phys stat sol (b) 38 No 1 10 16 (003) / DOI 10100/pssb00301805 Moving screw dislocations in piezoelectric bimaterials Xiang-Fa Wu *1 Yuris A Dzenis 1 and Wen-Sheng Zou 1 Department of Engineering Mechanics Center for Materials Research and Analysis University of Nebraska Lincoln NE 68588-056 USA Department of Vehicle Engineering Beijing Institute of Technology Beijing 100081 China Received 18 November 00 revised 4 March 003 accepted April 003 Published online 10 June 003 PACS 465Hf 617Lk 7765Ly An explicit solution for a moving screw dislocation in a piezoelectric bimaterial is given by means of complex variables The electroelastic fields and forces acting on the moving dislocation are obtained in closed form As special cases solutions for a moving screw dislocation in a piezoelectric half-plane with a free or rigid surface are obtained explicitly 1 Introduction Due to their intrinsic electromechanical coupling phenomena piezoelectric materials are finding extensive electromechanical applications in actuators sensors and transducers etc Piezoelectric materials such as ferroelectric ceramics are brittle and liable to cracking on all scales from domain to device Defects such as dislocations micro-cracks cavities and inhomogeneous inclusions etc may be embedded in piezoelectric materials during manufacturing process and service Under external electromechanical loadings the proper functions and service lifetime of piezoelectric devices may be dramatically reduced due to these defects interactions and resulting macro-cracking Theoretically dislocation models play an important role in the study of defects and macro-cracks in solid materials [1 4] In the past three decades a number of investigations have been conducted concerning dislocations in piezoelectric materials To mention a few Barnett and Lothe [5] first discussed the dislocation-induced electroelastic fields in anisotropic piezoelectric materials by means of the Stroh formalism Pak [6] determined the forces acting on a screw dislocation in transversely isotropic piezoelectric materials using the generalized Peach Koehler formula These results indicate the effect of the material electromechanical coupling property on the electroelastic fields Recently further investigations have been performed on a screw dislocation interacting with an elliptical inhomogeneity [7] a free boundary [8] in homogenous piezoelectric materials and an interface in piezoelectric bimaterials [9 10] In the case of a screw dislocation interacting with cracks various crack configurations in homogenous piezoelectric materials have been considered such as a semi-infinite crack [ 1] Griffith crack [13] and so on Furthermore detailed investigations have been achieved on the electroelastic fields and the fracture parameters of a screw dislocation interacting with a semi-infinite crack [8 14] Griffith crack [8] edge crack [8 15] and wedge crack [16 17] between two bonded dissimilar piezoelectric materials These results still keep the fundamental characteristics of dislocation crack interactions and also indicate the electromechanical coupling and geometric boundary effects on the electroelastic fields and the fracture parameters In the limiting cases the above interfacial solutions cover those for homogenous materials * Corresponding author: e-mail: xfwu@unlserveunedu wuxiangfa@yahoocom Fax: +1-40-47-89 003 WILEY-VCH Verlag GmbH & Co KGaA Weinheim 0370-197/03/3807-010 $ 1750+50/0
