Freund Publishing House Ltd. International Journal of Nonlinear Sciences Numerical Simulation 5(4), 34-346, 4 Screw Dislocation Interacting with Interfacial Edge-Cracks in Piezoelectric Bimaterial Strips Xiang-Fa Wu, Yuris A. Dzenis Department of Engineering Mechanics, Center for Materials Research Analysis University of Nebraska-Lincoln, Lincoln, NE 68588-56, USA Email: xfwu@unlserve.unl.edu (X.-F. Wu) Bradley D. Rinschen Department of Mechanical Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-656, USA Abstract his paper is concerned with the interaction between an interfacial edge-crack a screw dislocation under out-of-plane mechanical in-plane electric loading in a piezoelectric bimaterial strip. In addition to a discontinuous electric potential across the slip plane, the dislocation is subjected to a line-force a line-charge at the core. Under the framework of linear piezoelectricity, the out-of-plane displacement in-plane electric potentials are constructed in closed-form by means of conformal mapping technique the known solution for screw dislocation in cracked piezoelectric bimaterial. he intensity factors (IFs) energy release rate (ERR) are derived explicitly. Keywords: Screw dislocation; piezoelectric bimaterials; edge-crack; interfacial fracture; intensity factors (IFs); energy release rate (ERRs); strips; conformal mapping; butt joint. Introduction he study of dislocations in solid materials plays important role in fracture mechanics since dislocation solutions can serve as Green s functions to construct the solutions for various crack configurations under arbitrary loadings. In the last two decades, a number of investigations have been contributed to this field [-4]. With extensive applications of piezoelectric ceramics in actuators, sensors, transducers etc., the study of fracture failure of piezoelectric materials has attracted numerous researchers attention in recent years, especially the behaviors of various defects inclusions under the coupled electric mechanical loadings. For example, Deeg [5] first studied the response of piezoelectric solids with a dislocation, a crack, an inclusion under coupled electromechanical fields, Suo et al. [6] discussed the general solutions to collinear interfacial cracks in anisotropic piezoelectric bimaterials, etc. Detailed review of recent developments in fracture mechanics of piezoelectric materials can be found in the review paper by Zhang et al. [7]. In the study of screw dislocations in piezoelectric ceramics, Pak [8] first considered the Peach-Koehler forces acting on a screw dislocation under external electromechanical loadings. Lee et al. [9] Chen et al. [] dealt with a screw dislocation interacting with a semi-infinite crack in a transversely homogenous piezoelectric ceramics, Kwon Lee [] further considered the similar case of a finite crack. Recently, by means of superposition, Soh [,3], Wu et al. [4-6], Liu et al. [7,8] obtained the fundamental solutions for screw dislocations in cracked piezoelectric bimaterial media with finite dimensions. In reality, cracks more likely appear near free edges of bonded dissimilar blocks where stress
34 X.-F. Wu, Y. A. Dzenis, B. D. Rinschen. Screw Dislocation Interacting with Interfacial Edge-Cracks in Piezoelectric Bimaterial Strips concentration exists even without cracks. In this work, we consider a simple continuum model of screw dislocation interacting with an interfacial edge-crack in a piezoelectric bimaterial strip. By means of conformal mapping technique known fundamental solutions to screw dislocations in piezoelectric bimaterials [,5], the out-of-plane displacement in-plane electric potentials of the cracked strip are derived in closed-form. he intensity factors (IFs) the energy release rate (ERR) of the edge-cracks are obtained explicitly.. Problem statement solution procedure As discussed in [8,9], the piezoelectric medium is supposed to be transversely isotropic, which has an isotropic basal plane parallel to xyplane a poling direction perpendicular to xyplane. he boundary-value problem is treated as in the case of