Viscoelastic Modeling for Structural Analysis 9781119618331, 1119618339, 9781786304452

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Viscoelastic Modeling for Structural Analysis

Series Editor Yves Rémond

Viscoelastic Modeling for Structural Analysis

Jean Salençon

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2019 The rights of Jean Salençon to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019931575 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-445-2

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

List of Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Chapter 1. One-dimensional Viscoelastic Modeling . . . . . . .

1

1.1. Experimental observations . . . . . . . . . . . . . . . . . 1.2. Fundamental uniaxial tests . . . . . . . . . . . . . . . . . 1.2.1. Creep test, creep function . . . . . . . . . . . . . . . . 1.2.2. Stress relaxation test, relaxation function . . . . . . 1.2.3. First comments . . . . . . . . . . . . . . . . . . . . . . 1.2.4. Recovery . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5. Stress fading . . . . . . . . . . . . . . . . . . . . . . . 1.3. Functional description . . . . . . . . . . . . . . . . . . . . 1.4. Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1. Aging phenomenon . . . . . . . . . . . . . . . . . . . 1.4.2. Non-aging materials . . . . . . . . . . . . . . . . . . . 1.5. Linear behavior . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1. Superposition principle: Boltzmannian materials . . 1.5.2. Linear elasticity . . . . . . . . . . . . . . . . . . . . . 1.6. Linear viscoelastic material . . . . . . . . . . . . . . . . . 1.6.1. Instantaneous behavior . . . . . . . . . . . . . . . . . 1.6.2. Creep and relaxation functions . . . . . . . . . . . . 1.6.3. Recovery and stress fading . . . . . . . . . . . . . . . 1.6.4. Instantaneous behavior . . . . . . . . . . . . . . . . . 1.6.5. Validation of the linearity hypothesis: an example . 1.7. Linear viscoelastic constitutive equation . . . . . . . . . 1.7.1. Arbitrary stress history . . . . . . . . . . . . . . . . .

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1 2 2 5 7 8 9 10 11 11 12 14 14 15 15 16 16 18 19 19 20 20

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Viscoelastic Modeling for Structural Analysis

1.7.2. Arbitrary deformation history . . . . . . . . . . . . . . 1.7.3. Linear elastic material . . . . . . . . . . . . . . . . . . 1.7.4. Boltzmann’s formulas, integral operator . . . . . . . 1.7.5. Comments . . . . . . . . . . . . . . . . . . . . . . . . . 1.8. Non-aging linear viscoelastic constitutive equation . . . 1.8.1. Creep and relaxation functions . . . . . . . . . . . . . 1.8.2. Boltzmann’s formulas . . . . . . . . . . . . . . . . . . 1.8.3. Recovery and stress fading . . . . . . . . . . . . . . . 1.8.4. Operational calculus . . . . . . . . . . . . . . . . . . . 1.9. One-dimensional linear viscoelastic behavior . . . . . . . 1.9.1. Uniaxial viewpoint and one-dimensional modeling . 1.9.2. Structural elements . . . . . . . . . . . . . . . . . . . . 1.10. Harmonic loading process . . . . . . . . . . . . . . . . . 1.10.1. The loading process . . . . . . . . . . . . . . . . . . . 1.10.2. Asymptotic harmonic regime . . . . . . . . . . . . . 1.10.3. Complex modulus . . . . . . . . . . . . . . . . . . . . 1.10.4. Loss angle, specific loss . . . . . . . . . . . . . . . .

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23 24 24 26 27 27 28 30 30 33 33 36 39 39 40 41 43

Chapter 2. Rheological Models . . . . . . . . . . . . . . . . . . . . .

45

2.1. Rheological models . . . . . . . . . . . . . 2.2. Basic elements . . . . . . . . . . . . . . . . 2.2.1. Linear elastic element . . . . . . . . . 2.2.2. Linear viscous element . . . . . . . . . 2.3. Classical models . . . . . . . . . . . . . . . 2.3.1. Maxwell model . . . . . . . . . . . . . 2.3.2. Kelvin model . . . . . . . . . . . . . . 2.3.3. The standard linear solid . . . . . . . . 2.4. Generalized Maxwell and Kelvin models 2.4.1. Generalized Maxwell model . . . . . 2.4.2. Generalized Kelvin model . . . . . . . 2.4.3. Equivalence . . . . . . . . . . . . . . . 2.4.4. Continuous spectra . . . . . . . . . . .

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45 45 45 46 47 47 49 51 56 57 58 59 60

Chapter 3. Typical Case Studies . . . . . . . . . . . . . . . . . . . .

63

3.1. Presentation and general features . . . . . . . . 3.2. “Creep-type” problems . . . . . . . . . . . . . . 3.2.1. Homogeneous cantilever beam subjected to a uniformly distributed load . . . . . . . . . . . 3.2.2. Homogeneous cantilever beam subjected to a concentrated load . . . . . . . . . .

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63 64

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64

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68

Contents

3.2.3. Homogeneous statically indeterminate beam . 3.2.4. A statically indeterminate system . . . . . . . . 3.3. Prestressing of viscoelastic systems or structures . 3.3.1. Prestressed cantilever beam . . . . . . . . . . . 3.3.2. Prestressed hyperstatic system . . . . . . . . . . 3.3.3. Prestressed hyperstatic arc . . . . . . . . . . . . 3.3.4. The example of a rheological model . . . . . . 3.3.5. Practical applications . . . . . . . . . . . . . . . 3.4. A complex loading process . . . . . . . . . . . . . . 3.4.1. A practical problem . . . . . . . . . . . . . . . . 3.4.2. Mathematical treatment . . . . . . . . . . . . . . 3.4.3. Comments . . . . . . . . . . . . . . . . . . . . . . 3.5. Heterogeneous viscoelastic structures . . . . . . . .

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70 72 75 75 79 80 84 88 93 93 95 97 98

Chapter 4. Three-dimensional Linear Viscoelastic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

4.1. Multidimensional approach . . . . . . . . . . . . . . . . . . . 4.2. Fundamental experiments . . . . . . . . . . . . . . . . . . . . 4.2.1. The three-dimensional continuum framework . . . . . 4.2.2. General definition of the creep and relaxation tests . . 4.2.3. The linearity hypothesis. . . . . . . . . . . . . . . . . . . 4.2.4. Tensorial creep and relaxation functions . . . . . . . . . 4.2.5. Instantaneous elasticity . . . . . . . . . . . . . . . . . . . 4.3. Boltzmann’s formulas . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Integral operator . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Important identities . . . . . . . . . . . . . . . . . . . . . 4.4. Isotropic linear viscoelastic material. . . . . . . . . . . . . . 4.4.1. Material symmetries, principle of material symmetries 4.4.2. Isotropic linear viscoelastic material: creep test . . . . 4.4.3. Isotropic linear viscoelastic material: relaxation test . . 4.4.4. Boltzmann’s formulas . . . . . . . . . . . . . . . . . . . . 4.4.5. Uniaxial tension relaxation test . . . . . . . . . . . . . . 4.4.6. Constant Poisson’s ratio . . . . . . . . . . . . . . . . . . 4.5. Non-aging linear viscoelastic material . . . . . . . . . . . . 4.5.1. Boltzmann’s formulas . . . . . . . . . . . . . . . . . . . . 4.5.2. Operational calculus . . . . . . . . . . . . . . . . . . . . . 4.5.3. Isotropic material . . . . . . . . . . . . . . . . . . . . . .

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103 104 104 105 105 106 107 108 108 109 110 110 111 113 114 115 118 120 120 121 122

Chapter 5. Quasi-static Linear Viscoelastic Processes . . . . .

125

5.1. Quasi-static linear viscoelastic processes . . . . . . . . . . . . . .

125

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vii

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viii

Viscoelastic Modeling for Structural Analysis

5.1.1. Isothermal quasi-static processes . . . . . . . . . . . . . 5.1.2. Isothermal quasi-static linear viscoelastic processes . 5.1.3. Superposition principle . . . . . . . . . . . . . . . . . . 5.1.4. Loading parameters, kinematic parameters . . . . . . . 5.2. Solution to the linear viscoelastic quasi-static evolution problem . . . . . . . . . . . . . . . . . . . 5.2.1. Statically admissible stress histories, kinematically admissible displacement histories . . . . . . . . . . . . . . . . 5.2.2. Solution methods . . . . . . . . . . . . . . . . . . . . . . 5.3. Homogeneous isotropic material with constant Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Creep-type problems, creep-type evolutions . . . . . . 5.3.2. Relaxation-type problems, relaxation-type evolutions 5.3.3. Mixed data problems . . . . . . . . . . . . . . . . . . . . 5.3.4. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Non-aging linear viscoelastic material . . . . . . . . . . . . 5.4.1. Correspondence principle . . . . . . . . . . . . . . . . . 5.4.2. Comments . . . . . . . . . . . . . . . . . . . . . . . . . .

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125 128 129 129

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131

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131 132

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133 134 135 137 139 140 140 142

Chapter 6. Some Practical Problems . . . . . . . . . . . . . . . . .

143

6.1. Presentation . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Uniaxial tension–compression of a cylindrical rod . 6.2.1. Statement of the problem . . . . . . . . . . . . . 6.2.2. Solution . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Bending of a cylindrical rod . . . . . . . . . . . . . . 6.3.1. Statement of the problem . . . . . . . . . . . . . 6.3.2. Solution . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Twisting of a cylindrical rod . . . . . . . . . . . . . . 6.4.1. Preliminary comments . . . . . . . . . . . . . . . 6.4.2. Statement of the problem . . . . . . . . . . . . . 6.4.3. Solution . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4. Comment . . . . . . . . . . . . . . . . . . . . . . . 6.5. Convergence of a spherical cavity . . . . . . . . . . 6.5.1. Statement of the problem . . . . . . . . . . . . . 6.5.2. Solution . . . . . . . . . . . . . . . . . . . . . . . .

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143 143 143 145 148 148 149 152 152 153 154 157 158 158 160

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

Preface

It is commonly observed that, besides their instantaneous response (either reversible or irreversible), materials subjected to a mechanical action also demonstrate a delayed behavior that results, for example, in deformation increasing under the action of a constant load. The phenomenon is well known for fluids, with the concept of viscosity (either Newtonian or non-Newtonian), and is also observed for solid materials such as rocks, metals, polymers, etc. Acknowledging the fact that the corresponding timescales are of completely different orders of magnitude depending on the concerned material, this phenomenon may be considered as illustrating the popular aphorism by Heraclitus of Ephesus: “Panta rhei” – “Everything flows”1. The theory of viscoelasticity has been built up as a mechanical framework for modeling important aspects of the delayed behavior of a wide range of materials. The presentation proposed here is guided by standard practical applications in various domains of civil or mechanical engineering. It is therefore essentially devoted to linear viscoelastic behavior within the small perturbation framework and does not cover the case of large viscoelastic deformations. The first part of the book, namely Chapters 1–3, is dedicated to the one-dimensional viscoelastic behavior modeling with the meaning 1 Note that the term “Rheology” introduced in 1920 by Eugene C. Bingham to name the study of the flow of matter is said to have been inspired by this aphorism.

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Viscoelastic Modeling for Structural Analysis

that, for the considered constituent material element or the system under concern itself, both the action it is subjected to and the consequent response can be modeled as one-dimensional. Within this simple mechanical framework, Chapter 1 introduces the fundamental concepts of viscoelastic behavior from the phenomenological viewpoint of the basic creep and relaxation tests. The viscoelastic constitutive equation can be written as a functional relationship between the action and response histories. It is linear in the case of linear viscoelastic behavior, which is often defined through Boltzmann’s superposition principle, and takes the form of Boltzmann’s integral formulas whose kernels are derived from the creep and relaxation functions. For a non-aging material, these formulas can be identified as Riemann’s convolution products, which call for the use of Laplace or Laplace–Carson transforms, with operational calculus substituting computations in the convolution algebra with ordinary algebraic calculations. It is worth noting that the linearity and non-aging assumptions are here introduced separately and independently from each other, as it has been observed that, in many practical cases (e.g. civil engineering), linearity may be taken as a convenient simplifying assumption valid within a given range of applications, while material aging shall be taken into account. Rheological models are commonly used as a thought support when trying to write simple one-dimensional constitutive laws matching experimental results. The most classical ones are presented in Chapter 2 in the case of non-aging linear viscoelastic behavior. The typical viscoelastic response to a harmonic loading process is illustrated by the example of the standard linear solid. Chapter 3 is devoted to the analysis and solution of some illustrative quasi-static evolution problems. It is underscored that pre-eminence and priority must be given to an in-depth physical (and practical) understanding of the problem at hand before entering the mathematical treatment step. Stating the loading process and history properly is essential to reach a correct description and anticipation of the phenomena that will actually take place. It is shown that, for many practical problems, using Boltzmann’s integral operator makes it possible to straightforwardly derive the solution to the

Preface

xi

problem at hand from its counterpart within the linearized elastic framework. Particular attention is given to the potentially damaging consequences of creep and relaxation phenomena on prestressed structures if they are not correctly anticipated. This concludes the first part of the book, which may be sufficient for a first analysis of many practical cases, provided that the action and response variables are adequately defined. The second part of the book, namely Chapters 4–6, discusses the three-dimensional issue within the framework of classical continuum mechanics and the small perturbation hypothesis. In Chapter 4, relying on the physical concepts introduced in the one-dimensional case, fundamental creep and relaxation experiments are again introduced with the necessity of describing and defining them more precisely. The linear viscoelastic constitutive equation is then written in terms of tensorial integral operators, whose kernels are the tensorial creep and relaxation functions determined through the basic experiments. These functions must comply with material symmetries which specify them, reducing the number of their scalar components. In the isotropic case, it can be observed that two scalar creep functions, or conversely two scalar relaxation functions, are sufficient to completely define the constitutive law in the same way as for linear elasticity. The relationships between these functions bear some similarity with their elastic counterpart, but for the fact that they take the form of integral equations through Boltzmann’s operator. Quasi-static viscoelastic processes are stated in Chapter 5 within the three-dimensional context and the small perturbation hypothesis. They are defined in the same way as elastic equilibrium problems through field and boundary data depending on the time variable, which must be compatible with the quasi-static equilibrium assumption. It often happens that these data depend on a finite number of scalar loading or kinematic parameters, which may be used to express the global viscoelastic behavior of the studied system. As in the elastic case, no purely deductive method can be proposed for the solution of such problems in a systematic way. Based on intuition and experimental observations, or by analogy with similar linear elastic

xii

Viscoelastic Modeling for Structural Analysis

equilibrium problems, solutions are obtained following the methods that can be qualified as displacement history or stress history methods. Here again, and even more than in the one-dimensional case, an in-depth physical understanding of the problem at hand is necessary for a proper modeling of the process before any mathematical treatment. In connection with the typical case studies presented in Chapter 3, a few classical three-dimensional quasi-static viscoelastic processes are examined, which concern popular practical problems in the case of a homogeneous isotropic material. Without any non-aging assumption, explicit solutions are obtained, expressed in a simple way using well-chosen creep or relaxation functions of the material. As a result, the global viscoelastic behavior of the system under concern is expressed by creep and relaxation functions straightforwardly derived from the material ones. It will be observed that, but for a mere allusion in Chapter 2, thermodynamics is not mentioned anywhere in the book. We plead guilty for what may be considered an obvious shortcoming, especially as regards three-dimensional linear viscoelastic modeling, with the argument that this short book aims to make a first reader familiar enough with viscoelastic phenomena as to “feel” them. With the unfortunate experience that thermodynamics, if introduced too early, may act as a deterrent, we believe that the interested reader will refer to the many comprehensive textbooks that are listed in the bibliography. Furthermore, despite the book being only concerned with quasi-static processes, we hope that it may spur the reader towards the analysis of dynamic processes and wave propagation. Acknowledgment As a follow-up to lecture notes for a course at the City University of Hong Kong, this book was completed while the author was in residence at the Hong Kong Institute for Advanced Study (HKIAS), whose support is hereby gratefully acknowledged. Jean SALENÇON February 2019

List of Notations

List of main notations as they appear in the book Notation

Meaning

First cited

Υ (t )

Heaviside step function

[1.1]

σ0

Stress jump in the creep test

[1.3]

ε

Longitudinal stretch

[1.4]

J (t0 , t ;σ 0 )

Creep (or retardation) function

[1.4]

[[ J (t0 , t0 ;σ 0 )]]

Creep function jump

[1.6]

ε0

Longitudinal stretch in the stress relaxation test

[1.8]

R(t0 , t ; ε 0 )

Relaxation function

[1.9]

Functional of a stress history

[1.15]

R t [ε (τ )]

Functional of a deformation history

[1.17]

F ,R

Inverse functional relationships

[1.18]

t

Ft [σ (τ )] −∞ t

−∞

xiv

Viscoelastic Modeling for Structural Analysis

σ λ (t )

Translated stress history

[1.19]

ε λ (t )

Translated deformation history

[1.20]

E

Elastic modulus

[1.32]

J t0 (t ) = J (t0 , t )

Linear viscoelastic creep function [1.35]

Rt0 (t ) = R(t0 , t )

Linear viscoelastic relaxation function

[1.37]

δ

Dirac distribution

[1.48]

{ϕ }

Distribution defined by the function ϕ

[1.48]

< ϕ ,ψ >

Scalar product of functions ϕ and ψ

[1.60]

(×)

One-dimensional integral operator

[1.62]

f (τ )

Non-aging material linear viscoelastic creep function

[1.70]

r (τ )

Non-aging material linear viscoelastic relaxation function

[1.71]

*

Notation for Riemann’s convolution product

[1.77]

L ϕ ( p)

Laplace transform (LT) of function ϕ (t )

[1.87]

ϕ * ( p)

Laplace–Carson transform (LCT) [1.91] of function ϕ (t )

σ

Cauchy stress tensor

⊗

Notation for the tensorial product [1.93]

ex

Unit vector

[1.93]

[1.93]

List of Notations

Q

One-dimensional generic force variable

[1.94]

q

One-dimensional generic deformation variable

[1.94]

J (τ , t )

One-dimensional generic linear viscoelastic creep function

[1.94]

R (τ , t )

One-dimensional generic linear viscoelastic relaxation function

[1.95]

f

One-dimensional generic non-aging material linear viscoelastic creep function

[1.96]

r

One-dimensional generic non-aging material linear viscoelastic relaxation function

[1.97]

N

Normal force applied to a straight [1.100] rod

ε

Longitudinal deformation (stretch) of a straight rod

[1.100]

S

Cross-sectional area of a straight rod

[1.101]

M

Bending moment applied to a straight rod about a principal axis [1.103] of its cross-section

χ

Curvature of the director curve of [1.103] a straight rod

I

Principal moment of inertia of the [1.104] cross-section of a straight rod

C

Twisting moment applied to a straight rod

[1.106]

xv

xvi

Viscoelastic Modeling for Structural Analysis

α

Differential rotation about the axis of a straight rod

[1.106]

γ (τ , t )

Creep function in simple shear

[1.107]

J

Torsional inertia of the cross-section of a straight rod

[1.107]

μ (τ , t )

Relaxation function in simple shear

[1.108]

Re [

Real part

[1.110]

r * (i ω )

Complex modulus

[1.113]

M (ω )

Modulus of r * (i ω )

[1.118]

δ (ω )

Phase angle, lag angle, loss angle [1.118]

Re [r *(iω )]

Storage modulus

[1.120]

Im [

]

Imaginary part

[1.121]

Im[r *(iω)]

Loss modulus

[1.121]

f

Specific loss

[1.131]

η

Viscosity coefficient

[2.3]

τr

Relaxation characteristic time

[2.9]

τf

Creep characteristic time

[2.14]

v ( x, t )

Vertical deflection

[3.1]

p (t )

Uniformly distributed vertical force

[3.7]

F (t )

Concentrated vertical force

[3.14]

V (t )

Support reaction

[3.20]

]

List of Notations

E1

Elastic modulus of the cable element

[3.25]

u (t )

Horizontal displacement

[3.50]

f

Rise of a parabolic-shaped arc

[3.46]

Q (t )

Horizontal reactive force

[3.48]

Tr

Characteristic relaxation time of the prestressing force

[3.79]

a (t )

Inner radius of a cavity

[3.82]

ε xx

Stretch along the x -axis

[4.2]

ε

Linearized strain tensor

[4.4]

J (t0 , t )

Creep tensor

[4.10]

R(t0 , t )

Relaxation tensor

[4.13]

δi j

Kronecker symbol

[4.17]

1

1 = δ i hδ j k ei ⊗ eh ⊗ e j ⊗ ek

[4.18]

(× :)

Three-dimensional integral operator

[4.21]

1

1 = δ i j ei ⊗ e j

[4.30]

J (t0 , t )

Creep function in the uniaxial tension test

[4.31]

n (t0 , t )

Poisson’s ratio in the uniaxial tension creep test

[4.33]

μ (t0 , t )

Relaxation function in the simple [4.38] shear test

λ (t0 , t ) + 2μ (t0 , t )

Relaxation function in the simple [4.40] extension test

xvii

xviii

Viscoelastic Modeling for Structural Analysis

E (τ , t )

Uniaxial tension relaxation function

[4.47]

ν (τ , t )

Poisson’s ratio in the uniaxial tension relaxation test

[4.49]

f (τ )

Non-aging material creep tensor

[4.65]

r (τ )

Non-aging material relaxation tensor

[4.66]

F ( x, t )

Body forces

[5.1]

d

Upper index for volume and boundary data

[5.1]

Ω , ∂Ω

Volume, boundary of the system

S

[5.2]

T ( x, t )

Stress vector

[5.2]

ξ ( x, t )

Displacement vector

[5.2]

ξi ( x, t )

Component of the displacement vector

[5.2]

Ti ( x, t )

Component of the stress vector

[5.2]

Sξi (t )

Portion of the boundary where ξi ( x, t ) is prescribed

[5.2]

STi (t )

Portion of the boundary where Ti ( x, t ) is prescribed

[5.2]

[ Fe (t )]

Wrench of external forces

[5.3]

grad

Gradient of a vector field

[5.6]

div

Divergence of a tensor field

[5.7]

ρ ( x)

Mass per unit volume

[5.7]

List of Notations

Q (t ) ∈ n

Loading vector

[5.8]

q (t ) ∈  n

Kinematic vector

[5.10]

J (t0 , t )

Creep tensor of a system within [5.10] the loading parameter framework

(× .)

Integral operator within the loading parameter framework

[5.11]

R (t0 , t )

Relaxation tensor of a system within the loading parameter framework

[5.13]

S ( F , STi , Ti d )

Set of statically admissible stress [5.15] histories

C ( Sξi , ξi d )

Set of kinematically admissible displacement histories

d

J (t0 , t0 ) Λ = symmetric matrix of

Λ

instantaneous elastic compliances

E(t0 , t0 ) A = symmetric matrix of

A

instantaneous elastic moduli

Ξ k (ξi

d(0)

[5.16] [5.23] [5.28]

Displacement field solution to the problem with force data equal to [5.31] zero

;ν )

E (t0 ) Σ k (ξ i d(0) ;ν )

Stress field solution to the problem with force data equal to zero

[5.31]

Displacement field solution to the 1 Ξ s (Ti d(0) , F d(0) ;ν ) problem with kinematic data [5.32] E (t0 ) equal to zero

Σ s (Ti

d(0)

,F

d(0)

;ν )

Stress field solution to the problem with kinematic data equal to zero

[5.32]

xix

xx

Viscoelastic Modeling for Structural Analysis

R (τ , t )

Relaxation function of a rod subjected to tension–compression [6.8] within the kinematic parameter framework

J (τ , t )

Creep function of a rod subjected to tension–compression within [6.9] the loading parameter framework

R (τ , t )

Relaxation function of a rod subjected to bending within the kinematic parameter framework

J (τ , t )

Creep function of a rod subjected to bending within the loading [6.22] parameter framework

R (τ , t )

Relaxation function of a rod subjected to twisting within the kinematic parameter framework

J (τ , t )

Creep function of a rod subjected to twisting within the loading [6.39] parameter framework

J(τ , t )

Creep function of a converging spherical cavity

[6.51]

R (τ , t )

Relaxation function of a converging spherical cavity

[6.52]

[6.21]

[6.38]

1 One-dimensional Viscoelastic Modeling

1.1. Experimental observations Everyday practice, either domestic or industrial, provides examples of the delayed response of common materials to mechanical actions: the slow sagging of bookcase shelves subjected to the weight of knowledge, the progressive return1 of a polymer sample to its initial configuration after being subjected to a temporary loading process and creep of concrete and asphalt are just a few examples of evidence of this kind of behavior. That being said, it is necessary to go into some detail regarding the timescales that are referred to when describing the above-mentioned phenomena: the qualifiers “slow” and “progressive” are used here with the meaning that the characteristic times of these phenomena are long when compared with the propagation time associated with the “instantaneous” response of the same material in the studied system. Some consequences of these phenomena are just perceived as inconveniences either from an aesthetic viewpoint or regarding comfortable serviceability of a structure (e.g. an independent span bridge or similar isostatic structures). However, more damaging

1 This may be complete or incomplete.

Viscoelastic Modeling for Structural Analysis, First Edition. Jean Salençon. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

2

Viscoelastic Modeling for Structural Analysis

consequences may occur, such as delayed deformations that can be incompatible with the serviceability of a structure (a crane runway, for instance), and can even prove catastrophic when they generate, during the course of time, the redistribution of internal forces in a prestressed hyperstatic structure. On the other hand, bitumen, putties and polymers are used as dampers in industrial applications because of the dissipative effects caused by delayed deformations. Clearly, the effects of the delayed behavior of a material will be anticipated and mitigated inasmuch as a relevant mechanical model for this behavior is available. The aim of this chapter is to build such models within the framework of a one-dimensional continuum where the geometrical state of a material element is characterized by one scalar parameter. We will only consider models suitable for applications to the current practice of structural engineering under the small perturbation hypothesis (SPH) and within the isothermal framework. 1.2. Fundamental uniaxial tests 1.2.1. Creep test, creep function The isothermal uniaxial creep (or retardation) test is a simple experiment that makes it possible to demonstrate and quantify the delayed behavior of a material. As an example of such an experiment, we consider the case of a tension test performed on a specimen made from a homogeneous material whose geometrical characteristics are such that a homogeneous simple tension stress field can be assumed to be generated within the test piece, parallel to its axis, when a uniaxial traction (or compression) load is exerted. Temperature is considered to be constant in time and throughout the test piece. Within the small perturbation framework, the loading history is described as follows:

One-dimensional Viscoelastic Modeling

Figure 1.1. Creep test at time

3

t0

– initially, no load is applied to the test piece and in that equilibrium state, taken as a geometrical reference, the stress field is homogeneously zero; – at a given instant of time t0 , a load is applied “instantaneously” that results in a stress field σ 0 , which is thereafter held constant (Figure 1.1); – then, for t > t0 , the longitudinal stretch history, supposed to be homogeneous within the specimen, is recorded as a function of time ε (t ) with the following properties (Figure 1.1). By definition, ε (t ) is zero until t0 . At time t0 , a deformation ε (t0 ) is “instantaneously” produced, which reveals the instantaneous behavior of the material. Then, for t > t0 , ε (t ) increases continuously with t , the concavity of the curve being directed downwards, as typically shown in Figure 1.1. A mathematical description of this experiment can be written straightforwardly. Let Υ (t ) denote the Heaviside step function at time t = 0, which is defined as

Υ (t ) = 0 t < 0  Υ (t ) = 1 t ≥ 0+ ,

[1.1]

4

Viscoelastic Modeling for Structural Analysis

and let Υτ (t ) be the step function at time τ , which is defined as

Υ τ (t ) = Υ (t − τ ) ,

[1.2]

then, the history of σ , as shown in Figure 1.1, can be written as

σ (t ) = σ 0 Υ t (t ) .

