## 2 Method Proposed
### Verification of the Proposed Method  
To verify the developed UMAT, a median stress vs. number of cycles to failure (\(S\)-\(N\)) curve chosen from [34] is expressed as:  
\[\text{lg}N_{c}\ =\ 23.480-6.649\text{lg}S \tag{12}\]  
which is obtained from fatigue experiments of 12 high-strength bridge wire samples under four different cycle tensile forces with a constant stress range.  
According to Equation (3), \(D_{f}(N_{c})\ =\ 1\), where \(N_{c}\) is the fatigue life of a high-strength bridge wire, is chosen to fit the S-N curve to obtain the value of the model parameters. Therefore, by using the obtained parameters above, Equation (3) can be expressed as:  
\[\frac{dD_{f}}{dN}\ =\ \Big{(}1-D_{f}\Big{)}^{-1}\Big{[}\frac{S}{3772.53}\Big{]}^ {6.649} \tag{13}\]  
Based on Equation (13), an FE model of a high-strength bridge wire without initial damage is developed in ABAQUS software using UMAT to predict the fatigue life of the bridge wire under stress ranges ranging from 335 MPa to 670 MPa. Besides, the initial elastic modulus is 212,000 MPa and Poisson's ratio equals 0.3.  
The simulation results are shown in Figure 5 and are compared with the S-N curve given in [34]. These data show that the predicted results using UMAT agree well with the S-N curve, which means that the developed UMAT and fitted model parameters are reasonable and effective.  
In addition, since the main object of this paper is to simulate the fatigue damage evolution of corroded steel wires, verification against a structure with defects needs to be made. Based on Equation (13), an FE model of a high-strength bridge wire with a hemispherical pit is developed in ABAQUS software using UMAT to predict the fatigue life of the bridge wire under stress ranges ranging from 335 MPa to 670 MPa. The radius of the hemispherical pit equals 1 mm.  
The simulation results are compared with the results that are calculated by a numerical model developed in [7]. According to [7], the equation for predicting the fatigue life \(N_{c}\) of corroded steel wires can be deduced from the Paris law with an integral operation:  
\[N_{c}\ =\ \frac{1}{CS^{m}}\int_{x_{0}}^{x_{c}}\left(Y\cdot\sqrt{\pi x}\right)^{ -m}dx \tag{14}\]  
where, \(C\) and \(m\) are undetermined parameters that equal \(1.39\times 10^{-12}\) and 3.3 respectively for steel wire with the diameter of 7 mm when the stress ratio \(R\) is 0.5 according to [35]; \(x\) is the crack length, \(x_{0}\) is the initial crack length that equals 1 mm in this paper and \(x_{c}\) is the critical crack length that is calculated by the fracture toughness that equals 65.7 MPa\(\cdot\sqrt{m}\)[6]; \(S\) is the stress ranges ranging from 360 MPa to 670 MPa; \(Y\) is the crack shape factor. According to [35], \(Y\) can be expressed as:  
\[Y\ =\ 1.4835-2.9219\left(\frac{x}{D}\right)+5.9602\left(\frac{x}{D}\right)^{2}+1.6304\left(\frac{x}{D}\right)^{3} \tag{15}\]  
where \(D\) is the diameter of steel wire and equals 7 mm in this paper.  
The above two results are compared in Figure 6. The comparison shows that the UMAT written in this paper can achieve the expected goal and that the calculation results are also consistent well with the results calculated by the numerical model proposed in the literature [7; 35].  
Figure 5: S-N curve of steel wire without the initial defect under uniaxial tensile fatigue loads.