Coefficient of restitution and linear dashpot model revisited

0.08 MB·English

by  Thomas Schwager

GranularMattermanuscriptNo.
(willbeinsertedbytheeditor)
7 ThomasSchwager ThorstenPo¨schel
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0 Coefficient of restitution and linear dashpot model revisited
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f Received:February6,2008
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t. Abstract With the assumption of a linear-dashpot interac- istheunitvectorattheinstanceofthecollision.Forsimplic-
a
tion force, we compute the coefficient of restitution as a ity,weconsiderparticlesofidenticalmass-thegeneraliza-
m
function of the elastic and dissipative material constants, k tiontounequalmassesisstraightforward.
- andg byintegratingNewton’sequationofmotionforaniso- The coefficient of restitution is the most fundamental
d
latedpair of colliding particles.It turns out that an expres- quantity in the theory of dilute granular systems. This co-
n
o sione (k,g )widelyusedintheliteratureisevenqualitatively efficient together with the assumption of molecular chaos
c incorrectduetoanincorrectdefinitionofthedurationofthe is the foundation of the Kinetic Theory of granular gases,
[ collision. based on the Boltzmann or Enskog kinetic equation, see,

e.g., [1; 7; 8] and many references therein. On the other
1
Keywords particlecollisions coefficientofrestitution
v · hand,thecoefficientofrestitutionisalsoessentialforevent-
8 driven Molecular Dynamics simulations which allow for a
7 significant speed-up as compared to traditionalforce-based
2 1 Introduction simulations,see,e.g.,[9]foradetaileddiscussion.Themost
1 important precondition of Eq. (1) is the assumption of ex-
0
The(normal)coefficientofrestitutionofcollidingspherical clusivelybinary collision,that is, the system is assumed to
7
0 particlesrelatesthepost-collisionalvelocitiesv′i andv′j and bediluteenoughsuchthatmultiple-particlecontactscanbe
t/ thecorrespondingpre-collisionalvelocitiesvi andvj, hnoegwleevceter,di.tTwhaeslsahttoewrcnoinndmitaionnysseimemulsattoiobnesrtahtahteervreenstt-rdicrtiviveen,
a
m 1+e simulationsareapplicableuptoratherhighdensity[6].
- vi′=vi− 2 [(vi−vj)·eij]eij The coefficient of restitution is, however, not a funda-
d 1+e (1) mentalmaterialorparticleproperty.Instead,theparticlein-
n v =v + [(v v ) e ]e teractionforcesandNewton’sequationofmotiongovernthe
o ′j j 2 i− j · ij ij dynamicsofamechanicalmany-particlesystem.Therefore,
c
if we wish to use the concept of the coefficient of restitu-
: where
v tion and the simple equation (1) to compute the dynamics
Xi e ri−rj (2) of a granular system, we have to assure that Eq. (1) yields
ij≡ r r thesamepost-collisionalvelocitiesasNewton’sequationof
r i j
a (cid:12) − (cid:12) motionwoulddo.
(cid:12) (cid:12)
This research was supported by German Science Foundation (Grant
PO472/15-1).
2 CoefficientofrestitutionandNewton’sequationof
ThomasSchwager
Charite´ motionforfrictionlesssphericalparticles
AugustenburgerPlatz
10439Berlin Letusconsidertherelationbetweenthecoefficientofresti-
Germany
tutionandthesolutionofNewton’sequationofmotion.As-
E-mail:[email protected]
sumetwoapproachingparticlesatvelocitiesv andv come
1 2
ThorstenPo¨schel intocontactattimet =0.Fromthisinstantontheydeform
Charite´
one anotheruntil the contactislost attimet =t .We shall
AugustenburgerPlatz c
10439Berlin considertheendofthecollisionattimetc separatelybelow.
Germany Forsufficientlyshortcollisionswecantreattheunityvector
E-mail:[email protected] e as a constant. Then we can describe the collision by the
2
mutualdeformation withinitialconditions
x (t) max(0, 2R r r ) (3) x (0)=0; x˙(0)=v (9)
1 2
≡ −| − |
wherer (t)andr (t)arethetime-dependentpositionsofthe hastwosolutions.Withtheabbreviations
1 2
particlesandRistheirradius.Theinteractionforcebetween 2k g
theparticlesismodelspecific,e.g.[9;11],and,ingeneral,a w 2 ; b ; w w 2 b 2. (10)
functionofthematerialparameters,theparticlemassesand 0 ≡ m ≡ m ≡q 0−
radii,thedeformationx andthedeformationratex˙.Itstime wecanwritethesolutionforthecaseoflowdamping(b <
dependence is expressed via F =F x (t),x˙(t) . Newton’s w 0)as
equationofmotionforthecollidingp(cid:16)articlesrea(cid:17)ds x (t)= ve b tsinw t, (11)
w −
m
2x¨+F(cid:16)x˙,x (cid:17)=0; x (0)=0; x˙ =v (4) whileforthecaseofhighdamping(b >w 0)wehave
where v is the normal component of the initial relative ve- x (t)= v e b tsinW t, (12)
locity, W −
with
[v (0) v (0)] [r (0) r (0)]
1 2 1 2
v − · − . (5)
≡ 2R W = b 2 w 2. (13)
ThecoefficientofrestitutionfollowsfromthesolutionofEq. q − 0
(4), To determine the coefficient of restitutionaccording to Eq.
