Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
Proc. R. Soc. A (2009) 465, 937–960
doi:10.1098/rspa.2008.0221
Published online 9 December 2008
A semi-analytical model for oblique impacts
of elastoplastic spheres
B Y C HUAN -Y U W U 1, * , C OLIN T HORNTON 2
AND
L ONG -Y UAN L I 2
1
School of Chemical Engineering, and 2School of Civil Engineering,
University of Birmingham, Birmingham B15 2TT, UK
Results of finite-element analysis (FEA) of oblique impacts of elastic and elastic,
perfectly plastic spheres with an elastic flat substrate are presented. The FEA results are
in excellent agreement with published data available in the literature. A simple model is
proposed to predict rebound kinematics of the spheres during oblique impacts. In this
model, the oblique impacts are classified into two regimes: (i) persistent sliding impact,
in which sliding occurs throughout the impact, the effect of tangential (elastic or plastic)
deformation is insignificant and the model reproduces the well-established theoretical
solutions based on rigid body dynamics for predicting the rebound kinematics and
(ii) non-persistent sliding impact, in which sliding does not occur throughout the impact
duration and the rebound kinematics depends upon both Poisson’s ratio and the normal
coefficient of restitution (i.e. the yield stress of the materials). For non-persistent sliding
impacts, the variation of impulse ratio with impact angle is approximated using an
empirical equation with four parameters. These parameters are sensitive to the values of
Poisson’s ratio and the normal coefficient of restitution, but can be obtained by fitting
numerical data. Consequently, a complete set of solutions is obtained for the rebound
kinematics, including the tangential coefficient of restitution, the rebound velocity at the
contact patch and the rebound rotational speed of the sphere during oblique impacts.
The accuracy and robustness of this model is demonstrated by comparisons with FEA
results and data published in the literature. The model is capable of predicting complete
rebound behaviour of spheres for both elastic and elastoplastic oblique impacts.
Keywords: granular materials; contact mechanics; impact dynamics; oblique impact;
coefficient of restitution
1. Introduction
Impact between two colliding bodies is of fundamental importance in numerous
engineering applications and scientific studies. A binary collision may appear to
be a very simple problem but, in fact, it is a very complex event. This is due to
the short duration and the high localized stresses generated that, in most cases,
result in both frictional and plastic dissipation. In addition, if rigid body sliding
does not occur throughout the impact, then local elastic deformation of the two
bodies becomes significant.
* Author for correspondence ([email protected]).
Received 30 May 2008
Accepted 6 November 2008
937
This journal is q 2008 The Royal Society
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
938
C.-Y. Wu et al.
Many previous studies have been dedicated to understanding rebound
behaviour during normal impacts of spheres. The original pioneering work on
impact of spheres is due to Hertz (1896). Following directly from his theory of
elastic contact, Hertz analysed the impact of frictionless elastic bodies by
ignoring the effect of stress waves. From the theory of Hertz, it is possible to
obtain a good approximate solution for the normal impact of elastic bodies. For
instance, the duration of impact was determined (Johnson 1987) and was shown
that it is proportional to the radius of the sphere and inversely proportional to
1=5
Vni , where Vni is the initial normal impact velocity. The validity of the Hertz
theory was demonstrated by experiments reported by Andrews (1930), who
investigated the impact of two equal spheres of soft metal with low impact
1=5
velocities and confirmed that the duration of impact varies inversely as Vni and
the coefficient of restitution is very close to unity. The energy losses due to elastic
wave propagation during an elastic impact was analysed by Hunter (1957), who
showed that, for a steel ball impinging on a large block of steel or glass, less than
1 per cent of the kinetic energy of the ball is converted into elastic waves. The
energy dissipation during the normal impact of an elastic sphere with an elastic
substrate of finite size was analysed by the present authors using the finiteelement method (Wu et al. 2005), in which the effect of the substrate size on the
rebound behaviour of the sphere was investigated. By varying the substrate size,
the number of reflections of stress wave propagation within the contact duration
was altered. It was found that the energy dissipation due to stress waves is less
than 1 per cent of the total initial kinetic energy if there is more than one
reflection during the contact. If there is no reflection within the contact duration,
a significant amount of kinetic energy is dissipated due to stress wave
propagation, where the ratio of kinetic energy dissipated to the initial total
kinetic energy is proportional to the impact velocity with a power law of 3/5
(Wu et al. 2005), which is consistent with the analysis of Hunter (1957).
For the normal impact of elastoplastic spheres, kinetic energy may be dissipated
by stress wave motion and plastic deformation of the contacting bodies. The energy
dissipated by stress wave propagation during plastic impact was analysed by
Hutchings (1979), who showed that only a few per cent of the initial kinetic energy is
normally dissipated by stress waves. For instance, when a hard steel sphere collides
with a mild steel block at a velocity of approximately 70 m sK1, the measured
coefficient of restitution is approximately 0.4, but only approximately 3 per cent of
the kinetic energy is dissipated by stress waves. Hence, the fraction of the kinetic
energy dissipated by stress waves is very small, and plastic deformation is the
primary cause of kinetic energy dissipation during plastic impacts. Goldsmith and
co-workers (Goldsmith 1960; Goldsmith & Lyman 1960) reported some experimental results for plastic impacts of spheres and showed that the coefficient of
restitution is dependent on certain materials properties and, more significantly, on
the relative impact velocity. The impact of two nylon spheres was experimentally
studied by Labous et al. (1997) using high-speed video analysis. The velocity
dependences of the coefficient of restitution were investigated. It was suggested that
the basic energy dissipation mechanism at high impact velocities is plastic
deformation. More recently, accurate measurements of the coefficient of restitution
have been made by Kharaz et al. (2001) for the impact of 5 mm elastic (aluminium
oxide) spheres on thick plates of steel and aluminium alloy over a wide velocity range.
The variations in the coefficient of restitution with impact velocity were reported.
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
A model for oblique impacts of spheres
939
By ignoring the energy losses due to stress waves, several theoretical models
have been developed to predict the coefficient of restitution during the impact of
elastoplastic spheres. Johnson (1987) proposed a simplified model for fully plastic
impacts and showed that the coefficient of restitution is a power law function of
the impact velocity with an exponent of K1/4. Thornton (1997) developed a
theoretical model for the collinear impact of two elastic, perfectly plastic spheres,
accounting for the transition from elastic to fully plastic impacts, and an explicit
analytical solution for the coefficient of restitution was given. Li et al. (2002)
developed a more accurate and sophisticated model for the normal impact of an
elastic, perfectly plastic sphere, which was justified by experimental and finiteelement analysis (FEA) results (Li et al. 2002; Wu et al. 2003a). The coefficient of
restitution for normal impact of elastoplastic spheres of various material
properties over a wide range of impact velocities was reported by Wu et al.
