Homogenization and linearization in magnetoelasticity under small elastic response

TL;DR

This paper develops a homogenization and linearization framework for magnetoelastic materials, deriving a limiting quadratic energy functional that captures the combined effects of elasticity and magnetism in a complex mixed configuration setting.

Contribution

It presents the first homogenization result for manifold-valued mixed Eulerian-Lagrangian energies in magnetoelasticity, combining nonlinear and linearized analysis.

Findings

Characterized the asymptotic behavior via Gamma-convergence.

Identified the limiting energy as a sum of quadratic homogenized terms.

Established compactness of displacements and magnetizations.

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.