m o c c. ntifi e ci ds THE THERMONECHRNICS Of orl w w. w NONLINEAR IRREVERSIBLE BEHRVIORS w nloaded from nal use only. Hn Introduction wo aviors Do7. For pers h1 Be2/ ble 06/0 versi3 on nlinear Irre80.82.77.8 Noby of s c ni a h c e m o m er h T e h T m o c c. ntifi e ci s d orl w w. w w nloaded from nal use only. This page is intentionally left blank wo aviors Do7. For pers h1 Be2/ ble 06/0 versi3 on nlinear Irre80.82.77.8 Noby of s c ni a h c e m o m er h T e h T Series Editor: Leon O. Chua m o c c. ntifi THE THERMOMECHHNICS OF e ci s d orl w NONLINEAR IRREVERSIBLE REHRVIORS w. w w nloaded from nal use only. An Introduction wo aviors Do7. For pers h1 Be2/ ble 06/0 versi3 on Nonlinear Irreby 80.82.77.8 Gerard A Maugin of s CNRS & University Pierre & Marie Curie. c ni ha Paris c e m o m er h T e h T V World Scientific Singapore • New Jersey London • Hong Kong Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE m o c c. ntifi e ci s d orl w w. British library Cataloguing-in-Publication Data w A catalogue record for this book is available from the British Library. w nloaded from nal use only. wo aviors Do7. For pers h1 Be2/ ble 06/0 versi3 on THE THERMOMECHANICS OF NONLINEAR IRREVERSIBLE BEHAVIORS Nonlinear Irreby 80.82.77.8 AsCeylloesl cptretyirmgroih ngntihsoc tw ro©e rks nem1ro9evw9ce9nhd a.bo nTyrih cWtioas l ob,b reioln doic nklSvu,c eodinrien tnpegtadi pfr,ih tcswo Ptitothuhceboorlupiestyo hwifni,nr gigm,t tCaerenyoc .npo oPerrtdt embi.n eiLgs sr todieorp. n ra ofnrdyou micn eftodhr iemn P aautniboyln fios shrtmoerr .a ogre b ayn adn rye tmrieeavnasl , of s c ni a h c e m o m For photocopying of material in this volume, please pay a copying fee through the Copyright er Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to h T photocopy is not required from the publisher. e h T ISBN 981-02-3375-2 Printed in Singapore by Uto-Print m o c c. ntifi FOREWORD e ci s d orl w w. w w nloaded from nal use only. mPreo fDf. ouKrr ienasg t icnao 'smm mueneottniinm gpe lriyon j edHceata ifthac o ininnc e1trh9n9ei n2Sg J porasi negpb ho 1oK9k9 e3os tnia nn tdfhr eoarmnmo otBhdreyornw arnme aiascpospn. r omaBcahdoeethd, wo or would have made, such an enterprise impossible. The second reason is aviors Do7. For pers dthea tB rIo gwlaien,t ecdr etaot orr emofa iWn ainve frMieencdhlya nitcesr,m ws aws iaths kJeods ewphhy Khees thina.d Ocon-caeu tLhoorueisd Beh2/1 so few papers in his very long career (he died having reached the age of 96). ble 06/0 His answer was that he did it once with Maurice de Broglie, his own elder versi3 on brother, while he was in his twenties, and they almost went to the point of nlinear Irre80.82.77.8 trarunap l etuaxnrpeae lriyines nitsch eea inrad gb araiotnot.h meIircfh otthhoiedso, rasydo v itnehn atthut,er e b eeraienrflgyle ca1t s9g 2ew0nset,ll elt mhthea ens ,i sthiuteua tanitoeivnoe nrw wtoruiitelhdd sespvueeccnh­ Noby be more touchy with thermodynamics, a field for which it is, as we all know, s of an extremely difficult, perhaps even insuperable task, to find two individuals c ni who fully agree on all its basic aspects and the way it should be taught to a h ec students and researchers alike. In other words, therefore, this book had to be m o written by one author only, even though we cannot forget the immense debt m her that we owe to Joseph Kestin, Paul Germain and others, especially in France. T e My own pragmatic standpoint concerning thermodynamics, or