## 4 Control Threshold for Design Based on Dynamic Shakedown Characteristics  
### Criterion of Shakedown State Based on Dissipated Energy per Unit Volume  
For the design of subgrade structures for high-speed railways, in order to ensure the long-term durability and stability of subgrade filler under traffic loading, it is necessary to keep it in a shakedown state where the deformation can converge and stabilize in a short time. When the total strain is small enough, the plastic shakedown state (Range A) is allowed, while the Range C state is strictly prohibited. Therefore, from the perspective of controlling the shakedown state of the subgrade structure, it is necessary to clarify the critical criteria between the three ranges and then put forward the design control thresholds. Figure 9: Variation in dissipated energy per unit volume with the number of cycles. According to the concept of the \"kinematically admissible strain rate\" of the kinematic shakedown theorem [56; 57], if the external force power is not greater than the internal plastic dissipation power in a time period \(0<t<T\), then the object is stable, as shown in Equation (2). \[\int\limits_{0}^{T}P_{a}(t)\dot{u}_{a}^{*}(t)\mathrm{d}t\leq\int\limits_{0}^{T} \int\limits_{V}\sigma_{ij}^{*}\dot{\epsilon}_{ij}^{*p}\mathrm{d}V\mathrm{d}t \tag{2}\]  
where \(P_{a}(t)\) is the action of the external force, \(\dot{u}_{a}^{*}(t)\) is the kinematically admissible velocity field, \(\dot{\epsilon}_{ij}^{*p}\) is the plastic strain rate, and \(\sigma_{ij}^{*}\) is the stress associated with \(\dot{\epsilon}_{ij}^{*p}\) through the plastic flow law. Based on the above theory, the dissipated energy per unit volume is taken as the basis for judging the critical thresholds of shakedown states. Considering the corresponding relationship between unit volume dissipation energy and different long-term deformation development modes, the main influencing factors are the magnitude of unit volume dissipation energy and its variation trend of divergence or convergence. Therefore, the equivalent dissipated energy per unit volume \(\overline{U}_{\mathrm{eq}}^{\mathrm{d}}\) is proposed as an index to judge the shakedown critical state, which is expressed as  
\[\overline{U}_{\mathrm{eq}}^{\mathrm{d}}=\overline{U}^{\mathrm{d}}\cdot\left(1 +\frac{U_{m}^{\mathrm{d}}-U_{m/10}^{\mathrm{d}}}{U_{m}^{\mathrm{d}}}\right) \tag{3}\]  
where \(\overline{U}^{\mathrm{d}}\) is the mean value of dissipated energy per unit volume, which is the influence term reflecting the value of dissipated energy. \(U_{m}^{\mathrm{d}}\) is the final unit volume dissipative energy corresponding to the maximum cycle number \(m\); \(U_{m/10}^{\mathrm{d}}\) is the unit volume dissipative energy corresponding to the representative cycle number \(m/10\). The dimensionless influence term in brackets reflects the rising trend in dissipated energy. The values of \(\overline{U}_{\mathrm{eq}}^{\mathrm{d}}\) under different confining pressures and dynamic stresses are illustrated in Figure 10. The threshold lines were obtained by calculating the limit values \(\lambda_{\mathrm{A-B}}\) and \(\lambda_{\mathrm{B-C}}\), and data points of \(\overline{U}_{\mathrm{eq}}^{\mathrm{d}}\) were divided into three ranges according to different shakedown states. The mean values of the upper and lower limits of \(\overline{U}_{\mathrm{eq}}^{\mathrm{d}}\) in the critical region are taken as the critical lines to determine \(\lambda_{\mathrm{A-B}}\) and \(\lambda_{\mathrm{B-C}}\) and are represented by the red lines shown in the figure. The criterion for shakedown states based on the equivalent dissipated energy per unit volume can be given as follows:  
\[\left\{ \begin{array}{l}Range\ A:\overline{U}_{\mathrm{eq}}^{\mathrm{d}}< \lambda_{\mathrm{A-B}}\\ Range\ B:\lambda_{\mathrm{A-B}}<\overline{U}_{\mathrm{eq}}^{\mathrm{d}}< \lambda_{\mathrm{B-C}}\\ Range\ C:\overline{U}_{\mathrm{eq}}^{\mathrm{d}}>\lambda_{\mathrm{B-C}}\\ \end{array} \right.