## 5 Conclusions  
In this work, we used a novel procedure to identify CIKH model parameters. Three strain-controlled tests and one stress-controlled test were used to implement the procedure. Two strain-controlled tests were performed at \(R_{e}\approx-1\), whereas the third one was conducted at \(R_{e}\
eq-1\). From the strain-controlled tests, we extracted the stabilized cycles (Cycle I and Cycle II for the strain-controlled tests performed at \(R_{e}\approx-1\) and Cycle III for the strain-controlled test executed at \(R_{e}\
eq-1\)), and we determined the global properties of the stabilized cycles, namely, the slope at the inversion points, the hysteresis area, the stress amplitude, the average stress and the average plastic strain. The experimental stress-controlled test provided the ratcheting rate. The procedure validations are shown in Figures 10 and 11, and an accurate matching can be seen between the modeled curves and the experimental curves. A fourth backstress was added to improve the prediction near the elastic limit, as shown in Figure 14, resulting in a high level of local accuracy. The model was then completed by the determination of the VOC isotropic hardening model parameters, eventually obtaining the CIKH model. A pointwise fitting was used to determine the parameters of the CKH model with three backstress components. Figure 6 can better reproduce the experimental data than Figure 7. This is due to the wider range of experimental total strain, on which the fitting of Figure 6 was developed. However, the parameters obtained by fitting Cycle I cannot reproduce the extreme points of Cycle II, as in Figure 6b, and the numerical values of parameters \(C_{3}\) and \(\gamma_{2}\) are not physically meaningful. Therefore, it was not possible to reproduce the ratcheting rate. The Bouc-Wen model was also considered to reproduce the cyclic plastic behaviour, and the parameters obtained to fit the entire cyclic loading I showed a better ability to reproduce the stabilized cycles of strain-controlled tests than those obtained that fit the entire cyclic loading II and the \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Extracted From** & \(\mathbf{\alpha}\) **(MPa)** & \(\mathbf{\beta}\) **(MPa1\({}^{-n}\))** & \(\mathbf{\gamma}\) **(MPa1\({}^{-n}\))** & \(\mathbf{n}\) & \(\mathbf{\delta}\) **(MPa)** & \(\mathbf{c}\) & \(k_{1}\) **(MPa)** & \(k_{2}\) **(MPa)** & \(k_{3}\) **(MPa)** \\ \hline Strain-controlled test I & 15.4 & 0.391 & 0.184 & 0.858 & \(-\)0.041 & 0 & 0.674 & 0 & 0 \\ Strain-controlled test II & 14.7 & 0.011 & 0.005 & 1.940 & \(-\)0.086 & 0 & 2.94 & 0 & 0 \\ Stress-controlled test & 15.5 & 0.068 & 0.036 & 1.290 & \(-\)0.026 & 0 & 0.479 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Obtained values for the parameters of B-W model. Figure 20: Stress-controlled test used for calibration of B-W model parameters with its corresponding modeled trend. stress-controlled test. This is shown in Figures 17-19, and is due to the wider range of total mechanical strain in Cycle I. The parameters of the Bouc-Wen model, obtained by fitting the stress-controlled test, allow for the reproduction of the ratcheting rate, but not at the quality level obtained by following the proposed procedure. To conclude, the proposed procedure to calculate CIKH model parameters is physics-based, and is only formulated on closed-form expressions, which show how the parameters govern each global property. The equations are easily manageable, whereas, when a pointwise fitting is performed, the optimization dynamic behind the software is not fully understandable. The proposed technique involves four experimental cyclic loadings, which are still a reasonable number to determine the model parameters in an efficient way. This last statement allows for one to cover more experimental cases instead of completing the fitting on only one cycle, which clearly results in a good prediction for that cycle but can lack accuracy if validated on a cycle with a different \(R\) from that used for the calibration.