Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

Abstract

We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation whose velocity is a non-local quantity depending on the whole shape of the dislocation line. We study the special cases where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for HamiltonJacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model.

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

“…In the case where there is no exterior stress, this force is simply the self-force created by the elastic field generated by the dislocation line itself. In [5], [4], Alvarez, Hoch, Le Bouar and Monneau proposed to rewrite this model as a non-local HamiltonJacobi equation. Using viscosity solutions (we refer to the monographs of Barles [7] and Bardi and Capuzzo-Dolcetta [6] and to the paper of Crandall, Ishii and Lions [21] for a good introduction to this theory), Alvarez et al [5], [4] proved a short time existence and uniqueness result.…”

“…In [5], [4], Alvarez, Hoch, Le Bouar and Monneau proposed to rewrite this model as a non-local HamiltonJacobi equation. Using viscosity solutions (we refer to the monographs of Barles [7] and Bardi and Capuzzo-Dolcetta [6] and to the paper of Crandall, Ishii and Lions [21] for a good introduction to this theory), Alvarez et al [5], [4] proved a short time existence and uniqueness result. Then Alvarez, Cardaliaguet and Monneau [1] and Barles and Ley [10] proved a long time result under certain assumptions.…”

“…In this paper, we consider a simplification of the model proposed by Alvarez et al [5], [4]. We assume that the negative part of the kernel c 0 is concentrated at one point, i.e., c 0 = c 0 − ( R 2 c 0 )δ 0 where c 0 is now a positive kernel.…”

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

“…Let us mention the application to dislocations dynamics where the velocity depends on the front itself via a convolution term. The corresponding level set model leads to a non-local Hamilton-Jacobi equation which has been analyzed in [3], a numerical approximation has been proposed in [2]. Let us underline some differences between the classical FMM method and our algorithm.…”

“…Here the support of the discontinuities of the function θ localizes the front we are interested in. This work is motivated by the applications to dislocations dynamics where the velocity of the front depends on an integral term and can change sign (see Alvarez, Hoch, Le Bouar, Monneau [3]). …”