Coarse-Graining of a Discrete Model for Edge Dislocations in the Regular Triangular Lattice

We consider a discrete model of planar elasticity where the particles, in the reference configuration, sit on a regular triangular lattice and interact through nearest-neighbor pairwise potentials, with bonds modeled as linearized elastic springs. Within this framework, we introduce plastic slip fields, whose discrete circulation around each tri-angle detects the possible presence of an edge dislocation. We provide a gamma-convergence analysis, as the lattice spacing tends to zero, of the elastic energy induced by edge dislocations in the energy regime corresponding to a finite number of geometrically necessary dislocations.