Hertz theory

Contact mechanics, in the classical sense, describes the behavior of solids in contact under the action of an external load. The first studies in the area of contact mechanics date back to the seminal publication "On the contact of elastic solids of Heinrich Hertz in 1882 [ 1 ]. The original Hertz theory was applied to frictionless non-adhering surfaces of perfectly elastic solids. Lee and Radok [2], Graham [3], and Yang [4] developed the theories of contact mechanics of viscoelastic solids. None of these treatments, however, accounted for the role of interfacial adhesive interactions. 

Dutrowski [5] in 1969, and Johnson and coworkers [6] in 1971, independently, observed that relatively small particles, when in contact with each other or with a flat surface, deform, and these deformations are larger than those predicted by the Hertz theory. Johnson and coworkers [6] recognized that the excess deformation was due to the interfacial attractive forces, and modified the original Hertz theory to account for these interfacial forces. This led to the development of a new theory of contact mechanics, widely referred to as the JKR theory. Over the past two decades or so, the contact mechanics principles and the JKR theory have been employed extensively to study the adhesion and friction behavior of a variety of materials. 

The Hertz theory of contact mechanics has been extended, as in the JKR theory, to describe the equilibrium contact of adhering elastic solids. The JKR formalism has been generalized and extended by Maugis and coworkers to describe certain dynamic elastic contacts. These theoretical developments in contact mechanics are reviewed and summarized in Section 3. Section 3.1 deals with the equilibrium theories of elastic contacts (e.g. Hertz theory, JKR theory, layered bodies, and so on), and the related developments. In Section 3.2, we review some of the work of Maugis and coworkers. 

The JKR theory, much like the Hertz theory, assumes a parabolic approximation for the profile of sphere, which is valid for small ratios of contact radii to the sphere s radius. Maugis [34] has shown that for small particles on a soft substrate, this ratio could be so large that such parabolic approximation is no longer valid. Under such conditions, the use of exact expression for the sphere profile is necessary for the applicability of the JKR theory, which is expressed as... 

SFA measurements on mica. Horn et al. [68] studied the deformation of mica surfaces in contact. In these studies, Horn et al. established the applicability of Hertz theory of contact mechanics to non-adhering layered solids by measuring... 

At present, there are a variety of theoretical models to describe the mechanical contact between the two bodies under external load and many of such theories have been used to analyze force-distance curves.Among them. Hertz theory has been the most widely used because of its... 

The deformation of a specimen during indentation consists of two parts, elastic strain and plastic deformation, the former being temporary and the latter permanent. The elastic part is approximately the same as the strain produced by pressing a solid sphere against the surface of the specimen. This is described in detail by the Hertz theory of elastic contact (Timoshenko and Goodier, 1970). 

Hertz Theory of Impact is described in the Mathematical Theory of Elasticity by A.E.H Love, Dover Publications, NY (1944), p 198. This theory was applied by M.P. Murgai, JChemPhys 22, 1687-9 (1954) CA 49,... 

Here ri denotes a constant velocity parameter that was interpreted as the velocity of the ether. Hertz theory was discarded and forgotten at that time, because it spoiled the spacetime symmetry of Maxwell s equations. 

Chubykalo and Smirnov-Rueda [2,56] have presented a renovated version of Hertz theory, that is in accordance with Einstein s relativity principle. For a single point-shaped charged particle moving at the velocity v, the displacement current in Maxwell s equation is modified into a convection displacement current ... 

The interaction between fine particles is often dominated by their mechanical properties such as Young s modulus. This was first considered by Hertz theory. Adhesion between spherical particles increases with the radius of the particles and is described by JKR and DMT theories. 

Elastic deformation. For small loads we can use the Hertz model as a simple approximation. The microcontacts are thereby assumed to be spherical. Hertz theory predicts an actual contact surface for an individual sphere on a plane (see Eq. 6.64) ... 

