ELECTRONIC APPENDIX TO: ELASTIC-PLASTIC CONTACT OF A ROUGH SURFACE WITH WEIERSTRASS PROFILE. Yan-Fei Gao and A. F. Bower

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1 ELECTRONIC APPENDIX TO: ELASTIC-PLASTIC CONTACT OF A ROUGH SURFACE WITH WEIERSTRASS PROFILE Ya-Fi Gao ad A. F. Bowr Divisio of Egirig, Brow Uivrsity, Providc, RI 9, USA Appdix A: Approximat xprssios for cotact rspos of a siusoidal surfac I this appdix, w list th xprssios that charactriz th rspos of a lastic-prfctly plastic solid with siusoidal roughss to idtatio by a rigid flat puch. Th substrat is a lastic-prfctly plastic solid with Youg s modulus E, Poisso s ratio ν ad yild strss σ Y. Th surfac roughss has wavlgth λ ad amplitud g. I trms of ths paramtrs, w dfi a matrial ad gomtrical paramtr ψ = ge /[( ν ) λσ Y], which charactrizs th rsistac of th surfac to plastic flow. Th solid is idtd by a flat, rigid surfac, udr th actio of a omial prssur p. Th loadig iducs a cotact width a, prssur distributio p ( x ) ad ma prssur p ma, giv by fuctioal rlatioships p a/ λ = A, ψ σy p ( x) x p =Φ,, ψ σy λ σy ma p ma p =Φ, ψ σy σy W also giv xprssios for th prssur distributio fuctio q ( p/ σ ), which is dfid so that q ( p/ σ )( dp / σ ) is th fractio of th siusoidal surfac that is subjctd to ormalizd Y Y prssur btw p / σ Y ad ( p + dp)/ σ Y. Th rsult is spcifid by th fuctioal rlatioship p p p q = q,, ψ () σy σy σy W obsrv four gral rgims of bhavior, dpdig o th ormalizd cotact prssur ξ = p / σy, as outlid i th xt sctios. Y (A)

2 3. Elastic Rgim < ξ < πψ si ( π / ψ) ma I th lastic rgim, A, Φ ad Φ ar giv by th Wstrgaard solutio (Wstrgaard 939) si ξ, ξ < πψ A( ξψ, ) = π πψ, ξ πψ (A3) ξ cos( πη) { si ( πa /) si ( πη) } / ξ < πψ Φ ( ηξψ,, ) = si ( π A /) ξ + πψ cos( πη) ξ > πψ (A4) ( ξψ, ) πξ ma Φ = si ( ξ / πψ ) ξ πψ ξ ξ πψ η / π η < / q( ηξψ,, ) {( πψ ξ) η }{ ξ( πψ ξ ( πψ ξ) η ) η } = η > (A5) πψξ (A6) πψξ Not that th lastic solutio dpds oly o th ormalizd omial prssur ξ = p ad surfac proprty ψ, ad is idpdt of g / λ. Th critical load ξ = p / σy = πψ corrspods to full cotact ( a/ λ = ). This suggsts a scod physical itrprtatio for th paramtrψ : a surfac with a low valu for ψ ca asily b flattd. This solutio is valid blow th lastic limit: Gao t al (5) suggst that a covit (but approximat) stimat for th yild poit is giv by th coditio ψa / λ <., which rquirs that ξ = / σ < πψ si ( π / ψ). p Y / σy 3. Elastic-Plastic rgim πψ π ψ ξ ψ si ( / ) < < H / Th lastic-plastic rgim of bhavior is obsrvd for applid loads that slightly xcd yild. Udr ths coditios, th lastic ad plastic strais udr th idtr ar comparabl, ad ma xact xprssios for A, Φ ad Φ caot b foud. Dtaild umrical rsults ar giv i Gao t al (5). For th problm at had, howvr, w fid that at most a sigl scal of roughss is i th lastic-plastic rgim, so it is ot cssary to charactriz this rgim i ma dtail. Cosqutly, w hav costructd simpl fuctios for A, Φ ad Φ by usig a slfcosistt itrpolatio btw th lastic ad fully plastic rgims. Th itrpolatio is costructd to prdict corrctly: (i) th lastic limit, (ii) th critical load to trasitio to fully plastic bhavior; (iii) th cotact prssur distributio at th lastic limit; ad (iv) th cotact

