# A Damage Model for Concrete under Fatigue Loading
## 4 Verification and Discussions  
### Solution Procedure of CDM Model for Concrete under Fatigue Loading  
In order to characterize the three-stage behaviors of concrete under fatigue loading considering different stress level \(S\), based on the concept of the fatigue failure surface [4], a normalized fatigue damage variable \(D_{n}{}^{0}\) is defined in this work as follows:  
\[D_{n}^{0\pm}=\frac{D_{n}^{\pm}-D_{1}^{\pm}}{D_{N}^{\pm}-D_{1}^{\pm}} \tag{59}\]  
where \(D_{1}\) and \(D_{N}\) denote the damage variable of the joint points of the static stress-strain curve and the level line \(\sigma=\sigma_{\text{max}}\), respectively, and they are able to be calculated by the model in [22] as follows:  
\[\sigma^{\pm}=(1-D^{\pm\pm})E_{0}\varepsilon^{\pm} \tag{60}\]  
where \(\sigma=\sigma_{\text{max}}\), \(\varepsilon=\varepsilon_{1}\) or \(\varepsilon=\varepsilon_{N}\), \(\varepsilon_{1}\) and \(\varepsilon_{N}\) denote the strain of the joint points of the static stress-strain curve and the level line \(\sigma=\sigma_{\text{max}}\), respectively, \(D^{\text{s}}\) denotes the damage variable of concrete under monotonic uniaxial loading, and it is able to be predicted by the model in [22] as follows:  
\[D^{\pm}=\frac{A_{1}^{\pm}-A_{2}^{\pm}}{1+\left(\varepsilon-\varepsilon_{0}^{ \pm}\right)^{\pm\pm}}+A_{2}^{\pm} \tag{61}\]  
where \(A_{1}\), \(A_{2}\), \(\varepsilon_{0}\), and \(p\) denote the parameters related to the damage evolution in concrete under monotonic uniaxial loading, and they are able to be calibrated by experimental results [22].  
Thus, the fatigue damage variable of concrete is derived as follows:  
\[D_{n}^{\pm}=D_{1}^{\pm}+D_{n}^{0\pm}\cdot\left(D_{N}^{\pm}-D_{1}^{\pm}\right) \tag{62}\]  
By a set of trial and error procedures, the normalized fatigue damage variable is assumed to be modeled by the equation:  
\[D_{n}^{0\pm}=A_{3}^{\pm}\cdot\left(-\frac{n/N}{n/N-A_{4}^{\pm}}\right)^{\frac {1}{A_{5}^{\pm}}} \tag{63}\]  
where \(A_{3}\), \(A_{4}\), and \(A_{5}\) denote the parameters related to the damage in concrete during fatigue loading, and they are able to be calibrated by the experimental results.  
The literature [54; 55] presumed that the reciprocal of irreversible deformation damage variable \(1/D^{\text{i}}\) is linearly dependent on the total damage variable \(D\). However, by applying the above linearly analysis, the resulted value of irreversible deformation damage variable \(D^{\text{i}}\) is overestimated in certain cases. For instance, it is found that the calculated irreversible deformation damage \(D^{\text{i}}\) is greater than the total damage \(D\) when the total damage \(D\) is approximately lower than 0.2, which is unreasonable (see Equation (40) and Figure 7). Hence, the relation between \(1/D^{\text{i}}\) and \(D\) is assumed to be corrected as follows,  
\[D_{n}^{\text{i}}=1/\left(B_{1}+B_{2}\cdot D_{n}+B_{3}\cdot D_{n}^{2}\right) \tag{64}\]  
where \(B_{1}\), \(B_{2}\) and \(B_{3}\) denote the parameters related to the coupling of irreversible and elastic deformation damage in concrete under loading, and they are able to be calibrated by the experimental results of concrete under cyclic loading [54; 55].  
The fatigue life is estimated by the convenient method [56], such that,  
\[\log N=14.7-13.5\frac{\sigma_{\max}-\sigma_{\min}}{f_{\rm cu}^{\pm}-\sigma_{\min}}. \tag{65}\]  
Therefore, this CDM model is able to characterize the mechanical behaviors of concrete under fatigue loading including following characteristics: the progressive stiffness degradation, development of cyclic creep and residual strain in a three-stage process, and the post-fatigue stress-strain response under monotonic loading.