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Sunmm1ry-ThoproblemofanelaBticBtrippassingbetweentworo1lersi8conBidemdby meanBofanumericGLImOthod・ＡＢａｎｒＢｔＢｔｅｐｉｔｉＢａｓｓｕｍｅｄｔｈａｔｔｈｅｃontactingBurmcesmovetogetherwithoutmutua1slippingBothataBhe＆rBtrEssiBdevelopedatthecontact interf配０．Compari8onoftheseBheGuPstreBseBwiththenomnalcontactstre88eBBhowthab，ｍｇｅｎｏｍｌ，micro･B1ipmuBtoccurwithinth⑧nip，albeitconfinedtotheextrcmeedgｅｏｆ ●

contact・Ar6gimeofslippingiBBuggested・moreBultBprovidedetail8ofthecontactBtre8ses,theindentationofthoBtripbythe

roUerathecontactwidth，ａｎｄｔｈｏＢｐｅedatwhichtheBtrippaB8Gsfhroughthenipin relationtothemtationalBpeedoftheI℃lle頭．

ＮＯＴＡＴＩＯＮ

ＳｕｆＩｉｘｌ祀屈岡totheStrip,BuHix2mfbr8totheindentingmU⑧r･ａhalfcontactwidth

amhalfconmctwidthdeiinedbytheIIertz8olutioniftheBtripisinnniいｌｙｔｈｉｃｋ

Ａ＝qkUo bhalf8tripthiclmessfbrfreestrip,Btripthicknessfbrtyre B＝ｂ化

chalfLwidthoftriangular⑧lement ●

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ｊＤ(工ＬＰ(Ｘ）nomnalBurfncostrcssdiBtribution９(苑),9(Ｘ）BhearBurfnce8trC四diBtribution

pavemgecontactpr巴ssum

P⑩ｍａｍｍｕｍｎｏｒｍａｌＢｔＩもssdennedbytheHertz8olutionifthe8tripisinBnitelythick

P(Ｘ)＝ｐ(ＸＶｐ Ｑ(Ｘ)＝９(Ｘ)/何

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２ＳnumberofBtraightmne8bywhichtheBtrCs8diBtributionBaToapproximcLted Bign（）＝＋lifargumentiBpositivo，

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THJspallerdescribcsatheoreticalinvestigationofthestressesanddelhrnmtions prodｕｃｅｄｉｎａｓｔｒｉｐｏｒｓｈｅｅｔｏｆｅｌａｓｔｉｃｍａterialasitpassesthroughthenip

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ＦＩＧ･LBaSiCBtriI】con5gumtions．

betweenapairofrollers，TwodiBbrentarrangemel1t8havebeenstudied (Fig.１(a)ａｎｄ(b))．Inthenrstthestrippassesｎ℃elybetweenidenticalrolleIs lnthesecondarrangementthestripiBattachedtothesurfnceofonerollerlike atyre,acircumstancewhichiscommonlyfbumdinprocessingmachinery、Ｉ、

AnelasticBtripinplanerol1ingcontactO39

theIattercasetheanalysishasbeencon6nedtorollerswhicharerelativelV ⑪

rigidcomparedwiththeeIastictvre・ForbotharrangementsithasbeeluU

as8umedthatroIlingproceedswitholxtthetransmissionofanettractivefbrce， aIthoughthemethodofsolutionisnotrestrictedtothi8assumption

ThestripisnippedbetweentherollersbyanormalfbrceZValongtheline ofcentres、Therollersandstripareassumedtobewidesothatthedefbrmationmaybｅｔａｋｅｎｔｏｂｅｐｌａｎｅ，Ｗｅｓｅｔｏｕｔｔｏｆｉｎｄｔｈｅａｍｏｕｎｔｂｙｗｈｉｃｈtherollers indentthestrip(`),thewidthoftheregionofcontactbetweentheroUersand thestrip(2α)andthedi8tributionofstressestranBmittedacrosstheinterfnce・WhenthestressesattheinterftLcｅａｒｅｋｎｏｗｎ，thestressesanddelbrmation月

throughoutbothstripandroIIerscanbefbund、Finally，duetothecontactdefbrmation,thevelocityofthestripdiHbrs81ightIyfTomtheperipheralspeed oftheundefbrmedsurfbLceofthecontactingrollers・ThefractionaldiHbrel1ceinspeeｄｉ８ｃａｌｌＧｄｔｈｅ“oreepratio，，（：)．Itsvaluei8ofinterestandmaybe fbundfi･omtheanalysis・

Theapproachfbllowsthatusualincontactstresstheoryinwhichthc contactwidthremainsBmallcomparedwiththeradiioftherollers、WithinthisapproximationthetyreproblemshowluinFigl(b)isequivalenttoanelastic stripofthicknes8battachedtoarigidplaneindentedbyarollerofradius 〃＝E1IT2/(H1＋庇2)(Fig.1(c))．Thesimilaritybetweenthetyreandthefreestripisnowapparent，Thecontactgeometryisidentical，but，whilstthe centralplaneofthefree8trip,bysymmetry,isfreefromshcarstress,thelower surfnceofthetyre,attachedtoitsrigidbase,haSzerodispIacement、

Itisclearthatthenom1alstress(pressure)transmittedacrossthecontact interfacemustbalancethecompressiveload、Sllcarstressesattheinterfncearisethroughfriction、Athinstrip,compressedbetweentheroUers,attemptstoeIongatewithinthenip・ItfbllowsfromtheHertztheorythattherollers，

