Mathematical modeling in continuum mechanics

354 Pages·2005·1.113 MB·English

by  Roger Temam, Alain Miranville

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
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Published in the United States of America by Cambridge University Press, New York
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First published in print format 2005
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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
Preface
Thisbookisanextendedversionofacourseoncontinuummechanicstaught
bytheauthorstojuniorgraduatestudentsinmathematics.Besidesathorough
descriptionofthefundamentalpartsofcontinuummechanics,itcontainsram-
ificationsinanumberofadjacentsubjectssuchasmagnetohydrodynamics,
combustion,geophysicalfluiddynamics,andlinearandnonlinearwaves.As
is,thebookshouldappealtoabroadaudience:mathematicians(studentsand
researchers) interested in an introduction to these subjects, engineers, and
scientists.
This book can be described as an “interfacial” book: interfaces between
mathematicsandanumberofimportantareasofsciences.Itcanalsobede-
scribed by what it is not: it is not a book of mathematics: the mathematical
language is simple, only the basic tools of calculus and linear algebra are
needed.Thisbookisnotatreatiseofcontinuummechanics:althoughitcon-
tainsathoroughbutconcisedescriptionofmanysubjects,itleavesasidemany
developmentswhicharefundamentalbutnotneededinpracticalapplications
andutilizationsofmechanics,e.g.,theintrinsic–frameinvariance–character
ofcertainquantitiesorthecoherenceofcertaindefinitions.Thereaderinter-
estedbytheseissuesisreferredtothemanyexcellentmechanicsbookswhich
areavailable,suchasthosequotedinthelistofreferencestoPartI.Finally,by
itssizelimitations,thisbookcannotbeencyclopedic,andmanychoiceshave
beenmadeforthecontent;anumberofsubjectsintroducedinthisbookcan
bedevelopedthemselvesintoafullbook.Allinall,webelievethatthisbook,
benefitingfromprolongedeffortsandteachingexperienceoftheauthors,can
beveryusefultoscientistswhowanttoreducethegapbetweenmathematics
andsciences,agapusuallyduetothelanguagebarrierandthedifferencesin
thinkingandreasoning.
Thecoreofthebookcontainsthefundamentalpartsofcontinuummecha-
nics:descriptionofthemotionofacontinuousbody,thefundamentallawof
ix
x Preface
dynamics,theCauchyandthePiola-Kirchhoffstresstensors,theconstitutive
laws,internalenergyandthefirstprincipleofthermodynamics,shocksandthe
Rankine–Hugoniotrelations,anintroductiontofluidmechanicsforinviscid
and viscous Newtonian fluids, an introduction to linear elasticity and the
variational principles in linear elasticity, and an introduction to nonlinear
elasticity.
Besides the core of continuum mechanics, this book also contains more
orlessdetailedintroductionstoseveralimportantrelatedfieldsthatcouldbe
themselvesthesubjectsofseparatebooks:magnetohydrodynamics,combus-
tion,geophysicalfluiddynamics,vibrations,linearacoustics,andnonlinear
wavesandsolitonsinthecontextoftheKorteweg–deVriesandthenonlinear
Schro¨dingerequations.Thewholebookissuitableforaone-yearcourseatthe
advancedundergraduateorbeginninggraduatelevel.Partsofitaresuitable
foraone-semestercourseeitheronthefundamentalsofcontinuummechanics
oronacombinationofselectedtopics.
Thissecondeditionofthebookhasbeenaugmentedbytheintroductionof
exercisesandhintsatsolutionsmakingitmoresuitableforclassutilization,by
anewchapteronnonlinearelasticity,andbyseveraladditionsandcorrections
suggestedbythereadersofthefirstedition.Inparticularithasbenefitedfrom
the comments of the anonymous and non-anonymous reviewers of the first
edition, especially J. Dunwoody and J.J. Telega. The authors want also to
thankP.G.Ciarletforhiscomments;thenewchapteronnonlinearelasticity
borrowed very much from his classical book on the subject. Finally they
gratefullyacknowledgeessentialhelpintheproductionofthevolumefrom
TeresaBunge,JacquesLaminie,EricSimonnet,andDjokoWirosoetisno.
RogerTemam
AlainMiranville
June2004