A 2-D model for friction of complex anisotropic surfaces Journal of the Mechanics and Physics of Solids

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Journal of the Mechanics and Physics of Solids 112 (2018) 50–65

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Journal of the Mechanics and Physics of Solids

journalhomepage:www.elsevier.com/locate/jmps

A 2-D model for friction of complex anisotropic surfaces

Gianluca Costagliola^a, Federico Bosia^a, Nicola M.Pugno^b^,^c^,^d^,^∗

a Department of Physics and Nanostructured Interfaces and Surfaces Centre, University of Torino, Via Pietro Giuria 1, Torino, 10125, Italy

b Laboratory of Bio-Inspired & Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano, 77, Trento 38123, Italy

c School of Engineering and Materials Science, Queen Mary University of London, Mile End Road, London E1 4NS, UK

d Ket Labs, Edoardo Amaldi Foundation, Italian Space Agency, Via del Politecnico snc, Rome 00133, Italy

a rt i c l e i n f o

Article history:

Received 24 July 2017 Revised 7 November 2017 Accepted 16 November 2017 Available online 20 November 2017 Keywords:

Friction Numerical models Microstructures Anisotropic materials

a b s t ra c t

The friction force observedat macroscale is the result of interactionsat various lower lengthscalesthataredifficulttomodelinacombinedmanner.Forthisreason,simplified approaches arerequired,dependingonthe specificaspectto beinvestigated.In partic- ular,thedimensionalityofthesystemisoftenreduced,especiallyinmodelsdesignedto provideaqualitativedescriptionoffrictionalpropertiesofelasticmaterials,e.g.thespring- blockmodel.Inthispaper,weimplementforthefirsttimeatwodimensionalextension ofthespring-blockmodel,applyingittostructuredsurfacesandinvestigatingbymeansof numericalsimulationsthefrictionalbehaviourofasurfaceinthepresenceoffeatureslike cavities,pillarsorcomplexanisotropicstructures.Weshowhowfrictioncanbeeffectively tunedbyappropriatedesignofsuchsurfacefeatures.

1. Introduction

Thefrictionalbehaviorofmacroscopicbodiesarisesfromvarioustypesofinteractionsoccurringatdifferentlengthscales betweencontactsurfacesinrelative motion.Whileitisclearthat theirultimateoriginliesininter-atomicforces,itisdif- ﬁcult to scale theseup tothe macroscopic levelandto includeother aspects such asdependenceon surfaceroughness, elasticityorplasticity,wearandspeciﬁcsurfacestructures (NosonovskyandBhushan,2007;Persson,2000).Moreover,the dependenceon“externalparameters”,e.g.relative slidingvelocity ofthesurfacesandnormalpressure,isneglectedinap- proximatemodelssuchasthefundamentalAmontons–Coulomblaw,andviolationsofthelatterhavebeenobserved(Deng etal.,2012;Katanoetal.,2014).

Forthesereasons,simplifiedmodelsarerequiredintheoreticalstudiesandnumericalsimulations,andfrictionproblems canbe addressedindifferentwaysdependingon thespecificaspectsunderconsideration.Inordertoimprovetheoretical knowledgeoffriction,ortodesignpracticalapplications,itisnotnecessarytosimulateallphenomenasimultaneously,anda reductionistapproachcanbeusefultoinvestigateindividualissues.Thus,despitetheimprovementincomputational tools, inmostcases itis still preferableto developsimplified models todescribe specific aspects,aiming to providequalitative understandingofthefundamentalphysicalmechanismsinvolved.

One ofthe mostusedapproaches todeal withfriction of elasticbodies consistsinthe discretization ofa materialin springsandmasses,asdonee.g.intheFrenkel–Kontorovamodel(BraunandKivshar,2004),ortheBurridge–Knopoff model

∗ Corresponding author.

E-mail addresses: [email protected] (G. Costagliola), [email protected] (F. Bosia), [email protected] (N.M. Pugno).

