Fatigue, Stress, and Strain of Rubber Components: 'A Guide for Design Engineers

173 Pages·2008·3.947 MB·English

by  Judson Bauman

Judson T. Bauman
Fatigue, Stress, and
Strain of
Rubber Components
A Guide for Design Engineers
Hanser Publishers, Munich Hanser Publications, Cincinnati
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Bauman,JudsonT.
Fatigue,stress,andstrainofrubbercomponents:aguidefordesign
engineers/JudsonT.Bauman.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-56990-431-2
1. Rubber--Testing.2. Reliability(Engineering)3. Engineeringdesign.
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TA455.R8B382008
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Contents
1 Introduction................................................................................................................... 1
1.1 Objective................................................................................................................ 1
1.2 Discovery............................................................................................................... 1
1.2.1 FirstVulcanization.................................................................................... 2
1.2.2 EarlyManufactureofRubberProducts................................................... 2
1.2.3 DiscoveryofReinforcement..................................................................... 2
1.2.4 ProductionofRubber............................................................................... 2
1.3 TheRubberMolecule........................................................................................... 3
1.4 Synthetics.............................................................................................................. 3
1.4.1 CuringandCrosslinking........................................................................... 4
1.4.2 FillersandReinforcement......................................................................... 5
1.4.3 CuringIngredients.................................................................................... 5
1.4.4 OtherAdditives......................................................................................... 6
1.5 PrincipalUsesofSeveralElastomers................................................................... 6
Bibliography.......................................................................................................................... 7
2 RubberStress-StrainBehavior...................................................................................... 9
2.1 ChallengesofRubberBehavior............................................................................ 9
2.2 CharacteristicsofStress-StrainBehavior............................................................. 9
2.2.1 LowElasticModulus,HighElongationatBreak,andNon-Linearity.... 9
2.2.2 Hysteresis................................................................................................... 10
2.2.3 StressRelaxation....................................................................................... 11
2.2.4 Creep.......................................................................................................... 11
2.2.5 MullinsEffect............................................................................................ 12
2.2.6 Reinforcement........................................................................................... 13
2.2.7 CyclicFrequencyandStrainRate............................................................. 14
2.2.8 Temperature.............................................................................................. 15
2.2.9 ImmersionEffects..................................................................................... 15
2.2.10 StrainCrystallization................................................................................ 16
2.2.11 PermanentSet........................................................................................... 17
2.2.12 Recovery.................................................................................................... 17
Bibliography.......................................................................................................................... 18
3 ATheoryof theElastomerStress-Strain Curve.......................................................... 19
3.1 Introduction.......................................................................................................... 19
3.2 TheInternalStructureoftheVulcanizedElastomer.......................................... 20
3.3 AssumptionsandHypotheses.............................................................................. 21
3.3.1 TheCoilSpringAnalogy........................................................................... 21
3.3.2 ChainSegmentsandTerminations.......................................................... 24
VI
3.3.3 StatisticalDistributionofChainsinLength
andEndPoint Separation........................................................................ 24
3.3.4 ThePresenceofvanderWaalsBonds...................................................... 25
3.3.5 ReinforcementbyParticleRotation......................................................... 28
3.3.6 MigrationofEntanglements..................................................................... 31
3.3.7 Temperature-InducedChainVibration.................................................. 32
3.3.8 BondBreakingandRemakinginDeformation....................................... 33
3.3.9 Parallelism-InducedCrystallization......................................................... 33
3.4 ElastomerBehaviors............................................................................................. 34
3.4.1 TheNon-LinearStress-StrainCurve........................................................ 34
3.4.2 TheMullinsEffect..................................................................................... 34
3.4.3 LowElasticModulusandHighElongationatBreak.............................. 36
3.3.4 Hysteresis................................................................................................... 37
3.4.5 StiffeningbyReinforcingFillers............................................................... 37
3.4.6 StrainRateStiffening................................................................................ 37
3.4.7 TemperatureResponse............................................................................. 38
3.4.8 StressRelaxationandCyclicStressRelaxation........................................ 38
3.4.9 CreepandCreepunderCyclicConditions.............................................. 38
3.4.10 PermanentSet........................................................................................... 39
3.4.11 Recovery.................................................................................................... 39
3.4.12 StrainCrystallization................................................................................ 39
Acknowledgements............................................................................................................... 39
References............................................................................................................................. 40
4 Stress-StrainTesting...................................................................................................... 43
4.1 Introduction.......................................................................................................... 43
4.2 TensileTesting...................................................................................................... 43
