Magnetoelectric coupling and cross-property connections in a square array of a binary composite

Effective elastic, dielectric, magnetic, piezoelectric, piezomagnetic and magnetoelectric properties for the anti-plane shear magnetoelectroelastic state are determined. This properties are calculated by means of the Asymptotic Homogenization Method for a composite material with unidirectional cylindrical fiber periodically distributed in a square array. Closed-form formulae are obtained for the effective properties. The formulae exhibit explicitly the dependence on (i) the geometry through fiber volume fraction, (ii) the periodicity of the array through its lattice sums, and finally (iii) the material properties of the phases. Anti-plane property calculations are carried out for BaTiO₃/CoFe₂O₄ composite. The local problem is solved using potential methods of a complex variable. The solution involves double periodic Weierstrass elliptic and related functions. The shear modulus experiments a stiffening due to the coupling piezoelectric and piezomagnetic effect. Both exact and empirical cross-property connections are found for these composites. For the empirical case, the knowledge of the anti-plane effective dielectric property, ε₁₁, experimentally or otherwise, yields the remaining ones, “plane and anti-plane”, with good accuracy within a wide range of fiber volume fractions. The magnetoelectric effect results from the mechanical interaction between constituents. Hence, in order to maximize the magnetoelectric coefficient during composite design, matrix plane shear modulus must be as higher as possible than plane bulk modulus.