(2003a), in which elastoplastic impacts were classified into two regimes:
elastoplastic impacts and finite, plastic deformation impacts. It was found
that, for elastoplastic impacts, the coefficient of restitution is mainly dependent
on the ratio of the impact velocity Vni to the yield velocity Vy; while for impacts
of finite, plastic deformation, it is also dependent on the ratio of the
representative Young’s Modulus E to the yield stress Y.
The situation becomes more complicated for the oblique impact of particles, as
tangential reaction plays an important role in the rebound behaviour and, as
pointed out by Mindlin & Deresiewicz (1953), the response at any instant
depends not only on the present value of the normal and tangential forces, but
also on the history of such loadings. Rigid body dynamics was first developed as
an initial simple approach to predict the impact behaviour of objects (Goldsmith
1960) and has been extensively used (Keller 1986; Brach 1988, 1991; Smith 1991;
Wang & Mason 1992; Stronge 1993; Brogliato 1996) However, it has been shown
by Maw et al. (1976, 1981) and Johnson (1983, 1987) that this assumption cannot
accurately predict the rebound behaviour at small impact angles, in which the
tangential surface velocity reverses its direction during the impact (for example,
when Poisson’s ratio has a value of 0.3). Since rigid body dynamics is, by its
nature, based on the impulse–momentum law and does not include material
properties, the influence of the contact deformation is ignored and the tangential
compliance of the bodies is not taken into account. Hence it cannot predict the
stresses and contact forces induced during the impact. Its accuracy in predicting
the rebound behaviour of bodies is limited (Johnson 1987), and it cannot account
for the plastic deformation that is the primary energy dissipation mechanism. In
order to accurately predict the oblique impact behaviour of two bodies, it is
essential that the contact deformation is taken into account.
By considering the elastic deformation of contacting bodies, Mindlin (1949)
and Mindlin & Deresiewicz (1953) analysed the contact of elastic spheres under
tangential loadings. Their analysis showed that the contact response at any
instant depends not only upon the value of the normal and tangential contact
forces, but also upon the previous loading history. Therefore, changes in contact
radius, contact pressure and tangential traction must be calculated step by step.
This methodology was employed by Maw et al. (1976, 1981) to analyse the
oblique impact of two elastic spheres in conjunction with the Hertz theory of
elastic normal contact. The variation of the rebound tangential surface velocity
of the contact patch with impact angle was obtained. Their analysis has been
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
940
C.-Y. Wu et al.
substantiated by experiments (Maw et al. 1976, 1981; Foerster et al. 1994;
Labous et al. 1997; Kharaz et al. 2001). In the analysis of Maw et al. (1976, 1981),
all the normal effects are handled by the Hertz theory, so that the normal impact
is assumed to be purely elastic and the elastic wave effects are neglected. It does
not account for any normal energy loss during the impact and assumes a
coefficient of restitution of unity. Therefore, it cannot be used to predict the
rebound behaviour of oblique impact involving plastic deformation. An attempt to
deal with plastic oblique impact has been made by Stronge (1994), who developed a
lumped parameter model of contact between colliding bodies, in which it is assumed
that both colliding bodies are rigid, except for an infinitesimally small deformable
region that separates the bodies at the contact points. The lumped parameter
model allowed the effect of the normal coefficient of restitution to be taken into
account and the rebound tangential surface velocity at the contact patch was
obtained as functions of the impact angle and the normal coefficient of restitution.
In this paper, using FEA, we investigate how the complete rebound kinematics
depends on the impact velocity, impact angle and the degree of plastic
deformation. From an examination of the results, we develop a simple semianalytical model for predicting the complete rebound kinematics for both elastic
and elastoplastic spheres.
2. Theoretical aspects
We consider an oblique impact of a sphere with a target wall in the y–z plane by
ignoring the spins around the y- and z -axes and corresponding moment impulses,
and suppose that the sphere approaches the wall with an initial translational
velocity Vi and angular velocity ui at an impact angle qi (figure 1). After
interaction with the wall, the sphere rebounds with a rebound translational
velocity Vr and rebound angular velocity ur. Note that Vi and Vr are the
velocities of the sphere centre. The corresponding translational velocities at the
contact patch are denoted by vi and vr. We introduce normal and tangential
coefficients of restitution e n and e t,
e n ZKVnr =Vni
ð2:1aÞ
e t Z Vtr =Vti ;
ð2:1bÞ
and
where Vni and Vnr are the normal components of the impact speed and rebound
speed, respectively, and Vti and Vtr are the corresponding tangential velocity
components. It should be noted that it is necessary in (2.1a) to introduce the
negative sign since the normal component of the velocity reverses its direction after
the impact (Vnr is in the opposite direction to Vni) and the normal coefficient of
restitution is usually quoted as a positive value. The tangential coefficient
of restitution can be negative because, with initial spin, under certain conditions the
sphere can bounce backwards (Batlle 1993; Batlle & Cardona 1998).
The coefficients e n and e t can be used to represent the recovery of translational
kinetic energy in the normal and tangential directions, respectively. The recovery
of total translational kinetic energy during the impact can be obtained by
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
A model for oblique impacts of spheres
941
i
r
i
r
vr
cr
Figure 1. Diagram of the oblique impact of a sphere with a plane surface.
defining a total coefficient of restitution e as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
Vr
Vnr
Vtr2
Z e 2n cos2 qi C e 2t sin2 qi :
Z
C
eZ
Vi
Vni2 =cos2 qi Vti2 =sin2 qi
ð2:2Þ
It follows from (2.2) that the total coefficient of restitution e is dominated by e n
at small impact angles (qi/08) and by e t at large impact angles (qi/908).
The correlation between the tangential and normal interactions during the
impact can be characterized by an impulse ratio, which is defined as
Ð
F dt
Pt
fZ
ZÐ t ;
ð2:3Þ
Pn
Fn dt
where Pn and Pt are the normal and tangential impulses, respectively, and Fn
and Ft are the normal and tangential components of the contact force. It is clear
that the impulse ratio f is different to the interface friction coefficient m; it may or
may not be equal to m (Brach 1988).
According to Newton’s second law, Pn and Pt can be expressed in terms of the
incident and rebound velocities as
ð2:4aÞ
Pn Z mðVnr K Vni Þ
and
Pt Z mðVtr K Vti Þ;
ð2:4bÞ
where m is the mass of the particle. Substituting (2.4a) and (2.4b) into (2.3) and
using (2.1a) and (2.1b), we obtain
ð2:5Þ
e t Z 1Kf ð1 C e n Þ=tan qi :
Similarly, a rotational impulse Pu can be defined by
ð2:6Þ
Pu Z I ður K u i Þ;
where I is the moment of inertia of the sphere, and ui and ur are the initial
and rebound rotational angular velocities, respectively. According to the
conservation of angular momentum about point C (figure 1), we have
Pu Z RPt ;
ð2:7Þ
where R is the radius of the sphere.
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
942
C.-Y. Wu et al.
