Calculus of Variations and Geometric Measure Theory

R. Alicandro - L. De Luca - G. Lazzaroni - M. Palombaro - M. Ponsiglione

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

created by deluca on 19 Jan 2022
modified by lazzaroni on 27 Jan 2023


Accepted Paper

Inserted: 19 jan 2022
Last Updated: 27 jan 2023

Journal: J. Nonlinear Sci.
Year: 2022

ArXiv: 2201.05901 PDF


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.

Keywords: $\Gamma$-convergence, dislocations, plasticity, Discrete-to-continuum limits, Topological singularities