[1.3]

0

The response ε (t ) can be conveniently written in the form

ε (t ) = σ 0 J (t0 , t ;σ 0 )

[1.4]

 J (t0 , t ;σ 0 ) = 0 if t < t0   J (t , t ;σ 0 ) > 0 for t ≥ t 0  0   ∂J (t , t ;σ 0 ) ≥ 0 for t ≥ t 0  ∂t 0   ∂2 J  2 (t0 , t ;σ 0 ) ≤ 0 for t ≥ t0 .  ∂t

[1.5]

with

Corresponding to the jump imposed on σ (t ) at time t = t0 ,

J (t0 , t ;σ 0 ) is discontinuous at time t = t0 with a jump denoted by [[ J (t0 , t0 ;σ 0 )]], which corresponds to the instantaneous response of the material at time t0 in that experiment. As a general rule, the notation J (t0 , t0 ;σ 0 ) will be adopted to denote the right-hand side +

value of function J (t0 , t;σ 0 ) at time t0 , i.e. for t = t0 , so that +

J (t0 , t0 ;σ 0 ) = J (t0 , t0 ;σ 0 ) = [[ J (t0 , t0 ;σ 0 )]] ≥ 0.

[1.6]

One-dimensional Viscoelastic Modeling

5

If the constituent material is not subject to any physical or chemical transformation that would affect its mechanical properties instantaneously, the function J (t0 , t ;σ 0 ) is continuous with respect to t0 as a variable and ∂J (t0 , t ;σ 0 ) ≤ 0 for t ≥ t0 . ∂t0

[1.7]

The experiment described above is the creep (or retardation) test performed at time t0 with σ 0 as a stress jump. The function J (t0 , t;σ 0 ) is the corresponding creep (or retardation) function. 1.2.2. Stress relaxation test, relaxation function

The isothermal uniaxial stress relaxation test (or relaxation test) is the counterpart of the preceding experiment where the longitudinal deformation ε (t ) of the test piece is prescribed while the stress σ (t ) is the observed variable. Starting from the same initial equilibrium state as in section 1.2.1 taken as a geometrical reference, the deformation history is imposed as follows (Figure 1.2):

ε (t ) = ε 0 Υ t (t ). 0

[1.8]

The corresponding stress history σ (t ) , shown in Figure 1.2, can be written in the form

σ (t ) = ε 0 R(t0 , t ; ε 0 ) where

[1.9]

6

Viscoelastic Modeling for Structural Analysis

   R (t0 , t ; ε 0 ) = 0 for t < t0   [[ R (t , t ; ε 0 )]] > 0 0 0   0  R (t0 , t ; ε ) > 0 for t ≥ t0   ∂R 0  ∂t (t0 , t ; ε ) ≤ 0 for t ≥ t0   ∂2 R  (t , t ; ε 0 ) ≥ 0 for t ≥ t0 .  ∂t 2 0

[1.10]

The jump +

R(t0 , t0 ; ε 0 ) = R(t0 , t0 ; ε 0 ) = [[ R(t0 , t0 ; ε 0 )]]

[1.11]

is the instantaneous response of the material at time t = t0 . Then, it is observed that ε (t ) decreases continuously with t and the concavity of the curve is oriented upwards, as typically shown in Figure 1.2.

Figure 1.2. Stress relaxation test at time

t0

The function R(t0 , t ; ε 0 ) is called the relaxation function for the stress relaxation test performed at time t0 with magnitude ε 0 for the deformation jump. It is continuous with respect to t0 as a variable in the absence of physical or chemical transformation that would affect the mechanical properties of the material instantaneously and

One-dimensional Viscoelastic Modeling

∂R (t0 , t ; ε 0 ) ≥ 0 for t > t0 . ∂t0

7

[1.12]

1.2.3. First comments 1.2.3.1. Creep and relaxation

Despite their apparent similarity, the creep and relaxation isothermal tests do differ from one another with regard to their effective feasibility. While the creep test can be performed whatever the constituent material of the specimen, it turns out that the stress relaxation test can only be realized if it is indeed possible to impose an instantaneous deformation jump. It is known from actual practice that the characteristic timescales of these two phenomena are often significantly different, with stress relaxation evolving faster than creep (see Chapter 2, section 2.3.3.1). 1.2.3.2. Time and chronology

Obviously, the descriptions given in the preceding sections do not depend on the origin chosen for the time variable. From a practical viewpoint, as the goal of such experiments is to identify and characterize the instantaneous and delayed behaviors of a given material, it is advisable to attach the chronology to an origin that bears a significant meaning for this very material, which may be, for instance, the instant of time it was produced or some conventional origin related to it. Obviously, when analyzing a system where different materials are involved, a common chronology will be adopted. The discussion initiated in section 1.1 regarding the timescales that are referred to when qualifying delayed material behaviors must be revisited here in relation to the concept of instantaneous stress or deformation jumps being applied to the test piece at time t0 . The actual loading time, which is the time required to reach the plateaus σ 0 or ε 0 shown in Figures 1.1 or 1.2, must be sufficiently short to be regarded as instantaneous and, at the same time, sufficiently long to allow the evolution to be considered as quasi-static.

8

Viscoelastic Modeling for Structural Analysis

1.2.4. Recovery 1.2.4.1. The recovery experiment

Figure 1.3. Recovery experiment

The isothermal recovery experiment2 is nothing but the loading–unloading test associated with the creep test. It consists of imposing a stress history in the form of a “crenel”. The plateau shown in Figure 1.1 is reached at time t1 > t0 , and the specimen is instantaneously unloaded down to σ (t1+ ) = 0, then σ (t ) = 0 for t > t1 , an history that can be written as:

σ (t ) = σ 0 [Υ t (t ) −Υ t (t )] . 0

1

[1.13]

As shown in Figure 1.3, the deformation history is obviously identical to [1.4] as long as t < t1 ; then, at time t1 , an instantaneous decreasing response is observed followed by a decreasing evolution for t > t1 . The corresponding phenomenon is called “recovery”. It is said to be total when ε (t ) falls down to zero after a limited time span or tends to zero as t → ∞. 1.2.4.2. A crucial experiment Obviously, the recovery experiment described above recalls the loading–unloading test that is often referred to as a crucial experiment when defining the elastoplastic behavior. In fact, if the recovery experiment is instantaneous (Figure 1.4), with the meaning that t1 is infinitely close to t0 and does not allow 2 Also called the “creep-recovery” experiment (Mase 1970).

One-dimensional Viscoelastic Modeling

9

delayed deformation to develop, it reveals the elastic component in the behavior of the material through the deformation that is instantaneously recovered while the residual deformation (if any), which will not be recovered, is the plastic deformation.

Figure 1.4. Instantaneous “recovery” experiment

When no delayed deformation is observed in the creep test, and recovery is instantaneous and total whatever t1 > t0 , the material is purely elastic. 1.2.5. Stress fading The isothermal stress fading experiment is associated with the stress relaxation test in the same way as the recovery experiment is derived from the creep test. As shown in Figure 1.5, a deformation crenel is imposed on the specimen between the instants of time t0 and t1 :

ε (t ) = ε 0 [Υ t (t ) −Υ t (t )]. 0

1

[1.14]

As long as t < t1 , the stress history is identical to [1.9]. At time t = t1 , an instantaneous stress jump occurs which may even result in a negative value for σ (t1 ) as shown in Figure 1.5. Then, for t > t1 ,

σ (t ) decreases monotonously. This phenomenon is often called stress fading and said to be total when σ (t ) falls down to zero after a limited time span or tends to zero as t → ∞.

10

Viscoelastic Modeling for Structural Analysis

Figure 1.5. Stress fading experiment

1.3. Functional description The fundamental experiments that were described above make it possible to understand the delayed behavior of a material, but are not sufficient to fully determine it. In fact, each isothermal stress history brings out a new experiment. At any instant of time t , the deformation ε (t ) depends on the whole stress history up to that time and can be written as t

ε (t ) = Ft [σ (τ )],

[1.15]

−∞

t

where [σ (τ )] is a symbolic representation of the stress history up to −∞

time t . The fact that t stands for the “upper bound” that limits the stress history in this expression is just a consequence of the causality principle: the stress history subsequent to t does not affect ε (t ), which is the deformation at time t . Without going into mathematical details, we can say that equation [1.15], when applied to the set of all stress histories, generates a set of deformation histories. Given a history σ , we denote by F the functional relationship that associates σ with the corresponding deformation history ε through: I

t

F σ ⎯⎯ → ε ⇔ ∀t , ε (t ) = Ft [σ (τ )]. −∞

[1.16]

One-dimensional Viscoelastic Modeling

11

Conversely, for all deformation histories that are actually feasible, we can write in a similar way3: t

σ (t ) = R t [ε (τ )]

[1.17]

−∞

and denote by R the corresponding functional relationship, inverse of F , defined on the set of deformation histories generated by [1.16] such that4: I

I

R F ε ⎯⎯ →σ ⇔ σ ⎯⎯ →ε .

[1.18]

1.4. Aging 1.4.1. Aging phenomenon

Once a convenient origin has been chosen for the time variable, usually based on experiments performed to identify the behavior of the studied material, it follows that its physical properties, including its mechanical characteristics, may change with time independently of any mechanical action it may be subjected to. This phenomenon may result from various causes such as temperature changes, hygrometry, radiations (UV, γ), chemical reactions, phase changes, crystallizations or defect propagation. The phenomenon is called aging, a term that often suffers from a negative perception, being seen as accompanied by a degradation of mechanical properties, for instance in the case of polymers. However, aging should not always be considered as harmful since it turns out that it has quite positive effects in the case of some materials, of which concrete is a good example, when comparing its mechanical characteristics at the ages of 1, 7 or 28 days.

3 In fact, the roles of σ and ε are not fully interchangeable. 4 It is implicitly assumed here that F and R are one-to-one relationships (bijections).

12

Viscoelastic Modeling for Structural Analysis

1.4.2. Non-aging materials Although aging may be considered as a general phenomenon, it appears that it does not develop in the same way for all materials. For a given material, it depends on its “age” and often exists a significant period of time when mechanical properties can be considered as stabilized or consolidated and do not change with time. The model of a non-aging material proceeds from this observation. It characterizes a material whose properties (mechanical properties in the present case) do not depend on the age counted from its origin. This means that when considering two identical specimens of the material and applying the stress histories σ to the first one and σ λ to the second one, such that

σ λ (t ) = σ (t − λ ),

[1.19]

the response will be deformation histories ε and ε λ respectively, with

ε λ (t ) = ε (t − λ ).

[1.20]

In other words, if the stress history is translated with the time span λ , the corresponding deformation history is just translated with the same time span.

Figure 1.6. Creep tests on a non-aging material

One-dimensional Viscoelastic Modeling

13

This definition is presented in Figure 1.6 in the case of creep tests performed respectively at time t0 and (t0 + λ ), which shows that the creep function defined in [1.4] is such that

∀t , J (t0 + λ , t ;σ 0 ) = J (t0 , t − λ ;σ 0 ),

[1.21]

which implies

∀t , J (t0 + λ , t + λ ;σ 0 ) = J (t0 , t ;σ 0 )

[1.22]

and proves that the creep function only depends on the time arguments t and t0 through their difference (t − t0 )

J (t0 , t ;σ 0 ) = J (0, t − t0 ;σ 0 ) = J (t − t0 ;σ 0 ),

[1.23]

with, as a result of [1.5],  J (t − t0 ;σ 0 ) = 0 if t − t0 < 0   J (t − t ;σ 0 ) > 0 for t − t ≥ 0 0 0   0  ∂J (t − t0 ;σ ) ≥ 0 for t − t0 ≥ 0  ∂ (t − t0 )   ∂ 2 J (t − t ;σ 0 ) 0  ≤ 0 for t − t0 ≥ 0. 2 ( t t ∂ −  0)

[1.24]

In the general case of the stress histories σ and σ λ defined by [1.19], referring to the functional description [1.15], we can write for a non-aging material: t +λ

t

−∞

−∞

Ft+ λ [σ (τ )] = Ft [σ (τ )]. λ

[1.25]

Considering stress relaxation tests, we will write in the same way

R(t0 , t ;σ 0 ) = R(t − t0 ;σ 0 )

[1.26]

14

Viscoelastic Modeling for Structural Analysis

with  R(t − t0 ;σ 0 ) = 0 if t − t0 < 0   R(t − t ;σ 0 ) > 0 for t − t ≥ 0 0 0   0  ∂R (t − t0 ;σ ) ≤ 0 for t − t0 ≥ 0  ∂ (t − t0 )   ∂ 2 R (t − t ;σ 0 ) 0  ≥ 0 for t − t0 ≥ 0. 2  ∂ (t − t0 )

[1.27]

and finally t +λ

t

−∞

−∞

R t+ λ [σ (τ )] = R t [σ (τ )]. λ

[1.28]

Incidentally, it is worth mentioning that some authors (e.g. Mandel 1978) refer to the stress fading test (section 1.2.5) as the crucial experiment for defining the “rheological viscoelastic behavior” in the case of non-aging materials. The criterion they retain is that stress fading should be total. 1.5. Linear behavior 1.5.1. Superposition principle: Boltzmannian materials

Within the small perturbation framework, experimental evidences about the behavior of various materials subjected to complex stress or deformation histories lead to the statement of Boltzmann’s superposition principle and the definition of Boltzmannian materials through the linearity of the constitutive equation that quantifies the behavior of the material through a mathematical relationship. In the present case, let σ (1) and σ (2) be two stress histories and ε (1) , ε (2) the corresponding deformation histories associated through [1.16]. The linearity of functional F implies that for any linear

One-dimensional Viscoelastic Modeling

15

combination of these stress histories σ = λ1 σ (1) + λ 2 σ (2) , the associated deformation history is the corresponding linear combination of deformation histories ε (1) , ε (2) , which can be written as

∀λ1 , λ 2 , F (λ1 σ (1) + λ 2 σ (2) ) = λ1 F (σ (1) ) + λ 2 F (σ (2) )

[1.29]

and, for the inverse functional relationship5,

∀λ1 , λ 2 , R (λ 1 ε (1) + λ 2 ε (2) ) = λ1 R (ε (1) ) + λ 2 R (ε (2) ).

[1.30]

1.5.2. Linear elasticity

Linear elastic behavior is the most classical example of a Boltzmannian behavior. It corresponds to the case of a purely elastic material, as defined in section 1.2.4.2, when the reversible deformation jump ε (t0 ) is proportional to the stress jump σ 0 . The material is non-aging and functional relationships F and R reduce to the linear expressions I

t

F σ ⎯⎯ → ε ⇔ ∀t , ε (t ) = Ft [σ (τ )] = σ (t ) / E

[1.31]

−∞

I

t

R ε ⎯⎯ →σ ⇔ ∀t , σ (t ) = R t [ε (τ )] = E ε (t ),

[1.32]

−∞

where E is the elastic modulus and 1 E is called the elastic compliance. 1.6. Linear viscoelastic material

From now on, it will be assumed that the behavior of the material under consideration conforms to Boltzmann’s superposition principle. 5 The linearity hypothesis ensures that F and R are one-to-one relationships (bijections).

16

Viscoelastic Modeling for Structural Analysis

This assumption proves highly simplifying, but it must obviously be assessed in each practical case through experimental evidences in order to determine its domain of relevance (see section 1.6.5). 1.6.1. Instantaneous behavior

Only materials with elastic instantaneous behavior (section 1.2.4.2) and whose elastic modulus has a finite value will now be considered. This means that performing the instantaneous recovery experiment at any instant of time t0 reveals that the jump ε (t0 ) is non-zero and instantaneously totally recovered. 1.6.2. Creep and relaxation functions

The linearity hypothesis implies straightforward consequences on the expressions of the creep and relaxation functions defined in section 1.2. 1.6.2.1. Creep function

Regarding the creep function at time t0 with σ 0 as a stress jump, equation [1.29] implies that ε (t ) is proportional to σ 0 whatever its sign may be (traction or compression). It follows that the creep function defined in [1.4] is independent of σ 0 , hence:  σ (t ) = σ 0 Υ t0 (t )    ε (t ) = σ 0 J (t0 , t ), ∀σ 0 ,

[1.33]

also written as I

F σ = σ 0 Υ t ⎯⎯ → ε = σ 0 J t , ∀σ 0

[1.34]

J t0 (t ) = J (t0 , t ).

[1.35]

0

0

with

One-dimensional Viscoelastic Modeling

17

1.6.2.2. Relaxation function

From [1.30] and in the same way as for J t0 (t ), we derive that the relaxation function at time t0 is independent of ε 0 :  ε (t ) = ε 0 Υ t 0 ( t )    σ (t ) = ε 0 R (t0 , t ), ∀ε 0 ,

[1.36]

also written as I

R  ε = ε 0 Υ t0 ⎯⎯ →σ = ε 0 R t0 , ∀ε 0    R t0 (t ) = R (t0 , t ).

[1.37]

1.6.2.3. Mathematical properties of the creep and relaxation functions

The mathematical properties of creep and relaxation functions [1.35] and [1.37] in the case of linear behavior are derived from [1.5], [1.10] and associated comments:

 J (τ , t ) = 0 for τ > t   J (τ ,τ ) = J (τ ,τ + ) > 0   J (τ , t ) > 0 continuous and continuously differentiable   with respect to τ and t for τ < t    ∂J (τ , t ) ≥ 0 and ∂J (τ , t ) ≤ 0 for τ < t  ∂t ∂τ   ∂2 J  2 (τ , t ) ≤ 0 for τ < t  ∂t and

[1.38]

18

Viscoelastic Modeling for Structural Analysis

 R (τ , t ) = 0 for τ > t   R (τ ,τ ) = R (τ ,τ + ) > 0   R (τ , t ) > 0 continuous and continuously differentiable   with respect to τ and t for τ < t    ∂R (τ , t ) ≤ 0 and ∂R (τ , t ) ≥ 0 for τ < t  ∂t ∂τ   ∂2 R  2 (τ , t ) ≥ 0 for τ < t .  ∂t

[1.39]

In all the following sections, the mathematical conditions for the existence (convergence) of the integrals that will be introduced will be considered as satisfied even though they are not explicitly specified for simplicity’s sake. 1.6.3. Recovery and stress fading

As a first, straightforward consequence of the superposition principle, we can now write the expressions of the response in the recovery and stress fading experiments from the knowledge of the creep and relaxation functions. For the recovery experiment, the response to the stress history σ = σ 0 (Υ t0 −Υ t1 ) results from [1.34] in the form

ε = σ 0 ( Jt − Jt )

[1.40]

ε (t ) = σ 0 [ J (t0 , t ) − J (t1 , t )].

[1.41]

0

1

that is

In the same way, for the stress fading experiment, the response to the deformation history ε = ε 0 (Υ t0 −Υ t1 ) can be written as

One-dimensional Viscoelastic Modeling

 σ = ε 0 ( Rt0 − Rt1 )    σ (t ) = ε 0 [ R (t0 , t ) − R (t1 , t )].

19

[1.42]

1.6.4. Instantaneous behavior

Making t1 → t0 in [1.41] yields the response to the instantaneous recovery experiment. From the continuity of J (τ , t ) stated in [1.38], we note that recovery is complete. Conversely, the instantaneous stress fading experiment where t1 → t0 in [1.42] shows that stress fading is complete. The instantaneous behavior of the material is linear elastic with the elastic compliance J (t0 , t0 ) and elastic modulus R(t0 , t0 ) at time t0

σ 0 / ε (t0 ) = σ (t0 ) / ε 0 = 1 J (t0 , t0 ) = R(t0 , t0 ),

[1.43]

which implies that

∀τ , J (τ ,τ ) R (τ ,τ ) = 1.

[1.44]

1.6.5. Validation of the linearity hypothesis: an example

As an example, we consider the case of concrete subjected to a compressive stress in order to assess whether and to what extent its uniaxial behavior can be described through a linear viscoelastic model. While assessing the validity of the basic formula [1.33], we find that J (t0 , t;σ 0 ) is not independent of σ 0 but is actually an increasing function of σ 0 . Moreover, regarding the validity of [1.41] to describe the result of a recovery test, we find that it overestimates the recovery phenomenon. Nevertheless, in view of the substantial simplification thus provided, it is usually considered that, as long as σ 0 does not exceed 70% of the compressive limit stress of the material,

20

Viscoelastic Modeling for Structural Analysis

writing J (t0 , t ;σ 0 ) = σ 0 J (t0 , t ) is a good approximation for practical applications in structure design, at least as a first run. 1.7. Linear viscoelastic constitutive equation 1.7.1. Arbitrary stress history

Considering an arbitrary stress history σ , it is anticipated that linearity of the material behavior will make it possible to write the corresponding deformation history ε once the complete set of creep functions has been determined. From a mathematical viewpoint, this result may be considered as classical. Nevertheless, in order to make it better understood, we will adopt an intuitive approach, which also points out the specificity introduced here by the fundamental mechanical tests we are relying on. Let σ be a stress history with the following properties:  σ (t ) = 0 for t < t0   σ piecewise continuous and differentiable for t > t0 ,

[1.45]

and let τ i denote the instants of time when σ is discontinuous, which may include t0 , with [[σ ]] i being the corresponding stress jumps. As in section 1.2.1, σ (τ i ), ε (τ i ) … will denote the right-hand side +

values of the corresponding functions at time τ i : σ (τ i ) = σ (τ i ),

ε (τ i ) = ε (τ i + ) … This convention includes, as a particular case, the present instant of time t if it should be counted among the τ i . Hence, we can write

σ (t ) =  dσ (τ ) +  [[σ ]] i t

t0+

τ i ≤t

=  +Υ τ (t ) dσ (τ ) +  [[σ ]] i Υ τ i (t ), t

t0

τ i ≤t

[1.46]

One-dimensional Viscoelastic Modeling

21

which can be interpreted as the result of the sum of an infinite sequence of infinitesimal creep tests Υ τ (t ) dσ (τ ) performed at time τ and a countable sequence of finite creep tests performed at time τ i ≤ t , as shown in Figure 1.7. Note that, within the framework of the mathematical distribution theory6, equation [1.46]7 can also be written as t

σ (t ) =  σ ′(τ )dτ ,

[1.47]

−∞

where the generalized derivative σ ′ under the integral sign is the distribution obtained as the sum of the distribution { σ ′ } defined by the function σ ′, the derivative of σ when it is differentiable, and Dirac distributions [[ σ ]] i δτ i , τ i ≤ t :

σ ′ = { σ ′ } +  [[ σ ]] i δτ . τ i ≤t

i

[1.48]

Figure 1.7. Creep tests on a non-aging material

The response to the stress history σ defined by [1.46] results immediately from the superposition principle in the form

6 Schwartz (1965, 1966). See also Artola et al. (2000), Blanchard and Brüning (2015). 7 Also known as a Stieltjes integral.

22

Viscoelastic Modeling for Structural Analysis

ε (t ) =  J (τ , t ) dσ (τ ) +  [[σ ]] i J (τ i , t ), t

t0+

[1.49]

τ i ≤t

which, taking [1.38] into account, can also be written as a Stieltjes integral t

∞

−∞

−∞

ε (t ) =  J (τ , t ) σ ′(τ ) dτ =  J (τ , t ) σ ′(τ ) dτ

[1.50]

within the framework of the distribution theory. The integral in [1.50] admits integration by parts, taking into account the condition σ (−∞) = 0 and the jump [[ J (τ , t )]]τ =t of J (τ , t ) when τ increases beyond t 8. With the convention noted earlier, we have [[ J (τ , t )]]τ =t = J (t + , t ) − J (t , t ) = 0 − J (t , t ),

[1.51]

hence, Boltzmann’s formula: t ∂J  t t J t t ε ( ) σ ( ) ( , ) σ (τ ) (τ , t )dτ = −   −∞ ∂τ   t ∂J   ε (t ) = σ (t ) J (t , t ) − t0 σ (τ ) ∂τ (τ , t ) dτ .

