Effects of the impact angle on the coefficient of restitution based on a medium-scale laboratory test

The reliability of a computer program simulating rockfall trajectory depends on the ascertainment of reasonable values for the coefficients of restitution, which typically vary with the kinematic parameters and terrain conditions. The effects of the impact angle with respect to the slope on the coefficients of restitution have been identified and studied using 10 small scale laboratory tests. To investigate whether the existing conclusion based on small scale laboratory tests is valid when the test scale changes and the role of rotation in the effect of the impact angle on the coefficients of restitution, this study performed a medium-scale laboratory test using spherical limestone polyhedrons impacting concrete slabs. Free fall test are conducted and the velocities before and after the impact are obtained by a 3D motion capture system. The results comparison between our test and the existing small scale tests verified that several general laws occur when accounting for 15 the effect of the impact angle, regardless of the test scales and conditions. Increasing the impact angle will induce reductions in the normal coefficient of restitution Rn, the kinematic coefficient of restitution Rv and the kinetic energy coefficient of restitution RE, whereas it will lead to increases in the tangential coefficient of restitution Rt. The rotation plays an important role in the effect of the impact angle. A higher percentage of kinetic energy converted to rotational energy always induces a higher normal coefficient of restitution Rn and a lower tangential coefficient of restitution Rt. As the impact angle decreases, 20 the ratio between the rebound angle β and the impact angle α increases, and the percentage of kinetic energy dissipated in rotation during the collision becomes higher. Considering that the effect of block shape and the detailed impact orientations are not involved in the present study, the test results is valid for trajectory simulation codes based on a lumped-mass model, and can be referenced in the trajectory predication of spherical rocks impacting hard surface using a rigid body model.


Introduction 25
In mountain areas, rockfall is a frequent natural disaster that endangers human lives and infrastructure. Numerous examples of fatalities or infrastructure damage due to rockfall have been reported (Guzzetti, 2003;Pappalardo, 2014). Various protective measures, such as barrier fences, cable nets and rockfall shelters, have been widely used to reduce rockfall hazards.
To ensure the efficiency of mitigation techniques, the motion trajectory of the rockfall must be estimated. The trajectory can 2 provide important information, such as the travel distances of possible rockfall events, the bouncing height and kinetic energy level of the rockfall at various positions along the slope.
Numerous algorithms have been developed to solve this problem, and the progress up to the end of the last century has been summarized by Dorren (2003) and Heidenreich (2004). Due to these efforts, computer simulation codes, such as RockFall (Stevens, 1998), CRSP (Jones et.al, 2000) and Stone (Guzzetti et.al, 2002), RAMMS::Rockfall (Christen et al, 2007), 5 Rockyfor3D (Dorren, 2010) and Pierre (Valentin et al., 2015;Andrew and Oldrich, 2017), are developed to acquire motion information for rockfall. A main feature that allows one to distinguish between different rockfall trajectory codes is the representation of the objective rock. The first approach, a lumped-mass model, treats the rock as a single and dimensionless point, and assigns all of the properties of the rock to that point. The second one is a rigid body model, which considers the rock as a body with its own shape and volume, and accounts for all types of block movement, including rotation. Finally, a 10 hybrid model adopts a lumped mass model to calculate the free fall of the rock, and simulates other types of block motion using a rigid body model. In most codes the trajectory of falling rocks was described as combinations of four types of motion: free fall, rolling, sliding and rebound. The rebound motion, a succession of rockfalls impacting the slope surface, is the least understand and the most difficult to predict of the four types of motion (Volkwein A et.al, 2011), which is controlled by the coefficients of restitution in computer simulation. Thus, the reliability of the estimation of the coefficient of restitution must 15 be ensured.

Definition for the coefficient of restitution
The coefficient of restitution is a dimensionless value representing the ratio of velocities or energies of a boulder before and after it impacts the slope. Various definitions for the coefficient of restitution have been proposed in previous studies, but no consensus was reached on which definition is more appropriate for rockfall prediction. As shown in Fig. 1, when one boulder 20 impacts the slope surface, the impact velocity vi can be resolved into a normal component vni and a tangential component vti according to the slope angle θ. Then, the boulder leaves the surface with a rebound velocity vr, which similarly has a vnr and a vtr. The angular velocities of the boulder before and after impact are denoted as ωi and ωr, respectively. The impact angle α and rebound angle β are drawn in Fig. 1. (1) 5 Another common definition is the kinematic coefficient of restitution, Rv, representing the ratio between the magnitudes of the rebound and impact velocities: This definition originated from Newton's theory of particle collision, and had been used by Habib (1976), Paronuzzi (1989) and other scholars. When Rv is used in the trajectory predication, an assumption regarding the rebound direction is necessary 10 to fully determine the velocity vector after impact.
In addition, the ratio of kinetic energies before and after impact is used to define the kinetic energy coefficient of restitution RE, which is written as in which Ei and Er are the kinetic energy before and after the impact, respectively. Eir and Err are the rotational energy before 15 and after the impact; Eit and Ert denote the translational energy before and after the impact.
Here, m is the mass, I is the moment of inertia. RE can reflect the kinetic energy loss caused by the impact, and had been used by Bozzolo and Pamini (1986), Azzoni et al. (1995) and Chau et al. (2002). 20 In these definitions, Rn and Rt get more popularity in engineering practice for the simplicity in computer simulation software.
Rn and Rt are used conjointly and characterize the variation in the tangential and normal components of the boulder velocity, respectively. Given an impact velocity, the rebound velocity and direction can be completely determined using this definition without any further assumption. So, Rn and Rt attracted most attentions in the previous studies, and some typical values of Rn and Rt had been summarized (Agliardi and Crosta, 2003;Heidenreich, 2004;Scioldo, 2006).

