# Thermal Fatigue Life Prediction of Thermal Barrier Coat on Nozzle Guide Vane via Master-Slave Model
## 3 Life Modeling Process  
The TFL test data of the ceramic metal tube were adopted to build the phenomenological TFL model (Equation (2)) by finding parameters \(a\) and \(b\). The inner diameter, outer diameter, and length of the ceramic metal tube were 11 mm, 15 mm, and 85 mm, respectively. In the TFL test experiment, the central region of the tube with the length of 50 mm was heated for 160 s by an electromagnetic induction coil. After heating, the inner surface was cooled for 260 s by high pressure air [13]. The measured interface temperature of the TBC in a single cycle are shown in Figure 5. The TFL of a typical NGV was tested under six operating conditions (case 1, case 2,..., case 6). The results of the tests are shown in Table 5.  
The 2D axisymmetric FE model was established by adopting a micro-segment with the length of 0.125 mm in the center of the heating part, as shown in Figure 6. In the 2D FE model, the size, structure, and material of the TBC layers were consistent with the slave model, and the thermal cyclic load and heating/cooling period were also the same as those in the test. We separated the measured temperature to the cooling temperature curve and heating temperature curve, which were assigned to the inner boundary and outer boundary of the 2D axisymmetric finite element model, respectively. Displacement constraints in the \(z\) direction and \(x\) direction were applied to the lower edge and inner edge, respectively. A fully coupling element, ANSYS-Plane 223, was used to simulate the transient temperature and thermal stress based on the thermoelastic damping matrix, i.e.,  
\[\left[ \begin{array}{cc}[M]&[0]\\ 0&[0]\end{array} \right]\left\{ \begin{array}{c}\{\tilde{u}\}\\ \{\tilde{T}\}\end{array} \right\}+\left[ \begin{array}{cc}[C]&[0]\\ \begin{array}{cc}C^{tt}\end{array}&[C^{t}]\end{array} \right]\left\{ \begin{array} []{c}\{\tilde{u}\}\\ \{\tilde{T}\}\end{array} \right\}+\left[ \begin{array}{cc}[K]&\begin{array}[ ]{c}K^{ut}\\ 0\end{array}\end{array} \right]\left\{ \begin{array}{c}\{\tilde{u}\}\\ \{\{\tilde{T}\}\end{array} \right\}\right\}=\left\{ \begin{array}{c}\{F\}\\ \{[Q]\end{array} \}\right\} \tag{10}\]  
Figure 5: Interface temperature of the TBC in a single cycle [13]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multicolumn{3}{c}{**Test Condition**} & \multirow{2}{*}{**Cycle**} \\ \cline{2-3} \cline{5-3}  & **Preheating Time, H** & & **Heat Preservation Time, S** & \\ \hline Case 1 & 0 & 0 & 298 \\ Case 2 & 0 & 670 s & 505 \\ Case 3 & 50 h & 0 & 480 \\ Case 4 & 100 h & 0 & 441 \\ Case 5 & 200 h & 0 & 400 \\ Case 6 & 300 h & 0 & 206 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Test results of the thermal fatigue experiment under a preheating temperature of 1323 K [14]. where \([M]\), \([C]\), and \([K]\) are the metric of element mass, element structural damping, and element stiffness, respectively; \([u]\) and \([T]\) are the vector of displacement and temperature, respectively; \([F]\) indicates the um of the element nodal force; \([C^{t}]\) and \([K^{t}]\) are the matrix of element specific heat, element thermal conductivity, respectively; \([Q]\), \([K^{ut}]\), and \([C^{tu}]\) denote the sum of the element heat generation load and element convection surface heat flow, the matrix of element thermoelastic stiffness, and the matrix of element thermoelastic damping, respectively. Six typical TGO thicknesses (0, 2 \(\upmu\)m, 4 \(\upmu\)m, 6 \(\upmu\)m, 8 \(\upmu\)m, and 10 \(\upmu\)m) were selected in the simulation. To find the risk time and point in the thermal cycle, the von-Mises stress was selected, i.e.,  
\[\sigma_{\text{von}}=\left[\frac{\left(\sigma_{\text{x}}-\sigma_{\text{y}} \right)^{2}+\left(\sigma_{\text{x}}-\sigma_{x}\right)^{2}+\left(\sigma_{\text {y}}-\sigma_{x}\right)^{2}}{2}\right]^{\frac{1}{2}} \tag{11}\]  
The maximum thermal stress of the TC layer is shown in Figure 7a.