Plastic contact of self-affine surfaces

Delft University of Technology

Plastic contact of self-affine surfaces

Persson's theory versus discrete dislocation plasticity

Venugopalan, S. P.; Irani, N.; Nicola, L.

DOI

10.1016/j.jmps.2019.07.019

Publication date

2019

Document Version

Final published version

Published in

Journal of the Mechanics and Physics of Solids

Citation (APA)

Venugopalan, S. P., Irani, N., & Nicola, L. (2019). Plastic contact of self-affine surfaces: Persson's theory

versus discrete dislocation plasticity. Journal of the Mechanics and Physics of Solids, 132, [103676].

https://doi.org/10.1016/j.jmps.2019.07.019

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ContentslistsavailableatScienceDirect

Journal

of

the

Mechanics

and

Physics

of

Solids

journalhomepage:www.elsevier.com/locate/jmps

Plastic

contact

of

self-affine

surfaces:

Persson’s

theory

versus

discrete

dislocation

plasticity

S.P.

Venugopalan

a

_,

_N.

_Irani

a

_,

_L.

_Nicola

a,b,∗

a Department of Material Science and Engineering, Delft University of Technology, the Netherlands b Department of Industrial Engineering, University of Padova, I-35131 Padua, Italy

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 10 February 2019 Revised 19 July 2019 Accepted 28 July 2019 Available online 29 July 2019

Keywords: Persson’s theory Contact mechanics Self-affine surfaces Plasticity Dislocation dynamics

a

b

s

t

r

a

c

t

Persson’stheoryallowsforafastandeffectiveestimateofcontactareaandcontactstress distributionswhenaflatandaself-affineroughsurfacearepressedintocontact.For elas-ticbodies,theresultsofthetheoryhavebeenshowntobeinverygoodagreementwith rathercostlysimulations.Thetheoryhasalsobeenextendedtoplasticbodies.Inthiswork, theresultsofPersson’stheoryforplasticbodiesarecomparedwiththoseofdiscrete dis-locationplasticity.Thearea–loadcurvesobtainedbytheoryandsimulationsarefoundto beingoodagreement whentherough surfacehasaverysmallroot-mean-square(rms) height.Forlargerrmsheights,whicharemorerealisticformetalsurfaces,theagreement isnolongergoodunless inthetheory,insteadofasize-independent materialstrength, oneuses armsheight- and resolution-dependentyield strength. Amodification ofthis type,i.e.,theuseofayieldstrengthdependentonsize,doeshowevernotleadto agree-mentbetweentheprobabilitydistributionsofthecontactstress,whichismuchbroaderin thesimulationsthaninthetheory.Themostlikelyreasonforthisdiscrepancyisthatthe theory,apartfromneglectingplasticitysizedependence,onlyapplies toelastic-perfectly plasticbodiesandtherefore,neglectsstrainhardening.

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

1. Introduction

Inthepastdecades,numerousexperimentshaveconfirmedthatsurfaceshaveaself-affinefractalcharacterdowntothe nanoscale(Bouchaud, 1997;Bouchaud et al., 1990;Dauskardt etal., 1990; Imreetal., 1992; Krim andPalasantzas, 1995; Lechenaultetal., 2010;MajumdarandTien,1990;Mandelbrot etal., 1984;Plouraboué and Boehm, 1999).Toaccount for this, Persson (2001b)developed acontactmodelthat includesthepresence ofroughnessonsuccessivelength scales. At agivennominalpressure, histheory canpredict thecontactarea, contactstressdistribution andinterfacialseparationof elasticbodiesingood agreementwithexperiments(LorenzandPersson, 2009a; 2009b;Persson, 2001b). Persson’stheory hasalso beenextended to studyplasticity(Persson, 2001a). However, to the bestof the authorsknowledge, the validity ofPersson’stheoryhasneverbeentestedformetal surfacesthat deformplastically.Here,weintendtotestthetheory by comparingitsresultswiththoseoftwo-dimensionaldiscretedislocationplasticitysimulations.

∗ _{Corresponding author at: Department of Material Science and Engineering, Delft University of Technology, the Netherlands.}

E-mail addresses: syam.venugopal@gmail.com (S.P. Venugopalan), nil.irani@gmail.com (N. Irani), l.nicola@tudelft.nl (L. Nicola). https://doi.org/10.1016/j.jmps.2019.07.019

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

Fig. 1. Schematic representation of the metal slab indented by a rough surface.

