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IntroductiontotheVariationalFormulationinMechanics

IntroductiontotheVariationalFormulation inMechanics

FundamentalsandApplications

EdgardoO.Taroco,PabloJ.BlancoandRaúlA.Feijóo

HeMoLab-HemodynamicsModeling Laboratory

LNCC/MCTIC-NationalLaboratoryfor ScientificComputing,Brazil

INCT-MACC-NationalInstituteofScience andTechnologyinMedicineAssistedby ScientificComputing,Brazil

Thiseditionfirstpublished2020

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Contents

Preface xv

PartIVectorandTensorAlgebraandAnalysis 1

1VectorandTensorAlgebra 3

1.1PointsandVectors 3

1.2Second-OrderTensors 6

1.3Third-OrderTensors 17

1.4ComplementaryReading 22

2VectorandTensorAnalysis 23

2.1Differentiation 23

2.2Gradient 28

2.3Divergence 30

2.4Curl 32

2.5Laplacian 34

2.6Integration 35

2.7Coordinates 38

2.8ComplementaryReading 45

PartIIVariationalFormulationsinMechanics 47

3MethodofVirtualPower 49

3.1Introduction 49

3.2Kinematics 50

3.2.1BodyandDeformations 50

3.2.2Motion:DeformationRate 55

3.2.3MotionActions:KinematicalConstraints 61

3.3DualityandVirtualPower 66

3.3.1MotionActionsandForces 67

3.3.2DeformationActionsandInternalStresses 69

3.3.3MechanicalModelsandtheEquilibriumOperator 71

viii Contents

3.4BodieswithoutConstraints 74

3.4.1PrincipleofVirtualPower 75

3.4.2PrincipleofComplementaryVirtualPower 80

3.5BodieswithBilateralConstraints 81

3.5.1PrincipleofVirtualPower 81

3.5.2PrincipleofComplementaryVirtualPower 86

3.6BodieswithUnilateralConstraints 87

3.6.1PrincipleofVirtualPower 89

3.6.2PrincipleofComplementaryVirtualPower 92

3.7LagrangianDescriptionofthePrincipleofVirtualPower 94

3.8ConfigurationswithPreloadandResidualStresses 97

3.9LinearizationofthePrincipleofVirtualPower 100

3.9.1PreliminaryResults 101

3.9.2KnownSpatialConfiguration 102

3.9.3KnownMaterialConfiguration 102

3.10InfinitesimalDeformationsandSmallDisplacements 103

3.10.1BilateralConstraints 104

3.10.2UnilateralConstraints 105

3.11FinalRemarks 106

3.12ComplementaryReading 107

4HyperelasticMaterialsatInfinitesimalStrains 109

4.1Introduction 109

4.2UniaxialHyperelasticBehavior 109

4.3Three-DimensionalHyperelasticConstitutiveLaws 113

4.4EquilibriuminBodieswithoutConstraints 116

4.4.1PrincipleofVirtualWork 117

4.4.2PrincipleofMinimumTotalPotentialEnergy 117

4.4.3LocalEquationsandBoundaryConditions 118

4.4.4PrincipleofComplementaryVirtualWork 120

4.4.5PrincipleofMinimumComplementaryEnergy 121

4.4.6AdditionalRemarks 122

4.5EquilibriuminBodieswithBilateralConstraints 123

4.5.1PrincipleofVirtualWork 125

4.5.2PrincipleofMinimumTotalPotentialEnergy 125

4.5.3PrincipleofComplementaryVirtualWork 126

4.5.4PrincipleofMinimumComplementaryEnergy 127

4.6EquilibriuminBodieswithUnilateralConstraints 128

4.6.1PrincipleofVirtualWork 128

4.6.2PrincipleofMinimumTotalPotentialEnergy 128

4.6.3PrincipleofComplementaryVirtualWork 129

4.6.4PrincipleofMinimumComplementaryEnergy 130

4.7Min–MaxPrinciple 131

4.7.1Hellinger–ReissnerFunctional 131

4.7.2Hellinger–ReissnerPrinciple 133

4.8Three-FieldFunctional 134

4.9CastiglianoTheorems 136

4.9.1FirstandSecondTheorems 136

4.9.2BoundsforDisplacementsandGeneralizedLoads 139

4.10ElastodynamicsProblem 144

4.11ApproximateSolutiontoVariationalProblems 148

4.11.1ElastostaticsProblem 148

4.11.2Hellinger–ReissnerPrinciple 154

4.11.3GeneralizedVariationalPrinciple 156

4.11.4ContactProblemsinElastostatics 158

4.12ComplementaryReading 162

5MaterialsExhibitingCreep 165

5.1Introduction 165

5.2PhenomenologicalAspectsofCreepinMetals 165

5.3InfluenceofTemperature 168

5.4Recovery,Relaxation,CyclicLoading,andFatigue 170

5.5UniaxialConstitutiveEquations 173

5.6Three-DimensionalConstitutiveEquations 182

5.7GeneralizationoftheConstitutiveLaw 188

5.8ConstitutiveEquationsforStructuralComponents 191

5.8.1BendingofBeams 192

5.8.2Bending,Extension,andCompressionofBeams 195

5.9EquilibriumProblemforSteady-StateCreep 199

5.9.1MechanicalEquilibrium 199

5.9.2VariationalFormulation 201

5.9.3VariationalPrinciplesofMinimum 205

5.10CastiglianoTheorems 209

5.10.1FirstandSecondTheorems 209

5.10.2BoundsforVelocitiesandGeneralizedLoads 211

5.11ExamplesofApplication 214

5.11.1DiskRotatingwithConstantAngularVelocity 214

5.11.2CantileveredBeamwithUniformLoad 217

5.12ApproximateSolutiontoSteady-StateCreepProblems 219

5.13UnsteadyCreepProblem 225

5.14ApproximateSolutionstoUnsteadyCreepFormulations 227

5.15ComplementaryReading 228

6MaterialsExhibitingPlasticity 229

6.1Introduction 229

6.2Elasto-PlasticMaterials 229

6.3UniaxialElasto-PlasticModel 235

6.3.1ElasticRelation 235

6.3.2YieldCriterion 236

6.3.3HardeningLaw 238

6.3.4PlasticFlowRule 240

6.4Three-DimensionalElasto-PlasticModel 243

6.4.1ElasticRelation 244

x Contents

6.4.2YieldCriterionandHardeningLaw 246

6.4.3PotentialPlasticFlow 249

6.5DruckerandHillPostulates 253

6.6Convexity,Normality,andPlasticPotential 255

6.6.1NormalityLawandaRationaleforthePotentialLaw 255

6.6.2ConvexityoftheAdmissibleRegion 257

6.7PlasticFlowRule 258

6.8InternalDissipation 260

6.9CommonYieldFunctions 262

6.9.1ThevonMisesCriterion 263

6.9.2TheTrescaCriterion 264

6.10CommonHardeningLaws 266

6.11IncrementalVariationalPrinciples 267

6.11.1PrincipleofMinimumfortheVelocity 268

6.11.2PrincipleofMinimumfortheStressRate 269

6.11.3UniquenessoftheStressField 270

6.11.4VariationalInequalityfortheStress 270

6.11.5PrincipleofMinimumwithTwoFields 271

6.12IncrementalConstitutiveEquations 272

6.12.1ConstitutiveEquationsforRates 273

6.12.2ConstitutiveEquationsforIncrements 275

6.12.3VariationalPrincipleinFiniteIncrements 278

