3 FTFBSDI SUJDMF 'JOJUF &MFNFOU #BTFE.VMUJQMF/PSNBM … · 2019. 7. 31. · fs fgpsf uif psjh jobm vo efgpsnfe tqifsjdbm hfpnfusz jt opu gvmmz sfdpwfsfe f psjhjobm qsp mf uif efgpsnfe

Hindawi Publishing CorporationISRN TribologyVolume 2013 Article ID 871634 13 pageshttpdxdoiorg1054022013871634

Research ArticleFinite-Element-BasedMultiple Normal Loading-Unloading ofan Elastic-Plastic Spherical Stick Contact

Biplab Chatterjee and Prasanta Sahoo

Department of Mechanical Engineering Jadavpur University Kolkata 700032 India

Correspondence should be addressed to Prasanta Sahoo psjumegmailcom

Received 30 October 2012 Accepted 19 November 2012

Academic Editors J Antunes and F Findik

Copyright copy 2013 B Chatterjee and P Sahoo is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

e repeated normal elastic plastic contact problem of a deformable sphere against a rigid at under full stick contact condition isinvestigated with a commercial nite element soware ANSS Emphasis is placed on the effect of strain hardening and hardeningmodel with themaximum interference of load ranging from elastic to fully plastic which has not yet been reported Different valuesof tangent modulus coupled with isotropic and kinematic hardening models are considered to study their inuence on contactparameters Up to ten normal loading-unloading cycles are applied with a maximum interference of 200 times the interferencerequired to initiate yielding Results for the variation of mean contact pressure contact load residual interference and contactarea with the increasing number of loading unloading cycles at high hardening parameter as well as for low tangent modulus withtwo different hardening models are presented Results are compared with available nite element simulations and in situ resultsreported in the literature It is found that small variation of tangent modulus results in same shakedown behavior and similarinterfacial parameters in repeated loading-unloading with both the hardening rules However at high tangent modulus the strainhardening and hardening rules have strong inuence on contact parameters

1 Introduction

Multiple repeated normal loading unloading is common inengineering applications Several researchers have identiedthe use of repeated normal loading unloading in variouselds of engineering applications such as contact resistancein MEMS micro switches [1 2] head-disk interaction inmagnetic storage systems [3] ultrasonic interfacial stiffnessmeasurement [4] stamping mechanism and in rolling ele-ment bearings gears cams and so forth Earlier the prob-lems of multiple loading-unloading were solved assuming aspecic pressure distribution e contact region related tothe analysis of Kapoor et al [5] Williams et al [6] did notexceed greatly the elastic limit hence they assumed Hertzian[7] contact Merwin and Johnson [8] Kulkarni et al [9]Bhargava et al [10 11] used Hertzian or modied Herzianpressure distribution even for elastic plastic contacts In mostengineering applications the contact deformation occurredin elastic plastic as well as in plastic region e arbitrary

selection of pressure distribution where the plasticity effectsare dominant eventually would provide inaccurate solutionBeyond the Hertzian assumption of nonadhesive friction-less contact within elastic limit commercial nite elementsoware is best suited to calculate accurately the contactparameters like contact load contact area and pressure inthe elastic plastic and plastic region [12]e contact analysisof the rough surfaces involves the study of single asperitycontact Kogut and Etsion [13] rst provided an accurateresult of elastic plastic loading of a hemisphere against arigid at using commercial nite element soware ANSSey used nite element analysis under frictionless conditionfor a wide range of material elastic properties and spheresize to present generalized empirical relations for contactarea contact load and mean contact pressure as a functionof dimensionless interference eir investigations in elasticelastic plastic and plastic regions also include the effect ofplastic properties of the material Kogut and Etsion inferredthat the variation of tangent modulus plastic property of the

2 ISRN Tribology

material up to 10 percent of modulus of elasticity yieldedin maximum 20 percent difference of contact parametersEtsion et al [14] extended their investigation for unloadingunder perfect slip contact condition with tangent modulusof 2 percent of modulus of elasticity ey used bilinearisotropic hardening Jackson et al [15] studied the effect ofelastic properties of material on residual deformation andconcentrated on the residual stresses of the sphere underperfect slip contact condition Kadin et al [16] offered amultiple loading-unloading model with isotropic hardeningusing 002119864119864 tangent modulus under perfect slip contactconditioney found that the incipient interference requiredfor the secondary plastic ow in the peripheral zone increaseswith the increase in tangent modulus Kral et al [17]performed repeated normal indentation of a half space bya rigid sphere under perfect slip contact condition usingisotropic hardening It was shown that the effect of plasticproperties like strain hardening is more pronounced ratherthan the elastic properties of the material on the contactparameters during loading unloading in the elastic plasticand plastic region ey used strain-hardening exponent(119899119899) up to 05 Sahoo et al [18] studied the effect of strainhardening for elastic plastic loading of a deformable sphereagainst a rigid at for frictionless and non-adhesive contactby varying the hardening parameter (119867119867) up to 05 Variationof hardening parameter up to 05 enabled them to study theeffect of tangent modulus as high as 33 percent of modulus ofelasticity ey inferred that the increase in strain hardeningincreases the resistance to deformation of a material and thematerial becomes capable of carrying higher amount of loadin a smaller contact area However Chatterjee and Sahoo[19] observed that the contact loads aer rst and secondloading were same under perfect slip contact condition fornon-hardening material as well as material with hardeningparameter of 05

e aforementioned reviews are related to frictionlesscontact condition Friction is useful in many engineeringapplications such as brakes clutches bolts nuts drivingwheels on automobiles and so forth whereas unproductivefriction takes place in engines gears cams bearings and seals[20] Brizmer et al [21] found that the pressure distributionand the stress eld near the contact surfaces at low contactloads are different for perfect slip and full stick contactcondition Chatterjee and Sahoo [22] studied the effect ofelastic properties of material for an elastic plastic loadingof a deformable sphere against a rigid at under full stickcontact condition Zait et al [23] performed the unloadingof an elastic plastic spherical contact under full stick contactcondition ey found a substantial variation in load areacurve during unloading under full stick contact conditioncompared to that under perfect slip contact condition Zaitet al [24] expanded the previous single complete unloadingunder full stick contact condition to include multiple normalloading unloading ey used bilinear kinematic hardeningwith tangent modulus as 2 percent of modulus of elastic-ity ey concluded identical shakedown behavior for bothisotropic and kinematic hardening Chatterjee and Sahoo[25] studied the effect of strain hardening on the elastic plasticcontact loading of a deformable sphere against a rigid at

under full stick contact condition with two different hard-ening rules It was observed that the material with isotropichardening has higher amount of load carrying capacity thanthat of kinematic hardening particularly for higher strainhardening Johnson [26]mentioned that the surface tractionsand nonlinear material properties are the factors to inuencethe reversibility of cyclic normal loading in spherical contactproblems us under a specic contact condition (slip orstick) the plastic properties of thematerial and the hardeningrule have a particular importance inmultiple normal loading-unloading ough under perfect slip (frictionless) contactcondition the effect of strain hardening on loading unloadingof a deformable sphere against a rigid at is available inliterature the effect of plastic properties and hardeningmodelfor multiple normal loading-unloading for a deformablespherical contact under full stick contact condition is stillmissing

e present investigation therefore aims at studying theeffect of strain hardening and hardening models on contactparameters during multiple normal loading-unloading of adeformable sphere against a rigid at under full stick contactcondition using commercial nite element soware ANSS

2 Theoretical Background

Figure 1 shows the contact of a deformable sphere and arigid at e dashed line presents the original contour ofthe sphere having a radius of 119877119877 and the rigid at before thedeformatione solid line shows the loading phase with theinterference (120596120596) corresponding to the contact radius (119886119886) andthe contact load (119875119875)

Multiple loading unloading consists repeated normalloading and complete unloading During loading phase theinterference (120596120596) is increased up to the specied maximumdimensionless interference 120596120596lowastmax During unloading theinterference (120596120596) is gradually reduced At the end of theunloading under zero contact load and contact area thesphere has a residual interference (120596120596res) erefore the orig-inal un-deformed spherical geometry is not fully recoverede original prole the deformed shape aer loading phaseand the sphere prole at the end of unloading in singleloading unloading are represented in Figure 2 Stick contactcondition is applied at the interface of the deformed sphereand the rigid at e stick contact condition prevents thecontact point of the deformed sphere with the rigid at fromthe relative displacement in the radial direction but it allowsthe axial detachment of the sphere surface from the rigid atduring unloading

In the present study multiple normal loading-unloadingis simulated considering both the isotropic and kinematichardening models Figure 3 describes the difference betweenisotropic hardening and kinematic hardening by describ-ing the development of the yield surface with progressiveyielding In isotropic or work hardening the yield surfaceis uniformly spread out from the center while in kinematichardening the yield surface translates in stress space withconstant size e variation of different contact parameters isconsidered with the increase in loading or unloading cycles

ISRN Tribology 3

F 1 A deformable sphere pressed by a rigid at

in the present study e contact loads contact area meancontact pressure in repeated normal loading are normalizedby their corresponding values aer rst loading Similarlyresidual interference aer repeated unloading is normalizedby its value aer rst unloading e contact parameters andthe interferences in some cases are normalized with theircritical values in a manner as described in Chatterjee andSahoo [25] e following empirical relations are used toevaluate the critical interference (120596120596119888119888) which initiates the yieldinception at rst loading and the corresponding critical load(119875119875119888119888) and critical contact area (119860119860119888119888) under full stick condition

