Numerical study of wheel-rail impact contact solutions at an insulated rail joint

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Delft University of Technology

Numerical study of wheel-rail impact contact solutions at an insulated rail joint

Yang, Zhen; Boogaard, Anthonie; Wei, Zilong; Liu, Jinzhao; Dollevoet, Rolf; Li, Zili

DOI

10.1016/j.ijmecsci.2018.02.025

Publication date

2018

Document Version

Final published version

Published in

International Journal of Mechanical Sciences

Citation (APA)

Yang, Z., Boogaard, A., Wei, Z., Liu, J., Dollevoet, R., & Li, Z. (2018). Numerical study of wheel-rail impact

contact solutions at an insulated rail joint. International Journal of Mechanical Sciences, 138–139, 310-322.

https://doi.org/10.1016/j.ijmecsci.2018.02.025

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International Journal of Mechanical Sciences 138–139 (2018) 310–322

ContentslistsavailableatScienceDirect

International

Journal

of

Mechanical

Sciences

journalhomepage:www.elsevier.com/locate/ijmecsci

Numerical

study

of

wheel-rail

impact

contact

solutions

at

an

insulated

rail

joint

Zhen

Yang

a,b

_,

_Anthonie

_Boogaard

a

_,

_Zilong

_Wei

a

_,

_Jinzhao

_Liu

c

_,

_Rolf

_Dollevoet

a

_,

_Zili

_Li

a,∗

a Delft University of Technology, Section of Railway Engineering, Stevinweg 1, Delft, 2628 CN, the Netherlands b MOE Key Laboratory of High-Speed Railway Engineering, Southwest Jiaotong University, Chengdu, Sichuan, China c Infrastructure Inspection Research Institute, China Academy of Railway Sciences, Beijing, China

a

r

t

i

c

l

e

i

n

f

o

Keywords:

Insulated rail joint (IRJ) Explicit FEM

Wheel-rail impact contact Transient solution Wave

a

b

s

t

r

a

c

t

Thispaperpresentsananalysisofthetransientcontactsolutionsofwheel-railfrictionalrollingimpactscalculated byanexplicitfiniteelementmodelofthewheel-insulatedrailjoint(IRJ)dynamicinteraction.Theabilityofthe modeltosimulatethedynamicbehaviorofanIRJhasbeenvalidatedagainstacomprehensivefieldmeasurement inarecentpaper(Yangetal.,2018).Inadditiontothemeasuredrailheadgeometryandbi-linearelastoplastic materialmodelusedinYangetal.(2018),thisstudyadoptsanominalrailheadgeometryandanelasticmaterial modelforthesimulationstoprovideanoverallunderstandingofthetransientcontactbehaviorofwheel-IRJ impacts.Eachsimulationcalculatestheevolutionofthecontactpatcharea,stressmagnitudeanddirection, micro-slipdistribution,andrailheadnodalvibrationvelocityinthevicinityofthejointduringthewheel-IRJ impacts.Thesimulationsapplysmallcomputationalandoutputtimestepstocapturethehigh-frequencydynamic effectsatthewheel-IRJimpactcontact.Regularwavepatternsthatindicatewavegeneration,propagationand reflectionareproducedbythesimulations;thishasrarelybeenreportedinpreviousresearch.Thesimulated wavesreflectcontinuumvibrationsexcitedbywheel-railfrictionalrollingandindicatethatthesimulatedimpact contactsolutionsarereliable.

1. Introduction

Arecentpaper[1]establishedanexplicitﬁniteelement(FE) wheel-insulated railjoint (IRJ) dynamic interactionmodeltocalculate the high-frequencyimpactvibrationandnoisegeneratedbyatypicalIRJ intheDutchrailwaynetwork.Thedynamicbehaviorofthewheel-IRJ systemreproduced bytheexplicitFE modelwas validatedagainsta comprehensivehammertestandapass-bymeasurement.Thispaper ap-pliesthevalidateddynamicinteractionmodeltoinvestigatethecontact characteristicsofwheel-IRJimpacts.

Because thefiniteelement method (FEM) is ableto handle non-linearmaterialpropertiesandarbitrarydiscontinuouscontact geome-tries,ithasbeen widely appliedtostudywheel-railcontactandthe subsequenttrackdeteriorationatIRJs.A3DFEanalysisperformedby ChenandKaung[2]indicatedthatthetraditionalHertziancontact the-ory[3]isinadequateforpredictingwheel-railcontactpressure distri-butionsaroundIRJs.ChenandChen[4]establisheda2DFEmodelto studytheeffectsofanIRJonthewheel-railcontactstressdistributions underpartialslipconditionsandsuggestedthatCarter’ theory [5]is nolongereffectiveforpredictingthetangentialstressdistributionsof

∗_{Corresponding}_author.

E-mailaddress:Z.Li@tudelft.nl(Z.Li).

wheel-railcontactatIRJs.Wenetal.[6]appliedanexplicitFEMfor thecontact-impactstressanalysisatrailjointregions,andthismodel canaccountfordynamiceffects.TheexplicitFEmodelwasthen devel-opedbyCaietal.[7]tocalculatethedynamicimpactforce,stresses andstrainsimposedwhenawheelpassesanIRJwithaheight differ-ence.SandstromandEkberg[8]employeda3DelastoplasticFEMmodel topredicttheplasticdeformationandfatigueresultingfromwheel-IRJ impacts bycapturingtheaccumulationof plasticstrain. Mandaland Dhanasekar[9]proposeda sub-modelingFEstrategy toexamine the ratchetingfailureofIRJs,andthesamestrategywasadoptedby Man-dal[10]tostudytheinfluencesofend-postmaterialsonrailhead dete-riorationatIRJs.Zongetal.[11]appliedanimplicit-explicitFEmodel tosimulaterail/wheel dynamiccontactimpact andrailhead damage inthevicinityofIRJ.ZongandDhanasekar[12,13]employedgenetic algorithmscoupledwithFEMstoreduceimpactstressthroughshape optimizationofrailheadatjoints.Onthebasisofthecoupledgenetic algorithmandaparametricFEM,ZongandDhanasekar[14]also devel-opedanewdesignofIRJ;thedesignreliedonembeddinggappedrails withinonesleepertoprovidesufficientrigiditytotherailends,andmay eliminateanumberofcomponentsofIRJandtheirassociatedmodesof failure.

