The equipment (Figure 3) required for impact tests in this study includes a block ejection system and a slope angle control system. The former system is developed to include a flipping barrel, electric tension meter, limiting rope, slideway, slider, pulling rope, electromagnetic lock, base, and hydraulic support. The block is installed in the groove at the top of the steel barrel through a hexagonal axis. The incident velocity of the block can be controlled by applying different tensile forces to the slider, and the rotational speed of the block is applied using an adjustable speed motor through the hexagonal axis. The slider is released using an infrared remote control when the applied tension and rotational speed meet the preset values, and the block is released when the restraining rope connecting the front end of the barrel bottom and rear end of the slideway is tightened. A simple slope angle control system is assembled using a rock plate, two steel plates, a restraining rope and a hinge. The slope angle is controlled by adjusting the length of the restraining rope.

According to the statistical results presented in Table 1 and classified statistical results of rockfall shapes in alpine gorge areas (Fityus et al., 2013), cubes, cuboids, cylinders and discs were selected as the shapes of the blocks used in this experiment (Figure 4). The blocks had the same volume and were made of natural limestone. The material of the slope was the same as that of the blocks. The slope was prepared through infrared fine cutting, and the surface was extremely flat (the friction coefficient was 0.45). Detailed parameters of the samples are given in Table 2. The physical and mechanical properties of the material, presented in Table 2, were determined according to ISRM recommended methods (ISRM, 2007).

In the experiment, the four blocks with different shapes rotated reversely around the X and Y axes (based on the occurrence conditions of RR, the impact test between the cuboid-shaped block rotating around the Y-axis and slope surface was also performed) at 100–500 revolutions per minute (RPM). The incident speed was set as 3.5 m/s, the impact angle was set as 45°, and every test was repeated fivefold. Details of the experiment are presented in Table 3.

A high-speed camera (Model Phantom 710, 1280 × 800 pixels, 7,400 frames per second complete resolution) with a 1,000 fps operating speed was used to capture the impacts and bounces of the blocks. The velocities before and after impact were determined using the image-tracking program Tracker V2.0 and the abovementioned series of images. During the tests, the block was initially sprayed with a colour that was significantly different from the background, and thus, its pixel value in the image was distinguished from other background colours. After the tracking object (test block) was circled in the batch images, the software (Tracker V2.0) automatically tracked it according to the pixel value of the block colour and calculated the velocity of the block before and after the impact through the displacement of the centre point of the test block (automatically locked by the program) in two or several adjacent images and the elapsed time. The rebound angle of the block after the impact was determined using the software’s own measurement tool based on its post-impact trajectory. Eqs 9, 10 were used to calculate the RC values, which correspond to the definitions most commonly employed in the literature (Giani et al., 2004; Wyllie, 2014; Asteriou and Tsiambaos, 2016) as R n = V n ′ V n (9) and R t = V t ′ V t (10) where V n ′ and V n are the normal components, and V t ′ and V t are the tangential components of the block velocity after and before the impact, respectively.

(1) Influence of the RRS on R _n

As shown in Figure 5, the R _n values of the four blocks with different shapes did not exhibit a certain trend with the increase in the RRS, and the R _n values were not sensitive to the changes in the RRS. This phenomenon is consistent with the conclusions derived by Spadari et al. (2012), although their study was aimed at testing the influence of different forward rotation speeds on the R _n of the block. In different RRS conditions, the R _n value distributions of the blocks were relatively discrete, and the differences in the values were large. The narrowest value ranges of R _n for disc, cylinder, cuboid, cuboid (Y) and cube-shaped blocks were 0.036–0.275, 0.111–0.172, 0.039–0.132, 0.21–0.47 and 0.095–0.3, respectively, and the maximum value ranges were 0.085–0.56, 0.065–0.41, 0.024–0.29, 0.049–0.41 and 0.03–0.51, respectively. Buzzi et al. (2012) used the average value of five test results to analyse the effect of the forward rotation speed on the RC to alleviate the influence of the random error. However, this setting is inappropriate because it can lead to the loss of important information and is not conducive to the accurate prediction of the rockfall motion path. Therefore, the results of each test must be treated with caution.

