## 5 Verification of Modified Coefficient of the Plastic Modulus \(H_{m}\) in Cyclic Loadings  
There are two types of cyclic undrained triaxial data based on the Banding sand and the Shahrekord sand that were used to verify the coefficient HM in the improvement of the generalized constitutive model. These data show sand behavior under various conditions and loadings. The PZ model [15], which was modified to consider the critical state soil mechanics through use of the equations presented by [PERSON] and [PERSON] [27], has been calibrated for these two sands (i.e., Banding sand and Shahrekord sand), and the calibrated parameters are shown in Table 1. The values of these parameters are the same for both constitutive models (i.e., the original and the modified model). Figure 1: Schematic changes of the parameter n in terms of the value of the deviatoric plastic strain during cyclic loading based on two equations: (**a**) Equation (18) and (**b**) Equation (17). Then, the parameters associated with the coefficient \(H_{M}\) have been calibrated in order to adjust the plastic modulus during the unloading and reloading cycles. Table 2 presents the values of parameters related to this coefficient separately for the original model and the modified model. It should be noted that Equation (18) has been applied to calculate the parameter \(n\) in the presented predictions of the developed constitutive model. Now, given the generalized constitutive model parameters obtained for the Banding sand, the cyclic loading simulation of the Banding sand is presented in Figures 2-5 in accordance with the values calibrated based on laboratory data. Figures 2 and 4 are related to the use of Equation (6) for the coefficient of \(H_{M}\) in the generalized model, while Figures 3 and 5 show the generalized model predictions when the modified equation of \(H_{M}\) (Equation (12)) is used. In these figures, the presentation method based on the stress ratio q/p' makes it easier to see the process of reaching the critical (steady) state of soil samples. As seen from Figures 2 and 4, the parameter \(H_{M}\) calculated by using Equation (6) causes a very high plastic modulus to be created at the beginning of the loading, and the soil suddenly reaches the liquefaction state. In other words, the deformation and stress change process during the cyclic loading is not displayed well, and the steep slope of the diagram during unloading/reloading causes loading cycles to be focused on the beginning parts of the diagram and the liquefaction suddenly occurs. It should be underlined that predictions similar to these shown in Figures 2 and 4 can also be seen in the studies by [PERSON] and [PERSON] [6], [PERSON], [PERSON], and [PERSON] [30], and so on. On the other hand, based on Figures 3 and 5, which are related to the modified equation of \(H_{M}\) (Equation (12)), it can be concluded that the predictions of the constitutive model are more appropriate than in the previous case, and they are compatible with the results observed in the laboratory as the ascending and descending trend in the plastic modulus is exhibited in accordance with the cyclic triaxial laboratory observations. As it can be seen, using the modified coefficient \(H_{M}\), the model can predict the number and amplitude of cycles until reaching the critical (steady) and the liquefaction states. Therefore, in a nutshell, it can be concluded that the modified equation of \(H_{M}\) proposed in this study has affected the results of the generalized model predictions and promoted its power to predict the sand behavior.