# Experimental Evidence of Specimen-Size Effects on EN-AW6082 Aluminum Alloy in VHCF Regime
## 3 Results
### Theoretical Interpretation of the Specimen-Size Effect on Fatigue Resistance  
Almost two decades ago, the concept of fractality was successfully exploited by [PERSON] to explain the size effect on Fracture Mechanics parameters [36; 37; 38; 39]. In this framework, the material disorder due to the presence of a random distribution of flaws, inclusions, and micro-cracks [40], is accounted for adopting a fractal, rather than Euclidean, geometrical model [41; 42]. In the following, the concept of fractality is adopted to provide a theoretical explanation for the decrease in fatigue resistance with the specimen size. From a practical point of view, the disorder due to inherent defects is taken into account by assuming that the reacting section, or ligament, of a disordered material is a lacunar fractal set with non-integer dimension lower than 2. Consequently, the following scaling law for the stress range can be assumed [23; 43; 44; 45]:  
\[\Delta\sigma=\Delta\sigma^{*}b^{-d_{\sigma}} \tag{6}\]  
where \(\Delta\sigma^{*}\) is the fractal stress range, with physical dimensions given by \([\mathrm{F}][\mathrm{L}]^{-(2-d_{\sigma})}\), whereas \(d_{\sigma}\) is the dimensional decrement of the ligament due to the presence of cracks and voids distribution, which ranges from 0 up to 0.5. Anagonously, a similar scaling law for the intercept with the ordinate axis of the mean S-N curve, \(\Delta\sigma_{0,50\%}\), can be put forward:  
\[\Delta\sigma_{0,50\%}=\Delta\sigma_{0,50\%}^{*}b^{-d_{\sigma}} \tag{7}\]  
where the term \(\Delta\sigma_{0,50\%}^{*}\) is the coefficient of the power-law for the median fractal Wohler's curve. Substituting Equation (7) in Equation (4), yields to a vertical downward translation of P-S-N curves for increasing specimen sizes, i.e., the larger the specimen dimension, the lower the fatigue strength. Thus, a set of P-S-N curves as a function of the specimen size \(b\) can be obtained with the following analytical expression:  
\[\log N=\tilde{\beta}(b)[-\ln(1-P)]^{\mathrm{\bar{\alpha}}(b)}\left(\frac{ \Delta\sigma_{0,50\%}^{*}}{\Delta\sigma}\right)^{n}b^{-nd_{\sigma}} \tag{8}\]  
The terms \(\tilde{\beta}(b)\) and \(\bar{\alpha}(b)\) are the scale and shape parameters, which are not constant with the specimen size, as shown in Table 4. In other words, the scatter of experimental data is not constant with the structural size, but it decreases with it for a given stress range. Figure 16 plots the experimental data obtained by specimens of 3 mm, 6 mm, and 12 mm in diameter, together with the fitting median P-S-N curves expressed by Equation (8) in correspondence with a 50% failure probability. From a comparison between Figure 16 and Figures 13-15 it emerges clearly that the decrease in fatigue resistance with the specimen size is more pronounced than the scatter of experimental data, for the whole investigated dimensional range. Figure 15: P-S-N curves for the specimens of 12 mm in diameter. On the other hand, if the fractal stress range is used to represent experimental results, i.e. substituting Equation (6) in Equation (8),the set of specimen-size dependent P-S-N curves collapse onto a single fractal specimen-size independent P-S\({}^{*}\)-N curve, which is represented by the following equation:  
\[\log N=\bar{\beta}^{*}[-\ln(1-P)]^{\pi^{*}}\left(\frac{\Delta\sigma_{0,50\%}^{* }}{\Delta\sigma^{*}}\right)^{n} \tag{9}\]  
The terms \(\bar{\beta}^{*}\) and \(\bar{\alpha}^{*}\) entering Equation (9) are the scale and shape parameters of the fractal P-S\({}^{*}\)-N curves, which are specimen-size independent. The five free-parameters to be estimated in Equations (8) and (9) are: the dimensional decrement, \(d_{\sigma}\); the coefficient of the power-law for the median fractal Wohler's curve, \(\Delta\sigma_{0,50\%}^{*}\); the exponent \(n\); and the Weibull distribution coefficients \(\bar{\beta}^{*}\) and \(\bar{\alpha}^{*}\). The estimation is carried out in two steps.