Homogenization and linearization in magnetoelasticity under small elastic response
Mikhail Cherdantsev, Elisa Davoli, Lorenza D'Elia, Samuele Riccò
TL;DR
The paper develops a rigorous Gamma-convergence framework for simultaneous homogenization and linearization of a magnetoelastic energy with mixed Eulerian-Lagrangian structure under small elastic response. By leveraging two-scale convergence and a Beppo-Levi formulation for the magnetostatic field, it derives a homogenized energy $F_{hom}$ consisting of a quadratic magnetoelastic term with a homogenized exchange energy density $Q_{hom}$ and a homogenized exchange term $T_{hom}$, plus the magnetostatic self-energy. The results include compactness, two-scale limits for magnetizations, a magnetostatic limit, a sharp lower bound, recovery sequences, and a demonstration that homogenization and linearization commute in this setting. This constitutes the first homogenization result for manifold-valued mixed Eulerian-Lagrangian energies and provides foundational steps toward more general multiscale magnetoelastic theories with geometric constraints.
Abstract
We perform a simultaneous homogenization and linearization analysis for a magnetoelastic energy functional featuring a mixed Eulerian-Lagrangian structure. Neglecting Zeeman and anisotropic contributions, we characterize the asymptotic behavior in the sense of Gamma-convergence for the sum of a nonlinear magnetoelastic energy, a symmetric exchange term defined on the actual configuration, and for the associated magnetostatic self-energy. After establishing compactness of displacements and magnetizations with equibounded energy, we identify the limiting energy functional as the sum of a quadratic homogenized magnetoelastic contribution with a limiting homogenized exchange and magnetostatic term. This is, to the authors' knowledge, the first homogenization result for manifold-valued mixed Eulerian-Lagrangian energies.