[1.52]

From a physical viewpoint, this formula is of utmost importance as it splits ε (t ), the deformation at time t , into two contributions with different physical meanings: – the first term, namely σ (t ) J (t , t ), is the instantaneous response at time t to the stress σ (t ) imposed at this very instant of time:

8 It amounts to integrating ∑τi  ≤ t σ i  J τi , t terms.

t  J t+ 0

τ, t  dσ(τ) on each interval  τi-1 ,τ  and adding the

One-dimensional Viscoelastic Modeling

23

J (t , t ) = J (t , t + ), the elastic compliance, expresses the instantaneous response at time t to a unit stress jump; ∂J (τ , t ) dτ , is the memory term that ∂τ expresses the result at time t of the delayed deformation due to the stress history preceeding t . It should be noted that, as a result of the regularity hypotheses on Jτ specified in [1.38], this term does not exhibit any singularity. In addition, the response observed at time t to ∂J the force unit impulsion δτ imposed at time τ < t is − (τ , t ), which ∂τ is non-negative9. t

– the second term, −  σ (τ ) t0

1.7.2. Arbitrary deformation history

Starting from an arbitrary deformation history ε with the same properties as [1.45] and same notations as for σ  ε (t ) = 0 for t < t0   ε piecewise continuous and differentiable for t > t0 ,

[1.53]

we may follow the same track of reasoning to obtain the expression of the response in the form of the stress history σ . The corresponding equations are entirely similar to the previous ones, with σ and ε , Jτ and Rτ being substituted for each other. Hence, t

σ (t ) =  R(τ , t )dε (τ ) +  [[ε ]] i R(τ i , t ) t0+

[1.54]

τ i ≤t

t

∞

−∞

−∞

σ (t ) =  R(τ , t ) ε ′(τ )dτ =  R(τ , t ) ε ′(τ ) dτ

[1.55]

9 It may be observed that in the present case, the response to the unit stress jump plays the essential role while usually linear behavior analyses refer to the response to the unit impulsion. This is obviously a result of the creep test being taken as a basis to describe the viscoelastic material behavior.

24

Viscoelastic Modeling for Structural Analysis

within the framework of the distribution theory. Integration by parts yields Boltzmann’s formula: t ∂R   σ (t ) = ε (t ) R(t , t ) − −∞ ε (τ ) ∂τ (τ , t ) dτ   t ∂R   σ (t ) = ε (t ) R(t , t ) − t0 ε (τ ) ∂τ (τ , t ) dτ

[1.56]

with similar comments as in section 1.7.1. 1.7.3. Linear elastic material

The linear elastic material described in section 1.5.2 can be considered as a particular case of a linear viscoelastic material without any delayed behavior. Bringing together equations [1.31] and [1.32] on the one side, and [1.52] and [1.56] on the other, with conditions J (τ , t ) = 0 and R(τ , t ) = 0 for τ > t , we find that the constitutive equations [1.31] and [1.32] are expressed through Boltzmann’s formulas with  J (τ , t ) = Y (t − τ ) E = Yτ (t ) E   R (τ , t ) = E Y (t − τ ) = E Yτ (t )

[1.57]

as the creep and relaxation functions and obviously no memory term. 1.7.4. Boltzmann’s formulas, integral operator 1.7.4.1. Integral operator

Taking into account conditions J (τ , t ) = 0 and R (τ , t ) = 0 for τ > t on the creep and relaxation functions, the partial derivatives of J (τ , t ) and R (τ , t ) can be written, within the framework of the distribution theory, as:

One-dimensional Viscoelastic Modeling

25

∂J  ∂J  (τ , t ) =  (τ , t )  − [[ J (t , t )]]δ t ∂τ  ∂τ 

[1.58]

∂R  ∂R  (τ , t ) =  (τ , t )  − [[ R (t , t )]]δ t . τ ∂τ ∂  

[1.59]

Hence, equations [1.52] and [1.56] can be written in the compact forms ∞

ε (t ) = −  σ (τ ) −∞

∞

σ (t ) = −  ε (τ ) −∞

∂J ∂J (τ , t ) dτ = − < ,σ > (t ) ∂τ ∂τ

[1.60]

∂R ∂R (τ , t ) dτ = − < , ε > (t ). ∂τ ∂τ

[1.61]

These equations are formally identical, having been built up from the sets of creep and relaxation functions respectively. They are the explicit expressions, under the linearity hypothesis, of the inverse functional relationships F and R introduced in section 1.3. Denoting by (×) the integral operator defined explicitly in [1.52] and [1.56], we can write symbolically I

∂J , σ > = J (×) σ ∂τ

[1.62]

I

∂R , ε > = R (×) ε ∂τ

[1.63]

F σ ⎯⎯ →ε = −
=  ϕ (t ) e − p t dt , −∞

[1.87]

where the integral is understood within the distribution theory framework. In all the applications presented in the following, the concerned distributions will be defined by piecewise continuous and continuously differentiable functions or by derivatives of such functions. The following tables summarize the essential properties of Laplace transforms that should be retained for practical applications in the present context:  L ( a ∗ b) = L a L b   L δ =1  L δ′= p   L ϕ ′ = L (ϕ ∗ δ ′) = pL ϕ   − pu  L δu = e  L ϕ = L (ϕ ∗ δ ) = e − p u L ϕ , u u 

[1.88]

with ϕu = ϕ ∗ δ u being the translated distribution defined from a distribution ϕ through

32

Viscoelastic Modeling for Structural Analysis

ϕu (t ) = ϕ (t − u ) ,

[1.89]

and additional results

d   L [t ϕ (t )] = − dp (L ϕ )   ∞  L [ ϕ (t ) ] = L ϕ (u ) du  p  t  ∞  ∞ ϕ (t )  0 t dt = 0 L ϕ ( p ) dp . 

[1.90]

1.8.4.2. Laplace–Carson transform (LCT)

The first line in [1.88] explains how, through the use of Laplace transform (LT), a convolution product of two distributions is transformed into the algebraic product of their Laplace transforms. This procedure, quite convenient usually, seems not to be fully adequate in the present case because Boltzmann’s formulas [1.77] and [1.78] are expressed in terms of convolution products involving one distribution and the derivative of the other12. As already remarked in section 1.7.2, this is due to the specificity of the chosen fundamental experiments. It is therefore convenient here to refer to the Laplace–Carson transform (LCT) of a distribution ϕ , denoted by ϕ * ( p ), which is nothing but the Laplace transform of its derivative:

ϕ * = L ϕ ′ = pL ϕ .

[1.91]

With this definition, equations [1.77], [1.78] and [1.82] are transformed into simple algebraic products involving the Laplace– Carson transforms of the concerned functions

12 Also known as a Stieltjes convolution product.

One-dimensional Viscoelastic Modeling

 ε * ( p) = f * ( p) σ * ( p)   σ * ( p) = r * ( p) ε * ( p)   f * ( p ) r * ( p) = 1,

33

[1.92]

where r * ( p) is currently called the operational modulus. The profit derived from the use of LCT comes from the fact that algebraic equations [1.92] are identical to the equations obtained in the case of a one-dimensional linear elastic material. Theoretically, this makes it possible to derive the solution to a linear viscoelastic problem from the solution to the corresponding linear elastic one. Obviously, a major difficulty lies in the inversion of LCT that is necessary in order to express the solution to the original problem in terms of functions of the time variable13. In addition, it may be objected that, formally, the structure of the algebraic calculations for the elastic material is identical to that of the calculations necessary to solve the original problem in the convolution algebra, but this obviously requires a rather good practice to be performed, which explains the interest of using LCT. Nevertheless, it is advisable to refrain from having recourse to LCT immediately and systematically without having first gone into the original problem as deeply as possible. In fact, the danger in shifting to LCT too early lies in losing the physical feeling of the problem – e.g. answering a question such as “is it a creep- or relaxation-type phenomenon?” – as it practically disappears once the time variable t has been substituted by p. 1.9. One-dimensional linear viscoelastic behavior 1.9.1. Uniaxial viewpoint and one-dimensional modeling

The experiment chosen as an example in sections 1.2.1 and 1.2.2 when demonstrating the concept of delayed behavior of a material 13 See the Appendix for Laplace and Laplace–Carson transforms of usual functions.

34

Viscoelastic Modeling for Structural Analysis

enables us now to explain the difference between the uniaxial experimental viewpoint and one-dimensional behavior modeling14. In section 1.2.1, a creep tension test was performed on a three-dimensional specimen assumed to be such that a homogeneous uniaxial tensile stress field would develop under the action of an external tension load, which can be written in the form15

σ (t ) = σ 0 e x ⊗ e x Υ t (t ). 0

[1.93]

In fact, the response of the constituent material was three-dimensional in the form of a homogeneous deformation field ε (t ) but its longitudinal component ε x x (t ) = ε (t ), associated with

σ 0 = σ xx (t ) in the expression of the work by external forces, was the only observed variable for the definition of the corresponding one-dimensional creep function. In section 1.2.2, for the definition of the relaxation function, an experiment supposed to be the dual of the preceding one was imagined: on the same specimen and under the same simple tension conditions, the deformation component was imposed in the form ε x x (t ) = ε (t ) = ε 0 Υ t0 (t ) while the uniaxial response σ (t ) was observed. Clearly, these uniaxial analyses do not describe the behavior of the constituent material itself but the behavior of the specimen considered as a straight structural element subjected to tension or compression, e.g. a straight rod. We can say that the analyses developed in the preceding sections, besides introducing the general concepts about delayed behavior, provide the prototype of any one-dimensional viscoelastic constitutive equation, in the same way as Hooke’s law in its original form ut tensio, sic vis (“as the extension, so the force”) does in the case of linear elasticity. 14 This issue will be reconsidered in Chapter 4. 15 σ denotes the second-rank Cauchy stress tensor, ⊗ is the symbol for the tensorial product and e x is a unit vector.

One-dimensional Viscoelastic Modeling

35

Then, in order to write a general statement of a one-dimensional linear viscoelastic modeling, let Q denote the “force variable” and q its associated geometrical variable in the expression of the work by external forces Q q . The creep and relaxation functions, J (τ , t ) and R ( τ,t ) respectively, are introduced with the same properties as in [1.38] and [1.39], and the functional relationships [1.62] and [1.63] can now be written as: I

∂J , Q > = J (×) Q ∂τ

[1.94]

I

∂R , q > = R (×) q. ∂τ

[1.95]

F →q = − < Q ⎯⎯

R q ⎯⎯ →Q = −
0 for τ ≥ 0   f ′(τ ) ≥ 0, f ′′(τ ) ≤ 0 for τ ≥ 0

[1.98]

with

and

36

Viscoelastic Modeling for Structural Analysis

 r (τ ) = 0 if τ < 0   r (τ ) > 0 for τ ≥ 0   r ′(τ ) ≤ 0, r ′′(τ ) ≥ 0 for τ ≥ 0.

[1.99]

1.9.2. Structural elements 1.9.2.1. Traction–compression of a straight rod Within the framework of curvilinear one-dimensional continua (Salençon 2018) for a straight rod subjected to tension or compression, the “force variable” Q is the normal force classically denoted by N and the associated geometrical variable q is the longitudinal deformation (stretch) ε , and equation [1.94] can thus be written as:

Q = N , q = ε = J (×) N .

[1.100]

Figure 1.8. Traction–compression of a straight rod

In the particular case, when the one-dimensional modeling can be matched with the three-dimensional one, if the original threedimensional rod with cross-section S is made of a homogeneous isotropic material with J (τ , t ) as the uniaxial tension creep function and E (τ , t ) as the uniaxial tension relaxation function16, we then obtain17 J (τ , t ) =

J (τ , t ) N , ε = J (×) . S S

16 See Chapter 4, sections 4.4.2 and 4.4.5. 17 See Chapter 6, section 6.2.2.

[1.101]

One-dimensional Viscoelastic Modeling

37

and the relaxation function is

R (τ , t ) = S E (τ , t ).

[1.102]

1.9.2.2. Bending of a straight rod Similarly, if the bending of a straight rod about a principal axis of its cross-section is considered within the framework of a curvilinear one-dimensional continuum, the “force variable” Q is the bending moment classically denoted by M and the associated geometrical variable q is the curvature χ of the director curve18. We have:

Q = M , q = χ = J (×) M .

[1.103]

Figure 1.9. Bending of a straight rod

In the same particular case described above, when the one-dimensional model can be matched with the three-dimensional one, if bending is imposed along a principal axis of the cross-section with I denoting the corresponding moment of inertia, we then obtain19 J (τ , t ) =

J (τ , t ) M , χ = J (×) I I

[1.104]

and the relaxation function is

R (τ , t ) = I E (τ , t ).

[1.105]

1.9.2.3. Twisting of a straight rod When the straight rod is subjected to torsion, the “force variable” Q is the twisting moment C and the associated geometrical variable 18 See Chapter 7 in Salençon (2018). 19 See Chapter 6, section 6.3.2.

38

Viscoelastic Modeling for Structural Analysis

q is the differential rotation about the x -axis denoted by α . Hence, we can write

Q = C , q = α = J (×) C.

[1.106]

Figure 1.10. Twisting of a straight rod

In the particular case when the one-dimensional model can be matched with the three-dimensional one, if the original three-dimensional rod with cross-section S is made of a homogeneous isotropic material with γ (τ , t ) as the creep function in the simple shear test and μ (τ , t ) as the associated relaxation function20, we then obtain21

J (τ , t ) =

γ (τ , t ) J

, α = γ (×)

C , J

[1.107]

where J is the torsional inertia of the cross-section. The relaxation function is

R (τ , t ) = J μ (τ , t ).

[1.108]

1.9.2.4. Multi-parameter loading of a straight rod The response of a rod subjected to a multi-parameter loading process in the form of a linear combination of the loads studied above can be obtained through the superposition principle. With

20 See Chapter 4, sections 4.4.3 and 4.4.4. 21 See Chapter 6, section 6.4.3.

One-dimensional Viscoelastic Modeling

 Q 1 = N , q1 = ε   Q 2 = M, q2 = χ   Q 3 = T , q3 = α , 

39

[1.109]

and assuming the absence of any coupling effect where a creep test performed on Q i would induce a non-zero response in q j , j ≠ i, the response is the homolog multidimensional combination of [1.101], [1.103] and [1.106]. The bending of the rod about an axis that is not a principal inertia axis will be treated that way. As in the classical strength of materials theory and practice, these equations obtained for a homogeneous straight rod are used as constitutive equations for the infinitesimal curvilinear element of a beam or arc. 1.10. Harmonic loading process 1.10.1. The loading process

On a one-dimensional non-aging linear viscoelastic system, described as in section 9.1, we impose a loading process where the history of the geometrical variable q , starting from the instant of time t0 = 0, is prescribed in the form of a sinusoidal function q (t ) = q0 Υ (t ) cos ω t = q0 Re Y (t ) ei ω t  .

[1.110]

The response of the system describing the evolution of the force variable proceeds from [1.97], where the time derivative r ′ is understood within the framework of the distribution theory:

40

Viscoelastic Modeling for Structural Analysis

I

R  q ⎯⎯ → Q = r′ ∗ q   ∞ t  Q (t ) = q0 Re   r ′(τ ) Y (t − τ ) ei ω (t −τ ) dτ  = q0 Re   r ′(τ ) ei ω (t −τ ) dτ  , −∞ −∞      [1.111]

which is zero for t < 0 as a result of [1.99]. 1.10.2. Asymptotic harmonic regime

For t ≥ 0, the integral in the second line of [1.111] can be split into two terms: ∞ ∞ Q (t ) = q0 Re ei ω t  r ′(τ ) e− i ωτ ) dτ  − q0 Re ei ω t  r ′(τ ) e− i ωτ ) dτ  . t −∞     [1.112]

In the first integral, we can identify



∞

−∞

r ′(τ ) e− i ωτ ) dτ as the LCT

of r (t ) where i ω plays the role of the variable p



∞

−∞

r ′(τ ) e−i ωτ ) dτ = r * (i ω )

[1.113]

which is called the complex modulus. Hence, [1.112] takes the form ∞ Q (t ) = q0 Re ei ω t r * (iω )  − q0 Re ei ω t  r ′(τ ) e − i ωτ dτ  .   t

[1.114]

Finally, [1.111] can be written as ∀t , Q (t ) = q0 Y (t ) Re  ei ω t r * (iω )  − q0 Y (t ) Re  ei ω t 



∞ t

r ′(τ ) e − i ωτ dτ  . 

[1.115]

One-dimensional Viscoelastic Modeling

41

Looking at the second term on the right-hand side of this equation, we note that, from [1.99], r ′(τ ) has a constant sign, from which it follows that:



∞ t

r ′(τ ) e− i ωτ dτ ≤ r (∞) − r (t ) ,

[1.116]

if r (t ) has a finite limit r (∞) when t → ∞. Hence,



∞ t

r ′(τ ) e

− i ωτ

equations

[1.115]

and

[1.116]

prove

that

dτ → 0 when t → ∞ and Q (t ) is then described by the

harmonic regime t → ∞ Q (t ) ≈ q0 Re  ei ω t r * (iω )  .

[1.117]

This result characterizes the accommodation phenomenon when, with increasing time, the response of the system tends to become periodic with a period identical to that of the imposed variable. 1.10.3. Complex modulus

The complex modulus r * (i ω ) can be written as

r * (iω ) = M (ω ) ei δ (ω ) with M (ω ) > 0,

[1.118]

thus providing the explicit expression for Q (t )

Q (t ) = q0 M (ω ) cos[ ω t + δ (ω )],

[1.119]

where δ (ω ) is the phase angle of the response harmonic regime with respect to the loading process.

42

Viscoelastic Modeling for Structural Analysis

The real and imaginary parts22 of r * (iω ), from which δ (ω ) and M (ω ) are defined, are derived from [1.113]: ∞

Re [r * (iω ) ] = r (0) +  r ′(τ ) cos ωτ dτ 0

∞

Im [r * (iω ) ] = −  r ′(τ ) sin ωτ dτ . 0

[1.120] [1.121]

Taking advantage of [1.99], we can determine the signs of these two quantities. Regarding [1.121], we can write the integral as the result of the converging23 infinite sum of positive elementary integrals π ω

Im [r * (iω ) ] = − 

0

(r ′(τ ) − r ′(τ + π )) sin ωτ dτ

3π ω

−

2π ω

[1.122]

(r ′(τ ) − r ′(τ + π )) sin ωτ dτ − ...

thus proving that

Im [r *(iω )] > 0.

[1.123]

A similar decomposition for [1.120] can be written as

Re [r * (iω ) ] = r (0) + 

π 2ω

0

r ′(τ ) cos ωτ dτ

+

3π 2ω

+

7 π 2ω

π 2ω

5π 2ω

(r ′(τ ) − r ′(τ + π )) cos ωτ dτ

[1.124]

(r ′(τ ) − r ′(τ + π )) cos ωτ dτ + ...

22 Sometimes known respectively as the storage modulus and the loss modulus (Mase 1970). 23 Note that the convergence mathematical conditions must be satisfied (section 1.6.2).

One-dimensional Viscoelastic Modeling

43

where r (0) + 

π 2ω

0

r ′(τ ) cos ωτ dτ > r (π 2ω ) > 0

[1.125]

and the other integrals are positive. Hence

Re [r * (iω )] > 0

[1.126]

and, as a result of [1.123] and [1.126], we have tan δ (ω ) > 0 , 0 ≤ δ (ω )
t0    q (t ) = Q 0 (t1 − t0 ) η for t > t1

[2.10]

and stress fading is total. This model is called a Maxwell fluid.

2.3.2. Kelvin model 2.3.2.1. Description The Kelvin model is the combination of a linear elastic element and a linear viscous element connected in parallel (Figure 2.4).

Figure 2.4. Kelvin model

50

Viscoelastic Modeling for Structural Analysis

In any experiment performed on this model, the stretch q of the model is the stretch of both constituent elements, while the force Q exerted on the model is the result of the sum of the forces that are exerted in the linear elastic element and the linear viscous element respectively. It then follows that the relaxation function is obtained from the general rule, independent of any linearity or non-aging assumption, which states that the relaxation function for a model that is made up of elements connected in parallel is the result of the sum of the relaxation functions of the constituent elements. From [2.2] and [2.5], we derive the expression of the relaxation function in the form of the sum of a step function and a Dirac distribution:

r (t ) = EY (t ) + η δ (t )

[2.11]

r * ( p) = E + η p .

The creep function is obtained using the inverse LCT of f * ( p) f * ( p) r * ( p) = 1  f * ( p) =

1η , ( p + E η)

[2.12]

hence f (t ) = Υ (t )

1 (1 − exp (−t E η )) . E

[2.13]

2.3.2.2. Comments Equation [2.13] shows that f (0) = 0 , which illustrates the lack of instantaneous elasticity. When t → ∞ , the creep function admits a finite limit f (∞ ) = 1 / E because the response of the model is only governed by the linear elastic element while the linear viscous element is completely

Rheological Models

51

unloaded. The creep function f (t ) tends exponentially to that limit with a characteristic time

τ f =η E .

[2.14]

Recovery is total. The stress fading test cannot be performed (see section 2.2.2). This model is commonly known as a Kelvin solid or Kelvin–Voigt solid. 2.3.3. The standard linear solid 2.3.3.1. Description In order to provide the Kelvin solid with the instantaneous elasticity that is necessary for its practical relevance as a model for the viscoelastic behavior of solid materials, the model is connected in series with a linear elastic element as shown in Figure 2.5.

Figure 2.5. Kelvin–Voigt model with instantaneous elastic behavior

With the notations specified in Figure 2.5 and applying the general rule stated in section 2.3.1.1, we derive the creep function of that model from [2.2] and [2.13] in the form: f (t ) = Υ (t )[

1 1 + (1 − exp (− t E1 η1 ))] . E E1

[2.15]

52

Viscoelastic Modeling for Structural Analysis

The relaxation function is obtained as in section 2.3.1.1 using the LCT:

f * ( p) =

1 1 E1 η1 + E E1 p + E1 η1

p + E1 η1 f * ( p) r * ( p) = 1  r * ( p) = E p + ( E + E1 ) η1

[2.16]

hence r (t ) = Υ (t )

E [ E1 + E exp (− t ( E + E1 ) η1 )] . E + E1

[2.17]

2.3.3.2. Comments Creep function [2.15] increases exponentially from f (0) = 1 E

[2.18]

f (∞) = 1 E + 1 E1 = 1 K1

[2.19]

to

with a characteristic time

τ f = η1 E1 .

[2.20]

From a physical viewpoint, E is the instantaneous elastic modulus of the model and 1 K1 , defined in [2.19], is the elastic compliance when t → ∞ with K1 = E E1 ( E + E1 ) .

[2.21]

With these notations, the creep function can also be written as f (t ) = Υ (t )[

1 1 1 + ( − ) exp (− t τ f )] K1 E K1

[2.22]

Rheological Models

53

and the relaxation function becomes r (t ) = Υ (t )[ K1 + ( E − K1 ) exp (−t ( E + E1 ) η1 )] ,

[2.23]

which decreases exponentially from r (0) = E = 1 / f (0) to r (∞ ) = K1 = 1 / f (∞) and complies with [1.84] in Chapter 1, section 1.2.3.1. The characteristic time of the force relaxation phenomenon is

τr =

η1 E + E1

=

K1 τf , E

[2.24]

which is obviously less than the creep characteristic time and confirms that force relaxation evolves faster than creep as described in Chapter 1. Finally, we note that recovery and stress fading are total.

2.3.3.3. Zener model In order to master the unlimited creep of a Maxwell model when t → ∞ , the Zener model presented in Figure 2.6 introduces a parallel connection with a linear elastic element.

Figure 2.6. Zener model

With the notations specified in Figure 2.6 and applying the general

54

Viscoelastic Modeling for Structural Analysis

rule stated in section 2.3.2.1, we derive the relaxation function of that model from [2.2] and [2.8] as r (t ) = [ K + E2 exp (−t E2 η 2 )]Υ (t ) ,

[2.25]

which decreases exponentially from r (0) = K + E2 to r (∞) = K , with τ r = η 2 E2 being a characteristic time. When comparing this equation with [2.23], we find that they are identical to each other and describe the same behavior if the parameters in [2.25] are written in the form K = K1 , E2 = E − K , η 2 = η1 (1 − K E ) 2 .

[2.26]

2.3.3.4. The standard linear solid These two equivalent models are said to describe the behavior of the standard linear solid. It is the simplest one-dimensional model for a non-aging linear viscoelastic material with instantaneous elasticity and total recovery and stress fading. From the equations written in section 2.3.3.1, we note that it can be characterized by three parameters, namely the instantaneous elastic modulus E and the creep and relaxation characteristic times τ f and τ r < τ f . Putting together [2.17], [2.20], [2.21], [2.23] and [2.24], we can write its relaxation function as

r (t ) =Υ (t ) E [

τr τ + (1 − r )exp (− t τ r )] τf τf

[2.27]

This model appears in some technical codes as a possible candidate for describing the delayed behavior of concrete when the non-aging hypothesis can be considered as relevant from the material proper chronology viewpoint. 2.3.3.5. Harmonic loading process Following the same track as in Chapter 1 (section 1.10), we impose a harmonic loading process

ε (t ) = ε 0 cos ω tΥ (t )

[2.28]

Rheological Models

55

on a standard linear solid characterized by the instantaneous elastic modulus E and the creep and relaxation characteristic times τ f and

τ r < τ f . From [2.27], we derive r * ( p) = E

τr τ p + E (1 − r ) . τf τ f p +1 τr

[2.29]

The complex modulus defined in Chapter 1 (section 1.10.2) is

r * (iω ) = E

τr τ iω + E (1 − r ) = M (ω ) ei δ (ω ) τf τ f iω + 1 τ r

[2.30]

and response ε (t ) can be written explicitly as

 τr

ε (t ) = ε 0 Re  E

 τ f

+ E (1 −

 τr iω ei ω t Υ (t ) ) τ f iω + 1 τ r 

 1 1  1 + ε 0 Re  E ( − ) e − t τ r Υ (t ),  τ r τ f iω + 1 τ r 

[2.31]

where the first line represents the asymptotic harmonic regime and the second line represents the vanishing transitory term, as in equation [1.116] in Chapter 1. Computing M (ω ) and δ (ω ) from [2.30], we obtain M (ω ) = E

ω 2 + (1 τ f ) 2

[2.32]

ω 2 + (1 τ r ) 2

and tan δ (ω ) = ω

1 τ r −1 τ f

ω2 +1 τr τ f

as shown in Figure 2.7.

, 0 ≤ δ (ω ) < π 2

[2.33]

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Viscoelastic Modeling for Structural Analysis

As discussed in Chapter 1 (section 1.10.3), we observe that

ω = 0  δ (ω ) = 0 and M (ω ) = r (∞) = K ω → ∞  δ (ω ) → 0 and M (ω ) → r (0) = E .

[2.34]

The maximum value of the loss angle is reached when ω = 1 τ rτ f . From a practical viewpoint, this result highlights the

most favorable frequency range for taking advantage of the damping effect of the material. It also emphasizes the importance of controlling and mastering temperature changes resulting from the energy dissipated along the loading process, in order to prevent any undesirable alteration of the material’s physical properties.

