Polymer engineering science and viscoelasticity : an introduction

453 Pages·2008·10.585 MB·English

by  H F Brinson, L C Brinson

Hal F. Brinson L. Catherine Brinson
•
Polymer Engineering Science
and Viscoelasticity

An Introduction
Hal F. Brinson L. Catherine Brinson
University of Houston Northwestern University
Department of Mechanical Department of Mechanical
Engineering Engineering
4800 Calhoun Road Department of Materials
Houston, TX 77204 Science and Engineering
2145 Sheridan Road
Evanston, IL 60208-3111
Library of Congress Control Number: 2007936603

ISBN 978-0-387-73860-4 e-ISBN 978-0-387-73861-1

Printed on acid-free paper.

© 2008 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the written
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Preface
This text is an outgrowth or organized compilation of the notes the authors
have used to teach an introductory course on the viscoelasticity of poly-
mers for more than thirty years for the senior author and about fifteen
years for the junior author. Originally, the course was taught only to
graduate students but in recent years an effort has been made to teach a
modification of the course to senior level mechanical engineering students.
The authors have long held the view that the lack of knowledge of the fun-
damental aspects of the time and temperature behavior of polymer materi-
als is a serious shortcoming in undergraduate as well as graduate engineer-
ing education. This is especially important in our present society because
the use of polymeric materials pervades our experience both in our daily
lives and in our engineering profession. Still the basic thrust of under-
graduate education and even graduate education to some degree in the ar-
eas of mechanical and civil engineering is toward traditional materials of
metal, concrete, etc. Until about twenty-five years ago, elementary under-
graduate textbooks on materials contained little coverage of polymers. To-
day many elementary materials texts have several chapters on polymers
but, in general, the thrust of such courses is toward metals. Even the poly-
mer coverage that is now included treats stress analysis of polymers using
the same procedures as for metals and other materials and therefore often
misleads the young engineer on the proper design of engineering plastics.
Thus, it is not surprising that some structural products made from polymers
are often poorly designed and do not have the durability and reliability of
structures designed with metallic materials.
For the above reasons, the view of the authors is that specific courses on
polymer materials as well as associated stress analysis and engineering de-
sign need to be offered to every engineer. The present text has been devel-
oped with this in mind. The intent is to have sufficient coverage for a two
semester introductory sequence that would be available to upper class un-
dergraduates and first year graduate students. The level is such that only
basic knowledge of solid mechanics and materials science are needed as
prerequisites. The book is intended to be self sufficient even for those that
have little formal training in solid mechanics and therefore chemical engi-
neers, materials, forestry, chemistry, bio-engineering, etc. students as well
as mechanical and civil engineering students can use this text successfully.
Similarly, because chemistry background is often weak for non-chemical
engineers, introductory material is provided on the chemical basis of
viii Preface
polymers, which is essential for proper appreciation of the thermome-
chanical response.
Another major objective is for the text to be readable by recent engi-
neering graduates who have not had the advantage of a formal course on
polymer science or viscoelasticity. The reason, of course, is that today’s
engineering curricula, both undergraduate and graduate, have few extra
hours such that new courses can be accommodated in degree plans. There-
fore, a book such as this one should be of great value to the young engineer
who finds him/herself in a position heavily involved with the engineering
design and use of polymer based materials. In addition a text such as this
should be invaluable to those cross-disciplinary scientists such as biologist,
bio-chemist, etc. that need to understand the basic background to rigorous
mechanics approaches to the design of structures made with polymer based
materials.
The first chapter gives insight to the historical aspects of the subject. A
review of basic mechanics of materials (strength of materials) and materi-
als science is given in Chapter 2. Chapter 3 gives an introduction to the
mechanical properties of polymers and how they are determined as well as
general information on optical, electrical and other properties. Chapter 4 is
an introduction to the general character of polymers from a molecular
viewpoint and is valuable in assessing the mechanisms associated with vis-
coelastic deformations. Chapters 5 and beyond speak to the formal mathe-
matics and experimental methods associated with the relationship between
stress and strain in viscoelastic solids, both linear and nonlinear, as well as
stress analysis and failure.
Acknowledgements
Perhaps the first to receive our grateful acknowledgement are all the
authors of books and articles that we have referenced as well as those
authors from whom we have gained information that we have internalized
so well that the original source of inspiration is forgotten and therefore we
cannot properly acknowledge. We know that this text would not have been
possible without the guidance from professors and fellow students over our
formative years while a student at our respective universities; N.C. State,
Northwestern and Stanford (HFB) and Virginia Tech and Caltech (LCB).
In addition, we surely have greatly benefited from the council, encourage-
ment and friendship of fellow faculty at the universities where we have had
the pleasure to be a faculty member; N.C. State, Virginia Tech, UTSA and
University of Houston (HFB) and Northwestern (LCB). Certainly, we have
Preface ix
had the privilege to guide and be guided by our graduate students and
postdocs over the years who deserve enormous credit for their hard work
and the many papers and reports that we have jointly published. As any
professor must admit, our success would be small if not for the dedication,
energy and resourcefulness of our students – both graduate and under-
graduate. Both authors would also like to thank colleagues and staff at
University of Poitier in France and the Free University of Brussels in Bel-
gium (HFB) and the Helmut-Schmidt-Universität in Hamburg, Germany
(LCB), respectively, where each of us have spent many pleasant hours
with colleagues discussing the subject of this book plus many other issues.
We especially wish to thank Ms Susanne Pokossi for her skillful, cheerful
and tireless assistance in the preparation of the final version of this book.
Foremost of all we gratefully acknowledge those closest to us – our
families, our life partners and, of course, our children. We owe them much
for their patience, guidance, camaraderie and most of all their love.