ｏｎｔｈｅｏｔｈｅｒｈｍｄ，exhibitcircumfbrenti紐lcompression・Theseopposingtendenciesofthetwocontactingsurfncesareresistedbyfriction・However,ifatanyPointtheinterfncialshearstressreacIlesalimitingvaluegivenbythe productofthecoefHcientoffiPiction仏)andthenormalstress,ｔｈｅｎ“micro‐

sIip,，ｗｉｌｌoccur、CmtactstressprobIemsinvolvingmicro-slipduringrollinghavereceivedconsiderableattentionrecentlyintlleliterature・lItmaybeaPPreciatedfromquaIitativeargumentsthat，ｗｈｅｎｔｈｅｓｔｒｉｐｉｓｔｈｍ，the stresseswithintheBtriparesensitivetoftictionalstressesactingovercontact、Itwas，therefbre，ｒｅｓｏｌｖｅｄａｔｔｈｅｏｕｔｓｅｔｔｏａｔｔｅｍｐｔｔｏdeviseamethodof analysiswhichwouldenablefiPictionalstressestobeiIDcluded．（Pmviouswork ontheproblemhadnotexaminedslipandfrictionatthecontactinterfnce.）

IfthecontactcomprisesregionsofbotllB1ipandno-slip，theboundary conditionsintheseregionsaredifTbre1lt、IntheslipregionsthesurfnceshearstressisrelatedtothenormaIstressthroughthecoeIYicientofffiction；inthc no-slipregionstherelationshipbetweenthetangentialdisplacemcntsinthe twocontactingsurfncesisprescribed・Thesecondition8havebeendiscussedin

detailpreviously2andthere8uItsarereproducedinSection2below・TheprobleminelasticitycannowbedeIined，Aninfinitestripisloaded

overafiniteregionwllerethedisplacementsareprescribed；ｔｈｅｒｅｍａｉｎｄｅｒｏｆ

藍－０７仏白び、■ⅢⅡ？⑪ⅡⅡいⅡ曰

。■０■引口－０００■ＵａＩ０曰０■■■■Ⅱ■勺●口■■→凸。■■９Ｉ！■凸■ＵｄｐＵ９Ｌ■一二ＣＤＶ■‐■■ｉ■Ｑ４９６ｐＩｑ６ｑＰＩｒ●４ＤｄｒｈＰｌｂｌｒ△■｜、！■。ｒ｜Ⅱ‐Ⅲ

■Ｉ｜■ｑ●０ｍ〃００ＣＧＯいりｏＨ４Ｊ即いら■■■■且■■■■、Ｐ１ケ■田■●■８０・ＪＤＵＳＯⅡ■ＢＰｑ６Ｂ■００勺１－．０００日Ⅱ．９４ｊ・ＦＤ０－０●－０－０Ｆ・可‐・‐‐‐●ｉｌ６ｐ

Ｉ ＮＩ

６４０ Ｒ､Ⅱ.Ｅ庇ＮＴＡＴＬｍＭｌＫ.Ｌ・JomVBoN

thesurfnceofthestripisBtre8sftee・Withafreestrip,itscentre-p1aneisoneofsymmetryBothatthenorma1displacementBandthe8hemBtre88e8therearも

Zero、WithatyreattachedtoarigidroUerbothnorma1andtangentialdispIacementsmezeroatthelowerBurfhce、ＩｔｉｓｒｅｑｕｉｒｅｄｔｏｈｎｄｔｈｅｓｔｒｃｓｓｅｓｔhroughouttheBtripand,ｉｎparticular,thoseactmgatthecontactmterface・

TheproblemcanbefbrmulatedeasilyintermBofFourierintegralB (Appendixl)fbllowingamethoddescribedbySneddC､３．１１０we▼er,obtaining numericalrcsultsfbractualca8e8hasledtorealdi価cultiesduetothemixed

natureoftheboundalycondition8andtheintractabnityoftheintegralswhich desoribethedefbrmation・Variousapproximateteohniqueshavebeentried・

WhentheBtripi8thiokcomparedwiththecontactwidth，the8olution

approache8thatlbrasemi-inlinite8oUdandapproximationsgeneraUygive rapidconvergence、Ｗｈｅｎｔｈｅｓｔｒｉｐｉｓｔｈｉｎ，itispossibletoapproximatethedefbrmationthroughthethicknessoftheBtripFeng4hasadaptedMindlin's second-orderpIatetheory5tothepresentProblem・

FormoregeneralvaluesofBtripthickneBse8theu8ualapproachhasbeento approximatethedistributionofstressatthemterfnceandtoBatiBfythe boundaIycondition8ｔｈｅｒｅｏｎｌｙａｔ＆numberofdiscretepomts、Hannah6，PariBh7andMillelsapproximatothedistributionofnormalstressbymodifying theHertzBolutionwithFouriercosinePerturbations,thus

Ⅳ

，(x)…[ルェ曇)ｗ､…獺-1);刮ｗｈｅｒｅｔｈｅＡ祁mFeRmknowncoefYicientsdeterminedbytheboundaｪycondition8・Los８鉢αJ,ＤａｎｄＣｏｎｗａｙｅＵａＪ.]ohaveapproximatedthesurfncestreBsdistribu-tionsbyelementsofunifbrm8trcssappIiedtoanumberofdiscreteintewalsof thecontactregionAgainthedispIacementboundaryconditionsa””tisned (ie・matchedbetweenroUerandstrip)atdi8cretepoint8attheinterfnce､１IOrerecentlyTulnhasobtamednumericalsolutionsfbrtheaxi8ymmetricproblems ofthefreestrip，ｗｈｅｎｔｈｅｓｔｒｉｐｉｓｎｉｐｐｅｄｂｅｔｗｅｅｎｔｗｏsmoothspheres， prwiouslyconBideredbyTuandGazisn2and,ｍｏrecompletely,ｂｙＯ,COnnorⅡ3． Tu'8fbrmofapproximatione8sentiaUycoincideswiththepresentmethodof solution・

