Coarse-graining of a discrete model for edge dislocations in the regular triangular lattice

TL;DR

This paper analyzes a discrete elastic model on a triangular lattice, introducing plastic slip fields to detect edge dislocations and performing a b3-convergence analysis as the lattice spacing approaches zero.

Contribution

It introduces a novel discrete model with plastic slip fields for edge dislocations and provides a rigorous b3-convergence analysis of the elastic energy in the continuum limit.

Findings

Identification of plastic slip fields for edge dislocation detection

Rigorous b3-convergence of elastic energy as lattice spacing decreases

Characterization of energy regime with finite dislocations

Abstract

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 triangle 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.