# 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
Figure 7: Relationship between \(\varphi\) and corrosion pit depth. When processing the fatigue lifespan of the steel wire using Equation (12), the fatigue behavior of the wire is simplified as the integral form of the Paris equation. When considering the corrosion-fatigue coupling effect, further corrections can be made to the crack propagation rate. The modified equation is as follows:  
\[N=\varphi(a_{p})\cdot\int_{a_{0}(a_{p})}^{a_{c}}\frac{1}{(1+\gamma)\frac{da}{dN}}da \tag{15}\]  
where \(da/dN\) represents the crack propagation rate for long cracks, and \(\gamma\) is the coupling coefficient. The value of \(\gamma\) reflects the influence of the corrosion-fatigue coupling effect on the lifespan. The calculation results of comparative cases with different initial crack value and pit depth can be found in Table 2.  
It should be noted that the \(\gamma\) value in the above equation does not represent the specific influence on a particular crack propagation rate. Instead, it introduces the \(\gamma\) value in the form of the Paris integral equation to comprehensively represent the reduction effect of corrosion fatigue on the calculated lifespan. As shown in Table 2, the \(\gamma\) value exhibits significant variability and does not remain within a certain range. The maximum \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Case** & \multirow{2}{*}{**Case  Information**} & \multirow{2}{*}{**Case 1**} & \multirow{2}{*}{**Case 2**} & \multirow{2}{*}{**Case 3**} & \multirow{2}{*}{**Case 4**} & \multirow{2}{*}{**Case 5**} & \multirow{2}{*}{**Case 6**} & \multirow{2}{*}{**Case 7**} & \multirow{2}{*}{**Case 8**} & \multirow{2}{*}{**Case 9**} \\  & & & & & & & & & \\ \hline \multirow{4}{*}{Single fatigue source expansion} & \(a_{p}\)/mm & 0.136 & 0.277 & 0.167 & 0.434 & 0.219 & 0.337 & 0.116 & - \\  & \(a_{0}\)/mm & 0.003 & 0.005 & 0.003 & 0.008 & 0.004 & 0.006 & 0.002 & - \\  & \(\varphi\) & 1.212 & 1.194 & 1.241 & 1.154 & 1.216 & 1.176 & 1.276 & - \\  & \(\gamma\) & 0.160 & 0.049 & 0.125 & 0.051 & 0.013 & 0.000 & 0.037 & - \\ \hline \multirow{4}{*}{Multi-fatigue source expansion} & \(a_{p}\)/mm & 0.395 & 0.105 & 0.177 & 0.118 & 0.181 & 0.155 & 0.152 & 0.238 \\  & \(a_{0}\)/mm & 0.008 & 0.004 & 0.005 & 0.005 & 0.006 & 0.005 & 0.005 & 0.006 \\ \cline{1-1}  & \(\varphi\) & 1.157 & 1.226 & 1.198 & 1.219 & 1.197 & 1.205 & 1.206 & 1.183 \\ \cline{1-1}  & \(\gamma\) & 0.368 & 0.247 & 0.024 & 0.206 & 0.097 & 0.267 & 0.000 & 0.428 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation value of coupling coefficient. Figure 9: Lifespan reduction. value is 0.428, and the minimum value is approximately 0.013, with 0 indicating no effect from corrosion fatigue and symbol of \"-\" in the table indicating blank. In theory, the \(\gamma\) value should remain constant. However, in practice, it is influenced by crack closure effects (especially for small cracks) and the passivation phenomenon on the steel surface. The corrosive medium may not always affect crack propagation. The \(\gamma\) value shows larger values for multiple fatigue crack growth types compared to a single fatigue source growth. This is believed to be primarily due to the promotion of fatigue sources by the corrosive medium.