1. bookVolume 6 (2021): Issue 1 (January 2021)
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Two equal collinear cracks in magneto-electro-elastic materials: A study of electric and magnetic poling influences

Published Online: 26 Jun 2020
Page range: 403 - 420
Received: 22 Aug 2020
Accepted: 04 Jan 2020
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

In this paper, the problem of two equal collinear cracks is analytically studied for two-dimensional (2D) arbitrarily polarized magneto-electro-elastic materials. The electric and magnetic poling directions make arbitrary angles with the crack line. The Stroh's formalism and complex variable methodology is utilized to reduce the problem into non-homogeneous Riemann Hilbert problem. This numerical problem is then comprehended with the Riemann Hilbert way to obtain the intensity factors for stress, electric displacement and magnetic induction. A numerical contextual analysis is displayed for the BaTiO3CoFe2O4 composite. The numerical examination demonstrates that the change in electric/magnetic poling directions influences the intensity factors.

Keywords

Introduction

Magneto-electro-elastic(MEE) composite materials prepared by combining the ferrite and ferroelectric materials. The coupling interaction among ferroelectric and ferrite produces an interesting effect which has drawn the enthusiasm of scientists. MEE composites have a wide scope of utilization in electronics industries. Ferrite ferroelectric composites could create both piezomagnetic and piezoelectric effects. Use of a magnetic field could incite electric polarization and an electric field could instigate magnetization. This is because of the strain or mechanical deformation caused by the ferrite and ferroelectric phase of the composite.

It is well-known that a wide range of poled MEE materials retains their aligned electric and magnetic dipole fields in the material microstructure below Curie temperature. Alignment of poling directions of piezomagnetic and piezoelectric materials can add to the complexity of the MEE material behavior to which this study will be concerned with. Various investigations have demonstrated that the electric/magnetic poling directions incredibly impact crack growth in piezoelectric/MEE media [1, 2, 3, 4, 5, 6, 7, 8].

In analogous to the piezoelectric materials, Sih et al. [7] explore the piezomagnetic and piezoelectric poling influence on mode I and II crack initiation behavior of MEE materials. They found that the alignment of the poling directions of piezomagnetic and piezoelectric materials influences the various fracture parameters. Sih and Song [8] demonstrated that the poling directions presented magnetically and electrically could be different notwithstanding those for the applied magnetic and electric field. Their choices could influence the character of crack growth which could be enhanced or retarded. In the proposed model [8] they found that crack growth in the MEE materials could be stifled by expanding the magnitude of the piezomagnetic constants about those for piezoelectricity. Song and Sih [9] investigated the crack initiation behavior in MEE composite under in-plane deformation. They found that the magnetostrictive effect could have an influence on crack initiation as the prescribed field directions changed.

The earlier research work on the fracture in MEE materials was mainly on single crack, while in this paper we have investigated the problem for two collinear cracks in MEE media.

Considering the significance of the electric poling and magnetic poling in MEE materials, we propose and investigate the influence of the electric/magnetic poling on two equal collinear cracks in MEE media.

General equations

In-rectangular Cartesian coordinate system xi(i = 1,2,3) the stress, σij, electric displacement, Di, and magnetic induction, Bi, are linearly related to the strain, ɛij, electric field, Ei, and magnetic field, Hi,. A set of constitutive equations for a MEE material would thus be able to be written in the form: σij=CijksεksesijEshsijHsDi=eiksεks+αisEs+βisHsBi=hiksεks+βisEs+γisHs \matrix{{{\sigma _{ij}} = {C_{ijks}}{\varepsilon _{ks}} - {e_{sij}}{E_s} - {h_{sij}}{H_s}} \hfill \cr {{D_i} = {e_{iks}}{\varepsilon _{ks}} + {\alpha _{is}}{E_s} + {\beta _{is}}{H_s}} \hfill \cr {{B_i} = {h_{iks}}{\varepsilon _{ks}} + {\beta _{is}}{E_s} + {\gamma _{is}}{H_s}} \hfill \cr}

The conditions of equilibrium in the absence of body forces are given by σij,j=0,Di,i=0,Bi,i=0. {\sigma _{ij,j}} = 0,\quad {D_{i,i}} = 0,\quad {B_{i,i}} = 0.

