Screw Dislocation Interacting with Interfacial Edge-Cracks in Piezoelectric Bimaterial Strips
|
|
- Garey Wheeler
- 6 years ago
- Views:
Transcription
1 Freund Publishing House Ltd. International Journal of Nonlinear Sciences Numerical Simulation 5(4), , 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 , USA xfwu@unlserve.unl.edu (X.-F. Wu) Bradley D. Rinschen Department of Mechanical Engineering, University of Nebraska-Lincoln, Lincoln, NE , 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
2 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< ),
3 ISSN International Journal of Nonlinear Sciences Numerical Simulation5(4), , 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
4 344 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)
5 ISSN International Journal of Nonlinear Sciences Numerical Simulation5(4), , 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].
6 346 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) [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) [7] Zhang,.-Y, Zhao, M., ong, P. Fracture of piezoelectric ceramics, Advances on Applied Mechanics, 38 () [8] Pak, Y.E. Force on a piezoelectric screw dislocation, ASME Journal of Applied Mechanics, 57 (99) [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 () [] 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) [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) [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) [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) [9] Li, X.F., Fan,.Y. Mode-III interface edge crack between bonded quarter-planes of dissimilar piezoelectric materials, Archive of Applied Mechanics 7 () [] 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
7 Freund Publishing House Ltd. International Journal of Nonlinear Sciences Numerical Simulation 5(4), , 4
Moving screw dislocations in piezoelectric bimaterials
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
More informationScrew dislocation interacting with interface and interfacial cracks in piezoelectric bimaterials
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications from the Department of Engineering Mechanics Mechanical & Materials Engineering, Department of 4-2003
More informationTwo semi-infinite interfacial cracks between two bonded dissimilar elastic strips
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications from the Department of Engineering Mechanics Mechanical & Materials Engineering, Department of 9-2003
More informationA Piezoelectric Screw Dislocation Interacting with an Elliptical Piezoelectric Inhomogeneity Containing a Confocal Elliptical Rigid Core
Commun. Theor. Phys. 56 774 778 Vol. 56, No. 4, October 5, A Piezoelectric Screw Dislocation Interacting with an Elliptical Piezoelectric Inhomogeneity Containing a Confocal Elliptical Rigid Core JIANG
More informationElastic behaviour of an edge dislocation near a sharp crack emanating from a semi-elliptical blunt crack
Chin. Phys. B Vol. 19, No. 1 010 01610 Elastic behaviour of an edge dislocation near a sharp crack emanating from a semi-elliptical blunt crack Fang Qi-Hong 方棋洪, Song Hao-Peng 宋豪鹏, and Liu You-Wen 刘又文
More informationHomework Problems. ( σ 11 + σ 22 ) 2. cos (θ /2), ( σ θθ σ rr ) 2. ( σ 22 σ 11 ) 2
Engineering Sciences 47: Fracture Mechanics J. R. Rice, 1991 Homework Problems 1) Assuming that the stress field near a crack tip in a linear elastic solid is singular in the form σ ij = rλ Σ ij (θ), it
More informationAnalysis of Order of Singularity at a Vertex in 3D Transversely Isotropic Piezoelectric Single-Step Bonded Joints
American Journal of Engineering Research (AJER) 203 American Journal of Engineering Research (AJER) e-issn : 2320-047 p-issn : 2320-0936 Volume-02, Issue-09, pp-7-99 www.ajer.org Research Paper Open Access
More informationINFLUENCE OF TEMPERATURE ON BEHAVIOR OF THE INTERFACIAL CRACK BETWEEN THE TWO LAYERS
Djoković, J. M., et.al.: Influence of Temperature on Behavior of the Interfacial THERMAL SCIENCE: Year 010, Vol. 14, Suppl., pp. S59-S68 S59 INFLUENCE OF TEMPERATURE ON BEHAVIOR OF THE INTERFACIAL CRACK
More informationAnalytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals
PRAMANA c Indian Academy of Sciences Vol. 70 No. 5 journal of May 008 physics pp. 911 933 Analytical solutions for some defect problems in 1D hexagonal and D octagonal quasicrystals X WANG 1 and E PAN