At shorter distances the repulsive forces start to dominate. The repulsive interaction between two molecules can be described by the power-law potential l/rn (n>9) caused by overlapping of electron clouds resulting in a conflict with the Pauli exclusion principle. For a completely rigid tip and sample whose atoms interact as 1/r12, the repulsion would be described by W-l/D7. In practice, both the tip and the sample are deformable (Fig. 3d). The tip-sample attraction is balanced by mechanical stress which arises in the contact area. From the Hertz theory [77,79], the relation between the deformation force Fd and the contact radius a is given by ... 

Mathematical modelling of the compression of single particles 2.5.2.7 Hertz model. The mechanics of a sphere made of a linear elastic material compressed between two flat rigid surfaces have been modelled for the case of small deformations, normally less then 10% strain (Hertz, 1882). Hertz theory provides a relationship between the force F and displacement hp as follows ... 

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. 

One of the most simple approaches to analyze indentation measurements is based on Hertz theory. We discuss below an example reported by the Radmacher group on gelatin films (Fig. 4.20) [46]. 

Another technique widely used in the estimation of the properties of cells and their components is atomic force microscopy, where the sample is probed by a sharp tip located at the end of a cantilever of a prescribed stiffness, and the corresponding indentation is tracked with a laser. The force/indentation relationship is a characterization of the cell (cellular component) properties. A traditional interpretation of this experiment is based on Hertz theory of a frictionless contact of a rigid tip with an elastic isotropic half-space [Radmacher et al, 1996]. The finite thickness of the cell can be taken into account by considering an elastic layer adhered to a substrate. More details of cell geometry and rheology can be considered by using the finite element method. 

H-PDLC 28 Hamalcer constant 49 Hamiltonian 221,228,239,241 hard contact 59 Heisenberg model 216 Helmholtz layer 187 Hertz theory 47 heterodyne 102... 

Fig. 3.4. The thickness of the adsorbed trilayer of the liquid crystal 5CB on DMOAP covered BK7 glass substrate. The full line is a fit to the Hertz theory with E = E/(l — v ) = 1.2 X 10 (1 0.3)Nm. R om point B to the point C, the surface adsorbed molecular trilayer is elastically deformed. At the instability point C, the layer ruptures and the AFM tip is in hard contact with the surface at D. The inset shows the linear relation between the thickness of the adsorbed layer and the length of fully extended liquid crystal molecule.

Roberts,while observing the contact of rubber windscreen wiper blades on glass, had noticed that the contact spot was much larger than he expected from Hertz theory under dry conditions, yet approached the Hertz predictions rather precisely when wetted with soapy water. Kendall had been measuring the contact spot size between polymer, glass, and metal surfaces using optical and... 

Contact mechanics of engineering surfaces, especially layered surfaces has seen significant progress in the last decade [31], The main achievement has been the removal of the continuum assumptions in the Hertz theory. The application of numerical techniques including finite element methods has successfully solved many contact problems, which are difficult to solve analytically. 

The JKR theory, similar to the Hertz theory, is a continuum theory in which two elastic semi-infinite bodies are in a non-conforming contact. Recently, the contact of layered solids has been addressed within the framework of the JKR theory. In a fundamental study, Sridhar et al. [32] analyzed the adhesion of elastic layers used in the SFA and compared it with the JKR analysis for a homogeneous isotropic half-space. As mentioned previously and depicted in Fig. 5, in SFA thin films of mica or polymeric materials ( i, /ji) are put on an adhesive layer Ej, I12) coated onto quartz cylinders ( 3, /i3). Sridhar et al. followed two separate approaches. In the first approach, based on finite element analysis, it is assumed that the thickness of the layers and their individual elastic constants are known in advance, a case which is rare. The adhesion characteristics, including the pull-off force are shown to depend not only on the adhesion energy, but also on the ratios of elastic moduli and the layers thickness. In the second approach, a procedure is proposed for calibrating the apparatus in situ to find the effective modulus e as a function of contact radius a. In this approach, it is necessary to measure the load, contact area... 

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