Weyl Geometry and the Nonlinear Mechanics of Distributed Point Defects
In this paper we obtain the residual stress field of a nonlinear elastic solid with a spherically-symmetric distribution of point defects. The material manifold of a solid with distributed point defects – where the body is stressfree – is a flat Weyl manifold, i.e. a manifold with an affine connecti...
| Main Authors: | , |
|---|---|
| 格式: | Journal article |
| 出版: |
2012
|
| _version_ | 1826302791429128192 |
|---|---|
| author | Yavari, A Goriely, A |
| author_facet | Yavari, A Goriely, A |
| author_sort | Yavari, A |
| collection | OXFORD |
| description | In this paper we obtain the residual stress field of a nonlinear elastic solid with a spherically-symmetric distribution of point defects. The material manifold of a solid with distributed point defects – where the body is stressfree – is a flat Weyl manifold, i.e. a manifold with an affine connection that has non-metricity but both its torsion and curvature tensors vanish. Given a spherically-symmetric point defect distribution, we construct its Weyl material manifold using Cartan’s moving frames. Having the material manifold the anelasticity problem is transformed to a nonlinear elasticity problem; all one needs to calculate residual stresses is to find an embedding into the Euclidean ambient space. In the case of incompressible neo-Hookean solids we calculate the residual stress field. We finally consider the example of a finite ball of radius Ro and a point defect distribution uniform in a ball of radius Ri and vanishing elsewhere. We show that the residual stress field inside the ball of radius Ri 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. |
| first_indexed | 2024-03-07T05:52:49Z |
| format | Journal article |
| id | oxford-uuid:e97b51ad-b83f-4d4d-bb6a-f7bafdee8231 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T05:52:49Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:e97b51ad-b83f-4d4d-bb6a-f7bafdee82312022-03-27T10:54:34ZWeyl Geometry and the Nonlinear Mechanics of Distributed Point DefectsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:e97b51ad-b83f-4d4d-bb6a-f7bafdee8231Mathematical Institute - ePrints2012Yavari, AGoriely, AIn this paper we obtain the residual stress field of a nonlinear elastic solid with a spherically-symmetric distribution of point defects. The material manifold of a solid with distributed point defects – where the body is stressfree – is a flat Weyl manifold, i.e. a manifold with an affine connection that has non-metricity but both its torsion and curvature tensors vanish. Given a spherically-symmetric point defect distribution, we construct its Weyl material manifold using Cartan’s moving frames. Having the material manifold the anelasticity problem is transformed to a nonlinear elasticity problem; all one needs to calculate residual stresses is to find an embedding into the Euclidean ambient space. In the case of incompressible neo-Hookean solids we calculate the residual stress field. We finally consider the example of a finite ball of radius Ro and a point defect distribution uniform in a ball of radius Ri and vanishing elsewhere. We show that the residual stress field inside the ball of radius Ri 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. |
| spellingShingle | Yavari, A Goriely, A Weyl Geometry and the Nonlinear Mechanics of Distributed Point Defects |
| title | Weyl Geometry and the Nonlinear Mechanics of Distributed Point
Defects |
| title_full | Weyl Geometry and the Nonlinear Mechanics of Distributed Point
Defects |
| title_fullStr | Weyl Geometry and the Nonlinear Mechanics of Distributed Point
Defects |
| title_full_unstemmed | Weyl Geometry and the Nonlinear Mechanics of Distributed Point
Defects |
| title_short | Weyl Geometry and the Nonlinear Mechanics of Distributed Point
Defects |
| title_sort | weyl geometry and the nonlinear mechanics of distributed point defects |
| work_keys_str_mv | AT yavaria weylgeometryandthenonlinearmechanicsofdistributedpointdefects AT gorielya weylgeometryandthenonlinearmechanicsofdistributedpointdefects |