Elementary Continuum Mechanics for Everyone: With Applications to Structural Mechanics

600 Pages·2013·12.298 MB·English

by  Esben Byskov (auth.)

Solid Mechanics and Its Applications
Volume 194
Series Editor
G.M.L. Gladwell
Department of Civil Engineering, University of Waterloo, Canada
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The aim of this series is to provide lucid accounts written by authoritative
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The scope of the series covers the entire spectrum of solid mechanics. Thus it
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Esben Byskov
Elementary Continuum
Mechanics for Everyone
With Applications to Structural Mechanics
123
Esben Byskov
Department of CivilEngineering
Aalborg University
Aalborg
Denmark
ISSN 0925-0042
ISBN 978-94-007-5765-3 ISBN 978-94-007-5766-0 (eBook)
DOI 10.1007/978-94-007-5766-0
SpringerDordrechtHeidelbergNewYorkLondon
LibraryofCongressControlNumber:2012954065
(cid:2)SpringerScience?BusinessMediaDordrecht2013
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Preface
Why This Book?
Itismyhopethatthepresentbookisthefinaleditionofmylecturenotes
ElementaryContinuumMechanicsforEveryone. Theveryfirsteditionwas
writteninconnectionwithmyteachingafirstcourseoncontinuummechan-
icsattheTechnicalUniversityofDenmark. Itriedtofindatextthatwould
suitmyintentions,butfoundthatthebooksonthemarketeitherwerefilled
with tensor analysis, including e.g. curvilinear coordinates and Christoffel
Symbols,orthey wereofthe oldmechanicstraditions,meaningthatevery
new example was treated separately with the result that universally valid
principles,suchasthePrincipleofVirtualWork,P.1neverappeared. Inmy
opinion, if teaching continuum mechanics at any level is justified it must
contain a strong element of general statements. Then, of course, there is
theriskthatthetreatmentbecomesmathematicallysodifficultthatitcan-
notserveas an introductionto the subject. Therefore,this book contains
anelementary,butquitegeneral,expositionofthesubject,anditismysin-
cere expectation that most students—with some effort, of course—should
be able to get a feel for the important concepts of (generalized) strains,
(generalized)stresses,andthePrincipleofVirtualWork. Judgingfromthe
experienceofmanyofmypredecessorsImayhavesettoohighagoal,but
stillIshalltrytoreachit.
Ifeelthatasolidunderstandingofalmostanysubjectmayonlybeob-
tainedthroughexamples,andthisisoneofthereasonswhyIhaveincluded
PartsII–V,whichmaybeviewedasapplicationsofthetheoryofPartI. Of
course,I considerthe topics of PartsII–V to be importantby themselves,
but in the present context it is just as pertinent that they may be based
onandillustrateconceptsfromPartI. Thus,anessentialideabehindthis
bookis topresentcontinuummechanics,notonlyasavaluable subjectin
its ownright, but as the foundationfor allour theories andmethods gov-
erning the behavior of solids and structures. Although I endorse the use
of examples it has been my experience that they tend to obscure the real
subject by their sheer number and length. Therefore, I have limited the
numberofworkedexamples,exceptinPartIIIbecausetheretheexamples
provide useful information about properties of a number of cross-sections.
IhopethatIhavefoundareasonablebalanceinthisregard.
P.1Later,weshallseethattherearemorethanjustoneprincipleofvirtualwork,but
atthispointthisisnotimportant.
August14, 2012 ContinuumMechanicsfor Everyone EsbenByskov
v
vi Preface
Who Should Read this Book?
Thepresentbookismeantasanintroductiontocontinuummechanicswith
applications in solid and structural mechanics and, obviously, engineering
studentsarepotentialreaders. Itis,however,myhopethatengineerswho
wouldliketoachieveabetterunderstandingofthetheoriestheyapplywill
appreciatethisbook.
Other Possible Topics
AsregardstopicsnottreatedherethereisacouplethatIwouldhaveliked
very much to include. An important one is Elastic Fracture Mechanics.
The reason why I have left it out is that in order to do the job right one
has to introduce solutionsto somefundamentalelasticityproblems,which
can almostonly be found by use of complex functional analysis. If I were
to introduce the methods of Muskhelishvili (1963) that particular subject
wouldhavetakenupmuchmorespacethanIconsiderfeasibleinthepresent
context. Thepossibilityofpresentingthesolutionswithoutproofandthen
utilizing them as a basis for the theory of linear fracture mechanics may
bejustifiedunderothercircumstances,butnothere,wheretheemphasisis
different,namelyonfullderivationsofallP.2formulas.
