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Mechanics of Incremental Deformations Theory of Elasticity and Viscoelasticity of Initially StressedSolidsand Fluids, Including Thermodynamic Foundations and Applications to Finite Strain MAURICEA.BlOT New York,NewYork John Wiley&Sons,Inc., NewYork' London' Sydney MechanicsofIncrementalDeformationTheoryofElasticityandViscoelasticityofInitiallyStressedSolidsandFluids,IncludingThermodynamicsFoundationandApplicationstoFiniteStrainByMauriceA.Biot(19051985)OriginallypublishedbyJohnWiley&Sons,Inc.,NewYork/London/Sydney,1965.CopyrightreleasedtoMadameM.A.BiotbyJohnWiley&Sons,Inc.(LetterdatedJune4,2008)Mme.M.A.BiotAvenuePaulHymans117Bte.341200BruxellesBelgiumFree access and redistribution of the book is granted to students andresearchersbythecourtesyofMme.Biot.Noalteringorresellingofthebookispermittedwithouttheexplicitconsentofthecopyrightowner.ScannedbookpreparedbyAlexCheng,Nov.2008.PREFACE Thisbookembodiesanapproachtonon-linearelasticitywhich marksafundamentaldeparturefromclassicalandcurrenttrends. Thebasictheorywasfirstpublishedbetweentheyears1934and 1940 in seven papers listed at the end of this Preface.In addition to asystematictreatmentofthegeneraltheoryandextensionsto viscoelasticity,thebookincludescomprehensivenewdevelopments and applications, many of which are presented here for the first time. The work ischaracterized by the useof cartesian concepts and of elementary mathematical methodsthat donot requireaknowledge ofthetensorcalculusorothermorespecializedtechniques.The explicit introduction of alocal rotation field in the three-dimensional equationsleadstoatheorywhichseparatesthephysicsfromthe geometryand isequallyvalidforelasticand non-elasticmaterials, usingeither rectangular orcurvilinearcoordinates. Asthis book demonstrates,the scope of problems solved by these new methodsgoesfarbeyond the results which it has been possible to obtain by the more elaborate and less general traditional approach. New insights, leading to many discoveries and aunified outlook have beenbrought into such widely diversifiedareasas rubber elasticity, internal gravity waves in a fluid and tectonic folding in geodynamics. Thetheoryprovidesrigorousandcompletelygeneralequations governingthedynamicsandstabilityofsolidsandfluidsunder initialstressinthecontextofsmallperturbations.It doesnot requirethat themediumbeelasticorisotropicbut isapplicableto anisotropic,viscoelastic,orplasticmedia.Noassumptionsare introduced regarding the physical process by which the initial stress hasbeengenerated.Thetreatmentofviscoelasticity,which constitutesasubstantialportionof thebook,incorporatessomeof theresultsestablishedinmypreviousworkonnon-equilibrium thermodynamics. Non-lineartheoriesof deformationandapplicationstoproblems of finite strain are obtained by extension ofthe concept of incremental deformationinamediumunderinitialstress.Incontrasttothe v viPreface presentationinthepaperslistedattheendofthisPreface,the concepts and methods are developed primarily in the context of the linearizedmechanicsof continuousmediaunderinitialstressasan independent theory. InitsearlierphasethisworkwasinterruptedbytheSecond WorldWar.Myinterestinthesubjectwasrevivedsomefifteen yearsagoinconnectionwithgeologicalproblems.Becausethe theoryisvalidfornon-elasticmedia,itwasfoundapplicableto problemsingeodynamicswhereithasopenedanewphaseand provided newand fruitfulmethods of analysis.Although the basic theoryhasbeenavailablein thescientificliterature formorethan twenty-fiveyearsandhasbeenusedoccasionallyby afewinvesti-gatorsintechnologicalandgeophysicalproblems,itspotentialities seemtohavebeen generallyoverlooked.Thisisperhapsduetoa prevalentemphasisontensorformalism.Formany. yearsithas been my feeling that, between the formalistic approach of the mathe-maticianandthemorepragmatictreatmentofproblemsbythe engineer, there is aneed for a rigorous but intermediate theory based oncartesianconcepts.Itwouldextendtothree-dimensionaf deformationstheviewpointsandmethodsof whathascometobe known as"Strength of Materials." Classicalapproachestonon-linearelasticityhavebeenhandi-cappedintechnologicalapplicationsbyarigidformalismwhich obscuresthephysicalsignificanceof theanalyticalresults.In the solutionof complex problemsencountered in practicean important requirement isthe possibility of recognizing those factorswhich add considerablytothe mathematicalcomplexity and at the same time arenotrelevanttothephysicalproblemandmaybeneglected. Thiscannotbeachievedunlesstheanalyticalformulationitself is sufficiently simple and physically clear.One of the basic difficulties arising fromthetensor theory isdue to the useof the metric tensor as a measure of the finite strain.This requires the physical properties tobeexpressedintermsofthesquaresofthedistancesbetween material points.By its very nature this definition of the strain leads toaformulationwhich doesnot provide acleardistinction between thegeometryofthedeformationfieldandthosepropertieswhich represent the physics of the material.Because it contains quadratic termstobeginwith,themetrictensorisalsothesourceof much confusionregardingthesignificanceofsecondandhigherorder Prefacevii elastic coefficients.In this respect illuminating contrast is provided bythe simplifiedtreatmentof secondorderelasticity inChapter2 (section9). The formalconciseness'of the tensor calculus js deceptive,since it leavestotheengineerandthephysicisttheburdenofexpressing physical propertiesof materials by meansof non-cartesianconcepts whichare not essential and generallycomplicate the task.In fact, it may be said that the overemphasis on tensor inethods in thiscase provides aprime example of mathematical techniques which in some areashaveslowedprogressandledoccasionallytofalsephysical interpretations. Theapproachpresentedinthisbookisessentiallyfreeof these limitationsanddifficulties.Asmallregionof themediumiscon-sidered to undergo a"pure deformation" followed by a solid rotation. The order in which these two transformations are applied is important andischosensothatthestraincomponentsarereferredtoaxes which have been rotated with the material.Thus alocal rotation is introduced which varies from point to point and provides a separation of the purely geometric properties of the deformation field from those whichdependonthe physicsof thematerial.Correspondingly the stressisalsodefinedrelativetotheselocallyrotatedaxes.Adual representation is introduced by referring the stress to areas before or afterdeformation.Thisprovidesconsiderablefreedominthe formulationof physicalpropertiesand permitsthe incorporationof thermodynamicprinciplesinthestress-strainrelations.Onthe otherhand,problemsmay be formulatedwith equal easewhen,for example, it isnecessary to introduce ahydrostatic stress. To be sure, the separation between rotation and pure deformation isnotunique.Mathematicallyspeaking,norestrictionisimposed onhowthisseparationistobemade.It dependsentirelyonthe natureof the problemconsidered.Althoughin thegeneraltheory ofChapter1thepuredeformationisdefinedbyalineartrans-formationwithsymmetriccoefficientswhichimpliesnorotationof thestrainaxes,theformulationisbynomeansrestrictedtothis choice. Onthe other hand,this very arbitrariness in the definitionof the pure deformation and, at the same time, the use of adual representa-tionofthestressleadtogreaterflexibility.Thisisparticularly importantinapplicationswheresubstantialsimplificationsare viiiPreface achievedbydirectandadhocsolutionsspecificallytailoredtothe problem.In manycasesitispreferabletocarryouttheanalysis byaspecializedapproachwhichisnothandicapped fromthestart byrigidmethodsandbytheburdenofinvarianceandexcessive generality.Specializedandsimplemethodsappliedtotypical problemswhichembodyallessentialfeatureswillgenerallybring out moreclearly the fundamental physical properties. These points are illustrated by the treatment of plates and rods in Chapter2(section10)andChapter3(sections2and3)where remarkable simplificationand physicalclarity areachieved through achoiceof variables. whicharenottensorsandaretailoredtothe specificasymmetry of the physicsand the geometry.On theother hand, the analysis of isotropic. media and rubber elasticity in Chapter 2(section8)providesagoodexampleoftheuseofalternative definitionsof the stress which lead to new results and insights.For the purpose of comparison Ihave alsoderived these new results in a separate paper (see page 95) by a method oftensor invariants showing that the latter procedure is considerably more elaborate and tends to conceal the physicsas wellaspotential algebraicsimplifications. Thegeneralanalysisofstabilityinthepresenceofhydrostatic stresswhich isdeveloped inChapter3clearsupsomefundamental paradoxesandfurtherillustratestheadvantagesprovidedbythe alternative representationsof the stress. In later yearsit becameapparent that themethodswhichIhad developedearlierin thecontextof the theory of elasticitycould be extendedtostabilityproblemsofviscousandviscoelasticmedia. In fact, this realization has opened an entirely new phase in problems of deformation of the earth's crust and tectonic foldingof geological structures.A similar extension is applicable to problems of acoustic propagation in viscoelasticmedia underinitial stress.Incremental deformationsof amediuminitiallyatrestandinagivenstateof stressmaybeconsideredasthermodynamicperturbationsofan equilibriumstate.Hencethemechanicsof suchamedium may be analyzed by introducing the thermodynamics of irreversible processes asaunifyingbackground.Thissystematictheory,whichis developedinChapter6,includesmanynewresultsandtheorems whicharepresentedhereforthefirsttime.Forvanishinginitial stress,newresultsarealsoobtainedinlinearviscoelasticityasa particularcase.Thesimultaneoustreatmentofelasticityand Prefaceix viscoelasticityofinitiallystressedmediaunderconditionswhich include the most generalcasesof anisotropy is aconsequenceof the separation of the physics fromthe geometry,in combination with a verygeneral"principleofviscoelasticcorrespondence"(seepages 359 and 490).Some ofthe results may also be extended to plasticity by addingappropriate stress-strain relations to the equations which expressonly geometricand equilibrium properties. Fluids at rest under initial stress are treated as aparticular case of elasticityandviscoelasticity.Thisincludesthetheoryof internal gravitywavesandproblemsofstabilityanddynamicsofviscous fluidsin agravity field. The case of viscous fluids,which are not at rest under initial stress, requiresspecialtreatment.Inparticular,theconditionswhich determinethevalidityofviscoelasticcorrespondenceinthiscase havebeenexamined.At thesametimeanumberof rather subtle difficultiesassociated with fundamentalkinematicpropertiesof the strain rate have been clarified. A good deal of attention has been given in this book to variational methods and the principle of virtual work.They lead to the concept ofgeneralizedcoordinates,generalizedstressesandtoLagrangian equations.Theyareapplicabletobothelasticandnon-elastic media and may be used to derive approximate solutions forcomplex problems.In addition,an important application of the principle of virtualwork isitsusetoformulategeneraldynamicalequationsin