## 1 Introduction
\(G_{d}\) and \(\lambda\) can be easily extracted from the evolution of hysteresis curves. The present study on hysteresis curves mainly focuses on frozen clay, peat soils, swelling soils, etc. It is noted that a series of works on the hysteresis curves of frozen clay has been completed by [PERSON] et al. [26]. They first considered the effect of stress loading conditions on the hysteresis curve morphology [26] and then investigated the negative temperature after realizing that the melting of ice crystals under negative temperature affects the mechanical properties [27]. Later, they also took into account the confining pressure and loading frequency [28]. In the study on peat soils, [PERSON] et al. [29] considered the effect of confining pressure and loading frequency on the hysteresis curve. [PERSON] et al. [30] took into account the stress amplitude on this basis and the factors affecting the hysteresis curve of expansive soils, which mainly include the confining pressure and loading frequency [31; 32]. Different test factors can greatly affect the dynamic elastic modulus \(E_{d}\) and damping ratio \(\lambda\). For example, [PERSON] et al. [33] considered the effects of loading frequency, stress amplitude, initial partial stress and confining pressure on the dynamic elastic modulus \(E_{d}\) of soft clays and showed that \(E_{d}\) increases with the increase of cyclic stress amplitude, initial partial stress, and confining pressure; the lower the vibration frequency, the faster the decay of \(E_{d}\). [PERSON] et al. [34] analyzed the effect of dynamic stress ratio and confining pressure on \(E_{d}\) of saturated gravel sands and found that \(E_{d}\) increases with increasing confining pressure and dynamic stress ratio. [PERSON] et al. [35] studied the effects of sea sand admixture, loading frequency and dynamic stress on \(E_{d}\) and \(\lambda\) of cured silts, which demonstrated that \(E_{d}\) increases with the growing sand admixture and loading frequency. It first increases and then decreases with the increase of dynamic stress, while \(\lambda\) follows the opposite relationship. Hysteresis curves have strong variability for different types of soils in different regions, and so do the dynamic parameters. [PERSON] et al. [36] investigated the effect of \(G_{d}\) and \(\lambda\) of GRS under different confining pressure and stress amplitude and found that \(G_{d}\) decreases with the increase of stress amplitude and \(\lambda\) inversely. In addition, increasing the surrounding pressure can suppress the decay of \(G_{d}\) with dynamic stress. The results of [PERSON] et al. [37] on the effect of confining pressure and dynamic stress amplitude on \(G_{d}\) showed the same trends as those of [PERSON] et al. [36]. Furthermore, [PERSON] et al. [37] also found that the greater the water content in the soil, the faster \(G_{d}\) decays. The development of a dynamic constitutive model to quantitatively describe the dynamic deformation characteristics of soil under cyclic loading is a key issue [38]. The common soil models can be divided into three categories. The first type is the Masing model [39], which assumes that the hysteresis curve is obtained by doubling the skeleton curve, and defines the \"upper skeleton curve criterion\" and the \"upper great circle criterion\". The second type is the equivalent linear model, which assumes that the soil is a viscoelastic material, and introduces \(E_{d}\) and \(\lambda\) as dynamic model parameters into the dynamic stress-strain model, for example, the Hardin-Drnevich model [40] and Ramberg-Osgood model [41], etc. The third type is the Iwan model [42], which describes the dynamic stress-strain relationship of the soil by combining different elastoplastic units either in series or in parallel. Except for the above three categories, there are also empirical models of \(E_{d}\) with the number of load cycles, often denoted by \(N\), confining pressure and deviator stress dynamic intrinsic relationship proposed by scholars. [PERSON] et al. [43] and [PERSON] et al.