# Experimental Evidence of Specimen-Size Effects on EN-AW6082 Aluminum Alloy in VHCF Regime
## 3 Results
### P-S-N Curves  
In this sub-section, an estimation of the probabilistic S-N curves for the three different sizes is performed. In fact, fatigue test results are affected by a large scatter in the fatigue life, especially in the ultra-long fatigue regime [22]. Therefore, it is not sufficient to fit the experimental data with the least square method [31], since in this way only a mean life curve is obtained, whereas the influence of fatigue life statistical scatter is disregarded. In other words, it is strictly needed to use deterministic-stochastic models in order to provide reliable estimates of design S-N curves [32]. Since the aluminum cast alloys have no evident fatigue limit, the expression of the generic P-S-N curve can be properly interpolated by the following power-law equation:  
\[N=\left(\frac{\Delta\sigma_{0}(P)}{\Delta\sigma}\right)^{n} \tag{1}\]  
Equation (1),in the bi-logarithmic diagram, is represented by a straight line with the slope equal to \(1/n\), and the intercept with the ordinate axis equal to \(\Delta\sigma_{0}(P)\), which depends on the considered probability of failure. In other words, the fatigue life is considered as a stochastic variable described by the following two-parameter Weibull cumulative distribution function:  
\[P(N,\Delta\sigma)=1-\mathrm{e}^{\left[-\left(\frac{N}{P}\right)^{\alpha} \right]} \tag{2}\]  
The two constants \(\alpha\) and \(\beta\) are the shape and scale parameters, respectively, which are considered constant for a given stress range and a specimen size. It follows that, in order to obtain the probabilistic S-N curves, it should be necessary to fit the two parameters of the model with the available experimental data for each stress level. This procedure is not straightforward, and at least five samples should be considered at each stress range to provide a good estimate of the model parameters [32]. In the present experimental campaign, no more than three specimens for each stress range level are available. Fortunately, a specific statistical procedure proposed in [33; 34] can be adopted in order to overcome the problem of limited available data. The procedure prescribes to consider the random variable \(N=N/N_{50\%}(\Delta\sigma)\), which is obtained by normalizing the specimen fatigue life \(N\)  
Figure 9: SEM images details of the propagation zone and striation marks of the specimen of 12 mm in diameter. with respect to the mean fatigue life \(N_{50\%}\) obtained from each sample subjected to a certain stress range \(\Delta\sigma\). In this way, the Weibull CDF of failure probability is made independent of the stress range. Therefore, the shape and scale parameters, \(\bar{\alpha}\) and \(\bar{\beta}\) become constant with respect to the stress range as well. Finally, the failure probability is expressed by the following equation:  
\[P(\bar{N})=1-\mathrm{e}^{\left[-\left(\frac{\bar{\alpha}}{\bar{\beta}}\right)^ {\bar{\alpha}}\right]} \tag{3}\]  
The best-fitting parameters \(\bar{\alpha}\) and \(\bar{\beta}\) can be assessed, according to [35], by the Maximum Likelihood Method (MLM), thanks to its high-reliability and flexibility. Once that the fitting parameters are obtained, it is suitable to perform goodness-of-fit statistic tests, which detect the discrepancy between the estimated adopted distribution and the empirical one. In fact, these statistical tests measure the distance between observed and expected values, describing the level of adherence of the selected CDF to the sample, which is represented by the fatigue test data. Once that the selected cumulative distribution is checked, it is possible to compute S-N curves for a different probability of failure, \(P\), by using the following expression according to [33]:  
\[\log N=\bar{\beta}[-\ln(1-P)]^{\bar{\alpha}}\left(\frac{\Delta\sigma_{0,50 \%}}{\Delta\sigma}\right)^{n} \tag{4}\]  
where \(\Delta\sigma_{0,50\%}\) and \(n\) are the best-fitting parameters of the mean life curve, whereas \(\bar{\alpha}\) and \(\bar{\beta}\) are the two parameters of Weibull distribution.