curvilinearcoordinates.Thisprovidesasimpletechniquebased only on cartesian concepts which is applicable to all media regardless of their physical properties. The methods and concepts used in the linearized theory of initially stressedmediaaredirectlyapplicabletonon-lineartheoriesand largedeformations.Theequationsobtainedinbothof thesecases are analogous.Abrief outline of this is given in the Appendix.It issometimesnecessarytodistinguishinfinitesimalquantitiesof variousorders in the mathematical senseand quantities whichmay" besmallbutarenotnegligibleinthephysicalcontext.Thisis particularly true in certain problems of elastic stability of thin plates and shells where some components of strain are very small whilethe corresponding stresses are not negligible.In such cases the linearized theorymay notbeadequatetodeterminepractical stability.The so-called"post-buckling"behaviorwheresimilareffectsmustbe xPreface takenintoaccountmayalsobeanalyzedbyapplyingtheresults outlined in the Appendix. Such limitations of the linearized perturbation theory inproblems ofelasticstabilityalongwithothersofpurelymathematicalor academic interest arediscussedin Chapter3.Thereit isindicated how they may beclarified by consideringnon-elasticpropertiesand non-linearity. TheequationsdiscussedintheAppendixanddevelopedinthe paperslistedattheendofthisPrefaceprovideasimpletoolfor non-linearanalysis.Inparticular,Ihaveshown(1934-38)that the separation of the deformation from the rotation leads to important simplifications when the strain remains small relative to the rotations. For the same reason it is possible to separate the non-linearity due to physical properties of the material fromthat due to the geometry of thedeformationfield.Thistypeof non-linearanalysisshowsthat in the vast majority of problems the essential features are adequately represented by expressionswhich involveadiscriminatingchoiceof suitablesecondandthirdorderterms.Theseconsiderationsare very important in theoriesof plates and shells. Equationsapplicabletofinitestrainandexpressedin termsof a velocityfieldandratevariablesarereadilyobtainedfromthe mechanicsof incremental deformationsby atrivial limitingprocess whichintroducesinfinitesimalincrements.Thisamountstocon-sideringfinitestraintobegeneratedbyacontinuoussequenceof incremental deformations. It should be borne in mind that Applied Mathematics isan art as muchasitisascience. *Inphysicaltheoryitisofparamount importancetoacquireanintimategraspof therealitybehindthe mathematicalsymbols.Theformalismaloneorevennumerical solutions do not by themselves bring to light the significant qualitative features which lead to deeper insight and constitute an essential part * See the author's papers, Applied Mathematics an Art and aScience, Journal pf the Aeronautical Sciences,Vol.23,No.5,pp.406-410,p.489,1956;Are WeDrowning in Complexity?, Mechanical Engineering, Vol.85, No.2, pp.26-27,1963; and Science andtheEngineer,AppliedMechanicsReviews,Vol.16,No.2,pp.89-90,1963. The last paper hasbeen reprinted in the followingjournals:Journalof Engineering Education, Vol.54,No.5, pp.169-170,1964; ScientificWorld, Vol.7,No.4, pp. 9-10, 1963;Bulletin of Mechanical Engineering Education, Vol.2,No.3, pp.149-151,1963; Engineering and Science,Vol.26,No.4, pp.30-36,1963; Ciencia yTecnica,(Spanish translation),Vol.133, No.673,1964. Prefacexi of any trulycomprehensivetheoreticaltreatment.Thecommonly acceptednotionthatallproblemsaresolvedonceexactequations have been established whichcan be fed into automatic computers is afundamentalfallacy.Thecriterionofadequacyofaphysical theoryisnotnecessarilybasedonpurelogicalstructureand generality.The theory must be associated with other advantages of aconceptualandpragmaticnature.Thisobviouslyinvolvesa judgmentofvalueswhichliesbeyondthescopeofmathematical principles. There is no mathematical synthesis which guarantees the simplest and most direct solution to every type of problem.Some exceptional casesof largedeformationmay requiretheuseof morespecialized techniques.Manysuchproblems. aremainlyof academicinterest. They constitute asmall fraction of the vast field of technological and physicalproblemswhichcanbehandledbymoreappropriate methods. Whilestressingthe practical limitationsof the tensorcalculusin problemsof appliedmechanics,oneshouldof courserecognizethe well-established valueof the tensorconcept itself,particularly in its simplercartesian form.Theconceptof cartesian tensor isimplicit throughoutthepresentwork.However,asintheclassicaltreat-ment of linearelasticity,it has not been foundnecessarytodepend on the rulesof tensor algebra as aseparate mathematical discipline. Myeffortshavebeendirectedtowardgivingtheengineerand physicist adequate tools with asound mathematical foundation,and aminimumrequirementinmathematicaltechniques.Procedures and viewpointswhich tend to build up the mechanicsof continuous mediaasanexercisein tensorformalismhavebeenavoided.The emphasishasbeenputonmethodswhichachieveacompromise betweensimplicity,generality,andusefulness.It isnotintended to exclude other methods provided that the difference in emphasis is clearly understood and proper balance ismaintained. ThisbookisdividedintosixchaptersandanAppendix.The firstchapter,partsofChapters2and5,andtheAppendixare concernedmainlywiththematerialoriginallydevelopedduring theyears1934to1940inthesevenpaperslistedattheendof this Preface.The presentation here is given in quitedifferent form: the non-linearand largedeformation theoriesare treated separately xiiPreface intheAppendixasanextensionof thelinearizedequationsfora mediumwithinitialstress.Chapter2dealsprimarilywiththe general theory of elasticity.The next twochapters aredevoted to problems of elastic stability of isotropic and anisotropic media.The general dynamics of elastic media under initial stress is developed in Chapter5;itincludesproblemsofacousticpropagation,dynamic stability,and thetheoryof internalgravity wavesinafluid.The lastchapter,whichisbyfarthelongest,isdevoted exclusively to viscousandviscoelasticmediaunderinitialstressandincludesa discussionandapplicationsofthethermodynamicsofirreversible processes. Foradetaileddescriptionofthecontentsandinterrelationsof variouspartsof the bookthereader isreferred to the introductory sectionsat the beginning of each chapter. Thebasictheorycontainedin thesevenpaperslistedbelowwas developedwhileIwas a member of the Applied ScienceDepartment oftheUniversityofLouvain,andofthePhysicsDepartmentof Columbia University. Thepreparationof thebookitself and mostof the researchcon-nectedwiththenewdevelopments,someofwhichhavebeen presented in separate publications,weresupported by the AirForce OfficeofScientificResearchundercontractsAF49(638)-266, AF 49(638)-837,and AF49(638)-1329. Other original contributions incorporated in thisbook result from worksponsoredbytheShellDevelopmentCompanyaspartofa general research program in geodynamics. IamindebtedtoDr.A.Winzer forvaluableassistance in proof-reading, in the preparation of the Index, and in some of the analytical derivations in section8 of Chapter3 and section7 of Chapter 5. Theearlierpapers(1934--1940)on which thisbookisbasedare: 1.M.A.Biot, Sur la stabilite de l'equilibre elastique.Equations de l'elasticite d'un milieusoumisa tensioninitiale,AnnalesdelaSocieteScientifiquedeBruxelles, Vol.54,Ser.B, part I, pp.18-21,1934. 2.M.A.Biot,TheoryofElasticitywithLargeDisplacementsandRotations,in Proceedingsof theFifth International Congress forApplied Mechanics(Cambridge, Mass.,September1938),pp.117-122,JohnWiley&Sons,Inc.,NewYork, Chapman&Hall Ltd., London 1939. 3.M.A.Biot, Theorie de l'elasticite du second ordre avec application a la theorie du Prefacexiii flambage, Annales dela Societe Scientifique deBruxelles, Vol. 59, Ser. I, pp. 104-112, 1939. 4.M.A.Biot,NonlinearTheoryof Elasticity and theLinearizedCaseforaBody unde: Initial Stress,Philosophical Magazine,Vol.27,Ser.7,pp.468-489,1939. 5.M.A.Biot,ElastizitatstheoriezweiterOrdnungmitAnwendungen,Zeitschrift furAngewandte Mathematik und Mechanik,Vol.20,No.2, pp.89-99,1940. 6.M.A.Biot,IncreaseofTorsionalStiffnessofaPrismaticalBarduetoAxial Tension,Journalof AppliedPhysics,Vol.10,No.12,pp.860-864,1939. 7.M.A.Biot,The Influenceof Initial Stresson ElasticWaves,Journalof Applied Physics,Vol.11,No.8, pp.522-530,1940. In order to avoid undue repetition, these papers are referred to in the book by their numbers as listed here. NewYork,NewYork October,1964 MAURICEA.BlOT CONTENTS 1.StaticsandKinematicsofIncrementalStressesand Strains,1 1.Introduction,1 2.The Kinematicsof Two-Dimensional Strain,6 3.The Kinematicsof Three-Dimensional Strain,15 4.Incremental Stresses in Two Dimensions,23 5.Incremental Stresses in Three Dimensions,28 6.EquilibriumEquationsfortheStressFieldinTwo Dimensions,33 7.EquilibriumEquationsfortheStressFieldinThree Dimensions,44 2.Elasticity Theory of aMedium under InitialStress, 56 1.Introduction,56 2.The Incremental Stresses Referred toInitial Areas,58 3.Two-Dimensional Relations between Strain and Incremental Stress,63 4.Three-DimensionalRelationsbetween Strain and Incremental Stress,67 5.Variational Principles,73 6.Incremental Elastic Coefficients for an Orthotropic Medium, 82 7.Incremental Elastic Coefficients for an Isotropic Medium, 89 8.Incremental Stresses in Incompressible Media;Application toRubber Elasticity,96 9.Elastic Coefficients in Second Order Elasticity,108 10.Torsional Stiffnessof aBar under Axial Tension,112 3.Theory of Elastic Stability and Its Application to Isotropic Media, 122 1.Introduction,122 xv xvi 2. 3. 4. 5. 6. 7. 