out-of-plane mechanical displacement in-plane electric fields such that u = u =, u = u ( x, y), () x y z z E = E( xy, ), E = E( xy, ), E =. x x y y z In this case the constitutive relations reduce to σ xz = c44 uz, x + e 5 φ, x, Dx = e5 uz, x ε φ, x, σ = c u + e φ, D = e u ε φ, yz 44 zy, 5, y y 5 zy,,y () (3) where σ kz, D k (k=x, y), c 44, e 5, ε φ are the stress tensor, electric displacement vector, elastic modulus at constant electric field, piezoelectric constants, electric potential, respectively. he electric field is given by E x = φ, E = φ. (4), x y,y he governing equations are σ σ =, D + D. (5) xz, x + yz, y x, x y, y = Substitution of (3) into (5) leads to c u + e φ =, 44 z 5 e u ε φ =. 5 z he above equations may be reduced as (6) u =, φ =. (7) z As a result, the out-of-plane displacement in-plane electric potentials can be expressed as the imaginary parts of two analytic functions U(z) Φ(z), i.e. u z = Im[U( z, φ = Im[ Φ( z, (8) where Im( ) denotes the imaginary part of an analytic function. It is convenient to introduce the complex stress electric displacement: σ + iσ = c ( u + iu ) + e ( φ + iφ ), zy zx 44 z, y z, x 5, y, x D + id = e ( u + iu ) ε ( φ + iφ ). y x 5 z, y z, x, y,x Using (8), relation (9) may be rewritten as σ + iσ = c U( z) + e Φ ( z), zy zx 44 5 D + id = e U( z) ε Φ ( z). y x 5 (9) () Reserving the xy-coordinate system for the problem to be solved, we first determine the outof-plane displacement in-plane electric potentials for a screw dislocation located in a cracked bimaterial, as shown in Fig.. he crack surfaces are assumed to be traction-free electric impermeable. he screw dislocation is assumed located at ζ (ζ =ξ +iη ) in the lower half-plane characterized by Burgers vector b, line-force p, line-charge q, electric potential jump φ. he complex potentials for this problem have been determined [,5] as ( L+ L) L ( ς - ς) + L h( ς) ς D i..( e η > ), U( ς ) ( L L) ( L L) ( ς - ς) Φς ( ) = + () + ( ς - ς) + L h( ς), ς D i..( e η< ),
ISSN 565-339 International Journal of Nonlinear Sciences Numerical Simulation5(4), 34-346, 4 343 where h( ς ) = B { ( ς - ς )[ ( ς + ( ς - ς ) ( ς ς ) }. ς ) ] () decoupled Laplace equations (7), conformal mapping technique is applicable for solving some interfacial edge-cracks in piezoelectric bimaterial strips. Here the prime denotes the derivative of an analytic function with respect to ζ=ξ+iη, D D denote the half-planes above below respectively, L L represent the material matrices of the half-planes above below, B is the bimaterial matrix, is the complex quantity of the screw dislocation: c44 e5 L =, B= ( L + L ), e5 ε [ ] [ ] = ( π) b, φ + L ( πi) p, q. (3) In the xy-coordinate system as shown in Figs. 3, the IFs are defined as [5] K = π lim x [ h( x) + h( x, (4) x where K = {K III,K D}, KD is the electric intensity factor. he traction electric displacement at the interface a distance r ahead of the crack tip the displacement electric potential jumps a distance r behind of the crack tip are expressed as t( r ) = K πr, d( r) = BK πr, (5) with σ yz uz ( x,+ ) uz ( x, ) t =, d =. (6) Dy φ ( x,+ ) φ ( x, ) he ERR for a unit crack growth along the interface can be evaluated as lim( ) t ( r) d( r) dr (7) = ( 4) K BK. Due to the out-of-plane displacement inplane electric potentials satisfying two
344 X.-F. Wu, Y. A. Dzenis, B. D. Rinschen. Screw Dislocation Interacting with Interfacial Edge-Cracks in Piezoelectric Bimaterial Strips 3. Edge-cracked piezoelectric bimaterial strips Consider a screw dislocation interacting with an interfacial edge-crack between two sidebonded dissimilar piezoelectric strips of equal width, as shown in Fig., in which a H denote the crack length the strip width, respectively. he crack surfaces the edge surfaces are assumed to be traction-free electric impermeable. he screw dislocation is located at z (z =x +iy ) in the lower strip. Introduce the conformal mapping ζ = f ( z + a) f ( a) (8) with f ( z) = sinh[ πz (H, (9) which maps the bonded piezoelectric strips (Fig. ) with an interfacial edge-crack onto two bonded half-planes with a semi-infinite cut along