On Low-Velocity Collisions of Viscoelastic Particles

J Phys France II (1995) 5 1725-1738 NOVEMBER1995, 1725 PAGE Classification Phj,sics Abstracts 83.10-y 46 10+2 62 20-x Low-Velocity On Collisions Hertzsch(~), Jan-Martin Frank Viscoelastic of Spahn(~) and Nikolai Particles V. Brilliantov(~) Max-Planck-Arbeitsgruppe ~'Nichtlmeare Dynamik", Universitat Potsdam, Potsdam, Germany 15 53, D-14415 State Moscow University, Moscow 119899, Russia (~) Physics Department. (~) PF Am Neuen Palais, 60 (Received 7 October1994, July 1995) reiised 7 1995, Alarch received m final form 10 July1995, accepted 28 developp6e par Hertz [1] est g6n6corps visqueux I la tension totale Une dquation difl6rentielle nonlin6aire derivAe des particles dont les surfaces courbure est ont pour une arElle est r6solue bitraire. nurnAriquement dans le cas des particles sphdriques. La d6pendence du coefficient de riormale de la vitesse d'impact est calcu16e et compar6e avec des restitution donn6es exp6rimentales la glace aux obtenues temp6ratures basses [2,3]. Un bon accord pour trouv6 qui permet l'estimation des du mat6riel dans certains Une applicaest constantes cas. astrophysique de nos tion r6sultats discutAe brAvement dans un cas d'interAt particuher: des est particles de glace dans des plan6taires anneaux R4sumd. ralisde La th60rie compte tenant du contact 61astique contribution de la des de deux effects The theory of the elastic of two bodies developed by Hertz [1] is genercontact effects to the total including the contribution of nonlinear differential A stress viscous equation for the derived for particles arbitrary with of their and surfaces curvature compression is numerically for spherical particles The resulting dependence of the is solved normal restitution coefficient the calculated and compared with experimental data for ice at impact velocity is on low unknown material [2,3]. A good agreement is found which allows to estimate temperatures in certain An astrophysical application of the briefly results discussed for the constants cases is especially interesting case of icy particles in planetary rings Abstract, alized Introduction 1. Hertz's theory Unfortunately, account in a of the up to widely used in contact mechanics. theory taking into to exist no seems such effects like viscosity, although inelasticity effects in collisions physical processes and can affect seriously the development of manyof particular planetary rings, e.g. those of Saturn, which interest are contact now realistic way role in of elastic satisfying bodies play an important particle systems. One case composed of particles mainly consisting are influenced by several factors, especially the Q Les Editions de Physique 1995 [I] has extension of water been of this ice [4]. collisional The lifetimes properties of the of these particles. systems Wiesel are [5] JOURNAL 1726 PHYSIQUE DE II N°11 already found that the stability of planetary rings is not only strongly dependent on their coefficient of the material, I-e the ratio of the velocity distribution, but also on the restitution simulations postcollisional relative velocity of two particles to the precolhsional one. In these [2,3,6] Bridges, al. value assumed, but by Hatzes have shown experiments et constant a was coefficient the form dependence of the Therefore restitution the velocity strong impact a on of this dependence is liable to affect the stability and thus the evolution of a planetary ring which detailed mechanism Thus, we have started investigations of the collision to play seems role the gravitational satellites related of important action structures to [7]. an in had Poschl [8] proposed the attempts to generalize the theory of collisions of of the dissipative proportional the velocity compression. term to a power a dependency of quadratic he able this velocity in form In the special to express case was a achieved of a the Although qualitatively right results, his series in compression. he power motivated proposition had the disadvantage that the choice of the exponent of the velocity was Thus, no of the by mathematical rather than by physical relation arguments. convenience As of coefficients bodies found be collision introducing of results effects some (coefficients of be priori. found In a this propose bodies by Hertz [1,13,14] Its coefficients derived the on due contact m of curvature spherical theory "by spheres [3] ice and velocity, of part a elastic their to depend surfaces. their the on planetary coefficient the on Because and material