Previous studies on the effects of the impact angle on the coefficient of restitution
Various techniques, such as laboratory tests (Buzzi et.al, 2012;Asteriou et.al, 2012), field tests (Dorren et.al, 2006;Spadari et al. 2012), back analysis of field evidence (Paronuzzi, 2009) and theoretical estimation (He et.al, 2008), have been used to 5 determine the coefficient of restitution. Variations in the impact conditions, e.g., the material properties of both the rocks and slopes (Wu, 1985;Fornaro et.al, 1990;Robotham et.al, 1995;Richards et.al, 2001;Chau et.al, 2002;Asteriou et.al, 2012), the shape of the rocks (Chau et.al, 1999;Buzzi et.al, 2012), the roughness of the slope surface (Giani et.al, 2004) and the impact angle, influence the coefficient of restitution considerably. While, in those existing summaries for typical values, the coefficients of restitution were determined mainly accounting for the terrain conditions. 10 The impact angle, the angle between the directions of the impact velocity and the slope segment, is a kinematic parameter of the falling rock, indicating only that the terrain conditions involved in estimating the value of the coefficient of restitution may be unreliable. Since Broili (1973) first identified this problem, numerous experiments have been performed to acquire a comprehensive picture of the effects of the impact angle. In situ tests are expensive and not suitable for statistical and parameter analysis; thus, existing studies have largely been performed in the laboratory. In some literatures, the impact angle 15 was referred to as the slope angle θ (or the impact surface angle) in free-fall tests. While, the impact surface angle is only another expression because the slope angle θ and impact angle α sum up to 90° under these conditions. Wu (1985) conducted laboratory tests using rock blocks on a wooden platform and rock slope, and suggested that there is a linear correlation between the impact surface angle and the mean value of the restitution coefficient. He proposed that increasing the angle of the impact surface causes the normal coefficient Rn to increase regardless of the block mass and 20 causes the tangential coefficient Rt to decrease slightly. Richards et al. (2001) executed free-falling tests considering different types of rock and slope conditions and established a correlation between the coefficient of restitution and the Schmidt hammer rebound hardness. The impact surface angle was added to the correlation to reflect its linear improvement effect on the normal coefficient Rn. Chau et al. (2002) conducted experiments using spherical boulders and a rock slope platform, both made of dental plaster. 25 The free-falling tests indicated that the normal coefficient increases with increases in the impact surface angle, whereas there was no clear correlation with the tangential coefficient. Cagnoli and Manga (2003) studied oblique collisions of lapilli-size pumice cylinders on flat pumice targets and determined that the impact angle can influence the rebound angle, the kinetic energy loss and the ratios of the velocity components. The normal coefficient decreases as the impact angle approaches 90°. 30 Asteriou et al. (2012) performed laboratory tests using five types of rocks from Greece. The result of the parabolic drop tests indicated that the kinematic coefficient of restitution Rv was more appropriate than the normal coefficient of restitution for use in correlations with the impact angles. Then, the normal coefficient of restitution could be estimated accounting for the rebound-impact angle ratio. Buzzi et al. (2012) conducted experiments using flat concrete blocks in four different forms and determined that a combination of low impact angle, rotational energy and block angularity may result in a normal coefficient of restitution in excess of unity. James (2015) evaluated restitution coefficients using milled aluminium blocks and a planar wooden slope. Three different shapes of blocks were custom made and the slope surface was carpeted. Both the first impact under free fall conditions and 5 the series impacts during runout were recorded. It was observed that Rn shows a positive correlation with increasing slope angle while Rt shows a negative correlation.
These efforts have highlighted the importance of the impact angle with regard to the coefficient of restitution. Most of the existing tests indicated that increasing the impact angle induces a reduction in the normal coefficient of restitution Rn, but an improvement in the tangential coefficient of restitution Rt. However, there are two issues still unsolved. In the first place, 10 whether the laws regarding the effect of the impact angle on the coefficient of restitution are influenced by the test scale is uncertain. Up to now, the existing laboratory tests commonly captured the trajectory of small samples using a high-speed video camera, which means that the existing results are based on small scale laboratory tests. As Heidenreich (2004) noticed, the matured similarity theory regarding the model test on the coefficient of restitution is still absent, for the influence factors are much more than the material properties and sizes. Therefore, laboratory tests with larger scales should be performed to 15 confirm the validity of the existing conclusions, which is benefit for further interpreting the results of small-scale laboratory tests. Secondly, in free fall tests the rotation after impact plays an important role in kinetic energy dissipation of the falling block (Chau et al., 2002), and it was supposed to affect the variation of Rn and Rt (Broili, 1973;Cagnoli and Manga, 2003).
But, few work has been done to reveal the effect of the rotational motion on the coefficient of restitution through quantitative analysis. Whether a correlation occurs between the rotation and the effect of the impact angle deserves our attention, which 20 may offer some insights into the effect of the impact angle on the coefficients of restitution.
Hence, this study employs a 3D motion capture system and a special releasing device to perform a medium-scale laboratory experiment. Spherical polyhedrons made of limestone were selected as samples, with a maximum diameter of 20 cm. The landing plate consisted of C25 concrete slabs. To address the effect of the impact angle, different inclined plate angles and releasing heights were used in free fall tests. The resulting coefficients of restitution, Rn, Rt, Rv and RE, were calculated, and 25 their trends in terms of the impact angle were explored to provide a complete picture. Then, the results are compared with three existing small scale experiments to determine whether the test scale affects the law that the impact angle influences the coefficients of restitution. The percentage of the total kinetic energy before impact converted to rotational energy was investigated, and the role of rotation in the effect of the impact angle on the coefficient of restitution was analysed. For only spherical polyhedrons are taken as the samples in this study, the test results may have some limitations. 30 6 2 Laboratory investigation 2.1 Rock specimens and concrete slabs All falling rock specimens in this study were natural limestone from the China Three Gouges area and were customized in accordance with the required sizes. As shown in Fig. 2a, irregular artificial cutting facets constituted the surface of the specimens, and the edges were not smoothed; thus, the shape is called a spherical polyhedron in this study to distinguish it 5 from the standard sphere used in other research studies. To appraise the effect of rock size on the rebound characteristics, two different diameters were considered (D=10 cm and D=20 cm), with corresponding average masses of 1.2 kg and 10 kg. The C25 concrete slabs came from a prefabricated concrete factory. As shown in Fig. 2b, each concrete slab had dimensions of 1,200 mm×500 mm×150 mm. The mechanical properties of the materials adopted in test are determined beforehand. The limestone has the Young's modulus E=41 GPa, the Poisson's ratio ν=0.21 and Schmidt Hardness R=36.0. And the concrete 10 has E=28 GPa, ν=0.20 and R=32.5.

