Hertzian Pressure Distribution Research Articles

A continuum damage mechanics finite element model for investigating effects of surface roughness on rolling contact fatigue

In this study, a continuum damage mechanics (CDM) finite element (FE) model was developed to investigate the effects of surface roughness on rolling contact fatigue (RCF) life of non-conformal contacts. In order to assess the surface roughness of tribo-components, twelve deep groove rolling element bearings from various companies in different sizes were procured and measured using an optical surface profilometer. The roughness average (Ra) and the root mean square of surface roughness (RMS, σ) ranged from a low of 0.03, 0.05 µm to a high of 0.14, 0.20 µm, respectively. The number of peaks and valleys per 400 µm were measured and calculated. The number of peaks ranged from 11 to 31 (greater than99.5% Confidence Interval). The measured surfaces also revealed that a sinusoidal pattern can be used to accurately represent the surface patterns. The sinusoidal surface pattern was used to determine the elastohydrodynamic lubrication (EHL) pressure distribution between an equivalent rough surface in contact with a smooth surface. Four roughness amplitude were used to generate specific film thicknesses (λ-ratios) resulting in full to mixed EHL lubrication regimes. The EHL pressure distributions were replaced with representative symmetric Hertzian pressure distributions in order to remove the effect of asymmetry of an EHL pressure distribution. The resulting symmetric pressure distributions were used in a finite element continuum damage mechanics model to determine RCF life of machine elements operating in specific film thickness range of 1 < λ < 10. The RCF results from the FE model indicate that as roughness amplitude increases or lambda ratio decreases, the fatigue lives decrease for the various frequencies. Additionally, subsurface failure fatigue lives are reduced as roughness frequency increases regardless of amplitude or Hertzian pressure. The RCF results also indicate that for the low frequency pressure distribution the contact is most susceptible to surface failure, whereas for high frequency pressure distribution the contact resists surface failure. The results from this investigation were used to develop surface roughness effects for various RCF life equations commonly used in rolling element bearing application.

Response to Dr Greenwood\u2019s Comments on \u201cExtending the Double-Hertz Model to Allow Modeling of an Adhesive Elliptical Contact\u201d

© 2018, The Author(s). An adhesive elliptical contact is normally found in microscale applications that involve cylindrical solids, crossing at an angle between 0° and 90°. Currently, only one model is available to describe the elliptical contact’s surface interaction: the approximate Johnson–Kendall–Roberts (JKR) model which is limited to soft materials. In this paper, a new adhesive elliptical model is developed for a wide range of adhesive contacts by extending the double-Hertz theory, where adhesion is modeled by the difference between two Hertzian pressure distributions. Both Hertzian pressures are assumed to have an equivalent shape of contact areas, the only difference being in size. Assuming that the annular adhesive region is obtained by the area difference between the two Hertzian contact areas, the pull-off force curves can be calculated. In the limiting case of an adhesive circular contact, the results are very close to results from the existing models. However, for an adhesive elliptical contact in the JKR domain, lower pull-off forces are predicted when compared to the JKR values. Unlike the developed model, the shape of the JKR contact area varies throughout contact. Results show, particularly for conditions close to the JKR domain, that it is important to take into account that the adhesive region is the result of the two Hertzian contact areas having a non-equivalent shape.

Extending the Double-Hertz Model to Allow Modeling of an Adhesive Elliptical Contact

An adhesive elliptical contact is normally found in microscale applications that involve cylindrical solids, crossing at an angle between 0° and 90°. Currently, only one model is available to describe the elliptical contact’s surface interaction: the approximate Johnson–Kendall–Roberts (JKR) model which is limited to soft materials. In this paper, a new adhesive elliptical model is developed for a wide range of adhesive contacts by extending the double-Hertz theory, where adhesion is modeled by the difference between two Hertzian pressure distributions. Both Hertzian pressures are assumed to have an equivalent shape of contact areas, the only difference being in size. Assuming that the annular adhesive region is obtained by the area difference between the two Hertzian contact areas, the pull-off force curves can be calculated. In the limiting case of an adhesive circular contact, the results are very close to results from the existing models. However, for an adhesive elliptical contact in the JKR domain, lower pull-off forces are predicted when compared to the JKR values. Unlike the developed model, the shape of the JKR contact area varies throughout contact. Results show, particularly for conditions close to the JKR domain, that it is important to take into account that the adhesive region is the result of the two Hertzian contact areas having a non-equivalent shape.

