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Holm Altenbach Andreas Öchsner Editors Encyclopedia of Continuum Mechanics Encyclopedia of Continuum Mechanics Holm Altenbach • Andreas Öchsner Editors Encyclopedia of Continuum Mechanics With1092Figuresand82Tables 123 Editors HolmAltenbach AndreasÖchsner LehrstuhlTechnischeMechanik FakultätfürMaschinenbau InstitutfürMechanik EsslingenUniversityofAppliedSciences FakultätfürMaschinenbau Esslingen,Germany Otto-von-Guericke-UniversitätMagdeburg Magdeburg,Germany ISBN978-3-662-55770-9 ISBN978-3-662-55771-6(eBook) ISBN978-3-662-55877-5(printandelectronicbundle) https://doi.org/10.1007/978-3-662-55771-6 ©Springer-VerlagGmbHGermany,partofSpringerNature2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewhole orpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof illustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway, andtransmissionorinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:HeidelbergerPlatz3,14197Berlin,Germany Preface Continuum mechanics can be considered as one of the classical branches of mechanics and influences, in a broader sense, all engineering designs and structures of our daily life independent of the scale and taking into account solids and fluids in the same manner. Without a solid foundation in continuummechanics,itisnotpossibletomasterengineeringdisciplinessuch as mechanical, civil, or aerospace and engineering progress requires a solid theoreticalfoundation. In this Encyclopedia, continuum mechanics will be presented focusing on classical theories and new developments. In addition, information on numericalandexperimentalmethodswillbeprovided.Forthecompleteness, somebiographicalsurveysarepresented.Itseemsthatcontinuummechanics is a purely phenomenological approach, but one can discuss the problems of continuum mechanics on various scales preserving the fundamentals and applying some modifications only. After all, the common starting point for allbranchesofcontinuummechanicsisthecontinuityassumptioninsteadof discreteparticles. The state of the art of continuum mechanics is given in thousands of journal papers and various monographs and textbooks. However, with this Encyclopedia the reader will be able to find short articles on continuum mechanics providing a first insight into the complex structure of continuum mechanics and references for further reading. The authors of the entries present their individual point of view and by this the Encyclopedia will stimulatethescientificdiscourse. WithintheEncyclopedia, onereceives theminimumrequiredknowledge of continuum mechanics, which means, in the traditional sense, material- independent and material-dependent equations. In addition to examples of special continua, historical remarks and some topics beyond continuum mechanics like mathematical and numerical methods are included. The internationally recognized team of contributors employs a consistent and systematic approach, offering readers a user-friendly reference that is ideal forfrequentconsultation. ThepresentEncyclopediacoversmostareasofclassicalandnonclassical continuum mechanics including theoretical, experimental, and numerical aspects.Additionalinformationisgivenconcerningsomemathematicaltools (e.g.,tensorcalculus),numericalmethods(FEM,BEM),orbasicknowledge on material behavior. Any article belonging to continuum mechanics in thenarrowersensewillfollowthestructure:material-independent equations v vi Preface (kinematics, loadings, balances) and material-dependent equations charac- terizing the individual response of the given material on external loadings (constitutiveandevolutionequations).Finally,forsolvingpracticalproblems, boundary and initial conditions should be introduced. The Encyclopedia is notrestrictedtothree-dimensionalcontinuumproblemsandtheories.Shells, plates,beams,androdsarediscussedaswell.Inaddition,notonlymechanical loadings are taken into account that means the influence of temperature, electrical,magnetic,chemical,andotherfieldsinsomeentriesisconsidered. Severalentriespresentnewtrendswithrespecttobiologicalsystems. This three-volume Encyclopedia comprises approximately 350 entries now. Please note that it is a “living system” with possible online improve- ments