The cover illustrations portray the fundamental subdivision of viscoelastic behavior into rheodictic and arrheodictic response. Somewhat whimsically, they represent, on the left, an uncrosslinked and, on the right, a crosslinked polymer. The strings of beads simulate polymer chains. The small figures are "Maxwell's Demons": imaginary sentient, intelligent beings of atomic dimensions. In the illustration on the right the chains form a three-dimensionally crosslinked network. If a Demon finds itself on any portion of any chain, it can reach any other point on the network simply by walking along. In the illustration on the left the chains do not form a network. Here, the Demon is forced to jump or climb from one chain to the next. A crosslinked network cannot exhibit steady-state flow: therefore, the behavior is arrheodictic. An uncross-linked network can flow: hence, the response is rheodictic. Both illustrations also contain 'entanglements'. The Demon is puzzled: are these crosslinks or not? Can it walk across them?

Cover illustrations: Christopher A. Tschoegl

Nicholas W. Tschoegl

The Phenomenological Theory of Linear Viscoelastic Behavior

An Introduction

With 227 Figures and 25 Tables

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Nicholas W. Tschoegl

Professor of Chemical Engineering, Emeritus California Institute of Technology 1201 East California Blvd. Pasadena, CA 91125/USA

ISBN-13:978-3-642-73604-9 e-ISBN-13:978-3-642-73602-5 DOl: 10.1007/978-3-642-73602-5

Library of Congress Cataloging-in· Publication Data. Tschoegl, Nicholas w., 1918- . The phenomenological theory of linear viscoelastic behavior: an introduction / Nicholas W. Tschoegl. p. em. . Includes index. ISBN-13:978-3-642-73604-9 1. Viscoelasticity. 2. Rheology. I. Title. QA931.T765 1989 532'.053--dc19 88-30800

This work is subject to copyright. AlJ rights are reserved, whether the whole or part of the material is concerned, specificalJy the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1989 Softcover reprint of the hardcover 1 st edition 1989

The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typesetting: Asco Trade Typesetting Ltd., Hong Kong

2152/3020-543210 - Printed on acid-free paper

This book is dedicated in deep gratitude to Sophie

without whom I would not have become a scientist, and to those who made me one:

Alex, Eric, John and Thor

[Gott .. . j

Tu mir kein Wunder zulieb. Gib deinen Gesetzen recht die von Geschlecht zu Geschlecht sichtbarer sind.

Das Stundenbuch R.M. Rilke

Preface

One of the principal objects of theoretical research in any department of knowledge is to find the point of view from which the subject appears in its greatest simplicity.

J. Willard Gibbs

This book is an outgrowth of lectures I have given, on and off over some sixteen years, in graduate courses at the California Institute of Technology, and, in abbreviated form, elsewhere. It is, nevertheless, not meant to be a textbook. I have aimed at a full exposition of the phenomenological theory of linear viscoelastic behavior for the use of the practicing scientist or engineer as well as the academic teacher or student. The book is thus primarily a reference work.

In accord with the motto above, I have chosen to describe the theory of linear viscoelastic behavior through the use of the Laplace transformation. The treatment oflinear time-dependent systems in terms of the Laplace transforms of the relations between the excitation add response variables has by now become commonplace in other fields. With some notable exceptions, it has not been widely used in viscoelasticity. I hope that the reader will find this approach useful.

Elementary calculus and the rudiments of complex variable theory is the basic mathematical apparatus required for a profitable use of this book. The elements of transformation calculus are summarized in an Appendix. It introduces the notation used in this book and serves as a convenient reference. It also contains a discussion of those special functions, the delta, step, slope, ramp, and gate functions, which are indispensable in the theory. The reader is advised to scan the Appendix before he sets out on the book itself. This will show him whether he feels comfortable with what he knows about transformation calculus. Ifhe does not, he should read one or the other of the many texts on the subject.

The compass of the book is outlined on the pages entitled Scope which follow this Preface. As pointed out there, the linear theory of viscoelastic behavior is a specialized form of general linear response theory. The theory has much beauty owing to the symmetry resulting from the interchangeability of the variables which are considered excitation and response, respectively. I placed much emphasis on the development of this duality, sometimes at the risk of being repetitive. I did this because there are pitfalls in interchanging the excitation and response, and there is the tendency, all too prevalent, of arguing by analogy. This can easily lead to erroneous statements or equations. A further advantage of fully working out both sides ofthe theory, i.e. stating it in terms of relaxation behavior (response to strain) as well as in terms of retardation (or creep) behavior (response to stress) is that the book becomes a compendium in which most ofthe important relations are readily available. A quite detailed subject index should further aid this purpose. I attempted to include in it all those terms that the reader - in my idio-syncratic opinion - might be most likely to wish to look up. It is clearly impossible to do this so that it will satisfy everyone. My apologies if I slipped too

VIII Preface

often. Clearly, every occurrence of every term could not be referenced. Particular-ly with 'bread-and-butter' terms such as behavior, excitation, model, response, viscoelastic, etc., I had to concentrate on listing primarily those occurrences which refer to matters I thought important. I hope I did not miss too many items that should have been included. The entries are fairly extensively cross-indexed to minimize irritating 'see-under . . .'-s but I could not avoid these completely.

