Critical grain sizes and generalized flow stress - grain size dependence

The generalized eguation that describes the yield stress dependence upon the grain size in the wide range of grain sizes has been obtained. There are two critical grain sizes that corresponds to the changes in strengthening mechanism. The equation includes Hall - Petch relation in the range of d>dcr1. For dcr1>d>dcr2, the power in the Hall - Petch relation varies from -1/2 to -1. In a nanometer (nm) range (d<dcr2) there are posibileties of softerning (in case of “weak” boundaries) as well as significant strengthening (in case of “srtong” boundaries).