Crack branch in piezoelectric bimaterial...

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Crack branch in piezoelectric bimaterial system Qing-Hua Qin * , Yiu-Wing Mai Centre for Advanced Materials Technology, Department of Mechanical and Mechatronic Engineering, University of Sydney, Sydney, NSW 2006, Australia Received 17 February 1999; accepted 12 May 1999 Abstract A solution is presented for a class of two-dimensional electroelastic branched crack problems in various bimaterial combinations. The combinations may be piezoelectric–piezoelectric, or one is piezoelectric and the other is not. Explicit GreenÕs function for an interface crack subject to an edge dislocation is developed within the framework of two-dimensional electroelasticity, allowing the branched crack problem to be expressed in terms of coupled singular integral equations. The integral equations are obtained by the method that models a kink as a continuous distribution of edge dislocations, and the unknown dislocation density functions are defined on the line of branch crack only. Competition between crack extension along the interface and kinking into the substrate is investigated using the integral equations and the maximum energy release rate criterion. Numerical results are presented to study the eect of electric field on the path of crack extension. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Crack branching and bifurcation; Energy release rate; Piezoelectric material 1. Introduction Due to the intrinsic coupling eect between electrical and mechanical fields, piezoelectric ma- terials have been widely used as electromechanical devices in which interfaces between materials are ubiquitous. As such the interfacial failure mechanism has caused researchers to devote a great deal of attention to study the selection of crack path. Generally fracture along and adjacent to bima- terial interfaces has several morphological manifestations. In some cases the fracture occurs at the interface, while in others fracture occurs in the more brittle constituent [6]. Various hypotheses have been made concerning the crack kinking both in homogeneous material and bimaterials [7,9,19], such as maximum energy release rate [7], the minimum strain energy density theory [19]. In this context, there are many analytical studies of crack kinking by using the above criteria, such as www.elsevier.com/locate/ijengsci International Journal of Engineering Science 38 (2000) 673–693 * Corresponding author. Tel.: +61-2-9351-2392; fax: +61-2-9351-3760. E-mail address: [email protected] (Q.-H. Qin). 0020-7225/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII:S0020-7225(99)00061-0

Transcript of Crack branch in piezoelectric bimaterial...

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Crack branch in piezoelectric bimaterial system

Qing-Hua Qin *, Yiu-Wing Mai

Centre for Advanced Materials Technology, Department of Mechanical and Mechatronic Engineering, University of

Sydney, Sydney, NSW 2006, Australia

Received 17 February 1999; accepted 12 May 1999

Abstract

A solution is presented for a class of two-dimensional electroelastic branched crack problems in variousbimaterial combinations. The combinations may be piezoelectric±piezoelectric, or one is piezoelectric andthe other is not. Explicit GreenÕs function for an interface crack subject to an edge dislocation is developedwithin the framework of two-dimensional electroelasticity, allowing the branched crack problem to beexpressed in terms of coupled singular integral equations. The integral equations are obtained by themethod that models a kink as a continuous distribution of edge dislocations, and the unknown dislocationdensity functions are de®ned on the line of branch crack only. Competition between crack extension alongthe interface and kinking into the substrate is investigated using the integral equations and the maximumenergy release rate criterion. Numerical results are presented to study the e�ect of electric ®eld on the pathof crack extension. Ó 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Crack branching and bifurcation; Energy release rate; Piezoelectric material

1. Introduction

Due to the intrinsic coupling e�ect between electrical and mechanical ®elds, piezoelectric ma-terials have been widely used as electromechanical devices in which interfaces between materials areubiquitous. As such the interfacial failure mechanism has caused researchers to devote a great dealof attention to study the selection of crack path. Generally fracture along and adjacent to bima-terial interfaces has several morphological manifestations. In some cases the fracture occurs at theinterface, while in others fracture occurs in the more brittle constituent [6]. Various hypotheseshave been made concerning the crack kinking both in homogeneous material and bimaterials[7,9,19], such as maximum energy release rate [7], the minimum strain energy density theory [19]. Inthis context, there are many analytical studies of crack kinking by using the above criteria, such as

www.elsevier.com/locate/ijengsci

International Journal of Engineering Science 38 (2000) 673±693

*Corresponding author. Tel.: +61-2-9351-2392; fax: +61-2-9351-3760.

E-mail address: [email protected] (Q.-H. Qin).

0020-7225/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S0020-7225(99)00061-0

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the work of Atkinson [1], Lo [12], Hayashi and Nemat-Nasser [7, 8], He and Hutchinson [10],Miller and Stock [13], Hutchinson and Suo [11], Wang et al. [25], Atkinson et al. [2], and others.

The present paper deals with the problems of crack kinking in bimaterial system with variousmaterial combinations. The combinations may be of piezoelectric±piezoelectric (PP), or piezo-electric±anisotropic conductor (PAC), or piezoelectric±isotropic dielectric (PID) as well as pi-ezoelectric±isotropic conductor (PIC). The focus is on the e�ect of electric ®eld on the pathselection of crack extension. The geometrical con®guration analyzed is depicted in Fig. 1. A maincrack of length 2c is shown lying along the interface between the two dissimilar piezoelectricmaterials, with a branch of length a and angle h kinks upward into material 1 or downward intomaterial 2. The length a is assumed to be small compared to the length 2c of the main crack. Tosolve the branch problem, we ®rst derive the GreenÕs function satisfying the traction-charge freeconditions on faces of the main crack by way of the Stroh formalism and the solution forbimaterials due to an edge dislocation. Competition between crack extension along the interfaceand kinking into the substrate is investigated with the aid of the integral equations developed andthe maximum energy release rate criterion. The formulation for the energy release rate and thestress and electric displacement (SED) intensity factors to the onset of kinking are expressed interms of dislocation density function and branch angle. The numerical results for the energyrelease rate vs loading condition and branch angle are presented to study the relationship amongthe kinking angle, loading condition and the energy release rate.

