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.
2005
Dislocation dynamics; Peach-Koehler force; Hamilton-Jacobi equations; viscosity solutions; level sets method; non-local equations
01 Pubblicazione su rivista::01a Articolo in rivista
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]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/145610
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