# Study on Characteristic Strength and Constitutive Model of Red Sandstone under Hydraulic Coupling
## 5 Damage Constitutive Model Considering Seepage Pressure
### Construction of the Constitutive Model  
According to the hypothesis of equivalent strain [40; 41; 42],  
\[\widetilde{\sigma}_{ij}=\frac{\sigma_{ij}}{1-D} \tag{6}\]  
By combining Formulas (5) and (6), it can be obtained that the damage constitutive relationship of porous elastic materials under hydraulic coupling is  
\[\widetilde{\sigma}_{ij}^{{}^{\prime}}=\frac{\sigma_{ij}^{{}^{\prime}}}{1-D}= \frac{\sigma_{ij}-\alpha P_{w}\delta_{ij}}{1-D} \tag{7}\]  
Figure 10: [PERSON]’s circle and envelope line under different confining pressures and seepage pressures: (**a**) seepage pressure is 0 MPa, (**b**) seepage pressure is 1 MPa, (**c**) seepage pressure is 2 MPa; (**d**) seepage pressure is 3 MPa.  
Assume that a large number of microelements constitute the rock, and the microelement strength is randomly distributed; therefore, the probability density of the Weibull distribution [43] is  
\[P\Big{(}f_{(\sigma)}\Big{)}=\frac{m}{k}\bigg{(}\frac{f_{(\sigma)}}{k}\bigg{)}^{m -1}\exp\left[-\bigg{(}\frac{f_{(\sigma)}}{k}\bigg{)}^{m}\right] \tag{8}\]  
where \(k\) and \(m\) are parameters of the model, and \(f_{(\sigma)}\) is the rock yield criterion.  
Therefore, the damage variable \(D\)[44] can be expressed as  
\[D=1-\exp\left[-\bigg{(}\frac{f_{(\sigma)}}{F_{0}}\bigg{)}^{m}\right] \tag{9}\]  
The value range of \(D\) is (0-1), \(D=0\) represents the initial state or no damage state, and \(D=1\) represents the complete damage state.  
The microelement strength function is established based on the strength criterion. The rock microelement strength described by the M-C strength criterion [45; 46] is  
\[f_{(\sigma)}=\sigma_{1}-\sigma_{3}\frac{1+\sin\varphi}{1-\sin\varphi}-\frac{2 c\cos\varphi}{1-\sin\varphi} \tag{10}\]  
The traditional M-C criterion is insufficient to describe the rock stress state under hydraulic coupling; therefore, the seepage pressure is introduced. In combination with Formulas (5) and (10), an improved M-C criterion reflecting the effect of hydraulic coupling is obtained:  
\[f_{(\sigma^{\prime})}=\sigma^{\prime}_{1}-\sigma^{\prime}_{3}\frac{1+\sin \varphi}{1-\sin\varphi}+\frac{2\alpha\sin\varphi}{1-\sin\varphi}P_{w}-\frac{2 c\cos\varphi}{1-\sin\varphi} \tag{11}\]  
Combining Equations (7) and (11), the microelement strength function based on the improved M-C criterion was obtained:  
\[f_{(\sigma)}=\frac{\big{(}(1-\sin\varphi)\sigma_{1}-(1+\sin\varphi)\sigma_{3} +2\alpha\sin\varphi P_{w}\big{)}E\varepsilon_{1}}{(1-\sin\varphi)(\sigma_{1} -2\mu\sigma_{3}-(1-2\mu)\sigma P_{w})}-\frac{2c\cos\varphi}{1-\sin\varphi} \tag{12}\]  
where \(\frac{2c\cos\varphi}{1-\sin\varphi}\) is the damage threshold, which means that when \(f_{(\sigma)}\geq 0\), the rock enters the yield stage and the damage starts to appear.  
It can be seen that the existence of seepage pressure accelerates the process of sandstone's failure. Axial stress and seepage pressure play a positive role in rock yield, while confining pressure plays a restraining role in rock yield.  
[PERSON]'s law is expressed as  
\[\tilde{\epsilon}^{\prime}_{1}=\frac{1}{E}\Big{[}\tilde{\sigma}^{\prime}_{1} -\mu\Big{(}\tilde{\sigma}^{\prime}_{2}+\tilde{\sigma}^{\prime}_{3}\Big{)} \Big{]} \tag{13}\]  
By substituting Equations (6) and (12) into Equation (13), the rock hydraulic coupling damage constitutive model is obtained:  
\[\sigma_{1}=E\varepsilon_{1}\exp\left[-\bigg{(}\frac{f_{(\sigma)}}{k}\bigg{)}^ {m}\right]+2\mu\sigma_{3}+(1-2\mu)\alpha P_{w} \tag{14}\]  
This is the general form of the hydraulic coupling damage constitutive model, which can degenerate into the classical linear elastic constitutive model in the nondamage state; whether there is damage is determined according to Equation (12). When seepage water pressure is not considered, it can be reduced to the constitutive model based on the M-C criterion [47].