Weyl geometry and the nonlinear mechanics of distributed point defects

A. Yavari, A. Goriely

Research output: Contribution to journalArticlepeer-review

39 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

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