Lima;b/0 QðxKxr ;a;bÞ Z Qðx Kxr ; 0;0Þ ZpsgnðbÞdðxKxr Þ: (2.4)

Hða; bÞHða;bÞ ZKIða Ca; jbj CjbjÞ;

(2.11)

Hða; bÞIða;bÞ ZsgnðbÞHða Ca; jbj CjbjÞ;

(2.12)

Iða;bÞHða;bÞ ZsgnðbÞHðaCa; jbj CjbjÞ;

(2.13)

Iða;bÞIða; bÞ Z sgnðbÞsgnðbÞIða Ca; jbj CjbjÞ:

(2.14)

Eqs (2.7)–(2.14) can be proved by residue calculus ([15]) and have also been verified numerically. We note that H(0,0)and I(0,0G) are respectively the well-known Hilbert transform and the identity transform. In particular, Eq. (2.11) implies that H(0,0)$H(0,0)ZKI(0,0C), i.e., the inverse of the Hilbert transform is the negative of itself, and Eqs (2.12)–(2.14) imply the identity relations for aZaZ0 and b,b/0G.

3. Problem formulation and solution Fig. 1(a) shows the problem of a screw dislocation in a film-substrate system. The film A, of thickness h, and the substrate B are elastically anisotropic and dissimilar. Here we consider the elastic constants C14ZC15ZC24ZC25Z C46ZC56 of the materials in the global x–y–z frame to be

626

H.Y. Wang, M.S. Wu / Engineering Analysis with Boundary Elements 29 (2005) 624–635

(a)

(b) screw dislocations: density ρ (x)

free surface

y

y screw dislocation

screw dislocation h

h (x0, y0) A B

(x0, y0) A

bimaterial interface

x

B

bimaterial interface

x

Fig. 1. (a) The problem of a discrete screw dislocation at (x0,y0) in an anisotropic film-substrate (A–B). The film thickness is denoted by h. (b) The equivalent infinite bimaterial problem containing the screw dislocation at (x0,y0) and a continuous screw dislocation distribution with density r(x) on yZh.

zero. This condition allows the decoupling of a mixed dislocation into a pure screw and a pure edge dislocation for analysis ([16]). In the present problem, this means that the pure screw dislocation considered will not induce the stress components syx and syy anywhere in the materials. Also, the screw dislocation is situated at (x0,y0) and has the Burgers vector magnitude bz. The objective is to determine the stress field in the entire solid. The semi-infinite film-substrate problem is transformed into an infinite bimaterial problem, in which the interface boundary conditions are satisfied exactly a priori. Since syxZsyyZ0 everywhere, it is only necessary to require that syz vanish on the free surface, i.e.: syz ðx; y Z hÞ Z 0:

(3.1)

Thus, the original problem of Fig. 1(a) is transformed into the problem of Fig. 1(b), which consists of an infinite bimaterial A–B loaded by the screw dislocation at (x0,y0) and by infinite, continuously distributed image screw dislocations on yZh. The screw dislocation density r(x) is the unknown of the problem, and is determined on the basis of Eq. (3.1). The problem is thus formulated in the bimaterial domain. Since the strategy is to satisfy the interface conditions a priori, the bimaterial problem involving a screw dislocation must be solved first, and this is done in Section 3.1. The actual film-substrate problem is formulated in Section 3.2.

decomposed into the two homogeneous subproblems A and B as shown in Fig. 2. In each subproblem, the source dislocation is located at (xr,yr). Furthermore, in Subproblem A (or B), image screw dislocations with density rA(x) (or rB(x)) on yZ0 are introduced to satisfy the continuity conditions across the interface, i.e.: syz ðx; 0CÞ Z syz ðx; 0KÞ;

(3.2)

vw=vxðx; 0CÞ Z vw=vxðx; 0KÞ;

(3.3)

where w denotes the displacement in the z- direction. The elastic field in A or B of the bimaterial is determined from Subproblem A or B. For instance, syz(x,0C) and syz(x,0K) are determined from Subproblems A and B, respectively. This decomposition implies that the solutions for a screw dislocation in a homogeneous domain are required for the formulation. The stress and displacement gradient components of a general straight dislocation located at the origin of a rectangular frame x1Kx2 in a homogeneous anisotropic solid can be written in Stroh’s [17] formalism, see Hirth and Lothe [16]. In terms of the rational functions P and Q of Eqs (2.1) and (2.2), they can be written as: 3 X R I tI R I sij Z 2 ½xtR aij Pðx1 ; pa x2 ; pa x2 Þ C xaij Qðx1 ; pa x2 ; pa x2 Þbt ; aZ1

(3.4) 3 X vuk R I tI R I Z2 ½ztR akl Pðx1 ; pa x2 ; pa x2 Þ Czakl Qðx1 ; pa x2 ; pa x2 Þbt ; vxl aZ1

3.1. Stress field due to discrete screw dislocation in infinite bimaterial

(3.5)

The bimaterial problem of Fig. 1(b) is solved exactly by the image method. Suppose that the screw dislocation is located at (xr,yr) in the bimaterial. The problem is then

where i, j, k, lZ1, 2, 3, tZ1, 2, 3 denote a general dislocation with Burgers vector components b1, b2 and b3 along the x1, x2 and x3 directions respectively, and xtR aij ,

screw dislocation

A

ρ (x) A

A

screw dislocation

(xr , yr) B

ρ (x) B

(xr , yr)

B Subproblem A

Subproblem B

Fig. 2. Decomposition of the bimaterial problem into two homogeneous subproblems. The dislocation is located at (xr, yr), the image screw dislocation densities rA(x) and rB(x)are located on yZ0 in Subproblems A and B, respectively.

