Continuum Mechanics for Engineers, Third Edition

550 Pages·2·17.536 MB·English

by  X. Oliver C. , Agelet de Saracibar

CONTINUUM MECHANICS
FOR ENGINEERS

THEORY AND PROBLEMS

Second Edition

X. Oliver
C. Agelet de Saracibar

Xavier Oliver
Carlos Agelet de Saracibar
Continuum Mechanics
for Engineers
Theory and Problems
TranslationbyEsterComellas
X.OliverandC.AgeletdeSaracibar ContinuumMechanicsforEngineers.TheoryandProblems
doi:10.13140/RG.2.2.25821.20961
ContinuumMechanicsforEngineers.TheoryandProblems
Firstedition:September2016
Secondedition:March2017
Citeas:
X. Oliver and C. Agelet de Saracibar, Continuum Mechanics
for Engineers. Theory and Problems, 2nd edition, March 2017,
doi:10.13140/RG.2.2.25821.20961.
TranslationandLATEX compilationbyEsterComellas.
© XavierOliverandCarlosAgeletdeSaracibar.
OpenAccessThisbookisdistributedunderthetermsoftheCreativeCommons
Attribution Non-Commercial No-Derivatives (CC-BY-NC-ND) License, which
permits any noncommercial use, distribution, and reproduction in any medium
of the unmodified original material, provided the original author(s) and source
arecredited.
X.OliverandC.AgeletdeSaracibar ContinuumMechanicsforEngineers.TheoryandProblems
doi:10.13140/RG.2.2.25821.20961
Foreword
This book was born with the vocation of being a tool for the training of engi-
neersincontinuummechanics.Infact,itisthefruitoftheexperienceinteaching
this discipline during many years at the Civil Engineering School of the Tech-
nical University of Catalonia (UPC/BarcelonaTech), both in undergraduate de-
grees(CivilEngineeringandGeologicalEngineering)andpostgraduatedegrees
(Master and PhD courses). Unlike other introductory texts to the mechanics of
continuous media, the work presented here is specifically aimed at engineering
students. We try to maintain a proper balance between the rigor of the math-
ematical formulation used and the clarity of the physical principles addressed,
althoughalwaysputtingtheformerattheserviceofthelatter.Inthissense,the
essential vector and tensor operations use simultaneously the indicial notation
(moreusefulforrigorousmathematicalproof)andthecompactnotation(which
allowsforabetterunderstandingofthephysicsoftheproblem).However,asthe
textprogresses,thereisacleartrendtowardscompactnotationinanattemptto
focusthereader’sattentiononthephysicalcomponentofcontinuummechanics.
Thetextcontentisintentionallydividedintotwospecificparts,whicharepre-
sentedsequentially.Thefirstpart(Chapters1-5)introducesfundamentalandde-
scriptive aspects common to all continuous media (motion, deformation, stress
and conservation-balance equations). In the second (Chapters 6 to 11), specific
families of the continuous medium are studied, such as solids and fluids, in an
approachthatstartswiththecorrespondingconstitutiveequationandendswith
the classical formulations of solid mechanics (elastic-linear and elasto-plastic)
and fluid mechanics (laminar regime). Finally, a brief incursion into the varia-
tionalprinciples(principleofvirtualworkandminimizationofpotentialenergy)
is attempted, to provide the initial ingredients needed to solve continuum me-
chanicsproblemsusingnumericalmethods.Thisstructureallowstheuseofthis
text for teaching purposes both in a single course of about 100 teaching hours
orastwodifferentcourses:thefirstbasedonthefirstfivechaptersdedicatedto
the introduction of the fundamentals of continuum mechanics and, the second,
specificallydedicatedtosolidandfluidmechanics.Thetheoreticalpartinevery
chapter is followed by a number of solved problems and proposed exercises so
v
vi
astohelpthereaderintheunderstandingandconsolidationofthosetheoretical
aspects.
Finally,theauthorswishtothankDr.EsterComellasforhertranslationwork,
frompreviousversionsofthetheoreticalpartofthebookinSpanishandCatalan
languages, as well as for her compilation of the book’s problems and exercises
fromtheauthors’collection.
Barcelona,September2016
XavierOliver
and
CarlosAgeletdeSaracibar
X.OliverandC.AgeletdeSaracibar ContinuumMechanicsforEngineers.TheoryandProblems
doi:10.13140/RG.2.2.25821.20961
Contents
Foreword....................................................... v
1 DescriptionofMotion........................................ 1
1.1 DefinitionoftheContinuousMedium ....................... 1
1.2 EquationsofMotion...................................... 1
1.3 DescriptionsofMotion.................................... 6
1.3.1 MaterialDescription ................................ 6
1.3.2 SpatialDescription ................................. 6
1.4 TimeDerivatives:Local,MaterialandConvective ............. 8
1.5 VelocityandAcceleration ................................. 11
1.6 Stationarity ............................................. 14
1.7 Trajectory............................................... 15
