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

We consider an evolution equation arising in the Peierls-Nabarro model for crystal dislocation. We study the evolution of such a 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.