2. Problem formulations

2.1. Stroh's formalism

Based upon StrohÕs formalism [20] in anisotropic elasticity, the general solutions for steady-state plane piezoelectricity has been presented by [3] as

Fig. 1. Geometry of the kinked crack system.

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u � 2 Re�Af�z��; / � 2 Re�Bf�z��; P1 � ÿ/;2; P2 � /;1; �1�

where

f�z� � ff1�z1� f2�z2� f3�z3� f4�z4�gT; zi � x1 � pix2; �2�

�x1; x2� is a ®xed rectangular coordinate system, a comma (or a prime later) followed by an ar-gument represents the di�erentiation with respect to the argument, ``Re'' stands for the real partof a complex number, u� {u1 u2 u3 u}T, Pj� {r1j r2j r3j Dj}

T, j � 1; 2; i � �������ÿ1p

, ui and u are theelastic displacement and electric potential (EDEP), rij and Di are stress and electric displacement,pi are the electroelastic eigenvalues of the material whose imaginary parts are positive, A and B arewell-de®ned in the literature (see [4, 17] for example), fi�zi� are arbitrary functions with complexarguments zi. Following the one-complex-variable approach given by Suo [21], introduce a vectorfunction

f�z� � ff1�z� f2�z� f3�z� f4�z�gT; z � x1 � px2; Im�p� > 0: �3�

The solution to any given boundary value problem can be expressed in terms of such a vector,regardless of the precise value of p [22]. Once a solution of f�z� is obtained, the variable z in fi�z�should be replaced with the argument zi, to compute the related ®elds. This approach allows thestandard matrix algebra to be used in conjunction with the techniques of analytic functions of onevariable, and thus by-pass some complexities arising from the use of four complex variables.

2.2. Crack-dislocation interaction in a homogeneous material (HM)

Consider the problem of piezoelectric branched crack in a two-dimensional in®nite medium (seeFig. 1). The interaction between crack and dislocation can be analyzed by ®rst studying theGreen's function due to a dislocation line, since a crack may be viewed as a continuous distri-bution of dislocation singularities. For an in®nite plane subject to a line dislocation, say b, locatedat z0�x10; x20�, the electroelastic solution is [16]:

f0�z� � ln�zÿ z0�2pi

BTb: �4�

The expression for f0�z� is obtained by imposing zero net traction-charge in a circuit around thedislocation, and by demanding that the jumps in EDEP be given by b � fDu1 Du2 Du3 DugT

,where Du is the jump for the quantity u across the dislocation line. In the presence of crack,however, the solution for an edge dislocation is no longer given by f0�z� alone. In addition to theconditions imposed at z0, the boundary conditions for the crack have to be satis®ed. This can beaccomplished by evaluating the traction-charge on crack face due to the dislocation singularityand introducing an appropriate function fI�z� to cancel the traction-charge. Substituting for f0�z�from Eq. (4) into Eq. (1), the SED on the crack surface induced by the edge dislocation alone at z0

is of the form

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t�x� � Bf 00�x� � Bf 00�x� �1

2piB

1

xÿ z0a

� �BTbÿ 1

2piB

1

xÿ z0a

� �B

Tb; �5�

where overhead bar denotes the complex conjugate, h��ai � diag���1 ��2 ��3 ��4� represents a di-agonal matrix. The prescribed traction-charge, ÿt�x1�, leads to the following Hilbert problem [14]:

Bf 0I�z� � ÿv�z�2pi

Z c

ÿc

t�x�dxv��x��xÿ z� � cv�z�; �6�

where v�z� is the basic Plemelj function de®ned as

v�z� � �z2 ÿ c2�ÿ1=2: �7�

Using contour integration one is ®nally led to

f 0I�z� � ÿ1

2piF��z; z0�BTb� 1

2piBÿ1BF��z; z0�BT

b; c � ÿLb

4p; �8�

where

L � ÿ2iBBT; F��z; z0� � 1

2

1

za ÿ z0a1

��ÿ v�za�

v�z0a���: �9�

The functions de®ned in Eqs. (4) and (8) together provide a GreenÕs function for the crackproblem:

f 0�z� � f 00�z� � f 0I�z� �1

2pi1

za ÿ z0a

� �F��z; z0�

� �BTb

�� Bÿ1BF��z; z0�BT

b

�: �10�

It is obvious that the singularity is contained in f0�z� and the crack interactions are accounted forby fI�z�.