Metallicroughsurfacesspanvariousordersoflengthscales,withasperitiesassmallasafewnanometers.Alreadyatthe micro-scale,experimentshaveshownthatplasticityissizedependent(Flecketal.,1994;Greeretal.,2005;MaandClarke, 1995;VolkertandLilleodden,2006). Thesizedependenceentailsthat,differentlyfromtheelasticresponse,theplastic re-sponsedoesnotscalewithsize.Althoughthesizeeffectcannot becapturedbyclassicalplasticity,whichdoesnotcontain anymateriallengthscale,itiscapturedbynon-localplasticitytheories(NixandGao,1998;Gurtin,2000)andbynumerical simulationsofthetype ofdiscrete dislocationplasticity(DDP)(El-Awady,2015;Kraftetal.,2010;Nicolaetal., 2003).The lattercontainsvariouslength scales,includingthedislocation Burgersvector,andtheaveragespacingbetweendislocation sources anddislocationobstacles. Recently,VenugopalanandNicola(2019) performeddislocation dynamicssimulationsto studyindentationofmetalcrystalsbyself-affinerigidindenterswithvariousroughnessparameters.Resultsshowedindeed asizedependencewhenscalingthedimensionsoftheroughbody:thepressure-displacementresponsedoesnotscaleand smaller bodiesare stronger. Thisphenomenon is mostlycausedby limiteddislocation availability,i.e., when thestressed subsurface regions becometoo small,they nolonger containa sufficient amountofdislocation sources to sustainplastic deformation.ThesamephenomenonisobservedinVenugopalanandNicola(2019)whenindentingacrystalwithconstant sizeandasurfacewithsmallerrootmeansquare roughness:ifpressureisnormalizedontherootmeansquare slopeand interferenceis normalizedby therms height, theresultsare elasticallyidentical,butnot plastically.The crystalindented bythelargestrms-heightissubjectedtoabroaderstressedregionunderneaththesurfaceandismoresusceptibleto plas-ticdeformation and, therefore,is softer.The characteristicscale-independent materiallength scale isthe averagespacing betweendislocationsources.

In order to include size-dependent plasticityin his theory, Persson suggested to replace the constant material yield strength witha resolution-dependent yieldstrength, i.e., a yield strengththat increases withdecreasing theshort wave-lengthcut-off

λ

s(Persson,2006). Theexactdependenceofyieldstrengthonresolutionwasnotspecifiedanditmightbe inferredbycomparisonwithdislocationdynamicssimulations,iftheresultsareotherwiseinagreement.

BeforeproceedingwiththecomparisonitisimportanttonotethattheGFDDsimulationsarebasedonthesmallstrain andsmallslope approximation.Thismeans that forrms heightsrealistic formetalsone canonly reachpartialclosureof thecontact.Persson’soriginaltheoryisinsteadexactatfullcontactandrequiresacorrectingfactoratpartialcontact(Dapp etal., 2014;Manners andGreenwood,2006; WangandMüser, 2017).Therefore,forthesake ofcomparison,we willstart by showingtheresults ofGFDDsimulationsfora very smallrmsheight, whichis notobserved intypical metalsurfaces butallowstoreachclosure.Thisisthecaseforwhichthebestagreementisfoundbetweensimulationandtheory.Wewill thenseethatforsurfaceswithrmsheightsmoretypicalformetals,theagreementisnotgood.

2. Formulationoftheproblem

Arigidindenterwithself-affineroughnessispressedintocontactagainstametallicslaboffiniteheight,seeFig.1.The powerspectrumoftheindenteris

C

(

q

)

=C0



_q q0



−2(H+1 2) ifq0<q<qs, C

(

q

)

=0 else, (1)

whereq0=2

π

/

λ

l andqs=2

π

/

λ

s are thewavenumberscorresponding tothelong andshortwavelengths cut-offs,i.e.

λ

l and

λ

s,respectively.C0isdeterminedbytheroot-mean-square(rms)height ¯has

¯h2₌₂

 qs q0

Assumingqs>>q0 impliesthat

C0=

H¯h2

q0 ,

(3)

whereHistheHurst exponent.Thepower-lawscaling forthesurfaceheight spectrumapplies towavenumbersbetween cut-offsatlongandshortwavenumbersq0<q<qs.

TheslabistakentobeelasticallyisotropicwithYoung’smodulusE=70GPaandPoisson’sratio

ν

=0.33,representative valuesforaluminum.Moreover,ithasafiniteheightzm.Thetopsurfaceofthecrystalisfrictionlessatthepointsofcontact,

σ

xz

(

xc,zcm

)

=0,andtraction-freeelsewhere,

σ

xz

(

xnc,zmnc

)

=

σ

zz

(

xnc,zncm

)

=0Thesuperscripts‘c’and‘nc’standforpoints‘in contact’andpoints‘notincontact’,respectively.Furthermore,thebottomofthesubstrateiskeptfixed:ux

(

x,0

)

=uz

(

x,0

)

= 0.

