Continuum Mechanics, Volume 1: Foundations and Applications of Mechanics

877 Pages·2016·48.847 MB·English

by  Chandrashekhar S Jog

CONTINUUM MECHANICS
Foundations and Applications of Mechanics
Volume I, Third Edition
C. S. Jog
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Continuummechanics/C.S.Jog. –Thirdedition.
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Summary: "Presentsseveraladvancedtopicsincludingfourth-ordertensors,
differentiationoftensors,exponentialandlogarithmictensors,
andtheirapplicationtononlinearelasticity"–Providedbypublisher..
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Contents
ListofFigures x
ListofTables xv
Preface xvii
Notation xxi
1 IntroductiontoTensors 1
1.1 VectorSpaces 1
1.2 Vectorsin(cid:60)3 8
1.3 Second-OrderTensors 12
1.3.1 Thetensorproduct 15
1.3.2 Principalinvariantsofasecond-ordertensor 17
1.3.3 Inverseofatensor 20
1.3.4 Eigenvaluesandeigenvectorsoftensors 27
1.4 Skew-SymmetricTensors 29
1.5 OrthogonalTensors 31
1.6 SymmetricTensors 40
1.6.1 Principalvaluesandprincipaldirections 40
1.6.2 Positivedefinitetensorsandthepolardecomposition 48
1.6.3 Isotropicfunctions 53
1.7 Higher-OrderTensors 60
1.8 IsotropicTensors 77
1.9 DifferentiationofTensors 80
1.9.1 Thedirectionalderivative 81
1.9.2 Productrule 83
1.9.3 Chainrule 84
1.9.4 Gradient,divergenceandcurl 85
1.9.5 Examples 97
1.10 TheExponentialandLogarithmicFunctions 99
1.11 DivergenceandStokes’Theorems 117
1.12 Groups 119
2 Kinematics 137
2.1 LagrangianandEulerianDescriptions 137
2.2 Length,AreaandVolumeElementsintheDeformedConfiguration 139
vi Contents
2.2.1 Lengthelementinthedeformedconfiguration: Straintensors 140
2.2.2 Areaelementinthedeformedconfiguration: ThePiolatransform 146
2.2.3 Volumeelementinthedeformedconfiguration 149
2.3 VelocityandAcceleration 149
2.4 RateofDeformation 151
2.5 ExamplesofSimpleMotions 160
2.5.1 Pureextension 160
2.5.2 Rigidbodymotion 162
2.5.3 Simpleshear 162
3 BalanceLaws 166
3.1 TheFirstTransportTheorem 167
3.2 ConservationofMass 168
3.2.1 Lagrangianversion 168
3.2.2 Eulerianversion 169
3.3 TheSecondTransportTheorem 170
3.4 GeneralizedTransportTheorems 170
3.5 BalanceofLinearMomentum 172
3.6 BalanceofAngularMomentum 178
3.7 PropertiesoftheCauchyStressTensor 181
3.8 TheEquationsofMotionintheReferenceConfiguration 185
3.9 VariationalFormulations 188
3.10 EnergyEquation 191
3.11 ControlVolumeFormoftheBalanceLaws 199
4 ConstitutiveEquations 203
4.1 FrameofReference 203
4.2 TransformationofKinematicalQuantities 206
4.3 PrincipleofFrame-Indifference 208
4.4 PrincipleofMaterialFrame-Indifference 219
4.5 ConstitutiveRelationsforSimpleMaterials 222
4.6 MaterialSymmetry 226
4.7 ClassificationofMaterials 231
5 NonlinearElasticity 240
5.1 IsotropicElasticParticles 240
5.2 TheConstitutiveEquationofanIsotropicSolidfor’Small’Deformations 244
5.3 BoundsontheLame´ Constants 246
5.4 HyperelasticSolids 247
Contents vii
5.5 IsotropicHyperelasticSolids 253
5.6 StVenant–KirchhoffMaterial 257
5.7 ExamplesofNonlinearCompressibleHyperelasticModels 259
5.8 TheElasticityTensors 263
5.9 ElasticandMaterialStability 270
5.10 NonuniquenessofSolutionsinElasticity 282
5.11 ExactSolutionsforHomogeneous,Compressible,IsotropicElasticMaterials 283
5.11.1 Uniaxialstretch 284
5.11.2 Pureshear 285
5.11.3 Purebendingofaprismaticbeammadeofaparticular
StVenant–Kirchhoffmaterial 287
5.11.4 TorsionofacircularshaftmadeofaStVenant–Kirchhoffmaterial 289
5.12 ExactSolutionsforHomogeneous,Incompressible,
IsotropicElasticMaterials 290