Although R _n does not vary with the RRS in Figure 5, the RRS may still influence R _n . In theory, the change in the RRS influences the contact velocity of the block, which is directly related to the normal impact force, and thus, the RRS influences R _n . The preceding results just demonstrate that the RRS is not the only factor affecting R _n , and the RRS does not play a dominant role in R _n . In other words, to establish an effective relationship between the RRS and R _n , other factors must be comprehensively considered.

According to the comparison of the R _n values of four blocks with different shapes, the R _n values of cube- and cuboid (Y)-shaped blocks are higher, while the R _n values of the disc-, cylinder- and cuboid-shaped blocks are lower. This phenomenon occurs because the α ranges of cube and cuboid (Y)-shaped blocks are naturally smaller than those of the disc, cylinder, and cuboid-shaped blocks, which leads to a higher probability of normal or nearly normal impact between the cube and cuboid (Y)-shaped blocks and the slope in multiple tests. This finding highlights the influence of the block shape on R _n , and thus, this factor cannot be ignored when examining the influence of RRS on R _n . In addition, the contact attitudes of the same block are different even under the same RRS, and the influence mechanisms of the impact force on R _n are also considerably different under various contact attitudes. Therefore, it may be inappropriate to summarize the mechanisms with a simple and general rule, and classification and refinement may help solve this problem.

(2) Influence of the RRS on R _t

The relationships between R _t and RRS of four differently shaped blocks are shown in Figure 6. R _t is independent of RRS. The R _t value distributions of blocks of different shapes under various RRSs are relatively concentrated, exhibiting similar distribution ranges. The R _t value ranges of the disc, cylinder, cuboid, cuboid (Y) and cube-shaped blocks are 0.79–1.00, 0.80–1.06, 0.77–1.00, 0.78–1.00 and 0.86–0.99, respectively. This finding is more consistent with the conclusions derived by Asteriou (2019) and Spadari et al. (2012) in tests pertaining to the effect of the forward rotation speed on the R _t value of blocks having different shapes. Compared with the normal impulse, the tangential impulse is extremely small, while the RRS directly affects the normal impact force, which leads to an extremely small change in the tangential impulse in an extremely small impact time Consequently, the effect of the RRS on R _t is not significant. Moreover, under this test condition, the contact point of the block has a forward movement speed relative to the slope surface, and the direction of the frictional force is backwards, which decreases the tangential velocity such that the overall R _t values are less than 1. However, in the test, the block may rotate forward after the first impact and later contact the slope surface. At this time, the direction of the frictional force may be consistent with that of the tangential velocity, and the friction force may increase the tangential velocity such that a few R _t values of the blocks are greater than 1.

The effect of the RRS on R _t has been essentially identified based on the analysis of the test results. However, the effect of the RRS on R _n is unclear due to the diversity of contact attitudes of the blocks and sophisticated mechanical mechanism. Notably, this problem is not impossible to solve. The findings of the preceding study provide certain information and tools that may aid in the resolution of this problem, such as categorization and discussion based on various mechanical mechanisms and joint examination of the effects of block shape and contact attitudes. These aspects are thoroughly considered and adopted, as described in the following sections.

The impact-bounce behaviours of the blocks may be divided into two categories based on the observations of the impact-bounce processes of the blocks in each test: rebound following a single impact [I(1)] and rebound following two consecutive impacts [I(2)]. Buzzi et al. (2012) also observed these two circumstances and categorized them. In addition, according to the preliminary theoretical analysis, the rebound behaviours of the block with the initial RRS are closely related to the impulse moment equation. The position of the normal impact force relative to the mass centre (C) is different, and the effect of the impulse moment of the normal impulse of the contact point to the mass centre on the motion of the block after impact is different. Considering this aspect, I(1) can be subdivided into two subtypes, depending on whether the MC is in front of [F(1)] (Figure 7A) or behind the CP [B(1)] (Figure 7B). Similarly, I(2) may be subdivided into two subtypes, depending on whether the MC is in front of or behind the CP at the initial impact [F(2)/B(2)] (Figures 7C, D).

