Truesdell · Noll the Non-Linear Field Theories of Mechanics C

Truesdell · Noll The Non-Linear Field Theories of Mechanics C. Truesdell· W. Noll

The Non-Linear Field Theories of Mechanics

Edited by Stuart S. Antman

Third Edition

~Springer Professor Clifford Truesdell Deceased

Professor Walter Noll Carnegie Mellon University, Pittsburg, Pennsylvania 15213, USA

Professor Stuart S. Antman University of Maryland Department of Mathematics and Institute of Physical Science and Technology College Park, Maryland 20742-4015, USA

Library of Congress Cataloging-in-Publication Data Truesdell, C. (Clifford),1919- The non-linear field theories of mechanics I C. Truesdell, W. Noll- 3rd ed./ edited by StuartS. Antman. p. em. Includes bibliographical references and index. ISBN 978-3-642-05701-4 ISBN 978-3-662-10388-3 (eBook) DOI 10.1007/978-3-662-10388-3 1. Continuum mechanics. 2. Nonlinear mechanics. I. Noll, W. (Walter),1925- II. Antman, S. S. (StuartS.) III. Title QA8o8.2.T69 2004 531- dc22

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springeronline.com © Springer-Verlag Berlin Heidelberg 1965, 1992, 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of the hardcover 3rd edition 2004 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro• tective laws and regulations and therefore free for general use. Product liability: The publishers cannot guarantee the accuracy of any information about dosage and application contained in this book. In every individual case the user must check such information by consulting the relevant literature. Printed on acid-free paper 55/3141XO- 54 3 2 I 0 Publisher's Note

This Third Edition is based on the historic edition of the Encyclopedia of Physics, Volume 11113: The Non-Linear Field Theories of Mechanics, Edited by S. Fliigge/ Handbuch der Physik, Band 11113: Die Nicht-Linearen Feldtheorien der Mechanik, Herausgegeben von S. Fliigge, published in 1965. We are sincerely grateful to Ms. Sandra di Majo at the library of the Scuola Normale Superiore in Pisa, Italy, who lent us C. Truesdell's personal copy of The Non-Linear Field Theories of Mechanics. The original layout was maintained while handwritten corrections and additions made by C. Truesdell in his personal copy of The Non-Linear Field Theories of Me• chanics were incorporated. For this reason, readers will notice different fonts in some parts of this edition.

Heidelberg, September 2003 Springer-Verlag The Genesis of the Non-Linear Field Theories of Mechanics *

Clifford Truesdell (C.T.) was only in his early 30s in the early 1950s when he had already established himself as perhaps the world's best informed person in the field of continuum mechanics, which included not only the classical theories of fluid mechan• ics and elasticity but also newer attempts to mathematically describe the mechanical behavior of materials for which the classical theories were inadequate. In 1952 he published a 175-page account entitled The Mechanical Foundations of Elasticity and Fluid Dynamics in the first volume of the Journal ofRational Mechanics and Analysis. Springer-Verlag, with headquarters in Heidelberg, Germany, has been an important publisher of scientific literature since the beginning of the 20th century. In the 1920s it published the Handbuch der Physik, a multi-volume encyclopedia of all the then known knowledge in Physics, mostly written in German. By the 1950s the Handbuch had become hopelessly obsolete, and Springer-Verlag decided to publish a new version, written mostly in English and called The Encyclopedia of Physics. Here is C.T.'s own account, in a 1988 editorial in the Archive for Rational Mechanics and Analysis, of how he became involved in this enterprise: "In 1952 Siegfried Fliigge and his wife Charlotte were organizing and carrying through the press of Springer-Verlag the vast, new Encyclopedia of Physics (Handbuch der Physik), an undertaking that was to last for decades. Joseph Meixner in that year had called Fltigge's attention to my article The Mechanical Foundations of Elasticity and Fluid Dynamics .... On October 1, 1952, Fltigge invited me to compose for the Encyclopedia a more extensive article along the same lines but presenting also the elementary aspects of continuum mechanics, which my paper ... had presumed to be already known by the reader. In the second week of June 1953, Fltigge spent some days with us in Bloomington, mainly discussing problems of getting suitable articles on fluid dynamics for the Encyclopedia .... He asked me to advise him, and by the end of his visit I had agreed to become co-editor of the two volumes under discussion." It was C.T., in his capacity as co-editor, who invited James Serrin to write the basic article on fluid mechanics. C.T. himself decided to make two contributions dealing

*This description is an excerpt from a lecture given by the author at the Meeting of the Society for Natural Philosophy in the memory of Clifford Truesdell, held in Pi sa, Italy, in November 2000. © 2004 by Walter Noll VIII The Genesis of the Non-Linear Field Theories of Mechanics with the foundations of continuum mechanics, the first to be called The Classical Field Theories of Mechanics and the second The Non-Linear Field Theories of Mechanics. Here is what he wrote about this decision, in the preface to a 1962 reprint of his Mechanical Foundations:

"By September, 1953, I had agreed to write, in collaboration with others, a new exposition for Fliigge's Encyclopedia of Physics. It was to include everything in The Mechanical Foundations, supplemented by fuller development of the field equations and their properties in general; emphasis was to be put on principles of invariance and their representations; but the plan was much the same. As the work went into 1954, the researches of Rivlin & Ericksen and of Noll made the underlying division into fluid and elastic phenomena, indicated by the very title The Mechanical Foundations of Elasticity and Fluid Dynamics, no longer a natural one. It was decided to split the projected article in two parts, one on the general principles, and one on constitutive equations. The former part, with the collaboration of Mr. Toupin and Mr. Ericksen, was completed and published in 1960 ... In over 600 pages it gives the full material sketches in Chapters II and III of The Mechanical Foundations. The second article, to be called "The Non-linear Field Theories of Mechanics", Mr. Noll and I are presently engaged in writing. The rush of fine new work has twice caused us to change our basic plan and rewrite almost everything. The volume of grand and enlightening discoveries since 1955 has made us set aside the attempt at complete exposition of older things. It seems better now to let The Mechanical Foundations stand as final for most of the material included in it, and to regard the new article not only as beginning from a deeper and sounder basis than could have been reached in 1949 but also as leaving behind it certain types of investigation that no longer seem fruitful, no matter what help they may have provided when new."

In 1951 I started employment as a "Wissenschaftlicher Assistent" (scientific assis• tant) at an Institute of Engineering Mechanics (Lehrstuhl fiir Technische Mechanik) at the Technical University of Berlin. In late 1952 I came across a leaflet from Indiana University offering "research assistantships" at the Graduate Institute of Mathematics and Mechanics. I applied and was accepted to start there in the Fall of 1953. After arriving, I found out that I had become a "graduate student". At first I did not know what that meant, because German universities have nothing corresponding to it. But the situation was better than I expected, because no work was assigned to me, and I could pursue work towards a Ph.D. degree full time. I asked C.T. to be my thesis advisor, because he was the local expert on mechanics and I thought that would helpme after my return to Germany. At the time I did not know that C.T. had already become the global expert on mechanics. I learned from him not only the basic principles and the important open questions of mechanics, but also how to write clear English. As a thesis topic, C.T. gave me a paper written by S. Zaremba in 1903 and asked me to make sense of it. I found that, in order to do so, a general principle was needed, which I called "The Principle of Isotropy of Space". This principle also served to clarify many assertions in mechanics that had been obscure before (at least to me). C.T. had a wonderful sense of humor. One day he put up his son, then about 10 years old, to ask me: "Mr. Noll, please explain the Principle of Isotropy of Space to The Genesis of the Non-Linear Field Theories of Mechanics IX me". Actually, I think I have found a good explanation only recently, about 40 years later. I had a very good background in the concepts of coordinate-free linear algebra and used it in my thesis. At the time, C.T. was uncomfortable with this approach and forced me to add a coordinate version to all formulas. However, he did not have a closed mind and later, in a letter to me dated August 4, 1958, he wrote:

"I must also admit that the direct notations you use are better suited to funda• mental questions than are indicia! notations. Your present mathematical style is smoother and simpler than that in your thesis."

