Normal and tangential stiffnesses of rough surfaces in contact via an imperfect interface model

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Normal and tangential stiffnesses of rough surfaces in contact via an imperfect interface model

Maria Letizia Raffa, Frédéric Lebon, Giuseppe Vairo

To cite this version:

Maria Letizia Raffa, Frédéric Lebon, Giuseppe Vairo. Normal and tangential stiffnesses of rough

surfaces in contact via an imperfect interface model. International Journal of Solids and Structures,

Elsevier, 2016, 87, pp.245-253. �10.1016/j.ijsolstr.2016.01.025�. �hal-01308423�

Normal and tangential stiffnesses of rough surfaces in contact via an imperfect interface model

Maria Letizia Raffa

^a,b,∗

, Frédéric Lebon

^a

,

Giuseppe Vairo

^b

aLMA, Aix-Marseille University, CNRS, Centrale Marseille, 4 Impasse Nikola Tesla CS 40 0 06, 13453 Marseille Cedex 13, France

bDepartment of Civil Engineering and Computer Science Engineering (DICII), University of Rome “Tor Vergata”, via del Politecnico 1, 00133 Rome, Italy

Keywords: Interface contact stiffness, Contact area, Imperfect interface approach Homogenization of a microcracked interphase, Asymptotic analysis

In this paper a spring-like micromechanical contact model is proposed, aiming to describe the mechan- ical behavior of two rough surfaces in no-sliding contact under a closure pressure. The contact region between two elastic bodies is described as a thin damaged interphase characterized by the occurrence of non-interacting penny-shaped cracks ( internal cracks). By combining a homogenization approach and an asymptotic technique, tangential and normal equivalent contact stiffnesses are consistently derived. An analytical description of evolving contact and no-contact areas with respect to the closure pressure is also provided, resulting consistent with theoretical Hertz-based asymptotic predictions and in good agreement with available numerical estimates. Proposed model has been successfully validated through comparisons with some theoretical and experimental results available in literature, as well as with other well-established modeling approaches. Finally, the influence of main model parameters is addressed, proving also the model capability to catch the experimentally-observed dependence of the tangent-to- normal contact stiffness ratio on the closure pressure.

1. Introduction

Analyticalandnumericalmodelingofcontactproblemsrelated to rough surfacescan be surelyconsidered asan open andchal- lenging research topic, strictlyassociated to manyapplicationsin differentengineeringfields.Fromacomputationalpointofview,it ispossibletoidentifyaclass ofmodeling problemsinwhichitis neitherpossiblenorconvenienttoaccountforafineanddetailed description ofthecontactregions,although localcontactfeatures maystronglyaffecttheoverall mechanicalresponse fortheprob- lemunderinvestigation.Inthesecases,apossiblestrategyisbased onmodelingcontactscenariosbyintroducingsuitablestiffnessand dashpotdistributionsatthecontactnominalinterface,allowingto upscaleatthemacroscaletheinfluenceofdominantcontactmech- anismsoccurringattheroughnessscale.Inthiscontextandasre- viewedbyBaltazaretal.(2002),startingfromfundamentalresults ofclassiccontacttheoriesandaccountingformainmicrogeometric featuresatthecontactinterface,severaltheoreticalandnumerical

∗ Corresponding author at: LMA - CNRS UPR 7051, Aix-Marseille Univer- sity, 4 Impasse Nikola Tesla CS 40 0 06 -13453 Marseille Cedex 13, France.

Tel.: +33 6 34 14 13 59.

E-mail addresses: [email protected] , [email protected] (M.L. Raffa), [email protected] (F. Lebon),[email protected] (G. Vairo).

modelshave beenproposed inthe specializedliterature (namely, spring-likemodels),aimingtoconsistentlyderivesomeequivalent stiffnesses.

One of the earliest contact model for elastic rough surfaces wasproposedbyGreenwoodandWilliamson(1966).Thismodelis basedonthe Hertzcontactsolutionforcurvedelasticnominally- flat surfaces(Mindlin, 1949) andit accounts fora statistical dis- tribution of non-interacting asperities. Moreover, Yoshioka and Scholz (1989) proposed an elastic contact model via a statis- tical approach that allows to describe possible oblique contact conditionsamongasperities.BycombiningtheHertz–Mindlinthe- ory (Mindlin, 1949) and the previously-introduced Greenwood- Williamsoncontactmodel,SherifandKossa(1991)andKrolikowski andSzczepek(1993)providedan analytical descriptionofnormal andtangentialcontactstiffnesses, inorderto establisha theoret- ical interpretation for the experimental results they obtained. In thiscase,thecontactbetweentwonominally-flatroughsurfacesis modeledasthecontactbetweentwoelasticsurfaces,oneofwhich is ideally flat and the other is nominally flat but covered with manyspherically-shapedasperities.Ageneralizationofsuchanap- proachwasdevelopedbyBaltazaretal.(2002),accountingalsofor apossiblecontactmisalignment.Nevertheless,apossiblecommon drawbackofalltheaforementionedcontactmodelsisthattheyare generallybasedonastochastic approach.Accordingly,in orderto

makethempracticallyapplicable,theidentificationofanumberof statisticparameters, oftennot easily estimable (McCool, 1986), is required.

Acrucialaspectinderiving reliablecontactsolutionsisrelated to the description ofthe contact area and its evolution with re- spect to the closure pressure (Johnson, 1987). Starting from the analyticalsolutionofWestergaard(1939),Johnsonetal.(1985)de- velopedamodelfortheelasticcontactbetweenatwo-dimensional wavysurfaceandarigidflatplane,proposingananalyticdescrip- tion of the contact area in the asymptotic limit cases of early contact(namely, forsmall valuesof the closurepressure) and of nearly-fullcontactconditions(highvaluesoftheclosurepressure).

