## 3 Materials and Methods
### Efficient Procedure to Calculate CKH Parameters  
In this work, a CKH model with three backstress components was considered, and the dynamics of each component, according to our procedure for a qualitative loading with \(R_{\rm c_{\rm p}}=0\), are explained in Figure 4, where the stabilized cycles are indicated in red. \(R_{\rm c_{\rm p}}\) is defined as the ratio between the minimum and maximum imposed plastic strain during the plastic strain-controlled test. \(\gamma_{1}\) was taken as the highest gamma value, so the first backstress has a rapid dynamic. As shown in Figure 4a, this is responsible for the hysteresis area of the stabilized cycle of the plastic strain-controlled test. \(\gamma_{2}\) was modeled as being much lower than \(\gamma_{1}\), and the trend of the second backstress is shown in Figure 4b. Its contribution to the hysteresis area of the stabilized cycle is almost null. \(\gamma_{3}\) was imposed as exactly null; thus, according to Equation (11), the third backstress evolves linearly, as in Figure 4c. Note that the quantity of the x-axis of Figure 4 is the plastic strain, since the numerical simulations were plastic-strain-controlled. This last situation is experimentally challenging, and the experimental tests are always strain-controlled. As explained in [49], all the expressions shown in the rest of the paper regarding suggested procedure to identify CKH parameters, which were, in principle, obtained from plastic-strain-controlled tests, were generalized to strain-controlled tests without loss of meaning. To calculate CKH constants, the procedure of [49] is briefly recalled. The values of \(C_{3}\) and \(\chi_{3,0}\) were first determined using the stabilized cycles of strain-controlled tests executed  
Figure 4: Qualitative explanation of the behaviour of each backstress component according to the proposed procedure: (**a**) rapid backstress, (**b**) slow backstress, (**c**) linear backstress. at \(R_{\varepsilon}\approx-1\) and \(R_{\varepsilon}\
eq-1\). The linear system that must be solved is Equation (17), where \(\sigma_{\text{m}}\Big{(}\sigma_{\text{m}}=\frac{\sigma_{\text{max}}+\sigma_{ \text{min}}}{2}\Big{)}\) and \(\varepsilon_{\text{p,m}}\Big{(}\varepsilon_{\text{p,m}}=\frac{\varepsilon_{ \text{p,max}}+\varepsilon_{\text{p,min}}}{2}\Big{)}\) are the experimental inputs. \[ \begin{cases}\chi_{3,0}+C_{3}\varepsilon_{\text{p,m,l}}=\sigma_{\text{m,l}}\\ \chi_{3,0}+C_{3}\varepsilon_{\text{p,m,l}}=\sigma_{\text{m,l}}\end{cases}  \tag{17}\]  
Equations (18)-(20), used to model the global properties of the stabilized cycles of strain-controlled tests, have a general meaning and, in this case, they were all applied to the loadings performed at \(R_{\varepsilon}\approx-1\). \(C_{2}\) and \(C_{1}\) were determined as a function of \(\gamma_{1}\), using the slope at the inversion points of the experimental stabilized cycles of strain-controlled tests. Assuming \(\gamma_{2}\Delta\varepsilon_{\text{p}}\ll 1\), which is reasonable considering the low value of \(\gamma_{2}\), the nonlinear system (18) was obtained. \[ \begin{cases}C_{1}\Big{(}1-\tanh\Big{(}\frac{\gamma 1\Delta\varepsilon_{ \text{p,l}}}{2}\Big{)}\Big{)}+C_{2}=-C_{3}+\left.\frac{\text{d}\sigma}{\text{d }\varepsilon_{\text{p}}}\right|_{\sigma=\sigma_{\text{max}}^{\text{sub,l}}}\\ C_{1}\Big{(}1-\tanh\Big{(}\frac{\gamma 1\Delta\varepsilon_{\text{p,l}}}{2}\Big{)}\Big{)}+C_{2}=-C_{3}+\left.\frac{ \text{d}\sigma}{\text{d}\varepsilon_{\text{p}}}\right|_{\sigma=\sigma_{\text {max}}^{\text{sub,l}}}\end{cases}  \tag{18}\]  
In Equation (18), the values of \(\left.\frac{\text{d}\sigma}{\text{d}\varepsilon_{\text{p}}}\right|_{\sigma= \sigma_{\text{max}}^{\text{sub}}}\) and \(\Delta\varepsilon_{\text{p}}\) are the experimental inputs. The elastic limit was also determined as a function of \(\gamma_{1}\), in Equation (19), again assuming \(\gamma_{2}\Delta\varepsilon_{\text{p}}\ll 1\). \[