vi

ary2TT,168

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asingly viable tool to analyser owevelopompacr a widde Tsujbeds, Cand w

simulated partic

Powder Technology 210 (2011) 189197

Contents lists available at ScienceDirect

Powder Te

e lsadopted and large numbers of application papers have beenpublished. These examples all assume a soft-sphere approach inwhich particle overlaps are allowed and are used in the calculation ofcontact forces. An aspect of soft-sphere modelling whose effects areunknown is the nature and quality of the characterisation of theparticleparticle interactions, i.e. quantifying the evolution of theforces, velocities and displacements during a collision. Various contactforce models have been proposed and implemented in DEM codes tocater for this interaction.

At the macroscopic scale, particle systems may exhibit gas-like,

The work reported in this paper is part of an ongoing projectdesigned to rigorously assess the signicance of the detailedformulation of particleparticle interactions. Clearly the signicancemay depend on whether the problem involves collisions, enduringcontacts or both. Therefore, as a start, this paper examines thesimplest of problems, which is the oblique impact of an elastic spherewith an elastic target wall.

It is now well established that for interactions between two elasticspheres or an elastic sphere and a planar surface (whether elastic orrigid) the normal interaction follows the theory of Hertz [9] and theliquid-like or solid-like behaviour. Corresponmay involve particle collisions, particles witboth. It is therefore extremely difcultsimulations. Attempts to quantitatively valid

Corresponding author.E-mail address: sharen.cummins@csiro.au (S.J. Cumm

0032-5910/$ see front matter 2011 Elsevier B.V. Aldoi:10.1016/j.powtec.2011.01.013ear in ball mills. Hopperfui and Thornton [7] andethod has been broadly

imposed rather than the physics at the particle scale. Therefore, suchcomparisons may provide good correlations rather than rigorousvalidations of the simulations when compared with experiments.owsweremodelled by Langston et al. [6], Kachute ows by Walton [8]. Since then the mNumerical modelling is an increindustrial and environmental granulaMethod (DEM), whichwas originally d[1] for quasi-static deformation of cpopular technique that is now used foFor example, early contributions inclu[3] who used DEM to model uidisedRajamani [5] to simulate segregations. The Discrete Elemented by Cundall and Strackt particle systems, is ae variety of applications.i et al. [2] and Xu and Yuleary [4] and Mishra and

non-spherical. Even with the advent of very sophisticated experi-mental techniques, such as PIV, MRI and PEPT, comparisons arenormally restricted to the spatial and temporal particle velocitydistributions and not detailed local events; in other words the generaloverall behaviour. However, the overall behaviour in these systems isdictated by the applied external eld and the boundary conditionsdingly, DEM simulationsh enduring contacts, orto truly validate DEMate DEM simulations by

tangential interDeresiewicz [10referred to as thSection 3). The selastic sphere imnumerical impleDEM simulationexperiments by Fins).

l rights reserved.les are spheres and the experimental particles are1. Introduction comparing with experimental data are often frustrated by uncertain-ties in terms of the experimental data and the fact that frequently theAn investigation of the comparative behaduring elastic collisions

Colin Thornton a, Sharen J. Cummins b,, Paul W. Clea School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham B15b CSIRO Division of Mathematics, Informatics and Statistics, Private Bag 33, Clayton, Vic, 3

a b s t r a c ta r t i c l e i n f o

Article history:Received 21 May 2010Received in revised form 12 January 2011Accepted 27 January 2011Available online 3 February 2011

Keywords:Oblique impactContact interaction lawsDiscrete element method

In this paper the reboundsimple problem of an elaappropriately calibrated nostiffnesses, excellent resuldemonstrates that for non-cannot be used as the sprregarding the limitations of

j ourna l homepage: www.our of alternative contact force models

b

UK, Australia

matics obtained using different contact force models are compared for thesphere impacting obliquely with a target wall. It is shown that, for anal spring stiffness and a realistic ratio of the tangential to normal springan be obtained by using a simple linear spring model. The paper alsoar contact models, integral equations for the tangential forcedisplacementstiffness varies during the collision. Finally some comments are providedlinear spring model and alternatives are discussed.

