## 2 Methodology
### Equivalent Fatigue Damage Constitution
#### 2.3.2 Fatigue Residual Strength  
If there is no direct experimental constitution describing the material behavior, the constitutive equations from various codes can be employed to describe the envelope curve of fatigue residual strength. According to the Chinese code for the design of concrete structures [35], the descending branch of the monotonic constitutive equation can be expressed as:  
\[\frac{\sigma_{\rm r}}{f_{\rm r}}=\frac{x}{\alpha_{\rm r}[x-1]^{\beta}+x},\ x= \frac{\varepsilon}{\varepsilon_{\rm p,r}}\geq 1, \tag{22}\]  
Figure 10: Sketch of fatigue residual strength envelope curve that depicts the degradation behavior of concrete under fatigue loading. The abscissa \(x(n)\) is the relative strain defined in Equation (23). where \(x\) is defined as the ratio of the strain in the descending branch to the peak strain in the constitutive curve; \(\sigma_{\rm r}\) is stress in the descending branch; \(f_{\rm r}\) is the axial compressive or tensile strength; \(\alpha_{\rm r}\) is the parameter of the descending branch, which can be taken with reference to the code [35]; \(\beta\) is the equation parameter, which is taken as 2 for compression and 1.7 for tension, respectively. Transforming from Equation (22), the envelope curve of fatigue residual strength is derived by replacing \(\sigma_{\rm r}\) with the residual strength \(\sigma_{\rm r}(n)\) after \(n\) loading cycles and representing \(x\) with \(x(n)\). Based on the physical meaning of the envelope model (Figure 10), \(x(n)\) is defined as the following function of fatigue loading cycles:  
\[x(n)=\left\{ \begin{matrix}1&(n=1)\\ f(n,N_{\rm f})[x(N_{\rm f})-x(1)]+x(1)&(1\leq n\leq N_{\rm f})\end{matrix} ,\right. \tag{23}\]  
where _f(n,N\({}_{\rm f}\))_ is the shape function of \(x(n)\) during fatigue loading; \(x(N_{\rm f})\), as the maximum fatigue strain at fatigue failure \(\varepsilon_{\rm max}^{\rm f}\) divided by the monotonic peak strain \(\varepsilon_{\rm p,r}\), is a constant for a given fatigue specimen. Due to the irreversibility of energy dissipation, the fatigue residual strength envelope is a monotonically decreasing function. The envelope curve must also satisfy the following boundary conditions: \(\sigma_{\rm r}(x(1))=f_{\rm r}\); \(\sigma_{\rm r}(x(N_{\rm f}))=\sigma_{\rm max}\); \(\sigma_{\rm r}(n)\leq 0\). Combining these boundary conditions and Equations (22) and (23), the fatigue residual strength of concrete can be derived and is expressed in Equation (24) as:  
\[\sigma_{\rm r}(n)=f_{\rm r}\frac{f(n,N_{\rm f})[x(N_{\rm f})-x(1)]+x(1)}{ \alpha_{\rm r}(f(n,N_{\rm f})[x(N_{\rm f})-x(1)])^{a}+f(n,N_{\rm f})[x(N_{\rm f })-x(1)]+x(1)}. \tag{24}\]  
It is worth noting that the shape function in the form of cycle ratio \(f(n,N_{\rm f})=\frac{x(n)-x(1)}{x(N_{\rm f})-x(1)}=\frac{n}{N_{\rm f}}\) or the relative ratio of logarithmic life \(f(n,N_{\rm f})=\frac{x(n)-x(1)}{x(N_{\rm f})-x(1)}=\frac{\rm lgn}{\rm lgnN_{\rm f}}\) has been used in the literature [10; 16; 17; 18]. The shape function of the cycle ratio indicates linear accumulation of fatigue damage. For the shape function of relative logarithmic life, the second-order derivative of _f(n,N\({}_{\rm f}\))_ with respect to \(n\) yields \(f(n,N_{\rm f})^{\prime\prime}=-\frac{1}{n^{2}\ln 10\log N_{\rm f}}<0\), indicating that _f(n,N\({}_{\rm f}\))_ is a convex function, as illustrated by the dash-dot line in Figure 11. The dash-dot line shows that the rate of fatigue strain increase gradually decreases as loading cycles elapse. This phenomenon might be the case when maximum fatigue stress is less than the fatigue threshold. However, for specimens that failed under fatigue, residual strain gradually accumulated in the cyclic loading process, and fatigue strain grew rapidly in stage III [19], as shown by the solid line in Figure 11. The cycle ratio or relative logarithmic life as shape function, therefore, deviates from experimental observations and fails to describe the fatigue residual strength correctly.