6.13ComplementaryReading 279

PartIIIModelingofStructuralComponents 281

7BendingofBeams 285

7.1Introduction 285

7.2Kinematics 285

7.3GeneralizedForces 289

7.4MechanicalEquilibrium 290

7.5TimoshenkoBeamModel 294

7.6FinalRemarks 298

8TorsionofBars 301

8.1Introduction 301

8.2Kinematics 301

8.3GeneralizedForces 304

8.4MechanicalEquilibrium 305

8.5DualFormulation 309

9PlatesandShells 315

9.1Introduction 315

9.2GeometricDescription 316

9.3DifferentiationandIntegration 320

9.4PrincipleofVirtualPower 323

9.5UnifiedFrameworkforShellModels 326

9.6ClassicalShellModels 332

9.6.1NaghdiModel 332

9.6.2Kirchhoff–LoveModel 335

9.6.3LoveModel 340

9.6.4KoiterModel 342

9.6.5SandersModel 344

9.6.6Donnell–Mushtari–VlasovModel 346

9.7ConstitutiveEquationsandInternalConstraints 347

9.7.1PreliminaryConcepts 348

9.7.2ModelwithNaghdiHypothesis 350

9.7.3ModelwithKirchhoff–LoveHypothesis 357

9.8CharacteristicsofShellModels 360

9.8.1RelationBetweenGeneralizedStresses 360

9.8.2EquilibriumAroundtheNormal 361

9.8.2.1Kirchhoff–LoveModel 361

9.8.2.2LoveModel 362

9.8.2.3KoiterModel 363

9.8.2.4SandersModel 363

9.8.3ReactiveGeneralizedStresses 364

9.8.3.1ReactionsintheNaghdiModel 364

9.8.3.2ReactionsintheKirchhoff–LoveModel 366

9.9BasicsNotionsofSurfaces 369

9.9.1Preliminaries 369

9.9.2FirstFundamentalForm 370

9.9.3SecondFundamentalForm 372

9.9.4ThirdFundamentalForm 375

9.9.5ComplementaryProperties 375

PartIVOtherProblemsinPhysics 377

10HeatTransfer 379

10.1Introduction 379

10.2Kinematics 379

10.3PrincipleofThermalVirtualPower 381

10.4PrincipleofComplementaryThermalVirtualPower 386

10.5ConstitutiveEquations 388

10.6PrincipleofMinimumTotalThermalEnergy 390

10.7PoissonandLaplaceEquations 390

11IncompressibleFluidFlow 393

11.1Introduction 393

11.2Kinematics 394

11.3PrincipleofVirtualPower 396

xii Contents

11.4Navier–StokesEquations 403

11.5StokesFlow 405

11.6IrrotationalFlow 407

12High-OrderContinua 411

12.1Introduction 411

12.2Kinematics 412

12.3PrincipleofVirtualPower 418

12.4Dynamics 425

12.5MicropolarMedia 427

12.6SecondGradientTheory 429

PartVMultiscaleModeling 435

13MethodofMultiscaleVirtualPower 439

13.1Introduction 439

13.2MethodofVirtualPower 439

13.2.1Kinematics 439

13.2.2Duality 442

13.2.3PrincipleofVirtualPower 445

13.2.4EquilibriumProblem 446

13.3FundamentalsoftheMultiscaleTheory 447

13.4KinematicalAdmissibilitybetweenScales 449

13.4.1MacroscaleKinematics 449

13.4.2MicroscaleKinematics 451

13.4.3InsertionOperators 453

13.4.4HomogenizationOperators 456

13.4.5KinematicalAdmissibility 458

13.5DualityinMultiscaleModeling 462

13.5.1MacroscaleVirtualPower 462

13.5.2MicroscaleVirtualPower 464

13.6PrincipleofMultiscaleVirtualPower 467

13.7DualOperators 468

13.7.1MicroscaleEquilibrium 468

13.7.2HomogenizationofGeneralizedStresses 470

13.7.3HomogenizationofGeneralizedForces 472

13.8FinalRemarks 473

14ApplicationsofMultiscaleModeling 475

14.1Introduction 475

14.2SolidMechanicswithExternalForces 475

14.2.1MultiscaleKinematics 476

14.2.2CharacterizationofVirtualPower 479

14.2.3PrincipleofMultiscaleVirtualPower 480

14.2.4EquilibriumProblemandHomogenization 482

14.2.5TangentOperators 487

14.3MechanicsofIncompressibleSolidMedia 490

14.3.1PrincipleofVirtualPower 491

14.3.2MultiscaleKinematics 493

14.3.3PrincipleofMultiscaleVirtualPower 495

14.3.4IncompressibilityandMaterialConfiguration 497

14.4FinalRemarks 500

PartVIAppendices 501

ADefinitionsandNotations 503

A.1Introduction 503

A.2Sets 503

A.3FunctionsandTransformations 504

A.4Groups 507

A.5Morphisms 509

A.6VectorSpaces 509

A.7SetsandDependenceinVectorSpaces 512

A.8BasesandDimension 513

A.9Components 514

A.10SumofSetsandSubspaces 516

A.11LinearManifolds 516

A.12ConvexSetsandCones 516

A.13DirectSumofSubspaces 517

A.14LinearTransformations 517

A.15CanonicalIsomorphism 522

A.16AlgebraicDualSpace 523

A.16.1OrthogonalComplement 524

A.16.2PositiveandNegativeConjugateCones 525

A.17Algebrain V526

A.18AdjointOperators 528

A.19TranspositionandBilinearFunctions 529

A.20InnerProductSpaces 532

BElementsofRealandFunctionalAnalysis 539

B.1Introduction 539

B.2Sequences 541

B.3LimitandContinuityofFunctions 542

B.4MetricSpaces 544

B.5NormedSpaces 546

B.6QuotientSpace 549

B.7LinearTransformationsinNormedSpaces 550

B.8TopologicalDualSpace 552

B.9WeakandStrongConvergence 553

xiv Contents

CFunctionalsandtheGâteauxDerivative 555

C.1Introduction 555

C.2PropertiesofOperator �� 555

C.3ConvexityandSemi-Continuity 556

C.4GâteauxDifferential 557

C.5MinimizationofConvexFunctionals 557

References 559 Index 575

Preface

Thisbookwaswrittenintermittentlyovertheperiodbetween1980and2016withan aimtoprovidestudentsattendingthecoursesorganizedbytheauthors,particularly forthegraduatestudentsattheNationalLaboratoryforScientificComputing(LNCC), withthefoundationalmaterialofMechanicsusingavariationaltapestry.Itistheresult oftheknowledgeacquiredanddivulgedbyE.O.TandR.A.F.sincetheLNCCwasestablished,whichwasinitiatedwiththecreationoftheLaboratoryofComputing(LAC)of theBrazilianCenterforResearchinPhysics(CBPF)in1977,throughthefoundationof theLaboratoryforScientificComputing(LCC)in1980,itsconversionintothecategory ofanationallaboratory(LNCC)in1982andthedefinitivemovetothecityofPetrópolis in1998.

Partofthematerialpresentedherewasusedinvariouscoursesoftheoreticaland appliedmechanicsorganizedbyE.O.T.andR.A.F.Thesecoursewere

• 1stCourseonTheoreticalandAppliedMechanics:TheoryofShellsandtheirApplicationsinEngineering(ModuleI–BasicPrinciples,July5to30,1982;ModuleII–MechanicalModels,January3toFebruary11,1983;andModuleIII–Instabilityof Shells,July4to30,1983).

