## 3 Materials and Methods  
### Experimental Data  
The material involved in this work was 42 NiCrMo4 quenched and tempered steel. The technical drawing of the plain specimen engaged to perform both strain-controlled tests and stress-controlled test is shown in Figure 1, where dimensions and tolerances are reported. The experimental data were sampled with a frequency of 10 Hz, and the strain-rate was approximately \(\pm 2.5\times 10^{-3}\) s\({}^{-1}\). The yield and the ultimate strengths, extracted from the tensile test performed according to ASTM E8/E8M-11 [45], are \(S_{\mathrm{Y}}=500\) MPa and \(S_{\mathrm{U}}=700\) MPa, respectively. \(R_{e}\) is defined as the ratio between the minimum and the maximum imposed total strain during the strain-controlled test. Regarding the strain-controlled tests performed at \(R_{e}\approx-1\), the sampling was continuous due to the relatively low number of cycles, whereas, during the strain-controlled test executed at \(R_{e}\
eq-1\), the sampling was carried out by blocks, as the number of cycles was much higher. The choise of material investigated in this study does not represent any limitation to the proposed procedure, as no hypothesis about the material was introduced during the procedure and the CIKH model is generally engaged to model the plasticity of metals. The proposed technique can be extended to any metal with a sufficient ductile behaviour. Three strain-controlled tests and one stress-controlled test were performed on plain specimens. Two of the strain-controlled tests were conducted at \(R_{e}\approx-1\) and are shown in Figure 2, where the stabilized cycles are indicated in red (Cycle I and Cycle II), while the rest of the loading history is indicated in grey. Some useful experimental data about the stabilized cycles Cycle I and Cycle II, which are needed to determine the constants of the CKH model, are reported in Table 1. The extreme stress values of the stabilized cycles of strain-controlled tests are indicated with \(\sigma_{\mathrm{max}}^{\mathrm{stab}}\) and \(\sigma_{\mathrm{min}}^{\mathrm{stab}}\). Figure 1: Technical drawing of the specimen used to perform both strain-controlled tests and stress-controlled test. In Figure 3a, a qualitative trend for a stress-controlled test on a plain specimen is reported in the \(\sigma\)-\(\epsilon_{\text{p}}\) coordinates, and the meaningful experimental inputs regarding the stress-controlled test are indicated. The red line in Figure 3a indicates an open ratcheting cycle, which ranges from the maximum plastic strain of cycle \(N\) to the minimum plastic strain, and then again reaches the maximum plastic strain of cycle \(N+1\). Using the experimental inputs from the stress-controlled test, two important quantities for further analysis are defined in Equations (15) and (16). \[\Delta\varepsilon_{\text{p,}N}^{\text{a}}=\frac{\epsilon_{\text{p,}N+1}^{\text {max}}+\epsilon_{\text{p,}N}^{\text{max}}}{\epsilon_{\text{p,}N}^{\text{min}}} -\epsilon_{\text{p,}N}^{\text{min}} \tag{15}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stabilized** & \(\Delta\varepsilon_{\text{p}}\) & \(\Delta\sigma\) **(MPa)** & \(A\) **(m)/mm\({}^{3}\)** & \(\frac{\text{d}\varepsilon}{\text{d}\varepsilon_{\text{p}}}\Big{|}_{\sigma=\sigma _{\text{min}}^{\text{min}}}\) **(GPa)** \\ \hline Cycle I & 1.43\% & \(1.03\times 10^{3}\) & 12.0 & 5.81 \\ Cycle II & 0.50\% & 918 & 3.61 & 20.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Useful experimental data extracted from Cycle I and Cycle II. Figure 3: (**a**) Example of a stress-controlled test for a plain specimen with meaningful quantities reported; (**b**) experimental trend of the maximum and minimum plastic strain per cycle of the stress-controlled test involved in the work. Figure 2: Strain-controlled tests performed at \(R_{\varepsilon}\approx-1\): (**a**) loading I, (**b**) loading II.