Electro-elastic interaction between a piezoelectric screw dislocation and collinear rigid lines

International Journal of Engineering Science 44 (2006) 422–435 www.elsevier.com/locate/ijengsci

Electro-elastic interaction between a piezoelectric screw dislocation and collinear rigid lines B.J. Chen *, D.W. Shu, Z.M. Xiao School of Mechanical and Aerospace Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 30 August 2005; accepted 29 October 2005 Available online 27 June 2006 (Communicated by E.S. SUHUBI)

Abstract Electro-elastic stress investigation on the interaction between a piezoelectric screw dislocation and collinear rigid lines under anti-plane mechanical and in-plane electrical loading is carried out. The lines are considered, respectively, as dielectrics or conductors. The screw dislocation is subjected to a line charge and a line force at the core. Closed-form analytical solutions are derived by means of complex variable method. Explicit expressions for the field variables, the singularity of the field variables at the line tip and the force on the dislocation are obtained for a single rigid line.  2006 Elsevier Ltd. All rights reserved. Keywords: Piezoelectric; Screw dislocation; Rigid line; Interaction

1. Introduction Piezoelectric materials are widely used in the device applications such as sensors and actuators. When subjected to mechanical and electric loads, these piezoelectric materials can fail prematurely due to defects produced during their manufacturing process. It is therefore important to know how the defects, such as cracks, dislocations and inhomogeneities, disturb the field variables and how the stress concentration arises due to the existence of the defects. For a Mode III crack problem in a homogenous piezoelectric material, Pak [1] found that the crack growth could be either enhanced or retarded depending on the magnitude, the direction, and the type of the applied mechanical and electrical loads. Especially for certain ratios of the applied electrical load to mechanical load, crack arrestment can be observed. Pak [2] obtained the closed-form solutions for a screw dislocation in a piezoelectric solid subjected to external loads and calculated the generalized Peach–Koehler forces acting on the screw dislocation.

*

Corresponding author. Fax: +65 67936763. E-mail address: [email protected] (B.J. Chen).

0020-7225/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2005.10.008

B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435

423

In the case of a circular inclusion in a homogenous piezoelectric material, Pak [3] analyzed and showed that in modeling cavities, imposing an impermeable boundary condition is a good approximation provided that the piezoelectric material has high dielectric constant and strong electro-elastic coupling. Stress and electric field concentrations are also studied. It is shown that a high electric field can be induced in the inclusion under a mechanical load when the matrix and the inclusion are poled in the opposite directions. Later, Meguid and Zhong [4] provided a theoretical treatment for the elliptical inhomogeneity problem in piezoelectric materials under remote non-uniform anti-plane shear and in-plane electric field using the complex variable method. There are also a number of research works on the interaction problems among cracks, dislocations and inclusions in piezoelectric solids published in open literature for the anti-plane deformation case. Meguid and Deng [5,6] studied the interaction between a screw dislocation and an elliptical inhomogeneity in piezoelectric media. Lee et al. [7] examined the interaction between a semi-infinite crack and a screw dislocation under anti-plane mechanical and in-plane electrical loading. Chen et al. [8] investigated the case of a wedge shaped crack interacting with a screw dislocation. Zhang et al. [9] derived the electrical and mechanical fields induced by a screw dislocation near an electrically insulating elliptical cavity in a piezoelectric material through the image dislocation approach by considering the electric field inside the cavity and found that the difference in the electric boundary conditions leads to great differences in the image force acting on the dislocation, in the intensity factors and in the J integral for crack propagation induced by the dislocation. Li and Weng [10] investigated the problem of a finite crack in a strip of functionally graded piezoelectric material and shown that an increase in the gradient of the material properties can reduce the magnitude of the stress intensity factor. When a flat inclusion is much harder that the matrix, it is reasonable to be considered as a rigid line. Shi [11] investigated the collinear rigid lines under anti-plane deformation and in-plane electric field in piezoelectric media. Chen et al. [12] studied the interaction between a semi-infinite rigid line and a piezoelectric screw dislocation. Liu and Fang [13] studied the interaction problem between a piezoelectric screw dislocation and circular interfacial rigid lines. The rigid line problem is a counterpart of the conventional crack problem, and thus sometimes called as anti-crack problem. It is therefore the purpose of this work to study the problem of how a piezoelectric screw dislocation interacting with collinear rigid lines under anti-plane deformation and in-plane electric field. 2. Formulation of the problem The physical problem considered is shown in Fig. 1. A charged screw dislocation is located at the point zd(rd, hd) near some rigid lines embedded in an infinite piezoelectric medium under remote anti-plane strain deformation and in-plane electric field. The rigid lines are assumed to be collinearly located along the x-axis of a Cartesian coordinate system xyz. The dislocation is assumed to be straight and infinitely long in the z-direction, suffering a finite discontinuity in the displacement bz and electric potential bu across the slip plane. The dislocation has a line force p and a line charge q along its core. In a linear piezoelectric medium, the governing field equations and constitutive relations at constant temperature can be written as rij;j ¼ 0; Di;i ¼ 0;