3 prssur at th fully plastic limit. I additio, w itrpolat th cotact siz btw th lastic ad fully plastic limits as follows.9 ( ξ ξp ) ( ξ ξ) A( ξψ, ) = A + A.9 p (A7) ( ξ ξ ) ξ ξ whr ( ) p p π H ξ = πψ ξ ψ = = ψ A = /(5 ψ) A = / ψ si p H.8 p dfi dimsiolss valus of th critical omial prssur at yild; th critical omial prssur at full plasticity, ad th ormalizd cotact widths at yild ad th fully plastic limit, rspctivly. Th cotact prssur distributio is approximatd as αξψ (, )cos( πη) Φ ( ηξψ,, ) = { si ( πa/) si ( πη) } / + βξψ (, ) (A9) si ( π A/) whr th scalig factors α ad β ar itroducd to sur that th cotact prssur has th corrct avrag ad imum valus. This rquirs ( Aξ ξ ) si( πa/) ξ ξ si( π A/) αξψ (, ) = βξψ (, ) = (A) A si( π A/) A si( πa/) whr th imum cotact prssur is itrpolatd btw its valus at yild ad full plasticity as ( A A p ) ξ H( A A) ξ = + (A) A A A A A whr ( p) ( p ) ( ξ / ) + ( ξ / πψξ ) A H A Ap A A = H πψξ is a costat chos to sur that th pak cotact prssur has th corrct valu ξ = at th lastic limit ( A= A, ξ = ξ ). Th ma cotact prssur follows as πψξ (A8) (A) ma Φ ( ξψ, ) = ξ / A (A3) I additio, th cotact prssur distributio fuctio may b computd as ( η βh)/ π q( ηξψ,, ) = / { ω + ( η βh) }{ α( ω+ ω + ( η βh) ) ( η βh) } (A4) ω = αcot ( πa/) 3

4 3.3 Fully plastic rgim o-itractig aspritis H/ ψ < ξ < H/3 I th fully plastic rgim, th plastic strais udr th idtr gratly xcd th lastic strais, ad th strss filds approach th rigid-plastic limit (giv.g. by slip li fild solutios). Hr, it is cssary to distiguish btw two distict typs of bhavior. As log as th cotact spots ar small (umrical rsults suggst a/ λ <.67 ), thr is o itractio btw ma ighborig asprity cotacts, ad A, Π ad Π ca b stimatd from slip-li fild solutios (or umrical simulatios) for a flat puch idtig a rigid-plastic half-spac as A( ξ) = ξ / H Φ ( x) = H ( x / a< ) (A5) ma Φ = H whr H.9 is th hardss of th matrial ormalizd by its tsil yild strss. Numrical simulatios ad slip- li fild solutios show that H varis wakly with g / λ, but this variatio is isigificat for our prst purposs. Not that i this rgim th cotact prssur distributio is prfctly uiform, ad th solutio is idpdt of ψ. Fially, th prssur distributio fuctio follows as q( ηξψ,, ) = A( ξψδη, ) ( H) (A6) whr δ ( x) dots th Dirac dlta distributio. 3.4 Fully plastic rgim itractig aspritis H /3< ξ Wh th siz of th cotact spots bcoms comparabl to th wavlgth, th plastic strai filds associatd with ighborig aspritis bgi to itract. Numrical simulatios show that i this rgim, th cotact prssur distributio rmais approximatly uiform, but th magitud of th prssur icrass substatially, ad i som cass rachs twic th hardss of th matrial. This bhavior has a dramatic ifluc o th bhavior of fi roughss scals. As i Sctio 3. w hav dvisd approximat aalytical approximatios to umrical simulatios to simplify our calculatios. It is simplst to spcify th rlatioships btw ξ, ψ, A ad ma Φ paramtrically as ( ( ψ ) )( 3 ) 3 ma Φ = H + H H A (A7) ma ξ = AΦ (A8) whr H ( ) ψ is th limitig valu of th ormalizd cotact prssur px ( )/ σ Y at full cotact. Th followig fuctio givs a approximat fit to th rsults show i Fig. 8 of Gao t al (4) H 3/ ( ψ ) = 5.8 6/(6 + ψ ) (A9) It is straightforward to solv ths quatios to dtrmi th fuctios dfid i Eqs (A). Th rsults ivolv th solutio to a cubic quatio, howvr, ad so ar too lgthy to b rcordd hr. Fially, th cotact prssur distributio fuctio i th itractig asprity rgim has th sam form as that giv i Eq. (A6). 4

