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
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© 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
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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. 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 Contents
Foreword G./. Barenblatt and D. D. Joseph v
List of Collaborators ix
Autobiographical Postscript R.S. Rivlin xvii
Bibliography of the Publications of R.S. Rivlin xlvii
A. Isotropic Finite Elasticity I. Torsion of a Rubber Cylinder 3 2. Some Applications of Elasticity Theory to Rubber Engineering 9 3. A Uniqueness Theorem in the Theory of Highly-Elastic Materials 17 4. A Note on the Torsion of an Incompressible Highly-Elastic Cylinder 20 5. Large Elastic Deformations of Isotropic Materials. I. Fundamental Concepts 23 6. Large Elastic Deformations of Isotropic Materials. II. Some Uniqueness Theorems for Pure, Homogeneous Deformation 55 7. Large Elastic Deformations of Isotropic Materials. Ill. Some Simple Problems in Cylindrical Polar Coordinates. 73 8. Large Elastic Deformations of Isotropic Materials. IV. Further Developments of the General Theory 90 9. Large Elastic Deformations of Isotropic Materials. V. The Problem of Flexure 109 10. Large Elastic Deformations of Isotropic Materials. VI. Further Results in the Theory of Torsion, Shear and Flexure 120 11. An Informal Discussion on the Theory of Large Elastic Strains. I. On the Definition of Strain 143 12. Large Elastic Deformations of Isotropic Materials. VII. Experiments on the Deformation of Rubber 157 13. Large Elastic Deformations of Isotropic Materials. VIII. Strain Distribution About a Hole in a Sheet 195 14. Large Elastic Deformations of Isotropic Materials. IX. The Deformation of Thin Shells 205 15. Experiments on the Mechanics of Rubber. I. Eversion of a Tube 232 16. Experiments on the Mechanics of Rubber. II. The Torsion, Inflation, and Extension of a Tube 236 17. Experiments on the Mechanics of Rubber. Ill. Small Torsions of Stretched Prisms 251
xi xii Contents
18. The Free Energy of Deformation for Vulcanized Rubber 255 19. Departures of the Elastic Behaviour of Rubbers in Simple Extension from the Kinetic Theory 262 20. The Solution of Problems in Second-Order Elasticity Theory 273 21. A Theorem in the Theory of Finite Elastic Deformations 302 22. Stress-Relaxation in Incompressible Elastic Materials at Constant Deformation 311 23. Large Elastic Deformations 318 24. Dimensional Changes in Crystals Caused by Dislocations 352 25. Some Topics in Finite Elasticity 360 26. Energy Propagation for Finite Amplitude Shear Waves 390 27. Energy Propagation in a Cauchy Elastic Material 394 28. Stability of Pure Homogeneous Deformations of an Elastic Cube Under Dead Loading 398 29. The Strain-Energy Function for Elastomers 405 30. Some Research Directions in Finite Elasticity Theory 418 31. Reflections on Certain Aspects of Thermomechanics 430
B. Anisotropic Finite Elasticity, Kinematic Constraints
1. Large Elastic Deformations of Homogeneous Anisotropic Materials 467 2. Large Elastic Deformations of Isotropic Materials. X. Reinforcement by Inextensible Cords 488 3. Plane Strain in a Net Formed by Inextensible Cords 511 4. Stress-Deformation Relations for Anisotropic Solids 535 5. The Strain-Energy Function for Anisotropic Elastic Materials 541 6. Minimum Weight Design for Pressure Vessels Reinforced with Inextensible Fibers 560 7. Networks of lnextensible Cords 566 8. Constitutive Equation for a Fiber-Reinforced Lamina 580
c. Superposition of Small Deformations on Finite Deformations in Elastic Materials, Stability
1. General Theory of Small Elastic Deformations Superposed on Finite Elastic Deformations 589 2. Propagation of a Plane Wave in an Isotropic Elastic Material Subjected to a Pure Homogeneous Deformation 617 3. Surface Waves in Deformed Elastic Materials 625 4. A Note on the Secular Equation for Rayleigh Waves 649 5. Seismic Wave Propagation in a Self-Gravitating Anisotropic Earth 653 6. Energy Propagation in a Deformed Elastic Material 692 7. Instability of an Elastic Material 702 8. Stability Criteria for Elastic Materials 709 9. Bifurcation Conditions for a Thick Elastic Plate Under Thrust 720 10. The Flexural Bifurcation Condition for a Thin Plate Under Thrust 738 11. On the Speed of Propagation of Waves in a Deformed Elastic Material 743 12. On the Speed of Propagation of Waves in a Deformed Compressible Elastic Material 756 13. A Note on the Hadamard Criterion for an Incompressible Elastic Material 763 14. The Incremental Shear Modulus in an Incompressible Elastic Material 767 Contents xiii
15. Some Stability Conditions for a Compressible Elastic Material 772 16. Stability of a Thick Elastic Plate Under Thrust 784 17. Bifurcation in an Elastic Plate on a Rigid Substrate 809 18. Some Thoughts on Material Stability 817 19. Further Results on the Stability of a Thick Elastic Plate Under Thrust 835 20. The Incremental Shear Modulus in a Compressible Isotropic Elastic Material 871 21. Stability of an Elastic Material 883
D. Constitutive Equations, Invariants
1. Stress-Deformation Relations for Isotropic Materials 911 2. Further Remarks on the Stress-Deformation Relations for Isotropic Materials 1014 3. Tensors Associated with Time-Dependent Stress 1036 4. The Anisotropic Tensors 1042 5. The Mechanics of Non-Linear Materials with Memory. Part 1 1049 6. Note on a Paper "Further Remarks on the Stress-Deformation Relations for Isotropic Materials" 1070 7. The Theory of Matrix Polynomials and its Application to the Mechanics of Isotropic Continua 1071 8. Finite Integrity Bases for Five of Fewer Symmetric 3 X 3 Matrices 1099 9. The Formulation of Constitutive Equations in Continuum Physics. I 1111 10. The Mechanics of Non-Linear Materials with Memory. Part 2 1127 11. The Constitutive Equations for Certain Classes of Deformations 1137 12. Further Results in the Theory of Matrix Polynomials 1146 13. The Formulation of Constitutive Equations in Continuum Physics. II 1164 14. Constitutive Equations for Classes of Deformations 1176 15. The Mechanics of Non-Linear Materials with Memory. Part III 1192 16. Stress-Relaxation for Biaxial Deformation of Filled High Polymers 1210 17. Small Deformations Superposed on Large Deformations in Materials with Fading Memory 1225 18. Constitutive Equations Involving the Functional Dependence of One Vector on Another 1237 19. Isotropic Integrity Bases for Vectors and Second-Order Tensors. Part I 1243 20. Constraints on Flow Invariants Due to Incompressibility 1262 21. Integrity Bases for Vectors-the Crystal Classes 1265 22. Integrity Bases for a Symmetric Tensor and a Vector- the Crystal Classes 1318 23. On Cauchy's Equations of Motion 1359 24. Mechanics on Rate-Independent Materials 1362 25. Nonlinear Viscoelastic Solids 1377 26. Simple Deformations of Materials with Memory 1395 27. Some Remarks on the Mechanics of Nonlinear Viscoelastic Materials 1416 28. On the Principles of Equipresence and Unification 1425 29. Materials with Memory 1427 30. A Note on the Onsager-Casimir Relations 1447 31. The Elements of Non-Linear Continuum Mechanics 1451 32. Some Restrictions on Constitutive Equations 1476 33. Finite Elasticity Theory as a Model in the Mechanics of Viscoelastic Materials 1505 34. Functional Constitutive Equations: Mathematical Formalism and Physical Reality 1525 XIV Contents
35. On the Foundations of the Theory of Non-Linear Viscoelasticity 1532 36. On Identities for 3 X 3 Matrices 1550 37. The Thermodynamics of Materials with Fading Memory 1559 38. Material Symmetry and Constitutive Equations 1580 39. Some Comments on the Endochronic Theory of Plasticity 1592 40. A Note on the Simple Fluid 1610 41. Integral Representations of Constitutive Equations 1615 42. Two-Dimensional Constitutive Equations 1623 43. On Constitutive Functionals 1638 44. The Description of Material Symmetry in Materials with Memory 1654 45. The Passage from Memory Functionals to Rivlin-Ericksen Constitutive Equations 1664 46. A Note on Material Frame Indifference 1670 47. Some Remarks Concerning Material Symmetry 1675 48. Comments on the Paper "On the Derivation of the Constitutive Equation of a Simple Fluid from That of a Simple Material" by R.R. Huilgol 1683 49. The Simple Fluid Concept 1690 50. An Isotropic Solid is a Simple Fluid 1702 51. Objectivity of the Constitutive Equation for a Material with Memory 1710 52. Frame Indifference 1713
E. Internal Variable Theories
1. Simple Force and Stress Multipoles 1725 2. Multipolar Continuum Mechanics 1754 3. Generalized Continuum Mechanics 1789 4. The Mechanics of Materials with Structure 1804 5. The Relation Between Director and Multipolar Theories in Continuum Mechanics 1812 6. The Passage from a Particle System to a Continuum Model 1824
F. Non-Newtonian Fluids
1. Hydrodynamics of a Non-Newtonian Fluids 1839 2. The Hydrodynamics of Non-Newtonian Fluids. I 1842 3. The Hydrodynamics of Non-Newtonian Fluids. II 1864 4. The Normal-Stress Coefficient in Solutions of Macromolecules 1868 5. Some Flow Properties of Concentrated High-Polymer Solutions 1878 6. Measurement of the Normal Stress Effect in Solutions of Polyisobutylene 1887 7. The Hydrodynamics of Non-Newtonian Fluids. III. The Normal Stress Effect in High-Polymer Solutions 1891 8. Solutions of Some Problems in the Exact Theory of Viscoelasticity 1921 9. Steady Flow of Non-Newtonian Fluids Through Tubes 1931 10. The Relation Between the Flow of Non-Newtonian Fluids and Turbulent Newtonian Fluids 1941 11. Correction to My Paper "The Relation Between the Flow of Non-Newtonian Fluids and Turbulent Newtonian Fluids" 1945 12. Slow Steady-State Flow of Visco-Elastic Fluids Through Non-Circular Tubes 1946 13. Normal Stresses in Flow Through Tubes of Non-Circular Cross-Section 1963 14. Second and Higher-Order Theories for the Flow of a Visco-Elastic Fluid in a Non-Circular Pipe 1968 Contents XV
15. Stability in Couette Flow of a Viscoelastic Fluid. Part I 1978 16. Nonlinear Continuum Mechanics of Viscoelastic Fluids 2002 17. Stability in Couette Flow of a Viscoelastic Fluid. Part II 2032 18. Flow of a Viscoelastic Fluid Between Eccentric Cylinders. I. Rectilinear Shearing Flow 2058 19. Flow of a Viscoelastic Fluid Between Eccentric Cylinders. II. Fourth-Order Theory for Longitudinal Shearing Flow 2067 20. Flow of a Viscoelastic Fluid Between Eccentric Rotating Cylinders 2087 21. A Note on Secondary Flow of a Non-Newtonian Fluid in a Non-Circular Pipe 2124 22. Secondary Flows in Viscoelastic Fluids 2129 23. Some Superposition Theorems for Second-Order Fluids 2141 24. Flow of a Viscoelastic Fluid Between Eccentric Rotating Cylinders and Related Problems 2148 25. The Influence of Vibration on Poiseuille Flow of a Non-Newtonian Fluid. I 2159 26. The Effect of Vibration on the Flow of a Turbulent Newtonian Fluid 2176 27. The Effect of a Longitudinal Vibration on Poiseuille Flow in a Non-Circular Pipe 2178 28. The Influence of Vibration on Poiseuille Flow of a Non-Newtonian Fluid. II 2188 29. Flow of a Viscoelastic Fluid Between Eccentric Cylinders. III. Poiseuille Flow 2200 30. Run-Up and Spin-Up in a Viscoelastic Fluid I 2212 31. Superposition of Longitudinal and Plane Flows of a Non-Newtonian Fluid Between Eccentric Cylinders 2229 32. Run-Up and Spin-Up in a Viscoelastic Fluid II 2236 33. Run-Up and Spin-Up in a Viscoelastic Fluid III 2241 34. Spin-Up in Couette Flow 2251 35. Run-Up and Spin-Up in a Viscoelastic Fluid IV 2278 36. Run-Up and Decay of Plane Poiseuille Flow 2287 37. Decay of Shear Layers and Vortex Sheets 2302 38. The Propagation of Vorticity in a Viscoelastic Fluid 2330
G. Electromagnetism
1. Electrical Conduction in Deformed Isotropic Materials 2349 2. Linear Functional Electromagnetic Constitutive Equations and Plane Waves in an Isotropic Material 2353 3. Galvanomagnetic and Thermomagnetic Effects in Isotropic Materials 2364 4. Electrical Conduction in a Non-Circular Rod 2369 5. Electro-Magneto-Optical Effects 2373 6. Electrical Conduction in a Stretched and Twisted Tube 2389 7. Non-Rectilinear Current Flow in a Straight Conductor 2392 8. Propagation of Electromagnetic Waves in Circular Rods in Torsion 2396 9. The Faraday and Allied Effects in Circular Wave-Guides 2424 10. Maxwell's Equations in a Deformed Body 2436 11. Transverse Electric and Magnetic Effects 2439 12. Electrical, Thermal, and Magnetic Constitutive Equations for Deformed Isotropic Materials 2443 13. Electro-Optical Effects. I 2471 14. Electro-Optical Effects. II 2475 15. Electro-Optical Effects. III 2479 XVI Contents
16. Magneto-Optical Effects 2492 17. Photoelasticity with Finite Deformations 2526 18. Birefringence in Viscoelastic Materials 2541 19. Propagation of First-Order Electromagnetic Discontinuities in an Isotropic Medium 2556 20. Propagation of an Electromagnetic Shock Discontinuity in a Non-Linear Isotropic Material 2567 21. Harmonic Generation in an Electromagnetic Wave 2574 22. Electro-Optical and Magneto-Optical Effects 2589
H. Fracture
1. The Effective Work of Adhesion 2611 2. Rupture of Rubber. Part I. Characteristic Energy for Tearing 2615 3. The Trousers Test for Rupture 2643 4. The Incipient Characteristic Tearing Energy for an Elastomer Cross-Linked Under Strain 2649 5. The Effect of Stress Relaxation on the Tearing of Vulcanized Rubber 2657
I. Waves in Viscoelastic Materials
1. Propagation of Sinusoidal Small-Amplitude Waves in a Deformed Viscoelastic Solid. I 2673 2. Propagation of Sinusoidal Small-Amplitude Waves in a Deformed Viscoelastic Solid. II 2680 3. Longitudinal Waves in a Linear Viscoelastic Material 2692 4. A Class of Waves in a Deformed Viscoelastic Solid 2696 5. Plane Waves in Linear Viscoelastic Materials 2701
J, Crystal Physics
1. Grinding and Scratching Crystalline Surfaces 2713
K. General
1. Mathematics and Rheology: the 1958 Bingham Medal Address 2717 2. The Fundamental Equations of Nonlinear Continuum Mechanics 2721 3. Red Herrings and Sundry Unidentified Fish in Nonlinear Continuum Mechanics 2765 4. Forty Years of Non-Linear Continuum Mechanics 2783
L. Miscellaneous
1. Thermo-Elastic Similarity Laws 2815 Autobiographical Postscript R.S. Rivlin
Some time ago I agreed to a suggestion by my friends Dan Joseph and G.l. (Grisha) Baren• blatt that a collection of my scientific papers be republished with themselves as editors. In short order Dan negotiated with Springer-Verlag a contract for the publication of the vol• umes. The question arose of which of my papers should be omitted. While there were anum• ber of obvious candidates for inclusion and for exclusion, there was inevitably a significant gray area. I have taken a generous view in opting for inclusion. I also agreed, at the suggestion of the editors, to write this "autobiographical postscript," in which I attempt to give some sense of how the ideas in my research evolved and the set• ting in which this evolution took place. I have dealt mainly with work done prior to about 1970. This is primarily because my later work was largely an extension of ideas developed in the earlier period. Moreover, the change in the climate of research at about that time and the emphasis on so-called "relevance" led to a change in the motivation for my choice of problems. While this had an inhibiting effect on my research, I doubt that it did, in fact, make it more "relevant." This essay is not a comprehensive account of my research. Some papers that may have merit in their own right but have not been important in the development of my ideas are not mentioned. During the course of my career I have been fortunate in having had a number of ex• tremely able graduate students and other collaborators from whom I learnt a great deal. In most cases they also became my friends. I am grateful to them both for their scientific col• laboration and for their friendship. * * * I was born in London in 1915. I have very few recollections of my early childhood and suppose that it was quite uneventful. My sister, three years my senior, confirms this. From the age of five to eight I attended a small private school at which I was taught to read and write, as well as the elements of arithmetic. When I was eight I moved to a grammar school where I remained until I was eighteen. Throughout my school career I was a "good" student. This may have been due as much to family influence as to innate ability. Both my father and mother set great store by my do• ing well academically. Books were very much a part of the house we lived in and as a child and throughout my teens I read voraciously, without much discrimination, almost every• thing that came to hand, whether fiction or nonfiction. At the beginning of the 1933-34 academic year I entered St. John's College in the Uni• versity of Cambridge with the intention of obtaining a degree in mathematics. My choice of mathematics as my field of study, which had been made some years earlier, seems in ret• rospect to have been almost frivolous. During my school years I had been uniformly at the top, or near the top, of my class. But mathematics was only one of several subjects in which I did well. From the age of twelve onwards I fell under the influence of an extremely gifted teacher of mathematics, Frederick Swan, who felt that I had some mathematical tal• ent and encouraged me to specialize in mathematics.
xvii XVlll Autobiographical Postscript
* * * At Cambridge, in view of the level I had reached at school, I was excused from the re• quirement to take Part I of the Mathematical Tripos and embarked immediately on a two• year program of study for Part II. I did less well in the final examination than was ex• pected, achieving second rather than first class honors. This was probably due, at any rate in part, to my having spent much of my time in cultivating interests in a variety of cultural fields, in discussion, and in social development. Instead of going on to read for Part III of the Mathematical Tripos, I decided to change my field. My first choice was anthropology. My interest in the subject dated from child• hood. From the time I was about ten years old I had an intense interest in geography and in exotic cultures, whether geographically or historically distant. My parents and other fam• ily members encouraged these interests, mainly by gifts of books. In my later teens I pur• sued this interest further. Despite this background, my advisors felt that it would not be wise for me to read for Part II of the Archaeology and Anthropology Tripos, to which my years of mathematical study would contribute virtually nothing. One of them suggested that I read instead for Part II of the Physics Tripos. Largely for lack of any better idea I agreed to do so. In retrospect it is quite surprising to me that both the initial decision to study mathematics and the subsequent change to physics were made rather casually and without any significant discussion of the type of career to which these studies might lead. Attempts by my parents to elicit some statement of my plans in this direction always re• sulted in vague generalities. Although initially I planned to take Part II of the Physics Tripos at the end of a year, it soon became obvious that I would not be ready to do so, and accordingly my stay at Cam• bridge was extended for a fourth year. * * * The curriculum for Part II of the Mathematics Tripos consisted almost equally of pure and applied mathematics. I had a natural inclination to the applied mathematics courses-rigid body mechanics, fluid dynamics, electromagnetic theory, thermodynamics. I recall with particular appreciation courses on mechanics and electromagnetic theory given by Sydney Goldstein, who was also my supervisor for two terms. My other supervisors were E. Cun• ningham, M.H.A. Newman, and F.P. White. Many years later, in the summer of 1957, S. Goldstein, F. John, L. Bers, M. Kac, and I were the principal lecturers at an American Mathematical Society summer school at the University of Colorado in Boulder. Since Goldstein did not have a car with him, I invited him and his wife to accompany me and my wife on scenic drives in the Rockies. During one of these I asked him whether he remembered that he had been my supervisor at Cam• bridge. He said that he did not, but I should not be offended at this, since "there were so many of you and only one of me." I was not offended. At the time I was studying physics, the physics department, centered in the Cavendish Laboratory, was without question the world center for modern physics. During the two• year period, I attended courses by such luminaries as Rutherford, Dirac, Born, Cockcroft, and Appleton, and short series of lectures by J.J. Thomson, C.T.R. Wilson, and Aston, to mention just a few. Rutherford's course on atomic and nuclear physics was particularly memorable. In it the subject was treated historically and with great simplicity. More remarkably, Rutherford made me share the excitement of the discoveries as though T had actually had a part in them. The course stressed the importance of a highly critical approach in establishing the validity of a theory and the importance of not ignoring even minor discrepancies between the predictions of the theory and experimental observations. It is just from the pursuit of such discrepancies that major advances may follow. Rutherford also emphasized the im• portance of testing hypotheses independently of each other by critical experiments. To Rutherford the important element of a theory in physics was the physical model rather than the mathematical analysis based on it. He expressed this by saying, "The important thing is Autobiographical Postscript XIX to get the physics right; you can always find someone who will do the mathematics for half-a-crown." I find myself to this day contrasting Rutherford's approach with the uncrit• ical attitude that prevails in much of contemporary engineering science. The laboratory courses, taught by Searle and Dee, were less happy. Searle had a consid• erable reputation, both as a teacher and as an eccentric. Many of the stories about him were, no doubt, apocryphal. One, which smacks of gender discrimination and an obsolete familiarity with female attire, was his complaining to the female students that the steel ribs in their corsets disturbed the magnets. Searle's impatient attitude, and to a lesser extent Dee's, did much to reinforce my lack of confidence in my ability to carry out experiments. This has remained with me to this day and has been a great handicap in my research. My supervisors during my two years of physics were J.D. Cockcroft, M. Oliphant, and R. Peierls. Both Cockcroft and Peierls were very stimulating and I regard myself as highly privileged to have been their student. I regret not having taken more advantage of the op• portunities that this presented. Oliphant, on the other hand, seemed at that time to be ob• sessed with the problems arising in the design and fabrication of diffraction gratings. No matter where our discussions started they moved quickly to this topic.
