Elliptical Hertzian Contact Research Articles

Applicability of an Hertzian-type contact model for wheel-rail pairings as seen by an improved post-processing scheme for ultrasonic data

The contact between wheel and rail crucially affects the management of railways, since the vehicle dynamics, safety, and performance, for example, are all dependent on this. Therefore, in this contribution the load-dependence of nominal contact area – a broadly not considered aspect of the wheel-rail contacts within the literature – is addressed. We are applying an ultrasonic technique for the detection of the contact zone and an improved post-processing scheme for the measured ultrasonic reflection data. Accordingly, we found that the nominal contact area shows a power-law dependence on load which only on average is predicted by the elliptical Hertzian contact theory.

On Boussinesq's Problem for a Power-Law Graded Elastic Half-Space on Elliptical and General Contact Domains.

The indentation of a power-law graded elastic half-space by a rigid counter body is considered in the framework of linear elasticity. Poisson's ratio is assumed to be constant over the half-space. For indenters with an ellipsoidal power-law shape, an exact contact solution is derived, based on the generalizations of Galin's theorem and Barber's extremal principle for the inhomogeneous half-space. As a special case, the elliptical Hertzian contact is revisited. Generally, elastic grading with a positive grading exponent reduces the contact eccentricity. Fabrikant's approximation for the pressure distribution under a flat punch of arbitrary planform is generalized for power-law graded elastic media and compared with rigorous numerical calculations based on the boundary element method (BEM). Very good agreement between the analytical asymptotic solution and the numerical simulation is obtained for the contact stiffness and the contact pressure distribution. A recently published approximate analytic solution for the indentation of a homogeneous half-space by a counter body, whose shape slightly deviates from axial symmetry but is otherwise arbitrary, is generalized for the power-law graded half-space. The approximate procedure for the elliptical Hertzian contact exhibits the same asymptotic behavior as the exact solution. The approximate analytic solution for the indentation by a pyramid with square planform is in very good agreement with a BEM-based numerical solution of the same problem.

Analytic contact solutions of the Boussinesq and Cattaneo problems for an ellipsoidal power-law indenter

Based on Galin’s theorem for the indentation of an elastic half-space by a polynomial punch and Barber’s theorem for the determination of the contact area in elastic normal contact problems, an exact contact solution is developed for the frictionless indentation of an elastic half-space by an ellipsoidal power-law punch. Within the Cattaneo–Mindlin-approximation of tangential contacts, the tangential contact problem with friction can be reduced to the frictionless normal contact via the Ciavarella–Jäger principle, based on which also the tangential contact solution for the ellipsoidal power-law indenter is given. All known solutions for special cases (axisymmetric contact, elliptical Hertzian contact) are exactly recovered. A comparison of the pressure distribution obtained analytically for the 4-th power ellipsoidal indenter with a corresponding numerical solution based on the boundary element method shows no noticeable differences. The proposed solution procedure can also be used exactly for an arbitrary finite superposition of ellipsoidal power-law profiles, and, in an approximate sense, is applicable for arbitrary monotonous ellipsoidal profiles.

Subsurface stresses in an elliptical Hertzian contact

The traditional solution for the stresses below an elliptical Hertzian contact expresses the results in terms of incomplete Legendre elliptic integrals, so are necessarily based on the length of the semi-major axis a and the axis ratio k. The result is to produce completely different equations for the stresses in the x and y directions; and although these equations are now well-known, their derivation from the fundamental, symmetric, integrals is far from simple. When instead Carlson elliptic integrals are used, they immediately match the fundamental integrals, allowing the equations for the stresses to treat the two semi-axes equally, and so providing a single equation where two were needed before. The numerical evaluation of the Carlson integrals is simple and rapid, so the result is that more convenient answers are obtained more conveniently. A bonus is that the temptation to record the depth of the critical stresses as a fraction of the length of the semi-major axis is removed. Thomas and Hoersch’s method of finding all the stresses along the axis of symmetry has been extended to determine the full set of stresses in a principal plane. The stress patterns are displayed, and a comparison between the answers for the planes of the major and minor semiaxes is made. The results are unchanged from those found from equations given by Sackfield and Hills, but not previously evaluated. The present equations are simpler, not only in the simpler elliptic integrals, but also for the “tail” of elementary functions.

