Table Of Content

Galilean Mechanics and Thermodynamics of Continua Series Editor Noël Challamel Galilean Mechanics and Thermodynamics of Continua Géry de Saxcé First published 2016 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 2016 The rights of Géry de Saxcé 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: 2015957652 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-642-6 Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi Part1.ParticlesandRigidBodies . . . . . . . . . . . . . . . . . . . . . . 1 Chapter1.Galileo’sPrincipleofRelativity . . . . . . . . . . . . . . . . . 3 1.1.Eventsandspace–time . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.Eventcoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1.When? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2.Where? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.Galileantransformations . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1.Uniformstraightmotion . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2.Principleofrelativity . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3.Space–timestructureandvelocityaddition . . . . . . . . . . . . . . 10 1.3.4.Organizingthecalculus . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.5.Abouttheunitsofmeasurement . . . . . . . . . . . . . . . . . . . . 12 1.4.Commentsforexperts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Chapter2.Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.Staticaltorsor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1.Two-dimensionalmodel . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.2.Three-dimensionalmodel. . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.3.Staticaltorsorandtransportlawofthemoment . . . . . . . . . . . 18 2.3.Staticsequilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1.Resultanttorsor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.2.Freebodydiagramandbalanceequation . . . . . . . . . . . . . . . 20 vi GalileanMechanicsandThermodynamicsofContinua 2.3.3.Externalandinternalforces . . . . . . . . . . . . . . . . . . . . . . 23 2.4.Commentsforexperts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter3.DynamicsofParticles . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.Dynamicaltorsor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.1.Transformationlawandinvariants . . . . . . . . . . . . . . . . . . . 27 3.1.2.Boostmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.Rigidbodymotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.1.Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.2.Rigidmotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.Galileangravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.1.Howtomodelthegravitationalforces? . . . . . . . . . . . . . . . . 36 3.3.2.Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.3.Galileangravitationandequationofmotion . . . . . . . . . . . . . 40 3.3.4.Transformationlawsofthegravitationandacceleration. . . . . . . 42 3.4.Newtoniangravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5.Otherforces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5.1.Generalequationofmotion. . . . . . . . . . . . . . . . . . . . . . . 51 3.5.2.Foucault’spendulum . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5.3.Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.6.Commentsforexperts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter4.StaticsofArches,CablesandBeams . . . . . . . . . . . . 57 4.1.Staticsofarches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.1.Modelingofslenderbodies . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.2.Localequilibriumequationsofarches. . . . . . . . . . . . . . . . . 59 4.1.3.Corotationalequilibriumequationsofarches . . . . . . . . . . . . . 62 4.1.4.EquilibriumequationsofarchesinFresnet’smovingframe. . . . . 63 4.2.Staticsofcables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.Staticsoftrussesandbeams . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3.1.Tractionoftrusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3.2.Bendingofbeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Chapter5.DynamicsofRigidBodies . . . . . . . . . . . . . . . . . . . . 75 5.1.Kineticco-torsor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1.1.Lagrangiancoordinates . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1.2.Euleriancoordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1.3.Co-torsor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2.Dynamicaltorsor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2.1.Totalmassandmass-center. . . . . . . . . . . . . . . . . . . . . . . 80 5.2.2.Therigidbodyasaparticle . . . . . . . . . . . . . . . . . . . . . . . 81 5.2.3.Themomentofinertiamatrix . . . . . . . . . . . . . . . . . . . . . 84 Contents vii 5.2.4.Kineticenergyofabody . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3.Generalizedequationsofmotion . . . . . . . . . . . . . . . . . . . . . . 88 5.3.1.Resultanttorsoroftheotherforces . . . . . . . . . . . . . . . . . . 88 5.3.2.Transformationlaws. . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3.3.Equationsofmotionofarigidbody . . . . . . . . . . . . . . . . . . 91 5.4.Motionofafreerigidbodyaroundit . . . . . . . . . . . . . . . . . . . 93 5.5.Motionofarigidbodywithacontactpoint(Lagrange’stop) . . . . . . 95 5.6.Commentsforexperts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Chapter6.CalculusofVariations . . . . . . . . . . . . . . . . . . . . . . . 105 6.