For linear problems, the accompanying relations can be utilized: εij=12(ui,j+uj,i),Ei=ϕ,i,Hi=φ,i {\varepsilon _{ij}} = {1 \over 2}\left({{u_{i,j}} + {u_{j,i}}} \right),\quad {E_i} = - {\phi _{,i}},\quad {H_i} = - {\varphi _{,i}}

In-plane extension and solution methodology

For a transversely isotropic MEE materials, the constitutive relations in the ox1x3-plane under plane strain (i.e., D2 = B2 = 0, σ12 = σ23 = 0) are given in Song and Sih [8].

Let the magnetic poling H and electric poling E are oriented arbitrarily with the x1-axis by the respective angles of θH and θE. In this case, the constants Cijks, eiks, hiks, αis and γis referred to the x1x3 (or xy)-coordinate system are given in Appendix A.

To find the generalized 2D deformations, the generalized displacement vector u depending on two independent variables x1 and x3 (or x and y) are considered as follows: u=(u1,u3,ϕ,φ)T=af(x1+px3) {\bf{u}} = {\left({{u_1},{u_3},\phi,\varphi} \right)^T} = {\bf{a}}f({x_1} + p{x_3})

Since u is the generalized 2D solution of the problem, it must satisfy the field Eqs.(1), (2), (3) and given by: (c11+c44p2(c13+c44)p(e31+e15)p(h31+h15)p(c13+c44)pc44+c33p2e15+e33p2h15+h33p2(e31+e15)pe15+e33p2(α11+α33p2)0(h31+h15)ph15+h33p20(γ11+γ33p2))a=0. \left({\matrix{{{c_{11}} + {c_{44}}{p^2}} & {({c_{13}} + {c_{44}})p} & {({e_{31}} + {e_{15}})p} & {({h_{31}} + {h_{15}})p} \cr {({c_{13}} + {c_{44}})p} & {{c_{44}} + {c_{33}}{p^2}} & {{e_{15}} + {e_{33}}{p^2}} & {{h_{15}} + {h_{33}}{p^2}} \cr {({e_{31}} + {e_{15}})p} & {{e_{15}} + {e_{33}}{p^2}} & {- ({\alpha _{11}} + {\alpha _{33}}{p^2})} & 0 \cr {({h_{31}} + {h_{15}})p} & {{h_{15}} + {h_{33}}{p^2}} & 0 & {- ({\gamma _{11}} + {\gamma _{33}}{p^2})} \cr}} \right){\bf{a}} = {\bf{0}}.

This is an eigenvalue problem consisting of four equations; a non-trivial a exists if p is a root of the determinant polynomial det((c11+c44p2(c13+c44)p(e31+e15)p(h31+h15)p(c13+c44)pc44+c33p2e15+e33p2h15+h33p2(e31+e15)pe15+e33p2(α11+α33p2)0(h31+h15)ph15+h33p20(γ11+γ33p2)))=0 {\rm det} \left({\left({\matrix{{{c_{11}} + {c_{44}}{p^2}} & {({c_{13}} + {c_{44}})p} & {({e_{31}} + {e_{15}})p} & {({h_{31}} + {h_{15}})p} \cr {({c_{13}} + {c_{44}})p} & {{c_{44}} + {c_{33}}{p^2}} & {{e_{15}} + {e_{33}}{p^2}} & {{h_{15}} + {h_{33}}{p^2}} \cr {({e_{31}} + {e_{15}})p} & {{e_{15}} + {e_{33}}{p^2}} & {- ({\alpha _{11}} + {\alpha _{33}}{p^2})} & 0 \cr {({h_{31}} + {h_{15}})p} & {{h_{15}} + {h_{33}}{p^2}} & 0 & {- ({\gamma _{11}} + {\gamma _{33}}{p^2})} \cr}} \right)} \right) = 0

Note that p are the eigenvalues. Only the four roots of p in upper half complex plane are considered such that the general solution (according to Stroh's formalism [11] and Jangid [10]) is given by: u,1=AF(z)+A¯F(z)¯ {{\bf {u}}_{,1}} = {\bf {AF}}(z) + \overline {\bf {A}} \overline {{\bf {F}}(z)} Φ,1=BF(z)+B¯F(z)¯ {\Phi _{,1}} = {\bf {BF}}(z) + \overline {\bf {B}} \overline {{\bf {F}}(z)} where A = (a1,a2,a3,a4), B = (b1,b2,b3,b4), F(z) = df(z)/dz, f(zα) = [f1(z1), f2(z2), f3(z3), f4(z4)]T, zα = x1 + pαx3.

