Non-Metricity and the Nonlinear Mechanics of Distributed Point Defects

We discuss the relevance of non-metricity in a metric-affine manifold (a manifold equipped with a connection and a metric) and the nonlinear mechanics of distributed point defects. We describe a geometric framework in which one can calculate analytically the residual stress field of nonlinear elastic solids with distributed point defects. In particular, we use Cartan's machinery of moving frames and construct the material manifold of a finite ball with a spherically-symmetric distribution of point defects. We then calculate the residual stress field when the ball is made of an arbitrary incompressible isotropic solid. We will show that an isotropic distribution of point defects cannot be represented by a distribution of purely dilatational eigenstrains. However, it can be represented by a distribution of radial eigenstrains. We also discuss an analogy between the residual stress field and the gravitational field of a spherical mass.