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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: [email protected]
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: [email protected]
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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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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