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Nonlocal Continuum Field Theories A. Cemal Eringen SPRINGER Nonlocal Continuum Field Theories Springer NewYork Heidelberg Barcelona HongKong London Milan Paris Singapore Tokyo A. Cemal Eringen Nonlocal Continuum Field Theories With72Figures A.CemalEringen EmeritusProfessor,PrincetonUniversity 15RedTailDrive Littleton,CO80126-5001 USA LibraryofCongressCataloging-in-PublicationData Eringen,A.Cemal. Nonlocalcontinuumfieldtheories/A.Cemal.Eringen. p. cm. Includesbibliographicalreferencesandindex. ISBN0-387-95275-6(alk.paper) 1.Fieldtheory(Physics).2.Continuummechanics. I.Title. QC173.7.E76 2001 530–dc21 200102044 ©2002Springer-VerlagNewYork,Inc. 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PrintedintheUnitedStatesofAmerica. 9 8 7 6 5 4 3 2 1 ISBN0-387-95275-6 SPIN10833382 Springer-Verlag NewYork Berlin Heidelberg AmemberofBertelsmannSpringerScienetBusinessMediaGmbH Preface Thisbookisdevotedtothedevelopmentofthenonlocalcontinuumfieldtheories ofmaterialbodieswherenonlocalintermolecularattractionsareimportant.There exists a large class of problems in classical physics and continuum mechanics (classicalfieldtheories)thatfalloutsidetheirdomainofapplications.Fractureof solids, stress fields at the dislocation core and at the tips of cracks, singularities present at the point of application of concentrated loads (forces, couples, heat, etc.), sharp corners and discontinuities in bodies, the failure in the prediction of shortwavelengthbehaviorofelasticwaves,andseveraldecadesoftheviscosity riseinfluidsflowinginmicroscopicchannelsarebutfewmajoranomaliesthatdefy classical treatment. In the field of electromagnetic theory, polaritons, gyrotropic effects,andsuperconductivitycannotbetreatedbytheclassicalfieldtheories.For example, several branches of high-frequency waves are not predicted and short wavelengthregionsdeviategrosslyfromexperimentalobservations.Tocastlight on the materials behavior in these areas, physicists often seek help from atomic latticedynamics. Themainpurposeofthisbookistopresentaunifiedfoundationforthedevelop- mentofthebasicfieldequationsofnonlocalcontinuumfieldtheories.Tothisend, wehavereliedonthenaturalextensionsofthetwofundamentallawsofphysics tononlocality:(i)theenergybalancelawispostulatedtoremaininglobalform; and (ii) a material point of the body is considered to be attracted by all points of the body, at all past times. By means of these two natural generalizations of thecorrespondinglocalprinciples,theoriesofnonlocalelasticity,fluiddynamics, andelectromagneticfieldtheoriesareformulatedthatincludenonlocalityinboth spaceandtime(memory-dependence). vi Preface Solutions of the field equations are given for many critical problems in these fields,displayingunexpectedfavorableagreementswiththeatomictheoriesand/or theexperimentalobservations. Presently,thereexistsnopublishedtreatiseinthisfield,exceptsomepapersand limitedreviewsthatarescatteredintheliterature.Webelievethisbookoffersrich materialforresearchideasinnonlocalsolidandfluidmechanics,electromagnetic theory,andthenonlocaltheoriesofmediawithmicrostructureandnonlocaltheory ofliquidcrystals.Itprovidesmathematicalmethodsfortheexplorationofmany failuresofclassicalfieldtheoriesandforthediscoveriesofnewphysicalphenom- enaand/orexplanationsoftheoldones.Itcanbeusedforgraduatestudyinthe departments of applied physics, applied mathematics, engineering sciences, and forresearch. WeexpressourthankstoJenniferPagewhotypedthenonmathematicalpartsof themanuscript.MydaughterMevawhofreedmuchtimeformebyundertaking manychoresoflifeandlifeitself.Tohergoesmyeverlastinglove. Littleton,Colorado A.CemalEringen December2000 Introduction Nonlocal continuum field theories