Elastically inhomogeneous multilayer films were being exploited for use as ultra-hard coatings. These films exhibited a strong dependence between the compositional wavelength of the film, Λ, and the hardness, H=KΛ-a+H0 where the scaling exponent a depended on the elastic properties of the individual layers (shear moduli and Poisson ratios). The dislocation pile-up model could explain this trend and form a bridge between the microscopic strength of multilayer interfaces and the macroscopic strength of the multilayer. A semi-analytic solution to the pile-up model of multilayer strength was presented. All parameter dependencies were solved analytically except a single dimensionless coefficient which was found from numerical simulation. The predictions were compared to data from a 2D discrete dislocation model and to experimental Cu/Ni data.

Exponent for Hall-Petch Behaviour of Ultra-Hard Multilayers. L.H.Friedman: Philosophical Magazine, 2006, 86[11], 1443-81