Table Of Content

THE FOUNDATIONS OF MECHANICS AND THERMODYNAMICS SELECTED PAPERS BY WNOLL WITH A PREFACE BY C. TRUESDELL SPRINGER-VERLAG BERLIN HEIDELBERG NEW YORK 1974 WALTER NOLL Professor of Mathematics Carnegie-Mellon University, Pittsburgh, Pennsylvania, USA CLIFFORD AMBROSE TRUESDELL, III Professor of Rational Mechanics The Johns Hopkins University, Baltimore, Maryland, USA ISBN-13: 978-3-642-65819-8 e-ISBN-13: 978-3-642-65817-4 DOl: 10.1007/978-3-642-65817-4 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under§ 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1974. Library of Congress Catalog Card Number 74-1651. Softcover reprint of the hardcover 1s t edition 1974 Prd'ace WALTER NOLL'S influence upon research into the foundations of mechanics and thermodynamics in the past twenty years is plain, everywhere acknowledged. Less obvious is the wide effect his writings have exerted upon those who apply mechanics to special cases, but it is witnessed by the now common use of terms, concepts, and styles of argument he introduced, use sometimes by young engineers who have learnt them in some recent textbook and hence take them for granted, often with no idea whence they come. The purpose of this volume of reprints is to put into general hands those of NOLL'S works which presently promise broadest and most fertile service to stu dents of the thermomechanics of deformable bodies. This purpose explains the selection of the contents, about half of the pages he has published in periodicals or proceedings. The complete list of his works, which follows this preface, serves also as Table, the articles reprinted here being indicated by their respective page numbers at the right. Had influence already manifest been the basis of the choice, it would have dictated inclusion of No.2, which was NOLL'S thesis, but by now that work is in part obsolete, in part available through the intermediacy of a dozen books and a hundred papers. As a single example, I may adduce the fact that in Volume 10f A Supplement to the Oxford English Dictionary, 1972, that thesis is the earliest source quoted in the article "constitutive equation", although, with a lapse of accuracy scarcely to be expected, NOLL'S name is misspelled there. Equally certainly, on the basis of influence I should have had to exclude his latest devel opment of the foundations, presented in No. 35, since its impact is not yet widely felt. In the past NOLL'S major works have often seemed forbiddingly abstract and remote at first sight, and five to ten years have gone by before they came to be widely understood and applied. Had quality been the criterion, it would have forbidden me to exclude No.3, his paper on statistical mechanics, or No. 33, his study of GIBBS' phase rule, but their subjects are peripheral to the field of this volume. Instead, the reader will find below two largely expository papers, Nos. 24 and 25, which present in typical simplicity and elegance propositions in algebra and geometry now frequently applied in continuum mechanics. This preface is no place for an obituary of a man not yet fifty and in full power of work, but I will record the circumstances in which NOLL began to study the foundations of mechanics. Students well trained in mathematics, as mathematicians understand that discipline, rarely take any interest in mechanics. Students of mechanics, all too often aping their teachers, neglect the prime lesson the history of that discipline affords, namely, that "the paradise of the mathematical sciences" flowers and bears its best fruit when mathematicians cultivate it with their most powerful tools - indeed, design those tools for that purpose. After the war the relict V Preface German scholars, against odds now not only forgotten but also hard to imagine, were striving to revivify the life of the mind which the mental and physical barbarity preached and practised by the -isms and -acies of 1933-1946 had all but eradicated. Thinking that among the disciples of these elders, restorers rather than progressives, I might find a student or two who would wish to master new mathematics but grasp it and use it with the wholeness of earlier times, in 1952 I wrote to Mr. HAMEL, one of the few then remaining mathematicians from the classical mould, to ask him to name some young men fit to study for the doc torate in The Graduate Institute for Applied Mathematics at Indiana University, flourishing at that time though soon to be destroyed by the jealous ambition of the local, stereotyped pure. Having just retired from the Technische Universitat in Charlottenburg, he passed my inquiry on to Mr. SZABO, in whose institute there NOLL was then an assistant. Although Mr. NOLL informs me that he had attended only one course by HAMEL, and that "there were very few students, and none of us understood what he was talking about", nevertheless I like to think of WALTER NOLL as our link with the great Gottingen school, with the HILBERT who wrote: What a vital nerve would be cut off from mathematics by rooting out geometry and mathe matical