# Corrosion-Fatigue Evaluation of Uncoated Weathering Steel Bridges
## 4 Model of Corrosion-Fatigue
### Service Life of Corrosion-Fatigue Crack Initiation  
In solving the service life of C-F crack initiation it is crucial to determine the critical flaw size, beyond which the fatigue can become the major driving force. There are generally two approaches proposed to determine the critical size: (1) The equal rate method, assuming that the critical size reaches at the time when the growth rates of corrosion equals to the one of fatigue [16; 45] and (2) the fatigue limit method, which determines the critical size based on the fatigue limit in terms of SIFs [10; 46]. In the type (1) method, the growth rate of fatigue should be solved at the beginning of the C-F process when the flaw is extremely small. However, the propagation behavior of small cracks is still a controversial issue that can hardly be illustrated. Alternatively, the type (2) method only requires the solution on the well-initiated flaws, which is more concise and practical but may compromise the accuracy to a small extent. To determine the critical size with appreciable accuracy, the above two types of methods were combined in determining the crack initiation life in the present study.  
By setting the SIF same as the fatigue limit, the critical size could be solved using Equation (8). It is worth stating that the fatigue limit here is not the threshold value for the long crack, which will be discussed in detail later. Based on Equation (7), the service life of initiation could be solved, as shown in Equation (10):  
\[t_{i}=\frac{a_{c}^{3}}{B^{3}m_{asp}^{2}I_{p}C^{3a_{c}}C_{p}^{3a_{c}}}+t_{c}, \tag{10}\]  
where \(t_{i}\) is the service life of the C-F crack initiation; \(a_{c}\) stands for the critical depth and \(t_{c}\) is the service life spent during the competition, which is calculated from the moment when critical depth is reached to the moment when the growth rate of the fatigue effect exceeds that of pitting corrosion.  
Verification has been carried out using two typical materials [35], including the 2024-T3 aluminum alloy and the 12% Cr stainless steel. The material properties are listed in Table 2.  
The prediction value is plotted against the test data [22; 23] under various stress ranges, as shown in Figure 5. In the prediction, the critical depth was solved by the fatigue limit listed in Table 2, while the Paris' law was employed to determine the crack growth rate induced by the fatigue effect. The result shows that the prediction curve matched well with the test data, indicating the effectiveness of the proposed method in solving the C-F crack initiation life. It is worth noting that a similar verification was done by [PERSON] et al. [47].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & \(\mathbf{B}\) & \(\mathbf{C_{p}}\) & \(\mathbf{I_{p}}\) (C/s) & \(\mathbf{m_{asp}}\) & \(\Delta K_{th}\) (\(MPa\sqrt{m}\)) \\ \hline Aluminum alloy & \(2.55\times 10^{-4}\) & 1.001–1.02 & \(1.83\times 10^{-9}\) & 1.0 & 2.32 \\ Stainless steel & \(2.61\times 10^{-4}\) & 1.005–1.01 & \(1.51\times 10^{-10}\) & 1.0 & 8.14 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters for the selected aluminum alloy and stainless steel.