j06503 imeche 2004 proc. instn mech. engrs vol. 218 part j

Elasto-plastic hemispherical contact models for variousmechanical properties

J J Quicksall R L Jackson and I Green

Georgia Institute of Technology Atlanta Georgia USA

Abstract This work uses the finite element technique to model the elastoplastic deformation of ahemisphere contacting a rigid flat for various material properties typical of aluminium bronzecopper titanium and malleable cast iron Additionally this work conducted parametric finite elementmethod (FEM) tests on a generic material in which the elastic modulus and Poissonrsquos 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 several carbon steels with various yield strengths they did not test materials with variousPoissonrsquos 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 contact

NOTATION

A area of contactC critical yield stress coefficientey ratio of yield strength to elastic modulus

frac14 SyEE elastic modulusH hardnessHG hardness geometric limitK hardness factorP contact forceR radius of hemispherical asperitySy yield strength

n Poissonrsquos ratioo interference between the hemisphere and

the surface

Superscripts

0 equivalent dimensionless

Subscripts

c critical value at the onset of plasticdeformation

E elastic regimeP fully plastic regimet transitional value from elastic to

elastoplastic behaviour

1 INTRODUCTION

Contact between a deformable 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 asperities

The 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 Georgia30332ndash0405 USA

313

J06503 IMechE 2004 Proc Instn Mech Engrs Vol 218 Part J J Engineering Tribology

modelled as hemispheres dispersed across two surfacesforced together constitute a similar form of contact(although actual asperities may have shapes differinggreatly from this hemispherical approximation) Thecontact area and pressure between two surfaces isimportant on both scales For the macroscopic scaleunacceptable deformations on a ball bearing may be theresult of excessive load For the microscopic scalecontact 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 ZhaondashMalettandashChang [1] and ChangndashEtsionndashBogy(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 materials but the carbon steelsmodelled to create them have fixed values of Poissonrsquosratio and modulus of elasticity The present workutilizes the FEM to conduct similar deformation testsof 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 whichPoissonrsquos ratio and the elastic modulus are indepen-dently varied Neither this work nor the JG modelcreated consider strain hardening in their analysis andalthough 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 Poissonrsquos ratio

and elastic modulus typical of a wide variety ofmaterials with the results of this FEM study

Both JG and KE modelled the frictionless contact ofan elasticndashperfectly plastic hemisphere pressed against arigid flat by some specified distance known as theinterference (Fig 1) The hemisphere is described aselasticndashperfectly plastic because 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 loadanywhere within 684o 4 110 (depending on thematerial properties) the plastic region covers the entirecontact area and occupies a large portion of the space inthe hemisphere This is known as the fully plasticregime Through a large range of applied interferencesthe hemisphere deformation is subject to fully elasticelastoplastic and fully plastic behaviours JG and KEcalculated non-dimensional contact area and force foran applied interference using the FEM They then fittedempirical formulations to the resulting non-dimensionalcontact area and force as functions of non-dimensionalinterference and material properties Because of thechanging contact geometry and the transition fromelastic to plastic behaviour the non-dimensional contactarea and force between a rigid flat and a hemisphere isnot linearly related to the non-dimensional interferenceWhile for the fully plastic regime KE assumed a fixedratio of hardness to yield strength JG found that theaverage contact pressure (ie lsquohardnessrsquo) varied with thedeformed contact geometry a result consistent with thatobtained by Mesarovic and Fleck [6] It should be notedthat the term hardness as used in this work relates to a

Fig 1 Spherical contact model (a) before contact (b) during mostly elastic deformation and (c) duringmostly plastic deformation

J J QUICKSALL R L JACKSON AND I GREEN314

Proc Instn Mech Engrs Vol 218 Part J J Engineering Tribology J06503 IMechE 2004

deformable hemisphere in contact with a rigid flat unliketypical hardness tests where the hemisphere is consideredrigid and the flat deformable Other major differencesbetween the JG and KE models are as follows

