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Elasto-plastic hemispherical contact models for various mechanical properties

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Elasto-plastic hemispherical contact models for variousmechanical propertiesJ J Quicksall, R L Jackson and I Green*Georgia Institute of Technology, Atlanta, Georgia, USAAbstract: This work uses the finite element technique to model the elastoplastic deformation of ahemisphere contacting a rigid flat for various material propert ies typical of aluminium, bronze,copper, titanium and malleable cast iron. Additionally, this work conducted parametric finite elementmethod (FEM) tests on a generic material in which the elastic modulus and Poisson’s ratio are variedindependently while the yield strength is held constant. A larger spectrum of material properties arecovered in this work than in most previous studies. The results from this work are compared with twopreviously formulated elastoplastic models simulating the deformation of a hemisphere in contactwith a rigid flat. Both of the previously formulated models use carbon steel mechanical properties toarrive at empirical formulations implied to pertain to various materials. While both modelsconsidered severa l carbon steels with various yield strengths, they did not test materials with variousPoisson’s ratios or elastic moduli. The previously generated elastoplastic models give fairly goodpredictions when compared with the FEM results for various material properties from the currentwork, except that one model produces more accurate predictions overall, especially at largedeformations where other models neglect important trends due to decreases in hardness withincreasing deformation.Keywords: asperity, Hertzian contact, hemispherical contact, contact mechanics, surfacedeformation, elastoplastic contactNOTATIONA area of contactC critical yield stress coefficienteyratio of yield strength to elastic modulus¼ Sy/EE elastic modulusH hardnessHGhardness geometric limitK hardness factorP contact forceR radius of hemispherical asperitySyyield strengthn Poisson’s ratioo interference between the hemisphere andthe surfaceSuperscripts0equivalentdimensionlessSubscriptsc critical value at the onset of plast icdeformationE elastic regimeP fully plastic regimet transitional value from elastic toelastoplastic behaviour1 INTRODUCTIONContact between a deformab le hemisphere and a rigidflat surface is commonplace in modelling many engi-neering applications. On a macroscopic scale, a ballbearing forced against the race of a bearing can beapproximated by a hemisphere in contact with a rigidflat surface. On a microscopic scale, small asperitiesThe MS was received on 17 October 2003 and was accepted after revisionfor publication on 4 June 2004.*Corresponding author: The George W. Woodruff School of Mechan-ical Engineering, Georgia Institute of Technology, Atlanta, Georgia30332–0405, USA.313J06503 # IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part J: J. Engineering Tribol ogymodelled as hemispheres dispersed across two surfacesforced together constitute a similar form of contact(although actual asperities may have shapes differinggreatly from this hemispherica l approximation). Thecontact area and pressure between two surfaces isimportant on both scales. For the macroscopic scale,unacceptable deformations on a ball bearing may be theresult of excessive load. For the microscopic scale,contact area between asperities affects friction, wear andconduction between two surfaces. General empiricalapproximations for relating contact area and contactforce with hemisphere deformation are desired foraccurate solutions to many engineering problems.Many previous models have been formulated, such asthe Zhao–Maletta–Chang [1] and Chang–Etsion–Bogy(CEB) [2] models, but recent findings have proven themto be inadequate [ 3 , 4]. Thus these models will not beconsidered in this work.Recently, two independent studies by Jackson andGreen (JG) [3] and Kogut and Etsion (KE) [4] haveutilized the finite element method (FEM) to modelhemispherical contact with a rigid flat. Both models usethe Hertz solution [5 ] to non-dimensionalize their resultsfor interference, contact area and contact force (see theAppendix), so that these non-dimensional values equalunity at the onset of yielding. The models simulatedeformation of carbon steels to arrive at empiricalformulations for non-dimensional contact area andcontact force. The formulations imply that they areapplicable to all ductile mate rials, but the carbon steelsmodelled to create them have fixed values of Poisson’sratio and modulus of elasticity. The present workutilizes the FEM to conduct similar deformation test sof a hemisphere in contact with a rigid flat, but itconsiders material properties typical of five other uniquemetals: aluminium, bronze, copper, titanium and malle-able cast iron. Additionally, this work conducts para-metric FEM tests on a generic material for whichPoisson’s ratio and the elastic modulus are indepen-dently varied. Neither this work nor the JG modelcreated consider strain hardening in their analysis and,although KE investigated strain hardening, their modeldoes not include it either. The objective of this work is tocompare the empirical formulations proposed by JGand KE, as they apply to various sets of Poisson’s ratioand elastic modulus typical of a wide variety ofmaterials, with the results of this FEM study.Both JG and KE modelled the frictionless contact ofan elastic–perfectly plastic hemisphere pressed against arigid flat by some specified distance known as theinterference (Fig. 1). The hemisphere is described aselastic–perfectly plastic becau se at low interferences ahigh-stress region starts to form below the contactinterface. Eventually the material yields in this high-stress region and a plastic core forms. The plastic core issurrounded by elastic material, which diminishes as thehemisphere is subjected to larger interferences. At higherinterferences the plastic core expands in a three-dimensional fashion to the surface, and also inwardstowards the centre of the hemisphere. The reason thatthe plastic region expands is because the material in thehemisphere that is flowing plastically can no longerresist additional load. Therefore, any additional load iscarried by the surrounding elastic regions. At a value ofobetween six and ten the plastic core reaches thesurface near the edge of contact. Then there is an elasticcore below the contact area that is surrounded byplastically deforming material. At a much higher load,anywhere within 68 4o4 110


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