Dislocation dynamics described by non-local Hamilton–Jacobi equations

• Published in: Materials science & engineering. A, Structural materials : properties, microstructure and processing Vol. 400; pp. 162 - 165
• Main Authors: Alvarez, O., Carlini, E., Hoch, P., Le Bouar, Y., Monneau, R.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 25.07.2005
• Subjects: Dislocation dynamics; Hamilton–Jacobi equations; Level sets method; Non-local equations; Peach–Koehler force; Viscosity solutions; Level sets method; Peach–Koehler force; Hamilton–Jacobi equations; Dislocation dynamics; Non-local equations; Viscosity solutions
• ISSN: 0921-5093; 1873-4936
• DOI: 10.1016/j.msea.2005.01.062

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.