• 2ndCourseonTheoreticalandAppliedMechanics:FundamentalsoftheFiniteElementMethodanditsApplicationsinEngineering(ModuleI–Fundamentalsofthe FiniteElementMethod,July2to27,1984;ModuleII–ApplicationsoftheFinite ElementMethodinSolidMechanics,January7toFebruary1,1985,ModuleIII–ModernAspectsoftheFiniteElementMethod,July1to26,1985).

• 3rdCourseonTheoreticalandAppliedMechanics:Optimization:Fundamentalsand ApplicationsinEngineering(ModuleI–OptimizationintheModelingandAnalysisofEngineeringProblems,July7toAugust1,1986;ModuleII–OptimalDesign: FoundationsandApplications,July6to31,1987).

Unfortunately,andlargelyduetothelongperiodoftimethatthisdocumenttookto befinished,oneoftheauthors,Prof.EdgardoO.Taroco,leftus(passedawayinJanuary 2010).Nevertheless,wedecidedtokeephisnameamongtheauthorsasanacknowledgementtohisdeepcontributionsandinhonortohismemory,aswellastothefriendship andgenerositythathealwaysofferedusthroughoutalltheseyears.Therefore,allerrors (ofanykind)arethesoleresponsibilityofP.J.B.andR.A.F. PartofthismaterialwasusedbyR.A.F.inthecoursedictatedduringthePost-Doctoral LatinAmericanSeminaronContinuumMechanicsandMicrostructure,organizedby theNationalAtomicEnergyCommissioninArgentina,sponsoredbytheOrganization

xvi Preface

ofAmericanStates(OAS),andheldinBuenosAires,July-August1984.Itwasonlyafter 1998thattheLNCCwasmovedtoPetrópolis,andtheLNCCGraduateProgramwas initiated.Then,wedecidedtoconsolidatetheaforementionedtextsintoamonograph thatwouldgiveemphasistotheformulationofthemechanicswithinapurelyvariational structure.Morerecently,sinceP.J.B.joinedtheLNCCin2009,theideaoffinishingthis document,alsoincludingtopicsinotherareasofphysicsaswellastheextensionofthe variationalframeworktoamulti-scaleparadigm,resurfaced.Currently,thismaterialis beingusedinseveralcoursesinthegraduateprograminComputationalModelingat LNCC,Brazil,andalsointhegraduateprograminMechanicalEngineeringatNational UniversityofMardelPlata,Argentina.

Suchvariationalframeworkusedtopresentthemechanicswaschosennotonly forbeingoneofourmainareasofresearch,butalso,andfundamentally,becausewe stronglybelievethatthiswayoflookingintotherootsofmechanicsisthemostsuitable andconvenientperspectivetoapproachthemathematicalmodeling.Infact,andasit willbecomeincreasinglyclearasweproceed,thefoundationalpillarsuponwhichthe wholemodelingjourneyrestsusingthisvariationaltapestryarethefollowing

• Thefirstpillarisrelatedtothedescriptionofthekinematics,thatistheformalization ofthekinematicalhypotheseswhichprovidesthedefinitionofthegeneralizedmotion actionsandadmissiblegeneralizedstrainrateactionsforthemodelunderstudy.

• Thesecondpillarconsistsofthemathematicaldualitypostulatedbetweenquantitiesrelatedtosuchmotionactionsandgeneralizedstrainrateswith,respectively, theexternalgeneralizedforcesandinternalgeneralizedstresses.Inthisway,forces andstressesareconstructsfullyshapedbylinearcontinuousfunctionalswhoseargumentsarekinematicalentities.Thisaspectestablishesacleardifferencebetweenthe approachdevelopedinthisbookandtheprocedurefollowedbymostoftheliterature inthefieldofcontinuummechanics,whereforcesandstressesaremalleableentities introducedapriori,regardlessofthekinematicsdefinedforthephysicalsystem.

• ThethirdpillaristhePrincipleofVirtualPower(oritsgeneralization,thePrinciple ofMulti-scaleVirtualPowerinthecontextofmodelingproblemswithmorethan onescale).Thisprincipleallowstoestablish,forthephysicalsystemunderstudy,the conceptofmechanicalequilibriumbetweenexternalforcesandinternalstressesand, whenproperconstitutiverelationsaregiven,thisprinciplecharacterizesthegeneralizeddisplacementfieldforwhichtheassociatedgeneralizedstressstateequilibrates theexternalforces.

Thesethreepillarssupporttherealmofso-calledMethodofVirtualPower(MVP), whichestablisheswell-definedbasicstepstargetingafullyconsistentmodellingtechnique.Suchvariationalstructurewasproposed,althoughwithsubtlemodifications, byProf.PaulGermain[114–118].Particularly,Prof.Germain(lifetimememberofthe FrenchAcademyofScience)wasinvitedbyusin1982toteachthecourseFourLectures ontheFoundationofShellTheorywithinModuleIofthe1stCourseonTheoreticaland AppliedMechanicsthatweorganized,precisely,todisseminatetheseconceptsamong studentsandprofessorsfromdifferentLatinAmericancountries.Anotherprofessor whogreatlycontributedtoconsolidatethisapproachinthesecourseswasProf.Giovanni Romano(FacoltàdiIngegneriadell’UniversitàdegliStudidiNapoli,Italy). Theorganizationofthisbookcloselyfollowsthisspirit.InPartIwepresentthebasic conceptsofvectorandtensoralgebraandanalysis,wherethereaderisintroducedto

Preface xvii theubiquitoususeofcompactvectorandtensornotation.Thiscompactnotationallows togothroughthebasicprinciplesandconceptsofthemechanicsinaclearandconcise way,withoutbeingobscuredbythepresenceofindices,componentsandmetric-related entities,whichshouldberelegatedtotheirspecificroleatthetimeofthecalculation. PartIIisdedicatedtopresentingtheMethodofVirtualPower(MVP),fromthekinematics,throughthedualityandtothePrincipleofVirtualPower.ThisPartpresentsthe variationalgroundworkwhichisusedasaguidingthemeinallthatfollows.TheMVPis thenappliedtothemostgeneralcaseinthemechanicsofdeformablebodies,whileits applicationtothecaseofhyperelasticmaterials,andmaterialswhichmayexperience creepandplasticityphenomenaisalsodiscussed.InPartIIIwepresenttheapplicationoftheMVPtothemodellingofstructuralcomponentssuchasbeams,platesand shells.Forthesecomponents,itwillbeclearhowforcesandstressesemergeasnaturaloutgrowthofthekinematicalhypotheses.InPartIV,theapplicationoftheMVPto otherproblemsfromphysicsisaddressed,includingheatconduction,incompressible fluidflow,andhighordercontinua.Thiswillhelpthereadertoillustratetheuseofthis unifiedtheoreticalframeworkinproblemsinwhichapurelyvariationalapproachisseldomencounteredinthetextbooks.Finally,inPartV,weexpandthisvariationalrealm toembraceproblemswhichrequireamulti-scaleparadigm.Thisextendedvariational structurehasbeencalledtheMethodofMulti-scaleVirtualPower(MMVP).Pursuing thesamestandard,theMMVPallowstoprovideaconvenientandsafetooltosubstantiatemulti-scalemodelsofcomplexphysicalsystems,allowingtoconsistentlyprovide thegroundworkontopofwhichmulti-scalehomogenizationshouldtakeplace.Finally, intheAppendices,wepresentvariousmathematicalconceptsandresultstomakethe documentself-contained.