Analytical Modeling of an Oblique Edge Crack in Rolling Contact Fatigue

Surface cracks represent a frequent cause of damage and even failure in rolling contacts, observed in gears, cams, rails, and so on. In the literature, different approaches have been applied to describe the crack behaviour by means of Fracture Mechanics parameters, such as the stress intensity factors (SIFs) and the J-integral. In this paper, a general procedure for dealing with plane problems is presented, which is based on Linear Elastic Fracture Mechanics hypotheses. It combines the Weight Function Method for evaluating the SIFs in a loading cycle with the Kolosov-Muskhelishvili complex variable approach for estimating the nominal stress field. In this way, a completely analytical procedure can be applied for a general loading condition, assuming that the real geometry can be simplified in a half-plane with an oblique edge crack. As test case, a travelling load has been considered representing a combination of three contributions: Hertzian pressure distribution, traction force due to friction, and pressurization of the crack faces. A comparison with literature results proved that the proposed approach can be an efficient tool for SIFs estimation and crack growth description.

Shaping the trajectory of the billiard ball with approximations of the resultant contact forces

This paper presents mathematical models of the contact pressure distribution on a circular contact area and the corresponding rolling resistance. Hertzian pressure distribution is distorted in a special way in order to move rolling centre outside the geometrical centre of the contact area. With the assumption of fully developed sliding and classical Coulomb friction law on each element of the contact, integral models of the total friction force and moment reduced to the contact centre are given. In order to improve the convenience of use of the contact models in numerical simulations of rigid body dynamics and decrease their computational cost, special approximations of the integral models of friction force and moment are proposed. Moreover, special modifications of the corresponding expressions for friction forces and rolling resistance are proposed, which allows avoiding their singularities for vanishing relative motion of the contacting bodies. The application of the proposed contact models in mathematical modelling of a rigid ball rolling and sliding over a deformable table is presented. Furthermore, possibilities of use of the developed simulation models in shaping the billiard ball's trajectory are presented.

Residual Stresses Due to Rigid Cylinder Indentation and Rolling at a Very High Rolling Load

In this paper, residual stresses due to indentation and rolling of a rigid cylinder on a finite plate at a very high rolling load with a relative peak pressure of 22 are examined by two-dimensional plane strain finite element analyses using abaqus for the first time. In the finite element analyses, the roller is modeled as rigid and has frictionless contact with the finite plate. The geometry of the finite plate and its boundary conditions are assigned to correspond to those of fillet rolling of crankshafts with the constraint in the rolling direction. Finite element analyses with different meshes for single indentation on an elastic flat plate under plane strain conditions are first carried out, and the results are benchmarked with those of the elastic Hertzian solutions to establish the requirement of the finite element meshes for acceptable numerical results. The results show that the accuracy of computational results is limited by the discretization of the finite element analysis by a plot of the contact width as a function of the load. For accurate peak pressure, a total of at least eight linear elements are needed. Finite element analyses with different meshes for single indentation on an elastic–plastic flat plate under plane strain conditions are then carried out. The plate material is modeled as an elastic–plastic power-law strain hardening material with a nonlinear kinematic hardening rule for loading and unloading. The computational results are compared to establish the requirement of the finite element meshes for acceptable numerical results within 4 mm distance to the rolling surface for the crankshaft fatigue analyses. The computational results for rolling at the relative peak pressure of 22 show that the symmetric Hertzian or modified Hertzian pressure distribution should not be used to represent the contact pressure distribution for rolling simulation, while the computational results for rolling at the relative peak pressure of 5 show that the symmetric Hertzian or modified Hertzian pressure distribution may be used to represent the contact pressure distribution for rolling simulation. The computational results for the rolling case also show a significantly higher longitudinal compressive residual stress and a lower out-of-plane compressive residual stress along the contact surface when compared to those for the single indentation case. The results suggest that the effects of rolling must be accounted for when two-dimensional finite element analyses of crankshaft sections are used to investigate the residual stresses due to fillet rolling of the crankshafts under the prescribed roller loads. Due to the boundary conditions of the finite plate, the compressive residual stresses are larger when compared to those when the boundary conditions of the finite plate are fully relaxed.