H.Y. Wang, M.S. Wu / Engineering Analysis with Boundary Elements 29 (2005) 624–635 tI xtIaij and ztR akl , zakl are respectively the real and imaginary parts of

xtaij Z

Cijkl ½ml Cpa nl Aak Lat ; 2pi

(3.6)

627

where

Ds Z 2p

3 X

x3R a23 ;

3 X

z3R a31 ;

(3.12)

1 z3I a31 Z K ; 2 aZ1

(3.13)

Es Z 2p

aZ1

aZ1

and ztakl

and the relations

½m Cpa nl Aak Lat : Z l 2pi

(3.7)

pffiffiffiffiffiffiffi In Eqs (3.4)–(3.7), iZ K1, the repeated subscript t implies summation, Cijkl are the components of the elasticity tensor, mlZdl1 and nlZdl2 are the Kronecker deltas, and Aak and Lat are the k-th and t-th components of a certain normalized six-dimensional eigenvector {Aa1,Aa2, Aa3,La1,La2,La3} with the associated eigenvalues pa (real and imaginary parts pRa and pIa ), where aZ1, 2,., 6. The eigenvalues pa and the quantities xtaij and ztakl occur in complex conjugate pairs so that only three values of a need be considered. In Eqs (3.4) and (3.5), a ranges from 1 to 3 and the pas chosen are those such that sgnðpIa ÞO0). Furthermore, pa, xtaij and ztakl assume two sets of values corresponding to Materials A and B, e.g., pAa, pBa, etc. In applying Eqs (3.4) and (3.5) to the problem of an infinite bimaterial, note that x1hx, x2hy, s23hsyz, vu3/ vx1hvw/vx. According to the decomposition scheme of Fig. 2, we need to consider a source screw dislocation with Burgers vector component b3Zbz at (xr,yr) and a generic image screw dislocation at (x 00 ,y 00 Z0) with b3Zbiz, where x 00 and y 00 are respectively dummy x- and y- variables. Due to the source dislocation, the relevant stress and displacement gradient on the line yZ0 in either Material A or B are: syz ðx; 0Þ Z 2

3 h X R I bz x3R a23 Pðx K xr ; Kpa yr ; Kpa yr Þ

2p

3 X

x3I a23 Z 0;

aZ1

3 X

have been used. Eq. (3.13) arises from the physical requirement that a line integral of the stress syz in Eq. (3.8) along a closed loop around the dislocation must yield a zero force per unit thickness in the z- direction, whereas a similar line integral of the displacement gradient vw/vx in Eq. (3.9) must yield the Burgers vector magnitude of biz. For a distributed image dislocation with total Burgers vector r(x 00 )dx 00 between x 00 and x 00 Cdx 00 , the corresponding stress and displacement gradient are obtained by replacing biz with r(x 00 )dx 00 in Eqs (3.10) and (3.11) and then integrating:

syz ðx;0Þ Z

Ds p

ðN

Pðx Kx00 ; 0;0Þrðx00 Þdx00 ZDs Hð0; 0Þ½rðxÞ;

KN

(3.14)

vw E ðx; 0Þ Z s vx p

ðN

Pðx Kx00 ; 0; 0Þrðx00 Þdx00 K

KN

ðN !

aZ1

2p

1 2p

Qðx Kx00 ; 0; 0Þrðx00 Þdx00

KN

i

R I C x3I a23 Qðx K xr ; Kpa yr ; Kpa yr Þ ;

(3.8)

3 h X vw R I ðx; 0Þ Z 2 bz z3R a31 Pðx K xr ; Kpa yr ; Kpa yr Þ vx aZ1

i R I Qðx K x ; Kp y ; Kp y Þ C z3I r a31 a r a r ;

(3.9)

where Eqs (3.4) and (3.5), with x1 replaced by (xKxr) and x2 by yKyrZKyr, have been used. Furthermore, the related stress and displacement component on yZ0 due to the image screw dislocation can be deduced from Eqs (3.8) and (3.9): Db syz ðx; 0Þ Z s iz Pðx K x 00 ; 0; 0Þ; p

1 Z Es Hð0; 0Þ½rðxÞ K rðxÞsgnðyÞ; 2

(3.15)

where Eqs (2.5) and (2.6) have been used to identify the integrals as the Hilbert and the identity transforms. The signum function in Eq. (3.15) is reduced from sgn(yKy00 )Z sgn(y) since the image is located at y00 Z0. For subproblems A and B, sgn(y)Zsgn(0C)Z1 and sgn(y)Zsgn(0K)ZK1, respectively. Substituting Eqs (3.8), (3.9), (3.14) and (3.15) into Eqs (3.2) and (3.3), the interface continuity conditions become: DAs Hð0; 0Þ½rA ðxÞ K DBs Hð0; 0Þ½rB ðxÞ Z R1 ðxÞ;

(3.16)

EAs Hð0; 0Þ½rA ðxÞ K EBs Hð0; 0Þ½rB ðxÞ 1 1 K rA ðxÞ K rB ðxÞ Z R2 ðxÞ; 2 2

(3.17)

(3.10)

vw Eb b ðx; 0Þ Z s iz Pðx K x 00 ; 0; 0Þ K iz Qðx K x 00 ; 0; 0Þ; vx p 2p (3.11)