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G. Costagliola et al. / Journal of the Mechanics and Physics of Solids 112 (2018) 50–65 51

(BurridgeandKnopoff,1967), thelatteralsoknownasthespring-blockmodel.Forsimplicity,thesemodels areoftenfor- mulatedinonedimension alongthesliding direction,invarious versionsdependingonthe speciﬁcapplication. Inrecent years,interesting resultshave beenobtained withthese models, explainingexperimental observations (Amundsen etal., 2012;Bouchbinderetal.,2011;Maegawaetal.,2010;Rubinsteinetal.,2004;ScheibertandDysthe,2010;Trømborgetal., 2011).Theextensiontotwodimensionsisthestraightforwardimprovementtobetterdescribeexperimental resultsandto correctlyreproducephenomenaoccurringintwodimensions.Thishasalreadybeendoneforsomesystems,liketheFrenkel–

Kontorovamodel(Mandellietal.,2015;Norelletal.,2016)andthespring-blockmodelappliedtogeology(Andersen,1994;

Brownetal.,1991;Giaccoetal., 2014;Mori andKawamura,2008a;Mori andKawamura,2008b; Olamietal., 1992),but muchworkremainstobedonetodescribethefrictionofcomplexandstructuredsurfaces.

Theinterestofthisstudyliesnotonlyinthenumericalmodelingoffrictioninitself,butalsointhepracticalaspectsthat canbeaddressed:therearemanystudies relativetobio-inspiredmaterials(Baumetal.,2014;Lietal.,2016;Murarashet al.,2011;Yurdumakanetal.,2005)orbiologicalmaterials(Autumnetal.,2000;Labonteetal.,2014;StempﬂeandBrendle, 2006; Stempﬂe et al., 2009; Varenberg et al., 2010) that involve non-trivial geometries that cannot be reducedto one- dimensionalstructuresinordertobecorrectlymodeled.

Theonedimensionalspring-blockmodelwasoriginallyintroducedtostudyearthquakes(CarlsonandLanger,1989;Carl- son etal., 1994; Xiaet al., 2005) andhasalso been usedto investigatemany aspects ofdryfriction of elasticmaterials (Amundsen et al., 2015; Braun etal., 2009; Capozza and Pugno, 2015; Capozza et al., 2011; Capozza and Urbakh, 2012;

Pugnoetal.,2013;Trømborgetal.,2015).InCostagliolaetal.(2016),wehaveextensivelyinvestigatedthegeneralbehav- iorofthemodelandtheeffectsoflocalpatterning(regularandhierarchical)onthemacroscopicfriction coeﬃcients,and inCostagliolaetal.(2017a) wehaveextendedthestudytocompositesurfaces,i.e.surfaceswithvaryingmaterialstiffness androughness;ﬁnallyinCostagliolaetal.(2017b)wehaveintroducedthemultiscaleextension ofthemodeltostudythe statisticaleffectsofsurfaceroughnessacrosslengthscales.

Inthis paper,we propose a 2-D extension of the spring-block modelto describe the frictional behavior ofan elastic materialslidingona rigidsubstrate.Ourmain aimistocomparetheresultswiththoseobtainedinthe one-dimensional caseandto extendthe studytomorecomplexsurfacestructures,e.g. isotropicoranisotropic arrangementsofcavities or pillars,simulatingthosefoundinbiological materials.Thetwo-dimensional spring-blockmodelallowstoconsidera more realisticcasesandcapturesavarietyofbehaviorsthatcanbeinterestingforpracticalapplications.

Thepaperisorganizedasfollows:inSection 2,wepresentthemodel;inSection 3.1,wediscussthemaindifferences withtheone-dimensionalcaseandweexploretheroleoftheparameterswithoutsurfacestructures,highlightingthephe- nomenologyofthe model;inSection3.2,wepresenttheresultsforstandard1-D and2-D surfacestructureslikegrooves andcavities;inSections3.3and3.4,weconsidermorecomplexcasesofanisotropicsurfacepatterning;ﬁnally,inSection4, conclusionsandfuturedevelopmentsarediscussed.

2. Model

Theequationofmotionforanisotropiclinearelasticbodydrivenby aslideronaninfinitelyrigidplane withdamping andfriction can be written as:ρû^¨=μ∇²û+(λ+μ)∇(∇· u)−γ ρû^˙, where uis the displacementvector, ρ îs^the ^mass

density,γ ^is^the^damping^frequency,λ^,μ^are^theLamé constants.Thefollowingboundaryconditionsmustbeimposed:the topsurfaceofthebodyisdrivenatconstantvelocityv,thebottomsurfaceissubjectedtoaspatially variablelocalfriction force, whichwe discussbelow, representingthe surfaceinteractions betweentheelastic body andthe rigidplane, while freeboundaryconditionsaresetontheremainingsides.

