## 3 The Trend of Change in Dynamic Characteristics
### Equivalent Damping Ratio Variation Pattern  
The damping ratio is an important concept in the study of the dynamics of soil structures and expresses the significance of the damping magnitude of a soil structure, reflecting its ability to absorb energy and resist earthquakes [26]. The equivalent damping ratio \(\lambda\) of a soil is the proportion of energy dissipated by the soil under dynamic loads. The formula is as follows:  
\[\lambda=\frac{1}{4\pi}\frac{\Delta W}{W}=\frac{1}{4\pi}\frac{S}{S_{tri}} \tag{7}\]  
where \(W\) is the total energy obtained under each dynamic load, \(\Delta W\) is the energy consumed under each dynamic load [27], \(S\) is the area of the hysteresis curve that indicates the energy consumption of the soil due to the damping ratio, and \(S_{tri}\) is the area of the triangle. Figure 11 shows a typical hysteresis curve. The area enclosed by the hysteresis curve [13] can be approximated as \(\Delta W\) under each dynamic load, and the area of triangle \(OAA^{\prime}\) is \(W\) under each dynamic load [8]. The formula for the total energy stored, W, is the following:  
\[W=\frac{1}{2}\sigma_{\Delta}\epsilon_{\Delta} \tag{8}\]  
where \(\sigma_{\Delta}\) is the difference in value from the origin to the point of maximum dynamic stress in the hysteresis loop, and \(\epsilon_{\Delta}\) is the difference in value from the origin to the point of maximum dynamic strain. Figure 12 shows the stress-strain hysteresis curve at one point in the test where 10 cyclic loads were performed at each stage of the test. The number of loads reaching the deformation protection value was approximately eight to twelve loads each time, and all the maximum deformation values at the end of the test were less than 15%. Figure 13 shows the \(\sigma_{d}-\Delta W\) and \(\sigma_{d}-W\) curves for the soil samples with different sial admixtures at 5% EPS volume fraction. Figure 13 shows that, when the sisal admixture was less than 1.2%, the hysteresis curve area increased continuously with increasing admixture. This was because, at this time, the sisal fibre and soil particles could be well cemented  
Figure 11: Sample of the stress–strain hysteresis curve. Figure 12: Stress–strain hysteresis curves during the test. together, and this enhanced the ability of the soil to resist deformation. The soil energy consumption became large, the SSI doping was greater than 1.2%, the hysteresis curve area was constantly shrinking, and energy consumption became smaller. This indicated that the SSI doping was too large. As a result, the soil particles did not have a good combination of SSI fibre, the soil body was relatively soft, and the consumption of energy decreased. Figure 14 shows the variation in the damping ratio for normal clay and for the EPS-modified lightweight soil with different SSI dosages at 5% EPS volume fraction. Figure 14 shows that, at the beginning of the stress change, i.e., when the stress was less than 40 kPa, the damping ratio of all test groups, except the ordinary clay soil, showed a decreasing trend. This may have been because, under the action of the dynamic load, the internal pores of the soil and EPS particles were first and largely compressed. The overall shape of the modified soil remained almost unchanged at the beginning of the application of stress. The soil deformation under the action of dynamic loading was smaller, and less energy was consumed through the extrusion deformation This resulted in a smaller damping ratio. When the stress was greater than 40 kPa, the damping ratio increased with increasing stress in either test group for the following reasons. For the EPS-modified lightweight soils, EPS responded more to dynamic loads, consumed less energy through elasto-plastic deformation and had a denser internal structural connection. As the dynamic load increased, the soil sample gradually underwent greater elasto-plastic deformation, and the energy loss increased, which macroscopically expressed the increasing damping ratio of the soil.