The collision of particles in granular systemsNikolai
چکیده
Collisions between granular particles are irreversible processes which cause dissipation of mechanical energy by fragmentation or heating of the colliders. The knowledge of these phenomena is essential for the understanding of the behaviour of complex systems of granular particles. We have developed a model for inelastic collisions of granular particles and calculated the velocity restitution coe cients, which describe all possible collisions in the system. The knowledge of these coe cients allows for event-driven many-particle simulations which cannot be performed in the frame of molecular dynamics. This approach has the advantage that very large particle numbers can be treated which are necessary for the understanding of intrinsic large-scale phenomena in granular systems. Systems of granular particles are very common in nature, from terrestrial sands and gravels to the dust and planetary rings in space. These systems are also of importance in modern technology. Striking properties of the granular materials, which exhibit a solid-like behavior in unperturbed and uid-like in perturbed states caused much interest in these systems in the past few years. The numerical studies of the granular materials carried out recently (e.g. [1]) were based on the Molecular Dynamics technique, where all forces and moments acting between each pair of colliding particles must be calculated explicitly (e.g. [2]). These simulations are extremely time consuming, and thus only granular systems with a relatively small number of particles could be studied so far. In order to understand phenomena such as the origin of structures in planetary rings (e.g. [3]) one has to simulate much larger systems than it would be possible by using Molecular Dynamics. Alternative simulation methods are based on \event driven" algorithms (EDA). Using an EDA the computer time is proportional to the number of events, i.e. collisions. The simple propagation of particles in the time between collisions does not require computational e ort. 2 In EDA the collisions are not considered in detail by numerically integrating repulsive forces during a collision where each collision needs tens or hundreds of time steps and hence force evaluations. Instead one assumes the idealization that the particles are hard and therefore there are only pairwise interactions and the duration of a collision is assumed to be zero. These idealizations are well justi ed for the case of dense granular systems where the relative velocities are small. For molecular dynamics one has to provide the proper expressions for the interaction forces. Similarly for EDA one needs the transformation rules for the relative velocities after a collision as a function of the impact velocities. The aim of the present paper is to derive these rules for granular gases from the viscoelastic properties of the grain material. We describe the collision of grains { modelled as rough spheres { using the concept of restitution coe cients. This relates the relative velocities of a pair of particles after the collision (vN ;T)0 to those ones before the impact without considering the details of the collision themselves [4]: (vN)0 = N vN (0 N 1) (1a) (vT )0 = T vT ( 1 T 1) : (1b) The indicesN and T label the normal and tangential components with respect to the contact plane. The term \impact velocity" means the relative velocity of the surfaces of the bodies at the point of contact. Its components are functions of the relative translational and rotational velocities of the particles and of their radii (e.g. [5]). The normal impact velocity ~vN is in parallel with the normal unit vector ~n of the contacting surfaces, whilst the tangential impact velocity ~vT is in perpendicular to ~n. Thus, the investigation of the dynamics of the system reduces to the determination of the functions N(vN ; vT ) and T (vN ; vT ), which describe entirely the collisions and which will be discussed in the following. Knowing the functions N(vN ; vT ) and T (vN ; vT ) one can set up an event driven simulation which is much less time consuming than Molecular Dynamics. A very e ective implementation of such an algorithm was described by Rapaport [6]. 3 Experimental results (e.g. [7,8]) show that the coe cients of restitution depend sensitively on the impact velocity. Furthermore, these coe cients determine the further evolution of the velocities ~v and angular velocities ~!. In the framework of visco{elasticity we derive ordinary di erential equations for the compression = Ri + Rj (j ~ri(t) ~rj(t) j) (in normal direction ~n) and the tangential deformation (direction perpendicular to ~n) of both spheres in contact [4]. For the de nition of the used symbols we refer to g. 1. In the case of elastic deformation the Hertz contact law [9] FN (el)( ) = 2 Y 3 (1 2)qR e 3=2 (2) describes the restoring force. Re = RiRj=(Ri + Rj), Y , and are the e ective radius, the Young modulus, and the Poisson ratio, respectively. Under quasistatic conditions where the impact velocity is small compared with the speed of sound of the material, the dissipative part of the normal force can be written as [4] FN (dis) = Y (1 2)qR e Aq _ (3) Then, the normal component