# Lifetime Prediction of Single Crystal Nickel-Based Superalloys
## 2 Experimental Procedure and Methodology
### Lifetime Assessment  
#### 2.3.1 Critical Stress Determination  
Selecting an equivalent stress that accounts for material and loading type specifics is crucial for accurate life estimations, especially for complex geometries, such as those involving triaxially. For most metals, maximum principle stress and von Mises stress are considered to be the correct approach for estimating multiaxial fatigue. For single crystals, however, it is better to include the slip planes to cover the anisotropic response of every possible cut orientation of single crystals.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Slip System Family** & **System s** & **Slip Normal n\({}^{s}\)** & **Slip Direction m\({}^{s}\)** \\ \hline \multirow{8}{*}{Octahedral} & 1 & & [\(\overline{1}\)01] \\  & 2 & (111) & [\(\overline{0}\overline{1}\)1] \\  & 3 & & [\(\overline{1}\)10] \\ \cline{2-4}  & 4 & & [\(\overline{1}\)01] \\  & 5 & (1\(\overline{1}\)1) & [\(\overline{0}\overline{1}\)1] \\  & 6 & & [\(\overline{1}\)10] \\ \cline{2-4}  & 7 & & [\(\overline{0}\overline{1}\)1] \\  & 8 & (1\(\overline{1}\)1) & [\(\overline{1}\)10] \\  & 9 & & [\(\overline{1}\)01] \\ \cline{2-4}  & 10 & & [\(\overline{1}\)10] \\  & 11 & (11\(\overline{1}\)) & [\(\overline{1}\)10] \\  & 12 & & [\(\overline{0}\overline{1}\)10] \\ \hline \multirow{4}{*}{Cubic} & 1 & (001) & [\(\overline{1}\)10] \\  & 2 & (001) & [\(\overline{1}\)10] \\ \cline{2-4}  & 3 & (100) & [\(\overline{0}\overline{1}\)1] \\ \cline{1-1} \cline{2-4}  & 4 & (010) & [\(\overline{0}\overline{1}\)1] \\ \cline{1-1} \cline{2-4}  & 5 & (010) & [\(\overline{1}\)01] \\ \cline{1-1} \cline{2-4}  & 6 & (010) & [\(\overline{1}\)01] \\ \hline \end{tabular}
\end{table}
Table 2: FCC crystal slip system list.  
FCC single crystals have four active octahedral slip planes and three active cubic slip planes, as presented in Figure 2a,b. The effect of cubic slip systems is observed at elevated temperatures due to a higher activation energy requirement for plasticity, while octahedral systems are active even at room temperature, thanks to their densely packed structure. Motivated by the fact that fracture occurs on active slip planes, stress along cleavage stress directions is determined to be the deciding factor for fatigue life estimations.  
Experimental studies show that mixed-mode loadings are apparent in multi-axial fatigue fractures for many different specimens and loading types. Figure 3 illustrates the crack opening modes, with cleavage systems referred to as Mode I, and accommodation systems as Modes II and III. Mode I is observed to be the most dominant among the other modes, even under pure shear loadings, since the appearance of shearing and tearing modes of the crack is observed to be dependent on crack incline and material orientation. The cleavage system is always present, and the accommodation systems are observed to be dependent on fatigue loadings. The growth of fatigue micro-cracks is under the influence of the cleavage system more than the others [24; 25; 26; 27]. Under these findings, critical stress is chosen to be the cleavage stresses acting on the slip planes where cracks are initiated.