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Graduate Texts in Physics • G T I P RADUATE EXTS N HYSICS Graduate Texts in Physics publishes core learning/teaching material for graduate- and advanced-level undergraduate courses on topics of current and emerging fields within physics,bothpureandapplied.ThesetextbooksservestudentsattheMS-orPhD-leveland their instructors as comprehensive sources of principles, definitions, derivations, exper- imentsandapplications(asrelevant)fortheirmasteryandteaching,respectively.Inter- nationalinscopeandrelevance,thetextbookscorrespondtocoursesyllabisufficientlyto serveasrequiredreading.Theirdidacticstyle,comprehensivenessandcoverageoffun- damental material also make them suitable as introductions or references for scientists entering,orrequiringtimelyknowledgeof,aresearchfield. SeriesEditors ProfessorRichardNeeds CavendishLaboratory JJThomsonAvenue CambridgeCB30HE,UK E-mail:[email protected] ProfessorWilliamT.Rhodes FloridaAtlanticUniversity ImagingTechnologyCenter DepartmentofElectricalEngineering 777GladesRoadSE,Room456 BocaRaton,FL33431,USA E-mail:[email protected] ProfessorH.EugeneStanley BostonUniversity CenterforPolymerStudies DepartmentofPhysics 590CommonwealthAvenue,Room204B Boston,MA02215,USA E-mail:[email protected] Forfurthervolumes: http://www.springer.com/series/8431 Peter Hertel Continuum Physics With 37 Figures 123 Peter Hertel F achbereich Physik Universität Osnabrück Osnabrück, Germany • ISSN 1868-45 1 3 ISSN 1868-45 21 (electronic) ISBN 978-3-642-29499-0 ISBN 978-3-642-295 00-3 (eBook) DOI 10.1007/978-3-642-295 00-3 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012944985 © Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Thesubjectofthissmallbookisavastfield:continuumphysics.Quitenaturally,we cannotgointoallthedetailsofitsmanysubdomains.Instead,we provideabird’s eyeviewofthesubject. Acontinuumismadeupofmaterialpoints.Eachmaterialpointisatinyregion: tinyfromahumanpointofview,wherethenaturalscalearemeters,kilograms,and seconds. The same material point, however, contains very many particles, atoms or molecules, such that the statistical laws for infinitely many particles apply. In particular,externalparameterschangerelativelyslowlysuchthateachmaterialpoint isalwaysveryclosetothermodynamicequilibrium. Thus we deal with fields f D f.t;x/, where f stands for a property of the materialpointlocatedatxattimet,suchasmassdensity,temperature,tension,and soforth. Chapter 1 discusses additiveand transportablequantitiessuch as mass, electric charge, momentum, kinetic, potential and internal energy as well as entropy. For eachsuchquantity,abalanceequationisformulatedandrelatedwithotherbalance equations.Eachquantityisdescribedbydensitiesforcontent,flow,andproduction. The balance equations must be respected in any case; they apply to all kinds of matter in any state. The highlights of this chapter are the precise formulationsof the first and second main laws of thermodynamicsin terms of partial differential equations. Chapter2describesthespecialcontinuumunderdiscussionwithadditionalfield equations,suchasHooke’slaw,Ohm’slaw,opticaloracousticalproperties,theideal gaspressurerelation,andsoforth.Eachclosedsetoffieldequationscharacterizesa subdomainofcontinuumphysics:elasticity,hydrodynamics,aerodynamics,optics, rheology,thermo-electricity,diffusion,heatconduction,andsoforth. Next we study Linear Response Theory (Chap.3). The reaction of a system in thermodynamicequilibriumtorapidvariationsofexternalparametersisworkedout infirstorderperturbationtheory.Linearmaterialequationsandexplicitexpression fortheircoefficientsarearrivedat,suchasthedielectricsusceptibilityasafunction of frequency.Once a tractablemodelforthe system underconsiderationhas been v vi Preface established, these coefficientscan be calculated. Evenmore importantly,there are general relations, such as the Kramers-Kronig or Onsager relations, that can be shown to be true irrespective of a particular model. A highlight of this chapter is thederivationofthefluctuation-dissipationtheorem,whichisveryclosetoaproof ofthesecondmainlawofthermodynamics. In Chap.4, some standard topics of continuum physics are discussed. These examples serve to familiarize the reader with what she or he may already know, but now on a higher level, such as Archimedes’ principle, the first article. Some topics, however,may be new to the