as we prefer h T to say, thermomechanics, is expressed at length in Chapter 1 which has some historical and controversial flavor. Here, however, we simply want to point out a drastic change in the physical view of the World that took place during the last two or three decades, and which somewhat justifies, if there is any need for that, the existence of this book. We are all aware of the arrogant reductionism brought by elementary particle physics in the physical landscape after World V VI Foreword War II. But it is our view, shared by some prominent physicists (e.g. P. W. An­ derson) that condensed-matter physics, through both its object and methods, have had a radical influence that, to some extent, has helped to counteract this type of intellectual imperialism. In particular, if many good fundamental equations are definitely known, it is the solutions to the equations, and not the m co equations themselves, that provide a mathematical description of the physical c. ntifi phenomena. Furthermore, although no one would be so ridiculous as to reject cie any attempt at looking for some ultimate building block of matter — a search s orld which closely parallels the childish ambition to reach the horizon line — and w we certainly gain a better understanding of our World, if not a higher spiritual w. w status, through this quest, we must also recognize that the behavior of a large w nloaded from nal use only. dotaenuerdrrm s sosccn oao)ml.fe p ssPlieemaxer mpoaldseg y gteiornxe gthgr aaatvthpeeeo olbaFfe trcie"oonenmlc eheom fme eptnohitsreaet re ypma"r oospceloineegrtnititsicietee s s Go ofia fs se ftfneofeownct tsBpt oaat rhcbthaiecne l leuaosnrf"d d f,ea (rcPpsth.st o.yW1os id.c TsA h"aininst­ wo haviors Do17. For pers wGioaerssn ngpeoaser.ts i Ocauluolran rcglo ytn hcwiese pldlt iiocrenacp ottifuo rtneh,d eb etbhiyne rgLm .co oDmn.ec ceLhiavanenddi cafsur o, moJfo cthhonem Bpstlaearxrdt eiraersne ,av eafrnisedilb dlP eo. fbG ec.hr oadsves­ versible Be3 on 06/02/ fdbelarettniioldin zo aaft niPod.n D porufe hcmeimsaen y(tfi omsru ebh diasins rdci igpsolpiran)c,e esP ,. s acGca.cl eoBsm roipdfa gnomibeasdne rbv(yfao tari ohdniesse .o p pTeehxripaset ircoiomnmaelbniistnmael s) ,f oianun nda­ nlinear Irre80.82.77.8 Je.i thKeers etilne m(feonrt ahriys- pinaqrtuiicsliet ipvheynseiscsi)s, tsa nwdh oo trheejercst. sTuhchis dmevaeyl ospememen utsn bine athraeb dlea rkto­ Noby ness of macroscopic science, or to applied mathematicians who cannot stand of the view of a thermometer and who presently have the tendency to mistake s nic the proof of existence for a solution. We hope that this book (perhaps nothing a h more than a pamphlet), with all its defects and shortcomings, can bridge the c e m gap between these two communities. But our natural naivete is also bounded o m er by our own pragmatism. h T e A paradox exists in the thermodynamic literature. Practically all the books h T and treatises dealing with irreversible thermodynamics simultaneously claim an unbounded generality and present a very restricted range of applications only. These are essentially linear irreversible thermodynamic processes lim­ ited to the phenomena of viscosity in Newtonian fluids, diffusion-like processes 1But the joke works only in French ("d'effets" and "de faits", practically with the same pronounciation). We could say that physics has become a science of artifacts more than of facts. This