Calculus of Variations and Geometric Measure Theory

M. Bresciani - M. Kruzik

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

created by bresciani on 14 Sep 2021
modified on 23 Sep 2022


Submitted Paper

Inserted: 14 sep 2021
Last Updated: 23 sep 2022

Year: 2021


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 Kàrmàn 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 result previously 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 sequences of almost minimizers 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 energetic solutions for the bulk model converge to energetic solutions for the reduced model and we establish a similar result for solutions of the approximate incremental minimization problem. Both these results provide a further justification of the reduced model in the spirit of the evolutionary $\Gamma$-convergence.