* * * I spent the summer before I graduated as an unpaid assistant to N.R. Campbell at the re• search.laboratories ofthe General Electric Company (G.E.C.) at Wembley, near London. I was asked to make some measurements on the time-lag in gas-filled photocells. Apparently my performance was satisfactory, since at the end of the summer Campbell told me he thought that I had a natural gift for research and that if after graduation I needed a job, there would be one for me in the General Electric Company. He added that he did not see why I would want it. Since my father died the January before my graduation it became even more important than it otherwise would have been that I get a job. Accordingly, I applied to the G.E.C. and was appointed as a research physicist, thanks to the good offices of Campbell at a some• what higher salary than the standard for a beginner. For the first year I was given the task of designing the intermediate frequency stages of what was intended to be the first com• mercial television receiver. (Of course, as a result of the outbreak of the Second World War, the project was stillborn.) This gave me no opportunity for exercising my mathemat• ical skills, and accordingly, at the end of a year I was transferred to work on telephony, re• porting to Leslie Farren, with whom I maintained a happy and fruitful collaboration until I left the G.E.C. in 1942. At the time, the introduction of coaxial cables for multichannel long distance telephony created many problems in the design of passive circuits, such as filters and equalizers. One of the problems I was asked to tackle was that of optimizing the design of band-pass filters. In order to obtain low attenuation in the pass-band, high attenuation outside it, and steep transition from the region of low to that of high attenuation, coupled with stable perfor• mance, quartz piezoelectric elements were used as circuit elements. The fabrication of such piezoelectric elements raised a number of problems that led me to an interest in crys• tal physics. With the advent of World War II these problems became more urgent, since piezoelectric elements involving similar fabrication problems were required in large num• bers for frequency control in radio communications and in radar. At the same time it had become very difficult to obtain a sufficient supply of crystalline quartz. However, the Sal• ford Electric Company, the subsidiary of the G.E.C. that manufactured piezoelectric ele• ments, had a large stock of quartz crystals that had been set aside in more ample times be• cause they did not have defined facets that could be used as a starting point in cutting crystal plates with a specified orientation. I was given the problem of devising an easily implemented procedure that would enable us to use the stocks of unfaceted crystals. Dur• ing the course of this work I made an interesting observation [Jl]. A slab of quartz crystal with parallel surfaces normal to its optic axis has one of these surfaces coarsely abraded, and a point source of light is viewed through it. A hexagonal XX Autobiographical Postscript patch of light reflecting the hexagonal symmetry of the quartz structure is observed. Each point of this pattern corresponds to a specific orientation of surface elements such that light from this point is refracted into the eye of the observer. The vertices of the hexagonal patch have a slightly greater luminosity than the remainder of the patch and correspond to surface elements having the orientations (lOil) and (Olll). This implies that these planes are weakly defined cleavage planes. I then looked, in a similar manner, at a number of crystals that had more defined cleavage and slip planes than quartz and that had a surface abraded either by grinding on a lap or by drawing parallel scratches. In each case I found that the roughening of the surface was due mainly to fracture on cleavage planes and on zones of surfaces containing the slip planes. Microscopic examination showed roughening that was consistent with this observation. Most of this work was never published. I men• tion it here mainly because it led me some years later to suggest a point of departure for work on the abrasion of rubber, which turned out to be very fruitful. This work on the abrasion of crystals proved fortuitously to be formative in quite an• other way. When I wrote a note reporting some of this work for publication in Nature, I originally called the luminous patterns produced by the abraded surfaces refractograms. I showed the note to Campbell, whose somewhat angered response was "I am sure that the language of Shakespeare and Milton is rich enough to express any ideas that you may have, without the need to invent new words." I was so impressed by this remark, overstated though it was, that to this day I have an antipathy to the (increasingly prevalent) custom of inventing new words. Another area in which my interest in crystal physics proved valuable later was that it intro• duced me to linear elasticity theory, particularly in its application to anisotropic materials. During the whole of my stay at the G.E.C. ( 1937 -42) I benefited greatly from the inter• est of Campbell, who was always ready to discuss my ideas critically and to share with me his extensive knowledge of physics. Of even greater influence was the relationship that I established with Marcello Pirani, the inventor of the vacuum gauge that bears his name. Pi• rani was a refugee from Hitler's Germany, where he had been, I believe, a director of re• search of the Osramkonzern. Apart from being a man of great personal charm and broad culture, he had an extraordinary knowledge of physics, which he was prepared to discuss with me for hours on end. These discussions centered for the most part on the relation be• tween the macroscopic physical properties of materials and their structures. I feel that it is through these discussions that I really began to understand physics. It was while I was at the G.E.C. that I first met L.R.G. Treloar, who was to have later a considerable influence on my career, although at the time my contacts with him were only superficial. While my stay at the G.E.C. was a fairly happy one, it was frustrating in two respects. In the field of telephony, in which I was working, the position of the Bell Laboratories was so dominant that I had the sense of being in a backwater. A great deal of the research and de• velopment at the G.E.C. seemed to be directed more to honing its technical expertise and to improving its bargaining position in licensing negotiations than to making significant technical breakthroughs. Also, it was not possible to devote more than a minimum amount of time to pursuing an interest beyond the point at which it was perceived as having poten• tial for increasing the company's profits in the near future; the G.E.C. was not in the busi• ness of fundamental research. I was told some years later by Treloar that he had experi• enced similar frustrations while working for the G.E.C. When he expressed these to Clifford Paterson, the director of research, he was told, "Your trouble, Treloar, is that you want to be a Faraday. We have had one Faraday already and that is quite enough." Paterson liked to express his views of the contributions of mathematicians (which paralleled to some extent those of Rutherford) by the dictum "Mathematicians are like manure. Spread thinly over the countryside they are wholesome and fruitful, but in the mass they are nauseous." Of course, during the war years, which covered about half of the time I worked for the G.E.C., the short-term policy adopted by management was inevitable. To what extent a similar policy is appropriate in peacetime depends of the nature of the corporation's busi• ness and on its competitive position. I have noticed that, with a few notable exceptions, Autobiographical Postscript XXI when a corporation embarks on an ambitious program of fundamental research, its enthu• siasm is short-lived and rapidly founders on the absence of clearly identifiable financial benefit.
* * * In the spring of 1942, motivated partly by restlessness and partly by a desire to get closer to the war effort, I moved to the Telecommunications Research Establishment (T.R.E.), the central radar research and development establishment of the Ministry of Aircraft Produc• tion. This was located at Swanage, on the south coast of England, for the first few months after I joined it. However, the fall of France had left it exposed to air attack and to the pos• sibility of commando raids. Indeed, on one occasion intelligence reports that a commando raid was imminent led to the whole establishment being put on trucks and moved to the New Forest for a few days. As a result it was decided to move the establishment to a more secure inland location. The spa resort and retirement town of Malvern, situated in the west of England on the eastern slopes of the Malvern Hills and overlooking the beautiful Vale of Evesham, was chosen. The establishment was located in Malvern College, a prestigious boys' school. During the almost two and a half years that I spent at T.R.E. I reported to J.A. Ratcliffe. My office was located in the T.R.E. radar school, the head of which was L.G.H. Huxley, who also reported to Ratcliffe, and with whom he appeared to be in conflict much of the time. One of my more vivid recollections of this period is that of Huxley emerging from an interview with Ratcliffe, his face purple with rage. Ratcliffe thought that it would be useful if a series of reports was produced summariz• ing, from a scientific standpoint, the advances that had been made in various areas as a re• sult of the development of radar during the war and in the years immediately preceding. He made this my main assignment at T.R.E. This was hardly what I had in mind when I moved from the G.E.C. to T.R.E., but I was reconciled by the obvious interest of the project and the opportunity it presented to obtain a very wide knowledge of radar. Indeed, much of my activity consisted in visiting research units at universities and elsewhere and discussing their activities with the principal participants. In addition to this assignment, I was asked to look, from a theoretical point of view, into the use of nonlinear circuitry as a means of im• proving signal-to-noise ratios in radar systems. I also had the opportunity on a number of occasions for sharing with Ratcliffe short courses on radar developments for senior officers in the Navy, Army, and Air Force. This was particularly interesting since it gave me an opportunity for understanding Ratcliffe's engaging style of lecturing better than I had when I attended his lectures as a student at Cambridge.
* * * Towards the end of my stay at T.R.E. I was able to renew my acquaintance with Treloar. He had some years earlier left the G.E.C. to join an organization, the British Rubber Produc• ers' Research Association (B.R.P.R.A.), that had been started a year or so before the war in the Colloid Science Department at Cambridge and then moved to Welwyn Garden City, about twenty miles north of London. The organization was financed, through the Colonial Office, by a levy on rubber exported from the rubber-growing colonies, mainly Malaysia. It was controlled by a board whose membership consisted of representatives of the rubber growers, the Colonial Office, and two eminent scientists, the organic chemist and Nobel laureate Sir Norman Howarth and the physical chemist Sir Eric Rideal. Sometime in 1943, Treloar appeared at T.R.E., having been moved there, along with a number of other mature scientists, by government directive. While he was there, we saw a good deal of each other. In common with many of the others who were brought to T.R.E. at that time, Treloar felt that his talents were not being adequately used by T.R.E. and that his recruitment had been largely the result of empire building. In my conversations with Treloar, he consistently expressed his total satisfaction with xxii Autobiographical Postscript the B.R.P.R.A. and his intention of returning there as soon as he could. He suggested to me that I consider joining the organization when the war was over. With this in mind, in the summer of 1944 he set up a meeting with John Wilson, the director of the B.R.P.R.A., at an inn in a nearby town. Wilson turned out to be a bluff, outgoing, middle-aged Yorkshire• man. After a lavish meal, some drinks, and a game of pool that I played very badly, he of• fered me a job as research physicist for as soon as I could take it up. The war was evidently winding to a close and I was somewhat nervous at the prospect of being thrown onto the job market, along with thousands of others, in the dislocations that would inevitably follow. Moreover, it was evident that nothing that I was doing at the time could have any effect on the war. I therefore decided to explore the possibility of making the move right away. It emerged that this would be possible if I could persuade Sir Robert Watson Watt of its propriety. This I was able to do at a meeting with him in Whitehall. Ac• cordingly, in August 1944 I took up an appointment as research physicist at the B.R.P.R.A. The move involved a complete change in the direction of my research activities, a fact that appeared to me as a positive factor. Although I had, in the seven years since my graduation, published a number of papers in reputable scientific journals and was the inventor or co• inventor on a number of patents, I had not carved out an area of research with which I felt identified and to which I was fully committed. * * * During the War, although Scotch tape was used extensively in the packaging of supplies, the manufacturers lacked an understanding of the desirable rheological properties of the adhesive component. I was asked to look into this question and, more generally, into the question of what is meant when a material is said to be "sticky." The qualitative answer to these questions turned out to be rather simple [H1]. I quickly realized that although very little force is required to detach Scotch tape from an adherend, the work expended in so doing is very large-many orders of magnitude larger than the interfacial energy of the adhesive-adherend interface. This work evidently results from the fact that before an ele• ment of the adhesive can be detached from the adherend, it has to undergo a large defor• mation. Though the force may be small, it is the magnitude of this deformation, whether elastic or inelastic, that yields a large value for the work. This trivial observation, sup• ported by a few measurements, seemed to delight the manufacturers. Pursuing this train of thought, I wanted, as an ultimate objective, to calculate the manner in which the work required to peel the tape from the adherend depends on the rheological properties of the adhesive, the adhesive-adherend interfacial energy, and other factors. Of course, I realized that this would be very difficult and perhaps impossible. However, I thought that it might be possible by appropriate idealizations to illuminate the matter. I supposed that the adhesive detaches from the adherend when the release of energy stored in the adhesive exceeds the increase in interfacial energy. I quickly discovered that even if one idealized the adhesive as a perfectly elastic material, there appeared to be no body of mathematical theory that would provide a basis for carrying out such calculations, even in principle, in view of the large deformations to which the adhesive is subjected. What was lacking was a body of theory that would parallel, for an elastic material that is capable of undergoing large deformations, the classical elasticity theory for infinitesimal deformations. I therefore embarked on the construction of such a theory, that would be ap• plicable to vulcanized rubber. In this project I was enthusiastically encouraged by John Wilson, the director of the B.R.P.R.A., as well as by senior staff members. While this led me away from the study of soft adhesives, the seeds I had sown bore fruit many years later in the elegant and insightful work of Alan Gent at the University of Akron. (Of course, the point of view underlying my approach and that of Alan Gent has its antecedent in the criterion for brittle fracture of hard elastic solids, advanced by Griffith in 1920.) It is inter• esting also that a rather similar model was used much later by Roberts in his work, at the B.R.P.R.A., on the rolling friction of rubber on a hard surface. * * * Autobiographical Postscript xxiii
When I embarked on the task of constructing a phenomenological theory of rubber elastic• ity I had only a limited experience with classical (linear) elasticity theory, resulting from my interest in piezoelectric elements. I had some familiarity with, and very much admired, Love's famous treatise. Indeed, in the earlier stages I viewed my task as that of rewriting the early chapters of Love without assuming infinitesimal displacement gradients. Also, at the time there was a good deal of interest at the B.R.P.R.A. in the kinetic theory of rubber elasticity. In view of this rather limited background, it is not surprising that my earlier pa• pers on finite elasticity involved a good deal of rediscovery. The relatively high value of the compression modulus of vulcanized rubber enabled me in my theory to adopt the idealization of incompressibility, as had been done in the kinetic theory. The fact that I did this ab initio proved to be extremely fortunate, as it was respon• sible for much of the tractability of the theory. One of the necessary ingredients of the theory is some intrinsic characterization of the relevant material properties. This is provided in classical elasticity theory by a stress-strain relation-the generalized Hooke's law-or by the corresponding expression for the strain• energy. In the earlier papers [A5-7] of a series in the Philosophical Transactions of the Royal Society, I adopted the expression for the strain-energy that had been derived by Tre• loar in 1943 on the basis of the kinetic theory model. I called this the neo-Hookean strain• energy function. At the time, relations between the load and deformation derived from it for a number of simple types of deformation were thought to agree well with experiment. However, more critical interpretation of such experiments, based on a more general theory, showed that this agreement was not nearly as good as had been thought. By the Fall of 1945, when I had substantially completed the work published later in [A5], my attention was drawn to the work of E. and F. Cosserat and of L. Brillouin, pub• lished in 1896 and 1925 respectively. Some of the ground I had covered in my paper was forestalled in these, but only for the case when the material is compressible. Since the pas• sage from the compressible case to that of incompressibility appeared at the time to be nontrivial, this did not cause any significant change in my work. However, Brillouin's paper was written in general tensor notation, and I did consider whether to rewrite my pa• per in a similar notation. I was sufficiently ambivalent to seek the advice of a number of people, and finally decided not to do so. In retrospect, I think this was the correct decision for the time. Although this made the paper, and subsequent ones written in a similar nota• tion, less attractive to some mathematicians, had I written them in tensor notation, it would have closed them to rheologists, polymer chemists, and others who might be interested in the application of the theory rather than its formalism. One of the conclusions that emerged from the theory was that in order to maintain a simple shear in a block of the material it is necessary to apply to the faces of the block nor• mal forces proportional to the square of the amount of shear, in addition to shearing forces proportional to the amount of shear, and any hydrostatic pressure one may choose to apply. In the third paper [A 7] of the Philosophical Transactions series I solved some simple prob• lems involving inhomogeneous deformations that illustrated the implications of this fact and were designed to provide a basis for experimentation. The forces necessary to produce simple torsion in a circular cylinder were found to consist of a couple and a longitudinal thrust. For a tube, it was found that if radial contraction is to be avoided, a thrust must ad• ditionally be applied to the inner surface of the tube. At the time, these conclusions appeared to be new. However, some time later my atten• tion was drawn to two papers, published by Poynting in 1911 and 1913, in which experi• ments were described that showed that a steel wire and a rubber rod elongated when sub• jected to a torque by an amount proportional to the square of the torque. Poynting was led to look for these effects by his erroneous belief that light is a transverse vibration in a lu• miniferous ether that has elastic properties and by the experimental fact that light exerts a pressure when it impinges on a mirror. The association of a thrust with torsion of a circu• lar cylinder and the lengthening of a cylinder as a result of the application of a torque are called Poynting effects. Sometimes this name is also applied to any effect that arises from the requirement that in simple shear of an elastic material the normal components of the XXIV Autobiographical Postscript stress are other than a hydrostatic pressure or tension. However, it seems to me that the term normal stress effect is more appropriate for general use, since somewhat analogous effects can occur in inelastic solids and in non-Newtonian fluids. * * * Up to this point the work on rubber elasticity was based on the assumption that the appro• priate strain-energy function is neo-Hookean. It occurred to me that this assumption is not essential to the development of the phenomenological theory. From the fact that the mate• rial is isotropic and that locally any deformation can be resolved into a pure, homogeneous deformation followed by a rotation, I argued that the strain-energy function must depend on the deformation gradients through the three basic scalar invariants of the Cauchy strain, although this was not the terminology used. With this strain-energy function a canonical expression for the Cauchy stress was obtained, in which the characterization of the mate• rial properties arises through the first derivatives of the strain-energy function with respect to the strain invariants. A simplification is introduced if the idealization of incompressibil• ity is made. Then one of the invariants is necessarily unity. The strain-energy function is now a function of only two of the strain invariants, and in the canonical expression for the Cauchy stress it appears through its first derivatives with respect to these two invariants. However, the Cauchy stress resulting from a specified deformation is now undetermined to the extent of an arbitrary hydrostatic pressure. It was pointed out by Truesdell in 1952 that the constitutive equation I had obtained for a compressible material had been obtained much earlier, in 1894, in the largely forgotten papers of J. Finger. It appears that the modifications introduced by incompressibility and the recognition of the relative tractability of the resulting theory were new. Adopting the constitutive equation for an isotropic incompressible material that I had formulated, I solved, in the period 1946-49, a number of simple problems. Some of these involved homogeneous deformations, others inhomogeneous deformations. In most of them the strain-energy function was left as an arbitrary function of the two strain invari• ants. That this could be done resulted from the fact that the incompressibility, and the con• sequent isochoric character of the deformations, led to the deformations being kinemati• cally determined, while they would not have been if the material considered had been compressible. * * * In September 1946 I left for a year in the United States. Shortly before, John Wilson had visited the United States and arranged for an exchange with the Rubber Section of the Na• tional Bureau of Standards in Washington, D.C., headed by L.A. Wood. After my year there, one of their people was to spend a year at the B.R.P.R.A. In fact, the return visit never took place. A few weeks before my departure an episode occurred that affected my subsequent thoughts on finite elasticity and, to some extent, on the academic community. This arose as a residue of my work at the G.E.C. on quartz piezoelectric elements. In connection with this work, I had enlisted as a consultant a crystal physicist at Cambridge, W.A. Wooster, who had some standing as an experimentalist and as the author of a text on crystal physics. After I left the G.E.C. he continued as a consultant to my successor. Shortly before my departure for the United States, Wooster spent an evening with me following a visit to the G.E.C. research laboratories. He told me of some experimental work he had done for the G.E.C., and published in Nature, on stress-induced twinning in quartz. He had found that a plate consisting of quartz of one polarity could be transformed, by a torque, into one of opposite polarity, the magnitude of the required torque depending on the orientation of the plate relative to the crystal axes. He told me that he had spent a good deal of effort over the previous year in an attempt to construct a theory for the effect, but without success. I suggested that it should be possible to predict his experimental re• sults by calculating the change in the total energy that would result from a change in the Autobiographical Postscript XXV polarity of the plate under constant torque, for plates of various orientations. The transfor• mation will or will not take place accordingly as this is negative or positive. Wooster dis• missed this suggestion in a cavalier fashion with "It can't be as simple as that." The fol• lowing weekend I actually did the calculations I had suggested and a few days later went to Cambridge to discuss them with Wooster. After a good deal of discussion, Wooster agreed, with some reluctance, that my theory appeared to be sound and predicted the experimental results in some detail. I left my few pages of calculations with him and returned to London. However, as we parted I suggested that we might write a note for Nature on the theory. He felt that this was premature. Shortly afterwards I left for the United States and thought no more about the matter un• til a few months later I visited WP. Mason at the Bell Laboratories. When I told him about the experiments of Wooster and the theory I had formulated, he advised me to publish my theory as soon as possible. I wrote to Wooster reporting this conversation, but received no reply. I found this odd, but began to understand the situation some time later, when I went to Princeton to visit a young colleague, Donald Booth, who was interested in digital com• puters and had come to Princeton to spend a year with von Neumann at the Institute for Advanced Study. He asked me if I had seen a note in Nature by Wooster in which my the• ory was described. We went to the library and I saw the note for the first time. It acknowl• edged "discussions" with me. Some months after this I received a letter from Wooster in which he accused me of having written to him reporting Mason's suggestion in order to embarrass him. He also explained that he had omitted my name as a coauthor because the note related to work he had done as consultant to the G.E.C. While I did not understand this rationale, I replied that I had not seen the Nature note prior to my letter and that more• over, I did not wish to be an unwelcome intruder on his preserve. This was the last contact I had with Wooster. At the time, I shrugged the matter off. However, much later, when in• terest developed in the possibility of using finite elasticity theory for deformations beyond the onset of material instability, I found it rather annoying to have to ascribe my own the• ory to someone else.
