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

In this paper we study homogenization of a linear system of elastodynamics in an elastic body with soft inclusions, which is embedded in a highly oscillating magnetic field. We show two limits behaviors according to the magnetic field. On the one hand, if the magnetic field has at least two different directions at 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 infinite effective mass. On the other hand, if the magnetic field has a constant direction ξ at the interface, then the limit of the displacement in the hard phase involves an additional displacement in the direction ξ which 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 field has the same direction ξ 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 ξ is solution to a viscoelasticity equation.