Hertz Equation Research Articles

Approximation of contact stress for a compressed and laterally one side restrained O-ring

This paper revisits the classical O-ring problem by using an experimental, finite element analysis and conventional theories. Especially, the elastic moduli, deformation shape, friction coefficient and extrusion behaviour were investigated in detail during the experiments. A computed tomography was used to detect the deformed shape of the O-ring. The used finite element method results were validated by comparing them with the experimental results. For an application to the present study, an actual case of a compressed and laterally one side restrained condition was analyzed experimentally and numerically. As a result of the finite element analysis, the friction coefficient affects the contact stress profile and magnitude considerably. Lindley’s formulae for a contact (compressive) force underestimated the measured force but it showed similar results to the finite element analysis results. Applicability of the Hertz theory to the contact stress field was discussed by comparing it with the finite element analysis results in the case of a compressed and laterally one side restrained condition. Consequently, it was found that the normalized results with respect to the peak stress and contact width followed the typical Hertzian profile. So an implementation of the appropriate factors to the Hertz equation can yield an approximate solution of the contact stresses.

Application of the Na+ recirculation theory to ion coupled water transport in low- and high resistance osmoregulatory epithelia

The theory of Na+ recirculation for isosmotic fluid absorption follows logically from Hertz's convection-diffusion equation applied to the exit of water and solutes from the lateral intercellular space. Experimental evidence is discussed indicating Na+ recirculation based upon the following approaches: (i) An isotope tracer method in small intestine. Simultaneous measurement of water flow and ion transport in toad skin epithelium demonstrating, (ii) occasional hyposmotic absorbates, and (iii) reduced fluid absorption in the presence of serosal bumetanide. (iv) Studies of the metabolic cost of net Na+ absorption demonstrating an efficiency that is lower than the 18 Na+ per O2 consumed given by the stoichiometry of the Na+/K+-pump. Mathematical modeling predicts a significant range of observations such as isosmotic transport, hyposmotic transport, solvent drag, anomalous solvent drag, the residual hydraulic permeability in proximal tubule of AQP1(-/-) mice, the adverse relationship between hydraulic permeability and the concentration difference needed to reverse transepithelial water flow, and in a non-contradictory way the wide range of metabolic efficiencies from above to below 18 Na+/O2. Certain types of observations are poorly or not at all reproduced by the model. It is discussed that such lack of agreement between model and experiment is due to cellular regulations of ion permeabilities that are not incorporated in the modeling. Clarification of these problems requires further experimental studies.

Elasticity of rubber-like materials measured by AFM nanoindentation

We exploit the force spectroscopy capabilities of the atomic force microscope in characterizing the local elastic- ity of rubber-like materials. Extraction of elastic properties from force curves usually relies on the linear theory pioneered by Hertz. While the Hertzian force-indentation relationships have been shown to be accurate in modeling the contact mechanics at sufficiently shallow indentation depths, the linear deformation regime of the probed material is exceeded in many practical applications of nanoindentation. In this article, a simple, nonlinear force-indentation equation based on the Mooney-Rivlin model is derived and used to fit data from the indentation of lightly crosslinked poly(vinyl alcohol) gels in equilibrium with water. The extracted values of Young's modulus show good agreement with those obtained by both macroscopic compression testing and by fitting truncated portions of the force curves with the Hertz equation.

Study on the prediction technique of the wear amount of polyoxymethylene

AbstractThe influence of the sliding velocity and atmospheric temperature on the specific wear amount of polyoxymethylene was evaluated with the ring‐on‐ring apparatus. The specific wear amount showed the variation of S‐character type with maximum value, according to the change in the sliding velocity and the atmospheric temperature. When those results are plotted according to the sliding surface temperature, they have been found to be arranged as the same dependence. The specific wear amount of polyoxymethylene was assumed to be proportional to the product of actual contact area and reciprocal of the fracture energy. Then, the temperature dependency of the product of the fracture energy, calculated from stress–strain curve, and the actual contact area, calculated from Hertz's equation with elastic modulus, was able to explain the sliding surface temperature dependency of the specific wear amount well. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 100: 4212–4218, 2006

A Model of Damage Initiation in Singly Oriented Ply Fiber Metal Laminate under Concentrated Loads