4.2.1 Specimens.................................................................................................. 43
4.2.2 TestingwiththeDumbbellSpecimen...................................................... 44
4.2.3 TestingwiththePlanarStressSpecimen.................................................. 49
4.2.4 TestingwiththeLoopSpecimen.............................................................. 52
4.3 ShearTesting......................................................................................................... 54
4.3.1 Stress-StrainState...................................................................................... 54
4.3.2 Specimens.................................................................................................. 54
4.4 BiaxialStrainTesting............................................................................................ 57
4.4.1 TheBubbleTest........................................................................................ 57
4.4.2 TheCrossSpecimen.................................................................................. 63
4.5 CompressionTesting............................................................................................ 64
4.6 Summary............................................................................................................... 66
References............................................................................................................................. 66
VII
5 DesignEquations........................................................................................................... 69
5.1 Introduction.......................................................................................................... 69
5.1.1 UseofDesignEquations........................................................................... 69
5.1.2 ElasticConstants....................................................................................... 69
5.2 DesignEquationsforVariousGeometries.......................................................... 71
5.2.1 PadsinShear............................................................................................. 71
5.2.2 PadsinTorsion......................................................................................... 73
5.2.3 Bushings..................................................................................................... 74
5.2.4 PadsinCompression................................................................................ 76
5.2.5 CompressionofaLongStrip.................................................................... 80
5.2.6 SolidRubberRollers................................................................................. 81
5.2.7 Rubber-CoveredRollers........................................................................... 82
5.2.8 CompressionofaRubberSphere............................................................. 82
5.2.9 CompressionofSolidRubberTire.......................................................... 83
5.2.10 CompressionofSolidRubberRingofCircularCross-Section.............. 84
5.2.11 SolidRubberRingwithRectangularCross-Section................................ 84
5.2.12 Indenter,FlatEndedCylinder.................................................................. 85
5.2.13 Indenter,SphericalHead.......................................................................... 86
5.2.14 Indenter,Conical...................................................................................... 86
5.2.15 Indenter,LongNarrowFlatEnd.............................................................. 86
5.2.16 ProtrusionThroughaRoundHole.......................................................... 87
5.2.17 ProtrusionThroughLongNarrowGap................................................... 87
5.3 Summary............................................................................................................... 87
References............................................................................................................................. 88
6 CalculationMethodsforSpherical ElastomerBearings............................................. 89
6.1 Introduction.......................................................................................................... 89
6.2 HistoryoftheSphericalBearing.......................................................................... 89
6.3 MathematicalDescriptionoftheBearing............................................................ 91
6.3.1 OverallBearingParameters...................................................................... 92
6.3.2 ParametersofParticularPads................................................................... 92
6.3.3 AngularMoment....................................................................................... 95
6.4 ShearStrainofPadsunderAngularDeflection................................................... 95
6.5 AxialLoads............................................................................................................ 99
6.5.1 CompressionofPadsunderAxialForce..................................................100
6.5.2 BulgeShearStrain.....................................................................................101
6.5.3 SummaryofCalculations.........................................................................103
6.6 TorsionalLoads.....................................................................................................103
6.6.1 ShearStrainofPadsunderTorsionalRotation.......................................104
6.6.2 ComputationalProcedure........................................................................104
6.6.3 Limitations................................................................................................105
References.............................................................................................................................105
VIII
7 FiniteElementAnalysis.................................................................................................107
7.1 Introduction..........................................................................................................107
7.2 Procedure..............................................................................................................107
7.2.1 Symmetry...................................................................................................108
7.2.2 LoadsandBoundaryConditions.............................................................108
7.2.3 ElementSelectionandMeshing...............................................................108
7.3 MaterialModelorConstitutiveEquations..........................................................109
7.3.1 SimplerConstitutiveEquations...............................................................110
7.3.2 HigherOrderConstitutiveEquations......................................................110
7.4 FittingEquationstoTestData.............................................................................111
7.5 O-RingSealwithPressure....................................................................................112
7.6 RubberBoot..........................................................................................................114
7.7 Summary...............................................................................................................115
Acknowledgements...............................................................................................................115
