Various deformation models which incorporated higher-order gradients were analyzed with regard to situations ranging from the determination of the size of dislocation cores or elastic dislocation interactions, to the determination of the wavelengths of dislocation patterns or heterogeneous dislocation distributions and the structures of solid interfaces and localized strain zones during adiabatic shear deformation. Different scales were involved in each of these situations: nano-scale for single dislocations, micro-scale for dislocation patterning and macro-scale for adiabatic shear banding. Similarly, differing gradient models applied to each case, differing types of gradient term were involved and differing expressions for the gradient coefficients had to be assumed.

Gradient Deformation Models at Nano, Micro, and Macro Scales. E.C.Aifantis: Journal of Engineering Materials and Technology, 1999, 121[1], 189-96