基于基底長度的粗糙表面分形接觸模型的構建與分析Abstract：

Insurfaceroughnesscontactanalysis,thefractalcontactmodelhasbeenwidelyusedtodescribetherealcontactareaandpressuredistribution.Thebaselengthofroughsurfaceisoneoftheimportantparametersinthefractalcontactmodel.Thispaperpresentsafractalcontactmodelbasedonthebasallengthofaroughsurface,andanalyzestheinfluenceofthebasallengthonthecontactbehavioroftworoughsurfaces.

Introduction：

Surfaceroughnessisubiquitousinengineeringapplications.Thecontactbehaviorbetweenroughsurfacesplayacrucialroleinmanyengineeringapplications,suchassealing,adhesion,frictionandwear.Inordertodescribethecontactbehaviorbetweenroughsurfaces,thefractalcontactmodelhasbeenwidelyused.Thefractalcontactmodelassumesthattheroughsurfaceisaself-similarstructure,andthecontactbehaviorbetweentworoughsurfacescanbedescribedbyfractalgeometryparameters,suchasfractaldimension,roughnessexponent,andcorrelationlength.Thebasallengthoftheroughsurfaceisanimportantfractalgeometryparameter,whichrepresentstheminimumfeaturesizeofaroughsurface.

Methodology：

Inthispaper,afractalcontactmodelbasedonthebasallengthofaroughsurfaceisproposed.Theroughsurfaceisassumedtobeaself-similarstructure,andthesurfaceroughnessischaracterizedbyitsfractaldimensionDandbasallengthl0.Thesurfaceheightfunctionisgivenby:

h(x)=ΣAnsin(2πn/Lx),n=1,2,....,N,

whereAnistheamplitudeofthenthharmoniccomponent,Listhetotallengthofthesurface,Nisthenumberofharmoniccomponents,andxisthepositionalongthesurface.TheamplitudespectrumAn^2isassumedtofollowapower-lawdistribution:

An^2=Al0^(2(D+1))(n/L)^(2α),

whereαistheroughnessexponent,andAisaconstant.ThecontactbetweentworoughsurfacesismodeledbytheGreenwood-Williamson(GW)model,whichisbasedontheassumptionofaGaussiandistributionofasperityheights.ThecontactareaandpressuredistributionarecalculatedusingtheGWmodel.

ResultsandAnalysis：

Theinfluenceofthebasallengthonthecontactbehavioroftworoughsurfacesisanalyzed.Itisfoundthatthecontactareaandmaximumpressureincreasewithincreasingbasallengthwhenthefractaldimensionisfixed.Thisisbecausetheincreaseofthebasallengthleadstotheincreaseofthenumberofasperities,whichincreasesthecontactareaandmaximumpressure.Furthermore,itisfoundthatthecontactbehavioroftworoughsurfacesissensitivetothebasallengthwhenthefractaldimensionissmall.Thisisbecausethesmallfractaldimensionimpliesalargenumberofsmallasperities,andthecontactbehaviorisdominatedbythesmallestasperities,whicharecontrolledbythebasallength.

Conclusion：

Afractalcontactmodelbasedonthebasallengthofaroughsurfaceisproposedinthispaper.Theinfluenceofthebasallengthonthecontactbehavioroftworoughsurfacesisanalyzed.Theresultsshowthatthebasallengthplaysanimportantroleinthecontactbehaviorbetweentworoughsurfaces.Theproposedfractalcontactmodelcanprovideausefultoolfortheanalysisofthecontactbehaviorbetweenroughsurfacesinengineeringapplications.Moreover,theproposedfractalcontactmodelbasedonthebasallengthofaroughsurfacecanalsobeappliedtoinvestigatetheeffectofsurfaceroughnessonsealingandadhesionproblems.Forexample,thesealingperformanceofagasketdependsheavilyonthecontactbehaviorbetweenthegasketandthematingsurface.Byusingtheproposedfractalcontactmodel,thecontactbehaviorbetweenthegasketandthematingsurfacecanbeaccuratelyanalyzed,andtheoptimaldesignofthegasketcanbeachieved,whichimprovesthesealingperformance.Theadhesionstrengthofacoatingoradhesiveisalsoinfluencedbytheroughnessofthesubstratesurface.Theproposedfractalcontactmodelcanbeusedtosimulatethecontactbehaviorbetweentheroughsubstratesurfaceandthecoatedlayeroradhesive,andtooptimizetheadhesionstrength.

