The Mathematical Institute, University of Oxford, Eprints Archive

Asymptotic analysis of a system of algebraic equations arising in dislocation theory

Hall, C. L. and Chapman, S. J. and Ockendon, J. R. (2010) Asymptotic analysis of a system of algebraic equations arising in dislocation theory. SIAM Journal of Applied Mathematics . (Submitted)

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Abstract

The system of algebraic equations given by

$\sum_{j=0, j \neq i}^n sgn(x_i - x_j) / |x_i - x_j|^a = 1, i = 1, 2, \ldots n, x_0 = 0,$

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 only valid for i not 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.

Item Type:Article
Subjects:D - G > General
Research Groups:Oxford Centre for Collaborative Applied Mathematics
ID Code:994
Deposited By:Peter Hudston
Deposited On:28 Oct 2010 14:29
Last Modified:09 Feb 2012 15:59

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