## 5 Energy Dissipation Analysis  
Thermodynamic principles indicate that energy transformation is an essential factor for the physical change process of matter, and matter destruction is a state instability phenomenon driven by energy [43]. In discrete element theory, the energy dissipation during the erosion failure process of a slope is mainly reflected in the damage and fracturing  
Figure 16: Unbalanced force curves. (**a**) Monitoring point A1 (**b**) Monitoring point A2 (**c**) Monitoring point A3 (**d**) Monitoring point C1 (**e**) Monitoring point C2 (**f**) Monitoring point C3 (**g**) Monitoring point C4 (**h**) Monitoring point C5 (**i**) Monitoring point C6.  
of the cohesion model between a gully and nearby soil particles, which reflects the internal mechanism of erosion and erosion evolution. Assuming a closed system without heat exchange with the surroundings, the following equation can be obtained for the slope scour system, according to the first law of thermodynamics:  
\[U=U_{d}+U_{e}=U_{pre}+\sum_{N_{p}}\gamma^{(p)}V^{(p)}\Delta x \tag{19}\]  
where \(U\) is the total input energy generated by the work of the slope body force during the erosion process, \(U_{d}\) is the dissipated energy in the dynamic response process of the slope, \(U_{e}\) is the elastic strain energy stored in the slope particles that can be released, \(U_{pre}\) is the upper total input energy accumulated in a calculation time step, \(\gamma^{(p)}\) and \(\Delta x\) are the particle weight and displacement in the current time step, respectively, \(N_{p}\) is the number of particles, and \(V^{(p)}\) is the particle volume. The particle contact model in the PFC model is a parallel bonding model. The elastic strain energy, \(U_{e}\), is calculated from the particle strain energy \(U_{c}\) and the parallel bonding strain energy \(U_{pb}\), according to the equations:  
\[U_{e}=U_{c}+U_{pb} \tag{20}\]  
\[U_{c}=\frac{1}{2}\sum_{N_{c}}\left(\mid F_{i}^{n}\mid^{2}/k^{n}+\mid F_{i}^{s} \mid^{2}/k^{s}\right) \tag{21}\]  
\[U_{pb}=\frac{1}{2}\sum_{N_{pb}}\left[\mid\overline{F}_{i}^{n}\mid^{2}/(A \overline{k}^{n})+\mid\overline{F}_{i}^{s}\mid^{2}/(A\overline{k}^{s})+\mid \overline{M}\mid^{2}/(I\overline{k}_{n})\right] \tag{22}\]  
where \(N_{c}\) and \(N_{pb}\) are the number of contact force chains and the number of parallel bonds, respectively; \(i\) denotes the \(i\)th contact chain; \(F_{in}\) and \(F_{is}\) are the normal contact force and tangential contact force, respectively; \(\overline{F}_{i}^{n}\), \(\overline{F}_{i}^{s}\), and \(\overline{M}\) are the normal parallel bonding force, the tangential parallel bonding force, and the parallel bonding moment, respectively; \(k_{n}\) and \(k^{s}\) are the contact normal stiffness and contact tangential stiffness, respectively; \(\overline{k}^{n}\), \(\overline{k}^{s}\), and \(\overline{k}_{n}\) are the stiffness corresponding to each parallel bonding force; and \(A\) and \(I\) are the cross-sectional area and the moment of inertia of the parallel bond, respectively.  
The dissipated energy \(U_{d}\) of the final model can be obtained from Equation (19) by substituting and solving various equations. According to the first law of thermodynamics and the particle flow with an actual energy monitoring system, the instantaneous kinetic energy of the gully can be determined.