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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
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Official URL: http://dx.doi.org/10.1007/s00205-017-1076-5
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 | ||||||
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| 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: |
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| 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 | ||||||
| Funder: | France. Agence nationale de la recherche (ANR) | ||||||
| Grant number: | ANR-10-LABX-0098 |
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