英文弹性力学之绪论

ElasticityABriefLectureNote(Incomplete)GuidelinesThelectureswillbegiveninChinese,buttheLectureNoteswillbewritteninEnglish(well,ratherinformalEnglish).ThebilingualapproachisanticipatedtooffersurprisingadvantagessuchasloweringthelanguagebarrierandgettingfamiliarwithEnglishvocabulariesthatwillbeindesperateuseintheyearstocome.Ialsoforeseethepotentialtoexpandthelecturenotestoaformallywritingtextbook.Thecontentsareselectedtohighlightthephysicalinsightsofelasticity,topromotethereadabilitybyfrequentlycitingthehistoriceventsandperspectives,enumeratingthenewapplicationsofelasticity,andaddressingthenewresearchthrustsofelasticityinthepast50years.Thelectureswillbegivenmainlyinatraditionalwayofchalksplusblackboard.ForacoursewithaperfectlogicsystemsuchasElasticity,Icannotfindabetterwayotherthantheoldtradition.Lettheimaginativemindsofthestudentsfollowthetraceofachalktiponblackboard.Besidelectures,thecourseshouldbecomplementedthroughintensivereadings.Twelvereferencesareappointed.Discussionsamongstudentsabouttheircontentsareencouraged.Thehomeworkwilltake20%oftheoverallscore,themid-termexamanother20%,whilethefinalexam(openbook)willtaketherest60%.Forthosestudentswhoareeagertoexploremorechallengingissuesofelasticity,atotalof10to15advancedproblemswillbeproposed.Astudentmaysolveanyoneoftheseproblems,writeintheformofareport,andsubmitittotheinstructor.Thesereportswillbegradedbytheinstructor,andthosescoredhigherthan90/100canbeusedassubstitutesfortheirfinalexams.Onlythereportsthatsolvetheproblemearliestamongtheentireclassorsolvetheprobleminauniqueapproacharequalifiedasthesubstitutesofthefinalexams.WhoCares?Elasticityisthemostimportantcourseinthefieldofsolidmechanics,andalsotopsthelistforthekeycoursesinabroaderareaofengineeringmechanics.Elasticitygoesalongwayinitsapplications,spanninglengthscalesfromnanometersforacarbonnanotubetokilometersofageologicalformation.Withoutthemasteryofelasticity,youareincompetentindoinganyseriousworksasaprofessionalinengineeringmechanics,evennotrankedasacapableengineer.Besideitscontents,elasticityisacoursethatexemplifiesthebeautyofmathematicalphysics.Manyconceptsandmethodsinmodernscienceandengineeringareoriginatedfromelasticity.However,elasticityisalsoaquitedifficultcoursetolearn,pleasebepreparedforit.Youdefinitelyneedtospend5timesasthelecturehourstograspthematerials.Withoutcompetitors,itwillbeTheCoursethatburdensyouthemostinthissemester.AsoneoftheelitecoursesinTsinghuaUniversity,weareplanningafeastofmaterialsthatwouldbeunparallelinthemostuniversitiesintheworld(theexceptionsareMITandEcolePolytechnique).SomeofthecontentspreparedinthislecturenotewillsurpassthecoverageofthetextbookbyMingwanLUandXuefuLUO.Pre-requestThestudentsarerequiredtohavefirstcoursesonCalculus,MathematicalPhysics(includingtheintroductiononComplexAnalysis,OperationalCalculus,andPDE)andStrengthofMaterials.WealsoassumeyouhaveapreliminaryknowledgeonContinuumMechanics,familiarwiththeconceptsofstress,strainandbalancelaws.Theknowledgeontheindicialnotationoftensorsisalsoassumed.Forthosestudentswhoarelackofthispreliminaryknowledge,wesuggestthemtoreadthefirstfourchaptersandtherelevantappendicesinthetextbookbyLuandLuo.ContentsIntroduction贤1.1冷溜 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(1.1)whereisthestiffnessofaBernoullibeam,Pthecompressiveload,andxandydelineatetheprofileofthebar.Asaclassicalexampletoillustratethehistoricimportanceofelasticity,wementionthattwoimportantmathematicalconceptswerebornastheby-productsofthiselasticityanalysis.Thefirstoneisthe“VariationalCalculus”whichEulerusedtoderivetheequation.Thesecondistheconceptof“Bifurcation”thatinitiatesthewholethemeofnonlinearanalysis.Thesolutionof(1.1),knownasthe“elastica”intheliterature,wasfoundbyEuler.AsshowninFig.1.2,underdifferentstagesof“barcompression”,thebarwouldflipovertobecomeafoldedbarinstretching!SeveralpossiblefoldingpatternsshowninFig.1.2existfordifferentloadingparameters.Figure1.2Euler’selasticaShortlyafterthetimeofEuler,mostgiftedscientistsweregatheredinFrance.OneactiveresearchthrustintheAcademyofFranceisthepursuitofelasticity.Severalscientificgiantsgotinvolved.ThelistincludesNavier,Poisson,Columb,CauchyandSaintVenant.In1821,Claude-Louis-Marie-HenriNavierpublishedanessayentitled“Equilibriumandmotionofelasticsolids”,inwhichthegoverningequationofelasticityisfirstformulatedas (1.2)whereisthedisplacementvector,Cameasureofelasticmodulusandthebodyforcevector.Thisequationiscalledsinceasthedisplacementequationofelasticity,orsimplythe“Navierequation”.Equation(1.2)isnotexactlyinthesameformasweareusingtoday,see(3.27).Itwasonlyvalidforaspecialmaterial,namelywhentwoLamèconstantsareequal.ItwasSimonDenisPoisson(1829)whobroughttheissueoflateralcontraction,andnamedafterhimisthePoisson’sratiothatrepresentsthenegativeratiobetweenextensionandtheinducedlateralcontraction.Initsoriginalform(1.2),theNavierequationisvalidonlyforaPoisson’sratioofonequarter.Poissonalsocontributedonfindingthelongitudinalandtransversewavesthatopenedthegatetoelastodynamics.ItwasthegreatFrenchmathematicianAugustin-LouisCauchy,originallyeducatedasanengineer,whoin1822formalizedtheconceptofstressinthecontextofageneralizedthree-dimensionaltheory,showeditspropertiesasconsistingofathreebythreesymmetricarrayofnumbersthattransformasatensor.CauchyisresponsibleforrelatingtractionvectorandstresstensorintheformofCauchy’sstressprinciple,theconceptsofprincipalstressesandprincipalstrains,thegeneralizedHooke’slaw,andtheequationsofmotionforacontinuumintermsofcomponentsofstresswiththeirboundaryconditions.Cauchyalsointroducedtheequationsthatexpressthesixcomponentsofstrain(threeextensionalandthreeshear)intermsofderivativesofdisplacementsforthecasewhereallthosederivativesaremuchsmallerthanunity;thoughsimilarexpressionshadbeengivenearlierbyEulerinexpressingratesofstrainingintermsofthederivativesofthevelocityfieldinafluid.Notonlyarigorousmathematician,Cauchyalsoexploredtheatomisticaspectofelasticityandderived,byusingthepairedinter-particlepotential,theso-calledCauchy’srelationoftheelasticitytensor.Thatrelationinferredacompletesymmetryoftheelasticitytensorwhoselimitationwillbediscussedinthenextchapter.ForthespecialcaseofisotropicsolidsexaminedindetailbyCauchy,thetheoryoflinearelasticresponseonlyrequirestheknowledgeofoneelasticconstant.ControversiesconcerningthemaximumpossiblenumberofindependentelasticmoduliinthemostgeneralanisotropicsolidweresettledbytheBritishmathematicianGeorgeGreenin1837.Greenpointedoutthattheexistenceofelasticstrainenergyrequiredthatofthe36elasticconstantsrelatingthe6symmetricstresscomponentsandthe6strains,atmost21couldbeindependent.TheScottishphysicistLordKelvinputthisconsiderationonsoundergroundin1855aspartofhisachievementofmacroscopicthermodynamics,showingthatastrainenergyfunctionmustexistforreversibleisothermaloradiabatic(isentropic)response.Themiddleandlate1800swereaperiodinwhichmanybasicelasticsolutionswerederivedandappliedtotechnologyandtotheexplanationofnaturalphenomena.AdhemerJeanClaudeBarredeSaint-Venant,astudentofNavier,contributedsignificantlytothisendeavor.Heinvented,amongotherthings,themethodofsemi-inversesolution(1853)andsolvedtheproblemsofbeambendingandtorsionofanon-circularprismaticrodinanaccuratemanner.WewilldevotethewholeChapter4onthissubject.HissolutionsevaluatedtheaccuracyofthebendingandtorsionsolutionsobtainedearlierfromasimplifiedmethodologyofStrengthofMaterials.Moreover,heproposedthefamousSaint-VenantPrinciple,whichgivesnumeroustasksforthemathematiciansandmechanicianstoengage.ThecontributionsfromtheGermanscientistsattheendofthe19thcentury,thenreplacedFranceastheintellectualcenteroftheworld,arealsoworthofmentioning.TheversatilePrussianphysicistGustovRobertKirchhoff,thefoundingmasterofelectro-magnetism,lefthismarksonelasticity.Inhisbook“Mechanik”publishedin1876,Kirchhoffextendedtheapplicationdomainofelasticitytoanewtypeofgeometry,plates.Kirchhoffappliedvirtualworkandvariationalcalculusprocedureintheframeworkofsimplifyingthekinematicalassumptionsthatfibersinitiallyperpendiculartotheplatemiddlesurfaceremainsoafterdeformationofthatsurface.Inonedimensionalversion,theKirchhoffplatetheoryassemblestheEuler-Bernoullibeamtheory.Itfindsimmenseapplicationsincivilandmechanicalengineeringasthestructuresintheformofplatesandshellsemerge.Anotherfoundingmasterofelectro-magnetismtheory,Helmholtz,alsocontributedtoelasticitybyincludingtheconceptofelasticfreeenergy,namedafterhimastheHelmholtzfreeenergy,andthesolutionforstresswavesintermsofHelmholtztransformation.FormingoftheMansion(1880-1950)Allpiecesofunderstandingsonelasticitywereassembledandsystemizedinthatperiodoftimetoformaformidablemansionofelasticityknowledge.BesidehisowncontributionsonthepointsourcetheoryandLovewave,Lovecompletedacomprehensivebook“ATreatiseontheMathematicalTheoryofElasticity(1892-1893)”.Thesymbolisclear;elasticitytookthecenter-stageamongallbranchesofmathematicalphysicsattheendthe19thcentury.Theintroductionofelasticitytovariousareasofengineeringwasgreatlybenefitedbytheenthusiasticeffortbyoshenko.TheformerlyRussiannobletyusedtoworkwithPrandtl,thefatherofaerodynamics.Timoshenkowasparticularlykeentotheengineeringaspectsofelasticity,andmadeenormouscontributionstotheengineeringelasticity,suchasbeamsonelasticfoundation,Timoshenkobeamtheory,mechanicsofplatesandshellsandelasticvibration.Timoshenkoisnotonlyascientistandanengineer,butalsoagreateducator.Hewrotemanybooksthatdominatedtheengineeringcollegeeducationfordecades.AlongwithVonKarman,TimoshenkowasresponsiblefortheprosperityofAppliedMechanicsintheUnitedStatesofAmerica.Twoadditionaldevelopmentsinthatperiodoftimeareworthmentioning.Thefirstisalongthelineofstructuresinlargedeflectionandbuckling,tackledbythegreatTheodereVonKarman,alongwithhisstudents,noticeablyH.S.TsienandW.Z.Chien.Oneofthefoundingmastersofquantummechanics,WernerHeisenberg,alsousedthesubjectofbucklingashisPh.D.thesis.ThesecondmajordevelopmentcamefromtheformerlySovietUnion,intheschoolrepresentedbyKolosovandMuskhelishivilli.Theydevelopedthecomplexpotentialmethodofelasticity.ThatsubjectwasbrilliantlysummarizedintwomonographsofMuskhelishivilli,namely“SomeBasicProblemsinMathematicalTheoryofElasticity”and“SingularIntegralEquations”.Themethodsofanalyticalfunction,Cauchyintegral,singularintegralequation,conformalmapping,Riemann-Hilbertlinearrelationshipproblemwereconciselyputtogethertogetarationalefortheplaneandanti-planeproblemsoflinearelasticity.Thecomplexpotentialmethodwasextendedlateronfromtheisotropicelasticitytoanisotropicelasticity.ExpandingtheHorizonofElasticity(Since1950)Variousbranchesofelasticitywereprosperedinthepasthalfcentury.ThefieldofelasticstabilitywasadvancedbytheDutchappliedmechanicianandengineerK.T.Koiter(1950),theconceptsofstatic,kinematicalanddynamicstabilitiesareproposed.Theissueofdefectsensitivityisextensivelyexplored.ThefieldoffracturemechanicsispioneeredbytheBritishaeronauticalengineerffith(1921).Griffithmadehisfamouspropositionthataspontaneouscrackgrowthwouldoccurwhentheenergyreleasedfromtheelasticfieldjustbalancedtheworkrequiredtoseparatesurfacesinthesolid.Thefieldbecamethecenter-stageinsolidmechanicssincethemid-20thcentury,largelyduetothere-evaluationforthenavallossinWorldWarIIandtheenthusiasticeffortbyGeorgeR.Irwin,anAmericanengineerandphysicist.Irwin(1957)proposedanalternativemeasureofthestressintensityfactorfortheseveritynearacracktip.LargelyundertheimpetusofIrwin,amajorbodyofworkonthemechanicsofcrackgrowthandfracture,includingfracturebyfatigueandstresscorrosioncracking,startedinthelate1940sandiscontinuingintothe21stcentury.ThefoundationofnonlinearfracturemechanicswaslaiddownlargelyduetotheworksofAmericanmechanicianandgeologiste(1968).Thecriticalparametersinfracturemechanics,suchastheenergyreleaserateG,thestressintensityfactorK,andtheJ-integralJarenamedafterGriffith,IrwinandRice,respectively.AnotherimportantdevelopmentwhichnowformsthebasicsolutionstrategyinengineeringistheinventionoftheFiniteElementMethod(FEM).ThismethodwasoutlinedbythemathematicianRichardCourant(1943)andwasdevelopedindependently,andputtopracticaluseoncomputers,from50sto60s,bytheaeronauticalengineersM.J.Taylor,RayW.CloughandothersinUnitedStates,thecivilengineersJ.H.ArgyrisandO.C.ZienkiewiczinBritain,andthemathematicianFENGKanginChina.Themethodoriginatesinsolvingelasticityproblems,andexpandsbeyondimaginationtoformthebasicbuildingblocksofcomputationalmechanics.ThenewestapplicationsofFEMincludematerialsmicrostructures,biologicalstructuresandmedicalprocedures.Thetheoreticalaspectofelasticityhasalsobeenenrichedintherecentyears.Theclassicaldevelopmentofelasticityneverfullyconfrontedtheproblemoffiniteelasticstraining,inwhichmaterialfiberschangetheirlengthsbyotherthanverysmallamount.Thewidespreaduseofnaturalpolymericmaterials,suchasnaturalrubbersputtheanalysisoffinitedeformationelasticityinlargedemand.TheworksofBritish-bornengineerandappliedmathematicianRonaldS.Rivlin(1960)forfinitedeformationelasticityprovidedsolutionsfortension,torsion,bendingandinflectionatextremelylargedeformation.Healsoparticipatedinintroducingthetensorrepresentationtheorem(Rivlin-Ericksentheorem)forisotropicelasticity.Theso-calledtheMooney-Rivlintheorygivesafairdescriptionforthesubjectofrubberelasticity.Anotherimportantarenaforthedevelopmentofelasticityistargetedforanisotropicmaterials.Inthisarea,theworksofJ.D.Eshelby,S.G.LehnitskiiandA.N.Stroh(1959－1962)madearevolutionarychangeofthefield.ThetragiclifeofStrohandhisbrilliantcontributioninashortprofessionalperiodoftenyearswasprescribedintheeulogyinthecomprehensivebookofT.C.T.Ting(1996).Thescopeofthislecturenoteisunfortunatelynotabletocoverthissubject,andthereadersmayconsultthebookbyTingasareference.1.2ApplicationsofElasticityConstructionElasticityfindsapplicationseverywhere.Itsapplicationsforbasicinfrastructurearethefirstonthelist.Thestudentsmayreadaninterestingbook“MechanicsandEngineering”(2000)editedbyWuYoushengtoexplorethewondersofelasticity.TheexamplesincludetheintegrityforThreeGorgesDam,criticalrotatingspeedforanelectricpowergenerator,andthedesigntosuppressthewind-inducedwrigglingoftheTVantennapole,knownasthe“EastPearl”,inShanghai.OfparticularinterestaretheelasticityproblemsinthefourbasicconstructionprojectsforChinatoexploitherwest.Thefirstprojectconcernsthegas-lineconstructionfromthewesttotheeast.Thecreepofthesanddomesmayinduceunpredictablehighstressonthegas-linepipes,causingtheirdynamicrupture.Anotherissueconcernswiththebuildingofsuper-sizegastanksthatimposeseverestructurereliabilityproblem.ThesecondprojectconcernstherailwayfromQinghaiProvincetoTibet,agoodportionofitwillpassthroughthehighlandwithaltitudehigherthan5000mabovesealevel.Thedifficultylargelyliesonthemechanicsofsoilunderextremelylowtemperature.Theelasticityoftheproblemisintertwinedwiththecapillaryfluidasoneconsidersthemazeofwater-icemixture.Thethirdprojectconcernsthepowertransmissionfromthewesttotheeast,involvingelasticityproblemssuchasthewrigglingofhighvoltageelectriccablesandthecriticalrpmoftheelectricgenerators.Thefourthprojectconcernshighwaysconstruction,wheretheelasticityoftheroadlayer,incombiningwithareliabletestmethodbystresswaves,willsafeguardthequalitycontroloftheroad.EarthquakeThecivilengineerscanquantifytheeventofearthquakes,aswellastheireffectonstructuresviaelasticity.Anearthquakemanifestsitselfasstresswavesofdilatationalandshearfeatures.Elasticitycanpredictthesourceandtheamplitudeofthequake,aswellasitspowerofdestruction.Investigationonfaultingdynamics(anexcellentreviewwasgivenbyJ.R.RiceastheOpeningLectureofICTAM2000,withaChinesetranslationbyGaofengGuoinAdvancesinMechanics,2001.8.25).Mostearthquakesaresubsonic.The1999earthquakeinTurkey,however,isintersonic.Thest