## 3 Life Evaluation for High-Temperature Steam Pipes
### Low-Cycle Fatigue Life Prediction  
For high-temperature components, it is well-known that creep is the dominant damage mechanism, and low cycle fatigue (LCF) damage is relatively negligible for boiler components such as pipes and tubes. However, in this study, the LCF life was calculated to quantify the LCF damage rate and to check whether the 3 DDMS data were used to calculate the LCF life. The LCF damage was calculated by conducting curve fitting through the LCF test data [38]. It was observed that the cyclic strain process of the P91 steels proceeded with a dominant participation of the elastic strain component. In the elastic range (below 450 MPa), the maximum stress can be directly converted to strain values, since the strain is proportional to stress according to [PERSON]'s law. The fatigue life of the P91 material described by the Smith, Hirschberg, and Manson dependence [1; 2; 3; 4] is expressed as:  
\[\Delta\varepsilon_{t}=\Delta\varepsilon_{e}+\Delta\varepsilon_{p}=(G/E)N_{f}^{ ep}+MN_{f}^{2} \tag{6}\]  
where \(G\), \(E\), and \(N_{f}\) indicate the shear modulus, elastic modulus, and fatigue durability, respectively. In addition, \(v\) and \(z\) are the material constants. All parameters of the fatigue model are presented in Table 7. The fatigue durability can be determined from Equation (6) by using the Newton-Raphson method. The LCF life is determined as follows:  
\[\phi_{LCF}=\frac{N_{op}}{N_{t}} \tag{7}\]  
where \(N_{op}\) and \(N_{t}\) indicate the number of start-ups and the number of cycles to failure, respectively.  
In this study, the LCF life was calculated using two types of cycle-counting methods, which were number of cycle counting (NCC) and rainflow cycle counting (RCC). In the case of the NCC method, the maximum stress or strain during start-ups (i.e., cold) is used to calculate the fatigue life by only considering the number of starts \(N_{op}\). In contrast, the RCC method determines the load cycles for closed hysteresis loops in a loading history, and all 3 DDMS data are used as loading history for the count cycle. Among the two results  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(C\) & \(B_{0}\) & \(B_{1}\) & \(B_{2}\) \\ \hline
280 & 11.834 & 13.67 & \(-\)4.51161 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Larson–Miller parameters of P91.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(G/E\) & \(v\) & \(M\) & \(z\) \\ \hline
0.009 & \(-\)0.1 & 1.205 & \(-\)0.78 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Parameters of fatigue models for P91 steel.  
by each method, the maximum LCF life is used to calculate the total life with maximum creep life. The total life consumption after the \(N_{op}\) of start-ups and an operating time \(t_{op}\) is determined by using a nonlinear accumulation damage rule [7] as follows:  
\[\phi_{Total}=\phi_{CREEP}+\phi_{LCF}+e[\phi_{CREEP}\cdot\phi_{LCF}]^{r} \tag{8}\]  
The hyper-parameter e and r that determine the nonlinear interaction effects are assumed to be fixed to certain values under conservative assumptions. As a result, appropriate \(e\) and \(r\) values should be used to evaluate the life under actual operating conditions. In this study, we assumed that \(e\) was 0.75 and \(r\) was 0.5 when considering the creep and fatigue interactions in high chromium alloy steel, which is a representative material of power-plant facilities [39; 40; 41].