## 3 Results
### Plastic Deformation Evolution
To calculate the elastic energy of rock materials, a correction is needed. The elastic energy of rock can be expressed as:  
\[ \begin{split} U=\frac{k}{2E}\left[\sigma_{1}^{2}+2\sigma_{3}^{2}- 2\
u(2\sigma_{1}\sigma_{3}+\sigma_{3})\right]\\ \text{d}U=\frac{k}{2E}[2\sigma_{1}\text{d}\sigma_{1}-2\
u(2\sigma_{3} \text{d}\sigma_{1})]\\ \end{split}  \tag{6}\]  
where \(k\) is the elastic energy correction parameter, which is dimensionless. 3. Dissipated energy According to the energy conservation law, the dissipated energy during the process of rock damage is equal to the difference between the input energy generated by the external force doing work and the internal elastic energy of the rock. The dissipated energy can be calculated as: \[A=W-U\] (7)To summarize, the energy of the rock at any given point can be determined by employing Equations (2), (6) and (7) based on the stress-strain curve. In this study, the proposed energy calculation method was applied to the collected stress-strain data obtained from experiments, with the elastic energy correction parameter set to 0.6. The calculated energy evolution curves of the rock under different confining pressures are presented in Figure 12. The horizontal axis represents the axial strain, while the vertical axis represents the energy density. Regarding the input energy derived from external forces using Equation (2), its calculation involves integrating the area under the stress-strain curve. The results of this calculation exhibit a high degree of agreement with experimental measurements and do not necessitate further comparison. The comparison between the calculated results of elastic energy and the corresponding experimental results is illustrated in Figure 13, where the scattered points represent the experimental data, and the curve represents the calculated results. Similarly, the comparison between the calculated results of dissipated energy and the corresponding experimental data is depicted in Figure 14, where the scattered points represent the experimental results, and the curve represents the calculated results. Figure 12: Predicted energy density curves under different confining pressures ((**a**): 5 MPa, (**b**): 10 MPa, (**c**): 20 MPa, (**d**): 30 MPa). Based on the findings presented in Figures 12-14, the computed energy evolution curves of the rock provide valuable insights into the relationship between energy transformation and dissipation during the analyzed rock damage process. Specifically, the following observations can be made: (1) Comparison between Figures 4 and 12 reveals that the proposed energy calculation method accurately captures the evolution relationship among input energy, elastic energy, and dissipated energy, demonstrating consistency with the experimental results. This indicates that the proposed energy estimation method can to some extent achieve a quantitative description of the internal energy distribution in the rock during the loading process. (2) Figure 13 demonstrates that the proposed energy calculation method effectively represents the evolution of internal elastic energy in the rock. Prior to reaching the peak stress, the computed elastic energy results align well with the experimental results. However, during the post-peak softening and residual stress stages, the computed elastic energy results exhibit some deviation from the experimental results, predominantly showing a lower overall trend. Moreover, the proposed energy estimation method exhibits inadequate accuracy in calculating the peak value of elastic energy stored in the rock. (3) Figure 14 illustrates that the proposed energy calculation method effectively portrays the evolution of dissipated energy in the rock. Prior to reaching the peak stress, the computed dissipated energy results closely match the experimental results. Nevertheless, during the post-peak softening and residual stress stages, the computed dissipated energy results show a certain degree of deviation from the experimental results, generally exhibiting a higher overall trend.