of the relative motion is described by the equation = B 3=2 + 3 2 Aq _ B +A _ 3=2 (4) with the initial conditions _ (0) = vN and (0) = 0 (relative velocity and penetration depth at the beginning of the collision). A and B are well de ned material constants (see [4]): A = 1 3 (3 2 1)2 (3 2 + 2 1) (1 2) (1 2 ) Y 2 ! (5a) B = 2 YpR e 3 me (1 2) ; (5b) where 1=2 being the elastic and the viscous constants of the particle material. Surprisingly, the approximate part of eq. (4) has a similar form as the Hertz law (2) if the condition A _ holds, which is the case for the entire collision process. With the solution of eq. (4) the restitution coe cient reads 4 N = _ (tc) = _ (0) ; (6) where tc is the duration of the collision. Figure 2 shows the comparison of the restitution coe cient obtained by eqs. (4) (6) (for ice at low temperatures: Y = 10 GPa, = 0:3, R = 10 2 m, = 103 kg m 3 [10]), and the results of experiments by Bridges et. al. [7]. The experimental data agree well with the theoretical results. To derive the equation for the tangential motion one has to provide an expression for the tangential force FT between the colliding particles. The rough surfaces of of the bodies in contact are modelled by asperities which may vary in shape and size by several decades (see e.g. [11]). The transmission of momentum is caused by the contact of the asperities with the mean size L. The relative tangential motion _ = vT of the particle surfaces causes a shear deformation / =L of the asperities, resulting in a tangential stress T . The force FT = ST T causing this stress is transmitted by the portion ST (t) = fT ( N)S(t) T N S(t) of the nominal contact area S. For small deformations the surface ST is approximately a linear function of S and of the mean pressure N exerted on the surface, and the shear results in almost elastic stresses. When these stresses exceed some critical value T which is a speci c material constant, the asperity which hinders the tangential relative motion of the surfaces is assumed to break, i.e. to dissipate the energy which was stored in elastic stresses. This results in a sudden release of the shear stress, and the particle surfaces are shifted by a distance 0 with respect to each other. This yields the shear deformation ( ) = 0 $ 0%! (7) where bxc denotes the integer part of x and is the maximal elastic shear deformation. Thus, one obtains the tangential force FT = FN 0 $ 0%! : (8) 5 where FN = NS = me (t) and = T . Then, the maximum possible tangential force given by Fmax T = FN : (9) reproduces the Coulomb friction law [12] and explains the friction coe cient as a function of the mesoscopic properties of the surfaces, i.e. their roughness. The idea that the contact breaks by a permanent dissipative deformation of the asperities when the shear stress exceeds a critical value is closely related to the extensively investigated one dimensional model of Burridge and Knopo [13]. Summarising our model covers both this case and Coulomb's law.With the tangential force (8) we obtain the di erential equation for the motion in shear direction (t) 0 $ 0%! = 0 (10) with the initial conditions _ (0) = vT and (0) = 0. The coe cient here is expressed in terms of masses and moments of inertia of colliding particles [4]. The tangential restitution coe cient can be calculated from T = _ (tc)= _ (0) : (11) Equation (11) has been solved numerically for initial normal and tangential velocities in the range jvN ;T j 5cm=sec, which is, for instance, appropriate for collisions in planetary rings. Figures 3 and 4 shows the tangential restitution coe cient T as a function of the impact velocities for smooth (upper part) and rough spheres (lower part). A complex, nonlinear dependence of the tangential restitution coe cient on the impact velocity can be observed. In particular, a transition from sliding to rolling behaviour occurs if the asperities are not too small in comparison to the size of the colliding bodies. This behaviour which is due to the Coulomb friction law is very well known from every day experience. 6 In this paper we have presented an analytical model of collisions of granular grains which accounts for the energy-dissipation in granular systems. The description of the normal motion is a generalisation of Hertz-theory [9]. The results are con rmed by a good agreement with experimental data. For the tangential coe cient we nd that our description covers both the Coulomb friction and the widely used friction model by Burridge and Knopo . Furthermore, our model enables theoretical investigations of much larger systems than it would be possible by using Molecular Dynamics. Among the most interesting granular systems are planetary rings. It is known that they reveal complex structures in radial as well as in azimuthal direction whose origins are not su ciently explained so far. Of particular interest are transport processes and the formation of dissipative structures in the rings. We hope to get further insight into the principles of planetary ring genesis from numerical results using an event driven algorithm with the transition rules derived in this paper. This work is in progress. 7 REFERENCES [1] J. A. C. Gallas, H. J. 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