## 1 Introduction  
For rockfill dams made of rockfill particles, as the dam height increases, there are stricter requirements for slope stability, deformation control, and seismic safety [1, 2]. To address these requirements, the peak stress ratio and shear expansion-stress ratio are used to establish functional relationships within an intrinsic structure model [3], which is a significant step in optimizing the intrinsic constitutive relationship. The phase change, peak, and critical states are three significant characteristic states of rock particles. The phase change state was initially proposed to represent the state of sandy soil, where the pore water pressure decreases or the deviatoric stress increases under under under strained stress [4], signifying the densest state. This is crucial in constructing an elastic-plastic intrinsic model [5, 6]. The concept of a characteristic state is further extended to describe the stress level at the transition from volume compression to volume expansion in sandy soils under drainage conditions, which is the turning point from body contraction to body expansion [7]. The peak state is relatively well understood, representing the condition corresponding to the maximum stress or stress ratio value in the soil during loading [8]. The critical state is the limit achieved during soil deformation when the effective mean positive stress, shear stress, and volume cease to change [9], and it serves as a stable reference state for analysis. Researchers have analyzed and studied the three characteristic states of rockfill materials through physical tests and numerical simulations. Large-scale indoor triaxial tests are the most commonly used physical tests. [PERSON] et al. [10] carried out a series of large-scale triaxial tests on rockfill materials with different maximum grain sizes to investigate the effect of grain size distribution on the critical state of rockfill materials. They found that the intercept and gradient of the critical state line along the plane decreased as the gradient became wider with increasing grain size, while the gradient of the critical state line along the plane increased with increasing grain size. [PERSON] et al. [11] investigated the effect of stress paths on the evolution of the critical state and the particle crushing gradation of rockfill materials using large-scale triaxial compression tests. The average effective stress \(p^{\prime}\), deviatoric stress \(q\), and pore ratio \(e\) along different stress paths reached the same critical state line (CSL). Differences in the small and medium principal stress ratios in a true triaxial compression test also have a greater effect on the mechanical properties of the rockfill materials [8]. The medium principal stress ratio has a greater effect on the critical state friction angle, peak state friction angle, and maximum shear expansion of the simulated rockfill materials. [PERSON] et al. [12] conducted a series of large-scale triaxial compression tests on a Tacheng Rock Mound (TRM). They found that its deformation behavior was affected by density and stress and that the critical state friction angle of the TRM decreased with increasing initial confining pressure. The integration of histogram anisotropy into the description of critical state damage revealed that the orientation of the histogram tensor is isotropic concerning the stresses in the critical state [13]; this technique provides a unique value related to the shear mode or Lode angle. [PERSON] et al. [14] investigated the mechanical properties of rockfill materials with different wetting-drying cycles and wetting durations and found that deterioration occurred in the form of a decrease in yield stress and an increase in compressibility, which varied nonlinearly with the number of wetting-drying cycles and wetting duration. The critical state of a rockfill material is significantly influenced by its density, grade, and confining pressure.