628

H.Y. Wang, M.S. Wu / Engineering Analysis with Boundary Elements 29 (2005) 624–635

where

0 0 C DBs cB2 Z 0; KDAs cA2

3 h X R I R1 ðxÞ Z 2 bz x3R Ba23 Pðx K xr ; KpBa yr ; KpBa yr Þ

1 0 0 C cB1 Þ Z 0; EAs cA1 K EBs cB1 K ðcA1 2 1 0 0 EAs cA2 K EBs cB2 K ðcA2 C cB2 Þ Z 0; 2

aZ1 R I C x3I Ba23 Qðx K xr ; KpBa yr ; KpBa yr Þ R I K x3R Aa23 Pðx K xr ; KpAa yr ; KpAa yr Þ

i R I K x3I Aa23 Qðx K xr ; KpAa yr ; KpAa yr Þ ;

(3.18)

3 h X R I R2 ðxÞ Z 2 bz z3R Ba31 Pðx K xr ; KpBa yr ; KpBa yr Þ aZ1 R I C z3I Ba31 Qðx K xr ; KpBa yr ; KpBa yr Þ R I K z3R Aa31 Pðx K xr ; KpAa yr ; KpAa yr Þ

i R I K z3I Aa31 Qðx K xr ; KpAa yr ; KpAa yr Þ :

(3.19)

For reasons which will be made clear later, R1(x) and R2(x) are rewritten as: Ri ðxÞ Z 2

3  X 3R bz rBbi Pðx K xr ; KpRBb yr ; KpIBb yr Þ bZ1

3I C rBbi Qðx K xr ; KpRBb yr ; KpIBb yr Þ 3R Pðx K xr ; KpRAb yr ; KpIAb yr Þ K rAbi

 3I Qðx K xr ; KpRAb yr ; KpIAb yr Þ ; K rAbi where the new notations are defined as ( 3R ( 3I xAb23 ; i Z 1 xAb23 ; i Z 1 3R 3I ; rAbi Z rAbi Z 3R zAb31 ; i Z 2 z3I Ab31 ; i Z 2

1 0 0 C EBs cB1 K ðcA1 C cB1 Þ Z 0; KEAs cA1 2

(3.30)

1 0 0 C EBs cB2 K ðcA2 C cB2 Þ Z hAB ; KEAs cA2 2

(3.31)

where hABZ0 if AhB and 1 otherwise. If Materials A and B are identical, DAsZDBs and EAsZEBs, and Eqs (3.24)– 0 0 (3.27) predict that cA1ZcB1, cA2ZcB2, cA1 ZcB1 and 0 0 cA2 ZcB2 respectively, which by Eqs (3.28)–(3.31) 0 0 0 0 must lead to cA1 ZcB1 Z0, cA2 ZcB2 Z0, cA1ZcB1Z0 and cA2ZcB2Z0 respectively. All eight constants are zero and no images are needed to satisfy the interface conditions, as expected for the case of a homogeneous solid. In the general case when the materials are dissimilar, the eight coefficients can be solved exactly from Eqs (3.24)–(3.31). The stress components at any position (x, y) in the bimaterial due to the source at (xr, yr) can now be derived exactly from the known image dislocation densities. Suffice to consider syz. In Material A, the contribution from the source is, by Eq. (3.4): 3 X 3I sð1Þ bz ðx3R (3.32) yz ðx; yR 0Þ Z 2 Aa23 PA C xAa23 QA Þ; where the rational functions are given by:

(3.21)

QA hQðx K xr ; pRAa ðy K yr Þ; pIAa ðy K yr ÞÞ:

(3.23)

iZ1

which are substituted back into Eqs (3.16) and (3.17), yielding eight algebraic equations with eight unknowns: DAs cA1 K DBs cB1 Z 0;

(3.24)

DAs cA2 K DBs cB2 Z 0;

(3.25)

0 KDAs cA1

(3.26)

0 C DBs cB1

Z hAB ;

(3.33)

The contribution from the image dislocation distribution with density rA(x 00 ) can be obtained from Eq. (3.32) by replacing bz with rAdx 00 , the arguments of PA and QA by xKx 00 , pRAa y and pIAa y (since for the images y 00 Z0), and then integrating along x 00 . Using the density expression of Eq. (3.22) and noting the definition of the H and I transforms in Eqs (2.5) and (2.6), the contribution of the image distribution can be written as: sð2Þ yz ðx; yR 0Þ

0 ðcBi Ri ðxÞ C cBi Hð0; 0Þ½Ri ðxÞÞ;

(3.29)

PA hPðx K xr ; pRAa ðy K yr Þ; pIAa ðy K yr ÞÞ;

iZ1

rB ðxÞ Z

(3.28)

aZ1

(3.20)

3R 3I and rBbi . Eqs (3.16) and (3.17) are two and similarly for rBbi Cauchy singular integral equations with the two image densities as unknown. Exact solutions are constructed from the basic functions Ri(x) and H(0,0)[Ri(x)] with eight 0 unknown coefficients cA1, cA1 , etc.: 2 X 0 rA ðxÞ Z ðcAi Ri ðxÞ C cAi Hð0; 0Þ½Ri ðxÞÞ; (3.22)

2 X

(3.27)

Z 2p " !