reader.For instance, the last article presentsa simplifiedmodelofawhitedwarf.Therearearticlesoncrystaloptics,theelectro- andmagneto-opticeffect,and onoptical activity as wellas dielectric waveguides. Metamaterials and photonic crystals are also introduced. Standard problems of elasticity such as beam-bending and buckling, stress enhancement, and vibrating strings and membranes are discussed. The propagation and attenuation of sound wavesinelastic bodiesandinair aswellassurfaceacousticwavesaredealtwith. There is an article on surface plasmon polaritons, and we discuss Ohm’s law and the Hall effect, introducing the Drude model. The reader will find sections on the laws of Bernoulli, Hagen-PoiseuilleandStokes and on Reynold’snumber.An article on localchemicalreactionsand diffusionintroducesthe interestingsubject ofpatternformation.Fourier’sclassical treatmentof a heatconductionproblemis presented and thermoelastic effects explained (Thomson, Seebeck, Peltier). Two articles illustrate the dissipation-fluctuation theorem: one on the thermal noise of anOhmicresistor,andtheotheronvariousaspectsofBrownianmotion.Diffusion, a standard subject in continuum physics, is nothing else but mass-wise Brownian motion.Amodeloftheearth’satmosphereisalsodiscussed. Chapter 4 could also have been organized into a series of methods of solving partialdifferentialequations.Somecanbesolvedanalyticallyexploitingsymmetry, othersbyrewritingthemintosystemsofordinarydifferentialequations.Thearticles onStokes’lawandonstressconcentrationintroducethepotentialmethod.Various Computationalschemessuchasthefinitedifferencemethodandthefiniteelement methodarealsodiscussed.ThesampleprogramspresentedareinMATLABbecause thissoftwarepackageistailoredtotheneedsofphysics. An appendix covers Fields: their transformation properties, how they are dif- ferentiated and how they can be integrated over paths, areas, and volumes. After all,continuumphysicsisformulatedintermsoffieldsandrelationsbetweenthem. ThereisalsoaGlossary,whichmayserveasasummaryofthisbird’seyeviewof continuumphysics.Itprovidesbriefdescriptionsofthemaintopicsofthisbookand listsmostofthepeoplewhohaveenteredthestage. Thebooksystematicallypresentscontinuumphysicstheory(balanceequations, material equations, linear response) and substantiates the theory with selected examples.Bothpartsareofcomparablesize.Ontheonehand,theexamplescannot be understoodwithout a solid theoretical foundation.On the other hand, physical laws as such are withoutpertinenceif they cannotbe shown to be usefulfor real- worldproblems—fromArchimedes’principletowhitedwarfs. Preface vii This book can be read in practically any order. It is more like a web than an exposition where the next step cannot be taken without going over all the previoussteps.However,thefirstfewsectionsonbalanceequationsshouldberead first and consecutivelybecause they introducea precise system of notation that is neverthelessmoreorlesscompatiblewithacentury-oldtradition. ThisbookonContinuumPhysicsaddressesstudentsofphysics,engineering,and relatedbranchesofsciencewithageneralbackgroundincalculusandbasicphysics. It will also serve graduatestudentsand lecturerswho wantto embed their special fieldinawidercontext. The book presents the physics of continuously distributed matter in a unified way.Wehavebuiltanddevelopedastringenttheoreticalframeworkanddiscussed carefullychosenexamples.Thiswillenablethereadertoassemblethemultitudeof principles,rules,laws,effects,andtheoriesofcontinuumphysics—thepiecesofa jigsawpuzzle—intoacoherentwhole. Osnabru¨ck,Germany PeterHertel February2012 • Contents 1 BalanceEquations ........................................................... 1 1.1 MaterialPoints......................................................... 2 1.1.1 Summary ..................................................... 3 1.2 DensitiesforContent,Flow,andProductionRate ................... 4 1.2.1 Densities...................................................... 4 1.2.2 CurrentDensities............................................. 4 1.2.3 Production.................................................... 5 1.2.4 GenericBalanceEquation................................... 6 1.2.5 Summary ..................................................... 7 1.3 ParticleNumbers,Mass,andElectricCharge........................ 8 1.3.1 ParticleNumbersandCurrents.............................. 