# Corrosion Fatigue Assessment of Bridge Cables Based on Equivalent Initial Flaw Size Model
## 2 Corrosion Fatigue Lifespan Model Based on EIFS Theory
### Corrosion Fatigue Lifespan of Steel Wire  
The crack nucleation lifespan (\(N_{n}\)) can be calculated by Equation (1) [32]:  
\[N_{n}=\frac{9\Delta K_{th}^{2}G}{2E(\sigma-\sigma_{f})^{2}a_{0}} \tag{1}\]  
where \(G\) represents the shear modulus and \(E\) represents the elastic modulus, with a relationship of \(G=E/(2(1+v))\), where \(v\) is the Poisson's ratio and is taken as 0.3. When considering the local stress concentration effect of the corrosion pit, as the corrosion test does not alter the material's elastic region constitutive relationship, from the perspective of linear elastic fracture mechanics, the threshold value of the material \(\Delta K_{th}\) for crack initiation does not change with the fatigue limit \(\sigma_{f}\). For high-strength steel wire, the fatigue limit \(\sigma_{f}\) is taken as  
Figure 2: EIFS schematic of corroded wire cross-section. 247.5 MPa [37]. Since the fatigue initiation often occurs at the bottom of the corrosion pit, the effect of stress concentration should be considered when calculating the stress \(\sigma\) in the above equation. It can be calculated using the following equation:  
\[\sigma=K_{t}\cdot\sigma_{s} \tag{2}\]  
\[\sigma_{s}=\Delta F_{a}/S_{corr} \tag{3}\]  
\[K_{t}=\frac{1+6.6\cdot a_{p}/c_{p}}{1+2\cdot a_{p}/c_{p}} \tag{4}\]  
where \(K_{t}\) is the stress concentration coefficient, \(\Delta F_{a}\) is the load value of the pulsatile stress amplitude, \(\sigma_{s}\) is the stress of the corroded steel wire, \(S_{cord}\) is the cross-section area of the corroded steel wire, and \(a_{p}\) and \(c_{p}\) are the depth and width of the pit. Due to the large number of small cracks and no obvious propagation law, it is difficult to quantify the evolution mode. In some studies in the literature [38; 39], an equivalent empirical model based on the Paris formula is given considering the calculation method of fatigue lifespan:  
\[N_{sc}=\frac{a_{0}^{(1-m_{sc}/2)}-a_{sc}^{(1-m_{sc}/2)}}{C_{sc}(2\sigma_{s})^{ m_{sc}}\beta^{m_{sc}}\pi^{m_{sc}}/2(\frac{m_{sc}}{2}-1)} \tag{5}\]  
where \(\beta\) is the shape parameter of small crack propagation, and the value is taken as \(0.5\sqrt{\eta}\)[40]. \(C_{sc}\) and \(m_{sc}\) are material constants for short crack propagation, which are \(9\times 10^{-11}\) and \(3.3\), respectively [41]. When the initial defect depth \(a_{0}\) reaches the critical depth of small crack \(a_{sc}\), the small crack will be transformed into a long crack. \(a_{sc}\) refers to the crack depth when the crack depth reaches the threshold value of the stress intensity factor, which is calculated by Equation (6):  
\[a_{sc}=\frac{\Delta K_{th}^{2}}{\pi Y^{2}{(2\sigma_{s})}^{2}} \tag{6}\]  
The solution of the form factor formula for the arc crack \(Y\) is given by [PERSON] et al. [42]:  
\[Y(\frac{a}{D})=0.92\cdot\frac{2}{\pi}\cdot\sqrt{\frac{2D}{\pi a}\tan\frac{ \pi a}{2D}}\cdot\frac{0.752+1.286(\frac{a}{D})+0.37(1-\frac{7a}{2D})^{3}}{ \cos\frac{7a}{2D}} \tag{7}\]  
where \(a\) is the crack size and \(D\) is the wire diameter. The fatigue long crack growth lifespan \(N_{lc}\) is calculated by the Paris formula:  
\[N_{lc}=\int_{a_{sc}}^{a_{c}}\frac{1}{C_{lc}(\Delta K)^{m_{lc}}}da \tag{8}\]  
\[\Delta K=K_{\max}-K_{\min}=Y(2\sigma_{s})\sqrt{\pi a} \tag{9}\]  
where \(C_{lc}\) and \(m_{lc}\) are the material constants of long crack propagation, which are \(6\times 10^{-11}\) and \(3.3\), respectively [41]. The critical crack depth \(a_{c}\) is calculated using the observed fracture value. \(\Delta K\) is the equivalent stress intensity factor at the pit bottom. The fatigue lifespan \(N_{F}\) of corroded steel wire is shown as follows:  
Figure 2:0  
The procedure for calculating the EIFS of corroded steel wires is illustrated in Figure 2:1 The detailed calculation steps are listed below:  
1. The fracture morphology characteristics of steel wire were obtained through a corrosion fatigue test, e.g., depth of erosion pit \(a_{p}\), width of erosion pit \(c_{p}\), area of fracture \(S_{coor}\), depth of main crack \(a_{c}\), and depth of fracture \(D_{coor}\).