A list of symbols and an author index precede the subject index. The main body of the book is subdivided into eleven chapters and the Appendix. The various parts of the book are headed by one or more mottos. In addition there is also a prefatory quotation, a prologue and an epilogue. Notes and literary references to these follow the Appendix.

The chapters and the Appendix are divided into sections, and the latter are often further divided into subsections. In numbering these divisions, the chapter number is separated by a decimal point from the section number in the first, and the subsection number in the second, decimal position. The numbering of equations, figures, and tables begins anew with each section. The number is preceded by the chapter and section number and is separated from them by a dash. Thus, Eq. (8.2-5) is Equation 5 of Section 2 of Chapter 8.

When two equations are placed on the same line they are assigned the same number and are distinguished from each other in references by numerical subcripts appended to the equation number. The same device is used to distinguish portions of concatenated equations. Thus, when an equation takes the form

f(x) = g(x) = hex) (1)

Equation (1)1 refers to f(x) = g(x). Equation (1)2 may be g(x) = hex) or f(x) = hex). The context always makes clear which is meant.

In the interest of brevity certain abbreviations were used routinely. Thus, 'step strain' is simply short for 'strain as a step function of time'. Similary, 'harmonic stress excitation' stands for 'excitation consisting of a stress in the sinusoidal steady-state'. I trust that these shortcuts will be self-explanatory everywhere.

Each chapter contains several fully worked problems. These are collected at the end of the chapter for easier cross-referencing and to avoid interrupting the flow of the exposition. Many are essential to a full understanding of the theory. Others clarify or amplify mathematical details. Still others are designed to develop and test the manipulative skill of the reader.

References to the work of others, indicated by numbers in brackets, and compiled at the end of each chapter, have been used sparingly. I would have liked greatly to follow the historical development of the subject in detail. However, the book was long in writing and would have taken even longer if this kind of scholarly research had been added. Therefore, I made reference only to the earliest work whenever this seemed appropriate. Otherwise, the literature is referred to merely when I thought it necessary for the sake of further clarification or an extension of the text. This restraint applies equally to my own papers. I apologize to all that feel left out.

I have added, in footnotes, short biographical comments to the names (capitalized in the subject index) of the more often quoted scientists, physicists, mathematicians, and engineers where they are first mentioned in the text. The way

Preface IX

foreign names are commonly pronounced by English speakers unfamiliar with the spelling conventions (and their aberrations) of foreign tongues is all too often horrifying to others. I have therefore tried to render these names in the footnotes by their nearest American English phonemes, indicating the accent by an underscore. I hope that I have not erred here myself too often. Concerning my own name, for those who care, the letter combination 'Tsch' should be pronounced as the 'Ch' in Churchill. As for the rest of the name, the pronounciation has been clarified by Professor R. B. Bird of the University of Wisconsin, in the limerick:*

An eminent linguist called Tschoegl at an age when he barely could gurgle

knew Turkish and Frisian and Old Indonesian

and that the German for birds is Vogel.

During the writing of this book I was more then once reminded of the bewildered cry for help of a young warrior during Hungary's struggle against the Turks: "torokot fogtam, de nem ereszt"-'I caught a Turk but he doesn't let go of me'. It is thus with pleasure and pride that I acknowledge the contributions made by many of my students and collaborators who have taken part in working out specific details, checking problems, and reading and correcting parts of the manuscript. I would like to mention particularly (and in alphabetical order) R. Bloch, W. V. Chang, R. E. Cohen, M. Cronshaw, D. G. Fesko, R. W. Fillers, C;igdem Gurer, L. Heymans, K. Jud, W. K. Moonan, S. C. Sharda, G. Ward, and K. Yagii.

Heartfelt thanks go also to several of my colleagues at Cal tech who helped me with specific problems. These are especially professors Tom Apostol, Paco Lagerstrom, Willem Luxembourg, Charles de Prima, and John Todd.