2.3. Crack-dislocation interaction in a PP bimaterial

Consider a crack lying along the interface between material 1 and material 2 (see Fig. 1), subjectto a line dislocation at z0�x10; x20�. For convenience, we assume that z0 is in material 1. Thetreatment for z0 located in material 2 requires only a slight modi®cation. The electroelastic so-lution for the bimaterial due to the dislocation, b, can be assumed in the form [23]

u1 � 1

pIm A1 ln z�1�a

�Dhÿ z�1�0a

�EBT

1 bi� 1

pIm

X4

b�1

A1 ln z�1�a

�D"ÿ z�1�0b

�Eq�1�b

#; �11�

/1 �1

pIm B1 ln z�1�a

�Dhÿ z�1�0a

�EBT

1 bi� 1

pIm

X4

b�1

B1 ln z�1�a

�D"ÿ z�1�0b

�Eq�1�b

#�12�

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for material 1 in x2 > 0 and

u2 � 1

pIm

X4

b�1

A2 ln z�2�a

�D"ÿ z�1�0b

�Eq�2�b

#; �13�

/2 �1

pIm

X4

b�1

B2 ln z�2�a

�D"ÿ z�1�0b

�Eq�2�b

#�14�

for material 2 in x2 < 0. If the two materials are rigidly bonded we have

u1 � u2; /1 � /2 at x2 � 0: �15�

The substitution of Eqs. (11)±(14) into Eq. (15), leads to

f�1�0 �z�1�� �

1

2piln z�1�a

�D(ÿ z�1�0a

�EBT

1 b�X4

k�1

ln z�1�a

�Dÿ z�1�0k

�EB�kb

)�16�

for material 1 �x2 � y > 0�, and

f�2�0 �z�2�� �

X4

k�1

ln z�2�a

�Dÿ z�1�0k

�EB��k b �17�

for material 2 �y < 0�, where the superscripts (1) and (2) [or subscripts 1 and 2 below] label thequantities relating to the material 1 and 2, respectively, Ik � diag�d1k d2k d3k d4k�; dij � 1 for i� j;dij � 0 for i 6� j, and

B�k � Bÿ11 I

�ÿ 2 Mÿ1

1

��M

ÿ1

2

�ÿ1

Lÿ11

�B1IkB

T

1 ;

B��k �1

2piBÿ1

2 Mÿ1

1

��Mÿ1

2

�ÿ1

Lÿ11 B1IkBT

1 ;

Mj � ÿiBjAÿ1j ; �j � 1; 2�

�18�

the interface SED induced by the dislocation b is, then, of the form

t�x� � B2f0�2�0 �x� � B2f

0�2�0 �x� � �F���x� � F���x��b; �19�

where

F���x� �X4

k�1

B2B��kxÿ z�1�0k

: �20�

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This traction-charge vector, t�x� can be removed by superposing a solution with the trac-tion-charge, ÿt�x�, induced by the potential fI�z�. The solution for fI�z� can be obtained bysetting

h�z� � B1f0�1�I �z� in material 1;

Hÿ1HB2f0�2�I �z� in material 2;

(�21�

where H � ÿM1 ÿM2.The function h�z� de®ned above together with prescribed traction-charge, ÿt�x�, on crack faces,

yields the following non-homogeneous Hilbert problem:

h��x� �Hÿ1

Hhÿ�x� � ÿt�x� �22�

writing the quantities, h�x� and t�x� in their components in the sense of the eigenvector w, w3 andw4 as [22]

h�x� � h1�x�w� h2�x�w� h3�x�w3 � h4�x�w4;

t�x� � t�x�w� t�x�w� t3�x�w3 � t4�x�w4

�23�

one obtains

h1�z� � ÿ v�z�2pi

Z c

ÿc

t�x�dxv��x��xÿ z� � c1v�z�; h2�z� � ÿ v�z�

2pi

Z c

ÿc

t�x�dxv��x��xÿ z� � c2v�z�;

h3�z� � ÿ v3�z�2pi

Z c

ÿc

t3�x�dxv�3 �x��xÿ z� � c3v�z�; h4�z� � ÿ v4�z�

2pi

Z c

ÿc

t4�x�dxv�4 �x��xÿ z� � c4v�z�;

�24�

where w, w3 and w4 are eigenvectors and have been de®ned in [22]:

v�z� � �z� c�ÿ1=2ÿie�zÿ c�ÿ1=2�ie;

v3�z� � �z� c�ÿ1=2ÿj�zÿ c�ÿ1=2�j;

v4�z� � �z� c�ÿ1=2�j�zÿ c�ÿ1=2ÿj:

�25�

The functions hi and ti are evaluated by taking inner products with w, w3 and w4, respectively,i.e.

h1�x� � wTHh�x�wTHw

; h2�x� � wTHh�x�wTHw

; h3�x� � wT4 Hh�x�

wT4 Hw3

; h4�x� � wT3 Hh�x�

wT3 Hw4

;

t � wTHt�x�wTHw

; t3 � wT4 Ht�x�

wT4 Hw3

; t4 � wT3 Ht�x�

wT3 Hw4

:

�26�

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The contour integration of Eq. (24) provides

hi�z� � h�i �z� � b �i � 1; 2; 3; 4� �27�

h�1�z� �wTH

wTHw

X4

k�1

F e; v; z; z�1�0k ;B2;B��k

� �h� F e; v; z; z�1�0k ;B2;B

��k

� �i; �28�

h�2�z� �wTH

wTHw

X4

k�1

F�hÿ e; v; z; z�1�0k ;B2;B

��k

�� F

�ÿ e; v; z; z�1�0k ;B2;B

��k

�i; �29�

h�3�z� �wT

4 H

wT4 Hw3

X4

k�1

F�hÿ ij; v3; z; z

�1�0k ;B2;B

��k

�� F

�ÿ ij; v3; z; z

�1�0k ;B2;B

��k

�i; �30�

h�4�z� �wT

3 H

wT3 Hw4

X4

k�1

F ij; v4; z; z�1�0k ;B2;B

��k

� �h� F ij; v4; z; z

�1�0k ;B2;B

��k

� �i; �31�

c1 � ÿ wTH

wTHw

X�1� eÿ2pe

; c2 � ÿ wTH

wTHw

X�1� e2pe

;

c3 � ÿ wT4 H

wT4 Hw3

X�1� e2ipj

; c4 � ÿ wT3 H

wT3 Hw4

X�1� eÿ2ipj

;