1 As mandated by mesh convergence the mesh used byJG is at least an order of magnitude finer than thatused by KE (the mesh by JG is also used in this work)

2 The deformation in the JG model continues to morethan twice the deformation in the KE model

3 The empirical formulations produced by the KEmodel use a piecewise fit that contains discontinuitiesat six times the critical interference whereas the JGfit is continuous throughout

4 The KE model assumes a truncation model beyond110 times the critical interference (see the Appendixequations (21) and (22) while the JG model does not[The truncation model has been used by variousresearchers in the field (see for example references[1 2 4] and [7]) It has been extracted perhapsunsuitably from the work by Abbott and Firestone[8] who never made any suggestions about load ormean pressure they simply assumed that progressivewear would truncate the surface height distributionHence from this point forward the truncation modelwill be referred to without attributing it to Abbottand Firestone [8]]

5 The critical interference used by KE is based on afixed relationship between hardness and yieldstrength (as if hardness is a material property) wherethe critical interference derived by JG does not relyon such a relationship at all These differences arenow detailed mathematically

The KE solution applies for non-dimensional inter-ferences between o frac14 1 and o frac14 110 It assumes theHertz solution for o lt 1 and the truncation model foro gt 110 KE formulated piecewise non-dimensionalcontact area and contact force as follows for 14o 4 6

PKE frac14 103 oeth THORN1425

AKE frac14 093 oeth THORN1136

P

ASy

KE

frac14 119 oeth THORN0289

eth1THORN

and for 64o 4 110



P

ASy

KE


eth2THORN

In equation (1) note that at o frac14 1 the model isdisjointed from the Hertz solution the force is over-

estimated by 3 per cent while the area is underestimatedby 7 per cent Note also that equations (1) and (2) aredisjointed at o frac14 6 Beyond o frac14 110 the contacthemisphere is assumed to be well into the fully plasticregime and at this point the truncation model isassumed where equation (2) does not asymptoticallyapproach the truncation model it merely intersects withit

The JG model finds that contact area increasesbeyond the truncation model for interferences in thefully plastic regime JG formulated nondimensionalcontact area for o519 as

A frac14 o o

19

B

eth3THORN

where

B frac14 014 expeth23eyTHORN eth4THORN

ey frac14Sy

E0 eth5THORN

A geometric limit to hardness as explained in detail byJG is calculated to be

HG

Syfrac14 284 1 exp 082

a

R

07

eth6aTHORN

where a is the radius of the contact patch of thedeformed hemisphere and R is the nominal radius of thehemisphere Here the hardness changes as a function ofaR or with the evolving geometry of contact The trendmay be explained by the progression schematicallyshown in Fig 2 As the interference increases and thecontact geometry changes the limiting ratio HGSy ofaverage pressure to yield strength must change fromTaborrsquos predicted value of three to a theoretical value ofone when afrac14R It should be noted that equation (6a) isonly valid for 0 lt a=R4 0412 where

a

Rfrac14 pCey

2o o

ot

B 1=2

eth6bTHORN

and is derived from equation (3) in the paper by JGCaution should thus be taken when using equation (6a)outside this range This range is acceptable for manyapplications particularly tribological applications where

Fig 2 Diagram of progression of change in hardness withgeometry

ELASTO-PLASTIC HEMISPHERICAL CONTACT MODELS FOR MECHANICAL PROPERTIES 315


deformations above this range are either unlikely orunacceptable By substituting equation (6b) into equa-tion (6a) Jackson and Green found that

HG


pCey2

ffiffiffiffiffiffio

p o

19

B=2 07

0

1A

24

35

eth6cTHORN

where HG is the hardness geometric limit and C is thecritical yield stress coefficient [equation (13)] Equation(6c) incorporates the effect that the deformed geometryhas on hardness which is defined as average contactpressure JG found that the ratio HGSy for fully plasticyielding approaches 284 Using equation (6c) the JGmodel calculates the non-dimensional contact force foro519 as

