Table Of ContentBoundary Value Problems
in Linear Viscoelasticity
J.M. Golden G.A. C. Graham
Boundary Value Problems
in Linear Viscoelasticity
With 13 Figures
Springer-Verlag Berlin Heidelberg GmbH
Dr. John M. Golden
Roads Division, National Institute for Physical Planning
and Construction Research, St. Martin's House, Waterloo Road,
Dublin 4, Ireland
Professor Dr. George A. C. Graham
Department of Mathematics and Statistics, Simon Fraser University,
Burnaby, B.C. Canada VSA 1S6
ISBN 978-3-662-06158-9 ISBN 978-3-662-06156-5 (eBook)
DOI 10.1007/978-3-662-06156-5
Library of Congress Cataloging-in-Publication Data. Golden, J. M. (John M.), 1945 - . Boundary value problems
in linear viscoelasticity. Bibliography: p. Includes index. 1. Viscoelasticity. 2. Boundary value problems. I. Graham,
G.A.C. (George A.C.), 1939-. II. Title. QA931.G585 1988 531'.3823 88-4915
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© Springer-Verlag Berlin Heidelberg 1988
Originally published by Springer-Verlag Berlin Heidelberg New York in 1988.
Softcover reprint of the hardcover 1s t edition 1988
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It is a pleasure to dedicate this book to
Professor I.N. Sneddon, O.B.E., F.R.S.
It would not have come into existence had it
not been for his encouragement and advice
on general content.
Preface
The classical theories of Linear Elasticity and Newtonian Fluids, though trium
phantlyelegant asmathematicalstructures, do not adequatelydescribe the defor
mation and flow ofmost real materials. Attempts to characterize the behaviour
of real materials under the action of external forces gave rise to the science of
Rheology. Early rheological studies isolated the phenomena now labelled as
viscoelastic. Weber (1835, 1841),researching the behaviour ofsilk threats under
load, noted an instantaneous extension, followed by a further extension over a
long period oftime. On removal of the load, the original length was eventually
recovered. Healso deduced thatthe phenomenaofstressrelaxationand damping
of vibrations should occur. Later investigators showed that similar effects may
be observed in other materials. The German school referred to these as
"Elastische Nachwirkung" or "the elastic aftereffect" while the British school,
includingLord Kelvin,spoke ofthe "viscosityofsolids".The universal adoption
of the term "Viscoelasticity", intended to convey behaviour combining proper
ties both of a viscous liquid and an elastic solid, is of recent origin, not being
used for example by Love (1934), though Alfrey (1948)usesit in the context of
polymers.
The earliest attemptsat mathematicallymodelling viscoelasticbehaviourwere
those of Maxwell (1867)(actually in the context of his work on gases; he used
this model for calculating the viscosity of a gas) and Meyer (1874). The model
proposed by Meyer is generally associated with the names of Kelvin and Voigt
who however made theircontributionsmuch later. These differentialconstitutive
relations are discussed in section 1.6. Boltzmann(1874)proposed the general he
reditary integral form of the constitutive relations which is the basis of most
theoretical work on viscoelastic materials of the past three decades, though,
historically, it was slow in gaining acceptance against the more specialized and
cumbersome differential constitutive relations based on mechanical models,
which essentially generalize the work of Maxwell and Meyer. Boltzmann gives
the form applicable to isotropic bodies while Volterra (1909) gives the general
anisotropic form. Volterra's theory of functionals (1959)is also at the basis of
the modern formulation of Viscoelasticityin the general non-linearcase. Further
discussionofthe early historyofthe subject may be found in the interesting arti
cle by Markovitz (1977),and in the book by Love (1934). Until the nineteen fif
ties, development of the subject was slow. The emergence into common use of
a large variety of polymeric materials in the post-war years focussed increasing
attention on the topic. References which focus on polymeric materials and their
behaviour include Eirich (1956), Staverman and Schwarzl (1956), Ferry (1970)
and (more recently) Doi and Edwards (1986). The application of the theory to
VIII Preface
metals has been surveyed byZener (1948),whileArutyunyan (1952)and Bazant
(1975)survey literature that treats concrete as an aging viscoelastic material.
