# Experimental Study on the Compression Behavior of Ground Rock with Bi-Directional Penetrating Crack
## 3 Results
### Energy Analysis  
The process of uniaxial compression is the process of energy accumulation and release in the sample. It is assumed that there is no energy exchange with the outside world except the compression energy input. Before the samples reach the peak stress, energy is stored in two forms, one is in the form of strain energy, and the other is consumed by the initiation and propagation of microcracks and macro cracks. When the input energy exceeds the capacity of the samples, the strain energy will be released suddenly, accompanied by the failure of the samples. According to the first law of thermodynamics, before the samples reach the peak stress, the total energy input \(U_{0}\) is the sum of elastic strain energy \(U_{\rm e}\) and dissipation energy \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Crack Width/mm** & \(\beta\)/\({}^{\circ}\) & **Strength/MPa** & **Peak Strain/\%** & **Elastic Modulus/GPa** \\ \hline \multirow{4}{*}{2} & 0 & 24.42 & 1.61 & 2.09 \\  & 30 & 3.69 & 0.59 & 0.76 \\  & 45 & 16.80 & 1.49 & 1.70 \\  & 60 & 26.74 & 2.03 & 1.94 \\  & 90 & 27.12 & 2.04 & 2.34 \\ \hline \multirow{4}{*}{5} & 0 & 25.74 & 1.77 & 2.42 \\  & 30 & 4.22 & 0.45 & 1.21 \\  & 45 & 19.48 & 1.42 & 1.71 \\  & 60 & 31.37 & 2.09 & 2.01 \\  & 90 & 35.11 & 2.13 & 2.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical parameters of samples. Figure 12: Change of strength and elastic modulus. \(U_{\text{d}}\). \(U_{0}\), \(U_{\text{e}}\), and \(U_{\text{d}}\) can be calculated from Equations (1)-(3). The typical stress-strain curves and energy relationships of natural intact rock samples are shown in Figure 13.  
\[U_{0}=\int_{0}^{\epsilon_{1}}\sigma d\varepsilon \tag{1}\]  
\[U_{\text{e}}=\frac{\sigma_{1}^{2}}{2E} \tag{2}\]  
\[U_{\text{d}}=U_{0}-U_{\text{e}} \tag{3}\]  
where \(\sigma_{1}\) is peak stress, \(\epsilon_{1}\) is peak strain, and \(E\) is elastic modulus. As shown in Figure 14, in the uniaxial compression process, when \(\beta=0^{\circ}\) and t = 5 mm, two obvious stress drops appeared during the process of adding stress. Correspondingly, when t = 2 mm, the proportion of energy dissipated in the total energy during compression is larger. When the sample was compressed, the larger the volume ratio of the slurry was, the more obvious the deformation disharmony was, and the interface between the slurry and the rock was more prone to crack and tensile failure. When the slurry and the rock were separated, the rock and the slurry could continue to bear the pressure independently. After the peak stress of the two samples, the elastic energy decreased rapidly and the dissipated energy increased rapidly, which indicated that the brittle failure occurred finally. As shown in Figure 15, when the angle between the crack and the load direction was \(30^{\circ}\), the stress-strain curve was smooth and nearly linear before peak stress. At the same time, the compressive strength and energy storage capacity of the sample were not significantly related to the thickness of the slurry. Figure 14: Energy change diagram. Figure 13: Energy composition diagram. As shown in Figures 16-18, when \(\beta\geq 45^{\circ}\) and t = 5 mm, the peak strength and energy storage capacity of the grouted rock were greater than that with the thickness of 2 mm. With the increase of the angle between load direction and crack from \(45^{\circ}\) to \(90^{\circ}\), the peak strength and energy storage capacity of the grouted rock increased with the same crack thickness, and the reinforcement effect of slurry on the whole sample was obvious. In the process of uniaxial compression, from the view of energy input, accumulation, dissipation, and release, the damage and fracture evolution process of grouted rock is similar to that of intact rock sample. The damage and fracture evolution process of grouted rock sample in uniaxial compression process can be divided into four stages: (1) Initial  
Figure 16: Energy change diagram. Figure 17: Energy change diagram. Figure 12:0 Energy change diagram. damage stage.