The strongest size

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 drops into the nanometre range resulting in a peak plastic resistance at a grain size of about 12–15 nm, is 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 operate. In the model for a given applied strain rate it is shown that the plastic resistance reaches a maximum at a grain size of about 12.2 nm in Cu when the two mechanisms contribute to the overall strain rate equally, defining the so-called strongest size.