A dislocation density measure was proposed which was able to account for the evolution of systems of 3-dimensional curved dislocations. The definition and evolution equation of this measure arose as direct generalizations of the definition and kinematic evolution equation of the classical dislocation density tensor. The evolution of this measure permitted determination of the plastic distortion rate in a natural fashion and therefore yielded a kinematically closed dislocation-based theory of plasticity. A self-consistent theory was built upon the measure which accounted for both the long-range interactions of dislocations and their short-range self-interaction which was incorporated via a line-tension approximation. A 2-dimensional kinematic example illustrates the definitions and their relations to the classical theory.

A Three-Dimensional Continuum Theory of Dislocation Systems - Kinematics and Mean-Field Formulation. T.Hochrainer, M.Zaiser, P.Gumbsch: Philosophical Magazine, 2007, 87[8-9], 1261-82