Hertz problem

SC Hunter. The Hertz problem for a rigid spherical indentor and a viscoelastic half-space. J Mech Phys Solids 8 219, 1960. 

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. 

Let us present our rather simplified scheme of the application of a thermodynamic approach, following the Gibbs-Volmer-Griffith model, to the Hertz problem, described in Section 5.2. Let an indenter leave a hemispherical pit hole with a diameter equal to la and a depth equal to h in the plane of the sample surface (Figure 7.39). 

FIGURE 7.39 A simplified approach to the Hertz problem, (a) The indentation with a sphere results in an elastic deformation of the substrate (b) a portion of the elastic energy can be used to create a dislocation. 

Fogden, A. and White, L. R., Contact elasticity in the presence of capillary condensation. The nonadhesive Hertz problem, JCIS, 138, 414-430 (1990). 

In chapter 1, the properties of the viscoelastic functions are explored in some detail. Also the boundary value problems of interest are stated. In chapter 2, the Classical Correspondence Principle and its generalizations are discussed. Then, general techniques, based on these, are developed for solving non-inertial isothermal problems. A method for handling non-isothermal problems is also discussed and in chapter 6 an illustrative example of its application is given. Chapter 3 and 4 are devoted to plane isothermal contact and crack problems, respectively. They utilize the general techniques of chapter 2. The viscoelastic Hertz problem and its application to impact problems are discussed in chapter 5. Finally in chapter 7, inertial problems are considered. 

Hertz [27] solved the problem of the contact between two elastic elliptical bodies by modeling each body as an infinite half plane which is loaded over a contact area that is small in comparison to the body itself. The requirement of small areas of contact further allowed Hertz to use a parabola to represent the shape of the profile of the ellipses. In essence. Hertz modeled the interaction of elliptical asperities in contact. Fundamental in his solution is the assumption that, when two elliptical objects are compressed against one another, the shape of the deformed mating surface lies between the shape of the two undeformed surfaces but more closely resembles the shape of the surface with the higher elastic modulus. This means the deformed shape after two spheres are pressed against one another is a spherical shape. 

Often, Hertz s work [27] is presented in a very simple form as the solution to the problem of a compliant spherical indentor against a rigid planar substrate. The assumption of the modeling make it clear that this solution is the same as the model of a rigid sphere pressed against a compliant planar substrate. In these cases, the contact radius a is related to the radius of the indentor R, the modulus E, and the Poisson s ratio v of the non-rigid material, and the compressive load P by... 

Many electrical problems, or problems associated with the quality of the incoming power and internal to the motor, can be isolated by monitoring the line frequency. Line frequency refers to the frequency of the alternating current being supplied to the motor. In the case of 60-cycle power, monitoring of the fundamental or first harmonic (60 Hertz), second harmonic (120 Hz), and third harmonic (180 Hz) should be performed. 

Even if the length and power problem is solved, the integration time cannot be reduced well below 10 ms so the detection is limited to signals with frequencies below a few hundreds of hertz (in fact a few kilohertz). 

Including capillary condensation with the Hertz approximation, as considered by Fogden and White [20], introduces pressure outside the contact area i.e., adhesion enters the problem nonenergetically through the tensile normal stress exerted by the condensate in an annulus around the contact circle. The resulting equations cannot be solved analytically however, their asymptotic analysis may be summarized as follows. 

In the early work of Bewick and Robinson (1975), a simple monochromator system was used. This is called a dispersive spectrometer. In the experiment the electrode potential was modulated between two potentials, one where the adsorbed species was present and the other where it was absent. Because of the thin electrolyte layer, the modulation frequency is limited to a few hertz. This technique is referred to as electrochemically modulated infrared reflectance spectroscopy (EMIRS). The main problem with this technique is that data acquisition time is long. So it is possible for changes to occur on the electrode surface. 

By pressing a sphere upon a planar surface, a deformation occurs near the original point of contact, see Fig. F.8. This problem was first solved by Hertz in 1881 (see Landau and Lifshitz, 1986 Timoshenko and Goodier, 1970). The derivations are complicated. We state the results without proof. 

Heinrich Hertz long ago solved the problem of a thermal wave in front of a heated surface which is moving with constant velocity. This solution, first applied to a flame by Michelsohn, has the form... 

PROBLEM 10.5.4. To what wavelength X does that frequency v = 540 x 1012 hertz correspond ... 

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