## 1 Introduction
An incident reported by [PERSON] and [PERSON] [17] on the Shinkansen line in Japan illustrates a typical case where cold joints and initial micro-cracks in the lining expanded and eventually interconnected under the aerodynamic loading generated by high-speed train operation, leading to the overall detachment of concrete blocks in the lining and causing substantial economic losses. Similarly, numerous similar incidents have occurred in high-speed railway tunnels in China. Hence, scholars have begun to pay attention to initial defects in tunnel linings, especially cold joint defects. Our research team conducted fatigue fracture experiments on concrete components containing cold joints under low stress level and low stress amplitude fatigue loading [18]. The experimental background was the damage and cracking situation of linings containing cold joints caused by aerodynamic fatigue loading from high-speed trains. The experimental results indicate that under conditions of low stress level and low fatigue amplitude, aerodynamic fatigue loading significantly reduces the load-bearing capacity of concrete components containing cold joints. This suggests that aerodynamic fatigue loading induces fatigue damage in cold joint defects, and with an increase in the number of fatigue loading cycles, concrete components containing cold joints will eventually fracture. This experiment proves that aerodynamic fatigue loading generated by high-speed train operation will lead to a decrease in the load-bearing capacity of linings containing cold joint defects and eventually cause them to crack. This research conclusion can serve as an explanation for the cracking of plain concrete sections in high-speed railway tunnels, as plain concrete linings sections usually have stable surrounding rock conditions and external environments, making it difficult to determine the cause of their cracking. However, there is currently no method to analyze the damage and cracking status of linings containing cold joint defects under the aerodynamic fatigue loading generated by high-speed trains. Analyzing the damage and cracking status of linings containing cold joint defects under aerodynamic fatigue loading from high-speed trains through indoor experiments requires a significant amount of time and financial support [19]. However, numerical simulation methods can greatly improve analysis efficiency based on the powerful computational capabilities of computers, thereby overcoming the challenges faced by indoor experiments. In previous research work, the researchers of this study have already verified the applicability of using cohesive force elements to simulate cold joints. However, the cohesive force elements used in previous research are only applicable to simulate the fracture of concrete containing cold joint defects under monotonic loading conditions. Currently, there is no research on numerical simulation of fatigue damage in concrete containing cold joint defects. Some studies have proposed and applied cohesive force fatigue damage elements, providing reference experience for simulating fatigue damage in concrete containing cold joint defects. [PERSON] et al. [20] first proposed a fatigue cohesive force model, describing the evolution of fatigue damage as the softening of the cohesive zone envelope and assuming that a series of softened envelopes are independent of each other. Based on the exponential monotonic cohesive force model proposed by [PERSON] [21], [PERSON] and [PERSON] [22] further proposed an improved cyclic cohesive force model, in which they redefined the loading and unloading paths in the irreversible model and introduced an external variable D to describe fatigue damage. [PERSON] et al. [23] proposed an improved Roe Siegmund model to solve the problem of fatigue crack propagation in elastic fracture mechanics, and conducted mixed mode fracture fatigue tests on austentic stainless steel to verify the proposed improved cyclic cohesion model. [PERSON] et al.