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

Authors: Bogdan Rogowski
Journal: ISRN Materials Science
 (2012)

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## Abstract

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 lawr−α, r 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.


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## Body

## 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 relevant2×6 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 displacementw:(1)ux=0,uy=0,uz=w(x,y),
strain components γxz and γyz:(2)γxz=∂w∂x,γyz=∂w∂y
stress components τxz and τyz, in-plane electrical and magnetic potentials ϕ and ψ, which define electric and magnetic field components Ex,Ey,Hx, and Hy:(3)Ex=-∂ϕ∂x,Ey=-∂ϕ∂y,Hx=-∂ψ∂x,Hy=-∂ψ∂y
and electrical displacement components Dx,Dy, and magnetic induction components Bx, and By with all field quantities being the functions of coordinates x and y.The relations (2) and (3) have the following form:(4)γαz=w,α,Eα=-ϕ,α,Hα=-ψ,α,
where α=x,y and w,α=∂w/∂α.For linearly Magnetoelectroelastic medium, the coupled constitutive relations can be written in the matrix form as follows:(5)[ταz,Dα,Bα]T=C[γαz,-Eα,-Hα]T,
where the superscript T denotes the transpose of a matrix and(6)C=[c44e15q15e15-ε11-d11q15-d11-μ11],
where c44 is the shear modulus along the z-direction, which is direction of poling and is perpendicular to the isotropic plane (x,y),ε11 and μ11 are dielectric permittivity, and magnetic permeability coefficients, respectively, e15,q15, 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 as(7)τzα,α=0,Dα,α=0,Bα,α=0,α=x,y.
Subsequently, the Euler and Maxwell equations take the following form:(8)C[∇2w,∇2ϕ,∇2ψ]T=[0,0,0]T,
where ∇2=∂2/∂x2+∂2/∂y2 is the two-dimensional Laplace operator.Since|C|≠0, one can decouple the (8) as follows:(9)∇2w=0,∇2ϕ=0,∇2ψ=0.If we introduce, for convenience of mathematics in some boundary value problems, two unknown functions(10)[χ-e15w,η-q15w]T=C0[ϕ,ψ]T,
where(11)C0=[-ε11-d11-d11-μ11],
then(12)[ϕ,ψ]T=C0-1[χ-e15w,η-q15w]T,
where(13)C0-1=1ε11μ11-d112[-μ11d11d11-ε11]=[e1e2e2e3].The governing field variables are(14)τzk=c̃44w,k-αDk-βBk,ϕ=αw+e1χ+e2η,ψ=βw+e2χ+e3η,Dk=χ,k,Bk=η,k,k=x,y,(15)∇2w=0,∇2χ=0,∇2η=0,
where(16)c̃44=c44+αe15+βq15,α=μ11e15-d11q15ε11μ11-d112=-(e1e15+e2q15),β=ε11q15-d11e15ε11μ11-d112=-(e3q15+e2e15).Note thatc̃44 is the piezo-electro-magnetically stiffened elastic constant.Note also that the inverse of a matrixC is defined by parameters α,β,c̃44 and e1,e2,e3 as follows:(17)C-1=1c̃44[1αβαα2+c̃44e1αβ+c̃44e2βαβ+c̃44e2β2+c̃44e3].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 thez direction. For convenience, we denote the PEMO-elastic ceramics occupying the right and left half-planes x≥0 and x≤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,a](a>0) in the positive x-direction in ceramic I. For an antiplane shear crack having no thickness (so-called “mathematical 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,(18)Dy(x,0+)=Dy(x,0-),By(x,0+)=By(x,0-),ϕ(x,0+)=ϕ(x,0-),ψ(x,0+)=ψ(x,0-).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:(19a)χ,y(x,0+)=χ,y(x,0-),η,y(x,0+)=η,y(x,0-),(19b)χ=e15w,η=q15wforx,y=0±.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:(20)τyzI(x,y)=τ0I,DyI(x,y)=D0I,ByI(x,y)=B0IorEyI(x,y)=E0I,HyI(x,y)=H0I,x>0,y⟶±∞τyzII(x,y)=τ0II,DyII(x,y)=D0II,ByII(x,y)=B0IIorEyII(x,y)=E0II,HyII(x,y)=H0II,x<0,y⟶±∞,
where τ0I(τ0II),D0I(D0II),B0I(B0II) or E0I(E0II),H0I(H0II) 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:(21)τyzJ=τ0J,DyJ=DJ={D0J,caseI[e15c44τ0+(ε11+e152c44)E0+(d11+e15q15c44)H0]J,caseIIByJ=BJ={B0J,caseI[q15c44τ0+(d11+e15q15c44)E0+(μ11+q152c44)H0]J,caseII
with J=I,II.In addition the crack surfaces are traction-free, that is,(22)τyzI(x,y)=0;y=0±,0<x<a,
and owing to the symmetry one can directly write following conditions:(23)wI(x,0)=0,x>a,wII(x,0)=0,x<0.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 continuous(24)wI(0,y)=wII(0,y),τxzI(0,y)=τxzII(0,y);-∞<y<∞,ϕI(0,y)=ϕII(0,y),DxI(0,y)=DxII(0,y);-∞<y<∞,ψI(0,y)=ψII(0,y),BxI(0,y)=BxII(0,y);-∞<y<∞.


## 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 displacementw:(1)ux=0,uy=0,uz=w(x,y),
strain components γxz and γyz:(2)γxz=∂w∂x,γyz=∂w∂y
stress components τxz and τyz, in-plane electrical and magnetic potentials ϕ and ψ, which define electric and magnetic field components Ex,Ey,Hx, and Hy:(3)Ex=-∂ϕ∂x,Ey=-∂ϕ∂y,Hx=-∂ψ∂x,Hy=-∂ψ∂y
and electrical displacement components Dx,Dy, and magnetic induction components Bx, and By with all field quantities being the functions of coordinates x and y.The relations (2) and (3) have the following form:(4)γαz=w,α,Eα=-ϕ,α,Hα=-ψ,α,
where α=x,y and w,α=∂w/∂α.For linearly Magnetoelectroelastic medium, the coupled constitutive relations can be written in the matrix form as follows:(5)[ταz,Dα,Bα]T=C[γαz,-Eα,-Hα]T,
where the superscript T denotes the transpose of a matrix and(6)C=[c44e15q15e15-ε11-d11q15-d11-μ11],
where c44 is the shear modulus along the z-direction, which is direction of poling and is perpendicular to the isotropic plane (x,y),ε11 and μ11 are dielectric permittivity, and magnetic permeability coefficients, respectively, e15,q15, 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 as(7)τzα,α=0,Dα,α=0,Bα,α=0,α=x,y.
Subsequently, the Euler and Maxwell equations take the following form:(8)C[∇2w,∇2ϕ,∇2ψ]T=[0,0,0]T,
where ∇2=∂2/∂x2+∂2/∂y2 is the two-dimensional Laplace operator.Since|C|≠0, one can decouple the (8) as follows:(9)∇2w=0,∇2ϕ=0,∇2ψ=0.If we introduce, for convenience of mathematics in some boundary value problems, two unknown functions(10)[χ-e15w,η-q15w]T=C0[ϕ,ψ]T,
where(11)C0=[-ε11-d11-d11-μ11],
then(12)[ϕ,ψ]T=C0-1[χ-e15w,η-q15w]T,
where(13)C0-1=1ε11μ11-d112[-μ11d11d11-ε11]=[e1e2e2e3].The governing field variables are(14)τzk=c̃44w,k-αDk-βBk,ϕ=αw+e1χ+e2η,ψ=βw+e2χ+e3η,Dk=χ,k,Bk=η,k,k=x,y,(15)∇2w=0,∇2χ=0,∇2η=0,
where(16)c̃44=c44+αe15+βq15,α=μ11e15-d11q15ε11μ11-d112=-(e1e15+e2q15),β=ε11q15-d11e15ε11μ11-d112=-(e3q15+e2e15).Note thatc̃44 is the piezo-electro-magnetically stiffened elastic constant.Note also that the inverse of a matrixC is defined by parameters α,β,c̃44 and e1,e2,e3 as follows:(17)C-1=1c̃44[1αβαα2+c̃44e1αβ+c̃44e2βαβ+c̃44e2β2+c̃44e3].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 thez direction. For convenience, we denote the PEMO-elastic ceramics occupying the right and left half-planes x≥0 and x≤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,a](a>0) in the positive x-direction in ceramic I. For an antiplane shear crack having no thickness (so-called “mathematical 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,(18)Dy(x,0+)=Dy(x,0-),By(x,0+)=By(x,0-),ϕ(x,0+)=ϕ(x,0-),ψ(x,0+)=ψ(x,0-).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:(19a)χ,y(x,0+)=χ,y(x,0-),η,y(x,0+)=η,y(x,0-),(19b)χ=e15w,η=q15wforx,y=0±.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:(20)τyzI(x,y)=τ0I,DyI(x,y)=D0I,ByI(x,y)=B0IorEyI(x,y)=E0I,HyI(x,y)=H0I,x>0,y⟶±∞τyzII(x,y)=τ0II,DyII(x,y)=D0II,ByII(x,y)=B0IIorEyII(x,y)=E0II,HyII(x,y)=H0II,x<0,y⟶±∞,
where τ0I(τ0II),D0I(D0II),B0I(B0II) or E0I(E0II),H0I(H0II) 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:(21)τyzJ=τ0J,DyJ=DJ={D0J,caseI[e15c44τ0+(ε11+e152c44)E0+(d11+e15q15c44)H0]J,caseIIByJ=BJ={B0J,caseI[q15c44τ0+(d11+e15q15c44)E0+(μ11+q152c44)H0]J,caseII
with J=I,II.In addition the crack surfaces are traction-free, that is,(22)τyzI(x,y)=0;y=0±,0<x<a,
and owing to the symmetry one can directly write following conditions:(23)wI(x,0)=0,x>a,wII(x,0)=0,x<0.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 continuous(24)wI(0,y)=wII(0,y),τxzI(0,y)=τxzII(0,y);-∞<y<∞,ϕI(0,y)=ϕII(0,y),DxI(0,y)=DxII(0,y);-∞<y<∞,ψI(0,y)=ψII(0,y),BxI(0,y)=BxII(0,y);-∞<y<∞.


