## 1 Introduction  
The subgrade structure of high-speed railways provides an important foundation for the safe and smooth operation of high-speed trains, and the scientific design of the subgrade structure is a fundamental guarantee of durability and technical economy. Finding an effective method for designing high-speed railway subgrade structures is a challenge that must be overcome to achieve high-precision deformation control and durability of service. The railway and soil system is a classical soil structure, and several studies have been implemented to clarify its mechanism [1; 2; 3; 4]. Subgrade filling is used as the basic raw material for constructing the subgrade structure. Its mechanical characteristics and deformation evolution laws under the action of driving vibration are directly related to the overall durability and stability of the subgrade structure. Therefore, in the design of high-speed railway subgrade structures, the main focus is on the reasonable control of the structure under a working state and its long-term deformation under periodic dynamic loads. The current design method for high-speed railway subgrade structures in China adopts indicators such as the average dynamic strain of the subgrade bottom layer and the dynamic deformation of the surface layer, and the critical strain value is used as the design judgment criterion. Although the critical strain control value in current design mainly comes from typical test results and empirical values for fine-grained soil, it still essentially follows this design concept. In recent years, research on the design control threshold of subgrade filling materials has mainly focused on the cumulative deformation of filling materials under cyclic loading, the influence of factors including the stress level [5; 6], the number and frequency of cyclic actions [7; 8; 9], compactness [10; 11], stress history [12; 13; 14], and different development modes and convergence characteristics. [PERSON] et al. [15] and [PERSON] et al. [16] classified the relationship curve between the cumulative plastic strain and the number of load actions as one of two types, the development type and decay type, based on the concavity and convexity of the curve in semi-logarithmic coordinates. [PERSON] et al. [17] classified the cumulative plastic deformation curve as a stable, decay, or failure type and provided the critical cumulative strain between different states. [PERSON] [18] also classified the cumulative plastic strains under cyclic loading into three states based on the volumetric plastic strain. [PERSON] et al. [19] explored the concept of critical dynamic stress in soil and classified the cumulative strain characteristics into three categories: stable, failure, and critical. [PERSON] et al. [20] also provided these three types of cumulative strain forms. Based on the linear relationship between cumulative strain and cyclic vibration frequency, an empirical formula for critical dynamic stress with confining pressure, moisture content, and static strength as variables was proposed. According to the characteristics of the cumulative deformation of subgrade filling under cyclic loading, [PERSON] et al. [21] and [PERSON] et al. [22] proposed four types: rapid stability, slow stability, slow failure, and rapid failure. They established a power law discrimination method based on the development of the cumulative deformation rate and explored the control principles and technical conditions of the deformation state of railway subgrade structures. [PERSON] et al. [23] and [PERSON] et al. [24] proposed a high-speed railway subgrade structure analysis and strain control design method based on critical volume effect strain to control the deformation and pore pressure accumulation under repeated loads. During the hundred-year service life of the high-speed railway, it will face the long-term impact of billions of dynamic load cycles.