A model was developed to describe the grain size dependence of hardness (or strength) in nanocrystalline materials by combining the Hall–Petch relationship for larger grains with a coherent polycrystal model for nanoscale grains and introducing a log-normal distribution of grain sizes. The transition from the Hall–Petch relationship to the coherent polycrystal mechanism was shown to be a gradual process. The hardness in the nanoscale regime was observed to increase with decreasing grain boundary affected zone (or effective grain boundary thickness, Δ) in the form of Δ−1/2. The critical grain size increased linearly with increasing Δ. The variation of the calculated hardness value with the grain size was observed to be in agreement with the experimental data reported in the literature.
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