�32�

where

F�r; v; z; z0k;X;Y� � ÿ XY

�1� eÿ2pr� 1

�ÿ v�z�

v�z0k��

1

zÿ z0k;

X� �X4

k�1

B2B��kÿ � B2B

��k

�b:

�33�

Once hi�z� has been obtained, the functions f0�j�I �z� can be evaluated through use of Eqs. (21)

and (23).It should be pointed out that the above solutions do not apply when the dislocation is located

on the interface. For an edge dislocation b located on the interface, say x0, the solution can beobtained by setting

f�1�0 �z� � ln z�1�a

ÿ ÿ x0

��q1; for y > 0;

f�2�0 �z� � ln z�2�a

ÿ ÿ x0

��q2; for y < 0:

�34�

The continuity condition on the interface and the condition related to the dislocation give

q1 �1

4piA1

�ÿ A2B

ÿ1

2 B1

�b; q2 � ÿBÿ1

2 B1q1: �35�

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The corresponding interface traction-charge vector is

t�x� � 1

xÿ x0

�B1q1 � B1q1� �1

xÿ x0

�Y� � Y��b; �36�

where Y� � �B1=4pi��A1 ÿ A2Bÿ1

2 B1�.As was done previously, let

t�x� � 1

xÿ x0

�tw� tw� t3w3 � t4w4�: �37�

Similarly the related functions hi�x� are as follows

h�1�z� �wTH

wTHwF�e; v; z; x0; I;Y� � Y��; �38�

h�2�z� �wTH

wTHwF�ÿe; v; z; x0; I;Y� � Y��; �39�

h�3�z� �wT

4 H

wT4 Hw3

F�ÿij; v3; z; x0; I;Y� � Y��; �40�

h�4�z� �wT

3 H

wT3 Hw4

F�ij; v4; z; x0; I;Y� � Y��; �41�

c1 � ÿ wTH

wTHw

�Y� � Y��b1� eÿ2pe

; c2 � ÿ wTH

wTHw

�Y� � Y��b1� e2pe

;

c3 � ÿ wT4 H

wT4 Hw3

�Y� � Y��b1� e2ipj

; c4 � ÿ wT3 H

wT3 Hw4

�Y� � Y��b1� eÿ2ipj

:

�42�

2.4. Crack-dislocation interaction in a PAC bimaterial

In Section 2.3 the solutions for a PP bimaterial due to a single dislocation are studied. Here weextend them to the case of a PAC bimaterial. When one of the two materials, say material 2, is ananisotropic conductor, the interface condition (15) should be changed to

u1

u2

u3

u

8>><>>:9>>=>>;�1�

�u1

u2

u3

0

8>><>>:9>>=>>;�2�

;

/1

/2

/3

/4

8>><>>:9>>=>>;�1�

�/1

/2

/3

0

8>><>>:9>>=>>;�2�

: �43�

Further inside the conductor, we have

D�2�i � E�2�i � 0: �44�

680 Q.-H. Qin, Y.-W. Mai / International Journal of Engineering Science 38 (2000) 673±693

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The formulations presented in Section 2.3 are still available if the below modi®cation is made:

A2 �A�3�3� 0

0 1

� �; B2 �

B�3�3� 0

0 1

� ��45�

p�2�4 � i; q�2�b � fq�2�b1 q�2�b2 q�2�b3 0g �b � 1; 2; 3; 4�; �46�

where A�3� 3� and B�3� 3� are well-de®ned material matrices for anisotropic elastic materials(see [23]for example).

2.5. Crack-dislocation interaction in a PID bimaterial

In this subsection we consider a PID bimaterial for which the upper half-plane (x2 > 0) is oc-cupied by piezoelectric material (material 1), and the lower half-plane (x2 < 0) is occupied byisotropic dielectric material (material 2). For simplicity a plane strain deformation is only con-sidered in this subsection. Thus the related interface condition is reduced to

u1

u2

u

8<:9=;�1�

�u1

u2

u

8<:9=;�2�

;/1

/2

/3

8<:9=;�1�

�/1

/2

/3

8<:9=;�2�

at x2 � 0: �47�

2.5.1. General solution for isotropic dielectric materialThe general solutions for isotropic dielectric material can be assumed in the form

ui � 2Re�vif �z��; /i � 2Re�hif �z��; z � x1 � px2; �48�

where v�� fvig� and h�� fhig� are determined by

�Q� p�R� RT� � p2T�v � 0; h � �RT � pT�v: �49�

For isotropic dielectric material, the 3 ´ 3 matrices Q, R and T are in the form

Q �k� 2l 0 0

0 l 00 0 ÿ e0

24 35; R �0 k 0l 0 00 0 0

24 35; T �l 0 00 k� 2l 00 0 ÿ e0

24 35; �50�

where k and l are the Lame constants [24] de®ned by

k � Et�1� t��1ÿ 2t� ; l � E

2�1� t�

here E and t are called YoungÕs modulus and PoissonÕs ratio.Since p� i is a double root, a second independent solution can be written as

ui � 2Reoop�vif �z��

� �; /i � 2Re

oop�hif �z��

� �; �51�

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where

v � vp � 1

2Gf1 i0g; h � hp � fi10g �52�

for plane elastic deformation, and

v � ve � ÿ1

e0

f001g; h � he � f00 ig �53�

for electric ®eld. Following the way of [24], the complete general solutions can be expressedas

u � Aqf �z� � zÿ z2i

vpq2f 0�z�; / � Bqf �z� � zÿ z2i

hpq2f 0�z�; �54�

A �1

2l�1ÿ2k�i

2l 0i

2lÿ2k2l 0

0 0 ÿ1e0

264375; B �

i 0 0ÿ1 ÿ i 00 0 i

24 35; �55�

where e0 is the dielectric constant, q� {q1 q2 q3}T are constants to be determined, andk � �k� 2l�=�k� l�.

