A three-dimensional continuum theory of dislocation systems: Kinematics and mean-field formulation

We propose a dislocation density measure which is able to account for the evolution of systems of three-dimensional curved dislocations. The definition and evolution equation of this measure arise as direct generalizations of the definition and kinematic evolution equation of the classical dislocation density tensor. The evolution of this measure allows us to determine the plastic distortion rate in a natural fashion and therefore yields a kinematically closed dislocation-based theory of plasticity. A self-consistent theory is built upon the measure which accounts for both the long-range interactions of dislocations and their short-range self-interaction which is incorporated via a line-tension approximation. A two-dimensional kinematic example illustrates the definitions and their relations to the classical theory.