Hal F. Brinson L. Cate Brinson
University of Houston Northwestern University
Table of Contents
1. Introduction .............................................................................................1
1.1. Historical Background....................................................................1
1.1.1. Relation between Polymer Science and Mechanics.........6
1.1.2. Perspective and Scope of this Text.................................10
1.2. Review Questions.........................................................................14
2. Stress and Strain Analysis and Measurement..................................15
2.1. Some Important and Useful Definitions......................................15
2.2. Elementary Definitions of Stress, Strain
and Material Properties................................................................17
2.3. Typical Stress-Strain Properties...................................................23
2.4. Idealized Stress-Strain Diagrams.................................................27
2.5. Mathematical Definitions of Stress, Strain and
Material Characteristics................................................................28
2.6. Principal Stresses..........................................................................40
2.7. Deviatoric and Dilatational Components of Stress and Strain...42
2.8. Failure (Rupture or Yield) Theories............................................46
2.9. Atomic Bonding Model for Theoretical Mechanical
Properties.......................................................................................49
2.10. Review Questions.........................................................................52
2.11. Problems........................................................................................53
3. Characteristics, Applications and Properties of Polymers.............55
3.1. General Classification and Types of Polymers...........................55
3.2. Typical Applications....................................................................61
3.3. Mechanical Properties of Polymers.............................................66
3.3.1. Examples of Stress-Strain Behavior of
Various Polymers.............................................................68
3.4. An Introduction to Polymer Viscoelastic Properties and
Characterization............................................................................75
3.4.1. Relaxation and Creep Tests.............................................75
3.4.2. Isochronous Modulus vs. Temperature Behavior..........79
3.4.3. Isochronous Stress-Strain Behavior – Linearity.............82
3.5. Phenomenological Mechanical Models.......................................84
3.5.1. Differential Stress-Strain Relations and Solutions for a
Maxwell Fluid.....................................................................86
3.5.2. Differential Stress-Strain Relations and Solutions for a
Kelvin Solid........................................................................91
xii Contents
3.5.3. Creep of a Three Parameter Solid and a Four Parameter
Fluid.....................................................................................93
3.6. Review Questions.........................................................................95
3.7. Problems........................................................................................96
4. Polymerization and Classification......................................................99
4.1. Polymer Bonding..........................................................................99
4.2. Polymerization............................................................................103
4.3. Classification by Bonding Structure Between Chains and
Morphology of Chains................................................................108
4.4. Molecular Configurations..........................................................111
4.4.1. Isomers............................................................................111
4.4.2. Copolymers.....................................................................114
4.4.3. Molecular Conformations..............................................115
4.5. Random Walk Analysis of Chain End-to-End Distance..........118
4.6. Morphology.................................................................................122
4.7. Molecular Weight.......................................................................131
4.8. Methods for the Measurement of Molecular Weight...............139
4.9. Polymer Synthesis Methods.......................................................146
4.10. Spectrography.............................................................................153
4.11. Review Questions.......................................................................155
4.12. Problems......................................................................................157
5. Differential Constitutive Equations.................................................159
5.1. Methods for the Development of Differential Equations
for Mechanical Models...............................................................160
5.2. A Note on Realistic Creep and Relaxation Testing..................165
5.3. Generalized Maxwell and Kelvin Models.................................168
5.3.1. A Caution on the Use of Generalized Differential
Equations........................................................................176
5.3.2. Description of Parameters for Various Elementary
