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- About
- Editorial board
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- For authors
- For reviewers
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**Research article**
19 Nov 2018

**Research article** | 19 Nov 2018

Effects of the impact angle on the coefficient of restitution in rockfall analysis

^{1}Hubei Key Laboratory of Disaster Prevention and Mitigation, China Three Gorges University, Yichang, Hubei, 443002, People's Republic of China^{2}Department of Civil, Structural, and Environmental Engineering, University at Buffalo, Buffalo, NY 14260, USA

Abstract

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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
small-scale laboratory tests. To investigate whether the existing conclusion
based on small-scale laboratory tests is valid when the test scale changes
as well as 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 to impact concrete slabs. Free-fall tests are conducted, and the velocities before and after the impact are
obtained by a 3-D motion capture system. The comparison of results between our
test and the existing small-scale tests verified that several general laws
occur when accounting for 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 *R*_{n}, the
kinematic coefficient of restitution *R*_{v} and the kinetic energy
coefficient of restitution *R*_{E}, whereas it will lead to increases
in the tangential coefficient of restitution *R*_{t}. 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 *R*_{n} and a lower
tangential coefficient of restitution *R*_{t}. As the impact angle
decreases, the ratio between the rebound angle *β* and the impact
angle *α* increases, and the percentage of kinetic energy dissipated in
rotation as the collision became higher. Considering that the effect of block
shape and the detailed impact orientations are not involved in the present
study, the test results are valid for trajectory simulation codes based on a
lumped-mass model and can be referenced in the trajectory predication of
spherical rocks impacting hard surfaces using a rigid body model.

How to cite

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How to cite.

Wang, Y., Jiang, W., Cheng, S., Song, P., and Mao, C.: Effects of the impact angle on the coefficient of restitution in rockfall analysis based on a medium-scale laboratory test, Nat. Hazards Earth Syst. Sci., 18, 3045-3061, https://doi.org/10.5194/nhess-18-3045-2018, 2018.

1 Introduction

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In mountainous 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 et al., 2003; Pappalardo et al., 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 provide important information, such as the travel distances of possible rockfall events, the bouncing height and the 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 RocFall (Stevens, 1998), CRSP (Jones et al., 2000), STONE (Guzzetti et al., 2002), RAMMS::Rockfall (Christen et al., 2007), Rockyfor3D (Dorren, 2016) and Pierre 2 (Valentin et al., 2015; Andrew and Oldrich, 2017), are developed to acquire motion information regarding 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 model 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 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 understood and the most difficult to predict of the four types of motion (Volkwein et al., 2011) and is controlled by the coefficients of restitution in computer simulations. Thus, the reliability of the estimation of the coefficient of restitution must be ensured.

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 impacts the slope surface, the impact velocity *v*_{i} can
be resolved into a normal component *v*_{ni} and a tangential component *v*_{ti}
according to the slope angle *θ*. Then, the boulder leaves the
surface with a rebound velocity *v*_{r}, which similarly has a *v*_{nr} and
a *v*_{tr}. The angular velocities of the boulder before and after impact are
denoted as *ω*_{i} and *ω*_{r}, respectively. The impact
angle *α* and rebound angle *β* are drawn in Fig. 1.

The normal and tangential coefficients of restitution are the most commonly
used definitions, and the two coefficients of restitution are typically
denoted as *R*_{n} and *R*_{t}, respectively. The mathematical expressions
of *R*_{n} and *R*_{t} are

$$\begin{array}{}\text{(1)}& {\displaystyle}{\displaystyle}{R}_{\mathrm{n}}={v}_{\mathrm{nr}}/{v}_{\mathrm{ni}},\phantom{\rule{0.25em}{0ex}}{R}_{\mathrm{t}}={v}_{\mathrm{tr}}/{v}_{\mathrm{ti}}.\end{array}$$

Another common definition is the kinematic coefficient of restitution, *R*_{v},
representing the ratio between the magnitudes of the rebound and impact velocities:

$$\begin{array}{}\text{(2)}& {\displaystyle}{\displaystyle}{R}_{\mathrm{v}}={v}_{\mathrm{r}}/{v}_{\mathrm{i}}.\end{array}$$