2.1. Persson’stheoryforlinecontactsinsolidswithfiniteheight

Undertheassumptionthat atresolutionq thecontactisfull,Persson (2001a,b)statesthatonalllength scalesthe dis-tributionofcontactstress

σ

is

P

(

σ

,q

)

=



δ

(

σ

−

σ

nom

)



. (4)

Here,

σ

nomisthenominalcontactstresswhenthesurfaceroughnesswithwavenumberslargerthanqhavebeensmoothed outand



_...



standsforensembleaveragingoverdifferentsurfaceroughnessprofiles.Asfinerroughnessfeaturesareadded, thecontactstressdistributionbecomesP

(

σ

,q+



q

)

=



δ

(

σ

−

(

σ

nom+



σ

))



.Byexpandingthisequationtolinearorderin



q

∂

P

∂

q=k

(

q

)

∂

2_P

∂σ

2, (5) where k

(

q

)

=





σ

2



2



q . (6)

ThepartialdifferentialEq.(5)canbesolvedbyimposingthefollowingboundaryconditions:

P

(

0,q

)

=0, (7a)

P

(

σ

,0

)

=

δ

(

σ

−

σ

0

)

, (7b)

P

(

σ

Y,q

)

=0. (7c)

Theboundaryconditionsenforcethat: (7a)whenthelocalcontactstress reacheszero,contactingsurfacesdetach;(7b)at thelowestresolution,thestressdistributionisadeltafunction;(7c) thecontactstress doesnot exceedtheyieldstrength

σ

Y.ThelatterconditionentailsthatPersson’stheoryforplasticityappliestoidealelasto-plasticsolidsthatdisplaynowork hardening.Noticealsothatsolvinganelasticcontactproblemisequivalenttoimposing

σ

Y→∞inEq.(7c).

FollowingPersson(2001a,b),thesolutiontoEq.(5)withboundaryconditions(7a)–(7c)canbewrittenas

P

(

σ

,q

)

= ∞  n=1 An

(

q

)

sin



_n

_πσ

σ

Y



. (8)

SubstitutingtheexpressionaboveinEq.(5)leadstothefollowingpartialdifferentialequation:

dAn dq =−k

(

q

)



_n

_π

σ

Y



2 An, (9)

whichaftersolutionleadsto

An

(

q

)

= 2

σ

Y sin

(

αn

)

exp



−



n

_σ

π

Y



2q q0 k

(

q

)

dq



. (10)

Thefullexpressionofthecontactstressdistributionisthengivenby

P

(

σ

,q

)

= 2

σ

Y ∞  n=1 sin

(

α

n

)

exp



−

α

2 nL

(

q

)

sin



n

πσ

σ

Y



, (11) where

α

n= n

πσ

0

σ

Y andL

(

q

)

=q q₀ k

(

q

)

σ

2 0

dq. Inorderto solveEq.(5),one mustobtain k(q) from



σ

(q). WangandMüser (2017)showedthatforelasticsubstrateswithafiniteheight,



σ

isgivenby



σ

(

q

)

=



W

(

ar

)



qE∗_f

₍

_q

₎

2

|

_h˜

₍

_q

₎

_|

_{, (12)}

wherear istherelativecontactarea,andE∗ istheeffectiveelasticmodulus.Moreover,W

(

ar

)

isthecorrectionintroduced byWangandMüser(2017)forthelowloadregimes,i.e., whencontactispartial.Furthermore,forfrictionlesselastic con-tacts,asubstrateofheightzm,andafixedbottom,f(q)isgivenby Venugopalanetal.(2017b)as

f

(

q

)

=cosh_sinh

(

2qz

₍

m_2qz

)

+2

(

qzm

)

2+1 m

)

− 2qzm , (13) and k

(

q

)

=





₂

_

σ

_q2



= 1₂W[ar

(

q

)

]



qE∗f

(

q

)

2

2

|

˜_h

₍

_q

₎

_|

2_{. (14)}

Finally,forthepowerspectrumC(q)inthiswork

k

(

q

)

=H¯h2q0 8 W

(

ar

)

(

E∗f

(

q

)

)

2



q q0



1−2H . (15)

Havingfoundk(q),wemaythenproceedtosolvethepartialdifferentialEq.(5).

Subsequently, from the contact stress distribution P(

σ

, q), following Persson (2001a,b), one can obtain the following quantities:

(i) Thefractionofmacro-contactarea that isnotinrealcontactanon

r , thatformsa plasticcontactaplasr ,andanelastic contactaelas r : anon r = q q0 k

(

q

)

∂

P

∂σ

(

0,q

)

dq =

_π

2 ∞  n=1 sin

(

αn

)

n



1− exp



−

α

2 nL

(

q

)



, aplasr =− q q0 k

(

q

)

∂

P

∂σ

(

σ

Y,q

)

dq =−

_π

2 ∞  n=1

(

−1

)

nsin

(

α

n

)

n



1− exp



−

α

2 nL

(

q

)



, aelas r =1− aplasr − anonr . (16)

(ii)Therelativecontactareaar,comprisingboththeareainelasticandplasticcontact:

ar=1− anonr . (17)

2.2. Green’sfunctiondislocationdynamics

Ateachtimestepofthesimulation,thesolutionoftheboundaryvalueprobleminFig.1isobtainedbythesuperposition oftwolinearelasticsolutions:Theelasticanalyticalfieldsfordislocationsinahomogeneousinfinitesolid,andthesolution tothecomplementaryelasticboundary-valueproblem,whichcorrectsfortheboundaryconditions.Themethodologyis sim-ilartoVanderGiessenandNeedleman(1995),however,thesolutiontothecomplementaryelasticboundary-valueproblem isobtainedthroughGreen’sfunctionmoleculardynamics(GFMD)(Venugopalanetal.,2017a).

