Situating the Vector Density Approach Among Contemporary Continuum Theories of Dislocation Dynamics

For the past century, dislocations have been understood to be the carriers of plastic deformation in crystalline solids. However, their collective behavior is still poorly understood. Progress in understanding the collective behavior of dislocations has primarily come in one of two modes: the simulation of systems of interacting discrete dislocations and the treatment of density measures of varying complexity which are considered as continuum fields. A summary of contemporary models of continuum dislocation dynamics is presented. This includes, in order of complexity, the two-dimensional statistical theory of dislocations, the field dislocation mechanics treating the total Kroner-Nye tensor, vector density approaches which treat geometrically necessary dislocations on each slip system of a crystal, and high-order theories which examine the effect of dislocation curvature and distribution over orientation. Each of theories contain common themes, including statistical closure of the kinetic dislocation transport equations and treatment of dislocation reactions such as junction formation. An emphasis is placed on how these common themes rely on closure relations obtained by analysis of discrete dislocation dynamics experiments. The outlook of these various continuum theories of dislocation motion is then discussed.