Weyl geometry and the nonlinear mechanics of distributed point defects

A. Yavari, A. Goriely

Research output: Contribution to journalArticlepeer-review

57 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)3902-3922
Number of pages21
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume468
Issue number2148
DOIs
StatePublished - Sep 5 2012
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK C1-013-04
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

Fingerprint

Dive into the research topics of 'Weyl geometry and the nonlinear mechanics of distributed point defects'. Together they form a unique fingerprint.

Cite this