Substituting (2.4b) and (2.6) into (2.7) yields
ur Z u i KmRðVti K Vtr Þ=I :
ð2:8Þ
For a solid sphere, I Z 2mR2 =5. Hence,
5ðVti K Vtr Þ
5V ð1K e t Þ
Z u i K ti
:
ð2:9Þ
2R
2R
Substituting (2.5) into (2.9), we obtain
5f ð1 C e n ÞVni
:
ð2:10Þ
ur Z u i K
2R
The tangential component of the rebound surface velocity at the contact patch,
vtr, can be expressed as
ð2:11Þ
vtr Z Vtr C Rur :
Substituting (2.10) into (2.11), we obtain
ur Z u i K
5
vtr Z Vtr C Ru i K f ð1 C e n ÞVni :
2
Combining (2.1b), (2.5) and (2.12), we obtain
7
vtr Z vti K f ð1 C e n ÞVni
2
ð2:12Þ
ð2:13aÞ
or
7
vtr Z vti K ð1K e t ÞVti :
2
Equation (2.13b) can be rewritten as
5 2v
2Ru i
:
e t Z C tr K
7 7Vti
7Vti
ð2:13bÞ
ð2:14Þ
From figure 1, the rebound angle qr can be obtained from
tan qr Z
Vtr
e
ZK t tan qi :
Vnr
en
ð2:15Þ
It can be seen from (2.5), (2.10), (2.13a) and (2.13b) that all the kinematics of the
rebounding sphere depend upon the impact angle, the initial impact speed and
particle spin, the normal coefficient of restitution e n and the impulse ratio. In
other words, for a given impact angle and impact speed, the rebounding
kinematics of the sphere can be determined once e n and f are known (Brach 1988,
1991). Many studies have been carried out to investigate the normal coefficient of
restitution e n during elastoplastic impacts, and the rebound behaviour of
elastoplastic spheres during normal impacts is well established (Johnson 1987;
Thornton 1997; Thornton & Ning 1998; Kharaz et al. 2001; Li et al. 2000, 2002;
Thornton et al. 2001; Wu et al. 2003a). The impulse ratio can be determined by
measuring the initial and rebound velocities at the sphere centre (Brach 1988,
1991; Cheng et al. 2002). Accurate determination of the impulse ratio becomes
more challenging when the impact angle is very small or very large. A close
examination of (2.5), (2.10), (2.13a) and (2.13b) reveals that the rebound
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
A model for oblique impacts of spheres
943
parameters are not independent, but are correlated with each other. For
instance, rewriting (2.10), we have
2Rður K u i Þ
f ZK
:
ð2:16Þ
5ð1 C e n ÞVni
Substituting (2.16) into (2.5) and (2.13a), we obtain
2Rður K u i Þ
2 Rður K u i Þ m
et Z 1 C
Z1C
ð2:17Þ
5Vni tan qi
5
mVni
tan qi
and
7
ð2:18Þ
vtr Z vti C Rður K u i Þ:
5
Equation (2.18) can be rewritten as
vtr
v
7 Rður K u i Þ tan qi 7 Rur 2 Ru i
C
Z ti C
Z
K
:
ð2:19Þ
mVni
5 mVni 5 mVni
mVni
mVni 5
m
From (2.16)–(2.19), it is clear that both the tangential coefficient of restitution e t
and the tangential rebound velocity at the contact patch can be expressed as a
function of the rebound rotational angular velocity ur. Furthermore, recent
experimental studies have shown that the rebound rotational angular velocity
could be measured with high accuracy (Foerster et al. 1994; Kharaz et al. 2001).
Therefore, the rebounding kinematics can be determined without recourse to the
impulse ratio f if the rebound rotational angular speed ur can be predicted.
By taking into account the energy loss in the normal direction during plastic
impacts (i.e. the normal coefficient of restitution) and referring to (2.10) and
(2.12), we introduce dimensionless angular velocities Ur and Ui, a dimensionless
rebound tangential surface velocity at the contact patch Jr and a dimensionless
impact angle Q as follows:
2R
Ur Z
u;
ð2:20aÞ
5ð1 C e n ÞmVni r
2R
u;
ð2:20bÞ
Ui Z
5ð1 C e n ÞmVni i
Jr Z
2
2J2
vtr Z
ð1 C e n ÞmVni
ð1 C e n Þm
ð2:20cÞ
and
2
2J1
tan qi Z
;
ð1 C e n Þm
ð1 C e n Þm
where J1 and J2 are the parameters used by Foerster et al. (1994).
Hence, (2.10), (2.17) and (2.19) can be rewritten as
f
Ur Z Ui K ;
m
QZ
et Z 1 C
2ðUr K Ui Þ
2f
Z 1K
Q
Qm
and
Jr Z Q C 7Ur K2Ui Z Q C 5Ui K
Proc. R. Soc. A (2009)
ð2:20dÞ
ð2:21Þ
ð2:22Þ
7f
:
m
ð2:23Þ
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
944
C.-Y. Wu et al.
It can be seen from (2.22) and (2.23) that the tangential coefficient of restitution
e t , the dimensionless rebound tangential velocity at the contact patch Jr and the
dimensionless rebound rotational angular speed Ur are related to each other and
are functions of Ui and Q. If any one of these parameters can be experimentally
measured, the value of f/m can then be determined from (2.21), (2.22) or (2.23),
and the other two parameters can also be determined. Note that the derivations
given above are general formulations, which means that the equations are
applicable either for elastic oblique impacts (e nZ1) or elastoplastic oblique
impacts (e ns1).
3. The finite-element model
The oblique impact of a sphere with a substrate was simulated using the
DYNA3D code (Whirley & Engelmann 1993) The three-dimensional finiteelement model is shown in figure 2. Owing to geometrical and loading
symmetries, only half of the model is considered and discretized. The sphere
has a radius RZ10 mm. The substrate is selected as 10 mm in both the x - and
z -directions and 20 mm in the y-direction. Since further increasing the size of the
substrate does not produce any difference in the results (Wu 2001), the size of the
substrate is considered large enough to represent a half-space for the velocities
considered in this study. The meshes consist of 18 632 eight-node solid elements
with 20 097 nodes in the sphere and 21 896 elements with 26 236 nodes in the
substrate. Fine meshes are used in the vicinity of initial contact points in order to
accurately describe the localized deformation.