Damage initiation is modeled in singlyoriented ply(SOP) Fiber Metal Laminate (FML) under concentrated loading conditions. The finite element method (FEM) is used based on the first order shear deformation theoryfor modeling damage initiation in SOP FML. The failure indices of the fiber prepreg and the metal laminate were calculated using the Tasi–Hill failure criterion and the Miser yield criterion respectively. To verify the model, failure analysis was conducted under uniaxial loading and cylindrical bending, then under concentrated loading. Results show that the model is good. An indentation test was then conducted to compare the damage initiation load with the calculated failure index. The test was conducted under two side clamped conditions in order to studythe fiber orientation effect. The indentation curve was fitted using the Hertz equation, and the damage initiation load was defined as the point of divergence of the fitted curve from the real indentation curve. The damage initiation loads were obtained for various fiber orientations and were compared with the calculated failure index. The experiment results closelymatched the calculated failure index. It follows that the present method accuratelycaptures damage initiation of SOP FML.

粒子状物質の硬度評価法の開発

Hardness measurement of particulate matters has been carried out using the compression and stress relaxation test. In order to discuss the influence of particle size and particle shape, seven types of glass powder were used in the experiment. In this method, there was a region in which the applied pressure was linearly related to the displacement at the initial stage of compression. We applied Rumpf equation and Hertz equation to the proportional region and determined the constant k′ in Hertz equation. The constant k′ is independent of the particle size, showing that the hardness of particles can be reasonably evaluated by this method.

粒子状物質の硬度評価法の開発 実験値と推算値の比較

A new evaluation technique has been developed to measure the hardness of particulate matter by using a compression and stress relaxation test. In this paper we determined the constant k' in Hertz equation from the analysis of the proportional region and compared it with k' calculated from Young's modulus and Poisson's ratio for the verification of this method. Furthermore the effect of particle size distribution on the experimental results was discussed. It was shown that the constant k' obtained from the analysis of the proportional region was well correlated with the calculated k'. It was also found that the value of k' was almost independent of the particle size distribution. These results indicate that the constant k' determined by this method can be a new hardness index of particulate matter.

The Hertzian contact surface

It is tempting to accept the predictions regarding indentation depth and radius of circle of contact between two elastic bodies in contact given by the well-known Hertz equations at face value. However, it is nevertheless of interest to examine these predictions either by experiment or by independent computation. Indentation depth may be readily compared using standard experimental apparatus but in this paper, attention is given to the radius of curvature of the indented surface for a condition of full load. The conclusion arising from the Hertz equations, that contact between a flat surface and a non-rigid indenter of radius R is equivalent to that between the flat surface and a perfectly rigid indenter of a larger radius, has not thus far been examined in detail in the literature, possibly because of the difficulty in measuring such a radius of curvature in situ while load is applied to the indenter. This feature of contact between two solids is of interest since it has been often used as the basis for various hardness theories which involve an elastic–plastic contact. This paper addresses the issue by utilizing the finite-element method to compute the radius of curvature of the contact surface for both elastic and elastic–plastic contacts. It is shown that indentations involving elastic–plastic deformations within either or both the specimen and the indenter are equivalent to indentations with a perfectly rigid spherical indenter whose radius is somewhat smaller than that calculated using the Hertz equations for elastic contact. An experimental compliance response is used to indirectly validate the finite-element results.

Adhesion of elastic spheres

Bradley (1932) showed that if two rigid spheres of radii R 1 and R 2 are placed in contact, they will adhere with a force 2πΔ R γ, where R is the equivalent radius R 1 R 1 /( R 1 + R 2 ) and Δγ is the surface energy or ‘work of adhesion’ (equal to γ1+γ2-γ12). Subsequently Johnson et al. (1971) (JKR theory) showed by a Griffith energy argument (assuming that contact over a circle of radius a introduces a surface energy -π a 2 Δγ) how the Hertz equations for the contact of elastic spheres are modifed by surface energy, and showed that the force needed to separate the spheres is equal to (3/2)πΔ R γ, which is independent of the elastic modulus and so appears to be universally applicable and therefore to conflict with Bradley9s answer. The discrepancy was explained by Tabor (1977), who identified a parameter 3 Δγ 2 / 3 / E * 2 / 3 \e governing the transition from the Bradley pull-off force 2π R Δ|γ to the JKR value (3/2)π R Δγ. Subsequently Muller et al. (1980) performed a complete numerical solution in terms of surface forces rather than surface energy, (combining the Lennard–Jones law of force between surfaces with the elastic equations for a half-space), and confirmed that Tabor9s parameter does indeed govern the transition. The numerical solution is repeated more accurately and in greater detail, confirming the results, but showing also that the load–;approach curves become S-shaped for values of μ greater than one, leading to jumps into and out of contact. The JKR equations describe the behaviour well for values of μ of 3 or more, but for low values of μ the simple Bradley equation better describes the behaviour under negative loads.

