## 3 Results and Discussion
### Flow Stress Behavior Modeling Experiments
#### 3.3.1 ACE Model  
As follows (Equation (4)), the flow stress (\(\sigma\)) varies with the deformation temperature (\(T\)), strain rate (\(\dot{\epsilon}\)), and strain (\(\epsilon\)). \[\sigma=f\big{(}T,\dot{\epsilon},\ \dot{\epsilon}\big{)} \tag{4}\]  
In an exponent-type equation, the Zener-Holloman parameter (\(Z\)) was used to identify the relationship between \(T\), \(\dot{\epsilon}\) and \(\epsilon\) (Equations (5) and (6)) [67, 68]. The equation constants are collected in Table 1.  
\[Z=\ \dot{\epsilon}(exp^{(\frac{Q}{RT})}) \tag{5}\]  
\[\dot{\epsilon}=Af(\sigma)\Big{(}exp^{(-\frac{Q}{RT})}) \tag{6}\]  
Figure 4: Experimental flow behavior at temperatures of (**a**) 420 \({}^{\circ}\)C, (**b**) 460 \({}^{\circ}\)C, (**c**) 500 \({}^{\circ}\)C, and (**d**) 540 \({}^{\circ}\)C with various constant strain rates. In general, due to its high efficiency in large ranges of stress levels, the hyperbolic sine law was used to create the proposed constitutive model [69]. Constants Determination  
An example was taken to implement the solution procedures to determine the equation constants at a strain of 50%. By taking the logarithm of both sides of Equation (6), the following Equations (7)-(9) were expressed. \[\ln\ \dot{\varepsilon}=\ln\ A_{1}+\ n_{1}\ln\sigma-\ \frac{Q_{1}}{RT} \tag{7}\]  
\[\ln\ \dot{\varepsilon}=\ln\ A_{2}+\beta\sigma-\ \frac{Q_{2}}{RT}\ \Rightarrow \tag{8}\]  
\[\ln\ \dot{\varepsilon}=\ln\ A_{3}+\ n_{2}\ln\sinh(\alpha\sigma)\ -\frac{Q_{3}}{RT} \tag{9}\]  
By partial differentiation of Equations (7)-(9) and Equations (7)-(15) were derived. \[n_{1}=\ \left[\frac{\partial\ln\dot{\varepsilon}}{\partial\ln\sigma}\right]_{T} \tag{10}\]  
\[\beta=\ \left[\frac{\partial\ln\dot{\varepsilon}}{\partial\sigma}\right]_{T} \tag{11}\]  
Table 1.0  
Table 1.1  
Table 1.2  
Table 1.3  
Figure 5 illustrates the \(\ln\dot{\varepsilon}-\ln\sigma\) (Figure 5a), \(\ln\dot{\varepsilon}-\sigma\) (Figure 5b), \(\ln\dot{\varepsilon}-\ln\sinh(\alpha\sigma)\) (Figure 5c), and \(\ln\sinh(\alpha\sigma)-\frac{1000}{RT}\) (Figure 5d) linear plots. The \(n_{1}\), \(\beta\), and \(Q_{1,2}\) constants were computed as the correlation coefficients of the slopes of the \(\ln\dot{\varepsilon}-\ln\sigma\) and \(\ln\dot{\varepsilon}-\sigma\) plots, respectively (Figure 5a,b). While \(n_{2}\) and \(Q_{3}\) were obtained from the slope of the plots of \(\ln\dot{\varepsilon}-\ln\sinh(\alpha\sigma)\) and \(\ln\sinh(\alpha\sigma)-\frac{1000}{RT}\) (Figure 5c,d). The equations constants at a strain of 50% for the investigated alloy are collected in Table 1.4 \begin{table}
\begin{tabular}{c l} \hline A\({}_{1,2,3}\), \(\alpha\) (\(\alpha=\beta/n_{1}\)), \(\beta\), \(n_{1}\), \(n_{2}\) & the material constants that vary with the strain \\ \hline Q\({}_{1,2,3}\) & the activation energy of the deformation, J/mol \\ \hline T & the absolute temperature, K \\ \hline R & the universal gas constant, 8.314 J/(mol.K) \\ \hline \end{tabular}
\end{table} 
Table 1: The equations constants. The strain rate and the flow stress at a strain of 50% dependent on the hyperbolic sine law can be calculated as follows (Equations (16) and (17)) once the ACE constants were determined:  
\[\dot{\epsilon}=A_{3}[\sinh\!\left(\alpha\sigma\right)]^{n_{2}}exp^{\left(-\frac{ Q}{RT}\right)}=\ 12160\ [\sinh\!\left(0.04\sigma\right)]^{1.99}exp^{\left(-\frac{86\times 1 000}{RT}\right)} \tag{16}\]  
\[\sigma=\frac{1}{\alpha}\ln\!\left\{\left(\frac{z}{A_{3}}\right)^{\frac{1}{n_{2 }}}+\ \left[\left(\frac{z}{A_{3}}\right)^{\frac{2}{n_{2}}}+1\right]^{\frac{1}{2}} \right\}=\ \frac{1}{0.04}\ln\!\left\{\left(\frac{\dot{\epsilon}exp^{\frac{86\times 1 0^{3}}{RT}}}{12160}\right)^{\frac{1}{159}}+\ \left[\left(\frac{\dot{\epsilon}exp^{\frac{86\times 1 0^{3}}{RT}}}{12160}\right)^{\frac{2}{159}}+1\right]^{\frac{1}{2}}\right\} \tag{17}\]