## 3 Test Results and Discussion
### Stress--Dilatancy Relationship  
In the axisymmetric stress space of the triaxial test, the dilatancy rate can be expressed as \(D=\mathrm{d}\epsilon_{\mathrm{v}}^{\mathrm{p}}/\mathrm{d}\epsilon_{\mathrm{d}}^ {\mathrm{p}}\), where \(\mathrm{d}\epsilon_{\mathrm{v}}^{\mathrm{p}}\) and \(\mathrm{d}\epsilon_{\mathrm{d}}^{\mathrm{p}}\) represent the plastic volume strain increment and plastic deviatoric strain increment, respectively. The critical state theory shows that the peak strength of the soil can be described by two parts, the dilatancy and critical state strength; that is, it can be expressed as a stress-dilatancy equation. The triaxial test results in this study (Figure 3) show that the peak stress during shearing corresponds to the minimum dilatancy rate. Based on this observation, the relationship between \(\eta^{\prime}_{\mathrm{max}}\) and \(D_{\mathrm{min}}\) is taken as the starting point to explore the stress-dilatancy equation of the unsaturated soils in the different volume states and water content states. Figure 8 shows the relationships between the maximum effective stress ratio and minimum dilatancy rate for the varying water contents and confining pressures. It can be observed from the figure that \(\eta^{\prime}_{\mathrm{max}}\) and \(D_{\mathrm{min}}\) approximate a linear relationship, which can be expressed as follows:  
\[\eta^{\prime}_{\mathrm{max}}=\eta_{\mathrm{r}}+\mu\cdot D_{\mathrm{min}} \tag{6}\]  
where \(\eta_{\mathrm{r}}\) and \(\mu\) represent the friction and dilatancy parameters, respectively. In Figure 8, for the peak state, the representative fitted lines with the mass water contents of 4, 16, and 36% indicate that \(\eta_{\mathrm{r}}\) remains unchanged at the critical state stress ratio \(M\) (same value for the saturated and unsaturated states) when the water content changes; moreover, \(\mu\) is a function of the mass water content. In addition, at the same mass water content, the linear correlation of the test points under the different confining pressures implies that the value of \(\mu\) is the same when the confining pressure changes. Therefore, the maximum effective stress ratio and minimum dilatancy rate in the unsaturated state develops along the same slope as the confining pressure changes, and the slope changes accordingly as the water content changes. Figure 7: Critical state for the different water contents: (**a**) \(q\)-\(p^{\prime}\); (**b**) \(e-\ln p^{\prime}\). Figures 9 and 10 demonstrate the relationships between the dilatancy rate and effective stress during shearing. Figure 9 shows the results when the water content is 4, 16, and 36% under the different confining pressures. At the same mass water content, the dilatancy rate-effective stress relationship can be approximated with the same straight line under the different confining pressures. In other words, the zero dilatancy rate point and the slope of the dilatancy rate change are the same. Figure 10 shows the results at a confining pressure of 25, 50, and 100 kPa under changing water contents, respectively. At the same confining pressure, the dilatancy rate can be observed to change approximately linearly with the effective stress along different slopes under the different water contents, and the zero dilatancy rate point can be approximated as the same. Based on the above analysis, under the different compaction states and water content states, the stress-dilatancy relationship for the unsaturated soil based on the mass water content can be expressed as follows:  
\[D=\frac{M-\eta^{\prime}}{-\mu(w)} \tag{7}\]  
This is because the change in the water content state of the unsaturated soil has altered the soil fabric, thereby changing the slope of the change in the dilatancy rate with the current effective stress ratio. In other words, it changes the shear deformation  
Figure 8: Relationship between the maximum effective stress ratio and minimum dilatancy rate.