Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity

TL;DR

This paper uses Gamma-convergence to analyze the asymptotic behavior of single-slip elastoplastic bodies in large deformations, revealing that limit deformations can freely bend under certain conditions and identifying the occurrence of a Lavrentiev phenomenon.

Contribution

It provides a rigorous dimension reduction analysis for elastoplastic models with single-slip, highlighting the effects of differential constraints and the occurrence of Lavrentiev phenomena.

Findings

Limit deformations can freely bend despite constraints.

A Lavrentiev phenomenon occurs with smooth deformations.

Softening differential constraints removes the Lavrentiev phenomenon.

Abstract

We perform via $\Gamma$-convergence a 2d-1d dimension reduction analysis of a single-slip elastoplastic body in large deformations. Rigid plastic and elastoplastic regimes are considered. In particular, we show that limit deformations can essentially freely bend even if subjected to the most restrictive constraints corresponding to the elastically rigid single-slip regime. The primary challenge arises in the upper bound where the differential constraints render any bending without incurring an additional energy cost particularly difficult. We overcome this obstacle with suitable non-smooth constructions and prove that a Lavrentiev phenomenon occurs if we artificially restrict our model to smooth deformations. This issue is absent if the differential constraints are appropriately softened.