## 3 Experimental Results and Analysis
### Dynamic Impact Test Results
#### 3.2.1 Dynamic Stress-Strain Behavior
At the same time, the enhancement effect of the frozen samples is less than the weakening effect at low temperatures when the saturation increases from 0 to 100% (Figure 10b). In general, the dynamic strength of frozen rock at low temperatures still has the strain rate effect, but it increases and then decreases with saturation. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Samples**} & **Impact** & **Dynamic** & **Dynamic** & **Peak** & **Brittleness** \\  & **Velocity (m/s)** & **Strength (MPa)** & **Modulus (GPa)** & **Strain (\%)** & **Index** \\ \hline D-0-4 & 4 & 22.1 & 1.95 & 1.167 & 0.728 \\ D-0-5 & 5 & 25.7 & 2.04 & 1.247 & 0.694 \\ D-0-6 & 6 & 29.3 & 2.28 & 1.286 & 0.657 \\ D-0-7 & 7 & 34.6 & 2.59 & 1.353 & 0.602 \\ D-30-4 & 4 & 24.6 & 2.39 & 1.015 & 0.753 \\ D-30-5 & 5 & 31.3 & 2.83 & 1.094 & 0.721 \\ D-30-6 & 6 & 35.5 & 3.03 & 1.176 & 0.697 \\ D-30-7 & 7 & 42.8 & 3.42 & 1.218 & 0.665 \\ D-50-4 & 4 & 26.4 & 3.26 & 0.812 & 0.862 \\ D-50-5 & 5 & 33.8 & 3.8 & 0.891 & 0.819 \\ D-50-6 & 6 & 39.7 & 4.07 & 0.961 & 0.760 \\ D-50-7 & 7 & 46.5 & 4.47 & 1.041 & 0.682 \\ D-80-4 & 4 & 21.3 & 2.02 & 1.049 & 0.788 \\ D-80-5 & 5 & 26.2 & 2.27 & 1.154 & 0.745 \\ D-80-6 & 6 & 32.6 & 2.58 & 1.238 & 0.703 \\ D-80-7 & 7 & 40.3 & 3.06 & 1.318 & 0.675 \\ D-100-4 & 4 & 19.4 & 1.71 & 1.134 & 0.744 \\ D-100-5 & 5 & 23.6 & 1.83 & 1.292 & 0.712 \\ D-100-6 & 6 & 25.5 & 1.92 & 1.331 & 0.684 \\ D-100-7 & 7 & 28.8 & 2.11 & 1.411 & 0.659 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dynamic mechanical parameters of sandstone under different conditions. Figure 9: Dynamic stress versus time for frozen samples: (**a**) \(S_{\rm r}\) = 0%; (**b**) v= 6 m/s. In addition, as shown in Figure 11, a dynamic strength increase factor (DIF) is introduced to evaluate the strength increase rate for the frozen samples. The DIF is determined using Equation (3), where \(f_{v}\) (\(d\)) is the dynamic strength at the impact velocity \(v\) and \(f(s)\) is the static strength [30]. From this, it can be seen that the DIF of the sample increases with increasing saturation and then decreases. For an impact velocity of 6 m/s, DIF increases from 2.11 to 2.87 as the saturation increases from 0 to 50%, while it decreases to 2.31 as saturation increases to 80%. As a result, it is reasonable to infer a critical saturation (S\({}_{\rm rc}\)) between 50% and 80% for the samples at low temperatures. When the saturation is greater than the critical saturation, the strength enhancement caused by the pore ice is gradually inhibited by the damage caused by the volume expansion. As the samples approach full saturation, the strength increase is further suppressed by the cracking and expansion of the pores under the action of the frost heave force. \[DIF=\frac{f_{v}(d)-f(s)}{f(s)} \tag{3}\]  
Indeed, the volume expansion (9%) caused by the pore ice exerts a frost heave force on the rock skeleton, thereby affecting cracking and expansion in the rock. [PERSON] et al. [7]  
Figure 11: DIF versus impact velocity and saturation of the sample. Figure 10: Dynamic peak strength versus impact velocity (**a**) and saturation (**b**) of the sample. found that the sample saturation is a crucial factor for controlling the frost heave force. The relationship between frost heave force \(P\) and saturation \(S_{\text{r}}\) is expressed in Equation (4) [31]. \[\left\{ \begin{array}{l}P=\frac{[\beta nS_{\text{r}}u-n(1-S_{\text{r}})]K_{ \text{r}}K_{\text{i}}}{K_{\text{r}}S_{\text{r}}+K_{\text{i}}}H[u-\chi]\\ \\ H[u-\chi]=\left\{\begin{array}{ll}0&(u\leq\chi)\\ 1&(u>\chi)\\ \end{array}\right.\\ \\ \chi=\frac{1-S_{\text{r}}}{\beta S_{\text{r}}}\end{array} \right. \tag{4}\]  
where \(\beta\) is the volume expansion coefficient of ice; \(n\) represents the porosity; \(u\) denotes the freezing rate; \(K_{\text{r}}\) and \(K_{\text{i}}\) are the bulk modulus of the rock skeleton and the ice, respectively; and \(H\left[u-\chi\right]\) is the step function.