and extensions anytime. Comments, critical remarks, etc. are highly appreciated. The editors will do their best to make the Encyclopedia better andbetter.Atthesametime,anextensionwillberealizedsystematically. Wewouldliketoexpressoursincereappreciationtoalltheauthors,section editors, and the representatives of Springer, who made this major reference workinmechanicspossible. Magdeburg,Esslingen,Germany HolmAltenbach November2019 AndreasÖchsner List of Topics AdaptiveStructures Descartes,René Drucker,DanielCharles SectionEditor:HansPeterMonner Engesser,Friedrich ActiveControlofSound,Applicationsof Eringen,AhmedCemal ActiveControlofVibration,Applicationsof Euclid AdaptiveStructures,Principlesof Euler,Leonhard MorphingStructures,Applicationsof Föppl,AugustOtto Föppl,Ludwig Gauß,JohannCarlFriedrich BiographiesofScientists Geiringer,Hilda Germain,Marie-Sophie SectionEditor:HolmAltenbach Germain,Paul Aero,EronLyuttovich Gol’denveizer,AlexeiL’vovich Airy,GeorgeBiddell Green,George Ambartsumian,Sergey(Aleksandr) Grioli,Giuseppe Argyris,JohnHadji Hayes,MichaelAlphonsus Arutyunyan,NagushKhachaturovich Hencky,Heinrich Bauschinger,Johann Hill,Rodney Bernoulli,Daniel Hooke,Robert Bernoulli,JakobI Huber,MaksymilianTytus Bernoulli,Johann Ilyushin,AlexeiAntonovich Bingham,EugeneCook Johnson,William Bolotin,VladimirVasilyevich Kachanov,LazarMarkovich Bridgman,PercyWilliams Kármán,Theodorevon Budiansky,Bernard Kirchhoff,GustavRobert Burgers,JohannesMartinus Knowles,JamesKenyon Burzyn´ski,WłodzimierzStanisławTrzywdar Koiter,WarnerTjardus Carathéodory,Constantin Kudo,Hideaki Carroll,MichaelMary Kupradze,Victor Cauchy,AugustinLouis Lagrange,Joseph-Louis Christoffel,ElwinBruno Lee,ErastusHenry Codazzi,Delfino Lehmann,JohannesTheodor Coleman,BernardDavid Levi-Civita,Tullio Cosserat,EugèneandFrançois Levy,Maurice deCoulomb,Charles-Augustin Lippmann,Horst Crandall,StephenHarry Love,AugustusEdwardHough vii viii ListofTopics Ludwik,Paul Biomechanics Lurie,AnatoliIsakowitsch SectionEditor:DanielBalzani Mainardi,Gaspare Marguerre,Karl ComputationalModelsforHemodynamics Maugin,GérardA. ContinuumMechanicsofSoftTissueGrowth Mindlin,RaymondDavid andRemodeling vonMises,Richard CoupledProblemsinBiologicalSystems Murakami,Sumio DamageinSoftBiologicalTissues Mushtari,KhamidMuzafarovich ModelingFrameworkforComputational Muskhelishvili,Nikoloz Physiology Naghdi,PaulMansour ModelingofBoneAdaptionProcesses Neuber,HeinzAugustPaul ModelingofHeartMuscles Newton,Isaac ModelingoftheHumanCornea Nikuradze,Johann(Ivane) Patient-SpecificSimulationofAbdominalAortic Noll,Walter Aneurysms Novozhilov,Valentin SkeletalMuscleModelling Nádai,Arpád Oniashvili,Otar Composites Orowan,Egon SectionEditor:ErasmoCarrera Palmov,VladimirAlexandrovich Papkovich,PetrFedorovich Axiomatic/AsymptoticMethod Piola,Gabrio andBestTheoryDiagramforComposite Pipkin,AllenComporte PlatesandShells Pobedrya,BorisEfimovich BoundaryElementMethodforComposite Poisson,SiméonDenis Laminates Polanyi,Mihály(Michael) BucklingandPostbucklingofComposite Prager,William PlatesandShells Prandtl,Ludwig ClassicalandAdvancedModelingof Rabotnov,YuriNikolaevich RotatingCompositeBladesandRotors Reissner,Eric HierarchicalModelingofFunctionally Reuss,András GradedBeams Rychlewski,Jan HigherOrderTheoryofFunctionally deSaint-Venant,Adhémar-Jean-Claude GradedShells Barré HygrothermoelasticStressinOrganic Siebel,Erich MatrixCompositeMaterials Sneddon,IanNaismith Mechanicsof3DFibreReinforced Stokes,GeorgeGabriel PolymerComposites Taylor,GeoffreyIngram MethodstoDeterminetheMechanical Timoshenko,StepanProkofievitch PropertiesofCompositeMaterials Truesdell,CliffordAmbroseIII withContinuousFibers Vekua,Ilia ModelingDamageInitiationandPropagation Voigt,Woldemar inCompositeStructures Vorovich,IosifIzrailevich Node-DependentKinematics,Multilayered Westergaard,HaroldMalcolm Beam,Plate,andShellElements Young,Thomas Quasi-3DVibrationAnalysisofLaminated Zerna,Wolfgang CompositeShells Zhilin,PavelAndreevich Variable-Kinematics,MeshlessAnalysis Zienkiewicz,OlgierdCecil ofCompositeBeams ListofTopics ix ContinuumDamageMechanics ThermalEffectsbyMeansofTwo-Component CosseratContinuum SectionEditor:ArturGanczarski ThermoelasticDiffusionTheoryfor ContinuumDamageModelfor PiezoelectricMaterials