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

Handle URI:
http://hdl.handle.net/10754/597618
Title:
Asymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theory
Authors:
Hall, Cameron L.; Chapman, S. Jonathan; Ockendon, John R.
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.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Applied Mathematics
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
Jan-2010
DOI:
10.1137/090778444
Type:
Article
ISSN:
0036-1399; 1095-712X
Sponsors:
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).
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Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorHall, Cameron L.en
dc.contributor.authorChapman, S. Jonathanen
dc.contributor.authorOckendon, John R.en
dc.date.accessioned2016-02-25T12:43:09Zen
dc.date.available2016-02-25T12:43:09Zen
dc.date.issued2010-01en
dc.identifier.citationHall 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.en
dc.identifier.issn0036-1399en
dc.identifier.issn1095-712Xen
dc.identifier.doi10.1137/090778444en
dc.identifier.urihttp://hdl.handle.net/10754/597618en
dc.description.abstractThe 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.en
dc.description.sponsorshipReceived 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).en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectApproximation of sumsen
dc.subjectAsymptotic analysisen
dc.subjectDensityen
dc.subjectDiscrete-to-continuumen
dc.subjectDislocationsen
dc.titleAsymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theoryen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Applied Mathematicsen
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
kaust.grant.numberKUK-C1-013-04en
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