Lastbutnotleast,wewouldliketothankallthosewhosomehowcontributedtoour excursiontowardsthisbook.InparticularwethankGonzaloR.Feijóoforhiscontributioninthefirstdraftingofsomechaptersofthisbook,andtoProfessorsEnzoA.Dari (BarilocheAtomicCentre,Argentina),SebastiánGiusti(NationalTechnologicalUniversity,FacultyofCórdoba,Argentina),SantiagoA.Urquiza(NationalUniversityofMardel Plata,Argentina),PabloJ.Sánchez(NationalTechnologicalUniversity,RegionalFaculty ofSantaFe,Argentina),AlejandroClausse(NationalUniversityofCentralBuenosAires, Argentina)andEduardoA.deSouzaNeto(ZienkiewiczCenterforComputationalEngineering,SwanseaUniversity,UnitedKingdom)fortheircommentsanddiscussionsthat definitelyenrichedusand,therefore,ourwork.WewouldalsoliketothankourPh.D. students,particularlytoGonzaloD.Ares,GonzaloD.MasoTalou,CarlosA.Bulant, AlonsoM.AlvarezandFelipeFigueredoRochawho,withtheircriticismsandobservations,havealsohelpedtoimprovethistext.

Petrópolis,Brazil PabloJ.Blanco March2018 RaúlA.Feijóo

VectorandTensorAlgebraandAnalysis

Thegoalofthefollowingtwochapters,whichcomprisePartIofthisbook,istoprovide thereaderwiththebasicconceptsandtrainingintheareaoftensoralgebra(Chapter1) andtensoranalysis(Chapter2)whichwillbeomnipresentthroughoutthisbook.Itis importanttohighlightheretheuseofcompactnotation(alsocalledintrinsicnotation) wheninvokingvectorandtensorsaswellasalgebraicanddifferentialoperationsamong them.Thatis,vectorsandtensorsarewrittenindependentlyfromtheadoptedcoordinatesystem.Incontrasttotheuseofindicialnotation,whichputsinevidencethe componentsofvectorsandtensorsinthegivencoordinatesystem,thecompactnotationappearsasacleanandelegantformthatallowstheconceptstobepresentedwithout beingobnubilatedbythesometimesoverwhelmingpresenceofindexes.

Attheendofthesetwochapters,somefurtherreadingmaterialissuggestedtocomplementanddeepentheconceptspresentedhere.However,wehighlightthatforthe studyoftherestofthebookthisreadingisnotmandatory.Moreover,thereaderwill alsofindinAppendicesAandBamoredetailedexpositionofthetopicsaddressedin whatfollows,inanattempttomakethebookself-contained.

VectorandTensorAlgebra

1.1PointsandVectors

Considerthethree-dimensionalEuclideanspace,denotedby ℰ ,whosegeometryisbuilt uponasetofprimitiveelementscalledpoints.Notethat ℰ isnotavectorspaceinthe senseofthealgebrabecausetheadditionofpointsisaconceptwithoutmeaning.

Thedifferencebetweentwopoints X and Y ofspace ℰ isdefinedby v = X Y , (1.1) where v isthevectorwhoseoriginisin Y andendsat X .Allthevectorswhichcan bedeterminedthroughthedifferencesbetweenpointsbelongingto ℰ formtheset V associatedwith ℰ .Theset V isa(real)vectorspace,wherethetwobasicoperations inherenttothenotionofvectorspacearedefined,whichare(i)theadditionofvectors and(ii)theproductofavectorbyarealnumber.

Theadditionoperationbetweenapoint Y ∈ ℰ andthevector v ∈ V definesthepoint X ∈ ℰ suchthat(1.1)isverified.Thisoperationallowsustoestablishabiunivocalcorrespondencebetweenpointsof ℰ andvectorsof V .Infact,wecanarbitrarilypicka point O from ℰ ,andthenforeachpoint X ∈ ℰ thereexistsauniquevector v ∈ V such that v = X O

Space V iscalledthree-dimensionalprovidedthereexistin V setsofthreevectors {ei }={e1 , e2 , e3 } whicharelinearlyindependent1 andcanspantheentirevectorspace V ,thatis,theycangenerateanyvector v ∈ V throughthelinearcombination v = ��i ei . 2

Anyoftheselinearlyindependentsetsofvectors {ei } iscalledabasisforthe(real)vector space V

Beyondbasicoperations,multiplicationbyarealnumber,andadditionofvectors, thevectorspace V associatedwith ℰ isendowedwithaninnerproduct,alsocalledthe scalarproduct,operationbetweenvectorsof V .For u, v ∈ V ,thisoperationisdenoted by u ⋅ v ,whichisgeometricallydefinedbytheproductofthelengthsofthevectorsmultipliedbythecosineoftheanglebetweenthem.Thisoperationsatisfiestheproperties oftheinnerproductinthesenseofthealgebra.

1Vectors {ei } arecalledlinearlyindependentif ��i ei = �� implies ��i = 0for i = 1, 2, 3. 2Throughoutthisbook,unlessstatedotherwise,Einsteinnotationisadoptedtoshortensummation notations,resulting,forexample,inthefollowinglumpednotationwhenindexesarerepeated: v = ∑3 i 1 ��i ei = ��i ei

IntroductiontotheVariationalFormulationinMechanics:FundamentalsandApplications,FirstEdition. EdgardoO.Taroco,PabloJ.BlancoandRaúlA.Feijóo. ©2020JohnWiley&SonsLtd.Published2020byJohnWiley&SonsLtd.

4

1VectorandTensorAlgebra

Also,andthroughtheintroductionofthisoperation,thespace V hasatopological structureinducedbytheinnerproductthroughthedefinitionofthenormoperation foravector v ∈ V

||v || = √v v . (1.2)

Makinguseoftheseoperations,vectors u, v ∈ V aresaidtobeorthogonalif u v = 0. Similarly,abasis {ei } of V iscalledorthogonalif ei ej = ||ei ||||ej ||��ij isverified, i, j = 1, 2, 3,where ��ij istheKroneckersymbol.3 Finally,abasisof V iscalledorthonormal ifitisorthogonalandthenormofthevectorsinthebasisisunitary,thatis, ||ei || = 1, i = 1, 2 , 3.

Thematrix [gij ], i, j ∈{1, 2, 3},definedby

gij = ei ⋅ ej , i, j = 1, 2, 3, (1.3)

isnotsingularwhentheset {ei } isabasisfor V .Inturn,fromthedefinitionitfollows that gij = gji ,thatis, [gij ] isasymmetricmatrix.Letuscall

[g ij ]=[gij ] 1 , (1.4)

theinverseofmatrix [gij ].Fromthedefinitionweobtain

g ik gkj = gik g kj = ��ij , i, j = 1, 2, 3 (1.5)

Withtheseresults,wecanintroducethedualbasis {ei } associatedwith {ei } astheimage ofthelineartransformation

[g ij ]∶V → V , ej → ei = g ij ej (1.6)

Itcanbeprovedthatthistransformationproducesabasisfor V .Reciprocally,theapplicationof [gij ] over {ei } yieldstheoriginalbasis {ei }.Infact

gij ej = gij g jk ek = ��ik ek = ei . (1.7)

Anotherusefulresultisthefollowing

ei ⋅ ej = g ik ek ⋅ ej = g ik gkj = ��ij (1.8)

Acoordinatesystemconsistsofabasis {ei } for V ,notnecessarilyorthogonal,and anarbitrarypoint O of ℰ calledtheoriginofthecoordinatesystem.Whenthebasisis orthonormal,thecoordinatesystemiscalledCartesian.

