# Corrosion Fatigue Assessment of Bridge Cables Based on Equivalent Initial Flaw Size Model
## 3 Corrosion Fatigue Performance of Steel Wire
### Fatigue Lifespan Evaluation of Steel Wire
 \tag{11}\]  
The relationship between \(a_{0}\) and corrosion pit depth follows a power function, and the fitting results are satisfactory, with respective coefficients of determination of 0.90963 for single fatigue source propagation and 0.83048 for multiple fatigue source propagation. In Equation (11), when the corrosion pit depth is zero, the value of \(a_{0}\) represents the equivalent initial crack depth of material micro-defects, such as polishing, scratching, and local residual stress. Under the same corrosion pit depth, the value of \(a_{0}\) for multiple-source propagation is larger than that for single-source propagation, which explains that the fatigue lifespan under multiple-fatigue source propagation is smaller than that under single fatigue source propagation. As the corrosion pit depth increases, the values of \(a_{0}\) for both types of propagation approach each other. Therefore, after the corrosion pit depth reaches approximately 0.4 mm, the fatigue lifespan of the two types of damaged steel wires converges. Figure 6 depicts the variation of crack depth with fatigue lifespan for single fatigue source propagation and multiple fatigue source propagation types, considering the influ  
Figure 5: Relationship of EIFS with the corrosion pit depth. ence of corrosion pits. It can be observed that the EIFS values for both propagation types show minor fluctuations with different corrosion pit depths. The maximum and minimum values are within the same order of magnitude. This pattern is also reflected in the critical short crack depth \(a_{sc}\) and the critical long crack depth \(a_{c}\). However, there are significant differences in the crack initiation lifespan \(N_{n}\) corresponding to EIFS values. For single fatigue source propagation, the minimum \(N_{n}\) value is 635, and the maximum is 8210, while for multiple fatigue source propagation, the minimum \(N_{n}\) value is 300, and the maximum is 12,145. Regarding the crack initiation lifespan, as the corrosion pit depth increases, the initiation lifespan decreases for both propagation types. For the crack propagation lifespan, the lifespan is relatively similar for different corrosion pit depths. This is because factors influencing the calculation of the long crack lifespan include the crack growth threshold value \(\Delta K_{th}\), applied stress amplitude, and critical crack depth \(a_{c}\), where \(\Delta K_{th}\) determines the maximum depth \(a_{sc}\) of short cracks. When considering the material's linear elastic fracture mechanics behavior under the influence of corrosion surface pits, the main crack propagation plane already encompasses the two-dimensional pit surface during the propagation  
Figure 6: Relationship between crack depth and lifespan for different expansion types. (**a**) single fatigue source expansion; (**b**) multi-fatigue source expansion. of long cracks. The impact of corrosion pits is only manifested in the weakening of the cross-section, leading to increased stress amplitude.