## 3 Critical State Model for Clean Sand
### Yield Function  
The yield function of the MCC model is represented by an ellipse in \(p\)-\(q\) plane, as shown in Equation (10). To account for the behavior of loose sand, [PERSON] et al. [14] introduced the initial state parameter \(\chi\) (\(0\leq\chi\leq 1\)) into the MCC model, resulting in a Lemniscate-shaped yield function [30], as shown in Equation (11). The value of \(\chi\) is zero when sand is on the RCL, and \(\chi\) increases with distance above the RCL, i.e., \(\chi\) links to both the shape of the yield function and the distance above the RCL. Hence, \(\chi\) can describe the degree of looseness of sand. Figure 10a shows the yield functions of Equations (10) and (11).  
\[f=ln\bigg{(}\frac{p}{p_{0}}\bigg{)}+ln\bigg{(}1+\eta^{2}/M^{2}\bigg{)}=\int dH \tag{10}\]  
\[f=\ln\bigg{(}\frac{p}{p_{0}}\bigg{)}+\ln\bigg{(}1+\frac{\eta^{2}/M^{2}}{1- \chi\eta^{2}/M^{2}}\bigg{)}=\int dH \tag{11}\]  
According to [14], the original value of \(\chi\) is derived by the stress path of a constant mean effective stress (\(p=p_{0}\)), as expressed in Equation (12). The \(\chi\) modified by incorporating the yield function in Equation (11) and the new hardening in Equation (9) using the same method as [14] is shown in Equation (13).  
\[\chi=\frac{EXP\big{(}\frac{\mathrm{e_{0}}-\mathrm{e_{RCL}}}{\lambda-\kappa} \big{)}-2}{EXP\big{(}\frac{\mathrm{e_{0}}-\mathrm{e_{RCL}}}{\lambda-\kappa} \big{)}-1}\;;\;\chi\geq 0 \tag{12}\]  
\[\chi=\frac{EXP\big{(}\frac{\mathrm{e_{0}}-\mathrm{e_{RCL}}}{\lambda_{\mathrm{ t}}-\kappa}\big{)}-2}{EXP\big{(}\frac{\mathrm{e_{0}}-\mathrm{e_{RCL}}}{ \lambda_{\mathrm{t}}-\kappa}\big{)}-1}\;;\;\chi\geq 0 \tag{13}\]  
where \(e_{RCL}\) is the void ratio on the RCL at the same mean stress of the initial state, \(p_{0}\). In Equation (13), \(\overline{\lambda}_{t}\) is calculated using the tangential slope of the RCL in Equation (5) at the initial mean effective stress \(p_{0}\).  
The state parameter \(\chi\), in Equation (13), is constant during shearing and depends on both the initial mean effective stress \(p_{0}\) and the initial void ratio \(e_{0}\). For sand on the RCL, the \(\overline{\lambda}_{t}\) and \(\overline{\lambda}_{s}\) in Equations (5) and (6), respectively, are applicable. For loose sand, the \(e_{0}\) is located above the RCL, the \(\overline{\lambda}_{t}\) can be assumed by the interpolation function as shown in Equation (14). Figure 10 illustrates the effect of the initial parameter \(\chi\) to the yield function and stress-strain curves from the drained triaxial compression simulation. Figure 11 shows the comparison between the test results and the predictions from the MCC, Sand model by [PERSON] et al. [14] and the proposed model. For the analysis, seven soil parameters for loose sand are listed in Table 1[11].  
\[\overline{\lambda}_{t}=\overline{\lambda}_{(RCL)}+\chi\Big{(}2\alpha- \overline{\lambda}_{(RCL)}\Big{)} \tag{14}\]  
Figure 11: Comparison of the predictions from the MCC model, sand model [14] and the proposed model to the test data [11].  
Figure 10: Effect of parameter \(\chi\) on the MCC model: (**a**) yield function of the MCC (Equation (7) vs. yield function of loosed sands Equation (8) (\(\chi=0\) to 1); (**b**) stress–strain curves for loose sands from drained triaxial compression simulation using both parameter \(\chi\) and \(\overline{\lambda}\).  
In Figure 11, the predictions from both the MCC model and sand model by [PERSON] at al. [14] overestimate the shear strength of sand. However, the results from [14] provides a better result than the MCCs because of the improvement of the plastic strain calculation through the parameter \(\chi\). For the proposed model with both the initial parameters \(\chi\) and \(\overline{\lambda}\), the prediction agrees well with test results.