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Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics

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Abstract

We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold—where the body is stress free—is a Weitzenböck manifold, that is, a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan’s moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields, assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance.

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Authors and Affiliations

  1. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Arash Yavari

  2. OCCAM, Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK

    Alain Goriely

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  1. Arash Yavari
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  2. Alain Goriely
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Correspondence to Arash Yavari.

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Communicated by D. Kinderlehrer

Dedicated to the memory of Professor Jerrold E. Marsden (1942–2010)

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Yavari, A., Goriely, A. Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics. Arch Rational Mech Anal 205, 59–118 (2012). https://doi.org/10.1007/s00205-012-0500-0

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