## 2 Basic Theory  
### MCC Model  
The MCC model is a relatively well-developed and extensively applicable elastic-plastic constitutive model in geotechnics. [PERSON] [22] improved the Cambridge model based on the critical state theory by modifying the flow rules with the hardening parameter, which can effectively reflect the soil's bulk compression and rebound properties. The model captures the hardening and softening of the soil by the size of the critical state surface.  
In p-q space, the yield surface of the MCC model presents as an elliptical curve, as shown in Figure 1. It can be denoted as  
\[q^{2}+M^{2}p(p+p_{c})=0 \tag{1}\]  
where \(p\) denotes the mean effective pressure, \(q\) denotes the deviatoric stress determines the size of the yield surface, \(p_{c}\) denotes the value of the consolidation pressure, and M is the slope of the critical state line (CSL) that passes through the origin. It is assumed that the soil hardening and softening are only related to the plastic volumetric strain, which dominates the size of the plastic yield surface. When plastic volumetric contraction occurs, the yield surface becomes larger, and the soil stress falls on the left side of the CSL (\(q<Mp\)), resulting in softening of the soil. Otherwise, when plastic volumetric expansion occurs, the yield surface reduces, and the soil stress falls on the right side of the CSL (\(q>Mp\)), resulting in the hardening of the soil.  
As the plastic volume changing, the updated consolidation pressure \(p^{\text{N}}_{\text{c}}\) can be computed using the following equation.  
\[P_{c}^{\text{N}}=P_{c}\bigg{(}1+\Delta e_{p}^{p}\frac{\
u}{\lambda-\kappa} \bigg{)} \tag{2}\]  
where \(\Delta e^{p}_{p}\) denotes the increment of plastic volumetric strain, \(\
u\) is the specific volume, \(\lambda\) is the slope of the normal consolidation line, and \(\kappa\) is the slope of an elastic swelling line for an unloading-reloading excursion.  
The MCC model includes eight parameters: (1) rebound index, \(\kappa\); (2) compression index, \(\lambda\); (3) critical stress ratio, \(M\); (4) Poisson's ratio, \(\
u\); (5) \(\beta\), a parameter controlling the shape of the yield surface, generally taken as 1; (6) void ratio \(e_{1}\) on the NCL line when \(p=1\), which is used to obtain the parameter \(\alpha_{0}\) that determines the location of initial yield surface; (7) initial void ratio \(e_{0}\), which represents the current compact state of the soil; (8) initial stress \(p_{0}\), which can be calculated by the ground stress balance. Among them, the first four parameters (\(\lambda\), \(\kappa\), \(M\), and \(\upsilon\)) are directly associated with the deformation. The parameters \(\beta\) and \(e_{1}\) determine the location and shape of the initial yield surface. The parameters \(e_{0}\) and \(p_{0}\) determine the initial state of the soil conjointly but have no impact on the variation of the void ratio, i.e., the volumetric strain. Besides, the preconsolidation pressure \(p_{c}\) defines the size of the yield surface [41].  
Generally, \(\lambda\), \(\kappa\) and \(M\) are regarded as the three primary parameters for the MCC model. The former two parameters are ideally obtained from the isotropically loading  
Figure 1: Yield surfaces in the MCC model.  
unloading triaxial test. However, they can be derived in terms of empirical relations (Equations (3) and (4)) with compression index (\(C_{c}\)) and swelling coefficient (\(C_{s}\)) obtained by an oedometer test, which will be presented in the following section. \(M\) can be derived according to the empirical relation (Equation (5)) with the effective friction angle, \(\phi^{\prime}\), obtained from a triaxial compression test [42].  
\[\lambda=C_{c}/\ln(10) \tag{3}\]  
\[\kappa\approx\ C_{s}/\ln(10) \tag{4}\]  
\[M=6\text{sin}\phi^{\prime}/(3-\text{sin}\phi^{\prime}) \tag{5}\]