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

Materials Science and Engineering A 430 (2006) 46–58

Electro-elastic interaction between a moving screw dislocation and collinear interfacial rigid lines Q.H. Fang, B. Li, Y.W. Liu ∗ Department of Engineering Mechanics, Hunan University, Changsha 410082, China Received 23 March 2006; accepted 23 May 2006

Abstract This paper attempts to investigate the problem for the electro-elastic interaction between a piezoelectric moving dislocation and interfacial collinear rigid lines under combined longitudinal shear and in-plane electric field. Using Riemann–Schwarz’s symmetry principle integrated with the analysis singularity of complex functions, we present the general elastic solution of this problem and the closed form solution for interface containing single rigid line. The expressions of electro-elastic fields and image force acting on moving dislocation are derived explicitly. The results show that the velocity of moving dislocation has significant effect on the image force. The present solutions contain previously known results as the special cases. © 2006 Elsevier B.V. All rights reserved. Keywords: Piezoelectric materials; Interfacial rigid lines; Moving screw dislocation

1. Introduction Due to this intrinsic coupling behavior, piezoelectric materials are used widely in modern technology, such as high power sonar transducers, electro-mechanical actuator, piezoelectric power supplies and micropositioner. These devices are designed to work under combined electro-mechanical loads. The presence of various defects, such as dislocations, cracks and inclusions, can greatly influence their characteristics and coupled behavior. Rigid lines and stationary dislocations problems in piezoelectric materials have been discussed by various investigators. To name a few, Shi [1] studied the problems of the rigid line inclusions under anti-plane deformation and in-plane electric field in piezoelectric materials by using the complex variable method. Wu and Du [2] investigated the elastic field and electric field of a rigid line in a confocal elliptic piezoelectric inhomogeneity embedded in an infinite piezoelectric medium. The electro-elastic stress investigation on the interaction between a semi-infinite anti-crack and a screw dislocation under anti-plane mechanical and in-plane electrical loading was carried out by Chen et al. [3]. The special mixed boundary value problem in which a debonded conducting rigid line inclusion is embedded at the interface of two piezoelectric half-planes was solved analytically by Wang and Shen [4]. Liu and Fang [5] dealt with the electro-elastic interaction between a piezoelectric screw dislocation located either outside or inside inhomogeneity and circular interfacial rigid lines under remote loads in linear piezoelectric materials. Based on Zener–Stroh crack initiation mechanism, Xiao et al. [6,7] proposed the solutions for micro-crack initiation at the tip of finite and semi-infinite rigid line inhomogeneity. Fang et al. [8] considered the interaction of a generalized screw dislocation with circular-arc interfacial rigid lines under remote anti-plane shear stresses, in-plane electric and magnetic loads in linear magneto–electro-elastic materials by employing the complex variable method. There are also some works reported in the literature which treat moving dislocation problems. Khannanov [9] obtained the integral representations for the electro-elastic fields of moving dislocations and disclinations in piezoelectric crystals in terms of a four-dimensional formalism of dynamic Green’s functions. A moving screw dislocation in a transversely isotropic piezoelectric material was dealt with by Wang and Zhong [10]. Woo [11] investigated the problem of dislocations moving at supersonic speeds via molecular dynamics simulation. Wu et al. [12,13] and Liu and Fang [14] ∗

Corresponding author. Tel.: +86 731 8821889; fax: +86 731 8822330. E-mail address: [email protected] (Y.W. Liu).

0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.05.114

Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58

47

had a hot discussion about the problem of a moving screw dislocation in piezoelectric bimaterials. Yang [15] and Liu et al. [16] considered the problem of a moving dislocation in a magneto–electro-elastic solid. All the above investigations did not considered the influence of imperfect interfaces on moving dislocation in piezoelectric bimaterials. The present paper provides an analysis to the electro-elastic interaction between a piezoelectric moving dislocation and interfacial rigid lines under combined longitudinal shear and in-plane electric field. Using Riemann–Schwarz’s symmetry principle integrated with the analysis singularity of complex functions, we present the general elastic solution of this problem and the closed form solution for interface containing single crack. The expressions of electro-elastic fields and image force acting on dislocation are derived explicitly. Results presented in this paper contain the previous known solutions as special cases. 2. Basic formulation and problem description Assuming that the transversely isotropic piezoelectric media which has been poled along the x3 direction with an isotropic x1 Ox2 plane, is subjected to remote longitudinal shear and in-plane electric field, then only coupled out-of-plane displacement and in-plane electric field need be considered so that there are only nontrivial displacement u3 , stresses σ 13 and σ 23 , electric potential φ, electrical field components E1 and E2 , electric displacement components D1 and D2 in the Cartesian coordinates. All components are only functions of variables x1 and x2 . The mechanical and electric coupled constitutive equations can be expressed as: σ13 = c44 u3,1 + e15 φ,1 ,

σ23 = c44 u3,2 + e15 φ,2

(1)

