Modeling adhesive contacts under mixed-mode loading

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2019

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Journal of the Mechanics and Physics of Solids

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Khajeh Salehani, M., Irani, N., & Nicola, L. (2019). Modeling adhesive contacts under mixed-mode loading.

Journal of the Mechanics and Physics of Solids, 130, 320-329. https://doi.org/10.1016/j.jmps.2019.06.010

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Journal

of

the

Mechanics

and

Physics

of

Solids

journalhomepage:www.elsevier.com/locate/jmps

Modeling

adhesive

contacts

under

mixed-mode

loading

M.

Khajeh

Salehani

a,∗

_,

_N.

_Irani

a

_,

_L.

_Nicola

a,b,∗

a Department of Materials Science and Engineering, Delft University of Technology, CD Delft 2628, the Netherlands b Department of Industrial Engineering, University of Padova, Padua 35131, Italy

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 12 February 2019 Revised 17 June 2019 Accepted 17 June 2019 Available online 18 June 2019 Keywords:

Adhesive contacts

Green’s function molecular dynamics Contact area

Onset of sliding Reattachment

a

b

s

t

r

a

c

t

Experimentsshow that when anadhesive contact is subjectedtoatangential load the contact area reduces,symmetrically orasymmetrically, dependingonwhether the con-tactis under tensionor compression. What happensafter the onsetof slidingis more difficulttobeassessedbecauseconducting experimentsisrather complicated,especially under tensileloading. Here,weprovidethroughnumerical simulations,acomplete pic-ture ofhowthecontactareaand tractionsofanadhesivecircular smoothpunchevolve under mixed-modeloading,beforeand after sliding.First,the Green’sfunction molecu-lardynamicsmethodisextendedtoincludethedescriptionoftheinterfacialinteractions betweencontactingbodiesbymeansoftraction–separationconstitutivelawsthatenforce couplingbetweentension(orcompression)andshear.Next,simulationsareperformedto modelslidingofacircularsmoothpunchagainstaflatrigidsubstrate,undertensionand compression.Inlinewiththeexperimentalobservations,thereductioninthecontactarea duringshearloadingisfoundtobesymmetricundertensionandasymmetricunder com-pression. Inaddition,under tensileloading, full detachmentisobserved atthe onsetof slidingwithanon-zerovalueofthetangentialforce.Aftertheonsetofslidingandthe oc-currenceofslipinstability,thecontactareaabruptlyincreases(reattachment),underboth tensionandcompression. Forinterfaces withhighfriction,the reattachmentoccursonly partially.However,afullreattachmentisattainablewhenfrictionislow.

© 2019TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

IntheseminalworkbySavkoorandBriggs(1977) mixednormalandtangentialloadingwasappliedtoarubber hemi-spherein adhesivecontactwithaglassplate. Itwasfound thatasa resultofincreasingthe tangentialload, thesurfaces tend topeel apart andhence,thecontactarea decreases progressively. WatersandGuduru (2010) performedsimilar ex-periments fora wide range ofnormalloads (tensile andcompressive)while continually recordingimages ofthe contact area evolution.These imagesdemonstrateda symmetric contactareareduction undertensile loadingandan asymmetric reductionundercompressiveloading,beforetheonsetofsliding.Moreover,theycapturedapartialreattachment withthe adventofslipinstabilityattheinterfaceundercompressiveloading.However,undertensileloadingthisreattachmentwas notobserved.Recently,Sahlietal.(2018)carriedoutexperimentsonanasperityslidingonaflatplateundercompression

∗ _{Corresponding authors.}

E-mail addresses: m.khajehsalehani@tudelft.nl (M. Khajeh Salehani), l.nicola@tudelft.nl (L. Nicola).

https://doi.org/10.1016/j.jmps.2019.06.010

0022-5096/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Fig. 1. Sketch of the two-dimensional contact problem.

andfoundthatbothsmoothandroughasperitiesfollowsimilarcontactarea–forceequationsduringshearloadinguptothe onsetofsliding.