Morerecently, Yastrebov etal. (2014) proposed a refined numer- icalapproachconsistingina FFT-basedboundary-element formu- lation,andtheyobtainedanaccuratenumericaldescriptionofthe contact-areaevolutionwiththeclosurepressureinthecaseofthe elasticcontactbetweenawavysurfaceandaflatplane.

Severalexperimentalstudiescanbefoundintheliteraturead- dressing themechanical behavior of rough surfaces in no-sliding contactunderclosure-pressureconditions (e.g.,Krolikowskietal., 1989; Sherif and Kossa, 1991; Krolikowski and Szczepek, 1993;

Baltazar et al., 2002; Dwyer-Joyce and Gonzalez-Valadez, 2003;

Gonzalez-Valadez et al., 2010), providing also estimates for nor- mal and tangential contact stiffnesses. For instance, Krolikowski andcoworkers(KrolikowskiandSzczepek,1993;Krolikowskietal., 1989) proposed contact-stiffness measures through an ultrasonic method,basedonthemeasurementofthereflectioncoefficientof ultrasonicwavesatthecontactinterface. Sherif andKossa(1991) employed an experimental technique based on the evaluation of the local natural frequencies at the contact region. Gonzalez- Valadezetal.(2010)proposedresultsbasedonultrasonictestsand accountingalso for loading-unloading cycles ofthe closure pres- sure.As a matter offact, experimental resultsconfirm that: high stressconcentrationsappearatthecontactregion,andtheyresult practicallyunaffected bytheshape ofbodiesincontactatasuit- abledistancefromthecontactarea(Johnson,1987;Johnsonetal., 1985);hysteresisphenomenaoccurattheinterface(asaresultof theplastic deformationlocalized atthe asperity tips)inthe case ofcyclingloads(Gonzalez-Valadezetal., 2010);nullvaluesofin- terfacestiffnessesareachievedwhentheclosurepressuretendsto zero(Gonzalez-Valadezetal.,2010).

In this paper a novel spring-like theoretical model for no- sliding contact under a closure pressure is proposed. Incremen- tal normal and tangential equivalent stiffnesses at the nominal contact interface are derived, by assuming contact microgeom- etry be described by isolated internal cracks (Sevostianov and Kachanov, 2008a; 2008b) occurring in a thin interphase region.

Indetail, effective mechanicalproperties at the contactzone are consistentlyderivedfollowingtheimperfectinterfaceapproachre- centlyadoptedbyLebonandcoworkers(Fouchaletal.,2014;Rekik andLebon, 2010; 2012), by coupling a homogenizationapproach formicrocrackedmedia based onthe non-interactingapproxima- tion (Kachanov, 1994; Kachanov and Sevostianov, 2005; Sevos- tianovandKachanov,2013;TsukrovandKachanov,2000)andthe matched asymptotic expansion method, introduced by Sanchez- Palencia (1987)and Sanchez-Palencia andSanchez-Hubert (1992) andrecentlyemployed by Lebon andRizzoni(2011), Rizzoniand Lebon(2013),Rizzonietal.(2014)andLebonandZaittouni(2010). The proposedmodelisdetailedinSection2,andits validation isprovidedbycomparingnumericalresultswithavailabletheoret- icalandexperimentalfindings(Section3.1).Modeleffectivenessis alsoprovedforawiderangeofclosure-pressurevaluesbycompar- ingproposedresultswiththoseobtainedviathecontactmodelin- troducedbySherifandKossa(1991)(Section3.2).Afterwards,the influenceofmainmodelparameters isinvestigatedinSection3.3, andfinallysomeconclusionsaretracedinSection4.

2. Contactmodel 2.1. Generalframework

Let the contact problem C be introduced by considering two continuousbodies1 and2,comprisinglinearly-elasticisotropic materials(E_iand

ν

i,withi=1,2,beingYoungmodulusandPois- son ratio, respectively), in no-sliding contact via non-conforming rough surfaces under a closure pressure condition (Fig. 1). Let S⊂R²bethenominalcontactinterface,belongingtotheinterface plane

π

^{.LetaCartesian frame(O,e}1,e₂,e₃) beintroduced, with x₁,x₂andx₃ thecorrespondingcoordinates.TheoriginObelongs to

π

^,ande3isorthogonalto

π

^{anddirectedoutwardfrom}2.

Normal andtangentialincrementalcontactstiffnesses (K_N^C and K_T^C, respectively) per unit nominal contact area in S are defined as:

K_N^C=dFN

dw , K_T^C= dFT

ds ⁽¹⁾

wheredwanddsaretheincrementsoftherelativedisplacements atthe contactinterface regionin normal(i.e.,along e₃) andtan- gential(i.e.,parallelto

π

⁾directions,anddFN anddFT arethein- crementsofthenormalandtangentialforcestransmittedthrough the unit contact area, respectively. Contact microgeometryis as- sumedtobeisotropicinSandtherebythetangentialcontactstiff- nessK_T^C can bepostulatedasindependentfromthetangentialdi- rection.

InagreementwiththeapproachadoptedbyWestergaard(1939) and by Johnson et al. (1985),contact microgeometry is modeled by describing no-contact regions as parallel penny-shaped inter- nalcracks(SevostianovandKachanov, 2008a;2008b)lyingonthe interface plane

π

^{.Coplanarmechanical}interactions amongcracks areconsiderednegligible,resultinginthenon-interactingapproxi- mation(Kachanov,1994;SevostianovandKachanov,2013).Accord- ingly, the region close to the nominal contact interface S is re- garded as an imperfect interphaseB^ε, defined as the thin layer havingS asthemiddlesectionand

ε

^{astheuniformsmallthick-}

ness,andweakenedbyidenticalandparallelpenny-shapedcracks ofradiusb(Fig.1).