TheschematicsoftheindentedsinglecrystalisshowninFig.2.Indentationisperformedbyapplyingthedisplacement Uz ontopoftherigidindenterfromwhichthevalueoftheequivalentappliedpressure

σ

0 isobtained.

FollowingVanderGiessenandNeedleman(1995),thedislocationdynamicsarecontrolledbyconstitutiverulesinspired byatomicscalephenomenathatcontrolthenucleationandglideofthedislocations.Thecrystalisinitiallydislocationfree, and containsa given densityof slip planes, dislocation sources, andobstacles that are randomly distributed. When the stressinthebodyreachesthenucleationstrength

τ

¯nuconadislocationsourceforagivenamountoftimetnuc,adislocation dipoleisnucleatedfromthesources andglidesontheslipplane resultinginplasticdeformation.Thevelocitywithwhich thedislocationsglideiscontrolledbythePeach-Koehlerforceactingonthem.Dislocationsarestoppedbytheobstacles,but releasedwhentheresolvedshearstressonthemexceedsthecriticalstrengthassociatedtotheobstacle,

τ

obs.

2.2.1. Choiceofparametersforthesimulations

Dislocationsarenucleatedfromrandomlydistributednucleationsourcesonslipplanesorientedat

φ

=60◦,−60◦_,and 90◦ withrespecttotheloadingdirection. Thesimulationsareperformedforanucleation sourcedensity

ρ

nuc=40

μ

m−2. Inboth cases,thesources haveaGaussian strengthdistribution withthemeanstrength being

τ

¯nuc=50MPa.The nucle-ation timetnuc=10ns. Thedensityofobstacles is

ρ

obs=40

μ

m−2 andtheobstaclestrength is

τ

obs=150MPa.The drag coefficient for glideis B₌10−4Pas andthe critical distancefor annihilationis Le=6b, whereb=2.5× 10−4

μ

m is the magnitudeofthe Burgersvector. Moreover,inall calculationsa timestep of



t=2.5nsisemployed. The GFDD simula-tionsareperformedfor10realizationsofnucleationsourceandobstacledistributions.Inthefollowing,thepresentedresults areobtainedbyaveragingoverthese10realizations.

Fig. 2. Schematic representation of the metal crystal indented by a rough surface. Table 1

Default simulation parameters.

Parameters Notation Value

Angle between slip planes and loading direction φ 60 ◦_{, −60} ◦_{, 90} ◦

Source density ρnuc 40 μm −2

Mean nucleation strength τ¯nuc 50 MPa

Obstacle density ρobs 40 μm −2

Obstacle strength τobs 150 MPa

Drag coefficient B 10 −4 Pas

Length of the Burger’s vector b 2.5 × 10 −4_μm

Critical annihilation length Le 6 b

Time step t 2.5 ns

Thermodynamic discretization t 2 −1

Fractal discretization f 16 −1 , 32 −1 , 64 −1

Continuum discretization c 32 −1

Hurst exponent H 0.8

Long wavelength cut-off λl 10 μm

Rms height ¯h 0.001, 0.01, 0.1 μm

Height of the crystal zm 10 μm

Theroughnessoftheindenterisobtainedusingthepowerspectraldensitymethod(Campañá etal.,2008). Thepower spectrumC(q)isusedtoconstructaperiodicself-affinesurfacewithaGaussian heightdistribution.TheFouriertransform oftheheightprofileh(r)oftheindenterisgivenas:

˜ h

(

q

)

=h0



˜G

(

q

)



C

(

q

)

=h0 ˜



G

(

q

)

q ₁ 2+H , (18)

where h0 is a real-valued constant which can be adjusted to obtain the required rms slope of the surface,



˜G

(

q

)

is a Gaussianrandomvariablewithrandomphasesuchthat





˜G

(

q

)



=0,andHistheHurstexponent.Fordifferentrealizations oftheroughsurface,allparameters,includingthecut-off values,arekeptfixedexcepttheGaussianrandomvariable



˜G

(

q

)

whosephase is randomlyvaried. Furthermore,before startingthe simulations thesurfaces sogenerated are shiftedsuch that the lowest point touchesthe substrate atzero interfacialpressure. The fractaldiscretization,

f=

λ

s/

λ

l, defines the rangeofwavelengthsusedtodescribethesurface.Here,thelongwavelengthcut-off iskeptconstant,i.e.

