## 6 Conclusions  
This study developed cohesive fatigue damage elements and subsequently constructed a BP neural network to invert the parameters of cohesive fatigue damage elements corresponding to cold joints with different casting intervals. Finally, the fatigue fracture process of concrete components with cold joints under low stress amplitude fatigue loading was simulated. The research results demonstrate that:  
1. The developed cohesive fatigue damage elements are suitable for simulating the fatigue fracture behavior of concrete components with cold joints under low stress amplitude fatigue loading. The small error between the fatigue life calculated based on the fatigue damage element parameters obtained from the inverse process using the BP neural network and the experimental results verifies the reliability of the cohesive fatigue damage elements and the inverse parameters using the BP neural network. The developed cohesive fatigue damage elements and the corresponding fatigue damage parameters can be used to analyze the damage of concrete lining with cold joints under aerodynamic fatigue loading and predict the timing of cold joint cracking.
2. For cold joints with casting intervals of 2, 4, and 8 h, the corresponding initial normal strengths are 0.58 MPa, 0.36 MPa, and 0.28 MPa, respectively. The fatigue limits are 0.48, 0.42, and 0.34, respectively. The initial cohesive lengths are 0.0007 mm, 0.0004 mm, and 0.0002 mm, respectively. The initial strength and fatigue dam  
Figure 11: Fatigue cracking process and cold joint damage status.  
Figure 10: Reasons for discrete fatigue life of different pouring intervals.  
age initiation conditions of the cold joint interface decrease with the increase of pouring interval.
3. The variability of fatigue life of concrete components with cold joints decreases with the increase of casting interval. The thickness of the cold joint interface is the main factor affecting the variability of fatigue life. Subsequent research can investigate the influence of interface thickness on the fatigue life of concrete components with cold joints. Cold joint interface elements first suffer damage near the pre-existing cracks. With the increase in the number of fatigue loading cycles, the damage value of cohesive elements gradually increases to 1. At this point, the cohesive elements completely fail and macroscopic cracks occur. When the cracks extend to a certain extent, the component reaches its fatigue life and cannot withstand the load anymore.  
H.-L.F.: Methodology, Software, Data processing, Writing, original draft. H.-S.D.: Conceptualization, Methodology, Supervision, Project administration. Y.-M.W.: Validation, Formal analysis, Writing--review & editing. Y.-B.Z.: Validation, Formal analysis, Writing--review & editing. C.-D.X.: language problem check. All authors have read and agreed to the published version of the manuscript.  
This study was financially supported by the National Natural Science Foundation of China (Grant Nos. 51978668, 52108367 and 52478424) and the Key Science and Technology Projects of Hunan Provincial Department of Education (23A0014).  
Not applicable.  
Data Availability Statement: Data will be made available on request.  
The authors thank the Advanced Research Center, Central South University, for providing the experiment conditions. The authors also express special thanks to the editors and anonymous reviewers for their constructive comments.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.