# Corrosion Fatigue Assessment of Bridge Cables Based on Equivalent Initial Flaw Size Model
## 3 Corrosion Fatigue Performance of Steel Wire
### Quantitative Analysis of Corrosion Fatigue Coupling Effect  
The impact of corrosion fatigue coupling on the fatigue lifespan of the steel wire manifests in multiple stages. Before the coating fails, corrosion fatigue accelerates the electrochemical reactions of the coating. After the coating fails, the appearance of corrosion pits and the continuous action of the corrosive medium lead to an earlier initiation period of cracks and a reduction in the crack growth threshold, resulting in a decrease in the proportion of the crack initiation lifespan. The influence of corrosion fatigue on crack propagation rate is more evident with the propagation rate of long cracks increasing with the intensification of corrosion. To facilitate a comparative analysis of the impact of corrosion-fatigue coupling on fatigue lifespan, without considering the multi-stage lifespan, the corrosion pits can be simplified as single-edge cracks, and the fatigue lifespan \(N_{F}\) of the corroded steel wire can be calculated using Equation (12):  
\[N_{F}=\varphi\cdot\int_{a_{0}}^{a_{c}}\frac{1}{C(\Delta K)^{\pi}}da \tag{12}\]  
where the unilateral crack depth uses the EIFS value, and \(\varphi\) is a correction factor that takes into account the form nucleation lifespan as well as the small crack extension lifespan. In the above equation, \(a_{0}\) is simplified as a parameter that is affected by the depth of the corrosion pit, while \(\Delta K\) is mainly affected by the area of the corroded section. For ease of application, \(C\) and \(m\) are treated as initial crack depths in accordance with the long crack extension parameter. The above equation can be transformed into the following form:  
\[N_{F}=\varphi(a_{p})\cdot\int_{a_{0}(a_{p})}^{ac}\frac{1}{C_{lc}(\Delta K)^{m _{lc}}}da \tag{13}\]  
\[\left\{ \begin{aligned} \varphi(a_{p})&=\frac{1.08234}{a _{p}^{2}\varphi^{0.64}},\text{for single source extension type}\\ \varphi(a_{p})&=\frac{1.11154}{a_{p}^{2}\varphi^{ 0.6451}},\text{for multi source extension type}\end{aligned} \right. \tag{14}\]  
The detailed relationship between \(\varphi\) and the variation in corrosion pit depth can be seen in Figure 7. As the corrosion pit depth tends toward zero, the value of \(\varphi\) shows an exponential increase. This is because in the absence of corrosion pits, the stress amplitude applied to the smooth specimen (180 MPa) is lower than the fatigue limit, and its lifespan is calculated as infinite cycles. Therefore, the value of \(\varphi\) tends toward infinity. When the corrosion pit depth approaches the critical crack depth \(a_{c}\), the value of \(\varphi\) gradually tends toward zero. In this case, the steel wire fractures immediately upon fatigue, resulting in a fatigue lifespan of zero. The relationship between \(\varphi\) and the variation in corrosion pit depth is fitted using a power function. The coefficients of determination (R\({}^{2}\) values) for the fitted curves of the two types of \(\varphi\) values are 0.85387 and 0.73303, respectively. As shown in Figure 8, Equations (13) and (14) are used to calculate the parameters of seven fatigue specimens. The calculated fatigue lifespan values correspond well to the experimental values with an error of around 10%. Since the specimens used for corrosion fatigue in this study are from the same batch of corroded steel wire as those used by [PERSON] et al. [26], the equation mentioned above can be used to calculate the mechanical fatigue lifespan \(N_{\text{{cal,air}}}\) of the steel wire with the same fracture dimensions as the corroded fatigue specimens. The calculated result is shown in Figure 9, where the length of the descending straight-line segment represents the reduction in lifespan caused by the coupling effect of corrosion fatigue. This value is equal to \(N_{\text{{cal,air}}}\) minus the corrosion fatigue test value. It can be seen that for the multi-source extended steel wire, the influence of corrosion-fatigue coupling effect on its lifespan is more pronounced. Figure 8: Comparison of model calculation results.