Plasticity as the $\Gamma$-Limit of a Two-Dimensional Dislocation Energy: the Critical Regime without the Assumption of Well-Separateness

TL;DR

This paper derives a strain-gradient plasticity model from a mesoscopic dislocation model without assuming well-separated dislocations, using advanced mathematical techniques to establish lower bounds and address challenges in rigidity.

Contribution

It introduces a novel derivation of a plasticity model without the well-separateness assumption, adapting the ball construction technique for this context.

Findings

Established a new strain-gradient plasticity model from dislocation energies.

Extended the ball construction technique to handle non-well-separated dislocations.

Provided rigorous lower bounds in the critical regime for dislocation energies.

Abstract

In this paper, a strain-gradient plasticity model is derived from a mesoscopic model for straight parallel edge dislocations in an infinite cylindrical crystal. The main difference to existing work is that in this work the well-separateness of disloactions is not assumed. In order to prove meaningful lower bounds the ball construction technique, which was developed in the context of Ginzburg-Landau by Jerrard and Sandier, is adapted and modified. To overcome the difficulty of a loss of rigidity on thin annuli during the ball construction a combination of combinatorial arguments and local modifications of the occurring elastic strains is presented.