Finite deformations govern the anisotropic shear-induced area reduction of soft elastic contacts

Finite deformations govern the anisotropic shear-induced area reduction of soft elastic contacts

J. Lengiewicz

^a,b

, M. de Souza

^c

, M.A. Lahmar

^c,d

, C. Courbon

^d

, D. Dalmas

^c

, S. Stupkiewicz

^b,∗

, J. Scheibert

^c,∗

a Department of Engineering, Faculty of Science, Technology and Medicine, University of Luxembourg, Luxembourg

b Institute of Fundamental Technological Research (IPPT), Polish Academy of Sciences, Pawi ´nskiego 5B, 02-106, Warsaw, Poland

c Univ Lyon, Ecole Centrale de Lyon, ENISE, ENTPE, CNRS, Laboratoire de Tribologie et Dynamique des Systèmes LTDS, UMR 5513, F-69134 Ecully, France

d Univ Lyon, Ecole Centrale de Lyon, ENISE, ENTPE, CNRS, Laboratoire de Tribologie et Dynamique des Systèmes LTDS, UMR 5513, F-42100, Saint-Etienne, France

a r t i c l e i n f o

Article history:

Received 4 May 2020 Revised 9 June 2020 Accepted 9 June 2020 Available online 14 June 2020 Keywords:

Contact mechanics Friction Contact area Elastomer

Full-field measurement

a b s t r a c t

Solidcontactsinvolvingsoftmaterialsareimportantinmechanicalengineeringorbiome- chanics.Experimentally,suchcontactshavebeenshowntoshrinksignificantlyundershear, aneffectwhichisusuallyexplainedusingadhesionmodels.Hereweshowthatquantita- tiveagreementwithrecenthigh-loadexperimentscanbeobtained,withnoadjustablepa- rameter,usinganon-adhesivemodel,providedthatfinitedeformationsaretakenintoac- count.Analysisofthemodeluncoversthebasicmechanismsunderlyinganisotropicshear- inducedareareduction,localcontactliftingbeingthedominantone.Weconfirmexperi- mentallytherelevanceofallthosemechanisms,bytrackingtheshear-inducedevolution oftracersinsertedclosetothe surfaceofasmooth elastomersphere incontact witha smoothglassplate.Ourresultssuggestthatfinitedeformationsareanalternativetoadhe- sion,wheninterpretingavarietyofshearedcontactexperimentsinvolvingsoftmaterials.

© 2020TheAuthors.PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBYlicense.

(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Roughcontactsareubiquitousinbothnaturalandengineeringsystems,andindeed,roughcontactmechanicshavebeen activelyinvestigatedinthelastdecades(seee.g.Vakisetal.(2018)forarecentreview).Mostoftheefforthasbeendevoted tothenormalcontactoffrictionlessinterfaces(see Müseretal.(2017) fora comparisonofvariousmodellingapproaches tosuchaproblem).However,recentexperimentsinvolvingsoftmaterialslikepolymersorhumanskinhaverevealedcom- plexchangestothecontactmorphologywhenafrictionalroughcontactissubmittedtoanadditionalshearload.Notonly isthe overall realcontact area significantly reduced(Sahliet al., 2018;Weber et al., 2019), butit also becomesincreas- ingly anisotropic (Sahli etal., 2019), two effectsthat have not been satisfactorilyexplained yet. Forboth effects,smooth sphere/planecontacts havebeen shownto obey similar behaviour laws asrough contacts (Sahli etal., 2018, 2019). It is

∗ Corresponding authors.

E-mail addresses: sstupkie@ippt.pan.pl (S. Stupkiewicz), julien.scheibert@ec-lyon.fr (J. Scheibert).

https://doi.org/10.1016/j.jmps.2020.104056

0022-5096/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license.

( http://creativecommons.org/licenses/by/4.0/ )

energydissipationdueto,forinstance,micro-slipwithintheshearedcontact.

Fine-tuning ofthe amplitudeand shape offallowed to quantitatively reproduce experimental results onthe area re- ductionofsmoothPDMS(PolyDiMethylSiloxane)spheresincontactagainstsmoothglassplates(Ciavarella,2018;Papangelo andCiavarella,2019).Accounting forasecond mode-mixityfunctiondescribingtheinteractionbetweenmodeIandmode III (antiplane shear) furtherallowed toreproduce the anisotropic propertiesofthe area changes (Papangelo et al., 2019).

However, such agreement required atbest to fitthe amplitudeof f(Ciavarella, 2018;Papangelo andCiavarella, 2019), at worst tointerpolatethe wholeffunctionsfromone experimentout oftheset tobe reproduced (Papangelo etal., 2019).

Thislimitationarisesbecause ourcurrentunderstanding ofthephysical mechanismslumped intothe mode-mixityfunc- tions, andmoregenerallyoftheinteractions betweenadhesionandfriction, remains unsatisfactory.Forthesame reason, otheradhesionmodels,likethenumericalonesofMergel etal.(2019),KajehSalehanietal.(2019),Mergeletal.(2020)or the theoretical one of McMeekinget al.(2020), also requiread hoc descriptions of thelocal interfacialbehaviour under couplednormalandtangentialloading.

Surprisingly,simpler,adhesionlessmodelsbasedonlyonelasticityhavenotbeenproposed.Thisispartlyhistorical,be- causethefirstexperimentsonshear-inducedcontactreduction(SavkoorandBriggs,1977;WatersandGuduru,2010)were mainlyperformedundersmall(orevennegative)normalloads,forwhichadhesivestressesareexpectedtodominatethose dueto indentation.Recent experiments atmuch highernormalloads (Sahli etal., 2018), forwhich adhesive stressesare expectedtobelessprominent,suggestthattheareareductionmayalsooccurintheabsenceofadhesion.Anotherpossible reasonforhavingoverlookedelasticmodelsisthenatureofthematerialsusedfortheexperiments.Theymainlyconsidered elastomers,which arenearly incompressible, in contactwitharigidsubstrate. In thoseconditions,the normalandshear stressesatthecontactinterfacebetweenlinearelasticsolidsareuncoupled(Johnson,1985),sothattheobservedeffectof thetangentialloadonthecontactareaisunexpected.

Here, we investigate the hypothesis that the shear-induced contactarea reduction is an elastic effectenabled by the nonlinear,finite-deformationbehaviouroftheelastomer.Recently,thispossibilitywassuggestedinWangetal.(2020),and some preliminarysupport wasbrought byMergel etal.(2020) using2D simulations,butit hasneverbeentestedon 3D sphere/planecontacts.Tofullytestthehypothesis, wehavedevelopeda computational3D modelthat combinesahyper- elastic bulk and a non-adhesive butfrictional interface. The model suitablyfits the macroscopic experimental data from Sahlietal.(2018,2019)and quantitatively reproduces the anisotropic evolution of contact area therein,with no adjustable parameter (Section 2). Ourresults stronglysuggest that thecontact area reductionis governed bythe finite-deformation mechanics,witha dominatingeffectoflocal contactliftingandlesspronounced effectofmicro-slip-inducedin-plane de- formation. We then presentoriginal experiments in which those two mechanisms can indeed be directly observed and quantified(Section3).

2. Finite-strainmodellingofshear-inducedcontactareareduction

The finite-strain framework and theTresca friction modelare the two essential features ofour model,and theseare combined withthe finite-element methodas a suitable spatialdiscretization scheme.Since the individual ingredients of themodel areratherstandard, perhapsexceptforsome detailsof thecomputationaltreatment, themodelisonly briefly summarizededinSection2.1,anditsdetaileddescriptionisprovidedinAppendixA.