and Φ is the generalized stress function such that σ1=(σ11,σ13,D1,B1)T=Φ,3 {\sigma _1} = ({\sigma _{11}},{\sigma _{13}},{D_1},{B_1}{)^T} = - {\Phi _{,3}} σ3=(σ31,σ33,D3,B3)T=Φ,1 {\sigma _3} = ({\sigma _{31}},{\sigma _{33}},{D_3},{B_3}{)^T} = {\Phi _{,1}}

Statement of the problem

An infinitely transversely isotropic MEE plate occupying the x1x3-plane is considered. The magnetic and electric poling are arbitrarily oriented with the crack line and make the respective angles of θH and θE. The plate is cut with two equal collinear cracks symmetrically situated about the origin and occupying the intervals [−c,−d] and [d,c] along the x1-axis. The plate is subjected to under the effect of the remote uniform tensile, in-plane electric displacement and magnetic induction loadings, i.e., σ33=σ33 {\sigma _{33}} = \sigma _{33}^\infty , D3=D3 {D_3} = D_3^\infty and B3=B3 {B_3} = B_3^\infty . The entire configuration of the problem is schematically presented in Fig. 1.

Fig. 1

Schematic representation of the problem

The physical boundary conditions of the problem can be mathematically written as follows:

σ3j+=σ3j \sigma _{3j}^ + = \sigma _{3j}^ - , D3+=D3=0 D_3^ + = D_3^ - = 0 , B3+=B3=0 B_3^ + = B_3^ - = 0 , Φ,1+=Φ,1=(0,σ33,D3,B3)T \Phi _{,1}^ + = \Phi _{,1}^ - = - {(0,\sigma _{33}^\infty,D_3^\infty,B_3^\infty)^T} , d < |x1| < c,

σ33=σ33 {\sigma _{33}} = \sigma _{33}^\infty , D3=D3 {D_3} = D_3^\infty , B3=B3, {B_3} = B_3^\infty , |x3| → ∞

uj+=uj u_j^ + = u_j^ - , σ3j+=σ3j \sigma _{3j}^ + = \sigma _{3j}^ - , D3+=D3 D_3^ + = D_3^ - , B3+=B3 B_3^ + = B_3^ - , |x1| > c

Solution of the problem

The continuity of Φ1 on x1-axis yield [BF(x1)B¯F(x1)¯]+[BF(x1)B¯F(x1)¯]=0 {\left[ {{\bf {BF}}({x_1}) - \overline {\bf {B}} \overline {{\bf {F}}({x_1})}} \right]^ +} - {\left[ {{\bf {BF}}({x_1}) - \overline {\bf {B}} \overline {{\bf {F}}({x_1})}} \right]^ -} = 0

Following Muskhelishvili [12], the solution of Eq.(11) can be written as BF(z)=B¯F(z)¯=h(z)(say) {\bf{BF}}(z) = \overline {\bf{B}} \overline {{\bf{F}}(z)} = h(z)\;{\rm{(say)}}

Using boundary condition (i) into the Eqs.(8), (12), we get h+(x1)+h(x1)=(0,σ33,D3,B3)T {{\bf{h}}^ +}({x_1}) + {{\bf{h}}^ -}({x_1}) = - {(0,\sigma _{33}^\infty,D_3^\infty,B_3^\infty)^T}

Introducing a new complex function vector as Ω(z)=[Ω1(z),Ω2(z),Ω3(z),Ω4(z)]T=HBF(z) \Omega (z) = {\left[ {{\Omega _1}(z),{\Omega _2}(z),{\Omega _3}(z),{\Omega _4}(z)} \right]^T} = {\bf{HBF}}(z) such that h(z) = ϒΩ(z), ϒ = H−1, H = 2Re(Y) and Y = iAB−1.