InnoneoftheseBtudieswerotheboundaryconditionsoftangentialdiBplace‐ mentappropriatetoroUingcontactappliedtodeterminetheshearstI℃ssat theinterfnce，

Thetechniqueｕｓｅｄhere，whichisdiBcuB8edmreflandmdetailby Bentalll`，approximatesthe8urface8tressdistributionbyoverlappmgtri‐ angularelement8,charaCterizedbytheircentrcordinate8(Rn,Q伽)asshowninFig、２．TheBuperpositionOfalltheelementsresultsinapiecewise-】ineardistributionofBtressasBhownTheuseoftriangulareIementsgivesanorder ofimprovementuponunifbrmconstantBtresselementsmrepresentinga continuousdistribution･Alsoitsatisne8theconditionthatthecontactstIEsses

I繰鰯鰯、

‐１Ｆ１■０■０■ＢＰｌｇ００Ⅱ０凸■０９１０９■Ｉ■■０Ｉ０ｌ－－Ｌ■ＦＤ－．ⅡⅡｑ・口

ｌＩｌｌｌ

Ｉ１００ＩＵｄＩ０－９０ⅡⅡＫ００ｌ００ｂ■■■曰■■■■■Ⅱ■■□■ＪⅡｉ０５■■１‐■・■■■■１０ⅡⅡ７０■■■■口０‐‐１１’０１１．‐

(綴,､.、

Ｉｉ１ＩＩ－ＩＩｌ１ＩＩ１ｌｌＩｌｌ１．１１Ｉ

ＢｈｏｕｌｄｆｎＵｔｏｚｅｒｏａｔｔｈｅｅｄｇｅｓｏｆｔｈecontactregion、Thi8requirementiBnecessarysmce(a)atensUenormal8tres8isun8upportableand(b)interfbrence betweenthecontactingBurfnceBoutsidethecontactregionmustbeavoided．

AnelasticStripinplaneroUmgcontact６４１

ThediBplacementboundaryconditionsarethen8atisfiedatthepointsonthe contactmterfncewhichcoincidewiththeapexofeachstrEsseIement．

。‐今‐ｈ】。●て、□０『８同ざ■■｜■ロ用ａｑ４岾・■可！‐

ＳＴＲＥＳＳ ｡

、０．２．ApproximationtoBtr℃ｓｓｄ由tribution．

２．ＦＯＲＨＵＬＡＴＩＯＮＯＦｒＨＥＰＲＯＢＬＥＭ

(＆)Bow1cdmVcooO(ZiUio1DB

ThocontacCdefbrmationcommontobothBtripproblemBiBBhowninFi９．３．Inthe BymmetricaMreeBtripproblem,thothickneB8ofth⑧Btrip由２６．Thetyr℃problemhnsboenrcducedgeometricaUytocontactbetweenarolleraｎｄａｓｔｒｉｐ，thiclmess6，Btucktoa hB1fLSpace・IhecontactiB8howneccentricwith、８pecttothorollercentre・lines,aUowingfbrtheaSymmetryassociatｅｄｗｉｔｈＢｌｉｐ．

Ⅷ剛ⅧⅧⅧⅧⅧ緬螂州岬叩剛ⅧⅧ細川Ⅷ川棚剛洲”“”洲叩椰州Ⅷ靴馴馴岬Ⅶｈｈｈトル孔Ⅲ小川

Ｘ

ｙＯＩ ＬｌＮＥＣＦ

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ｒＩＣ、３．Contactdefbrmation．

TheboundaryconditionsdOnnmgthedelbrmationimposodbyBtcadyroningcontact a8derivedinrcflapplycquaIlyweUtothoBtrippmblｅｍｓｅｘｃｅｐｔｔｈａｔｈｏｍｗＧｃｏｎＢｉｄｅｒ djBPlacementB(凹,ｕ)relativeto(0,0)andamthusabletodeterminethepenctrationoftheBtripbytheroller･

Ｆｉ函t,ｗｅｈａｖｅｔｈａｔ

：lfl二:”'夢！山wherGp(Ｘ)ａｎｄ｡(Ｘ)arethe8tressesactingonthe8urfnceorthestrip,ａｎｄｗｈｅｍＸ＝勿/α・

mBidecontact,thenormaldiBplacements(Ｕ)arodet0rminedbythepenotrntionofthe roUer(｡)anditsundefbrmedpronle,hence

H+昭一､+Ｘｅ;-x鈴 （２）

Sub8criPtSland2 whereH,JB＝Ｕ,ノロ,tlg〃,、＝。〃,and“由theeccentricityorcontact・rBfbrtotheBtripandroUerrCspectively．

642皿.Ｈ･ＢｍｍＡＬＬａｎｄＫ.Ｌ･ＪｏｍｑｓｏＮ

ＡＢｔｈｅｔｗｏ８ｕｒｆｎｃｅｓｍｌｌｔｏｇ⑧ther，thocontactiB，ingenerLl，dividcdintorcgionBof no･B1ipandmicro･Ｂｌｉｐ

ｌｎｎｏ･BliprGgionB,thodiflbrCnceinBtIPainBofthetwosurlncesrcmaingataconBtant value(ど)detGrminedbytheirrelativevelooitiesJland尾．

