## 3 Long-Term Deformation and Energy Dissipation Characteristics
### Energy Dissipation Characteristics  
The long-term stability or performance deterioration of high-speed railway subgrade structures under dynamic action is closely related to the change in its internal energy. Energy dissipation is an essential property of geotechnical materials and reflects the process of weakening and the eventually loss of intrinsic strength during cyclic deformation [54; 55]. The dissipated energy of the filler in each cycle represents the unrecoverable strain energy under the action of external force. While the dissipated energy is generated, plastic deformation and residual stress and strain gradually increase from the initial state. This change can be understood as irreversible deterioration or damage caused by energy dissipation, which causes the filler to change from a stable state to an unstable state. As shown in Figure 7, in the stress-strain hysteresis loop, the area surrounded by the loading section, the vertical dashed line, and the transverse axis is the unit volume energy, indicating the work carried out by the external force on the sample. The area around the unloading section and the transverse axis is the elastic energy per unit volume. The area difference in the loading and unloading curve is the dissipation energy per unit volume, that is, the area of the hysteresis loop. The dissipated energy per unit volume \(U^{\rm d}\) is defined as  
\[ \begin{array}{ll}U^{\rm d}&=U-U^{\rm e}=\int\sigma\rm d\varepsilon-\int \sigma^{\prime}\rm d\varepsilon^{\prime}=\\ &\sum\limits_{loading}(\sigma_{i}+\sigma_{i+1})(\varepsilon_{i+1}-\varepsilon _{i})/2-\\ &\sum\limits_{unloading}(\sigma^{\prime}_{i}+\sigma^{\prime}_{i+1})(\varepsilon ^{\prime}_{i+1}-\varepsilon^{\prime}_{i})/2\end{array}  \tag{1}\]  
where \(U\) is the total unit volume energy, \(U^{\rm e}\) is the unit volume elastic energy, \(\sigma\) and \(\varepsilon\) are the stress and strain of the loading section, \(\sigma^{\prime}\) and \(\varepsilon^{\prime}\) are the stress and strain of the unloading section, and \(\sigma_{i}\) and \(\varepsilon_{i}\) are the stress and strain of segment \(i\) after discretization. The typical stress-strain hysteresis curves are in accordance with the three dynamic shakedown states, as shown in Figure 8. For Range A, the shape of the hysteretic loop at each stage was approximately linear, and the stress-strain relationship showed elastic behavior. This indicated that there was almost no new plastic dissipated energy generated in each cycle. For Range B, the shape of the hysteresis loop presented a thin crescent shape with a small hysteresis loop area. A certain amount of plastic dissipated energy was generated in each cycle, but the overall shape of the hysteresis loop did not change with the  
Figure 7: Hysteresis loop and energy dissipation under cyclic load. development of the loading stage, indicating that the cumulative incremental rate of plastic strain was a relatively stable constant. For Range C, the hysteresis loops at each stage exhibited a wide crescent shape, with a strongly nonlinear stress-strain relationship. The hysteresis loop area was large, and the deformation of the specimen accumulated rapidly. The variation curves of the dissipated energy per unit volume \(U^{\text{d}}\) obtained by Formula (1) with the number of cycles are shown in Figure 9. Taking Group B as an example, the values of \(U^{\text{d}}\) in Range A remained at a low level, about 0-0.01 kJ/m\({}^{3}\). In Range B, the values of \(U^{\text{d}}\) were approximately 0.01-0.05 kJ/m\({}^{3}\), showing a slight upward trend after a certain number of cycles. The values of \(U^{\text{d}}\) in Range C were greater than 0.05 kJ/m\({}^{3}\), with significant fluctuations or a significant increase with the increase in loading cycles. The final failure reached around 0.15-0.25 kJ/m\({}^{3}\). The results showed a good correspondence between the energy dissipation characteristics of fillers and the shakedown states exhibited by stress-strain behavior.