Fundamentals of the Mechanics of Solids

439 Pages·2016·7.877 MB·English

by  Paolo Maria Mariano, Luciano Galano

Paolo Maria Mariano
Luciano Galano
Fundamentals
of the Mechanics
of Solids

Paolo Maria Mariano • Luciano Galano
Fundamentals of the
Mechanics of Solids
PaoloMariaMariano LucianoGalano
DICeA DICeA
UniversityofFlorence UniversityofFlorence
Firenze,Italy Firenze,Italy
EserciziariodiMeccanicadelleStrutture
OriginalItalianeditionpublishedby©EdizioniCompoMat,Configni,2011
ISBN978-1-4939-3132-3 ISBN978-1-4939-3133-0 (eBook)
DOI10.1007/978-1-4939-3133-0
LibraryofCongressControlNumber:2015946322
MathematicsSubjectClassification(2010):7401,7402,74A,74B,74K10
SpringerNewYorkHeidelbergDordrechtLondon
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Preface
This book emerged from an “experiment in didactics” that has been under devel-
opment at the University of Florence (Italy) since the academic year 2006/2007:
teachingthemechanicsofsolidstoundergraduatestudentsinmechanicalengineer-
ing in a deductive way from a few first principles, including essential elements of
thedescriptionoffinite-strainbehaviorandpayingattentiontotheroleofinvariance
propertiesunderchangesinobservers.
We do not claim originality for this program and are also conscious of the
advantages of other approaches: ours is just a description of the origin of a
choice dictated by personal taste and history. It has been also motivated by the
consciousness that a merely descriptive style can risk reducing the treatment to a
list of special examples or formulas with unexplained origin, while an inductive
approachcouldgive,evenindirectly,prominencetoreasoningbyanalogy,blurring
insomewaythelogicalstructureofthetheory.
Since its inception, about 170 have taken the 84-hour course each year. All
students had previous training in analysis and geometry, including basic elements
oflinearalgebra.Theywerealsotrainedinrationalmechanicsofmasspointsand
rigid bodies, which is taught in a course of the same length. Their mathematical
background had been enlarged during the course by requisite notions from tensor
algebra and analysis. We collect the pertinent material in an appendix, where we
clarify further the notation adopted in the text. One of us (PMM) developed the
“experiment” varying the course every year according to student response and the
changes in his perception. The other (LG) began later to transfer appropriate por-
tionsofthespiritofthecoursetoanalogouscoursesinotherfieldsofengineering.
There are several introductory textbooks on the mechanics of solids. Some of
themfollowastrictlydeductiveprogramratherthanbeingprimarilydescriptiveor
havinganinductiveapproach.Wehavewrittenthisbookbyfollowingourpersonal
taste, with the goal of organizing the subject matter in a way that prepares the
readersforfurtherstudy,beingconsciousthatthedevelopmentofmechanicscould
requireevenmodificationsofthefirstprinciples.Beyondthetechnicalaspects,our
conviction is that the subject must be presented in a critical way, without giving
v
vi Preface
the reader the impression that it has been constructed as an immutable structure
crystallizedonceandforall.Infact,itseemstousthatamerelydogmaticapproach
to mechanics does not contribute to the possibility of deeply investigating the
foundationalaspectsofthesubject.Incontrast,theattentiontofoundationalaspects
is the primary tool for constructing new models, even new theories: families of
interconnected models. The interest for the analysis of the theoretical foundations
is not a mere interest for the formal structure of the theories; rather, it has to be
stimulatedinthestudentsinuniversitycourses,eventhoughperhapsonlyafewof
themwillbeinvolvedinresearchactivitiesafterthecompletionoftheireducation.