References.............................................................................................................................115
8 FatigueTesting...............................................................................................................117
8.1 Introduction..........................................................................................................117
8.2 ParametersAffectingtheStrain-LifeCurve.........................................................117
8.2.1 ParameterstoBeSpecified........................................................................118
8.2.2 SelectingStrainAmplitude.......................................................................118
8.3 FailureCriteria......................................................................................................118
8.4 R-Ratio..................................................................................................................119
8.5 CombinedStrainState..........................................................................................119
8.6 WaveForm............................................................................................................121
8.7 CreepandStressRelaxation.................................................................................122
8.8 FrequencyandStrainRate....................................................................................122
8.9 EffectofTemperature...........................................................................................123
8.10 LiquidImmersion.....................................................................................124
8.11 Recovery....................................................................................................125
8.12 Scragging....................................................................................................125
8.13 BatchVariation.........................................................................................125
8.14 Storage.......................................................................................................126
Acknowledgements...............................................................................................................126
References.............................................................................................................................126
9 FittingtheStrain-LifeCurve.........................................................................................127
9.1 Introduction..........................................................................................................127
9.2 DevelopmentofanEquationforNin εa,RandT............................................127
9.3 TheStrain-LifeCurveEquationwithNagel’sEquation forTemperature........130
9.4 EmployingtheSimpleEmpiricalFormula forTemperature.............................131
Acknowledgements...............................................................................................................132
References.............................................................................................................................133
IX
10 FatigueLifeEstimation..................................................................................................135
10.1 Introduction..........................................................................................................135
10.2 SingleWaveForm,theε-NMethod....................................................................135
10.3 TheMiner’sNumber............................................................................................136
10.4 TheDeterministicFatigueSpectrum...................................................................136
10.5 SampleCalculationoftheMiner’sNumber........................................................137
10.6 WhiteNoise...........................................................................................................138
10.6.1 RainflowCounting....................................................................................139
11 FatigueCrackGrowthandTearingEnergy..................................................................143
11.1 Introduction..........................................................................................................143
11.2 GriffithStrainEnergyReleaseRate......................................................................143
11.2.1 GriffithCriterion.......................................................................................143
11.2.2 Derivation..................................................................................................143
11.2.3 GriffithConditionforFracture................................................................146
11.2.4 CriticalAssumptions.................................................................................146
11.3 RivlinandThomasandTearingEnergy..............................................................147
11.3.1 ModificationofGriffith’sCriterionforFractureofMetals....................147
11.3.2 ApplicationtoRubber..............................................................................147
11.3.3 StateofCriticalAssumptions...................................................................149
11.4 ShortcutFormulasforT.......................................................................................150
11.5 TearingEnergyAppliedtoFatigueCrackGrowth..............................................151
11.5.1 PioneeringDevelopmentsinFatigue.......................................................151
11.5.2 TheChangeinDefinitionofTearingEnergy..........................................151
11.6 Limitations............................................................................................................152
11.6.1 FatigueCrackGrowthParameter.............................................................152
11.6.2 CyclestoFailurebyTor εa?...................................................................154
11.7 SummaryandConclusions..................................................................................156
Acknowledgements...............................................................................................................157
References.............................................................................................................................157
AppendixI. RubberNomenclature....................................................................................159
Appendix2. FatigueTerminology......................................................................................167
Appendix3. EnglishtoMetricConversion........................................................................177
Appendix4. FittingtheStrain-LifeCurve..........................................................................179
Appendix5. DerivationofTearingEnergy Equations......................................................187
Appendix6. DerivationofEquations forSphericalElastomerBearings.........................193
Index ......................................................................................................................................213
1.4 Synthetics 1
1 Introduction
1.1 Objective
Ifyouareamechanicalengineerworkingeasilywithsteelpartsbutarechallengedbythe
behavior of rubber in the components you design, then this book is for you. Similarly, if
you are a chemist comfortable with the concept of rubber’slong chain molecules but are
requiredtoestimateitsmechanicalbehaviorinacomponent,thisbookisforyou.Previ-
ousknowledgeofrubberbehaviormaybehelpfulbutisnotnecessary.