More informationClosed-form solution for the size of plastic zone in an edge-cracked strip
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications from the Deartment of Engineering Mechanics Mechanical & Materials Engineering, Deartment of May 2002
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body
More informationOn the singularity of temperature gradient near an inclined crack terminating at bimaterial interface
International Journal of Fracture 58:319-324, 1992. 1992 Kluwer Academic Publishers. Printed in the Netherlands. 319 On the singularity of temperature gradient near an inclined crack terminating at bimaterial
More informationResearch Article Analysis of Mode I Periodic Parallel Cracks-Tip Stress Field in an Infinite Orthotropic Plate
Mathematical Problems in Engineering Volume 3, Article ID 47, 8 pages http://dx.doi.org/.55/3/47 Research Article Analysis of Mode I Periodic Parallel Cracks-Tip Stress Field in an Infinite Orthotropic
More informationTIME HARMONIC BEHAVIOUR OF A CRACKED PIEZOELECTRIC SOLID BY BIEM. Marin Marinov, Tsviatko Rangelov
Serdica J. Computing 6 (2012), 185 194 TIME HARMONIC BEHAVIOUR OF A CRACKED PIEZOELECTRIC SOLID BY BIEM Marin Marinov, Tsviatko Rangelov Abstract. Time harmonic behaviour of a cracked piezoelectric finite
More informationScienceDirect. Hamiltonian approach to piezoelectric fracture
Available online at www.sciencedirect.com ScienceDirect Procedia Materials Science 3 ( 014 ) 318 34 0th European Conference on Fracture (ECF0) Hamiltonian approach to piezoelectric fracture J.M. Nianga
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Surface and body forces Tensors, Mohr circles. Theoretical strength of materials Defects Stress concentrations Griffith failure
More informationShijiazhuang, P.R. China. Online Publication Date: 01 June 2008 PLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by:[feng, W. J.] On: 5 June 28 Access Details: [subscription number 793822887] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 172954
More informationJournal of Theoretical and Applied Mechanics, Sofia, 2015, vol. 45, No. 1, pp
Journal of Theoretical and Applied Mechanics, Sofia, 2015, vol. 45, No. 1, pp. 69 86 SOLID MECHANICS DOI: 10.1515/jtam-2015-0005 PROPAGATION OF SURFACE WAVES IN A HOMOGENEOUS LAYER OF FINITE THICKNESS
More informationIntegral equations for crack systems in a slightly heterogeneous elastic medium
Boundary Elements and Other Mesh Reduction Methods XXXII 65 Integral equations for crack systems in a slightly heterogeneous elastic medium A. N. Galybin & S. M. Aizikovich Wessex Institute of Technology,
More information3D Elasticity Theory
3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.
More informationStress intensity factors for a mixed mode center crack in an anisotropic strip
International Journal of Fracture 108: 367 381, 2001. 2001 Kluwer Academic Publishers. Printed in the Netherlands. Stress intensity factors for a mixed mode center crack in an anisotropic strip HAIYING
More informationFundamental Solution of 3D Isotropic Elastic Material
Page 6 Fundamental Solution of 3D Isotropic Elastic Material Md. Zahidul Islam, Md. Mazharul Islam *, Mahesh Kumar Sah Department of Mechanical Engineering, Khulna University of Engineering & Technology,
More informationLecture 8. Stress Strain in Multi-dimension
Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]
More informationAn Energy Dissipative Constitutive Model for Multi-Surface Interfaces at Weld Defect Sites in Ultrasonic Consolidation
An Energy Dissipative Constitutive Model for Multi-Surface Interfaces at Weld Defect Sites in Ultrasonic Consolidation Nachiket Patil, Deepankar Pal and Brent E. Stucker Industrial Engineering, University
More informationMode III Stress Singularity Analysis of Isotropic and Orthotropic Bi-material near the Interface End
JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 0 6 Mode III Stress Singularity Analysis of Isotropic and Orthotropic Bi-material near the Interface End Junlin Li, Jing Zhao, Wenting Zhao, Caixia Ren, and
More informationGreen s Functions for Anisotropic/Piezoelectric Bimaterials and Their Applications to Boundary Element Analysis
Copyright 200 Tech Science Press CMES, vol.57, no., pp.3-50, 200 Green s Functions for Anisotropic/Piezoelectric Bimaterials and Their Applications to Boundary Element Analysis Y.C. Chen and Chyanbin Hwu
