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

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.