https://doi.org/10.1016/j.ijmecsci.2018.02.025

Received 14 November 2017; Received in revised form 26 January 2018; Accepted 10 February 2018 Available online 12 February 2018

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Althoughmanyresearchersbelievethatdynamiceffectsplaycertain rolesduringwheel-railimpacts[4,6–8,10,15,16],quasi-static wheel-railcontactisstillgenerallyassumedintheFEmodels[2,4,8–10]. Tran-sientsolutionsthatcanreflectdynamiceffectsinthewheel-railimpact contacthaverarelybeenstudied,whichmaybebecausewell-accepted methods[3,5,17–19]capableofresolvingcommonwheel-railcontact problemsaregenerallybased onthequasi-staticcontactassumption, andthedynamiceffectsduringcontactarethereforenotnecessarily con-sideredinmanysituations.Furthermore,nodirectexperimental meth-odsforaccuratelymeasuringthetransientcontactsolutionsare avail-able[20];therefore,althoughthetransientcontactsolutionshave al-readybeencalculated,accurateexperimentalvalidationscannotbe per-formed.

Reasonable transient wheel-rail contact solutions have been ob-tainedbyexplicitFEMsandusedtostudythecompression-shift-rolling contact[21],non-steady-statetransitionfromsingle-pointtotwo-point contact[22],anddynamiccrackingbehavior[20].Thisstudyalso cal-culatesthetransientsolutionsofwheel-IRJimpactcontactbyemploying anexplicitFEM.BecausetheexplicitFEMfullycouplesthecalculationof wheel/raildynamicresponseswiththecalculationofwheel-railcontact, thevalidityofthetransientimpactcontactsolutionssimulatedinthis papermaybeconﬁrmedbyseparatelyvalidatingthequasi-steady con-tactsolutionsandwheel/raildynamicresponses.Quasi-staticfrictional rollingcontactsolutionscalculatedbytheexplicitFEMincluding con-tactarea,pressure,surfaceshearstressandmicro-sliphavebeenshown tobeaccurateviacomparisonswithHertzianandCONTACTsolutions

[23,24],whereasthesimulatedstructuraldynamicresponsesto wheel-IRJimpacthavebeenvalidatedbyacomprehensiveﬁeldmeasurement in[1].Inaddition,bypresentingtherailsurfacevibratingvelocities cal-culatedatconsecutivetimesteps,regularwavepatternswereobserved inthisstudy,whichprovidemoreevidenceforthereliabilityofthe re-sults.

ComparedwiththeimplicitFEM,theexplicitintegrationschemeis morerobust inhandlingdiﬃcultcontactproblemsbecauseitavoids theconvergencediﬃcultiescausedby demandingcontactconditions

[25]andtheregularizationofthefrictionlawrequiredtotreatthe no-slipconditionintheadhesionarea[26].Moreover,thecomputational efficiencyis significantlyimprovedwhenconsidering high-frequency dynamics.Thisstudycalculatedtheevolutionofthecontactpatcharea, stressmagnitude anddirection,micro-slip distributionsandrailhead nodalvibrationvelocitiesinthevicinityofthejointusingsmalltime stepstocapturethehigh-frequencydynamiceffectsduringthe wheel-IRJimpact.

2. Wheel-IRJimpactcontactmodel

A3DexplicitFE wheel-IRJdynamicinteractionmodelwas estab-lishedinthisstudytosimulatethewheel-railimpactatatypicalIRJ intheDutchrailwaynetwork.Detaileddescriptionsofthemodelcan befoundin[1].InformationonthetargetIRJandFEmodelrelevant tocalculatingthetransientsolutionsofwheel-IRJimpactcontactare givenhere.Fig.1showstheinsituconditionofthetargetIRJselected forthestudy,anditislocatedintheAmsterdam-Utrechttrunklineof theDutchrailwaynetwork.UIC54railswithaninclinationof1/40are supportedbyNS90sleepersevery0.6mexceptintheproximityofthe IRJ,whereapairofadjacenttimbersleeperswithadistanceof0.24m isemployedtoreducethedeﬂectionofthejointandabsorbthe vibra-tionscausedbywheel-IRJimpacts.Aclose-upviewofthetargetIRJin

Fig.1(b)showsthattheIRJdoesnotpresentvisibledeteriorationbut showsanasymmetricrunningbandwithrespecttothejoint;in addi-tion,abroaderrunningband andabrighterspotcan beseenonthe railheadjustafterthejointalongthetraﬃcdirection,whichiswhere thewheel-IRJimpactsareexpectedtooccur.