Digital twin by DEM for ball bearing operating under EHD conditions

In this paper we investigate the elastohydrodynamic (EHD) lubrication in ball bearings by means of a digital twin. Based on an original description involving the discrete element method (DEM), the digital twin integrates all the components of ball bearings and enables realistic behavior under mechanical loading and kinematic conditions. In order to check the standard indicators recommended by most ball bearing manufactures, a stiffness model for elliptical Hertzian contact and an improved EHD formulation for lubricated contact are implemented in the numerical tool. In addition, we have introduced into the discrete modeling an electrical capacitance model correlated to the fluid film thickness and the contact pressure. The numerical predictions of lubricant film capacitance provided by digital twin are in good accordance, both qualitatively and quantitatively, with the experimental data available in the literature. The coupling of the discrete method with the electrical approach enables efficient solutions to be provided in terms of lubrication regime in relation to the lubricant properties to optimize the bearing lifetime.

Numerical Simulation of Wheel-rail Interfaces in Heavy Freight Corridors using Hybrid Rigid-deformable Model

Design and operation of dedicated railway freight corridors require detailed analysis of the arduous loading conditions imposed by trains during heavy load transit. A hybrid rigid-deformable numerical model is formulated for estimating the increase in wheel-rail frictional stresses, contact pressures, equivalent von-Mises stresses and strains produced in the axle, corresponding to an increase in the applied axle load. The hybrid model considers the wheel-rail system to be a combination of deformable wheel-set running on rigid rails, which considerably reduces the computational demands and time associated with the analysis of large number of nodes and elements in the numerical model. Validation of the model was done by analyzing a section of Italian rail UNI-60 and contact pressure results were correlated with the data available from literature. Wheel-rail contact analyses were then performed on Indian Railways IRS-52 track section, for heavy axle loads upto 32.5 tonnes, as proposed by Indian Railways for dedicated freight corridors. A Hertzian elliptical contact patch was observed at the wheel-rail interface. Assessment of contact pressure and frictional stresses were then done at the wheel-rail interfaces for a load range of 20 to 32.5 tonnes, and it was noted that the increase in pressures were non-linear, with a quadratic trend. The variation in stresses and strains induced in the axle during loading could also be estimated, since wheel-set was modelled to be deformable in nature. The model analyses are expected to be useful to the railway authorities for designing rail and wheel sections to cater the intense loading conditions in dedicated freight corridors.

Wheel-rail Interactions in High Speed Railway Networks during Rapid Train Transit

Implementation of high speed railway networks in India requires a thorough investigation of the contact pressure and frictional stresses induced in wheel-rail interfaces during rapid train transit. In this study, 3-D numerical moving load simulations of wheel-rail interactions in broad gauge railway networks in India were performed for fast and heavy train operations, and the trends in variation of contact pressure and frictional stresses with increase in speed of the train were quantified. Simulations were performed in ANSYS, with a hybrid model incorporating a flexible wheel-set running over a couple of rigid rails, which aided in the estimation of frictional stresses at the interface of wheel-rail contact during train run, while maintaining a reduced number of elements associated with lengthy railway track sections. Analyses were done for train operations ranging from maximum speeds of 160 to 200 km/h and an axle load of 32.5 tonnes, as recommended by Indian Railways for operation of freight carriers in the existing broad gauge tracks. It was noted that with increase in speeds of trains, there was a reduction in the area of wheel-rail contact. The elliptical Hertzian contact patch of the static train wheels were found to transform into a more rounded patch during rapid train transit, causing concentration of stresses at the centre and effectively increasing the maximum stresses at the points of contact. Contact pressure and frictional stresses were found to increase quadratically, with an increase in the speed of the train. It was ascertained that the rate of wheel and rail wear during high speed train operations would be significant, and that high speed networks would require new wheel and rail sections, and the adoption of materials with very high frictional wear capacity.

