# Experiments on the State Boundary Surface of Aeolian Sand for Road Building in the Tengger Desert
## 3 Test Results
### State Boundary Surface
At low and medium effective confining pressure, the void ratio increases rapidly to a critical state after reaching the phase transformation point. At high effective confining pressures, the void ratio increases and \(q\) decreases, and the specimen show strain softening behaviours. At low effective confining pressures, and the dilatation characteristics are significant. The higher the effective confining pressure, the greater the peak shear stress. The critical state lines and phase transformation lines in \(e\)-\(q\) space are nonlinear. The critical state line and phase transformation line can be used to predict the development trend of voids under different test conditions. According to Equations (3) and (5), the critical state line in \(e\)-\(q\) space can be expressed as follows:  
\[q=M_{\rm CSL1}p=M_{\rm CSL1}\exp\frac{e_{\rm CSL}-e_{\rm C}}{M_{\rm CSL2}} \tag{8}\]  
Based on Equations (4) and (6), the phase transformation line in \(e\)-\(q\) space can be expressed as follows:  
\[q=M_{\rm PTL1}p=M_{\rm PTL1}\exp\frac{e_{\rm PTL}-e_{\rm P}}{M_{\rm PTL2}} \tag{9}\]  
Figure 14 shows the critical state characteristic curve of aeolian sand in \(e\)-\(p\)-\(q\) space. The three-dimensional surface composed of \(e\), \(p\) and \(q\) is the state boundary surface or Roscoe surface. The curve connected by the transformation points where the void ratio first decreases and then increases at different stress states is defined as the phase transformation state line, which reflects the state transformation from contraction to dilation. While the granular material is sheared to failure, the curve composed of the effective stress path, void  
Figure 12: \(e\)-ln \(p\) relationships under drained conditions. ratio and average effective stress is defined as the critical state line. The three-dimensional critical state line can be expressed in Equations (5) and (7), and reads:  
\[\left\{ \begin{array}{l}e_{\text{CSL}}=M_{\text{CSL2}}\ln p+e_{\text{C}}\\ q=M_{\text{CSL1}}\exp\frac{e_{\text{CSL}}-e_{\text{C}}}{M_{\text{CSL2}}}\end{array} \right. \tag{10}\]  
The three-dimensional phase transformation line can be expressed in Equations (6) and (8), and reads:  
\[\left\{ \begin{array}{l}e_{\text{PTL}}=M_{\text{PTL2}}\ln p+e_{\text{P}}\\ q=M_{\text{PTL1}}\exp\frac{e_{\text{PTL}}-e_{\text{P}}}{M_{\text{PTL2}}}\end{array} \right. \tag{11}\]  
The test results take into account the dilatancy characteristics for the establishment of the state boundary surface, so the state boundary surface is quite different from that of remodelled clay proposed by [PERSON] [1]. The state boundary surface describes the unique relationship among the stress state, strength and void ratio of aeolian sand. The spatial critical state line and phase transformation line shown in Figure 14 and their descriptive Equations (9) and (10) can better describe and predict the quantitative relationship between stress state characteristics and volume state characteristics in the three-dimensional space. Figure 14: State boundary surface in three-dimensional space. Figure 13: \(e\)-\(q\) relationships under drained conditions.