Introduction to Linear Elasticity

395 Pages·2018·11.219 MB·English

by  Phillip L. Gould, Yuan Feng

Phillip L. Gould · Yuan Feng
Introduction
to Linear
Elasticity
Fourth Edition
Introduction to Linear Elasticity
(cid:129)
Phillip L. Gould Yuan Feng
Introduction to Linear
Elasticity
Fourth Edition
PhillipL.Gould YuanFeng
DepartmentofMechanicalEngineering SchoolofBiomedicalEngineering
andMaterialsScience ShanghaiJiaoTongUniversity
WashingtonUniversity Shanghai,China
St.Louis,MO,USA
ISBN978-3-319-73884-0 ISBN978-3-319-73885-7 (eBook)
https://doi.org/10.1007/978-3-319-73885-7
LibraryofCongressControlNumber:2018935965
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Dedication from P.L. Gould
To my children
Elizabeth Sue Gould (of blessed memory)
Nathan Charles Gould
Rebecca Blair Carlisle
Joshua Robert Gould
Dedication from Yuan Feng
To my family
mother Huiling Li
father Zijin Feng
wife Long Huang
and my son Samuel Baisheng Feng
Preface
The fourth edition of this book is inspired by the enduring importance of classical
elasticity in the emerging fields of solid mechanics, especially biomechanics and
materials engineering, as well as in the traditional fields of structural, mechanical,
andaerospaceengineering.
To repeat a quote from the preface of earlier editions, “Elasticity is one of the
crowningachievementsofWesternculture”byourlatecolleagueProfessorGeorge
Zahalak. It is this sentence, eloquently expressing my admiration for the creative
efforts of the premier physicists, mathematicians, and mechanicians of the nine-
teenth and twentieth centuries, that led us to try and popularize the basis of solid
mechanics in this introductory text. Although most of the names mentioned in the
courseofthedevelopmentofthetheoryintheearlychapters,suchasHooke,Euler,
Lagrange, St. Venant, Poisson, Boussinesq, Lamé, and Rayleigh, date to the nine-
teenth century and before, at least two Nobel Prize–winning physicists in the
twentieth century made contributions to elasticity. They are Albert Einstein with
thesummationconventionandMaxBornwhowrotehisdoctoralthesisonatopicin
elasticity.
Thebookisintendedtoprovideathoroughgroundingintensor-basedtheoryof
elasticity,whichisrigorous intreatmentbutlimitedinscope. Whilethetraditional
audienceforelasticityisgraduatestudentsinthefieldsmentionedabove,advanced
undergraduateshaveincreasinglybecomeinterestedinthesubject.Thisisreflected
bytheemergenceofabasiccourseinelasticityasareplacementforthetraditional
Advanced Strength of Materials in some curriculums. The authors are indebted to
many historical and some contemporary contributors to this subject, and have
endeavoredtoselect,organize,andpresentthematerialinaconcisemanner.
As before, the text is strongly influenced by a new generation of students who
soughtoutmodernapplicationsofelasticitytoemergingfields.EricClayton,Kevin
Kilgallon, Kevin Derendorf, Rebecca Veto, Chris Ward, and Michael Van
Dusseldorp have made significant contributions. Also the careful collating of the
textbyLindaBuckinghamisappreciated.
vii
viii Preface
The larger blocks of new materials in the third edition of this text dealt with
functionallygradedmaterialsandviscoelasticity.Thesematerialsweredrawnfrom
the works of Professors B.V. Sakkar of the University of Florida and David
Roylance of MIT, respectively, and their active assistance is appreciated. Many
corrections to the second edition were submitted by Professor Charles Bert of the
UniversityofOklahomaandbyourlatecolleagueProfessorS.Sridharanwhoalso
class-tested the book over several years. To broaden the scope of the book to
introduce some topics not traditionally covered in an introductory text but of
importance to contemporary researchers, a brief discussion of nonlinear elasticity
is presented along with the aforementioned chapter on viscoelasticity. Also the
chapter on plates was expanded to include an introduction to thin shell theory
basedonearlierworksoftheseniorauthor.
Computational tools have been playing an important role in the research and
applications of elasticity theories, both analytically and numerically. Therefore, in
this edition, we introduce the usage of MATLAB in solving classical elasticity
problems. The basics of using MATLAB are introduced in the first chapter with
common commands summarized in appendices. To illustrate the application of the
computationaltools,sampleproblemsaresolvedandpresentedasthelastsectionof
eachchapter.Wehopethatthisadditionofcomputationalmaterialswillenhancethe
understanding oftheelasticity theoriesand providean opportunitytolearnnumer-
icalmethodsandtoolsviasolvingclassicalelasticityproblems.
We continue to believe that calibration with classical analytical solutions is
essentialtoestablishconfidence,togainefficiency,andtoquantifyerrorsinnumer-
icallybasedresultsandtrustthatthestudentswhoundertakethestudyofthissubject
willfinditstimulatingandrewarding.
St.Louis,MO,USA PhillipL.Gould
Shanghai,China YuanFeng
Contents
1 IntroductionandMathematicalPreliminaries. . . . . . . . . . . . . . . . . 1
1.1 Scope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 VectorAlgebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 ScalarandVectorFields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.2 Gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.3 Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.4 Divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.5 Curl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.6 IntegralTheorems. . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 IndicialNotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 CoordinateRotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 CartesianTensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 AlgebraofCartesianTensors. . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 OperationalTensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.9 ComputationalExamples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.10 FundamentalsofMATLAB. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.11 IntroductiontoMuPAD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.12 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Traction,Stress,andEquilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 StateofStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 TractionandCouple-StressVectors. . . . . . . . . . . . . . . 17
