DOI 10.1140/epje/i2007-10356-3

Eur. Phys. J. E 27, 107–114 (2008) THE EUROPEAN

PHYSICAL JOURNAL E

Coefficient of tangential restitution for viscoelastic spheres

T. Schwager1, V. Becker1, and T. Poschel2,a

1 Charite, Augustenburger Platz 1, 13353 Berlin, Germany2 Physikalisches Institut, Universitat Bayreuth, D-95440 Bayreuth, Germany

Received 19 September 2007 and Received in final form 12 April 2008Published online: 27 August 2008 – c© EDP Sciences / Societa Italiana di Fisica / Springer-Verlag 2008

Abstract. The collision of frictional granular particles may be described by an interaction force whosenormal component is that of viscoelastic spheres while the tangential part is described by the model byCundall and Strack (Geotechnique 29, 47 (1979)) being the most popular tangential collision model inMolecular Dynamics simulations. Albeit being a rather complicated model, governed by 5 phenomenologicalparameters and 2 independent initial conditions, we find that it is described by 3 independent parametersonly. Surprisingly, in a wide range of parameters the corresponding coefficient of tangential restitution, εt,is well described by the simple Coulomb law with a cut-off at εt = 0. A more complex behavior of thecoefficient of restitution as a function on the normal and tangential components of the impact velocity, gn

and gt, including negative values of εt, is found only for very small ratio gt/gn. For the analysis presentedhere we neglect dissipation of the interaction in normal direction.

PACS. 45.70.-n Granular systems – 45.50.Tn Collisions – 45.50.-j Dynamics and kinematics of a particleand a system of particles

1 Introduction

The collision of granular particles may be characterized bytwo complementary descriptions: first, the collision maybe described by the time-dependent interaction forces innormal and tangential direction with respect to the con-tact plane, Fn and Ft, being the foundation of MolecularDynamics simulations. The components of the relative ve-locity in normal and tangential direction after the collisionare obtained by integrating Newton’s equation of motion.Second, the collision may be described by the coefficientsof restitution, εn and εt, quantifying the ratios of theincoming and outgoing relative velocities. The latter de-scription is the basis of event-driven Molecular Dynamics.

As both models describe the same physical phe-nomenon, the coefficients of restitution should be deriv-able from the forces, including material parameters, andthe impact velocity.

In this paper we consider the collision of frictional vis-coelastic spheres. The interaction force in normal directionwith respect to the contact plane, that is, in the directionof the unit vector between the centers of the particles,en ≡ (ri − rj)/|ri − rj | is described by the Hertz contactforce [1] extended to viscous deformations [2],

Fn = max

[0,

2Y√

Reff

3(1 − ν2)

(ξ3/2 + A

√ξξ

)], (1)

a e-mail: thorsten.poeschel@uni-bayreuth.de

where Reff ≡ RiRj/(Ri + Rj), with Ri and Rj be-ing the radii of the particles. The dynamical argumentof this force is the time-dependent mutual deformationξ ≡ max(0, Ri + Rj − |ri − rj |) of the particles and themax[. . .] takes into account that there are no attract-ing forces between granular particles. The material is de-scribed by the Young modulus Y and the Poisson ratioν and by the dissipative constant A being a function ofthe material viscosity, for details see [2]. The compressionobeys, thus, Newton’s equation

meff ξ + Fn

(ξ, ξ

)= 0; ξ(0) = gn ; ξ(0) = 0, (2)

with the normal component of the impact velocity gn =−(vi − vj) · en and meff ≡ mimj(mi + mj). Note thatthe normal and tangential force components Fn and Ft

are signed quantities. Their combination forms the vectorF = Fnen + Ftet of the total force.

The coefficient of normal restitution, defined as theratio of the post-collisional and the pre-collisional valuesof the normal component of the relative velocity followsfrom the solution of equation (2) through

εn ≡ −g′ngn

≡ ξ(tc)

ξ(0), (3)

where tc is the duration of the collision. From its defini-tion, obviously, 0 ≤ εn ≤ 1, where εn = 1 describes an

108 The European Physical Journal E

elastic collision. The analytical solution of equation (2)and, thus, equation (3) is technically complicated [3,4]and the computation of tc needed in equation (3) is evenmore complicated [5].

