The well known break-down of the Hall-Petch effect of the rise of the plastic resistance with decreasing grain size in polycrystalline metals, when the grain size dropped into the nm range, resulting in a peak plastic resistance at a grain size of some 12 to 15nm, was explained by considering two alternative and complementary rate mechanisms of plasticity: grain boundary shear and dislocation plasticity; each contributing to the overall strain rate in proportion to the volume fraction of the material in which they operated. In this model, for a given applied strain-rate, it was shown that the plastic resistance attained a maximum at a grain size of 12.2nm in Cu when the two mechanisms contributed equally to the overall strain rate; thus defining the so-called strongest size.

The Strongest Size. A.S.Argon, S.Yip: Philosophical Magazine Letters, 2006, 86[11], 713-20