Fig.2showsthe3DFEwheel-IRJdynamicinteractionmodel estab-lishedin[1].This modeliscomposedofa10-m-longhalf-trackwith anIRJinthemiddleandahalf-wheelsetwiththesprungmassofthe

car bodyandbogie.Thewheel,rails,andsleepersweremodeled us-inghexahedralelements.TheexplicitFEwheel-railinteractionanalysis adoptedtheone-pointquadratureschemeforthesakeofcomputational eﬃciency,which,however,leadedto‘hourglass’ modesforhexahedral elements.AnorthogonalFlanagan-Belytschkohourglasscontrolscheme

[27]wasthususedtoavoidtheundesirablehourglassmodesfrom grow-inglargeanddestroyingsolutions.Thewheelgeometrycorrespondsto thatofapassengercarwheeloftheDutchrailwaywiththestandard profileofS1002.Becausethevalueofelasticmodulusoftheend-post (insulationlayerbetween tworailends)is muchlowerthanthoseof therailsandthepresenceofairgap(showninFig.1(b))mayresultin freerail-end[28],theend-postlayerwasomittedinthemodeland sim-plifiedasagap.TheIRJwitha6-mmgapwasmodeledindetailwith thenominalgeometryandfinemeshes.Non-uniformmeshingwasused, andregulardiscretizationwithameshsizeof1mmwasallocatedatthe initialwheel-railcontactareaandwithinthe0.2-m-longsolutionzone aroundthejoint(Fig.2(b)).Freeboundarieswereusedontherailends atthejoint,whereasnon-reflectingboundariesweredefinedatthefar endsoftherails.

BecausetheexplicitFEMislessefficientthantheimplicitFEMfor staticequilibriumanalyses,animplicit-explicitsequentialapproachwas appliedinthisstudytominimizeboththesolutiontimeandthedynamic effectsinducedbytheinitializationofwheel-railinteractionanalysis. Theimplicit-explicit sequentialapproachinvolvesperformingan im-plicitstaticequilibriumanalysisfollowedbyanexplicittransient dy-namicsanalysis,asusedin[11].Thesimulationfirstemployedan im-plicitFEdynamicrelaxationtoallowthewheel-tracksystemtoreach an equilibriumstateundergravity,which provided theinitial nodal displacements tothe explicit wheel-rail transient rolling simulation. Theinitialpositionofthewheelmodelwas1.32mawayfromtheIRJ (twostandardsleeperspansandhalfatimbersleeperspanasshownin

Fig.2(a)).Therotationandforwardtranslationmovementsofthewheel wereappliedastheinitialnodalvelocitiesofthetransientexplicit anal-ysis.

Intheexplicittransientdynamicsanalysis,wecalculatedthe wheel-rail frictional rollingcontact witha penalty contactalgorithm[29], which isdirectlyimplemented intheexplicitFEMandiscalledasa subroutineateachtimesteppriortotheupdatesofthestructural dy-namicresponses.Consideringthattheexplicitintegrationschemeis con-ditionallystable:theintegrationisonlystableifthetimestepsizeused issmallerthanthecriticaltimestepsize,andthatthecriticaltimestep mayvaryinthenonlinearwheel-raildynamicinteractionanalysis be-causeofchangesinthematerialparametersand/orgeometry,ascale factorofcriticaltimestep0.9wasadoptedtocontrolthecomputational timestep andguaranteethe stabilityof theexplicit integration.The stabilityofthepenaltycontactalgorithmcanbecontrolledbyscaling down thepenalty contactstiﬀness;however,this wasunnecessaryin thisstudy.

Coulomb’slawoffrictionwasimplementedforthewheel-rail con-tactpairwithafrictioncoefficientof0.35(atypicalintermediatevalue oftherailtopfriction[30]).Thewheelwassubsequentlydrivenbya torqueappliedontheaxletorollalongtherailfromtheinitialposition towardsthejoint,thusgeneratingalongitudinalcreepforcebetweenthe wheelandrailthatsatisfiestherequirementthatthetractioncoefficient isbelowthefrictioncoefficient.

Fig.3(a)and(b)showthemeshesoftherailtopsurfacein prox-imitytothejoint before andafterapplyingthemeasuredgeometry, respectively.TheinsitugeometryofthetargetIRJwasmeasuredusing aHandySCAN3Dlaserscanner.Becausetherailheadsurface geome-triesvarywiththeoperationaltimeandeachIRJ,themeasured geome-trymayimposerandomnessonthesimulatedimpactcontactsolutions. Moreover,thematerialpropertiesofcontactbodiesalsoinﬂuencethe wheel-railcontactsolutions[24,31].Therefore,inadditiontothe sim-ulationconductedbythemeasured-geometrymodelin[1],two addi-tionalsimulationswereperformedinthispaperusingnominal-geometry models,withonesimulatingelasticmaterialsandtheothersimulating

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Z. Yang et al. International Journal of Mechanical Sciences 138–139 (2018) 310–322

Fig.1. InsituconditionofthetargetIRJ.

Fig.2. FEwheel-IRJinteractionmodel.

Fig.3. ApplyingtherealisticgeometrytotheIRJ(thesizeoftheirregularityisexaggerated)[1].

elastoplasticmaterials, togainanoverallunderstanding ofthe tran-sientcontactbehaviorofwheel-IRJimpacts.Thesetupparametersof thethreesimulationsconductedinthisstudyarelistedinTable1,and simulation3wasperformedin[1].Table2liststhevaluesofthe mate-rialparametersusedinthesimulations.Theelastoplasticmaterialmodel appliedinthepapercorrespondstotheR260Mnrailsteelthatiswidely usedintheDutchrailway.

3. Impactcontactsolutions

Toprovideabroadoverviewofthewheel-railimpactcontactatan IRJ,thetimehistoriesofthenormalcontactforcescalculatedbythe threesimulationsarepresentedatthebeginningofthissection. Subse-quently,typicaltransientcontactsolutionscalculatedwithinthe solu-tionzoneareanalyzed,includingthecontactpatcharea,stress magni-tudeanddirection,micro-slipandadhesion-slipdistributions.The inﬂu-enceoftherailsurfacegeometriesandthematerialmodelsarediscussed

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Fig.4. Timehistoryofthewheel-railcontactforce.

Table1

Parametersof thesimulationswithdiﬀerent setups.

