Table of Contents
ISRN Materials Science
VolumeΒ 2012, Article IDΒ 659352, 19 pages
Research Article

Antiplane Shear Crack Normal to and Terminating at the Interface of Two Bonded Piezo-Electro-Magneto-Elastic Materials

Department of Mechanics of Materials, Technical University of Lodz, Al. Politechniki 6, 93-590 Lodz, Poland

Received 15 December 2011; Accepted 13 February 2012

Academic Editor: V.Β Sglavo

Copyright Β© 2012 Bogdan Rogowski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The magnetoelectroelastic analysis of two bonded dissimilar piezo-electro-magneto-elastic ceramics with a crack perpendicular to and terminating at the interface is made. By using the Fourier integral transform (in perpendicular directions in each materials), the mixed boundary conditions and continuity conditions are transformed to a singular integral equation with generalized Cauchy kernel, the solution of which has been well studied, and classical methods are directly applicable here to obtain the closed form solution. The results are presented for a permeable crack under anti-plane shear loading and in-plane electric and magnetic loadings, as prescribed electric displacement and magnetic inductions or electric and magnetic fields. Obtained results indicate that the magnetoelectroelastic field near the crack tip in the homogeneous PEMO-elastic ceramic is dominated by a traditional inverse square-root singularity, while the coupled field near the crack tip at the interface exhibits the singularity of power law π‘Ÿβˆ’π›Ό, π‘Ÿ being distant from the interface crack tip and 𝛼 depending on the material constants of a bimaterial. In particular, electric and magnetic fields have no singularity at the crack tip in a homogeneous solid, whereas they are singular around the interface crack tip. Numerical results are given graphically to show the effects of the material properties on the singularity order, field intensity factors and energy release rates. Results presented in this paper should have potential applications to the design of multilayered magnetoelectroelastic structures.

1. Introduction

The newly emerging materials named magnetoelectroelasticity, which exhibit piezoelectric, piezomagnetic, and electromagnetic properties, have found increasing wide engineering applications, particularly in aerospace and automotive industries. Magnetoelectroelastic solids have been widely used as transducer, sensors, and actuators in smart structures. Because of the brittleness of PEMO-elastic materials, a high possibility of material debonding and cracking or sliding of the interface exists. Consequently, this problem has been the subject of research and discussion in the literature on elasticity theory of coupled fields. Li and Kardomateas [1] investigated the mode III interface crack problem for dissimilar piezo-electromagnetoelastic bimaterial media. The extended Stroh’s theory and analytic continuation principle of complex analysis have been used to obtain the solution for interfacial cracks between two dissimilar Magnetoelectroelastic half-planes by Li and Kardomateas [2]. The problem for an antiplane interface crack between two dissimilar PEMO-elastic layers was analyzed by Wang and Mai [3]. Gao et al. [4] derived the exact solution for a permeable interface crack between two dissimilar Magnetoelectroelastic solids under general applied loads. Gao et al. [5] derived also the static solution related to antiplane crack problem. The antiplane shear cracks are a class of simple problems. But, for the case of a crack perpendicular to the interface, the problem becomes more complicated. This problem has been subject of research in the classical literature of elasticity theory. Cook and Erdogan [6] and Erdogan and Cook [7] were apparently the first to publish the solution of this problem for two bonded dissimilar isotropic half-planes. For piezoelectric biceramics an arbitrarily oriented plane crack terminating at the interface was extended by Qin and Yu [8]. The antiplane shear crack normal to and terminating at the interface of two piezoelectric ceramics was extended later by Li and Wang [9]. Although the above studies deal strictly with piezoelectric, it is reasonable to assume that the extension of the findings to electromagnetoelastic materials is valid.

To the best of author knowledge, the behaviour of interfacial cracks normal to and terminating at the interface of two bonded piezo-electromagnetoelastic materials has not been addressed yet. Motivated by these considerations, the author investigates the antiplane deformations and in-plane electric and magnetic fields of a PEMO-elastic bi-material with Mode-III interface crack normal to and terminating at the interface.

The crack is assumed to be electrically and magnetically permeable. Under applied electric, magnetic, and mechanical loading, electric, magnetic, and elastic behaviours near both crack tips are obtained. Two kinds of loading conditions are adopted. By using Fourier integral transform, in perpendicular directions in each materials, the associated boundary value problem is transformed to a singular integral equation with generalized Cauchy kernel. Similar types of equations have been studied, and classical methods of their solutions are directly applicable here to obtain the solution in closed form. The results indicate that magnetoelectroelastic field near the crack tip in a homogeneous PEMO-elastic ceramic exhibits an inverse square-root singularity, while singular field near the interface crack tip is dominant by a singularity of power law. The singularity order is dependent on relevant 2Γ—6 material constants of two ceramics. The effects of magneto-electro-mechanical parameters on the field intensity factors are evaluated by numerical analysis, which could be of particular interest to the analysis and design of smart sensors/actuators constructed from Magnetoelectroelastic composite laminates.

2. Formulation of the Problem

2.1. Basic Equations

For a linearly Magnetoelectroelastic medium under antiplane shear coupled with in-plane electric and magnetic fields, there are only the nontrivial antiplane displacement 𝑀:𝑒π‘₯=0,𝑒𝑦=0,𝑒𝑧=𝑀(π‘₯,𝑦),(1) strain components 𝛾π‘₯𝑧 and 𝛾𝑦𝑧:𝛾π‘₯𝑧=πœ•π‘€πœ•π‘₯,𝛾𝑦𝑧=πœ•π‘€πœ•π‘¦(2) stress components 𝜏π‘₯𝑧 and πœπ‘¦π‘§, in-plane electrical and magnetic potentials πœ™ and πœ“, which define electric and magnetic field components 𝐸π‘₯,𝐸𝑦,𝐻π‘₯, and 𝐻𝑦:𝐸π‘₯=βˆ’πœ•πœ™πœ•π‘₯,𝐸𝑦=βˆ’πœ•πœ™πœ•π‘¦,𝐻π‘₯=βˆ’πœ•πœ“πœ•π‘₯,𝐻𝑦=βˆ’πœ•πœ“πœ•π‘¦(3) and electrical displacement components 𝐷π‘₯,𝐷𝑦, and magnetic induction components 𝐡π‘₯, and 𝐡𝑦 with all field quantities being the functions of coordinates π‘₯ and 𝑦.

The relations (2) and (3) have the following form:𝛾𝛼𝑧=𝑀,𝛼,𝐸𝛼=βˆ’πœ™,𝛼,𝐻𝛼=βˆ’πœ“,𝛼,(4) where 𝛼=π‘₯,𝑦 and 𝑀,𝛼=πœ•π‘€/πœ•π›Ό.

For linearly Magnetoelectroelastic medium, the coupled constitutive relations can be written in the matrix form as follows:ξ€Ίπœπ›Όπ‘§,𝐷𝛼,𝐡𝛼𝑇𝛾=C𝛼𝑧,βˆ’πΈπ›Ό,βˆ’π»π›Όξ€»π‘‡,(5) where the superscript 𝑇 denotes the transpose of a matrix andβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘C=44𝑒15π‘ž15𝑒15βˆ’πœ€11βˆ’π‘‘11π‘ž15βˆ’π‘‘11βˆ’πœ‡11⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,(6) where 𝑐44 is the shear modulus along the 𝑧-direction, which is direction of poling and is perpendicular to the isotropic plane (π‘₯,𝑦),πœ€11 and πœ‡11 are dielectric permittivity, and magnetic permeability coefficients, respectively, 𝑒15,π‘ž15, and 𝑑11 are piezoelectric, piezomagnetic and magneto-electric coefficients, respectively.

The mechanical equilibrium equation (called as Euler equation) and the charge and current conservation equations (called as Maxwell equations), in the absence of the body force electric and magnetic charge densities, can be written asπœπ‘§π›Ό,𝛼=0,𝐷𝛼,𝛼=0,𝐡𝛼,𝛼=0,𝛼=π‘₯,𝑦.(7) Subsequently, the Euler and Maxwell equations take the following form:Cξ€Ίβˆ‡2𝑀,βˆ‡2πœ™,βˆ‡2πœ“ξ€»π‘‡=[]0,0,0𝑇,(8) where βˆ‡2=πœ•2/πœ•π‘₯2+πœ•2/πœ•π‘¦2 is the two-dimensional Laplace operator.

