Calculus of Variations and Geometric Measure Theory

S. Almi - M. Caponi - M. Friedrich - F. Solombrino

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

created by caponi on 12 Jun 2024
modified on 13 Jun 2024


Submitted Paper

Inserted: 12 jun 2024
Last Updated: 13 jun 2024

Year: 2024

ArXiv: 2406.08023 PDF


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.

Keywords: Gamma-convergence, edge dislocations, strain-gradient plasticity, Fractional model of discrete dislocations, Fractional and nonlocal gradients, Riesz potentials