physics! On the contrary, I think that wherever mathematical ideas come up, whether from the theory of knowledge or in geometry, or out of the theories of natural science, mathe matics ought to investigate the principles underlying these ideas and by means of a simple and complete system of axioms establish them in such a way that in deduction the new ideas shall be no whit inferior to the old arithmetic concepts. In Berlin NOLL had followed courses by E. SCHMIDT and HASSE, the former, like HAMEL, once a doctorand under HILBERT and the latter once a colleague. The volume of the Journal of Rational Mechanics and Analysis in which NOLL'S thesis is published is dedicated in memoriam to HAMEL. The present volume of reprints will mark the twentieth anniversary of the defense of that thesis on August 9, 1954, in my office. The room itself, like all that we valued of our circumstances then, has since been destroyed. Nowadays, when the common student seeks a secure berth by grafting him self upon some modest little professor whom he regards as prone to foster painlessly his limaceous glide toward a dissertation not too strenuous or, even better, to draught it for him, tradition is moribund, and we lightly disregard it. Mathematics once was transmitted almost like a priesthood, through novitiate, trial, and the laying on of hands. The burden lay upon the candidate, not upon his professor. Before NOLL came to Indiana, a year at the Sorbonne had given him some of the abstract, direct mathematics of BOURBAKI, which then seemed to him unrelated to natural science, but from which he afterward created what is now become the common dialect of continuum mechanics. That year brought him into contact with living exponents of another tradition, even more power ful than HILBERT'S: the didactic, systematic rationalism of DESCARTES. C. TRUESDELL VI Publications of Walter Noll The numbers prefixed to the papers reprinted in this volume are set in bold type in the list below. The pages on which they appear here are indicated at the right. In the texts reprinted, both such errata as have been published before and also some further ones are included. Whenever it was feasible to do so, the text itself has been corrected, but in some few cases errata are listed at the ends of the works to which they pertain. BOOKS I. The Non-Linear Field Theories of Mechanics, Encyclopedia of Physics, Volume III/3, 602 pages, Berlin-Heidelberg-New York, Springer-Verlag, 1965 (with C. TRUESDELL) II. Viscometric Flows of Non-Newtonian Fluids, Theory and Ex periment, Springer Tracts in Natural Philosophy, Volume 5, 130 pages, Berlin-Heidelberg-New York, Springer-Verlag, 1966 (with B. D. COLEMAN and H. MARKOVITZ) PAPERS AND NOTES 1. Eine Bemerkung zur Schwarzschen Ungleichheit, Mathemati sche Nachrichten, Volume 7, pp. 55-59 (1952) (with E. MOHR) 2. On the· Continuity of the Solid and Fluid States, Journal of Rational Mechanics and Analysis, Volume 4, pp. 3-81 (1955) 3. Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistischen Mechanik, Journal of Rational Mechanics and Analysis, Volume 4, pp. 627-646 (1955) 4. Verschiebungsfunktionen flir elastische Schwingungsprobleme, Zeitschrift fUr angewandte Mathematik und Mechanik, Volume 37, pp. 81-87 (1957) 5. On the Uniqueness and Non-existence of Stokes Flow, Archive for Rational Mechanics and Analysis, Volume 1, pp. 97-106 (1957) (with R. FINN) 6. On the Rotation of an Incompressible Continuous Medium in Plane Motion, Quarterly of Applied Mathematics, Volume 15, pp. 317-319 (1957) 7. On Exterior Boundary Value Problems in Linear Elasticity, Archive for Rational Mechanics and Analysis, Volume 2, pp. 191-196 (1958) (with R. J. DUFFIN) 8. A Mathematical Theory of the Mechanical Behavior of Con tinuous Media, Archive for Rational Mechanics and Analysis, Volume 2, pp. 197-226 (1958) 1-30 VII Publications of WALTER NOLL 9. The Foundations of Classical Mechanics in the Light of Recent Advances in Continuum Mechanics, pp. 266-281 of The Axio- matic Method, with Special Reference to Geometry and Physics (Symposium at Berkeley, 1957), Amsterdam, North-Holland Publishing Co., 1959 32-47 10. On Certain Steady Flows of General Fluids, Archive for Ra- tional Mechanics and Analysis, Volume 3, pp. 289-303 (1959) (with B. D. COLEMAN) 49-63 11. Helical Flow of General Fluids, Journal of Applied Physics, Volume 30, pp. 1508-1512 (1959) (with B. D. COLEMAN) 12. Conditions for Equilibrium at Negative Absolute Temperatures, The Physical Review, Volume 115, pp. 262-265 (1959) (with B. D. COLEMAN) 13. On the Thermostatics of Continuous Media, Archive for Ra tional Mechanics and Analysis, Volume 4, pp. 97-128 (1959) (with B. D. COLEMAN) 65-96 14. An Approximation Theorem for Functionals with Applications in Continuum Mechanics, Archive for Rational Mechanics and Analysis, Volume 6, pp. 355-370 (1960) (with B. D. COLEMAN) 97-112 15. Recent Results in the Continuum Theory of Viscoelastic Fluids, Annals of the New York Academy of Science, Volume 89, pp. 672-714 (1961) (with B. D. COLEMAN) 16. Foundations of Linear