Calculus of Variations and Geometric Measure Theory

G. A. Bonaschi - P. van Meurs - M. Morandotti

Dynamics of screw dislocations: a generalised minimising-movements scheme approach

created by morandott on 03 Sep 2015
modified on 19 Jul 2017


Published Paper

Inserted: 3 sep 2015
Last Updated: 19 jul 2017

Journal: Eur. J. Appl. Math.
Volume: 28
Number: 4
Pages: 636-655
Year: 2017
Doi: 10.1017/S0956792516000462

Preprint SISSA 382015MATE


The gradient flow structure of the model introduced in [CG99] for the dynamics of screw dislocations is investigated by means of a generalised minimising-movements scheme approach. The assumption of a finite number of available glide directions, together with the "maximal dissipation criterion" that governs the equations of motion, results into solving a differential inclusion rather than an ODE. This paper addresses how the model in [CG99] is connected to a time-discrete evolution scheme which explicitly confines dislocations to move each time step along a single glide direction. It is proved that the time-continuous model in [CG99] is the limit of these time-discrete minimising-movement schemes when the time step converges to 0. The study presented here is a first step towards a generalization of the setting in [AGS08, Chap. 2 and 3] that allows for dissipations which cannot be described by a metric.

Keywords: Motion of dislocations, Generalised gradient flows, energy dissipation inequality, minimising-movements scheme