Derivation of Linearized Polycrystals from a Two-Dimensional System of Edge Dislocations

Abstract

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.