ð2:1aÞ ð2:1bÞ

rij ¼ cijkl uk;l  ekij Ek ;

ð2:2aÞ

Di ¼ eikl uk;l þ eik Ek ;

ð2:2bÞ

where rij, ui, Di and Ei are stress, displacement, electric displacement and electric fields, respectively. cijkl, ekij and eij are the corresponding elastic, piezoelectric and dielectric constants which satisfy the following relations cijkl ¼ cklij ¼ cijlk ¼ cjikl ;

ekij ¼ ekji ;

eik ¼ eki :

ð2:3Þ

As for the current an anti-plane problem, the anti-plane displacement w is coupled with the in-plane electric field Ex and Ey, those variables are independent of the longitudinal coordinate z, such that

424

B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435 ∞ ∞ τ zy or γ zy

zd (rd ,θ d )

y rd Dx∞ or E x∞

∞ ∞ τ zx or γ zx

θd

aj

Lj

bj

x

D y∞ or E y∞ Fig. 1. A piezoelectric screw dislocation near collinear rigid lines.

w ¼ wðx; yÞ;

Ex ¼ Ex ðx; yÞ;

Ey ¼ Ey ðx; yÞ:

ð2:4Þ

The governing field equations and constitutive relations in (2.1) and (2.2) are reduced to orzx orzy þ ¼ 0; ox oy

ð2:5aÞ

oDx oDy þ ¼ 0; ox oy

ð2:5bÞ

rzx ¼ c44

ow ou þ e15 ; ox ox

rzy ¼ c44

ow ou þ e15 ; oy oy

ð2:6aÞ

Dx ¼ e15

ow ou  e11 ; ox ox

Dy ¼ e15

ow ou  e11 ; oy oy

ð2:6bÞ

where u = u(x, y) is the electric potential and Ex ¼ 

ou ; ox

Ey ¼ 

ou : oy

ð2:7Þ

Substituting (2.6) into (2.5), we have c44 r2 w þ e15 r2 u ¼ 0;

ð2:8Þ

e15 r2 w  e11 r2 u ¼ 0;

ð2:9Þ

where $2 is the two-dimensional Laplacian operator. The above equations can be satisfied if we choose r2 w ¼ 0;

ð2:10Þ

r2 u ¼ 0:

ð2:11Þ

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425

If we let the harmonic functions w, u be the imaginary parts of some complex potentials of the complex variable z = x + iy = reih, or w ¼ ImwðzÞ;

ð2:12Þ

u ¼ ImuðzÞ;

ð2:13Þ

then we have czy þ iczx ¼ W ðzÞ;

ð2:14Þ

Ey þ iEx ¼ UðzÞ;

ð2:15Þ

szy þ iszx ¼ c44 W ðzÞ þ e15 UðzÞ;

ð2:16Þ

Dy þ iDx ¼ e15 W ðzÞ  e11 UðzÞ;

ð2:17Þ

where W ðzÞ ¼ w0 ðzÞ;

ð2:18Þ

UðzÞ ¼ u0 ðzÞ;

ð2:19Þ

the prime denotes the derivative with respect to the argument z. Assume that there is no external loads acting on the lines, the mechanical boundary condition and the equilibrium condition of the lines are wþ ðxÞ ¼ w ðxÞ ¼ w0j on L; Z  ½sþ on L; zy ðxÞ  szy ðxÞdx ¼ 0

ð2:20Þ ð2:21Þ

Lj

where Pthe superscripts + and  refer, respectively, to the upper and lower line surfaces, w0j is constant and L ¼ j Lj . Eq. (2.20) can be rewritten as þ



w0 ðxÞ ¼ w0 ðxÞ ¼ 0

on L;

ð2:22Þ

where w0 ðxÞ ¼

dwðxÞ : dx

ð2:23Þ

Assume that zero thickness of lines, the electric boundary conditions can be described as uþ ðxÞ ¼ u ðxÞ ¼ u0j on L; Z  ½Dþ on L y ðxÞ  Dy ðxÞdx ¼ 0

ð2:24Þ ð2:25Þ

Lj

for the rigid conducting lines, and uþ ðxÞ ¼ u ðxÞ

on L;

ð2:26Þ

Dþ y ðxÞ

on L

ð2:27Þ

¼

D y ðxÞ

for the rigid dielectric lines, where u0j is constant. Eq. (2.24) can also be rewritten as u0 ðxÞþ ¼ u0 ðxÞ ¼ 0

on L

ð2:28Þ

where u0 ðxÞ ¼

duðxÞ : dx

ð2:29Þ

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B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435

