**4. Results**

As mentioned in the previous section, numerical and experimental results have been compared. On the one hand, in STBF specimens, it has been possible to identify both the point where the crack nucleated and the direction of crack propagation for each tooth that failed during the test. On the other hand, through the elaboration of numerical results, for each node within the *ρf P* it has been possible to evaluate the damage parameter (it indicates the criticality of the node in question) and the direction of the critical plane (it indicates the direction of the initial crack propagation if the crack nucleates in the studied node).

The comparison has allowed for assessing the effectiveness of each criterion to correctly predict the failure. More specifically, each criterion has been evaluated based on its attitude to:


In Table 3, the minimum *SF* calculated according with the investigated criteria and the relative node location is reported for the two gears. With respect to the parameter χ, all the criteria show a congruence in identifying the critical node (χ = 0.400 for Gear A and χ = 0.435 for Gear B). In addition, it is interesting to highlight that, according to the standard [14], the critical node should be located in χ = 0.508 for Gear A and χ = 0.430 for Gear B (the standard [14] defines the critical point as the point of the fillet tangent to a straight line having 30◦ inclination with respect to the axis of the tooth). Therefore, on the one hand, numerical results lead to individuate the critical node in the same position of the standard for Gear B and in a different position for Gear A. On the other hand, experimental results show a greater variability in the nucleation point that, in turn, are not in agreemen<sup>t</sup> with the standard in either case but in very good agreemen<sup>t</sup> with the numerical results for Gear A.

**Table 3.** Minimum *SF* calculated through different fatigue criteria and associated critical node location χ.


With respect to the value of *SF*, in Table 3 it emerges that the implementation of the Findley criterion leads to values of *SF* closer to the unity for both the gears. Comparable values emerge even when implementing the Papadopoulos criterion. Moreover, while for Gear B all *SF* values are close to unity (ranges from 0.94 to 1.23), for Gear A Matake and McDiarmid criteria lead to very high values of *SF* i.e., 1.96 and 2.14, respectively. In both the cases it is possible to assert that the Susmel criteria is the most conservative one.

In Figures 10 and 11, experimental and numerical results are graphically compared. Figure 10 is related to the *ρf P* of Gear A, Figure 11 concerns the *ρf P* of Gear B. In particular, for each of the criteria investigated, the direction of the critical plane calculated in different nodes of the fillet are shown through blue lines. The length of the segments is proportional to the damage parameter. The thicker blue line represents the critical plane having the higher damage parameter. The red lines represent the experimental results and have length as if it was a critical plane having a unit *SF*. For each criterion, only the *ρf P* and the tooth axis have been reported. The results can be represented graphically in 2D since the critical planes are all perpendicular to the views in the figures.

**Figure 10.** Direction of the critical planes according to the different criteria studied at different nodes for Gear A. Numerical results in blue (segment length proportional to the damage parameter) and experimental results in red (segment length proportional to the damage parameter that lead to a unitary *SF*).

**Figure 11.** Direction of the critical planes according to the different criteria studied at different nodes for Gear B. Numerical results in blue (segment length proportional to the damage parameter) and experimental results in red (segment length proportional to the damage parameter that lead to a unitary *SF*).

Naturally, the direction of the critical planes only varies between Findley and the other criteria, which, in turn, identify the critical plane in the same way. What changes between the various criteria is the value of the damage parameter associated with each node and, therefore, the length of the blue segments.

With respect to Gear A, it is possible to notice that most of the experimentally measured cracks are located in the proximity of the most critical node (i.e., the intersection between the thickest blue line and the radius). However, only the Findley criterion is capable of identifying, with very good approximation, the direction of early propagation of the crack. In addition, Findley's criterion also allows for identifying the direction of cracks even when these nucleate in different positions of the radius, i.e., χ = 0.576, χ = 0.574, χ = 0.775 (most likely due to minor manufacturing or material defects in those positions).

The other criteria lead to very different angles with respect to the ones observed experimentally, e.g., 59◦ difference between the plane with the maximum *<sup>τ</sup>c*,*<sup>a</sup>* and the crack observed in its proximity. Therefore, the Findley criterion is the criterion that better models crack nucleation (and early propagation) at the *ρf P* of Gear A.

With respect to Gear B, most of the experimentally observed cracks are not in the proximity of the most critical plane calculated numerically. However, also in this case, the Findley criterion is capable of better estimating the crack propagation direction within the whole *ρf P*. Indeed, the other criteria suffer from errors ranging from 15◦ to 25◦ while Findley approximates the direction with an error of less than 5◦.