5 Appdix B: Numrical simulatios of plastic cotact of a trucatd Wirstrass profil A ctral assumptio that udrlis th Ciavarlla t al () modl is that succssiv scals of roughss itract oly through th rquirmt that th tru cotact prssur actig o th th scal must qual th omial cotact prssur xrtd o th +th scal, ad th dformatio filds othrwis fully dcoupl. Sait Vat s pricipl ca b usd to justify this assumptio for lastic solids. For matrials that dform plastically, howvr, thr is o rigorous basis for this approximatio ad idd thr is good raso to bliv that thr ar circumstacs whr th dformatio filds btw succssiv roughss scals do ot dcoupl. W hav thrfor coductd fiit lmt simulatios of lastic-plastic cotact btw surfacs with two or mor scals of roughss, to ivstigat th validity of this assumptio. A fw rprstativ rsults of ths simulatios will b outlid hr. Ma cotact prssur p m /σ Y λ Approximatio Asprity Asprity Asprity 3 Asprity 4 Asprity λ / Cotact fractio a /λ Fig B: Rsults of a fiit lmt simulatio of th idtatio rspos of a lastic-prfctly plastic solid with two scals of surfac roughss. Th figur shows th variatio of ma prssur with cotact fractio for ach asprity cotact o th fir scal as th load is progrssivly icrasd. Our approximatio to th bhavior of a sigl roughss scal is show for compariso. Fig B shows th bhavior of a lastic-prfctly plastic solid with two roughss scals, which is idtd by a rigid flat idtr. Rsults ar show for γ = λ/ λ = E σ Y =, g λ =, g λ = 4 ad ν =.3. Ths paramtrs corrspod to ψ =, ψ = 5, ad D =.4. W hav computd th ma cotact prssur as a fuctio of cotact siz for ach asprity i th fir roughss scal, ad compard th rsults with our approximatio to th bhavior of a surfac with a purly siusoidal roughss. For small valus of a/ λ th two cass ar idistiguishabl. For valus of a/ λ >.6 (whr aspritis o th fir scal bgi to itract), w fid that th prssur o th two-scal surfac riss somwhat mor gradually tha that for a pur siusoid, but th limitig bhavior as a/ λ is corrctly modld by a sigl-scal surfac. 5

6 Th rsults of a scod st of simulatios ar illustratd i Figs B ad B3. Hr, w compar dirctly th bhavior of surfacs with o, two ad thr scals of roughss, wh idtd by a rigid flat surfac. Rsults ar show for γ =, E σ Y =, ν =. 3, g λ =, g λ = 4, g λ = 6, which corrspod to D =. 4, ψ =., ψ = 5.5, ψ = 3.7. All thr scals of roughss ar thrfor i th fully plastic rgim for sufficitly high loads. Fig 5 shows th distributio of cotact prssur for th thr scals of roughss, for two valus of omial load. Ths rsults clarly agr qualitativly with th prdictios of our approximat modl. I particular, th computatios dmostrat th progrssiv icras i cotact fractio ad cotact prssur o succssiv scals of roughss, ad show also that asprity itractios caus th avrag cotact prssur o scals = ad to xcd th matrial hardss. Cotact prssur p/σ Y Scal Scal Scal Cotact prssur p/σ Y Scal Scal Scal Positio x/λ Positio x/λ (a) (b) Fig. B Th cotact prssur distributio prdictd by fiit lmt computatios for trucatd Wirstrass surfacs with o, two ad thr scals of roughss that ar idtd by a rigid, flat surfac. Rsults ar show for two valus of omial prssur: (a) p / σ Y =.64 (b) p / σ Y =. A mor quatitativ compariso of our umrical ad aalytical prdictios is show i Fig 6. Hr, w hav plottd th total tru ara of cotact prdictd by fiit lmt computatios for ach of th thr roughss scals, ad compard th rsults with our approximat aalytical xprssio. Th aalytical rsults clarly provid a good approximatio to th fiit lmt computatios. 6