This definition, originating from Newton's law of restitution, has
been used by Habib (1976), Paronuzzi (1989) and other scholars. When
*R*_{v} is used in the trajectory predication, an assumption regarding the
rebound direction is necessary 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 *R*_{E}, which is
written as

$$\begin{array}{}\text{(3)}& {\displaystyle}{\displaystyle}{R}_{\mathrm{E}}={E}_{\mathrm{r}}/{E}_{\mathrm{i}}=\left({E}_{\mathrm{rr}}+{E}_{\mathrm{rt}}\right)/\left({E}_{\mathrm{ir}}+{E}_{\mathrm{it}}\right),\end{array}$$

in which *E*_{i} and *E*_{r} are the kinetic energy before and after the
impact, respectively. *E*_{ir} and *E*_{rr} are the rotational energy before and
after the impact; *E*_{it} and *E*_{rt} denote the translational energy before
and after the impact. *E*_{ir}, *E*_{it}, *E*_{rr} and
*E*_{rt} are computed as

$$\begin{array}{}\text{(4)}& {\displaystyle}{\displaystyle}{E}_{\mathrm{ir}}=\mathrm{0.5}I{\mathit{\omega}}_{\mathrm{i}}^{\mathrm{2}},\phantom{\rule{0.25em}{0ex}}{E}_{\mathrm{it}}=\mathrm{0.5}m{v}_{\mathrm{i}}^{\mathrm{2}},\phantom{\rule{0.25em}{0ex}}{E}_{\mathrm{rr}}=\mathrm{0.5}I{\mathit{\omega}}_{\mathrm{r}}^{\mathrm{2}},\phantom{\rule{0.25em}{0ex}}{E}_{\mathrm{rt}}=\mathrm{0.5}m{v}_{\mathrm{r}}^{\mathrm{2}}.\end{array}$$

Here, *m* is the mass, *I* is the moment of inertia. *R*_{E} can reflect the
kinetic energy loss caused by the impact and has been used by Bozzolo and
Pamini (1986), Azzoni et al. (1995) and Chau et al. (2002).

In these definitions, *R*_{n} and *R*_{t} become more popular in engineering
practice for simplicity in computer simulation software. *R*_{n} and
*R*_{t} 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. Therefore, *R*_{n} and
*R*_{t} attracted the most attention in previous studies, and some typical
values of *R*_{n} and *R*_{t} have been summarized (Agliardi and Crosta, 2003;
Heidenreich, 2004; Scioldo, 2006).

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 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. However, in those existing summaries for typical values, the coefficients of restitution were determined mainly accounting for the terrain conditions.

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 studies, the impact angle was referred to as the slope angle *θ* (or
the impact surface angle) in free-fall tests. However, 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 *R*_{n} to increase regardless of the block mass and causes the
tangential coefficient *R*_{t} 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 *R*_{n}.

Chau et al. (2002) conducted experiments using spherical boulders and a rock slope platform, both made of dental plaster. 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^{∘}.

Asteriou et al. (2012) performed laboratory tests using five types of rocks
from Greece. The results of the parabolic drop tests indicated that the
kinematic coefficient of restitution *R*_{v} 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 aluminum
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 the series impacts during runout were recorded. It
was observed that *R*_{n} shows a positive correlation with increasing slope
angle, while *R*_{t} 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 *R*_{n} but an improvement in the tangential
coefficient of restitution *R*_{t}. However, there are two issues that remain
unsolved. First, whether the laws regarding the effect of the impact angle
on the coefficient of restitution are influenced by the test scale is
uncertain. Until 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) noted, the mature similarity theory regarding the model
test on the coefficient of restitution is still absent because the influence
factors are much more than the material properties and sizes. Therefore,
laboratory tests with larger scales should be performed to confirm the
validity of the existing conclusions, which is beneficial for further
interpreting the results of small-scale laboratory tests. Second, 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 *R*_{n} and *R*_{t} (Broili, 1973;
Cagnoli and Manga, 2003). However, few studies have been performed to reveal
the effect of 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 and may offer some
insights into the effect of the impact angle on the coefficients of restitution.

Hence, this study employs a 3-D 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, *R*_{n}, *R*_{t}, *R*_{v} and *R*_{E}, were calculated, and
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
whereby 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 analyzed. For only
spherical polyhedrons that are taken as the samples in this study, the test
results may have some limitations.

2 Laboratory investigation

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All falling rock specimens in this study were natural limestone from the
China Three Gorges 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 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 and 10 kg. The C25 concrete slabs came from a prefabricated concrete
factory. As shown in Fig. 2b, each concrete slab had dimensions of
120 cm × 50 cm × 15 cm. The mechanical properties of the
materials adopted in the test are determined beforehand. The limestone has
Young's modulus *E*=41 GPa, Poisson's ratio *ν*=0.21 and Schmidt
hardness *R*=36.0. The concrete has *E*=28 GPa, *ν*=0.20 and *R*=32.5.