1.7.1 DifferentialEquationoftheTrajectories ............... 16
1.8 Streamline .............................................. 18
1.8.1 DifferentialEquationoftheStreamlines................ 18
1.9 Streamtubes ............................................. 20
1.9.1 EquationoftheStreamtube .......................... 20
1.10 Streaklines .............................................. 21
1.10.1EquationoftheStreakline ........................... 22
1.11 MaterialSurface ......................................... 23
1.12 ControlSurface .......................................... 26
1.13 MaterialVolume ......................................... 26
1.14 ControlVolume.......................................... 27
ProblemsandExercises ................................... 29
2 Strain ...................................................... 41
2.1 Introduction ............................................. 41
2.2 DeformationGradientTensor .............................. 41
2.2.1 InverseDeformationGradientTensor.................. 43
2.3 Displacements ........................................... 45
2.3.1 MaterialandSpatialDisplacementGradientTensors ..... 46
vii
viii Contents
2.4 StrainTensors ........................................... 47
2.4.1 MaterialStrainTensor(Green-LagrangeStrainTensor)... 48
2.4.2 SpatialStrainTensor(AlmansiStrainTensor)........... 49
2.4.3 StrainTensorsintermsoftheDisplacement(Gradients) .. 51
2.5 VariationofDistances:StretchandUnitElongation............ 51
2.5.1 Stretches,UnitElongationsandStrainTensors .......... 53
2.6 VariationofAngles....................................... 55
2.7 PhysicalInterpretationoftheStrainTensors .................. 57
2.7.1 MaterialStrainTensor .............................. 57
2.7.2 SpatialStrainTensor................................ 60
2.8 PolarDecomposition ..................................... 62
2.9 VolumeVariation ........................................ 64
2.10 AreaVariation ........................................... 66
2.11 InfinitesimalStrain ....................................... 67
2.11.1StrainTensors.InfinitesimalStrainTensor.............. 68
2.11.2Stretch.UnitElongation............................. 71
2.11.3PhysicalInterpretationoftheInfinitesimalStrains ....... 71
2.11.4EngineeringStrains.VectorofEngineeringStrains....... 73
2.11.5Variation of the Angle between Two Differential
SegmentsinInfinitesimalStrain ...................... 74
2.11.6PolarDecomposition................................ 74
2.12 VolumetricStrain ........................................ 79
2.13 StrainRate .............................................. 80
2.13.1VelocityGradientTensor ............................ 81
2.13.2StrainRateandSpinTensors ......................... 82
2.13.3PhysicalInterpretationoftheStrainRateTensor......... 82
2.13.4PhysicalInterpretationoftheRotationRateTensor ...... 84
2.14 MaterialTimeDerivativesofStrainandOtherMagnitudeTensors 85
2.14.1DeformationGradientTensoranditsInverseTensor ..... 85
2.14.2MaterialandSpatialStrainTensors.................... 86
2.14.3VolumeandAreaDifferentials ....................... 87
2.15 MotionandStrainsinCylindricalandSphericalCoordinates .... 89
2.15.1CylindricalCoordinates ............................. 90
2.15.2SphericalCoordinates............................... 92
ProblemsandExercises ................................... 95
3 CompatibilityEquations .....................................109
3.1 Introduction .............................................109
3.2 Preliminary Example: CompatibilityEquations ofa Potential
VectorField .............................................110
3.3 CompatibilityConditionsforInfinitesimalStrains .............112
3.4 IntegrationoftheInfinitesimalStrainField ...................116
3.4.1 PreliminaryEquations ..............................116
3.4.2 IntegrationoftheStrainField ........................118
3.5 CompatibilityEquationsandIntegrationoftheStrainRateField .121
X.OliverandC.AgeletdeSaracibar ContinuumMechanicsforEngineers.TheoryandProblems
doi:10.13140/RG.2.2.25821.20961
Contents ix
ProblemsandExercises ...................................123
4 Stress ......................................................127
4.1 ForcesActingonaContinuumBody ........................127
4.1.1 BodyForces.......................................127
4.1.2 SurfaceForces .....................................129
4.2 Cauchy’sPostulates ......................................130
4.3 StressTensor ............................................132
4.3.1 ApplicationofNewton’s2nd LawtoaContinuousMedium132
4.3.2 StressTensor ......................................133
4.3.3 GraphicalRepresentationoftheStressStateinaPoint ...138
4.4 PropertiesoftheStressTensor .............................141
4.4.1 CauchyEquation.InternalEquilibriumEquation ........141
