## 4 Development of Constitutive Relationship for Superalloy Nimonic 80A
### Arrhenius-Type Constitutive Model
The mean value and standard deviation calculated by Equations (16) and (17) [6], respectively, reflect the central tendency and discrete degree of a set of data, and smaller value of \(w\) and \(\mu\) close to 0 hint that better errors distribution is achieved. As shown in Figure 8a,b, the mean value (\(\mu\)) and the standard deviation (\(w\)) of the Arrhenius-type model is 3.0012 and 2.0533, respectively, while the mean value (\(\mu\)) and the standard deviation (\(w\)) of the BP-ANN model is 0.0714 and 0.2564, respectively, which indicate that the distribution of relative percentage errors obtained by the BP-ANN model is more centralized. It suggests that the BP-ANN model has a good generalization capability. \[y=y_{0}+Ae^{-\frac{\left(\delta_{i}-\mu\right)^{2}}{2w^{2}}} \tag{15}\]  
\[\mu=\frac{1}{N}{\sum_{i=1}^{N}\delta_{i}} \tag{16}\]  
\[w=\sqrt{\frac{1}{N-1}{\sum_{i=1}^{N}\left(\delta_{i}-\mu\right)^{2}}} \tag{17}\]  
where \(\delta_{i}\) is a value of the relative error; \(\mu\), \(w\), and \(y\) are the mean value, standard deviation and probability density of \(\delta\) respectively; \(y_{0}\) and \(A\) are constants, and \(N\) is the number of relative errors, here \(N=36\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Strain** & **Temperature** & & \multicolumn{2}{c}{**True Stress (MPa)**} & \multirow{2}{*}{**Equation**} & \multicolumn{2}{c}{**Relative Error (\%)**} \\
**Rate (s\({}^{-1}\))** & & & **Experimental** & & **BP-ANN** & & **BP-ANN** & **Equation** \\ \hline \hline \multirow{8}{*}{1} & & 0.05 & 91.67 & 87.76 & 90.72 & \(-4.26\) & \(-1.04\) \\  & & 0.1  
In addition, two commonly used statistical indicators of the correlation coefficient (\(R\)) and average absolute relative error (_AARE_) are introduced to check the ability and the predictability of the BP-ANN model and Arrhenius-type model, which are expressed by Equations (18) and (19) [6]. \(R\) is a numerical value between \(-1\) and \(1\) that expresses the strength of the linear relationship between two variables. A high \(R\)-value close to \(1\) illustrates that the predicted values conform to the experimental ones well. A value of \(0\) indicates that there is no relationship. Value close to \(-1\) signal a strong negative relationship between the two variables. The average absolute relative error (_AARE_) is also computed through a term-by-term comparison of the relative error and thus is an unbiased statistical parameter to measure the predictability of a model [6; 15]. Meanwhile, a low _AARE_-value close to \(0\) indicates that the sum of the errors between the predicted and experimental values tends to be \(0\). \[R=\frac{\sum_{i=1}^{N}{(E_{i}-\overline{E})(P_{i}-\overline{P})}}{\sqrt{\sum_{ i=1}^{N}{(E_{i}-\overline{E})^{2}}{\sum_{i=1}^{N}{(P_{i}-\overline{P})}^{2}}}} \tag{18}\]  
\[AARE\left(\%\right)=\frac{1}{N}{\sum_{i=1}^{N}{\left|\frac{P_{i}-E_{i}}{E_{i }}\right|}}\times 100\% \tag{19}\]  
where \(E_{i}\) is the sample of experimental value and \(P_{i}\) is the sample of predicted value. \(\overline{E}\) is the mean value of experimental sample values, \(\overline{P}\) is the mean value of predicted sample values, and \(N\) is the number of data which were employed in the investigation. The correlation relationships between the experimental and respectively predicted true stress by BP-ANN model and modified Arrhenius-type constitutive equation were illustrated in Figure 9. It is discovered that the points in Figure 9, which take experimental true stress as horizontal axis and predicted true stress as vertical axis, lie fairly close to the best linear fitted line, suggesting that the predicted stress-strain values conform very well to the homologous experimental ones. Besides, the \(R\)-values for the predicted true stress of BP-ANN and modified Arrhenius are \(0.9998\) and \(0.9899\), respectively, from another quantitative perspective proving the strong linear relationships between the predicted and experimental true stress.