## 2 Background
### General Framework to Calculate \(f_{korr}\) by Means of FEM Simulation and Findley's Criterion  
As mentioned in the introduction, the load/stress histories in the MG and STBF conditions are different and therefore the damage caused by the two tests (for each loading cycle) is different. For each test, the damage can be quantified through the implementation of fatigue criteria based on the critical plane, in this specific case the Findley criterion which was proven to be the most accurate for such analyses [62; 63]. Indeed, it can be proven that \(f_{korr}\), defined as the ratio of \(\sigma_{FlimMG}\) and \(\sigma_{FlimSTBF}\), is equivalent to the ratio between the maximum damages (calculated through Findley) in the STBF condition and the MG condition [26]. Therefore, a general framework to estimate \(f_{korr}\) for a specific combination of gear geometry and material involves two steps. 1. The first step relies on setting up FEM simulations of MG and STBF tests on the same gear. The loading conditions (force applied to the anvil in STBF simulation and torque applied to the MG simulation) have to lead to the same \(\sigma_{F}\) according to [16]. In addition, to model a critical condition, \(\sigma_{F}\) should be similar to a presumed \(\sigma_{FP}\). In this way, the results of the FEA related to the STBF and the ones related to MG are equivalent from the standard's perspective [16]. 2. The second step involves the calculation of the maximum damage reached in the fillet region according to the Findley criterion. With this respect, the stress history tensor \(\overline{\mathbf{\sigma}}(t)\) (Equation (1)) has to be extracted for each position along the fillet region for both the FEM analysis performed. In this way, the points of the fillet region can be analyzed through the Findley criterion. Indeed, elaborating the various \(\overline{\mathbf{\sigma}}(t)\) it is possible to detect the most critical point and calculate the maximum damage parameter for each type of test. \[\overline{\mathbf{\sigma}}(t)=\left[ \begin{array}{ccc}\sigma_{\mathbf{xx}}(\mathbf{t}) &\mathbf{\tau_{xy}}(\mathbf{t})&\mathbf{\tau_{xz}}(\mathbf{t})\\ \mathbf{\tau_{yx}}(\mathbf{t})&\mathbf{\sigma_{yy}}(\mathbf{t})&\mathbf{\tau_{yz}}(\mathbf{t})\\ \mathbf{\tau_{xx}}(\mathbf{t})&\mathbf{\tau_{xy}}(\mathbf{t})&\mathbf{\sigma_{zz}}(\mathbf{t})\\ \end{array} \right]\] (1)  
In the following of this section, the concepts behind the fatigue criteria based on the critical plane are presented together with the procedure to elaborate \(\overline{\mathbf{\sigma}}(t)\) according to the Findely's one to acquire the maximum damages parameters and, therefore, to calculate \(f_{korr}\). The mathematical modelling behind this procedure is explained in more detail in [62]. In Figure 1, a plane defined by its normal vector n (having spherical coordinates \(\phi_{n},\theta_{n}\)) is shown. This plane passes through the point to be evaluated in terms of damage due to fatigue. To this regard, \(\overline{\mathbf{\sigma}}(t)\) has to be extracted in the above-mentioned point. The total stress acting on this plane can be represented by a vector having time-dependent modulus and direction (\(\mathbf{P_{n}}\)) that, in turn, can be defined through Equation (2). \[\mathbf{P_{n}}(\phi_{n},\theta_{n},t)=\overline{\mathbf{\sigma}}(t)\;\mathbf{n}(\phi_{n}, \theta_{n}) \tag{2}\]  
\(\mathbf{P_{n}}\) can be decomposed into \(\mathbf{\sigma_{n}}\) and \(\mathbf{\tau_{n}}\). The former has fixed direction (normal to the plane) and time-dependent modulus. The latter is tangential to the plane and have a time-dependent modulus and direction. In Figure 2, it is possible to note that, when periodic stresses are in play, \(\mathbf{P_{n}}\) describes a three-dimensional closed curve whose projection on the plane is the positions of \(\mathbf{\tau_{n}}\) assumed in a load cycle (\(\Gamma_{n}\) in Figure 2). In addition, along a cycle, it is possible to individuate \(\sigma_{n,min}\) and \(\sigma_{n,max}\) (Figure 2).