Thermal Analyses for Electrophotographic Toner During Fusing.

A discrete particle model for simulating an accurate temperature field for a toner particle packed bed during fusing has been proposed. Several toner particle arrays consisted of one to three particles are separately built on a paper surface. Particle-particle and particle-paper are thermally connected via contact resistance. Thermal contact conditions are theoretically examined for obtaining the contact resistance. In the examination, a contact area is derived from electrostatic force acting to the particle and Hertz's equation. The calculations of the temperature field have been carried out by a finite difference method. The calculated result, under a condition that the contact resistance decreases during fusing, indicates excellent agreement with experimental result. It is revealed that the contact resistance and its decrease during fusing play an important role for the heat conduction in the toner packed bed, which has been less considered in the current analysis.

Numerical computation of free boundary problems in elastohydrodynamic lubrication

An alternative algorithm has been developed for computing the behavior of thin fluid films in two elastohydrodynamic lubrication problems. The presence of elasticity, lubrication, and cavitation leads to a nonlinear coupled system of partial differential equations. The hydrodynamic part of both problems is governed by the well-known Reynolds equation combined with the cavitation model of Elrod-Adams, which motivates the appearance of a free boundary. Elastic deformations are taken into account by means of the Hertz equation in rolling ball contact problems or the elastic hinged plate biharmonic equation in the case of journal-bearing devices with thin bearing. A numerical method decoupling the hydrodynamic part of the problem and the elastic one is presented. This method also involves an upwind scheme to discretize the lubrication model and finite element approximations.

Viscoelastic characterization of materials using a dynamic hardness tester

The Shore or IRHD hardness test has been widely used for many years and has become one of the standards for characterizing materials in the rubber and plastics industries. The obtained hardness scale values, ASTM or IRHD, are often related to the static elastic properties of rubber‐like materials. The classical Hertz equation has also been used to correlate the hardness scale values to the static elastic properties. Recently, progress has been made on the problem of obtaining dynamic viscoelastic properties through the use of a vibrating spherical indentor and modeling the measured input impedance by the radiation impedance of a sphere radiating into a half‐space. In the limiting case where the sphere radius is small, such that ka≪1 for longitudinal waves, it is shown that the measured shear modulus and loss factor are in good agreement with that found using other dynamic methods such as the DMTA or the resonance method. The model is most appropriate for relatively soft materials or very thick hard materials where the thickness of the material can be ignored. The problem of very hard materials and finite thickness where the terminating boundary is important will be discussed.

The Normal Approach and Stick-Slip Phenomena at the Interface of Two Rough Bodies

A solution for the dry rolling/sliding contact of rough bodies under an applied normal load has been obtained. Both the deflection of the asperities and the bulk material are considered. The conservation of volume approach was used to model elastic-plastic deformation of the asperities. The bulk material was allowed to deform elastically. The stick-slip phenomen of the rolling contact has also been investigated, including asperity deformation and wear as a function of time. The contact width between the two rough cylinders is shown to be approximately 20 percent larger than the width predicted for smooth surfaces by the Hertz equations for the same loading conditions. Similarly, the maximum surface pressure on the bulk material decreases approximately 25 percent when rough surfaces are considered.

Elastic modulus determined by Hertzian indentation

A new method for determining the elastic modulus from Hertzian contact experiments has been developed. The test surfaces are polished and coated with a thin layer of gold/palladium and then indented with a small sphere. The contact removes the coating from the contact area and the size of this can, therefore, be readily measured in an optical microscope. A simple analysis based on Hertz equations is then used to derive the modulus. The elastic modulus of glass and several ceramics is reported.

Optimization of work roll taper for extremely-thin strip rolling.

