## 3 Explicit Formula for the Characterization of the Nonlinear Response  
The soil dynamic response is often characterized by the shear modulus and damping ratio of soils using a resonant column and torsional shear (RCTS) testing apparatus. It is a typical practice to express the nonlinear stress-strain behavior of soils in terms of the shear modulus and the damping ratio. The secant type shear modulus (\(G_{\text{sec}}=\tau_{\text{upper}}/\gamma\)), which is the slope of the secant drawn from the origin to any specified point on the stress-strain curve, can be easily obtained using the analytical form of the backbone curve in Equation (3a):  
\[\frac{G_{\text{sec}}}{G_{\text{max}}}=1-\frac{\alpha}{n}(2\gamma)^{n-1} \tag{4}\]  
Note that the secant modulus is always larger than the tangent modulus, because \(n\) is a positive integer. Figure 2 presents the evolution of the shear modulus along the loading path. The individual part in the hysteresis loop is formulated by mechanical features whose maximum shear stress switches between two configurations at the imposed strain amplitude. The phenomenon of discrete memory is illustrated in the A \(\rightarrow\) B \(\rightarrow\) C \(\rightarrow\) D loading path. This loop begins and ends at the same stress-strain point. This process creates bow-tie stiffness in the entire hysteresis loop [24; 32]. The results of this simple analysis indicate that the dynamic stiffness has strong dependency on the loading path and imposed strain amplitude. In addition, the secant-type shear stiffness at the given shear strain amplitude is superimposed on its tangent counterpart. It was found that \(G_{\text{sec}}(\Delta\gamma)\) is equal to the average of the two tangent moduli at \(\pm\Delta\gamma\) according to Equations (2) and (4), such that \(G_{\text{sec}}(\Delta\gamma)=G_{\text{tan}}(\gamma=0)\) with the second model order (\(n=2\)). Figure 1: Nonlinear stress-strain behavior of soil subjected to three different shear strain amplitudes. The hysteretic loops and backbone curve are obtained using Equations (1) and (3). The model parameters are \(G_{\text{max}}=180\) MPa, \(\alpha=370\), and \(n=2\). The inherent viscous characteristics of soils dissipate the energy against the imposed loads. Thus, this relationship allows an estimation of the nonlinear damping ratio with closed hysteresis loop:  
\[\xi^{NL}(\gamma)=\frac{W_{D}}{4\pi W_{S}} \tag{5}\]  
where \(W_{S}\) (\(=G_{\text{sec}}\)\(\Delta\gamma^{2}/2\)) is the maximum strain energy stored during the cycle and \(W_{D}\) is the closed area related to the hysteresis loop in one cycle of loading. The hysteresis loop is constructed by the upper and lower parts depending on the sign of the strain rate. For example, the upper part is the positive strain rate from \(+\Delta\gamma\) to \(-\Delta\gamma\). The energy dissipation per cycle can be calculated using Equation (1) with two integration sections:  
\[ \begin{split} W_{D}=\int\tau d\gamma&=\int_{- \Delta\gamma}^{+\lambda\gamma}G_{\text{max}}\big{\{}\gamma-\frac{\pi}{\#} \big{[}(\Delta\gamma+\gamma)^{\pi}-2^{n-1}(\Delta\gamma)^{\pi}\big{]}\big{\}}d \gamma+\int_{+\lambda\gamma}^{-\Delta\gamma}G_{\text{max}}\big{\{}\gamma-\frac {\pi}{\#}\big{[}-(\Delta\gamma-\gamma)^{\pi}+2^{n-1}(\Delta\gamma)^{\pi}\big{]} \big{\}}d\gamma\\ &=\frac{2^{n+1}(n-1)}{n(n+1)}G_{\text{max}}(\Delta\gamma)^{n+1} \end{split}  \tag{6}\]  
Figure 2: Evolution of the shear modulus along the A \(\rightarrow\) B \(\rightarrow\) C \(\rightarrow\) D loading path for the given shear strain amplitudes of \(\Delta\gamma\) = 1.2 \(\times\) 10\({}^{-3}\). (**a**) Hysteresis loop and (**b**) Shear modulus. The tangent shear modulus is calculated from Equation (2). Circles indicate the secant shear modulus. Note that \(G_{\text{sec}}(\Delta\gamma)\) is equal to the average of the tangent moduli at \(\pm\)\(\gamma\).