# Shear Mechanical Properties of Rock Joints Under Non-Uniform Load Based on DEM
## 4 Simulation Results and Analysis
### Energy Evolution Law  
The essence of the mechanical behavior of rocks under loading lies in the process of external energy input, storage, and dissipation. Examining the rock damage process from an energy perspective can effectively reveal the underlying mechanical mechanisms of rocks. Assuming that no energy exchange occurs between the rock loading system and the external environment, according to the law of the conservation of energy, the relationship between absorbed energy (\(U\)), elastic energy (\(U_{e}\)), and dissipated energy (\(U_{d}\)) generated by microdefect closure or expansion can be expressed as  
\[U=U_{e}+U_{d} \tag{6}\]  
where \(U\) is the energy absorbed in loading; \(U_{e}\) is the elastic energy stored inside the rock and released during the unloading process; \(U_{d}\) is the energy consumed during the plastic deformation of the rock, which is irreversible.  
Equation (6) can be expressed in the integral form of energy density as  
\[\int udV=\int u_{e}dV+\int u_{d}dV \tag{7}\]  
Figure 13: Crack percentage under different load distributions (\(G_{T}\) = 20 kN; standard deviations: UD = 0.096; NFC = 0.083; FOC = 0.078; FFC = 0.073).  
where \(u\) represents the input energy density, \(U_{e}\) denotes the elastic energy density, \(U_{d}\) signifies the dissipative energy density, and \(dV\) represents the sum of the energy density corresponding to the unit volume.  
These parameters could be calculated by integrating the curves [30], as shown in Figure 14. The lower stress point could be set to 30% of shear strength, and the area of trapezoid ABCF represents the input energy density for the loading process. Trapezoid ABCED indicates the elastic energy density that can be released. Trapezoid CDEF represents the dissipated energy density. The calculation of these areas can be obtained from Equation (8):  
\[\left\{\begin{array}{l}u^{n}=\int_{\gamma_{1\mu}^{n}}^{\gamma_{n}}\tau_{max}d \gamma\\ u_{e}^{n}=\int_{\gamma_{1\mu}^{n}}^{\gamma_{n}}\tau_{low}d\gamma\\ u_{d}^{n}=u^{n}-u_{e}^{n}=\int_{\gamma_{1\mu}^{n}}^{\gamma_{n}}\tau_{max}d \gamma-\int_{\gamma_{1\mu}^{n}}^{\gamma_{n}^{n}}\tau_{low}d\gamma\end{array} \right., \tag{8}\]  
where \(\tau_{max}\) and \(\tau_{low}\) are used to calculate the maximum and minimum shear stresses, respectively.  
The evolution relationships of absorbed energy (\(U\)), elastic energy (\(U_{e}\)), and dissipated energy (\(U_{d}\)) are presented in Figure 15 to analyze the energy level of the joint surface under different load distribution methods. As the stress level increased, both \(U\) and \(U_{e}\) showed a gradual rise, except under UD conditions, while \(U_{d}\) did not display a discernible trend. The average values of \(U\) and \(U_{e}\) for FFC were larger than those for FOC and NFC, and the average values of \(U_{d}\) for FFC were greater than those for NFC and FOC. These differences stemmed from the varying degrees of crack evolution under different forms of load distribution; FFC was characterized by shear cracks with higher strength and greater energy dissipation, whereas NFC and FOC were dominated by tensile cracks with lower strength and less energy dissipation. In practical engineering, it is advisable to avoid placing load concentration locations too close together. Based on the above results and analysis, it can be observed that as the load concentration location becomes closer to the far end along the shear direction, there is greater deformation in rock mass during failure, requiring more strength and energy for failure. Increasing the distance between load concentration locations makes the rock mass structure less prone to damage; therefore, it should be avoided when load concentration locations are too close together but rather placed in positions with large load concentrations.  
Figure 14: Schematic diagram of energy calculation.