# Modified Equation of Shear Strength with Respect to Saturation
## 4 Results and Discussion  
### Comparison with Experimental Data  
For the curve fitting, the nonlinear least squares method was applied to get the parameters of the vG numerical model and the Equation (4). Table 2 lists four parameters that were calculated under three various dry densities (\(\rho_{\mathrm{d}}=1.5\) t/m\({}^{3}\), \(\rho_{\mathrm{d}}=1.6\) t/m\({}^{3}\), \(\rho_{\mathrm{d}}=1.7\) t/m\({}^{3}\)). The soil-water characteristic curve of the soil measured at various dry densities and fitted curve is plotted in Figure 8. As shown in Figure 8, the predicted curve provided good consistency with the experimental points because the correlation coefficient was 0.95. This also verified the vG numerical model. For a rising degree of saturation, the matrix suction declined. Further analysis revealed that when the degree of saturation was less than 50%, the suction rose with increasing dry density. When the soil became increasingly compacted, the voids became smaller and the bonding situation of the soil particles became increasingly tight. This is why matric suction decreased with an increasing degree of saturation. There was no significant variation in the matric suction when the saturation was more than 50%. In the boundary effect stage, as the number of large pores containing free water reduces with increasing dry density, the declining amplitude of the volumetric water content decreases with the increasing matric suction. In the transition stage, the higher the dry density of silty sand, the greater the air entry value of the sample. It is really difficult for soil samples of higher dry density to evolve from a saturated state to an \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \(\mathbf{\xi_{2}}\) & \(\mathbf{\xi_{3}}\) & \(\mathbf{\xi_{4}}=\mathbf{\xi_{1}}\)**-**P\({}_{0}\)** & **m** \\ \hline \(\rho_{\mathrm{d}}=1.5\) t/m\({}^{3}\) & 40,009.57 & \(-4.21\) & 20,295.99 & 0.24 \\ \(\rho_{\mathrm{d}}=1.6\) t/m\({}^{3}\) & 40,017.51 & \(-4.51\) & 11,675.79 & 0.21 \\ \(\rho_{\mathrm{d}}=1.7\) t/m\({}^{3}\) & 64,735.01 & \(-5.01\) & 9886.80 & 0.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of the model parameter fitting for the experimental points. Figure 7: Test system of the triaxial shear apparatus. unsaturated state. The larger dry density corresponds to the larger pores with free water, fewer pores as well as poorer connectivity of water, as a result, it is difficult for water to drain out of the sample. Therefore, if the sample wanted to evolve from the quasi-saturated status to unsaturated status, it would need more matric suction. In the residual stage, the residual volumetric water content of the sample increased with increasing dry density. There were more soil particles in the samples with higher dry density, which could provide more adsorption space (such as pores on the surface of particles and small pores between particles) for capillary water and bound water. Figure 9 shows that the effective stress parameter of the unsaturated soil increased with a rising degree of saturation and the shape of this curve looked like the letter 'S'. The correlation coefficient between the modeled and experimental data was 0.98, which can also be regarded as a validation of Equation (4). The relationship between the principal stress difference and axial strain was plotted with the principal stress difference as the vertical axis and axial strain as the horizontal  
Figure 8: SWCC based on the experimental points and fitted curve at various dry densities. Figure 9: The effective stress parameter \(\chi\) of the calculated points and fitted curve at various dry densities. axis (Figure 10a). As outlined in the Chinese Standards for Soil Test Methods (GB/T50123-1999), the test should be stopped when the total axial strain is 20%. If there is no peak, we considered a shear peak at 15% of the axial strain as the principal stress difference of the soil sample.