## 4 Effect of Fault Size Extension on Bearing Dynamic Characteristics
### Bearing Outer Ring Stress Change and Crack Extension Trend
#### 4.2.1 Maximum Shear Stress  
In this part, we investigated the variations in shear stress as the rolling elements enter and exit the fault point. Then, the impact of continued operation on the crack propagation direction in the outer ring of the faulty bearing is determined. Taking the fault-free bearing and bearing fault type 2 (0.5 mm \(\times\) 0.3 mm) as examples, the study investigates the maximum shear stress trends in the rolling elements in the load  
Figure 8: Equivalent stress distribution. Figure 9: Schematic diagram of failure points when rolling elements roll in and roll out. zone of a normal bearing as they roll on the inner raceway of the outer ring, as well as during the fault entry, fault traversal, and fault exit phases. Figure 10a,c shows the variation curves of the maximum shear stress for adjacent shell elements in fault-free bearings made from GCr15 and AISI304 materials, respectively. Figure 10b,d presents the maximum shear stress variation curves for fault type 2 in GCr15 and AISI304 bearings. In cases where the middle element consists of multiple shell elements, the curve for the middle element represents the average stress of these multiple units. From Figure 10a-d, it can be observed that in the rolling process along the raceway of fault-free bearings, the maximum shear stress for the GCr15 material reaches 387 MPa, while for AISI304, it is 380 MPa. When the faulty bearings enter and exit the fault state, both materials exhibit a sharp increase in maximum shear stress. Specifically, the maximum shear stresses for the rolling elements exiting the fault (Element 2) are 763 MPa for GCr15 and 750 MPa for AISI304, both of which are higher than the maximum shear stresses of 732 MPa and 711 MPa, respectively, for the rolling elements entering the fault (Element 1). Throughout the process of entering the fault, rolling over the fault, and exiting the fault, the maximum shear stress of the middle element at the 'pit' initially increases gradually, then experiences a sudden drop and stabilizes, followed by a sharp increase before gradually declining. However, the maximum shear stress at the middle element of the 'pit' remains consistently lower than that of normal bearings. This is attributed to the rolling elements not contacting the bottom of the 'pit' while rolling over the fault. Figure 10: Max shear stress curve. (**a**) Material GCr15 trouble-free bearing; (**b**) material GCr15 fault type 2; (**c**) material AISI304 trouble-free bearing; (**d**) material AISI304 fault type 2. Therefore, when a bearing fails, the alternating shear stress caused by the rolling elements repeatedly passing through the fault point will lead to crack propagation along the path aligned with the rotational direction of the inner ring of the bearing. In other words, crack propagation occurs earlier when exiting the fault point (Unit 2) than when entering the fault point (Unit 1). A further investigation of the stress fluctuations for fault-free bearings and fault types 1-5 during fault entrance and exit was performed to analyze how crack propagation speed changes with increasing fault severity, as shown in Figure 11.  
From Figure 11a-d, it is evident that as the fault size increases, there is a significant upward trend in the maximum shear stress for both materials during the entry and exit of the fault. Notably, the maximum shear stress curves for fault types 3-5 in GCr15 and for fault types 4-5 in AISI304 exhibit a serrated fluctuation pattern, which is attributed to the occurrence of macro-slipping in the rolling elements. The appearance of macro-slipping in fault type 3 for GCr15, while not observed in fault type 3 for AISI304, indicates that the material properties influence the timing of macro-slipping occurrence.