While the normal interaction force originates from theviscoelastic properties of the particle material and can be,therefore, mathematically derived from considering thedeformation of the bulk of the material [2], the tangen-tial force is mainly determined by the surface properties.In the literature, there are several models for the descrip-tion of the tangential force between rough spheres, for areview see [6]. In this paper we refer to the model by Cun-dall and Strack [7]. This model was used in many Molec-ular Dynamics simulations of granular systems and yieldsvery realistic results [8–16]. Therefore, it is the most pop-ular tangential collision model for the simulation of roughspheres. In the next section we will discuss the tangentialforce in more detail.

2 Tangential forces and the coefficient of

tangential restitution

The collision of frictional particles changes not only thenormal component of the relative velocity but also itstangential component as well as the particles’ rotationalvelocity.

The relative velocity of the surfaces of the particles atthe point of contact is

g ≡ vi − vj − (RiΩi + RjΩj) × en , (4)

with Ωi/j and Ri/j being the angular velocities and radiiof the particles. Its components in normal and tangentialdirection read

gn = (en · g)en , (5)

gt = (en × g) × en . (6)

The direction of the initial tangential velocity definesthe unit vector et in tangential direction. The tangen-tial component gt of the impact velocity is defined asgt(t) = g(t) · et, similar to the normal component gn. Bydefinition, the initial value of gt = gt(0) is non-negative,gt(0) ≥ 0. Similar to the coefficient of normal restitutionfor the change of the relative velocity in normal direction,equation (3), the coefficient of tangential restitution de-scribes the change of the tangential component1,

εt ≡ g′tgt

=gt(tc)

gt(0). (7)

The coefficient of tangential restitution has two elasticlimits, εt = ±1. In the case εt = 1 the tangential relativevelocity keeps conserved, that is, this limit correspondsto perfectly smooth particles. The other case, εt = −1,

1 Note that the definition of εt differs from the correspondingdefinition, equation (3), of εn by the sign. In the literature bothdefinitions are used.

corresponds to reversal of the relative tangential velocityas it would occur for the collision of very elastic gearwheels. Consequently, −1 ≤ εt ≤ 1. The value εt = 0describes the total loss of relative tangential velocityafter a collision. Let us elaborate now how the coefficientof tangential restitution relates to the correspondingcomponent of the interaction force.

From a theoretical point of view, the description of thetangential force is problematic: If the particles slide oneach other (dynamic friction), the tangential force adoptsthe value

Ft = − sgn[gt(t)]μFN , (8)

called the Coulomb friction law with the friction coeffi-cient μ. The function sgn(·) is the sign function. If theparticles do not slide, obviously, the tangential force mustbe |Ft| < μFn. So, which value is adopted then? In caseof no sliding, the tangential force adopts the value whichkeeps the particles from sliding. This tautological state-ment reflects the fact that the tangential force is not welldefined unless we make assumptions on the microscopicdetails of the deformation of the asperities at the contactsurface, see e.g. [17] for such a model.

For practical applications, that is, Molecular Dynamicssimulations, however, we need the tangential componentof the interaction force as a function of the particles’ rela-tive position and velocity and possibly of the history of thecontact. Therefore, for the application in particle simula-tions, interaction force models were elaborated to mimicstatic friction between contacting particles, see [6].

One of the most realistic models used in many Molec-ular Dynamics simulations is the model by Cundall andStrack [7] where static friction is described by means of aspring acting in the contact plane. The spring is initial-ized at the time of first contact, t = 0, and exists until thesurfaces of the particles separate from one another afterthe collision. The elongation

ζ(t) =

∫ t

0

gt(t′)dt′ (9)

quantifies the restoring tangential force, limited byCoulomb’s friction law, equation (8). The tangential in-teraction force reads then

Ft = − sgn(ζ)min(μFn, kt|ζ|), (10)

where kt is a phenomenological constant which character-izes the elastic resistance of the surface asperities againstdeformation in tangential direction. Note that the tangen-tial force at some time t∗, Ft(t

∗), does not only depend onthe relative positions and velocities of the particles at thesame time, ξ(t∗) and ξ(t∗), but on the full history of thecontact, 0 ≤ t ≤ t∗.

From equation (10) there follows the elongation of thespring |ζ| = μFn/kt in the regime of sliding, accordingto the Coulomb criterion. The spring acts as a reservoirof energy which may be released in a later stage of thecollision. As shown below, this reservoir is the reason whythe model by Cundall and Strack is able to model alsonegative values of the coefficient of tangential restitution.