Referring to a simplistic idealization of the contacting rough surfacesviabi-sinusoidalwavy-likesmoothsurfaces,bothofthem withwavelength

λ

^{andamplitude(suchthat}

λ

^),a

ε

^-thick

representative elementary volume (REV) at the contact interface, and occupying the region ^ε⊂B^ε, can be conveniently intro- ducedassketchedinFig.1.

Accordingly, thecontact problemC is faced by introducingan auxiliary model problem A, defined on the microcracked inter- phaseB^εanddescribedviatheREV.

2.2. Imperfectinterfaceapproach

Referring to the auxiliary model problem A, and as a nota- tion rule, the following symbols will be adopted: ^ε₊=1

\

B^ε and ^ε₋=2

\

B^ε, with ^ε_± ^{also referred to asadherents;} S_±^ε= ^ε_±∩B^ε identifyingtheplane interfacesparallelto

π

^betweenin-

terphaseandadherents.Itisassumedthat^ε_±^andB^εareperfectly bonded, sothatdisplacement andstress vectorfieldsare ensured tobecontinuousacrossS_±^ε.

2.2.1. Homogenizationofthemicrocrackedinterphase

Let ⊂S be the crack middle surface for a penny-shaped crack in B^ε, and let u⁺ and u⁻ be the displacement vectors at the parallel-to-S crack boundaries. Denote also with u_cod= (^u+−u⁻)^d/

|

|

^{theaveragemeasureofthe}displacementjump across the crack, in the following referred to as the crack open- ing displacement vector. In agreement with a well-established

Fig. 1. Contact problem Cand auxiliary model problem A . Notation.

homogenizationtechnique(Kachanov, 1994; KachanovandSevos- tianov, 2005; Tsukrov and Kachanov, 2000), and considering a plane-stress assumption under a frictionless condition along the crack faces, u_cod can be expressed in terms of the stress vector t₃=

σσσ

^e3as(Kachanov,1994):

u_cod=

β

_E^b

0

P₃t₃+

γ

_E^b

0

(

^I−P3

)

^t3 (2)

where

σσσ

^isthesecond-order Cauchystress tensor,P₃=e₃⊗e₃ is the projector operatoralong e₃ (symbol’⊗’indicating thedyadic product),Iistheidentitysecond-ordertensor,andwhere

β

^{= 16}

⁽

¹⁻

ν

0²

)

3

π

^,

γ

^{= 32}

⁽

¹⁻

ν

0²

)

3

π (

²⁻

ν

⁰

)

⁽³⁾

E₀and

ν

0 being,respectively,theYoungmodulusandthePoisson ratio oftheundamaged interphase, assumedtobe isotropic. Itis worth pointing out that E₀ and

ν

0 can be obtained in terms of the elasticproperties(E_iand

ν

i,withi=1,2)ofthe twomateri- als incontact,astheresultofapreliminaryhomogenizationstep performedontheundamaged

ε

^{-thick REV(e.g.,}Sanchez-Palencia, 1980;Nemat-NasserandHori,2013;TemizerandWriggers,2011).

Following the Eshelbyapproach (Eshelby, 1961), the complemen- taryelasticpotential

ψ

(

σσσ

)characterizingtheeffectivematerialof themicrocrackedinterphaseresultsin:

ψ ( σσσ )

⁼

ψ

0+

ψ

⁼¹₂

σσσ

^:S0:

σσσ

⁺

| |

2

|

^ε

| σσσ

^:

(

^e3⊗u_cod

)

⁽⁴⁾

where symbol ‘: ’ denotes the double-dot product,

ψ

0(

σσσ

) ^{is the} complementary potential expressed in terms of the fourth-order compliance tensor S0=S0(^E0,

ν

0) ^{of the undamaged} interphase, and

ψ

^isaperturbationterm,dependingonbothmicrostructural crackfeaturesandthecrackopeningdisplacement.

By introducing Eqs. (2) and (3) in Eq. (4), the complemen- taryelasticpotential

ψ

(

σσσ

)^{ofthecrackedmaterialin}B^ε readsas (KachanovandSevostianov,2005):

ψ

⁼

ψ

^{0+ 16}

⁽

¹⁻

ν

0²

)

3

(

²⁻

ν

⁰

)

^E0

( σσσ σσσ )

^:

ααα

⁻

ν

⁰

2

σσσ

^:

( ααα

^⊗P3

)

^:

σσσ

(5) wherethe complementaryelastic potential oftheundamagedin- terphaseisexpressedby

ψ

0

( σσσ )

^{= 1+}

ν

0

2E₀

σσσ

^:

σσσ

⁻

ν

0

2E₀

σσσ

^{:I (6)}

and

ααα

=

ρ

^P3isthesecond-ordercrack-densitytensor,expressedin termsofthe scalardensity

ρ

^{. Thelatteris defined,in agreement}

withBristow(1960),by(seeFig.1):

ρ

= b³

|

^ε

|

^{= 2b}

3

λ

²

ε

⁽⁷⁾

DuetoEq.(5),theeffectivecompliancetensorSofthemicroc- rackedinterphasecanbederivedcomponent-wiseas:

(

S

)

i jkl=

(

S0

)

i jkl+

(

S

)

i jkl=

∂

²

ψ

∂ σ

i j

∂ σ

kl

(8) whereSisthepartofthecompliance tensorassociatedto

ψ

^.

As a result, the damaged interphase behaves as a transversely isotropic material, with symmetry plane coincident with the in- terphasemid-plane(namely,

π

^{).Therefore,by}discriminatingwith indexesNandTquantitiesreferredtonormalandtangentialdirec- tionsto

π

^,respectively,theeffectivemoduli ofthecrackedinter- phaseresultin:

EN=E0

1+16

(

¹−

ν

0²

)

3

ρ

−1

, ET=E0

1+16

ν

0

(

¹−

ν

0²

)

3

(

²⁻

ν

⁰

) ρ

−1

GNT=G0

1+16

(

¹−

ν

⁰

)

3

(

²⁻

ν

0

) ρ

−1

,

ν

NT=

ν

0

1+16

ν

⁰

(

¹⁻

ν

0²

)

3

(

²⁻

ν

0

) ρ

−1

ν

TN=

ν

0

1+16

(

¹−

ν

0²

)

3

ρ

−1

(9) wheresymbolG_·indicatesshearmoduli.

