# Influence of Uneven Subgrade Frost Heave on Deformation and Damage of CRTSIII Slab Track
## 2 Concrete Plastic Damage Model Based on Damage Mechanics  
### Theory of Concrete Plastic Damage  
The linear elastic constitutive model of concrete can be expressed by using the generalized Hooke's law:  
\[\sigma=E_{0}\varepsilon \tag{1}\]  
Here, \(E_{0}\) is the non-destructive elastic modulus.  
When a concrete plastic damage constitutive model is adopted, the concrete uniaxial stress-strain curves can be converted into stress versus plastic-strain curves, and the concrete tensile and compressive stresses can be expressed as follows:  
\[\sigma_{t}=\sigma_{t}(\tilde{\varepsilon}_{t}^{pl},\tilde{\varepsilon}_{t}^{ pl},\theta,f_{i}) \tag{2}\]  
\[\sigma_{c}=\sigma_{c}(\tilde{\varepsilon}_{c}^{pl},\tilde{\varepsilon}_{c}^{ pl},\theta,f_{i}) \tag{3}\]  
Here, the subscripts t and c refer to tension and compression, respectively; \(\tilde{\varepsilon}_{t}^{pl}\) and \(\tilde{\varepsilon}_{c}^{pl}\) are the tensile and compressive equivalent plastic strains, respectively; \(\tilde{\varepsilon}_{t}^{pl}\) and \(\tilde{\varepsilon}_{c}^{pl}\) are the tensile and compressive equivalent plastic strain rates, respectively; \(\theta\) is the temperature; and \(f_{i}\), (\(i=1,2,\,\dots\) ) are other predefined field variables.  
The stiffness degradation of concrete materials can be described by the following two damage variables: tensile damage and compression damage. When the damage factor is 0, there is no damage. When the damage factor approaches 1, it indicates that macroscopic damage occurs. When the damage factor is 1, it means that concrete has completely lost its stiffness.  
\[d_{t}=d_{t}(\tilde{\varepsilon}_{t}^{pl},\theta,f_{i});\qquad\quad 0\leq d_{t}\leq 1 \tag{4}\]  
\[d_{c}=d_{c}(\tilde{\varepsilon}_{c}^{pl},\theta,f_{i});\qquad\quad 0\leq d_{c}\leq 1 \tag{5}\]  
The elastic modulus of concrete material after damage is expressed by the non-destructive elastic modulus and the stiffness damage factor, whose expression is as follows:  
\[E=(1-d)E_{0} \tag{6}\]  
In this case, the constitutive relationship of the concrete plastic damage model can be expressed as the following equations, and its stress-strain curves are shown in Figures 2 and 3.  
\[\sigma_{t}=(1-d_{t})E_{0}(\varepsilon_{t}-\tilde{\varepsilon}_{t}^{pl}) \tag{7}\]\[\sigma_{c}=(1-d_{c})E_{0}(\varepsilon_{c}-\widetilde{\varepsilon}_{c}^{pl}) \tag{8}\]  
The inelastic strain is defined as the total strain minus the elastic strain corresponding to the undamaged material, then, the tensile inelastic strain and compressive inelastic strain are shown as follows:  
\[\widetilde{\varepsilon}_{t}^{kl}=\varepsilon_{t}-\varepsilon_{0t}^{el}= \varepsilon_{t}-\frac{\sigma_{t}}{E_{0}} \tag{9}\]  
\[\widetilde{\varepsilon}_{c}^{in}=\varepsilon_{c}-\varepsilon_{0c}^{el}= \varepsilon_{c}-\frac{\sigma_{c}}{E_{0}} \tag{10}\]  
By substituting Equations (9) and (10) into the constitutive relations (7) and (8) of the concrete plastic damage model, the tensile equivalent plastic strain and compressive equivalent plastic strain can be expressed as follows:  
\[\widetilde{\varepsilon}_{t}^{pl}=\widetilde{\varepsilon}_{t}^{ck}-\frac{d_{t} }{1-d_{t}}\frac{\sigma_{t}}{E_{0}} \tag{11}\]  
\[\widetilde{\varepsilon}_{c}^{pl}=\widetilde{\varepsilon}_{c}^{in}-\frac{d_{c }}{1-d_{c}}\frac{\sigma_{c}}{E_{0}} \tag{12}\]  
Figure 3: Uniaxial compressive stress-strain curve.  
Figure 2: Uniaxial tensile stress-strain curve.