Thistextemphasizesthemechanicalbehavioroftheelastomer.Itdiscussesthemolecular
and micro configuration of the rubber matrix and how they produce the observed
mechanical behavior. But in this text chemical behavior, chemistry of formulation, and
role of various compounds in the formula are not included. The analysis of the compo-
nentbehavioristheobjective.
1.2 Discovery
On his second voyage to the New World, it is said that Christopher Columbus observed
Native Americans in present day Haiti playing a game with a rubber ball. The earliest
written record by Europeans was made in 1530, by Pietro Martire d’Anghiera, an Italian
attached to the Spanish court, who described “gummi optimum” in his writings. Other
early visitors to the New World mentioned a rubber ball in ceremonial games played by
theAztecsanddescribedtheapplicationofthelatextoclothforwaterresistance.
Inlatercenturies,onlyoccasionalmentionwasmadeofrubberinEuropeanliterature.But
inthe18thcentury,twoFrenchmenbroughtrubbertotheattentionofthegrowingscien-
tific communities of their time. In 1735, Charles Marie Condamine on behalf of the
FrenchAcademyledanexpeditiontoSouthAmerica.Ayearlater,inQuito,hesentback
samples of a black resinous substance which he called caoutchouc in imitation of the
Quechuawordmeaningweepingwood.In1747,anotherFrenchman,Fresneau,wholived
in Cayenne, explored the forests of Guiana in search of the source of the native manu-
factured rubber items he had encountered. There he discovered several latex producing
trees including Hevea braziliensis and actually fashioned a pair of boots from the latex.
Fresneau reported his findings to the Academy in 1751. However, it was Joseph Priestly,
theEnglishchemist,whonameditrubber.In1770,Priestlynoticedthatitremovedpencil
markingsfrompaperwhenrubbedoverthem.Thusthetermrubberwasborn.
2 1 Introduction
1.2.1 FirstVulcanization
Therewereveryfewproductsmadeofrubberinthecenturiesaftertheinitialdiscovery.It
wasoccasionallyusedasawaterproofcoatingforclothandperhapsasaneraserbutlittle
else. Unvulcanized rubber is similar to chewing gum. It does not hold its form well and
becomesstickywhenwarm.Then,in1839,CharlesGoodyeardiscoveredvulcanization.He
found that when raw rubber was mixed with sulfur and then heated, the puttylike sub-
stance lost its stickiness and was made elastic. It would spring back to its original shape
afterdeformation.Thiselasticbehaviorandbettertolerancetoelevatedtemperatureledto
rubberbeingusedinamuchwidervarietyofconsumerproducts.
1.2.2 EarlyManufactureofRubberProducts
For over a century after Goodyear’s discovery, all rubber goods were manufactured from
natural crepe vulcanized with sulfur. In the 20th century, man-made synthetic crude
rubbers were developed and became available in commercial quantities. They were pre-
pared by reacting certain low-molecular weight monomers to form long chain polymers.
Naturalrubberandthesyntheticstakentogetherarecalledelastomers,atermderivedfrom
elasticpolymers.
1.2.3 DiscoveryofReinforcement
Itwasnotuntilthefirstdecadeofthe20thcenturythatreinforcementbycarbonblackwas
discovered.Itwasa majorstepforward,particularlyforthemanufacturersoftires.Rein-
forcementgreatlyincreasesthestrengthandstiffnessoftheelastomer,allowingthetireto
lastmanytimeslonger.Today,nearlyallnaturalrubberproductsarereinforcedwithcar-
bonblack.
