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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
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. This book, primarily intended for civil and mechanical engineering students, is devoted specifically to linear viscoelastic behavior within the small perturbation framework. The fundamental concepts of viscoelastic behavior are first presented from the phenomenological viewpoint of the basic creep and relaxation tests within the simple one-dimensional framework. The linearity and non-ageing hypotheses are introduced successively, with the corresponding expressions of the constitutive law in the form of Boltzmann's integral operators and Riemann's convolution products respectively. Applications to simple quasi-static processes underline the dramatic and potentially catastrophic consequences of not taking viscoelastic delayed behavior properly into account at the design stage. Within the three-dimensional continuum framework, the linear viscoelastic constitutive equation is written using compact mathematical notations and takes material symmetries into account. The general analysis of quasi-static linear viscoelastic processes enhances similarities with, and differences from, their elastic counterparts. Simple typical case studies illustrate the importance of an in-depth physical understanding of the problem at hand prior to its mathematical analysis.
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Veröffentlichungsjahr: 2019
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Table of Contents
Cover
Preface
List of Notations
1 One-dimensional Viscoelastic Modeling
1.1. Experimental observations
1.2. Fundamental uniaxial tests
1.3. Functional description
1.4. Aging
1.5. Linear behavior
1.6. Linear viscoelastic material
1.7. Linear viscoelastic constitutive equation
1.8. Non-aging linear viscoelastic constitutive equation
1.9. One-dimensional linear viscoelastic behavior
1.10. Harmonic loading process
2 Rheological Models
2.1. Rheological models
2.2. Basic elements
2.3. Classical models
2.4. Generalized Maxwell and Kelvin models
3 Typical Case Studies
3.1. Presentation and general features
3.2. “Creep-type” problems
3.3. Prestressing of viscoelastic systems or structures
3.4. A complex loading process
3.5. Heterogeneous viscoelastic structures
4 Three-dimensional Linear Viscoelastic Modeling
4.1. Multidimensional approach
4.2. Fundamental experiments
4.3. Boltzmann’s formulas
4.4. Isotropic linear viscoelastic material
4.5. Non-aging linear viscoelastic material
5 Quasi-static Linear Viscoelastic Processes
5.1. Quasi-static linear viscoelastic processes
5.2. Solution to the linear viscoelastic quasi-static evolution problem
5.3. Homogeneous isotropic material with constant Poisson’s ratio
5.4. Non-aging linear viscoelastic material
6 Some Practical Problems
6.1. Presentation
6.2. Uniaxial tension–compression of a cylindrical rod
6.3. Bending of a cylindrical rod
6.4. Twisting of a cylindrical rod
6.5. Convergence of a spherical cavity
Appendix
Laplace transforms of usual functions and distributions
References
Index
End User License Agreement
List of Illustrations
Chapter 1
Figure 1.1. Creep test at time t
0
Figure 1.2. Stress relaxation test at time t
0
Figure 1.3. Recovery experiment
Figure 1.4. Instantaneous “recovery” experiment
Figure 1.5. Stress fading experiment
Figure 1.6. Creep tests on a non-aging material
Figure 1.7. Creep tests on a non-aging material
Figure 1.8. Traction–compression of a straight rod
Figure 1.9. Bending of a straight rod
Figure 1.10. Twisting of a straight rod
Chapter 2
Figure 2.1. Linear elastic rheological element
Figure 2.2. Linear viscous rheological element
Figure 2.3. Maxwell model
Figure 2.4. Kelvin model
Figure 2.5. Kelvin–Voigt model with instantaneous elastic behavior
Figure 2.6. Zener model
Figure 2.7. Modulus and loss angle for the standard linear solid
Figure 2.8. Generalized Maxwell model
Figure 2.9. Generalized Kelvin model
Chapter 3
Figure 3.1. Cantilever beam
Figure 3.2. Uniformly distributed load applied to a cantilever beam
Figure 3.3. Concentrated load applied at the endpoint of a cantilever beam
Figure 3.4. Statically indeterminate structure
Figure 3.5. Statically indeterminate system
Figure 3.6. Prestressed cantilever beam
Figure 3.7. Bending moment diagrams
Figure 3.8. Parabolic-shaped hyperstatic arc
Figure 3.9. Prestressed rheological system
Figure 3.10. The model with (τ,t)=
2
(τ,t)/E
1
as a creep function
Figure 3.11. Internal elastic prestressing of a concrete block in compression
Figure 3.12. Internal prestressing of a concrete beam
Figure 3.13. Chazey Bridge: soffit tension cracks due to prestress loss
Figure 3.14. Palau Bridge before collapse (Burgoyne and Scantlebury 2006)
Figure 3.15. Palau Bridge geometry as built (after Burgoyne and Scantlebury 2006...
Figure 3.16. Excavating the cavity
Figure 3.17. Installing a concrete or metallic cladding
Figure 3.18. The cavity after the cladding has been installed
Figure 3.19. The loading process
Figure 3.20. Heterogeneous cantilever beam
Chapter 5
Figure 5.1. Evolution of a mechanical system
Figure 5.2. Boundary data
Figure 5.3. Schematic view of the quasi-static evolution problem
Chapter 6
Figure 6.1. Uniaxial tension–compression of a cylindrical rod
Figure 6.2. Bending of a cylindrical rod about a principal axis of inertia
Figure 6.3. Twisting of a cylindrical rod about a longitudinal axis
Figure 6.4. Spherical cavity within an infinite medium
Guide
Cover
Table of Contents
Begin Reading
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Series EditorYves 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
www.iste.co.uk
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
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
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 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 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 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
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.
