Collected Papers of R.S. Rivlin Volume I Springer Science+ Business Media, LLC Ronald S. Rivlin. Photo by Fella Studios, Inc. G.I. Barenblatt D.D. Joseph Editors Collected Papers of R.S. Rivlin Volume I With 323 Illustrations Springer Grigori Isaakovich Barenblatt G.l. Taylor Professor of Fluid Mechanics, Emeritus Department of Applied Mathematics and Theoretical Physics University of Cambridge Silver Street Cambridge CB3 9EW United Kingdom Daniel D. Joseph Department of Aerospace Engineering 107 Ackerman Hali University of Minnesota Il O Union Street, SE Minneapolis, MN 55455, USA Collected Papers of R.S. Rivlin Volume 1: 1-1424 Volume II: 1425-2828 Library of Congress Cataloging-in-Publication Data Rivlin, Ronald S. [Works. 1996] Collected papers of R.S. Rivlin 1 G.!. Barenblatt and D.D. Joseph, editors. p. cm. Includes bibliographical references. ISBN 978-1-4612-7530-5 ISBN 978-1-4612-2416-7 (eBook) DOI 10.1007/978-1-4612-2416-7 1. Continuum mechanics. 2. Nonlinear theories. 1. Barenblatt, G.l. II. Joseph, Daniel D. III. Title. QA3.R588 1996 531-cd20 96-22080 Printed on acid-free paper. © 1997 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1997 Softcover reprint of the hardcover 1st edition 1997 Ali rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Lesley Poliner; manufacturing supervised by Joe Quatela. 9 8 7 6 5 4 3 2 SPIN 10541642 Foreword Ronald S. Rivlin is an unusually innovative scientist whose main researches cover a range of topics that have become known as nonlinear continuum mechanics. His work includes a number of seminal contributions that qualify him as one of a small number of workers who may be credited with establishing this relatively new branch of theoretical physics. Al­ though many scientists are aware of those contributions of Rivlin that relate to their partic­ ular field of interest, fewer realize the full range and significance of his researches. This is, no doubt, partly due to the fact that he has never assembled his work as a treatise. It is to remedy this situation, at least partially, that we persuaded Rivlin to agree to the publication of this collection of most of his scientific papers. At our urging, Rivlin has prepared the autobiographical postscript in volume 1, in which he gives some account of how his ideas evolved and of the social setting in which this evolution took place. In view of this, our foreword is restricted to a brief mention of some of Rivlin's more important contributions. Inevitably there is some overlap with the postscript. Rivlin was born in London, England in 1915. After graduating from the University of Cambridge, from which he has B.A., M.A., and Sc.D. degrees, he spent seven years in re­ search on various aspects of electrical communications, first at the research laboratories of the General Electric Company (from 1937 to 1942) and then at the Telecommunications Research Establishment of the Ministry of Aircraft Production (from 1942 to 1944). He then joined the British Rubber Producers' Research Association and almost immediately became interested in the mechanics of materials. At the time, linear elasticity theory and the mechanics of Newtonian fluids were well­ developed mathematical disciplines that had, over a period of almost two centuries, at­ tracted the attention of many highly sophisticated mathematicians. Indeed, the develop­ ment of analysis was intimately interwoven with the development of these theories and their implications. Also, some attention had been directed spasmodically to the mechanics of viscoelastic materials, in which the stress depends on the history of the displacement gradients. In all these theories the stress is a linear function of the displacement gradients or velocity gradients, and it is this linearity that is largely responsible for the tractability of the theories. Although during the nineteenth and first half of the twentieth century the emphasis in elasticity theory was largely directed to infinitesimal deformations, some of the results ob­ tained applied also to deformations of any magnitude and even to inelastic materials. Also, in the early 1940s, there were some attempts, in the so-called kinetic theory of elastomers, to calculate the load-deformation behavior of vulcanized rubber, for some simple defor­ mations, from highly idealized models of its molecular structure. This was the situation when in 1944 Rivlin embarked on the development of a phenomenological theory of the mechanics of vulcanized rubber that would be valid for large elastic deformations. The theory he developed, which has become known as finite elasticity theory, has had a great influence on research on the mechanics of