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OXFORDSERIESONMATERIALSMODELLING SeriesEditors AdrianP.Sutton,FRS DepartmentofPhysics,ImperialCollegeLondon RobertE.Rudd LawrenceLivermoreNationalLaboratory OXFORDSERIESONMATERIALSMODELLING Materials modelling is one of the fastest growing areas in the science and engi- neering of materials, both in academe and in industry. It is a very wide field covering materials phenomena and processes that span ten orders of magni- tude in length and more than twenty in time. A broad range of models and computational techniques has been developed to model separately atomistic, microstructural, and continuum processes. A new field of multiscale modeling has also emerged in which two or more length scales are modeled sequentially or concurrently. The aim of this series is to provide a pedagogical set of texts spanningtheatomisticandmicrostructuralscalesofmaterialsmodeling,written byacknowledgedexperts.Eachbookwillassumeatmostarudimentaryknowl- edge of the field it covers and it will bring the reader to the frontiers of current research.Itishopedthattheserieswillbeusefulforteachingmaterialsmodeling atthepostgraduatelevel. APS,London RER,Livermore,California 1.M.W.Finnis:InteratomicForcesinCondensedMatter 2.K.Bhattacharya:MicrostructureofMartensite—WhyItFormsandHowItGives RisetotheShape-MemoryEffects 3.V.V.Bulatov,W.Cai:ComputerSimulationsofDislocations 4.A.S.Argon:StrengtheningMechanismsinCrystalPlasticity 5.L.P.Kubin:Dislocations,MesoscaleSimulationsandPlasticFlow 6.A.P.Sutton:PhysicsofElasticityandCrystalDefects 7.D.Steigmann:ACourseonPlasticityTheory Forthcoming: D.N.Theodorou,V.Mavrantzas:MultiscaleModellingofPolymers A Course on Plasticity Theory David J. Steigmann DepartmentofMechanicalEngineering,UniversityofCalifornia,Berkeley GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©DavidJ.Steigmann2022 Themoralrightsoftheauthorhavebeenasserted 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:2022942494 ISBN978–0–19–288315–5 DOI:10.1093/oso/9780192883155.001.0001 Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. Tomyfamily Preface Thetheoryofplasticityhasalongandinterestinghistorydatingbackabouttwoanda halfcenturies.Activityinthefieldexpandedrapidlyoverthecourseofthepastcentury inparticular,givingrisetoarapidpaceofadvancement.Duringmuchofthelatterphase ofitsmoderndevelopment,thefieldwasbesetbyambiguityandcontroversyconcerning someofitsconceptualfoundations.Unsurprisingly,thisledtotheemergenceofdiffer- ent, often incompatible, schools of thought on the subject. A comprehensive survey of the state of plasticity theory during this period may be found in the review article by Naghdi. Meanwhile, great strides were being made by applied mathematicians in laying the foundations of modern continuum mechanics. Their emphasis on permanence andrigormeantthattheunsettledsubjectofplasticitytheorywaslargelyavoided,however, with the result that this lacuna in the panoply of continuum theories began to be filledquiterecently,aroundtheturnofthemillennium,aftertheantagonismoftheolder schoolshadbeguntofade. Practicallyeverythingknownaboutplasticitythroughthemiddleofthepastcentury isdocumentedinthesuperbtreatisesbyPragerandHodge,Nadai,Hill,andKachanov, whichshouldbecarefullyreadbyanyseriousstudentofoursubject.Ataroundthesame time,newdevelopmentsweretakingplaceintheapplicationofdifferentialgeometryto thecontinuumtheoryofdefectsassociatedwithplasticity.Thishasbecomealargeand active discipline in its own right, and a substantial part of this book