After I received my Ph.D. in September 1954 and returned to Germany, C.T. and I stayed in contact by mail, and I saw him again when he gave a lecture in Berlin in June 1955 ("Das ungeloste Hauptproblem der endlichen Elastizitatstheorie"). The term of my position in Berlin ran out in the Fall of 1955 and I needed a new job. On C.T.'s recommendation I was offered an Associate Professorship at Carnegie Institute of Technology (now Carnegie Mellon University), where I moved in the Fall of 1956 after having spent a year at the University of Southern California. Later, in December of the same year, he wrote me the following letter:

"Dear Noll, Ericksen and I have long promised to write a second article, The Non-linear Field Theories of Mechanics, to cover exact work on elastic, fluid, and plastic phenomena. Ericksen is far behind on the first article, and therefore we have not even outlined this second one. Although I have not yet consulted Ericksen, I feel sure that he would be relieved if you would consent to join us and do a major part of the second article. The order of authors would be according to the amount of work provided, and if you could do most of the work, you would be the senior author. You know both the old Mechanical Foundations and the new Classical Field Theories thoroughly. Therefore you have as much background as anyone else, and you have made important contributions yourself. I have not been able to think of a good organization. All I can say is that I should prefer to cut down on the amount of space given to special proposals prior to 1948 and to emphasize general work such as you and Rivlin have done. Whether it is better to treat very general visco-elastic theories first or to give a definitive presentation of classical finite elasticity first is not clear to me. The article should be exhaustive, but for the work prior to 1948 I think we need only condense the Mechanical Foundations or change the emphasis, as I believe the list of sources cited there is virtually complete. The presentation should be concise, but the length is not critical. If this proposal is acceptable to you, I will put it up to Ericksen at once. Also, if you could start right away on the organization and suggest a distribution of material among the three authors it would help. I believe I can begin active work within two months, but I do not expect to be able to prosecute exclusively this one project in the near future." X The Genesis of the Non-Linear Field Theories of Mechanics

Here is an excerpt from my answer, written on January 2, 1957: "I shall be glad to accept your proposal and join you and Ericksen as co-author of "the non-linear field theories of mechanics". It will be a challenging task for me, but I shall do the best I can. I have thought a little bit about the organization. Here is very roughly what I would like to propose: The first section should contain general remarks concerning constitutive equa• tions, their classification, invariance requirements, constraints, definitions of terms like "material", "isotropic", "aelotropic", etc. I have been thinking about this subject for quite a while, and I plan to include my ideas on it in a paper on the axiomatic foundation of mechanics, which I plan to write in the near future. I think it is best to give then an account of finite elasticity and non-linear fluid theory (Reiner-Rivlin). These are the simplest and best known non-linear theories. Next, more general stress-strain relations (as those proposed by Rivlin and Ericksen) could be treated. Finally, constitutive equations involving stress rates (as those proposed by Oldroyd, Cotter and Rivlin, and in my thesis) and their special cases (hypo-elasticity) could be worked out, and the connection of these general theories with the simpler ones could be established. I would be willing to do the general remarks and also most of the treatment of the various constitutive equations in general. However, I would appreciate some help with the special cases and solutions. I would think, for instance, that Ericksen would have little trouble to present all special solutions valid for an arbitrary strain energy function in finite elasticity by taking his two papers (Z. angew. Math. Phys. 5, 466-489 and J. Math. Phys. 2,126-128) as a basis. Also, it is probably easier for one of you to deal with finite deformations of shells, anisotropic elastic bodies, and similar material. I would think, too, that you would want to do the section on hypo-elasticity yourself. I am open to alternative and more detailed proposals concerning the organization as well as the distribution of the material to the authors, and I am looking forward to have your opinion." The new task of working on the Non-Linear Field Theories of Mechanics (NLFT) changed my professional life forever, because it focussed my attention on the founda• tions of continuum mechanics. C.T. was very good at reading, digesting, and summa• rizing existing literature. I am not. I cannot help rethinking and recreating whatever subject I wish to understand. This had perhaps the effect of improving the NLFT in the end, but it also slowed down the progress towards completion. Sometimes C.T. became very impatient with me, and with good reason. The low point was reached when C.T wrote the following letter to me: "Dear Walter, Upon my return from the West, I did not find any manuscript of our article. As I said in Pittsburgh, I feel now that you should withdraw from the article. This is the third period of half a year or more in which you have not sent me a line, and again the subject is slipping away from us. My having had to disturb you every few weeks or days for the past six months so as to get your assurance that a large section of manuscript would be ready in a few days has been very painful, the more so since fruitless. I have put in this treatise the better part of The Genesis of the Non-Linear Field Theories of Mechanics XI

all my work in the past five years, and as I prepare for another trip to Europe, I cannot drag this weight of responsibility any longer. I plan to finish the article by intensive work in January and February, and I request you to send me all material I gave you before I went to Europe last spnng. Surely, you know that I feel this loss deeply. The parts written by you are far better than anything I could write, and your criticism and correction of the parts I wrote have been of the highest value. There is no one else who could do the job you are capable of doing, but you have not done it, and some sort of article must be published nevertheless. There is too much invested in it already."

Well, I pulled myself together and I helped to finish NLFT. During our collaboration we were in constant contact by mail or in person. I visited Bloomington in November 1958. C.T. came to Pittsburgh in June 1959 and gave lectures at the Mellon Institute (later a part of Carnegie Mellon University). In August 1961, C.T. left Indiana University and he and Charlotte moved to Baltimore where he joined the Johns Hopkins University with the title Professor of Rational Mechanics. In September 1961, C.T. visited Pittsburgh again. During the academic year 1962/63, C.T. arranged for me to come to Johns Hopkins University as a visiting professor. Occasionally, C.T. and I had disagreements on terminology. Perhaps the most interesting concerned what we finally called "The Principle of Material Frame-Indifference". Here are some excerpts from letters that we exchanged on this subject:

C.T. to W.N., August 1, 1958

"I was just planning to write something mentioning your old "isotropy of space". It seems to me the term "principle of objectivity" shares one of the undesirable features of the old term, namely, it is too vague. I was going to use "the principle of material indifference". However, it would be bad for there to be three names running around for the same principle. If you find my suggestion appealing, let me know right away; otherwise, I will use your present name."

W.N. to C.T., August 5, 1958:

"I have switched from "isotropy of space" to "objectivity" because the former suggests only that there are no preferred directions in static space, which is much less than is implied by the principle of objectivity. If "objectivity" means independence of the observer, as I believe it does, then it seems to me that it is the correct term. One could remove some of the vagueness you mentioned by using "principle of objectivity of material properties" (or perhaps, "principle of material objectivity"). If you think that this is advisable, please insert "of material properties" after "principle of objectivity" in the table of contents (section II), in line 2 of page 3, in the title of section II on page 20, and in line 15 from the bottom of mage 21. I think that no additional changes are necessary because the other references to the "principle of objectivity" cannot lead to confusion." XII The Genesis of the Non-Linear Field Theories of Mechanics