* * * I traveled to the United States on the Queen Mary, which had not yet been converted from its wartime use as a troopship. We were therefore sleeping in bunks, six to a cabin. By the oddest chance two of the other occupants of my cabin were ex-colleagues at the G.E.C. Since the dockers in New York were on strike, we docked at Halifax and traveled to New York on a special train. There was a heat wave in progress and the journey took well over twenty-four hours. There were no sleepers or club cars on the train, and only one of the coaches was air-conditioned. This was reserved for a contingent who were on their way to join the United Nations in New York. Fortunately I had become friendly with some of these people on the transatlantic crossing and was able to spend most of the journey in rel• ative comfort. We arrived at Grand Central in New York at about one o'clock on a Saturday morning and were installed at the Commodore Hotel, which was located at the terminal. Despite my fatigue, I could not resist the temptation to spend an hour or so exploring the immediate neighborhood of the hotel. I think I wandered as far as Times Square. The bright lights and the people still on the streets at this hour created an unforgettable atmosphere of excite• ment, enhanced by the steamy heat and the contrast with the drabness of wartime and post• war England. After a night's sleep, I was whisked off by a family friend to spend my first few days in the United States in Larchmont, just north of New York City. My host, who had been brought up in England, spent many hours telling me about American folkways-social, cultural, linguistic-and their differences from those of Great Britain. I also spent many hours exploring Manhattan.
* * * XXVI Autobiographical Postscript
It was during my year in Washington that I first met W. Prager. Hugh Dryden, the deputy director of the National Bureau of Standards, suggested that I would find it interesting to visit the applied mathematics group at Brown University. Accordingly, in due course I trav• eled to Providence on an overnight train and in the morning presented myself at the office of R.G.D. Richardson, dean of the graduate school at Brown. Richardson received me cor• dially and after half an hour of conversation, suggested that I talk to Prager. He telephoned Prager to tell him of my imminent arrival at his office. Some minutes later I knocked on the door of Prager's office, and following his command to enter, I did so. He was seated at his desk facing the door and with a slight inclination of his head and pronouncing each word in a staccato manner that emphasized its monosyl• labic character, said, "And what can I do for you?" I explained that I was spending a year at the National Bureau of Standards and had come to Providence, following an arrange• ment made by Dryden and Richardson, to learn something of the applied mathematical ac• tivities at Brown. He responded, still staccato, "At Brown we do research on elasticity, plasticity, theory of structures, fluid mechanics." Then, with a slight and somewhat ingra• tiating inclination of the head, "Is there anything else I can do for you?" I thanked him for this information, returned to Richardson's office, and reported the conversation. Richard• son was apologetic and muttered words to the effect that Prager was "rather peculiar." He then sent me over to talk to W. Marchant, who was chairman of the Division of Engineer• ing and occupied an office next to Prager's. Marchant was very courteous, and we had a long conversation, after which I explored the campus and then made my way back to Washington. I did not meet Prager again untill953. But more about that later.
* * * The first project on which I embarked at the Bureau of Standards was an experimental study of the Poynting effect in a circular cylinder of vulcanized rubber. The experiment was completed in January 1947 and published in May [Al]. It showed that in order to pro• duce simple torsion in the cylinder it is necessary to exert thrusts proportional to the square of the amount of torsion, distributed parabolically over the ends, in addition to the torque. This result is in accord with a form for the strain-energy function proposed by Mooney in 1940. The fact that the thrust did not vanish at the periphery of the cylinder showed the in• adequacy of the neo-Hookean (i.e., kinetic theory) strain-energy function. I did no further experiments at the Bureau of Standards, since a chance meeting with the director of extramural research at the Office of the Rubber Reserve, Oliver Burke, led me to turn my attention to viscoelastic fluids. My interest in the subject was aroused early in 1946, when I first made the acquaintance of Karl Weissenberg, who was working at Impe• rial College. During a visit to the B.R.P.R.A. he told me about some experimental work in which he was involved, in which viscoelastic fluids, subjected to torsional or Couette flows, exhibited effects that are now often called Weissenberg effects. Weissenberg cor• rectly interpreted these as evidence of the association of (nonhydrostatic) normal stresses with steady shearing flow. The similarity of my predictions for an elastic solid, such as vul• canized rubber, subjected to simple shear was immediately evident. When I mentioned them to Weissenberg, he offered to show me the experiments at Imperial College. He also offered, for the future, to direct my work. I accepted the first of these offers and in due course, visited Imperial College where I saw, for the first time, the very striking normal stress effects in viscoelastic fluids. During our chance meeting, I told Burke about these experiments. He was sufficiently intrigued to suggest that I set up a torsional flow experi• ment with the financial support of his agency. It was arranged that this work would be done at the Mellon Institute in Pittsburgh in conjunction with an existing program funded by the Office of the Rubber Reserve. This resulted in my spending about half of my time at the Mellon Institute from February 1947 to September, when I returned to England. Burke also used me as a general consultant on his program, and in this connection we visited to• gether a number of laboratories in which work was being funded by the Office of the Rub• ber Reserve. Autobiographical Postscript xxvii
The object of the experiment that I set up at the Mellon Institute was the measurement of the normal thrusts associated with torsional flow in a viscoelastic fluid. The arrange• ment was essentially the same as the one I had seen at Imperial College. It soon became apparent, however, that in order to obtain qualitatively meaningful results much more care would have to be taken in the design and construction of the apparatus, in controlling the temperature and other experimental conditions, and in the actual conduct of the experi• ments. When the time came for me to return to England, the Mellon Institute offered me a permanent appointment to continue the work on viscoelastic fluids. However, I decided to remain in England. There we constructed an apparatus that embodied the lessons I had learnt at the Mellon Institute. The Office of the Rubber Reserve decided to continue the work I had started, and T.W de Witt was appointed to take charge of the program. Afterwards, H. Markovitz joined the group and some years later, B. Coleman. At Markovitz's request I sent him blueprints of the apparatus I had constructed in England, and I believe he constructed a similar appara• tus at the Mellon Institute. Concurrently with my experimental work at the Mellon Institute, I tried to formulate a phenomenological theory for the flow of viscoelastic fluids that would lead naturally to the existence of normal stress effects. In this I was much influenced by my work on finite elas• ticity. While for elastic materials it was evident what primitive constitutive assumption should be adopted, this was not the case for viscoelastic fluids. However, I argued correctly that in torsional, Couette, and Poiseuille flows each material element is undergoing a simple shearing flow whose magnitude, but not necessarily direction, remains unchanged as the element executes its path; consequently, idealizing the material as incompressible, the stress in the element must depend, apart from an arbitrary hydrostatic pressure, on the velocity gradient only. However, I concluded erroneously, as was pointed out by Oldroyd in 1950, that for these flows, the (Cauchy) stress matrix, referred to a fixed rectangular Cartesian coordinate system, must depend only on the velocity gradient matrix referred to that coor• dinate system, apart from an arbitrary hydrostatic pressure. I then obtained the restrictions on this dependence implied by rotation invariance and by the assumption that the fluid is isotropic in its rest state. The canonical form for the stress so obtained has since become known as the Reiner-Rivlin equation. It was only when the paper [F2] presenting this result was substantially written that I found that Reiner had obtained earlier a somewhat similar canonical representation for the stress matrix, from the assumption that it is an isotropic matrix polynomial in the velocity gradient matrix, without however introducing the as• sumption of incompressibility. He did this in formulating a theory for dilatancy (increase in volume of an isotropic fluid produced by shearing forces-the wet-sand phenomenon). After my return to England in September 1947, I applied the Reiner-Rivlin equation to the calculation, without further assumption, of the forces associated with various viscometric flows [F2,3]. Concurrently with my interest in the phenomenological theory, it was natural to investi• gate the aspects of the structure of polymer solutions that give rise to normal stress effects. It was, of course, evident that they arise from the orientation of the polymer molecules when a velocity gradient exists in the fluid. In an attempt [F5] to quantify this model I adopted the so-called "pearl necklace" idealization of a polymer molecule that had been used earlier by Kramers in his calculation of the viscosity of a polymer solution. I as• sumed, as did Kramers, that the contributions to the normal stress of the individual poly• mer molecules are additive. Also, the velocity gradients in the fluid were assumed to be sufficiently small so that terms of higher degree than the second in them could be neglected in the expression for the stress (just as in the calculations of Kramers terms of higher de• gree than the first were neglected). Moreover, the calculations were interpreted against the background of the Reiner-Rivlin constitutive equation. With realistic values adopted for the parameters in the "pearl necklace" model, the cal• culations yielded values for both the viscosity and normal stress many orders of magnitude lower than those observed experimentally. I suggested that this results from the fact that at the polymer concentrations at which measurable normal stresses are observed, the poly- xxviii Autobiographical Postscript mer chains interact so that the tension in one chain is transmitted to many chains. I also suggested that it might, in a rough sense, be possible to take account of these interactions by considering the solute to consist effectively of fewer much longer chains. The measured value of the viscosity could then be used to obtain the effective length of the chains and their number per unit volume of solution. With these values the formula for the normal stress yielded realistic values. Viewed against later developments in the phenomenological theory of constitutive equa• tions, my calculation yields one combination of the two normal stress coefficients in the second-order approximation to the constitutive equation for a material with memory, and agrees with the much later work of Bird in which using a similar "pearl necklace" model, both of the second-order coefficients were calculated.
* * * My stay in the United States, and particularly my association with the Mellon Institute, was memorable in other respects than the purely scientific, since it was at the Mellon Insti• tute, where she was employed as a research chemist, that I met the lady who became my wife in England some nine months after my return, and remains so to this day. This left as an open question whether we would make our life in Great Britain or the United States, un• til finally we decided on the latter about five years later. My return journey to England, mercifully not on an unreconstructed troopship, was con• siderably more comfortable than my outward journey and gave me ample opportunity for reflection regarding my year in the United States. I realized that prior to my visit I had given almost no thought to what was just one faraway country among many. Such impres• sions as I had, vague as they were, were formed largely from films and novels, with those from the films, glitzy and chromium-plated, dominating. The reality that I found was rather different. Although the expected characteristics were present, I was impressed by the preva• lence, outside New York, of an old-fashioned and understated quality in the American scene. This was evidenced in the ubiquitous colonial architecture and in the personal inter• actions that reminded one of the agrarian background of much of the population. * * * On arriving in England I found that Treloar, who had returned to the B.R.P.R.A. in the summer of 1945 from his stint at T.R.E., had been joined by two young physicists, D.W. Saunders and A.G. Thomas. Saunders was a fairly recent graduate of Imperial College, and Thomas was a recent graduate of Oxford University. Although Saunders was working under Treloar's direction, I was asked almost immediately to take over responsibility for both of them. Treloar was very much a hands-on experimentalist and seemed, in general, to prefer to work alone. It should not be concluded from this that he was a loner. On the contrary, he was a warm person whose favorite relaxation was to retire, with one or more colleagues, to a local hostelry, the Cherry Tree, and, over a few beers, discuss physics or any other topic that happened to arise. Shortly afterwards, I was given the title "senior physicist" and mandated to develop a physics research group. There was at the time one other research physicist at the B.R.P.R.A., A. Schallamach, a Ph.D. of the University of Breslau, who was several years older than I and was already at the B.R.P.R.A. when I appeared on the scene in 1944. He had for many years been engaged in measuring dielectric losses in elastomers. Despite the meticulous manner in which his experiments were carried out, they did not seem to lead to any particularly interesting conclusions. My first task was to propose experimental programs for Saunders, Thomas, and Schalla• mach. I accordingly asked Saunders to undertake the measurement of the load-deforma• tion behavior of vulcanized rubber in various types of deformation with a view to deter• mining experimentally the dependence of the strain-energy function W on the strain invariants, usually denoted I 1 and h, and then using the expression for W so obtained to Autobiographical Postscript XXIX predict the results of other experiments. In order to provide a richer theoretical background for this work, I continued my calculations of the load-deformation relations in various sit• uations that might be realized experimentally, with arbitrary dependence of Won I 1 and I 2• In our program, Saunders and I had the advantage of close contact with Treloar and with the outstanding group of physical chemists headed by Geoffrey Gee. This enabled us to draw on their experience in carrying out meaningful experiments on rubber. Shortly after my return from the U.S.A., the director of the B.R.P.R.A., John Wilson, re• signed to become the founding director of the British Rayon Research Association, and was succeeded by Gee. There had always been a close relationship between Wilson and Treloar, so no one was surprised when shortly afterwards Treloar resigned to join Wilson. Then early in 1949, Saunders in turn joined Treloar, to my regret. Although my relations with Saunders had always been good, it is probable that he had temperamentally a more natural affinity with Treloar than with me. His work was continued by AN. Gent, a young physicist who had joined the B.R.P.R.A. in April1949, with the mandate to act as a bridge between me and the engineering activities. Then, early in 1950, our group was greatly strengthened by the arrival of Leonard Mullins, who had been head of the physics group at the British Rubber Manufacturers' Research Association. In a paper published in April 1950, A.E. Green and W Zerna gave a tensor formulation of some of the basic equations of finite elasticity theory in convected coordinates. Then in August 1950 Green and his graduate student R.T. Shield rederived the results I had ob• tained for torsion of a circular cylinder. They also discussed two further problems that had not been discussed by me-the deformation of a circular cylinder of compressible elastic material rotating about its axis and that of a thick spherical shell the surfaces of which are subjected to spherically symmetric systems of forces. One of their avowed objectives was to show that their tensor formulation provided a simpler basis for the solution of such problems than that which I had presented. Which of two methods of obtaining a result is simpler is very often subjective; it depends on one's background and involves an element of personal taste. Indeed, there are ways of obtaining my results and those of Green and Shield that would undoubtedly appear to many as simpler than either of the methods we used. As I have mentioned earlier, the decision not to formulate my earlier papers in tensor notation was a deliberate one. In retrospect I think it was the correct decision for the time. Finite elasticity theory, and a fortiori nonlinear continuum mechanics generally, lends itself to many notations and formulations. My own preference is for those that admit of the most direct physical interpretations and employ the most elementary mathematical appara• tus. For example, in order to derive the point equations of balance from the global equa• tions, there seems little merit, from a pedagogical standpoint, in carrying out the discus• sion in general tensor notation, as is often done in textbooks. The- equations referred to a rectangular Cartesian coordinate system can be obtained more simply using Cartesian ten• sor notation. It is then a trivial matter to obtain the corresponding equations, referred to an arbitrary curvilinear coordinate system, by appropriately raising and lowering indices in accordance with the conventions of tensor analysis. However, I digress. I responded to the paper by Green and Shield by inviting Green, who was at the time professor of mathematics at the University of Newcastle, to become a consultant to the B.R.P.R.A. It appeared to me that if we were both going to pursue an interest in finite elas• ticity theory, it would be preferable that we do so cooperatively rather than in competition. Green accepted my invitation, and thus was initiated a collaboration that persisted inter• mittently for fifteen years. The first fruits of our collaboration soon appeared. In a paper [A I 0] published in De• cember 1949 I had obtained a formula for the torsional modulus of a rod of circular cross• section, in which infinitesimal torsions are superposed on finite extension. The material of the rod was assumed to be elastic, incompressible, and isotropic. The striking feature of this result was that the expression for the torsional modulus depended on the tensile force and extension ratio, but did not contain the strain-energy explicitly. With Saunders, the formula had also been verified experimentally using rods of vulcanized rubber [A12]. Pol- XXX Autobiographical Postscript lowing my suggestion, Green and Shield then extended this result to rods with noncircular cross-section, and their formula was verified experimentally by Gent and myself using rods of vulcanized rubber with rectangular cross-sections of various aspect ratios [Al7]. Then Green, Shield, and I formulated the general theory for the superposition of infinites• imal deformations on finite deformations and applied it to the indentation of a homoge• neously deformed elastic half-space by a flat-ended cylindrical punch. At Green's insis• tence the final version of the paper [C1] was written in tensor notation using convected coordinates. Green seemed to have an almost religious attachment to convected coordi• nates, which persisted for a number of years. * * * Two of the most technologically important properties of vulcanized rubber are its resis• tance to tearing and its resistance to abrasion. I therefore proposed to Thomas that he work on the first of these topics and to Schallamach that he work on the second. I chose these topics, rather than other technologically important ones, because in each case I saw a point of entry into the problem radically different from any previous investigation of which I was aware. With regard to the problem of tearing, my first idea was to examine the validity of a cri• terion of the Griffith type, according to which an existing tear in a test-piece held at con• stant extension will grow if the elastic energy thereby released exceeds the increase in the surface energy. Griffith was concerned with glass, for which the elastic deformations are small, so that classical elasticity theory can be used to calculate the energy release. In the case of vulcanized rubber the large deformations involved rendered similar calculations impossible at the time, although such calculations were carried out much later by Lindley using finite-element methods. We therefore devised experiments in which the necessity for elaborate calculations was avoided. My (not very firm) expectation was that the Griffith criterion would not apply. I was therefore somewhat surprised when we found it possible, by measuring the force required to tear a test-piece with a preexisting cut, to calculate a characteristic tearing energy that could be used to predict the tearing forces for test-pieces for which these forces were orders of magnitude different. This characteristic energy was, however, many orders of magnitude greater than one would expect for a surface energy. We interpreted it as the energy expended in the irreversible processes that take place in a neighborhood of the crack prior to its formation. Although this work [H2] was not published until1953, it was substantially complete by the summer of 1950, and I lectured on it in George Irwin's department at the Naval Re• search Laboratory in the fall of 1950. The 1953 paper with Thomas represented my last ex• cursion into the question of tearing until many years later, when I returned to the subject briefly on two occasions [H3,4,5]. However, the work on tearing and related problems such as fatigue failure was continued for many years, with great ingenuity and insight, by Thomas and his colleagues, notably E.H. Andrews and A.N. Gent. My thoughts on the subject of abrasion stemmed from my unpublished experiments at the G.E.C. research laboratories on the scratching and grinding of crystals. I suggested to Schallamach that he examine visually the effect of scratching the surface of vulcanized rubber with a pin, or with a comb, and the effect of abrading it by unidirectional motion over a rough surface. Schallamach did in fact follow some of my suggestions, but beyond this initiation I was not involved in his work, apart from sharing occasionally in the excite• ment of a discovery and cheering his progress. Schallamach continued his work on abra• sion, which has become a classic in rubber science, long after I left the B.R.P.R.A. * * * Sir Eric Rideal, who in addition to being a member of the governing board of the B.R.P.R.A. was director of the Royal Institution, suggested that since the Davy-Faraday Laboratory was recognized as a graduate institute of London University, appropriate mem• bers of the B.R.P.R.A. staff could work there as graduate students, under my direction. In Autobiographical Postscript XXXI accord with this suggestion, in 1949 J.E. Adkins and H.W. Greensmith were appointed as graduate students at the Davy-Faraday Laboratory, on the B.R.P.R.A. payroll. Tragically, shortly after his appointment Adkins was diagnosed as having Hodgkin's disease. Accord• ing to his wife, he was not expected to live longer than five years. In fact, with many re• missions and relapses, he survived for about seventeen years, in which time he built a dis• tinguished career and at the time of his death was founding head of the Department of Theoretical Mechanics at Nottingham University.
