# Dynamic Characteristics of Post-Cyclic Saturated Loess
## 3 Results
### Damping Ratio of Saturated Loess before and after the PSR Process  
The \(D\) of the soil is reflected as the hysteresis of the \(\sigma_{d}-\varepsilon_{d}\) or \(\tau_{d}-\gamma_{d}\) relationship hysteresis circle performance under the periodic dynamic load. It is caused by the energy consumed by frictional action during soil deformation, reflecting the energy loss resulting in the inner resistance of the soil under cyclic load. Thus, the \(D\) is one of the essential dynamic properties of soil and an essential parameter for the analysis of ground motion. The \(D\) can be obtained by the \(\sigma_{d}-\varepsilon_{d}\) or \(\tau_{d}-\gamma_{d}\) hysteresis curve method, as seen in Figure 10. The \(D\) of soil is obtained by calculating the area \(\Delta W\) and area \(W\) of \(\Delta AOB\) of the area enclosed by the hysteresis curve, and then calculated in accordance with the Equation (9). \[D=\frac{1}{4\pi}\frac{\Delta W}{W} \tag{9}\]  
Figure 10: Calculation method of the damping ratio. Figure 9: Curves of the dynamic shear modulus ratio versus dynamic shear strain of the saturated loess under different test conditions. Based on Figure 10 and Equation (9), the \(D\) of the saturated loess is determined. The curves of the \(D\) versus the dynamic strain are drawn, as shown in Figure 11. According to the figure, the \(D\) of the saturated loess under different test conditions has a tendency to grow with an increase in the \(\varepsilon_{d}\), and the \(D-\varepsilon_{d}\) curve is steep when the \(\varepsilon_{d}\) is small; the curves gradually tend to flatten after a certain dynamic strain. The \(D\) of the saturated loess after pre-shock tends to increase more obviously with an increase in the \(\varepsilon_{d}\). The inflection point of the curve corresponds to a larger dynamic strain. Combined with the definition of the \(D\), the energy loss caused by frictional action during the soil deformation after the pre-shock is enhanced. The deformation hysteresis effect is more obvious, and its deformation stability tends to be weakened under the dynamic load. In addition, Figure 11 also shows the \(D-\varepsilon_{d}\) curves of the post-seismic saturated loess with a different OCR. The impact of the OCR on the trend of the \(D-\varepsilon_{d}\) curves is more obvious. The \(\varepsilon_{d}\) corresponding to the inflection point of the \(D-\varepsilon_{d}\) curves decreases continuously with an increase in the OCR. When the OCR reaches two, the \(D\) of the post-seismic saturated loess at the same level of \(\varepsilon_{d}\) is lower than that of the original saturated loess, indicating that the deformation capability of the loess under cyclic load tends to be weakened after the over-consolidation effect. The ratio of the \(D\) to the \(\gamma_{d}\), \(\gamma_{d}/D\), and the \(\gamma_{d}\), has a linear relationship as shown in Equation (10):  
\[\frac{\gamma_{d}}{D}=A+B\gamma_{d} \tag{10}\]  
where \(A\) and \(B\) are the fitting parameters. Equation (9) could be adapted as:  
\[D=\frac{\gamma_{d}}{A+B\gamma_{d}} \tag{11}\]  
Based on Figure 11, the normalized curves of the \(D\) versus the \(\gamma_{d}\) of the saturated loess before and after the PSR process are calculated, as shown in Figure 12. According to Figure 12, the \(D-\gamma_{d}\) curves of the saturated loess under different test conditions show a characteristic of moderate growth--sharp growth--moderate growth: the shape of the curve is like a capital \"S\". Under the same consolidation conditions, the growth trends  
Figure 11: Curves of the damping ratio versus the dynamic strain of the saturated loess under different test conditions. of the \(D-\gamma_{d}\) curves before and after pre-shock are different. However, the difference is mainly manifested in the large shear strain stage (\(\gamma_{d}\) > 2 \(\times\) 10\({}^{-4}\)). When the \(\gamma_{d}\) is 0.0005, 0.001, 0.005, and 0.01, the \(D\) of the pre-seismic saturated loess increases by 15.76%, 20.24%, 29.89%, and 31.48%, respectively, compared with that of the original saturated loess.