2.2.2. Asymptoticanalysis

Letthe interphasethickness

ε

^{(see Fig. 1) be considered as a}

small parameter. Accordingly, an asymptotic expansion with re- spectto

ε

^canbeconvenientlyperformedreferringtothecompos- itesystem^ε=^ε₊∪^ε₋∪B^ε,wherethemechanicalpropertiesof theadherentsare thoseofthebodies 1 and2 in contact,and withthe effective material propertiesof the interphaseB^ε given byEq.(9).

InagreementwithCiarlet(1997),letthechangeofvariablegˆ: (^x1,x₂,x₃)→(^z1,z₂,z₃)^{beintroducedin}B^ε,withz₁=x₁,z₂=x₂, z3=x3/

ε

^{. Moreover, let the change of variable g¯:}(^x1,x2,x3)→ (^z1,z₂,z₃) ^{be introduced in ε}_±, with z₁=x₁, z₂=x₂, z₃=x₃± (¹−

ε

)/2,wheretheplus(resp.,minus)signappliesfor^ε₊^(resp.,

^ε₋^{).As aresult, interphase}B^ε andadherents^ε_± ^{arerescaled in} B=

(^z1,z₂,z₃)∈R³

|

(^z1,z₂)∈S,

|

^z3

|

<1/2

and±=^ε_±±(¹−

ε

)^e3/2, respectively. In what follows, symbols ‘¯·’ and ‘ˆ·’ refer to rescaled quantities for B and ±, respectively. In detail, uˆ^ε= u^ε◦gˆ⁻¹ and ˆ

σσσ

^ε=

σσσ

^ε◦gˆ⁻¹ denotedisplacement and stress fields forB,andu¯^ε=u^ε◦g¯⁻¹ and

σσσ

¯^ε=

σσσ

^ε◦g¯⁻¹ aredisplacementvec- torandstress tensorfor±, u^ε and

σσσ

^{ε beingthe}corresponding fieldsonthecompositesystem^ε.Accordingly,displacementand stressasymptoticexpansionswithrespecttothesmallthickness

ε

result,respectively,in:

u^ε=u⁰+

ε

^u1+o

( ε )

ˆ

u^ε=uˆ⁰+

ε

^uˆ1+o

( ε )

u¯^ε=u¯⁰+

ε

^u¯1+o

( ε )

⁽¹⁰⁾

σσσ

^ε=

σσσ

⁰+

ε σσσ

¹+o

( ε ) σσσ

ˆ^ε=

σσσ

ˆ⁰+

ε σσσ

^ˆ1+o

( ε )

σσσ

¯^ε=

σσσ

¯⁰+

ε σσσ

^¯1+o

( ε )

⁽¹¹⁾

and, dueto Eq.(10), strain tensor in therescaled domains takes theform(Rizzonietal.,2014):

e

(

^uˆε

)

⁼

ε

^{−1eˆ−1+eˆ0+O}

( ε )

⁽¹²⁾

e

(

^u¯ε

)

=

ε

^{−1e¯−1+e¯0+O}

( ε )

⁽¹³⁾

wherethelower-ordertermsare ˆ

e⁻¹=uˆ⁰_,3⊗e3+e3⊗uˆ⁰_,3

2 , e¯⁻¹=0 (14)

f_,3denotingthepartialderivativeoffwithrespecttoz₃,withf_,3=

ε ∂

^f/

∂

^x3intheinterphasedomain.

OwingtheperfectbondedassumptionbetweenB^εand^ε_±,the continuityconditionatS_±^ε forthefieldsu^εand

σσσ

^{εleadstomatch-}

ingrelationshipsbetweenexternalandinternalexpansions.Inpar- ticular,zero-ordertermshavetosatisfythefollowingconditions:

u⁰ x1,x2,0^±

=uˆ⁰

z1,z2,±1 2

=u¯⁰

z1,z2,±1 2

(15)

t⁰₃ x1,x2,0^±

=ˆt⁰₃

z1,z2,±1 2

=¯t⁰₃

z1,z2,±1 2

(16) whereu^ε→u⁰ andt^ε₃=

σσσ

^εe3→t⁰₃=

σσσ

^0e3when

ε

→0.

It isworthpointingout that,neglectingbody forcesinthein- terphaseB^ε,andasaresultoftherescaledequilibriumproblemin B(Rizzonietal., 2014),thezero-orderstressfield

σσσ

ˆ^{0 issuchthat} thestressvectorˆt⁰₃=

σσσ

ˆ^0e3 doesnotdependonz₃.

