In the present paper, the fractal nature of the material microstructure [5,6] and the renormalization group theory [7-9] can be considered to analyse the interactions between the two above levels (micro and macro), as has been proposed in Ref.[10]. In other words, the reacting cross section of a given structure shows a self-similar weakening due to the material heterogeneity, cracks, defects, etc., and therefore the fractal dimension of such a surface can be assumed to be lower than 2 [11,12]. Consequently, the damaged ligament of a heterogeneous solid may be modelled through a “lacunar” fractal set, analogous to the mathematical middle-third Cantor set, which presents Hausdorff dimension lower than that of the domain where it is contained. Then, new mechanical properties can be defined with physical dimensions depending on the fractal dimension of the damaged heterogeneous ligament (renormalization procedure), and such properties are scale-invariant constants. According to this approach, the renormalized fatigue strength could be represented by a force amplitude acting on a surface with a fractal dimension lower than 2, as is discussed in the following. On the other hand, Mandelbrot [13] pointed out a non-uniform (multifractal) scaling of the natural fractals (different from the uniform one of the mathematical fractals), i.e. in the physical reality a transition occurs from a fractal (heterogeneous) regime for small structures to a Euclidean (homogeneous) one for structures large enough with respect to a characteristic microstructural size. In other words, the effect of the microstructural heterogeneity and/or damage (disorder) of a given material on the macroscopic mechanical behaviour gradually vanishes by increasing the structure size [14]. A monofractal scaling law for fatigue limit of metals is herein proposed, and some experimental results [15] are analysed to show how to apply the theoretical approach adopted. FRACTAL NATURE OF FATIGUE FAILURE OF STRUCTURES According to the concepts previously discussed, the reacting cross section of a disordered material is herein assumed to present a fractal dimension d = −2 α , with 0 1 ≤ < d , where the decrement depends on a self-similar microstructural weakening (heterogeneity and/or damage) [10-12], the value of d being higher when such a weakening is more significant. Let us consider two geometrically similar cylinders ( and d A B), made up of the same material, subjected to cyclic axial loading (Fig.1). On the basis of the theoretical approach proposed for static loading [10], the renormalized fatigue strength (the subscript standing for amplitude) may be assumed as a material constant with physical dimensions given by * aσ a [ ] [ ] d L −2 F , and the following expression can be written : DA DB Figure 1: Geometrically similar cylinders under cyclic axial loading 2
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