## 2 Materials and Methods
### Maximum Shear Modulus \(G_{max}\) and Shear Modulus Degradation Curves \(G/G_{max}\)  
The typical description of nonlinear soil behavior are models that rely on hysteresis stress-strain relations. With NERA, the hysteresis curve is modeled using such parameters as maximum shear modulus \(G_{max}\) and so-called shear modulus degradation curves \(G/G_{max}\) (\(\gamma\)) (Figure 3).  
The easiest way to obtain \(G_{max}\) for each sublayer of the soil deposit is to use known values of the averaged density \(\rho\) of saturated porous medium and shear wave velocity \(V\)s derived from the geophysical and geotechnical investigations:  
\[G_{max}=\rho V_{s}^{2} \tag{13}\]  
However, there are some empirical expressions derived from laboratory tests that demonstrate the dependency of \(G_{max}\) on different parameters, including a void ratio. The following example is for seabed claye soils [38]:  
\[G_{max}=\frac{358-3.8 PI}{0.4+0.7e}\left(\sigma_{0}^{\prime}\right) \tag{14}\]  
In turn, the following example is for sandy soils with round particles [19]:  
\[G_{max}=700\frac{\left(2.17-e\right)^{2}}{1+e}\left(\sigma_{0}^{\prime}\right) ^{0.5} \tag{15}\]  
Figure 3: Hysteresis curve (**a**), shear modulus degradation curves \(G/G_{max}\) (\(\gamma\)), and damping ratio curves \(\xi(\gamma)\) (**b**) models for cyclic loading.  
where \(Pl\) is a plasticity index, \(\sigma_{0}^{\prime}\) is the average effective mean stress level, which can be expressed as [39]:  
\[\sigma_{0}^{\prime}=\ \frac{(1+2K_{0})}{3}\sigma_{v}^{\prime}\ \ \ \ K_{0}=0.0042 PI+0.44 \tag{16}\]  
where \(\sigma_{v}^{\prime}\) is effective vertical stress.  
Many data on nonlinear dynamic deformation features of soils were obtained from drained tests, while the effects of drainage are not negligible in the real ground [19]. In this regard, the following expression was obtained for undisturbed undrained samples [40]:  
\[\frac{G}{G_{max}}=(A_{1}+A_{2}lgD_{50})\sigma_{0}^{\prime}(B_{1}+B_{2}lgD_{50}) \tag{17}\]  
where \(D_{50}\) is a mean particle size of sand (mm) and \(A_{1}\), \(A_{2}\), \(B_{1}\), \(B_{2}\) are strain-dependent empirical parameters.  
For cohesive soils, the following empirical expression is more suitable that includes a plasticity index \(PI\)[41]:  
\[\frac{G}{G_{max}}=K(\gamma,PI)\big{(}\sigma_{0}^{\prime}\big{)}^{m(\gamma,PI) -m_{0}} \tag{18}\]  
where \(K(\gamma,PI)\) and \(m(\gamma,PI)\) are functions that depend on strain \(\gamma\) and plasticity index \(PI\).  
One can conclude that maximum shear modulus \(G_{max}\) depends on a void ratio either implicitly (Equation (13) or explicitly (Equations (14) and (15)). Meanwhile, shear modulus degradation curves \(G/G_{max}\) (\(\gamma\)) under undrained conditions considerably depend on the mean effective stress, as well as the plasticity index or mean size of soil particles (Equations (17) and (18)), but not on a void ratio. Special comparative analyses also showed conflicting results on the influence of the void ratio on the form of the shear modulus degradation curves of cohesive soils [42]. Hence, it is assumed here that shear modulus degradation curves do not require special modifications to consider a porous medium. The same is true to the damping ratio curves, as additional damping is not considered due to the lack of relative motion of the soil matrix and pore fluids. Therefore, in this study, the classic relationships from the literature were used.