Interaction between the sphere and half-space is modelled by employing a
sliding interface defined as ‘sliding with separation and friction’ (Whirley &
Engelmann 1993), which allows two bodies to be either initially separate or in
contact and permits large relative motions with friction. In the present study,
Coulomb’s law of dry friction is used, and coefficients of static and dynamic
friction are assumed to be identical and remain constant with mZ0.3 for all
impact cases considered here, since all results for the impacts of spheres with
various friction coefficients coalesce onto a single curve using the dimensionless
parameters proposed in this study (Wu 2001). Nodes on the symmetry plane
(xZ0) are restricted in the x -direction. Nodes on boundaries (planes xZ10,
yZG10 and zZK10) are fixed. The half-space is assumed to be elastic and the
sphere to be either elastic or elastic, perfectly plastic, so two different impact
cases were considered: an impact of an elastic sphere with an elastic half-space
(EE impact) and an impact of an elastic, perfectly plastic sphere with an elastic
half-space (PE impact). The corresponding material properties are listed in
table 1. These properties represent a typical steel material. Additionally, impacts
of a rigid sphere with an elastic half-space (RE impacts) are also analysed in
order to explore the effect of Poisson’s ratio on the rebound kinematics. The
impact is modelled by applying an initial velocity Vi to every node within the
sphere at an angle qi. Different impact angles varying from 08 (normal impact) to
858 (close to glancing) are considered. In this study, different impact angles are
specified by keeping the normal component of initial velocity fixed, so that the
change in impact angle will only change the tangential component of the initial
velocity and the effect of tangential response on the normal response can thus be
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
945
A model for oblique impacts of spheres
(a)
(b)
A
i
B
C
D
Figure 2. FE model for the oblique impact of a sphere with a half-space (unit: mm): (a) the model
and (b) the finite-element meshes.
Table 1. Material properties in oblique impacts.
material property
E (GN mK2)
n
Y (GN mK2)
r (Mg mK3)
elastic
elastic, perfectly plastic
208.0
208.0
0.3
0.3
—
1.85
7.85
7.85
explored more directly. Two different values of the normal initial velocity are
specified (VniZ2.0 m sK1 and 5.0 m sK1) for EE and PE impacts, while only
VniZ5.0 m sK1 is chosen for RE impacts. In addition, no initial rotation is
considered for all impact cases, i.e. uiZ0.
4. Impact of a rigid sphere with an elastic substrate (RE impact):
effect of Poisson’s ratio
It can be seen from §2 that knowing the value of impulse ratio f for an oblique
impact is a key to obtaining the complete rebound kinematics, which is not a
trivial task for experimentalists. However, this can be readily obtained from
numerical analysis with the finite-element models presented in §3. The impulse
ratio is calculated using (2.3), in which the normal and tangential impulses are
obtained by integrating the normal and tangential contact forces over time,
respectively. The impulse ratios for RE impacts with various Poisson ratios are
shown in figure 3. It is clear that the impulse ratio increases with dimensionless
impact angle until a critical angle is reached, above which f/m is essentially equal
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
946
C.-Y. Wu et al.
1.2
1.0
0.8
0.6
equation (4.5)
0.4
0.2
0
2
4
6
8
10
12
14
Figure 3. The variation of f/m with dimensionless impact angle for elastic impacts with various
Poisson ratios. Squares, nZ0.0; diamonds, nZ0.3; circles, nZ0.49.
to unity, indicating that sliding persists throughout the whole duration of the
impact. This is referred to as a persistent sliding impact. The critical normalized
impact angle above which sliding occurs throughout the impact is given by
Qc Z
7k K 1
;
k
ð4:1Þ
which corresponds to the criterion of Maw et al. (1976, 1981). In (4.1), k is the
ratio of the initial tangential contact stiffness (FtZ0) to the normal contact
stiffness and is defined by
ð1K n1 Þ=G1 C ð1K n2 Þ=G2
:
kZ
n n 1K 21 =G1 C 1K 22 =G2
ð4:2Þ
For RE impacts, (4.2) can be rewritten as
kZ
2ð1K n1 Þ
:
ð2K n1 Þ
ð4:3Þ
The data shown in figure 3 were re-plotted against kQ=ð7k K1Þ in figure 4. It is
clear that, when kQ=ð7k K 1ÞR 1, all three cases coalescence and
f =m Z 1;
ð4:4Þ
for QR Qc :
For Q! Qc , sliding does not occur throughout the impact. We refer to this as a
non-persistent sliding impact. Hence, Qc marks the transition from nonpersistent to persistent sliding impacts. For non-persistent sliding impacts, f/m
is clearly a function of the dimensionless impact angle Q and Poisson’s ratio.
Rigorous prediction of the dependence of f/m on Q appears to be intractable.
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
947
A model for oblique impacts of spheres
1.2
1.0
0.8
0.6
0.4
0.2
0
0.5
1.0
1.5
2.0
2.5
/(7 –1)
Figure 4. The variation of f/m with kQ=ð7k K 1Þ for elastic impacts with various Poisson ratios.
Squares, nZ0.0; diamonds, nZ0.3; circles, nZ0.49.
In this study, a close examination of the data in this region suggests that, for a
given Poisson ratio, the correlation between f/m and Q can be given by the
following expression:
f =m Z c1 C c2 tanhðc3 C c4 QÞ;
ð4:5Þ
where c1, c2, c3 and c4 are parameters related to the properties of the colliding
bodies. The parameters for RE impacts with different Poisson ratios are
determined by curve fitting of the FEA data presented in figure 3 and are given
in table 2. Equating (4.4) and (4.5) gives the critical impact angle Qc , which is
listed in the last column of table 2. It can be seen that Qc is comparable with the
Qc given by (4.1).
Substituting (4.4) and (4.5) into (2.21), (2.22) and (2.23), the complete
rebound kinematics can be obtained as follows:
(
K½c1 C c2 tanhðc3 C c4 QÞ ðQ! Qc Þ
Ur Z
ð4:6Þ
K1
ðQR Qc Þ;
8
2
>
1K ½c1 C c2 tanhðc3 C c4 QÞ ðQ! Qc Þ
>
>
<
Q
et Z
ð4:7Þ
2
>
>
>
ðQR Qc Þ
: 1K Q
and
(
Jr Z
QK7½c1 C c2 tanhðc3 C c4 QÞ
ðQ! Qc Þ
QK7
ðQR Qc Þ:
ð4:8Þ
The rebound kinematics for RE impacts with different Poisson ratios are presented
in figures 5–7. In these figures, the open symbols denote the FEA results and the
solid lines are predictions using (4.6)–(4.8). In figure 5, the data of Johnson (1983)
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
948
C.-Y. Wu et al.
6
4
2
0
–2
0
2
4
6
8
10
12
14
Figure 5. The variation of the dimensionless tangential rebound velocity at the contact patch with
the dimensionless impact angle for elastic impacts with various Poisson ratios. Squares, nZ0.0;
diamonds, nZ0.49; circles, nZ0.3; triangles, nZ0.5 ( Johnson 1983).
Table 2. Parameters for RE impacts with different Poisson ratios.
Poisson ratio c1
nZ0.0
nZ0.3
nZ0.49
0.4627
0.4459
0.4268
c2
c3
c4
Qc
Qc
K0.6053
K0.6112
K0.6250
1.0000
0.9288
0.8297
K0.3832
K0.4050
K0.4263
6.0
5.786
5.520
6.1277
5.8363
5.4440
for the collision of a ‘superball’ with a Poisson ratio of 0.5 are also superimposed. It is
clear that the FEA results are in good agreement with Johnson’s experimental data.
It can be seen from these figures that the above equations accurately predict the
rebound kinematics of particles during oblique impacts.
