Introduction to computational contact mechanics: a geometrical approach

305 Pages·2015·7.525 MB·English

by  Izi, Ridvan;Konyukhov, Alexander

INTRODUCTION TO
COMPUTATIONAL
CONTACT MECHANICS
WILEY SERIES IN COMPUTATIONAL MECHANICS
SeriesAdvisors:
RenédeBorst
PerumalNithiarasu
TayfunE.Tezduyar
GenkiYagawa
TarekZohdi
IntroductiontoComputationalContact Konyukhov April2015
Mechanics:AGeometricalApproach
ExtendedFiniteElementMethod: Khoei December2014
TheoryandApplications
ComputationalFluid-Structure Bazilevs,Takizawa January2013
Interaction:MethodsandApplications andTezduyar
IntroductiontoFiniteStrainTheoryfor Hashiguchiand November2012
ContinuumElasto-Plasticity Yamakawa
NonlinearFiniteElementAnalysisof DeBorst,Crisfield, August2012
SolidsandStructures,SecondEdition Remmersand
Verhoosel
AnIntroductiontoMathematical Oden November2011
Modeling:ACourseinMechanics
ComputationalMechanicsof Munjiza,Knightand November2011
Discontinua Rougier
IntroductiontoFiniteElement Szabo´ andBabusˇka March2011
Analysis:Formulation,Verificationand
Validation
INTRODUCTION TO
COMPUTATIONAL
CONTACT MECHANICS
A GEOMETRICAL APPROACH
Alexander Konyukhov
KarlsruheInstituteofTechnology(KIT),Germany
Ridvan Izi
KarlsruheInstituteofTechnology(KIT),Germany
Thiseditionfirstpublished2015
©2015JohnWiley&SonsLtd
Registeredoffice
JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,United
Kingdom
Fordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapply
forpermissiontoreusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com.
Therightoftheauthortobeidentifiedastheauthorofthisworkhasbeenassertedinaccordancewith
theCopyright,DesignsandPatentsAct1988.
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or
transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingor
otherwise,exceptaspermittedbytheUKCopyright,DesignsandPatentsAct1988,withouttheprior
permissionofthepublisher.
Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprint
maynotbeavailableinelectronicbooks.
Designationsusedbycompaniestodistinguishtheirproductsareoftenclaimedastrademarks.All
brandnamesandproductnamesusedinthisbookaretradenames,servicemarks,trademarksor
registeredtrademarksoftheirrespectiveowners.Thepublisherisnotassociatedwithanyproductor
vendormentionedinthisbook.
LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbestefforts
inpreparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor
completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof
merchantabilityorfitnessforaparticularpurpose.Itissoldontheunderstandingthatthepublisheris
notengagedinrenderingprofessionalservicesandneitherthepublishernortheauthorshallbeliable
fordamagesarisingherefrom.Ifprofessionaladviceorotherexpertassistanceisrequired,theservices
ofacompetentprofessionalshouldbesought
LibraryofCongressCataloging-in-PublicationData
Konyukhov,Alexander.
Introductiontocomputationalcontactmechanics:ageometricalapproach/AlexanderKonyukhov,
KarlsruheInstituteofTechnology(KIT),Germany,RidvanIzi,KarlsruheInstituteofTechnology
(KIT),Germany.
pagescm.–(Wileyseriesincomputationalmechanics)
Includesbibliographicalreferencesandindex.
ISBN978-1-118-77065-8(cloth:alk.paper)1.Contactmechanics.2.Mechanics,Applied.I.Izi,
Ridvan.II.Title.III.Title:Computationalcontactmechanics.
