## 2 The Cohesive Force Fatigue Damage Constitutive Model
### Damage Calculation
According to the cyclic damage evolution model of cohesive force elements proposed by [PERSON] and [PERSON] [22], the calculation formula for damage \(D_{c}\) induced by cyclic loading is as follows:  
\[D_{C}=\frac{|\Delta\vec{u}|}{\delta_{\Sigma}}\bigg{[}\frac{\overline{T}}{ \sigma_{\text{max}}}-C_{f}\bigg{]}H(\Delta\overline{u}-\delta_{0}) \tag{2}\]  
where: \(\Delta\vec{u}\) represents the increment of effective separation displacement at a certain time, \(\Delta\vec{u}=\Delta\overline{u}_{t}-\Delta\overline{u}_{t-\Delta t}\), \(\Delta\overline{u}_{t}\) represents the effective separation displacement at time \(t\), \(\Delta\overline{u}_{t-\Delta t}\) represents the effective separation displacement at time \(t\)-\(\Delta t\); \(\delta_{\Sigma}\) denotes the accumulated cohesive force length, used for normalizing the increase in effective displacement of cohesive force elements, typically being a multiple of the initial cohesive force length \(\delta_{0}\); H represents the Heaviside function, which takes a value of \(1\) when the cumulative displacement \(\Delta\overline{u}\) exceeds the initial cohesive force length \(\delta_{0}\), and \(0\) when \(\Delta\overline{u}\) is less than \(\delta_{0}\). Through the Heaviside function, we define the onset of damage accumulation, namely when the cumulative displacement \(\Delta\overline{u}\) exceeds the initial cohesive force length \(\delta_{0}\), fatigue damage begins to accumulate; The initial cohesive force length \(\delta_{0}\) corresponds to the separation displacement of the cohesive force element when the load reaches \(\sigma_{max,0}\); \(\overline{T}\) represents the equivalent traction force of the cohesive force element model, and its calculation formula is:  
\[\overline{T}=\sqrt{{T_{n}}^{2}+{T_{t}}^{2}/(2 eq^{2})} \tag{3}\]  
\(C_{f}\) represents the fatigue limit, which defines the ratio of the cohesive force fatigue limit strength \(\sigma_{f}\) to the initial cohesive strength \(\sigma_{max,0}\). Below this level, the material can withstand an infinite number of cyclic loads without failure, while damage accumulates only when the stress level exceeds the fatigue endurance limit. The calculation formula for the fatigue limit \(C_{f}\) is:  
Figure 3: Bilinear traction separation law under cyclic loading. \[C_{f}=\frac{\sigma_{f}}{\sigma_{\max,0}} \tag{4}\]  
Equation (4) illustrates that the fatigue limit \(C_{f}\) falls within the range of 0 to 1. The calculation formula for damage \(D_{m}\) under monotonic loading is:  
Figure 2.0  
The total damage of the cohesive force element under cyclic loading is composed of the damage induced by cyclic loading, \(D_{c}\), and the damage induced by monotonic loading, \(D_{m}\). Its calculation is given by:  
Figure 2.1  
In fatigue experiments, due to the low stress levels and cyclic loading intensity, the cumulative fatigue damage caused by each loading is relatively small. If calculated according to actual conditions, the slow accumulation of fatigue damage will result in very low computational efficiency. [PERSON] et al. [27] proposes the use of a damage amplification factor \(A\) to accelerate the process of fatigue accumulation, allowing N/\(A\) cycles of fatigue loading to substitute for \(N\) cycles. Relevant studies have verified the applicability of the damage amplification factor [28]. Therefore, in the subsequent research, this paper uses the damage amplification factor to improve computational efficiency. The calculation formula for damage \(D_{c}\) induced by cyclic loading with the addition of the damage amplification factor is:  
Figure 2.2  
Under fatigue loading, the working states of cohesive force elements include four types: loading, unloading, contact, and compression, as shown in Figure 2.3 Here, \"contact\" refers to the condition where the cohesive force element undergoes complete damage after a certain number of cycles, and consequently, the cohesive force element no longer functions, leading neighboring elements to transition into a state of pure contact.