phys stat sol (b) 38 No 1 (003) Although stationary dislocations in piezoelectric materials have been discussed thoroughly moving dislocation problems have not yet attracted much attention The simplest case of a moving screw dislocation in homogenous piezoelectric materials was considered recently [18] In this paper we study the fundamental solution of a moving screw dislocation in a piezoelectric bimaterial by means of complex variables The electroelastic fields and forces acting on the moving dislocation are obtained explicitly As special cases solutions for a moving screw dislocation in a piezoelectric half-plane with a free or rigid surface are derived explicitly Basic equations Consider two dissimilar piezoelectric half-planes bonded along the x 1 -axis as shown in Fig 1 The piezoelectric materials are assumed to be transversely isotropic with the isotropic basal planes parallel to the (x 1 x )-plane and the poling directions along the x 3 -axis The dynamic boundary value problem of the piezoelectric materials is constructed assuming only nontrivial out-of-plane mechanical displacement and in-plane electric field such that u = u = 0 u = u ( x x t) (1) 1 3 3 1 E = E ( x x t) E = E ( x x t) E = 0 () 1 1 1 1 3 In this case the constitutive relations become σ = c u + e φ D = e u ε φ σ = c u + e φ D = e u ε φ (3) 13 44 31 15 1 1 15 31 1 3 44 3 15 15 3 σ k3 is the stress tensor D k (k = 1 ) the electric displacement vector c 44 the elastic modulus measured at constant electric field e 15 and ε the piezoelectric and dielectric constants and φ the electric potential The electric field is given by E = φ E = φ (4) 1 1 The governing equations are σ + σ = ρu D + D = 0 (5) 131 3 3 () = t and ρ is the mass density of the related material Substituting (3) into (5) yields c u + e φ = ρu e u ε φ = 0 (6) 44 3 15 3 15 3 = / x1 + / x is the Laplace operator When c ε + e the governing equations in (6) can be rewritten as 44 15 0 ρε ρe u = u φ = u (7) 15 3 3 3 c44ε + e15 c44ε + e15 x (y) Material 1 V x 1 (x) Material Fig 1 Moving screw dislocation in a piezoelectric bimaterial
1 Xiang-Fa Wu et al: Moving screw dislocations in piezoelectric bimaterials Following Bleustein [19] we introduce an auxiliary function ϕ such that e ϕ φ ε Substitution of (8) into (7) leads to two decoupled equations as = 15 u 3 (8) 1 u3 = u 3 ϕ = 0 (9) c c44ε + e15 c = (10) ρε The stresses and electric displacements can be expressed in terms of two independent functions u 3 and ϕ as 13 u31 u 3 3 = C = C D1 ϕ1 D ϕ C c ε + e 44 15 e15 = ε 0 ε The analysis presented above shows that the electroelastic solutions for piezoelectric materials subjected to dynamic antiplane mechanical and in-plane electrical loadings can be represented by two independent functions u 3 and ϕ satisfying the boundary conditions 3 Moving screw dislocation in two bonded dissimilar piezoelectric materials Consider an infinitely long screw dislocation moving constantly in a piezoelectric bimaterial Without loss of generality at time t = 0 assume the transient location of the screw dislocation z 0 (z 0 = x 10 + ix 0 x 0 < 0) in the lower half-plane The dislocation is characterized by a Burgers vector b an electric potential jump φ and moving parallel to the x 1 -axis with constant speed V (V < c) as shown in Fig 1 By means of the superposition scheme [1 3] the overall solution of this problem can be constructed from the solution for a moving screw dislocation in an infinite homogenous piezoelectric medium and an auxiliary potential (interface image) satisfying the interface continuity conditions The electroelastic solutions u 3 (z) and ϕ(z) for a moving screw dislocation in an infinite homogenous piezoelectric medium have been obtained by Wang and Zhong [18] By a coordinate translation we introduce a new coordinate system (x y) such that x = x Vt y = x (13) 1 In this new coordinate system Eq (9) can be rewritten as α u3 xx u3 yy ϕ xx ϕ yy + = 0 + = 0 (14) V α = 1 ( V < c) (15) c () (1)