Theaimofthisstudyistodevelopa computationaltechniqueto studyadhesivefrictionalcontactsundermixed-mode (normalandtangential)loadingthatcanreproducethefeaturesobservedintheabove-mentionedexperimentsandpredict thecontactbehaviorinconditionsthataredifficulttobeachievedexperimentally,i.e.aftertheonsetofsliding.Bythatwe intendtoprovideacompletepictureoftheadhesivefrictionalproblemforsmoothcontacts.

In the past decades, several adhesive contact models such as Johnson–Kendall–Roberts (JKR) Johnson et al. (1971), Derjaguin–Muller–Toporov (DMT) Derjaguin et al. (1975), Maugis–Dugdale (MD) Maugis (1992), and more recently

GreenwoodandJohnson(1998)andBarthelandHaiat(2002)havebeenproposedandwidelyused.However,thesemodels arenotapplicable ifthereistangentialloading.Adhesivecontactsundermixed-modeloadinghavebeenextensively stud-iedinSavkoorandBriggs(1977),WatersandGuduru(2010),Thornton(1991),Johnson(1997),ChenandWang(2006)and

Ciavarella(2018)byusinglinearelasticfracturemechanics(LEFM)orthemodified(mixed-mode)JKRadhesiontheory,but thesestudiestackledexclusivelynon-slippingcontacts.Adams(2014)investigatedadhesivecontactofelasticcylinders un-der mixed-mode loading where both stick and slip regions can be presentat the interface. Adams’ analysisworks well forcompressive loading,butnot fortensileloading: his formulationleadsto negativecontactsizes. Recently, Huangand Yan(2017) introduced a modelfor slidingof adhesivecontacts withthe aimof analysing shearing problemsunder both compressiveandtensileloading.Withthesameaim,Mergeletal.(2018) developedacontinuumcontactmodelbasedon thefiniteelementmethod(FEM).However,bothworks(HuangandYan,2017;Mergeletal.,2018)arelimitedtostudythe contactareaevolutiononlyuptotheonsetofsliding.

Inthiswork, thestaticelastic deformationofthebodyismodeled usingGreen’sfunctionmolecular dynamics(GFMD) (KhajehSalehanietal.,2018).Theinteractions betweenthesurfacesare describedthroughconstitutivetraction–separation laws,ofthetype usedincohesivezone models(CZMs). Among thewideclass ofCZMs(Van denBosch etal.,2006; Mc-Garry et al., 2014; Sun et al., 1993; Xu and Needleman, 1993), we have selected the so-calledNP1 model proposed by

McGarryetal.(2014).TheNP1isanon-potentialbasedCZM,thereforelessrigorousinformulationthanthepotentialbased modelsintheliterature.However,ithastheadvantagethatitworkswell bothundercompressionandtensionwhen cou-pledtoshear.Morespecifically,thephysicsofoverclosurearewellcaptured:theresistancetotangentialseparationincreases withincreasingoverclosure.

While the presented formulationis applicable to linearly elastic solids witha generic surfaceroughness obeying the smallslopeapproximation,thefocusofouranalysisisonthecontactshearingofaflatrigidsubstratewitha deformable solid.First,simulations areperformedforflat-on-flatcontactshearingto assesthecapability ofourmodeltocapturethe typicalstick-slipbehaviorduringsliding(Dikkenetal.,2017;Gao,2010;Socoliucetal.,2004).Then,simulationresultsare presentedfortheadhesivefrictionalcontactbetweenadeformablecircularprotrusionandaflatrigidsubstrate.

2. Method

2.1. Problemformulation

Aflat rigidsubstrate (body1) is adjacentto an elastically deformable solid (body2), having periodiccircular protru-sionswithradiusR,asschematicallyshowninFig.1.Periodicityisan intrinsicpropertyoftheGreens’function molecular dynamicsmodel,which reliesontheperiodicityofFouriertransforms.In thiswork, wefocusonthe behaviorofasingle protrusion,thereforewetake thewidthoftheperiodicunitcell wtobelarge enoughtoguaranteethattheasperitydoes notinteractwithitsreplicaswithintherangeofappliedloads.

