## 3 Results
### Arrhenius Constitutive Model
#### 3.6.2 Compensation of Strain Effect  
The magnitude of strain has a noticeable impact on the material constants. Hence, it is imperative to take into account the influence of strain while figuring out flow stress. The material constants were obtained for flow stress values in strain ranging from 0.05 to 0.6. The constants (\(\alpha\), \(n_{2}\), \(Q_{3}\) and \(\ln(A_{3})\)) strain dependances are illustrated in Figure 8. The values of the constants (\(\alpha\), \(n_{2}\), \(Q_{3}\) and \(\ln(A_{3})\)) were obtained through the utilization of polynomial functions of strain, as described in Equation (16). The coefficients of Equation (16) are listed in Table 4. The dependances of the constants on strain were fitted using third-order polynomials fit based on the Adj. R-Square values (Table 4); there were no significant differences in the Adj. R-Square values when using higher order polynomials, such as fifth-order polynomials (Table 4). The number of constants when using the fifth-order polynomials is more than that of third-order polynomials fitting. Thus, the third-order polynomials exhibited the required accuracy with a lower number of constants than in fifth-order polynomials.  
\[\left\{\begin{array}{l}\alpha=Y_{0}+B_{1}\epsilon^{1}+B_{2}\epsilon^{2}+B_{3 }\epsilon^{3}\\ n_{2}=Y_{0}+B_{1}\epsilon^{1}+B_{2}\epsilon^{2}+B_{3}\epsilon^{3}\\ A_{3}=Y_{0}+B_{1}\epsilon^{1}+B_{2}\epsilon^{2}+B_{3}\epsilon^{3}\\ Q_{3}=Y_{0}+B_{1}\epsilon^{1}+B_{2}\epsilon^{2}+B_{3}\epsilon^{3}\\ \end{array}\right. \tag{16}\]  
Figure 8: The dependence of (**a**) \(Q\), (**b**) \(n\), (**c**) \(\ln(A_{3})\), and (**d**) \(\alpha\) constants vs. true strain (\(\epsilon\)). The square box symbols are the calculated constants, and the red lines are the third-order polynomials fits.  
In order to assess the appropriateness of the acquired constitutive model, the correlation coefficient (R), average absolute relative error (AARE), and root mean square error (RMSE) were calculated using Equations (17)-(19) and accounted for 0.93, 6.6%, and 1.15, respectively. The comparison between the experimental and predicted values of flow stress is illustrated in Figure 9.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Parameter** & \(Y_{0}\) & \(B_{1}\) & \(B_{2}\) & \(B_{3}\) & **Adj. R-Square (Third Order)** & **Adj. R-Square (Fifth Order)** \\ \hline \(\alpha\) & 0.161 & \(-\)0.729 & 1.792 & \(-\)1.478 & 0.96 & 0.99 \\ \hline \(n_{2}\) & 2.176 & \(-\)2.210 & 16.755 & \(-\)20.177 & 0.98 & 0.98 \\ \hline \(\ln(A_{\,3})\) & 22.178 & \(-\)64.456 & 190.923 & \(-\)200.369 & 0.97 & 0.99 \\ \hline \(\mathcal{Q}\) & 103.819 & \(-\)394.307 & 1143.484 & \(-\)1192.050 & 0.98 & 0.99 \\ \hline \end{tabular}
\end{table}
Table 4: Coefficients in Equation (20).  
Figure 9: The experimental and the predicted stress–strain from the constructed model at different temperatures: (**a**) 460 \({}^{\circ}\)C, (**b**) 480 \({}^{\circ}\)C, and (**c**) 500 \({}^{\circ}\)C; and (**d**) the verification of the constitutive equation (through points) between experimental and theoretical data. The square box symbols in (**d**) are the experimental data, and the red lines in (**d**) are the linear fits.  
The patterns collected from the data demonstrate a satisfactory level of consistency between the experimental results and support the model's high level of prediction (Figure 9d).  
\[R=\frac{\sum_{i=1}^{N}(E_{i}-\overline{E})\left(P_{i}-\overline{P}\right)}{\sqrt{ \sum_{i=1}^{N}\left(E_{i}-\overline{E}\right)^{2}\sum_{i=1}^{N}\left(P_{i}- \overline{P}\right)^{2}}} \tag{17}\]  
\[AARE=\frac{1}{N}\overline{\sum_{i=1}^{N}\left|\frac{E_{i}-P_{i}}{E_{i}}\right|} \tag{18}\]  
\[RMSE=\sqrt{\frac{1}{N}\overline{\sum_{i=1}^{N}\left(E_{i}-P_{i}\right)^{2}}} \tag{19}\]