2011 Elsevier B.V. All rights reserved.

chnology

ev ie r.com/ locate /powtecaction is provided by the theory of Mindlin and]. This combination of interaction rules will bee HMD model (details of the model are provided inolution for the rebound kinematics in the case of anpacting a target wall has been demonstrated by (i)mentation of the HMD model by Maw et al. [11], (ii)s using the HMD model by Thornton et al. [12], (iii)oerster et al. [13] and Kharaz et al. [14] and (iv) nite

element analyses by Wu [15]. The excellent agreement between thevarious data sets has been illustrated by Wu et al. [16,17].

The HMD model involves complex algorithms and requires loadreversal points to be stored in memory. Consequently, manyresearchers elect to use simpler contact force models in order toreduce computing time requirements. The most common contactforce model, used for both normal and tangential interactions, is thelinear springdashpot model introduced by Walton [8]. The linearspringdashpot models are widely used due to their ease ofimplementation in numerical codes and can readily be applied tomultiple interactions with particles of differing shape, [18].

In DEM simulations, the contact interaction laws invariablyincorporate some form of energy dissipation, which can be eitherviscous or plastic. In a separate paper, currently being prepared,such inelastic impacts are examined. In this paper, we study elasticoblique impacts using linear and non-linear spring models. The aimof the paper is to show how the rebound characteristics depend onthe choice of model used and to provide some explanations for thedifferences found. Previous comparisons of the effect of differentelastic particle interaction models were reported by Di Renzo andDi Maio [19]. However, the simplication of the HMD modelproposed in that paper signicantly alters the collisional behaviour,

with the wall, the sphere rebounds at an angle r with a rebound

190 C. Thornton et al. / Powder Technology 210 (2011) 189197translational velocity Vr and a rebound angular velocity r. Note thatVi and Vr are the velocities of the sphere centre. The correspondingand in some cases leads to unphysical results. We discuss this inSection 5.

2. Description of the problem

DEM simulations involve millions of particleparticle collisionsand, generally, the particles are rotating prior to the collision. Inthis paper we examine the simpler problem of a single sphericalparticle impacting a planar target wall at different impact angles. Afurther simplication is that the sphere approaches the wallwithout any rotation and there is no gravitational eld. This simpleproblem is sufcient to achieve the objectives of the paper, i.e. tocompare the consequences of using different contact interactionrules.

Fig. 1 illustrates diagrammatically a typical oblique impact of asphere with a target wall. The sphere approaches the wall with aninitial translational velocity Vi at an impact angle i. After interactionFig. 1. Diagram of the oblique impact of a sphere with a plane surface.tangential surface velocities at the contact patch are denoted by viand vr.

2.1. Rigid body dynamics

The simplest theoretical approach to the oblique impact of asphere with a target wall is that of rigid body dynamics [20,21,22].However, the approach is only valid if the impact angle issufciently large that sliding occurs throughout the impactduration. At smaller impact angles, the theory predicts that thetangential surface velocity is zero at the end of the impact event.That this is not the case was demonstrated using both theory andexperiment by Maw et al. [11,23] who showed that the tangentialsurface velocity reverses its direction due to the tangential elasticcompliance. Nevertheless, rigid body dynamics does provideappropriate dimensionless groups that characterise the kinematicbehaviour of oblique impacts.

In rigid body dynamics, which is based on the impulsemomentum equations, the correlation between the tangential andnormal interactions is characterised by an impulse ratio, which isdened as

f =PtPn

=FtdtFndt

1

where Pn and Pt are the normal and tangential impulses and Fn andFt are the normal and tangential components of the contact force. Ifsliding occurs throughout the impact duration then f= ,otherwise fb. Here is the interface friction coefcient. Reboundvelocities are normally related to impact velocities by empiricalcoefcients of restitution in the normal and tangential directiondened by

en =VnrVni

and et =VtrVti

2

Assuming f= and no initial particle spin, the followingequations are obtained to dene the complete rebound kine-matics when sliding occurs throughout the impact, (see forexample [17])

vtr = Vti72

1 + en Vni 3

r = 5 1 + en Vni

2R4

Vtr = vtrRr: 5

Rearranging Eqs. (3), (4) and (5) we obtain

2vtr1 + en Vni

=2tani1 + en

7 6

2Rr1 + en Vni

= 5 7

et =VtrVti

= 1 1 + en tani

: 8

FromEqs. (6) and (7)we identify three dimensionless groups, namely

2tani1 + e ;

2vtr1 + e V and

2Rr1 + e V : 9n n ni n ni

C. Thornton et al. / Powder TechnMaw et al. [11] introduced the following parameter, whichdenes the ratio of the initial tangential contact stiffness (Ft=0) tothe normal contact stiffness, to normalise their oblique impactdata,

=2 1 2 10

where, for the impact of two bodies with similar mechanicalproperties, is Poisson's ratio. For the case of dissimilar bodies seeJohnson [24]. If en=1 and the rst two groups in Eq. (9) aremultipliedby then we obtain the parameters 1 and 2 introduced by Maw etal. [11]. It has, however, been demonstrated [25] that the relationshipbetween 2 and 1 is different for different values of Poisson's ratioand therefore these parameters are not appropriate for normalisingdifferent data sets. Consequently, in this paper, results are presentedin terms of the dimensionless groups given in Eq. (9).