More information17th European Conference on Fracture 2-5 September,2008, Brno, Czech Republic. Thermal Fracture of a FGM/Homogeneous Bimaterial with Defects
-5 September,8, Brno, Czech Republic Thermal Fracture of a FGM/Homogeneous Bimaterial with Defects Vera Petrova, a, Siegfried Schmauder,b Voronezh State University, University Sq., Voronezh 3946, Russia
More informationOn the uniformity of stresses inside an inhomogeneity of arbitrary shape
IMA Journal of Applied Mathematics 23) 68, 299 311 On the uniformity of stresses inside an inhomogeneity of arbitrary shape Y. A. ANTIPOV Department of Mathematics, Louisiana State University, Baton ouge,
More informationAn Interface Anticrack in a Periodic Two Layer Piezoelectric Space under Vertically Uniform Heat Flow
Mechanics and Mechanical Engineering Vol., No. 3 018 667 681 c Lodz University of Technology An Interface Anticrack in a Periodic Two Layer Piezoelectric Space under Vertically Uniform Heat Flow Andrzej
More informationINFLUENCE OF THE LOCATION AND CRACK ANGLE ON THE MAGNITUDE OF STRESS INTENSITY FACTORS MODE I AND II UNDER UNIAXIAL TENSION STRESSES
INFLUENCE OF THE LOCATION AND CRACK ANGLE ON THE MAGNITUDE OF STRESS INTENSITY FACTORS MODE I AND II UNDER UNIAXIAL TENSION STRESSES Najah Rustum Mohsin Southern Technical University, Technical Institute-Nasiriya,
More informationStress intensity factors for an inclined and/or eccentric crack in a finite orthotropic lamina
1886 Stress intensity factors for an inclined and/or eccentric crack in a finite orthotropic lamina Abstract Stress intensity factors (SIF) are determined for an inclined and / or eccentric crack in a
More informationStress intensity factors for a crack in front of an inclusion
Stress intensity factors for a crack in front of an inclusion Johan Helsing Department of Solid Mechanics and NADA, Royal Institute of Technology, SE-100 44 Stockholm, Sweden Email: helsing@nada.kth.se,
More informationS.P. Timoshenko Institute of Mechanics, National Academy of Sciences, Department of Fracture Mechanics, Kyiv, Ukraine
CALCULATION OF THE PREFRACTURE ZONE AT THE CRACK TIP ON THE INTERFACE OF MEDIA A. Kaminsky a L. Kipnis b M. Dudik b G. Khazin b and A. Bykovtscev c a S.P. Timoshenko Institute of Mechanics National Academy
More informationMechanics PhD Preliminary Spring 2017
Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n
More informationA Green s function for steady-state two-dimensional isotropic heat conduction across a homogeneously imperfect interface
A Green s function for steady-state two-dimensional isotropic heat conduction across a homogeneously imperfect interface W. T. Ang*, K. K. Choo and H. Fan Division of Engineering Mechanics, School of Mechanical
More informationMingHao Zhao, HaiTao Liu, CuiYing Fan, Ernian Pan & Tong-Yi Zhang
A nonlinear bilayer beam model for an interfacial crack in dielectric bimaterials under mechanical/electrical loading MingHao Zhao HaiTao Liu CuiYing Fan Ernian Pan & Tong-Yi Zhang International Journal
More informationHEAT TRANSFER MATERIALS. in COMPOSITE SEIICHI NOMURA A. HAJI-SHEIKH. The University of Texas at Arlington
HEAT TRANSFER in COMPOSITE MATERIALS SEIICHI NOMURA A. HAJI-SHEIKH The University of Texas at Arlington Heat Transfer in Composite Materials D EStech Publications, Inc. 439 North Duke Street Lancaster,
More informationTesting and analysis of high frequency electroelastic characteristics of piezoelectric transformers
Arch. Mech., 59, 2, pp. 119 131, Warszawa 2007 Testing and analysis of high frequency electroelastic characteristics of piezoelectric transformers F. NARITA, Y. SHINDO, F. SAITO, M. MIKAMI Department of
More informationDYNAMIC WEIGHT FUNCTIONS FOR A MOVING CRACK II. SHEAR LOADING. University of Bath. Bath BA2 7AY, U.K. University of Cambridge
DYNAMIC WEIGHT FUNCTIONS FOR A MOVING CRACK II. SHEAR LOADING A.B. Movchan and J.R. Willis 2 School of Mathematical Sciences University of Bath Bath BA2 7AY, U.K. 2 University of Cambridge Department of
More informationSolving the torsion problem for isotropic matrial with a rectangular cross section using the FEM and FVM methods with triangular elements
Solving the torsion problem for isotropic matrial with a rectangular cross section using the FEM and FVM methods with triangular elements Nasser M. Abbasi. June 0, 04 Contents Introduction. Problem setup...................................