Mechanical Models..........................................................177
5.4. Alfrey’s Correspondence Principle...........................................180
5.5. Dynamic Properties - Steady State Oscillation Testing...........181
5.5.1. Examples of Storage and Loss Moduli and Damping
Ratios.................................................................................191
5.5.2. Molecular Mechanisms Associated with Dynamic
Properties..........................................................................196
5.5.3. Other Instruments to Determine Dynamic Properties....198
5.6. Review Questions.......................................................................199
5.7. Problems......................................................................................199
Contents xiii
6. Hereditary Integral Representations of Stress and Strain...........201
6.1. Boltzman Superposition Principle.............................................201
6.2. Linearity......................................................................................208
6.3. Spectral Representation of Viscoelastic Materials...................208
6.4. Interrelations Among Various Viscoelastic Properties............211
6.5. Review Questions.......................................................................217
6.6. Problems......................................................................................217
7. Time and Temperature Behavior of Polymers...............................221
7.1. Effect of Temperature on Viscoelastic Properties of
Amorphous Polymers.................................................................222
7.2. Development of Time Temperature-Superposition-Principle
(TTSP)Master Curves.................................................................225
7.2.1. Kinetic Theory of Polymers..........................................228
7.2.2. WLF Equation for the Shift Factor...............................230
7.2.3. Mathematical Development of the TTSP.....................235
7.2.4. Potential Error for Lack of Vertical Shift.....................241
7.3. Exponential Series Representation of Master Curves..............242
7.3.1. Numerical Approach to Prony Series Representation.245
7.3.2. Determination of the Relaxation Modulus
from a Relaxation Spectrum..........................................251
7.4. Constitutive Law with Effective Time......................................254
7.5. Molecular Mechanisms Associated with Viscoelastic
Response......................................................................................256
7.6. Entropy Effects and Rubber Elasticity......................................257
7.7. Physical and Chemical Aging....................................................264
7.8. Review Questions.......................................................................271
7.9. Problems......................................................................................271
xiv Contents
8. Elementary Viscoelastic Stress Analysis for Bars and Beams.....275
8.1. Fundamental Concepts...............................................................275
8.2. Analysis of Axially Loaded Bars...............................................278
8.3. Analysis of Circular Cylinder Bars in Torsion.........................282
8.4. Analysis of Prismatic Beams in Pure Bending.........................284
8.4.1. Stress Analysis of Beams in Bending...........................284
8.4.2. Deformation Analysis of Beams in Bending................285
8.5. Stresses and Deformation in Beams for Conditions
other than Pure Bending.............................................................288
8.6. Shear Stresses and Deflections in Beams..................................296
8.7. Review Questions.......................................................................297
8.8. Problems......................................................................................297
9. Viscoelastic Stress Analysis in Two and Three Dimensions.........299
9.1 Elastic Stress-Strain Equations..................................................299
9.2 Viscoelastic Stress-Strain Relations..........................................301
9.3 Relationship Between Viscoelastic Moduli (Compliances).....303
9.4 Frequently Encountered Assumptions in Viscoelastic Stress
Analysis.......................................................................................304
9.5 General Viscoelastic Correspondence Principle.......................306
9.5.1 Governing Equations and Solutions for Linear
Elasticity.........................................................................306
9.5.2 Governing Equations and Solutions for Linear
Viscoelasticity................................................................308
9.6 Thick Wall Cylinder and Other Problems.................................311
9.6.1 Elasticity Solution of a Thick Wall Cylinder...............311
9.6.2 Elasticity Solution for a Reinforced Thick Wall
Cylinder (Solid Propellant Rocket Problem)................314
9.6.3 Viscoelasticity Solution for a Reinforced Thick Wall
Cylinder (Solid Propellant Rocket Problem)................316
9.7 Solutions Using Broadband Bulk, Shear and Poisson’s
Ratio Measured Functions..........................................................322
9.8 Review Questions.......................................................................324
9.9 Problems......................................................................................325