## 2 Numerical Simulation Calculation Model
### Model Control Equations
#### 2.1.1 Cohesive Pore Pressure Unit
When the stress value in any one direction of the cohesive cell reaches the critical stress value, the cell starts to break:  
\[\max\left\{\frac{\left\langle\frac{t_{n}}{t_{n}^{0}}\right\rangle},\frac{t_{s}}{t _{s}^{0}},\frac{t_{t}}{t_{t}^{0}}\right\}=1 \tag{2}\]  
where \(t_{n}^{0}\) is the tensile strength of the rock, and the units are MPa; \(t_{s}^{0}\) and \(t_{t}^{0}\) are the shear strengths of the rock in the other two directions, and the units are MPa; and \(<\) indicates that the unit is subjected to tensile stress damage only, and no damage occurs for compressive stress or compressive deformation units. * Secondary stress criterion The cohesive unit begins to break under the condition that the unit is subjected to stress in three directions, and its corresponding critical value of the square of the ratio is equal to 1:  
\[\left(\frac{\left\langle t_{n}\right\rangle}{t_{n}^{0}}\right)^{2}+\left(\frac {t_{s}}{t_{s}^{0}}\right)^{2}+\left(\frac{t_{t}}{t_{t}^{0}}\right)^{2}=1 \tag{3}\]  
where \(t_{n}^{0}\) is the unit normal stress, and the units are Pa, and \(t_{s}^{0}\) and \(t_{t}^{0}\) are the critical stresses in the first all direction and second tangential direction of the cohesive unit, respectively, and the units are Pa.
* Maximum stress criterion When the strain value in any direction of the cohesive cell reaches a critical value, the cell starts to break:  
\[\max\left\{\frac{\left\langle\varepsilon_{n}\right\rangle}{\varepsilon_{n}^{0 }},\frac{\varepsilon_{s}}{\varepsilon_{s}^{0}},\frac{\varepsilon_{t}}{ \varepsilon_{t}^{0}}\right\}=1 \tag{4}\]  
where \(\varepsilon_{n}^{0}\) is the critical strain normal to the rock and is dimensionless, and \(\varepsilon_{s}^{0}\) and \(\varepsilon_{t}^{0}\) are the critical strains in the other two directions of the rock and are dimensionless. * Secondary strain guidelines The cohesive cell begins to break under the condition that the sum of the squares of the strains in the three directions of the cell and its corresponding critical strain ratio is equal to 1:  
\[\left(\frac{\left\langle\varepsilon_{n}\right\rangle}{\varepsilon_{n}^{0}} \right)^{2}+\left(\frac{\varepsilon_{s}}{\varepsilon_{s}^{0}}\right)^{2}+ \left(\frac{\varepsilon_{t}}{\varepsilon_{t}^{0}}\right)^{2}=1 \tag{5}\]  
In practice, cracks are generally a combination of two modes, tensile and shear, and in Abaqus, a mixed mode ratio of these two types is simulated in stress mode and energy mode, respectively. * Hybrid mode based on the stress mode ratio The mixed mode of fracture extension is defined by the relationship between the given total rupture displacement \(\phi_{1}\) and the mixed mode ratio \(\phi_{2}\): \[\left\{ \begin{array}{l}\phi_{1}=\left(\frac{2}{\pi}\right)\tan^{-1}\left( \frac{\sqrt{t_{s}^{2}+t_{t}^{2}}}{\left\langle t_{n}\right\rangle}\right)\\ \phi_{2}=\left(\frac{2}{\pi}\right)\tan^{-1}\left(\frac{t_{t}}{t_{s}}\right) \end{array} \right.\] (6) where \(\phi_{1}\) and \(\phi_{2}\) are the mixed\(-\)mode ratios and are dimensionless. * Hybrid mode based on energy mode The mixed\(-\)mode ratio using the energy mode definition is \[\left\{ \begin{array}{l}m_{2}+m_{3}=\frac{G_{s}+G_{t}}{G_{s}+G_{t}+G_{t}}\\ \frac{m_{3}}{m_{2}+m_{3}}=\frac{G_{t}}{G_{s}+G_{t}}\end{array} \right.\] (7)where \(G_{n}\) is the work performed by the normal displacement of the cohesive cell, and the units are J; \(G_{s}\) and \(G_{t}\) are the work performed by the displacements in the remaining two directions of the cell, and the units are J.