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 modeled 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 $n\to\infty$. The main aim of the paper is to study the convergence of the evolution of the empirical measure as $n\to\infty$. 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 quasi-static 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 self- or 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 $n\to\infty$.
Mora, M. G., Peletier, M. A., & Scardia, L. (2017). Convergence of interaction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane confinement. SIAM Journal on Mathematical Analysis, 49(5), 4149–4205. https://doi.org/10.1137/16M1096098