ABSTRACT
In order to compute the effective response of the ME composite we analyze a representative volume element (RVE) of the actual body, which has to be imagined as periodically embedded into a higher scale macroscopic continuum. The RVE is driven by boundary conditions, derived by exploring a scale transition between the scales. For this purpose, we first decompose the microscopic fields into constant macroscopic fields •¯ and fluctuation parts •˜ as
Periodic boundary conditions that are
where x+ and x− denote points on the opposite faces of a periodic unit cell. After the solution of the boundary value problem, we can compute the homogenized macroscopic response by averaging over the microscopic fields over the RVE. For simplicity we make use of the volume averages
2.2 Constitutive framework and effective ME coefficient
For the analysis of the effective ME behavior of the composite, we suppose linear transversely isotropic material response for the piezoelectric and piezomagnetic phases on the microscale. The constitutive behavior for each of the two phases is given by
where C, , µ, e, q, and α denote the constant material moduli of the individual phase. Here, it has to be emphasized that for each of the two phases the ME coupling modulus is zero: α≡ 0. However, the overall macroscopic ME modulus of the composite α is in general non-zero since it is activated by the electric-or magnetic-field induced deformation of the microstructure. This macroscopic property is defined as
3 NUMERICAL EXAMPLES
In the following example we consider a composite consisting of a piezoelectric matrix (BaTiO3) and piezomagnetic inclusions (CoFe2O4), based on a microscopy of a experimental composite obtained from Etier et al. (2012).
