# A Theoretical Model with the Effect of Cracks in the Local Spalling of Full Ceramic Ball Bearings
## 1 Introduction  
Compared with steel bearings, full ceramic bearings have been widely studied because of their lighter weight and higher hardness. The performances of full ceramic bearings and hybrid ceramic bearings have been compared with those of steel bearings and have showed advantages in speed [1] and service life [2; 3; 4; 5]. Full ceramic bearings have been widely used in high-speed lathes, aero-engines, chemical machinery, and other fields. In the special environment of high-speed heavy-duty, low noise, and high precision, full ceramic bearings have gradually taken the main positions. The main failure modes of full ceramic bearings are spalling and cracking. Compared with steel, the ceramic material is higher in brittleness and lower in fracture toughness, so spalling occurs more frequently on the contact surface under high Hertz pressure, which leads to small cracks around the surface defects [6; 7; 8; 9; 10]. The study of the effect of the local spalling cracks on the vibration response of full ceramic bearings is valuable for the health monitoring of full ceramic bearings during operation. The spalling defect is more complicated on the rings of full ceramic bearings than on steel bearings as the failure modes of ceramic material are different from the steel modes. [11] Cracks usually appear in spalling areas, and as a result, the model with the effect of cracks in spalling is more accurate. At the same time, considering that the ceramic material is more sensitive to the local damage, the analysis on the vibration signals of the outer ring can be of great help for the fault detection on the outer ring. The majority of these models [12; 13; 14] are designed to model bearings with line spalling defects that occur at the early stages of bearing failure. The extended spalling defects that occur at later stages due to the successive rolling element that passes over the defect have received very little attention [15]. Most models of bearing outer ring spalling adopt the method of establishing the geometric model of flaw and simulating the deformation of flaw to calculate the time-varying rigidity of bearing outerring fault [16; 17; 18; 19]. Because the hardness and the fracture toughness of ceramic materials are very different from those of steel materials, the contact deformation of ceramic bearing outer ring cannot be accurately simulated by establishing the geometric model of spalling defects. Previous research [20; 21; 22] has deeply discussed the cracks in ceramic ball bearings, and it considered that the cracks were caused by the contact stress between the rolling elements and the bearing rings during the operation of the equipment. However, the bending stress exerted by the rolling elements on the outer ring during the process of passing through the spalling position of the outer ring was not taken into account in the study, and the bending stress may also be the reason for the initiation and propagation of the cracks. Researchers [23; 24; 25; 26; 27; 28] have conducted deep analyses on the crack formation mechanism of the cracks under contact stress. On the basis of the principle of fracture mechanics, the influence of cracks caused by bending moment on stiffness can be analyzed by calculating the strain energy caused by cracks [29; 30; 31]. According to the failure principles of ceramic materials, when the spalling occurs on the ring of the full ceramic ball bearing, cracks also appear in the spalling area. The generation and propagation of cracks are significant to the dynamic characteristics of the full ceramic bearing. However, studies on the effect of the cracks in spalling are quite rare so far, to the author's knowledge. In this paper, an improved nonlinear dynamic model with the effect of cracks in spalling is presented, which can be used to analyze the vibration responses of bearings with cracks and spalling on the outer ring.