Much of Chap. 5 was written during my two months tenure of a visiting professorship at the Technische Hogeschool, Delft, The Netherlands. Chapters 6, 7, and 9 as well as several sections in other chapters were worked out during a six month stay at the Johannes Gutenberg University in Mainz, Federal Republic of Germany, as the recipient of a U.S. Senior Scientist Award from the Alexander von Humboldt Foundation. It is a pleasure to mention that this award, also called the Humboldt Prize, was instituted by the German Federal Government in recognition of aid received from the United States after World War II.

Further work was done during my tenure of a visiting professorship at the Eidgenossische Technische Hochschule in Zurich, Switzerland, and during two months spent at the Centre de Recherche sur les Macromolecules in Strasbourg, France. Chapters 10 and 11 were drafted largely during a stay of three months at Edvard Kardelj University in Ljubljana, Yugoslavia. My sincere thanks go to all

* Not to be outdone, I offer advice on the writing of Professor Bird's name as follows: Another great linguist called Bird by his friends was once overheard

to mutter: in Chinese for my name the sign is

the same as for bird. How absurd!

x Preface

those who made these stays possible, particularly professors Hermann J aneschitz-Kriegl, Erhard Fischer, Joachim Meissner, Henri Benoit, and Igor Emri.

Last but not least I wish to acknowledge the dedicated work of a succession of very able secretaries, particularly (and, this time, in chronological order) the late Mrs. Eileen Walsh-Finke, Mrs. Kim Engel, Mrs. Lorraine Peterson, Mrs. Helen Seguine, Mrs. Rita Mendelson.

A great deal is being said in this book about models. I though it appropriate, therefore, to append as a Prologue Jorge Luis Borge's delightful little piece Del rigor en la sciencia, dealing with mankind's original model, the map.

The Epilogue at the end of the text admirably expresses my own feelings at the completion of my labor of many years.

Pasadena, January 1989 Nicholas W. Tschoegl

Scope

II concetto vi dissi. Or, ascoltate com'egli e svolto.

Leoncavallo: I Pagliacci

This book is concerned with the phenomenological description of the behavior of materials when these are deformed mechanically. Its subject matter is therefore a particular aspect of the science of rheology, that branch of mechanics which deals with the deformation and flow of matter. Material behavior is governed by rheological equations of state or constitutive equations. A constitutive equation links a dynamic quantity, the stress, with a kinematic quantity, the strain, through one or more parameters or functions which represent the characteristic response of the material per unit volume regardless of size or shape. In dealing with material behavior we may be concerned primarily with one or the other of two complementary aspects of the constitutive equation. Thus, we may be interested primarily in the stress-strain relations taking the material properties as given, or we may be concerned primarily with the material properties and not with the particular stresses and strains to which a given body of matter is subjected. The prediction of stresses and strains resulting from the imposition of prescribed tractions and/or displacements on a material body is the subject of stress analysis and is discussed in texts on solid and fluid mechanics. In this book we shall be concerned with the alternative way of viewing the constitutive equation, that of material behavior. An example may make the distinction clearer. The extension of a rod and the deflection of a cantilever beam fashioned of the same material represent different stress analysis problems. Both deformations, however, depend on the same material property. For a purely elastic material this is simply its Young's modulus. The determination of the modulus from either deformation assumes the stress-strain relations to be known. Conversely, the prediction of the extension of the rod or the imposed force requires that the modulus be known. Clearly, material behavior and stress analysis cover complementary aspects of deformation and flow.

Material behavior is termed viscoelastic if the material stores part of the deformational energy elastically as potential energy, and dissipates the rest simultaneously through viscous forces. We shall distinguish the theory of viscoelastic behavior as a discipline concerned with material behavior, from the theory of viscoelasticity which is concerned with stress analysis problems involving materials that are neither purely viscous nor purely elastic.

The rheological properties of a viscoelastic material are time-dependent. Although, in principle, all real materials are viscoelastic, this property becomes most prominent when the time required for the full development of a response is comparable with the time scale of the experiment performed to elicit it. The condition is notably present in polymeric materials which are thus the viscoelastic

XII Scope

materials par excellence. Although we shall deal here more specifically with the theory as developed for polymeric materials, most of it is applicable, mutatis mutandis, to other materials such as metals and ceramics, inasmuch as they exhibit viscoelastic behavior. Furthermore, the theory, with suitable modifications in notation, is applicable also to time-dependent material behavior other than rheological. In particular, the theory of dielectric behavior is quite closely related to that of viscoelastic behavior. The two theories have, in fact, developed in close parallel and allusions to this will be made several times.

