# Homogenization in $BV$ of a model for layered composites in finite crystal plasticity

## TL;DR

This paper investigates the effective behavior of layered composites in finite crystal plasticity using Gamma-convergence, characterizing deformation limits and deriving an explicit homogenized energy formula.

## Contribution

It provides a rigorous Gamma-convergence analysis for a layered composite model in finite crystal plasticity, including deformation characterization and explicit homogenization formula.

## Key findings

- Deformation limits involve horizontal splitting into shear and rotation
- Homogenized energy is identified as a lower bound of the Gamma-limit
- Complete homogenization formula derived for a regularized anisotropic model

## Abstract

In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials arranged into periodically alternating thin horizontal strips of an elastically rigid component and a softer one with one active slip system. The energies arising from these modeling assumptions are of integral form, featuring linear growth and non-convex differential constraints. We approach this non-standard homogenization problem via Gamma-convergence. A crucial first step in the asymptotic analysis is the characterization of rigidity properties of limits of admissible deformations in the space $BV$ of functions of bounded variation. In particular, we prove that, under suitable assumptions, the two-dimensional body may split horizontally into finitely many pieces, each of which undergoes shear deformation and global rotation. This allows us to identify a potential candidate for the homogenized limit energy, which we show to be a lower bound on the Gamma-limit. In the framework of non-simple materials, we present a complete Gamma-convergence result, including an explicit homogenization formula, for a regularized model with an anisotropic penalization in the layer direction.