is devoted to it. TheworksofBloomandWang,andthevolumeeditedbyKröner,arerecommendedto those interested in learning about its foundations, while those by Clayton, Epstein and El anowski,Epstein,andSteinmanncovermanyofthemorerecentdevelopments.The modernengineeringtheory,asdistinctfromthegeometricaltheory,isablysummarized inthebooksbyLubliner,BesselingandvanderGiessen,andBigoni.ThebooksbyHan andReddyandbyGurtinetal.arerecommendedformathematicaldevelopmentsand someofthemorerecentthinkingonthesubject. While writing this book I have been guided by the belief that one can always learn somethingfromanythoughtfulperson.Accordinglythecontentsreflectmyunderstand- ingoftheworkofresearchersandscholarsspanningalargeanddiverserangeofviews onthesubjectofplasticity.Inthecourseofsurveyingthemodernliterature,Ihavebeen struck by the continuing isolation of the various schools from one another, with scant evidence of cross-fertilization. Particularly glaring, from my perspective, is the lack of acknowledgment of the efforts of Noll in laying the foundations of the modern theory. This has been rectified to a great degree by Epstein and El anowski, and I follow their lead in giving primacy to Noll’s perspective. In fairness, Noll is not an easy read, and muchstudyisneededtograspthefullimportofhiswork. Preface vii The book is certainly not self-contained. Readers are presumed to have had prior exposure to a good introductory course on basic continuum mechanics at the level of the excellent books by Chadwick and Gurtin, for example. Aspects of this basic back- groundaresummarizedasneeded,butnotdevelopedinanydetail.Theemphasishere isonconceptualissuesconcerningthefoundationsofplasticitytheorythathaveproved challenging,tomeatleast.Thesehaveledmetotheviewthatthetimehascometoseek ameasureofconsolidationandunificationinthefield.Idonotignoretheclassicalthe- ory,butratherdevelopitfromtheperspectiveofthemoderntheory.Forexample,the classicaltheoryofperfectlyplasticsolidswaspresentedhistoricallyinawaythatledto itsnaturalinterpretation,fromthevantagepointofmoderncontinuummechanics,asa theoryofnon-Newtonianfluidsratherthanasamodelofthebehaviorofcertainsolids. The resolution of this dilemma is a prime example of the clarity that can be achieved onceasecurelogicalfoundationforthegeneraltheoryhasbeenestablished. Someexplicitsolutionstotheequationsofplasticitytheoryarecoveredinthisbook, but not nearly to the extent found in the older books. The reason for this omission is, firstly, that the small collection of explicit solutions that are known is ably covered elsewhere, so that duplication is hardly justified, and secondly, that due to the advent of modern computing, they are not nearly as relevant as they once were. I devote the remainderofthebooktothetheoreticalfoundationsofthesubject,inaccordancewith myownpredilections,ratherthantomattershavingtodowithcomputation.Thereason forthisemphasisismybeliefthatstudentsaretypicallynotaswellversedintheconcep- tualfoundationsastheyshouldbeiftheyaretorealizethefullpotentialofcomputational mechanics. A number of exercises of varying degrees of difficulty appear throughout. Theseservetoreinforceunderstandingandtoencouragethereadertofillinanygapsin thedevelopment.Comprehensivesolutionstoselectedexercisesareincludedattheend ofthebook. Thosewhomighthavereadmypreviousbook,FiniteElasticityTheory,willfindthe styleandpresentationofthisonetobequitefamiliar.Thepresentbookisperhapsabit moredemanding,however,insofarasvariousconceptsfromnon-Euclideandifferential geometry are covered in detail. I gratefully acknowledge the small group of dedicated graduate students at the University of California, Berkeley, whose interest and persis- tenceprovidedtheimpetusforthedevelopmentofagraduatecourseonwhichthebook is based. I am especially grateful to one of them, Milad Shirani, for his critical reading ofthemanuscriptandforpreparingthefigures. DavidSteigmann Berkeley,2021 References Besseling,J.F.,andvanderGiessen,E.