Homogenization of high-contrast media in finite-strain elastoplasticity

TL;DR

This paper analyzes the effective behavior of high-contrast elastoplastic composites with microstructure, deriving a macroscopic model via $ ext{Gamma}$-convergence that accounts for degeneracy in soft inclusions.

Contribution

It provides a rigorous $ ext{Gamma}$-convergence analysis for high-contrast elastoplastic media, explicitly handling degeneracy and deriving a combined macroscopic energy.

Findings

$ ext{Gamma}$-limit is the sum of contributions from stiff matrix and soft inclusions.

Convergence of energy-minimizing configurations is established.

The analysis handles degeneracy due to high contrast in material properties.

Abstract

This work is devoted to the analysis of the interplay between internal variables and high-contrast microstructure in inelastic solids. As a concrete case-study, by means of variational techniques, we derive a macroscopic description for an elastoplastic medium. Specifically, we consider a composite obtained by filling the voids of a periodically perforated stiff matrix by soft inclusions. We study the $\Gamma$-convergence of the related energy functionals as the periodicity tends to zero. The main challenge is posed by the lack of coercivity brought about by the degeneracy of the material properties in the soft part. We prove that the $\Gamma$-limit, which we compute with respect to a suitable notion of convergence, is the sum of the contributions resulting from each of the two components separately. Eventually, convergence of the energy minimizing configurations is obtained.