A reduced model for plates arising as low energy $\Gamma$-limit in nonlinear magnetoelasticity

TL;DR

This paper derives a simplified model for thin magnetoelastic plates by analyzing the low-energy limit using $ ext{Gamma}$-convergence, accounting for applied loads and quasistatic evolution, extending previous incompressible results.

Contribution

It extends the $ ext{Gamma}$-limit analysis of magnetoelastic plates to the compressible case and incorporates applied loads and dynamic evolution.

Findings

Reduced model obtained via $ ext{Gamma}$-convergence as thickness tends to zero.

Sequences of almost minimizers converge to the reduced model's minimizers.

Solutions to incremental problems converge to energetic solutions, justifying the reduced model.

Abstract

We investigate the problem of dimension reduction for plates in nonlinear magnetoelasticity. The model features a mixed Eulerian-Lagrangian formulation, as magnetizations are defined on the deformed set in the actual space. We consider low-energy configurations by rescaling the elastic energy according to the linearized von K\'{a}rm\'{a}n regime. First, we identify a reduced model by computing the $\Gamma$-limit of the magnetoelastic energy, as the thickness of the plate goes to zero. This extends a previous result obtained by the first author in the incompressible case to the compressible one. Then, we introduce applied loads given by mechanical forces and external magnetic fields and we prove that, under clamped boundary conditions, sequences of almost minimizes of the total energy converge to minimizers of the corresponding energy in the reduced model. Subsequently, we study quasistatic evolutions driven by time-dependent applied loads and a rate-independent dissipation. We prove that solutions to the approximate incremental minimization problem at the bulk converge to energetic solutions to the reduced model. This result provides a further justification of the latter in the spirit of evolutionary $\Gamma$-convergence.