Dynamics for Systems of Screw Dislocations

Abstract

The goal of this paper is the analytical validation of a model of Cermelli and Gurtin [12] for an evolution law for systems of screw dislocations under the assumption of antiplane shear. The motion of the dislocations is restricted to a discrete set of glide directions, which are properties of the material. The evolution law is given by a "maximal dissipation criterion", leading to a system of differential inclusions. Short time existence, uniqueness, cross-slip, and fine cross-slip of solutions are proved.

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).…”

“…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.…”

“…In Section 3, we state and prove the main theorems about the attracting behaviour of free boundaries, and the estimates on the collision times of one dislocation with the boundary and of two dislocations hitting each other. These results will be compared in Section 4 to some explicit cases also discussed in [BFLM15]. We also include numerical plots for domains (namely the square and the cardioid) which exhibit interesting symmetries.…”

Homogenization of vector-valued partition problems and dislocation cell structures in the plane

“…This has been done in several situations, ranging from point singularities in two dimensions [6,36,40] to a general three-dimensional setting [16]. Most of these results require the additional assumption that the admissible dislocation densities correspond to well separated dislocations in order to obtain compactness and to guarantee that sequences with bounded regularized elastic energy concentrate on dislocation lines with finite length and multiplicity, i.e., μ has finite total variation.…”

Minimising movements for the motion of discrete screw dislocations along glide directions

“…According to this criterion the velocity field might be not uniquely determined and the formulation needs to be relaxed. The effective dynamics is then described by a differential inclusion rather than a differential equation (as analysed in [6]). …”