Strain-Gradient Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy with Mixed Growth

Abstract

In this paper, we derive by means of $\Gamma$-convergence a macroscopic strain-gradient plasticity theory from a semidiscrete model for edge dislocations in an infinite cylindrical crystal. We consider an energy with subquadratic growth close to the dislocations. This allows us to treat the stored elastic energy without the need to introduce an ad hoc cut-off radius. As the main tool to prove a complementing compactness statement, we present a generalized version of the geometric rigidity result for fields with nonvanishing $\operatorname{curl}$. One of the main ingredients for the proof is a fine decomposition result for $L^1$-functions whose divergence is in certain critical Sobolev spaces.