Duetothedependenceon

ε

^{oftheeffectivemoduliintroduced}

inEq.(9),theinterphaseresultstobecharacterizedbyasoftma- terial(Rizzonietal., 2014).Thereby,theinterphaseelasticmoduli arelinearly-rescaled withrespect to thethickness

ε

^asC^ε=

ε

C, whereC^ε=S⁻¹ results from Eq.(9) and wherethe fourth-order tensorCdoesnot depend on

ε

^.Accordingly, the interphasecon- stitutivelaw

σσσ

ˆ^ε=C^ε:e(^uˆε)^{can be recastwithinthe asymptotic} frameworkas

σσσ

ˆ⁰⁺

ε σσσ

^ˆ1=C:

(

^eˆ−1+

ε

^eˆ0

)

^+o

( ε )

⁽¹⁷⁾

previousequationhavingtobesatisfiedforanyvalueof

ε

^.Thereby,

thefollowingrelationshipholds

σσσ

ˆ⁰=C:

(

^eˆ−1

)

⁽¹⁸⁾

and,owingEq.(14),itfollowsthat

σσσ

ˆ^0e3=Kuˆ⁰_,3 (19)

wherethematrixK isdefinedcomponent-wiseas(^K)i j=(C)i3j3, anditresultsinK=diag

K_T^A K_T^A K_N^A

with K_T^A=3E0

λ

²

(

²−

ν

0

)

64b³

(

¹−

ν

0²

)

^{, KNA=}

3E0

λ

²

32b³ 1−

ν

0²

⁽²⁰⁾

ByintegratingEq.(19)withrespect toz₃ from−1/2to+1/2, andinthelimitof

ε

^{tendingtozero,thefollowingzero-ordersoft-}

interfacelawisstraightobtained

t3=K[u], [t3]=0 across S ⁽²¹⁾ where[·]denotesthejumpacrosstheinterface.

Asaresult,quantitiesK_T^A andK_N^AintroducedinEq.(20)repre- senttangentialandnormalinterfacestiffnesses,respectively,asso- ciatedtotheauxiliarymodelproblemA.Inparticular,theydepend ontheeffectiveelasticpropertiesoftheundamagedinterphase,as wellasonthecharacteristiclength(namely,

λ

^{)oftheplaneregion ε}∩

π

^{and on the microcrack radius b. It is worth remarking}

that,thesequantitiesareintroducedintheauxiliaryproblemAas descriptorsof themicrogeometryfeaturesinthe contactproblem C,andthey correspondtotheaverage roughnesswavelengthand totheaverageno-contactradius,respectively.

2.3. Effectiveincrementalcontactstiffnesses

As a matter of fact, contact microgeometry and, in particu- lar,the characteristiclength ofno-contact regions dependon the valueofthenominalcontactpressurep¯.Thereby,foragivenpres- sureconditioncharacterizingthecontactproblemC(^p¯),thecorre- spondingauxiliarymodelproblemA(^p¯)^{ismatchedtotheprevious} one by considering the microcracks radius b asdependent itself on p¯, such that 0≤2b≤

λ

/√

2.In detail, in agreement withthe approach proposed by Johnson et al. (1985)and Johnson (1987), 2b≤

λ

/√

2when p¯<p^∗ andb→0⁺ when p¯→p^∗,wherep^∗isa measure of the nominal pressure which brings the surfaces into complete contact in C, and referred to as the complete contact pressure.

InagreementwiththeHertzelasticcontacttheory,anestimate forp^∗inC canbedefinedas(Johnsonetal.,1985):

p^∗_H=√

2

π

^E

λ

^{0 (22)}

(^E)⁻¹=[(¹−

ν

1²)/E₁+(¹−

ν

2²)/E₂]beingthereducedHertzmod- ulusand0 beingtheamplitudeofthebi-sinusoidalshapeideal- izingthecontactingroughsurfacesatthereferenceconfiguration.

It is worth observing that, Eq. (22) strictly holds in an elas- tic regime. Nevertheless, due to localization mechanisms associ- ated to tips plastic deformation and fatigue effects, the pressure thatbringsthesurfacesintoacompletecontactconditionissurely lowerthanp^∗_H anditcanbeconsideredasafunctionoftheelasto- plasticmaterialbehavior,ofthepressureloadinghistory,aswellas ofthenumberofloadingcycles.Accordingly,apossibledescription oftheactualclosurepressurecanbepostulatedas:

p^∗=hp^∗_H=h√

2

π

^E

λ

^{0 (23)}

whereh≤1isasuitablehistory-basedcorrectionterm.Inthecase ofloadingcyclescharacterizedbythesamemaximumvalueofthe closurepressure,hcanberetainedintheorderof^/0,where isameasureoftheamplitudeforthewavygeometrythatidealizes thecontactingsurfacesattheactualconfiguration.

Referringtothe contactproblemC,the contactareaslying on S aredescribedbyintroducingtheaveragecontactradiusa^C,such that a^C→0⁺ when p¯→0⁺ and a^C→

λ

/√

2 when p¯→p^∗. With referencetothesketch depictedinFig.2,andinagreement with both indications provided by Johnson et al. (1985) and numeri- cal evidence by Yastrebov et al. (2014), when p¯→0⁺ then con- tact areas are describedas circular(i.e., contact point condition)

Fig. 2. Contact (light red) and no-contact (light blue) areas in ^ε∩ π⊂S. Outline of the simplified behavior modeled in the contact problem Cfor p ¯→ 0 ⁺and p ¯→ p ^∗. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

and no-contact areas are almost square-shaped. Onthe contrary, inthenearcompleteclosurecondition(p¯→p^∗),thecontactareas areassumedtobealmostsquare-shapedandtheno-contactareas asquasi-circular.

Leta₀ andb₀(respectively,a₁ andb₁)bethevaluesofthecon- tact and no-contact radii, respectively, when p¯→0⁺ (resp., p¯→ p^∗).Inagreement withasymptoticestimatesprovidedby Johnson et al. (1985), strictly valid for the elastic contact between a bi- sinusoidalsurfaceandarigidflatplane,thefollowingrelationships areassumedtohold:

a0

p¯ p^∗

=_√

λ

2

3 8

π

¯ p p^∗

1/3

(24)

b0

p¯ p^∗

=

λ

2√ 2

1−

π

3

8

π

¯ p p^∗

2/3

(25)

a1

p¯ p^∗

=

λ

2√ 2

1− 3 2

π

1− p¯ p^∗

(26)

b1

p¯ p^∗

=

λ

2

π

3

1− p¯ p^∗

(27)