120596120596119888119888 = 10076551007655119862119862119907119907120587120587 100765010076501 minus 12058412058421007666100766621007652100765211988411988411986411986410076681007668100766810076682

times 119877119877 10076501007650682120584120584 minus 1205848120584 100765010076501205841205842 + 005861007666100766610076661007666

119875119875119888119888 =120587120587120584119884119884611986211986211990711990712058410076521007652119877119877 100765010076501 minus 120584120584210076661007666 10076521007652

11988411988411986411986410076681007668100766810076682

times 10076501007650888120584120584 minus 101120584 100765010076501205841205842 + 00891007666100766610076661007666

119860119860119888119888 = 120587120587120596120596119888119888119877119877

(1)

Here 119862119862119907119907 is a parameter that depends on Poissonrsquos ratio as119862119862119907119907 = 121205844 + 1256120584120584 e parameters 119884119884 119864119864 and 120584120584 are thevirgin yield stress the Young modulus and Poissonrsquos ratio ofthe sphere material respectively and 119877119877 is the radius of thesphere e sphere size used for this analysis is 119877119877 = 1 119877119877me material properties used here are Youngrsquos Modulus (119864119864)= 70GPa Poissonrsquos Ratio (120584120584) = 03 and Yield stress (119884119884) =100MPa For full stick contact condition innite frictioncondition is adopted

3 Finite Element Model

A commercial nite element package ANSYS 110 is usedin the present study in order to get the accurate solutionof a complicated subject like multiple loading-unloadingunder full stick contact condition e sphere is representedby a quarter of a circle due to its axisymmetry A linemodels the rigid at e elements as shown in Figure 4consists the mesh of maximum 18653 number of plane183nite elements Plane 183 is six-node triangular axisymmetricelement e mesh density at the bottom of the sphereis coarsest one and is made gradually ner towards the

Sphere profile

Fully unloaded residual profile

Fully loaded profile

res

max

F 2 ree dierent proles of the sphere

sphere summit e nest mesh density near the contactregion simultaneously allows the spherersquos curvature to becaptured and accurately simulated during deformation witha reduction in computation time Window 2 of Figure 4presents the enlarged view of the nestmesh density at spheresummite detail description and boundary condition of FEmodel can be found in [25]

4 Results and Discussion

Engineering stress-strain curves are used within elastic limite dimension of the specimen changes substantially in theregion of plastic deformatione increment of strain in con-junction with true stress can be termed as strain hardeningStrain hardening causes an increase in strength and hardnessof themetal Strain hardening is expressed in terms of tangentmodulus (119864119864119905119905) which is the slope of the stress-strain curveBelow the proportional limit the tangentmodulus is the sameas the Youngrsquos modulus (119864119864) Above the proportional limit thetangent modulus varies with the straine tangent modulusis useful in describing the behavior of materials that havebeen stressed beyond the elastic region In elastic perfectlyplastic cases the tangent modulus becomes zero Very fewmaterial exhibit elastic perfectly plastic behaviors generallyall the materials follow the multilinear behavior with sometangent modulus is multi-linear behavior can be assumedas bilinear behavior for analysis purpose in elastic-plasticcases In this analysis a bilinear material property as shownin Figure 5 is used for the deformable sphere Shankarand Mayuram [27] mentioned that the tangent modulusfor the most practical materials is less than 005119864119864 whereasKadin et al [16] found the tangent modulus for mostpractical materials below 002119864119864 However both the authorsused tangent modulus up to 01119864119864 for analytical purpose Onthe other hand Ovcharenko et al [28] used stainless steelspecimen with tangent modulus of 026119864119864 (Figure 6(b)) intheir in situ investigation) It is also available in literature thatstructural steel aluminum alloys have signicant amount ofstrain hardening

Figure 6(a) represents the load interference hystereticloop during ten repeated loading unloading e maximumdimensionless interference for loading is 120596120596lowast = 100 with

4 ISRN Tribology

Spread out yielding surface

1

2

(a)

Translated constant size yielding surface

1

2

(b)

F 3 (a) Isotropic and (b) kinematic hardening models for two-dimensional stress eld

1

2 Elements

Elements

2 Elements

F 4 Finite element mesh of a sphere generated by ANSYS

Stress

Strain

F 5 Stress-strain diagram for a material with bilinear proper-ties

tangent modulus 119864119864119905119905 = 0025119864119864 using kinematic hardeningis simulation was performed to validate the present resultswith Zait et al [24] Zait et al furnished the results (Figure4) with maximum dimensionless interference of 60 using

kinematic hardening ey have shown that with small tan-gent modulus the materials result in elastic shakedown evenunder the inuence of kinematic hardening e agreementbetween the present simulation and the results of Zait et al[24] is appreciably good Figures 6(b) and 6(c) are the detailsof Figure 6(a) Figure 6(b) shows the decrease of maximumcontact load with the increase in loading cycles for multipleloading unloading for the same dimensionless interference ofloading Figure 6(c) presents the increase of residual interfer-enceswith the increase in unloading cycles during ten loadingunloading cycles Zait et al [24] also observed the decreaseof maximum contact load and increase of dimensionlessinterference during multiple loading unloading

41 Interfacial Parameters with Low Tangent Modulus Inthe rst part of present analysis materials are chosen aselastic perfectly plastic material and the materials withtangentmodulus of 0025119864119864 005119864119864 and 009119864119864 for studying theeffect of strain hardening with both isotropic and kinematichardening rule Figure 7(a) presents the normalized contactload 1198751198751198751198751198750 during ten loading-unloading cycles with bilinearisotropic hardening 1198751198750 is the contact load aer the comple-tion of rst loading e maximum loading was performedup to a dimensionless interference of 120596120596lowastmax = 100 It is foundthat the contact load decreases with each loading-unloadingcycle Chatterjee and Sahoo [19] found identical contactload in second loading (Figures 9(a) and 9(b)) with bilinearisotropic hardening under perfect slip contact condition Sothis decrease in contact load is due to the contact conditionswhich was also inferred by Zait et al [24] e decrease ofcontact loadwith increase in loading cycles depends on strainhardening e decrease in contact load is less with increasein tangent modulus e decrease of contact loads aer tenloading-unloading cycles are 95 773 7 and 51 percent formaterials with tangent modulus of 00119864119864 0025119864119864 005119864119864 and009119864119864 respectively

Figure 7(b) represents the normalized contact load 119875119875119875119875119875119900119900during ten loading unloading in case of bilinear kinematichardening with maximum loading up to the dimensionless

ISRN Tribology 5

0 20 40 60 80 1000

100

200

300

400

500

1st loading

Subse

quent l

oadin

g

unload

ing

cycle

s

Dimensionless normal interference ( )

Dim

ensi

on

less

co

nta

ct lo

ad (

)

= 0025 kinematic

(a)

88 90 92 94 96 98 100

430

440

450

460

470

480


Dim

ensi

on

less

co

nta

ct l

oad

(

)

= 0025 kinematic

(b)

67 68 69 70 71 72 730

5

10

15

20

25

30


Dim

ensi

on

less

co

nta

ct l

oad

(

)

= 0025 kinematic

(c)

F 6 (a) e load-interference hysteretic loop during ten loading-unloading cycles following maximum loading to a dimensionlessinterference of 120596120596lowast = 100 (b) Increase of residual interferences and decrease of contact load (c) Increase of residual interferences

interference of 120596120596lowastmax = 100 e decrement trend of contactload is same for both the hardening rule e decrease ofcontact loads aer ten loading-unloading cycles are 95 75565 and 28 percent for materials with tangent modulus of00119864119864 0025119864119864 005119864119864 and 009119864119864 respectively It is seen from theresults of both the hardening rules that the decrease of contactload is less with kinematic hardening than that of isotropichardening Elastic perfectly plasticmaterials are insensitive tothe hardening rule however the contact load using isotropicand kinematic hardening with the increase in loading cyclehas signicant variation for higher tangent modulus

Figure 8(a) is the plot of normalized residual displace-ment 120596120596lowastres120596120596

lowastmax during ten loading unloading cycles with

bilinear isotropic hardening e residual displacement orinterferences are found aer complete unloading e con-tact load and area becomes zero aer complete unloadinge residual interferences are normalized by the maximumdimensionless interference of loadingemaximumdimen-sionless loading interference is 120596120596lowastmax = 100 It can be seen

from the plot that the normalized residual interference duringten loading-unloading cycles is dependent on tangent mod-ulus e normalized residual displacement decreases withthe increase of tangent modulus e normalized residualdisplacement in case of elastic perfectly plastic material is32 64 and 1152 percent higher aer ten loading-unloadingwhen compared with the materials having tangent modulusof 0025119864119864 005119864119864 and 009119864119864 respectively Kadin et al [16]also observed that higher hardening results in lower residualstrain

