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

TL;DR

This paper derives a strain-gradient plasticity model as a $\Gamma$-limit from a dislocation energy with mixed growth, avoiding ad-hoc cut-offs and introducing advanced mathematical tools for compactness.

Contribution

It introduces a novel derivation of strain-gradient plasticity from a semi-discrete dislocation model with subquadratic energy growth, expanding the mathematical framework.

Findings

Derived a macroscopic strain-gradient plasticity model via $\Gamma$-convergence.

Handled energies with subquadratic growth near dislocations without cut-offs.

Developed a generalized geometric rigidity result for fields with non-zero curl.

Abstract

In this paper a we derive by means of $\Gamma$-convergence a macroscopic strain-gradient plasticity from a semi-discrete model for dislocations in an infinite cylindrical crystal. In contrast to existing work, we consider an energy with subquadratic growth close to the dislocations. This allows 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 non-vanishing $\operatorname{curl}$. A main ingredient is a fine decomposition result for $L^1$-functions whose divergence is in certain critical Sobolev spaces.