D1 = e15 u3,1 − ε11 φ,1 ,

D2 = e15 u3,2 − ε11 φ,2

(2)

c44 is the longitudinal shear modulus at a constant electric field, e15 the piezoelectric modulus and ε11 is the dielectric modulus. The electric field is given by E1 = −φ,1 ,

E2 = −φ,2

(3)

In the absence of body forces and body charges, the governing field equations are σ13,1 + σ23,2 = ρu¨ 3 ,

D1,1 + D2,2 = 0

(4)

where (¨) = ∂2 ( )/∂t2 , and ρ is the mass density of a piezoelectric material. Substitution Eqs. (1) and (2) into (4) will yield c44 ∇ 2 u3 + e15 ∇ 2 φ = ρu¨ 3 ,

e15 ∇ 2 u3 − ε11 ∇ 2 φ = 0

(5)

where ∇ 2 = ∂2 /∂x12 + ∂2 /∂x22 . When c44 ε11 + e215 = 0, (5) may be converted to the following equivalent system of partial differential equations ∇ 2 u3 =

ρε11 u¨ 3 , c44 + e215

∇ 2φ =

ρe15 u¨ 3 c44 + e215

Following Bleustein [17], a new function ϕ is introduced as e15 u3 ϕ=φ− ε11

(6)

(7)

Consequently, (6) can be rewritten into the canonical form ∇ 2 u3 = where c=




1 ∂ 2 u3 , c2 ∂t 2

∇ 2ϕ = 0

c44 ε11 + e215 ρε11

(9)

The stresses and electric displacements can be expressed in terms of independent variables u3 and ϕ as follows         u3,1 σ23 u3,2 σ13 =M , =M D1 ϕ,1 D2 ϕ,2 where

⎡

c44 ε11 + e215 ⎣ M= ε11 0

(8)

⎤ e15 ⎦ −ε11

(10)

(11)

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Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58

Fig. 1. Interaction of piezoelectric moving dislocation and interfacial rigid lines model.

The problem presented here is shown in Fig. 1. Let piezoelectric medium I with electro-elasticity property M1 occupy the upper half-plane, while piezoelectric medium II with electro-elasticity property M2 occupy the lower half-plane. Longitudinal shear stress ∞ , σ ∞ and σ ∞ as well as electric displacements D∞ , D∞ and D∞ are applied at infinite. It will be seen, σ ∞ = σ ∞ = σ ∞ σ13(1) 13(2) 23 1(1) 1(2) 2 23(1) 23(2) 23 ∞ = D∞ = D∞ , but in general σ ∞ = σ ∞ and D∞ = D∞ . A series of electrically conducting interfacial collinear and D2(1) 2(2) 2 13(1) 13(2) 1(1) 1(2) rigid lines lie along a part L of the interface between two materials, where L is a union of rigid lines segments Lj with the end points aj and bj (j = 1, 2, . . ., n). L is the remainder of the interface which two dissimilar piezoelectric materials are perfectly bonded. Without loss of generality, at time t = 0, assume the transient location of the screw dislocation z0 (z0 = x10 + ix20 , x20 > 0) T in the upper half-plane. The dislocation b = [ bz bϕ ] is characterized by a Burgers vector bz , an electric potential jump φ (bϕ = (ε11 φ − e15 bz )/ε11 ), and moves in arbitrary orientation with constant speed v (v < c), as shown in Fig. 1. The boundary conditions of displacements and electric potentials, stresses and electric displacements can be expressed as +   − +  −  u3(1) u3(2) σ23(1) σ23(2) (12) = , = on L ϕ(1) ϕ(2) D2(1) D2(2) 

u3(1)

+

 =

ϕ(1)

u3(2)

−

 =

ϕ(2)

δj



ϕj

on L

(13)

where the subscripts 1 and 2 denote the quantities defined in the upper and lower half-planes, with the superscripts + and − used to denoting the boundary values of the physical quantities as they approached the interface from the upper and lower half-planes, respectively. In addition to determine the unknown coefficients in the solution for such a problem, the equilibrium condition of each rigid line must be considered. Assuming that rigid lines are external traction and charge free, we have + −  bj   bj  σ23(1) σ23(2) dx1 − dx1 = 0, j = 1, 2, . . . , n (14) D2(1) D2(2) aj aj After taking the derivatives with respect to x1 , the addition and subtraction of (13) yield   +   − u3(1) u3(2) + = 0 on L   ϕ(1) ϕ(2) 

u3(1)  ϕ(1)

+

 −

u3(2)  ϕ(2)

(15)

− =0

on L

(16)

Since the velocity can be decomposed, thus, we consider the motion directions of dislocation parallel and perpendicular to the x1 -axis with velocities v1 (v1 = v cos θ) and v2 (v2 = v sin θ), respectively.

Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58

49

3. General solution of problem The above problem can be reduced to a Riemann–Hilbert problem, which can be solved particularly simply and effectively. We consider the case of the motion direction parallel to the x1 -axis with velocity v1 first. By a coordinate translation, we introduce a new coordinate system (x, y) [10] as x = x1 − v1 t,

y = x2

(17)

then Eq. (8) can be transformed to the following equations in the new coordinate system α2

∂ 2 u3 ∂ 2 u3 + = 0, 2 ∂x ∂y2

where




α=

1−

∂2 ϕ ∂2 ϕ + 2 =0 ∂x2 ∂y

v21 c2

(18)

(19)

The general solutions to Eq. (18) can be immediately arrive at u3 = ReU(z1 ),

ϕ = ReΦ(z) (20) √ where z1 = x + iαy, z = x + iy, i = −1, and Re denotes the real part. In the new coordinate system, the displacement and electric potential (derivative), the stress and electric displacement can be rewritten as       u3 (z1 ) σ13 σ23 ∗ ∗ = MReF(z ) = −MAImF(z∗ ) = ReF(z ) (21) ϕ (z) D1 D2 where

 ∗

F(z ) =

U  (z1 ) Φ (z)



 ,

A=

α 0

0 1

 (22)

Referring to the work by Pak [18], Jiang et al. [19] and Wang and Zhong [10], the generalized analytical function vectors F1 (z* ) and F2 (z* ) in the upper and lower half-planes, respectively, under considerations can be written as F1 (z∗ ) =

1 BG(z∗ ) +
1 + F10 (z∗ ), 2πi

F2 (z∗ ) =
2 + F20 (z∗ ), where B=



bz

0 bϕ

0

y<0

y>0

(23) (24)

 (25)

⎤ 1 ⎢ z1 − z10 ⎥ ⎥ G(z∗ ) = ⎢ ⎦ ⎣ 1 z − z0 ⎡

(26)

F10 (z* ) and F20 (z* ) are holomorphic in the upper and lower half-planes, respectively, and they vanish at infinity.
1 and
2 are identified with the stresses and electric displacements of the upper and lower half-planes at infinity, respectively.  ∞   ∞  σ13(k) σ23(k) −1 −1 −1 − iAk Mk , k = 1, 2 (27)
k = Mk ∞ ∞ D1(k) D2(k) Applying the Riemann–Schwarz’s symmetry principle, a new analytic function vector 1 (z* ) in the lower half-plane can be introduced 1 (z∗ ) = F¯ 1 (z∗ ) = F1 (z∗ ),

y<0

(28)

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Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58

Substituting (23) into (28), we get 1 (z∗ ) = −

1 ¯ ∗ ¯ 1 + 10 (z∗ ), BG(z ) +
2πi

y<0

(29)

where 10 (z* ) is holomorphic in the lower half-plane, and it vanishes at infinity. Similarly, we have ¯ 2 + 20 (z∗ ), 2 (z∗ ) = F¯ 2 (z∗ ) =


y>0

(30)

Replacing Fk (z∗ ) by k (z∗ ), (21) can be expressed further as    u3(k) 1 = [Fk (z∗ ) + k (z∗ )], k = 1, 2  2 ϕ(k)   σ13(k) 1 = Mk [Fk (z∗ ) + k (z∗ )], k = 1, 2 2 D1(k)   σ23(k) i = Mk Ak [Fk (z∗ ) − k (z∗ )], k = 1, 2 2 D2(k)

(31)

(32)

(33)

From (16) and the first equation of (12), it is seen that   +   − u3(2) u3(1) = on L∗ + L∗   ϕ(1) ϕ(2)

(34)

where L* and L∗ in the new coordinate system are relative to L and L in the old coordinate system, respectively. Noting that z1 = z on x-axis, the solution procedure of the function vector with argument z* can be translated into the corresponding function vector with argument z. Substituting (31) into (34) and rearranging, we obtain [F1 (x) − 2 (x)]+ = [F2 (x) − 1 (x)]− ,

x ∈ L∗ + L∗

(35)

which implies that F1 (z) − 2 (z) and F2 (z) − 1 (z) are a mutual direct analytic continuation through the real axis, and they approach the same limit at infinity, which requires ¯ 2 =
2 −
¯1
1 −


(36)

According to the generalized Liouville’s theorem, (35) leads to  F1 (z) − 2 (z), y > 0 ␰(z) = F2 (z) − 1 (z), y < 0

(37)

It can be seen that 1 ¯ ¯2 ␰(z) = B[G(z) + G(z)] +
1 −
2πi Substituting (33) into the second equation of (12), and rearranging, we obtain [M1 A1 F1 (x) + M2 A2 2 (x)]+ = [M2 A2 F2 (x) + M1 A1 1 (x)]− ,

x ∈ L∗

(38)

(39)

which implies that M1 A1 F1 (z) + M2 A2 2 (z) and M2 A2 F2 (z) + M1 A1 1 (z) are a mutual direct analytic continuation through L∗ and they approach the same limit at infinity, which requires ¯ 2 = M2 A2
2 + M1 A1
¯1 M 1 A 1
1 + M 2 A2
From (36) and (40), it is seen that  ∞   ∞  σ13(1) σ13(2) −1 −1 M1 = M2 , ∞ ∞ D1(1) D1(2) Letting ␩(z) =





∞ σ23(1) ∞ D2(1)