Inordertosimulatethissystem,weextendthespring-blockmodeltothetwo-dimensionalcase:thecontactsurfaceis discretizedintoelementsofmassm,eachconnectedbyspringstotheeightﬁrstneighborsandarrangedinaregularsquare mesh(Fig. 1)withNx contactpointsalong thex-axisandNy contactpointsalongthe y-axis. Hence,the totalnumberof blocksisN_b≡NxN_y.Thedistancesontheaxisbetweentheblocksare,respectively,l_xandl_y.Themeshadoptedinprevious studiesofthe2-Dspring-block model,e.g. Olamietal.(1992)andGiaccoetal.(2014),doesnotincludediagonalsprings, butweaddthemtoaccountforthePoissoneffect(ourmeshinsimilartothatusedinTrømborgetal.(2011)).

Inordertoobtaintheequivalenceofthisspring-masssystemwithahomogeneous elasticmaterialofYoung’smodulus E,thePoisson’sratiomustbeﬁxedtoν=1/3(AbsiandPrager,1975),whichcorrespondstotheplanestresscase,lx=ly≡ l andK_int=3/8Elz, where lz is the thickness ofthe 2-D layer andK_int is the stiffness of the springs connectingthe four nearestneighborsofeachblock,i.e.thosealignedwiththeaxis.ThestiffnessofthediagonalspringsmustbeK_int/2.Hence, the internal elastic force on the block i exerted by the neighbor j is F⁽_int^{i j}⁾=k_{i j}(^ri j− li j)(^rj− ri)/ri j, wherer_i, r_j are the positionvectorsofthetwoblocks,r_ijisthemodulusoftheirdistance,l_ijisthemodulusoftheirrestdistanceandk_ij isthe stiffnessofthespringconnectingthem.

All the blocks are connected, through springs of stiffness Ks, to the slider that is moving at constant velocity v in the xdirection, i.e. theslider vector velocity is v=(v,0)^. ^Given^the înitial ^rest^position ^r⁰i of block i,the shear force is F⁽_sⁱ⁾=K_s(^v^t+r⁰_i− ri)^. ^We^define ^the^total^driving ^forceônⁱ âs^F⁽motⁱ⁾ =

jF⁽_int^{i j}⁾+F⁽_sⁱ⁾.ThestiffnessK_s canbe relatedtothe macroscopicshearmodulusG=3/8E,sincealltheshearspringsareattachedinparallel,sothatbysimplecalculationswe obtainKs=K_intl²/lz².Inthefollowing,forsimplicityweﬁxlz=l.Thisformulation,commonlyusedinspring-blockmodels, neglectsthelong-rangeinteractions thatmayarisefromwavepropagationthroughthe bulk(Elbanna 2011;Hulikaletal.,

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Fig. 1. Discretization of a square surface into a 2-D spring-bock model, showing the mesh of the internal springs. The shear springs K s modeling the block interaction with the slider are not shown.

2015;Rice,1993;RiceandRuina,1983).Here,wesupposethatthelocalinteractionsaredominating,whichisareasonable assumptionforslowslidingvelocitiestypicaloftheexperimentswe useasbenchmarks(Baumetal.,2014;Lietal., 2016;

Murarashetal., 2011).Thisassumptionhasalreadyallowed toobtaincorrectdescriptions ofthephenomenaoccurringat thetransitionfromstatictodynamicfriction(Braunetal.,2009;Trømborgetal.,2015).

Theinteractions betweentheblocksandtherigidplanecanbe introducedinmanyways:intheoriginal paperonthe springblock model(BurridgeandKnopoff,1967) andinearthquakerelatedpapers,e.g. Olamietal.(1992)andMori and Kawamura(2008a), itisintroduced by meansofan effectivevelocity-dependent force(Dieterich,1979;Rice etal.,2001), infrictionstudies,e.g.Amundsenetal.(2012)andBraunetal.(2009),byspringsthatattachanddetachduringmotion,in Trømborgetal.(2011)andGiaccoetal.(2014)bymeansoftheclassicalAmontons–Coulomb(AC)frictionforce.Thesevari- ousapproachesgiverisetoslightlydifferentquantitativeresults,butiftheyareimplementedunderreasonableassumptions, theydonotsigniﬁcantlyaffecttheoverallpredictionsofthemodel,whichisthoughttoprovideaqualitativeunderstanding ofthebasicmechanismsoffriction.Thisistrueatleastforthesmallslidingvelocitiesweareconsideringcomparedtothe characteristicvelocityscalesofthesystem,i.e.l