Observethatthenotionofvectorwasintroducedindependentlyfromtheadopted basis,or,equivalently,fromthecoordinatesystem.Whenabasis {ei },andthenitsdual basis {ei },ischosen,theneachvector u ∈ V canbeassociatedwithatripleofrealnumbers {u1 , u2 , u3 } calledcomponentsof u withrespecttothebasis {ei },whicharedefined asfollows

ui = u ⋅ ei i = 1, 2, 3 (1.9)

Thesearethecomponentswithrespectto {ei } because,as {ei } isabasis,itresultsin u = ��i ei andthen

ui = u ei = ��k ek ei = ��k ��ki = ��i . (1.10)

3TheKroneckersymbol ��ij issuchthat ��ii = 1and ��ij = 0for i ≠ j

1.1PointsandVectors 5

Inparticular,givenabasis {ei },anditsdual {ei },thecomponentsof u withrespectto thesebasesareusuallynamedasfollows

• Componentsof u withrespectto {ei } arecontravariantcomponents,andare definedby

ui = u ei i = 1, 2, 3. (1.11)

Withthesecomponentsthevector u canberepresentedthroughthelinearcombination u = ui ei .

• Componentsof u withrespectto {ei } arecovariantcomponents,andaredefinedby

ui = u ei i = 1, 2, 3 (1.12)

Withthesecomponentsthevector u canberepresentedthroughthelinearcombination u = ui ei .

Ifthebasisisorthonormal,itiseasytoshowthatitisidenticaltoitsdual basis,andthereforethereisnodistinctionbetweencovariantandcontravariant components.

Likewise,givenacoordinatesystemin ℰ ,characterizedbythebasis {ei } andthepoint O ∈ ℰ ,wecandefinethecoordinatesofanarbitrarypoint X ∈ ℰ asthecovariantcomponentsofthevector X O from V ,thatis

=(X O) ⋅ ei (1.13)

Thus,thesamevector u ∈ V ,orthesamepoint X ∈ ℰ ,canbeassociatedwithdifferenttriplesofcomponentsandrepresentationsdependinguponthechosencoordinate system.4

Tounderlinethedifferencebetweenintrinsicandindicialnotation,notethat theinnerproductofvectors u, v ∈ V asafunctionoftheirdifferentcomponents resultsin

thatis,severalexpressionsarepossibleforthesameconcept,whichmaydelaytheunderstandingprogress.

Exercise1.1 Verifythevalidityofidentitiesgivenby(1.14).

4Asstatedabove,wewillemploycompactnotation,whichmeansthatwehighlighttheentityindetriment ofitscomponents,inthiscasevector u,orpoint X .Notationemphasizingcomponentsisalsocalledindicial notation,anditshouldonlybeemployedduringtheanalysisofagivenspecificproblem,specificallywhen calculationsarerequired.Inthisway,wecanpresenttheconceptsandoperationsinaclearandconcise manner,withoutbeingconfinedtoacertaincoordinatesystem.Thisalsohelpstocalltheattentiontothe factthat,duringcalculations,weshouldchoosethatbasiswhichmakesthetreatmentoftheproblemas simpleaspossible.Tosumup,indicialnotation isnotrelatedtoconcepts,buttocalculus.

6 1VectorandTensorAlgebra

1.2Second-OrderTensors

Wewillemploythetermtensor,orsecond-ordertensor,asasynonymforthelinear transformationbetween V and V 5 Then,wehave

T ∶V → V , u → v = Tu, (1.15) verifying

T(u + v )= Tu + Tv ∀u, v ∈ V , (1.16) T(�� u)= �� Tu ∀�� ∈ ℝ and ∀u ∈ V . (1.17)

Clearly,expressions(1.16)and(1.17)areequivalentto T(�� u + �� v )= �� Tu + �� Tv ∀��,�� ∈ ℝ and ∀u, v ∈ V (1.18)

Taking �� and �� zero,weobtainanexpressionthatwillbefrequentlyusedhereafter,

T�� = ��, (1.19)

where �� representsthenullelementin V .

Thesetofallsecond-ordertensors,thatis,thesetofalllineartransformationsbetween V and V ,willbecalled Lin

Lin ={T; T ∶ V → V , linear} (1.20)

Introducingtheadditionandmultiplicationbyrealnumbersin Lin definedby

(T + S)u = Tu + Su ∀u ∈ V , (1.21)

(�� T)u = �� (Tu)∀u ∈ V , (1.22)

turns Lin intoavectorspace,wherethenulltensor O transformseveryvector u ∈ V intothenullvector �� ∈ V ,thatis,

Ou = �� ∀u ∈ V (1.23)

Theidentitytensorisdenotedby I andisdefinedby

Iu = u ∀u ∈ V (1.24)

Given T ∈ Lin,thesetofvectors v ∈ V satisfying Tv = �� isdenotedby �� (T),thatis,

�� (T)={v ∈ V ; Tv = ��} (1.25)

Itiseasytoprovethat �� (T) isalsoavectorsubspaceof V ,calledthenullspaceof T (alsothekernelof T).

Given A, B ∈ Lin,thecompositionofthesetensors(composedtransformation)is anothertensor(lineartransformation) T ∈ Lin suchthat

Tu =(AB)u = ABu ∀u ∈ V (1.26)

5Inthischapterwelimitthepresentationtosecond-andthird-ordertensors,andsomeassociated operationsbecausethesearethemostusedelementsalongthebook.However,inAppendicesAandBthe readerwillfindmaterialrelatedtolineartransformationsbetweenvectorspacesofpossiblydifferent dimensions,whichnaturallyembracesthecaseofthird-andalsosecond-ordertensors.

1.2Second-OrderTensors 7

Sinceingeneral AB ≠ BA,whentheidentityisverifiedwesaythat A and B arecommutative.

Frompreviousdefinitions,andgivenarbitrary T, S, D ∈ Lin and �� ∈ ℝ,thenwehave

T(SD)=(TS)D = TSD, (1.27)

T(S + D)= TS + TD, (1.28)

�� (TS)= T(�� S), (1.29)

IT = TI = T (1.30)

Indeed,forarbitrary v ∈ V ,itis

T(SD)v = T(S(Dv ))=(TS)(Dv )=(TS)Dv , (1.31) fromwhich(1.27)isverified.Also,wehave

T(S + D)v = T((S + D)v )= T(Sv + Dv )= TSv + TDv =(TS + TD)v , (1.32) whichverifies(1.28).Similarly,

�� (TS)v = �� (T(Sv ))= T(�� Sv ))= T(�� S)v , (1.33) and(1.29)holds.Notefinallythat

= I(Tv )= Tv , (1.34) andthenwehaveproved(1.30). Thefollowingnotationwillalsobeused

n = n ⏞⏞⏞ TT T n ∈ N, (1.35) with T0 = I. Thetransposeofatensor T istheuniquetensor TT satisfying Tu v = u TT v ∀u, v ∈ V (1.36)

Uniquenessisprovedassumingthattherearetwotensortransposesfor T,denotedby TT 1 and TT 2 ,andwhichwillbeassumedtobedifferent.Fromdefinition(1.36),eachtensor satisfies Tu ⋅ v = u ⋅ TT 1 v ∀u, v ∈ V , (1.37) Tu ⋅ v = u ⋅ TT 2 v ∀u, v ∈ V (1.38)

Subtractingbothexpressionsyields

Recallingthat a b = 0, ∀a ∈ V implies b = ��,wehave

(TT 1 TT 2 )v = �� ∀v ∈ V , (1.40) andfromthedefinitionofthenulltensor(1.23),wearriveat TT 1 = TT 2 , (1.41)

8 1VectorandTensorAlgebra whichcontradictsthefactthatbothtensorsweredifferent.Therefore,thereexistsa uniquetransposeofatensor.

