Calculus of Variations and Geometric Measure Theory

M. G. Mora - M. Peletier - L. Scardia

Convergence of interaction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane confinement

created by mora on 15 Sep 2014
modified by scardia on 17 Jan 2018


Published Paper

Inserted: 15 sep 2014
Last Updated: 17 jan 2018

Journal: SIAM J. Math. Anal.
Year: 2017


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 $n\to\infty$.

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 $n\to\infty$.