Transducer Misalignment and Contact Pressure Distributions as Error Sources in Friction Measurement on Small-Diameter Pin-on-Disk Experiments

This paper explores the effects of wear track diameter, contact area and pressure distributions (Hertzian and uniform circular) on errors in measuring friction coefficients on rotating pin-on-disk experiments. Integral solutions of the measured friction coefficient are compared to the actual friction coefficient. Errors associated with angular misalignments of the force transducer are also evaluated. It is shown that the errors for small-diameter wear tracks relative to the contact width are lower for sphere-on-flat (Hertzian) contact geometries than circular flat-on-flat contact geometries. Small-diameter experiments result in surprisingly small errors for Hertzian contact pressure distribution. Angular misalignments further collude to increase the error in measured friction coefficient for small track-diameter-to-contact-width ratios, converging to the cosine error for large track-diameter-to-contact-width ratios.

Adhesion between elastic cylinders based on the double-Hertz model

A cohesive zone model for two-dimensional adhesive contact between elastic cylinders is developed by extending the double-Hertz model of Greenwood and Johnson (1998). In this model, the adhesive force within the cohesive zone is described by the difference between two Hertzian pressure distributions of different contact widths. Closed-form analytical solutions are obtained for the interfacial traction, deformation field and the equilibrium relation among applied load, contact half-width and the size of cohesive zone. Based on these results, a complete transition between the JKR and the Hertz type contact models is captured by defining a dimensionless transition parameter μ, which governs the range of applicability of different models. The proposed model and the corresponding analytical results can serve as an alternative cohesive zone solution to the two-dimensional adhesive cylindrical contact.

Development of Mathematical Model for Control Wear in Backup Roll for Hot Strip Mill

The precision of strip flatness depends on several factors; wear of rolls is one of the main variables that have influence on the surface quality of the strip. The wear of the rolls represents a complex friction condition, sometimes the wear in the backup roll is not analyzed because the strip is not in contact with the backup roll; however, after several campaigns of rolling, the wear in the backup roll becomes dangerous because the pressure distribution is not uniform. Investigation of mechanism of the surface deterioration of the backup roll for the hot strip rolling is very important for the development of the automatic strip shape control system used in hot strip mills. A mathematical model is developed considering the Hertzian pressure distribution between two cylinders with parallel axes. It is used in real time for calculating the wear in the backup roll and in this manner take decisions for preventing finished product reworking or damage of equipments, which result in accidents caused by excessive wear in the backup rolls.

Effects of Assumed Pressure Distributions on the Fundamental Relation Used in Nanoindentation Data Analysis

This short note shows that contact stiffness under a non-Hertzian pressure distribution is 33% higher than that under a Hertzian pressure distribution. This is due to the fact that there is no physical indenter that gives a non-Hertzian pressure distribution during elastic contact, and the fundamental relation used in nanoindentation data analysis does not apply.

Development of simple equation for calculating average wear of hot strip mill work rolls

Plastic deformation of the strip, thermal crown of rolls, roll deflection, flattening and roll wear are the main factors that have influence on the surface quality of strip. Roll wear represents complex friction conditions that, after some specific tonnage rolled, become critical. This is because the pressure distribution is not uniform, so the stress between the work and the back-up roll, plus the contact between the strip and the work roll affects the performance of the work rolls. Investigation of the mechanism of the surface deterioration of the work roll is very important for the development of an average wear control system used in real time in hot strip mills. This project has developed a simple equation that considers the Hertzian pressure distribution and mechanical contact between two cylinders with parallel axes in order to calculate the average wear in the work roll.

Influence of Process Parameters on Surface Hardening in Hammer Peening and Deep Rolling

This paper focusses on the statistical evaluation of process parameters in the mechanical surface treatments deep rolling (DR) and machine hammer peening (MHP) on the hardness increase. In MHP process a spherical hard metal tool is repeatedly accelerated onto the material surface. Just as the shot peening process MHP is an impact treatment although in MHP the impact area can be controlled, leading to the desired impact density. In DR the contact between spherical tool and work piece is quite different to MHP as the spherical is in sliding contact as it is moved along the surface. Although the material loading of both surface treatments vary, the resulting surface structure is the same. Both lead to a cold worked, smooth surface including compressive residual. Technically DR and MHP parameters have been part of researches but there still is a lack of statistical validation of every single process parameter leading to a hardened surface. This paper tries to close this gap. DR and MHP are conducted on different materials, containing tool steel 1.2379 and grey cast iron EN-JS-2070. Using a fractional factorial test design an experimental matrix was created able to examine the influence of every single process parameter. Which were for DR: rolling pressure, line spacing between hammer traces, diameter of roller ball and the travelling speed. For MHP the influence of the following process parameters was investigated: angle between hammering direction and surface normal, line spacing between hammering traces, diameter of hammering ball, hammering energy, travelling speed and hammering frequency. On every single sample ten Brinell hardness indents are made which give the statistical coverage needed to calculate the effect of every single process parameter within a confidence interval of at least 95 %. For all mentioned materials the effect of every single process parameter has been calculated with respect to hardening. It could be shown that especially the loading of the cast iron is quiet complex as a high amount of impact energy (MHP) or contact pressure (DR) can lead to overloading of the material leading to a degradation of the surface. At least an explanatory approach which describes the different influence of the tool diameter on the surface hardness is given using FEM simulations. These FEM simulations contain an advanced material model in which the Bauschinger-effect of 1.2379 is implemented. It can be clearly shown that a larger tool diameter in DR produces a higher amount of cold working in the material surface leading to harder surfaces compared to the smaller tool diameter. In contrast to DR the contact pressure in MHP is determined by the Hertzian pressure distribution. Here smaller tool diameters create larger Hertzian pressure and therefore a higher amount of cold working.

