Continuum Mechanics of Solids PDF
Preview Continuum Mechanics of Solids
OUPCORRECTEDPROOF – FINAL,15/6/2020,SPi Continuum Mechanics of Solids OUPCORRECTEDPROOF – FINAL,15/6/2020,SPi OUPCORRECTEDPROOF – FINAL,15/6/2020,SPi Continuum Mechanics of Solids Lallit Anand DepartmentofMechanicalEngineering,MassachusettsInstituteofTechnology, Cambridge,MA02139,USA Sanjay Govindjee DepartmentofCivilandEnvironmentalEngineering,UniversityofCalifornia, Berkeley,Berkeley,CA94720,USA 1 OUPCORRECTEDPROOF – FINAL,15/6/2020,SPi 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©LallitAnandandSanjayGovindjee2020 Themoralrightsoftheauthorshavebeenasserted FirstEditionpublishedin2020 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2020938849 ISBN 978–0–19–886472–1 DOI:10.1093/oso/9780198864721.001.0001 Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. OUPCORRECTEDPROOF – FINAL,15/6/2020,SPi Preface ContinuumMechanicsisausefulmodelofmaterialswhichviewsamaterialasbeingcontin- uously divisible, making no reference to its discrete structure at microscopic length scales well below those of the application or phenomenon of interest. Solid Mechanics and Fluid Mechanics are subsets of Continuum Mechanics. The focus of this book is Solid Mechanics, which is concerned with the deformation, flow, and fracture of solid materials. A study of SolidMechanicsisofusetoanyonewhoseekstounderstandthesenaturalphenomena,and toanyonewhoseekstoapplysuchknowledgetoimprovethelivingconditionsofsociety.This isthecentralaimofpeopleworkinginmanyfieldsofengineering,e.g.,mechanical,materials, civil,aerospace,nuclear,andbio-engineering.Thisbook,whichisaimedatfirst-yeargraduate studentsinengineering,offersaunifiedpresentationofthemajorconceptsinSolidMechanics. InaddressinganyprobleminSolidMechanics,weneedtobringtogetherthefollowingmajor considerations: • Kinematics or the geometry of deformation, in particular the expressions of stretch/strainandrotationintermsofthegradientofthedeformationfield. • Balanceofmass. • Thesystemofforcesactingonthebody,andtheconceptofstress. • Balance of forces and moments in situations where inertial forces can be neglected, andcorrespondinglybalanceoflinearmomentumandangularmomentumwhenthey cannot. • Balance of energy and an entropy imbalance, which represent the first two laws of thermodynamics. • Constitutiveequationswhichrelatethestresstothestrain,strain-rate,andtemperature. • Thegoverningpartialdifferentialequations,togetherwithsuitableboundaryconditions andinitialconditions. Whilethefirstfiveconsiderationsarethesameforallcontinuousbodies,nomatterwhatmate- rialtheyaremadefrom,theconstitutiverelationsarecharacteristicofthematerialinquestion, thestresslevel,thetemperature,andthetimescaleoftheproblemunderconsideration. The major topics regarding the mechanical response of solids that are covered in this book include: Elasticity, Viscoelasticity, Plasticity, Fracture, and Fatigue. Because the consti- tutive theories of Solid Mechanics are complex when presented within the context of finite deformations, in most of the book we will restrict our attention to situations in which the deformationsmaybe adequatelypresumedtobe small—a situationcommonlyencountered inmoststructuralapplications.However,inthelastsectionofthebookonFiniteElasticityof elastomericmaterials,wewillconsiderthecompletelargedeformationtheory,becauseforsuch materialsthedeformationsintypicalapplicationsareusuallyquitelarge. OUPCORRECTEDPROOF – FINAL,15/6/2020,SPi vi PREFACE In addition to these standard topics in Solid Mechanics, because of the growing need for engineeringstudentstohaveaknowledgeofthecoupledmulti-physicsresponseofmaterials in modern technologies related to the environment and energy, the book also includes chapters on Thermoelasticity, Chemoelasticity, Poroelasticity, and Piezoelectricity—all within