Foundations of Engineering Mechanics Jeremiah J. Rushchitsky Nonlinear Elastic Waves in Materials Foundations of Engineering Mechanics Series Editors: V. I. Babitsky, Jens Wittenburg For furthervolumes: http://www.springer.com/series/3582 ThiSisaFMBlankPage Jeremiah J. Rushchitsky Nonlinear Elastic Waves in Materials JeremiahJ.Rushchitsky DepartmentofRheology S.P.TimoshenkoInstituteofMechanics TheNationalAcademyofSciences Kyiv,Ukraine Series Editors: V.I.Babitsky J.Wittenburg Department of Mechanical Engineering Karlsruhe Instituteof Technology(KIT) LoughboroughUniversity Instituteof EngineeringMechanics LE113TU Loughborough Kaiserstr. 10 Leicestershire 76131Karlsruhe Great Britain Germany ISSN1612-1384 ISSN1860-6237(electronic) ISBN978-3-319-00463-1 ISBN978-3-319-00464-8(eBook) DOI10.1007/978-3-319-00464-8 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014937315 ©SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerpts inconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeing enteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthe Publisher’s location, in its current version, and permission for use must always be obtained from Springer.PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter. ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface The main goal of the book is a coherent treatment of the theory of propagating inmaterialsnonlinearlyelasticwavesofdisplacements,whichcorrespondstoone modernlineofdevelopmentofthenonlineartheoryofelasticwaves. The book can be conditionally divided on five basic parts: the necessary informationonwavesandmaterials;thenecessaryinformationonnonlineartheory of elasticity and elastic materials; analysis of one-dimensional nonlinear elastic waves of displacement—longitudinal, vertically and horizontally polarized trans- verse plane nonlinear elastic waves of displacement; analysis of one-dimensional nonlinear elastic waves of displacement—cylindrical and torsional nonlinear elastic waves of displacement; analysis of two-dimensional nonlinear elastic waves of displacement—Rayleigh and Love nonlinear elastic surface waves. In addition, the book includes all the necessary components of scientific book: the contents,foreword,thereferencelistineachchapter,afterword. Theauditoryofthisbookisassumedasthemoderatelyeducatedinthefieldof mechanics and mathematics. Sometimes the presence of elementary knowledge onlywillbeinsufficientforunderstandingthebook.Inthefieldofmechanics,the knowledgeoffundamentalsofcontinuummechanicswillberequired,whichinturn areavailableonconditionsthatelementsofarowofotherdivisionsofmechanics are known. In the field of mathematics, the elements of knowledge of the full university course (mathematical analysis, analytical and differential geometry, theory of functions of complex variable, vector and tensor calculation, higher algebra)willberequired. The book is addressed first of all to people working in solid mechanics—from studentsatanadvancedundergraduateandgraduateleveltoscientists,profession- allyinterestinginwaves.Butmechanicsisunderstoodinthebroadsense,whenit includesmechanicalandotherengineering,materialscience,appliedmathematics andphysicsandsoforth. v vi Preface The genesis of this book can be found in author’s years of research and teaching while a head of department at SP Timoshenko Institute of Mechanics (National Academy of Sciences of Ukraine), a member of Center for Micro and NanomechanicsatEngineeringSchoolofUniversityofAberdeen(Scotland)anda professor at Physical–mathematical Faculty of National Technical University of Ukraine“KPI.” Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 NonlinearityinMaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 ThreeFragmentsfromtheHistoryoftheNonlinear TheoryofWaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 RiemannSimpleWaves:TransitionfromLinearPlane WavestoNonlinearOnes. . . . . . . . . . . . . . . . . .. . . . . . 4 1.2.2 Earnshow’sSolutionoftheBasicEquationsof Hydrodynamics:TheOldestExampleofUsingthe SuccessiveApproximationsMethod. . . . . . . . . . . . . . . . 9 1.2.3 ToNonlinearWavesinOptics. . . . . . . . . . . . . . . . . . . . 11 1.3 StructureofBook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Audience. . . . .. . . .. . . .. . . . .. . . .. . . .. . . . .. . . .. 15 1.3.2 FiveBasicPartsoftheBook. . . . . . . . . . . . . . . . . . . . . 16 1.3.3 DetailedStructureofBook. . . . . . . . . . . . . . . . . . . . . . . 16 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 PreliminaryInformationAboutWavesandMaterials. . . . . . . . . . 25 2.1 AboutWaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.1 ObservationsandDefinitions. . . . . . . . . . . . . . . . . . . . . 25 2.1.2 ClassificationsofWaves. . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.3 FromHistoryofStudyingtheWaves. . . . . . . . . . . . . . . 28 2.2 AboutMaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.1 GeneralDefinitionsandClassifications. . . . . . . . . . . . . . 