## 2 Dynamic Models Containing Subsurface Mesoscale Fault  
### Stiffness Weakening Coefficient and Strain Energy Release Rate  
Different from steel bearings, FCBBs are usually made through hot isostatic pressing processes, and therefore contain submillimeter or even millimeter subsurface mesoscale cracks. The mesoscale crack extends continuously in the uneven stress field, and finally leads to breakdowns. Considering the load distribution at the crack area, each subsurface mesoscale crack is located within the elliptical region. The wing cracks in the brittle ceramic material are divided into two categories, the transverse crack leading to material cracking and the longitudinal crack leading to weakening of the bearing outer ring stiffness. Here, the outer ring with mesoscale cracks is taken as an example, and the mesoscale cracks are shown in Figure 1.  
Due to the presence of wing crack clusters in the outer ring, this paper discusses the weakening of the outer ring stiffness of FCBBs caused by mesoscale cracks, mainly for longitudinal cracks. The longitudinal cracks are difficult to observe or detect directly, but have significant impacts on the stiffness of related components. It was found in previous studies [1; 27] that the momentum inertia with mesoscale faults was proportional to the original inertia, as shown below  
\[EI_{C}=\mu_{c}EI \tag{1}\]  
where \(E\) is the modulus of elasticity of ceramic material, \(I\) is the section moment of inertia for a healthy outer ring, and \(I_{C}\) is the section moment of inertia for an outer ring with cracks. Combined with the different weakening of the bearing outer ring stiffness by different crack lengths, the composite stiffness weakening coefficient \(\mu_{\rm c}\) is obtained for mesoscale faults as follows:  
\[\mu_{\rm c}=\left[\frac{1}{\left(1/\mu_{1}\right)^{2/3}+\left(1/\mu_{2}\right)^{ 2/3}}\right]^{3/2} \tag{2}\]  
where \(\mu_{1}\), \(\mu_{2}\) represent the stiffness weakening coefficients of an FCBB outer ring containing crack and defects, respectively, where \(\mu_{1}=EI_{\rm C1}/EI\), \(\mu_{2}=EI_{\rm C2}/EI\), \(I_{\rm C1}\) and \(I_{\rm C2}\) represent the moment of inertia of FCBB outer ring with cracks and defects, respectively. Since the main cause of subsurface mesoscale crack generation in the rotation of FCBBs is the bending load, the applied bending moment not only promotes the occurrence of tip crack expansion, but also increases the strain energy of the structure. The cracks mentioned in this paper are cracks of opening mode. A schematic diagram of subsurface cracks and local sections of the outer ring is shown in Figure 2.  
The strain energy release rate of the outer ring of the FCBB under external additional load is expressed as  
\[SERR=\frac{K_{\rm I}{}^{2}}{E} \tag{3}\]  
where \(K_{\rm I}\) is the strength coefficient of the crack calculated by the open displacement method. \(K_{\rm I}\) is expressed as  
\[K_{\rm I}=M_{\rm e}S_{\rm b}\sqrt{\frac{\pi d}{Q}} \tag{4}\]  
where \(M_{\rm e}\) is the surface correction factor, \(S_{\rm b}=3M/ba^{2}\), \(d\) is the radial depth of the subsurface cracks, \(a\) is the length from the top of the crack to the outer surface of the outer ring, and \(b\) is the length from the center of the crack to the side of the outer ring. In general,  
Figure 1: Schematic diagram of subsurface mesoscale cracks and local cross-sections of the outer ring. Figure 2: Schematic diagram of subsurface mesoscale crack of outer ring (**a**): radial section of mesoscale crack; (**b**): axial section of mesoscale crack. the surface correction factor \(M_{\rm e}\) is related to the crack-outer ring thickness ratio \(d/a\), the crack length-outer ring width ratio \(c/b\), and the crack tip angle \(\varphi\). [PERSON] and [PERSON] gave an expression for the surface correction factor [28]. \[M_{e}= \begin{cases}H\left(\frac{d}{a},\varphi\right)\\ F\left(\frac{d}{a},\frac{c}{b},\varphi\right)\end{cases}  \tag{5}\]  
The functions \(H(\cdot)\) and \(F(\cdot)\) in the formula can be calculated using the reference [29].