## 2 Numerical Investigation on the Correlation of the COR and Damping Parameter
### Verification of the Numerical Model  
A field test [38] was numerically modelled to investigate the correspondence between damping and COR, and its numerical simulation [20] was used to for comparison. In this field test (Figure 4a), nine selected cylindrical blocks with almost identical shapes and masses, a density of 2630 kg/m\({}^{3}\), an average diameter of 14 cm, and an average length of approximately 11 cm were considered. A slope with uneven surfaces and three plateaus s selected. The slope surface is composed of exposed weathered mudstones and accumulated gravel layers. The nine blocks are freely released one by one under same conditions with only slightly random variations in initial orientation and position at the top of the slope. The rockfall trajectories, horizontal run-out distances, lateral offsets, and arrival time were recorded for nine single runs of each block in the test [20,38]. Figure 4a shows the randomness of nine rockfall trajectories under nearly identical release conditions. Cylindrical blocks and flat slopes were generated in numerical simulations on the basis of the actual size and shape of blocks determined from field tests. According to the existing case [9], the normal and shear joint stiffnesses of the rockfall and slope were set to be \(k_{\mathrm{n}}=4.5\times 10^{6}\) Pa/m and \(k_{\mathrm{s}}=3.0\times 10^{6}\) Pa/m, respectively, on the basis of trial and error to match the ratio of \(k_{\mathrm{n}}/k_{\mathrm{s}}=2/3\) suggested by [PERSON] and [PERSON] [21]. The friction angle of the block was set to \(35^{\circ}\) according to a field test [38]. When the optimal contact stiffness was obtained, there was no block penetration or abnormal block movement during the rockfall simulation. The 3 DEC manual states that problems involving free fall and the bouncing of blocks on fixed bases require accurate modeling using the COR [35]. The COR is typically used to define the velocity change before and after a collision in a rockfall movement (Figure 5):  
\[R_{\mathrm{n}} =V_{\mathrm{n}}^{\prime}/V_{\mathrm{n}} \tag{5}\] \[R_{\mathrm{t}} =V_{\mathrm{t}}^{\prime}/V_{\mathrm{t}} \tag{6}\]  
where \(V_{\mathrm{n}}\) and \(V_{\mathrm{n}}^{\prime}\) are the normal components of the block velocity before and after a collision, respectively. \(V_{\mathrm{t}}\) and \(V_{\mathrm{t}}^{\prime}\) are the tangential components of the block velocity before  
Figure 4: Comparisons of trajectories between test and numerical predictions. (**a**) Test observation [38]; (**b**) previously numerical prediction [20]; (**c**) numerical prediction in this study. and after a collision, respectively. Table 1 lists the COR's field test results and theoretical calculations for various slope materials. Considering the rock mass case detailed in subsequent chapters, a typical COR \(R_{\text{n}}=0.35\) was selected as the simulation benchmark. As shown in Figure 5, a falling rock is released at a certain height above the slope to allow it to fall freely. Only the velocity in the first collision between the rockfall and the slope are considered to prevent the influence of rockfall rotation on the velocity component. Then, the normal and tangential CORs along the slope are calculated from the velocity component. After repeated tests, when the Rayleigh damping parameters \(f_{\text{min}}=0.015\) and \(\xi_{\text{min}}=0.95\), the normal COR \(R_{\text{n}}=0.35\) and the tangential COR \(R_{\text{t}}=0.521\) are obtained, which conform to the principles of rockfall movement. In this model, the initial direction and position of the rockfall at the top of slope slightly change, and nine rockfall trajectories are obtained. Figure 4b is the previous numerical prediction by [PERSON] [20], and Figure 4c shows the numerical prediction with the above parameters. The proposed approach showed a more suitable for the parabolic trajectory of the field test compared with that of [PERSON]'s model, which had large energy dissipation in the second plateau, so that the trajectory shows poor agreement with the field test.