Increase of mass and nonlocal effects in the homogenization of magneto-elastodynamics problems

TL;DR

This paper investigates how oscillating magnetic fields affect the homogenized equations in magneto-elastodynamics, revealing increased effective mass and nonlocal Lorentz forces with complex limit behaviors.

Contribution

It introduces a detailed analysis of the homogenization process for magneto-elastodynamics with time-space dependent magnetic fields, highlighting nonlocal effects and mass increase in the limit equations.

Findings

Homogenized equations show increased effective mass due to oscillating magnetic fields.

Nonlocal Lorentz force terms appear in the limit, confined to a light cone.

The limit involves a decomposition of the oscillating Lorentz force into two distinct terms.

Abstract

The paper deals with the homogenization of a magneto-elastodynamics equation satisfied by the displacement $u\_\varepsilon$ of an elastic body which is subjected to an oscillating magnetic field $B\_\varepsilon$ generating the Lorentz force $\partial\_t u\_\varepsilon\times B\_\varepsilon$.When the magnetic field $B\_\varepsilon$ only depends on time or on space, the oscillations of $B\_\varepsilon$ induce an increase of mass in the homogenized equation. More generally, when the magnetic field is time-space dependent through a uniformly bounded component $G\_\varepsilon(t,x)$ of $B\_\varepsilon$, besides the increase of mass the homogenized equation involves the more intricate limit $g$ of $\partial\_t u\_\varepsilon\times G\_\varepsilon$ which turns out to be decomposed in two terms. The first term of $g$ can be regarded as a nonlocal Lorentz force the range of which is limited to a light cone at each point $(t,x)$. The cone angle is determined by the maximal velocity defined as the square root of the ratio between the elasticity tensor spectral radius and the body mass. Otherwise, the second term of $g$ is locally controlled in $L^2$-norm by the compactness default measure of the oscillating initial energy.