2.5.2. Green's function for homogeneous isotropic materialsFor an in®nite domain subject to a line dislocation b located at z0�� x10 � ix20�, the function

f �z� in Eq. (54) can be assumed as

f �z� � ln�zÿ z0�: �56�

The conditionZC

du � b; for any closed curve C enclosing the point z0 �57�

yields

q � q0 � fq10 q20 q30gT � 1

2piA�b; �58�

where

A� �ÿ4kl1ÿ4k

ÿ�2ÿ4k�il1ÿ4k 0

ÿ2il1ÿ4k

2l1ÿ4k 0

0 0 ÿ e0

264375: �59�

682 Q.-H. Qin, Y.-W. Mai / International Journal of Engineering Science 38 (2000) 673±693

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2.5.3. Green's function for PID bimaterials(a) z0 located in isotropic dielectric material (material 2). Consider a PID bimaterial for which a

line dislocation b is applied at z0�� x10 � ix20; x20 < 0�. The interface condition (47) suggests thatthe solution can be assumed in the form:

u1 � 1

pIm�A1 ln�zah ÿ z0iq1�; /1 �

1

pIm�B1 ln�zah ÿ z0iq1� �60�

for material 1 in x2 > 0 and

u2 � 1

pIm A2A�b ln�z�

ÿ z0� � p�zÿ z�vpq20

�zÿ z0� � A2q2 ln�zÿ z0��;

/2 �1

pIm B2A�b ln�z�

ÿ z0� � p�zÿ z�hpq20

�zÿ z0� � B2q2 ln�zÿ z0�� �61�

for material 2 in x2 < 0, where q1 and q2 are unknowns to be determined. The substitution of Eqs.(60) and (61) into Eq. (47), yields

A1q1 � A2q2 � A2A�b; B1q1 � B2q2 � B2A

�b: �62�

Solving Eq. (62), we have

q1 � Bab; q2 � Bbb; �63�

where

Ba � �Bÿ1

2 B1 ÿ Aÿ1

2 A1�ÿ1�Bÿ1

2 B2 ÿ Aÿ1

2 A2�A�;Bb � �Bÿ1

1 B2 ÿ Aÿ1

1 A2�ÿ1�Bÿ1

1 B2 ÿ Aÿ1

1 A2�A�:�64�

(b) z0 located in piezoelectric material (material 1). When the single dislocation b is applied at z0

with x20 > 0, the electroelastic solution can be assumed in the form

u1 � 1

pIm A1 ln�zah� ÿ z0a�iBTb

�� 1

pImX3

b�1

A1 ln�za

hÿ z0b�

�q�1�b

i; �65�

/1 �1

pIm B1 ln�zah� ÿ z0a�iBTb

�� 1

pImX3

b�1

B1 ln�za

hÿ z0b�

�q�1�b

i�66�

for material 1 in x2 > 0 and

u2 � 1

pImX3

b�1

A2 ln�zhÿ z0b�

�q�2�b

i; �67�

Q.-H. Qin, Y.-W. Mai / International Journal of Engineering Science 38 (2000) 673±693 683

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/2 �1

pImX3

b�1

B2 ln�zhÿ z0b�

�q�2�b

i�68�

for material 2 in x2 < 0. Substituting Eqs. (65)±(68) into Eq. (47), we found the solution has thesame form as that given in Section 2.3 except that all matrices here are 3 ´ 3 ones.

The interface SED induced by the dislocation b is, then, of the form

t�x� � /1;1 � 2Re�B1f0�1�0 � �

1

pIm

1

xÿ z0

B1Bab

� ��69�

for z0 in material 2 and

t�x� � /2;1 � 2Re�B2f0�2�0 � �

1

pIm

X3

b�1

B2

1

xÿ z0b

� �q�2�b

" #� �F���x� � F���x��b: �70�

Similarly, t(x) can be removed by superposing a solution with the traction-charge, ÿt�x�, inducedby the potential fI�z�. The solution for fI�z� can be also obtained by setting

h�z� � B1f0�1�I �z� in material 1;

Hÿ1HB2f0�2�I �z� in material 2:

(�71�

The subsequent derivation is the same as that in subsection 2.3 except that

v�z� � �z� c�ÿ1=2ÿie�zÿ c�ÿ1=2�ie;

v3�z� � �z� c�ÿ1=2�zÿ c�ÿ1=2;�72�

where e is determined by a similar way to that of [22].

