A dimensional analysis interpretation to grain size and loading frequency dependencies of the Paris and Wöhler curves

In this paper, a mathematical model based on dimensional analysis and incomplete self-similarity is proposed for the interpretation of the grain size and loading frequency effects on the Paris and Wohler regimes in metals. In particular, it is demonstrated that these effects correspond to a violation of the physical similitude hypothesis underlying the simplest Paris’ and Wohler power-law fatigue relationships. As a consequence, generalized representations of fatigue have to be invoked. From the physical point of view, the incomplete similarity behaviour can be regarded as the result of the multiscale character of the problem, where the crack length and the grain size are the two length scales interacting together. Moreover, it will be shown that the relationship between strength and grain size (Hall–Petch relationship) has also to be considered in order to consistently interpret the two opposite effects of the grain size on the Paris and Wohler regimes within a unified framework. The incomplete similarity exponents are suitably quantified according to experimental results for Aluminum, Copper, Titanium and Nickel. The derived scaling laws are expected to be of paramount importance today, especially after the advent of ultra fine grained materials that offer unique mechanical properties owing to their fine microstructure.