# Research on Multi-Directional Spalling Evolution Analysis Method for Angular Ball Bearing
## 2 Spalling Region Contact FEA Model  
To accurately compute the spalling evolution rate of a bearing while accounting for temperature and lubrication effects, the finite element simulation method is essential. In bearings where the inner ring is mobile, the position of the maximum stress point on the inner ring raceway shifts over time. Conversely, the location of the maximum stress point on the outer ring raceway remains static. This also makes the outer ring of the bearing more susceptible to spalling. Therefore, in the development of a fatigue spalling simulation model for a bearing's outer raceway, simplifications can be made by selectively modeling only those areas identified as potential weak points. Figure 3 illustrates a simplified finite element model of the interaction between a roller and the outer raceway in the spalling region. In reference [20], a three-dimensional crack propagation model is established. In this model, the center of the spalling region aligns with the maximum stress point on the outer ring of the bearing. The findings indicate that the crack torsion angle resulting from mode III crack propagation is generally within 10 degrees. Consequently, it is feasible to develop a two-dimensional crack propagation model that excludes considerations of mode III cracks. This simplified model can effectively analyze the evolution process of spalling faults in angular contact ball bearings. In order to further simplify the analysis process and simulate the subsurface crack propagation process, a two-dimensional analytical model is developed. This model combines  
Figure 3: Finite element model of contact stress simulation at spalling. The dotted line in the figure is the line between the roller and its center of rotation. Figure 2: Flowchart of multi-direction bearing spalling expansion analysis. contact stress analysis of the spalling region with subsurface crack propagation analysis, further improving the computational efficiency. As shown in Figure 4, the constructed spalling simulation model is shown. In the actual calculation, the normal deformation of the bearing roller on the contact surface is much smaller than that of the raceway, so the roller is set as a discrete rigid body in the model. The three faces of the raceway in the model are set to a completely fixed state. The size parameters and constraints of the model are shown in Figure 5.  
In order to perform crack propagation analysis on the model, static analysis methods are employed to calculate the stress intensity factor (SIF) at the crack front. This approach allows for a significant reduction in simulation time while also enabling the model to be more easily integrated with the crack propagation module. The mesh division of the model is shown in Figure 6. In the model, CPS4R cells are used for meshing, and the cell length is refined to 0.002 mm at the contact point. The model uniformly employs quadrilateral meshes to facilitate model iteration and updating. Figure 4: Spalling region contact FEA model. In the figure, the outline of the roller appears as a polygon. Figure 5: Size parameters and constraints of the model. In the figure, the red line represents the position as a completely fixed constraint. The model was constructed for a 7208 AC angular contact ball bearing, and the stress response when the bearing roller rolls over the edge of the spalling region was simulated and calculated, as shown in the diagram in Figure 7.  
The model obtains the contact parameter settings and normal force \(Q_{\mathrm{e}}\) between the roller and raceway based on the calculation results of the elastic hydrodynamic lubrication (EHL) model [21, 22, 23, 24]. The force variation curve \(Q_{\mathrm{e}}\) of the roller calculated by the EHL model is shown in Figure 8.  
Figure 6: Mesh division of spalling region contact FEA model. In the figure, the polygonal lines are the outer outline of the roller before grid division. The cyan lines are the outer outline of the roller after grid division.