# Exact Solution for an Anti-Plane Interface Crack between Two Dissimilar Magneto-Electro-Elastic Half-Spaces

Authors: Bogdan Rogowski
Journal: Smart Materials Research
 (2012)

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## Abstract

This paper investigated the fracture behaviour of a piezo-electro-magneto-elastic medium subjected to electro-magneto-mechanical loads. The bimaterial medium contains a crack which lies at interface and is parallel to their poling direction. Fourier transform technique is used to reduce the problem to three pairs of dual integral equations. These equations are solved exactly. The semipermeable crack-face magneto-electric boundary conditions are utilized. Field intensity factors of stress, electric displacement, magnetic induction, cracks displacement, electric and magnetic potentials, and the energy release rate are determined. The electric displacement and magnetic induction of crack interior are discussed. Obtained results indicate that the stress field and electric and magnetic fields near the crack tips exhibit square-root singularity.


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## Body

## 1. Introduction

Mechanics of magneto-electro-elastic solid has gained considerable interest in the recent decades with increasable wide application of piezoelectric/piezomagnetic composite materials in engineering, particularly in aerospace and automotive industries. Recently, much attention has been paid to dislocation crack and inclusion problems in magneto-electro-elastic solids, which simultaneously possess piezoelectric, piezomagnetic, and magnetoelectric effects. Therefore, it is of vital importance to investigate the magneto-electro-elastic fields as a result of existence of defects, such as cracks, in these solids.The mode III interface crack solution for two dissimilar half-planes has been analyzed by Li and Kardomateas [1] while for two dissimilar layers by Wang and Mai [2] and by Li and Wang [3]. Li and Kardomateas [1] solved corresponding plane problem by means of Stroh’s formalism and complex variable methods. Li and Wang [3] investigated the problems involving an antiplane shear crack perpendicular to and terminating at the interface of a bimaterial piezo-electro-elastic ceramics. Wang and Mai [2] investigated the mode III-crack problem for a crack at the interface between two dissimilar magneto-electro-elastic layers. Extension of those investigation problems interested in fracture theory, on dielectric and magnetostrictive crack behaviour, is very important, and results and conclusions could have applications in the failure of PEMO-elastic devices and in smart intelligent structures [4].However, to the authors’ best knowledge, no researches dealing with the interface-dielectric crack in PEMO-elastic two-phase composite have been reported in literature. When subjected to mechanical, electrical, and magnetic loads in service, these magneto-electro-elastic composites can fail prematurely due to some defects, namely, cracks, holes and others, arising during their manufacturing process. Therefore, it is of great importance to study the magneto-electro-elastic interaction and fracture behaviours of PEMO-elastic materials.In mechanic where two-phase composites have twelve material constants only exact solutions are useful. This is motivation for this study. For electrical, magnetic, and mechanical loads (two cases of electrical and magnetic excitations) and semipermeable electrical and magnetic boundary conditions at the interface crack, exact analytical solutions are obtained here for full field interesting in fracture mechanics.


## 2. Basic Equations

For a linearly magneto-electro-elastic 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 electrical and magnetic field components Ex, Ey, Hx and Hy(3)Ex=-∂ϕ∂x,Ey=-∂ϕ∂y,Hx=-∂ψ∂x,Hy=-∂ψ∂yand electrical displacement componentsDx, Dy, and magnetic induction components Bx, By with all field quantities being the functions of coordinates x and y.The generalized strain-displacement relations (2) and (3) have the form(4)γαz=w,α,Eα=-ϕ,α,Hα=-ψ,α,
where α=x,y and w,α=∂w/∂α.For linearly magneto-electro-elastic medium, the coupled constitutive relations can be written in the matrix form(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]
is the material property matrix, 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), 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 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(8)(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 the matrix C0, a principal submatrix of C, is(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]=[c1c2c2c3].The relevant field variables are(14)τzk=c̃44w,k-αDk-βBk,ϕ=αw+c1χ+c2η,ψ=βw+c2χ+c3η,Dk=χ,k,Bk=η,k,k=x,y,(15)∇2w=0,∇2χ=0,∇2η=0.The two last equations (15) are equivalent to (9) since c1c3-c22=1/(ε11μ11-d112)≠0. In (14) the material parameters are defined as follows:(16)c̃44=c44+αe15+βq15,α=μ11e15-d11q15ε11μ11-d112=-(c1e15+c2q15),β=ε11q15-d11e15ε11μ11-d112=-(c3q15+c2e15).Note thatc̃44 is the piezo-electro-magnetically stiffened elastic constant.Note also that the inverse of a matrixCis defined by parameters α, β, c̃44 and c1, c2, c3 as follows:(17)C-1=1c̃44[1αβαα2+c̃44c1αβ+c̃44c2βαβ+c̃44c2β2+c̃44c3]
and is the matrix generalized compliances of PEMO-elastic material. These material parameters will be appear in our solutions.


