# Deformation Law and Control Limit of CRTSIII Slab Track under Subgrade Frost Heave
## 2 Plastic Damage Analysis Model of Slab Tracks
### Plastic Damage Constitutive Relationship of Slab Track Concrete  
In order to simulate the process of material damage under an external load, the concrete plastic damage constitutive relationship was deduced, after which a foundation for the research of damage characteristics of the CRTSIII slab track structure could be created. Referring to The Code for Design of Concrete Structures [42], the stress-strain curve equations of concrete under uniaxial tension and compression are as follows:  
\[\sigma=(1\ -\ d_{t})E_{\mathrm{c}}\varepsilon \tag{1}\]  
\[d_{\mathrm{t}}=\left\{ \begin{array}{l}1-\rho_{\mathrm{t}}\big{[}1.22\ -\ 0.2x^{5}\big{]}\ (x\ \leq\ 1)\\ 1-\frac{\rho_{\mathrm{t}}}{\alpha_{\mathrm{t}}(x\ -1)^{1/2}+x}\ (x>1)\end{array} \right. \tag{2}\]  
\[x=\frac{\varepsilon}{\varepsilon_{\mathrm{t,r}}}\,\ \rho_{\mathrm{t}}=\frac{f_{ \mathrm{t,r}}}{E_{\mathrm{c}}\varepsilon_{\mathrm{t,r}}} \tag{3}\]  
\[\sigma=(1-d_{\mathrm{c}})E_{\mathrm{c}}\varepsilon \tag{4}\]  
\[d_{\mathrm{c}}=\left\{ \begin{array}{l}1-\frac{\rho_{\mathrm{c}}n}{n-1+x^{n} }\ (x\ \leq\ 1)\\ 1-\frac{\rho_{\mathrm{c}}}{\alpha_{\mathrm{c}}(x\ -1)^{1/2}+x}\ (x>1)\end{array} \right. \tag{5}\]  
\[x=\frac{\varepsilon}{\varepsilon_{\mathrm{c,r}}}\,\ \rho_{\mathrm{c}}=\frac{f_{ \mathrm{c,r}}}{E_{\mathrm{c}}\varepsilon_{\mathrm{c,r}}}\,\ n=\frac{1}{1-\rho_{\mathrm{c}}} \tag{6}\]  
\[\varepsilon_{\mathrm{c}}=\varepsilon_{\mathrm{z}} \tag{7}\]  
\[\varepsilon^{\mathrm{in}}=d_{\mathrm{c}}\varepsilon \tag{8}\]  
\[\varepsilon_{\mathrm{ca}}=\frac{\varepsilon_{\mathrm{c}}}{\varepsilon_{ \mathrm{c}}+\varepsilon}\sqrt{\varepsilon_{\mathrm{c}}\varepsilon} \tag{9}\]  
\[\varepsilon_{\mathrm{ca}}=\frac{\varepsilon_{\mathrm{c}}}{\varepsilon_{ \mathrm{c}}+\varepsilon}\sqrt{\varepsilon_{\mathrm{c}}\varepsilon} \tag{10}\]  
\[\varepsilon^{\mathrm{ck}}=d_{\mathrm{c}}\varepsilon \tag{11}\]  
where \(\alpha_{\mathrm{t}}\) and \(\alpha_{\mathrm{c}}\) are parameter values of a downward-facing section of the concrete stress-strain curve under uniaxial tension and compression, respectively; \(f_{\mathrm{tr}}\) and \(f_{\mathrm{cr}}\) are representative values of concrete strength under uniaxial tension and compression, respectively; \(\varepsilon_{\mathrm{tr}}\) and \(\varepsilon_{\mathrm{cr}}\) are the peak strain of tensile and compressive strength, respectively; \(d_{\mathrm{t}}\) and \(d_{\mathrm{c}}\) are the damage evolution parameters of concrete under uniaxial tension and compression, respectively; \(\varepsilon_{\mathrm{p}}\) and \(\varepsilon_{\mathrm{z}}\) are plastic strain and residual stress, respectively; and \(\varepsilon^{\mathrm{in}}\) and \(\varepsilon^{\mathrm{ck}}\) are inelastic strain and cracking strain, respectively. By combining Equations (1)-(6), the tension-compression constitutive stress-strain relationship of concrete is obtained. We used C40 concrete as an example; its material parameters are shown in Table 1 and Figure 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Name** & **Symbol** & **Parameter Value** \\ \hline Modulus of elasticity (GPa) & \(E_{\mathrm{c}}\) & 32.5 \\ Representative value of tensile strength (MPa) & \(f_{\mathrm{tr}}\) & 2.39 \\ Tensile peak strain & \(\varepsilon_{\mathrm{t}}\) & 0.000104 \\ Parameters of the falling section of the tensile stress–strain curve & \(\alpha_{\mathrm{t}}\) & 1.796 \\ Representative value of compressive strength (MPa) & \(f_{\mathrm{cr}}\) & 26.8 \\ Compressive peak strain & \(\varepsilon_{\mathrm{c}}\) & 0.001589 \\ Parameters of falling section of compressive stress–strain curve & \(\alpha_{\mathrm{c}}\) & 1.168 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relevant parameters of C40 concrete. Based on the concrete stress-strain curve and the concrete plastic damage model, concrete damaged plasticity (CDP), the failure mechanisms of concrete materials are mainly tensile cracking and compression failure.