* * * From the work on the load-deformation relations for vulcanized rubber, a number of sur• prising results emerged. It was evident from the calculations that the form of the tensile force vs. extension curve for simple extension and the shearing force vs. amount of shear curve for simple shear are not very sensitive to the dependence of the strain-energy func• tion Won the strain invariants / 1 and / 2. The nonlinearity of the former and the near• linearity of the latter had been adduced as support for both the neo-Hookean (i.e., kinetic theory) strain-energy function and that proposed by Mooney. A more sensitive manner of plotting the experimental results for simple extension, which has become known as the Mooney-Rivlin plot, showed very clearly that the neo-Hookean strain-energy function pro• vided significantly poorer agreement than had been supposed [Al2].* The fact that pro• vided the extension ratios are not too large, the Mooney-Rivlin plot yielded a straight line seemed superficially to lend strong support to the validity of a strain-energy function of the Mooney form. However, continuation of the simple extension experiments into the regime
*If the principal extension ratios are denoted by A1 , A2, A3, then I 1 and I 2 are given by
I, =AI+ A}+ A~, I2 = A!2 + Az 2 + x3 2. The neo-Hookean strain-energy function is given by
W= C(l1 - 3), and the strain-energy function suggested by Mooney (now often called the Mooney-Rivlin strain• energy function for reasons that will appear below) is given by
W = C1(I1 - 3) + Cz(I2 - 3), where C, C" and C 2 are constants. If, for simple extension, the extension ratio is denoted by A, the tensile force, f per unit initial cross-sectional area, is given by
where
In effect, in the Mooney, Rivlin plot,
is plotted against IIA. If W has the Mooney form this yields a straight line. If W is neo-Hookean, the straight line is parallel to the abscissa. For a simple shear of amount K, the shearing force, a per unit area, is given by
a= zK(aw + _!_ aw) A where ai, ai2 xxxii Autobiographical Postscript in which the extension ratios were less than unity contradicted this conclusion. Rather, we concluded that an expression for W of the form
W = C(/1 - 3) + f(/2- 3), where Cis a constant andfis a monotonically decreasing function of / 2 - 3, agrees well with the experimental results. In this form for W the departure from that given by the ki• netic theory is carried by the termf(/2 - 3). Strong support for this conclusion was provided by experiments carried out by Gent [A16] in which a tube is subjected to torsion and prevented from contracting radially by a pressure applied to its inner surface. The calculations showed that this pressure depends on Wthrough aw/a/ 1 only and is otherwise proportional to 1/1 2, where 1/J is the amount of tor• sion. In the experiments the pressure was found to be accurately proportional to 1/12 , yield• ing strong support, although not exactly proof, for aw/a/ 1 being independent of / 1 and h These experiments yielded another result, which was unexpected and suggestive. Al• though my calculations were based on the assumption that the material is ideally elastic, the experiments were performed on vulcanized rubbers, which inevitably have some hystere• sis; the load-deformation curves for unloading are, in general, different from those for load• ing. However, it was found in the experiments with tubes that while this was the case for the torsional couple, the pressure vs. 1/12 curves obtained were substantially the same for loading and unloading. This was the case even when the rubber used was one that showed a high degree of hysteresis in the torsional couple vs. 1/J curves. The natural implication is that whatever mechanism, not envisaged in the kinetic theory calculations, leads to the termf(/2 - 3) in the expression for W also leads to the irreversibility evidenced by the hysteresis. * * * We obtained two further results that bear on the departure of the actual strain-energy func• tion from the neo-Hookean form predicted by the kinetic theory [A19]. Tensile force vs. extension ratio measurements were made on a natural rubber vulcanizate swollen to differ• ent extents with organic solvents. Mooney-Rivlin plots of the results yielded a series of straight lines whose inclinations to the abscissa decrease linearly with decrease in the vol• ume fraction of dry rubber. Moreover, for any specified fraction of swelling agent the slope was substantially independent of the swelling agent used. This led to the conclusion that as the amount of swelling agent increased (and the hysteresis decreased) the strain-energy function approached the neo-Hookean form. We also found that provided the extension ra• tios were not too large, the slopes of the Mooney-Rivlin plots were insensitive to the degree of vulcanization and to some extent, to the choice of polymer. That the latter result is not general is evident from a later experiment of Ciferri and Flory in which they found that for certain silicone rubbers the slope of the linear portion of the Mooney-Rivlin plot is nearly zero, indicating the validity of the neo-Hookean strain-energy function. While most of the principal contributors to the kinetic theory-Guth, Gee, Treloar• readily accepted our conclusion that the kinetic theory model required significant modifica• tion, Flory was extremely reluctant to do so. He alleged that the departure of our experimen• tal results from neo-Hookean behavior was due to our experiments not being equilibrium experiments. This appeared to me to be a strange contention in view of the fact that our simple extension experiments were similar to his own experiments, which according to his claim supported the kinetic theory predictions. The research program that I initiated at the B.R.P.R.A. was continued for many years af• ter my departure. One indication of its success is that of the people who participated in it while I was at the B.R.P.R.A., five have been recipients of the Charles Goodyear Medal, which is awarded annually by the Rubber Division of the American Chemical Society on an international basis for contributions to rubber science or technology. * * * Autobiographical Postscript XXX111
It was, I think, in 1949 that I first met Clifford Truesdell. While on a visit to England he called on me at the Royal Institution and told me that he had been studying the foundations of continuum mechanics and was interested in my work. During his visit I described my work in progress and we discussed, at some length, our views on scientific research. We found that we had a good deal in common, although there were significant areas of dis• agreement. I recall that he showed great interest in the exhibit of historical scientific apparatus at the Royal Institution, and I was very much impressed by his knowledge of the history of sci• ence. More generally, I found him to be a memorable person, who was well informed, had given thought to a wide variety of topics, and could enlist a sophisticated command of lan• guage, unusual in a scientist, to express his views. He also displayed a well-developed sense of humor that I enjoyed immensely. He had a tendency to express sensible, and even perceptive, views in terms so extravagant as to give them a humorous effect. Later, as our relationship developed, I found it difficult at times to know where to draw the line between the hyperbole and what he really believed. Following his visit to the Royal Institution I en• joyed a friendship with Clifford that lasted for more than fifteen years, but ultimately foundered, to my great regret, on his inability to accept my criticism of some of the work of his proteges.
* * *
In the Fall of 1950, I embarked on a tour of the United States during which I gave talks at numerous universities and research organizations. I was invited by Truesdell to speak at Indiana University. I was also invited by George Irwin, I suspect on Truesdell's recom• mendation, to visit the Naval Research Laboratory (N.R.L.) in Washington, D.C. and to give two lectures there. One of these was on the work we were doing at the B.R.P.R.A. on the tearing of vulcanized rubber; the other was on finite elasticity. Irwin was particularly interested in the first of these, so much so that he raised tentatively the possibility of my spending a year at N.R.L. Some months after my return to England I received a formal in• vitation to spend a year at N.R.L. as consultant GS-15 to the Mechanics Division, of which Irwin was the head. To enable me to do so I obtained a year's leave of absence from the B.R.P.R.A. and flew to the United States on a military plane, arriving on Aprill, 1952. My wife joined me shortly thereafter, having traveled by sea. One of the units of the Mechanics Division was an applied mathematics group headed by a cordial gentleman, Horace Trent. He asked me to provide some guidance to a group of three people that had been led by Paul Nemenyi prior to his recent death. This was the group of which Clifford Truesdell had been a member until his departure for Indiana Uni• versity in 1950. It consisted of A. W. Saenz, Jerald Ericksen, and Richard Toupin. Saenz was involved with a problem in general relativity theory with which he and Trent seemed perfectly happy; so I did not interfere. Toupin was working on his Ph.D. thesis under the direction of M. Lax of Syracuse University, so again no change was indicated. Ericksen had joined N.R.L. in June 1951 from Indiana University, where he had obtained a Ph.D. degree under the direction of D. Gilbarg with a thesis on some problems in classical fluid mechanics. With Ericksen I embarked on the extension of finite elasticity theory for isotropic mate• rials to transversely isotropic materials (i.e., materials that have fiber symmetry). In the pa• per covering this work [B 1] we also advanced a general theory of kinematic constraints. While Ericksen was occupied with the details of this work, I busied myself developing constitutive equations for isotropic viscoelastic materials. I took as my starting point the assumption that the Cauchy stress at any instant, t say, is a function of the deformation gradients and the gradients of the velocity and its material time derivatives of all orders, all measured at time t. (For incompressible materials the stress is, of course, also undeter• mined to the extent of an arbitrary hydrostatic pressure.) Although this was not spelled out in my paper with Ericksen [D 1] in which this work was reported, I was motivated by the xxxiv Autobiographical Postscript consideration that provided the deformation gradients depend sufficiently smoothly on time, their history up to time tis determined by their values at time t and those of their time derivatives of all orders through a Taylor expansion. In this basic constitutive assumption, the deformation gradients were taken with respect to the undeformed configuration, while the gradients of the velocity and its time derivatives were taken with respect to the current configuration. By omitting dependence on the deformation gradients, we arrive at a consti• tutive assumption appropriate to a fluid. The manner in which the stress can depend on the various kinematic gradients may be restricted by the consideration that the superposition on the assumed deformation of an ar• bitrary time-dependent rigid rotation causes the stress matrix to be rotated by the current amount of the superposed rotation. Further restrictions can be imposed if the material is isotropic. We were able to give canonical form to the effect of these restrictions. We con• cluded that the stress matrix must depend on the kinematic variables through the Finger (left Cauchy-Green) strain matrix and a sequence of matrices that have become known as the Rivlin-Ericksen matrices, and is an isotropic function of these. If dependence on the Finger strain matrix is omitted, the constitutive equation describes fluid-like behavior. Meanwhile Toupin was becoming increasingly discontented with his thesis topic. So I suggested that he change it with, of course, the agreement of his advisor. In my work with Ericksen on transversely isotropic, elastic materials, the strain-energy is taken, in effect, to be a function of the deformation gradient matrix and a unit vector normal to the plane of isotropy. In our paper we obtained the restrictions that can be placed on this dependence. It occurred to me that a mathematically similar problem arises in the theory of electrostric• tion (or magnetostriction) in an isotropic material. The stored energy must now depend on the deformation gradient matrix and the polarization vector. I therefore suggested to Tou• pin that he adopt as his thesis problem the development of a phenomenological theory of electrostriction in isotropic materials. This he did, with the agreement of his advisor. His successful thesis formed the basis for a now well-known paper, "The Elastic Dielec• tric," in which he dealt with complexities beyond anything that I had envisaged when I suggested the topic. Toupin recognized that in his theory the polarization could be re• garded as an internal variable, and this led to his seminal papers on internal variable theo• ries. I had no part in this development beyond my original suggestion. However, later, in 1963, stimulated by Toupin's papers, I took up in collaboration with Albert Green the for• mulation of internal variable theories from a more general standpoint [E1,2]. After I had left N.R.L., and while Toupin was still engaged in writing his Ph.D. thesis, I collaborated with him for a number of years. One of the main results of this collaboration was a study of electro-optical and magneto-optical effects [G5]. This formed the starting point for M.M. Carroll's Ph.D. thesis at Brown University in the early 1960s and for my papers with him [G13-15]. * * * When I accepted the invitation to spend a year at N.R.L., my wife and I agreed that we would take advantage of our stay to decide whether we wished to make our home in Great Britain or the United States. After some months in Washington we decided on the latter, and I informed Geoffrey Gee, the director of the B.R.P.R.A., of our decision. I told him that I felt bound to return to the B.R.P.R.A. for a year or so, unless he freed me from this obligation. Gee was not surprised by our decision, since he was well aware of my ambiva• lence, and generously did not insist on my return. It was generally known at the B.R.P.R.A. that, since my first visit in 1946-47, I had considered the possibility of eventually moving to the United States. My marriage evidently made it more likely. I had discussed this with Sir Eric Rideal, who was particularly sympathetic since his own wife was originally Amer• ican. During the first of our conversations on the subject he asked me why I was consider• ing such a move. I replied that I saw my future at a university rather than the B.R.P.R.A.; I felt that it would be easier to achieve this in the United States than in England. In the British academic community, in any given field, jobs, recognition, and advancement were Autobiographical Postscript XXXV controlled by a very few people. In the United States there seemed to be many foci of influence resulting in a much more fluid situation. Rideal conceded that my observation was valid. However, he felt that I was well positioned in this respect since I would have his strong support and that of Sir Geoffrey Taylor. He told me that it was their intention to pro• pose my election to the Royal Society in a very few years and to promote my claim to a good chair in applied mathematics when one opened up. Alternatively, I could count on be• ing the next director of the B.R.P.R.A., if I so wished, when, as was expected, Gee moved to a university. In fact, shortly before my departure for N.R.L. I was told by Andrade, who by this time had replaced Rideal as director of the Royal Institution, that I had been nomi• nated for fellowship of the Royal Society by Rideal and seconded by Taylor. He indicated that he was somewhat offended at not having been brought into the nomination process, but that he would nevertheless support the nomination. When after a few months in Washington, I told George Irwin that I wished to stay in the United States, he offered me a permanent position at N.R.L., but explained that for bu• reaucratic reasons, it would have to be at the GS-14 level. I accepted this but told him that my real objective was to obtain an appropriate university appointment. With this in mind he put me in touch with Mina Rees of the Office of Naval Research, who drew my avail• ability to the attention of a few universities that might be interested in employing me. As a result of this-and possibly also of the fact that E. H. Lee of the Division of Applied Math• ematics at Brown University had attended a well-received talk I gave at a meeting of the Society of Rheology-in the early summer I received an invitation to have lunch with Prager in Washington. He told me that they would like to discuss with me the possibility of my joining the Division of Applied Mathematics at Brown and to this end invited me to give a talk in their seminar. I of course accepted the invitation readily. Prager's extreme cordiality contrasted sharply with his behavior at our previous meeting. Indeed, I do not know whether he was aware that we had met previously, and I did not see fit to remind him of the fact. Shortly afterwards, Violet and I made our way to Providence, where I gave a talk on finite elasticity and was offered an appointment as professor of applied mathe• matics with immediate tenure. I accepted the offer without hesitation. While other possi• bilities had emerged, none was at as prestigious an institution, or offered as much potential interest. During the time I was at N.R.L. there was a good deal of experimental work in progress on the fracture of metals, under George Irwin's direction. I found this educative, but al• though I had several conversations on the subject with Irwin, we never seemed to be on the same wavelength. Despite this, during the whole of my stay at N.R.L., Irwin was uniformly friendly and helpful, as were Horace Trent and the other members of the Mechanics Divi• sion. I still think of them with gratitude. But above all I feel that I was fortunate to have been thrown into contact with two people as able as Ericksen and Toupin in a situation in which I was able to have some influence on the direction of their research. * * * We moved to Providence in August 1953. E. H. Lee was in the process of taking over from Prager as chairman of the Division of Applied Mathematics, but I soon discovered that Prager seemed to regard the Division as his Lehrstuhl-a situation that seemed to be ac• cepted by all concerned. Shortly after my arrival at Brown I acquired two graduate students, Barbara Cotter and Gerald Smith. With Smith I took up the extension of finite elasticity theory to anisotropic materials with the various crystal symmetries [B4,5]. The mathematical problem was the determination of irreducible integrity bases for functions of a symmetric matrix with re• spect to the crystallographic point groups. Then, if the strain-energy function is a polyno• mial in the elements of the Cauchy strain matrix, it is expressible as a polynomial in the elements of the appropriate irreducible integrity basis. It is heuristically evident that this irreducible integrity basis also provides a function basis, although not necessarily an irre• ducible one. (This was formally proven by Wineman and Pipkin much later.) Accordingly XXXVI Autobiographical Postscript if the strain-energy function is a single-valued function of the elements of the Cauchy strain matrix, it is expressible as a single-valued function of the elements of the irreducible integrity basis, but some of these elements may be redundant. My paper with Ericksen [Dl] on isotropic viscoelastic materials also raised questions in the theory of invariants. In it the stress in a viscoelastic material is a symmetric matrix• valued function of an arbitrary number of symmetric matrices. The canonical expression that was obtained for the stress depended on any degeneracies that might exist in the argu• ment matrices. To avoid difficulties of this kind, in obtaining canonical forms I concen• trated on the situation when dependence on the arguments is polynomial and this poly• nomial character is preserved in the canonical form. For the viscometric flows, only the first two Rivlin-Ericksen matrices-the strain• velocity and strain-acceleration matrices-are nonzero. This confirms the validity of Oldroyd's criticism of the argument that led me to formulate the Reiner-Rivlin equation, which tacitly assumed that only the first of these is nonzero. I therefore obtained the irreducible canonical form for an isotropic symmetric 3 X 3 matrix-valued polynomial function of two symmetric 3 X 3 matrices, as well as an irre• ducible integrity basis for two symmetric 3 X 3 matrices [D2]. I also obtained correspond• ing results for symmetric 2 X 2 matrices. The results for 3 X 3 matrices were used to obtain the solutions for a number of viscometric flows in an incompressible viscoelastic fluid [F8]. * * * In the Fall of 1955, A.J.M. Spencer, who had recently obtained his doctorate with LN. Sneddon as advisor, came to Brown to spend two years as a research associate. I realized that canonical representations of matrix-valued or vector-valued polynomial functions of matrices or vectors could be obtained by a polarization process from the integrity basis for an augmented set of independent variables. Specifically, the canonical representation of a symmetric 3 X 3 matrix-valued polynomial function of N symmetric 3 X 3 matrices can be obtained by polarization, with respect to one of the matrices, of a polynomial in ele• ments of an integrity basis for N + 1 symmetric 3 X 3 matrices. Spencer therefore embarked on the problem of determining an irreducible integrity ba• sis for an arbitrarily large number of symmetric 3 X 3 matrices with respect to the full or• thogonal group [D7,8,12]. This turned out to be a problem of considerable complexity. However, with much labor and some contribution from G.F. Smith the task was satisfacto• rily completed. Drawing on the experience gained with this problem, it was natural to de• termine an irreducible integrity basis for an arbitrary number of matrices and vectors un• der the full and proper orthogonal groups [D19]. This work was not completed until long after Spencer's return to England. The general problem of the restrictions on the constitutive equations of continuum mechanics and of other branches of continuum physics, which are imposed by form• invariance under superposed rotation of the material system and by symmetry of the mate• rial, were discussed in two papers [D9,13] by my graduate student A.C. Pipkin and myself. This raised the question of the integrity bases for sets of tensors under orthogonal groups that are not necessarily the full or proper orthogonal group. Accordingly, G.F. Smith and I applied the methods used in our paper [D4] on the irreducible integrity bases for a single symmetric 3 X 3 matrix under the crystallo• graphic point groups to the determination of the corresponding integrity bases for a sym• metric 3 X 3 matrix and a vector [D22] and for an arbitrary number of vectors [D21]. This work was done after Smith had obtained his doctorate and gone on to various academic appointments. In the program of work on the determination of integrity bases my main motivation was not an intention to apply the results to any particular physical problem. Rather, it seemed desirable to have the results in the literature so that they could be drawn on, when appro• priate, by myself and others. Autobiographical Postscript xxxvii
