## 3 Results and Analysis
### Analysis of the Variation Law of Compression Modulus  
The compression modulus is an important characteristic parameter that reflects the deformation resistance of granular materials. Referring to the definition of the compression  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Axial Stress/MPa** & **Optimal Water Content/\%** & **Maximum Axial Strain** \\ \hline
16 MPa & 5.11 & 0.387 \\
20 MPa & 4.91 & 0.398 \\
24 MPa & 4.79 & 0.403 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Fitting results of the relationship between axial strain and water content.  
Figure 7: Relationship curves between axial strain and water content.  
modulus of soil, the compression modulus of crushed gangue is simply defined as the slope of the line connecting the point on the axial stress-strain curve with the origin, as shown in Equation (5).  
\[E_{c}=\frac{\sigma_{c}}{\varepsilon_{c}} \tag{5}\]  
where \(\sigma_{c}\), \(\varepsilon_{c}\) are the ordinate and abscissa of a point on the stress-strain curve, respectively.  
Figure 8 shows the relationship between the compression modulus and the axial stress of the crushed gangue sample. During the confined compaction process, the compression modulus has a linear relationship with the stress, and the compression modulus increases continuously with the increase in the stress.  
Table 4 shows the fitting analysis of the compression modulus and axial stress relationship curves under different water contents using a linear function. The linear fitting relationship between the compression modulus and axial stress is good, with R-squared values all above 0.99.  
The axial stress and water content are important factors affecting the variation pattern of the compression modulus. It can be seen from Figure 8 that when the axial stress is 4 MPa, the maximum variation in the compression modulus under different water contents is 9.9 MPa. When the axial stress is 24 MPa, the maximum variation in the compression modulus under different water contents is 19.3 MPa. The greater the axial stress, the more obvious the effect of the water content on the compression modulus. Under the same axial stress, the compression modulus of crushed gangue is the largest when the water content is 0, and the variation range is about 23-79 MPa; the compression modulus is the smallest when the water content is 4.5%, and the variation range is about 13.5-60 MPa. With the increase in the water content, the compression modulus shows a trend of first decreasing and then increasing, and the influence of the water content on the compression modulus gradually weakens. This is consistent with the change in the slope of the first-order fitting equation in the table.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Water Content/\%** & **Fitting Equation** & \(\mathbf{R^{2}}\) \\ \hline
0 & \(E_{s}=13.5178+2.7657\sigma\) & 0.9988 \\
1.5 & \(E_{s}=9.1559+2.4796\sigma\) & 0.9993 \\
3.0 & \(E_{s}=6.3365+2.3501\sigma\) & 0.9994 \\
4.5 & \(E_{s}=5.0527+2.2935\sigma\) & 0.9996 \\
6.0 & \(E_{s}=5.3232+2.3695\sigma\) & 0.9996 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Fitting equations for the relationship between compression modulus and axial stress under different water content conditions.  
Figure 8: Relationship curves between compression modulus and axial stress.