Existence and stability of a screw dislocation under anti-plane deformation

TL;DR

This paper develops a variational model for screw dislocations in anti-plane lattice structures, proving the existence and stability of dislocation solutions despite non-coercivity challenges.

Contribution

It introduces a new variational framework accounting for lattice symmetries and establishes the global stability of dislocations in this setting.

Findings

Existence of a global energy minimiser for screw dislocations.

Dislocations are proven to be globally stable equilibria.

The model handles non-coercivity through coercivity and concentration compactness.

Abstract

We formulate a variational model for a geometrically necessary screw dislocation in an anti-plane lattice model at zero temperature. Invariance of the energy functional under lattice symmetries renders the problem non-coercive. Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove existence of a global energy minimiser and thus demonstrate that dislocations are globally stable equilibria within our model.