2.1. Computationalmodel:finite-strainframeworkandTrescafriction

Thecontactproblemunderconsideration,sketchedinFig.1,correspondstothenormalandtangentialloadingofahyper- elasticsphericalsolid in(adhesionless) frictionalunilateralcontactwitharigidplate.Thefinite-strainframework employs thegeometrically exactkinematics offinitedeformations(AppendixA.1)andcontact(AppendixA.2),aswell asadequate

Fig. 1. 2D sketch of the (3D) soft-sphere/rigid-plane contact under study. The initial (dashed lines) and deformed (solid lines) configurations are shown for two stages: (a) under pure normal load P and (b) when an additional tangential displacement d is applied, giving rise to a tangential load Q . In all subsequent figures, the leading (resp. trailing) edge is always on the left (resp. right) side.

constitutivedescriptionsofboththebulkelasticityofthesphere(AppendixA.1)andthefrictionalbehaviouratthecontact interface(AppendixA.3).

HyperelasticityistreatedusingtheMooney–Rivlinmodel,inthenearlyincompressibleversion.Disregardingcompress- ibility andtherelatedbulk modulus, themodel (Eq.(A.3)) involves two parameters,

μ

1 and

μ

2,such that

μ

=

μ

¹+

μ

², with

μ

^{theshearmodulus.Inthefollowing,wewillconsidertwoparticularcasesreducingtoasingleparameter,}

μ

^:(i)the

neo-Hookeanmodel,forwhich

μ

2=0and

μ

1=

μ

,and(ii)anarbitrarilychosenother combinationofparameters,namely

μ

1=

μ

2=

μ

/2(simplydenotedasMooney–Rivlinfromnow).Notethat

μ

^{canbereadilydeterminedforanyparticularex-}

perimentalsphere/planecontactfromtheknowledgeofthecontactarea,A₀=A(^Q=0),underapure,knownnormalload, P.

Fora widerangeoftribologicalmaterialpairs,thestaticfrictionforce ofsphere/planecontactsismeasured tobepro- portionaltothecontactarea (see Sahlietal.(2018)andreferencestherein). Itisthus appealingto examinethesimplest frictionmodelthat automaticallyproducessuch adependence,namelytheTrescafrictionmodel,whichis employedhere (AppendixA.3).IntheTrescamodel,noslipoccurslocallyuntilthecontactshearstrength,

σ

^{,isreached,and}

σ

^isassumed

constantandindependentofthenormalcontacttraction.TheTrescamodel,evenifoversimplified,isexpectedtoprovidea reasonableapproximation ofthetangentialstress distribution(atleast) whenthestaticfrictionforce,Qs,isreached. Note that

σ

^{canbemeasuredinall}experimentsthatmonitorthecontactarea(e.g.,Mergeletal.,2019;Sahlietal.,2018;Savkoor andBriggs,1977;WatersandGuduru,2010),astheratioofQsoverAs=A(^Q=Qs)^.

The finite-element treatment of the contact problem at hand is described in Appendix A.4 and in Section S.1.1 inSupplementaryInformation¹.Notethat theTrescafrictionmodelmayleadtosignificant convergenceproblems,particu- larlywhentheinterfacialshearstrainsarelarge,whichwillbethecaseinoursimulations.Tocircumventthoseproblems, aPrakash-Clifton-likeregularizationscheme(AppendixA.3)hasbeenemployed,whichallowedouractualcomputationsto besuccessfullycarriedoutforarelativelyfinefinite-elementmesh.

WehaveappliedourmodeltoperformadirectquantitativecomparisonwiththePDMS-sphere/glass-plateexperiments reportedin Sahli et al.(2018, 2019).In particular, the geometry, boundary conditions andloading conditions match the experimental ones. The elastomersample(identical to that in insetof Fig.10) isa cylinder, the topof which features a sphericalcapwitharadius ofcurvatureof9.42mm.Itisfixed ona rigidsupport.Thesphericalcap isfirstbroughtinto normalcontact with a rigidplate under constant normal loadP (we chose four representative valuesof P covering the wholerangeexplored inSahli etal.(2018):0.27, 0.55,1.65and2.12 N).The contactisthen shearedby pullingtheplate horizontallywithaconstantvelocityV=0.1mm/s.

A value of

μ

=0.60 MPa was derived in a unique manner using the experimental values of P and A(^Q=0) ^(see AppendixA.4). Similarly,a value of

σ

=0.41MPawasderived usingthe experimentalvalues ofAs andQs.Notethat the ratio

σ

^/

μ

^issignificant,suchthat shearstrainsexceeding50% areexpected.Thisisfarbeyondtherangeofvalidityofthe small-straintheory,andlinearelasticityinparticular,sothatthefinite-strainframeworkusedhereisactually essential.At thoselargestrains,themechanicalbehaviouroftheelastomerusedintheexperimentsofSahlietal.(2018,2019)(Sylgard 184)isalreadywellbeyonditslinearrange(Maraghechietal.,2020;Nguyenetal.,2011).

Tofurthercharacterizethecontactconditions,letusintroducetheHertziancontactradiusa_H=(^3PR/(¹⁶

μ

))^1/3,where thePoisson’sratio

ν

=0.5hasbeenassumedforsimplicity.Inthe presentconditions,thedimensionlessratioa_H/Rvaries between0.098and0.195forP=0.27NandP=2.12N,respectively,thusexceedingtheusualrangeofvalidityoftheHertz contacttheory,a_H R(Johnson,1985).Fortheshearloading,therelevantdimensionlessparameteris

σ

^/

μ

^.

Fig.2showsthe finite-elementmeshused inthecaseof P=2.12N (notethat thesize oftherefined-mesh regionis adjustedtothenormalload),andzoom-insforselectconfigurations.Inparticular,panel(d)illustratesthedeformationpat- ternforQ=Qs,showingasignificantsheardeformationinthesubsurfacelayerandanon-uniformin-planedeformationof

1 The paper is accompanied by Supplementary Information (SI) available online. The sections in SI are referred to as S.x throughout the paper.

Fig. 2. Finite-element mesh used in the case of the highest normal load P = 2 . 12 N: (a) undeformed mesh, (b) zoom-in of the undeformed mesh, (c) zoom-in of the deformed mesh after initial normal loading, (d) zoom-in of the deformed mesh at full sliding. The leading edge is on the left.

Fig. 3. Shear-induced contact area reduction predicted for various elastic models. (a) Concurrent evolution of the tangential force, Q (dots), and the contact area, A (squares), as a function of the rigid plate displacement, d , for three different elastic models: linear elastic (dotted blue), neo-Hookean (solid purple) and Mooney–Rivlin (dashed green). P = 0 . 27 N. (b) A vs Q , for the three elastic models and for the four normal loads. The markers correspond to selected instants; the computations were carried out with a much finer time stepping. Solid straight red line: Q = σA . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

thecontactsurface.Thecorrespondinghighmeshdistortion,visibleattheleadingedge,isduetothejumpofthetangen- tialcontacttraction,introducedbytheTrescamodel,andalsoduetothenon-conformingdiscretizationofthecontactarea boundary(seerelateddiscussioninSectionS.1.2).

2.2. Shear-inducedcontactareareduction

Using the computational model described above, we simulated the shear-induced contact area reduction in the sphere/planeexperimentsof Sahlietal.(2018,2019),forthetwo hyperelasticmodels(neo-Hookean andMooney–Rivlin).

