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Nonlinear Continua Eduardo N. Dvorkin Marcela B. Goldschmit Nonlinear Continua SPIN 10996775 — Monograph — May 6, 2005 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo To the Argentine system of public education Preface This book develops a modern presentation of Continuum Mechanics, oriented towards numerical applications in the fields of nonlinear analysis of solids, structures and fluids. Kinematics of the continuum deformation, including pull-back/push-forward transformations between different configurations; stress and strain measures; objective stress rate and strain rate measures; balance principles; constitutive relations, with emphasis on elasto-plasticity of metals and variational princi- ples are developed using general curvilinear coordinates. Being tensor analysis the indispensable tool for the development of the continuum theory in general coordinates, in the appendix an overview of ten- soranalysisisalsopresented. Embedded in the theoretical presentation, application examples are devel- oped to deepen the understanding of the discussed concepts. Even though the mathematical presentation of the different topics is quite rigorous; an effort is made to link formal developments with engineering phys- ical intuition. This book is based on two graduate courses that the authors teach at the Engineering School of the University of Buenos Aires and it is intended for graduate engineering students majoring in mechanics and for researchers in the fields of applied mechanics and numerical methods. VIII Preface Preface IX I am grateful to Klaus-Jürgen Bathe for introducing me to Computational Mechanics, for his enthusiasm, for his encouragement to undertake challenges and for his friendship. I am also grateful to my colleagues, to my past and present students at the University of Buenos Aires and to my past and present research assistants at the Center for Industrial Research of FUDETEC because I have always learnt from them. I want to thank Dr. Manuel Sadosky for inspiring many generations of Argentine scientists. I am very grateful to my late father Israel and to my mother Raquel for their efforts and support. Last but not least I want to thank my dear daughters Cora and Julia, my wife Elena and my friends (the best) for their continuous support. Eduardo N. Dvorkin I would like to thank Professors Eduardo Dvorkin and Sergio Idelsohn for introducing me to Computational Mechanics. I am also grateful to my students at the University of Buenos Aires and to my research assistants at the Center for Industrial Research of FUDETEC for their willingness and effort. I want to recognize the permanent support of my mother Esther, of my sister Mónica and of my friends and colleagues. Marcela B. Goldschmit Contents 1 Introduction ............................................... 1 1.1 Quantificationofphysicalphenomena...................... 1 1.1.1 Observationofphysicalphenomena.................. 1 1.1.2 Mathematicalmodel............................... 2 1.1.3 Numericalmodel.................................. 2 1.1.4 Assessmentofthenumericalresults.................. 2 1.2 Linearandnonlinearmathematicalmodels ................. 2 1.3 Theaimsofthisbook.................................... 4 1.4 Notation............................................... 5 2 Kinematics of the continuous media ....................... 7 2.1 The continuous media and its configurations ............... 7 2.2 Massofthecontinuousmedia............................. 9 2.3 Motionofcontinuousbodies.............................. 9 2.3.1 Displacements .................................... 9 2.3.2 Velocities and accelerations ......................... 10 2.4 Material and spatial derivatives of a tensor field............. 12 2.5 Convectedcoordinates ................................... 13 2.6 Thedeformationgradienttensor........................... 13 2.7 Thepolardecomposition................................. 21 2.7.1 TheGreendeformationtensor ...................... 21 2.7.2 Therightpolardecomposition...................... 22 2.7.3 TheFingerdeformationtensor ..................... 25 2.7.4 Theleftpolardecomposition........................ 25 2.7.5 Physical interpretation of the tensors t R , t U and t V 26 2.7.6 Numericalalgorithmforthepolardecomposition......◦ ◦ ◦ 28 2.8 Strainmeasures......................................... 33 2.8.1 TheGreendeformationtensor ...................... 33 2.8.2 TheFingerdeformationtensor...................... 33 2.8.3 TheGreen-Lagrangedeformationtensor.............. 34 2.8.4 TheAlmansideformationtensor .................... 35 XII Contents 2.8.5 TheHenckydeformationtensor..................... 35 2.9 Representation of spatial tensors in the reference configuration(“pull-back”) ............................... 36 2.9.1 Pull-backofvectorcomponents ..................... 