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v1 (2006-06-28)

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Homogenization of first order equations with $u/\epsilon$-periodic Hamiltonians. Part II: application to dislocations dynamics

Cyril Imbert 1, Régis Monneau 2, Elisabeth Rouy 3

For the ACI JC 1025 collaboration(s)

(2006-06-16)

This paper is concerned with a result of homogenization of a non-local first order Hamilton-Jacobi equations describing the dislocations dynamics. Our model for the interaction between dislocations involve both an integro-differential operator and a (local) Hamiltonian depending periodicly on $u/\eps$. The first two authors studied in a previous work homogenization problems involving such local Hamiltonians. Two main ideas of this previous work are used: on the one hand, we prove an ergodicity property of this equation by constructing approximate correctors which are necessarily non periodic in space in general; on the other hand, the proof of the convergence of the solution uses here a twisted perturbed test function for a higher dimensional problem. The limit equation is a nonlinear diffusion equation involving a first order Lévy operator; the nonlinearity keeps memory of the short range interaction, while the Lévy operator keeps memory of long ones. The homogenized equation is a kind of effective plastic law for densities of dislocations moving in a single slip plane.

1:	Institut de Mathématiques et de Modélisation de Montpellier (I3M)
	CNRS : UMR5149 – Université Montpellier II - Sciences et techniques
2:	Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS)
	INRIA – Ecole des Ponts ParisTech
3:	École Centrale de Lyon (ECL)
	Ministère de l'Enseignement Supérieur et de la Recherche Scientifique

Subject

:

Mathematics/Analysis of PDEs

Keyword(s): periodic homogenization – Hamilton-Jacobi equations – integro-differential operators – dislocations dynamics – non-periodic approximate correctors

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From: Cyril Imbert <>

Submitted on: Friday, 16 February 2007 14:13:11

Updated on: Friday, 16 February 2007 15:04:06