## 3 Results and Discussion
### Flow Stress Behavior Modeling Experiments
#### 3.3.1 ACE Model
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**In(A\({}_{1}\))** & **n\({}_{1}\)/m * &
\begin{tabular}{c} **Q\({}_{1}\)** \\ **[K]/mol]** \\ \end{tabular} & **In(A\({}_{2}\))** & \(\beta\) & **Q\({}_{2}\)** & **Q\({}_{3}\)** \\  & & **[KJ/mol]** & **[MPa\({}^{-1}\)]** & **[kJ/mol]** & **\(\alpha\)** & **In(A\({}_{3}\))** & **n\({}_{2}\)** & **[kJ/mol]** \\ \hline
1.6 & 2.6/0.4 & 86 \(\pm\) 5 & 6.1 & 0.11 & 80 \(\pm\) 5 & 0.04 & 9.4 & 1.99 & 86 \(\pm\) 5 \\ \hline \hline \end{tabular}
* The strain-rate sensitivity (m) was calculated by \(m=\frac{1}{n_{1}}\).  
\end{table} 
Table 2: The determined constants of the Arrhenius-type constitutive hyperbolic-sine equation model (ACE) at a strain of 50% of the investigated alloy. To model the flow behavior, the strain impact on the ACE model constants should be regarded. Thus, the ACE model constants \(\alpha\), \(Q_{3}\), \(ln(A_{3})\), and \(n_{2}\) versus a different strain were computed (Figure 6a-d). The 5 th polynomial order was the best relevant order, depending on the determination coefficient (R\({}^{2}\)), for all ACE model constants, the polynomial regression coefficients were as follows (Equations (18)-(21)):  
\[\alpha=\text{a}+\text{b}\left(\varepsilon\right)+\text{c}\left(\varepsilon^{2 }\right)+\text{d}\left(\varepsilon^{3}\right)+\text{e}\left(\varepsilon^{4} \right)+\text{f}\left(\varepsilon^{5}\right) \tag{18}\]  
\[\text{Q}_{3}=\text{a}+\text{b}\left(\varepsilon\right)+\text{c}\left( \varepsilon^{2}\right)+\text{d}\left(\varepsilon^{3}\right)+\text{e}\left( \varepsilon^{4}\right)+\text{f}\left(\varepsilon^{5}\right) \tag{19}\]  
\[\text{Ln}(\text{A}_{3})=\text{a}+\text{b}\left(\varepsilon\right)+\text{c} \left(\varepsilon^{2}\right)+\text{d}\left(\varepsilon^{3}\right)+\text{e} \left(\varepsilon^{4}\right)+\text{f}\left(\varepsilon^{5}\right) \tag{20}\]  
\[n_{2}=\text{a}+\text{b}\left(\varepsilon\right)+\text{c}\left(\varepsilon^{2 }\right)+\text{d}\left(\varepsilon^{3}\right)+\text{e}\left(\varepsilon^{4} \right)+\text{f}\left(\varepsilon^{5}\right) \tag{21}\]  
where \(\text{a}\), \(\text{b}\), \(\text{c}\), \(\text{d}\), \(\text{e}\), and f are the polynomial regression coefficients, R\({}^{2}\) is the coefficient of determination, which is shown in Table 3.  
Once the ACE model's constants were computed through Equations (18)-(21), the flow stress at an effective strain rate was predicted using Equation (17). Figure 7 compares the experimental and the predicted stress vs. strain via the ACE model at different temperatures and strain rates. The ACE modeled stress-strain data well fitted the experimental stress-strain superplastic behavior for the alloy studied. The values for the effective (apparent) activation energy of about 75-90 kJ/mol were close to the value of the activation energy of the grain boundary self-diffusion for pure aluminum, Q\({}_{\rm GB}\) = 84-86 kJ/mol [70,71]. Thus, the effective activation energy values suggested that the superplastic deformation was controlled by the grain boundary diffusion. The same effective activation energy of the superplastic deformation with the activation energy of the grain boundary diffusion was also found to be \(\approx\) 1.5 kJ/mol [70,71]. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\alpha\) & \(Q_{3}\) & \(Ln(A_{3})\) & \(n_{2}\) \\ \hline a & 0.07 & 52.5 & 4.33 & 2.69 \\ b & \(-0.001\) & 1.62 & 0.26 & \(6.91\times 10^{-4}\) \\ c & \(2.42\times 10^{-5}\) & \(-0.02\) & \(-0.004\) & \(4.30\times 10^{-5}\) \\ d & \(-1.99\times 10^{-7}\) & \(1.59\times 10^{-4}\) & \(2.61\times 10^{-5}\) & \(-1.26\times 10^{-6}\) \\ e & \(7.94\times 10^{-10}\) & \(-4.88\times 10^{-7}\) & \(-8.22\times 10^{-8}\) & \(8.49\times 10^{-9}\) \\ f & \(-1.22\times 10^{-12}\) & \(5.75\times 10^{-10}\) & \(1.00\times 10^{-10}\) & \(-1.74\times 10^{-11}\) \\ R\({}^{2}\) & 0.997 & 0.999 & 0.998 & 0.997 \\ \hline \end{tabular}
\end{table} 
Table 3: The calculated polynomial regression coefficients and R\({}^{2}\) in Equations (18)–(21).