## 2 The Plastic Modulus in the Generalized Model
\(H_{u0}\) and \(\gamma_{u}\) are the other two parameters of the model that are used in the definition of the unloading plastic modulus. Based on cyclic laboratory results, it is observed that during unloading and reloading, a considerable reduction occurs in the plastic behavior of soil, which can result from a significant increase in the plastic modulus. Accordingly, in the generalized model [5; 15], the coefficient \(H_{M}\) is introduced to take into account the history of past events during the cyclic loading, which is called the discrete memory factor. The coefficient \(\text{H}_{\text{M}}\) should be defined so that it is equal to one in the initial cycle, and, after that, it increases to a value greater than one in subsequent cycles, based on the stress ratio \(\eta\). Therefore, the coefficient \(\text{H}_{\text{M}}\) will lead to a higher plastic modulus during reloading compared to the plastic modulus at the first cycle. This is consistent with experimental observations (e.g., [PERSON] [23] and [PERSON], [PERSON], and [PERSON] [3]). [PERSON], [PERSON], and [PERSON] [30] defined the coefficient \(H_{M}\) in the main constitutive model of generalized plasticity [5; 15]. However, it was observed in accordance with laboratory results that the defined equation cannot exhibit the soil behavior in cyclic loadings. Therefore, a new and suitable equation for the definition of the coefficient \(H_{M}\) is proposed in this paper so as to enhance the ability of the generalized model in the prediction of soils' cyclic behavior, especially sandy soils, and the simulation of the liquefaction phenomenon. 3. Criticism of the Loading History Coefficient (Memory Parameter) \(H_{M}\) in Generalized Models  
In the basic generalized model [5; 15], the equation of \(H_{M}\) is equal to:  
\[H_{M}=\bigg{(}\frac{\zeta_{\max}}{\zeta}\bigg{)}^{\gamma}, \tag{6}\]  
where \(\gamma\) is a parameter of the model. According to the definition, the coefficient \(H_{M}\) should be equal to or greater than unity. In particular, the coefficient \(H_{M}\) is equal to unity during loading (the first cycle of the cyclic loading), and its value is larger than unity in the next cycles. For this purpose, it is necessary to consider an additional parameter, \(\zeta\), in unidirectional loading; thus, the maximum value of \(\zeta\), that is, \(\zeta_{\max}\), is equal to the current \(\zeta\), and, as a result, in accordance with Equation (6), \(H_{M}\) will be equal to unity. During unloading, the current value of \(\zeta\) should be smaller than \(\zeta_{\max}\); therefore, the coefficient \(H_{M}\) will be higher than unity. As a result, the coefficient may cause the plastic modulus to increase after the first loading cycle, which follows the trend observed in the laboratory (e.g., [PERSON] [23] and [PERSON], [PERSON], and [PERSON] [3]). Thus, the parameter \(\zeta\) should theoretically be defined in such a way that as the deviatoric stress (or the stress ratio \(\eta=\
icefrac{{q}}{{p}}\)) increases, the value of \(\zeta\) constantly increases. However, the definition of parameter \(\zeta\) in the PZ constitutive models is as follows [5; 15]:  
\[\zeta=p^{\prime}\bigg{[}1-\bigg{(}\frac{1+\alpha}{\alpha}\bigg{)}\frac{\eta}{ M_{f}}\bigg{]}^{-1/\alpha}, \tag{7}\]  
Evaluating Equation (7) shows that defined in this way, parameter \(\zeta\) cannot theoretically meet the aim of the definition of coefficient \(H_{M}\), and the equation for \(\zeta\) is not an increasing function in terms of the deviatoric stress \(q\) (or the stress ratio \(\eta\)) because Equation (7) can be written in the following way for the typical value of \(\alpha\) (0.45):  
\[\zeta\approx p^{\prime}\bigg{(}1-3.22\frac{\eta}{M_{f}}\bigg{)}^{-2.22}=\frac{ p^{\prime}}{\Big{(}1-3.22\frac{\eta}{M_{f}}\Big{)}^{2.22}}, \tag{8}\]  
which introduces a hyperbolic function. The maximum value of this function is equal to one at \(\eta=0\), and, as \(\eta\) gradually increases, it approaches zero.