Eq.(13) can be written in its components Ω2(z), Ω3(z) and Ω4(z) as follows: ϒ22[Ω2+(x1)+Ω2(x1)]+ϒ23[Ω3+(x1)+Ω3(x1)]+ϒ24[Ω4+(x1)+Ω4(x1)]=σ33 {\Upsilon _{22}}\left[ {\Omega _2^ + ({x_1}) + \Omega _2^ - ({x_1})} \right] + {\Upsilon _{23}}\left[ {\Omega _3^ + ({x_1}) + \Omega _3^ - ({x_1})} \right] + {\Upsilon _{24}}\left[ {\Omega _4^ + ({x_1}) + \Omega _4^ - ({x_1})} \right] = - \sigma _{33}^\infty ϒ32[Ω2+(x1)+Ω2(x1)]+ϒ33[Ω3+(x1)+Ω3(x1)]+ϒ34[Ω4+(x1)+Ω4(x1)]=D3 {\Upsilon _{32}}\left[ {\Omega _2^ + ({x_1}) + \Omega _2^ - ({x_1})} \right] + {\Upsilon _{33}}\left[ {\Omega _3^ + ({x_1}) + \Omega _3^ - ({x_1})} \right] + {\Upsilon _{34}}\left[ {\Omega _4^ + ({x_1}) + \Omega _4^ - ({x_1})} \right] = - D_3^\infty ϒ42[Ω2+(x1)+Ω2(x1)]+ϒ43[Ω3+(x1)+Ω3(x1)]+ϒ44[Ω4+(x1)+Ω4(x1)]=B3 {\Upsilon _{42}}\left[ {\Omega _2^ + ({x_1}) + \Omega _2^ - ({x_1})} \right] + {\Upsilon _{43}}\left[ {\Omega _3^ + ({x_1}) + \Omega _3^ - ({x_1})} \right] + {\Upsilon _{44}}\left[ {\Omega _4^ + ({x_1}) + \Omega _4^ - ({x_1})} \right] = - B_3^\infty

Following Muskhelishvili [12], the solution of above equations can be written as: Ω2(z)=Π12Π{z2c2λ12X(z)1} {\Omega _2}(z) = {{{\Pi _1}} \over {2\Pi}}\left\{{{{{z^2} - {c^2}\lambda _1^2} \over {X(z)}} - 1} \right\} Ω3(z)=Π22Π{1z2c2λ12X(z)} {\Omega _3}(z) = {{{\Pi _2}} \over {2\Pi}}\left\{{1 - {{{z^2} - {c^2}\lambda _1^2} \over {X(z)}}} \right\} Ω4(z)=Π32Π{1z2c2λ12X(z)} {\Omega _4}(z) = {{{\Pi _3}} \over {2\Pi}}\left\{{1 - {{{z^2} - {c^2}\lambda _1^2} \over {X(z)}}} \right\} where X(z)=(z2d2)(z2c2) X(z) = \sqrt {({z^2} - {d^2})({z^2} - {c^2})} , λ12=E(k1)/F(k1) \lambda _1^2 = E({k_1})/F({k_1}) , k12=(c2d2)/c2 k_1^2 = ({c^2} - {d^2})/{c^2} . Π, Π1, Π2 and Π3 are given in Appendix B.

Derivation of the fracture parameters

In this section, the explicit expressions are determined for the stress, electric displacement, and magnetic induction intensity factors.

Stress intensity factor(SIF)

Open mode stress intensity factor, KI, at the crack tips x1 = d and x1 = c, may be calculated using the following formulae KI(d)=limx1d2π(dx1)σ33(x1), {K_I}(d) = \mathop {\lim}\limits_{{x_1} \to {d^ -}} \sqrt {2\pi (d - {x_1})} {\sigma _{33}}({x_1}), KI(c)=limx1c+2π(x1c)σ33(x1). {K_I}(c) = \mathop {\lim}\limits_{{x_1} \to {c^ +}} \sqrt {2\pi ({x_1} - c)} {\sigma _{33}}({x_1}).

Substituting σ33(x1) obtained from Eqs.(8, 17, 18 and 19) into above equations and simplifying one obtains KI(d)=πd(c2d2)(d2c2λ12)(ϒ22Π1ϒ23Π2ϒ24Π3Π), {K_I}(d) = - \sqrt {{\pi \over {d({c^2} - {d^2})}}} ({d^2} - {c^2}\lambda _1^2)\left({{{{\Upsilon _{22}}{\Pi _1} - {\Upsilon _{23}}{\Pi _2} - {\Upsilon _{24}}{\Pi _3}} \over \Pi}} \right), KI(c)=πc(c2d2)(c2c2λ12)(ϒ22Π1ϒ23Π2ϒ24Π3Π). {K_I}(c) = \sqrt {{\pi \over {c({c^2} - {d^2})}}} ({c^2} - {c^2}\lambda _1^2)\left({{{{\Upsilon _{22}}{\Pi _1} - {\Upsilon _{23}}{\Pi _2} - {\Upsilon _{24}}{\Pi _3}} \over \Pi}} \right).