Anothersubjectwhichmighthavebeencoveredisperfectplasticitybe-
cause results from computations by hand based on this assumption have
beenusedoverandoverduringthepastsixtyyearsorso. However,withfew
exceptions,computeranalysesthatconsiderthemorerealisticcaseofstrain
hardeninghavebeenmadepossibledue tothesmallcostofpersonalcom-
putersandhavethereforemadetheoldermethodsredundant. Inmyopin-
ion,todaythemostimportantapplicationofperfectplasticityisJohansen’s
Yield Line Theory,P.3 see e.g.(Johansen 1963)and (Johansen 1972). The
beautyofhistheoryisthatitmayprovideusefulresultswithonlyalimited
effort, even for rather complicated situations. On the other hand, deriva-
tion of the Yield Line Theory would take up more space than I consider
reasonableinthepresentcontext.
My Writing Style
Readersofthisbookwillsoondiscoverthatitsstyleissomewhatunusual—
probably an older reader will find it appalling, see also the Introduction,
but I have my reasons for writing the way I do. I have tried to write in
an informal style without sacrificing accuracy. A main idea behind my
P.2Thereisoneexceptiontothis,namelytheformulasforimperfectionsensitivityin
Chapter20. Thereasonforthisisthattheformulasbythemselvesareshort,whiletheir
derivation is long and very complicated, see e.g. (Budiansky 1974) or (Christensen &
Byskov2010).
P.3Johansenhimselfnever acknowledged thathisyieldlinetheoryactuallycouldbe
viewed as an example of upper bound solutions of the moregeneral theory of perfect
plasticity.
EsbenByskov ContinuumMechanicsfor Everyone August14,2012
Preface vii
writingisthatIattemptnottointroduceanyconceptwithoutfirstgivinga
motivationforitsusefulnessandrelevance. Imaynothavesucceeded,but
thiswasmygoal.
ManyyearsagowhenIwasanundergraduateattheTechnicalUniversity
ofDenmarkIencounteredthe subjectofmechanicsinaphysicscourse. It
wasneverexplainedto us why the subjectasa whole wasimportant, and
in particular there never was an indication of the reason behind the way
thedifferenttopicsweredealtwith—otherthanthecomment:“Ifyoudon’t
understand this, you don’t belong here.” In my case they may have been
right,butatthattimeIdidnotfeelthatastatementlikethatshowedmuch
insight into how the human mind works, and today I think that teachers
likethatshouldnothavebeenallowedtostandinfrontofaclass. Iintend
toavoidsucharrogance,butcautionthattheconsequenceinsomecasesis
thatthepresentationwillseemunnecessarilylengthytosomereaders.
You might say that my ambition has been to explain“why—not just
how.”
InallprobabilityitisnotnecessarytomentionthatEnglishis notmy
mothertongue,buthopefullythemeaningofmyeffortsisclearinspiteof
that.
If, on the other hand, the language of this book bears some resem-
blance to (American) English, my wonderful secretaryat AalborgUniver-
sity, Kirsten Aakjær, who has read all the pagesP.4 of a shorter, previous
edition and has suggested many substantial improvements to my writing,
deservestobementioned. KirstenAakjærfavorsBritishEnglish,andmany
timesshetriedtoconverteveryphraseintothatlanguage. Overtheyears,
wehavehaddiscussionsaboutthevirtuesofthesetwokindsofEnglish,but
haveneverobtainedacompleteagreement—onlyafriendlyceasefire.P.5
ItisalsoworthmentioningthatKirstenAakjærinmanycasessuggested
more fundamental improvements to my writing, such as telling me that a
particularexamplehadatooabruptending.
Being a very stubborn person, I have sometimes chosen to follow my
owninstincts in spite of the advice by KirstenAakjær, so blame me—not
her—fortheerrors.
As you will see later, in order that you may have a quick view of the Manygrayboxesin
topicsIhaveputalotofgrayboxesinthemargin. margin
P.4Itmusthave beenaboringjobconsideringthefactthat the contents musthave
beenalientoKirstenAakjær.
P.5Inthemid-SeventiesmyfamilyandIstayedinMassachusettsonasabbaticalleave
atHarvardUniversity. Myson,Torben,whowasfiveorsixyearsoldatthattime,once
proclaimed: “British English is bad English,”which is a statement that I have often
quotedsince.
Tobefair,ouryearinMassachusettswastheBicentennialyear,andtheBritishwere
alwaysthebadguyswhenthekindergartenkidsplayed.
August14, 2012 ContinuumMechanicsfor Everyone EsbenByskov
viii Preface
Open Source Programs
Icouldnothavewrittenthisbookwithouttheuseofopensourceandfree
programssuchasLATEX2ε,BibTEX, makeindex,texindy,P.6 gnuplot,maxima,
Octave,xfig,gcc,andg++.
Possible Errors in This Book
I am sure that there must be some errors even in this edition. There is,
unfortunately,nobodyelsebutmyselftoblame: Ihavepressedthekeyson
thekeyboard,Ihavemovedandclickedthemouse,Ihavewrittenthetext,
andIhavedrawnallfiguresandproducedalltheplots.