Figure 2.7. Modulus and loss angle for the standard linear solid

2.4. Generalized Maxwell and Kelvin models The standard linear solid is characterized by one parameter, E , which defines the instantaneous elastic behavior of the non-aging material, and two parameters, τ f and τ r < τ f , that characterize its delayed behavior. It is easily anticipated that such a model may be too constrained for practical applications. Increasing the number of parameters introduced to describe the delayed behavior of the considered material seems likely to make fitting with experimental data easier.

Rheological Models

57

2.4.1. Generalized Maxwell model In the form of a Zener model, the linear standard solid appears as a particular case of a generalized Maxwell model, also known as the Maxwell–Wiechert model, as shown in Figure 2.8. Such a model is built up from m Maxwell models connected in parallel to each other and a linear elastic element. With the notations introduced in the figure, it is defined by (2m + 1) parameters E , E1 ,η1 ,...E j ,η j ,... Em ,ηm .

Figure 2.8. Generalized Maxwell model

The relaxation function for this model proceeds from the general rule stated in section 2.3.2.1 and can be written as j =m

r (t ) = EΥ (t ) + Υ (t ) E j exp (− t [τ r ] j ) ,

[2.35]

j =1

where [τ r ] j = η j E j denotes the relaxation characteristic time of the elementary Maxwell model with number j . The [τ r ] j can be ordered as a discrete spectrum.

58

Viscoelastic Modeling for Structural Analysis

j =m

The instantaneous elastic modulus is r (0) = E +  E j

while

j =1

r (∞ ) = E . Recovery and stress fading are total. The operational modulus of the model is j =m

r * ( p) = E +  ( E j j =1

p ), p + 1 / [τ r ] j

[2.36]

hence f * ( p) = 1 r * ( p ) . The creep function f (t ) is obtained as the inverse LCT of f * ( p) and brings out the discrete spectrum of creep characteristic times: [τ f ]1 ,...[τ f ] m . Finally, since [τ r ] j = η j E j , the model shown in Figure 2.8 can also be considered as being described by the set of (2m + 1) parameters E , E1 ,[τ r ]1 ,...E j ,[τ r ] j ,... Em ,[τ r ] m . 2.4.2. Generalized Kelvin model

In the same way as for a Maxwell model, the linear standard solid presented in Figure 2.5 is a particular case of a generalized Kelvin model. Such a model is composed of n Kelvin models connected in series with each other and a linear elastic element (Figure 2.9). It is defined by (2n + 1) parameters E , E1 ,η1 ,...Ei ,ηi ,... En ,η n . Through the general rule stated in section 2.3.1.1, we obtain the creep function of this model in the form f (t ) =

i=n 1 1 Υ (t ) + Υ (t ) (1 − exp ( − t [τ f ] i )) , E E i =1 i

[2.37]

Rheological Models

with [τ f ] i = ηi Ei

59

being the creep characteristic times of the

constituent Kelvin models, which also enables us to describe the model with parameters E , E1 ,[τ f ]1 ,...E i ,[τ f ] i ,... En ,[τ f ] n .

Figure 2.9. Generalized Kelvin model

The instantaneous elastic modulus is 1 f (0) = E 1 i=n 1 compliance when t → ∞ is f (∞) = +  . E i =1 Ei

and the

Recovery and stress fading are total.

The LCT of [2.37] can be written as

f * ( p) =

1 i =n 1 1 ), + ( E i =1 ηi p + 1 / [τ f ] i

[2.38]

from which the relaxation function r (t ) is derived as the inverse LCT of r * ( p ) = 1 f * ( p ) and brings out the discrete spectrum of the relaxation characteristic times [τ r ]1 ,...[τ r ] n .

2.4.3. Equivalence The (tedious) algebraic inversions of f *( p) in section 2.4.1 or r * ( p) in section 2.4.2 prove that the two models considered above are equivalent to each other. Consequently, any set of linear elastic and linear viscous basic elements connected to each other either

60

Viscoelastic Modeling for Structural Analysis

in series or in parallel, with finite instantaneous elasticity modulus and total recovery, is equivalent to a generalized Maxwell model or a generalized Kelvin model. 2.4.4. Continuous spectra

In the definition of a generalized Maxwell model, each E j can be considered as the value of a given function of the corresponding [τ r ] j written in the form: E j = E (τ ) τ =[τ ] .

[2.39]

r j

Hence, relaxation function [2.35] can be written as j =m

r (t ) = EΥ (t ) + Υ (t ) ( E (τ ) τ =[τ j =1

r ]j

exp (− t [τ r ] j ))

[2.40]

which may be given the form of an integral within the framework of the distribution theory:

r (t ) = EΥ (t ) +Υ (t ) 

∞

0

j =m

 E (τ )exp(− t τ ) δ (τ − [τ

] ).

r j

[2.41]

j =1

This expression leads to the concept of a continuous spectrum of the relaxation characteristic times [0, ∞[ , where the discrete Dirac measures E (τ ) δ (τ − [τ r ] j ), j = 1,...m are substituted by a density g (τ r ) dτ r in the form ∞

r (t ) = EΥ (t ) +Υ (t )  exp (− t τ r ) g (τ r )dτ r . 0

[2.42]

The same reasoning can be followed with respect to the generalized Kelvin model starting from [2.37], which leads to the

Rheological Models

61

concept of a continuous spectrum of the creep characteristic times with an expression of f (t ) similar to [2.42]: f (t ) =

∞ 1 Υ (t ) +Υ (t )  exp ( − t τ f ) h (τ f ) dτ f . 0 E

[2.43]

Both approaches are equivalent, as stated in section 2.4.3. Distributions g (τ r ) or h (τ f ) must be determined from experimental data on r (t ) or f (t ) respectively.

3 Typical Case Studies

3.1. Presentation and general features Within the framework of the small perturbation hypothesis (SPH)1, we will now present simple examples of problems that highlight significant aspects of the response of a homogeneous or heterogeneous system whose constituent material exhibits linear viscoelastic behavior. The constituent material is described by a one-dimensional geometrical modeling and its behavior follows a one-dimensional linear elastic or viscoelastic constitutive equation, as presented in Chapter 1. The systems considered here are subjected to quasi-static loading processes which depend on one scalar loading parameter Q with q as its associated geometrical parameter in the expression of the work by external forces. One important point is that, in practical circumstances, such a one-parameter loading process is often defined by a complex history of both Q (t ) and q (t ). More precisely, it may be a sequence of “creep-type” periods where the evolution of Q (t ) is prescribed and “relaxation-type” periods where the history of q (t ) is assigned. 1 The SPH is taken here with its whole meaning, including the fact that displacements are considered small enough to allow all equations to be written on the initial geometry at any instant of time along the loading process of the considered systems (see Salençon 2001).

Viscoelastic Modeling for Structural Analysis, First Edition. Jean Salençon. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Viscoelastic Modeling for Structural Analysis

In Chapter 1, we emphasized the significant differences between the two phenomena as regards, for instance, their characteristic times; therefore, great care should be taken when giving a mathematical formulation to the original practical problem before starting to solve it. This is one reason why we insist on an in-depth understanding of the physical problem before shifting to mathematical techniques. Chapter 5 will offer a general presentation of quasi-linear viscoelastic processes, which can obviously be applied to the particular cases to be treated here. Nonetheless, we prefer to tackle them in a naive straightforward way, making them simple illustrative examples of ulterior general results. 3.2. “Creep-type” problems 3.2.1. Homogeneous cantilever beam subjected to a uniformly distributed load 3.2.1.1. Description of the system under study The system chosen as an example, on which typical cases of “creep-type” problems will be presented, is a cantilever beam OA with length  (Figure 3.1) subjected to vertical loads to be defined later. Within the framework of a curvilinear continuum, the bending moment M ( x, t ) at point x and time t will be considered as the only internal force, and the corresponding geometrical variable in the expression of the work by internal forces is the curvature of the beam at the current point P at time t:

χ ( x, t ) =

d2v ( x, t ), dx 2

[3.1]

where v ( x, t ) represents the deflection at that point, counted positive upwards. The behavior of the one-dimensional constituent element dx at the current point P is defined by the constitutive equation presented in Chapter 1 (section 1.9.2.2), where J ( x;τ , t ) and R ( x;τ , t ) are the

Typical Case Studies

65

creep and relaxation functions at that point, and can be written in the form (distribution theory framework) ∞

χ ( x, ⋅) = −  M ( x,τ ) −∞

∞

M ( x, ⋅) = −  χ ( x,τ ) −∞

∂J ( x;τ , ⋅) dτ = [ J( x) (×) M ( x)] ∂τ

[3.2]

∂R ( x;τ , ⋅) dτ = [R( x) (×) χ ( x)] ∂τ

[3.3]

with the explicit meaning t ∂J   χ ( x, t ) = M (x, t ) J( x; t , t ) − −∞ M (x,τ ) ∂τ ( x;τ , t ) dτ   t ∂R   M ( x, t ) = χ (x, t ) R( x; t , t ) − −∞ χ (x,τ ) ∂τ ( x;τ , t ) dτ .

[3.4]

Figure 3.1. Cantilever beam

3.2.1.2. The quasi-static evolution problem The first loading process that we consider consists in applying a uniformly distributed load with line density history p (t ), counted positive downwards, to the cantilever beam, which is here supposed to be made from a homogeneous linear viscoelastic material (Figure 3.2). Hence, the functions J ( x;τ , t ) and R ( x;τ , t ) do not depend on the space variable x , and equations [3.2] and [3.3] reduce to

66

Viscoelastic Modeling for Structural Analysis

χ ( x, ⋅) = [J (×) M ( x)] ⇔ M ( x, ⋅) = [R (×) χ ( x)],

[3.5]

t ∂J   χ ( x, t ) = M (t ) J(t , t ) − −∞ M (x,τ ) ∂τ (τ , t )dτ   t ∂R   M ( x, t ) = χ (t ) R(t , t ) − −∞ χ ( x,τ ) ∂τ (τ , t ) dτ .

[3.6]

i.e.

From a practical viewpoint, this description may correspond to the case when the beam is made from homogeneous (reinforced) concrete. The history of the active force variable is defined by

p (t ) = Q (t ),

[3.7]

with the associated geometrical variable 

q (t ) = −  v ( x, t ) dx. 0

Figure 3.2. Uniformly distributed load applied to a cantilever beam

[3.8]

Typical Case Studies

67

The solution of this quasi-static evolution problem proceeds from the equilibrium equations written at any instant of time, from which we classically derive

( − x ) 2 p(t ) 2

[3.9]

( − x ) 2 J (×) p 2

[3.10]

M ( x, t ) = − hence, from [3.5],

χ ( x, ⋅) = −

and, integrating [3.1],

v ( x, ⋅) = −

x2 2  x x2 ( − + ) J (×) p. 2 2 3 12

[3.11]

In terms of the associated force and geometrical variables:

q=

5 J (×) Q. 20

[3.12]

3.2.1.3. Comments As a first important comment, we note that the results obtained above are only derived from the linearity of the constitutive law, and do not require any non-aging assumption as would have been necessary if we had wished to rely on the LCT method. Equation [3.12] shows that, considered as a system, the beam follows a linear viscoelastic constitutive law, which is straight-forwardly derived from that of its constituent material. As a particular case, [3.11] and [3.12] provide the creep response of the beam to the action of its own weight supposed to be applied at time t = 0 (when the shuttering of the concrete beam is removed):

68

Viscoelastic Modeling for Structural Analysis

 5  Q (t ) = pY (t ) q (t ) = p J (t ) 20    x2 2  x x2 v ( x , t ) p ( − = − + ) J (t ).  2 2 3 12 

[3.13]

3.2.2. Homogeneous cantilever beam subjected to a concentrated load 3.2.2.1. The quasi-static evolution problem The same homogeneous cantilever beam as presented in section 3.2.1.2 is now subjected to the loading process defined by a vertical concentrated force F (t ), counted positive upwards, being applied at endpoint A (Figure 3.3) with the history:

F (t ) = Q (t ). The associated geometrical displacement at endpoint A :

q (t ) = v (, t ).

[3.14] variable

is

just

the

vertical [3.15]

Figure 3.3. Concentrated load applied at the endpoint of a cantilever beam

Typical Case Studies

69

The same reasoning as in section 3.2.1.2, based on the equilibrium equation and the homogeneous linear constitutive law, yields:   M ( x, t ) = ( − x) F (t )   x2  v ( x, ⋅) = (3 − x) J (×) F 6   3  q =  J (×) Q .  3

[3.16]

3.2.2.2. Comments The comments to be made are strictly identical to those in section 3.2.1.3, and the response to the creep experiment in the present case is given by  3  Q (t ) = F Y (t ) q (t ) = F J (t ) 3    x2 v ( x , t ) F (3 − x) J (t ). =  6 

[3.17]

Conversely, we derive from [3.16] or [3.17] the response of the beam, considered as a system, to the relaxation experiment when q (t ) = v0 () Y (t ) is imposed 3 v0 ()   q (t ) = v0 () Y (t ) Q (t ) = 3 R (t )   3 v0 () (  − x ) R (t )  M ( x, t ) = 3   2  v ( x, t ) = x (3 − x) v (). 0  2 3

Then, when an arbitrary history q (t ) = v (, t ) is imposed,

[3.18]

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Viscoelastic Modeling for Structural Analysis

3   q = v(, ⋅) Q = 3 R (×) v(, ⋅)   3  M ( x, ⋅) = 3 ( − x) R (×) v(, ⋅)    2  v ( x, t ) = x (3 − x) v(, t ).  2 3

[3.19]

3.2.3. Homogeneous statically indeterminate beam

Figure 3.4. Statically indeterminate structure

Figure 3.4 shows a statically indeterminate structure that consists of the homogeneous cantilever beam (studied in section 3.2.2) that is now resting on an unyielding support at endpoint A. The loading process is defined as in section 3.2.1.2 by the history of the active force variable p (t ) = Q (t ).

3.2.3.1. The quasi-static evolution problem Due to the statical indeterminacy of the system, it is necessary to specify the initial value of the self-equilibrated internal force field, namely the initial bending moment distribution along OA before the beginning of the loading process. We here assume that this distribution is identically zero, which means that there is no prestressing of the beam. This implies that the initial value of the support reaction at endpoint A is zero:

Typical Case Studies

V (t ) = 0 for t < 0.

71

[3.20]

In view of the linearity of the equilibrium equation and constitutive law, the solution to this problem takes advantage of the preceding results. At any instant of time t ≥ 0, deflection v (, t ) at endpoint A due to the histories of the active force variable p (t ) and reactive force V (t ) up to that time must be equal to zero. Hence, from [3.11] and [3.16],

J (×)V =

3 J (×) p 8

[3.21]

and the solution to this equation is simply V (t ) =

3 p (t ). 8

[3.22]

The response of the beam is then described by the evolution of the deflection at the current point:

v ( x, ⋅) = −

x 2 ( − x)(3 − 2 x) J (×) p, 48

[3.23]

while the bending moment distribution follows p (t ) according to M ( x, t ) = −

( − x)( − 4 x) p (t ). 8

[3.24]

3.2.3.2. Comments

In the preceding sections, it was no surprise that the problems presented were “creep-type” examples since the corresponding data defining the loading processes were only concerned with force variables. In the present case, we may wonder about encountering a “creep-type” problem again, although the support at endpoint A introduces an additional data to the description of the loading process that is concerned with the geometrical variable v (, t ). As we will find in section 3.3.1 and can already be guessed from [3.19], this is due to

72

Viscoelastic Modeling for Structural Analysis

the fact that the value assigned to v (, t ) by this condition is just zero and does not induce any prestressing.

3.2.4. A statically indeterminate system 3.2.4.1. The system

Figure 3.5. Statically indeterminate system

The same cantilever beam as in section 3.2.1 is now part of the system presented in Figure 3.5 where its endpoint A is connected at time t = t0 to a homogeneous perfectly supple cable AB , with length  cos α , which is anchored at a fixed point B. The linear elastic behavior of the cable element ds is defined by

ε ( s, t ) = N ( s, t ) E1

[3.25]

where N ( s, t ) is the tension at time t at the current point on AB and E1 is the elastic modulus. This system is statically indeterminate. In the absence of any force exerted on AB , the tension force N ( s, t ) is constant along the cable: N ( s, t ) = N (t ). Equilibrium of the node at endpoint A shows that the action of the cable on the beam results in a vertical force V (t ) and a horizontal force H(t):

Typical Case Studies

 V (t ) = N (t )sin a   H (t ) = − N (t ) cos α .

73

[3.26]

As stated in section 3.2.1.1, the analysis of the beam is performed with M ( x, t ), the bending moment at the current point, being considered as the only internal force. The cantilever beam is only considered as subjected to bending, and the stretching effect of the horizontal force H (t ) is not taken into account. At any instant of time t ≥ t0 , the bending moment along OA can be written as

M ( x, t ) = N (t )( − x)sin α − p(t )(l − x)2 2

[3.27]

and from [3.11] and [3.16], we derive

v (, ⋅) = −

4 3 J (×) p + sin α J (×) N . 8 3

[3.28]

Meanwhile, maintaining the connection between the beam and the cable at point A implies the compatibility condition between the vertical displacement of the endpoint of the cable and the vertical displacement induced by the bending of the beam:

∀t ≥ t0 ,  (ε (t ) − ε (t0 )) tan α = v(, t0 ) − v(, t ).

[3.29]

3.2.4.2. A creep-type problem

It is now supposed that at time t = t0 , when the uniformly distributed load history p(t ) = pYt0 (t ) is applied and the connection is realized, tension N (t0 ) is adjusted in such a way that the vertical displacement of endpoint A is zero, which implies from [3.28] that

N (t0 ) = 3 p  8sin α .

[3.30]

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Viscoelastic Modeling for Structural Analysis

In order to determine the evolution of tension N (t ) for t > t0 , we can write that the compatibility condition expressed in [3.29] is maintained with v (, ⋅) obtained from [3.28], which yields the functional equation (sin α

tan α p 3 3p 1 2 Y ) (×) N = Yt . J+ Jt0 + 3 8 8cos α E1 0 E1

[3.31]

Recalling that J (×) Yt0 = Jt0 , the solution to this equation is simply N (t ) =

3p Yt (t ). 8sin α 0

[3.32]

This means that tension N (t ) in the cable remains constant and consequently, the deflection at endpoint A remains equal to zero. The beam behaves exactly as in section 3.2.3.1. The deflection is governed by [3.23]. It is a creep-type problem:

v ( x, t ) = − p

x 2 ( − x)(3 − 2 x) J (t0 , t ). 48

[3.33]

3.2.4.3. Comments

This final result could have been straightforwardly derived from the analysis performed in the preceding section that could be applied to the particular case when p(t ) = pYt0 (t ). Indeed, we know from section 3.2.3.1 that the value of the vertical reactive force exerted at endpoint A of the cantilever beam by a fixed support is V (t ) = 3 p  8. Substituting this fixed support by cable AB whose extension is initially adjusted to produce the required value of the tension N (t0 ) = 3 p  8sin α is equivalent to the reaction of the fixed support at time t0 . Then, the tension remaining equal to N (t ) = 3 p  8sin α for t > t0 does not change the extension of the cable while exerting the required vertical force on the cantilever beam in order to prevent any deflection at the endpoint, hence assuring that the beam and the cable stay connected to each other.

Typical Case Studies

75

3.3. Prestressing of viscoelastic systems or structures 3.3.1. Prestressed cantilever beam 3.3.1.1. The problem As an introduction to the concept of prestressing, we consider the same beam OA as in section 3.2 being now subjected to the loading process defined as follows.

Figure 3.6. Prestressed cantilever beam

Starting from the “natural” unloaded initial state2, where the bending moment distribution over OA is zero for t < 0 and the position of endpoint A is taken as a reference, we suppose that (Figure 3.6): – at time t = 0+ , a vertical displacement v0 (), maintained for t > 0, is imposed at endpoint A :

v (, t ) = v0 () Y (t );

which is

[3.34]

– at the same time (or immediately afterwards), the uniformly distributed load p is applied and maintained for t > 0:

p(t ) = pY (t ).

2 Sometimes called the “dead” initial state (Mase 1970).

[3.35]

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Viscoelastic Modeling for Structural Analysis

From a physical viewpoint, it is clear that the first step in this sequence generates a bending moment distribution that can be calibrated in such a way that it mitigates the bending moment distribution due to the second step. In other words, the first step produces a prestressing of the beam before it is loaded. However, analyzing the sequence, we see that, as time passes, the first step amounts to a relaxation experiment while the second one defines a creep-type problem. Hence, due to the difference in the nature of the two phenomena involved (relaxation and creep), it can be anticipated that the mitigation effect as calibrated at time t = 0+ will not remain constant, which may result in unforeseen consequences. More precisely, the solution to the quasi-static evolution problem defined by [3.34] and [3.35] is obtained from sections 3.2.1.2 and 3.2.3.1 using the superposition principle. Putting together equations [3.18], [3.23] and [3.24] with equations [3.34] and [3.35], we obtain: 3 v0 () ( − x)( − 4 x)  Y (t )  M ( x, t ) = 3 ( − x) R (t ) − p 8   x 2 (3 − x) x 2 ( − x)(3 − 2 x)  v x t v  Y t p J (t ) [3.36] ( , ) = ( ) ( ) −  0 3  2 48    V (t ) = p 3 Y (t ) + 3v0 () R (t ).  3 8 These equations show how both creep and relaxation phenomena are involved in the evolution of the system in this case, where the uniformly distributed load may, for instance, represent the beam’s own weight. The reactive force V (t ) at the support and the bending moment distribution are governed by the relaxation phenomenon as a result of the displacement imposed on the support. Deflection evolves as the creep phenomenon generated by the uniformly distributed active force.

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77

To illustrate the consequences of this result from a practical viewpoint, let us assume that the beam is made from reinforced concrete and that the described loading process corresponds to the removal of the shuttering of the beam. We may imagine that prestressing is calibrated in such a way that the bending moment distribution due to the beam’s own weight does not exhibit any negative value along OA at that very instant of time. From [3.36], this condition determines the minimum value of v0 () by writing that M (0,0) = 0, i.e.: v0 () =

p 4 , 24 R (0)

[3.37]

where R (0) is the instantaneous elastic modulus of the beam material element. The evolution of M ( x, t ) is then given by M ( x, t ) =

R (t ) p ( − x)( − 4 x) −p Y (t ) ( − x ) R (0) 8 8

[3.38]

and, as R (t ) is a decreasing function of the time variable, the embedding moment at point O becomes negative and decreases steadily to M (0, ∞) = −

p 2 R (∞ ) (1 − ). 8 R (0)

[3.39]

Equation [3.38] describes a particular example of the general phenomenon known as the redistribution of internal forces due to the delayed behavior of the beam constituent material, after the initial prestressing defined by [3.37]. From a practical viewpoint, this phenomenon may have damaging consequences if it has not been anticipated.

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Figure 3.7. Bending moment diagrams

Figure 3.7 shows how bending moments, which are initially positive, progressively become negative along the beam length as t → ∞. Since we find in [3.36] that the prestressing effect on the bending moment distribution results from the part of the support reactive force generated by the imposed vertical displacement v0 (), the phenomenon can be monitored by measuring the value of this reactive force V (t ). Prestressing can thus be controlled and adjusted by increasing the value of v (, t ) either initially, for instance, by choosing deliberately v0 () =

p 4 , 24 R (∞)

[3.40]

as an initial value for the vertical displacement of endpoint A, or when it becomes critical during the monitored evolution.

3.3.1.2. Comments The simple description presented above can easily be generalized to the case when the prestressing and loading processes are not simultaneous. For an arbitrary loading history p (t ) with v (, t ) = v0 () Y (t ), equations [3.36] become:

Typical Case Studies

3 v0 () ( − x)( − 4 x)  p (t )  M ( x, t ) = 3 ( − x) R (t ) − 8   x 2 (3 − x) x 2 ( − x)(3 − 2 x)  v  Y J (×) p ( ) −  v ( x, ⋅) = 0 2 3 48    V (t ) = 3 p(t ) + 3v0 () R (t ).  3 8

79

[3.41]

3.3.2. Prestressed hyperstatic system

The statically indeterminate system examined in section 3.2.4 (Figure 3.5) also provides a typical example of prestressing of a viscoelastic system. We now consider the case when the tension force in the cable is initially adjusted in such a way that the bending moment in the embedment support is equal to zero:

M (0, t0 ) = 0 ⇔ N (t0 ) = p  2sin α .

[3.42]

With arguments similar to those developed in section 3.2.4.2, the functional equation governing the evolution of the prestressing effect on the embedment moment is obtained in the form (

3 p 2 Y + J ) (×) M (0, ⋅) = (J (t0 , t0 ) Yt0 − Jt0 ). 8 E1 cos α 2

[3.43]

The first term in this equation obviously complies with equations [1.38] in Chapter 1, which list the mathematical properties of creep functions, and is therefore a creep function J. Its associated relaxation function will be denoted by R , defined by3 J=(

3 Y + J ), R (×) J t0 = Yt0 E1 2 cos α

and the solution to [3.43] can then be formally written as 3 See Chapter 1, section 1.7.4.2.

[3.44]

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Viscoelastic Modeling for Structural Analysis

M (0, ⋅) =

p 2 (J (t0 , t0 ) R t0 − R (×) Jt0 ). 8

[3.45]

3.3.3. Prestressed hyperstatic arc 3.3.3.1. The problem

Figure 3.8. Parabolic-shaped hyperstatic arc

A simple example of a statically indeterminate arc is shown in Figure 3.8 in the form of a symmetrical, parabolic-shaped arc AB with a 2 span and an f rise:

y = f (1 − x 2  2 ).