3 X

( R I 3I R I ðx3R Aa23 HðpAa y; pAa yÞ C xAa23 IðpAa y; pAa yÞÞ

aZ1 2 X

#)

0 ðcAi Ri ðxÞ C cAi Hð0; 0Þ½Ri ðxÞÞ

iZ1

Z 2p

3 X 2 n X 3I 0 R I ðx3R Aa23 cAi C xAa23 cAti ÞHðpAa y; jpAa yjÞ aZ1 iZ1

o 0 3I R I K ðx3R Aa23 cAi K xAa23 cAi ÞIðpAa y; jpAa yjÞ ½Ri ðxÞ; ð3:34Þ

H.Y. Wang, M.S. Wu / Engineering Analysis with Boundary Elements 29 (2005) 624–635

where the iterative properties of H and I as given in Eqs (2.11)–(2.14) have been used to arrive at the last line. The ð2Þ sum of sð1Þ yz and syz in Eqs (3.32) and (3.34) gives an exact expression for syz in Material A of the bimaterial, see Fig. 2. For Material B, pIBa y! 0 and it can be similarly shown that: sð1Þ yz ðx; y% 0Þ Z 2

3 X

629

where PAA hPðx K xr ; pRAa y K pRAb yr ; jpIAa yj C j K pIAb yr jÞ; (3.41) PAB hPðx K xr ; pRAa y K pRBb yr ; jpIAa yj C j K pIBb yr jÞ;

3I b3 ðx3R Ba23 PB C xBa23 QB Þ;

(3.42)

(3.35)

aZ1

PBA hPðx K xr ; pRBa y K pRAb yr ; jpIBa yj C j K pIAb yr jÞ; sð2Þ yz ðx; y% 0Þ Z 2p

(3.43)

3 X 2 n X

3I 0 ðx3R Ba23 cBi K xBa23 cBi Þ! aZ1 iZ1   0 3I HðpRBa y; pIBa yÞ K ðx3R Ba23 cBi C xBa23 cBi Þ!

 o IðpRBa y; pIBa yÞ ½Ri ðxÞ;

ð3:36Þ

PBB hPðx K xr ; pRBa y K pRBb yr ; jpIBa yj C j K pIBb yr jÞ; (3.44)

where PB and QB are similarly defined as PA and QA in Eq. (3.33). The H and I transforms of Ri (and hence P and Q) in Eqs (3.34) and (3.36) can be explicitly determined in closed form using Eqs (2.7)–(2.10):

and the arguments of QAA, QAB, QBA and QBB are respectively the same as those of PAA, etc. Substituting Eqs (3.37)–(3.40) into Eqs (3.34) and (3.36) ð1Þ ð2Þ yields sð2Þ yz in the entire bimaterial. Adding syz and syz , the exact expressions for syz in Materials A and B can be obtained:

HðpRAa y; jpIAa yjÞ½Ri ðxÞ

syz ðx; yR 0Þ Z 2

3 X

3I bz ðx3R Aa23 PA C xAa23 QA Þ

aZ1

Z2

3 X

3R 3I bz ðKrBbi QAB K sgnðyr ÞrBbi PAB

C 4p

bZ1

C cab3 AA23 QAA

(3.37)

syz ðx; y% 0Þ Z 2

IðpRAa y; jpIAa yjÞ½Ri ðxÞ Z2

C 4p

3R 3I 3R bz ðrBbi PAB K sgnðyr ÞrBbi QAB K rAbi PAA

3I C sgnðyr ÞrAbi QAA Þ;

ð3:45Þ

3I bz ðx3R Ba23 PB C xBa23 QB Þ 3 X 3 X

ab3 bz ðJab3 BA23 PBA K JBB23 PBB

aZ1 bZ1

C cab3 BA23 QBA (3.38)

K cab3 BB23 QBB Þ;

ð3:46Þ

where the eight constants are given by Jab3 AA23 Z

HðpRBa y; jpIBa yjÞ½Ri ðxÞ Z2

3 X

K cab3 AB23 QAB Þ;

aZ1

bZ1

3 X

ab3 bz ðJab3 AA23 PAA K JAB23 PAB

aZ1 bZ1

3R 3I C rAbi QAA C sgnðyr ÞrAbi PAA Þ;

3 X

3 X 3 X

2 X

3I 0 3I ½sgnðyr Þðx3R Aa23 cAi C xAa23 cAi ÞrAbi

iZ1 0 3I 3R C ðx3R Aa23 cAi K xAa23 cAi ÞrAbi ;

3R 3I bz ðKrBbi QBB K sgnðyr ÞrBbi PBB

(3.47)

bZ1 3R 3I C rAbi QBA C sgnðyr ÞrAbi PBA Þ;

(3.39)

Jab3 AB23 Z

2 X

3I 0 3I ½sgnðyr Þðx3R Aa23 cAi C xAa23 cAi ÞrBbi

iZ1 0 3I 3R C ðx3R Aa23 cAi K xAa23 cAi ÞrBbi ;

IðpRBa y; jpIBa yjÞ½Ri ðxÞ Z2

3 X

cab3 AA23 Z

3R 3I 3R bz ðrBbi PBB K sgnðyr ÞrBbi QBB K rAbi PBA

2 h X 3I 0 3R ðx3R Aa23 cAi C xAa23 cAi ÞrAbi K sgnðyr Þ iZ1

bZ1 3I C sgnðyr ÞrAbi QBA Þ;

(3.48)

(3.40)

i 0 3I 3I !ðx3R Aa23 cAi K xAa23 cAi ÞrAbi ;