8. Contents PhysicalSignificanceof theStabilityEquationsinPlane Strain,124 Special Equations for the Stability of Rods and Plates,128 Variational Formulation of Stability,135 Stability in the Presenceof Hydrostatic Stress,150 Surface Instability,159 Bucklingof aThick Slab,166 Instability of aNon-homogeneousHalf-Space,174 4.Elastic Stability of Anisotropic Media,182 1.Introduction,182 2.ALaminated Mediumasan Example of Anisotropy,184 3.Internal Instability,192 4.Surface Instability of the AnisotropicHalf-Space,204 5.General Equations foraPlate under Initial Stress,216 6.BucklingofaFreeandEmbeddedPlate;Interfacial Instability,227 7.StabilityTheoryofMultilayeredMediaIncludingthe Effectof Gravity,243 5.Dynamics of Elastic Media under Initial Stress, 260 1.Introduction,260 2.Dynamical Equations foran Elastic Medium underInitial Stress,202 3.The Influenceof Gravity on Rayleigh Waves,272 4.SomeFundamentalPropertiesofAcousticPropagation under Initial Stress,281 5.Theory of Acoustic-Gravity Waves in aFluid,291 6.Variational Principles for Acoustic-Gravity Waves,304 7.Dynamicsof ElasticPlatesand Multilayered Mediaunder Initial Stress,320 6.Mechanics of ViscoelasticMedia underInitialStress, 337 1.Introduction,337 Contentsxvii 2.Thermodynamicsof Viscoelasticity with Initial Stress,340 3.OperationalExpressionsforIncrementalStresses.Corre-spondence Principle,349 4.Properties of Characteristic Solutions,365 5.SmallDeformationsSuperposedonanInitialStateof Flow,375 6.Internal Instability in AnisotropicViscoelasticity,397 7.Surface Instability of Viscoelastic Media,405 8.Folding Instability of Layered Media,414 9.Dynamicsof Viscoelastic Media under Initial Stress,438 10.Instability and Small Motion Dynamics of aViscousFluid in aGravity Field,459 Appendix:Non-linear Theories andFiniteStrain, 481 Indexes, 499 CHAPTERONE StaticsandKinematics of IncrementalStressesandStrains 1.INTRODUCTION It is well known that a state of initial stress in adeformable medium induces mechanical properties which depend mainly on the magnitude of thestressandarequitedistinctfromthoseassociatedwiththe rigidity of the material itself. Thisisbest illustratedby theexampleof astringundertension. A perfectly flexible string is stretched under a tension Tbetween two fixed points Aand B(Fig.1.1).If a vertical load Fis applied to the string at point 0, the deflection w at that point is determined entirely by the laws of static equilibrium. The static analysisissimplifiedby assuming that the slopeof the string issmalland may be treated mathematicallyasaquantityof the firstorder.The deflectionof the string is F (1.1) w=k with k=T(A10 + O ~ ) (1.2) Although no elastic properties of the material itself are involved, the deflectionisthesameasif the loadwereactingonaspringwhose rigidity ismeasured by amodulus k. 1 2StaticsandKinematicsof Incremental Stressesand StrainsCh.1 Thesameanalogyextendstovariationalandenergymethods. Whenthestringdeflects,theelasticpotentialenergyinthestring increasesby the amount W=T(AD+DB- AB) Tothe secondorder wemay write Hence 1w2 AD =V (AC)2+ w2 =AC+ 2 AC 1w2 DB=V(CB)2+ w2 =CB+ 2CB W=1T (_1_+_1_)w2 =1kw2 2ACCB2 (1.3) (1.4) (1.5) Thisexpressionalsorepresentsthepotentialenergystoredina spring of modulus k.The deflection of such a spring under a forceF may beobtained by the principle of virtual work: Fow=oW=kwow(1.6) Thedeflectionwderived from(1.6)coincideswith thevalue(1.1). Ar==-__ __________T;______DW Figure 1.1Deflection of astring under tension as an example of the apparent rigidity of asystem under initial stress. Thissimpleexamplebringsforththeimportantpointthatthe equations derived fromdirect balance of the forcesinvolve only first ordertermsinthegeometryof thedeformation.Bycontrastthe correspondingvariational principle involvessecondorder geometry. The next step which comesto mind is represented by problems in which the elastic properties of the material and the initial stress both contribute to the over-all rigidity of the structure.For example, we mayaskwhathappenswhenthestringconsideredintheprevious exampleisnot perfectly flexibleand possessesanelasticrigidityin bending. Sec.1Introduction3 Problemsof thistypehavebeentreatedextensivelyinthepast mostlyinthecontextof engineeringandtheparticularbranchof scienceusually referredtoas"Strength of Materials."Oneof the early developments isEuler's theory of buckling of athin rod under axial compression.The presence of an initial stress may increase or decrease the over-all rigidity of an elastic structure.In arod under axialcompressiontheinitialstressproducesadecreaseinlateral stiffness.For increasing values of the compression this decrease will overcomethenaturalbendingrigidityof therod,producingan in-stabilityknownasbuckling.Ontheotherhand,acablehanging under itsownweightisunderan initialtensionwhichincreasesits rigidity.This effect is used in the design of suspension bridges. These problems have usually been treated by approximate methods restricted tospecial structures suchasslender rodsand thin plates. The viewpoints and methods used in these approximate theories lead toaformulationwhichbringsoutexplicitlyintheequationsthe particular terms whichcontain the initial stressand are responsible forthecharacteristicfeaturesduetothepresenceofthisstress. The same viewpointcan be maintained to developasystematic and rigorous three-dimensional theory of the initially stressed continuum using elementary methods and following exactly the same procedures asin the classical theory of linear elasticity. Thepresentationof thistheoryalongthe linesdevelopedbythe author isthe objective of thischapter.It isessentially an analysis ofthestaticsandkinematicsforincrementaldeformationsinthe presence of initial stress.The concepts and equations are developed entirely with referencetothe geometryof thedeformationand the equilibrium of the stresses.At no time is any reference made to the physical propertiesof the material.The resultsof thischapter are therefore applicable to any type of continuum,whether it be afluid or asolid,with elastic,plastic,or viscoelastic properties. As shown in the simple example of the string under tension, acom-pleteanalysisoftheproblemrequiresanunderstandingofthe geometryof thedeformationwhichincludesbothfirstandsecond order terms.The second order terms are required in the linear theory in order toformulatethecorresponding variationalprinciples.We have therefore analyzed the concept of strain fromthis viewpoint in the two initial sections of this chapter.This is done by considering firstastate of finitestrain.Asmall region around amaterial point 4StaticsandKinematicsof Incremental Stressesand StrainsOk.1 undergoesatranslation,asolidbody rotation,and a"pure"homo-geneous strain.The pure strain is defined by using the property that therearethreeorthogonaldirectionsinthemediumwhichremain orthogonal after deformation.This also leads to aunique definition ofthesolidrotationofanelement.Thisparticulardefinitionof finitestrainiswellknownandleadsgenerallytotranscendental equations due to the introduction of asolid body rotation, equations whichin threedimensionsinvolveintricaterelationsforthetrans-formationof the coordinate axes. However,thisdifficultydisappears in twoimportantcases.One ofthemisrepresentedbytheclassicaltheoryofinfinitesimal deformationsofthefirstorder.Theother,whichistheonecon-sidered in thischapter,involvesan evaluation of the strain with an approximationofthesecondorder.Thissecondorderanalysis provides immediately the required quadratic expression for the strain energyleadingtoavariationalformulationof thetheorywhichis developed in detail in the next chapter.The non-linear expressions derived forthe strain alsoclarifythe validity of firstorderapproxi-mations.Inaddition,asshownbytheauthor,theyleadtoa simplified non-linear theory of elasticity. * Wehavetreated separatelythetwo-dimensionaldeformation(in section 2) and the three-dimensional deformation (in section 3).This separationof thetwocasesismaintainedthroughoutbecausethe physicalsignificanceismoreeasilyexplainedand illustrated in two dimensions,whereas the mathematical symmetry of the equations is morereadilyemphasizedinthreedimensions.Inconnectionwith the three-dimensional kinematics of strain we have used the so-called "dummy index" rule.This is anotation of considerable conciseness usedasastandardprocedureinthetensorcalculus.It hasbeen usedextensivelyin thisbookwheneverneededeithertoavoidun-necessary writingor tobringout the mathematical structure of the equations.It should be remembered that, although this is a notation of the tensor calculus, no use is made of the tensor calculus itself and the mathematical procedures remain entirely elementary. Thefollowingsectionsaredevotedtothelinearmechanicsofa continuum under initial stress under conditions of static equilibrium. * Seereferences3,4,and5attheendofthePreface.Abrief discussionof the nonlinear theory isalsogiven in the Appendix. Sec.1Introduction5 Theoriginalcoordinatesrefertothemediuminthestateof initial stress.Asmalldisplacementfieldisthensuperimposed.The strainassociatedwiththisdeformationisinfinitesimalandisde-scribedby the classical components for small strain.These classical considerations do not apply to the stress.The significanceof incre-mental stresses is analyzed in sections 4 and 5.Of importance is the introduction of incremental stress components referred to axes whose directionsareobtainedbyrotatingtheoriginalcoordinatesbyan amount equal to the local rotation of the material.The stress is thus referred toaxeswhoseorientation variesfrompoint to point.The purpose of representing the stress in this way isthat itscomponents now depend only on the physical propertiesof the material;thus the physicsisseparated fromthegeometryand the solidbody rotation is eliminatedfromtherelationsbetweenstressandstrain.This featureisparticularlyusefulinstudiesofviscoelasticandplastic materials. It should be kept in mind that the linear theory is strictly applic-able only if the stress variation is asmall fraction of the initial stress. Smallnessof thedeformation in thephysical sensedoesnotalways guarantee this condition to be fulfilled,as in certain types of problems of thin plates and shells.Such problems must be handled by special-izedmethods.Thenon-lineartheoriesdevelopedearlierbythe author*alsoprovideabasisforafundamentalbutelementary approach to such problems which issimilar to the linear theory. The last two sections of this chapter are devoted to the derivation ofequilibriumequationsandboundaryconditionsusingthefore-goingdefinitionof theincrementalstresses.Oneimportantchar-acteristic of these equations is that they are intrinsic, i.e., they depend on the local geometry of the deformation and at the same time retain the cartesian representationof the stress.This has the advantage of clarifying the physical significance of the mathematics and constitutes the reason forthe usefulnessof this formof the equations. Theequationsderivedinthischapterarerestrictedtocartesian coordinates.Theirformulationforcurvilinearcoordinateshasbeen relegated to Chapter 2 as an application of the variational principles. Finallyitshouldalsoberemarkedthatthisapproachdoesnot requireany knowledgeof thephysicalprocessby whichtheinitial * Seereferences3,4,and 5at the end of the Preface. 6StaticsandKinematicsof Incremental Stressesand StrainsCh.1 stress has been generated.We should remember that the physics of initial stresses can be very different from that of incremental stresses. For example,agasmay be in isothermalequilibrium under gravity and incrementalacousticwavesmay propagate through it adiabati-cally.The sameconsiderationsapply to rapid elasticdeformations in the earth where the initial stress isassociated with aslow process of creepdue to viscousand plasticdeformations. 