2.6. Crack-dislocation interaction in a PIC bimaterial

As treated in Section 2.4, the solutions for a PIC bimaterial due to a single dislocation can beobtained based on the results given in Section 2.5. When one of the two materials, say material 2,is an isotropic conductor, the interface condition (15) should be changed to

u1

u2

u

8<:9=;�1�

�u1

u2

0

8<:9=;�2�

;/1

/2

/3

8<:9=;�1�

�/1

/2

0

8<:9=;�2�

at x2 � 0: �73�

The formulations presented in Section 2.5 are still available if the below modi®cation is made:

q2 � fq21 q22 0g �b � 1; 2; 3�: �74�

684 Q.-H. Qin, Y.-W. Mai / International Journal of Engineering Science 38 (2000) 673±693

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3. Singular integral equations

For the sake of conciseness, all formulations developed in this section are only for the materialcombinations of HM, PP and PAC. The related singular integral equations for PID and PICbimaterials can be obtained straightforwardly, as we need only to choose the appropriate dislo-cation density function.

3.1. Electroelastic ®elds induced by remote uniform load

To satisfy the boundary conditions for the branch crack, one needs to know the electroelastic®elds induced by the remote uniform load P22 fty, as the GreenÕs functions derived above do notsatisfy the condition of uniform SED at in®nity. In consequent we set

f 0�z� � f 00�z� � f 0I�z� � f 01�z�; �75�

where f1�z� is the solution corresponding to the remote load. It is convenient to represent thesolution due to P12 as the sum of a uniform SED in an un¯awed solid and a corrective solution inwhich the main crack subject to ÿP12 . The solution to the latter is [22, 26]:

f 01�z� �1

2�z�z2 ÿ c2�ÿ1=2 ÿ 1�Bÿ1P12 �76�

for homogeneous piezoelectric material, and

f 0�1�1 �z� � Bÿ11 �h11�z�w� h21�z�w� h31�z�w3 � h41�z�w4�;

f 0�2�1 �z� � Bÿ12 H

ÿ1H�h11�z�w� h21�z�w� h31�z�w3 � h41�z�w4�

�77�

for PP or PAC bimaterials, where

h11�z� � 1

1� eÿ2pe

zÿ cz� c

� �ie

�z2

"ÿ c2�ÿ1=2�z� 2ice� ÿ 1

#wTHP12wTHw

;

h21�z� � 1

1� e2pe

zÿ cz� c

� �ÿie

�z2

"ÿ c2�ÿ1=2�zÿ 2ice� ÿ 1

#wTHP12wTHw

;

h31�z� � 1

1� e2pij

zÿ cz� c

� �j

�z2

�ÿ c2�ÿ1=2�z� 2jc� ÿ 1

�wT

4 HP12wT

4 Hw3

;

h41�z� � 1

1� eÿ2pij

zÿ cz� c

� �ÿj

�z2

�ÿ c2�ÿ1=2�zÿ 2jc� ÿ 1

�wT

3 HP12wT

3 Hw4

:

�78�

So far the function f �z� satis®es the conditions at in®nity and on the main crack face. It remainsonly to insure that the traction-charge free conditions for the branch crack be met.

Q.-H. Qin, Y.-W. Mai / International Journal of Engineering Science 38 (2000) 673±693 685

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3.2. Singular integral equations and solution methods

The remaining boundary conditions for the branch can be satis®ed by rede®ning the discretedislocation b, in terms of distributed dislocation densities, b�n� de®ning along the line, z� c + gz�,z0� c + nz�, where z� � cosh� p sinh emanating from the main crack tip. For simplicity we ®rstconsider the case of a branch in a homogeneous solid. Enforcing the satisfaction of traction-charge free condition on the branch, a system of singular integral equations for the dislocationb�n� is obtained as

L

2p

Z a

0

b�n�dngÿ n

� 1

p

Z a

0

Y0�g; n�b�n�dn�Q1�g� � 0; �79�

where Y0�g; n� is a kernel matrix function of the singular integral equation and is Holder-con-tinuous along 06 n6 a, Q1�g� is a known function vector corresponding to the SED ®eld inducedby the external load. They are

Y0�g; n� � ImfB z�a �

F��z; z0�BT ÿ B z�a �

F��z; z0�BTg; �80�

Q1�g� � Re B z�aza

�z2a ÿ c2�1=2

"*(ÿ 1

#+Bÿ1P12

)�81�

in which z�a � cosh� pa sinh.For the purpose of numerical solution, the following normalised quantities are introduced

s0 � 2gÿ aa

; s � 2nÿ aa

: �82�

If we retain the same symbols for the new functions caused by the change of the variables, thesingular integral Eq. (79) can, then, be rewritten in the form

L

2p

Z 1

ÿ1

b�s�dss0 ÿ s

� 1

p

Z 1

ÿ1

Y0�s; s0�b�s�ds�Q1�s0� � 0: �83�

Similarly for a branch in a PP or PAC bimaterial solid, the corresponding singular integralequation can be obtained as

