## 2 Background
### General Framework to Calculate \(f_{korr}\) by Means of FEM Simulation and Findley's Criterion
However, for implementing fatigue criteria, it is necessary to identify the value of the alternate tangential stress \(\tau_{n,n}\) and its average value \(\tau_{n,m}\). These values represent the entire load cycle in terms of shear acting on the studied plane. With this respect, the most common method to summarize \(\Gamma_{n}\) into two scalars (\(\tau_{n,a}\) and \(\tau_{n,m}\)) is the Minimum Circumscribed Circle (MCC) [64]. In this approach, represented in Figure 3, \(\tau_{n,a}\) is calculated as the radius of the smallest circle that can entirely contains the curve \(\Gamma_{n}\), while \(\tau_{n,m}\) can be defined as the distance between the center of the circle and the origin of \(\mathbf{\tau_{n}}\). Hence, by varying the parameters (\(\phi_{n}\), \(\theta_{n}\)) it is possible to define multiple planes passing through the studied point and, for each of these planes, the relevant stress values, i.e. ; \(\tau_{n,a}\), \(\tau_{n,m}\), \(\sigma_{n,max}\), \(\sigma_{n,min}\) can be calculated through the aforementioned methods. At this point, establishing specific criteria it is possible to individuate (among the defined planes) the critical one. In the present paper, the spherical coordinates and the related stress' values are labelled with the subscript c (critical) as in Equations (3) and (4). \[\sigma_{c,max} = \sigma_{n,max}(\phi_{C},\theta_{C}) \tag{3}\] \[\tau_{c,a} = \tau_{n,a}(\phi_{C},\theta_{C}) \tag{4}\]  
Figure 1: Components of the vector \(P_{n}(\phi_{n},\theta_{n},t)\) on the plane defined through \(n(\phi_{n},\theta_{n})\). The plane (in green) is represented in a Cartesian space (x, y, z) and the polar coordinates (\(\phi_{n},\theta_{n}\)) identify the direction of the vector normal to the plane \(n\). Figure 2: Values and vectors related to periodic \(P_{n}\) on a plane defined by \(n\). \(\Gamma_{n}\). is the projection on the plane (having normal vector \(n\)) of all the values assumed by \(P_{n}\) in the loading cycle, i.e., all the values assumed by \(\tau_{n}\) in a loading cycle. The principle to establish the critical plane (\(\phi_{c}\), \(\theta_{c}\)) varies according with different criteria. In most of the cases, the critical plane corresponds with the plane having the maximum value of \(\tau_{c,a}\), e.g., [48; 50; 53; 65]. However, according to the Findley criterion (Equation (5)), the critical planes can be found through Equation (6) by varying \(\phi\) and \(\theta\) in the range \([0,\pi]\). In Equations (5) and (6), \(r_{\tau/\sigma}\) is the ratio between the material fatigue limit at symmetrical alternating torsional loading (\(\tau_{f}\)) and the limit at symmetrical alternating bending loading (\(\sigma_{f}\)) (Equation (7)). The Findley criterion states that the studied point withstands the loading history \(\overline{\sigma}(t)\) for an infinite lifetime, if the left-hand side of Equation (5) is less than a constant related to a material property obtained by fatigue testing with pure bending and pure torsion. This left-hand of Equation (5) can be considered the Damage Parameter (DP) of the studied point that, in turn, its maximum value determines the critical plane (Equation (6)). Indeed, the Findley criterion defines the critical planes as the plane in which the maximum DP is present. This definition of the critical plane differentiates it from the other criteria. Indeed, the Findley criterion exploits a linear combination of \(\tau_{n,a}\) and \(\sigma_{n,max}\) for the individuation of the critical plane while, as mentioned above, the other criteria consider only \(\tau_{n,a}\).