36 2.9.2 Pull-backoftensorcomponents ..................... 40 2.10 Tensors in the spatial configuration from representations in the reference configuration(“push-forward”)................ 42 2.11Pull-back/push-forwardrelationsbetweenstrainmeasures.... 43 2.12Objectivity............................................. 44 2.12.1Referenceframeandisometrictransformations ....... 45 2.12.2 Objectivity or material-frame indifference ............ 47 2.12.3Covariance ....................................... 49 2.13Strainrates............................................. 50 2.13.1Thevelocitygradienttensor........................ 50 2.13.2 The Eulerian strain rate tensor and the spin (vorticity) tensor............................................ 51 2.13.3 Relations between differentratetensors.............. 53 2.14TheLiederivative....................................... 56 2.14.1ObjectiveratesandLiederivatives .................. 58 2.15 Compatibility ........................................... 61 3StressTensor.............................................. 67 3.1 Externalforces.......................................... 67 3.2 TheCauchystresstensor................................. 69 3.2.1 Symmetry of the Cauchy stress tensor (Cauchy Theorem) ........................................ 71 3.3 Conjugatestress/strainratemeasures...................... 72 3.3.1 The Kirchhoff stresstensor......................... 74 3.3.2 The first Piola-Kirchhoff stresstensor ............... 74 3.3.3 The second Piola-Kirchhoff stresstensor ............. 76 3.3.4 A stress tensor energy conjugate to the time derivative oftheHenckystraintensor ........................ 79 3.4 Objectivestressrates.................................... 81 4 Balance principles ......................................... 85 4.1 Reynolds’transporttheorem.............................. 85 4.1.1 GeneralizedReynolds’transporttheorem............. 88 4.1.2 Thetransporttheoremanddiscontinuitysurfaces ..... 90 4.2 Mass-conservationprinciple............................... 93 4.2.1 Eulerian (spatial) formulation of the mass-conservation principle ......................................... 93 4.2.2 Lagrangian (material) formulation of the mass conservationprinciple.............................. 95 4.3 Balanceofmomentumprinciple(Equilibrium) .............. 95 Contents XIII 4.3.1 Eulerian (spatial) formulation of the balance of momentumprinciple............................... 96 4.3.2 Lagrangian (material) formulation of the balance of momentumprinciple...............................103 4.4 Balanceofmomentofmomentumprinciple(Equilibrium) ....105 4.4.1 Eulerian (spatial) formulation of the balance of momentofmomentumprinciple ....................105 4.4.2 Symmetry of Eulerian and Lagrangian stress measures 107 4.5 Energybalance(FirstLawofThermodynamics) ............109 4.5.1 Eulerian (spatial) formulation of the energy balance . 109 4.5.2 Lagrangian (material) formulation of the energy balance112 5 Constitutive relations ......................................115 5.1 Fundamentalsforformulatingconstitutiverelations..........116 5.1.1 Principleofequipresence...........................116 5.1.2 Principle of material-frame indifference ..............116 5.1.3 Application to the case of a continuum theory restrictedtomechanicalvariables....................116 5.2 Constitutive relations in solid mechanics: purely mechanical formulations............................................120 5.2.1 Hyperelasticmaterialmodels .......................121 5.2.2 Asimplehyperelasticmaterialmodel................122 5.2.3 Othersimplehyperelasticmaterialmodels............128 5.2.4 Ogdenhyperelasticmaterialmodels..................129 5.2.5 Elastoplastic material model under infinitesimal strains 135 5.2.6 Elastoplastic material model under finitestrains ......155 5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations............................................167 5.3.1 Theisotropicthermoelasticconstitutivemodel .......167 5.3.2 Athermoelastoplasticconstitutivemodel ............170 5.4 Viscoplasticity..........................................176 5.5 Newtonian fluids........................................180 5.5.1 Theno-slipcondition..............................181 6 Variational methods .......................................183 6.1 ThePrincipleofVirtualWork ............................183 6.2 The Principle of Virtual Work in geometrically nonlinear problems ..............................................186 6.2.1 IncrementalFormulations ..........................189 6.3 ThePrincipleofVirtualPower............................194 6.4 ThePrincipleofStationaryPotentialEnergy ...............195 6.5 Kinematicconstraints....................................207 6.6 Veubeke-Hu-Washizuvariationalprinciples
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