A homogenization result in finite plasticity

TL;DR

This paper proves a homogenization result for the energy functionals of heterogeneous finite-strain elastoplastic materials with periodic microstructure, addressing the complex plastic deformation constraints.

Contribution

It establishes the $ ext{Gamma}$-convergence of energies in finite plasticity with hardening, considering the plastic deformation constraint as a Finsler manifold.

Findings

$ ext{Gamma}$-convergence of energy functionals established

Homogenized energy characterized for periodic composites

Plastic deformation constraint handled via Finsler manifold approach

Abstract

We carry out a variational study for integral functionals that model the stored energy of a heterogeneous material governed by finite-strain elastoplasticity with hardening. Assuming that the composite has a periodic microscopic structure, we establish the $\Gamma$-convergence of the energies in the limiting of vanishing periodicity. The constraint that plastic deformations belong to $\mathsf{SL}(3)$ poses the biggest hurdle to the analysis, and we address it by regarding $\mathsf{SL}(3)$ as a Finsler manifold.