3. Elastic impact

For an impact of an elastic sphere with a wall, the impact duration tc isgiven by

tc =2maxVni

1

0

d =max 1 =max 5=2 1=22:94maxVni 11

where is the relative displacement (approach) of the centre of thesphere in the normal direction and max is the maximum relativeapproach (overlap) given by

max =15mV2ni16R1=2E4

!2=512

[24]. Substituting Eq. (12) into Eq. (11) therefore gives

tc2:865m2

RE42Vni

!1=5: 13

where

E =E

2 12 14

3.1. The HMD model

For elastic spheres, the theory of Hertz (see [24]) is used to modelthe normal forcedisplacement relationship and the theory of Mindlinand Deresiewicz [10] is used for the tangential forcedisplacementrelationship.

For the case of an elastic sphere impacting a wall with the sameelastic properties, the normal forcedisplacement relationship is

Fn =43ER1=23=2 15

The normal stiffness is

kn = 2Ea 16

which varies throughout the collisionwith the radius of the contact area

a =R

p: 17

In the original Mindlin and Deresiewicz paper, solutions were

provided in the form of instantaneous compliances which, due tothe dependence on both the current state and the previous loadinghistory, could not be integrated a priori. However, several loadingsequences involving variations of both normal and tangentialforces were examined from which general procedural rules wereidentied. Adopting an incremental approach, the procedure is toupdate the normal force and contact area radius, using Eqs. (15)and (17), followed by calculating the tangential force incrementFtusing the new values of Fn and a. By re-analysing the loading casesconsidered by Mindlin and Deresiewicz [10], it was shown byThornton and Randall [26] that, for all cases, the tangentialincremental displacement may be expressed as

=1

8G4aFn +

FtFn

18

except when; for Fn N 0; then jjbFn8G4a

19

where G =G

2 2 : 20

Rearranging Eq. (18) denes the tangential stiffness as

kt =Ft

= 8G4a 1 Fn

21

where

3 = 1 Ft + Fn Fn

N 0 loading 22a

3 = 1F4tFt + 2Fn

2Fn b 0 unloading 22b

3 = 1FtF44t + 2Fn

2Fn N 0 reloading 22c

and the negative sign in Eq. (21) is only invoked during unloading.The forces Ft*andFt* dene the load reversal points and need to becontinuously updated

F4t = F4t + Fn and F

44t = F

44t Fn: 23

If Eq. (19) is true then a problem occurs since FtbFn and, as aconsequence, the newly calculated value of Ft does not lie on theforce displacement curve corresponding to the new value of Fn. Asatisfactory solution to this problem is obtained by setting =1 inEq. (21) until the following condition is satised

8G4a N Fn 24

at which point the tangential force lies on the corresponding tangentialforce displacement curve. For further details of the tangential interac-tions of elastic spheres see Thornton and Yin [27] and Thornton [28].

3.2. The LS model

The simplest elastic contact force model is to assume that,during contact, the two interacting bodies are connected, bothnormally and tangentially, by linear springs. In this linear spring(LS) model, the normal and tangential contact forces can becalculated from the following equations

Fn = kn 25

Fnew = Fold + k except if Fnew F then Fnew = F 26

191ology 210 (2011) 189197t t t t n t n

If || this leads to Ft=Fn where Fn is the updated value. Zhouet al. [30] adopted the same model but, to account for b0, used

Ft = Fn 1 1min jj;

n o

0@

1A

3=2264

375 jj 34

Therefore, in this paper, we use Eq. (32) and the followingequation

192 C. Thornton et al. / Powder Technology 210 (2011) 189197where kn and kt are the normal and tangential spring stiffnesses and is the relative tangential surface displacement at the contact.