the negative ξ-axis, as shown in Fig.. Substitution of (8) (9) into () () leads to the potential of the strip. Its main contributor () may be expressed as h B H z a H (z) = π (4 )sinh[ π( + ) ] { [f a f a (z + ) - (z + f ( z + a) f ( a) f ( z+ a) f ( a) + f + a f + a [ (z ) - (z f ( z + a) f ( a) f ( z+ a) f ( a). () By the definition (4), the IFs are evaluated as K = πb H sinh( Re{ i cosh( cosh[ π ( z + a) H ]}, () where is the complex quantity of the screw dislocation defined in (3). By letting H in (), we obtain the IFs for an interfacial edgecrack between two bonded dissimilar piezoelectric quarter-planes as K = π B Re{ -i a a ( a+ z ) }. () 4 Furthermore, by letting a in (), we get the IFs for a semi-infinite crack between two infinite side-bonded piezoelectric strips as K = πb H Re[ i exp( πz H. (3) As shown in Fig., here we further consider two special loading cases of a line-force P a line-charge Q located at z =-b-i (<b<a) a line-force P a line-charge Q located at z =- a-ih (h<h), respectively. hen, may be rewritten as = L ( πi)[ P, Q ], = L (πi)[ P, Q ]. (4) With the aid of (7) (), the IFs ERR are evaluated respectively as K = B L H sinh( cosh[ (H cosh[ π ( a b) H ][ P ], (5) sinh( H cosh[ (H cosh[ π ( a b) (H [ P ] L B L [ P for z =-b-i, K = B [ P H L ], ] H sinh( (6) cosh( cos( πh sinh( cosh( cos( πh (7) [ P ] L B L [ P ], for z =-a-ih. If letting H, results (5)-(8) cover those given by Li Fan [9], who used the method of dual integral equations. Furthermore, integration of (5) with respect to b in the interval [, a] leads to the solutions for a Griffith crack embedded at the mid-plane of a piezoelectric layer under out-of-plane mechanical in-plane electric loading as (8)
ISSN 565-339 International Journal of Nonlinear Sciences Numerical Simulation5(4), 34-346, 4 345 those discussed by Li Duan [,] Li []. 4. Interfacial edge-crack in piezoelectric bimaterial butt joint Now let us consider the second case of a screw dislocation interacting with an interfacial edge-crack in a piezoelectric bimaterial butt joint, as shown in Fig. 3. Here a H denote the crack length the strip width, respectively. As aforementioned, the crack surfaces the edge surfaces are assumed to be traction-free electric impermeable. he screw dislocation is supposed located at z (z =x +iy ) in the lower strip. In this case, we choose the conformal mapping ζ = g ( z + a) g ( a) (9) with g( z) = tan[ πz (H, (3) which maps the cracked butt joint (Fig. 3) onto two bonded half-planes with a semi-infinite cut along the negative ξ-axis, as shown in Fig.. Substituting (9) (3) into () (), we obtain the potential for this problem. Its main contributor () may be expressed in terms of h (z) = πb (H )sec [ π ( z + a) (H { g( z + a) [ g g g + g( z + a) [ g ( z + a) g ( z + a) g ( z + a) -g ( ) ( ) g z + a g a. ( ) ( ) g z + a g a (3) With the help of definition (4), we obtain the IFs as K = 4πB Re {- i (z+ a) - g sec[ (H tan [ (H tan [ π ( z ( z ( a) ( a) (H ) tan[ (H (z + a + a + a) (H }. (3) By letting H, (3) covers the IFs for an interfacial edge-crack between two bonded dissimilar piezoelectric quarter-planes as (). Furthermore, by letting a simultaneously keeping c=(h-a) constant, we have the IFs for a semi-infinite interfacial crack heading towards a free surface such that K = 4π B πc Re[ ( c z) z(c z) ]. (33) Similar to Section 3, here we consider the same loadings shown in Fig. 3. With the aid of (4) (3), the IFs ERR are evaluated respectively as K = sec[ (H (H ) tan[ (H (34) tan [ (H tan [ π ( a b) (H ] B L [ P tan[ (H sec [ (H H tan [ (H tan [ π ( a b) (H [ P ] L for z =-b-i, K = B L [ P ] B sec[ (H L [ P (H ) tan[ (H tan [ H ] + tanh H ] [ πh (H tan[ (H sec [ (H tan [ (H + tanh [ πh (H ) ] (35) (36) [ P ] L B L [ P (37) for z =-a-ih. By letting H, results (34)-(37) again return to those given by Li Fan [9] Wu et al. [5]. Since the general potentials for the above crack configurations have been obtained, the entire electroelastic fields of the strips the forces acting at the screw dislocation can be extracted explicitly based on the method discussed by Pak [8].
