Viscoelastic spheres

Rolling friction is often found to be proportional to the velocity, but more complex relationships may be observed, depending on the combination of the bodies. For a soft, viscoelastic sphere on a hard substrate, Brilliantov et al. [464] predicted a linear dependence of rolling friction on speed. For a hard cylinder on a viscous surface, a much more complex behavior was found [465,466], At lower speeds, the rolling friction increases with speed to reach a maximum value and then decreases at higher speeds. The reason is an effective stiffening of the substrate at higher speeds. 

In the first part of this chapter we studied the radial vibrations of a solid or hollow sphere. This problem was considered an extension to the dynamic situation of the quasi-static problem of the response of a viscoelastic sphere under a step input in pressure. Let us consider now the simple case of a transverse harmonic excitation in which separation of variables can be used to solve the motion equation. Let us assume a slab of a viscoelastic material between two parallel rigid plates separated by a distance h, in which a sinusoidal motion is imposed on the lower plate. In this case we deal with a transverse wave, and the viscoelastic modulus to be used is, of course, the shear modulus. As shown in Figure 16.7, let us consider a Cartesian coordinate system associated with the material, with its X2 axis perpendicular to the shearing plane, its xx axis parallel to the direction of the shearing displacement, and its origin in the center of the lower plate. Under steady-state conditions, each part of the viscoelastic slab will undergo an oscillatory motion with a displacement i(x2, t) in the direction of the Xx axis whose amplitude depends on the distance from the origin X2-... 

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. 

Several researchers have commented on the need for a mechanics model for the adhesion of viscoelastic solids to assist in the interpretation of microprobe adhesion experiments. This paper is a progress report on work in Cambridge to extend the JKR theory to the adhesion of viscoelastic spheres. The present state of play will be reviewed in a predominantly qualitative way analytical details will be presented in a separate publication. 

Figure 8. Simulation of a ramp loading cycle of viscoelastic spheres in contact (k = 0.1), showing hystereras loops for varying values of rate parameter A..

Roscoe, R. 1967. On rheology of a suspension of viscoelastic spheres in a viscous liquid. J. Fluid Meek 28 273-293. 

The radiation and temperature dependent mechanical properties of viscoelastic materials (modulus and loss) are of great interest throughout the plastics, polymer, and rubber from initial design to routine production. There are a number of laboratory research instruments are available to determine these properties. All these hardness tests conducted on polymeric materials involve the penetration of the sample under consideration by loaded spheres or other geometric shapes [1]. Most of these tests are to some extent arbitrary because the penetration of an indenter into viscoelastic material increases with time. For example, standard durometer test (the "Shore A") is widely used to measure the static "hardness" or resistance to indentation. However, it does not measure basic material properties, and its results depend on the specimen geometry (it is difficult to make available the identity of the initial position of the devices on cylinder or spherical surfaces while measuring) and test conditions, and some arbitrary time must be selected to compare different materials. 

Many materials of practical interest (such as polymer solutions and melts, foodstuffs, and biological fluids) exhibit viscoelastic characteristics they have some ability to store and recover shear energy and therefore show some of the properties of both a solid and a liquid. Thus a solid may be subject to creep and a fluid may exhibit elastic properties. Several phenomena ascribed to fluid elasticity including die swell, rod climbing (Weissenberg effect), the tubeless siphon, bouncing of a sphere, and the development of secondary flow patterns at low Reynolds numbers, have recently been illustrated in an excellent photographic study(18). Two common and easily observable examples of viscoelastic behaviour in a liquid are ... 

FIGURE 6.9 Dependence of viscoelastic parameters on solvent quality. The (A) static force, (B) drag coefficient at 10 kHz, (C) dynamic spring constant, and (D) dispersion parameter are shown as a function of the surface-sphere distance. The results for water, propanol, and a 50/50 water/propanol mixture are given. Reprinted with permission from Benmouna and Johannsmann (2004). 

There are not a great number of studies on the viscoelastic behaviour of quasi-hard spheres. The studies of Mellema and coworkers13 shown in Figure 5.5 indicate the real and imaginary parts of the viscosity in a high-frequency oscillation experiment. Their data can be normalised to a characteristic time based on the diffusion coefficient given above. 

Also shown is the dimensionless energy density for hard spheres, 3kBTf2 per particle. The viscoelastic liquid zone is difficult to define... 

There is a wealth of microstructural models used for describing nonlinear viscoelastic responses. Many of these relate the rheological properties to the interparticle forces and the bulk of these consider the action of continuous shear rate or stress. We will begin with a consideration of the simplest form of potential, a hard rigid sphere. 

By virtue of its yield stress, an unsheared viscoelastic material is capable of supporting the immersed weight of a particle for an indefinite period of time, provided that the immersed weight of the particle does not exceed the maximum upward force which can be exerted by virtue of the yield stress of the fluid. The conditions for the static equilibrium of a sphere are now discussed. 

One conclusion from this study is that although the hard-sphere fluid has been very successful as a reference fluid, for example, in developing analytical equations of state, it is unrealistic in representing the dynamical relaxation processes in real systems, even with very steeply repulsive potentials. Owing to the discontinuity in the hard-sphere potential, this fluid, in fact, is not a good reference fluid for the short time (fast or j9 ) viscoelastic relaxation aspects of rheology. 

Globular proteins form close-packed monolayers at fluid interfaces. Hence a large contribution to the adsorbed layer viscoelasticity arises from short-range repulsive interactions between hard-sphere particles. In addition to, or instead of, this glass-like5 structure from hard spheres densely packed in two dimensions, many adsorbed proteins can exhibit attractive interactions leading to a more gel-like5 network structure. Hence the mechanical properties of an adsorbed layer depend on many... 

Single-molecule theories originated in early polymer physics work (45) to describe the flow behavior of very dilute polymer solutions, which are free of interpolymer chain effects. Most commonly, the macromolecular chain, capable of viscoelastic response, is represented by the well-known bead-spring model or cartoon, shown in Fig. 3.8(a), which consists of a series of small spheres connected to elastic springs. 

Coalescence of mesophase is often said to be determined by the mesophase viscosity. This aspect requires much further investigation. However, it is clear that, amongst other factors, the rheological behaviour (including viscoelastic effects) of each phase is important in mesophase growth and coalescence. Diffusion of molecular species through the isotropic pitch to the mesophase spheres is likely to be related to the viscosity of the isotropic med i urn. 

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