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2.ParticlesubjectedtotheGalileangravitation . . . . . . . . . . . . . . . 109 6.2.1.GuessingtheLagrangianexpression. . . . . . . . . . . . . . . . . . 109 6.2.2.ThepotentialsoftheGalileangravitation . . . . . . . . . . . . . . . 110 6.2.3.Transformationlawofthepotentialsofthegravitation . . . . . . . 113 6.2.4.Howtomanageholonomicconstraints?. . . . . . . . . . . . . . . . 116 Chapter7.ElementaryMathematicalTools . . . . . . . . . . . . . . . . 117 7.1.Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.2.Matrixcalculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.2.1.Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.2.2.Rows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.2.3.Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.2.4.Blockmatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.3.VectorcalculusinR3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.4.Linearalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.4.1.Linearspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.4.2.Linearform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.4.3.Linearmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.5.Affinegeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.6.Limitandcontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.7.Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.8.Partialderivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.9.Vectoranalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.9.1.Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.9.2.Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.9.3.VectoranalysisinR3andcurl . . . . . . . . . . . . . . . . . . . . . 139 Part2.ContinuousMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Chapter8.Staticsof3DContinua . . . . . . . . . . . . . . . . . . . . . . 143 8.1.Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.1.1.Stresstensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 viii GalileanMechanicsandThermodynamicsofContinua 8.1.2.Localequilibriumequations . . . . . . . . . . . . . . . . . . . . . . 148 8.2.Torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.2.1.Continuumtorsor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.2.2.Cauchy’scontinuum. . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.3.Invariantsofthestresstensor . . . . . . . . . . . . . . . . . . . . . . . . 155 Chapter9.ElasticityandElementaryTheoryofBeams . . . . . . . . 157 9.1.Strains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 9.2.Internalworkandpower . . . . . . . . . . . . . . . . . . . . . . . . . . 162 9.3.Linearelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 9.3.1.Hooke’slaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 9.3.2.Isotropicmaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.3.3.Elasticityproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 9.4.Elementarytheoryofelastictrussesandbeams . . . . . . . . . . . . . 171 9.4.1.Multiscaleanalysis: fromthebeamtothe elementaryvolume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.4.2.Transverselyrigidbodymodel . . . . . . . . . . . . . . . . . . . . . 176 9.4.3.Calculatingthelocalfields . . . . . . . . . . . . . . . . . . . . . . . 179 9.4.4.Multiscaleanalysis: fromtheelementary volumetothebeam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Chapter10.Dynamicsof3DContinuaandElementary MechanicsofFluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 10.1.Deformationandmotion. . . . . . . . . . . . . . . . . . . . . . . . . . 187 10.2.Flash-back: Galileantensors . . . . . . . . . . . . . . . . . . . . . . . 192 10.3.Dynamicaltorsorofa3Dcontinuum. . . . . . . . . . . . . . . . . . . 196 10.4.Thestress–masstensor. . . . . . . . . . . . . . . . . . . . . . . . . . . 198 10.4.1.Transformationlawandinvariants . . . . . . . . . . . . . . . . . . 198 10.4.2.Boostmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 10.5.Euler’sequationsofmotion . . . . . . . . . . . . . . . . . . . . . . . . 202 10.6.Constitutivelawsindynamics . . . . . . . . . . . . . . . . . . . . . . 206 10.7.Hyperelasticmaterialsandbarotropicfluids . . . . . . . . . . . . . . . 210 Chapter11.DynamicsofContinuaofArbitraryDimensions . . . . . 215 11.1.Modelingthemotionofone-dimensional(1D) materialbodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 11.2.Groupofthe1DlinearGalileantransformations . . . . . . . . . . . . 217 11.3.Torsorofacontinuumofarbitrarydimension . . . . . . . . . . . . . . 219 11.4.Force–masstensorofa1Dmaterialbody . . . . . . . . . . . . . . . . 220 11.5.Fulltorsorofa1Dmaterialbody . . . . . . . . . . . . . . . . . . . . . 222 11.6.Equationsofmotionofacontinuumofarbitrary dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

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This title proposes a unified approach to continuum mechanics which is consistent with Galilean relativity.  Based on the notion of affine tensors, a simple generalization of the classical tensors, this approach allows gathering the usual mechanical entities − mass, energy, force, moment, stresse
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This title proposes a unified approach to continuum mechanics which is consistent with Galilean relativity.  Based on the notion of affine tensors, a simple generalization of the classical tensors, this approach allows gathering the usual mechanical entities − mass, energy, force, moment, stresse

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