## 3 Results and Discussion
### Damping Ratio  
According to the conventional definition [26], the damping ratio is related to the energy loss rate in a certain time, and the formula can be expressed as:  
\[y=\alpha\sigma_{3}^{2}+\beta\sigma_{3}+\gamma \tag{7}\]  
where \(\lambda\) is the damping ratio, \(\Delta W\) is the loss energy within a loading-unloading cycle, and \(W\) is the total energy of a complete loading-unloading cycle. In this study, the damping ratio is calculated by the energy loss rate in a loading-unloading cycle. The typical stress-strain relationship of the tested specimen can be illustrated in Figure 10.  
\begin{table} \begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{\(\mathbf{a_{w}}\) = 14\%, \(\mathbf{\sigma_{3}}\) = 50 kPa} & \multicolumn{3}{c}{\(\mathbf{a_{w}}\) = 14\%, \(\mathbf{\sigma_{3}}\) = 150 kPa} \\ \cline{2-9} \(\mathbf{e_{a}\)/\%} & \multicolumn{2}{c}{**Strength/kPa**} & \multicolumn{2}{c}{\(\mathbf{\varepsilon_{h}}\)/\%} & \multicolumn{2}{c}{**Strength/kPa**} & \multicolumn{2}{c}{\(\mathbf{\varepsilon_{h}}\)/\%} \\ \cline{2-9}  & **LSES** & **LCES** & **LSES** & **LCES** & **LSES** & **LCES** & **LSES** & **LCES** \\ \hline
2 & 26.71 & 22.00 & 0.0528 & 0.0537 & 70.77 & 64.71 & 0.0809 & 0.0964 \\
3 & 30.03 & 26.11 & 0.0696 & 0.0707 & 74.26 & 68.27 & 0.0893 & 0.1286 \\
4 & 31.73 & 30.41 & 0.0801 & 0.0926 & 78.36 & 72.33 & 0.1005 & 0.1507 \\
6 & 32.03 & 31.91 & 0.0911 & 0.0926 & 82.93 & 76.16 & 0.1526 & 0.1806 \\
8 & 33.17 & 33.09 & 0.1254 & 0.1258 & 86.15 & 80.38 & 0.1718 & 0.2006 \\
10 & 34.74 & 33.95 & 0.1600 & 0.1370 & 90.86 & 84.83 & 0.1925 & 0.2177 \\
12 & 38.58 & 36.47 & 0.1845 & 0.1535 & 94.42 & 88.04 & 0.2319 & 0.2445 \\
14 & 41.90 & 39.54 & 0.2116 & 0.1796 & 98.31 & 92.79 & 0.2432 & 0.2641 \\
16 & 44.69 & 42.30 & 0.2431 & 0.1956 & 102.84 & 96.44 & 0.2792 & 0.2844 \\ \hline \hline \end{tabular} 
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{\(\mathbf{a_{w}}\) = 20\%, \(\mathbf{\sigma_{3}}\) = 50 kPa} & \multicolumn{3}{c}{\(\mathbf{a_{w}}\) = 20\%, \(\mathbf{\sigma_{3}}\) = 150 kPa} \\ \cline{2-9} \(\mathbf{\varepsilon_{a}\)/\%} & \multicolumn{2}{c}{**Strength/kPa**} & \multicolumn{2}{c}{\(\mathbf{\varepsilon_{h}}\)/\%} & \multicolumn{2}{c}{**Strength/kPa**} & \multicolumn{2}{c}{\(\mathbf{\varepsilon_{h}}\)/\%} \\ \cline{2-9}  & **LSES** & **LCES** & **LSES** & **LCES** & **LSES** & **LCES** & **LSES** & **LCES** \\ \hline
2 & 45.90 & 35.41 & 0.0517 & 0.0521 & 100.23 & 83.75 & 0.0803 & 0.0893 \\
3 & 50.29 & 42.30 & 0.0665 & 0.0670 & 107.24 & 92.63 & 0.0882 & 0.1223 \\
4 & 53.49 & 49.19 & 0.0774 &In a complete loading-unloading cycle, the work that is done by extra loads can be defined as: at the beginning OA section, most of the work done by external load is converted into elastic potential energy, indicating that the specimen's deformation is mainly attributed to the elastic deformation. Subsequently, in the second AB section, plastic deformation is produced in the specimen and it becomes larger and larger. Although, the work done by external load on the specimen is mainly consumed by the plastic deformation and the viscous resistance, there will still be a small part stored as the elastic potential energy. Within the BC section, when the external load is removed, the elastic potential energy that is stored in sections of OA and AB will be gradually released and absolutely consumed by viscous resistance. According to the general form of stress-strain relationship and energy composition analyses, the EPS beads-mixed lightweight soil belongs to the viscoelastic plastic material. The \(\Delta W\) and \(W\) in Equation (7) can be calculated by the following expression:  
\[\Delta W=S_{0 DA}+S_{AEB}+S_{BFC}+S_{AGIH} \tag{8}\]  
\[W=S_{0 DAEBFC0} \tag{9}\]  
where \(S_{0 DA}\), \(S_{AEB}\), \(S_{BFC}\), \(S_{AGIH}\), and \(S_{0 DAEBFC0}\) are the areas of different regions in Figure 10. In Equation (8), the former three items are the energy consumed by damping, the last one is the energy consumed by plastic deformation.