## 3 Simulation Results and Analysis
### Energy Characteristics of Rock-like Samples  
According to the law of thermodynamics, the destruction of matter is a kind of instability phenomenon caused by energy evolution. Under the condition of uniaxial compression, the external force does work on the rock, and part of the energy is stored in the rock, which is called strain energy. The other part of the energy is used for the friction of micro-defect closure, crack propagation and relative dislocation of flaw surface in rock, which is called dissipative energy. When the accumulated strain energy of rock reaches a certain threshold, it will be released and transformed into dissipative energy, leading to rock failure. Figure 4: Variation of UCS and elastic modulus of a sample model with different flaw inclination angles. Figure 5: Result comparison of laboratory tests and numerical simulation of crack initiation stress. In the parallel bonding model, the boundary energy is the total input energy [48]. According to the first law of thermodynamics, there are  
\[E=E_{\text{d}}+E_{\text{e}} \tag{5}\]  
\[E_{\text{e}}=E_{\text{p}}+E_{\text{pb}} \tag{6}\]  
where \(E\) is the boundary energy, \(E_{\text{d}}\) is the dissipative energy of the rock, \(E_{\text{e}}\) is the strain energy of the rock, \(E_{\text{p}}\) is the grain strain energy, and \(E_{\text{pb}}\) is the parallel bond strain energy. The system energy monitoring in PFC2D can be realized by the built-in fish language. Figure 7 shows the relationship between the variation law of meso energy and the variation law of stress in samples with different flaw inclination angles. It can be found that the meso energy evolution of samples with different flaw inclination angles has similar characteristics. Combined with the characteristic points of the stress-strain curve, it can be roughly divided into three stages. The analysis was performed taking the sample model with the flaw inclination angle \(\beta\) = 45\({}^{\circ}\) as an example, as shown in Figure 7d. According to the analysis of micro energy curve, in stage I (section OA), that is, before the crack initiation stress, the boundary energy and strain energy curves almost overlap, and the dissipated energy is at a low value. It indicates that only a small number of micro-cracks are generated, the internal damage of the sample is small, and the boundary energy is basically stored in the sample in the form of strain energy. In stage II (section AB), when the crack initiation stress is reached, the boundary energy and strain energy continue to increase, but the curves of boundary energy and strain energy begin to separate obviously. Part of the boundary energy is transformed into dissipative energy, and the dissipative energy increases steadily. It indicates that with the continuous work of external force, the bond between particles in the sample is destroyed, which leads to the continuous generation of micro-cracks and the increase of damage degree in the sample. In stage III (section BC), after the peak stress, the boundary energy continues to increase, and the strain energy is released rapidly and transformed into dissipative energy, which leads to the rapid growth of dissipative energy. When the curves of strain energy and dissipative energy intersect, the dissipative energy occupies the dominant position. It indicates that with the rapid increase of the number of micro-cracks, the micro-cracks further expand and connect, and form a macro flaw surface, resulting in the complete loss of the bearing capacity of the sample. To further analyze the energy evolution characteristics of samples with different flaw inclination angles, the boundary energy, strain energy and dissipated energy of samples with different flaw inclination angles under peak stress are presented in Figure 8. It can be seen that there is an obvious relationship between the flaw inclination angle of the sample and the accumulation and dissipation of energy.