Collected Papers of R.S. Rivlin: Volume I and II

2868 Pages·1997·117.594 MB·English

by  R. S. Rivlin (auth.), Grigori Isaakovich Barenblatt, Daniel D. Joseph (eds.)

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 1s t edition 1997
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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.
Foreword vii
(ii) In the 1950s, Rivlin and his collaborators, J.E. Adkins and J.L. Ericksen, published
a number of papers on the continuum mechanics of finitely deformed elastic mate
rials with intrinsic kinematic constraints other than that imposed by incompressibil
ity. This work was mainly directed to the study of elastomers reinforced by inexten
sible filaments. Some years later, with the growth of interest in fiber-reinforced
materials, this work was taken up by a number of workers.
(iii) From the canonical constitutive equation for the stress in an isotropic material it is
a simple matter to obtain the constitutive equation for the infinitesimal incremental
stress when an infinitesimal deformation is superposed on an underlying finite de
formation. This involves the strain-energy function through its first and second de
rivatives with respect to the strain invariants and provides a rational basis for the so
lution of initial-stress problems. An example is the calculation by Hayes and Rivlin
of the effect of an underlying finite pure homogeneous deformation on the propa
gation of waves of infinitesimal amplitude, which has been applied by Rivlin and
Sawyers, and by others, to the determination of conditions on the strain-energy func
tion to ensure stability of the material modeled. Rivlin (with Sawyers) has also ap
plied the constitutive equation for the incremental stress to the determination of the
existence of bifurcation solutions in a slab under thrust, or tension, without limita
tion on the thickness of the slab or on the strain-energy function.
In the last half century there has been a considerable amount of activity in the develop
ment of the continuum mechanics of both viscoelastic solids and fluids, particularly the lat
ter. Rivlin was one of the pioneers in this development and with various collaborators has
made numerous contributions to it throughout the whole period. Among his more signifi
cant contributions was the development, with J.L. Ericksen, of the Rivlin-Ericksen consti
tutive equations and with A.E. Green, of the Green-Rivlin constitutive equations, as well as
the establishment of the relation between them. These equations are canonical expressions
for the stress in a solid or fluid. The Rivlin-Ericksen equation for a solid stems from the as
sumption that the stress is a function of the deformation gradients and the gradients with
respect to the current configuration of the velocity, acceleration, ... ; for a fluid the depen
dence on the deformation gradients is omitted. The Green-Rivlin equations stem from the
assumption that the stress is a functional of the history of the deformation gradients. The
equations express the restrictions that are imposed on the stress by its invariance under su
perposed rigid motion of the system and by material symmetry, particularly isotropy.
In giving explicit expression to the material symmetry restrictions, theorems concern
ing the integrity bases for arbitrary numbers of symmetric second-order tensors, derived
by Rivlin, with A.J.M. Spencer and G.F. Smith, are used. The Rivlin-Ericksen equation for
an isotropic fluid has been used by Rivlin to obtain general solutions for the viscometric
flows, providing the first rational theoretical basis for the Weissenberg normal stress ef
fects. It has also been used as the basis for simple approximations, such as the slow flow and
slightly non-Newtonian approximations. Rivlin, with Langlois, was the first to use these to
predict new flow phenomena in non-Newtonian fluids.
Rivlin was also the first to create, in 1947, a plausible semiquantitative theory that pre
dicted, from the structure of a polymer solution, normal stresses of the observed order of
magnitude, anticipating further advances in this direction by roughly two decades.
It was realized quite early by Rivlin that some of the considerations involved in giving
canonical form to the constitutive equations of continuum mechanics had wide applicabil
ity in other areas of continuum physics. Once the vector or tensor variables that are related
in the constitutive equation are chosen, a canonical form for the equation, which expresses
any symmetry which it may possess, can be easily obtained if the integrity basis is known for
an appropriate set of tensors and vectors and the transformation group describing the sym
metry. Accordingly, with G.F. Smith and A.J.M. Spencer, Rivlin obtained integrity bases
for an arbitrary number of vectors and symmetric second-order tensors for the full and
proper orthogonal groups and the various crystal classes. Apart from their application in
Foreword
continuum mechanics already mentioned, these results have been applied to a limited ex
tent by Rivlin and his collaborators, and by others, to a few areas of continuum physics.
However, the potential of the methods and the viewpoint that Rivlin pioneered appear to be
far from exhausted.
In 1953, Rivlin, with A.G. Thomas, published a paper on the rupture of vulcanized rub
ber in which by simple but ingenious experiments and by drawing freely on the results in his
earlier papers on finite elasticity theory, they established the application of the Griffith cri
terion for fracture to vulcanized rubber. This paper is the seminal paper in a research field,
now highly developed, in which the rupture, fatigue failure, tensile strength, and other prop
erties of elastomers and other high-polymeric materials are interrelated.
In this foreword we have mentioned some ofRivlin's most important contributions. Many
more are mentioned by him in his autobiographical postscript. One may obtain an appreci
ation of the full range of his researches from a glance at the Bibliography, which includes
both the papers reprinted in these volumes and those that are not.
Quite apart from its scientific and technological value, Rivlin's work has had a consider
able influence on the teaching of continuum mechanics. He has also exerted a considerable
influence through the graduate students and other collaborators who were introduced by
him to the fields of his research and then went on to make distinguished contributions to
these fields.
Rivlin's contributions to science have been recognized by the award of honorary doctor
ates by the National University of Ireland, Nottingham University, Tulane University, and
the Aristotelian University of Thessaloniki and by his election to membership of the Na
tional Academy of Engineering and the American Academy of Arts and Sciences, and
honorary (foreign) membership in the Academia Nazionale dei Lincei and the Royal Irish
Academy. He has also received a number of prestigious awards, including the Bingham
Medal of the Society of Rheology, the Timoshenko Medal of the American Society of Me
chanical Engineers, the von Karman Medal of the American Society of Civil Engineers, the
Charles Goodyear Medal of the American Chemical Society (Rubber Division), and the
Panetti Prize and Medal of the Accademia delle Scienze di Torino.
G. I. Barenblatt and D.D. Joseph
List of Collaborators
J.E. Adkins F.J. Lockett
B.Y. Ballal D.C. Messersmith
J.T. Bergen L. Mullins
N.R. Campbell A.C. Pipkin
M.M. Carroll J.R.M. Radok
R.V.S. Chacon H.P. Rooksby
E.C. Cherry D.W. Saunders
B.A. Cotter K.N. Sawyers
J.L. Ericksen R.T. Shield
L.I. Farren G.F. Smith
S.M. Genensky M.M. Smith
A.N. Gent A.J.M. Spencer
A.E. Green A.G. Thomas
W.A. Green C. Topakoglu
H.W. Greensmith R.A. Toupin
S.M. Gumbrell R. Venkataraman
M.A. Hayes J.B. Walker
J. Y. Kazakia K. Wilmanski
W.E. Langlois W.A. Wooster
G. Lianis