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


 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 BaTiO
 3 – CoFe
 2
 O
 4 composite. The numerical examination demonstrates that the change in electric/magnetic poling directions influences the intensity factors.


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. Fer-rite=erroelectric 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 inplane 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. material constants transform matrix of the inclusion M C material constants transform matrix of the matrix V f volume fraction of the inclusion α is dielectric permittivities β is electromagnetic constants γ is magnetic permeabilities u 4 = φ electric potential u 5 = ϕ magnetic potential χ Jacobi amplitude

General equations
In-rectangular Cartesian coordinate system x i (i = 1, 2, 3) the stress , σ i j , electric displacement , D i , and magnetic induction , B i , are linearly related to the strain , ε i j , 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: The conditions of equilibrium in the absence of body forces are given by For linear problems, the accompanying relations can be utilized:
Let the magnetic poling H and electric poling E are oriented arbitrarily with the x 1 -axis by the respective angles of θ H and θ E . In this case, the constants C i jks , e iks , h iks , α is and γ is referred to the x 1 x 3 (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 x 1 and x 3 (or x and y) are considered as follows: Since u is the generalized 2D solution of the problem, it must satisfy the field Eqs.
This is an eigenvalue problem consisting of four equations; a non-trivial a exists if p is a root of the determinant polynomial 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: and is the generalized stress function such that

Statement of the problem
An infinitely transversely isotropic MEE plate occupying the x 1 x 3 -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 x 1 -axis. The plate is subjected to under the effect of the remote uniform tensile, in-plane electric displacement and magnetic induction loadings, i.e., The entire configuration of the problem is schematically presented in Fig.1. The physical boundary conditions of the problem can be mathematically written as follows: The continuity of ,1 on x 1 -axis yield Following Muskhelishvili [12], the solution of Eq.(11) can be written as Using boundary condition (i) into the Eqs. (8), (12), we get Introducing a new complex function vector as Eq.(13) can be written in its components Ω 2 (z), Ω 3 (z) and Ω 4 (z) as follows: Following Muskhelishvili [12], the solution of above equations can be written as: where Π, Π 1 , Π 2 and Π 3 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, K I , at the crack tips x 1 = d and x 1 = c, may be calculated using the following formulae K I (c) = lim Substituting σ 33 (x 1 ) obtained from Eqs. (8, 17, 18 and 19) into above equations and simplifying one obtains

Electric displacement intensity factor(EDIF)
The EDIF, K IV , at the crack tips x 1 = d and x 1 = c, may be calculated using the following formulae

Magnetic induction intensity factor(MIIF)
Analogous to SIF and EDIF, the MIIF, K V , at the crack tips x 1 = d and x 1 = c, may be calculated using the following formulae

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 BaTiO 3 − CoFe 2 O 4 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 BaTiO 3 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, σ ∞ 33 = 5MPa, electric displacement, D ∞ 3 = 2(e 33 /c 33 )σ ∞ 33 , and magnetic induction B ∞ 3 = 2(h 33 /c 33 )σ ∞ 33 . The behaviors of the fracture parameters obtained in section 6 are demonstrated graphically with electric and magnetic poling directions. The material constants of the BaTiO 3 −CoFe 2 O 4 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 = 0 0 , 30 0 , 90 0 ).  (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 x 1 = d are higher than that at outer tip x 1 = 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 = 0 0 and 30 0 .

Effect of Magnetic poling direction
In this section, the effect of θ H is shown on the various fracture parameters for fixed θ E = 0 0 , 30 0 , and 90 0 . 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 0 0 to 90 0 . 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 = 0 0 . 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 K 1 and K V demonstrate that the if the electric poling is along the crack line, then magnetic poling does not influence the behaviors of K 1 and K V .

Appendix(A)
Consider a composite BaTiO 3 − CoFe 2 O 4 made of piezoelectric BaTiO 3 barium titanate as the inclusions and magnetostrictive CoFe 2 O 4 cobalt iron oxide as the matrix. The volume fraction of the inclusions is denoted by V f .
The mixture rule is given by where the superscripts c, i and m denote the composite, the inclusion, and the matrix. Moreover, k i j stand for the elastic constants c i j , piezoelectric constants e i j , piezomagnetic constants h i j , dielectric permittivity α i j and dielectric permeability γ i j . The above rule is not applicable for the determination of electromagnetic constants, because no electromagnetic coupling is present in any of the single-phase. Above mixture rule in matrix form can be written as (Sih et al. [7]):  T h i s p a g e i s i n t e n t i o n a l l y l e f t b l a n k