Sphere frictionless

Consider a collinear collision of two rigid, frictionless, and nonrotating spheres, as shown in Fig. 2.1. Neither sphere has a tangential momentum component in this system. Therefore, conservation of the normal momentum component of the two-ball system yields... 

When two elastic and frictionless spheres are brought into contact under compressional forces or pressures, deformation occurs. The maximum displacement and contact area depend not only on the compressional force but also on the elastic material properties and radii of the spheres. The contact between two elastic and frictionless spherical bodies under compression was first investigated by Hertz (1881) and is known as the Hertzian contact. 

Consider two frictionless spheres in contact as shown in Fig. 2.9. M is a point on the surface of sphere 1 with a distance r from the Z -axis. N is the opposite point on the surface of sphere 2 with the same distance r from the Z2-axis. O is the contact point and also serves as the origin of both Zi-axis and Z2-axis. N, M, and O are in the same plane. The distances from M or N to the tangential plane which is normal to the Z i -axis and Z2-axis are denoted by i and Z2, respectively. In Fig. 2.9, the triangle O2AO is similar to the triangle NBO. Hence, the theorem of similarity of triangles gives... 

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. 

Consider a collision between two frictionless elastic spheres so that only normal force and normal velocities are involved i. e., no tangential forces or tangential velocities need to be accounted for in this case. A general case is shown in Fig. 2.15, where two spheres with different sizes, velocities, and material properties collide with each other. Only the collisional force is considered. 

To model the elastic properties of dispersions of soft particles, we consider a dispersion of N spheres in a periodic box, as shown in Fig. 6. The particles are either monodisperse with radius R or polydisperscd with a Gaussian distribution around a mean radius R. The concentration of particles is above the random close-packed volume fraction of 0c = 0.64 so that the particles are jammed together and form facets at contact. The contacts are assumed to be purely repulsive and frictionless and hence exert only a normal repulsive force at contact. The total elastic energy stored in the structure is the summation of the pairwise contact energies. Even at the highest volume fraction at near-equilibrium conditions, i.e., without flow, deformation of a particle is no more than 10% of its radius. Thus, the particle deformation is small compared to the size of the undeformed sphere and the contacts obey the Hertzian contact potential given by (1). 

In the dumbbell model, a polymer chain in a solvent is pictured as two massless spheres of equal size connected by a frictionless spring. The spheres experience a hydrodynamic drag proportional to their size, characterized by the Flory radius. Assume that the displacement of the spring generated by the thermal energy is also characterized by the Flory radius. Write the equation of motion for the dumbbell and show that the characteristic relaxation time for the chain deformation is that given by Eq. (9.2.1). 

FIGURE 103 Mechanical resistance and reactance of soft thigh tissue (2 cm in diameter) in vivo from 10 Hz to I MHz. The measured values (open circles—resistance diamonds— reactance) are compared with the calculated resistance and reactance of a 2-cm-diameter sphere vibrating in a viscous, elastic compressible medium with properties similar to soft human tissue (continuous lines, curves A). The resistance is also shown for the sphere vibrating in a frictionless compressible fluid (acoustic compression wave, curve B) and an incompressible viscous fluid (curve Q. (von Gierke el at., 1952.)... 

A more rigorous cell model, inasmuch as it involves a solution of the Navier-Stokes equation for creeping flow rather than a modification of that equation as in the case of Brinkman, is that of Happel (1958). In this case the basic cell is that of a single sphere surrounded by a concentric spherical envelope of fluid, the volume of which bears the same ratio to the volume of the cell as bed voidage does to unity. The crucial feature of this model is that the outer surface of the fluid envelope is frictionless (zero shear stress), so that it is often referred to as the free surface model. The solution is... 

Figure 6.2. Flow of ideal (frictionless) fluid past a sphere.