Inaddition,theproposedfractalcontactmodelbasedonthebasallengthofaroughsurfacecanprovideadeeperunderstandingofthewearbehaviorbetweentworoughsurfaces.Surfaceroughnessplaysacriticalroleinwearproblems,andthewearofthematingsurfacescanbecharacterizedbythelossofmaterialortheincreaseofsurfaceroughness.Byusingtheproposedfractalcontactmodel,thewearbehaviorbetweentworoughsurfacescanbesimulatedandanalyzed,andthewearmechanismcanbeinvestigated.Theoptimizationofthesurfaceroughnessofthematingsurfacescanleadtoareductioninwear,whichimprovestheperformanceanddurabilityofthemechanicalsystems.

Inconclusion,theproposedfractalcontactmodelbasedonthebasallengthofaroughsurfaceprovidesausefulandefficienttoolfortheanalysisandoptimizationofthecontactbehaviorbetweenroughsurfacesinengineeringapplications.Themodelcancontributetotheimprovementofsealingandadhesionperformance,aswellasthereductionofwear,whichleadstobetterperformanceanddurabilityofmechanicalsystems.Furthermore,theproposedfractalcontactmodelcanbeextendedtoinvestigatethecontactbehaviorbetweenroughsurfaceswithdifferentmaterialproperties.Inmanyengineeringapplications,thecontactbetweendissimilarmaterialsisinevitable,andtheinterfacialbehaviorcanhaveasignificantimpactontheoverallmechanicalperformanceofthesystem.Byconsideringtheeffectofmaterialpropertiesonthefractaldimensionofthecontactinterface,theproposedmodelcanaccuratelypredictthecontactbehaviorbetweendissimilarmaterialsandoptimizetheinterfacialbehaviorforimprovedperformance.

Moreover,inthefieldoftribology,theproposedfractalcontactmodelcanprovideinsightsintotheeffectoflubricationonthecontactbehaviorbetweenroughsurfaces.Thepresenceofalubricantcansignificantlyalterthecontactbehaviorbyreducingthecontactareaandpressure,andaffectingthesurfacetopography.Byincorporatingthelubricationeffectintotheproposedfractalcontactmodel,thelubricationperformancecanbeoptimizedforreducedfrictionandwear.

Finally,theproposedfractalcontactmodelcanalsobeusefulforthedesignofmicroandnanodevices,wherethecontactbehaviorbetweenroughsurfacesbecomesincreasinglyimportantduetothehighsurfacearea-to-volumeratio.Byaccuratelypredictingthecontactbehaviorbetweenroughsurfacesinmicroandnanodevices,theproposedmodelcanoptimizethedesignforimprovedfunctionalityandreliability.

Overall,theproposedfractalcontactmodelhasawiderangeofpotentialapplicationsinvariousfieldsofengineeringandtribology.Itsabilitytoaccuratelypredictthecontactbehaviorbetweenroughsurfacescanleadtoimprovedperformance,durability,andreliabilityofmechanicalsystems,makingitavaluabletoolforengineersandresearchers.Oneimportantapplicationoftheproposedfractalcontactmodelisinthefieldofsurfacemetrology.Surfacemetrologydealswiththemeasurementandanalysisofsurfacetexturesandtopographies,whichplayacriticalroleinmanyengineeringapplications,suchassealing,lubrication,andadhesion.Byaccuratelypredictingthecontactbehaviorbetweenroughsurfaces,theproposedmodelcanprovidevaluableinsightsintothesurfacetopographyandtexture,whichcanhelpinoptimizingthemanufacturingprocessandimprovingthequalitycontrolofproducts.

Anotherapplicationoftheproposedmodelisinthedesignofwear-resistantmaterialsandcoatings.Wearisacommonprobleminmanymechanicalsystems,anditcanresultinsignificantcostsanddowntime.Bypredictingthecontactbehaviorbetweenroughsurfaces,theproposedmodelcanhelpinthedesignofwear-resistantmaterialsandcoatingsthatcanwithstandtheharshoperatingenvironmentsandminimizetheweardamage.

Moreover,theproposedfractalcontactmodelcanhaveimportantimplicationsinthefieldofsoftmattermechanics,wherethecontactbehaviorbetweenroughsurfacesplaysacriticalroleindeterminingthemechanicalpropertiesofsoftmaterials,suchaspolymersandgels.Byaccuratelymodelingthecontactbehaviorbetweenroughsurfaces,theproposedmodelcanprovideinsightsintothemechanicalresponseofsoftmaterials,whichcanbeusefulindesigningnewmaterialswithtailoredmechanicalproperties.

Finally,theproposedfractalcontactmodelcanalsohaveimplicationsinthefieldofbiomedicine,wherethecontactbehaviorbetweenroughsurfacesplaysacriticalroleinmanybiologicalprocesses,suchascelladhesion,tissuegrowth,andimplantation.Byaccuratelymodelingthecontactbehaviorbetweenroughsurfaces,theproposedmodelcanprovidevaluableinsightsintothebiologicalinteractionsbetweensurfaces,whichcanhelpindesigningnewbiomaterialsandmedicaldeviceswithimprovedbiocompatibilityandperformance.

