A probabilistic approach to polycrystalline plasticity part I: theory

Abstract

A set of constitutive equations for polycrystalline plasticity is derived using arguments based directly on the dislocation processes involved. Distributed glide-plane orientations and Burgers-vector directions facilitate handling of the polycrystalline structure, and they yield equations involving probability distributions for variables that are directly related to measurable dislocation quantities. When the motion of the dislocations is isochoric, the tensorial character of the plastic strain rate is shown to be entirely determined by a second-rank symmetric tensor directly related to ordinary elements of crystallographic glide. This same tensor is also shown to determine the resolved shear stress acting on a dislocation in the direction of its Burgers vector, a quantity critical to the determination of the dislocation speed. Evolutionary equations for the dislocation density and the mobile fraction of dislocations are developed to complete the material description. The resulting theory, which allows for the production and interaction of non-uniform dislocation distributions, can model such phenomena as the development of anisotropy with plastic deformation, and material hardening or softening.