Grain size dependence of the strength of metals: The Hall–Petch effect does not scale as the inverse square root of grain size

The classic data in the literature for the grain size dependence of the strength in many metals are reviewed. The exponent x relating strength to grain size d–x is not often the eponymous inverse square-root relationship (known as the Hall-Petch effect), but is widely scattered from values as low as x = 0.2 to values as high as x = 1. These exponents for individual datasets are shown to be largely meaningless. For an ensemble of n selected datasets, the fit to the functional form ln[d]/d + const with n + 1 free fitting parameters is found to be almost as good as the fit to 1/Sqrt[d] + const with 2n fitting parameters (the Hall-Petch fit). The probability that the former is the preferable fit is high. Some data sets do not agree with the ln[d]/d fit, but their deviation is readily explained on simple physical grounds. Moreover, even when they are included in the fit, statistical tests still show that the ln[d]/d form is preferable by a wide margin. The conclusion is that the Hall-Petch effect is not another size effect sui generis but is the same size effect as that observed in epitaxial thin film growth and in micromechanical testing of small specimens. Consequently we propose that grain size strengthening of metals is driven by constraints on stress and dislocation curvature according to the space available