C.T. to W.N., August 9, 1958 (handwritten): "Dear Noll, I have to confess that my dislike of "objective" is subjective. Psychologists, sociologists, and all sorts of other charlatans and fakers use "objective" when the only sense I can find for what they claim is "nonsensical" or "thoughtless". Thus I exclude, banish and ostracize "objective" as the opposite of "subjective" in my own usage, but of course there is no reason why this should influence you. Cordial regards, CT" C. T.' s objection did influence me and I am now glad it did. I suggested that we change "indifference" to "frame-indifference". C.T. accepted that and I believe now that the term we finally used expresses the idea better than any of the ones used before. The entire task of writing the NLFT turned out more extensive than was expected, and C. T. called it the "monsterino". (He called the earlier Classical Field Theories the "monster".) The monsterino was finally published in 1965. There were many people who helped us with the work by reading parts of the manuscript and offering corrections, critique, and suggestions. They include B. D. Coleman, J. Ericksen, M. E. Gurtin, R. Toupin, K. Zoller, C. C. Wang, D. C. Leigh, C. C. Hsiao, W. Jauzemis, and A. J. A. Morgan. On June 25, 1961 C.T. wrote to me: "I now have an excellent secretary, so that the preparation of the manuscript should be easier." That helped, too. Clifford Truesdell was very meticulous about attributions. I believe he bent over backwards in my favor when he described, in a footnote to the Introduction to NLFT, how the work was divided between us: "Acknowledgment. This treatise, while it covers the entire domain indicated by its title, emphasizes the reorganization of classical mechanics by Noll and his associates. He laid down the outline followed here and wrote the first drafts of most sections in Chaps. B, C, and E and of a few in Chap. D. Among the places where he has given new results not published elsewhere, shorter proofs, or major simplifications of older ideas may be mentioned (21 items). The larger part of the text was written by Truesdell, who also took the major share in searching the literature. While Noll revised many of the sections drafted by Truesdell, it is the latter who prepared the final text and must take responsibility for such oversights, crudities, and errors as may remain."

Walter Noll December 2002 Preface to the Third Edition

The period following the Second World War saw an explosive growth in the science of non-linear continuum mechanics. This growth was sparked by the appearance of (i) a beautiful series of papers by RIVLIN [1947, 9,10] [1948, 11-16] [1949, 15-19] 1 showing that illuminating problems for incompressible non-linearly elastic solids and for incompressible non-Newtonian fluids could be solved for arbitrary non-linear con• stitutive equations, and (ii) the magisterial synthesis, clarification, and simplification by TRUESDELL [ 1952, 20] [ 1953, 25] of the available theories of elasticity, fluid dynamics, thermodynamics, and kinetic theory. In the period 1952-1965, these two streams co• alesced to deliver a true science of continuum mechanics, equipped with fundamental principles capable of describing a rich class of material response including memory effects, with an enlarged collection of exact solutions for arbitrary families of materials, and with studies of the stability of these exact solutions on the basis of linearizations of the governing equations about them. The first definitive treatment of part of this work appeared in the The Classical Field Theories, henceforth abbreviated CFT, by TRUESDELL and ToUPIN in the Handbuch der Physik Vol. 11111 of 1960.) This treatise, an elaborate outgrowth of part of TRUESDELL's [1952, 20] paper, emphasized kinematics of deformation. It employed a general, flex• ible, precise, and complicated indicia) notational scheme involving two-point tensor fields (explained in the Appendix by ERICKSEN), developed by TRUESDELL [1952, 20] to replace a notational morass. The treatment of constitutive equations, the heart of continuum mechanics, was intentionally superficial; a thorough treatment was deferred to The Non-Linear Field Theories of Mechanics, henceforth abbreviated NFTM, the treatise reprinted here. The aims of TRUESDELL and NoLL for NFTM, stated in Sect. 5, were to render continuum mechanics "simple, easy, and beautiful". They sought permanence, clarity, and finality. In this endeavor they were remarkably successful. (This success explains both the continued esteem with which NFTM is held and the demand for a reprinting.) Under the influence of NoLL, they based their exposition on abstract linear algebra, direct in both notation and concept, relegating the indicia) notation of CFT to a sub• sidiary role. Consequently, their essential formulae are far shorter, visibly simpler, and more geometrically and physically illuminating than those of CFT. In this setting, many hitherto complicated developments are made transparent, elegant, and brief.

1 References with dates enclosed in brackets appear in the main list of references starting on page 542. XIV Preface to the Third Edition

The heart of NFTM is in Parts C, D, E, treating respectively the general theory of material behavior, elasticity, and fluidity, with the most space devoted to elasticity. The last two parts have an inviting combination of theory and specific solutions of inverse and semi-inverse problems. TRUESDELL and NoLL were fortunate in the timing of the publication of NFTM: The general theory was well established and the collec• tion of specific solutions was broad. Since then, there are but few places where the general theory has undergone significant refinement and there have been relatively few additions to the collection of specific solutions. As pointed out below, areas that TRUES• DELL and NoLL did not treat in depth, like materials with memory, have subsequently resisted treatments that have permanence, clarity, and finality. Research in contin• uum mechanics since the appearance of NFTM has largely focused on three endeavors: (i) The determination of effective constitutive equations for specific materials or classes of materials on the basis of experiment, microscopic modelling, or physical intuition. In many cases no consensus has been reached on either specific or general forms of these equations. (ii) Numerical treatment of specific problems, which requires the selection of specific constitutive equations. To handle the typical complexity of problems of continuum mechanics by numerical methods has often required great innovation. This same complexity has generally precluded the justification of numerical methods with useful error bounds. (iii) Mathematical analysis of the governing equations. This work is technically difficult, and consequently has been cultivated by far fewer researchers than those in areas (i) and (ii). As we shall see below, analysis nevertheless has provided important insights into material behavior. In the remainder of this Preface I briefly discuss some of the subsequent develop• ments in continuum mechanics in the spirit of the work of TRUESDELL and NoLL. For reasons of space, I artificially limit the references wherever feasible to recent works that themselves have extensive reference lists. Thus the list of references, which omits many seminal papers, cannot be used to establish priorities; it merely offers an entree to a voluminous literature.

Representation theorems. The representation theorems in Chap. 8.11, more than adequate for the purposes of NFTM, have been greatly extended to tensor functions, which need not be polynomials, invariant under a wide collection of isotropy subgroups. For references, see SMITH { 1994} 2 and XrAo { 1998}. Foundations of mechanics. Hilbert's Sixth Problem is to axiomatize those physi• cal sciences in which mathematics plays an important role. In response, NOLL { 1959} [1963, 58] {1973} (all these works were reprinted in {1974}) formulated an axiomatic foundation for continuum mechanics by defining bodies as certain kinds of measurable set-like objects (so that they could have mass and sustain force) and by defining forces as certain kinds of vector-valued measures on bodies. From his primitive axioms, NoLL was able to derive a complete theory of stress (exempt from the ambiguity that in the classical equations of motion it is defined only to within a divergence-free tensor). This theory was not included in NFTM. perhaps because the authors thought it to be too esoteric at the time. (If so, they were right.) Later TRUESDELL {1991} gave it an extensive exposition.

