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# 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:
- 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:
- Journal:
- KAUST Grant Number:
- 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).

- Appears in Collections:
- Publications Acknowledging KAUST Support

# Full metadata record

DC Field | Value | Language |
---|---|---|

dc.contributor.author | Hall, Cameron L. | en |

dc.contributor.author | Chapman, S. Jonathan | en |

dc.contributor.author | Ockendon, John R. | en |

dc.date.accessioned | 2016-02-25T12:43:09Z | en |

dc.date.available | 2016-02-25T12:43:09Z | en |

dc.date.issued | 2010-01 | en |

dc.identifier.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. | en |

dc.identifier.issn | 0036-1399 | en |

dc.identifier.issn | 1095-712X | en |

dc.identifier.doi | 10.1137/090778444 | en |

dc.identifier.uri | http://hdl.handle.net/10754/597618 | en |

dc.description.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. | en |

dc.description.sponsorship | 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). | en |

dc.publisher | Society for Industrial & Applied Mathematics (SIAM) | en |

dc.subject | Approximation of sums | en |

dc.subject | Asymptotic analysis | en |

dc.subject | Density | en |

dc.subject | Discrete-to-continuum | en |

dc.subject | Dislocations | en |

dc.title | Asymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theory | en |

dc.type | Article | en |

dc.identifier.journal | SIAM Journal on Applied Mathematics | en |

dc.contributor.institution | University of Oxford, Oxford, United Kingdom | en |

kaust.grant.number | KUK-C1-013-04 | en |

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