3. Solution of the problem The solution for the problem is to find two analytic functions W(z) and U(z) which should satisfy the boundary conditions (2.21), (2.22) either with (2.25), (2.28) for rigid conducting lines or with (2.26), (2.27) for rigid dielectric lines, respectively. For the current problem, W(z) can be chosen as W ðzÞ ¼ Cw þ W d ðzÞ þ W 1 ðzÞ; UðzÞ ¼ Cu þ Ud ðzÞ þ U1 ðzÞ;

ð3:1Þ ð3:2Þ

where 1 1 1 ðe11 s1 zy þ e15 Dy Þ þ iðe11 szx þ e15 Dx Þ ; c44 e11 þ e215 1 1 1 ðe15 s1 zy  c44 Dy Þ þ iðe15 szx  c44 Dx Þ ; Cu ¼ c44 e11 þ e215 A1 þ iA2 bz 1 e11 p  e15 q ; A2 ¼  W d ðzÞ ¼ ; A1 ¼ ; 2p c44 e11 þ e215 z  zd 2p B1 þ iB2 bu 1 e15 p þ c44 q ; B2 ¼  Ud ðzÞ ¼ ; B1 ¼ ; 2p c44 e11 þ e215 z  zd 2p

Cw ¼

ð3:3Þ ð3:4Þ ð3:5Þ ð3:6Þ

and W1(z), U1(z) correspond to the disturbed fields caused by the lines which vanish at infinity. With reference to (2.18), condition (2.22) yields W ðxÞþ  W ðxÞ ¼ 0;

W ðxÞ  W ðxÞþ ¼ 0

on L

ð3:7Þ

which leads to ½W ðxÞ þ W ðxÞþ  ½W ðxÞ þ W ðxÞ ¼ 0 þ



½W ðxÞ  W ðxÞ þ ½W ðxÞ  W ðxÞ ¼ 0

on L;

ð3:8Þ

on L;

ð3:9Þ

where over-bar denotes conjugate. The substitution of (3.1) into (3.8) and (3.9) yields þ



þ



½W 1 ðxÞ þ W 1 ðxÞ  ½W 1 ðxÞ þ W 1 ðxÞ ¼ 0

on L;

½W 1 ðxÞ  W 1 ðxÞ þ ½W 1 ðxÞ  W 1 ðxÞ ¼ 2f 0w ðxÞ on L;

ð3:10Þ ð3:11Þ

where f0w ðxÞ ¼ Cw  Cw þ

A1  iA2 A1 þ iA2  : x  zd x  zd

ð3:12Þ

Based on the theory of Muskhelishvili [14], the solution of boundary problems (3.10) and (3.11) can be obtained as W 1 ðzÞ þ W 1 ðzÞ ¼ 0;

ð3:13Þ

W 1 ðzÞ  W 1 ðzÞ ¼ 2X 0 ðzÞ½I w ðzÞ þ P w ðzÞ;

ð3:14Þ

where X 0 ðzÞ ¼

n Y 1 1 ðz  aj Þ 2 ðz  bj Þ 2 ; j¼1

I w ðzÞ ¼ P w ðzÞ ¼

Z

f0w ðxÞdx ; þ L X 0 ðxÞðx  zÞ cn1 zn1 þ    þ c0 :

1 2pi

ð3:15Þ ð3:16Þ ð3:17Þ

B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435

427

The addition of Eqs. (3.13) and (3.14) results in W 1 ðzÞ ¼ X 0 ðzÞ½I w ðzÞ þ P w ðzÞ:

ð3:18Þ

Condition (2.21), together with (2.16), (3.1) and (3.2) arrives at Z f½c44 ðW 1  W 1 Þ þ e15 ðU1  U1 Þþ  ½c44 ðW 1  W 1 Þ þ e15 ðU1  U1 Þ gdx ¼ 0;

ð3:19Þ

L

which can be rewritten into the closed path integral as I fc44 ½W 1 ðzÞ  W 1 ðzÞ þ e15 ½U1 ðzÞ  U1 ðzÞgdz ¼ 0;

ð3:20Þ

K

where K is the closed path around each line. 3.1. The rigid conducting lines In this case, the boundary conditions (2.25) and (2.28) are applied. Following the similar derivation as above, condition (2.28) results in U1 ðzÞ  U1 ðzÞ ¼ 2X 0 ðzÞ½I u ðzÞ þ P u ðzÞ; U1 ðzÞ ¼ X 0 ðzÞ½I u ðzÞ þ P u ðzÞ; where

Z

f0u ðxÞdx ; þ L X 0 ðxÞðx  zÞ d n1 zn1 þ    þ d 0 ;

