## 3 Flow Stress Modeling  
### Arrhenius Type Constitutive Equation  
In the present work, the Arrhenius constitutive equation based on the Zener-Hollomon theory [18] was used to model the hot deformation flow stress of 5005 aluminum alloy. \(\mathrm{Q}\left(\mathrm{J}\ \mathrm{mol}^{-1}\ \right)\) is the activation energy, while \(\mathrm{T}\left(\mathrm{K}\right)\) is the temperature and denotes the gas constant of 8.314 J mol\({}^{-1}\) K\({}^{-1}\). The Zener-Hollomon parameter Z was defined according to Equation (1). \[\mathrm{Z}=\ \dot{\epsilon}\ \exp\left(\frac{\mathrm{Q}}{\mathrm{RT}}\right) \tag{1}\]  
In (1), \(\dot{\epsilon}\ \left(\mathrm{s}^{-1}\right)\) denotes the strain rate defined in Equation (2) below,  
\[\dot{\epsilon}\ =\mathrm{AF}(\sigma)\exp\left(\frac{-\mathrm{Q}}{\mathrm{RT}}\right) \tag{2}\]  
where A is the material constant and \(\mathrm{F}(\sigma)\) is the function, defined as (3):  
\[\mathrm{F}(\sigma)=\left\{ \begin{array}{l}\sigma^{n^{\prime}}\ \alpha\sigma<0.8\\ \exp\left(\beta\sigma\right)\ \alpha\sigma>1.2\\ \left[\sinh(\alpha\sigma)\right]^{n}\ \mathrm{for\ all}\ \alpha\sigma\end{array} \right. \tag{3}\]  
In (3), \(\sigma\) is the stress and n\({}^{\prime}\), \(\alpha\), \(\beta\), and n are the materials constants. After substituting the first, second, and third equations of (3) into Equation (2) and taking the logarithms of the three formulas, one can respectively obtain Equations (4)-(6):  
\[\mathrm{ln}\ \sigma=\frac{1}{n^{\prime}}\mathrm{ln}\ \dot{\epsilon}\ -\ \frac{1}{n^{\prime}}\mathrm{ln}\mathrm{B} \tag{4}\]  
\[\sigma=\frac{1}{\beta}\mathrm{ln}\ \dot{\epsilon}\ -\ \frac{1}{\beta}\mathrm{ln} \mathrm{C} \tag{5}\]  
\[\mathrm{ln}\left[\sinh(\alpha\sigma)\right]=\frac{\mathrm{ln}\ \dot{\epsilon}}{\mathrm{n}}+\frac{\mathrm{Q}}{\mathrm{nRT}}-\frac{\mathrm{ln }\mathrm{A}}{\mathrm{n}} \tag{6}\]  
After obtaining the equations, n\({}^{\prime}\) can be obtained by calculating the slope of \(\mathrm{ln}\ \sigma-\mathrm{ln}\ \dot{\epsilon}\) plots, and \(\beta\) can be obtained from the slope of the \(\sigma-\mathrm{ln}\ \dot{\epsilon}\) plots. The material constant n is obtained by calculating the slope of the \(\mathrm{ln}\left[\sinh(\alpha\sigma)\right]-\mathrm{ln}\ \dot{\epsilon}\) plots, and the constant Q can be obtained by calculating the slope of the \(\mathrm{ln}\left[\sinh(\alpha\sigma)\right]-\left(\frac{1}{\mathrm{T}}\right)\) plots. After n\({}^{\prime}\) and \(\beta\) are obtained, the material constant \(\alpha\) can be determined as in (7):  
\[\alpha=\frac{\beta}{n^{\prime}} \tag{7}\]  
Figure 2: Test equipment (MTS-810). Equation (8) is obtained using (3) and (6) as below. \[\ln Z=\ln A+\ln[\sinh(\alpha\sigma)] \tag{8}\]  
Then, the material constant A is calculated using the y-axis intercept of the \(\ln Z\ -\ \ln[\sinh(\alpha\sigma)]\) plot. The obtained material constants n, \(\alpha\), Q, and A are the functions of strain. In this research, the constants are described as 5 th order polynomials, as indicated in (9)-(12). The coefficients of the polynomials obtained by regression are presented in Table 2.  
\[\mathrm{n} =\mathrm{n}_{5}\varepsilon^{5}+\mathrm{n}_{4}\varepsilon^{4}+ \mathrm{n}_{3}\varepsilon^{3}+\mathrm{n}_{2}\varepsilon^{2}+\mathrm{n}_{1} \varepsilon^{1}+\mathrm{n}_{0} \tag{9}\] \[\alpha =\mathrm{a}_{5}\varepsilon^{5}+\mathrm{a}_{4}\varepsilon^{4}+ \mathrm{a}_{3}\varepsilon^{3}+\mathrm{a}_{2}\varepsilon^{2}+\mathrm{a}_{1} \varepsilon^{1}+\mathrm{a}_{0}\] (10) \[\mathrm{Q} =\mathrm{q}_{5}\varepsilon^{5}+\mathrm{q}_{4}\varepsilon^{4}+ \mathrm{q}_{3}\varepsilon^{3}+\mathrm{q}_{2}\varepsilon^{2}+\mathrm{q}_{1} \varepsilon^{1}+\mathrm{q}_{0}\] (11) \[\sigma=\frac{1}{\beta}\mathrm{ln}\ \dot{\epsilon}\ -\ \frac{1}{\beta}\mathrm{ln} \mathrm{C} \tag{5}\]0  
The flow stress of the hot deformation can be derived using (2) and (3), as in (13).