[3.46]

The endpoint A is connected to a fixed hinged support, while the endpoint B can be moved horizontally. The arc is supposed to be homogeneous with a constant cross-section. With the notations introduced in Chapter 1 (sections 1.9.2.1 and 1.9.2.2), the viscoelastic behavior of the arc element ds is defined by the constitutive equations

 ε ( s, ⋅) = ρ 2 J (×) N ( s, ⋅)   χ ( s, ⋅) = J (×) M ( s, ⋅),

[3.47]

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81

where J is the creep function in bending and ρ 2 J is the creep function in tension–compression, with ρ 2 being a constant with dimension 2 , ρ 0  2   AB    D1 =  x sin θ ( s) cos θ ( s) ds < 0   AB  2  D2 =  cos 2 θ ( s) ds > 0,  f   AB

[3.51]

and putting together equations [3.48], [3.50] and [3.51], we derive the expression of the functional equation for the horizontal displacement u(t): p f f u = J (×) C ( Yt0 − Q 2 ) + ρ 2 J (×) ( D1 pYt0 − D2 Q 2 ). 2   Making Q(t0 ) = p  2 2 f determine the value of u(t0):

[3.52]

at time t = t0 in this equation, we

u (t0 ) = ρ 2 J (t0 , t0 ) p ( D1 − D2 2) < 0.

4 This is called the Bresse–Navier equation (see Salençon 2001).

[3.53]

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83

Then, for t > t0 , the support being fixed, the evolution of the arc is determined by the endpoint condition at point B, which can be written as u (t ) = u (t0 ) Yt0 (t ) = ρ 2 J (t0 , t0 ) p ( D1 − D2 2) Yt0 (t ),

[3.54]

to be combined with [3.52] in the form p f f J (×) C ( Yt0 − Q 2 ) + ρ 2 J (×)( D1 pYt0 − D2 Q 2 )   2 2 = ρ J (t0 , t0 ) p ( D1 − D2 2) Yt0 .

[3.55]

Hence, the functional equation that governs the evolution of thrust Q(t ) is given by f C (C + ρ 2 D2 ) − p ( + ρ 2 D1 ) Yt0 ] 2  2 2 = − ρ J (t0 , t0 ) p ( D1 − D2 2) Yt0 , J (×)[ Q

[3.56]

with an explicit solution written as Q (t ) =

p 2 1 C [ ( + ρ 2 D1 ) Yt0 (t ) − ρ 2 ( D1 − D2 2) J (t0 , t0 ) R(t0 , t ) ]. 2 f C + ρ D2 2

[3.57] Noting that D1 < 0 and D2 > 0, this equation describes a relaxation-type evolution of the horizontal thrust Q (t ) = Q (t0 ) (Yt0 (t ) − ρ 2

R(t0 , t ) D2 − 2 D1 (1 − ) ), 2 C + ρ D2 R (t 0 , t 0 )

[3.58]

which decreases from Q(t0 ) = p  2 2 f at time t = t0 to the final value Q (t∞ ) = Q (t0 ) (1 − ρ 2

R (t 0 , t ∞ ) D2 − 2 D1 (1 − ) ). 2 C + ρ D2 R (t 0 , t 0 )

[3.59]

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If a permanent prestressing effect is required despite this relaxation-type evolution, then the initial horizontal displacement of the support at endpoint B shall be back-calculated from [3.59]. 3.3.3.2. Comments

From [3.53], it is clear that if the analysis of the arc is performed without taking the normal force effect into account using the first equation in [3.47], which amounts to making ρ 2 = 0, then the horizontal displacement required to obtain the desired prestressing effect would be equal to zero. This means that it would only be necessary to fix the support at endpoint B at time t0 when the arc centering is removed. Furthermore, [3.52] shows that, in this case, the prestressing effect would be permanent, which is not surprising as the arc that is not subjected to any bending moment would not deform. This is obviously due to the fact that [3.46] describes a funicular arc5 for the uniformly distributed load p. In such cases, it is necessary to take the normal force effect into account to determine the relaxationtype evolution of the prestressing effect. 3.3.4. The example of a rheological model 3.3.4.1. The problem

The system presented in Figure 3.9 consists of a linear elastic element, denoted by 1 , and a linear viscoelastic element, denoted by 2 , connected to each other in parallel. The force and deformation

variables in these elements are respectively Q 1 , q 1 and Q 2 , q 2 where the reference for q 1 and q 2 is the “natural” initial state, i.e.:

q 1 = 0 when Q 1 = 0, q 2 = 0 when Q 2 = 0. The force variable acting on the system is denoted by Q :

5 See (Salençon 2018).

[3.60]

Typical Case Studies

Q = Q1 + Q 2

85

[3.61]

with the associated geometrical variable q.

Figure 3.9. Prestressed rheological system

With these definitions, the linear elastic constitutive law for element 1 can be written as:

1   q 1 = E Y (×) Q 1 1    Q 1 = E1 Y (×) q 1

[3.62]

while with J 2 (τ , t ) and R 2 (τ , t ) being the creep and relaxation functions, the linear viscoelastic constitutive law for element 2 can be expressed as

 q 2 = J 2 (×) Q 2    Q 2 = R 2 (×) q 2 . The prestressing of viscoelastic element obtained as follows.

[3.63]

2

in this model is

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Viscoelastic Modeling for Structural Analysis

At time t = t0 , the parallel connection of 1 with 2 is realized in the absence of an external force, Q (t0 ) = 0, and with deformations in the elements respectively equal to q 1 (t0 ) and q 2 (t0 ) counted with reference to their “natural” initial state. Hence:  Q 1 (t0 ) = − Q 2 (t0 )    Q 1 (t0 ) = E1 q 1 (t0 )   Q 2 (t0 ) = R 2 (t0 , t0 ) q 2 (t0 ). 

[3.64]

Q 2 (t0 ) is the initial prestressing force in the viscoelastic element. From t = t0 onwards, no external force is exerted on the system. The elements remain connected to each other, which implies that their deformations at any instant of time satisfy the geometric compatibility equation

q 1 (t ) − q 1 (t0 ) = q 2 (t ) − q 2 (t0 ),

[3.65]

while the absence of an external force imposes

Q 1 (t ) = − Q 2 (t ).

[3.66]

From [3.62] and [3.63], this results in −

Q 2 (t0 ) 1 Yt0 (×) Q 2 + Yt0 = J 2 (×) Q 2 − J 2 (t0 , t0 ) Q 2 (t0 ) Yt0 , [3.67] E1 E1

hence (J 2 +

1 1 Yt0 ) (×) Q 2 = ( J 2 (t0 , t0 ) + ) Q 2 (t0 ) Yt0 . E1 E1

[3.68]

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87

In the same way as in section 3.3.2, J (τ , t ) defined by 1 J = ( J 2 + Yτ ) complies with equations [1.38] in Chapter 1, which E1 list the mathematical properties of a creep function. Let R (τ , t ) denote its inverse with respect to the operator (×) 6: J = (J 2 +

1 Yτ ), R (×) Jτ = Yτ , E1

[3.69]

equation [3.68] can be solved as Q2 =

R (×) Q 2 (t0 ) Yt0 R (t 0 , t 0 )

[3.70]

and finally Q 2 (t ) = Q 2 (t0 )

R (t0 , t ) . R (t0 , t0 )

[3.71]

It follows that as a result of the viscoelastic behavior of element 2 , the magnitude of the prestressing force in this element decreases from its initial value Q 2 (t0 ) to the final value: Q 2 (t ) ⎯⎯⎯ → Q 2 (t 0 ) t →∞

R (t 0 , ∞ ) . R (t0 , t0 )

[3.72]

3.3.4.2. Comments

Equation [3.71] gives the evolution of the prestressing force in the viscoelastic element as time passes and shows that the problem is of the relaxation type. From a physical viewpoint, it is worth noting that J (τ , t ) defined in [3.69] is just the creep function of the model shown in Figure 3.10, 6 See Chapter 1, section 1.7.4.2.

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where the same elastic element 1 is connected in series to the same viscoelastic element 2 , which amounts to developing the “loop” in Figure 3.9. The function R (τ , t ) is the relaxation function of that model.

Figure 3.10. The model with J (τ , t ) = J 2 (τ , t ) + Yτ (t ) as a creep function

E1

Hence, from [3.68], we find that the prestressing process described in Figure 3.9 is equivalent to the relaxation process defined on the model in Figure 3.10 by the stretch history q (t ) = ( J 2 (t0 , t0 ) +

1 1 Yt , ) Q 2 (t0 ) Yt0 = Q 2 (t0 ) E1 R (t0 , t0 ) 0

[3.73]

which induces the response Q (t ) = Q 2 (t ) = Q 2 (t0 )

R (t0 , t ) . R (t0 , t0 )

[3.74]

3.3.5. Practical applications 3.3.5.1. Prestressing of a concrete block in compression Although very simple, the one-dimensional system presented in Figure 3.9 is a good prototype of many examples of elastic prestressing applied to structural elements such as stones, concrete blocks and beams.

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89

Figure 3.11. Internal elastic prestressing of a concrete block in compression

Figure 3.11 shows a concrete block, whose behavior is modeled as linearly viscoelastic, being prestressed by means of an axial elastic steel rod pulled in tension and anchored onto two plates that are pressing the block at both ends. In this internal elastic prestressing process, the concrete cylindrical block plays the role of element 2 , shown in Figure 3.9, while the elastic rod is modeled as element 1 . If the standard linear solid model is adopted to describe the viscoelastic behavior of element 2 supposed to be non-aging, with

J 2 (t ,τ ) = (

1 1 1 t −τ ) exp ( − ))Υ τ (t ), +( − [τ f ] 2 K2 E2 K 2

[3.75]

where [τ f ] 2 is the creep characteristic time of the concrete material, then the creep function J (τ , t ) in [3.69] can be written as J (τ , t ) = (

1 1 1 1 t −τ + +( − ) exp (− ))Υ τ (t ), K 2 E1 E2 K 2 [τ f ] 2

[3.76]

which is also a standard linear solid creep function with

K = K 2 E1 ( K 2 + E1 ) and E = E1 E2 ( E1 + E2 )

[3.77]

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Viscoelastic Modeling for Structural Analysis

and the same creep characteristic time J (τ , t ) = (

1 1 1 t −τ + ( − ) exp (− ))Υ τ (t ). K E K [τ f ] 2

[3.78]

The corresponding relaxation function can be written as R (τ , t ) = (K + (E − K) exp ( −

(t − τ ) )) Yτ (t ), Tr

[3.79]

with Tr being the characteristic relaxation time that governs the evolution of the prestressing force. It is worth noting that Tr is larger than the proper relaxation characteristic time of the material itself: Tr =

K ( E + E2 ) ( E + E2 ) K [τ f ] 2 = 2 1 [τ f ] 2 = 1 [τ r ] 2 . E2 ( K 2 + E1 ) ( K 2 + E1 ) E

[3.80]

The ultimate value of the elastic prestressing force is given by

Q 2 (t ) ⎯⎯⎯ → Q 2 (t0 ) t →∞

K . E

[3.81]

3.3.5.2. Prestressing of a concrete beam The same reasoning may be adopted to analyze other examples of elastic internal prestressing, such as the beam presented in Figure 3.12, which is prestressed by means of a cable in tension, in order to mitigate the bending moment distribution due to a uniformly distributed load.

Figure 3.12. Internal prestressing of a concrete beam

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91

3.3.5.3. Prestressed concrete slab bridges

As shown in section 3.3.1 (Figure 3.8), delayed behavior of concrete as a constituent material is at the origin of internal force redistribution in prestressed systems. This may result in dramatic consequences when prestressed concrete beams that are subjected to bending are designed without taking the creep phenomenon into account accurately. Two illustrative examples will now be briefly outlined to emphasize the practical relevance of the topic: – the Chazey Bridge (France) was one of the first prestressed concrete slab bridges. It was built in 1955 and completed in July 1957 as a substitute for the old suspension bridge built in 1829 by the Seguin brothers, which had been destroyed during World War II (1944). Unfortunately, as soon as November 1971, important disorders were observed as a result of an evolution of the bending moment distribution, similar to the one described in Figure 3.7: soffit tension cracks were quite visible, as shown in Figure 3.13. The bridge was considered unserviceable, closed for fear of a sudden collapse and finally destroyed. A new bridge was built on the same piers and opened for use in 1977.

Figure 3.13. Chazey Bridge: soffit tension cracks due to prestress loss

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– the prestressed concrete slab Palau Bridge (Koror–Babelthuap Bridge) was completed in April 1977 as a 240 m single span bridge (Figure 3.14), then the longest concrete girder bridge in the world7. The span consisted of two cantilever slabs extending on the water and meeting in the center (Figure 3.15).

Figure 3.14. Palau Bridge before collapse (Burgoyne and Scantlebury 2006)

In 1990, the cantilevers deflected due to creep and prestress loss, yielding a 1.2 m sag at mid span. This induced the Palauan government to “retrofit” the bridge through remediation works in order to prevent further inconveniences. Unfortunately, despite this reinforcement, the bridge collapsed on September 26, 1996 with dramatic consequences: two fatalities, four people injured and all services that used to pass through the bridge between the concerned islands were suspended, leaving thousands of people without fresh water and electricity.

Figure 3.15. Palau Bridge geometry as built (after Burgoyne and Scantlebury 2006) 7 All data and pictures related to this topic are obtained from Chris Burgoyne and Richard Scantlebury (2006) which the reader should refer to for an exhaustive analysis.

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93

3.4. A complex loading process 3.4.1. A practical problem The physical problem to be dealt with in this section is related to civil or mining engineering. It is concerned with the convergence of a cavity, which can be geometrically modeled as cylindrical (e.g. a tunnel) or spherical (e.g. a storage facility), after it has been bored and a supporting concrete or metallic cladding has been installed. The loading process history can be sketched out as follows.

Figure 3.16. Excavating the cavity

−

Initially, at time t = t0 , the soil or rock mass where the cavity should be excavated is at rest. Modeling the corresponding initial local stress state at the level of the cavity as a uniform isotropic pressure field, denoted by p0 (Figure 3.16), can be considered as a good approximation to perform the cavity convergence analysis. At time t = t0 , the cavity is assumed to be instantaneously excavated (Figure 3.16). Due to this excavation process, the pressure at the boundary of the cavity decreases from p0 to p (t0 ) = 0. This induces an instantaneous elastic convergence of the soil or rock − boundary from its original radius a0 at time t0 to the initial radius of the cavity denoted by a(t0 ) 8. +

8 As in Chapter 1, we adopt the convention a (t0 ) = a (t0 ).

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Then, for t > t0 , the pressure at the boundary of the cavity remains equal to zero, p (t ) = 0, and delayed convergence of the cavity due to the viscoelastic behavior of the soil or rock material takes place, which reduces the radius of the cavity to a(t ) (Figure 3.17). At time t = t1 , a metallic or concrete cladding is installed, whose mechanical behavior can be modeled either as undeformable or elastic (or even viscoelastic). It is assumed that this setting is done instantaneously without forcing in such a way that p(t1 ) = 0 and a (t1 ) suffers no discontinuity (Figure 3.17).

Figure 3.17. Installing a concrete or metallic cladding

Let us assume, as an example, that the cladding is undeformable. Then, for t > t1 , the convergence of the cavity can no longer occur and the internal radius a(t ) remains unchanged: a(t ) = a(t1 ) Yt1 (t ). Consequently, due to the delayed behavior of the soil or rock mass, the pressure acting on the cladding at the boundary of the cavity, i.e. p (t ), starts increasing from its zero value at time t1 (Figure 3.18).

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95

Figure 3.18. The cavity after the cladding has been installed

3.4.2. Mathematical treatment Despite its obvious three-dimensional physical nature, the problem can be treated within a one-dimensional framework. Taking the state − of the system at time t0 , when the system is at rest, as a geometrical

( p(t ) − p0 ) as the force variable with the displacement (a(t ) − a0 ) as its associated geometrical variable9. With

reference, we choose

these definitions, we denote by J (τ , t ) and R (τ , t ) the creep and relaxation functions of the cavity involved in Boltzmann’s formulas:

 Q (t ) = p (t ) − p0   q (t ) = a(t ) − a0    q = J (×) Q   Q = R (×) q .

[3.82]

Then, the history of the loading process described in the preceding section can be written as follows (Figure 3.19):

9 It should be noted that the analysis is performed within the small perturbation hypothesis framework. Furthermore, it is clear that the correct geometrical variable would incorporate the area of the cavity boundary, but this is of no importance in the present analysis.

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Viscoelastic Modeling for Structural Analysis

 Q (t ) = 0   q (t ) = 0

[3.83]

 Q (t0 ) = Q (t0 + ) = − p0    q (t0 ) = q (t0 + ) = a (t0 ) − a0 = J (t0 , t0 ) Q (t0 )

[3.84]

 Q (t ) = Q (t0 ) = − p0 Yt0 (t )   q (t ) = a (t ) − a0 = J (t0 , t ) Q (t0 )

[3.85]

t = t1

 Q (t1 ) = Q (t0 ) = − p0   q (t1 ) = a (t1 ) − a0 = J (t0 , t1 ) Q (t0 )

[3.86]

t > t1

 Q (t ) = p (t ) − p0 Yt1 (t )   q (t ) = a (t ) − a0 = J (t0 , t1 ) Q (t0 ).

[3.87]

t < t0

−

t = t0

+

t > t0

+

Figure 3.19. The loading process

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97

The goal of the game is to determine Q (t ) in this latter equation in order to assess the evolution of the pressure on the cladding and to monitor the behavior of the cavity in service. We may remark from [3.83] to [3.87] that the complete history of the geometrical parameter can be written as

q (t ) = J (t0 , t ) Q (t0 ) − J (t0 , t ) Q (t0 ) Y (t − t1 ) + J (t0 , t1 ) Q (t0 ) Y (t − t1 ),

[3.88]

i.e.

q = Jt0 Q (t0 ) − Jt0 Q (t0 ) Yt1 + J (t0 , t1 ) Q (t0 ) Yt1 .

[3.89]

With Q (t0 ) = − p0 , we derive the complete history of the force parameter through Boltzmann’s formula: Q = − p0 (Yt0 + J (t0 , t1 ) R t1 ) + p0 R (×) ( Jt0 Yt1 ).

[3.90]

It follows that the pressure exerted by the soil or rock mass on the cladding for t ≥ t1 is

p = Q + p0 = − p0 J (t0 , t1 ) Rt1 + p0 R (×)(Jt0 Yt1 )

[3.91]

In the particular case, when it can be considered that the cladding is installed instantaneously and immediately after the boring process has been completed, equation [3.91] can be explicitly written and yields: p (t ) = p0 (1 −

R (t0 , t ) ). R (t0 , t0 )

[3.92]

3.4.3. Comments

The functions J (τ , t ) and R (τ , t ) involved in these equations are usually determined from experimental results obtained through direct

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Viscoelastic Modeling for Structural Analysis

experiments performed on test cavities by means of specific devices. A theoretical analysis is sometimes possible based on the knowledge of the constitutive equation of the soil or rock where the cavity is bored. For instance, in the case of a spherical cavity, assuming that the material is isotropic and homogeneous and denoting by γ (τ , t ) and μ (τ , t ) its simple shear creep and relaxation functions respectively, J (τ , t ) and R (τ , t ) can be written as:  J (τ , t ) = γ (τ , t ) a0 4   R (τ , t ) = 4 μ (τ , t ) a0

[3.93]

and equations [3.91] and [3.92] become respectively

p = Q + p0 = − p0 γ (t0 , t1 ) μt1 + p0 μ (×)(γ t0 Yt1 ) p (t ) = p0 (1 −

μ (t0 , t ) ). μ (t 0 , t 0 )

[3.94] [3.95]

3.5. Heterogeneous viscoelastic structures

In fact, some structures and systems that we examined in the preceding sections were already examples of heterogeneous viscoelastic systems with the specificity that one of their constituent elements was linearly elastic. In order to better explain the possible effects induced in a system whose elements exhibit different viscoelastic behaviors, we consider the simple case of the hyperstatic cantilever beam examined in section 3.2.3, assuming now that it consists of two geometrically identical spans denoted by 1 and 2 , with length a =  2, which are rigidly connected to each other at midpoint B (Figure 3.20).

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99

Figure 3.20. Heterogeneous cantilever beam

With a common chronology being adopted, let

J1(τ , t) and

J 2 (τ , t) denote the creep functions in bending of beam spans 1 and 2 respectively, so that the constitutive equations can be written as:

on OB : χ ( x, ⋅) =

d2v ( x, ⋅) = J 1 (×) M ( x, ⋅) dx 2

[3.96]

on BA : χ ( x, ⋅) =

d2v ( x, ⋅) = J 2 (×) M ( x, ⋅). 2 dx

[3.97]

From a practical viewpoint, this may correspond to a beam whose spans are made from the same reinforced concrete material at different ages10. The active load history consists of a uniformly distributed vertical load being applied at time t0 and maintained for t ≥ t0 : p(t ) = pYt0 (t ). As discussed in section 3.2.3, solving the quasi-static evolution problem reduces to the determination of the vertical reactive force 10 This example Salençon (1983).

was

originally

proposed

by

Professor

B.

Halphen

in

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V (t ) exerted by the support at endpoint A. With V (t ) chosen as a redundant unknown, the bending moment history over the beam is statically determined as ∀t ≥ t0 , ∀0 < x <  = 2a, M ( x, t ) = −p

(2a − x) 2 Yt0 (t ) + V (t ) (2a − x). 2

[3.98]

Hence, integrating [3.96] with the embedment conditions at endpoint O, we obtain along OB:

(2a − x) 4 4 3 2 − a x + a 4 ) Jt10 24 3 3 3 (2a − x) 4 +( + 2a 2 x − a 3 ) J 1 (×)V . 6 3

v( x, ⋅) = p (−

[3.99]

Along BA, [3.97] is integrated with the fixed support condition at endpoint A and yields

v( x, ⋅) = − p

(2a − x) 4 2 (2a − x)3 2 Jt0 + J (×)V − (2a − x) J 2 (×) C , [3.100] 24 6

where C (t ) is a function of time that must be determined, together dv ( x, t ) for with V (t ), from the continuity of v( x, t ) and dx x = a, ∀t ≥ t0 . After rather tedious calculations, the functional equation for V (t ) can be written as (7 J 1 + J 2 ) (×)V =

3pa (15 Jt10 + Jt20 ). 8

[3.101]

The first term in this equation is a creep function J = (7J 1 + J 2 ) with the associated relaxation function R :

Typical Case Studies

J = (7 J 1 + J 2 ), R (×) J t0 = Yt0

101

[3.102]

and the solution to [3.101] can then be formally written as: V=

3pa R (×) (15 Jt10 + Jt20 ). 8

[3.103]

Obviously, from [3.101], we find that when spans 1 and 2 are made from the same material at the same age, or the same non-aging material at different ages, the result obtained in section 3.2.3 is retrieved: the reactive force V (t ) and the distribution of bending moments remain constant for t ≥ t0 , since in both cases Jt10 (t ) = J(t − t0 ) = Jt20 (t ).

[3.104]

In addition to this particular circumstance, it is worth finding out under what conditions on J 1 and J 2 this property remains valid, i.e.:

V (t ) = V0 Yt0 (t ).

[3.105]

Putting together [3.101] and [3.105], we find ∀t ≥ t0 , Jt20 =

56V0 − 45 p a 1 Jt 0 , 8V0 − 3 p a

[3.106]

as a necessary condition, which states that J 1 and J 2 must be proportional to each other. This necessary condition is easily proven to be sufficient. When, as evoked initially, the beam is made from the same material (e.g. concrete) with different ages, condition [3.106] is satisfied for any aging material whose creep function depends on the age only through a multiplying factor. This is the case for

Jt0 = ϕ (t0 ) f (t − t0 ) Yt0 ,

[3.107]

which has been proposed as a possible formulation for a viscoelastic constitutive equation for concrete, as in Quagliaroli (2011).

4 Three-dimensional Linear Viscoelastic Modeling

4.1. Multidimensional approach Chapters 1–3, devoted to the one-dimensional approach of viscoelastic behavior, made it relatively easy to introduce the basic concepts of viscoelastic modeling. Simple fundamental uniaxial experiments were presented, namely the creep and relaxation tests, which provide the basic functions that describe the response of the concerned one-dimensional material. Then, within the linearity hypothesis, the response to any loading process history can be written in terms of an integral operator whose kernel is a derivative of one of these functions (taken within the mathematical theory of distribution framework). All these results are established within the small perturbation hypothesis (SPH) that will be retained here throughout the chapter. Considering the example chosen to introduce the creep phenomenon, namely the uniaxial tension test performed on a well-designed homogeneous specimen, it can be interpreted: – either as the creep test identifying the delayed behavior of a one-dimensional rod subjected to tension, as in Chapter 1, section 1.9.1;

Viscoelastic Modeling for Structural Analysis, First Edition. Jean Salençon. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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– or as a test performed on the three-dimensional constituent material subjected to the uniaxial stress tensor history defined by

σ (t ) = σ x0x e x ⊗ e x Yt ,

[4.1]

0

where only the extension ε xx (t0 , t;σ xx ) is recorded as the observable variable, which can be written as 0

ε xx (t0 , t;σ xx0 ) = J (t0 , t;σ xx0 )σ xx0 .

[4.2]

The definition of the converse experiment, which is described by 0 the relaxation function R(t0 , t;ε xx ) , is quite clear from the first point of view; however, when considered from the second point of view, it corresponds to a “mixed” experiment defined by simultaneous stress and strain tensor histories:

ε xx (t ) = ε x0x Yt (t ) , σ (t ) − σ x x (t ) e x ⊗ e x = 0 . 0

[4.3]

A suitable interpretation of the second point of view calls for the description and formulation of the linear viscoelastic behavior of a three-dimensional continuum. 4.2. Fundamental experiments 4.2.1. The three-dimensional continuum framework Shifting from a one-dimensional to a three-dimensional viewpoint, the force variable σ introduced in Chapter 1 (section 1.2.1) is now substituted by the Cauchy stress tensor acting on the material element, which is denoted by σ with components σ i j in an orthonormal basis1. The natural initial state where the stress tensor is zero will be taken as the geometrical reference state (unless stated otherwise).

1 For basic notions regarding tensor calculus, the reader may refer to Appendix 1 in Salençon (2018).

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105

The linearized strain tensor denoted by ε with components ε i j , counted from this reference state, is the geometrical variable associated with the stress tensor (SPH framework). 4.2.2. General definition of the creep and relaxation tests Only isothermal experiments will be considered. Strictly following the same line of reasoning as in Chapter 1, the general definition of a creep test consists in the description of the response of the three-dimensional continuum material element to the force variable history σ (t ) = σ 0 Yt0 (t ) in the form

σ (t ) = σ 0 Yt (t )  ε (t0 , t ;σ 0 ) .

