We study a mathematical model describing dislocation dynamics in crystals. This phase-field model is based on the introduction of a core tensor which mollifies the singular field on the core of the dislocation. We present this model in the case of the motion of a single dislocation, without cross-slip. The dynamics of a single dislocation line, moving in its slip plane, is described by an Hamilton-Jacobi equation whose velocity is a non-local quantity depending on the whole shape of the dislocation line. Introducing a level sets formulation of this equation, we prove the existence and uniqueness of a continuous viscosity solution when the dislocation stays a graph in one direction. We also propose a numerical scheme for which we prove that the numerical solution converges to the continuous solution.
Dislocation dynamics described by non-local Hamilton-Jacobi equations / O., Alvarez; Carlini, Elisabetta; P., Hoch; Y., LE BOUAR; R., Monneau. - In: MATERIALS SCIENCE AND ENGINEERING A-STRUCTURAL MATERIALS PROPERTIES MICROSTRUCTURE AND PROCESSING. - ISSN 0921-5093. - 400-4001:(2005), pp. 162-165. [10.1016/j.msea.2005.01.062]
Dislocation dynamics described by non-local Hamilton-Jacobi equations
CARLINI, Elisabetta;
2005
Abstract
We study a mathematical model describing dislocation dynamics in crystals. This phase-field model is based on the introduction of a core tensor which mollifies the singular field on the core of the dislocation. We present this model in the case of the motion of a single dislocation, without cross-slip. The dynamics of a single dislocation line, moving in its slip plane, is described by an Hamilton-Jacobi equation whose velocity is a non-local quantity depending on the whole shape of the dislocation line. Introducing a level sets formulation of this equation, we prove the existence and uniqueness of a continuous viscosity solution when the dislocation stays a graph in one direction. We also propose a numerical scheme for which we prove that the numerical solution converges to the continuous solution.File | Dimensione | Formato | |
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