## 3.  Method of Solution

From the symmetry of the problem, it is sufficient to consider the upper half-plane of the bi-ceramic. Consequently, fory≥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:(25)[wI(x,y)χI(x,y)ηI(x,y)]=[γIDIBI]y+[1e15Iq15I]∫0∞A1(ξ)e-yξcos⁡(ξx)dξ+∫0∞[B1(ξ)C1(ξ)D1(ξ)]e-ξxsin⁡(ξy)dξ,
for x≥0 and(26)[wII(x,y)χII(x,y)ηII(x,y)]=[γIIDIIBII]y+[1e15IIq15II]∫0∞A2(ξ)e-yξcos⁡(ξx)dξ+∫0∞[B2(ξ)C2(ξ)D2(ξ)]eξxsin⁡(ξy)dξ
for x≤0, where Aj,Bj,Cj, and Dj(j=1,2) are unknowns to be determined from given boundary conditions and where(27)γJ=τ0J+αJDJ+βJBJc̃44J;J=I,II.Furthermore with the aid of (14), one can give the components of stress, electric displacement, magnetic induction, and electric and magnetic potentials(28)[τyzI(x,y)DyI(x,y)ByI(x,y)]=[τ0IDIBI]-[c44Ie15Iq15I]∫0∞ξA1(ξ)e-yξcos⁡(ξx)dξ+∫0∞ξ[c̃44IB1(ξ)-αIC1(ξ)-βID1(ξ)C1(ξ)D1(ξ)]×e-ξxcos⁡(ξy)dξ,(29)[ϕI(x,y)ψI(x,y)]=CI-1[τ0IDIBI]y+0+[αe1e2βe2e3]I×∫0∞ξ[B1(ξ)C1(ξ)D1(ξ)]e-ξxsin⁡(ξy)dξ
for x≥0 and(30)[τyzII(x,y)DyII(x,y)ByII(x,y)]=[τ0IIDIIBII]-[c44IIe15IIq15II]∫0∞ξA2(ξ)e-yξcos⁡(ξx)dξ+∫0∞ξ[c̃44IIB1(ξ)-αIIC2(ξ)-βIID2(ξ)C2(ξ)D2(ξ)]×eξxcos⁡(ξy)dξ,(31)[ϕII(x,y)ψII(x,y)]=CII-1[τ0IIDIIBII]y+0+[αe1e2βe2e3]II×∫0∞ξ[B2(ξ)C2(ξ)D2(ξ)]eξxsin⁡(ξy)dξ
for x≤0 and(32)[τxzI(x,y)DxI(x,y)BxI(x,y)]=-[c44Ie15Iq15I]∫0∞ξA1(ξ)e-yξsin⁡(ξx)dξ-∫0∞ξ[c̃44IB1(ξ)-αIC1(ξ)-βID1(ξ)C1(ξ)D1(ξ)]×e-ξxsin⁡(ξy)dξ
for x≥0 and(33)[τxzII(x,y)DxII(x,y)BxII(x,y)]=-[c44IIe15IIq15II]∫0∞ξA2(ξ)e-yξsin⁡(ξx)dξ+∫0∞ξ[c̃44IIB2(ξ)-αIIC2(ξ)-βIID2(ξ)C2(ξ)D2(ξ)]×eξxsin⁡(ξy)dξ
for x≤0.Now, application of the continuity conditions (24), at the interface x=0 to (25) to (33), yields
(34)τ0I+αIDI+βIBIc̃44I=τ0II+αIIDII+βIIBIIc̃44II,(35)CI-1[τ0IDIBI]=CII-1[τ0IIDIIBII],CJ-1=1c̃44J[αα2+c̃44Ie1αβ+c̃44Ie2βαβ+c̃44Ie2β2+c̃44Ie3]J
(36)-[c̃44IB1(ξ)-αIC1(ξ)-βID1(ξ)]=[c̃44IIB2(ξ)-αIIC2(ξ)-βIID2(ξ)],C1(ξ)=-C2(ξ),D1(ξ)=-D2(ξ)
(37)[αIIe1IIe2IIβIIe2IIe3II][B2(ξ)C2(ξ)D2(ξ)]=[αIe1Ie2IβIe2Ie3I][B1(ξ)C1(ξ)D1(ξ)].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, τ0II, DII, and BII by means of loadings of ceramics I, namely, τ0I, DI, and BI. 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
(38)A2(ξ)=0.Continuity ofw(x,y) at the interface x=0 requires(39)∫0∞[B2(ξ)-B1(ξ)]sin⁡(ξy)dξ=∫0∞A1(ξ)e-ξydξ
so that(40)B2(ξ)-B1(ξ)=2π∫0∞A1(η)ξξ2+η2dη,
since(41)∫0∞e-ηysin⁡(ξy)dy=ξξ2+η2.The result (40) in connection with (36) and (37) yields(42)B1(ξ)=-2πIc̃44IIΔ+(e3II+e3I)αII(αII-αI)+(e1II+e1I)βII(βII-βI)(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2+2πI(e2II+e2I)(αIβII+βIαII+2αIIβII)(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2,B2(ξ)=2πIc̃44IΔ-(e3II+e3I)αI(αII-αI)-(e1II+e1I)βI(βII-βI)(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2+2πI(e2II+e2I)(αIβII+βIαII+2αIIβII)(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2,C1(ξ)=B1(ξ)βI(e2II+e2I)-αI(e3II+e3I)Δ-B2(ξ)βII(e2II+e2I)-αII(e3II+e3I)Δ,D1(ξ)=-B1(ξ)βI(e1II+e1I)-αI(e2II+e2I)Δ+B2(ξ)βII(e1II+e1I)-αII(e2II+e2I)Δ,where(43)Δ=(e1II+e1I)(e3II+e3I)-(e2II+e2I)2,(44)I=∫0∞A1(η)ξξ2+η2dη.In the special cases, we obtain thatfor both piezoelectric materials(45)B1(ξ)=-2πIc44II(ε11I+ε11II)+e15II(e15I+e15II)(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2,B2(ξ)=2πIc44I(ε11I+ε11II)+e15I(e15I+e15II)(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2,C1(ξ)=-2πIe15I(c44IIε11II+(e15II)2)+e15II(c44Iε11I+(e15I)2)(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2,D1(ξ)=0,
for both piezomagnetic materials(46)B1(ξ)=-2πIc44II(μ11I+μ11II)+q15II(q15I+q15II)(c44I+c44II)(μ11I+μ11II)+(q15I+q15II)2,B2(ξ)=2πIc44I(μ11I+μ11II)+q15I(q15I+q15II)(c44I+c44II)(μ11I+μ11II)+(q15I+q15II)2,C1(ξ)=0,D1(ξ)=-2πIq15I(c44IIμ11II+(q15II)2)+q15II(c44Iμ11I+(q15I)2)(c44I+c44II)(μ11I+μ11II)+(q15I+q15II)2.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 that(47)g(x)=∂wI(x,0+)∂x.From the boundary conditions (23), g(x) should satisfy the single-value displacement constraint condition, that is,
(48)∫0ag(x)dx=0.Utilizing (25)1 in (23) leads to(49)wI(x,0)=∫0∞A1(ξ)cos⁡(ξx)dξ=0,x>a
from which together with (47), by using the inverse Fourier transform, can be deduced(50)A1(ξ)=-2πξ∫0ag(t)sin⁡(ξt)dt.Now, we calculate the following:(51)∫0∞ξξ2+η2A1(η)dη=-2πξ∫0ag(t)dt∫0∞sin⁡(ηt)η(ξ2+η2)dη.Using the result(52)∫0∞sin⁡(ηt)η(ξ2+η2)dη=π(1-e-ξt)2ξ2,
we find with the use of (48) that(53)∫0∞ξξ2+η2A1(η)dη=∫0ae-ξtξg(t)dt.Substitution of (53) into (42) yields the expressions for B1(ξ),B2(ξ),C1(ξ), and D1(ξ) in terms of g(x).From fraction-free condition (22) from (28)1, one can derive(54)∫0∞ξ[c44IA1(ξ)cos⁡(ξx)-(c̃44IB1(ξ)-αIC1(ξ)-βID1(ξ))e-ξx]dξ=τ0I.Substituting (50) and (42) with the use of (53) into (54), we have with the help of known integrals(55)2π∫0∞sin⁡(ξt)cos⁡(ξx)dξ=1π(1t-x+1t+x),∫0∞e-ξ(t+x)dξ=1t+x;t+x>0
the following singular integral equation with generalized Cauchy kernel for g(t):(56)1π∫0a(1t-x+λt+x)g(t)dt=-τ0Ic44I;0<x<a,
where(57)λ=1-2c̃44IIΔ+(e3II+e3I)αII(αII-αI)+(e1II+e1I)βII(βII-βI)-(e2II+e2I)(αIβII+βIαII+2αIIβII)(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2×[c̃44Ic44I-(e3II+e3I)αI(αII-αI)+(e1II+e1I)βI(βII-βI)-(e2II+e2I)(αIβII+βIαII-2αIβI)c44IΔ]+2(e3II+e3I)αIαII+(e1II+e1I)βIβII-(e2II+e2I)(αIβII+βIαII)c44IΔ.For both piezoelectric materials,λ is obtained as follows:(58)λ=(c44I-c44II)(ε11I+ε11II)+(e15I)2-(e15II)2+2e15I(e15II-e15I(c44II/c44I))(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2.The value ofλ for both piezomagnetic materials is obtained from formula (58) if we replace ε11 by μ11 and e15by q15. 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/(t-x) as t→x in (56), the other term λ/(t+x) is also unbonded as t,x→0 simultaneously. Particularly for two elastic dielectric, meaning e15=0, and diamagnetic, meaning q15=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, λ reduces to(59)λ=c44I-c44IIc44I+c44II.
Then the integral equation is simplified to(60)1π∫0a(1t-x+λt+x)g(t)dt=-τ0c44.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(t) of (56) subjected to (48) can be obtained as follows:(61)g(x)=τ0I2c44Isin⁡(πα/2)×[(xa+a2-x2)α(αaa2-x2+1)+(xa+a2-x2)-α(αaa2-x2-1)]
for 0<x<a with(62)cos⁡(πα)=-λ,
where 0<α<1.Onceg(t) is determined the crack tearing displacement can be obtained by the following integrations:(63)wI(x,0+)=∫0xg(x)dx=-τ0Ix2c44Isin⁡(πα/2)×[(xa+a2-x2)α-(xa+a2-x2)-α].0≤x≤a