In discussing nonlinear constitutive equations, physicists generally suppose them to be polynomials in the independent variable vectors and tensors. These are truncated at some chosen degree, and the restrictions on the coefficients that are imposed by the assumed symmetry of the material are then obtained in the form of linear algebraic equations. The solution of these equations is then substituted for the coefficients in the polynomial, yield• ing the desired constitutive equation. This procedure has two basic shortcomings. Gener• ally, it becomes increasingly difficult, and eventually prohibitively so, as the degree of the polynomial increases. Also, it is totally inapplicable unless the constitutive equation is polynomial. Both of these shortcomings are avoided by using the appropriate integrity bases. Moreover, the constitutive equations are then obtained in a form in which the in• variant character is transparent. In view of the evident advantages of this approach it seems strange that it has been almost entirely neglected by the solid-state physicists. Although I have from time to time discussed problems in branches of continuum physics other than mechanics, I have been limited by an insufficient feel for what problems would be of interest. * * * Sometime in 1954 I revived my interest in the mechanics of materials in which there are di• rections of inextensibility. Although in the next few years I authored or coauthored three papers on the subject, only one of these seems now to be worth mentioning. This is a paper [B3] on planar deformations of a trellis-like network of ideally inextensible and flexible fibers, which appears to have had considerable influence on the work of Pipkin, Rogers, and Spencer. The paper, as it appears in the Archive for Rational Mechanics and Analysis, has an amusing publication history. I originally submitted it to Prager for publication in the Quar• terly ofApplied Mathematics, of which he was the editor. A short time later he told me that he found it interesting, but saw a way of obtaining geometrically some of the results that I had obtained analytically. He outlined for me the geometric argument and suggested that we rewrite the paper using his approach and publish it jointly. Since his argument appeared to me to be less attractive than my almost trivial analysis, I replied that I would prefer to stay with my version. However, I suggested that he might then publish his geometric method as an alternative procedure. He responded that he could not then publish my paper in the Quarterly ofApplied Mathematics, justifying this with the astonishing statement that "The Quarterly is dedicated to the publication of geometrical approaches to problems." I accordingly published my paper in the Archive. Prager never published his geometrical approach. * * * In the Fall of 1955, A.E. Green came to Brown to spend a sabbatical year with me. We col• laborated in developing mechanical constitutive equations for materials with memory. We took as our starting point the assumption that the Cauchy stress matrix at time t depends on the history of the deformation gradient matrix from some initial time in which the mate• rial is undeformed up to and including time t. The form of the constitutive equation is re• stricted by invariance under superposed rotation and by any symmetry the material may possess. In the first paper [D5] that emerged, it was pointed out that subject to appropriate continuity conditions, the constitutive functional may be expressed as the sum of multiple integrals. Also, the manner in which the Rivlin-Ericksen constitutive equation emerges from these constitutive equations of the memory (i.e., functional) type was spelled out explicitly. In two further papers [D I 0, 15] resulting from this collaboration (in one of which Spencer was a coauthor) these ideas were developed with particular emphasis on the possibility of particular dependence on the current values of the deformation gradients-for example, the possibility of instantaneous elastic response. xxxviii Autobiographical Postscript
* * * About a year after the publication of the first of these papers, Noll published a paper that has had a good deal of influence on the subsequent literature, particularly from the point of view of style and idiom. The paper deals with essentially the same topic as do the papers by Green and myself, but reads as though it is concerned with a branch of pure mathemat• ics. It consists of a sequence of definitions and theorems couched in the language of point set topology and seems to distance itself as far as possible from any physical meaning or insight. Since I regard mechanics as a branch of physics, I was naturally repelled by this aspect of the paper. However, I recognized that this was a question of background and taste and that the important issue was whether the idiom did in fact yield new results or insights. For many years I thought that Noll's simple derivation of the restrictions resulting from invariance of the constitutive equation under rotation of the reference system, i.e., frame indifference (or equivalently, under superposition of a rotation on the assumed deforma• tion), was fundamentally superior to that of Green and myself. However, I was disturbed by the manner in which he gave mathematical expression to the restrictions imposed by material symmetry and above all, by his characterization as an incompressible simple fluid of any material for which the expression for the Cauchy stress (apart from an arbitrary hydrostatic pressure) is invariant under all unimodular transformations of the reference system. Noll concluded from this definition that the constitutive equation for the Cauchy stress models a simple fluid if and only if the constitutive functional depends on the defor• mation only through the history of the deformation gradient matrix referred to the current configuration and in addition satisfies an isotropy condition. This immediately struck me as absurd since any functional of the deformation gradients referred to the initial configu• ration can also be expressed as a functional of the deformation gradients referred to the current configuration. However, apart from mentioning this curiosity, I did not look into the matter in depth until fairly recently. A few years ago, with my colleague (and erstwhile student) Gerald Smith, I made a critical study of the matter. We concluded that Noll's definition actually implies material symmetry, as normally understood by physicists, only if frame indifference is also introduced [D46]. Also, his definition of a simple fluid by an invariance condition has no physical basis and, in the mathematical argument based on it, Noll fails to recognize that in variance with respect to a group of transformations is mean• ingless unless the elements of the group are constant, independent of the entities trans• formed. It is therefore not surprising that by using Noll's argument one can deduce anum• ber of mutually contradictory results. Our recognition of the error in Noll's argument led us to realize that a similar logical er• ror arises in his determination of the implications of frame indifference by an argument based on the polar decomposition theorem, although in this case the conclusion reached is correct. It is, however, a trivial matter to formulate a logically correct argument based on the polar decomposition theorem. It is curious that despite the unsatisfactory nature of Noll's theory it has been extensively repeated in the secondary literature. That this is not a unique phenomenon is evidenced by the thermomechanical theory for materials with memory advanced by Coleman. In it the second law of thermodynamics is expressed in the form of the Clausius-Duhem inequality, in which the entropy is regarded as a functional of the histories of the deformation gradi• ent matrix and the temperature. However, no indication is given of the manner in which this functional dependence could be determined for a particular material by experiment, or thought experiment, or calculation from a model. In certain of the calculations based on the theory, the envisaged thermomechanical processes are, in fact, reversible-although this fact could be easily missed in the irrele• vant elaboration contained in the presentations. Accordingly, it is not surprising that in these cases the results obtained conform with well-known results in classical thermo• dynamics. Before I fully appreciated the difficulties inherent in the application of the Clausius• Duhem inequality to thermomechanical processes in materials with memory, Green and I Autobiographical Postscript XXXIX made the mistake of quoting it in such contexts [El,2]. Fortunately this was not in situa• tions in which our conclusions were seriously affected. Since then, I and others have drawn attention to these difficulties and our criticisms have not been answered. Nevertheless, many papers have been published, and continue to be published, in which the Clausius• Duhem inequality is used in irreversible situations, without any meaning being assigned to the entropy. I have mentioned two examples of theories that have been presented with a promise of mathematical rigor but lead to physically meaningless conclusions. More examples could have been given. Indeed, over the years an extensive literature of this type has emerged, sometimes involving mathematical errors. However, not all of the observations and con• clusions in these papers are without value, and consequently I could not totally ignore them, while pretending to be au courant with advances in continuum mechanics. Typically the difficulty I faced, and still face, in trying to extract what is worthwhile from papers of this type is that I first have to make their mathematical definitions and assumptions mean• ingful in physical terms. I then have to wade through a tedious succession of lemmas and theorems, analyzing in each case their physical implications. All too often, when I have fully understood the conclusions, I find them, where correct, more intuitively acceptable than the primitive assumptions on which the theorems are based. In my student days I was told that the main object of publication is the communication of one's ideas, throwing them open to critical evaluation by one's peers. Publication thus becomes a contribution to a debate out of which emerges an approximation to truth that may serve as a basis for further advances.* It was largely in this spirit that I presented at a meeting sponsored by the Batelle Institute, and subsequently published in the proceedings, a paper with the title "Red Herrings and Sundry Unidentified Fish in Nonlinear Continuum Mechanics." The somewhat strident tone of the paper, for which I have on occasion been criticized, was a response to the fact that previous more moderately worded criticisms were either not answered, or answered in a manner that was frivolous to the point of being offensive. The paper has the distinction that I received a request for a reprint from a marine biological library in La Jolla, California.
* * * In 1956 Ericksen published a paper in which he considered the flow pf a Reiner-Rivlin fluid through a straight pipe of noncircular cross-section under a constant pressure head. He showed that if it is assumed that the particles of the fluid flow in rectilinear paths, the equations for the determination of the velocity distribution over the cross-section of the pipe do not, in general, have a solution. Green and I then showed [F9] that for a Reiner• Rivlin fluid in which the departure from Newtonian behavior is sufficiently small, the flow consists of a transverse flow superposed on the longitudinal flow. This was calculated ex• plicitly when the cross-section of the pipe is elliptical. It was found that the stream lines for the transverse flow form four eddies, one in each of the quadrants of the cross-section; the particles of the fluid follow helical paths. At about this time W.E. Langlois joined me as a graduate student and I suggested that for his thesis, he calculate a number of flows for a slightly non-Newtonian fluid whose consti• tutive equation is of the Rivlin-Ericksen type. The first of the problems that he considered was that which Green and I had already discussed in the context of the Reiner-Rivlin con• stitutive equation. He found, as we had, that for a straight pipe of elliptical cross-section a transverse flow is superposed on the longitudinal flow that obtains in the Newtonian limit• ing case. In this as in the earlier work, the departure of the velocity field from that corre• sponding to Newtonian behavior is assumed to be sufficiently small so that the equations of motion that it satisfies can be linearized. If the non-Newtonian terms in the constitutive
*I have no doubt that these "pure" motives for publication never represented the total reality. How• ever, in continuum mechanics and perhaps in other areas of scientific research they seem to have been largely forgotten. xl Autobiographical Postscript equation are polynomial in the velocity gradients, the transverse flow first appears with the inclusion of fourth-degree terms and is then similar to that obtained by Green and myself. Inclusion of terms of higher (even) degree results in a breakup of each of the eddies in the transverse flow into two or more. These results, and corresponding results for convergent flows between infinite planes and in cones, are contained in Langlois's Ph.D. thesis. However, in our publication [F12] of the pipe flow results* we adopted a slightly different starting point, in which a hierarchy of constitutive equations for slow steady flows, based on the assumption that the velocity is small, is first set up, and the linear equations for successive non-Newtonian corrections to the flow field are solved. It was, of course, found that the transverse flow first arises in the fourth-order theory. The publication of these results was delayed by the fact that while our paper was being written, Langlois had already left Brown to take up employment with I.B.M. in California. Meanwhile, Coleman had proposed a slightly different hierarchy of slow flow approxima• tions based on the expansion of the time scale, which applies to nonsteady as well as steady flows.
* * * In connection with a research program funded by the Armstrong Cork Company, I was asked to study the characterization with respect to mechanical behavior of a class of mate• rials in which they were interested-polyvinyl chloride-clay mixtures of the type used mainly,as flooring materials. The behavior of these materials is highly nonlinear, even for small deformations, and they exhibit instantaneous elasticity and marked creep and stress• relaxation effects. Quite early in my consideration of the constitutive equations for materials with memory it was apparent that the canonical forms that emerge involve far too many functions or functionals to allow for their experimental determination in a manner analogous to that in which the strain-energy function for vulcanized rubber had been determined. The canoni• cal forms are to be regarded as frameworks in which the constitutive equations for par• ticular materials must reside. In view of the impracticality of obtaining experimentally a constitutive equation which would be valid for all deformations of the material, I decided to tackle first the simpler problem of obtaining the constitutive equations valid for a few simple classes of deformations. It was my hope that the particular behavior exhibited in these experiments would lead me to simplifying restrictions that could be placed on the constitutive equations for wider classes of deformations. With this motivation I embarked on the problem of obtaining the constitutive equation appropriate for stress-relaxation experiments, i.e., for experiments in which the body is de• formed instantaneously and then held at constant deformation, while the change with time of the forces necessary to maintain the deformation is measured. The appropriate canoni• cal form can be obtained by specialization of the general constitutive equation for materi• als with memory. Alternatively, and more simply, it can be obtained from the assumption that the stress matrix at time t after the deformation is produced depends on the deforma• tion gradient matrix and on t [Dll]. The restrictions imposed on the form of this depen• dence by form in variance under superposed rotation and isotropy of the material in its un• deformed configuration then yield a canonical form for the Cauchy stress very similar to that obtained in finite elasticity theory. There, for an incompressible material, the material properties enter into the constitutive equation for the Cauchy stress through awta/1 and awta/2, derivatives of the strain-energy function Wwith respect to two strain invariants, / 1 and h. For stress relaxation in materials with memory, the derivatives of W with respect to the strain invariants are replaced by functions of the strain invariants, which are not neces• sarily derivable from a potential function, and oft. By making these substitutions in the ex-
*Our results on convergent flows were never published in the open literature. However, in 1963 Lan• glois discussed the flow of a non-Newtonian fluid between concentric rotating spheres. Autobiographical Postscript xli pressions for the forces associated with controllable deformations of an incompressible, isotropic, elastic material, it is possible to obtain corresponding results for stress-relaxing materials. In this way expressions were obtained for the tensile force and torque necessary to main• tain specified simultaneous simple extension and torsion in a rod of incompressible, isotropic, stress-relaxing material from the corresponding expressions for a rod of elastic material. These expressions, simplified somewhat by introducing the assumption that the displacement gradients are small while the expression for the stress remains nonlinear in them, were used successfully to model the results of a series of experiments on circular rods of the PVC-clay mixture in which the Armstrong Cork Company was interested. In these experiments, which were carried out in collaboration with J.T. Bergen and D.C. Messersmith of the Armstrong Cork Company, the rods were subjected simultaneously to various amounts of simple extension and torsion and held in these deformations while the dependence of the torque and tensile force on time was measured [Dl6]. One of the inter• esting results that emerged was the fact that the dependence on time was separable from the dependence on the amounts of torsion and extension. We then embarked on analogous experiments in which the rods were subjected simulta• neously to various constant rates of extension and torsion while the dependence of the ten• sile force and torque on time t was measured. The preliminary experiments indicated that our results could be modeled by a constitutive equation restricted to a class of deforma• tions in which the displacement vector is proportional to t. The Armstrong Cork Company management appeared to be quite interested in the re• sults we were obtaining, at any rate to the extent of sponsoring, in Lancaster in the summer of 1958, a two-day conference on related matters at which our work could be featured. However, while matters of more immediate commercial concern forced discontinuance of the experimental program, they continued their modest financial support of my theoretical research activities for many years. * * * Throughout the 1950s, although my theoretical research had developed an inner life of its own, it still drew to a significant extent on my experimental work at the B.R.P.R.A. However, more and more towards the end of the decade, I felt the lack of experimental ba• sis for my research and the danger of losing contact with reality-a disease common enough in continuum mechanics. While I had a good deal of confidence in my ability to design meaningful experiments, I had very little confidence in my abilities as a hands-on experimentalist. Also, I did not relish having to deal with the logistics of organizing a lab• oratory. These considerations pointed to collaboration with a good experimentalist. Ac• cordingly, Kolsky and I agreed to collaborate on an experimental program, the primary ob• jective of which would be to provide a physical basis for and critical evaluation of my theories. In order to implement this plan we would have required the cooperation of the Engineering Division in the provision of laboratory space, graduate students, workshop fa• cilities, and other services. It quickly became apparent that our proposed activity was sufficiently unwelcome to some members of the Division of Engineering, so that we might expect continual difficulty in this respect. We therefore abandoned our plan. * * * Early in 1957, I was told by Truesdell that Eli Sternberg was seriously interested in mov• ing from his position at the Illinois Institute of Technology. Although I was not familiar with Sternberg's work in detail, I had a favorable impression of it, and since in addition Truesdell praised it highly, I suggested to Prager and Lee that he be invited to join our department. Prager's immediate reaction was "he is much too difficult." I urged, rather strongly, that a university should be able to accommodate somewhat difficult people if they had sufficient creativity to justify the effort. On the basis of this argument Prager withdrew his opposition, and in due course, Sternberg was offered and accepted a professorship in xlii Autobiographical Postscript the Division of Applied Mathematics at Brown. He joined us at the beginning of the 1957- 58 academic year. At the time I had only limited experience-my four years at Brown-as a university faculty member and had not even begun to appreciate how unreasonable faculty members could be. I discovered this only after I assumed the chairmanship of the division at the beginning of the 1958-59 academic year. By then Lee had served successfully as chair• man for five years and gave every indication of relief at being able to hand over the chairmanship.
* * * In 1810, Wilhelm von Humboldt, who was at the time engaged in the organization of the German university system, wrote to his wife, "Mit wie vielen Schwierigkeiten ich bei dem allen zu kampfen habe, wie die Gelehrten-die unbandigste und am schwersten zu be• friedigende Menschenklasse-mit ihren sich ewig durchkreuzenden Interessen, ihrer Eifersucht, ihrem Neid, ihrer Lust zu regieren, ihren einseitigen Unsichten, wo jeder meint, dass nur sein Fach Unterstiitzung und Berforderung verdiene, mich umlagern, wie dann noch jetzt Unannehmlichkeiten und Zantereien mit andern Kollegien und Menschen hinzukommen, davon hast Du, teures Kind, keinen Begriff."* Allowing that this is, no doubt, an exaggeration induced by frustration, it remains a valid assessment of a substan• tial minority of faculty members. During my career as a faculty member, particularly while I was chairman of D.A.M., my colleagues provided ample examples of the charac• teristics enumerated by von Humboldt. It is interesting that the distance in time and space should have left unchanged these traits endemic to academics. When I assumed the chairmanship of D.A.M., my first indication of things to come was the reception of my appointment by two of my colleagues. One came to my office to con• gratulate me in the following terms: "Congratulations on becoming chairman! But you know what I think of chairmen; they are really glorified office boys. You are much too good a scientist for that." The next day another came to my office and congratulated me in nearly identical words. Leaving aside the ungraciousness of this statement, it did reflect some elements of my own view that a major role of the chairman is to provide for the needs of the faculty mem• bers in their activities as scholars and educators and to implement their wishes with regard to the conduct of the department. Decisions should be made by consensus, with particular weight attached to the views of the senior faculty members. When I took over the chairmanship of D.A.M., I expected that problems would arise as a result of conflicts of personality and opinion. However, in my belief that the people in• volved were intelligent and reasonable, I assumed that where such difficulties arose they would be resolved in a spirit of collegiality. I did not anticipate the cruelty with which such conflicts were sometimes given expression. Nor did I appreciate the extent to which jeal• ousy is endemic in the academic community, or the extremes to which it could lead some faculty members. I was therefore shocked when I encountered the readiness of some fac• ulty members to destroy a colleague with no better motivation than jealousy or pique at some slight, real or imagined.
