871634
TRANSCRIPT
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Hindawi Publishing CorporationISRN TribologyVolume 2013, Article ID 871634, 13 pageshttp://dx.doi.org/10.5402/2013/871634
Research ArticleFinite-Element-BasedMultiple Normal Loading-Unloading ofan Elastic-Plastic Spherical Stick Contact
BiplabChatterjee andPrasanta Sahoo
Department of Mechanical Engineering, Jadavpur University, Kolkata 700032, India
Correspondence should be addressed to Prasanta Sahoo; [email protected]
Received 30 October 2012; Accepted 19 November 2012
Academic Editors: J. Antunes and F. Findik
Copyright 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 hasnot yetbeen 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 with
two 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 resistance
in 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 solution.Beyond 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 in
the 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 ANSS.ey 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 elastic,elastic 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
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material, up to 10 percent of modulus of elasticity yieldedin maximum 20 percent difference of contact parameters.Etsion 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 of
elastic 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 0.02 tangent modulus under perfect slip contactcondition. ey found thatthe incipientinterference requiredforthe 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() up to 0.5. 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 () up to 0.5. Variationof hardening parameter up to 0.5 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 0.5.
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 stick
contact 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 theeffect of strainhardening 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 inuence
the reversibility of cyclic normal loading in spherical contactproblems. us under a specic contact condition (slip orstick), the plastic properties of the material and the hardeningrulehavea particularimportance in multiplenormal 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 effectof 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, and the rigid at before thedeformation. e solid line shows the loading phase with theinterference (), corresponding to the contact radius (), andthe contact load ().
Multiple loading unloading consists repeated normalloading and complete unloading. During loading phase, theinterference () is increased up to the specied maximumdimensionless interference, max. During unloading, theinterference () is gradually reduced. At the end of theunloading, under zero contact load and contact area, thesphere has a residual interference (res). erefore the orig-inal un-deformed spherical geometry is not fully recovered.e 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 from
the 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
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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
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, = 0.025using kinematic hardening.is 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-ences with theincreasein unloadingcycles duringten loadingunloading cycles. Zait et al. [24] also observed the decreaseof maximum contact load and increase of dimensionlessinterference during multiple loading unloading.
4.1. Interfacial Parameters with Low Tangent Modulus. Inthe rst part of present analysis, materials are chosen aselastic perfectly plastic material and the materials withtangent modulus of0.025, 0.05, and 0.09 for studying theeffect of strain hardening with both isotropic and kinematichardening rule. Figure 7(a) presents the normalized contactload,0during ten loading-unloading cycles with bilinearisotropic hardening.0is the contact load aer the comple-tion of rst loading. e maximum loading was performedup to a dimensionless interference ofmax = 100. It is foundthat the contact load decreases with each loading-unloadingcycle. Chatterjee and Sahoo [19] found identical contact
load 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 conditions,which 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 9.5, 7.73, 7, and 5.1 percent formaterials with tangent modulus of0.0, 0.025, 0.05, and0.09, respectively.
Figure 7(b) represents the normalized contact load,during ten loading unloading in case of bilinear kinematichardening with maximum loading up to the dimensionless
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0 20 40 60 80 1000
100
200
300
400
500
1st loading
Subs
eque
ntloading
unlo
adin
gcy
cles
Dimensionless normal interference ( / )
Dimension
lesscontactloa(
/
)
= 0.025 , kinematic
(a)
88 90 92 94 96 98 100
430
440
450
460
470
480
Dimensionless normal interference ( / )
Dimension
lesscontactloa
d(/
)
= 0.025 , kinematic
(b)
67 68 69 70 71 72 730
5
10
15
20
25
30
Dimensionless normal interference ( / )
Dimension
lesscontactloa
d(/
)
= 0.025 , kinematic
(c)
F 6: (a) e load-interference hysteretic loop during ten loading-unloading cycles following maximum loading to a dimensionlessinterference of = 100(b) Increase of residual interferences and decrease of contact load. (c) Increase of residual interferences.
