## 5 Conclusions  
This paper presents the completion of the joint simulation using Franc3D and ABAQUS based on the measured friction coefficient obtained from the rolling friction experiment. The study focuses on the variation in the SIFs along the crack front with crack propagation at five load contact positions during cyclic motion. Additionally, it investigates the determination of the maximum effective SIFs and the impact of the initial crack depth on the cycle life and direction of crack propagation. The following conclusions can be drawn:  
1. The SIFs associated with the surface cracks on the silicon nitride ring channel exhibit similar orders of magnitude throughout the propagation process, providing evidence of their compound crack nature. The \(K_{I}\) and \(K_{II}\), corresponding to five loading contact positions, demonstrate an axisymmetric distribution along the line positioned at the midpoint (0.5) of the crack front. In contrast, \(K_{III}\) exhibits central and symmetrical distribution with the midpoint (0.5) of the crack front serving as the symmetry center.
2. During the progression of ring channel surface crack propagation, the \(K_{I}\) and \(K_{II}\) maximum values at five load contact positions exhibit an initial increase followed by a subsequent decrease. Nevertheless, after two instances of crack propagation, there is a noticeable alteration in both the load contact positions and crack front points where the \(K_{I}\) and \(K_{II}\) maximum values are situated, manifesting prominently during the cyclic movement. Concurrently, the \(K_{I}\) and \(K_{II}\) maximum values gradually decrease as the crack continues to propagate.
3. The maximum effective SIFs at the crack front can indicate the amplitude of the effective SIFs. However, considering various stages of initial depth crack propagation, it is not necessarily determined by the same load contact position for the maximum effective SIFs at the ring channel surface crack front.
4. For crack propagation with the same maximum propagation size, the deeper the initial crack depth, the longer the cycle life of the silicon nitride ring channel surface crack. Additionally, the cycle life increases with the growth in the number of crack propagations.
5. The deflection angle of ring channel surface crack propagation decreases with the increase in the initial crack depth, and the difference decreases gradually. The deflection direction is towards the inside of the ring channel. Consequently, surface cracks of three different initial depths are less likely to cause material spalling.  
Figure 15: Comparison of propagation angle of cracks with initial depths of (**a**) 0.1 mm, (**b**) 0.15 mm and (**c**) 0.2 mm.  
**Author Contributions:** Conceptualization, P.W. and S.L.; methodology, P.W.; software, P.W.; validation, P.W., Y.W. ([PERSON]), C.W. and Y.W. ([PERSON]); formal analysis, P.W.; investigation, P.W., Y.Z. and Y.W. ([PERSON]); resources, P.W.; data curation, P.W.; writing--original draft preparation, P.W.; writing--review and editing, P.W.; visualization, [PERSON] ([PERSON]) and [PERSON] ([PERSON]); supervision, S.L.; project administration, P.W.; funding acquisition, S.L. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the National Natural Science Foundation of china under grant U23A20631, and by the Sichuan Provincial Department of Education university platform project under grant GK201920 and GK201915.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author due to privacy.  
**Conflicts of Interest:** The authors declare no conflict of interest.