(3.49)

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H.Y. Wang, M.S. Wu / Engineering Analysis with Boundary Elements 29 (2005) 624–635

cab3 AB23 Z

2 h X 3I 0 3R ðx3R Aa23 cAi C xAa23 cAi ÞrBbi K sgnðyr Þ

as: fyz ðxÞ Z syz ðx; y Z hÞ

iZ1

i 0 3I 3I !ðx3R Aa23 cAi K xAa23 cAi ÞrBbi ;

(3.50)

Z2

3 X

3I bz ðx3R Aa23 PA1 C xAa23 QA1 Þ

aZ1

Jab3 BA23 Z

2 h X

3I 0 3I sgnðyr Þðx3R Ba23 cBi K xBa23 cBi ÞrAbi

C 4p

iZ1

i 0 3I 3R C ðx3R Ba23 cBi C xBa23 cBi ÞrAbi ; Jab3 BB23 Z

2 h X

(3.52)

(3.53)

2 h X 3I 0 3R ðx3R Ba23 cBi K xBa23 cBi ÞrBbi K sgnðyr Þ

PAA1 hPðx K x0 ; pRAa h K pRAb y0 ; jpIAa hj C j K pIAb y0 jÞ: and similarly for QA1, PAB1, QAA1 and QAB1. Note that sgn(yr) in Jab3 AA23 , etc. becomes sgn(y0). In an analogous manner, syz on yZh due to the image rdx 0 at (x 0 ,h) can be obtained using Eq. (3.45). For the entire distribution, integration with respect to x 0 yields: syz ðx; y Z hÞ Z 2

iZ1

i 0 3I 3I c C x c Þr !ðx3R Ba23 Bi Ba23 Bi Bbi :

(3.56)

(3.57)

iZ1

cab3 BB23 Z

(3.55)

PA1 hPðx K x0 ; pRAa ðh K y0 Þ; pIAa ðh K y0 ÞÞ;

2 h X 3I 0 3R Z ðx3R Ba23 cBi K xBa23 cBi ÞrAbi K sgnðyr Þ

i 0 3I 3I !ðx3R Ba23 cBi C xBa23 cBi ÞrAbi ;

ab3 C cab3 AA23 QAA1 K cAB23 QAB1 Þ;

where by Eqs (3.33) and (3.41)

iZ1

cab3 BA23

ab3 bz ðJab3 AA23 PAA1 K JAB23 PAB1

aZ1 bZ1

(3.51)

3I 0 3I sgnðyr Þðx3R Ba23 cBi K xBa23 cBi ÞrBbi

i 0 3I 3R C ðx3R Ba23 cBi C xBa23 cBi ÞrBbi ;

3 X 3 X

(3.54)

For the stress component sxz, only the subscripts 23 in ab3 x3R Aa23 , JAA23 , etc. need be changed to 13. The subscripts of 3R the terms in rAbi , etc. are not changed and this is why x3R Ab23 , 3R 3R zAb31 , etc. were redefined as rAbi , etc. in Eq. (3.21). Eqs (3.45) and (3.46) show that the stress components due to a screw dislocation in a bimaterial are in fact rational functions (P and Q) multiplied by appropriate material coefficients. The summations over b and a reflect the effect of anisotropic elasticity on the image and source dislocations, and sgn(yr) distinguishes the two cases in which the source dislocation lies in the film and in the substrate, respectively.

3 ðN X

0 ðx3R Aa23 Pðx K x ; 0; 0Þ

aZ1 KN

0 0 0 C x3I Aa23 Qðx K x ; 0; 0ÞÞrðx Þdx C 4p

aZ1 bZ1 KN

0 R R I fJab3 AA23 Pðx K x ; ðpAa K pAb Þh; jpAa hj 0 R R I C j K pIAb hjÞ K Jab3 AB23 Pðx K x ; ðpAa K pBb Þh; jpAa hj 0 R R I C j K pIBb hjÞ C cab3 AA23 Qðx K x ; ðpAa K pAb Þh; jpAa hj 0 R R I C j K pIAb hjÞ K cab3 AB23 Qðx K x ; ðpAa K pBb Þh; jpAa hj

C j K pIBb hjÞgrðx 0 Þdx 0 ;

ð3:58Þ

where sgn(yr)Zsgn(h)Z1 in Jab3 AA23 , etc. Noting Eqs (3.13), (2.5) and (2.6), Eq. (3.58) can be written in terms of the Hilbert transform and the H and I transforms: syz ðx;y ZhÞ Z DAs Hð0;0Þ½rðxÞ C4p2

3 X 3 n X

Jab3 AA23 HAA2

aZ1 bZ1

3.2. Governing equation for screw dislocation in film-substrate The results of Section 3.1 are now used to derive the integral equation for the problem of a screw dislocation in a film-substrate. According to Fig. 1(b) and the traction-free condition in Eq. (3.1), it is necessary to obtain expressions for syz due to the source dislocation located at (x0,y0) and due to the screw image dislocations with density r(x) distributed on the line yZh. By setting xrZx0 and yrZy0 (which can be positive or negative) in Eq. (3.45), the stress component at (x,yZh) due to the source with Burgers vector component bz can be written

3 X 3 ðN X

ab3 ab3 KJab3 AB23 HAB2 CcAA23 IAA2 KcAB23 IAB2

o

½rðxÞ;

ð3:59Þ where HAA2 hHððpRAa KpRAb Þh;ðpIAa CpIAb ÞhÞ;