A subtle feature of figure 5 is the detail at very small values of Q. For nZ0 and
0.3, the values of jr are positive for very small values of Q, but not in the case of
nZ0.49. This is in agreement with fig. 2 of Maw et al. (1976) and fig. 3 of Maw
et al. (1981). Furthermore, this has also been demonstrated by the experimental
data of Johnson (1983) and Kharaz et al. (2001). Although the differences in jr at
very small impact angles are small, as a consequence of equation (2.14), they
have a significant effect on the magnitude of the tangential coefficient of
restitution, as can be seen in figure 7.
5. Impact of an elastic sphere with an elastic substrate (EE impact)
Using nZ0.3, the variation of f/m with Q is shown in figure 8 for EE impacts with
different initial velocities. The figure clearly shows that, for elastic collisions, the
impulse ratio is independent of the magnitude of initial velocity for a given
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
949
A model for oblique impacts of spheres
0
– 0.2
– 0.4
– 0.6
– 0.8
–1.0
0
4
2
6
8
10
12
14
Figure 6. The variation of the dimensionless rebound angular velocity with the dimensionless
impact angle for elastic impacts with various Poisson ratios. Squares, nZ0.0; diamonds, nZ0.49;
circles, nZ0.3.
1.0
0.9
0.8
0.7
0.6
0.5
0
2
4
6
8
10
12
14
Figure 7. The variation of e t with the dimensionless impact angle for elastic impacts with various
Poisson ratios. Squares, nZ0.0; diamonds, nZ0.49; circles, nZ0.3.
impact angle. The value of f/m is found to increase as the normalized impact
angle Q increases when Q!Qc, and is constant at a value of unity when QRQc.
Fitting the data for Q!Qc with (4.5) gives the parameters listed in table 3. The
corresponding rebound kinematics for EE impacts are shown in figures 9–11. In
these figures, data reported in the literature (Maw et al. 1976; Kharaz et al. 2001;
Thornton et al. 2001) are also superimposed. It is clear that our FEA results are
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
950
C.-Y. Wu et al.
1.0
0.8
equation (4.5)
0.6
0.4
0.2
0
2
4
6
8
10
12
14
Figure 8. The variation of f/m with the dimensionless impact angle for EE impacts with various
initial impact velocities. Squares, VniZ2.0 mm sK1; circles, VniZ5.0 mm sK1.
Table 3. Parameters for EE and PE impacts.
impact cases
c1
c2
c3
c4
Qc
EE VniZ2 mm sK1 and 5 mm sK1
PE VniZ2 mm sK1; e n
PE VniZ5 mm sK1; e n
0.4458
0.4768
0.4868
K0.6024
K0.6030
K0.6414
0.9522
1.0658
0.9841
K0.4341
K0.4245
K0.3669
5.6582
5.5799
5.5005
in good agreement with the published data. The predictions using (4.6)–(4.8) are
also superimposed in figures 9–11. It is clear that the predictions are in excellent
agreement with the FEA results, indicating that equations (4.6)–(4.8) can be
used to accurately predict the rebound behaviour for impacts of elastic spheres
with an elastic substrate.
It is worth noting that Walton (1992) and Louge and co-workers (Foerster
et al. 1994; Lorenz et al. 1997) proposed a simplified model for the rebound
behaviour. In their model, it was assumed that the variation of the dimensionless
rebound tangential surface velocity with the dimensionless impact velocity can
be represented using a bilinear model, i.e.
Jr ZKbQ
ðQ! Qc Þ
ð5:1Þ
Jr Z QK7
ðQR Qc Þ;
ð5:2Þ
and
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
951
A model for oblique impacts of spheres
7
6
5
4
3
2
1
0
–1
–2
0
2
4
6
8
10
12
14
Figure 9. The variation of the dimensionless tangential rebound velocity at the contact patch with
the dimensionless impact angle for EE impacts with various initial impact velocities. Squares, EE
(VniZ2.0 mm sK1); circles, EE (VniZ5.0 mm sK1); diamonds, discrete element method (DEM)
( Thornton et al. 2001); uptriangles, experimental (Kharaz et al. 2001); downtriangles, Maw et al.
(1976); solid line, present model; dashed line, bilinear model; dot-dashed line, rigid body dynamics.
0
– 0.2
– 0.4
– 0.6
– 0.8
–1.0
0
2
4
6
8
10
12
14
Figure 10. The variation of the dimensionless rebound angular velocity with the dimensionless
impact angle for EE impacts with various initial impact velocities. Squares, EE (VniZ2.0 mm sK1);
circles, EE (VniZ5.0 mm sK1); diamonds, DEM ( Thornton et al. 2001); uptriangles, experimental
(Kharaz et al. 2001); solid line, present model; dashed line, bilinear model; dot-dashed line, rigid
body dynamics.
where b is a constant. Using (4.1), (5.1) and (5.2), we identify that
1
bZ
:
7k K1
Substituting (5.1) and (5.2) into (2.23) gives
8
>
< ð1 C bÞ Q ðQ! Qc Þ
7
f =m Z
>
:
1
ðQR Qc Þ:
Proc. R. Soc. A (2009)
ð5:3Þ
ð5:4Þ
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
952
C.-Y. Wu et al.
1.0
0.9
0.8
0.7
0.6
0.5
0
2
4
6
8
10
12
14
Figure 11. The variation of e t with the dimensionless impact angle for EE impacts with various
initial impact velocities. Squares, EE (VniZ2.0 mm sK1); circles, EE (VniZ5.0 mm sK1);
uptriangles, experimental (Kharaz et al. 2001); solid line, present model; dashed line, bilinear
model; dot-dashed line, rigid body dynamics.
Substituting (5.4) into (2.21) and (2.22) leads to
8
>
<K ð1 C bÞ Q ðQ! Qc Þ
7
Ur Z
>
:
K1
ðQR Qc Þ
and
et Z
8
2ð1 C bÞ
>
>
>
1K
>
<
7
ðQ! Qc Þ
>
>
>
>
:
ðQR Qc Þ:
2
1K
Q
ð5:5Þ
ð5:6Þ
In addition, rigid body dynamics (Brach 1988) assumes that the rebound
tangential surface velocity at the contact patch can be either zero, i.e. vtrZ0 for
f!m or vtrR0 if fZm. This implies that
8
>
< Q ðQ! Qc Þ
f =m Z 7
ð5:7Þ
>
:
1 ðQR Qc Þ;
with QcZ7.
Substituting (5.7) into (2.21) and (2.22) gives
8
>
<K Q
7
Ur Z
>
:
K1
and
Proc. R. Soc. A (2009)
ðQ! Qc Þ
ðQR Qc Þ
ð5:8Þ
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
A model for oblique impacts of spheres
et Z
85
>
>
>
<7
ðQ! Qc Þ
2
>
>
>
: 1K Q
ðQR Qc Þ:
953
ð5:9Þ
The predictions of the bilinear model and rigid body dynamics are also
superimposed in figures 9–11. It can be seen that rigid body dynamics
overestimates the critical impact angle above which sliding occurs throughout
the impact. For impacts with sliding throughout, all three models give identical
predictions and agree with experimental and numerical results. However, for the
impacts where sliding does not occur throughout the impact, both the bilinear
model and rigid body dynamics give constant tangential coefficients of restitution
e t , which is not supported by experimental and numerical results (figure 11).