Since |C|β‰ 0, one can decouple the (8) as follows:βˆ‡2𝑀=0,βˆ‡2πœ™=0,βˆ‡2πœ“=0.(9)

If we introduce, for convenience of mathematics in some boundary value problems, two unknown functionsξ€Ίπœ’βˆ’π‘’15𝑀,πœ‚βˆ’π‘ž15𝑀𝑇=C0[]πœ™,πœ“π‘‡,(10) whereC0=βŽ‘βŽ’βŽ’βŽ£βˆ’πœ€11βˆ’π‘‘11βˆ’π‘‘11βˆ’πœ‡11⎀βŽ₯βŽ₯⎦,(11) then[]πœ™,πœ“π‘‡=C0βˆ’1ξ€Ίπœ’βˆ’π‘’15𝑀,πœ‚βˆ’π‘ž15𝑀𝑇,(12) whereC0βˆ’1=1πœ€11πœ‡11βˆ’π‘‘211βŽ‘βŽ’βŽ’βŽ£βˆ’πœ‡11𝑑11𝑑11βˆ’πœ€11⎀βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ£π‘’1𝑒2𝑒2𝑒3⎀βŽ₯βŽ₯⎦.(13)

The governing field variables areπœπ‘§π‘˜=̃𝑐44𝑀,π‘˜βˆ’π›Όπ·π‘˜βˆ’π›½π΅π‘˜,πœ™=𝛼𝑀+𝑒1πœ’+𝑒2πœ‚,πœ“=𝛽𝑀+𝑒2πœ’+𝑒3π·πœ‚,π‘˜=πœ’,π‘˜,π΅π‘˜=πœ‚,π‘˜βˆ‡,π‘˜=π‘₯,𝑦,(14)2𝑀=0,βˆ‡2πœ’=0,βˆ‡2πœ‚=0,(15) wherẽ𝑐44=𝑐44+𝛼𝑒15+π›½π‘ž15,πœ‡π›Ό=11𝑒15βˆ’π‘‘11π‘ž15πœ€11πœ‡11βˆ’π‘‘211𝑒=βˆ’1𝑒15+𝑒2π‘ž15ξ€Έ,πœ€π›½=11π‘ž15βˆ’π‘‘11𝑒15πœ€11πœ‡11βˆ’π‘‘211𝑒=βˆ’3π‘ž15+𝑒2𝑒15ξ€Έ.(16)

Note that ̃𝑐44 is the piezo-electro-magnetically stiffened elastic constant.

Note also that the inverse of a matrix C is defined by parameters 𝛼,𝛽,̃𝑐44 and 𝑒1,𝑒2,𝑒3 as follows:Cβˆ’1=1̃𝑐44⎑⎒⎒⎒⎒⎣1𝛼𝛽𝛼𝛼2+̃𝑐44𝑒1𝛼𝛽+̃𝑐44𝑒2𝛽𝛼𝛽+̃𝑐44𝑒2𝛽2+̃𝑐44𝑒3⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.(17)

These material parameters will appear in our solutions.

2.2. Boundary Conditions

Consider a crack terminating at the interface of two bonded dissimilar PEMO-elastic ceramics polarized in the 𝑧 direction. For convenience, we denote the PEMO-elastic ceramics occupying the right and left half-planes π‘₯β‰₯0 and π‘₯≀0 as piezoceramics I and II, respectively, shown in Figure 1.

Figure 1: Two bonded dissimilar PEMO-elastic ceramics with a crack perpendicular to and terminating at the interface.

Let a crack be perpendicular to the interface and be situated at [0,π‘Ž](π‘Ž>0) in the positive π‘₯-direction in ceramic I. For an antiplane shear crack having no thickness (so-called β€œmathematical crack”), the crack surfaces contact each other, in reality; so the crack is electrically and magnetically contacted. Consequently, the electric and magnetic boundary conditions at the crack surfaces can be described according to so-called double permeable conditions, namely,𝐷𝑦π‘₯,0+ξ€Έ=𝐷𝑦(π‘₯,0βˆ’),𝐡𝑦π‘₯,0+ξ€Έ=𝐡𝑦(π‘₯,0βˆ’πœ™ξ€·),π‘₯,0+ξ€Έ=πœ™(π‘₯,0βˆ’ξ€·),πœ“π‘₯,0+ξ€Έ=πœ“(π‘₯,0βˆ’).(18)

Note that besides the crack surfaces, the above conditions, in fact, certainly hold at the crack-absent parts of the crack plane. Using the relations (14), it can be shown that the condition (18) may be replaced by conditions as follows:πœ’,𝑦π‘₯,0+ξ€Έ=πœ’,𝑦(π‘₯,0βˆ’),πœ‚,𝑦π‘₯,0+ξ€Έ=πœ‚,𝑦(π‘₯,0βˆ’),(19a)πœ’=𝑒15𝑀,πœ‚=π‘ž15𝑀forπ‘₯,𝑦=0Β±.(19b)

Let the constant mechanical loads and uniform electric displacement and magnetic induction or electric field and magnetic field be applied at infinity (two cases of electric and magnetic loads), and the following:𝜏I𝑦𝑧(π‘₯,𝑦)=𝜏I0,𝐷I𝑦(π‘₯,𝑦)=𝐷I0,𝐡I𝑦(π‘₯,𝑦)=𝐡I0or𝐸I𝑦(π‘₯,𝑦)=𝐸I0,𝐻I𝑦(π‘₯,𝑦)=𝐻I0𝜏,π‘₯>0,π‘¦βŸΆΒ±βˆžII𝑦𝑧(π‘₯,𝑦)=𝜏II0,𝐷II𝑦(π‘₯,𝑦)=𝐷II0,𝐡II𝑦(π‘₯,𝑦)=𝐡II0or𝐸II𝑦(π‘₯,𝑦)=𝐸II0,𝐻II𝑦(π‘₯,𝑦)=𝐻II0,π‘₯<0,π‘¦βŸΆΒ±βˆž,(20) where 𝜏I0(𝜏II0),𝐷I0(𝐷II0),𝐡I0(𝐡II0) or 𝐸I0(𝐸II0),𝐻I0(𝐻II0) are prescribed constants, a quantity with superscribes I or II that specifies the one in the PEMO-ceramic I or II, respectively. To solve the crack problem in linear elastic solids, the superposition technique is usually used. Thus, we first solve the Magnetoelectroelastic field problem without the cracks in the medium under electric, magnetic, and mechanical loads. This elementary solution is the following:πœπ½π‘¦π‘§=𝜏𝐽0,𝐷𝐽𝑦=𝐷𝐽=⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩𝐷𝐽0𝑒,caseI15𝑐44𝜏0+ξƒ©πœ€11+𝑒215𝑐44ξƒͺ𝐸0+𝑑11+𝑒15π‘ž15𝑐44𝐻0𝐽𝐡,caseII𝐽𝑦=𝐡𝐽=⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩𝐡𝐽0ξ‚Έπ‘ž,caseI15𝑐44𝜏0+𝑑11+𝑒15π‘ž15𝑐44𝐸0+ξƒ©πœ‡11+π‘ž215𝑐44ξƒͺ𝐻0𝐽,caseII(21) with 𝐽=I,II.

In addition the crack surfaces are traction-free, that is, 𝜏I𝑦𝑧(π‘₯,𝑦)=0;𝑦=0Β±,0<π‘₯<π‘Ž,(22) and owing to the symmetry one can directly write following conditions:𝑀I(π‘₯,0)=0,π‘₯>π‘Ž,𝑀II(π‘₯,0)=0,π‘₯<0.(23)

We further consider the situation when the interface under consideration is perfectly bonded, across which the displacement, stress, electric and magnetic potentials, electric displacement, and magnetic induction are continuous𝑀I(0,𝑦)=𝑀II(0,𝑦),𝜏Iπ‘₯𝑧(0,𝑦)=𝜏IIπ‘₯π‘§πœ™(0,𝑦);βˆ’βˆž<𝑦<∞,I(0,𝑦)=πœ™II(0,𝑦),𝐷Iπ‘₯(0,𝑦)=𝐷IIπ‘₯πœ“(0,𝑦);βˆ’βˆž<𝑦<∞,I(0,𝑦)=πœ“II(0,𝑦),𝐡Iπ‘₯(0,𝑦)=𝐡IIπ‘₯(0,𝑦);βˆ’βˆž<𝑦<∞.(24)

