## 3 Results
### P-S-N Curves  
The marginal P-S-N curves are finally estimated from the experimental failures. The model proposed by the authors [41] is considered. The cumulative distribution function (CDF) of the logarithm of the fatigue life random variable is given by the product of the distribution of the conditional fatigue life and the statistical distribution of \(\sqrt{a_{c}}\). The conditional fatigue life is assumed normal, with constant standard deviation and with a mean that linearly depends on \(s_{local}\) and \(\sqrt{a_{c}}\). The defect size \(\sqrt{a_{c}}\) is assumed to follow the largest extreme value distribution, according to [29]. A continuously decreasing linear trend (i.e., without a horizontal asymptote at the end of the curve, the fatigue limit) is considered for modelling the P-S-N curves. The maximum likelihood principle was applied for parameter estimation, by considering both failures and runout data. Figure 8 shows the S-N plot of the experimental data, together with the estimated median and 0.1% (99% reliability) P-S-N curves. The VHCF response obtained in [22] by testing hourglass specimens with similar process parameters (batch B) is also shown in Figure 8 (light grey markers). According to Figure 8, the estimated P-S-N curves fit the experimental data well. In particular, 5 out of 12 (about the 50%) of the experimental failures are above the median curve. The 0.1% P-S-N curve is below all the experimental failures. On the other hand, for horizontally as-built AlSi10 Mg specimens, the 1% P-S-N is below all the experimental failures. Therefore, the large VHCF scatter of  
Figure 8: Marginal probabilistic S-N (P-S-N) curve: Median and 0.1%-quantile marginal P-S-N curves. Figure 7: Stress intensity factor (SIF) threshold, with respect to \(N_{f}\). vertically built specimens implies a more conservative choice for the design curve to avoid unexpected failures due to rare critical defects. The VHCF strength is also compared with literature results. For the same \(N_{f}\), the VHCF strength in [22] (light grey markers in Figure 8) is generally larger than that obtained in the paper, but the difference is limited. On the other hand, if the design curve is considered, the difference is significantly larger. For instance, the runout limit at \(10^{8}\) cycles in [22] is equal to 90 MPa, whereas the VHCF strength at \(10^{8}\) estimated from the 0.1% P-S-N curve is about 40 MPa. Therefore, it can be inferred that the VHCF strength obtained in the present paper is averagely close to the literature results obtained by testing the specimen manufactured with similar process parameters. However, the scatter found experimentally is significantly larger, thus lowering significantly the P-S-N curves with smaller failure probability. The reason for this difference is mainly related to the specimen conditions. The tested Gaussian specimens are not machined and, therefore, surface defects due to incomplete fusion or improper layer bonding, whose size is characterized by a large dispersion, are not removed and contribute to increase the experimental scatter of the VHCF response. Moreover, the tested \(V_{90}\) of the Gaussian specimens is significantly larger than the largest \(V_{90}\) tested in the literature, thus increasing the probability of the presence of rare large defects in the risk volume (the size effect, according to [29]). Therefore, the experimental results confirm that the surface region is critical for the VHCF response. In particular, the removal of a layer of surface material has not had a significant effect on the average VHCF strength, but it permits us to reduce the experimental scatter. In this respect, the optimization of the process parameters to reduce the formation of large defects would be more effective for enhancing the VHCF response. On the other hand, in horizontally built specimens [20], the fatigue crack mainly originated from defects formed in the region where the support structures were removed.