estimation is of the advantage an intended the assumption power Our are of -1/4 that constant a of the one better Finally, we required than of impact to check mean with model is of granular on [12] we exerted stresses Following the algorithm dependence of the compression properties of the colliding bodies and also solved numerically for the simple case of for ice at of temperatures low collisions inelastic for the velocity curves dependencies coefficients have properties of the to be led to the [2,3], an dynamics restitution restitution coefficient chosen for this particles, this viscosity. This ability of of the restitution a for dependent of the yet unkno~vn discussed theory of inelastic impact of metallic bodies. It was based on a and material of the time models material pressure appear m coefficient [14] purpose allows our u.hich the model was analysis and proportional to static the solutions [8]. Our of other rigorous differential model is found equations for the compresexperimental certain to fit ones. possible our derivative time velocity. P6schl's other the collisions compared the discuss to of by Dilley ill]. He results presented and of energy. First, the properties. the available are two the magnitude comparison treatment model is also sion, e-gresults in for the the of the order Only to its consideration the viscous for theoretical rings [15]. A good fit of the different with the experimental with ones impact velocity is achieved. directly related these are and investigations equation The experimental dissipation viscous with equation spheres where experimentally obtained data example of importance for further applications of colliding of the velocity is made dependent on the shape of the particles is taken into account by This procedure allows for a good fit to hand". but the right choice of three constants requires of the impact velocity) which exponent cannot starting differential article recent a deformation general more theory Hertz's material deformation The clean is of extension of the the of in mentioned Hertz deformation which paper an the used actual Both masses law with experiments above assumption the coefficient of the discussed were the well point as of the constants viscous model. on and the power, velocity via a power impact is based first the and elastic his describe not model treated are the to from approaches [9,10] they do that particles on equation derived theoretical new in the in could Other a first of the one introduction results. extensions of the model and experimental investigations which N°11 COLLISIONS STRESSES 2. I DISPLACEMENTS AND following the ~"~ ~~~ " The and the + dxk 2 deformation ulus. The two LamA Kronecker the u, b~k latter being dx~ related are to which constants the further elastic use analogy the stress tensor the notation in between the formulae be can written viscous stress tensor is the viscosity bulk a)~ " yields (remember for equation by m the (~i that hi (V V the of the ~IUllbik force ~~~ Lam4 ~~~ which constants elastic + and allows to us phenomena. viscous benefit Then the ~IUllbJ " + velocity fi~ki (4) Thus, ~Ii 2~iiU~k has for one l/lUilbik + expressed be can displacement 2~llUzk + viscosity (3) 2~lIU~k of the terms m ~lUllb~k " ~~~ ~~ ~ + the total stress (~) 2~ilUik as dark/dxk) the following dynamical V (V ii Au medium: continuum + a~( the from derived is form: " shear the + which ii and >II via with describing the expressed and ~I a~k which given is (1) displacements 2(1~ u) terms a~( with here (1~ 2u) a)~ The body solid a 3 of the denote ~~ will on b~kuiij + 2~t* tensor will we 3 from acting stress symbol, K the compression modulus and ~t* the shear modthe Young modulus E and the Poisson respectively ratio u ~ We elastic u~k I(uiib~k = u~k 1727 expression: a]) with PARTICLES Theory Collision 2. VISCOELASTIC OF u) hi + (~i Au + ii) + h) + (6) pfi = introduce the characteristic scale R and the characteristic velocity vu of the problem as R/uo will be the particle's radius and velocity, respectively Then T characteristic cl, where cl and ct are the Taking into that iii/p c) and Iii + 2111)/P time. account longitudinal and transversal variables and sound speed in the material, one can rescale the the equation in the following way (for simplicity we keep the for the rewrite notations same variables for the ones). rescaled original as We the = " (V~ VV.) " 2 u ~~ + (VV + u c (fl (V~ VV +'f u (VV c~fi u) (7) = ct where c = characteristic e. if of the c (vo/ct), and fl = ~i/(pRct), velocity of the