Homogenization of layered materials with rigid components in single-slip finite crystal plasticity

TL;DR

This paper derives explicit formulas for the effective behavior of layered composite materials in finite-strain crystal plasticity, revealing how orientation affects macroscopic properties through homogenization and advanced variational analysis.

Contribution

It introduces a novel homogenization approach for layered composites with rigid and slip-active components, including explicit formulas and analysis of orientation-dependent regimes.

Findings

Explicit homogenization formulas obtained via Γ-convergence.

Identification of three regimes based on slip direction orientation.

Demonstration of anisotropic effects on macroscopic behavior.

Abstract

We determine the effective behavior of a class of composites in finite-strain crystal plasticity, based on a variational model for materials made of fine parallel layers of two types. While one component is completely rigid in the sense that it admits only local rotations, the other one is softer featuring a single active slip system with linear self-hardening. As a main result, we obtain explicit homogenization formulas by means of {\Gamma}-convergence. Due to the anisotropic nature of the problem, the findings depend critically on the orientation of the slip direction relative to the layers, leading to three qualitatively different regimes that involve macroscopic shearing and blocking effects. The technical difficulties in the proofs are rooted in the intrinsic rigidity of the model, which translates into a non-standard variational problem constraint by non-convex partial differential inclusions. The proof of the lower bound requires a careful analysis of the admissible microstructures and a new asymptotic rigidity result, whereas the construction of recovery sequences relies on nested laminates.