30 2.2.2 OnStructuralMechanicsofMaterials. . . . . . . . . . . . . . . 32 2.2.3 AFewWordsonNanotechnologyandNanomechanics ofMaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 35 2.2.4 ToStructuralNanomechanicsofCompositeMaterials. . . 37 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 vii viii Contents 3 ElasticMaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 BasicConceptsintheNonlinearTheoryofElasticity. . . . . . . . . 45 3.1.1 BasicConcepts:Body.Motion.Configuration, VectorofDisplacements. . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.2 BasicNotions:StrainTensors,Invariants, ChristoffelSymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.1.3 BasicNotions:Forces,Moments,StressTensors. . . . . . . 51 3.1.4 BasicNotions:BalanceLaws. . . . . . . . . . . . . . . . . . . . . 53 3.2 NonlinearElasticIsotropicMaterials:ThreeTypes ofElasticMaterials.. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 56 3.2.1 NonlinearElasticIsotropicMaterials:Generally ElasticMaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.2 NonlinearElasticIsotropicMaterials:Hypoelastic Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.3 NonlinearElasticIsotropicMaterials;Hyperelastic Materials:SethandSignoriniPotentials;Treloar, Mooney,Rivlin–Saundersmodels;John HarmonicMaterial.. . . .. . . .. . . .. . . .. . . .. . .. . . .. 61 3.2.4 NonlinearElasticIsotropicMaterials:Hyperelastic Materials(CubicPotentialsandMurnaghanPotential, andItsVariants). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 TheSimplestLinearWavesinElasticMaterials. . . . . . . . . . . . . . . 79 4.1 ClassicalLinearWavesintheTheoryofElasticity. . . . . . . . . . . 79 4.1.1 BasicEquations:VolumeandShearWaves. . . . . . . . . . 79 4.1.2 ClassicalWaveEquation:BasicFacts. . . . . . . . . . . . . . . 81 4.1.3 ClassicalWaveEquation:WavesastheResult ofBreakingtheCorrectness. . . . . . . . . . . . . . . . . . . . . . 84 4.1.4 ClassicalWaveEquation:BasicCharacteristics andTerminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.1.5 ClassicalWaveEquation:PlaneWaves. . . . . . . . . . . . . 90 4.1.6 ClassicalWaveEquation:CylindricalWaves. . . . . . . . . 95 4.2 ClassicalLinearWavesintheTheoryofElasticMixtures. . . . . . 97 4.2.1 SomeKnownMicrostructuralTheoriesofMaterials. . . . 97 4.2.2 BasicEquations:VolumeandShearWaves inMixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2.3 ClassicalWaveEquation:PlaneWavesinMixtures. . . . 109 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Contents ix 5 NonlinearPlaneLongitudinalWavesinElasticMaterials (MurnaghanModel,Five-ConstantModel). . . . . . . . . . . . . . . . . . . 121 5.1 QuadraticallyNonlinearElasticPlaneLongitudinalWaves. . . . . 122 5.1.1 QuadraticallyNonlinearWaveEquationsDescribing PlaneWaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.1.2 MethodofSuccessiveApproximationsasApplied toStudyingPlaneHyperelasticHarmonicWaves. . . . . . 124 5.1.3 NumericalModelingasanAddendumtothePrevious Subsection.. . . .. . . .. . . .. . . .. . . .. . . . .. . . .. . . .. 132 5.1.4 ProblemonTripletsofQuadraticallyNonlinear ElasticPlanePolarizedWaves. . . . . . . . . . . . . . . . . . . . 142 5.1.5 MethodofSlowlyVaryingAmplitudesasApplied totheStudyofPlaneHyperelasticHarmonic LongitudinalWaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.1.6 MethodofSlowlyVaryingAmplitudes.Self-Switching ofTwoLongitudinalElasticHarmonicPlaneWaves. . . . 151 5.2 CubicallyNonlinearElasticPlaneLongitudinalWaves. . . . . . . . 161 5.2.1 BasicNonlinearWaveEquations. . . . . . . . . . . . . . . . . . 161 5.2.2 GenerationofNewHarmonicsbyLongitudinalPlane CubicallyNonlinearElasticHarmonicWave (FirstStandardProblem). . . . . . . . . . . . . . . . . . . . . . . . 164 5.2.3 InfluenceofThirdHarmonicsProgressonEvolution ofLongitudinalPlaneElasticWaveProfile. . . . . . . . . . . 166 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6 NonlinearPlaneLongitudinalWavesinElasticMaterials (JohnModel,Two-ConstantModelandSignoriniModel, Three-ConstantModel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.1 NonlinearPlaneLongitudinalElasticHarmonicWaves (JohnModel,Two-ConstantModel,Geometrically NonlinearOnlyModel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.1.1 QuadraticallyNonlinearElasticPlane LongitudinalWaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.1.2 CubicallyNonlinearElasticPlaneLongitudinalWaves. . . 179 6.2 NonlinearPlaneLongitudinalElasticHarmonicWaves (SignoriniModel—Three-ConstantModel). . . . . . . . . . . . . . . . 182 6.2.1 UtilityofUniversalDeformationsinanAnalysis ofSignoriniNonlinearModel. . . . . . . . . . . . . . . . . . . . . 182 6.2.2 TransitionfromMurnaghan-BasedNonlinear WaveEquationstoSignorini-BasedNonlinear WaveEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.2.3 LongitudinalNonlinearWavesintheSignoriniModel. . . 194 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196