The apparatus used in this study consisted of a ramp, landing plate and release 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 on the inclined surface to form the landing plate. One device was designed and manufactured specifically 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 to 25 cm. A wireless receiver and electromagnetic 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 crossbar 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. The surface of the concrete slabs becomes worn with successive impacts. Once the damage to the surface is excessive, as shown in Fig. 3d, the slabs used are replaced with new slabs.

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 according to Eqs. (1)–(4) for subsequent analysis.

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 inclined
angle *θ* of the landing plate under free-fall test conditions.
Limestone specimens were released at three different heights of 2.5, 3.5 and
4.5 m upon the inclined concrete slabs. However, 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 may differ slightly in any two tests.

In Table 1, the release conditions of our experiment are presented in
addition to the resulting impact velocities and angles, *R*_{n}, *R*_{t},
*R*_{v}, *R*_{E} 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 did not exceed 3 rad s^{−1}, and the
rotational energy of the rock only accounted for 0.01 %–0.03 % of the
total kinetic energy in this study.

3 Analysis of the results and comparison with existing studies

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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^{∘}, 23^{∘} < *α* < 30^{∘} and
6^{∘} < *α* < 15^{∘}. The mean values 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.

The effect of impact angle on the normal coefficient of restitution *R*_{n}
is shown in Fig. 5a. When the impact angle is smaller than
15^{∘}, the values of *R*_{n} range from 0.709 to 1.989, and more
than 60 % of the values of *R*_{n} are larger than 1.0. A larger impact
angle tends to produce a smaller value of *R*_{n} and reduce the
discreteness. Initially, the solid line is above the dashed line, although
the gap narrows with increasing 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 *R*_{n}
than large specimens with small impact angles, and the effect of the
rock size on *R*_{n} 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 *R*_{t} compared to *R*_{n}. First, the
discreteness of data points has not been reduced as the impact angle
increases. Then, *R*_{t} increases slightly with increasing impact angle. In
the first impact angle interval, the solid line is below the dashed line,
which implies that small specimens gain a lower *R*_{t} than large specimens
with small impact angles. Until the impact angle reaches 23^{∘}, the
two lines have no distinct difference to be distinguished. Overall, the mean
value lines in Fig. 5b seem to accord with the linear correlation between *R*_{t}
and the impact angle *α* (Wu, 1985).

Furthermore, the kinematic coefficient of restitution *R*_{v} versus the
impact angle is plotted in Fig. 5c. As the impact angle increases, the peak
values of *R*_{v} of the four impact angle intervals decrease gradually, and
*R*_{v} becomes concentrated. However, the mean values present a more
complicated trend in Fig. 5c. Except where the solid line rises from the
first impact angle interval to the second, the mean values generally
decline. 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, larger specimens obtain a small *R*_{v}
more easily 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 *R*_{E} is illustrated in Fig. 5d. Similar to Fig. 5c, the peak
values of *R*_{E} of the four impact angle intervals decrease with increasing
impact angle. However, the discreteness of data points does not disappear
clearly until the fourth impact angle interval. The trends for the mean
value of *R*_{E} are similar to *R*_{v}. However, the decline in the mean
values of *R*_{E} is more intuitive than *R*_{v} from the second impact angle
interval to the third. The gap between the mean lines of *R*_{E} is narrower
than *R*_{v} with larger impact angles. Although some scholars (Chau et al., 2002;
Asteriou et al., 2012) suggested that smaller impact angles induce less kinetic
energy loss and higher *R*_{E}, the deduction may not be suitable for small
impact angles in this study.

In addition to the effect of the impact angle on the four coefficients of
restitution, another 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 a smaller impact angle, the gaps
between the two mean lines in Fig. 5 are much smaller compared to the
magnitudes of the 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 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 *R*_{t} become
concentrated as the impact angle increases. In the first impact angle
interval, the diversity of *R*_{n} is clearly larger than the other three
coefficients. However, when the impact angle exceeds 23^{∘}, the
lowest diversity occurs for *R*_{n}, and the second is for *R*_{v}. 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 *R*_{t},
*R*_{v} and *R*_{E} for all options considered. The power function provides
the best *R*^{2} in matching data points of *R*_{n}, which reaches 0.80.
Therefore, the scaling law to describe data points is abandoned in this
study. Although Asteriou et al. (2012) suggested that *R*_{v} was more
suitable than *R*_{n} for use in correlations with the impact angles, it is
invalid in this study due to the variations in test conditions.

