Asymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theory

Type
Article

Authors
Hall, Cameron L.
Chapman, S. Jonathan
Ockendon, John R.

KAUST Grant Number
KUK-C1-013-04

Date
2010-01

Abstract
The system of algebraic equations given by σn j=0, j≠=i sgn(xi-xj )|xi-xj|a = 1, i = 1, 2, ⋯ , n, x0 = 0, appears in dislocation theory in models of dislocation pile-ups. Specifically, the case a = 1 corresponds to the simple situation where n dislocations are piled up against a locked dislocation, while the case a = 3 corresponds to n dislocation dipoles piled up against a locked dipole. We present a general analysis of systems of this type for a > 0 and n large. In the asymptotic limit n→∞, it becomes possible to replace the system of discrete equations with a continuum equation for the particle density. For 0 < a < 2, this takes the form of a singular integral equation, while for a > 2 it is a first-order differential equation. The critical case a = 2 requires special treatment, but, up to corrections of logarithmic order, it also leads to a differential equation. The continuum approximation is valid only for i neither too small nor too close to n. The boundary layers at either end of the pile-up are also analyzed, which requires matching between discrete and continuum approximations to the main problem. © 2010 Society for Industrial and Applied Mathematics.

Citation
Hall CL, Chapman SJ, Ockendon JR (2010) Asymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theory. SIAM Journal on Applied Mathematics 70: 2729–2749. Available: http://dx.doi.org/10.1137/090778444.

Acknowledgements
Received by the editors November 30, 2009; accepted for publication (in revised form) June 7, 2010; published electronically August 10, 2010. This publication is based on work supported by award KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
SIAM Journal on Applied Mathematics

DOI
10.1137/090778444

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