Sphere perfect elastic spheres

It is clear that e = 1 for the normal impact of perfect elastic spheres, while e = 0 for the normal impact of perfect plastic spheres. 

Particles can be regarded as point masses which exert no force on each other beyond a separation distance r > R (R is known as the interactive radius). At r < R there is an infinite repulsive force between the particles involved. They therefore behave like perfectly elastic spheres of radius R/2. 

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). 

Equation (2.1) can be derived from the kinetic theory of gases assuming the gas molecules to behave as perfectly elastic spheres having negligible volume with no intermol-ecular attraction or repulsion. 

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... 

Gases consist of particles (atoms and molecules) which are perfectly elastic spheres. The particles are in a complete and continous state of agitation. The system is in equilibrium. External forces such as gravity and magnetism are ignored. 

In this simple model [1], the contact force between two perfectly elastic spheres is resolved into normal and shear components with respect to the contact plane ... 

Based on this model [13], the normal contact force between two perfectly elastic spheres in contact is given by [14]... 

Continuum mechanics models for the adhesion of perfectly elastic spheres under the action of reversible surface forces are well develops. The essential features of the JKR model are shown in Figure la. The surface traction acting on a contact area of radius a comprises two terms (i) a Hertz pressurepi(r), caused by the compressive force Pi, which flattens the spherical surfaces and (ii) an adhesive tension pa( ) hich gives rise to the adhesive force Pa. The net contact force P can be expressed ... 

Maxwell, J.C. (1860). Illustrations of the dynamical theory of gases. Part 1. On the motions and collisions of perfectly elastic spheres. Phil. Mag., 19, 19-32. 

Up to an error proportional to the square of quantities assumed to be small, these expressions are identical to those for perfectly elastic spheres as given, for example, in Section 16.34 of Chapman and Cowling (1970). Consequently, so also are the expressions for the pressure tensor and the energy flux vector calculated by employing the velocity distribution function (4.7) and Enskog s extension of the assumption of molecular chaos. 

The first molecular dynamics simulation of a condensed phase system was performed by Alder and Wainwright in 1957 using a hard-sphere model [Alder and Wainwright 1957]. In this model, the spheres move at constant velocity in straight lines between collisions. All collisions are perfectly elastic and occur when the separation between the centres of... 

To be more precise, let us assume, as Boltzman first did in 1872 [boltz72], that we have N perfectly elastic billiard balls, or hard-spheres, inside a volume V, and that a complete statistical description of our system (be it a gas or fluid) at, or near, its equilibrium state is contained in the one-particle phase-space distribution function f x,v,t) ... 

The present author has performed computer simulations to examine the transition of pressure distributions and shear response from a hydrodynamic to boundary lubrication. Figure 4(a) shows an example of a smooth elastic sphere in contact with a rigid plane, the EHL pressure calculated at a very low rolling speed coincides perfectly with the... 

In the previous sections, only the normal contact of two elastic spheres with perfect smooth surfaces (i.e., no tangential force) is considered. However, for oblique contact between two frictional spheres, tangential forces are encountered, and, consequently,... 

It is therefore remarkable that 100 years or so before the laws of thermodynamics were formulated, Daniel Bernoulli developed a billiard ball model of a gas that gave a molecular interpretation to pressure and was later extended to give an understanding of temperature. This is truly a wonderful thing, because all it starts with is the assumption that the atoms or molecules of a gas can be treated as if they behave like perfectly elastic hard spheres—minute and perfect billiard balls. Then Newton s laws of motion are applied and all the gas laws follow, together with a molecular interpretation of temperature and absolute zero. You have no doubt... 

Kolev [46] discussed the validity of these relations for fluid particle collisions considering the obvious discrepancies resulting from the different nature of the fluid particle collisions compared with the random molecular collisions. The basic assumptions in kinetic theory that the molecules are hard spheres and that the collisions are perfectly elastic and obey the classical conservation laws do not hold for real fluid particles because these particles are deformable, elastic and may agglomerate or even coalescence after random collisions. The collision density is thus not really an independent function of the coalescence probability. For bubbly flow Colella et al [15] also found the basic kinetic theory assumption that the particles are interacting only during collision violated, as the bubbles influence each other by means of their wakes. 

Calculating X Assuming that atoms or molecules in a gas can be modeled as spheres that interact only when they collide and that all collisions are perfectly elastic, the kinetic theory of gases can also be used to predict the frequency at which gas atoms will collide and hence the average distance (or mean free path, X) between collisions. 

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

The case of a perfectly elastic contact between the solid surface and the absolnte solid ball is known as the Hertz Problem of contact mechanics. The Hertz Problem has a rather cumbersome solution. With the application of dimensional analysis (Section 5.2), one can get a characteristic nonlinear dependence of the size of the impression on the indenting ball s diameter, the applied force and the Young s modulus of the material. In its reverse version, that is, for the case of a contact between a compliant sphere and a solid surface (bottom of a 15 g weight), this method was used for a long time to measure the internal eye pressure of the eye. 

It has recently become common to use the JKR theory (Johnson, Kendall Roberts, 1971) to extract the surface and inteifacial energies of polymeric materials from adhesion tests with micro-probe instruments such as the Surface Force Apparatus and the Atomic Force Microscope. However the JKR theory strictly applies only to perfectly elastic solids. The paper will review progress in extending the JKR theory to the contact mechanics and adhesion of linear viscoelastic spheres. The observed effects of adhesion hysteresis and rate-dependent adhesion are predicted by the extended eory. 

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