problem is 'f = much Ii less + than 2~ii)/(pRct). the speed 1, and if the dissipation in the bulk is low, that means order of unity, one can use the quasistatic approximation that « (V~ VV.) u + ~i) ct One of the can sound the that see m the coefficients if the material, fl, +f are ~ (VV u = 0 (8) JOURNAL 1728 PHYSIQUE DE II N°11 approximation the displacement field u(r, t) in the problem u(r) Note that m the static onl» static one case (elastic) The problem had already been solved by static present. contact is Before we deal with the generalization on the viscoelastic 188? [ii case, u,e the main results of this classical theory (for details see e.g. [13]) ive for simplicity that only normal forces ~vith respect to the contact assume Thus, iii the quasistatic material ~vith for the the bodies solid two coordinate a labeled centered system the in u~i material the " ii=i of the centre I-r, y) bodies of the of the surfaces and on contact in uz2 contact z = + i~=2 the is " of of the sum and ill N displacements the of compressions related are to the of radii + R2 Ii = + Ri RI ~ /~ /[ ~ Ri Y, o) = // ~h2 / / ~ hi /[ ~ ~ + Ri R2 Uz2(/,Y') [10) R2 and simply ~')2 related + the to y')~, Iv ~~~"~'~ " total and dJ"d§' ~(i (~( = + ~(i) i sion and b are be may ~ the found " ~ the ~[~ %~ F~'D ~~ of the semiaxes from j ~~~ j°° o~ of set /ia2 + 2 ib2 + + The normal pressure P~ F~' ~(12) fi " ellipse. contact + b2 The latter q)/ia2 + )~/ia~~) values as well as the compres- [1,13,14] equations I[b2 a~ q) ~~l~~ l~~l~~ " q a~I((kj 2F~'D b~E(k) dq ia2 (11) = 3F~~ a the dz'dy' ~~ ~ ~~~~ 11 ~'z(~>Y) is of u~i between the r force normal qJ values r 7r j(x ~'~ ~'~" 7r = in cun~ature , + Ri = Ri> R2 and R[, R[ are the principal radii of of the tu.o bodies curvature between the planes corresponding to the radii Ri and R[ The curvature fix, y), that acts and uz2 of the normal terms may be expressed m pressure bodies m the plane z 0 uzilx, the in bodies both where h can (9) angle where one fi z-components ii-c +2cos2~ is Using (cf. [1,13] ). ~~~ ~~~~ r between act area in briefly sl;etch will contact contact u=i constants relations via + 0, fi the while area ivy~ + u=2(r, y) = plane the 2(fit+N) Here stress Hertz Hemrich flattened region will be where set z = 0. region we the in of the middle JI~~ ~vhere surfaces Their and 2 coincides elastic following equation the write the q) ib2 + q) q ~~ q) ib2 + q) q a2bib2 ~ ~ ~~~ n2) ~~~ ~~~~ N°11 COLLISIONS VISCOELASTIC OF PARTICLES (3/4) (hi + h2) and Elk) and I<(k) being the @fi16 is the eccentricity of [16,17]. k for generality). The size of this ellipse depends restrictions with D elliptic functions m usual ellipse (b > a without the normal force, its semiaxes a on and third of the above to F~' via the second Using this dependence, equations. equations (13) Hertz's famous solution of the elastic problem can be contact = notation and b related first the of [1,13,14]: derived by related power a bodies all for the contact in of constant a surfaces the material, in the contact elastic total force and compression the are law F~'(h) with Jacobian the = are from 1729 h~/~ const = (14) depending on the elastic of the materials and on constants of the colliding bodies. If we that both bodies assume of spherical particles the above equation reads: case the local curvatures of consist the same 2E4 ~ fl where RiR2/(Ri = The displacement fields value of the ui velocities (r) solution completely are emphasize this we To depends the on quasistatic the m we that the using turn colliding particles appropriate an Using (16) we write °~( Here emphasize we constants equal We the normal transform the for the in contact. elastic problem contact that is write = We the as one by the dissipative in for the m at the part of the 2l/IIUIk) + elastic Pz coordinates (1G) of the The material. same expression pressure ~~ ~~(j h = We stress. general also may case for simplicity be considered assume the following stress z = tensor (~I ~+ brackets with that Note plane (°~~ curled elastic ones. the h stress the 0, which l/I> in ~II the ~+ right-hand only difference component a][ the that of the given above is (l~) l/II) side of the viscous elastic Namely, we with ° obtain- It ~/~ + ) 1) + /~i ~~~ + ~~/~~t ~ the stress have way x=az',g=oy',z=z' and the by the value of F~', and thus by the u(r) u(r, fi) so that the displacement obtain for the field of the displacement rescaling. the same substituted are to are ( (l/IUllbik that equation is the above is ~l particles of dissipative