Thebottom boundaryofbody 1 isfixed. Theminimumdistance betweenthebodies isinitially

δ

0. Auniformnormal

displacementUf

Fig. 2. (a) Schematic representation of the bodies in contact. (b) The interfacial interactions are represented by springs for the region highlighted by a dashed rectangle. (c) The updated set of interaction springs, in a snapshot after sliding.

Next,auniformtangentialdisplacementUx isexertedincrementallyonthesameboundary,whileitsverticaldisplacement

isconstrained.

Body2 istakento be elasticallyisotropic.Plane-strain conditionsare assumedforthe two-dimensional problemwith deformationsrestricted tothe x-zplane (see Fig.1). Thesurfaces ofthe contactingsolidsare discretized witha number ofequispacednodes.IntheGFMD technique,theresponse ofthematerialto theexternalloadisobtainedusingdamped dynamicsbyonlyconsidering theinteractionsofthesurfacenodeswiththeir degreesoffreedom coupledtotheexternal force(Prodanovetal.,2014).Bytakingadvantageofthetranslationalsymmetryoftheproblem,theequilibriumsolutionis obtainednumericallyinreciprocalspace.

Inordertokeep trackofthe surfaceevolutionduringloading,thecalculation iscarriedoutinan incrementalmanner (Spence,1968)withtheappliedloadincreasing monotonically.Formoredetails ontheincremental scheme,the readeris referredtothepreviouspaperbytheauthors(KhajehSalehanietal.,2018).Eventually,basedonthesolutionforthesurface, bodyfieldsarecalculatedbymeansofanalyticalrelations(Venugopalanetal.,2017).

2.2. Interfacialinteractions

The interfacial interactions are modeled through traction-separation laws. To visualize the interactions between dis-cretizationnodesatthe interface,we willrepresentthemthrough “springs”,assketchedinFig.2b.Red(or blue)springs representtheinteractionsbetweennodei(orj)belongingtothesurfaceofabody andthenodesoftheoppositesurface.

Fig.2c illustratesa snapshotofthe sameregionshown inFig.2bafter sliding,alongwiththe updated setofinteraction springsbetweenthetwosurfaces.Inprinciple,eachnodecanbeconnectedthroughspringstoallothernodesofthesurface ofthecounter-body.However,tolimitthecomputationalcostofthesimulation,sincetheinteractionswithfarawaynodes isveryweak,weconsiderthateachnodereachesoutwithspringsonlytothenodesthatfallwithinawindow,thatmoves withthenodeitself.InFig.2cthedashedspringsarethosethatfalloutsideofthewindow.

The interface model is implemented in the Green’s function molecular dynamics (GFMD) technique described in

Khajeh Salehanietal. (2018) asfollows.When the two solidsare in mechanicalequilibriumat time t, they exchange at theinterfaceequalandoppositeforces:

T(_if1)=−T(2)

if

∀

t, (1)

whereT_ifistheinterfacialtraction.Theinterfacialtractionsactingoneachnodeiarecomputedasasumoverthemsprings inthewindow: Tif

(

i

)

= m  α Tcz[



t

(

i,

α

)

,



n

(

i,

α

)

], (2)

whereTcz istheconstitutiverelationofeach individualspringasafunction ofthetangentialandnormalgapvalues,i.e.,

theend-to-enddistanceofthesprings:



t

(

i,

α

)

=x

(

j

)

− x

(

i

)

,



n

(

i,

α

)

=z

(

j

)

− z

(

i

)

, (3)

Fig. 3. Graphical representation of normal and tangential tractions versus normal and tangential gap values, as given by the cohesive-zone constitutive relations of Eq. (4) .