phys stat sol (b) 38 No 1 (003) 13 The general solutions to Eqs (14) can be expressed in terms of two modified holomorphic functions such that u = Im [ U( z )] ϕ = Im [ Φ( z)] (16) 3 1 z 1 = x + iαy z = x + iy and Im[ ] stands for the imaginary part of an analytic function In the new coordinate system by using the Cauchy-Euler conditions for analytic functions the stress and electric displacement relations in () can be rewritten as 13 u31 U ( z1) 3 u3 U ( z1) = C = CIm C LRe D1 ϕ = = 1 Φ () z D ϕ Φ () z C is defined in (1) L is defined as α( c44ε + e15) α 0 e15 L = C ε 0 1 = 0 ε and Re[ ] is the real part of an analytic function The screw dislocation is defined as d u3 = b d φ = φ (19) 0 0 or in the form of u 3 and ϕ as e u b ϕ φ b ε (0) 15 d 3 = d = 0 0 Γ is a closed contour surrounding the dislocation core z = z 0 in the physical plane The electroelastic solutions U(z 1 ) and Φ(z) for a moving screw dislocation with constant speed V parallel to the x 1 -axis in an infinite homogenous piezoelectric medium can be expressed as [18] b ε φ eb Uz ( 1) = ln( z1 z10) Φ( z) = ln( z z0) (1) π πε z 10 = x 0 + iαy 0 and z 0 = x 0 + iy 0 Now let us construct the solution for the current bimaterial problem In the following procedure indices 1 and denote material constants displacements electric potentials stresses and electric displacements of the upper and lower half-planes respectively The x 1 (x)-axis is directed along the interface as shown in Fig 1 The electroelastic solutions U(z 1 ) and ϕ(z) can be expressed as (17) (18) and U1( z1) z D1 (ie y > 0) Uz () = U( z1) + U0( z1) z D (ie y< 0) Φ1() z z D1 (ie y > 0) Φ() z = Φ() z + Φ0() z z D (ie y< 0) () (3) Here U 1 (z 1 ) U (z 1 ) Φ 1 (z) and Φ (z) are unknown analytic functions to be determined and U 0 (z 1 ) and Φ 0 (z) are solutions for a moving screw dislocation in an infinite homogenous piezoelectric material as expressed in (1) with material constants related to the lower half-plane
14 Xiang-Fa Wu et al: Moving screw dislocations in piezoelectric bimaterials and and In order to simplify the derivation we introduce the five vectors U ( z ) U ( z ) U ( z ) f z f z f z 1 1 1 0 1 1() = () 0() Φ1() z = Φ() z = Φ0() z u3 3 u = t = φ D With the aid of (8) () and (16) (5) can be rewritten as T { Φ } { } u= B Im[ U ] = ib/[ f() z f ()] z (6) t = L/ [ f () z + f ()] z (7) 1 0 B = e15 / ε 1 (8) and ()denotes the conjugate of an analytic function Now let us determine the unknown U 1 (z 1 ) U (z 1 ) Φ 1 (z) and Φ (z) by enforcing continuity of the outof-plane displacement electric potential mechanical force and electric displacement across the interface (x = 0) It should be noted that U 1 (z 1 ) and Φ 1 (z) are analytic in the upper half-plane while U (z 1 ) and Φ (z) are analytic in the lower half-plane By using the relations (4) (7) the continuity of the out-ofplane displacement and electric potential across the interface requires B[ f ( x) f ( x)] = B [ f ( x) f ( x) + f ( x) f ( x)] (9) 1 1 1 0 0 Rearranging (9) we obtain B f ( x) + B f ( x) B f ( x) = B f ( x) + B f ( x) B f ( x) (30) 1 1 0 1 1 0 Equation (30) holds along the whole x-axis Moreover the functions on the left-hand side are analytic in the upper half-plane while those on the right-hand side are analytic in the lower half-plane With the standard analytic continuity arguments we have B f () z + B f () z B f () z = 0 z D (31) 1 1 0 1 With the same arguments the stress and electric force continuity across the interface leads to L f () z L f () z L f () z = 0 z D (3) 1 1 0 1 Relations (31) and (3) yield (4) (5) f () z = ( B B + L L ) f () z z D 1 1 1 1 1 1 0 1 f () z = ( I + L L B B ) ( I L L B B ) f () z z D 1 1 1 1 1 1 1 1 1 1 0 (33) I is a second-order identity matrix and the subscripts 1 and behind B and L stand for the material matrices corresponding to the upper and lower half-planes respectively In the above by letting L 1 = L and B 1 = B results in (33) cover the known solutions in (1) Furthermore the moving screw dislocation in a half-plane (y < 0) with a traction-free and electrically im-
