Comparison of closure approximations for continuous dislocation dynamics

We discuss methods to describe the evolution of dislocation systems in terms of a limited number of continuous field variables while correctly representing the kinematics of systems of flexible and connected lines. We show that a satisfactory continuum representation may be obtained in terms of only four variables. We discuss the consequences of different approximations needed to formulate a closed set of equations for these variables and propose a benchmark problem to assess the performance of the resulting models. We demonstrate that best results are obtained by using the maximum entropy formalism to arrive at an optimal estimate for the dislocation orientation distribution based on its lowest-order angular moments.