Usingthedefinitionofthetransposeofatensoritisstraightforwardtoconcludethat, forarbitrary A, S ∈ Lin and �� ∈ ℝ,weobtain

(S + A)T = ST + AT , (1.42)

(�� S)T = �� ST (1.43)

Infact,takingarbitrary u, v ∈ V ,then

(S + A)T u v = u (S + A)v = u Sv + u Av = ST u v + AT u v =(ST u + AT u) ⋅ v =(ST + AT )u ⋅ v , (1.44) andso(1.42)isproved.Analogously,

(�� S)T u v = u (�� S)v = �� u Sv = �� ST u v , (1.45) andso(1.43)isdemonstrated.

Asitiseasytosee,thetransposeoperationisalineartransformationbetween Lin and Lin.Inaddition,givenarbitrary A, S ∈ Lin,itis

(SA)T = AT ST , (1.46)

(ST )T = S (1.47)

Infact,forarbitrary u, v ∈ V ,wehave

(SA)T u v = u (SA)v = u S(Av )= ST u Av = AT ST u v , (1.48) andwearriveat(1.46).Analogously

(ST )T u v = u (ST )v = Su v , (1.49) andthen(1.197)isverified.

Atensor S iscalledsymmetricif

Su v = u Sv ∀u, v ∈ V , (1.50) andinsuchacaseweconcludethat S = ST .Atensoriscalledskew-symmetricif

Su ⋅ v =−(u ⋅ Sv )∀u, v ∈ V , (1.51)

whichimplies S =−ST .

Thesetofallsymmetrictensorswillbedenotedby Sym andthesetofall skew-symmetrictensorswillbedenotedby Skw.Inparticular,thenulltensor O issymmetricandskew-symmetric.

Anytensor S ∈ Lin canbeunivocallyrepresentedbytheadditionofasymmetrictensor(called Ss )andaskew-symmetrictensor(called Sa ),thatis,

S = Ss + Sa , (1.52) where

Ss = 1 2 (S + ST ), (1.53)

Sa = 1 2 (S ST ), (1.54)

1.2Second-OrderTensors 9 are,respectively,calledthesymmetriccomponentandtheskew-symmetriccomponentof S.

Sincethetransposeoperationyieldsauniquetensortranspose,itfollowsthatthe linearcombinationofasymmetric(skew-symmetric)tensorresultsinasymmetric (skew-symmetric)tensor.Then, Sym and Skw arevectorsubspacesof Lin.Moreover, fromtheuniquenessofthedecompositionintosymmetricandskew-symmetric components,itisconcludedthat Lin canbewrittenasthedirectsumofthesetwo subspaces

Lin = Sym ⊕ Skw (1.55)

Consideranarbitrary W ∈ Skw andanarbitrary T ∈ Lin.Then,itispossibletoshow thatforany u ∈ V thefollowingholds

u Wu = 0, (1.56)

u Tu = u Ts u. (1.57)

Infact,forarbitrary u ∈ V wehave u ⋅ Wu = WT u ⋅ u =−Wu ⋅ u, (1.58) fromwhere(1.56)isproved.Similarly,andusing(1.56),wehave u Tu = u (Ts + Ta )u = u Ts u + u Ta u = u Ts u, (1.59) and(1.57)follows.

Thetensorproductbetweentwovectors a, b ∈ V isthesecond-ordertensor a ⊗ b thattransformsanyvector v ∈ V intovector (b ⋅ v )a,thatis,

(a ⊗ b)v =(b ⋅ v )a ∀v ∈ V (1.60)

Fromthepreviousdefinition,andgivenarbitrary a, b, c, d ∈ V ,weobtainthefollowingresults

(a ⊗ b)T = b ⊗ a, (1.61) (a ⊗ b)(c ⊗ d)=(b c)(a ⊗ d). (1.62)

Inaddition,given T ∈ Lin,itcanbeshownthat T(a ⊗ b)=(Ta) ⊗ b, (1.63)

(a ⊗ b)T = a ⊗ (TT b) (1.64)

Indeed,considerarbitrary u, v ∈ V ,usingdefinition(1.60)forwardandbackward, wehave

(a ⊗ b)T u v = u (a ⊗ b)v = u (b v )a =(u a)(b v )=(b ⊗ a)u v , (1.65) andthen(1.61)holds.Now,observethat

(a ⊗ b)(c ⊗ d)u =(a ⊗ b)(d ⋅ u)c =(d ⋅ u)(b ⋅ c)a =(b c)(d u)a =(b c)(a ⊗ d)u, (1.66) andwearriveat(1.62).Similarly,wehave T(a ⊗ b)u =(b ⋅ u)Ta =((Ta) ⊗ b)u, (1.67)

1VectorandTensorAlgebra

(n ⊗ n)v π

Figure1.1 Geometricconceptoforthogonal projectionovertheplane �� whosenormalvectoris n.

(I n ⊗ n)v v

whichyields(1.63).Lastly,notethat

(a ⊗ b)Tu =(b Tu)a =(TT b u)a =(a ⊗ (TT b))u, (1.68) andthen(1.64)isproved.

Analogouslytothedefinitionoftheinnerproductin V ,itisalsopossibletodefine theinnerproductin Lin.Considertwoelements T, S ∈ Lin whichcanbewritteninthe forms T = t1 ⊗ t2 and S = s1 ⊗ s2 ,respectively,with t1 , t2 , s1 , s2 ∈ V .Then,wedefine theinnerproduct T S in Lin as

T ⋅ S =(t1 ⊗ t2 ) ⋅ (s1 ⊗ s2 )=(t1 ⋅ s1 )(t2 ⋅ s2 ) (1.69)

Withdefinition(1.69),itisstraightforwardtoprovethat,forarbitrary T ∈ Lin and u, v ∈ V ,thefollowingresultholds

T ⋅ (u ⊗ v )= u ⋅ Tv (1.70)

Infact,putting T = t1 ⊗ t2 ,andmakinguseofdefinitions(1.69)and(1.60)weobtain

T ⋅ (u ⊗ v )=(t1 ⊗ t2 ) ⋅ (u ⊗ v )=(t1 ⋅ u)(t2 ⋅ v ) = u ⋅ [(t2 ⋅ v )t1 ]= u ⋅ [(t1 ⊗ t2 )v ]= u ⋅ Tv (1.71)

Letusdenote n theunitnormalvectortotheplane �� (seeFigure1.1).Thetensor n ⊗ n appliedoveranyvector v ∈ V gives

(n ⊗ n)v =(n ⋅ v )n, (1.72)

whichistheorthogonalprojectionof v ∈ V overthedirection n.Inturn,the second-ordertensor P = I n ⊗ n appliedoveranyvector v ∈ V yields

Pv =(I n ⊗ n)v = v −(n ⋅ v )n, (1.73)

whichistheorthogonalprojectionof v overtheplane �� . Itcanbeappreciatedthattensor P issymmetricandalsoverifies P2 = P.Infact

PT =(I n ⊗ n)T = IT −(n ⊗ n)T = I n ⊗ n = P, (1.74) and,forarbitrary v ∈ V ,itis

P2 v = P(Pv )= P[v −(n ⋅ v )n]= Pv −(n ⋅ v )Pn = Pv , (1.75) andthepreviousstatementshold.

