## 2 The Cohesive Force Fatigue Damage Constitutive Model  
### Damage Calculation  
The cohesive force model is established on the basis of damage mechanics, assuming the existence of small-sized cohesive zones near the crack tip, within which micro-cracks initiate, propagate, and eventually coalesce, promoting crack tip advancement and eventually forming macroscopic cracks [25; 26]. The cohesive force model does not fully simulate the actual physical cracking process but rather idealizes the crack tip as a region where cohesive forces prevent crack propagation, as illustrated in Figure 1. This model defines the relationship between the cohesive force \(\sigma\) on the crack surface and the crack opening displacement \(\delta\) as the Traction-Separation Law (TSL), while the energy released by the cohesive zones during the cracking process is defined as fracture energy. The Traction-Separation Law encompasses various forms such as bilinear traction-separation law, exponential traction-separation law, polynomial traction-separation law, etc. The choice of traction-separation law depends on the application. Among these, the bilinear traction-separation law has the widest applicability. The complete process of the bilinear traction-separation law can be divided into three stages: elastic, damage, and failure, as illustrated in Figure 2.  
In the Figure 2, \(\sigma_{n}\) represents the normal cohesive forces, \(\sigma_{max}\) represents the maximum normal cohesive forces, \(k_{0}\) represents the normal initial stiffness, \(k_{n}\) initial unloading stiffness, \(G_{n\!f}\) represents the normal fracture energy, \(\delta_{max}\) represents the normal critical separation, \(\delta_{0}\) represents the normal opening displacement. The constitutive relationship between cohesive force and opening displacement under cyclic loading for the bilinear traction-separation law is depicted in Figure 3. Under cyclic loading, the initial stiffness of the cohesive force element continuously degrades. This phenomenon is reflected in Figure 3 as the decreasing slope of the loading curve, eventually leading to the complete failure of the cohesive force element when the stiffness degrades to 0. During unloading in the cyclic loading phase, the cohesive force element model  
Figure 1: Cohesive zone model (CZM). Figure 2: Bilinear traction separation law. exhibits a linear relationship between traction and unloading deformation. Additionally, the unloading curve points towards the loading origin, indicating the absence of plastic deformation in the cohesive force element. According to the relationship in Figure 3, the initial cohesive strength parameter \(\sigma_{max}\) was used to establish the constitutive relationship of CZM for cumulative damage under cyclic loading. Within this framework, the current cohesion strength is defined as:  
\[\sigma_{\text{max}}=\sigma_{\text{max,0}}(1-D) \tag{1}\]  
where: \(\sigma_{max}\) is the strength of the cohesive force unit in the current state; \(\sigma_{max,0}\) is the initial strength of the cohesive unit; \(D\) represents the amount of damage in the current state. During the cyclic loading process, the cohesive force element experiences both cyclic loading-induced damage \(D_{c}\) and monotonic loading-induced damage \(D_{m}\).