Table Of Content

Discrete Mechanics Series Editor Roger Prud’homme Discrete Mechanics Concepts and Applications Jean-Paul Caltagirone First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2019 The rights of Jean-Paul Caltagirone to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2018962477 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-283-0 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii ListofSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi Chapter1.FundamentalPrinciplesofDiscreteMechanics . . . . . . 1 1.1.Definitionsofdiscretemechanics . . . . . . . . . . . . . . . . . . . . . 1 1.1.1.Notionofdiscretespace–time . . . . . . . . . . . . . . . . . . . . . 1 1.1.2.Notionofadiscretemedium . . . . . . . . . . . . . . . . . . . . . . 4 1.2.Propertiesofdiscreteoperators. . . . . . . . . . . . . . . . . . . . . . . 6 1.3.Invarianceundertranslationandrotation . . . . . . . . . . . . . . . . . 9 1.4.Weakequivalenceprinciple . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.Principleofaccumulationofstresses . . . . . . . . . . . . . . . . . . . 13 1.6.Duality-of-actionprinciple . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.7.Physicalcharacteristicsofamedium . . . . . . . . . . . . . . . . . . . 16 1.8.Compositionofvelocitiesandaccelerations . . . . . . . . . . . . . . . 20 1.9.Discretecurvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.10.Axiomsofdiscretemechanics . . . . . . . . . . . . . . . . . . . . . . 28 Chapter2.ConservationofAcceleration . . . . . . . . . . . . . . . . . . 31 2.1.Generalprinciples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.Continuousmemory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.Modelingthecompressionstress. . . . . . . . . . . . . . . . . . . . . . 37 2.3.1.Compressionexperiment . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.2.Modelingthestressinasolid. . . . . . . . . . . . . . . . . . . . . . 39 2.3.3.Modelingthestressinafluid . . . . . . . . . . . . . . . . . . . . . . 39 2.3.4.Compressionwithsmalltimeconstants . . . . . . . . . . . . . . . . 41 vi DiscreteMechanics 2.3.5.Modelingtheaccumulationofthenormalstress . . . . . . . . . . . 42 2.3.6.Theenergyformula,e=mc2 . . . . . . . . . . . . . . . . . . . . 43 2.4.Modelingtherotationstress . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4.1.Couette’sexperiment . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4.2.Behaviorovertime . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.3.Rotationstressinsolids . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4.4.Rotationstressinfluids . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4.5.Stressesinaporousmedium,Darcy’slaw . . . . . . . . . . . . . . 47 2.4.6.Modelingtheaccumulationoftherotationstress. . . . . . . . . . . 48 2.4.7.RotationinCouetteandPoiseuilleflows . . . . . . . . . . . . . . . 49 2.5.Modelingothereffects . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.5.1.Gravitationaleffects . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5.2.Inertialeffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.6.Discreteequationsofmotion . . . . . . . . . . . . . . . . . . . . . . . . 56 2.6.1.Geometricdescription. . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.6.2.Derivationoftheequationsofmotion . . . . . . . . . . . . . . . . . 58 2.6.3.Dissipationofenergy . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.7.Couplingconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.8.Formulationoftheequationsofmotionatadiscontinuity. . . . . . . . 65 2.9.Otherformsoftheequationsofmotion . . . . . . . . . . . . . . . . . . 66 2.9.1.Curlandvectorpotentialformulation . . . . . . . . . . . . . . . . . 67 2.9.2.Conservativeformoftheequationsofmotion . . . . . . . . . . . . 69 2.10.Incompressiblemodelsderivedfromthediscreteformulation. . . . . 70 2.10.1.Kinematicprojectionmethods . . . . . . . . . . . . . . . . . . . . 70 2.10.2.Incompressibilityindiscretemechanics . . . . . . . . . . . . . . . 74 2.11.Consequencesonthedynamicsofthevorticity . . . . . . . . . . . . . 74 Chapter3.ConservationofMass,FluxandEnergy . . . . . . . . . . . 77 3.1.Conservationofmassinahomogeneousmedium . . . . . . . . . . . . 77 3.1.1.Incontinuummechanics . . . . . . . . . . . . . . . . . . . . . . . . 78 3.1.2.Indiscretemechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2.Transportwithinmulticomponentmixtures . . . . . . . . . . . . . . . . 81 3.2.1.Classicalapproach. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.2.2.Discretemodelforthetransportofchemicalspecies . . . . . . . . 84 3.2.3.Equilibriuminabinarymixture . . . . . . . . . . . . . . . . . . . . 86 3.3.Advection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4.Conservationofflux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4.1.Generalremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4.2.Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5.Conservationofenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.5.1.Conservationoftotalenergy . . . . . . . . . . . . . . . . . . . . . . 93 3.5.2.Conservationofkineticenergy . . . . . . . . . . . . . . . . . . . . . 94 3.5.3.Conservationofinternalenergy . . . . . . . . . . . . . . . . . . . . 95 Contents vii 3.5.4.Monotonicallydecreasingkineticenergy . . . . . . . . . . . . . . . 97 3.6.Acompletesystemofequations . . . . . . . . . . . . . . . . . . . . . . 98 3.7.Asimpleheatconductionproblem. . . . . . . . . . . . . . . . . . . . . 99 3.7.1.Caseofanisotropicmaterials . . . . . . . . . . . . . . . . . . . . . . 101 3.8.Phasechange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.8.1.TheStefanproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.8.2.Condensation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Chapter4.PropertiesoftheDiscreteFormulation . . . . . . . . . . . 