7 Total cotact ara A /λ.8 Fiit lmt rsults Approximatio A A A Nomial cotact prssur p - /σ Y Fig B3: Compariso of th tru ara of cotact prdictd by fiit lmt simulatios with, ad 3 scals of roughss with th approximat aalytical xprssios prdictd by assumig ach scal bhavs as a isolatd siusoid. 7

8 Appdix C: Numrical computatios of prssur distributio fuctios. Elmtary calculatios show that th prssur distributio fuctio for a lastic-plastic solid will hav th form ( ) ( ) q( ξ) = q( ξ) + Qj δξ ( ξj ) (C) whr j= q dots a cotiuous fuctio, ad th sum cosists of a sris of dirct Dlta ( ) ( ) distributios with magitud Q j, which occur at ormalizd prssurs ξ j. Giv th th trm i th sris, w sk to comput th xt trm. Th gral xprssio is giv by ( ) ( ) q η = q ηξψ,, q ( ξ) (C) This itgral must b dividd ito sparat cotributios that corrspod to th lastic rgim, lastic plastic rgim, ad fully plastic rgim p ( ) = ( ) + ( ) q η q ηξψ,, q ( ξ) q ηξψ,, q ( ξ) p ξmi ξmi ξmi i ( ) ( ) + q ηξψ,, q ( ξ) + q ηξψ,, q ( ξ) i ξmi (C3) whr th limits ar ξ = η /4πψ ξ = mi( ξ, ξ ) mi ξ = ( p ( η), ξ ) ξ = mi( ξ, β ( η)) p p mi ξ = ( ξ, H / ψ ) ξ = mi( ξ, H /3) mi ξ = ( ξ, H /3) ξ = mi( ξ, H ( ψ )) i mi (C4) whr scal; ξ dots th critical valu of dimsiolss omial prssur to iduc yild i th th ξ is th imum dimsiolss omial prssur actig o th th scal; p dots th valu of ξ that satisfis q (A) with ξ = η ; β ( η) is th valu of ξ that satisfis th scod of Eq. (A) with β = η. Nxt, rcall that that Substitut ito (C) to s that q( ξη, ) = A( ξψδη, ) ( H ) ξ < ξ < ξ mi q( ξη, ) = A( ξψδη, ) ( Φ ( ξ)) ξ < ξ < ξ ma i i mi (C5) 8

9 p ( ) = ( ) + ( ) + ( ) q η q ηξψ,, q ( ξ) q ηξψ,, q ( ξ) δη ( H ) A ξψ, q ( ξ) p ξmi ξmi ξmi i i ξmi ( ) ma ( ) ma j j j j= + A ξψ, δη ( Φ ( ξ)) q ( ξ) + Q A( ξ, ψ ) δη ( Φ ( ξ, ψ )) (C6) Hc = ξmi Q A ( ξψ, ) q ( ξ) Q = Q A ( ) ( ) j j i j ma j j ( ) = ( ) + ( ) ( ) p p ξmi ξmi i i ξmi ( ξ, ψ ) ξ =Φ ( ξ, ψ ) q η q ηξψ,, q ( ξ) q ηξψ,, q ( ξ) ma + A ξψ, δη ( Φ ( ξ)) q ( ξ) (C7) 9