Testing apparatus 15
The apparatus used in this study consisted of a ramp, landing plate and releasing device (Fig. 3a). The ramp was built by compacting gravelly soil and had an inclined surface with planned angles produced by artificial excavation. Then, two concrete slabs were placed upon the inclined surface to form the landing plate. One device was designed and manufactured specially to catch and release specimens of various sizes. As shown in Fig. 3b, the device had four adjustable tongs at the bottom, which could grasp spherical blocks with diameters from 8 cm to 25 cm. A wireless receiver and electromagnetic 20 relay were installed in the upper portion of the device, offering a wireless method of altering the tong status, grasping or loosing. The device could be connected to an indoor mobile crane using the top ring, which means that the device could go up and down by managing the crane. A free-fall test was performed in the experiment, and the complete process of one test is as follows. First, when one spherical polyhedron is prepared to be tested, the tongs are adjusted to accommodate the polyhedron by moving the cross bar up and down. After the sample is in the tongs, the grasping state is selected. Then, the device hanging on the indoor crane is moved to the position above the landing plate and lifted to the planned height. Next, by operating the wireless switch, the tongs are loosened, and the sample begins to fall (Fig. 3c). Finally, the sample impacts the landing plate, and its motion is recorded. 10 The surface of the concrete slabs becomes worn with successive impacts. Once the surface occurs excessive damage as shown in Fig. 3d, the used slabs are replaced with new slabs. 8

Data acquisition
The spatial motion information of falling samples was obtained by the Doreal DIMS-9100(8c) Motion Capture System. This system has eight near-infrared cameras (see Fig. 4a) with an operating speed of 60 fps and can capture the spatial trails of markers attached to the surface of the sample, as shown in Fig. 4b. Then, the motion analysis program provides the spatial motion information of the sample, e.g., its position and velocity. Finally, the coefficient of restitution can be calculated 5 according to Eq. (1-4) for subsequent analysis.

Experimental program 10
Four different inclined angles θ of the landing plate (30°, 45°, 60° and 75°) were considered in this study to determine the effect of the impact angle on the coefficients of restitution. The impact angles are approximately related to the incline angle θ of the landing plate under free-fall test conditions. Limestone specimens were released at three different heights of 2.5 m, 3.5 m and 4.5 m upon the inclined concrete slabs. While, two tests do not necessarily have identical release conditions even if they have the same release height and use the same specimen because the positions on which the tongs catch the specimen 15 may differ slightly in any two tests.
In Table 1, the initial conditions of our experiment are presented, in addition to the resulting impact velocities and angles, Rn, Rt, Rv, RE and the rebound angles. The inertia moment of the sample was approximated to a full sphere in the calculation of rotational energy. Before impact, the angular velocities didn't exceed 3 rad/s, and the rotational energy of the rock only took up 0.01%-0.03% of the total kinetic energy in this study. 20

Effect of the impact angle on the coefficients of restitution based on our tests
Although the mean values and standard deviations have been calculated in terms of various release conditions, data points are considered in this section to provide a broad perspective for an evaluation of the effect of the impact angle. Four different inclined angles of the landing plate (θ=30°, 45°, 60° and 75°) induce four intervals of impact angles, 55°<α<60°, 36°<α<44°, 5 23°<α<30° and 6°<α<15°. The mean value of the coefficients of restitution are computed for the four intervals. In Fig. 5, solid lines are adopted to represent the mean values for samples with size D=10 cm, and dashed lines represent the mean values for size D=20 cm.

