Elliptical Contact Research Articles

Elliptic Elastic Contact Between High Order Symmetrical Surfaces

This paper proves that a generalized Hertz pressure (the product of Hertz square root and an even polynomial of degree 2n with respect to coordinates) applied over elastic half-space boundary generates a polynomial normal displacement of degree 2n+2. Polynomial surface coefficients are combinations of elliptical integrals. The equation of rigid punch surface generating this pressure is derived, as well as the conditions in which an elliptical contact occurs. For second order surfaces, n=0, these results yield all Hertz formulas, whereas new formulas are derived for contact parameters between fourth, sixth, and eight order surfaces.

A study of the mechanical properties of nanowires using nanoindentation

A nanoindenter XP with scanning capabilities was used to perform nanoindentations on GaN and ZnO nanowires with radii in the range of 20–50nm, positioned on a silicon substrate and bonded to the substrate at their ends with platinum. Since the geometry of indentation of a nanowire differs significantly from the indentation of a half-space, the standard Oliver-Pharr method [W. C. Oliver and G. M. Pharr, J. Mater. Res. 7, 1564 (1992)] of analysis may not be used. A two interface contact model has been developed for the nanoindentation of a nanowire on a flat substrate, with the two interfaces, indenter/nanowire and nanowire/substrate, being in a series. The contact at the indenter/nanowire interface is modeled as an elliptical contact at the sphere (indenter)/cylinder interface. The contact at the nanowire/substrate interface is modeled as a contact at the cylinder/half-space interface under some concentrated forces applied on top of the cylinder. Under these latter conditions the cylinder may be expected to recede from the half-space when the load is applied. In order to predict the contact stiffness for the two interfaces, the theories of Hertzian contacts and receding contacts have been reviewed, generalized, and used. Considering the possible adhesion at the nanowire/substrate interface and the fixed ends of the nanowire, we have considered two limits for the contact at the nanowire/substrate interface: one with and one without separation at the interface; thus, we obtain two bounds for the contact stiffness and hardness. The model has been used to analyze the nanoindentation data for GaN and ZnO nanowires. We found that the hardness of the GaN nanowire is 46.7±5.6GPa, which is much higher than that of the ZnO nanowire, 3.4±0.9GPa. We also found that the Oliver-Pharr hardness [W. C. Oliver and G. M. Pharr, J. Mater. Res. 7, 1564 (1992)] may be the rough lower bound of the hardness and the Joslin-Oliver hardness [D. L. Joslin and W. C. Oliver, J. Mater. Res. 5, 123 (1990)] may be the rough upper bound of the hardness.

Load Distribution in Spiral Bevel Gears

Abstract A new approach for the computerized simulation of load distribution in mismatched spiral bevel gears with point contact is presented. The loaded tooth contact is treated in a special way: it is assumed that the point contact under load spreads over a surface along the “potential” contact line (Simon, 2006, Mech. and Machine Theory, in press), which line is made up of the points of the mating tooth surfaces in which the separations of these surfaces are minimal, instead of assuming the usually applied elliptical contact area. The bending and shearing deflections of gear teeth, the local contact deformations of mating surfaces, gear body bending and torsion, the deflections of supporting shafts, and the manufacturing and alignment errors of mating members are included. The tooth deflections of the pinion and gear teeth are calculated by the finite element method. As the equations governing the load sharing among the engaged tooth pairs and load distribution along the tooth face are nonlinear, an approximate and iterative technique is used to solve this system of equations. The method is implemented by a computer program. By using this program the load and tooth contact pressure distributions, the angular displacements of the driven gear and the stresses in the pinion and gear teeth are calculated. The influence of design data and transmitted torque on load distribution parameters and fillet stresses is investigated and discussed.