2 References with dates enclosed in braces appear at the end or this preface. Preface to the Third Edition XV

In certain quarters, NoLL's theory was scornfully dismissed as lacking any intrinsic value for mechanics. RIVLIN {1976} wrote "While few will find that [Noll's presenta• tions in the idiom of elementary set theory] enhance their understanding of the concepts of particle, configuration, and mass-distribution, the approach might be justifiable if significant new results emerged from the formalism. This is rarely, if ever, the case." In the second half of the twentieth century, however, it became painfully clear that so• lutions of the partial differential equations of three-dimensional non-linear continuum mechanics may suffer from severe singularities. (This is especially true for problems involving shocks, irregular boundaries, phase changes, or contact.) At such singularities any traditional concept of stress and any traditional concept of differential equation is lost. Increasingly, the sophisticated techniques of geometric measure theory (see, e.g., EVANS and GARIEPY {1992}), generalizing those of NoLL, are being used to treat such equations and to formulate the fundamental theory of continuum mechanics in a way that is compatible with both natural physical concepts and modem methods of analysis. I regard the non-technical part of NoLL's theory as giving the most attractive available axiomatic foundation for continuum mechanics, and the technical part as foreshadowing subsequent sophisticated elaborations dictated by the needs of analysis. Among these elaborations is the paper of ANTMAN and OSBORN {1979} relating the principle of virtual power to integral laws of motion, and the following papers treat• ing the Cauchy stress theorem under very weak hypotheses: GuRTIN, WILLIAMS, and ZIEMER {1986}, NOLL and VIRGA {1988}, SILHAVY {1991}, DEGIOVANNI, MARZOCCHI, and MUSESTI {1999}, and RODNAY and SEGEV {2003}. Internal constraints. In Sect. 30, TRUESDELL and NOLL laid down a (local) con• straint principle stating how Lagrange multipliers should enter the constitutive equations for a material subject to internal constraints. For conservative problems for hyperelastic solids and inviscid fluids, whose governing equations can be characterized as Euler• Lagrange equations for a suitable Lagrangian functional, their principle agrees with the multiplier rule of the calculus of variations. Their constraint principle is intended to apply to all constrained materials, however. In positing their constraint principle, TRUESDELL and NoLL did not face the possibility that a constraint on the Cauchy• Green deformation tensor C could prevent it from corresponding to a globally defined position field, i.e., prevent C from being compatible and accordingly from having its Riemann-Christoffel curvature tensor vanish. To establish a global version of the constraint principle, ANTMAN and MARLOW {1991} treated constrained materials as very general limits of unconstrained materials. They found two different classes of constraints: those that could and those that could not be reduced to the local constraint principle of NFTM. The former includes the important constraint of incompressibility and the latter those that deliver rod and shell theories. In each case, however, the extra stress may depend on the multipliers, except when this dependence can be precluded, as in hyperelasticity. Such dependence appears to be useful in plasticity theory and lubrication theory. Constitutive restrictions in non-linear elasticity. If an elastic body is pulled by equal and opposite tensile forces, we expect it to elongate along the line of action of these forces and to contract in transverse directions. For other simple experiments we have similar expectations. To translate such intuitive ideas into general mathematically XVI Preface to the Third Edition precise and physically natural restrictions on the constitutive equation TR = ~ (F, X) of nonlinear elasticicty is by no means a routine task because there are several commonly used tensors for both strain and stress and because three-dimensional material behavior can be very subtle. In Sects. 51-53, Truesdell and Noll effected such a translation (based largely on the work of TRUESDELL and TOUPIN and of COLEMAN and NOLL) for several such intuitive restrictions. They compared these restrictions with those for linear elasticity. They determined the logical structure of which of their restrictions implied others, finding that conditions of the Coleman-Noll type imply many of the others. Of course, such strong conditions are necessarily more restrictive than those they imply and their imposition might prevent a theory from describing interesting phenomena. Versions of the Strong Ellipticity Condition (44.21) on the constitutive function for the stress (which differ in the strictness of inequalities and in the smoothness of the function) arise naturally in the theories of hyperbolic equations and the calculus of variations. (The specializations of these conditions to hyperelastic materials are the Legendre-Hadamard conditions on the stored-energy function.) These conditions, which are not comparable to the Coleman-Noll conditions, were not given a whole• hearted endorsement by Truesdell and Noll, possibly because their restrictions to linear elasticity are too weak to support the standard theory, and possibly because various formulations of it, like ( 44.21 ), seem to lack both a simple mathematical structure and a physically obvious interpretation. Such features, however, are exposed if we rewrite (44.21) in direct notation as (A. 0 JL) : ~F : (A. 0 JL) > 0 for all vectors A., JL. (Here'":" is the inner product for tensors.) A simple proof shows that if A. and JL are independent ofF, then this inequality is equivalent to

a( A.. ~ JL) > o iJ(A. . FJL) for all such A., JL. This condition says that if the dyadic (or tensor) product A. 0 JL is regarded as a member of a basis for the space of tensors, then the (A. 0 JL)-component of TR is an increasing function of the corresponding component ofF when all the other components are held fixed. Since the publication of NFTM, the Strong Ellipticity Condition has played an increasing role (and the Coleman-Noll conditions have played scarcely any role) in the analysis of the partial differential equations of nonlinear elasticity. Using the Strong Ellipticity Condition, HUGHES, KATO, and MARSDEN {1977}, CHEN and VON WAHL {1982}, and DAFERMOS and HRUSA {1985} for unconstrained materials, and HRUSA and RENARDY {1988} and EBIN and S!MANCA {1992} for incompressible materials proved the existence (local in time) of solutions to initial-value problems for the dy• namical equations. (See MARSDEN and HUGHES {1983 }.) The treatment of equilibrium problems has turned out to be very difficult. Studies of equilibrium states near the undeformed states were made by STOPELLI [1955, 271 [1957, 25, 26] [1958, 451, VAN BUREN {1968}, and WAN and MARSDEN {1984} (see VALENT {1988}). HEALEY and SIMPSON {1998} developed a very attractive global continuation theory for such solu• tions. BALL {1977} made a major breakthrough in proving the existence of minimizers of the total-energy functional under constitutive assumptions intimately related to the Strong Ellipticity Condition. (The Euler-Lagrange equations for this functional are the equilibrium equations for a hyperelastic body that is subject to conservative loads.) Preface to the Third Edition XVII

The most notable of the constitutive hypotheses introduced by BALL is the polycon• vexity condition (a concrete special case ofMoRREY's {1966} quasiconvexity condition), which says that the stored-energy function a introduced in (82.1 0) has the form

