Homogenization of an elastodynamics system with a strong magnetic field and soft inclusions inducing a viscoelastic effective behavior

Abstract

In this paper we study the homogenization of a linear elastodynamics system in an elastic body with soft inclusions, which is embedded in a highly oscillating magnetic field. We show two limit behaviors according to the magnetic field. On the one hand, if the magnetic field has two different directions on the interface between the hard phase and the soft phase, then the limit of the displacement in the hard phase is independent of time, so that the magnetic field induces an effective infinite mass. On the other hand, if the magnetic field has a constant direction $\xi$ on the interface, then the limit of the displacement in the hard phase and in the direction $\xi$ is solution to an elastodynamics equation with a memory mass, a memory stress tensor and memory external forces depending on the initial conditions, which read as time convolutions with some kernel. When the magnetic has the same direction $\xi$ in the soft phase with smooth inclusions, we prove that the space-average of the kernel is regular and that the limit of the overall displacement in the direction $\xi$ is solution to a viscoelasticity equation.