λ

l=10

μ

mand theshortwavelengthcut-off isvariedwhile

_fis varied.Thethermodynamic discretizationisdefinedas

t=

λ

l/Lx=1/2, whereLx isthe widthof theperiodic unit cell. Inthe limitingcaseof

t→0, whichcorresponds to the thermodynamic limit,thesurfaceisno longerperiodicsince Lx→∞.Finally,the continuumdiscretization isdefinedas

c=a0/

λ

s=1/32 wherea0isthespacingbetweenthegridpointsthatdiscretizethesurfaceofthesubstrate.Inthelimitingcaseof

c→0, thegridspacinga0→0andhencethesurfacehasacontinuumrepresentation,thereforethesolutionmustconvergetothe continuummechanicssolution.WeselectedfortheHurstexponentthevalue H=0.8,sinceitistypicalformanymetallic surfaces(Bouchaudetal.,1990;Dauskardtetal.,1990).Thelongwavelengthcut-off

λ

_liskeptconstantandequalto10μm. Theshortwavelengthcut-off ischangedinthesimulationstorepresentachangeintheresolutioninPersson’stheory.The rmsheightvaluesconsideredare ¯h=0.001, 0.01, and0.1

μ

m,thelatterbeingthemostrealisticformetalsurfaces.

Fig. 3. (a) Schematics of the uniaxial tensile test. (b) Uniaxial tensile stress–strain (σ−ε) curve from discrete dislocation plasticity.

Fig. 4. (a) Relative contact area a r against reduced applied pressure σ0∗ for indenters with ¯h = 0 . 001 μm and three fractal discretizations f . (b) Probability

distribution of reduced contact stress σ∗_{for an indenter with}

f = 16 −1 at three instances of applied pressure σ0∗ = 0 . 1 , 0.5 and 1.3.

2.3. Theyieldstrength

WhileinPersson’stheorytheyieldstressisaninputtothecalculation,indiscretedislocationplasticityitisanoutput. Tocalculateit,weperformauniaxialtensileloadingsimulationonasinglecrystalwiththematerialpropertiesmentioned above.

The dimensionsofthecrystalareselected tobe large enoughto notexperience sizeeffects andareL=12.5

μ

m and W=5

μ

m. The schematicsof theuniaxial tensile test isshown inFig. 3(a).The predictednominal tensilestress versus appliedstrain

ε

₌2Ux/LispresentedinFig.3(b).Thisfigureshowsthatthetensileyieldstrengthofthecrystalis

σ

_YTensile= 60MPa.Itisimportanttohighlightthattheyieldstrengthidentifiedwith

σ

Tensile

Y ,hereandthroughoutthemanuscriptisa size-independentquantity.

3. Persson’stheory:correctingfactoratlowloadsforvariousfractaldiscretizations

TheexpressionfortheelasticenergyinPersson’stheorywascorrectedby WangandMüser(2017)inordertoholdat lowloads.Thecorrectingfactorinthefractallimit,

λ

s→0,wasgivenbytheauthorsas

W[ar

(

σ

0∗

)

]=1+c1

(

1− ar

(

σ

0∗

)

2

)

+c2

(

1− ar

(

σ

0∗

)

4

)

, (19) where

σ

₀∗=

σ

0

E∗¯g isthereducedpressure,

σ

0istheappliedpressureand ¯gistheroot-mean-squaregradientoftheindenter. Selectingthe valuesof2/9 and−2/3forthe constantsc1 andc2 leadsto goodcorrespondencebetweenPersson’stheory andelasticGFMDsimulations.

Inthiswork,weintendtoconsiderfractaldiscretizationsalsofarfromthelimit,namely

_f=64−1_,32−1_,or16−1.Tothis end,wefirstproceedtochecktowhichextentthecorrectionfactorisindependentoffractaldiscretization.Thisisdoneby comparinginFig. 4(a)theresultsofPersson’stheorywithourelastic GFMDcalculationsforanindenter withrms height ¯h=0.001

μ

m.

Table 2

Coefficients c 1 and c 2 in Eq. (19) . f = 16 −1 f = 32 −1 f = 64 −1

c1 −2.3 −0.6 0.19

c2 1.9 0.2 −0.6

Fig. 5. (a) The correction factor W (ar) for three different fractal discretizations f . (b) Probability distribution of the reduced contact stress σ∗ for an

indenter with f = 16 −1 at three instances of applied pressure σ0∗ = 0 . 1 , 0.5 and 1.3.

Theresultsoftherelativecontactarea versusappliedloadinFig.4(a)showthat thereisasmalldiscrepancybetween GFMDandPersson’stheoryatintermediateload.Thiscorrespondstoanon-negligibledifferenceintheprobability distribu-tionofthecontactstressP(

σ

∗)forlargerfractaldiscretizations,asshowninFig.4(b).

Toassurethat atleastourelasticsimulations agreewellwithPersson’stheory,weproceedto searchforthec1 andc2 coefficientsthatminimize thedifference betweenthearea–load curvesfortheoryandsimulations. Thevalues,whichare non-unique, arelisted in Table2 andlead to thecorrection factors inFig. 5(a)and tothe contactstress distributions in

Fig.5(b).

