## 3 Results
### Plastic Deformation Evolution
(4) In the residual stage, the dissipated energy-plastic strain curve shows a generally linear increasing trend, and the slope of the curve can be regarded as a constant. Figure 9: Correspondence between plastic strain and axial strain (**(a)**: 5 MPa, **(b)**: 10 MPa, **(c)**: 20 MPa, **(d)**: 30 MPa). The dissipated energy-plastic strain curves are depicted in Figure 11. These curves exhibit a consistent evolution pattern and can be categorized into four stages: compaction, elastic, post-peak softening, and residual stress. From Figure 11, it is evident that, under identical plastic strain conditions, higher confining pressures result in greater dissipated energy. Conversely, under equivalent dissipated energy conditions, higher confining pressures lead to smaller plastic strains. All dissipated energy-plastic strain curves under different confining pressures undergo a progression characterized by a slow initial increase, rapid subsequent increase, a gradual decrease in the rate of increase, and eventual convergence toward a constant value. In the residual stress stage, the slopes of the dissipated energy-plastic strain curves display minimal variation, exhibiting no conspicuous relationship with the confining pressure. The calculation of energy in different parts of a rock requires the collection of strain and stress data from cyclic loading-unloading tests. However, due to current experimental limitations, the number of loading-unloading cycles that can be performed in each test is limited. As a result, the obtained energy data for rocks suffer from problems such as a limited number of samples, high sample variability, and sample discontinuity, which hinder the study of rock damage evolution from an energy perspective. Therefore, it is urgent to calculate the rock energy at any time during the damage evolution process. Based on this,  
Figure 11: Dissipated energy–plastic strain curves under different confining pressures. Figure 10: Correspondence between dissipated energy and axial plastic strain. this paper explores the calculation of input energy, elastic energy, and dissipated energy in rock from a theoretical perspective. As expounded in the previous section, the rock damage process involves input energy, elastic energy, and dissipated energy. By leveraging the stress-strain curve of the rock obtained from experimentation, it becomes feasible to estimate the internal energy of the rock through the subsequent theoretical calculation process [39]. 1. Input energy In conventional triaxial compression tests, the input energy of the rock during the loading process includes two parts, namely, axial positive work and circumferential negative work, which can be calculated by integrating the stress-strain curve. \[W=W_{1}+W_{3}=\int_{0}^{\varepsilon_{1}^{t}}\sigma_{1}d\varepsilon_{1}+2\int_ {0}^{\varepsilon_{3}^{t}}\sigma_{3}d\varepsilon_{3}\] (2) where \(\varepsilon_{1}^{t}\) represents the axial strain at time \"\(t\)\"; \(\varepsilon_{3}^{t}\) represents the circumferential strain at time \"\(t\)\"; \(\sigma_{i}(i=1,2,3)\) represents the stress in different directions; and \(\varepsilon_{i}(i=1,2,3)\) represents the strain in different directions. 2. Elastic Energy According to the theory of elasticity, the elastic energy can be expressed as: Figure 7.0 (3) According to the generalized Hooke's law: \[ \begin{cases}\varepsilon_{1}=\frac{1}{E}[\sigma_{1}-\
u(\sigma_{2}+\sigma_{ 3})]\\ \varepsilon_{2}=\frac{1}{E}[\sigma_{2}-\
u(\sigma_{1}+\sigma_{3})]\\ \varepsilon_{3}=\frac{1}{E}[\sigma_{3}-\
u(\sigma_{1}+\sigma_{2})]\\ \end{cases} \] (4) where \(E\) represents the elastic modulus, MPa; and \(\
u\) represents the Poisson's ratio. In conventional triaxial compression tests, \(\sigma_{2}=\sigma_{3}\). By combining Equations (3) and (4), the elastic energy can be expressed as: \[U=\frac{1}{2E}\left[\sigma_{1}^{2}+2\sigma_{3}^{2}-2\
u(2\sigma_{1}\sigma_{3} +\sigma_{3})\right]\] (5)  
However, Equation (5) is only applicable to the calculation of elastic energy for ideal elastic materials.