Inverse Square Root Of Grain Size Research Articles

Factors determining the magnitude of grain-size strengthening in polycrystalline metals

Grain-size strengthening of polycrystalline metals, the Hall-Petch effect, has been described for the past sixty years as a dependence of the strength of polycrystalline metals on the inverse square-root of grain size, d. The value of the coefficient of the dependence has been the subject of discussion throughout. Here, we find what known factors in the experiments may determine its value, by meta-analysis using maximum-likelihood methods of the literature values of the coefficient in sixty-one datasets. No dependence of the coefficient is found on plastic strain, and a strong dependence is found on the average grain size of each study. Combining these and other factors accounts for the reported values of about 80% of the sixty-one coefficients. The grain-size dependence of the Hall-Petch coefficient is an artefact arising from fitting the data to an incorrect expression. An alternative grain-size effect described by a lnd/d function is consistent with the theory of dislocations dynamics and generation. The corresponding analysis of the coefficients of fits based on this theory shows that none of the factors investigated are statistically significant, confirming the correctness of this approach.

The Hall-Petch effect as a manifestation of the general size effect.

The experimental evidence for the Hall-Petch dependence of strength on the inverse square-root of grain size is reviewed critically. Both the classic data and more recent results are considered. While the data are traditionally fitted to the inverse square-root dependence, they also fit well to many other functions, both power law and non-power law. There have been difficulties, recognized for half-a-century, in the inverse square-root expression. It is now explained as an artefact of faulty data analysis. A Bayesian meta-analysis shows that the data strongly support the simple inverse or lnd/d expressions. Since these expressions derive from underlying theory, they are also more readily explicable. It is concluded that the Hall-Petch effect is not to be explained by the variety of theories found in the literature, but is a manifestation of, or to be underlain by the general size effect observed throughout micromechanics, owing to the inverse relationship between the stress required and the space available for dislocation sources to operate.