## 4 Constitutive Modelling
### Model Parameters  
The current expression for the effective stress of the unsaturated soil cannot accurately describe the change in the contact stress between the soil particles that are caused by the physical and chemical effects [31; 32]. Consequently, the inter-particle effective stress parameter \(\sigma^{s}\) of the unsaturated soil in the model can be preliminarily determined by applying the uniaxial tensile strength test.  
Parameters \(M\), \(\lambda\), \(\Gamma(w)\), \(\mu(w)\), and \(X(w)\) can be determined from the shear tests of the unsaturated soils with the different water contents. In addition, based on the accumulation of the test results, they can be provided directly based on the water content from the fitting relationships with the water content. As demonstrated by the tests in this study, the water content varies in a wide range, from saturated to almost completely dried, involving the whole moisture content variation range; the variations of the parameters \(\Gamma(w)\), \(\mu(w)\), and \(X(w)\) exhibit a unimodal function form, which can be fitted with commonly used polynomials. For example, for the parameters \(\mu(w)\) (Figure 11) and \(X(w)\) (Figure 13), the following fitting functions can be obtained:  
\[\mu=-1.14\times 10^{-3}w^{2}+0.03w-0.839 \tag{15}\]  
\[X=4.3\times 10^{-4}w^{3}-0.043w^{2}+0.988w+5.517 \tag{16}\]  
Figure 13: Relationship between the minimum dilatancy rate coefficient \(X\) and the mass water contents \(w\).  
When considering most of the currently available literature on unsaturated soil, engineering practice analyses, and some clays with a strong adsorption effect, the research on shear characteristics has mostly only involved (or been expressed in terms of) the enhancement of dilatancy from saturation to gradual water loss. That is, the stage where the parameter increases with the decrease in the water content. From the test results in this study, it can be observed that the dilatancy enhancement stage of the unsaturated soil--as the water content decreases from saturation--can be fitted with a relatively simple linear form. For example, the parameter \(\mu(w)\) in Figure 11 and \(X(w)\) in Figure 13 can be, respectively, fitted as follows:  
\[\mu =-0.029w-0.106 \tag{17}\] \[X =-0.376w+18.750 \tag{18}\]  
This type of linear relationship is simpler, satisfying most requirements. The variation of the parameter \(\Gamma(w)\) with the water content, as described in Section 3.3, exhibits similar characteristics as described above. In subsequent studies, by accumulating test results of the different soils, the model parameter prediction equation based on the water content can be further verified or provided, thus facilitating its promotion and application.  
Parameters \(H_{\text{min}}\) and \(\delta_{H}\) (i.e., hardening modulus \(H\)) can be obtained by fitting the deviatoric stress-strain curves in the triaxial tests (that is, trial calculation of the test results).  
For the elastic parameters, \(A\), \(n\), and \(p_{\text{ref}}\) can be obtained from the unloading-reloading cycle test, and \(v\) can be obtained from the local deformation measurement test. In practice, the elastic deformation of the soil during shearing is negligible, and the total deformation can be considered as plastic deformation. Thus, the elastic deformation is neglected in the model calculation. However, starting from the theory, the elastic part is included to complete the model.  
The initial condition for the calculation--that is, the void ratio \(e_{0}\) of the sample before shearing--can be obtained from the volume deformation record during consolidating and air drainage by applying the confining pressures. Suction or degree of saturation is not directly introduced into the model; instead, the mass water content is used, which is more convenient for practical engineering applications.