interference of max
= 100. e decrement trend of contactload is same for both the hardening rule. e decrease ofcontact loads aer ten loading-unloading cycles are 9.5, 7.55,6.5 and 2.8 percent for materials with tangent modulus of0.0, 0.025, 0.05 and 0.09 respectively. It is seen from theresults of both thehardening rules that the decrease of contactload is less with kinematic hardening than that of isotropic
hardening. Elastic perfectly plastic materials 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, res/
max, during ten loading unloading cycles withbilinear isotropic hardening. e residual displacement orinterferences are found aer complete unloading. e con-tact load and area becomes zero aer complete unloading.e residual interferences are normalized by the maximumdimensionless interference of loading. e maximum dimen-sionless loading interference is
max = 100. It can be seen
from the plotthatthe normalizedresidual 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 is3.2, 6.4, and 11.52 percent higher aer ten loading-unloadingwhen compared with the materials having tangent modulus
of 0.025, 0.05, and 0.09, respectively. Kadin et al. [16]also observed that higher hardening results in lower residualstrain.
Figure 8(b) shows the normalized residual displacement,
res/
max, 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 3.61, 8.26, and 20.57percent higher than that with the materials having tangentmodulus of0.025, 0.05, and 0.09, respectively. Compar-ing the results with isotropic and kinematic hardening, it isobserved that using kinematic hardening yields less residual
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0 2 4 6 8 100.9
0.92
0.94
0.96
0.98
1
Loading cycles
Norma
lize
dconta
ctloa
d(/
)
= 0= 0.025
= 0.05
= 0.09
(a)
0 2 4 6 8 100.9
0.92
0.94
0.96
0.98
1
Loading cycles
Norma
lize
dcontactloa
d(/
)
= 0= 0.025
= 0.05
= 0.09
(b)
F 7: Normalized contact load, 0, as a function of loading cycles with (a) bilinear isotropic hardening, (b) bilinear kinematichardening.
0 2 4 6 8 100.62
0.64
0.66
0.68
0.7
0.72
0.74
0.76
Loading cycles
Norma
lize
dresi
dua
ldisp
lacement,
res
/max
= 0
= 0.025
= 0.05
= 0.09
(a)
0 2 4 6 8 100.55
0.6
0.65
0.7
0.75
Loading cycles
Norma
lize
dresi
dua
ldisplacement,
res
/max
= 0= 0.025
= 0.05
= 0.09
(b)
F 8: Normalized residual interference, res/max, as a function of loading cycles with (a) bilinear isotropic hardening, (b) bilinear
kinematic hardening.
displacement than that with isotropic hardening. is is morepronounced for the materials with high tangent modulus.Zolotarevskiy 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 interferenceofres with respect to
res0 (residual interference aer rst
unloading) 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 4.5 percent irrespective of hardening typeand tangent modulus except the simulation with tangentmodulus of0.09 using kinematic hardening. e simulation
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0 2 4 6 8 101
1.01
1.02
1.03
1.04
1.05
1.06
Isotropic hardening
Kinematic hardening
= 0
= 0.025
= 0.05
= 0.09
Loading cycles
Norma
lize
dresi
dua
linterference,
res
/res0
F 9: Normalized residual interference, resx/res0, asa function
of loading cycles for maximum loading,max= 100.
of material with tangent modulus of 0.09 and kinematichardeningresulted in 1.8 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 0.09 and two different hardening type clearlyindicates that the interfacial parameters depend predomi-nantly on the extent of strain hardening and hardening rule
with the increase in tangent modulus.
4.2. 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 steel,structural steel and aluminum alloys in repeated loadingunloading. In our forth-coming simulations, the tangentmodulus () is varied according to a hardening parameter(). e hardening parameter is dened as = /().We have considered four different values of, 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. e
values of used in this analysis are within range 0 0.as most of the practical materials falls in this range [30]. e
value of equals to zero indicates elastic perfectly plasticmaterial behavior, which is an idealized material behavior.e hardening parameters used for this analysis and theircorrespondingvalues are shown in Table 1.