(3.60)

HAB2 hHððpRAa KpRBb Þh;ðpIAa CpIBb ÞhÞ;

(3.61)

IAA2 hIððpRAa KpRAb Þh; ðpIAa CpIAb ÞhÞ;

(3.62)

IAB2 hIððpRAa KpRBb Þh; ðpIAa CpIBb ÞhÞ:

(3.63)

H.Y. Wang, M.S. Wu / Engineering Analysis with Boundary Elements 29 (2005) 624–635

The sum of Eqs (3.55) and (3.59) gives syz due to the source and image dislocations and should be equated to zero according to Eq. (3.1), resulting in: DAs Hð0;0Þ½rðxÞ ZK4p2

3 X 3 n X ab3 Jab3 AA23 HAA2 KJAB23 HAB2 aZ1 bZ1

o ab3 Ccab3 AA23 IAA2 KcAB23 IAB2 ½rðxÞ Kfyz ðxÞ; ð3:64Þ which can be regarded as a Cauchy integral equation with rational kernels and unknown density function r(x). The details for solving Eq. (3.64) are given in the next subsection. 3.3. Solution of integral equation The Cauchy term H(0,0)r(x) in Eq. (3.64) is transformed by applying the Hilbert operator to all terms. This requires the evaluation of H(0,0)$HAA2, H(0,0)$IAA2, etc. which can be done exactly using Eqs (2.11)–(2.14), yielding a Fredholm integral equation of the second kind: rðxÞ Z K½rðxÞ C GðxÞ;

(3.65)

iterative solution of r. One can set the first solution r0(x)Z G(x), which would be the exact solution for a screw dislocation in a homogeneous half space AhB. This can be observed from Eq. (3.66), which shows that K would ab3 be a zero operator (since Jab3 AA23 IAA2 Z JAB23 IAB2 , ab3 ab3 cAA23 HAA2 Z cAB23 HAB2 ). Eq. (3.65) then confirms that r(x)ZG(x) is the exact solution to Eq. (3.64), which is the simple Cauchy integral equation with all the terms under the double summation sign eliminated. Thus, in the case of dissimilar materials the subsequent terms r1(x), r2(x), etc. are images which account for the material dissimilarity. Furthermore, Eq. (3.68) is the exact solution of Eq. (3.65) when n tends to infinity. In practice, only a few terms of r(x) may be needed and the stress field in the whole solid can be obtained by using expressions such as Eqs (3.45) and (3.46). Taking r0(x)ZG(x), we present here the expression for the stress component syz in the thin film 0%y%h: syz ðx; 0% y% hÞ Z2

3 X

3I bz ðx3R Aa23 PA C xAa23 QA Þ

aZ1

C 4p

where

3 X 3 X

ab3 ab3 bz ðJab3 AA23 PAA K JAB23 PAB C cAA23 QAA

aZ1 bZ1

3 X 3 n 4p2 X ab3 KZ K Jab3 AA23 IAA2 C JAB23 IAB2 DAs aZ1 bZ1

o ab3 C cab3 AA23 HAA2 K cAB23 HAB2 ;

K cab3 AB23 QAB Þ C 2p

Z

aZ1

(3.66)

C2p

ab3 ab3 ðJab3 AA23 HAA K JAB23 HAB C cAA23 IAA

aZ1 bZ1

Kcab3 AB23 IAB Þ

1 Hð0; 0Þ½fyz ðxÞ DAs

3 X 3 X

Km ½GðxÞ ;

ð3:69Þ

HA hHðPRAa ðy K hÞ; PIAa ðy K hÞÞ;

aZ1 bZ1 ab3 C cab3 AA23 PAA1 K cAB23 PAB1 Þ:

(3.67)

rnC1 ðxÞ Z Krn ðxÞ C GðxÞ Z K½K½rnK1 ðxÞ C GðxÞ C GðxÞ Z . Km ½GðxÞ;

IA hIðPRAa ðy K hÞ; PIAa ðy K hÞÞ; (3.70)

The kernels in IAA2, IAB2, HAA2 and HAB2 are all rational functions. Various methods are available for solving the integral equation. Here, we consider the solution to be in the form of a Neumann series ([18]):

n X

#

where the arguments of PA, QA and those of PAA, PAB, etc. are as given in Eq. (3.33) and Eqs (3.41)–(3.44) respectively, with xrZx0 and yrZy0, and the transforms are defined as:

ab3 bz ðKJab3 AA23 QAA1 C JAB23 QAB1

Z KnC1 r0 ðxÞ C

n X mZ0

3 2 X 4p 3I bz ðKx3R Aa23 QA1 C xAa23 PA1 Þ C DAs aZ1 DAs

!

3 X 3I ðx3R Aa23 HA C xAa23 IA Þ

3 X 3 X

!"

GðxÞ Z

631

(3.68)

mZ0

where nZ0, 1, 2,., N, K0 is the identity matrix, Km is the K matrix operating upon itself m times, and rn is the n–th

HAA Z HðPRAa y K PRAb h; jPIAa yj C j K PIAb hjÞ; HAB Z HðPRAa y K PRBb h; jPIAa yj C j K PIBb hjÞ; and similarly for IAA and IAB. Eq. (3.69) shows that these transforms operate on K0[G], K1[G],., Kn[G], and all these evaluations can be done exactly using the properties of the H and I transforms as given in Eqs (2.7)–(2.14). Note that the first two terms on the right-hand side of Eq. (3.69) are the stress due to the screw dislocation in an infinite homogeneous solid without boundaries and interfaces and the remaining term is the image stress arising from the interface and the free surface. The equation can be

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H.Y. Wang, M.S. Wu / Engineering Analysis with Boundary Elements 29 (2005) 624–635

used as the Green’s function for various problems of thin film composites. For instance, the problem of a mode III crack in a film-substrate can be formulated using Eq. (3.69) by setting bzZrcdx and integrating, where rc is the density of the screw dislocations modeling the crack.

