## 1 Introduction  
A precise analysis and design need accurate knowledge and definitions of materials' behavior, which necessitate using an appropriate constitutive model. Accordingly, in recent decades, many researchers in the field of geotechnical engineering have tried to propose and develop suitable and relatively accurate constitutive models to appropriately show the behavior of soils (see, for example, [1, 2, 3, 4]). On the other hand, one of the situations that may result in the failure of structures is related to dynamic and cyclic loading of soils and the related problems, such as liquefaction observed during earthquakes [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Therefore, in addition to precise simulation of the soil behavior under monotonic loadings, a suitable constitutive model for the soil behavior under cyclic loadings is essential. Among different elasto-plastic constitutive models that have been developed in recent decades, the generalized plasticity model has acquired a special position in modeling the mechanical behavior of different types of soil. The constitutive models of this framework are among very powerful constitutive models for the simulation of sand behavior. The generalized plasticity theory and the constitutive model of Pastor-Zienkiewicz (PZ) were developed by [PERSON], et al. [15] to predict the behavior of a variety of soils. The basic generalized model of PZ that had been presented as a generalized plasticity framework was successfully used for the simulation of soils' behavior [5]. However, some limitations of this model restrict its wide application for the analysis of complex soil behavior in various conditions. Therefore, various researchers have proposed different modified models on the basis of the generalized plasticity theory for the improvement of the capability of the main model (see [1; 6; 15; 16; 17; 18; 19; 20; 21; 22]). As a modification, the concept of the state parameter has been used as a tool to associate the sand behavior to density (or the void ratio) and the mean confining stress, and, as a result, to enter the concept of critical state soil mechanics into the constitutive models. Some researchers have entered state parameters into the PZ model using similar methods (e.g., [6; 20]). However, the major drawback of the constitutive models that still exists is the appropriate accuracy of the simulation of sand cyclic behavior, especially in the case of undrained cyclic experiments. It seems that the problem is related to the equation presented for the loading history, which considers the memory parameter \(\mathrm{H_{M}}\) (also known as the discrete memory factor). The main reason that the coefficient \(\mathrm{H_{M}}\) 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, while it seems that 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. Accordingly, in this research, a study has been conducted on the loading history equation (memory parameter), \(\mathrm{H_{M}}\), and a new equation is proposed for it. In other words, in order to accurately simulate the behavior under cyclic loading, an equation is proposed to consider loading history with a new and innovate approach in such a way that the plastic modulus is modified. The performance accuracy of the proposed coefficient has been evaluated in accordance with experimental data. For this purpose, two types of cyclic data based on two different sands have been provided. First, one-way cyclic triaxial data based on Banding sand from Castro [23] have been used for the validation. These data are based on undrained Banding sand in different initial conditions.