Elliptical Hertzian Contact Research Articles

Lateral contact of slender prismatic bodies

Many nominally two‐dimensional contact problems, such as the contact of parallel rollers or spur gears, exhibit slow axial variation in contact conditions due to manufacturing errors, misalignment or generalized beam deflections of the contacting bodies. In this paper, an asymptotic method is used to develop a solution to such problems, the inner problem being the two‐dimensional Hertzian contact theory and the outer problem the bending of the contacting bodies as beams separated by a nonlinear elastic foundation. The method is illustrated in the case of two cylindrical rollers that are slightly misaligned and pressed together by forces applied at their ends. The classical Hertzian elliptical contact area is obtained at low loads, but with increasing load the contact area evolves first into a dog‐bone shape and then into two separated contact areas surrounding the point forces that would be predicted by an elementary beam theory solution of the problem.

The Elastic Field Resulting From Elliptical Hertzian Contact of Transversely Isotropic Bodies: Closed-Form Solutions for Normal and Shear Loading

This analysis presents the elastic field in a half-space caused by an ellipsoidal variation of normal traction on the surface. Coulomb friction is assumed and thus the shear traction on the surface is taken as a friction coefficient multiplied by the normal pressure. Hence the shear traction is also of an ellipsoidal variation. The half-space is transversely isotropic, where the planes of isotropy are parallel to the surface. A potential function method is used where the elastic field is written in three harmonic functions. The known point force potential functions are utilized to find the solution for ellipsoidal loading by quadrature. The integrals for the derivatives of the potential functions resulting from ellipsoidal loading are evaluated in terms of elementary functions and incomplete elliptic integrals of the first and second kinds. The elastic field is given in closed-form expressions for both normal and shear loading.

Analysis of elliptical Hertzian contacts

The paper compares three approximate methods of finding the area of contact, the contact pressure and the deformation in elliptical Hertzian contacts.

Contact of surface asperities in wear

Stresses have been examined under the elliptical Hertzian contact area, which appear during a real contact between two asperities in the process of sliding friction. The stresses are found according to the Mises-Hencky criterion and characterize the ability of the material to change its shape. Relationships have been found of such a maximum stress under the surface of the contact area depending on the geometric parameters of the contact area, on the friction coefficient and the maximum pressure in the centre of contact area. The processing of the calculation data was conducted according to a special program on a computer and yielded an approximate relationship to calculate the maximum reduced stress. The distribution of dangerous stresses under the contact surface has been shown. The most critical points may be both within the surface layer of the material and on the contact area itself.

The Influence of Local Slip in Hertzian Contacts on the Subsurface Elastic Material Stresses of Rolling Element Components

AbstractRolling element bearing components are subjected to high loads. These loads are transmitted from one bearing component to the next through Hertzian contacts. Since the bearing components are in motion as they transmit load, the elliptical contact region is subjected to normal as well as tangential pressure. The direction of the tangential pressure distribution is dependant on the direction of the local slip vectors at each point within the elliptical contact. The purpose of the work presented here, is to study the influence of the local slip vectors in highly loaded elliptical Hertzian contacts. Instead of applying the conventional method of using a constant traction coefficient within the contact region, the author, for the first time, employs a variable traction coefficient depending on the local slip velocity. The paper points out the importance of including traction‐slip relationships in material stress analysis to obtain realistic stress distributions of rolling element bearing components, especially in the mixed lubrication region.

The Stress Field Induced by a General Elliptical Hertzian Contact

Simple, concise formulas are given for the components of stress induced by a general elliptical Hertzian contact in terms of elementary functions. Explicit, readily evaluated formulas are given.

A Simple, Accurate Method for Calculation of Stresses and Deformations in Elliptical Hertzian Contacts

The note describes a simple and highly accurate numerical routine for calculating contact pressure and dimensions in elliptical Hertzian contacts. A FORTRAN implementation of the routine is given.

Propagation of Semi-Elliptical Surface Cracks in Lubricated Rolling/Sliding Elliptical Contacts

In this paper the three-dimensional mixed mode stress intensity factors for semi-elliptical inclined cracks formed on elliptical Hertzian contact surfaces are calculated. This analysis accounts for both the frictional forces between the contact surfaces and between the crack faces, as well as including fluid pressure acting on the crack faces. Various factors affecting the crack growth are discussed and possible growth modes are investigated. In particular, the analytical results predict that the crack has a faster growth rate when the major axis of the contact ellipse is in the direction of motion of the load as compared to the minor axis being in the direction of motion of the load.

Rolling contact fatigue testing of PM high speed steels (HSS): J A Whitehead, et al. (Powdrex Ltd, tonbridge, Kent, UK), Int. J. Powder Metallurgy, Vol 26, No 4, 1990, 345–349

The large diameter bearings (diameter >400 mm) are of great importance in complex engineering mechanisms such as Aircraft gas turbines, Rolling Mills and Nuclear Reactor etc, in which varying load and critical environmental conditions leads to the failure of such bearings. Majorly spalling failure occurs due to contact mechanism. The contact mechanism of ball and raceway of bearing behaves highly non-linear and it reduces the service life of component considerably. This paper presents to determine the contact stress of large diameter ball bearings using analytical and numerical methods. In analytical method the contact stress is found out using the Hertzian Elliptical Contact Theory. The calculation procedure consists of calculation of the maximum contact pressure on the rolling element, which is done by the in-house developed program, and detailed finite element analysis of the contact between the ball and the raceway. The finite element analysis also performed to predict contact pressure in ball and raceway. Comparison of the results is made at the end.

Axial sliding of cylinders

We consider the stress state induced by frictional sliding parallel to the axis of two aligned cylinders pressed together by a normal force. This is the limiting case of a sliding elliptical Hertzian contact, greatly elongated in the direction of relative motion. A generalized form of the result is also presented, which relates to the skew sliding of two dissimilar circular cylinders.

Solution of the Multiple Wheel and Rail Contact Dynamic Problem

SUMMARY An unconventional method for calculating the forces developing in the wheel and rail contact patches of a railway vehicle has been implemented at the New Technology Laboratory of INRETS. It takes into account the elastic deformations of the materials in the Hertzian elliptical contact areas; the possibility of having simultaneously several contact patches on each wheel, is introduced in the simulation of the dynamic phenomena. The theory is applied for a high speed bogie running on a perfectly straight track.

Formulas for Moderately Elliptical Hertzian Contacts

For small ellipticities the Hertzian contact pressure and approach can be obtained to a good approximation by using the formulae for circular contact with an equivalent radius of curvature (AB) −1/2, where A and B are the principal relative curvatures. For higher ellipticities (AB(A + B)/2)−1/3 should be used to find the contact area and pressure; an equivalent radius for finding the approach is also given. The ellipticity of the contact can be estimated from the asymptotic equation (a/b) ≈ (B/A)2/3.

Rheological and Thermal Effects in Lubricated E.H.D. Contacts

Traction in lubricated elliptical Hertzian contacts with colinear speeds is studied. A method is described to calculate the stresses, temperatures and shear strain rates distribution along and across the film. Visco elasto-plastic behavior of the lubricant, convection, and conduction effects and pressure distributions are taken into account in this method. Elastic energy release is discussed. Shear modulus and limiting shear stresses are determined from experiment by curve fitting. Correlation between theory and experiment is analyzed.