T. Schwager et al.: Coefficient of tangential restitution for viscoelastic spheres 109

Albeit rather successful in Molecular Dynamics simula-tions, we would like to mention that the addressed modelis a phenomenological one which cannot be directly re-lated to microscopic properties of the material. Insofar,the constant kt of the above-introduced tangential springcannot be computed from material and surface proper-ties. Nevertheless, using a rather crude estimate we canobtain the order of magnitude of the phenomenologicalparameter kt if we relate the imaginary tangential springto the spheres’ surface roughness2. A more realistic two-dimensional model for the interaction forces in normal andtangential direction, which is based on the microscopic de-tails of the surfaces can be found in [18]. Here, a particle(disk) is modeled by 800 mass points which are connectedto their neighbors via non-linear springs. Because of itsgreat complexity, this model is, however, not suited forthe simulation of a many-particle system.

3 Limiting case of pure Coulomb sliding

Let us assume that during the entire collision the par-ticles slide on each other, that is, the friction force isnot sufficient to stop the tangential relative motion. Theforce is therefore described by equation (8), i.e. Ft =− sgn(gt)μFn.

We assume nearly instantaneous collisions, that is, theunit vector en does not change during the collision [19]and obtain from equations (5) and (6) Newton’s law forthe components of the relative velocities,

dgn(t)

dt= − 1

meff

Fn ;dgt(t)

dt=

1

αmeff

Ft , (11)

2 The order of magnitude of kt can be estimated from thefollowing argument: The maximal elastic force (neglecting thedissipative part) during an impact of spheres,

Fmax

n =μ

3

»250 Y 2π3ρ3

(1 − ν2)

–1/5

R2g6/5

n ,

can be obtained by equating the elastic energy of the com-pressed spheres (Hertz force) and the initial kinetic energy ac-cording to the normal component of the relative velocity gn.We assume that at least in the moment of maximum compres-sion, the particles do not slide, due to the Coulomb criterion,equation (8). The maximum tangential force, Ft = μFmax

n , cor-responds to the deformation ζ = Ft/kt of the spring in tangen-tial direction. With the realistic material constants used later(see caption of Fig. 2), μ = 0.5, Y = 210 MPa, ρ = 7850 kg/m3,R = 0.02 m, ν = 0.3 and assuming the initial normal velocitygn = 20m/s, we obtain ζ = 1.109 · 105 m s2/kg k−1

t . Thus,the value kt = 1012 kg/s2 used in the simulations below cor-responds to ζ = 1.109 · 10−7 m = 0.1109 μm. In a microscopicinterpretation, the tangential force originates from the defor-mation ζ of microscopic asperities of the surface (in tangen-tial direction). Obviously, the deformation of the asperities, ζ,should not exceed the size of the asperities themselves. Thenumerical value for ζ given above corresponds to the surfaceroughness of non-polished steel, therefore, the assumed valuekt = 1012 kg/s2 seems to be the right order of magnitude.

with the effective mass, meff ≡ mimj/(mi + mj), and

α ≡[1 +

meffR2i

Ji+

meffR2j

Jj

]−1

, (12)

where Ji and Jj are the moments of inertia of the parti-cles. The parameter α characterizes the mass distributionwithin the particles. For homogeneous spheres it adoptsthe value α = 2/7.

During the collision the normal component of the ve-locity changes by

g′n − gn = − 1

meff

∫ tc

0

Fn(t)dt = −(1 + εn)gn (13)

and the tangential component after the collision is

g′t − gt =1

αmeff

∫ tc

0

Ftdt = − μ

αmeff

∫ tc

0

Fndt

= −μ(1 + εn)

αgn. (14)

Using the definition of the coefficient of tangential resti-tution, we obtain

εCt = 1 − μ(1 + εn)

α

gn

gt, (15)

where the superscript C stands for pure Coulomb regime.Note that this result was derived independently from thefunctional form of the normal and tangential force laws(see [20,21]). Moreover, εC

t depends always significantlyon both the normal and the tangential components of therelative velocity.

According to our basic assumption of pure Coulombfriction, the particles slide on each other during the en-tire contact, that is, g′t ≥ 0 and εC

t ≥ 0. Consequently,equation (15) must be cut off:

εCt = max

(0, 1 − μ(1 + εn)

α

gn

gt

). (16)

Figure 1 shows the coefficient of tangential restitution,equation (16), as it follows from the simple Coulomb law.

