## 3 Joint Simulation of Crack Propagation on Channel Surface of Silicon Nitride Ceramic Ball Bearing
### Theory of Crack Propagation and Calculation Method of Franc3D-V8.4  
In structural materials, cracks propagate under the influence of external forces through three primary modes, as illustrated in Figure 5. Mode I, often referred to as \"the opening mode\", is predominantly induced by normal stress, resulting in the displacement of the crack surface perpendicular to the crack plane. Mode II, also known as \"the sliding mode\", is generated by inplane shear. Mode III, or \"the tearing mode\", is produced by out-of-plane shear, wherein the crack surface displacement occurs within the crack plane and runs parallel to the edge of the crack on the leading surface. \(K_{I}\), \(K_{II}\), and \(K_{III}\) represent the SIFs corresponding to mode I, mode II, and mode III, respectively. In the working process of a silicon nitride ceramic ball bearing, the surface of the ring channel experiences positive pressure and frictional force due to the rolling contact of the ceramic ball. If a surface crack is present on the ring channel's surface, the crack becomes subject to contact stress caused by positive pressure and frictional force, resulting in SIFs at the crack front. When the amplitude of the effective stress intensity factor, \(\Delta K_{eff}\), surpasses the threshold value, \(\Delta K_{th}\), of the silicon nitride material but remains below its fracture toughness, the crack will expand steadily. If it is below the threshold value, the crack will propagate extremely slowly. Conversely, when it exceeds the threshold value, the crack will propagate unstably, leading to rapid fracture. Ref. [23] indicates that when the contact load is distant from the crack, then \(\left(K_{eff}\right)_{\text{min}}=0\). Therefore, \(\Delta K_{\text{eff}}\) can be determined using \(\left(K_{\text{eff}}\right)_{\text{max}}\) and expressed as follows:  
\[\Delta K_{\text{eff}}=\left(K_{\text{eff}}\right)_{\text{max}}-\left(K_{\text {eff}}\right)_{\text{min}}=\left(K_{\text{eff}}\right)_{\text{max}}-0=\left[ \left(K_{I}+\left|K_{\text{III}}\right|\right)^{2}+2K_{II}\right]^{2}\right]^{ 1/2} \tag{1}\]  
The stable expansion interval is called the Paris region, and the formula of crack growth rate in this region can be calculated by the Paris formula, and its expression is as follows:  
\[\frac{d_{a}}{d_{N}}=c\left(\
abla K_{\text{eff}}\right)^{m} \tag{2}\]  
where a is the crack propagation length, N is the cycle movement life, c and m are the material constants of silicon nitride. In earlier studies, [PERSON] et al. [36] gave the HIP silicon nitride ceramic material \(c=1.01\times 10^{-21}m=8\). Therefore, the cyclic movement life can be expressed as:  
\[N=\int_{a_{e}}^{a}\frac{d_{a}}{\frac{d_{a}}{d_{N}}}=\int_{a_{e}}^{a_{f}}\frac {d_{a}}{C\left(\
abla K_{eff}\right)^{m}} \tag{3}\]  
where \(a_{e}\) and \(af\) is the initial length of the crack and the length after propagation, respectively. Figure 5: Basic crack model diagram. (**a**) Mode I; (**b**) Mode II; (**c**) Mode III. Franc3D-V8.4 software uses the M-integral method to calculate the SIFs(\(K_{I},K_{II},K_{III}\)).