## 4 Control Threshold for Design Based on Dynamic Shakedown Characteristics
### Shakedown Threshold of Typical Filling Materials  
The relationship curves between the equivalent dissipated energy per unit volume and dynamic stress amplitude are shown in Figure 11, and the criterion lines for judging the critical value are marked in the figure (shown by the dotted lines). The intersection of the curve and the critical line represents the critical point of the shakedown critical state, that is, the thresholds of stress amplitude for Range A, B, and C under the corresponding confining pressures. Similarly, for the dynamic strain amplitude under different test conditions, the relationship between its variation curves and the critical lines of unit volume dissipation energy are shown in Figure 12. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Subgrade Filler** & \(\lambda_{\mathbf{A-B}}\) & \(\lambda_{\mathbf{B-C}}\) \\  & **k/m3** & **k/m3** \\ \hline Group A of bottom layer & 0.01328 & 0.05336 \\ Group B of bottom layer & 0.01198 & 0.05590 \\ Group C of bottom layer & 0.01135 & 0.05450 \\ Group D of bottom layer & 0.01029 & 0.05679 \\ Type I of surface layer & 0.01494 & 0.07479 \\ Type II of surface layer & 0.01206 & 0.06697 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Critical values of equivalent dissipated energy per unit volume of different subgrade fillers. Figure 10: Partition of dissipated energy per unit volume based on shakedown states. Figure 11: Variation curves for equivalent dissipated energy per unit volume with dynamic stress amplitude. Figure 12: Variation curves for equivalent dissipated energy per unit volume with dynamic strain amplitude. In order to define the dynamic stress shakedown critical threshold of coarse-grained soil filler for high-speed railway subgrade, the dimensionless parameter of the cyclic stress ratio (\(CSR\)) is used, which is expressed as follows:  
\[CSR=\sigma_{\text{d}}/2\sigma_{3} \tag{7}\]  
By substituting the critical value of dynamic stress amplitude \(\sigma_{\text{d}}\) in Figure 9 into Equation (7), the threshold curve of \(CSR^{*}\)A-B and \(CSR^{*}\)B-C can be obtained. As shown in Figure 13, the data points \(CSR_{\text{A}}\) at the lower left of the \(CSR^{*}\)A-B threshold curve are in the plastic shakedown state, the data points \(CSR_{\text{C}}\) at the upper right of the \(CSR^{*}\)B-C threshold curve are in the incremental collapse state, and the data points \(CSR_{\text{B}}\) in the shaded area between \(CSR^{*}\)A-B and \(CSR^{*}\)B-C are in the plastic creep state. Figure 13: Shakedown threshold of dynamic stress amplitude. The threshold curves of \(CSR^{*}\) are fitted by using the power function form, which can be expressed as  
\[CSR^{*}=A\sigma_{3}^{\ast}{}^{b} \tag{8}\]  
where \(A\) is mainly related to the variation amplitude of critical \(CSR\) with confining pressure, and \(b\) is related to the inclination of the threshold curve. For the typical high-speed railway subgrade fillers studied in this paper, the parameter values are shown in Table 3.  
As shown in Figure 11, the curves of \(CSR^{*}\)A-B and \(CSR^{*}\)B-C showed a downward trend; however, the threshold of stress amplitude still increased with the increase in confining pressure. Taking the subgrade filler of Group A as an example, according to the conversion relationship in Equation (7), the thresholds of stress amplitude corresponding to \(CSR^{*}\)A-B and \(CSR^{*}\)B-C at \(\sigma_{3}=30\) kPa were increased by about 46% and 15% compared with the values at \(\sigma_{3}=15\) kPa, respectively. This indicated that the increase in confining pressure significantly enhanced the dynamic stability of subgrade fillers. Therefore, the filling near the surface layer of the high-speed railway subgrade was more likely to induce a reduction in stability and the accumulation of long-term deformation under a high external dynamic load.