Propagation of Sub-surface Cracks in Railway Wheels for Wear-induced Conformal Contacts

The ability to predict the criticality of internal cracks in railway wheels requires accurate knowledge of stress intensity factors KI, KII, KIII under contact loads. These factors depend both on the total load acting on the wheel and on how the load is transmitted through the wheel/rail interface, that is to say, they depends on the pressure distribution between wheel and rail. However, till today the solutions commonly used consider a theoretical (Hertzian) pressure distribution, even though the real contact patch may be far different for most of the lifespan of the wheel due to wear or to the dynamic conditions. In this paper an approach is developed with the aim of solving the case of an internally cracked wheel subjected to an arbitrary contact patch and pressure distribution. In particular, attention is focused on the case of hollow wear, which makes it possible to obtain very conformal contacts. The results are discussed and compared with the analytical solutions that consider Hertzian pressure distributions with the same total load.

A study of multiple crack interaction at rolling contact fatigue loading of rails

The objective of this paper is to investigate geometric and material parameters that affect the interaction of multiple pre-existing (initiated) cracks due to rolling contact fatigue (RCF) loading conditions. In particular, the behaviour of short cracks (head checks) in railway rail is studied. Parameters of interest are initial crack angle and spacing, distribution of initial crack lengths, width of the load/contact zone, and the material properties (in particular, the friction coefficient). Furthermore, the sensitivity of the finite element (FE)-mesh density is investigated. An open question of particular interest is the effect of crack interaction, e.g. shielding, on the prevailing crack spacing. In view of all uncertainties of the material model as well as the three-dimensional geometric complexity of the RCF problem, it is important to obtain a good understanding of the sensitivity of parameter changes. This is achieved in the presented investigation, although it is carried out using simplifying assumptions such as plane strain, linear elasticity, and Hertzian pressure distribution.

347101 PROPAGATION OF SUB-SURFACE CRACKS IN RAILWAY WHEELS FOR WEAR-INDUCED CONFORMAL CONTACTS(Safety,Technical Session)

The ability to predict the criticality of internal cracks in railway wheels requires accurate knowledge of stress intensity factors K_I, K_<II>, K_<III> under contact loads. These factors depend both on the total load acting on the wheel and on how the load is transmitted through the wheel/rail interface, that is to say, they depends on the pressure distribution between wheel and rail. However, till today the solutions commonly used consider a theoretical (Hertzian) pressure distribution, even though the real contact patch may be far different for most of the lifespan of the wheel due to wear or to the dynamic conditions. In this paper an approach is developed with the aim of solving the case of an internally cracked wheel subjected to an arbitrary contact patch and pressure distribution. In particular, attention is focused on the case of hollow wear, which makes it possible to obtain very conformal contacts. The results are discussed and compared with the analytical solutions that consider Hertzian pressure distributions with the same total load.

