Existence results in large-strain magnetoelasticity

TL;DR

This paper establishes the existence of solutions in large-strain magnetoelasticity models, addressing both static and quasistatic cases with a mixed formulation and introducing a regularized approach for energetic solutions.

Contribution

It provides the first existence results for large-strain magnetoelasticity with a mixed Eulerian-Lagrangian formulation, including a regularized model for energetic solutions.

Findings

Existence of minimizers in static large-strain magnetoelasticity.

Solvability of incremental minimization in quasistatic setting.

Existence of energetic solutions for the regularized model.

Abstract

We investigate variational problems in large-strain magnetoelasticity, both in the static and in the quasistatic setting. The model contemplates a mixed Eulerian-Lagrangian formulation: while deformations are defined on the reference configuration, magnetizations are defined on the deformed set in the actual space. In the static setting, we establish the existence of minimizers. In particular, we provide a compactness result for sequences of admissible states with equi-bounded energies which gives the convergence of the composition of magnetizations with deformations. In the quasistatic setting, we consider a notion of dissipation which is frame-indifferent and we show that the incremental minimization problem is solvable. Then, we propose a regularization of the model in the spirit of gradient polyconvexity and we prove the existence of energetic solutions for the regularized model.