More informationSurface/interface Energy Effect on Electromechanical Responses Around a Nanosized Elliptical Inclusion under Far-field Loading at an Arbitrary Angle
Copyright 2014 Tech Science Press CC, vol.40, no.2, pp.145-163, 2014 Surface/interface Energy Effect on Electromechanical Responses Around a Nanosized Elliptical Inclusion under Far-field Loading at an
More informationEngineering Fracture Mechanics 74 (2007) Elsevier Ltd, doi: /j.engfracmech
Engineering Fracture Mechanics 74 (007) 771-781 Elsevier Ltd, doi: 101016/j.engfracmech.006.06.015 A new fracture mechanics theory of orthotropic materials like wood T.A.C.M. van der Put, TU-Delft, CiTG,
More informationThe Effects of Transverse Shear on the Delamination of Edge-Notch Flexure and 3-Point Bend Geometries
The Effects of Transverse Shear on the Delamination of Edge-Notch Flexure and 3-Point Bend Geometries M. D. Thouless Department of Mechanical Engineering Department of Materials Science & Engineering University
More informationThe Effect of Cohesive-Law Parameters on Mixed-Mode Fracture
The Effect of Cohesive-Law Parameters on Mixed-Mode Fracture R. B. Sills * and M. D. Thouless * Abstract * Department of Mechanical Engineering Department of Materials Science & Engineering University
More informationSingularity characteristics for a lip-shaped crack subjected to remote biaxial loading
International Journal of Fracture 96: 03 4, 999. 999 Kluwer Academic Publishers. Printed in the Netherlands. Singularity characteristics for a lip-shaped crack subjected to remote biaxial loading YU QIAO
More informationElastic Fields of Dislocations in Anisotropic Media
Elastic Fields of Dislocations in Anisotropic Media a talk given at the group meeting Jie Yin, David M. Barnett and Wei Cai November 13, 2008 1 Why I want to give this talk Show interesting features on
More informationOn the study of elastic wave scattering and Rayleigh wave velocity measurement of concrete with steel bar
NDT&E International 33 (2000) 401 407 www.elsevier.com/locate/ndteint On the study of elastic wave scattering and Rayleigh wave velocity measurement of concrete with steel bar T.-T. Wu*, J.-H. Sun, J.-H.
More informationStatic Deformation Due To Long Tensile Fault Embedded in an Isotropic Half-Space in Welded Contact with an Orthotropic Half-Space
International Journal of cientific and Research Publications Volume Issue November IN 5-5 tatic Deformation Due To Long Tensile Fault Embedded in an Isotropic Half-pace in Welded Contact with an Orthotropic
More informationSEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by
SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e
More informationNDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.
CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo
More informationOn the generalized Barnett Lothe tensors for monoclinic piezoelectric materials
International Journal of Solids and Structures 44 (00) 508 51 www.elsevier.com/locate/ijsolstr On the generalized Barnett Lothe tensors for monoclinic piezoelectric materials J.Y. Liou a, *, J.C. Sung
More informationPEAT SEISMOLOGY Lecture 2: Continuum mechanics
PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a
More informationFracture Mechanics, Damage and Fatigue Linear Elastic Fracture Mechanics - Energetic Approach
University of Liège Aerospace & Mechanical Engineering Fracture Mechanics, Damage and Fatigue Linear Elastic Fracture Mechanics - Energetic Approach Ludovic Noels Computational & Multiscale Mechanics of
More informationStress intensity factor analysis for an interface crack between dissimilar isotropic materials
Stress intensity factor analysis for an interface crack between dissimilar isotropic materials under thermal stress T. Ikeda* & C. T. Sun* I Chemical Engineering Group, Department of Materials Process
More informationCRACK INITIATION CRITERIA FOR SINGULAR STRESS CONCENTRATIONS Part I: A Universal Assessment of Singular Stress Concentrations
Engineering MECHANICS, Vol. 14, 2007, No. 6, p. 399 408 399 CRACK INITIATION CRITERIA FOR SINGULAR STRESS CONCENTRATIONS Part I: A Universal Assessment of Singular Stress Concentrations Zdeněk Knésl, Jan
More informationStudies of Bimaterial Interface Fracture with Peridynamics Fang Wang 1, Lisheng Liu 2, *, Qiwen Liu 1, Zhenyu Zhang 1, Lin Su 1 & Dan Xue 1
