Many-particle limits and non-convergence of dislocation wall pile-ups

Abstract

The starting point of our analysis is a class of one-dimensional interacting particle systems with two species. The particles are confined to an interval and exert a nonlocal, repelling force on each other, resulting in a nontrivial equilibrium configuration. This class of particle systems covers the setting of pile-ups of dislocation walls, which is an idealised setup for studying the microscopic origin of several dislocation density models in the literature. Such density models are used to construct constitutive relations in plasticity models. Our aim is to pass to the many-particle limit. The main challenge is the combination of the nonlocal nature of the interactions, the singularity of the interaction potential between particles of the same type, the non-convexity of the the interaction potential between particles of the opposite type, and the interplay between the length-scale of the domain with the length-scale $\ell_n$ of the decay of the interaction potential. Our main results are the $\Gamma$-convergence of the energy of the particle positions, the evolutionary convergence of the related gradient flows for $\ell_n$ sufficiently large, and the non-convergence of the gradient flows for $\ell_n$ sufficiently small.