1.2Second-OrderTensors 11

Tensorssatisfyingthesetwoproperties,thatis, P ∈ Sym and P2 = P,arecalled orthogonalprojectiontensors.Examplesofthiskindoftensorare

I, I n ⊗ n, n ⊗ n (1.76) Itispossibletoshowthat

dim(Lin)= dim V × dim V = 9, (1.77) andifwetakeabasis {ei } for V ,thesets

{(ei ⊗ ej ); i, j = 1, 2, 3}, (1.78)

{(ei ⊗ ej ); i, j = 1, 2, 3}, (1.79)

{(ei ⊗ ej ); i, j = 1, 2, 3}, (1.80)

{(ei ⊗ ej ); i, j = 1, 2, 3}, (1.81) aredifferentpossiblebasesfor Lin.Inthisway,anytensor T ∈ Lin canbeexpressedbya uniquelinearcombinationoftheelementofthechosenbasis.Thecomponentsoftensor T inthechosenbasisaredefinedinananalogousmannertothatforvectors.Hence,we have

Tij = T (ei ⊗ ej )= ei Tej , (1.82)

T ij = T ⋅ (ei ⊗ ej )= ei ⋅ Tej , (1.83)

T i j = T ⋅ (ei ⊗ ej )= ei ⋅ Tej , (1.84)

T j i = T (ei ⊗ ej )= ei Tej , (1.85)

wheretheinnerproductbetweenelementsof Lin isdefinedin(1.69).Also,thisinner productcanbeintroducedintermsofthetraceoperation,asshownbelow.Thisway, therepresentationof T ∈ Lin intermsofthesecomponentsisgivenby

T = Tij (ei ⊗ ej )= T ij (ei ⊗ ej )= T i j (ei ⊗ ej )= T j i (ei ⊗ ej ), (1.86) wheretheimplicitsummationofrepeatedindicesisconsidered.Coefficients Tij , T ij , T i j and T j i are,respectively,covariant,contravariant,andmixedcomponentsoftensor T. Inparticular,theidentitytensor I is

I = ei ⊗ ei , (1.87) againwithimplicitsummationoverindex i.Infact,givenarbitrary u = uj ej = uj ej ∈ V , wehave

Iu =(ei ⊗ ei )uj ej = uj ei ��ij = ui ei = u, (1.88) orequivalently,

Iu =(ei ⊗ ei )uj ej = uj g ij ei = uj ej = u (1.89)

ForaCartesianbasis {ei } for V ,thereisnodifferencebetweencomponentsof T.In thiscase,wesimplyrefertotheCartesiancomponentsofthetensor. Theadvantageofemployingcompactnotationisagainevidentwhencomparingto indicialnotation.Atensorisaconcept(lineartransformationin V )whichdoesnot

12 1VectorandTensorAlgebra

dependonthebasischosenfor V .Thesameisvalidforthetraceoperationandinner productbetweentensors.

Tofurtherillustratetheconceptualaspectshighlightedbytheintrinsicnotation,let usintroducesomeofthedefinitionsalreadypresentedintermsofbothnotations.

• Tensor.Compactnotation: T ∈ Lin.Indicialnotation

Tij = T kl gki glj = T k j gki = T k i gkj , (1.90)

T ij = Tkl g ki g lj = T i k g kj = T j k g ki , (1.91)

T i j = Tkj g ki = T ik gkj = T l k g ki glj , (1.92)

T j i = Tik g kj = T kj gki = T k l gki glj (1.93)

• Applicationofatensoroveravector.Compactnotation: u = Tv , u, v ∈ V and T ∈ Lin.Indicialnotation(justsomeofthepossibleexpressions)

ui = Tij vj = Tij g jk vk = T j i vj = T j i gjk vk , (1.94) ui = T ijvj = T ij gjk vk = T i j vj = T j i g jk vk . (1.95)

• Compositionoftensors.Compactnotation: T = AB, T, A, B ∈ Lin.Indicialnotation

Tij = Aik Bkj = A k i Bkj , (1.96)

T ij = Aik B j k = Aik Bkj , (1.97)

T i j = Aik Bkj = Aik Bkj , (1.98)

T j i = A k i B j k = Aik Bkj , (1.99) andvariantsincludingthetensor g ij or gij .

• Tensorproductbetweenvectors.Compactnotation: u ⊗ v , u, v ∈ V .Indicialnotation

(u ⊗ v )ij = ui vj , (1.100)

(u ⊗ v )ij = ui vj , (1.101)

(u ⊗ v )i j = ui vj , (1.102)

(u ⊗ v ) j i = ui vj (1.103)

• Symmetriccomponentofatensor.Compactnotation: S ∈ Sym ⇔ S = ST ,andequivalently Sa = O.Indicialnotation

Sij = Sji =⇒ [Sij ]=[Sij ]T , (1.104)

S ij = S ji =⇒ [S ij ]=[S ij ]T , (1.105)

S i j = S i j (1.106)

Notethatforasymmetrictensorthematrixofcovariantcomponentsisalsosymmetric,andthesameholdsforcontravariantcomponents.Incontrast,thematrix representationofasymmetrictensorinmixedcomponentsisnotsymmetricingeneral.Indeed,since S i j = S i j ,weconcludethatthesamecomponentsaresymmetrically placedinthetwo(different)matrixrepresentationsofthetensor.

1.2Second-OrderTensors 13

• Skew-symmetriccomponentofatensor.Compactnotation: W ∈ Skw ⇔ W =−WT , andthen Ws = O.Indicialnotation Wij =−Wji =

Thatis,foraskew-symmetrictensorthematrixrepresentationsincovariantand contravariantcomponentsareskew-symmetric,whilethematrixrepresentationsin mixedcomponentsarenotskew-symmetric.Asbefore,theexpression W i j =−W i j indicatesthatthesecoefficientsaresymmetricallyplaced,butinthetwodifferent matrixrepresentationsinmixedcomponents.

Thetraceoperationofasecond-ordertensor T ∈ Lin isalinearfunctionalwhichassociatestoeachtensor T ∈ Lin arealnumberdenotedbytrT,thatis, tr ∶ Lin → ℝ

withtheproperty

Fromthelinearityofthetraceoperationitfollowsthat trT = T ij tr(ei ⊗ ej )= Tij tr(ei ⊗ ej )= T i j tr(ei ⊗ ej )= T j i tr(ei ⊗ ej ) = T ij gij = Tij g ij

TheseexpressionsallowustoevaluatetrT intermsofthecomponentsofthetensor.

