A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations
Stefano Almi, Maicol Caponi, Manuel Friedrich, Francesco Solombrino
TL;DR
The article develops a core-radius-free, fractional-gradient framework for discrete dislocations with finite horizon, establishing Γ-convergence to a 2D strain-gradient plasticity energy as $oldsymbol{eta}$ and the dislocation measure $oldsymbol{ it}$ vary under well-separated conditions. A key step is the cell-formula analysis that yields a self-energy density $oldsymbol{oldsymbol{ extvarphi}}$ matching the core-radius model's density, enabling a unified treatment of critical, subcritical, and supercritical regimes. The critical regime yields a Γ-limit combining bulk elasticity and a convex, positively 1-homogeneous self-energy density $oldsymbol{ extvarphi}$; the subcritical and supercritical regimes recover a decoupled or purely elastic limit, respectively. Overall, the work provides a core-radius-free derivation of a macroscopic strain-gradient plasticity model from a nonlocal, fractional-dislocation description, with clear implications for the mathematical understanding and modeling of dislocation-mediated plasticity in two dimensions.
Abstract
We derive a strain-gradient theory for plasticity as the $Γ$-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 $1-α$. As $α$ goes to $0$, we show that suitably rescaled energies $Γ$-converge to the macroscopic strain-gradient model of Garroni, Leoni, and Ponsiglione of 2010.