A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations

TL;DR

This paper develops a nonlocal strain-gradient plasticity model using fractional gradients derived from discrete dislocation energies, connecting microscopic dislocation behavior to macroscopic continuum theories without a core-radius.

Contribution

It introduces a fractional gradient-based nonlocal model for dislocations and proves its convergence to classical strain-gradient plasticity as the fractional order approaches zero.

Findings

Derived a nonlocal dislocation model using fractional gradients.

Proved $b3$-convergence to classical strain-gradient plasticity.

Established a link between discrete dislocation energies and continuum models.

Abstract

We derive a strain-gradient theory for plasticity as the $\Gamma$-limit of discrete dislocation fractional energies, without the introduction of a core-radius. By using the finite horizon fractional gradient introduced by Bellido, Cueto, and Mora-Corral of 2023, we consider a nonlocal model of semi-discrete dislocations, in which the stored elastic energy is computed via the fractional gradient of order $1-\alpha$. As $\alpha$ goes to $0$, we show that suitably rescaled energies $\Gamma$-converge to the macroscopic strain-gradient model of Garroni, Leoni, and Ponsiglione of 2010.