4 Numerical integration of the equations of

motion

4.1 Example

Newton’s equations of motion, equations (11), can besolved numerically to obtain the coefficient of tangentialrestitution as a function of the components of the impactvelocity and the particle material constants. Before dis-cussing the more general scaling behavior of the problem,here we want to present an example of this integration inits phenomenological variables. Figure 2 shows εt(gn, gt)for material parameters of steel. The values of the corre-sponding system parameters are given in the caption ofthe figure.

110 The European Physical Journal E

0.5 1

1.5 2

0.51

1.52

0

0.2

0.4

0.6

0.8

1

εCt

gt gn[m/s] [m/s]

Fig. 1. The coefficient of tangential restitution as a functionof the components of the impact velocity, gn and gt, in the caseof pure Coulomb friction. The relevant parameters are μ = 0.4and α = 2/7.

50100

150200

050

100150

200

-0.6-0.4-0.2

0 0.2 0.4 0.6 0.8

1

εt

gt gn[m/s] [m/s]

Fig. 2. Coefficient of tangential restitution for the collisionof identical steel spheres of R = 2 cm (Y = 210 GPa, ρ =7850 kg/m3, and ν = 0.3, α = 2/7) as a function of the im-pact velocities, εt(gn, gt). Dissipation in normal direction wasneglected, the tangential spring constant was kt = 1012 kg/s2,the friction constant was μ = 0.4. For small tangential impactvelocities the coefficient of restitution may become negaitve.

It turns out that εt(gn, gt) resembles the coefficientεC

t (gn, gt) of the simplified model discussed in Section 3in a wide range of impact velocities. Significant deviationsare found only in the region of small gt where also negativevalues of εt are found. Differently from the model of pureCoulomb friction, here εt is not cut off at some value,but the plateau comes from the inherent dynamics of theCundall-Strack model and the corresponding equation ofmotion. The plateau at εt ≈ 0 as well as the oscillationsfor small gt can be understood from the analysis of thescaled equations of motion given below.

4.2 Scaling properties

In this paper we focus on the coefficient of tangential resti-tution, therefore, to derive scaling properties we neglect

the damping of the motion in normal direction and as-sume A = 0 for the dissipative constant in equation (1).At first glance, this assumption seems to be an importantsimplification, however, the complex behavior of the tan-gential motion discussed here is caused exclusively by thetangential interaction force. Thus, the choice of the in-teraction force in normal direction is less important. Themain difference between A = 0 and A > 0 is the du-ration of the contact of the colliding spheres. Except forthese quantitative changes, the system behavior will bethe same for A > 0 since (naturally) the normal and tan-gential forces are perpendicular to one another and, thus,do not interfere.

We write the equations of motion in a more convenientway,

ξ + βξ3/2 = 0,

ζ + min

(μβ

αξ3/2,

kt

αmeff

ζ

)= 0, (17)

ξ(0) = 0; ξ(0) = gn ; ζ(0) = 0; ζ(0) = gt ,

with

β ≡ 2Y

3(1 − ν2)

R1/2

eff

meff

. (18)

In the Coulomb regime, i.e. if Ft = − sgn(ζ)μmeffβξ3/2,the elongation of the tangential spring is ζ =sgn(ζ)μmeffβξ3/2/kt with the sign, sgn(ζ), taken from thevalue of ζ before applying the limiting rule.

The above problem contains 5 phenomenological pa-rameters: β, kt, μ, meff , α and 2 initial velocities gn and gt.Using suitable units of length and time, we can reduce thenumber of independent parameters to only 3. The properlength scale is the maximal compression,

ξmax =

(5

4

)2/5g4/5

n

β2/5. (19)

The proper time scale is the duration of the collision,

tc =C

β2/5g1/5

n

, (20)

with C being a pure number. We drop the numerical pref-actors and obtain the time and length scales

tscale = β−2/5g−1/5

n ,

ξscale = g4/5

n β−2/5 . (21)

We define the scaled compression and elongation of thetangential spring, x ≡ ξ/ξscale, z ≡ ζ/ξscale and writetime derivatives with respect to the scaled time t/tscale.The initial conditions of the collision in scaled variablesare

x(0) = gntscaleξscale

= 1,

z(0) = gttscaleξscale

=gt

gn, (22)

T. Schwager et al.: Coefficient of tangential restitution for viscoelastic spheres 111

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 5 10 15 20

0.5

1

1.5

2

2.5

3

3.5

tgng /

κ

Fig. 3. The coefficient of tangential restitution as obtainedfrom the numerical solution of the scaled equation of motion,equation (23). For explanation see text.