Therefore,contact(namely, a^C)andno-contact(b^C)radiusevo- lutionwithp¯inC ispostulatedtobesimplydescribedbythefol- lowingarea-basedweightedaverages:

a^C

p¯ p^∗

=

π

^a20

1− p¯ p^∗

+4a²₁ p¯ p^∗

π

1− p¯ p^∗

+4 p¯ p^∗

(28)

b^C

p¯ p^∗

=

4b²₀

1− p¯ p^∗

+

π

^b21

¯ p p^∗ 4

1− p¯ p^∗

+

π

_p^p¯_∗

(29)

Itisworthpointingoutthat,Eqs.(28)and(29)satisfythecon- sistencycondition:

Ac+Anc=An=

λ

²

2 ⁽³⁰⁾

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

p p AcAn

Fig. 3. Normalized contact area A c/ A nversus the pressure ratio p ¯/p ^∗. : Numeri- cal results by Krithivasan and Jackson (2007) . • : Experimental results by Johnson et al. (1985) . ◦: Numerical results by Johnson et al. (1985) . − −: Numerical results by Yastrebov et al. (2014) . − − −: Asymptotic limits by Johnson (1987) . − −: Pro- posed analytical description ( Eqs. (30) and (31) ).

wherethecontact(Ac) andtheno-contact (Anc) area in^ε∩

π

are:

Ac=

(

^aC

)

²

π

1− p¯ p^∗

+4 p¯ p^∗

(31)

Anc=

(

^bC

)

²

4

1− p¯ p^∗

+

π

_p^p¯_∗

(32) Moreover, Eqs. (28) and(29) recover the asymptotic relation- shipsintroduced inEqs.(24)–(27),andinparticularthefollowing limitsforp¯→0⁺hold:

a^C→a₀→O

¯ p p^∗

¹₃

, b^C→b₀→

λ

2√

2 ⁽³³⁾

In order to show consistency and accuracy of proposed esti- mates for contact and no-contact areas, the normalized contact area Ac/An computed via Eqs. (30) and (31) is successfully com- pared,inFig.3andwithrespecttothedimensionlessclosurepres- sure p¯/p^∗, with: numerical results proposed by Krithivasan and Jackson(2007) andby Yastrebov etal. (2014), experimental data reportedbyJohnsonetal.(1985),theoreticalasymptoticlimitspro- videdbyJohnson(1987).Itappearsthat,althoughsimple,thepro- posedapproachcanberetainedeffectivefordescribinginasatis- factorily accuratewaythe evolutionof thecontactarea versus p¯. Accordingly,Eq.(29)canbeconsideredasasuitableevolutionlaw fortheaverageno-contactradiusb^CinC.

Byadoptingb=b^Casthematchingconditionbetweenthecon- tactproblemC(^p¯)^andthecorrespondingauxiliarymodeloneA(^p¯) fora givenvalue ofthenominalclosurepressure,Eq.(29)canbe employedinEq.(20)fordefiningtheinterfacestiffnessesresulting fromtheproblemA.

It is worth remarking that, Eq. (29)has not to be considered as governing the physical evolution of the crack-closing process induced by a closure pressure condition in the problem A. In- deed,Eq.(29)furnishes asimpleandphysically-oriented descrip- tion of the pressure-based evolution of the no-contact radius in C(^p¯),adopted,foreachvalue ofp¯,fordefiningthecorresponding actualauxiliarymodelproblemA(^p¯)^.

Furthermore, Eq. (20) cannot be directly used as interfacial contact stiffnesses since, due to the limit conditions expressed in Eq.(33),stiffnesses introduced in Eq.(20) do not recover the physicalbehaviorassociatedtonullstiffnessvalueswhen p¯→0⁺. Suchanoccurrenceisdescribed,forinstance,bytheHertztheory.

0 100 200 300 400 0.0

0.2 0.4 0.6 0.8 1.0 1.2 1.4

p MPa KNCGPaΜm

KNth

KNexp

K_N^num

0 100 200 300 400

0.0 0.2 0.4 0.6 0.8

p MPa KTCGPaΜm

K_T^th K_T^exp KTnum

Fig. 4. Comparison between numerical (num, present model) and experimental (exp, Gonzalez-Valadez et al., 2010 ) results for normal (on the left) and tangential (on the right) contact stiffnesses vs. the nominal closure pressure ¯p . Hertz-based theoretical predictions (th, based on Eq. (34) ) are also provided.

Indeed,inthiscase,theoreticalestimatesofinterfacecontactstiff- nessescanbeexpressedas(SevostianovandKachanov,2008a) K_N^th=4E

λ

^{2 aC, KTth=KNth2}

⁽

¹⁻

ν

0

)

(

²−

ν

0

)

⁽³⁴⁾

andthereforeforp¯→0⁺itresults

{

^KN^th,K_T^th

}

→O((^p¯/p)^1/3)^. In order to recover such a behavior for p¯→0⁺, the effective contactstiffnessesarerecastas:

K_i^C=K_i^A

1−e^−γi0

(

^pp¯∗

)

^1/3

i=N,T (35)

whereK_N^A and K_T^A are obtainedfrom the imperfect interface ap- proach,thatisbyEq.(20),and

γ

N⁰and

γ

T⁰aremodeldimensionless parameters.

Byenforcingthat K_i^C=K_i^thwhen p¯→0⁺ (fori=N,T), thefol- lowingtheoreticalestimatesfor

γ

^{0-typemodelparameterscanbe}

obtained:

γ

N⁰,th=

(

⁹

π )

^−1/30.33,

γ

T⁰,th=

γ

N⁰,th

4

(

¹−

ν

⁰

)

(

²−

ν

⁰

)

^{2 (36)}

with

γ

T⁰,th0.32for

ν

0=0.3. 3. Resultsanddiscussion

In the following,some comparisons among modelresults and bothexperimentaldataavailableinliteratureandtheoreticalesti- matesarepresented,aimingtovalidatetheproposedapproachand toshow itssoundnessandeffectiveness.Afterwards,theinfluence ofthemodelparameters

γ

N⁰,

γ

T⁰andhontheevolutionofthein- terfacecontactstiffnessesintroducedinEq.(35)versustheclosure pressureisinvestigated.