## 3. Formulation of the Crack Problem

Let the medium I occupy the upper half-space and medium II be in the lower half-space; the interface crack is assumed to be located in the region from-a to +a along the x-axis. The two-phase composite is subjected to electric, magnetic, and mechanical loads applied at infinity. These are (τ0,D0,B0) or (τ0,E0,H0). Under applied external loading, the crack, filled usually by vacuum or air, accumulated an electric and magnetic field, denoted by d0 and b0, would be built up. By the superposition principle, the interface crack problem is equivalent to the one under the applied loading on the upper surface (Figure 1):(18)τzy(x,0±)=-τ0,Dy(x,0±)=-D+d0,By(x,0±)=-B+b0;|x|<a,
where(19)Dy=D={D0,caseIe15c44τ0+(ε11+e152c44)E0+(d11+e15q15c44)H0,caseII(20)By=B={B0,caseIq15c44τ0+(d11+e15q15c44)E0+(μ11+q152c44)H0,caseII.Figure 1
The interface crack under antiplane mechanical and in-plane electric and magnetic load. Inside the crack the unknown electromagnetic fieldsd0 and b0 appear. (a) Perturbation problem, (b) Elementary solution for bimaterial without the crack (19) and (20).Similarly is for lower crack surface, where the material parameters and electro-magnetic loadings are denoted by prime.To guarantee the continuity of physical quantities at the perfectly bonded interface, applied electro-magnetic loadingsE and H must obey the relations from which the loadings of upper material, namely, E0 and H0 may be determined by means of loading of lower material, namely, E0′ and H0′, using (19) and (20). Of course, D0=D0′ and B0=B0′ in Case I of loading and E0=E0′ and H0=H0′ for homogeneous medium only.At the interfacey=0±, we have the conditions(21)⌊|τzy|⌋=0,⌊|Dy|⌋=0,⌊|By|⌋=0,|x|<∞,[|w|]=0,[|ϕ|]=0,[|ψ|]=0,|x|≥a,
where the notation [|f|]=f+-f- and f+ denotes the value for 0+ while f- for 0-.The electric displacementd0 and magnetic induction b0 inside the crack are obtained from semipermeable crack-face boundary conditions [5]. For two different magnetoelectric media: PEMO-elastic material I and notch space, we have the continuity condition of electric and magnetic potential in both materials at interfaces, similarly for interface between second PEMO-elastic material and crack interior. The semi-permeable crack-face magnetoelectric boundary conditions are expressed as follows:(22)d0=-εc[|ϕ|]2δ(x),b0=-μc[|ψ|]2δ(x),
where δ(x) describes the shape of the notch, and εc, μc are the dielectric permittivity and magnetic permeability of crack interior. If we assume the elliptic notch profile such that(23)δ(x)=δ0aa2-x2,
where δ0 is the half-thickness of the notch at x=0, we obtain that(24)2d0δ0aεca2-x2=-[|ϕ|],2d0δ0aμca2-x2=-[|ψ|];|x|≤a.Equations (24) form two coupling linear equations with respect to d0 and b0 since [|ϕ|] and [|ψ|] depend linearly on these quantities as shown boundary conditions (18) and (21).