Figure 8(b) shows the normalized residual displacement120596120596lowastres120596120596

lowastmax as a function of loading cycles with bilinear

kinematic hardening It is revealed from the gure that aerten loading unloading the normalized residual displacementof elastic perfectly plastic material is 361 826 and 2057percent higher than that with the materials having tangentmodulus of 0025119864119864 005119864119864 and 009119864119864 respectively Compar-ing the results with isotropic and kinematic hardening it isobserved that using kinematic hardening yields less residual

6 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(b)

F 7 Normalized contact load 1198751198751198751198751198750 as a function of loading cycles with (a) bilinear isotropic hardening (b) bilinear kinematichardening

0 2 4 6 8 10062

064

066

068

07

072

074

076

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 10055

06

065

07

075

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0= 0025

= 005

= 009

(b)

F 8 Normalized residual interference 120596120596lowastres120596120596lowastmax as a function of loading cycles with (a) bilinear isotropic hardening (b) bilinear

kinematic hardening

displacement than that with isotropic hardeningis ismorepronounced for the materials with high tangent modulusZolotarevskiy et al [29] studied the elastic plastic sphericalcontact under cyclic tangential loading in presliding with2 hardening and found the tangential displacement at thecompletion of the rst cycle is less in the case of kinematichardening compared to that in the isotropic hardening eagreement between Zolotarevskiy et al and present results isexcellent

Figure 9 shows the increment of residual interferenceof 120596120596lowastres with respect to 120596120596lowastres0 (residual interference aer rstunloading) during ten unloading We have studied the effectof hardening type also It can be seen from the gure thatthe increase of residual interference aer tenth unloadingwith respect to the residual interference aer rst unloadingranges from 4 to 45 percent irrespective of hardening typeand tangent modulus except the simulation with tangentmodulus of 009119864119864 using kinematic hardeninge simulation

ISRN Tribology 7

0 2 4 6 8 101

101

102

103

104

105

106

Isotropic hardening

Kinematic hardening

= 0

= 0025

= 005

= 009

Loading cycles

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

F 9 Normalized residual interference120596120596lowastresx120596120596lowastres0 as a function

of loading cycles for maximum loading 120596120596lowastmax = 100

of material with tangent modulus of 009119864119864 and kinematichardening resulted in 18 percent increase of residual interfer-ence aer ten unloading with respect to residual interferenceaer rst unloading e study of contact load and residualinterferences during ten loading unloading with tangentmodulus up to 009119864119864 and two different hardening type clearlyindicates that the interfacial parameters depend predomi-nantly on the extent of strain hardening and hardening rulewith the increase in tangent modulus

42 Interfacial Parameters with High Tangent Modulus It isnecessary to investigate the effect of strain hardening andhardening rule on interfacial parameters with higher tangentmodulus to understand the response of the materials withsignicant amount of strain hardening such as stainless steelstructural steel and aluminum alloys in repeated loadingunloading In our forth-coming simulations the tangentmodulus (119864119864119905119905) is varied according to a hardening parameter(119867119867) e hardening parameter is dened as119867119867 = 119864119864119905119905(119864119864 119864 119864119864119905119905)We have considered four different values of119867119867 covering widerange of tangent modulus to depict the effect of strain hard-ening in single asperity multiple loading unloading contactanalysis with other material properties being constant evalues of119867119867 used in this analysis are within range 0 le 119867119867 le 0119867as most of the practical materials falls in this range [30] evalue of 119867119867 equals to zero indicates elastic perfectly plasticmaterial behavior which is an idealized material behaviore hardening parameters used for this analysis and theircorresponding 119864119864119905119905 values are shown in Table 1

421 Mean Contact Pressure Distribution Figure 10(a) isthe plot of dimensionless mean contact pressure (119901119901119901119901) as afunction of ten loading cycles e maximum dimensionlessinterference for loading is 120596120596lowastmax = 1198670 (119875119875

lowastmax = 196 for

T 1 Different 119867119867 and 119864119864119905119905 values used for the study of strainhardening effect

119867119867 119864119864119905119905 in 119864119864 119864119864119905119905 (GPa)0 00 0001 90 6303 230 16105 330 231

elastic perfectly plastic material aer rst loading) rizmeret al [21] used 2 linear hardening and found dimensionlessmean contact pressure around 27 for normalized contactload of 200 Our simulation resulted dimensionless meancontact pressure 25 aer rst loading for dimensionlessinterference of 50erefore the present results correlate wellwith the ndings of rizmer et al [21] e mean contactpressures increase with the increase of tangent modulus forboth the isotropic and kinematic hardening rule For thetangent modulus ranging from 009119864119864 to 033119864119864 the increaseof mean contact pressures with isotropic hardening are in therange of 4656 to 118 percent than that elastic perfectly plasticmaterial aer ten loading cycles whereas the increase ofmeancontact pressures with kinematic hardening ranges from 42to 113 percent for the same range of tangent modulus aerten loading cycles is indicates that the effect of tangentmodulus or strain hardening on the mean contact pressureis signicantly more pronounced than the hardening rule

Figure 10(b) shows the dimensionless mean contactpressure during ten loading cycles for the maximum dimen-sionless interference of 120596120596lowastmax = 100 It is clear from thegure that the mean contact pressure intensied with theincrease in dimensionless interference of loading Meancontact pressure using isotropic hardening increased from1598 to 345 percent aer ten loading cycles for the fourindicated tangent modulus in the gure with the increase ofdimensionless interference of loading from 50 to 100 It isalso observed from the gure that a marginal deviation ofmean contact pressure is resulted with kinematic hardeningAs already mentioned earlier that the elastic perfectly plasticmaterial is insensitive to the hardening rule the evolution ofmean contact pressure depends predominantly on the strainhardening characteristics and the intensity of loading ratherthan the hardening rule

Figure 10(c) represents the evolution of dimensionlessmean contact pressure at the end of each seven loading cyclesfor the maximum dimensionless interference of 120596120596lowastmax = 200e evolution of mean contact pressure clearly indicatesthe same qualitative trend of Figures 10(a) and 10(b) Itcan be seen from the gure that the evolution of meancontact pressure with isotropic hardening is higher thanthat with kinematic hardening e dimensionless meancontact pressures with isotropic hardening are 64 89 and88 percent higher aer rst loading than that with kinematichardening for thematerials having tangentmodulus of 009119864119864023119864119864 and 033119864119864 respectively

Kral et al [17] simulated repeated indentation of a halfspace by a rigid sphere with isotropic hardening ey usedthree strain hardening exponent as 0 03 and 05 and

8 ISRN Tribology

0 2 4 6 8 1015

2

25

3

35

4

45

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Loading cycles

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(a)

0 2 4 6 8 102

3

4

5

6

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(b)

1 2 3 4 5 6 72

3

4

5

6

7

8

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(c)

F 10 Dimensionless contact pressure 119901119901119901119901119901 as a function of loading cycles for maximum loading (a) 120596120596lowastmax = 50 (b) 120596120596lowastmax = 100 (c)

120596120596lowastmax = 200

indentation load up to 300 times the load necessary for initialyield Kral et al inferred that the distribution of contactpressure over the contact area increases with increasing loadand strain hardening characteristics (Figures 4(b) and 8)ere is no scope of comparison as we have simulated witha deformable sphere against a rigid at but the ualitativenatures of present results (Figures 10(a)ndash10(c)) are in goodagreement with the results of Kral et al

e deviation of mean contact pressure with the increaseof loading cycles for varying tangent modulus in case ofisotropic and kinematic hardening is shown in Figure 11emaximum dimensionless interference of loading is 120596120596lowastmax =100 1199011199010 represents the mean contact pressure aer rstloading It is found from the gure that the variation of mean

contact pressure aer every cyclic loading depends on thetangent modulus or the strain hardening characteristics andhardening type It can also be seen from the gure that themean contact pressure decreases with the increasing loadingcycle for the materials with isotropic hardening irrespectiveof the extent of tangent modulus though the decrease is notso prominent with kinematic hardening Kral et al [17] alsomentioned that the maximum contact pressure is lower insecond load half cycle in case of an indentation of a halfspace by a rigid spherewith isotropic hardeningedecreaseof mean contact pressures with isotropic hardening aersecond loading compared to rst loading are 3 8 38and 02 percent for the materials with tangent modulus of 0009119864119864 023119864119864 and 033119864119864 respectivelye decrement of mean

ISRN Tribology 9

0 2 4 6 8 10

08

085

09

095

1

105

Loading cycles

Isotropic

Kinematic= 0 = 033

= 009

= 023N

orm

aliz

ed m

ean

co

nta

ct p

ress

ure

(

)

F 11 Normalized contact pressure 1199011199011199011199011199010 as a function ofloading cycles for maximum loading 120596120596lowastmax = 100

contact pressure with isotropic hardening is consistent withincreasing loading cycles aer tenth loading the decreasesare 1574 125 65 and 03 percent compared to the meancontact pressure aer rst loading for the materials with tan-gentmodulus of same order asmentioned for second loadingIt is therefore imperative that the maximum decrement rateof mean contact pressure is occurred aer second loadingcycle and the decreases in subsequent cycles are insignicantKral et al also observed the same pressure prole distributionaer second load half cycle However the variations of meancontact pressure with the increasing loading cycles are notpronounced in case of kinematic hardening