the(restricted)frameworkofsmalldeformations. For pedagogical reasons, we have also prepared a companion book with many fully solvedexampleproblems,ExampleProblemsforContinuumMechanicsofSolids(Anandand Govindjee,2020). Thepresentbookisanoutgrowthoflecturenotesbythefirstauthorforfirst-yeargraduate students in engineering at MIT taking 2.071 Mechanics of Solid Materials, and the material taught at UC Berkeley in CE 231/MSE 211 Introduction to Solid Mechanics by the second author.Althoughthebookhasgrownoutoflecturenotesforaone-semestercourse,itcontains enoughmaterialforatwo-semestersequenceofsubjects,especiallyifthematerialoncoupled theoriesistobetaught.Thedifferenttopicscoveredinthebookarepresentedinessentially self-contained “modules”, and instructors may pick-and-choose the topics that they wish to focusonintheirownclasses. Wewishtoemphasizethatthisbookisintendedasatextbookforclassroomteachingorself- studyforfirst-yeargraduatestudentsinengineering.Itisnotintendedtobeacomprehensive treatiseonalltopicsofimportanceinSolidMechanics.Itisa“pragmatic”bookdesignedto provideageneralpreparationforfirst-yeargraduatestudentswhowillpursuefurtherstudyor conductresearchinspecializedsub-fieldsofSolidMechanics,orrelateddisciplines,duringthe courseoftheirgraduatestudies. Attributionsandhistoricalissues OuremphasisisonbasicconceptsandcentralresultsinSolidMechanics,notonthehistoryof thesubject.Wehaveattemptedtocitethecontributionsmostcentraltoourpresentation,and weapologizeinadvanceifwehavenotdonesofaultlessly. LallitAnand Cambridge,MA SanjayGovindjee Berkeley,CA OUPCORRECTEDPROOF – FINAL,15/6/2020,SPi Acknowledgments LallitAnandisindebtedtohisteacherMortonGurtinofCMU,whofirsttaughthimcontinuum mechanicsin1975–76,andonceagainduringtheperiod2004–2010,whenheworkedwith Gurtinonwritingthebook,TheMechanicsandThermodynamicsofContinua(Gurtinetal., 2010). He is also grateful to his colleague David Parks at MIT—who has used a prelimi- nary version of this book in teaching his classes—for his comments, helpful criticisms, and his contributions to the chapters on Fracture and Fatigue. Discussions with Mary Boyce, now at Columbia University, regarding the subject matter of the book are also gratefully acknowledged. Sanjay Govindjee is indebted to his many instructors and in particular to his professors at theMassachusettsInstituteofTechnologyandatStanfordUniversity,whonevershiedaway fromadvancedconceptsandphilosophicaldiscussions.Specialgratitudeisextendedtothelate JuanCarlosSimo,JeromeSackman,andEgorPopov.Additionalthanksaregiventocurrent colleagueRobertL.Taylorandcountlessstudentswhosepenetratingquestionshaveformed histhinkingaboutsolidmechanics. OUPCORRECTEDPROOF – FINAL,15/6/2020,SPi OUPCORRECTEDPROOF – FINAL,15/6/2020,SPi Contents I Vectorsandtensors 1 1 Vectorsandtensors:Algebra 3 1.1 Cartesiancoordinateframes.Kroneckerdelta.Alternatingsymbol 3 1.1.1 Summationconvention 4 1.2 Tensors 6 1.2.1 Whatisatensor? 6 1.2.2 Zeroandidentitytensors 7 1.2.3 Tensorproduct 7 1.2.4 Componentsofatensor 7 1.2.5 Transposeofatensor 8 1.2.6 Symmetricandskewtensors 9 1.2.7 Axialvectorofaskewtensor 9 1.2.8 Productoftensors 10 1.2.9 Traceofatensor.Deviatorictensors 11 1.2.10 Positivedefinitetensors 12 1.2.11 Innerproductoftensors.Magnitudeofatensor 12 1.2.12 Matrixrepresentationoftensorsandvectors 13 1.2.13 Determinantofatensor 14 1.2.14 Invertibletensors 15 1.2.15 Cofactorofatensor 15 1.2.16 Orthogonaltensors 15 1.2.17 Transformationrelationsforcomponentsofavectorandatensor underachangeinbasis 15 Transformationrelationforvectors 16 Transformationrelationfortensors 17 1.2.18 Eigenvaluesandeigenvectorsofatensor.Spectraltheorem 17 Eigenvaluesofsymmetrictensors 19 Otherinvariantsandeigenvalues 20 1.2.19 Fourth-ordertensors 20 Transformationrulesforfourth-ordertensors 22
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