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Reduced ODE dynamics as formal relativistic asymptotics of a Peierls–Nabarro model

Published online by Cambridge University Press:  08 April 2014

H. IBRAHIM  and
R. MONNEAU
H. IBRAHIM
Affiliation:
Mathematics Department, Faculty of Sciences, Lebanese University, Hadeth, Beirut, Lebanon Mathematics Department, School of Arts and Sciences, Lebanese International University (LIU), Beirut Campus, Al-Mouseitbeh, Beirut, Lebanon email: hassan.ibrahim@liu.edu.lb
R. MONNEAU
Affiliation:
CERMICS, Ecole des Ponts, Université Paris-Est, 6 et 8 Avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France email: monneau@cermics.enpc.fr
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Abstract

In this paper, we consider a scalar Peierls--Nabarro model describing the motion of dislocations in the plane (x1,x2) along the line x2=0. Each dislocation can be seen as a phase transition and creates a scalar displacement field in the plane. This displacement field solves a simplified elasto-dynamics equation, which is simply a linear wave equation. The total displacement field creates a stress which makes move the dislocation itself. By symmetry, we can reduce the system to a wave equation in the half plane x2>0 coupled with an equation for the dynamics of dislocations on the boundary of the half plane, i.e. on x2=0. Our goal is to understand the dynamics of well-separated dislocations in the limit when the distance between dislocations is very large, of order 1/ɛ. After rescaling, this corresponds to introduce a small parameter ɛ in the system. For the limit ɛ → 0, we are formally able to identify a reduced ordinary differential equation model describing the dynamics of relativistic dislocations if a certain conjecture is assumed to be true.

Keywords

Peierls–Nabarro modelDislocation dynamicsRelativitic dymamicsFormal asymptoticsReduced model
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 4 , August 2014 , pp. 511 - 529
Copyright
Copyright © Cambridge University Press 2014 

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