The foregoing has served to clarify provisionally the meaning of the words viscoelastic behavior in the title of the book. This will be enlarged upon in Chap. 2. However, some further comments appear in order.

By phenomenological theory I mean that I have tried to formulate a general framework which, in principle, applies to all viscoelastic materials regardless of their molecular structure. The viscoelastic behavior of polymers in relation to their structure has been described in several excellent books (see, e.g. [1,2]). Similar books exist in the field of metals (see e.g. [3]). These have generally made use, without developing it explicitly, of the underlying phenomenological theory.

Confining the discussions in this book to linear behavior restricts it to behavior in deformations which are small enough so that a doubling of the excitation will elicit twice the response from the material. This restriction results in very considerable simplification and allows formulation of the subject as a unified theory applicable to all aspects of deformation and flow within the limitation to linear response. Moreover, the theory thus becomes another branch of general linear response theory. This permits us to use results worked out in other fields of physics concerned with linear response (e.g., electric circuit theory, the most highly developed of all linear systems theories) by applying the appropriate analogies (cf. Chap. 3).

Another important restriction ist that viscoelastic behavior is discussed in this book under the assumption that the thermodynamic variables, temperature and pressure, are constant. Thus I am dealing here exclusively with isothermal and isobaric viscoelastic behavior. I hope to present a discussion of the effects of temperature and of pressure elsewhere at another time.

Finally, the word Introduction in the subtitle refers to the level of presentation. In accordance with my aims stated in the Preface, I have foregone mathematical rigor without, I hope, sacrificing precision and clarity. The emphasis is thus on an understanding of the structure of the theory as it is applied in practice. The reader wishing to go on to more exacting treatments is referred to the excellent axiomatic presentations of Gurtin and Sternberg [4], and of Leitman and Fischer [5].

References 1. J.D. Ferry, Viscoelastic Properties of Polymers, 3rd ed., Wiley, New York, 1980. 2. R. Byron Bird, Robert C. Armstrong, and Ole Hassager, Dynamics of Polymeric Liquids:

Volume I, Fluid Dynamics; the previous authors and C. F. Curtiss, Volume II, Kinetic Theory, Wiley, New York, 1977.

3. C. M. Zener, Elasticity and Anelasticity of Metals, University of Chicago Press, Chicago, 1948. 4. M. E. Gurtin and E. Sternerg, On the Linear Theory of Viscoelasticity, Arch. Rat. Mech. Anal.

11: 291-356 (1962). 5. M.J. Leitman and G.M.C. Fischer, The Linear Theory of Viscoelasticity, in: S. Flugge, ed.,

Encyclopedia of Physics, Vol. VIa/3, Springer, Berlin, Heidelberg, New York 1973, pp. 1-123.

Prologue

Del rigor en la cienca

... En aquel Imperio, el Arte de la Cartografia logro tal Perfeccion que el Mapa de una sola Provincia ocupaba toda una Ciudad, y el Mapa del Imperio, toda una Provincia. Con el tiempo, estos Mapas Desmesurados no satisficieron y los Colegios de Cartografos levantaron un Mapa del Imperio, que tenia el Tamano del Imperio y coincidia puntualmente con el. Menos Adictas al Estudio de la Cartografia, las Generaciones Siguientes entendieron que ese dilatado Mapa eta Inutil y no sin Impiedad 10 entregaron a las Inc1emencias del Sol y de los Inviernos. En los Desiertos del Oeste perduran despedazadas Ruinas del Mapa, habitadas por Animales y por Mendigos; en todo el Pais no hay otra reliquia de las Disciplinas Geognificas.

Of Exactitude in Science

... In that Empire, the craft of Cartography attained such Perfection that the Map of a Single province covered the space of an entire City, and the Map of the Empire itself an entire Province. In the course of Time, these Extensive maps were found somehow wanting, and so the College of Cartographers evolved a Map of the Empire that was of the same Scale as the Empire and that coincided with it point for point. Less attentive to the Study of Cartography, succeeding Generations came to judge a map of such Magnitude cumbersome, and, not without Irreverence, they abandoned it to the Rigours of sun and Rain. In the western Deserts, tattered Fragments of the Map are still to be found, Sheltering an occasional Beast or beggar; in the whole Nation, no other relic is left of the Discipline of Geography.

Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. VII

Scope ........................................................... XI

Prologue ........................................................ XIII

1. Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.0 Introduction................................................. 1

1.1 Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Stress....................................................... 5

1.3 Strain and Rate of Strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Purely Elastic Linear Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13 1.4.1 Single-Plane Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16 1.4.2 Three-Plane Symmetry or Orthotropy . . . . . . . . . . . . . . . . . . . .. 17 1.4.3 Axisymmetry, or Transverse Isotropy. . . . . . . . . . . . . . . . . . . . .. 19 1.4.4 Isotropy............................................... 21

1.4.4.1 The Generalized Hooke's Law and the Elastic Moduli 23 1.4.4.2 The Generalized Hooke's Law and the Elastic

Compliances ................................... 28

1.5 Purely Viscous Linear Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30 1.5.1 The Generalized Newton's Laws. . . . . . . . . . . . . . . . . . . . . . . . .. 30

1.6 Problems.................................................... 32

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34

2. Linear Viscoelastic Response ................................... 35

2.0 Introduction................................................. 35

2.1 Linear Time-dependent Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37 2.1.1 Differential Representations: The Operator Equation. . . . . . .. 39 2.1.2 Integral Representation. The Boltzmann Superposition Integrals 41 2.1.3 Excitation and Response in the Transform Plane. . . . . . . . . . .. 43

2.2 The Impulse Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44

2.3 The Step Response Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47

XVIII Contents

2.4 The Slope Response Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51

2.5 The Harmonic Response Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55

2.6 Excitation and Response in the Time Domain. . . . . . . . . . . . . . . . . . .. 63

2.7 Problems.................................................... 66

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 68

3. Representation of Linear Viscoelastic Behavior by Series-Parallel Models 69

3.0 Introduction................................................. 69

3.1 The Theory of Model Representation. . . . . . . . . . . . . . . . . . . . . . . . . .. 69 3.1.1 The Elements of Electric Circuit Analysis. . . . . . . . . . . . . . . . .. 70 3.1.2 The Elements of Mechanical Model Analysis. . . . . . . . . . . . . .. 72

3.2 Electromechanical Analogies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77

3.3 The Elementary Rheological Models. . . . . . . . . . . . . . . . . . . . . . . . . . .. 80 3.3.1 The Spring and Dashpot Elements. . . . . . . . . . . . . . . . . . . . . . .. 81 3.3.2 The Maxwell and Voigt Units. . . . . . . . . . . . . . . . . . . . . . . . . . .. 82

3.4 Models with the Minimum Number of Elements. . . . . . . . . . . . . . . . .. 88 3.4.1 The Standard Three- and Four-Parameter Series-Parallel Models 88

3.4.1.1 The Models of the Standard Linear Solid .......... 89 3.4.1.2 The Models of the Standard Linear Liquid. . . . . . . .. 93

3.4.2 The Non-Standard Three- and Four-Parameter Series-Parallel

3.4.3 3.4.4

3.4.5

Models............................................... 96 3.4.2.1 The Non-Standard Three-Parameter Models ....... . 3.4.2.2 The Non-Standard Four-Parameter Models ....... . Other Four-Parameter Models .......................... . Behavior of the Standard Models in the Complex Plane. Initial and Final Values ................................ . Response of the Standard Models to the Standard Excitations

97 99

101

103 110

3.5 Models with Large Numbers of Elements ........................ 118 3.5.1 The Generalized Series-Parallel Models .................... 119

3.5.1.1 The Wiechert and Kelvin Models ................. 119 3.5.1.2 Spectral Strength and Time Dependence in Wiechert

and Kelvin Models .............................. 127 3.5.1.3 The Canonical Models ........................... 134

3.5.2 Non-Standard Series-Parallel Models ...................... 134 3.5.3 Non-Series-Parallel Models with Large Numbers of Elements. 135

3.6 Model Fitting ................................................ 136 3.6.1 Procedure X ........................................... 136 3.6.2 Collocation Method .................................... 137 3.6.3 Multidata Method ...................................... 143 3.6.4 Algorithm of Emri and Tschoegl. ......................... 145

3.7 Series-Parallel Models and the Operator Equation ................ 145

Contents XIX

3.8 Problems .................................................... 149

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 156

4. Representation of Linear Viscoelastic Behavior by Spectral Response Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 157

4.0 Introduction................................................. 157

4.1 The Continuous Spectral Response Functions . . . . . . . . . . . . . . . . . . .. 158 4.1.1 Continuous Time Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 158 4.1.2 Continuous Frequency Spectra ........................... 163 4.1.3 Determination of the Continuous Spectra .................. 169

4.2 Methods for Deriving the Continuous Spectra from the Step Responses 170 4.2.1 The Transform Inversion Method: The Approximations of