4.2.1. Mean Contact Pressure Distribution. Figure 10(a) isthe plot of dimensionless mean contact pressure (/)as afunction of ten loading cycles. e maximum dimensionlessinterference for loading is max = 0 (
max = 196 for
T 1: Different and values used for the study of strainhardening effect.
in% (GPa)
0 0.0 0.0
0.1 9.0 6.3
0.3 23.0 16.10.5 33.0 23.1
elastic perfectly plastic material aer rst loading). rizmeret al. [21] used 2%linear hardening and found dimensionlessmean contact pressure around 2.7 for normalized contactload of 200. Our simulation resulted dimensionless meancontact pressure 2.5 aer rst loading for dimensionlessinterference of 50. erefore 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 the
tangent modulus ranging from 0.09to 0.33, the increaseof mean contact pressures with isotropic hardening are in therange of 46.56 to 118 percent than that elastic perfectly plasticmaterial aer ten loading cycles whereas the increase of meancontact 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 max = 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 from15.98 to 34.5 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 hardening.As 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 cycles
for the maximum dimensionless interference ofmax = 200.e 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 6.4, 8.9, and8.8 percent higher aer rst loading than that with kinematichardening forthe materialshaving tangent modulus of0.09,0.23, and0.33, 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, 0.3, and 0.5 and
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0 2 4 6 8 101.5
2
2.5
3
3.5
4
4.5
Isotropic hardening
Kinematic hardening
= 0.33
= 0.23
= 0.09
= 0
Loading cycles
Dimension
lessmeancontactpressure
(
/
)
(a)
0 2 4 6 8 102
3
4
5
6
Loading cycles
Isotropic hardening
Kinematic hardening
= 0.33
= 0.23
= 0.09
= 0
Dimension
lessmeancontactpressure
(
/
)
(b)
1 2 3 4 5 6 72
3
4
5
6
7
8
Loading cycles
Isotropic hardening
Kinematic hardening
= 0.33
= 0.23
= 0.09
= 0
Dimension
lessmeancontactpressure
(
/
)
(c)
F 10: Dimensionless contact pressure,, as a function of loading cycles for maximum loading, (a) max = 50, (b)
max = 100, (c)
max= 200.
indentation load up to 300 times the load necessary for initial
yield. 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)10(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 11. emaximum dimensionless interference of loading is max =100, 0 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 the
tangent 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 isotropichardening. e decreaseof mean contact pressures with isotropic hardening aersecond loading compared to rst loading are .3, .8, 3.8,and 0.2 percent for the materials with tangent modulus of 0,0.09, 0.23, and 0.33, respectively. e decrement of mean
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0 2 4 6 8 10
0.8
0.85
0.9
0.95
1
1.05
Loading cycles
Isotropic
Kinematic= 0 = 0.33
= 0.09
= 0.23
Norma
lize
dmeancontactpr
essure
(
/
)
F 11: Normalized contact pressure, 0, as a function ofloading cycles for maximum loading,
max = 100.
contact pressure with isotropic hardening is consistent withincreasing loading cycles, aer tenth loading the decreasesare 15.74, 12.5, 6.5, and 0.3 percent compared to the meancontact pressure aer rst loading for the materials with tan-gent modulus of same order as mentioned for secondloading.It is therefore imperative that the maximum decrement rateof mean contact pressure is occurred aer second loadingcycle and the decreases in subsequent cycles are insignicant.
Kral 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.
4.2.2. 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). 0 is the contact load aer rst loading andthe maximum dimensionless loading for the simulation ofrepeated loading unloading ismax = 100. e contact loaddecreases with the increase in loading cycle and the rateof decrease is the function of hardening rule and tangent
modulus. 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 max = 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 material.However the materials with high tangent modulus ( =0.33 and kinematic hardening produced no variation ofcontact load with the increase of loading cycle.
4.2.3. Residual Displacements. Figure 13(a) shows normal-ized residual interference, res/
max, aer each unloading
during ten loading unloading cycles for maximum dimen-sionless interference of loading, max = 100 (
max = 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 ( = 0.33 ) 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 0.5 compared
with the elastic perfectly plastic material aer rst unloadingunder perfect slip contact condition. ey simulated withmaximum dimensionless load, max = 300 and inferredthat above max = 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 of0.09,0.23, and0.33,respectively.
Figure 13(b) shows the normalized residual displacementaer each unloading for seven loading unloading cycles.e maximum dimensionless interference of loading forthis simulation is max = 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 unloading. res0is the residual
interference 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 max = 200. Fig-ures 14(a) and 14(b) indicates that higher dimensionlessinterference of loading produces marginally less increment
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0 2 4 6 8 100.9
0.92
0.94
0.96
0.98
1
Loading cycles
= 0= 0.09 ; isotropic
= 0.23 ; isotropic
= 0.33 ; isotropic
= 0.09 ; kinematic
= 0.23 ; kinematic
= 0.33 ; kinematic
Norma
lize
dcontactloa
d(
/
)
(a)
1 2 3 4 5 6 70.88
0.9
0.92
0.94
0.96
0.98
1
Loading cycles
= 0= 0.09 ; isotropic
= 0.23 ; isotropic
t= 0.33 ; isotropic
= 0.09 ; kinematic
= 0.23 ; kinematic
= 0.33 ; kinematic
Norma
lize
dcontact
loa
d(
/
)
(b)
F 12: Normalized contact load, 0, as a function of loading cycles for maximum loading, (a) max = 100, (b)
max = 200.