## 4. The Solution for Two-Phase Medium with the Discontinuity at Interface

Define the Fourier transform pair by the equations(25)f̂(s)=∫0∞f(x)cos⁡(sx)dx,f(x)=2π∫0∞f̂(s)cos⁡(sx)ds.Then (15) are converted to ordinary differential equations and their solutions(26)ŵ(s,y)=A(s)e-sy,y≥0,ŵ(s,y)=B(s)esy,y≤0,χ̂(s,y)=C(s)e-sy,y≥0,χ̂(s,y)=-C(s)esy,y≤0,ψ̂(s,y)=D(s)e-sy,y≥0,ψ̂(s,y)=-D(s)esy,y≤0.These solutions satisfy the regularity conditions at infinity and the conditions of vanishing jumps of electric displacement and magnetic induction at interface and crack surfaces.From (21)1 we obtain that(27)B(s)=-c̃44c̃44′A(s)+α-α′c̃44′C(s)+β-β′c̃44′D(s).The material parameters of the lower material are denoted by prime.The mixed boundary conditions on the crack plane and outside lead to(28)∫0∞s{A(s)C(s)D(s)}cos⁡(sx)ds={τ0+α(D-d0)+β(B-b0)c̃44D-d0B-b0},|x|<a∫0∞s{A(s)C(s)D(s)}cos⁡(sx)ds={000},|x|≥a.Using the integrals(29)∫0∞J1(as)cos⁡(sx)=1a{1;|x|<aa-|x|x2-a2,|x|≥a∫0∞J1(as)scos⁡(sx)={0;|x|≥ax2-a2a,|x|≤a
we see that the solutions of (28) are(30){A(s)C(s)D(s)}=asJ1(as){τ0+α(D-d0)+β(B-b0)c̃44D-d0B-b0}.We calculate that(31)[[|w|],[|ϕ|],[|ψ|]]T=2πa2-x2[C-1+C′-1]×[τ0,D-d0,B-b0]T,
where C-1 is defined by the matrix (17) and C′-1 by the same matrix with material parameters of second material.From the condition (24), we obtain that(32)[d0,b0]T=-12[ε000μ0][C1-1+C1′-1][τ0,D-d0,B-b0]T,
where(33)ε0=2πaδ0εc,μ0=2πaδ0μc,(34)C1-1=1c̃44[αα2+c̃44c1αβ+c̃44c2βαβ+c̃44c2β2+c̃44c3]
and similarly for C1′-1 (second material).


