# Performance of CRTS-II Ballastless Track-Bridge Structural System Rebars under Fatigue Loading Test
## 4 Finite Element Analysis Based on Material Fatigue Damage Model
### Constitutive Model of Material
#### 4.1.2 Concrete Constitutive Model  
In this paper, concrete damage plastic model is used for numerical simulation, while the fatigue damage of concrete is based on a uniaxial constitutive model under concrete static conditions. According to the constitutive relation recommended by the Chinese Design of Concrete Structures specifications [33], the static constitutive relation of concrete is determined as follows:  
\[\sigma=(1-\mathrm{d}_{t})E_{0}\varepsilon \tag{14}\]  
\[\sigma=(1-\mathrm{d}_{c})E_{0}\varepsilon \tag{15}\]  
where \(\sigma\) and \(\varepsilon\) are the tensile (compressive) stress and the strain of concrete, respectively; \(E_{0}\) is the initial elastic modulus of concrete; \(\mathrm{d}_{t}\) and \(\mathrm{d}_{c}\), the damage evolution parameters of concrete materials under tension and compression, selected according to the specification. Based on the above-mentioned uniaxial constitutive model of concrete, the time-varying and random nature of the concrete material damage after fatigue loading is comprehensively considered and the concrete static compressive stress-strain curve is recommended to be calculated according to the following formula:  
\[\sigma_{N}^{-}=(1-d_{C})\big{(}1-D_{k}^{-}\big{)}E_{0}\varepsilon_{N}^{-} \tag{16}\]  
where \(D_{k}^{-}\) is the degradation coefficient of the elastic modulus when concrete is compressed. \[D_{k}^{-}=(1-S)D_{0,N}^{-} \tag{17}\]  
where S is the stress level: the ratio of the maximum compressive stress to the compressive strength. The fatigue damage variable of concrete under tension (compression) \(D^{\pm}(N)\) can be solved by the following formula:  
\[D^{\pm}(N)=D_{1}^{\pm}+D_{0,N}^{\pm}\cdot\big{(}D_{N}^{\pm}-D_{1}^{\pm}\big{)} \tag{18}\]  
where \(D_{1}^{\pm}\) is the damage variable after the first fatigue loading cycle of concrete, \(D_{N}^{\pm}\) is the damage variable at fatigue failure of concrete and \(D_{0,N}^{\pm}\) is the normalized damage variable under fatigue tension (compression) of concrete under the \(N\)-th loading cycle. The values of the above variables are determined according to references [36; 37]. The evolution of \(D_{0,N}^{\pm}\) has a \"three-stage\" trend, characteristic of structural fatigue damage. Because the fatigue life of concrete beam is very large, the load cycle ratio (\(N_{i}/N_{f,i}\)) of each stage is very small. It can be seen from the formula of \(D_{0,N}^{\pm}\) that the increase in fatigue damage in the first 50,000 loading cycles is very large. If the multi-stage linear damage criterion is used, the fatigue damage increases sharply every time the load is changed, which is obviously inconsistent with the actual situation. Therefore, the accumulated damage criterion for multi-stage is revised in this paper. The starting point for calculating the fatigue damage value under the second, third and fourth stage loads is changed from 0 to the 50,000-th loading cycle. Based on this, the constitutive model of concrete compression fatigue under variable amplitude fatigue load is as follows:  
\[\sigma_{n}^{-}=\prod_{i=1}^{m}(1-d_{C})\Big{(}1-D_{k,i}^{-}\Big{)}E_{0} \varepsilon_{N}^{-} \tag{19}\]  
The concrete compressive damage variable under variable amplitude fatigue load \(D_{c}^{VF^{-}}(N)\) is as follows:  
\[D_{c}^{VF^{-}}(N)=1-\prod_{i=1}^{m}\Big{(}1-D_{k,i}^{-}\Big{)} \tag{20}\]  
where \(D_{k,i}^{-}\) is the damage variable of concrete under compression in the i-th stage. After fatigue loading, it is recommended that the static tensile stress-strain curve of concrete be calculated according to the following formula:  
\[\sigma_{N}^{+}=(1-\mathrm{d}_{t})\Bigg{(}1-\frac{D_{N}^{+}-D_{j,N}^{+}}{1-D_{j,N }^{+}}\Bigg{)}E_{0}\varepsilon_{N}^{+} \tag{21}\]  
Non-recoverable deformation damage variable \(D_{j,N}^{+}\) of tensioned concrete is taken as shown in reference [36].