2.THEKINEMATICSOFTWO-DIMENSIONALSTRAIN We consider a homogeneous deformation in the plane x,y such that a square S is transformed into a rectangle R while the sides keep fixed orientationsIand II (Fig.2.1).Such adeformation is represented by the linear transformation with symmetric coefficients: g =(1+ Bll)X+ B12Y (2.1 ) where (2.2) ApointPof coordinatesx, YistransformedintoapointP'of co-ordinatesg,7]'ThecoefficientsBijdefineapuredeformation.The reason for this appellation is the existence of two directions Iand II, y II Figure2.1Representation of apure deformation.The square S is transformed into the rectangleR. Sec.2Kinematicsof Two-Dimensional Strain7 perpendicular to each other, called principal directions, whose orienta-tionremainsfixedduringthetransformation.Thedeformation representedbyequations2.1isthereforealwaysequivalentto positive or negative elongationsin the principal directions. Theexistenceoftheprincipaldirectionsisaconsequenceof the symmetryproperty(2.2).Thiscanbeshownbyconsideringthe quadratic form rP=HI+ Bll)X2 + B12XY+ t(1+ B22)y2(2.3) Becauseofthesymmetryrelation(2.2)wemaywritethetrans-formation(2.1)as (2.4) Therefore the vectorg,TJisparallel to the gradient of rP,i.e.,normal to the conicsection whoseequation is rP=const. (2.5) and which passes through the point x, y.Obviously the axes of this conicsectionare the principal directionsof thedeformation. Let us now write the general linear homogeneous transformation in the plane x,y,i.e., g =(1+ all)x+ a12y TJ=a21x+(1+ a22)Y (2.6) wherethecoefficientsaij mayor may not besymmetric,i.e.,where in general In matrix formwe write a12][X] 1+ a22 Y (2.7) Apure rigid rotation is aparticular caseof transformation(2.6).A clockwiserotationthroughtheangle0transformsthecoordinates x',y'intog,TJby the linear relations: -sin 0][X'] cos0y' (2.8) 8StaticsandK inernaticsof Incremental Stressesand StrainsOk.1 The question immediately arises whether the transformation(2.6)is alwaysequivalenttotwosuccessivetransformations, namely, first a pure deformation [X:][1+ elle12] [X](2.9) Ye211+ e22Y withe12=e21followedbyapurerotation(2.8).Thesuccessive application of these two transformations leads to [g]=[ C ~ s0 YJsm 0 -sin 0][1+ ell cos0e21 (2.10) Thistransformationmustbeequivalenttotransformation(2.7). Performing the matrix multiplication and equating the corresponding matrixelementsinequations2.7and2.10,wederivethefour equations (1+ ell)cos0- e21sin 0=1+ all (1+ ell) sin 0+ e21cos0=a21 (1+ e22)cos0+ e12sin 0=1+ a22 - (1+ e22)sin 0+ e12cos0=a12 (2.11) We may solve the first two of these equations for 1+ ell and e21and the last two for1+ e22and e12.We find e21=a21 cos0- (1+ all) sin 0 e12=a12 cos0+(1+a22)sin 0 (2.12) 1 + ell= (1+ all) cos0+ a21 sin 0 1+ e22= (1+a22)cos0- a12 sin 0 Becausee12=e21expressionsontherightsideof thefirsttwoof relations(2.12)are equal,and wederive (2.13) Thisyieldsthemagnitudeof thepurerotationcontained intrans-formation(2.7).It isacounterclockwise rotation through an angle O.Knowing0,weareabletocalculatefromrelations(2.12)the Sec.2Kinematicsof Two-Dimensional Strain9 x Figure 2.2Superposition of apure deformation and arotationO. coefficients8ijof thepuredeformation(2.9)containedinequation 2.7.They are 821=812=. t(a21 + a12)cos(J+ t(a22 - all) sin(J 811=all cos(J+ a21 sin (J+ cos(J- 1 822=a22 cos(J- a12 sin (J+ cos(J- 1 (2.14) Thefirstofequations2.14isobtainedbyaddingthefirsttwoof equations2.12and dividing by 2. By transformation(2.6)theunitsquareOABOinFigure2.2is transformed intothe parallelogramOA' B'O'.Wehavejust shown that this is equivalent to rotating the square counterclockwise through anangle(J,thensubmittingittoapuredeformation.Thispure deformationisdefinedbythecoefficients81i'i.e.,byasymmetric transformation(2.1)wherethecoordinatesx, yare now referred toaxes 1, 2 which are rotatedby theangle(Jfromtheir original direction.It is important to note that this isaconsequence of the fact that we have firstappliedthetransformation(2.9)andthentherotation(2.8). The sequence of these two transformations isnot arbitrary sincethe multiplication of the two matrices in equation2.10 isnot commuta-tive;thatis,wefindadifferentresultif wereversetheorderof mUltiplication.If we reverse the order of transformations(2.8)and (2.9),i.e.,if we firstapply arotation and then perform apure defor-mation,wefindthe sameexpression(Jforthe angleof rotation but differentvaluesforthecoefficients8ij.There is,of course,nocon-tradictionherebecausethe8ijrepresentthe samepuredeformation, 10StaticsandKinematicsof Incremental Stressesand StrainsOh.1 referred this time to the original unrotated axes instead of the rotated axes1,2. In the present theory the definition(2.12)of the pure deformation isadopted;that is,weshall refer the pure deformation torotatedaxes. Until now we have considered finitedeformations.We now intro-duceanassumptionof"smallness";thatis,weshallarbitrarily considerthecoefficientsaij tobe smallquantitiesof the firstorder, (2.15) Letting (2.16) we may write to the firstorder ()~W(2.17) Furthermore weobtain expressionsforthecoefficients8ljwhichare correct to the second order if in relations(2.14)we replace sin ()byw cos()by1 1- cos()bytw2 Hence to the second order we find 821=812=t(a21 + a12)+t(a22 - ail)w 811=all+ a21w- tw2 822=a22 - a12w- tw2 (2.18) (2.19) In the text below wehave used an equivalent formof theseexpres-sionswhichintroducesexplicitlythequantityt(a21 + aI2).We may write the identities a21 =!(a21 + a12)+ w a12=t(a21 + a12)- w and the coefficients(2.19)become 821=812=!(a21+ a12)+ !(a22 - all)w 811=all+ !(a21 + al2)w+ tw2 822=a22 - t(a21+ al2)w+ !w2 (2.20) (2.21) Sec.2Kinematicsof Two-Dimensional Strain11 Thenextandlaststepintheanalysisistoconsideraninhomo-geneousdeformation,i.e.,suchthat apointPof coordinatesx, yis transformed into apointP' of coordinates (2.22) Thedisplacementfieldisrepresentedbythevectorof components u=u(x, y) v=v(x, y) bothfunctionsoftheinitialcoordinatesx, y. relations dg=1+ - dx+ - dy ( OU)ou oxoy ov(OV) dTJ=ox dx+1+oydy (2.23) Thedifferential (2.24) represent alinear transformation of the infinitesimal vector of com-ponents dx, dy in the vicinity of point Pinto an infinitesimal vector of componentsdg,dTJin thevicinityof pointP'.In otherwords, relations(2.24)defineahomogeneoustransformationof theinfini-tesimal area around P into an infinitesimal area around P' (Fig.2.3). Suchatransformationisidenticalwiththehomogeneoustrans-formation(2.6),and the previous analysis is immediately applicable. The coefficientsaij become the partial derivatives, OU all=ox OV a21 =ox OU a12 =-oy OV a22 =-oy (2.25) It ispossiblethereforetodefinealocalrotationBofthematerial whichvariesfrompointtopoint;it isgivenbyexpression(2.13). Thepuredeformationof the infinitesimal regionaroundpointPis representedby thecoefficientsBijgivenby expressions(2.14).We must remember that thesecoefficientsrepresent apure deformation which isdefinedrelativetothelocallyrotateddirections1,2(Fig. 2.3). 12StaticsandKinematicsof Incremental Stressesand StrainsOh.1 y d ~ oL---------------------------------x Figure2.3Deformation in the vicinity of apointP' originally at P. From equation 2.16 we see that the magnitude of the local rotation is given to the firstorder by W=~( : ~- :;) (2.26) For convenienceweintroduce the notation auov exx axeyy =oy I (OVaU) eXY =eyX ="2ax+oy (2.27) To the second order the coefficients ejj or strain components are given by equations2.21.With the notation(2.27)wemay write* e12=eXY + t(eyy- exx)w ell=exx + eXYw+ tw2 e22=eyy- eXYw+ tw2 (2.28) The quantities exx,eyy,eXY are the first order strain components of the classical theory of elasticity. Tothe firstorder wemay write (2.29) * Expressions (2.28) were derived by the author in 1938 (reference2 at the end of the Preface). Sec.2Kinel1w,ticsof Two-Dimensional Strain \ \ "-dy -Figure2.4Uniformdilatationrepresentedby linear transformation(2.32). 13 Another interesting consequence of relations (2.28)arises when the classical strain components are zero: exx =e)J)J=eX)J=0 In thiscase there isasecond order strain, which corresponds to an isotropic extension of magnitude lw2. iseasilyverifieddirectlybyconsideringthetransformation which in this case becomes dg=dx- wdy dYJ=wdx+ dy (2.30) (2.31 ) This (2.24) (2.32) All points on aunit circle centered at the origin are transformed by a displacement win a direction tangent to the circle.Hence the radius of the circle is enlargedby afactor The circle,of course,alsorotates through an angleB given by tanB=w This is illustrated in Figure2.4. (2.33) (2.34) 14StaticsandKinematicsof Incremental Stressesand StrainsOk.1 An interesting feature of the pure deformation as defined above is that two successive pure deformations do not combine to give apure deformation.In mathematical language we say that pure deforma-tions do not constitute a group.In order to show this let us consider the two followingpure deformations [ ~ ~ ] 812][dX] 1+ 822dy (2.35) [dX'] dy' (2.36) The latter transformation is equivalent to a resultant transformation: [dX:]=' [Cll dyC21 C12][dX] C22 dy (2.37) Thematrix of this resultant transformation isobtainedby multi-plicationofthematricesdefiningthetransformations(2.35)and (2.36);hence The elementsof this matrix are Cll=:=1+ 8 ~ 1+ 811+ 8 ~ 1811+ 8 ~ 2 8 1 2 C12=8 ~ 2+ 812+ 8 ~ 1812+ 8 ~ 2 8 2 2 C21=8 ~ 2+ 812+ 8 ~ 2 8 l l+ 8 ~ 2 8 1 2 C22=1+ 8;2+ 822+ 8;2822+ 8 ~ 2 8 1 2 (2.38) (2.39) The resultant transformation(2.37)will represent apure deforma-tion only if (2.40) This relation which in general willnot be fulfilledisequivalent to (2.41) The significanceof this relationappears if weconsider the principal directionsofstrain.Thesedirectionsaregivenbytheprincipal Sec.3Kinematicsof Three-Dimensional Strain15 axes of the conic represented by equation 2.5.By asimplecalcula-tion the angle aof these principal directions with the xaxis is found tosatisfy the relation tan 2a=(2.42) Weconcludethatthenecessaryandsufficientconditionfortwo successive pure deformations to represent alsoapure deformation is that their principaldirectionscoincide. Condition(2.40)isalwaysfulfilledif weneglectthesecondorder quantitiessuchas8 ~ 1812and8 ~ 2 8 2 2 'Hence forinfinitesimalstrain thecombinationof twopuredeformationsyieldsa pure deformation. It isalsointerestingtonotethat if relation(2.41)isnot satisfied the resultanttransformation(2.37)containsahigherorder rotation of angleB definedbyequation2.13.Forinstance,thesuccessive applicationof twopuredeformationsof thefirstorderproducesa rotation of the second order.The sign of this rotation is reversed if we reversethesequenceof thepuredeformations(2.35)and(2.36). 3.THEKINEMATICSOFTHREE-DIMENSIONALSTRAIN Theconceptsdevelopedintheprecedingsectionfortwo-dimen-sional strain may be extended readily to three dimensions.It is not necessarytorepeatalltheargumentsindetail,andwestartim-mediately with the general non-homogeneoustransformation.The pointPof initial coordinatesx, y,Z,istransformedintoapointP' of coordinates g=x+u T)=y+v '=z+w (3.1) Thedisplacementfieldisrepresentedbythevectorof components u=u(x, y,z) v=v(x,y, z) w=w(x,y,z) (3.2) 16Staticsand Kinematicsof Incremental Stressesand StrainsOk.1 InthevicinityofpointPthecontinuumundergoesthelinear transformation (3.3) Thisisahomogeneoustransformation.Inthistransformationa pointofcoordinatesdx,dy,dzinthevicinityof pointPistrans-formedintoapointofcoordinatesdg,dYj,d, inthevicinityofP'. Asin the two-dimensionalcasediscussedabove,weshall showthat the transformation (3.3) is equivalent to apure deformation followed byapuresolidrotation.Inthreedimensionsthekinematicsof solidrotationisconsiderably more involved,and weshalltherefore approach the analysis fromadifferent viewpoint. Let us introduce the symmetric and linear transformation with de=(1+ell) dx+e12dy+e13dz dr/=e21dx+(1+e22)dy+e23dz. de=e31dx+e32dy+(1+e33)dz (3.4) That this transformation representsapure deformation can be seen byusingthesameargumentsasfortwo-dimensionalstrainand writing thequadratic form ~=t(l +eu)X2 +t(l +edy2+t(l + edz2 + e23YZ+ e31ZX+ e12XY(3.5) The threeaxesof thequadric ~=const.