## 4 Results and Validation
### Determination of CKH with Pointwise Fitting  
The determination of CKH parameters was also carried out using a pointwise fitting between the experimental data represented by the stabilized cycles of the strain-controlled tests performed at \(R_{x}\approx-1\) and the modeled curve obtained by Equations (9) and (11), with three backstress components imposing \(\gamma_{3}=0\). The fitting was conducted on cycle I, then validated on cycle II and vice versa. For this operation, the Matlab software was employed and the command fit [50] was engaged, which provides a nonlinear least squares fit with the default Matlab algorithm. The results are shown in Figures 6 and 7. Figure 6: Comparisons between experimental and modeled stabilized cycles of strain-controlled tests using a pointwise fitting: (**a**) cycle I, used for calibration, (**b**) cycle II, used for validation  
Considering Figure 6, whose results present a better forecast than those of Figure 7, the ratcheting was validated with the parameters obtained from the first computation. In this case, an in-depth discussion should occur regarding the validation of the stress-controlled test. Starting from Equation (11), the evolution of the nonlinear backstress components can be obtained, as in Equation (29). \[ \begin{cases}\dot{\chi}_{\rm i}=\frac{\dot{\rm c}_{\rm i}}{\dot{\gamma}_{\rm i}} +(\dot{\chi}_{\rm i,0}-\frac{\dot{\rm c}_{\rm i}}{\dot{\gamma}_{\rm i}})\text{ exp}(-\gamma_{\rm i}(\varepsilon_{\rm p}-\varepsilon_{p,0})&\text{if} \ \ \varepsilon_{\rm p}-\varepsilon_{p,0}>0\\ \dot{\chi}_{\rm i}=-\frac{\dot{\rm c}_{\rm i}}{\dot{\gamma}_{\rm i}}+(\dot{\chi }_{\rm i,0}+\frac{\dot{\rm c}_{\rm i}}{\dot{\gamma}_{\rm i}})\text{ exp}(-\gamma_{\rm i}(\varepsilon_{\rm p,0}-\varepsilon_{p})&\text{if}\ \ \ \varepsilon_{\rm p,0}-\varepsilon_{p}>0\end{cases}  \tag{29}\]  
The maximum value of the first of Equation (29) is given by \(\lim_{\varepsilon_{\rm p}\to\infty}\dot{\chi}_{\rm i}=\chi_{\rm i}^{\rm max}= \frac{\dot{\rm c}_{\rm i}}{\dot{\gamma}_{\rm i}}\), whereas the minimum value of the second of Equation (29) is given by \(\lim_{\varepsilon_{\rm p}\to-\infty}\dot{\chi}_{\rm i}=\chi_{\rm i}^{\rm min} =-\frac{\dot{\rm c}_{\rm i}}{\dot{\gamma}_{\rm i}}\). Some other considerations are needed to analyze the linear backstress, where the maximum and the minimum value reached during the entire cycle loading are provided by \(\chi_{3}^{\rm max}=\chi_{3,0}+C_{3}\left(\varepsilon_{\rm p}^{\rm max}- \varepsilon_{\rm p,0}\right)\) and \(\chi_{3}^{\rm min}=\chi_{3,0}+C_{3}\left(\varepsilon_{\rm p}^{\rm min}- \varepsilon_{\rm p,0}\right)\), respectively. Recalling Equation (9), the maximum and the minimum values of the stress obtained during the cyclic loading can be expressed by \(\sigma_{xx}^{\rm max}=\sigma_{0}+\lambda_{1}^{\rm max}+\lambda_{2}^{\rm max} +\lambda_{3}^{\rm max}\) and \(\sigma_{xx}^{\rm min}=-\sigma_{0}+\chi_{1}^{\rm min}+\chi_{2}^{\rm min}+\chi_ {3}^{\rm min}\), respectively. It is clear that once the parameters are obtained from the fitting, the values of \(\chi_{1}^{\rm max}\), \(\chi_{2}^{\rm max}\), \(\chi_{1}^{\rm min}\) and \(\chi_{2}^{\rm min}\) can be determined. Regarding stress-controlled tests, the values obtained from \(\sigma_{0}+\lambda_{1}^{\rm max}+\lambda_{2}^{\rm max}\) and from \(\sigma_{0}+\chi_{1}^{\rm min}+\chi_{2}^{\rm min}\) are far from the maximum and the minimum values of the imposed stress, and can only be reached by exploiting the third backstress. Depending on the values of \(C_{3}\) and \(\chi_{3,0}\), these can be obtained at very high plastic strain values. In Figure 8, a piece of the first modeled ramp of the stress-controlled test is shown. The parameters used to reproduce the blue curve in Figure 8 were those obtained by fitting Cycle I.