1 I u ðzÞ ¼ 2pi P u ðzÞ ¼

ð3:21Þ ð3:22Þ

f0u ðxÞ ¼ Cu  Cu þ

ð3:23Þ ð3:24Þ

B1  iB2 B1 þ iB2  ; x  zd x  zd

ð3:25Þ

and the condition (2.25) yields I fe15 ½W 1 ðzÞ  W 1 ðzÞ  e11 ½U1 ðzÞ  U1 ðzÞgdz ¼ 0:

ð3:26Þ

K

The functions W1(z) and U1(z) are thus determined, respectively, by (3.18) and (3.22). While the constants cn1, . . . , c0 and dn1, . . . , d0 therein can be calculated by (3.20) and (3.26) together with (3.14) and (3.21). 3.2. The rigid dielectric lines In this case, the boundary conditions (2.26) and (2.27) are applied. Introducing (2.13) into (2.26) with the derivation with respect to x gives ½U1 ðxÞ þ U1 ðxÞþ  ½U1 ðxÞ þ U1 ðxÞ ¼ 0

ð3:27Þ

on L;

where the relationship (2.19) has been used. According to the Liouville theorem, the following equation holds: U1 ðzÞ þ U1 ðzÞ ¼ 0:

ð3:28Þ

On the other hand, the condition (2.27) with the aid of (2.17) arrives at þ



½e15 ðW 1  W 1 Þ  e11 ðU1  U1 Þ  ½e15 ðW 1  W 1 Þ  e11 ðU1  U1 Þ ¼ 0

on L;

ð3:29Þ

where W 1 ¼ W 1 ðxÞ; U1 ¼ U1 ðxÞ: Using the Liouville theorem, we obtain e15 ½W 1 ðzÞ  W 1 ðzÞ  e11 ½U1 ðzÞ  U1 ðzÞ ¼ 0:

ð3:30Þ

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B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435

zd (rd ,θ d )

y r1 rd

θ1 −a

r2

θd L

θ2 a

x

Fig. 2. A piezoelectric screw dislocation near a rigid lines.

Solving (3.28) and (3.30), we get, e15 ½W 1 ðzÞ  W 1 ðzÞ; 2e11 e15 U1 ðzÞ ¼  ½W 1 ðzÞ  W 1 ðzÞ: 2e11

U1 ðzÞ ¼

The substitution of (3.31) and (3.32) into (3.20) yields I ½W 1 ðzÞ  W 1 ðzÞdz ¼ 0;

ð3:31Þ ð3:32Þ

ð3:33Þ

K

or

I

X 0 ðzÞ½I w ðzÞ þ P w ðzÞdz ¼ 0:

ð3:34Þ

K

After determining W1(z) from (3.18) and (3.34), the function U1(z) is obtained by (3.31). The problem is thus solved. It is readily shown that the current solution reduces to that of Shi [11] in the case of the dislocation vanishes. In the following sections, the solution will be given in more explicit forms for a single rigid line as shown in Fig. 2. No remote loads are added. 4. Singularity of field variables In this section, the singularity of the field variables around the tip of the single rigid line due to a piezoelectric screw dislocation is discussed. For this case, we have 1

X 0 ðzÞ ¼ ðz2  a2 Þ2 ; A1  iA2 A1 þ iA2  ; x  zd x  zd B1  iB2 B1 þ iB2  ; f0u ðxÞ ¼ x  zd x  zd

f0w ðxÞ ¼

ð4:1Þ ð4:2Þ ð4:3Þ

P w ðzÞ ¼ c0 ;

ð4:4Þ

P u ðzÞ ¼ d 0 :

ð4:5Þ

B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435

429

4.1. The rigid conducting line Substituting (4.2), (4.3), respectively, into (3.16), (3.23), and applying the residue theorem, we obtain "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # z2  a2  z2d  a2 z2  a2  z2d  a2 A1  iA2 A1 þ iA2 I w ðzÞ ¼ 1  1 ; ð4:6Þ z  zd z  zd 2 2 "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # B1  iB2 B1 þ iB2 z2  a2  z2d  a2 z2  a2  z2d  a2 1  1 : ð4:7Þ I u ðzÞ ¼ z  zd z  zd 2 2 The substitution of (3.14), (3.21) into (3.20) and (3.26), together with (4.4) and (4.5) results in c0 = d0 = 0. The functions W(z) and U(z) expressed in (3.1) and (3.2) with Cw and Cu vanished for the rigid conducting line are thus written as A1 þ iA2 þ W 1 ðzÞ; z  zd B1 þ iB2 UðzÞ ¼ þ U1 ðzÞ; z  zd

