Dislocation Dynamics in Crystals: A Macroscopic Theory in a Fractional Laplace Setting

Abstract

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.

Getting Acquainted with the Fractional Laplacian

“…Equations like the one in (2.31) naturally arise, for instance, in long-range phase coexistence models and in models arising in atom dislocation in crystals, see e.g. [52,110].…”

Heteroclinic connections for nonlocal equations

“…and 20) where I N is the N × N identity matrix. Here and henceforth C denotes various positive constants independent of the parameters.…”

Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium

“…The presence of a fractional exponent s ∈ (0, 1) is motivated by models which try to take into account long-range particle interactions (as a matter of fact, these models may produce either a local or non-local tension effect, depending on the value of s, see [SV12,SV14]; see also [PSV13] for the variational analysis of the different scales of energy that are involved in the model) . We also recall that equations of this type naturally occur in other areas of applied mathematics, such as the Peierls-Nabarro model for crystal dislocations when s = 1/2, and for generalizations of this model when s ∈ (0, 1) (see e.g. [Nab97,DPV15,DFV14]). Related problems also arise in models for diffusion of biological species (see e.g.…”