Hertz’s theory

When the particle deformation is small compared to the size of the undeformed spheres, the contacts obey Hertzian contact mechanics. According to Hertz s theory, the elastic energy associated with a single contact is [118] ... 

Fig-6 Periodic box of concentrated dispersion of soft spherical particles. Each pair of particles at contact forms a facet, as shown in Fig. 4, that deforms according to Hertz s theory or similar law... 

If a circle with radius a (m) is regarded as the contact area between a stainless steel ball and substrate under a normal load P (0.49 N), Hertz s theory affords the following relationship using Young s modulus of stainless steel and silicon wafer, (1.96 x 10" Pa) and (1.30 x 10" Pa), and Poisson s ratio D (0.30) and Ub (0.28), respectively ... 

In 1885 Joseph Boussinesq (6), trying to extend the validity of these results to the case of axi-symetrical rigid convex punches indenting a flat semi-infinite elastic medium, demonstrates that, without an adequate boundary comlition, the size of the contact area is generally unknown. In ordo to overcome this difficulty, he imposes that normal stresses vanish on the border of the contact area. In other words, the profile of the distorted medium must be tangent to the surfiice of the punch on the border of the contact area. Note that this condition is the same as the condition presupposed by the Hertz s theory. With this assumption, the size oh of the contact area and the penetration depth 5h are completely defined (Figure 1). 

The subj t of adhesive contact mechanics may be said to have started when Kendall (//), solving the problem of the adhesive contact of a rigid flat cylinder punch indenting the smooth plane surface of an elastic medium, demonstrated that the border of the contact area can be considered as a crack tip. The more complex problem of a spherical punch was solved in 1971 by Johnson, KendaU and Roberts (72). The JKR theory predicts the existence of contact area greater then that ven by the elastic contact Hertz s theory. The molecular attractive forces are responsible for this increase and, even in the absence of external compressive loading, the contact area has a finite size. Separating the two solids requires the application of an adherence force despite the existence of infinite normal stresses in the border of the contact area. 

Figure 8. lUf-width of the equilibrium contact area between a rigid cylindw and the smooth surface of an elastic solid as a fimction of the normal plied load per unit axial length, in reduced coordinates. E q)erimental data M in the immediate vicinity of the theoretical curve (heavy line). The curve deduced from the classical Hertz s theory (non-adhesive elastic contact) is given for conq>arison. 

The surface of an asperity of rubber is taken to be spherical. The increase in the area of contact with the time of contact may be expressed in terms of creep properties, on the assumption that normal load is borne by the asperity as soon as it makes contact with the rigid countersurface. It is convenient to relate the area of contact Ap (t) after a time t, reckoned from the instant of first contact to the area A corresponding to the rubber elastic state. The latter may be determined from considerations of surface statistics and Hertz s theory. 

A few results could be found in the literature for the relation between F/N and the radius of a hemispherical or spherical slider. When this radius R was [20] 0.285, 1-33, and 10.0 cm, respectively, F/N was 0.4, 0.46, and 0.57- The slider was of glass, the polymer was Nylon-66, and N was constant at 10 dynes. From Hertz s theory for ideally elastic materials, the product wd is independent of R (w is proportional to the cubic root of R and d is inversely proportional to this root). It is seen that F/N also was little affected by the radius when R rose in the ratio of 35 to 1, F/N increased only in the ratio 1.4 to 1. Presumably it was not quite constant because Nylon-66 was not a Hookean solid. The change of the track width with R also seems to confirm the poor applicability of Hertz s equation to nylon this width increased less steeply than did the cubic root of R. 

The exponent m is mostly in the order of m 1 /6 — 1 /4. An exponent m= /6 results from Hertz s theory (see Section 6.8.2) for spherical grain contacts under pressure thus, this type of equation is preferred for sediments with a granular structure (sand, sandstone). 

The factor (6.143) also covers the pressure influence on the elastic properties. Hertz s theory applied on the grain-grain contact results in a power law ... 

According to Hertz s theory of elastic collision, the change rate of the contact area Af during the collision is given by... 

