Internal stresses in a homogenized representation of dislocation microstructures

To develop a continuum theory based on the evolution of dislocation microstructures, two challenges have to be resolved: the correct representation of the kinematics of dislocation motion in terms of dislocation density and the formulation of a mobility law reflecting an effective description of the physical behavior of the discrete many-body problem. Kröner's classical continuum theory has inspired different approaches to model plasticity based on the motion of dislocations. Amongst them, the Continuum Dislocation Dynamics (CDD) theory was formulated as a generalization of the classical theory. The \{CDD\} theory allows for a continuous representation of the evolution of dislocation microstructures and is found to be kinematically complete. Here, a numerical formulation of the \{CDD\} theory is presented and constitutive laws for the incorporation of dislocation interactions are derived based on the representation of the dislocation microstructure in two dimensions. An error measure is introduced to analyze the constitutive law and the results are compared to discrete dislocation dynamics simulations. Important aspects for the implementation of a 3D theory are discussed.