# Ultrasonic VMCF Tests on Very Large Specimens with Risk-Volume Up to 5000 mm+
## 2 Materials and Methods
### Gaussian Specim Design  
The geometry of Gaussian specimens with a risk-volume of 5000 mm\({}^{3}\) was designed through an iterative optimization procedure. At first, a preliminary geometry was analytically defined according to [6]. Thereafter, an iterative numerical procedure was developed in order to define the specimen geometry, ensuring a resonance frequency (f\({}_{0}\)) of 20 kHz, a stress concentration factor (k\({}_{t,0}\)) smaller than 1.07 (smooth specimen), a risk-volume (V\({}_{90,0}\)) larger than 5000 mm\({}^{3}\) and a specimen amplification factor (M\({}_{\sigma,0}\)), defined according [6; 10], larger than 1.4 (to ensure the application of a sufficient stress amplitude in the Gaussian profile). A model of the specimens was created by using the commercial software Ansys. Axisymmetric elements, Plane82 in Ansys, were used for the model. Due to its symmetry, only half of the specimen was modeled. A modal analysis was then carried out and the resonance frequency (f\({}_{\text{ris,FEA}}\)) the stress concentration factor (k\({}_{\text{t,FEA}}\)) [9], the risk-volume (V\({}_{90,\text{FEA}}\)) and the stress amplification factor (M\({}_{\sigma,\text{FEA}}\)) were assessed. A Matlab routine was written in order to iteratively vary the specimen geometry, with the aim of minimizing an objective function defined as the sum of the percentage difference between the computed FEA values (f\({}_{\text{ris,FEA}}\), k\({}_{\text{t,FEA}}\), V\({}_{90,\text{FEA}}\) and M\({}_{\sigma,\text{FEA}}\)) and the desired nominal values (f\({}_{\text{ris,0}}\), k\({}_{\text{t,0}}\), V\({}_{90,\text{FEA}}\) and M\({}_{\sigma,0}\)). The final geometry, characterized by f\({}_{\text{ris,FEA}}\) equal to 20.028 kHz, k\({}_{\text{t,FEA}}\) equal to 1.042, V\({}_{90,\text{FEA}}\) equal to 5067 mm\({}^{3}\) and M\({}_{\sigma,\text{FEA}}\) equal to 1.42, is shown in Figure 1a. The geometry of the investigated hourglass specimens, with 3 mm diameter at the gage section (a typical value for ultrasonic tests in the VHCF literature [10; 11; 12]) and characterized by a significantly smaller risk-volume equal to 55 mm \({}^{3}\) is depicted in Figure 1b. Dimensions in Figure 1 are in mm.  
The hourglass and the Gaussian specimens were produced through a machining process, starting from round bars provided in the annealed condition. Thereafter, the specimens were subjected to a heat treatment involving preheating at 750 \({}^{\circ}\)C for 1 h, austenitization at 1030 \({}^{\circ}\)C for 1 h, gas quenching and triple tempering (first tempering at 520 \({}^{\circ}\)C for 1 h; second tempering and third tempering at 540 \({}^{\circ}\)C for 1 h). The heating phase was carried out in a vacuum environment, whereas the cooling phase was carried out in a furnace filled with inert gas. The Vickers hardness after the heat treatment was experimentally measured (INNOVATEST(r) Nemesis 9000) at the center of the cross-section at the specimen free end; the measured hardness was 615 HV, with a limited variation within the cross-section (smaller than 2%), thus confirming the uniformity of the microstructure. Before the experimental tests, the specimens were finely mechanically polished [13] with sandpapers, with increasing grit, to enhance the crack nucleation from defects randomly distributed within the risk-volume and avoid premature failures in the HCF region due to cracks propagating directly from large defects that originated during the machining process.  
Figure 1: Geometry of the specimens subjected to ultrasonic tests: (**a**) Gaussian specimen; (**b**) hourglass specimen (not in scale, dimensions in mm).