In order to compare the predictions of the linear spring modelswith those of HMD the normal spring stiffness kn is chosen to givethe same contact duration as the HMD model (13) for a normalimpact (i=0). For a simple mass-spring system

tc = mkn

1=2: 27

Equating the collision times for the models in Eqs. (13) and (27)gives the desired normal linear spring stiffness which is

kn = 1:2024 m1=2E4

2

RVni

2=5: 28

3.3. The HM model

Another possible elastic model is to combine the Hertzian modelfor the normal stiffness with the no-slip theory of Mindlin [29] forthe tangential stiffness. This combination of interaction rules will bereferred to as the HM model, in which the normal and tangentialstiffness can be written as

kn = 2E4R

pand kt = 8G

4R

p29

and, using Eqs. (14) and (20), it can be shown that the stiffness ratio isgiven by Eq. (10).

The HM model would appear to be an attractive simplication ofthe HMD model. Like the HMD model, the stiffness ratio is xed byPoisson's ratio and, like the LS model, there are no memory/storageissues. However, unlike the LS model, the normal and tangentialsprings are non-linear since is continuously changing during animpact. This model was also examined in the paper by Di Renzo and DiMaio [19] where it was referred to as the H-MDns model.

3.4. The LTH model

In their simulations of hopper ow, Langston, Tzn and Heyes [6]used Eq. (15) for the normal elastic interactions and a simpliedversion of the HMD model to model the tangential interactions. Theyignored the hysteretic nature of the theory of Mindlin andDeresiewicz [10] and simply used the equation for tangential loading,which can be written as

Ft = Fn 1 1jj

!3=2" #30

where is the displacement at which sliding commences, given by

=3Fn

16G4R

p : 31

Substituting for Fn using Eq. (15) and then Eqs. (14), (20) and (10)leads to

= 32

asnoted by Langston et al. [6],whoalso stated that if ||N then ||= .Consequently, rewriting Eq. (30)

Ft = Fn 1 1min jj;

n o

0@

1A

3=2264

375: 33Ft = Fn 1 1minjj

;1

( ) !3=2" #jj 35

4. Results

In this section, we examine elastic oblique impacts using linear andnon-linear springmodels. For all models we consider an elastic sphereof radius R=25 mm, density =2650 kg/m3, and hence a massm=0.1734 kg, impacting a target wall at various impact angles i andat a constant speed Vi=5m/s. In the HM and HMDmodels, the sphereand the wall have identical properties namely Young's modulusE=70 GPa, Poisson's ratio =0.3 and interface friction coefcient=0.1. The same properties are used for the HM model. For the LSmodel, we rst need to examine the effect of the ratio of tangential tonormal stiffness , dened by Eq. (10), on the rebound characteristics.

4.1. Linear spring models

As recognised by Schfer et al. [31], Di Renzo and Di Maio [19] andWu et al. [16], the kinematic rebound characteristics are sensitive tothe magnitude of dened by Eq. (10), which is the ratio of the initialtangential contact stiffness to the normal contact stiffness. Therefore,in the context of linear springmodels, the choice of the value of =kt/kn is important. In many DEM simulations, e.g. Xu and Yu [3] andKawaguchi et al. [32], the value of has been arbitrarily set to unity. Atthe other extreme, Landry et al. [33] used a value of =2/7 for whichthe period of oscillation of the tangential force equals that of thenormal force. Luding [34] even suggested the possibility of b2/7.However, it is noted that since 00.5 the range of realistic valuesof for elastic impacts is 12/3. Values of the normal andtangential linear spring stiffnesses used for different values of (including the case of =0.3 studied in Section 4.2) are provided inTable 1.

Simulations of an elastic sphere impacting a wall with identicalelastic properties have been performed for values of =1, 2/3, 1/2and 2/7. The results in terms of the dimensionless groups given byEq. (9) are shown in Figs. 2 and 3 and the tangential coefcient ofrestitution et is shown in Fig. 4. Fig. 2 shows the normalised tangentialsurface velocity at the end of the collision plotted against thenormalised impact angle. The gure shows that, for compressiblematerials (b0.5) the rebound tangential surface velocity is in thesame direction as the initial tangential surface velocity for very smallangles of impact. This is in agreement with the data presented by Wu

Table 1Normal and tangential spring stiffness for different values of .