To us, even those who will work as professional engineers can take meaningful
advantageofthistypeofprogramsothattheymighteventuallyhavetheflexibility
to learn and (perhaps) manage new models and techniques, those that might be
developedtosatisfyfuturetechnologicalneedsor,aboveall,forgivingusabetter
knowledge of nature. Moreover, an attitude that favors the comprehension and
analysisofthefoundationalaspectsofmechanicaltheoriesencouragesonetosearch
forthephysicalmeaningofeveryformalstepwedo,onthebasisofouranalytical,
geometric,and/orcomputationalskills.
Inthisspirit,webeginwiththedefinitionofbodiesanddeformation,recovering
thekinematicsoftherigidonesasaspecialcase.Inthisway,weestablishalinkwith
the basic courses in rational mechanics of mass points and rigid bodies, showing
how the subject matter we present is a natural continuation of the previous topics.
Wedistinguishbetweenthespaceinwhichweselectthereferencepointforabody
and the one in which we record shapes that we consider deformed. The second
space is what we consider the physical one, the first being just a “room” used
for comparing lengths, areas, volumes, with their prototypical counterparts that
we declare to be undeformed. This unusual distinction allows us to clarify some
statementsconcerningchangesinobserversandrelatedinvarianceproperties.
We distinguish also between material and spatial metrics, each defined in the
pertinent space. Then finite-strain measures emerge from the comparison between
onemetricandthepullbackoftheotherinthespacewherewedecidetocompare
thetwo.Small-straindeformationtensorsarisefromthelinearizationprocess.This
isthetopicofChapter1.
Chapter 2 deals with the definition of observers and a class of their possible
changes, those determined by rotating and translating frames (i.e., coordinate
systems)intheambientphysicalspace.Wecallthesechangesinobserversclassical.
Wesuggestoptionsforthem,allpertainingtothewayinwhichwealterframesin
space, indeed, irrespectively of the type of body considered; in fact, the class of
changesinobserversisnottobeconfusedwiththeclassofadmissiblemotionsfor
abody,althoughthetwoclassesintersect.
In Chapter 3, we tackle the representation of bulk and contact actions in terms
ofthepowertheydevelop.Wewritejusttheexternalpoweronagenericpartofthe
bodyandrequireitsinvarianceunderclassesofisometricchangesinobservers.The
integral balances of forces and couples emerge as a result. Then they are used to
derive the action–reaction principle, the existence of the stress tensor, the balance
Preface vii
equation in Eulerian and Lagrangian descriptions, the expression of the internal
(orinner)powerinbothrepresentations.Theapproachfollowsthespiritofa1963
proposalbyWalterNoll.1
Chapter 4 deals with constitutive issues. We discuss the way of restricting a
priori the set of possible constitutive structures on the basis of the second law
of thermodynamics—here presented as a mechanical dissipation inequality—and
on requirements of objectivity. Our attention is essentially focused on nonlinear
and linearized elasticity. We discuss also the notion of material isomorphism.
Incidentally, when we foresee changes in observers in the reference (material)
space,therequirementthattheobserversrecordthesamematerialforcesthechange
in observer itself to preserve the volume, according to the definition of material
diffeomorphism, irrespectively of the type of body under scrutiny. Such classes of
changesinobserversbecomecrucialinthedescriptionofmaterialmutations,atopic
nottreatedhere,sinceitgoesbeyondthescopeofthisbook.
In Chapter 5, we discuss variational principles in linearized elasticity. Among
them, the Hellinger–Prange–Reissner and Hu–Washizu principles are additional
to the material constituting the course mentioned repeatedly above. The chapter
includes also Kirchhoff’s uniqueness theorem, and the Navier and Beltrami–
Donati–Michell equations. The latter equations are essential tools for the analyses
developed in the subsequent chapter. We end the chapter with some remarks on
two-dimensionalequilibriumproblems.
Chapter6dealswiththedeSaint-Venantproblem:thestaticsofalinearelastic
slender cylinder, free of weight, loaded just on its bases. There are two ways of
discussing such a problem: in terms of displacements or stresses. We follow the
second approach and are indebted to the 1984 treatise in Italian on the matter by
RiccardoBaldacci.2ThechapterendswithaproofofthebasicToupin’stheoremon
thedeSaint-Venantprinciple.