3.1.Modelvalidation

The experimental results obtained by Gonzalez-Valadez et al.

(2010) are herein chosen to validate the proposed model.In de- tail,theyprovidedexperimental measures,bymeansofultrasonic pulser-receivers,ofinterfacecontactstiffnessesforsteelspecimens incontact through rough nominally-flat surfaces under a closure pressure.The specimenswere subjected toloading-unloading cy- cles of compressive pressure in a hydraulic frame operating in loadingcontrolmode.Theloadwasappliedinaquasi-staticway, uptothe nominal-closure-pressurevalue of400MPa.Steel spec- imenswerecharacterized bythefollowingmechanicalproperties:

E₁=E₂=E₀=200GPaand

ν

1=

ν

2=

ν

0=0.3.Accordingly,Hertz- based reduced modulus value results in E=109.89 GPa. More- over,inagreementwithdatafurnishedbyGonzalez-Valadezetal.

(2010), contactrough surfaces in the referenceconfiguration can beidealizedasregularized wavy-shapes(see Fig.1)characterized by

λ

=130

μ

^mand0=1.58

μ

^m.

0 100 200 300 400

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

p MPa KCGPaΜm

Fig. 5. Numerical (present model) and experimental ( Gonzalez-Valadez et al., 2010 ) contact stiffnesses vs. the nominal closure pressure ¯p , computed considering: p ^∗= p ^∗_H(i.e., for h = 1 , the notation rule adopted in Fig. 4 applies), and a corrected com- plete contact pressure p ^∗= 0 . 75 p ^∗_H(i.e., for h = 0 . 75 , - - - -: K ^C_N; — —: K _T^C).

Acomparisonprocedureamongnumericalresultsbasedonthe proposedmodelandexperimentaldatarelevanttothe11-thload- ingcyclehasbeencarriedout,computing(viaa least-squaresfit- tingalgorithm)theoptimalvaluesforparameters

γ

N⁰and

γ

T⁰.Asa result,thebest-fittingvaluesare

γ

N⁰,num=0.96and

γ

T⁰,num=0.58, and the comparison betweenthe corresponding results obtained viatheproposedmodelandtheavailableexperimentaldataisde- pictedinFig.4.

Itisworthpointingoutthat,theaforementionednumericales- timatesfor

γ

N⁰ and

γ

T⁰ areinthesameorderofmagnitudeofthe correspondingHertz-basedtheoreticalones(seeEq.(36)),andthey clearlyallowtoobtainagoodagreementwithexperimentaldata.

Previouslyresultshavebeenobtainedbyconsideringthecom- pletecontactpressure p^∗ asequal tothe Hertz-based one p^∗_H in- troducedinEq.(22)(i.e.,forh=1). Nevertheless,byassuming as a coarse first approximation that h=/0, andreferring to the measures provided by Gonzalez-Valadez etal.(2010), the follow- ingestimatescanbeconsistentlyconsidered:=1.18(atthe11- thloading cycle)andthereby h=0.75.Byadopting p^∗=0.75p^∗_H, thenumericalpredictionsofthecontactstiffnessesremainingood agreementwiththebenchmarkexperimentalresults,asshownin Fig.5,andinthiscasethebest-fittingvaluesof

γ

⁰-parametersre- sultin

γ

N⁰,num=0.83and

γ

T⁰,num=0.51.Therefore,ahistory-based correction ofthe complete contactpressure leadsto a consistent reductionofmodelparameters

γ

N⁰ and

γ

T⁰ towardstheirtheoreti- calvalues.

Table 1 summarizes the best-fitting values of

γ

^{0-type model}

parameters(normalizedwithrespecttotheirHertz-basedtheoret- icalpredictions)fordifferentvaluesofthehistoryparameterh.In detail,proposedresultsclearlyshow thatamonotonic decreaseof hinducesamonotonicdecreaseofthebest-fittingvaluesforboth

Table 1

Influence of the history-based correction quantity h on the best-fitting values (num) of the γ⁰-type model parameters, normalized with respect to the theoreti- cal (th) Hertz-based predictions.

h 0.15 0.2 0.25 0.3 0.35 0.4 0.5 0.75 1

γ_N,⁰num/γ_N,⁰_th 0.7 1.0 1.2 1.4 1.6 1.8 2.0 2.5 2.9 γ_T,⁰num/γ_T,⁰th 0.5 0.7 0.8 0.9 1.0 1.1 1.3 1.6 1.8

(1) (2)

(3) (4)

0 100 200 300 400

0.2 0.4 0.6 0.8 1.0

p MPa KTCKNC

Fig. 6. Tangential-to-the-normal stiffness ratio vs. the nominal closure pressure p.¯ Comparison among present results (- - - -, computed with p ^∗= p ^∗_H) and both experimental (by Gonzalez-Valadez et al. (2010) , ) and theoretical predictions: (1) Mindlin (1949) , (2) Sherif and Kossa (1991) , (3) Baltazar et al. (2002) , (4) Yoshioka and Scholz (1989) . Results relevant to the model by Baltazar et al. (2002) have been computed considering ξ= 0 . 65 and κ= 1 .

γ

^0-type parameters, attaining the Hertz-based predictions differ- ently fortangentialandnormal quantities.In all thecorrespond- ingcasestherelativeerrorsamongmodelstiffnesspredictionsand benchmarkingexperimentaldataremainalwayslessthan12%.