## 5. Field Intensity Factors

The singular behaviour ofτzy, Dy, and By at y=0, |x|→a+ are:(35)[τzy,Dy,By]T=2π|x|x2-a2[τ0,D-d0,B-b0]T.Defining the stress, electric displacement and magnetic induction intensity factors as follows:(36)[Kτ,KD,KB]T=lim⁡|x|→a+2(|x|-a)[τzy,Dy,By]T,
we obtain that(37)[Kτ,KD,KB]T=2πa[τ0,D-d0,B-b0]T.Furthermore, we obtain the displacement, electric and magnetic potentials intensity factors(38)[Kw,Kϕ,Kψ]T=lim⁡|x|→a-122(a-|x|)[[|w|],[|ϕ|],[|ψ|]]T.In view of results (31) and (37), we have(39)[Kw,Kϕ,Kψ]T=12[C-1+C′-1][Kτ,KD,KB]T.The energy release rate is defined as(40)G=12(KτKw+KDKϕ+KBKψ)
and is the following:(41)G=14[Kτ,KD,KB][C-1+C′-1][Kτ,KD,KB]T
or in explicit form(42)G=1π2a{(1c̃44+1c̃44′)τ02+(α2c̃44+α′2c̃44′+c1+c1′)(D-d0)2+(β2c̃44+β′2c̃44′+c3+c3′)(B-b0)2+2(αc̃44+α′c̃44′)τ0(D-d0)+2(βc̃44+β′c̃44′)τ0(B-b0)+2(αβc̃44+α′β′c̃44′+c2+c2′)(D-d0)(B-b0)}.For fully impermeable case, we haved0=0 and b0=0, and the solutions are obtained from (37) and (39). For fully permeable case, we have ε0→∞ and μ0→∞ and(43)D-d0=12(e15c44+e15′c44′)τ0,B-b0=12(q15c44+q15′c44′)τ0.The energy release rate is(44)G=1π2(1c44+1c44′)τ02a.Note that energy release rate (44) for fully permeable crack problem is defined by the harmonic mean of the shear moduli of both materials, that is,(44a)G=2π2τ02ac44*;1c44*=12(1c44+1c44′).The remaining field intensity factors are obtained, in this case, as follows:(45)Kτ=2πτ0a,KD=1π(e15c44+e15′c44′)τ0a,KB=1π(q15c44+q15′c44′)τ0a.In particular, for fully permeable crack between two PEMO-elastic materials polarized in opposite directions, we haveKD=0 and KB=0, since e15=-e15′ and q15=-q15′ in this case.For electrically impermeable and magnetically permeable crack, the solutions are independent of the applied magnetic field (d0=0 and B-b0 is independent on B for ε0→0 and μ0→∞ as shown in (32)).Alternatively, the solutions for the electrically permeable and magnetically impermeable crack are independent on the applied electric displacement.In practical applications the following cases appear:(i)
Letε0 tends to infinity and μ0 is finite.Then(46)KDperm⋅μc=KDperm⋅imp[1-f1(μ̅)]+KDperm⋅permf1(μ̅),KBperm⋅μc=KBimp⋅imp[1-f1(μ̅)]+KBperm⋅permf1(μ̅),
where
(47)f1(μ̅)=11+μ̅,μ̅=π2δ0a1μc12(μ11+q152c44+μ11′+q′152c44′).(ii)
Letμ0 tends to infinity and ε0 is finite.Then(48)KDεc⋅perm=KDimp⋅imp[1-f2(ε̅)]+KDperm⋅permf2(ε̅),KBεc⋅perm=KBimp⋅perm[1-f2(ε̅)]+KBperm⋅permf2(ε̅),
where(49)f2(ε̅)=11+ε̅,ε̅=π2δ0a1εc12(ε11+e152c44+ε11′+e′152c44′).The functions of permittivity (εc) and permeability (μc) approaches zero as εc and μc tend to zero and are unity as εc and μc tend to infinity. The solution perfectly matches exact solution in both limiting cases, namely, permeable and/or impermeable electric and/or magnetic boundary conditions.In above equations the notationKperm·imp denotes the intensity factor for electrically permeable and magnetically impermeable crack boundary conditions.The electric displacementd0 and magnetic induction b0 in the crack region depend on the matrix(50)C̅1-1=12(C1-1+C1′-1)
as well as KD, KB, Kϕ, Kψ, and Kw depend on the matrix(51)C̅-1=12(C-1+C′-1),
where again “−1” denotes the inverse matrix.Thus,(52)C̅=[c̅44e̅15q̅15e̅15-ε̅11-d̅11q̅15-d̅11-μ̅11]=[12(C-1+C′-1)]-1
is the matrix of material property of equivalent homogeneous material after homogenization in our problem. The generalized effective electroelastic compliances of bi-material system are obtained as arithmetically mean of compliances of single materials constituents. If the lower medium and the upper medium have the same properties but are poled in opposite directions, then α=-α′ and β=-β′ (see (16)). In consequence from (17), we have(53)C̅-1=1c̃44[1000α2+c1c̃44αβ+c2c̃440αβ+c2c̃44β2+c3c̃44].Then(54)C̅=c̃44Δ[Δ000β2+c3c̃44-(αβ+c2c̃44)0-(αβ+c2c̃44)α2+c1c̃44],Δ=(α2+c1c̃44)(β2+c3c̃44)-(αβ+c2c̃44)2.Magneto-electro-elastic materials usually comprise alternating piezoelectric material and piezomagnetic material. If upper material is piezomagnetic and lower material is piezoelectric (or otherwise), we have(55)C=[c440q150-∞0q150-μ11],C′=[c44′e15′0e15′-ε11′000-∞].The bi-material matrixC̅-1 defined by (51) has the form(56)C̅-1=12[μ110q15000q150-c44]c44μ11+q152+12[ε11′e15′0e15′-c44′0000]c44′ε11′+e′152.In the solutions also appears electric and magnetic field components(57)[Ey,Hy]T=-C1-1[τ0,D,B]TincaseI,Ey=E0,Hy=H0incaseII,
where the matrix C1-1 is defined by (34). Of course, for lower material, we have C1′-1 matrix (the material parameters are denoted by prime). This states that, in general, electric and magnetic fields are also singular. The electric and magnetic field intensity factors KE and KH are related to Kτ, KD, and KB, as shown (57)1. In particular, for a fully permeable crack between two materials polarized in opposite directions, we have KD=0=KB and(58)KE=αc̃44Kτ,KH=βc̃44Kτ.The particular solutions(59)w(y)=γy,w′(y)=γ′y,ϕ(y)=-Eyy,ϕ′(y)=-Ey′y,ψ(y)=-Hyy,ψ′(y)=-Hy′y
with(60)γ=1c̃44(τ0+αD+βB),γ′=1c̃44′(τ0+α′D+β′B)
complete the full fields in both materials.


