Mathematical and computational modelling of correlations in dislocation dynamics

This work focuses on the mathematical and computational modelling of dislocation correlation in the context of dislocation dynamics modelling of mesoscale plasticity. A hierarchical system of kinetic equations describing the evolution of dislocation densities of various orders is presented to illustrate the role of correlation in dislocation dynamics, which is followed by a mathematical description of the spatial and line-orientation statistics of dislocation systems within the framework of stochastic fiber processes. In this framework, the pair correlation is related to the second moment measure of the dislocation distribution in the crystal and dislocation line-tangent spaces. The stochastic fiber process description of correlation is used in conjunction with the method of dislocation dynamics simulation to compute the pair correlation in both face-centered cubic and body-centered cubic crystals undergoing small homogeneous plastic deformation at a constant rate. An edge correction scheme is employed as part of the overall methodology for computing the pair correlation. The main characteristics of correlations, including its oscillatory behavior, anisotropy and symmetry, are studied in detail. The self-correlation and cross correlation of dislocations of various species are compared. The implications of the present results for the development of kinetic theory of dislocations are discussed.