WehaveverifiedthatthecorrectionfactorsW

(

ar

)

foundforindenterswithrmsheight ¯h=0.001

μ

marealso appropri-atefortheotherrmsheightsusedinthiswork,andarethereforeusedthroughoutthemanuscript.

4. Comparisonbetweentheoryandsimulationsforindenterswithsmallrmsheight

WestartbycomparingPersson’stheorywithGFDDsimulationsfortheindentationofametalcrystalbyarigidindenter withsurfaceroughnesswithh₌0_.001

μ

m andfractaldiscretization

−1_{f =}64.Thisallows toreachnearfull closurewith dislocationdynamicssimulationswhilestillobeyingthesmallstrainandsmallslopeapproximations.

InPersson’splasticitytheory,theyieldstrength

σ

Yisaninputparameter,whichhasbeeninterpretedaseitherthetensile yieldstrengthofthematerial

σ

Tensile

Y orthemacroscopicindentationhardness,estimatedbyJohnson(1987)tobe3

σ

YTensile. Bycomparisonbetweentheoryandsimulationsweassesswhichofthetwodefinitionsismostappropriate:Fig.6(a),shows amuchbetteragreementbetweenthearea–loadcurveswhenhardnessisusedforthedefinitionof

σ

Y.

Fig.6(b)shows,therelativecontactareatogetherwithhowtheelasticaelas

r andplasticaplasr fractionsofcontactchange withload.Rememberthataelas

r andaplasr arecalculatedseparatelyinPersson’stheory(seeEq.(16)).Thus,itispossibletosee thattheportionofcontactundergoingelasticdeformations,aelas

r ,initiallyincreaseswithload,

σ

0∗,andthendecreaseswith increasing plasticity.Instead,the relative plasticcontactarea, i.e.aplas_r ,continues toincrease withloaduntil the external loadreaches

σ

₀∗=1.3.Thisloadcorrespondstothepointatwhichthecontactstressiseverywhereplasticandequalto

σ

Y. Noticethatthecontactindislocationdynamicssimulationsneverreachfullclosure,ascanbeseenfromthedecreaseofthe interfacialgapinFig.7(a)anda snapshotoftheinterfaceat

σ

₀∗₌1_.3inFig.7(b).Thedepthofthevalleysformed during deformationcanbeaslargeasthermsheight ¯hoftheindenter.ThisisinagreementwiththeobservationofBowdenand Tabor(2001)accordingtowhomfullclosureisimpossibleforroughmetalsurfacesduetoworkhardening.

Fig.8presentsthedistributionofcontactstress forthreeinstancesoftheapplied pressure,

σ

₀∗=0.1,0.5, and1.3. The probability distribution function representing the plastic partof the contact isa deltapeak at

σ

∗₌

σ

_Y∗. The other delta peakat

σ

∗=0representsthepartofthesurfacewhichisnotincontact.Theareaundertheprobabilitydistributioncurve inPersson’stheoryistheelasticfractionofthecontactarea aelas

r .Thisisdifferentfromthedistributionfunctionobtained throughGFDDsimulations,wheretheareaunderthecurverepresentstherelativecontactareaar:elasticandplasticcontact

Fig. 6. Relative contact area a r calculated using GFDD and modified-Persson’s theory. In the latter, yield strength σY = σYTensileand 3 σYTensilewere applied

as an input.

Fig. 7. (a) The average interfacial separation ¯u / ¯h are plotted against reduced applied pressure σ∗

0 for the case of an indenter with h = 0 . 001 μm and −1

f = 64 . The input yield strength to Persson’s theory is σY = 3 σYTensile. (b) The snapshot of the contact between the indenter and the substrate at σ0∗ = 1 . 3 .

Fig. 8. Probability distribution of the contact stress σ∗_{at three instances of loading, σ}∗

0 = 0 . 1 , 0.5, and 1.3, for an indenter with h = 0 . 001 μm and f−1 = 64 . areasarenotdistinguished,nordistinguishable,andthedeformation only‘partiallyplastic’.Therefore,agreementbetween thecontactstressdistributionobtainedbyPersson’stheoryandbytheGFDDsimulationsceasestobegoodwhenplasticity becomesrelevant.Thesimulationspredictamuchbroaderstressdistribution,withcontactstresseslargerandsmallerthan themacroscopichardness.ThereasonforthisdiscrepancycanbepartlyattributedtothefactthatPersson’splasticitytheory doesnot accountformaterial hardening.In ouropinion,a better agreementwiththesimulations wouldbe found,ifthe

Fig. 9. (a) Reduced nominal pressure and (b) mean contact pressure are plotted against reduced displacement for three different rms heights.

Fig. 10. (a) Relative contact area a r against reduced applied pressure σ0∗ for an indenter with f = 64 −1 and three rms heights ¯h . The input yield strength

to Persson’s theory is 3 σTensile

Y . (b) The input yield strength to Persson’s theory is the indentation yield strength obtained through GFDD calculations.

theorywouldbeslightlymodifiedby making

σ

_Y∗increasewithplasticdeformation.Theplasticpeak willthenshifttothe rightduringindentationandtheelasticcontributionwouldbecomemorepronouncedthanitisnow.