ð4:8Þ

W ðzÞ ¼

ð4:9Þ

where

"pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # z2  a2  z2d  a2 z2  a2  z2d  a2 A1  iA2 A1 þ iA2 W 1 ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ; ð4:10Þ z  zd z  zd 2 z 2  a2 2 z 2  a2 "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # B1  iB2 B1 þ iB2 z2  a2  z2d  a2 z2  a2  z2d  a2  1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 : ð4:11Þ U1 ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi z  zd z  zd 2 z 2  a2 2 z 2  a2 pffiffi It is seen from (2.14)–(2.17) that the field variables show 1= r type of singularity near the line tip. If we define the strain and electric field intensity factors at the right tip as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K czy þ iK czx ¼ lim 2pðz  aÞðczy þ iczx Þ ¼ lim 2pðz  aÞW ðzÞ; ð4:12Þ z!0 z!0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K Ey þ iK Ex ¼ lim 2pðz  aÞðEy þ iEx Þ ¼  lim 2pðz  aÞUðzÞ; ð4:13Þ z!0

z!0

then, the stress and electrical displacement field intensity factors can be obtained as K szy þ iK szx ¼ c44 ðK czy þ iK czx Þ  e15 ðK Ey þ iK Ex Þ;

ð4:14Þ

K Dy þ iK Dx ¼ e15 ðK czy þ iK czx Þ þ e11 ðK Ey þ iK Ex Þ:

ð4:15Þ

The substitution of (4.8) and (4.9), respectively, into (4.12) and (4.13) with the aids of (3.5) and (3.6) yields   rffiffiffiffi 1 r1 h1  h2 e11 p  e15 q h1  h2   1 ; ð4:16 Þ K czx ¼  pffiffiffiffiffiffi bz sin cos c44 e11 þ e215 2 2 2 pa r2   rffiffiffiffi 1 r1 h1  h2 e15 p þ c44 q h 1  h2 p ffiffiffiffiffi ffi  1 ; ð4:17 Þ cos bu sin K Ex ¼ 2 pa r2 c44 e11 þ e215 2 2   rffiffiffiffi 1 r1 h1  h2 h1  h2  p cos 1 ; ð4:18 Þ K szx ¼  pffiffiffiffiffiffi ðc44 bz þ e15 bu Þ sin 2 2 2 pa r2   rffiffiffiffi 1 r1 h1  h2 h 1  h2 þ q cos 1 ; ð4:19 Þ K Dx ¼  pffiffiffiffiffiffi ðe15 bz  e11 bu Þ sin 2 2 2 pa r2 K szy ¼ K czy ¼ K Dy ¼ K Ey ¼ 0; where the following relationships have been used (refer to Fig. 2): rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi zd þ a r1 h1  h2 zd þ a r1 h1  h2 Re ; Im : cos sin ¼ ¼ zd  a zd  a r2 2 r2 2

ð4:20Þ

ð4:21Þ

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B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435

4.2. The rigid dielectric line Substituting (4.2) into (3.16), and applying the residue theorem, we obtain "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # z2  a2  z2d  a2 z2  a2  z2d  a2 A1  iA2 A1 þ iA2 I w ðzÞ ¼ 1  1 : z  zd z  zd 2 2

ð4:22Þ

The substitution of (4.4) and (4.22) into (3.34) results in c0 = 0. The function W1(z) is thus obtained as "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # z2  a2  z2d  a2 z2  a2  z2d  a2 A1  iA2 A1 þ iA2 W 1 ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 : ð4:23Þ z  zd z  zd 2 z2  a2 2 z 2  a2 The function U1(z) is then obtained from (3.31) as ( "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # "pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi #) z2  a2  z2d  a2 z2  a2  z2d  a2 e15 A1  iA2 A1 þ iA2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi U1 ðzÞ ¼  1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 : z  zd z  zd 2e11 z 2  a2 z 2  a2

ð4:24Þ

Thus the final functions W(z) and U(z) for a single rigid dielectric line are expressed in (4.8) and (4.9), where W1(z) and U1(z) are given in (4.23) and (4.24), respectively. The field intensity factors at the right tip are then obtained as   rffiffiffiffi 1 r1 h1  h2 e11 p  e15 q h1  h2 K czx ¼  pffiffiffiffiffiffi   1 ; ð4:25 Þ cos bz sin 2 2 2 pa r2 c44 e11 þ e215   rffiffiffiffi  1 r1 e15 h1  h2 e11 p  e15 q h1  h2   1 ; ð4:26 Þ bz sin cos K Ex ¼ pffiffiffiffiffiffi c44 e11 þ e215 2 2 2 pa r2 e11    rffiffiffiffi 1 r1 c44 e11 þ e215 h1  h2 e11 p  e15 q h1  h2   1 ; ð4:27 Þ bz sin cos K szx ¼  pffiffiffiffiffiffi 2 pa r2 c44 e11 þ e215 e11 2 2 K Dx ¼ 0;

K szy ¼ K czy ¼ K Dy ¼ K Ey ¼ 0:

