## 3 Materials and Methods
### Efficient Procedure to Calculate CKH Parameters
\[\sigma_{\rm m}=\frac{\Delta\varepsilon_{\rm p}^{\rm r}}{2}\sum_{i=1}^{2}\frac{ C_{\rm i}}{\sinh\!\left(\gamma_{\rm i}\Delta\varepsilon_{\rm p}^{\rm a} \right)} \tag{23}\]  
After obtaining \(\gamma_{1}\), Equation (23) can easily be inverted and \(\gamma_{2}\) can be found. It is interesting to highlight that, according to Equation (22), plastic shakedown \(\left(\Delta\varepsilon_{\rm p}^{\rm r}=0\right)\) can occur only for null values of the average stress. The superscript of the sum in Equation (23) is 2, but the model involved in the work employed three backstress components. The maximum value of the third backstress component, due to its linear trend, evolves cycle per cycle, as in Equation (24). \[\chi_{3,\,{\rm i+1}}^{\rm max}=\chi_{3,\,{\rm i}}^{\rm max}+C_{3}\Delta \varepsilon_{\rm p}^{\rm r} \tag{24}\]  
According to Equation (24), it is clear that the only achievable equilibrium, according to the CKH with three backstress components, is given by \(\Delta\varepsilon_{\rm p}^{\rm r}=0\), which means that only plastic shakedown can occur as equilibrium. Therefore, \(\Delta\varepsilon_{\rm p}^{\rm a}\) and \(\Delta\varepsilon_{\rm p}^{\rm r}\) cannot be considered constant anymore. In Figure 4, it is qualitatively shown, according to the full demonstration in [49], that the maximum and the minimum values of the first and second backstress components are exactly opposite for the stabilized cycle of a strain-controlled test. This last statement can be further extended to stress-controlled tests [49]. The average stress of the stabilized cycle of the stress-controlled test can be related to its corresponding average  
Figure 5: Qualitative trend of \(\psi(\gamma_{1})\) for different values of \(\alpha\). plastic strain, similarly to Equation (17), and the average plastic strain value of the stabilized cycle is formalized in Equation (25). \[\varepsilon_{\rm P}^{\rm m}=\frac{\sigma_{\rm m}-\chi_{3,0}}{C_{3}} \tag{25}\]  
If the value of \(C_{3}\) is high, then the average plastic strain of the stabilized cycle is low, and can be reached after few cycles. Despite this, by exploiting a fully relaxed stabilized cycle arising from a test at \(R_{\varepsilon}\
eq-1\), the obtained value of \(C_{3}\) is generally smaller than the values of \(C_{1}\) and \(C_{2}\); due to this, a certain number of cycles are needed to reach the plastic shakedown. Therefore, \(\Delta_{\rm p}^{\rm a}\) and \(\Delta c_{\rm P}^{\rm r}\) can still be considered constant without any loss of accuracy, and Equation (23) continues to be a valid expression to calculate the value of \(\gamma_{2}\). The determination of the parameters of the CKH model with three backstress components, according to the proposed procedure, is finally resumed in the steps below:  
* The determination of \(C_{3}\) and \(\chi_{3,0}\), extracted from the average stress and average plastic strain of the experimental stabilized cycles of two strain-controlled tests;
* The slope at the inversion points was used to determine \(C_{1}\) and \(C_{2}\);
* The elastic limit was computed as the average of the elastic limit of Cycle I and Cycle II, exploiting the stress amplitude and plastic strain amplitude of the stabilized cycles of strain-controlled tests;
* The hysteresis areas of the stabilized cycles I and II were modeled and used the same experimental inputs as the elastic limit computation;
* Three error functions based on the elastic limit and the hysteresis area of the stabilized cycle I and II were introduced and combined in a global error function (21);
* The value of \(\gamma_{1}\) was found to be that which minimized the global error function (21), and \(C_{1}\), \(C_{2}\) and \(\sigma_{\rm L}\) were then updated following Equations (18) and (19);
* \(\gamma_{2}\) was the last parameter to be computed by exploiting the experimental ratcheting rate and according to Equation (23). The first step considered one strain-controlled test executed at \(R_{\varepsilon}\
eq-1\) and one at \(R_{\varepsilon}\approx-1\) (Loading II).