* * * I spent the 1961-62 academic year at the University of Rome, where Gaetano Fichera was my gracious host. During that time, we and our wives established a warm friendship,
*You have no notion, dear child, with how many difficulties I always have to contend; how the schol• ars-the most intractable and most difficult people to please-beset me with their ever-conflicting interests, their jealousy, their grudges, their desire to rule, their one-sided vision where each one thinks that only his specialty is worthy of support and encouragement. To this must be added their unpleasantnesses and squabbles with colleagues and other people. Autobiographical Postscript xliii which has persisted to the present. In the preceding few years and during my stay in Rome I experienced more unreasonable behavior on the part of my colleagues at Brown than I felt was compensated by the satisfactions that I could derive from the chairmanship of D.A.M. Accordingly, by the time I returned to the United States, I had decided to resign from it. However, I delayed my resignation for about a year since it appeared desirable that I remain chairman until after the Fourth International Congress of Rheology, which I hosted at Brown in August 1963. On my relinquishing the chairmanship, I was appointed to an L. Herbert Ballou University Professorship, one of the few endowed chairs at Brown. Shortly after my return from my sabbatical year in Rome I became aware of Toupin's pa• per on continuum theories with internal variables and spent some time looking into their meaning from a physical point of view. In the Fall of 1963 A.E. Green came to Brown for a semester. I introduced him to Toupin's paper and to my own reflections on the subject. The subject seemed to intrigue him and together we embarked on a more general and, I think, more powerful approach to the formulation of internal variable theories [El ,2]. The essential element in our approach was the derivation of the field equations from the conservation of energy principle and the manner in which the internal variables involved transform under rigid motion. In subse• quent papers I underlined the importance that these invariance assumptions be based on the physical meaning of the internal variables. The same material may be modeled by dif• ferent choices of the internal variables, leading to different field equations. The form taken by the field equations for a given system may depend on the manner in which its total en• ergy is apportioned between internal and kinetic energies. I was concerned mainly with internal variable theories as they purport to model aspects of the structure of the material. Since the theories considered are continuum theories, the internal variables must, in general, be continuous functions of position, except perhaps at material discontinuities. The manner in which the passage is made from variables de• scribed on the material regarded as a system of discrete entities (e.g., atoms or molecules in a crystal, crystallites in a polycrystalline material) may be critical. At the end of his stay at Brown, Green went to Berkeley for a semester at the invita• tion of Paul Naghdi. There he established a collaboration that persisted until Naghdi's re• cent death. In the course of this they developed many implications of the internal variable concept. In the year or two after my resignation from the chairmanship of D.A.M. a number of senior faculty members, led by Prager, having maneuvered themselves into positions in which they could hardly stay at Brown without considerable loss of face, had either left Brown or were about to do so. In 1966 I received a tempting offer from Lehigh University. It was proposed that I as• sume the position of director of a "center for the application of mathematics," with the aim of strengthening and gaining distinction for Lehigh in theoretical research. The center was to be well financed by the university. This and the fact that the proposed compensation, fringe benefits, and conditions of employment generally were unusually generous led me, after much soul-searching, to accept the offer. However, I could not take up the appoint• ment until the beginning ofthe 1967-68 academic year, since I was already committed to spending a sabbatical year at the University of Paris. Unfortunately, shortly after I took up the appointment at Lehigh, it became apparent to me that the promised financial support for the center and the promised cooperation of var• ious departments would not materialize. However, the conditions of the appointment that applied to me personally-salary, fringe benefits, etc.-were meticulously honored. So, the major effect of the failure of the university to fund the center to the extent promised was to deprive me of a great deal of work and of such satisfactions and frustrations as might have resulted therefrom. Moreover, it freed me to take a much more active part in scientific organizations, such as the ASME and the Society of Rheology, than I had hitherto.
* * * xliv Autobiographical Postscript
The theory of the superposition of infinitesimal deformations on finite deformations of elastic materials, discussed, for example, by Green, Shield, and myself in 1951 [C 1], is a linear theory. Consequently many problems for which the th(!ory is relevant are tractable. The effect of the underlying finite static deformation on the propagation of stress waves of infinitesimal amplitude in an elastic material was studied by my graduate student Michael Hayes and me in the late 1950s [C2,3]. The theory for the superposition of infinitesimal vi• brations on finite deformations is, of course, closely related. We resumed our collaboration in 1968 when Hayes spent six months at Lehigh Univer• sity, on leave from the University of East Anglia. We extended our discussion of the effect of an underlying finite static deformation on the propagation of small amplitude waves to the case when the material is a viscoelastic solid [11 ,2]. From the early 1970s onwards, largely in collaboration with my colleague and erstwhile graduate student K.N. Sawyers, I used the theory developed with Hayes in the late 1950s to study two types of stability problem-the necessary and sufficient conditions on the strain• energy function for an isotropic, elastic material to ensure material stability and the condi• tions for the (structural) stability of a rod, not necessarily slender, of isotropic, elastic ma• terial under longitudinal thrust (or tension). The material stability problem proved to be surprisingly difficult. We translated it into mathematical terms as the determination of the necessary and sufficient conditions on the strain-energy function for the acoustic tensor to be positive definite for all directions of propagation of waves of infinitesimal amplitude superposed on finite, homogeneous defor• mations [C7, 11-14, 15, 18, 20, 21]. Surprisingly, the algebraic problem so posed turned out to be quite different for compressible and incompressible materials. While we obtained some previously undiscovered necessary conditions, we failed to complete the determina• tion of the desired necessary and sufficient conditions. It was left to Zee and Sternberg to achieve this in the incompressible case and to Simpson and Spector in the compressible case, although the latter work was presented in the context of Cauchy elastic materials. The problem of determining material stability conditions for viscoelastic fluids and solids remains largely an open problem. * * * K.N. Sawyers and I also studied the stability of a thick rectangular plate under thrust within the framework of finite elasticity for an incompressible, isotropic, elastic material, in which the strain-energy function is an arbitrary function of two basic strain invariants. The principle of exchange of stabilities reduces the problem to the determination of the critical conditions on a pure, homogeneous deformation of the plate for which a bifurca• tion solution of either the flexural or barreling (i.e., antisymmetric or symmetric) type ex• ists [C9, 16, 17, 19]. One of the interesting results that emerged when the analysis was ap• plied to a thin plate is the insensitivity of the critical condition for a flexural bifurcation to the choice of the strain-energy function [CIO]. This work led naturally to the consideration of the post-buckling behavior of a (possibly thick) elastic plate under thrust [C 16, 19]. Our point of view was similar to that adopted by Koiter in his well-known work on the question from 1945 onwards. Whether the post• buckling behavior is of the stable or snap-through type depends on whether the critical pure, homogeneous deformation at which a bifurcation solution occurs is or is not stable. This involves determining whether the stored energy is or is not greater than the stored en• ergy in the critical state of pure, homogeneous deformation for all kinematically possible states in its neighborhood. In our work we were concerned with an incompressible, isotropic, elastic material. Since the calculations are inherently cumbersome, we made in our first paper [C 16] the simplifying assumption that the strain-energy function is of the neo-Hookean type, but dealt with the stability of all the critical states, flexural and barrel• ing. In our second paper [Cl9], we employed the general strain-energy function for an incompressible, isotropic, elastic material, but restricted the analysis to flexural critical states and assumed that the aspect ratio of the plate is small. Autobiographical Postscript xlv
* * * Although by the end of the 1960s we had arrived at a good understanding of most of the problems involved in formulating constitutive equations in continuum mechanics, the modeling of finite deformation, rate-independent behavior, such as is involved in the form• ing of metals, remained an open question. Despite the important contributions of E.H. Lee and P.M. Naghdi, it is still, I believe, not fully resolved. More important, from my point of view, was the nonexistence of specific constitutive equations appropriate to specific viscoelastic solids or fluids. However, the slow flow ap• proximation to the Rivlin-Ericksen constitutive equation seemed to have validity for some viscoelastic fluids. Accordingly, I took these as the basis for many of the boundary-value problems I addressed in the following years. The general Rivlin-Ericksen equation also provides a basis from which linear constitu• tive equations can be derived for the superposition of infinitesimal flows on viscometric flows. With F.J. Lockett and M.M. Smith [Fl5,17] I applied these to obtain the conditions for a bifurcation solution to exist in Couette flow of a viscoelastic fluid and to the determi• nation of the aspect ratios of the Taylor cells in this flow. Most of the work that has been done on the numerical solution of boundary-value prob• lems for viscoelastic fluids by other workers has been based on constitutive equations of the evolution type, i.e., the rate of change of stress at any instant is a given function of the stress and kinematic gradients at that instant. Usually the equation adopted is a frame• indifferent modification of the equation for a Maxwellian fluid to which are added more or less arbitrarily chosen nonlinear terms with associated physical constants. Such equations, like the Maxwell equation on which they are based, involve only one relaxation time. Since viscoelastic fluids generally-and perhaps always-involve a spectrum of relax• ation times, it seems unlikely that equations of this type will provide quantitative predic• tions. However, by appropriate choice of the physical constants they may provide useful qualitative predictions. I am told by the computing experts that the widespread use of constitutive equations of the evolution type, rather than those of the memory type, is due to the fact that use of the latter would raise serious computational difficulties. If this is indeed the case, I hope that advances in computer technology may remove this limitation, in view of the greater flexi• bility of calculations based on constitutive equations of the memory type. To illustrate this we observe that in order to model a fluid with n relaxation times by a constitutive equation of the evolution type, an nth-order evolution equation, or n first -order equation, is required. The computer program that would be required in order to solve problems would depend radically on the number of relaxation times. In contrast, in a constitutive equation of the memory type, changing the number of relaxation times, or replacing a discrete spectrum of relaxation times by a continuous one, merely involves changing the kernels in the inte• grals, and hence very little change in the computer program. * * * As I reflect on my career, I realize that the time I spent as a faculty member coincided, for the most part, with a period in which universities were ascendant and, at any rate in the bet• ter universities, creativity and scholarly achievement were valued. This was, no doubt, due to some extent to the spectacular contributions of scientists in the Second World War and, in the competitive atmosphere of the Cold War, to the high esteem in which scientists were held in the Soviet Union. As I watch the changes that have taken place in the universities since my retirement, my sense of good fortune is reinforced. The liberal funding of research by the various government agencies, particularly the de• fense departments and the National Science Foundation, made a major contribution to the research activities of the Division of Applied Mathematics while I was at Brown and to the Center for the Application of Mathematics during the earlier part of my stay at Lehigh. During that time I was able to obtain, without much difficulty, more than adequate funding xi vi Autobiographical Postscript for my own research activities and those of some of my colleagues who found the process of writing proposals and reporting more annoying than I did. However, as the years passed most people have found it increasingly difficult and frustrating to obtain government sup• port for their research. This is due, in part, to the increased demands of an expanding uni• versity community. More importantly, from the mid 1960s on there has been an increasing emphasis on the part of the funding agencies on "mission relevance," first in the defense departments and then in the National Science Foundation. To make matters worse, the mis• sions seemed to change every few years, if not more often. Of course, "mission" has al• ways been a factor in funding by the defense departments, but until the mid 1960s it was, for the most part, rather liberally interpreted. Concurrently with the increasing difficulties surrounding the funding process, univer• sities have exerted increasing pressure on their faculty members to obtain increasing amounts of contract or grant support. As I view the situation from the vantage point of many years of retirement, I find it sad when I see young assistant professors, who should be spending their time more creatively, devoting six years awaiting tenure or dismissal to writing twenty, say, research proposals in the hope that one will be funded. Unfortunately scientific creativity is not generally accompanied by an aptitude for writing research pro• posals and a willingness to devote one's life to doing so-although it may be in particular cases. As the funding process became more restrictive, the influence of program managers in determining what proposals should be funded inevitably increased. In the fields of interest to me, and no doubt in many others, this has greatly enhanced the influence of the program managers in the various agencies in determining what areas of research should be devel• oped and the style of research that should be encouraged. It also leads to a significant de• crease in the quality of research, as it leads faculty members to undertake research in areas that, although they satisfy the "relevance" criterion, are not those in which they have their best ideas, interests, and experience. Any experienced researcher is aware that one of the most important ingredients for suc• cess in research lies in the choice of the problem to be addressed: the identification and for• mulation of a problem that is both significant and likely to be tractable. It is ironic that the final arbiters in this critical stage of research should be bureaucrats in Washington, whose personal involvement in research and whose aptitude for it is, for the most part, minimal. Of course, there are notable exceptions, who, although not themselves steeped in research, have sufficient vision as administrators to have made positive contributions. While these developments were taking place in Washington, changes were taking place in the universities, due in part, but only in part, to them. In my early days at Brown, while faculty members were encouraged to obtain external financial support for their research, they were not regarded as delinquent in their duties if they failed to do so, whether this was due to their own inclination or to lack of interest in their program on the part of the fund• ing agencies. However, it appears that as the years have passed, grant and contract support has become an addiction in many universities. They seem to have passed from a situation in which contracts and grants were regarded as a means of facilitating the research ambi• tions of faculty members, to one in which the main objective of a faculty member's re• search is the attraction of contracts and grants, without any concern as to whether or not they have any valid academic purpose. At the same time as these changes were taking place, the size, intrusiveness, and influence of university administrations has greatly in• creased. One can only hope that this is a passing phase. The Complete Bibliography of the Publications of R.S. Rivlin*
A. Isotropic Finite Elasticity I. Torsion of a rubber cylinder. R.S. Rivlin, Journal ofApplied Physics 18,444-449 (1947). 2. Some applications of elasticity theory to rubber engineering. R.S. Rivlin. Proc. Rubber Technol• ogy Conference, London, June 23-25, ed. T.R. Dawson, publ. Heffer, Cambridge 1948, pp. 1-8. 3. A uniqueness theorem in the theory of highly-elastic materials. R.S. Rivlin, Proceedings of the Cambridge Philosophical Society 44,595-597 (1948). 4. A Note on the torsion of an incompressible highly-elastic cylinder. R.S. Rivlin, Proceedings of the Cambridge Philosophical Society 45,485-487 (1948). 5. Large elastic deformations of isotropic materials. I. Fundamental concepts. R.S. Rivlin, Philo• sophical Transactions of the Royal Society of London, A 240, 459-490 (1948). 6. Large elastic deformations of isotropic materials. II. Some uniqueness theorems for pure, ho• mogeneous deformation. R.S. Rivlin, Philosophical Transactions of the Royal Society of Lon• don, A 240,491-508 (1948). 7. Large elastic deformations of isotropic materials. III. Some simple problems in cylindrical po• lar coordinates. R.S. Rivlin, Philosophical Transactions of the Royal Society of London, A 240, 509-525 (1948). 8. Large elastic deformations of isotropic materials. IV. Further developments of the general theory. R.S. Rivlin, Philosophical Transactions ofthe Royal Society ofLondon, A 241,379-397 (1948). * Cylindrical shear mountings. R.S. Rivlin, D.W Saunders, Transactions of the Institution of the Rubber Industry, 25,296-306 (1949). 9. Large elastic deformations of isotropic materials. V. The problem of flexure. R.S. Rivlin, Pro• ceedings of the Royal Society of London, A 195, 463-473 ( 1949). 10. Large elastic deformatiuns of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. R.S. Rivlin, Philosophical Transactions of the Royal Society of London, A 242, 173-195 (1949). 11. An informal discussion on the theory of large elastic strains. I. On the definition of strain. R.S. Rivlin, Some Recent Developments in Rheology, ed. British Rheologists' Club, pub!. United Trade Press, London 1950, pp. 125-129 * Theoretical aspects of dynamical experiments on rubber. R.S. Rivlin, Transactions of the Insti• tution of the Rubber Industry, 26,78-85 (1950). * Mechanics of large elastic deformations with special reference to rubber. R.S. Rivlin, Nature, 167,570-595 (1951). 12. Large elastic deformations of isotropic materials. VII. Experiments on the deformation of rub• ber. R.S. Rivlin, D.W Saunders, Philosophical Transactions of the Royal Society of London, A 243,251-288 (1951). 13. Large elastic deformations of isotropic materials. VIII. Strain distribution about a hole in a sheet. R.S. Rivlin, A. G. Thomas, Philosophical Transactions of the Royal Society of London, A 243, 289-298 (1951).
*An asterisk implies that the publication is not included in this collection.
xlvii xlviii Bibliography
14. Large elastic deformations of isotropic materials. IX. The deformation of thin shells. J.E. Adkins, R.S. Rivlin, Philosophical Transactions ofthe Royal Society ofLondon, A 244, 505-531 (1952). 15. Experiments on the mechanics of rubber. I. Eversion of a tube, A.N. Gent, R.S. Rivlin, Pro• ceedings of the Physical Society of London, B 65, 118-121 (1952). 16. Experiments on the mechanics of rubber. II. The torsion, inflation, and extension of a tube. A.N. Gent, R.S. Rivlin, Proceedings of the Physical Society of London, B 65, 487-501 (1952). 17. Experiments on the mechanics of rubber. III. Small torsions of stretched prisms. A.N. Gent, R.S. Rivlin, Proceedings of the Physical Society of London, B 65,645-648 (1952). 18. The free energy of deformation for vulcanized rubber. D.W. Saunders, R.S. Rivlin, Transac• tions of the Faraday Society 48, 200-206 (1952). 19. Departures of the elastic behaviour of rubbers in simple extension from the kinetic theory. L. Mullins, S.M. Gumbrell, R.S. Rivlin, Transactions of the Faraday Society, 49, 1495-1505 (1953). 20. The solution of problems in second-order elasticity theory. R.S. Rivlin, Journal ofRational Me• chanics and Analysis 2, 53-81 (1953). 21. A theorem in the theory of finite elastic deformations. R.S. Rivlin, C. Topakoglu, Journal ofRa• tional Mechanics and Analysis 3, 581-589 (1954). 22. Stress-relaxation in incompressible elastic materials at constant deformation. R.S. Rivlin, Quarterly ofApplied Mathematics 14, 83-89 (1956). 23. Large elastic deformations. R.S. Rivlin, Rheology, Theory and Applications, Vol. 1, ed. F.R. Eirich, publ. Academic Press, New York 1956, pp. 351-385. 24. Dimensional changes in crystals caused by dislocations. R.A. Toupin, R.S. Rivlin, Journal of Mathematical Physics 1, 8-15 (1960). 25. Some topics in finite elasticity. R.S. Rivlin, Structural Mechanics, eds. J.N. Goodier, N.J. Hoff, publ. Pergamon, New York 1960, pp. 169-198. 26. Energy propagation for finite amplitude shear waves. M.A. Hayes, R.S. Rivlin, Zeitschrift fiir Angewandte Mathematik und Physik 22, 1173-1176 ( 1971 ). * Review of "Large elastic deformations" by A.E. Green and J.E. Adkins. R.S. Rivlin, Physics To• day, 57 (1971). 27. Energy propagation in a Cauchy elastic material. M.A. Hayes, R.S. Rivlin, Quarterly ofApplied Mathematics 30,219-222 (1972). 28. Stability of pure homogeneous deformations of an elastic cube under dead loading. R.S. Rivlin, Quarterly of Applied Mathematics 32,265-271 (1974). 29. The strain-energy function for elastomers. K.N. Sawyers, R.S. Rivlin, Transactions of the Soci• ety of Rheology 20, 545-557 (1976). 30. Some research directions in finite elasticity theory, Rheologica Acta 16, 101-112 (1977). * Rubberlike elasticity: phenomenological theory. R.S. Rivlin, Encyclopedia ofMaterials Science and Engineering, ed. M.B. Bever, publ. Pergamon 1986, pp. 4286-4288. 31. Reflections on certain aspects of thermomechanics. R.S. Rivlin, Finite Thermoelasticity, Con• tributi del Centro Interdisciplinare di Scienze Matematiche e Loro Applicazione No. 76, ed. G. Grioli, Accademia Nazionale dei Lincei, Rome 1986, pp. 11-44. * The elasticity of rubber. R.S. Rivlin, Rubber Chemistry and Technology 65, G51-G66 (1992).
