Geometric rigidity for incompatible fields in the multi-well case and an application to strain-gradient plasticity

We derive a quantitative rigidity estimate for a multi-well problem in nonlinear elasticity with dislocations. Precisely, we show that the $L^{1^{\ast }}$-distance of a possibly incompatible strain field $\beta$ from a single well is controlled in terms of the $L^{1^{\ast }}$-distance from a finite set of wells, of $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\beta$, and of $\mathrm{d}\mathrm{i}\mathrm{v}\beta$. As a consequence, we derive a strain-gradient plasticity model as $\mathrm{\Gamma }$-limit of a nonlinear finite dislocation model, containing a singular perturbation term accounting for the divergence of the strain field. This can also be seen as a generalization of the result of (Alicandro et al. 2018) to the case of incompatible vector fields.