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This page intentionally left blank MATHEMATICALMODELINGINCONTINUUMMECHANICS SECONDEDITION Temam and Miranville present core topics within the general themes of fluid andsolidmechanics.Thebriskstyleallowsthetexttocoverawiderangeoftop- icsincludingviscousflow,magnetohydrodynamics,atmosphericflows,shock equations, turbulence, nonlinear solid mechanics, solitons, and the nonlinear Schro¨dingerequation. Thissecondeditionwillbeauniqueresourceforthosestudyingcontinuum mechanicsattheadvancedundergraduateandbeginninggraduatelevelwhether in engineering, mathematics, physics, or the applied sciences. Exercises and hints for solutions have been added to the majority of chapters, and the final partonsolidmechanicshasbeensubstantiallyexpanded.Theseadditionshave now made it appropriate for use as a textbook, but it also remains an ideal referencebookforstudentsandanyoneinterestedincontinuummechanics. MATHEMATICAL MODELING IN CONTINUUM MECHANICS SECOND EDITION ROGER TEMAM Universite´Paris-Sud,OrsayandIndianaUniversity ALAIN MIRANVILLE Universite´dePoitiers    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521617239 © Cambridge University Press 2000, 2005 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2005 - ---- eBook (EBL) - --- eBook (EBL) - ---- paperback - --- paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface pageix Afewwordsaboutnotations xi PARTI FUNDAMENTALCONCEPTSINCONTINUUM MECHANICS 1 Describingthemotionofasystem:geometryandkinematics 3 1.1 Deformations 3 1.2 Motionanditsobservation(kinematics) 6 1.3 Descriptionofthemotionofasystem:Eulerianand Lagrangianderivatives 10 1.4 Velocityfieldofarigidbody:helicoidalvectorfields 13 1.5 Differentiationofavolumeintegraldependingonaparameter 18 2 Thefundamentallawofdynamics 24 2.1 Theconceptofmass 24 2.2 Forces 30 2.3 Thefundamentallawofdynamicsanditsfirstconsequences 32 2.4 Applicationtosystemsofmaterialpointsandtorigidbodies 34 2.5 Galileanframes:thefundamentallawofdynamicsexpressed inanon-Galileanframe 38 3 TheCauchystresstensorandthePiola-Kirchhoff tensor.Applications 42 3.1 Hypothesesonthecohesionforces 42 3.2 TheCauchystresstensor 45 3.3 Generalequationsofmotion 48 3.4 Symmetryofthestresstensor 50 3.5 ThePiola-Kirchhofftensor 52 v vi Contents 4 Realandvirtualpowers 57 4.1 Studyofasystemofmaterialpoints 57 4.2 Generalmaterialsystems:rigidifyingvelocities 61 4.3 Virtualpowerofthecohesionforces:thegeneralcase 63 4.4 Realpower:thekineticenergytheorem 67 5 Deformationtensor,deformationratetensor, constitutivelaws 70 5.1 Furtherpropertiesofdeformations 70 5.2 Thedeformationratetensor 75 5.3 Introductiontorheology:theconstitutivelaws 77 5.4 Appendix.Changeofvariableinasurfaceintegral 87 6 Energyequationsandshockequations 90 6.1 Heatandenergy 90 6.2 ShocksandtheRankine–Hugoniotrelations 95 PARTII PHYSICSOFFLUIDS 7 GeneralpropertiesofNewtonianfluids 103 7.1 Generalequationsoffluidmechanics 103 7.2 Staticsoffluids 109 7.3 Remarkontheenergyofafluid 114 8 Flowsofinviscidfluids 116 8.1 Generaltheorems 116 8.2 Planeirrotationalflows 120 8.3 Transsonicflows 130 8.4 Linearaccoustics 134 9 Viscousfluidsandthermohydraulics 137 9.1 