Inthiswork,Tczisspecifiedintermsofthecohesive-zoneconstitutiverelationsinnormalandtangentialdirections,Tcz,n

andTcz,t,whichrepresenttheadhesiveandfrictionalforcesperunitarea,respectively.FollowingMcGarryetal.(2014),the

constitutiverelationsareexpressedas

Tcz,n =

φ

n

δ

n





n

δ

n



exp



−



_δ

n n



exp



−



2t

δ

2 t



, Tcz,t =2

φ

t

δ

t





t

δ

t



exp



−



n

δ

n



exp



−



2t

δ

2 t



, (4)

where

(

φ

n,

φ

t

)

and

(

δ

n,

δ

t

)

are, respectively, the normalandtangential worksofseparation andcharacteristic lengthsof

eachindividual spring.Consequently,the maximumvaluesofTcz,n andTcz,t,i.e., thenormalandtangentialcohesive-zone

strengths

σ

maxand

τ

max,aregivenas

σ

max= 1 exp

(

1

)

φ

n

δ

n exp



−



2t

δ

2 t



,

τ

max= 1



0.5exp

(

1

)

φ

t

δ

t exp



−



n

δ

n



, (5)

as graphically shown in Fig. 3. These relations describe well the interface interaction between common materials (Khajeh Salehanietal., 2018). Ifdesired, it ispossibleto employ adifferent “traction-separation” law for modeling other systems,suchasbiologicalstructures.

2.3.Choiceofparameters

Thecircularprotrusiononthesurfaceofbody2hasaradiusofR/w=2.5,whichissufficientlylargetoobeythesmall slopeapproximation.The chosen materialandinterface propertiesarepresentedinterms ofnon-dimensionalparameters  =/ref_{, wherethe following “reference” parameters are used: E}ref_{=1 GPa,}

_φ

ref

n =

φ

tref=1 N/m,and

δ

nref=

δ

tref=

1nm. The referenceparameters are chosen to be unity andare onlyused fornormalization. Unless otherwise specified,

δ

n=

δ

t=1,

φ

n=2,the ratiooftangential-to-normal work ofseparation

φ

t/

φ

n=1,andE =E(2)=70.Moreover, E(1)=

103_E(2)_{(body1 isassumed tobe rigid)and}

_ν

₌

_ν

(1)₌

_ν

(2)_{=0.45.Here,weoptfor}

_ν

_{=0.45sincemostoftheexperiments}

havebeenperformedforalmostincompressiblematerials.

Theinitial distancebetweenthetwo solidsis

δ

0=10

δ

n, whichislarge enough tohavezerointerface interactions at

thebeginningofthecalculations(see Fig.3a).Simulations areperformedforawide rangeofnormaldisplacements

δ

:=

δ

/

δ

0=[−1,1]inordertostudybothtensileandcompressiveloading.Whenconsideringflat-on-flatcontact(noprotrusion),

wetake

δ

=0,sothat



n=0allovertheinterface,providingtheopportunitytoassessourmodelinpure-shearmode(see Eq.(4)).

TheappliedtangentialdisplacementUxisnormalizedonthewidthoftheunitcellUx=Ux/w.Thetangentialandnormal

contactforcesare,respectively,calculatedas

F = w 0 T_if,t(1)dx=−  w 0 T_if,t(2) dx, L=− w 0 T_if,n(1)dx=  w 0 T_if,n(2) dx, (6)

Fig. 4. Tangential contact force ¯F versus applied tangential displacement U x for various values of elastic modulus E and tangential work of separation φt .

A transition is observed from stick-slip motion (a) to continuous sliding (b) by increasing E or decreasing φt . In all cases, the elastic modulus E = 70 is

used in normalizing the tangential contact force.

wherewehaveconsideredthattheout-of-planethicknessofthesolidsisunity,andchosenthatanattractivenormalforce isnegative.ThesecontactforcesarenormalizedasF¯=

(

F/w

)

/E andL =

(

L/w

)

/E.Thecontactarea Aisdefinedasthesize oftheinterfaceunderrepulsionandisnormalizedasA¯=A/w.

Thenumericalconvergenceisguaranteedbyemployinganincrementaldisplacement



Uz=

δ

n/10and



Ux=

δ

t/10.To

makethesimulationscomputationallymoreefficient,theinteractionbetweentwooppositenodesisconsideredonlyifthey fallwithinaninteraction“window”.Thesizeofthiswindowisindependentofthesurfacediscretizationandisdetermined by the rangeof thecohesive-zone interactions. Aconverged solutionis achievedby setting thewindow-size to 10

δ

n in

normaland5

δ

t intangentialdirections.