0 2 4 6 8 100.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Isotropic
Kinematic
= 0
= 0.23
= 0.33
= 0.09
Loading cycles
Norma
lize
dresi
dua
ldisplacement,
res
/max
(a)
1 6 70.2
0.4
0.6
0.8
1
2 4 53
Norma
lize
dresi
dua
ldisplacement,
res
/max
Isotropic hardening
Kinematic hardening
= 0
= 0.23
= 0.33
= 0.09
Loading cycles
(b)
F 13: Normalized residual interference,res/
max, as a function of loading cycles for maximum loading, (a)
max = 100, (b)
max = 200.
of residual interference with increasing loading unloadingcycles. is may cause early shakedown at higher maximumdimensionless interference of loading.
4.2.4. Contact Area with Loading Cycles. Figure 15(a)presents the normalized contact area at the end of eachloading. 0 is the contact area at the end of rst loading.e present simulation consists ten loading unloading cycles.Two different maximum dimensionless loadings ( = 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 5.3 %and 8%of0 with elastic perfectly plastic copper sphere( = 400) using maximum dimensionless loading ()of 48 and 116 at the end of tenth loading. Zait et al. [24]estimated 8-9%growth of0 aer sixth loading with thedeformable sphere ( = 1000) against a rigid at usingmaximum dimensionless loading () of 40. We have useddeformable sphere of E/Y ratio 700, our simulation resulted
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ISRN Tribology 11
0 2 4 6 8 101
1.01
1.02
1.03
1.04
1.05
1.06
Norma
lize
dresi
dua
linterference,
res
/res0
Loading cycles
Isotropic hardening
Kinematic hardening
= 0
= 0.23
= 0.33
= 0.09
(a)
1 2 3 4 5 6 71
1.005
1.01
1.015
1.02
1.025
1.03
1.035
Norma
lize
dresi
dua
linterference,
res
/res0
Loading cycles
Isotropic hardening
Kinematic hardening
E = 0
= 0.2
= 0.33
= 0.09
(b)
F 14: Normalized residual interference, res/
res0, as a function of loading cycles for maximum loading, (a)
max = 100, (b)
max = 200.
0 2 4 6 8 101
1.05
1.1
1.15
1.2
1.25
Normaliz
edcontactarea,
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 8 100.95
1
1.05
1.1
1.15
Norma
lize
dco
ntactarea,
Loading cycles
Isotropic hardening
Kinematic hardening
= 0
= 0.23
= 0.33
= 0.09
(b)
F 15: Normalized contact area, 0, as a function of the number of loading cycles; (a) comparison of different studies; (b) comparisonof hardening models for maximum loading,
max = 100.
12%and 23.5%increase of0 for the maximum loading() of 48 and 116, respectively, aer tenth loading. eagreement between thepresent results andthe ndings of aitet al.[24] is appreciablygood. e variationof ourresultswiththat 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.
Figure15(b) shows the growth of normalized contact areaaer each loading cycle with four different tangent modules
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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 of
loading
= 100. Small discrepancy was observed for theresults of higher tangent modulus ( = 0.23and 0.33)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 against
a 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 pressure,contact 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 due
to 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 model. e 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 kinematic
hardening 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 condition.e analysis of hysteretic loops at high tangent modulus is aninteresting eld for future study.
Nomenclature
: Contact area radius: Modulus of elasticity of the sphere: Yield Strength of the sphere material: Real contact area: Radius of the sphere: Contact load: Interference: Poissons ratio of sphere: Mean contact pressure: Tangent modulus of the sphere
: Dimensionless contact load,, in stick contact
: Dimensionless contact area,, in stick contact
: Dimensionless interference,/, in stick contact.
Subscripts
: Critical valuesres: Residual values following unloadingmax: Maximum values during loading-unloading process.
Superscripts
Dimensionless.
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