10
The effect of impact angle on the normal coefficient of restitution Rn is shown in Fig. 5a. When the impact angle is smaller than 15°, the values of Rn range from 0.709 to 1.989, and more than 60% of the values of Rn are larger than 1.0. A larger impact angle tends to produce a smaller value of Rn and reduce the discreteness. Initially the solid line is above the dashed line, although the gap narrows with increasing in the impact angle. When the impact angle is larger than 30°, these two lines do not exhibit a clear difference. Therefore, small specimens are more likely to have a higher Rn than large specimens with 5 small impact angles, and the effect of the rock size on Rn can be neglected when the impact angle is more than 30°. As Fig. 5b shows, the impact angle has a different effect on the tangential coefficient of restitution Rt compared to Rn. In the first place, the discreteness of data points hasn't been reduced as the impact angle increases. Then, Rt increases slightly with increasing in the impact angle. In the first impact angle interval, the solid line is below the dashed line, which implies that small specimens gain a lower Rt than large specimens with small impact angles. Until the impact angle reaches 23 o , the two 15 lines have no distinct difference to be distinguished. Overall, the mean value lines in Fig. 5b seem to accord with the linear correlation between Rt and the impact angle α (Wu, 1985).
Furthermore, the kinematic coefficient of restitution Rv versus the impact angle is plotted in Fig. 5c. As the impact angle increases, the peak values of Rv of the four impact angle intervals fall down gradually, and Rv become concentrated. However, the mean values present a more complicated trend in Fig. 5c. Except that the solid line rises from the first impact angle 20 interval to the second, the mean values have a decline in general. The decline is tiny from the second impact angle interval to the third, while it is apparent from the third to the fourth. Taking the small gap between the mean lines into account, bigger specimens are easier to gain a small Rv than small specimens when the impact angle is more than 23°.
Finally, the effect of the impact angle on the coefficient of kinetic energy restitution RE is illustrated in Fig. 5d. Similar to Fig. 5c, the peak values of RE of the four impact angle intervals decrease with increasing in the impact angle. But the 5 discreteness of data points does not disappear clearly until the fourth impact angle interval. The trends for the mean value of RE are similar to Rv. However, the decline in the mean values of RE is more intuitive than Rv from the second impact angle interval to the third. And the gap between the mean lines of RE is narrower than Rv with larger impact angles. Although some scholars (Chau 2002;Asteriou 2012) suggested that smaller impact angles induce less kinetic energy loss and higher RE, the deduction may be not suitable for small impact angles in this study. 10 Besides the effect of the impact angle on the four coefficients of restitution, other interesting phenomenon can be observed in Fig. 5. Two sizes are adopted in this experiment to evaluate the effect of rock size on the rebound characteristics. Except for smaller impact angle, the gaps between the two mean lines in Fig. 5 are much tiny compared to the magnitudes of restitution coefficients. Therefore, the four coefficients of restitution seem to be independent of the sample sizes in our test when the impact angle exceeds 23°, which could be attributed to the test conditions. As Farin et al. (2015) noted, the thickness of the 15 impacted objective is an important factor in determining whether the coefficients of restitution change with the boulder size.
When the impacted objective has a large thickness compared to the boulder size, the coefficient of restitution is independent of the boulder size. In this study, the impacted objective is concrete slabs fixed in the ground, which has enough thickness to eliminate the effect of rock size on the coefficients of restitution in case of large impact angles.
In addition, data points for all coefficients except Rt become concentrated as the impact angle increases. In the first impact 20 angle interval, the diversity of Rn is clearly larger than the other three coefficients. However, when the impact angle exceeds 23°, the lowest diversity occurs for Rn, and the second is for Rv. Various functions were considered to match data points, but no function can provide a correlation coefficient R 2 more than 0.60 in terms of Rt, Rv and RE for all options considered. Power function provides the best R 2 in matching data points of Rn, which reaches 0.80. So, the scaling law to describe data points is abandoned in this study. Although Asteriou et al. (2012) suggested that Rv was more suitable than Rn for use in correlations 25 with the impact angles, it is invalid in this study, which is caused by the variations of test conditions.