DuctileMaterialsBased ThermoelasticWavesinaMediumwith onStress-State-DependentDamageFunctions Heat-FluxRelaxation DeactivationofDamageEffects ThermomechanicalProcessingofSteels EffectiveVariablesandDamage andAlloys:MultilevelModeling EffectFunctions Truesdell’sandZhilin’sApproaches:Derivation MetalFormingSimulationBased ofConstitutiveEquations onAdvancedMechanicalModel Zhilin’sMethodandItsModifications StronglyCoupledwithDuctileDamage PhenomenologicalModelofOrthotropic DamageinPolymerMatrixComposites ContinuumWaves SectionEditor:JüriEngelbrecht ContinuumMechanicsBasics ElasticWavesinMicrostructured Solids SectionEditor:RainerGlüge NonlinearWavesinContinuousMedia ContinuumMechanicsBasics, SurfaceWaves IntroductionandNotations ThermoelasticWaves FrameworksforMaterialModeling WavesinContinuousMedia:ClassicalTheory Kinematics Material-IndependentBalances PrinciplesofMaterialModeling CreepMechanics SectionEditor:Błaz˙ejSkoczen´ ContinuumThermodynamics AnisotropyofLinearCreep CreepatExtremelyLowTemperatures SectionEditor:ElenaA.Ivanova CreepDeformation ChemicalAffinityTensorin CreepFatigue CoupledProblemsofMechanochemistry CreepinModernMaterials Coleman-NollProcedureforClassical CreepinStructures andGeneralizedContinuumTheories IntroductiontoCreepMechanics ContinuumMechanicswithSpontaneous MicromechanicsofCreep ViolationsoftheSecondLaw Viscoelasticity ofThermodynamics Viscoplasticity DiscreteandContinuumThermomechanics DispersionRelationsfortheCoupled FractionalCalculusinContinuum HyperbolicThermoelasticity Mechanics EntropyPrincipleExploitedbyLagrange SectionEditor:YuryA.Rossikhin Multipliers andMarinaV.Shitikova KineticTheoryandThermodynamics, Non-equilibriumReactingGasFlows ClassicalBeamsandPlatesinaFractional NonlocalTransportEquationsforSmallSystems DerivativeMedium,ImpactResponse andFastProcesses CollisionofTwoSphericalShells,Fractional SystemofSymmetricHyperbolicEquations, OperatorModels ExtendedThermodynamicsofGases FractalViscoelasticModels x ListofTopics FractionalCalculusinBiomechanics HomogenizationMethodsandGeneralized FractionalCalculusinThermoelasticity ContinuainLinearElasticity FractionalDerivativeTimoshenkoBeams MicromorphicApproachtoMaterials andUflyand-MindlinPlates,ImpactResponse withInternalLength FractionalOperatorModelsofViscoelasticity NonlocalTheories Fractional-OrderOperatorsinFracture StrainGradientPlasticity Mechanics WavesandGeneralizedContinua LinearandNonlinearVibrations:Fractional Oscillators NumericalMethodsfortheFractional ImpactMechanics DifferentialEquationsofViscoelasticity SectionEditor:FilipeTeixeira-Dias ThinBodiesEmbeddedinFractional DerivativeViscoelasticMedium,Dynamic AnalyticalandEnergy-Based Response MethodsforPenetrationMechanics VibrationsofSuspensionBridges:Fractional Crashworthiness DerivativeModel ElasticandInelasticStressWaves VibrationsofViscoelasticRods,Fractional HighStrainRateMetalPlasticity OperatorModels ImpactofCellularMaterials WavePropagationinViscoelasticRods, ImpactofTextileStructures FractionalOperatorModels ImpactonCeramicMaterials ImpactonReinforcedConcreteStructures NumericalMethodsandModelingin ImpulsiveDynamics FractureMechanics PenetrationMechanicsofRigidProjectiles inMetallicTargets SectionEditor:Eduard-MariusCraciun SoftImpact AnalysisofCracksinPiezoelectric SolidswithConsideration ofElectricFieldandStrainGradients LightweightStructures DamageandFailureoftheVascularWall SectionEditor:FranzG.Rammerstorfer, DynamicalContactProblemsofFracture MelanieTodt,andIsabellaC.Skrna-Jakl Mechanics DynamicsandThermodynamicsofFracture AnalyticalStabilityConsiderationsin Mechanics LightweightDesign FractureMechanicsofComposites Beams,Plates,andShells FractureMechanicsofElasticComposites DamageandFailureinLayeredComposite NonlinearFractureMechanics Structures FractureMechanicsinLightweightDesign GeneralizedContinua MacromechanicsofLaminates MethodsofPlasticityTheoryinLightweight SectionEditor:SamuelForest Design ComputationalMechanics Micromechanics ofGeneralizedContinua Numerical Treatment of Stability Problems in CosseratMedia LightweightDesign DislocationsandCracksinGeneralized PureTorsionofThin-WalledCross Continua Sections HigherGradientTheoriesandTheir ShearWebTheory Foundations StaticandDynamicBifurcations