Hence,fbrno･ｓｌｉｐ Ｕ１－Ｕ画＝：Ｘ＋０（３）

WhelもＵ,,Ｕ２＝“､〃,叱位,ｆ＝(奥一喝)/ＦａｎdOdenote8thocumulativOBliPfmmtheorigintothepomtinque8tion．

Inadditiontheshear8tr巳ssiB1imitedbythecoelYicientoffriction,ｔｈｕｓＪ１

(4) 19(Ｘ)|≦lLlp(Ｘ)１

Ifmicro-BlipoccurU，theBhearBtressattheBurfncewillm⑥chitB1imitingvalueina dircctionBuchthatthefrictionfbrCeopposesthGBlip・TheTclativevOIocity(〃)ｂｅｔｗｅｅｎｔｈｅｔｗｏＢｕｒｍｃｅＢｉｓｇｉｖｅｎｂｙｔhoequation

豐臺`+儂一叢）に’,撫爾、

Hence,inaBliPrCgionDworequim

Q(Ｘ)＝Big、(〃)”(Ｘ）

whcreBign<△/)＝＋1,-lfbrムノpositive,negativere8pectively．

(6)

(b）DC7i…ioDLq'Zhemu8CJbi”eql“imo8 TheboundaryconditionsofdiBplacemente(equations(2)ａｎｄ(3))willbeBatis6ed(i､⑧．

ｍLLchedbetweenrollerandBtrip)atanumberofdi8cretepointsonthointerfbucc・Thoderivationofthemmtchingequationsfbrthefrco8tripprobleｍｏｎｌｙｗｍｂｅｐｒｃｓｅｎｔｅｄ ｈ０rG；thosefbrthetyrUproblemmaybeobtainedmthesameway・

ThegurfncediBplacementBofafYe⑧Btrip(rclativetoエ'＝０，Ｊ＝Ｏ)duetoevenapp1iedBurfncestr巴sgdistributionsP(⑰'）GLndg(〃')ｈａｖｅｂ⑧enderivedinAppendiｴﾋﾟ１（equationB(Ａ､6)-(Ａ､9))．Ｗｉｔｈｔｈｅ８ｅｅ｡Epre8BionBwecandefinetheinfluenceofanindividualBtreBs eleme､ｔａｔ垂＝勿向ondi8placemeHutat⑰＝Ｔｍ,Ｆｉｇ．２．ThuB,⑰'beingmeHLBuredlromaFn，thenormaldisplacemcntat⑰'is

oF-鶚IillF(.`:鶚塗J’１．１｡…砦

-21鶚｣川1-…('-,`)[蔬::毎l)小)…等（ｱIForatriangularstresselement(equation(Ａ､14))ofbasewidth2c,ｐ(α)ｉｓｇｉｖｅｎｂｙ

ｐ(α)-警｡in圏(｡．/2）（８）Wenowdefinethenon-dimenBionaltｅｒｍ８；

β＝麩６，Ｂ＝&ぬｓ＝．/c，ｚ＝c/４６＝1/4ＳＥ（９）

wherCSchamcterizesthedegrceorrBfinementoftheapproximation･nneBtmssdistribu．

１１１ し

tioniBmddOupof2S1inearmcrements･ Notingthat年'=c(nD-”)wheremandooamintegeraweobtainthonormaldiBPlace-

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IARIm-"］ａｎｄＩＥ几[ｍ－，０］rep…entthecomparablod垣p1acemontBofLhalr､Bpace，relativOto(工'＝0,J＝－６)andJA[伽_犯]andIE[、－，z]rCpI℃sentthemodincationtothatrelativediBplacementduetot】uennitethicknes8ofthestrip・LmrepresontBtheabsolutedeHOxｉｏｎａｔＺ`＝0,Ｊ＝－６．

TheintegmndsorI,０，Ｊ｣[p】ａｎ〔lIE[j､]tqketheIbrmofo8cillationswithinadocayingenvOlope・Asthewidtho｢thestI℃sselementcompamdwiththethicknes3ofthe8trip(z）increaseBpsodoestheIrequencyoftheoBcillation，worBoningtheconvergenceofthe integraLDimculticsofconvergencｏｃａｎｂｅｏｖｅｒＣｏｍｅｂｙｅｍｐｌｏｙｍｇａtechniquodesc『ibedmAppendix3andintegmtionsprcsentfbwproblemstoLmodernhigh-Bpeedcomputer･ ReaBonableandconB鱈tentrcsuItshavebeenobtainedｄｏｗｎｔｏｔｌＩｉｃｌｍｃ５ｓｅｓｏｆＢ＝ＯＯＯ１ｕＢｉｎｇＳ＝２０．

Inasimilarwayweobtainthotang･mtiald由placementsatamatchingpointX＝WSduotoa8tmsse】cmentaｔＸ＝'0/８．Relativeto(0,0)theseａｒｅ

ひ`…膳i'１Ｊ`[''1-,Ｗ`[凧､＋ 4(l＋Ｐｕ)(1-2"】）

蓬umM"`-"]+恥廠[蝿])］￣￣．.Ｌ万必，｡￣ヨーニーj,．斤囮，-,-jJnL…

＋9阿Ｂ２(L二ﾋﾟｺ(('"["１－，０]－１⑩])＋4z(I“[,"-"]－１』jw[,､])）７TEI who”

(12）

”[`鰯-鯛]-=r(1-誌辮)……-鰍'率：’1`）FollowingHertz，theroUerdisplacement8amtho8cofahalfspace･Theycanbo

obtainodeithorfromfi応tprinciples,”『.（１）,orfromeqlMLtion8（１０)ａｎｄ（12)asthestripthicknessB→ＣＯ．