Chapter 7 includes a description of some yield criteria and a discussion of
their role in the representation of the material behavior. There are several criteria,
introduced for various reasons, not all of the same importance. Our choice is to
includeinthisbookjusttheclassicalones,andnothingmore.
In one aspect, Chapter 8 is separate from the program followed in the course
mentionedabove.Thechapterincludesdirector-basedmodelsofrods,atermused
here in a broad sense for rods themselves, beams, shafts, columns, etc. Their
description is a revisitation in terms of invariance of the external power under
changesinobservers—theviewfollowedforthethree-dimensionalcontinuum—of
1NollW.(1963),LaMécaniqueclassique,baséesuruneaxiomed’objectivité,pp.47–56ofLa
MéthodeAxiomatiquedanslesMécaniquesClassiquesetNouvelles(ColloqueInternational,Paris,
1959),Gauthier-Villars,Paris.
2BaldacciR.(1984),Scienzadellecostruzioni,vol.I,UTET,Torino.
viii Preface
a 1985 proposal by Juan Carlos Simo.3 In the chapter, we include both the finite-
strain and linearized treatments; the course that we taught involved just the latter
one.
Chapter9isanoverviewofsomebifurcationphenomena.Attentionisessentially
focusedontheEulerrod.
Thisbookcanbeusedvariouslyforacourseinthemechanicsofsolids,withthe
instructorselectingsomepartsandneglectingothers.Oursisjustaproposal.
In ending this work, we have to express our gratitude to the Birkhäuser team
fortheirhelpandinparticularfortheirunderstandingofthereasonsforourfalling
behindtheoriginalschedule.Amongothers,wementionandthankAllenMannand
Christopher Tominich for the care they have taken in following our work during
differentportionsofitsdevelopment.Also,wethankthecopyeditor,DavidKramer,
forhiswork.
Firenze,Italy PaoloMariaMariano
LucianoGalano
April2015
3Simo J.C.(1989), Afinite-strainbeam formulation. The three-dimensional dynamic problem.
PartI,Comp.Meth.Appl.Mech.Eng.49,55–70.
Contents
ListofFigures ................................................................... xv
ListofTables..................................................................... xxiii
1 Bodies,Deformations,andStrainMeasures............................. 1
1.1 RepresentationofBodies............................................. 1
1.2 Deformations.......................................................... 2
1.3 TheDeformationGradient ........................................... 4
1.4 FormalAdjointF(cid:2)andTransposeFTofF.......................... 6
1.5 HomogeneousDeformationsandRigidChangesofPlace ......... 8
1.6 LinearizedRigidChangesofPlace.................................. 9
1.7 KinematicConstraintsonRigidBodies ............................. 10
1.8 Kinematicsofa1-DimensionalRigidBody......................... 11
1.9 KinematicsofaSystemof1-DimensionalRigidBodies........... 17
1.10 TheFlat-LinkChainMethod......................................... 19
1.11 ExercisesontheKinematicsof1-DimensionalRigidBodies...... 23
1.12 ChangesinVolumeandtheOrientation-PreservingProperty...... 36
1.13 ChangesinOrientedAreas:Nanson’sFormula..................... 39
1.14 FiniteStrains.......................................................... 40
1.15 SmallStrains.......................................................... 45
1.16 FiniteElongationofCurvesandVariationsofAngles.............. 49
1.17 DeviatoricStrain...................................................... 50
1.18 Motions ............................................................... 51
1.19 ExercisesandSupplementaryRemarks ............................. 53
1.20 Furtherexercises...................................................... 65
2 Observers .................................................................... 69
2.1 ADefinition........................................................... 69
2.2 ClassesofChangesinObservers .................................... 70
2.3 Objectivity ............................................................ 72
2.4 RemarksandGeneralizations........................................ 73
ix