  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.
