## 2 Finite Element Analysis  
FE analysis was performed using three types of 2D models and a 3D model for the turbine blade. The nonlinear FE program ADINA 9.3.4 (ADINA R&D, Inc., Watertown, MA, USA) was used for the FE analysis. Figure 1 shows the form and dimensions of the 2D analysis model. Figure 1a shows Model 2D and Model 2D-Anisotropy, and Figure 1b shows Model 2D-TBC. Table 1 shows the material properties of the four models. As shown in Table 1, the material properties of Model 2D and Model 2D-Anisotropy are different. These were determined based on the material properties of the Ni-based superalloy NCF625 (isotropic material) and a Ni-based directionally solidified (DS) alloy (anisotropic material), respectively. As these Ni-based superalloys have excellent high-temperature strength, they are widely used as materials for gas turbine blades [10; 11; 12]. The Ni-based DS alloy is a material with a controlled crystal orientation and grain growth direction (hence anisotropic [12]), and has high strength at high temperatures. Therefore, in order to investigate the effect of anisotropy on the deformation behavior of  
the turbine blade, analysis was performed using Model 2D-Anisotropy. For Model 2D-TBC, the authors considered a heat-resistant coating (bond coat layer and top coat layer). The material properties of the bond coat and top coat were determined with reference to the ceramic coating [13]. The materials of the bond coat and the top coat were CoNiCrAlY alloy and yttria-stabilized zirconia, respectively. In this study, we focused on the materials that may be used in the actual turbine blade. A plane strain condition was used for the 2D models. The actual turbine blade is large in the depth direction. Therefore, the plane strain condition can be more suitable than the plane stress condition. Figure 2 shows the form and dimensions of the 3D analysis model. The material properties of the 3D model were determined based on the Ni-based superalloy NCF625 (isotropic material). The load values in the case of the 2D and 3D models are 1916 N. The load range is determined by the following conditions. The lower limit of the load should provide a large enough displacement difference for DHM to be distinguishable, and the higher limit of the load should limit the elastic deformation regime. If a several millimeter-sized crack can be detected in the actual turbine blade, the crack detection method can be practically useful. Based on the ideal, the size and depth of the crack were determined. Figure 3 shows the elements of Model 2D and Model 3D. Table 2 shows the number of nodes and number of elements in each model. In order to determine the element size, FE analysis was performed with a simple shape before analyzing the actual model, and the analysis values of the stress distribution near the crack were compared with theoretical values. To conduct the FE analysis calculation with high accuracy and efficiency, two types of elements were used for the models. A square eight-node \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Young’s Modulus, \(E\) [GPa]** & **Poisson’s Ratio, \(\
u\)** \\ \hline Model 2D & 207 & 0.3 \\ Model 2D-Anisotropy & \(E_{x}=125.5\), \(E_{x}=168.7\), \(G_{xy}=133.9\), \(G_{xy}=64.5\) & \(v_{xx}=0.333\), \(v_{xx}=0.447\) \\ Model 2D-Thermal barrier coating & 207 (base material), 200 (bond coat), 40 (top coat) & 0.3 (base material), 0.3 (top coat) \\ Model 3D & 207 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the four types of models. Figure 1: Shape and dimensions of the two-dimensional models: (**a**) Model 2D and Model 2D-Anisotropy; (**b**) Model 2D-Thermal barrier coating (TBC). Figure 2: Shape and dimensions of the three-dimensional model. element was used to model the element near the crack tip, and square four-node elements were used for other areas. The weight of the beam was not considered in the analysis because we focused on the difference of the deformation before and after the loading. There is no contact element in the FE analysis.