  We derive a quantitative rigidity estimate for a multi-well problem in
nonlinear elasticity with dislocations. Precisely, we show that the
$L^{1^{*}}$-distance of a possibly incompatible strain field $\beta$ from a
single well is controlled in terms of the $L^{1^{*}}$-distance from a finite
set of wells, of ${\rm curl}\beta$, and of ${\rm div}\beta$. As a consequence,
we derive a strain-gradient plasticity model as $\Gamma$-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.
