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

TL;DR

This paper investigates the gradient flow structure of screw dislocation dynamics using a generalized minimising-movements scheme, connecting discrete and continuous models and extending to non-metric dissipations.

Contribution

It establishes the convergence of a time-discrete scheme to the continuous model and generalizes the framework to include non-metric dissipations.

Findings

Proved the limit of discrete schemes converges to the continuous model.

Connected the model to a differential inclusion with finite glide directions.

Extended the model to dissipations beyond metric descriptions.

Abstract

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 generalisation of the setting in [AGS08, Chap. 2 and 3] that allows for dissipations which cannot be described by a metric.