## 4 Development of Constitutive Relationship for Superalloy Nimonic 80A
### Arrhenius-Type Constitutive Model  
Generally, Arrhenius type equation is expressed as Equation (3) [27], which correlates the flow stress (\(\sigma\)) with temperature (\(T\)) and strain rate (\(\dot{\epsilon}\)). \[\dot{\epsilon}=AF(\sigma)\text{exp}(-Q/RT) \tag{3}\]  
where \(F(\sigma)=\left\{ \begin{array}{cc}\sigma^{n^{\prime}}&\alpha\sigma<0.8\\ \text{exp}(\beta\sigma)&\alpha\sigma>1.2\\ \left[\sinh(\alpha\sigma)\right]^{n}\text{ }\text{ }\text{for all }\sigma\end{array} \right.\),  
and where \(\dot{\epsilon}\) is the strain rate (s\({}^{-1}\)), \(R\) is the universal gas constant (8.31 J\(\cdot\) mol\({}^{-1}\cdot\) K\({}^{-1}\)), \(T\) is the absolute temperature (K), \(Q\) is the activation energy of deformation (kJ\(\cdot\) mol\({}^{-1}\)), \(\sigma\) is the flow stress (MPa) for a given stain, \(A\), \(\alpha\), \(n^{\prime}\) and \(n\) are the material constants, \(\alpha=\beta/n^{\prime}\). For the low stress level (\(\alpha\sigma<0.8\)), taking natural logarithms on both sides of Equation (3), the following equation can be obtained:  
\[\text{ln}\dot{\epsilon}=\text{ln}A+n^{\prime}\text{ln}\sigma-Q/RT \tag{4}\]  
Figure 3: The comparison of the BP-ANN prediction with experimental values at different temperatures and strain rates (a) 0.01 s\({}^{-1}\); (b) 0.1 s\({}^{-1}\); (c) 1 s\({}^{-1}\); (d) 10 s\({}^{-1}\). For the high stress level (\(\alpha\sigma>1.2\)), taking natural logarithms on both sides of Equation (3) gives:  
\[\ln\dot{\epsilon}=\ln A+\beta\sigma-Q/RT \tag{5}\]  
According to Equations (4) and (5), \(n^{\prime}=d\ln\dot{\epsilon}/d\ln\sigma\) and \(\beta=d\ln\dot{\epsilon}/d\sigma\). Then, the linear relationships of \(\ln\sigma\)-\(\ln\dot{\epsilon}\) and \(\sigma\)-\(\ln\dot{\epsilon}\) for strain of 0.5 at the temperatures of 1050-1250 \({}^{\circ}\)C were fitted out as shown in Figure 4. The adjusted coefficient of determination \(R^{2}\) for each condition was calculated and showed in Figure 4, which was used to prove the reliability of fitting curves. The inverse of the slopes of straight lines in \(\ln\sigma\)-\(\ln\dot{\epsilon}\) and \(\sigma\)-\(\ln\dot{\epsilon}\) plots is accepted as the values of material constants \(n^{\prime}\) and \(\beta\) at each tested temperature, respectively. Thus the values of \(n^{\prime}\) and \(\beta\) at strain of 0.5 were obtained by averaging the inverse of slopes under different temperatures, which were found to be 5.3625 MPa\({}^{-1}\) and 0.0336 MPa\({}^{-1}\), respectively. Furthermore, the value of another material constant \(\alpha=\beta/n^{\prime}=0.0063\) MPa\({}^{-1}\) was also obtained. For all the stress level (including low and high-stress levels), Equation (3) can be rewritten as the following:  
\[\dot{\epsilon}=A[\sinh(\alpha\sigma)]^{n}\exp(-Q/RT) \tag{6}\]  
Taking natural logarithms on both sides of Equation (6), the following equation can be obtained:  
\[\ln\dot{\epsilon}=\{\ln A+n\left[\ln\sinh(\alpha\sigma)\right]-Q\}/RT \tag{7}\]  
For the given deformation temperature (\(T\)), the stress exponent (\(n\)) is expressed as Equation (8):  
\[n=\frac{\partial\ln\dot{\epsilon}}{\partial\ln(\sinh(\alpha\sigma))}|\tau \tag{8}\]  
When the strain rate (\(\dot{\epsilon}\)) is a constant, the activation energy (\(Q\)) can be expressed as Equation (9):  
\[Q=nR\frac{\partial\ln(\sinh(\alpha\sigma))}{\partial\dot{\epsilon}(1/T)}|_{ \dot{\epsilon}} \tag{9}\]  
According to Equations (8) and (9), the linear relationship between \(\ln(\sinh(\alpha\sigma))\) and \(\ln\dot{\epsilon}\) and the relationship between \(\ln(\sinh(\alpha\sigma))\) and \(1/T\) were fitted out as shown in Figure 5 and the determination coefficient \(R^{2}\) have been exhibited in each figure. Consequently, the value of constant parameter \(n\) and the activation energy \(Q\) can be derived from the mean slope of lines in Figure 5a,b respectively, here, \(n\) is 3.7806 and \(Q\) is 403.81 kJ\(\cdot\)mol\({}^{-1}\). In addition, the material constant \(A\) can be calculated as 4.5496 \(\times\) 10\({}^{14}\) s\({}^{-1}\).