It was pointed out that classical plasticity theory had reached its limit in describing crystalline material behavior at the micron level and below. Its inability to predict size-dependent effects at this length-scale had therefore encouraged the use of higher-order gradients in order to model material behavior. The basis of the use of strain gradients was a framework of geometrically-necessary dislocations. A new definition of Nye's dislocation tensor, which was a measure of the density of geometrically-necessary dislocations, was proposed here on the basis of the integrated properties of dislocation lines within a volume. A discrete form of the definition was applied to redundant crystal systems, and methods for associating the dislocation tensor with realizable crystallographic dislocations were presented. As a result, 2 types of 3-dimensional dislocation structure were found. These were open periodic networks, which had long-range geometrical effects, and closed 3-dimensional dislocation structures which self-terminated and had no geometrical effect. The implications of these structures for the presence of geometrically-necessary dislocations in polycrystalline materials led to the introduction of a Nye factor which related the geometrically-necessary dislocation density to plastic strain gradients.

Crystallographic Aspects of Geometrically Necessary and Statistically-Stored Dislocation Density. A.Arsenlis, D.M.Parks: Acta Materialia, 1999, 47[5], 1597-611