## 3 Experimental Results Analysis
### The Impact of the Number of Freze-Thaw Cycles and Freezing Temperature on the Elastic Modulus  
In this study, the slope of the linear segment of the stress-strain curve is taken as the elastic modulus (\(E\)) for analysis, with this linear segment approximately corresponding to 40% to 60% of the peak compressive stress [27]. Table 2 presents the data set with the highest coefficient of variation among the experimental groups, with coefficients ranging from 3.94% to 14.589%, indicating the reliability of the experimental data. Under room temperature conditions (\(T\) = 20 \({}^{\circ}\)C), the average elastic modulus of the sandstone specimens is determined to be 2.096 GPa; this will serve as a control group for the analysis. As shown in Figure 6, the elastic modulus of coal-measure sandstone exhibits an exponential decrease with a reduction in the freezing temperature. For instance, at \(N\) = 10, when the freezing temperature decreases from \(-\)5 \({}^{\circ}\)C to \(-\)20 \({}^{\circ}\)C, the elastic modulus decreases by 0.294 GPa (X\(\rightarrow\)B1), 0.201 GPa (B1\(\rightarrow\)B2), 0.05 GPa (B2\(\rightarrow\)B3), and 0.112 GPa (B3\(\rightarrow\)B4), with reduction rates of 14.03%, 11.15%, 3.12%, and 7.22%, respectively. This indicates that, as the freezing temperature decreases, the rate of reduction in the elastic modulus gradually diminishes, demonstrating a clear interval effect of freezing temperature on the elastic modulus of coal-measure sandstone specimens. When the freezing temperature is constant, an increase in the number of freeze-thaw cycles leads to a significant decrease in the elastic modulus of the sandstone specimens. Taking \(T\) = \(-\)15 \({}^{\circ}\)C as an example, when the number of freeze-thaw cycles increases from five to twenty, the elastic modulus decreases by 0.218 GPa (X\(\rightarrow\)A3), 0.327 GPa (A3\(\rightarrow\)B3), 0.17 GPa (B3\(\rightarrow\)C3), and 0.123 GPa (C3\(\rightarrow\)D3), with reduction rates of 10.40%, 17.41%, 10.96%, and 8.91%, respectively. This indicates that the number of freeze-thaw cycles has a significant impact on the elastic modulus of the specimens, with reductions exceeding 8%. Thus, it is evident that the effect of the freeze-thaw cycle count is more sensitive than that of the freezing temperature on the deformation resistance of coal-measure sandstone.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \begin{tabular}{c} **Test** \\ **Scenario** \\ \end{tabular} & **Test Case** & \begin{tabular}{c} **Elastic** \\ **Modulus** \\ **\(E\) (GPa)** \\ \end{tabular} & \begin{tabular}{c} **Mean** \\ **Value** \\ **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Absolute** \\ **Deviation** \\ **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Relative** \\ **Deviation** \\ **(\%)** \\ \end{tabular} & \
\begin{tabular}{c} **Standard** \\ **Deviation** \\ **(GPa)** \\ \end{tabular} &
\begin{tabular}{c} **Coefficient of** \\ **Variation** \\ **(\%)** \\ \end{tabular} \\ \hline \multirow{3}{*}{A3} & Case1 & 2.070 & & 0.192 & 10.224 & \multirow{3}{*}{0.243} & \multirow{3}{*}{12.939} \\  & Case2 & 1.534 & 1.878 & 0.344 & 18.317 & & \\ \cline{1-1}  & Case5 & 2.030 & & 0.151 & 8.040 & & \\ \hline \multirow{3}{*}{B1} & Case2 & 1.885 & & 0.083 & 4.606 & \multirow{3}{*}{0.071} & \multirow{3}{*}{3.940} \\  & Case3 & 1.712 & 1.802 & 0.090 & 4.994 & & \\ \cline{1-1}  & Case5 & 1.808 & & 0.006 & 0.333 & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Elastic modulus values, \(E\), of the specimens.  
Figure 5: A typical stress–strain relationship curve. The compaction stage (i); the approximately linear elastic stage (ii); the non-linear crack propagation stage (iii); the post-peak stage (iv).