Evolution equations for the scalar density and orientation of the fields of curved dislocations, as formulated within the framework of the continuum theory of moving dislocations, was used as a starting point for the development of a non-local dislocation-based constitutive relationship for crystal plasticity. This result was applicable to a length scale which was intermediate between the phenomenological hardening laws of strain-gradient crystal plasticity, and the explicit treatment of 3-dimensional discrete dislocation dynamics. A key feature of this approach was a refined averaging, in the continuum theory, which was based upon the separation of single-valued dislocation fields. Another feature was the account taken of the line energy of bowed dislocations, which made the theory non-local.

The Importance of Being Curved - Bowing Dislocations in a Continuum Description. R.Sedláček, J.Kratochvíl, E.Werner: Philosophical Magazine, 2003, 83[31], 3735-52