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

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.