TypicalresultsarereportedinFig.3(a)whichshowstheconcurrent evolutions ofthe tangentialforce, Q,andthecontact area,A,asafunctionofthedisplacementoftherigidplate,d.Forcomparison,theresultscorrespondingtolinearelasticity arealsoincluded inFig.3,even iftherangeofstrainsencounteredinourconditionsdoesnotreallyadmit applicationof thesmall-strainframework.

Fig.3(a)showsthatthethreemodelsyielddrasticallydifferentpredictions,bothforthetangentialforceandforthecon- tactarea.Whiletheinitialcontactareaatzerotangentialforceishardlyaffectedbythemodel,majordifferencesappearas

Fig. 4. Contact zone (shown in red in the frame attached to the moving rigid plate) at the onset of full sliding for P = 0 . 27 N. (a) Experiment ( Sahli et al., 2018 ) (this specific diagram has not been published in the reference). (b) Neo-Hookean model. (c) Mooney–Rivlin model. (d) Small-strain linear elastic model. Dashed circles indicate the boundary of the initial contact zone, green regions denote the lifted area in the current (deformed) configuration. The leading edge is on the left. The mesh-dependent features and secondary contact regions visible in panels (b)–(d) are commented in Appendix A.5 and in Section S.1.2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

thetangentialforceincreases.Asignificantareareductionispredictedforbothhyperelasticmodels,butitremains negligi- bleforthelinearelasticmodel.Thisresultalreadyindicatesthatthefinite-deformationframeworkisactuallyaprerequisite toreproduceanarea reductionofseveraltensofpercents,asobservedexperimentally.Notethat theamplitudeofthere- ductionislargerfortheMooney–Rivlinmodel(37%forP=0.27N)thanfortheneo-Hookeanmodel(28%),showingthatthe effectofthematerialmodelthatgovernsthehyperelasticbehaviour ofthesampleissignificant.Those verydifferentfinal contactareas,combinedwiththevery samecontactshearstrength,

σ

^{,explainwhythefinal friction forceis}significantly largerforthelinearelasticmodel,andisalsodifferentforbothhyperelasticmodels.

Fig. 3(b) synthesizes all results by showing,for all normal loads and all three elastic models, the contact area A as a function ofthe tangentialforce Q. The absence ofarea reduction forthe linear elasticmodel andthe difference in its amplitudeforthetwo hyperelasticmodels isclearlyevidenced.The factthatall curvesendontheredlineindicatesthat ahomogeneous shear stress distribution equalto thecontactshear strength

σ

^{issuitably enforcedat theonset ofgross}

sliding.

Thedifferencesbetweenthevariouselasticmodels concernalsotheshape ofthecontactzone.InFig.4(b)–(d), typical modelpredictionsforthefinalcontactshapeareshownforthethreeelasticmodels(redregions).Althoughallmodelsstart withanalmostidenticalcircularcontactwhenQ=0(dashedlines),thefinalcontactshapehassignificantdifferences.The linearelasticmodelpredictsa finalcontactwhichremains circular,withalmostthesamearea asintheinitialconfigura- tion,consistentlywiththeresultsofFig.3.Thelargeareareduction intheMooney–Rivlinmodeloccurswhilekeepingan essentiallycircularcontactshape.Incontrast,theneo-Hookeanmodelleadstoan ellipse-likecontactshape,withthesize reductionoccurringalmostexclusivelyalongthesheardirection.

Wearenowinapositionto comparequantitatively ourmodelresultstotheexperimentalresultsofSahlietal.(2018, 2019).InFig.4,panel(a)showstheexperimentalcounterpartofpanels(b)–(d).Itclearlyappearsthatthemodelthatmost closelymatchesthefinal shapeintheexperimentsistheneo-Hookeanmodel.Inparticular,boththesizeandeccentricity oftheellipse-likecontactshape arevery-well captured.Inthe experiment,the radiusofcurvature ofthetrailingedge is visiblysmallerthanthatoftheleadingedge.Inthemodel,thiseffectisalsovisible,butlesspronounced.Alsonotethat,in theexperiments,thefinalcontactregionextendsbeyondtheinitialcontactattheleadingedge,aneffectwhichisweaker intheneo-Hookeanmodel,andthatwillbefurthercommentedinSection3.

Theexcellentagreementoftheneo-HookeanmodelwithexperimentsisfurtherdemonstratedinFig.5.Fig.5(a)directly comparesthepredictedA(Q) curveswiththose ofSahli etal.(2018) (seetheir Fig.2C),whileFig.5(b)compares thepre- dictedcontactsizesalongandorthogonaltoshear,_and⊥respectively,withthoseofSahlietal.(2019)(seetheirFig.3b).

Inbothcases,theamplitudesandshapesofthecurvesarewellcaptured,althoughthemodelslightlyunderestimatesAand doesnot capture the slightincrease of ⊥ observed in the experiments.Note that a similarlygood agreement has been obtainedforanalternativemodelwithdifferentregularizationsandnumericalimplementations(seeAppendixA.5andSec- tionS.1.3),thusshowingtherobustnessofourresults.

2.3.Elementarymechanismsofcontactareareduction

InSection 2.2,we have shownthat ourneo-Hookean modelprovides, without anyadjustableparameter, a very good quantitativepredictionof theshear-inducedcontactareareduction observedinexperiments.Onthismodelonly, we will nowperformathoroughanalysisofthesimulationresultstounderstandwhataretheelementarymechanismsresponsible forsuchareduction.

Important qualitative insights can already be obtained from a careful inspection of Fig. 4(b). First, the green region around thefinal contact zone corresponds to all nodesthat were initially in contact, andwhichare not anymore at the onsetof macroscopicsliding.Those nodeshave beenlifted out ofcontactduringthe incipientloading phase,dueto the

Fig. 5. Comparison of the model predictions obtained for the neo-Hookean model, for all normal loads, to the experimental results of Sahli et al. (2018, 2019) . (a) A vs Q . Solid straight red line: Q = σA . (b) Contact size along (  _ ) and perpendicular to (  ⊥ ) the shear loading direction vs Q . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

shear-induceddeformationsoftheelasticsphere.Thisfirstelementarymechanismofareavariationswillbenamedbelow

“contactlifting” (orsimply“lifting”).Second,smallparts(hardlyvisibleinFig.4(b))ofthefinalcontactarefound outside thedeformedinitialcontactregion(shown asa greenregioninthecurrentconfiguration).Thesenodeswereinitially out ofcontact,butcameintocontactduringtheincipientloadingphase.Thissecondelementarymechanismofareavariations willbenamed“contactlaying” (or“laying”).Finally,thethirdelementarymechanismisrelatedtoinhomogeneousslipand in-planedeformationwithinthecontactzoneandleadstoeither“in-planecompression” or“in-planedilation”.Suchdefor- mationsextendbeyondthecontactregion,whichexplainswhytheboundaryofthegreenzonedoesnotcoincidewiththe dashedcircle.

Theexistence ofmicro-slip inthemodelcanbeunambiguouslyidentified inFig.6,wheretheevolutionofthecontact duringincipientloadingisshownatfiveselecttangentialdisplacementsoftherigidplate:d₀=0,dsattheonsetofgross slidingandthreeintermediateconfigurations(d_i).Thenodesoftheelasticsurfacearelabeledinrediftheyhaveundergone some local slip, andin blueif they are still stuck tothe same point ofthe rigidplate. It can be seen inFig. 6 that the slip zone advances fromthe contact periphery towards the centreat the expense ofthe stick zone until the stick zone vanishesandfullslidingoccurs.ThecorrespondingMovieM1isprovidedasasupplementarymaterial.Asalsoobservedin theexperiments(Sahlietal.,2019),thecontactzoneandthestickzonehavean ellipticalshape,therelatedshear-induced anisotropyincreaseswithincreasingdisplacementdandishigherforlowernormalloads.

