# Experimental Study on Small-Strain Shear Modulus of Unsaturated Sility-Fine Sand
## 4 Shear Modulus of Sand with Different Saturations
### The Relationship between \(G_{max}\) and \(S_{r}\) and the Prediction Model
It can be seen that they are in good agreement and the variation trend of the \(G_{max}\) with saturation at different stress levels is almost similar; the \(G_{max}\) increases slowly from the steady state as the saturation decreases and reaches the maximum value at  
Figure 10: The prediction of \(G_{max}\) under different net stresses, Stage 1: unsaturated residual stage; Stage 2: transition stage; Stage 3: boundary effect stage. the optimum saturation and decreases rapidly when the saturation becomes lower. Finally, the \(G_{max}\) of dry sand decays to almost the same level as that of saturated sand. The fitted parameters under different net stresses were derived from the improved prediction model, as shown in Table 3. Furthermore, values of optimal saturation b were obtained as 13.98%, 14.63%, 14.68% and 15.71% under the net stresses of 400 kPa, 300 kPa, 200 kPa and 100 kPa, respectively. This indicates that the optimal saturation of unsaturated sand tends to decrease as the net stress increases. Based on the distribution morphology of pore water and the internal force mechanism in sand particle aggregates under different saturations, the macroscopic \(G_{max}\)-\(S_{r}\) curves could be divided into three stages: the boundary effect stage, the transition stage and the unsaturated residual stage [40]. In the boundary effect stage (saturation of sand 100-70%), the capillary action has not yet appeared in sand particles with a decrease in saturation. In the transition stage (saturation 70-15%), the shear resistance property of sand is dominated by the capillary action and the combined force caused by the capillary action, which reaches the maximum value at optimum saturation. When the saturation further decreases to be lower than the optimum saturation, it enters the unsaturated residual stage, while the capillary action disappears and the short-range action force is activated. Here, the short-range action force refers to the combined force resulting from the action of the electric and van der Waals force fields near the solid-liquid interface. Figure 11 is a schematic diagram of pore water distribution around particles. The structural water of a granular crystal has no effect on the interaction forces between particles under normal temperature conditions. The surface of a particle from near to far in order exists the strong adsorption water, the weak adsorption water, capillary water and pore gravity water. Adsorption water is primarily a water film attached to the particle surface, which is formed by the electrostatic gravitational force and the van der Waals force. The water film closest to the particle is subjected to adsorption forces up to several thousand atmospheres and its properties are similar to those of solids; the water film is called strong adsorption water. The remote layer of water film is called weak adsorption water, which is less affected by electro-molecular forces due to its distance from the particle. When the thickness of the weak adsorption water film reaches the maximum value, the redundant water forms a meniscus under the action of surface tension due to interaction between the water and air, thus, forming capillary water. Once the capillary water fills all the contact corners between granular particles until the curved liquid surface disappears, the excess water forms free-flowing gravity water, gradually filling all the remaining inter-particle pore space. Figure 11: Schematic diagram of pore water distribution around sand particles. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \(\sigma\) **= 400 kPa** & \(\sigma\) **= 300 kPa** & \(\sigma\) **= 200 kPa** & \(\sigma\) **= 100 kPa** \\ \hline a & 1.131 & 1.195 & 1.305 & 1.364 \\ b & 13.98 & 14.63 & 14.68 & 15.71 \\ R\({}^{2}\) & 0.98417 & 0.9869 & 0.9949 & 0.97877 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The fit parameters of predictive model.