Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T23:47:00.918Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Al Haj, M., Forcadel, N. & Monneau, R. (2013) Existence and uniqueness of traveling waves for fully overdamped Frenkel-Kontorova models. Arch. Ration. Mech. Anal. 210 (1), 45–99.CrossRefGoogle Scholar
[2]Cabré, X. & Solà-Morales, J. (2005) Layer solutions in a half-space for boundary reactions. Comm. Pure Appl. Math. 58 (12), 1678–1732.Google Scholar
[3]Denoual, C. (2004) Dynamic dislocation modeling by combining Peierls-Nabarro and Galerkin methods. Phys. Rev. B 70, 024106.Google Scholar
[4]Denoual, C. (2007) Modeling dislocation by coupling Peierls-Nabarro and element-free Galerkin methods. Comput. Methods Appl. Mech. Eng. 196, 1915–1923.CrossRefGoogle Scholar
[5]El Hajj, A., Ibrahim, H. & Monneau, R. (2009) Dislocation dynamics: From microscopic models to macroscopic crystal plasticity. Contin. Mech. Thermodyn. 21, 109–123.CrossRefGoogle Scholar
[6]Eshelby, J. D. (1949) Uniformly moving dislocations. Proc. Phys. Soc. A 62, 307–314.Google Scholar
[7]Fino, A., Ibrahim, H. & Monneau, R. (2012) The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model. J. Differ. Equ. 252, 258–293.CrossRefGoogle Scholar
[8]Forcadel, N., Imbert, C. & Monneau, R. (2009) Homogenization of fully overdamped Frenkel-Kontorova models. J. Differ. Equ. 246 (3), 1057–1097.CrossRefGoogle Scholar
[9]Gonzalez, M. & Monneau, R. (2012) Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete Contin. Dyn. Syst. A 32 (4), 1255–1286.Google Scholar
[10]Hirth, J. R. & Lothe, L. (1992) Theory of Dislocations, 2nd ed.Krieger, Malabar, FL.Google Scholar
[11]Hirth, J. R., Zbib, H. M. & Lothe, L. (1998) (Forces on high velocity dislocations. Model. Simul. Mater. Sci. Eng. 6 (2), 165–169.CrossRefGoogle Scholar
[12]Monneau, R. & Patrizi, S. (2012) Homogenization of the Peierls-Nabarro model for dislocation dynamics. J. Differ. Equ. 253 (7), 2064–2105.Google Scholar
[13]Nabarro, F. R. N. (1997) Fifty-year study of the Peierls-Nabarro stress. Mater. Sci. Eng. A 234–236, 67–76.Google Scholar
[14]Pellegrini, Y. P. (2010) Dynamic Peierls-Nabarro equations for elastically isotropic crystals. Phys. Rev. B 81, 024101.Google Scholar
[15]Pillon, L. (2008) Modélisations du Mouvement Instationnaire et des Interactions de Dislocations, PhD thesis, Pierre et Marie Curie University, Paris, France.Google Scholar
[16]Pillon, L. & Denoual, C. (2009) Inertial and retardation effects for dislocation interactions. Phil. Mag. 89 (2), 127–141.Google Scholar
[17]Pillon, L., Denoual, C., Madec, R. & Pellegrini, Y. P. (August 2006) Influence of inertia on the formation of dislocation dipoles. J. Phys. IV (Proc.) 134 (1), 49–54.Google Scholar
[18]Pillon, L., Denoual, C. & Pellegrini, Y. P. (2007) Equation of motion for dislocations with inertial effects. Phys. Rev. B 76, 224105.Google Scholar