International Power, Electronics and Materials Engineering Conference (IPEMEC 2015) Studies of Bimaterial Interface Fracture with Peridynamics Fang Wang 1, Lisheng Liu 2, *, Qiwen Liu 1, Zhenyu Zhang 1,
More informationStresses and Displacements in Functionally Graded Materials of Semi-Infinite Extent Induced by Rectangular Loadings
Materials 2012, 5, 210-226; doi:10.3390/ma5020210 Article OPEN ACCESS materials ISSN 1996-1944 www.mdpi.com/journal/materials Stresses and Displacements in Functionally Graded Materials of Semi-Infinite
More informationA Note on Suhir s Solution of Thermal Stresses for a Die-Substrate Assembly
M. Y. Tsai e-mail: mytsai@mail.cgu.edu.tw C. H. Hsu C. N. Han Department of Mechanical Engineering, Chang Gung University, Kwei-Shan, Tao-Yuan, Taiwan 333, ROC A Note on Suhir s Solution of Thermal Stresses
More informationExercise: concepts from chapter 8
Reading: Fundamentals of Structural Geology, Ch 8 1) The following exercises explore elementary concepts associated with a linear elastic material that is isotropic and homogeneous with respect to elastic
More informationContinuum Mechanics. Continuum Mechanics and Constitutive Equations
Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform
More informationFig. 1. Different locus of failure and crack trajectories observed in mode I testing of adhesively bonded double cantilever beam (DCB) specimens.
a). Cohesive Failure b). Interfacial Failure c). Oscillatory Failure d). Alternating Failure Fig. 1. Different locus of failure and crack trajectories observed in mode I testing of adhesively bonded double
More informationDRIVING FORCES. and INCLUSION BOUNDARIES with transformation strain. Xanthippi Markenscoff
DRIVING FORCES ON MOVING DEFECTS: DISLOCATIONS and INCLUSION BOUNDARIES with transformation strain Xanthippi Markenscoff UCSD Energy release rates (Driving forces) for moving defects: dislocations and
More information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More informationChapter 2 Governing Equations
Chapter Governing Equations Abstract In this chapter fundamental governing equations for propagation of a harmonic disturbance on the surface of an elastic half-space is presented. The elastic media is
More informationRocking behaviour of a rigid foundation with an arbitrary embedment
Rocking behaviour of a rigid foundation with an arbitrary embedment H. Moghaddasi Islamic Azad University, Qazvin, Iran. M. Kalantari University of Tehran, Tehran, Iran N. Chouw The University of Auckland,
More informationA Simplified Eigenstrain Approach for Determination of Sub-surface Triaxial Residual Stress in Welds
1 A Simplified Eigenstrain Approach for Determination of Sub-surface Triaxial Residual Stress in Welds Michael R. Hill Mechanical and Aeronautical Engineering Department University of California, Davis
More informationJournal of Solid Mechanics and Materials Engineering
and Materials Engineering Analysis of In-Plane Problems for an Isotropic Elastic Medium with Many Circular Holes or Rigid Inclusions Mutsumi MIYAGAWA, Jyo SHIMURA, Takuo SUZUKI and Takanobu TAMIYA Dept.of
More informationELASTICITY AND FRACTURE MECHANICS. Vijay G. Ukadgaonker
THEORY OF ELASTICITY AND FRACTURE MECHANICS y x Vijay G. Ukadgaonker Theory of Elasticity and Fracture Mechanics VIJAY G. UKADGAONKER Former Professor Indian Institute of Technology Bombay Delhi-110092
More informationLecture M1 Slender (one dimensional) Structures Reading: Crandall, Dahl and Lardner 3.1, 7.2
Lecture M1 Slender (one dimensional) Structures Reading: Crandall, Dahl and Lardner 3.1, 7.2 This semester we are going to utilize the principles we learnt last semester (i.e the 3 great principles and
More informationFinite element analysis of longitudinal debonding between fibre and matrix interface
Indian Journal of Engineering & Materials Sciences Vol. 11, February 2004, pp. 43-48 Finite element analysis of longitudinal debonding between fibre and matrix interface K Aslantaş & S Taşgetiren Department
More information12. Stresses and Strains
12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)
More informationStress transformation and Mohr s circle for stresses
Stress transformation and Mohr s circle for stresses 1.1 General State of stress Consider a certain body, subjected to external force. The force F is acting on the surface over an area da of the surface.
