## 1 Introduction  
With the advantages of high hardness, low thermal deformation and low abrasive grain, full ceramic ball bearings (FCBBs) have important applications and prospects in fields such as aero engines, food processing machinery and artificial replacement hip joints [1; 2; 3]. However, when FCBBs are subjected to long-term uneven stress fields as well as alternating temperature conditions, internal early failures usually evolve into surface faults such as wear and spalling in the loaded zones [4; 5; 6], which not only weaken the overall stiffness characteristics of the bearing [7] but also affect the service life of the whole system. While the time-varying contact stiffness is an important index to characterize the operating condition of the bearing, and when subsurface cracks occur in the bearing, it leads to a reduction in contact stiffness [8; 9], which is not conducive to the accurate operation of the ceramic bearing rotor system. Since the fault location and damage mechanism of FCBBs are different from those of steel bearings, the traditional steel bearing outer ring fault dynamics model cannot be directly transferred and applied to FCBBs. Considering that ceramic materials are more sensitive to local damage, it is significant to investigate the damage mechanism of subsurface mesoscale cracks in FCBBs. For the study of the failure modes of various components of bearings, the bearing dynamic models have been widely adopted. [PERSON] et al. [10] proposed a method to calculate the quasi-static load distribution and changing stiffness of bearings with raceway defects of different depths, lengths, and surface roughness, and verified the correlation between low- and high-frequency events of ball entering and exiting defects and the dynamic characteristics of bearings. [PERSON] et al. [11] investigated the wear-loss behavior of bearings using a whole-life test method to reveal the bearing surface damage and failure evolution process. For the study of subsurface failures in steel bearings, [PERSON] et al. [12] studied subsurface fatigue crack expansion under rolling contact fatigue based on stress intensity factor calculations to assess fatigue crack expansion. [PERSON] et al. [13] proposed a finite element model to study the effect of horizontal and inclined subsurface cracks on the contact characteristics of roller bearings, and obtained the relationship between contact deformation and crack dimensions (length, depth, and angle). [PERSON] et al. [14] investigated the damage pattern of white etched cracks on ceramic bearings, and the test results showed that the stresses were significantly concentrated near the defect area. [PERSON] et al. [15] carried out research on FCBB, and proposed a stiffness evaluation model to obtain a mapping relationship between subsurface cracking and dynamic response. The mechanical properties of ceramic materials have also been extensively investigated using the relevant theories of fracture mechanics [16; 17; 18]. [PERSON] et al. [19] considered the nonlinear contact stiffness and damping of ceramic bearings in terms of energy conservation. In this way, a more accurate differential equation for the vibration of angular contact ball bearings was developed. Our previous studies [20; 21; 22] also provided insight into cracking in FCBBs, and the cracks were proved to be related to contact stresses between the rolling element and the bearing collar during the operation of the equipment. However, the bending stresses exerted on the outer ring by the rolling element during its passage through the outer ring failure location was not taken into consideration, which was also a main cause for the generation and extension of internal defects. Researchers [23; 24] have carried out in-depth analyses on the mechanism of crack formation under contact stresses. Based on the principles of fracture mechanics, the effect of cracking on stiffness due to bending moments can be analyzed through changes in strain energy [25; 26].