part of the of the coordinate can of approximation: calculation the to (15) defined compression. i1[r, t) Now radii are the of j~3/2 u2) 3 (1 u2(r) and h ~~~ Ri, R2 and property compression parametrically field R2) + important most ~~ )i ~~ i (19) ())~ ~~°~ ~~ ~~i ~~~ Ill II Ill + ~~2 + /~2 + ~~~~t, ~~~~ JOURNAL 1730 Applying the viscous operator d/dh fi the on last preceding the in expression N°11 II equation obtain we the stress °~~~~'~'~~ The PHYSIQUE DE total force viscous may be result- yielding the following ~~i ~i ~ by obtained ~~2 ~~~ integrating ~~~~ viscous ~~~~ stress the over contact area, h~~~~~~ ~ ~~ the ~~2 ~~~~ Wilere ~ ~ ~ '~ ~q~~ ~~~ ~~~ ~~ ~~~ (24) ~I the From equation last viscoelastic two one find can colliding bodies in F The As e-g one can the elastic efficiently On collision force is ower For of interpretation an particles move the Tv,~ the is relaxation time-dependent same of A for both that A A constant one viscous can write lliding that so (25) equation particles exceeds the llision when > h thus less in h an lastic o, < yields and the and 0, than and case lower a we (27j that notice the viscous constants relaxation time total force we can is much in the see order the h~/~h = ~I/II re 1281 of from equation magnitude of Ah~/~/tc shoI.ter this m than h is as well h/t~ accompanying material relaxation times from as Thus, the one can write (Tv,~/tc) h~/~ the duration (29) of the collision- (Tv,~ /tc) )~~~ a familiarity power series between of the equations latter. We the are definition « 1, following form: F The be can Tv,sir/II dissipative in the processes for simplicity that the assume the one Furthermore, Tv,~. the If coefficients, viscous m for time deformation. A If the the force elastic follows: as where force the hjt~ if hjoj = ~I/II the for each in than the between acts (25) maximal compression is other away from the elastic Their one. e~ written which fi~/~h of The than A one total stage case. less is velocity postcollisional the when force ~ + that with the uations, first elastic the hand, other the the above the in in than the see from force total 2 in for the regime- h~/~ const = for relation quasistatic this equation coincides spherical particles: constant reads general the the const = (25) note and that (h (30) + A (30) can also be the approximative demonstrated formula (30) by can an be expansion in obtained by N°11 COLLISIONS considering and u (6) equation fi in VISCOELASTIC OF independent as PARTICLES 1731 [7j, ~vhich is in reality variables the not case. We conclude results malized one With the above. 2.2, wide relaxation regime be may the in in the used if of case bulk viscosity all the compression h the renor- moderate of the instead Section In relations. one OF 3 will we solutions the compare the estimate fl and coefficients )/R (assuming ii * )II ). condition (Tv,sct) /R introduced 7 That that means the r- COLLIDING radii, and collisional Particles PARTICLES. curvatures of the evolution their the to THE masses, of the influences also can (Tv,~ct r- corresponds of simulations particles time ~I/(pRct) = SHAPE THE OF distribution practical for substituted g. e moderate a of the theory fl viscosity obtain INFLUENCE have quasistatic h + Ah is viscous We of case the contact in equations both for that elastic the for granular behaviour point of the at gases should one know shapes of ho~v the and granular in gases Therefore, contact. the shape particles are how the distributed We will briefly discuss now vicinity of the point of contact (13) equations influence the analogy in ~ ~ i on curvature their material the eccentricity of the of the bodies the that see the relation two only affect &~~ill force between obtained, be and F With for the reduced collision the the the initial + (25) ~~~~ $ = 2D ~ in the in [14j From Nk' ~~~~ the on The contact. size of the figure contact Jf/N, ratio acting force finally e them between and equations (13) also From for 2~D the fi~~~ + force viscoelastic fi(0) (Ii(k))~~/~ gN = be can rewritten. 