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

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

The normal and tangential coefficients of restitution are the most commonly
used definitions, and the two coefficients of restitution are typically
denoted as

Related quantities adopted in the definitions for the coefficient of restitution.

In addition, the ratio of kinetic energies before and after impact is used
to define the kinetic energy coefficient of restitution

In these definitions,

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

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

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

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

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

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

Materials used in this experiment:

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,

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 (

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.

The testing apparatus in this study:

The data acquisition system:

Four different inclined angles

Release conditions and results of our experiments.

In Table 1, the release conditions of our experiment are presented in
addition to the resulting impact velocities and angles,

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
(

The effect of impact angle on the normal coefficient of restitution

As Fig. 5b shows, the impact angle has a different effect on the tangential
coefficient of restitution

The variations of four coefficients of restitution:

Furthermore, the kinematic coefficient of restitution

Finally, the effect of the impact angle on the coefficient of kinetic energy
restitution

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

In addition, data points for all coefficients except

Test conditions of previous studies and our experiment.

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

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

Next, Fig. 6b indicates that the trends of

Results comparison between this study and existing small-scale
experiments in terms of the effect of the impact angle on the four coefficients
of restitution:

Finally, the trends of

In conclusion, various experimental conditions induce different results
for

In addition, the Asteriou et al. (2012) test provided the highest trend lines of

The mean value lines of the normal coefficient of restitution

Taking the ratio between the rebound angle and the impact angle

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

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.

The variation of

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

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

Direction transitions of translational velocities induced by impacts
under different impact angle conditions:

The effect of

Figure 10a shows the effect of the impact angle on

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

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

Figure 11 plots the coefficients of restitution versus

The variations of four 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

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

When the impact angle is small, the rebound angle easily exceeds the impact
angle and causes a high

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

Another important factor in generating a high

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

Of the various consequences of the rebound angle being greater than the
impact angle, high values of the normal coefficient of restitution

Considering that the rotation before impact is small in this study, the
normal coefficient of restitution

Figure 13b plots the relationship between

A smaller impact angle more easily obtains a high

The variation of the kinetic energy coefficient of restitution

Increasing

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

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).

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

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

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.

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

The supplement related to this article is available online at:

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.

The authors declare that they have no conflict of interest.

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