*3.1. Individuation of Cracks Characteristic through Experimental Images*

The crack propagation just after nucleation can be characterized by the two parameters χ and β (Figure 4). Using these coordinates, it has been assumed that the early propagation plane is always perpendicular to the view in Figure 4.

**Figure 4.** Cracks characteristics (χ, β) in experimental tests.


In Figure 4, χ and β are reported for generic cracks (highlighted in red) in Gear A and Gear B. In the figure, the point in which χ assumes its minimum and its maximum value are indicated.

In Figures 5 and 6, experimental images on which χ and β have been identified are shown. More specifically, Figure 5 shows six images of different teeth belonging to Gear A. The same, referred to as Gear B, is shown in Figure 6.

**Figure 5.** Individuation of cracks characteristics (χ, β) in experimental tests performed on Gear A.

In the figures, the yellow dashed line represents the tooth profile (before the test) while the red solid line represents the direction of early propagation of the crack. It is worth noting that while in Gear B the crack always led to the complete detachment of the tooth, as far as Gear A is concerned, the tests were interrupted when the crack was detected via the variation of the stiffness of the system (even if it did not lead to the complete breakage of the tooth). Therefore, in some images of Gear A, the crack is of limited size and is hidden by the red line, which, however, represents its initial propagation direction.

With respect to Gear A (Figure 5), all the cracks nucleated in 0.382 ≤ χ ≤ 0.775 having a direction 54.5◦ ≤ β ≤ 65◦. With respect to Gear B (Figure 6), all the cracks nucleated in 0.550 ≤ χ ≤ 0.664 having a direction 42◦ ≤ β ≤ 51.5◦. It is interesting to notice that, for Gear A, three cracks nucleated in the proximity of χ = 0.400, while the other three cracks nucleated in different points. The latter cracks may be nucleated at different locations due to micro defects in the material. Moreover, in Gear B, the nucleation points have a lower dispersion, but are located in the proximity of the end of the grinding zone where, most likely, a micro notch has formed between the root radius and the beginning of the involute tooth profile.

### *3.2. Numerical Elaboration Aimed to Characterize Cracks within Tooth Root Radius*

The FEM has been set up into the open-source software, Salome-Meca/Code\_Aster. In Figures 7 and 8, it is possible to see the STBF test modeling for Gear A and Gear B, respectively. In the present study, 3D simulations have been performed to also consider the boundary effects.

**Figure 6.** Individuation of cracks characteristics (χ, β) in experimental tests performed on Gear B.

**Figure 7.** Finite Element Model of the STBF of Gear A.

**Figure 8.** Finite Element Model of the STBF of Gear B.

To reduce the computational effort, only a quarter of each gear has been modeled exploiting symmetries. More specifically, on the one hand, half of the face width has been modeled. On the other hand, gears are symmetric on a plane parallel to the contact-face of the anvil and positioned at half of the Wildhaber distance (yellow line in Figures 7 and 8).

The models have been created through extruded meshes. Linear elements having typical isotropic steel properties have been used i.e., a Young modulus equal to 205,000 MPa and a Poisson's ratio of 0.3. In each model, hexahedral elements have been exploited to model the loaded tooth while TRIA6 elements, i.e., triangular base prisms, have been used to model the remaining volume of the gear. The mesh density has been increased in the loaded tooth after a sensitivity analysis. More specifically, the mesh density has been increased by 10% until the results of the simulations present a variation of less than 1%. The final models have the mesh characteristics listed in Table 2.

**Table 2.** Mesh characteristics of the simulated gears.


Non-linear simulations have been performed to simulate the contact between the anvil and the tooth flank for each gear. While the analyses are non-linear due to the contacts, the state of stress never exceeded the yielding. In Figures 7 and 8, the contact faces are indicated with green lines and the theoretical contact point is indicated with a green circle. It is located in the intersection between the horizontal line tangent to the base circle (represented in the figures) and the tooth flank. With respect to Gear A, a pulsating compressive force varying sinusoidally from a minimum value of 3700 kN to a maximum value of 37,000 kN has been applied to the anvil. With respect to Gear B, the minimum and maximum value of the force applied result 1498 kN and 14,980 kN, respectively. Through the above-mentioned loading configuration, taking into consideration the symmetries exploited, it has been possible to replicate the experimental conditions, i.e., ratio between the minimum and maximum force of 0.1 (applied in the experimentation). Those levels of force are the loads that averagely lead to a failure in 10<sup>6</sup> cycles.

The stress tensor *σ*(*t*) was extracted for both gears in the most critical areas where fracture is expected to nucleate, i.e., within the *ρf P* (nodes highlighted with the red line in Figures 7 and 8). At this point, the approaches presented in Section 2 have been applied by defining the material properties ( *σf* , *τf* , *<sup>σ</sup>R*). In particular, Gear A has been manufactured with VAR 9310 having a bending fatigue limit *σf* = 1400 MPa, a torsional fatigue limit *τf* = 1100 MPa, and an ultimate tensile strength *σR* = 2700 MPa. On the other hand, Gear B has been manufactured through 20MnCr5 having *σf* = 516 MPa, *τf* = 303 MPa, and *σR* = 1028 MPa.

Therefore, for each gear and for each point within the *ρf P*, it has been possible to elaborate the stress tensor *σ*(*t*) implementing the different fatigue criteria presented in the previous section. In the present paper, the studied points are the nodes of the computational grid belonging to the *ρf P*. This choice was made in order to avoid the need of interpolation. The workflow followed is graphically explained in Figure 9. For each gear, the workflow is structured with four FOR loops. The innermost one analyses data for each simulated time step (in these cases *T* = 40). The FOR loops on *ϑ* and *ϕ* aim to discretize the space by defining the direction of different planes varying by 0.5◦ each cycle (from 0◦ to 180◦). The FOR loop on the nodes within the *ρf P* i.e., *Nmax* = 31 for Gear A and *Nmax* = 51 for Gear B, aims to study the most critical positions. Indeed, for each node *<sup>N</sup>*(*<sup>θ</sup>cϕc*), belonging to the symmetry section of the tooth (i.e., the most critical), the critical plane has been individuated through the presented framework. This allowed for achieving a twofold objective. First, it allows us to calculate the damage parameters for each node and each criteria (through Equations (11)–(14) and (17)). Therefore, it has been possible to calculate *SF* for each node and each criteria (Equations (18)–(22)) (green boxes in Figure 9). In this way, it has been possible to estimate the differences between nodes in terms of criticality. Moreover, the most critical node according to the different criteria implemented has been established. Second, it has allowed us to identify the direction of the crack propagation (at least in in the proximity of the studied nodes) if it nucleates in any of them (by differentiating between the various fatigue criteria) (blue boxes in Figure 9).

The above-mentioned direction of the critical plane corresponds to the direction of early propagation of the crack after nucleation (evaluated for each node and each criterion). In addition, *SF* is representative of the criticality of the node (according to the criteria in question). The combination of these two results, i.e., direction of critical plane and *SF*, allowed for obtaining an overview of possible crack propagation scenarios in the *ρf P* according to the various criteria. These results have been compared with the experimental ones in terms of crack positions and paths observed after performing STBF tests. The comparison has allowed for assessing the effectiveness of each criterion to correctly predict the failure.

**Figure 9.** Framework to elaborate the time-dependent stress tensor on each node implementing different fatigue criteria.
