# 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
The geometry of the meniscus between two equal-size particles can be described by the radius \(r_{1}\) and \(r_{2}\), the particle radius \(R\) and the filling angle \(\theta\), as shown in Figure 12a. Interparticle force analysis is shown in Figure 12b. If the air pressure \(u_{a}\) is the only contribution to the external force, the intergranular stress \(F_{e}\) consists of three parts when the meniscus is formed. The total force generated by isotropic pore air pressure \(u_{a}\) acts at the gas-solid interface \(F_{a}\), the projection of total force due to water pressure \(u_{w}\) acts on the water-solid interface in the vertical direction \(F_{w}\) and the combined force of surface tension \(\gamma\) acts on the circumference of the meniscus \(F_{\gamma}\), described by Equations (8)-(11). The combined forces of capillary action \(F_{cap}\) consist of matrix suction and surface tension, as shown in Equation (12). Additionally, the intergranular stress \(F_{e}\) generated by the capillary phenomenon acts on particles in the form of tensile force [41,42]. \[F_{a}=u_{a}\Big{(}\pi R^{2}-\pi r_{2}^{2}\Big{)} \tag{8}\]  
\[F_{w}=u_{w}\pi r_{2}^{2} \tag{9}\]  
\[F_{\gamma}=-\gamma 2\pi r_{2} \tag{10}\]  
\[F_{e}=F_{a}+F_{w}+F_{\gamma}=u_{a}\pi R^{2}-(u_{a}-u_{w})\pi r_{2}^{2}-\gamma 2 \pi r_{2} \tag{11}\]  
\[F_{cap}=(u_{a}-u_{w})\pi r_{2}^{2}+\gamma 2\pi r_{2} \tag{12}\]  
As can be seen from Figure 10, in the initial stage of the boundary effect (saturation 100-92%), the voids between sand particles are all filled with pore gravity water connecting to each other and there is only a small amount of air bubbles, which can freely flow with pore water. No capillary effect exists between the granular particles at this time; thus, the intergranular stress is unchanged. Only small changes in saturation occur and the \(G_{max}\) of sand is not affected. When the saturation of sand is gradually reduced from 92 to 70%, the pore gravity water in voids is displaced due to the entry of air. Although the curved liquid surface between the two phases of water-air is formed, the contact between the gas phase and the particle surface is not yet formed because the gravity water still exists around particles, while there is no effect of capillary action. Due to the fact that matrix suction causes little variation in intergranular stress, the ability of the particle system to resist shear deformation hardly changes; therefore, the \(G_{max}\) of sand remains unchanged. With a further reduction in saturation (from 70 to 15%), it enters the transition stage. The contact angle is formed with the solid particle-liquid-gas contact and capillary action is observed. According to Equations (8)-(12), although the surface tension action \(F_{\gamma}\) on the meniscus decreases with the decrease in the principal radius of curvature \(r_{2}\), the matrix suction \(\psi\) gradually increases, leading to a gradual increase in the combined force of capillary action \(F_{cap}\). This contribution to the intergranular stress \(F_{e}\) is enhanced, meaning the \(G_{max}\) of the sand begins to increase gradually. The intergranular stress reaches the maximum value when the curvature radius \(r_{2}\) of residual capillary water meniscus in the contact areas of particles reaches the minimum value. At this time, the macroscopic saturation is at the  
Figure 12: Gas–liquid–solid interaction mode between two particles and meniscus:(**a**) sketch of interaction between two sand particles and meniscus; (**b**) illustration of inter-particle forces. optimum saturation and the capability of sand to resist shear deformation reaches its peak. If the saturation is below the optimum saturation, it enters into the residual saturation phase. With the decrease in saturation, the capillary water, which only exists at corners near the particle contact point, disappears rapidly, meaning the combined force of capillary action \(F_{cap}\) is lost. Particles only connect through weak adsorption water.