115 4.1.Fundamentalproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.1.1.Limitationsonthevelocity . . . . . . . . . . . . . . . . . . . . . . . 115 4.1.2.InvertingtheformulasV =∇φandV =∇×ψ . . . . . . . . 118 φ ψ 4.1.3.Materialframe-indifference . . . . . . . . . . . . . . . . . . . . . . 121 4.1.4.Fundamentalinvariants . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.2.Systemofequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.3.Differencesfromcontinuummechanics . . . . . . . . . . . . . . . . . . 126 4.3.1.DifferencesfromtheNavier-Laméequations . . . . . . . . . . . . . 126 4.3.2.DifferencesfromtheNavier-Stokesequations . . . . . . . . . . . . 127 4.3.3.Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.3.4.CompatibilityconditionsfortheNavier-Stokesequations. . . . . . 133 4.4.Examplesofanalyticsolutionsoftheequationsofmotion . . . . . . . 136 4.4.1.Rigidrotationalmotion . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.4.2.PlanarCouetteflow . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.4.3.Poiseuilleflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.4.4.Radialflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.5.Incompressiblemotion . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.5.1.TheGreen-Taylorvortex . . . . . . . . . . . . . . . . . . . . . . . . 145 4.5.2.Lid-drivencavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.6.Compressiblefluidsandperfectfluids . . . . . . . . . . . . . . . . . . . 150 4.6.1.GeneralizedBernoulliequation . . . . . . . . . . . . . . . . . . . . 151 4.6.2.Propagationoflinearwaves . . . . . . . . . . . . . . . . . . . . . . 152 4.6.3.Sodshocktube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.7.Staticsoffluidsandsolids . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.8.Conditionsformodelingarigidsolid . . . . . . . . . . . . . . . . . . . 159 4.9.Flowsinaporousmedium . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.10.Stretchingofspace-timeandHugoniot’stheorem . . . . . . . . . . . 165 Chapter5.Two-PhaseFlows,CapillarityandWetting . . . . . . . . . 169 5.1.Formulationoftheequationsattheinterfaces . . . . . . . . . . . . . . 169 5.1.1.Modelingthecurvature . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.1.2.Formulationoftheequationsofmotion . . . . . . . . . . . . . . . . 174 5.2.Two-phaseflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.2.1.Two-phasePoiseuilleflow . . . . . . . . . . . . . . . . . . . . . . . 179 viii DiscreteMechanics 5.2.2.Sloshingoftwoimmisciblefluids . . . . . . . . . . . . . . . . . . . 181 5.3.Capillarity-dominatedflows . . . . . . . . . . . . . . . . . . . . . . . . 187 5.3.1.TheLaplaceproblem . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.3.2.Oscillatingellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 5.3.3.Marangoni-typeflowinadroplet . . . . . . . . . . . . . . . . . . . 190 5.3.4.Interactingbubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 5.3.5.Simulatingfoaminequilibrium . . . . . . . . . . . . . . . . . . . . 194 5.4.Partialwetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.4.1.Dropletinequilibriumonaplane . . . . . . . . . . . . . . . . . . . 198 5.4.2.Spreadingofadroplet. . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.4.3.Dropletacteduponbygravity . . . . . . . . . . . . . . . . . . . . . 204 5.4.4.Flowswithinalens . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5.4.5.Capillaryascensioninatube . . . . . . . . . . . . . . . . . . . . . . 206 Chapter6.StressesandStrainsinSolids . . . . . . . . . . . . . . . . . 209 6.1.Discretesolidmedium . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.2.Stressesinsolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.2.1.Discreteequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.2.2.Materialframe-indifference . . . . . . . . . . . . . . . . . . . . . . 214 6.2.3.Solidstaticsequations. . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.2.4.Calculatingthedisplacement . . . . . . . . . . . . . . . . . . . . . . 217 6.3.Propertiesofsolidmedia . . . . . . . . . . . . . . . . . . . . . . . . . . 219 6.3.1.Incontinuummechanics . . . . . . . . . . . . . . . . . . . . . . . . 220 6.3.2.Indiscretemechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.4.Boundaryconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 6.5.Rigidmotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 6.6.Validationofthemodelonexamples . . . . . . . . . . . . . . . . . . . 230 6.6.1.Simpleexampleofamonolithicfluid–structureinteraction . . . . . 230 6.6.2.Mechanicalequilibriumofsloshing . . . . . . . . . . . . . . . . . . 233 6.6.3.Beamunderextension. . . . . . . . . . . . . . . . . . . . . . . . . . 235 6.6.4.Multimaterialcompression . . . . . . . . . . . . . . . . . . . . . . . 237 6.6.5.Planarshearing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 6.6.6.Flexingbeam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 6.6.7.Settlingofablockundergravity . . . . . . . . . . . . . . . . . . . . 240 6.6.8.Mechanicalequilibriumofasolidobject . . . . . . . . . . . . . . . 242 6.6.9.Extensiontootherconstitutivelaws . . . . . . . . . . . . . . . . . . 243 6.7.Towardaunificationofsolidandfluidmechanics . . . . . . . . . . . . 246 Chapter7.MultiphysicalExtensions . . . . . . . . . . . . . . . . . . . . 249 7.1.Deflectionoflight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 7.1.1.Descriptionofthephysicalphenomenon . . . . . . . . . . . . . . . 250 7.1.2.DeflectionoflightbytheSuninNewtonianmechanics . . . . . . . 252

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