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MECHANICS OF SOLIDS AND MATERIALS
MechanicsofSolidsandMaterialsintendstoprovideamodernandintegratedtreat-
mentofthefoundationsofsolidmechanicsasappliedtothemathematicaldescription
ofmaterialbehavior.Thebookblendsbothinnovativetopics(e.g.,largestrain,strain
rate, temperature, time-dependent deformation and localized plastic deformation in
crystallinesolids,anddeformationofbiologicalnetworks)andtraditionaltopics(e.g.,
elastictheoryoftorsion,elasticbeamandplatetheories,andcontactmechanics)ina
coherenttheoreticalframework.This,andtheextensiveuseoftransformmethodsto
generatesolutions,makesthebookofinteresttostructural,mechanical,materials,and
aerospace engineers. Plasticity theories, micromechanics, crystal plasticity, thin films,
energeticsofelasticsystems,andanoverallreviewofcontinuummechanicsandther-
modynamicsarealsocoveredinthebook.
RobertJ.AsarowasawardedhisPhDinmaterialssciencewithdistinctionfromStanford
Universityin1972.HewasaprofessorofengineeringatBrownUniversityfrom1975
to1989,andhasbeenaprofessorofengineeringattheUniversityofCalifornia,San
Diegosince1989.Dr.Asarohasledprogramsinvolvedwiththedesign,fabrication,and
full-scalestructuraltestingoflargecompositestructures,includinghigh-performance
shipsandmarinecivilstructures.Hislistofpublicationsincludesmorethan170research
papersintheleadingprofessionaljournalsandconferenceproceedings.Hereceivedthe
NSFSpecialCreativityAwardforhisresearchin1983and1987.Dr.Asaroalsoreceived
the TMS Champion H. Mathewson Gold Medal in 1991. He has made fundamental
contributionstothetheoryofcrystalplasticity,theanalysisofsurfaceinstabilities,and
dislocation theory. He served as a founding member of the Advisory Committee for
NSF’sOfficeofAdvancedComputingthatfoundedtheSupercomputerPrograminthe
UnitedStates.HehasalsoservedontheNSFMaterialsAdvisoryCommittee.Hehas
beenanaffiliatewithLosAlamosNationalLaboratoryformorethan20yearsandhas
servedasconsultanttoSandiaNationalLaboratory.Dr.Asarohasbeenrecognizedby
ISIasahighlycitedauthorinmaterialsscience.
VladoA.LubardareceivedhisPhDinmechanicalengineeringfromStanfordUniver-
sityin1980.HewasaprofessorattheUniversityofMontenegrofrom1980to1989,
Fulbright fellow and a visiting associate professor at Brown University from 1989 to
1991,andavisitingprofessoratArizonaStateUniversityfrom1992to1997.Since1998
hehasbeenanadjunctprofessorofappliedmechanicsattheUniversityofCalifornia,
SanDiego.Dr.Lubardahasmadesignificantcontributionstophenomenologicalthe-
oriesoflargedeformationelastoplasticity,dislocationtheory,damagemechanics,and
micromechanics.Heistheauthorofmorethan100journalandconferencepublications
andtwobooks:StrengthofMaterials(1985)andElastoplasticityTheory(2002).Hehas
served as a research panelist for NSF and as a reviewer to numerous international
journalsofmechanics,materialsscience,andappliedmathematics.In2000Dr.Lubarda
waselectedtotheMontenegrinAcademyofSciencesandArts.Heisalsorecipientof
the2004DistinguishedTeachingAwardfromtheUniversityofCalifornia.
