## 4 Calculation Results and Discussion  
To facilitate the representation of the crack front, normalization is employed, wherein each point on the crack front is assigned a value between 0 and 1. Figure 8 illustrates a comparison of SIFs at five distinct load contact positions during cyclic motion, while the crack with an initial depth of 0.1 mm remains non-propagated. Observing the SIFs at these positions, it becomes apparent that the magnitudes of \(K_{I},K_{II}\), and \(K_{III}\) are all within the same order of magnitude. This observation substantiates the categorization of the surface crack in the silicon nitride ring channel as a composite crack. Additionally, \(K_{I}\) and \(K_{II}\) of the five load contact positions exhibit an axisymmetrical distribution along the line where the midpoint 0.5 of the crack front is situated. \(K_{III}\) is centrally and symmetrically distributed, with the midpoint 0.5 of the crack front serving as the symmetry center. The \(K_{I}\) maximum values of the five load contact positions are as follows: 0.186 \(\mathrm{Mpa}\sqrt{\mathrm{m}},0.799\)\(\mathrm{Mpa}\sqrt{\mathrm{m}},1.211\)\(\mathrm{Mpa}\sqrt{\mathrm{m}},2.252\)\(\mathrm{Mpa}\sqrt{\mathrm{m}}\) and 0.218 \(\mathrm{Mpa}\sqrt{\mathrm{m}}\), respectively. Notably, all these maxima occur at the midpoint 0.5 of the crack front, demonstrating a pattern of initial increase followed by a subsequent decrease. Based on Figure 6 and Formula (8), it is evident that the midpoint (0.5) of the crack front lies on the midline where \(X=0\). This location corresponds to a region with a substantial load pressure, leading to the maximum value for each load contact position at the midpoint (0.5). The reason behind the highest value among the maximum values of the five load contact positions occurring at position (4) instead of position (3), which is directly above the crack, is the initial inclination of the crack at \(30^{\circ}\). When the load contact is directly above the crack, the load center at \(X=0,Y=0\) remains relatively distant from the crack front point (0.5). However, as the load center shifts to the load contact position (4), it comes into closer proximity to the crack  
Figure 6: Schematic diagram of the working motion of the silicon nitride ceramic ball bearing. Figure 7: Diagram of five positions next to the crack. front point (0.5). Consequently, the load contact position of \(K_{I}\) yielding the maximum value is position (4). The \(K_{I}\) of the five load contact positions are all greater than \(K_{II}\) and \(K_{III}\), except for the load contact position (3). It can be concluded that model I dominates the fatigue crack propagation behavior. With the exception of the load contact position (3), the maximum value of \(K_{II}\) is still observed at the crack front point (0.5). The maximum values of \(K_{II}\) are 0.129 \(\mathrm{Mpa}\sqrt{\mathrm{m}}\), 0.742 \(\mathrm{Mpa}\sqrt{\mathrm{m}}\), 2.662 \(\mathrm{Mpa}\sqrt{\mathrm{m}}\), 1.949 \(\mathrm{Mpa}\sqrt{\mathrm{m}}\) and 0.1247 \(\mathrm{Mpa}\sqrt{\mathrm{m}}\), respectively. These values exhibit an initial increase followed by a subsequent decrease. It is evident that for \(K_{\mathrm{III}}\), almost all five load contact positions in the crack's initial state are less than \(K_{I}\) and \(K_{II}\). In Figure 9, it is evident that after the crack with an initial depth of 0.1 mm undergoes propagation once, the symmetry characteristics of \(K_{I}\) and \(K_{II}\) remain consistent with those observed when the crack is not propagated. However, the maximum value among the five load contact positions no longer resides at the midpoint (0.5) of the crack front; instead, it is situated at points  
In Figure 11a, when the crack at the initial depth of 0.1 mm does not propagate, the maximum effective SIFs of the crack front are determined by the effective SIFs of the load contact positions (3) and (4).