A Course on Plasticity Theory

209 Pages·2023·6.98 MB·English

by  David J. Steigmann

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
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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 lay-
ing the foundations of modern continuum mechanics. Their emphasis on permanence
andrigormeantthattheunsettledsubjectofplasticitytheorywaslargelyavoided,how-
ever, 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