3. Resultsanddiscussion 3.1. Flat-on-flatcontact

Simulations are first performedfor flat-on-flat contactunder simpleshearing to asses thecapability of ourmodel to predictthestick-slipmotionobservedexperimentally(Socoliucetal.,2004),viaatomisticsimulations(Dikkenetal.,2017), andmacroscopicmodelssuchasGao(2010).Besides,weobservethetransitionfromstick-slipmotiontocontinuoussliding, wherethedissipatedenergybecomesnegligible(KrylovandFrenken,2014)andhence,anultralowfrictionregimecanbe achieved.

ResultsforvariousvaluesofelasticmodulusE andtangentialworkofseparation

φ

t inFig.4aindeed showthetypical

sawtoothbehavior(Dikkenetal.,2017;Socoliucetal.,2004)ofthetangentialcontactforceF¯versustheappliedtangential displacementUx.Inallcases,intheinitialstickingstage,theforceincreaseslinearlywithappliedtangentialdisplacement,

witha slope controlled by E.The force then drops abruptlyas thetwo solids slipover each other.For stiffer materials, the stored elasticenergy is larger andis released by a larger drop in force at each slipinstability. This can be seen by comparing the blue solid line forE ₌70and the dotted red line forE ₌140 in Fig. 4a. Moreover, as expected, asthe interfaceinteractions becomeweaker(bydecreasingthetangentialworkofseparation

φ

t) theonsetofslidingoccursata

lowertangentialcontactforceF¯.Thiscanbeobservedbycomparingthebluesolid linefor

φ

t=0.2andthedashedgreen

linefor

φ

t=0.1inFig.4a.

AsshowninFig.4b,atransitionfromstick-slipmotiontocontinuousslidingisobservedeitherwhen

φ

t isdecreasedor

whenE isincreased.Bycontinuousslidingwemeanthatthetangentialcontactforceoscillatesaroundzerowithnoinitial stickingstage.Inthiscase,thedissipatedenergyduringslidingbecomesnegligibleandhence,anultralowfrictionregime is achieved.Thisisin linewiththe predictionsof thePrandtl–Tomlinsonmodel forcasesin whichthe contactpotential corrugation islow enough(low

φ

t in thiswork)and/orthe stiffnessofthesystemis highenough(highE in thiswork)

(KrylovandFrenken,2014).

Here,thefrictionforce



F¯



iscalculatedastheaverageofthemaximumandtheminimumtangentialcontactforceduring sliding(KrylovandFrenken,2014),F¯sandF¯k:



F¯



= F¯s+F¯k

Fig. 5. Friction reduces by increasing E / τmax . The shaded area indicates the ultra low friction regime. The dashed line is a guide for the eye.

Fig. 6. Tangential and normal contact forces, ¯F and ¯L , and contact area ¯A with increasing tangential displacement U x . Results are shown for the normal

displacement δ= 0 and φn = φt = 2 .

Fig.5showsamonotonicreductionin



F¯



withincreasingEand/or decreasing

τ

max.AnincreaseinE/

τ

max,correspondsto

theCZMlawdescribing linearelasticfracture mechanics behavior(Gaoetal., 2003; GaoandYao, 2004;Liu etal.,2015), withnegligibleadhesivebehavior,andleadsthereforetoanegligiblefrictionforce,suchthatanultralowfrictionregimeis achieved(seeFig.5).

3.2.Circularprobeonflatbody

Here,weperformsimulationsforadhesivefrictionalcontactofadeformablesolidwithcircularprotrusiononaflatrigid substrateundermixed-modeloading.Inthiscase, besidethetangentialcontactforce,wealsoinvestigatetheevolutionof thenormalcontactforceandfocusonthechangeinthecontactarea.

ThetangentialcontactforceF¯versusappliedtangentialdisplacementUx,fornormaldisplacement

δ

=0andaninterface

with

φ

n=

φ

t=2,isshown withthesolid blue lineinFig. 6.In thisfigure,similar to thecaseof flat-on-flatcontactin Section3.1,asawtoothcurveforF¯versusUxisobservedduetothestick-slipmotion.Theevolutionofthenormalcontact

forceL is also includedin Fig.6 (see the dashed blue line).Here, the negativevalue ofL indicates tensile loading.This meansthatthesummationoftheattractiveforces(withnegativesign)ontheinterfaceislarger thantherepulsiveforces (withpositivesign).Byapplyingalarger

δ

thesolidsarepushedharderagainsteachotherandthenormalcontactforceL

maybecomepositive(compressiveloading).

