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

TL;DR

This paper investigates how a strong magnetic field influences the homogenized behavior of an elastodynamics system with soft inclusions, revealing conditions that lead to viscoelastic effective behavior and infinite mass effects.

Contribution

It introduces a novel analysis of elastodynamics with magnetic fields, showing how different magnetic configurations induce viscoelasticity or infinite mass in the homogenized limit.

Findings

Magnetic field with two directions causes infinite mass in the limit.

Constant magnetic direction leads to a viscoelastic effective behavior.

Space-average of the kernel in the viscoelastic limit is regular.

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.