a(F, x) = 'E(F, cof F, det F, x) where 'E is strictly convex in (the 19 components of) its first three arguments for these arguments lying in the half space where the third argument is positive. Here cof F denotes the tensor of cofactors of F. The strict convexity of a with respect to F is unacceptable because this condition is incompatible with the requirements that (i) a, restricted to the nonconvex setofF's where det F > 0, goes to oo as det F -+ 0, (ii) certain equilibrium problems admit nonunique solutions (buckled states, which would be prohibited by strict convexity), and (iii) a is frame-indifferent (see Sect. 52). BALL's work initiated a rich and technically difficult study of variational problems of nonlinear elasticity, in which several challenging problems remain. For example, it has not yet been shown that the energy minimizer is even a weak solution of the Euler-Lagrange equations in the natural case when the stored energy becomes infinite as det F goes to zero, although substantial progress has been made in this surprisingly difficult problem. See MARSDEN and HUGHES {1983}, CIARLET {1988}, DACOROGNA {1989}, and PEDREGAL {2000} for accounts of BALL's work, for refinements thereof, and for references. See BALL {2002} for the status of open problems and for references. Coexistent phases. One of the triumphs of continuum mechanics since 1965 has been the development of theories of coexistent phases in elastic solids, which may be regarded as continuum mechanical theories for certain problems of metallurgy. Some of these theories were inspired by BALL's work and by ERICKSEN's {197 5} study of the equilibrium of bars in which the tension is not an increasing function of the elongation, so that the energy is not a convex function of the elongation. In the three-dimensional theory, the stored-energy function is taken to have multiple local minima (energy wells) and thus lacks even the limited convexity ensured by the Legendre-Hadamard condition. In this case, there can be a multitude of equilibrium states, most of which are irregular in the mathematical sense. In the setting of the calculus of variations, where much of the analysis is carried out, a nonconvex energy functional need not have a minimum on its domain, although it typically has an infimum. When such a functional does have a local minimum, the minimizer is usually reckoned to characterize a stable equilibrium configuration. (For caveats about such interpretations, see KNOPS and WILKES {1973}.) A remarkable property of an energy functional (with multiple energy wells) that does not attain a minimum is that its infimizing sequences can describe (infinitely fine) microstructure, like that occurring at interfaces (see BALL and JAMES {1987}). Anal• ternative to treating such energy functionals directly is to regularize them by adding (i) a small higher-order term (e.g., one that contributes a biharmonic operator to the Euler-Lagrange equations) or (ii) a small interfacial energy to the stored energy func• tional, e.g., one that penalizes the area of sets over which the strain is discontinuous. Such regularizations often lead to problems having minimizers that can be readily identified with deformations having coexistent phases. (The challenging problem of constructing admissibility conditions as the size of these regularizations goes to zero is largely open.) GuRTIN {2000} has developed a very attractive general thermome• chanical theory of interfaces in which appropriate physical fields are defined on (and XVIII Preface to the Third Edition confined to) the interfaces themselves. This work gives physical interpretations to many of the analytic regularizations that have been introduced. For discussions of all these matters, consult the following works and the references cited therein: FRIED {1999), AMBROSIO, Fusco, and PALLARA {2000), GURTIN {2000}, BALL {2002}, DOLZMANN {2002), PITTERI andZANZOTTO {2003}. Homogenization. In the last twenty years, considerable progress has been made on the mathematically challenging determination of effective bulk properties of materials from their microscopic properties, although almost all this work is devoted to materials with linear constitutive equations. See DAL MAso {1993}, BRA IDES and DEFRANCESCHI {1998), MILTON {2001}, ToRQUATO {2002), and the works cited therein. Materials with memory. The simplest models for material behavior are those of elasticity, viscoelastic of differential type, and Newtonian fluids. In each of these theories the stress at time t depends on the deformation and deformation rate at that time. Thus the governing equations are pure (though very formidable) partial differ• ential equations. The constitutive equations of these materials do not account for the dependence on the past history of deformation and temperature that is observed in many real materials. Solids exhibiting memory effects are often termed elastoplastic or viscoplastic; fluids exhibiting memory effects belong to the class of non-Newtonian fluids (which, strictly speaking, consists of any fluid for which the deviator of the Cauchy stress is not a linear function of the symmetric part of the spatial velocity gradient). Truesdell and Noll give a very elegant discussion of the theory of materials with memory in Chaps. C.III-C. V, D.IV, E.I and in Sects. 96bis and 96ter. Chapter D.IV treats TRUESDELL's theory of hypoelastic materials, which was de• signed to account for effects associated with plasticity. This is an invariant theory differ• ing from classical theories of small-strain plasticity in that it accounts for large strains, dynamical effects, and yielding (without introducing a yield function). Some simple and illuminating problems are worked out for this theory. This theory, while never seri• ously adopted as adequate for the description of the kinds of memory effects observed in solids, inspired efforts to create properly invariant dynamical theories for large-strain plasticity, beginning with GREEN and NAGHDI [ 1965, 18]. Most of these theories em• ploy some sort of yield criterion, which TRUESDELL wished to avoid. In the ensuing years a variety of such theories have been developed, several based on the use of internal variables ( cf. CoLEMAN and GURTIN {1967}, for which RrcE {1971 } gave a microscopic motivation). See NAGHDI {1990}, SIMO {1998), and SIMO and HUGHES {1998} for dis• cussions and references. The development of many of these theories were closely corre• lated with methods for their numerical treatment. There is, however, scant mathematical analysis of the governing equations. Another invariant approach to large-strain plastic• ity are theories of materials with elastic range, developed by PIPKIN and RIVLIN [ 1965, Z] , OWEN {1970), SILHAVY {1977), and LuccHESI, OwEN, and Pooro-Gumuau {1992}. The governing equations have attracted neither numerics nor analysis. There is no consensus as to the general form of material response for many solids other than metals and for many non-Newtonian liquids. The very pretty special flow problems treated in Chap. E.I have both the virtue and the defect that only a limited range of material response intervenes. (See COLEMAN, MARKOVITZ, and NoLL {1966} and TRUESDELL and RAJAGOPAL {2000}.) For references to a voluminous literature on Preface to the Third Edition XIX non-Newtonian fluids, see JosEPH {1990}. For the difficult mathematical treatment of problems for materials with memory, see RENARDY, HRUSA and NoHEL { 1987}. Liquid crystals. Ericksen's continuum theory ofliquid crystals, presented in Sects. 127-129, has undergone a considerable mathematical development, partly due to the relationship of the theory to the mathematical theory of harmonic maps. For modem theory and references see ERICKSEN and KrNDERLEHRER {1987} and VIRGA {1994}. Thermodynamics. The beautiful method oftreating the entropy inequality (second law of thermodynamics) as a restriction on constitutive behavior when the thermody• namic fields are sufficiently regular was introduced by CoLEMAN and NoLL [ 1963, 18]. An account of some of the applications of this method is given in Sects. 96 and 96bis, but because these ideas were being elaborated at the time of publication of NFTM, a full account could not be incorporated. The favorite version of the entropy condition culti• vated in continuum mechanics is the Clausius-Duhem inequality. A physical motivation for it (beyond the fact that it delivers the usual inequalities for linearly heat-conducting viscous fluids) was given by TRUESDELL {1984}. The relationship of it to other versions of the entropy inequality has been the object of several studies. (The method of CoLE• MAN and NoLL can be applied to any version.) SERRIN {1986} gave a probing analysis of the foundations of thermodynamics. Thus today there is a rich theory of rational thermodynamics consistent with the approach of NFTM, various aspects of which are treated in the texts of DAY { 1972}, OWEN {1984}, TRUESDELL {1984} ERICKSEN { 1991 }, SILHAVY {1997} and MuLLER and RuGGERI {1998}. Nevertheless, there remains the fundamental issue of precisely what the form of the entropy inequality should be. One place where this issue is crucial is in the study of shocks in the partial differential equations of continuum physics. A weak formulation of the governing equations (the Principle of Virtual Power) does not give enough conditions to extend a solution uniquely past the time of shock. Admissibility conditions, which are like the entropy condition, are used to select a unique continuation. For other than the simplest equations corresponding to the simplest physical situations, there is a lack of agreement between admissibility conditions based on (i) purely mathematical considerations, (ii) versions of the entropy inequality, (iii) formal asymptotics of equations for materials with dissipative mechanisms as the dissipation goes to zero, (iv) asymptotics, usually formal but occasionally rigorous, of the equations as artificial dissipative mechanisms go to zero, or (v) numerical regularization. Part of the problem is that the mathematical theory is notoriously difficult and is far from complete. An ideal resolution would be to find admissibility conditions compatible with (i)-(iii). See DAFERMOS {2000}. Although there is the beginning of a mathematical analysis of thermo-visco• elasticity (see ZHENG {1995} and RACKE and SHIBATA {2000}), much of the work is restricted to one-dimensional problems or to problems in which the equations have been simplified, e.g., by taking some constitutive functions to be linear. There have been no comprehensive studies of what would happen to solutions if some of the consequences of the CLAUsrus-DUHEM inequality, e.g., were suspended for nonlinear constitutive equations. Non-simple materials. The treatment of coexistent phases by regularizing non• convex functionals by adding to them a small higher-order term effectively makes the constitutive equation be that for a material of strain-gradient type, which is not simple XX Preface to the Third Edition

in the sense of Sect. 28. Such theories were formulated in the 1960's. but without a deep consideration of constitutive restrictions. They were but briefly mentioned in CFT and NFTM in Sects. 28 and 98. The treatment of strain-gradient theories by analysts has been largely ad hoc, certainly not in the spirit of NFTM. But the work on coexis• tent phases using such regularizations, work on plasticity theory using strain-gradient theories (see GAo, HUANG, Nrx, and HuTCHINSON { 1999}), and work on shock the• ory in which some regularizations have this character suggest that it is time that the constitutive theory for nonsimple materials were given a more intensive examination. Conclusion. The advances beyond the permanent scientific conquests collected in NFTM proved to be slow and painful, many requiring technically difficult mathematics or refined numerical techniques (few of which have yet been justified with convergence theorems or error estimates). Consequently there are numerous open problems of ratio• nal continuum mechanics awaiting analysis. Truesdell and Noll have established a firm foundation for such research. It is interesting to note that their system of notation, sym• bols, and terminology has been widely adopted, even by the scientific descendents of critics of their enterprise. The availability of powerful computers has made it feasible to conduct numerical studies of precisely formulated nonlinear problems of contin• uum mechanics, and has thus endowed theoretical works like NFTM with a practical permanence that its authors may not have intended.

