Non-Riemannian geometry and the theory of lattice imperfections

Abstract

Creation and development of continuum theory of imperfections of a crystal structure (dislocations and disclinations) is closely associated with ideas and methods of non-Euclidean and non-Riemannian geometry. In the geometrical interpretation of the continuum theory of lattice defects Kondo [1] and Bilbi et at. [2] identified the Cartan torsion tensor with the dislocation density, Anthony [3] used the Riemann-Christoffcl curvature tensor to describe disclination. The above-mentioned authors considered static imperfections. To give a differential- geometrical interpretation of the imperfection kinematics it is necessary to take time into consideration. We discuss a three-dimensional space of affine connection with time as a parameter and introduce the material time derivative using the “time-connection" tensor.