# 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 has been shown that the influence of confining pressure and the initial void ratio on the \(G_{max}\) is larger. Scholars have previously proposed empirical models to predict the \(G_{max}\) based on these two parameters [35; 36; 37; 38; 39]. The \(G_{max}\) generally decreases monotonically with the increase in saturation in clayy soils; in contrast, there is a threshold value for the \(G_{max}\) with increasing saturation in cohesionless soils. [PERSON] et al. [22] studied the effect of capillary forces on the small-strain modulus of sand under different stresses and different saturation states based on RC tests and they found that the \(G_{max}\) gradually increases when the saturation is below a certain threshold, while the \(G_{max}\) decreases once the saturation exceeds the threshold, which is called the optimum saturation. They also proposed an empirical formula for determining the optimal saturation (\(S_{r}\))\({}_{opt}\) based on the effective particle size \(D_{10}\), as shown in Equation (3):  
\[\left(S_{r}\right)_{opt}=-6.5\log_{10}(D_{10})+1.5 \tag{3}\]  
The particle size \(D_{10}\) for the silty-fine sand from Hangzhou Bay used in this work is 0.014 mm. Based on the calculation of Equation (3) and the optimal saturation in the silty-fine sand, we predicted ((\(S_{r}\))\({}_{opt}\)\(\approx\)13.55%). The measured \(G_{max}\)\(\sim\)\(S_{r}\) curve test showed that the saturation at the peak of the \(G_{max}\) is between 10% and 20%, which is generally  
Figure 9: Relationship between \(G\)/\(G_{max}\) and shear strain at different net stress (**a**) 100 kPa; (**b**) 200 kPa; (**c**) 300 kPa; (**d**) 400 kPa. consistent with the findings of [PERSON] et al. [22]. Additionally, a prediction model for the \(G_{max}\) considering the variation in saturation was given:  
\[G_{max}=[1+H(S_{r})]G_{dry} \tag{4}\]  
where \(H(Sr)\) is a function of the saturation \(S_{r}\), \(G_{dry}\) is the maximum shear modulus of the dry soil and \(H(Sr)\) is defined as shown in Equation (5):  
\[H(S_{r}) \begin{cases}(a-1)\sin\left(\frac{\pi S_{r}}{2b}\right)&S_{r}\leq b\\ (a-1)F(S_{r})&S_{r}>b\end{cases}  \tag{5}\]  
where \(a\) is the maximum value of the ratio of the maximum shear modulus of unsaturated soil to the maximum shear modulus of dry soil; \(b\) is the optimum saturation; and \(F(S_{r})\) is defined as shown in Equation (6):  
\[F(S_{r})=\frac{1}{2}\left(\frac{S_{r}-b}{100-b}\right)^{2}\cdot\sin\left\{ \frac{\pi}{100-b}\times\left[S_{r}+50-\frac{3}{2}b\right]\right\} \tag{6}\]  
From Equation (5), it can be found that the prediction model is not continuous at the point of optimal saturation (\(S_{r}=b\)); the \(G_{max}\) at the optimum saturation converges to \(a\) times the value of the dry soil (i.e., \(G_{max}(opt)\to aG_{dry}\)) when the left side of the equation is closed to the optimal saturation; however, the \(G_{max}\) at the optimum saturation converges to the value of the dry soil (i.e., \(G_{max}(opt)\to G_{dry}\)) when the right side of the equation is close to the optimal saturation. Thus, in order to solve the discontinuity, \(F(S_{r})\) is proposed to be modified in Equation (7):  
\[F(S_{r})=\left(1-\frac{S_{r}-b}{100}\right)^{2}\cdot\sin\left\{\frac{\pi}{100- b}\times\left[S_{r}+50-\frac{3}{2}b\right]\right\} \tag{7}\]  
Equation (7) overcomes the problem of discontinuity in predicting the \(G_{max}\) of sand with different saturations by Equation (4). Thus, the \(G_{max}\) of silty-fine sand from Hangzhou Bay tested was predicted, as shown in Figure 10, where the solid line is the predicted value and the scatter point is the test value. Figure 10 shows a comparison between the predicted \(G_{max}\) of the proposed model and the tested result.