3. Method of Solution

From the symmetry of the problem, it is sufficient to consider the upper half-plane of the bi-ceramic. Consequently, for 𝑦β‰₯0, it is easily found that an appropriate solution of the problem, which satisfies the boundary conditions (19a) and (20), takes of the following form:βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘€Iπœ’(π‘₯,𝑦)I(πœ‚π‘₯,𝑦)I⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π›Ύ(π‘₯,𝑦)I𝐷I𝐡I⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣1𝑒𝑦+I15π‘žI15⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ€œβˆž0𝐴1(πœ‰)π‘’βˆ’π‘¦πœ‰+ξ€œcos(πœ‰π‘₯)π‘‘πœ‰βˆž0⎑⎒⎒⎒⎒⎣𝐡1𝐢(πœ‰)1𝐷(πœ‰)1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘’(πœ‰)βˆ’πœ‰π‘₯sin(πœ‰π‘¦)π‘‘πœ‰,(25) for π‘₯β‰₯0 andβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘€IIπœ’(π‘₯,𝑦)II(πœ‚π‘₯,𝑦)II⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π›Ύ(π‘₯,𝑦)II𝐷II𝐡II⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣1𝑒𝑦+II15π‘žII15⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ€œβˆž0𝐴2(πœ‰)π‘’βˆ’π‘¦πœ‰+ξ€œcos(πœ‰π‘₯)π‘‘πœ‰βˆž0⎑⎒⎒⎒⎒⎣𝐡2𝐢(πœ‰)2𝐷(πœ‰)2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘’(πœ‰)πœ‰π‘₯sin(πœ‰π‘¦)π‘‘πœ‰(26) for π‘₯≀0, where 𝐴𝑗,𝐡𝑗,𝐢𝑗, and 𝐷𝑗(𝑗=1,2) are unknowns to be determined from given boundary conditions and where𝛾𝐽=𝜏𝐽0+𝛼𝐽𝐷𝐽+𝛽𝐽𝐡𝐽̃𝑐𝐽44;𝐽=I,II.(27)

Furthermore with the aid of (14), one can give the components of stress, electric displacement, magnetic induction, and electric and magnetic potentials⎑⎒⎒⎒⎒⎒⎣𝜏I𝑦𝑧𝐷(π‘₯,𝑦)I𝑦(𝐡π‘₯,𝑦)Iπ‘¦βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=⎑⎒⎒⎒⎒⎣𝜏(π‘₯,𝑦)I0𝐷I𝐡I⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘I44𝑒I15π‘žI15⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ€œβˆž0πœ‰π΄1(πœ‰)π‘’βˆ’π‘¦πœ‰+ξ€œcos(πœ‰π‘₯)π‘‘πœ‰βˆž0πœ‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£Μƒπ‘I44𝐡1(πœ‰)βˆ’π›ΌI𝐢1(πœ‰)βˆ’π›½I𝐷1𝐢(πœ‰)1𝐷(πœ‰)1⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦(πœ‰)Γ—π‘’βˆ’πœ‰π‘₯βŽ‘βŽ’βŽ’βŽ£πœ™cos(πœ‰π‘¦)π‘‘πœ‰,(28)Iπœ“(π‘₯,𝑦)I(⎀βŽ₯βŽ₯⎦π‘₯,𝑦)=πΆβˆ’1I⎑⎒⎒⎒⎒⎣𝜏I0𝐷I𝐡I⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£π‘¦+0+𝛼𝑒1𝑒2𝛽𝑒2𝑒3⎀βŽ₯βŽ₯⎦IΓ—ξ€œβˆž0πœ‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΅1𝐢(πœ‰)1𝐷(πœ‰)1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘’(πœ‰)βˆ’πœ‰π‘₯sin(πœ‰π‘¦)π‘‘πœ‰(29) for π‘₯β‰₯0 and⎑⎒⎒⎒⎒⎣𝜏II𝑦𝑧𝐷(π‘₯,𝑦)II𝑦(𝐡π‘₯,𝑦)IIπ‘¦βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦=⎑⎒⎒⎒⎒⎣𝜏(π‘₯,𝑦)II0𝐷II𝐡II⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘II44𝑒II15π‘žII15⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ€œβˆž0πœ‰π΄2(πœ‰)π‘’βˆ’π‘¦πœ‰+ξ€œcos(πœ‰π‘₯)π‘‘πœ‰βˆž0πœ‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£Μƒπ‘II44𝐡1(πœ‰)βˆ’π›ΌII𝐢2(πœ‰)βˆ’π›½II𝐷2𝐢(πœ‰)2𝐷(πœ‰)2⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦(πœ‰)Γ—π‘’πœ‰π‘₯βŽ‘βŽ’βŽ’βŽ£πœ™cos(πœ‰π‘¦)π‘‘πœ‰,(30)IIπœ“(π‘₯,𝑦)II⎀βŽ₯βŽ₯⎦(π‘₯,𝑦)=πΆβˆ’1II⎑⎒⎒⎒⎒⎣𝜏II0𝐷II𝐡II⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£π‘¦+0+𝛼𝑒1𝑒2𝛽𝑒2𝑒3⎀βŽ₯βŽ₯⎦IIΓ—ξ€œβˆž0πœ‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΅2𝐢(πœ‰)2(π·πœ‰)2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘’(πœ‰)πœ‰π‘₯sin(πœ‰π‘¦)π‘‘πœ‰(31) for π‘₯≀0 and⎑⎒⎒⎒⎒⎣𝜏Iπ‘₯𝑧𝐷(π‘₯,𝑦)Iπ‘₯(𝐡π‘₯,𝑦)Iπ‘₯⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘(π‘₯,𝑦)=βˆ’I44𝑒I15π‘žI15⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ€œβˆž0πœ‰π΄1(πœ‰)π‘’βˆ’π‘¦πœ‰βˆ’ξ€œsin(πœ‰π‘₯)π‘‘πœ‰βˆž0πœ‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£Μƒπ‘I44𝐡1(πœ‰)βˆ’π›ΌI𝐢1(πœ‰)βˆ’π›½I𝐷1𝐢(πœ‰)1𝐷(πœ‰)1⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦(πœ‰)Γ—π‘’βˆ’πœ‰π‘₯sin(πœ‰π‘¦)π‘‘πœ‰(32) for π‘₯β‰₯0 and⎑⎒⎒⎒⎒⎣𝜏IIπ‘₯𝑧𝐷(π‘₯,𝑦)IIπ‘₯(𝐡π‘₯,𝑦)IIπ‘₯⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘(π‘₯,𝑦)=βˆ’II44𝑒II15π‘žII15⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ€œβˆž0πœ‰π΄2(πœ‰)π‘’βˆ’π‘¦πœ‰+ξ€œsin(πœ‰π‘₯)π‘‘πœ‰βˆž0πœ‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£Μƒπ‘II44𝐡2(πœ‰)βˆ’π›ΌII𝐢2(πœ‰)βˆ’π›½II𝐷2𝐢(πœ‰)2𝐷(πœ‰)2⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦(πœ‰)Γ—π‘’πœ‰π‘₯sin(πœ‰π‘¦)π‘‘πœ‰(33) for π‘₯≀0.

Now, application of the continuity conditions (24), at the interface π‘₯=0 to (25) to (33), yields 𝜏I0+𝛼I𝐷I+𝛽I𝐡Ĩ𝑐I44=𝜏II0+𝛼II𝐷II+𝛽II𝐡IĨ𝑐II44,𝐢(34)βˆ’1I⎑⎒⎒⎒⎒⎣𝜏I0𝐷I𝐡I⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=πΆβˆ’1II⎑⎒⎒⎒⎒⎣𝜏II0𝐷II𝐡II⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,πΆπ½βˆ’1=1̃𝑐𝐽44βŽ‘βŽ’βŽ’βŽ£π›Όπ›Ό2+̃𝑐I44𝑒1𝛼𝛽+̃𝑐I44𝑒2𝛽𝛼𝛽+̃𝑐I44𝑒2𝛽2+̃𝑐I44𝑒3⎀βŽ₯βŽ₯⎦𝐽(35) β€‰βˆ’ξ€ΊΜƒπ‘I44𝐡1(πœ‰)βˆ’π›ΌI𝐢1(πœ‰)βˆ’π›½I𝐷1ξ€»=ξ€Ί(πœ‰)̃𝑐II44𝐡2(πœ‰)βˆ’π›ΌII𝐢2(πœ‰)βˆ’π›½II𝐷2ξ€»,𝐢(πœ‰)1(πœ‰)=βˆ’πΆ2(πœ‰),𝐷1(πœ‰)=βˆ’π·2(πœ‰)(36) β€‰βŽ‘βŽ’βŽ’βŽ£π›ΌII𝑒II1𝑒II2𝛽II𝑒II2𝑒II3⎀βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣𝐡2𝐢(πœ‰)2(π·πœ‰)2⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ£π›Ό(πœ‰)I𝑒I1𝑒I2𝛽I𝑒I2𝑒I3⎀βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣𝐡1𝐢(πœ‰)1(π·πœ‰)1⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦(πœ‰).(37)

The first two equations, that is, (34) and (35), give three constraints for applied remote electro-magneto-mechanical loadings, from which we may determine the loadings of ceramics II, namely, 𝜏II0, 𝐷II, and 𝐡II by means of loadings of ceramics I, namely, 𝜏I0, 𝐷I, and 𝐡I. In other words, in order to guarantee the continuity of all physical quantities at the perfectly bonded interface, applied electro-magneto-mechanical loadings must obey the relations (34) and (35). The five equations (36) and (37) give the constraints with respect to unknown functions; that is, the disturbed field, due to the presence of a cracks, requires to satisfy those equations.

