## 4 Discussion  
The combined analytical-numerical approach based on the inverse Bogy formulation and FEA investigations was effectively applied as a design methodology for producing a fatigue test configuration. Indeed, the absence of analytically predicted stress singularities was confirmed by FEA investigations for both test conditions. Notably, the FEA studies demonstrated that even though the overall stress distribution was different for the test-representative experimental conditions (Figure 7a,b), the recovered longitudinal stress levels within the adhesive were equal under fixed applied explorative stress conditions (Figure 8a,b) of 15 MPa. From a design perspective, performing FEA studies was particularly important to carefully approach the test phase, as the joint was thus tested in its  
Figure 12: Strength distributions for 10%, 50%, and 90% probability levels at N = 2 \(\times\) 10\({}^{6}\) cycles. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Failures at 5 Hz** & **Failures at 50 Hz** & **Failures at 20 kHz** \\ \hline Standard deviation (SD) & 0.3038 & 0.5763 & 0.4687 & 0.7691 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Standard deviations of data processed with ML and optimization. physical best condition (i.e., no stress singularity effects) and under the same test levels (i.e., no damping, resonance, or visible frequency-induced effects). The experimental campaign executed using a commonly adopted fatigue machine and an ultrasonic testing machine highlighted different aspects of fatigue testing, both in general and regarding adhesive joint testing in particular: (i) the use of an ultrasonic testing machine was effectively applied for testing specimens up to the VHCF range. This fatigue range is relatively new for material characterization, and several studies have been performed in recent years; (ii) performing ultrasonic tests confirmed the capability of the investigated adhesive to sustain loads for a higher number of cycles. This evidence is of particular importance and interest as modern structures that are composed by innovative and engineered materials are required to sustain acting loads for much more time than before; (iii) performing commonly executed fatigue tests enabled the investigation of the loading frequency effects, and the experimental outcomes have shown an extreme sensibility to this parameter for the investigated adhesive. For example, at 14 MPa, there is a difference in the range of five orders of magnitude between the ultrasonic and low-frequency fatigue lives. Specifically, joints subjected to ultrasonic fatigue tests failed in the VHCF range, whereas the remaining classes failed in the HCF range. Similarly, the run-out threshold decreases from values close to 15 MPa in the case of ultrasonic testing to 10 MPa when specimens are subjected to commonly adopted frequencies. Indeed, as illustrated by other research (e.g., [14]), loading frequency effects and strain rate effects have a tendency to increase material properties (e.g., the strength or elasticity modulus) and enhance material responses. Information such as that extracted in this work is of fundamental importance in the structural design of equipment, as modern structures should sustain loads over a broad frequency spectrum that is commonly characterized by complex stress/strain histories at different stress ratios; (iv) the use of a statistical tool was particularly useful for both the precise quantification and visualization of frequency effects, as well as to manage data scattering and uncertainties. For the special case of the selected adhesive, the ML technique was employed for extracting the P-S-N curves, while SD studies quantified data scattering.