4. Numerical results for strip and film-substrate systems To verify the Green’s function such as Eq. (3.69), we consider film-substrate composites and numerically investigate the stress components of a screw dislocation and the image force on the dislocation resulting from the interface/ free surface. Specifically, various film-substrate systems of germanium-silicon alloy on silicon (GexSi1Kx/Si) or the reverse (Si/GexSi1Kx), indium nitride on silicon carbide (InN/SiC), zinc on zinc (Zn/Zn) and silicon carbide on silicon carbide (SiC/SiC) are considered. Convergence tests and back-substitution tests are carried out to judge if the obtained solutions are acceptably accurate. The convergence tests show that the numerical results are satisfactory for nZ2 in Eq. (3.69), i.e., r2 Z K2 GðxÞC KGðxÞC GðxÞ. All results are generated with nZ2 unless stated otherwise. In the back-substitution tests, the solved image density is substituted back into the governing integral equation to check how well the free surface condition is satisfied. The elastic constants of the cubic and hexagonal materials are listed in Table 1, where the data are compiled from various sources: Sheleg and Savastenko [19], Kamitani et al. [20], Hirth and Lothe [16], and Jain et al. [21]. Consider first an interfacial screw dislocation with the Burgers vector magnitude b at the origin in an InN/SiC film-substrate of thickness hZ500b. The in-plane c-axis orientations of the InN and SiC crystals with respect to the x-axis are respectively qAZ158 and qBZ1308. To show the convergence of the results to the correct values, the component syz on the free surface yZ500b is plotted for different values of the iterative index nZ0, 1, 2 in Fig. 3. For comparison, syz on yZ500b in an infinite InN/SiC bimaterial is also plotted. It can be seen that the solutions converge as n increases, and they converge to the value of zero, which is correct since syz must vanish on the free

surface. The curve syz(hZN) shows that the stress in the infinite bimaterial at the position corresponding to the free surface in the film-substrate is rather large, of the order of 0.01 GPa. For completeness, the stress component sxz for nZ0, 1, 2 in the film-substrate and sxz in the infinite bimaterial on the line yZ500b are also plotted in Fig. 4. The adjustment of the sxz curve is evident as n increases from 0 to 2. Moreover, the marked difference between the nZ2 and the hZN solutions illustrates the significant influence of the free surface on the elastic field. The results in Figs. 3 and 4 show that the stress components converge with the inclusion of more, albeit progressively weaker, image components nZ0, 1, 2, etc., and that the governing integral equation is satisfied quite well by the image solution for nZ2. The latter check is important since although the convergence of solutions has been studied in previous works, how well the converged solution satisfies the governing equation has rarely been reported. We consider next the GexSi1Kx/Si and Si/GexSi1Kx systems, where xZ0.5. Specifically, the image force in the y-direction on a screw dislocation with Burgers vector bZb(0,0,1) in the film is calculated as a function of position y0/h. For the {1 0 0} plane epitaxy, the x-, y- and  z-axes coincide with the crystallographic directions ½101, [010] and [101], respectively. Here, F2 ZKsxx bx Ksxy by K sxz bz ZKsxz b. Fig. 5 plots F2/Ffree against y0/h for the range 0!y0/h!0.8, where Ffree is the image force on the dislocation at the distance h from the free surface in a half-space of the substrate material. In the present case, the ratio of Ffree for a dislocation in silicon and germanium silicon half spaces is 1.1. The results show that the screw dislocation is attracted towards the free surface. If the substrate is harder than the film as in GexSi1Kx/Si, the dislocation is repelled by the interface. Conversely, if the substrate is softer as in Si/GexSi1Kx, the dislocation is attracted towards the interface. This example demonstrates the importance of the details of material inhomogeneity on the image force. In the next example, we investigate how the normalized image force F2/Ffree on a screw dislocation varies with

Table 1 Elastic constants of hexagonal and cubic crystals in the crystal frame x10 K x20 K x30 . For the hexagonal crystals, the x10 -axis is perpendicular to the basal plane Hexagonal

InNa

SiCb

Znc

Cubic

Sid

GexSi1Kxe

0 C11 (GPa) 0 C33 0 C12 0 C23 0 C66

182 190 121 104 9.9

553 501 52 111 163

61 161 50.1 34.2 38.3

0 C11 0 C12 0 C44

165.7 63.9 79.6

128.9xC165.7(1Kx) 48.3xC63.9(1Kx) 67.1xC79.6(1Kx)

a b c d e

Sheleg and Savastenko [19]. Kamitani et al. [20]. Hirth and Lothe [16]. Jain et al. [21]. Jain et al. [21].

