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
appears in dislocation theory in models of dislocation pileups. 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 firstorder 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 pileup 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 13:29 
Last Modified:  29 May 2015 18:41 
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