P frac14 exp 1

4oeth THORN5=12

oeth THORN3=2

thorn 4HG

CSy1 exp 1

25oeth THORN5=9

o eth7THORN

For o lt 19 the JG model assumes the Hertz solutionTheoretically the Hertz solution is valid only up too frac14 1 However the FEM results obtained by JG findthat the Hertz solution can be extended to o lt 19without a noticeable sacrifice in accuracy This isprobably due to the fact that although plastic deforma-tion does form at o frac14 1 and beyond it is small enoughand contained in a predominantly elastic region The JGformulation for non-dimensional contact area andcontact force provides continuous functions throughoutthe entire range of deformation 04o 4 250

2 PROCEDURE

The validity of the empirical formulations for contactwith an elastoplastic hemisphere presented by the KEand the JG models is tested by conducting the FEM ofhemispherical contact with varied material propertiesTwo sets of tests were conducted using an FEM similarto that used by JG (who detailed the numericalprocedure and convergence of the FEM) The first testgenerated contact area and contact force data for fivehypothetical metals with Poissonrsquos ratios yield strengthsand elastic modulus typical of aluminium bronzecopper titanium and malleable cast iron (see Table 1for material properties) Non-dimensional interferencesbetween o frac14 5 and o frac14 250 were used to generate datain both the elastoplastic and the fully plastic regimesexcept for lsquoaluminiumrsquo at o frac14 250 and lsquomalleable castironrsquo above o frac14 10 because of difficulties in obtainingsolution convergence at these interferences

The second set of tests generated non-dimensionalcontact area and contact force data for a genericmaterial in which the elastic modulus and Poissonrsquosratio were independently varied with the yield strengthheld constant at Syfrac14 200MPa Firstly Poissonrsquos ratiowas varied between 028 and 036 with the elasticmodulus held constant at 200GPa Then the elasticmodulus was varied between 160 and 240GPa withPoissonrsquos ratio held constant at 032 The non-dimen-sional interference was set at o frac14 20 80 and 250 foreach test iteration

The FEM used in this work is similar to that used byJG The mesh consists of a minimum of 11 100 elementsand may increase depending on the expected region ofcontact The contact region is meshed by 100 contactelements The non-dimensional area is calculated byfinding the number of elements that form the radius of acontact patch Some small error exists due to theresolution of a single contact element The error dueto contact patch resolution measuring the contact radiusranges from as much as 19 per cent for lsquocast ironrsquo ato frac14 2 to as little as 08 per cent for lsquobronzersquo ato frac14 250

3 NUMERICAL RESULTS

The relative difference between the models and the FEMconducted in this study is referred to as the percentageerror and calculated as

Percentage error frac14 100model FEM

FEM

For the first test modelling the deformation of fivehypothetical materials the average error for the JGformulations is 310 per cent for non-dimensionalcontact area and 449 per cent for non-dimensionalcontact force The average error for the KE model is526 per cent for non-dimensional contact area and 734per cent for the non-dimensional contact force It shouldbe noted that the KE model assumes the truncation

Table 1 Material properties

Sy EMaterial n (GPa) (GPa)

Aluminium 7075-T6 033 0505 72Bronze SAE 40 033 0125 93Copper UNS C15500 034 0124 115Titanium IMI 834 031 0925 120Malleable cast iron typical 025 0250 170Carbon steel 032 0210 200

matwebcom From reference [9]



model above o frac14 110 The truncation model is usedwhen calculating the average errors for the KE model ato frac14 250 The higher average error associated with theKE modelrsquos prediction of non-dimensional contact forceis mostly due to the large error associated withtruncation model values assumed at o frac14 250 (see theAppendix equations (21) and (22))

