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

TL;DR

This paper extends previous homogenization results for layered composite materials by considering multiple slip directions in crystal plasticity, deriving the Gamma-limit for orthogonal slips and providing partial results for non-orthogonal cases.

Contribution

It introduces a Gamma-convergence analysis for layered materials with two slip directions, advancing the understanding of complex plastic deformation in composites.

Findings

Derived the Gamma-limit for orthogonal slip systems.

Provided a partial homogenization result for non-orthogonal slip directions.

Addressed nonconvex constraints in the microstructure analysis.

Abstract

This paper is an extension of the result by Christowiak and Kreisbeck (2017), which addresses the Gamma-convergence approach to a homogenization problem for composite materials consisting of two distinct types of parallel layers. In Christowiak et al. (2017), one of the layers of the material undergoes only local rotations while the other allows local rotation and plastic deformation along a single slip system. On the other hand, real materials show an interplay of multiple directions of slip. Here we obtain the Gamma-limit for the problem where the number of slip directions is increased to two. When the slip systems are orthogonal, we derive the full homogenized energy based on generalized convex envelopes of the original energy density. Since these envelopes are not completely known for angles between slip directions differing from the right angle, we present a partial, conditional homogenization result in this general case. The analysis is based on a modification of the classical construction of laminate microstructures but several nontrivial difficulties arise due to nonconvex constraints being present in the composite energy.