Acknowledgment. I thank J. M. BALL, C. M. DAFERMOs, and M. E. GuRnN for helpful comments. The preparation of this preface was supported in part by the NSF. List of Works Cited.

1959 NOLL, W.: The foundations of classical mechanics in the light of recent advances in continuum mechanics. The Axiomatic Method, with Special Reference to Geometry and Physics. pp. 266-281. North-Holland. 1966 COLEMAN, B. D., N. MARKOVITZ, and W. NOLL: Viscomctric Flows of Non-Newtonian fluids. Springer. MORREY, C. B., JR.: Multiple Integrals in the Calculus of Variations. Springer. VAN BuREN, W.: On the existence and uniqueness of solutions to boundary value problems in finite elasticity. Res. Rpt. 68-1 D7-MEKMA-RI, Westinghouse Research Labs, Pittsburgh. 1970 OWEN, D. R.: A mechanical theory of materials with clastic range. Arch. Rational Mcch. Anal. 37. 85-110. 1971 RtcE, J. R.: Inelastic constitutive relations for solids: An internal-variable theory and its applications to metal plasticity. J. Mech. Phys. Solids 19,433-455. 1972 DAY, W. A.: The Thermodynamics of Simple Materials with Fading Memory, Springer. 1973 KNOPS, R. J .. and E. W. WILKES: Theory of elastic stability. Handbuch der Physik. Vol. Vla/3. pp. 125-302. Springer. NoLL, W.: Lectures on the foundations of continuum mechanics and thermodynamics. Arch. Rational Mech. Anal. 52, 62-92. 1974 NOLL, W.: The Foundations of Mechanics and Thennodynamics. Springer. 1975 ERICKSEN, J. L.: Equilibrium of bars. J. Elasticity 5 191-202. 1976 RtVLIN. R. S.: Review of NOLL {1974}. American Scientist 64, 100-101. 1977 HUGHES, T. J. R., T. KATO, and J. E. MARSDEN: Well-posed quasilinear second-order hyperbolic systems with applications to nonlinear clastodynamics and general relativity. Arch. Rational Mcch. Anal. 63, 273-294. StLHA VY, M.: On transformation laws for plastic deformations of materials with clastic range. Arch. Rational Mech. Anal. 63, 169-182. 1979 ANTMAN, S. S .. and J. E. OsBORN: The principle of virtual work and integral laws of motion. Arch. Rational Mech. Anal. 69. 231-262. 1982 CHEN, C., and W. voN W MIL: Die Rand-Anfangswcrtproblemc riir quasilinearc Wdlcngleichungcn in Sobolewraumen niedriger Ordnung. J. rcine angew. Math. 337. 77-112. 1983 MARSDEN. J. E., and T. J. R. HuGHES: Mathematical Foundations of Elasticity. Prcntiec-1-lall. 1984 OwEN, D. R.: A First Course in the Mathematical Foundations of Thermodynamics. Springer. TRUESDELL, C.: Rational Thermodynamics. 2nd ed. Springer. WAN. Y. H., and J. E. MARSDEN: Symmetry and bifurcation in three-dimensional elasticity. Ill. Arch. Rational Mech. Anal. 84, 203-233. Preface to the Third Edition XXI

1985 DAFERMOS, C. M., and W. J. HRUSA: Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics. Arch. Rational Mech. Anal. 87,267-292. 1986 GuRTIN, M.E., W. 0. WILLIAMS, and W. P. ZIEMER: Geometric measure theory and the axioms of continuum thermodynamics. Arch. Rational Mech. Anal. 92, 1-22. SERRIN, J. B.: An outline of thermodynamic structure. New Perspectives in Thermodynamics, pp. 3-32. Springer. 1987 BALL, J. M., and R. D. JAMES: Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100, 13-52. ERICKSEN, J. L., and D. KINDERLEHRER, eds.: Theory and Applications of Liquid Crystals. Springer. RENARDY, M., W. S. HRUSA, and J. NoHEL: Mathematical Problems in Viscoelasticity. Longman. 1988 CIARLET, P. G.: Mathematical Elasticity, Vol. I, North Holland. NoLL, W., and E. G. VIRGA: Fit regions and functions of bounded variation. Arch. Rational Mech. Anal. 102, 1-21. VALENT, T. Boundary Value Problems of Finite Elasticity. Springer. 1989 DACOROGNA, B.: Direct Methods in the Calculus of Variations. Springer. 1990 JosEPH, D. D.: Fluid Dynamics of Viscoelastic Liquids. Springer. 1991 ANTMAN, S. S., and R. S. MARLow: Material constraints, Lagrange multipliers, and compatibility. Arch. Rational Mech. Anal. 116, 257-299. ERICKSEN, J. L.: Introduction to the Thermodynamics of Solids. Chapman & Hall. SILHAVY, M.: The existence of the flux vector and the divergence theorem for general Cauchy fluxes. Arch. Rational Mech. Anal. 116, 223-255. TRUESDELL, C.: A First Course in Continuum Mechanics, Vol. I, 2nd ed. Academic Press. 1992 EBIN, D. G., and S. R. SIMANCA: Deformation of incompressible bodies with free boundary. Arch. Rational Mech. Anal. 119,61-97. EvANS, L. C., and R. F. GARIEPY: Measure Theory and Fine Properties of Functions. CRC Press. LuccHESI, M., D. R. OWEN, and P. Pomo-GumuGu: Materials with elastic range: A theory with a view toward applications. Part III: Approximate constitutive relations. Arch. Rational Mech. Anal. 117,53-96. 1993 DAL MASO, G.: An Introduction tor-Convergence. Birkhiiuser. 1994 SMITH, G. F.: Constitutive Equations for Anisotropic and Isotropic Materials. Elsevier. VIRGA, E. G.: Variational Theories for Liquid Crystals. Chapman & Hall. 1995 ZHENG, SoNGMU: Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems. Long• man. 1997 SILHA vv, M.: The Mechanics and Thermodynamics of Continuous Media. Springer. 1998 BRAIDES, A., and A. DEFRANCESCHI: Homogenization of Multiple Integrals. Oxford Univ. Pr. HEALEY, T., and H. C. SIMPSON: Global continuation in nonlinear elasticity. Arch. Rational Mech. Analysis. 124, 1-28. MOLLER, I., and T. RUGGERI: Rational Extended Thermodynamics, 2nd edn. Springer. SIMO, J. C.: Numerical analysis and simulation of plasticity. Handbook of Numerical Analysis, Vol. VI, pp. 183-499. North Holland. SIMO, J. C., and T. J. R. HUGHES: Computational Inelasticity. Springer. XIAO, H.: On anisotropic invariants of a symmetric tensor: crystal classes, quasi-crystal classes and others, Proc. Roy. Soc. Lond. 454, 1217-1240. 1999 DEGIOVANNI, M., A. MARZOCCHI, and A. MusEsTI: Cauchy fluxes associated with tensor fields having divergence measure. Arch. Rational Mech. Anal. 147, 197-223. FRIED, E.: Fifty years of research on evolving phase interfaces. Evolving Phase Interfaces in Solids, pp. 1-29. Springer. GAO, H., Y. HUANG, W. D. NIX, and J. W. HUTCHINSON: Mechanism-Based Strain Gradient Plasticity. I. Theory. J. Mech. Phys. Solids. 47, 1239-1263. 2000 AMBROSIO, L., N. Fusco, and D. PALLARA: Functions of Bounded Variation and Free Discontinuity Problems, Birkhiiuser. DAFERMOS, C. M.: Hyperbolic Conservation Laws in Continuum Physics. Springer. GuRTIN, M. E.: Configurational Forces as Basic Concepts in Continuum Physics. Springer. JIANG, S., and R. RACKE: Evolution Equations in Thermoelasticity, Chapman & Hall. PEDREGAL, P.: Variational Methods in Nonlinear Elasticity. SIAM. TRUESDELL, C. and K. R. RAJAGOPAL: An Introduction to the Mechanics of Fluids. Birkhiiuser. 2001 MILTON, G.: The Theory of Composites. Cambridge Univ. Press. 2002 BALL, J. M.: Some Open Problems in Elasticity. Geometry. Dynamics, and Mechanics: 60th Birthday Volume for J. E. Marsden, pp. 3-59. Springer. DoLZMANN, G.: Variational Methods for Crystalline Microstructure-Analysis and Computation. Springer. ToRQUATO, S.: Random Heterogeneous Media. Springer. 2003 PITTERI, M. and G. ZANZOTTO: Continuum Models for Phase Transitions and Twinning in Crystals. Chapman & Hall. RoDNAY, G. and R. SEGEv: Cauchy's flux theorem in light of geometric integration theory. in press.