Partialslipconfigurations asthoseseeninFig.6areclassically found(withacircularstickzone)inmodelsofsheared frictionallinearelasticsphere/planecontact(Barber,2018;Johnson,1985).Theytypicallycorrespondtoheterogeneousslip fields,thuscausingin-plane deformations.Fig.7provides adetailedinsightintothismechanism, byshowingthe fieldof localsurfacedilation/compression.Foreachcontactnodei,itstributaryarea(i.e.theareaassociatedwiththenode,seealso SectionS.1.1)iscomputedbothintheinitialconfiguration(atd₀),Aⁱ₀,andinthecurrentconfiguration,Aⁱ.Thecolormapin Fig.7corresponds tothelocalsurfacedilation/compression,(^Ai− A₀ⁱ)/Aⁱ₀.Surfacedilationisobservedcloseto theleading edge whilecompressionisobserved closetothe trailingedge,whichis consistentwiththe distributionofthe

σ

^{xx stress}

componentshowninFig.A.2(c).Thedecreasinggreenellipticalregioninthemiddleofthecontactzone,inwhichthelocal surfacedilation/compressionisequaltozero,correspondstothestickzoneinFig.6.

Fig. 8compares the individual contributions ofall mechanisms (calculated asdescribedin Appendix A.4) tothe total relativecontactareachangeintheneo-Hookeanmodel.Forallnormalloads,themodelpredictsthatliftingrepresentsabout 90–95%ofthetotalshear-inducedareareduction,andisthustheprimarymechanismresponsibleforit.Thecontributionsof dilationandcompressionareindividuallysignificant,however,theyessentiallycancelout,sothatthein-planedeformation isresponsibleonlyforabout5–10% ofthetotalareareduction.Contactlayingisrarelyobserved(ifso,attheleadingedge only),sothatitscontributiontothetotalareareductionisessentiallynegligibleinthemodel.

We emphasize that the relevant liftingin our neo-Hookeanmodel is a locallifting due toshear deformations ofthe elastic sample,rather thana shear-inducedgloballiftingof therigidplate. Such agloballifting, althoughexisting inour constant normalloadsetting,hasbeenfound tobeonly about4% ofthenormaldisplacementassociatedwiththeinitial, purelynormalloading.Thus,itisfarfromsufficienttoexplainthe15–30%areareductionobservedinFig.8.

Fig. 6. Evolution of the contact zones for (a) P = 0 . 27 N ( d 1 , d 2 , d 3 : 35, 61, 81% of d s = 0 . 45 mm) and (b) P = 2 . 12 N ( d 1 , d 2 , d 3 : 29, 58, 80% of d s = 1 . 04 mm) shown in the frame attached to the moving rigid plate. Dashed circles indicate the boundary of the initial contact zone (at d 0 ). Blue and red regions denote the stick and slip zones, respectively. Green regions indicate the current (deformed) location of the initial contact zone that has been lifted. The leading edge is on the left. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. Evolution of the field of surface dilation/compression, (A ⁱ − A ⁱ₀) /A ⁱ₀ , shown in the frame attached to the moving rigid plate for (a) P = 0 . 27 N and (b) P = 2 . 12 N, and for the same select displacements as in Fig. 6 . Dashed circles indicate the boundary of the initial contact zone. The leading edge is on the left.

2.4.Qualitativeroleoffinitedeformations:half-spaceloadedbyatangentialtraction

Toreach ageneral understanding ofhow finitedeformations enhanceanisotropic contact area reductionwith respect tothelinearelasticcase,westudiedasimplifiedauxiliaryproblem:anearly incompressiblehyperelastichalf-spaceloaded by a homogeneous tangential traction qx (to mimick our Tresca model), applied over a circulararea of radius r. In the finite-elementmodel,thehalf-spaceistruncated,andthecomputationaldomainisacylinderofradius20randheight20r. ThedisplacementsatthetruncationboundaryareprescribedtothoseoftheBoussinesq–Cerrutisolutionforalinear-elastic half-spaceloadedbyaconcentratedtangentialforce(Johnson,1985).

Fig.9(a)showsthenormalsurfacedisplacementsinducedbythetangentialloading,forboththelinearelasticandneo- Hookeancases.Thosedisplacements arethesignatureofanormal/tangentialcoupling.Anon-vanishingcouplingisfound inthe linearelastic casebecausethe material isonly nearly incompressible. Still,the couplingremains weak, explaining whyasignificantcontactareareductionisnotfoundforlinearelasticity.Intheneo-Hookeancase,forshearstressessimilar tothoseusedinthecontactproblem(qx/

μ

=0.66correspondstoqx=0.4MPa,closetothevalueof

σ

^{usedinourTresca}

model),thenormalizeddisplacementsaremuchlarger,indicatingastrongnormal/tangentialcoupling.Thesymmetry,which is characteristicfor the linear-elastic model, is broken due to the finite-deformation effects. In particular, large negative displacementsoccuratthetrailingedge,explainingwhyliftinghasbeenfoundsignificantandlocalizedatthetrailingedge ofthecontact.Finally,the2DmapofthedisplacementfieldinFig.9(b)clearlyshowspositivedisplacementsinthedirection orthogonaltoshear.Suchpositivedisplacementsfoundintheneo-Hookeancase(resp.almostabsentintheMooney–Rivlin case, Fig.9(c))are presumably responsibleforthe absence ofliftingin thetransverse direction, andthus forthecontact anisotropyseeninFig.4(b)(resp.absentinFig.4(c)).

(a) (b)

Fig. 8. Evolution of the individual contributions of the various elementary mechanisms to the total relative area variation, A / A 0 , as a function of the rigid plate displacement, d , for (a) P = 0 . 27 N and (b) P = 2 . 12 N.

Fig. 9. Normalized vertical surface displacement, ¯u z = u zμ/ (q x r) , for a nearly incompressible ( ν= 0 . 49 ) half-space loaded by a homogeneous tangential traction q x along the x -axis, over a circular area of radius r . (a) Profile along the symmetry plane. Solid lines: neo-Hookean model for various q x / μ^{. Dashed} line: ( q x -independent) linear elastic case. (b,c) 2D maps within the ( x, y )-plane for q ^x /μ= 0 . 66 : (b) neo-Hookean and (c) Mooney–Rivlin model. Spatial coordinates ( x, y ) refer to the current configuration.

3. Illustrationexperiment

Inthissection,we performan illustrationexperiment, similarto thoseofSahlietal.(2018,2019)used to validate the modelintheprevious section,inordertotestwhethertheelementarymechanismsresponsibleforcontactareareduction inthemodel(lifting,laying andin-planedeformation)canalsobeidentifiedexperimentally.Thestrategyistoincorporate particlesclosetothesurfaceoftheelastomersphereandusethemastracersofthelocalmotionofthefrictionalinterface duringincipienttangentialloading.

