Weyl Geometry and the Nonlinear Mechanics of Distributed Point Defects

Journal article

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.

Authors: Yavari, A; Goriely, A

• Publication date: 2012-01-01