## 2 Fatigue Constitutive Model of Materials
### Fatigue Constitutive Model of Concrete
#### 2.1.3 Fatigue Residual Strain  
Under fatigue loading, the residual strain of concrete will accumulate with the increase of fatigue load cycles. The residual strain of concrete is related to the number of fatigue load cycles and stress levels. The formula for calculating the longitudinal residual strain of concrete under compression proposed by [PERSON] [14] is shown in Equation (5):  
\[\Delta\varepsilon_{\text{r}}(N)=\Delta\varepsilon_{\text{r}}(1)+\frac{0.00105 \varepsilon_{\text{c,max}}^{1.98}(1-\frac{\varepsilon_{\text{c,min}}}{ \varepsilon_{\text{c,max}}})^{5.27}}{\varepsilon_{\text{unstab}}^{1.41}}N^{0.395} \tag{5}\]  
Figure 1: Fatigue residual strength envelope of concrete.  
where \(\Delta\varepsilon_{\text{r}}(N)\) is the fatigue residual strain after \(N\) cycles of fatigue loading. \(\Delta\varepsilon_{\text{r}}(1)\) is the residual strain after the first stress cycle, where \(\Delta\varepsilon_{r}(1)=0.25(\frac{\varepsilon_{\text{c,max}}}{\varepsilon_{ \text{unstgb}}})^{2}\). \(\varepsilon_{\text{c,min}}\) and \(\varepsilon_{\text{c,max}}\) are the initial instantaneous strains of concrete under the lower and upper limits of fatigue loading, respectively. \(\varepsilon_{\text{unstab}}\) is the total longitudinal strain of concrete at the beginning of the third stage, and its value is very close to the strain \(\varepsilon_{0}\) corresponding to the peak stress during a single loading and failure, therefore it can be approximately taken as \(\varepsilon_{0}\).  
Based on changes of stiffness, residual strength and residual strain of concrete under fatigue loading with an increasing number of loading cycles, and referring to the literature [11], the fatigue constitutive model for concrete under any fatigue loading cycles is established based on the concrete uniaxial stress-strain curve in the Code of GB50010-2010 [14].  
Fatigue principal structure of concrete under compression is calculated using the following:  
\[\sigma_{\text{c}}(N)=\left\{\begin{array}{ll}E_{\text{N}}[\varepsilon(N)- \Delta\varepsilon_{\text{r}}(N-1)]&\Delta\varepsilon_{\text{r}}(N-1)\leq \varepsilon(N)<\varepsilon_{\text{e}}(N)\\ \frac{\frac{\pi_{\text{c}}(N)}{\Gamma_{\text{N}}\varepsilon_{\text{max}}(N) -\sigma_{\text{c}}(N)}E_{\text{N}}[\varepsilon(N)-\Delta\varepsilon_{\text{ r}}(N-1)]}{n-1+\left[\frac{\left(\frac{N}{\Gamma_{\text{N}}}\right)-\Delta\varepsilon_{ \text{r}}(N-1)}{\varepsilon_{\text{c}}(N)}\right]}&\varepsilon_{\text{e}}(N) \leq\varepsilon(N)\leq\varepsilon_{\text{c,max}}(N)\\ \frac{\varepsilon(N)-\Delta\varepsilon_{\text{r}}(N-1)}{\varepsilon_{\text{c,max}}(N)}&\varepsilon(N)>\varepsilon_{\text{c,max}}(N)\end{array}\right. \tag{6}\]  
Fatigue principal structure of concrete under tension is calculated using the following:  
\[\sigma_{\text{r}}(N)=\left\{\begin{array}{ll}E_{\text{N}}[\varepsilon(N)- \Delta\varepsilon_{\text{r}}(N-1)]&\Delta\varepsilon_{\text{r}}(N-1)\leq \varepsilon(N)<\varepsilon_{\text{t,max}}(N)\\ \frac{\varepsilon(N)-\Delta\varepsilon_{\text{r}}(N-1)}{\varepsilon_{\text{t,max}}(N)}&\varepsilon(N)\geq\varepsilon_{\text{t,max}}(N)\end{array}\right. \tag{7}\]  
where \(n=\frac{E_{N}\varepsilon_{\text{c,max}}(N)}{E_{N}\varepsilon_{\text{c,max}}(N) -\sigma_{\text{r,c}}(N)}\). \(\varepsilon_{\text{c,max}}(N)\) and \(\varepsilon_{\text{t,max}}(N)\) are the peak compressive and tensile strains of concrete after \(N\) cycles of fatigue loading, respectively, where \(\varepsilon_{k,\text{max}}(N)=\frac{\sigma_{\text{r,k}}(N)}{E_{N}}+\Delta \varepsilon_{r}(N-1)(k=c,t)\). \(\varepsilon_{\text{e}}(N)\) is the compressive elastic strain of concrete after \(N\) cycles of fatigue loading; generally, take \(\varepsilon_{\text{e}}(N)=0.3\)-\(0.4\varepsilon_{\text{c,max}}(N)\).