422 Dimensionless Load with Loading Cycles e variationof normalized contact load up to ten loading cycles forboth isotropic and kinematic hardening rule is plotted inFigure 12(a) 1198751198750 is the contact load aer rst loading andthe maximum dimensionless loading for the simulation ofrepeated loading unloading is 120596120596lowastmax = 100 e contact loaddecreases with the increase in loading cycle and the rateof decrease is the function of hardening rule and tangentmodulus e decrease of contact load with the increase ofloading cycle is more pronounced in isotropic hardeningthan the kinematic hardening e material with less tangentmodulus produced more variation in contact load with theincrease in each load cycle

Figure 12(b) shows the variation of normalized contactload up to seven cyclic loading for the maximum loadingof dimensionless interference 120596120596lowastmax = 200 Figures 12(a)and 12(b) demonstrate that the variation of contact loadwith each loading cycle signicantly depends on the extentof maximum loading interference e decrease of contactload with maximum dimensionless loading interference of200 aer seventh cyclic loading is 2 more even aer the

tenth cyclic loading with maximum dimensionless loadinginterference of 100 for the elastic perfectly plastic materialHowever the materials with high tangent modulus (119864119864119905119905 =033119864119864119864 and kinematic hardening produced no variation ofcontact load with the increase of loading cycle

423 Residual Displacements Figure 13(a) shows normal-ized residual interference 120596120596lowastres120596120596

lowastmax aer each unloading

during ten loading unloading cycles for maximum dimen-sionless interference of loading 120596120596lowastmax = 100 (119875119875

lowastmax = 459

for elastic perfectly plastic material) It reveals from thegure that the residual displacement depends predominantlyon the tangent modulus and hardening rule e residualdisplacement for the materials with the largest tangentmodulus (119864119864119905119905 = 033119864119864 ) using isotropic hardening is about61 that of the elastic perfectly plastic material aer rstunloading Kral et al [17] found half residual displacementfor material with strain hardening exponent 05 comparedwith the elastic perfectly plastic material aer rst unloadingunder perfect slip contact condition ey simulated withmaximum dimensionless load 119875119875lowastmax = 300 and inferredthat above 119875119875lowastmax = 50 the contact load increases signif-icantly with the increase of strain hardening for the sameindentation depth Chatterjee and Sahoo [25] reported thesame qualitative results while studying the effect of strainhardening during the loading of a deformable sphere againsta rigid at under full stick contact condition e residualdisplacement with isotropic hardening is signicantly higherthan that of with kinematic hardening for higher tangentmodulus er rst unloading the residual displacementswith isotropic hardening are 9 52 and 86 percent higher thanthe residual displacement with kinematic hardening for thematerials with tangent modulus of 009119864119864 023119864119864 and 033119864119864respectively

Figure 13(b) shows the normalized residual displacementaer each unloading for seven loading unloading cyclese maximum dimensionless interference of loading forthis simulation is 120596120596lowastmax = 200 e normalized residualdisplacements aer rst unloading increased up to 12 forthe increase of dimensionless interference of loading from100 to 200 Hardening rule and the extent of loading has asignicant effect on residual displacement

Figure 14(a) presents the increment of residual interfer-ence with the increase of unloading cycles compared with theresidual interference aer rst unloading120596120596lowastres0 is the residualinterference aer rst unloading It reveals from the gurethat the increases in residual interferences with isotropichardening are signicantly more than that with kinematichardening e gure also demonstrates that the incrementof residual interference with the increase in unloading cyclesdepends on tangent modulus Higher tangent modulus yieldsslightly less increment of residual interference with theincrease in unloading cycles

Figure 14(b) shows the normalized residual interferenceduring seven loading unloading cycles for the maximumdimensionless interferences of loading 120596120596lowastmax = 200 Fig-ures 14(a) and 14(b) indicates that higher dimensionlessinterference of loading produces marginally less increment

10 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

= 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(a)

1 2 3 4 5 6 7088

09

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

t = 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(b)

F 12 Normalized contact load 1198751198751198751198751198750 as a function of loading cycles for maximum loading (a) 120596120596lowastmax = 100 (b) 120596120596lowastmax = 200

0 2 4 6 8 1002

03

04

05

06

07

08

09

Isotropic

Kinematic

= 0

= 023

= 033

= 009

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

(a)

1 6 702

04

06

08

1

2 4 53

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

Loading cycles

(b)

F 13 Normalized residual interference 120596120596lowastres120596120596lowastmax as a function of loading cycles for maximum loading (a) 120596120596lowastmax = 100 (b) 120596120596

lowastmax = 200

of residual interference with increasing loading unloadingcycles is may cause early shakedown at higher maximumdimensionless interference of loading

424 Contact Area with Loading Cycles Figure 15(a)presents the normalized contact area at the end of eachloading 1198601198600 is the contact area at the end of rst loadinge present simulation consists ten loading unloading cyclesTwo different maximum dimensionless loadings (119875119875lowast = 48

and 116) are used in this simulation and the results arecompared with the available literature Ovcharenko et al[28] found in their experiments an increase of around 53and 8 of 1198601198601198751198601198600 with elastic perfectly plastic copper sphere(119864119864119875119864119864 = 400) using maximum dimensionless loading (119875119875lowast)of 48 and 116 at the end of tenth loading Zait et al [24]estimated 8-9 growth of 1198601198601198751198601198600 aer sixth loading with thedeformable sphere (119864119864119875119864119864 = 1000) against a rigid at usingmaximum dimensionless loading (119875119875lowast) of 40 We have useddeformable sphere of EY ratio 700 our simulation resulted

ISRN Tribology 11

0 2 4 6 8 101

101

102

103

104

105

106

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 0 23

= 033

= 009

(a)

1 2 3 4 5 6 71

1005

101

1015

102

1025

103

1035

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

E = 0

= 0 2

= 033

= 009

(b)

F 14 Normalized residual interference 120596120596lowastres120596120596lowastres0 as a function of loading cycles for maximum loading (a) 120596120596lowastmax = 100 (b) 120596120596

lowastmax = 200

0 2 4 6 8 101

105

11

115

12

125

No

rmal

ized

co

nta

ct a

rea

Present work = 48

Present work = 116

Ovcharenko et al 2007 = 48

Ovcharenko et al 2007 = 116Zait et al 2010 = 40

Loading cycles

(a)

0 2 4 6 8 10095

1

105

11

115

No

rmal

ized

co

nta

ct a

rea

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

(b)

F 15 Normalized contact area1198601198601198601198601198600 as a function of the number of loading cycles (a) comparison of different studies (b) comparisonof hardening models for maximum loading 120596120596lowastmax = 100

12 and 235 increase of 1198601198601198601198601198600 for the maximum loading(119875119875lowast) of 48 and 116 respectively aer tenth loading eagreement between the present results and the ndings of aitet al [24] is appreciably goodevariation of our resultswiththat of Ovcharenko et al [28] is due to the fact they have usedelastic at contrary to our assumption of rigid at oreoverthe variation can be attributed to some extent to the discrete

meshing of contact elements with nite resolution and themagnitude of the contact element stiffness used in the niteelement soware during evaluating contact radius It is alsoevident from the gure that the growth of normalized contactarea increases with the increase of maximum loading

Figure 15(b) shows the growth of normalized contact areaaer each loading cycle with four different tangent modules

12 ISRN Tribology

using isotropic and kinematic hardening It is evident fromthe plot that the contact area increases slightly aer eachloading cycles indicating marginal effect of strain hardeningin isotropic hardening e materials with kinematic hard-ening exhibit very less increase of contact area during tenloading cycles with maximum dimensionless interference ofloading 120596120596lowast = 100 Small discrepancy was observed for theresults of higher tangent modulus (119864119864119905119905 = 023119864119864 and 033119864119864)in kinematic hardening probably due to the discretization ofcontact elements as discussed earlier

5 Conclusion

e effect of strain hardening and hardening models onthe contact parameters during the multiple normal loadingunloading under full stick contact were analyzed with niteelement soware ANSYS e sphere was deformed againsta rigid at with the maximum interference up to 200 timesthe interference necessary for initial yield Maximum tenrepeated loading unloading cycles were performed withdifferent plastic properties using isotropic and kinematichardening models e contact load decreases and theresidual interference increases with the increase in loadingunloading cycles e variations of contact parameters withthe increase in loading unloading cycles are predominantlydue to the contact conditions Under full stick contactcondition the effect of hardening model on the variationof contact parameters is more pronounced at large tangentmodulus us small variation of tangent modulus resultedsame shakedown behavior and similar interfacial parametersin repeated loading unloading with both the hardening ruleas reported in literature However at high tangent modu-lus the strain hardening and hardening rules have stronginuences on contact parameters like mean contact pressurecontact load residual interference and contact area

