## 3 Examples
### Influence of the Initial Pitting Morphology on the Fatigue Life  
To verify the accuracy and advantage of the proposed numerical simulation method, the fatigue life of the corresponding steel wires with a regular pit is calculated and compared with the above  
Figure 11: Damage evolution process and stress distribution of pre-corroded steel wire on a cross section. (**a**) step = 51; (**b**) step = 151; (**c**) step = 195. Figure 12: Time-varying curves of the stress intensity factor and defect size in steel wire. results. First, the pit shape factor \(\lambda\) proposed by [PERSON] et al. [40] is used to determine the morphology of the regular pit, which can be defined as:  
\[\lambda\ =\ c/a \tag{19}\]  
where \(c\) and \(a\) represent the width and depth of the corrosion pit, respectively. Then, hemi-ellipsoidal pit models with shape factors of 0.25, 0.5, 1.0, 1.5 and 2.0 are established. The mass loss rate of all hemi-ellipsoidal pits is 0.25%, which is consistent with the random pit generated by CA technology in Figure 8a. The example of established models of the pre-corroded steel wire with a regular pit is shown in Figure 13. Finally, the UMAT developed in Section 2.3 is used to calculate the fatigue life of each model, and the results are shown in Table 2.  
From Table 2, at the same stress range, even if the mass loss rate is the same, the fatigue life of the pre-corroded steel wire increases gradually with an increase in the pit shape factor, that is, a reduction in the pit depth. In addition, compared with wide-shallow pits, narrow and deep pits have a greater impact on the fatigue performance of steel wires, which is consistent with the conclusions given in [17; 41]. From a qualitative point of view, this behavior is because the stress concentration of hemi-ellipsoidal pit gradually decreases with increasing of \(\lambda\), which reduces the fatigue damage accumulation rate and eventually leads to an increase in the fatigue life of the corroded steel wire. Figure 14 shows the S-N curves of the steel wires with a hemi-ellipsoidal pit. Compared with the corresponding S-N curve of the steel wire with a random corrosion pit, the fatigue life of the steel wire with a random pit is less than that of the other two hemi-ellipsoidal pit models, even though the \begin{table}
\begin{tabular}{c c c c c} \hline \hline S/MPa & \(n\) & \(\lambda\) & \(x\)/mm & \(N_{\epsilon}\) \\ \hline \multirow{4}{*}{335} & \multirow{4}{*}{100,000} & 0.25 & 0.90246 & 4,900,000 \\  & & 0.5 & 0.5685 & 5,000,000 \\  & & 1.0 & 0.358 & 5,100,000 \\  & & 1.5 & 0.27331 & 5,200,000 \\  & & 2.0 & 0.2256 & 5,400,000 \\ \hline \multirow{4}{*}{418} & \multirow{4}{*}{10,000} & 0.25 & 0.90246 & 1,060,000 \\  & & 0.5 & 0.5685 & 1,080,000 \\ \cline{1-1}  & & 1.0 & 0.358 & 1,090,000 \\ \cline{1-1}  & & 1.5 & 0.27331 & 1,100,000 \\ \cline{1-1}  & & 2.0 & 0.2256 & 1,200,000 \\ \hline \multirow{4}{*}{520} & \multirow{4}{*}{5000} & 0.25 & 0.90246 & 235,000 \\  & & 0.5 & 0.5685 & 245,000 \\ \cline{1-1}  & & 1.0 & 0.358 & 250,000 \\ \cline{1-1}  & & 1.5 & 0.27331 & 255,000 \\ \cline{1-1}  & & 2.0 & 0.2256 & 280,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Life of pre-corroded steel wire with a regular corrosion pit (\(\eta\ =\ 0.25\%\)). Figure 13: Pre-corroded steel wire model with regular a corrosion pit (\(\lambda\) = 1.0). maximum depth of the random pit is less than that of the hemi-ellipsoidal pit with a shape factor of 0.25. Based on the simulative results of these models, the reason why the random pit model has a small depth but a low fatigue life is investigated. Figure 15a,b are stress contours of random pit model and the hemi-ellipsoidal pit model. From the point of view of stress, the maximum stress of the random pit model is 1.162 times that of the hemi-ellipsoid pit model with shape factor of 0.25. The excessive stress concentration of the random pit model is caused by the irregular sharp angle in the pit, such as point A in Figure 15a.