Equationsofviscousincompressiblefluids 137 9.2 Simpleflowsofviscousincompressiblefluids 138 9.3 Thermohydraulics 144 9.4 Equationsinnondimensionalform:similarities 146 9.5 Notionsofstabilityandturbulence 148 9.6 Notionofboundarylayer 152 10 Magnetohydrodynamicsandinertialconfinementofplasmas 158 10.1 TheMaxwellequationsandelectromagnetism 159 10.2 Magnetohydrodynamics 163 10.3 TheTokamakmachine 165 11 Combustion 172 11.1 Equationsformixturesoffluids 172 Contents vii 11.2 Equationsofchemicalkinetics 174 11.3 Theequationsofcombustion 176 11.4 Stefan–Maxwellequations 178 11.5 Asimplifiedproblem:thetwo-speciesmodel 181 12 Equationsoftheatmosphereandoftheocean 185 12.1 Preliminaries 186 12.2 Primitiveequationsoftheatmosphere 188 12.3 Primitiveequationsoftheocean 192 12.4 Chemistryoftheatmosphereandtheocean 193 Appendix.Thedifferentialoperatorsinsphericalcoordinates 195 PARTIII SOLIDMECHANICS 13 Thegeneralequationsoflinearelasticity 201 13.1 Backtothestress–strainlawoflinearelasticity:the elasticitycoefficientsofamaterial 201 13.2 Boundaryvalueproblemsinlinearelasticity:the linearizationprinciple 203 13.3 Otherequations 208 13.4 Thelimitofelasticitycriteria 211 14 Classicalproblemsofelastostatics 215 14.1 Longitudinaltraction–compressionofacylindricalbar 215 14.2 Uniformcompressionofanarbitrarybody 218 14.3 Equilibriumofasphericalcontainersubjectedto externalandinternalpressures 219 14.4 Deformationofaverticalcylindricalbodyunderthe actionofitsweight 223 14.5 Simplebendingofacylindricalbeam 225 14.6 Torsionofcylindricalshafts 229 14.7 TheSaint-Venantprinciple 233 15 Energytheorems,duality,andvariationalformulations 235 15.1 Elasticenergyofamaterial 235 15.2 Duality–generalization 237 15.3 Theenergytheorems 240 15.4 Variationalformulations 243 15.5 Virtualpowertheoremandvariationalformulations 246 16 Introductiontononlinearconstitutivelawsand tohomogenization 248 16.1 Nonlinearconstitutivelaws(nonlinearelasticity) 249 viii Contents 16.2 Nonlinearelasticitywithathreshold (Henky’selastoplasticmodel) 251 16.3 Nonconvexenergyfunctions 253 16.4 Compositematerials:theproblemofhomogenization 255 17 Nonlinearelasticityandanapplicationtobiomechanics 259 17.1 Theequationsofnonlinearelasticity 259 17.2 Boundaryconditions–boundaryvalueproblems 262 17.3 Hyperelasticmaterials 264 17.4 Hyperelasticmaterialsinbiomechanics 266 PARTIV INTRODUCTIONTOWAVEPHENOMENA 18 Linearwaveequationsinmechanics 271 18.1 Returningtotheequationsoflinearacousticsand oflinearelasticity 271 18.2 Solutionoftheone-dimensionalwaveequation 275 18.3 Normalmodes 276 18.4 Solutionofthewaveequation 281 18.5 Superpositionofwaves,beats,andpacketsofwaves 285 19 Thesolitonequation:theKorteweg–deVriesequation 289 19.1 Water-waveequations 290 19.2 Simplifiedformofthewater-waveequations 292 19.3 TheKorteweg–deVriesequation 295 19.4 ThesolitonsolutionsoftheKdVequation 299 20 ThenonlinearSchro¨dingerequation 303 20.1 Maxwellequationsforpolarizedmedia 304 20.2 Equationsoftheelectricfield:thelinearcase 306 20.3 Generalcase 309 20.4 ThenonlinearSchro¨dingerequation 313 20.5 SolitonsolutionsoftheNLSequation 316 Appendix. Thepartialdifferentialequationsofmechanics 319 Hintsfortheexercises 321 References 332 Index 337