Derivation of strain-gradient plasticity from a generalized Peierls-Nabarro model

TL;DR

This paper derives a strain-gradient plasticity model from a nonlocal phase-field dislocation model, connecting microscopic dislocation interactions to macroscopic plastic behavior through mathematical scaling limits.

Contribution

It introduces a novel derivation of strain-gradient plasticity from a phase-field model, incorporating microstructure formation via $ ext{Gamma}$-convergence.

Findings

Derivation of a continuous energy depending on dislocation density

Identification of a nonlocal effective energy for dislocation interactions

Automatic inclusion of microstructure formation in the limiting process

Abstract

We derive strain-gradient plasticity from a nonlocal phase-field model of dislocations in a plane. Both a continuous energy with linear growth depending on a measure which characterizes the macroscopic dislocation density and a nonlocal effective energy representing the far-field interaction between dislocations arise naturally as scaling limits of the nonlocal elastic interaction. Relaxation and formation of microstructures at intermediate scales are automatically incorporated in the limiting procedure based on $\Gamma$-convergence.