On static and evolutionary homogenization in crystal plasticity for stratified composites

TL;DR

This paper investigates static and evolutionary homogenization in layered crystal plasticity, analyzing minimizers, uniqueness, and conditions for rigid responses, and extends results to rate-independent systems with decoupled effects.

Contribution

It provides a detailed analysis of minimizers in static homogenization and extends the theory to evolutionary systems, revealing conditions for energy-only limits in layered composites.

Findings

At least one key deformation quantity is uniquely determined.

Conditions for trivial rigid-body responses are identified.

Homogenization can lead to purely energetic limits with no dissipation.

Abstract

The starting point for this work is a static macroscopic model for a high-contrast layered material in single-slip finite crystal plasticity, identified in [Christowiak & Kreisbeck, Calc. Var. PDE (2017)] as a homogenization limit via {\Gamma}-convergence. First, we analyze the minimizers of this limit model, addressing the question of uniqueness and deriving necessary conditions. In particular, it turns out that at least one of the defining quantities of an energetically optimal deformation, namely the rotation and the shear variable, is uniquely determined, and we identify conditions that give rise to a trivial material response in the sense of rigid-body motions. The second part is concerned with extending the static homogenization to an evolutionary {\Gamma}- convergence-type result for rate-independent systems in specific scenarios, that is, under certain assumptions on the slip systems and suitable regularizations of the energies, where energetic and dissipative effects decouple in the limit. Interestingly, when the slip direction is aligned with the layered microstructure, the limiting system is purely energetic, which can be interpreted as a loss of dissipation through homogenization.