and the equations of motion are

x + x3/2 = 0,

z + min(μ

αx3/2, κz

)= 0, (23)

x(0) = 0; x(0) = 1; z(0) = 0; z(0) =gt

gn,

with

κ ≡ kt

αmeffβ4/5g2/5

n

. (24)

If the Coulomb limit is active, the tangential elongationis determined by z = μx3/2/(ακ). Hence, the scaled prob-lem contains only three parameters: μ/α, κ, and the ratiogt/gn. Figure 3 shows the solution of Newton’s equationof motion in the reduced variables. As shown in the ex-ample, Figure 2, for sufficiently large values of gt or gn weobtain a very small coefficient of tangential restitution,εt ≈ 0. For small gt/gn or large κ corresponding to a stifftangential spring, the plot reveals the typical oscillations.The limit of large κ is discussed in Section 4.4.

4.3 Switching between the modes of the tangentialforce

Depending on the parameters of the motion κ, μ and gt/gn

the motion of the particles during contact can be compli-cated. For sufficiently high scaled elasticity κ during thecollision, the tangential force, equation (10), may switchmultiple times between the Coulomb regime, Ft = ±μFn,and the Cundall-Strack regime, Ft = −ktζ. This is shownin Figure 4.

For large values of gt/gn � 3 we see that the collisiontakes place entirely in the Coulomb regime, except for verysmall values of κ. This corresponds to the upper region in

g /g t

n

8 12 16o

4 κ

1

3

2

ab

c

d

ef

h

...

2

46

0

8

Fig. 4. During the collision the tangential force according toequation (10) may switch multiple times between the Coulombregime, Ft = ±μFn and the Cundall-Strack regime, Ft = −ktζ.The figure shows the number of switching events during thecollision in dependence on the system variables κ and gt/gn

for identical homogeneous spheres (α = 2/7) and μ = 0.4. Thenumbers in circles indicate the number of switching events.For κ = 18 and the indicated values of gt/gn (“a” . . . “o” (notall letters are shown)) the time-dependent forces are drawn inFigure 5.

Figure 3 where the resulting coefficient of restitution isalmost zero.

For larger values of κ, depending on gt/gn the forceswitches repeatedly between the regimes. For the exampleκ = 18, the time-dependent forces are drawn in Figure 5,where the letters on the right-hand side of Figure 4 indi-cate the ratio of the impact velocity components. Theseletters correspond to the same letters in Figure 5.

First one notes that the motion always starts in theCoulomb regime (Fig. 5a-o). This is due to the fact thatat the beginning of the collision the normal force increasesroughly as t3/2 while the tangential force increases as t.Thus, in the beginning of the contact, t � 0, the tangentialforce is always larger than μFn for a certain period oftime. The same is true for the final part of the collision:with the same line of reasoning, one can convince oneselfthat the collision must terminate in the Coulomb regime—except for the trivial case gt ≡ 0. Therefore, the numberof switches between Cundall-Strack regime and Coulombregime is always even.

Furthermore, for very small values of κ the system can-not remain in the Coulomb regime. It can only remain inthe regime of pure Coulomb friction if μ/αx3/2 is smallerthan κz (see Eq. (23)). Hence for sufficiently small valuesof κ we will alway observe a switch to the Cundall-Strackregime (and back to the Coulomb regime at the end of thecollision). This explains the vertical line on the left-handside of Figure 4 which indicates this region of small κ witha non-zero number of switches.

112 The European Physical Journal E

a b c

d e f

g h i

j k l

m n o

Fig. 5. The normal and tangential components of the inter-action force as a function of time for κ = 18 and the valuesgt/gn indicated in Figure 4. The full lines show the tangentialcomponent, Ft/μ, due to equation (10) and the dashed linesshow the normal component, ±Fn, due to equation (1) (withA = 0).

For a large ratio of the velocity components, gt/gn

(Fig. 5a) we observe the regime of pure Coulomb friction,that is, the number of switches is zero, except for verysmall values of κ.

For a velocity slightly smaller than the limit of pureCoulomb friction (Fig. 5b) the friction force is sufficientto stop the particle (in tangential direction) in the courseof the collision. Hence, the force switches to the Cundall-Strack regime. The direction of the relative motion of theparticles at the contact is reversed and the force switchesback to the Coulomb regime eventually. Since we have onemotion reversal, the coefficient of tangential restitution isnegative.