Comparison with existing small scale experiments
In this section, the effect of the impact angle on the coefficient of restitution obtained in this study is compared with some existing small scale experiments, to determine the effect of the test scale. Tests conducted by Chau et al. (2002), Cagnoli and Manga (2003), Asteriou et al. (2012) are selected here for the data availability. The test conditions of those studies are 30 provided in Table 2 in comparison with this study. This study mainly differs from the other studies in terms of the size and mass of the samples.  (2003), Asteriou et al. (2012) are plotted in Fig. 6 with this study. In Chau's results, the specimen with a mass of 204.33 g was selected because that mass is closest to those of our samples. In Asteriou's results, we chose a marble specimen as the reference because marble and limestone have nearly the same hardness. In the absence of detailed data, only the trend line 5 results in the related literature are extracted and redrawn in Fig. 6 to make a comparison. Different line styles are adopted for trend lines in Fig. 6. The lines with data markers are the mean value lines, while those lines without data markers are fitting lines. Two lines, Rv versus the impact angle a for Cagnoli and Manga's test, and RE versus the impact angle a for Asteriou's test, are absent in Fig 6 because the literatures didn't provide them. In addition, ①a and ①b are used to represent results for D=10 cm and D=20 cm in this study, respectively. 10 Although results of the previous studies and our tests are quite different when they are plotted in one figure, some general trends could be observed. First, all of the trends in Rn versus the impact angle are consistent (see Fig. 6a): Rn decreases with increasing in the impact angle. Asteriou's tests offered the maximum Rn in most cases, which can be attributed to the lighter mass and lower impact velocities adopted in the tests. The small Rn values in Cagnoli and Manga's tests were due to the weak strength of pumice whose damage upon impact dissipates kinetic energy, and the impact velocity in this test is much 15 higher than the others. Compared to the other tests, our results produce the steepest descent in the beginning of the trend line.
Linear function had been suggested to be used to describe the correlation of Rn and the impact angle α in several reports (Wu, 1985;Richards et al., 2001), although we cannot make a definitive conclusion that linear functions are the best choice. In Fig.   6a, the fitting curve ③ is a second-order polynomial, and the fitting curve ④ is a power function. As mentioned above, it is also found that the best correlation coefficient R 2 is provided by power function when matching Rn in this study. 20 Finally, the trends of Rv and RE versus the impact angle are shown in Fig. 6c and 6d, respectively. Cagnoli and Manga's result is not involved in Fig. 6c for its absence, and for the same reason Asteriou's result is not involved in Fig. 6d. Four unique trend lines are plotted in Fig. 6c, although Rv exhibits a descending trend overall, which means that Rv is reduced in 15 most cases as the impact angle α increases. Similar to Rv, all experiments produce downward trend lines for RE, except the initial ascent stage in line ①a, which implies that increasing the impact angle induce more kinetic energy dissipation. But, the trend lines in Fig. 6d are scattering. Clearly, the trend lines for Rv and RE are more likely to be influenced by the test conditions than Rn and Rt. In Fig. 6c the fitting curve ④ is a power function, and in Fig. 6d the fitting curve ③ is a linear function. The difference in the trend lines is apparent for the listed experiments, and we cannot have a conclusion which type of functions should be recommended to match Rv and RE.
In conclusion, various experimental conditions induce different results for Rn, Rt, Rv and RE, although there are certain trends that occur regardless of the test conditions. The normal coefficient of restitution Rn, kinematic coefficient of restitution Rv 5 and kinetic energy coefficient of restitution RE all decrease with increasing in the impact angle, while the tangential coefficient of restitution Rt increases as the impact angle increases in most cases. Power function appears suitable to be used in fitting data points of Rn, while its validity is worth to be further verified by other studies.

10
In addition, Asteriou's test provided the highest trend lines of Rn and Rv in Fig. 6, while Cagnoli and Manga's test provided the lowest trend lines of Rn, Rt and RE. Asteriou's experiment was conducted using the lowest impact velocity in Table 2, while the highest impact velocity are adopted by Cagnoli and Manga. And Cagnoli and Manga employed pumice, which has a much weaker strength compared to sample materials in other tests. Asteriou and Tsiambaos (2018) has just noticed that Rn reduces when increasing the impact velocity, and increases as the material become harder, which partly accounts for the 15 difference between Asteriou's and Cagnoli and Manga's test. However, we cannot make a definitive conclusion which factor in Table 2 is the main reason for the magnitude difference in the coefficients of restitution between the tests compared. The tests compared differ from each other in multiple test conditions as Table 2 lists, therefore the estimation of the effect of one specific factor on the magnitude of the coefficient of restitution is unreasonable using their data together. To evaluate the effect of the impact velocity in this study, Fig. 7 plots the mean value of Rn versus the impact velocity with different slope 20 angles. No determined trend of Rn appears, for the limited variation range of the impact velocity.

Direction transitions of translational velocities
Taking the ratio between the rebound angle and the impact angle β/α as a reference, the direction transition of the translational velocity versus the impact angle are illustrated in Fig. 8. Assuming that the falling rock is spherical and no 25 energy dissipation occurs during the collision, the rebound angle should theoretically be equal to the impact angle, which would result in the data points lying on the red line β/α=1 in Fig. 8.

Fig. 8. The ratio between the rebound angle and the impact angle β/α versus impact angle
However, the test results are almost entirely located above the line in the first impact angle interval, and nearly 50% of the 5 test results are above the line in the second interval. The data points are stably located below the line until the impact angle reaches to 36°. As the impact angle increases, the ratio between the rebound angle and the impact angle β/α appears a clear reduction, and the discreteness of data points decreases. The mean values of β/α are still represented by a solid line for samples with size D=10 cm, and a dashed line for size D=20 cm. The two mean values line have little difference from the second interval to the fourth. But, under small impact angle conditions, a smaller sample is more likely to have a larger β/α 10 than a bigger sample.
A rebound angle greater than the impact angle was also observed by Cagnoli and Manga (2003), which does not violate the energy dissipation rule. The experimental results presented in Section 3.1 demonstrated that in this study the kinetic energy loss constituted 40-65% of the total kinetic energy for many data points in the first impact angle interval, and constituted 35-55% in the third interval. Therefore, the ratio between the rebound angle and the impact angle cannot be directly used as a 15 reference in estimating whether the energy loss level is high or low.
This phenomenon implies that the rebound motion may have an unexpected direction of translational velocity.