EsbenByskov
JægersborgandAalborg
August14,2012
P.6Justthethoughtofusingoneofthewysiwig,sometimescalledwysiaigfor“What
YouSeeIsAllYouGet,”texteditingprogramsbringssweattomyface.
EsbenByskov ContinuumMechanicsfor Everyone August14,2012
Contents
Preface v
Contents ix
Introduction xxv
WhatIsContinuumMechanics? . . . . . . . . . . . . . . . . . . xxv
“I Need Continuum Mechanics Like I Need Another Hole in My
Head”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi
TheMainEmphasisofthisBook . . . . . . . . . . . . . . . . . xxvii
HowtoReadthisBook . . . . . . . . . . . . . . . . . . . . . . xxviii
ExpectedPrerequisites . . . . . . . . . . . . . . . . . . . . . . . xxviii
WhatthisBookIsAbout—andwhatitIsNot . . . . . . . . . xxviii
PartI,ContinuumMechanics . . . . . . . . . . . . . . . xxviii
WhatAretheseOtherPartsAbout? . . . . . . . . . . . xxviii
PartII,SpecializedContinua. . . . . . . . . . . . . . . . xxix
PartIII,BeamswithCrossSections . . . . . . . . . . . . xxix
PartIV,Buckling . . . . . . . . . . . . . . . . . . . . . . xxix
PartV,IntroductiontotheFiniteElementMethod . . . xxix
PartVI,MathematicalPreliminaries . . . . . . . . . . . xxx
SomeCommentsonNotation . . . . . . . . . . . . . . . . . . . xxx
SomeCommentsonLength . . . . . . . . . . . . . . . . . . . . xxx
I Continuum Mechanics 1
1 The Purpose of Continuum Mechanics 3
2 Large Displacements and Large Strains 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 KinematicsandDeformation . . . . . . . . . . . . . . . . . . 7
2.2.1 KinematicsandStrain . . . . . . . . . . . . . . . . . 7
2.2.2 KinematicFieldEquations—LagrangeStrain . . . . 9
ix
x Table of Contents
2.2.2.1 “Fiber”Elongation. . . . . . . . . . . . . . 11
2.2.2.2 ChangeofAngle . . . . . . . . . . . . . . . 12
2.2.3 InfinitesimalStrainsandInfinitesimalRotations . . 13
2.2.4 CompatibilityEquations . . . . . . . . . . . . . . . . 14
2.2.5 KinematicBoundaryConditions . . . . . . . . . . . 15
2.3 EquilibriumEquations . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 StaticFieldEquations . . . . . . . . . . . . . . . . . 15
2.3.2 Properties of the Stress Vector—Static Boundary
Conditions . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 PrincipleofVirtualDisplacements. . . . . . . . . . . . . . . 21
2.4.1 TheBudiansky-HutchinsonDotNotation . . . . . . 25
2.4.2 GeneralizedStrainsandStresses . . . . . . . . . . . 26
2.5 PrincipleofVirtualForces . . . . . . . . . . . . . . . . . . . 27
2.6 ConstitutiveRelations . . . . . . . . . . . . . . . . . . . . . 27
2.6.1 HyperelasticMaterials . . . . . . . . . . . . . . . . . 27
2.6.2 PlasticMaterials . . . . . . . . . . . . . . . . . . . . 28
2.7 PotentialEnergy. . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7.1 LinearElasticity . . . . . . . . . . . . . . . . . . . . 30
2.8 ComplementaryEnergy . . . . . . . . . . . . . . . . . . . . . 30
2.9 StaticEquationsbythePrincipleofVirtualDisplacements . 30
3 KinematicallyModerately NonlinearTheory 33
4 Infinitesimal Theory 35
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 KinematicsandDeformation . . . . . . . . . . . . . . . . . . 35
4.2.1 KinematicsandStrain . . . . . . . . . . . . . . . . . 35
4.2.2 StrainCompatibilityEquations . . . . . . . . . . . . 36
4.2.3 KinematicBoundaryConditions . . . . . . . . . . . 39
4.2.4 InterpretationofStrainComponents . . . . . . . . . 39
4.2.4.1 BothIndicesEqual . . . . . . . . . . . . . 40
4.2.4.2 DifferentIndices . . . . . . . . . . . . . . . 41
4.2.5 TransformationofStrain. . . . . . . . . . . . . . . . 41
4.2.5.1 TransformationofCoordinates . . . . . . . 42
4.2.5.2 TransformationofComponentsofDisplace-
mentandComponentsofStrain . . . . . . 42
4.2.6 PrincipalStrains . . . . . . . . . . . . . . . . . . . . 43
4.3 EquilibriumEquations . . . . . . . . . . . . . . . . . . . . . 47
EsbenByskov ContinuumMechanicsfor Everyone August 14, 2012