## 2 The Critical State Line in \(e\)-In(\(p\)) Plane  
As mentioned earlier, the MCC model adopted a straight line of the CSL, but a curved line is valid for sand and silty sand. To address this, the proposed CSL should have the following features:  
1. The equation should provide both the curved portion and the connecting straight line. 2. One of the fitting parameters should be the desired slope of the straight portion. 3. The equation should consist of parameters that can adjust to fit various curvatures and locations. 4. All parameters should be determined from a conventional test. By incorporating these features, the proposed CSL can better capture the behavior of sand and silty sand and provide more accurate predictions for engineering analyses. Equation (3) presents the new CSL with four fitting parameters: \(\Gamma\), the apparent maximum void ratio which controls the vertical translation of the curve; \(p_{ref}\), the reference pressure which is a positive value used to normalize the mean effective stress controlling the horizontal translation; \(\theta\), the curvature index which controls the curvature of the upper portion; and \(\alpha\), the half-slope of the straight portion which controls the slope of the lower portion. The term \"apparent maximum\" is used for the \(\Gamma\) because no data point from the test result can be greater than this value. It is recommended that the value of \(\lambda/2\) or a value close to \(\lambda/2\), where \(\lambda\) is the slope of CSL in Equation (1). To investigate Equation (3), variations of all fitting parameters were demonstrated in Figures 3-6. Notably, when \(\theta=0\), the curvature of the CSL vanishes and Equation (3) becomes a straight line with a slope of \(2\alpha\), as shown in Equation (4). Furthermore, when \(\theta=0\) and \(p_{ref}=1\), Equation (3) reduces to Equation (1) with \(\alpha=\lambda/2\). In Figure 7, the test results of Toyoura sand [11] were plotted against Equations (1)-(3) to evaluate their performance. Equations (5) and (6) present the slope \(\overline{\lambda}\) of Equation (3) in terms of tangential and secant, respectively. \[e=\Gamma-\alpha\ln\left(\frac{p}{p_{ref}}\right)-\sqrt{\theta+\left(\alpha \ln\left(\frac{p}{p_{ref}}\right)\right)^{2}} \tag{3}\]  
\[e=\Gamma-2\alpha\ln\left(\frac{p}{p_{ref}}\right) \tag{4}\]Figure 3: Effects of parameter \(\Gamma\) in Equation (3) for \(0.6\leq\Gamma\leq 1.4\). All four fitting parameters can be determined by the conventional isotropic compression test. While Equation (3) requires more parameters than Equations (1) and (2), all four parameters can be determined by fitting the same test data used for Equations (1) and (2), without requiring additional testing. Therefore, Equation (3) satisfies all the aforementioned features. \[\overline{\lambda}_{t}=\frac{\partial e}{\partial(Inp)}=\alpha\left[1+\frac{ \alpha\ln\left(\frac{p}{p_{ref}}\right)}{\sqrt{\theta+\left(\alpha\ln\left( \frac{p}{p_{ref}}\right)\right)^{2}}}\right] \tag{5}\]  
\[\overline{\lambda}_{s}=-\frac{\Delta e}{\Delta ln(p)}=\alpha+\left\{\sqrt{ \theta+\left(\alpha ln\left(\frac{p_{1}}{p_{ref}}\right)\right)^{2}}-\sqrt{ \theta+\left(\alpha ln\left(\frac{p_{2}}{p_{ref}}\right)\right)^{2}}\right\}/ ln\left(\frac{p_{2}}{p_{1}}\right) \tag{6}\]  
In the MCC model and its family, the CSL and NCL are both assumed to be a straight line with a constant slope \(\lambda\) (see Figure 1) and the vertical distance of \((\lambda-\kappa)\ln(2)\)[14]. In the proposed model, once the curve of NCL is obtained using Equation (3), the CSL is assumed to locate below the NCL with a distance of \(\langle\overline{\lambda}-\kappa\rangle\ln\left(2\right)\), where \(<>\) represents the Macualey brackets (i.e., \(\overline{\lambda}-\kappa=\overline{\lambda}-\kappa\) when \(\overline{\lambda}-\kappa\geq 0\) and zero otherwise). The shape and location of NCL compared to the CSL for this case are shown in Figure 8a. It is evident that both the CSL and NCL contain both curved and straight portions.