are concerned with the physics of material bodieswhosebehavioratamaterialpointisinfluencedbythestateofallpointsof thebody.Followingtheclassicalnotions,materialpointsofabodyareconsidered tobecontinuousandareassignedsomephysicallyindependentobjects(variables) (e.g.,mass,charge,electricfield,magneticfield).Thestateofthebody,atamaterial point,isdescribedbytherelationsoftheresponseobjectsthatconstituteanother class (e.g., stress, internal energy, heat) as functions of the independent objects. Theserelationsarecalledconstitutiveequations.Thenonlocaltheorygeneralizes theclassicalfieldtheoryintworespects:(i)theenergybalancelawisconsidered valid globally (for the entire body); and (ii) the state of the body at a material pointisdescribedbytheresponsefunctionals.Thismeansthatweneedcomplete knowledgeoftheindependentvariablesatallpointsofthebodytodescribethe state of the body at each point. In this book the terminology of nonlocal field theories is to be understood in this sense. In fact, we have also included, in this description,nonlocalityintimewhichisknownasmemory-dependence. Asasubclass,thepresentdefinitionincludesgradienttheorieswhichpossess limitednonlocality.Herevariousordergradientsoftheindependentobjects,ata materialpointandtheirtimerates,enterintoresponsefunctionsatthesamematerial points,Eringen[1966].Athirdcategoryofnonlocalityinvolvesconstitutiveersatzs between the spatial statistical moments of dependent and corresponding sets of independent variables. Such a group of theories are known as director theories, micromorphic,microstretch,andmicropolartheories.Thesetheoriesareexplored elsewhere,Eringen[1999],[2001]. viii Introduction Thepresent bookisdevotedtothedevelopment ofthenonlocal fieldtheories describedbythefundamentalnotions(i)and(ii)above. Thequestionarises,whynonlocality?Thedomainoftheapplicabilityofclas- sical field theories is intimately connected to the length and time scales. If L denotestheexternalcharacteristiclength(e.g.,cracklength,wavelength,alength overwhichappliedloadsaresmooth)andltheinternalcharacteristiclength(e.g., granular distance, lattice parameter), then in the region L/l (cid:1) 1, classical field theories predict sufficiently accurate results. On the other hand, when L/l ∼ 1, localtheoriesfailandwemustresorttoeitheratomicornonlocaltheoriesthatcan accountforthelong-rangeinteratomicattractions.Forthedynamicalcase,there will be a similar scale T/τ where T is the external characteristic time (e.g., the time scale of the applied loads) and τ is the internal characteristic length (e.g., the time scale of transmission of signal from one molecule to the next).Again, classicaltheoriesfailwhenT/τ ∼1.Thusweseethatthephysicalphenomenon inspace–timerequiresnonlocalityandmemoryeffectsscaledbyL/landT/τ. Theconceptofnonlocalityisinherentinsolidstatephysicswherethenonlocal attractions of atoms are prevalent. Here the material is considered to consist of discreteatomsconnectedbydistantforcesfromotherneighboringatoms.Hence, weexpectrelationsbetweenthenonlocalfieldtheoriesandthelatticedynamics (seeSection6.9). Realmaterialsdonotresembleperfectcrystallatticesorlatticeswithfewim- perfections (e.g., dislocations, foreign atoms, etc.) which enable a physicist to describethestateofthebody.Unfortunately,amorphousandengineeringmateri- alsdonotlendthemselvestosuchidealdescriptions,whilenonlocalcontinuum theories make it possible to describe material properties very well. In fact, this description is faithful in microscopic scales all the way to the size of the lattice parameter(seeSections6.10to6.17). Thus, nonlocal continuum theory should be considered a viable discipline in physicsandappliedscience. Prior to the 1960s some elements of continuum theory had