## 3 Results Furthermore, Discussion  
In the present work, the effect of shoot peening on the tooth root fatigue resistance of gears has been studied via Finite Element Simulations in combination with the Findley  
Figure 5: (**a**): Components of vector \(\vec{P}_{n}\) in plane defined by \(\vec{n}\). (**b**): Definition of curve \(\Gamma_{n}\) that is the projection of positions of \(\vec{P}_{n}\) assumed in a load cycle in plane defined by \(\vec{n}\) or positions of \(\theta_{n}\) assumed in a load cycle. Individuation values of maximum and minimum normal stress and definition of \(\tau_{n,a}\) and \(\tau_{n,m}\) according to the MCC method.  
fatigue criterion. The gear geometry is the standard geometry used for the majority of experimental tests on gears (STBF). The loading cycle and the corresponding time-dependent stress state in the tooth root region were obtained with non-linear FE simulations where the STBF configuration was reproduced. A reference case without shot peening was simulated in the wrought condition. In a second model, the residual stress state induced by a shot peening process (\(d=0.8\) mm, \(v=50\) m/s) was simulated with a separate pre-conditioning simulation and successively subjected to the same loading cycle as the reference non-treated gear. The most critical point and the severity of the damage were evaluated for both conditions exploiting the Findley fatigue criterion (Figure 6).  
The results for the reference non-treated gear are coherent with the experimental findings [57] showing a nucleation site in the \"critical section\" as defined by the ISO standard and with an early propagation direction of \(\gamma_{NP}=43^{\circ}\). The shot-peened gear shows quite a different behaviour. The nucleation position lies significantly above the \"critical section\" and the early propagation direction results in \(\gamma_{SP}=74^{\circ}\) (angle between the propagation direction and the tooth axis). This difference could be justified by the residual compression state which reduces the maximum normal stresses in the filter region (the one that was shot-peened). In this regard, the damage parameter result was about 5.9 times lower with respect to the reference gear (\(DP_{SP}=0.16\cdot f\) vs. \(DP_{NP}=1\cdot f\)). The most critical region, is therefore not in the filter, but slightly above, where, despite a bigger section-to-bending ratio, the loading history results were more critical than in the filter area where the compressive stresses mitigate the stress peaks. The authors are not aware of available data in terms of crack position for shot-peened gears. However, the phenomenon seems to be comparable to the effect of surface hardening. In this regard, some authors claim that the case-hardening protects the fillet leading to other failure mechanisms such flank fracture [71], which represents damage similar to the tooth tooth bending failure where cracks start far from the root. The shift of the nucleation position to a larger diameter is a typical phenomenon of the case-hardened and the nitrided gears [10] where the increased material performances in the fillet region act as the compressive residual stresses in the shot peening, protecting the material from incipient damages. A suggestion could be to extend the shot-peened area as much as possible and also outside the fillet region.