## 3 Results and Discussion  
### Two-Dimensional Models  
#### 3.1.1 Model 2D  
Figure 4 shows the \(z\)-direction displacement of the surface of Model 2D. Macroscopically, no difference is observed between the four types of test specimens. Therefore, in order to reveal the influence of the existence and size of the crack on displacement in the \(z\)-direction, the difference in the \(z\)-direction displacement of the models with and without a crack is shown, as illustrated by Figure 5. In addition, it is found that the difference increases as the crack grows. Furthermore, it is found that there is a sharp change in the inclination near the position where \(x=247.5\) mm, where the crack is introduced. These height distribution differences are in the order of several microns to several tens of microns, which are sufficiently large with respect to the vertical DHM resolution of 0.2 nm. Therefore, it may be possible to detect cracks by ascertaining this change by DHM. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Crack Size, \(a\) (mm)** & **Total Number of Nodes** & **Total Number of Elements** \\ \hline \multirow{3}{*}{Model 2D} & 0.5 & 8044 & 6479 \\  & 1.0 & 7600 & 6403 \\  & 1.5 & 8043 & 6479 \\ \hline \multirow{3}{*}{Model 2D-Anisotropy} & 0.5 & 8044 & 6479 \\  & 1.0 & 8057 & 6491 \\  & 1.5 & 7683 & 6359 \\ \hline \multirow{3}{*}{Model 2D-TBC} & 0.5 & 20,181 & 14,559 \\  & 1.0 & 20,195 & 14,571 \\  & 1.5 & 20,181 & 14,559 \\ \hline \multirow{3}{*}{Model 3D} & 0.5 & 318,023 & 189,422 \\  & 1.0 & 158,950 & 151,767 \\ \cline{1-1}  & 1.5 & 318,118 & 189,891 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Total number of nodes and elements of models. Figure 3: Mesh of the finite element (FE) models: (**a**) Model 2D, \(a=0.5\) mm; (**b**) Model 3D, \(a=0.5\) mm. The effects of internal cracks on the difference in the change of the top surface height are discussed based on the results of stress and strain analysis. First, regarding the stress around the crack, it is required to determine the dominant stress among the normal stress (\(\sigma_{x}\)), which causes Mode I type deformation, and the shear stress (\(\tau_{xz}\)), which leads to Mode II type deformation. Figure 6 shows the distribution of each stress above the cooling passage for each crack length. As shown in Figure 6a, when there is no crack, it can be observed that the proportion comprising the part with large tensile stress in the \(x\)-direction is higher than the stress in the \(z\)-direction and shear stress. As shown in Figure 6b-d, due to the introduction of the crack, the region where the tensile stress in the \(x\)-direction is 150 MPa or more is found to extend over a broad range compared to the tensile stress and shear stress in the \(z\)-direction in the vicinity of the crack tip. The ratio of \(\sigma_{x}/\tau_{xz}\) at the same location near the crack tip in the same model, which roughly corresponds to the ratio of \(K_{\text{I}}/K_{\text{II}}\), is approximately 2.3-2.4. \(K_{\text{I}}\) and \(K_{\text{II}}\) are the stress intensity factors for Model I and Mode II, respectively. From this, Mode I deformation is considered to have a greater effect than Mode II deformation. Additionally, as the crack length increases, the region in which each stress value is large expands. Subsequently, to clarify the influence of stress on the top surface height, Figure 7 shows the strain distribution in the \(z\)-direction above the cooling passage at each crack length. It is found that the strain distribution changes significantly around the crack in all the models. Additionally, in the range \(x=240\)-245 mm and 250-255 mm, the compressive strain in the \(z\)-direction is found to increase as the crack length increases. This is considered to be the cause of the increase in the difference in change with the increase in the crack length.