Collision of elastic spheres

The basic theories of elastic deformations associated with various contact forces under static contact conditions have been introduced in the last section. Assuming that an impact process of two solids can be regarded as quasi-static, the theories given in 2.3 are used directly to link the dynamic deformations of the colliding solids with the impact forces. In this section, the collisions of elastic spheres are described. 

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ZJiang J, Fan LS, ZJiu C, Pfeifer R, Qi D. Dynamic behavior of collision of elastic spheres in viscous fluids. Powder Technology 106 98 109, 1999. 

Any solid material has its own upper limit of elastic deformation under either normal or tangential stresses. Once the stresses exceed this limit, plastic deformation will occur. In this section, collisions of inelastic spheres are presented. The degree of inelastic deformation is characterized by the restitution coefficient. 

For simplicity, it is assumed that the impact is a Hertzian collision. Thus, no kinetic energy loss occurs during the impact. The problem of conductive heat transfer due to the elastic collision of solid spheres was defined and solved by Sun and Chen (1988). In this problem, considering the heat conduction through the contact surface as shown in Fig. 4.1, the change of the contact area or radius of the circular area of contact with respect to time is given by Eq. (2.139) or by Fig. 2.16. In cylindrical coordinates, the heat conduction between the colliding solids can be written by... 

The physical condition of the kinetic theory of gases can be described by elastic collisions of monodispersed spheres with the Maxwellian velocity distribution in an infinite vacuum space. Therefore, for an analogy between particle-particle interactions and molecular interactions to be directly applicable, the following phenomena in gas-solid flows should not be regarded as significant in comparison to particle-particle interactions the gas-particle... 

Example 52 On the basis of the kinetic theory, which is used to model collision-dominated gas-solid flows, derive a general expression of solid stresses of elastic spheres in a simple shear flow. 

Thornton, C. (1997) Coefficient of restitntion for collinear collisions of elastic-perfectly plastic spheres. Journal of Applied Mechanics, Transactions ASME 64,383—386. 

POSTULATE 4 When molecules collide with one another, the collisions are elastic. In an elastic collision, the total kinetic energy remains constant no kinetic energy is lost. To understand the difference between an elastic and an inelastic collision, compare the collision of two hard steel spheres with the collision of two masses of putty. The collision of steel spheres is nearly elastic (that is, the spheres bounce off each other and continue moving), but that of putty is not (Figure 5.22). Postulate 4 says that unless the kinetic energy of molecules is removed from a gas—for example, as heat— the molecules will forever move with the same average kinetic energy per molecule. 

Boltzman s H-Theorem Let us consider a binary elastic collision of two hard-spheres in more detail. Using the same notation as above, so that v, V2 represent the velocities of the incoming spheres and v, V2 represent the velocities of the outgoing spheres, we have from momentum and energy conservation that... 

The angle of deflection for the collision of rigid elastic spheres may be obtained from Fig. 1-3. The minimum distance of approach is... 

A 2D soft-sphere approach was first applied to gas-fluidized beds by Tsuji et al. (1993), where the linear spring-dashpot model—similar to the one presented by Cundall and Strack (1979) was employed. Xu and Yu (1997) independently developed a 2D model of a gas-fluidized bed. However in their simulations, a collision detection algorithm that is normally found in hard-sphere simulations was used to determine the first instant of contact precisely. Based on the model developed by Tsuji et al. (1993), Iwadate and Horio (1998) incorporated van der Waals forces to simulate fluidization of cohesive particles. Kafui et al. (2002) developed a DPM based on the theory of contact mechanics, thereby enabling the collision of the particles to be directly specified in terms of material properties such as friction, elasticity, elasto-plasticity, and auto-adhesion. 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

Collisions between particles with smooth surfaces may be reasonably approximated as elastic impact of frictionless spheres. Assume that the deformation process during a collision is quasi-static so that the Hertzian contact theory can be applied to establish the relations among impact velocities, material properties, impact duration, elastic deformation, and impact force. 

It can be proved from Eq. (2.156) that, for materials with Poisson s ratio of 0.3 (which is true for most solids), the maximum shear stress oz — or occurs at z/rc = 0.48. Consequently, according to Tresca s criterion, the yield stress Y in a simple compression is 0.62 p0. Therefore, when the hardness or the yield stress Y of the particle material is less than 0.62 times the maximum contact pressure, the sphere will, most likely, undergo plastic deformation. From the elastic collision of two solid spheres, the maximum contact pressure is given by Eq. (2.134). Thus, the relation between the critical normal collision velocity, Ui2Y. and the yield stress is given by... 

The recoverability or restitution of the kinetic energy during a normal collision between two objects can be represented by the coefficient of restitution defined by Eq. (2.3). Note that the coefficient of restitution cannot be used as a criterion to judge whether a collision is elastic or not unless the collision is solely considered as a normal collision. For example, the sliding at contact for the collision between two elastic spheres will make the collision inelastic while the value of the coefficient of restitution in this case is equal to 1. 

Figure 5.8. Impaction due to shear motion (a) Elastic collision between two particles (b) Elastic collision of a single sphere with a cloud of particles.

J. Cl. Maxwell [l] Illustrations of the dynamical theory of gases, Part 1 On the motions and collisions of perfectly elastic spheres, Phil. mag. (4) 19 (1860), p. 19 Part 2 On the process of diffusions of two or more kinds of moving particles among one another, Phil. Mag. (4) 20 (1860), p. 1 (also Scientific Papers 1, Cambridge 1890, p. 377). 

For example, in the particular case when the molecules are rigid elastic spheres the apse-line becomes identical with the line of center at collision. In this case the distance d 2 between the centers of the spheres at collision is connected with their diameters d, d/2 by the relation [77] ... 

Assuming that the molecules are rigid elastic spheres a typical value for the mean free path for a gas, say oxygen, can be calculated from (2.516). Consider a typical room temperature at 300 (K) and a pressure of 101325 (Pa). The collision diameter of molecular oxygen can be set to 3.57 x 10 ° (m) in accordance with the data given by [51], example 1.4. We can then calculate the mean free path / for oxygen ... 

Maxwell JC (1860) lUnstrations of the Dynamical Theory of gases. - Part I. On the Motions and Collisions of Perfectly Elastic Spheres. Phil Mag 19 19-32 Maxwell JC (1867) On the Dynamical Theory of gases. Phil Trans Roy Soc London 157 49-88... 

Particle behavior often depends on the ratio of particle size to some other characteristic length. The mechanisms of heat, mass, and momentum transfer between particle and carrier gas depend on the Knudsen number. 2J /d , where Ip is the mean fr. e. P ath of the gas molecules. The mean free path or mean distance traveled by a molecule between successive collisions can be calculated from the kinetic theory of gases. A good approximation for a single-component gas composed of molecules that act like rigid elastic spheres is... 

Df that appears in the collision kernel. In ihe Tree molecule range, the basic form of the coiltston kernel is assumed lo be the same as [he kinclic theory expression for collision of rigid elastic spheres (Chapter 7)... 

The maximum energy transferred can be calculated assuming purely elastic collisions between hard spheres ( 12.1). Thus for a 1.5 MeV fission neutron, (max) is 425 keV in C, 104 keV in Fe, and 25 keV in U. With E 25 eV, up to 8500, 2080 and 500 displacements, respectively, occur in these metals due to the absorption of a fission neutron (neglecting nuclear reactions). In practice the numbers are somewhat smaller, especially at... 

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