## 3 Experimental Results
A further investigation on the frequency effect, especially for low-frequency tests, was carried out by estimating the probability-stress-number of cycles (P-S-N) curves. P-S-N estimation was carried out with the maximum likelihood (ML) principle to account for both failures and run-outs [60; 61]. In Equation (5) the compact formulation of the likelihood function that must be maximized is reported. \[\mathrm{L}[\theta]=\prod_{i=1}^{n_{f}}\mathrm{f}_{Y|X=x}[y_{i};x_{i},\theta] \cdot\prod_{j=1}^{n_{f}}\left(1-\mathrm{F}_{Y|X=x}[y_{j};x_{i},\theta]\right), \tag{5}\]  
Here, \(\mathrm{L}[\cdot]\) is the likelihood operator, \(\theta\) is the set of parameters to be estimated, \(\mathrm{f}_{Y|X=x}\) is the probability density function of the fatigue life, \(\mathrm{F}_{Y|X=x}\) is the cumulative distribution function, \(\mathrm{y}_{i}\) and \(\mathrm{x}_{i}\) are the fatigue life and the logarithm of the applied stress of the i-th specimen, respectively, and \(\mathrm{n}_{f}\) and \(\mathrm{n}_{r}\) are the number of failures and run-outs, respectively. According to [19], the random variable \(Y\), i.e., the logarithm of the fatigue life, is assumed to follow a normal distribution, with a constant standard deviation and mean that is linearly dependent on the logarithm of applied stress. In Equation (6) the mathematical relationship describing the mean is reported:  
\[\mu_{Y}(x)=c_{Y}+m_{Y}\cdot x, \tag{6}\]  
where \(c_{Y}\) and \(m_{Y}\) are the parameters to be determined as well as the standard deviation within the normal distribution. The unknown parameters were estimated practically, using a Matlab(r) R2023a script and an optimization toolbox that exploits the Nelder-Mead simplex algorithm in order to identify the entire \(\theta\) vector of the unknowns. The 50% P-S-N curves associated with each tested condition were estimated. The resulting curves are shown in Figure 11, where the selected probability level aims at representing the averaged behaviours of the tested specimens. The P-S-N curves confirm the frequency rate effect, with a recursive increase in the joint life as the applied frequency increases. This effect can be observed at every investigated test level, namely, between the ultrasonic and low frequencies and among the three selected low-frequency levels. Figure 11: S–N curves at P = 50% for VHCF and HCF data. The impact of the data scattering was analysed through the analysis of standard deviations (SD), whose values are reported in Table 3. Indeed, low frequencies tend to produce a slightly lower variability in failure distribution, whereas this quantity tends to increase as the applied frequency passes into the ultrasonic range. For the sake of completeness, even though this effect is present, it is limited without a relevant impact on the experimental outcomes. Finally, as shown in Figure 12, fatigue strengths at N = 2 \(\times\) 10\({}^{6}\) cycles were collected for three probability levels, namely, 10%, 50%, and 90%. Data were interpolated through a power law scheme, i.e., \(\sigma=A\cdot f^{b}\), where \(\sigma\) is the experimental fatigue strength, \(f\) is the applied test frequency, and \(A\), \(b\) are the model parameters to be determined. The calculation was performed with the least squares method. The R\({}^{2}\) factor was higher than 0.995 for all investigated cases. As demonstrated by the interpolation, in a bi-log chart, the data were quite well approximated. This information is particularly useful, and the results demonstrated a power law increase in the frequency effect. Moreover, this experimental evidence can be considered when predicting strength values not covered by the experimentation.