The second set of tests is conducted by firstindependently varying Poissonrsquos ratio and then inde-pendently varying the elastic modulus for a genericmaterial and was run using non-dimensional interfer-ences of o frac14 20 80 and 250 For this test the KEmodel has a slightly lower average error for non-dimensional contact area than the JG model and the JGmodel has a slightly lower average error for non-dimensional contact force than the KE model Bothmodels have average errors below 30 per cent forcontact area and contact force

Tests performed by holding Poissonrsquos ratio constantat 032 and varying the elastic modulus also yield smallerrors for the JG and KE formulations Again the KEmodel has slightly lower average error for non-dimen-sional contact area and the JG model produces aslightly lower average error for non-dimensional contactforce Neither model has an average error greater than33 per cent for either non-dimensional contact force ornon-dimensional contact area

4 DISCUSSION

Figures 3 and 4 show the errors from tests conducted onvarious hypothetical materials Generally comparingthe JG and KE models with the FEM results from thisstudy reveals small errors for most tests Nearly allresults fall below 10 per cent error and most results fallbelow 5 per cent error There is a general trend for theerrors of both the JG and the KE formulations todecrease with higher non-dimensional contact interfer-ence (except for the fully plastic case for the KE modelin which the truncation model value is assumed) This isprobably due to the increased accuracy of the FEM athigher interferences because the relative uncertainties ofnodal position are higher at lower displacements TheJG formulations have an average error about 2 per centless than the KE average error for both non-dimensionalcontact area and non-dimensional contact force Part ofthis reduced error may be attributed to the highernumber of degrees of freedom incorporated into the JGcurve fit the finer mesh used in the JG FEM than theKE FEM and the similarity between the JG mesh andthe mesh used in this model Additionally the JG modelincorporates the changing hardness of the deforminghemisphere into its non-dimensional contact forceformulations while the KE model assumes that the ratioof hardness to yield strength is fixed For some

Fig 3 Error of non-dimensional area as a function of non-dimensional interference between current FEMand values from JG and KE models



materials eg lsquoaluminiumrsquo or lsquotitaniumrsquo the errors inthe truncation model are significant (above 20 per cent)at large interferences In Figs 3 and 4 it can be seen thatgenerally the open symbols of JG are closer to zero thanthe full symbols of KE

Errors for both models are even lower for theparametric tests which varied Poissonrsquos ratio and elasticmodulus independently for a generic materialethSy frac14 200MPaTHORN Figures 5 and 6 show the errors fornon-dimensional contact area and contact force whenvarying Poissonrsquos ratio and holding the elastic modulusconstant The FEM conducted in the current workshows that the non-dimensional contact area increaseswith increasing Poissonrsquos ratio at a non-dimensionalinterference of o frac14 250 The FEM results also show adependence of the normalized load P on Poissonrsquosratio (Fig 7) At small interferences of o frac14 20 ando frac14 80 the dependence of P on n is weak however atinterferences of o frac14 250 P decreases with increasingn The JG model captures these trends while the KEmodel assumes that there is no dependence between P

and n although the errors of the KE model are still lessthan the JG model at some values of o and n

Finally Fig 8 shows the error of the JG and the KEmodels when Poissonrsquos ratio is held constant at 032and the elastic modulus is varied The FEM results fromthis study indicate that non-dimensional contact area

decreases slightly with increased elastic modulus andnon-dimensional contact force increases with increasedelastic modulus These trends are captured by the JGmodel but not by the KE model although again thereare some values of o and E for which the KE model hassmaller error

5 CONCLUSION

This work presents the results of elasticndashperfectly plastichemispherical contact for a variety of material proper-ties These results are compared with the empiricalformulations by JG and KE Both the JG and the KEstudies created formulations dependent on materialproperties without testing their models on materialswith varied Poissonrsquos ratio and elastic modulus Thiswork verifies these formulations against an FEM thatdid test varied material properties Both sets offormulations show small relative errors compared withthe FEM results of this test These low errors offer anacceptable range of uncertainty for most engineeringapplications The KE formulations are each segmentedinto two regions with a discontinuity at each intersec-tion while the JG formulations are continuous Thismay be advantageous when these equations need to be