3.1. Experimentalmethods

Theillustrationexperimentwasperformedonalaboratory-builtexperimentalsetup(seedetailsinSectionS.2.1)adapted fromthoseusedinMergeletal.(2019),Papangeloetal.(2019),Sahlietal.(2018,2019),andsketchedinFig.10.Thetwo mainimprovementsconsistedin(i)replacingthesinglebeamcantileverwithadoublebeamcantileverand(ii)addingforce sensors tomeasurethe normalload. Thissetupis usedtoshear theinterface betweenasmooth glassplateanda cross- linkedPDMSsample(nearlyincompressible,E=1.5± 0.1MPa)withasmoothsphericalcapseededwithalayerofparticles about16μmbelowthesurface(seedetailsinSectionS.2.2).Theexperimentpresentedherewasperformedunderconstant

Fig. 10. Sketch of the opto-mechanical setup. The lower glass plate supporting the elastomer sample (inset) is attached to an optical table, while the substrate (upper glass plate) is driven tangentially at constant velocity V , through a horizontal double cantilever. The tangential force along x is measured at the right extremity of the cantilever. The contact interface is illuminated from the top (lighting system not shown) and imaged in reflection with the camera. Inset: Sketch of the tracer-seeded, spherically-topped elastomer sample (blue, top). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. Concurrent evolution of the tangential force Q (solid black curve, left axis) and of the contact area A (solid grey curve, right axis) as a function of the imposed tangential displacement of the glass substrate. P = 1 . 85 N. V = 0 . 1 mm/s. Four values of the displacement are indicated, which will be used in further figures.

normalforceP=1.85N.19safterthecontacthasbeencreated,aconstantdrivingvelocityV=0.1mm/swasimposedto theglass,overa totaldistanceof2mm. Theevolutionofthe tangentialforce Q asa functionofthe displacementofthe glassplateisshowninblackinFig.11.Theincipienttangentialloadingoftheinterfacereachesamaximum,denotedasthe staticfrictionforceQ_s,beforeafullslidingregimeduringwhichQ<Q_s.

Typicalrawimagesofthe contactinterfaceare showninFig. 12(a–d)for fourdifferentdisplacements (alreadyshown inFig.11):d₀ beforeanyshear,ds thedisplacementatthestaticfriction peakQs,andtwo intermediatedisplacementsd₁ andd₂.AfullmovieisavailableasMovieM2providedasasupplementarymaterial.Thecontactregioncorrespondstothe biggestregionofdarkpixels.Figs.12(e–h)showtypicalcontoursofthecontact,theinnerareaofwhichdefinesthecontact area,A.TheevolutionofAasafunctionofthedisplacementimposedtotheglassplateisshowninFig.11(greycurve).Note thatbothcurvesinFig.11showthesamequalitativebehaviourasthatinpreviousexperimentsintheliterature(Mergelet al.,2019;Sahlietal.,2018;WatersandGuduru,2010),indicatingthatthe introductionoftheparticlesintotheelastomer sphereisnotsignificantly affectingthemechanicalresponseoftheinterface.The oscillationsobservableintheA(d)curve inFig.11originatefromso-calledre-attachmentfolds(Barquins,1985)occurringatthetrailingedgeofthecontact.

images. (a, e): d 0 , (b, f): d ¹ , (c, g): d ² , (d, h): d ^s . In (a-d), the main dark region is the contact, bright spots are the particles, and the top right dark region is a tracer drawn on the glass substrate to monitor its macroscopic motion. In (e-h), the contours are those of the contact (same colors as in Fig. 11 ), the inner area of which defines the contact area; black spots correspond to the particles and are the tracked objects. Images (a-d) are in the frame of the camera, while (e-h) are in the frame of the (moving) glass plate. The leading edge is on the left.

The contactregionisnot uniformlydark,asinour previoussphere/planeexperiments (Mergel etal., 2019;Papangelo etal., 2019; Sahliet al., 2018,2019), butis sprinkledwith randombright spotswhich are dueto light reflectiononthe particlesintroducedclosetotheelastomersurface. Byusingahomemadetrackingprocedure(seedetailsinSectionS.2.3), we were able to use those spots as tracers of the contact evolution and to generate a dataset in which the x- and y- positionsofeach tracerisgivenforeach image/time-step.Notethat onlythetracersattheverticalofthecontactregions arevisible,sothattheywillbeusednotonlytomeasurein-planedisplacementsbutalsotolookforliftingandlaying(see Sections3.2.1and3.2.2respectively).

3.2. Resultsandanalysis

Inthissection,basedontheresultsofthetrackingprocedure,weperformathoroughanalysisofthetracers’behaviour (appearance/disappearanceandin-plane motion)totest theexistence,inthe experiments,oftheelementarymechanisms foundtoberesponsibleforcontactarea evolutioninthemodel(seeSection2.3).Wefirstdemonstratethatcontactlifting andcontactlayingdooccur,atoppositesidesofthecontactzone,andwequantifytheiramountusingVoronoitessellation.

Second, using Delaunaytriangulation, we illustrate the progressive developmentof a heterogeneous in-plane strain field withinthecontactarea.

3.2.1. Contactlifting

Thefactthatonlytheparticlesattheverticalofthecontactregioncanbeseenopensthepossibilitytocheckwhether liftingoccursattheinterface.Indeed,thesignatureoflocalcontactliftingisaparticlethatdisappearsfromtheimagewhen reachedbythemovingcontactperiphery,meaningthatapointoftheelastomerinitiallyincontactwiththeglasshasbeen liftedout-of-contact.

Inpractice,welookedfortracertrajectoriesthatendedatalocationcloserthan20pixelstothecontactperiphery.Such tracersarerepresentedin Fig.13,eitherattheirinitial location(at d₀,filleddisks)orwherethey disappear(opendisks).

Asignificant numberoftracersare indeedlifted duringtheincipienttangentialloadingofthecontact. Thelargemajority ofthemarefound atthe trailingedge ofthe contact,wherepointsoftheglassleave thecontact.Twoofthemarefound attheleadingedgeofthecontact,butareprobablytracersinitiallyclosetothecontactperipheryanddisappearingdueto noiseintheimages.

Thecolorofeach diskinFig.13corresponds tothe glassdisplacementatwhichthetracer disappears.It appearsthat the colorsof the filleddisks are spatially organized, fromblue (earlydisappearance)for therightmost disks to red(late disappearance) for the leftmost disks. Such a pattern indicates that lifting occurs through an inward front propagation, startingfromthetrailingedge.

Inordertoquantifytheamountofcontactareathatislostduetolifting,weassignedarepresentativeindividualareato eachtracer.Forthis,weperformed,intheinitialimage,aboundedVoronoitessellationonthecentroidsofthetracers,the boundarybeingthecontactcontour. Eachtracerwasthusassignedthearea ofits Voronoicell.Then,foreach image,the liftedareaiscomputedasthesumoftheareasoftheVoronoicellsofallliftedtracers.InFig.13,thegreycellscorrespond to all Voronoicellsassignedto liftinginthe final image (atds). Theevolution ofthe liftedarea along theexperimentis showninblueinFig.14.Theevolutionisstair-like,becauseeachliftingeventincreasesabruptlytheliftedareabyafinite amount(thecellarea).Theestimatedfinalliftedarearepresentsabout31%oftheinitialcontactarea.

Fig. 13. Measurement of the lifted and laid areas. Solid colored lines: contours of the contact at the four selected displacements (same colors as in Figs. 11 and 12 ). Open (resp. filled) disks: position of the lifted particles when they disappear (resp. at d 0 ). The symbol color corresponds to the displace- ment at which the particle is lifted (same color code as for the contours). Grey cells: cells of the Voronoi tessellation at d 0 associated to lifted particles at d s . Filled (resp. open) diamonds: position of the laid particles at d s (resp. when they appear). The symbol color corresponds to the displacement at which the particle is laid. Red cells: cells of the Voronoi tessellation at d s associated to laid particles at d s . The leading edge is on the left. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 14. Contributions of the various mechanisms to the area variation. Relative area variation, A / A 0 , vs the imposed displacement, d . Black: reduction of the total contact area. Blue (resp. pink): area lost (resp. gained) by lifting (resp. laying), measured as in Fig. 13 . Brown: area variations due to slip-induced in-plane deformation, measured as in Fig. 15 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3.2.2. Contactlaying

Themechanismoflaying,i.e.,pointsoftheelastomergettingintocontactwiththeglassuponshearing,canbeanalyzed inasimilarwayaslifting.Weperformedthesameanalysisasintheprevioussection,butbackwardintime:thefirstimage consideredwastheoneatthestaticfrictionpeak(atd_s),whilethelastonewastheinitialimage(atd₀).Doingso,tracers appearingclosetothecontactperipheryinforwardtimearedetectedasdisappearinginbackwardtime.