From the condition (23)2 along with (26), one gets 𝐴2(πœ‰)=0.(38)

Continuity of 𝑀(π‘₯,𝑦) at the interface π‘₯=0 requiresξ€œβˆž0𝐡2(πœ‰)βˆ’π΅1ξ€»ξ€œ(πœ‰)sin(πœ‰π‘¦)π‘‘πœ‰=∞0𝐴1(πœ‰)π‘’βˆ’πœ‰π‘¦π‘‘πœ‰(39) so that𝐡2(πœ‰)βˆ’π΅12(πœ‰)=πœ‹ξ€œβˆž0𝐴1πœ‰(πœ‚)πœ‰2+πœ‚2π‘‘πœ‚,(40) sinceξ€œβˆž0π‘’βˆ’πœ‚π‘¦πœ‰sin(πœ‰π‘¦)𝑑𝑦=πœ‰2+πœ‚2.(41)

The result (40) in connection with (36) and (37) yields𝐡12(πœ‰)=βˆ’πœ‹Ĩ𝑐II44𝑒Δ+II3+𝑒I3𝛼II𝛼IIβˆ’π›ΌIξ€Έ+𝑒II1+𝑒I1𝛽II𝛽IIβˆ’π›½I̃𝑐I44+̃𝑐II44𝑒Δ+II3+𝑒I3𝛼IIβˆ’π›ΌIξ€Έ2+𝑒II1+𝑒I1𝛽IIβˆ’π›½Iξ€Έ2+2πœ‹I𝑒II2+𝑒I2𝛼I𝛽II+𝛽I𝛼II+2𝛼II𝛽II̃𝑐I44+̃𝑐II44𝑒Δ+II3+𝑒I3𝛼IIβˆ’π›ΌIξ€Έ2+𝑒II1+𝑒I1𝛽IIβˆ’π›½Iξ€Έ2,𝐡22(πœ‰)=πœ‹Ĩ𝑐I44ξ€·π‘’Ξ”βˆ’II3+𝑒I3𝛼I𝛼IIβˆ’π›ΌIξ€Έβˆ’ξ€·π‘’II1+𝑒I1𝛽I𝛽IIβˆ’π›½I̃𝑐I44+̃𝑐II44𝑒Δ+II3+𝑒I3𝛼IIβˆ’π›ΌIξ€Έ2+𝑒II1+𝑒I1𝛽IIβˆ’π›½Iξ€Έ2+2πœ‹I𝑒II2+𝑒I2𝛼I𝛽II+𝛽I𝛼II+2𝛼II𝛽II̃𝑐I44+̃𝑐II44𝑒Δ+II3+𝑒I3𝛼IIβˆ’π›ΌIξ€Έ2+𝑒II1+𝑒I1𝛽IIβˆ’π›½Iξ€Έ2,𝐢1(πœ‰)=𝐡1𝛽(πœ‰)I𝑒II2+𝑒I2ξ€Έβˆ’π›ΌI𝑒II3+𝑒I3ξ€ΈΞ”βˆ’π΅2𝛽(πœ‰)II𝑒II2+𝑒I2ξ€Έβˆ’π›ΌII𝑒II3+𝑒I3ξ€ΈΞ”,𝐷1(πœ‰)=βˆ’π΅1𝛽(πœ‰)I𝑒II1+𝑒I1ξ€Έβˆ’π›ΌI𝑒II2+𝑒I2ξ€ΈΞ”+𝐡2𝛽(πœ‰)II𝑒II1+𝑒I1ξ€Έβˆ’π›ΌII𝑒II2+𝑒I2ξ€ΈΞ”,(42)where𝑒Δ=II1+𝑒I1𝑒II3+𝑒I3ξ€Έβˆ’ξ€·π‘’II2+𝑒I2ξ€Έ2,(43)ξ€œI=∞0𝐴1πœ‰(πœ‚)πœ‰2+πœ‚2π‘‘πœ‚.(44)

In the special cases, we obtain that

for both piezoelectric materials 𝐡12(πœ‰)=βˆ’πœ‹I𝑐II44ξ€·πœ€I11+πœ€II11ξ€Έ+𝑒II15𝑒I15+𝑒II15𝑐I44+𝑐II44πœ€ξ€Έξ€·I11+πœ€II11ξ€Έ+𝑒I15+𝑒II15ξ€Έ2,𝐡22(πœ‰)=πœ‹I𝑐I44ξ€·πœ€I11+πœ€II11ξ€Έ+𝑒I15𝑒I15+𝑒II15𝑐I44+𝑐II44πœ€ξ€Έξ€·I11+πœ€II11ξ€Έ+𝑒I15+𝑒II15ξ€Έ2,𝐢12(πœ‰)=βˆ’πœ‹I𝑒I15𝑐II44πœ€II11+𝑒II15ξ€Έ2+𝑒II15𝑐I44πœ€I11+𝑒I15ξ€Έ2𝑐I44+𝑐II44πœ€ξ€Έξ€·I11+πœ€II11ξ€Έ+𝑒I15+𝑒II15ξ€Έ2,𝐷1(πœ‰)=0,(45) for both piezomagnetic materials𝐡12(πœ‰)=βˆ’πœ‹I𝑐II44ξ€·πœ‡I11+πœ‡II11ξ€Έ+π‘žII15ξ€·π‘žI15+π‘žII15𝑐I44+𝑐II44πœ‡ξ€Έξ€·I11+πœ‡II11ξ€Έ+ξ€·π‘žI15+π‘žII15ξ€Έ2,𝐡22(πœ‰)=πœ‹I𝑐I44ξ€·πœ‡I11+πœ‡II11ξ€Έ+π‘žI15ξ€·π‘žI15+π‘žII15𝑐I44+𝑐II44πœ‡ξ€Έξ€·I11+πœ‡II11ξ€Έ+ξ€·π‘žI15+π‘žII15ξ€Έ2,𝐢1𝐷(πœ‰)=0,1(2πœ‰)=βˆ’πœ‹Iπ‘žI15𝑐II44πœ‡II11+ξ€·π‘žII15ξ€Έ2+π‘žII15𝑐I44πœ‡I11+ξ€·π‘žI15ξ€Έ2𝑐I44+𝑐II44πœ‡ξ€Έξ€·I11+πœ‡II11ξ€Έ+ξ€·π‘žI15+π‘žII15ξ€Έ2.(46)

The formulae (45) are equivalent to that derived by Li and Wang [9] who solved the problem of two bonded dissimilar piezoelectric media with an antiplane shear crack perpendicular to and terminated at the interface. Next, we denote that𝑔(π‘₯)=πœ•π‘€Iξ€·π‘₯,0+ξ€Έπœ•π‘₯.(47)

From the boundary conditions (23), 𝑔(π‘₯) should satisfy the single-value displacement constraint condition, that is, ξ€œπ‘Ž0𝑔(π‘₯)𝑑π‘₯=0.(48)

Utilizing (25)1 in (23) leads to𝑀Iξ€œ(π‘₯,0)=∞0𝐴1(πœ‰)cos(πœ‰π‘₯)π‘‘πœ‰=0,π‘₯>π‘Ž(49) from which together with (47), by using the inverse Fourier transform, can be deduced𝐴12(πœ‰)=βˆ’ξ€œπœ‹πœ‰π‘Ž0𝑔(𝑑)sin(πœ‰π‘‘)𝑑𝑑.(50)

Now, we calculate the following:ξ€œβˆž0πœ‰πœ‰2+πœ‚2𝐴12(πœ‚)π‘‘πœ‚=βˆ’πœ‹πœ‰ξ€œπ‘Ž0ξ€œπ‘”(𝑑)π‘‘π‘‘βˆž0sin(πœ‚π‘‘)πœ‚ξ€·πœ‰2+πœ‚2ξ€Έπ‘‘πœ‚.(51)

Using the resultξ€œβˆž0sin(πœ‚π‘‘)πœ‚ξ€·πœ‰2+πœ‚2ξ€Έπœ‹ξ€·π‘‘πœ‚=1βˆ’π‘’βˆ’πœ‰π‘‘ξ€Έ2πœ‰2,(52) we find with the use of (48) thatξ€œβˆž0πœ‰πœ‰2+πœ‚2𝐴1ξ€œ(πœ‚)π‘‘πœ‚=π‘Ž0π‘’βˆ’πœ‰π‘‘πœ‰π‘”(𝑑)𝑑𝑑.(53)

Substitution of (53) into (42) yields the expressions for 𝐡1(πœ‰),𝐡2(πœ‰),𝐢1(πœ‰), and 𝐷1(πœ‰) in terms of 𝑔(π‘₯).

