## 3 Critical State Model for Clean Sand
### Peak Strength of Dense Sand  
In [14], the yield function of the MCC model can be used for dense sand without the modifications, i.e., \(\chi=0\) in Equations (11) and (12). The goal then becomes identifying the peak in the stress-strain relationship of dense sand. A beautiful derivation for another state parameter \(\chi_{2}\) controlling \(M_{f}\) during shearing is introduced to the MCC model in [PERSON] et al. [14], as shown in Equation (15). The additional soil parameter \(M_{fmax}\) and \(N_{d}\) are required to work with its state parameter. Although this formulation works, it required the densest condition for sand in the calculation. Furthermore, [PERSON] et al. [14] assumed the slope of the densest condition to be the same value of the slope of the unloading-reloading curve with a constant slope \(\kappa\). The computation for \(M_{f}\) needs to be changed in order to simplify the model for sand and silty sand. [PERSON] and [PERSON] [31] introduced the effective state parameter for sand \(\Psi=e-e_{cs}\), where \(e\) and \(e_{cs}\) are the void ratio and the critical state void ratio at the same mean effective stress, respectively. In [32; 33], the relationships of \(M_{f}\) are proposed relating to \(\Psi=e-e_{cs}\). Some researchers have hypothesized that the ratio \(e/e_{cs}\) could function as a state parameter, e.g., [34]. However, other researchers, such as [35; 36], favor using the mean effective stresses ratio \(p/p_{cs}\) where \(p\) and \(p_{cs}\) are the current and critical state mean effective stress. According to a comprehensive review by [PERSON] [37], two of the most effective state variables which can provide a reasonable peak strength \(M_{f}\) of granular soil are \(\Omega=I_{D}\text{ln}(p_{cs}/p)\) and \(\Psi=e-e_{cs}\), where \(I_{D}=(e_{max}-e)/(e_{max}-e_{min})\) is the relative density, \(e_{max}\) the maximum void ratio and \(e_{min}\) the minimum void ratio. The expression for \(M_{f}\) from \(\Omega\) and \(\Psi\) are shown in Equations (16) and (17), respectively. In this research, we utilized the equation of [PERSON] and [PERSON] [34] in Equation (17) in order to achieve the basic model without requiring many extra parameters. The parameter \(n_{b}\) can be determined from the traiaxial compression test of the dense sand. \[M_{f}=(M_{fmax}-M_{c})\sqrt{\chi_{2}}+M_{c} \tag{15}\]  
\[M_{f}=M_{c}[1+n_{b}I_{D}\text{ln}(p_{c}/p)] \tag{16}\]  
\[M_{f}=M_{c}[1+n_{b}\langle-\Psi\rangle] \tag{17}\]  
Before moving forward to the next section of modeling silty sand, the predictive capability of clean sand for both loose and dense conditions are demonstrated by the comparison between the model prediction and two sets of the test results from [PERSON] [38] and [PERSON] and Ishihara [39]. Adding \(n_{b}\) to the model, eight soil parameters listed in Table 2 from [37; 38; 39] are used in the analyses of dense sand. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Soil Model** & \(\mathbf{\Gamma}\) & \(\mathbf{\alpha}\) & \(\mathbf{\theta}\) & \(p_{ref}\) & \(\mathbf{\lambda}\) & \(\mathbf{\kappa}\) & \(M_{c}\) & \(\mathbf{\
u}\) \\ \hline MCC and Sand & \multirow{2}{*}{1.97} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{0.141} & \multirow{2}{*}{0.009} & \multirow{2}{*}{1.305} & \multirow{2}{*}{0.3} \\ Model [14] & & & & & & & & \\ \hline Proposed Model & 0.864 * & 0.0795 * & 0.016 * & 7.5 * & - & 0.009 & 1.305 & 0.3 \\ \hline \end{tabular}
* Fitting parameters from existing data of the NCL (see Figure 7).  
\end{table} 
Table 1: Soil parameters for drained analyses. Figures 12 and 13 show the comparison of the test results and the calculation of the proposed model. In Figure 12, the results of the undrained triaxial tests at the initial mean effective stress of 490 kPa and three different initial void ratios (\(e_{0}\)) of 0.861, 0.883, and 0.910 are demonstrated. Figure 12a depicts the stress path, while Figure 12b illustrates the stress-strain curves. The peak can be observed in the stress-strain curves of \(e_{0}\) = 0.883 and 0.910 on both the