15 Xiang-Fa Wu et al: Moving screw dislocations in piezoelectric bimaterials permeable surface y = 0 can be described by f() z = f () z f ( z) z D (34) 0 0 Another interesting case is the moving screw dislocation in the half-plane (y < 0) with a rigid and electrically impermeable surface y = 0 The corresponding solutions are Uz () = U() z+ U() z Φ() z = Φ () z Φ () z z D (35) 0 0 0 0 4 Electroelastic fields The electroelastic fields excited by the moving screw dislocation can be obtained by substituting () (3) (4) and (33) into (17) as b 1 V / c ( y y0 ) 13 1 1 1 ( x x0 Vt) (1 V / c )( y y0) C1( B B1 L L1) π + = + D1 ε φ eb ( y y0 ) πε ( x x0 Vt) + ( y y0) = L ( B B + L L ) D 3 1 1 1 1 1 1 b x x0 Vt π ( x x0 Vt) + (1 V / c )( y y0) ε φ eb x x0 Vt πε ( x x0 Vt) + ( y y0) in the upper half-plane and 13 1 1 1 1 1 = C[( I + L L1B1 B) ( I L L1B1 B) I] D1 b 1 V / c ( y y0 ) π ( x x0 Vt) + (1 V / c )( y y0) ε φ eb ( y y0 ) πε ( x x0 Vt) + ( y y0) (36) (37) (38) 3 1 1 1 1 1 = L[( I + L L1B1 B) ( I L L1B1 B) + I] D b x x0 Vt π ( x x0 Vt) + (1 V / c )( y y0) ε φ eb x x0 Vt πε ( x x0 Vt) + ( y y0) in the lower half-plane B j C j and L j (j = 1 ) are the material matrices defined in (1) (18) and (8) By letting B 1 = B B 1 = B and L 1 = L the above solutions cover those in the literature [18] From the electroelastic solutions (36) (39) it is found that the electroelastic solutions are dependent 1 1 on the ratios B1 B and L1 L When the electromechanical properties of two component materials are highly mismatched the electroelastic fields excited by the moving screw dislocation are modified significantly in which two limiting cases are the moving screw dislocation in a half-plane with rigid or free surface as discussed in Section 3 (39)
16 Xiang-Fa Wu et al: Moving screw dislocations in piezoelectric bimaterials The above solutions also cover the static ones by letting the dislocation speed V 0 Furthermore if letting e 15 = 0 and φ = 0 the aforementioned solutions will reduce to those for a moving screw dislocation in two bonded dissimilar elastic media The image forces acting on the screw dislocation can be obtained straightforwardly using the generalized Peach Koehler formula [6] and the whole electroelastic field solutions (36) (39) 5 Conclusion The electroelastic solutions for a moving screw dislocation in a piezoelectric bimaterial are obtained in this paper The obtained solutions indicate the effects of material mismatch and coupling electromechanical property on the electroelastic fields These fundamental solutions can be further used as the Green s functions to solve the dynamic electroelastic problems in piezoelectric materials with defects The given solutions can be also used to determine the dynamic electroelastic fields of defects in a piezoelectric/non-piezoelectric material/strip bonded to another piezoelectric/non-piezoelectric material/strip which is very common in piezoelectric applications such as acoustic sensors Acknowledgements The authors would like to thank the anonymous reviewers for their careful reading and helpful comments for the improvement of this paper X-F Wu would like to express his grateful acknowledgement to the Milton E Mohr Research Fellowship of the Engineering College at the University of Nebraska-Lincoln NE 001 00 References [1] R Thomsom Solid State Phys 39 1 (1986) [] Z Suo Int J Solids Struct 5 33 (1989) [3] Z Suo Proc R Soc Lond A 47 331 (1990) [4] J Weertman Dislocation Based Fracture Mechanics (World Scientific Singapore 1996) [5] D M Barnett and J Lothe phys stat sol (b) 67 105 (1975) [6] Y E Pak J Appl Mech (ASME) 57 863 (1990) [7] S A Meguid and W Deng Int J Solids Struct 35 1467 (1998) [8] X F Li and X Y Duan phys stat sol (b) 7 613 (001) [9] J X Liu S Y Du and B Wang Mech Res Commun 6 415 (1999) [10] X F Wu S Cohn and Y A Dzenis Int J Eng Sci 41 667 (003) [] K Y Lee W G Lee and Y E Pak J Appl Mech (ASME) 67 165 (000) [1] B J Chen Z M Xiao and K M Liew Int J Solids Struct 39 1505 (00) [13] K K Kwon and K Y Lee J Appl Mech (ASME) 69 55 (00) [14] A K Soh J X Liu and D N Fang phys stat sol (b) 3 73 (00) [15] X F Li and T Y Fan Arch Appl Mech 71 703 (001) [16] B J Chen Z M Xiao and K M Liew Int J Eng Sci 40 1665 (00) [17] X F Wu Y A Dzenis and W S Zou Int J Fract 7 L9 (00) [18] X Wang and Z Zhong Mech Res Commun 9 45 (00) [19] J L Bleustein Appl Phys Lett 13 41 (1969)