An approach to size effect in fatigue of metals using fractal theories

As was experimentally observed by several authors, the fatigue strength of metallic materials decreases with increasing the specimen size. Such a decrease can be remarkable for very large structures like, for example, big cargo ships (some hundred meters long) transporting oil or other goods. Size effect in fatigue is herein explained by considering the fractal nature of the reacting cross sections of structures, that is, the renormalized fatigue strength is represented by a force amplitude acting on a surface with a fractal dimension lower than 2, where such a dimensional decrement depends on a self-similar weakening of the material ligament, owing to the presence of cracks, defects, voids and so forth (microscopic level). However, this decrement tends to progressively disappear with increasing the structure size (macroscopic level), i.e. the effect of the material microstructure on the macroscopic fatigue behaviour gradually vanishes for structures large enough with respect to a characteristic microstructural size, this phenomenon being defined as multifractality. A multifractal scaling law for fatigue limit of metals is proposed, and some experimental results are examined in order to show how to apply the theoretical approach presented.