Two equal collinear cracks in magneto-electro-elastic materials: A study of electric and magnetic poling influences

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.

In-rectangular Cartesian coordinate system x_i(i = 1,2,3) the stress, σ_ij, electric displacement, D_i, and magnetic induction, B_i, are linearly related to the strain, ɛ_ij, electric field, E_i, and magnetic field, H_i,. A set of constitutive equations for a MEE material would thus be able to be written in the form: (1) σij=Cijksεks−esijEs−hsijHsDi=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 (2) 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: (3) 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}}

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 θ_H and θ_E. In this case, the constants C_ijks, e_iks, h_iks, α_is and γ_is 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: (4) 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: (5) (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 (6) 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: (7) u,1=AF(z)+A¯F(z)¯ {{\bf {u}}_{,1}} = {\bf {AF}}(z) + \overline {\bf {A}} \overline {{\bf {F}}(z)} (8) 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(z_α) = [f₁(z₁), f₂(z₂), f₃(z₃), f₄(z₄)]^T, z_α = x₁ + p_αx₃.

and Φ is the generalized stress function such that (9) σ1=(σ11,σ13,D1,B1)T=−Φ,3 {\sigma _1} = ({\sigma _{11}},{\sigma _{13}},{D_1},{B_1}{)^T} = - {\Phi _{,3}} (10) σ3=(σ31,σ33,D3,B3)T=Φ,1 {\sigma _3} = ({\sigma _{31}},{\sigma _{33}},{D_3},{B_3}{)^T} = {\Phi _{,1}}

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 θ_H and θ_E. 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

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

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,

The continuity of Φ₁ on x₁-axis yield (11) [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 (12) 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 (13) 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: (14) 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 (15) 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 (16) 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: (17) Ω2(z)=Π12Π{z2−c2λ12X(z)−1} {\Omega _2}(z) = {{{\Pi _1}} \over {2\Pi}}\left\{{{{{z^2} - {c^2}\lambda _1^2} \over {X(z)}} - 1} \right\} (18) Ω3(z)=Π22Π{1−z2−c2λ12X(z)} {\Omega _3}(z) = {{{\Pi _2}} \over {2\Pi}}\left\{{1 - {{{z^2} - {c^2}\lambda _1^2} \over {X(z)}}} \right\} (19) Ω4(z)=Π32Π{1−z2−c2λ12X(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)=(z2−d2)(z2−c2) 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=(c2−d2)/c2 k_1^2 = ({c^2} - {d^2})/{c^2} . Π, Π₁, Π₂ and Π₃ are given in Appendix B.

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

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 BaTiO₃ – CoFe₂O₄ 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 V_f 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 BaTiO₃ – CoFe₂O₄ 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 θ_H = 0⁰, 30⁰, 90⁰).

7.1

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 θ_E with fixed θ_H. It is observed that the values of K_I, K_IV and K_V increases as θ_E increased. Also, the values of K_I, K_IV and K_V 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 K_I, K_IV and K_V at both the crack tips get coincide as the inter-crack distance increases. Moreover, the effect of θ_E is negligible over K_I in Figs. (2,3) for θ_H = 0⁰ and 30⁰.

Fig. 2

Fig. 3

Fig. 4

Fig. 5

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Fig. 9

Fig. 10

7.2

Effect of Magnetic poling direction

In this section, the effect of θ_H is shown on the various fracture parameters for fixed θ_E = 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 θ_H with fixed θ_E. It is observed that K_I, K_IV and K_V increases as the fixed θ_E increased from 0⁰ to 90⁰. Also from the Figs. (11,17), it can be seen that the effect of magnetic poling is negligible over K_I and K_V for fixed θ_E = 0⁰.

Fig. 11

Fig. 12

Fig. 13

Fig. 14

Fig. 15

Fig. 16

Fig. 17

Fig. 18

Fig. 19

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.