An efficient model of load distribution for helical gears with modification and misalignment

An efficient model is introduced for evaluating the load distribution of helical gears, considering tooth modifications and misalignment errors. Instant contact points are obtained by unloaded meshing simulation. Combined with the full numerical method for elliptical Hertzian contact, the contact line under load is determined. Then the load distribution is derived from the minimization of potential energy. The proposed model is numerically implemented by a fixed-point iteration method based computational scheme, assuring high efficiency and superior versatility for tackling modifications and misalignments. And it is verified by finite element analysis. The effects of profile crowning, lead modification, misalignment and input torque on load distribution are investigated. Results indicate that contact pattern shrinks with increasing magnitude of profile crowning and decreasing input torque, resulting in abrupt load transitions between two and three meshing tooth pairs. While lead modification can transfer load from the edge of tooth surface to the center by inclining contact pattern to longitudinal direction, providing desirable load distribution. In addition, misalignment deviates contact pattern and sharing load respectively towards the ends of tooth profile and meshing cycle.

A Novel Modeling Approach to Simulate Rolling Contact Fatigue and Three-Dimensional Spalls

In this study, a new approach has been developed to simulate three-dimensional (3D) experimental rolling contact fatigue (RCF) spalls using a two-dimensional (2D) finite element (FE) model. The model introduces a novel concept of dividing the 3D Hertzian pressure profile into 2D sections and utilizing them in a 2D continuum damage mechanics (CDM) RCF model. The distance between the two sections was determined by the size of the grains in the material microstructure. The 2D RCF model simulates characteristics of case carburized steels by incorporating hardness gradient and residual stress (RS) distribution with depth. The model also accounts for the topological randomness in the material microstructure using Voronoi tessellation. In order to define the failure criterion for the current model, sub-surface stress analysis was conducted for the Hertzian elliptical contact. It was predicted that the high shear stress region near the end of the major axis of the contact is the cause of catastrophic damage and spall formation. This prediction was validated by analyzing the spalls observed during RCF experiments using a surface profilometer. The model was implemented to predict RCF lives for 33 random material domains for different contact geometry and maximum Hertzian pressures. The model results were then compared to the RCF experiments conducted on two different test rigs, a three-ball-on-rod and a thrust bearing test apparatus (TBTA). It was found that the RCF lives obtained from the model are in good agreement with the experimental results. The results also demonstrated that the spalls generated using the analytical results resemble the spalls observed in experiments.

Design of dimpled engineering surfaces for improving lubrication performance in rolling-sliding contacts

Functional performance depends largely on the surface characteristics of critical components. This paper describes the design of dimpled engineering surfaces and the fabrication of micro-patterns on steel samples. The designed engineering surfaces are analysed and evaluated in terms of their mixed EHL film thicknesses and pressure distributions in the elliptical Hertzian contact zone.

High accuracy complete elliptic integrals for solving the Hertzian elliptical contact problems

Complete Elliptic integrals are widely used for solving different types of real problems. For instance, it is required by the Hertzian elliptical contact model, which is the fundament of contact mechanics and establishes the relationship between the stress and the deformation for two elastic solids. Since there is a singular point in the definition domain, traditional approaches often behave badly there. Moreover, no research has noticed the importance of the IEEE quadruple-precision standard on the high accuracy calculation of elliptic integrals. This paper presents a fast high accuracy method that incorporates the 128-bit quadruple-precision floating-point (ISO/IEC/IEEE 60559:2011, IEEE Std 754-2008). The method is used for solving the Hertzian elliptical contact problems. The experiments show that the method can achieve very high accuracy results.

A numerical and effective method for the contact stress calculation of elliptical partial slip

A general elliptical Hertzian contact, such as a ball rolling in a non-conforming groove or two non-orthogonal cylinders, is a universal phenomenon in engineering. A partial elliptical contact is formed by contact bodies under a normal force and a transverse force that is insufficient to cause complete sliding. A computational model based on the semi-analytical method is established to calculate the von Mises stress distribution. Furthermore, a parametric study on factors influencing the location of the maximum von Mises stress point, such as the coefficient of friction and stick zone ratio, is conducted. Results show that the location of the maximum von Mises stress point depends on the coefficient of friction f and stick zone ratio c. The von Mises stress increases significantly with the increasing coefficient of friction f. Moreover, the maximum von Mises stress point tends to occur at a subsurface when the stick ratio c increases.

