Nonlinear Solid Mechanics A Continuum Approach for Engineering Gerhard A. Holzapfel GrazUniversityofTechnology,Austria JOHNWILEY &SONS, LTD Chichester (cid:127) Weinheim (cid:127) NewYork (cid:127) Brisbane (cid:127) Singapore (cid:127) Toronto Copyright©2000 by JohnWiley&SonsLtd, BaffinsLane,Chichester, WestSussexP0191UD,England National 01243779777 International (+44)1243779777 e-mail(forordersandcustomerserviceenquiries):[email protected] VisitourHomePageonhttp://www.wiley.co.uk or http://www.wiley.com Reprinted,withcorrections,December2001 Allrightsreserved.Nopartofthispublicationmaybereproduced,stored inaretrievalsystem,ortransmitted,inanyformorbyanymeans,electronic, mechanical,photocopying,recording,scanningorotherwise,exceptundertheterms oftheCopyright,DesignsandPatentsAct1988orunderthetermsofalicenceissued bytheCopyrightLicensingAgency,90TottenhamCourtRoad,London,UKW1P9HE, withoutthepermissioninwritingofthepublisher. OtherWileyEditorialOffices JohnWiley&Sons,Inc.,605ThirdAvenue, NewYork,NY10158-0012,USA Wilcy-VCHVerlagGmbH,Pappelallee3, D-69469Weinheim,Germany JacarandaWileyLtd,33ParkRoad,Milton, Queensland4064,Australia JohnWiley&Sons(Asia)PteLtd,2ClementiLoop#02-01, JinXingDistripark,Singapore0512 JohnWiley&Sons(Canada)Ltd,22WorcesterRoad, Rexdale,OntarioM9W1L1,Canada LibraryofCongressCataloginginPublicationData Holzapfel,GerhardA. Nonlinearsolidmechanics:acontinuumapproachforengineering/GerhardA.Holzapfel. p.cm. Includesbibliographicalreferencesandindex. ISBN0-471-82304-X—ISBN0-471-82319-8 1.Continuummechanics.I.Title. QA8—08.2.H6552000 531 dc21 00-027315 BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN0-471-82304-X(ppc) 0-471-82319-8 (pbk) Producedfromcamera-readycopysuppliedbytheauthor PrintedandboundinGreatBritainbyAntonyRoweLtd,Chippenham,Wiltshire Thisbookisprintedonacid-freepaperresponsiblymanufacturedfromsustainable forestry,inwhichatleasttwotreesareplantedforeachoneusedforpaperproduction. > 1 Contents Preface u IX Acknowledgements 47927 u£ X1U 1 Introduction to Vectorsand Tensors 1 1.1 AlgebraofVectors 1 1.2 AlgebraofTensors 9 1.3 Higher-orderTensors 20 1.4 Eigenvalues,EigenvectorsofTensors 24 1.5 TransformationLawsforBasisVectorsandComponents 28 1.6 GeneralBases 32 1.7 Scalar,Vector,TensorFunctions 40 1.8 GradientsandRelatedOperators 44 1.9 IntegralTheorems 52 2 Kinematics 55 2.1 Configurations,andMotionsofContinuumBodies 56 2.2 Displacement,Velocity,AccelerationFields . . . . 61 2.3 Material,SpatialDerivatives 64 2.4 DeformationGradient 70 2.5 StrainTensors 76 2.6 Rotation,StretchTensors 85 2.7 RatesofDeformationTensors 95 2.8 LieTimeDerivatives 106 V x ,IN/ vi Contents 109 3 The Concept of Stress 3.1 Traction Vectors,andStressTensors 109 119 3.2 ExtremalStressValues 3.3 Examplesof Statesof Stress . . . . 123 127 3.4 AlternativeStressTensors 131 4 Balance Principles 131 4.1 Conservationof Mass 138 4.2 Reynolds’ TransportTheorem 141 4.3 MomentumBalancePrinciples 152 4.4 Balance of MechanicalEnergy 4.5 Balanceof Energy inContinuumThermodynamics 161 166 4.6 Entropy InequalityPrinciple 174 4.7 Master BalancePrinciple 5 Some Aspects of Objectivity 179 5.1 ChangeofObserver,andObjectiveTensorFields 179 187 5.2 Superimposed Rigid-bodyMotions 192 5.3 ObjectiveRates 5.4 Invarianceof ElasticMaterial Response 196 205 6 Hyperelastic Materials 6.1 GeneralRemarksonConstitutiveEquations 206 212 6.2 IsotropicElyperelasticMaterials 6.3 IncompressibleHyperelasticMaterials. . . 222 6.4 CompressibleHyperelasticMaterials . . . 227 6.5 SomeFormsof Strain-energy Functions . . 235 252 6.6 ElasticityTensors 265 6.7 TransverselyIsotropicMaterials Contents (cid:127)(cid:127) Vll 6.8 CompositeMaterialswithTwoFamiliesofFibers 272 6.9 ConstitutiveModelswithInternalVariables . . . 278 6.10 ViscoelasticMaterialsatLargeStrains 282 6.11 HyperelasticMaterialswithIsotropicDamage . 295 7 ThermodynamicsofMaterials 305 7.1 PhysicalPreliminaries 306 7.2 ThermoelasticityofMacroscopicNetworks 311 7.3 ThermodynamicPotentials 321 7.4 Calorimetry 325 7.5 Isothermal,IsentropicElasticityTensors 328 7.6 EntropicElasticMaterials 333 7.7 ThermodynamicExtensionofOgden’sMaterialModel 337 7.8 SimpleTensionof EntropicElasticMaterials 343 7.9 ThermodynamicswithInternalVariables 357 8 Variational Principles 371 8.1 VirtualDisplacements,Variations 372 8.2 Principleof VirtualWork 377 8.3 PrincipleofStationaryPotentialEnergy . . . 