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(theexceptionsareMITandEcolPre-requestThestudentsarerequiredtohavefirstcoursesonCalculus,MathematicalPhysics(includingtheintroductiononComplexAnalysis,OperationalCalculus,andPDE)andStrengthofMaterials.WealsoassumeyouhaveapreliminaryknowledgeonContinuumMechanics,familiarwiththeconceptsofstress,strainandbalancelaws.Theknowledgeontheindicialnotationoftensorsisalsoassumed.Forthosestudentswhoarelackofthispreliminaryknowledge,wesuggestthemtoreadthefirstfourchaptersandtherelevantappendicesinthetextbookbyLuandLuo.

ContentsIntroduction1.1 HistoryofElasticity1.2 ApplicationsofElasticity2． ElasticityofSolids2.1 DefinitionofElasticity2.2 TwoPhysicalOriginsofElasticity2.3 TensorDescriptionofElasticity2.4 PhysicalFoundationofElasticSymmetry3． FieldEquationsofElasticity-DifferentialFormulation3.1 BalanceEquationsofMomentumandMoment3.2 CompatibilityEquation3.3 FieldEquationsofDynamicElasticity3.4 Quasi-staticFieldEquations3.5 Constraints3.6 BoundaryConditions3.7 FormulationofElasticity4. PrismaticRods4.1 FormulationforPrismaticRods4.2 UniaxialTensionandPureBending4.3 CorrectiveSolution(SaintVenantDecay)4.4 FreeTorsionofPrismaticRods4.5 InverseandSemi-inverseSolutions4.6 FormulationofAnti-planeProblems5． PlaneProblems–TheoryandMethods5.1 PlaneStrainandPlaneStress5.2 PlanarAnisotropicCase5.3 PlanarIsotropicCase–BiharmonicEquation5.4 SolutioninCartesianCoordinates5.5 SolutioninPolarCoordinates5.6 Kolosov-MuskhelishvilliMethod6． PlaneProblems–Applications6.1 StressConcentration6.2 CurvedBeams6.3 Wedges6.4 PointForceandCouples6.5 ContactandIndentation6.6 Dislocations6.7 VoidsandInclusions6.8 Mismatch6.9 Singularities6.10 Interfaces7． VariationalFormulationofElasticity7.1 BasicConceptsofVariation7.2 WeakSolutions7.3 PrincipleofVirtualWork7.4 VariationalPrinciples7.5 NumericalMethodsBasedonEnergyPrinciples8． ThreeDimensionalProblemsinElasticity8.1 SolutionsbyDisplacementPotentials8.2 BoussinesqSolution8.3 SingularitySolution8.4 EshelbySolution8.5 DislocationLoopsReviewReferences陸明萬、羅學(xué)富《彈性理論基礎(chǔ)》，清華大學(xué)出版社，1990J.R.Barber,Elasticity,KluwerAcademicPublisher,Boston,1992S.P.TimoshenkoandJ.N.Goodier,TheoryofElasticity,3rdEdition,McGraw-Hill,1969H.Love,ATreatiseontheMathematicalTheoryofElasticity,4thDoverEdition,1944A.E.GreenandW.Zerna,TheoreticalElasticity,OxfordN.I.Muskhelishvilli,SomeBasicProblemsinMathematicalTheoryofElasticity,NoordhoffInt.,1977M.E.Gurtin,ContinuumMechanics,AcademicPress,1981J.E.MarsdenandT.J.R.Hughes,MathematicalFoundationsofElasticity,PrenticeHallInc,1983武際可,力學(xué)史,重慶大學(xué)出版社,1999楊衛(wèi)，宏微觀斷裂力學(xué)，1995，國防工業(yè)出版社，第1章W.YangandW.B.Lee,MesoplasticityandItsApplications,1993,Springer-VerlagT.C.T.Ting,AnisotropicElasticity:TheoryandApplications,1996,Oxford

Introduction1.1 HistoryofElasticityReferencesS.P.Timoshenko,Historyofstrengthofmaterial,Dover,1953R.Dugas,Ahistoryofmechanics,Dover,1955武際可,力學(xué)史,重慶大學(xué)出版社,1999Mechanicssymbolizesthefirstglimpseofscientificunderstandingofthehumanbeingforthephysicalworld.Mechanicsformsthebackboneofscienceandengineering.Mechanicspavesthefoundationfortheinfrastructuresofnumerouscitiesintheworld. LIUQi,MayorofBeijing,InvitationLetterforICTAM2008BeijingEraoftheExploration(1600－1700)ElasticitydevelopedintheoutpouringofmathematicalandphysicsstudiesfollowingtheeraofNewton