B. Anisotropic Finite Elasticity, Kinematic Constraints 1. Large elastic deformations of homogeneous anisotropic materials. J.L. Ericksen, R.S. Rivlin, Journal of Rational Mechanics and Analysis 3, 281-301 (1954). 2. Large elastic deformations of isotropic materials, X. Reinforcement by inextensible cords. J.E. Adkins, R.S. Rivlin, Philosophical Transactions of the Royal Society of London, A 248, 201-223 (1955). 3. Plane strain in a net formed by inextensible cords. R.S. Rivlin, Journal of Rational Mechanics and Analysis 4, 951-974 (1955). 4. Stress-deformation relations for anisotropic solids. R.S. Rivlin, G.F. Smith, Archive for Ratio• nal Mechanics and Analysis 1, 107-112 (1957). 5. The strain-energy function for anisotropic elastic materials. G.F. Smith, R.S. Rivlin, Transac• tions of the American Mathematical Society 88, 175-193 (1958). * The deformation of a membrane formed by inextensible cords. R.S. Rivlin, Archive for Ratio• nal Mechanics and Analysis, 2, 447-476 (1959). * Infinitesimal plane strain in a network of elastic cords. S.M. Genensky, R.S. Rivlin, Archive for Rational Mechanics and Analysis, 4, 30-44 (1959). Bibliography xlix
6. Minimum weight design for pressure vessels reinforced with inextensible fibers. A.C. Pipkin, R.S. Rivlin, Journal ofApplied Mechanics 36, 103-108 (1963). 7. Networks of inextensible cords. R.S. Rivlin, Nonlinear Problems of Engineering, ed. W.F. Ames, pub!. Academic Press, New York 1964, pp. 51-64. * Plane stress in fiber-reinforced composites. R.S. Rivlin, Constitutive Laws for Engineering Ma• terials: Recent Advances and Industrial and Infrastructure Applications, eds. C.S. Desai, E. Krempl, G. Frantziskonis, H. Saadatmanesh, ASME Press, New York 1991, pp. 71-79. 8. Constitutive equation for a fiber-reinforced lamina. R.S. Rivlin, IUTAM Symposium on Ani• sotropy, Inhomogeneity and Nonlinearity in Solid Mechanics, eds. D.F. Parker, A.H. England, publ. Kluwer Academic Publishers, Dordrecht 1995, pp. 379-384.
C. Superposition of Small Deformations on Finite Deformations in Elastic Materials, Stability 1. General theory of small elastic deformations superposed on finite elastic deformations. A.E. Green, R.S. Rivlin, R.T. Shield, Proceedings of the Royal Society of London, A 211, 128-154 (1951). 2. Propagation of a plane wave in an isotropic elastic material subjected to a pure homogeneous deformation. M. Hayes, R.S. Rivlin, Archive for Rational Mechanics and Analysis, 8, 15-22 (1961). 3. Surface waves in deformed elastic materials. M. Hayes, R.S. Rivlin, Archive for Rational Me• chanics and Analysis, 8, 358-380 (1961). 4. A note on the secular equation for Rayleigh waves. M. Hayes, R.S. Rivlin, Zeitschrift for Ange• wandte Mathematik und Physik 13, 80-83 (1962). 5. Seismic wave propagation in a self-gravitating anisotropic earth. K.N. Sawyers, R.S. Rivlin, Philosophical Transactions of the Royal Society of London, A 263, 615-655 (1969). 6. Energy propagation in a deformed elastic material. M.A. Hayes, R.S. Rivlin, Archive for Ratio• nal Mechanics and Analysis 45, 54-62 (1972). 7. Instability of an elastic material. K.N. Sawyers, R.S. Rivlin, International Journal of Solids and Structures 9, 607-613 (1973). 8. Stability criteria for elastic materials. K.N. Sawyers, R.S. Rivlin, Developments in Mechanics, Vol. 7, eds. J.I. Abrams, T.C. Woo, School of Engineering, University of Pittsburgh 1973, pp. 321-331. 9. Bifurcation conditions for a thick elastic plate under thrust. K.N. Sawyers, R.S. Rivlin, Interna• tional Journal of Solids and Structures 10,483-501 (1974). * On the invalidity of certain material stability conditions. R.S. Rivlin, Proceedings of the Sec• ond National Congress of Theoretical and Applied Mechanics, Varna, Bulgaria, Vol. 1, ed. G. Brankov, publ. Bulgarian Academy of Sciences 1975, pp. 168-176. 10. The flexural fiburcation condition for a thin plate under thrust. K.N. Sawyers, R.S. Rivlin, Me• chanics Research Communications 3, 203-207 (1976). 11. On the speed of propagation of waves in a deformed elastic material. K.N. Sawyers, R.S. Rivlin, Zeitschrift for Angewandte Mathematik und Physik 28, 1045-1058 ( 1977). 12. On the speed of propagation of waves in a deformed compressible elastic material. K.N. Sawyers, R.S. Rivlin, Zeitschriftfiir Angewandte Mathematik und Physik 29,245-251 (1978). 13. A note on the Hadamard criterion for an incompressible elastic material. K.N. Sawyers, R.S. Rivlin, Mechanics Research Communications 5, 211-214 (1978). 14. The incremental shear modulus in an incompressible elastic material. K.N. Sawyers, R.S. Rivlin, Meccanica 13, 225-229 (1978). * Some reflections on material stability. R.S. Rivlin, Mechanics Today, Vol. 5, ed. S. Nemat• Nasser, Pergamon, Oxford 1980, pp. 409-425. 15. Some stability conditions for a compressible elastic material. K.N. Sawyers, R.S. Rivlin, Zeitschriftfor Angewandte Mathematik und Physik 32, 255-266 (1981). 16. Stability of a thick elastic plate under thrust. K.N. Sawyers, R.S. Rivlin, Journal ofElasticity 12, 101-125 (1982). 17. Bifurcation in an elastic plate on a rigid substrate. R.S. Rivlin, International Journal of Solids and Structures 18,411-418 (1982). 18. Some thoughts on material stability. R.S. Rivlin, Finite Elasticity, eds. D.E. Carlson, R.T. Shield, Martinus NijhoffPublishers, The Hague 1982, pp. 105-122. 19. Further results on the stability of a thick elastic plate under thrust. K.N. Sawyers, R.S. Rivlin, Journal de Mecanique Theorique et Appliquee 2, 663-698 (1983). Bibliography
20. The incremental shear modulus in a compressible isotropic elastic material. R.S. Rivlin, Zeitschrift fiir Angewandte Mathematik und Physik 35, 1-12 ( 1984 ). * Stability of an elastic material. R.S. Rivlin, Zeitschrift fiir Angewandte Mathematik und Physik, 35, 1-12 (1984). 21. Stability of an elastic material. R.S. Rivlin, Problemi Attuali dell'Analisi e della Fisica Mate• matica, ed. P.E. Ricci, publ. Dipartimento di Matematica, Universita di Roma "La Sapienza," Rome 1993, pp. 201-206.
D. Constitutive Equations, Invariants 1. Stress-deformation relations for isotropic materials. R.S. Rivlin, J.L. Ericksen, Journal of Ra• tional Mechanics and Analysis 4, 323-425 (1955). 2. Further remarks on the stress-deformation relations for isotropic materials. R.S. Rivlin, Journal of Rational Mechanics and Analysis 4, 681-701 ( 1955). 3. Tensors associated with time-dependent stress. B.A. Cotter, R.S. Rivlin, Quarterly of Applied Mathematics 13, 177-182 (1955). 4. The anisotropic tensors. G.F. Smith, R.S. Rivlin, Quarterly of Applied Mathematics 15, 308-314 (1957). 5. The mechanics of non-linear materials with memory. Part 1. A.E. Green, R.S. Rivlin, Archive for Rational Mechanics and Analysis 1, 1-21 (1957). 6. Note on a paper "Further remarks on the stress-deformation relations for isotropic materials." A. C. Pipkin, R.S. Rivlin, Archive for Rational Mechanics and Analysis 1, 469 (1958). 7. The theory of matrix polynomials and its application to the mechanics of isotropic continua. A.J.M. Spencer, R.S. Rivlin, Archive for Rational Mechanics and Analysis 2, 309-336 (1959). 8. Finite integrity bases for five of fewer symmetric 3 X 3 matrices. A.J.M. Spencer, R.S. Rivlin, Archive for Rational Mechanics and Analysis 2, 435-446 ( 1959). 9. The formulation of constitutive equations in continuum physics. I. A.C. Pipkin, R.S. Rivlin, Archive for Rational Mechanics and Analysis 4, 129-144 (1959). 10. The mechanics of non-linear materials with memory. Part 2. A.E. Green, A.J.M. Spencer, R.S. Rivlin, Archive for Rational Mechanics and Analysis 3, 82-90 (1959). 11. The constitutive equations for certain classes of deformations. R.S. Rivlin, Archive for Rational Mechanics and Analysis 3, 304-311 (1959). 12. Further results in the theory of matrix polynomials. A.J.M. Spencer, R.S. Rivlin, Archive for Rational Mechanics and Analysis 4, 214-230 (1960). 13. The formulation of constitutive equations in continuum physics. II. A.C. Pipkin, R.S. Rivlin, Archive for Rational Mechanics and Analysis 4, 262-272 (1960). 14. Constitutive equations for classes of deformations. R.S. Rivlin, Viscoelasticity, Phenomenolog• ical Aspects, ed. J.T. Bergen, Academic Press, New York 1960, pp. 93-108. 15. The mechanics of non-linear materials with memory. Part 3. A.E. Green, R.S. Rivlin, Archive for Rational Mechanics and Analysis 4, 387-404 (1960). 16. Stress-relaxation for biaxial deformation of filled high polymers. J.T. Bergen, D.C. Messer• smith, R.S. Rivlin, Journal ofApplied Polymer Science 3, 153-167 (1960). 17. Small deformations superposed on large deformations in materials with fading memory. A.C. Pipkin, R.S. Rivlin, Archive for Rational Mechanics and Analysis 8, 297-308 (1961). 18. Constitutive equations involving the functional dependence of one vector on another. R.S. Rivlin, Zeitschrift fiir Angewandte Mathematik und Physik 12, 44 7-452 (1961 ). 19. Isotropic integrity bases for vectors and second-order tensors. Part I. A.J.M. Spencer, R.S. Rivlin, Archive for Rational Mechanics and Analysis 9, 45-63 ( 1962). 20. Constraints on flow invariants due to incompressibility. R.S. Rivlin, Zeitschriftfiir Angewandte Mathematik und Physik 8, 589-591 (1962). 21. Integrity bases for vectors-the crystal classes. G.F. Smith, R.S. Rivlin, Archive for Rational Mechanics and Analysis 15, 169-221 (1964). 22. Integrity bases for a symmetric tensor and a vector-the crystal classes. G.F. Smith, M.M. Smith, R.S. Rivlin, Archive for Rational Mechanics and Analysis 12, 93-133 (1963). 23. On Cauchy's equations of motion. A.E. Green, R.S. Rivlin, Zeitschrift fiir Angewandte Mathe• matik und Physik 15,290-292 (1964). * Representation theorems in the mechanics of materials with memory. R.V.S. Chacon, R.S. Rivlin, Zeitschriftfiir Angewandte Mathematik und Physik, 15,444-447 (1964). * A note on the mechanical constitutive equations for materials with memory. R.S. Rivlin, Zeitschriftfiir Angewandte Mathematik und Physik, 15,652-654 (1964). Bibliography li
24. Mechanics of rate-independent materials. A. C. Pipkin, R.S. Rivlin, Zeitschrift fiir Angewandte Mathematik und Physik 16, 313-327 (1965). 25. Nonlinear viscoelastic solids. R.S. Rivlin, SIAM Review 7, 323-340 (1965). * Fundamental tensor relations of non-linear continuum mechanics. R.S. Rivlin, Theory of Plates and Shells, eds., J. Brilla, J. Balas, pub!. Slovak Academy of Sciences 1966, pp. 107-118. 26. Simple deformations of materials with memory. W.A. Green, R.S. Rivlin, Acta Mechanica 5, 254-273 (1968). 27. Some remarks on the mechanics of nonlinear viscoelastic materials. R.S. Rivlin, Proceedings of the Fifth International Congress on Rheology, Vol. 1, ed. S. Onogii, pub!. Tokyo University Press 1969, pp. 147-155. * Orthogonal integrity basis for N symmetric matrices. R.S. Rivlin, Contributions to Mechanics, ed. D. Abir, pub!. Pergamon, Oxford 1970, pp. 121-141. 28. On the principles of equipresence and unification. R.S. Rivlin, Quarterly ofApplied Mathemat• ics 30, 227- 228 (1972). 29. Materials with memory. R.S. Rivlin, Fracture of High Polymers, eds. H. Kausch, J.A. Hassell, R.I. Jaffee, pub!. Plenum Press, New York 1973, pp. 71-90. 30. A note on the Onsager-Casimir relations. R.S. Rivlin, Zeitschrift fur Angewandte Mathematik und Physik 24, 897-900 (1973). 31. The elements of non-linear continuum mechanics. R.S. Rivlin, Continuum Mechanics Aspects of Geodynamics and Rock Fracture Mechanics. ed. P. Thoft-Christensen, pub!. D. Reidel, Dor• drecht 1974, pp. 151-175. 32. Some restrictions on constitutive equations. R.S. Rivlin, Foundations of Continuum Thermo• dynamics, eds. J.J. Domingos, M.N.R. Nina, J.H. Whitelaw, pub!. Macmillan, London 1974, pp. 229-258. * Symmetry in constitutive equations. R.S. Rivlin, Symmetry, Similarity and Group-Theoretic Methods in Mechanics, eds. P.G. Glockner, M.C. Singh, pub!. University of Calgary 1974, pp. 23-44. 33. Finite elasticity theory as a model in the mechanics of viscoelastic materials. R.S. Riv• lin, J. Polymer Science, Symposium No. 48, ed. A.S. Dunn, pub!. Wiley, New York 1974, pp. 125-144. 34. Functional constitutive equations: mathematical formalism and physical reality. R.S. Rivlin, Proceedings of Workshop on Inelastic Constitutive Equations for Metals, Rensselaer Polytech• nic Institute, ed. E. Krempl, pub!. Rensselaer Polytechnic Institute 1974, pp. 162-168. 35. On the foundations of the theory of non-linear viscoelasticity. R.S. Rivlin, Mechanics of Visco• elastic Media and Bodies, ed. J. Hult, pub!. Springer, Berlin 1975, pp. 26-43. 36. On identities for 3 X 3 matrices. G.F. Smith, R.S. Rivlin, Rendiconti di Matematica 8, 345-353 (1975). 37. The thermodynamics of materials with fading memory. R.S. Rivlin, Theoretical Rheology, eds. J.F. Hutton, J.R.A. Pearson, K. Walters, pub!. Applied Science Publishers, London 1975, pp. 83-103. * The application of the theory of invariants to the study of constitutive equations. R.S. Rivlin, Trends in the Application of Pure Mathematics to Mechanics, ed. G. Fichera, pub!. Pitman, London 1976, pp. 299-310. * Notes on the theory of constitutive equations. R.S. Rivlin, Materials with Memory, ed. D. Graffi, pub!. Liguori Editore, Naples 1979, pp. 184-294. 38. Material symmetry and constitutive equations. R.S. Rivlin, Ingenieur-Archiv 49, 325-336 (1980). 39. Some comments on the endochronic theory of plasticity. R.S. Rivlin, International Journal of Solids and Structures 17, 231-248 (1981 ). * Comments on "On the Substance of Rivlin's Remarks on the Endochronic Theory, by K.C. Valanis." R.S. Rivlin, International Journal of Solids and Structures, 17, 267-268 (1981 ). 40. A note on the simple fluid. R.S. Rivlin, Journal of Non-Newtonian Fluid Mechanics 11, 209-213 (1982). 41. Integral representations of constitutive equations. R.S. Rivlin, Rheologica Acta 22, 260-267 (1983). 42. Two-dimensional constitutive equations. R.S. Rivlin, Rendiconti del Seminario Matematico dell'Universita di Padova 68,279-293 (1983). * On the fallacy in a paper by A.l. Murdoch. G.F. Smith, R.S. Rivlin, Journal of Non-Newtonian Fluid Mechanics, 12, 393-394 (1983). Iii Bibliography
* A reply to notes by A.I. Murdoch and C. Truesdell. R.S. Rivlin, Journal of Non-Newtonian Fluid Mechanics, 15,253-254 (1984). * Constitutive equations for materials with memory under restricted classes of deformations. R.S. Rivlin, Mechanics of Engineering Materials, eds. C.S. Desai, R.H. Gallagher, pub!. Wiley, New York 1984, pp. 535-546. 43. On constitutive functionals. R.S. Rivlin, Wissenschaftskolleg zu Berlin Jahrbuch I984/I985, pp. 163-178. 44. The description of material symmetry in materials with memory. G.F. Smith, R.S. Rivlin, Inter• national Journal of Solids and Structures 23, 325-334 ( 1987). 45. The passage from memory functionals to Rivlin-Ericksen constitutive equations. K. Wilmanski, R.S. Rivlin, Zeitschriftfiir Angewandte Mathematik und Physik 38,624-629 (1987). · 46. A note on material frame indifference. G.F. Smith, R.S. Rivlin, International Journal of Solids and Structures 23, 1639-1643 (1987). 47. Some remarks concerning material symmetry. R.S. Rivlin, Constitutive Laws for Engineering Materials: Theory and Applications, eds. C.S. Desai, E. Krempl, P.D. Kiousis, T. Kundu, pub!. Elsevier, New York 1987, pp. 213-220. 48. Comments on the paper "On the derivation of the constitutive equation of a simple fluid from that of a simple material" by R.R. Huilgol. G.F. Smith, R.S. Rivlin, Rheologica Acta 28, 246-252 (1989). 49. The simple fluid concept. G.F. Smith, R.S. Rivlin, Fluid Dynamics, eds. W.F. Ballhaus Jr., M.Y. Hussaini. pub!. Springer 1989, pp. 228-239. 50. An isotropic solid is a simple fluid. R.S. Rivlin, Rend. di Matematica 10,733-740 (1990). 51. Objectivity of the constitutive equation for a material with memory. R.S. Rivlin, International Journal of Solids and Structures 27,395-397 (1991). * Some restrictions on the mechanical constitutive equations for materials with memory. R.S. Rivlin, Atti dei Convegni Lincei, 92, 211-223 ( 1992). 52. Frame indifference. R.S. Rivlin, Rend. Mat. Accad. Lincei 3, 51-59 (1992).
E. Internal Variable Theories 1. Simple force and stress multipoles, A.E. Green, R.S. Rivlin, Archive for Rational Mechanics and Analysis 16, 325-353 (1964 ). 2. Multipolar continuum mechanics. A.E. Green, R.S. Rivlin, Archive for Rational Mechanics and Analysis 17, 113-147 (1964). * Directors and multipolar displacements in continuum mechanics. A.E. Green, P.M. Naghdi, R.S. Rivlin, International Journal of Engineering Science, 2, 611-620 (1965). * Multipolar continuum mechanics: functional theory I. A.E. Green, R.S. Rivlin, Proceedings of the Royal Society of London. A., 284, 303-324 (1965). 3. Generalized continuum mechanics. A.E. Green, R.S. Rivlin, Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids, ed. H. Parkus, L.I. Se• dov, pub!. Springer, New York 1966, pp. 132-145. 4. The mechanics of materials with structure. A.E. Green, R.S. Rivlin, Rheology and Soil Me• chanics, eds. J. Kravtchenko, P.M. Sirieys, pub!. Springer, Berlin 1966, pp. 1-7. 5. The relation between director and multipolar theories in continuum mechanics. A.E. Green, R.S. Rivlin, Zeitschriftfiir Angewandte Mathematik und Physik 18,208-218 (1967). * Generalized mechanics of continuous media. R.S. Rivlin, Mechanics of Generalized Continua, ed. E. Kroner, pub!. Springer, Berlin 1968, pp. 1-17. * The formulation oftheories in generalized continuum mechanics and their physical significance. R.S. Rivlin, The Formulation of Theories in Generalized Continuum Mechanics and their Phys• ical Significance, Istituto Nazionale di Alta Matematica, Symposia Mathematica I, ed., Istituto Nazionale di Alta Matematica, pub!. Academic Press, London 1969, pp. 357-373. 6. The passage from a particle system to a continuum model. R.S. Rivlin, Proceedings of Sympo• sium on Continuum Models of Discrete Systems, Archives of Mechanics, Vol. 20, 1975, pp. 549-561.