ð4:28Þ

5. Image force on the dislocation According to the generalized Peach–Koehler formula derived by Pak [2], the image forces on the piezoelectric screw dislocation with a line force and a line charge are F x ¼ bz sTzy þ b/ DTy þ pcTzx þ qETx ; Fy ¼

bz sTzx



b/ DTx

þ

pcTzy

þ

qETy ;

ð5:1Þ ð5:2Þ

where rTzy , rTzx , DTy , DTx , cTzy , cTzx , ETy and ETx are calculated from the functions W1(z) and U1(z) with z ! zd: cTzy þ icTzx ¼ lim W 1 ðzÞ;

ð5:3Þ

ETy þ iETx ¼  lim U1 ðzÞ;

ð5:4Þ

z!zd

z!zd

sTzy þ isTzx ¼ c44 lim W 1 ðzÞ þ e15 lim U1 ðzÞ;

ð5:5Þ

DTy þ iDTx ¼ e15 lim W 1 ðzÞ  e11 lim U1 ðzÞ:

ð5:6Þ

z!zd

z!zd

z!zd

z!zd

5.1. The rigid conducting line Substituting (4.10) and (4.11) into (5.3)–(5.6), with the aids of (3.5) and (3.6), we have   1 e11 p  e15 q cTzy ¼ ðcs1 þ cs2 Þbz  ðcs3  cs4 Þ ; 2pa c44 e11 þ e215

ð5:7Þ

B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435

  1 e11 p  e15 q ðcs3 þ cs4 Þbz þ ðcs1  cs2 Þ ; ¼ 2pa c44 e11 þ e215   1 e15 p þ c44 q T ðcs1 þ cs2 Þbu þ ðcs3  cs4 Þ Ey ¼ ; 2pa c44 e11 þ e215   1 e15 p þ c44 q T ðcs3 þ cs4 Þbu  ðcs1  cs2 Þ ; Ex ¼ 2pa c44 e11 þ e215

cTzx

431

ð5:8Þ ð5:9Þ ð5:10Þ

sTzy ¼

1 ½ðcs1 þ cs2 Þðc44 bz þ e15 bu Þ  ðcs3  cs4 Þp; 2pa

ð5:11Þ

sTzx ¼

1 ½ðcs3 þ cs4 Þðc44 bz þ e15 bu Þ þ ðcs1  cs2 Þp; 2pa

ð5:12Þ

DTy ¼

1 ½ðcs1 þ cs2 Þðe15 bz  e11 bu Þ þ ðcs3  cs4 Þq; 2pa

ð5:13Þ

DTx ¼

1 ½ðcs3 þ cs4 Þðe15 bz  e11 bu Þ  ðcs1  cs2 Þq; 2pa

ð5:14Þ

where cs1, cs2, cs3 and cs4 are functions of dislocation location only, and are expressed explicitly as " # 1 sinðh1 þ h2 Þ cosðh1 =2 þ h2 =2Þ pffiffiffiffiffiffiffiffi cs1 ¼  ; 2 r1 sin h1 þ r2 sin h2 r1 r2 " # 1 r1 cos h2 þ r2 cos h1 cosðh1 =2 þ h2 =2Þ pffiffiffiffiffiffiffiffi  þ cs2 ¼ ; 2 2r1 r2 r1 r2 " # 1 cosðh1 þ h2 Þ  1 sinðh1 =2 þ h2 =2Þ pffiffiffiffiffiffiffiffi þ cs3 ¼ ; 2 r1 sin h1 þ r2 sin h2 r1 r2 " # 1 r1 sin h2 þ r2 sin h1 sinðh1 =2 þ h2 =2Þ pffiffiffiffiffiffiffiffi  cs4 ¼ 2 2r1 r2 r1 r2

ð5:15Þ

ð5:16Þ

ð5:17Þ

ð5:18Þ

with r1 ¼

r1 ; a

r2 ¼

r2 : a

Substituting (5.7)–(5.14) into (5.1) and (5.2) yields  1 ðcs1 þ cs2 Þðc44 b2z þ 2e15 bz bu  e11 b2u Þ þ 2cs4 ðpbz  qbu Þ Fx ¼ 2pa  cs1  cs2 2 2 þ ðe p  2e pq  c q Þ ; 11 15 44 c44 e11 þ e215  1 ðcs3 þ cs4 Þðc44 b2z  2e15 bz bu þ e11 b2u Þ þ 2cs2 ðpbz  qbu Þ Fy ¼ 2pa  cs3  cs4 2 2 þ ðe p þ 2e pq þ c q Þ : 11 15 44 c44 e11 þ e215

ð5:19Þ

ð5:20Þ

ð5:21Þ

5.2. The rigid dielectric line For this case, rTzy , rTzx , DTy , DTx , cTzy , cTzx , ETy and ETx are calculated from the functions W1(z) and U1(z) expressed in (4.23) and (4.24) with z ! zd:

