Some applications of the Hall-Petch Relationship for single-phase imperfect polycrystals

In this article, the modified Hall-Petch Relationship (HPR) theory is considered further to explain an anomaly that arises in the case of nanocrystals. Strength of the material decreases as grain size decreases if the grain size is below a critical dimension. This is referred to as the “negative slope effect” in the HPR (i.e., K < 0). This effect is rationalized with the proposed revision to the HPR. A new modified HPR equation to characterize microhardness has also been derived. The new approach has been applied to films of TiN, TiC, TaC, and WxC on glass substrates and to α-brass. The calculated microhardnesses of the films are compared with experimentally measured data.