5.12.1 Bendingandstretchingofarectangularblock 292
5.12.2 Straightening,stretchingandshearingasectorofacylinder 294
5.12.3 Torsion,inflation,bending,etc. ofanannularwedge 295
5.12.4 Inflation/eversionofasphericalshell 300
6 LinearizedElasticity 308
6.1 Kinematics 308
6.2 GoverningEquations 315
6.3 EnergyandVariationalFormulationsinLinearizedElasticity 322
6.3.1 Single-fieldvariationalformulation 323
6.3.2 Two-fieldandthree-fieldvariationalformulations 329
6.4 UniquenessofSolution 331
6.5 ExactSolutionsofsomeSpecialProblemsinElasticity 333
6.5.1 Torsionofacircularcylinder 333
6.5.2 Torsionofnon-circularbars–Saint-Venanttheoryoftorsion 335
6.5.3 GeneralizationoftheSaint-Venanttorsiontheorytoan
anisotropic,inhomogeneousbar 357
6.5.4 Torsionofcircularshaftsofvariablediameter 371
6.5.5 Purebendingofprismaticbeams 387
6.5.6 Bendingofprismaticbeamsbyterminalloads 392
6.5.7 Hollowspheresubjectedtouniformpressure/Gravitatingsphere 425
6.6 GeneralSolutionsforElastostaticsusingPotentials 427
6.6.1 Cylindrical/ellipticalelasticinclusioninaninfinitedomainwith
auniformstateofstressatinfinity 463
6.6.2 Rectangulardomain(e.g.,cantileverbeam)loadedbytractions
onitsedges 479
viii Contents
6.6.3 Circulardiscloadedbyatractiondistributiononitsrim 506
6.6.4 Wedgeloadedbytractiondistributionsonitsedges 510
6.6.5 Thermalstresses 537
6.6.6 Thickhollowcylindersubjectedtoalinearlyvaryingpressure
ontheinnerandoutersurfaces 540
6.6.7 Sphere/sphericalsegmentspinningaboutitsaxis 542
6.6.8 Circularcylinderwithloadingonitsendfaces,andlateralsurfaces
traction-free 546
6.6.9 Circularcylinderwithloadingonthelateralsurface,andend
surfacestraction-free(’Filon’sproblem’) 553
6.6.10 Clampedandsimplysupportedcircularcylinders 560
6.6.11 Circularcylinderspinningaboutitsaxis 565
6.6.12 Circularcylinderonafrictionlesssurfaceloadedunderits
ownweight 568
6.6.13 Pointloadinaninfiniteelasticbody(Kelvinproblem) 572
6.6.14 Pointloadactingnormaltotheboundaryofaninfinitehalf-space
(Boussinesqproblem) 574
6.6.15 Truncatedcone 575
6.6.16 Contactproblemsonafinitedomain 589
6.6.17 Sphericalcavityinaninfinitedomainwithauniaxialstateof
stressatinfinity 594
6.6.18 Prolateoroblatespheroidalcavityinaninfinitedomain
withauniformstressstateatinfinity 599
6.6.19 Pointloadactingtangentialtotheboundaryofaninfinite
half-space(Cerrutiproblem) 603
6.7 Elastodynamics 605
6.7.1 Progressivewaves 605
6.7.2 Solutiontospecialproblems 607
7 Thermomechanics 686
7.1 Thermoelasticmaterials 686
7.2 Viscoplasticmaterials 695
7.3 Restrictionsontheconstitutiverelationsforfluids 699
7.3.1 Thermodynamicrelationsforaperfectgas 713
7.3.2 TheNavier–Stokesandenergyequations 715
7.3.3 SummaryofthegoverningequationsforaNewtonianfluid 715
8 Rigid-BodyDynamics 718
Example1: Central-forcemotion 725
Example2: Cylinderrollingdownaplane 727
Example3: Slidingrod 728
Contents ix
Example4: Motionofasleigh 730
Example5: Spinningdisc 732
Example6: Spinningtop 734
Example7: Forceonabar 736
Example8: Rotatingbar 737
Example9: Slider-crankmechanism 738
Appendices 742
A OrthogonalCurvilinearCoordinateSystems 742
B CylindricalCoordinateSystem 751
C SphericalCoordinateSystem 756
D EllipticCylindricalCoordinateSystem 760
E BipolarCylindricalCoordinateSystem 764
F ProlateSpheroidalCoordinateSystem 768
G OblateSpheroidalCoordinateSystem 772
H ArbitraryCurvilinearCoordinateSystems 776
I MatrixRepresentationsofTensorsforEngineeringApplications 789
J SomeResultsinn-DimensionalEuclideanSpaces 802
K ANoteonBoundaryConditions 820
Bibliography 825
Index 847