### 4.2.  Crack Tearing Displacement

Expanding the expression (63) near the crack tips yields the asymptotic crack tearing displacement as(64)wI(x,0)=τ0Ic44Iαsin⁡(πα/2)2a(a-x)+O(r);r=a-x≈0wI(x,0)=τ0Iaα2c44Isin⁡(πα/2)x1-α+O(r);r=x≈0
at the right and left crack tip.HereO(r) denotes the infinitesimal terms compared to r, r 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:(65)τyzI(x,0)=1πc44I∫0a(1t-x+λt+x)g(t)dt+τ0I,DyI(x,0)=1πe15I∫0a(1t-x+1-2λD1+t)g(t)dt+DI,ByI(x,0)=1πq15I∫0a(1t-x+1-2λB1+t)g(t)dt+BI,
for x>a and(66)τyzII(x,0)=1-λπc44I∫0ag(t)t-xdt+τ0II,DyII(x,0)=2λDπe15I∫0ag(t)t-xdt+DII,ByII(x,0)=2λBπq15I∫0ag(t)t-xdt+BII
for x<0, where 1-λ is defined by (57) and(67)e15IλD={(e2II+e2I)2Δ[(c̃44IIαI+c̃44IαII)(e3II+e3I)-(c̃44IIβI+c̃44IβII)(e2II+e2I)]+(e3II+e3I)×(e2II+e2I)(αII-αI)(αIβII+βIαII+2αIIβII)+(e3II+e3I)(e1II+e1I)(βII-βI)(αIβII-βIαII)+(e2II+e2I)2(βII-βI)(αIβII+βIαII+2αIIβII)}×{Δ[(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2]}-1,q15IλB={(e2II+e2I)2Δ[(c̃44IIβI+c̃44IβII)(e1II+e1I)-(c̃44IIαI+c̃44IαII)(e2II+e2I)]-(e1II+e1I)×(e2II+e2I)(βII-βI)(αIβII+βIαII+2αIIβII)-(e2II+e2I)(e1II+e1I)(βII-βI)(αIβII-βIαII)-(e2II+e2I)2(αII-αI)(αIβII+βIαII+2αIIβII)}×{Δ[(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2]}-1.For both piezoelectric or piezomagnetic materials, (67) give(68)e15IλD=e15I(c44IIε11II+(e15II)2)+e15II(c44Iε11I+(e15I)2)(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2,q15IλB=0
or(69)e15IλD=0,q15IλB=q15I(c44IIμ11II+(q15II)2)+q15II(c44Iμ11I+(q15I)2)(c44I+c44II)(μ11I+μ11II)+(q15I+q15II)2.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(x) near the point x=0 and x=a by the following asymptotic expressions:(70)g(x)=-τ0I2c44Iαsin⁡(πα/2)2aa-x+O(1);x≈a-0,(71)g(x)=-τ0I2c44Iα-1sin⁡(πα/2)[2(a-x)x]α+O(1);x≈0+0,
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 as(72)Khomq=lim⁡x→a+2π(x-a)qI(x,0+),Kint⁡q=lim⁡x→0-(-2πx)αqII(x,0+),
respectively, where q stands for one of τyz,γyz,Dy,By,Ey, and Hy.


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

Using the integral(73)1π∫0a1(t-x)a-tdt=-2πx-atan-1ax-a,x>a,
we obtain from (65)1 that(74)τyzI(x,0)=1πc44I∫0ag(t)t-xdt+O(1)=Khomτ2π(x-a),
where(75)Khomτ=αsin⁡(πα/2)τoIπa
the stress intensity factor at the right crack tip. Other field intensity factors are related to Khomτ as follows:(76)Khomγ=1c44IKhomτ,KhomD=e15Ic44IKhomτ,KhomB=q15Ic44IKhomτ,Khomϕ=Khomψ=KhomE=KhomH=0.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 Khomγ,KhomD, and KhomB are related to Khomτ 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],(77)1π∫0a1(t-x)(a-tt)αdt=1sin⁡(πα)[(x-ax)α-1],x<0
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:(78)[Kint⁡τ;Kint⁡D;Kint⁡B;Kint⁡γ;Kint⁡E;Kint⁡H]=2(1-α)(1+λ)1-λτ0Ic44I(4πa)α×[c44I1-λ2;e15IλD;q15IλB;λγ;λE;λH],
where the identity is used as follows:(79)sin⁡(πα)sin⁡(πα2)=(1+λ)1-λ2
and where(80)λγ=[c̃44IΔ-(e3II+e3I)αI(αII-αI)-(e1II+e1I)βI(βII-βI)+(e2II+e2I)(αIβII+βIαII+2αIIβII)]×[(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2]-1λE={λγ[αIΔ+(αII-αI)(e1I(e3II+e3I)-e2I(e2II+e2I))-(βII-βI)(e1Ie2II-e2Ie1II)]-αII(e1I(e3II+e3I)-e2I(e2II+e2I))+βII(e1Ie2II-e2Ie1II)}Δ-1λH={λγ[βIΔ+(βII-βI)(e3I(e1II+e1I)-e2I(e2II+e2I))-(αII-αI)(e2Ie3II-e3Ie2II)]-βII(e2I(e2II+e2I)-e3I(e1II+e1I))+αII(e2Ie3II-e3Ie2II)}Δ-1
for PEMO-elastic bimaterial and(81)λγ=c44I(ε11I+ε11II)+e15I(e15I+e15II)(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2,λE=c44IIe15I-c44Ie15II(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2,λH=0
for piezoelectric bi-material and(82)λγ=c44I(μ11I+μ11II)+q15I(q15I+q15II)(c44I+c44II)(μ11I+μ11II)+(q15I+q15II)2,λE=0,λH=c44IIq15I-c44Iq15II(c44I+c44II)(μ11I+μ11II)+(q15I+q15II)2
for piezomagnetic bi-material.Note that for piezoelectric bi-material, we have(83)λτ=1-λ2=c44I(ε11I+ε11II)+(e15II)2+(e15I)2(c44II/c44I)(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2.The material parameters for piezoelectric ceramics coincide, in general, with the ones derived by Li and Wang [9]. But in λ, 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 λ 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 λ=0 and α=1/2. The parameter λE in this paper has opposite sign to that presented by Li and Wang. This gives that for ρc>1(c44II>c44I) meaning that piezoelectric ceramic II is more stiffer that piezoelectric ceramic I (e15II=e15I), in this case λE>0, so stands also KintE>0, and KintE increases with ρc. Also it is seen that KintE decreases with the ratio ρe of e15II to e15I. In the paper Li and Wang [9], the conclusions, associated with KintE, are inverse. The presented conclusions are consistent with physical consideration. The field intensity factors must satisfy the constitutive equations(84)Kτ=c44IIKγ-e15IIKE,KD=e15IIKγ+ε11IIKE,
or material parameters must satisfy the equivalent equations(85)c44Iλτ=c44IIλγ-e15IIλE,e15IλD=e15IIλγ+ε11IIλE.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(86)[Kint⁡τKint⁡ϕKint⁡ψ]=[c̃44-α-βαe1e2βe2e3][KwKDKB],
as shown in (14). Of course, we have Kintϕ=-KintE and Kintψ=-KintH.


### 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 as(87)G=12c44I[(Khomτ)2+(πa2)1-2α(Kint⁡τ)2]=Ghom12[(khomτ)2+(kint⁡τ)2],
where(88)Ghom=πa(τ0I)22c44I,(89)khomτ=Khomττ0Iπa/2,(90)kint⁡τ=Kint⁡ττ0I(πa/2)α
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 ared0 and b0, respectively. Then from permeable crack boundary conditions (18) and solutions (78), it follows that the quantities d0 and b0 are as follows:(91)d0={D0I-e15Iτ0Ic44I2λD1-λ,caseIe15Iτ0Ic44I(1-2λD1-λ)+(ε11I+(e15I)2c44I)E0I+(d11I+e15Iq15Ic44I)H0I,caseIIb0={B0I-q15Iτ0Ic44I2λB1-λ,caseIq15Iτ0Ic44I(1-2λB1-λ)+(μ11I+(q15I)2c44I)H0I+(d11I+e15Iq15Ic44I)E0I.caseIIThen, using (21), we obtain that(92)DI-d0=e15Iτ0Ic44I2λD1-λ,BI-b0=q15Iτ0Ic44I2λB1-λ
in both cases of loading conditions.The electric displacement and magnetic induction intensity factors are proportional toD0I-d0 and B0I-b0, respectively [13], and the following relations hold(93)Kint⁡D=Kint⁡τe15Ic44I2λD1-λ,Kint⁡B=Kint⁡τq15Ic44I2λB1-λ
which are in agreement with the solutions (78). For piezoelectric bi-materials or piezomagnetic bi-materials we have, for instance,(94)Kint⁡D=Kint⁡τc44IIε11IIe15I+c44Iε11Ie15II+e15Ie15II(e15I+e15II)c44IIc44I(ε11I+ε11II)+c44I(e15II)2+c44II(e15I)2,Kint⁡B=Kint⁡τc44IIμ11IIq15I+c44Iμ11Iq15II+q15Iq15II(q15I+q15II)c44IIc44I(μ11I+μ11II)+c44I(q15II)2+c44II(q15I)2.In particular, for a fully permeable crack considered here, and two identical magneto or electroelastic planes polarized in opposite directions we have (from (94))
(95)Kint⁡D=Kint⁡B=0.Note that the crack tip electric displacementKintD and the electric displacement inside the crack d0 exist only in the piezoelectric plane. Alternatively the crack tip magnetic induction intensity factor KintB and the magnetic induction inside the crack b0 exist only in the piezomagnetic plane. All quantities occur in the PEMO-elastic bimaterial.