F. Non-Newtonian Fluids 1. Hydrodynamics of non-Newtonian fluids. R.S. Rivlin, Nature 160, 611 (1947). 2. The hydrodynamics of non-Newtonian fluids. I.R.S. Rivlin, Proceedings of the Royal Society of London, A 193, 260-281 (1948). Bibliography !iii
3. The hydrodynamics of non-Newtonian fluids. II. R.S. Rivlin, Proceedings of the Cambridge Philosophical Society 45, 88-91 (1948). * Normal stress coefficient in solutions of macromolecules. R.S. Rivlin, Nature, 161, 567-569 (1948). 4. The normal-stress coefficient in solutions of macromolecules. R.S. Rivlin, Transactions of the Faraday Society 45, 739-748 (1949). 5. Some flow properties of concentrated high-polymer solutions. R.S. Rivlin, Proceedings of the Royal Society of London, A 200, 168-176 (1950). 6. Measurement of the normal stress effect in solutions of polyisobutylene. R.S. Rivlin, H.W. Greensmith, Nature 168,664-667 (1951). 7. The hydrodynamics of non-Newtonian fluids. III. The normal stress effect in high-polymer so• lutions. H.W. Greensmith, R.S. Rivlin, Philosophical Transactions of the Royal Society of Lon• don, A 245,399-428 (1953). 8. Solutions of some problems in the exact theory of viscoelasticity. R.S. Rivlin, Journal ofRatio• nal Mechanics and Analysis 5, 179-188 (1956). 9. Steady flow of non-Newtonian fluids through tubes. A.E. Green, R.S. Rivlin, Quarterly of Ap• plied Mathematics 14, 299-308 (1956). * Some flow properties of non-Newtonian fluids. R.S. Rivlin, Proceedings of the 9th. Interna• tional Congress ofApplied Mechanics, Vol. 3, pub!. University of Brussels 1957, pp. 187-195. 10. The relation between the flow of non-Newtonian fluids and turbulent Newtonian fluids. R.S. Rivlin, Quarterly ofApplied Mathematics 15, 212-215 (1957). 11. Correction to my paper "The relation between the flow of non-Newtonian fluids and turbulent Newtonian fluids." R.S. Rivlin, Quarterly ofApplied Mathematics 17, 447 (1959). * Some reflections on non-linear viscoelastic fluids. R.S. Rivlin, Phenomenes de Relaxation at de Fluage en Rheologie Non-Lineaire, ed. Colloque international de rheologie, pub!. Editions du Centre National de Ia Recherche Scientifique, Paris 1961, pp. 83-93. 12. Slow steady-state flow of visco-elastic fluids through non-circular tubes. W.E. Langlois, R.S. Rivlin, Rendiconti di Matematica 22, 169-185 (1963). 13. Normal stresses in flow through tubes of non-circular cross-section. A.C. Pipkin, R.S. Rivlin, Zeitschriftfiir Angewandte Mathematik und Physik 14,738-742 (1963). 14. Second and higher-order theories for the flow of a visco-elastic fluid in a non-circular pipe. R.S. Rivlin, Second-Order Effects in Elasticity, Plasticity, and Fluid Dynamics, ed. M. Reiner, pub!. Macmillan, New York 1964, pp. 668-677. * Viscoelastic fluids. R.S. Rivlin, Research Frontiers in Fluid Dynamics, eds. R.J. Seeger, G. Temple, pub!. Interscience, New York 1965, pp. 144-170. * Viscometric flows of non-Newtonian fluids. R.S. Rivlin, Procedings of the First International Conference on Hemorheology, ed. A.L. Coply, pub!. Pergamon, Oxford 1966, pp. 157-171. 15. Stability in Couette flow of a viscoelastic fluid. Part I. F.J. Lockett, R.S. Rivlin, Journal de Me• canique 1, 475-498 (1968). 16. Nonlinear continuum mechanics of viscoelastic fluids. K.N. Sawyers, R.S. Rivlin, Annual Re• view of Fluid Mechanics 3, 117-146 (1971). * Some comments on the mechanics of viscoelastic fluids. R.S. Rivlin, Proceedings of the Third Canadian Congress ofApplied Mechanics, ed. P.G. Glockner, pub!. 1971, pp. 39-58. 17. Stability in Couette flow of a viscoelastic fluid. Part II. M.M. Smith, R.S. Rivlin, Journal de Me• canique 11,69-94 (1972). 18. Flow of a viscoelastic fluid between eccentric cylinders. I. Rectilinear shearing flow. B.Y. Bal• la1, R.S. Rivlin, Rheologica Acta 14, 484-492 (1975). 19. Flow of a viscoelastic fluid between eccentric cylinders. II. Fourth-order theory for longitudinal shearing flow. B.Y. Balla!, R.S. Rivlin, Rheologica Acta 14, 861-880 (1975). * Some results on the flow of viscoelastic fluids. R.S. Rivlin, Metodi Valutativi nella Fisica• Matematica, Problemi Attuali di Scienza No. 217, pub!. Accadamia Nazionale dei Lincei Rome 1975, pp. 119-141. 20. Flow of a viscoelastic fluid between eccentric rotating cylinders. B.Y. Balla!, R.S. Riv1in, Transactions of the Society of Rheology 20,65-101 (1976). 21. A note on secondary flow of a Non-Newtonian fluid in a non-circular pipe. R.S. Rivlin, Journal of Non-Newtonian Fluid Mechanics 1, 391-395 ( 1976). * Flow of a Newtonian fluid between eccentric rotating cylinders: Inertial effects. B.Y. Balla!, R.S. Rivlin, Archive for Rational Mechanics and Analysis, 62, 237-294 (1976). 22. Secondary flows in viscoelastic fluids. R.S. Rivlin, Theoretical and Applied Mechanics, ed. W.T. Koiter, pub!. North-Holland, Amsterdam 1977, pp. 221-232. liv Bibliography
23. Some superposition theorems for second-order fluids. J.Y. Kazakia, R.S. Rivlin, Journal of Non-Newtonian Fluid Mechanics 2, 151-157 (1977). * Flow of a Newtonian fluid between eccentric rotating cylinders. R.S. Rivlin, Problemi Attuali di Meccanica Teorica e Applicata, Supplement to Vol. 3 Atti della Accademia delle Scienze di Torino, Turin 1977, pp. 87-102. 24. Flow of a viscoelastic fluid between eccentric rotating cylinders and related problems. J.Y. Kazakia, R.S. Rivlin, Rheologica Acta 16, 229-239 (1977). 25. The influence of vibration on Poiseuille flow of a non-Newtonian Fluid. I. J.Y. Kazakia, R.S. Rivlin, Rheologica Acta 17,210-226 (1978). * Flow of a Newtonian fluid between eccentric rotating cylinders and related problems. J.Y. Kazakia, R.S. Rivlin, Studies in Applied Mathematics, 58, 209-247 (1978). 26. The effect of vibration on the flow of a turbulent Newtonian fluid. J.Y. Kazakia, R.S. Rivlin, Mechanics Research Communications 5, 383-384 (1978). 27. The effect of a longitudinal vibration on Poiseuille flow in a non-circular pipe. J.Y. Kazakia, R.S. Rivlin, Journal of Non-Newtonian Fluid Mechanics 6, 145-154 ( 1979). * Some recent results on the flow of non-Newtonian fluids. R.S. Rivlin, Journal of Non• Newtonian Fluid Mechanics, 5, 79-101 (1979). 28. The influence of vibration on Poiseuille flow of a non-Newtonian fluid. II. J.Y. Kazakia, R.S. Rivlin, Rheologica Acta 18, 244-255 (1979). 29. Flow of a viscoelastic fluid between eccentric cylinders. III. Poiseuille flow. B.Y. Balla!, R.S. Rivlin, Rheologica Acta 18, 311-322 (1979). * The mechanics of non-Newtonian fluids. R.S. Rivlin, Proc. 25th. Conference of Army Mathe• maticians, pub!. Army Research Office, Durham 1980, pp. 1-22. 30. Run-up and spin-up in a viscoelastic fluid. I. J.Y. Kazakia, R.S. Rivlin, Rheologica Acta 20, 111-127 (1981). 31. Superposition of longitudinal and plane flows of a non-Newtonian fluid between eccentric cylinders. J.Y. Kazakia, R.S. Rivlin, Journal of Non-Newtonian Fluid Mechanics 8, 311-317 (1981). 32. Run-up and spin-up in a viscoelastic fluid. II. R.S. Rivlin, RheologicaActa 21, 107-111 (1982). 33. Run-up and spin-up in a viscoelastic fluid. III. R.S. Rivlin, Rheologica Acta 21, 213-222 (1982). 34. Spin-up in Couette flow. R.S. Rivlin, Applicable Analysis 15, 227-253 (1983). 35. Run-up and spin-up in a viscoelastic fluid. IV. R.S. Rivlin, RheologicaActa 22,275-282 (1983). 36. Run-up and decay of plane Poiseuille flow. R.S. Rivlin, Journal of Non-Newtonian Fluid Me• chanics 14,203-217 (1984). 37. Decay of shear layers and vortex sheets. R.S. Rivlin, Journal of Non-Newtonian Fluid Mechan• ics 15, 199-226 (1984). 38. The propagation of vorticity in a viscoelastic fluid. R.S. Rivlin, Journal of Non-Newtonian Fluid Mechanics 17, 313-329 (1985).
G. Electromagnetism 1. Electrical conduction in deformed isotropic materials. A.C. Pipkin, R.S. Rivlin, Journal of Mathematical Physics 1, 127-130 ( 1960). 2. Linear functional electromagnetic constitutive equations and plane waves in a hemihedral isotropic material. R.A. Toupin, R.S. Rivlin, Archive for Rational Mechanics and Analysis 6, 188-197 (1960). 3. Galvanomagnetic and thermomagnetic effects in isotropic materials. A.C. Pipkin, R.S. Rivlin, Journal of Mathematical Physics 1, 542-546 (1960). 4. Electrical conduction in a non-circular rod. A. C. Pipkin, R.S. Rivlin, Journal of Mathematical Physics 2, 865-868 (1961). 5. Electro-magneto-optical effects. R.A. Toupin, R.S. Rivlin, Archive for Rational Mechanics and Analysis 7, 434-448 (1961). 6. Electrical conduction in a stretched and twisted tube. A. C. Pipkin, R.S. Rivlin, Journal ofMath• ematical Physics 2, 636-638 (1961). 7. Non-rectilinear current flow in a straight conductor. R.S. Rivlin, A.C. Pipkin, Journal of Math• ematical Physics 3, 368-371 (1962). 8. Propagation of electromagnetic waves in circular rods in torsion. J.E. Adkins, R.S. Rivlin, Philosophical Transactions of the Royal Society of London, A 255, 389-416 (1963). Bibliography lv
9. The Faraday and allied effects in circular wave-guides. J.E. Adkins, R.S. Rivlin, International Journal of Engineering Science 1, 187-198 (1963). 10. Maxwell's equations in a deformed body. J.B. Walker, A.C. Pipkin, R.S. Rivlin, Atti dell'Ac• cademia Nazionale dei Lincei 38,674-676 (1965). 11. Transverse electric and magnetic effects. M.M. Carroll, R.S. Rivlin, Quarterly of Applied Mathematics 23, 365-368 (1966). 12. Electrical, thermal, and magnetic constitutive equations for deformed isotropic materials. A.C. Pipkin, R.S. Rivlin, Memorie dell 'Accademia Nazionale dei Lincei 8(1 ), 1-29 (1966). 13. Electro-optical effects. I. M.M. Carroll, R.S. Rivlin, Journal of Mathematical Physics 8, 2088-2091 (1967). * Phenomenological theory of magnetic hysteresis. A.C. Pipkin, R.S. Rivlin, Journal of Mathe• matical Physics, 8, 878-883 (1967). 14. Electro-optical effects. II. M.M. Carroll, R.S. Rivlin, Journal of Mathematical Physics 9, 1701-1704 (1968). 15. Electro-optical effects. III. M.M. Carroll, R.S. Rivlin, Journal of Mathematical Physics 9, 2267-2278 (1968). 16. Magneto-optical effects. R.S. Rivlin, Memorie dell'Accademia Nazionale dei Lincei 9, 233-266 (1969). 17. Photoelasticity with finite deformations. G.F. Smith, R.S. Rivlin, Zeitschrift fiir Angewandte Mathematik und Physik 21, 101-115 (1970). 18. Birefringence in viscoelastic materials. G.F. Smith, R.S. Rivlin, Zeitschrift fiir Angewandte Mathematik und Physik 22, 325-339 (1971). 19. Propagation of first-order electromagnetic discontinuities in an isotropic medium. R. Venkatara• man, R.S. Rivlin, Archive for Rational Mechanics and Analysis 40, 373-383 (1971). 20. Propagation of an electromagnetic shock discontinuity in a non-linear isotropic material. R. Venkataraman, R.S. Rivlin, Zeitschrift fiir Angewandte Mathematik und Physik 23, 43-49 (1972). 21. Harmonic generation in an electromagnetic wave. R. Venkataraman, R.S. Rivlin, Zeitschriftfiir Angewandte Mathematik und Physik 24, 661-675 (1973). * Some comments on a paper by M.F. McCarthy. R.S. Rivlin, Archive for Rational Mechanics and Analysis, 46, 61-65 ( 1972). 22. Electro-optical and magneto-optical effects. R.S. Rivlin, The Photoelastic Effect and its Appli• cations, ed. J. Kestens, pub!. Springer, Berlin 1974, pp. 406-424. * Birefringence in finitely deformed materials. R.S. Rivlin, Comportement Mecanique des So/ides Anisotropes, Colloques internationaux du CNRS No 295, pub!. Editions Scientifiques du CNRS 1979, pp. 123-131.
H. Fracture 1. The effective work of adhesion. R.S. Riv1in, Paint Technology 9, 1-4 (1944). 2. Rupture of rubber. Part I. Characteristic energy for tearing. A. G. Thomas, R.S. Rivlin, Journal of Polymer Science 10, 291-318 (1952). 3. The trousers test for rupture. K.N. Sawyers, R.S. Rivlin, Engineering Fracture Mechanics 6, 557-562 (1974). 4. The incipient characteristic tearing energy for an elastomer cross-linked under strain. R.S. Rivlin, A. G. Thomas, Journal of Polymer Science (Physics edition) 21, 1807-1814 (1983). 5. The effect of stress relaxation on the tearing of vulcanized rubber. R.S. Rivlin, A. G. Thomas, Engineering Fracture Mechanics 18, 389-401 (1983).
I. Waves in Linear Viscoelastic Materials I. Propagation of sinusoidal small-amplitude waves in a deformed viscoelastic solid. I. M.A. Hayes, R.S. Rivlin, Journal of the Acoustical Society ofAmerica 46,610-616 (1969). 2. Propagation of sinusoidal small-amplitude waves in a deformed viscoelastic solid. II. M.A. Hayes, R.S. Rivlin, Journal of the Acoustical Society ofAmerica 51, 1652-1663 (1972). 3. Longitudinal waves in a linear viscoelastic material. M.A. Hayes, R.S. Rivlin, Zeitschrift fiir Angewandte Mathematik und Physik 23, 153-156 (1972). 4. A class of waves in a deformed viscoelastic solid. M.A. Hayes, R.S. Rivlin, Quarterly of Ap• plied Mathematics 30,363-367 (1972). !vi Bibliography
5. Plane waves in linear viscoelastic materials. M.A. Hayes, R.S. Rivlin, Quarterly of Applied Mathematics32, 113-121 (1974).
J. Crystal Physics 1. Grinding and scratching crystalline surfaces. R.S. Rivlin, Nature 146, 806-807 (1940). * Refraction patterns of the surfaces of opaque and translucent solids. R.S. Rivlin, W.A. Wooster, Nature, 148, 372 (1941). * Optical refraction patterns, Part 1: Theory. R.S. Rivlin, Proceedings of the Physical Society of London, 53,409-417 (1941). * Optical methods for determining the orientation of quartz crystals. R.S. Rivlin, Journal of Sci• entific Instruments, 22, 11 (1945). * A simple X-ray spectrometer. R.S. Rivlin, H.P. Rooks by, Journal of Scientific Instruments, 23, 148-150 (1946).
K. General 1. Mathematics and rheology: the 1958 Bingham Medal Address. R.S. Rivlin, Physics Today 12, 32-34 (1959). 2. The fundamental equations of nonlinear continuum mechanics. R.S. Rivlin, Dynamics of Fluids and Plasmas, eds. S.I. Pai eta!., pub!. Academic Press, New York 1966, pp. 83-126. * Review of "The non-linear field theories of mechanics" by C. Truesdell and W. Noll. R.S. Rivlin, Journal of the Franklin Institute, 282,338-340 (1966). * Review of "The non-linear field theories of mechanics" by C. Truesdell and W. Noll. R.S. Rivlin, Journal of the Acoustical Society, 40, 1213 (1966). * Review of "The non-linear field theories of mechanics" by C. Truesdell and W. Noll. R.S. Rivlin, Quarterly of Applied Mathematics, 25, 119-120 (1967). 3. Red herrings and sundry unidentified fish in nonlinear continuum mechanics. R.S. Rivlin, In• elastic Behavior of Solids, eds. Kanninen, Adler, Rosenfeld, Jaffee, pub!. McGraw-Hill, New York 1970, pp. 117-134. * An introduction to non-linear continuum mechanics. R.S. Rivlin, Non-linear Continuum Theo• ries in Mechanics and Physics and their Applications, ed. R.S. Rivlin, pub!. Edizioni Cre• monese, Rome 1970, pp. 151-309. * Relativistic equations of balance in continuum mechanics. G. Lianis, R.S. Rivlin, Archive for Rational Mechanics and Analysis, 48,64-82 (1972). * Review of "Stress analysis of polymers" by J.G. Williams. R.S. Rivlin, Journal of Polymer Sci• ence: Polymer Letters Edition, 11, 294 (1973). * Comments on some recent researches in thermomechanics. Plenary lecture presented at the tenth anniversary meeting of the Society for Engineering Science, North Carolina State Univer• sity, 1973. R.S. Rivlin, Recent Advances in Engineering Science, pub!. Scientific Publishers, Boston 1977, pp. 1-23. * Review of "The Foundations of Mechanics and Thermodynamics-Selected Papers" by W. Noll. R.S. Rivlin, American Scientist, 64, 100-101 (1976). * Review of "Selected papers on rheology by M. Reiner." R.S. Rivlin, Journal of Fluid Mechan• ics, 6, 429-432 (1976). 4. Forty years of non-linear continuum mechanics. R.S. Rivlin, Advances in Rheology, eds. B. Mena, A. Garcia-Rejon, C. Rangel-Defaille, pub!. Universidad Nacional de Mexico, Mexico City 1984, pp. 1-29. * Convocation for award of honorary doctorate to Ronald S. Rivlin, Aristotelian University of Thessaloniki. R.S. Rivlin,
L. Miscellaneous * The effect of hydrogen on the time-lag of argon-filled photoelectric cells. N.R. Campbell, R.S. Rivlin, Proceedings of the Physical Society of London, 49, 12-13 ( 1937). 1. Thermo-elastic similarity laws. A.E. Green, J.R.M. Radok, R.S. Rivlin, Quarterly of Applied Mathematics 15, 381-393 (1958). * Presentation of the Bingham Medal to Clifford A. Truesdell. R.S. Rivlin, Proceedings of the Fourth International Congress on Rheology, Vol2, ed. E.H. Lee, Interscience, New York 1965, pp. 1-2. Bibliography lvii
* Biography-Gaetano Fichera. R.S. Rivlin, Applicable Analysis 15, 3-5 (1983). * Archimedes: father of statics. R.S. Rivlin, Archimede, Mito Tradizione Scienza, ed. C. Dollo, pub I. Leo S. Olschki, Firenze 1992, pp. 395-414.
M. Electrical circuits * Impedance networks. R.S. Rivlin, The Physical Society Reports on Progress in Physics, 6, 389 (1939). * The stability of regenerative circuits. R.S. Rivlin, The Wireless Engineer, 17,298-302 (1940). * Stray capacitances-their influence on the effective inductance of a coil in a metal container. L.I. Farren, R.S. Rivlin, The Wireless Engineer, 18, 313-323 (1941). * Non-linear distortion, with particular reference to the theory of frequency modulated waves• Part I. E.C. Cherry, R.S. Rivlin, Philosophical Magazine, 32,265-281 (1941). * Non-linear distortion, with particular reference to the theory of frequency modulated waves• Part II. E.C. Cherry, R.S. Rivlin, Philosophical Magazine, 32, 272-293 (1942). * An extension of Campbell's theorem of random fluctuations. R.S. Rivlin, Philosophical Maga• zine, 34, 688-693 (1945). G.l. Barenblatt D.D. Joseph Editors
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