(3.6) representthethreeprincipaldirectionsofstrain. *Thesethree * Amore extended discussion of homogeneous strain will be found in Love's treatise, TheMathematicalTheoryofElasticity.FourthEdition,pp.66-73,Cambridge University Press(reprinted by Dover Publications,New York,1944). Sec.3Kinematicsof Three-Dimensional Strain17 directionsaremutually perpendicularanddonotchangewhenthe mediumundergoesthesymmetrictransformation(3.4).Acube whoseedgesareoriginallyorientedalongthesethreedirections becomesarectangularparallelepiped,withitsedgesoriented along the same directions.We are therefore justified in defining the sym-metric transformation(3.4)asapure deformation.The coefficients in the transformation (3.4)are represented by the symmetric matrix 812831] 822823 823833 (3.7) Thesecoefficientsarecalledthestraincomponentsofthepure deformation. Immediately therearisestheproblemof findingoutunderwhat condition the more general transformation (3.3) will contain the same deformation as the symmetric transformation(3.4).The necessary and sufficient condition for this to occur is obviously that the distance between any pairs of points remain the same for both transformations. It maybeexpressedmathematicallyasfollows.Apairof points whosevectorialdistanceisrepresentedbydx,dy,dzacquirean absolutedistancedsbythetransformation(3.3)andanabsolute distance ds'by the transformation(3.4).Thesedistancesare given by ds2 =de + dYJ2+ d ~ 2 dS'2=dg'2+ dYJ'2+ d ~ ' 2 (3.8) Theconditionthatthetwotransformationscontainthesame deformation isthat the relation ds2 =dS'2(3.9) be verified identically forallvalues dx,dy,dz. In order to carry out this identification we introduce the notation OU 1(OWOV)1(OWOV) exx =oxeyZ =eZy ='2oy+ozWx='2oy- oz OV 1(OUOW) 1euOW) (3.10) eyy =oyezx =exz ='2oz+oxWy='2oz- ox ow 1(OVOU)1(OVOU) ezz =OZeXY =eyX ='2ox+oy w=----z20xoy 18Staticsand Kinematicsof Incremental Stressesand StrainsCh.1 With this notation the transformation(3.3)becomes dg=(1+ exx) dx+(eXY - wz) dy+(ezx + wy) dz dTJ=(eXY + wz) dx+(1+ eyy ) dy+ (eyZ - Wx)dz(3.11) =(ezx - wy) dx+(eyZ + Wx)dy+ (1+ ezz) dz The length element forthis transformation iswritten ds2 =(1+ 2flxx)dx2 + (1+ 2flyy)dy2+(1+ 2flzz)dz2 + 4flyzdy dz+ 4flzxdz dx+ 4flxydx dy(3.12) with the definitions flxx=exx +ie;x+ i(exy + wz)2+ i(ezx - wy)2 flyy=eyy++ i(eyz + wx)2+ i(exy - wz)2 flzz=ezz + ie;z+ i(ezx ++i(eyz - wx)2 (3.13) flyz=eyZ + i(eXY - wz)(ezx + wy)+ ieyy(eyZ - Wx)+ -!ezAeyz + Wx) flzx=ezx + i(eyZ- wx)(eXY + wz)+ iezAezx - wy)+ iexAezx + wy) flxy=eXY + i(ezx - wy)(eyZ + Wx)+ iexAexy - wJ+ ieyy(exy + wJ Thesequantitiesrepresenttheclassicaldefinitionoffinitestrain. The length element forthe pure deformation(3.4)is dS'2=(1+ 2Y11)dx2 + (1+ 2Y22)dy2+(1+ 2Y33)dz2 + 4y23dy dz+ 4y31dz dx+ 4y12dx dy(3.14) with Y11=811+ i(8i1+ 8i2+Y22=822++ + 8i2) Y33=833+ + + Y23=823+ i(812831+ 822823+ 833823) Y31=831+ i(823812+ 833831+ 811831) Y12=812+ i(831823+ 811812+ 822812) (3.15) Now,as already pointed out, the pure deformation (3.4)can be made to represent exactly the same state of strain as that produced by the transformation(3.3)providedthelengthelementsdsandds'are identical after the transformation, i.e.,provided that relation (3.9)is satisfied identically.This condition is expressed analytically by the six equations flxx=Y11 flyy=Y22 flzz=Y33 flyz=Y23 flzx=Y31 flxy=Y12 (3.16) Sec.3Kinematicsof Three-Dimensional Strain19 Theseequationsdeterminethesixstraincomponents(3.7)as functionsof the ninecoefficients(3.10)appearingin transformation (3.3). Transformations(3.3)and(3.4)thusrelatedrepresentthesame stateof strainandcandifferonlybyarigidbodyrotation.The rigid body rotation that we must add to transformation (3.4) in order to obtain transformation(3.3)willbecalled thelocalrotationof the material.Transformation(3.3)containsnineindependentcoeffi-cients,while the state of strain isdetermined by only sixquantities. There are therefore threedegreesof freedomleaving unchanged the lengthelementdsandcorrespondingtotherigidbodyrotation contained in the general transformation(3.3). Thefinitestraincomponents(3.7)havetheadvantagethat they arelinearly related totheactualchangesof lengthin thematerial, whereastheclassicalcomponents(3.13)arelinearlyrelatedtothe changeof thesquareof thelength.Ontheotherhand,thecom-ponents(3.7)havethedisadvantagethat theycannotbeexpressed rationallybymeansoftheninequantities(3.10).However,this disadvantagevanisheswhenweassumetheninequantities(3.10)to besmall of the first order and when weconsider only the firstandsecond ordertermsintheexpressionsforthestraincomponents(3.7)asa function of the ninequantities(3.10).Asolution of equations3.16 isobtained immediately as follows. Wenoticefromequations3.16thatexx andelldifferonlybya second order quantity; the same is true for eXY and e121etc., so that we may write with an error of only the third order + + e;x=erl + er2++ + = + + er2 e;z + e;x+ =+ +(3.17) eXyeZX + eyyeyZ+ eZZeyZ=812831+ 822823+ 833823 eYZeXY +ezzezx +exxezx823812+ 833831+ 811831 eZXeyZ+ eXXeXY + eyyeXY 831e23 + 811812+ 822812 Introducing the approximate relations (3.17) intoequations3.16,we find for the strain components with an error of only the third order* * Equations3.18werederivedby theauthorin1939,inreferences3and4atthe end of the Preface.They were applied subsequently in reference5. 20StaticsandKinematicsof Incremental Stressesand StrainsCk.1 dr 3 2 dz J'------ d7J d ~ r------dy (]x Figure 3.1Local rotated coordinate system (1,2,3) and unrotated coordinate system (dg,dT],d ~ )in the vicinity of apointP' initially at P. 811=exx + eXYwZ - ezxwy+ !(wz 2 + Wy 2) 822=eyy+ eyZwX -:- eXYwZ + !(wx 2 + Wz 2) 833=ezz + eZXwy- eyZwX + !(wy2 + wx2) 823=eyz+ !wx(ezz - eyy )+ !wyeXY - !wzezx 831=ezx + !wy(exx - ezz)+ !wZeyZ 812=eXY + !wieyy- exx)+ !wxezx !wyWZ iwxexy!wzwx !wyeyZ - !WXwy (3.18) At this point it is important to stress the physical significance ofthese componentsof strain.If welookat the homogeneoustransforma-tion(3.3)of asmall region in the vicinity of apoint attached to the material,we see that it can be obtained asfollows(Fig.3.1). l. ThematerialistranslatedasarigidbodysothatpointP coincideswith P'. 2.Werotate thisregionasarigidbody.(Weshowbelowthat this rotation isdefined to the firstorder by the vector wx,WY'wz.) 3.Asystem of rectangular coordinates with its origin at pointpi and parallel with the x, y,z directions is rigidly rotated by the same amount as the material and becomes thereby a system we call (1, 2, 3). With respect to this coordinate system(1,2,3)wethen perform the pure deformation(3.4)with strain components(3.18). Sec.3Kinematicsof Three-Dimensional Strain21 Thereforewemaylookuponstraincomponents(3.18)asrepre-senting the pure deformation referred to arectangular frame(1,2,3) originally parallel with the x, y,z directions and undergoing the same rotation asthe material.The strain fieldisthus referredtoafield ofrectangularaxeswhoseorientationvariesfrompointtopoint accordingtothe localrotationof thematerial.It isimportantto bearthisin mind whenconsideringthestress,becausetocorrelate stressand strain wemust refer them to the same set of axes. Let us now examine the solid body rotation.We have mentioned abovethattothefirstorderit isrepresentedbyavectorof com-ponentswx,WY'WzThismayeasilybeverifiedasfollows.If we denotethecomponentsell, e12,etc.,byeijandexx,eyy,etc.,by eli' wederivefromrelations(3.18)thatthecomponentsofthepure deformationare represented by eljif we neglect second order terms. In other words,to the firstorder Thereforeto the same order the pure deformation isrepresented by the transformation in matrix form :::][::] 1+ ezz dz (3.19) On the other hand, let us add asecond transformation of de', dr/, d,' into dg,d'Y},d, (3.20) By substituting transformation (3.19) into (3.20), we must perform thematrixmultiplication.If wedothisandkeeponlythefirst order terms, we obtain transformation (3.11).Hence transformation (3.20)representstothefirstorderthesolidrotation.Thematrix which represents this rotation may be written by introducing double indices as follows. o WY][Wll -Wx =W21 oW31 (3.21) 22Staticsand Kinematicsof Incremental Stressesand StrainsOk.1 with With general indicesthesematrix elementsare written Wo. W!j=0fori=j Wo=-Wj!foriof:j (3.22) Hence (3.23) It will also be found convenient to introduce general indices for the coordinatesand displacementsby putting Z=X3 W=U3 We may then write the moreconcise general expressions elf=! (au!+ aUf) 2oXjax! 1(aU!OUj) Wij="2OXj - oXi (3.24) (3.25) By suchdefinitionwemay writerelations(3.18)forthestrain in a formwhich iscompletely symmetric and alsomuch moreconcise: (3.26) This form may be further abbreviated, using the so-called dummy indexruleby whichsummationsignsaredroppedaltogether.We then write eij=eij+ !(eillw/lJ+ ejllWlli)+ !WillWjll(3.27) By this notation, which isstandard procedure in the tensor calculus, summationsaretakenforallpossiblevaluesof theindiceswhich appear more than once in the sameterm. Another formof the strain components isfounddirectly in terms of the gradientsof U!by substituting expressions(3.25)in equation 3.27.This yields e ..=! (OUi + OUj)+ ! (3oUIl OUIl 112oXjax!8ax!oXf oUj oUi oUIl oUj oUIl OUi) - aXilaXil- ax!oXIl - oXj aXil(3.28) Sec.4Incremental StressesinTwoDimensions23 Thisform,whichisconvenientinsomemathematicalderivations, obscuresthe physical significanceof the expression. DummyIndexRuleNotToBeOonfv:sedwiththeTensor Oalculus.Useofadummyindexasaconventionalnotationto replacethesummationsignisextremelyhelpfulnotonlyforthe purposeofabbreviationbutalsobecauseitbringsoutthehidden symmetry in the formulas.It isused throughoutthisbookwhen-ever convenient.Although the dummy index is generally associated withthetensorcalculus,it isin factquiteindependentof it.The treatment of continuum mechanics in this book is carried out without recoursetothe tensor calculus at any time. 4.INCREMENTALSTRESSESINTWODIMENSIONS We now turn our attention tothe analysisof the stressfield.It differsessentiallyfromthestrainanalysis.Thefactthat thecon-tinuum isalreadydeformed in the initialstateisirrelevant forthe definitionof theincrementalstrain.Thisisnotsofortheincre-mental stress, and we shall see that the state of initial stress must be considered in the analysis. In ordertobringoutmoreclearlytheconceptsandmethodswe considerfirstatwo-dimensionalstress field.We start by recalling someelementarydefinitionsandproperties.Thetwo-dimensional stressat apoint in the plane isdefined by the three components (4.1) referredtoorthogonalaxesxandy.Thephysicalsignificanceof these components isobtained by considering the plane x,yto repre-sentaslabof unit thickness.The stresscomponents represent the forcesinthex, yplane,actingperunitareaonthesidesofan infinitesimalelementofsizedx, dycutoutof theslab.Thecon.: dition that the tangential component UXY be the same on both sides dx anddyof theelementisaconsequenceof thefactthatthetotal torque resulting from the stresses on the element must be zero.This featureofthestresscomponentsisreferredtoasthesymmetry property. However,thereareexceptionalcasesin whichthispropertywill 24Staticsand Kinematicsof Incremental Stressesand Strai'MOh.1 y A L - - - - - ~ x Figure4.1Variation of normal stress Uaa and tangential stress Ua{J with the normaldirection. not be verified.This occurs if the body forcecontains amoment per unitvolumeorifadequaterepresentationoftheinternalstresses requirestheintroductionof couplesperunitarea.Suchcasesare excluded fromthe present treatment. If wecutasmall right-angled triangleOAB out of the slab(Fig. 4.1), the normal and tangential forces per unit area acting on the side AB are found by writing the equation of equilibrium of this element in the xand ydirections.We derive Uaa =Uxx cos2 a+ Uyy sin2 a+ UXy sin 2a UafJ=t(uyy - UXX)sin 2a+ UXy cos2a (4.2) The angle a measures the inclination of the normal to AB with the x direction.Relations(4.2)yield immediately the stresscomponents with respect toaxes1,2, which are rotated clockwiseby an anglea fromtheoriginaldirectionsx, y. *Thenewcomponents(Fig.4.2) (4.3) are found by substituting the values aand a+ TTj2in relations(4.2). They are Uu=Uxx cos2 a+Uyy sin2 a+ UXy sin 2a U22=Uxx sin2 a+ Uyy cos2 a- UXy sin 2a U12=l(uyy - uxx) sin 2a+ UXy cos2a (4.4) * For furtherdiscussion see,forexample,S.Timoshenko,Theoryof Elasticity, p. 16, McGraw-Hill Book Co.,New York,1934. Sec.4Incremental StressesinTwoDimensions y 2 x Figure4.2Representationof the same stress fieldrelative to the directionsx,yand the rotated axes1,2. 