PNNL

Abstract

Plastic deformation of a single crystal is a manifestation of the motion and interactions of dislocations in the crystal bulk. Typically, crystal dislocations are very large in number and their motion and interactions are of statistical nature. Therefore, the concepts of statistical mechanics can be applied to develop models for the spatio-temporal evolution of dislocations. The starting point is to represent dislocations by phase space densities and develop equations of motion for these densities. Here, we use a topological invariant given by the integral of the dislocation density tensor over the crystal volume to develop a set of Liouville-type kinetic equations describing the evolution of dislocation phase densities on all slip systems in a deforming crystal. Under the condition of finite deformation, two variants of these kinetic equations are developed: in the reference and spatial frames. In either case, the kinetic equations are similar to the classical Boltzmann equation. Hence, we discuss the possibility of development of hydrodynamic equations governing the transport of dislocations and derivation of crystal properties relevant to plastic deformation via a treatment of the kinetic equations in the spirit of Chapman-Enskog expansion of Boltzmann equation of the classical kinetic theory.