Schwarzl and Staverman ................................ 170 4.2.2 The Differential Operator Method ........................ 176 4.2.3 The Power Law Method: The Approximations of Williams

and Ferry ............................................. 180 4.2.4 The Finite Difference Operator Method: The Approximations

of Yasuda and Ninomiya, and of Tschoegl ................. 181

4.3 Methods for Deriving the Continuous Spectra from the Harmonic Responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 191 4.3.1 Differential Operator Method: The Approximations of Schwarz I

and Staverman, and of Tschoegl .......................... 191 4.3.1.1 Approximations Derived from the Storage Functions 192 4.3.1.2 Approximations Derived from the Loss Functions ... 200

4.3.2 The Transform Inversion Method ......................... 204 4.3.3 The Power Law Method: The Approximations of Williams

and Ferry ............................................. 207 4.3.4 The Finite Difference Operator Method: The Approximations of

Ninomiya and Ferry, and of Tschoegl ..................... 210 4.3.4.1 Approximations Derived from the Storage Functions 210 4.3.4.2 Approximations Derived from the Loss Functions ... 215

4.4 Comparison of the Approximation to the Continuous Spectra . . . . .. 220

4.5 The Discrete Spectral Response Functions ....................... 224

4.6 The Viscoelastic Constants .................................... 229

4.7 Problems .................................................... 237

References ....................................................... 243

5. Representation of Linear Viscoelastic Behavior by Ladder Models . . .. 244

5.0 Introduction ................................................. 244

5.1 General Ladder Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 244

xx Contents

5.2 Regular Ladder Models with a Finite Number of Elements: The Gross-Marvin Models ..................................... 249 5.2.1 The Respondances of the Gross-Marvin Models ............ 250 5.2.2 The Equivalent Series-Parallel Models ..................... 254

5.2.2.1 Limit Values ................................... 254 5.2.2.2 Poles and Residues .............................. 256 5.2.2.3 Respondances .................................. 259

5.3 Regular Ladder Models with a Finite Number of Elements: The Regular Converse Ladder Models .......................... 262 5.3.1 The Respondances of the Regular Converse Ladder Models .. 263 5.3.2 The Equivalent Series-Parallel Models ..................... 266

5.4 Comparison of the Obverse and Converse Regular Ladder Models. Model Fitting ................................. 267

5.5 Regular Ladder Models with an Infinite Number of Elements ...... 270 5.5.1 The Extended Gross-Marvin Models ...................... 270 5.5.2 The Extended Marvin-Oser Model ........................ 276 5.5.3 The Extended Regular Converse Ladder Model ............. 282 5.5.4 Other Extended Regular Ladder Models ................... 283

5.6 The Continuous Ladder or Material Transmission Line ............ 286 5.6.1 The Continuous Gross-Marvin Ladder or Inertialess Material

Transmission Line ...................................... 287 5.6.2 The Material Transmission Line with Inertia ............... 293

5.6.2.1 The Lossless Line ............................... 295 5.6.2.2 The Lossy Line ................................. 300

5.7 Problems .................................................... 307

References ....................................................... 313

6. Representation of Linear Viscoelastic Behavior by Mathematical Models 314

6.0 Introduction ................................................. 314

6.1 Modelling by the Use of Matching Functions .................... 315 6.1.1 Matching Functions of the Z- and S-Type ................. 317 6.1.2 Matching Functions of the A-Type ........................ 322 6.1.3 Modelling of the Experimental Response Functions ......... 326

6.1.3.1 The Step Responses ............................. 326 6.1.3.2 The Slope Responses ............................ 330 6.1.3.3 The Harmonic Responses ........................ 331

6.1.4 Modelling of the Respondances .......................... 333 6.1.4.1 Harmonic Response Models from Respondance

Models ........................................ 333 6.1.4.2 Step Response Models from Respondance Models ... 337

6.2 Models Based on Fractional Differentiation (Power Laws) ......... 338

6.3 Modelling of the Spectral Response Functions .................... 342

Contents XXI

6.3.1 Direct Modelling ....................................... 342 6.3.1.1 The Box Distribution ............................ 343 6.3.1.2 The Wedge and Associated Power Law Distribution. 346

6.3.2 Spectra Derived from Models for the Experimental Responses 355 6.3.2.1 Spectra Derived from Models for the Step Responses 356 6.3.2.2 Spectra Derived from Models for the Harmonic