25 Thelastequationshowsimmediatelythatthereisalwaysa directionex=exlforwhichthetangentialcomponentUl2orshear stressvanishes.The angleexlisgiven by (4.5) The stresscomponents referred tothisdirection reduceto normal components U11and U22and are called principal stresses. Inversely,by replacingexby- exwemay expressthe stressesUxx, etc.,in termsof the components U11,etc.We find Uxx =U11cos2 ex+ U22sin2 ex- Ul2sin 2ex Uyy=U11sin2 ex+ U22cos2 ex+ Ul2 sin 2ex UXy =!(U11- U22)sin 2ex+ Ul2cos2ex We consider nowan initial stress field (4.6) (4.7) These components define the initial stress at apoint Pof coordinates x,y in the plane (Fig.4.3).If the plane continuum is deformed,any 26StaticsandKinematicsof Incremental Stressesand StrainsOh.1 2 P(x,y) tS12 rS11 L, Figure 4.3Representation of the initial stresses.S11' S22'S12 and the incremental stresses 811,822,812. pointPisdisplaced toapointP' of coordinatesg,Yj,and the stress at this point P' acquiresanew valuedefinedby the components a ~ ~=811 + 8 ~ ~ an1/=822 + 8n1/ a ~ n=812 + 8 ~ n (4.8) Thesecomponentsarereferredtothefixeddirectionsx, y.Thecom-ponents8W 8n1/'8 ~ 1 /representtheincrementof thetotalstressatthe displaced pointP' of coordinatesg and TJafterdeformation. Weintroducenowan importantconsiderationinthewholepro-cedure,namely,that the incremental components 8 ~ ~ ,8n1/'8 ~ naredue not only to the strain but alsoto the factthat the initial stress field hasbeenrotatedby acertain anglewhen movingfromPtoP'.In other words,if there were nodeformation at all,but simply atrans-----+ lationequaltothevectorP P'followedbyasolidrotation,there wouldbeincrementalstresscomponents8 ~ ~ ,81/1/'8 ~ nduetothis Sec.4Incremental StressesinTwoDimensions27 rotation, hence of purely geometric origin.In addition, if the material undergoesastrain thereisan incremental stressof purelyphysical nature.It isthereforeessential to separate the geometry fromthe physics in expressing the incremental stress components.This can be accomplishedif,insteadofreferringthestresscomponentstothe original directions x, y,werefer them to new directions1,2.These new directionsare rotated with respect to the original directionsby an anglee which isequal to the local rotation of the material.This anglehasbeenevaluatedinsection2andisgivenbyexpressions (2.13)and(2.17).Its approximate value tothe firstorder is e ~w=~(OV_ OU) - 20xoy The stresscomponents referred to these rotated axesare all=811 + 811 a22= 822 + 822 a12= 812 + 812 (4.9) (4.10) Thequantities811,822,812aretheincrementsof stressreferredto axeswhichrotatewiththemedium.It ispossibletoexpressthe stressesa ~ ~ ,al1n, a ~ nin termsof the stressesall, a22'a12by using the transformationformulas4.6inwhichwereplaceau, ayy,aXY by a ~ ~ ,ann'and a ~ n 'and the angleaby w;;;;;a.We write a ~ ~=all cos2 w+ a22sin2 w- a12sin 2w al1l1 =all sin2 w+ a22cos2 w+ a12sin 2w a ~ 1 1=t(all- a22)sin 2w+ a12cos2w (4.11 ) Weshallassumethattheincremental8tres8e8andtherotationare quantitie8of the firstorder. To the firstorder weput cosw=cos2w;;;;;1 sin w=t sin 2w~w (4.12) Substitutingexpressions(4.8)and(4.10)inequations4.11and retainingonly firstorderquantities,we find 8 ~ ~=811 - 2812W 81111 =822 + 2812w 8 ~ n=812 + (811 - 822)W (4.13) 28StaticsandKinematicsof Incremental Stressesand StrainsOk.1 These equations bring out the terms representing that portion of the incrementalstresseswhichisduetotherotationalone.Thefirst terms S11'S22,and S12represent the stress due to the deformation and depend only on the physical propertiesof thematerial. 5.INCREMENTALSTRESSESINTHREEDIMENSIONS Weshallextendtheprecedingdefinitionsto athree-dimensional stressfieldandconsiderastateofstressrepresentedbythe components (5.1) Let uscut out of the mediumasmall tetrahedron of sidesOA,OB, 00 parallel to the axes x,y,z,and such that the triangular face ABO has aunit area(Fig.5.1). z nF(n) B y Figure 5.1Force F(n)acting per unit area on asurfaceof normal direction nin astress field. The orientation of the triangular face ABO is defined by avector n of unit length directedpositivelyoutward of the tetrahedron.The vectorniscalledaunitvector.Thecartesiancomponentsof this unitvectorarethedirectionalcosinesofthedirectionn.These Sec.5Incremental StressesinThreeDimensions29 directionalcosinesare thecosinesof theanglebetween the positive direction of nand the threecoordinate axes,i.e., cos(n,x),cos(n,y),cos(n,z)(5.2) If thestressfield(5.1)isactingin thetetrahedralelement,from the condition of equilibrium of the element wemay derive the force F(n)acting per unit area on the faceABC. The cartesian components of this forceare axx cos(n,x)+ aXY cos(n,y)+ azx cos(n,z) aXY cos(n,x)+ ayy cos(n,y)+ ayZ cos(n,z) azx cos(n,x)+ ayZ cos(n,y)+ azz cos(n,z) (5.3) Because of the symmetry of the stress system (5.1)we may associate with these relationsaquadratic form cp=axxx2+ ayyy2+ azzz2 + 2aXYxy+ 2ayzyz+ 2azxzx(5.4) If we identify the coordinates x,y,z with the directional cosines (5.2), weseethat the vectorFisparallel tothe gradient of cp,i.e.,to the vector, grad cp=(OCP,oCP,OCP) aXoyoz (5.5) ThisgradientISnormaltothequadricsurface,calledthestress quadric,* cP=const.(5.6) If the unit vector nis directed along anyone of the three axes of this quadric, the force F(n) is parallel to the vector n, hence normal to the faceABC.Wederivefromthistheexistenceofthreeprincipal directions of stress, i.e., directions for which the tangential components ofstressaXY'ayZ'azx vanish.Thecorrespondingnormalstress componentsaxx,ayy,azz are the principal stresses. The preceding equations also lead to expressions for the stress for a systemofcoordinateaxeswhicharedifferentfromx, y, z.Let us considerasystemof rectangularaxes1,2, 3withitsoriginat the same point asthe original system x, y,z. * Forotherpropertiesof thestressquadricseeLove'streatise,TheMathematical TheoryofElasticity,FourthEdition,pp.80-81,CambridgeUniversityPress (reprinted by Dover Publications, New York,1944). 30StaticsandKinematicsof Incremental Stressesand StrainsOh.1 The directional cosinesof axes1,2,3relative tox,y,z are cos(1,x)cos(1,y)cos(1,z) cos(2,x)cos(2,y)cos(2,z) cos(3,x)cos(3,y)cos(3,z) Wedenote the stresscomponents referred tothe new axesby (5.7) (5.8) The stress component Ull' for instance, may be found by orienting the normal direction nof the ,face ABO along axis1.The force F(I) acting on this faceisthen projected on axis1..We find Ull=FAl) cos(1,x)+Fy(l) cos(1,y)+Fz(l) cos(1,z)(5.9) Substituting the values(5.3)forthecomponentsof F(I) yields Ull=Uxx cos2 (1, x)+ Uyycos2 (1,y)+ uzz cos2 (1,z) +2uyZ cos(1,y)cos(1,z)+2uzx cos(1,z)cos(1,x) +2uXY cos(1,x)cos(1,y)(5.10) ! Theothercomponentsateexpressedin thesameway;usingthe dummy index rule,wemay write UIlV =Uticos(ft,i) cos(v, j)(5.11) We put Ulj=Ujl,hence alsoUIlV =UVIl'Note that these expressions yieldjustaswellthestresscomponents(5.1)intermsof thecom-ponents(5.8).Thisamountstocommutingthe indicesx, y,zwith 1,2,3 in relations(5.10)and(5.11).Relations(5.11)then become Uu=ullV cos(i,ft)cos(j, v) Explicitly this iswritten Uxx =Ull cos2 (x,1)+ U22cos2 (x,2)+ U33cos2 (x,3) + 2U23cos (x,2)cos (x,3)+ 2U31cos (x,3)cos (x,1) + 2U12cos (x,1)cos (x,2), etc. (5.12) (5.13) Wenowgobacktoathree-dimensionaldeformation.The kinematicswasanalyzedinsection3.ApointPoriginallyof co-ordinatesx,y,zistransportedtoapointpiof coordinatesg,TJ,,. Sec.5Incremental StressesinThreeDimensions31 Themediumisunderastateof initialstress.Thecomponentsof initial stressat point Pare 811 812 831 812 822 823 831 823 833 (5.14) AtpointP'(g, 7],')afterdeformationthestressesreferredtoaxes parallel to x,y,z become a ~ ~=811 + s ~ ~ ann=822 + snn a(C= 833 + s1;1; anc=823 + sn1; a 1 ; ~=831 + s 1 ; ~ a ~ n=812+ s ~ n (5.15) Following the procedureof section 4 fortwo-dimensional stresses, weshallreferthestressestorectangularaxes1,2,3obtainedby rotatinglocallywiththematerialarectangularsystemg,7],, originally parallel to x,y,z and with its origin at the displaced point P'.The rotation is defined to the first order by the vector wx,Wy,Wz asgivenby expressions(3.10).Asin section3wherewediscussed thekinematicsofthree-dimensionalstrain,aninfinitesimalvector dg,d7],d,intheunrotatedcoordinatesystemg,7],,isrepresented bythecomponentsde, d7]',d,'intherotatedaxes1,2,3.The relation between those twovectorsisgiven tothe firstorderby the equations dg=de- Wz d7]'+ Wyd,' d7]=Wz de+ d7]'- Wxd,' d,=- Wyde+ Wxd7]'+ d,' (5.16) Theseequationsarederivedfromthekinematicsof rigidbodies. They yield the displacement field for a small solid rotation represented by the vectorwx,Wy,WzOn the other hand,transformation(5.16) mayalsobeconsideredacoordinatetransformationfromtheaxes 1,2,3 toaxesg,7],,.Thechangeof coordinatesisrepresented by the equations dg=de'cos(g,1)+ dr/cos(g,2)+ dr cos(g,3) d7]=de cos(7],1)+ d7]'cos(7],2)+ dr cos(7],3)(5.17) d,=de cos(', 1)+ d7]'cos(', 2)+ d,'cos(', 3) 32Static8and Kinematic8of Incremental Stre88e8and StrainsCh.1 Comparingrelations(5.16)and(5.17),wederivethefollowingfirst order approximation forthe directional cosines. cos1)=1 cos(T},1)=Wz cos(T},2)=1cos(T},3)- Wx(5.18) cos1)=-wy cos2)=Wxcos3)=1 The stress components referred to the locally rotated axes are denoted by Ull=Sll+ 811 U22=S22+ 822 U33=S33+833 U23=S23+ 823 U31=S31+ 831 U12=S12+ 812 (5.19) Thequantities811,822,etc.,' nowdesignatethestressincrements relative to the rotated axes.The transformation relations fromone setofstressestotheotherareeasilyestablishedbyapplyingthe results obtained above.We use relations (5.12) and (5.13), replacing x,y,zby T},and UXXUXy,etc., byetc.We then substitute in theserelationstheapproximatevalues(5.18)forthedirectional cosines,and expressions(5.19)for the stresses Ull,U22'etc.Retain-ingonly quantitiesof the firstorder in 811,822,etc.,and WX,Wy,wz, wefind =Sll+ 811 +2S31W y- 2S12Wz ann=S22+ 822 + 2S12wz - 2S23WX a{{=S33+ 833 + 2S23WX - 2S31 Wy an{=S23+823 +(S22- S33)WX - S12Wy+S31Wz =S31+831 +(S33- Sl1)Wy- S23WZ+S12WX =S12+812 +(Sl1- S22)Wz - S31WX+S23Wy In abbreviated notation equations5.20 may be written (5.20) (5.21) In thisexpression wedesignateby Sij the initialstresscomponents (5.14)with theconvention Sij=Sjt(alsoaij=ajt).Note that the subscriptsi,jstandfor T},ontheleftsideandfor1,2,3on the right side.The quantities Wjjaredefined by the elements of the matrix(3.21). The term Sp,jWiJl+ SiJlWjJlin equation 5.21represents that portion Sec.6Equilibrium Equations in TwoDimensions33 of the incremental stressesduetothe rotationalone.The termSjj represents the stress increment due to the deformation and depends thereforeonly on the physical propertiesof thematerial. 