## 1 Introduction
### Establishment of Damage Model Considering Microstructure Weakening  
The parallel wire cable is generally composed of several parallel wires, and its structural characteristics are similar to fiber bundles. [PERSON] et al. [41] studied the carrying capacity of non-damaging slings based on the fiber bundle strength theory. In the theory of fiber bundle strength, as shown in Figure 2, it is assumed that the fiber bundle is composed of a large number of parallel fibers of the same length, each fiber is independent of the other, and the mechanical properties of the fiber bundle are completely dependent on the properties of each fiber. In addition, all fibers are considered to deform equally under stretching due to the clamping of the two ends of the bundle [42]. Therefore, on the premise of not considering fretting wear between wires, the whole cable is simplified into a parallel system based on the fiber bundle strength theory; that is, the cable is regarded as a parallel system composed of multiple wires. Since the cable is carried as a whole in service, the overall deformation of the wires in the cable body should be consistent. It is also assumed that the elongation of the wires in the cable body is equal when the cable is subjected to fatigue load. In addition, considering the balance between calculation efficiency and calculation accuracy, the finite element model of the cable corrosion fatigue failure process is simulated by embedding the corrosion fatigue damage evolution model into local steel wire, considering the specific structural details of local corrosion pits and microcracks (shown in Figure 3). At the stage of short crack growth, the damage variable D can describe the damage failure process of steel wire at the macro scale and reflect the evolution behavior of the short crack population at the micro scale. Therefore, the selected representative body element (RVE) contains enough micro short cracks to reflect the average damage state of the steel wire at the macro scale. Therefore, the multiscale homogenization method establishes the relationship between the microscopic short crack density and the macroscopic continuous  
Figure 3: Short crack propagation in the representative unit. (**a**) Microcrack length at time \(t\); (**b**) Microcrack length at \(t+\Delta t\); (**c**) short cracks in the unit. Figure 2: The model diagram fiber bundle strength. damage field. According to the concept of effective bearing area, the micro-damage \(D_{i}\) of the \(i\)th crystalline grain in the representative bulk element can be expressed as Formula (2):  
\[D_{i}=\frac{\delta\widetilde{S}_{i}}{\delta S_{i}}=\frac{x_{i}d_{0}}{2d_{0}^{2}} \tag{2}\]  
where \(\delta S_{i}\) is the nominal bearing area of fine grain, \(\delta\widetilde{S}_{i}\) is the defect area of fine grain, \(x_{i}\) is the internal crack length in the fine grain, \(d_{0}\) is the grain size. The experimental results show that the pitting pit size remains almost constant after the short crack propagation stage [42]. Therefore, the steel wire damage evolution equation at the stage of short crack growth can be expressed as Formula (3):  
\[D=\frac{3\sqrt{3}Kd_{0}^{2}}{8P}\left(1-\frac{S_{pit}}{S}\right)\cdot\left( \frac{exp(2 Pd_{0}(t-t_{0}))-1}{P^{2}(t-t_{0})^{2}}-\frac{2d_{0}}{P(t-t_{0})}-2 d_{0}^{2}\right)+\frac{S_{pit}}{S} \tag{3}\]  
where \(S\) is the cross-sectional area of the wire, \(S_{pit}\) is the defect area caused by corrosion at time \(t_{0}\). The fact that the weakening of the microstructure is projected to the macrostructure is the weakening of the effective bearing area of the section. In the computational model, the structure is weakened by the initial damage. The impact of the corrosion pit and microcrack on structural points was considered in the initial damage. From the corrosion fatigue failure form of the cable, the failure of the cable is caused by the failure of the steel wire inside the cable, and the failure of the steel wire is caused by the weakening of the local microstructure of the steel wire.