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Mechanics of Solids
and Materials
ROBERT J. ASARO
UniversityofCalifornia,SanDiego
VLADO A. LUBARDA
UniversityofCalifornia,SanDiego
iii
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru,UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Informationo nthi stitle :www.cambri dge.org/9780521859790
© Robert Asaro and Vlado Lubarda 2006
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format
isbn-13 978-0-511-14707-4 eBook (NetLibrary)
isbn-10 0-511-14707-4
eBook (NetLibrary)
isbn-13 978-0-521-85979-0 hardback
isbn-10 0-521-85979-4 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
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Contents
Prefacepage xix
PART1: MATHEMATICALPRELIMINARIES
1 VectorsandTensors 1
1.1. VectorAlgebra 1
1.2. CoordinateTransformation:Rotation
ofAxes 4
1.3. Second-RankTensors 5
1.4. SymmetricandAntisymmetricTensors 5
1.5. PreludetoInvariantsofTensors 6
1.6. InverseofaTensor 7
1.7. AdditionalProofs 7
1.8. AdditionalLemmasforVectors 8
1.9. CoordinateTransformationofTensors 9
1.10. SomeIdentitieswithIndices 10
1.11. TensorProduct 10
1.12. OrthonormalBasis 11
1.13. EigenvectorsandEigenvalues 12
1.14. SymmetricTensors 14
1.15. PositiveDefinitenessofaTensor 14
1.16. AntisymmetricTensors 15
1.16.1. EigenvectorsofW 15
1.17. OrthogonalTensors 17
1.18. PolarDecompositionTheorem 19
1.19. PolarDecomposition:PhysicalApproach 20
1.19.1. LeftandRightStretchTensors 21
1.19.2. PrincipalStretches 21
1.20. TheCayley–HamiltonTheorem 22
1.21. AdditionalLemmasforTensors 23
1.22. IdentitiesandRelationsInvolving
∇ Operator 23
1.23. SuggestedReading 25
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2 BasicIntegralTheorems 26
2.1. GaussandStokes’sTheorems 26
2.1.1. ApplicationsofDivergenceTheorem 27
2.2. VectorandTensorFields:PhysicalApproach 27
2.3. SurfaceIntegrals:GaussLaw 28
2.4. EvaluatingSurfaceIntegrals 29
2.4.1. ApplicationoftheConceptofFlux 31
2.5. TheDivergence 31
2.6. DivergenceTheorem:RelationofSurfacetoVolume
Integrals 33
2.7. MoreonDivergenceTheorem 34
2.8. SuggestedReading 35
3 FourierSeriesandFourierIntegrals 36
3.1. FourierSeries 36
3.2. DoubleFourierSeries 37
3.2.1. DoubleTrigonometricSeries 38
3.3. IntegralTransforms 39
3.4. Dirichlet’sConditions 42
3.5. IntegralTheorems 46
3.6. ConvolutionIntegrals 48
3.6.1. EvaluationofIntegralsbyUseofConvolution
Theorems 49
3.7. FourierTransformsofDerivativesof f(x)49
3.8. FourierIntegralsasLimitingCasesofFourierSeries 50
3.9. DiracDeltaFunction 51
3.10. SuggestedReading52
PART2: CONTINUUMMECHANICS
4 KinematicsofContinuum 55
4.1. Preliminaries 55
4.2. UniaxialStrain 56
4.3. DeformationGradient 57
4.4. StrainTensor 58
4.5. StretchandNormalStrains 60
4.6. AngleChangeandShearStrains 60
4.7. InfinitesimalStrains 61
4.8. PrincipalStretches 62
4.9. EigenvectorsandEigenvaluesofDeformationTensors 63
4.10. VolumeChanges 63
4.11. AreaChanges 64
4.12. AreaChanges:AlternativeApproach 65
4.13. SimpleShearofaThickPlatewithaCentralHole 66
4.14. Finitevs.SmallDeformations 68
4.15. Referencevs.CurrentConfiguration 69
4.16. MaterialDerivativesandVelocity 71
4.17. VelocityGradient 71
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4.18. DeformationRateandSpin 74
4.19. RateofStretchingandShearing 75
4.20. MaterialDerivativesofStrainTensors: E˙ vs.D76
4.21. RateofFinTermsofPrincipalStretches 78
4.21.1. SpinsofLagrangianandEulerianTriads 81
4.22. AdditionalConnectionsBetweenCurrentandReference
StateRepresentations 82
4.23. TransportFormulae 83
4.24. MaterialDerivativesofVolume,Area,andSurfaceIntegrals:
TransportFormulaeRevisited 84
4.25. AnalysisofSimpleShearing 85
4.26. ExamplesofParticleandPlaneMotion 87
4.27. RigidBodyMotions 88
4.28. BehaviorunderSuperposedRotation 89
4.29. SuggestedReading 90
5 KineticsofContinuum 92
5.1. TractionVectorandStressTensor 92
5.2. EquationsofEquilibrium 94
5.3. BalanceofAngularMomentum:Symmetryofσ 95
5.4. PrincipalValuesofCauchyStress 96
5.5. MaximumShearStresses 97
5.6. NominalStress 98
5.7. EquilibriumintheReferenceState 99
5.8. WorkConjugateConnections 100
5.9. StressDeviator 102
5.10. FrameIndifference 102