Electric displacement intensity factor(EDIF)

The EDIF, KIV, at the crack tips x1 = d and x1 = c, may be calculated using the following formulae KIV(d)=πd(c2d2)(d2c2λ12)(ϒ32Π1ϒ33Π2ϒ34Π3Π), {K_{IV}}(d) = - \sqrt {{\pi \over {d({c^2} - {d^2})}}} ({d^2} - {c^2}\lambda _1^2)\left({{{{\Upsilon _{32}}{\Pi _1} - {\Upsilon _{33}}{\Pi _2} - {\Upsilon _{34}}{\Pi _3}} \over \Pi}} \right), KIV(c)=πc(c2d2)(c2c2λ12)(ϒ32Π1ϒ33Π2ϒ34Π3Π). {K_{IV}}(c) = \sqrt {{\pi \over {c({c^2} - {d^2})}}} ({c^2} - {c^2}\lambda _1^2)\left({{{{\Upsilon _{32}}{\Pi _1} - {\Upsilon _{33}}{\Pi _2} - {\Upsilon _{34}}{\Pi _3}} \over \Pi}} \right).

Magnetic induction intensity factor(MIIF)

Analogous to SIF and EDIF, the MIIF, KV, at the crack tips x1 = d and x1 = c, may be calculated using the following formulae KV(d)=πd(c2d2)(d2c2λ12)(ϒ42Π1ϒ43Π2ϒ44Π3Π), {K_V}(d) = - \sqrt {{\pi \over {d({c^2} - {d^2})}}} ({d^2} - {c^2}\lambda _1^2)\left({{{{\Upsilon _{42}}{\Pi _1} - {\Upsilon _{43}}{\Pi _2} - {\Upsilon _{44}}{\Pi _3}} \over \Pi}} \right), KV(c)=πc(c2d2)(c2c2λ12)(ϒ42Π1ϒ43Π2ϒ44Π3Π). {K_V}(c) = \sqrt {{\pi \over {c({c^2} - {d^2})}}} ({c^2} - {c^2}\lambda _1^2)\left({{{{\Upsilon _{42}}{\Pi _1} - {\Upsilon _{43}}{\Pi _2} - {\Upsilon _{44}}{\Pi _3}} \over \Pi}} \right).

Numerical results and discussion

In this section, numerical study for arbitrarily polarized MEE composite is presented. A problem of two equal collinear semi-permeable cracks in a 2D infinite BaTiO3CoFe2O4 plate is considered. The plate is infinite in the sense that the length of the crack is small as compared to the plate. For the whole numerical study, the length of the crack and volume fraction Vf of the inclusion BaTiO3 are taken fixed as 10mm and 0.5, respectively, unless otherwise stated. The plate is subjected to the far-field loadings with the mechanical loading, σ33=5MPa \sigma _{33}^\infty = 5MPa , electric displacement, D3=2(e33/c33)σ33 D_3^\infty = 2({e_{33}}/{c_{33}})\sigma _{33}^\infty , and magnetic induction B3=2(h33/c33)σ33 B_3^\infty = 2({h_{33}}/{c_{33}})\sigma _{33}^\infty . The behaviors of the fracture parameters obtained in section 6 are demonstrated graphically with electric and magnetic poling directions. The material constants of the BaTiO3CoFe2O4 taken for the study are taken from Jangid [10].

In the first analysis, the effect of electric poling direction is shown on the various fracture parameters such as SIF, EDIF, and MIIF with fixed magnetic poling direction (such as θH = 00, 300, 900).

Effect of electric poling direction

Figs. (2,3,4), Figs. (5,6,7) and Figs. (8,9,10), respectively, represent the behaviors of SIF, EDIF and MIIF versus inter-crack distance for different θE with fixed θH. It is observed that the values of KI, KIV and KV increases as θE increased. Also, the values of KI, KIV and KV at the inner crack tip x1 = d are higher than that at outer tip x1 = c, because of the mutual interaction of the two collinear cracks on the inner crack tips. And the values of KI, KIV and KV at both the crack tips get coincide as the inter-crack distance increases. Moreover, the effect of θE is negligible over KI in Figs. (2,3) for θH = 00 and 300.

