## 2 The Plastic Modulus in the Generalized Model  
In the generalized models, the loading/reloading plastic modulus, \(H_{L}\), was proposed as follows (e.g., [6; 15]):  
\[H_{L}=H_{0}H_{M}p^{\prime}(1-\frac{\eta}{\eta_{f}})^{4}\left(1-\frac{\eta}{M_ {S}}+B_{0}B_{1}\exp(-B_{0}\zeta)\right), \tag{1}\]  
where \(\eta\) is the stress ratio \(\
icefrac{{q}}{{p^{\prime}}}\) in the current stress situation, \(\alpha\) is one of the model parameters that plays a role in the definition of the strain direction, and \(B_{0}\), \(B_{1}\), and \(H_{0}\) are other parameters of the model that are calibrated in the definition of the loading/reloading plastic modulus for the desired soil. The parameter \(H_{M}\) is a loading history coefficient (memory parameter). The parameter \(\zeta\) is called the mobilized stress function [15], which is discussed in the next section. The parameter \(M_{S}\) is the slope of the phase transformation line in the space of \(q-p^{\prime}\) stresses. At the phase transformation (PT) state, the sands exhibit a transition from contractive to dilatant behavior. In the undrained triaxial test, the phase transformation is often defined as the 'knee' of the effective stress path, i.e., on the plane \(q-p^{\prime}\)[25]. At the 'knee' of the effective stress path, the increment of the mean effective stress change will be zero, i.e., \(dp^{\prime}=0\)[26]. The stress path turns its direction on the plane \(q-p^{\prime}\), i.e., the point where the effective stress has a 'knee' and the effective mean stress reaches a minimum value [25]. A line on the plane \(q-p^{\prime}\) can exhibit the location of this state or point. This position is known as a quantity dependent on the state parameter. The parameter \(\eta_{f}\) is defined in the basic generalized model as follows [5; 15]:  
\[\eta_{f}=(1+\frac{1}{\alpha})M_{f}, \tag{2}\]  
where \(M_{f}\) is another model parameter that is used for the definition of the main loading direction. In order to enter the concept of the critical state soil mechanics into the basic generalized model, the relations proposed by [PERSON] and [PERSON] [27] can be used to define \(M_{g}\) and \(\eta_{f}\):  
\[\eta_{f}=M_{c}\exp(-n_{b}\psi), \tag{3}\]  
\[M_{g}=M_{c}\exp(n_{d}\psi), \tag{4}\]  
where \(n_{b}\) and \(n_{d}\) are the parameters of the model, \(M_{C}\) is the slope of the failure line in the \(q-p^{\prime}\) space, and \(\psi\) is the state parameter introduced by [PERSON] and [PERSON] [28]. These equations were extensively used to modify the PZ generalized model (e.g., [PERSON] and [PERSON] [6]). The critical state (CS) is defined as the ultimate condition in which shearing could continue indefinitely without changes in volume or effective stresses [29]. The critical state line is a unique line in the \(q-p^{\prime}-e\) (\(q-p^{\prime}-v\)) space. It is common to project it onto the planes \(q-p^{\prime}\) and \(e-p^{\prime}(v-p^{\prime})\). The CS line on the plane \(q-p^{\prime}\) is defined as \(q=M_{C}p^{\prime}\), and it is defined as \(e=e_{\Gamma}-\lambda\left(\frac{p^{\prime}}{p^{\prime}_{0}}\right)^{\xi}\) on the plane \(e-p^{\prime}\). The concept of the critical state was first proposed for clays; however, a similar concept is used for sand, and it is usually called the steady state. The parameter \(\eta_{f}\) represents the failure stress ratio, depending on the state parameter. When the soil reaches the critical (steady) state condition (\(\psi\to 0\)), the value of this parameter becomes equal to \(\text{M}_{\text{C}}\) (the slope of the critical state line on the plane \(q-p^{\prime}\)). On the other hand, a separate equation was suggested for the plastic modulus during unloading [5; 15; 30]:  
\[H_{u}= \begin{cases}H_{u0}\left(\frac{M_{g}}{\eta_{u}}\right)^{\gamma_{u}}&for \quad\left|\frac{M_{g}}{\eta_{u}}\right|>1\\ H_{u0}&for\quad\left|\frac{M_{g}}{\eta_{u}}\right|\leq 1\end{cases} , \tag{5}\]  
where \(\eta_{u}\) is the stress ratio from which unloading starts.