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    Weyl geometry and the nonlinear mechanics of distributed point defects

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    Type
    Article
    Authors
    Yavari, A.
    Goriely, A.
    KAUST Grant Number
    KUK C1-013-04
    Date
    2012-09-05
    Online Publication Date
    2012-09-05
    Print Publication Date
    2012-12-08
    Permanent link to this record
    http://hdl.handle.net/10754/600192
    
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    Abstract
    The residual stress field of a nonlinear elastic solid with a spherically symmetric distribution of point defects is obtained explicitly using methods from differential geometry. The material manifold of a solid with distributed point defects-where the body is stress-free-is a flat Weyl manifold, i.e. a manifold with an affine connection that has non-metricity with vanishing traceless part, but both its torsion and curvature tensors vanish. Given a spherically symmetric point defect distribution, we construct its Weyl material manifold using the method of Cartan's moving frames. Having the material manifold, the anelasticity problem is transformed to a nonlinear elasticity problem and reduces the problem of computing the residual stresses to finding an embedding into the Euclidean ambient space. In the case of incompressible neo-Hookean solids, we calculate explicitly this residual stress field. We consider the example of a finite ball and a point defect distribution uniform in a smaller ball and vanishing elsewhere. We show that the residual stress field inside the smaller ball is uniform and hydrostatic. We also prove a nonlinear analogue of Eshelby's celebrated inclusion problem for a spherical inclusion in an isotropic incompressible nonlinear solid. © 2012 The Royal Society.
    Citation
    Yavari A, Goriely A (2012) Weyl geometry and the nonlinear mechanics of distributed point defects. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468: 3902–3922. Available: http://dx.doi.org/10.1098/rspa.2012.0342.
    Sponsors
    This publication was based on work supported in part by Award No KUK C1-013-04, made by King Abdullah University of Science and Technology (KAUST) and by the National Science Foundation under grant DMS-0907773 (A.G.), CMMI-1130856 (A.Y.) and AFOSR, grant no. FA9550-10-1-0378. A.G. is a Wolfson Royal Society Merit Holder and acknowledges support from a Reintegration Grant under EC Framework VII.
    Publisher
    The Royal Society
    Journal
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
    DOI
    10.1098/rspa.2012.0342
    ae974a485f413a2113503eed53cd6c53
    10.1098/rspa.2012.0342
    Scopus Count
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