## 4 Characteristics and Models of Dynamic Parameters of GRS  
### Dynamic Parameters of GRS  
(1) Dynamic shear modulus \(G_{d}\)  
Since the dynamic shear modulus (\(G_{d}\)) can not be directly determined in the dynamic triaxial tests, it is usually calculated indirectly, as follows:  
\[G_{d}=\frac{E_{d}}{2(1+\mu)} \tag{5}\]  
where \(E_{d}\) is elasticity modulus, \(\mu\) is [PERSON]'s ratio. According to [58], the [PERSON]'s ratio of GRS ranges from 0.3 to 0.33, thus \(\mu\) is taken as 0.3 in this study.  
The dynamic elasticity modulus (\(E_{d}\)) is obtained from the dynamic triaxial test. It can be obtained as:  
\[E_{d}=\frac{\sigma_{A}-\sigma_{B}}{e_{A}-\epsilon_{B}} \tag{6}\]  
where \(\sigma_{A}\), \(\sigma_{B}\) and \(\epsilon_{A}\), \(\epsilon_{B}\) are the stress and strain values corresponding to the intersection points of A and B, respectively.  
(2) Damping ratio \(\lambda\)  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{Range of Parameters under Different Load Levels} \\ \hline  & \multicolumn{8}{c}{\(\Delta S_{\rm ave}\) (N3\(-\)N1)} & \multicolumn{8}{c}{\(\Delta k_{\rm ave}\) (N3\(-\)N1)} & \multicolumn{8}{c}{\(\Delta d_{\rm ave}\) (N2\(-\)N1)/\%} & \multicolumn{8}{c}{\(\Delta\epsilon_{\rm grave}\) (N2\(-\)N1)/\%} \\ \cline{2-13}  & C0 & C10 & C20 & C30 & C0 & C10 & C20 & C30 & C0 & C10 & C20 & C30 & C0 & C10 & C20 & C30 \\ \hline T0 & \(-1.86\) & \(-2.86\) & \(-3.26\) & \(-5.30\) & 254.3 & 178.7 & 254.6 & 267.9 & \(-1.6\) & \(-2.82\) & \(-3.89\) & \(-8.05\) & \(-2.06\) & \(-3.83\) & \(-4.84\) & \(-9.82\) \\ T1 & \(-1.92\) & \(-2.63\) & \(-2.55\) & \(-3.56\) & 214.6 & 189.4 & 162.2 & 178.2 & \(-1.38\) & \(-2.63\) & \(-2.29\) & \(-4.32\) & \(-1.80\) & \(-3.38\) & \(-2.92\) & \(-5.36\) \\ T2 & \(-1.42\) & \(-3.03\) & \(-3.27\) & \(-3.57\) & 191.4 & 166.3 & 179 & 188.2 & \(-0.88\) & \(-2.84\) & \(-3.82\) & \(-4.49\) & \(-1.16\) & \(-3.68\) & \(-4.48\) & \(-5.60\) \\ T3 & \(-1.39\) & \(-2.58\) & \(-1.98\) & \(-3.47\) & 188.6 & 162.8 & 158 & 155.2 & \(-0.95\) & \(-2.79\) & \(-1.84\) & \(-4.16\) & \(-1.24\) & \(-3.58\) & \(-2.36\) & \(-5.17\) \\ T4 & \(-1.43\) & \(-2.27\) & \(-2.16\) & \(-3.52\) & 174.0 & 146.9 & 178 & 171.9 & \(-1.01\) & \(-2.31\) & \(-2.54\) & \(-4.56\) & \(-1.26\) & \(-2.87\) & \(-3.18\) & \(-5.64\) \\ T5 & \(-3.05\) & \(-3.02\) & \(-2.67\) & \(-3.49\) & 205.3 & 162.3 & 121.6 & 150.3 & \(-3.41\) & \(-3.42\) & \(-3.09\) & \(-4.41\) & \(-4.27\) & \(-4.27\) & \(-3.91\) & \(-5.42\) \\ T6 & \(-2.85\) & \(-2.74\) & \(-3.00\) & \(-3.49\) & 161.5 & 141.4 & 159.6 & 170.9 & \(-2.34\) & \(-3.11\) & \(-3.19\) & \(-4.43\) & \(-3.06\) & \(-3.9\) & \(-4.11\) & \(-5.45\) \\ T7 & \(-2.67\) & \(-3.49\) & \(-2.51\) & \(-3.11\) & 151.1 & 136.8 & 127.7 & 143.4 & \(-1.78\) & \(-3.86\) & \(-2.86\) & \(-3.79\) & \(-2.37\) & \(-4.8\) & \(-3.55\) & \(-4.72\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Range of parameters under different factors.  
The damping ratio is the ratio of the dissipated energy and the elastic strain energy in one load cycle. According to the Standard for geotechnical testing method (GB/T 50123-2019) [59], the formula is given as follows:  
\[\lambda=\frac{1}{4\pi}\frac{A_{z}}{A_{s}} \tag{7}\]  
where \(A_{z}\) is the area of the hysteresis curve and \(A_{s}\) is the area of the triangle.