## 3 Long-Term Deformation and Energy Dissipation Characteristics  
### Long-Term Cumulative Deformation and Shakedown Characteristics  
Based on shakedown theory, the stress-strain cycles of subgrade filler present different development laws under different dynamic loadings, as shown in Figure 4. The curve with an arrow indicates the load-strain relationship with the loading and unloading cycle, its direction indicates the load path, and the dashed section indicates several cycles that are not shown. When the stress-strain cycle stabilized in the initial few cycles, the accumulated strain rate continuously decreased, while the hysteresis curve closed or showed an approximate linear relationship; thus, no new deformation accumulation occurred, and it was called the \"plastic shakedown state\" (Range A). When the hysteresis loop gradually developed in the direction of strain accumulation, as the strain rate still increased in a high cycle, it exhibited a \"plastic creep state\" (Range B). When the hysteresis curve did not close and rapidly developed towards strain accumulation, the strain rate exhibited an increasing trend even in a low cycle, and a sudden increase in deformation occurred in a short duration, which was called \"incremental collapse state\" (Range C). The long-term deformation development of subgrade filler under different cyclic stress levels was obtained. Taking the cumulative deformation and strain rate curves of group B under different dynamic stress amplitudes as an example, the results under the three confining pressures of 15 kPa, 30 kPa, and 60 kPa are shown in Figures 5 and 6. The test conditions correspond to the typical confining pressure at the surface, center, and bottom position in the bottom layer of subgrade. Figure 4: Stress–strain relationship and shakedown characteristics under cyclic loading. Figure 5: Cumulative deformation curves under different dynamic stress amplitudes (Group B). Under different dynamic stress amplitudes, there are obvious differences in the cumulative law of the permanent strain of the filler. To be specific, for Range A, when the dynamic stress amplitude was small enough, the accumulation curve remained stable for a long time after a small amount of strain accumulation in the initial several cycles, and no new deformation accumulation occurred. For Range B, with the gradual increase in the dynamic stress amplitude, the development rate of the strain increased gradually. In this process, new deformation accumulation occurred in each external load cycle, and the permanent deformation showed a continuous and slow growth trend. For Range C, when the dynamic stress amplitude exceeded a certain limit, the permanent deformation curve showed an upward trend, a large cumulative strain occurred in a very short time (\(\epsilon^{p}>5\%\)), and the sample was damaged. According to the development trend of cumulative strain, even at small stress levels, all samples showed plastic strain, which indicated that the actual working state of railway subgrade filler was usually not a purely elastic state. Corresponding to the cumulative strain, the strain rate can further reflect the development trend and different stable states of the above three types of strain accumulation. The variation in strain rate with cumulative strain under different dynamic stress amplitudes is shown in Figure 5. The strain rates of specimens were also divided into three ranges. Taking the result of confining pressure \(\sigma_{3}=30\) kPa as an example, the strain rate in Range A decreased rapidly with the increase in accumulated strain, reaching below \(1\times 10^{-5}\)/cycle, without continuous deformation accumulation. The strain rate in Range B decreased as the accumulated strain increased, with a certain deflection at the end of the curve, and  
Figure 6: Variation in strain rate with cumulative strain under different dynamic stress amplitudes (Group B). finally stayed between 1 \(\times\) 10\({}^{-3}\)-1 \(\times\) 10\({}^{-4}\)/cycle.