Computational Fluid and Solid Mechanics pot

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 dierent 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 dierent topics is quite rigorous; an eort 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 eorts 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 indierence 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 indierence 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 [...]... cause it and without introducing into the analysis the behavior of the material Some reference books for this chapter are: (Truesdell & Noll 1965, Truesdell 1966, Malvern 1969, Marsden & Hughes 1983) 2.1 The continuous media and its congurations Continuum mechanics is the branch of mechanics that studies the motion of solids, liquids and gases under the hypothesis of continuous media This hypothesis... momentum and energy In the fth chapter we develop an extensive presentation of constitutive relations for solids and uids, with special focus on the elastoplasticity of metals Finally, in the sixth chapter we develop the variational approach to continuum mechanics, centering our presentation on the principle of virtual work and discussing also the principle of stationary potential energy and the VeubekeHu-Washizu... nonlinearities are introduced in the mathematical model, the more computational resources will be necessary to solve the numerical model and in many cases it may happen that the necessary computational resources are much larger than the available ones, making the analysis impossible Example 1.1 JJJJJ In the analysis of a solid under mechanical and thermal loads some of the nonlinearities that we may encounter... g and are Kronecker deltas 5 ) To prove the rst equality ( ) we start from Eq (2.40) and get R = 1 Xã U = 1 ( ) g (2.41) g The tensor R dened by the above equation is, in the same sense as X , a two-point tensor The components of R are: = 1 ( ) (2.42a) and using an similar equation to (2.28c) we have ( ) = 1 ( ) (2.42b) hence, R = 1 ( ) g (2.42c) g Considering that = X g (2.42d) g and = ( U ã U and. .. tensor analysis Objective and covariant strain and strain rate measures are derived In the third chapter we discuss di erent stress measures that are energy conjugate to the strain rate measures presented in the previous chapter Objective stress rate measures are derived In the fourth chapter we present the Reynolds transport theorem and then we use it to develop Eulerian and Lagrangian formulations... Eqs.(2.1) and (2.3) we obtain the bijective mapping conguration at time and the reference conguration, = ( ) ; = Ê 1 In a regular motion the inverse mapping 1 (Marsden & Hughes 1983), where with continuous derivatives up to the order ular motion agrees with the intuitive concept of interpenetration From Eqs (2.2) and (2.4) we get h i = ( ) = ( ) The mapping the the the the 1 Ô ( ) (2.4) exists and. .. Concentrated masses do not belong to the eld of Continuum Mechanics (therefore, Rational Mechanics is not part of Continuum Mechanics) We dene the density corresponding to the conguration at time as Z = d (2.6) where, : mass of body B, : volume of B in the conguration at time Equation (2.6) incorporates an important postulate of Newtonian mechanics: the mass of a body is constant in time 2.3 Motion... reference conguration and d is the corresponding di erential volume in the spatial conguration Since in a regular motion, a nonzero volume in the reference conguration cannot be ( )= 20 Nonlinear continua collapsed into a point in the spatial conguration and vice versa (Aris 1962), 1 and cannot be zero In a xed Cartesian system we can dene, for a motion , = ( ) (2.32) The vectors (Hildebrand 1976) t = (... modern and rigorous exposition of nonlinear continuum mechanics and even though it does not deal with computational implementations it is intended to provide the basis for them In the second chapter of the book we present a consistent description of the kinematics of the continuous media In that chapter we introduce the concepts of pull-back, push-forward and Lie derivative requiring only from the reader... interpretation of the tensors R, t U and t V In this Section, we will discuss a physical interpretation of the second-order tensors introduced by the polar decomposition The rotation tensor Assuming a motion in which U = g and therefore X = V = g ,we get (2.49) R and considering in the reference conguration, at the point under analysis, two arbitrary vectors dx1 and dx2 we have, in the spatial conguration: . continuous m edia and its configurations Con t inuum mec hanics is the branch of mechanics that studies the motion of solids, liquids and gases under the hypothesis of continuous media.This h ypothesis. 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. Computational Fluid and Solid Mechanics Series Editor: Klaus-Jürgen Bathe Massachusetts Institute of Technology Cambridge,