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), and Asteriou et al. (2012) are selected here for the data availability. The test conditions of those studies are 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.

Although previous studies imposed various test conditions, they provided
references for us to evaluate the effect of the test scale. The effects of
the impact angle on *R*_{n}, *R*_{t}, *R*_{v} and *R*_{E} provided by Chau et
al. (2002), Cagnoli and Manga (2003) and Asteriou et al. (2012) are plotted in
Fig. 6 with this study. In the Chau et al. (2002) results, the specimen with a mass of 204.33 g was selected because that mass is closest to those of our samples.
In the Asteriou et al. (2012) 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 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, *R*_{v} versus the impact angle *a* for the Cagnoli and Manga (2003) test and
*R*_{E} versus the impact angle *a* for the Asteriou et al. (2012) test, are absent in Fig. 6
because the literature did not provide them. In addition, (1)a and b in Fig. 6 are
used to represent the results for *D*=10 cm and *D*=20 cm in this study, respectively.

Although the 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 *R*_{n} versus the impact angle are
consistent (see Fig. 6a): *R*_{n} decreases with increasing impact angle.
The Asteriou et al. (2012) tests offered the maximum *R*_{n} in most cases, which can be
attributed to the lighter mass and lower impact velocities adopted in the
tests. The small *R*_{n} values in the Cagnoli and Manga (2003) 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 higher than the others.
Compared to the other tests, our results produce the steepest descent at the
beginning of the trend line. Linear function has been suggested to be used
to describe the correlation of *R*_{n} and the impact angle *α* in
several reports (Wu, 1985; Richards et al., 2001), although we cannot offer
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 the power function when
matching *R*_{n} in this study.

Next, Fig. 6b indicates that the trends of *R*_{t} versus the impact angle
are scattered. Except for the Cagnoli and Manga (2003) result, the *R*_{t} obtained in the tests
are all in the range from 0.5 to 1.0. Wu (1985) suggested that *R*_{t} may
decrease linearly with increasing slope angle *θ*. In other words,
*R*_{t} may experience an improvement with increasing impact angle *α*,
which is in line with the results of all experiments except for Chau et al. (2002). Cagnoli
and Manga (2003) matched *R*_{t} using a linear function, which resulted in the
fitting curve (4) in Fig. 6b. Therefore, the improvement effect of the impact
angle *α* on *R*_{t} is valid in most cases. In addition, if the impact
angle is less than 45^{∘}, there is an apparent gap between the Cagnoli and Manga (2003) result and the other tests. Although the results of the other three
experiments are different, the variation ranges of *R*_{t} occur regardless
of the test conditions.

Finally, the trends of *R*_{v} and *R*_{E} versus the impact angle are shown
in Fig. 6c and d, respectively. The Cagnoli and Manga (2003) result is not involved
in Fig. 6c for its absence, and for the same reason, the Asteriou et al. (2012) result is
not involved in Fig. 6d. Four unique trend lines are plotted in Fig. 6c,
although *R*_{v} exhibits a descending trend overall, which means that
*R*_{v} is reduced in most cases as the impact angle *α* increases.
Similar to *R*_{v}, all experiments produce downward trend lines
for *R*_{E}, except the initial ascent stage in line (1)a, which implies that
increasing the impact angle induces more kinetic energy dissipation.
However, the trend lines in Fig. 6d are scattered. Clearly, the trend lines
for *R*_{v} and *R*_{E} are more likely to be influenced by the test
conditions than *R*_{n} and *R*_{t}. In Fig. 6c, the fitting curve (4) is a
power function, and in Fig. 6d, the fitting curve (4) is a linear function.
The difference in the trend lines is apparent for the listed experiments,
and we cannot conclude which type of functions should be recommended to
match *R*_{v} and *R*_{E}.

In conclusion, various experimental conditions induce different results
for *R*_{n}, *R*_{t}, *R*_{v} and *R*_{E}, although there are certain trends that
occur regardless of the test conditions. The normal coefficient of
restitution *R*_{n}, kinematic coefficient of restitution *R*_{v} and kinetic
energy coefficient of restitution *R*_{E} all decrease with increasing impact
angle, while the tangential coefficient of restitution *R*_{t} increases as
the impact angle increases in most cases. A power function appears suitable
for use in fitting data points of *R*_{n}, while its validity should be
further verified by other studies.