[4.4]

0

Note that this definition confirms the force variable history [4.1] as that of a true creep test from a three-dimensional viewpoint. In the same way as [4.4], a relaxation test (if feasible) is defined as

ε (t ) = ε 0 Yt (t )  σ (t0 , t ; ε 0 ) .

[4.5]

0

4.2.3. The linearity hypothesis With notations similar to those introduced in Chapter 1 (section 1.3), we can write the relationships between the histories of σ and ε in a functional form as: I

t

σ ⎯⎯→ ε ⇔ ∀t , ε (t ) = Ft [σ (τ )] F

[4.6]

−∞

I

t

R ε ⎯⎯ → σ ⇔ ∀t , σ (t ) = R t [ε (τ )] . −∞

[4.7]

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We now assume that the material behavior complies with Boltzmann’s superposition principle as stated in Chapter 1, which means that these functional relationships are linear ∀λ 1 , λ 2 , F ( λ 1 σ ∀λ 1 , λ 2 , R ( λ 1 ε

(1)

(1)

+ λ2 σ + λ2 ε

(2)

( 2)

) = λ 1 F (σ

(1)

) + λ 2 F (σ

(1)

) = λ 1 R (ε ) + λ 2 R (ε

(2)

(2)

).

)

[4.8] [4.9]

4.2.4. Tensorial creep and relaxation functions 4.2.4.1. Tensorial creep function As a result of the linearity hypothesis, ε (t0 , t ; σ 0 ) in [4.4] can be expressed as a linear function of σ 0 through a fourth-rank tensor

J (t0 , t ) in the form2

σ (t ) = σ 0 Yt (t )  ε (t0 , t ;σ 0 ) = J (t0 , t ):σ 0 0

[4.10]

with the explicit meaning3

ε i j (t0 , t ;σ 0 ) = J i j h k (t0 , t ) σ k0h ,

[4.11]

where J (t0 , t ) = 0 for t < t0 . In view of the symmetry of ε (t ) , tensor J (t0 , t ) is symmetric with respect to its two first indices. Moreover, as σ 0 is symmetric, it is convenient, in the same way as in the theory of linear elasticity,

2 The symbol “:” denotes the doubly contracted product of two second-rank tensors. 3 Unless stated otherwise, the “dummy index” convention will be adopted with the meaning that summation is performed on repeated indices.

Three-dimensional Linear Viscoelastic Modeling

107

to adopt for J (t0 , t ) a symmetrical expression with respect to its two last indices. Hence:  J i j h k (t 0 , t ) = J j i h k (t 0 , t )    J i j h k (t0 , t ) = J i j k h (t0 , t ).

[4.12]

Tensor J (t0 , t ) is the creep tensor. 4.2.4.2. Tensorial relaxation function Similarly, when the relaxation experiment is feasible (material with instantaneous elastic behavior), we will write:

ε (t ) = ε 0 Yt (t )  σ (t0 , t ; ε 0 ) = R (t0 , t ):ε 0

[4.13]

σ i j (t0 , t ; ε 0 ) = Ri j h k (t0 , t ) ε k0h

[4.14]

 Ri j h k (t0 , t ) = R j i h k (t0 , t )    Ri j h k (t0 , t ) = Ri j k h (t0 , t ).

[4.15]

0

Tensor R(t0 , t ) is the relaxation tensor. 4.2.5. Instantaneous elasticity Denoted by J (t0 , t0 ) and R(t0 , t0 ) , the values taken by J (t0 , t ) and

R(t0 , t ) for t = t0+ correspond to the instantaneous linear elastic behavior of the material at time t0 . They are respectively the instantaneous elastic compliance tensor and the instantaneous elastic modulus tensor. In addition to the symmetries written in [4.12] and [4.15], the existence of an elastic potential4 implies that these 4 See Salençon (2001).

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tensors are also symmetric with respect to the pairs of indices (i, j) and (h, k):  J i j h k ( t 0 , t 0 ) = J h k i j (t 0 , t 0 )    Ri j h k (t0 , t0 ) = Rh k i j (t0 , t0 ).

[4.16]

Finally, from [4.11] and [4.14] written at time t = t0 , we derive5 1 J i j  m (t0 , t0 ) Rm  h k (t0 , t0 ) = δ i hδ j k = (δ i hδ j k + δ i k δ j h ) 2

[4.17]

which can also be written as

J (t0 , t0 ) : R(t0 , t0 ) = 1 .

[4.18]

4.3. Boltzmann’s formulas 4.3.1. Integral operator Through the same reasoning process as in Chapter 1, the knowledge of the whole catalogue of tensorial creep functions or tensorial relaxation functions of a linear viscoelastic material makes it possible to write down the response to any stress tensor or strain tensor history in the form of an integral equation similar to the one-dimensional Boltzmann’s formula: I

F σ ⎯⎯ → ε ⇔ ∀t , ε (t ) = J (t , t ):σ (t ) − 

t

−∞

I

ε ⎯⎯→ σ ⇔ ∀t , σ (t ) = R (t , t ):ε (t ) −  R

t

−∞

∂J ∂τ ∂R ∂τ

(τ , t ):σ (τ ) dτ

[4.19]

(τ , t ):ε (τ ) dτ . [4.20]

5 The Kronecker symbol is δ i j defined as: δ i j = 1 if i = j , δ i j = 0 if i ≠ j.

Three-dimensional Linear Viscoelastic Modeling

109

Not surprisingly, the integral operator is formally the same as in the one-dimensional modeling but with tensorial notations and doubly contracted products: I

F →ε = − < σ ⎯⎯

I

R →σ = − < ε ⎯⎯

∂J ∂τ

∂R ∂τ

(:) σ > = J (× :) σ

[4.21]

(:) ε > = R (× :) ε

[4.22]

ε = J (× :) σ ⇔ σ = R (× :) ε .

[4.23]

4.3.2. Important identities Putting together equation [4.10], defining the tensorial creep function, and [4.21], we have I

0 0 F σ 0 Yt ⎯⎯ → ε = J t :σ = J (× :) Yt σ 0

0

0

[4.24]

and similarly, from [4.13] and [4.22], we have I

0 0 R ε 0 Yt ⎯⎯ → σ = R t :ε = R (× :) Yt ε , 0

0

0

[4.25]

from which we derive the following identities:

J (×) Yt0 = J t0 R (×) Yt0 = R t0

[4.26] [4.27]

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Viscoelastic Modeling for Structural Analysis

Inversion identities through [4.23] result in

R (× :) J t0 = 1 Yt0 J (× :) R t0 = 1 Yt0

[4.28] [4.29]

4.4. Isotropic linear viscoelastic material 4.4.1. Material symmetries, principle of material symmetries We now assume that the material has material symmetries that can be considered as maintained all along the evolution within the SPH framework. These symmetries are defined through a group of isometries called the material symmetry group G . In the present case, the material symmetry principle implies that the constitutive equation of the material must be invariant under any isometric transformation belonging to G . Physically, this principle may be interpreted as saying: – either that it is not possible to distinguish two elements related by an isometry belonging to G , as they respond exactly in the same way when subjected to the same action; – or, equivalently, that if we apply, to a given element, actions that are related to each other by an isometry in the group G , the corresponding responses are related by the same isometry belonging to G . An immediate consequence of the principle of material symmetries is clearly that, if the group of material symmetries is known, the expression of the constitutive law will have to comply with the corresponding mathematical constraints and therefore be specified. In view of its practical importance and considerable impact on the form of the constitutive law, we will focus on the case of an isotropic material, for which the material symmetry group G is defined as the

Three-dimensional Linear Viscoelastic Modeling

111

group of all isometries (both direct and indirect). For the sake of simplicity, we will take advantage of well-established mathematical results that are commonly implemented in the theory of linear elasticity and therefore refer to that theory. Nevertheless, it is necessary at this stage to make it clear that the viscoelastic analysis cannot, by any means, be straightforwardly transposed from the elastic one. 4.4.2. Isotropic linear viscoelastic material: creep test The general description of the creep test and introduction of the creep tensor given in [4.1] is now revisited in order to comply with the principle of material symmetries. At any instant of time t ≥ t0 , the linear relationship between the second-rank symmetric tensors ε (t0 , t ; σ 0 ) and σ 0 must be invariant under any direct or indirect isometry. We observe that this is exactly the condition that is imposed on the linear elastic constitutive law in the case of an isotropic material. Referring to the general representation theorem about isotropic tensorial functions6, we derive, in the same way as in the elastic case, that the second-rank symmetric tensor ε (t0 , t ; σ 0 ) , which is an isotropic linear function of the second-rank symmetric tensor σ 0 , must be written as:

ε (t0 , t ;σ 0 ) = A(t0 , t ) σ 0 + B (t0 , t ) 1 (tr σ 0 )

[4.30]

where A(t0 , t ) and B (t0 , t ) are scalar material functions of t and 1 denotes the second-rank unit tensor with components δ i j . Implementing this equation in the particular case of the uniaxial tension creep test described in [4.1], we observe from [4.2] that A(t0 , t ) + B(t0 , t ) is the creep function in the uniaxial tension test

6 See, for example, Rivlin (1955).

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Viscoelastic Modeling for Structural Analysis

performed at time t0 on the isotropic material, which is therefore independent of the chosen x -axis:

 σ (t ) = σ x0 x e x ⊗ e x Yt0  .   A(t0 , t ) + B (t0 , t ) = ε xx (t0 , t ;σ xx0 ) σ xx0 = J (t0 , t )

[4.31]

Moreover, [4.30] shows that, in that same creep test, ε (t0 , t;σ xx0 ) reduces to

ε (t0 , t;σ xx0 ) = J (t0 , t )σ xx0 e x ⊗ e x + B(t0 , t ) σ xx0 (e y ⊗ e y + e z ⊗ e z ) . [4.32] Hence, noting that the scalar ratio n (t0 , t ) defined as n (t 0 , t ) = −

ε yy (t0 , t ;σ xx0 ) B (t 0 , t ) Yt (t ) = − 0 ε xx (t0 , t ;σ xx ) J (t 0 , t ) 0

[4.33]

0 is independent of σ xx , we can write [4.30] in the form

ε (t0 , t ;σ 0 ) = (Yt (t ) + n (t0 , t )) J (t0 , t ) σ 0 − n (t0 , t ) J (t0 , t ) 1 (tr σ 0 ) . [4.34] 0

In that equation, n (t0 , t ) is the Poisson ratio determined in the uniaxial tension creep test performed at time t0 :  σ (t ) = σ x0 x e x ⊗ e x Yt0   0 0  J (t0 , t ) = ε xx (t0 , t ;σ xx ) σ xx   n (t0 , t ) = −Yt (t ) ε yy (t0 , t ;σ xx0 ) ε xx (t0 , t ;σ xx0 ). 0 

[4.35]

This test is commonly realized in practice (either in tension or compression).

Three-dimensional Linear Viscoelastic Modeling

113

4.4.3. Isotropic linear viscoelastic material: relaxation test

Referring to the general definition of a relaxation test given in [4.5], using the same arguments as for [4.30], we can state that the response σ (t0 , t ; ε 0 ) must be written in the form

σ (t0 , t ; ε 0 ) = λ (t0 , t ) 1 (tr ε 0 ) + 2 μ (t0 , t ) ε 0 ,

[4.36]

where λ (t0 , t ) and μ (t0 , t ) are scalar material functions of t that can be interpreted as genuine relaxation functions for the three-dimensional continuum by: – the simple shear relaxation test

ε (t ) =

ε0 2

(e x ⊗ e y + e y ⊗ e x ) Yt0 (t )

[4.37]

that yields from [4.36]

μ (t0 , t ) = Yt (t ) σ xy (t0 , t; ε 0 ) ε 0 , 0

[4.38]

which is called the relaxation function in the simple shear test performed at time t0 ; – the simple extension relaxation test

ε (t ) = ε 0 e x ⊗ e x Yt (t ) 0

[4.39]

where λ (t0 , t ) + 2μ (t0 , t ) is the relaxation function in the simple extension test performed at time t0

λ (t0 , t ) + 2μ (t0 , t ) = Yt σ xx (t0 , t; ε 0 ) ε 0 . 0

[4.40]

Practically, the simple shear relaxation test is commonly used to determine μ (t0 , t ) while the simple extension (or compression) test is

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Viscoelastic Modeling for Structural Analysis

unrealistic. The function λ (t0 , t ) may be determined through the isotropic compression relaxation test performed at time t0 that yields the value of 3 λ (t0 , t ) + 2 μ (t0 , t ) :  ε (t ) = Yt0 (t ) ε 0 1  σ (t0 , t ; ε 0 ) = − p (t0 , t ; ε 0 ) 1    3 λ (t0 , t ) + 2 μ (t0 , t ) = −Yt0 p (t0 , t ; ε 0 ) ε 0 .

7

[4.41]

4.4.4. Boltzmann’s formulas

Boltzmann’s formulas given in section 4.3 are now simplified within the context of an isotropic material and become: I

F σ ⎯⎯ → ε = ((Yτ + n) J ) (×) σ − 1 (n J )(×) tr σ

I

R ε ⎯⎯ →σ = 1 λ (×) tr ε + 2 μ (×) ε

[4.42] [4.43]

The similarity between these equations and the classical expressions of the linear elastic constitutive law cannot be missed here, as it is enhanced by the compact notation adopted for the integral operator. It should also be noted that [4.36] and [4.43] use notations that are similar to the linear elastic ones – namely the Lamé coefficient and the shear modulus – which is not the case with [4.34] and [4.42]. This important difference is explained by the relationships between J (τ , t ) , n (τ , t ) on the one side and λ (τ , t ) , μ (τ , t ) on the other side, which can be derived from the inversion process in the following way. With [4.42] applied to the simple shear relaxation test [4.38], we obtain

((Yτ + n) J ) (×) 2 μτ = Yτ ,

[4.44]

7 The Heaviside step function Yτ is systematically introduced here and in other equations further on, although it could sometimes be omitted.

Three-dimensional Linear Viscoelastic Modeling

115

which shows that 2 (Yτ + n (τ , t )) J (τ , t ) is the creep function in the simple shear test denoted by γ (τ , t ) , the inverse of μ (τ , t ) through the integral operator  2 (Yτ + n (τ , t )) J (τ , t ) = γ (τ , t ) .   γ (×) μτ = Yτ .

[4.45]

Similarly, we derive from [4.41] and [4.42]

((Yτ − 2 n) J )(×)(3λτ + 2 μτ ) = Yτ

[4.46]

which proves that ((Yτ (t ) − 2 n (τ , t )) J (τ , t ) is the creep function in the isotropic compression test8. 4.4.5. Uniaxial tension relaxation test 4.4.5.1. Uniaxial tension relaxation function

In the preceding section, we stated that λ (τ , t ) and μ (τ , t ) were genuine relaxation functions according to definition [4.5], while equation [4.3], which describes the converse experiment of the uniaxial tension creep test, corresponds to a “mixed” experiment that we may call the “uniaxial tension relaxation test”. Despite its ambiguous nature, this experiment deserves to be considered in view of its numerous practical applications. Starting from [4.3] and making t0 = τ , we define the function Eτ (t ) = E (τ , t ) , the uniaxial tension relaxation function, by  ε xx (t ) = ε x0 x Yt0 (t ) , σ (t ) − σ x x (t ) e x ⊗ e x = 0    σ xx (t ) = Eτ (t ) ε xx0

8 To be compared to (1 − 2ν ) E in linear elasticity.

[4.47]

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Viscoelastic Modeling for Structural Analysis

which shows that E (τ , t ) is actually R (τ , t ) , the one-dimensional relaxation function that is referred to in section 4.1, as confirmed by Boltzmann’s formula [4.42]:

ε xx0 Yτ = J (×)σ xx = ε xx0 J (×) Eτ  Yτ = J (×) Eτ .

[4.48]

From a three-dimensional viewpoint, considering the evolution of components ε yy (t ) and ε zz (t ) in that same experiment, it is convenient to introduce the corresponding Poisson ratio in the form

ν τ (t ) = −Yτ (t ) ε yy (t ) ε xx0 ,

[4.49]

which results from [4.42] and [4.47] as

ν τ = (n J )(×) Eτ .

[4.50]

Finally, the uniaxial tension relaxation test [4.3] is described by  ε xx (t ) = ε x0x Yt0 (t ) , σ (t ) = σ x x (t ) e x ⊗ e x   0  σ x x (t ) = E (t0 , t ) ε x x   ε (t ) = ε 0 Y (t ) − ν (t , t ) ε 0 (e ⊗ e + e ⊗ e ). 0 x x t0 xx y y z z 

[4.51]

Then, for any “mixed” history defined by

history ε xx (t ) and condition σ (t ) = σ x x (t ) e x ⊗ e x

[4.52]

the response is written as  σ = E (×) ε x x e x ⊗ e x    ε = ε x x e x ⊗ e x − ν (×) ε x x (e y ⊗ e y + e z ⊗ e z ).

[4.53]

Three-dimensional Linear Viscoelastic Modeling

117

Bringing together [4.42] and [4.53], we also obtain, as a “converse” of [4.50]

ν (×) Jτ = (n J )τ .

[4.54]

We may wonder at the introduction of so many functions since it is clearly established that just λ (τ , t ) and μ (τ , t ) , or J (τ , t ) and n (τ , t ) , are necessary and sufficient to completely define the linear viscoelastic behavior of an isotropic three-dimensional continuum. The reason for such “inflation” lies in the implementation of theoretical results in practical applications, where it turns out that the problem at hand will determine the proper choice of a particular set of functions among those presented above depending on the loading process history. 4.4.5.2. Comments

Here, it is important to emphasize a risk of confusion that may result from a notation frequently adopted in practice. More precisely, it happens that the uniaxial tension creep test is described not by means of the corresponding creep function J (t0 , t ) as in [4.2] but through the algebraic inverse of a function unfortunately denoted by E (t0 , t ) in the form

ε xx (t0 , t ;σ xx0 ) =

1 1 (1 + ϕ (t0 , t )) σ xx0 . σ xx0 = E (t 0 , t ) Ec ( t 0 )

[4.55]

This notation may look like a natural straightforward extension of the linear elastic constitutive law expressed in terms of Young’s modulus and Poisson’s ratio in order to take the delayed behavior of the considered material into account. However, it is in fact physically inconsistent. Basically, the experiment Young modulus E referred to in elasticity, although it may be overshadowed, is the counterpart of a relaxation test in viscoelasticity. For practical applications, no confusion arises using [4.55] as long as the only problems to be considered are creep-type problems (Chapter 5, section 5.3.1).

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On the contrary, the notation reveals itself to be misleading when relaxation phenomena are involved, which is quite frequent in practice (Chapter 5, section 5.3.2), and E (t0 , t ) is taken from [4.55] and considered as a relaxation function! In simple words, algebraic inversion should not be confused with inversion with respect to the integral operator. Hence, we consider it advisable to ban notation [4.55] and stick to denoting the uniaxial tension creep function by J (t0 , t ) . 4.4.6. Constant Poisson’s ratio 4.4.6.1. Boltzmann’s formulas

We often find from experimental results that the Poisson ratio in the uniaxial tension relaxation test, as defined by [4.29], can be considered to have a constant value ν throughout the evolution and can be written as

ν τ (t ) = ν Yτ (t ) .

[4.56]

When this hypothesis is valid, the formulas written in the preceding sections can be drastically simplified: – bringing together [4.50] and [4.56] with [4.48], we have

(n J )(×) Eτ = ν Yτ = ν J (×) Eτ

[4.57]

n (τ , t ) = ν Yτ (t ) ,

[4.58]

hence

which means that the Poisson ratio in the uniaxial tension creep test is constant and equal to the Poisson ratio in the uniaxial tension relaxation test;

Three-dimensional Linear Viscoelastic Modeling

119

– with this result, [4.44] becomes

(1 + ν ) J (×) 2 μτ = Yτ

[4.59]

and, with [4.48],

2 μτ = Eτ (1 + ν ) ;

[4.60]

– similarly, from [4.46],

(3 λτ + 2 μτ ) = Eτ (1 − 2ν ) .

[4.61]

In other words, under the assumption that the Poisson ratio is constant, the relaxation functions E (t0 , t ) , λ (t0 , t ) , μ (t0 , t ) and Poisson’s ratio ν Y (t0 , t ) are linked to each other by the same algebraic relationships as in isotropic linear elasticity. Moreover, Boltzmann’s formula [4.42] now takes the form I

F σ ⎯⎯ → ε = (1 + ν ) J (×) σ − 1ν J (×)(tr σ ) ,

[4.62]

which, taking [4.48] into account, can also be written as

E (×) ε = (1 + ν ) σ −ν (tr σ ) 1 .

[4.63]

4.4.6.2. Comments

Under the initial hypothesis of a constant Poisson ratio, whose value ν is obviously determined from the instantaneous elastic behavior of the material, the isotropic linear viscoelastic law of the material is defined by only knowing the uniaxial tension creep or relaxation function, J (τ , t ) or E (τ , t ) . Equations [4.62] and [4.63] are transposed from their isotropic linear elastic counterparts by substituting the integral operator (×) for the algebraic product.

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4.5. Non-aging linear viscoelastic material 4.5.1. Boltzmann’s formulas

Assuming the material to be non-aging results in the same mathematical properties on functional relationships F and R as in Chapter 1: t +λ t   Ft+ λ [σ λ (τ )] = Ft [σ (τ )]  −∞ −∞  t +λ t   Rt+ λ [ε λ (τ )] = Rt [ε (τ )].  −∞ −∞

[4.64]

Hence  J (t0 , t ) = f (t − t0 )    f (τ ) = 0 if τ < 0 

[4.65]

 R (t0 , t ) = r (t − t0 )    r (τ ) = 0 if τ < 0. 

[4.66]

and

It follows that the integral operator (× :) in [4.23] can now be expressed in terms of a Riemann convolution product within the framework of the distribution theory

ε = f ′ (∗ :)σ = f (∗ :)σ ′ ⇔ σ = r ′ (∗ :) ε = r (∗ :) ε ′

[4.67]

Three-dimensional Linear Viscoelastic Modeling

121

Identities [4.28] and [4.29] become

f ′ (∗ :) r = f (∗ :) r ′ = 1 Y

[4.68]

as a result,

f (0): r (0) = 1

[4.69]

and assuming that f (τ ) and r (τ ) have finite limits when τ → ∞ ,

f (∞): r (∞) = 1 .

[4.70]

4.5.2. Operational calculus

Operational calculus can be implemented using the Laplace–Carson transforms (LCT) of tensor functions, which reduce equations [4.67] and [4.68] to simple algebraic tensor equations:

ε ∗ ( p) = f ∗ ( p ):σ ∗ ( p) ⇔ σ ∗ ( p ) = r ∗ ( p ): ε ∗ ( p ) ∗

∗

[4.71]

f ( p): r ( p) = 1 . These equations are formally identical (but for the adopted notations) to the equations that express the linear elastic constitutive law. Laplace–Carson transforms ε ∗ ( p ) and σ ∗ ( p ) are symmetric ∗

second-rank tensors while f ( p ) and r ∗ ( p ) are fourth-rank tensors whose symmetries are those of J (τ , t ) and R(τ , t ) described in [4.12] and [4.15]. We must note that the symmetry between the pairs of indices (i, j ) and (h, k ) , which could not be evoked in [4.12] and [4.15], can be written for non-aging materials as a result of Onsager’s reciprocity principle. Finally:

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Viscoelastic Modeling for Structural Analysis

 f i ∗j h k ( p ) = f j∗i h k ( p )   ∗ ∗  fi j h k ( p) = fi j k h ( p)   f ∗ ( p) = f ∗ ( p) hki j  i j hk

[4.72]

 ri∗j h k ( p ) = rj∗i h k ( p )   ∗ ∗  ri j h k ( p ) = ri j k h ( p )   r ∗ ( p) = r ∗ ( p) hk i j  i jhk

[4.73]

and [4.71] is the exact counterpart of the linear elastic constitutive law where the variable p plays the role of a parameter. 4.5.3. Isotropic material 4.5.3.1. Boltzmann’s formulas

When the material is isotropic, Boltzmann’s formulas are straightforwardly derived from [4.42] and [4.43] using [4.67] I

F → ε = ((Yτ + n) J ) ∗σ ′ − 1 (n J ) ∗ (tr σ ′) σ ⎯⎯

I

R ε ⎯⎯ →σ = 1 λ ∗ (tr ε ′) + 2 μ ∗ε ′

[4.74] [4.75]

with corresponding inversion formulas  ((Y + n) J ) ∗ 2 μ ′ = γ ∗ μ ′ = Y τ τ τ  τ   ((Yτ − 2 n) J ) ∗ (3 λτ ′ + 2 μτ ′ ) = Yτ   J ∗ E ′ =Y τ τ    ν τ = ( n J ) ∗ Eτ ′

[4.76]

Three-dimensional Linear Viscoelastic Modeling

123

which make it sometimes possible to directly solve the problems in the convolution algebra. Using the LCT of the functions defined in section 4.4.5, Boltzmann’s formulas become formally identical to the classical equations obtained in isotropic linear elasticity9

ε* =

1 +ν * ν* σ *− tr σ * 1 E* E*

σ * = λ *tr ε * 1 + 2 μ *ε *

[4.77] [4.78]

4.5.3.2. Constant Poisson’s ratio

If the Poisson ratio can be assumed to be constant

ν τ (t ) = ν Yτ (t ) , ν * ( p) = ν

[4.79]

equation [4.77] ultimately simplifies to

ε* =

1 +ν ν σ *− tr σ * 1 , E* E*

[4.80]

with E*   2 μ* = 1 + ν    3 λ * +2 μ * = E * .  1 + 2ν 

9 The variable p plays the role of a parameter.

[4.81]

5 Quasi-static Linear Viscoelastic Processes

5.1. Quasi-static linear viscoelastic processes 5.1.1. Isothermal quasi-static processes Figure 5.1 presents a schematic outline of the concept of mechanical evolution, where a system S is defined in an initial equilibrium state by a set of initial data and then subjected to a loading process characterized by given histories of geometrical variables and applied forces. The problem to be solved is how to determine the current state of the system at any instant of time along the loading process.

