## 4 Effect of Fault Size Extension on Bearing Dynamic Characteristics
### Bearing Outer Ring Stress Change and Crack Extension Trend
#### 4.2.2 Principal Stresses  
Analyzing the maximum shear stresses at different failure severities indicates that the direction of crack expansion is consistent with the direction of rotation of the inner raceway. Under the action of alternating shear stress, the crack gradually extends and branches at the crack edges, eventually leading to spalling. Therefore, further research into the trend of crack branching based on crack propagation is necessary. As noted from the maximum shear stress analysis, the maximum shear stress during fault exit is higher than during fault entry. Thus, this section investigates the trend of principal stress changes when the rolling element exits the fault. The variation in principal stress at the crack edges influences the direction of crack bifurcation. To investigate the crack bifurcation direction in faulty bearings, the trends in the first principal stress (X direction), second principal stress (Y direction), and third principal stress (Z direction) for both materials under different fault degrees were analyzed. The trends of these three principal stresses are illustrated in Figure 13.  
Figure 12: Maximum shear peak stress curve of different types of faults. (**a**) Material GCr15; (**b**) material AISI304. Analyzing Figure 13a-f, the variation rule of principal stress is obtained as follows:  
1. Both materials exhibit tensile stress in the first principal stress when the rolling elements exit the fault, showing similar patterns. During the repetitive rolling of the rolling elements along the outer ring raceway, alternating tensile-compressive stress  
Figure 13: Trend in three principal stresses. (**a**) Material GCr15 first principal stress (in X direction). (**b**) Material AISI304 first principal stress (in X direction). (**c**) Material GCr15 second main stress (Y direction). (**d**) Material AISI304 second main stress (Y direction). (**e**) Material GCr15 third principal stress (Z direction). (**f**) Material AISI304 third principal stress (Z direction). regions are generated at the crack edges. The influence of the superimposed stresses leads to crack bifurcation in the X direction at the crack propagation edge. 2. The second principal stress for both materials is compressive stress, which increases after the outer ring experiences a fault compared to that of normal bearings. This compressive stress can suppress crack bifurcation, with the second principal stress demonstrating a strong ability to inhibit bifurcation in the Y direction. 3. The third principal stress for both materials is also compressive stress, which decreases after the outer ring experiences a fault compared to that of normal bearings. While this compressive stress can suppress crack bifurcation, the third principal stress exhibits a weaker ability to inhibit bifurcation in the Z direction. From the above analysis of the three principal stresses, it can be concluded that under alternating shear stress, cracks on the outer ring's inner raceway extend in the direction of rotation towards the inner race. Under the action of tensile-compressive stress, the likelihood of crack branching in the direction of rotation of the inner race (X direction) is higher compared to branching in the Y and Z directions. To provide a clearer visualization of the principal stress variation trends under different fault types, the peak values of the three principal stresses at the point of fault exit for various fault severities were compared, as shown in Figure 14.  
From Figure 14, it can be observed that after the bearing experiences a fault, the first principal stress exhibits tensile stress, which increases with the expansion of the bearing crack, accelerating crack bifurcation in the X direction while the rolling element is in a micro-slipping state. However, when macro-slipping occurs, the tensile stress fluctuates significantly as the crack expands, indicating that noticeable slipping greatly impacts the stress during the rolling element's exit from the fault.