## 5 Results and Discussion
### Hardening Parameters
Once the parameters of the isotropic hardening model had been determined, we determined the parameters of the kinematic hardening of the Chaboche model. The parameters \(C_{i}\) and \(\gamma_{i}\) were determined by fitting the Equation (16) to the cyclic Ramberg-Osgood Equation (13). The fitting model followed an unconstrained numerical optimization scheme using the least-squares method, such that the objective function needed to be minimized according to the optimization problem in Equation (24). The initial solution was generated using a pseudo-random number generator named Mersenne Twister, which is a 32-bit twisted GFSR random number generator MT19937 [58; 59]. The side bounds for the parameters \(C_{i}\) and \(\gamma_{i}\) were set to limits of \(10^{-7}\) and \(10^{7}\). The Ramberg-Osgood fit curve, \(\mathbf{y}\), was defined up to the limiting condition defined by the strain amplitude for 1 cycle. Different Chaboche models with 1, 2, 3, and 4 back stresses were used and compared in terms of the objective function value. The non-linear models with 4 back stresses and the model with 3 non-linear and 1 linear back stresses were the ones that obtained the best results. Owing to [PERSON]'s statement that 3 non-linear back stresses are sufficient to describe the initial hardening zone, and because only 1 linear back stress is used to describe the hardening for large strains, the model with 3 non-linear and 1 linear stresses was considered.Table 7 presents the values of the parameters obtained for this model. Using the values obtained in \(C_{i}\) and \(\gamma_{i}\), we verified that the model was coherent with other works; that is, \(C_{1}>C_{2}>C_{3}\) and \(\gamma_{1}>\gamma_{2}>\gamma_{3}\) and \(C_{i}>\gamma_{i}\). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(\sigma_{y}\) (MPa) & \(R_{\infty}\) (MPa) & \(b_{\infty}\) & \(\sigma_{y}^{\prime}\) (MPa) & \(n_{a}\) & \(C_{1}\) (MPa) & \(\gamma_{1}\) & \(C_{2}\) (MPa) & \(\gamma_{2}\) & \(C_{3}\) (MPa) & \(\gamma_{3}\) & \(C_{4}\) (MPa) \\ \hline
938.79 & \(-260.90\) & 0.8397 & 578.49 & 4 & 83,773.67 & 582.52 & 16,778.05 & 101.71 & 3013.29 & 15.97 & 854.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Hardening parameters that defined the [PERSON] isotropic and kinematic hardening model. Figure 10: Evolution of the amount of cyclic softening with the total strain amplitude applied. Notice that in Figure 11-right, there was a good agreement between the evolution of decomposition components of the back stress with [PERSON]'s model assumptions. The back stresses \(\alpha_{1}\) and \(\alpha_{2}\) made a greater contribution to the initial hardening of the model up to 2% plastic deformation. On the other hand, \(\alpha_{3}\) contributed gradually until a value of 18% plastic strain, saturating from there. Last, the linear component, \(\alpha_{4}\), increased linearly from 0% to 30%, and then \(\alpha_{4}\) had a higher contribution in the large strain region. Figure 11-left compares the decomposition of the evolution of the cyclic yield stress, \(\sigma_{y}\), given by the sum of the back stress, \(\alpha\), with the size of the elasticity limit surface, \(R^{\prime}_{y}\), validating the fit of the Chaboche model to that of Ramberg-Osgood.