期刊
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
卷 468, 期 2148, 页码 3902-3922出版社
ROYAL SOC
DOI: 10.1098/rspa.2012.0342
关键词
geometric elasticity; point defects; residual stresses
资金
- King Abdullah University of Science and Technology (KAUST) [KUK C1-013-04]
- National Science Foundation [DMS-0907773]
- AFOSR [FA9550-10-1-0378]
- Reintegration Grant under EC Framework VII
- [CMMI-1130856]
- Div Of Civil, Mechanical, & Manufact Inn
- Directorate For Engineering [1130856] Funding Source: National Science Foundation
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.
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