2 dynamical following the h(0) with equation = 0 the gN + force is equation Ah~/~h) h~/~ precollisional the (26) have both (34) 0 = 2 and (33) ~Ah~"h) + write ~~~ ~~~~ h~/~ (I((k))~~/~ m2) we can particles: )~~~ Nk ~~~~ ~Ii(k~~E(k) viscoelastic ~~~~~ conditions of point mim2/(mi = 2) I~(k)~~ ~~j~k~) ~ ~~~ nonspherical fi the ~~~~~ velocity. the same The form last except factor 2pD Nk ~~i takes the value (~ph = 3 ii ~~~ ~~~~ ~ (= which case surfaces discussed compression equation (34) and the dynamical for particle the already ellipse depends only contact in absolute and equation mass of ii with of curvature elastic 11 the ~ can the follows~~ We of the to in the fi~~~ case (35) (IV(k))~~/~ of spherical particles This u surfaces problem of the collision of particles with non-spherically curved can mapped onto the corresponding collision problem for spherical particles after JOURNAL DE FHYSIQIJE J1 T 3, N° 11, shows the that successfully an be appropriate NOVEMBER 1W3 33 JOURNAL 1732 rescahng of particles We is also for as be can a the calculation of the restitution iii) evaluation of the distribution As calculation of the velocity, impact To find by noticed, just have we the spherical stochastic solve ( variable Therefore for problem twofold the f((). problem be can coefficient after reduced rescaling appropriate an colliding spherical for particles as function a function useful. k in in approximations of high precision they cannot be easily solved terms usual logarithmic contain factor (34) equation elliptic integrals for the approximation an the Because also fit. (, variable for the elementary functions is for the elliptic function [17] Jacobian k Thus, tried the to in we express equation (31) by a simple numerical of the terms in 3f/N ratio ~ eccentricity k of the ellipse contact " 1125ji be can ~( We factor the express now can m k 1- re tei in from of deviations approximations fits for the Ii +ai al " the we ratio k~) the In We + one a2(1 of out 0.0725296 into 1 125 (37) from the follows as 1315 the values %. This k~) and bo 0.5, " ~~~~ ~~~ b2(1 + hi for k~)~) one can the solve the using use of these "polynomial" simple rather I((k) (cf. [17j). ln 0.1213478, " equation, by obtained approximations bill dynamic (()° justifies non-spherical bodies "polynomial" of the k~ )~) + (bo + " ~~ b2 it " 0 with ao 0288729. numerically~ Ii(k) (ao = 3862944, " Substituting and find the coefficient following and we besides, interest for Comparison have and '~ to be less than of collisions of treatment give 1119723, a2 above expressions model, 3. turn 0 restitution are [17] further Finally (36) fits our approximately Elk) k~ The JI/N ratio j37) &?~ iv K(k) the ~~~ of this ins l~~) k2)o66s estimated as k in found: We ~~~~~i~~~~ The to of the (gN distribution of means the (gN, () EN function first the of EN e the coefficient restitution problem for binary collision granular of not ii) of treatment only of the impact velocity, but also of the dynamical description of granular material one has to function N°11 collision the the evolution The variable II processes. distribution of the gases the by the distribution of the value of (, which may coefficient restitution then be calculated can as a of the characterized stochastic that see problem for simulations particles considered We velocities. "benchmark" a that see shapes of the be particle initial the actually PHYSIQUE DE solved only deal only case applications. will our of Collision numerically the spherical with it is the ~vhere Models up particles because this experimental data now Experimental with differential to is are a "benchmark" available which Results equation fi FN = P (39) COLLISIONS N°11 for FN forces h(0) of the ~vith The been the results of diameter a sphere of infinite apparatus We the The evacuated the equations already mentioned, nonelasticity of the material formulate of the equation an this equation, obtained [8] form of in Aih~/~ affect as one and lo have the on a ~ 3 p(1- u2) by in form of be to temperature second per a sphere. ice was less, the or of the and velocity relative ~~oung the which ~vill we briefly dissipative effects relative velocitj~, I-e- include the of the law port,er A21~ due to With respect in and 2 = choice can would contact in deformation com- behm~iour This = of the a also the examined 3/2 [15]. namely a shape of the surfaces derivative the integrability tool< P6schl convenience, curved time denotes analytical the to We have series. exponents, the above, h As parameters. power a (40) o = mathematical the on scaled the to according time models collision to similar a way itself. we using with = force convenience determined been spheres of the characteristic other a + ~~ motivated (fi) of fi deformation numerical For of the the rings. deformation proposed fitting as equation of dependence the the Ai a (40) for other heuristically by arguing that solution motivated be exponent an only thus dependence a of the radius form abbreviated have ~Ve pression of A2 and constant a