# Experimental and Empirical Studies to Evaluate Liquefaction Resistance of Partially Saturated Sands
## 5 Proposed Empirical Functions
### Maximum Volumetric Strains  
The maximum volumetric strain (\(\epsilon_{\text{v}}^{*}\)) incorporates the effects of several parameters that govern the liquefaction response of partially saturated sands. Establishing a direct correlation between the liquefaction-resistance ratio (\(LRR\)) and maximum volumetric strain is of great importance in engineering. The maximum volumetric strains were calculated using Equation (3) for different initial conditions that were considered during the experimental work in the present study. The calculated \(\epsilon_{\text{v}}^{*}\) values are plotted against the liquefaction-resistance ratios (\(LRR\)) on the left-hand side of Figure 14; a power function describing the general trend was proposed as follows:  
\[LRR=0.9\cdot(\epsilon_{\text{v}}^{*})^{0.45}+1 \tag{10}\]  
In this equation, the positive exponent (namely, 0.45) indicates the growth of the liquefaction-resistance ratio (\(LRR\)) with an increase in the normalized potential volumetric strains (\(\epsilon_{\text{v}}^{*}\)). The amplitude of the increase in the liquefaction-resistance ratio with increasing normalized potential volumetric strains is characterized by a scaling factor of 0.9. As the exponent is smaller than 1, the slope of the curve is expected to continuously decrease.  
The right-hand side of Figure 14 compares the predictions of the power law relationship in Equation (10) with previously published experimental data obtained for different types of sand tested under a wide range of initial conditions (e.g., \(S_{\text{r}}\) and \(D_{r}\)) and stress levels (e.g., \(\sigma_{\text{c}}^{\prime}\) and \(u_{0}\)). It appeared that the normalized liquefaction resistance of partially saturated sands increased with the maximum volumetric strains and followed a non-linear trend. Most of the experimental data points were successfully captured by the proposed empirical power function.  
Figure 14: Variation in liquefaction-resistance ratio with maximum volumetric strain [17; 35; 36].  
By considering a fully submerged sand element, the liquefaction-resistance ratios were estimated using Equation (10) and plotted against the degrees of saturation for five different field cases (Figure 15). In this analysis, back pressures corresponding to the pore water pressure in the field and the effective confining stresses used during the experimental studies were considered. For a buoyant unit weight of 10 kN/m\({}^{3}\), Case 1 represents a condition in which the water table was at the ground surface, whereas the groundwater table was located at 5-10 m above the ground surface for other cases.  
The evidence in Figure 15 indicates that irrespective of the sand's depth and the position of the water table, the liquefaction-resistance ratios increased with decreasing degrees of saturation. In the right plot in the figure, it is apparent that the liquefaction-resistance ratio of sand under an equivalent depth (e.g., 10 m) slightly decreased as the water table rose, which caused an increase in the pore water pressure that acted on the sand element. This may be ascribed to the decrease in the pore-fluid compressibility with increasing back or pore water pressure [41]. It is also evident in the left plot that under a constant initial pore water pressure, the liquefaction resistance of the sand increased quite significantly with an increasing depth or effective confining stress. These results may imply that in comparison with the initial pore water pressure, the degree of saturation and effective confining stress played a more important role and were the controlling factors in the liquefaction resistance of the partially saturated sands.