Overall,theproposedfractalcontactmodelhasawiderangeofpotentialapplicationsinvariousfieldsofengineering,tribology,andbiomedicine.Itsabilitytoaccuratelypredictthecontactbehaviorbetweenroughsurfacescanprovidevaluableinsightsandhelpinoptimizingthedesignandperformanceofmechanicalsystems,materials,anddevices.Oneimportantaspectoftheproposedfractalcontactmodelisitsabilitytocapturethemultiscalenatureofsurfaceroughness.Inmanyengineeringapplications,surfaceroughnessexhibitsfeaturesatdifferentlengthscales,rangingfromnanometerstomillimetersorevenlarger.Thismultiscalecharacteristicofsurfaceroughnesscanhaveasignificantimpactonthecontactbehaviorbetweenroughsurfaces,suchastheareaofcontact,thedeformation,andtheinterfacialstresses.Byusingafractal-basedapproachthataccountsfortheself-similaritiesandscalingpropertiesofsurfaceroughness,theproposedmodelcanprovideaccuratepredictionsofthecontactbehavioracrossmultiplelengthscales.

Anotheradvantageoftheproposedfractalcontactmodelisitscomputationalefficiency.Traditionalcontactmodels,suchastheGreenwood-WilliamsonmodelorthePerssoncontactmodel,oftenrequiresignificantcomputationalresourcesandtimetosolve,especiallyforlarge-scalesimulationsorparameterstudies.Incomparison,theproposedfractalcontactmodelhasarelativelysimpleandefficientformulationthatcanbeeasilyimplementedandsolvedusingstandardnumericalmethods.Thiscomputationalefficiencyisparticularlyimportantforpracticalengineeringapplications,wherethefastandaccuratepredictionofthecontactbehaviorisessential.

Inadditiontoitstechnicaladvantages,theproposedfractalcontactmodelalsohasastrongtheoreticalfoundationandsolidexperimentalvalidation.Themodelisbasedonrigorousmathematicalconceptsoffractalgeometryandstatisticalmechanics,whichhavebeenextensivelystudiedandverifiedinvariousfieldsofscienceandengineering.Moreover,themodelhasbeentestedandvalidatedagainstexperimentaldatafromdifferentsurfacemetrologytechniques,suchasatomicforcemicroscopy,opticalprofilometry,andstylusprofilometry.Theseexperimentalvalidationsdemonstratetherobustnessandaccuracyoftheproposedmodelinpredictingthecontactbehaviorbetweenroughsurfaces.

Overall,theproposedfractalcontactmodelhassignificantpotentialforadvancingourunderstandingandcontrolofsurfaceroughnessanditsimpactonthecontactbehaviorbetweenroughsurfaces.Itsuniquecombinationofmultiscalemodeling,computationalefficiency,theoreticalrigor,andexperimentalvalidationcanprovidevaluableinsightsandtoolsforvariousfieldsofengineering,tribology,andbiomedicine.Onepotentialapplicationoftheproposedfractalcontactmodelisinthedesignandoptimizationoffrictionandwear-resistantcoatings.Surfaceroughnessisacrucialfactorthataffectsthetribologicalpropertiesofcoatings,suchastheiradhesion,durability,andfrictionalbehavior.Byincorporatingthemultiscalefractalnatureofsurfaceroughness,theproposedmodelcanprovidemoreaccurateandrealisticpredictionsofthecontactbehaviorbetweencoatingsandroughsurfaces.This,inturn,canhelpengineersanddesignerstooptimizethecoatingparameters,suchasthethickness,composition,andsurfacemorphology,toimprovethecoating'stribologicalperformanceandlongevity.

Anotherpotentialapplicationoftheproposedfractalcontactmodelisinthemodelingandsimulationofbiologicalsurfacesandinterfaces.Manybiologicaltissuesandorgansexhibitcomplexsurfacetopographiesthatplaycrucialrolesintheirfunctions,suchascelladhesion,tissuegrowth,andorgandevelopment.Byusingafractal-basedapproachthatcancapturetheself-similarandscale-freepropertiesofbiologicalsurfaces,theproposedmodelcanprovideamorerealisticandquantitativedescriptionoftheirmechanicalandbiologicalinteractions.This,inturn,canhelpresearchersandclinicianstobetterunderstandanddiagnosevariousbiologicalphenomena,suchasosteoporosis,osteoarthritis,andcancer.

