# Study on Elastoplastic Damage Coupling of Reservoir Mudstone Considering Permeability Change
## 3 Elastoplastic Damage Coupling Mechanical Model  
### Elastoplastic Damage Coupling  
Assuming that mudstone is a small strain and isotropic material, the total strain increment \(\dot{\epsilon}\) is composed of an elastic strain increment \(\dot{\epsilon}^{e}\) and a plastic strain increment \(\dot{\epsilon}^{p}\):  
\[\dot{\epsilon}=\dot{\epsilon}^{e}+\dot{\epsilon}^{p} \tag{4}\]  
Under the condition of an isothermal and static load, the Helmhotz free energy \(\varphi\) is composed of elastic free energy \(\varphi^{e}\) and plastic free energy \(\varphi^{p}\)[24]:  
\[\varphi\left(\epsilon^{e},\gamma_{p},\omega\right)=\varphi^{e}(\epsilon^{e}, \omega)+\varphi^{p}\big{(}\gamma_{p},\omega\big{)} \tag{5}\]  
where \(\epsilon^{e}\) is the second-order elastic strain tensor, and \(\gamma_{p}\) and \(\omega\) are the plastic internal variable and the damage internal variable, respectively. Based on the Clausius-Duhan inequality of thermodynamics, the entropy increment \(\dot{\eta}\) is as follows:  
\[\dot{\eta}=\sigma:\dot{\epsilon}-\dot{\varphi}\geq 0 \tag{6}\]  
where \(\sigma\) is the second-order stress tensor. Bringing \(\mathrm{d}\varphi\) into Equation (6):  
\[\left(\sigma-\frac{\partial\varphi^{e}}{\partial\epsilon^{e}}\right):\dot{ \epsilon}^{e}+\left(\sigma:\dot{\epsilon}^{p}-\frac{\partial\varphi^{p}}{ \partial\gamma_{p}}\dot{\gamma_{p}}\right)-\frac{\partial\varphi}{\partial \omega}\dot{\omega}\geq 0 \tag{7}\]  
The total equation of the stress-strain relation is obtained:  
\[\sigma=\frac{\partial\varphi^{e}}{\partial\epsilon^{e}} \tag{8}\]  
The damage driving forces related to plasticity and damage are:  
\[K=-\frac{\partial\varphi^{p}}{\partial\gamma_{p}} \tag{9}\]  
\[Y=-\frac{\partial\varphi}{\partial\omega} \tag{10}\]  
Figure 5: Damage evolution curve of mudstone.  
It is assumed that the plastic potential function g\({}^{p}\) and the damage potential function g\({}^{\omega}\) are both functions of the damage driving force and the damage variable, and the plastic strain \(\epsilon^{p}\) is closely related to stress, so the plastic potential function g\({}^{p}\) is also a function of stress:  
\[\text{g}^{p}=\text{g}^{p}(\text{\boldmath$\sigma$},\,K;\text{\boldmath$\omega$}) \tag{11}\]  
\[\text{g}^{\omega}=\text{g}^{\omega}(Y;\text{\boldmath$\omega$}) \tag{12}\]  
The increment of the plastic internal variable \(\dot{\gamma}_{p}\) and the increment of the damage internal variable \(\dot{\omega}\) are as follows:  
\[\dot{\gamma}_{p}=f\Big{(}\dot{\epsilon}^{p}\Big{)},\,\,\dot{\epsilon}^{p}=\dot {\lambda}\frac{\partial\text{g}^{p}}{\partial\text{\boldmath$\sigma$}} \tag{13}\]  
\[\dot{\omega}=\dot{\mu}\frac{\partial\text{g}^{\omega}}{\partial Y} \tag{14}\]  
where \(\dot{\lambda}\) and \(\dot{\mu}\) are a non-negative plastic factor and a damage factor, respectively. Combining the consistency conditions of the yield surface equation \(f^{p}=f^{p}(\text{\boldmath$\sigma$},K;\text{\boldmath$\omega$})\) and damage evolution equation \(f^{\omega}=f^{\omega}(Y;\text{\boldmath$\omega$})\), it can be found that:  
\[\dot{f}^{p}=\frac{\partial f^{p}}{\partial\text{\boldmath$\sigma$}}:\dot{ \text{\boldmath$\sigma$}}+\frac{\partial f^{p}}{\partial K}:\dot{K}+\frac{ \partial f^{p}}{\partial\text{\boldmath$\omega$}}\dot{\omega}=0 \tag{15}\]  
\[\dot{f}^{\omega}=\frac{\partial f^{\omega}}{\partial Y}:\dot{Y}+\frac{ \partial f^{\omega}}{\partial\text{\boldmath$\omega$}}\,\,\dot{\omega}=0 \tag{16}\]  
The plastic internal variable \(\gamma_{p}\) and damage internal variable \(\omega\) can be determined, and the elastoplastic damage coupling calculation of mudstone can be realized.