Weyl geometry and the nonlinear mechanics of distributed point defects

Handle URI:
http://hdl.handle.net/10754/600192
Title:
Weyl geometry and the nonlinear mechanics of distributed point defects
Authors:
Yavari, A.; Goriely, A.
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.
Publisher:
The Royal Society
Journal:
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
KAUST Grant Number:
KUK C1-013-04
Issue Date:
5-Sep-2012
DOI:
10.1098/rspa.2012.0342
Type:
Article
ISSN:
1364-5021; 1471-2946
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.
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Full metadata record

DC FieldValue Language
dc.contributor.authorYavari, A.en
dc.contributor.authorGoriely, A.en
dc.date.accessioned2016-02-28T06:44:52Zen
dc.date.available2016-02-28T06:44:52Zen
dc.date.issued2012-09-05en
dc.identifier.citationYavari 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.en
dc.identifier.issn1364-5021en
dc.identifier.issn1471-2946en
dc.identifier.doi10.1098/rspa.2012.0342en
dc.identifier.urihttp://hdl.handle.net/10754/600192en
dc.description.abstractThe 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.en
dc.description.sponsorshipThis 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.en
dc.publisherThe Royal Societyen
dc.subjectGeometric elasticityen
dc.subjectPoint defectsen
dc.subjectResidual stressesen
dc.titleWeyl geometry and the nonlinear mechanics of distributed point defectsen
dc.typeArticleen
dc.identifier.journalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciencesen
dc.contributor.institutionGeorgia Institute of Technology, Atlanta, United Statesen
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
kaust.grant.numberKUK C1-013-04en
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