Algorithmic two-scale transition for magneto-electro-mechanically coupled problems: FE2-scheme: Localization and homogenization

In this paper we present a computational homogenization procedure for the simulation of magneto-electro-mechanically coupled boundary value problems (bvp)s on two scales. We derive the basic equations for the localization and the homogenization of the individual field variables and give an algorithmic expression for the effective tangent moduli. The resulting algorithmic two-scale transition procedure is implemented into an FE2-method, which allows us to compute macroscopic boundary value problems in consideration of attached microscopic representative volume elements. The challenge in the simulation of magneto-electro-mechanically coupled materials is the modeling of the complicated interactions between magnetical, electrical and mechanical quantities on both scales. A primer example for such interactions is given by magneto-electric composites. Thus, we apply the presented method to the two-scale simulation of a variety of versions of magneto-electric composites. In detail, we consider two-phase composites composed of (i) piezomagnetic and piezoelectric, (ii) piezomagnetic and non-linear electrostrictive, as well as (iii) piezomagnetic and non-linear dissipative ferroelectric phase. Based on numerical examples, we discuss aspects of the applicability, the numerical stability as well as the predictive capability of the proposed method.