Nonlinear Elastic Waves in Materials

445 Pages·2014·10.532 MB·English

by  Jeremiah J. Rushchitsky (auth.)

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
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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