6.EQUILIBRIUMEQUATIONSFORTHESTRESSFIELD INTWODIMENSIONS Wenowestablishtheequationswhichmustbeverifiedbythe incremental stress fieldunder the condition of static equilibrium. For the sake of clarity weproceed first with the analysis of atwo-dimensional field.In the plane x,y we consider a body outlined by a contour O.The initial stresses in thebodyare Sll' S22' S12.If X andYare the componentsof the body forceperunitmass,and if p isthemassdensityof themediumbeforedeformation,theinitial stress components must satisfy the well-known equilibrium conditions 8Sll 8S12 ax +81/+ pX(x, y)=0 (6.1) 8S12 8S22 Y()_0 ax +8y+ Px, Y-We have assumed here that the body force per unit mass isafixed fieldinspace,afunctiononlyof thecoordinatesx,y.Inpractice thisisgenerallythecase;however,thereareexceptionsas,for instance,insomegeophysicalproblemswherethegravityfield dependsalsoon thedeformationitself.Forsimplicityweexclude y c x Figure6.1Forces on the boundary 0' of adeformed body. 34Staticsand Kinematicsof Incremental Stressesand StrainsOh.1 thiscaseforthepresent.It may,however,beincludedinthe equations,and weshall indicate briefly in the next section howthis may be done. Apoint Pof the material originally of coordinates x, ymoves to a point P' of coordinatesg,Y)after deformation (Fig.6.1).We denote by bx and bythe xand y components of the force bacting at point P' of the boundary per unit areaafterdeformation.We lookupon this forceas that acting on the solid inside the contour 0'.The forceon a line element ds' of the contour 0' is b ds', where ds' is chosen positive inacounterclockwisedirectiononthecontour0'.Withthese definitions and by considering the equilibrium of a triangular element adjacent tothe boundary asshown in Figure6.2,wemay write bx ds'~a ~ ~dY)- a ~ ndg byds'=a ~ ndY)- anndg (6.2) Figure6.2Boundaryforces andstressesonaboundary element. This force is the external forceacting attheboundaryonthesolidlying insidethecontour0'.Theother external forceactingonthissolidis the body force.Consider an element of the solid of area dS at point x, yand of mass density pbefore deformation. Afterdeformation it has moved toa point of coordinatesg,y),its area has becomedS',anditsnewdensityis nowp'.Becauseof the lawof con-servation of mass we may write p dx dy=p dS=p'dS'(6.3) Letusnowwritetheconditionof equilibriumforthesolidinsidethecontourbystatingthatthe resultant of the boundary forcesandthebodyforcesactingonthe solid vanishes.Thiscondition is [bx ds'+ Ii X(g, Y))p'dS'=0 Jc'S' (6.4) [byds'+ Ii Y(g, Y))p'dS'=0 Jc'S' The contour integrations are performed counterclockwise.We may Sec.6Equilibrium EquationsinTwoDimensions 35 changethevariablesof integration in theseintegralstoxandyby using transformation(2.24);that is, (aU)au =1+ - dx+- dy axoy dT)=OVdx+(1+OV)dy axoy (6.5) With these expressions,relations(6.2)become bxds'= :: - (1+ ::)] dx+ (1+:;) - ::] dy (6.6) byds'=- ann(1+ ::)1 dx+ (1+ :;) - ann::] dy Wesubstitute theseexpressionsin the equilibriumconditions(6.4), alsoreplacingp'dS'byp dxdyaccordingtorelations(6.3).By applyingtheoremtothecontourintegrals,theyaretrans-formed to surface integralsand equations6.4become Ii

(1+OV)-aU] saxoynay - :y- (1+ ::)]. +T))p}dxdy=0 I Is {:x[al;n(1+ :;) - ann::] (6.7) - :y [al;n:: - ann(1+ ::)] +=0 Since these relations must be verified forany arbitrary contour 0, i.e.,forany arbitrary domainof integration S,the integrandsmust vanish.Thereforetheequilibriumconditionofthestressfield becomes the differential equations a(_ov_aU) ax+oy+axoy- oy (6.8) o(OVaU) - oyax- annax+ T))p=0 36StaticsandKinematicsof Incremental Stressesand StrainsOk.1 Theseequationsdonotinvolveanyapproximation.They include theconditionsof equilibrium(6.1)fortheinitialstresssincethey reduce to these equations for U=v=O.Note also that the quantity p in these equations is the mass density before deformation,hence is a given function of xand y. Atthispointweareinterested in introducingstresscomponents which depend only on the strain and donot change when wesuper-imposeasolidrotationonthemedium.Suchstresscomponents were introduced in section 4.From equations 4.8 and 4.13 we write a ~ ~= 811 + 811 - 2812W a1/1/=822 + 822 + 2812W a ~ 1 /=812 +.812 +(811 - 822)W (6.9) Thestresscomponents811,822,812arethe incremental stressespro-jected on axeswhich rotate with the material by an anglewdefined byexpression2.26.Theequations6.9areapproximatetothefir8t order.Weshallnowsubstitutethesevaluesinequations6.8and take into account the equilibrium conditions (6.1) for the initial stress. Afterthesesubstitutions,keepingonlythetermsof the firstorder, we find 8811 8812 8x+8y+ p[X(g, 1))- X(x, y)] 88 - 2 8x (812w)+ 8y [(811 - 8dw] + ~ ( 8 1 18v_812 8U)_~(811 8v_812 8U) 8x8y8y8y8x8x 88128822 8x+8y+ p[Y(g,1))- Y(x, y)] =0 (6.10) 88 + 8x [(811- 8dw] + 2 8y (812W) + ~(812 8v_822 8U)_~(812 8v_822 8U)=0 8x8y8y8y8x8x In these equations we may write the incremental body forceas ..::IX=X(g,1))- X(x, y) ..::IY=Y(g,1))- Y(x, y) (6.11) Sec.6Equilibrium Equations in TwoDimensions 37 Equations6.10maybetransformedbymakinguseofidentities derived fromequations2.26and 2.27: ov ov Oy=eyy OU (6.12) ox=eXY+ woy=eXY- w Whenweintroducetheseexpressionsintoequations6.10,they become 08110812AX0S0S ox+oy+ p ~- ox(12w)- oy(22W) o,0 +ox (Sl1eyy- S12exy)- oy(Sl1eXY- S12exx)=0 0812 0822 00 ox+oy+ p Ll Y+ ox(Suw)+oy (S12W) (6.13) o0 +ox (S12eyy- S22exy)- oy (S12eXY- S22eXX)=0 These equations may be further simplified if we take into account the followingidentities derived from(6.12). (6.14) Introducingtheseidentitiesintoequations6.13andagaintaking intoaccounttheequilibriumconditions(6.1)fortheinitialstress, they are transformed to 0811 0812 owSow ox+oy+ pLlX+ pwY(x, y)- 2S12ox+(11- Sd oy OS11(OS11OS/2)OS/2_0 +oxeyy - oy+ axeXY +oyexr - (6.15) 0812 0822 OwSow ox+oy+pLlY- pwX(x, y)+ 2S12Oy+ ( 11- S22)ox OS22(OS22OS/2)OS/2_0 +oyexx - ox+oyeXY + ax eyy -38Stnticsnnd Kinemnticsof Incremental Stressesnnd StminsCk.1 Theseequationsaretheequilibriumconditionsforthetwo-dimen-sionalstressfieldexpressedintermsoftheincrementalstresses 811,822,812'Equations6.15werefirstderivedbytheauthorin 1938.*It wasalsoshownthatthevarioustermshaveasimple physical interpretation,asexplained at the end of this section. If weagaintake intoaccounttheequilibriumconditions(6.1)of the initial stress field, equations 6.15 may be written in an alternative and moresymmetric form: 0811 0812 AX ox+oy+ P LJ+ pW Y(x,y)- peX(x, y) owow - 2S12 - +(S11- S22)-ox.oy _OSll_OS/2_(OSl1OS/2)_0 oxexx oyeyy oy+oxeXY-0812 0822 A - + - +PLJY- pwX(x,y)- peY(x,y) oxoy (6.16) OS/2_(OS22+ OS/2)e=0 oxexx oxoyXY We have put e= exx + eyy Someinterestingpropertiesoftheseequationsareimmediately apparent.If there isnobody force(X=Y=0)and if the initial state of stress is uniform, i.e., independent of xand y,equations 6.16 assume the simpler form (6.17) If the initial stress ishydrostatic,i.e.,if * In reference2at the end of the Preface. Sec.6EquilibriumEquationsinTwoDimensions39 equations6.17reduce to OS11+ OS/2=0 oxoy OS/2+ OS22=0 oy (6.18) whicharetheclassicalconditionsofequilibriumforastressfield whenthereisnoinitialstress.Thesameequations(6.18)are obtained forasolid body rotation,i.e.,for w=const.(6.19) Inthesolutionof specificproblemswemustbeabletoexpress certainboundaryconditions.Letusthereforeturnourattention tothe forceactingat theboundary.Consider aportion AB of the BB By y L-________________________Figure6.3Integrated boundary forceBon afiniteportion A' B' of the deformedboundary. contour0(Fig.6.3).Afterdeformationitbecomesthepartof the contour 0' designated A' B'.The external forceBactingon the solid boundary A' B' after deformation has the cartesian components fB'fB' Bx=bx ds'=dYJ- dg) A'A' (6.20) Theseexpressionsarederivedfromequations6.2.The integration isperformedinthecounterclockwisedirectionalongthedeformed line A' B', and the solid is lying to the left when one moves from A' to 40StaticsandKinenwtics of Incremental Stressesand StrainsOh.1 B'.Byrelations(6.6)theseintegralsmaybetransformedtoline integralsalongthelineABlyingontheoriginalcontourbefore deformation.Substitutingrelations(6.6)intheintegrals,wefind Bx=f::: - (1+ dx + f:(1+ :;) - :;] dy By=LB :: - ann(1+ dx + f:(1 + :;) - ann:;] dy (6.21) If thelineABisanelementoftheinitialcontourofcomponents dx,dy,theforcedBactingonthiselementafterdeformationis represented by the components dBx=+ :: - dx + + :; - :;) dy (6.22) dBy =( -ann+ :: - anndx + + :; - anndy Theboundaryforcemaybeexpressedintermsofthestresses referredtotherotatedaxesbysubstitutingexpressions(6.9)into equations6.22.Indoingsoweretainonlythefirstorderterms. We also substitute expressions (6.12) for the partial derivatives.We find dBx =- (812 + 812 - 82200- 811eXY + 812eXX)dx + (811 + 811 - 81200+ 811eyy- 812eXY)dy dBy=- (822 + 822 + 81200- 812eXY + 822eXX)dx +(812 + 812 + 81100+ 812eyy- 822eXY)dy (6.23) We note that the differential dx,dymust represent an element of arc positivecounterclockwise,thesolidlyingontheleftsideofthe element.Theseexpressionsmayalsobegivenanequivalent form Sec.6Equilibrium EquationsinTwoDimensions 41 by introducingaunitvectornnormaltotheoriginalcontourand chosenpositiveinadirectionawayfromthesolid(Fig.6.3).We have the relations -dx =dscos(n,y) dy=dscos(n,x) (6.24) where cos(n,x),cos(n,y)are the directional cosines of the direction nnormal to the original contour and dsisthe absolute value of the element of arc, ds=Vdx2 + dy2 Furthermore wedefineaforcefof components f = .dBx xds f=dBy yds This is the boundary force per unit initial area of the boundary. these definitions wemay write the boundary forceas fx=(Sl1+ Sl1- S12W + Sl1eyy- S12eXY)cos(n,x) + (S12+ S12- S22W - Sl1exy+ S12exx)cos(n,y) fy=(S12+ S12+ Sl1W + S12eyy- S22exy)cos(n,x) +(S22+ S22+ S12W - S12exy+ S22eXX)cos(n,y) (6.25) With (6.26) It isalsopossibletointroduceincrementalboundary forces,i.e.,the difference between the actual boundary forcesafter deformation and theirinitialvaluebeforedeformation.Expressedperunitinitial area,these incremental boundary forcesare ,dfx=(S11- S12W + S11eyy- S12exy)cos(n,x) + (S12- S22W - Sl1eXY+ S12eXX)cos(n,y) ,dfy=(S12+ Snw + S12eyy- S22exy)cos(n,x) +(S22+ S12W - S12eXY+ S22exx)cos(n,y) (6.27) We shall end this section with a few remarks on the significance of these results.Let us first look at the incremental body force,dX, ,dY definedby equations6.11.Theseexpressionsrepresent thechange inbody forceper unit masswhenwemovefromtheoriginalpoint x,y to the displaced point g,TJafter deformation.If this body force 42StaticsandKine-Inaticsof Incremental Stressesand StrainsOk.1 isrepresentedby afixedfield,functiononlyof thecoordinates,we may linearize expressions(6.11)and write LlX=(u ! + v:y) X(x,y) LlY=(u~+ v:y)Y(x,y) (6.28) Atthebeginningof thissectionwereferredtothepossibilitythat thefieldmaydepend notonlyon thefixedcoordinatesbut alsoon thedeformationitself.Thismightoccur,forinstance,if wewere interested in thedeformationof alargegravitationalbodysuchas aplanet.In thiscasetheconfigurationof thegravitationalfield would depend on the deformation itself.Additional terms Ll' X, Ll' Y mustthenbeaddedtorepresentthelattercontributiontothe incremental body force,and weshould write LlX=(u! + v~ )X(x,y)+ Ll'X (. 