L1

2p

Z 1

ÿ1

b�s�dss0 ÿ s

� 1

p

Z 1

ÿ1

fY0�s; s0�b�s� � Y1�s; s0�gds�Q1�s0� � 0; �84�

686 Q.-H. Qin, Y.-W. Mai / International Journal of Engineering Science 38 (2000) 673±693

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where

Y0�s; s0� � ImX4

k�1

B1

z�1��a

�s0 � 1�z�1��a ÿ �s� 1�z�1��k

* +B�k

" #; �85�

Y1�s; s0� � paRe B1 z�1��a

�Bÿ1

1 h�s; s0��

; �86�Q1�s; s0� � 2Re B1 z�1��a

�f�1�1 �s; s0�

� : �87�

3.3. Numerical scheme

Both the integral Eqs. (83) and (84) can be solved numerically using a method developed by [5].First we write the unknown density function, b(s) in Eq. (83) in the form [10]:

b�s� � b�s�������������1ÿ s2p �

Pmj�1 bjTj�s�������������

1ÿ s2p ; �88�

where b�s� is a regular function vector de®ned in the interval jsj6 1, and b�ÿ1� � 0 [10], since theSED behaves like dÿ1=2, here d measures distance from the branched crack tip, and the SED at thevertex behaves as dÿe, (e < 1/2) for a SED-free wedge with a wedge angle less than p. bj are the realunknown constant vectors, and Tj�s� the Chebyshev polynomials of the ®rst kind. Thus the dis-cretized form of Eq. (83) [or Eq. (84)] together with the condition, b�ÿ1� � 0, may be written as[5]:

Xm

k�1

1

nL

2�s0r ÿ sk��

� Y0�s0r; sk�b�sk���Q1�s0r� � 0

Xm

k�1

bkTk�ÿ1� � 0

�89�

for a homogeneous piezoelectric solid, and

Xm

k�1

1

nL1

2�s0r ÿ sk���

� Y0�s0r; sk�b�sk��� Y1�s0r; sk�

��Q1�s0r� � 0;

Xm

k�1

bkTk�ÿ1� � 0

�90�

for a PP or PAC bimaterial solid, where

Y1�x; y� � Y1�x; y�������������1ÿ s2p ; �91�

Q.-H. Qin, Y.-W. Mai / International Journal of Engineering Science 38 (2000) 673±693 687

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sk � cos�2k ÿ 1�p

2m

� �; �k � 1; 2; . . . ;m�;

s0r � cos�rp=m�; �r � 1; 2; . . . ;mÿ 1�:�92�

Eq. (89) [or (90)] provides a system of 4m equations for the determination of the 4m-values of bkj.Once the function b�s� has been found from Eq. (89) [or (90)], the SED, P�1�n �g�, in a coordinatesystem local to the crack branch line can be expressed in the form

P�1�n �s0� � X�h� L

2p

Z 1

ÿ1

b�s�s0 ÿ s

ds�

�Z 1

ÿ1

Y0�s; s0�b�s�ds�Q1�s0��

�93�

for a homogeneous solid, and

P�1�n �s0� � X�h� L1

2p

Z 1

ÿ1

b�s�dss0 ÿ s

�� 1

p

Z 1

ÿ1

fY0�s; s0�b�s� � Y1�s; s0�gds�Q1�s0��

�94�

for a PP or PAC bimaterial solid, where the 4 ´ 4 matrix X�h� whose components are the cosine ofthe angle between the local coordinates and the global coordinates is in the form

X�h� �cosh sinh 0 0ÿ sinh cosh 0 0

0 0 1 00 0 0 1

26643775: �95�

4. SED intensity factors and energy release rate

The SED intensity factors at the right tip of the branch crack are of interest and can be derivedby ®rst considering the traction and surface charge (TSC) on the direction of the branch line andconsidering the TSC very near the tip (s0 ! 1) which is given from Eq. (91)

P�1�n �r� � X�h� L�b�1�������������������8�s0 ÿ 1�p � 1

4

�������arX

r�h�L�b�1�; �96�

where r is a distance ahead of the branch tip, L� �L for a homogeneous material, and L� �L1 fora bimaterial.

Using Eq. (96) we can calculate the SED intensity factors at the right tip of the branch by thefollowing usual de®nition

K � fKII KI KIII KDgT � limr!0

�������2prp

P�1�n �r�: �97�

688 Q.-H. Qin, Y.-W. Mai / International Journal of Engineering Science 38 (2000) 673±693

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Combining with the results of Eq. (96), one then leads to

K �������ap8

rX�h�L�b�1�: �98�

Thus the solution of the singular integral equation enables the direct determination of the stressintensity factors.

The energy release rate G can be computed by the closure integral [15]

G � limx!0

1

2x

Z x

0

P�1�Tn �r�Dun�xÿ r�dr; �99�

where x is the assumed crack extension, Du is a vector of elastic displacement and electric potentialjump across the branch crack. Noting that

Du�r� �Z r

0

b�aÿ n�dn �100�

we have

G � limx!0

1

2x

Z x

0

U�1�Tn �r� oDun�xÿ r�ox

dr; �101�

where

U�1�n �r� �Z r

P�1�n �x�dx: �102�

During the derivation of Eq. (101), the following conditions have been employed

U�1�n �0� � 0; Du�xÿ r� r�xj � 0: �103�

Substituting Eq. (88) into (100), and Eq. (96) into (102), later into Eq. (101), one obtains

G � ap16

bT�1�L��XT�h��2b�1�: �104�

5. Criterion for crack kinking

An interface crack can advance either by continued growth in the interface or by kinking out ofthe interface into one of the adjoining materials. This competition can be assessed by comparingthe ratio of the energy release rate for kinking out of the interface and for interface cracking, Gkink/Gi, to the ratio of substrate toughness to interface toughness, Cs=Ci. In the concrete, if

Gkink

Gi>

Cs

Ci�105�

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kinking is favoured. Conversely, if the inequality in Eq. (105) is reversed, the ¯aw meets thecondition for continuing advance in the interface at an applied load lower than that necessary toadvance the crack into the substrate, where Cs and Ci are substrate and interface toughness, re-spectively. For a given load condition, P12 , the fracture toughness, C, is de®ned as the energyrelease rate G at the onset of the crack growth and has discussed in detail elsewhere [11]. Gkink andGi are the energy release rates for kinking out of the interface and for interface cracking, re-spectively, in which Gi can be derived by setting h� 0 in the previous sections, and Gkink is relatedto SED intensity factors, K, by

Gkink � 1

2KDK; �106�

where D � XT�h�L�ÿ1XT�h�.