E (GPa) kn (MN/m) kt (MN/m)

1 0 35.0 100 1002/3 1/2 46.7 126 841/2 2/3 63.0 160 802/7 5/6 114.5 259 740.8235 0.3 38.5 108 89

0 2 4 6 8 102tani/(1+en)

-4

-3

-2

-1

0

1

2

32v

tr/(1

+en)V

ni

kappa=1kappa=2/3kappa=1/2kappa=2/7

Fig. 2. Normalised rebound surface velocity (LS model).

0.4

0.5

0.6

0.7

0.8

0.9

1.0

e t

kappa=1kappa=2/3kappa=1/2kappa=2/7

0 2 4 6 8 102tani/(1+en)

Fig. 4. Tangential coefcient of restitution (LS model).

193C. Thornton et al. / Powder Technology 210 (2011) 189197et al. [17]. Sliding occurs throughout the impact when the followingcondition is satised

2tani 1 + en

71= : 36

Therefore the range of impact angles for which sliding occursthroughout the collision depends on the value of , as seen in Figs. 2and 3.

The case of =2/7 is worthy of further comment. As stated above,for =2/7, the period of oscillation of the tangential force equals thatof the normal force and, as a consequence, the obliquity of theresultant contact force is constant throughout the collision. If slidingdoes not occur throughout the collision Ft = 27 Fn tani otherwiseFt=Fn. For the case of =2/7, Figs. 2 and 3 show a bilinearrelationship that distinguishes between the cases when sliding occursthroughout the impact and Eq. (36) is true and when it is not true. It isalso shown in Fig. 4 that when condition Eq. (36) is not true then thetangential coefcient of restitution has a constant value of 3/7. Theresults for =2/7 are unrealistic since it has been demonstrated bytheory [11], DEM simulations [27], FEM modelling [15] and experi-ments [35] that, if sliding does not occur throughout the impactduration, then the tangential force reverses direction during theimpact.-6

-5

-4

-3

-2

-1

0kappa=1kappa=2/3kappa=1/2kappa=2/7

2R

r/(1+e

n)V

ni

0 2 4 6 8 102tani/(1+en)

Fig. 3. Normalised rebound angular velocity (LS model).4.2. Comparison of linear and nonlinear spring models

It follows from the results presented in the previous section that, inorder to provide a rigorous comparison of a linear spring (LS) modelwith the HMD and HMmodels, it is necessary to select a stiffness ratiocorresponding to the same value of Poisson's ratio, i.e. =0.8235 for=0.3, as given in Table 1.

The predictions of the LS, HM and HMD models are compared interms of the dimensionless groups given by Eq. (9) in Figs. 5 and 6. Thecorresponding tangential coefcients of restitution are shown inFig. 7. It can be seen from all three gures that there is excellentagreement between the predictions of the LS and HMD models. Thepredictions of et for the twomodels only deviatewhen i5 and evenwhen i=1 the value of et predicted by the LS model is only 4%higher than that predicted by the HMD model. At very small impactangles, the data points in Fig. 7 are very sensitive to the location of thecorresponding data points in Fig. 5 since it can be shown that for thecase of no initial spin, i=0,

et =57+

2vtr7Vti

: 37

Eq. (37) also demonstrates that, since all the data sets in Figs. 2 and5 pass through the origin, the tangential coefcient of restitution isindeterminate when i=0.0 2 4 6 8 102tani/(1+en)

2vtr/(1

+en)V

ni

-3

-2

-1

0

1

2

3HMDLSHMH-MDns

Fig. 5. Normalised rebound surface velocity comparison of different models.

0 2 4 6 8 102tani/(1+en)

HMDLSHMH-MDns

-6

-5

-4

-3

-2

-1

02R

r/(1

+en)V

ni

Fig. 6. Normalised rebound angular velocity comparison of different models.

HMDLTH

0 2 4 6 8 102tani/(1+en)

-4

-3

-2

-1

0

1

2

3

2vtr/(1

+en)V

ni

194 C. Thornton et al. / Powder Technology 210 (2011) 189197It can be seen from Figs. 5, 6 and 7 that the predictions of the HMmodel are signicantly different from those of the other two modelswhen sliding does not occur throughout the impact duration. The HMmodel predicts a signicantly smaller reversal of the surface velocity(see Fig. 5) with less spin imparted to the sphere (see Fig. 6) leading toa higher tangential coefcient of restitution (see Fig. 7). Di Renzo andDi Maio [19] also found excellent agreement between the LS and HMDmodels. However, our HM model predictions differ from and arebetter than those of their H-MDns model, which is also superimposedon Figs. 5, 6 and 7. Since both models are in principle the same, i.e.they both combine Hertzian theory for the normal interaction withMindlin's no-slip theory for the tangential interaction, the reasons forthe different predictions were explored and are discussed in the nextsection.