Material Proﬁle Simulation1 Elastic Nominal Simulation2 Elastoplastic Nominal Simulation3 Elastoplastic Measured

Table2

Valuesofthematerials.

Elastoplasticmaterialparameters Values Elasticpart Young’smodulus 210GPa

Poisson’sratio 0.3 Density 7800kg/m3

Plasticpart Yieldstress 500MPa Tangentmodulus 21GPa

bycomparingthesolutionsobtainedbythethreesimulations.The tran-sientcontactsolutionsobtainedinthisstudyalsocapturedwave phe-nomena,whichwillbepresentedinSection4.

3.1. Wheel-IRJimpactcontactforce

Thetimehistoriesofthenormalwheel-railcontactforcescalculated bythethreesimulationsareplottedinFig.4(a),whichshowsthat ob-viousimpactcontactoccurredwhenthewheelrolledoverthejointat approximately47ms.Thedampinginthesystemdissipatedtheinitial kineticandpotentialenergyoriginatingfromanyinitialinequilibrium ofthesystemsuchthattheoscillationsweredampedouttolessthan 10%ofthestaticvaluesuponarrivingatthesolutionzone.The close-up viewofthetimehistoriesaroundtheimpactplotted in Fig.4(b) showsthesimulatednormalimpactloadswithinthesolutionzone.The ﬁgureshows thattheimpactsimulatedwiththemeasuredgeometry (simulation3)wasmuchlargerthanthosesimulatedwiththenominal geometry.Inaddition,acomparisonofthenormalloadscalculatedby simulations1and2,whichonlydiﬀerinthematerialproperties,shows thattheelastoplasticmaterialmodel(simulation2)providedaslightly higherimpactmagnitudethantheelasticmodel(simulation1), proba-blyduetothatduringtheimpactswiththesameduration(seeFig.4(b)), largercontactcompressionoccurredintheelastoplasticsolutionto bal-ancethesamequasi-staticwheelload[31].

Asreportedin[1],thewheel-IRJdynamicinteractionsimulationin thisstudyemployedasmallcomputationaltimestep(49ns).By

apply-inganexplicitcentraldifferencetimeintegrationandapenaltycontact algorithm[29],nodalforcesandmotionsinthesolutionzonewere cal-culatedforeachtimestep.Certainnodalforcesandmotionswere sub-sequentlyoutputandusedtocalculatethetransientsolutionsofimpact contact.Asmalloutputtimestep(1μs)wasusedinthisstudytocapture high-frequencydynamiceffectsupto500kHzinthetransientsolutions ofimpactcontact.Transientcontactsolutionsof6300outputtimesteps, specificallyfrom43.5msto49.8ms(abscissarangeofFig.4(b)),were calculatedforeachsimulation.

Anexampleofatransientcontactsolutioncalculatedbysimulation 3thatwasoutputatinstant45.613ms(outputtimestep=2113)is dis-playedin Fig.5.InFig.5(a),fromleft toright,thegraphsshow the simulatedstressdistributionsalongthelongitudinalcenterlineofthe contactpatch,stressdistributionswithinthecontactpatch,micro-slip distributionsandrailsurfacenodalvelocities.Thecontactstress distri-butionsweredeterminedbythecalculatedrailsurfacenodalforces;and therailsurfacenodalvelocitiescanbedirectlyoutput.Asforthe micro-slip,orthewheel-railrelativevelocity,becausearailsurfacecontact nodeisactuallyincontactwiththe‘contactpoint’ ratherthanawheel surfacenode[32],interpolationswasusedtoconverttheoutput veloc-itiesofwheelnodesintothevelocitiesofthe‘contactpoints’.

Thetransientwheel-railcontactpositionatthisinstantortimestep canbemoreeasilyidentiﬁedinFig.5(b).Bydisplayingcontactsolutions ofacertainamountofconsecutivetimestepsasthoseshowninFig.5(b), animations[33–35]werecreated,whichclearlyshowtheevolutionof thecontactsolutionsalongwithcertainhigh-frequencydynamiceﬀects. Typicaltransientcontactsolutionsareselectedandanalyzedinthe fol-lowingsectionstodemonstratethecharacteristicsofthetransient solu-tionsofimpactcontact.

3.2. Contactareasandstressdistributions

3.2.1. Evolutionofthecontactareasandstressdistributions

Theevolution ofthecontactpressuretogether withthedirection andmagnitudeofthesurfaceshearstresscalculatedbythethree simu-lationsareplottedinthecontour/vectordiagramsinFig.6.Thecontact patchareacanbedeterminedviathecontactpressure:anelementisin contactifthecontactpressureisnon-zero.Thus,theevolutionofthe contactpatchareacanalsobeobservedfromFig.6.Eighttimesteps (t1-t8)withaﬁxedintervalof0.77ms(770timestheoutputtimestep) betweentwoconsecutivetimestepsaredisplayedforeachsimulationto showthemaincharacteristicsoftheimpactcontactareaandthestress evolution.Theoriginofthecoordinatesystemwasatthecenterofthe railbottomsurfaceattheinitialpositionofthewheel-railcontact. Be-causethecoordinatesystemincludedtherailinclinationofthetrack, thelongitudinalcenterlinesofthecontactpatchesshowninFig.6areat

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Fig.5.Exampleofthetransientcontactsolution.

approximately−3mminthelateraldirectionratherthanat0mm.The contactpressuremagnitudecorrespondstothedepthofcolorwithinthe contactpatchasindicatedbythecolorbar.Thesurfaceshearstresses areindicatedbybluearrows.Thearrowspointinthedirectionofthe shearstress,andthearrowlengthisproportionaltothemagnitude.

