 Yielding occurs when the stress due to dislocation pileup at a grain boundary becomes sufficient to activate slip of dislocations in the adjacent grain.This critical stress increases as the grain size decreases, and these physics are empirically captured by the Hall-Petch relationship, 
where formula_5 is the yield stress, formula_6 is a material-specific constant that accounts for the effects of all other strengthening mechanisms, formula_7 is a material-specific constant that describes the magnitude of the metal’s response to grain size strengthening, and formula_8 is the average grain size .Additionally, because nanocrystalline grains are too small to contain a significant number of dislocations, nanocrystalline metals undergo negligible amounts of strain-hardening , and nanocrystalline materials can thus be assumed to behave with perfect plasticity.