Responses ..................................... 357

6.4 Problems .................................................... 361

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 363

7. Response to Non-Standard Excitations . .......................... 365

7.0 Introduction ................................................. 365

7.1 Response to the Removal or the Reversal of a Stimulus ........... 366 7.1.1 Creep Recovery ........................................ 366 7.1.2 Response to the Removal of a Constant Rate of Strain ...... 370 7.1.3 Response to the Reversal of Direction of a Constant Rate

of Strain .............................................. 376

7.2 Response to Repeated Non-Cyclic Excitations .................... 377 7.2.1 Staircase Excitations .................................... 378 7.2.2 Pyramid Excitations .................................... 380

7.3 Response to Cyclic Excitations ................................. 380 7.3.1 Non-Steady-State Response .............................. 382 7.3.2 Steady-State Response .................................. 385

7.3.2.1 In Terms of the Spectral Response Functions ....... 385 7.3.2.2 In Terms of the Harmonic Response Functions ..... 387

7.4 Approximations to the Spectra from Responses to Non-Standard Excitations .................................................. 391

7.5 Problems .................................................... 392

References ....................................................... 395

8. Interconversion of the Linear Viscoelastic Functions . . . . . . . . . . . . . . .. 396

8.0 Introduction ................................................. 396

8.1 Interconversion Between Relaxation and Creep Response Functions. 399 8.1.1 Interconversion Between the Respondances ................ 399 8.1.2 Interconversion Between the Harmonic Responses .......... 400 8.1.3 Interconversion Between the Step Responses ............... 401

8.1.3.1 Theoretical Interrelations ........................ 401 8.1.3.2 Numerical Evaluation of the Convolution Integrals .. 403 8.1.3.3 Empirical Interconversion Equations .............. 406

8.1.4 Interconversion Between the Spectral Functions ............ 407 8.1.4.1 Theoretical Interrelations ........................ 407 8.1.4.2 Approximate Interconversion . . . . . . . . . . . . . . . . . . . .. 409

XXII Contents

8.1.4.3 Approximate Calculation of the Spectra from the Step Responses ..................................... 410

8.2 Interconversion Between Time- and Frequency-Dependent Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 413 8.2.1 Theoretical Interrelations ................................ 413 8.2.2 Interconversions Requiring Numerical Integration .......... 417 8.2.3 Interconversion by Kernel Matching ...................... 418 8.2.4 Empirical Interconversion Equations ...................... 425

8.3 Interconversion Within the Frequency Domain ................... 425 8.3.1 Relations Between the Real and Imaginary Parts of the

Harmonic Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . .. 426 8.3.1.1 The Kronig-Kramers Relations ................... 426 8.3.1.2 Approximations to the Kronig-Kramers Relations ... 430 8.3.1.3 Using Mathematical Models ...................... 433

8.3.2 Relations Between the Absolute Modulus or Compliance and the Loss Angle ......................................... 435

8.4 Problems .................................................... 439

References ....................................................... 441

9. Energy Storage and Dissipation in a Linear Viscoelastic Material .... 443

9.0 Introduction ................................................. 443

9.1 Strain Excitation (Stress Relaxation Behavior) .................... 445 9.1.1 Step Response ......................................... 448 9.1.2 Slope Response ........................................ 450 9.1.3 Harmonic Response .................................... 452

9.2 Stress Excitation (Creep Behavior) .............................. 460 9.2.1 Step Response ......................................... 463 9.2.2 Slope Response ........................................ 464 9.2.3 Harmonic Response .................................... 466

9.3 Hysteresis ................................................... 471 9.3.1 Response to Harmonic Excitations ........................ 472 9.3.2 Response to Triangular Excitations ....................... 475

9.3.2.1 Single Cycle Response ........................... 476 9.3.2.2 Steady-State Response ........................... 482

9.4 Problems .................................................... 484

9.5 References ................................................... 488

10. The Modelling of Multimodal Distributions of Respondance Times . ... 489

10.0 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 489

10.1 Bimodal Distributions ........................................ 490 10.1.1 Series-Parallel Models ................................... 491

Contents XXIII

10.1.2 Models for the Spectral Response Functions ............... 493 10.1.3 Ladder Models ......................................... 494 10.1.4 Mathematical Models ................................... 496

10.2 Prolongated Unimodal Distributions ............................ 500 10.2.1 Series-Parallel Models ................................... 501 10.2.2 Models for the Spectral Response Functions ............... 502 10.2.3 Ladder Models ......................................... 503 10.2.4 Mathematical Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 506

10.3 Problems .................................................... 507

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 507

11. Linear Viscoelastic Behavior in Different Modes of Deformation ..... 508

11.0 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 508

11.1 The General (Anisotropic) Viscoelastic Stress-Strain Relations ...... 508 11.1.1 The Correspondence Principle. . . . . . . . . . . . . . . . . . . . . . . . . . .. 508 11.1.2 The General Stress-Strain Relations ....................... 512