InFig.6(a)and(b),otherthanthediscontinuouscontactatthejoint att4,thecontactpatchareascalculatedbyboththeelastic(simulation 1)andelastoplasticmodels(simulation2)correspondwellwiththose reportedin[24,31].Thewheel-railcontactareassimulatedbythe elas-ticmodelhaveellipticalshapes,whereasthosesimulatedbythe elasto-plasticmodelhave‘egg’ shapes,withthetrailingpartsofthecontact patchesenlargedbecauseplasticdeformationhasoccurredintherear

[31].

Although simulation 1 produced the smallest impact force (Fig.4(b)),theamplitudesofthecontactpressurescalculatedby simu-lation1arelargerthanthosecalculatedbytheothertwosimulations asindicatedbyFig.6becausesimulation1hadthesmallestcontact areas.Themagnitudesofthecontactpressurelocatedapproximately inthemiddleofthecontactpatchinFig.6(a)butintheleading sec-tioninFig.6(b)arealsoconsistentwiththeresultsreportedin[24,31].

ThesimulatedcontactpatchareasinFig.6(a)and(b)basicallyremain steadyandincreasetosomeextentduringtheimpactatt4andt5.This phenomenonismoreevidentundertheelastoplasticmaterialcondition. Simulation3providedmoreobviouslynon-steady-statecontact so-lutionsasshowninFig.6(c).Thecontactareasandstressdistributions varyconsiderablywiththetimestepduetothegeometricirregularity oftherailtopsurfaceandsigniﬁcantimpact.Thecontactpatchesatt1 andt2inFig.6(c)aresimilartothoseatthesametimestepsinFig.6(b). Subsequently,theareaofthecontactpatchdecreasesatt3andt4 be-cause ofthegeometricdeclivitybeforethejoint (seeFig.3(b)). The contactareaincreasesremarkablyduringtheimpactcontactatt5and thenshrinksatt6,whenthewheelhasatendencytobounce.Therail surfacegeometricirregularitycontributestotheirregularshapesofthe contactpatches,whichareneitherellipticalnor‘egg’ shaped,aswellas theirregularstressdistributionsinFig.6(c).

Byplottingatrailoftransientcontactareas,the‘footprints’ ofthe contactpatchcalculatedbysimulation3arepresentedinFig.7(a).The intervalbetweeneachtwoconsecutivecontactpatchesis0.3ms(300 timestheoutputtimestep).Goodcorrespondencecanbeobtainedby comparingthesimulated‘footprints’ totheinsiturunningbandofthe

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Fig.6. Evolutionoftheimpactcontactareaandstressdistribution.(Forinterpretationofthereferencestocolorinthisﬁgurelegend,thereaderisreferredtothe webversionofthisarticle.)

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Fig.7. Comparisonofthesimulatedcontactpatch‘footprints’ andtheinsiturunningband.(Forinterpretationofthereferencestocolorinthisﬁgurelegend,the readerisreferredtothewebversionofthisarticle.)

Fig.8. Exampleofthesurfaceshearstressdistribution.

targetIRJshowninFig.7(b).The‘footprints’ becomenarrowatthe regionjustbeforethejoint(1310–1317mm)inFig.7(a),which corre-spondstothecut-oﬀ oftherunningband(atapproximately1310mm) inFig.7(b).Wheel-IRJimpacts(theﬁrstpeakofthecontactforce in

Fig.4(b))occuratapproximately1330–1350mm,wherethe‘footprints’ arelargerinFig.7(a),andabroaderrunningbandandabrighterspot canbe foundin Fig.7(b).Thesecond wheel-railimpact(the second peakofthecontactforceinFig.4(b))occursatapproximately1380– 1390mm,wherelargerthanusualcontactpatchesandabrighterspot canbeobservedinFig.7(a)and(b),respectively,althoughtheyareless pronouncedthanthoseintheﬁrstimpact.Thegoodcorrespondence be-tweenthesimulatedcontactpatch‘footprints’ andinsiturunningband impliesthatsimulation3(withthemeasuredgeometry)canmore ac-curatelyreproducethetransientimpactcontactsolutionsatthetarget IRJ,whichislikelyinanon-steadystate.

3.2.2. Positivesurfaceshearstress

Whenawheelrollsalongaraildrivenbyatorqueattheaxle,the shearstressdirection on therailsurfaceisgenerallyoppositetothe

wheelrollingdirection.AnexampleisshowninFig.8(a),inwhichthe railsurfaceshearstressdistributionwascalculatedviasimulation2at instant46ms(outputtimestep=2500).Byextractingthesurfaceshear stressalongthelongitudinalcenterlineofthecontactpatchand com-paringittothetractionbound(theproductofthecontactpressureF_n

andfrictioncoeﬃcientf),thedistributionoftheadhesion-slipregions canbeobtained,asshowninFig.8(b).Themaximumamplitudeofthe surfaceshearstressislocatedatthejunctureoftheadhesionandslip regions.Theadhesion-slipdistributionwithinthecontactpatchwillbe analyzedindetailinthenextsection.

Inthispaper, thedirection ofthesurfaceshearstressoppositeto thedirectionofwheelrollingisdeﬁnedasnegativeandthatalongthe directionofwheelrollingisdeﬁnedaspositive.Areviewofthe evolu-tionofstressdistributionshowninFig.6indicatesthatasthe counter-forceofthetractioncausesthewheeltomoveforward,thesimulated surfaceshearstresspointsinthenegativedirectionforthemajorityof graphsexceptattheregionsimmediatelyafterthejoint(roughly1323– 1327mm)att4.Toshowthisphenomenonmoreclearly,anevolutionof thesurfaceshearstressdistributionwithinthecontactpatchcalculated

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Fig.9. Evolutionofthesurfaceshearstresscalculatedbysimulation2.

bysimulation2withasmallertimestep(0.1ms,100timesoftheoutput timestep)thanthatusedinFig.6(0.77ms)isdepictedinFig.9.The evolutioncalculatedbysimulations1and3(notpresentedhere)shows thesametrend.