Relativeto(０，－６>theroUer《li8placementBduotop励,９，mmre,Fig.２，

１１

'…驫鶚尖('"[純一耐]-J"Ｍ１+，鳳些LL空』き('"[…]+虹"M)汀Ｂ２ａｎｄ

U曇－，銅(…ル2いき('厨､[噸-鑓]+恥鳳[総]１－，薊砦巽(j釧廠[,腿-”]-z"M)万Ｅ２

Ｕ４１

Equationg（１０),（12)ａｎｄ（14)givethodi8placementBduetosingIＣｓn℃sselementB・Todcscribethedefbrmationa8awhole，thecontTibutionofeKLchelemeｎｔｉｎｔｈｅＢｔｒＣ８ｓ

ヨ

n.Ｈ･ＢＥＮＴＡＬＬａｎｄＫ．Ｌ,JoImsox ６４４

地唾⑭dmtributionmmuｮｔｂｏｃｏｎｓｉｄｏ唾｡、ThomatChingequatjonBarBobtamｅｄｗｈｅｎ孟鱗HiHi戯､h5hmJ5idinlgP…｡…bsｻﾞPHlofPnP99B聖iRF(2)"｡(3)．

IftheBtripi8infinitolythick,thecontcLctwidth,２Cl四,泊givら、ｂｙ

｡:一半(等十署)-:…(等十署） (15）

wherePiBthemeanpressuuPBN/”． ＤｅｍｉｎｇＡ＝αノロロ,thematchingequationfbrnomaldispIacemcnt8become8

。｣翼L幽聡|圃膨(Ⅱ"+'』[,鰯-"])+き…-"]-匹"Ｍ１+:…-鰯]｝‐燗｣蔦Lm9(固…-“]一念(聡風[…]÷`"[輝ﾙ鳥豐ｚＥ風[総]｝

＝幽劉+竿一等a

qndfbrtangentialdi8plaCementsinaregionofno-slip

oA鷺Lu展|…[瀬‐…回)-:(Ⅱ,…]+n厘mＭ１｝

＋鯛｣覚LIP蝿I…[…]-ⅢＤＭＩ÷き(z"[…]-Ｊ"[風])｝‐2鍔…･；』。

ｗｈｅｍ

(16）

'1鰯､、

(17）

L

脆'＝丁干Ｚ

雌‐署/署 (18）

臆-P'竺豊二2塗'-11生理豊二塾']/(豐十署）２，＝ハノヮ，Ｑ鰯＝ｑ凧ﾉグ

ToobtainaBolutionfbrthevaluesofBurfHce8tr巴sscs凪､andQnthematchingequation(16>muBbbesatiBHedthroughoutthecontactrogion,thatisat(２Ｓ＋1)point８．Ｉｎａｎｏ・BliprcgionthesecondmatchingeqUation（17）mugtbeBati86edBimultaneouBly,ｗｉｔｈｔｈｅ 型ditionnlconditionthat

qDﾐﾉﾙＲｕ（19）

nlaBlipregiontheaB8umeddimctionofBlipmustbecDnBistentwith

Q同＝Big、(△/)lLB、（20）

,`總恩、

' AtboIdersbetweenBlipnndno､BlipregionB,equationB（19)ａｎｄ(20)ambo6happ1ied・

Finally，fbrequilibrium,thenormalprcBBurBmuBtbalancothe1oadowhenco （Ｓ－１Ｉ

ＺＩﾖ､＝－２８ ，－－(Ｓ－ﾕ》

(21）

ApplyingthematChmgeqUationB，andtheequnibriumequation，ｐｒｏｖｉｄｅｓａｓｅｔｏｆ ４Ｓ＋３BimultaneouBeqUGLtionB・SolutionoftheseequationsprovidesvalueBfbr(２８－１)Ｂ,，

￣

645 AncIasticstripinplnneroUingcontacb

(2S－１)ｏｎ,、,e,Ａ,どand０.IfmorctlMmoncno-BIipregioni8involvedinthedefbrmation，additionalunlmowns5'andＣ',ど"andc",etc・omuBtbointmduced,andasolutionmuBtbesoughtwhereﾋﾟｰｾﾞ'＝ど",etc．

(c）GouejPm打gpammcte7u/b『仇eルee8IｱｺﾞPExaminBtionofequations（16)，（17)，〈20)ａｎｄ（21）ghowBth醜the8olutionBdepend

upontheratioofBtripthicknesstocontactwidtｈａｎｄｕｐｏｎｔｈｅｎｏｎ･dimensionalmaterial pammeter8比，ＫａｎｄＰ１･InadditiontheextentoftheBIipregionBdependBuponthecoefYicieｎｔｏｆ金ictionlL．

IｔｍａｙｂｅｓｈｏｗＴ１ｔｈａｔｔｈｅＢｔｒｅｓｓｄｉBtributionBaredepondenton〃,KandlLLonly；theabsolutevalueo｢"nonlyinnuencesthevalueoftheindentationD･rnrther,Ｐ,onlyappeaｴ洵aBanindependantpammeterinthetermiIwolvingQ伽inequation（１６),andnotatallinequation(17)．Theinnu⑧nceoftheshear8trBssQ回uponthenomma1stms8BQisnotverygreat,aBBuner，sworkl5indicGLtes・ILaBiscommonincontactBt唾8theory,weneglectit，theterminquestiondiBappeBIB･WOmcLyBtatethat,therefbre,tocLclosGapproximation Che801uCionstothematChingoquationsarefimction8offburindOpendentpGLrfLmeterBD B,ル,虻ａｎｄ脾．