8 1.3.2 ChemicalReactions.......................................... 8 1.3.3 Mass .......................................................... 9 1.3.4 ElectricCharge............................................... 10 1.3.5 DigressiononElectromagnetism............................ 11 1.3.6 Summary ..................................................... 12 1.4 ConvectionandConduction........................................... 13 1.4.1 Conduction................................................... 13 1.4.2 TransformationPropertiesofCurrents...................... 14 1.4.3 MaterialTimeDerivative.................................... 15 1.4.4 SpecificQuantities........................................... 16 1.4.5 Summary ..................................................... 17 1.5 LinearandAngularMomentum ...................................... 17 1.5.1 StressandExternalForces................................... 17 1.5.2 AngularMomentum ......................................... 18 1.5.3 NormalandShearForces.................................... 19 1.5.4 Summary ..................................................... 20 1.6 EnergyandtheFirstLawofThermodynamics....................... 20 1.6.1 KineticEnergy ............................................... 20 1.6.2 PotentialEnergy.............................................. 21 1.6.3 InternalEnergy............................................... 22 ix x Contents 1.6.4 DigressiononTimeReversal................................ 23 1.6.5 FirstLawofThermodynamics .............................. 24 1.6.6 Summary ..................................................... 26 1.7 EntropyandtheSecondLawofThermodynamics................... 26 1.7.1 ThermodynamicEquilibrium................................ 27 1.7.2 BalanceEquationforEntropy............................... 27 1.7.3 SecondLawofThermodynamics........................... 29 1.7.4 Summary ..................................................... 30 2 MaterialEquations .......................................................... 31 2.1 FluidMedia............................................................ 32 2.1.1 IncompressibleFluids ....................................... 32 2.1.2 IdealGas...................................................... 33 2.1.3 NewtonianFluidMedium ................................... 34 2.1.4 Hydrodynamics .............................................. 35 2.1.5 Aerodynamics................................................ 36 2.1.6 Summary ..................................................... 37 2.2 SolidMedia............................................................ 37 2.2.1 Strain ......................................................... 38 2.2.2 Hooke’sLaw ................................................. 39 2.2.3 StructuralMechanics ........................................ 40 2.2.4 Elastodynamics............................................... 42 2.2.5 Summary ..................................................... 42 2.3 HeatConduction....................................................... 42 2.3.1 Fourier’sLaw................................................. 43 2.3.2 MoreonHeatConduction................................... 44 2.3.3 HeatEquation................................................ 44 2.3.4 Summary ..................................................... 45 2.4 Diffusion............................................................... 45 2.4.1 DiffusionCurrents ........................................... 46 2.4.2 ChemicalPotential........................................... 46 2.4.3 DiffusionEquation........................................... 47 2.4.4 DigressiononOpenSystems................................ 47 2.4.5 Summary ..................................................... 48 2.5 ChargeTransport ...................................................... 49 2.5.1 TheElectromagneticFieldinMatter........................ 49 2.5.2 ContributionstotheElectricCurrentDensity.............. 51 2.5.3 Ohm’sLaw................................................... 51 2.5.4 Summary ..................................................... 52 2.6 ThermoelectricEffects ................................................ 52 2.6.1 SimultaneousHeatandChargeTransport .................. 53 2.6.2 ForcesandFluxes............................................ 53 2.6.3 KineticCoefficientsforHeatandChargeTransport ....... 54 2.6.4 Summary ..................................................... 55