H.Y. Wang, M.S. Wu / Engineering Analysis with Boundary Elements 29 (2005) 624–635

Fig. 3. Plots of the stress syz on the free surface yZ500b due to an interfacial screw dislocation with Burgers vector magnitude b in an InN/SiC film-substrate (solid lines) and at the corresponding position due to the same dislocation in an infinite bimaterial (dashed line). The three solid lines indicate the iterative index n used for the dislocation image density rn(x). For nZ2, syz is essentially zero, as required for the free surface.

the normalized film thickness h/b, where 60 ! h/b ! 500 and Ffree is the image force on a screw dislocation at a depth of 500b in a half space of the substrate material. The dislocation has the Burgers vector bZb(0,0,1) and is located at the fixed position (0, 5b). Thus, the dislocation is always closer to the interface than to the free surface in this simulation. Fig. 6 plots the variation for the GexSi1Kx/Si and Si/GexSi1Kx (xZ0.5) systems. Again, the x-, y-, z- directions coincide  [010] and [101], with the crystallographic directions ½101, respectively. The results show that in the case of GexSi1Kx/Si the dislocation is always repelled by the interface. For Si/ GexSi1Kx, the dislocation is attracted towards the interface for thick films h/b O 120 but repelled by it for thin films h/b ! 120. This shows the influence of the free surface overriding that of the soft substrate as the film becomes thin. To verify the continuity of syz across the interface yZ0, Fig. 7(a) and (b) plot the interfacial stresses due to a screw dislocation with Burgers vector magnitude b at (x0,y0)Z (0, 0.5h) in a Zn/Zn and a SiC/SiC film-substrate, respectively. The film thickness is hZ500b and the c-axis

Fig. 4. Plots of the stress sxz on the free surface yZ500b due to an interfacial screw dislocation with Burgers vector magnitude b in an InN/SiC film-substrate (solid lines) and at the corresponding position due to the same dislocation in an infinite bimaterial (dashed line). The three solid lines indicate the iterative index n used for the dislocation image density rn(x). The influence of the free surface is evident from the marked difference between the nZ2 and hZN curves.

633

Fig. 5. Plot of the normalized image force F2/Ffree versus normalized dislocation position y0/h in GexSi1Kx/Si and Si/GexSi1Kx (xZ0.5) systems.  The x-, y-, z- directions coincide with the crystallographic directions ½101, [010] and [101], respectively. The Burgers vector bZb(0,0,1). The image force propels the dislocation towards the free surface, repels it from the interface in GexSi1Kx/Si but attracts it towards the interface in Si/GexSi1Kx, where the substrate is softer than the film.

orientations are qAZ158, qBZ1308. It can be seen that syz(x,0) is indeed continuous across the Zn/Zn or the SiC/ SiC interface, as the interface continuity conditions are exactly satisfied in the formulation. In contrast, sx(x,0C) differs considerably from sx(x,0K) in the Zn/Zn bicrystal due to the strong elastic anisotropy of the Zn crystal. The difference is somewhat less for the SiC/SiC due to the weaker anisotropy of the SiC crystal. Finally, Fig. 8 plots the whole field for syz due to an interfacial screw dislocation located at the origin in InN/ SiC. Again, the c-axis orientations of the InN and SiC crystals are taken to be q AZ158 and q BZ1308, respectively. Part (a) of the figure refers to an infinite bimaterial. Part (b) refers to a film-substrate with the film thickness hZ500b. The results are shown in the region K500!x/b!500, K500!y/b!500. The contours are

Fig. 6. Plot of the normalized image force F2/Ffree versus normalized film thickness h/b in GexSi1Kx/Si and Si/GexSi1Kx (xZ0.5) systems. The image force assumes both positive and negative values in the case of Si/GexSi1Kx, demonstrating the interaction between the bimaterial interface and the free surface.

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H.Y. Wang, M.S. Wu / Engineering Analysis with Boundary Elements 29 (2005) 624–635

Fig. 7. Variation of interfacial stresses with normalized distance x/b due to a screw dislocation at (0, 0.5h) in (a) Zn/Zn and (b) SiC/SiC bicrystal with a free surface. The film thickness is hZ500b, and the c-axis orientations are qAZ158 and qBZ1308. Note the jump in sxz and the exact continuity of syz across the interface.

Fig. 8. Contour plots of syz due to an interfacial screw dislocation in the InN/SiC (a) infinite bicrystal and (b) film-substrate. The c-axis orientations are qAZ158, qBZ1308, and the thickness of the InN film is hZ500b. The contours are continuous with kinks across the interface.

continuous, albeit with kinks, across the interface. The zero stress contours merge at the film surface, in agreement with the free surface condition. The contour loops in the film-substrate reduce in size compare to the loops in the infinite bimaterial, showing the influence of the free surface.

5. Conclusions In this paper, the image method is used to derive the Green’s function for a screw dislocation in an anisotropic film-substrate composite. The semi-infinite film-substrate problem is transformed to an infinite bimaterial problem in which the interface boundary conditions are satisfied exactly a priori. By using the properties of the H and I transforms, the image formulation leads to a Fredholm integral equation of the second kind corresponding to the free surface condition. Numerical investigations are conducted for the GexSi1Kx/Si, Si/GexSi1Kx, InN/SiC, Zn/Zn, and SiC/SiC film-substrate systems. Using only three terms of the series for the solved image density, the free surface boundary condition is satisfied very well. The numerical results demonstrate the importance of elastic anisotropy,

film-substrate inhomogeneity and the free surface in determining the various stress components and the image forces.

Acknowledgements The support of the Agency for Science, Technology and Research, Singapore (Grant No: 032 101 0019) in this investigation is greatly appreciated.

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