Table Of ContentOUPCORRECTEDPROOF – 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
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3
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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.
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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
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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.
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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