Finally, in Figs. 8, 9 and 10, we compare the predictions of the LTHmodel with those of the HMD model. It can be seen that the resultsobtained with the LTH model are very different from the resultspredicted by all the other models examined. At intermediate angleswhen 10bib30, the LTH model predicts much larger reversals ofthe surface velocity (Fig. 8) with much less spin imparted to thesphere (Fig. 9) leading to much lower values of the tangentialcoefcient of restitution (Fig. 10). Furthermore, sliding throughoutthe impact occurs for i20, which is signicantly less than i30as predicted by Eq. (36). It is also noted that the LTH model uses anintegral equation for the tangential interaction and that, for impact

angles of 5 and 10, the tangential forcedisplacement curvesexhibited the same strange type of hysteresis loop as shown in

HMDLSHMH-MDns

0 2 4 6 8 102tani/(1+en)

0.5

0.6

0.7

0.8

0.9

1.0

e t

Fig. 7. Tangential coefcient of restitution comparison of different models.Fig. 12 and discussed in the next section. Consequently, we concludethat the tangential force model used by Langston et al. [6] and Zhouet al. [30] is not satisfactory and should not be used.

5. Discussion

Fig. 11 shows the tangential forcedisplacement curves predictedby our HM and HMD models for an impact angle of 5. In this case,sliding does not occur until the end of the impact. At the start of theimpact, as the normal force increases, FtbFn. For this condition, inthe HMDmodel the tangential stiffness is given by Eq. (21) with =1.This is identical to the tangential stiffness used in the HMmodel givenby Eq. (35). Consequently, as can be seen in Fig. 11, the twopredictions are identical when the tangential force is increasing.When the tangential displacement reverses direction, b0, thetangential stiffness continues to increase since the normal force, andtherefore is still increasing. Consequently, during the initialtangential unloading, the tangential force is less than that duringtangential loading for a given value of . During the second half of theimpact duration, when the normal force is decreasing, the HM modelcontinues to use Eq. (29) to dene the tangential stiffness. In contrast,for the HMDmodel, the tangential stiffness is now dened by Eq. (21)but using the negative sign since b0 indicating unloading, togetherwith the substitution for given by Eq. (22b). Consequently, when thenormal force is decreasing the predictions of the two models are

Fig. 8. Normalised rebound surface velocity (LTH model).expected to differ. The corresponding prediction of the LS model isalso superimposed on Fig. 11. In the case of the LS model the

0 2 4 6 8 102tani/(1+en)

-6

-5

-4

-3

-2

-1

0

2R

r/(1+e

n)V

ni

HMDLTH

Fig. 9. Normalised rebound angular velocity (LTH model).

[37] who suggested that when sliding occurred, i.e. FtFn, not onlyshould the tangential force be reset to the Coulomb limit F =F but

0.3

0.5

0.4

0.6

0.7

0.8

0.9

1.0e t

0 2 4 6 8 102tani/(1+en)

HMDLTH

Fig. 10. Tangential coefcient of restitution (LTH model).

195C. Thornton et al. / Powder Technology 210 (2011) 189197tangential forcedisplacement is linear during tangential loading andunloading until the tangential force reaches its maximum negativevalue. At this point sliding starts to occur and continues until the endof the collision and, therefore, the tangential forcedisplacementcurve deviates from the linear elastic curve dened by kt.

Di Renzo and Di Maio [19] also considered the HMmodel which, intheir paper, was referred to as the H-MDns model. In Fig. 12 of theirpaper, they show the forcedisplacement curve obtained by their H-MDns model for an impact angle of 5. Their results are very differentfrom the HM results shown in Fig. 11. According to their paper thefollowing integral equation is used to calculate the tangential force

Ft = 8G4R

p 38

instead of the incremental approach i.e.

