## 4 Development of Constitutive Relationship for Superalloy Nimonic 80A
### Arrhenius-Type Constitutive Model
It can be seen that the proposed constitutive equation gives an accurate estimation on the flow stress of superalloy irimonic 80A in most of the experimental conditions. Figure 6: Relationships between: (**a**) \(\alpha\); (**b**) ln\(A\); (**c**) \(n\); and (**d**) \(Q\) and true strain \(\varepsilon\) by polynomial fit. Figure 7: _Cont._  
5 Prediction Capability Comparison between the BP-ANN Model and Arrhenius Type Constitutive Equation  
Depending on the improved Arrhenius type constitutive equation, in this study, the true stresses of 32 points under the conditions of 1100 \({}^{\circ}\)C and 0.01 s\({}^{-1}\) and 1200 \({}^{\circ}\)C and 1 s\({}^{-1}\) at a strain range of 0.05 to 0.9 with a strain interval of 0.05 were calculated to compare with the true stress predicted by BP-ANN model and obtained from the isothermal compression tests. For the sake of the contrast of prediction accuracy between these two models, the relative error (\(\delta\)) is introduced, which is expressed by Equation (14). \[\delta\left(\%\right)=\frac{P_{i}-E_{i}}{E_{i}}\times 100\% \tag{14}\]  
where \(E_{i}\) is the sample of experimental value and \(P_{i}\) is the sample of predicted value. The \(\delta\)-values relative to the experimental true stress was calculated by Equation (14) and listed in Table 2. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Strain Rate (s\({}^{-1}\))** & **Temperature** & \multirow{2}{*}{**Strain**} & \multicolumn{2}{c}{**True Stress (MPa)**} & \multirow{2}{*}{**Equation**} & \multicolumn{2}{c}{**Relative Error (\%)**} \\ \cline{4-7}  & & & **Experimental** & & **BP-ANN** & & **BP-ANN** & **Equation** \\ \hline \multirow{9}{*}{0.01} & & 0.05 & 65.08 & 63.59 & 64.37 & \(-\)2.29 & \(-\)1.08 \\  & & 0.10 & 71.80 & 71.58 & 71.18 & \(-\)0.31 & \(-\)0.86 \\  & & 0.15 & 75.24 & 74.75 & 72.79 & \(-\)0.64 & \(-\)3.25 \\  & & 0.20 & 76.09 & 75.31 & 72.42 & \(-\)1.03 & \(-\)4.83 \\  & & 0.25 & 75.35 & 74.99 & 72.30 & \(-\)0.47 & \(-\)4.04 \\  & & 0.30 & 73.81 & 74.05 & 71.59 & 0.33 & \(-\)3.01 \\  & & 0.35 & 72.06 & 72.75 & 71.23 & 0.96 & \(-\)1.15 \\  & & 0.40 & 70.57 & 71.31 & 70.43 & 1.04 & \(-\)0.20 \\  & & 0.45 & 69.47 & 69.93 & 71.33 & 0.67 & 2.68 \\  & & 0.50 & 68.73 & 68.77 & 70.15 & 0.05 & 2.06 \\  & & 0.55 & 68.32 & 67.90 & 70.40 & \(-\)0.62 & 3.04 \\  & & 0.60 & 68.17 & 67.36 & 70.40 & \(-\)1.18 & 3.28 \\  & & 0.65 & 68.20 & 67.13 & 69.34 & \(-\)1.56 & 1.68 \\  & & 0.70 & 68.34 & 67.13 & 70.24 & \(-\)1.78 & 2.77 \\  & & 0.75 & 68.54 & 67.26 & 69.70 & \(-\)1.87 & 1.69 \\  & & 0.80 & 68.72 & 67.39 & 68.91 & \(-\)1.93 & 0.28 \\  & & 0.85 & 68.80 & 67.37 & 69.76 & \(-\)2.08 & 1.39 \\  & & 0.90 & 68.73 & 67.08 & 67.71 & \(-\)2.41 & \(-\)1.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relative errors of the predicted results by the back-propagational artificial neural network (BP-ANN) model and constitutive equation to experimental results under the condition of 1100 \({}^{\circ}\)C and 0.01 s\({}^{-1}\) and 1200 \({}^{\circ}\)C and 1 s\({}^{-1}\). Figure 7: Comparisons between predicted and measured under different deformation temperatures with strain rates of (**a**) 0.01 s\({}^{-1}\); (**b**) 0.1 s\({}^{-1}\); (**c**) 1 s\({}^{-1}\) and (**d**) 10 s\({}^{-1}\). It is found in Table 2 that the relative percentage error obtained from BP-ANN model varies from \(-4.26\%\) to \(3.24\%\), whereas it is in the range from \(-4.83\%\) to \(11.13\%\) for the improved Arrhenius-type constitutive model. As shown in Figure 8a,b, the relative percentage errors have been summarized in which the height of the histogram expresses the relative frequency of the relative percentage errors. Through nonlinear curve fitting, the distributions of relative percentage errors obtained from Arrhenius-type model and BP-ANN model present a typical Gaussian distribution, which was expressed as in Equation (15). In the function, the two parameters of \(\mu\) and \(w\) represent the mean value and standard deviation, respectively, which are two of the most important indexes in statistical work.