## 1 Introduction  
In recent years, modeling problems with complex geometries and challenging soil conditions has become widely used by engineers and researchers due to significant advances in numerical tools and computational technologies. A major challenge in numerical modeling is the determination of parameters related to the stress-strain behavior of soils affected by both static and dynamic loading conditions. Soils under geotechnical applications such as foundations and embankment structures are subjected to a relatively slow loading rate. The stress-strain behavior of such soils can be described by the static elasticity modulus (E\({}_{\rm stat}\)) [1]. On the other hand, the behavior of soils under cyclic and dynamic loading such as earthquakes, winds, explosions, and machine vibrations can be represented and modeled by the dynamic modulus of elasticity (E\({}_{\rm dyn}\)), which is considered as a fundamental parameter in geotechnical engineering applications [2]. At very small deformation levels (\(\gamma\leq 10^{-5}\%\)), where the elastic behavior remains essentially constant, the modulus is defined as the initial shear modulus (G\({}_{0}\)) or \"dynamic\" stiffness, whether obtained by laboratory tests or by measurement or calculation by seismic in situ methods. In contrast, the \"static\" stiffness is derived from the initial loading curve of oedometer or monotonic triaxial tests. Although the difference between static and dynamic stiffness was initially attributed to the difference in loading rates, later studies have shown that these differences are mainly determined by the deformation levels. The effect of drainage conditions on the stiffness of sandy soils at small to moderate deformations is another important issue that needs to be clarified. By the 1970s, a comprehensive experimental database of various soil types had been established in the literature, thanks to the more consistent results of monotonic experiments performed under stress or deformation control. However, the acquisition of dynamic parameters was very limited in this period. To overcome this deficiency, ref. [3] proposed an empirical relationship derived from the solution of a differential equation subject to boundary conditions and thus successfully introduced the concept of dynamic-static modulus ratio into the principles of soil mechanics. A rheological model considered by [3] depends on the Kelvin-Voigt rigid body [4; 5]. Static elastic modulus (E\({}_{\rm stat}\)) is defined as a ratio of constant normal stress, \(\sigma_{0}\), to time-dependent axial strains \(\varepsilon(\mathfrak{t})\), as presented in Equation (1). \[\mathrm{E}_{\rm stat}=\frac{\sigma_{0}}{\varepsilon}=\frac{\mathrm{E}}{1- \exp\Bigl{(}-\frac{\mathfrak{t}}{\tau\mathfrak{t}}\Bigr{)}} \tag{1}\]  
where T\({}_{\rm ret}\), denotes the retardation time of the force applied in rheologic model; E represents the modulus of elasticity for the springs-within model; and t refers to the duration of the static test. In this context, the value of E is employed in the static elasticity formula. T\({}_{\rm ret}\) may be interpreted as the characteristic duration over which the stress induced by dynamic or static loading is transferred to the soil skeleton. For dynamic loading states, the Kelvin-Voigt rigid body exhibits dynamic elastic modulus (E\({}_{\rm dyn}\)) influenced by the periods (T) of cyclic loading, given by Equation (2). \[\mathrm{E}_{\rm dyn}=\frac{\sigma_{0}}{\varepsilon_{0}}=\mathrm{E}\Bigl{[}1+ \Bigl{(}2\pi\mathrm{T}_{\rm ret}/\mathrm{T}^{2}\Bigr{)}\Bigr{]}^{1/2} \tag{2}\]  
where T \(=\) 2\(\pi\)/\(\omega\) denotes the vibration period of dynamic stress. [PERSON]'s concept is still widely accepted and applied by researchers and engineers today. This relationship allows a direct transition between E\({}_{\rm dyn}\)/E\({}_{\rm stat}\) and E\({}_{\rm stat}\). However, in [PERSON]'s work, it is unclear whether the E\({}_{\rm stat}\) is equal to E\({}_{0}\), E\({}_{\rm ur}\) or E\({}_{50}\). This relation needs to be clarified.