## 2 Materials and Methods
### Finite Element Simulations
#### 2.1.1 Shot Peening
While a numerical approach provides an effective way for quantitatively determining relationships between shot and target parameters and residual stress characteristics, it cannot model a full stream of shots without an unmanageable computation effort. Some scholars [20; 31; 32; 34; 36] simulated multiple impacts, but not a realistic stream of shots: the pattern was specified a priori while the interaction between the spheres was not taken into consideration. Alternatively to FE, the Discrete Element Method (DEM) could be employed for the modelling of the shoot peening [37]. Some scholars demonstrated the possibility of combining FEM and DEM [38]. In the latter, the interaction between the spheres was extensively studied. Numerical studies dealing with the impact of the shot diameter, the friction and sphere size were studied by different scholars [39; 40; 41]. The shot impact locations were thus assumed a priori and the stream of shots was not modelled. In the present paper, the shot peening process was computationally modelled assuming a random (a priori) distribution of the shots. The shots were modelled as infinitely rigid. Their diameter was assumed to be \(d=0.8\) mm. The speed of the shots was assumed to be \(v=50\) m/s. For this purpose, a non-linear dynamic explicit algorithm was used. The direct method consists of solving the problem resulting from the discretization by finite elements of displacement's formulation. The discretization of the virtual work of the inertial forces, in a field \(\delta\vec{v}\in\hat{V}^{0}_{adm}\) is written as:  
\[\int_{\Omega}\rho\vec{u}\delta\vec{v}d\Omega=\delta\vec{V}^{T}[M]\vec{U} \tag{2}\]  
Figure 1: Mesh: about 1M second-order wedge elements. where \(\vec{u}\) is the field absolute displacement, \([M]\) the matrix of inertia and \(\vec{U}\) and \(\vec{\dot{U}}\) the vector degrees of freedom its second-order derivative. The discretization of the virtual work of the interior efforts in linear elasticity is written:  
\[\int_{\Omega}\varepsilon(\vec{u})[A]\varepsilon(\delta\vec{v})d\Omega=\delta \vec{V}^{T}[K]\vec{U} \tag{3}\]  
\([K]\) is the stiffness matrix. Lastly, \([L]\) designates the second member resulting from the discretization of the virtual work of the external forces. For \(\vec{U}\in\mathbb{R}\)  
\[[M]\vec{\dot{U}}+[K]\vec{U}=[L] \tag{4}\]  
accompanied by boundary conditions. For all non-linear and dynamic analyses, incremental loads are needed. Implicit or explicit methods can be used to solve these problems. Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later state. For the time integration, the centred differences were used. This is an explicit scheme. This scheme, contrary to the implicit one, is conditionally stable. In this method, the accelerations and velocities are approximated in terms of the displacements using the finite difference expressions and substituted in the equation of motion in order to solve for the displacements. The final expression for the equation of motion in terms of displacement alone was obtained as:  
\[\Big{(}\frac{1}{\Delta t^{2}}[M]\Big{)}_{|t-\Delta t}\vec{V}=[L]^{T}-\Big{(}[ K]-\frac{2}{\Delta t^{2}}M\Big{)}\vec{\dot{U}}^{T}-\Big{(}\frac{1}{\Delta t^{2}} M\Big{)}\vec{\dot{U}}_{|t-\Delta t} \tag{5}\]  
from which it is possible to solve for \(\vec{\dot{U}}_{|t+\Delta t}\). The initial time step was set to \(1.3\cdot 10^{-8}\) s but, during the simulation, it was reduced up to \(4.0\cdot 10^{-12}\) s. The simulation was stopped after \(1\cdot 10^{-4}\) s when all the spheres had hit the surface. Through the finite element method, the elastic-plastic dynamic process of shot peening on the gear-sample was simulated.