Many studies on contacting rough surfaces (Greenwood and Williamson,1966;Johnson etal.,1985;KrolikowskiandSzczepek, 1993; Mindlin, 1949) revealed that the ratio of tangential-to- normal stiffnessissolely dependent on thePoisson ratioof con- tacting materials. In those cases and under the assumption that the two bodies incontact are madeby the same material, itre- sultsthat:

K_T^C

K_N^C =

χ φ ( ν )

⁽³⁷⁾

where0.5≤

χ

≤2isaconstantand

φ

⁽

ν

^{)isafunctionofthePois-}

son ratio

ν

^{.In thecaseofthe classic} Hertz–Mindlincontactthe- ory(GreenwoodandWilliamson,1966;KrolikowskiandSzczepek, 1993;Mindlin,1949),

χ

=2and

φ

(

ν

)=(¹−

ν

)/(²−

ν

)^.Manyap- proaches areavailableinliterature,providingrelationshipsidenti- cal tothat inEq.(37),characterizedbythesamefunction

φ

⁽

ν

^)as

intheHertz–Mindlintheorybutwithdifferentvaluesof

χ

^.Sherif

andKossa(1991) found

χ

=

π

/2.YoshiokaandScholz (1989)ob- tained the approximated estimate

χ

=0.71. Moreover, following the contactmodel by Baltazaret al.(2002),coefficient

χ

^{can be}

expressed as

χ

= ²_κ^ξ, where

ξ

^and

κ

^{are correction factors ac-}

countingforthegeometricalmisalignments withrespecttoshear andlongitudinaldirections,respectively.Theresultingvaluesofthe tangential-to-normalstiffnessratiofortheabovecitedmodelsare plottedinFig.6,incomparisonwiththebenchmarkingexperimen- tal data(Gonzalez-Valadez etal.,2010) andwithresults obtained via the presentmodel (referringto the best-fittingvaluesof

γ

^0-

typemodelparameterscomputedforh=1).

It is worth noting that, the available experimental data sug- gest asignificant dependenceofthestiffnessratioonthe closure pressure,especiallyforlowvaluesofp¯.Thisisproperlydescribed by the proposed model, whereas the aforementioned theoretical

predictionsfail,providingaconstantvalueoftheratioK_T^C/K_N^C,with asuitabledescriptionforvaluesofp¯greaterthan50MPaonlyin thecaseoftheformulationbyBaltazaretal.(2002).

3.2.ComparisonwiththecontactmodelbySherifandKossa SherifandKossa(1991),andsimilarlyKrolikowskiandSzczepek (1993), in order to elucidate they experimental findings, pro- videda mathematical formulation of normaland tangentialcon- tactstiffnesses,bycombiningthecontactmodel(GW)formulated byGreenwoodandWilliamson(1966)andtheHertz–Mindlinthe- ory(Mindlin,1949).Indetail,contactingroughsurfacesweremod- eled as elastic surfaces covered withelastic asperities which are assumedtobe,atleastneartheir summits,assphericalandchar- acterizedbythesameradiusofcurvatureRs.FollowingSherifand Kossa(1991)andtheirmainreferences,thecorrespondingnormal (denotedasK_N^sk) andtangential(K_T^sk) contactstiffnesses per unit areacanbewrittenintheform

K_N^sk=2Ds

E0

1−

ν

0²

(

^Rs

σ

s

)

^12F1₂

(

^t

)

⁽³⁸⁾

K_T^sk=

π

^Ds

₍

₂₋

ν

0^E

)(

⁰¹+

ν

0

) (

^Rs

σ

^s

)

^12F1₂

(

^t

)

⁽³⁹⁾

whereDsisthedensityofasperitiesperunit area,

σ

^sisthestan-

darddeviationoftheheightdistributionofasperities,tisthenor- malized(to

σ

^{s)meanseparationbetweencontactingsurfaces,and}

functionF1

2(^t)^resultsin F¹

2

(

^t

)

=

_∞

t

(

^r−t

)

¹²

(

^r

)

^{dr (40)}

^{(r) beingdefined as the height} distribution, taken asGaussian and scaled to make its standard deviation equal to the unity (GreenwoodandWilliamson,1966).

In order to provide a consistent comparison among the pro- posedcontactstiffnessesinEq.(35),thoserecalledinEqs.(38)and (39), andavailable experimental data by Gonzalez-Valadez et al.

(2010), the normalized separation t is assumed to be dependent onthepressureratiop¯/p^∗andexpressedby(Johnsonetal.,1985):

t=1− p¯ p^∗

1−log

p¯ p^∗

(41)

Moreover, in agreement with the experimental characteriza- tion provided by Gonzalez-Valadez et al. (2010), stiffnesses in Eqs. (38) and (39) are computed by considering E₀=200 GPa,

ν

0=0.3,andby settingstatisticalparameters viathecorrelations givenbyMcCool(1986):Rs=6.34

μ

^m,

σ

^s=2

μ

^m,Ds=7.32·10⁻³ summits/

μ

^m2.

Results proposed in Fig. 7 clearly highlight that the model herein proposed is much more effective than the statistical de- scriptionadoptedbySherifandKossa(1991)inreproducingbench- marking experimental data, both in terms of quantitative values ofcontactstiffnesses and asregardstheir evolutionwith respect the closure pressure, especially for small values of p¯. In detail, proposed comparison confirms, in agreement withevidence pro- videdby Buczkowskiet al. (2014), that the model by Sherif and Kossa (1991) leads to underrated values ofthe incremental con- tactstiffnesses.Moreover,sucha GW-basedapproach,contraryto theherein-proposedtheoreticalformulation,isnotabletorecover thepressure-dependentdifferencebetweennormalandtangential stiffnesses,andthereby itisexpectedtobe inaccurateindescrib- ingthetangential-to-normalstiffnessratioversustheclosurepres- sure.