M1 A1 F1 (z) + M2 A2 2 (z),

y>0

M2 A2 F2 (z) + M1 A1 1 (z),

y<0

(40) 

 =

∞ σ23(2) ∞ D2(2)

 (41)

(42)

Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58

51

from the analysis of the singularity of ␩(z), it is seen that ␩(z) =

1 ¯ ¯ 2 + ␩0 (z) M1 A1 B[G(z) − G(z)] + M 1 A1
1 + M 2 A 2
2πi

(43)

where ␩0 (z) is holomorphic in the whole plane cut along L* and it vanishes at infinity. From (37) and (42), we have F1 (z) = (M1 A1 + M2 A2 )−1 ␩(z) + (M1 A1 + M2 A2 )−1 M2 A2 ␰(z),

y>0

(44)

2 (z) = (M1 A1 + M2 A2 )−1 ␩(z) − (M1 A1 + M2 A2 )−1 M1 A1 ␰(z),

y>0

(45)

F2 (z) = (M1 A1 + M2 A2 )−1 ␩(z) + (M1 A1 + M2 A2 )−1 M1 A1 ␰(z),

y<0

(46)

1 (z) = (M1 A1 + M2 A2 )−1 ␩(z) − (M1 A1 + M2 A2 )−1 M2 A2 ␰(z),

y<0

(47)

Substituting (44)–(47) into (15), we get ␩+ (x) + ␩− (x) = 0,

x ∈ L∗

According to [20], the general solution of (48) can be written as ⎤ ⎡ X0 (z) X0 (z)  − ⎢ X0 (z0 )(z − z0 ) X0 (¯z0 )(z − z¯ 0 ) ⎥ X0 (z) 1 ⎥+ M 1 A1 B ⎢ ␩(z) = ⎦ ⎣ X0 (z) X0 (z) 2πi 0 − X0 (z0 )(z − z0 ) X0 (¯z0 )(z − z¯ 0 ) where

 n

Pn (z) =

X0 (z) =

(u3 ) n−j j=0 Cj z n (ϕ) n−j j=0 Cj z

n

(48)

0 X0 (z)

 Pn (z)

(49)



(z − aj∗ )−1/2 (z − bj∗ )−1/2

(50)

(51)

j=1

with aj∗ and bj∗ related to aj and bj , respectively. Namely aj∗ = aj − v1 t, bj∗ = bj − v1 t. X0 (z) is a single-valued branch in the plane cut along with L∗ and for which lim zn X0 (z) = 1

|z|→∞

(52)

By a comparison of the expansions of (43) and (49) at infinity, we obtain [ Cn(u3 )

(ϕ) T ¯2 Cn ] = M1 A1
1 + M2 A2


(53)

There are remaining 2n unknown constants in Eq. (49) are determined from the equilibrium conditions, (14), of the rigid lines. Noting Eq. (33), we obtain 2n closed contour integrals  ␩(z) dz = 0, j = 1, 2, . . . , n (54) ∧j

where ∧j is a closed contour encircling each rigid line Lj∗ with (z0 , z¯ 0 ) outside the contour. The set of 2n linear algebraic equations given by (54) determines solely 2n remaining integration constants. Once ␩(z) is available, Fk (z) and k (z) (k = 1, 2) will be simply obtained. In the end, replacing z with z* , we get the function vectors Fk (z* ) and k (z* ) of the original problem. 4. Closed form solution for typical case As a typical case, the interface is considered to contain a single rigid line. Without loss in generality, assuming that an interfacial rigid line symmetrically placed with respect to the x2 -axis whose ends are located at a and −a. From (51), we have 1

X0 (z) = 

(z + v1 t)2 − a2

(55)

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Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58

From (53), and noting (27), we obtain [ C1(u3 )

T

(ϕ) ∞ ¯ 2 = M1 A1 M−1 [ σ13(1) C1 ] = M1 A1
1 + M2 A2
1

∞ ∞ D1(1) ] + M2 A2 M−1 2 [ σ13(2) T

∞ D1(2) ]

T

(56)

Substituting (55) and (56) into (54), we get [ C0(u3 )

T

(ϕ) C0 ] = [ 0

T

0]

Thus, from (44) and (46), and replacing z with z* , it is seen that ⎧ ⎤ ⎡ 1 1 ⎪ ⎪ + ⎨ ⎢ z1 − z10 1 z1 − z¯ 10 ⎥ ⎥ F1 (z∗ ) = (M1 A1 + M2 A2 )−1 M2 A2 B ⎢ ⎦ ⎣ 1 1 ⎪ 2πi ⎪ ⎩ + z − z0 z − z¯ 0   ⎤⎫ ⎡ 1 (z10 + v1 t)2 − a2 (¯z10 + v1 t)2 − a2 ⎪ ⎪  − ⎪ ⎥⎪ ⎢ 2 2 z1 − z10 z1 − z¯ 10 ⎥⎬ ⎢ (z + v t) − a 1 1 ⎥ ⎢   + M1 A1 B ⎢   ⎥⎪ 1 (z0 + v1 t)2 − a2 (¯z0 + v1 t)2 − a2 ⎦⎪ ⎣ ⎪ ⎪  − ⎭ 2 z − z0 z − z¯ 0 (z + v1 t) − a2 ⎧ ⎡ z1 ⎪  0 ⎪ ⎨ ⎢ (z1 + v1 t)2 − a2 −1 ⎢ ¯ + (M1 A1 + M2 A2 ) M2 A2 (
1 −
2 ) + ⎣ z ⎪ ⎪  0 ⎩ (z + v1 t)2 − a2

