# 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  
#### 4.1.1 Constitutive Model of Steel Rebar  
With reference to the \"Specification for Design of Concrete Structures\" [33], the static constitutive model of steel bars adopts the elastic, perfectly plastic model:  
\[\sigma_{\text{s}}=\left\{ \begin{array}{cc}E_{\text{s}}\varepsilon_{\text{s}} &\varepsilon_{\text{s}}\leq\varepsilon_{y}\\ f_{y,r}+k\big{(}\varepsilon_{\text{s}}-\varepsilon_{y}\big{)}&\varepsilon_{y}< \varepsilon_{\text{s}}\leq\varepsilon_{u}\\ 0&\varepsilon_{\text{s}}>\varepsilon_{u}\end{array} \right. \tag{3}\]  
\[k=\frac{f_{st,r}-f_{y,r}}{\varepsilon_{u}-\varepsilon_{uy}} \tag{4}\]  
where \(E_{\text{s}},\sigma_{\text{s}},\ f_{y,r}\,f_{st,r}\) are [PERSON]'s modulus, the stress yield strength, and the ultimate strength of the steel rebar, respectively; \(\varepsilon_{y}\) is the yield strain of the steel rebar corresponding to \(f_{y,r}\); \(\varepsilon_{u}\) is the peak strain of the steel bar corresponding to \(f_{st,r}\); and \(k\) is the slope of the hardened section of the steel rebar. Under normal conditions, it is considered that the elastic modulus of the steel bar is almost unchanged under a fatigue load, so the fatigue constitutive model of the steel rebar is the same as the static constitutive model, and the fatigue damage of the steel rebar is reflected by changing the remaining area of the steel rebar. For fatigue damage under variable amplitude fatigue, the hypothesis of \"deformation uniqueness\" can be used. [PERSON] [34] first proposed the hypothesis of the \"uniqueness of deformation\" and applied it to concrete materials. It is believed that no matter what the previous repeated load history is, as long as the residual deformation is the same, when the same repeated load is applied again, the load-deformation relationship will remain unchanged, regardless of the previous load-deformation history. This is the so-called multi-stage cumulative damage criterion in this paper. [PERSON] uses the fatigue characterization model of tensile steel continuity to describe the gradual materials degradation [35], which refers to the ratio of the effective bearing area in the damaged state to the bearing area in the undamaged state. In the bending fatigue test, the fatigue failure process of longitudinal steel rebars is a process of fatigue crack accumulation and propagation, and ultimately, transient fractures occur because the residual effective area of the section cannot bear the load. Therefore, using the residual area of damage as the characterization parameter of the fatigue damage of longitudinally stressed steel bars can well reflect the whole process from fatigue damage accumulation to fracture. It is assumed that the fatigue fracture of steel bars occurs after \(N_{\text{f}}\) fatigue cycles under a constant amplitude fatigue load. Assuming that the initial area of reinforcement is and the effective area after \(N_{\text{f}}\) loading cycles is \(A^{\text{f}}_{\text{s}}\) (\(N\)/\(f\)), the following formula should be met when fatigue fracture occurs in the rebar:  
\[A_{s}\sigma_{s,max}=A^{f}_{\text{s}}\Big{(}N_{f}\Big{)}f_{y} \tag{5}\]  
where \(\sigma_{s,max}\) is the maximum value of the initial stress of steel rebars. According to the above formula, we can get the effective area of fatigue fracture:  
\[A^{f}_{\text{s}}\Big{(}N_{f}\Big{)}=A_{s}\sigma_{s,max}/f_{y} \tag{6}\]  
Then the area damage \(\Delta A_{S,Cr}\) at the critical point of fatigue fracture is as follows:  
\[\Delta A_{S,Cr}=A_{s}\big{(}1-\sigma_{s,max}/f_{y}\big{)} \tag{7}\]  
It is assumed that the damage accumulation of the effective area of steel bars under constant amplitude fatigue stress conforms to the linear development trend.