The Lubrication Regime at Pin-Pulley Interface in Chain CVTs

This paper deals with the strongly nonstationary squeeze of an oil film at the interface between the chain pin and pulley in chain belt continuously variable transmission. We concentrate on the squeeze motion as it occurs as soon as the pin enters the pulley groove. The duration time to complete the squeeze process compared with the running time the pin takes to cover the entire arc of contact is fundamental to understand whether direct asperity-asperity contact occurs between the two approaching surfaces to clarify what actually is the lubrication regime (elastohydrodynamic lubrication (EHL), mixed, or boundary) and to verify if the Hertzian pressure distribution at the interface can properly describe the actual normal stress distribution. The Hertzian pressure solution is usually taken as a starting point to design the geometry of the pin surface; therefore, it is of utmost importance for the designers to know whether their hypothesis is correct or not. Taking into account that the traveling time, the pin spends in contact with the pulley groove, is of about 0.01 s, we show that rms surface roughness less than 0.1 μm, corresponding to values adopted in such systems, guarantees a fully lubricated EHL regime at the interface. Therefore, direct asperity-asperity contact between the two approaching surfaces is avoided. We also show that the Hertzian solution does not properly represent the actual pressure distribution at the pin-pulley interface. Indeed, after few microseconds a noncentral annular pressure peak is formed, which moves toward the center of the pin with rapidly decreasing speed. The pressure peak can grow up to values of several gigapascals. Such very high pressures may cause local overloads and high fatigue stresses that must be taken into account to correctly estimate the durability of the system.

An experimental–numerical approach for the analysis of internally cracked railway wheels

In this paper the problem of the evaluation of the criticality of an internal crack in a railway wheel is dealt with. In particular this paper considers pressure distributions experimentally measured with a technique based on the ultrasonic reflection at the contact interface. A numerical approach is developed to calculate the stress intensity factors of internal cracks under the action of the experimental pressure distributions. Some cases are described and critically discussed: the differences with respect of the corresponding Hertzian pressure distribution are evidenced and the importance of the transversal eccentricity of the crack with respect of the contact patch is underlined.

Numerical solution of elastic bodies in contact by FEM utilising equilibrium displacement fields

In this paper a new effective formulation of the computation of the statics of elastic bodies in contact is described and demonstrated. Line contact, plane strain, and negligible friction are assumed. The formulation is an extension of the standard finite element method (FEM). With the aim to utilise the Hertz theory directly, we use the exact solution of the elastic 2-dimensional halfspace loaded by Hertzian pressure distribution and enforce the contact condition by the method of Lagrange multipliers. In numerical examples we have focussed on the demonstration and evaluation of the accuracy of the new formulation for selected applications compared with the state of the art node-to-segment contact algorithm implemented in the software system ADINA. Proposed formulation is more accurate for problems where Hertz contact dominates the strain state, especially for small number of elements, whereas we obtained a fairly good agreement with ADINA for a more general bending problem.

Annular Component Transient Thermoelastic Analysis Using a State Space Approach

Annular components are used widely in engineering systems and include bearing bushes and races, which may be exposed to extreme operating conditions. A method to establish the localized transient thermoelastic deformation of a homogeneous two-dimensional annular component is developed. The analysis is based on solving the thermoelasticity equations using a state space formulation for the Fourier components of the radial and tangential displacements. Two boundary conditions are considered, namely, rigid and resiliently mounted outer boundaries, both associated with stress free inner boundary conditions. The thermoelastic solution is then demonstrated for a transient temperature distribution induced by inner boundary frictional heating due to rotor contact, which is derived from a dynamic Hertzian pressure distribution. The application is to a relatively short auxiliary bearing for which a state of plane stress is appropriate. However, the thermoelastic analysis is generalized to cover cases of plane strain and plane stress.

Analysis of internal cracks in railway wheels under experimentally determined pressure distributions

In recent years much effort has been devoted to the definition of design approaches of railway systems based on the analysis of the system itself and on accurate knowledge of its effective working conditions. In this paper, the attention is focused on railway wheels. This latter component is subjected to different types of damage: sub-surface crack propagation is considered. The prediction of the evolution of this process depends on the knowledge of the stress intensity factors concerning modes I, II and III, which are dependant both on the total load acting on the wheel and on how the load is transmitted through the wheel/rail interface. However, until now the solutions commonly used consider a theoretical (Hertzian) pressure distribution, even if, due to wear or to the dynamic phenomena, the actual contact patch can be strongly different for most of the lifespan of the wheel. An approach is developed with the aim to solve the case of an internally cracked wheel subjected to an arbitrary contact patch and pressure distribution. It is based on Boussinesq's formulae and utilises a three-dimensional finite element model of the part of the wheel close to the crack to calculate the stress intensity factors along a curvilinear crack front. Pressure distributions experimentally determined by means of a technique based on the reflection of high-frequency ultrasonic waves from the wheel–rail interface were applied to internal cracks in wheels: the results were critically compared with the those obtained by considering Hertzian pressure distributions, the aim being the assessment of the influence of the contact conditions in respect of damage cause by internal crack propagation.