# Experimental Evidence of Specimen-Size Effects on EN-AW6082 Aluminum Alloy in VHCF Regime
## 2 Materials and Methods
### Specimens Design  
In order to study the specimen-size effect on the fatigue resistance of aluminum alloy EN-AW6082, three different sets of hourglass specimens, respectively, with diameters of 3 mm, 6 mm, and 12 mm in the middle cross-section, were designed to be tested with the ultrasonic fatigue testing machine MU90. In particular, the specimens of 3 and 6 mm in diameter were obtained by the round bar of 20 mm in diameter, whereas the 30 mm round bar was used to obtain the largest specimens of 12 mm in diameter. The shape of the hourglass-shaped specimens was obtained, analytically, according to [3], as a function of the physical and mechanical properties of the adopted round bar. In addition, the normal stress \(k_{\sigma}\) in the middle cross-section of the specimen subjected to unit amplitude mechanical vibration can also be assessed analytically. Subsequently, the obtained geometry for all three different specimens was checked by the commercial finite element code, ANSYS(r) Workbench. The axis-symmetry of the problem can be exploited to set up a 2D reduced model, discretized with 8-node axisymmetric quadrilateral elements, so that the computational cost of the analysis has been drastically reduced compared to a fully 3D finite element simulation. At first, a modal analysis was carried out to check that the resonance frequency of the specimens was as close as possible to the working frequency of the machine, i.e., 20 kHz. Subsequently, an harmonic analysis was performed to determine the stress concentration factor, \(k_{t}\), which is evaluated according to [14]. In addition, the risk-volume has been evaluated, \(V_{90}\), which is usually defined as the region where the stress amplitude is higher than the 90% of the peak value [9; 11; 28]. The above-mentioned dynamic mechanical properties of three different specimens, as well as the risk-volume, are summarised in Table 3, whereas Figure 2 shows a photograph of the specimens. Figure 1: Ultrasonic fatigue testing machine MU90. Table 3 shows that, the larger the diameter of the specimen, the lower the value of \(k_{\sigma}\), whereas the stress concentration factor \(k_{l}\) increases for larger specimen dimensions, while remaining less than 1.15. Moreover, it is relevant to observe that the largest specimen investigated in the present experimental campaign has dimensions larger than the ones commonly used for ultrasonic fatigue testing, which have the diameter in the middle-section ranging from 3 to 6 mm. On the other hand, the risk-volume obtained for the specimen of 12 mm in diameter is not so large if compared to the gaussian specimens tested in [17; 19; 29], although the diameter in the middle-section is similar enough. [PERSON] et al. [17; 19; 29], proposed to relate the specimen-size effect on the fatigue strength to the increase in the probability to find defects with larger size within the risk-volume, when the latter increases. Conversely, according to the present authors [23], the specimen-size effect is due to the lacunarity of the reacting cross-section, and can be explained by exploiting the concept of fractal medium. As a consequence, the decrease in the fatigue strength with the specimen size is not directly connected to the risk-volume, but rather to the specimen-characteristic size, e.g., the diameter, and to the fractal dimension of the middle cross-section. As mentioned above, the values of \(k_{\sigma}\) for the specimens of 6 mm and 12 mm in diameter have been also validated experimentally by means of strain gauges measurements, which have confirmed the theoretically determined values reported in Table 3 and the numerical simulations. The specimens geometries used for the ultrasonic fatigue are shown in Figures 3a-5a, while in Figures 3b-5b the results of the numerical analysis with ANSYS Workbench are reported for the three different specimens. Eventually, in Figure 6a-c the photographs of the three specimens connected to the UFTM are reported.