(6)weneedthedurationofthecollisiont .Anaturalchoice
c
e = x˙(t ) v, (6) seemstobe
c
− .
x t0 =0; t0>0, (14)
wheret isthedurationofthecollision. c c
c
(cid:0) (cid:1)
Thus,thecoefficientofrestitutionduetoaspecifiedin-
thatis,thecollisionisfinishedwhen r r =2R.Withthis
teractionforce lawcanbe determinedby solving the equa- i− j
tionofmotion[10;11;13].Ingeneral,e isafunctionofthe conditionweobtaintc0=p /w andth(cid:12)(cid:12)ecoeffi(cid:12)(cid:12)cientofrestitu-
tion
material parameters, the particle masses and radii and the
impactvelocityv. x˙ t0
e 0 c =e bp /w , (15)
d ≡ x˙(cid:0)(0)(cid:1) −
3 Lineardashpotmodel for the case of low damping which is widely used, e.g. in
[11]andmanyothers.
Considerthelineardashpotforce, Unfortunately, condition(14)cannot be correct sinceat
timet0=p /w theinteractionforceisnegative,
F x ,x˙ = kx g x˙. (7) c
(cid:16) (cid:17) − − mx¨ p = 2mvb e bp /w <0, (16)
Thisforceisproblematicasaparticleinteractionmodelsince (cid:16)w (cid:17) − w −
particlesmadeofalinear-elasticmaterialdonotrevealalin-
seeFig. 1, whichcontradictsthe assumptionof exclusively
earrepulsiveforce,neitherintreedimensions[4]norin2D
repulsive interaction between granular particles. Moreover,
[3].Thesubsequentanalysiscanbealsoperformedformore
for the case of high damping Eq. (14) has no real solution,
realisticforce models, such as viscoelasticforces in 3d [2]
that is, the particles sticktogether after a head-on collision
and2d[12]whichwillbepublishedelsewhere.
(seeFig.2).Onecancallthissituationdissipativecapture.
Thelineardashpotmodelisofinterestherebecausethis
Toavoidtheseartifacts,insteadofEq.(14),wenowex-
forceleadstoaconstantcoefficientofrestitution,e.g.[11],
plicitely take into account the condition of no attraction.
that is, e does not depend on the impact velocity but only
That means, instead of Eq. (7) we have to use the force
onthematerialparameterskandg .Theconstantcoefficient
F =max(0, F), with F given by Eq. (7). This expression
of restitution in turn is the preferred model in both the Ki- ∗
as is in common use in (force-based) Molecular Dynamics
neticTheoryofgranulargasesandalsoinevent-drivensim-
simulations,assuresthatonlyrepulsiveforcesact.Itisclear
ulationsofgranularmatter.Incontrast,morerealisticforce
thatoncetheforcebecomeszero,thecollisionisfinished,
models lead to an impact-velocitydependent coefficient of
restitution[5;10;13;14;15]. x¨(t )=0; t >0; x˙(t )<0, (17)
c c c
TheequationofmotioncorrespondingtoEq.(7)
thatis,thecollisioniscompletewhentherepulsiveforcebe-
mx¨+2g x˙+2kx =0 (8) tweentheparticlesvanishesandthesurfacesoftheparticles
3
x <0 x >0 x max>0 x >0 x >0 x >0 x <0
x˙=g>0 x˙>0 x˙=0 x˙<0 x˙∗<0 xx˙˙><x0˙∗ x˙<0
F=0 F>0 F>0 F>0 F=0 F<0 F=0
F=0 F>0 F>0 F>0 F=0 F=0 F=0
Fig.1 Sketchofaparticlecollision.Theupperpartofthefigurevisualizesthecondition(14)fortheendofthecollision.Asanartifactofthe
model,thereappearattractingforcestowardstheendofthecollision(boxednotation).Lowerpart:Thecondition(17)fortheendofthecollision
avoidsthisartifact.
separate from one another. This condition takes, thus, into (21)
accountthatEq.(7)appliestoparticlesincontact.
