## Abstract

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.

## 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.

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 , where *V*_{ni} 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 velocities and confirmed that the duration of impact varies inversely as 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 finite-element 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 s^{−1}, 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.

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 −1/4. 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 finite-element analysis (FEA) results (Li *et al*. 2002; Wu *et al*. 2003*a*). 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*. (2003*a*), 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 *V*_{ni} to the yield velocity *V*_{y}; 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 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 semi-analytical 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 *V*_{i} and angular velocity *ω*_{i} at an impact angle *θ*_{i} (figure 1). After interaction with the wall, the sphere rebounds with a rebound translational velocity *V*_{r} and rebound angular velocity *ω*_{r}. Note that *V*_{i} and *V*_{r} are the velocities of the sphere centre. The corresponding translational velocities at the contact patch are denoted by *v*_{i} and *v*_{r}. We introduce normal and tangential coefficients of restitution *e*_{n} and *e*_{t},(2.1a)and(2.1b)where *V*_{ni} and *V*_{nr} are the normal components of the impact speed and rebound speed, respectively, and *V*_{ti} and *V*_{tr} are the corresponding tangential velocity components. It should be noted that it is necessary in (2.1*a*) to introduce the negative sign since the normal component of the velocity reverses its direction after the impact (*V*_{nr} is in the opposite direction to *V*_{ni}) 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 defining a total coefficient of restitution *e* as(2.2)It follows from (2.2) that the total coefficient of restitution *e* is dominated by *e*_{n} at small impact angles (*θ*_{i}→0°) and by *e*_{t} at large impact angles (*θ*_{i}→90°).

The correlation between the tangential and normal interactions during the impact can be characterized by an impulse ratio, which is defined as(2.3)where *P*_{n} and *P*_{t} are the normal and tangential impulses, respectively, and *F*_{n} and *F*_{t} 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 *μ*; it may or may not be equal to *μ* (Brach 1988).

According to Newton's second law, *P*_{n} and *P*_{t} can be expressed in terms of the incident and rebound velocities as(2.4a)and(2.4b)where *m* is the mass of the particle. Substituting (2.4*a*) and (2.4*b*) into (2.3) and using (2.1*a*) and (2.1*b*), we obtain(2.5)Similarly, a rotational impulse *P*_{ω} can be defined by(2.6)where *I* is the moment of inertia of the sphere, and *ω*_{i} and *ω*_{r} are the initial and rebound rotational angular velocities, respectively. According to the conservation of angular momentum about point *C* (figure 1), we have(2.7)where *R* is the radius of the sphere.

Substituting (2.4*b*) and (2.6) into (2.7) yields(2.8)For a solid sphere, . Hence,(2.9)Substituting (2.5) into (2.9), we obtain(2.10)The tangential component of the rebound surface velocity at the contact patch, *v*_{tr}, can be expressed as(2.11)Substituting (2.10) into (2.11), we obtain(2.12)Combining (2.1*b*), (2.5) and (2.12), we obtain(2.13a)or(2.13b)Equation (2.13*b*) can be rewritten as(2.14)From figure 1, the rebound angle *θ*_{r} can be obtained from(2.15)It can be seen from (2.5), (2.10), (2.13*a*) and (2.13*b*) 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*. 2003*a*). 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), (2.13*a*) and (2.13*b*) reveals that the rebound parameters are not independent, but are correlated with each other. For instance, rewriting (2.10), we have(2.16)Substituting (2.16) into (2.5) and (2.13*a*), we obtain(2.17)and (2.18)Equation (2.18) can be rewritten as(2.19)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 *ω*_{r}. 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 *ω*_{r} 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 *Ω*_{r} and *Ω*_{i}, a dimensionless rebound tangential surface velocity at the contact patch *Ψ*_{r} and a dimensionless impact angle *Θ* as follows:(2.20a)(2.20b)(2.20c)and (2.20d)where *Ψ*_{1} and *Ψ*_{2} are the parameters used by Foerster *et al*. (1994).

Hence, (2.10), (2.17) and (2.19) can be rewritten as(2.21)(2.22)and(2.23)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 *Ψ*_{r} and the dimensionless rebound rotational angular speed *Ω*_{r} are related to each other and are functions of *Ω*_{i} and *Θ*. If any one of these parameters can be experimentally measured, the value of *f*/*μ* 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*_{n}=1) or elastoplastic oblique impacts (*e*_{n}≠1).

