Computational Fluid and Solid Mechanics Series Editor: Klaus-Jürgen Bathe Massachusetts Institute of Technology Cambridge, MA, USA Advisors: Franco Brezzi University of Pavia Pavia, Italy Olivier Pironneau Université Pierre et Marie Curie Paris, France Available Volumes D. Chapelle, K.J. Bathe The Finite Element Analysis of Shells - Fundamentals, 2003 D. Drikakis, W. Rider High-Resolution Methods for Incompressible and Low-Speed Flows 2005 M. Kojic, K.J. Bathe Inelastic Analysis of Solids and Structures 2005 E.N. Dvorkin, M.B. Goldschmit Nonlinear Continua 2005 Eduardo N. Dvorkin · Marcela B. Goldschmit Nonlinear Continua With 30 Figures Authors: Eduardo N. Dvorkin, Ph.D. Marcela B. Goldschmit, Dr. Eng. Engineering School University of Buenos Aires and Center for Industrial Research TENARIS Dr. Simini 250 B2804MHA Campana Argentina ISBN-10 3-540-24985-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-24985-6 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifi cally the rights of translation, reprinting, reuse of illustrations, recitation, broad- casting, reproduction on microfi lm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Sprin- ger. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specifi c statement, that such names are exempt from the relevant pro- tective laws and regulations and therefore free for general use. Typesetting: Dataconversion by authors Final processing by PTP-Berlin Protago-T E X-Production GmbH, Germany Cover-Design: deblik, Berlin Printed on acid-free paper 62/3141/Yu – 5 4 3 2 1 0 Library of Congress Control Number: 2005929275 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 dierent configurations; stress and strain measures; objective stress rate and strain rate measures; balance principles; constitutive relations, with emphasis on elasto-plasticity of metals a nd 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 t he dierent topics is quite rigorous; an eort 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 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 eorts and support. Last but not least I w ant 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 A ires and to my research assistants at the Center for Industrial Research o f FUDETEC for their willingness and e ort. I want to recognize the permanent support of my mother Esther, of my sister Mónica and of my friends and collea gues. Marcela B. Goldschmit Conten ts 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 Assessment ofthenumericalresults 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 Mass ofthecontinuousmedia 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 Theright polar decomposition 22 2.7.3 TheFingerdeformationtensor 25 2.7.4 Theleftpolardecomposition 25 2.7.5 Physical interpretation of the tensors w R > w U and w 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 XContents 2.8.5 TheHenckydeformationtensor 35 2.9 Represent ation of spatial tensors in the reference configuration(“pull-back”) 36 2.9.1 Pull-back of vectorcomponents 36 2.9.2 Pull-back of tensorcomponents 40 2.10 Tensors in the spatial configuration from representations in the reference configuration(“push-forward”) 42 2.11 Pull-back/push-forward relations betweenstrainmeasures 43 2.12 Objectivity 44 2.12.1 Referenceframe andisometrictransformations 45 2.12.2 Objectivity or material-frame indierence 47 2.12.3 Covariance 49 2.13 Strainrates 50 2.13.1 The velocity gradienttensor 50 2.13.2 The Eulerian strain rate tensor and the spin (vorticity) tensor 51 2.13.3 Relations between di erentratetensors 53 2.14 TheLiederivative 56 2.14.1 ObjectiveratesandLiederivatives 58 2.15 Compatibility 61 3 Stress Tensor ============================================== 67 3.1 Externalforces 67 3.2 TheCauchystresstensor 69 3.2.1 Symmetry of the Cauchy stress tensor (Cauchy Theorem) 71 3.3 Conjugatestress/strain ratemeasures 72 3.3.1 The Kirchho stresstensor 74 3.3.2 The first Piola-Kirchho stresstensor 74 3.3.3 The second Piola-Kirchho 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’transport theorem 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 XI 4.3.1 Eulerian (spatial) formulation of the balance of momentumprinciple 96 4.3.2 Lagrangian (material) formulation o f 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 (FirstLaw ofThermodynamics) 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 Fundamentals forformulating constitutiverelations 116 5.1.1 Principleofequipresence 116 5.1.2 Principle of material-frame indierence 116 5.1.3 Application to the case of a continuum theory restrictedtomechanicalvariables 116 5.2 Constitutive relations in solid mech anics: purely mechanical formulations 120 5.2.1 Hyperelasticmaterial models 121 5.2.2 Asimple hyperelasticmaterial model 122 5.2.3 Other simplehyperelasticmaterial models 128 5.2.4 Ogden hyperelastic materialmodels 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 Theisotropicthermoelastic constitutive model 167 5.3.2 Athermoelastoplasticconstitutive model 170 5.4 Viscoplasticity 176 5.5 Newtonian fluids 180 5.5.1 Theno-slipcondition 181 6 Variational methods =======================================183 6.1 ThePrinciple ofVirtual Work 183 6.2 The Principle of Virtual Work in geometrically nonlinear problems 186 6.2.1 IncrementalFormulations 189 6.3 ThePrinciple ofVirtual Power 194 6.4 ThePrinciple ofStationaryPotentialEnergy 195 6.5 Kinematicconstraints 207 6.6 Veubeke-Hu-Washizuvariationalprinciples 209 6.6.1 KinematicconstraintsviatheV-H-Wprinciples 209 6.6.2 ConstitutiveconstraintsviatheV-H-W principles 211 [...]