Theresults ofthisanalysisare shownin Fig.13,where thelocationsof thelaid tracersare showneither inthefinal configuration(atd_s,filleddiamonds)orwheretheyappear(opendiamonds).Alllaidtracersarefoundattheleadingedgeof thecontact,beyondtheinitialcontactregion,consistentlywithFig.4(a).Theircolor,evolvingfrombluetored,corresponds tothestageatwhichtheyappearandisagainspatiallyorganized,indicatingthatcontactlayingoccursthroughanoutward front propagationstartingat theleadingedge of thecontact. The amount ofcontact area gainedvia laying is estimated usingaboundedVoronoitessellationperformedinthe finalcontactarea (atds).Ateach instant,the laidarea iscounted asthesumof theareasof all laid tracers. InFig.13,thered cellscorrespond tothe area assignedto laying inthe final image(atds).TheevolutionoftheareagainedthankstolayingisshowninpinkinFig.14.Thefinalamountoflaidareais

Fig. 15. Measurement of in-plane deformations. (a) Grey network: Delaunay tessellation based on all tracers surviving from d 0 to d s . (b-d) Snapshots of the evolution of the Delaunay cells (defined at d 0 ) at: d 1 (b), d 2 (c) and d s (d). Solid lines are the contours of the contact at d 0 , d 1 , d 2 and d s with the same color code as in Figs. 11 , 12 (e-h) and 13 . The color of each cell corresponds to its relative area change with respect to the situation at d 0 (see colorbar).

Colder (resp. warmer) colors mean in-plane compression (resp. dilation). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

about9%oftheinitialcontactarea.Unlikelifting,thelayingmechanismseemstoinitiateonlyafterafiniteshearhasbeen applied.

3.2.3. In-planedeformation

Interestingly,intheprevious analysesofboth contactliftingandlaying (seeFig.13), thefilledandopenmarkers have differentpositionswithrespecttotheglasssubstrate.Thisisaclearindicationthatslipoccursbetweenthetwoinstants:

localliftingispreceded byslip, whilelocallayingis followedbyslip. Slipisfoundtobe roughlyparallel, butopposedto theglassmotion,andoccursinboththeleadingandtrailingregionsofthecontact.

This observationis consistentwith a scenarioof a micro-slip front propagating inward thecontact regionas shearis increased,whichisclassicalinshearedsphere/planecontacts,eitherrough(Prevostetal.,2013)orsmooth (Chateauminois et al., 2010). In this scenario, which is apparent in Movie M2 and also observed in the model results of Section 2.3, a peripheralslipregionprogressivelyinvadesthecontact,replacingashrinkingcentralstuckregion.Fromthecombinationof backwardslipatthetrailingedgeandastuckzoneatthecontactcenter,oneexpectsin-planecompressionoftheelastomer inthetrailinghalfofthecontact.Symmetrically,in-planedilationisexpectedintheleadinghalf.Theoveralleffectofboth typesofin-planedeformationmustleadtoachangeincontactareathatweestimatedasfollows.

We onlyconsidered thetracerspresentin theinitial image andthat couldbe followed duringthewhole experiment, fromd₀ tods,thusexcludingtheliftedandlaidones.WeperformedaDelaunaytriangulationtomeshallthosetracersin theinitialimage,asshownasagraynetworkinFig.15.Asshearincreases,thetracersmoverelativetoeachother.Keeping the samemesh,we follow therelative changeof area ofeach initial Delaunaytriangle,asan estimatorof localin-plane deformation.

Fig.15showstheevolutionofthefieldofin-planedeformationasshearisincreased(coldcolorsforcompression,warm onesfordilation).Befored₁nodeformationisobserved,presumablybecausetheslipfronthasnotreachedyetthefraction ofinitialcontactcoveredbytheDelaunaytriangulation.Betweend₁andd₂,acompressedregionsetsinatthetrailingside ofthe contact;the absenceof adetectable symmetricaldilated region onthe leadingside suggeststhat the propagation of theslip front is not axisymmetrical,as usually considered in models ofthe incipient shearloading of circularlinear- elastic contacts (Barber, 2018). At ds, a heterogeneous slip-induced in-plane deformation field is fully developed,with a largecompression regiononthetrailingside anda smallerdilationregion ontheleadingside. Thosefield measurements are qualitativelyconsistent withprevious measurements ona similarsystem, butmadeonlyalongthe centrallineofthe contactinsteady-statesliding(Barquins,1985).

Theevolution undershearofthesumofthetotalarea containedinthe deformingDelaunaytriangulationisshownin brownin Fig.14.Thisarea remains essentially unchangedduringthefirst halfofthe experiment, andthenprogressively decreasesbyuptoabout7%oftheinitialcontactarea.

4. Discussion

ThemodelpresentedinSection2hasbeenshowntocapturequantitativelytheexperimentalresultsofSahlietal.(2018, 2019) about the anisotropic shear-inducedarea reduction of sphere/plane elastomer contacts for relatively large normal loads(Fig.5). Weemphasizethatthisagreementisobtainedwithoutanyadjustableparameter,inthesense thatvirtually all modelparametersaresetby independentmeasurementson thesameexperimentalsystem. Thoseparameters are:the

ofourmodelresultstoallimplementationdetails.

Thedirectimplicationofourmodelisthatfinite-deformationeffectsandthenon-linearelasticityofelastomersarepre- sumablythekeyingredientsexplainingthoseexperimentalresults,ratherthanviscoelasticityoradhesion.Nevertheless,our modelhasonlybeenappliedtocontactssubmittedtorelativelylargenormalload,inthenewtonrange(Sahlietal.,2018), whileother experimentshaveusedmuchsmallernormalloads,inthemillinewtonrange(Mergeletal.,2019;Savkoorand Briggs,1977;WatersandGuduru,2010).Forthoselighterloads,adhesivestressesmaybeoftheorderof,orevenexceed, contactstresses, and non-linear elasticitymay be not the only dominantingredient forshear-induced contactshrinking anymore.Identifyingthenormalloadregimesinwhichadhesionneedstobeaccountedforisanimportantgoalforfuture contactmechanicsmodels.Inthefollowing,wewillrestrictourdiscussiontothehigh-loadregimeexploredinthepresent study.

Bycomparing models assuming either small or finite strains(Fig. 3), we have shown that finite deformations is the singlemodelingredientresponsibleforthesignificantshear-inducedcontactareareductionobservedinsphere/planeelastic contacts.Althoughtheamplitudeofthereductionsignificantly dependsontheparticularhyperelasticmodelused(Fig.3), therespectiverolesofhyperelasticityandoftheexactgeometricallynon-linearkinematicsaredifficulttodisentangle.What isclear isthat finite deformationsinduce a significant couplingofthe normaland tangentialdeformation modes, which islackinginthe small-strainincompressibleelasticity.Such acouplinggeneratessubstantial verticaldisplacements which accountbothforasignificantliftinglocalizedatthetrailingedgeandforthemarkedanisotropyofthefinalcontactshape (Fig.9).

