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Upscaling a model for the thermally-driven motion of screw dislocations

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Hudson, Thomas (2017) Upscaling a model for the thermally-driven motion of screw dislocations. Archive for Rational Mechanics and Analysis, 224 (1). pp. 291-352. doi:10.1007/s00205-017-1076-5 ISSN 0003-9527.

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Official URL: http://dx.doi.org/10.1007/s00205-017-1076-5

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Abstract

We formulate and study a stochastic model for the thermally-driven motion of interacting straight screw dislocations in a cylindrical domain with a convex polygonal cross-section. Motion is modelled as a Markov jump process, where waiting times for transitions from state to state are assumed to be exponentially distributed with rates expressed in terms of the potential energy barrier between the states. Assuming the energy of the system is described by a discrete lattice model, a precise asymptotic description of the energy barriers between states is obtained. Through scaling of the various physical constants, two dimensionless parameters are identified which govern the behaviour of the resulting stochastic evolution. In an asymptotic regime where these parameters remain fixed, the process is found to satisfy a Large Deviations Principle. A sufficiently explicit description of the corresponding rate functional is obtained such that the most probable path of the dislocation configuration may be described as the solution of Discrete Dislocation Dynamics with an explicit anisotropic mobility which depends on the underlying lattice structure.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Mathematics, Markov processes, Large deviations, Monte Carlo method
Journal or Publication Title: Archive for Rational Mechanics and Analysis
Publisher: Springer
ISSN: 0003-9527
Official Date: April 2017
Dates:
DateEvent
April 2017Published
5 January 2017Accepted
Volume: 224
Number: 1
Page Range: pp. 291-352
DOI: 10.1007/s00205-017-1076-5
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access (Creative Commons)
Date of first compliant deposit: 2 October 2017
Date of first compliant Open Access: 2 October 2017
Funder: France. Agence nationale de la recherche (ANR)
Grant number: ANR-10-LABX-0098

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