# Thermal Fatigue Life Prediction of Thermal Barrier Coat on Nozzle Guide Vane via Master-Slave Model
## 2 Modeling and Simulation Methods
### Material Parameters and Boundary Conditions  
In the above FE model, each layer of the NGV and TBC is considered as an isotropic and homogeneous material. The TC is modeled as an elastic body, whereas the BC, TGO, and substrate are regarded as elastic-perfectly plastic materials [29; 30]. The temperature-dependent material properties are listed in Tables 1 and 2. It should be noted that to limit stress to the experimental level [31; 32], the TGO is allowed to undergo stress relaxation at high temperatures, which is realized by introducing the yield strength of TGO at peak temperatures [31]. The multilinear kinematic hardening model was selected to simulate the elastic-plastic behaviors. The back-stress tensor for multilinear kinematic hardening evolves such that the effective stress versus effective strain curve is multilinear with linear segments defined by the stress-strain-temperature points [33]. The yield stress \(\sigma_{yi}\) (\(i=1,2,\,\ldots,7\)) for the \(i\)th temperature point in Table 2 is  
\[\sigma_{yi}=\frac{1}{2(1+\
u)}(3\varepsilon\varepsilon_{i}-(1-2\
u)\sigma_{i}) \tag{4}\]  
where (\(\varepsilon_{i},\,\sigma_{i}\)) is stress and strain at the \(i\)th temperature point, respectively; \(E\) is the Young modulus; and \(v\) is the Poisson ratio. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **SUB** & **BC** & **TGO** & **TC** \\ \hline Temperature, K & 20–1100 & 20–1100 & 20–1100 & 20–1100 \\ Thermal expansion coefficient, \(10^{-5}\)/K & 1.48–1.80 & 1.36–1.76 & 0.80–0.96 & 0.90–1.22 \\ Young modulus \(E\), GPa & 220–120 & 200–110 & 400–320 & 48–22 \\ Poisson ratio \(v\) & 0.31–0.35 & 0.30–0.33 & 0.23–0.25 & 0.10–0.12 \\ Shear modulus, \(\chi 10^{11}\) Pa & 0.84–0.44 & 0.77–0.41 & 1.66–1.28 & 0.21–0.01 \\ Heat transfer coefficient, W/m-K & 88–69 & 5–8–17 & 10–4.1 & 2–1.7 \\ Density, kg/m\({}^{3}\) & 8500 & 7380 & 3984 & 3610 \\ Specific heat, J/kg K & 440 & 450 & 755 & 505 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Temperature dependent material parameters for different layers [30]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Temperature** & **SUB** & **BC** & **TGO** \\ \hline
1 & 300 K & 800 MPa & 426 MPa & 10,000 MPa \\
2 & 473 K & 800 MPa & 412 MPa & 10,000 MPa \\
3 & 673 K & 800 MPa & 396 MPa & 10,000 MPa \\
4 & 873 K & 800 MPa & 362 MPa & 10,000 MPa \\
5 & 1073 K & 800 MPa & 284 MPa & 10,000 MPa \\
6 & 1273 K & 800 MPa & 202 MPa & 1000 MPa \\
7 & 1373 K & 800 MPa & 114 MPa & 1000 MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The variations of yield strength with temperature for different layers [31]. Table 3 reveals the performance parameters of the axial flow turbojet engine, which are employed to derive the simulated boundary conditions of the master model. In line with the power balance principle of the aeroengine rotor shaft, the relationship between the inlet Mach number \(Ma_{\text{eng}}\) and the engine inlet total pressure \(P^{*}_{\text{eng}}\) is  
\[P^{*}_{\text{eng}}=P_{\text{eng}}\big{(}1+0.2\cdot Ma_{\text{eng}}2\big{)}^{3.5} \tag{5}\]  
where \(P_{eng}\) is the inlet static pressure. With respect to the efficiency of compressor \(\eta_{\text{com}}\) = 0.89 [34], the outlet temperature of compressor \(T^{*}_{\text{com}}\) is  
\[T^{*}_{\text{com}}=T^{*}_{\text{eng}}\bigg{(}1+\frac{\pi^{0.2857}-1}{\eta_{ \text{com}}}\bigg{)} \tag{6}\]  
where \(\pi\) = \(P^{*}_{\text{com}}\)/\(P^{*}_{\text{eng}}\) is the pressure ratio of the compressor and \(T^{*}_{\text{eng}}\) is the total temperature of engine inlet. The temperature of the cooling air is determined by the outlet temperature of the compressor because cooling air is sucked up directly from the end of compressor.