Continuum theory of moving dislocations was 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 permitted the keeping-track of the curvature of the continuously distributed gliding dislocations with line tension was used. The theory was formulated in the Eulerian as well as in the so-called dislocation-Lagrangian forms. The general theory was then specialized to a form which was appropriate to the formulation and solution of plane-strain continuum mechanics problems. The key equation of the specialized theory was identified as a transport equation of diffusion-convection type. The numerical instabilities resulting from the dominating convection were eliminated by resorting to the dislocation-Lagrangian approach. Several examples illustrate the application of the theory.

Continuum Theory of Evolving Dislocation Fields. R.Sedláček, C.Schwarz, J.Kratochvíl, E.Werner: Philosophical Magazine, 2007, 87[8-9], 1225-60