Convergence of Interaction-Driven Evolutions of Dislocations with Wasserstein Dissipation and Slip-Plane Confinement

We consider systems of n parallel edge dislocations in a single slip system, represented by points in a two-dimensional domain; the elastic medium is modelled as a continuum. We formulate the energy of this system in terms of the empirical measure of the dislocations, and prove several convergence results in the limit n → ∞. The main aim of the paper is to study the convergence of the evolution of the empirical measure as n → ∞. We consider rate-independent, quasi-static evolutions, in which the motion of the dislocations is restricted to the same slip plane. This leads to a formulation of the quasistatic evolution problem in terms of a modified Wasserstein distance, which is only finite when the transport plan is slip-plane-confined. Since the focus is on interaction between dislocations, we renormalize the elastic energy to remove the potentially large selfor core energy. We prove Gamma-convergence of this renormalized energy, and we construct joint recovery sequences for which both the energies and the modified distances converge. With this augmented Gamma-convergence we prove the convergence of the quasi-static evolutions as n → ∞.