On the Dang Van fatigue limit in rolling contact fatigue

Recently, various methods have been proposed to assess the risk of rolling contact fatigue failure and in particular, the Dang Van multiaxial fatigue criterion has been suggested in a simple approximate formulation by Ekberg, Kabo and Andersson. In a recent note by Ciavarella and Maitournam, it was found that the approximation is only valid in a restricted range of cases. Here, a much larger range of conditions including elliptical contact and partial slip conditions are considered and analytical formulae are also derived. The Ekberg, Kabo and Andersson calculation is shown to be a good approximation only for nearly circular contacts, high Poisson's ratio and high Dang Van constant. The Dang Van fatigue limits are very high, particularly for line contact: however, under those conditions ratcheting deformations also are likely to occur unless perhaps for very hard materials showing cyclic yield limit much higher than fatigue limit (these materials in turn could then show very low wear and be prone to fatigue crack propagation). Classical findings about the RCF fatigue suggest nearly twice higher fatigue limit in point contact with respect to line contact conditions, and this is apparently in contradiction to the Dang Van criterion. A possible qualitative explanation is that in point contact above elastic shakedown there is a regime of cyclic plasticity, rather than the direct transition to ratcheting regime as in line contact. However, the Dang Van criterion has been found to be possibly too conservative under RCF also by other authors recently, and further investigations are required.

The directional contact distance of two ellipsoids: Coarse-grained potentials for anisotropic interactions

We obtain the distance of closest approach of the surfaces of two arbitrary ellipsoids valid at any orientation and separation measured along their intercenter vector. This directional distance is derived from the elliptic contact function. The geometric meaning behind this approach is clarified. An elliptic pair potential for modeling arbitrary mixtures of elliptic particles, whether hard or soft, is proposed based on this distance. Comparisons with Gay-Berne potentials are discussed. Analytic expressions for the forces and torques acting on the elliptic particles are given.

A simplified elliptic model of rough surface contact

Nayak's analysis of isotropic surface roughness as a random field has been extended to show that most summits are only mildly ell, the most common ratio of principal summit curvatures being near 2:1. Also from Nayak's theory, the distribution of the geometric-mean summit curvature with height has been obtained. By using an approximate solution for elliptical Hertzian contacts based on the geometric-mean summit curvature, the full elliptical solution of Bush, Gibson and Thomas can be reproduced more conveniently. Their values for the area of contact are accurately reproduced, and it is argued that the present values for load and contact pressure are more plausible: unlike the original numerical values, the present values converge smoothly to the BGT asymptote p ¯ ∼ Ω ≡ E ∗ m 2 / π . Once again, it is found that elastic contact models can explain the proportionality between contact area and load, although at realistic loads the proportionality is merely very good, not exact. The model shows that a plasticity index ψ m ≡ ( E * / H ) σ m (closely related to Mikic's index) can be used to predict the behaviour of surfaces in contact.

Pressure of a paraboloidal die on a thin elastic layer

The nonclassical contact problem for an elasticlayer is analyzed under the assumption that the thick-ness of the layer is small as compared to the a prioriunknown contact area. The asymptotic behavior isfound for the internal stressed state. Explicit expres-sions are derived for the boundary layer and variation ofthe elliptic contact domain. The asymptotic behavior isfound for the contact pressure and its resultant.FORMULATION OF THE PROBLEMLet us constructively consider the nonlinear prob-lem concerning the pressure of an elliptic-paraboloidaldie on an elastic layer with thickness

Film pressure distributions in point contacts predicted by thermal EHL analyses

In general, lubricated rolling/sliding contact fatigue problems have been investigated by assuming Hertzian contact pressure distributions. In this paper, thermal EHL analyses in consideration of the variations in oil properties in all directions within the film have been carried out under conditions of circular and elliptical contacts. It has been found that the actual film pressure distributions differ markedly from the Hertzian pressure distribution depending on the thermal conductivities of both contacting surfaces, slide–roll ratios and viscosity–pressure coefficients.

Shakedown analyses for rolling and sliding contact problems

There is a range of problems where repeated rolling and sliding contact occurs over a half space of an elastic–perfectly plastic material. For such problems shakedown and limit analysis provide significant advantages over other forms of analysis when a global understanding of deformation behaviour is required. In this paper, a recently developed numerical upper bound method, the Linear Matching Method (LMM), for shakedown analyses is applied to the solution of a problem previously considered by Ponter et al. [Ponter, A.R.S., Hearle, A.D., Johnson, K.L., 1985. J. Mech. Phys. Solids 33 (4), 339–362] for a moving Hertzian contact, with sliding friction. This semi-analytic solution is an upper bound based on certain specific kinematic assumptions. We show that the Ponter, Hearle and Johnson solution is a reasonable approximate solution for a circular contact area but is less accurate for an elliptic contact area. For an elliptic contact area LLM solutions converge to the line contact solution. The effect of the non-coincidence of the direction of travel and slide is also investigated.