4.1. Sizedependence

ItwasshownbyVenugopalanandNicola(2019)thatwhenindentingametalcrystal,indenterswithdifferentrmsheights giverise tothesamereducedpressure forequal reducedinterferencevalues.However,they inducea differentplastic re-sponse.The smallerthe rms heightthe later theonset ofplasticdeformation. Thisplasticitysize dependenceoccurs be-causethe sizeofthesubsurfaceregion wherethedislocationnucleation strengthisexceeded scales withrms height,but theavailabilityofdislocationsources doesnotscaleaccordingly:thespacingbetweendislocationsources isamaterial pa-rameterwhichisscaleindependent.Thisiswhyalargerreducedpressureisrequiredtoinducenucleationinthecaseofa smallrmsheight.

Fig.9presents curvesofreducedpressure versus reducedinterference,obtainedthrough GFMD simulations,for three indentersofrms heightsh=0.001,0.01and0.1

μ

m,together withthecorresponding curvesofmeancontactpressure as afunctionofdisplacement.Here,themeancontactpressureiscalculatedaspm≡ F/A,whereFisthetotalinterfacialforce andAisthetruecontactarea.NoticethatinFig.9thedifferencebetweenthecurvesissolelycausedbyplasticity.

Therelativecontactareaar,ascalculatedbyGFDDandmodifiedPersson’stheory,areshowninFig.10forindenterswith

f=64−1andvariousrms-heights.Forallindenters,theinputyieldstrengthtoPersson’stheoryisassumedtobe3

σ

YTensile. Itcan beseen thatunderthisassumption,thecontactarea aspredictedby thetheory andGFDDare verydifferentfrom eachotherwhen¯h=0.01

μ

m.

It isnoteworthy that scaling rms height corresponds to a vertical shiftin thepower spectrum ofthe roughness, and hasnoinfluenceontherangeofwavelengthsconsideredintheproblem:bothlargeandsmallwavelengtharethesameas before.Alreadythisobservationissufficienttoconcludethatconsideringaresolution-dependentyieldstrengthinPersson’s theory,i.e.

σ

(

λ

s),wouldnotimprovetheagreementbetweensimulations andtheoryinthiscase.Instead,itispossibleto seeinFig.10bthatifthesize-dependentyieldstrength obtainedthrough dislocationdynamicssimulations

σ

GFDD

Fig. 11. Probability distribution of the contact stress σ∗ _{for different yield strength as input to Persson’s theory compared to GFDD for indentation using}

h = 0 . 01 μm at σ∗ 0 = 0 . 06 .

Fig. 12. (a) Dislocations, stress distribution and (b) contact tractions σ∗_{for an indenter with ¯h = 0 . 01 μm at an applied load σ}∗ 0 = 0 . 06 .

astheinputyieldstrength

σ

YinPersson’stheory,amuchbetteragreementisfoundfortheload–displacementcurves.The yieldstrengthiscalculatedat0.2%offsetstrain,asindicatedinFig.9b.

However,thistypeoffixisnotsufficienttoobtainagreementbetweentheprobabilitydistributionsofthecontactstress, asonecanseefromFig.11,wheretheprobabilitydistributionisshownforthesimulations andforPersson’stheory with andwithoutcorrectionfor

σ

Y,fortheindenterwithh=0.01

μ

m.Whilethefixgivesagreementbetweentheareasthatare not incontact,the simulationsshow a muchhigherprobability ofhavingsmalleraswell aslargercontactstresseswhen comparedtothetheory.ThecontactandbodystressesobtainedthroughGFDDinasinglesimulationareshowninFig.12.

4.2. Effectofshort-wavelengthcut-off

Inthissection,weperformsimulationsfordifferentroughnessresolutionbychangingthesmallwavelength,while keep-ingthelargewavelengthconstant.Thiscorrespondstochangingthefractaldiscretization

f.Itisnoteworthythatchanging resolutiondoesnotcorrespond toscalingthecontactproblem.Changingresolutioninvolvesextendingthe rangebetween smalland largewavelength, i.e.addingsmaller wavelengthsto the surface, andthereforeresults ina differentboundary valueproblem, withadifferentelasticandthusplasticresponse.Suchsimulationsarethereforenotsuitabletohighlighta plasticitysizedependence.Theymighthowevergiveanindicationonhowappropriateitistoreplace

σ

YinPersson’stheory with

σ

Y

(

λ

s

)

,assuggestedinPersson(2006).

Fig. 13. Relative contact area a r against reduced applied pressure σ0∗ for indenters with h = 0 . 001 μm and three fractal discretizations f .

Fig. 14. Relative contact area a r against reduced applied pressure σ0∗ for indenters with h = 0 . 01 μm and three fractal discretizations f . The input yield

strength for Persson’s theory is taken as (a) σY = 3 σYTensile(b) σY = σYGFDD .

