## 3 The Indoor Tests on Loess Materials
### Tests Result  
The reference tangent modulus \(E_{\text{oed}}^{\text{ref}}\) and compression modulus \(E_{\text{s1}-2}\) of the soil sample were obtained through standard consolidation tests. Three parallel tests were conducted, and the relationship between the axial load and axial strain of the soil sample is presented in Figure 3. The fitted curve function \(R^{2}\) for all three tests exceeded 0.99, resulting in \(E_{\text{oed}}^{\text{ref}}\) values of 6.78, 6.86, and 6.41 MPa for the three soil samples, respectively, with an average value of 6.68 MPa.  
Figure 4 illustrates the relationship between the void ratio and axial load in the standard consolidation test. Through parallel testing of samples a, b, and c, the compression moduli \(E_{\text{s1}-2}\) of the soil samples were determined to be 7.08 MPa, 6.93 MPa, and 6.84 MPa, respectively, with an average value of 6.95 MPa.  
In Figure 5, the stress\(-\)strain relationship curves for soil samples a, b, and c obtained from the triaxial consolidation-drained shear tests are presented. From these curves, the failure values \(q_{\text{f}}\) and the reference secant modulus \(E_{\text{50}}^{\text{ref}}\) at the reference stress can be determined. The \(q_{\text{f}}\) values were set to the peak stress values, which were 268.0 kPa, 264.3 kPa, and 266.6 kPa, respectively. The value of \(E_{\text{50}}^{\text{ref}}\) is equal to the slope of the line connecting the origin of the coordinate system with the point on the curve corresponding to 0.5\(q_{\text{f}}\). These values were 8.75 MPa, 8.19 MPa, and 9.22 MPa, respectively, yielding an average \(E_{\text{50}}^{\text{ref}}\) value of 8.72 MPa.  
Figure 4: Relationship between void ratio and axial load diagram, as determined by the standard consolidation test.  
Figure 3: Relationships between axial load and axial strain, as determined by the standard consolidation test.  
In the definition of the HSS model, the axial strain \(\varepsilon_{1}\) has a hyperbolic relationship with the deviatoric stress \(q\), as shown in Formula (1):  
\[\varepsilon_{1}=\frac{1}{2E_{50}}\cdot\frac{q}{1-q/q_{\rm a}}(q<q_{\rm f}) \tag{1}\]  
After transformation, a linear relationship between \(\varepsilon_{1}/q\) and \(\varepsilon_{1}\) can be obtained, as shown in Formula (2):  
\[\frac{\varepsilon_{1}}{q}=\frac{\varepsilon_{1}}{q_{\rm a}}+\frac{1}{2E_{50}} (q<q_{\rm f}) \tag{2}\]  
Due to the difficulty of perfectly aligning the actual stress\(-\)strain relationship with a hyperbolic curve, the transformed \(\varepsilon_{1}/q-\varepsilon_{1}\) relationship curve will also deviate from linearity. To reduce human factors and perform a better linear fitting, the fitting line was made to pass through the points where the stress levels \(q/q_{\rm f}\) were 70% and 95%, and the \(\varepsilon_{1}/q-\varepsilon_{1}\) relationship line was plotted [11] (as shown in Figure 6); by taking the reciprocal of the slope of this line, the asymptotic value \(q_{\rm a}\) could be obtained, and the failure ratio \(R_{\rm f}=q_{\rm f}/q_{\rm a}\), resulting in \(R_{\rm f}\) values of 0.75, 0.84, and 0.86, with \(R_{\rm f}\) set to 0.85.  
Figure 5: Stress\(-\)strain relationship based on the triaxial test.  
Figure 6: Curve graph of the relationship between \(\varepsilon_{1}/q\) and \(\varepsilon_{1}\).  
To obtain the reference loading and unloading modulus \(E_{\text{ur}}^{\text{ref}}\), triaxial consolidation drained loading and unloading tests were required. The stress\(-\)strain curve obtained during the loading and unloading test at the reference confining pressure \(p^{\text{ref}}=100\) kPa is shown in Figure 7. During the loading and unloading process, the test curve of the soil sample exhibited a hysteresis loop. Connecting the two endpoints of the hysteresis curve, the slope of the resulting straight line is the reference loading and unloading modulus of the soil sample. The \(E_{\text{ur}}^{\text{ref}}\) values obtained from three sets of tests were 44.1 MPa, 41.7 MPa, and 43.4 MPa, with an average value of 43.1 MPa.