In addition, the Asteriou et al. (2012) test provided the highest trend lines of *R*_{n}
and *R*_{v} in Fig. 6, while the Cagnoli and Manga (2003) test provided the lowest trend
lines of *R*_{n}, *R*_{t} and *R*_{E}. The Asteriou et al. (2012) experiment was conducted
using the lowest impact velocity in Table 2, while the highest impact
velocity was adopted by Cagnoli and Manga (2003). Cagnoli and Manga (2003) employed
pumice, which has a much weaker strength compared to sample materials in
other tests. Asteriou and Tsiambaos (2018) noted that *R*_{n} reduces when
increasing the impact velocity and increases as the material becomes harder,
which partly accounts for the difference between the Asteriou et al. (2012) and Cagnoli and
Manga (2003) test. However, we cannot make a definitive conclusion regarding
which factor in Table 2 is the main reason for the magnitude difference in
the coefficients of restitution between the tests compared. The tests 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 *R*_{n} versus the impact velocity with different slope angles.
No determined trend of *R*_{n} appears for the limited variation range of the
impact velocity.

4 Direction transitions of translational velocities and rotation

Back to toptop
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 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 $\mathit{\beta}/\mathit{\alpha}=\mathrm{1}$ in Fig. 8.

However, the test results are almost entirely located above the line in the
first impact angle interval, and nearly 50 % of the test results are above
the line in the second interval. The data points are stably located below
the line until the impact angle reaches 36^{∘}. As the impact angle
increases, the ratio between the rebound angle and the impact angle *β*∕*α*
appears to be 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 value lines have little difference from the second
interval to the fourth. However, under small impact angle conditions, a
smaller sample is more likely to have a larger *β*∕*α* than a
larger 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 Sect. 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 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. Figure 9 plots the direction transition
of translational velocity caused by the impact, in which diagrams are
individually drawn for four impact angle intervals. For a uniform
expression, the landing plate is denoted as the bottom black line. Although
the impact velocity directions are concentrated for each impact angle
interval, the rebound velocity directions vary considerably. The variation
for the interval 36^{∘} < *α* < 44^{∘} is
the smallest of all intervals.

Except for the direction transition of translational velocity, the rotation
is another significant consequence of the impact. Despite little rotation
before impact, the samples experienced 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 *R*_{E} in
Table 1. Considering that the magnitudes of kinetic energy before impact varied in
this study, the percentage between *E*_{rr} and *E*_{i} is used to denote how
much kinetic energy is dissipated in rotation after the impact. As mentioned
in Sect. 1.1, *E*_{i} is the total kinetic energy before impact, and
*E*_{rr} is the rotation energy after impact.

Figure 10a shows the effect of the impact angle on *E*_{rr}∕*E*_{i}. A solid line
represents the mean values for specimens with size *D*=10 cm, while a dashed
line is for size *D*=20 cm. First, the difference is most remarkable between
the two sample sizes in Fig. 10a. *E*_{rr}∕*E*_{i} 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
*E*_{rr}∕*E*_{i} than larger samples. Next, *E*_{rr}∕*E*_{i} decreases as the
impact angle increases. For size *D*=10 cm, *E*_{rr}∕*E*_{i} experiences a
steep decline from the first impact angle interval to the third and then
decreases gently to the fourth. For size *D*=20 cm, *E*_{rr}∕*E*_{i} has a
gradual reduction over time, which may be attributed to its small variation
range. Finally, the improvement of the impact angle results in more
concentrated data points, although data points for size *D*=10 cm are always
more scattered than size *D*=20 cm. Both the impact angle and the sample
size have an important impact on *E*_{rr}∕*E*_{i} in this study. In
conclusion, larger samples are more likely to have a stead and small
*E*_{rr}∕*E*_{i} than small samples, and a large impact angle
leads to a small *E*_{rr}∕*E*_{i}.