1.2.4 ProductionofRubber
Rubber latex is gathered from cuts made in the bark of the tree Hevea brasiliensis. The
treeiscultivatedintheplantationsofMalaysiaandothertropicalcountries.Rubbertrees
are tapped for the latex every other day. The cuts produce a milky fluid, which is coagu-
latedandthenprocessedintosheetsorcrepeforshipping.Atthisstageitisstillrawrubber
andhaslittlepracticaluse.Itflowsunderloadasthelongmoleculesslideovereachother,
thus it remains in its deformed shape when the load is removed. It crystallizes readily at
temperaturesaround0°Candbelow,anditbecomessoftandstickyinhotweather.
Most natural rubber latex today is produced in Thailand and Southeast Asia. In the year
2007,9.9millionmetrictonsofnaturalrubberwereproducedworldwide.Ofthistonnage,
about¾wentintotireproduction.
1.4 Synthetics 3
1.3 TheRubberMolecule
Tothenineteenthcenturychemist,naturalmaterialssuchaswood,leather,rubber,andso
forth, were puzzles since they did not behave as their chemical formulas suggested. As
earlyas1826,MichaelFaradaydeterminedthattheformulaforrubberwasCH.In1838,
5 8
JeanBaptisteAndreDumasconfirmeditandin1860,GrevilleWilliamsfirstisolatediso-
prene. Then in 1879, Gustave Bouchardt determined that isoprene was, in fact, natural
rubber. But the evidence of its physical behavior indicated that the molecule was much
larger than the modest size implied by CH. Finally, in the 1920’s, Hermann Staudinger
5 8
putforwardthethenrevolutionaryideathattherubbermoleculewasagiantmolecule,or
macromolecule.Today,itiswellknownthatthenaturalmaterialswhichformusefulfibers
suchaswood,cotton,andleather,aswellasplasticsandrubberareallcomposedofsuch
macromolecules.
Besides being macromolecules these natural substances are also polymers. A polymer is
simplyasubstancemadeupofaseriesofrepeatingunitsjoinedinalongchain.Soinstead
oftheformulaforrubberbeingCH,itshouldberepresentedas(CH),wherenisonthe
5 8 5 8 n
order of 10,000. The repeat unit CH is the monomer and when formed into a chain, it
5 8
becomes a polymer. Natural rubber consists of chains of the isoprene monomer and is
calledpolyisoprene.Innaturalrubberaswellasmostofthesynthetics,thebackboneofthe
chain is made up of carbon atoms. They are linked together mostly by single bonds with
animportantminorityofdoublebonds.
The configuration of the isoprene unit has a definite and unique geometrical pattern
(Fig.1.1.).Anyotherarrangementwouldresultinamaterialofquitedifferentmechanical
properties. Innatural rubber thearrangement is designated cis, hence the chemical name
for natural rubber – cis,1,4-polyisoprene. The other arrangement of the isoprene unit is
the trans configuration found in gutta percha which is trans-polyisoprene. When two
polymersofthesameformulaaredistinguishedfromoneanotherbyonlythegeometrical
arrangement within the monomer, they are called stereo-isomers. Thus, polyisoprene has
twoisomers,cis-polyisopreneandtrans-polyisoprene.
1.4 Synthetics
After Straudinger’s explanation of rubber as a macromolecule was advanced, efforts to
synthesize natural rubber or a replacement began in earnest. In 1929, Thiokol A became
thefirstcommerciallyproducedsyntheticrubber.Inthelate1920’sand1930’s,companies
especially in Germany and the U.S. began developing rubberlike compounds. Among
these were nitrile (NBR) by I.G. Farben in Germany and Butyl rubber (IIR) by Standard
OilintheU.S.WorldWarIIalsoaddedtotheurgencytofindrubbersubstitutesbecause
most of the world was cut off from Southeast Asia where most natural rubber was
produced. After the war, inspired by the pioneering work, discoveries and development
continued.