Fig.6alsopresentstheevolutionofthecontactareaA¯duringtangentialloading,shownbythedottedredline.During thestickingstages,areductioninA¯isobservedwithincreasingUx,abehaviorwerefertoas“shear-peeling”.Subsequently,

attheonsetofslidingwheretheslipinstabilityoccurs,thecontactareaabruptlyincreases(reattachment).Thiscycleofarea reduction → slipinstability → reattachmentpersistsforcontinuedtangentialdisplacement.Thedetailsoftheseeventsare examinedinthefollowingsection.

Fig. 7. (a) Contact area ¯A versus applied tangential displacement U x for different normal loading and ratio of tangential-to-normal work of separation. (b)

Snapshots (A–E) of the contact region during tangential loading for the three cases shown in (a). Snapshots are denoted by the same symbols in both (a) and (b). Note that in (b), contact regions are shown in the original configuration. The horizontal dotted lines are a visual aid to distinguish between symmetry and asymmetry of the contact area. Results are shown for φn = 2 .

3.2.1. Shear-peelingandreattachment

WatersandGuduru(2010)recordedimagesofthecontactareaevolutionthroughouttheircontactshearingexperiments forarangeofnormalloads.Beforetheonsetofsliding,theseimagesdemonstratedsymmetricandasymmetriccontactarea reductionundertensileandcompressive loading,respectively.Aftertheonsetofsliding,theycould captureapartial reat-tachmentundercompressiveloading.Here,weperformsimulationswiththeaimofcapturingtheseexperimental features andtoinvestigatewhetherthereattachmentoccursalsoundertensileloading.Moreover,wealsostudytheroleoffriction ontheevolutionofcontactarea.

Fig.7ashowstheevolutionofthecontactareaA¯versustheappliedtangentialdisplacementUxatvariousnormal

load-ings,tensileandcompressive,andtheratioofthetangential-to-normalworkofseparation,

φ

t/

φ

n=1and0.1.Forthe

cho-sen parameters in thisfigure,

δ

0.2 and

δ

<0.2 representthe compressive andtensile loading,respectively. Moreover,

φ

t/

φ

n=1and0.1are employedinordertoinvestigatetherole ofhighandlowfriction, respectively,onthecontactarea

evolution:loweringthevalueof

φ

t/

φ

n,forexample,resemblesintroducinglubricantintotheinterface.

Snapshotsofthecontactregion,labeledasA–E,areshowninFig.7bforthreecasesinFig.7aatvariousvaluesofthe tangentialdisplacement.These snapshotsare denotedby thesame symbolsinboth figures. Inall cases,snapshotAis at

Ux=0, B andC areduring theshear-peelingstage, andD andEare beforeand afterthe firstslipinstability event. The

followingkeyfeaturesemergefromthisfigure:

• A → B: Thecontactareashrinks symmetrically.Thissymmetricalpeelingisobservedinall cases:tensile/compressive loadingandlow/highfriction.

• B _→ C:Undertensileloading(caseI),thecontactareacontinuestoshrinksymmetrically.Forcompressiveloading(case IIandIII),however,thecontactareabecomesasymmetric,withmorepeel occurringatthetrailingedgeofthecontact thanattheleadingedge.Thehorizontaldottedlinesareavisualaidtodistinguishbetweensymmetryandasymmetry ofthecontactarea.

• C → D:Under tensileloadingin caseI,a fullseparation occursduringtangentialloading attheonset ofsliding.On thecontrary, forcompressiveloading (caseII andIII), thereisno fullseparation andcontactarea continuestoshrink asymmetrically.

• D → E:Asslipprogresseswithinthecontactarea,slipinstabilityoccursandimmediatelyreattachmentfollows.Under tensile/compressive loadingwithhighfriction (case IandII),the reattachmentoccurs onlypartially. However, forthe interfacewithlowfriction(caseIII),afullreattachmentisobserved(comparesnapshotsEandA).