346 X.-F. Wu, Y. A. Dzenis, B. D. Rinschen. Screw Dislocation Interacting with Interfacial Edge-Cracks in Piezoelectric Bimaterial Strips 5. Concluding remarks Explicit solutions to screw dislocation interacting with interfacial edge-crack between two bonded dissimilar piezoelectric strips have been determined in this work. Conformal mapping technique has been shown to be a powerful tool in solving some interfacial edgecracks in piezoelectric strips. Closed-form solutions () (3) can be employed as a useful theoretical base for the assessment of numerical analyses, especially for estimating the effect of ah ratio on the IFs ERR of cracks in bonded piezoelectric structures. Furthermore, these explicit solutions can serve as Green s functions to construct the IFs ERRs of piezoelectric strips with multiple cracks under arbitrary out-of-plane mechanical in-plane electric loadings. Acknowledgement Partial support of this work by the U. S. Air Force Office of Scientific Research the U. S. Army Research Office is gratefully acknowledged. References [] homson, R. Physics of fracture, Solid State Physics, 39 (986) -9 [] Suo, Z. Singularities interacting with interface cracks, International Journal of Solids Structures, 5 (989) 33-4 [3] Suo, Z. Singularities, interfaces cracks in dissimilar anisotropic media, Proceedings of the Royal of Society of London, A47 (99) 33-358 [4] Weertman, J. Dislocation Based Fracture Mechanics, World Scientific Publishing, Singapore, 996 [5] Deeg, W.F. he Analysis of Dislocation, Crack, Inclusion in Piezoelectric Solids, Ph.D. thesis, Stanford University, Stanford, CA, 98 [6] Suo, Z., Kuo, C.-M., Barnett, D.M., Willis, J.R. Fracture mechanics for piezoelectric ceramics, Journal of the Mechanics Physics of Solids, 4 (99) 739-765 [7] Zhang,.-Y, Zhao, M., ong, P. Fracture of piezoelectric ceramics, Advances on Applied Mechanics, 38 () 47-89 [8] Pak, Y.E. Force on a piezoelectric screw dislocation, ASME Journal of Applied Mechanics, 57 (99) 863-869 [9] Lee, K.Y., Lee, W.G., Pak, Y. E. Interaction between a semi-infinite crack a screw dislocation in a piezoelectric material, ASME Journal of Applied Mechanics, 67 () 65-7 ] Chen, B.J, Xiao, Z.M., Liew, K.M., On the interaction between a semi-infinite anti-crack a screw dislocation in piezoelectric solid, International Journal of Solids Structures, 39 () 55-53 [] Kwon, K.K., Lee, K.Y. Electromechanical effects of a screw dislocation around a finite crack in a piezoelectric material, ASME Journal of Applied Mechanics, 69 () 55-6 [] Soh, A.K., Liu, J.X., Fang, D.N., A screw dislocation interacting with an interfacial crack in two dissimilar piezoelectric media, Physica Status Solidi B- Basic Research, 3 () 73-8 [3] Soh, A.K., Liu, J.X., Hoon, K.H., 3-D Green s functions for transversely isotropic magnetoelectroelastic solids, International Journal of Nonlinear Sciences Numerical Simulation, 4 (3) 39-48 [4] Wu, X.F., Dzenis Y. A., Zou W.S. Screw dislocation interacting with an interfacial edge crack between two bonded dissimilar piezoelectric wedges, International Journal of Fracture, 7 () L9-L4. [5] Wu, X.F., Cohn, S., Dzenis, Y.A. Screw dislocation interacting with interface interfacial cracks in piezoelectric bimaterials, International Journal of Engineering Science, 4 (3) 667-68 [6] Wu X.F., Dzenis, Y.A., Fan,.Y. Screw dislocation interacting with twin interfacial edge cracks between two bonded dissimilar piezoelectric strips, Mechanics Research Communications, 3(3) 547-555 [7] Liu, X.L., Liu, J.X., Liu, J. Green s function for a semi-infinite piezoelectric bimaterial strip with an interfacial edge crack. International Journal of Nonlinear Sciences Numerical Simulation, 5 (4) 6-66 [8] Liu, J.X., Screw dislocation in two dissimilar piezoelectric layers, Physica Status Solidi B- Basic Research, 4 (4) 98-34 [9] Li, X.F., Fan,.Y. Mode-III interface edge crack between bonded quarter-planes of dissimilar piezoelectric materials, Archive of Applied Mechanics 7 () 73-74 [] Li, X.F, Duan, X.Y. Electroelastic analysis of a piezoelectric layer with electrodes, International Journal of Fracture, (a) L73-L78 [] Li, X.F, Duan, X.Y. Closed-form solution for a mode-iii crack at the mid-plane of a piezoelectric layer, Mechanics Research Communications, 8 (b) 73-7 [] Li, X.F. Electroelastic analysis of an anti-plane shear crack in a piezoelectric ceramic strip, International Journal of Solids Structures, 39 () 97-7
Freund Publishing House Ltd. International Journal of Nonlinear Sciences Numerical Simulation 5(4), 34-346, 4