... tensor analysis 213 A .1 Coordinates transformation 213 A .1. 1 Contravariant transformation rule 214 A .1. 2 Covariant transformation rule 215 A.2 Vectors 215 A.2 .1 Base vectors 216 A.2.2 Covariant base vectors 216 A.2.3 Contravariant... -configuration ) (2 .10 ) assuming that the time derivatives in Eq (2 .10 ) exist The material velocity vector is defined in the spatial configuration (see Fig 2.2) We can have, alternatively, the following functional dependencies: v = v( v = v( ) ) (2 .11 a) (2 .11 b) Equation (2 .11 a) corresponds to a Lagrangian (material) description of motion, while Eq (2 .11 b) corresponds to a Eulerian (spatial) description of ... of the particle to the spatial configuration is, 1 See Appendix (2.7) )e (2.8) from the reference configuration 10 Nonlinear continua Fig 2 .1 Motion of continuous body u( ) = x( ) x( ) (2.9a) and the Cartesian components of this vector are, ( ) = ( ) ( ) (2.9b) 2.3.2 Velocities and accelerations During the motion is , the material velocity of a particle v( ) = x( ) = u( in the -configuration ) (2 .10 ) assuming... the Appendix we present a review of this topic 1. 4 Notation Throughout the book we shall use the summation convention; that is to say, in a Cartesian coordinate system = 3 X =1 = 3 X =1 2 3 =1 and in a general curvilinear system = 3 X =1 = 3 X =1 2 3 =1 Also, our notation is compatible with the notation introduced in continuum mechanics by Bathe (Bathe 19 96) We shall define all notation at the point... study 1. 1 .1 Observation of physical phenomena This is a crucial step that conditions the next three Making an educated observation of a physical phenomenon means establishing a set of concepts and relations that will govern the further development of the mathematical model At this stage we also need to decide on the quantitative output that we shall require from the model 2 Nonlinear continua 1. 1.2... Laplacian of a tensor 239 A .10 Rotor of a tensor 240 A .11 The Riemann-Christo el tensor 240 A .12 The Bianchi identity 243 A .13 Physical components 244 References 247 Index 255 1 Introduction The quantitative description of the deformation... value of the time coordinate We can establish a bijective mapping (Oden 19 79) between each point of space occupied by a material particle at and an arbitrary curvilinear coordinate system { = 1 2 3} The fact that at each instant the set { } defines one and only one particle implies that in a continuum medium, di erent material particles 1 The requirement of an open subset is introduced in order to eliminate... define an arbitrary curvilinear coordinate system { = 1 2 3}: the material coordinates For the reference configuration Eqs.(2 .1) are = ( ) ; = ( ) (2.3) between the From Eqs.(2 .1) and (2.3) we obtain the bijective mapping configuration at time and the reference configuration, = ( ) ; = £ 1 In a regular motion the inverse mapping 1 (Marsden & Hughes 19 83), where with continuous derivatives up to the order... model comparing its predictions with experimental observations 1. 2 Linear and nonlinear mathematical models When deriving the PDE system that constitutes the mathematical model of a physical phenomenon there are normally a number of nonlinear terms that appear in those equations Considering always all the nonlinear terms, even if 1. 2 Linear and nonlinear mathematical models 3 their influence is negligible... priori unknown to the analyst • Material nonlinearities: elastoplastic materials (e.g metals); creep behavior of metals in high-temperature environments; nonlinear elastic materials (e.g polymers); fracturing materials (e.g concrete); etc JJJJJ 4 Nonlinear continua Example 1. 2 JJJJJ In the analysis of a fluid flow under mechanical and thermal loads some of the nonlinearities that we may encounter when . Goldschmit Conten ts 1 Introduction =============================================== 1 1 .1 Quantificationofphysicalphenomena 1 1 .1. 1 Observationofphysicalphenomena 1 1 .1. 2 Mathematicalmodel 2 1. 1.3 Numericalmodel. ===================================== =11 5 5 .1 Fundamentals forformulating constitutiverelations 11 6 5 .1. 1 Principleofequipresence 11 6 5 .1. 2 Principle of material-frame indierence 11 6 5 .1. 3 Application to the. ============================ 213 A .1 Coordinatestransformation 213 A .1. 1 Contravarianttransformationrule 214 A .1. 2 Covarianttransformationrule 215 A.2 Vectors 215 A.2 .1 Basevectors 216 A.2.2 Covariantbase vectors 216 A.2.3