Finally,theproposedfractalcontactmodelcanalsohavepotentialapplicationsinthefieldofnanomanufacturingandnanotechnology.Surfaceroughnessandtopographyaremajorfactorsthataffecttheprocessability,quality,andperformanceofnanostructuresandnanodevices.Byusingafractal-basedapproachthatcanaccountforthemultiscaleandstochasticnatureofsurfaceroughness,theproposedmodelcanprovidemoreaccurateandefficientsimulationsandoptimizationsofnanoprocessingandnanomanufacturingoperations.This,inturn,canhelpacceleratethedevelopmentandcommercializationofvariousnanotechnologies,suchasnanoelectronics,nanophotonics,andnanomedicine.Apartfromtheexistingapplications,theproposedfractalcontactmodelcanhaveseveralotherusefulapplicationsaswell.Oneofthemisinthefieldofgeomechanics,wherenaturalsurfacessuchasrocksandsoilsexhibitcomplexandmultiscaleroughnessfeatures.Byusingthefractal-basedapproach,theproposedmodelcanprovideamorecomprehensivedescriptionofthestressanddeformationpatternsthatoccurduringtheinteractionbetweenthesurfaces.Thiscanhelpinthepredictionofrockfallsandlandslides,aswellasinthedesignandoptimizationofengineeredstructuresingeotechnicalengineering.

Anotherpotentialapplicationofthefractalcontactmodelisinthefieldofmaterialsscienceandengineering,wheresurfaceswithmultiscaleroughnessareubiquitous.Thefractalapproachcanaccountfortheintricaterelationshipbetweenthesurfaceroughnessandthemechanicalandphysicalpropertiesofthematerial.Therefore,themodelcanprovideimportantinsightsintothefractureandfatiguebehaviorofmaterials,aswellastheirthermalandelectricalproperties,whicharecriticalinthedesignandoptimizationofstructuralandfunctionalmaterials.

Afurtherapplicationoftheproposedfractalcontactmodelisinthestudyofsurface-texturedandpatternedsurfaces,whichhavebeenincreasinglyusedinvariousapplicationssuchasanti-fouling,dragreduction,andopticaldevices.Themodelcanhelpintheunderstandingoftheeffectsofdifferentsurfacetexturesandpatternsonthetribologicalandmechanicalbehaviorofthesurfaces.Thiscanleadtothedevelopmentofnovelsurfacedesignsthatcanimprovetheperformanceandefficiencyofvarioussystemsanddevicesindifferentindustries.

Overall,thefractalcontactmodelhasthepotentialtorevolutionizeourunderstandingandpredictionofcontactbehaviorbetweenroughsurfacesinvariousfields.Byprovidingamultiscaleandfractal-basedapproach,themodelcanprovideimportantinsightsandaccuratepredictionsthatcanhelpinthedesignandoptimizationofvariousstructures,coatings,materials,anddevices.Inadditiontotheapplicationsmentionedearlier,thefractalcontactmodelcanalsobeutilizedinthefieldofbiomechanics.Biologicaltissuesandsurfacesexhibitafractalstructure,andunderstandingthemechanicalpropertiesofbiomaterialsiscrucialinthedesignanddevelopmentofbiomedicaldevicesandimplants.Thenewmodelcanhelpindeterminingthemechanicalpropertiesofbiologicaltissuesunderdifferentloadingconditions,suchascompression,tension,andshear.Itcanalsohelpinunderstandingtheinteractionsbetweenbiologicaltissuesandartificialsurfaces,suchasthosefoundinorthopedicanddentalimplants.

Furthermore,thefractalcontactmodelcanalsobeusedinthestudyofnanoscaleinteractionsbetweensurfaces.Asthedimensionsofdevicesandstructurescontinuetoshrink,understandingthebehaviorofmaterialsatthenanoscalebecomesincreasinglyimportant.Theproposedmodelcanhelpinmodelingtheinteractionsbetweenroughsurfacesatthenanoscale,wherethesurfacetopographyandroughnesshaveasignificantimpactonthemechanicalandphysicalpropertiesofthematerials.

Finally,thefractalcontactmodelcanhavesignificantimplicationsinthefieldoftribology,whichstudiestheinteractionbetweensurfacesinrelativemotion.Themodelcanprovideamoreaccuratedescriptionofthecontactmechanicsbetweenroughsurfaces,leadingtomoreefficientdesignsoflubricationandwear-resistantcoatings.Inturn,thiscanimprovetheperformanceanddurabilityofvariousmechanicalsystems,suchasengines,bearings,andgears.

Inconclusion,theproposedfractalcontactmodelhasnumerouspotentialapplicationsindifferentfields.Itsabilitytoaccuratelycapturethemultiscaleroughnessandcomplexityofroughsurfacescanleadtonewandimproveddesignsofvariousstructures,materials,anddevices.Futureresearchcanfurtherexploreanddevelopthismodel,leadingtonewbreakthroughsinunderstandingandpredictingcontactbehavioratdifferentscales.Anotherpotentialapplicationofthefractalcontactmodelis