Issue 10

M. Paggi, Frattura ed Integrità Strutturale, 10 (2009) 43-55 ; DOI: 10.3221/IGF-ESIS.10.06 45 Following an independent line of research, the use of the concepts of dimensional analysis and incomplete self-similarity originally proposed by Barenblatt and Botvina [22,23] seems to be very promising in understanding the phenomenon of fatigue. Applying these concepts to metals, the present author has determined the relationship between the Paris’ law parameters C and m on the basis of theoretical arguments [18]. Then, the Barenblatt and Botvina’s analysis of the size- scale effects on the Paris’ law parameter m has been extended in [24] to the Paris’ law parameter C . Finally, a dimensional analysis of the cumulative fatigue damage and of the damage tolerant approaches has been proposed in [25], interpreting the observed deviations from the Paris and Wöhler regimes within a unified theoretical framework. This approach promises to bring simplicity in what has been up to now considered as an unexplained set of data, putting into evidence the role of the most influential variables on the fatigue behaviour of materials. The aim of the present study is to extend the dimensional analysis approach to quasi-brittle materials, determining the main dimensionless numbers influencing the d d / a N - K  and the S-N curves. More specifically, the possibility of incomplete self-similarity in the dimensionless numbers related to the microstructural size, the crack size and the structural size is explored in details. As a result, different aspects that have always been so far treated separately are herein interpreted within a unified mathematical framework. The corresponding incomplete self-similarity exponents and their dependencies on the dimensionless numbers are also quantified on the basis of relevant experimental data taken from the literature. The new proposed correlations provide useful information for design purposes. G ENERALIZED MATHEMATICAL REPRESENTATIONS OF FATIGUE The generalized Paris law ccording to the pioneering work b y Barenblatt and Botvina [22], the following functional dependence can be put forward for the analysis of the phenomenon of fatigue crack growth:   IC th d , , ; , , , , , ;1 , d u a F K K K h d a E R N       (1) where the governing variables are summarized in Tab. 1 , along with their physical dimensions expressed in the Length- Force-Time class (LFT). Variable definition Symbol Dimensions Ultimate tensile strength u  2 FL  Fracture toughness IC K 3/2 FL  Frequency of the loading cycle  1 T  Stress-intensity factor range K  3/2 FL  Threshold stress-intensity factor range th K  3/2 FL  Stress range   2 FL  Fatigue limit fl   2 FL  Structural size h L Microstructural size (aggregate or fibre size) d L Crack length a L Elastic modulus E 2 FL  Loading ratio R  Table 1 : Governing variables of the fatigue crack growth phenomenon. A