The rotation caused by the impact
Except the direction transition of translational velocity, the rotation is another significant consequence of the impact. Despite 5 little rotation before impact, the samples occurred an observable rotation after impact in this study, and the angular velocities were recorded and involved in the calculation of the kinetic energy coefficient of restitution RE in Table 1. Considering that the magnitudes of kinetic energy before impact varied in this study, the percentage between Err and Ei is used to denote how much kinetic energy is dissipated in rotation after the impact. As mentioned in Section 1.1, Ei is the total kinetic energy before impact, and Err is the rotation energy after impact. 10 Fig. 10a shows the effect of impact angle on Err/Ei. A solid line represents the mean values for specimens with size D=10 cm, while a dashed line is for size D=20 cm. At firstly, the difference is most remarkable between two sample sizes in Fig. 10a. 15

Fig. 10. Rotational energy caused by the impact
Err/Ei ranges from 3.3% to 13.7% for size D=10 cm, and ranges from 0.7% to 4.5% for size D=20 cm, which means that small samples are more likely to have a high Err/Ei than larger samples. Next, Err/Ei reduces as the impact angle increases.
For size D=10 cm, Err/Ei experiences a steep decline from the first impact angle interval to the third, then decreases gently to the fourth. For size D=20 cm Err/Ei has a gradual reduction all the time, which may be attributed to its small variation range.
At last, the improvement of the impact angle results in more concentrated data points, although data points for size D=10 cm 20 are always more scattering than size D=20 cm. Both the impact angle and the sample size have an important impact on Err/Ei in this study. In conclusion, larger samples are more likely to have a stead and small Err/Ei than small samples, and a large impact angle leads to a small Err/Ei.
As mentioned in Section 4.1, the unexpected direction transition of translational velocity always happens when the impact angle is small. The correlation between the direction transition of translational velocity and Err/Ei is investigated by using β/α (the ratio between the rebound angle and the impact angle) as a reference. Fig. 10b illustrated the trends for Err/Ei versus β/α. 5 Err/Ei increases as the ratio β/α increases. If β/α is smaller than 1.0, Err/Ei are much concentrated, which ranges from 3.3% to 6.5% for size D=10 cm and ranges from 0.7% to 2.6% for size D=20 cm. With increasing in β/α, the data points become scattering. Moreover, the improvement of Err/Ei appears being terminated when β/α reaches a specific value. So, we can have a conclusion that a strong correlation occurs between Err/Ei and β/α. For a given impact angle, larger rebound angles means that more kinetic energy be converted to rotational energy during the collision. 10

The correlation between the coefficients of restitution and the rotation
The rotation plays an important role in energy dissipation during impact, especially for small samples. The percentage between the resulting rotational energy and the original total kinetic energy decreases as the impact angle increases. The correlation between Err/Ei and the coefficients of restitution is investigated in this section, to evaluate the effect of rotation. Fig. 11 plots the coefficients of restitution versus Err/Ei for this study. The four coefficients fall into two categories according 15 to their responses to Err/Ei. The first category includes Rn and Rt, two most commonly used coefficients of restitution, which appears a strong correlation with Err/Ei. As Err/Ei increases, Rn increases but Rt decreases, which verified Broili's deduction (1973). The rotation generated from impact results in an increased normal velocity and reduced tangential velocity.
Furthermore, more kinetic energy being converted to rotational energy during the collision, higher Rn and lower Rt we have.
In Fig. 11a and 11b, data points for two sizes are not mixed, which can be attributed to the effect of sample sizes on the 20 magnitude of Err/Ei. Besides that, data points become more scattering with increasing in Err/Ei. Rv and RE belong to the second category. There is no remarkable correlation between them and Err/Ei as shown in Fig. 11c and 11d, so Rv and RE are independent of the rotation motion in this study. In conclusion, the improvement in the percentage of kinetic energy converted to rotational energy leads to a larger Rn and a smaller Rt, while it has no distinct influence on Rv and RE.

Fig. 11. The influence of Err/Ei on the coefficients of restitution
As illustrated in Fig. 10a, more kinetic energy is converted to rotational energy during the collision with a smaller impact angle. Considering the effect of Err/Ei on Rn and Rt, a smaller impact angle is more likely to have a high Rn and a low Rt than 5 a larger impact angle. Therefore, Rn typically decrease with increasing in the impact angle, and Rt increases as the impact angle increases. When the impact angle is small, two sample sizes appear a clear distinction in Err/Ei as shown in Fig. 10a, which results in the difference in the mean values of Rn and Rt between two sizes in the first impact angle interval in Fig. 5a and 5b.