Fig 4 Error of non-dimensional force as a function of non-dimensional interference between current FEMand values from JG and KE models



Fig 5 Error of non-dimensional area as a function of Poissonrsquos ratio between current FEM and values fromJG and KE models

Fig 6 Error of non-dimensional force as a function of Poissonrsquos ratio between current FEM and values fromJG and KE models



Fig 7 Error of non-dimensional area as a function of elastic modulus between current FEM and values fromJG and KE models

Fig 8 Error of non-dimensional force as a function of elastic modulus between current FEM and valuesfrom JG and KE models



integrated such as using the GreenwoodndashWilliamson[10] integrals For its continuity the JG formulation issomewhat more complex compared with the KEformulation Overall however the JG model has aslightly smaller error than the KE model whencompared with the current FEM results This workfinds that at large interferences the errors in thetruncation model postulated by KE are significant(above 20 per cent)

REFERENCES

1 Zhao Y Maletta D M and Chang L An asperity

microcontact model incorporating the transition from

elastic deformation to fully plastic flow Trans ASME J

Tribology 2000 122 86ndash93

2 Chang W R Etsion I and Bogy D B An elasticndashplastic

model for the contact of rough surfaces Trans ASME J

Tribology 109 257ndash263

3 Jackson R L and Green I A finite element study of

elasto-plastic hemispherical contact In Proceedings of the

2003 ASMEndashSTLE International Tribology Conference

Preprint 2003-TRIB268

4 Kogut L and Etsion I Elasticndashplastic contact analysis of a

sphere and a rigid flat Trans ASME J Appl Mechanics

2002 69(5) 657ndash662

5 Timoshenko S and Goodier J N Theory of Elasticity

1951 (McGraw-Hill New York)

6 Mesarovic S D and Fleck N A Frictionless indentation

of dissimilar elasticndashplastic spheres Int J Solids Structs

2000 37 7071ndash7091

7 Greenwood J A and Tripp J H The contact of two

nominally flat rough surfaces Proc Instn Mech Engrs

1971 185 625ndash633

8 Abbott E J and Firestone F A Specifying surface

qualitymdasha method based on accurate measurement and

comparison Mech Engr 1933 55 569ndash572

9 Shigley J E and Mischke C R Mechanical Engineering

Design 5th Edition 1989 (McGraw-Hill New York)

10 Greenwood J A and Williamson J B P Contact of

nominally flat surfaces Proc R Soc Lond A 1966 295

300ndash319

11 Johnson K L Contact Mechanics 1985 (Cambridge

University Press Cambridge)

12 Tabor D The Hardness of Materials 1951 (Clarendon

Oxford)

APPENDIX

Although the Hertz solution is formulated for contactbetween two hemispheres an equivalent modulus ofelasticity and radius can be calculated to apply tocontact between a hemisphere and a rigid flat [5] TheHertz theory of elastic contact calculates the contact

area and contact force during fully elastic contact

AE frac14 pRo eth8THORNPE frac14 4

3E0 ffiffiffiffi

Rp

ethoTHORN3=2 eth9THORN

where AE is the circular contact area PE is the totalcontact load and o is the interference E 0 the equivalentelastic modulus and R the equivalent radius arecalculated as

1

E0 frac141 n22E1

thorn 1 n22E2

eth10THORN

1

Rfrac14 1

R1thorn 1

R2eth11THORN

where E1 n1R1E2 n2 and R2 are the elastic moduliPoissonrsquos ratios and radii of hemispheres 1 and 2respectively The Hertz solution approximates thecontact hemisphere as a parabolic curve with the radiusof curvature at the tip of the parabola equal to theradius of the contact hemisphere It also assumes thatthe deformation (interference) of the contact hemisphereis much less than the radius of the hemisphere and thatno friction is present between the hemisphere and rigidflat