As mentioned in Sect. 4.1, the unexpected direction transition of
translational velocity always occurs when the impact angle is small. The
correlation between the direction transition of translational velocity and *E*_{rr}∕*E*_{i}
is investigated using *β*∕*α* (the ratio between the rebound angle
and the impact angle) as a reference. Figure 10b illustrates the trends for
*E*_{rr}∕*E*_{i} versus *β*∕*α*. *E*_{rr}∕*E*_{i} increases as the
ratio *β*∕*α* increases. If *β*∕*α* is smaller than 1.0, the
variation range of *E*_{rr}∕*E*_{i} is rather limited, 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 *β*∕*α*, the data points become
scattered. Moreover, the improvement of *E*_{rr}∕*E*_{i} appears to be
terminated when *β*∕*α* reaches a specific value. Therefore, we
can conclude that a strong correlation occurs between *E*_{rr}∕*E*_{i} and
*β*∕*α*. For a given impact angle, larger rebound angles indicate
that more kinetic energy is converted to rotational energy during the collision.

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 *E*_{rr}∕*E*_{i} and the
coefficients of restitution is investigated in this section to evaluate the
effect of rotation.

Figure 11 plots the coefficients of restitution versus *E*_{rr}∕*E*_{i} for this
study. The four coefficients fall into two categories according to their
responses to *E*_{rr}∕*E*_{i}. The first category includes *R*_{n}
and *R*_{t}, the two most commonly used coefficients of restitution, which
appear to be strongly correlated with *E*_{rr}∕*E*_{i}. As *E*_{rr}∕*E*_{i}
increases, *R*_{n} increases but *R*_{t} decreases, which verified the Broili (1973)
deduction. The rotation generated from impact results in an increased
normal velocity and reduced tangential velocity. Furthermore, in case that
more kinetic energy is converted to rotational energy during the collision,
the collision yields a higher *R*_{n} and lower *R*_{t}. In Fig. 11a and b,
data points for two sizes are not mixed, which can be attributed to the
effect of sample sizes on the magnitude of *E*_{rr}∕*E*_{i}. In addition, the
data points become more scattered with increasing *E*_{rr}∕*E*_{i}.
*R*_{v} and *R*_{E} belong to the second category. There is no remarkable correlation
between them and *E*_{rr}∕*E*_{i}, as shown in Fig. 11c and d, so *R*_{v} and
*R*_{E} 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 *R*_{n} and a smaller *R*_{t}, while it has no
distinct influence on *R*_{v} and *R*_{E}.

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 *E*_{rr}∕*E*_{i} on *R*_{n} and *R*_{t}, a smaller impact angle is
more likely to have a high *R*_{n} and a low *R*_{t} than a larger impact
angle. Therefore, *R*_{n} typically decreases with increasing impact angle,
and *R*_{t} increases as the impact angle increases. When the impact angle is
small, two sample sizes show a clear distinction in *E*_{rr}∕*E*_{i}, as shown
in Fig. 10a, which results in the difference in the mean values of *R*_{n}
and *R*_{t} between two sizes in the first impact angle interval in Fig. 5a and b.

5 Discussion

Back to toptop
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 *R*_{n} and a lower *R*_{t}. The
percentage can be associated with the ratio between the rebound angle and
the impact angle *β*∕*α*. As the impact angle decreases, the
ratio *β*∕*α* increases, and more kinetic energy is converted to
rotational energy. In this section, the reason why a small impact angle
achieves a high *β*∕*α* more easily and its consequences are discussed.

When the impact angle is small, the rebound angle easily exceeds the impact
angle and causes a high *β*∕*α*, which can be associated with the
impact orientation and the damage 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 lower hardness compared to the
falling samples. Supposing that the spherical polyhedron impacts the landing
plate with a corner or an edge, damage will occur. Figure 12a shows the
indentations on the surface caused by the impacts and the rough areas
resulting from the repeated damages. For an individual indentation, the
diameter *d* and the depth *h* are measured and noted in Fig. 12a.

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 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. However, when the impact angle is
small and generates a default rebound angle as shown by the dashed arrow,
rotation motion must be involved to overcome the constraint, and an
unexpectedly larger rebound angle occurs. Thus, the penetration caused by
the impact may contribute to the high *β*∕*α* in the case of small impact angles.

Another important factor in generating a high *β*∕*α* is the macro-roughness of the landing plate, which comes from repeated damage to 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. In the first stage, the sample impacts the surface before the stair and leaves the surface with the original rebound velocity. Then, the sample contacts the stair, and the rebound velocity changes in the second stage. The time interval between the two stages is so short
that the two stages appear to finish simultaneously. The probability that
two-stage interactions occur 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 where 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 the impact
angle must improve the risk of the sample contacting the stair. When the
impact angle is small, the sample has more of a possibility of having a two-stage
interaction and a high *β*∕*α*.