K_int/m.Adifferentqualitativebehaviormayariseforhigherslidingveloc- ities,asshownforrate-and-state frictionlaws (Elbanna andHeaton,2012;Heaton,1990; ZhengandRice,1998).In these cases,acarefulevaluationoftheinterplaybetweenthefrictionlawandslidingvelocityofthesystemmustbeperformed.

Inthisstudy,weadoptaspring-blockmodelbasedontheACfrictionforce andastatisticaldistributiononthefriction coeﬃcients (Costagliola et al., 2016; 2017a;2017b): while the block i is at rest,the friction force F⁽_{f r}ⁱ⁾ is opposedto the total drivingforce, i.e.F⁽_{f r}ⁱ⁾=−F⁽_motⁱ⁾,up to a thresholdvalue F_{f r}⁽ⁱ⁾=μ^siF_n⁽ⁱ⁾,where μ^si isthe staticfriction coeﬃcient and F_n⁽ⁱ⁾ isthenormalforceactingoni.Whenthislimitisexceeded,aconstantdynamicfrictionforceopposesthemotion,i.e.

F⁽_{f r}ⁱ⁾=−μdiF_n⁽ⁱ⁾r˙_i,whereμdiisthestaticfrictioncoeﬃcientandr˙_iisthevelocitydirectionoftheblock.Inthefollowing,we willdropthesubscriptss,deverytimetheconsiderationsapplytoboththecoeﬃcients.

Thefrictioncoefficientsareextractedfroma Gaussianstatisticaldistributiontoaccountfortherandomnessofthesur- faceasperities,i.e.p(μi)=(^√²πσ)⁻¹êxp^[−(μi−(μ)m)²/(²σ²)^],where(μ⁾^m^denotes^the^meanôf^themicroscopicfriction coefficientsandσ îsîts^standard^deviation.Înôrder^to^simulate^the^presenceôf^patterningôrôf^structuresôn^the^surface,

wesettozerothefrictioncoefficientsoftheblockslocatedonzonesdetachedfromtherigidplane.Themicroscopicstatic anddynamicfrictioncoefficientsarefixed conventionallyto (μ^s)m=1.0(¹)ând(μd)m=0.50(⁵),respectively,wherethe numbersinbracketsdenotethestandarddeviationsoftheirGaussiandistributions.

The macroscopic friction coefficientsare denoted with (μ⁾M. The staticfriction coefficient is calculated fromthe first maximum ofthetotal friction force, whilethe dynamicone asthetime average over thekinetic phase. Tocalculatethe frictioncoefficientsasratiobetweenlongitudinalforceandnormalforce,thenormofthelongitudinalforcevectormustbe calculated.When calculatingtime averages,care mustbe takenintheorderofthe operations:ifthereisan inversion of thefrictionforce(i.e.someblocksexceedtherestposition,asintheanalyticalcalculationsofCostagliolaetal.(2016))ora periodicmotiontakesplace,switchingthe operationsofnormandtimeaverageproducesdifferentresults.Inthesecases, the calculation closerto the realistic experimental procedure must be adopted. However, in the results below, we have checkedthatthe aboveconditionsdonotoccur andtheorderoftheoperationsisirrelevant.The modeldoesnot include roughnessvariationsduringslidingorotherlongtermeffects,sothattheresultsfordynamicfrictionaretobeconsidered withinthelimitsofthisapproximation.

