Calculus of Variations and Geometric Measure Theory

S. Dipierro - G. Palatucci - E. Valdinoci

Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting

created by palatucci on 23 Sep 2013
modified on 20 Apr 2015


Published Paper

Inserted: 23 sep 2013
Last Updated: 20 apr 2015

Journal: Comm. Math. Phys.
Volume: 3333
Number: 2
Pages: 1061-1105
Year: 2015
Doi: 10.1007/s00220-014-2118-6


Abstract. We consider an evolution equation arising in the Peierls-Nabarro model for crystal dislocation. We study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior repulsive potential.

Keywords: particle systems, fractional Laplacian, fractional Sobolev spaces, nonlocal Allen-Cahn, reaction-diffusion, Peierls-Nabarro model, dislocation dynamics