# Lifetime Prediction of Single Crystal Nickel-Based Superalloys
## 3 Results and Discussion  
### Lifetime Prediction of the Notched Specim at 750 \({}^{\circ}\)C  
The plastic slip threshold parameter K and the rate sensitivity parameter were chosen to mimic the stress-strain behaviour of CMSX-4 at 750 \({}^{\circ}\)C. The elastic moduli constants of CMSX-4 provided in Table 3 were taken from the literature [31]. The temperature-dependent material parameters of the lifetime assessment model, yield strength (\(\sigma_{yield}\)), and ultimate tensile strength (\(\sigma_{uts}\)) of CMSX-4 at 600 \({}^{\circ}\)C, 750 \({}^{\circ}\)C, and 900 \({}^{\circ}\)C were taken from the literature [32]. The amplitude stress and mean stress were calculated from HCF results, and the temperature-dependent model parameters were adjusted for a proper lifetime estimation on both R ratios. The Table 4 demonstrates the parameter set of both the crystal plasticity and lifetime assessment models. Since CMSX-4 does not exhibit an actual fatigue limit due to its FCC microstructural nature, the endurance limit (\(\sigma_{l_{0}}\)) was determined to be 0 instead of choosing a specific lifetime as the infinite lifetime and corresponding stresses as the fatigue limit for better visualization of lifetime throughout the specimen. Since fatigue limit computed for different mean stress would also yield zero, the parameter, \(b_{1}\) becomes obsolete in this case. The remaining parameters were manually fitted to the experimental data, which includes at most two R-ratios per temperature. Optimization algorithms, such as gradient descent or genetic algorithms, were not used as they offered no significant advantage in this case. The maximum cyclic stress-strain distribution of the CMSX-4 notched specimen with a cut orientation of (100)[001] is represented in Figure 6. Stress on the notch tip remains elastic throughout the whole cyclic loading for R ratios of 0 and 0.6, as observed in HCF loadings. Figure 6 suggests that the maximum applied stress distribution and lifetime distribution differ according to the dominant cleavage stress directions along active slip systems. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\sigma_{\text{uts}}\) (MPa) & \(\sigma_{\text{yield}}\) (MPa) & M\({}_{0}\) (MPa) & \(\sigma_{\text{lo}}^{\text{clv}}\) (MPa) & b\({}_{1}\) (MPa\({}^{-1}\)) & b\({}_{2}\) (MPa\({}^{-1}\)) & a & \(\beta\) \\ \hline
1200 & 1100 & 1800 & 0 & \(1.12\times 10^{-4}\) & \(1.12\times 10^{-4}\) & 0.65 & 7.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of anisotropic lifetime assessment model for CMSX-4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**K (MPa\({}^{1/n}\))** & **n** & **Q** & **b** & **C\({}_{11}\) (GPa)** & **C\({}_{12}\) (GPa)** & **C\({}_{44}\) (GPa)** \\ \hline
2200 & 7.5 & 100 & 250 & 296 & 204 & 125 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of crystal plasticity model for CMSX-4. For (100)[001] cut orientation, octahedral slip planes are observed to be more dominant compared to cubic systems, even at elevated temperatures. Therefore, it is only natural to expect crack initiation on [111] planes and crack growth paths along cleavage (Mode I opening) stress directions. As a result, crack initiation points are completely dependent on the orientation of the specimen with the same applied elastic stress distribution as in Figure 6.  
Lifetime distributions for different cut orientations are determined by adjusting the local coordinate system of the crystal plasticity model to align with the tensile loading direction. A comparison of crack initiation points with simulation results for different cut orientations of the notch specimen is provided in Figures 7 and 8. Fatigue crack initiation of the notched specimen with crystal orientation of <001> takes place above the notch tip in spite of the higher geometric stress concentration on the notch tip, as represented in Figure 8a. This result is due to the high accumulation of slip in these regions caused by the elastic and plastic anisotropy of the cut orientation [33].