6. Numerical results

Since the main purpose of this paper is to outline the basic principles of the proposed method,the assessment has been limited to a PP bimaterial plate with an interface crack of length 2c, abranch crack of length a and its interface coincided with x1-axis shown in Fig. 1. In all calcula-tions, the ratio of a/c is taken to be 0.1. The upper and lower materials are assumed to be PZT-5Hand PZT-5 [18], respectively. The material constants for the two materials are as follows:1. material properties for PZT-5H:

c11 � 117 GPa; c12 � c13 � 53 GPa; c22 � c33 � 126 GPa; c44 � 35:5 GPa;

c23 � 55 GPa; c55 � c66 � 35:3 GPa; e12 � e13 � ÿ6:5 C=m2; e11 � 23:3 C=m

2;

e35 � e26 � 17 C=m2; j11 � 130� 10ÿ10 C=Vm; j22 � j33 � 151� 10ÿ10 C=Vm:

2. material properties for PZT-5:

c11 � 111 GPa; c12 � c13 � 75:2 GPa; c22 � c33 � 121 GPa;

c44 � 22:8 GPa; c23 � 75:4 GPa; c55 � c66 � 21:1 GPa;

e12 � e13 � ÿ5:4 C=m2; e11 � 15:8 C=m

2; e35 � e26 � 12:3 C=m2;

j11 � 73:46� 10ÿ10 C=Vm; j22 � j33 � 81:7� 10ÿ10 C=Vm:

The ratio Gkink=Gi versus kink angle h with r123 � D12 � 0 is shown in Fig. 2 for a number ofloading combinations measured by w � tanÿ1�K1II =K1I �, where KI and KII are the conventionalmode I and mode II stress intensity factors. The curve clearly shows that the maximum energyrelease rate occurs at di�erent kink angle h for di�erent loading phase w, where h is the criticalkink angle for which the maximum energy release rate will occur under a given loading condition.For example, h � 0� for curves 4 and h � 11� for curve 3, h � 13� for curve 4 and h � 18� forcurve 1.

690 Q.-H. Qin, Y.-W. Mai / International Journal of Engineering Science 38 (2000) 673±693

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The ratio Gkink=Gi is plotted in Fig. 3 as a function of kink angle h for several values of remoteloading D12 with r122 � 4� 106 N=m

2and r112 � r123 � 0. The calculation indicates that the energy

release rate can have negative value depending on the direction and magnitude of the remoteloading D12 . For this particular problem, the energy release rate Gi will be positive whenÿ1:8� 10ÿ3 C=m

2 < D12 < 1:8� 10ÿ3 C=m2. Fig. 3 also shows that the critical kink angle is af-

fected by the load D12 . Generally, the critical kink angle will increase along with the increase ofabsolute value of D12 .

Fig. 4 shows the variation of Gkink=G0 as a function of D12 for several values of kink angle h andagain for r122 � 4� 106 N=m

2and r112 � r123 � 0. Here G0 is the value of G at zero electric loads. It

Fig. 3. Variation of Gkink=Gi with kink angle h for several loadings D12 �r122 � 4� 106 N=m2�.

Fig. 2. Variation of Gkink=Gi with kink angle h for several loading combinations speci®ed by w � tanÿ1�KII=KI�.

Q.-H. Qin, Y.-W. Mai / International Journal of Engineering Science 38 (2000) 673±693 691

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can be seen from this ®gure that the ratio of Gkink=G0 will mostly decrease along with the increaseof the magnitude of electric load.

7. Conclusions

An integral equation solution has been presented for determining whether cracks extend along,or kink out of the interface between various bimaterials. The combinations of the bimaterial maybe of PP, or PAC, or PID as well as PIC. In this investigation, explicit GreenÕs function for aninterface crack subject to an edge dislocation is derived allowing a branched crack problem to beexpressed in terms of coupled singular integral equations. The resulting integral equations involvethe dislocation density function de®ned on the line of branch only. As a result, the formulation forthe energy release rate and SED intensity factors can be expressed in terms of the dislocationdensity function and the branch angle. A particular example of a PP bimaterial containing aninterface crack and a branch was examined to see the application of the proposed formulation. Byway of the maximum energy release rate criterion, the numerical solution developed can be usedto assess whether an interface crack will extend in the interface or whether it will kink out of theinterface. The analysis also shows that the energy release rate can have either positive or negativevalues in the presence of electric load, depending on the direction and the magnitude of the ap-plied electric displacement. In other words, for a given mechanical load, presence of the appliedelectric displacement can either promote or retard crack extension. This conclusion may be usefulin designing piezoelectric components with interface.

Acknowledgements

The authors wish to thank the Australian Research Council (ARC) for the continuing supportof this work. Q.-H. Qin is supported by a Queen Elizabeth II fellowship from the ARC and theJ.G. Russell Award by the Australian Academy of Science.

Fig. 4. Variation of Gkink=Gi with remote loading D12 for several kink angles h�r122 � 4� 106 N=m2�.

692 Q.-H. Qin, Y.-W. Mai / International Journal of Engineering Science 38 (2000) 673±693

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