## 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 finite-element 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 *R*=10 μm. The substrate is selected as 10 μm in both the *x*- and *z*-directions and 20 μm 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 *μ*=0.3 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 (*x*=0) are restricted in the *x*-direction. Nodes on boundaries (planes *x*=10, *y*=±10 and *z*=−10) 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 *V*_{i} to every node within the sphere at an angle *θ*_{i}. Different impact angles varying from 0° (normal impact) to 85° (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 explored more directly. Two different values of the normal initial velocity are specified (*V*_{ni}=2.0 m s^{−1} and 5.0 m s^{−1}) for EE and PE impacts, while only *V*_{ni}=5.0 m s^{−1} is chosen for RE impacts. In addition, no initial rotation is considered for all impact cases, i.e. *ω*_{i}=0.

## 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*/*μ* is essentially equal 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(4.1)which corresponds to the criterion of Maw *et al*. (1976, 1981). In (4.1), *κ* is the ratio of the initial tangential contact stiffness (*F*_{t}=0) to the normal contact stiffness and is defined by(4.2)For RE impacts, (4.2) can be rewritten as(4.3)The data shown in figure 3 were re-plotted against in figure 4. It is clear that, when , all three cases coalescence and(4.4)for

For , sliding does not occur throughout the impact. We refer to this as a non-persistent sliding impact. Hence, *Θ*_{c} marks the transition from non-persistent to persistent sliding impacts. For non-persistent sliding impacts, *f*/*μ* is clearly a function of the dimensionless impact angle *Θ* and Poisson's ratio. Rigorous prediction of the dependence of *f*/*μ* on *Θ* appears to be intractable. In this study, a close examination of the data in this region suggests that, for a given Poisson ratio, the correlation between *f*/*μ* and *Θ* can be given by the following expression:(4.5)where *c*_{1}, *c*_{2}, *c*_{3} and *c*_{4} 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 , which is listed in the last column of table 2. It can be seen that is comparable with the *Θ*_{c} 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:(4.6)(4.7)and(4.8)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) 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 *Θ*. For *ν*=0 and 0.3, the values of *ψ*_{r} are positive for very small values of *Θ*, but not in the case of *ν*=0.49. 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 *ψ*_{r} 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 *ν*=0.3, the variation of *f*/*μ* with *Θ* 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 impact angle. The value of *f*/*μ* is found to increase as the normalized impact angle *Θ* increases when *Θ*<*Θ*_{c}, and is constant at a value of unity when *Θ*≥*Θ*_{c}. Fitting the data for *Θ*<*Θ*_{c} 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 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.(5.1)and(5.2)where *β* is a constant. Using (4.1), (5.1) and (5.2), we identify that(5.3)Substituting (5.1) and (5.2) into (2.23) gives(5.4)Substituting (5.4) into (2.21) and (2.22) leads to(5.5)and (5.6)In addition, rigid body dynamics (Brach 1988) assumes that the rebound tangential surface velocity at the contact patch can be either zero, i.e. *v*_{tr}=0 for *f*<*μ* or *v*_{tr}≥0 if *f*=*μ*. This implies that(5.7)with *Θ*_{c}=7.

Substituting (5.7) into (2.21) and (2.22) gives(5.8)and(5.9)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, *V*_{ni}=5.0 mm s^{−1}). 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*. 2003*a*). 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 , sliding occurs at the start of the impact and the normal coefficient of restitution decreases at an increasing rate until 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 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 (i.e. 6.6% for *V*_{ni}=5.0 mm s^{−1} and 3.0% for *V*_{ni}=2.0 mm s^{−1}). A similar phenomenon is also observed for oblique impacts of an elastic, perfectly plastic sphere with a rigid wall (Wu *et al*. 2003*a*). 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*. 2003*b*), 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 *θ*_{i} 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*/*μ* with normalized impact angle *Θ* for PE impacts with *V*_{ni}=2.0 mm s^{−1} and 5.0 mm s^{−1}. It can be seen that, similar to the RE and EE impacts shown in figures 3 and 8, *f*/*μ* increases as the impact angle increases until the normalized impact angle *Θ* reaches *Θ*_{c}; thereafter *f*/*μ* 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.

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 (*θ*_{i}=0°) 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 *θ*_{i} 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 *V*_{ni}=2.0 mm s^{−1} and dashed lines for *V*_{ni}=5.0 mm s^{−1}, 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 *Ω*_{r} is plotted against the dimensionless impact angle *Θ* 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 *Θ*, 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.

## 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 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 *Θ* is expressed in terms of the incident angle of the contact patch *θ*_{ci}, i.e.(7.1)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 and .

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.

The oblique impacts have been classified into two regimes: (i) persistent sliding impact, in which sliding persists throughout the impact and (ii) non-persistent 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 non-persistent 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.

## Footnotes

- Received May 30, 2008.
- Accepted November 6, 2008.

- © 2008 The Royal Society