## 2 Materials and Methods
### Ultrasonic Fatigue Tests  
Fatigue tests at a loading frequency of 20 kHz were carried out with an ultrasonic testing machine. Constant stress amplitude tension-compression fatigue tests (stress ratio equal to \(-1\)) were performed up to \(10^{9}\) cycles (runout specimens). For details on the testing setup, on the closed loop system control developed to maintain the stress amplitude constant and on the limits on temperature increment, the reader was referred to [20; 21; 27]. Gaussian specimens [25; 27; 28] with large risk volume \(V_{90}\)[29] of 2300 mm\({}^{3}\) were tested. According to the VHCF literature [23; 29; 30; 31; 32], the size effect influences the VHCF response and, as shown in [31], the prediction of the VHCF response of components with very large risk volume can be dangerously not conservative through tests on small risk volume specimens. The Gaussian specimen profile was analytically designed by the research group of the Politecnico di Torino to investigate the size effect in VHCF through ultrasonic fatigue tests. In particular, by solving the Webster's equation for a plane wave, it was found that the profile ensuring a uniform stress distribution is described by a Gaussian function [25]. The Gaussian specimen was thereafter numerically verified and experimentally validated and finally adopted by the authors for several experimental tests [20; 21; 27; 28]. For more details on the Gaussian specimen design, the reader is referred to [25; 27] and references therein. Following a standard procedure [20; 21], specimens were mechanically polished with sandpapers with grit in the range (240-1000) in order to investigate the influence of both surface and internal defects on the VHCF behavior and avoid fatigue failures from large superficial scratches at a number of cycles below \(10^{5}\). In particular, specimens were polished in order to smooth the external surface and avoid premature failures at a number of cycles below \(10^{4}\) and, therefore, not in the VHCF region. Indeed, through a mechanical polishing process with sandpapers, only a thin layer of material was removed (about 0.1 mm) and, therefore, it was possible to investigate the role of both sub-surface and internal defects. On the other hand, with a machining process, a larger layer of material (i.e., more than 1 mm) was removed and the influence of critical sub-surface defects could not be properly assessed. The final surface roughness was equal to \(1.66\pm 0.14\)\(\upmu\)m. The Gaussian specimen geometry is shown in Figure 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & & & & \\ \hline Si & Mg & Cu & Ni & Fe & Mn & Ti & Al \\ \hline
10 & 0.4 & 0.25 & 0.05 & 0.25 & 0.1 & 0.15 & bal. \\ \hline \multicolumn{8}{c}{**SLM Process Parameters**} \\ \hline Laser power & \begin{tabular}{c} Plate \\ temperature \\ \end{tabular} & \begin{tabular}{c} Scanning \\ speed \\ \end{tabular} & Spot size & \begin{tabular}{c} Hatch \\ distance \\ \end{tabular} & \
\begin{tabular}{c} Layer \\ thickness \\ \end{tabular} &
\begin{tabular}{c} Average \\ powder size \\ \end{tabular} \\ \hline
350 W & \(150\)\({}^{\circ}\)C & \(1.15\) m/s & \(80\)\(\upmu\)m & \(170\)\(\upmu\)m & \(50\)\(\upmu\)m & \(45\)\(\upmu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: AlSi10 Mg powder (wt.%) chemical composition and selective laser melting (SLM) process parameters. After the fatigue tests, fracture surfaces were investigated with the scanning electron microscope (SEM) to analyze the defects originating the fatigue crack. The feasibility of ultrasonic tests on large Gaussian specimens, which was the main objective of the paper published by the authors in [20], is further attested in the present paper. Probabilistic S-N (P-S-N) curves are estimated through a statistical model proposed by the authors. The experimental results are compared with those obtained through tests on smaller hourglass AlSi10 Mg specimens vertically built and produced with similar process parameters [22].