### Determining the Sensitive Stress
#### 3.1.1 Data Expansion  
The accuracy of the nonlinear fitting of BP neural networks requires large numbers of valid data. However, the data sample size of this test is small. To solve the problem, we first need to establish the mathematical relationship between the amount of degradation and the sensitive stress according to the material property data (Table 2). Then, combining the above established mathematical function with the test date, the degradation model under each test condition can be derived.  
The endurance performance is assessed throughout the life cycle of the whole test. The cumulative amount of stress \(\sigma\) at the action time t is recorded as the accumulated load \(L(t)\). The relationship between the three can be expressed as follows:  
\[\sigma_{t}=f(L(t),\theta) \tag{1}\]  
where \(\theta\) is the temperature and \(L(t)=\sigma_{\text{f}}t\) or \(\sigma_{\text{f}}\) is the persistence limit.  
The model accuracy of the neural network greatly depends on the sample data. To reduce the influence of the randomness of the original data, the information of regularity in the data is highlighted and improves the model's stability. The Larson-Miller formula is used to expand the existing data, and is given as follows:  
\[P(\sigma)=T(C+\text{lg}t)\times 10^{-3} \tag{2}\]  
where \(\sigma(\text{MPa})\) is the stress on the material, \(T(K)\) is the temperature, \(t(h)\) is the time, \(C\) is the material constant, and \(P(\sigma)\) is the Larson-Miller parameter, in which the relationship between \(P\) and \(\sigma\) is given as follows:  
\[\text{lg}\sigma=a_{2}P^{2}+a_{1}P+a_{0} \tag{3}\]  
where \(a_{i}(i=0,1,2)\) is the coefficient.  
For the selection of \(C\), we use TAE (trial-and-error) to take values in the appropriate range. Using the data in Table 2 and combining Equations (2) and (3), the variation relationship between the GOF (goodness-of-fit) of \(R^{2}\) and \(C\) is as shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c} \hline
**RPM** & **Turbine Blade Root Stress/MPa** & **Turbine Blade Root Temperature/K** \\ \hline \hline
**AE: 64,000 Static Structural** & **AD: 64,000 Steady State Thermal** \\  & **Type Equivalent Stress** & **Temperature** \\  & **Uint: 30\%** & **Type Temperature** \\  & **Time: 1** & **Time: 1** \\ \hline
**114.8** & **1016.8** & **84.95** \\
**64,000** & **762.9** & **73.8** \\
**63.79** & **748.75** & **748.75** \\
**508.78** & **723.7** & **73.7** \\
**381.78** & **609.65** & **673.6** \\
**254.77** & **648.55** **33n** & **648.55** **33n** \\ \hline \hline \end{tabular}
\end{table}
Table 6: _Cont._When \(C=20.3\), we can obtain \(\text{MAX}\big{\{}{R_{i}}^{2}\big{\}}=0.9962\) and \(\text{RMSE}=0.01708\) (root mean square error). We can see the GOF of this model is very close to 1 and the RMSE is small, which can better reflect the relationship between the parameter \(P\) and the stress \(\sigma\). Then, from Equations (2) and (3), we can derive:  
\[\text{lg}\sigma=-0.007822P^{2}+0.2744P+0.6002 \tag{4}\]  
\[P(\sigma)=T(20.3+\text{lg}t)\times 10^{-3} \tag{5}\]  
Because test results rely on low-temperature data, the temperature is expanded toward the lower range. Taking \(T/^{\circ}\text{C}=[500,1000]\) and \(t/h=[50,1000]\), and keeping the initial data and the step size as 10, we then combine Equation (4) with Equation (5), and the data can be expanded. The expanded sample has 3492, which is approximately 50 times the original data. We choose these data as the neural network training data.