(57)

⎤

⎫ ⎪ ⎪ ⎬

⎥ ⎥ (M1 A1
1 + M2 A2
¯ 2) ⎦ ⎪ ⎪ ⎭ (58)

⎧⎡ ⎤ 1 1 ⎪ ⎪ + ⎨⎢ 1 z1 − z10 z1 − z¯ 10 ⎥ ⎥ F2 (z∗ ) = (M1 A1 + M2 A2 )−1 M1 A1 B ⎢ ⎣ ⎦ 1 1 ⎪ 2πi ⎪ ⎩ + z − z0 z − z¯ 0   ⎡ ⎤⎫ 1 (z10 + v1 t)2 − a2 (¯z10 + v1 t)2 − a2 ⎪ ⎪  − ⎪ ⎢ ⎥⎪ 2 2 z1 − z10 z1 − z¯ 10 ⎢ ⎥ (z1 + v1 t) − a ⎥⎬ ⎢     +⎢ ⎥⎪ 1 (z0 + v1 t)2 − a2 (¯z0 + v1 t)2 − a2 ⎣ ⎦⎪ ⎪ ⎪  − ⎭ 2 2 z − z0 z − z¯ 0 (z + v1 t) − a ⎧ ⎡ z1 ⎪  0 ⎪ ⎨ 2 2 ⎢ (z + v t) − a 1 1 −1 ¯ 2) + ⎢ + (M1 A1 + M2 A2 ) M1 A1 (
1 −
z ⎣ ⎪ ⎪  0 ⎩ (z + v1 t)2 − a2

⎤

⎫ ⎪ ⎪ ⎬

⎥ ⎥ (M1 A1
1 + M2 A2
¯ 2) ⎦ ⎪ ⎪ ⎭ (59)

4.1. Electro-elastic fields The electro-elastic fields excited by the moving dislocation can be obtained by substituting (58) and (59) into (32) and (33) as 

σ13(1) D1(1)



⎤ ⎧ ⎡  1 1 ⎪ ⎪ + Im ⎨ ⎢ 1 z − z10 z1 − z¯ 10 ⎥ 1  ⎥ = M1 (M1 A1 + M2 A2 )−1 M2 A2 B ⎢ ⎣ ⎦ 1 1 ⎪ 2π ⎪ ⎩ Im + z − z0 z − z¯ 0

Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58

  ⎤⎫  ⎡   1 (z10 + v1 t)2 − a2 (¯z10 + v1 t)2 − a2 ⎪ ⎪  − ⎪ ⎢ Im ⎥⎪ 2 2 z1 − z10 z1 − z¯ 10 ⎢ ⎥ (z1 + v1 t) − a ⎥⎬ ⎢      +M1 A1 B ⎢ ⎥⎪ 1 (z0 + v1 t)2 − a2 (¯z0 + v1 t)2 − a2 ⎣ ⎦⎪ ⎪ ⎪  Im − ⎭ 2 2 z − z0 z − z¯ 0 (z + v1 t) − a ⎧ ⎡ z1 ⎪  0 ⎪ ⎨ 2 2 ⎢ (z + v t) − a 1 1 −1 ¯ 2 ) + Re ⎢ + M1 (M1 A1 + M2 A2 ) M2 A2 Re(
1 −
z ⎣ ⎪ ⎪  0 ⎩ (z + v1 t)2 − a2 ⎫ ⎪ ⎪ ⎬ ¯ 2) × (M1 A1
1 + M2 A2
⎪ ⎪ ⎭ 

σ23(1)



D2(1)

53

⎤ ⎥ ⎥ ⎦

(60)

⎤ ⎧ ⎡  1 1 ⎪ ⎪ ⎨ ⎢ Re z1 − z10 + z1 − z¯ 10 ⎥ 1   ⎥ = M1 A1 (M1 A1 + M2 A2 )−1 M2 A2 B ⎢ ⎣ ⎦ 1 1 ⎪ 2π ⎪ ⎩ Re + z − z0 z − z¯ 0   ⎤⎫  ⎡   1 (z10 + v1 t)2 − a2 (¯z10 + v1 t)2 − a2 ⎪ ⎪ ⎪  − ⎪ ⎥ ⎢ Re 2 2 ⎥⎬ z1 − z10 z1 − z¯ 10 ⎢ (z + v t) − a 1 1 ⎥ ⎢    ⎥   + M1 A1 B ⎢ 1 (z0 + v1 t)2 − a2 (¯z0 + v1 t)2 − a2 ⎪ ⎦⎪ ⎣ ⎪ ⎪  Re − ⎭ 2 2 z − z0 z − z¯ 0 (z + v1 t) − a ⎧ ⎡ z1 ⎪  0 ⎪ ⎨ ⎢ (z1 + v1 t)2 − a2 −1 ⎢ ¯ − M1 A1 (M1 A1 + M2 A2 ) M2 A2 Im(
1 −
2 ) + Im ⎣ z ⎪ ⎪  0 ⎩ (z + v1 t)2 − a2 ⎫ ⎪ ⎪ ⎬ ¯ × (M1 A1
1 + M2 A2
2 ) ⎪ ⎪ ⎭