StuartS. Antman November 2002 Preface to the Second Edition

This volume is a second, corrected edition of The Non-Linear Field Theories of Mechanics, which first appeared as Volume III/3 of the Encyclopaedia of Physics, 1965. Its principal aims were to replace the conceptual, terminological, and notational chaos that existed in the literature of the field by at least a modicum of order and coherence, and second, to describe, or at least to summarize, everything that was both known and worth knowing in the field at the time. Inspecting the literature that has appeared since then, we conclude that the first aim was achieved to some degree. Many of the concepts, terms, and notations we introduced have become more or less standard, and thus communication among researchers in the field has been eased. On the other hand, some ill-chosen terms are still current. Examples are the use of "configuration" and "deformation", for what we should have called, and now call "placement" and "transplacement", respectively. (To classify translations and rotations as deformations clashes too severely with the dictionary meaning ofthe latter.) We believe that the second aim was largely achieved also. We have found little published before 1965 that should have been included in the treatise but was not. However, a large amount of relevant literature has appeared since 1965, some of it important. As a result, were the treatise to be written today, it should be very different. On p. 12 of the Introduction we stated " ... we have subordinated detail to importance and, above all, clarity and finality". We believe now that finality is much more elusive than it seemed at the time. The "General theory of material behavior" presented in Chapter C, although still useful, can no longer be regarded as the final word. The "Principle of Determinism for the Stress" stated on p. 56 has only limited scope. It should be replaced by a more inclusive principle, using the concept of "state" rather than a history of infinite duration, as a basic ingredient. In fact, forcing the theory of "materials of the rate type" into the general framework of the treatise as is done on p. 95 must now be regarded as artificial at best, and unworkable in general. This difficulty was alluded to in footnote I on p. 98 and in the discussion of B. Bernstein's concept of a "material" on p. 405. This major conceptual issue was first resolved in 1972 r"A New Mathematical Theory of Simple Materials" by W. Noll, Archive for Rational Mechanics and Analysis, Vol. 48, pp. 1-50], and then only for simple materials. The new concept of material makes it possible also to include theories of "plasticity" in the general framework, and one can now do much more than "refer the reader to the standard treatises", as we suggested on p. II of the Introduction. Here is a list of topics that pertain to the field and have seen major development since 1965: XXIV Preface to the Second Edition

The theory of large elastic deformations treated by the tools of modern non-linear analysis, especially those created to solve problems posed by elasticity, a pilot example. Solutions of problems arising in the mechanics of fluids displaying non-linear response. Application of modern analysis to the theory of material with memory. Mathematical theories intended to describe the behavior of liquid crystals. Theories of materials with microstructure. Theories of therrnomechanics. Here is a list of books appearing after 1965 on subjects cognate with NLFT. We regret having overlooked any that should have been included. DAY, W. A., The Thermodynamics of Simple Materials with Fading Memory, Springer Tracts in Natural Philosophy, Vol. 22, 1972. WANG, C.-C., & TRUESDELL, C., Introduction to Rational Elasticity, Leyden, Nordhoff, 1977. WANG, C.-C., Mathematical Principles of Mechanics and Electromagnetism, Vol. I. N.Y. etc., Plenum, 1979. GURTIN, M. E., An Introduction to Continuum Mechanics, N.Y. etc., Academic Press, 1981. OwEN, D. R., A First Course in the Mathematical Foundations of Thermodynamics, N.Y. etc., Springer, 1983. TRUESDELL, C., Rational Thermodynamics, N.Y. etc., Springer, second edition, 1984. KRA VIETZ, A., Materialtheorie, mathematische Beschreibung des phlinomenologischen thermodynamischen Verhaltens, Heidelberg etc., Springer, 1986. RENARDY, M., HRUSA, J., & NoHEL, J., Mathematical Problems in Viscoelasticity, Pittman Monographs and Surveys, Vol. 35, Longman, 1987. C!ARLET, P. G., Mathematical Elasticity, Vol. 1, Amsterdam etc., North Holland, 1988. VALENT, T., Boundary Value Problems of Finite Elasticity, Springer Tracts in Natural Philosophy, Vol. 31. 1988. CAPRIZ, G., Continua with Microstructure, Springer Tracts in Natural Philosophy, Vol. 35, 1989. ERICKSEN, J. L., Introduction to the Thermodynamics of Solids, London etc., Chapman & Hall, 1991.

VIRGA, E. G., Variational Theories for Liquid Crystals, London etc., Chapman & Hall, 1993.

Note on the corrigenda and addenda. Typographical errors and minor infelicities of style have been silently emended. The few inserted sentences in square brackets remark false conjectures, omitted conditions, and possibly confusing statements; some are also cancelled by rules and emphasized by new footnotes, marked with an asterisk. We have not tried to bring the book up to date. C. T. &W. N. 1992 Contents.

Page

Publisher's Note v The Genesis of the Non-Linear Field Theories of Mechanics Walter Noll ...... VII Preface to the Third Edition Stuart S. Antman XIII Preface to the Second Edition XXIII

The Non-Linear Field Theories of Mechanics. By C. Truesdell, Professor of Rational Mechanics, The Johns Hopkins University, Baltimore, Maryland (USA) and W. Noll, Professor of Mathematics, Carnegie Institute of Technology, Pittsburgh, Pennsyl• vania (USA). (With 28 Figures) A. Introduction ...... I I. Purpose of the non-linear theories I 2. Method and program of the non-linear theories 3 3. Structure theories and continuum theories 5 4. General lines of past research on the field theories of mechanics 8 5. The nature of this treatise ...... I I 6. Terminology and general scheme of notation 13 6 A. Appendix. Cylindrical and spherical co-ordinates 19 B. Tensor functions 20 I. Basic concepts 20 7. Definitions 20 8. Invariants and isotropic tensor functions 22 9. Gradients of tensor functions 24 II. Representation theorems . . . . . 27 10. Representation of invariants of one symmetric tensor 27 II. Representation of simultaneous invariants 29 12. Tensor functions of one variable ...... 32 13. Tensor and vector functions of several variables 34 C. The general theory of material behavior 36 14. Scope and plan of the chapter 36 I. Basic principles ...... 37 15. Bodies and motions 37 16. Forces and Stresses . 39 17. Changes of frame. Indifference 41 18. Equivalent processes 43 19. The principle of material frame-indifference 44 19 A. Appendix. History of the principle of material frame-indifference 45 20. The main open problem of the theory of material behavior 47 II. Kinematics . . . . 48 21. Deformation 48 XXVI Contents