Figure 5.1. Evolution of a mechanical system

Viscoelastic Modeling for Structural Analysis, First Edition. Jean Salençon. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Viscoelastic Modeling for Structural Analysis

As in the preceding chapters, we stay within the small perturbation hypothesis (SPH), with respect to its entire meaning, including the fact that displacements are considered small enough to allow all equations to be written regarding the initial geometry along the whole loading process. Within this context, denoting respectively by Ω and ∂Ω the volume and boundary of S defined in the initial equilibrium state that is taken as a reference (Figure 5.2), the data at any instant of time consist of1: – the history of applied body forces F ( x, t ) given in Ω d

F ( x, t ) = F ( x, t );

[5.1]

– the history of the boundary data on ∂Ω that can be described as follows. At any instant of time, at the current point M with position vector x on ∂Ω , the boundary data take the form of three prescribed orthogonal components of the stress vector T ( x, t ) and the displacement vector ξ ( x, t ) . Most frequently, the corresponding orthogonal directions lie along the normal and in the plane tangent to the boundary ∂Ω at point M , as shown in Figure 5.2.

Figure 5.2. Boundary data

1 The upper index “d” denotes given quantities (data).

Quasi-static Linear Viscoelastic Processes

127

Let Sξi (t ) and STi (t ) , i = 1,2,3 , with Sξi (t ) ∪ STi (t ) = ∂Ω and

Sξi (t ) ∩ STi (t ) = ∅ respectively denote the portions of the boundary d

d

where ξ i ( x, t ) and Ti ( x, t ) are prescribed as ξi ( x, t ) and Ti ( x, t ) respectively. We write the boundary data in the form:  ξi ( x, t ) = ξi d ( x, t ) on Sξ (t ) i   d  Ti ( x, t ) = Ti ( x, t ) on STi (t )   Sξ (t ) ∪ ST (t ) = ∂Ω i  i  S (t ) ∩ S (t ) = ∅ . Ti  ξi

[5.2]

Only quasi-static evolutions will be considered, meaning that throughout the loading process, the system is assumed to be in a state of (quasi-)equilibrium, where no inertia terms or propagation phenomena must be taken into account. This obviously imposes conditions on the data, both initially and throughout the loading process. First, data concerning the applied body and boundary forces d d F ( x, t ) and Ti ( x, t ) should be compatible with the system global equilibrium, i.e. with the equation

[ Fe (t )] = 0,

[5.3]

where [ Fe (t )] represents the wrench2 of all external forces exerted on the system at time t . Then, the loading process itself must be sufficiently slow to allow inertia phenomena to be considered as negligible. Under these conditions, the field equations of motions can be substituted by the field equilibrium equations. Finally, only isothermal evolutions will be considered.

2 See Salençon (2018).

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5.1.2. Isothermal quasi-static linear viscoelastic processes The natural initial state of the system, where the stress field is zero over Ω , is taken as the mechanical state of reference with the field d d and boundary force data F and Ti equal to zero. To complete the set of equations defining the process, we assume that the system is made from a linear viscoelastic material whose constitutive equation can be written in the form established in Chapter 4:

ε = J (× :) σ ⇔ σ = R (× :) ε .

[5.4]

Tensors J and R depend on the spatial variable x if the system is not homogeneous. It is also assumed that, during the considered loading process, boundary conditions retain the same pattern, which means that, as allowed by the SPH, Sξi (t ) and STi (t ) do not depend on the time variable:

Sξi (t ) = Sξi , STi (t ) = STi .

[5.5]

In summary, the equations governing the problem consist of: – over ∂Ω : boundary conditions [5.2] with [5.5]; – in Ω : constitutive equation [5.4]; – in Ω : definition of the linearized strain tensor

ε ( x, t ) = (grad ξ ( x, t ) + t grad ξ ( x, t )) 2;

[5.6]

– in Ω : field equation of equilibrium d

divσ ( x, t ) + ρ ( x) F ( x, t ) = 0.

[5.7]

Quasi-static Linear Viscoelastic Processes

129

Note that, consistently with the SPH, the mass per unit volume of the material, ρ ( x, t ) , is considered as independent of t in [5.7]: ρ ( x, t ) = ρ ( x ) . 5.1.3. Superposition principle This set of equations defines a linear problem. It follows that the relationship that exists between a solution to the problem in the form of stress and displacement field histories σ ( x, t ) and ξ ( x, t ) on the d

d

d

one side, and data histories F ( x, t ) , ξi ( x, t ) , Ti ( x, t ) , Sξi and STi on the other, is linear. As a result, the superposition principle now holds at the global level of the system and can be stated as follows: (1)

(1)

let σ ( x, t ) , ξ ( x, t ) be a solution to the problem with data d

d

d

F ( x, t )(1) , ξi ( x, t )(1) and Ti ( x, t )(1) , let σ d

(2)

(2)

( x, t ) , ξ ( x, t ) be a solution to the problem with data d

d

F ( x, t )(2) , ξi ( x, t )(2) and Ti ( x, t )(2) , (1)

then λ (1) σ ( x, t ) + λ (2) σ

(2)

(1)

( x, t ) , λ (1) ξ ( x, t ) + λ (2) ξ d

(2)

( x, t ) is a d

solution to the problem with data λ (1) F ( x, t )(1) + λ (2) F ( x, t )(2) ,

λ (1)ξi d ( x, t )(1) + λ (2)ξi d ( x, t )(2) and λ (1)Ti d ( x, t )(1) + λ (2)Ti d ( x, t )(2) . Obviously, this principle holds only as long as the hypotheses it is based on remain satisfied. 5.1.4. Loading parameters, kinematic parameters 5.1.4.1. Loading parameters Let us consider the particular case when the boundary data on the displacement are fixed to zero while the field and boundary data on the forces depend linearly on n scalar parameters Q j (t ) , the components of a loading vector Q (t ) ∈  n

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Viscoelastic Modeling for Structural Analysis

ξi d ( x, t ) = 0 on Sξ , i

d

d

Ti ( x, t ) on STi and F ( x, t ) depend linearly on Q (t )

[5.8]

and let q (t ) ∈  n denote the associated kinematic vector such that

Q (t ).q (t ) represents the work by external forces. A creep-type evolution problem for such a system is defined by the data history 0

Q (t ) = Q Yt0 (t ),

[5.9]

for which it follows from the superposition principle that the global response of the system can be written as 0

q (t ) = J (t0 , t ).Q ,

[5.10]

which defines the creep tensor of the system within this loading parameter framework. Hence, whatever may be the data history in the form [5.8], the history of parameter q can be written as3:

q = J (×.) Q.

[5.11]

5.1.4.2. Kinematic parameters

Similarly, we consider the case when the force data are fixed to zero and the displacement data depend linearly on n scalar parameters q j (t ) , the components of a kinematic vector q (t ) ∈  n , with Q (t ) being the associated loading vector such that Q (t ).q (t ) represents the work by external forces:

3 (×.) denotes the integral operator within the loading parameter framework.

Quasi-static Linear Viscoelastic Processes

d

131

d

Ti ( x, t ) = 0 on STi , F ( x, t ) = 0,

[5.12]

ξi d ( x, t ) on Sξ depend linearly on q (t ) ∈  n . i

A relaxation-type evolution problem is then defined by 0 q (t ) = q Yt0 (t ) and the response of the system within this framework can be written as 0

0

q (t ) = q Yt0 (t )  Q (t ) = R (t0 , t ).q ,

[5.13]

where R (t0 , t ) is the relaxation tensor of the system within the kinematic parameter framework. Hence, for any data history in the form [5.12], the response of the system can be written as:

Q = R (×.) q. 5.2. Solution to evolution problem

[5.14] the

linear

viscoelastic

quasi-static

5.2.1. Statically admissible stress histories, kinematically admissible displacement histories

Figure 5.3. Schematic view of the quasi-static evolution problem

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A solution to the quasi-static evolution problem defined in the preceding section consists of a stress history σ and a displacement history ξ related to each other by the constitutive equation [5.4], which satisfy equations [5.2] and [5.4] to [5.7]. This is schematically represented in Figure 5.3 with the following definitions: d

d

– S ( F , STi , Ti ) is the set of statically admissible stress histories for the problem, such that

 div σ ( x, t ) + ρ ( x) F d ( x, t ) = 0  σ ∈ S ( F , STi , Ti ) ⇔ ∀t ≥ t0 ,  [5.15]  Ti ( x, t ) = Ti d ( x, t ) on ST i  d

d

d

– C ( Sξi , ξi ) is the set of kinematically admissible displacement histories for the problem, such that

ξ ∈ C ( Sξ , ξi d ) ⇔ ∀t ≥ t0 , ξi ( x, t ) = ξi d ( x, t ) on Sξ . i

i

[5.16]

Mathematical regularity conditions that must be satisfied by the corresponding stress and displacement fields are similar to those encountered in the theory of linear elasticity regarding the spatial variable; piecewise continuity and continuous differentiability is assumed with respect to the time variable. 5.2.2. Solution methods

As in the case of thermoelastic equilibrium problems, no purely deductive systematic method can be proposed to lead from a set of data ([5.1] and [5.2]) to a solution. The same two approaches that are classically referred to when solving thermoelastic equilibrium problems can be transposed on to the present case. They both appeal to intuition and acquired experience. 5.2.2.1. Displacement history method

Referring to Figure 5.3, we may present the displacement history method saying that it treats the displacement history as the principal

Quasi-static Linear Viscoelastic Processes

133

unknown. It starts from an intuitive idea about a possible form for a kinematically admissible displacement history ξ to be a solution and then, going “clockwise” through the schematic outline in Figure 5.3, ends up specifying the form to be retained for ξ allowing the corresponding stress history σ be statically admissible:

ξ ∈ C (Sξ , ξi d )  σ ∈ S ( F d , ST , Ti d ). i

[5.17]

i

5.2.2.2. Stress history method

Similarly, the stress history method starts from a possible form for a statically admissible stress history σ to be a solution and then goes “anticlockwise” through the outline in Figure 5.3 up to a displacement history ξ . It first states that at any instant of time, the strain field

ε ( x, t ) must be geometrically compatible in order to be derived from a displacement field ξ ( x, t ) and that the displacement history ξ must be kinematically admissible, which ends up specifying the form to be retained for σ :

σ ∈ S ( F d , ST , Ti d )  ξ ∈ C ( Sξ , ξi d ). i

5.3. Homogeneous Poisson’s ratio

[5.18]

i

isotropic

material

with

constant

This section is devoted to the specific analysis of the quasi-static evolution problem when the considered system is homogeneous and made from an isotropic linear viscoelastic material with constant Poisson’s ratio denoted by ν , whose constitutive law [5.4] takes the reduced form established in Chapter 4 (section 4.4.6): I

F σ ⎯⎯ → ε = (1 + ν ) J (×) σ − 1ν J (×)(tr σ ),

[5.19]

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where J (τ , t ) is the creep function determined in the uniaxial tension test. It should be recalled that, denoting the uniaxial tension relaxation function as E (τ , t ) , [5.19] can also be written as

E (×) ε = (1 + ν ) σ −ν (tr σ ) 1.

[5.20]

5.3.1. Creep-type problems, creep-type evolutions 5.3.1.1. General creep-type problem

The general creep-type problem is defined as per section 1.2 with the specific data histories:  i = 1, 2,3   ξ d ( x, t ) = 0 on S ξi  i  d d (0)  Ti ( x, t ) = Ti ( x) Yt0 (t ) on STi   F d ( x, t ) = F d ( x)(0) Y (t ) in Ω . t0 

[5.21]

The solution to this problem is obtained through the stress history 0 0 method from the instantaneous elastic solution at time t0 . Let σ , ξ denote the stress and displacement field solution to the linear elastic 1 being the equilibrium problem, with E (t0 , t0 ) = R (t0 , t0 ) = J (t 0 , t 0 ) Young modulus. Referring to Figure 5.3, it is easy to verify that the solution to the creep-type problem defined by [5.21] is given by

 σ ( x, t ) = σ 0 ( x) Yt (t ) 0    ξ ( x, t ) = ξ 0 ( x) J (t0 , t ) J (t0 , t0 ). 

[5.22]

It means that the internal forces in the system are constant throughout the quasi-static loading process while displacements and strains follow the rule of the uniaxial tension creep function.

Quasi-static Linear Viscoelastic Processes

135

5.3.1.2. Creep-type problem within the loading parameter framework

In the case of loading parameters (section 5.1.4.1), for the 0 creep-type evolution problem defined by Q (t ) = Q Yt0 (t ) , the instantaneous elastic response at time t = t0 can now be written as 0

0

q (t0 ) = J (t0 , t0 ).Q = J (t0 , t0 ) Λ .Q , where

Λ

is

a

J (t0 , t0 ) = J (t0 , t0 ) Λ

purely

geometric

[5.23] symmetric

matrix

and

is nothing but the symmetric matrix of

instantaneous elastic compliances for the system at time t0 within the loading parameter framework. Then, for t ≥ t0 , equation [5.10] takes the form 0

q (t ) = J (t 0 , t ) Λ . Q =

J (t 0 , t ) q (t0 ). J (t 0 , t 0 )

[5.24]

From [5.23], we also derive the solution to any creep-type evolution problem defined by the history of the loading vector Q

q = J (×.) Q = J (×) Λ .Q.

[5.25]

5.3.2. Relaxation-type problems, relaxation-type evolutions 5.3.2.1. General relaxation-type problem

We consider the general relaxation-type problem defined by  i = 1, 2,3   ξ d ( x, t ) = ξ d ( x)(0) Y (t ) on S i t0 ξi  i  d  Ti ( x, t ) = 0 on STi   d  F ( x, t ) = 0 in Ω .

[5.26]

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The solution to this problem is obtained by the displacement history method from the instantaneous elastic solution at time t0 . With

σ 0 , ξ 0 the stress and displacement field solution to the linear elastic equilibrium problem where E (t0 , t0 ) is the Young modulus of the material, the solution to problem [5.26] is simply

 ξ ( x, t ) = ξ 0 ( x) Yt (t ) 0    σ ( x, t ) = σ 0 ( x) E (t0 , t ) E (t0 , t0 ). 

[5.27]

The displacement and strain fields remain constant throughout the loading process while the stress field follows the rule of the uniaxial tension relaxation function. 5.3.2.2. Relaxation-type problem within the kinematic parameter framework

With kinematic parameters (section 5.1.4.2), the relaxation-type 0 problem is defined by q (t ) = q Yt0 (t ) and the instantaneous elastic response at time t = t0 can be written as: 0

0

Q (t0 ) = R (t0 , t0 ).q = E (t0 , t0 ) A.q .

[5.28]

In this equation, A is a purely geometric symmetric matrix and

R (t0 , t0 ) = E (t0 , t0 ) A is the symmetric matrix of instantaneous elastic moduli for the system at time t0 within the kinematic parameter framework. For t ≥ t0 , equation [5.13] can now be written as 0

Q (t ) = R ( t 0 , t ) A . q =

R (t 0 , t ) Q (t 0 ) R (t 0 , t 0 )

[5.29]

and the solution to any relaxation-type evolution problem defined by the history of the kinematic vector takes the form

Q = R (×.) q = E (×) A.q.

[5.30]

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137

5.3.3. Mixed data problems

We now consider the problems that are stated in the general form through equations [5.2] and [5.4] to [5.7], with the natural initial state of the system being taken as the geometrical and mechanical state of reference. From the superposition principle in linear elasticity, the instantaneous elastic response of the system at time t0 is written as the result of the sum of: a) the solution to the linear elastic problem with force data equal to zero and kinematic data equal to the prescribed ones at time t0 and denoted by ξi

d (0)

;

b) the solution to the linear elastic problem with kinematic data equal to zero and force data equal to the prescribed ones and denoted by Ti

d (0)

and F

d (0)

.

The displacement and stress field solution to problem a) can be written in the form (0)

(0)

ξ k (t0 ) = Ξ k (ξi d ;ν ) and σ k (t0 ) = E (t0 ) Σ k (ξi d ;ν ), where the fields Ξ k (ξ i depend linearly on ξi

d (0)

d (0)

;ν ) and Σ k (ξ i

d (0)

[5.31]

;ν ) are functions of ν and

.

Similarly, the displacement and stress field solution to problem b) can be written as

ξ s (t 0 ) =

(0) (0) (0) (0) 1 Ξ s (Ti d , F d ;ν ) and σ s (t0 ) = Σ s (Ti d , F d ;ν ) [5.32] E (t 0 )

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where the fields Ξ s (Ti

d (0)

,F

d (0)

;ν ) and Σ s (Ti

functions of ν and depend linearly on Ti

d (0)

d (0)

and F

,F

d (0)

d (0)

;ν ) are the

.

Hence, formally in terms of displacement and stress fields, the instantaneous elastic response of the system to the problem stated by [5.2] and [5.4] to [5.7] with data ξi time t0 can be written as:

d (0)

( x), Ti

d (0)

( x) and F

d (0)

( x) at

1  d (0) d (0) d (0)  ξ (t0 ) = Ξ k (ξi ;ν ) + E (t ) Ξ s (Ti , F ;ν ) 0   d (0) d (0) d (0)  σ (t0 ) = E (t0 ) Σ k (ξi ;ν ) + Σ s (Ti , F ;ν ).

[5.33]

Recalling condition [5.5] on surfaces Sξi (t ) and STi (t ) , the d

solution to the evolution problem for t > t0 with data ξi ( x, t ) , d

d

Ti ( x, t ) and F ( x, t ) is written as the result of the sum of: – the solution to the relaxation-type evolution problem associated with problem a), defined by  i = 1, 2,3   ξ ( x, t ) = ξ d ( x, t ) on S i ξi  i   Ti ( x, t ) = 0 on STi   F ( x, t ) = 0 in Ω 

[5.34]

which consists of the displacement and stress fields

ξ k = Ξ k (ξi d ;ν ), σ k = E (×) Σ k (ξi d ;ν );

[5.35]

– the solution to the creep-type evolution problem associated with problem b), defined by

Quasi-static Linear Viscoelastic Processes

 i = 1, 2,3   ξi ( x, t ) = 0 on Sξi   d  Ti ( x, t ) = Ti ( x, t ) on STi   F ( x, t ) = F d ( x, t ) in Ω , 

139

[5.36]

which is written as

ξ s = J (×) Ξ s (Ti d , F d ;ν ), σ s = Σ s (Ti d , F d ;ν ). d

[5.37] d

Finally, the response to the histories F ( x, t ) , Ti ( x, t ) and

ξi d ( x, t ) can be formally written as  ξ = Ξ k (ξi d ;ν ) + J (×) Ξ s (Ti d , F d ;ν )    σ = Σ (Ti d , F d ;ν ) + E (×) Σ (ξi d ;ν ), s k 

[5.38]

where ν is a constant. 5.3.4. Comments

When introduced in Chapter 4 (section 4.4.6), the constant Poisson ratio assumption was considered somewhat anecdotic, although it turns out to be often realistic in practice. In fact, its core significance is that the viscoelastic behavior of the considered isotropic material is then governed by one and only one function of time, namely the uniaxial tension creep function or, conversely, the uniaxial tension relaxation function. This explains the similarity that may be observed between the general results presented above and the particular solutions to the typical cases studied in Chapter 3. Furthermore, as in the one-dimensional framework seen in Chapter 3, practical problems are often concerned with loading processes that consist of a sequence of creep-type and relaxation-type evolutions.

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Viscoelastic Modeling for Structural Analysis

The superposition principle is an efficient tool to build up solutions in such cases. 5.4. Non-aging linear viscoelastic material 5.4.1. Correspondence principle

Without any homogeneity or specific material symmetry properties, we now assume that the isothermal quasi-static process defined in section 5.1.1 concerns a non-aging linear viscoelastic material. Recalling that J (t0 , t ) = f (t − t0 ) and R (t0 , t ) = r (t − t0 ) for such a material, the constitutive law can be written as tensorial Riemann’s convolution products

ε = f ′ (∗ :) σ = f (∗ :) σ ′ ⇔ σ = r ′ (∗ :) ε = r (∗ :) ε ′.

[5.39]

Using the Laplace–Carson transforms (LCT), these equations yield

ε ∗ ( p ) = f ∗ ( p ):σ ∗ ( p) ⇔ σ ∗ ( p) = r ∗ ( p): ε ∗ ( p) ∗

∗

[5.40]

f ( p ): r ( p ) = 1 ,

which, as noted previously, is the exact counterpart of the linear elastic constitutive law with the variable p playing the role of a parameter. Taking into account the SPH framework and assumption [5.5] about the boundary conditions, the LCT may also be applied to the field and boundary equations of the problem: – from [5.6]

ε ∗ ( x, p ) = (grad ξ ∗ ( x, p ) + t grad ξ ∗ ( x, p )) 2, – from [5.7]

[5.41]

Quasi-static Linear Viscoelastic Processes

∗

d∗

div σ ( x, p ) + ρ ( x) F ( x, p ) = 0,

141

[5.42]

– from [5.2] and [5.5]  ξ ∗ ( x, p) = ξ d ∗ ( x, p) on S i ξi  i   T ∗ ( x, p ) = T d ∗ ( x, p ) on S , i Ti  i

[5.43]

– while [5.40], taking the possible heterogeneity of the material into account, can be written as  ε ∗ ( x, p) = f ∗ ( x, p):σ ∗ ( x, p)   ∗ ∗ ∗  σ ( x, p) = r ( x, p): ε ( x, p). 

[5.44]

This set of equations, with p as a parameter, defines a linearized ∗

elastic equilibrium problem where the tensor functions f ( x, p ) and ∗

r ( x, p) play the role of the the tensors of elastic compliances and

elastic moduli respectively, with the same symmetries as in linear elasticity4:

 fi ∗j h k ( x, p) = f j∗i h k ( x, p )  ri∗j h k ( x, p ) = rj∗i h k ( x, p )    ∗  ∗ ∗ ∗  fi j h k ( x, p) = f i j k h ( x, p ) and  ri j h k ( x, p ) = ri j k h ( x, p ) [5.45]    fi ∗j h k ( x, p) = f h∗k i j ( x, p )  ri∗j h k ( x, p ) = rh∗k i j ( x, p ).   It follows that the functions of the parameter p that are the solution to the linear elastic equilibrium problem set by equations [5.40] to [5.44] simply provide the solution to the original evolution problem, using the inversion of the Laplace–Carson transforms. 4 See Chapter 4, section 5.2.

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This result is known as the Lee–Mandel correspondence principle (Lee 1995; Mandel 1955). 5.4.2. Comments

Formally, the Lee–Mandel correspondence principle is a spectacular result. From a practical viewpoint, it may turn out that passing from the algebraic explicit solution to the linear elastic equilibrium problem, to the solution of the viscoelastic evolution problem with t as a variable using the inversion of the LCT proves quite difficult. Furthermore, as has been highlighted in the previous chapters, “jumping” onto the LCT before or without analyzing the physical nature of the phenomena involved during the loading process may lead to misunderstandings when algebraic calculations substitute physical convolution products. In fact, it often turns out, at least at the first stage of the analysis of a problem, that assumptions necessary to implement the methods described in sections 5.2 and 5.3, such as homogeneity, isotropy and a, constant Poisson’s ratio, can be considered as acceptable and thus make it possible to derive an explicit solution in terms of the integral operator.

6 Some Practical Problems

6.1. Presentation This brief chapter presents some examples of quasi-static linear viscoelastic processes (as defined in Chapter 5) within the SPH framework, which are frequently referred to in engineering practice. These amount to a three-dimensional vision of the problems that were previously evoked or analyzed in Chapters 1 and 3 from a one-dimensional viewpoint. The constituent material of the considered systems is supposed to be homogeneous and isotropic. With the exception of section 6.5, the initial state of the system, which is taken as a geometrical and mechanical reference, is supposed to be the “natural” state with the meaning that no external forces are acting and the stress field is uniformly zero. 6.2. Uniaxial tension–compression of a cylindrical rod 6.2.1. Statement of the problem Figure 6.1 shows a schematic representation of a uniaxial tension–compression quasi-static loading process for a cylindrical rod with length  , cross-section S and lateral cylindrical boundary S L . The data are specified as listed in Chapter 5:

Viscoelastic Modeling for Structural Analysis, First Edition. Jean Salençon. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Figure 6.1. Uniaxial tension–compression of a cylindrical rod

– no body forces d

F ( x, t ) = 0 in Ω

[6.1]

– boundary conditions  ξ x ( x, t ) = ξ x d (t ) on S   ξ d ( x, t ) = 0 on S 0  x  d  T ( x, t ) = 0 on S L  d  d  Ty ( x, t ) = Tz ( x, t ) = 0 on S0 and S .

[6.2]

This problem depends on one kinematic parameter, namely d q (t ) = ξ x (t ) , the given uniform value of the displacement x-component over S . The associated loading parameter Q (t ) is the component along Ox of the resultant of the external forces exerted on S :

 q (t ) = ξ x d (t )    Q (t ) = S σ xx (, t ) da = N (t ).  

[6.3]

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145

d

We now consider the relaxation-type problem where ξ x (t ) is specified as:

ξ x d (t ) = q 0 Yt (t ) . 0

[6.4]

6.2.2. Solution As in Chapter 5 (section 5.3.2.1), the solution to this problem is derived from its elastic counterpart and obtained through the stress history method. The instantaneous elastic solution at time t = t0 is a classical result

z  0 x 0 y  ξ ( x, t0 ) = q  e x − ν (t0 , t0 ) q (  e y +  e z )   q0 q0 = ⊗ − ε ( , ) ν ( , ) (e y ⊗ e y + e z ⊗ e z ) x t e e t t  0 0 0 x x     0  σ ( x , t ) = E (t , t ) q e ⊗ e , 0 0 0 x x  

[6.5]

from which we can write the solution to the relaxation-type problem [6.4] in the form z  0 x 0 y  ξ ( x, t ) = q  e x − ν (t0 , t ) q (  e y +  e z )   q0 q0 e x ⊗ e x − ν (t0 , t ) (e y ⊗ e y + e z ⊗ e z )  ε ( x, t ) =     0  σ ( x, t ) = E (t , t ) q e ⊗ e . 0 x x  

[6.6]

The second and third lines in [6.6] describe a uniaxial tension relaxation test (Chapter 4, section 4.4.5) performed on material elements in Ω . It follows that, referring to equation [4.53] in

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Viscoelastic Modeling for Structural Analysis

Chapter 4, we can write the response to any relaxation-type evolution d defined by a loading process ξ x (t ) = q (t ) as: x y z   ξ = q  e x − ν (×) q (  e y +  e z )   q q ε = e x ⊗ e x − ν (×) (e y ⊗ e y + e z ⊗ e z )     q   σ = E (×)  e x ⊗ e x   q  Q = S E (×) .  