entered into the discussion of various physical phenomena that required mathematical descrip- tionsintheformsoftheintegro-partialdifferentialequations.Forexample,Chan- drasekhar’s[1950]radiativetransferequationisofthistype.Otherhistoricalref- erencesinthiscategoryaregiveninseveralarticlesinContinuumPhysics,Vol.IV, edited by Eringen [1976]. The formulation of the nonlocal elastic constitutive equationsbymeansoflatticedynamics,wasgivenbyKrumhansl[1965],[1968], Kröner and Datta [1966], and Kunin [1966]. The continuum approach to memory-dependent nonlocal elasticity was made by Eringen [1966] and to nonlocal micropolarelasticitybyEringen[1965].Theseearlytheoriesdidnotincludethe aspect(i),namely,theglobalnatureofthebalancelaws.Thisaspectalongwiththe nonlocalconstitutiveequationswasformulatedbyEringenandEdelen[1972]and Eringen[1972a,b].Areviewanddevelopmentofthetheory,andhistoricreference upto1976,arecontainedinContinuumPhysics,Vol.IV,editedbyEringen[1976]. Introduction ix Thetheoryofnonlocalelasticityhasattractedtheattentionofmanywriters,per- hapsbecauseofitsearlysuccessinresolvinganoldprobleminfracturemechanics. Itiswellknownthatclassicalelasticitypredictsinfinitestressatthetipofasharp crack.Thisledresearcherstoinventseveralfracturecriteria(e.g.,Griffith,fracture toughness,energyreleaserate,J-integral,etc.)independentofthestateofstress atthetipofacrack.ThenonlocalelasticitysolutionofEringenandKim[1974a,b] and Eringen et al. [1977] showed that the stress at the tip of a crack is finite, it risestoamaximumandthendiminisheswiththedistancefromthecracktip(cf. Section6.14).Similarly,thenonlocalsolutionofthediscreetdislocationproblem (Eringen[1977a,b])leadstovanishingstressattheeyeofthedislocation,risingto amaximumintheinteriorofthebodyandapproachestheclassicalsolutionwith the distance.These results led to the introduction of a natural fracture criterion, basedonthemaximumstresshypothesisandtheaccuratepredictionofthecohe- sive stress (cf. Sections 6.12 to 6.17). Moreover, nonlocal solutions also lead to dispersionrelationsforharmonicwavesthatarefaithfultothelatticedynamical resultsthroughouttheentireBrillouinzone. Thenonlocalcontinuumtheoryoffluentmedia,includingthemicrostructural effects, were discussed by Eringen [1972b], [1973b], applications to turbulence bySpezialeandEringen[1981],thediffusionofgasesbyDemirayandEringen [1978],andmagnetohydrodynamicsbyEringen[1986].McCayandNarasimhan [1981]discussedtheoryofnonlocalelectromagneticfluids,andNarasimhanand McCay[1981]appliedtheirtheorytodispersivewavesindielectricfluids.Eringen [1973c],[1984a],[1984b],[1984c],[1986],[1990],[1991]introducedthetheories ofnonlocalelectromagneticelasticsolidsandgavesolutionsofseveralproblems: pointcharges,opticalwaves,polaritons,Eddycurrents,gyrotropicmedia,super- conductivity,Alfvenwaves,etc.AbookbyAgranovichandGinsburg[1984]covers manyaspectsofthenonlocaleffectsoncrystaloptics.Thenonlocalityhereappears undertheterminologyof“SpatialDisperson." Inspiteofitsage,theliteratureonnonlocalfieldtheoriesisnotextensive.There existseveralreviews,conferencesymposia,andreports.Amongthemwemention: ContinuumPhysics,Vol.IV,editedbyA.C.Eringen,AcademicPress,1976. Thisbookcontainsaccountsofthepolarandnonlocalpolartheoriesofelastic solids. NonlocalTheoriesofMaterialMedia,ThePolishAcademy,1976. Thisbookcontainsreviewsonnonlocalelasticity. NonlocalTheoryofMaterialMedia,CISMCoursesandLectures,No.268,edited byD.Rogula,Springer-Verlag,1982. Here, lattice defects, nonlocal elasticity, and some soluble problems are dis- cussedbyseveralauthors. ElasticMediawithMicrostructure,Vols.IandII,byI.A.Kunin,Springer-Verlag, 1982/1983. Inthesebooksthelatticedynamicalapproachisemployedforthediscussionof microstructuralproblems.