３．ＲＥＳＵＬＴＳＯＦＴＨＥＣＯｎｌＰＵＴＡＴＩＯＮＳ（noSlip）

ThemethodofanalyBi8developedinthepmvious8ectioniB,Ｂｏｍｒ,perfbctlygenemU･ ThecoexistenceofBlipandno-B1ipmguonBcanbehandled；theinterfnco…ytrnnBmita tractivefbrUevaryingfiomlimitingfrictionwhencomp1eteBIipoccmBinonGdirもctiontocomplete81ipintheother･

Ｈｏｗwer,thecomputationBpr℃sentcdinthiBpaperamrG3trictedtoooIteeroUing，，,ｉ､e・whonthe”iBnorcsultamttangentialfb配etranBmittedbythecontact・Intheseclrcum‐BtancesitiggenercUIlytruethaMbrpracticalvalue8ofthecce団cientoffrictiontheextentoftho81iprcgion8isnotverygmat．Forthi8rea8onthecomputationhaBconc⑧ntⅡmedon solutionsfbrnoBlipanywh…(lL÷ＣＯ)．Thetmldencyto8Uipcanthenbesppreciatedby examiningtheratioofBheartonormalBtI℃ＳＳＱ(Ｘ)/p(Ｘ)Boobtained・Theinfluencoofs1ip垣conRid⑨【℃dinthenextBectiolL

Cmnputationshavebeencarriedoutoverarangeofstripthicknes8eB(Ｂ)．Whenthe stripisthick(Ｂ→｡｡)iCbecome8anelastichalfgp“ofbrwhichBolutionsinclosedfbrm exist・ForvBrythinBtripB（Ｂ→Ｏ>approximateBimpIeBolutionsmaybefbundinthemanneriⅡndicatedmAppendix2・Thocomputedm8ultBmUbetweenthesｅｂｗｏｅｘｔｍｍｅｓ．

ｌｌｌｌ

ＩｉＩ，１１１１１

：卜’１１１１，１１１‐１－。‐

（&）ｍｅ介cc8ケ中ｐｍｂｌｅｍ

(1)E9umJclaBliCco加8ｍ"６８．ルー1,Ｋ＝0．（1,,＝0.3）

Ifstripandrollerhavethesameela8ticcom3tants,ｔｈｅlimitingSolutionfbraninfinitely thickstripbecomesthatofHertz；

p(Ｘ)＝ルイ(ｌ－Ｘｇ）

９(Ｘ）＝0

2,②＝－２Ｎ/汀α－

．選-.:=竿儒+響）ｄルー｡｡

鼻－．＝鼻一＝0

０

(22）

１１

IⅡ

IntheotheTlimitingBolutionoraninnnite8imaUythinBtrip，althoughBhearBtresses amdevelopedduetoami8matchintangentialdisplacement8，ｔｈｅｙａ“vani8hingly BmaII，ＢｏｔｈａｔｔｈｅＢｕｒｆｎ幻econmct8tTCssesaro昭alnessontiaUydOfinedbytheHertzBOlution･Ｔｈ画thinBtripsolutionisdevelopedinAppendix2、

４２

Ｒ､Ｈ･ＥＥｗＴＡＺＬａｎｄＫ．Ｌ･JoIKJIsoN ６４６

ThIn民

鱒Ｉｘ１臺弐命｛u-x圏’

@(x１－觜【…'十剛鎧Ｘ

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α圏＝竺匹l二二；７ＴＥＢ (23）

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４_｡＝[ん＋K(1＋化)]ｂ化庇

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ThenumericalresultBgivSninFi9.4BhowthevariationoftheBurfncocontactstⅢも目已esG

wiChstmpthiCImess,ＰｌｏｔｔｅｄｉｎｔｅｒｍＢｏｆＰの。ＷｅＢｅｅｔｈａｔ,⑩ｒｎｏｒｍａｌＢｔｚもsseB,themaUorpartoftmnBitionbetweenthickandthin8tripbehcwiourliｅｓｉｎｔｈｅｒａｎｇｅ２＞Ｂ＞0.1, whenthestripthiclmes8iBtheBamｅｏｒｄｅｒｏｒmagnitudoaBthecontactwidth･TheBhear BtrcsBes，however，ａppm⑥chtheirlowerlimibmoreB1owly，ｂｅｍｇｍｏＢｔｄｅｖＯｌｏｐｅｄａｔａ ｔｈ６ｎＬｍＰ２写ｏｆａｂｏｕｔＢ＝0.3.

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ＦＩＣ､４．StripandroI1erofequalela8ticconstant8ルー１，Ｋ＝0．（＆)Stressdi8tributiong；(b)Stressratio・

Ap1otoftheratioofBheartonomnalBurlnceBtress,Fig.４(bLteBteChetendencyfbr B1iptooccur･Iheratioapproache8amaximumclosetotheedgeofcontactandadiHiculty arisesfromtheinGUbnityofthenuⅡnericaltechniquBtoresolvethevarmtionBinBtr℃ssratio

－●ＣＧ～ －－

■□‐□Ⅱ日山Ｏｂ０ＰＪ

Ａｎｅ】nsticstripinpIanemUingcontuctO47

iluthc唾giono「highstI℃ssgradicntclosetothecdge･ItiBnotclearfmmthenumerical

anmysifBwhetherthestre&ﾖratiofmUstozoToorriBeBtoinnnityatX＝±１．Toanswerthis qucstionwoconsidertheBingulariWattheedgGofcontact･Whatcvertheactualthickness o「thestrip，ｉｔｗｉｌｌａｐｐｅａｒｔｏｂｅｔhickwhenconBideringthoBtateof8tressontheBurfnceBulYicientlyclosetotheedgeofcontact･Weconclude，ｔｈｅ”fb麺，thattheBinglllarityatjY＝±ｌｗｉｌｌｂｅｔｌｈｏｓａｍｅａ８１ｂｒａｔｈｉｃｋＢｔｒｉｐＤｉ.e･IbranelaBtichalfLBpnca