A simulation model was developed for analyzing the ''roll-kiss'' phenomenon, i.e., a situation where the upper and lower work rolls contact each other in the roll barrel. As a result of the analysis, a theoretical method of providing taper on the work rolls was found. The applicability of the method was experimentally confirmed in a laboratory mill, and the following findings were obtained.(1) The accuracy of the load of roll-kiss calculation by Hertz's equation proved satisfactory for the practical use.(2) Analysis using the simulation model revealed two different configurations of roll-kiss.(3) The experimental results showed that for both configurations of roll-kiss, it is possible to prevent roll-kiss, by giving work rolls the optimum taper predicted from the simulation model.

Some properties of Hertz vectors for source-free electromagnetic fields

Abstract In this paper we discuss some properties of solutions of the Hertz equation for source-free electromagnetic fields. First we give an operational form of the solution at any time in terms of the field at a particular time. Then we discuss the use of the Hertz equation to obtain conservation laws for various expressions involving the field vectors. Finally we discuss the variational formulation of the Hertz equation giving a Lagrangian and two possible Hamiltonians.

Effect of imposed strain and moisture content on some viscoelastic characteristics of cowpeas ( Vigna unguiculata)

Cowpeas ( Vigna unguiculata) were equilibrated at relative humidities of 33, 57, 64·5, 67, 75, 79 and 86% over saturated salt solutions at 25°C. Uniaxial compressive loading and relaxation tests were done on a Universal testing machine at these relative humidities. The modified Hertz equation was employed to calculate the modulus of elasticity and the viscoelastic parameters. The modulus of elasticity of cowpeas was found to be independent of the imposed strain levels up to 2%. It was also found that the viscoelastic behaviour of the cowpea samples tested can be represented by two Maxwell models in parallel.

Contact width and compliance between cylinders and rough plates

The contact mechanism between a cylinder and a rough plate is theoretically analysed for mixed, elastic and plastic contacts of asperities. The analysis leads to the result that the contact pressure, the contact width and the compliance between the cylinder and plate differ considerably from those calculated from the Hertz equation and the Lundberg equation when the surface roughness in contact is greater and the normal load is lower. It is also found that the difference between the calculated contact width and the compliance based on mixed asperity contacts and those based on elastic or plastic asperity contacts is small. To confirm the analysed results, the contact width between the cylinder and the rough steel or rough copper plate was measured by means of evaporated carbon and lamp black film coatings on the rough surfaces. The compliance between the surfaces was also measured using differential transformers. Little difference was found between the analysed results and the experimental results.

Energy losses due to elastic wave propagation during an elastic impact

During the impact of an elastic sphere on an elastic, massive, substrate some of the kinetic energy is radiated into the massive substrate as elastic waves and is not available for subsequent recovery as kinetic energy. This loss of recoverable energy then determines the maximum possible value for the coefficient of restitution for any impact. In evaluating the radiated energy loss Hunter (1957) used an approximation to the displacement-time relationship in a Hertzian impact so as to obtain an analytic solution to the problem. However, in doing so he derived a force-time relationship which had a different power dependence from that of the Hertz equations. In this paper the energy loss is re-evaluated using Hunter's approximation for the displacement-time relationship but applying it in such a way as to maintain similarity with Hertz equations. This re-evaluation predicts the energy loss as being some 4.5 times greater than predicted by Hunter's analysis. It is shown that the present analysis results in a generally better fit to experimental data.

Dispersion Forces and Plastic Deformation in Tablet Bond

A model for tablet bond formation is developed. It assumes that attraction forces arise from dispersion forces between surfaces with spherical curvature. The radius of these surfaces depends on the amount of plastic deformation produced during compression. The radius of curvature is calculated by using Hertz's equation for elastic deformation, which is assumed to hold during decompression. The number of contact points per unit cross section is taken as a simple multiple of a number deduced from the particle size, the indentation hardness values, and the solid fraction. A similar approach is used to replace the area per contact point produced when under compression. The resulting equation is used to estimate the bonding index. Values corresponding to the lower range of experimentally determined bonding indices are obtained. Arguments are presented to suggest that large values of the bonding index (tensile strength/indentation hardness) probably result principally from plastic (including viscoelastic) deformation during decompression. Nondispersion forces may contribute but are not believed to be of sufficient magnitude to account for the total bond for materials that exhibit a large value of the bonding index.