Elastic solutions Hertz

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. 

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

Contact problems have their origins in the works of Hertz (1881) and Boussinesq (1885) on elastic materials. Indentation problems are an important subset of contact problems (17,18). The assessment of mechanical properties of materials by means of indentation experiments is an important issue in polymer physics. One of the simplest pieces of equipment used in the experiments is the scleroscope, in which a rigid metallic ball indents the surface of the material. To gain some insight into this problem, we consider the simple case of a flat circular cylindrical indentor, which presents a relatively simple solution. This problem is also interesting from the point of view of soil mechanics, particularly in the theory of the safety of foundations. In fact, the impacting cylinder can be considered to represent a circular pillar and the viscoelastic medium the solid upon which it rests. 

Fig. 5 Variations of the interaction energy between particles versus the center-to-center distance, (a) Hertzian potentials for particles with bulk elasticity the dashed line represents the usual Hertz potential (1), and the solid line the generalized Hertzian potential (2). (b) Potentials for emulsions with surface elasticity the dashed line denotes the approximate solution for small compression ratios (3), and the solid line the general solution (4). (c) Ultrasoft potentials for star polymers (where a 1.3Rg, with Rq being the radius of gyration of the star, following [123]) (5) dashed line f = 256 solid line f = 128 dashed and dotted line f = 64. (d) Hard-sphere potential...

We have also measured the surface compression elasticity and viscosity of DTAB-PAMPS mixed surface layers. These two coefficients describe the resistance of the layer to a uniaxial compression in the surface plane. They were measured with a device in which surface waves are excited at a frequency of a few hundred hertz. It was found that as expected, the layers start to exhibit a measurable elasticity at surfactant concentrations much less than with pure surfactant solutions (Figure 5). The elasticity (both real and imaginary parts, r and j respectively) exhibits a maximum around CAC and decreases to zero around CMC. 

Further information about the possible range of mechanical effects that occur within thin synthetic gel substrates has been provided in a study that used gel indentation with the cantilever of an atomic force microscope (AFM). Gels of varying thickness, H, were made at the same time with the same polyacrylamide gel solutions to maintain a constant E of tissue-like ( kPa) elasticities, and the gels were all indented by l-2 pm at forces that bend the cantilever in the nano-Newton (nN) range. An apparent elasticity E pp was obtained in this AFM experiment by fitting the force/versus indentation depth d with a generalized Hertz model ... 

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 is to be expected that three-dimensional boundary value problems will present greater difficulties than plane problems. In particular, with the far wider choice of boundary regions on which to specify displacement and stress, one rapidly meets problems that are unsolvable - at least analytically. This is true even for elastic materials. In fact, the contact problem with an elliptical contact area is the most general problem that allows an explicit analytic solution - for elastic materials [Galin (1961), Lur e (1964)], in the case of half-space problems. This corresponds to an ellipsoidal indentor, according to classical Hertz theory. The theory can be extended to cover contact between two gently curved bodies. The solution is valid only for quasi-static conditions. 

The solution for an elastic contact was given by Hertz (1881) and Sneddon (1965) (Chapter 8) for a blunt and sharp indenter respectively. The solutions can be written in the general form (Chapter 8 we suppose the indenter is much more rigid than the specimen)... 

It should be noted that the Hertz model assumes the contact of elastic half-spaces. In other words, the bodies in contact are assumed to be infinitely thick. However, when studying thin films, this may not necessarily be tme. The ramifications and prescribed solution to this problem were worked out by Akhremitchev and Walker. ... 

The most basic configuration for an elastic contact is the indentation of an elastic halfspace by a rigid axisymmetric frictionless punch. Frictionless means that we assume that no shear stress can develop between the punch and the half-space. While historically the first solution of such a problem was given by Hertz for the case of a spherical indenter [846], we will start with a flat rigid cylindrical punch (Figure 8.3) that was first worked out by Boussinesq in 1885 [847] and solved in all details by Sneddon in 1946 [848]. 

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