# A Thermomechanical Coupling Constitutive Model of Concrete Including Elastoplastic Damage
## 2 Model Development
### Formulation of Developed Constitutive Model
 \tag{33}\]  
The deviatoric derivative terms of the hardening parameter to the strain components are written as follows  
\[\begin{split}\frac{\partial K}{\partial\
u_{11}^{2}}& =-\frac{\sqrt{2}}{3}\frac{A}{\left(A+B\epsilon_{d}^{p}\right)^{2}} \frac{2\
u_{11}^{p}-\epsilon_{22}^{p}-\epsilon_{33}^{p}}{\sqrt{\left(\
u_{11} ^{p}-\epsilon_{22}^{p}\right)^{2}+\left(\epsilon_{22}^{p}-\epsilon_{33}^{p} \right)^{2}+\left(\epsilon_{33}^{p}-\epsilon_{11}^{p}\right)^{2}+6\left(\epsilon _{12}^{p}+\epsilon_{13}^{p}+\epsilon_{23}^{p}\right)}}\\ \frac{\partial K}{\partial\
u_{22}^{2}}&=-\frac{ \sqrt{2}}{3}\frac{A}{\left(A+B\epsilon_{d}^{p}  
transducer to measure the deformation of the specimen. At a specified test temperature, the compressive load was applied to the tested concrete specimen. The corresponding stress-strain curve was obtained with the X-Y function recorder. The calculation model for the uniaxial compression of a concrete cubic element is shown in Figure 2. The concrete element was simulated with the C3D8 element with eight nodes in ABAQUS. The side length of the element is 1 mm. The bottom surface of the element is constrained, and vertical displacement is applied to the top surface to simulate the uniaxial compression test. The numerical simulation of the uniaxial compression tests of concrete at different temperatures were implemented with the thermodynamic properties of concrete being characterized by the proposed thermomechanical coupling constitutive model. The concrete strength is C20 and C40, and the corresponding model parameters are listed in Tables 1 and 2, respectively. \(E_{(T)}\) is the elastic modulus of concrete corresponding to the temperature \(T\). The comparison of the numerical results of the stress-strain relationship simulated by the developed constitutive model with the test results is shown in Figures 3 and 4. It is illustrated that the simulation results accord well with the test results at different temperatures. The simulation results show that the stress-strain curves tend to grow more gently with the rise of temperature. With the rise of temperature, the peak stress of concrete decreases, but the strain corresponding to the peak stress increases. This accords with the essential mechanical properties of concrete at different temperatures. The above comparison indicates that the constitutive model develop \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \(E_{(20)}\) & \(E_{(100)}\) & \(E_{(300)}\) & \(E_{(500)}\) & \(E_{(700)}\) & \(v\) & \(\alpha\) & \(\epsilon_{u}\) & \(f_{c}\) & \(M_{1}\) & \(M_{2}\) & \(N_{1}\) & \(N_{2}\) \\ (**GPa**) & (**GPa**) & (**GPa**) & (**GPa**) & (**GPa**) & & & & (**MPaa**) & & & & \\ \hline
21.781 & 18.732 & 11.696 & 5.793 & 1.481 & 0.15 & 0.01 & 0.002 & 20.435 & 0.0189 & 0.9102 & 0.0296 & 0.6225 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Constitutive model parameters for C20 concrete. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(E_{(20)}\) & \(E_{(100)}\) & \(E_{(300)}\) & \(E_{(500)}\) & \(E_{(700)}\) & \(v\) & \(\alpha\) & \(\epsilon_{u}\) & \(f_{c}\) & \(M_{1}\) & \(M_{2}\) & \(N_{1}\) & \(N_{2}\) \\ (**GPa**) & (**GPa**) & (**GPa**) & (**GPa**) & (**GPa**) & & & & (**MPaa**) & & & & \\ \hline
39.278 & 33.779 & 21.090 & 10.446 & 2.671 & 0.15 & 0.01 & 0.002 & 36.85 & 0.0189 & 0.9102 & 0.0296 & 0.6225 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constitutive model parameters for C40 concrete. Figure 2: Calculation model for uniaxial compression of the concrete cubic element. can characterize the mechanical performance of concrete at different temperatures with considerable accuracy. Figure 3: Comparison of simulation results and test results for C20 concrete. the stress-strain relationship curves are not presented, although they have been revealed in the original literature. The relationship between the damage variable value and the compressive strain at different temperatures for C20 and C40 concrete is presented in Figures 5 and 6.