On the mechanism of squat formation on train rails – Part I: Origination

A phenomenological investigation of squat defects on rail grade R260Mn is performed. The surface-breaking crack pattern, which is either linear or branched (V-shaped), shows a typical position and orientation in the running band. Particular characteristics are asymmetry of this pattern, with the presence of a leading and a trailing branch, and crack reflection or deviation at the running band border. Bending tests reveal a 3D internal crack pattern, with a pair of crack planes or ‘wings’ enclosing a wedge at the surface. Microstructural analysis of the rail upper layer shows metallurgical principles of crack initiation: delamination and transverse fracture of white etching material at the surface. This analysis moreover reveals a 3D anisotropic texture of the upper layer under combined bi-directional tangential surface stresses. Mechanical interpretation of the crack morphology shows that the leading or single branch of the surface-breaking crack pattern is a shear-induced fatigue crack, following the anisotropic microstructure when growing into the rail. The trailing crack of a branched squat is explained as the result of a subsequent transverse, wedge-shaped brittle failure mechanism of the surface layer of the rail, developing within the actual elliptical Hertzian contact patch – or the envelope of potential contact ellipses at the leading crack position. It is driven by the transverse shear loading towards the rail gauge face.

Contact Stress Distribution of Large Diameter Ball Bearing Using Hertzian Elliptical Contact Theory

The large diameter bearings (diameter >400mm) 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.

Finite element analysis of nanowire indentation on a flat substrate

Abstract

Finite element modelling of the rail gauge corner and underhead radius stresses under heavy axle load conditions

Highly localized stresses generated at the gauge corner of the outer rail cause fatigue cracking. The longitudinal bending stresses in the rail underhead radius position are of special interest, as tension spikes have been identified at this location during in-track measurements under high axle load conditions. The tension spike is a result of vertical and lateral head bending on the web. This effect is highly localized and is additional to the stresses generated due to vertical and lateral bending of the whole rail profile (the so-called global bending). This study examined contact and bending stresses by modelling the rail on a discrete foundation using finite elements and by considering the loading to be a fully slipping elliptical Hertzian contact patch. The analysis revealed that the tensile longitudinal stress was highly dependent on several service conditions: the contact patch offset from the rail centreline, the ratio of lateral ( L) to vertical ( V) loads, the direction of lateral traction, foundation stiffness and seasonal temperature variations. The tension spike increases, and the depth below the contact surface at which the stresses become tensile reduces, as the contact patch offset and/or the L/ V ratio increases. Both these enhance the tendency for rolling contact fatigue cracks to turn downwards and become transverse defects. This is because an increase in tensile bending stresses together with both residual and thermally induced stresses can facilitate cracks to turn perpendicular to the tensile stresses once they reach a critical length. Shear traction towards the gauge corner was found to be the most damaging phenomenon.

Rapid method of stress intensity factor calculation for semi-elliptical surface breaking cracks under three-dimensional contact loading

Fast methods of stress intensity factor calculation for inclined surface breaking cracks under contact loading are presented. Cracks are loaded in normal and tangential traction by a three-dimensional Hertzian elliptical contact patch, and friction between the crack faces is considered. Stress intensities are calculated from Green's functions originally developed for two-dimensional cracks through application of stresses on a plane below the three-dimensional contact patch, in place of those previously considered for a two-dimensional contact. This approach gives the method great speed advantages over fully three-dimensional methods. Both semi-circular and semi-elliptical cracks are examined. The validity of the approximations and the results are judged by validation with results from alternative fully three-dimensional cases. Very good agreement is found between trends in stress intensity factor with changes in crack size and applied tractions. Absolute values of stress intensity factor agree well for semicircular and shallow semi-elliptical cracks, but values were below those of the reference case for deep, narrow semi-elliptical cracks. Calibration of the model to overcome this under-prediction is discussed. A case of special value in railway rail-wheel contact modelling is that of a contact offset to the side of a crack, representing the wheel running alongside rather than directly across an existing crack. This configuration results from the common procedure of grinding the rail to change or maintain its cross-sectional profile. The three-dimensional contact patch methods presented here enable this case to be modelled while retaining very fast running times for the calculations.

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.

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.

The elliptical Hertzian contact of transversely isotropic magnetoelectroelastic bodies

First, the general solution for transversely isotropic magnetoelectroelastic media is given concisely in form of five harmonic functions. Second, the extended Boussinesq and Cerruti solutions for the magnetoelectroelastic half-space are obtained in terms of elementary functions by utilizing this general solution. Third, the coupled fields for elliptical Hertzian contact of magnetoelectroelastic bodies are solved in smooth and frictional cases. At last, the graphic results are presented.