More informationStress Intensity Factors Analyses of a Three-Dimensional Interface Crack Between Anisotropic Dissimilar Materials Under Thermal Stress
Stress Intensity Factors Analyses of a Three-Dimensional Interface Crack Between Anisotropic Dissimilar Materials Under Thermal Stress Masaki NAGAI, Toru IKEDA*3 and Noriyuki MIYAZAKI Department of Mechanical
More informationInteraction of newly defined stress intensity factors for angular corners in two diamondshaped
Transactions on Engineering Sciences vol 3, 996 WIT Press, www.witpress.com, ISSN 743-3533 Interaction of newly defined stress intensity factors for angular corners in two diamondshaped inclusions N.-A.
More informationInteraction between elliptic hole and crack in thin plate under uniform bending heat flux
Interaction between elliptic hole and crack in thin plate under uniform bending heat flux J. J. Han & N. Hasebe Department of Civil Engineering, Nagoya Institute of Technology, Japnli Abstract The interaction
More informationResearch Article The Characteristic Solutions to the V-Notch Plane Problem of Anisotropy and the Associated Finite Element Method
Mathematical Problems in Engineering Volume 2013, Article ID 593640, 11 pages http://dx.doi.org/10.1155/2013/593640 Research Article The Characteristic Solutions to the V-Notch Plane Problem of Anisotropy
More informationFinite Element Method in Geotechnical Engineering
Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps
More information1 Stress and Strain. Introduction
1 Stress and Strain Introduction This book is concerned with the mechanical behavior of materials. The term mechanical behavior refers to the response of materials to forces. Under load, a material may
More informationDynamics of Glaciers
Dynamics of Glaciers McCarthy Summer School 01 Andy Aschwanden Arctic Region Supercomputing Center University of Alaska Fairbanks, USA June 01 Note: This script is largely based on the Physics of Glaciers
More informationBasic Equations of Elasticity
A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ
More informationFree transverse vibrations of cracked nanobeams using a nonlocal elasticity model
Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model J. Loya, J. López-Puente, R. Zaera, and J. Fernández-Sáez a Department of Continuum Mechanics and Structural Analysis,
More informationQing-Hua Qin. Advanced Mechanics of Piezoelectricity
Qing-Hua Qin Advanced Mechanics of Piezoelectricity Qing-Hua Qin Advanced Mechanics of Piezoelectricity With 77 figures Author Prof. Qing-Hua Qin Research School of Engineering Australian National University
More informationDynamic Stress Intensity Factors of Collinear Cracks under a Uniform Tensile Stress Wave 1
Copyright 2013 Tech Science Press CMES, vol.93, no.2, pp.133-148, 2013 Dynamic Stress Intensity Factors of Collinear Cracks under a Uniform Tensile Stress Wave 1 K.-C. Wu 2, S.-M. Huang 2, S.-H. Chen 3
More informationExplicit models for elastic and piezoelastic surface waves
IMA Journal of Applied Mathematics (006) 71, 768 78 doi:10.1093/imamat/hxl01 Advance Access publication on August 4, 006 Explicit models for elastic and piezoelastic surface waves J. KAPLUNOV AND A. ZAKHAROV
More informationAnalytical solutions for two penny-shaped crack problems in thermo-piezoelectric materials and their finite element comparisons
International Journal of Fracture 117: 113 18,. Kluwer Academic Publishers. Printed in the Netherlands. Analytical solutions for two penny-shaped crack problems in thermo-piezoelectric materials and their
More informationEnergy of a Prismatic Dislocation Loop in an Elastic Cylinder
Mathematics and Mechanics of Solids, in press (8) Energy of a Prismatic Dislocation Loop in an Elastic Cylinder Wei Cai and Christopher. Weinberger Department of Mechanical Engineering, Stanford University,
More informationLASER GENERATED THERMOELASTIC WAVES IN AN ANISOTROPIC INFINITE PLATE