Wequantified,forthefirsttime,thevariouselementarymechanismsbywhichareavariationscanoccur:contactlifting, contactlayingandin-planedeformation.Ourmodelresultssuggestthat,intheexperimentalconditionsusedinSahlietal.

(2018,2019),localliftingisthedominantmechanismexplainingtheobservedsignificantarea reductionanditsanisotropy (Fig.8). Thisconclusionistrueforall normalloadsexplored, withthe totalreductionamplitudebeingdependent onthe normalload (as alsoobserved experimentally). In futureworks, it wouldbe interesting to varysystematically all model parametersbeyondtheexperimentalrange,toclarifytheirrespectiverolesintheareavariationsdueto eachofthethree mechanisms.

Theincipient shear-loading ofsmooth sphere/plane elastic contactsis characterized inthe modelby two propagating fronts:aliftingfrontatthecontactperiphery,andanon-circularmicro-slipfrontwithinthecontact.Theexistenceofthose twodifferentfronts,althoughexplicitlyacknowledgedbysomeauthors(Johnson,1997;McMeekingetal.,2020),hasnever beenproperlydescribed infracture-based adhesivemodels. Indeed,inthose models, theadditional energydissipation in thecontactduetofrictionalmicro-slip,anddescribedviathemode-mixityfunction,isassumedtobelocatedatthecontact periphery,andnotwithinagrowing,finiteregionofthecontact.

The relevance of our model observations has been qualitatively confirmed by the original experimental results of Section 3. Using an elastomer sphere seeded withtracers, we demonstrated that shear-inducedanisotropic contactarea reductionisindeedaccompaniedbyallthreepossiblemechanisms.Althoughahigherarealdensityoftracerswouldbede- sirabletoreduce themeasurementuncertainties(AppendixB),wecould concludethatliftingispresumablythedominant areareductionmechanism,alsointheexperiments.Whileexperimentsrevealedapotentiallylargercontributionofthelay- ingandin-plane mechanisms,adefinitivequantitative comparisonwiththemodelishinderedby thefiniteexperimental resolution.Alsonotethat,duetothehighercontactshearstrength(0.53ratherthan0.41MPa)andsmallerYoung’smodulus (1.5ratherthan1.8MPa) ofthepresentexperimentscomparedtothose ofSahlietal.(2018,2019),ournumericalmodel could not convergein those more severe conditions, thus impedingdirect comparison between model andtracer-based experiments.Suchacomparisonisalsoanimportantchallengeforfutureworks.

Finally,thesimplicityandgenerality ofourmodelassumptions suggest that ourresultsmaybe alsorelevant toother systemsthan the elastomersphere/planecontacts studiedhere. First, non-linear elasticitybeinga generic feature ofsoft materials, fromgels to human skin,we expect it to be a likely mechanismfor contactarea reduction in all studies in- volvingsuchmaterials. Itwouldthusbe interestingtore-interpretrecentexperimentslikethoseofe.g.Das andChasiotis (2020) onpolyacrylonitrile orthose ofSahlietal. (2018),Sirinetal. (2019)on humanfingertips, fromthestandpoint of finite-deformation mechanics. Second, Sahliet al.(2018, 2019)argued that themechanisms of shear-inducedanisotropic contactareareductionmaybethesameinmillimetricsphere/planecontactsandinindividualmicrometricjunctionswithin roughcontactinterfaces.Such asimilarityacrossscalessuggeststhatthepresentconclusionsmayalsobe usedtofurther understandthe shear behaviour ofsoft material multicontacts, the modellingof whichmay requirenon-linear elasticity.

wesuggestthatfinite-deformationeffectsandnon-linearelasticitycanbeequallyimportant,allthemoresoaslargenormal loadsareconsidered.Clarificationofthevaliditydomainswhereadhesionand/orfinitedeformationsneedtobeaccounted forremainsasamajoropenissueonthetopic.

DeclarationofCompetingInterest

The authors declare that they have noknown competing financial interests orpersonal relationshipsthat could have appearedtoinfluencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

J. Lengiewicz: Conceptualization, Methodology, Software, Investigation, Writing - original draft, Visualization, Funding acquisition. M. deSouza:Investigation,Visualization, Data curation,Writing - review& editing.M.A.Lahmar: Methodol- ogy,Investigation.C.Courbon:Conceptualization,Methodology,Software,Investigation,Writing-review&editing,Funding acquisition. D.Dalmas:Conceptualization,Methodology, Data curation,Writing- review& editing,Fundingacquisition. S.

Stupkiewicz:Conceptualization,Writing-originaldraft,Writing-review&editing.J.Scheibert:Conceptualization,Method- ology,Writing-originaldraft,Writing-review&editing,Fundingacquisition,Supervision.

Acknowledgement

This work waspartially supported by the EU Horizon 2020 Marie Sklodowska Curie Individual Fellowship MOrPhEM (H2020-MSCA-IF-2017,projectno.800150).DD,MdSandJSareindebtedtoInstitutCarnotIngénierie@Lyonforsupportand funding.TheyalsothankCristóbalOliverVergaraforhishelpintheinitialdevelopmentoftheparticletrackingprocedure.

WethankananonymousreviewerforsuggestingthesimulationsdescribedinSection2.4.

AppendixA. Descriptionofthefinite-deformationmodel A.1. Finite-strainframework

Thefinite-strainframework isstandard,soonlythebasicnotionsareprovidedbelowforcompleteness,withthescope limited tohyperelasticity. Twoconfigurations ofthe body are considered:the referenceconfiguration ^{and thecurrent} (deformed) configuration

ω

^{. The} deformation that brings ^to

ω

^{is describedby the}deformation mapping

ϕ

^{such that}

x=

ϕ

(^X),whereX∈^andx∈

ω

^{denotethepositionofamaterialpointintherespective}configuration.Thedeformation gradientisdefinedasF=Grad

ϕ

,whereGradisthegradientwithrespecttothereferenceconfiguration^.

Equilibriumequationinthestrongformiswritteninthereferenceconfiguration^{,and,intheabsenceofbodyforces,} reads

DivP=0, P=

∂

^W

∂

^{F, (A.1)}

whereP isthefirst Piola–Kirchhoff stresstensor,Div isthedivergenceoperator relativeto thereferenceconfiguration^, andtheelasticstrainenergyW=W(^F)^{specifiesthe}constitutiveresponseofahyperelasticbody.

Thereferencematerialmodelusedinthisworkisthenearly-incompressibleisotropicneo-Hookeanmodelspecifiedby W=WnH,

W_nH

(

^F

)

=1

2

μ (

^¯I1− 3

)

+ Wvol

(

^J

)

, W_vol

(

^J

)

= 1 4

κ 

(

^J− 1

)

²+

(

^logJ

)

²



, (A.2)

whereW_vol(J)isthevolumetricpartoftheelasticstrainenergy,J=detF,and¯I1=tr¯bistheinvariantof ¯b=J^−2/3b,b=FF^T. Materialpropertiesareherespecifiedbytheshearmodulus

μ

^{andbulkmodulus}

κ

^.

Theneo-HookeanmodelisaspecialcaseoftheMooney–RivlinmodelspecifiedbyW=W_MR,

WMR

(

^F

)

⁼¹₂

μ

¹

(

^{¯I1− 3}

)

⁺¹₂

μ

²

(

^{¯I2− 3}

)

^+Wvol

(

^J

)

^{, (A.3)}

where¯I₂=¹₂(^¯I2₁− tr¯b²)^and

μ

=

μ

1+

μ

2.Theneo-Hookeanmodelisrecoveredfor

μ

2=0.Theotherparticularcaseofthe Mooney–Rivlinmodelusedin thisstudycorresponds to

μ

1=

μ

2=

μ

/2, andissimplycalledMooney–Rivlininall model results.

