## 3 Results
### Results of Cyclic Tests  
The results of the experiments demonstrate the effects of particle shape changes on the damping ratio of sand. The hysteresis loops of shear stress-strain were plotted after each cyclic test, and the damping ratio was estimated using Equation (4). Figure 7 illustrates a typical example of the results obtained during cyclic tests, where the CSR and vertical stresses were 0.3 and 150 kPa, respectively. The results demonstrate that the volumetric strain exhibits a negative trend with an increasing number of cycles. This indicates that the sample experienced settlement and, hence, its density increased after cycling; the constant vertical pressure loading in the cyclic tests is the primary reason for this phenomenon. Figure 6: Hysteresis loop in cyclic tests. Figure 7: Results of the cyclic tests for CSR = 0.3 and vertical stress = 150 kPa: **(a)** shear stress–strain hysteresis loops; and **(b)** volumetric strain by number of cycles. To find the worst-case scenario in this study, it is evident from the results in Section 3.1 that the behavior of dry sand grains is greatly influenced by three parameters: the CSR, number of cycles and vertical stress. The damping ratio decreases with an increase in the cyclic shear ratio (CSR) and the number of cycles. For the studied sand, the worst-case scenario could be when the CSR is 0.5, the number of cycles is around 35 and the vertical stress is 250 kPa. In this vertical pressure and CSR, particles showed the most change in their shapes and damping ratios. In Figures 11 and 12, three-dimensional images, two-dimensional cross-sectional areas and circles used to determine the values of three shape parameters, \(S\), \(R\) and \(\rho\), are displayed for two different particles. The first particle was examined before cyclic testing, while the second particle was examined after undergoing four cyclic tests, each consisting of 30 loading cycles with CSR = 0.2 and vertical stress = 150 kPa. It is evident from the figures  
paring the values between these tables allows one to draw some conclusions on how these shape descriptors and the damping ratio are affected by cycling. Referring to Table 1, it can be seen that the mean values for \(S\), \(R\), \(\rho\) and \(D\) before cycling were 0.713, 0.518, 0.616 and 16.050, respectively. The SDs for \(S\), \(R\) and \(\rho\) were relatively small (i.e., 0.014, 0.011 and 0.011, respectively), indicating that the samples were fairly consistent in terms of particle shape characteristics/variations. For \(D\), however, the SD was found to be 5.903, suggesting greater variability in the damping ratio. Considering Table 2, it can be seen that after 30 cycles, the mean values for \(S\), \(R\), \(\rho\) and \(D\) were 0.760, 0.557, 0.658 and 13.392, respectively. The mean values for the three shape descriptors were higher compared to those reported in Table 1 (before cycling), suggesting that the particles became more spherical, rounder and more regular (in shape) after 30 cycles. Furthermore, the damping ratio decreased compared to its pre-cycling value, indicating that the samples became less stiff and more flexible over the course of cycling. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Minimum** & **Maximum** & **Mean** & **SD** \\ \hline \(S\) (-) & 0.690 & 0.743 & 0.713 & 0.014 \\ \(R\) (-) & 0.498 & 0.534 & 0.518 & 0.011 \\ \(\rho\) (-) & 0.594 & 0.638 & 0.616 & 0.011 \\ \(D\) (\%) & 8.500 & 25.500 & 16.050 & 5.903 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Shape descriptors and damping ratio before cycling. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Minimum** & **Maximum** & **Mean** & **SD** \\ \hline \(S\) (-) & 0.733 & 0.793 & 0.760 & 0.017 \\ \(R\) (-) & 0.529 & 0.572 & 0.557 & 0.014 \\ \(\rho\) (  
Finally, after 120 cycles, it can be seen that the mean values for \(S\), \(R\), \(\rho\) and \(D\) were 0.774, 0.570, 0.672 and 12.425, respectively (see Table 3).