⎤ ⎥ ⎥ ⎦

(61)

in the upper half-plane, and as 

σ13(2) D1(2)



⎤ ⎧⎡  1 1 ⎪ ⎪ Im + ⎨⎢ 1 z − z10 z1 − z¯ 10 ⎥ 1  ⎥ = M2 (M1 A1 + M2 A2 )−1 M1 A1 B ⎢ ⎣ ⎦ 1 1 ⎪ 2π ⎪ ⎩ Im + z − z0 z − z¯ 0   ⎤⎫  ⎡   1 (z10 + v1 t)2 − a2 (¯z10 + v1 t)2 − a2 ⎪ ⎪  − ⎪ ⎢ Im ⎥⎪ 2 2 z − z z − z ¯ ⎢ ⎥⎬ 1 10 1 10 (z + v t) − a 1 1 ⎢ ⎥     ⎥  +⎢ 1 (z0 + v1 t)2 − a2 (¯z0 + v1 t)2 − a2 ⎪ ⎣ ⎦⎪ ⎪ ⎪  Im − ⎭ 2 2 z − z0 z − z¯ 0 (z + v1 t) − a ⎧ ⎡ z1 ⎪  0 ⎪ ⎨ 2 2 ⎢ (z + v t) − a 1 1 −1 ¯ 2 ) + Re ⎢ + M2 (M1 A1 + M2 A2 ) M1 A1 Re(
1 −
z ⎣ ⎪ ⎪  0 ⎩ (z + v1 t)2 − a2 ⎫ ⎪ ⎪ ⎬ ¯ 2) × (M1 A1
1 − M2 A2
⎪ ⎪ ⎭

⎤ ⎥ ⎥ ⎦

(62)

54

Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58



σ23(2)



D2(2)

⎤ ⎧⎡  1 1 ⎪ ⎪ + ⎨⎢ Re 1 z1 − z10 z1 − z¯ 10 ⎥   ⎥ = M2 A2 (M1 A1 + M2 A2 )−1 M1 A1 B ⎢ ⎣ ⎦ 1 1 ⎪ 2π ⎪ ⎩ Re + z − z0 z − z¯ 0  ⎤⎫   ⎡   1 (z10 + v1 t)2 − a2 (¯z10 + v1 t)2 − a2 ⎪ ⎪  − ⎪ ⎢ Re ⎥⎪ 2 2 z − z z − z ¯ ⎢ ⎥⎬ 1 10 1 10 (z + v t) − a 1 1 ⎢ ⎥     ⎥  +⎢ 1 (z0 + v1 t)2 − a2 (¯z0 + v1 t)2 − a2 ⎪ ⎣ ⎦⎪ ⎪ ⎪  Re − ⎭ 2 z − z0 z − z¯ 0 (z + v1 t) − a2 ⎧ ⎡ z1 ⎪  0 ⎪ ⎨ 2 2 ⎢ ¯ 2 ) + Im ⎢ (z1 + v1 t) − a − M2 A2 (M1 A1 + M2 A2 )−1 M1 A1 Im(
1 −
z ⎣ ⎪ ⎪  0 ⎩ (z + v1 t)2 − a2 ⎫ ⎪ ⎪ ⎬ ¯ (M1 A1
1 + M2 A2
2 ) ⎪ ⎪ ⎭

⎤ ⎥ ⎥ ⎦

(63)

in the lower half-plane. Assuming a = 0 and
k = 0 (k = 1, 2), namely the interfacial rigid line and loads at infinite vanish, we get the electro-elastic fields of a moving screw dislocation in two piezoelectric bimaterials as ⎤ ⎡ α1 (y − y0 )   ⎢ (x − x0 − v1 t)2 + α2 (y − y0 )2 ⎥ σ13(1) 1 1 ⎥ = − M1 B ⎢ ⎦ ⎣ y − y0 2π D1(1) 2 2 (x − x0 − v1 t) + (y − y0 ) ⎤ ⎡ α1 (y + y0 ) ⎢ (x − x0 − v1 t)2 + α2 (y + y0 )2 ⎥ 1 1 ⎥ + (64) M1 (M1 A1 + M2 A2 )−1 (M1 A1 − M2 A2 )B ⎢ ⎦ ⎣ y + y0 2π (x − x0 − v1 t)2 + (y + y0 )2 

σ23(1)



D2(1)