From fraction-free condition (22) from (28)1, one can deriveξ€œβˆž0πœ‰ξ€Ίπ‘I44𝐴1βˆ’ξ€·(πœ‰)cos(πœ‰π‘₯)̃𝑐I44𝐡1(πœ‰)βˆ’π›ΌI𝐢1(πœ‰)βˆ’π›½I𝐷1(ξ€Έπ‘’πœ‰)βˆ’πœ‰π‘₯ξ€»π‘‘πœ‰=𝜏I0.(54)

Substituting (50) and (42) with the use of (53) into (54), we have with the help of known integrals2πœ‹ξ€œβˆž01sin(πœ‰π‘‘)cos(πœ‰π‘₯)π‘‘πœ‰=πœ‹ξ‚€1+1π‘‘βˆ’π‘₯,ξ€œπ‘‘+π‘₯∞0π‘’βˆ’πœ‰(𝑑+π‘₯)1π‘‘πœ‰=𝑑+π‘₯;𝑑+π‘₯>0(55) the following singular integral equation with generalized Cauchy kernel for 𝑔(𝑑):1πœ‹ξ€œπ‘Ž0ξ‚€1+πœ†π‘‘βˆ’π‘₯ξ‚πœπ‘‘+π‘₯𝑔(𝑑)𝑑𝑑=βˆ’I0𝑐I44;0<π‘₯<π‘Ž,(56) whereπœ†=1βˆ’2̃𝑐II44𝑒Δ+II3+𝑒I3𝛼II𝛼IIβˆ’π›ΌIξ€Έ+𝑒II1+𝑒I1𝛽II𝛽IIβˆ’π›½Iξ€Έβˆ’ξ€·π‘’II2+𝑒I2𝛼I𝛽II+𝛽I𝛼II+2𝛼II𝛽II̃𝑐I44+̃𝑐II44𝑒Δ+II3+𝑒I3𝛼IIβˆ’π›ΌIξ€Έ2+𝑒II1+𝑒I1𝛽IIβˆ’π›½Iξ€Έ2×̃𝑐I44𝑐I44βˆ’ξ€·π‘’II3+𝑒I3𝛼I𝛼IIβˆ’π›ΌIξ€Έ+𝑒II1+𝑒I1𝛽I𝛽IIβˆ’π›½Iξ€Έβˆ’ξ€·π‘’II2+𝑒I2𝛼I𝛽II+𝛽I𝛼IIβˆ’2𝛼I𝛽I𝑐I44Δ𝑒+2II3+𝑒I3𝛼I𝛼II+𝑒II1+𝑒I1𝛽I𝛽IIβˆ’ξ€·π‘’II2+𝑒I2𝛼I𝛽II+𝛽I𝛼II𝑐I44Ξ”.(57)

For both piezoelectric materials, πœ† is obtained as follows:ξ€·π‘πœ†=I44βˆ’π‘II44πœ€ξ€Έξ€·I11+πœ€II11ξ€Έ+𝑒I15ξ€Έ2βˆ’ξ€·π‘’II15ξ€Έ2+2𝑒I15𝑒II15βˆ’π‘’I15𝑐II44/𝑐I44𝑐I44+𝑐II44πœ€ξ€Έξ€·I11+πœ€II11ξ€Έ+𝑒I15+𝑒II15ξ€Έ2.(58)

The value of πœ† for both piezomagnetic materials is obtained from formula (58) if we replace πœ€11 by πœ‡11 and 𝑒15by π‘ž15. It is noted that, in a usual integral equation with Cauchy kernel, other kernels except Cauchy kernel are continuous over the entire interval involved. In addition to the singularity of the Cauchy kernel terms 1/(π‘‘βˆ’π‘₯) as 𝑑→π‘₯ in (56), the other term πœ†/(𝑑+π‘₯) is also unbonded as 𝑑,π‘₯β†’0 simultaneously. Particularly for two elastic dielectric, meaning 𝑒15=0, and diamagnetic, meaning π‘ž15=0, elastic field and electric field, and elastic field and magnetic field are not coupled as well as when 𝑑11=0, the electromagnetic field does not occur. In this case, πœ† reduces toπ‘πœ†=I44βˆ’π‘II44𝑐I44+𝑐II44.(59) Then the integral equation is simplified to1πœ‹ξ€œπ‘Ž0ξ‚€1+πœ†π‘‘βˆ’π‘₯ξ‚πœπ‘‘+π‘₯𝑔(𝑑)𝑑𝑑=βˆ’0𝑐44.(60)

This equation is equivalent to that derived by Cook and Erdogan [6] and Erdogan and Cook [7], who were apparently the first to publish the solution of an antiplane shear crack terminating at the interface of two joined purely elastic media.

4. Magnetoelectroelastic Field

4.1. Solution of the Singular Integral Equation

Based on the result derived by Bueckner [10], the desired solution for 𝑔(𝑑) of (56) subjected to (48) can be obtained as follows:πœπ‘”(π‘₯)=I02𝑐I44Γ—π‘₯sin(πœ‹π›Ό/2)ξƒ¬ξƒ©βˆšπ‘Ž+π‘Ž2βˆ’π‘₯2ξƒͺπ›Όξƒ©π›Όπ‘Žβˆšπ‘Ž2βˆ’π‘₯2ξƒͺ+π‘₯+1βˆšπ‘Ž+π‘Ž2βˆ’π‘₯2ξƒͺβˆ’π›Όξƒ©π›Όπ‘Žβˆšπ‘Ž2βˆ’π‘₯2βˆ’1ξƒͺξƒ­(61) for 0<π‘₯<π‘Ž withcos(πœ‹π›Ό)=βˆ’πœ†,(62) where 0<𝛼<1.

Once 𝑔(𝑑) is determined the crack tearing displacement can be obtained by the following integrations:𝑀Iξ€·π‘₯,0+ξ€Έ=ξ€œπ‘₯0πœπ‘”(π‘₯)𝑑π‘₯=βˆ’I0π‘₯2𝑐I44Γ—π‘₯sin(πœ‹π›Ό/2)ξƒ¬ξƒ©βˆšπ‘Ž+π‘Ž2βˆ’π‘₯2ξƒͺπ›Όβˆ’ξƒ©π‘₯βˆšπ‘Ž+π‘Ž2βˆ’π‘₯2ξƒͺβˆ’π›Όξƒ­.0≀π‘₯β‰€π‘Ž(63)

4.2. Crack Tearing Displacement

Expanding the expression (63) near the crack tips yields the asymptotic crack tearing displacement as𝑀I𝜏(π‘₯,0)=I0𝑐I44π›Όβˆšsin(πœ‹π›Ό/2)𝑀2π‘Ž(π‘Žβˆ’π‘₯)+𝑂(π‘Ÿ);π‘Ÿ=π‘Žβˆ’π‘₯β‰ˆ0I𝜏(π‘₯,0)=I0π‘Žπ›Ό2𝑐I44π‘₯sin(πœ‹π›Ό/2)1βˆ’π›Ό+𝑂(π‘Ÿ);π‘Ÿ=π‘₯β‰ˆ0(64) at the right and left crack tip.

Here 𝑂(π‘Ÿ) denotes the infinitesimal terms compared to π‘Ÿ, π‘Ÿ being the distance from the crack tip. Only for 𝛼=1/2 the behaviours of the crack tearing displacement for both tips are the same.