ForthecaseoflowdampingEqs.(11)and(17)yield Inaccordancewithphysicalintuitionwehaveed >ed0 since
theresultgiveninEq.(21)doesnot sufferfrom theartifact
2bw ofunphysical attractiveforces impliedinEq.(15).The un-
tanw t = . (18)
c −w 2 b 2 physicaldissipativecapturedoes not occur–evenforlarge
− dampingconstanttheparticlesseparateeventually.Forvery
One has to take care to select the correct branch of the arc
high damping the coefficient of restitution can be approxi-
tangentforb <w /√2(orb <w )andforb >w /√2.The
0 0 matedas
solutionreads
w 2
w1 (cid:18)p −arctanw 22bw b 2(cid:19) for b < √w 02 ed ≈ 4b02 for b ≫w 0 (22)
tc= − (19) Fig. 2 shows both the (wrong) coefficient of restitution
1 2bw w
arctan for b > 0 as derived from criterion (14) (dashed line) and the coef-
w w 2−b 2 √2 ficient of restitution derived from the correct criterion (17)
 (fullline).AsthesolutionEq.(21)onlydependsontheratio
Forthecoefficientofrestitutionwefind
a =b /w we draw the coefficient of restitutionas a func-
0
ed =e−b tc (20) tion of this parameter. Figure 2 shows that the unphysical
attractionindeed yields a too low coefficient of restitution.
regardlessofthebranchofthesolutionofEq.(18).Together
Theinsetshowsalogarithmicplotofthesamecurveshow-
with the solution for the case of high damping which can
ingthatthereisnocaptureifthecorrectcriterionisapplied.
beobtainedbystraight-forwardcomputationsweobtainthe
coefficientofrestitution:
b 2bw w 4 Conclusion
exp p arctan forb < 0
 (cid:20)−w (cid:18) − w 2 b 2(cid:19)(cid:21) √2
ed=exp(cid:20)−wb arctanw 22bw b 2(cid:21)− forb ∈(cid:20)√w 02,w 0(cid:21) AatisesssaubfmyuniinncgtteioagnrlaiontifenagtrhNedaeeswlhatpsotonict’sianentqeduradactiistoisonipnoafftoimvrceoemtiwoaentecfrooiarmlapppuraoteiprdeoref-
−
b b +W colliding particles. The condition for the duration t of the
exp(cid:20)−W lnb −W (cid:21) forb >w 0 collision x (cid:0)tc0(cid:1)=0, that is, the interaction stops wchen the
4
1
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0
Springer,Berlin
0.8 10. Ram´ırezR,Po¨schelT,BrilliantovNV,SchwagerT(1999)Coef-
ficientofrestitutionofcollidingviscoelasticspheres.PhysRevE
60:4465–4472
eon 0.6 en -5 11. Scha¨ferJ,DippelS,WolfDE(1996)Forceschemesinsimulations
ti l ofgranularmaterials.JPhysI(France)6:5–20
u
t 12. SchwagerT(2006)Coefficientofrestitutionforviscoelasticdisks.
sti0.4 cond-mat/0701142
re -10 13. SchwagerT,Po¨schelT(1998)Coefficientofrestitutionofviscous
0 5 10 15 20 particlesandcoolingrateofgranulargases.PhysRevE57:650–
0.2 a
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Fig.2 Coefficientofrestitutionasfunctionofa =b /w 0.Thedashed
curveistheresultforthewrongterminationconditionx (t0)=0.Obvi-
c
ously,forhighdampinga >1wehavedissipativecapture.Theinlay
shows lne as a function of a for the correct termination condition.
Thereisnodissipativecaptureevenforveryhighdamping.
particleshavethedistance r r =2R,iswidelyusedin
i j
−
theliterature,however,itlea(cid:12)dstoer(cid:12)roneousattractingforces
(cid:12) (cid:12)
betweentheparticlesand,thus,towrong valuesfortheco-
efficientof restitution.In extremecases,the result maybe-
come even qualitativelyincorrect since despite the interac-
tionforcebeingpurelyrepulsive,theparticlesmayagglom-
erate.Inthisstateonewouldneedtosupplyenergytosepa-
ratetheparticlesfromoneanother.
Toaccountforthefactthattheinteractionforcecannever
become attractive, we use the condition x¨(t ) = 0 which
c
avoidsthementionedartifacts.Thisconditiontakesintoac-
countthatthecollisionmaybecompletedevenbeforex =0,
thatis,thesurfacesoftheparticleslosecontactslightlybe-
fore the distance of their centers exceeds the sum of their
radii.Thus,thedeformationoftheparticlesmaylastlonger
thanthe timeof contactand the particlesgraduallyrecover
theirsphericalshapeonlyaftertheylostcontact.
The resulting coefficient of restitution is always larger
than the value based on x t0 =0. In agreement with the
c
repulsivecharacterofthein(cid:0)ter(cid:1)actionforce,thecoefficientof
restitutionisalwayswelldefinedandnon-zero.
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