Statistical mechanics treatment of the evolution of dislocation distributions in single crystals

A statistical mechanics framework for the evolution of the distribution of dislocations in a single crystal is established. Dislocations on various slip systems are represented by a set of phase-space distributions each of which depends on an angular phase space coordinate that represents the line sense of dislocations. The invariance of the integral of the dislocation density tensor over the crystal volume is proved. From the invariance of this integral, a set of Liouville-type kinetic equations for the phase-space distributions is developed. The classically known continuity equation for the dislocation density tensor is established as a macroscopic transport equation, showing that the geometric and crystallographic notions of dislocations are unified. A detailed account for the short-range reactions and cross slip of dislocations is presented. In addition to the nonlinear coupling arising from the long-range interaction between dislocations, the kinetic equations are quadratically coupled via the shea-range reactions and linearly coupled via cross slip. The framework developed here can he used to derive macroscopic transport-reaction. models, which is shown for a special case of single-slip configuration. The boundary value problem of dislocation dynamics is summarized, and the prospects of development of physical plasticity models for single crystals are discussed.