4.3. Asymptotic Crack-Tip Field

Antiplane shear crack and in-plane electric displacement and magnetic induction may be deduced by evaluating the following integrals:𝜏I𝑦𝑧1(π‘₯,0)=πœ‹π‘I44ξ€œπ‘Ž0ξ‚€1+πœ†π‘‘βˆ’π‘₯𝑑+π‘₯𝑔(𝑑)𝑑𝑑+𝜏I0,𝐷I𝑦1(π‘₯,0)=πœ‹π‘’I15ξ€œπ‘Ž0ξ‚΅1+π‘‘βˆ’π‘₯1βˆ’2πœ†π·ξ‚Ά1+𝑑𝑔(𝑑)𝑑𝑑+𝐷I,𝐡I𝑦1(π‘₯,0)=πœ‹π‘žI15ξ€œπ‘Ž0ξ‚΅1+π‘‘βˆ’π‘₯1βˆ’2πœ†π΅ξ‚Ά1+𝑑𝑔(𝑑)𝑑𝑑+𝐡I,(65) for π‘₯>π‘Ž and𝜏II𝑦𝑧(π‘₯,0)=1βˆ’πœ†πœ‹π‘I44ξ€œπ‘Ž0𝑔(𝑑)π‘‘βˆ’π‘₯𝑑𝑑+𝜏II0,𝐷II𝑦(π‘₯,0)=2πœ†π·πœ‹π‘’I15ξ€œπ‘Ž0𝑔(𝑑)π‘‘βˆ’π‘₯𝑑𝑑+𝐷II,𝐡II𝑦(π‘₯,0)=2πœ†π΅πœ‹π‘žI15ξ€œπ‘Ž0𝑔(𝑑)π‘‘βˆ’π‘₯𝑑𝑑+𝐡II(66) for π‘₯<0, where 1βˆ’πœ† is defined by (57) and𝑒I15πœ†π·=Δ̃𝑐II44𝛼I+̃𝑐I44𝛼II𝑒II3+𝑒I3ξ€Έβˆ’ξ€·Μƒπ‘II44𝛽I+̃𝑐I44𝛽II𝑒II2+𝑒I2+𝑒II3+𝑒I3×𝑒II2+𝑒I2𝛼IIβˆ’π›ΌI𝛼I𝛽II+𝛽I𝛼II+2𝛼II𝛽IIξ€Έ+𝑒II3+𝑒I3𝑒II1+𝑒I1𝛽IIβˆ’π›½I𝛼I𝛽IIβˆ’π›½I𝛼IIξ€Έ+𝑒II2+𝑒I2ξ€Έ2𝛽IIβˆ’π›½I𝛼I𝛽II+𝛽I𝛼II+2𝛼II𝛽II×Δ̃𝑐I44+̃𝑐II44𝑒Δ+II3+𝑒I3𝛼IIβˆ’π›ΌIξ€Έ2+𝑒II1+𝑒I1𝛽IIβˆ’π›½Iξ€Έ2ξ‚„ξ‚‡βˆ’1,π‘žI15πœ†π΅=Δ̃𝑐II44𝛽I+̃𝑐I44𝛽II𝑒II1+𝑒I1ξ€Έβˆ’ξ€·Μƒπ‘II44𝛼I+̃𝑐I44𝛼II𝑒II2+𝑒I2βˆ’ξ€·π‘’ξ€Έξ€»II1+𝑒I1×𝑒II2+𝑒I2𝛽IIβˆ’π›½I𝛼I𝛽II+𝛽I𝛼II+2𝛼II𝛽IIξ€Έβˆ’ξ€·π‘’II2+𝑒I2𝑒II1+𝑒I1𝛽IIβˆ’π›½I𝛼I𝛽IIβˆ’π›½I𝛼IIξ€Έβˆ’ξ€·π‘’II2+𝑒I2ξ€Έ2𝛼IIβˆ’π›ΌI𝛼I𝛽II+𝛽I𝛼II+2𝛼II𝛽II×Δ̃𝑐I44+̃𝑐II44𝑒Δ+II3+𝑒I3𝛼IIβˆ’π›ΌIξ€Έ2+𝑒II1+𝑒I1𝛽IIβˆ’π›½Iξ€Έ2ξ‚„ξ‚‡βˆ’1.(67)

For both piezoelectric or piezomagnetic materials, (67) give𝑒I15πœ†π·=𝑒I15𝑐II44πœ€II11+𝑒II15ξ€Έ2+𝑒II15𝑐I44πœ€I11+𝑒I15ξ€Έ2𝑐I44+𝑐II44πœ€ξ€Έξ€·I11+πœ€II11ξ€Έ+𝑒I15+𝑒II15ξ€Έ2,π‘žI15πœ†π΅=0(68) or𝑒I15πœ†π·π‘ž=0,I15πœ†π΅=π‘žI15𝑐II44πœ‡II11+ξ€·π‘žII15ξ€Έ2+π‘žII15𝑐I44πœ‡I11+ξ€·π‘žI15ξ€Έ2𝑐I44+𝑐II44πœ‡ξ€Έξ€·I11+πœ‡II11ξ€Έ+ξ€·π‘žI15+π‘žII15ξ€Έ2.(69)

The analytical expressions for physical quantities may be obtained substituting the solution (61) into (65) and (66). We omit full solution and pay our attentions to the asymptotic crack-tip field. This is very interest from the view point of fracture mechanics. From (61), one can write out the singular behaviour of the function 𝑔(π‘₯) near the point π‘₯=0 and π‘₯=π‘Ž by the following asymptotic expressions:πœπ‘”(π‘₯)=βˆ’I02𝑐I44𝛼sin(πœ‹π›Ό/2)2π‘Žπœπ‘Žβˆ’π‘₯+𝑂(1);π‘₯β‰ˆπ‘Žβˆ’0,(70)𝑔(π‘₯)=βˆ’I02𝑐I44π›Όβˆ’1ξ‚Έsin(πœ‹π›Ό/2)2(π‘Žβˆ’π‘₯)π‘₯𝛼+𝑂(1);π‘₯β‰ˆ0+0,(71) where 𝑂(1) stands for nonsingular terms.

Now we define the intensity factor at the right crack tip in the homogeneous solid and the left crack tip at the interface of a bimedium asπΎπ‘žhom=limπ‘₯β†’π‘Ž+√2πœ‹(π‘₯βˆ’π‘Ž)π‘žIξ€·π‘₯,0+ξ€Έ,πΎπ‘žint=limπ‘₯β†’0βˆ’(βˆ’2πœ‹π‘₯)π›Όπ‘žIIξ€·π‘₯,0+ξ€Έ,(72) respectively, where π‘ž stands for one of πœπ‘¦π‘§,𝛾𝑦𝑧,𝐷𝑦,𝐡𝑦,𝐸𝑦, and 𝐻𝑦.

4.3.1. Magnetoelectroelastic Field near the Crack Tip in the Homogeneous PEMO-Elastic Ceramics

Using the integral1πœ‹ξ€œπ‘Ž01(βˆšπ‘‘βˆ’π‘₯)2π‘Žβˆ’π‘‘π‘‘π‘‘=βˆ’πœ‹βˆšπ‘₯βˆ’π‘Žtanβˆ’1ξ‚™π‘Žπ‘₯βˆ’π‘Ž,π‘₯>π‘Ž,(73) we obtain from (65)1 that𝜏I𝑦𝑧1(π‘₯,0)=πœ‹π‘I44ξ€œπ‘Ž0𝑔(𝑑)πΎπ‘‘βˆ’π‘₯𝑑𝑑+𝑂(1)=𝜏hom√2πœ‹(π‘₯βˆ’π‘Ž),(74) where𝐾𝜏hom=π›Όπœsin(πœ‹π›Ό/2)Iπ‘œβˆšπœ‹π‘Ž(75) the stress intensity factor at the right crack tip. Other field intensity factors are related to 𝐾𝜏hom as follows:𝐾𝛾hom=1𝑐I44𝐾𝜏hom,𝐾𝐷hom=𝑒I15𝑐I44𝐾𝜏hom,𝐾𝐡hom=π‘žI15𝑐I44𝐾𝜏hom,πΎπœ™hom=πΎπœ“hom=𝐾𝐸hom=𝐾𝐻hom=0.(76)

For the crack tip in homogeneous PEMO-elastic medium the elastic, electric, and magnetic fields still exhibit an inverse square-root singularity at the crack tip. Application of electric and magnetic fields does not alter the stress intensity factors. The stress intensity factor depends on the material properties of two PEMO-elastic ceramics involved since it is governed by (75) and 𝛼 by (62). The intensity factors 𝐾𝛾hom,𝐾𝐷hom, and 𝐾𝐡hom are related to 𝐾𝜏hom and also depend on the material properties, as shown in (76).