restitu- considered be can the centimeters some the for P6schl As h + with g[ normal container. dependence of time differential with done was here the of order and impact velocity has been spherical ice particles the plane ice block which h is then equal to typical for planetary the the 0 value = the collision. same discuss to those in were an in estimated also a radius From bodies two on h(0) conditions velocity. experiments, these In [2, 3j hit reflect to the between 1733 initial dependence eN(gN) reduced velocities mounted was have during chosen the given impact a contact Its centimeters The K, the impact 150 of the PARTICLES with and and experiments of some were end calculated radius. conditions ts the at has with deformation initial zero velocity coefficient tion T for relative compared by equations (25) and (30) given I-e gN, = VISCOELASTIC OF the results modulus E = differential (25) by a length characteristic toi. The scaling values lo and to equation lox and t to h of Hertz's theory for = Ei " = E2 and non-dissipative a Poisson the ratio u collision vi = " u2. of For two these cm/s, which has been a characteristic impact velocity in labeled by following velocities the scaled VN experiments are With these scalings and assuming typical values for ice at low temperatures [18]. the Young 10~ m~~, kg Poisson 0 3, and a modulus E m 10 GPa, the density (p) ratio m mean m u a typical particle size h m 10~~ m, one obtains the scaling values lo " 10~~R and to " 10~~ s and calculations have we the the Ai constant of taken [2, 3] about -3.5 °C m gN 1. = the In The above is quite which values too necessity of extended raises and thus, the above values temperatures, Analogous scalings were made for the is that it is not always easy to relate the This except the in the case of a 3/2 = Figure la the results (impact) velocity of VN Using the above model. 16 sho~vs the Figure o 7 In " for where the 1 for material time material have of to be differential constant constants with the the are material considered equations valid conditions as of the -42 to the physical for in temperatures planetary properties of ice approximations type (40). The properties of the low problem material is parameters, dependence rings at possible at least concerning the dimension deformation shown for au dependence of the are different models and compared with the results of this time these of the high compared investigations the of the constants relative take velocity. the We values of ~41 remember " ottr initial Hertz's and A2 discussion = JOURNAL 1734 PHiSIQUE DE II N°11 2 Herlz's model model model model exact approx Poeschl's I ~'~~,,, __------__ ,,-." 0 8 ,"' "' ~'~, '~', ." ",, ', o 6 C ° ', + ". ". ". E § # 0 4 02 o -0 2 0 OS I 2 5 2 5 3 5 3 time a) ._"~,,, Hertz's exact ',,, ',,, model model model model approx Poeschl's ',,, ",._ 0 5 "~_ ,,, .___~ "'~?.. 0 ( ~ 3 -05 -1 .1 5 0 05 2 15 2 5 3 3 5 time b) Fig (30) of1 a) 1 model Dots cm/s dashes dependence Long Time of the (elastic collision) Poschl's Solid Solution line of model Hertz's equation deforniation dashes b) Time model (30) Solution for of an impact equation velocity (25) deformation dependence of the (elastic collision). Long dashes Dots Poschl's model short of1 cm/s dashes velocity Solution for of Solid line Hertz's solution oi equation impact velocity an Short equation (25) COLLISIONS N°11 of the collision figures It can deceleration the is case which first especially in velocity (Fig similar 16). the to one The Iii latter dependence Figures the 2a-c well very the o of the for that results the experimental shows a coefficient to of the equation different an Thus, for results A2 have One model our to frost covered be estimated. problem experiments seems the in ice this balls to This be objects, not first velocity experimental experimental stage of the model ~vhich behaviour in gN the shown is [2, 3]. ones in Here to reached for indicate the and the shows achieved. is We ho~v the both best apparatus, that in model the found have constants of that However, exponents presence advantage Also is (a 2) gives decreasing qualitatively right. lJut experiments the mentioned collision. curves with values = curves. the matter no (already comparison mounted >.ere an on the imp~Lct the fit to the P6schl's It has although undergoes might be then models the preferable it and different that forces given by equations (25) and (30) agree for frost covered ice spheres, not~vithstanding contradiction is the with velocity, experimental might