Adhesive contact between an elliptical rigid flat-ended punch and an elastic half space

The adhesive contact between an elliptical rigid flat-ended punch and an elastic half space was analysed in this study. Non-self-similar growth in the contact area was obtained. The separation was found to initiate at the edges of the major axis, which would lead the elliptical contact to a circular contact. The pull-off force for the initiation of separation increases from the slim elliptical contact to the circular contact and is proportional to the square root of the interfacial energy and shear modulus of the elastic material. A relation between the pull-off force and the corresponding punch displacement was obtained, which is independent of the elastic properties of elastic materials.

An approximate JKR theory for elliptical contacts

JKR theory describes how surface energy modifies the Hertz theory of contact between elastic solids, giving rise to a larger contact area for positive loads, and to a finite tensile load 1.5πRΔγ in order to separate the solids. Here the original theory for circular contacts is extended to the general elliptical Hertzian contact. To a good approximation, the contact area remains elliptical, but the eccentricity varies continuously with the load, and at pull-off is substantially less than the Hertzian value.

Transient elastohydrodynamic analysis of elliptical contacts. Part 2: Thermal and Newtonian lubricant solution

Transient thermal elastohydrodynamic lubrication (EHL) of general elliptical point contacts was investigated numerically in this study. Both entrainment directions along the major and the minor axes of the contact ellipse were considered, together with a transient load impulse. In this study, a Newtonian lubricant was assumed to highlight the thermal influence. The transient solution was achieved at every instant, starting from a steady state thermal EHL solution. At each instant, a multilevel solver was used for pressure and surface deformation, whereas a column-by-column relaxation technique was used for solving temperature. The unknown rigid central distance between the contact bodies was adjusted after each iteration between the transient fields of pressure and temperature, so that in each iteration, only one W cycle was required for pressure and only a few relaxation cycles were required for temperature. With these numerical techniques, the computing time required for a typical transient case was reduced to ∼ 12 h on a personal computer with a 3.0 GHz central processing unit. The transient thermal results were compared with those corresponding to isothermal conditions presented in Part 1 of this series of papers. It was found that, in general, the transient behaviour under thermal conditions was similar to that under isothermal conditions, however, the former was weaker than the latter when the slide-roll ratio was large enough.

An extension of the Hertz theory for three-dimensional coated bodies

This paper presents a work on extending the Hertz theory for circular and elliptical point contact problems involving coated bodies. The extended form of the Hertzian formulae are adopted to express maximum contact pressure, contact radius, and contact approach in terms of applied load, equivalent radius, and an extended equivalent modulus that properly considers the presence of a coating. The extended equivalent modulus is a function of Young’s moduli and Poisson’s ratios of the coating and the substrate, coating thickness, and a parameter, which is obtained through substantial numerical simulation. The extended Hertzian formulae are easy to use and give accurate predictions of contact characteristics.

Analysis of Mixed Lubrication Effects in Simulated Gear Tooth Contacts

The paper presents results obtained using a transient analysis technique for point contact elastohydrodynamic lubrication (EHL) problems based on a formulation that couples the elastic and hydrodynamic equations. Results are presented for transverse ground surfaces in elliptical point contact that show severe film thinning and asperity contact at the transverse limits of the contact area. This thinning is caused by transverse leakage of the lubricant from the contact in the remaining deep valley features between the surfaces. A comparison is also made between the point contact results on the entrainment center line and the equivalent line contact analysis. The extent of asperity contact is shown to be dependent on the Hertzian contact aspect ratio. It is also shown that transverse waviness (superimposed on the roughness) of even relatively small amplitude can lead to large increases in asperity contact rates over all waviness peaks in the contact.

Elliptic crack in a transversely isotropic body revisited: New symbolism

It is shown that the introduction of two new parameters, similar to those used by the author earlier, allows to simplify dramatically the complete solution of the elliptic crack problem. We consider the cases of both normal and shear loading of the crack. The limiting case of an isotropic body is also considered. The developed results can be used for solving various elliptic contact problems as well.

