**5. Discussion and Conclusions**

In the present paper, a methodological approach for implementing five different fatigue criteria based on the critical plane is presented. This relies on the elaboration of the stress tensor *σ*(*t*) calculated via FE simulations on specific nodes modeling the *ρF* in STBF loading condition. With the aim of evaluating the accuracy of the different criteria, two different gear geometries have been studied. In both cases, the gears had been experimentally tested and, therefore, it has been possible to obtain the force values leading to the permissible stress, the crack nucleation points, and the crack propagation (just after nucleation) directions in multiple tests. Therefore, the numerical results have been compared with the experimental ones in terms of: (1) capability of the criteria to provide a *SF* equal to one; (2) identify the actual critical node; and (3) determine the actual crack direction.

With respect to the point (1), Findley and Papadopoulos are the criteria that lead to the expected outcome most effective in both gears. The Matake criterion leads to overestimating the material strength in both the gears. The Susmel et al. criterion tends to underestimate the material properties and, therefore, it results in being a conservative criterion. The McDiarmid criterion leads to two different results in the two gears, i.e., in Gear B the value of *SF* is close to unity while it is more than double for Gear A. This may be due to the high tensile strength of the Gear A material that, in turn, it is considered in the formulation of the damage parameter according to McDiarmid. Eventually, in terms of the point (1), the Findley and Papadopoulos criteria are the most appropriate ones to be applied on gears for estimating fatigue behavior.

With respect to the point (2), all criteria agree in identifying the most critical node in both gear geometries. However, the comparison with experimental results shows that in Gear A, numerical results correctly identify the nucleation point of the crack, while in Gear B, the nucleation point is not accurately identified. Nevertheless, the numerical results of Gear B agree with the standard [14]. In addition, it is worth noting that some cracks in Gear A are located in different points, probably due to micro defects of the material or in the manufacturing process, while in Gear B all the cracks nucleate in the proximity of the end of the grinding zone between the *ρf P* and the tooth flank. Moreover, it is possible to notice that the difference in the damage parameter between neighboring nodes is relatively low (less than 3% of difference in the proximity of the most critical node). Therefore, it is possible to state that about 25% of the studied area is subject to a damage parameter

above 90% of the maximum damage parameter. Eventually, it is possible to assert that the experimental cracks occur in different positions due to phenomena related to micro-defects that, in turn, were not reproduced with the present FE modeling.

With respect to the point (3), Findley's criterion is undoubtedly the most appropriate for identifying the direction of crack propagation in each possible nucleation point of both the studied gears. Therefore, it is possible to assert that the crack propagation direction at the *ρf P* does not follow the plane of maximum alternating shear stress but the plane of maximum damage parameter according to Findley. Indeed, all experimentally identified cracks follow a direction relevant to that indicated by the implementation of Findley's criterion. This result could open the door to the development of new fatigue criteria based on the critical plane for the study of gear. Indeed, an interesting future research direction would be to formulate and/or verify criteria defining the critical plane by the damage parameter (as currently done by Findley's criterion) and, therefore, taking into account also the stress normal to the critical plane for its definition.

Eventually, it is worth noting that the method proposed in this paper has a general validity since it models three-dimensional geometries. However, in the specific case studied in this article, two-dimensional models could also be used to speed up the simulations. In this case, to implement the elaboration of the stress history, it would have been possible to use cylindrical coordinates by setting an angle constant consistent with the simulated model.

**Author Contributions:** Conceptualization, F.C. and L.M.; methodology, F.C. and L.M.; software, L.F.; validation, L.B., L.M. and F.C.; formal analysis, F.C. and L.M.; investigation, F.C. and L.M.; resources, L.B., F.C. and L.M.; data curation, L.F., F.C. and L.M.; writing—original draft preparation, L.M.; writing—review and editing, F.C.; visualization, L.M., L.B.; supervision, F.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Open Access Publishing Fund of the Free University of Bozen-Bolzano.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