386 8.4 LinearizationofthePrincipleofVirtualWork 392 8.5 Two-fieldVariationalPrinciples 402 8.6 Three-fieldVariationalPrinciples 409 References 415 Index 435 Preface My desire in writing this textbook was toshow the fascination and beauty of nonlin- earsolid mechanicsand thermodynamicsfrom anengineeringcomputational point of view. My primary goal was not only to offer a modern introductory textbook using thecontinuum approach to be read with interest, enjoyment and curiosity, but also to offerareferencebookthatincorporatessomeof therecentdevelopmentsinthefield. I wanted tostimulateandinvitethereadertostudythisexcitingscienceand takehimon a pleasant journey in the wonderful world of nonlinear mechanics, which serves as a solidbasisforasurprisinglylarge varietyof problemsarisingin practicalengineering. Linear theories of solid mechanics are highly developed and are in a satisfactory stateof completion. Mostprocessesin nature, however,are highly nonlinear. Theap- proach taken has theaimof providinginsightin the basicconceptsof solid mechanics with particular reference to the nonlinear regime. Once familiar with the main ideas the reader will beabletospecializeindifferent aspectsof thesubject matter. Ifelt the need for a self-contained textbook intended primarily for beginners who want to un- derstand thecorrespondence between nonlinearcontinuum mechanics, nonlinearcon- stitutivemodels and variational principles as essential prerequisites for finiteelement formulations. Ofcourse,nosinglebookcancoverallaspectsofthebroadfieldofsolidmechanics, so that many topics are not discussed here at all. The selection of the material for inclusion is influenced strongly by current curricula, trends in the literature and the author’s particular interests in engineering and science. Here, a particular selection and style was chosen in order to highlight someof the more inspiring topics in solid mechanics. I hope that mychoice, which is of course subjective, will befound to be acceptable. My ultimateintention was to present an introduction to thesubject matterin a di- dacticallysoundmannerand asclearlyaspossible. Ihopethatthetextprovidesenough insightsfor understanding of the terminology used in scientific state-of-the art papers andtofindthe‘rightandstraightforwardpath’inthescientificworldthroughtheeffec- tiveuseof figures,whichare very importantlearningtools.Theyaredesignedinorder IX x Preface to attract attention and to be instructive and helpful to the reader. Necessary mathe- matics and physics are explained in the text. The approach used in each of theeight chapters will enable the reader to work through the chapters in order of appearance, each topic being presented in a logical sequence and basedon thepreceding topics. A proper understanding of the subject matter requires knowledge of tensor alge- bra and tensor calculus. For most of the derivations throughout the text I have used symbolic notation with those clear bold-faced symbols which give the subject matter adistinguished beauty. However, for higher-order tensors and for final resultsin most of thederivationsI have used index notation, which providesthe reader with more in- sight. Terminology is printed in bold-face whereit appearsfor thefirst time while the notation used in thetextisdefined at theappropriate point. Forthose whohavenot beenexposed tothenecessary mathematicsI haveincluded a chapter on tensor algebra and tensor calculus. It includes the essential ideasof lin- earization in the form of the concept of the directional derivative. Chapter 1 summa- rizes elementary properties which are needed for the vector and tensor manipulations performed in all subsequent chapters and which are necessary to many problems that arisefrequentlyinengineeringand physics. - It is the prime consideration of Chapter 2 to use tensor analysis for the descrip tion of the motion and finite deformation of continua. The continuum approach is - introduced along with the notion ‘Lagrangian’ (material) and ‘Eulerian’ (spatial) de scriptions. In asystematicway themostimportant kinematictensorsareprovided and their physical significance explained. The push-forward and pull-back operations for material and spatial quantities and the concept of the Lie time derivative are intro- duced. The concept of stress is the main topic of Chapter3. Cauchy’s stresstheorem is