Theaforementionedfeaturesagreewell withtheexperimentalobservationsofWatersandGuduru(2010).Besides,two extrafeaturesare alsocapturedcomparedtotheexperiments:First,thereattachment(D → E)wasnotobservedfor ten-sileloading(caseI)intheexperiments.As mentionedinWatersandGuduru(2010),thismightbe becausetheemployed feedbackloopcouldnotcorrectthenormalloadingfastenoughtomaintaincontactastheslipinstabilityoccurred.Second, forthechosenmaterialandinterfacepropertiesonlyapartialreattachmentwasobservedintheexperiments.However,our resultsshowthat attainingafullreattachment ispossiblewhenfrictionislow(caseIII). Inthiscase, ascanbe seenfrom the red lineinFig. 7a,the pre-slidingdistance issmall. Hence,the induced deformation inthe solid duringthe sticking

Fig. 8. (a) Contact area ¯A versus tangential contact force ¯F for various normal displacements δ. Solid curves are fits using Eq. (8) ( Sahli et al., 2018 ). The filled circles indicate contact area ¯A s at the maximum tangential contact force ¯F s . (b) ¯A s versus ¯F s for the case shown in (a) along with cases for φn = 1 ,

φt /φn = 1 and φn = 2 , φt /φn = 0 . 1 . Slopes of the solid lines in (b) represent the contact shear strength τc ( Sahli et al., 2018 ).

stagecanbefullyreleasedastheslipinstabilityoccurs.Consequently,forthecaseoflowfriction,theinitialconfiguration isre-attainableinthereattachmentcycle.

3.2.2. Lawsofareareductionandonsetofsliding

AcommonwayofidentifyingthelawofcontactareareductionistoplotthecontactareaA¯asafunctionofthetangential contactforceF¯. Tothisend, the evolutionofA¯ versusF¯ upto theonsetofsliding forvarious normaldisplacements

δ

is showninFig.8a.Forthechosenparametersinthisfigure,

δ

0.2and

δ

<0.2representthecompressiveandtensileloading, respectively.Itisobservedthatforall

δ

_,theinitial contactarea A¯0=A¯

(

Ux=0

)

decreasesuntilamaximumF¯isreached

attheonsetofsliding.

Recently,Sahlietal.(2018)carriedoutslidingexperimentsforarangeofcompressiveloadings.Intheirexperiments,for bothsmoothandroughasperities,thereductioninthecontactareawasfoundtobewell fittedby anempiricalquadratic lawoftheform:

¯

A=_A¯₀₋

_ζ

_F¯2_{, (8)}

with

ζ

beingafittingparameter.Wehavefittedthisequationtoourresultsforadhesivefrictionalcontactofasmooth as-peritywithvariousnormaldisplacements

δ

,seethesolidcurvesinFig.8a.Itisobservedthatagoodagreementisobtained forcompressive loading,similartotheexperiments inSahlietal.(2018).Fortensileloading,forwhichexperimental data isnotavailable,we findadeviationfromthequadraticlawofEq.(8).Itisconcludedthatundertensileloading,whenthe onsetofslidingisapproached,therateofcontactareareductionincreases.

InFig.8a,filledblackcirclesindicatecontactarea A¯s atthe maximumtangentialcontactforceF¯s justbeforetheonset

ofsliding,i.e.,thestaticfrictionforce.Fig.8billustratesthedatapoints(A¯s,F¯s)relatedtovarious

δ

forthecaseshownin Fig.8a(

φ

n=2,

φ

t/

φ

n=1)along withtwo othercaseswhere

φ

n and

φ

t/

φ

n are independentlyvaried.Interestingly,inall

cases,thepointsmarkingtheonsetofsliding(A¯s,F¯s)alignwellonastraightline.Thisshowsthat,

