## 2 Fatigue Constitutive Model of Materials
### Fatigue Constitutive Model of Corroded Steel Bar  
It was shown that under fatigue loading, the steel bar is at relatively low stress levels and most of the fatigue deformation is within the elastic phase [15], and the influence of corrosion on the elastic modulus of the steel bar are relatively small and can be neglected [16]. Therefore, it is assumed that the fatigue elastic modulus of a corroded steel bar remains unchanged. The principal structure model of the steel bar used in this study is based on the ideal elastoplastic model, and the principal structure relationship of an uncorroded steel bar under an arbitrary number of fatigue loading is calculated in Equation (8) [11]:  
\[\sigma_{\text{s}}(N)=\left\{\begin{array}{ll}E_{\text{s}}\varepsilon_{\text{ s}}(N)&\Delta\varepsilon_{\text{s,r}}(N-1)<\varepsilon_{\text{s}}(N)\leq \varepsilon_{\text{y}}(N)\\ f_{\text{y}}(N)&\varepsilon_{\text{s}}(N)>\varepsilon_{\text{y}}(N)\end{array}\right. \tag{8}\]  
where \(E_{\text{s}}\) is the initial elastic modulus of steel bar. \(\Sigma_{\text{s}}(N)\) and \(\varepsilon_{\text{s}}(N)\) are the stress and strain of the steel bar at any fatigue loading cycles \(N\). \(f_{\text{y}}(N)\) and \(\varepsilon_{\text{y}}(N)\) are the residual yield strength and yield strain of the steel bar, respectively, at any fatigue loading cycles \(N\). \(\Delta\varepsilon_{\text{s,r}}(N-1)\) is the residual strain of steel bar after \(N-1\) fatigue loading cycles, which can be calculated using Equation (9):  
\[\Delta\varepsilon_{\text{s,r}}(N-1)=\frac{\left[f_{\text{y}}(N)-f_{\text{y}}(N-1 )\right]}{E_{\text{s}}} \tag{9}\]The residual yield strength of steel bar after \(N\) fatigue loading cycles can be calculated using the formula in the literature [17], which is improved through combining the form of the \(S\)-\(N\) double logarithmic curve of steel bar, as shown in Equation (10):  
\[f_{y}(N)=f_{y}[1-\frac{\text{lg}N}{\text{lg}N_{\text{f}}}(1-\frac{\sigma_{\text {max}}}{f_{y}})] \tag{10}\]  
where \(f_{y}\) is the initial yield strength of the steel bar. \(\sigma_{\text{max}}\) is the stress of the steel bar at the upper limit of the fatigue load. \(N_{\text{f}}\) is the fatigue life of the steel bar.  
Due to a lack of relevant experimental data, this study only considers the effect of steel bar corrosion on its yield strength, and the degradation formula of the yield strength \(f_{\text{yc}}\) after corrosion of the steel bar was obtained according to the literature [18], as shown in Equation (11):  
\[f_{\text{yc}}=\frac{1-1.049\eta}{1-\eta}f_{y} \tag{11}\]  
where \(\eta\) is the mass corrosion degree of the steel bar, \(\eta=(m_{0}-m)/m_{0}\times 100\%\). \(m_{0}\) is the mass of the steel bar before corrosion, and \(m\) is the mass of the steel bar after corrosion.  
Based on Equations (10) and (11), the calculation formula for the residual yield strength of a corroded steel bar after \(N\) fatigue loading cycles is obtained as shown in Equation (12):  
\[f_{\text{yc}}(N)=(\frac{1-1.049\eta}{1-\eta})f_{y}\bigg{\{}1-\frac{\text{lg}N }{\text{lg}N_{\text{f}}}[1-\frac{(1-\eta)\sigma_{\text{max}}}{(1-1.049\eta)f _{y}}]\bigg{\}} \tag{12}\]  
where \(N_{\text{f}}\) can be calculated from Equation (13) [19], substituting Equation (12) into Equation (8) to obtain the fatigue constitutive relationship equation for a corroded steel bar.  
\[\text{lg}N_{\text{f}}=(15.138+0.086\eta)-(3.687+0.051\eta)\text{lg}(\Delta \sigma) \tag{13}\]