For even smaller initial tangential velocity (Fig. 5c)the motion is stopped a second time after the switch backto the Coulomb regime (second switch). Hence, a secondmotion reversal takes place and subsequent switch to theCoulomb regime. As we have two motion reversals (andfour switches) the coefficient of tangential restitution ispositive.

Note that the elongation of the Cundall-Strack springat the instant of the first switch from the initial Coulombregime increases with decreasing initial tangential velocity,as all switches take place in the final decompression phase.Therefore, the tangential oscillation cannot complete evenhalf a period —the switch to the Coulomb regime afteronly one motion reversal is inevitable. The number of mo-tion reversals is half the number of switches.

For further decreasing initial tangential velocity thenumber of switches increases to its maximum of eight (forthe chosen elasticity κ), see Figure 5e. In this case thecoefficient of tangential restitution is positive (due to fourmotion reversals). The initial switch from the Coulombregime takes place just before the moment of maximumcompression. Now, for further decreasing initial tangential

velocity (Fig. 5f) the tangential oscillation may completehalf a cycle once —the number of switches decreases to six.However, we have four motion reversals, the coefficient oftangential restitution is positive again.

For an even smaller velocity (Fig. 5g) we have againeight switches as there is an additional switch from theCoulomb regime just before the end of collision, accompa-nied by a motion reversal and a final switch back to theCoulomb regime. The coefficient of tangential restitutionis negative.

For further decreasing tangential velocity the initialelongation of the tangential spring decreases. The tan-gential oscillation performs more and more cycles beforeswitching back to the Coulomb regime. This switch takesplace later and later, reducing the number of switches be-tween the regimes that may take place. So, for decreas-ing tangential velocities we have a decreasing number ofswitches. This decrease is not monotonous, there are tan-gential velocities where the number of switches briefly risesby two due to short episodes of motion in the Cundall-Strack regime near the end of the collision (Fig. 5j).

Finally, for sufficiently small initial tangential veloc-ity (Fig. 5o) the motion switches from the Cundall-Strackregime once, the tangential spring performs a number ofundamped oscillations and finally switches back to theCoulomb regime. The number of switches is two.

Note that, due to the undamped nature of the tangen-tial spring there can only be energy dissipation during theCoulomb regime.

4.4 Limit of stiff tangential springs

For very stiff tangential springs, that is, large kt or largeκ in scaled variables, and a certain range of gt/gn we canestimate the absolute value of εt. We will show that inthis case |εt| is small with the limit limκ→∞ |εt| = 0.

Assume the ratio of the velocity components, gt/gn,sufficiently small such that the relative tangential motionof the particles stops at least once during the course ofthe collision (see Sect. 4.3). The largest elongation of thescaled tangential spring is zmax = μ/(ακ). Figure 6 (toppanel) shows a typical situation. Therefore, when the tan-gential motion of the particle ceases, the maximum energystored in the tangential spring is

Emax

tan =1

2κz2

max =μ2

2α2κ. (25)

This is also the maximum amount of energy of the tan-gential motion that can be recovered after the collision,provided there are no further switches to the Coulombregime3. Therefore, equation (25) is an upper limit of

3 As discussed above, close to the end of the collision theforce turns back to the Coulomb regime where further energy isdissipated. Without damping, this switch back to the Coulombregime is inevitable, except for the trivial case of vanishinginitial tangential velocity. For non-zero normal damping thereexists a range of initial conditions where the collision ends inthe Cundall-Strack regime. Nonetheless, the above estimate,equation (25), being an upper limit, stays correct.

T. Schwager et al.: Coefficient of tangential restitution for viscoelastic spheres 113

0 1 2 3rescaled time

0

0.1

0.2

0.3

rela

tive

tang

entia

l vel

ocity

gt(t

)/g n(t

)

0 1 2 3rescaled time

−1

0

1

Ft ,

+μ

Fn

, -μ

Fn

(sc

aled

uni

ts)

Fig. 6. Top panel: the normal (dashed lines) and tangen-tial (full line) component of the interaction force for the caseof very stiff tangential springs (κ = 100). The initial valuez(0) = gt/gn = 0.3. As discussed in the text, the tangentialmotion is stopped near the point of maximal normal compres-sion. For better presentation, the tangential force was multi-plied by 1/μ. After this stop of tangential motion there aremany subsequent episodes of Coulomb sliding. Bottom panel:the tangential velocity over time. For better orientation, thevalue of gt = 0 is marked (dashed line). As discussed in thetext, the velocity estimated by equation (26) (first minimumshortly after t = 1) grossly overestimates the final velocity.