## 4.1.  Solution of the Singular Integral Equation

Based on the result derived by Bueckner [10], the desired solution for g(t) of (56) subjected to (48) can be obtained as follows:(61)g(x)=τ0I2c44Isin⁡(πα/2)×[(xa+a2-x2)α(αaa2-x2+1)+(xa+a2-x2)-α(αaa2-x2-1)]
for 0<x<a with(62)cos⁡(πα)=-λ,
where 0<α<1.Onceg(t) is determined the crack tearing displacement can be obtained by the following integrations:(63)wI(x,0+)=∫0xg(x)dx=-τ0Ix2c44Isin⁡(πα/2)×[(xa+a2-x2)α-(xa+a2-x2)-α].0≤x≤a


## 4.2.  Crack Tearing Displacement

Expanding the expression (63) near the crack tips yields the asymptotic crack tearing displacement as(64)wI(x,0)=τ0Ic44Iαsin⁡(πα/2)2a(a-x)+O(r);r=a-x≈0wI(x,0)=τ0Iaα2c44Isin⁡(πα/2)x1-α+O(r);r=x≈0
at the right and left crack tip.HereO(r) denotes the infinitesimal terms compared to r, r 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:(65)τyzI(x,0)=1πc44I∫0a(1t-x+λt+x)g(t)dt+τ0I,DyI(x,0)=1πe15I∫0a(1t-x+1-2λD1+t)g(t)dt+DI,ByI(x,0)=1πq15I∫0a(1t-x+1-2λB1+t)g(t)dt+BI,
for x>a and(66)τyzII(x,0)=1-λπc44I∫0ag(t)t-xdt+τ0II,DyII(x,0)=2λDπe15I∫0ag(t)t-xdt+DII,ByII(x,0)=2λBπq15I∫0ag(t)t-xdt+BII
for x<0, where 1-λ is defined by (57) and(67)e15IλD={(e2II+e2I)2Δ[(c̃44IIαI+c̃44IαII)(e3II+e3I)-(c̃44IIβI+c̃44IβII)(e2II+e2I)]+(e3II+e3I)×(e2II+e2I)(αII-αI)(αIβII+βIαII+2αIIβII)+(e3II+e3I)(e1II+e1I)(βII-βI)(αIβII-βIαII)+(e2II+e2I)2(βII-βI)(αIβII+βIαII+2αIIβII)}×{Δ[(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2]}-1,q15IλB={(e2II+e2I)2Δ[(c̃44IIβI+c̃44IβII)(e1II+e1I)-(c̃44IIαI+c̃44IαII)(e2II+e2I)]-(e1II+e1I)×(e2II+e2I)(βII-βI)(αIβII+βIαII+2αIIβII)-(e2II+e2I)(e1II+e1I)(βII-βI)(αIβII-βIαII)-(e2II+e2I)2(αII-αI)(αIβII+βIαII+2αIIβII)}×{Δ[(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2]}-1.For both piezoelectric or piezomagnetic materials, (67) give(68)e15IλD=e15I(c44IIε11II+(e15II)2)+e15II(c44Iε11I+(e15I)2)(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2,q15IλB=0
or(69)e15IλD=0,q15IλB=q15I(c44IIμ11II+(q15II)2)+q15II(c44Iμ11I+(q15I)2)(c44I+c44II)(μ11I+μ11II)+(q15I+q15II)2.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(x) near the point x=0 and x=a by the following asymptotic expressions:(70)g(x)=-τ0I2c44Iαsin⁡(πα/2)2aa-x+O(1);x≈a-0,(71)g(x)=-τ0I2c44Iα-1sin⁡(πα/2)[2(a-x)x]α+O(1);x≈0+0,
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 as(72)Khomq=lim⁡x→a+2π(x-a)qI(x,0+),Kint⁡q=lim⁡x→0-(-2πx)αqII(x,0+),
respectively, where q stands for one of τyz,γyz,Dy,By,Ey, and Hy.


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

Using the integral(73)1π∫0a1(t-x)a-tdt=-2πx-atan-1ax-a,x>a,
we obtain from (65)1 that(74)τyzI(x,0)=1πc44I∫0ag(t)t-xdt+O(1)=Khomτ2π(x-a),
where(75)Khomτ=αsin⁡(πα/2)τoIπa
the stress intensity factor at the right crack tip. Other field intensity factors are related to Khomτ as follows:(76)Khomγ=1c44IKhomτ,KhomD=e15Ic44IKhomτ,KhomB=q15Ic44IKhomτ,Khomϕ=Khomψ=KhomE=KhomH=0.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 Khomγ,KhomD, and KhomB are related to Khomτ 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],(77)1π∫0a1(t-x)(a-tt)αdt=1sin⁡(πα)[(x-ax)α-1],x<0
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:(78)[Kint⁡τ;Kint⁡D;Kint⁡B;Kint⁡γ;Kint⁡E;Kint⁡H]=2(1-α)(1+λ)1-λτ0Ic44I(4πa)α×[c44I1-λ2;e15IλD;q15IλB;λγ;λE;λH],
where the identity is used as follows:(79)sin⁡(πα)sin⁡(πα2)=(1+λ)1-λ2
and where(80)λγ=[c̃44IΔ-(e3II+e3I)αI(αII-αI)-(e1II+e1I)βI(βII-βI)+(e2II+e2I)(αIβII+βIαII+2αIIβII)]×[(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2]-1λE={λγ[αIΔ+(αII-αI)(e1I(e3II+e3I)-e2I(e2II+e2I))-(βII-βI)(e1Ie2II-e2Ie1II)]-αII(e1I(e3II+e3I)-e2I(e2II+e2I))+βII(e1Ie2II-e2Ie1II)}Δ-1λH={λγ[βIΔ+(βII-βI)(e3I(e1II+e1I)-e2I(e2II+e2I))-(αII-αI)(e2Ie3II-e3Ie2II)]-βII(e2I(e2II+e2I)-e3I(e1II+e1I))+αII(e2Ie3II-e3Ie2II)}Δ-1
for PEMO-elastic bimaterial and(81)λγ=c44I(ε11I+ε11II)+e15I(e15I+e15II)(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2,λE=c44IIe15I-c44Ie15II(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2,λH=0
for piezoelectric bi-material and(82)λγ=c44I(μ11I+μ11II)+q15I(q15I+q15II)(c44I+c44II)(μ11I+μ11II)+(q15I+q15II)2,λE=0,λH=c44IIq15I-c44Iq15II(c44I+c44II)(μ11I+μ11II)+(q15I+q15II)2
for piezomagnetic bi-material.Note that for piezoelectric bi-material, we have(83)λτ=1-λ2=c44I(ε11I+ε11II)+(e15II)2+(e15I)2(c44II/c44I)(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2.The material parameters for piezoelectric ceramics coincide, in general, with the ones derived by Li and Wang [9]. But in λ, 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 λ 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 λ=0 and α=1/2. The parameter λE in this paper has opposite sign to that presented by Li and Wang. This gives that for ρc>1(c44II>c44I) meaning that piezoelectric ceramic II is more stiffer that piezoelectric ceramic I (e15II=e15I), in this case λE>0, so stands also KintE>0, and KintE increases with ρc. Also it is seen that KintE decreases with the ratio ρe of e15II to e15I. In the paper Li and Wang [9], the conclusions, associated with KintE, are inverse. The presented conclusions are consistent with physical consideration. The field intensity factors must satisfy the constitutive equations(84)Kτ=c44IIKγ-e15IIKE,KD=e15IIKγ+ε11IIKE,
or material parameters must satisfy the equivalent equations(85)c44Iλτ=c44IIλγ-e15IIλE,e15IλD=e15IIλγ+ε11IIλE.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(86)[Kint⁡τKint⁡ϕKint⁡ψ]=[c̃44-α-βαe1e2βe2e3][KwKDKB],
as shown in (14). Of course, we have Kintϕ=-KintE and Kintψ=-KintH.


