# Research on TBM Cutterhead Crack Damage and Fatigue Reliability
## 2 Analysis of TBM Cutterhead Crack Damage Characteristics
### Calculation of the Stress Intensity Factor and an Analysis of the Influencing Factors  
The stress intensity factor (SIF) is a description of the stress distribution in the crack tip region, which can be used to evaluate whether the structure is in a state of unstable growth [29] and is an important parameter for evaluating the growth life of structures with cracks [30]. There are many calculation methods that are used to find the SIF, including stress extrapolation, displacement extrapolation, and interactive integration [31]. For a simple structure, the SIF can be determined using classic analytical methods, but the cutterhead shape and stress are complex; thus, it requires numerical methods and sophisticated analysis. With the continuous improvement of computation capacity, the finite element method provides accurate and reliable calculations of the SIF [32]. However, due to the large volume of the cutterhead, it continues to require a lot of time to simulate the whole model since the calculation efficiency is low. To decrease the computational time, a widely used submodeling technique [33; 34] was applied in this study. The crack-prone area of the cutterhead was taken as the submodel, where the selected area and crack insertion point is shown in Figure 3a,b, respectively. The long half-axis of the crack was \(c_{1}\) and the short half-axis was \(a_{1}\), and the position angle and parameter diagram are shown in Figure 3c. The displacement calculated by the FEM analysis on the global model was applied on the submodel cut surfaces (Figure 3d) [35]. Then, the mesh was refined (Figure 3e). The numbers of grids and nodes were 187,113 and 132,995, respectively. Figure 3: Submodeling technical schematic: (**a**) schematic diagram of the submodeled area, (**b**) schematic diagram of the crack insertion point, (**c**) schematic diagram of the crack position angle and shape parameter, (**d**) the boundary conditions, and (**e**) the crack grid after remeshing. Based on the static strength analysis of the cutterhead, the SIF of the crack was calculated and the influence of the different variables on the SIF was analyzed. In the crack initiation stage, the position angle and the shape ratio were random; this study mainly analyzed the influence of the crack position angle and shape ratio on the SIF, and then investigated the damage characteristics of the crack. First, \(a_{1}\) = 6 mm and \(c_{1}\) = 12 mm were selected to calculate the SIF under different position angles, where the results of the stress intensity factors \(K_{\rm I}\), \(K_{\rm II}\), and \(K_{\rm III}\) are shown in Figure 4.  
It can be seen from Figure 4 that the absolute values of \(K_{\rm I}\) and \(K_{\rm III}\) at the deepest crack tip reached the maximum and \(K_{\rm II}\) was approximately zero, which indicates that the crack propagation in the crack tip was mainly of the open and tearing types and that there were three kinds of propagation modes on the crack surface. According to the fracture mechanics, the SIF at the front of the crack had a great influence on the crack propagation. Therefore, it is necessary to analyze the relationship between the equivalent SIF and the crack angle, which could be found using the expression of equivalent SIF:  
\[K_{eq}=\sqrt{\left(K_{\rm I}+K_{\rm II}\right)^{2}+K_{\rm III}^{2}/(1-2\lambda )}, \tag{1}\]  
where \(\lambda\) is the Poisson's ratio of the material, which was taken to be 0.3 (Table 1). Under the same loading and constraints, the equivalent SIF of the front end of the crack was calculated by changing the crack position angle and parameters. The calculation results are shown in Figure 5.  
The results (Figure 5) clearly show that with the change of the crack position angle, the equivalent SIF in the fore-end of the crack also changed, and the closer the position angle is to 80\({}^{\circ}\), the smaller the equivalent SIF was, which indicated that the crack growth trend was relatively weak.