# Ultrasonic VMCF Tests on Very Large Specimens with Risk-Volume Up to 5000 mm+
## 3 Results
### Size-Effect: Discussion  
One of the main objectives of the paper, besides experimentally validating the Gaussian specimens with very large risk-volume, was to verify how accurate and reliable is the prediction of the VHCF response of parts with large risk-volume through tests on small hourglass specimens, which are commonly adopted in the literature. The P-S-N curve for a risk-volume, \(V_{90,pred}\), larger than that experimentally tested, \(V_{90,exp}\), can be estimated by modifying Equation (1). According to [3; 15; 17], the pdf (\(f\sqrt{a_{4,pred}}\big{(}\sqrt{a_{4,0}}\big{)}\)) of the predicted defect size, \(\sqrt{a_{4,pred}}\), was computed as follows:  
\[f\sqrt{a_{4,pred}}\big{(}\sqrt{a_{4,0}}\big{)}=n\cdot\Big{[}F\sqrt{a_{4,exp}} \big{)}^{n-1}f\sqrt{a_{4,exp}}\big{(}\sqrt{a_{4,0}}\big{)}, \tag{2}\]  
where \(f\sqrt{a_{4,exp}}\) and \(F\sqrt{a_{4,exp}}\) are the pdf and the cdf of the experimental defect size and \(n=V_{90,pred}/V_{90,exp}\). Equation (2) permitted to predict \(\sqrt{a_{40}}\) for a risk-volume larger than that experimentally tested. It is worth noting that, for \(n=1\), the tested risk-volume was taken into account, whereas for \(n>1\), the critical \(\sqrt{a_{40}}\) for a larger risk-volume was predicted. By substituting \(f\sqrt{a_{40}}\) in Equation (1) with \(f\sqrt{a_{4,pred}}\), the P-S-N curve as a function of \(V_{90,pred}\) was easily obtained. Figure 8 plots the median P-S-N curves estimated from experimental tests on hourglass (\(V_{exp}=55\) mm\({}^{3}\) in Figure 8) and on Gaussian specimens (\(V_{exp}=5000\) mm\({}^{3}\) in Figure 8) and the median P-S-N curve predicted by considering the experimental results on hourglass specimens (\(V_{pred}=5000\) mm\({}^{3}\) in Figure 8). Figure 8: Median P–S–N curves estimated from the experimental data and predicted from test results on hourglass specimens. Figure 7: Marginal P–S–N curves (median, 0.05-th and 0.95-th quantiles): (a) hourglass with \(V_{90}=55\) mm\({}^{3}\); (b) Gaussian with \(V_{90}=5163\) mm\({}^{3}\). As shown in Figure 8, the predicted P-S-N curve is above the experimental P-S-N curve. By considering the fatigue limit, the difference is about 7%, but it increases up to 14% in the finite life region. This difference is mainly related to the capability of predicting the defect size in large \(V_{prnd}\). Figure 9 shows the Gumbel plot of the experimental dataset, together with the estimated LEVD. The LEVD for \(V_{prnd}=5000\) mm\({}^{3}\) was estimated, according to Equation (2), starting from the LEVD of hourglass specimens. According to Figure 9, the predicted LEVD underestimates the Gaussian dataset. This is one of the main reasons for the error in the prediction of the median P-S-N curve for \(V_{pred}=5000\) mm\({}^{3}\) (Figure 8). In particular, the definition of risk-volume as \"the region subjected to a stress amplitude larger than the 90% of the maximum stress\" could be not appropriate and could lead to conservative or non-conservative predictions. Indeed, even if a 10% stress-amplitude variation was shown in the literature [3; 4] to be appropriate to describe the \"region at risk\" and, therefore, to be considered as the reference volume for predictions, it cannot completely take into account the effect of stress gradients within that region. Indeed, the more uniform is the stress distribution in the risk-volume, the higher the probability of finding larger critical defects. In addition, defects subjected to a stress amplitude below the 90% of the maximum stress may be in any case critical in the VHCF region, since it is well-known that cracks can propagate even though the stress intensity factor is below the threshold for crack propagation [18; 19; 20]. This is confirmed by the fact that two fatigue failures originated at a stress amplitude below the 90% of the largest applied stress amplitude.