Generalized dependence of the yield stress on the grain size - Shut, O. O. (Reports of the National Academy of Sciences of Ukraine, 2014)
The generalized equation that describes the yield stress dependence upon the grain size in a wide range of grain sizes has been obtained. There are two critical grain sizes (dcr1,dcr2) that correspond to changes in the strengthening mechanism. The equation includes the Hall–Petch relation in the range d > dcr1. For dcr1 > d > dcr2, the power in the Hall–Petch relation varies from -1/2 to -1. In a nanometer (nm) range (dcr2 > d), there are the possibilities of softening (in case of "weak" boundaries), as well as significant strengthening (in case of "strong" boundaries).