## 3 Results and Discussion
### Three-Dimensional Model  
Figure 12 shows the \(z\)-direction displacement for trace line 1 of Model 3D. Macroscopically, no difference is observed between the four types of test specimens, the same as for Model 2D. Therefore, in order to reveal the influence of the existence and size of the crack on the \(z\)-direction displacement, the difference in the \(z\)-direction displacement of a model with and without a crack is shown in Figure 13. It is found that the difference increases as the crack grows. The difference in the \(z\)-direction displacement differs from that of the 2D model. In the 3D model, the height rises once at around \(x=200\) mm, and then decreases. The minimum height is at \(x=247.5\) mm, which is the location of the crack. Subsequently, it decreases gradually. These height distribution differences are in the order of several nanometers to several tens of nanometers, which are large with respect to the vertical DHM resolution of 0.2 nm. Therefore, it may be possible to detect cracks by ascertaining this change through DHM. 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, similar to the 2D Model, 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 (\(\tau_{xx}\)), which leads to Mode II type deformation. Figure 14 shows the distribution of each stress above the cooling passage for the model with no crack and the models with cracks. The top surface of the model is trace line 1. As shown in Figure 14a, when there is no crack, it can be observed that the proportion comprising the part with a large tensile stress in the \(x\)-direction is higher than the stress in the \(y\)- and \(z\)-directions and the shear stress. As shown in Figure 14b, due to the introduction of the crack, the region where the tensile stress in the \(x\)-direction is 40 MPa or more, is found to extend more widely compared to the tensile stress in the \(y\)- and \(z\)-directions and shear stress in the vicinity of the crack tip. From this, the Mode I deformation is considered to have a greater effect than the Mode II deformation. Next, to clarify the influence of stress on the top surface height, Figure 15 shows the strain distribution in the \(z\)-direction above the cooling passage for the model with no crack and the model with a crack. The top surface of the model is trace line 1. The strain distribution in the range \(x=230\)-\(265\) mm is shown. It is found that the strain distribution changes  
Figure 11: The strain distribution (\(\varepsilon_{z}\)) of Model 2D-TBC, \(a=0\), \(0.5\), \(1.0\), \(1.5\) mm. significantly around the crack in the range \(x=245\)-250 mm. Furthermore, at the upper part of the crack tip at \(x=247.5\) mm in Figure 15, tensile strain in the \(z\)-direction is confirmed. In addition, in the range \(x=240\)-245 mm and 250-255 mm, the part with a large compressive strain, which is lower than \(-0.0001050\), in the \(z\)-direction of the model with a crack is found to be slightly broader than that in the model without a crack. This is considered to be the reason for the descent in the range \(x=240\)-245 mm and 250-255 mm and the inclination changing sharply in Figure 13.  
Figure 14: The stress \(\sigma_{x}\), \(\sigma_{y}\), \(\sigma_{z}\), and shear stress \(\tau_{xz}\) distribution above the cooling passage of Model 3D, \(a=0\), 1.5 mm. (**a**) \(a=0\) mm (No crack); (**b**) \(a=1.5\) mm. Figure 12: The \(z\)-displacement of the surface of Model 3D along trace line \(1\). The \(z\)-displacement implies the change in \(z\)-direction due to loading. Figure 13: The difference in the \(z\)-displacement between the models with and without a crack of Model 3D along trace line \(1\).