## 2 Materials and Methods
### Fatigue Criteria  
Generally speaking, the determination of the fatigue performances of mechanical components under multiaxial loads is challenging. Several scholars have proposed plenty of fatigue criteria [27; 60; 61; 62; 63; 64; 65; 66]. The most advanced model relies on the critical plane concept. The fatigue failure is assumed to nucleate on the plane where both the normal and the shear stress components are considered critical. While the application of such advanced methods is already infrequent due to the need of certain coding to determine all the required parameters, their application to gears is very rare. Ref. [67] shows the application of such methods to hypoid gears combining the Liu and Mahadevan criterion [68] and an FEM. In [69], the Crossland fatigue criterion [61] was exploited to evaluate the state of stress in spur gears. In the present work, among all the available criteria, the Findley one was applied. The stress exerted on a plane defined by a normal vector \(\vec{n}\), having (spherical) coordinates \(\phi_{n}\) and \(\theta_{n}\), is defined by the vector \(\vec{P}_{n}\) characterized by both modulus and direction varying in time. The vector can be written as  
\[\vec{P}_{n}=[\sigma]\vec{n} \tag{6}\]  
\(\vec{P}_{n}\) has normal component \(\vec{\sigma}_{n}\), which varies in modulus but is characterized by a fixed direction, and a tangential one (\(\vec{\tau}_{n}\)) having both time-varying modulus and direction. In the particular case where the stresses are periodic, the vertex of \(\vec{P}_{n}\) describes a closed curve, whose minimum and maximum distance from the plane could be identified as \(\sigma_{n,min}\) and \(\sigma_{n,max}\), respectively. The projection of the curve \(\vec{P}_{n}\) into the plane, defines the two-dimensional curve \(\Gamma_{n}\) (Figure 5). The stress history can be obtained via the time-dependant FEM simulations where the entire loading-unloading cycle is simulated in 20 discrete steps (applying a force varying from \(-1150\) to \(-11,\)500 N). The stress tensor history (a symmetric matrix showed in Equation (1)) must be extracted for each point where fracture could nucleate, i.e., each point within the root fillet region for both the FE analyses. Thus, through the numerical analysis, it is possible to obtain the stress tensor \([\sigma]\) of all points belonging to the tooth root region (namely the most critical area). These are analysed with the fatigue criterion according to Findley which also considers non-proportional loads. Through the geometrical properties of the curve \(\Gamma_{n}\), the alternate \(\tau_{n,a}\) and the average \(\tau_{n,m}\) components of the tangential stress could be determined. Several methods observe this scope. The most used one is the so called Minimum Circumscribing Circle Method (MCC) [70]. \(\tau_{n,a}\) is assumed equal to the radius of the smallest circle that entirely contains  
Figure 4: STBF test configuration. the curve \(\Gamma_{n}\), \(\tau_{n,m}\) equal to the distance between the centre of the MCC and the origin (Figure 5). For each plane with a generic orientation, the shear stress amplitude \(\tau_{n,a}\) and the maximum normal stress \(\sigma_{n,max}\) should be carefully evaluated. The critical plane is determined, among all the possible planes having generic orientations, as the one where the damage parameter results maximum (Equation (7)). The damage parameter summarizes the severity of a stress cycle. It has to be compared with the fatigue limit of the material \(f\) as with any other equivalent stress (e.g., Sines, Gough-Pollard). Its units are MPa. However, instead of just combining some of the stress components of the stress tensor, the damage parameter is not only a function of the stress state, but also depends on the material through the parameter \(k\). For the present steel \(k=0.2569\). \[ \begin{split}(\phi_{C},\theta_{C})\to max_{\phi,\theta}\{ \tau_{n,a}(\phi,\theta)+k\sigma_{n,max}(\phi,\theta)\}\\ DP=\tau_{C,a}+k\sigma_{C,max}\leq f\end{split}