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

Using the integral(73)1π∫0a1(t-x)a-tdt=-2πx-atan-1ax-a,x>a,
we obtain from (65)1 that(74)τyzI(x,0)=1πc44I∫0ag(t)t-xdt+O(1)=Khomτ2π(x-a),
where(75)Khomτ=αsin⁡(πα/2)τoIπa
the stress intensity factor at the right crack tip. Other field intensity factors are related to Khomτ as follows:(76)Khomγ=1c44IKhomτ,KhomD=e15Ic44IKhomτ,KhomB=q15Ic44IKhomτ,Khomϕ=Khomψ=KhomE=KhomH=0.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 Khomγ,KhomD, and KhomB are related to Khomτ 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],(77)1π∫0a1(t-x)(a-tt)αdt=1sin⁡(πα)[(x-ax)α-1],x<0
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:(78)[Kint⁡τ;Kint⁡D;Kint⁡B;Kint⁡γ;Kint⁡E;Kint⁡H]=2(1-α)(1+λ)1-λτ0Ic44I(4πa)α×[c44I1-λ2;e15IλD;q15IλB;λγ;λE;λH],
where the identity is used as follows:(79)sin⁡(πα)sin⁡(πα2)=(1+λ)1-λ2
and where(80)λγ=[c̃44IΔ-(e3II+e3I)αI(αII-αI)-(e1II+e1I)βI(βII-βI)+(e2II+e2I)(αIβII+βIαII+2αIIβII)]×[(c̃44I+c̃44II)Δ+(e3II+e3I)(αII-αI)2+(e1II+e1I)(βII-βI)2]-1λE={λγ[αIΔ+(αII-αI)(e1I(e3II+e3I)-e2I(e2II+e2I))-(βII-βI)(e1Ie2II-e2Ie1II)]-αII(e1I(e3II+e3I)-e2I(e2II+e2I))+βII(e1Ie2II-e2Ie1II)}Δ-1λH={λγ[βIΔ+(βII-βI)(e3I(e1II+e1I)-e2I(e2II+e2I))-(αII-αI)(e2Ie3II-e3Ie2II)]-βII(e2I(e2II+e2I)-e3I(e1II+e1I))+αII(e2Ie3II-e3Ie2II)}Δ-1
for PEMO-elastic bimaterial and(81)λγ=c44I(ε11I+ε11II)+e15I(e15I+e15II)(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2,λE=c44IIe15I-c44Ie15II(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2,λH=0
for piezoelectric bi-material and(82)λγ=c44I(μ11I+μ11II)+q15I(q15I+q15II)(c44I+c44II)(μ11I+μ11II)+(q15I+q15II)2,λE=0,λH=c44IIq15I-c44Iq15II(c44I+c44II)(μ11I+μ11II)+(q15I+q15II)2
for piezomagnetic bi-material.Note that for piezoelectric bi-material, we have(83)λτ=1-λ2=c44I(ε11I+ε11II)+(e15II)2+(e15I)2(c44II/c44I)(c44I+c44II)(ε11I+ε11II)+(e15I+e15II)2.The material parameters for piezoelectric ceramics coincide, in general, with the ones derived by Li and Wang [9]. But in λ, 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 λ 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 λ=0 and α=1/2. The parameter λE in this paper has opposite sign to that presented by Li and Wang. This gives that for ρc>1(c44II>c44I) meaning that piezoelectric ceramic II is more stiffer that piezoelectric ceramic I (e15II=e15I), in this case λE>0, so stands also KintE>0, and KintE increases with ρc. Also it is seen that KintE decreases with the ratio ρe of e15II to e15I. In the paper Li and Wang [9], the conclusions, associated with KintE, are inverse. The presented conclusions are consistent with physical consideration. The field intensity factors must satisfy the constitutive equations(84)Kτ=c44IIKγ-e15IIKE,KD=e15IIKγ+ε11IIKE,
or material parameters must satisfy the equivalent equations(85)c44Iλτ=c44IIλγ-e15IIλE,e15IλD=e15IIλγ+ε11IIλE.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(86)[Kint⁡τKint⁡ϕKint⁡ψ]=[c̃44-α-βαe1e2βe2e3][KwKDKB],
as shown in (14). Of course, we have Kintϕ=-KintE and Kintψ=-KintH.


## 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 as(87)G=12c44I[(Khomτ)2+(πa2)1-2α(Kint⁡τ)2]=Ghom12[(khomτ)2+(kint⁡τ)2],
where(88)Ghom=πa(τ0I)22c44I,(89)khomτ=Khomττ0Iπa/2,(90)kint⁡τ=Kint⁡ττ0I(πa/2)α
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 ared0 and b0, respectively. Then from permeable crack boundary conditions (18) and solutions (78), it follows that the quantities d0 and b0 are as follows:(91)d0={D0I-e15Iτ0Ic44I2λD1-λ,caseIe15Iτ0Ic44I(1-2λD1-λ)+(ε11I+(e15I)2c44I)E0I+(d11I+e15Iq15Ic44I)H0I,caseIIb0={B0I-q15Iτ0Ic44I2λB1-λ,caseIq15Iτ0Ic44I(1-2λB1-λ)+(μ11I+(q15I)2c44I)H0I+(d11I+e15Iq15Ic44I)E0I.caseIIThen, using (21), we obtain that(92)DI-d0=e15Iτ0Ic44I2λD1-λ,BI-b0=q15Iτ0Ic44I2λB1-λ
in both cases of loading conditions.The electric displacement and magnetic induction intensity factors are proportional toD0I-d0 and B0I-b0, respectively [13], and the following relations hold(93)Kint⁡D=Kint⁡τe15Ic44I2λD1-λ,Kint⁡B=Kint⁡τq15Ic44I2λB1-λ
which are in agreement with the solutions (78). For piezoelectric bi-materials or piezomagnetic bi-materials we have, for instance,(94)Kint⁡D=Kint⁡τc44IIε11IIe15I+c44Iε11Ie15II+e15Ie15II(e15I+e15II)c44IIc44I(ε11I+ε11II)+c44I(e15II)2+c44II(e15I)2,Kint⁡B=Kint⁡τc44IIμ11IIq15I+c44Iμ11Iq15II+q15Iq15II(q15I+q15II)c44IIc44I(μ11I+μ11II)+c44I(q15II)2+c44II(q15I)2.In particular, for a fully permeable crack considered here, and two identical magneto or electroelastic planes polarized in opposite directions we have (from (94))
(95)Kint⁡D=Kint⁡B=0.Note that the crack tip electric displacementKintD and the electric displacement inside the crack d0 exist only in the piezoelectric plane. Alternatively the crack tip magnetic induction intensity factor KintB 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 on2×6-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 andc44II=ρcc44I,e15II=ρee15I, and ε11II=ρεε11I, and analyze the situations(a)
ρc changes and ρe=ρε=1, that is, not change,(b)
ρe changes and ρc=ρε=1,(c)
ρε changes and ρc=ρe=1.This states that it is analyzed that right half-plane is fixed, and left one contains a fictitious material with only changingρc or ρe or ρε.(a) The changes of the ratioρc of c44II to c44I: we have(96)λ=(1-ρc)(1+m)1+ρc+2m,m=((e15)2c44ε11)I,|λ|<1,ρc<3+2mforρe=ρε=1
or(97)λ=1-ρc1+ρc-m,λ=0forρc=1-m1+m,ρc<2m-1,0<m<1forρe=-1,ρε=1.Figure3 shows the effects of varying elastic stiffness ρc on λ and α with unchanging piezoelectric and piezomagnetic constants ρe=ρε=1 or ρe=-1 and ρε=1. Note that λ=0 and α=1/2 for ρe=-1,ρε=1 and if (c44I-c44II)c44Iε11I=(c44I+c44II)(e15I)2,e15II=-e15I or if ρc=1 and ρe=ρε=1. Note also that λ(ρe=1,ρε=1)>λ(ρe=-1,ρε=1) for all of ρc.The singularity orderα is larger for two of the same ceramics poled in opposite directions together since α(ρe=1,ρε=1)<α(ρe=-1,ρε=1).We take six kinds of particular piezoelectric ceramics as representatives, the relevant material constants and parametersm, and 1/m 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 m and 1/m.
c44 [109 N/m2]e15 [C/m2]ε11 [10−9 C/Vm]m1/mBaTiO343,011,6011,200,2794,348PZT-5H35,317,0015,100,5421,844PZT-425,613,446,001,1750,851P-725,013,5017,100,4302,325C-20587,013,597,950,2104,761PZT-PIC15120,012,009,820,7331,364(b) The changes of the ratioρe of e15II to e15I: we have(98)λ=-(1-ρe)24/m+(1+ρe)2,ρe>-1m=-(c44ε11(e15)2)I,λmax⁡=0forρe=1,λmin⁡=-1-mforρe=-1-2m<-1m,λ=-1forρe=-1m,λ=-m4+mforρe=0λ=-mforρe=-1.For-1/m<ρe<1, the singularity parameter λ increases from −1 to maximum λ=0 and for ρe>1 declines to −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 (-z)-direction, then to satisfy the condition λ>-1 must hold |e15II(-)|/e15I(+)<1/m or |e15II(-)|e15I(+)<c44Iε11I. If the selection of e15II violates the condition |λ|<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) Forρε varying and other parameter unchanged, it is easily found that λ=0 and α=0,5 for ρc=ρε=1 and varying ρε. But if ρe=-1, then(99)λ=-2m1+ρε;m=((e15)2c44ε11)I.Figure5 shows the variation of λ and α with the ratio ρε for e15II(-)/e15I(+)=-1.The parameterλ assumes negative values and increases from -2m to zero with ρε>0. The singularity parameter α is positive and increases from (1/π)arccos(2m) to 1/2 with ρε>0. Note that 2m must be less unity if ρε tends to zero or m<1 for ρε>1. Some materials shown in Table 1 limit the range of ρε; for example, PZT-4 has m=1,175, and ρε must be larger 1,35 to ensure that λ<-1. Of course this situation is addressed to two piezoelectrics poled in opposite directions.For piezomagnetic materials, the parameterm is(100)m=((q15)2c44μ11)I,
and for magnetostrictive material CoFe2O4 assumes the value m=0,0113.For CoFe2O4 we have(101)c44=45,3GPa,q15=550N/Am,μ11=590×10-6N/A2.The “relative sensitivity” analysis includes three cases:(a) The changes of ratioρc of c44II to c44I: we have(102)λ=1,0113(1-ρc)1,0226+ρc,ρc<20,7ρq=ρμ=1
or(103)λ=1-ρc1+ρc-0,0113,ρc<16,7ρq=-1,ρμ=1.
Approximately,(104)λ=1-ρc1+ρc,ρc<16.For Magnetoelectroelastic composite BaTiO3-CoFe2O4  (Vf=0,5)(q15)2/c44μ11=0,005 and (e15)2/c44ε11≈0,135.Figure6 shows the effect of ρc on λ and α for CoFe2O4 magnetostrictive ceramic.For both poling directions, the values ofλ and α are the same.(b) The changes of the ratioρq of q15II to q15I: we have(105)λ=-(1-ρq)235,4+(1+ρq)2,ρq>-8,85,λmax⁡=0forρq=1,λ=-1forρq=-8,85.
Figure 7 shows the effect of ρq on λ and α for CoFe2O4 ceramic.(c) The changes of the ratioρμ of μ15II to μ15I: we have(106)λ=-0,02261+ρμforρq=-1,λ=0forρq=1,alwaysρc=1.
Figure 8 shows the effect of ρμ on λ and α for CoFe2O4.