In conclusion, the restriction from the configuration of the indentation, as
well as the macro-roughness caused by repeated damage, is more likely to
affect the rebound motion when the impact angle is small. As a consequence,
the rebound angle easily exceeds the impact angle in the case of a small
impact angle, which ultimately results in a high *β*∕*α*.

Of the various consequences of the rebound angle being greater than the
impact angle, high values of the normal coefficient of restitution *R*_{n}
may be remarkable. Engineers usually take 1.0 as the upper bound
of *R*_{n} in computer codes, whereas several scholars have reported
*R*_{n} values larger than 1.0 (Azzoni et al., 1992; Paronuzzi, 2009; Spadari et al.,
2012). In this section, the relationship between *R*_{n} and the direction
transition of translational velocity is investigated.

Considering that the rotation before impact is small in this study, the
normal coefficient of restitution *R*_{n} can be expressed as Eq. (5) based
on the basic definition in Sect. 1.1:

$$\begin{array}{ll}{\displaystyle}{R}_{\mathrm{n}}& {\displaystyle}={v}_{\mathrm{nr}}/{v}_{\mathrm{ni}}=\sqrt{{E}_{\mathrm{rt}}/{E}_{\mathrm{it}}}\times (\mathrm{sin}\mathit{\beta}/\mathrm{sin}\mathit{\alpha})=\sqrt{{E}_{\mathrm{rt}}/{E}_{\mathrm{i}}}\\ \text{(5)}& {\displaystyle}& {\displaystyle}\times (\mathrm{sin}\mathit{\beta}/\mathrm{sin}\mathit{\alpha}).\end{array}$$

By introducing an angle coefficient

$$\begin{array}{}\text{(6)}& {\displaystyle}{\displaystyle}\mathit{\lambda}=\mathrm{sin}\mathit{\beta}/\mathrm{sin}\mathit{\alpha},\end{array}$$

Eq. (6) can be simplified as

$$\begin{array}{}\text{(7)}& {\displaystyle}{\displaystyle}{R}_{\mathrm{n}}=\mathit{\lambda}\sqrt{{E}_{\mathrm{rt}}/{E}_{\mathrm{i}}}.\end{array}$$

*E*_{rt}∕*E*_{i}, the ratio between the translational energy after impact and
the total kinetic energy before impact, is plotted in Fig. 13a with respect
to the impact angle. As the impact angle increases, the mean value of
*E*_{rt}∕*E*_{i} increases when the impact angles are smaller than 36^{∘} and
then decreases. The peak values of *E*_{rt}∕*E*_{i} of the four impact angle
intervals decrease gradually with increasing impact angles. In this study,
the values of *E*_{rt}∕*E*_{i} are located in the range of 0.20 to 0.60 in most
cases, which provides us with a reference to explore the conditions of *R*_{n} larger than 1.0.

Figure 13b plots the relationship between *R*_{n} and the angle
coefficient *λ* under different *E*_{rt}∕*E*_{i}. The value of *R*_{n} increases
when increasing the angle coefficient *λ*. Even if
*E*_{rt}∕*E*_{i} is only 0.2, *R*_{n} 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 *E*_{rt}∕*E*_{i} is unchanged, a case in which the
rebound angle is larger than the impact angle must lead to a higher *R*_{n}.
Although the value of *λ* corresponding to *R*_{n}=1.0
varies with *E*_{rt}∕*E*_{i}, the condition *λ*>1.0 is
required to obtain *R*_{n} greater than 1.0. As shown in Fig. 13b,
*R*_{n} cannot exceed 1.0 if the rebound angle is lower than the impact angle. As
discussed in the previous sections, small impact angles easily result in
unexpectedly large rebound angles. If the angle coefficient *λ* formed
by the rebound and the impact angle is sufficiently large, *R*_{n} will
exceed 1.0 even though *E*_{rt}∕*E*_{i} is small. Furthermore, assuming a
constant *E*_{rt}∕*E*_{i}, the reduction in the impact angle decreases the
threshold value of the rebound angle that should be satisfied to achieve
an *R*_{n} in excess of unity, which means that smaller impact angles are more
likely to yield *R*_{n} larger than 1.0.