Page 22. Localizations. Local configurations. Histories 51 23. Stretch and rotation ...... 52 24. Stretching and spin ...... 53 25. Polynomial expressions for the rates 55 Ill. The general constitutive equation 56 26. The principle of determinism. The general constitutive equation 56 27. Material isomorphisms. Homogeneity ...... 5R 28. Simple materials, materials of grade n, dimensional invariancc 60 29. Reduced constitutive equations ...... 66 30. Internal constraints, incompressibility, inextcnsibility 69 31. The isotropy group 76 32. Simple 11uids 79 33. Simple solids RI 33 his. Simple subfluids 86 34. Material connections, inhomogeneity, curvilinear aeolotropy, continuous dislocations 88 IV. Special classes of materials . . . . . 92 35. Materials of the differential type 93 36. Materials of the rate type 95 37. Materials of the integral type 98 V. Fading memory ...... 101 38. The principle of fading memory 101 39. Stress relaxation ...... 106 40. Asymptotic approximations 108 41. Position of the classical theories of viscosity and visco-cla,ticity 113 D. Elasticity ...... 117 42. Scope and plan of the chapter 117 I. Elastic materials 119 a) General considerations 119 43. Definition of an elastic material 119 43 A. Appendix. The Piola-Kirchhoff stress tensors 124 44. Formulation of boundary-value problems 125 45. The elasticities of an elastic material 131 46. Stoppeli's theorems of the existence, uniqueness, and analyticity of the solution of a class of traction boundary-value problems . . . 133 47. Isotropic elastic materials, l. General properties ...... 139 48. Isotropic elastic materials, II. The principal stresses and principal forces ...... 142 49. Incompressible isotropic elastic materials ...... 147 50. Elastic nuids and solids. Natural states ...... 148 51. Restrictions upon the response functions. I. Isotropic compressible materials ...... 153 52. Restrictions upon the response functions, II. Compressible materials in general ...... 162 53. Restrictions upon the response functions, III. Incompressible materials 171 b) Exact solutions of special problems of equilibrium ..... 171 54. Homogeneous strain of compressible clastic bodies 171 55. Incompressible elastic materials, I. Homogeneous strain 179 56. Incompressible clastic materials, II. Preliminaries for the non- homogeneous solutions for arbitrary materials ...... 183 57. Incompressible elastic materials, III. The non-homogeneous solutions for arbitrary isotropic materials ...... 186 Contents XXVII

Page 58. Incompressible elastic materials, IV. Non-homogeneous solutions for bodies with certain kinds of aeolotropy and inhomogeneity 197 59. Semi-inverse methods: Reduction of certain static deformations that depend upon material properties 199 60. Plane problems ...... 204 c) Exact solutions of special problems of motion . . . . . 208 61. Quasi-equilibrated motions of incompressible bodies 208 62. Radial oscillations of isotropic cylinders and spheres 214 d) Systematic methods of approximation ...... 219 63. Signorini's expansion ...... 219 64. Signorini's theorems of compatibility and uniqueness of the equilibrium solution ...... 223 65. Rivlin and Topakoglu's interpretation ...... 227 66. Second-order effects in compressible isotropic materials 229 67. Second-order effects in incompressible isotropic materials 241 68. Infinitesimal strain superimposed upon a given strain. I. General equations ...... 246 68 bis. Infinitesimal strain superimposed upon a given strain. II. In• finitesimal stability ...... 252 69. Infinitesimal strain superimposed upon a given strain, III. General theory for isotropic materials ...... 260 70. Infinitesimal strain superimposed upon a given strain, IV. Solutions of special problems ...... 263 e) Wave propagation ...... 267 71. General theory of acceleration waves 267 72. Waves of higher order ...... 272 73. General theory of plane infinitesimal progressive waves 273 74. Waves in isotropic compressible materials. I. General properties 278 75. Waves in isotropic compressible materials. II. The case of hydrostatic pressure ...... 284 76. Waves in isotropic compressible materials. Ill. Determination of the stress relation from wave speeds ...... 285 77. Waves in isotropic compressible materials. IV. Second-order effects. 288 78. Waves in incompressible materials 291 II. Hyperelastic materials 294 a) General considerations . . . . . 294 79. Thermodynamic preliminaries 294 80. Perfect materials ...... 296 81. Thermal equilibrium of simple materials 298 82. Definition of a hyperelastic material 30 I 82 A. Appendix. History of the theory of hyperelastic materials in finite strain ...... 304 83. Work theorems ...... 304 84. The strain-energy function ...... 307 85. The isotropy group of the strain-energy function. Isotropic hyper• elastic materials, hyperelastic fluids and solids ...... 310 85 his. Hyperelastic subfiuids ...... 314 86. Explicit forms of the stress relation for isotropic hyperelastic materials ...... 3 17 b) General theorems ...... 319 87. Consequences of restrictions upon the strain-energy function 319 88. Betti's theorem. Variational problems ...... 324 XXVIII Contents

Page 89. Stability ...... 328 90. Wave propagation 332 c) Solutions of special problems 336 91. Ericksen's analysis of the deformations possible in every isotropic hyperelastic body ...... 336 92. Special properties of some exact solutions for isotropic materials 342 93. Second-order effects in isotropic materials . 344 d) Special or approximate theories of hyperelasticity 347 94. The nature of special or approximate theories 347 95. The Mooney-Rivlin theory for rubber . . . 349 III. Various generalizations of elasticity and hyperelasticity 355 96. Thermo-elasticity ...... 355 96 bis. Coleman's general thermodynamics of simple materials 363 96 ter. Coleman and Gurtin's general theory of wave propagation in simple materials . . . . . 382 97. Electromechanical theories 385 98. Polar elastic materials 389 IV. Hypo-elastic materials ...... 401 99. Definition of a hypo-elastic material 401 100. Relation to elasticity 406 101. Work theorems ...... 411 102. Acceleration waves . 413 103. Solutions of special problems 415 E. Fluidity ...... 426 104. Scope and plan of the chapter 426 I. Simple fluids ...... 427 a) General classes of flows . . . . . 427 105. The concept of a simple fluid 427 106. Lineal flows, viscometric flows 429 107. Curvi1ineal flows . . . . . 432 I 08. Steady viscometric flows 435 109. Motions with constant stretch history 438 110. Dynamical preliminaries for the exact solutions of special flow problems ...... 440 b) Special flow problems ...... 441 Ill. Problems for lineal flows. Simple shearing, channel flow, lineal oscillations 441 112. Helical flows, I. General equations...... 445 113. Helical flows, II. Special cases: Flows in circular pipes 448 114. Normal-stress end effects ...... 452 115. Steady torsional flows ...... 458 116. Position of the general and Navier-Stokes theories of viscometry 465 117. Steady flow through tubes ...... 468 118. Steady extension ...... 472 118 bis. Special flow problems for subfluids 473 c) Special simple fluids ...... 475 119. The older theories of non-linear viscosity 475 119 A. Appendix. Truesdell's theory of the "Stokesian" fluid 485 120. Asymptotic approximations, I. Steady viscometric flows 488 121. Asymptotic approximations, II. General flows 490 122. Secondary flows in tubes 497 123. Fluids of second grade ...... 504 Contents XXIX

Page II. Other fluids ...... 513 124. Korteweg's theory of capillarity . . . . . 513 125. Truesdell's theory of the "Maxwellian" fluid 515 126. Anisotropic solids capable of flow 520 127. The anisotropic fluids of Ericksen, I. General theory 523 128. The anisotropic fluids of Ericksen, II. Statics . . . 528 129. The anisotropic fluids of Ericksen, III. Special flows 530 130. Theories of diffusion 537 General references . 542 List of works cited 542 Addendum 578 Sachverzeichnis (Deutsch-Englisch) 580 Subject-Index (English-German) 591