## 4 Development of Constitutive Relationship for Superalloy Nimonic 80A
### Arrhenius-Type Constitutive Model
However, the effect of temperature and strain rate on flow behavior cannot be considered in Equation (3). Zener-Hollomon parameter, \(Z\), in an exponent-type Equation (10) [28] has been introduced to model the comprehensive function of temperature and strain rate. \[Z=\dot{\epsilon}\text{exp}(\frac{Q}{RT}) \tag{10}\]  
Base on Equations (6) and (10), the stress \(\sigma\) at the strain of 0.5 can be written as a function of \(Z\) parameter:  
\[\sigma=\frac{1}{\alpha}\text{ln}\left\{(Z/A)\frac{1}{n}+\left[(Z/A)\frac{2}{n }+1\right]^{\frac{1}{2}}\right\} \tag{11}\]  
It is well known that the constitutive model is affected not only by the deformation temperature and strain rate, but also by the strain in the hot deformation of metal materials [7; 10; 12; 17; 29; 30; 31; 32]. Therefore, the values of material coefficients (_i.e._, \(\alpha\), \(A\), \(n\), \(Q\)) of the constitutive equation are calculated under different strains in a range of 0.05 to 0.9 with the interval of 0.05 by the same method used previously. These values were then used to fit the polynomial functions (Figure 6), and the variation of \(\alpha\), \(\text{ln}A\), \(n\) and \(Q\) with true strain \(\epsilon\) could be represented by a sixth order polynomial respectively, as shown in Equation (12). The coefficients of the sixth order polynomial functions are tabulated in Figure 3:0  
\[\left\{ \begin{array}{l}\alpha=B_{0}+B_{1}\epsilon+B_{2}\epsilon^{2}+B_{3} \epsilon^{3}+B_{4}\epsilon^{4}+B_{5}\epsilon^{5}+B_{6}\epsilon^{6}\\ \text{ln}A=C_{0}+C_{1}\epsilon+C_{2}\epsilon^{2}+C_{3}\epsilon^{3}+C_{4} \epsilon^{4}+C_{5}\epsilon^{5}+C_{6}\epsilon^{6}\\ n=D_{0}+D_{1}\epsilon+D_{2}\epsilon^{2}+D_{3}\epsilon^{3}+D_{4}\epsilon^{4}+D_ {5}\epsilon^{5}+D_{6}\epsilon^{6}\\ Q=E_{0}+E_{1}\epsilon+E_{2}\epsilon^{2}+E_{3}\epsilon^{3}+E_{4}\epsilon^{4}+E_ {5}\epsilon^{5}+E_{6}\epsilon^{6}\end{array} \right. \tag{12}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\alpha\) & & \(\text{ln}A\) & \(n\) & & \(Q\) \\ \hline \(B_{0}\) & 0.01006 & \(C_{0}\) & 40.29126 & \(D_{0}\) & 7.23447 & \(E_{0}\) & 467.28232 \\ \(B_{1}\) & \(-\)0.05783 & \(C_{1}\) & \(-\)144.12464 & \(D_{1}\) & \(-\)44.03632 & \(E_{1}\) & \(-\)1502.21622 \\ \(B_{2}\) & 0.33009 & \(C_{2}\) & 982.95072 & \(D_{2}\) & 234.75515 & \(E_{2}\) & 10438.62694 \\ \(B_{3}\) & \(-\)0.95142 & \(C_{3}\) & \(-\)3062.36904 & \(D_{3}\) & \(-\)653.05658 & \(E_{3}\) & \(-\)32768.83516 \\ \(B_{4}\) & 1.47519 & \(C_{4}\) & 4851.02761 & \(D_{4}\) & 984.16398 & \(E_{4}\) & 52125.14469 \\ \(B_{5}\) & \(-\)1.16554 & \(C_{5}\) & \(-\)3813.23640 & \(D_{5}\) & \(-\)756.88970 & \(E_{5}\) & \(-\)41.091.74522 \\ \(B_{6}\) & 0.36783 & \(C_{6}\) & 1183.77777 & \(D_{6}\) & 232.91924 & \(E_{6}\) & 12784.92195 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Polynomial fitting results of superalloy nimonic 80A. Thus, the improved Arrhenius type model with variable coefficients can be expressed as Equation (13). \[\sigma=\frac{1}{g\left(\varepsilon\right)}\ln\{(\frac{\dot{\varepsilon}\exp[j( \varepsilon)/8.314T]}{f(\varepsilon)})^{1/h(\varepsilon)}+[(\frac{\dot{ \varepsilon}\exp(j(\varepsilon)/8.314T)}{f(\varepsilon)})^{2/h(\varepsilon)}+ 1]^{1/2}\} \tag{13}\]  
where \(g\left(\varepsilon\right)\), \(f\left(\varepsilon\right)\), \(h\left(\varepsilon\right)\), \(j\left(\varepsilon\right)\) are polynomial functions of strain for \(\alpha\), \(A\), \(n\), \(Q\). Applying the aforementioned material constants to Equation (13), the true stress values are calculated for the experimental temperature, strain and strain rate ranges. Figure 7 shows comparisons between the experimental data and the predicted results calculated from the developed constitutive equations (considering the compensation of strain) at the temperatures of 1050 \({}^{\circ}\)C, 1100 \({}^{\circ}\)C, 1150 \({}^{\circ}\)C, 1200 \({}^{\circ}\)C, and 1250 \({}^{\circ}\)C, and the strain rates of 0.01 s\({}^{-1}\), 0.1 s\({}^{-1}\), 1 s\({}^{-1}\) and 10 s\({}^{-1}\).