The Hertz solution can be used to non-dimensionalizecontact area and contact force outside the truly elasticregime by obtaining the interference at the inception ofyielding JG calculated interference at the onset ofplastic yielding using the von Mises yield criterion as

oc frac14CpSy

2E0

2

R eth12THORN

where oc is the critical interference R is the contacthemisphere radius E0 is the equivalent elastic modulusSy is the yield strength and C is a function given in theJG study as

C frac14 1295 expeth0736nTHORN eth13THORN

The critical contact area at yielding Ac was found byJG by substituting the critical interference [equation(12)] into the Hertz solution for contact area [equation(8)]

Ac frac14 p3CSyR

2E0

2

eth14THORN

The critical contact force Pc was found by JG bysubstituting the critical interference [equation (12)] intothe Hertz solution for contact force

Pc frac144

3

R

E0

2CpSy

2

3

eth15THORN



KE used a critical interference given by CEB

oc frac14pKH2E0

2

R eth16THORN

Here a fixed ratio of hardness H to yield strength Sy isassumed Hfrac14 28Sy which results in the factor K givenas

K frac14 0454thorn 041n eth17THORN

The critical area and critical force are found bysubstituting the critical interference into the Hertzsolution Equations (12) and (13) produce almostidentical numerical results

With the calculation of critical values both JG andKE non-dimensionalized their interference contactforce and contact area as

o frac14 ooc

eth18THORN

P frac14 P

Pceth19THORN

A frac14 A

Aceth20THORN

where o P and A are the non-dimensional para-meters of interference contact force and contact arearespectively Values of o P and A equal to unityrepresent the onset of plastic deformation

At large deformations the contact hemisphere entersthe fully plastic regime The fully plastic regime wascharacterized by Johnson [11] as a state where the plasticstrains are large compared with the elastic strains whichcan be correlated to when the entire contact surfacedeforms plastically Tabor [12] gave a similar definitionand also provided a relationship of hardness to yieldstrength of Hamp3Sy Within the fully plastic regions ifstrain hardening is small the material will flowplastically at a constant stress while conserving volumeThe truncation model assumes that under fully plasticconditions the contact area of a hemisphere can beapproximated by the truncation of the rigid spheregeometry by the rigid flat The result is approximated as

AP frac14 2pRo eth21THORN

The contact force on the hemisphere can be calculatedas the contact area multiplied by the average pressurewhich in the fully plastic regime is the hardness Thefully plastic contact force is given as

PP frac14 2pRoH eth22THORN

where Hfrac14 28Sy as previously stated However it wasfound by JG that this result is unsubstantiated Green-wood and Tripp [7] also independently modelled fullyplastic contact between hemispheres using a similartruncation method



modelled as hemispheres dispersed across two surfacesforced together constitute a similar form of contact(although actual asperities may have shapes differinggreatly from this hemispherical approximation) Thecontact area and pressure between two surfaces isimportant on both scales For the macroscopic scaleunacceptable deformations on a ball bearing may be theresult of excessive load For the microscopic scalecontact 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 ZhaondashMalettandashChang [1] and ChangndashEtsionndashBogy(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 materials but the carbon steelsmodelled to create them have fixed values of Poissonrsquosratio and modulus of elasticity The present workutilizes the FEM to conduct similar deformation testsof 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 whichPoissonrsquos ratio and the elastic modulus are indepen-dently varied Neither this work nor the JG modelcreated consider strain hardening in their analysis andalthough 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 Poissonrsquos ratio

and elastic modulus typical of a wide variety ofmaterials with the results of this FEM study