A smaller impact angle more easily obtains a high *β*∕*α* and a
high percentage of kinetic energy converted to rotational energy, which then
induces a higher *R*_{n}. However, the kinetic energy coefficient of
restitution *R*_{E} appears independent of the percentage of kinetic energy
converted to rotational energy. Therefore, simply treating a
higher *R*_{n} as a symbol of lower kinetic energy loss may be unreasonable.
Stronge (1991) 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 *R*_{n} and *R*_{E} is more complicated in this study, which
verifies the argument of Stronge (1991).

Increasing *R*_{n} will increase *R*_{E} initially but decrease it overall. For
simplicity, two boundaries (*R*_{n}=0.65 and *R*_{n}=0.95) are added in
Fig. 14. The kinetic energy coefficient of restitution *R*_{E} increases with
increasing *R*_{n} when *R*_{n}<0.65, in agreement with the
relationship between *R*_{n} and the energy loss level based on the
elastic-plastic response analysis. If *R*_{n} is greater than 0.95, a larger *R*_{n}
indicates a smaller *R*_{E}. High values of *R*_{n} are associated with
unexpectedly large rebound angles in this study, which means that the
unexpectedly large rebound angles can be related to a higher level of
kinetic energy loss. *R*_{E} is disordered if *R*_{n} lies in the range of 0.65, 0.95,
which is caused by the two different trends meeting. Therefore, the normal
coefficient of restitution *R*_{n} cannot be directly used in the evaluation
of the kinetic energy dissipation level.

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 restitution in rockfall trajectory simulation. However, some problems cause a barrier to developing a reasonable way to account for the effect of the impact angle in computer simulation.

First, 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. Taking *R*_{n} as an example, the effect of the
impact angle on *R*_{n} has been formulated by several different functions,
such as the linear function (Wu, 1985; Richards et al., 2001), power
function (Asteriou et al., 2012) and second-order polynomial (Cagnoli and
Manga, 2003). In this study, the power function provides the best
correlation coefficient in fitting data points of *R*_{n}. 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 cannot conclude which type of function is
the best choice to describe the effect, and it is questionable whether a uniform expression
occurs.

Another problem arises 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 scattered under small impact angle conditions, which means that using a typical value in the simulation may be unreliable.

Therefore, further research should be carried out 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).

6 Conclusions

Back to toptop
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 has 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 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 3-D motion
capture system. Spherical limestone polyhedrons with diameters of 10 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
restitution were estimated under free-fall 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 *R*_{n}, the kinematic coefficient of restitution *R*_{v} and
the kinetic energy coefficient of restitution *R*_{E} all decrease when
increasing the impact angle, while the tangential coefficient of
restitution *R*_{t} increases as the impact angle increases in most cases. However, the
reason for the magnitude difference in the coefficients of restitution
between the tests compared is unidentified because the tests differ from
each other in multiple test conditions.

In the free-fall test, the rotation after impact dissipates part of the
kinetic energy of the sample and plays an important role in the effect of
the impact angle on the coefficient of restitution. The test results show
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 yield a high *β*∕*α* for
the indentations and 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 increases. Given *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 *R*_{n} and a
lower *R*_{t}. However, no correlations are observed in this study between the
rotation energy and the other two coefficients of restitution, *R*_{v}
and *R*_{E}. In addition, *R*_{n} being larger than 1.0 can be related to the
rebound angle being greater than the impact angle under small impact angle conditions.

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 still lack 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 conditions and the absence of a uniform and reasonable function describing the effect of the impact angle.

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 a boulder with other shapes should be further investigated through more elaborate experiments. In view of this, the test results are valid for trajectory simulation codes based on a lumped-mass model and can be referenced in the trajectory predication of spherical rocks impacting hard surfaces using a rigid body model.

Data availability

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Data availability.

The data sets of this experiment have been uploaded to the Supplement.

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/nhess-18-3045-2018-supplement.

Author contributions

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Author contributions.

YW conceived the idea of this article and carried out the tests. WJ proposed some crucial suggestions for the evaluation of the experimental results and wrote the main part of the manuscript. SC contributed to the detailed design of the experiment. Data collecting, chart sorting and editing were completed by PS and CM.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

Sponsored by the Research Fund for Excellent Dissertation of China Three Gorges
University (no. 2018BSPY008), the National Natural Science Funds of China
(no. 51409150), the CSC Scholarship (no. 201707620009) and the Open Research
Programme of the Hubei Key Laboratory of Disaster Prevention and Mitigation (no. 2017KJZ03).

Edited by: Jean-Philippe Malet

Reviewed by: Maxime Farin and one anonymous referee

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