Weyl Geometry and the Nonlinear Mechanics of Distributed Point Defects
Arash Yavari (Georgia Institute of Technolog)
SES Medal Symposium in honor of D.J. Steigmann
Mon 9:00 - 10:30
MacMillan 115
We show how to 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 stress-free -- 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 the method of 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 then 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 then compare the nonlinear and classical linear solutions. We also prove a nonlinear analogue of Eshelby's celebrated inclusion problem for a spherical inclusion in an isotropic incompressible nonlinear solid.