is what happened to chemistry in the recent past. Foreword VII such as in mixture theory, and the traditional linear coupling between heat and electricity conductions. Sometimes an application to plasmas (another exam­ ple of mixture) and a remark on electric relaxation are presented as ultimate examples. From this it would seem that only physico-chemists and chemical engineers may get some benefit from a thermodynamical framework. Our at­ m o titude is completely different as this book testifies and the title implies. Rich c c. ntifi in our experience in several fields of physics and engineering science, we do e present irreversible thermodynamics in its realm and splendor. The reader will ci s d find here the application of this inclusive science to both fluids and solids, to orl w viscous as well as viscoelastic, viscoplastic or purely plastic continua, to so­ w. w lutions of polymers and polyelectrolytes, to liquid crystals, to the phenomena w nloaded from nal use only. osrtoaifs m,dc dioacimmss,spa ialgpepeaxp t ialopincnhda e tinicnoor nedm eetpefoon irasnm h ioasncob klelie ld ewssc,aut vrptooeem rptcahroogenpn dfearutgaicacctt toiuborrosne,d , itoeehflse e s c(fotdrlraiiiedccl steua, crntteodr i copm fhr aeeagllsaneexce tattrirtcoai omnhnsay igtsiintnoe ecnrteesi­c,­ wo aviors Do7. For pers esriolcsl hiodnfse )ns sea rnavdne,d- pf iuwnlaisdlele y d,v yrinseiaaomcnti iocasnr -eod,r iw ftfhuee sbi openlrio epsvyaegs, tauetmniomsn a sotucfch hpe hdaa ssb etyh- toorsatehn eserix tihboionbo itkfersod. n itMns. o msTto hodif­s h1 versible Be3 on 06/02/ ettehrcaelh tnfiimieqleud esw sa esn uhdca hwv eea sth racievoden vtneoox tg aihvneeas lgiytesanitsee droa rlt otth hheea otvhreee mroerscy o wouhfr sigcehe tn ofei rnmadlo izadepedpr nlfi ucmnactatiitoohnnessm . iant Tiscehavils­ nlinear Irre80.82.77.8 itse ntshoer parnicael ywsies hoand ftloa tp maya ntiof orledasc,h a are s uufsfeicdi esnyt stdeemgraeteic oalfl yg.e nerality. Vector and Noby The origins of this book are to be found in the lectures which I deliv­ of ered at the Institut Prancais du Petrole in Paris in 1987 in a seminar devoted s c ni to irreversible thermodynamics. I have had several opportunities to improve, a h c polish, and elaborate upon those notes, especially through my yearly lectures e m o on plasticity and fracture at the University of Paris (Pierre et Marie Curie), m er during the writing of a long review paper on internal variables together with h T e Prof. Wolfgang Muschik of T. U. Berlin during my stay at the Wissenschaft- h T skolleg zu Berlin in 1991-1992, during our research on complex fluids together with Prof. Raymonde Drouot in the years 1980-1988, during our research in the field of material forces with Prof. Marcelo Epstein (Calgary, Canada) and Prof. Carmine Trimarco (Pisa, Italy) in the years 1989-1994, during our re­ search on the thermodynamics of nerve-pulse dynamics together with Prof. Jiiri Engelbrecht (Tallinn, Estonia), while writing my lecture notes for a course in Udine (C.I.S.M., Italy, 1992) on the applications of thermodynamics to vin Foreword electromagnetic solids, during the preparation of my previous two books, re­ spectively on the Thermomechanics of Plasticity and Fracture (C.U.P., 1992) and the Theory of Material Inhomogeneities in Elasticity (Chapman, 1993), and obviously in fruitful discussions with colleagues (in particular, D. Lhuillier) and students and