TY - JOUR

T1 - Convergence of interaction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane confinement

AU - Mora, Maria Giovanna

AU - Peletier, Mark

AU - Scardia, Lucia

PY - 2017

Y1 - 2017

N2 - 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 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→∞.

AB - 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 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→∞.

UR - https://doi.org/10.1137/16M1096098

U2 - 10.1137/16M1096098

DO - 10.1137/16M1096098

M3 - Article

VL - 49

SP - 4149

EP - 4205

JO - SIAM Journal on Mathematical Analysis (SIMA)

JF - SIAM Journal on Mathematical Analysis (SIMA)

SN - 0036-1410

IS - 5

ER -