Also, these two models overestimate the tangential rebound velocity at the
contact patch (figure 9) and the rebound rotational angular speeds (figure 10),
especially at intermediate impact angles.
6. Impact of an elastoplastic sphere with an elastic substrate
(PE impact)
In the impact cases discussed in the preceding §§4 and 5, the deformation is
elastic and energy dissipated in the normal direction is negligible, so that the
normal coefficient of restitution e n has a value close to unity. In this section,
we introduce plastic deformation. Consequently, a certain portion of energy will
be dissipated by plastic deformation of the sphere and the normal coefficient of
restitution e n will have a value less than unity. How this affects the rebound
behaviour during oblique impacts is reported below.
Figure 12 shows the normal coefficient of restitution at different impact angles
for various EE and PE impacts considered. As expected, for EE impacts, the
normal coefficients of restitution are very close to unity, regardless of the impact
angle and the impact speed. For PE impacts, lower normal coefficients of
restitution are obtained for higher values of the initial normal velocity (say,
VniZ5.0 mm sK1). This indicates that the normal coefficient of restitution is
velocity dependent (Johnson 1987; Thornton 1997; Thornton & Ning 1998; Li
et al. 2000, 2002; Kharaz et al. 2001; Thornton et al. 2001; Wu et al. 2003a). It is
found that, for PE impacts with a constant normal component of initial velocity,
the normal coefficient of restitution is essentially constant for small impact
angles, during which sliding does not occur until the very end of the impact. At
higher impact angles, when QO 1=k, sliding occurs at the start of the impact and
the normal coefficient of restitution decreases at an increasing rate until
QZ ð7kK3Þ=k when sliding occurs during the whole of the loading period
(Wu 2001). With further increases in impact angle, the normal coefficient
of restitution continues to decrease, but at a decreasing rate until QR QcZ
ð7k K1Þ=k when sliding occurs throughout impact, and the normal coefficient of
restitution then remains essentially constant at large impact angles. The overall
relative change in the normal coefficient of restitution between high and low
impact angles increases as the normal component of initial velocity increases
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
954
C.-Y. Wu et al.
1.1
1.0
0.9
0.8
0.7
0.6
0
20
60
40
impact angle q i
80
Figure 12. The variation of normal coefficient of restitution with impact angle. Open circles, EE
(VniZ2.0 mm sK1); open squares, EE (VniZ5.0 mm sK1); filled circles, PE (VniZ2.0 mm sK1);
filled squares, PE (VniZ5.0 mm sK1).
(i.e. 6.6% for VniZ5.0 mm sK1 and 3.0% for VniZ2.0 mm sK1). A similar
phenomenon is also observed for oblique impacts of an elastic, perfectly plastic
sphere with a rigid wall (Wu et al. 2003a). This feature is difficult to identify
from conventional experiments, which normally vary the impact angle with the
impact speed constant and the normal component of the initial velocity therefore
decreases as the impact angle increases. Consequently, the normal coefficient of
restitution will increase as the impact angle increases due to the decreasing
normal component of initial velocity. However, a close examination of the normal
coefficient of restitution for oblique impacts at constant speed reveals that there
is a slight kink in the normal coefficient of restitution curve at intermediate
impact angles (Wu et al. 2003b), which corresponds to a decrease in the normal
coefficient of restitution at intermediate impact angles when the normal
component of the initial velocity is constant, as shown in figure 12. Experimental
evidence of this phenomenon can be found in fig. 15 of Brauer (1980). This
indicates that, for the oblique impact of plastic particles, the normal coefficient of
restitution is not merely a function of the normal impact velocity, but also
depends on impact angle. It is believed that the dependency of e n on qi is due to
the subtle change in geometry (contact curvature) around the contact patch
when permanent plastic deformation occurs in one of the contacting bodies.
Figure 13 shows the variation of f/m with normalized impact angle Q for PE
impacts with VniZ2.0 mm sK1 and 5.0 mm sK1. It can be seen that, similar to
the RE and EE impacts shown in figures 3 and 8, f/m increases as the impact
angle increases until the normalized impact angle Q reaches Qc; thereafter f/m
has a constant value of unity. Although the results for the impacts with different
initial normal impact velocities are generally close, some differences exist at
intermediate impact angles. The data for these two impact cases were separately
fitted using (4.5), and the resultant parameters are given in table 3.
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
955
A model for oblique impacts of spheres
1.0
0.8 equation (4.5)
0.6
0.4
0.2
0
2
4
6
8
10
12
14
Figure 13. The variation of f/m with the dimensionless impact angle for PE impacts with various
initial impact velocities. Squares, VniZ2.0 mm sK1; circles, VniZ5.0 mm sK1.
The corresponding rebound kinematics are presented in figures 14–16.
Considering the variation of the normal coefficient of restitution with impact
angle, as shown in figure 12, the FEA data presented in the figures are normalized
by using (i) the actual normal coefficient of restitution e n, i.e. variable e n was used
in the normalization and (ii) the normal coefficient of restitution e n at the normal
impact (qiZ08) and ignoring the variation of e n with impact angle. The second
approach is denoted as ‘fixed e n0’ in figures 14–16. It can be seen, from these figures,
that the normalization using these two different approaches give generally the same
trend and the results are very close, with negligible deviation. This indicates that
the influence of the variation of e n with qi shown in figure 12 on the rebound
kinematics is insignificant and can be ignored. The predictions using (4.6)–(4.8) are
also superimposed in these figures using solid lines for VniZ2.0 mm sK1 and dashed
lines for VniZ5.0 mm sK1, respectively.
Figure 14 shows the variation of the dimensionless tangential rebound
velocity at the contact patch with the dimensionless impact angle. We have
also superimposed the DEM results of Thornton et al. (2001), the experimental
results presented in Gorham & Kharaz (2000) and the numerical results given
by Maw et al. (1976). The corresponding dimensionless rebound angular
velocity Ur is plotted against the dimensionless impact angle Q in figure 15, in
which the DEM results of Thornton et al. (2001) and the experimental results of
Gorham & Kharaz (2000) are also superimposed. Figure 16 shows the variation
of e t with Q, in which the experimental data of Gorham & Kharaz (2000) are
also superimposed. It can be seen that the FEA results are in good agreement
with the experimental results of Gorham & Kharaz (2000) and the DEM results
of Thornton et al. (2001). Furthermore, the predictions of (4.6)–(4.8) are in
excellent agreement with the FEA results for both impact cases. The difference,
for all rebound kinematics, between the two impact cases at intermediate
impact angles is clearly seen, indicating that the results are sensitive to e n in
this impact region.
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
956
C.-Y. Wu et al.