Both JG and KE modelled the frictionless contact ofan elasticndashperfectly plastic hemisphere pressed against arigid flat by some specified distance known as theinterference (Fig 1) The hemisphere is described aselasticndashperfectly plastic because 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 loadanywhere within 684o 4 110 (depending on thematerial properties) the plastic region covers the entirecontact area and occupies a large portion of the space inthe hemisphere This is known as the fully plasticregime Through a large range of applied interferencesthe hemisphere deformation is subject to fully elasticelastoplastic and fully plastic behaviours JG and KEcalculated non-dimensional contact area and force foran applied interference using the FEM They then fittedempirical formulations to the resulting non-dimensionalcontact area and force as functions of non-dimensionalinterference and material properties Because of thechanging contact geometry and the transition fromelastic to plastic behaviour the non-dimensional contactarea and force between a rigid flat and a hemisphere isnot linearly related to the non-dimensional interferenceWhile for the fully plastic regime KE assumed a fixedratio of hardness to yield strength JG found that theaverage contact pressure (ie lsquohardnessrsquo) varied with thedeformed contact geometry a result consistent with thatobtained by Mesarovic and Fleck [6] It should be notedthat the term hardness as used in this work relates to a

Fig 1 Spherical contact model (a) before contact (b) during mostly elastic deformation and (c) duringmostly plastic deformation












P

ASy

KE


eth1THORN

and for 64o 4 110



P

ASy

KE


eth2THORN




A frac14 o o

19

B

eth3THORN

where


ey frac14Sy

E0 eth5THORN


HG


a

R

07

eth6aTHORN


a

Rfrac14 pCey

2o o

ot

B 1=2

eth6bTHORN






HG


pCey2

ffiffiffiffiffiffio

p o

19

B=2 07

0

1A

24

35

eth6cTHORN


P frac14 exp 1

4oeth THORN5=12

oeth THORN3=2

thorn 4HG

CSy1 exp 1

25oeth THORN5=9

o eth7THORN


2 PROCEDURE




3 NUMERICAL RESULTS



FEM











4 DISCUSSION










5 CONCLUSION














REFERENCES














2002 69(5) 657ndash662





2000 37 7071ndash7091



1971 185 625ndash633








300ndash319




Oxford)

APPENDIX




3E0 ffiffiffiffi

Rp



1

E0 frac141 n22E1

thorn 1 n22E2

eth10THORN

1

Rfrac14 1

R1thorn 1

R2eth11THORN



oc frac14CpSy

2E0

2

R eth12THORN




Ac frac14 p3CSyR

2E0

2

eth14THORN


Pc frac144

3

R

E0

2CpSy

2

3

eth15THORN




oc frac14pKH2E0

2

R eth16THORN





o frac14 ooc

eth18THORN

P frac14 P

Pceth19THORN

A frac14 A

Aceth20THORN


















P

ASy

KE


eth1THORN

and for 64o 4 110



P

ASy

KE


eth2THORN




A frac14 o o

19

B

eth3THORN

where


ey frac14Sy

E0 eth5THORN


HG


a

R

07

eth6aTHORN


a

Rfrac14 pCey

2o o

ot

B 1=2

eth6bTHORN






HG


pCey2

ffiffiffiffiffiffio

p o

19

B=2 07

0

1A

24

35

eth6cTHORN


P frac14 exp 1

4oeth THORN5=12

oeth THORN3=2

thorn 4HG

CSy1 exp 1

25oeth THORN5=9

o eth7THORN


2 PROCEDURE




3 NUMERICAL RESULTS



FEM











4 DISCUSSION










5 CONCLUSION














REFERENCES














2002 69(5) 657ndash662





2000 37 7071ndash7091



1971 185 625ndash633








300ndash319




Oxford)