Viscoelasticboundary value problems have been activelyresearched now for
more than thirty years. During the nineteen fifties, attention centred mainly on
the Classical Correspondence Principle and problems to whichit was applicable
(e.g. see Hunter (1960) and Lee (1960». Extensions of this principle were
discovered during the sixties.Also, severalproblems not covered bythe classical
form of the principle received attention. These were of two kinds, extending
crack problems and contact problems where the load on the indentor isvarying
or the indentor ismoving across the surface. These problems wereof interest in
the context of polymer fracture, rebound testing of polymers and the
phenomenon of hysteretic friction, respectively. Methods werealso developedto
handle problems involving thermoviscoelastic behaviour where the dependence
on temperatue is non-linear. All of this work presupposed that inertial effects
could be neglected. Very little work on inertial boundary value problems was
published up to the end of the sixties.
Comprehensive surveys of the application of viscoelastic stress analysis to
design were prepared by Rogers (1965)and Lee (1966).
Duringtheseventies,the workon non-inertial problemswasconsolidated.The
main purpose of the presentvolumeisto present a coherent, unified development
of this topic, in particular of those problem classeswhichare not covered by the
ClassicalCorrespondence Principle.There hasalsobeensomeprogresson inertial
problems. Typicallyhowever, to make progress on such problems it is necessary
either to confine one's attention to the most idealized configurations or to in
troduce some approximation. Also, the mathematical techniques used have been
generallyrather sophisticated. Webrieflydiscussthisworkinthelastchapter, and
derive certain results by comparatively elementary methods.
The theory isdeveloped without any serious attempt at mathematical rigour.
However, we also avoid the use of merely heuristic arguments which are par
ticularly commonintheliteratureon fracture. The orientationofthe book isap
plied mathematical, though with the ultimate aim of extracting physically in
teresting results. Certain required techniques and results, notably the statement
and solution of the Hilbert problem and the use of Hilbert transforms, are
discussedinseveralmathematicalappendices.Shorttables of integrals and other
relations are included.
In chapter 1,the properties of the viscoelasticfunctions are explored insome
detail. Alsothe boundaryvalue problems of interest are stated. In chapter2, the
Classical Correspondence Principle and its generalizations are discussed. Then,
general techniques, based on these, are developed for solving non-inertial
isothermal problems. A method for handling non-isothermal problems is also
discussed and in chapter 6 an illustrative example of its application is given.
Chapter 3 and 4 are devoted to plane isothermal contact and crack problems,
respectively. They utilize the general techniques of chapter 2. The viscoelastic
Hertz problem and its application to impact problems are discussed in chapter
5. Finally in chapter 7, inertial problems are considered.
Exercisesare scatteredthroughout the text, one of their main purposes being
to allowthestatement, without detailed derivation, of fairlystandard orstraight-
Preface IX
forward results. The equations occurring in these problems are numbered
separately from the ordinary equations. They are distinguished by the letter "p"
occurring after the number.
One ofthe authors (JMG) wishes to acknowledge gratefully two most plea
santperiodsspentat Simon FraserUniversityfor the academic year 1983/84 and
during the Spring of 1986.It wasduring these periods that most ofhis contribu
tion to this work was made. He would also like to acknowledge the encourage
ment of P. O'Keefe, Head of Roads Division, J. Sheedy, Head of Road Con
structionSectionand A.J. Curran, HeadofRoadSafety Sectionofthe National
Institute for Physical Planning and Construction Research over the years, and
for their appreciation of the importance in some contexts, ofa fundamental ap
proach to applied research problems. The other author (GACG) acknowledges
generous use of the facilities ofthe Dublin Institute for Advanced Studies par
ticularly during his stay there during the academic year 1986/87.
Both authors are very grateful to Marion Jacques and Cindy Lister for their
careful typing ofthe manuscript and to DorothyCorr and Eva Wills for obtain
ing bibliographical information.
This project was supported by funds provided by the Natural Sciences and
Engineering Research Council of Canada and the Faculty of Science of Simon
Fraser University.