Although the impact-bounce behaviours of the block are categorized, the impact dynamic equation cannot be directly used to analyse the effect of the RRS on R _n based on this aspect. According to the findings in Section 3.2.1, the influencing variable must be re-determined. Considering the shape features of the block, RRS and contact posture, a new integrated variable, namely, the impact coefficient (I _c ) is proposed to evaluate the effect of the RRS on R _n , expressed as. I c = ω ⋅ d (11) where ɷ represents the angular velocity of the block (Figure 8) (rad/s), and d is the distance from the CP to the projection of the MC, reflecting the characteristics of the block shape and contact posture (m). The unit of I _c is m/s, which is essentially the change in the normal velocity of the CP relative to the slope caused by the rotation of the block. Changing the normal impact velocity of the CP leads to a change in the normal impact force, thereby influencing R _n .

Using the measurement module in the software tracker V2.0, the ω and d values of each test are measured, and the I _c values are calculated using Eq. 11. The relationships between the R _n values and I _c of the blocks with different impact-bounce behaviours are shown in Figures 9, 10.

(1) Influence of I _c on the Rn of blocks with I(1)-type impact-bounce behaviours

The relationships between I _c and R _n of blocks with I(1)-type impact-bounce behaviours are shown in Figure 9. Regardless of the F(1) or B(1) conditions, the R _n values of the four blocks with different shapes are more sensitive to changes in the I _c values. By fitting the data, under condition F(1), the R _n values of the disc, cylinder and cuboid-shaped blocks exhibit strong linear positive correlations with the I _c values (the data of cuboid (Y) and cube-shaped blocks are not collected under this condition), and the R ² values are 0.98, 0.96 and 0.94, respectively. Under this scenario, I _c contributes to an increase in the normal impact velocity (V _n ). As shown in Figure 8A, as I _c increases, the normal impact velocity of the CP relative to the slope surface and normal impact force increase, resulting in an increasing R _n . Under condition B(1), the R _n values of the disc, cylinder, cuboid and cube-shaped blocks exhibit strong linear negative correlations with the I _c values, and the R ² values are 0.86, 0.87, 0.96, 0.997 and 0.98, respectively. Under this scenario, I _c tends to decrease the normal impact velocity (Figure 8B). As I _c increases, the normal impact velocity of the CP relative to the slope surface decreases, thereby decreasing the normal impact force and R _n .

The linear correlation between I _c and R _n , as shown in Figure 9, established by straightforward data fitting, can be supported by the corresponding mechanical theory. According to the relevant theory of impact dynamics (Meyers, 1994), during impact, the normal impact force is approximately proportional to the normal impact velocity when the same material deforms in the linear elastic range. In this experiment, the mass centre of the blocks has the same initial velocity in each test, and a linear relationship must exist between the change in the initial normal impact velocity (I _c ) and change in the normal impact force. Because the value of R _n is directly determined by the normal impact force, the correlation between I _c and R _n is expected to be linear.

(2) Influence of I _c on the R _n of blocks with I(2)-type impact-bounce behaviours

When the included angle (α) between line L and the slope surface is smaller than a certain critical value at the moment of impact (Figure 11), the block may rebound after two consecutive impacts on the slope, which is essentially the result of the combined action of an impact in which the MC is in front of the CP and that in which the MC is behind the CP. As shown in Figure 10, for blocks that exhibit I(2)-type impact-bounce behaviours, the R _n values of four differently shaped blocks under conditions F(2) and B(2) increase with increasing I _c . By fitting the data, under the F(2) condition, the R _n values of the disc, cylinder, cuboid, cuboid (Y) and cube-shaped blocks exhibit power, logarithmic and linear functional relationships with I _c, respectively, and the R ² values are 0.90, 0.96, 0.93, 0.79 and 0.78, respectively, corresponding to a strong correlation. Under condition B(2), the R _n values of the disc, cylinder, cuboid, cuboid (Y) and cube-shaped blocks exhibit exponential, linear and power functional relationship with I _c , and the R ² values are 0.95, 0.96, 0.89, 0.95 and 0.85, respectively.

