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Beyond kinetic relations

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Abstract

We introduce the concept of kinetic or rate equations for moving defects representing a natural extension of the more conventional notion of a kinetic relation. Algebraic kinetic relations, widely used to model dynamics of dislocations, cracks and phase boundaries, link the instantaneous value of the velocity of a defect with an instantaneous value of the driving force. The new approach generalizes kinetic relations by implying a relation between the velocity and the driving force which is nonlocal in time. To make this relation explicit one may need to integrate a system of kinetic equations. We illustrate the difference between kinetic relation and kinetic equations by working out in full detail a prototypical model of an overdamped defect in a one-dimensional discrete lattice. We show that the minimal nonlocal kinetic description, containing now an internal time scale, is furnished by a system of two ordinary differential equations coupling the spatial location of defect with another internal parameter that describes configuration of the core region.

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Authors and Affiliations

  1. Laboratoire de Mechanique des Solides, CNRS-UMR 7649, Ecole Polytechnique, 91128, Palaiseau, France

    Lev Truskinovsky

  2. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Anna Vainchtein

Authors
  1. Lev Truskinovsky
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  2. Anna Vainchtein
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Correspondence to Anna Vainchtein.

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Communicated by Stefan Seelecke.

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Truskinovsky, L., Vainchtein, A. Beyond kinetic relations. Continuum Mech. Thermodyn. 22, 485–504 (2010). https://doi.org/10.1007/s00161-010-0167-4

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  • DOI: https://doi.org/10.1007/s00161-010-0167-4

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