A variational approach to the modeling of compressible magnetoelastic materials

TL;DR

This paper develops a variational framework for modeling the evolution of compressible magnetoelastic materials with non-convex energy, proving the existence of weak solutions using De Giorgi's minimizing movements scheme.

Contribution

It introduces a novel variational approach to handle non-convex energies and state spaces in magnetoelastic materials, extending the mathematical analysis of such models.

Findings

Existence of weak solutions for the model.

Application of De Giorgi's scheme to non-convex energies.

Unified energy and dissipation potentials for magnetic force balance.

Abstract

We analyze a model of the evolution of a (solid) magnetoelastic material. More specifically, the model we consider describes the evolution of a compressible magnetoelastic material with a non-convex energy and coupled to a gradient flow equation for the magnetization in the quasi-static setting. The viscous dissipation considered in this model induces an extended material derivative in the magnetic force balance. We prove existence of weak solutions based on De Giorgi's minimizing movements scheme, which allows us to deal with the non-convex energy as well as the non-convex state space for the deformation. In the application of this method we rely on the fact that the magnetic force balance in the model can be expressed in terms of the same energy and dissipation potentials as the equation of motion, allowing us to model the functional for the discrete minimization problem based on these potentials.