TA353.K662015
620.1(cid:2)05–dc23
2015005384
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
Typesetin11/13ptTimesLTStdbyLaserwordsPrivateLimited,Chennai,India
1 2015
Contents
SeriesPreface xiii
Preface xv
Acknowledgments xix
PartI THEORY
1 IntroductionwithaSpring-MassFrictionlessContactSystem 3
1.1 StructuralPart–DeflectionofSpring-MassSystem 3
1.2 ContactPart–Non-PenetrationintoRigidPlane 4
1.3 ContactFormulations 5
1.3.1 LagrangeMultiplierMethod 5
1.3.2 PenaltyMethod 6
1.3.3 AugmentedLagrangianMethod 8
2 GeneralFormulationofaContactProblem 13
2.1 StructuralPart–FormulationofaProbleminLinearElasticity 13
2.1.1 StrongFormulationofEquilibrium 14
2.1.2 WeakFormulationofEquilibrium 15
2.2 FormulationoftheContactPart(Signorini’sproblem) 17
3 DifferentialGeometry 23
3.1 CurveanditsProperties 23
3.1.1 Example:CircleanditsProperties 26
3.2 FrenetFormulasin2D 28
3.3 DescriptionofSurfacesbyGaussCoordinates 29
3.3.1 TangentandNormalVectors:SurfaceCoordinateSystem 29
3.3.2 BasisVectors:MetricTensoranditsApplications 30
3.3.3 RelationshipsbetweenCo-andContravariantBasisVectors 33
3.3.4 Co-andContravariantRepresentationofaVectoronaSurface 34
3.3.5 CurvatureTensorandStructureoftheSurface 35
vi Contents
3.4 DifferentialPropertiesofSurfaces 37
3.4.1 TheWeingartenFormula 37
3.4.2 TheGauss–CodazziFormula 38
3.4.3 CovariantDerivativesontheSurface 38
3.4.4 Example:GeometricalAnalysisofaCylindricalSurface 39
4 GeometryandKinematicsforanArbitraryTwoBody
ContactProblem 45
4.1 LocalCoordinateSystem 46
4.2 ClosestPointProjection(CPP)Procedure–Analysis 48
4.2.1 ExistenceandUniquenessofCPPProcedure 49
4.2.2 NumericalSolutionofCPPProcedurein2D 54
4.2.3 NumericalSolutionofCPPProcedurein3D 54
4.3 ContactKinematics 55
4.3.1 2DContactKinematicsusingNaturalCoordinatessandζ 58
4.3.2 ContactKinematicsin3DCoordinateSystem 59
5 AbstractFormofFormulationsinComputationalMechanics 61
5.1 OperatorNecessaryfortheAbstractFormulation 61
5.1.1 ExamplesofOperatorsinMechanics 61
5.1.2 ExamplesofVariousProblems 62
5.2 AbstractFormoftheIterativeMethod 63
5.3 FixedPointTheorem(Banach) 64
5.4 NewtonIterativeSolutionMethod 65
5.4.1 GeometricalInterpretationoftheNewtonIterativeMethod 66
5.5 AbstractFormforContactFormulations 69
5.5.1 LagrangeMultiplierMethodinOperatorForm 69
5.5.2 PenaltyMethodinOperatorForm 71
6 WeakFormulationandConsistentLinearization 73
6.1 WeakFormulationintheLocalCoordinateSystem 73
6.2 RegularizationwithPenaltyMethod 75
6.3 ConsistentLinearization 75
6.3.1 LinearizationofNormalPart 76
6.4 ApplicationtoLagrangeMultipliersandtoFollowingForces 79
6.4.1 LinearizationfortheLagrangeMultipliersMethod 80
6.4.2 LinearizationforFollowingForces:NormalForce
orPressure 80
6.5 LinearizationoftheConvectiveVariationδξ 81
6.6 NitscheMethod 81
6.6.1 Example:IndependenceoftheStabilizationParameter 83
Contents vii
7 FiniteElementDiscretization 85
7.1 ComputationoftheContactIntegralforVariousContactApproaches 86
