## 1 Introduction
### Finite Element Analysis on the Corrosion Fatigue Damage Evolution Process of Cable Strand  
The deterioration degree of cable fatigue performance is closely related to the damage state of the cable body, so the deterioration process of the cable fatigue performance is essentially the result of the coevolution of all the cable body damage. To study the damage evolution process of the cable under a corrosion environment, a section of parallel wire cable composed of 37 steel wires will be taken as the research object in this section. We simulate that the member in nominal mean stress is 400 MPa, nominal stress amplitude is 300 MPa, loading frequency is 5 Hz, and nominal stress ratio is 0.158 to simulate the corrosion environment. For convenience, the steel wires in the cable body are numbered 1-37, as shown in Figure 1. Assuming that the initial damage of the wire in the cable satisfies the Weibull distribution, Table 1 shows the initial damage value of the wire satisfying this distribution function. Figure 5 shows the change in the stress distribution and damage distribution of the steel wire in the cable body during the process of fatigue deterioration. The stress distribution is the state of the cable section under average stress. As can be seen from Figure 5, due to the changes in the effective bearing area of the cable section caused by damage, the phenomenon of stress redistribution will occur in the steel wire inside the cable. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & **Initial** & \multirow{2}{*}{**Number**} & **Initial** & \multirow{2}{*}{**Number**} & **Initial** & \multirow{2}{*}{**Number**} & **Initial** \\  & **Damage** & & **Damage** & & **Damage** & & **Damage** \\ \hline
1 & 0.00532 & 11 & 0.00753 & 21 & 0.00352 & 31 & 0.00625 \\
2 & 0.00375 & 12 & 0.00626 & 22 & 0.03827 & 32 & 0.00027 \\
3 & 0.00259 & 13 & 0.00377 & 23 & 0.00441 & 33 & 0.00352 \\
4 & 0.00337 & 14 & 0.00224 & 24 & 0.00173 & 34 & 0.00436 \\
5 & 0.00142 & 15 & 0.03755 & 25 & 0.00023 & 35 & 0.00516 \\
6 & 0.00398 & 16 & 0.00371 & 26 & 0.00493 & 36 & 0.00433 \\
7 & 0.00426 & 17 & 0.00611 & 27 & 0.00026 & 37 & 0.00082 \\
8 & 0.00152 & 18 & 0.00522 & 28 & 0.00125 & / & / \\
9 & 0.00392 & 19 & 0.00478 & 29 & 0.00368 & /The steel wire with less damage will bear more load, which promotes the development of corrosion fatigue damage. According to the theory of continuum damage mechanics, the stress of the steel wire in the cable body can be considered to be distributed based on the proportion of the effective bearing area of each steel wire to the total effective bearing area of the cable. Therefore, to explore the change law of stress redistribution in the cable body, the changes in the effective bearing area ratio of the steel wire under different loading times are counted here. The formula for calculating the effective bearing area ratio of the steel wire can be expressed as (14). \[R=\frac{S_{sf-i}}{\sum\limits_{k=1}^{n}S_{ef-k}}=\frac{1-D_{i}}{\sum\limits_{k =1}^{n}(1-D_{k})}=\frac{1-D_{i}}{n(1-D_{AVE})} \tag{14}\]  
where \(R\) is the proportion of the effective bearing area of the wire to the total effective bearing area of the cable, \(S_{sf-i}\) is the effective bearing area of the \(i\) wire, \(D_{i}\) is the damage  
Figure 5: Changes of stress and damage distribution in the cable body during damage evolution. (**a**) Initial damage state; (**b**) damaged state after under 6.5 \(\times\) 10\({}^{3}\) loads; (**c**) damaged state after under 1.3 \(\times\) 10\({}^{4}\) loads. to the \(i\) steel, \(D_{AVE}\) is the average damage to the cable, \(n\) is the number of wires in the body of the rope. The uneven stress distribution of the steel wire is directly related to the relative difference between the proportion of the effective bearing area of the steel wire. Figure 6 lists the proportion of the effective bearing area of the steel wire with maximum and minimum damage to the total bearing area of the cable under the current loading times.