Fnewt = Foldt + 8G

4R

p: 39

In order to reproduce the results from Di Renzo and Di Maio [19],we used Eq. (38) to calculate the tangential force in a non-incrementalmanner for an impact angle of 5. The results, labelled integral, areshown in Fig. 12. It can be seen that, initially, as the tangentialdisplacement increases the tangential force increases but at a rategreater than that predicted by the HMD model and the incrementalHM model. This is due to the fact that the stiffness kt = 8G4

R

pis

a secant stiffness in Eq. (38) but a tangent stiffness in Eq. (39).Furthermore, when the tangential displacement reverses direction-15 -10 -5 0 5 10 15tangential displacement (m)

-1000

-500

0

500

1000

tang

entia

l for

ce (N

)

HMDHMLS

Fig. 11. Tangential forcedisplacement curves for an impact of 5.the tangential stiffness reduces and becomes negative even thoughthe contact area is still increasing because the normal force is stillincreasing. Then as the tangential displacement continues to decreasethe tangential force reduces in a manner that results in a very strangehysteresis loop that has created new energy by the time that thetangential displacement rst returns to zero. This strange phenom-enon is exactly what was obtained by Di Renzo and Di Maio [19], seetheir Fig. 12. It is therefore incorrect to use the integral form Eq. (38)for the tangential force. For non-linear spring models, an incrementalapproach should always be used for the tangential interaction as thespring stiffness, the normal force and hence in Eq. (38) is alwayschanging. For the linear spring model, the results using anincremental approach are identical to those using a non-incrementalapproach because the spring stiffness is constant.

Fig. 12 also shows that, when using the non-incremental approach,the data follows another loop when the tangential force andtangential displacement are both negative. The loop is, however, notclosed since there is a nal residual negative tangential displacementat the end of the impact. This differs from Fig. 12 of Di Renzo and DiMaio [19] who obtained a second closed hysteresis loop with a naltangential displacement of zero. From their paper it is not clear howand why this zero nal tangential displacement occurred. Accordingto Kruggel-Emden et al. [36], however, Di Renzo and Di Maio [19]constrained the total tangential surface displacement when slidingoccurred. This idea appears to have originated in a paper by Tsuji et al.

incrementalintegralintegral+slidingconstraint

-15 -10 -5 0

0

5 10 15tangential displacement (m)

-1500

-1000

-500

500

1500

1000

tang

entia

l for

ce (N

)

Fig. 12. Hertz-Mindlin predictions for an impact angle of 5.t n

also the tangential displacement should be reset to

=Fnkt

: 40

The results obtained when using the integral Eq. (38) with thedisplacement constraint in Eq. (40) whenever the Coulomb frictionlimit was exceeded are also shown in Fig. 12. It can be seen that, as aconsequence of sliding occurring towards the end of the impact, asecond closed hysteresis loop is obtained leading to =0 at the end ofthe impact, which is in agreement with Di Renzo and Di Maio [19].However, the forcedisplacement curve obtained using Eqs. (38) and(40) is misleading in that when, at the end of the impact, sliding isoccurring and the tangential force is becoming less negative, theconstrained delta is in effect the elastic displacement (due to thespring) and not the total tangential displacement as implied in thegure. Further discussion is provided in Appendix A.

It should be noted that Di Renzo and DiMaio [38] recognised some ofthe problems with their H-MDns model, and proposed a modied

varying relative impact velocities. A further problem is the fact that,contrary to reality, the impact duration for the linear spring is

dependent. However, for systems with enduring contacts, furthercomparative studies are required.

Appendix A

The idea that, when sliding occurs, the tangential displacementshould be reset according to Eq. (40) was also suggested by Brendel andDippel [39]. Brendel and Dippel [39] suggested that the use of Eq. (40)was necessary to avoid excessive extension of the elastic spring whencontinued sliding occurred. However, this problem does not exist, asdemonstrated in Fig. A1.

In the absence of a dashpot, the tangential interaction can berepresented by a spring and a friction slider, as shown in Fig. A1. For asmall tangential displacement b, where is the displacement atwhich sliding commences, the spring is extended and =e, where e isthe extension of the spring. If N then an additional displacement foccurs as a result of the relative movement of the slider and the totaldisplacement is =e+f.

The problem is also illustrated in Fig. A2, in which the line OA denesthe tangential elastic stiffness kt. Using the linear springmodel Ft=kt, letthe tangential force be given by point B. If this exceeds the Coulomb limitthe tangential force is reset to Ft=Fn and the state is given by point C. IfEq. (40) is applied one obtains point D which corresponds to the elasticdisplacement e and lies on OA thereby permitting continued use of theintegral equation.Note that the actual tangential stiffnessk is givenby the

196 C. Thornton et al. / Powder Technology 210 (2011) 189197independent of the relative impact velocity. This simplication could bequite signicant for systems near the jamming transition when thecollision frequency is high or for systems where acoustic transmission ofwaves through the particle network is important.