Fig. 2

SIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 3

SIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 4

SIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 5

EDIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 6

EDIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 7

EDIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 8

MIIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 9

MIIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 10

MIIF versus inter-crack distance for different electric poling with fixed magnetic poling

Effect of Magnetic poling direction

In this section, the effect of θH is shown on the various fracture parameters for fixed θE = 00,300, and 900.

Figs. (11,12,13), Figs. (14,15,16), and Figs. (17,18,19), respectively, show the variations of SIF, EDIF, and MIIF versus inter-crack distance for different θH with fixed θE. It is observed that KI, KIV and KV increases as the fixed θE increased from 00 to 900. Also from the Figs. (11,17), it can be seen that the effect of magnetic poling is negligible over KI and KV for fixed θE = 00.

Fig. 11

SIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 12

SIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 13

SIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 14

EDIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 15

EDIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 16

EDIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 17

MIIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 18

MIIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 19

MIIF versus inter-crack distance for different magnetic poling with fixed electric poling

Conclusions

The following conclusions are made from the analytical and numerical studies presented for the proposed two equal collinear semi-permeable cracks model with arbitrarily oriented electric and magnetic poling in MEE media:

The complex variable methodology is effectively used to model and obtained the analytical solution for two equal collinear semi-permeable cracks weakening MEE media with arbitrarily electric and magnetic poling directions.

The closed form analytical expressions are obtained of the SIF, the EDIF, and the MIIF for the proposed problem.

The effect of electric and magnetic poling direction is graphically shown over the various fracture parameters.

The numerical investigations of K1 and KV demonstrate that the if the electric poling is along the crack line, then magnetic poling does not influence the behaviors of K1 and KV.

Fig. 1

Schematic representation of the problem
Schematic representation of the problem

Fig. 2

SIF versus inter-crack distance for different electric poling with fixed magnetic poling
SIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 3

SIF versus inter-crack distance for different electric poling with fixed magnetic poling
SIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 4

SIF versus inter-crack distance for different electric poling with fixed magnetic poling
SIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 5

EDIF versus inter-crack distance for different electric poling with fixed magnetic poling
EDIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 6

EDIF versus inter-crack distance for different electric poling with fixed magnetic poling
EDIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 7

EDIF versus inter-crack distance for different electric poling with fixed magnetic poling
EDIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 8

MIIF versus inter-crack distance for different electric poling with fixed magnetic poling
MIIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 9

MIIF versus inter-crack distance for different electric poling with fixed magnetic poling
MIIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 10

MIIF versus inter-crack distance for different electric poling with fixed magnetic poling
MIIF versus inter-crack distance for different electric poling with fixed magnetic poling

Fig. 11

SIF versus inter-crack distance for different magnetic poling with fixed electric poling
SIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 12

SIF versus inter-crack distance for different magnetic poling with fixed electric poling
SIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 13

SIF versus inter-crack distance for different magnetic poling with fixed electric poling
SIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 14

EDIF versus inter-crack distance for different magnetic poling with fixed electric poling
EDIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 15

EDIF versus inter-crack distance for different magnetic poling with fixed electric poling
EDIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 16

EDIF versus inter-crack distance for different magnetic poling with fixed electric poling
EDIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 17

MIIF versus inter-crack distance for different magnetic poling with fixed electric poling
MIIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 18

MIIF versus inter-crack distance for different magnetic poling with fixed electric poling
MIIF versus inter-crack distance for different magnetic poling with fixed electric poling

Fig. 19

MIIF versus inter-crack distance for different magnetic poling with fixed electric poling
MIIF versus inter-crack distance for different magnetic poling with fixed electric poling

Symbol Notation
d inner crack tip
c outer crack tip
eiks piezoelectric constants
f analytic function
hiks piezomagnetic constants
k1 elliptic modulus
mE coordinate transfer matrix of electric poling
mH coordinate transfer matrix of magnetic poling
u generalized displacement vector
ui (i = 1,2,3) mechanical displacement
xi (i = 1,2,3) material coordinate system
z complex variable
Cijks elastic constants
E(k1) complete elliptic integral of second kind
F(k1) complete elliptic integral of first kind
MB material constants transform matrix of the inclusion
MC material constants transform matrix of the matrix
Vf volume fraction of the inclusion
αis dielectric permittivities
βis electromagnetic constants
γis magnetic permeabilities
u4 = ϕ electric potential
u5 = φ magnetic potential
χ Jacobi amplitude

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