Crystal plasticity was governed by the motion of lattice dislocations. Although continuum theories of static dislocation assemblies date back to the 1950s, the line-like character of these defects posed serious problems for the development of a continuum theory of plasticity which was based on the averaged dynamics of dislocation systems. Only recently the geometrical problem of performing meaningful averages over systems of moving, oriented lines was solved. Such averaging leads to the definition of a dislocation density tensor of second order along with its evolution equation. This tensor could be envisaged as the analogue of the classical dislocation density tensor in an extended space which includes the line orientation as an independent variable. Here, consideration was given to the numerical implementation of a continuum theory of dislocation evolution that was based upon this dislocation density measure and this was applied to some simple benchmark problems as well as to plane-strain micro-bending.

Numerical Implementation of a 3D Continuum Theory of Dislocation Dynamics and Application to Micro-Bending. S.Sandfeld, T.Hochrainer, P.Gumbsch, M.Zaiser: Philosophical Magazine, 2010, 90[27-28], 3697-728