## 2 The Plastic Modulus in the Generalized Model
Therefore, this equation cannot theoretically be true because, as it was stated, \(\zeta\) should be an ascending function, and not a descending one. The typical value of \(\alpha\) should be equal to \(-0.45\), and, if it is substituted in Equation (7), we will have:  
\[\zeta\approx p^{\prime}\bigg{(}1+1.22\frac{\eta}{M_{f}}\bigg{)}^{2.22}, \tag{9}\]  
Equation (9) represents a parabolic function in terms of the variable \(\eta\), so that the value of the parameter \(\zeta\) increases with the start of the deviatoric loading and with the increase in the parameter \(\eta\). Therefore, according to the above explanations, it can be concluded that Equation (7) with a negative value for \(\alpha\) can play its role to define \(\zeta\) and, consequently, \(H_{M}\). [PERSON], [PERSON], and [PERSON] [30] used a similar method to define \(\zeta\), as in the following equation:  
\[\zeta=p^{\prime}\bigg{[}1-\bigg{(}\frac{\alpha}{1+\alpha}\bigg{)}\frac{\eta}{ M_{f}}\bigg{]}^{-1/\alpha}, \tag{10}\]If \(\alpha=-0.45\), Equation (10) takes the form  
\[\zeta\approx p^{\prime}\bigg{[}1+0.82\frac{\eta}{M_{f}}\bigg{]}^{2.22}, \tag{11}\]  
Thus, it is seen that Equation (11) and, consequently, Equation (10), are practically increasing functions, and there is, theoretically speaking, the possibility that the plastic modulus is modified during reloading using this equation. Therefore, the coefficient \(H_{M}\) is used to simulate the cyclic behavior of sand in accordance with the equation given by [PERSON], [PERSON], and [PERSON] [30]. However, the results obtained from the simulation of soil in cyclic loading is not very favorable, and similar results were observed by [PERSON] and [PERSON] [6] and [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] [5] for the undrained cyclic simulation of sand. Hence, an attempt is undertaken to present an appropriate and innovative equation for \(\zeta\) and \(H_{M}\) so that the predictions of the generalized constitutive model in cyclic loadings are improved in an acceptable manner. The main reason that the coefficient \(H_{M}\) in the previous equations (definitions) cannot play its role in correcting the plastic modulus during unloading and reloading is that this coefficient is only defined based on the stress ratio. Meanwhile, it seems that the extra stiffness caused by unloading and reloading in successive cycles is caused by the accumulated plastic strain. On this basis, in this paper, an attempt was made to define this coefficient based on the accumulated plastic strain during cyclic loading in an innovative way. Definition of a New Equation for Coefficient \(H_{M}\) and Modification of the Reloading Plastic Modulus  
According to cyclic drained and undrained triaxial laboratory results (e.g., [PERSON] [23] and [PERSON], [PERSON], and [PERSON] [3]), it is seen that a considerable reduction occurs in the plastic behavior of soils during unloading and reloading, which can be due to a significant increase in the plastic modulus. However, in high (great) cycles, especially during undrained cyclic loadings, when the soil approaches the critical (steady) state, significant plastic strains occur in the soil, and the soil's behavior severely exits the elastic state and approaches the plastic state. In other words, the soil situation approaches the critical (steady) state in the cyclic loading, particularly in the case of occurrence of the liquefaction phenomenon in the undrained sand, when the behavior becomes severely plastic, which can be attributed to a considerable decrease in the plastic modulus. Therefore, the following modification of coefficient \(H_{M}\) is proposed, which indeed introduces the loading history so as to increase the accuracy of the model's predictions in cyclic loadings.