7
6
5
4
3
2
1
0
–1
–2
0
2
4
6
8
10
12
14
Figure 14. The variation of the dimensionless tangential rebound velocity at the contact patch with
the dimensionless impact angle for PE impacts with various initial impact velocities. Circles, PE
(VniZ2.0 mm sK1); squares, PE (VniZ5.0 mm sK1); pluses, PE (VniZ2.0 mm sK1; fixed en0);
crosses, PE (VniZ5.0 mm sK1; fixed en0); diamonds, DEM ( Thornton et al. 2001); uptriangles,
experimental (Gorham & Kharaz 2000); downtriangles, Maw et al. (1976); solid line, present model
(VniZ2.0 mm sK1); dashed line, present model (VniZ5.0 mm sK1).
0
– 0.2
– 0.4
– 0.6
– 0.8
–1.0
0
2
4
6
8
10
12
14
Figure 15. The variation of the dimensionless rebound angular velocity with the dimensionless
impact angle for PE impacts with various initial impact velocities. Circles, PE (VniZ2.0 mm sK1);
squares, PE (VniZ5.0 mm sK1); pluses, PE (VniZ2.0 mm sK1; fixed en0); crosses, PE (VniZ
5.0 mm sK1; fixed en0); diamonds, DEM ( Thornton et al. 2001); uptriangles, experimental
(Gorham & Kharaz 2000); solid line, present model (VniZ2.0 mm sK1); dashed line, present model
(VniZ5.0 mm sK1).
7. Initial particle spin
In the above cases, it has been assumed that the spheres impact the target wall
with no initial spin. This is not realistic since, in general, particles will be
spinning prior to impact as a consequence of previous collisions. The effect of
initial particle spin has been addressed by Horak (1948) and Cross (2002). Using
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
957
A model for oblique impacts of spheres
1.0
0.9
0.8
0.7
0.6
0.5
0
2
4
6
8
10
12
14
Figure 16. The variation of e t with the dimensionless impact angle for PE impacts with various
initial impact velocities. Circles, PE (VniZ2.0 mm sK1); squares, PE (VniZ5.0 mm sK1); pluses,
PE (VniZ2.0 mm sK1; fixed en0); crosses, PE (VniZ5.0 mm sK1; fixed en0); uptriangles,
experimental (Gorham & Kharaz 2000); solid line, present model (VniZ2.0 mm sK1); dashed
line, present model (VniZ5.0 mm sK1).
results obtained from DEM simulations, Ning (1995) has demonstrated that, at
least for in-plane spin, the normalized rebound parameters shown in figures 3–11
and 13–16 remain unaltered, and equations (4.5)–(4.8) can be used to fit the
data, provided that Q is expressed in terms of the incident angle of the contact
patch qci, i.e.
QZ
2 tan qci
2vti
:
Z
ð1 C e n Þm ð1 C e n Þmvni
ð7:1Þ
This is because the contact reactions depend on the relative surface velocities of
the two bodies, as recognized by Maw et al. (1976). If there is no initial particle
spin then vti Z Vti and qci Z qi .
The general impact problem involving out-of-plane spin is more complicated
(Brach 1998). In this general case, although the direction of the surface velocity
and that of the tangential force do not change during an impact, the trajectory of
the sphere centre changes direction and the direction of the plane of spin rotates.
This problem is currently being examined.
8. Conclusions
FEA has been used to investigate the rebound characteristics resulting from
oblique collisions between a sphere and a flat substrate. Results have been
presented for the tangential rebound velocity of the contact patch and the
rebound angular velocity of the sphere, both of which have been normalized by
appropriate dimensionless groups. Consequently, at least for the values of
Poisson’s ratio and the normal coefficient of restitution used in this study, it is
possible to obtain all the rebound characteristics from the graphs presented in
the paper.
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
958
C.-Y. Wu et al.
The oblique impacts have been classified into two regimes: (i) persistent
sliding impact, in which sliding persists throughout the impact and (ii) nonpersistent sliding impact, in which sliding does not occur throughout the impact
duration. The transition between these two regimes is governed by a critical
dimensionless impact angle. For persistent sliding impacts, the effect of
tangential deformation does not play any role, and the present model follows
the well-established theoretical solutions based on rigid body dynamics. For nonpersistent sliding impacts, the rebound kinematics depend upon both Poisson’s
ratio and the normal coefficient of restitution (i.e. the yield stress of the
materials). There is no theoretical analytical solution due to the sensitivity to
the values of Poisson’s ratio and the normal coefficient of restitution. Therefore,
in the present model, the variation of impulse ratio with the impact angle is
approximated using an empirical equation with four parameters that are related
to Poisson’s ratio and the normal coefficient of restitution, and can be obtained
by fitting numerical data. Using this empirical equation, a complete set of
solutions to the rebound kinematics, including the tangential coefficient of
restitution, the rebound velocity at the contact patch and the rebound rotational
speed of the sphere, during oblique impacts is obtained. The accuracy and
robustness of this model (equations (4.5)–(4.8)) are also demonstrated by
excellent agreement with experimental data and FEA results for oblique impacts
of rigid, elastic and elastic, perfectly plastic spherical particles with an elastic flat
substrate. It is, therefore, concluded that the model is capable of accurately
predicting the complete rebound kinematics for both elastic and elastoplastic
oblique impacts.
References
Andrews, J. P. 1930 Experiment on impact. Proc. Phys. Soc. 43, 8–17. (doi:10.1088/0959-5309/43/
1/303)
Batlle, J. A. 1993 On Newton’s and Poisson’s rules of percussive dynamics. Trans. ASME J. Appl.
Mech. 60, 376–381. (doi:10.1115/1.2900804)
Batlle, J. A. & Cardona, S. 1998 The jamb (self-locking) process in three-dimensional rough
collisions. Trans. ASME J. Appl. Mech. 65, 417–423.
Brach, R. M. 1988 Impact dynamics with applications to solid particle erosion. Int. J. Impact Eng.
7, 37–53. (doi:10.1016/0734-743X(88)90011-5)
Brach, R. M. 1991 Mechanical impact dynamics—rigid body collisions. New York, NY: Wiley
Interscience.
Brach, R. M. 1998 Formulation of rigid body impact problems using generalized coefficients. Int.
J. Eng. Sci. 36, 61–72. (doi:10.1016/S0020-7225(97)00057-8)
Brauer, H. 1980 Report on investigations on particle movement in straight horizontal tubes,
particle/wall collisions and erosion of tubes and tube bends. J. Powder Bulk Solids Technol. 4,
3–12.
Brogliato, B. 1996 Nonsmooth impact mechanics. London, UK: Springer.