### 5.2. Effect of Material Constants on the Field Intensity Factors

The material constants also affect the intensity factors. Figure9 presents the variation of normalized SIFs khomτ and kintτ defined by (89) and (90) which depend on α and λ(107)khomτ=2α1+λ,kint⁡τ=23α-1/21-α1+λ1-λ.
For 0<α<1khomτ increasing monotonously from 22/π through 1 to 2 as α tends to zero and equals 1/2 and 1, respectively. From Figures 8 and 2, one can observe that the effect of ρc on kintτ is more evident than that on khomτ. Moreover, ρc increased the singularity parameter α that decreases (see Figure 3), and kintτ rises suddenly, while khomτ falls down slightly. For ρc<1 and ρe=1 or ρc<(1-m)/(1+m) and ρe=-1, we have α>1/2. This means that if piezoelectric II is more elastically complaisant than piezoelectric ceramic I, in this case khomτ>kintτ. On the other hand, for ρc>1, which gives α<1/2, meaning that piezoelectric II is stiffer than piezoelectric I in this case khomτ<kintτ. From the Figures 3, 4, and 5, we see that the range 0<α<1/2 corresponds to ρc>1 or ρc>(1-m)/(1+m) (in the case ρe=-1), ρe>-1/m, and ρε>0. Then always kintτ>khomτ. The range 1/2<α<1 is for 0<ρc<1 or 0<ρc<(1-m)/(1+m) (in the case ρe=-1). Then, kintτ<khomτ for all of α.Figure 2
The curveλ=-cos(πα);λ is the bi-material parameter, and α is singularity order parameter.Effect ofρc on λ and α with ρe=ρε=1 (Case I) and ρe=-1, ρε=1 (Case II).
(a)
Case I(b)
Case IIFigure 4
Effects ofρe on λ and α with ρc=ρε=1 (ρe>0 or ρe<0 denote piezoelectric ceramics poled parallel to or antiparallel to the z-axis, respectively, i.e., ρe=-1 denotes that e15I=-e15II=e15).Figure 5
Effect ofρε on λ and α with ρc=1 when ρe=e15II(-)/e15I(+)=-1; for ρe=1, we have λ=0 and α=0,5.Figure 6
Effect ofρc on λ and α for CoFe2O4, ρμ=1 and ρq=1 or ρq=-1.Figure 7
Effects ofρq on λ and α for CoFe2O4, ρc=ρμ=1.Figure 8
Effects ofρμ on λ and α for CoFe2O4, ρq=1 or ρq=-1 and ρc=1.Figure 9
Normalized SIFs as a function ofα.Note that the caseα=1,λ=1 gives the limiting values kintτ=0 and khomτ=2 which gives(108)Kint⁡τ=0,Khomτ=τ0Iπa.This is the solution for edge crack of lengtha.The normalized intensity factors for strain, electric displacement, magnetic induction, electric field, and magnetic field at the interface crack tip are defined by (78) and by the following formula:(109)kint⁡q=c44IτoIKint⁡q(πa/2)α,
where q stands for one of γ, D, B, E, and H.Then we have(110)[kint⁡γ;kint⁡D;kint⁡B;kint⁡E;kint⁡H]=2(1-α)8α(1+λ)1-λ[λγ;e15IλD;q15IλB;λE;λH],
respectively.Of course, the normalized intensity factors satisfy the constitutive equations (5), that is,
(111)[kint⁡τ;kint⁡D;kint⁡B]=CII[kint⁡γ;-kint⁡E;-kint⁡H]
with the matrix (6) or inverse form with the use of matrix (CII)-1, defined by (17).The analysis above implies that, for the magnetically (or electrically) permeable interfacial cracks, the applied magnetic (or electric) loadings have no influence on the fracture behaviours of the crack tips.Figures10 and 11 are devoted to the variation of kintτ and khomτ.Normalized SIFs as a function ofρc with ρe=ρε=1 (Case I) and ρe=-1, ρε=1 (Case II).
(a)
Case I(b)
Case IINormalized SIFs as a function ofρe with ρc=ρε=1.
(a)(b)We have(112)khomτ=22πforρc=3+2m(ρe=ρε=1)orρc=2m-1(ρe=-1,ρε=1).The figures show that the normalized stress intensity factor in homogeneous solid is only weakly dependent on the elastic constants and dielectric permeabilities. In contrastkintτ strongly depends on ρc and ρe. This is consistent with physical considerations; for large difference of piezocoefficients ρe<0 or ρe>1 the kintτ are larger than khomτ (Figure 11). From the Figure 10 it can be shown that the piezoelectric ceramic II is more complaisant than piezoelectric ceramic I (ρc<1), then khomτ>kintτ. In contrast if ρc>1 meaning that piezoelectric ceramic II is stiffer than piezoelectric ceramic I, in this case khomτ<kintτ.Other normalized field intensity factors are presented on Figures12 and 13.Variation ofkintγ, kintD (in C/m2) and kintE (in 106 kV/m) against ρc with ρe=ρε=1.
(a)(b)(c)Variation ofkintγ,kintD (in C/m2) and kintE (in 106 kV/m) against ρe with ρc=ρε=1.
(a)(b)(c)ThekintE is equal to zero for ρc=1 (Figure 12) and for ρe=1 (Figure 13). From (81) one finds that kintE=0 occurs only when c44II/c44I=e15II/e15I. In Figure 13 we see that ρe has a strong influence on kintD and kintE and kintγ if ρe<1, and when ρe=1,kintγ=1,kintE=0, and kintD=e15I, as expected.Figure14 presents the variation of normalized ERRs, G/Ghom obtained from (87) with the use of (107).Figure 14
Normalized ERRs,G/Ghom as a function of α: G/Ghom=(2α2/(1+λ))[1+82α-1(1-1/α)2(1-λ)/(1+λ)], λ=-cos(πα), α=3/4, λ=1/2, G/Ghom=(112-10)/8, α=1/4, λ=-1/2, G/Ghom=(67+472)/16.There are two states whereG=Ghom. The first state, in which α=1/2 and λ=0, that is, c44II=c44I, corresponds to crack in monolithic medium (no bi-material). The second state, in which α and λ tend to unity, corresponds to edge crack problem (the second material is air). For α>1/2 ERRs decrease weakly from 1 to 0,69 for α=3/4 and later increase to unity for α→1. In this case the piezoelectric ceramic II is more elastically complaisant. The range 0<α<1/2 corresponds to the following cases: ρc>1 or ρc>(1-m)/(1+m) (in the case ρe=-1);ρe>-1/m and ρε>0 (for any ε). Then always G>Ghom and piezoelectric II are stiffer than piezoelectric I. Similar conclusions may be formulated for magnetostrictive material, changing material parameters e15 and ε11 by q15 and μ11, respectively.


#### 5.2.1. A Crack between a Piezoelectric Material and a Piezomagnetic Material

Magnetoelectroelastic materials usually comprise alternating piezoelectric medium and piezomagnetic medium. Here, we consider a special case. This is a right medium I that is a piezoelectric and the left medium II is a piezomagnetic (Case I) or inversely (Case II). The material constants of the piezoelectric medium (No. I) and piezomagnetic medium (No. II) have the following values [16–18]:BaTiO3-piezoelectric (barium titanate)(113)c44I=43×109Nm-2,e15I=11,6Cm-2,q15I=0,ε11I=11,2×10-9CV-1m-1,d11I=0,μ11I=5,0×10-6NA-2CoFe2O4-piezomagnetic (cobalt iron oxide)(114)c44II=45,3×109Nm-2,e15II=0,q15II=550NA-1m-1,ε11II=0,08×10-9CV-1m-1,d11II=0,μ11II=590×10-6NA-2.The material parameter (57) assumes the values(115)λ=1-2c44J/(c44I+(e15I)2/(ε11I+ε11II))+c44J/(c44II+(q15II)2/(μ11I+μ11II));J=I,II.where c44J is the shear modulus of the cracked material, for Case I and Case II, respectively. We have(116)λ={-0,1618,CaseI-0,1028,CaseIIα={0,4483,CaseI0,4672,CaseII.The energy release rates are obtained as follows(117)G={πa(15,0)×10-12(τ0I)2×[m2/N],CaseIπa(12,9)×10-12(τ0I)2×[m2/N],CaseII.For “homogenous” composite BaTiO3/CoFe2O4 with the ratio roughly 50 : 50, we have with the use of arithmetic mean c44=44,15×109Nm-2, and Ghom assumes the value(118)Ghom=πa(11,4×10-12(τ0I)2)×[m2/N].We see that ERRs for bi-materials cannot be determined by the mixture rule since it is a significant new feature in interface crack problem considered in this paper.Obviously for piezoelectric/piezomagnetic composite (I/II) isμ11I≪μ11II and ε11II≪ε11I, and (115) reduces to the following formula:(119)λ=1-c̃44*c44J;J=I,II,
where c̃44* is the harmonic mean of the piezoelectric and piezomagnetic stiffened elastic constants c̃44I and c̃44II defined as follows:(120)1c̃44*=12(1c̃44I+1c̃44II),
where(121)c̃44I=c44I+(e15I)2ε11I,c̃44II=c44II+(q15II)2μ11II.Using (119) to (121), we obtain that(122)λ={-0,1626,CaseI-0,1036,CaseIIα={0,4480,CaseI,0,4670,CaseII,G={πa(15,0)×10-12(τ0I)2×[m2/N],CaseI,πa(12,9)×10-12(τ0I)2×[m2/N],CaseII.


