## 2 Test Material and Methods
### Experimental Procedure and Operational Conditions  
The experiment utilized the MTS-810 dynamic triaxial system manufactured by MTS Systems Corporation, located in Minneapolis, Minnesota, USA, to study the dynamic properties of frozen soil. The dynamic triaxial apparatus is shown in Figure 2. The testing system mainly consists of five parts: a refrigeration system, a vertical loading system, a triaxial chamber, an oil pressure system, and a computer interaction system. Initially, the prepared specimen was introduced into the triaxial cell and subjected to a preload  
Figure 1: Distribution pattern of particle sizes within frozen soil [14]. under initial stress for a duration of 10 min prior to the commencement of the experiment. Figure 3 illustrates the repetitive loading pattern applied to frozen soil in the form of a sine wave, alongside the corresponding deformation calculations, \(\sigma_{1,0}\), \(\sigma_{3,0}\), and \(\sigma_{\text{d}}\) represent the initial axial stress, initial confining pressure, and the amplitude of the repetitive stress, respectively. As a result of residual strain, the graph depicting the axial stress-strain relationship under repeated cyclic loading, is presented in Figure 3c. It is recognized that the hysteresis curve incorporates both the unloading and reloading curves. Therefore, the initial loop is formed by the conjunction of the unloading curve from the first loading cycle and the loading curve of the subsequent second loading cycle. Similarly, the Nth loop is constructed by combining the unloading curve from the Nth loading cycle with the loading curve of the (N + 1)th loading cycle. As a result, the accumulated axial strains \(\epsilon_{\text{acc,1}}\) for the first loop and \(\epsilon_{\text{acc,N}}\) for the Nth loop are depicted in Figure 3c. Scenarios involving cycles exceeding 1000 are classified as long-term loading conditions. The test termination criteria are either reaching 20,000 cycles or achieving a maximum vertical strain of 20%. The specific loading conditions are outlined in Table 1, \(\eta_{0}\) is the initial stress ratio, \(\text{p}_{0}\) is the initial mean stress, \(\text{q}_{0}\) is the initial bias stress and \(\omega\) is moisture content. As depicted in Figure 4, \(\text{p}_{0}\) and \(\eta_{0}\) remain constant, with the primary focus being on the key factors of sample moisture content and testing temperature. For each test series, the soil samples are subjected to varying amplitudes of repeated stress under stress-controlled  
Figure 3: Cyclic loading pattern applied to frozen soil and axial stress–strain relationship. (**a**) Sample under cyclic loading; (**b**) Repeated cyclic loading variation to frozen soil; (**c**) axial stress–strain relationship [14]. Figure 2: Experimental equipment of the dynamic triaxial test. conditions, while maintaining a constant loading frequency of 2 Hz, an initial mean stress of 0.9 MPa, and an initial stress ratio of 0.5. The loading frequency of 2 Hz falls within the range of long-term vibrations caused by railways and highways, and it is a commonly adopted frequency in research [14; 41; 42]. In the previous article, the mechanical properties of low-temperature stabilized remolded frozen soil from the Qinghai-Tibet Railway using a reference temperature of \(-5\)\({}^{\circ}\)C and an optimal moisture content of 14% are studied. To investigate the impact of different temperatures and moisture contents on the mechanical properties of frozen soil samples, we selected temperatures of \(-3\)\({}^{\circ}\)C, \(-7\)\({}^{\circ}\)C, \(-9\)\({}^{\circ}\)C, \(-11\)\({}^{\circ}\)C, and \(-13\)\({}^{\circ}\)C near the reference temperature to study the effects of varying temperatures. The liquid limit and plastic limit of the soil samples are 19.5% and 11.5%, respectively. Therefore, we selected several representative moisture contents within this liquid-plastic limit range for our research.