Here, we employ (4.5) and variants of it that are more appropriate for dense and very dissipative systems to describe the dependence of the transport coefficients on the volume fraction. The product G = vgo will be shown to arise prominently in the expression for the mean free path or, equivalently, the distance between the edges of the spheres, quantities that rapidly go to zero as the random close packed limit is approached (volume fraction v 0.64). We note that as the limit is approached, G increases rapidly (e.g., at v = 0.5, G = 3). In what follows, we make use of this to simplify the dependence of the theory on the volume fraction. Also, for volume fractions above 0.49, a more accurate form of 5 0 introduced by Torquato (1985) replaces the singularity of (4.5) at unity with a singularity at the value of 0.64 appropriate to random close packing of identical, frictionless spheres ... 

For shearing flows of more dissipative spheres at solid volume fractions less than 0.49 we adopt the constitutive relations obtained by Garzo and Dufty (1999) for identical frictionlesS inelastic spheres, but do not incorporate the small terms introduced by their function c of the coefficient of restitution. The magnitude of c is less than 0.4 and terms proportional to it are typically multiplied by a small numerical coefficient. The theory is linear in the first spatial gradients of the fields p, M, and T, as is the theory for nearly elastic spheres, and its derivation involves the tacit assumption that the deviatoric part A of the secoi moment is a small fraction of T, its trace. However, the determination (4.8) of A in the simplest theory, used with the solution (4.19) for T in steady, homogeneous shearing, indicates that A/T can become large as e becomes small. Consequently, the theory has to be used with some caution. 

In the context of a theory for frictionless, inelastic spheres, Woodhouse et al. (2010) have phrased the boundary-value problem for steady, uniform, flows driven by gravity down a bumpy incline and solved it numerically over a range of volume flow rates and angles of inclinations. They find as many as three solutions at a given inclination and volume flux and, in a subsequent analysis (Woodhouse et al. 2010), they characterize the stability of the solutions that they find. The average volume fraction in their solutions ranges from dilute to dense. Those solutions in the intermediate range of volume fraction indicate the features of the solutions, the fact of their multiplicity, and the details of their stability. Flows in their dilute limit are probably better treated by a theory that incorporates anisotropy in the velocity fluctuations flows in the dense limit are likely to require a theory with a somewhat more complicated structure. 

Amorphous static packings of frictionless disks in two dimensions and spheres in three dimensions generated from molecular dynamics methods are typically isostatic, with the minimal number of contacts required to constrain all nontrivial degrees of freedom. In periodic boundary conditions, a necessary condition for mechanical stability of isotropically compressed static packings is that the total number of contacts satisfies... 

These results raise several important questions concerning static packings of frictional spheres (1) What determines the characteristic p that separates frictionless and frictional behavior of the structural and mechanical properties of static sphere packings (2) Since isotropic compression yields a well-defined... 

The bubble model treats foams and emulsions as disordered collections of soft, frictionless spheres. Chapter 6 discusses the structure and mechanical response of frictionless sphere packings, which correspond to the bubble model (1) in the absence of driving or (2) subject to oscillatory driving in the limit where the driving frequency is vanishingly small. This latter case is referred to as quasistatic driving. Near the critical packing fraction 4), the quasistatic shear modulus of the bubble model is... 

Figure 13.18 Jamming diagram proposed by Liu and Nagel (1998), revised by O Hern et al. (2003), and experimentally determined by Trappe et al. (2001). The diagram illustrates that many disordered materials are in a jammed state for low temperature T, low load E, and large density ( ), but can become unjammed when these parameters are varied. For frictionless soft spheres, there is a well-defined jamming transition indicated by point 7 on the inverse density axis, which exhibits similarities to a critical phase transition.

The problem of the elastic contact of a flat cylindrical punch is less complex than that of a sphere due to the fact that for the flat punch, the contact radius is just given by the radius of the cylinder and thus known a priori. The problem of the elastic contact between a sphere and a planar surface and between two spheres was solved by Hertz in 1882 [846]. Under the assumption that the contact radius a is small compared to the sphere radii, that the contact is frictionless and no tensile stress exists within the area of contact. Hertz derived an equation for the contact radius a between the spheres ... 

Schmid (56) then analyzed the possibility of the macromolecules being broken by frictional forces. In a first case he considered the polymer molecule being represented by a frictionless thread, having spheres of radius r placed at regular intervals along its axis and rigidly held at one end in the solution. The total friction force / may be calculated by Stoke s law ... 

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