# A Thermomechanical Coupling Constitutive Model of Concrete Including Elastoplastic Damage
## 5 Conclusions  
A thermomechanical coupling constitutive model for the description of the mechanical behaviors of concrete at different temperatures is proposed. The expression of the elastic strain increment is derived with the free energy function including the temperature variable. The expression of the plastic strain increment is derived from the yield function based on the Drucker-Prager strength criterion. The elastoplastic damage effect is included in this constitutive model. The damage variable is included in the yield function to consider the effect of the damage on the elastoplastic mechanical behaviors of concrete. The damage variable evolution equation is expressed by the macro mechanical variables.  
The proposed constitutive model was coded and implemented into the finite element software ABAQUS through UMAT interface. The uniaxial compression tests of concrete at different temperatures were numerically simulated, and the results are compared with the test results to validate the proposed constitutive model. The simulation results accord well with the test results at different temperatures. This indicates that the proposed constitutive model can characterize the mechanical behaviors of concrete at different temperatures with  
Figure 10: Damage nephograms of axially compressive concrete column at different temperatures.  
considerable accuracy. The proposed constitutive model was then applied to simulate an axially compressive concrete column. The simulation results are consistent with the essential mechanical response behaviors of concrete members at different temperatures.  
The constitutive model proposed in the current study has relatively simple expressions, and it is convenient for the computer implementation of this model. Fewer model parameters are included in this model, and they all have explicit physical meanings. It can be applied to the numerical simulation of the mechanical performance of concrete members at different temperatures.  
The presented constitutive model has some vital disadvantages and limitations. It can only be applied to the uniaxial compression loading case and can only simulate the ascending segment of the stress-strain relationship of concrete. Some important factors, such as moisture, pore pressure, chemical action and loading rate, are not included in this model. On the other hand, the presented model was developed mainly based on the elastoplastic theory and damage mechanics. The constitutive model strictly based on invisible thermodynamics should be developed in future study.  
The proposed constitutive model in the current study cannot be applied to confined concrete, which is the actual working state of the concrete in structural members. The presented model should be revised and developed for application to confined concrete in future investigation.  
[PERSON] developed the thermomechanical coupling constitutive model and performed the formula derivation of the model. [PERSON] prepared the manuscript. [PERSON] performed the program composition and numerical simulation for the model validation. [PERSON] discussed the numerical results and provided suggestions on the overall organization of the paper. [PERSON] performed the numerical simulation for the model application and revised the manuscript. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of People's Republic of China [No. 52078288] and the Beijing Municipal Natural Science Foundation [No. 8212001].  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to the intellectual property.  
The authors declare no conflict of interest.