378 articles – 175 references  [version française]
 HAL: hal-00080397, version 2
 Available versions: v1 (2006-06-28) v2 (2007-02-16)
 Homogenization of first order equations with $u/\epsilon$-periodic Hamiltonians. Part II: application to dislocations dynamics
 Cyril Imbert 1, Régis Monneau 2
 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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 hal-00080397, version 2 http://hal.archives-ouvertes.fr/hal-00080397 oai:hal.archives-ouvertes.fr:hal-00080397 From: Cyril Imbert <> Submitted on: Friday, 16 February 2007 14:13:11 Updated on: Friday, 16 February 2007 15:04:06