It was found that the evolution of mean contact pressuredepends predominantly on the strain hardening characteris-tics and the intensity of loading e mean contact pressureincreases with the increase in tangent modulus and intensityof loading However the variation of mean contact pressureowing to the change of hardening model is not signicanteven at large load e mean contact pressure decreases withthe increase in loading cycles in isotropic hardening isdecrease is prominent aer second loading although dueto the increased interfacial conformity the decrease is notso pronounced in subsequent cycles Kinematic hardeningproduced insignicant variation of mean contact pressurewith increasing loading cycles e contact load decreaseswith the increase in loading cycles e rate of decreaseincreases with tangent modulus and the intensity of maxi-mum loading e decrease of contact load is large enoughin isotropic hardening for the materials with low tangentmodulus compared to kinematic hardening for the materialswith high tangent modulus

e residual interference varies with both tangent modu-lus and hardening modele residual interference decreaseswith the increase in tangent modulus and it is signicantly

higher in isotropic hardening compared to that with kine-matic hardening e increase in residual interference withthe increasing number of unloading cycles was severelyaffected by hardening model at high tangent modulus econtact area increases slightly aer each loading cycles inisotropic hardening whereas the materials with kinematichardening exhibit very less increase of contact area egrowth of contact area was found to be independent of strainhardening but increases with the increase in intensity of load-ing e effect of hardening model on contact parameters athigh tangent modulus clearly indicates different shakedownbehavior for isotropic and kinematic hardening for repeatednormal loading unloading under full stick contact conditione analysis of hysteretic loops at high tangent modulus is aninteresting eld for future study

Nomenclature

119886119886 Contact area radius119864119864 Modulus of elasticity of the sphere119884119884 Yield Strength of the sphere material119860119860 Real contact area119877119877 Radius of the sphere119875119875 Contact load120596120596 Interference120584120584 Poissonrsquos ratio of sphere119901119901 Mean contact pressure119864119864119905119905 Tangent modulus of the sphere119875119875lowast Dimensionless contact load 119875119875119875119875119875119888119888 in stick contact119860119860lowast Dimensionless contact area 119860119860119875119860119860119888119888 in stick contact120596120596lowast Dimensionless interference 120596120596120596120596119888119888 in stick contact

Subscripts

119888119888 Critical valuesres Residual values following unloadingmax Maximum values during loading-unloading process

Superscripts

lowast∶ Dimensionless

References

[1] S Majumder N E McGruer G G Adams et al ldquoStudy ofcontacts in an electrostatically actuated microswitchrdquo Sensorsand Actuators A vol 93 no 1 pp 19ndash26 2001

[2] S Majumder N E McGruer and G G Adams ldquoContact resis-tance and adhesion in a MEMS microswitchrdquo in Proceedingsof STLEASME Joint International Tribology Conference TRIB-270 pp 1ndash6 2003

[3] W Peng and B Bhushan ldquoTransient analysis of sliding contactof layered elasticplastic solids with rough surfacesrdquo Microsys-tem Technologies vol 9 no 5 pp 340ndash345 2003

[4] J Y Kim A Baltazar and S I Rokhlin ldquoUltrasonic assessmentof rough surface contact between solids from elastoplasticloading-unloading hysteresis cyclerdquo Journal of the Mechanicsand Physics of Solids vol 52 no 8 pp 1911ndash1934 2004

ISRN Tribology 13

[5] A Kapoor F J Franklin S K Wong and M Ishida ldquoSurfaceroughness and plastic ow in rail wheel contactrdquoWear vol 253no 1-2 pp 257ndash264 2002

[6] J A Williams I N Dyson and A Kapoor ldquoRepeated loadingresidual stresses shakedown and tribologyrdquo Journal of Materi-als Research vol 14 no 4 pp 1548ndash1559 1999

[7] HHertz ldquoUumlber die Beruumlhrung fester elastischer Koumlperrdquo Journalfuumlr die reine und angewandte Mathematik vol 92 pp 156ndash1711882

[8] J E Merwin and K L Johnson ldquoAn analysis of plasticdeformation in rolling contactrdquo Proceedings of Institution ofMechanical Engineers vol 177 pp 676ndash685 1963

[9] S M Kulkarni G T Hahn C A Rubin and V BhargavaldquoElastoplastic nite element analysis of three-dimensional purerolling contact at the shakedown limitrdquo Journal of AppliedMechanics Transactions ASME vol 57 no 1 pp 57ndash64 1990

[10] V Bhargava G T Hahn and C A Rubin ldquoAn elastic-plasticnite element model of rolling contact Part 1 analysis of singlecontactsrdquo Journal of AppliedMechanics TransactionsASME vol52 no 1 pp 67ndash74 1985

[11] V Bhargava G T Hahn and C A Rubin ldquoAn elastic-plasticnite element model of rolling contact Part 2 analysis ofrepeated contactsrdquo Journal of Applied Mechanics TransactionsASME vol 52 no 1 pp 75ndash82 1985

[12] P Sahoo D Adhikary and K Saha ldquoFinite element basedelastic-plastic contact of fractal surfaces considering strainhardeningrdquo Journal of Tribology and Surface Engineering vol 1pp 39ndash56 2010

[13] L Kogut and I Etsion ldquoElastic-plastic contact analysis of asphere and a rigid atrdquo Journal of Applied Mechanics Transac-tions ASME vol 69 no 5 pp 657ndash662 2002

[14] I Etsion Y Kligerman and Y Kadin ldquoUnloading of an elastic-plastic loaded spherical contactrdquo International Journal of Solidsand Structures vol 42 no 13 pp 3716ndash3729 2005

[15] R Jackson I Chusoipin and I Green ldquoA nite element study ofthe residual stress and deformation in hemispherical contactsrdquoJournal of Tribology vol 127 no 3 pp 484ndash493 2005

[16] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo Interna-tional Journal of Solids and Structures vol 43 no 22-23 pp7119ndash7127 2006

[17] E R Kral K Komvopoulos and D B Bogy ldquoElastic-plasticnite element analysis of repeated indentation of a half-spaceby a rigid sphererdquo Journal of Applied Mechanics TransactionsASME vol 60 no 4 pp 829ndash841 1993

[18] P Sahoo B Chatterjee andD Adhikary ldquoFinite element basedElastic-plastic contact behavior of a sphere against a rigid at-Effect of Strain Hardeningrdquo International Journal of Engineeringand Technology vol 2 no 1 pp 1ndash6 2010

[19] B Chatterjee and P Sahoo ldquoEffect of strain hardening onunloading of a deformable sphere loaded against a rigid at-a nite element studyrdquo International Journal of Engineering andTechnology vol 2 no 4 pp 225ndash233 2010

[20] P Sahoo Engineering Tribology PHI Learning Private LimitedNew Delhi India 2009

[21] V Brizmer Y Kligerman and I Etsion ldquoe effect of contactconditions and material properties on the elasticity terminusof a spherical contactrdquo International Journal of Solids andStructures vol 43 no 18-19 pp 5736ndash5749 2006

[22] B Chatterjee and P Sahoo ldquoElastic-plastic contact of adeformable sphere against a rigid at for varying material

properties under full stick contact conditionrdquo Tribology inIndustry vol 33 no 4 pp 164ndash172 2011

[23] Y Zait Y Kligerman and I Etsion ldquoUnloading of an elastic-plastic spherical contact under stick contact conditionrdquo Inter-national Journal of Solids and Structures vol 47 no 7-8 pp990ndash997 2010

[24] Y Zait V Zolotarevsky Y Kligerman and I Etsion ldquoMultiplenormal loading-unloading cycles of a spherical contact understick contact conditionrdquo Journal of Tribology vol 132 no 4Article ID 041401 7 pages 2010

[25] BChatterjee andP Sahoo ldquoEffect of strain hardening on elastic-plastic contact of a deformable sphere against a rigid at underfull contact conditionrdquo Advances in Tribology vol 2012 ArticleID 472794 8 pages 2012

[26] K L Johnson Contact Mechanics Cambridge University PressCambridge Mass USA 1985

[27] S Shankar and M M Mayuram ldquoEffect of strain hardening inelastic-plastic transition behavior in a hemisphere in contactwith a rigid atrdquo International Journal of Solids and Structuresvol 45 no 10 pp 3009ndash3020 2008

[28] A Ovcharenko G Halperin G Verberne and I Etsion ldquoInsitu investigation of the contact area in elastic-plastic sphericalcontact during loading-unloadingrdquo Tribology Letters vol 25no 2 pp 153ndash160 2007

[29] V Zolotarevskiy Y Kligerman and I Etsion ldquoElastic-plasticspherical contact under cyclic tangential loading in pre-slidingrdquoWear vol 270 no 11-12 pp 888ndash894 2011

[30] F Wang and L M Keer ldquoNumerical simulation for threedimensional elastic-plastic contact with hardening behaviorrdquoJournal of Tribology vol 127 no 3 pp 494ndash502 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