Dublin-Burnaby
The Authors
December, 1987
Contents
1. Fundamental Relationships 1
1.1 Stress and Strain 1
1.2 One-dimensional Linear Viscoelasticity 4
1.2.1 Linear Hereditary Constitutive Laws 4
1.2.2 The Operator Algebra 6
1.2.3 Alternative Notation 8
1.2.4 Non-aging Materials 9
1.3 Energy Considerations in the One-dimensional Case .......... 12
1.4 Creep and Relaxation 14
1.4.1 The Boltzmann Superposition Principle 17
1.5 The Frequency Representation 18
1.5.1 Dispersion Relations 20
1.5.2 Energy Considerations under Sinusoidal Deformation .. 22
1.5.3 Creep 23
1.6 Special Forms of the Viscoelastic Functions 25
1.6.1 Standard Linear Solid 25
1.6.2 Maxwell and Voigt Models 26
1.6.3 Spectrum Models 28
1.6.4 Continuous Spectra 31
1.6.5 Power Law Viscoelastic Functions 32
1.7 Temperature Dependence of the Viscoelastic Functions 34
1.7.1 Variable Temperature History .... ... ..... ..... .. .... 35
1.8 Three-dimensional Constitutive and Dynamical Equations 37
1.8.1 Isothermal Theory 37
1.8.2 The Non-inertial Approximation 39
1.8.3 Frequency Representation. .......................... 40
1.8.4 Proportionality Assumption 40
1.8.5 Non-isothermal Equations 41
1.8.6 Uniqueness and Other Theorems 42
1.9 Isotropic Media 43
1.9.1 Frequency Representation. .......................... 45
1.9.2 Proportionality Assumption 45
1.9.3 Non-isothermal Relations 46
1.9.4 Compatibility Equations 47
1.10 Causality ................................................ 48
1.11 Summary 50
XII Contents
2. General Theorems and Methods of Solution
of Boundary Value Problems 54
2.1 The Classical Correspondence Principle 54
2.1.1 Separation of Space and Time Variables 57
2.2 Time-dependent Boundary Regions 57
2.3 Elastic Solutions in Terms of Green's Functions 59
2.4 Decomposition of Hereditary Integrals 63
2.5 The Integral Equation 67
2.6 Expanding and Contracting Boundary Regions 68
2.6.1 The Extended Correspondence Principle 68
2.6.2 The Generalized Partial Correspondence Principle 69
2.6.3 Repetitive Expansion and Contraction 70
2.7 Viscoelastic Papkovich-Neuber Solution 73
2.8 Plane Strain in Linear Viscoelasticity 74
2.9 Contact between Viscoelastic Media 78
2.9.1 Inclusion of Inertial Affects 81
2.10 Receding Contact in Plane Viscoelasticity 81
2.11 Energy Loss in Moving Contact Problems 84
2.12 Solution of Problems Involving Aging Materials or
Non-isothermal Conditions 87
2.12.1 Aging Materials 87
2.12.2 Non-isothermal Problems 88
2.13 Summary 89
3. Plane Non-inertial Contact Problems 91
3.1 Kolosov-Muskhelishvili Equations Adapted to the Half-Plane . 92
3.2 The First and Second Boundary Value Problems. ............ 95
3.3 The General Mixed Boundary Value Problem 99
3.3.1 Single Contact Interval 102
3.3.2 Frictionless Contact 103
3.4 The General Integral Equation 104
3.5 Moving Load Problems 105
3.5.1 Steady-State Limit 106
3.6 Solution for a Single Load 109
3.6.1 Discrete Spectrum Model 112
3.7 Small Viscoelasticity Approximation 116
3.8 Hysteretic Friction 120
3.8.1 Small Viscoelasticity 122
3.9 Increasing and Decreasing Contact Area 123
3.10 The Plane Normal Contact Problem 126
3.11 The Steady-State Limit of the Normal Problem 130
3.11.1 The Standard Linear Model 131
3.12 Summary 137
4. Plane Non-inertial Crack Problems 140
4.1 Problem Formulation ..................................... 140
4.2 Fully Open Cracks that are Stationary or Growing 143