11.2 The Isotropic Viscoelastic Stress-Strain Relations ................. 513 11.2.1 The Viscoelastic Generalized Hooke's Laws ................ 513 11.2.2 The Viscoelastic Generalized Newton's Laws ............... 514

11.3 Linear Viscoelastic Behavior of Isotropic Materials in Different Modes of Deformation .............................................. 515 11.3.1 Behavior in Shear ...................................... 516 11.3.2 Behavior in Isotropic Compression ........................ 517 11.3.3 Behavior in Constrained Uniaxial Compression ............. 523 11.3.4 Behavior in Uniaxial Tension and Compression ............. 525 11.3.5 Lateral Contraction (Poisson's Ratio) ..................... 528 11.3.6 Wave Propagation ...................................... 532

11.3.6.1 Wave Propagation in an Extended Medium ......... 533 11.3.6.2 Wave Propagation in a Bounded Medium .......... 536

11.4 Interconversion of the Isotropic Response Functions in Different Modes of Deformation ........................................ 537 11.4.1 Behavior in Shear from Measurements in Uniaxial Tension

Including Lateral Contraction. . . . . . . . . . . . . . . . . . . . . . . . . . .. 539 11.4.1.1 Exact Interconversion ........................... 539 11.4.1.2 Numerical Interconversion ....................... 541 11.4.1.3 Approximate Interconversion ..................... 542

11.4.2 Behavior in Bulk from Measurements in Uniaxial Tension and in Shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 542

11.4.3 Behavior in Bulk from Measurements in Uniaxial Constrained Compression and Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 543

11.4.4 Behavior in Bulk from Measurements in Uniaxial Tension or Compression Including Lateral Contraction or Dilatation .... 543

XXIV Contents

11.4.4.1 Exact Interconversion ........................... 543 11.4.4,2 Numerical Interconversion ....................... 544

11.5 Problems ......................... : .......................... 544

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 548

Appendix: Transformation Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 549

A 1 Introduction ................................................. 549

A2 Special Functions ............................................. 550 A2.1 The Delta Function ...................................... 551 A2.2 The Step Function ....................................... 554 A2.3 The Gate Function ...................................... 556 A2.4 The Slope Function ...................................... 558 A2.5 The Ramp Function ..................................... 559

A3 The Laplace Transform ........................................ 560 A 3.1 Properties .............................................. 560

A3.1.1 Linearity ......................................... 561 A3.1.2 Uniqueness ...................................... 561 A3.1.3 Translation ....................................... 561 A 3.1.4 Real Differentiation ............................... 561 A 3.1.5 Real Integration .................................. 562 A3.1.6 Real Convolution ................................. 563 A3.1.7 Limit Values ..................................... 563 A 3.1.8 Cut-off. The Finite Laplace Transform ............... 564 A3.1.9 Repetition ....................................... 564

A3.2 Inversion ............................................... 565 A3.2.1 The Residue Theorem ............................. 565 A3.2.2 Inversion on the Real Axis ......................... 568

A3.3 Laplace Transform Pairs .................................. 568 A3.4 The s-Multiplied Laplace Transform

(Carson Transform) ...................................... 570

A4 The Fourier Transform ........................................ 570 A4.1 Properties .............................................. 571 A4.2 Inversion ............................................... 571 A4.3 Generalized Fourier Transforms ........................... 573 A4.4 Generalized Fourier Transform Pairs ....................... 574

A5 The Stieltjes Transform ........................................ 574 A5.1 Properties .............................................. 575 A5.2 Inversion ............................................... 575

A5.2.1 The Jump ........................................ 575 A 5.2.2 Inversion on the Real Axis ......................... 577

A5.3 The Fourier-Laplace Transform ............................ 580 A5.4 Stieltjes Transform Pairs .................................. 581

Contents xxv

A6 The Hilbert Transform 581 A 6.1 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 582

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 582

Solutions to Problems ............................................. 583 Chapter 1 ....................................................... 585 Chapter 2 ....................................................... 596 Chapter 3 ....................................................... 601 Chapter 4 ....................................................... 620 Chapter 5 ....................................................... 638 Chapter 6 ....................................................... 663 Chapter 7 ....................................................... 673 Chapter 8 ....................................................... 684 Chapter 9 ....................................................... 694 Chapter 10 ...................................................... 714 Chapter 11 ...................................................... 721

Epilogue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 735

Notes on Quotations .............................................. 739

List of Symbols . .................................................. 741

Author Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 749

SUbject Index .................................................... 751