Adampingforceisaddedto eliminateartiﬁcialblockoscillations: inBraunetal.(2009)andinthepapersbasedonit (e.g.CapozzaandPugno,2015)thisisdonebymeansofaviscousdampingforceproportionaltothevelocityoftheblock, i.e.F⁽_dⁱ⁾=−γ^m^r^˙i.Anotheroption,asinthe2-DmodelinTrømborgetal.(2011),istoimposethedampingontheoscillations betweeneachpairofblocksiandj,i.e.F⁽_d^{i j}⁾=−mγ (^r^˙i− ˙r_j),thusemulatingthedescriptionusuallyadoptedforviscoelastic

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Fig. 2. Time evolution for the total friction force and percentage of moving blocks for N = 20 , pressure P = 0 . 1 MPa, velocity v = 0 . 1 cm/s, γ/ω = 0 . 1 (a) or γ^/ω = 0 . 5 (b), where ω is the internal frequency ω^≡

K int /m . The other parameters are set to the default values. Greater damping enhances the dynamic friction coeﬃcient and reduces stick-slip oscillations.

materials(Banksetal.,2011;CarboneandPutignano,2013;Hunter,1961;Krushynskaetal.,2016;Persson, 2001;Tschoegl 1989).InSection 3.1,wediscussthe differentbehaviorobtainedwiththetwo approaches,butinthe followingweadopt theformerone,whichisthesimplesttoallowdampingofnon-physicaloscillations.

Thus,thecompleteequation ofmotionforblock i is:mr¨_i=

jF⁽_int^{i j}⁾+F⁽_sⁱ⁾+F⁽_{f r}ⁱ⁾+F⁽_dⁱ⁾.The overallsystemofdifferential equationsis solved using a fourth-order Runge–Kuttaalgorithm. Inorder to calculatethe average ofany observable, the simulationmust beiterated, extractingeachtime newrandom friction coeﬃcients. Inrepeatedtests, an integrationtime steph=10⁻⁸sprovestobesuﬃcienttoreduceintegrationerrorsunderthestatisticaluncertaintyintherangeadoptedfor theparametersofthesystem.

Weconsideronly asquare mesh,i.e.Nx=Ny≡ N,andwe willspecifythenumberofblocks foreach considered case.

ThedefaultnormalpressureisP=0.05MPa,sothatthenormalforceoneachblockisF_n⁽ⁱ⁾=Pl² andthetotalnormalforce isF_n=Pl²N².Theslidervelocity isv₌0.05cm/s.We willdiscussinSection3.1the motivationsforthesechoices, butin anycasetheresultsdisplaysmalldependenceontheseparameters.

Realisticmacroscopicelasticpropertiesarechosen,e.g.aYoung’smodulusE=10MPa,whichittypicalforasoftpolymer orrubber-like materialanda densityρ=1.2g/cm³. Thedistancebetweenblocks lin themodelis anarbitraryparame- ter representingthe smallestsurfacefeature that can be takeninto account andis chosen by defaultasl=10⁻³ cm,so thatthe orderofmagnitudematchesthose typicalofsurfacestructures usedin experiments(Baumetal., 2014;Lietal., 2016).

3. Results

3.1. Non-patternedsurface

Inthissection,wemodelfrictionproblemsrelativetohomogeneous,non-patternedsurfacesvaryingthefundamentalpa- rameterstounderstandtheoverallbehaviorandtocompareitwiththatofthe1-DmodelstudiedinCostagliolaetal.(2016). InFig.2,thefriction forcebehavior asa functionoftime isshownwiththedefaultsetofparameters:thereisalinearly growingstaticphase upto themacroscopicruptureevent, followedbya dynamicphaseinwhich thesystemslideswith smallstick-slip oscillationsatconstantvelocityv.Thepercentageofblocksinmotionasafunctionoftimeisalsoshown:

inthekinetic phase,singleblocksorsmallgroupsslipsimultaneouslybutnot ina synchronizedmannerwithrespectthe restofthesurface.

Thefirstdifferencewiththe1-D modelisthatthe2D arrayofspringsshowninFig.1allowstocorrectlysimulatethe Poissoneffect,i.e.adeformationinthetransversaldirectionduetothestretchinginthelongitudinalone.Secondly,dueto themodeldefinitionexplainedinSection2,thestiffnessesdonotdependonthetotalnumberofblocks,sothatincreasing Ndoesnot modifytheelasticproperties,butonlythesizeofthesystem.Sincethenumberofpointsgrowsasthesquare ofthe lateral size, N࣡100 can already be considered a large system, as shownin Fig. 3, where the size effects on the globalstaticfrictioncoefficientareshown.Similarresultsholdforthedynamicfriction.Intheleft(Fig.3a)andrightpanels (Fig.3b),theinfluenceoftheappliedpressurePandtheslidervelocityvisalsoshown,respectively.Inthetypicalrangesof