Continuum theory of evolving dislocation fields

Continuum theory of moving dislocations is used to set up a non-local constitutive law for crystal plasticity in the form of partial differential equations for evolving dislocation fields. The concept of single-valued dislocation fields that enables us to keep track of the curvature of the continuously distributed gliding dislocations with line tension is utilized. The theory is formulated in the Eulerian as well as in the so-called dislocation-Lagrangian forms. The general theory is then specialized to a form appropriate to formulate and solve plane-strain problems of continuum mechanics. The key equation of the specialized theory is identified as a transport equation of diffusion-convection type. The numerical instabilities resulting from the dominating convection are eliminated by resorting to the dislocation-Lagrangian approach. Several examples illustrate the application of the theory.