⎡

x − x0 − v 1 t ⎢ (x − x0 − v1 t)2 + α21 (y − y0 )2 1 = M1 A 1 B ⎢ ⎣ x − x 0 − v1 t 2π (x − x0 − v1 t)2 + (y − y0 )2

⎤ ⎥ ⎥ ⎦ ⎡

x − x0 − v 1 t ⎢ 1 (x − x − v1 t)2 + α21 (y + y0 )2 0 − M1 A1 (M1 A1 + M2 A2 )−1 (M1 A1 − M2 A2 )B ⎢ ⎣ x − x 0 − v1 t 2π (x − x0 − v1 t)2 + (y + y0 )2

⎤ ⎥ ⎥ ⎦

(65)

in the upper half-plane, and as ⎤ α2 y − α1 y0 ⎢ (x − x0 − v1 t)2 + (α2 y − α1 y0 )2 ⎥ σ13(2) 1 ⎥ = − M2 (M1 A1 + M2 A2 )−1 M1 A1 B ⎢ ⎦ ⎣ y − y0 π D1(2) 2 2 (x − x0 − v1 t) + (y − y0 ) ⎤ ⎡ x − x0 − v 1 t   ⎢ (x − x0 − v1 t)2 + (α2 y − α1 y0 )2 ⎥ σ23(2) 1 ⎥ = M2 A2 (M1 A1 + M2 A2 )−1 M1 A1 B ⎢ ⎦ ⎣ x − x 0 − v1 t π D2(2) 2 2 (x − x0 − v1 t) + (y − y0 ) 



⎡

in the lower half-plane. It can be seen that (64)–(67) are identical to the results in Wu and Dzenis [13] and Liu and Fang [14].

(66)

(67)

Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58

55

4.2. Image force The image force on the screw dislocation due to the presence of the interfacial rigid line can be calculated according to the generalized Peach–Koehler formula [18] as ⎤ ⎧ ⎡  1 ⎪ ⎪ Re ⎨ ⎢ 1 T z − z¯ 10 ⎥  10  ⎥ b M1 A1 (M1 A1 + M2 A2 )−1 M2 A2 B ⎢ Fx = ⎣ ⎦ 1 ⎪ 2π ⎪ ⎩ Re z0 − z¯ 0  ⎤⎫ ⎡  (¯z10 + v1 t)2 − a2 1 z10 + v1 t ⎪ ⎪ ⎪  + ⎪ ⎥ ⎢ Re 2 2 ⎬ 2 2 z10 − z¯ 10 (z + v t) − a ⎥ ⎢ (z + v t) − a 10 1 10 1 ⎥    − M1 A1 B ⎢ ⎥⎪ ⎢ (¯z0 + v1 t)2 − a2 1 z0 + v 1 t ⎦⎪ ⎣ ⎪ ⎪  Re + ⎭ 2 2 2 2 z0 − z¯ 0 (z + v t) − a (z0 + v1 t) − a 0 1 ⎧ ⎤ ⎡ z10 ⎪  0 ⎪ ⎨ 2 2 ⎥ ⎢ ⎥ ¯ 2 ) + Im ⎢ (z10 + v1 t) − a − bT M1 A1 (M1 A1 + M2 A2 )−1 M2 A2 Im(
1 −
z0 ⎦ ⎣ ⎪ ⎪  0 ⎩ 2 2 (z0 + v1 t) − a ⎫ ⎪ ⎪ ⎬ ¯ × (M1 A1
1 + M2 A2
2 ) (68) ⎪ ⎪ ⎭ ⎤ ⎧ ⎡  1 ⎪ ⎪ Im ⎨ ⎢ 1 z10 − z¯ 10 ⎥   ⎥ Fy = − bT M1 (M1 A1 + M2 A2 )−1 M2 A2 B ⎢ ⎣ ⎦ 1 ⎪ 2π ⎪ ⎩ Im z0 − z¯ 0  ⎤⎫ ⎡  1 z10 + v1 t (¯z10 + v1 t)2 − a2 ⎪ ⎪ ⎪  + ⎪ ⎢ Im 2 2 ⎥ ⎬ 2 2 z10 − z¯ 10 (z + v t) − a ⎢ ⎥ (z + v t) − a 10 1 10 1 ⎢ ⎥    − M1 A1 B ⎢ ⎥ (¯z0 + v1 t)2 − a2 1 z0 + v 1 t ⎪ ⎣ ⎦⎪ ⎪ ⎪  Im + ⎭ 2 2 2 2 z0 − z¯ 0 (z0 + v1 t) − a (z0 + v1 t) − a ⎧ ⎡ z10 ⎪  0 ⎪ ⎨ 2 2 ⎢ (z + v t) − a 10 1 T −1 ¯ 2 ) + Re ⎢ − b M1 (M1 A1 + M2 A2 ) M2 A2 Re(
1 −
z0 ⎣ ⎪ ⎪  0 ⎩ (z0 + v1 t)2 − a2 ⎫ ⎪ ⎪ ⎬ ¯ 2) × (M1 A1
1 + M2 A2
⎪ ⎪ ⎭

