## 2 Material and Methods  
### Sample Description  
All the samples used for laboratory experiments were provided by a professional rock manufacturing company located in Liuyang county of China. The lithologic character of the selected rock material was tufufaceous feldspar quartz sandstone, which was formed in Cambrian Period, and the color was reddish brown. Compared with other types of rock material, this sandstone has relatively good homogeneity, continuity and isotropy [54]. The mineralogical analysis demonstrates that this rock is mainly made up of five types of minerals, as presented in Figure 1[4]. Particles of the rock are medium-to-fine in size (0.15-0.50 mm), and the structure belongs to massive.  
To obtain the basic mechanical parameters of this rock, according to the specifications for rock testing issued by the ISRM (International Society for Rock Mechanics and Rock Engineering) [55], many kinds of specimens were prepared. These specimens include three cylindrical specimens with a diameter of 50 mm and a height of 100 mm for uniaxial compression tests, three disk specimens with dimensions of 25 mm \(\times\) 50 mm (thickness \(\times\) diameter) for Brazilian tests, 15 cuboid specimens with dimensions of 50 mm \(\times\) 50 mm \(\times\) 50 mm for inclined plane shear tests, and three semi-circular bend (SCB) specimens with a thickness and a radius of 25 mm for three-point bend tests. All tests were conducted under displacement-control mode at a rate of 0.1 mm/min. The test results are given in Table 1. Note that the values of the cohesion and angel of internal friction were fitted by Mohr-Coulomb criterion on the basis of the results of the inclined plane shear tests.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(v\)/(m/s) & \(\rho\)/(kg/m\({}^{3}\)) & \(\alpha_{\text{t}}\)/MPa & \(\sigma_{\text{p}}\)/MPa & \(E\)/GPa & \(\mu\) & \(\sigma\)/MPa & \(\phi\)/\({}^{0}\) & \(K_{\text{IC}}\)(MPa-m\({}^{1/2}\)) \\ \hline
3174.5 & 2472.2 & 5.3 & 99.3 & 24.4 & 0.26 & 19.0 & 40.4 & 0.6 \\ \hline \hline \end{tabular}  
* Notes: \(v\), \(\rho\), \(\alpha_{\text{p}}\), \(E\), \(\mu\), \(c\), \(\phi\) and \(K_{\text{IC}}\) denote the P-wave velocity, density, tensile strength, uniaxial compression strength, Young’s modulus, Poisson’s ratio, cohesion, angel of internal friction and fracture toughness, respectively.  
\end{table}
Table 1: Several rock mechanics parameters of brown sandstone.  
Figure 1: Type and content of minerals in sandstone slice (Cal–calcite; Kfs–K-feldspar; Pl–plagioclase; Qtz–quartz; Zeo–zeolite).  
Since three-dimensional cracks are difficult to form in thin plates, surface cracks can be devoted to characterizing their internal fracture. Thus, samples were designed to be a rectangular prismatic shape, and the length, thickness and width were set as 150 mm, 25 mm and 100 mm, respectively. All the surfaces of the samples shall be ground to ensure that the perpendicularity and parallelism meet the requirements of rock mechanics testing specifications [55]. A total number of 15 samples manufactured were equally categorized into five groups, as named C, S, H, D and V, respectively. Group C used as a reference contained three intact samples, while Group S denoted the samples with a single inverted U-shaped opening. In contrast, Groups H, D and V referred to the samples containing two inverted U-shaped openings with a rock bridge inclination of 0\({}^{\circ}\), 45\({}^{\circ}\) and 90\({}^{\circ}\), respectively. During sample preparation, a 50 HC cantilever waterjet 3-axis cutting machine was utilized to excavate the central holes, and the rock bridge length between the two openings was fixed to 20 mm. Figure 2 illustrates the detailed dimensions of the openings. Prior to loading, we adopted Vaseline to smooth the upper and lower surfaces of the samples before they were put on the loading platform. The purpose was to decrease the lateral friction between the samples and the compression platens.