[6.7]

The last equation in [6.7] reveals that R (τ , t ) , the relaxation function of the rod subjected to tension–compression, within the kinematic parameter framework, is proportional to the material’s uniaxial tension relaxation function E (τ , t ) through the geometrical parameter S  : Q = R (×) q , R (τ , t ) =

S E (τ , t ) . 

[6.8]

Using the inversion formula

J (×)Rτ = Yτ

[6.9]

we define J (τ , t ) , which is simply derived from J (τ , t ) , the material’s uniaxial tension creep function, as J (τ , t ) =

 J (τ , t ). S

[6.10]

Then, formally,

q = J (×) Q

[6.11]

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147

denotes the inverse relationship of [6.8] and expresses the response of the rod to any creep-type evolution where the loading process is prescribed by Q (t ) = N (t ) . The function J (τ , t ) is therefore called the creep function of the rod in tension–compression although, from a three-dimensional continuum mechanics viewpoint, this does not define a well-posed problem. This issue will be addressed and commented on in section 6.4.1, with reference to SaintVenant’s problem, “mixt” method and principle. Taking into account equations [4.48] and [4.54] in Chapter 4, the displacement, strain and stress fields corresponding to [6.11] are written as

x 1   ξ = S J (×) Q e x − S (n J ) (×) Q ( y e y + z e z )   1 1 ε = J (×) Q e x ⊗ e x − (n J ) (×) Q (e y ⊗ e y + e z ⊗ e z ) [6.12] S S    σ = Q ex ⊗ ex .  S These equations obviously become much simpler when the Poisson ratio is constant! As in the theory of strength of materials, equations [6.8] and [6.11] are taken as constitutive viscoelastic equations relating the normal force to the longitudinal deformation of a one-dimensional curvilinear element subjected to tension–compression (Chapter 1, section 1.9.2.1): N   ε = J (×) N = J (×) S    N = R (×) ε = S E (×) ε .

[6.13]

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Viscoelastic Modeling for Structural Analysis

6.3. Bending of a cylindrical rod 6.3.1. Statement of the problem

Figure 6.2. Bending of a cylindrical rod about a principal axis of inertia

Using Figure 6.2 and notations similar to the previous ones, we now investigate the loading process where the cylindrical rod is subjected to normal bending about a principal axis of inertia of its cross-section chosen as the z-axis, with I z denoting the corresponding moment of inertia. The problem is stated with the following data: – no body forces d

F ( x, t ) = 0 in Ω

[6.14]

– boundary conditions

 ξ x ( x, t ) = − χ d (t )  y on S   ξ d ( x, t ) = 0 on S 0  x  d  T ( x, t ) = 0 on S L   T d ( x, t ) = T d ( x, t ) = 0 on S and S . 0 z   y

[6.15]

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149

The problem depends on one kinematic parameter, namely q (t ) = χ d (t )  , the rigid body rotation about the z-axis of the plane of the end section S . The associated loading parameter Q (t ) is the moment about the z-axis in S of the external forces exerted on the end section:  q (t ) = χ d (t )     Q (t ) = S − y σ xx (, t ) da = M z (t ).  

[6.16]

With these data, we consider the relaxation-type problem defined by the loading process

ξ x d (t ) = −q 0 yYt (t ) . 0

[6.17]

6.3.2. Solution

As in section 6.2.2, we obtain the solution to problem [6.17] from the instantaneous elastic solution at time t = t0 that is classically written as  y2 − z2 q0 q 0 x2 ξ ( , ) ( ν ( , ) )ey x t x y e t t = − + + 0 0 0 x  2 2     q0  + ν (t0 , t0 ) y z e z   [6.18]   q0 q0 y e x ⊗ e x + ν (t0 , t0 ) y (e y ⊗ e y + e z ⊗ e z )  ε ( x, t0 ) = −     0  σ ( x, t0 ) = − E (t0 , t0 ) q y e x ⊗ e x ,   from which we derive the solution to problem [6.17] in the form

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Viscoelastic Modeling for Structural Analysis

 y2 − z2 q0 q 0 x2 ( + ν (t0 , t ) )ey x y ex +  ξ ( x, t ) = − 2   2   q0  + ν (t0 , t ) y z e z   [6.19]   q0 q0 y e x ⊗ e x + ν (t 0 , t ) y (e y ⊗ e y + e z ⊗ e z )  ε ( x, t ) = −     0  σ ( x, t ) = − E (t0 , t ) q y e x ⊗ e x .   As in section 6.2.2, we note that the last two lines of [6.19] describe a uniaxial relaxation test performed on the material elements in Ω , which enables us to express the response to any relaxation-type evolution defined by a loading process χ d (t )  = q (t ) as  xy x2 y2 − z2 yz = − + + × q q q e e e y + ν (×) q ez ξ ν ( ) x y    2 2    ε = −q y e ⊗ e + ν (×) q y (e ⊗ e + e ⊗ e ) x x y y z z    [6.20]   y  σ = − E (×) q  e x ⊗ e x   y2 I  Q = E (×) q S  da = z E (×) q .  From the last line of [6.20], we derive the relaxation function of the rod subjected to bending within the kinematic parameter framework: R (τ , t ) is proportional to the material relaxation function E (τ , t ) through the geometrical parameter I z  Q = R (×) q , R (τ , t ) =

Iz E (τ , t ). 

[6.21]

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151

As in section 6.2.2 and with the same comments to be found in section 6.4.1, we define J (τ , t ) as the creep function of the rod subjected to bending defined through the inversion formula J (×)Rτ = Yτ which implies J (τ , t ) =

 J (τ , t ). Iz

[6.22]

The inverse relationship of [6.22], written as

q = J (×) Q,

[6.23]

formally expresses the response of the rod to a creep-type evolution defined by Q (t ) = M z (t ) , with reference to Saint-Venant’s principle. The corresponding displacement, strain and stress fields can be written as  xy x2 = − × Q + × Q J e J ey ξ ( ) ( ) x  Iz 2 Iz   2 2  + (n J ) (×) Q y − z e + (n J ) (×) Q y z e y z  Iz 2 Iz   y   ε = − J (×) Q e x ⊗ e x Iz    + y (n J )(×) Q (e ⊗ e + e ⊗ e ) y y z z  Iz   y  σ = −Q e x ⊗ e x . Iz 

[6.24]

The constitutive viscoelastic equation adopted to relate the bending moment to the curvature of a one-dimensional curvilinear element subjected to bending about a principal axis of inertia is then derived from [6.21] and [6.23] in the form, stated in Chapter 1, section 1.9.2.2, of

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Viscoelastic Modeling for Structural Analysis

M   χ = J (×) M = J (×) I    M = R (×) χ = I E (×) χ .

[6.25]

6.4. Twisting of a cylindrical rod 6.4.1. Preliminary comments

The solutions to the elastic equilibrium problems we were referring to in the preceding sections were obtained by Saint-Venant using his mixt or semi-inverse method1, which can be outlined as follows. Facing a practical problem that is not well-posed from a mathematical viewpoint because its boundary conditions in the end sections can only be described in terms of exerted external resultant forces and moments, Saint-Venant introduces an associated well-posed problem whose data are precisely specified in a form such as [6.2] or [6.15], and to which an explicit solution can be built up. Then, based on experimental observations and theoretical arguments, he appeals to what is now currently called Saint-Venant’s principle to state that, setting aside very localized end effects, the way forces and moments are exerted does not affect the elastic equilibrium solution which can be relied on for practical applications. Implicitly assuming that Saint-Venant’s principle is also valid within the linear viscoelastic context, we derived solutions to the quasi-static processes that were examined and used to finally write equations [2.13] and [6.25] as global constitutive viscoelastic laws. The same procedure will now be implemented when studying the quasi-static twisting of a viscoelastic rod.

1 Barré de Saint-Venant (1855, pp. 2, 3, 65, 81, 112, 140).

Some Practical Problems

153

6.4.2. Statement of the problem

Figure 6.3. Twisting of a cylindrical rod about a longitudinal axis

The well-posed loading process that must be associated with the quasi-static twisting of a viscoelastic cylindrical rod is stated using the following data (Figure 6.3): – no body forces d

F ( x, t ) = 0 in Ω

[6.26]

– boundary conditions

 ξ x d ( x, t ) = ξ y d ( x, t ) = 0 on S0   d d d d  ξ x ( x, t ) = −α (t )  y , ξ y ( x, t ) = α (t )  x on S   Tz d ( x, t ) = 0 on S0 and S   d  T ( x, t ) = 0 on S L .

[6.27]

This problem depends on α d (t )  , the rigid body rotation of the end section plane S about the z-axis, as a kinematic parameter q (t ) , whose associated loading parameter Q (t ) is the twisting moment about the z-axis of the external forces exerted on the end section:

 q (t ) = α d (t )     Q (t ) = S (− y σ zx (, t ) + x σ zy (, t )) da = C (t ).  

[6.28]

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Viscoelastic Modeling for Structural Analysis

The relaxation-type problem is then defined by the loading process

q (t ) = q 0 Yt0 (t ).

[6.29]

6.4.3. Solution

The instantaneous elastic solution at time t = t0 , as established by Saint-Venant, introduces the warping function ϕ ( x, y ) solution to the purely geometrical Neumann problem2

 Δϕ ( x, y ) = 0 on S0   ∂ϕ ( x, y ) ∂n = (n ∧ OM ). e z on ∂S0 ,

[6.30]

from which the torsional inertia of the cross-section is derived as J =  (x ( S0

∂ϕ ∂ϕ + x) − y ( − y )) da. ∂y ∂x

[6.31]

Without going into more details, the solution consists of the displacement and strain fields3

 zy zx ϕ ( x, y ) e x + q0 e y + q0 ez  ξ ( x, t0 ) = −q0      q0 ∂ϕ  ( − y ) (e x ⊗ e z + e z ⊗ e x )  ε ( x, t 0 ) = 2  ∂x   q0 ∂ϕ  + + x ) (e y ⊗ e z + e z ⊗ e y ) (  2  ∂y  and stress field 2 The symbol ∧ denotes the vector product. 3 See Salençon (2001, p. 407).

[6.32]

Some Practical Problems

σ ( x, t0 ) = μ (t0 , t0 )

q0 ∂ϕ − y ) (e x ⊗ e z + e z ⊗ e x ) (  ∂x

q0 ∂ϕ + μ (t0 , t0 ) + x) (e y ⊗ e z + e z ⊗ e y ), (  ∂y

155

[6.33]

where μ (t0 , t0 ) is the instantaneous shear modulus of the material. The relationship between the kinematic parameter and the loading parameter can be written as Q (t0 ) = C (t0 ) =

J μ (t0 , t0 ) q0 . 

[6.34]

The solution to the relaxation-type problem [6.29] is then obtained from [6.32] and [6.33] in the form  zx ϕ ( x, y ) 0 z y e x + q0 e y + q0 ez  ξ ( x , t ) = −q      0  ε ( x, t ) = q ( ∂ϕ − y ) (e x ⊗ e z + e z ⊗ e x )  2  ∂x   q0 ∂ϕ + + x ) (e y ⊗ e z + e z ⊗ e y ) (  2  ∂y   q0 ∂ϕ  = − y ) (e x ⊗ e z + e z ⊗ e x ) σ ( , ) μ ( , ) ( x t t t 0   ∂x   q0 ∂ϕ  + μ (t0 , t ) + x) (e y ⊗ e z + e z ⊗ e y ), (  ∂y 

[6.35]

which corresponds to simple shear relaxation tests performed on material elements in the rod. Hence, as in the preceding examples, we can derive the response to any relaxation-type evolution defined by the loading process q (t ) = α (t )  , which only involves the simple shear relaxation function μ (τ , t ) in the form

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Viscoelastic Modeling for Structural Analysis

 zy zx ϕ ( x, y ) e x + q (t ) e y + q (t ) ez  ξ ( x, t ) = −q (t )      q (t ) ∂ϕ  − y ) (e x ⊗ e z + e z ⊗ e x ) (  ε ( x, t ) = 2  ∂x   q (t ) ∂ϕ  + + x ) (e y ⊗ e z + e z ⊗ e y ) (  2  ∂y  1 ∂ϕ − y ) (e x ⊗ e z + e z ⊗ e x )  ∂x

[6.36]

σ = μ (×) q (

1 ∂ϕ + μ (×) q ( + x ) (e y ⊗ e z + e z ⊗ e y )  ∂y

[6.37]

and then Q = R (×) q , R (τ , t ) =

J μ (τ , t ), 

[6.38]

where R (τ , t ) , the relaxation function of the rod subjected to twisting within the kinematic parameter framework, is proportional to the material’s simple shear relaxation function µ (τ , t ) through the geometrical parameter J  . As in sections 6.2.2 and 6.3.2, we define J (τ , t ) , the creep function of the rod subjected to twisting, using the inversion formula. With γ (τ , t ) the simple shear creep function, we obtain:

 J (×) Rτ =Yτ  J (τ , t ) = γ (τ , t ), J

[6.39]

which, with reference to Saint-Venant’s principle, provides the solution to any creep-type evolution defined by a loading process Q (t ) = C (t )

q = J (×) Q ,

[6.40]

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157

where the displacement, strain and stress fields are simply  ξ   ε       σ     

= −γ (×) Q

zy zx ϕ ( x, y ) e x + γ (×) Q e y + γ (×) Q ez J J J

=

1 ∂ϕ γ (×) Q ( − y ) (e x ⊗ e z + e z ⊗ e x ) 2J ∂x

+

1 ∂ϕ γ (×) Q ( + x) (e y ⊗ e z + e z ⊗ e y ) 2J ∂y

=

Q ∂ϕ ( − y ) (e x ⊗ e z + e z ⊗ e x ) J ∂x

+

Q ∂ϕ ( + x) (e y ⊗ e z + e z ⊗ e y ). J ∂y

[6.41]

Equations [6.38] and [6.40] yield the constitutive equation adopted for a one-dimensional curvilinear element when relating the twisting moment about its longitudinal axis to the differential rotation about that axis (Chapter 1, section 1.9.2.3): C   α = J (×) C = γ (×) J    C = R (×) α = J μ (×) α .

[6.42]

6.4.4. Comment

This problem, as well as the two preceding ones, provide examples that illustrate the importance of a proper statement of the problems at hand when they pertain to creep or relaxation, which does not immediately result from the usual presentation of their elastic counterparts. In the present case, the well-posed problems we had to solve were of the relaxation type and consequently solutions to them involved the uniaxial tension or simple shear relaxation functions, straightforwardly providing the response to any kinematic parameter history. Conversely, the response to loading parameter histories, with

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reference to Saint-Venant’s principle, is expressed in terms of the associated creep functions which should not be confused with the algebraic inverse of the relaxation functions.

6.5. Convergence of a spherical cavity 6.5.1. Statement of the problem The convergence of a spherical cavity with radius a, bored in an infinite medium, is studied within the framework of the schematic geometrical model and the loading process defined as follows (Figure 6.4).

Figure 6.4. Spherical cavity within an infinite medium

Before the cavity has been bored, it is assumed that the stress field in the medium is a uniform isotropic pressure field σ ( x, t ) = − p0 1 , which is supposed to be an acceptable model for the earth pressure acting around the cavity and hence makes it possible to set body forces to zero: d

F ( x, t ) = 0 in Ω .

[6.43] −

This prestressed initial state at time t = t0 is taken as a reference for the constitutive law of the material (soil or rock), both from the

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159

geometrical and the mechanical viewpoints, which means that it relates the deformation history ε to (σ + p0 1 ) , the difference between the stress history and the initial stress state, in the form I

R ε ⎯⎯ → (σ + p0 1 ) = 1 λ (×) tr ε + 2 μ (×) ε .

[6.44]

The cavity is supposed to be bored instantaneously at time t = t0 . It may be modeled as causing the inner pressure on ∂Sa (the boundary −

of the cavity) to drop from its pre-existing value p (t0 ) = p0 to +

p (t0 ) = 0 . Meanwhile, the stress field still complies with the boundary condition at infinity: using a spherical coordinate system centered on O ,

σ ( x, t0 ) → − p0 1 when r → ∞.

[6.45]

The loading process is then defined by p (t ) = 0 for t ≥ t0 and the corresponding data can be written as  F d ( x, t ) = 0 in Ω    (σ (a, t ) + p0 1 ) . e r = p0 e r   (σ (r , t ) + p0 1 ) → 0 when r → ∞. 

[6.46]

With reference to the prestressed initial state, the problem depends on the loading parameter Q (t ) = p(t ) − p0 . The associated kinematic parameter is q (t ) = ξ r (a, t ) 4. The convergence of the cavity is the creep-type problem where

Q (t ) = Q 0 Yt0 (t ) = − p0 Yt0 (t ).

4 In fact, 4 π a 2ξ r (a, t ).

[6.47]

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6.5.2. Solution

The instantaneous elastic response to [6.47] is obtained from the classical solution to the equilibrium of a cavity inside an infinite elastic medium (see Salençon, 2001) and yields

ξ r (a, t0 ) = p0

a 4 μ (t0 , t0 )

[6.48]

where μ (t0 , t0 ) is the instantaneous shear modulus of the medium, which is the value at time t0 of the simple shear relaxation function μ (t0 , t ) . In order to straightforwardly derive the solution to the creep-type problem [6.47] from [6.48], this result must be expressed in an equivalent form introducing the value at time t0 of the simple shear creep function γ (τ , t ) , i.e.

ξ r (a, t0 ) = γ (t0 , t0 ) p0

a 4

[6.49]

The response to [6.47] is then written as

a   ξ r (a, t ) = γ (t0 , t ) p0 4    0 a  q (t ) = γ (t0 , t ) Q 4 .

[6.50]

a as the creep function J (τ , t ) of 4 the cavity with respect to the loading and kinematic parameters Q and q .

This equation identifies γ (τ , t )

The response to any creep-type evolution defined by the loading parameter history is thus given by

Some Practical Problems

 Q (t ) = p (t ) − p0   q (t ) = ξ r (a, t )    q = J (×) Q   J (τ , t ) = γ (τ , t ) a 4

161

[6.51]

4 μ (τ , t ) is the corresponding relaxation function a R (τ , t ) and the response to any relaxation-type evolution defined by the kinematic parameter history can be written as

Conversely,

 Q = R (×) q   R (τ , t ) = 4 μ (τ , t ) a .

[6.52]

The functions J (τ , t ) and R (τ , t ) determined here are simply the expressions of the creep and relaxation functions J (τ , t ) and R (τ , t ) referred to in Chapter 3 (sections 3.4.2 and 3.4.3), when the soil or rock medium in which the cavity is bored is assumed to be homogeneous and isotropic.

Appendix

Laplace transforms of usual functions and distributions ∞

L ϕ ( p) =  ϕ (t ) e − p t dt

ϕ * ( p) = p L ϕ ( p)

δu

e− p u

p e− p u

C Yu (t )

e− p u C p

e− p u C

Y (t ) e − a t

1 p+a

p p+a

Υ (t ) t n

n! p n +1

n! pn

Y (t ) t ne − a t

n! ( p + a ) n +1

p n! ( p + a ) n +1

Υ (t ) t q

Γ( q + 1) p q +1

Γ(q + 1) pq

ϕ (t )

Y (t ) t

Υ (t )(1 − e − a t )

−∞

π

π

2 p3 2

2 p1 2

a p ( p + a)

a p+a

Viscoelastic Modeling for Structural Analysis, First Edition. Jean Salençon. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

164

Viscoelastic Modeling for Structural Analysis

b a

b a

Υ (t )[ − (1 − ) e − a t ] Υ (t )sin ω t

p+b p ( p + a)

p+b p+a

ω

pω p2 + ω 2

p2 + ω 2

( p + a)2 + ω 2

pω ( p + a)2 + ω 2

Υ (t ) cos ω t

p p + ω2

p2 p + ω2

Υ (t ) e − a t cos ω t

( p + a) ( p + a)2 + ω 2

p ( p + a) ( p + a)2 + ω 2

e − a t ϕ (t )

(L ϕ ) ( p + a )

p [(L ϕ )( p + a )]

Υ (t ) e − a t sin ω t

1 ϕ (t ) t

ω

2



t ϕ (t )

∞ p

( L ϕ ) (u ) d u

−

d Lϕ dp

2

∞

p  L ϕ (u ) du p

−p

d Lϕ dp

COMMENTS.– – t is a real number, typically denoting the time variable; – p is a complex number; – n is an integer; – q > −1 is real; – a > 0, b, u and ω are real; – δu is the translated Dirac distribution; – Yu is the translated Heaviside step function: Yu (t ) = Y (t − u ) ; – C is a constant;

Appendix

165

– Γ is the Gamma function, the Euler integral of the second kind, ∞

Γ( z ) =  x z −1e− x dx, Re ( z ) > 0 , Γ(n + 1) = n! ; 0

– ϕ (t ) is a distribution with support  + .

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Index

A, B, C accommodation, 40 aging, 11, 98 arc, 36, 80 asymptotic harmonic regime, 40 beam, 36 bending, 36, 148 moment, 36 Boltzmann formulas, 24, 28, 108, 114, 120, 122 materials, 14 superposition principle, 14, 105, 129 boundary conditions, 125, 143, 148, 153 cantilever beam, 64, 75 causality principle, 10 characteristic time scales, 7, 51, 56 Chazey Bridge, 91 complex modulus, 41, 51 concrete, 19, 88, 115 constitutive equation, 14, 20, 27, 36, 98, 145, 149, 154, 158

continuous spectra, 60 contracted product, 108 convergence of a cavity, 93, 158 correspondence principle, 140 creep function, 2, 16, 27, 47, 145, 149 recovery experiment, 8, 16, 18, 30, 46 tensor, 106, 129 test, 2, 105 -type evolution, 129, 134, 137, 139, 145, 149, 154, 160 -type problems, 30, 64, 129, 139, 158 crucial experiment, 8 curvature, 36 curvilinear medium, 145, 149, 154 cylindrical rod, 143, 148, 152 D, E, F damper, 46 delayed deformation, 8, 20 differential rotation, 36, 154 Dirac distribution, 20, 46

Viscoelastic Modeling for Structural Analysis, First Edition. Jean Salençon. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

176

Viscoelastic Modeling for Structural Analysis

displacement, 63, 68, 72, 80, 125, 129, 131, 132, 135, 139, 143, 154 dissipated energy, 43 distribution theory, 20, 28, 30, 39 dummy index convention, 106 elastic compliance, 15, 19, 20, 28, 51, 139 modulus, 15, 135 equivalence, 59 force parameter, 33 functional description, 10, 105 funicular arc, 80 G, H, I geometrical parameter, 33 global equilibrium, 125 harmonic loading process, 39, 51 Heaviside step function, 2 heterogeneity, 98 homogeneity, 2, 33, 36, 64, 75, 97, 133, 160 hyperstatic, 70, 72, 80, 98 inertia axis, 148 instantaneous behavior, 2, 16, 19 elasticity, 51, 93, 107 integral operator, 24, 108 isothermal process, 1, 2, 5, 8–10, 125, 128 isotropic compression creep function, 114 relaxation function, 113 isotropy, 36, 97, 110, 122, 133 K, L, M kinematic parameter, 129, 135, 143, 148, 153

kinematically admissible displacement histories, 131 lag angle, 41, 51 Laplace(-Carson) transforms, 30, 40, 49, 51, 57, 58, 121, 163 Lee-Mandel correspondence principle, 140 Linear elastic element, 45 elasticity, 15, 24 functional relationship, 14, 105 viscoelastic material, 15 viscous element, 46 linearity, 14, 105 loading parameter, 129, 139, 143, 148, 153 loss angle, 41, 51 modulus, 41 material symmetries, 110 model generalized Kelvin, 58 generalized Maxwell, 57 Kelvin, 49 Maxwell, 47 Maxwell-Wiechert, 57 rheological, 45, 84 Zener, 51 memory approach, 26 term, 20 moment of inertia, 36 N, O, P natural initial state, 75, 84, 104, 137 Neumann problem, 154 non-aging materials, 12, 27, 45, 120, 140

Index

Onsager reciprocity principle, 121 operational calculus, 30, 121 Palau Bridge, 92 parallel connection, 49 phase angle, 41 Poisson ratio, 111, 115, 118, 122, 133 prestressing, 75, 90 Q, R, S quasi-static processes, 125 reactive force, 75 recovery, 8, 16, 18, 30, 46 redistribution of internal forces, 75 redundant unknown, 80 relaxation, 5, 105 function, 5, 16, 27, 49, 145, 149 tensor, 106, 129 -type evolution, 129, 135, 137, 139, 145, 149, 154, 160 -type problems, 30, 80, 129, 135, 143, 148, 153 resultant, 143, 152 rheological viscoelastic behavior, 12 Riemann’s convolution product, 28 Saint-Venant’s mixt method, 145, 152 principle, 145, 149, 152 series connection, 47 simple extension relaxation function, 113 shear creep function, 36, 97, 114, 154 shear relaxation function, 36, 97, 113, 154

177

small perturbation hypothesis (SPH), 1, 103, 125, 143 specific loss, 41 spring, 45 standard linear solid, 51, 88 statically admissible stress histories, 131 indeterminate system, 70, 72, 80, see also hyperstatic Stieltjes integral, 20 convolution product, 30 storage modulus, 41 strain, 104, 105, 108, 128, 132, 135, 139 strength of materials, 36, 145 stress, 103–105, 108 fading, 9, 18, 30 relaxation test, 5 structural elements, 33 superposition approach, 26 principle, 14, 105, 129 T, U, V Tensorial creep function, 106 relaxation function, 106 torsion, 36 torsional inertia, 36 transitory term, 51 twisting, 36, 152 moment, 36, 153, 154 uniaxial tension creep function, 36, 111, 133 relaxation function, 36, 115, 133 test, 2, 33, 111, 115, 118 unit impulsion, 20 unloading, 8

178

Viscoelastic Modeling for Structural Analysis

validation, 19 viscosity coefficient, 46 Voigt solid, 49 Volterra integral equations, 1, 8

W, Z warping function, 154 Wiechert, 57 wrench, 125

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