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

We derive a strain-gradient theory for plasticity as the $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }$-converge to the macroscopic strain-gradient model of Garroni, Leoni, and Ponsiglione of 2010.