Universal Relations and Effective Coefficients of Magneto-Electro-Elastic Perforated Structures

Based on the two-scale convergence homogenization method, the limiting equations modelling the behaviour of a three-dimensional magneto-electro-elastic (MEE) composite made of periodically perforated microstructure are rigorously established. The homogenized problem is considered for the particular case of porous materials consisting of identical parallel empty cylinders periodically distributed in a transversely isotropic MEE and homogeneous medium. For these composites, universal relationships involving the MEE effective properties were derived without imposing any restrictions about their global behaviour. For the particular case of transversely isotropic (and tetragonal) effective behaviour, simple analytical expressions for the effective coefficients are explicitly given for two types of empty fibres periodical distribution (square and hexagonal arrays). An analytical formula to estimate the specific volume fraction where the magnetoelectric effective coefficient attains its minimum value is given. These relationships and formulae are used to check the accuracy of a numerical homogenization scheme based on Fourier transform technique. Comparisons with others micromechanical models are also included.