For the blocks under condition B(2) (Figure 11A), the restitution coefficients (R _n1 , which is equal to V _n1 /V _n , with V _n1 being the normal velocity of the MC after the first impact or before the second impact) after the first impact with the slope decrease with the increase in I _c (the influencing mechanism is similar to that under B(1)). However, according to the observation of the impact processes of the blocks under this condition, the intensity of the first impact is significantly less than that of the second impact (Figure 12), and the restitution coefficients (R _n2 , which is equal to V _n ′/V _n1 ) after the second impact dominantly influence the magnitude of R _n . After the first impact of the blocks on the slope, the RRSs of the blocks increase [as shown in Figure 11A (1), the normal impact force increases the RRS of the block], that is, ω ₁ >ω (ω ₁ is the angular velocity of the block following the first impact or before the second impact). Moreover, the α ₁ values at the second impact are slightly smaller than the α values at the first impact (Figure 12), that is, d ₁ >d [d ₁ is the distance from the CP to the projection of the MC during the second impact, as shown in Figure 11A (2)]. In this context, as I _c increases, ω ₁ and d ₁ during the second impact increase, and R _n2 increases [the influencing mechanism is similar to that under F(1)], owing to which, R _n increases.

For the blocks under condition F(2), after the first impact with the slope surface, the R _n1 values increase with increasing I _c [as shown in Figure 11B (1), and the influencing mechanism is similar to that under F(1)]. However, the relationship between I _c and R _n cannot be determined based on only this aspect. R _n eventually increasing or decreasing with the I _c also needs to consider the relationship between the R _n2 and I _c after the second impact. As the tangential velocities of the blocks change slightly before and after impact, the tangential velocities can be considered to be nearly unchanged. The energy conservation equation and momentum equation before and after the first impact can be defined as follows: 1 2 m V n 2 + 1 2 J ω 2 = 1 2 m V n 1 2 + 1 2 J ω 1 2 + Δ E (12) F N 1 ⋅ d ⋅ t = J ω 1 + ω (13) F N 1 ⋅ t = m V n − V n 1 (14) where m is the mass of the block; J is the moment of inertia of the block around the centre of mass; ΔE is the energy loss; t is the impact time; the rotation speed is positive in the clockwise direction; the directions of V _n1 and V _n are the same (both positive and downwards), consistent with the phenomenon observed in the experiment. As shown in Figure 13, the α value of each block is less than α ₁ .②

The results of simultaneous Eqs 12–14 are V n + V n 1 d = J ω 1 − ω ω 1 + ω + Δ E J ω 1 + ω = ① Δ E ⏞ J ω 1 + ω + ② ω 1 − ω ⏞ (15)

The value of V n + V n 1 d in Eq. 15 is positive, and the energy loss (ΔE) of the block in the impact process is extremely small relative to the total energy. Therefore, term ① can be ignored in the simplified analysis, and from this, it can be deduced that ω ₁ and ω are equivalent or ω ₁ is greater than ω. In this context, ω ₁ increases with the increase in ω, and d is less than d ₁ under this working condition (Figure 13). Therefore, R _n2 increases with the increase in ωd (the influencing mechanism is similar to that of F(1)). Taking into account the relationship between I _c and R _n1 determined in the aforementioned analysis, it can be considered that R _n = R _n1 · R _n2 is also positively correlated with I _c .

In addition, Figure 10 shows that the R _n values of the blocks under condition F(2) are generally greater than those of the blocks under condition B(2). This phenomenon occurs because I _c increases the normal velocity of the contact point in the two impacts of the block under condition F(2). In contrast, under condition B(2), although I _c affects the normal velocity of the contact point during the second impact of the block, I _c decreases the normal velocity of the contact point in the first impact. Consequently, the increase in R _n under this condition is smaller than that under condition F(2).

In this manner, the influence mechanism of RRS on R _n can be clarified according to the analysis of the influence mechanism of I _c on R _n . For blocks with I(1)-type impact-bounce behaviours, when the block shape and contact attitude are fixed, the RRS and R _n are linearly positively correlated and negatively correlated under working conditions F(1) and B(1), respectively. For blocks with I(2)-type impact-bounce behaviours, under working conditions B(2) or F(2), when the block shape and contact attitudes are fixed, R _n is positively correlated with the RRS. Nevertheless, maintaining the contact attitudes of the block consistent in every laboratory or outdoor test is challenging, and the varied contact attitudes of the irregular rockfall are an essential characteristic of the rockfall event. Therefore, investigating the influence of only the RRS on R _n is nearly impossible and meaningless, and it is more feasible to clarify the influence of the RRS on R _n in terms of I _c .