# Lifetime Prediction of Single Crystal Nickel-Based Superalloys
## 3 Results and Discussion
### Fatigue Curve Predictions with Artificial Neural Networks  
Training Dataset Configuration  
The temperature-dependent material parameter of the lifetime assessment model, ultimate tensile strength (\(\sigma_{uts}\)) of CMSX-4 at 600 \({}^{\circ}\)C and 900 \({}^{\circ}\)C, was taken from the literature [32]. This parameter also serves as model input, and parameter optimization was performed separately for each temperature, as represented in Table 5. Fatigue curves at these temperatures are producible with the damage model; however, this temperature dependency makes it impossible to predict temperature variation without making further  
Figure 8: Fatigue crack initiation at the surface of SC16 nickel-base single crystal with tensile crystal orientations <011> and <111> are represented in (**a**,**c**). Reproduced with permission from [PERSON], crystal plasticity and damage at cracks and notches in nickel-base single-crystal superalloys; published by Elsevier, 2022 [33]. Lifetime assessment of the same specimen orientations is represented in (**b**,**d**), respectively, in which the minimum lifetime on the FEA model is marked as the predicted crack initiation location. Figure 9: Comparison of numerical and experimental results of CMSX4 notched specimen under HCF loading at 750 \({}^{\circ}\)C with \(\mathrm{R}=0\) (**a**) and \(\mathrm{R}=0.6\) (**b**). modifications to the damage law. Artificial neural networks (ANNs) do not have this restriction since temperature is another variable to establish the nonlinear relation between input variable maximum stress, R ratio, and output variable lifetime result. ANNs require a wide range of loading information at these temperatures to accurately predict the lifetime of components under complex loadings. Since producing experimental results for single crystal superalloys such as CMSX-4 is not feasible, the combination of ANNs and the modified damage model enhances the performance of both approaches in lifetime assessment by providing the data range from the damage model and making predictions without parameter restrictions with ANNs. The parameter optimization of the modified damage model for 600 \({}^{\circ}\)C and 900 \({}^{\circ}\)C is demonstrated in Table 5. Although FCC materials do not generally demonstrate a clear fatigue limit, for this case, a target lifetime of \(10^{8}\) is considered as infinite life; the fatigue limit is adjusted by defining \(\sigma_{l_{0}}\) as the endurance limit for \(R=-1\) and establishing the relation between mean stress and fatigue limit with parameter, \(b_{1}\). Fatigue curves produced by Chaboche Model for 600 \({}^{\circ}\)C and 900 \({}^{\circ}\)C with R ratios from 0 to 0.4 are fed into ANNs to achieve temperature variance between 600 \({}^{\circ}\)C and 900 \({}^{\circ}\)C. These curves are illustrated in Figure 10.  
RMSE of training and validation set are monitored during training and RMSE of independent test set is evaluated to assess the general behavior of the model. In addition to this evaluation, mean absolute error (MAE) and \(R^{2}\) score for both sets are also evaluated to address the model's predictive accuracy. Since the evaluated lifetime values are in the \(log_{10}\)-domain, translating the errors, represented in Table 6, back to the original scale results in multiplicative factors for the  
Figure 10: Comparison of modified Chaboche model (CM) and experimental results of CMSX4 at 600 \({}^{\circ}\)C for R = 0 and R = 0.4 (a) and at 900 \({}^{\circ}\)C for R = 0 (b). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**T (\({}^{\circ}\)C)** & \(\sigma^{\text{clv}}_{\text{uts}}\) (MPa) & **M\({}_{0}\) (MPa)** & \(\sigma^{\text{clv}}_{\text{l_{0}}}\) (MPa) & **b\({}_{1}\) (MPa\({}^{-1}\))** & **b\({}_{2}\) (MPa\({}^{-1}\))** & **a** & \(\beta\) \\ \hline
600 & 840 & 1010 & 480 & \(1.24\times 10^{-3}\) & \(1.22\times 10^{-3}\) & 0.01 & 14.5 \\ \hline
900 & 675 & 800 & 375 & \(2.2\times 10^{validation and independent test sets of 1.419 and 1.429 for MAE, and 1.822, and 1.845 for RMSE, respectively.