# Influence of Uneven Subgrade Frost Heave on Deformation and Damage of CRTSIII Slab Track
## 2 Concrete Plastic Damage Model Based on Damage Mechanics
### Concrete Damaged Plasticity Parameters  
When defining a plastic damage model of concrete, the stress-strain curve of concrete can be defined according to the concrete constitutive relation given in the GB 50010-2010 Code for Design of Concrete Structures [28].  
When the concrete is under uniaxial tensile:  
\[\sigma_{t}=(1-d_{t})E_{t}\epsilon_{t} \tag{13}\]  
\[d_{t}=\left\{\begin{array}{l}1-\rho_{t}\big{[}1.2-0.2x^{5}\big{]}(x\leq 1) \\ 1-\frac{\rho_{t}}{\alpha_{t}(x-1)1.7+x}(x>1)\end{array}\right. \tag{14}\]  
\[x=\frac{\epsilon_{t}}{\epsilon_{t,r}},\ \rho_{t}=\frac{f_{t,r}}{E_{t}\epsilon_{t,r}} \tag{15}\]  
When the concrete is under uniaxial compression:  
\[\sigma_{\rm c}=(1-d_{c})E_{c}\epsilon_{c} \tag{16}\]  
\[d_{c}=\left\{\begin{array}{l}1-\frac{\rho_{t}n}{n-1+x^{n}}(x\leq 1)\\ 1-\frac{\rho_{c}}{\alpha_{c}(x-1)^{2}+x}(x>1)\end{array}\right. \tag{17}\]  
\[x=\frac{\epsilon_{c}}{\epsilon_{c,r}},\ \rho_{c}=\frac{f_{c,r}}{E_{c}\epsilon_{c, r}},\ n=\frac{1}{1-\rho_{c}} \tag{18}\]  
Here, \(\alpha_{t}\) and \(\alpha_{c}\) are parameter values of the descending section of the stress-strain curve under tension and compression, respectively; \(f_{t,r}\) and \(f_{c,r}\) are the representative values of uniaxial tensile and compressive strengths of concrete, respectively; \(\epsilon_{t,r}\) and \(\epsilon_{c,r}\) are the peak strains of tensile strength and compressive strength, respectively; \(x\rho_{t}\rho_{c}\), and \(n\) are the intermediate parameters set to facilitate the analytical expression and have no physical meanings themselves.  
The tensile and compressive inelastic strain can be expressed as:  
\[\widetilde{\epsilon}^{ck}=\epsilon-\frac{\sigma}{E_{t}}\ \widetilde{\epsilon}^{in}= \epsilon-\frac{\sigma}{E_{c}} \tag{19}\]  
Assuming that the proportion of tensile equivalent plastic strain in the tensile inelastic strain is \(\eta_{t}\), the expression of tensile stiffness damage factor can be obtained as follows:  
\[d_{t}=\frac{(1-\eta_{t})\widetilde{\epsilon}^{ck}_{t}E_{0}}{\sigma_{t}+(1- \eta_{t})\widetilde{\epsilon}^{ck}_{t}E_{0}} \tag{20}\]  
Similarly, assuming that the proportion of compressive equivalent plastic strain in the compressive inelastic strain is \(\eta_{c}\), the expression of compressive stiffness damage factor can be obtained as follows:  
\[d_{c}=\frac{(1-\eta_{c})\widetilde{\epsilon}^{in}_{c}E_{0}}{\sigma_{c}+(1- \eta_{c})\widetilde{\epsilon}^{in}_{c}E_{0}} \tag{21}\]  
Based on the stress-strain curve of concrete and the CDP model of ABAQUS, a nonlinear analysis of concrete structure can be conducted by considering four groups of data: tensile stress-inelastic strain (\(\sigma-\widetilde{\epsilon}^{ck}\)), compressive stress-inelastic strain (\(\sigma-\widetilde{\epsilon}^{in}\)), tensile damage-inelastic strain (\(d_{t}-\widetilde{\epsilon}^{ck}\)) and compressive damage-inelastic strain (\(d_{c}-\widetilde{\epsilon}^{in}\)).  
Four sets of data from various concrete grades are derived. Taking C60 as an example, its four curves are shown as follows in Figure 4.