¯

Fs=

τ

cA¯s+F¯s,nc, (9)

with

τ

candF¯s,ncbeingthecontactshearstrengthandthemaximumslidingforcewithoutcontact(i.e.withoutacontactarea

underrepulsion),respectively.ThelinearrelationinEq.(9)isinlinewiththeso-calledthresholdlawbySahlietal.(2018). SinceSahlietal.(2018)onlyconsideredcompressiveloading,theyconcludedthatthestraightlinegoesthroughtheorigin. Here,byconsideringalsotensileloading,wecanobservecaseswhereA¯siszeroatafinitevalueoftangentialcontactforce

dueto the adhesive interactions at the interface. For example,in Fig. 8a, this is the casefor the normal displacements

δ

=−0.2and−0.3.Thisbehavior canbebettercapturedbyourproposed‘extendedthresholdlaw’ (Eq.(9)), choosingthis nameisinspiredbytheextendedAmontons’law,whichincorporatessimilarlyanadhesivecontribution.

Finally,Fig.8bshowsthat thecontactshearstrength

τ

c andtheslidingforce whenthere isnorepulsive contactarea

¯

Fs,ncchange with interfacial properties, namely

φ

n and

φ

t/

φ

n.Thefollowingkeyfeaturesareidentified:

• DecreasingthenormalworkofseparationdecreasesF¯s,nc,yetithasnegligibleeffecton

τ

c.ThiscanbeseenfromFig.8b,

by comparing the linesmarked withblack circlesfor

φ

n=2and redsquares for

φ

n=1.A waytoreduce

φ

n is the

cases

φ

n=2).Thevalueof

φ

tcanbereducedbyintroducingalubricantintotheinterface.

4. Concludingremarks

Asimplecomputational modelisdevelopedtostudyadhesivefrictionalcontactsofelasticallydeformablesolidsunder mixed-modeloading. Thestrength ofthemodel liesinits capabilityof studyingthevariation ofcontactarea andofthe friction forcebefore andaftertheonsetof sliding,underacompressive ortensileloading.Thefull rangeofconditionsis difficulttobeaddressedexperimentally.

Thesimulationsareperformedforan elasticallydeformableadhesivecircularprotrusionslidingagainst arigidflat.The modelcancapturethefeaturesobservedintheexperimentsbyWatersandGuduru(2010)andbySahlietal.(2018):

• Under compressive loading, a tangentialdisplacement, first induces the contact area to shrink symmetrically. As the appliedtangentialdisplacementincreasesmorepeeloccursatthetrailingedge.Undertensileloading,onthecontrary, thecontactareacontinuestoshrinksymmetrically.

• Reductioninthe contactarea asafunction ofthe tangentialcontactforceis found tobe well fittedby theempirical quadraticlaw reportedin Sahlietal. (2018)for experimental resultsunder compressiveloading. Moreover, thereis a linearrelationbetweenthemaximumtangentialcontactforceattheonsetofslidinganditscorrespondingcontactarea atvariousnormalloadings.

Inadditionthemodelcanpredictthefollowingbehavior,notyetobservedexperimentally:

• Withslipinstabilitiesoccurringduringsliding,acycleofcontactareareductionandreattachmentpersistsforcontinued tangentialloading,evenundertension.Whetherthereattachmentis partialorfulldependsonfriction: The lowerthe friction,themorecompletethereattachment.

• Undertensileloadingtheempiricalquadraticlawrelatingareatotangentialforce breaksdown:thereisalargerrateof contactareareductionwhentheonsetofslidingisapproached.

• Alsoundertensileloadingalinearrelationholdsbetweenthemaximumtangentialcontactforceattheonsetofsliding andthecorrespondingcontactarea.However,undertensileloading,afullinterfaceseparationcantakeplaceattheonset ofslidingwithanon-zerotangentialforce:Thecontactareaisunderadhesivecontact.

Acknowledgments

LNreceivedfundingfromtheEuropeanResearchCouncil(ERC)undertheEuropean UnionsHorizon2020research and innovation programme(grantagreement no.681813). LNalso acknowledges supportby the Netherlands Organisationfor ScientificResearchNWOandDutchTechnologyFoundationSTW(VIDIgrant12669).

Supplementarymaterial

Supplementarymaterialassociatedwiththisarticlecanbefound,intheonlineversion,atdoi:10.1016/j.jmps.2019.06.010.

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