A.2.Contactwitharigidplane

Thecontactformulationbrieflydescribedbelowislimitedtothecaseofcontactwitharigidplane,whichissufficientfor thepurposeofthiswork,seee.g.Wriggers(2006)foramoregeneralpresentation.Theorientationoftheplaneisspecified bytheoutwardunitnormal

ν

^{,andthemotionoftherigidplaneisrestrictedtoa}translation,thus

ν

˙=0.

Contactkinematics isspecified bythe normalgapg_N andtangentialslipvelocity v_T that are definedforeach pointx onthepotentialcontactsurface

γ

^c=

ϕ

(c),wherecdenotesthecontactsurfaceinthereferenceconfiguration.Here,the kinematicquantitiesaresimplygivenby

gN=

(

^x− xR

)

·

ν

^,

v

T=

(

¹−

ν



ν )(

^x˙− ˙xR

)

, (A.4) wherex_Rdenotesthecurrentpositionofafixedpointontherigidplane,and denotesthediadicproduct.

The contacttraction vector tis definedasthe traction vector exerted by the body on therigid plane. Bythe action- reaction principle itopposes the traction vector actingon the body, sowe havet=−

σ

ⁿ, where

σ

^{isthe Cauchy stress}

tensor,nistheoutwardunitnormaltothecontactsurface

γ

^c,andn=−

ν

^{whenevercontactoccurs.Notethattisaspatial}

tractionvector,i.e.,itreferstoaunitareainthecurrentconfiguration.tisdecomposedintoitsnormalandtangentialparts relativetotherigid-planenormal

ν

^,namely

t_N=t·

ν

, t_T=

(

¹−

ν



ν )

^t, (A.5)

sothatt=t_N

ν

+t_T.

Withtheabovedefinitions,unilateralcontactconditionsareexpressedas:

gN≥ 0, tN≤ 0, gNtN=0. (A.6)

ThefrictionmodelisdiscussedinAppendixA.3.

Finally,thevirtual work principle,i.e., theweakform ofthemechanicalequilibrium, whichis thebasis forthefinite- elementimplementation,isexpressedas:



P· Grad

δϕ

^dV+

γc

(

^tN

δ

^gN+tT·

δ

^gT

)

^ds=0

∀ δϕ

^{, (A.7)}

where

δϕ

^{isthetestfunctionthatvanishesonthepartoftheboundaryof}^onwhichthedisplacementisprescribed,and wehave

δ

^gN=

ν

·

δϕ

^and

δ

^gT=(¹−

ν



ν

)

δϕ

^{(cf.Eq.(A.4)).Sincet}Nandt_Tarespatialtractions,thecontactcontribution, i.e.,thesecondterminEq.(A.7),isintegratedoverthecontactsurface

γ

^{cinthecurrent}configuration.

A.3.RegularizedTrescafrictionmodel

IntheTrescafrictionmodel,thelimitfrictionstress,calledherethecontactshearstrength,

σ

^{,isconstantandindepen-}

dentofthenormalcontacttractiont_N(Fig.A.1(b)).TheTrescamodelcanbewrittenas:



^tT



⁻

σ

^{≤ 0,}



^tT

 v

T=tT

 v

T



^,

( 

^tT



⁻

σ )  v

T



^{=0, (A.8)}

tangentialtractionvanishes,t_T=0.Notethat,inviewoftheunilateralcontactcondition(Eq.(A.6)),penetration(g_N<0)is notallowed.

The first condition in Eq.(A.8) is the limit friction condition. The second condition is the sliprule that implies that thedirectionofthetangential(slip) velocityv_Tisthatofthetangentialtractiont_T.Thethird(complementarity)condition controlsthestick/slipstatesuchthatstickingcontact(

v

T=0)occurswhen



^tT



<

σ

^andsliding(



^vT



>0)mayoccuronly when



^tT



=

σ

^.

OnefeatureoftheTrescamodelisthatthetangentialtractiont_Tmaysufferajumpattheinstantoftransitionbetween active contactandseparation.This,inturn,mayleadtosignificant convergenceproblemsintherespectivecomputational scheme, forinstance,in the framework ofthe finite-element method.Accordingly, a regularization scheme hasbeen de- veloped, as describedbelow, andemployed in the actual computations.Note that an alternative approach, basedon the Coulomb–Orowanfriction model (Fig.A.1(c)), hasalso beenexamined, andtherespective resultsare briefly presentedin AppendixA.5,seealsoSectionS.1.3.TheTrescamodelisthelimitcaseoftheCoulomb–Orowanmodelfor f→+∞.

Intheregularizationschemeadoptedhere,itisassumedthat thetangentialtractiont_T doesnotdroptozeroinstantly afterseparationoccursbutrequiressomecharacteristictime

τ

^{tograduallyvanish.}Accordingly,thefollowingsimpleevolu- tionlawisadoptedinthecaseofseparation(g_N>0):

t˙T=



−

σ

τ 

^ttTT



^for



^tT



>0,

0 otherwise, (A.9)

so that the magnitude of thetangential traction t_T decreases towards zeroat a constant rate of

σ

^/

τ

^{. A similar regular-}

ization, leadingto a kind ofrate-and-state friction law,has been adopted by Cochard andRice (2000) to regularize the abruptchangesinfrictionassociatedwithabruptchangesinthenormalcontacttraction,seealsoPrakashandClifton(1993), Prakash(1998)fortheexperimentaljustificationandrespectiveconstitutivemodelling.Here,theregularizationisintroduced purelyforcomputationalreasons.

A.4. Finite-elementmodel

Finite-elementimplementationhasbeenperformedusingtheAceGen/AceFEMsystem(KorelcandWriggers,2016).Eight- nodehexahedralTSCG12elements(Korelcetal.,2010)areusedforthesolid.TheaugmentedLagrangianmethod(Alartand Curnier,1991)combinedwithnodalintegrationisusedtoenforcetheunilateralcontactandfriction conditions(Eqs.(A.6) and(A.8)),seeLengiewicz etal.(2011)fortherespectiveautomationandAceGen-basedimplementation.Moredetails are providedinSectionS.1.1.

The geometric parameters andmaterial propertiesused in thesimulations are directly takenfrom one ofthe experi- mentaldatasetsreportedinSahlietal.(2018)(smoothPDMS-sphere/bare-glasscontact,seeFig.2Ctherein).Theboundary conditionsandtheloading sequencealsocorrespond tothose intheexperiment. Thedisplacements arefullyconstrained atthebottomsurfaceoftheelastomersample(cf.Figs.1and10),exceptforarigid-bodytranslationinthez-directionso that the normalloadP canbe applied against a rigidplane representing theupper glassplate. Subsequently, a constant tangentialvelocityoftherigidplane isprescribedalong thex-direction whilethenormalloadP iskeptconstant.Further, thesymmetrywithrespecttothey=0planeisexploitedsothatonlyonehalfofthesampleiseffectivelymodelledwith adequateboundaryconditionimposedonthesymmetryplane.

Theresultingfinite-elementmeshisshowninFig.2.Thesampleisdiscretizedwithabout100,000hexahedralelements, whichgivesapproximately380,000degreesoffreedom.Thehanging-nodetechniqueisusedtoconvenientlyrefinethemesh inthevicinityofthepotentialcontactsurface,andthesizeoftherefined-meshregionisadjustedtothenormalload.The contactsurfacecomprisesabout12,600nodeswiththesizeofthequadrilateralelementsrangingfrom13μmforP=0.27N to25μmforP=2.12N.