collision the bodies on E~V increasing type (40) behaviour a best the that sho~n.s relative the [15]. demonstrated (25) equation visible are the materials most obtained in Differences dependence of time This collision deformation perinanent models indicates results with these complete values somewhat that agreement with already had we n~ay curves with agreement in a behaviour quite good results our collision inelastic an these efficient more a very low at the beginning of the collision, 2) and (30) lead to curves where the a = colliding for numerical model our different a good [15j for compare chosen in of range of the of certain a three in the This coefficient restitution ~vith leads 1 = work fi.. been time end beginning of the at to stage of the the by illustrated leads second from is (40) (with value well terms At zero, obtained deceleration material quite to of all particular have we the unconventional the restitution provide fit have dropped in equations of the we abovementioned 3/2 = a of where constants We find a types to be seems case The which model: the in examined features collision, significant quite long a been has common contrary, different for are curve already has only little changes over acceleration have acceleration Hertz's deceleration suitable lower a 1735 very is dissipative of the presence which stage of the of later increases to course, first the and the when These, of seen the PARTICLES equation (26) ~vhich of context that models three reached is remains. but the in for all the in process easily be VJSCOELASTIC OF being equal mixed agreement only the in terms with two in choice previous a chosen are doesn't the to the 3/? both h best but and experimental the constants Ai and [11]) is the following: since in the dependence of EN on the particle the unknown dependence is kno~n.n, we are able to estimate material the experimental constants, in particular the viscosity, from fitting our to ones. curves the order of the magnitude With the above values of Ai and ~42 Here we can only estimate really soft of about 10 MPa s [15]. This quite low value led to indicates viscosities a ~ve are collision take place this surface like a frost layer Apparently, the main of the processes in ielativelj, thin layer This is plausible regarding the lo~v collision velocities and sniall masses material which of the bodies in the An improved kno~vledge of the constants experiments can will ~Lllow more conclusive only be obtained by new statements measurements Surprisingly, our model does not seem to &~-ork for ice spheres which &~,ere designed to have surfaces clean where an exponential fit instead of a power law had been found [3]. This very indicates that other viscoelastic phenomena ion n,hich our model is based) play a than purely model ~vith a P6schl's role in these collisions However, a cur~.e obtained from 2 shows in from model better perfect. but than the this got not agreement ones we our case a a observed, which caused of the frost layer was In the initial experiments [3) a compression found effect after smooth collisions, and a the surface of the particles to saturation was some these the of survived. Because in of number collisions particle had the dependence a in EN on mass could not be estimated If this = JOURNAL 1736 PHYSIQUE DE ~ 60 0 j , ~0 0 I 0 60 ~~ 0 50 0 40 ° 30 i j ~'l' ~~~l~' ~ ~ N°11 II '[ -~. ~ '~ / ~,, 0 30 l 0 ~ Z 0 3 5 4 Z 0 [cm/sl g» 3 5 4 [cm/sl g» b) a) i off ~ ', ~l ", j ~ o ~~ ~ '~' '~" 0 0 2 ~ h 4 icm/sl 9~ C) Fig a) 2 Dashes b) made obtained curve coefficient Restitution iuents made curve obtained coefficient clean, with A2 = smooth produce frost using EN covered us spheres [3j EN impact us. covered equation impact experimentally (30). line obtained spheres [3j Dashed of comparison gN Dash-dotted Solid velocity Solid velocity spheres [2j gN curve comparison obtained curve of Poschl's experimentally model our experimentally Dash-dotted line of comparison obtained obtained equation using model using with curve the Solid (25) model our obtained equation with curve (25) experi- Dashes curve c) experiments line the with Restitution performed Poschl's model 0 75 particles hit the ice brick only ~vith one and the same side, we believe that brittle fracture is likely reality more complicated. In particular, continued quite soft surface layer ~vhich, in turn, can protect the bull< material from further Thus, we do not consider this case to be relevant for our applications. the behaviour destruction coefficient with equation (30) using