## 2 Materials and Methods
### Finite Element Simulations
#### 2.1.1 Shot Peening  
Before running the simulation of the fatigue tests, a numerical simulation of the shot peening process was performed. Simulating the shot peening process is a very challenging task that involves dynamic analysis of moving parts (the shots--which have been modelled as rigid in the present study) impacting the metallic component to be shot peened. The number of parameters that will affect the shot peening results is quite high and frequently concealed by the industrial secrets of the manufacturer. The most relevant parameters include the size of the spheres, their density, the shooting velocity, the angle of impact, the hardness, the level of coverage, etc. By acting on these parameters, it is possible to affect the properties of the shot-peened component such as the initial yield stress, the work-hardening characteristics, the surface hardness, strain-rate dependence, etc. In order to gain better control over the residual stresses after peening, it is important to establish a relationship between these parameters and residual stress characteristics. In the past, this was achieved with empirical relations only, based on experimental evidence. Nowadays, with the increasing computational resources, the numerical approaches to this problem are becoming increasingly widespread (Figure 1). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Normal module [mm] & \(m_{n}\) & 5 \\ Normal pressure angle [\({}^{\circ}\)] & \(\alpha_{n}\) & 20 \\ Number of teeth & \(z\) & 24 \\ Face width [mm] & \(b\) & 30 \\ Profile shift coefficient & \(x^{*}\) & 0 \\ Dedendum coefficient & \(h^{*}_{P}\) & 1.25 \\ Addendum coefficient & \(h^{*}_{aP}\) & 1 \\ Root radius factor & \(\rho^{*}_{P}\) & 0.38 \\ Wildhaber & \(W\) & 3 \\ Angle STBF [\({}^{\circ}\)] & \(\alpha_{Pen}\) & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Gear geometry. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Pure Bending Fatigue Limit** & **Pure Torsion Fatigue Limit** & **Ultimate Tensile Strength** \\ \(\sigma_{f}\) & \(\tau_{f}\) & \(UTS\) \\ \hline
410 MPa & 256 MPa & 795 MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties. Considering that measuring the residual stresses could be very time demanding, establishing a relationship between the shot peening parameters and the induced stress fields would require an immense effort. In this regard, the Finite Element Method (FEM) represents a valid and more effective alternative to better understand this relationship. The impact of the shoot peens having a high velocity and double non-linearity of the problem (related on the one side to the contact of two bodies and on the other side of the elastic-plastic behaviour of the material) can be easily included in a unique numerical model capable of precisely determining the residual stress field for that configuration. [PERSON] et al. [29] was among the first to simulate the problem of a rigid body indenting an elastic-perfect plastic plate via FEM. An example of analysis exploiting a commercial FE software was shown by [PERSON] et al. [30]. [PERSON] et al. simulated the impact of a single sphere on visco-plastic and elasto-plastic materials. However, the combinations of parameters analysed did not represent the typical range used in modern shot peening treatments. More complete studies were performed in the late 90s by [PERSON] [31] and [PERSON] et al. [32]. Other examples of Finite Element simulations of shot peening considering the deformability of the shots were performed by [PERSON] and [PERSON] [33; 34]. In their studies, [PERSON] and [PERSON] highlighted the poor accuracy in predicting the stress strain field of the empirical models. Another paper dealing with shot peening and its simulation with FEM was published by [PERSON] et al. [35; 36]. They conducted a dynamic FE analysis of single and multiple shot impacts, investigating the effect of some shot and target parameters.