# Study on Composite Fracture Characteristics and Hydraulic Fracturing Behavior of Hard Rock
## 2 Composite Fracture Characteristics in the Fracture Toughness Test  
### The GMTS Criterion  
Considering that there are more I-II composite faults [35; 36; 37; 38] in rock engineering, and I-II composite faults only crack on the fracture cross-section [39], this study conducted theoretical analysis on the fracture cross-section. This study considered the influence of non-singular terms on the fracture of medium cracks and represented the stress field at the crack tip using the stress intensity factors \(K_{\text{I}}\), \(K_{\text{II}}\), and T stress, as shown below [40]:  
\[\left\{ \begin{array}{l}\sigma_{r}=\frac{1}{\sqrt{2\pi r}}\cos\frac{\theta}{2 }\left[K_{\text{I}}\left(1+\sin^{2}\frac{\theta}{2}\right)+K_{\text{II}}\left( \frac{3}{2}\sin\theta-2\tan\frac{\theta}{2}\right)\right]+T\cos^{2}\theta\\ \sigma_{\theta}=\frac{1}{\sqrt{2\pi r}}\cos\frac{\theta}{2}\left(K_{\text{I}} \cos^{2}\frac{\theta}{2}-K_{\text{II}}\frac{3}{2}\sin\theta\right)+T\sin^{2} \theta\\ \tau_{r\theta}=\frac{1}{\sqrt{2\pi r}}\cos\frac{\theta}{2}\left[K_{\text{I}} \sin\theta+K_{\text{II}}(3\cos\theta-1)\right]-T\sin\theta\cos\theta\end{array} \right. \tag{1}\]  
In this equation, \(\sigma_{r}\), \(\sigma_{\theta}\), and \(\tau_{r\theta}\) represent the radial stress, tangential stress, and shear stress at the crack tip, respectively, measured in megapascals (MPa). The variables \(r\) and \(\theta\) denote the radius and angle at the crack tip, measured in millimeters (mm) and degrees (\({}^{\circ}\)), respectively. \(K_{\text{I}}\) and \(K_{\text{II}}\) are the stress intensity factors for Mode I and Mode II cracks, measured in megapascals per square root meter (MPa\(\cdot\)m\({}^{-1/2}\)), respectively. \(T\) is the non-singular term of the stress field, i.e., T stress, representing the stress component parallel to the crack direction, measured in megapascals (MPa). The MTS theory was primarily employed to explain the fracture problems associated with Mode I, Mode II, and Mode I-II compound cracks. This criterion posits the following:Cracks initiate at the crack tip radially in the direction of the maximum tangential stress \(\sigma_{\Theta}\). Based on the tangential stress \(\sigma_{\Theta}\) achieving its maximum value \(\sigma_{\Theta\max}\), the angle of crack propagation \(\theta_{0}\) is determined. A fracture occurs at a radial distance \(r_{\text{c}}\) along the crack tip, simultaneously with the tangential stress \(\sigma_{\Theta}\) exceeding the tensile strength \(\sigma_{\text{t}}\). Therefore, the mathematical expression for the MTS criterion is [41]:  
\[\left\{ \begin{array}{l}\partial\sigma_{\theta}/\partial\theta|_{\theta=\theta _{0}}=0\\ \partial^{2}\sigma_{\theta}/\partial^{2}\theta|_{\theta=\theta_{0}}<0\\ \sigma_{\Theta\max}=\sigma_{\text{t}}\end{array} \right. \tag{2}\]  
Substituting Equation (1) into Equation (2) and defining \(\eta_{0}=\sqrt{2\pi r_{c}}\), the crack initiation criterion for the central straight Brazilian disk (CSTBD) splitting test specimen was obtained:  
\[\left\{\begin{array}{l}\left\{\begin{array}{l}\left\{-\frac{3}{4}\cos\frac{ \theta_{0}}{2}[K_{\text{I}}\sin\theta_{0}+K_{\text{II}}(3\cos\theta_{0}-1)]+ \eta_{0}T\sin 2\theta_{0}\right\}=0\\ \left\{ \begin{array}{l}\frac{3}{8}\left[\begin{array}{l}K_{\text{I}}\Big{(} \sin\frac{\theta_{0}}{2}\sin\theta_{0}-2\cos\frac{\theta_{0}}{2}\cos\theta_{0 }\Big{)}+\\ K_{\text{II}}\Big{(}3\cos\theta_{0}+6\cos\frac{\theta_{0}}{2}\sin\theta_{0}-1 \Big{)}\end{array}\right]+2\eta_{0}T\cos 2\theta_{0}\right\}<0\end{array} \right. \tag{3}\]  
After obtaining the crack propagation angle \(\theta_{0}\) using Equation (3), based on the boundary conditions of the CSTBD test, the tangential stress at the crack tip was determined. In Equation (3), there exists a critical distance \(r_{\text{c}}\) from the crack tip. For compound cracks, the critical break distance is variable and depends on the stress state at the crack tip.