## 5.1. Effect of Material Constants on the Singularity Order

We now consider the dependence of the singularity order on2×6-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 andc44II=ρcc44I,e15II=ρee15I, and ε11II=ρεε11I, and analyze the situations(a)
ρc changes and ρe=ρε=1, that is, not change,(b)
ρe changes and ρc=ρε=1,(c)
ρε changes and ρc=ρe=1.This states that it is analyzed that right half-plane is fixed, and left one contains a fictitious material with only changingρc or ρe or ρε.(a) The changes of the ratioρc of c44II to c44I: we have(96)λ=(1-ρc)(1+m)1+ρc+2m,m=((e15)2c44ε11)I,|λ|<1,ρc<3+2mforρe=ρε=1
or(97)λ=1-ρc1+ρc-m,λ=0forρc=1-m1+m,ρc<2m-1,0<m<1forρe=-1,ρε=1.Figure3 shows the effects of varying elastic stiffness ρc on λ and α with unchanging piezoelectric and piezomagnetic constants ρe=ρε=1 or ρe=-1 and ρε=1. Note that λ=0 and α=1/2 for ρe=-1,ρε=1 and if (c44I-c44II)c44Iε11I=(c44I+c44II)(e15I)2,e15II=-e15I or if ρc=1 and ρe=ρε=1. Note also that λ(ρe=1,ρε=1)>λ(ρe=-1,ρε=1) for all of ρc.The singularity orderα is larger for two of the same ceramics poled in opposite directions together since α(ρe=1,ρε=1)<α(ρe=-1,ρε=1).We take six kinds of particular piezoelectric ceramics as representatives, the relevant material constants and parametersm, and 1/m 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 m and 1/m.
c44 [109 N/m2]e15 [C/m2]ε11 [10−9 C/Vm]m1/mBaTiO343,011,6011,200,2794,348PZT-5H35,317,0015,100,5421,844PZT-425,613,446,001,1750,851P-725,013,5017,100,4302,325C-20587,013,597,950,2104,761PZT-PIC15120,012,009,820,7331,364(b) The changes of the ratioρe of e15II to e15I: we have(98)λ=-(1-ρe)24/m+(1+ρe)2,ρe>-1m=-(c44ε11(e15)2)I,λmax⁡=0forρe=1,λmin⁡=-1-mforρe=-1-2m<-1m,λ=-1forρe=-1m,λ=-m4+mforρe=0λ=-mforρe=-1.For-1/m<ρe<1, the singularity parameter λ increases from −1 to maximum λ=0 and for ρe>1 declines to −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 (-z)-direction, then to satisfy the condition λ>-1 must hold |e15II(-)|/e15I(+)<1/m or |e15II(-)|e15I(+)<c44Iε11I. If the selection of e15II violates the condition |λ|<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) Forρε varying and other parameter unchanged, it is easily found that λ=0 and α=0,5 for ρc=ρε=1 and varying ρε. But if ρe=-1, then(99)λ=-2m1+ρε;m=((e15)2c44ε11)I.Figure5 shows the variation of λ and α with the ratio ρε for e15II(-)/e15I(+)=-1.The parameterλ assumes negative values and increases from -2m to zero with ρε>0. The singularity parameter α is positive and increases from (1/π)arccos(2m) to 1/2 with ρε>0. Note that 2m must be less unity if ρε tends to zero or m<1 for ρε>1. Some materials shown in Table 1 limit the range of ρε; for example, PZT-4 has m=1,175, and ρε must be larger 1,35 to ensure that λ<-1. Of course this situation is addressed to two piezoelectrics poled in opposite directions.For piezomagnetic materials, the parameterm is(100)m=((q15)2c44μ11)I,
and for magnetostrictive material CoFe2O4 assumes the value m=0,0113.For CoFe2O4 we have(101)c44=45,3GPa,q15=550N/Am,μ11=590×10-6N/A2.The “relative sensitivity” analysis includes three cases:(a) The changes of ratioρc of c44II to c44I: we have(102)λ=1,0113(1-ρc)1,0226+ρc,ρc<20,7ρq=ρμ=1
or(103)λ=1-ρc1+ρc-0,0113,ρc<16,7ρq=-1,ρμ=1.
Approximately,(104)λ=1-ρc1+ρc,ρc<16.For Magnetoelectroelastic composite BaTiO3-CoFe2O4  (Vf=0,5)(q15)2/c44μ11=0,005 and (e15)2/c44ε11≈0,135.Figure6 shows the effect of ρc on λ and α for CoFe2O4 magnetostrictive ceramic.For both poling directions, the values ofλ and α are the same.(b) The changes of the ratioρq of q15II to q15I: we have(105)λ=-(1-ρq)235,4+(1+ρq)2,ρq>-8,85,λmax⁡=0forρq=1,λ=-1forρq=-8,85.
Figure 7 shows the effect of ρq on λ and α for CoFe2O4 ceramic.(c) The changes of the ratioρμ of μ15II to μ15I: we have(106)λ=-0,02261+ρμforρq=-1,λ=0forρq=1,alwaysρc=1.
Figure 8 shows the effect of ρμ on λ and α for CoFe2O4.


## 5.2. Effect of Material Constants on the Field Intensity Factors

The material constants also affect the intensity factors. Figure9 presents the variation of normalized SIFs khomτ and kintτ defined by (89) and (90) which depend on α and λ(107)khomτ=2α1+λ,kint⁡τ=23α-1/21-α1+λ1-λ.
For 0<α<1khomτ increasing monotonously from 22/π through 1 to 2 as α tends to zero and equals 1/2 and 1, respectively. From Figures 8 and 2, one can observe that the effect of ρc on kintτ is more evident than that on khomτ. Moreover, ρc increased the singularity parameter α that decreases (see Figure 3), and kintτ rises suddenly, while khomτ falls down slightly. For ρc<1 and ρe=1 or ρc<(1-m)/(1+m) and ρe=-1, we have α>1/2. This means that if piezoelectric II is more elastically complaisant than piezoelectric ceramic I, in this case khomτ>kintτ. On the other hand, for ρc>1, which gives α<1/2, meaning that piezoelectric II is stiffer than piezoelectric I in this case khomτ<kintτ. From the Figures 3, 4, and 5, we see that the range 0<α<1/2 corresponds to ρc>1 or ρc>(1-m)/(1+m) (in the case ρe=-1), ρe>-1/m, and ρε>0. Then always kintτ>khomτ. The range 1/2<α<1 is for 0<ρc<1 or 0<ρc<(1-m)/(1+m) (in the case ρe=-1). Then, kintτ<khomτ for all of α.Figure 2
The curveλ=-cos(πα);λ is the bi-material parameter, and α is singularity order parameter.Effect ofρc on λ and α with ρe=ρε=1 (Case I) and ρe=-1, ρε=1 (Case II).
(a)
Case I(b)
Case IIFigure 4
Effects ofρe on λ and α with ρc=ρε=1 (ρe>0 or ρe<0 denote piezoelectric ceramics poled parallel to or antiparallel to the z-axis, respectively, i.e., ρe=-1 denotes that e15I=-e15II=e15).Figure 5
Effect ofρε on λ and α with ρc=1 when ρe=e15II(-)/e15I(+)=-1; for ρe=1, we have λ=0 and α=0,5.Figure 6
Effect ofρc on λ and α for CoFe2O4, ρμ=1 and ρq=1 or ρq=-1.Figure 7
Effects ofρq on λ and α for CoFe2O4, ρc=ρμ=1.Figure 8
Effects ofρμ on λ and α for CoFe2O4, ρq=1 or ρq=-1 and ρc=1.Figure 9
Normalized SIFs as a function ofα.Note that the caseα=1,λ=1 gives the limiting values kintτ=0 and khomτ=2 which gives(108)Kint⁡τ=0,Khomτ=τ0Iπa.This is the solution for edge crack of lengtha.The normalized intensity factors for strain, electric displacement, magnetic induction, electric field, and magnetic field at the interface crack tip are defined by (78) and by the following formula:(109)kint⁡q=c44IτoIKint⁡q(πa/2)α,
where q stands for one of γ, D, B, E, and H.Then we have(110)[kint⁡γ;kint⁡D;kint⁡B;kint⁡E;kint⁡H]=2(1-α)8α(1+λ)1-λ[λγ;e15IλD;q15IλB;λE;λH],
respectively.Of course, the normalized intensity factors satisfy the constitutive equations (5), that is,
(111)[kint⁡τ;kint⁡D;kint⁡B]=CII[kint⁡γ;-kint⁡E;-kint⁡H]
with the matrix (6) or inverse form with the use of matrix (CII)-1, defined by (17).The analysis above implies that, for the magnetically (or electrically) permeable interfacial cracks, the applied magnetic (or electric) loadings have no influence on the fracture behaviours of the crack tips.Figures10 and 11 are devoted to the variation of kintτ and khomτ.Normalized SIFs as a function ofρc with ρe=ρε=1 (Case I) and ρe=-1, ρε=1 (Case II).
(a)
Case I(b)
Case IINormalized SIFs as a function ofρe with ρc=ρε=1.
(a)(b)We have(112)khomτ=22πforρc=3+2m(ρe=ρε=1)orρc=2m-1(ρe=-1,ρε=1).The figures show that the normalized stress intensity factor in homogeneous solid is only weakly dependent on the elastic constants and dielectric permeabilities. In contrastkintτ strongly depends on ρc and ρe. This is consistent with physical considerations; for large difference of piezocoefficients ρe<0 or ρe>1 the kintτ are larger than khomτ (Figure 11). From the Figure 10 it can be shown that the piezoelectric ceramic II is more complaisant than piezoelectric ceramic I (ρc<1), then khomτ>kintτ. In contrast if ρc>1 meaning that piezoelectric ceramic II is stiffer than piezoelectric ceramic I, in this case khomτ<kintτ.Other normalized field intensity factors are presented on Figures12 and 13.Variation ofkintγ, kintD (in C/m2) and kintE (in 106 kV/m) against ρc with ρe=ρε=1.
(a)(b)(c)Variation ofkintγ,kintD (in C/m2) and kintE (in 106 kV/m) against ρe with ρc=ρε=1.
(a)(b)(c)ThekintE is equal to zero for ρc=1 (Figure 12) and for ρe=1 (Figure 13). From (81) one finds that kintE=0 occurs only when c44II/c44I=e15II/e15I. In Figure 13 we see that ρe has a strong influence on kintD and kintE and kintγ if ρe<1, and when ρe=1,kintγ=1,kintE=0, and kintD=e15I, as expected.Figure14 presents the variation of normalized ERRs, G/Ghom obtained from (87) with the use of (107).Figure 14
Normalized ERRs,G/Ghom as a function of α: G/Ghom=(2α2/(1+λ))[1+82α-1(1-1/α)2(1-λ)/(1+λ)], λ=-cos(πα), α=3/4, λ=1/2, G/Ghom=(112-10)/8, α=1/4, λ=-1/2, G/Ghom=(67+472)/16.There are two states whereG=Ghom. The first state, in which α=1/2 and λ=0, that is, c44II=c44I, corresponds to crack in monolithic medium (no bi-material). The second state, in which α and λ tend to unity, corresponds to edge crack problem (the second material is air). For α>1/2 ERRs decrease weakly from 1 to 0,69 for α=3/4 and later increase to unity for α→1. In this case the piezoelectric ceramic II is more elastically complaisant. The range 0<α<1/2 corresponds to the following cases: ρc>1 or ρc>(1-m)/(1+m) (in the case ρe=-1);ρe>-1/m and ρε>0 (for any ε). Then always G>Ghom and piezoelectric II are stiffer than piezoelectric I. Similar conclusions may be formulated for magnetostrictive material, changing material parameters e15 and ε11 by q15 and μ11, respectively.


