Strongly Nonlocal Dislocation Dynamics in Crystals

Abstract

We consider the equation

where L _s is an integro-differential operator of order 2s, with s ∈ (0, 1), W is a periodic potential, and σ_ϵ is a small external stress. The solution v represents the atomic dislocation in the Peierls–Nabarro model for crystals, and we specifically consider the case s ∈ (0, 1/2), which takes into account a strongly nonlocal elastic term.

We study the evolution of such dislocation function for macroscopic space and time scales, namely we introduce the function

We show that, for small ϵ, the function v _ϵ approaches the sum of step functions. From the physical point of view, this shows that 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. We also show that the motion of these dislocation points is governed by an interior repulsive potential that is superposed to an elastic reaction to the external stress.

Keywords:

• Dislocation dynamics
• Fractional Laplacian
• Nonlocal Peierls-Nabarro model
• Oscillation and regularity results

2010 Mathematics Subject Classfication:

• 49N60
• 35B05
• 35Q99
• 35B40
• 35J25
• 35D30
• 35G25