,althoughithasearlierroots.Theneedtounderstandandcontrolthefractureseemstohavebeenafirstmotivation.LeonardodaVincisketchedinhisnotebooksapossibletestofthetensilestrengthofawirethatmightbecrucialinhanginghispaintings.Hewasawareofthepossiblelengthdependenceofthewirestrengthduetostatisticaldefectsdistribution.TheclassicalmechanicsissometimeslabeledGalileo-Newtonmechanics.Thereasonisquiteclear.Galileo,whodiesintheyearofNewton’sbirth(1642),proposedthePrincipleofInertia,whileNewtonextendedittoThreeBasicLawsofMechanics.TheclassicalworksofGalileo“ADialogueofTwoNewSciences(1683)”isamilestoneinthedevelopmentofmechanics.BesidethePrincipleofInertianowknownineveryhouse-hold,Galileoconductedalengthydiscussiononthedeformationandstrengthofsolidsinthatbook.Heinvestigatedthebreakingloadsofrodsundertensionandconcludedthattheloadwasindependentoflength(contrasttotheperceptionofdaVinciwhichwasbasedonthestatisticaldistributionofdefectsalongthewirelength)andproportionaltothecrosssectionarea.Itwasquiteunusualinanerawhenastronomydominatedthescience.Hetookonamaintargetofnowadaysstrengthofmaterials,thebendingofabeam.AhistoricaccountofGalileo’smethodologyisdescribedinthebook“HistoryofStrengthofMaterials”byTimoshenko.ThebasicschematicsareshowninFig.1.1.Abucketofwater(oranyformofweight)ishangedononeendofabeamoflengthLandarectangularcross-section.Theotherendofthebeamisclampedinawall.Galileoperformedthemechanicsanalysisforacantileverbeam.Thelatteristreated,thefirsttimeinthehistory,asadeformablebody.Theanalysisprovidedcorrectgeometricdependenceofthestrengthontheparameterssuchasthebeamlengthandthebendingresistanceofitscross-section.Unfortunately,thestressdistributionacrosstheheightofthebeamiswrong.ThelongitudinalstresswasassumedtobezerobyGalileoatthebottomofthebeam,ratherthanattheneutralplaneaswasfoundlater.Figure1.1Galileo’sperceptionofbeambendingTheconceptofanelasticrelationwaspioneeredbytheEnglishscientistRobertHooke.Thelawwasdiscoveredin1660,butpublishedonlyin1678.Inhispaperentitled“OfSpring”,theelasticrelation,initsmostprimitiveform,wasphrasedinananagramofLatin“ceiiiosssttuu”.Rearrangingtheletters,onefindstherealmeaningofso-calledHooke’sLaw“uttensiosicvis”,or“tensionisproportionaltothestretch”inEnglish.Hooke’slawestablishedthenotionof(linear)elasticitybutnotyetinawaythatwasexpressibleintermsofstressandstrain.StrokesoftheGenius(1700-1880)Earlydevelopmentsofelasticityrecordedmanyworksofscientificgiants.TheconceptsofstressandstrainwereintroducedbytheBernoullibrothers.ItwastheSwissmathematicianandmechanicianJakobBernoulliwhoobserved,inthefinalpaperofhislife,in1705,thattheproperwayofdescribingadeformationwastogiveforceperunitarea,orstress,asafunctionofelongationperunitlength,orstrain,ofamaterialfiberundertension.TheSwissmathematicianandmechanicianLeohardEuler,whowastaughtmathematicsbyJakob’sbrotherJohannBernoulli,proposedalinearrelationbetweenstressandstrain,in1727,oftheform.ThecoefficientEisnowgenerallycalledYoung’smodulusaftertheBritishnaturalistThomasYoung,whodevelopedarelatedideain1807.In1744,LeonardEuleranalyzedthebucklingofacompressedbar.Thedeflectionofthebarwasshowntoobeythefollowingequation: (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,CauchyIn1821,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.TheintroductionofelasticitytovariousareasofengineeringwasgreatlybenefitedbytheenthusiasticeffortbyS.P.Timoshenko.