## 5 Results and Discussion
### Hardening Parameters  
Isotropic and kinematic hardening parameters were determined. The unconstrained optimization model presented in Equation (24) was used to determine the regression parameters of the hardening models. Regarding the isotropic saturation rate parameter, \(\hat{b}_{\infty}\), it is advisable to determine it according to the recommendations in [29; 57]. Assuming a response parameter estimated as \(\hat{\mathbf{y}}_{i}=1+\exp(-\hat{b}_{\infty,i}\mathbf{P}_{i})\) and \(\mathbf{y}_{i}=R_{i}/R_{\infty}\), Figure 9 shows the normalized maximum stress variation throughout the cyclic test up to the half-life in the scheme presented in [29]. The average curve and the respective 1- and 2-times standard deviations are presented. The experimental results showed a good behavior throughout the entire test in all applied strain ranges. Comparing the average curve with the data obtained experimentally, we can see that there is a large variation between the average curve and the experimental data. This difference was greater in the initial test cycles. However, for higher strain gamma levels, the regression curve tends to describe the isotropic hardening behavior, as presented by the experimental data. Regarding the threshold saturation zone, it appears that there was not an exact saturation value for the experimental data, and therefore the isotropic model was unable to predict the real phenomenon.The value found by the optimization procedure was 0.8397, which is consistent with the values usually found for a wide range of materials, between 50 and 0.5 [57]. Regarding the variation of the amount of cyclic softening, Equation (20) was used to determine \(R_{\infty,max}\). The amount of change in the size of the stabilized yield strength surface for a given deformation, \(R_{\infty}\), was normalized using the yield strength, \(R_{\infty}^{*}\). Thus, we defined the response, which was estimated as \(\hat{\mathbf{y}}_{i}=\hat{R}_{\infty,max}^{*}\big{(}1-\exp\big{[}\hat{k}_{0}( \varepsilon_{a,j}-\hat{\varepsilon}_{0})\big{]}\big{)}\) and \(\mathbf{y}_{i}=R_{\infty}^{*}=|R_{\infty,i}|/\sigma_{y,j}\). The monotonic yield strength, \(\sigma_{y}\), considered was given by the arithmetic mean between \(\sigma_{p}\) and \(\sigma_{02}\). The regressors \(R_{\infty,max}\), \(k_{0}\) and \(\varepsilon_{0}\) were determined according to the unconstrained optimization scheme, with side bounds encompassing a pool permissible solution for the initial design vector of regressors, \(\mathbf{x}=\{R_{\infty,max},k_{0},\varepsilon_{0}\}\). The optimal solution was assumed to be that which had the lowest value of the objective function. Figure 10 presents the regression obtained using the power model of Equation (20). According to Figure 10, the material began to exhibit a cyclic softening behavior for higher strain amplitudes than the threshold \(\hat{\varepsilon}_{0}=0.2120\%\). Comparing the corresponding stress for \(\hat{\varepsilon}_{0}\) with the cyclic proportional stress value \(\sigma_{p}^{\prime}\), and cyclic yield strength, \(\sigma_{y}^{\prime}\), resulted  
Figure 9: Evolution of the normalized maximum stress during the cyclic test, until half of the total life was reached. Assessment of the isotropic model given by [PERSON] using the estimators. in differences of only 0.33 and 14.43%, respectively. The amount of softening tended to increase for strain amplitudes around 0.7 and 0.9%. This increase was given by a multiplicative factor, \(k_{0}=-466.01\). For higher strain amplitude levels than 0.9%, the softening gradient decreased significantly, until it stabilized. The limiting value obtained using the optimization procedure was \(\hat{R}_{\infty,max}^{*}=27.92\%\), which corresponds to \(-260.90\) MPa. This value, \(\hat{R}_{\infty,max}^{*}\), corresponds to a strain amplitude level of 30.49%, which is the strain level for a lifetime of 2 reversals, as previously shown in Table 5. Table 7 presents the parameters determined for the isotropic hardening model.