# A Thermomechanical Coupling Constitutive Model of Concrete Including Elastoplastic Damage
## 4 Model Application  
The proposed constitutive model was applied as the material model of concrete to simulate an axially compressive concrete column. The simulated concrete column is 3 m long. It has a square cross section with a side length of 500 mm. The bottom surface of the concrete column is fixed, and all the degrees of freedom are constrained. The top surface of the concrete column is subjected to an axial uniform load along the vertically downward direction to simulate the axially compressive loading case. The concrete strength is C40. The finite element discretization of column was implemented with the C3D8 element in ABAQUS. The element size is 100 \(\times\) 100 \(\times\) 100 mm. The calculation model consists of 750 elements and 1116 nodes. The total number of the degrees of freedom of the calculation model is 3240. The finite element calculation model is shown in Figure 7. The constitutive model parameters for the simulation are listed in Table 3.  
Figure 5: Damage variable vs. strain at different temperatures (C20 concrete).  
Figure 6: Damage variable vs. strain at different temperatures (C40 concrete).  
The vertical displacement nephograms of the simulated axially compressive concrete column at different temperatures are shown in Figure 8. The comparison of the load-displacement relationship of the simulated column at different temperatures is presented in Figure 9. It is indicated that under the action of the load with the same amplitude, the vertical deformation of the concrete column increases with the rise of temperature, especially when the temperature is higher than 400 \({}^{\circ}\)C. This results from the degradation of the mechanical performance of concrete at elevated temperatures. The above results simulated by the proposed constitutive model are consistent with the essential mechanical response behaviors of concrete members at different temperatures.  
The damage nephograms of the axially compressive concrete column at different temperatures are illustrated in Figure 10. It is shown that more serious damage to the concrete column can be observed with the rise of temperature, especially when the temperature is higher than 400 \({}^{\circ}\)C. This also characterizes the degradation of concrete strength at elevated temperatures. The above simulation results indicate that the developed constitutive model in the current study can be applied to the numerical simulation of the mechanical behaviors of concrete members at high temperatures.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
\begin{tabular}{c} \(E_{(T)}\) \\ **(GPa)** \\ \end{tabular} & \(\mathbf{v}\) & \(\mathbf{\alpha}\) & \(\varepsilon_{\mathbf{u}}\) & \(\begin{array}{c}f_{\mathbf{c}}\\ \textbf{(MPa)}\end{array}\) & \(M_{\textbf{1}}\) & \(M_{\textbf{2}}\) & \(N_{\textbf{1}}\) & \(N_  
Figure 8: _Cont._  
In the current numerical simulation, only the mechanical behaviors of a plain concrete column at different temperatures were investigated, but the practical structural columns have longitudinal and transverse reinforcements, which have an important effect on their mechanical behaviors. The interaction between the concrete and reinforcements and its variation with the temperature are not included in the proposed constitutive model and current numerical simulation. They should be taken into account in future investigation.  
Figure 8: Vertical displacement nephograms of axially compressive concrete column at different temperatures.  
Figure 9: Comparison of load–displacement relationship of axially compressive concrete column at different temperatures.  
Figure 10: _Cont._