On the hierarchy of plate models for a singularly perturbed multi-well nonlinear elastic energy

TL;DR

This paper extends the hierarchy of plate models for nonlinear elastic energies to multi-well structures, incorporating regularization to handle phase transitions and analyzing the convergence of minimizers under external forces.

Contribution

It introduces a regularized multi-well energy model for plates, recovering the full hierarchy of models with explicit dependence on wells, and studies minimizer convergence.

Findings

Hierarchy of plate models established for multi-well energies

Regularization ensures compactness despite incompatible wells

Convergence of energy minimizers under external forces demonstrated

Abstract

In the celebrated work of Friesecke, James and M\"uller '06 the authors derive a hierarchy of models for plates by carefully analyzing the $\Gamma$-convergence of the rescaled nonlinear elastic energy. The key ingredient of their proofs is the rigidity estimate proved in an earlier work of theirs. Here we consider the case in which the elastic energy has a multi-well structure: this type of functional arises, for example, in the study of solid-solid phase transitions. Since the rigidity estimate fails in the case of compatible wells, we follow Alicandro, Dal Maso, Lazzaroni and Palombaro '18 and add a regularization term to the energy that penalizes jumps from one well to another, leading to good compactness properties. In this setting we recover the full hierarchy of plate models with an explicit dependence on the wells. Finally, we study the convergence of energy minimizers with suitable external forces and full Neumann boundary conditions. To do so, we adapt the definition of optimal rotations introduced by Maor, Mora '21.