## 1 Introduction  
The geotechnical engineering problems can be solved by many types of solutions, e.g., analytical solution [1; 2; 3], semi-analytical solution [4; 5] and numerical approachs [6; 7; 8]. Recent technological advancements lead to sophisticated construction including phrased construction. The finite element method (FEM) is one of the powerful numerical methods applicable for the complex in both the geometry and loading of the problem. Nevertheless, the rigorousness of the solutions from FEM depends largely on the constitutive models incorporating the FEM-based analysis. Constitutive models are generally developed for specific types of geomaterial, such as clay and sand. The majority of the models are developed solely for each type of soil. However, in nature, the soil deposits are composed of more than one type, and silty sand is a good example. In order to accurately predict the behavior of silty sand, the critical state model's complexity rises to require twenty or more soil characteristics [9], thus making it unapproachable to general users. Therefore, it is crucial to develop a simple yet practical model for silty sand. The modified Cam-clay (MCC) model was originally developed to describe the stress-strain strength characteristics of normally consolidated clay based on [PERSON]'s critical state theory [10]. A number of modifications have been made to the MCC to improve its ability to model geomaterials, such as [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. Within the MCC and its family, the critical state void ratio (\(e_{cs}\)) defined by the critical state line (CSL) is assumed to be a straight line lower than the normally consolidated line (NCL) with the same slope \(\lambda\), as shown in Figure 1a. The distance between the NCL and CSL is assumed to be \((\lambda-\kappa)\ln(2)\), where \(\kappa\) is the slope of the unloading-reloading portion in the isotropic compression test result. Equation (1) contains two fitting parameters: \(\Gamma\), the limiting critical state void ratio, usually defined as \(e_{cs}\) at \(p=1.0\), and \(\lambda\), which represents the constant slope of the CSL. However, unlike clay, the literature shows that the CSL of granular soil is not a straight line, e.g., [22; 23; 24; 25; 26; 27; 28]. A power function curve shown in Equation (2) provides a better representation for granular soil. In Equation (2), \(\varGamma\) acts as the upper bound of \(e_{cs}\), while \(\lambda,\ p_{a}\)_and_\(\bar{\varsigma}\) are the fitting parameters. Although \(p_{a}\) is recommended, it is not necessary to have the atmospheric pressure (approximately 101 kPa). Thus, Equation (2) requires three fitting parameters. It should be noted that the slope of CSL in Equation (2) is not a constant but a varied value \(\overline{\lambda}\). A comparison between Equations (1) and (2) is demonstrated in Figure 1b. Using Equation (1) for granular soil would result in a significant error in the volumetric plastic strain (\(\Delta e_{p}^{p}=\Delta e^{p}/(1+e_{0})\)) between two points on the CSL such as points A and B in Figure 1b. Therefore, a power function curve is popularly used for sand or silty sand. However, this form of the power function can be problematic in sand models when dealing with high pressure because it cannot provide a straight portion after the curved portion. Figure 2a shows the concavity on the extension of Equation (2) after fitting the laboratory data [11], where the dashed line deviated from the trend of a straight line. To achieve a proper function of \(e_{cs}\) in the e-In(p) for granular soil, including clean sand and silty sand, without any error in the volumetric plastic strain calculation, a curve illustrated in Figure 2b is required. \[e_{cs}=\varGamma-\lambda\ln(p) \tag{1}\]  
\[e_{cs}=\varGamma-\lambda\left(p/p_{ref}\right)^{\bar{\varsigma}} \tag{2}\]  
Figure 1: Critical state line (CSL): (**a**) straight version of the Normally consolidation line (NCL) and the CSL in MCC and MCCs family; (**b**) comparison between CSL in Equations (1) and (2).