A geometrically linear formulation of the high-gradient plasticity of monocrystalline and polycrystalline material was based upon the continuum theory of dislocations and incompatibilities. General continuum dislocation densities and incompatibilities were introduced from the viewpoint of continuum mechanics by considering the spatial closure failure of arbitrary line integrals of the displacement differential. These findings were then related to the plastic part of the displacement gradient (plastic distortion) and the plastic strain, respectively, within an elasto-plastic solid. This defined the tensor fields of plastic dislocation densities and plastic incompatibilities. In the case of monocrystalline material, the plastic dislocation density and (in the case of polycrystalline material) the plastic incompatibility were considered in the light of the thermodynamic principle of positive dissipation. As a result, a phenomenological but physically motivated description of hardening was obtained. This incorporated the second spatial derivatives of the plastic deformation gradient for monocrystals, and the fourth spatial derivatives of the plastic strains of polycrystals, into the yield condition. These modifications reproduced the characteristic structure of kinematic hardening, in which the back-stress obeyed a non-local evolution law.

On the Formulation of Higher Gradient Single and Polycrystal Plasticity. A.Menzel, P.Steinmann: Journal de Physique IV, 1998, 8[8], 239-47