A continuum theory of curved dislocation-lines from a differential geomtric approach

Plastic deformation of crystals is mainly driven by the motion of line-like crystal defects called dislocations. Physically founded continuum descriptions of dislocation-based crystal plasticity must be formulated in terms of dislocation densities. Classical dislocation density measures, as the Kroener-Nye tensor, can account for the kinematic evolution of dislocation systems only if they are considered on the discrete dislocation level. Upon averaging, relevant information both on the orientation of dislocation segments and the presence of 'statistically stored' dislocations of zero net Burgers vector is lost, and therefore any theory of dislocation motion and plastic flow based on the averaged Kroener-Nye tensor is bound to be incomplete. At the beginning of a three-dimensional continuum theory of dislocation motion therefore stands the definition of a dislocation density measure which retains the macroscopically relevant information about the dislocation system, together with the derivation of a kinematic evolution equation for it. A 3D dislocation density measure is proposed as a differential form on the space of directions and curvatures at each point of a crystal viewed as Riemannian manifold with a metric connectiction which is not necessarily free of torsion. From the velocity of a curved dislcoation-line, a higher order velocity on the configuration space including line-directions and curvatures is derived in this general setting. This allows for a nice crystallographic interpretation of the torsion tensor. A kinematic evolution equation for the proposed density measure is derived from assuming the density to be invariant under the flow of the derived higher order velocity field. This evolution equation inherently accounts for line-element rotation and elongation as well as for curvature changes during the motion of curved dislocation-lines, as is shown at an exemplary numerical comparison with the motion of a single dislocation. Finally relations with other density-based models of the kinematic evolution of dislocation systems are discussed.