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B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435

  1 e11 p  e15 q ðcs1 þ cs2 Þbz  ðcs3  cs4 Þ ; ¼ 2pa c44 e11 þ e215   1 e11 p  e15 q T ðcs3 þ cs4 Þbz þ ðcs1  cs2 Þ czx ¼ ; 2pa c44 e11 þ e215   1 e15 e11 p  e15 q T ðcs1 þ cs2 Þbz  ðcs3  cs4 Þ ; Ey ¼  2pa e11 c44 e11 þ e215   1 e15 e11 p  e15 q T ðcs3 þ cs4 Þbz þ ðcs1  cs2 Þ ; Ex ¼  2pa e11 c44 e11 þ e215   1 c44 e11 þ e215 e11 p  e15 q T ðcs1 þ cs2 Þbz  ðcs3  cs4 Þ szy ¼ ; 2pa e11 c44 e11 þ e215   1 c44 e11 þ e215 e11 p  e15 q T ðcs3 þ cs4 Þbz þ ðcs1  cs2 Þ ; szx ¼ 2pa e11 c44 e11 þ e215 cTzy

ð5:22Þ ð5:23Þ ð5:24Þ ð5:25Þ ð5:26Þ ð5:27Þ

DTy ¼ 0;

ð5:28Þ

DTx ¼ 0:

ð5:29Þ

Substituting (5.22)–(5.29) into (5.1) and (5.2) yields " # 1 cs1  cs2 ðe11 p  e15 qÞ2 2 2 ðcs1 þ cs2 Þðc44 þ e15 =e11 Þbz þ 2cs4 ðp  qe15 =e11 Þbz þ Fx ¼ ; 2pa e11 c44 e11 þ e215 " # 2 1 cs3  cs4 ðe11 p  e15 qÞ 2 2 ðcs3 þ cs4 Þðc44 þ e15 =e11 Þbz þ 2cs2 ðp  qe15 =e11 Þbz  Fy ¼ : 2pa c44 e11 þ e215 e11

ð5:30Þ

ð5:31Þ

6. Discussions 6.1. Field intensity factors The closed-form analytical expressions for the field intensity factors induced by the dislocation Burgers vector bz, b/, line force p and line charge q are given, respectively, by (4.16)–(4.20) for a rigid conducting line, and by (4.25)–(4.28) for a rigid dielectric line. In this section, we aim to discuss how the parameters bz, b/, p and q affect the field intensity factors. 6.1.1. The rigid conducting line When all the parameters bz, b/, p and q are positive, (4.16)–(4.20) suggest that K czx is independent of bu, K Ex is independent of bz, K szx is independent of q, and K Dx is independent of p, respectively. In the case that the line defect is located along the real axis, all the singularities disappeared. If the line defect is located at the upper 2 half-plane, since sin h1 h 6 0, we may conclude that the Burger’s vector bz always increases (anti-shields) K czx , 2 K szx and K Dx ; the Burger’s vector bu always shields K Ex and K Dx , while anti-shields K szx ; The line force p always shields K czx and K szx , while anti-shields K Ex ; The line charge q always anti-shields K czx , K Dx and K Ex . These conclusions also hold for the line force and line charge when they are located at the lower half-plane, while reverse for the Burger’s vectors bz and bu. The above line-tip shielding effects are summarized in Table 1. 6.1.2. The rigid dielectric line In this case, the shielding effects from the parameters bz, b/, p and q on the field intensity factors K czx and K Ex are the same as those of the rigid conducting line case as discussed above. The electrical displacement Dx has no singularities at the line tip. Such shielding effect is summarized in Table 2.

B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435

433

Table 1 Line-tip shielding effect induced by a line force, a line charge and a screw dislocation with positive bz, bu, p, q for a rigid conducting line Position

K czx K Ex K szx K Dx

p < hd < 0

hd = 0

0 < hd < p

p

q

bz

bu

p

q

bz

bu

p

q

bz

bu

 +  ·

+ · + +

 ·  

· +  +

· · · ·

· · · ·

· · · ·

· · · ·

 +  ·

+ + · +

+ · + +

·  + 

+: Anti-shielding, : shielding.

Table 2 Line-tip shielding effect induced by a line force, a line charge and a screw dislocation with positive bz, bu, p, q for a rigid dielectric line Position

K czx K Ex K szx K Dx

p < hd < 0

hd = 0

0 < hd < p

p

q

bz

bu

p

q

bz

bu

p

q

bz

bu

 +  ·

+ · + ·

 ·  ·

· + · ·

· · · ·

· · · ·

· · · ·

· · · ·

 +  ·

+ + + ·

+ · + ·

·  · ·

+: Anti-shielding, : shielding.

Fig. 3. Force on the dislocation along the x-axis versus angular position (deg).