Withequalel四ticconBtant8，thehalfspacesolutiongivesm8etoｎｏｓｈｅａｒＢｔＩ℃&ｮ．Inthisca8eweexPoctthoBtres8ratiotolnlltozeroattheedge四indicatedinFig､４(b)．エtigthusnotunrCnBonabletoI℃ｇａｒｄｔｈｅｖａｌｕｅｏｆｔｈｅＢｔｒｅｓＳｍｔｉｏｇivenbythenumerical methodinthelastintorvala81LroughapproximationtothetruenMLximumvaluethroughout th⑧contact、ThusDfmmthOcurvefbrB＝0.75,themaximumvalueofq(Ｘ)/p(Ｘ)＝0.09,ｓｏｔｈａｔａｖａｌｕｅｏ｢lLgTcaterthanthiswouldpreventBlipentircly･

Thecms8eBinFig､４(＆)Bhowth⑧shearstrEssegpIもdictedbythinstriptheory(equation(23))ｆｂｒＢ＝0.025．TheseBtresseSagmewellwiththenum0ricalIもsultBoverthecentrallxwtofcontact，ｂｕｔｎｏｔｔｏｗａｒｄＢｔｈｅｅｄｇｅｗｈｅＩ℃thinBtTipaBsumptionBbecomeless aPWop｢iBte、

ThevariqtionBofmllingcI℃ｅｐ０ｃｏｎｔａｃｔｗｉｄｔｈａｎｄｉｎ〔1entGLtionwithBtlpipthickneBBa唾BhowninFig5．

！ⅧｉⅧ！ⅢＩⅧＩｉＩｌｉＩ１

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ＦＩＣ､５． tStripandrollerofequale】asticconBtants･ルー１，Ｋ＝0,シ，＝0.3．VaFiHLLionof5,DandAwithstripthickness．

(2)UDWuJ“eku8zjCcmwmoね．ルー3.0,Ｋ＝-0.286．（〃u＝0.3）Thematerialpammotc耐ｃｈｏ８ｃｍｆｂｒｔｈｉ８ｅｘａｍｐｌｏｒｃｐＪ℃sentcontactbetwccn8teel

roller臼andaluminium8trip・Ｔｈｅ配sultsaIもdisplayedintheBamewayinFigB･Ｏａｎｄ７・SincothoeIasticconstantBarediflcrentinrollerandBlip,theshearBtrCsBattheBurfkLce

nolongerwhnisheswhenthestripisvorythick･InBteadthestuでssesapproachasolutionduetoBuIlorl5fbrdiBsimiIqrelaBticro1leIB．

ＴｈｕｓｗｈｅｎＢ→CＯ

2pのＪｕ－ｘ２）

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－２ｱｰｲ(Ｉ－Ｘ２） 9(Ｘ）

１ ８１，

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鼻_．＝0.457Kα/丘

４－．－急１，(聟)=÷｡/歴

(24）

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〈a）（b）FIG､６．Stripandmll⑧rofunequalelaBticco噸tants･ルー３，Ｋ＝-0.286．

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[二。こ＝二|鰯…｡一ＦＩＧ､７．StripandrollerofunequalelasticconstantB･ルー３０Ｋ＝-0.286．

VariatiOnorf,DandAwithstripthickn…

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will 649 Auuelasticstripinplanorollingcontact

ｗｈｅｍ

β-鳶'､(譜）“ｄｗｈｅｍｕ四and必arogivOninequation<22)．

TheotherlimitingBolutiom,fbrinnnitesimallythinstrips，isgivenbyequation（23)． BGcGLusoK(equuLtion(22))isnegative,theBheGLrBtT℃ssactingontheBnrfnceoftheBtrip

change8dhもctiona8thoBtripthicknessdecreases･ThiBi8clearlydemonBtratedinFi９．６，ｗｈｅｍｗｅ８ｅｅｔｈａｔｔｈｅＢｈｅａｒｓｔｒｅＢｓｎ画tchangesBigmatthecentrBofcontact･ThehalfBpacosolutionfbrbodieswhosoelasticconstantearelmequalgiveBaｒａｔｉｏｏｆｑ(Ｘ)ｔｏ ｐ(Ｘ)whichbecomesinlmiteaCtheedgOofcontact，Wemayexpect,thorDfbrもthattheplotoftheBtrc8Bratio,Fig.６(b),willalBoh＆vOasingu1arityGLtX＝±１，andthat8omesUp iBinovitablewhatｅｖｅｒｔｈｅｔｈｉｃｌｍｅ８ｓｏｆｔｈｅｓｔｒｉｐ･

Thovariationsofcontactwidth，indentationandcreopwithBtｒｉｐｔｈｉｃｋｎｅｓｓａｒｅ８ｈｏｗｎ

ｉｎＦｉ9．７．

(3)ＡｄｾﾞﾉbPwOub妙Zripb卸…Pure”i鯵cJy殉jUroUUCmThiBBituationiBcommonintheproces8ingindustuPioBwhoretheexistenceofBhear

Btrcssandnnicro-BIipexistingwithinthenipareessentialtotheBurfEcennighoftheproduct， refl6･ItiBconvenientto8ubdividetheproblemintotwocategories,ｄｅpendimgonthe compI℃ssibilityofthe8trip．