Transient elastohydrodynamic analysis of elliptical contacts. Part 1: Isothermal and Newtonian lubricant solution

Transient elastohydrodynamic lubrication of general elliptical point contacts was analysed in the present study based on multi-level techniques. Both entrainment directions along the major and minor axes of the contact ellipse were considered, together with transient load and speed impulses. In this study, only a Newtonian lubricant and isothermal conditions were assumed. The computing time required for a typical case took only approximately 2 h on a personal computer with a 2.8 GHz central processing unit. The predicted film thickness and the pressure were found to be mainly governed by the squeeze-film or separation actions. A sudden load increase resulted in an increase in the contact conjunction and a significant development of the squeeze-film action. Consequently the lubricant film thickness was increased initially around the inlet region and then moved along in the entraining direction towards the exit. A significant increase in the predicted central film thickness was observed, long after the load impulse had ceased. Such a process was found to be mainly governed by the time period of the impulse, and the time taken for the lubricant to move through the contact conjunction in the entraining direction. The speed impulse had similar effects. However, a sudden speed decrease was found to have negligible effect on the predicted film thickness, due to the powerful squeeze-film action.

“The elliptical Hertizian contact of transversely isotropic magnetoelectroelastic bodies” by Hou et al., Vol. 40, 2003, pp. 2833–2850

“The elliptical Hertizian contact of transversely isotropic magnetoelectroelastic bodies” by Hou et al., Vol. 40, 2003, pp. 2833–2850

Starved Lubrication of Elliptical EHD Contacts

This paper focuses on the lubrication behavior of starved elliptical Elasto-HydroDynamic (EHD) contacts. Starvation is governed by the amount of lubricant available in the inlet region and can result in much thinner films than occurring under fully flooded conditions. Therefore, it would be desirable to be able to predict the onset and severity of starvation and to be able to relate film reduction directly to the operating conditions and lubricant properties. The aim of this work is to explore the influence of these parameters on starvation. A combined modeling and experimental approach has been employed. The numerical model has been developed from an earlier circular contact study [1]. In this model, the amount and distribution of the lubricant in the inlet region determines the onset of starvation and predicts the film decay in the contact. Numerical simulations for a uniform layer on the surface show that a single parameter, characteristic of the inlet length of the contact in the fully flooded regime, determines the starved behavior. Film thickness measurements under starved conditions were performed to validate this theory. For a circular contact excellent agreement was found. In theory the same mechanism applies to elliptic contacts, however, the behavior is more complicated.

Hertzian contact of orthotropic materials

Contact loading of orthotropic materials is of interest in impact problems and other applications with composite materials. Although solutions are available for special cases such as transversely isotropic materials, solutions for contact loading for more general orthotropic materials have appeared much less frequently in the literature. The present work shows that a general procedure for calculating stresses due to contact loading can be obtained by combining two previous solution techniques. The first is the procedure outlined by Willis, that uses numerical contour integration to determine the size and aspect ratio of the elliptical contact area, and the contact pressure distribution. Detailed stress fields are then obtained by using these parameters in the general solution for transverse pressure loading of laminated orthotropic materials due to Pagano and Srinivas and Rao. Comparisons with known results for special cases, along with comparisons of surface displacements and the indentor profile, indicate that good accuracy of the solution can be achieved.

Transient elastohydrodynamic point contact analysis using a new coupled differential deflection method Part 2: Results

The paper presents results obtained using a transient analysis technique for point contact elastohydrodynamic lubrication (EHL) problems based on a formulation that effectively couples the elastic and hydrodynamic equations. Results are presented for transverse ground surfaces in an elliptical contact that show severe film thinning at the transverse limits of the contact area. This thinning is caused by transverse (side) leakage of the lubricant from the contact in the remaining deep valley features. Comparison is made between the elliptical contact results on the entrainment centreline and the equivalent line contact analysis. This confirms the importance of edge effects as a likely cause of film collapse and scuffing failure. The surface profiles used in the analysis are taken from test discs used in scuffing experiments and from gears used in micropitting tests. Side leakage is found to be sufficiently severe to cause microasperity contact in the numerical examples presented. This contact mainly occurs close to the edges of the corresponding Hertzian area and correlates in position with the location at which scuffing is found to first occur in the earlier experiments. Comparisons are made with other numerical results for point contact configurations with sinusoidally varying surface features obtained by Zhu (2000) and considerable differences are seen in the calculated extent of asperity contact. The differences are thought to be due to the simplified treatment of the lubrication equation adopted by Zhu.