5.11. ContinuityEquationandEquationsofMotion 107
5.12. StressPower 108
5.13. ThePrincipleofVirtualWork 109
5.14. GeneralizedClapeyron’sFormula 111
5.15. SuggestedReading 111
6 ThermodynamicsofContinuum 113
6.1. FirstLawofThermodynamics:EnergyEquation 113
6.2. SecondLawofThermodynamics:Clausius–Duhem
Inequality 114
6.3. ReversibleThermodynamics 116
6.3.1. ThermodynamicPotentials 116
6.3.2. SpecificandLatentHeats 118
6.3.3. CoupledHeatEquation 119
6.4. ThermodynamicRelationshipswithp,V,T,ands 120
6.4.1. SpecificandLatentHeats 121
6.4.2. CoefficientsofThermalExpansionand
Compressibility 122
6.5. TheoreticalCalculationsofHeatCapacity 123
6.6. ThirdLawofThermodynamics 125
6.7. IrreversibleThermodynamics 127
6.7.1. EvolutionofInternalVariables 129
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6.8. GibbsConditionsofThermodynamicEquilibrium 129
6.9. LinearThermoelasticity 130
6.10. ThermodynamicPotentialsinLinearThermoelasticity 132
6.10.1. InternalEnergy 132
6.10.2. HelmholtzFreeEnergy 133
6.10.3. GibbsEnergy 134
6.10.4. EnthalpyFunction 135
6.11. UniaxialLoadingandThermoelasticEffect 136
6.12. ThermodynamicsofOpenSystems:Chemical
Potentials 139
6.13. Gibbs–DuhemEquation 141
6.14. ChemicalPotentialsforBinarySystems 142
6.15. ConfigurationalEntropy 143
6.16. IdealSolutions 144
6.17. RegularSolutionsforBinaryAlloys 145
6.18. SuggestedReading 147
7 NonlinearElasticity 148
7.1. GreenElasticity 148
7.2. IsotropicGreenElasticity 150
7.3. ConstitutiveEquationsinTermsofB 151
7.4. ConstitutiveEquationsinTermsofPrincipalStretches 152
7.5. IncompressibleIsotropicElasticMaterials 153
7.6. ElasticModuliTensors 153
7.7. InstantaneousElasticModuli 155
7.8. ElasticPseudomoduli 155
7.9. ElasticModuliofIsotropicElasticity 156
7.10. ElasticModuliinTermsofPrincipalStretches 157
7.11. SuggestedReading 158
PART3: LINEARELASTICITY
8 GoverningEquationsofLinearElasticity 161
8.1. ElementaryTheoryofIsotropicLinearElasticity 161
8.2. ElasticEnergyinLinearElasticity 163
8.3. RestrictionsontheElasticConstants 164
8.3.1. MaterialSymmetry 164
8.3.2. RestrictionsontheElasticConstants 168
8.4. CompatibilityRelations 169
8.5. CompatibilityConditions:Cesa`roIntegrals 170
8.6. Beltrami–MichellCompatibilityEquations 172
8.7. NavierEquationsofMotion 172
8.8. UniquenessofSolutiontoLinearElasticBoundaryValue
Problem 174
8.8.1. StatementoftheBoundaryValueProblem 174
8.8.2. UniquenessoftheSolution 174
8.9. PotentialEnergyandVariationalPrinciple 175
8.9.1. UniquenessoftheStrainField 177
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8.10. Betti’sTheoremofLinearElasticity 177
8.11. PlaneStrain 178
8.11.1. PlaneStress 179
8.12. GoverningEquationsofPlaneElasticity 180
8.13. ThermalDistortionofaSimpleBeam 180
8.14. SuggestedReading 182
9 ElasticBeamProblems 184
9.1. ASimple2DBeamProblem 184
9.2. PolynomialSolutionsto∇4φ =0 185
9.3. ASimpleBeamProblemContinued 186
9.3.1. StrainsandDisplacementsfor2DBeams 187
9.4. BeamProblemswithBodyForcePotentials 188
9.5. BeamunderFourierLoading 190
9.6. CompleteBoundaryValueProblemsforBeams 193
9.6.1. DisplacementCalculations 196
9.7. SuggestedReading 198
10 SolutionsinPolarCoordinates 199
10.1. PolarComponentsofStressandStrain 199
10.2. PlatewithCircularHole 201
10.2.1. FarFieldShear 201
10.2.2. FarFieldTension 203
10.3. DegenerateCasesofSolutioninPolarCoordinates 204
10.4. CurvedBeams:PlaneStress 206
10.4.1. PressurizedCylinder 209
10.4.2. BendingofaCurvedBeam 210
10.5. AxisymmetricDeformations 211
10.6. SuggestedReading 213
11 TorsionandBendingofPrismaticRods 214
11.1. TorsionofPrismaticRods 214
11.2. ElasticEnergyofTorsion 216
11.3. TorsionofaRodwithRectangularCrossSection 217
11.4. TorsionofaRodwithEllipticalCrossSection 221
11.5. TorsionofaRodwithMultiplyConnectedCrossSections 222
11.5.1. HollowEllipticalCrossSection 224
11.6. BendingofaCantilever 225
11.7. EllipticalCrossSection 227
11.8. SuggestedReading 228
12 Semi-InfiniteMedia 229
12.1. FourierTransformofBiharmonicEquation 229
12.2. LoadingonaHalf-Plane 230
12.3. Half-PlaneLoading:SpecialCase 232
12.4. SymmetricHalf-PlaneLoading 234
12.5. Half-PlaneLoading:AlternativeApproach 235
12.6. AdditionalHalf-PlaneSolutions 237