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B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435

Fig. 4. Force on the dislocation along the y-axis versus angular position (deg).

6.2. Force on the dislocation The image forces on the dislocation are given explicitly in (5.20) and (5.21) for a conductor case and in (5.30) and (5.31) for a dielectric case. They are functions of bz, bu, p, q, zd and the material constants of the piezoelectric solid. In order to have a clear and direct picture on how these parameters affect the image force, here, the piezoelectric material, namely, PZT-6B is used as an example to numerically demonstrate the force on the dislocation. The material properties are given by c44 ¼ 2:71  1010 N=m2 ;

e15 ¼ 4:6 C=m2 ;

e11 ¼ 3:6  109 C=V m:

ð6:1Þ

For the units of the material constants, N is the force in Newtons, C is the charge in coulombs, V is the electric potential in volts and m is the length in meters. Other parameters are taken to have the values below p ¼ 10 N=m;

q ¼ 108 C=m; bz ¼ 109 m; b/ ¼ 1:0 V:

ð6:2Þ

The forces on the dislocation are normalized by the parameter F0 ¼

c44 b2z : 4pa

ð6:3Þ

The normalized forces on dislocation versus the angular position around the right line tip under the above parameters are depicted in Figs. 3 and 4, respectively, where rd is fixed at rd = 1.2a. It is found that the figures are symmetric about the real axis for the conductor case, while this is not true for the dielectric case. Furthermore, the rigid dielectric line always repels the line defect in the x-axis when they are located near the line tip, while for he conductor case, it first attracts, then repels the line defect when increasing jhdj.

B.J. Chen et al. / International Journal of Engineering Science 44 (2006) 422–435

435

7. Conclusions The interaction between a piezoelectric screw dislocation and collinear rigid lines of either conducting or dielectric has been investigated. The closed-form solution has been obtained by using the complex potential method. The field intensity factors and the forces on the dislocation are derived explicitly for a single line. The crack tip shielding or anti-shielding effects from the dislocation, the line force and the line charge are discussed in details. A numerical example is given to show the influence of the dislocation angular location on the image force. Comparing the rigid conducting line with the rigid dielectric one, it is found that both the field intensity factors and the image forces are much more different. References [1] Y.E. Pak, Crack extension force in a piezoelectric material, ASME Journal of Applied Mechanics 57 (1990) 647–653. [2] Y.E. Pak, Force on a piezoelectric screw dislocation, ASME Journal of Applied Mechanics 57 (1990) 863–869. [3] Y.E. Pak, Circular inclusion problem in antiplane piezoelectricity, International Journal of Solids and Structures 29 (1992) 2403– 2419. [4] S.A. Meguid, Z. Zhong, On the elliptical inhomogeneity problem in piezoelectric materials under antiplane shear and inplane electric field, International Journal of Engineering Science 36 (1998) 329–344. [5] S.A. Meguid, W. Deng, Electro-elastic interaction between a screw dislocation and an elliptical inhomogeneity in piezoelectric materials, International Journal of Solids and Structures 35 (1998) 1467–1482. [6] W. Deng, S.A. Meguid, Analysis of a screw dislocation inside an elliptical inhomogeneity in piezoelectric solids, International Journal of Solids and Structures 36 (1999) 1449–1469. [7] K.Y. Lee, W.G. Lee, Y.E. Pak, Interaction between a semi-infinite crack and a screw dislocation in a piezoelectric material, ASME Journal of Applied Mechanics 67 (2000) 165–170. [8] B.J. Chen, Z.M. Xiao, K.M. Liew, Electro-elastic stress analysis for a wedge-shaped crack interacting with a screw dislocation in piezoelectric solid, International Journal of Engineering Science 40 (2002) 621–635. [9] T.Y. Zhang, T.H. Wang, M.H. Zhao, Interaction of a piezoelectric screw dislocation with an insulating crack, Philosophical Magazine A: Physics of Condensed Matter, Structure, Defects and Mechanical Properties 82 (2002) 2805–2824. [10] Chunyu Li, G.J. Weng, Antiplane crack problem in functionally graded piezoelectric materials, ASME Journal of Applied Mechanics 69 (2002) 481–488. [11] Weichen Shi, Rigid line inclusions under anti-plane deformation and in-plane electric field in piezoelectric materials, Engineering Fracture Mechanics 56 (1997) 265–274. [12] B.J. Chen, Z.M. Xiao, K.M. Liew, On the interaction between a semi-infinite anti-crack and a screw dislocation in piezoelectric solid, International Journal of Solids and Structures 39 (2002) 1505–1513. [13] Y.W. Liu, Q.H. Fang, Electro-elastic interaction between a piezoelectric screw dislocation and circular interfacial rigid lines, International Journal of Solids and Structures 40 (2003) 5353–5370. [14] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Leyden, 1975.