### 5.2.1. A Crack between a Piezoelectric Material and a Piezomagnetic Material

Magnetoelectroelastic materials usually comprise alternating piezoelectric medium and piezomagnetic medium. Here, we consider a special case. This is a right medium I that is a piezoelectric and the left medium II is a piezomagnetic (Case I) or inversely (Case II). The material constants of the piezoelectric medium (No. I) and piezomagnetic medium (No. II) have the following values [16–18]:BaTiO3-piezoelectric (barium titanate)(113)c44I=43×109Nm-2,e15I=11,6Cm-2,q15I=0,ε11I=11,2×10-9CV-1m-1,d11I=0,μ11I=5,0×10-6NA-2CoFe2O4-piezomagnetic (cobalt iron oxide)(114)c44II=45,3×109Nm-2,e15II=0,q15II=550NA-1m-1,ε11II=0,08×10-9CV-1m-1,d11II=0,μ11II=590×10-6NA-2.The material parameter (57) assumes the values(115)λ=1-2c44J/(c44I+(e15I)2/(ε11I+ε11II))+c44J/(c44II+(q15II)2/(μ11I+μ11II));J=I,II.where c44J is the shear modulus of the cracked material, for Case I and Case II, respectively. We have(116)λ={-0,1618,CaseI-0,1028,CaseIIα={0,4483,CaseI0,4672,CaseII.The energy release rates are obtained as follows(117)G={πa(15,0)×10-12(τ0I)2×[m2/N],CaseIπa(12,9)×10-12(τ0I)2×[m2/N],CaseII.For “homogenous” composite BaTiO3/CoFe2O4 with the ratio roughly 50 : 50, we have with the use of arithmetic mean c44=44,15×109Nm-2, and Ghom assumes the value(118)Ghom=πa(11,4×10-12(τ0I)2)×[m2/N].We see that ERRs for bi-materials cannot be determined by the mixture rule since it is a significant new feature in interface crack problem considered in this paper.Obviously for piezoelectric/piezomagnetic composite (I/II) isμ11I≪μ11II and ε11II≪ε11I, and (115) reduces to the following formula:(119)λ=1-c̃44*c44J;J=I,II,
where c̃44* is the harmonic mean of the piezoelectric and piezomagnetic stiffened elastic constants c̃44I and c̃44II defined as follows:(120)1c̃44*=12(1c̃44I+1c̃44II),
where(121)c̃44I=c44I+(e15I)2ε11I,c̃44II=c44II+(q15II)2μ11II.Using (119) to (121), we obtain that(122)λ={-0,1626,CaseI-0,1036,CaseIIα={0,4480,CaseI,0,4670,CaseII,G={πa(15,0)×10-12(τ0I)2×[m2/N],CaseI,πa(12,9)×10-12(τ0I)2×[m2/N],CaseII.


## 5.2.1. A Crack between a Piezoelectric Material and a Piezomagnetic Material

Magnetoelectroelastic materials usually comprise alternating piezoelectric medium and piezomagnetic medium. Here, we consider a special case. This is a right medium I that is a piezoelectric and the left medium II is a piezomagnetic (Case I) or inversely (Case II). The material constants of the piezoelectric medium (No. I) and piezomagnetic medium (No. II) have the following values [16–18]:BaTiO3-piezoelectric (barium titanate)(113)c44I=43×109Nm-2,e15I=11,6Cm-2,q15I=0,ε11I=11,2×10-9CV-1m-1,d11I=0,μ11I=5,0×10-6NA-2CoFe2O4-piezomagnetic (cobalt iron oxide)(114)c44II=45,3×109Nm-2,e15II=0,q15II=550NA-1m-1,ε11II=0,08×10-9CV-1m-1,d11II=0,μ11II=590×10-6NA-2.The material parameter (57) assumes the values(115)λ=1-2c44J/(c44I+(e15I)2/(ε11I+ε11II))+c44J/(c44II+(q15II)2/(μ11I+μ11II));J=I,II.where c44J is the shear modulus of the cracked material, for Case I and Case II, respectively. We have(116)λ={-0,1618,CaseI-0,1028,CaseIIα={0,4483,CaseI0,4672,CaseII.The energy release rates are obtained as follows(117)G={πa(15,0)×10-12(τ0I)2×[m2/N],CaseIπa(12,9)×10-12(τ0I)2×[m2/N],CaseII.For “homogenous” composite BaTiO3/CoFe2O4 with the ratio roughly 50 : 50, we have with the use of arithmetic mean c44=44,15×109Nm-2, and Ghom assumes the value(118)Ghom=πa(11,4×10-12(τ0I)2)×[m2/N].We see that ERRs for bi-materials cannot be determined by the mixture rule since it is a significant new feature in interface crack problem considered in this paper.Obviously for piezoelectric/piezomagnetic composite (I/II) isμ11I≪μ11II and ε11II≪ε11I, and (115) reduces to the following formula:(119)λ=1-c̃44*c44J;J=I,II,
where c̃44* is the harmonic mean of the piezoelectric and piezomagnetic stiffened elastic constants c̃44I and c̃44II defined as follows:(120)1c̃44*=12(1c̃44I+1c̃44II),
where(121)c̃44I=c44I+(e15I)2ε11I,c̃44II=c44II+(q15II)2μ11II.Using (119) to (121), we obtain that(122)λ={-0,1626,CaseI-0,1036,CaseIIα={0,4480,CaseI,0,4670,CaseII,G={πa(15,0)×10-12(τ0I)2×[m2/N],CaseI,πa(12,9)×10-12(τ0I)2×[m2/N],CaseII.


## 6.  Conclusions

A crack perpendicular to and terminating at the interface of two bonded dissimilar piezo-electromagnetoelastic media are studied in this paper. Analytical solutions and numerical simulations suggest the following conclusions.(i)
Closed form solution has been obtained for a crack between two dissimilar magneto electro-elastic ceramics. The crack is localized in one materials, and its one tip lies on the interface. Expressions for the crack-tip field intensity factors, the electromagnetic fields inside the crack, are given for electrically and magnetically permeable crack assumptions.(ii)
The energy release rate can be explicitly expressed in terms of the intensity factors. It is affected by electric-magnetic properties of the constituents of the bi-material media. The normalized energy release rate is unity for homogeneous medium (ρc=1) and for edge crack (ρc=0) and assumes minimum value 0,69 for ρc=3-22= 0,18. If ρc tends to infinity, also this quantity tends to infinity (the interface is clamped).(iii)
For two identical Magnetoelectroelastic planes polarized in opposite directions, we haveKintD=0=KintB.(iv)
At interface we haveKintE=0 when c44II/c44I=e15II/e15I, while KintH=0 if c44II/c44I=q15II/q15I.(v)
Application of electric and magnetic fields do not alter the stress intensity factors; they depend on the elastic, electric, and magnetic constants of bi-material ceramic.(vi)
The coupling between electromagnetic fields and mechanical field leads to existing electric displacement and magnetic induction intensity factors at the crack tip, which respond to the applied stress intensity factor.(vii)
If magnetic effects are neglected, the result of the stress intensity factors is the same as the solution for the piezoelectric materials given by Li and Wang [9], but kintE differs in sign.The results could be of particular interest to the analysis and design of smart sensors and actuators constructed from Magnetoelectroelastic composite laminates. Nowadays, electromagnetoelastic coupled multiphase composite has wide range applications in science and engineering such as space planes, supersonic air planes, rockets, missiles nuclear fusion, reactors, and submarines.



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*Source: 659352-2012-06-19.xml*