Derivation of Linearised Polycrystals from a 2D system of edge dislocations

TL;DR

This paper derives a mathematical model showing how polycrystalline structures emerge from elastic energy minimization in 2D dislocation systems, revealing the formation of grain structures through variational analysis.

Contribution

It introduces a new variational model for 2D edge dislocation systems and derives the Gamma-limit of the elastic energy functional, linking dislocation measures to strain fields.

Findings

Polycrystalline structures emerge as energy minimizers.

Dislocation measure is the curl of the skew part of strain.

Minimizers are piece-wise constant antisymmetric strains representing grains.

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.