88) LlY=u8x+ v8yY(x,y)+ Ll' Y (6.29) These additional terms can be evaluated only by solving the complete problem.In many applications when we are dealing with auniform gravity fieldthe incremental body forcevanishesaltogether. Our next remark deals with the physical significanceof equations 6.15.Let us rewrite the firstof equations6.15. 8s11 8S12 8x+8y+pLlX+pwY(x, y) 8w8w - 28128x+(811- 8d 8y 8811_(8811 8822)8812 - 0 +8xeyy 8y+8xeXY +8yexx -(6.30) Letuslookatthetermsonthesecondline.Theycontain8w/8x and8w/8yandaredifferentfromzeroonlyjf thedeformationis inhomogeneous.In theoriginalpublicationstheywerereferredto as the curvature terms. *By contrast the terms on the third line are * See references2,3,and 4at the end of the Preface. Sec.6EquilibriumEquationsin TwoDimensions43 811811 Figure6.4Physical interpret'l-tion of the"curvature terms." different fromzeroonly if the initial stress isinhomogeneous.The physical significanceof thecurvature termsisillustrated by Figure 6.4.The horizontal resultant of the forces indicated in the figure are duetothecurvaturesandchangesof areasof adeformedelement. Notealsothatequation6.30isanequilibriumconditionreferredto locallyrotatedaxes.Thisisconsistentwiththeappearanceof the term pw Ywhich represents aprojection of the body forceon rotated axes.Equations 6.15may therefore be considered an intrinsic form of the local equilibriumconditions. Attention shouldalsobecalledtothesignificanceofequations 6.8.They could have been derived exactly by writingtheequilib-riumconditionsforthe stressfieldin termsof the coordinatesg,7]. Theseconditions are (6.31) The unknown massdensity after deformation isp'. Wemaytransformtheseequationsbyusingthedifferential relations(6.5)and expressall partialderivativeso/N, 0/07]in terms ofa/axand%y.If weperformthistransformationinequations 6.31,weobtain equations6.8.This method of derivation wasused in some of the earlier work,and it further illustrates the significance 44StaticsandKinematicsof Incremental Stressesand StrainsOh.1 of the formulas.However, the method used above has the advantage of providing at the same time suitable expressions forthe boundary conditions. 7.EQUILIBRIUMEQUATIONSFORTHESTRESS FIELDINTHREEDIMENSIONS The analysis of the equilibrium conditions for the two-dimensional field presented above may be extended to the three-dimensional case by followingasimilar procedure. Westartwithastateofinitialstressrepresentedbythecom-ponents(5.14).They satisfy the equilibrium equations 8S118S12 8S31 X()- 0 ax +8y+ Tz +pX,y,Z-8S128S228S23Y()_0 ax + By + 8z +pX,y,Z-(7.1) 8S318S23 8S33 Z()- 0 ax +8y+ 8z +pX,y,Z-Thecomponentsof thebody forceperunitmassareX,Y,Z,and the massdensity at the point x,y,z,isp(x,y, z). ConsideravolumeVbounded by asurface Sbefore deformation. Afterdeformation this surfacebecomes S',and it enclosesavolume V'.The xcomponentBxof the forceactingon the boundary S'is Bx=fIs, dTJ +dg+dg dTJ)(7.2) The surface integral is extended to the boundary afterdeformation, andare stress components at a point of coordinates g, TJ,Werememberthatthesymmetryofstresscomponentsimplies ==and=We shall therefore pay no atten-tion tothe orderof the indicesandchooseit forconvenience.The variablesofintegrationinequation7.2maybechangedtothe original coordinates x, y,z.The two sets of variables are related by relations(3.1) g=x+u TJ=y+v (7.3) Sec.7Equilibrium Equations inThreeDimensions45 By known methodsof transformation the integralBxbecomes The surface integrals are now extended to the same material boundary Sbeforedeformation.In thisexpression[d(7],m/[d(y, z)],etc.,are thepartialJacobiansofthetransformationofx,y,zintog,7],For instance,we write 07]07] oyoz oyoz (7.5) These J acobians are the cofactors ofthe determinant of the differential transformation(3.3).In order toabbreviate the writing let usput The surface integral(7.4)is then written Bx=I1 (Axxdy dz+ Axy dz dx+ Axz dx dy)(7.7) The massdensitypat apoint x,y,zbeforedeformation becomesp' afterdeformationat thedisplacedpointg,Thexcomponent. of the resultant body forceactingon the volumeV' is IIL,X(g, 7], dV'=III X(g,7],dV(7.8) This equation results fromthe conservation of mass, namely, p'dV'=pdV(7.9) 46Staticsand Kinematicsof Incremental Stressesand StrainsOk.1 Thetotalexternalforceactinginthexdirectionisthesumof expressions(7.7)and(7.8).Forequilibriumit mustvanish;hence IL Au dy dz+ Axy dzdx+ Axz dx dy + III X(g, Y),')p dV=0 (7.10) The surface integral istransformed toavolume integral by Green's theorem,and equation7.10becomes fff [8Axx8Axy8AxzK( t.Y)]d V=0 ++ y),'"p vuxuyuZ (7.11) Since thisequation must be satisfied forany arbitrary volumeV,it impliesthe differential equation 8Axx8Axy8AxzX( t.Y)=0 8x+8y+8z+s, Y),'"P (7.12) There are twoother such equations fortheyandzdirections;they areobtained by cyclicpermutation of thecoordinate axes. These equilibrium conditions(7.12)forthe stress fieldcontain the initialstateasaparticularcase.Puttingu=v=w=0,they coincide with the equilibrium equations(7.1)forthe initial stress. Weshallnowintroducefirstorderapproximationsinequations 7.12.Todothisitisconvenienttointroducesomeabbreviated notation.We denote the Jacobians by M=d(y),') 11d(y,z) M_d(y),') 12- d(z,x) M_d(y),O 13- d(x,y) M_d("g) 21- d(y,z) M=d("g) 22d(z,x) Md("g) 23=d(x,y) (7.13) M_d(g,Y)) 31- d(y,z) M_d(g,Y)) 32- d(z,x) Md(g,Y)) 33=d(x,y) Wealsowrite Ali forAxx, Axy, etc. aij for etc. (aij=aj;) (7.14) Relations7.6may then be written (7.15) Sec.7 EquilibriumEquationsinThreeDimensions47 The summation isperformed forall valuesof the indexk in accord-ancewiththedummyindexruleexplainedinsection3.If we denotex,y,zby Xlandg,7],,bygland thebody forceX,Y,Zby Xl' the equilibrium equations7.12may be written (7.16) We keeponly firstorder terms,and the approximate values forthe Jacobians are Mll1+ eyy+ ezz Ml2= -eXy - Wz Ml3= -ezx + Wy M21 -eXy + Wz M22=1+ exx + ezz M23 =-eyZ - Wx M31= -ezx + WyM32=-eyZ + Wx M33= 1 + exx + e yy (7.17) In abbreviated notation these relations take the form (7.18) where eljand Wljare the matrix elements in equations 3.19and 3.21. We denote byOjjthe Kronecker symbol definedas and we put Otj=1fori=j otj=0fori=1=j (7.19) (7.20) Wealsomakeuseof relation(5.21)forthestresscomponentsatj expressed in termsof the initial stressand thestressincrementsSlj referred to rotated axes.These relations are (7.21) We now substitute into equation7.15the approximate values for Mlj and alj as given by expressions (7.18)and (7.21).In doing so we drop all terms of order higher than the first,namely, those which are squares and products of thequantities Sjj'ejj'and Wlj'We find Au=(Slk+ Sik+ SP.kWIP.+ SjP.WkP.)Okj + Slk(eOkj- ekj+ Wkj) (7.22) 48Staticsand Kinematicsof Incremental Stressesand StrainsOh.1 Becauseof the significanceof 8lei wemay write 8ile8lei =8ij Sjle81cj=Sjj SIlleWjll8lei =SlljWill SillWlell8lej= SillWjll (7.23) Furthermore,becausef1- isadummy indexwemay replaceit by k, and the last two expressionscan be written S/ljWill=SlejWile SIIlWjll=SjleWjle (7.24) Bytakingintoaccountidentities(7.23)and(7.24),relation(7.22) becomes Alj =Sjj+ 8 jj+ SlejWjle+ SileWjle+ Sije- Silee1cj+ SileWlei(7.25) Further simplification is obtained if we take into account the property of antisymmetry ofthe matrix Wile(equation 3.21).This property is expressed by Hence Taking the last identity into account, we finallyobtain Substitution of Aij into equation 7.16yields (7.26) (7.27) (7.28) Note that p isthe original massdensity at the point Xjbefore defor-mation.Equations7.29arethethree-dimensionalequilibrium equations for the incremental stress field8;j.Asinthe two-dimen-sional case examined in the previous section, they may be simplified by taking into account additional relations and identities.First we may take intoaccounttheequilibriumconditions(7.1)satisfiedby the initial stress field.They may alsobe written (7.30) Sec.7 Equilibrium Equationsin ThreeDimensions49 By introducing thiscondition into equations7.29they become 080 +[SkjWik+Sjje- SikekJ+p LlXj=0 uXj uXj (7.31) SinceSkj=Sjkandekj=ejk weimprovethesymmetryofthe notation by writing this equation as 08ij0S - +- [jkWik+Sjje- Sikejk]+p LlXj=0 OXj OXj (7.32) We have put (7.33) Hence LlXj represents the in body force per unit mass from the initial point to the displaced point.Equations7.32correspond to equations6.13forthe two-dimensional case.Asbefore,wemay furthersimplifytheseequationsby usingidentitiesbetweenstrain and rotation similar toequations6.14 forthe particular caseof two dimensions.Suchrelationsarederivedbystartingfromthe identities Because of definitions(3.25)these identities may be written Let us multiply this equation by olj.Since wederive eljolj=e wljolj=0 (elk+ Wlk)Olj=ejk+ Wjk MUltiplyingthe last equation by Slk'weobtain (7.34) (7.35) (7.36) (7.37) (7.38) 50Staticsand Kinematicsof Incremental Stressesand StrainsOh.1 Becausekisadummy index we may alsowrite this relation as Sikoexjk"=Sij ooe- SikoWjk 1XjXj (7.39) Introducing the last identity i n ~ oequations7.32,weobtain oStjSOWlkoW"kOSjk - +"k-- +S"k_,_+Wikox" OXj 1OXj IOXj 1 oStjOSik + e - - ejk -- + p LlXj=0 oXj oXj (7.40) Finally we again make use of the equilibrium condition(7.30)of the initial stressfieldwhich may' be written oSik - - -pXk(X1) OXj-OSI _, =- pXt(x1) oXj Substituting theseexpressions,equations7.40become* OSt" _, + P L I ~ t- PWtkXk(X1)- peXj(x1) oXj S OWjkSOWjkOSjk +jk-- +tk-- - e"k-- =0 oXj OXj 1OXj (7.41 ) (7.42) In twodimensionsthese- equationsreducetothe form(6.16).The same remarks may be made here as discussed for the two-dimensional case at the end of the previous section.The incremental body force may be expressed as (7.43) where the firstterm represents a linearizing of the increment of body forcedue to the displacement alone,andLl'Xt is the increment of the over-allfieldduetothedeformationof thebodyasawhole.As alreadystated,thelatterpartwouldariseforthegravityfield generated by alargedeformingmedium. * Equations7.42werederived in this particular formby the authorin references3 and 5at the end of the Preface.An alternative derivation was given in reference 7. Sec.7 Equilibrium Equationsin ThreeDimensions51 Variation of the Gravitational Field Due to Deformation.In problems of planetary and astrophysical dynamics dealing with large gravitational bodies wemust take intoaccounttheterm LI'X"which representsthevariationof the gravitational fielddue to the deformation itself.The initial gravitational potentialUsatisfies the equation V2U=47TGp(7,43a) where G is the gravitational constant and p the initial mass density distribution. The deformation changes the local massdistribution by the amount (7,43b) HencetheincrementU'of thegravitationalpotentialsatisfiestheequation V2U'=-47TG(OPu,+ pe) OX, and the corresponding incremental fieldis Llx'=_oU' ,ox, (7,43c) (7,43d) Densitydiscontinuitiesaretaken intoaccountbyaddingatthesurfacesof di