PHYSICAL REVIEW E
VOLUME 53, NUMBER 5
MAY 1996
Model for collisions in granular gases Nikolai V. Brilliantov* Moscow State University, Physics Department, Moscow 119 899, Russia
Frank Spahn and Jan-Martin Hertzsch University Potsdam, Am Neuen Palais, D-14 415 Potsdam, Germany
Thorsten Po¨schel† Institut fu¨r Physik, Humboldt-Universita¨t zu Berlin, Invalidenstraße 110, D-10115 Berlin, Germany ~Received 8 September 1994; revised manuscript received 27 February 1995! We propose a model for collisions between particles of a granular material and calculate the restitution coefficients for the normal and tangential motion as functions of the impact velocity from considerations of dissipative viscoelastic collisions. Existing models of impact with dissipation as well as the classical Hertz impact theory are included in the present model as special cases. We find that the type of collision ~smooth, reflecting or sticky! is determined by the impact velocity and by the surface properties of the colliding grains. We observe a rather nontrivial dependence of the tangential restitution coefficient on the impact velocity. PACS number~s!: 46.10.1z, 51.10.1y, 62.20.Fe, 83.70.Fn
*Also at Universita¨t Potsdam, Am Neuen Palais, D-14 415, Potsdam, Germany. † Also at The James Franck Institute, The University of Chicago, 5640 South Ellis Ave., Chicago, IL 60 637. Electronic address:
[email protected] http://summa.physik.hu-berlin.de:80/ thorsten/
have shown that the type of the collision ~sliding or sticking! depends on the ratio of g N and g T . The results were explained with different models for each type, and the coefficients in these models were independent of the velocity. On the other hand, laboratory experiments with ice balls @33# as well as with spheres of other materials ~for an overview see @34#! have shown that the normal restitution coefficient e N depends significantly on the impact velocity. As already seen, the tangential restitution coefficient depends on the impact parameters as well. The behavior of the sheared granular material may be significantly different if the restitution coefficients depend on the impact velocity. This dependence should be taken into account in order to get an adequate model of the stress distribution @35#. It is also known that the parameters e N and e T crucially influence the global dynamics of granular systems ~e.g., @36,37#!. In the present study we investigate how the restitution coefficients depend on the relative impact velocity. For the normal component of the relative motion we derive an expression for the normal force acting between the colliding particles, which accounts for the dissipation in the bulk of material. One particular application of the results presented here is the explanation of experiments with ice balls @33#, which are of importance for the investigation of the dynamics of planetary rings @38#. A static model for the inelastic impact of metal bodies was presented in @39#, which is based on the assumption of fully plastic indentation and constant mean contact pressure and leads analytically to a proportionality e N }(g N ) 21/4 for arbitrary material constants. On the contrary, our quasistatic approach does not request other additional assumptions and can be adapted to different experimental results by changing the coefficients in the differential equation that describes the time dependence of the deformation. From these coefficients, material coefficients can be estimated @40#. Our result contains the Hertz theory of elastic impact @41# and the theory of the viscoelastic impact by Pao @42# as
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I. INTRODUCTION
A rich variety of systems one encounters in nature may be considered as ‘‘granular gas’’ @1#. The most important difference between a ‘‘gas’’ of granular particles and a regular gas is the inelastic nature of the interparticle collisions. The steady removal of kinetic energy from the granular gas due to dissipative collisions causes a variety of nonequilibrium processes that have been subjects of experimental ~e.g., @2–10#! and theoretical ~e.g., @11–15#! interest. Particularly in recent time many of the experimental results have been reproduced and investigated using various techniques such as cellular automata ~e.g., @16 –18#!, Monte Carlo methods @19#, lattice-gas models @20#, and molecular dynamics in two @21– 24# and three @25–27# dimensions and hybrid methods @28 – 31#. The loss of kinetic energy of a pair of inelastically colliding grains can be described using the restitution coefficients for the normal and tangential components of the relative motion e N and e T ~ gW N ! 8 52 e N gW N ~ gW T ! 8 5 e T gW T
~ 0< e N
