  In this paper we show the emergence of polycrystalline structures as a result
of elastic energy minimisation. For this purpose, we introduce a variational
model for two-dimensional systems of edge dislocations, within the so-called
core radius approach, and we derive the $\Gamma$-limit of the elastic energy
functional as the lattice space tends to zero. In the energy regime under
investigation, the symmetric and skew part of the strain become decoupled in
the limit, the dislocation measure being the curl of the skew part of the
strain. The limit energy is given by the sum of a plastic term, acting on the
dislocation density, and an elastic term, which depends on the symmetric
strains. Minimisers under suitable boundary conditions are piece-wise constant
antisymmetric strain fields, representing in our model a polycrystal whose
grains are mutually rotated by infinitesimal angles.

Analysis of PDEs