AsshowninFig.9,whenthewheeljusttouchestherightrailafter thejoint(T1-T3),theshearstressvectorsontherightrailarepositive, whereasthoseontheleftrailarenegative.FromT4toT6,asthecontact patchmoves,theamplitudesofthepositiveshearstressesontheright railbetween1325mmand1329mmdecreasetozeroandthenbecome negative.GraphsofT4andT5alsoindicatethatthelateralshearstress playsanimportantroleatthesemoments,especiallyatlocationsclose tothetopandbottomedgesof thecontactpatch,wherethesurface shearstressvectorspointoutwards.Suchstressmayexacerbate mate-rialﬂowontherailheadandconsequentlywidentherunningbandat theimpactlocationasshowninFig.7(b).AttheinstancesofT7andT8, thecontactpatchhasexitedthepositivesurfaceshearstressregionand allthestressvectorspointinthenegativedirection.Theoccurrenceof transientpositivesurfaceshearstressontherightrailendisshownto resultfromtheimpactcontactattheIRJwithdiscontinuousgeometry. Intheregionimmediatelyafterthejoint,theamplitudesofthe posi-tiveshearstressescausedbythewheel-IRJimpactarelargerthanthe amplitudesoftheoriginalnegativeshearstresses(counterforceofthe traction),thusmakingtheresultantsurfaceshearstressespositive.

Thisstudymodeledawheeldrivenbyatorqueonitsaxle.Whena brakingwheelrollsonasectionoftrackwithanIRJ,theoriginal direc-tionoftherailsurfaceshearstressesisexpectedtobepositive (coun-terforceofthebrakingforce).Insuchcases,thepositiveshearstresses imposedbyimpactwillbeaddedtotheoriginalpositiveshearstresses. TowhatextentanimpactbetweenabrakingwheelandanIRJcan in-ﬂuencethesurfaceshearstressdistributionandtheconsequentwear behaviorontherailheadafterthejointshouldbestudiedinthefuture.

3.3. Adhesion-slipdistributionandmicro-slip

Thedivisionbetweenadhesionandslipregionsinthecontactpatch isanimportantfeatureoffrictionalrollingcontact.Thetransientcontact solutionsthatindicatetheadhesion-slipdistributionduringthe wheel-IRJimpactpredictedbysimulation3arepresentedinthissection.The adhesion-slipdistributioncanbedeterminedeitherbycomparingthe surfaceshearstresswiththetractionboundorbycalculatingthe micro-slipwithinthecontactpatch.Inthisstudy,thesimulatedadhesion-slip distributionsdeterminedbythesetwoapproachesareconsistentwith eachother,asshowninFig.10.

Fig.10(a)displayscomparisonsofthesurfaceshearstressand trac-tionboundalongthelongitudinalcenterlineofthecontactpatchat15 instants.Instantt1isequalto44.4ms,andtheintervalbetweeneach twoconsecutiveinstantsis0.3ms(300timesoftheoutputtimestep).

Fig.10(b)showstheevolutionofthemicro-slipdistributionwithinthe contactpatchatthesame15instants.Theredarrowspointinthe direc-tionofthemicro-slip,andthearrowlengthisproportionaltothe mag-nitude.Themicro-slipvectors(slipregion)occuratthetrailingpartof thecontactpatch,andtheirdirectionslargelycorrespondtothewheel rollingdirection.Thecolordepthwithinthecontactpatchindicatesthe magnitudeofthenormalwheel-railrelativelyvelocity.Thecolor out-sidethecontactpatchcorrespondstoazerorelativevelocity,whereas thatattheleadingandtrailingedges ofthecontactpatcharelighter anddarker,indicatingapositiveandnegativenormalrelativevelocity, respectively.

Theadhesion-slipdistributionsshowninFig.10arenon-steady dur-ingthewheel-IRJimpact,andtheproportionofcontactpatchoccupied bytheadhesionandslipregionsvarygreatlywiththetimestepfromt8 onwards.Themostsigniﬁcantvariationoccursfromt9tot13.Atinstant t9,theadhesionregionaccountsforalmosttheentirecontactpatch. Next,theadhesionzoneshrinksgraduallyandtheslipregionreaches itsmaximumoccupationatinstantt12,whenthebouncetendencyof thewheelcomestoanendandthewheelisnearlyatthesecondimpact, whichcorrespondstothecontactforcetroughatapproximately47.7ms inFig.4(b).Thegoodconsistencybetweentheadhesion-slip distribu-tionscalculatedbythecontactstressesandthemicro-slipssupportthe conclusionthattheexplicitFEMpresentedherecansolvethetransient impactcontactproblemwithnon-linearmaterialpropertiesand arbi-trarydiscontinuouscontactgeometries.

4. Wavephenomena

Comparedwiththecontactforceandstress,thesurfacenodal vibra-tionvelocityisfoundtobemoresensitivetodynamiceffectsexcitedby wheel-railcontact[22,36].Therailsurfacenodalvelocitycalculations inthisstudyrevealedwavephenomenaexcitedbywheel-railfrictional rollingimpactcontact,andtheresultswerebasedonthefinemeshof theFEmodel,smallcomputationalandoutputtimesteps,andfull cou-plingofthecontactanddynamicsintheexplicitintegration.Thesizeof thefinemeshisbasedontherequirementthatthesizeoftheelements shouldbenolargerthanhalfawavelength.Thesmallcomputational timestepenablesthecalculationstocapturehigh-frequencydynamic effects,andthesmalloutputtimestepfacilitatestheobservationofthe

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Fig.10. Evolutionoftheadhesion-slipdistributionscalculatedbysimulation3.(Forinterpretationofthereferencestocolorinthisﬁgurelegend,thereaderis referredtothewebversionofthisarticle.)

