# Lifetime Prediction of Single Crystal Nickel-Based Superalloys
## 3 Results and Discussion
### Fatigue Curve Predictions with Artificial Neural Networks
Small MAE and RMSE values in the \(log_{10}\)-domain correspond to reasonable multiplicative errors in the original domain, making the model reliable for practical fatigue life prediction. The high \(R^{2}\) score highlights the model's reliability and strong correspondence with the underlying data. Fatigue curves produced on the training data spectrum, Figure 11, demonstrate that it is possible to reproduce the lifetime results obtained by modified Chaboche models with two different parameter sets using ANNs. The smooth data flow suggests that training performed with a batch size of 3.5% of the training data for 250 epochs provides sufficient data to establish the effects of fatigue limit and temperature, as well as the convergence to ultimate tensile strength for each R ratio at the same temperature, without over-training the model. The uncertainty analysis indicates that the model performs reliably under small (1%) and moderate (5%) perturbations, resulting in multiplicative uncertainties of approximately 1.3 and 3.5, respectively, for both validation and test datasets, as represented in Table 7. These findings suggest that the model maintains consistent predictions within the trained input range. For larger perturbations (10%), however, the uncertainty increases significantly, highlighting the model's sensitivity to inputs beyond the training data distribution. The ultimate tensile strength (UTS) of metals typically exhibits a slight decrease with increasing temperature due to thermal relaxation effects up to a certain threshold \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset** & **RMSE** & **MAE** & \(\mathbf{R^{2}}\) \\ \hline Validation Set & 0.2605 & 0.1521 & 0.9919 \\ Independent Test Set & 0.2659 & 0.1549 & 0.9902 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance metrics of the model on validation and independent test sets. Figure 11: Fatigue curves produced on training data spectrum for different temperatures and R ratios and their comparison with the experimental data (**a**) and for different R ratios at the same temperature (**b**). temperature. Beyond this threshold, a rapid and nonlinear decrease in UTS is commonly observed. For nickel-based superalloys, this threshold temperature is often around 500 \({}^{\circ}\)C to 600 \({}^{\circ}\)C, although it may vary depending on the alloy composition [34]. Prediction results between 600 \({}^{\circ}\)C and 900 \({}^{\circ}\)C exhibit coherent behavior with respect to temperature, in which ultimate tensile strength (UTS) and fatigue limit decrease with increasing temperature since the material becomes softer. Since the predictions presented in Figure 12 are above the threshold temperature, material properties like yield strength and UTS are considered significantly affected, leading to pronounced changes in fatigue behavior. In contrast, the predictions represented in Figure 11b for different R-ratios at the same temperature still start from the same UTS, as the initial lifetime is always defined as 1 cycle. This results in the temperature influence being more dominant in the low-cycle region. Since UTS and fatigue limit values are not provided to ANNs for 700 \({}^{\circ}\)C and 800 \({}^{\circ}\)C due to the characterization of these properties not being performed at these temperatures, real-life values may differ from the represented data in Figure 12. However, this difference can be overcome by simply introducing the ANNs characterized fatigue limit and UTS values to further enhance the performance of the model.