## 3 Crack Propagation and Dual-Crack Model  
The diameter of the steel wires used in the parallel wire stay cables is 7 mm, and the corrosion pits and cracks on their surfaces are difficult to control and capture through manual means during the expansion process. To clarify the corrosion damage and crack propagation mechanisms of the steel wires under axial loading, a finite element method was employed. Based on the assumptions of isotropy and linear elasticity for the steel wires, this study utilized FRANC3D and ANSYS for numerical simulations. First, a model of uncracked steel wires was established, and the stress distribution before crack initiation was obtained using SOLID185, which is an eight-node solid element in ANSYS 19.0 This model includes both a global model and a local model, with local and global interaction analysis performed using FRANC3D 7.5.5. The local model, which contains only the cracks, will be re-meshed, as shown in Figure 4. Subsequently, FRANC3D automatically calculates the stress intensity factor at the crack tip based on linear elastic fracture mechanics. The crack model generated by FRANC3D will be computed using ANSYS to create a new model that meets the crack propagation requirements. The crack propagation process is based on Equation (1), which calculates the incremental crack growth at the nodes along the crack front. Once the local crack growth increments are obtained, the nodes at the crack front will be updated to new positions, and cubic spline interpolation will be used to generate the new crack front. Then, the updated node positions will be appropriately adjusted along the spline curve to avoid singularities during the computation, as shown in Figure 5. This process will continue until the maximum stress intensity factor reaches the fracture toughness. In the numerical analysis, a stress ratio of 0.5 was utilized, where the steel wire model was fixed at one end and subjected to tensile loading at the other end, resulting in a stress amplitude of 360 MPa. Figure 3: Fatigue fracture cross-section. (**a**) Mild corrosion. (**b**) Moderate corrosion. (**c**) Severe corrosion. In axial tensile load tests, Paris' law is widely used to describe the process of fatigue crack propagation. In this process, the crack propagation rate can be expressed as follows:  
\[\frac{da}{dN}=C(\Delta K)^{m} \tag{1}\]  
where \(\Delta K\) is the range of the stress intensity factor at the crack tip, \(C\) is a constant in Paris' law, and \(m\) is the material exponent, which is typically dependent on material properties and test conditions. [PERSON] [26] recommends the parameter values for \(C\) and \(M\) obtained from experiments on cold-drawn steel wire as 4.1 and 3.0, respectively, with units in MPa and m.  
Before conducting crack propagation simulations, it is necessary to determine the material stress intensity factor threshold and the critical crack length. [PERSON] and [PERSON] obtained the stress intensity factor threshold for high-strength steel wires through experiments and used the least squares method to establish the relationship between the threshold value and stress ratio [28], as shown in Equation (2). For a stress ratio of 0.5 in this study, \(\Delta K_{th}=87.67\) MPa-mm\({}^{0.5}\) can be calculated. \[\Delta K_{th}=-3.43R+5.54 \tag{2}\]  
After determining the threshold value for crack propagation in the steel wire, it is also necessary to determine the critical crack length \(a_{f}\) at which the wire ultimately fractures. In fracture mechanics, it is considered that when a structure experiences fracture failure, the stress intensity factor at the crack tip and the fracture toughness satisfy the following equation:  
\[K=Y\sigma\sqrt{\pi a_{f}}\leq K_{c} \tag{3}\]  
Figure 4: FEM crack insertion procedure. Figure 5: Continuous process analysis of fatigue crack propagation.