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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 ©SpringerScience+BusinessMediaNewYork2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper SpringerScience+BusinessMediaLLCNewYorkispartofSpringerScience+BusinessMedia(www. springer.com) 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