(1994).MathematicalModellingofInelasticDeformation. ChapmanandHall,London. Bigoni,D.(2012). NonlinearSolidMechanics:BifurcationTheoryandMaterialInstability. Cam- bridgeUniversityPress,Cambridge,UK. viii Preface Bloom,F.(1979).ModernDifferentialGeometricTechniquesintheTheoryofContinuousDistributions ofDislocations.LectureNotesinMathematics,Vol.733.Springer,Berlin. Chadwick,P.(1976).ContinuumMechanics:ConciseTheoryandProblems.Dover,NewYork. Clayton,J.D.(2011).NonlinearMechanicsofCrystals.Springer,Dordrecht. Epstein, M. (2010). The Geometrical Language of Continuum Mechanics. Cambridge University Press,Cambridge,UK. Epstein,M.,andElz˙anowski,M.(2007).MaterialInhomogeneitiesandTheirEvolution.Springer, Berlin. Gurtin,M.E.(1981).AnIntroductiontoContinuumMechanics.AcademicPress,Orlando. Gurtin,M.E.,Fried,E.,andAnand,L.(2010).TheMechanicsandThermodynamicsofContinua. CambridgeUniversityPress,Cambridge,UK. Han,W., and Reddy, B. D. (2013). Plasticity: Mathematical Theory and Numerical Analysis. Springer,N.Y. Hill,R.(1950).TheMathematicalTheoryofPlasticity.ClarendonPress,Oxford. Kachanov,L.M.(1974).FundamentalsoftheTheoryofPlasticity.MIRPublishers,Moscow. Kröner,E.(Ed)(1968).Proc.IUTAMSymposiumonMechanicsofGeneralizedContinua.Springer, N.Y. Lubliner,J.(2008).PlasticityTheory.Dover,N.Y. Nadai,A.(1950).TheoryofFlowandFractureofSolids.McGraw-Hill,N.Y. Naghdi,P.M.(1990).Acriticalreviewofthestateoffiniteplasticity.J.Appl.Math.Phys.(ZAMP) 41,315–394. Noll, W. (1967). Materially uniform simple bodies with inhomogeneities. Arch. Ration. Mech. Anal.27,1–32. Prager,W.,andHodge,P.G.(1951).TheoryofPerfectlyPlasticSolids.JohnWiley&Sons,N.Y. Steigmann,D.J.(2017).FiniteElasticityTheory.OxfordUniversityPress,Oxford. Steinmann, P. (2015). Geometrical Foundations of Continuum Mechanics: An Application to First andSecond-OrderElasticityandElasto-Plasticity. LectureNotesinAppliedMathematicsand Mechanics,Vol.2.Springer,Berlin. Wang,C.-C.(1979).MathematicalPrinciplesofMechanicsandElectromagnetism.PartA:Analytical andContinuumMechanics.PlenumPress,N.Y. Contents 1 Preliminaries 1 1.1 Phenomenology 1 1.2 Elementsofcontinuummechanics 4 2 Briefresuméofnonlinearelasticitytheory 15 2.1 Stressandstrainenergy 15 2.2 Conservativeproblemsandpotentialenergy 21 2.3 TheLegendre–Hadamardinequality 25 2.4 Materialsymmetry 29 3 Aprimerontensoranalysisinthree-dimensionalspace 34 3.1 Coordinates,bases,vectors,andmetrics 34 3.2 Second-ordertensors 41 3.3 Derivativesandconnections 44 3.4 TheLevi-Civitaconnection 50 3.5 Thecurl,Stokes’theorem,andcurvature 54 3.6 TorsionandtheWeitzenböckconnection 59 4 Deformationandstressinconvectedcoordinates 63 4.1 Deformationgradientandstrain 63 4.2 Straincompatibility 66 4.3 Stress,equationsofmotion 71 5 Elasticandplasticdeformations 74 5.1 Elastic–plasticdeformation,dislocationdensity 74 5.2 Differential-geometricconsiderations 79 5.3 Incompatibilityoftheelasticstrain 82 6 Energy,stress,dissipation,andplasticevolution 87 6.1 Materiallyuniformelasticbodies 87 6.2 Surfacedislocationsandstressrelaxation 93 6.3 Dissipationduetoplasticevolution 97 6.4 Superposedrigid-bodymotions 100