## 4 Particle Energy Dissipation Mechanism  
### Verification of Energy Dissipation in Particle Systems  
According to the law of conservation of energy, the energy input into the particle system (\(E_{\rm input}\)) during vibratory compaction should equal the total energy dissipated (\(E_{\rm total}\)) within the system. This principle has been verified in cyclic shear testing simulations by [PERSON] et al. [30]. \(E_{\rm total}\) comprises dashpot energy, slip energy, rolling slip energy, and strain energy (sliding and rolling). With the periodic cosine excitation on the load plate, \(E_{\rm input}\) is calculated from the force-displacement curve:  
\[E_{\rm input}=\sum_{i=1}^{T}S_{\rm T}=\sum_{i=1}^{T}\int_{\varepsilon_{\rm O }}^{\varepsilon_{\rm A}}F\mathrm{d}\varepsilon \tag{18}\]  
where \(T\) is the number of load cycles; \(F\) is the force magnitude; \(\varepsilon_{0}\) is the displacement at the start of the cycle; and \(\varepsilon_{\rm A}\) is the maximum displacement in the cycle.  
Studies by [PERSON] et al. [31] on input and dissipated energy relationships showed the calculated \(E_{\rm total}\) was always less than the actual \(E_{\rm input}\). This is because some energy (\(E_{\rm art}\)) is neglected by artificial definitions, mainly normal and tangential elastic work, \(W_{\rm n}\) and \(W_{\rm t}\) in Equations (15) and (16).  
By introducing a correction and defining the total dissipated energy as \(E^{\prime}_{\rm total}=E_{\rm total}+E_{\rm art}\), the comparison is presented in Figure 7. It was observed that \(E^{\prime}_{\rm total}\) closely matched \(E_{\rm input}\), confirming the dissipation calculations by PFC criteria appropriately represent the compaction mechanism energy pathways. This monitoring approach enhanced insights into meso-scale particle interactions and enabled a better understanding of macro-scale responses. However, it is crucial to verify that the calculated meso-scale energy dissipations correspond to the physical and mechanical behaviors observed at the macro-scale, as this is critical for mechanism analysis.  
Figure 8 presents the cumulative input energy over load cycles for different mixing degrees. The increments of input energy decreased over time but experienced a significant rise before t < 25T, during periods of rapid void ratio changes. Afterward, the increments slowly increased. This can be attributed to the easier structural adjustments and greater efficiency of dissipation in the initial looser states. As densification proceeded, the presence of more compact and uniform particle contacts reduced sliding contacts and tangential creep, resulting in a severe limitation of energy dissipation and input [32; 33]. The consistency observed with the void ratio-time patterns in Figure 3 supports the notion that energy input leads to void ratio reductions.  
Figure 6: Self-diffusion coefficient varies with mixing degree.  
Figure 9 further displays the aligned energy input and void ratio trends over mixing degrees. Their monotonic decline with improving homogeneity visually confirms void ratio changes had direct correspondence to particle system energy input and thus total dissipation. Therefore, analyzing the compaction mechanism via energy dissipation pathways is reasonable.  
Figure 8: Variations in input energy with load period.  
Figure 7: Comparison of the input energy with the corrected total dissipated energy.  
Figure 9: Input energy and void ratio changes with mixing degree.