By combining Hertz s contact theory (Eq. 1) and with Hamaker s functional form for the attractive force (Eq. 17), the Derjaguin model takes the form... 

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... 

Equation (3) reduces to Equation (1) if the second term on the right-hand side is ignored, which corrects the Hertz s displacement since Hertz s assumption of near sphericity is not valid at large deformations. For a given force, the displacement predicted by the Tatara model is always smaller than that from Hertz theory since the value of /( ) is positive. 

Takahashi s approach is diametrically opposed to Hertz s electrostatic model in that it assumes the source of the transient electric field gradients to arise from symmetry distortions in the first solvation shell. Moreover, the theory is of less general applicability since it requires a well defined solvation complex hence confining it to strongly hydrating cations. 

A further direct confirmation of Bohr s theory on the existence of discrete energy levels in the atom is given by the experime.nts of Franck and Hertz (1914). If the atoms are su])])Hed with energy in any way, e.g. by electronic collision, i.e. by bombarding the atom with (dectrons,... 

The wave theory of light reached its climax with Hertz s contributions around 1888. Physics was now in a state of turmoil, tom apart by the wave-particle duality caused by the quantum theory. As a result, research in photo- and opto-related areas was divided into two streams quantum electrodynamics and molecular spectroscopy. 

Baeyer s speech has considerable interest. After the expected preliminaries, he began by asking the rhetorical question Is Kekule s benzene theory a true depiction of the molecule, or is it simply a heuristically useful fiction This question evoked a consideration of molecular models. Van t Hoff was not the first to suggest a tetrahedral shape for the carbon atom, Baeyer noted it was Kekule who had introduced tetrahedral carbon models in 1867. Of course, van t Hoff had taken the idea further than Kekule, in particular by affirming that the four valence bonds emanating from each carbon atom were relatively fixed and could therefore be studied chemically. In this sense Kekule s tetrahedral models were analogous to Heinrich Hertz s famous comment about James Clerk Maxwell s equations of the electromagnetic field that they have almost an independent life, that they can appear wiser even than their creator and can yield more than was ever invested in them. ... 

To be sure, Baeyer said, what we are talking about are pictures or representations (Bilder), which must never be confused with real things themselves, but Hertz s comment applies whenever our theoretical pictures approach the unseen reality. These same considerations apply to the benzene theory. Baeyer proceeded to summarize recent research on the structure of benzene. The benzene formula proposed years earlier by James Dewar had quickly been ruled out as unable to explain the... 

The photoelectron effect was first discovered by Henrich Hertz [11] in early 1887 in order to verify the implications of Maxwell s theory and relations. Hertz noticed a spark of light on metal contacts in electrical units when exposed to light. The dawn of a new era actually came in 1905. Albert Einstein brilliantly utilized Planck s new quantum energy concept to explain how low radiation intensity and high frequency can actually eject electrons from a metal piece. The converse failed to produce any electrons. Max Planck received the Nobel Prize on quantization of energy [12] in 1918 and Einstein received the Nobel Prize on photoelectric effect in 1921. The single relationship proposed so long ago by Einstein is still today the fundamental basis of photoelectron spectroscopy,... 

Heinrich Rudolf Hertz (1857-1894). German physicist. He performed a number of experiments confirming Maxwell s theory of electromagnetic radiation. His discovery of radio waves led to the development... 

Note that the surface area is th( same whether the ball is in air or iiniiierscd in the liquid. TIk so-callcd JKR theory (.Johnsoii, Kendall, and Roberts) extends Hertz s calculation to the case when the rubber adheres to the glass.The theory takes Dupre s adhesion energy into account via the relation H q = jSL Isr = S. This energy depends on whether... 

According to the Hertz-Knudsen theory, the drop of vapor pressure depends on the rate of evaporation v (cm/s) ... 

The appropriateness of a theory, however, has a limited claim to reality. Thus the end-of-the-century theorising on scientific theories was, to a very large extent, based on an instrumental conception of theories with respect to observation. The American scheme was very similar to this. In a symposium in 1937, E.U. Condon made remarks almost identical to the opening sentence of Hertz s book ... 

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