co-workers (especially, M. Motogi, M. Sabir, E. Bassiouny m o and C. Dascalu, respectively from Japan, Morocco, Egypt, and Romania) c c. ntifi at the Laboratoire de Modelisation en Mecanique, Universite Pierre et Marie e Curie, Paris, over the last two decades. My early experience with continuum ci s d thermodynamics was deeply influenced by Prof. A. C. Eringen at Princeton orl w University. Private discussions with Joseph Kestin, wherever we could meet w. w in the World, have enriched this view to an extent that the reader can hardly w nloaded from nal use only. wcawsohr nioptcimlenei gavI se ou.h fr aeMt vhsyeih s ohubuneolajduor tsk at.lf lyewl ta fyothsr gaaonctkctseo nmg. op aWtnoy ea lswlt iutshdhey ainn. dTeivnhijidosuy iaasbl slw em htaeitnm tIei o fnteoel td o duaurn rdir negma datenhryes wo aviors Do7. For pers Paris, January 1997. G.A.M. h1 Be2/ ble 06/0 versi3 on nlinear Irre80.82.77.8 Noby of s c ni a h c e m o m er h T e h T m o c c. entifi CONTENTS ci s d orl w w. w w nloaded from nal use only. F1.O IRNETWROODRUD CTION: A post-Duhemian thermodynamics 1v wo aviors Do7. For pers 2. 2T.1H. ETRheMrmOoSdTynAaTmIicC SS ysAteNmDs THERMODYNAMICS 1177 h1 Be2/ 2.2. Thermodynamic States 23 ble 06/0 2.3. Thermostatics (Born-Caratheodory) 24 versi3 on A. Axioms of Thermostatics 24 nlinear Irre80.82.77.8 BDC... STTchhaeel rinmEgvo odolfyu ntTiaoemnm ipoc fe PrRaotetuaerln etS,i ayClssat ermnost; 'sC Tonhteionrueam 333614 Noby of s 3. VARIOUS THERMODYNAMICS 47 c ani 3.1. Preliminary Remarks 47 h c e 3.2. Theory of Irreversible Processes (T.I.P.) 48 m mo A. Axiom of Local State 48 her B. Application to Deformable Material Continua 51 T he C. The Compressible Newtonian Fluid 53 T D. The Linear Viscoelastic Solid 55 E. Finite-Strain Behavior of a Solid 57 F. Rubber-Like Materials 59 G. Anisotropic Elastic Materials 59 H. Onsager-Casimir Symmetry Relations 60 I. Dissipation Potential 62 ix X Contents 3.3. Rational Thermodynamics 63 A. General Features 63 B. Thermoelastic Materials 65 C. Comparison with T.I.P. 71 D. Further Improvements 71 m co 3.4. Extended Thermodynamics 72 c. ntifi 3.5. Thermodynamics with Internal Variables 74 e ci ds 4. THERMODYNAMICS WITH INTERNAL VARIABLES 77 worl 4.1. Nature and Choice of Internal Variables 77 w. w 4.2. Internal Variables and Functional Constitutive Equations 79 w nloaded from nal use only. 44..43.. ANAB..co cFnVo-oeEmrrmqbpuaaaliln l iySSbitrtnaaigutte emmPm reaoennnctdet soosEffe q sttu hhaieeln i bLdL r..iASAut.m.aSSt ..e Ss tates 88880743 wo aviors Do7. For pers 444...675... IPAnoptteperlnnytaiinla glVs Taor.fIi a.DPb.il setssoi paTan.tdIi. oVMn. icrostructure 899952 h1 ble Be06/02/ AB.. IHnitgehrnlya lH Vetaerriaobgleense ooru sI nBteordniaels Degrees of Freedom? 9975 versi3 on 4.8. Internal Variables and Phase Transitions 102 nlinear Irre80.82.77.8 5. 4A.9P. PCLoImCpAarTisIoOn NwSit:h GExEtNenEdeRdA TLh eFrmRoAdMynEamWicOs RK 110074 Noby 5.1. Summary 107 of s 5.2. Convexity of the Energy 110 c ni a 5.3. General Properties of Dissipation Potentials 113 h c me 5.4. Convex Pseudo-Potential of Dissipation 114 o m 5.5. Nonconvex Dissipation Potential 120 er Th 5.6. Reminder of Basic Equations 124 e h A. The Case Solids 124 T B. The Case of Fluids 127 6. VISCOSITY IN COMPLEX FLUIDS 129 6.1. Introductory Remarks 129 6.2. The Notion of Simple (Non-Newtonian) Fluid 131 6.3. Statistical Theory of Polymeric Fluids 133 A. Molecular Models 133