LASER GENERATED THERMOELASTIC WAVES IN AN ANISOTROPIC INFINITE PLATE H. M. Al-Qahtani and S. K. Datta University of Colorado Boulder CO 839-7 ABSTRACT. An analysis of the propagation of thermoelastic waves
More informationThe Effects of Cohesive Strength and Toughness on Mixed-Mode Delamination of Beam-Like Geometries
The Effects of Cohesive Strength and Toughness on Mixed-Mode Delamination of Beam-Like Geometries J. P. Parmigiani and M. D. Thouless,2 Department of Mechanical Engineering 2 Department of Materials Science
More informationANALYSIS AND NUMERICAL MODELLING OF CERAMIC PIEZOELECTRIC BEAM BEHAVIOR UNDER THE EFFECT OF EXTERNAL SOLICITATIONS
Third International Conference on Energy, Materials, Applied Energetics and Pollution. ICEMAEP016, October 30-31, 016, Constantine,Algeria. ANALYSIS AND NUMERICAL MODELLING OF CERAMIC PIEZOELECTRIC BEAM
More informationResearch Article Effect of Couple Stresses on the Stress Intensity Factors for Two Parallel Cracks in an Infinite Elastic Medium under Tension
Mathematical Problems in Engineering Volume 212, Article ID 575891, 14 pages doi:1.1155/212/575891 Research Article Effect of Couple Stresses on the Stress Intensity Factors for Two Parallel Cracks in
More informationModule 7: Micromechanics Lecture 34: Self Consistent, Mori -Tanaka and Halpin -Tsai Models. Introduction. The Lecture Contains. Self Consistent Method
Introduction In this lecture we will introduce some more micromechanical methods to predict the effective properties of the composite. Here we will introduce expressions for the effective properties without
More informationStructural Health Monitoring Using Smart Piezoelectric Material
Structural Health Monitoring Using Smart Piezoelectric Material Kevin K Tseng and Liangsheng Wang Department of Civil and Environmental Engineering, Vanderbilt University Nashville, TN 37235, USA Abstract
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting Lectures & 3, 9/31 Aug 017 www.geosc.psu.edu/courses/geosc508 Discussion of Handin, JGR, 1969 and Chapter 1 Scholz, 00. Stress analysis and Mohr Circles Coulomb Failure
More informationA time domain BEM for transient dynamic straight crack in piezoelectric solids
Boundary Elements and Other esh Reduction ethods XXXVI 219 A time domain BE for transient dynamic straight crack in piezoelectric solids Peng Zhao 2 & Taiyan Qin 1 1 College of Science China Agricultural
More informationJournal of Applied Mathematics and Physics (ZAMP) /88/ $ 2.70/0 Vol. 39, July BirkhS.user Verlag Basel, 1988
Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/88/004573-06 $ 2.70/0 Vol. 39, July 1988 9 BirkhS.user Verlag Basel, 1988 Cracked orthotropic strip with clamped boundaries By H. G. Georgiadis*)
More informationChapter 4. Higher-Order Differential Equations
Chapter 4 Higher-Order Differential Equations i THEOREM 4.1.1 (Existence of a Unique Solution) Let a n (x), a n,, a, a 0 (x) and g(x) be continuous on an interval I and let a n (x) 0 for every x in this
More informationStress, Strain, Mohr s Circle
Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected
More informationInternational Journal of Innovative Research in Science, Engineering and Technology Vol. 2, Issue 7, July 2013
ANALYSIS AND MITIGATION OF STRESS CONCENTRATION FACTOR OF A RECTANGULAR ISOTROPIC AND ORTHOTROPIC PLATE WITH CENTRAL CIRCULAR HOLE SUBJECTED TO INPLANE STATIC LOADING BY DESIGN OPTIMIZATION Shubhrata Nagpal
More informationPrincipal Stresses, Yielding Criteria, wall structures
Principal Stresses, Yielding Criteria, St i thi Stresses in thin wall structures Introduction The most general state of stress at a point may be represented by 6 components, x, y, z τ xy, τ yz, τ zx normal
More informationRayleigh waves in magneto-electro-elastic half planes
Acta Mech 202, 127 134 (2009) DOI 10.1007/s00707-008-0024-8 W. J. Feng E. Pan X. Wang J. Jin Rayleigh waves in magneto-electro-elastic half planes Received: 23 August 2007 / Revised: 17 February 2008 /
More information