2 ISRN Tribology

material up to 10 percent of modulus of elasticity yieldedin maximum 20 percent difference of contact parametersEtsion et al [14] extended their investigation for unloadingunder perfect slip contact condition with tangent modulusof 2 percent of modulus of elasticity ey used bilinearisotropic hardening Jackson et al [15] studied the effect ofelastic properties of material on residual deformation andconcentrated on the residual stresses of the sphere underperfect slip contact condition Kadin et al [16] offered amultiple loading-unloading model with isotropic hardeningusing 002119864119864 tangent modulus under perfect slip contactconditioney found that the incipient interference requiredfor the secondary plastic ow in the peripheral zone increaseswith the increase in tangent modulus Kral et al [17]performed repeated normal indentation of a half space bya rigid sphere under perfect slip contact condition usingisotropic hardening It was shown that the effect of plasticproperties like strain hardening is more pronounced ratherthan the elastic properties of the material on the contactparameters during loading unloading in the elastic plasticand plastic region ey used strain-hardening exponent(119899119899) up to 05 Sahoo et al [18] studied the effect of strainhardening for elastic plastic loading of a deformable sphereagainst a rigid at for frictionless and non-adhesive contactby varying the hardening parameter (119867119867) up to 05 Variationof hardening parameter up to 05 enabled them to study theeffect of tangent modulus as high as 33 percent of modulus ofelasticity ey inferred that the increase in strain hardeningincreases the resistance to deformation of a material and thematerial becomes capable of carrying higher amount of loadin a smaller contact area However Chatterjee and Sahoo[19] observed that the contact loads aer rst and secondloading were same under perfect slip contact condition fornon-hardening material as well as material with hardeningparameter of 05

e aforementioned reviews are related to frictionlesscontact condition Friction is useful in many engineeringapplications such as brakes clutches bolts nuts drivingwheels on automobiles and so forth whereas unproductivefriction takes place in engines gears cams bearings and seals[20] Brizmer et al [21] found that the pressure distributionand the stress eld near the contact surfaces at low contactloads are different for perfect slip and full stick contactcondition Chatterjee and Sahoo [22] studied the effect ofelastic properties of material for an elastic plastic loadingof a deformable sphere against a rigid at under full stickcontact condition Zait et al [23] performed the unloadingof an elastic plastic spherical contact under full stick contactcondition ey found a substantial variation in load areacurve during unloading under full stick contact conditioncompared to that under perfect slip contact condition Zaitet al [24] expanded the previous single complete unloadingunder full stick contact condition to include multiple normalloading unloading ey used bilinear kinematic hardeningwith tangent modulus as 2 percent of modulus of elastic-ity ey concluded identical shakedown behavior for bothisotropic and kinematic hardening Chatterjee and Sahoo[25] studied the effect of strain hardening on the elastic plasticcontact loading of a deformable sphere against a rigid at

under full stick contact condition with two different hard-ening rules It was observed that the material with isotropichardening has higher amount of load carrying capacity thanthat of kinematic hardening particularly for higher strainhardening Johnson [26]mentioned that the surface tractionsand nonlinear material properties are the factors to inuencethe reversibility of cyclic normal loading in spherical contactproblems us under a specic contact condition (slip orstick) the plastic properties of thematerial and the hardeningrule have a particular importance inmultiple normal loading-unloading ough under perfect slip (frictionless) contactcondition the effect of strain hardening on loading unloadingof a deformable sphere against a rigid at is available inliterature the effect of plastic properties and hardeningmodelfor multiple normal loading-unloading for a deformablespherical contact under full stick contact condition is stillmissing

e present investigation therefore aims at studying theeffect of strain hardening and hardening models on contactparameters during multiple normal loading-unloading of adeformable sphere against a rigid at under full stick contactcondition using commercial nite element soware ANSS

2 Theoretical Background

Figure 1 shows the contact of a deformable sphere and arigid at e dashed line presents the original contour ofthe sphere having a radius of 119877119877 and the rigid at before thedeformatione solid line shows the loading phase with theinterference (120596120596) corresponding to the contact radius (119886119886) andthe contact load (119875119875)

Multiple loading unloading consists repeated normalloading and complete unloading During loading phase theinterference (120596120596) is increased up to the specied maximumdimensionless interference 120596120596lowastmax During unloading theinterference (120596120596) is gradually reduced At the end of theunloading under zero contact load and contact area thesphere has a residual interference (120596120596res) erefore the orig-inal un-deformed spherical geometry is not fully recoverede original prole the deformed shape aer loading phaseand the sphere prole at the end of unloading in singleloading unloading are represented in Figure 2 Stick contactcondition is applied at the interface of the deformed sphereand the rigid at e stick contact condition prevents thecontact point of the deformed sphere with the rigid at fromthe relative displacement in the radial direction but it allowsthe axial detachment of the sphere surface from the rigid atduring unloading

In the present study multiple normal loading-unloadingis simulated considering both the isotropic and kinematichardening models Figure 3 describes the difference betweenisotropic hardening and kinematic hardening by describ-ing the development of the yield surface with progressiveyielding In isotropic or work hardening the yield surfaceis uniformly spread out from the center while in kinematichardening the yield surface translates in stress space withconstant size e variation of different contact parameters isconsidered with the increase in loading or unloading cycles

ISRN Tribology 3



120596120596119888119888 = 10076551007655119862119862119907119907120587120587 100765010076501 minus 12058412058421007666100766621007652100765211988411988411986411986410076681007668100766810076682

times 119877119877 10076501007650682120584120584 minus 1205848120584 100765010076501205841205842 + 005861007666100766610076661007666

119875119875119888119888 =120587120587120584119884119884611986211986211990711990712058410076521007652119877119877 100765010076501 minus 120584120584210076661007666 10076521007652

11988411988411986411986410076681007668100766810076682

times 10076501007650888120584120584 minus 101120584 100765010076501205841205842 + 00891007666100766610076661007666

119860119860119888119888 = 120587120587120596120596119888119888119877119877

(1)




Sphere profile



res

max






4 ISRN Tribology


1

2

(a)


1

2

(b)


1

2 Elements

Elements

2 Elements


Stress

Strain






ISRN Tribology 5

0 20 40 60 80 1000

100

200

300

400

500

1st loading

Subse

quent l

oadin

g

unload

ing

cycle

s


Dim

ensi

on

less

co

nta

ct lo

ad (

)

= 0025 kinematic

(a)

88 90 92 94 96 98 100

430

440

450

460

470

480


Dim

ensi

on

less

co

nta

ct l

oad

(

)

= 0025 kinematic

(b)

67 68 69 70 71 72 730

5

10

15

20

25

30


Dim

ensi

on

less

co

nta

ct l

oad

(

)

= 0025 kinematic

(c)










6 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(b)


0 2 4 6 8 10062

064

066

068

07

072

074

076

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 10055

06

065

07

075

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0= 0025

= 005

= 009

(b)


kinematic hardening



ISRN Tribology 7

0 2 4 6 8 101

101

102

103

104

105

106

Isotropic hardening

Kinematic hardening

= 0

= 0025

= 005

= 009

Loading cycles

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0






lowastmax = 196 for


119867119867 119864119864119905119905 in 119864119864 119864119864119905119905 (GPa)0 00 0001 90 6303 230 16105 330 231





8 ISRN Tribology

0 2 4 6 8 1015

2

25

3

35

4

45

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Loading cycles

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(a)

0 2 4 6 8 102

3

4

5

6

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(b)

1 2 3 4 5 6 72

3

4

5

6

7

8

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(c)


120596120596lowastmax = 200




ISRN Tribology 9

0 2 4 6 8 10

08

085

09

095

1

105

Loading cycles

Isotropic

Kinematic= 0 = 033

= 009

= 023N

orm

aliz

ed m

ean

co

nta

ct p

ress

ure

(

)









lowastmax = 459





10 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

= 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(a)

1 2 3 4 5 6 7088

09

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

t = 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(b)


0 2 4 6 8 1002

03

04

05

06

07

08

09

Isotropic

Kinematic

= 0

= 023

= 033

= 009

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

(a)

1 6 702

04

06

08

1

2 4 53

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

Loading cycles

(b)


lowastmax = 200




ISRN Tribology 11

0 2 4 6 8 101

101

102

103

104

105

106

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 0 23

= 033

= 009

(a)

1 2 3 4 5 6 71

1005

101

1015

102

1025

103

1035

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

E = 0

= 0 2

= 033

= 009

(b)


lowastmax = 200

0 2 4 6 8 101

105

11

115

12

125

No

rmal

ized

co

nta

ct a

rea

Present work = 48

Present work = 116



Loading cycles

(a)

0 2 4 6 8 10095

1

105

11

115

No

rmal

ized

co

nta

ct a

rea

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

(b)





12 ISRN Tribology


5 Conclusion





Nomenclature


Subscripts


Superscripts


References





ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









ISRN Tribology 3



120596120596119888119888 = 10076551007655119862119862119907119907120587120587 100765010076501 minus 12058412058421007666100766621007652100765211988411988411986411986410076681007668100766810076682

times 119877119877 10076501007650682120584120584 minus 1205848120584 100765010076501205841205842 + 005861007666100766610076661007666

119875119875119888119888 =120587120587120584119884119884611986211986211990711990712058410076521007652119877119877 100765010076501 minus 120584120584210076661007666 10076521007652

11988411988411986411986410076681007668100766810076682

times 10076501007650888120584120584 minus 101120584 100765010076501205841205842 + 00891007666100766610076661007666

119860119860119888119888 = 120587120587120596120596119888119888119877119877

(1)




Sphere profile



res

max






4 ISRN Tribology


1

2

(a)


1

2

(b)


1

2 Elements

Elements

2 Elements


Stress

Strain






ISRN Tribology 5

0 20 40 60 80 1000

100

200

300

400

500

1st loading

Subse

quent l

oadin

g

unload

ing

cycle

s


Dim

ensi

on

less

co

nta

ct lo

ad (

)