Discussion 10
The test results demonstrated the correlation between the rotation and the effect of the impact angle on the coefficients of restitution. Under free fall conditions, a higher percentage of kinetic energy converted to rotational energy always induces a higher Rn and a lower Rt. The percentage can be associated with the ratio between the rebound angle and the impact angle β/α. As the impact angle decrease, the ratio β/α increases, then more kinetic energy is converted to rotational energy. In this section, the reason why a small impact angle is easier to have a high β/α and its consequences are discussed. 15

The main reason for the high β/α in case small impact angles
When the impact angle is small, the rebound angle is easier to exceed the impact angle and causes a high β/α, which can be associated with the impact orientation and the damages caused by the impact. The sample has irregular cutting facets and rear edges in this study, while the landing plate was made of concrete slabs of a smaller hardness compared to the falling samples. Suppose that the spherical polyhedron impacts the landing plate with a corner or an edge, the damages will happen. 20 Fig. 12a shows the indentations on the surface caused by the impacts. The configuration of indentation is simplified as Fig.   12b to evaluate the effect of the indentation on the rebound direction.
Once the impact compression ends, the indentation is formed completely and the rebound motion starts. The rebound angle is constrained by the border of the resulting indentation. The restriction is less susceptible to a small rebound angle, because a translational motion along the dashed arrow indicates additional penetration. Theoretically, the rebound angle will be less 25 19 than the impact angle accounting for energy loss. When the impact angle is sufficiently large to generate a rebound angle as the solid arrow, the border imposes no constraints on the rebound motion, and the sample can leave with the default rebound angle. But, when the impact angle is small and generate a default rebound angle as the dashed arrow, rotation motion must be involved to overcome the constraint, then an unexpected larger rebound angle happens. Thus, the penetration caused by the impact may contribute to the high β/α in case of small impact angles. 5

10
Another important factor to generate a high β/α is the macro roughness of the landing plate, which comes from repeated damages on the slab surface. Assuming that the macro roughness of the landing plate is represented as a small stair in Fig.   12c, the interaction between the falling sample and surface may have two stages in certain situations. The sample impacts the surface before the stair and starts leaving in the first stage. Then, the sample contacts the stair and the velocity changes again in the second stage. The time interval between the two stages is so short that the two stages appear to finish simultaneously. 15 The probability that two stages interaction happens is related to the magnitude of the impact angle. As Fig. 12c illustrates, if the default rebound angle is 15°, the stair can affect the rebound motion if the sample contacts the surface within 3.73 times the stair height before the stair. As the default rebound angle increases, the surface region that the stair can affect the rebound motion decreases. Considering that a smaller impact angle will, in theory, induce a smaller rebound angle, the reduction of impact angle must improve the risk of the sample contacting the stair. When the impact angle is small, the sample has more 20 possibility to have a two stages interaction and leave the plate with a rebound angle larger than the impact angle, and a high β/α will happen.
In conclusion, the restriction from the configuration of the indentation, as well as the macro roughness caused by repeated damages, are more likely to affect the rebound motion when the impact angle is small. As a consequence, the rebound angle is easy to exceed the impact angle in case of small impact angle, which results in a high β/α ultimately. 5

Interpretation of normal coefficient of restitution Rn larger than 1.0
Of the various consequences of the rebound angle being greater than the impact angle, high values of the normal coefficient of restitution Rn may be remarkable. Engineers usually take 1.0 as the upper bound of Rn in computer codes, whereas several scholars had reported Rn values larger than 1.0 (Azzoni et al., 1992;Paronuzzi 2009;Spadari et al., 2012). In this section the relationship between Rn and the direction transition of translational velocity is investigated. 10 Considering that the rotation before impacting is little in this study, the normal coefficient of restitution Rn can be expressed as Eq. (5) based on the basic definition in Section 1.1.
By introducing an angle coefficient Eq. (6)  increasing the angle coefficient λ. Even if Ert/Ei is only 0.2, Rn is greater than 1.0 when λ>2.24. An extremely large rebound angle is not needed to generate such a λ when the impact angle is small. For example, when the impact angle is 12° and 15°, a rebound angle of 27.8° and 35.5° is sufficient to obtain λ>2.24. Assuming that Ert/Ei is unchanged, a case that the rebound angle is larger than the impact angle must lead to a higher Rn. Although the value of λ corresponding to Rn=1.0 varies with 5 Ert/Ei, the condition λ>1.0 is required to obtain Rn greater than 1.0. As shown in Fig. 13b, Rn cannot exceed 1.0 if the rebound angle is lower than the impact angle. As discussed in previous sections, small impact angles are easy to result in unexpected large rebound angles. If the angle coefficient λ formed by the rebound and the impact angle is sufficiently large, Rn will exceed 1.0 even though Ert/Ei is small. Furthermore, assuming a constant Ert/Ei, the reduction in the impact angle decreases the threshold value of the rebound angle that should be satisfied to achieve an Rn in excess of unity, which means that 10 smaller impact angles are more likely to yield Rn larger than 1.0.

Relation between the normal coefficient of restitution and the kinetic energy loss
A smaller impact angle is easier to have a high β/α and a high percentage of kinetic energy converted to rotational energy, then induces a higher Rn. However, the kinetic energy coefficient of restitution Rv appears independent of the percentage of kinetic energy converted to rotational energy. Therefore, simply treating a higher Rn as a symbol of lower kinetic energy loss 15 maybe unreasonable. Stronge (1991) had indicated that in the valuation of kinetic energy dissipation, the normal coefficient of restitution is only reliable for nonfrictional collisions. Under frictional collisions conditions, the total kinetic energy may have a paradoxical increase, if the normal coefficient of restitution is adopted as the unique reference. As shown in Fig. 14, the correlation between Rn and RE is more complicated in this study, which verifies Stronge's argument. which means that the unexpected large rebound angles can be related to higher level of kinetic energy loss. RE is disordered 22 if Rn lies in (0.65, 0.95), which is caused by the two different trends meeting. Therefore, the normal coefficient of restitution Rn cannot be directly used in the evaluation of the kinetic energy dissipation level.