7.1.1 NumericalIntegrationfortheNode-To-Node(NTN) 86
7.1.2 NumericalIntegrationfortheNode-To-Segment(NTS) 86
7.1.3 NumericalIntegrationfortheSegment-To-AnalyticalSegment
(STAS) 86
7.1.4 NumericalIntegrationfortheSegment-To-Segment(STS) 87
7.2 Node-To-Node(NTN)ContactElement 88
7.3 NitscheNode-To-Node(NTN)ContactElement 89
7.4 Node-To-Segment(NTS)ContactElement 91
7.4.1 ClosestPointProjectionProcedurefortheLinear
NTSContactElement 94
7.4.2 PeculiaritiesinComputationoftheContactIntegral 95
7.4.3 ResidualandTangentMatrix 96
7.5 Segment-To-Analytical-Surface(STAS)Approach 98
7.5.1 GeneralStructureofCPPProcedureforSTAS
ContactElement 98
7.5.2 ClosedformSolutionsforPenetrationin2D 100
7.5.3 DiscretizationforSTASContactApproach 102
7.5.4 ResidualandTangentMatrix 102
7.6 Segment-To-Segment(STS)MortarApproach 104
7.6.1 PeculiaritiesoftheCPPProcedurefortheSTSContact
Approach 106
7.6.2 ComputationoftheResidualandTangentMatrix 106
8 VerificationwithAnalyticalSolutions 109
8.1 HertzProblem 109
8.1.1 ContactGeometry 110
8.1.2 ContactPressureandDisplacementforSpheres:
3DHertzSolution 113
8.1.3 ContactPressureandDisplacementforCylinders:
2DHertzSolution 114
8.2 RigidFlatPunchProblem 114
8.3 ImpactonMovingPendulum:CenterofPercussion 116
8.4 GeneralizedEuler–EytelweinProblem 118
8.4.1 ARopeonaCircleandaRopeonanEllipse 119
9 FrictionalContactProblems 121
9.1 MeasuresofContactInteractions–StickingandSlidingCase:
FrictionLaw 121
9.1.1 CoulombFrictionLaw 123
viii Contents
9.2 RegularizationofTangentialForceandReturnMappingAlgorithm 123
9.2.1 Elasto-PlasticAnalogy:PrincipleofMaximumofDissipation 125
9.2.2 UpdateofSlidingDisplacementsintheCaseofReversible
Loading 127
9.3 WeakFormanditsConsistentLinearization 128
9.4 FrictionalNode-To-Node(NTN)ContactElement 129
9.4.1 RegularizationoftheContactConditions 130
9.4.2 LinearizationtheofTangentialPartfortheNTNContact
Approach 131
9.4.3 DiscretizationofFrictionalNTN 131
9.4.4 AlgorithmforaLocalLevelFrictionalNTNContactElement 133
9.5 FrictionalNode-To-Segment(NTS)ContactElement 134
9.5.1 LinearizationandDiscretizationfortheNTSFrictionalContact
Element 134
9.5.2 AlgorithmforaLocalLevelNTSFrictionalContactElement 135
9.6 NTSFrictionalContactElement 135
PartII PROGRAMMINGANDVERIFICATIONTASKS
10 IntroductiontoProgrammingandVerificationTasks 139
11 Lesson1NonlinearStructuralTruss–elmt1.f 143
11.1 Implementation 144
11.2 Examples 148
11.2.1 ConstitutiveLawsofMaterial 148
11.2.2 LargeRotation 149
11.2.3 Snap-ThroughBuckling 150
12 Lesson2NonlinearStructuralPlane–elmt2.f 151
12.1 Implementation 152
12.2 Examples 156
12.2.1 ConstitutiveLawofMaterial 156
12.2.2 LargeRotation 158
13 Lesson3PenaltyNode-To-Node(NTN)–elmt100.f 159
13.1 Implementation 160
13.2 Examples 161
13.2.1 TwoTrusses 161
13.2.2 ThreeTrusses 162
13.2.3 TwoBlocks 163
14 Lesson4LagrangeMultiplierNode-To-Node(NTN)–elmt101.f 165
14.1 Implementation 166