Variational models for plasticity by homogenization of discrete dislocations

Dislocations are topological defects in crystal that are considered responsible for plastic deformations. We consider a 2D model for edge dislocations, where the deformation has a singularity on points that represent dislocations, while the crystal behaves elastically far from the core. This model is very close to the 2D Ginburg-Landau model for the study of vortices in superconductors. We study, in a dilute regime, the limit as the number of points (dislocations) tends to infinity and we obtain as limit problem an elasto-plastic model, given by the elastic energy and a term depending on the Curl of the plastic deformation (the dislocations density).