The difficulty in introducing the effect of impact angle into trajectory simulation
This study, as well as the previous experiments, has demonstrated that the variation of the coefficients of restitution in terms of the impact angle are significant. For this reason, the impact angle should be involved in determining the coefficients of 5 restitution in rockfall trajectory simulation. However, some problems cause a barrier to develop a reasonable way to account for the effect of the impact angle in computer simulation.
First of all, although the test scales and conditions have little influence on the general laws that the impact angle affects the coefficients of restitution, it is difficult to construct a uniform formula to reflect the effect under various test conditions. Take Rn as an example, the effect of the impact angle on Rn had been formulated by several different functions, such as the linear 10 function (Wu, 1985;Richards et al., 2001), power function (Asteriou et al. 2012) and second-order polynomial (Cagnoli and Manga, 2003). In this study, power function provides the best correlation coefficient in fitting data points of Rn. Furthermore, the mathematic expression regarding the effect of the impact angle on the coefficients of restitution is abandoned in more experiments (Chau et al., 2002;James, 2015). Therefore, we can't have a conclusion which type of functions is the best choice to describe the effect, and whether a uniform expression occurs is questionable. 15 Another problem comes from the discreteness of data points. Given the impact angle, the discreteness of data points determines the reliability of the rebound velocity estimated by adopting a typical value of the coefficients of restitution. For all coefficients of restitution, the discreteness of data points experiences a reduction if the impact angle increases in this study. When the impact angle is large, it may be acceptable to predict the rebound using a typical value of the coefficients of restitution, e.g. the mean value. However, the data points are extremely scattering under small impact angle condition, which 20 means that using a typical value in the simulation maybe unreliable.
Therefore, further researches should be carried on to establish a reasonable and comprehensive method to reflect the effect of the impact angle on the coefficients of restitution in rockfall trajectory simulation. The stochastic model has more potential in achieving this target, because it accounts for the variation of the coefficients of restitution in terms of various factors based on data collection (Jaboyedoff et al., 2005;Frattini et al., 2008;Bourrier et al., 2009;Andrew and Oldrich, 2017). 25

Conclusions
The coefficients of restitution are critical parameters in the predication of rockfall trajectory by computer codes. Both the terrain characteristics and kinematic parameters can significantly affect the coefficients of restitution. The effect of the impact angles on the coefficients of restitution have been observed and some laws have been concluded in a series of tests.
Until now, the existing laboratory tests have largely been limited to small scale tests, and whether the previous conclusion is 30 23 valid for different scale tests is uncertain. The role of rotation is still unresolved in the effect of the impact angle on the coefficient of restitution.
In the present study, laboratory tests were performed using a 3D motion capture system. Spherical limestone polyhedra with diameters of 10 cm and 20 cm were taken as samples, and C25 concrete slabs were adopted to form the landing plate. By altering the release height and the inclined angle of the landing plate, the effects of the impact angle on the coefficients of 5 restitution were estimated under freefall test conditions. The result comparison between our test and the existing small scale tests indicated that several general laws occur when accounting for the effect of the impact angle, regardless of the test scales and conditions. The normal coefficient of restitution Rn, the kinematic coefficient of restitution Rv and the kinetic energy coefficient of restitution RE all decrease when increasing the impact angle, while tangential coefficient of restitution Rt increases as the impact angle increases in most cases. However, the reason for the magnitude difference in the coefficients of 10 restitution between the tests compared is unidentified, for the tests differ from each other in multiple test conditions.
In free fall test, the rotation after impact dissipates part of the kinetic energy of the sample, and play an important role in the effect of the impact angle on the coefficient of restitution. Test results shows that the percentage of kinetic energy converted to rotational energy can be associated with the ratio between the rebound angle β and the impact angle α. When the impact angle is small, the rebound angle is more likely to exceed the impact angle and yields a high β/α, for the indentations and 15 macro roughness caused by the impacts. As the impact angle decreases, the ratio β/α increases, and the percentage of kinetic energy converted to rotational energy become higher. Given a β/α, large samples are more likely to have a stead and small percentage than small samples. A higher percentage of kinetic energy converted to rotational energy always induces a higher Rn and a lower Rt. But no correlations are observed in this study between the rotation energy and the other two coefficients of restitution, Rv and RE. In additional, Rn being larger than 1.0 can be related to the rebound angle greater than the impact angle 20 under small impact angle condition.
Although it is verified in this study that several general laws regarding the effect of the impact angle on the coefficients of restitution are independent of the test scales and conditions, we are still lacking a reliable method to introduce the effect of the impact angle into rockfall trajectory simulation, which is caused by the discreteness of the measured data under small impact angle condition and the absence of a uniform and reasonable function describing the effect of the impact angle. 25 Last but not least, only the spherical limestone polyhedrons are taken as the samples, and the detailed impact orientations during impact are not involved in this study. Whether the conclusions are valid for the boulder with other shapes should be further investigated through more elaborated experiments. In views of this, the test results is valid for trajectory simulation codes based on a lumped-mass model, and can be referenced in the trajectory predication of spherical rocks impacting hard surface using a rigid body model. 30