Quasistatic evolution in magnetoelasticity under subcritical coercivity assumptions

TL;DR

This paper develops a variational model for magnetoelasticity incorporating a mixed formulation and discontinuous deformations, establishing existence results for static and quasistatic evolutions under subcritical coercivity assumptions.

Contribution

It introduces a novel compactness result for magnetoelastic deformations and magnetizations, enabling existence proofs for static minimizers and quasistatic evolutions.

Findings

Existence of minimizers in static magnetoelasticity

Existence of energetic solutions in quasistatic evolution

Compactness of magnetization-deformation compositions

Abstract

We study a variational model of magnetoelasticity both in the static and in the quasistatic setting. The model features a mixed Eulerian-Lagrangian formulation, as magnetizations are defined on the deformed configuration in the actual space. The magnetic saturation constraint is formulated in the reference configuration and involves the Jacobian determinant of deformations. These belong to the class of possibility discontinuous deformations excluding cavitation introduced by Barchiesi, Henao and Mora-Corral. We establish a compactness result which, in particular, yields the convergence of the compositions of magnetizations with deformations. In the static setting, this enables us to prove the existence of minimizers by means of classical lower semicontinuity methods. Our compactness result also allows us to address the analysis in the quasistatic setting, where we examine rate-independent evolutions driven by applied loads and boundary conditions. In this case, we prove the existence of energetic solutions.