## 6. Numerical Results

The basic data for the material properties selected here are similar to those in Sih and Song [6, 7]. These constants read as: c44=43,7×109N/m2, e15=8,12C/m2, ε11=7,86×10-9C/Vm, d11=0,0, q15=165,0N/Am, and μ11=180,5×10-6N/A2 for first material and c44=44,6×109N/m2, e15=3,48C/m2, ε11=3,42×10-9C/Vm, d11=0,0, q15=385,0N/Am, μ11=414,5×10-6N/A2 for second material.Using these properties of both materials, the material property matrixC̅-1 is obtained as (the matrix of generalized “compliances”):(61)C̅-1=[19,88×10-12m2/N20,38×10-3m2/C18,33×10-6Am/N20,38×10-3m2/C-1,89×108Vm/C18,78×103Am/C18,33×10-6Am/N18,78×103Am/C-3,96×103A2/N].The matrix of generalized stiffness is obtained as follows:(62)C̅=[45,08×109N/m24,89C/m2231,8N/Am4,89C/m2-4,77×10-9C/Vm1,36×10-12C/Am231,8N/Am1,36×10-12C/Am-251×10-6N/A2].Therefore, the properties of composite, obtained by averaging the properties of single-phase materials using its volume fractions, as in the literature (see [8]) gives erroneous result, since give (if ratio is roughly 50 : 50)(63)C̅aver=[44,15×109N/m25,80C/m2275N/Am5,80C/m2-5,64×10-9C/Vm0275N/Am0-297,5×10-6N/A2].The nonzero material constants for BaTiO3-piezoelectric and CoFe2O4-magnetostrictive medium are given in Table 1 [9].Table 1
The material constants for BaTiO3 and CoFe2O4.
propertiesBaTiO3 piezoelectricCoFe2O4 piezomagneticc44(109N/m2)43,0045,30e15(C/m2)11,600,00ε11(10-9C/Vm)11,200,08q15(N/Am)0,00550,00μ11(10-6N/A2)5,00590,00d11(10-9C/Am)0,000,00The bi-material matrixC̅-1 defined by (51) is (“compliance” matrix)(64)C̅-1=[20,00×10-12m2/N94,13×10-4m2/C10,17×10-6Am/N94,13×10-4m2/C-6,28×109Vm/C010,17×10-6Am/N0-1,01×105A2/N].The matrixC̅ is obtained as follows: (“stiffness” matrix)(65)C̅=[49,96×109N/m274,82×10-3C/m25,04N/Am74,82×10-3C/m2-1,59×10-10C/Vm7,55×10-12C/Am5,04N/Am7,55×10-12C/Am-9,91×10-6N/A2].Using the mixture rule [6], κc=κVf+κ′(1-Vf), for Vf=0,5, where κ with superscripts c without prime or prime denotes the corresponding constants c44, e, ε, q, μ, d of the composite, first and second material, respectively, and Vf is the volume fraction of the first material (piezoelectric), we obtain that(66)C̅*=[44,15×109N/m25,80C/m2275N/Am5,80C/m2-5,64×10-9C/Vm0275N/Am0-297,5×10-6N/A2]which completely differs fromC̅.Note that in both examples the sums of corresponding material parameters are constant. In consequence the matrix,C̅aver and C̅* have the same elements. Of course, the matrices of generalized stiffness are dissimilar in both bi-material composites.Due to the absence of magnetoelectric coupling coefficient in a single-phase piezoelectric and piezomagnetic material, the magnetoelectric constantd11, existing only in the piezoelectric/piezomagnetic composite as a significant new feature, cannot be determined by the above mixture rule. Therefore, based on the analysis of micromechanics, this coefficient is obtained as d11=1,36×10-12C/Am for first combination of materials and 7,55×10-12C/Am for barium titanate-cobalt iron oxide bi-material. This is magnetoelectric coupling effect in composite of piezoelectric and piezomagnetic phases.


## 7. Conclusions

The mode III interface crack in a bi-material magneto-electro-elastic medium subjected to mechanical, electrical, and magnetic loads on the surfaces is studied in this paper, and the following points are noted.(i)
Closed form solution has been obtained for a crack between two dissimilar PEMO-elastic materials. Expressions for the crack-tip field intensity factors, the electromagnetic fields inside the crack, are given. The semipermeable crack-face magneto-electric boundary conditions are investigated.(ii)
The energy release rate can be explicitly expressed in terms of the external loadings (by (42)). It is affected by electric-magnetic properties of the two constituents of the bi-material media.(iii)
Applications of electric and magnetic fields do not alter the stress intensity factor of mode III. The values of SIF are identical for any kind of crack-face electric and magnetic boundary condition assumptions. In other words, the crack-face electric and magnetic boundary conditions have no effects on SIF.(iv)
For electrically impermeable and magnetically permeable crack, the solutions for field intensity factors are independent of the applied magnetic field. Alternatively, these solutions for the electrically permeable and magnetically impermeable crack are independent on the applied electric displacement.(v)
For fully permeable crack between two PEMO-elastic materials polarized in opposite directions, the electric displacement and magnetic induction intensity factors vanish. In this case electric and magnetic field intensity factorsKE and KB are related to Kτ (by (58)).(vi)
The matrices of “generalized” compliances or stiffness cannot be determined by the mixture rule since it is a significant new feature in interface crack problem considered in this paper.(vii)
From the reviewing of literature dealing with interface crack, problem may be concluded that the characterization of bonded dissimilar materials with interface crack is still an open problem.



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*Source: 786190-2012-04-29.xml*