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Fig.11. Patternsofwavepropagationandreﬂectionproducedbysimulation3.(Forinterpretationofthereferencestocolorinthisﬁgurelegend,thereaderis referredtothewebversionofthisarticle.)

generationandpropagationprocessesofthesimulatedwaves.Full cou-plingmeansthatthecontactforceandwheel-raildynamicsareupdated simultaneouslyineverytimestep,therebyallowingthedynamiceﬀects tobefullyconsideredwhencalculatingthecontactsolutions.Certain wavepatternsthatindicatewavegeneration,propagationand reﬂec-tionwereexcitedbywheel-railcontactproducedinthesolutionzone, andtheyarepresentedandanalyzedinthissection.Moreevidentwave phenomenacanbeobservedintheanimations[33–35].

4.1. Wavesgeneratedbywheel-railcontact

Thewavephenomenondiscussedherewasﬁrstobservedbeforethe wheel-IRJimpact.Fig.11(a)and(b)showtherailnodalvelocitiesinthe solutionzonesimulatedbysimulation3atinstants43.615ms(output timestep=115)and43.640ms(outputtimestep=140),respectively. Thecolordepthin theﬁgureindicatesthemagnitude ofthenormal nodalvelocity,whichcanbeusedtoidentifytheapproximatepositionof thewheel-railcontactpatch.Thedarkerandlightersemi-ovalsindicate theleadingandtrailingpartsof thecontactpatch,respectively. The tangentialnodalvelocitiesareindicatedby bluearrows. Thearrows pointinthedirectionofthetangentialvelocity,andthearrowlengthis proportionaltothemagnitude.Thestripeinthemiddleofeachgraph showsthepositionofthejoint.

Aregularwavepatternpropagatingfromthewheel-railcontactarea towardsthejointcanbeobservedinFig.11(a).Thewavewitha wave-lengthofapproximately6mmisstrongeraroundthecontactpatchand dissipateswhenpropagating.Whenthewavefrontreachesthejoint,a reflectivewaveoccursunderthefreeboundaryconditiondefined on therailendatthejoint.Thereflectivewaveextendsfromthejointback tothecontactpatchandinterfereswiththeoriginalwaveasindicated inFig.11(b).Moreobviouswaveinterferencecanbeobservedinthe correspondinganimation[33].

Thewavephenomenadiscoveredinthisstudyaretransientand nor-mallytakelessthan0.1msfromgenerationtodisappearance.The gen-erationprocessofthewaveshowninFig.11(a)isdisplayedwithan outputtimestepof1μminFig.12.Theinstantoftheﬁrstgraph(T1) is43.603ms,whichis0.012ms(12outputtimesteps)earlierthanthat ofFig.11(a).AtT1,therailsurfacenodalvelocitiesappeartobe dis-tributedsymmetricallywithrespecttothelongitudinalcenterline.The leadingandtrailingpartsofthecontactpatchcanbeidentiﬁedbythe darkerandlightersemi-ovals,respectively.Theregionaheadofthe con-tactareaisslightlydarkerthanthatbehindthecontactarea.The tan-gentialvelocitiesaremainlyconcentratedon thetrailingpartof the

contactpatch,andtheirdirectionsarelargelyconsistentwiththewheel rollingdirection.Thelateralcomponentsofthetangentialvelocities in-creasewiththedistancetothelongitudinalcenterlineofthecontact patch.AtinstantT2,turbulenceofthenodalvelocitysuddenlyoccurs intheleadingpartofthecontactpatch.Theturbulencespreadsradially andconsequentlydevelopsintoawaveinthefollowinginstants.This turbulenceissuspectedtoberelatedtothewheel-railfriction-induced instability,andstudiestodetermineitscausearestillongoing.

Graphs ofinstantsT3andT6indicatethatthewavepatternsare embodiedinboththenormalandtangentialnodalvelocities,andthe directionofthetangentialvelocitiesareconsistentwiththewave prop-agationdirection.Thewaveisinitiatedatthelongitudinalpositionof 1244mm(T2),andwithin3timesteps,itsfrontreaches1254mm(T5). Neglectingthewheelrollingdistanceinsuchashortperiod(lessthan 0.1mm),thewavespeedisestimatedas3km/s.Boththepropagation formandthespeedoftheproducedwaveareconsistentwiththe prop-ertiesofRayleighwaves[37].

4.2. Waveexcitedbywheel-IRJimpact

Thisstudyalsorevealedthatanimpactwavecanbegeneratedwhen thewheel rolls overthejoint andjust touchestherail ontheother side. Fig.13 displaysan impact waveproduced by simulation 3at instants46.390ms(outputtimestep=2890),46.395ms(output time step=2895)and46.400ms(outputtimestep=2900).Attheseinstants, thewheelwasrollingfromtheleftrailtotherightrail,anditwasin con-tactwithbothrailends.AsshowninFig.13,anobviouswaveoccursand propagatesontherailafterthejoint.Comparedwiththewavepatterns displayedinFig.11,theimpactwavepatternsdisplayedinFig.13are mainlyformedbythenormalnodalvelocities,andthewavelengthis ap-proximately10mm.Thecontributionsofthetangentialnodalvelocities tothewavepatternsaremuchlesspronounced.Moreevidentimpact wavepropagationcanbeseenintheanimation[34].

In addition to the wave types displayed in Figs. 11 and 13, simulations 1 and2 also producedwaves with longerwavelengths.