4.3.2. Magnetoelectroelastic Field near the Crack Tip at the Interface

Using the known result [11],1πœ‹ξ€œπ‘Ž01(ξ‚€π‘‘βˆ’π‘₯)π‘Žβˆ’π‘‘π‘‘ξ‚π›Ό1𝑑𝑑=sin(πœ‹π›Ό)π‘₯βˆ’π‘Žπ‘₯ξ‚π›Όξ‚„βˆ’1,π‘₯<0(77) putting (71) into (66) and using (77), we obtain the asymptotic expressions for the antiplane shear stress and in-plane electric displacement and magnetic induction, as well as elastic strain, electric and magnetic field, ahead on the left crack tip at the interface as follows:ξ€ΊπΎπœint;𝐾𝐷int;𝐾𝐡int;𝐾𝛾int;𝐾𝐸int;𝐾𝐻intξ€»=√2(1βˆ’π›Ό)(√1+πœ†)𝜏1βˆ’πœ†I0𝑐I44(4πœ‹π‘Ž)𝛼×𝑐I441βˆ’πœ†2;𝑒I15πœ†π·;π‘žI15πœ†π΅;πœ†π›Ύ;πœ†πΈ;πœ†π»ξ‚„,(78) where the identity is used as follows:ξ‚€sin(πœ‹π›Ό)sinπœ‹π›Ό2=ξ‚™(1+πœ†)1βˆ’πœ†2(79) and whereπœ†π›Ύ=̃𝑐I44ξ€·π‘’Ξ”βˆ’II3+𝑒I3𝛼I𝛼IIβˆ’π›ΌIξ€Έβˆ’ξ€·π‘’II1+𝑒I1𝛽I𝛽IIβˆ’π›½Iξ€Έ+𝑒II2+𝑒I2𝛼I𝛽II+𝛽I𝛼II+2𝛼II𝛽II×̃𝑐I44+̃𝑐II44𝑒Δ+II3+𝑒I3𝛼IIβˆ’π›ΌIξ€Έ2+𝑒II1+𝑒I1𝛽IIβˆ’π›½Iξ€Έ2ξ‚„βˆ’1πœ†πΈ=ξ€½πœ†π›Ύξ€Ίπ›ΌI𝛼Δ+IIβˆ’π›ΌI𝑒I1𝑒II3+𝑒I3ξ€Έβˆ’π‘’I2𝑒II2+𝑒I2βˆ’ξ€·π›½ξ€Έξ€ΈIIβˆ’π›½I𝑒I1𝑒II2βˆ’π‘’I2𝑒II1ξ€Έξ€»βˆ’π›ΌII𝑒I1𝑒II3+𝑒I3ξ€Έβˆ’π‘’I2𝑒II2+𝑒I2ξ€Έξ€Έ+𝛽II𝑒I1𝑒II2βˆ’π‘’I2𝑒II1Ξ”ξ€Έξ€Ύβˆ’1πœ†π»=ξ€½πœ†π›Ύξ€Ίπ›½I𝛽Δ+IIβˆ’π›½I𝑒I3𝑒II1+𝑒I1ξ€Έβˆ’π‘’I2𝑒II2+𝑒I2βˆ’ξ€·π›Όξ€Έξ€ΈIIβˆ’π›ΌI𝑒I2𝑒II3βˆ’π‘’I3𝑒II2ξ€Έξ€»βˆ’π›½II𝑒I2𝑒II2+𝑒I2ξ€Έβˆ’π‘’I3𝑒II1+𝑒I1ξ€Έξ€Έ+𝛼II𝑒I2𝑒II3βˆ’π‘’I3𝑒II2Ξ”ξ€Έξ€Ύβˆ’1(80) for PEMO-elastic bimaterial andπœ†π›Ύ=𝑐I44ξ€·πœ€I11+πœ€II11ξ€Έ+𝑒I15𝑒I15+𝑒II15𝑐I44+𝑐II44πœ€ξ€Έξ€·I11+πœ€II11ξ€Έ+𝑒I15+𝑒II15ξ€Έ2,πœ†πΈ=𝑐II44𝑒I15βˆ’π‘I44𝑒II15𝑐I44+𝑐II44πœ€ξ€Έξ€·I11+πœ€II11ξ€Έ+𝑒I15+𝑒II15ξ€Έ2,πœ†π»=0(81) for piezoelectric bi-material andπœ†π›Ύ=𝑐I44ξ€·πœ‡I11+πœ‡II11ξ€Έ+π‘žI15ξ€·π‘žI15+π‘žII15𝑐I44+𝑐II44πœ‡ξ€Έξ€·I11+πœ‡II11ξ€Έ+ξ€·π‘žI15+π‘žII15ξ€Έ2,πœ†πΈπœ†=0,𝐻=𝑐II44π‘žI15βˆ’π‘I44π‘žII15𝑐I44+𝑐II44πœ‡ξ€Έξ€·I11+πœ‡II11ξ€Έ+ξ€·π‘žI15+π‘žII15ξ€Έ2(82) for piezomagnetic bi-material.

Note that for piezoelectric bi-material, we haveπœ†πœ=1βˆ’πœ†2=𝑐I44ξ€·πœ€I11+πœ€II11ξ€Έ+𝑒II15ξ€Έ2+𝑒I15ξ€Έ2𝑐II44/𝑐I44𝑐I44+𝑐II44πœ€ξ€Έξ€·I11+πœ€II11ξ€Έ+𝑒I15+𝑒II15ξ€Έ2.(83)

The material parameters for piezoelectric ceramics coincide, in general, with the ones derived by Li and Wang [9]. But in πœ†, defined exactly by (58), the fourth term in numerator of (58) is omitted in (46) of Li and Wang paper. In consequence, the conclusions in Table 2 of Li and Wang paper that πœ† vanishes also in the case of ceramics poled in opposite direction are incorrect. The formula (58) shows that only for two bonded piezoelectric ceramics with 𝑐44 unchanged poled in the same direction (not opposite) the field singularity at the interface crack tip maintains the inverse square root singularity, since in this case is πœ†=0 and 𝛼=1/2. The parameter πœ†πΈ in this paper has opposite sign to that presented by Li and Wang. This gives that for πœŒπ‘>1(𝑐II44>𝑐I44) meaning that piezoelectric ceramic II is more stiffer that piezoelectric ceramic I (𝑒II15=𝑒I15), in this case πœ†πΈ>0, so stands also 𝐾𝐸int>0, and 𝐾𝐸int increases with πœŒπ‘. Also it is seen that 𝐾𝐸int decreases with the ratio πœŒπ‘’ of 𝑒II15 to 𝑒I15. In the paper Li and Wang [9], the conclusions, associated with 𝐾𝐸int, are inverse. The presented conclusions are consistent with physical consideration. The field intensity factors must satisfy the constitutive equations𝐾𝜏=𝑐II44πΎπ›Ύβˆ’π‘’II15𝐾𝐸,𝐾𝐷=𝑒II15𝐾𝛾+πœ€II11𝐾𝐸,(84) or material parameters must satisfy the equivalent equations𝑐I44πœ†πœ=𝑐II44πœ†π›Ύβˆ’π‘’II15πœ†πΈ,𝑒I15πœ†π·=𝑒II15πœ†π›Ύ+πœ€II11πœ†πΈ.(85)

It is easily verified that both constitutive relations (85) are satisfied by the coefficients defined by (68), (81), and (83). In general, for Magnetoelectroelastic ceramic, the field intensity factors must satisfy the constitutive equations⎑⎒⎒⎒⎒⎣𝐾𝜏intπΎπœ™intπΎπœ“int⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£Μƒπ‘44βˆ’π›Όβˆ’π›½π›Όπ‘’1𝑒2𝛽𝑒2𝑒3⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πΎπ‘€πΎπ·πΎπ΅βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦,(86) as shown in (14). Of course, we have πΎπœ™int=βˆ’πΎπΈint and πΎπœ“int=βˆ’πΎπ»int.

4.4. The Energy Release Rate

For magnetoelectrically permeable crack, the energy release rates are very important to evaluate the behaviours of crack tips. In accordance with the definition of the energy release rate proposed by [12] (the virtual crack closure integral), the energy release rate can finally be derived as1𝐺=2𝑐I44ξ‚Έξ€·πΎπœhomξ€Έ2+ξ‚€πœ‹π‘Ž21βˆ’2π›Όξ€·πΎπœintξ€Έ2ξ‚Ή=𝐺hom12ξ‚ƒξ€·π‘˜πœhomξ€Έ2+ξ€·π‘˜πœintξ€Έ2ξ‚„,(87) where𝐺hom=ξ€·πœπœ‹π‘ŽI0ξ€Έ22𝑐I44π‘˜,(88)𝜏hom=𝐾𝜏hom𝜏I0βˆšπ‘˜πœ‹π‘Ž/2,(89)𝜏int=𝐾𝜏int𝜏I0(πœ‹π‘Ž/2)𝛼(90) are the energy release rate for homogeneous material (no bi-material) and normalized stress intensity factors at right and left crack tip. One interesting observation from equation (87) is that though the energy release rate, 𝐺, is independent on the applied electric-magnetic load, it is affected by electric-magnetic properties of two constituents of the bi-material media.

4.5. Electric Displacement and Magnetic Induction inside the Crack

Since the medium inside the crack (usually air or vacuum) allows some penetrations of the some electric and magnetic fields, these fields may not be zero. Suppose that the normal components of the electric displacement and magnetic induction inside the crack are 𝑑0 and 𝑏0, respectively. Then from permeable crack boundary conditions (18) and solutions (78), it follows that the quantities 𝑑0 and 𝑏0 are as follows:𝑑0=⎧βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺ⎩𝐷I0βˆ’π‘’I15𝜏I0𝑐I442πœ†π·π‘’1βˆ’πœ†,caseII15𝜏I0𝑐I44ξ‚΅1βˆ’2πœ†π·ξ‚Ά+ξƒ©πœ€1βˆ’πœ†I11+𝑒I15ξ€Έ2𝑐I44ξƒͺ𝐸I0+𝑑I11+𝑒I15π‘žI15𝑐I44ξƒͺ𝐻I0𝑏,caseII0=⎧βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺ⎩𝐡I0βˆ’π‘žI15𝜏I0𝑐I442πœ†π΅π‘ž1βˆ’πœ†,caseII15𝜏I0𝑐I44ξ‚΅1βˆ’2πœ†π΅ξ‚Ά+ξƒ©πœ‡1βˆ’πœ†I11+ξ€·π‘žI15ξ€Έ2𝑐I44ξƒͺ𝐻I0+𝑑I11+𝑒I15π‘žI15𝑐I44ξƒͺ𝐸I0.caseII(91)

Then, using (21), we obtain that𝐷Iβˆ’π‘‘0=𝑒I15𝜏I0𝑐I442πœ†π·,𝐡1βˆ’πœ†Iβˆ’π‘0=π‘žI15𝜏I0𝑐I442πœ†π΅1βˆ’πœ†(92) in both cases of loading conditions.

