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### The Closure Problem in Density-Based Modeling of Dislocation Dynamics

#### Abstract

Development of density-based models of dislocations is a critical step towards predicting the collective dislocation dynamics in deforming metals. In the recent years, a few approaches have been developed to accomplish this objective, one of which is the statistical approach to dislocation dynamics. This approach is based on the concepts of statistical mechanics, and it treats the evolution of the dislocation population using kinetic equations, the completion of which requires dealing with a closure problem similar to the case of classical kinetic theories of particle systems. Such a closure problem can be approached by modeling the statistics of dislocations and their internal fields. This communication focuses on this task. Specifically, we present modeling and simulation results for (a) the spatial and orientation statistics of dislocations, (b) the statistics of dislocation velocity, resolved shear stress on dislocations and resolved shear stress in the crystal, and (c) the temporal statistics of cross slip and short range reactions in both FCC and BCC crystals. The statistics of the velocity and resolved shear stress are required to develop a density-based dislocation mobility law for use in the kinetic equations. The spatial statistics is required to model the dislocation correlations, and the temporal statistics is needed to model all source (rate) terms in the kinetic equations for the dislocation density evolution. The method of dislocation dynamics simulation is used to conduct all numerical simulation of all relevant statistical measures. The utility of the statistical measures presented here in the context of density-based models of dislocations is discussed.

#### Keywords

mesocale deformation; dislocation dynamics; crystal plasticityText