# A Thermomechanical Coupling Constitutive Model of Concrete Including Elastoplastic Damage
## 2 Model Development
### Elastic Stress-Strain Model  
The uncoupling assumption of elasticity and plasticity for concrete was adopted for the development of the elastic stress-strain model. The elasticity and plasticity are unattached and do not impact each other. The free energy function consists of thermoelastic free energy and thermoplastic free energy. Based on the above assumption, the free energy function can be written as follows  
\[\psi\left(\varepsilon_{ij}^{e},\varepsilon_{d}^{p},T,D\right)=\psi^{e}\left( \varepsilon_{ij}^{e},T\right)+\psi^{p}\left(\varepsilon_{d}^{p},T,D\right) \tag{1}\]  
where \(\psi\) is the Helmholtz free energy function, \(\psi^{e}\) is the elastic Helmholtz free energy function and \(\psi^{p}\) is the plastic Helmholtz free energy function. \(\varepsilon_{ij}^{e}\) is the elastic strain tensor, \(\varepsilon_{d}^{p}\) is the plastic shear strain, \(T\) is the temperature and \(D\) is the damage variable.  
The thermodynamic law of the isothermal pure mechanical process can be expressed as follows  
\[\dot{\eta}=\sigma:\dot{\epsilon}-\dot{\psi}\geq 0 \tag{2}\]  
The derivative of the free energy function is  
\[\dot{\psi}=\frac{\partial\psi^{e}}{\partial\varepsilon_{ij}^{e}}\dot{ \varepsilon}_{ij}^{e}+\frac{\partial\psi}{\partial T}\dot{T}+\frac{\partial \psi}{\partial D}\dot{D}+\frac{\partial\psi^{p}}{\partial\varepsilon_{d}^{p }}\dot{\varepsilon}_{d}^{p} \tag{3}\]  
Substituting Equation (3) into Equation (2), we can get  
\[\left(\sigma_{ij}-\frac{\partial\psi^{e}}{\partial\varepsilon_{ij}^{e}}\right) \dot{\varepsilon}_{ij}^{e}-\frac{\partial\psi}{\partial T}\dot{T}-\frac{ \partial\psi}{\partial D}\dot{D}+\left(\sigma_{ij}:\dot{\varepsilon}_{ij}^{p} -\frac{\partial\psi^{p}}{\partial\varepsilon_{d}^{p}}\dot{\varepsilon}_{d}^{ p}\right)\geq 0 \tag{4}\]  
In order to let Equation (4) be satisfied for arbitrary \(\varepsilon_{ij}^{e}\), we use  
\[\sigma_{ij}=\frac{\partial\psi^{e}}{\partial\varepsilon_{ij}^{e}}=\frac{ \partial\psi^{e}\left(\varepsilon_{ij}^{e},T\right)}{\partial\varepsilon_{ij }^{e}} \tag{5}\]  
An elastic free energy function proposed by [PERSON] [40] was adopted. It includes both the mechanical damage variable and temperature variable. It can be written as follows  
\[\psi^{e}=\frac{1}{2}\varepsilon_{ij}^{e}C_{ijkl}^{e}\varepsilon_{kl}^{e}- \theta(T-T_{0})\varepsilon_{ij}^{e}+c_{k}\left[T-T_{0}-T\ln\frac{T}{T_{0}}\right] \tag{6}\]  
where \(\theta\) is the thermomechanical coupling tensor, \(\theta=C_{ijkl}^{e}\beta_{kl}\); \(\beta_{kl}\) is the thermal expansion coefficient tensor, \(\beta_{kl}=\beta\delta_{kl}\) for the isotropic case and \(\delta_{kl}\) is the Kronecker \(\delta\). \(C_{ijkl}^{e}\) is the elastic modulus tensor in the damaged state, \(C_{ijkl}^{e}=(1-D)C_{ijkl}^{0}\); \(C_{ijkl}^{0}\) is the initial elastic modulus tensor in the undamaged state. \(T_{0}\) is the initial temperature, \(T_{0}=20\)\({}^{\circ}\)C. \(c_{k}\) is the specific heat capacity, and it is assumed to be constant and independent of strain (or stress) and temperature.  
Substituting Equation (6) into Equation (5), we can get  
\[\sigma_{ij}=C_{ijkl}^{e}\varepsilon_{kl}^{e}-(T-T_{0})C_{ijkl}^{e}\beta\delta _{kl} \tag{7}\]  
The incremental elastic stress-strain model can be expressed as  
\[d\sigma_{ij}=C_{ijkl}^{e}d\varepsilon_{kl}^{e} \tag{8}\]