= 0025 kinematic

(a)

88 90 92 94 96 98 100

430

440

450

460

470

480


Dim

ensi

on

less

co

nta

ct l

oad

(

)

= 0025 kinematic

(b)

67 68 69 70 71 72 730

5

10

15

20

25

30


Dim

ensi

on

less

co

nta

ct l

oad

(

)

= 0025 kinematic

(c)










6 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(b)


0 2 4 6 8 10062

064

066

068

07

072

074

076

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 10055

06

065

07

075

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0= 0025

= 005

= 009

(b)


kinematic hardening



ISRN Tribology 7

0 2 4 6 8 101

101

102

103

104

105

106

Isotropic hardening

Kinematic hardening

= 0

= 0025

= 005

= 009

Loading cycles

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0






lowastmax = 196 for


119867119867 119864119864119905119905 in 119864119864 119864119864119905119905 (GPa)0 00 0001 90 6303 230 16105 330 231





8 ISRN Tribology

0 2 4 6 8 1015

2

25

3

35

4

45

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Loading cycles

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(a)

0 2 4 6 8 102

3

4

5

6

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(b)

1 2 3 4 5 6 72

3

4

5

6

7

8

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(c)


120596120596lowastmax = 200




ISRN Tribology 9

0 2 4 6 8 10

08

085

09

095

1

105

Loading cycles

Isotropic

Kinematic= 0 = 033

= 009

= 023N

orm

aliz

ed m

ean

co

nta

ct p

ress

ure

(

)









lowastmax = 459





10 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

= 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(a)

1 2 3 4 5 6 7088

09

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

t = 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(b)


0 2 4 6 8 1002

03

04

05

06

07

08

09

Isotropic

Kinematic

= 0

= 023

= 033

= 009

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

(a)

1 6 702

04

06

08

1

2 4 53

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

Loading cycles

(b)


lowastmax = 200




ISRN Tribology 11

0 2 4 6 8 101

101

102

103

104

105

106

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 0 23

= 033

= 009

(a)

1 2 3 4 5 6 71

1005

101

1015

102

1025

103

1035

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

E = 0

= 0 2

= 033

= 009

(b)


lowastmax = 200

0 2 4 6 8 101

105

11

115

12

125

No

rmal

ized

co

nta

ct a

rea

Present work = 48

Present work = 116



Loading cycles

(a)

0 2 4 6 8 10095

1

105

11

115

No

rmal

ized

co

nta

ct a

rea

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

(b)





12 ISRN Tribology


5 Conclusion





Nomenclature


Subscripts


Superscripts


References





ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









4 ISRN Tribology


1

2

(a)


1

2

(b)


1

2 Elements

Elements

2 Elements


Stress

Strain






ISRN Tribology 5

0 20 40 60 80 1000

100

200

300

400

500

1st loading

Subse

quent l

oadin

g

unload

ing

cycle

s


Dim

ensi

on

less

co

nta

ct lo

ad (

)

= 0025 kinematic

(a)

88 90 92 94 96 98 100

430

440

450

460

470

480


Dim

ensi

on

less

co

nta

ct l

oad

(

)

= 0025 kinematic

(b)

67 68 69 70 71 72 730

5

10

15

20

25

30


Dim

ensi

on

less

co

nta

ct l

oad

(

)

= 0025 kinematic

(c)










6 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(b)


0 2 4 6 8 10062

064

066

068

07

072

074

076

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 10055

06

065

07

075

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0= 0025

= 005

= 009

(b)


kinematic hardening



ISRN Tribology 7

0 2 4 6 8 101

101

102

103

104

105

106

Isotropic hardening

Kinematic hardening

= 0

= 0025

= 005

= 009

Loading cycles

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0






lowastmax = 196 for


119867119867 119864119864119905119905 in 119864119864 119864119864119905119905 (GPa)0 00 0001 90 6303 230 16105 330 231





8 ISRN Tribology

0 2 4 6 8 1015

2

25

3

35

4

45

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Loading cycles

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(a)

0 2 4 6 8 102

3

4

5

6

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(b)

1 2 3 4 5 6 72

3

4

5

6

7

8

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(c)


120596120596lowastmax = 200




ISRN Tribology 9

0 2 4 6 8 10

08

085

09

095

1

105

Loading cycles

Isotropic

Kinematic= 0 = 033

= 009

= 023N

orm

aliz

ed m

ean

co

nta

ct p

ress

ure

(

)









lowastmax = 459





10 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

= 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(a)

1 2 3 4 5 6 7088

09

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

t = 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(b)


0 2 4 6 8 1002

03

04

05

06

07

08

09

Isotropic

Kinematic

= 0

= 023

= 033

= 009

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

(a)

1 6 702

04

06

08

1

2 4 53

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

Loading cycles

(b)


lowastmax = 200




ISRN Tribology 11

0 2 4 6 8 101

101

102

103

104

105

106

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 0 23

= 033

= 009

(a)

1 2 3 4 5 6 71

1005

101

1015

102

1025

103

1035

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

E = 0

= 0 2

= 033

= 009

(b)


lowastmax = 200

0 2 4 6 8 101

105

11

115

12

125

No

rmal

ized

co

nta

ct a

rea

Present work = 48

Present work = 116



Loading cycles

(a)

0 2 4 6 8 10095

1

105

11

115

No

rmal

ized

co

nta

ct a

rea

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

(b)





12 ISRN Tribology


5 Conclusion





Nomenclature


Subscripts


Superscripts


References





ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









ISRN Tribology 5

0 20 40 60 80 1000

100

200

300

400

500

1st loading

Subse

quent l

oadin

g

unload

ing

cycle

s


Dim

ensi

on

less

co

nta

ct lo

ad (

)

= 0025 kinematic

(a)

88 90 92 94 96 98 100

430

440

450

460

470

480


Dim

ensi

on

less

co

nta

ct l

oad

(

)

= 0025 kinematic

(b)

67 68 69 70 71 72 730

5

10

15

20

25

30


Dim

ensi

on

less

co

nta

ct l

oad

(

)

= 0025 kinematic

(c)










6 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(b)


0 2 4 6 8 10062

064

066

068

07

072

074

076

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 10055

06

065

07

075

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0= 0025

= 005

= 009

(b)


kinematic hardening



ISRN Tribology 7

0 2 4 6 8 101

101

102

103

104

105

106

Isotropic hardening

Kinematic hardening

= 0

= 0025

= 005

= 009

Loading cycles

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0






lowastmax = 196 for


119867119867 119864119864119905119905 in 119864119864 119864119864119905119905 (GPa)0 00 0001 90 6303 230 16105 330 231





8 ISRN Tribology

0 2 4 6 8 1015

2

25

3

35

4

45

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Loading cycles

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(a)

0 2 4 6 8 102

3

4

5

6

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(b)

1 2 3 4 5 6 72

3

4

5

6

7

8

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(c)


120596120596lowastmax = 200




ISRN Tribology 9

0 2 4 6 8 10

08

085

09

095

1

105

Loading cycles

Isotropic

Kinematic= 0 = 033

= 009

= 023N

orm

aliz

ed m

ean

co

nta

ct p

ress

ure

(

)









lowastmax = 459





10 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

= 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(a)

1 2 3 4 5 6 7088

09

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

t = 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(b)


0 2 4 6 8 1002

03

04

05

06

07

08

09

Isotropic

Kinematic

= 0

= 023

= 033

= 009

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

(a)

1 6 702

04

06

08

1

2 4 53

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

Loading cycles

(b)


lowastmax = 200




ISRN Tribology 11

0 2 4 6 8 101

101

102

103

104

105

106

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 0 23

= 033

= 009

(a)

1 2 3 4 5 6 71

1005

101

1015

102

1025

103

1035

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

E = 0

= 0 2

= 033

= 009

(b)


lowastmax = 200

0 2 4 6 8 101

105

11

115

12

125

No

rmal

ized

co

nta

ct a

rea

Present work = 48

Present work = 116



Loading cycles

(a)

0 2 4 6 8 10095

1

105

11

115

No

rmal

ized

co

nta

ct a

rea

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

(b)





12 ISRN Tribology


5 Conclusion





Nomenclature


Subscripts


Superscripts


References





ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









6 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

No

rmal

ized

co

nta

ct lo

ad (

)

= 0

= 0025

= 005

= 009

(b)


0 2 4 6 8 10062

064

066

068

07

072

074

076

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0

= 0025

= 005

= 009

(a)

0 2 4 6 8 10055

06

065

07

075

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

= 0= 0025

= 005

= 009

(b)


kinematic hardening



ISRN Tribology 7

0 2 4 6 8 101

101

102

103

104

105

106

Isotropic hardening

Kinematic hardening

= 0

= 0025

= 005

= 009

Loading cycles

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0






lowastmax = 196 for


119867119867 119864119864119905119905 in 119864119864 119864119864119905119905 (GPa)0 00 0001 90 6303 230 16105 330 231





8 ISRN Tribology

0 2 4 6 8 1015

2

25

3

35

4

45

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Loading cycles

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(a)

0 2 4 6 8 102

3

4

5

6

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(b)

1 2 3 4 5 6 72

3

4

5

6

7

8

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(c)


