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

Description

The paper deals with the homogenization of a magneto-elastodynamics equation satisfied by the displacement uεuε of an elastic body which is subjected to an oscillating magnetic field BεBε generating the Lorentz force ∂tuε×Bε∂tuε×Bε. When the magnetic field BεBε only depends on time or on space, the oscillations of BεBε induce an increase of mass in the homogenized equation. More generally, when the magnetic field is time-space dependent through a uniformly bounded component Gε(t,x)Gε(t,x) of BεBε, besides the increase of mass the homogenized equation involves the more intricate limit g of ∂tuε×Gε∂tuε×Gε 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 L2L2-norm by the compactness default measure of the oscillating initial energy.

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