# Research on Multi-Directional Spalling Evolution Analysis Method for Angular Ball Bearing
## 3 Fatigue Crack FEA Model  
The established two-dimensional contact stress analysis model is already capable of accurately calculating the stress response in multiple directions around a spalling zone. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Value** \\ \hline Internal diameter of the bearing \(d/\)mm & 40 \\ External diameter \(D/\)mm & 80 \\ Number of rollers \(z\) & 12 \\ Roller diameter \(D_{\rm w}/\)mm & 11.1125 \\ Radial clearance/mm & 0.156 \\ Pitch diameter \(D_{\rm m}/\)mm & 60 \\ Nominal contact angle of balls and outer race \(\alpha/^{\circ}\) & 25 \\ Rotational speed/\(\)rpm & 7000 \\ Radial load \(F_{\rm z}/\)N & 2000 \\ Axial load \(F_{\rm x}/\)N & 10,000 \\ Modulus of elasticity \(E/\)MPa & 214,500 \\ Poisson’s ratio \(\mu\) & 0.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometry and material parameters of 7208 AC bearing. Figure 8: Contact force of the outer ring. Figure 9: Image of 7208 AC bearing. At the same time, the model can be effectively combined with crack propagation analysis methods. In the model, the propagation of cracks is primarily caused by the contact stress between the roller and the edge of the spalling region. Figure 10 shows the von Mises stress simulation of an initial spalling crack with an initial angle of 90\({}^{\circ}\) and a length of 0.01 mm at the edge of the spalling region after a propagation with a deflection angle of 55.92\({}^{\circ}\). At this time, the initial crack depth is 0.02 mm. In finite element simulation software, the contour integral method (CIM) is frequently utilized to calculate the SIF at the crack front. This method is a prevalent numerical technique specifically designed for determining the SIF, making it particularly well-suited for addressing crack problems in finite element analysis. The CIM involves the integration of the stress field along a contour that encircles the crack, thereby allowing computation of the SIF. This technique is versatile and can be applied to a wide range of crack types and loading conditions. For a square spalling zone as shown in Figure 11, the four cracks propagating in different directions can be denoted as F, B, L, and R according to the actual rolling direction of the roller. Through the established two-dimensional model, it is possible to analyze the subsurface crack extension in these four directions. In this model, the crack extension is mainly affected by the strain at the edge of the spalling region. To calculate the crack extension life, the propagation rate of the spalling crack can be obtained by the Paris formula [25]:  
\[\frac{da}{dN}=C(\Delta K)^{m} \tag{1}\]  
where \(da/dN\) is the crack extension rate, \(C\) and \(m\) are material constants, and \(\Delta K\) is the SIF range. In the model, \(\Delta K\) is mainly controlled by the extension of cracks of types I and II and can be defined as follows:  
\[\Delta K=K_{\text{max}}-K_{\text{min}} \tag{2}\]  
Figure 11: Four-direction crack locations. Figure 10: Von Mises stress at the edge of the spalling region. where \(K_{\rm max}\) is the maximum stress intensity factor and \(K_{\rm min}\) is the minimum stress intensity factor. The criterion for determining whether crack extension has begun is given by the following formula:  
\[\Delta K>K_{\rm th} \tag{3}\]  
where \(K_{\rm th}\) is the fatigue crack growth threshold. The propagation rate of spalling cracks is influenced by several factors, including the stress intensity factor (SIF). In the established subsurface crack propagation analysis model of bearings, the analysis is mainly focused on type I and type II cracks. By combining the stress intensity factors \(K_{\rm I}\) and \(K_{\rm II}\) of type I and type II cracks, the equivalent stress intensity factor \(K_{\rm eq}\) can be calculated [25; 26]. \[K_{\rm eq}=\left(K_{\rm I}{}^{2}+K_{\rm II}{}^{2}\right)^{0.5} \tag{4}\]  
By comprehensively considering the two crack types through \(K_{\rm eq}\), the crack propagation rate can be calculated by \(da/dN=C\big{(}\Delta K_{\rm eq}\big{)}^{m}\).