4.4.2 EquilibriumEquationattheBoundary .................142
4.4.3 SymmetryoftheCauchyStressTensor ................142
4.4.4 Diagonalization.PrincipalStressesandDirections .......144
4.4.5 MeanStressandMeanPressure ......................146
4.4.6 Decomposition of the Stress Tensor into its Spherical
andDeviatoricParts ................................147
4.4.7 TensorInvariants ...................................148
4.5 StressTensorinCurvilinearOrthogonalCoordinates...........149
4.5.1 CylindricalCoordinates .............................149
4.5.2 SphericalCoordinates...............................150
4.6 Mohr’sCirclein3Dimensions .............................151
4.6.1 GraphicalInterpretationoftheStressStates ............151
4.6.2 DeterminationoftheMohr’sCircles...................153
4.7 Mohr’sCirclein2Dimensions .............................157
4.7.1 StressStateonaGivenPlane.........................159
4.7.2 DirectProblem:DiagonalizationoftheStressTensor.....161
4.7.3 InverseProblem....................................162
4.7.4 Mohr’sCircleforPlaneStates(in2Dimensions) ........162
4.7.5 PropertiesoftheMohr’sCircle .......................164
4.7.6 ThePoleofMohr’sCircle ...........................166
4.7.7 Mohr’sCirclewiththeSoilMechanicsSignCriterion ....171
4.8 Mohr’sCircleforParticularCases ..........................172
4.8.1 HydrostaticStressState .............................172
4.8.2 Mohr’sCirclesforaTensoranditsDeviator ............173
4.8.3 Mohr’sCirclesforaPlanePureShearStressState .......173
ProblemsandExercises ...................................175
5 BalancePrinciples ...........................................193
5.1 Introduction .............................................193
5.2 MassTransportorConvectiveFlux .........................193
5.3 LocalandMaterialDerivativesofaVolumeIntegral ...........198
5.3.1 LocalDerivative ...................................198
X.OliverandC.AgeletdeSaracibar ContinuumMechanicsforEngineers.TheoryandProblems
doi:10.13140/RG.2.2.25821.20961
x Contents
5.3.2 MaterialDerivative .................................200
5.4 ConservationofMass.MasscontinuityEquation ..............203
5.4.1 SpatialFormofthePrincipleofConservationofMass.
MassContinuityEquation ...........................203
5.4.2 MaterialFormofthePrincipleofConservationofMass ..205
5.5 BalanceEquation.ReynoldsTransportTheorem ..............206
5.5.1 Reynolds’Lemma..................................206
5.5.2 Reynolds’Theorem.................................206
5.6 GeneralExpressionoftheBalanceEquations .................208
5.7 BalanceofLinearMomentum..............................212
5.7.1 GlobalFormoftheBalanceofLinearMomentum .......213
5.7.2 LocalFormoftheBalanceofLinearMomentum ........214
5.8 BalanceofAngularMomentum ............................215
5.8.1 GlobalFormoftheBalanceofAngularMomentum......216
5.8.2 LocalSpatialFormoftheBalanceofAngularMomentum 217
5.9 Power ..................................................219
5.9.1 MechanicalPower.BalanceofMechanicalEnergy.......220
5.9.2 ThermalPower ....................................222
5.10 EnergyBalance ..........................................225
5.10.1ThermodynamicConcepts ...........................225
5.10.2FirstLawofThermodynamics........................228
5.11 ReversibleandIrreversibleProcesses........................231
5.12 SecondLawofThermodynamics.Entropy ...................233
5.12.1SecondLawofThermodynamics.Globalform..........233
5.12.2Physical Interpretation of the Second Law of
Thermodynamics...................................234
5.12.3ReformulationoftheSecondLawofThermodynamics ...236
5.12.4Local Form of the Second Law of Thermodynamics.
Clausius-PlanckEquation............................238
5.12.5AlternativeFormsoftheSecondLawofThermodynamics 240
5.13 ContinuumMechanicsEquations.ConstitutiveEquations.......242
5.13.1UncoupledThermo-MechanicalProblem...............245
ProblemsandExercises ...................................247
6 LinearElasticity.............................................263
6.1 HypothesisoftheLinearTheoryofElasticity .................263
6.2 LinearElasticConstitutiveEquation.GeneralizedHooke’sLaw .265
6.2.1 ElasticPotential....................................266
6.3 Isotropy.Lame´’sConstants.Hooke’sLawforIsotropicLinear
Elasticity ...............................................269
6.3.1 InversionofHooke’sLaw.Young’sModulus.Poisson’s
Ratio .............................................270
6.4 Hooke’sLawinSphericalandDeviatoricComponents .........272
6.5 LimitsintheValuesoftheElasticProperties .................274
6.6 TheLinearElasticProblem ................................277
X.OliverandC.AgeletdeSaracibar ContinuumMechanicsforEngineers.TheoryandProblems
doi:10.13140/RG.2.2.25821.20961