# Study on Composite Fracture Characteristics and Hydraulic Fracturing Behavior of Hard Rock
## 2 Composite Fracture Characteristics in the Fracture Toughness Test
### The Composite Stress Intensity Factor of the CSTBD Specim  
The CSTBD sample splitting test is an effective method for testing fracture toughness, and the specimen processing is simple. By adjusting the \(\beta\) angle between the central straight crack and the point load line, the specimen was loaded by point load until the specimen ruptured. Based on the peak load at rupture and the corresponding shape parameters of the specimen, the I-II composite stress intensity factor of the specimen material could be obtained, as shown in Figure 1.  
Using the following equations to estimate the corresponding Mode I stress intensity factor, Mode II stress intensity factor, and T stress [43]:  
\[K_{\text{I}}=\frac{P\sqrt{\pi a}}{\pi RB}Y_{\text{I}} \tag{7}\]  
\[K_{\text{II}}=\frac{P\sqrt{\pi a}}{\pi RB}Y_{\text{II}} \tag{8}\]\[T=\frac{P}{\pi RB}\frac{1}{1-\alpha}T^{*} \tag{9}\]  
where \(K_{\rm{I}}\), \(K_{\rm{II}}\), and \(T\) represent the Mode I stress intensity factor, Mode II stress intensity factor, and T stress, respectively, at the initiation of rock cracking in the CSTBD specimen. \(P\) is the critical fracture load at the rock failure stage; \(R\) is the radius of the disk specimen; \(B\) is the thickness of the disk specimen; and \(a\) is the half-length of the central straight-crack in the disk specimen. \(Y_{\rm{L}}\), \(Y_{\rm{II}}\), and \(T^{*}\) are dimensionless intensity factors that depend on the shape and loading conditions of the test specimen. For the CSTBD splitting test, \(Y_{\rm{I}}\), \(Y_{\rm{II}}\), and \(T^{*}\) are defined as follows:  
\[Y_{\rm{I}}=f_{11}+2{\sum\limits_{i=1}^{n}{A_{1i}f_{i1}\alpha^{2(i-1)}}} \tag{10}\]  
\[Y_{\rm{II}}=2{\sum\limits_{i=1}^{n}{A_{2i}f_{2i}\alpha^{2(i-1)}}} \tag{11}\]  
\[T^{*}=(1-\alpha)g_{1}+2(1-\alpha){\sum\limits_{i=1}^{n}{(A_{1i}g_{i}-A_{1i}-A_ {3i})\alpha^{2(i-1)}}} \tag{12}\]  
In the equation, when \(i=100\), \(Y_{\rm{I}}\), \(Y_{\rm{II}}\), and \(T^{*}\) are negligible, so the range of \(i\) is from 1 to 151. \(\alpha\) is the ratio of the crack's half-length to the radius of the disk specimen, \(\alpha=a/R\). The coefficients \(f_{\rm{jj}}\), \(g_{i}\), and \(A_{\rm{ji}}\) (where \(j=1\),\(2\); \(i=1\),\(2\),\(3\), \(\ldots\), n) were determined by the following equations:  
\[f_{ji}=\frac{(2i-3)!!}{(2i-2)!!}\left[1+\frac{C_{j1}^{1}}{2i}+\frac{3C_{j2}^{1 }}{4i(i+1)}\right] \tag{13}\]  
\[g_{i}=\frac{2C_{1}^{2}}{(2i-1)(2i+1)}+\frac{8C_{2}^{2}}{(2i-1)(2i+1)(2i+3)} \tag{14}\]  
\[A_{1i}(\beta)=i\cos(2i\beta)-i\cos[2(i-1)\beta] \tag{15}\]  
\[A_{2i}(\beta)=i\sin(2i\beta)-(i-1)\sin[2(i-1)\beta] \tag{16}\]  
\[A_{3i}(\beta)=i\cos(2i\beta)-(i-2)\cos[2(i-1)\beta] \tag{17}\]  
where \(C_{11}^{1}\), \(C_{12}^{1}\), \(C_{21}^{1}\), \(C_{2}^{1}\), and \(C_{2}^{2}\) are the structure coefficients associated with \(\alpha\).  
To analyze the applicability of the MTS criterion to the stress intensity factors during the CSTBD specimen splitting test and investigate the characteristics of compound fractures under the conditions of \(K_{\rm{I}}>0\) and \(K_{\rm{II}}>0\) in practical scenarios, combining Equations (7)-(9), and by comparing Equations (1) and (6), \(K_{\rm{I}}/K_{\rm{IC}}\) and \(K_{\rm{II}}/K_{\rm{IC}}\) were obtained:  
\[\left\{\begin{array}{l}\frac{K_{\rm{L}}}{K_{\rm{IC}}}=\frac{g_{1}(\theta)-( \gamma_{\rm{II}}/\gamma_{\rm{I}})g_{2}(\theta)+(T^{*}/\gamma_{\rm{I}})g_{3}( \theta)}\\ \frac{K_{\rm{II}}}{K_{\rm{IC}}}=\frac{1}{(\gamma_{\rm{II}}/\gamma_{\rm{II}})g_{ 1}(\theta)-g_{2}(\theta)+(T^{*}/\gamma_{\rm{II}})g_{3}(\theta)}\end{array}\right. \tag{18}\]  
where \(g_{1}(\theta)=\cos^{3}(\theta_{0}/2),g_{2}(\theta)=(3/2)sin(\theta_{0})\cos( \theta_{0}/2)\), _and_\(g_{3}(\theta)=[\eta