2.3 ComponentsofStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 StressataPoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 StressonaNormalPlane. . . . . . . . . . . . . . . . . . . . . . 21
2.4.2 DyadicRepresentationofStress. . . . . . . . . . . . . . . . . 23
2.4.3 ComputationalExample. . . . . . . . . . . . . . . . . . . . . . . 24
ix
x Contents
2.5 Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.1 PhysicalandMathematicalPrinciples. . . . . . . . . . . . . 27
2.5.2 LinearMomentum. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.3 AngularMomentum. . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.4 ComputationalExample. . . . . . . . . . . . . . . . . . . . . . . 30
2.6 PrincipalStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6.1 DefinitionandDerivation. . . . . . . . . . . . . . . . . . . . . . 30
2.6.2 ComputationalFormat,StressInvariants,
andPrincipalCoordinates. . . . . . . . . . . . . . . . . . . . . . 33
2.6.3 ComputationalExample. . . . . . . . . . . . . . . . . . . . . . . 36
2.6.4 StressEllipsoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7 StressesinPrincipalCoordinates. . . . . . . . . . . . . . . . . . . . . . . 39
2.7.1 StressesonanObliquePlane. . . .. . . . . . .. . . . . . .. . 39
2.7.2 StressesonOctahedralPlanes. . . . . . . . . . . . . . . . . . . 40
2.7.3 AbsoluteMaximumShearingStress. . . . . . . . . . . . . . 42
2.7.4 ComputationalExample. . . . . . . . . . . . . . . . . . . . . . . 43
2.8 PropertiesandSpecialStatesofStress. . . . . . . . . . . . . . . . . . . 43
2.8.1 ProjectionTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.8.2 PlaneStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.8.3 LinearStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.8.4 PureShear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.8.5 HydrostaticStress. . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.9 ComputationalExampleswithMATLAB. . . . . . . . . . . . . . . . . 45
2.10 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Deformations. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . . .. . . . . . . .. 51
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 PhysicalInterpretationofStrainTensor. . . . . . . . . . . . . . . . . . 55
3.4 PrincipalStrain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 VolumeandShapeChange. . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6 Compatibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7 Incompatibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.8 NonlinearStrain–DisplacementRelations. . . . . . . . . . . . . . . . . 66
3.9 StrainEllipsoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.10 ComputationalExample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.11 ComputationalExampleswithMATLAB. . . . . . . . . . . . . . . . . 70
3.12 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4 MaterialBehavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 UniaxialBehavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 GeneralizedHooke’sLaw. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Contents xi
4.4 TransverseIsotropicMaterial. . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5 FunctionallyGradedMaterials. . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6 ViscoelasticMaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.7 ThermalStrains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.8 PhysicalData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.9 ComputationalExampleswithMATLAB. . . . . . . . . . . . . . . . . 93
4.10 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5 Formulation,Uniqueness,andSolutionStrategies. . . . . . . . . . . . . . 99
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 DisplacementFormulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 ForceFormulation. . .. . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. 101
5.4 OtherFormulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5 Uniqueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.6 MembraneEquation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.7 SolutionStrategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.8 ComputationalExamples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 Extension,Bending,andTorsion. . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 PrismaticBarUnderAxialLoading. . . . . . . . . . . . . . . . . . . . . 111
6.3 CantileverBeamUnderEndLoading. . . . . . . . . . . . . . . . . . . . 116
6.3.1 ElementaryBeamTheory. . . . . . . . . . . . . . . . . . . . . . 116
6.3.2 ElasticityTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.4 Torsion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.4.1 TorsionofCircularShaft. . . . . . . . . . . . . . . . . . . . . . 125
6.4.2 TorsionofSolidPrismaticShafts. . . . . . . . . . . . . . . . 128
6.4.3 TorsionofEllipticalShaft. . . . . . . . . . . . . . . . . . . . . . 134
6.4.4 MembraneAnalogy. . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.5 ComputationalExampleswithMuPAD. . . . . . . . . . . . . . . . . . 141
6.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7 Two-DimensionalElasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2 PlaneStressEquations. . . .. . . . . . .. . . . . .. . . . . .. . . . . .. . 150
7.3 PlaneStrainEquations. . . .. . . . . . .. . . . . .. . . . . .. . . . . .. . 153
7.4 CylindricalCoordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.4.1 GeometricRelations. . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.4.2 TransformationofStressTensor
andCompatibilityEquation. . . . . .. . . . . . .. . . . . . .. 156
7.4.3 AxisymmetricStressesandDisplacements. . . . . . . . . . 158
7.5 Thick-WalledCylinderorDisk. . . . . . . . . . . . . . . . . . . . . . . . 160
7.6 SheetwithSmallCircularHole. . . . . . . . . . . . . . . . . . . . . . . . 163