Fig.14(a)and(b)showtwoexamplesproducedbysimulation2at in-stants46.74ms(outputtimestep=3240)and46.762ms(outputtime step=3252),respectively. Thesetwowavesappearimmediatelyafter thewholecontactpatchistransferredtotherailafterthejoint,andtheir wavelengths,whichareshowninFig.14(a)and(b),areapproximately 40mmand20mm,respectively.Theanimation[35]showsamoreclear depictionofthesewaves.Thelonger-wavelengthwavewasnotobserved

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Fig.12. Generationprocessofawave.

Fig.13. Impactwavepatternproducedbysimulation3.

insimulation3.Thecauseoftheselonger-wavelengthwavesmustbe furtherstudied.

Asillustratedabove,wavephenomenawereproducedbythe ex-plicitFEMandobservedinthesimulatedrailnodalvibrationvelocities inthesolutionzone.Becausethecalculationsofthewheel-railcontact forceanddynamicsarefullycoupled,thewavephenomenawerealso capturedbythesimulatedwheel-railcontactforce.Fig.15depictsthe 100-kHzhigh-passﬁlteredsignalsofthewheel-railcontactforcesinthe solutionzonecalculatedbysimulations2and3.Eachpeakofthe sig-nalsshowninFig.15correspondstoawavephenomenonshownbythe

railnodalvelocities.Peaks1,2and3denotedinFig.15correspondto thewavephenomenashowninFigs.11,13and14,respectively.The wavephenomenacorrespondingtotheotherpeaksofFig.15werealso producedbythesimulationsbutarenotpresentedhere.Higherpeaks correspondtomoresigniﬁcantcorrespondingwavephenomena.Peak 2showninthelateral(uppergraph)andlongitudinal(bottomgraph) contactforcesignalsislessremarkablethanthatintheverticalforce sig-nal(middlegraph),anditcorrespondstotheresultsshowninFig.13, whichshowsthatthewavepatternsaremainlyformedbythenormal nodalvelocities.Thecauseofthesehigh-frequencycontactforcepeaks,

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Fig.14. Wavepatternswithlongerwavelengthsproducedbysimulation2.

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whichislikelythecauseofthewavephenomenaaswell,hasnotbeen clearlyidentiﬁedinthisstudyandwillbeinvestigatedinthefuture.

5. Conclusionsandfuturework

Thispaperinvestigatedthetransientcontactsolutionsofwheel-IRJ frictionalrollingimpactswithdynamiceffectssimulatedbyanexplicit FEwheel-IRJdynamicinteractionmodel,whichwasvalidatedagainsta comprehensivefieldmeasurementin[1].Thetransientsolutionsof im-pactcontactwithsmalltimestepscalculatedinthispaperincludethe contactpatcharea,stressmagnitudeanddirection,micro-slip distribu-tion,andrailsurfacenodalvelocityinthevicinityofajoint.The sim-ulatedcontactsolutionstendedtovarynoticeablywiththetimestep, indicatingthatdynamiceffectsplayimportantrolesinthewheel-IRJ impactcontact.Transientpositivesurfaceshearstress,whosedirection isoppositethatofshearstressunderordinarytractiverolling,was dis-coveredontherailsurfaceimmediatelyafterthejoint.Thesimulated transientadhesion-slipdistributionsdeterminedbythecontactstresses andmicro-slipsolutionswereverifiedbyeachother.

Inaddition,regularwavepatternswereproducedbothbeforeand duringthewheel-IRJimpactsinthesimulations,andthesepatterns re-flectcontinuumvibrationsexcitedbywheel-railfrictionalrollingand impactcontactandconfirmthatthesimulatedtransientcontact solu-tionsarereliable.Incombinationwiththeconclusionsdrawnfrom[1], thepresentedexplicitFEMissufficientandaccurateforsolving wheel-IRJimpactproblemsbyfullycouplingthehigh-frequencydynamicsof wheelandrailcontinuawiththecomplextransientimpactcontactin onesimulation.

Theinfluenceof thewheel-railcontactgeometryonthetransient contactsolutionswasalsoinvestigatedinthisstudy.Thecontact solu-tionscalculatedwiththenominalgeometrycorrespondwellwiththose reportedintheliterature,whereasthosesimulatedwiththemeasured geometry show obviously non-steady-state impact effects. The good agreementbetweenthesimulated‘footprints’ ofthecontactpatchand theinsiturunningbandimpliesthatthemodelwiththemeasured ge-ometryprovidesmorerealisticpredictionsofthetransientsolutionsof theimpactcontactatthetargetIRJ.Withoutconsideringtherealistic contactgeometries,theimpactcontactforceandimpactcontactarea fluctuationsmaybesignificantlyunderestimated.

Experimentalvalidationofthewavephenomenaproducedbythis study is planned to be conducted in the future by measuring high-frequencyrailsurfacevibrationsupto1MHz.Incombinationwiththese measurements,thesourceoftheinitiationofwaveswillbefurther in-vestigated.Becausethepropagationandreflectionofrailsurfacewaves areexpectedtobe influencedbygeometricdiscontinuities,thisstudy maybefurtherdevelopedandappliedtotrain-bornedetectionof early-stagecracksontherailhead.Inaddition,theinfluenceofthedynamic effectsofimpactsontrackdeteriorationinthevicinityofIRJscanbe studied,andtheresultsofsuchstudiesmaycontributetoasustainable IRJdesignandeffectivemaintenanceinpractice.

Acknowledgments

ThisworkwassupportedbytheChinaScholarship Council(grant No.201206260105),theDutchrailwayinfrastructuremanagerProRail, andtheopenresearchfundoftheMOEKeyLaboratoryofHigh-speed RailwayEngineering,SouthwestJiaotongUniversity.

Supplementarymaterials

Supplementarymaterialassociatedwiththisarticlecanbefound,in theonlineversion,atdoi:10.1016/j.ijmecsci.2018.02.025.

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