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

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 minimisingmovement 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.

Properties of Screw Dislocation Dynamics: Time Estimates on Boundary and Interior Collisions

“…It is commonly observed in numerical simulations that dislocations are attracted to free boundaries and that dislocations of opposite signs attract. In fact, as dislocations approach the boundary, or as dislocations with Burgers moduli of opposite sign approach one another, the renormalised energy diverges to −∞, and hence solutions of the evolution problem blow up and cease to exist, at least in the senses considered in [BFLM15,BvMM16]. In Lemma 2.4, we prove a gradient bound for the function h Ω for points in the vicinity of the boundary: this allows us to treat case (i).…”

“…Various suggestions for possible mobility functions can be found in [CG99]; we refer the reader to Section 5 for a discussion on other possible choices that are relevant in our context. For a specific choice of the mobility, (1.9) takes the form of a differential inclusion, and was studied both in [BFLM15] to obtain existence and uniqueness results, and in [BvMM16] from the point of view of gradient flows.…”

“…The model we consider was first proposed in full generality in [CG99], and studied extensively with specific choices of mobility in [BFLM15,BvMM16]. In particular, the latter two works prove existence of the evolution by differing methods, but acknowledge that blow-up of solutions appears to be a ubiquitous phenomenon.…”

Variational Evolution of Dislocations in Single Crystals

“…If one restricts to the motion of parallel dislocations, the problem becomes twodimensional, since the dislocations are modelled as points. In this case, the mathematical analysis of dynamics model already exist in the wake of Ginsburg-Landau vortices dynamics, and can be found for instance in [1,2,8,9]. The dynamics of this kind of dislocations have been studied by several authors, see for instance the important contributions [18] and [25] for a rate-independent evolution.…”

An existence result for Discrete Dislocation Dynamics in three dimensions

“…The fact that the dislocations alone are tracked in this approach has the advantage of drastically reducing the computational complexity in comparison with phase field approaches, and as such, DDD has been used as a simulation technique for studying plasticity since the early 1990s [5,6,8,17,33,44,77]. While a significant mathematical literature has developed which considers one-and two-dimensional DDD models for the motion of straight dislocations [1,2,12,13,15,23,24,46,47,78], few mathematical results concerning DDD in a three-dimensional setting exist to date, and in part, this appears to be due to the lack of a clear mathematical statement of what the evolution problem for DDD should be in this setting.…”