## 5 Conclusions  
Based on shakedown theory and adopting an energy perspective, this paper investigated the characteristics of plastic strain accumulation, cyclic stress-strain hysteresis, and the evolution of dissipation energy in six types of typical coarse-grained soil fillings used in subgrade construction. A criterion for identifying the shakedown critical state based on equivalent unit volume dissipation energy is proposed, along with a methodology for determining the corresponding control threshold. Different from the traditional method, the method proposed in this paper was more accurate in evaluating the shakedown states. It established a relationship among long-term deformation trends, the shakedown critical  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Filler Type** & **Threshold** & \(\mathbf{k}\) & \(\mathbf{\varepsilon_{\text{d0}}}\) & \(\mathbf{\varepsilon^{*}_{\text{d (10}}^{-3})\) \\ \hline Group A of bottom layer & \(\varepsilon^{*}_{\text{d A-B}}\) & \(-4.03\times 10^{-6}\) & 0.00131 & 0.119 \\  & \(\  
state, and dissipation energy. This method offers the advantage of comprehensively considering both the magnitude and the trend of strain energy dissipation. The findings of this research can provide a theoretical foundation for the shakedown design of high-speed railway subgrades. The main results are as follows:  
1. The three deformation development modes of high-speed railway subgrade fillers corresponded to distinct shakedown characteristics, strain accumulation rates, and variations in dissipated energy. Range A (Plastic Shakedown): the strain accumulation rate decreased rapidly with increasing permanent deformation, while the dissipated energy per unit volume stabilized at a low level. Range B (Plastic Creep): the strain accumulation rate declined gradually, accompanied by a slight increase in dissipated energy per unit volume after a certain number of cycles. Range C (Incremental Collapse): the strain accumulation rate remained elevated, with substantial fluctuations in the dissipated energy per unit volume as the cycle number increased. 2. A criterion of shakedown state based on dissipated energy per unit volume was proposed. For the six types of typical high-speed railway subgrade fillers, the values of \(\lambda_{\text{A-B}}\) and \(\lambda_{\text{B-C}}\) were observed to fall within the ranges of 0.01 to 0.015 kJ/m\({}^{3}\); and 0.05 to 0.08 kJ/m\({}^{3}\); respectively. 3. The thresholds of \(CSR\)* and \(\epsilon\)*d of the six types of high-speed railway subgrade fillers have been established. It was found that the critical \(CSR\) decreased as the confining pressure increased, while the threshold for dynamic stress shakedown exhibited an upward trend. Conversely, the critical threshold for dynamic strain was relatively unaffected by changes in confining pressure. In this study, the values of \(\epsilon\)*d A-B and \(\epsilon\)*d B-C were approximately in the range of 0.10% to 0.12% and 0.20% to 0.25%, respectively. 4. Control thresholds for dynamic shakedown design of high-speed railway subgrade were proposed. By regulating the thresholds for dynamic stress and dynamic strain, the subgrade can be maintained in a shakedown state, thereby preventing long-term accumulated deformation and incremental collapse. These results offer valuable theoretical support for the shakedown design of high-speed railway subgrades. This study primarily focuses on conducting experiments and analyzing data related to the types and compaction requirements of high-speed railway subgrade fillers as specified in current standards. The critical shakedown conditions and design thresholds proposed herein have limitations when applied to a broader range of filler types, varying compaction levels, and different moisture conditions. Therefore, further experimental analysis is necessary, to enhance the applicability of these results across diverse scenarios. Conceptualization, Z.B. and Y.Y. ; Methodology, Z.B., D.C. and K.S. ; Software, K.S. and M.O. ; Validation, Z.B. and H.Y.