APPENDIX




3E0 ffiffiffiffi

Rp



1

E0 frac141 n22E1

thorn 1 n22E2

eth10THORN

1

Rfrac14 1

R1thorn 1

R2eth11THORN



oc frac14CpSy

2E0

2

R eth12THORN




Ac frac14 p3CSyR

2E0

2

eth14THORN


Pc frac144

3

R

E0

2CpSy

2

3

eth15THORN




oc frac14pKH2E0

2

R eth16THORN





o frac14 ooc

eth18THORN

P frac14 P

Pceth19THORN

A frac14 A

Aceth20THORN










HG


pCey2

ffiffiffiffiffiffio

p o

19

B=2 07

0

1A

24

35

eth6cTHORN


P frac14 exp 1

4oeth THORN5=12

oeth THORN3=2

thorn 4HG

CSy1 exp 1

25oeth THORN5=9

o eth7THORN


2 PROCEDURE




3 NUMERICAL RESULTS



FEM











4 DISCUSSION










5 CONCLUSION














REFERENCES














2002 69(5) 657ndash662





2000 37 7071ndash7091



1971 185 625ndash633








300ndash319




Oxford)

APPENDIX




3E0 ffiffiffiffi

Rp



1

E0 frac141 n22E1

thorn 1 n22E2

eth10THORN

1

Rfrac14 1

R1thorn 1

R2eth11THORN



oc frac14CpSy

2E0

2

R eth12THORN




Ac frac14 p3CSyR

2E0

2

eth14THORN


Pc frac144

3

R

E0

2CpSy

2

3

eth15THORN




oc frac14pKH2E0

2

R eth16THORN





o frac14 ooc

eth18THORN

P frac14 P

Pceth19THORN

A frac14 A

Aceth20THORN












4 DISCUSSION










5 CONCLUSION














REFERENCES














2002 69(5) 657ndash662





2000 37 7071ndash7091



1971 185 625ndash633








300ndash319




Oxford)

APPENDIX




3E0 ffiffiffiffi

Rp



1

E0 frac141 n22E1

thorn 1 n22E2

eth10THORN

1

Rfrac14 1

R1thorn 1

R2eth11THORN



oc frac14CpSy

2E0

2

R eth12THORN




Ac frac14 p3CSyR

2E0

2

eth14THORN


Pc frac144

3

R

E0

2CpSy

2

3

eth15THORN




oc frac14pKH2E0

2

R eth16THORN





o frac14 ooc

eth18THORN

P frac14 P

Pceth19THORN

A frac14 A

Aceth20THORN














5 CONCLUSION














REFERENCES














2002 69(5) 657ndash662





2000 37 7071ndash7091



1971 185 625ndash633








300ndash319




Oxford)

APPENDIX




3E0 ffiffiffiffi

Rp



1

E0 frac141 n22E1

thorn 1 n22E2

eth10THORN

1

Rfrac14 1

R1thorn 1

R2eth11THORN



oc frac14CpSy

2E0

2

R eth12THORN




Ac frac14 p3CSyR

2E0

2

eth14THORN


Pc frac144

3

R

E0

2CpSy

2

3

eth15THORN




oc frac14pKH2E0

2

R eth16THORN





o frac14 ooc

eth18THORN

P frac14 P

Pceth19THORN

A frac14 A

Aceth20THORN


















REFERENCES














2002 69(5) 657ndash662





2000 37 7071ndash7091



1971 185 625ndash633








300ndash319




Oxford)

APPENDIX




3E0 ffiffiffiffi

Rp



1

E0 frac141 n22E1

thorn 1 n22E2

eth10THORN

1

Rfrac14 1

R1thorn 1

R2eth11THORN



oc frac14CpSy

2E0

2

R eth12THORN




Ac frac14 p3CSyR

2E0

2

eth14THORN


Pc frac144

3

R

E0

2CpSy

2

3

eth15THORN




oc frac14pKH2E0

2

R eth16THORN





o frac14 ooc

eth18THORN

P frac14 P

Pceth19THORN

A frac14 A

Aceth20THORN










oc frac14pKH2E0

2

R eth16THORN





o frac14 ooc

eth18THORN

P frac14 P

Pceth19THORN

A frac14 A

Aceth20THORN









j06503 imeche 2004 proc. instn mech. engrs vol. 218 part j

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