## 3 Evolution of the Plastic Zone under Osmotic Pressure
### Fracture Expansion under High Osmotic Pressures  
According to Equation (1), the main surface stress (\(\sigma_{ne}\)) of the fracture is negative under high osmotic pressures, indicating that the fracture surface is under a slip-tension compound stress. Under this circumstance, the following inequality holds:  
\[P_{p}\geq\frac{\sigma_{xx}}{2}+\frac{\sigma_{xx}}{2}\cos 2\theta \tag{19}\]  
During the opening of a fracture tip, the fracture width is usually perpendicular to the crack propagation direction. The angle at the tip of the fissure is a nonzero parameter, which is called the fracture opening degree (\(\rho\)). [PERSON] demonstrated that under this condition, the transverse compressive stress (\(\sigma_{ne}\)) at the fracture surface causes a vertical tensile stress (\(\sigma_{te}\)) and its maximum stress occurs at the apex [45]. The maximum vertical tensile stress can be obtained from the following expression:  
\[\sigma_{te}\ =\ \frac{\sigma_{xx}}{2}-\frac{\sigma_{xx}}{2}\cos 2\theta-P \tag{20}\]  
Moreover, the corresponding type I stress intensity is calculated as follows:  
\[K_{\text{I}(te)}\ =\ \sigma_{te}\sqrt{\frac{\rho}{a}}\sqrt{\pi a}\ =\ \sigma_{te}\sqrt{\pi\rho} \tag{21}\]  
Equation (20) holds when \(\rho/a\) approaches zero. Similarly, the stress intensity factor \(K_{\text{I}}\) caused by normal compressive stress (\(\sigma_{ne}\)) is calculated as follows:  
\[K_{\text{I}(ne)}\ =\ \sigma_{ne}\sqrt{\pi a} \tag{22}\]The corresponding \(K_{\rm I}\) of the open fracture can be obtained by superimposing the tensile and normal stresses at the fracture tip:  
\[ \begin{array}{ll}K_{\rm I}&=K_{\rm I(t)}+K_{\rm I(\
u n)}\\ &=\sigma_{\rm r}\sqrt{\pi\rho}-\sigma_{\rm r\sigma}\sqrt{\pi a}\\ &=(\frac{\sigma_{\rm x}}{2}-\frac{\sigma_{\rm x}}{2}\cos 2\theta-P_{p})\sqrt{\pi \rho}-(\frac{\sigma_{\rm x}}{2}+\frac{\sigma_{\rm x}}{2}\cos 2\theta-P_{p}) \sqrt{\pi a}\end{array}  \tag{23}\]  
The tensile fracture behavior of fractured rock bodies under high osmotic pressure is often accompanied by type I or type II fractures. Accordingly, the stress field at the fracture tip can be expressed in the form below:  
\[\sigma_{x} = \frac{K_{\rm I}}{\sqrt{2\pi r}}(\cos\frac{\theta}{2}-\cos\frac{ \theta}{2}\sin\frac{\theta}{2}\sin\frac{3\theta}{2})-\frac{K_{\rm II}}{\sqrt{2 \pi r}}(2\sin\frac{\theta  
low osmotic pressures. Under low osmotic pressures, \(r\) first decreases, then increases, and finally decreases with the increase in the fracture dip angle. Conversely, under high osmotic pressures, the radius \(r\) rises nonlinearly as \(\theta\) increases. However, a slight decrease is observed when \(\theta\) varies in the range of 30-45\({}^{\circ}\). Figure 2a indicates that at low osmotic pressures and a fracture dip angle of less than 45\({}^{\circ}\), the dominant fracture at the tip is of type II fractures, which occurs under shear stresses [43, 44]. The difference in \(r\) values at different osmotic pressures is significant, and it exhibits a decreasing-increasing trend. Under lower osmotic pressures, \(r\) decreases more rapidly in the earlier stages, and the rate of increase in later stages is less pronounced. At a fracture dip angle of 45\({}^{\circ}\), as the osmotic pressure increases gradually, \(r\) approaches an upward convex distribution. The plastic deformation caused by higher osmotic pressure is more significant and reaches the maximum \(r\). When the fracture dip angle is smaller than 45\({}^{\circ}\), the tip fracture is predominantly of type I fractures, occurring under shear stress, which reduces the plastic deformation degree of the permeable rock body. Furthermore, it is observed that as the fracture dip angle increases from 0\({}^{\circ}\) to 90\({}^{\circ}\), the plastic zone radius decreases rapidly. The results reveal that under low permeability pressures, the friction coefficient of the fracture surface is less sensitive to the softening effect of the seepage field.