# Coupled Elastic-Plastic Damage Modeling of Rock Based on Irreversible Thermodynamics
## 2 Basic Framework of the Elastoplastic Damage Coupled Model
### Stress-Strain Relationship  
In the derivation process of the stress-strain relationship for shale, derivation with respect to different parameters is involved. Therefore, in this section, using \(d\) and \(\partial\) to denote derivatives is more appropriate. Equation (18) is rewritten as:  
\[\sigma:d\varepsilon-d\phi\geq 0 \tag{19}\]  
Shale is a quasi-brittle rock, and the total strain in quasi-brittle materials is the sum of elastic strain \(\varepsilon^{e}\) and plastic strain \(\varepsilon^{p}\), with the total strain increment being the sum of the increments of elastic \(d\varepsilon^{e}\) and plastic strains \(d\varepsilon^{p}\):  
\[\left\{ \begin{array}{l}\varepsilon=\varepsilon^{e}+\varepsilon^{p}\\ d\varepsilon=d\varepsilon^{e}+d\varepsilon^{p}\end{array} \right. \tag{20}\]  
According to the principle of strain equivalence, the damage variable can be defined as:  
\[\omega=1-E/E_{0} \tag{21}\]  
where \(E\) and \(E_{0}\) are the elastic moduli of the damaged and undamaged states of the shale, respectively [81]. The thermodynamic potential in rocks can be decomposed into elastic potential energy \(\phi^{e}\) and plastic potential energy \(\phi^{p}\). Under isothermal conditions, the thermodynamic potential in the elastoplastic damage coupling model requires several state variables to describe:  
\[\phi=\phi(\varepsilon^{e},\gamma^{p},\omega)=\phi^{e}(\varepsilon^{e},\omega) +\phi^{p}(\gamma^{p},\omega) \tag{22}\]  
where \(\varepsilon^{e}\) is the elastic strain; \(\gamma^{p}\) is the equivalent plastic strain; \(\omega\) is the damage variable. Differentiating Equation (22) with respect to time and then substituting it along with Equation (20) into Equation (19), and upon rearrangement, we obtain:  
\[(\sigma-\frac{\partial\phi^{e}}{\partial e^{e}}):d\varepsilon+\frac{\partial \phi^{e}}{\partial e^{e}}:d\varepsilon^{p}-\frac{\partial\phi^{p}}{\partial \gamma^{p}}:d\gamma^{p}-\frac{\partial\phi}{\partial\omega}:d\omega\geq 0 \tag{23}\]  
The conditions of elastic strain \(\varepsilon^{e}\), equivalent plastic strain \(\gamma^{p}\), and damage variable \(\omega\) as arbitrary values always satisfy Equation (23); hence, we can conclude:  
\[\sigma=\frac{\partial\phi^{e}}{\partial\varepsilon^{e}} \tag{24}\]The thermodynamic forces related to plasticity \(R\) and damage \(Y\) can be represented as follows:  
\[\left\{ \begin{array}{l}R=-\frac{\partial\phi^{p}}{\partial\tau^{p}}\\ Y=-\frac{\partial\phi}{\partial\omega}\end{array} \right. \tag{25}\]  
In the thermodynamic framework, both \(\gamma^{P}\) and \(\omega\) are non-negative state variables; therefore, their increments \(d\gamma^{P}\) and \(d\omega\) are also non-negative during the process of inelastic dissipation, thus satisfying the non-negativity condition expressed in Equation (25). Previous research has shown that the damage-related thermodynamic force \(Y\) acts during both the elastic and plastic portions [67; 82], allowing \(Y\) to be decomposed as follows:  
\[Y=rY^{e}+(1-r)Y^{p} \tag{26}\]  
where \(Y^{e}\) and \(Y^{P}\) represent the damage-related thermodynamic forces of the elastic and plastic parts, respectively; \(r\) is a weighting coefficient controlling the contributions of the elastic and plastic parts. Many triaxial experimental studies with acoustic emission outcomes indicate that damage in quasi-brittle rocks significantly increases after the onset of plasticity, with a greater weight on the plastic component, i.e., the value of the weighting coefficient \(r\) ranges from 0 to 0.5 [83; 52; 84].