Consequently, although the HMmodel did not perform aswell as theLS model for the simplied problem examined in the paper, it maysometimes still be a more attractive alternative to the complicated HMDmodel. This is because (i) the model parameters are based on realmaterial properties, Young's modulus and Poisson's ratio, (ii) there is noneed for any calibration of the normal spring stiffness, (iii) the forceintegral model for the tangential interaction. This was achieved bymultiplying the RHS of Eq. (38) by 2/3. The consequence of this rescalingwas that their new integral model (the H-DD model in their paper)agreed with the HMD model during initial tangential loading. However,the ake-like forcedisplacement curves (e.g. the integral+slidingcurve in Fig. 12) were still obtained, see Figs. 10 and 11 of their paper,indicating that new energy was created during their simulated impacts.In addition, the tangential coefcients of restitution obtained from theimproved integral model were signicantly less than those obtainedwith the HMD model.

6. Conclusions

The paper has compared the results for four important classes ofcontact force models for an elastic sphere impacting obliquely with aplanar target wall. It was demonstrated that, at least in terms of therebound kinematics, the LSmodel can produce excellent agreementwiththe results of the much more complex HMD model provided that anappropriate value of the ratio of the tangential to normal stiffnesses isused. For an elastic sphere impacting a target wall with the same elasticproperties, the ratio of the initial tangential stiffness to the normalstiffness, as givenby theHMandHMDmodels, depends only on the valueof Poisson's ratio which is limited to values 0bb0.5. Consequently, inorder to obtain realistic resultswhen using the LSmodel it is necessary touse values of the stiffness ratio 1NN2/3. The actual value of the normalspring stiffness, or thevalueofYoung'smodulus in thenon-linearmodels,do not affect the rebound kinematics. These parameters only affect theimpactduration, as indicatedbyEqs. (13)and (27). TheHMmodel,whichmaybeconsidered tobeamore rational simplicationof theHMDmodel,performed less well; tending to result in a smaller reversal of the surfacevelocity, less spin imparted to the sphere and a higher tangentialcoefcient of restitution. The results of the HM model, however, werebetter than the corresponding results obtained by Di Renzo and Di Maio[19].

The initial objective of the paper was to compare the performance ofdifferent contact force models in the context of elastic impacts. The ideawas that this might provide useful information regarding validation ofDEM codes. However, as a consequence of examining this simple impactproblem it was discovered that the problem is, in fact, an excellentexample for code verication, i.e. irrespective ofwhichmodel is chosenis the implementation of themodel correct? In this context, the paper hasdemonstrated that, if a non-linear contact model is used, the tangentialforce must be updated incrementally. This is due to the fact that thespring stiffness changes during the contact as a consequence of thevarying normal force. For a linear model, the results are identicalregardless of whether an incremental or a non-incremental approach isused.

Although the paper has demonstrated that very accurate predictionsof the rebound kinematics can be obtained using the simple LS modelthere are other aspects of the impact problem that need to be considered.In general DEM simulations, even in the collisional regime, the springstiffness cannot be reliably calibrated for all possible collisions due to theevolution is more realistic, and (iv) the impact duration is velocityline OC.Ifwe consider anon-linear springwith Ft = kt and kt = 8G

R

p, as

was done by Di Renzo andDiMaio [19], the problem can be illustrated asshown in Fig. A3. The forcedisplacement curve for elastic loading is OA.If, for a given , the calculated force is given by point B and this exceedsthe Coulomb limit then the force is reset to Ft=Fn and the state is givenby point Cwith the elastic displacement indicated by point D. However, ifone tries to calculate the elastic displacement using e = Fn = 8G4

R

p

then one obtains point E. This is further evidence that, for a non-linearspring, an integral equation cannot be used for the tangential interaction.If an integral equation is used and no attempt is made to calculate theelastic displacement e then both the H-MDns model and the LS modelgive similar results to the LTHmodel shown in Figs. 8, 9 and 10,which areclearly incorrect.Fig. A1. Tangential springdashpot system.

Fig. A3. Tangential forcedisplacement (nonlinear spring).

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An investigation of the comparative behaviour of alternative contact force models during elastic collisionsIntroductionDescription of the problemRigid body dynamics

Elastic impactThe HMD modelThe LS modelThe HM modelThe LTH model

ResultsLinear spring modelsComparison of linear and nonlinear spring models

DiscussionConclusionsAppendix AReferences