the final energy of the tangential motion and with z =gt(t)/gn, defined in equation (22), we obtain

|g′t| ≤ gnμ

α√

κ. (26)

Hence, for very stiff tangential springs we have

εt ≤ gn

gt

μ

α√

κ. (27)

The above considerations are valid if the initial tangentialenergy is larger than Emax

tan , i.e.

gt ≥ gnμ

α√

κ. (28)

Otherwise the initial kinetic energy of the tangential mo-tion would not be sufficient to fully load the tangentialspring. Note that there is no contradiction between equa-tions (26) and (28), the former concerning g′t and the lat-ter concerning gt. Combining both equations leads to thetrivial but correct |g′t| ≤ gt. Furthermore, the result equa-tion (27) is only valid if the tangential motion ceases at

least once. As a rough estimate, there will be no stopping if

gt ≥ 2μ

αgn , (29)

see equation (15) for εn = 1 and εt ≥ 0. The actual lim-iting tangential velocity of pure Coulomb friction is evensmaller. However, the limit equation (27) still holds, as, ac-cording to equation (15), the coefficient of tangential fric-tion is zero for gt = 2μgn/α in the case of pure Coulombfriction. Note that in the whole discussion we assumedthat there is no further energy loss after the first switchfrom the Coulomb regime to the Cundall-Strack regime.However, this assumption is not realistic as for high tan-gential stiffness there will be many episodes of Coulombsliding with the accompanying energy loss (see Fig. 6, toppanel). The resulting final tangential velocity will actuallybe much smaller than the estimate equation (27). Figure 6,bottom panel, illustrates the mechanism. Hence, for largetangential stiffness κ there is a range

gnμ√ακ

� gt ≤ 2μ

αgn , (30)

where we can approximate the coefficient of tangentialrestitution in good accuracy by zero! Therefore, the pureCoulomb model, equation (15), as discussed before, is agood approximation for the velocity dependence of thetangential coefficient of restitution. This vindicates thepragmatic model εt = max[εconst

t , 1 − μ(1 + εn)/α · gn/gt](with εconst

t being a constant from the interval [−1, 1]) asused in the literature [20–23]. However, as we have seenin the course of the discussion, the only sensible choice ofεconstt for Cundall-Strack–like tangential interaction mod-

els is εconstt = 0.

5 Conclusions

We investigated the coefficient of tangential restitution,εt, for the dissipative collision of frictional viscoelasticspheres. The tangential force was described by the modelby Cundall and Strack [7] which was shown to deliverrather realistic results in many Molecular Dynamics sim-ulations of granular systems, e.g. [8–16]. Here the tangen-tial force is modeled by means of a linear spring in thedirection of the contact plane where the force is cut offdue to the Coulomb friction law. It was shown that thedynamics of the system depends on 3 independent vari-ables, including material constants and initial velocities ofthe collision. The detailed dynamics of the collision maybe rather complex, revealing multiple transitions between(dissipative) Coulomb friction and (conservative loadingand unloading) of the spring. Nevertheless, in a ratherwide range of parameters, the collision is well describedby simple Coulomb friction in tangential direction with acut-off at εt = 0, thus, 0 ≤ εt ≤ 1. Only for small ratio,gt/gn, of the tangential and normal components of the im-pact velocity the coefficient of tangential restitution showsa more complicated behavior, that is, it oscillates between

114 The European Physical Journal E

εt = −1 and εt = 1 in dependence on gt/gn and on κ beingmainly a function of the particle properties, in particularthe strength of the tangential spring. Only in this regionwe observe negative values of the coefficient of tangentialrestitution. For stiff tangential springs, asymptotically weobtain εt → 0.

As a consequence, we obtain that the coefficient of tan-gential restitution as obtained from pure Coulomb fric-tion, cut off at ε ≥ 0, equation (16), is a rather goodapproximate description for the colliding spheres. A sim-ilar approximation where, instead of the limit of zero inequation (16) a constant from the interval [−1, 1] was as-sumed, was already used in event-driven Molecular Dy-namics simulations, e.g. [20–23], where the justification ofthis approximation was not considered.

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