## 3 Test Results and Analysis  
The stress-strain relationship curve with centrosymmetric characteristics formed in the same load cycle is the hysteresis curve. The typical hysteresis curve form and the related dynamic characteristic parameters are shown in Figure 4.  
As can be seen from Figure 4, the backbone curve of dynamic stress-dynamic strain can be obtained by connecting the vertices of hysteresis loops at various levels, which reflects the nonlinear characteristics of soil deformation under the action of dynamic load.  
Currently, the nonlinear characteristics of the backbone curve can often be described by the Hardin-Drnevich hyperbolic model as follows:  
\[\sigma_{\text{d}}=\frac{\varepsilon_{\text{d}}}{a+be_{\text{d}}} \tag{1}\]  
Figure 4: Schematic diagram of kinetic parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Group** & **Test No.** & _uv/\%_ & **Freeze–Thaw Cycle** \\ \hline \multirow{6}{*}{1} & 1-1 & & 0 \\  & 2-1 & & 1 \\  & 3-1 & 0 & 3 \\  & 4-1 & & 6 \\  & 5-1 & & 9 \\ \hline \multirow{6}{*}{2} & 1-2 & & 0 \\  & 2-2 & & 1 \\  & 3-2 & 10 & 3 \\  & 4-2 & & 6 \\  & 5-2 & & 9 \\ \hline \multirow{6}{*}{3} & 1-3 & & 0 \\  & 2-3 & & 1 \\  & 3-3 & 20 & 3 \\  & 4-3 & & 6 \\  & 5-3 & & 9 \\ \hline \multirow{6}{*}{4} & 1-4 & & 0 \\  & 2-4 & & 1 \\ \cline{1-1}  & 3-4 & 30 & 3 \\ \cline{1-1}  & 4-4 & & 6 \\ \cline{1-1}  & 5-4 & & 9 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Test plans of dynamic triaxial.  
\[E_{\rm d}=\frac{\sigma_{\rm d}}{\varepsilon_{\rm d}}=\frac{1}{a+be_{\rm d}} \tag{2}\]  
where \(a>0\), \(b>0\), and both are material constants, and the ultimate dynamic stress amplitude \(\sigma_{\rm dmax}\) and the maximum dynamic modulus of elasticity \(E_{\rm dmax}\) have the following mathematical relationship:  
\[\sigma_{\rm dmax}=\lim_{\varepsilon_{\rm d}\to\infty}\frac{\varepsilon_{\rm d }}{a+be_{\rm d}}=\frac{1}{b} \tag{3}\]  
\[E_{\rm dmax}=\lim_{\varepsilon_{\rm d}\to 0}\frac{\varepsilon_{\rm d}}{a+be_{\rm d }}=\frac{1}{a} \tag{4}\]  
where \(\sigma_{\rm dmax}\) is the limit value of dynamic stress amplitude \(\sigma_{\rm d}\) when the dynamic strain \(\varepsilon_{\rm d}\) tends to be infinity. \(E_{\rm dmax}\) is the maximum dynamic modulus of elasticity, that is, the initial dynamic modulus of elasticity obtained when the dynamic strain \(\varepsilon_{\rm d}\) tends to be 0.  
According to [PERSON]'s related research on the dynamic properties of freeze-thaw sandy soils [19], when the traditional Hardin-Drnevich hyperbolic model is used to describe the mechanical properties of sandy soils after freeze-thaw, the fitted curve fits better with the actual test data in the strain range of 0.001-0.003, but outside this range, the Hardin model cannot reflect the experimental data relationship between stress and strain of the frozen compacted sand well. Therefore, this paper adopts the three-parameter modified Hardin-Drnevich hyperbolic model proposed by [PERSON] to describe the nonlinear characteristics of the soil dynamic properties. The Hardin hyperbolic model was modified as follows:  
\[\sigma_{\rm d}=\frac{\varepsilon_{\rm d}}{\left(a^{c}+b^{c}\varepsilon_{\rm d }c\right)^{\frac{1}{2}}} \tag{5}\]  
where \(a=\)1/\(E_{\rm dmax}\), \(b