Cheng, W., Brach, R. M. & Dunn, P. F. 2002 Three-dimensional modeling of microsphere
contact/impact with smooth, flat surfaces. Aerosol Sci. Technol. 36, 1045–1060. (doi:10.1080/
02786820290092203)
Cross, R. 2002 Measurements of the horizontal coefficient of restitution for a superball and a tennis
ball. Am. J. Phys. 70, 482–489. (doi:10.1119/1.1450571)
Foerster, S. F., Louge, M. Y., Chang, H. & Allia, K. 1994 Measurements of the collision properties
of small spheres. Phys. Fluids 6, 1108–1115. (doi:10.1063/1.868282)
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
A model for oblique impacts of spheres
959
Goldsmith, W. 1960 Impact. London, UK: Arnold.
Goldsmith, W. & Lyman, P. T. 1960 The penetration of hard-steel spheres into plane metal
surfaces. Trans. ASME J. Appl. Mech. 27, 717–725.
Gorham, D. A. & Kharaz, A. H. 2000 The measurement of particle rebound characteristics. Powder
Technol. 112, 193–202. (doi:10.1016/S0032-5910(00)00293-X)
Hertz, H. 1896 In Miscellaneous papers by H. Hertz (eds D. E. Jones & G. A. Schott). London, UK:
Macmillan and Co.
Hunter, S. C. 1957 Energy absorbed by elastic waves during impact. J. Mech. Phys. Solids 5,
162–171. (doi:10.1016/0022-5096(57)90002-9)
Horak, F. 1948 Impact of a rough ball spinning round its vertical diameter onto a horizontal plane.
In Transactions of the faculty of mechanical and electrical engineering. Czechoslovakia, Czech
Republic: Technical University of Prague.
Hutchings, I. M. 1979 Energy absorbed by elastic waves during plastic impact. J. Phys. D Appl.
Phys. 12, 1819–1824. (doi:10.1088/0022-3727/12/11/010)
Johnson, K. L. 1983 The bounce of ‘superball’. Int. J. Mech. Eng. Educ. 111, 57–63.
Johnson, K. L. 1987 Contact mechanics. Cambridge, UK: Cambridge University Press.
Keller, J. B. 1986 Impact with friction. Trans. ASME J. Appl. Mech. 53, 1–4.
Kharaz, A. H., Gorham, D. A. & Salman, A. D. 2001 An experimental study of the elastic rebound
of spheres. Powder Technol. 120, 281–291. (doi:10.1016/S0032-5910(01)00283-2)
Labous, L. R., Rosato, A. D. & Dave, R. N. 1997 Measurements of collisional properties of spheres
using high-speed video analysis. Phys. Rev. E 56, 5717–5725. (doi:10.1103/PhysRevE.56.5717)
Li, L.-Y., Thornton, C. & Wu, C.-Y. 2000 Impact behaviour of the elastoplastic sphere with a rigid
wall. Proc. I. Mech. E. C J. Mech. Eng. Sci. 214, 1107–1114. (doi:10.1243/0954406001523551)
Li, L.-Y., Wu, C.-Y. & Thornton, C. 2002 A theoretical model for the contact of elastoplastic
bodies. Proc. I. Mech. E. C J. Mech. Eng. Sci. 216, 421–431. (doi:10.1243/0954406021525214)
Lorenz, A., Tuozzolo, C. & Louge, M. Y. 1997 Measurements of impact properties of small, nearly
spherical particles. Exp. Mech. 37, 292–298. (doi:10.1007/BF02317421)
Maw, N., Barber, J. R. & Fawcett, J. N. 1976 The oblique impact of elastic spheres. Wear 38,
101–114. (doi:10.1016/0043-1648(76)90201-5)
Maw, N., Barber, J. R. & Fawcett, J. N. 1981 The role of elastic tangential compliance in oblique
impact. Trans. ASME J. Lub. Tech. 103, 74–80.
Mindlin, R. D. 1949 Compliance of elastic bodies in contact. Trans. ASME J. Appl. Mech. 16,
259–268.
Mindlin, R. D. & Deresiewicz, H. 1953 Elastic spheres in contact under varying oblique force.
Trans. ASME J. Appl. Mech. 20, 327–344.
Ning, Z. 1995 Elasto-plastic impact of fine particles and fragmentation of small agglomerates. PhD
thesis, University of Aston, Birmingham, UK.
Smith, C. E. 1991 Predicting rebounds using rigid-body dynamics. Trans. ASME J. Appl. Mech.
58, 754–758. (doi:10.1115/1.2897260)
Stronge, W. J. 1993 Two-dimensional rigid-body collisions with friction. Trans. ASME J. Appl.
Mech. 60, 564–566. (doi:10.1115/1.2900835)
Stronge, W. J. 1994 Planar impact of rough compliant bodies. Int. J. Impact Eng. 15, 435–450.
(doi:10.1016/0734-743X(94)80027-7)
Thornton, C. 1997 Coefficient of restitution for collinear collisions of elastic-perfectly plastic
spheres. Trans. ASME J. Appl. Mech. 64, 383–386. (doi:10.1115/1.2787319)
Thornton, C. & Ning, Z. 1998 A theoretical model for the stick/bounce behaviour of adhesive,
elastic–plastic spheres. Powder Technol. 99, 154–162. (doi:10.1016/S0032-5910(98)00099-0)
Thornton, C., Ning, Z., Wu, C.-Y., Nasrullah, M. & Li, L.-Y. 2001 Contact mechanics and
coefficients of restitution. In Granular gases (eds T. Poschel & S. Luding), pp. 56–66. Berlin,
Germany: Springer.
Walton, O. R. 1992 Numerical simulation of inelastic, frictional particle–particle interactions.
In Particulate two-phase flow (ed. M. C. Roco), ch. 25. Boston, MA: Butterworth-Heinemann.
Proc. R. Soc. A (2009)
Downloaded from http://rspa.royalsocietypublishing.org/ on June 18, 2017
960
C.-Y. Wu et al.
Wang, Y. & Mason, M. T. 1992 Two-dimensional rigid-body collisions with friction. Trans. ASME
J. Appl. Mech. 59, 635–642. (doi:10.1115/1.2893771)
Whirley, R. G. & Engelmann, B. E. 1993 DYNA3D-A nonlinear, explicit, three-dimensional finite
element code for solid and structural mechanics-user manual. Livermore, CA: Lawrence
Livermore National Laboratory.
Wu, C.-Y. 2001 Finite element analysis of particle impact problems. PhD thesis, University of
Aston, Birmingham, UK.
Wu, C.-Y., Li, L.-Y. & Thornton, C. 2003a Rebound behaviour of spheres for plastic impacts. Int.
J. Impact Eng. 28, 929–946. (doi:10.1016/S0734-743X(03)00014-9)
Wu, C.-Y., Thornton, C. & Li, L.-Y. 2003b Coefficient of restitution for elastoplastic oblique
impacts. Adv. Powder Technol. 14, 435–448. (doi:10.1163/156855203769710663)
Wu, C.-Y., Li, L.-Y. & Thornton, C. 2005 Energy dissipation during normal impact of elastic and
elastic–plastic spheres. Int. J. Impact Eng. 32, 593–604. (doi:10.1016/j.ijimpeng.2005.08.007)
Proc. R. Soc. A (2009)