# Study on Experimental and Constitutive Model of Granite Gneiss under Hydro-Mechanical Properties
## 4 Numerical Simulation
### Establishment of Constitutive Model  
Conventional linear failure surfaces cannot accurately indicate the nonlinear features of the rock sample. On account of the test analysis, an improved Durkcor-Prgaer criterion was used as the plastic yield function, and a hydro-mechanical properties variable was introduced into this equation to realize the characteristics of the sample hydro-mechanical properties [32; 33; 34]. This model simulates rocks as isotropic porous media. The improved equation is expressed as follows:  
\[f=q^{2}+A_{0}(1-\alpha\zeta)h(\theta)\alpha_{p}(p-C_{0})p_{0}=0, \tag{9}\]  
In Formula (9), parameters \(A\) and \(C_{0}\) in the plastic yield surface are obtained from different confining pressure conditions triaxial tests, \(\zeta\) is the coupling variable, and \(\alpha\) is the coefficient of the coupling variable. The plastic potential energy can determine the incremental characteristics of plastic strain under the permeate action. In line with the relevant research results of [PERSON] et al. [35]. The plastic potential energy equation in elastic-plastic damage constitutive model is improved by introducing the coupling variable:  
\[g=q+A_{0}(1-\alpha_{1}\xi)(1-d)\eta h(\theta)(p-C_{0})\ln\left(\frac{p-C_{0}} {I_{0}}\right)=0, \tag{10}\]  
In Formula (10), the intersection of the mean stress and plastic potential surface is \(I_{0}\), \(\eta\) is the volume compression and expansion points slope. Damage occurs and expands continuously under the action of damage-driving forces. This study introduces a damage criterion proposed by [PERSON] [36] to characterize the granitic gneiss damage evolution. \[f_{d}\Big{(}Y_{d}^{p},d\Big{)}=d_{c}\text{tanh}\Big{(}B_{d}Y_{d}^{p}\Big{)}-d \leq 0, \tag{11}\]  
In Equation (11), \(Y_{d}{}^{p}\) is the rock damage driving force, \(d_{c}\) is the maximum damage variable, \(B_{d}\) controls the damage evolution rate. The coupling effect on rocks could lead to degradations in the modulus of rock materials. Based on irreversible thermodynamic characteristics, the stiffness tensor can be obtained. The following expression of the stress-strain relationship is obtained:  
\[\mathbf{\sigma}=(1-\alpha_{2}\xi)(1-d)\mathbb{C}^{0}:\mathbf{\varepsilon}^{e}=\mathbb{C} (\xi,d):\mathbf{\varepsilon}^{e}, \tag{12}\]  
In Formula (12), it can be seen that both the coupled variables of seepage stress and damage can lead to the degradation of rock stiffness. In the test, degradations in the moduli are the result of the combined coupling effect, which is a non-increasing function. When coupling variables \(\xi\) and the damage variable d are both 0, the rock is in an initial state of no seepage and no damage, and the hydro-mechanical properties elastic stiffness matrix \(\mathbb{C}(\xi,d)\) is the same as the initial elastic stiffness matrix \(\mathbb{C}^{0}\). According to the consistency principle, considering that granite gneiss failure and nonlinear behavior is the result of damage, plastic deformation, and hydro-mechanical properties, listing the consistency conditions of damage, plasticity, and coupling. The damage multiplier d\(d\) can be obtained as shown in Equation (13):  
\[ \begin{split}\text{d}d&=-\frac{\partial f_{d}}{ \partial\gamma_{d}^{p}}\left(\frac{\partial\gamma_{d}^{p}}{\partial\mathbf{ \varepsilon}^{p}}:\text{d}\mathbf{\varepsilon}^{p}+\frac{\partial\gamma_{d}^{p}} {\partial\gamma_{p}}\text{d}\gamma_{p}\right)\Big{/}\frac{\partial f_{d}}{ \partial\text{d}}\\ &=-\lambda_{s}\frac{\partial f_{d}}{\partial\gamma_{d}^{p}}\left( \frac{\partial\gamma_{d}^{p}}{\partial\mathbf{\varepsilon}^{p}}:\frac{\partial \xi}{\partial\mathbf{\varepsilon}}+\frac{\partial\gamma_{d}^{p}}{\partial\gamma_{p }}\right)\Big{/}\frac{\partial f_{d}}{\partial\text{d}}\end{split}