（i”＜0.5：IfthBBtripiBverythick,thesurfncestrBssesaregivenbycquMion（24鵬theBhear8trCssesactoutwaⅡｄＢｏｎｔｈｅＢｕｒｆｍ心ｅｏｆｔｈｅｓｔｒｉｐ･Atdecrea且ingvaluesoTthe

Btripthiclmess,thenormal8tmBsbecomesmorcconcentrated･The8hearBtr℃sschanges direction,ＥⅡ召ｔａｂｔｈｅｃｃｎｔＩ②ofcontact,Ｂｏｔｈａｔ，fbrverythinBtTips，itisprcdominantlyinwardacting(ＣｆFig.６)．IfcoInpleteroUerrigidityiBas8umedHLthinBtripsolutioni8then appro“ｈｅｄｗｈｉｃｈｉｓｂｎｓｅｄｏｎａｐｑ｢abolicdistributionofnormalBtress。

p(Ｘｙｐ＝－３(ｌ－Ｘ２）

,Ｉｘ１ﾉｶﾞｰ豊国璽

ｶﾞｰ急島

．~…子ルーα/2況

鼻-．=－０"台論ソⅡα

鼻--=-－－２(ｌ－ｙＩ）ＢＩＦ

(25）

However，ｉｆｔｈｅｓｔｒｉｐｉＢｖｅｒｙｔｈｉｎｉｔｂｅｃｏｍｅsuntenGbblotonBsumothatthemUers 西mminumdefbrmed･Ｉｎｔｈｅｌｉｍｉｔ，whateverthemtioofelasticmoduli，thedefbrmatiom

oftherollerm四tprGdominateoverthatoftheBtripThenormalBtrcssdistributionwiⅡthenbecomoHoTtzianandCheBolutionwillbetlMLtgivenbyequations(23)．

ThismiBesthequestionofthepointatwhichtl1eagsumptionorrigidrollerBBhouldbe abandoncd、Simpleanaly8iBofmLthinBtripbetweenela8ticrolle函(Ｅ忠勇〃､)givesthcratiobetweenBtripandmuerdeflexionBa8

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MicTo,81iphasamarkedinnuenceonthestateｏｆｇｔｍｍｉｎｔｈｅｓｔｒｉｐｗｈｉｃｈｉｓｍａｎｉ化stbythoc…ｐratio：、Valuesor：h8webeencomputodfbrnosIipい＝｡◎ハｕｎ”strictedBlip中->O)and,intheparticuIarexample,quotedfbrl4=0.1.Rca8onableinteTpolationBbouldbepoBBibleinFig８．５，７，９andl4notingthatpIncticalcircumstancesamlikelyto lieclossrtocompleteadhe8ionthGLntocomploteBlip・

Micro-sliponlyinfluence8theBizeofthecontactarEa(Ａ）andthepenetHntion(Ｄ)toa BecondordelP･AJ1exceptiomtothiBBtqtementoccurBwhemathinincompressiblOstrip pa日Ｂｅ８ｂ⑧twecnIdativelyrigidmⅡemB,ＢｅｅFig.０.Ｉｎｔｈｉｓｃａｓｃｔｈｅ唾ｉｓ＆ImgcintemFLctionbetweonthesheaＴｓｔＩもssandnormalstmssattheintermce．

５．ＣＯＮＣＬＵＳＩＯＮ

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tiondiHbrsfiPompreviousworkinthatproperaccounthasbeentakenofthe shearstressesdevelopedbyfi｢ictionattheinte㎡ncebetweenthestripandtherolIe頤．

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particularlylbrthinstrips,thestresseswithinthestripsandtherateatwhich thestripfbedsthroughtherolIs,ｉ､e・ｔｈｅ‘`creep,,,arecriticalIvaIYｂｃｔｅｄｂｙｔｈｅ￣

臼'】｢filceshear・

DependingupontheelasticpropertiesoftheroIIerandstripmaterialｓａｎｄ ｔｈｅｃｏｅ缶cientoffiPiction，somemicro-slipmaytakeplaceattheinterfnoe・Ｉｆ

ｔｈｅｍＩｌｅｒａｎｄｓｔｒｉｐｍａにriaIsarethesame，ａｓufYicientIyhighcoeHicientofftictionwouldprevent81ipemtirely，butwithdifYbrentmaterials，Ｋ≠０，ｓｏｍｅ 8】ipisinevitable・Tllepattemofslipissimilarinmostcases；tl1rees】ipregionsexist,ｏｎｅａｔｅａｃｈｅｄｇｅｏｆｔｈｅｃｏｎｔａｃｔａｎｄａｔhird,inwhichtheslipdirectionis rever8ed，placedtowardstherearofthecontact・ForllighvaluesoflLthemicm-slipregionsaresｍａｌＩａｎｄｃｌｏｓｅｔｏｔｈｅｅｄｇｅｓ・ＡｓｌＬｉｓｒｅｄｕｃｃｄｔｈｅｃｅｎｔｒｅ

８１ｉｐｒｅgIonsprCadsacrossthecontactuntilasvmmetricalpatternofth頤ｅ●1

1℃gionsisfbrmedasshowninFi9.15(b)． WithmetalstripsincontactwithmetalrolIs,underconditionsofboundary

lubricatio､，ｔｈｅｖａｌｕｅｏｆ脾ｗｉｌｌｂｅabout0.1．Theextentofmicro-slipinthiscaBei88howninFi9.15(a).Itisapprecinblebutnotsolargethatanestimateof thestrcssesbaseduponnoslip(i､e､infinitefriction)wouldbegreatlyinerror．

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（1966)．

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