Evidently,theresultisindependentoftheadoptedbasis. Asthetransposeoperationisalinearoperation,wehave

trT = trTT , (1.113)

tr(AB)= tr(BA), (1.114)

trI = 3 (1.115)

Indeed,putting T = t1 ⊗ t2 , A = a1 ⊗ a2 ,and B = b1 ⊗ b2 ,andrecallingthat I = ei ⊗ ei ,itis

trT = tr(t1 ⊗ t2 )= t1 ⋅ t2 = t2 ⋅ t1 = tr(t2 ⊗ t1 )= trTT , (1.116) andthen(1.113)follows.Also,using(1.62)and(1.113)yields tr(AB)= tr[(a1 ⊗ a2 )(b1 ⊗ b2 )]= tr[(a2 ⋅ b1 )(a1 ⊗ b2 )]=(a2 ⋅ b1 )(a1 ⋅ b2 ) = tr[(a1 b2 )(a2 ⊗ b1 )]= tr[(a1 b2 )(b1 ⊗ a2 )] = tr[(b1 ⊗ b2 )(a1 ⊗ a2 )]= tr(BA), (1.117) and(1.114)isproved.Finally,withdefinition(1.87),wehave trI = tr(ei ⊗ ei )= ei ei = ��ii = 3, (1.118) thusproving(1.115).

14

1VectorandTensorAlgebra

Noticethatusingthetraceoperationmakesitpossibletodefineaninnerproductin Lin

S T = tr(ST T)∀S, T ∈ Lin. (1.119)

Withthisdefinition,thefollowingpropertiesaresatisfied

S T = T S, (1.120)

S S ≥ 0 ∀S ∈ Lin, (1.121)

S ⋅ S = 0 ⇔ S = O, (1.122)

andtheinnerproduct,inturn,inducesanormin Lin

‖S‖ = √S S = √tr(ST S) (1.123)

Expressionsoftheinnerproductintermsofsomeofthedifferenttensorcomponents are

A ⋅ T = Aij T ij = Aij Tij = A j i T i j = Aij T j i , (1.124)

andvariantsalsoinvolving g ij or gij .

Now,considerarbitrary a, b, c, d ∈ V , T, A, B ∈ Lin, S ∈ Sym and W ∈ Skw.Then, somepropertiesoftheinnerproductin Lin are

I T = trT, (1.125)

A BT = BT A T, (1.126)

T ⋅ (u ⊗ v )= u ⋅ Tv , (1.127)

(a ⊗ b) ⋅ (c ⊗ d)=(a ⋅ c)(b ⋅ d), (1.128)

S T = S Ts , (1.129)

W T = W Ta , (1.130)

S ⋅ W = 0, (1.131)

T ⋅ A = 0 =⇒ T = O, (1.132)

T S = 0 =⇒ T ∈ Skw, (1.133)

T W = 0 =⇒ T ∈ Sym, (1.134)

A B = As Bs + Aa Ba (1.135)

Infact,fromdefinition(1.119),expression(1.125)followsdirectlybecause

I T = tr(IT T)= trT. (1.136)

Identity(1.126)isalsoproveddirectly,since

A BT = tr(AT BT)= tr[(BT A)T T]= BT A T. (1.137)

Exercise1.2 Provethatidentities(1.127)–(1.135)holdusingthepropertiesofthetrace operation.

1.2Second-OrderTensors 15

Thedeterminantofatensor A ∈ Lin canbedefinedas

det A = 1 6 [2tr(A3 )+(trA)3 3(trA)tr(A2 )] (1.138)

Forarbitrary A, T ∈ Lin,and �� ∈ ℝ,thedeterminantoperationsatisfies

det(AT)= det(TA), (1.139)

det(AT)=(det A)(det T), (1.140)

det(�� T)= �� 3 det T, (1.141)

det I = 1 (1.142)

Theinverseofatensor T isalsoatensor T 1 withtheproperty

T 1 T = I. (1.143)

Atensor T iscalledinvertibleifdet T ≠ 0,andif T isinvertible,itstransposetensor TT isalsoinvertible,verifying

(TT ) 1 =(T 1 )T = T T . (1.144)

Tensor Q iscalledanorthogonaltensorifitpreservestheinnerproductbetween vectors,thatis,

Qu ⋅ Qv = u ⋅ v ∀u, v ∈ V (1.145)

Thenecessaryandsufficientconditionforatensor Q tobeorthogonalis

QQ T = Q T Q = I, (1.146) or,equivalently,

Q T = Q 1 . (1.147)

Thesetofallorthogonaltensorsisdenotedby Orth.

Exercise1.3 Showthatatensorisorthogonalifandonlyif QQ T = Q T Q = I.

Anyorthogonaltensorwithpositivedeterminantiscalledarotationtensor.Inparticular,thesetofallrotationtensorsisdenotedby Rot. Atensor T iscalledpositivedefiniteif v ⋅ Tv > 0 ∀v ∈ V v ≠ ��, (1.148) v ⋅ Tv = 0 ⇔ v = �� (1.149)

Givenabasis {ek },itissaidthatallthebases {ek } havethesameorientationas {ek } iftheycanbeobtainedfromarotationofthelatter.Thatmeans

ek = Rek k = 1, 2 , 3. (1.150)

Sinceanyorthogonaltensor Q isarotation R,oritisarotationmultipliedby 1,there existtwoclassesofbaseseachassociatedwithanorientation.Hereafterwewillassume thatoneoftheseorientationshasbeenchosen.

16

1VectorandTensorAlgebra

Thecrossproduct u × v betweenvectors u, v ∈ V ,whoseanglebetweenthemis �� ,is anothervector w suchthat w ⟂ totheplanedefinedby u and v , (1.151)

||w|| = ||u||||v || sin ��, (1.152)

{u, v , w} havethesameorientationastheadoptedbasis (1.153)

Then,itfollowsthat

u × v =−(v × u), (1.154)

u × u = ��, (1.155)

u v × w = w u × v = v w × u. (1.156)

Giventhreearbitraryelements u, v , w ∈ V ,thecrossproductsatisfies

u ×(v × w)=(u w)v −(u v )w, (1.157) orequivalently,

u ×(v × w)=(v ⊗ w)a u. (1.158)

Exercise1.4 Provethat(1.157)holds.

When u, v , w arelinearlyindependent,thevalueoftheproduct u (v × w) represents thevolumeoftheparallelepiped �� determinedbythevectors u, v , w.Then,givena tensor T,itisnotdifficulttoverifythat

det T = Tu Tv × Tw u v × w , (1.159) fromwhereitfollowsthat

det T = vol(T(�� )) vol(�� ) , (1.160)

whichprovidesageometricinterpretationofthedeterminantofatensor T,andwhere T(�� ) istheimageoftheparallelepiped �� underthelineartransformation(tensor) T andvol( ) standsforthevolume.

Givenasecond-ordertensorfield T ∈ Lin,weintroducetheprincipalinvariantsof T asfollows

I1 (T)= trT, (1.161)

I2 (T)= 1 2 (tr(T2 )−(trT)2 ), (1.162)

I3 (T)= det T (1.163)

Theinvariancepropertyisestablishedbythefactthat

Ii (T)= Ii (QTQ T )∀Q ∈ Orthi = 1, 2, 3 (1.164)

Then,anytensor T admitstherepresentationestablishedbytheCayley–Hamilton theorem[132]

det(�� I T)= �� 3 �� 2 I1 (T)− �� I2 (T)− I3 (T) (1.165)