Fig. 15. Probability distribution of contact stress σ∗ _{for different fractal discretizations f for indentation for h = 0 . 01 μm at σ}∗

0 = 0 . 06 . The input yield

strength for Persson’s theory is taken as (a) σY = 3 σYTensile(b) σY = σYGFDD .

Fig.13showstheincrease oftherelative contactareaar withloadforindenters withh=0.001

μ

mandthreefractal discretizations.Thecurvesareinsensitivetoachangein

λ

s suggestingagainsttheuseofaresolution-dependent

σ

Y.Forall resolutionsthereisagoodagreementbetweensimulationsandtheory.Ifoneinsteadconsidersindenterswithh₌0_.01

μ

m theagreementispoorforalldiscretizations(seeFig.14a).Theagreementimprovesifagain,insteadofusing

σ

Y=3

σ

YTensile oneuses

σ

Y=

σ

YGFDD(seeFig.14).

Giventhat

σ

GFDD

Y isresolution-dependent, we concludethat indeeda

σ

Y

(

λ

s

)

should beused, assuggestedinPersson

(2006).However, theuse ofa resolution-dependent

σ

Y isnot sufficientto obtain theagreement betweentheprobability distributions,asonecanseeinFig.15.Also,thedependenceof

σ

Yonresolutionappearstobemuchmoreimportantthan thedependenceof

σ

Yonrmsheight.

Table 3

Normalized yield strength σGFDD Y / 3 σYTensile.

rms height f = 4 −1 f = 8 −1 f = 16 −1 f = 32 −1 f = 64 −1

h = 0 . 01 μm 1.4 1.7 2.0 2.7 4.1

h = 0 . 1 μm 3 4.7 9.5 10.6 13.3

Here in Table 3 we provide the flow stress

σ

GFDD

Y found through GFDD simulations for indenters with h=0.01

μ

m and ¯h=0.1

μ

m that allowsforgood agreementforarea–load curvesinPersson’stheory.Data forh=0.001

μ

m arenot reported,giventhatnocorrectionisneededforagreementbetweenarea–loadcurvesinthatcase.

5. Concludingremarks

Inthispaper,wehaveshownacomparisonbetweendislocationdynamicssimulationsandPersson’stheoryinthestudy ofcontactbetween aflat metal body by arigidindenter withself-affine roughness. Althoughthere isa goodagreement betweensimulationsandtheorywhenthemetalbehaves elastically,theagreementceasestobegoodwhenthereis plas-ticity.Thebestagreementforthearea–loadcurvesisfoundforsmallvaluesofthermsheight,whentheresponseisclose toelasticuptolargecontactfractions.Thisisbecauseforsmallrmsheight,theonsetofplasticityindislocationdynamics simulationsoccursatlargerindentationdepththaninacontinuumtheory,becausethereareonlyfewdiscretedislocation sourcesinthesubsurfaceregionwherethestressconcentrationissufficientlylargetoinducedislocationnucleation.

A good agreementbetween the area–loadcurves ofsimulations andtheory is obtainedalso forlarger rms heights if the size independentyield strength inthe theory isreplaced by the size-dependent yieldstrength obtainedthrough the simulations.

The necessityof usinga resolution-dependent yieldstrength inhis plasticitytheory wasalready explicitlymentioned by Persson.The fixisindeedimportant,becausetheplasticresponse dependson resolution.However,ifonebelievesthe resultsofthesimulations(ofcoursetheyhavelimitationstoo,tomentionacoupleofthem:theyaretwodimensionaland smallstrain),thisfixisnotsufficient.Thisisbecauseitdoesnotaccountfortheplasticitysizedependencethatisobserved whenoneormoregeometricallengthintheproblemunderstudyarescaleddown:Toaccountforthis,theyieldstrength shouldbecomescale-dependent.

Anotherpointthatisimportanttoconsideristhat,althoughwefoundgoodagreementforarea–loadcurves,whenusing Persson’stheorywiththeyieldstrengthobtainedthroughdislocation dynamics(i.e.ayieldstrength thatdependsonrms heightandresolution)thecontactstressprobabilitydistributionwasstillmarkedlydifferent.Thesimulationspredictamuch broadercontactstressdistributioncomparedwiththetheory.Thisisbecauseinthetheorythematerialbehavesasperfectly plastic,i.e.,withoutanystrainhardening.Apossibleimprovementofthetheorymightbetouse,insteadofaconstantyield strength,ayield strengththatincreaseswithplasticdeformation.Thiswouldtranslateinaplasticpeak intheprobability distributionthatmovestowardslargerpressureswithincreasingclosureofthecontactandabroaderdistributioninstresses oftheelasticpartofthecontact.

Acknowledgement

Thisprojecthasreceived fundingfromtheEuropeanResearchCouncil(ERC)undertheEuropeanUnion’sHorizon2020 researchandinnovationprogramme(grantagreementno.681813).

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