## 3 Materials and Methods
### Efficient Procedure to Calculate CKH Parameters
\begin{cases}\sigma_{\text{L,l}}=\frac{\Delta\varepsilon_{\text{l}}}{2}-\frac {C_{1}}{\gamma_{1}}\tanh\Big{(}\frac{\gamma_{1}\Delta\varepsilon_{\text{p,l}} }{2}\Big{)}-\frac{C_{2}+C_{3}}{2}\Delta\varepsilon_{\text{p,l}}\\ \sigma_{\text{L,l}}=\frac{\Delta\varepsilon_{\text{l}}}{2}-\frac{C_{1}}{ \gamma_{1}}\tanh\Big{(}\frac{\gamma_{1}\Delta\varepsilon_{\text{p,l}}}{2} \Big{)}-\frac{C_{2}+C_{3}}{2}\Delta\varepsilon_{\text{p,l}}\end{cases}  \tag{19}\]  
In Equation (19), the experimental inputs are \(\Delta\sigma\) and \(\Delta\varepsilon_{\text{p}}\) and, in principle, \(\sigma_{\text{L,l}}\) and \(\sigma_{\text{L,l}}\) should be coincident. As this does not occur in practice, but the value of the elastic limit is unique for the material, it was defined as \(\sigma_{\text{L}}=\frac{\sigma_{\text{L,l}}+\sigma_{\text{L,l}}}{2}\). The last useful property of the stabilized cycle of the strain-controlled test is the hysteresis area, which was modeled, depending on \(\gamma_{1}\) and assuming \(\gamma_{2}\Delta\varepsilon_{\text{p}}\ll 1\), as in Equation (20) for Cycle I and Cycle II, respectively. \[ \begin{cases}A_{\text{I}}^{\text{mod}}=2\sigma_{\text{L}}\Delta\varepsilon_{ \text{p,l}}+2\Big{(}\frac{C_{1}}{\gamma_{1}}\Delta\varepsilon_{\text{p,l}}-2 \frac{C_{1}}{\gamma_{1}^{2}}\tanh\Big{(}\frac{\gamma_{1}\Delta\varepsilon_{ \text{p,l}}}{2}\Big{)}\Big{)}\\ A_{\text{II}}^{\text{mod}}=2\sigma_{\text{L}}\Delta\varepsilon_{\text{p,l}}+2 \Big{(}\frac{C_{1}}{\gamma_{1}}\Delta\varepsilon_{\text{p,l}}-2\frac{C_{1}}{ \gamma_{1}^{2}}\tanh\Big{(}\frac{\gamma_{1}\Delta\varepsilon_{\text{p,l}}}{2} \Big{)}\Big{)}\end{cases}  \tag{20}\]  
The experimental inputs in these latter two equations are the same as those brought into play during elastic limit computation. Different error functions were then introduced:  
* An error function that quantifies the discrepancy between the elastic limits deduced from the two independent cycles: \(\Sigma=\left|\frac{\sigma_{\text{L,l}}-\sigma_{\text{L,l}}}{\sigma_{\text{L,l} }}\right|\)
* The relative error function on the prediction of the hysteresis area of cycle I: \(\Lambda_{\text{I}}=\frac{A_{\text{I}}^{\text{mod}}-A_{\text{I}}}{A_{\text{I}}}\)
* The relative error function on the prediction of the hysteresis area of cycle II: \(\Lambda_{\text{II}}=\frac{A_{\text{II}}^{\text{mod}}-A_{\text{II}}}{A_{\text{II}}}\) These three functions were combined into a global error function defined in (21): \[\psi(\gamma_{1})=(1-\alpha)\Sigma^{2}+\alpha\Big{(}\Lambda_{1}^{2}+\Lambda_{2} ^{2}\Big{)}\] (21) where \(\alpha\) can balance the importance of considering the hysteresis area of the stabilized cycle of strain-controlled tests rather than the amplitude of the stabilized cycle of strain-controlled tests, or vice versa, and the value of \(\alpha\) is imposed in the range \([0,1]\). After obtaining the value of \(\gamma_{1}\) that minimized the global error function in (21), which is indicated with a red arrow in Figure 5, \(C_{1}\), \(C_{2}\) and \(\sigma_{\rm L}\) were updated following Equations (18) and (19). A qualitative graphic of the global error function (21) for different values of \(\alpha\) is reported in Figure 5.  
To calculate the value of \(\gamma_{2}\) the stress-controlled test was exploited and the experimental ratcheting rate was involved. Considering a CKH model with two backstress components, the relationship between \(\sigma_{\rm m}\) and \(\Delta\varepsilon_{\rm p}^{\rm a}\) and \(\Delta\varepsilon_{\rm p}^{\rm r}\) is provided in Equation (22), in accordance with [49]. \[\sigma_{\rm m}=\sum_{i=1}^{2}\frac{C_{\rm i}}{\gamma_{\rm i}}\frac{\sinh\! \left(\gamma_{\rm i}\Delta\varepsilon_{\rm p}^{\rm r}/2\right)}{\sinh\!\left( \gamma_{\rm i}\Delta\varepsilon_{\rm p}^{\rm a}\right)} \tag{22}\]  
As, generally, \(\gamma_{\rm i}\Delta\varepsilon_{\rm p}^{\rm r}\ll 1\) for each backstress component due to the small plastic strain increment per cycle, a simplified form of Equation (22) was used, as in Equation (23).