Viscoelasticity, Reviews of Modern Phys- ics, Volume 33, pp. 239-249 (1961) (with B. D. COLEMAN) 113-123 17. Normal Stresses in Second -order Viscoelasticity, Transactions of the Society of Rheology, Volume 5, pp. 41-46 (1961) (with B. D. COLEMAN) 18. Steady Extension of Incompressible Simple Fluids, The Physics of Fluids, Volume 5, pp. 840-843 (1962) (with B. D. COLEMAN) 19. Simple Fluids with Fading Memory, pp. 530-552 of Second Order Effects in Elasticity, Plasticity, and Fluid Dynamics (Sym posium at Haifa, 1962), Oxford etc., Pergamon Press, 1964 (with B. D. COLEMAN) 20. Motions with Constant Stretch History, Archive for Rational Mechanics and Analysis, Volume 11, pp. 97-105 (1962) 125-133 21. La Mecanique Classique, Basee sur un Axiome d'Objectivite, pp. 47-56 of La Methode Axiomatique dans les Mecaniques Classiques et Nouvelles (Colloque International, Paris. 1959), Paris, Gauthier-Villars, 1963 135 -144 22. The Thermodynamics of Elastic Materials with Heat Conduc tion and Viscosity, Archivefor Rational Mechanics and Analysis, Volume 13, pp. 167-178 (1963) (with B. D. COLEMAN) 145-156 VIII Publications of WALTER NOLL 23. Material Symmetry and Thermostatic Inequalities in Finite Elas tic Deformations, Archive for Rational Mechanics and Analysis, Volume 15, pp. 87-111 (1964) (with B. D. COLEMAN) 157-181 24. Euclidean Geometry and Minkowskian Chronometry, American Mathematical Monthly, Volume 71, pp. 129-144 (1964) 183-198 25. Proof of the Maximality of the Orthogonal Group in the Uni modular Group, Archive for Rational Mechanics and Analysis, Volume 18, pp. 100-102 (1965) 200-202 26. The Equations of Finite Elasticity, pp. 93-101 of Symposium on Applications of Nonlinear Partial Differential Equations in Mathematical Physics (1964), Providence, American Mathemati cal Society, 1965 27. The Foundations of Mechanics, pp. 159-200 of Non-Linear Continuum Theories (C. 1. M. E. Lectures, 1965), Rome, Cremo nesi,1966 28. Space·CI'ime Structures in Classical Mechanics, pp. 28-34 of Delaware Seminar in the Foundations of Physics, Berlin-Heidel- berg-New York, Springer-Verlag, 1967 204-210 29. Materially Uniform Simple Bodies with Inhomogeneities, Ar chive for Rational Mechanics and Analysis, Volume 27, pp. 1-32 (1967) 211-242 30. Inhomogeneities in Materially Uniform Simple Bodies, pp. 239-246 of IUTAM Symposium on Mechanics of Generalized Continua (1967), Berlin-Heidelberg-New York, Springer-Ver lag, 1968 31. Quasi-invertibility in a Staircase Diagram, Proceedings of the American Mathematical Society, Volume 23, pp. 1-4 (1969) 32. Representations of Certain Isotropic Tensor Functions, Archiv der Mathematik, Volume 21, pp. 87-90 (1970) 33. On certain Convex Sets of Measures and on Phases of Reacting Mixtures, Archive for Rational Mechanics and Analysis, Volume 38, pp. 1-12 (1970) 34. Annihilators of Linear Differential Operators, Journal d'Ana lyse, Volume 24, pp. 205-284 (1971) (with H. D. DOMBROWSKI) 35. A New Mathematical Theory of Simple Materials, Archive for Rational Mechanics and Analysis, Volume 48, pp. 1-50 (1972) 243-292 36. Lectures on the Foundations of Continuum Mechanics and Thermodynamics, Archive for Rational Mechanics and Analysis, Volume 52, pp. 62-92 (1973) 294-324 IX Publications of WALTER NOLL Several of the above-listed papers have been reprinted already in collective volumes, as follows: Nos. 2, 8, 10, 11, and 14 in Continuum Mechanics II, The Rational Mechanics of Materials, Interna tional Science Review Series, Volume VIII, Part 2, New York etc., Gordon & Breach, 1965. Nos. 13 and 16 in Continuum Mechanics III, Foundations of Elasticity Theory, International Science Review Series, Volume VIII, Part 3, New York etc., Gordon & Breach, 1965. Nos. 8, 29, and 30 in Continuum Theory of Inhomogeneities in Simple Bodies, New York, Sprin ger-Verlag, 1968. x A Mathematical Theory of of the Mechanical Behavior Continuous Media WALTER NOLL Communicated by C. TRUESDELL Contents 1. Introduction . . . . . . ............ 198 I. Local kinematics 2. Basic concepts . . . . . . . . . . . 200 3. Deformations and linear transformations 200 4. Local configurations. . . . 202 5. Gradients . . . . . . . . 203 6. Rotation and strain tensors 204 7. Histories. . . . . . . . . 205 8. Rate of strain and spin 205 9. Rational expressions for the rates 206 II. The general constitutive equation 10. Basic concepts . . . . . . . . . . . . . . . 207 11. The principle of objectivity of material properties 208 12. The principle of determinism for the stress 209 13. The general constitutive equation 210 14. Material isomorphisms . 211 15. Constitutive functionals 212 16. The local isotropy group 213 III. Simple materials 17. Simple constitutive functionals 214 18. Simple materials . . . . . . 216 19. The isotropy group . . . . . 217 20. Isotropic and anisotropic solids 218 21. Fluids .......... . 219 22. Constitutive equations involving the Cauchy-Green tensors 220 IV. Special classes of materials 23. Materials of the differential type 221 24. Materials of the rate type 223 References. . . . . . . . . 226 1