The electric displacement and magnetic induction intensity factors are proportional to 𝐷I0βˆ’π‘‘0 and 𝐡I0βˆ’π‘0, respectively [13], and the following relations hold𝐾𝐷int=𝐾𝜏int𝑒I15𝑐I442πœ†π·,𝐾1βˆ’πœ†π΅int=𝐾𝜏intπ‘žI15𝑐I442πœ†π΅1βˆ’πœ†(93) which are in agreement with the solutions (78). For piezoelectric bi-materials or piezomagnetic bi-materials we have, for instance,𝐾𝐷int=𝐾𝜏int𝑐II44πœ€II11𝑒I15+𝑐I44πœ€I11𝑒II15+𝑒I15𝑒II15𝑒I15+𝑒II15𝑐II44𝑐I44ξ€·πœ€I11+πœ€II11ξ€Έ+𝑐I44𝑒II15ξ€Έ2+𝑐II44𝑒I15ξ€Έ2,𝐾𝐡int=𝐾𝜏int𝑐II44πœ‡II11π‘žI15+𝑐I44πœ‡I11π‘žII15+π‘žI15π‘žII15ξ€·π‘žI15+π‘žII15𝑐II44𝑐I44ξ€·πœ‡I11+πœ‡II11ξ€Έ+𝑐I44ξ€·π‘žII15ξ€Έ2+𝑐II44ξ€·π‘žI15ξ€Έ2.(94)

In particular, for a fully permeable crack considered here, and two identical magneto or electroelastic planes polarized in opposite directions we have (from (94)) 𝐾𝐷int=𝐾𝐡int=0.(95)

Note that the crack tip electric displacement 𝐾𝐷int and the electric displacement inside the crack 𝑑0 exist only in the piezoelectric plane. Alternatively the crack tip magnetic induction intensity factor 𝐾𝐡int and the magnetic induction inside the crack 𝑏0 exist only in the piezomagnetic plane. All quantities occur in the PEMO-elastic bimaterial.

5. Results and Discussions

In studying the fracture behaviour of the PEMO-elastic material, the field intensity factors are of significance. In this section, examples are given to illustrate the effects of material properties on the field intensity factor and the order of singularity.

5.1. Effect of Material Constants on the Singularity Order

We now consider the dependence of the singularity order on 2Γ—6-constituent independent piezo-electromagnetoelastic constants. Although analytical evaluation of the relative sensitivities is possible, on the basis of the results presented above, it is rather cumbersome. Therefore, the sensitivity is evaluated here in other way.

Firstly, we assume that both materials are piezoelectric and 𝑐II44=πœŒπ‘π‘I44,𝑒II15=πœŒπ‘’π‘’I15, and πœ€II11=πœŒπœ€πœ€I11, and analyze the situations(a)πœŒπ‘ changes and πœŒπ‘’=πœŒπœ€=1, that is, not change,(b)πœŒπ‘’ changes and πœŒπ‘=πœŒπœ€=1,(c)πœŒπœ€ changes and πœŒπ‘=πœŒπ‘’=1.

This states that it is analyzed that right half-plane is fixed, and left one contains a fictitious material with only changing πœŒπ‘ or πœŒπ‘’ or πœŒπœ€.

(a) The changes of the ratio πœŒπ‘ of 𝑐II44 to 𝑐I44: we haveξ€·πœ†=1βˆ’πœŒπ‘ξ€Έ(1+π‘š)1+πœŒπ‘ξƒ©ξ€·π‘’+2π‘š,π‘š=15ξ€Έ2𝑐44πœ€11ξƒͺI,||πœ†||<1,πœŒπ‘2<3+π‘šforπœŒπ‘’=πœŒπœ€=1(96) orπœ†=1βˆ’πœŒπ‘1+πœŒπ‘βˆ’π‘š,πœ†=0forπœŒπ‘=1βˆ’π‘š1+π‘š,πœŒπ‘<2π‘šβˆ’1,0<π‘š<1forπœŒπ‘’=βˆ’1,πœŒπœ€=1.(97)

Figure 3 shows the effects of varying elastic stiffness πœŒπ‘ on πœ† and 𝛼 with unchanging piezoelectric and piezomagnetic constants πœŒπ‘’=πœŒπœ€=1 or πœŒπ‘’=βˆ’1 and πœŒπœ€=1. Note that πœ†=0 and 𝛼=1/2 for πœŒπ‘’=βˆ’1,πœŒπœ€=1 and if (𝑐I44βˆ’π‘II44)𝑐I44πœ€I11=(𝑐I44+𝑐II44)(𝑒I15)2,𝑒II15=βˆ’π‘’I15 or if πœŒπ‘=1 and πœŒπ‘’=πœŒπœ€=1. Note also that πœ†(πœŒπ‘’=1,πœŒπœ€=1)>πœ†(πœŒπ‘’=βˆ’1,πœŒπœ€=1) for all of πœŒπ‘.

The singularity order 𝛼 is larger for two of the same ceramics poled in opposite directions together since 𝛼(πœŒπ‘’=1,πœŒπœ€=1)<𝛼(πœŒπ‘’=βˆ’1,πœŒπœ€=1).

We take six kinds of particular piezoelectric ceramics as representatives, the relevant material constants and parameters π‘š, and 1/π‘š of which are listed in Table 1 (with materials poling axes aligned in the positive 𝑧-direction).

Table 1: Relevant material properties [14, 15] and values of material parameters π‘š and 1/π‘š.

(b) The changes of the ratio πœŒπ‘’ of 𝑒II15 to 𝑒I15: we haveξ€·πœ†=βˆ’1βˆ’πœŒπ‘’ξ€Έ2ξ€·4/π‘š+1+πœŒπ‘’ξ€Έ2,πœŒπ‘’1>βˆ’π‘šξƒ©π‘=βˆ’44πœ€11𝑒15ξ€Έ2ξƒͺI,πœ†max=0forπœŒπ‘’πœ†=1,min=βˆ’1βˆ’π‘šforπœŒπ‘’2=βˆ’1βˆ’π‘š1<βˆ’π‘š,πœ†=βˆ’1forπœŒπ‘’1=βˆ’π‘š,π‘šπœ†=βˆ’4+π‘šforπœŒπ‘’=0πœ†=βˆ’π‘šforπœŒπ‘’=βˆ’1.(98)

For βˆ’1/π‘š<πœŒπ‘’<1, the singularity parameter πœ† increases from βˆ’1 to maximum πœ†=0 and for πœŒπ‘’>1 declines to βˆ’1. Then the singularity parameter 𝛼 varies between (0,0,5), respectively. If both poling directions are opposite; that is, one is in the 𝑧-direction and second is in the (βˆ’π‘§)-direction, then to satisfy the condition πœ†>βˆ’1 must hold |𝑒II(βˆ’)15|/𝑒I(+)15<1/π‘š or |𝑒II(βˆ’)15|𝑒I(+)15<𝑐I44πœ€I11. If the selection of 𝑒II15 violates the condition |πœ†|<1, then the electroelastic field near the interface crack tip is dominant by either logarithmic singularity or is bonded. This situation seems unlikely, take place for realistic piezoelectric ceramics, and it is not beyond the scope of abilities of results of this paper.

(c) For πœŒπœ€ varying and other parameter unchanged, it is easily found that πœ†=0 and 𝛼=0,5 for πœŒπ‘=πœŒπœ€=1 and varying πœŒπœ€. But if πœŒπ‘’=βˆ’1, thenπœ†=βˆ’2π‘š1+πœŒπœ€ξƒ©ξ€·π‘’;π‘š=15ξ€Έ2𝑐44πœ€11ξƒͺI.(99)

Figure 5 shows the variation of πœ† and 𝛼 with the ratio πœŒπœ€ for 𝑒II(βˆ’)15/𝑒I(+)15=βˆ’1.

The parameter πœ† assumes negative values and increases from βˆ’2π‘š to zero with πœŒπœ€>0. The singularity parameter 𝛼 is positive and increases from (1/πœ‹)arccos(2π‘š) to 1/2 with πœŒπœ€>0. Note that 2π‘š must be less unity if πœŒπœ€ tends to zero or π‘š<1 for πœŒπœ€>1. Some materials shown in Table 1 limit the range of πœŒπœ€; for example, PZT-4 has π‘š=1,175, and πœŒπœ€ must be larger 1,35 to ensure that πœ†<βˆ’1. Of course this situation is addressed to two piezoelectrics poled in opposite directions.

For piezomagnetic materials, the parameter π‘š isξƒ©ξ€·π‘žπ‘š=15ξ€Έ2𝑐44πœ‡11ξƒͺI,(100) and for magnetostrictive material CoFe2O4 assumes the value π‘š