impact velocity gN us experiments this frost with EN with to Restitution experiments the a is in usually not as clean as it is surfaces of particles natural In addition, the systems are in material. possible to produce in a laboratory. They may be covered ~vith mineral dust or other from collisions survived particular, particles in planetary rings which have already In many suifaces. unlikely smooth have of the system several sides during the loitg existence to are N°11 COLLISIONS OF VISCOELASTIC PARTICLES 1737 Conclusions 4. study developed a theoretical model for the collision of viscoelastic quasistatic approximation which corresponds to the case where the collision velocity is much lower than the sound speed in the material The expression for the total viscoelastic force is then a generalization of the &~~elLknown Hertz relation 11,13] for the elastic problem. An explicit relation has been obtained for the force acting between contact spherical colliding particles. The general case of the collision of particles of arbitrary shape the collision problem for spherical particles The theoretical results onto may be mapped for the coefficient have been compared with experimental restitution data for spherical icy In the present particles. We have have we used a particles [2,3]. Our model based is experimental with low value of well with the simple on phenomena viscoelastic of the estimations provides It coefficient restitution of satisfying very a covered frost ice The estimated approximately from our viscosity which could be agrees that the main of the collision take place in this quite soft assumption, processes resemble We that this relevant is most properties to a fluid. assume case may the layer, whose for applications agreement spheres theory towards formation of these fractured their the systems surfaces. This planetary already of evolution the particles have and fact, the during the because rings, collisions survived many the that ring particles long have may covered be to seem of the time which with properties similar to those of the frost layer in the experiments. to ~vith the experimental results for spheres which were designed to have The agreement worse surfaces by the experience especially at very low temperclean be explained that ice can very brittle material. Consequently, instead of viscoelasticity, fracturing atures processes very is a likely fractal velocities. This produce will these collisions low impact is to at surgovern even faces Preliminary investigations jig] have shown that the model according to equation (40) is dust, will suitable more for the of the the viscoelasticity surfaces. results not This We different of the objects of icy influenced surfaces type of with collisions erning than pure is rection found e.g in therefore can that state to seem collisional the material, only by the properties of the bull< should be proved by a microscopic that shows experiments planetary rings the be processes behaviour but normal in di- also by the of examination gov- complicated more ice properties spheres used in experiments. such It also is we to clarify of the new mass perhaps experiments, of the using particles on the particles falling in evacuated an problems. these metal) of ice at temperatures low Topics of future equations of the ii the or should experiments interest type (40) which special materials problem without the of as theoretical of the outcome should also be made ~vith experiments with bodies consisting of other comparison Furthermore, the material and elastic with known properties. viscous A g. that think influence the estimate to necessary collisions, and tube, can help in this comparison The (e surface contribute the are may be be estimated physical exponents and constants meaning of the of bodies suitable for the treatment more particular surface geometries assumption of quasistationarity. having material constants and the solution consisting of the collision Acknowledgments We would readable. checked like to thank In particular very carefully the we our referees address for our calculations their comments thanks to helping to which Jean-Marc make the helped Petit, paper to make Observatoire moie the paper de Nice, convincing more ~vho JOURNAL 1738 PHYSIQUE DE II N°11 References iii Hertz [2j Bridges H Angeiv Reme J Math 92 (1882) D N C., Nature N C, Mon 156 , [3] [4] F A and G and Lin LmD , 333 231(1988)1091 Soc Astr Marouf E Lissauer J J Holberg J B Esposito L. 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