⎤ ⎥ ⎥ ⎦

(69)

By now, we consider nothing more than the case of the motion direction of dislocation parallel to the x1 -axis. 5. Dislocation motion perpendicular to x1 -axis For the case of the motion direction perpendicular to the x1 -axis, we introduce a coordinate translation as x = x1

y = x2 − v2 t

(70)

Eq. (8) can be transformed to the following form 1 ∂ 2 u3 ∂ 2 u3 + = 0, β2 ∂x2 ∂y2

∂2 ϕ ∂2 ϕ + 2 =0 ∂x2 ∂y

(71)

56

Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58

Fig. 2. The polar coordinate system.

where




β=

1 − v22 c2

(72)

Comparing Eqs. (18) and (71), we find that the governing field equations arising from the screw dislocation with velocity along the x2 direction can be obtained from that with velocity along the x1 direction if α in (18) is replaced with 1/β. Therefore, the electro-elastic fields and image force on moving screw dislocation along the x2 direction are obtained from the results of the moving screw dislocation along the x1 direction if α, z1 and z10 are replaced by 1/β, z2 and z20 , respectively, with z2 and z20 expressed as z2 = x +

iy β

z20 = x0 +

iy0 β

(73)

The solutions for uniformly moving dislocation of arbitrary orientation in piezoelectric media (in the upper-plane) can be derived from superposing the solutions of above two cases. 6. Numerical examples This section provides numerical examples that indicate how the velocity of dislocation affects the forces on the moving dislocation. Assuming
k = 0 (k = 1, 2), namely the loads at infinite are equal to zero. The materials I and II are assumed to be PZT-5H [21] and zinc oxide (ZnO) [22], respectively. The material properties are given by c44 = 2.3 × 1010 N/m2 ,

e15 = 17.0 C/m2 ,

ε11 = 150.4 × 10−10 C/V m,

ρ = 7.5 × 103 kg/m3 ,

(74)

for PZT-5H, c44 = 4.247 × 1010 N/m2 ,

e15 = −0.59 C/m2 ,

ε11 = 0.738 × 1010 C/V m,

ρ = 5.676 × 103 kg/m3 ,

(75)

for ZnO. For the units of the material constants, N is the force in Newtons, C the charge in coulombs, V the electric potential in volts and m is the length in meters. Other parameters are taken to have the values below bz = 1.0 × 10−9 m,

φ = 1.0 V.

(76)

Fig. 3. Fθ /F0 vs. θ for θ d = 0.5 rad.

Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58

57

Fig. 4. Fr /F0 vs. v/c for θ d = 0.5 rad.

The forces on the dislocation are normalized by the parameter (1)

F0 =

c44 bz2 . 2πa

(77)

For convenience of analyzing the image forces, a polar coordinate is introduced as shown in Fig. 2. It is assumed that rd = 0.4a, vt = 0.5a and a = 0.01 m. The radial and tangential parts of the image forces can be calculated as follows: Fr = Fx cos θd + Fy sin θd ,

Fθ = −Fx sin θd + Fy cos θd

(78)

The normalized force Fr /F0 on dislocation versus velocity direction θ is depicted in Fig. 3 with different v/c for θ d = 0.5 rad. It is seen that the radial component of the image force possesses identical trend for different velocity. With the increase of θ, the image force decreases for the case of θ < π/2 and increases for the case of θ > π/2. The influence of the velocity direction on radial image force is stronger when velocity is larger. The variation of the normalized force Fr /F0 on dislocation with v/c is plotted in Fig. 4 with different θ for θ d = 0.5 rad. Noticeable, the normalized image force Fr /F0 increases with the increasing magnitude of velocity as the dislocation moves along the x1 direction (θ = 0) and decreases with the increasing magnitude of velocity as the dislocation moves along the x2 direction (θ = π/2). In Fig. 5, we illustrate variation of normalized tangential image force Fθ /F0 versus θ with different velocity v/c for θ d = 0.5 rad. It is seen that the magnitude of velocity can alter the trend of tangential image force which is different from the radial image force. The variation of the normalized force Fθ /F0 with v/c is plotted in Fig. 6 with different θ for θ d = 0.5 rad. It is seen that the trend of tangential image force is similar to the radial image force.

Fig. 5. Fθ /F0 vs. θ for θ d = 0.5 rad.

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Q.H. Fang et al. / Materials Science and Engineering A 430 (2006) 46–58

Fig. 6. Fθ /F0 vs. v/c for θ d = 0.5 rad.

7. Conclusions Using Muskhelishvili’s complex variable method, the closed form complex potentials are obtained for a piezoelectric moving screw dislocation interacting with interfacial collinear rigid lines in this paper. Analytical expressions of stresses and electric displacements fields and the image force acting on the moving dislocation are given explicitly. The obtained explicit solutions can be used as Green’s functions to solve the problem of interaction between interfacial collinear rigid lines and moving crack in media under anti-plane mechanical and in-plane electric loads at infinite. Acknowledgement The authors would like to deeply appreciate the support by the National Natural Science Foundation of China (10472030). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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