materials. Although his theory involves to some extent the earlier work on mechanics with finite deformations, Rivlin is usually-and quite v VI Foreword correctly-credited with having supplied those missing elements that were necessary in order to render it useful as a valid theory for the prediction and description of mechanical phenomena in materials that undergo large elastic deformations. In Rivlin's theory the mechanical behavior of the material is characterized by a strain­ energy function that is assumed to be a function of the deformation gradients. The assump­ tion is made that the material is isotropic and incompressible. This is physically justified for vulcanized rubber and for many other materials that can undergo large elastic defor­ mations. With these assumptions the strain-energy function can be expressed as a function of two invariants of a finite strain tensor, which is itself a nonlinear function of the defor­ mation gradients. From this a canonical form for the constitutive equation for the stress tensor can be derived by a simple mathematical argument. In it the mechanical properties specific to the particular material considered appear only through the first derivatives of the strain-energy function with respect to the two strain invariants. Rivlin then solved a number of simple problems without further assumption regarding the dependence of the strain-energy function on the strain invariants. By comparing these re­ sults with experiments on vulcanized rubber test-pieces he and his collaborators were able to determine this dependence for the particular vulcanizate considered. From the strain­ energy function so determined, the load-deformation behaviors for other types of defor­ mation were predicted and verified experimentally. Concerning this work, J.F. Bell has re­ marked in the introduction to his encyclopedic article The Experimental Foundations of Solid Mechanics, which forms volume VIall of the Handbuch der Physik, "The experi­ ments of Rivlin in the 1950s on the finite elasticity of rubber stand as a classic model; they emphasize what may be achieved in solid mechanics when rare insight is simultaneously focused on both experiment and theory." Later in the same article, in a section headed Ex­ periments on the finite elasticity of rubber: From Joule to Rivlin ( 1850s to 1950s ), Bell again remarks, "Certainly the most important 20th-century experimental development in the finite elasticity of rubber was the experiments of Ronald S. Rivlin and D.W. Saunders in 1951. ... These experiments ofRivlin and Saunders are a landmark in the history of ex­ perimental mechanics, as Rivlin's theoretical counterpart is in rational mechanics. Few of the subjects considered in this treatise have seen anything approximating the successful confluence of experimental observation and theoretical observation which this mid-20th­ century study achieved." While mainly as a result of the work of Rivlin, the chief elements of finite elasticity the­ ory and its application to rubber-like materials were understood by about 1952, since that time that theory has formed the basis for a great deal of research by Rivlin and others into the solution of problems involving finite elastic deformations, and it has given rise to a dis­ tinct discipline in the mechanics of continua. It has also had a profound influence on the development of the continuum mechanics of solids that are not ideally elastic and fluids that are not ideally viscous. Rivlin's theory has formed the basis for stress-analysis com­ puter programs, which are widely used in the design of rubber components, while the strain-energy functions he and his collaborators determined experimentally and their de­ pendence on various structural parameters, such as degree of cross-linking and swelling of the vulcanizate, have had a considerable influence on research on the interpretation of the mechanical behavior of elastomers in terms of molecular structure. Beyond his seminal work on the mechanics of vulcanized rubber, most of the contribu­ tions of Rivlin and his collaborators to finite elasticity over the past forty years or so lie in three main areas: (i) Paralleling the canonical strain-energy functions for compressible and incompress­ ible isotropic materials, he obtained with J.L. Ericksen the corresponding strain­ energy functions for elastic materials having fiber symmetry (transverse isotropy) and with G.F. Smith, those for each of the crystal classes. From these the canonical expressions for the stress can be obtained very simply. The results for isotropic and transversely isotropic materials have been used extensively in biomechanical studies.