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.ThefieldoffracturemechanicsispioneeredbytheBritishaeronauticalengineerA.A.Griffith(1921).Griffithmadehisfamouspropositionthataspontaneouscrackgrowthwouldoccurwhentheenergyreleasedfromtheelasticfieldjustbalancedtheworkrequiredtoseparatesurfacesinthesolid.Thefieldbecamethecenter-stageinsolidmechanicssincethemid-20thcentury,largelyduetothere-evaluationforthenavallossinWorldWarIIandtheenthusiasticeffortbyGeorgeR.Irwin,anAmericanengineerandphysicist.Irwin(1957)proposedanalternativemeasureofthestressintensityfactorfortheseveritynearacracktip.LargelyundertheimpetusofIrwin,amajorbodyofworkonthemechanicsofcrackgrowthandfracture,includingfracturebyfatigueandstresscorrosioncracking,startedinthelate1940sandiscontinuingintothe21stcentury.ThefoundationofnonlinearfracturemechanicswaslaiddownlargelyduetotheworksofAmericanmechanicianandgeologistJ.R.Rice(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,incombiningwitharelEarthquakeThecivilengineerscanquantifytheeventofearthquakes,aswellastheireffectonstructuresviaelasticity.Anearthquakemanifestsitselfasstresswavesofdilatationalandshearfeatures.Elasticitycanpredictthesourceandtheamplitudeofthequake,aswellasitspowerofdestruction.Investigationonfaultingdynamics(anexcellentreviewwasgivenbyJ.R.RiceastheOpeningLectureofICTAM2000,withaChinesetranslationbyGaofengGuoinAdvancesinMechanics,2001.8.25).Mostearthquakesaresubsonic.The1999earthquakeinTurkey,however,isintersonic.Thestudyforsuchproblempromotestherecentresearchesofintersonicfracturemechanics.Thepredictionofearthquakesreliesonthedetectablesignalsbytheprecursoroftheearthquakes.Doestheprecursorexistanddetectable,ornotdetectableinprinciple?Thisisthecurrentresearchsubjectinelasticity.Thephenomenonoffailurewaveswasjustdiscoveredseveralyearsago.Thedispersionforthepropagationofthefailurewavewillshedlightstothepredictabilityofanearthquake.Anotherissueinwhichelasticitywillprovidesahelpinghandistosuppressthevibrationinducedbyearthquakes.AgoodexampleforsuppressedvibrationunderearthquakeinancientJapancanbefoundinthe33Palace,Kyoto,wherelayeredfoundationwasutilizedtoabsorbanddiffusetheshakenofearthquakesoverhundredsofyears.Themodernexampleconcernswiththeapproachofactivecontrol.Acloselyloopedsystemcouldfirstsensetheamplitudeandfrequencyofthevibration,thencounter-balanceitbyout-of-phasemotionoftheactuators.Largelyattributedtothistechnique,theskyscraperscannowbebuiltinJapan.AstronauticEngineeringThesametechniqueofvibrationsuppressionisinampledemandinaeronauticandastronauticengineering.Thevibrationofarocketanditspayloads(suchassatellites)duringlaunchingisacriticalissue.Theprooftestsonarealsizevibrationtableusuallyconsumealargeamountoftheexpensesofsatellitemakingandoccupyasubstantialdurationinitsproductionperiod.Uponair,theflutteringofair-frameisacriticalissuewhichendorsedtheestablishmentofareoelasticity.Furtherupinthespace,thegravityforcediminishes,leavingonlytheelasticityeffectforspace-platforms.Thevibrationforthesolar-energycellpaneldictatesthesuccessofaspacemission.Thefunctionalityofamissilealertsatellitedependsontheaccuracytotonedownitsframevibrationwhenagiganticradarantennarotatestoscanthesky.Forthescientificexplorationinthespace,theFASTprojectutilizestheuniquegeologicalformationinChina.Adownscaled(oneto