120596120596lowastmax = 200




ISRN Tribology 9

0 2 4 6 8 10

08

085

09

095

1

105

Loading cycles

Isotropic

Kinematic= 0 = 033

= 009

= 023N

orm

aliz

ed m

ean

co

nta

ct p

ress

ure

(

)









lowastmax = 459





10 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

= 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(a)

1 2 3 4 5 6 7088

09

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

t = 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(b)


0 2 4 6 8 1002

03

04

05

06

07

08

09

Isotropic

Kinematic

= 0

= 023

= 033

= 009

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

(a)

1 6 702

04

06

08

1

2 4 53

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

Loading cycles

(b)


lowastmax = 200




ISRN Tribology 11

0 2 4 6 8 101

101

102

103

104

105

106

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 0 23

= 033

= 009

(a)

1 2 3 4 5 6 71

1005

101

1015

102

1025

103

1035

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

E = 0

= 0 2

= 033

= 009

(b)


lowastmax = 200

0 2 4 6 8 101

105

11

115

12

125

No

rmal

ized

co

nta

ct a

rea

Present work = 48

Present work = 116



Loading cycles

(a)

0 2 4 6 8 10095

1

105

11

115

No

rmal

ized

co

nta

ct a

rea

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

(b)





12 ISRN Tribology


5 Conclusion





Nomenclature


Subscripts


Superscripts


References





ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









ISRN Tribology 7

0 2 4 6 8 101

101

102

103

104

105

106

Isotropic hardening

Kinematic hardening

= 0

= 0025

= 005

= 009

Loading cycles

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0






lowastmax = 196 for


119867119867 119864119864119905119905 in 119864119864 119864119864119905119905 (GPa)0 00 0001 90 6303 230 16105 330 231





8 ISRN Tribology

0 2 4 6 8 1015

2

25

3

35

4

45

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Loading cycles

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(a)

0 2 4 6 8 102

3

4

5

6

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(b)

1 2 3 4 5 6 72

3

4

5

6

7

8

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(c)


120596120596lowastmax = 200




ISRN Tribology 9

0 2 4 6 8 10

08

085

09

095

1

105

Loading cycles

Isotropic

Kinematic= 0 = 033

= 009

= 023N

orm

aliz

ed m

ean

co

nta

ct p

ress

ure

(

)









lowastmax = 459





10 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

= 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(a)

1 2 3 4 5 6 7088

09

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

t = 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(b)


0 2 4 6 8 1002

03

04

05

06

07

08

09

Isotropic

Kinematic

= 0

= 023

= 033

= 009

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

(a)

1 6 702

04

06

08

1

2 4 53

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

Loading cycles

(b)


lowastmax = 200




ISRN Tribology 11

0 2 4 6 8 101

101

102

103

104

105

106

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 0 23

= 033

= 009

(a)

1 2 3 4 5 6 71

1005

101

1015

102

1025

103

1035

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

E = 0

= 0 2

= 033

= 009

(b)


lowastmax = 200

0 2 4 6 8 101

105

11

115

12

125

No

rmal

ized

co

nta

ct a

rea

Present work = 48

Present work = 116



Loading cycles

(a)

0 2 4 6 8 10095

1

105

11

115

No

rmal

ized

co

nta

ct a

rea

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

(b)





12 ISRN Tribology


5 Conclusion





Nomenclature


Subscripts


Superscripts


References





ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









8 ISRN Tribology

0 2 4 6 8 1015

2

25

3

35

4

45

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Loading cycles

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(a)

0 2 4 6 8 102

3

4

5

6

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(b)

1 2 3 4 5 6 72

3

4

5

6

7

8

Loading cycles

Isotropic hardening

Kinematic hardening

= 033

= 023

= 009

= 0

Dim

ensi

on

less

mea

n c

on

tact

pre

ssu

re (

)

(c)


120596120596lowastmax = 200




ISRN Tribology 9

0 2 4 6 8 10

08

085

09

095

1

105

Loading cycles

Isotropic

Kinematic= 0 = 033

= 009

= 023N

orm

aliz

ed m

ean

co

nta

ct p

ress

ure

(

)









lowastmax = 459





10 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

= 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(a)

1 2 3 4 5 6 7088

09

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

t = 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(b)


0 2 4 6 8 1002

03

04

05

06

07

08

09

Isotropic

Kinematic

= 0

= 023

= 033

= 009

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

(a)

1 6 702

04

06

08

1

2 4 53

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

Loading cycles

(b)


lowastmax = 200




ISRN Tribology 11

0 2 4 6 8 101

101

102

103

104

105

106

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 0 23

= 033

= 009

(a)

1 2 3 4 5 6 71

1005

101

1015

102

1025

103

1035

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

E = 0

= 0 2

= 033

= 009

(b)


lowastmax = 200

0 2 4 6 8 101

105

11

115

12

125

No

rmal

ized

co

nta

ct a

rea

Present work = 48

Present work = 116



Loading cycles

(a)

0 2 4 6 8 10095

1

105

11

115

No

rmal

ized

co

nta

ct a

rea

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

(b)





12 ISRN Tribology


5 Conclusion





Nomenclature


Subscripts


Superscripts


References





ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









ISRN Tribology 9

0 2 4 6 8 10

08

085

09

095

1

105

Loading cycles

Isotropic

Kinematic= 0 = 033

= 009

= 023N

orm

aliz

ed m

ean

co

nta

ct p

ress

ure

(

)









lowastmax = 459





10 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

= 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(a)

1 2 3 4 5 6 7088

09

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

t = 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(b)


0 2 4 6 8 1002

03

04

05

06

07

08

09

Isotropic

Kinematic

= 0

= 023

= 033

= 009

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

(a)

1 6 702

04

06

08

1

2 4 53

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

Loading cycles

(b)


lowastmax = 200




ISRN Tribology 11

0 2 4 6 8 101

101

102

103

104

105

106

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 0 23

= 033

= 009

(a)

1 2 3 4 5 6 71

1005

101

1015

102

1025

103

1035

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

E = 0

= 0 2

= 033

= 009

(b)


lowastmax = 200

0 2 4 6 8 101

105

11

115

12

125

No

rmal

ized

co

nta

ct a

rea

Present work = 48

Present work = 116



Loading cycles

(a)

0 2 4 6 8 10095

1

105

11

115

No

rmal

ized

co

nta

ct a

rea

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

(b)





12 ISRN Tribology


5 Conclusion





Nomenclature


Subscripts


Superscripts


References





ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









10 ISRN Tribology

0 2 4 6 8 1009

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

= 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(a)

1 2 3 4 5 6 7088

09

092

094

096

098

1

Loading cycles

= 0= 009 isotropic

= 023 isotropic

t = 033 isotropic

= 009 kinematic

= 023 kinematic

= 033 kinematic

No

rmal

ized

co

nta

ct lo

ad (

)

(b)


0 2 4 6 8 1002

03

04

05

06

07

08

09

Isotropic

Kinematic

= 0

= 023

= 033

= 009

Loading cycles

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

(a)

1 6 702

04

06

08

1

2 4 53

No

rmal

ized

res

idu

al d

isp

lace

men

tre

sm

ax

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

Loading cycles

(b)


lowastmax = 200




ISRN Tribology 11

0 2 4 6 8 101

101

102

103

104

105

106

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 0 23

= 033

= 009

(a)

1 2 3 4 5 6 71

1005

101

1015

102

1025

103

1035

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

E = 0

= 0 2

= 033

= 009

(b)


lowastmax = 200

0 2 4 6 8 101

105

11

115

12

125

No

rmal

ized

co

nta

ct a

rea

Present work = 48

Present work = 116



Loading cycles

(a)

0 2 4 6 8 10095

1

105

11

115

No

rmal

ized

co

nta

ct a

rea

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

(b)





12 ISRN Tribology


5 Conclusion





Nomenclature


Subscripts


Superscripts


References





ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









ISRN Tribology 11

0 2 4 6 8 101

101

102

103

104

105

106

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 0 23

= 033

= 009

(a)

1 2 3 4 5 6 71

1005

101

1015

102

1025

103

1035

No

rmal

ized

res

idu

al i

nte

rfer

ence

re

sre

s0

Loading cycles

Isotropic hardening

Kinematic hardening

E = 0

= 0 2

= 033

= 009

(b)


lowastmax = 200

0 2 4 6 8 101

105

11

115

12

125

No

rmal

ized

co

nta

ct a

rea

Present work = 48

Present work = 116



Loading cycles

(a)

0 2 4 6 8 10095

1

105

11

115

No

rmal

ized

co

nta

ct a

rea

Loading cycles

Isotropic hardening

Kinematic hardening

= 0

= 023

= 033

= 009

(b)





12 ISRN Tribology


5 Conclusion





Nomenclature


Subscripts


Superscripts


References





ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









12 ISRN Tribology


5 Conclusion





Nomenclature


Subscripts


Superscripts


References





ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









ISRN Tribology 13






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

3 FTFBSDI SUJDMF 'JOJUF &MFNFOU #BTFE.VMUJQMF/PSNBM … · 2019. 7. 31. · fs fgpsf uif psjh jobm vo efgpsnfe tqifsjdbm hfpnfusz jt opu gvmmz sfdpwfsfe f psjhjobm qsp mf uif efgpsnfe