Autoparallel curves and Riemannian geodesics for materially uniform but inhomogeneous bodies

Abstract

According to the theory of materially uniform but inhomogeneous bodies two geometric structures can be defined on the body manifold when a uniform reference is known. The first is given by the material connection and the second by the intrinsic Riemannian metric. Two important classes of curves related with these structures are the geodesic and the autoparallel curves. The goal of the present contribution is to define and characterize mechanically these kind of curves for materially uniform but inhomogeneous bodies. We propose the use of these curves for constructing the stress-free non-Euclidean material manifold that plays the role of the reference configuration for the dislocated problem. A generic scheme for the construction of this manifold based on geodesics/autoparallel curves is given as well as a discussion related with the field of internal stresses. Attention is then focused on a continuous distribution of edge dislocations. We solve numerically the geodesic equation that corresponds to a solid body with a continuous symmetry group. By using the $L_2$ norm for the dislocation density tensor we conclude that the higher the dislocation density the greater the deviation from the straight line is. For the same distribution of dislocations, but for a solid body with a discrete symmetry group, we solve analytically the autoparallel curves.