introduced, along with the Cauchy and first Piola-Kirchhoff traction vectors, and theessential stress tensors are defined and their interrelationshipsdiscussed. InChap- ter 4 attention is focused on the discussion of the balance principles. Both statics and dynamics are treated. Based upon continuum thermodynamics theentropy inequality principle is provided and the general structure of all principles is summarized as the masterbalance(inequality) principle. Chapter5deals withimportantaspectsofobjec- tivity, which plays a crucial part in nonlinear continuum mechanics. A discussion of changeofobserverandsuperimposedrigid-bodymotionsisfollowedbyadevelopment of objective(stress)ratesandinvarianceof elasticmaterial response. Chapters 6 and 7 form the central part of the book and provide insight in the con- struction of nonlinear constitutiveequations for thedescription of the mechanical and thermomechanical behavior of solids. Thesetwochaptersshow theessentialrichness of thefield. They are written for those who want to gain experience in handling ma- terial modelsand derivingstress relationsand theassociated elasticity tensorsthatare fundamental for finite element methods. Several examples and exercises areaimed at enablingthereadertothinkintermsofconstitutivemodelsandtoformulatemorecom- Preface xi plexmaterialmodels. Allofthetypesofconstitutiveequationspresentedareaccessible for usewithinfiniteelementprocedures. ThebulkofChapter6isconcernedwithfiniteelasticityandfiniteviscoelasticity.It includes adiscussionof isotropic, incompressibleand compressible hyperelastic ma- terials and providesconstitutivemodels for transversely isotropic and composite ma- terials which are suitablefor a large number of applications in practical engineering. An approach toinelastic materials with internal variables is given along with instruc- tiveexamplesof hyperelasticmaterialsthatinvolverelaxationand/orcreepeffectsand isotropicdamagemechanismsatfinitestrains.ThemainpurposeofChapter7istopro- vide an introduction to the thermodynamicsof materials. Thischapter is devoted not onlytothefoundationofcontinuumthermodynamicsbutalsotoselected topicsofsta- tisticalthermodynamics.Itstartswithastatisticalapproach bysummarizingimportant physical aspects of the thermoelastic behavior of molecular networks (for example, amorphous solid polymers), based almost entirely on an entropy concept, and con- tinues with a systematic phenomenologicalapproachincludingfinitethermoelasticity and finite thermoviscoelasticity. The stress-strain-temperature response of so-called entropic elastic materials is discussed in more detail and based on a representative example which is concerned with the adiabatic stretching of a rubber band. Typical thermomechanicalcouplingeffectsarestudied. Chapter8 is designed tocover the essential features of the most important varia- tional principles thatare very useful informulating approximation techniquessuch as the finite element method. Although finite elements are not treated in this text, it is hoped that thischapter will be attractive tothose who approach the subjectfrom the computational side. It shows the relationship between the strong and weak formsof initialboundary-valueproblems,presentstheclassicalprincipleof virtual workinboth spatial and material descriptions and its linearized form. Two- and three-field varia- tionalprinciplesarealsodiscussed.Thepresenttextendswhereconventionallyabook onthefiniteelementmethodwould begin. There are numerous worked examples adjacent to the relevant text. These have the goal of clarifying and supplementing the subject matter. In many cases they are straightforward,but provide anessential part of thetext. Thesymbol isused tode- notetheend ofan exerciseoraproof.Theend-ofchapterexercisesareforhomework. The (almost)200exercises provided aredesigned to supplement the text and tocon- solidateconceptsdiscussedin thetext. Mostof them servethepurposeofstimulating the reader to further study and to reinforce and develop practical skills in nonlinear continuum and solid mechanics, towards the direction of computational mechanics. In manycasesthesolutionsof selectedexercisesaregivenandfrequently usedlaterin furtherdevelopments.Therefore,itshouldbeinstructiveforthereadertoworkthrough areasonablenumberofexercises. Numerousreferencestosupplementarymaterialaresuggestedanddiscussedbriefly