## 4 Results and Validation
### Introduction of the Fourth Backstress  
To better reproduce the zone near the elastic limit, a fourth backstress, which is more rapid than the first, was added and calibrated on Cycle II. In Figure 12, \(\epsilon_{\rm B}\) and \(\epsilon_{\rm D}\) indicate the strains that correspond to the elastic limit for the modeled curve; then, \(\epsilon_{\rm A}=\epsilon_{\rm B}\), \(\epsilon_{\rm C}=\epsilon_{\rm D}\), \(\sigma_{\rm A}\
eq\sigma_{\rm B}\) and \(\sigma_{\rm C}\
eq\sigma_{\rm D}\). After a high value of \(\gamma_{4}\) was imposed, the aim was to find the ratio of \(C_{4}\) and \(\gamma_{4}\), which minimized the error function defined in (30). This is equal to the imposition of a coincident point A with point B, and point C with point D, in Figure 12.  
\[\varphi\bigg{(}\frac{C_{4}}{\gamma_{4}}\bigg{)}=\left|\sigma_{\rm A}-\sigma_{ \rm B}\right|^{2}+\left|\sigma_{\rm C}-\sigma_{\rm D}\right|^{2} \tag{30}\]  
After the introduction of the fourth backstress, the value of \(\sigma_{\rm L}\) must be updated according to Equation (19); thus, the value of \(\sigma_{\rm L}^{{}^{\prime}}\) is ultimately given by (31)  
\[\sigma_{\rm L}^{{}^{\prime}}=\sigma_{\rm L}-\frac{C_{4}}{\gamma_{4}} \tag{31}\]  
Another interesting consideration about the evolution of the modeled hysteresis area of the stabilized cycle of strain-controlled tests, due to the introduction of the fourth backstress, is explained in Figure 13a where the hysteresis area decreases with the increasing value of \(\frac{C_{4}}{\gamma_{4}}\). Observing Equation (20) and updating the value of \(\sigma_{\rm L}\) following (31), it is evident that the area must decrease. Nevertheless, the prediction of the cycle amplitude, provided by Equation (19), remains unchanged because of the high value of \(\gamma_{4}\). In Figure 13b, a comparison between the experimental Cycle II and its modelling is shown. Dashed line was obtained through the procedure with the three backstress components, while the solid line was obtained by adding the fourth backstress component. Figure 13b highlights the improvement provided by the fourth backstress near the elastic limit zone. Figure 12: Graphical explanation of the procedure used to introduce the fourth backstress component. \begin{table}
\begin{tabular}{c c c c c c} \hline \(C_{1}\) (MPa) & \(\gamma_{1}\) & \(C_{2}\) (MPa) & \(\gamma_{2}\) & \(C_{3}\) (MPa) & \(\chi_{3,0}\) (MPa) & \(\sigma_{\rm L}\) (MPa) \\ \hline \(69.21\times 10^{3}\) & 426 & \(2.836\times 10^{3}\) & 4.63 & \(2.669\times 10^{3}\) & \(-4.86\) & 316 \\ \hline \end{tabular}
\end{table} 
Table 3: Parameters obtained following the proposed procedure for CKH model with three backstress components. Definitive comparisons between the simulated stabilized cycles (with four backstress components) and the experimental stabilized cycles of strain-controlled tests performed at \(R_{\epsilon}\approx-1\) and \(R_{\epsilon}\
eq-1\), are shown in Figure 14.  
The addition of the fourth backstress did not change the constant values reported in Table 3, except for the new elastic limit \(\sigma_{\rm L}^{{}^{\prime}}\). The numerical values of the parameters are all gathered in Table 4. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(C_{1}\) (MPa) & \(\gamma_{1}\) & \(C_{2}\) (MPa) & \(\gamma_{2}\) & \(C_{3}\) (MPa) & \(\chi_{3,0}\) (MPa) & \(\sigma_{\rm L}^{{}^{\prime}}\) (MPa) & \(C_{4}\) (MPa) & \(\gamma_{4}\) \\ \hline \(69.21\times 10^{3}\) & 426 & \(2.836\times 10^{3}\) & 4.63 & \(2.669\times 10^{3}\) & \(-4.86\) & 240 & \(381.8\times 10^{3}\) & \(5\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Numerical values of the parameters obtained by following the proposed procedure for CKH model with four backstress components  
Figure 14: Comparisons between the experimental and simulated stabilized cycles after adding the fourth backstress component for Cycle I, Cycle II and Cycle III. Figure 13: (**a**) Evolution of the modeled hysteresis area of stabilized cycle with respect to a change in \(\frac{C_{4}}{\gamma_{4}}\).