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1 An Introduction to the Mechanical Properties of Solid Polymers Second Edition I. M. Ward IRC in Polymer Science and Technology, School of Physics and Astronomy, University of Leeds, UK and J. Sweeney IRC in Polymer Science and Technology, School of Engineering, Design and Technology, University of Bradford, UK

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3 An Introduction to the Mechanical Properties of Solid Polymers Second Edition

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5 An Introduction to the Mechanical Properties of Solid Polymers Second Edition I. M. Ward IRC in Polymer Science and Technology, School of Physics and Astronomy, University of Leeds, UK and J. Sweeney IRC in Polymer Science and Technology, School of Engineering, Design and Technology, University of Bradford, UK

6 Copyright # 2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on or All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or ed to permreq@wiley.co.uk, or faxed to (+44) This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA , USA Wiley-VCH Verlag GmbH, Boschstr. 12, D Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging-in-Publication Data Ward, I. M. (Ian Macmillan), An Introduction to the mechanical properties of solid polymers / I. M. Ward and J. Sweeney. 2nd ed. p. cm. Includes bibliographical references and index. ISBN (cloth : alk. paper) ISBN X (pbk. : alk. paper) 1. Polymers Mechanical properties. I. Sweeney, John, 1952-II. Title. TA455,P58W dc British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN hardback X paperback Typeset in /121 2 pt Times by Keytec Typesetting, Bridport Printed and bound in Great Britain by TJ International Ltd., Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

7 Contents Preface xi 1 Structure of Polymers Chemical compostion Polymerization Cross-linking and chain-branching Average molecular mass and molecular mass distribution Chemical and steric isomerism and stereoregularity Liquid crystalline polymers Blends, grafts and copolymers Physical structure Rotational isomerism Orientation and crystallinity 11 References 16 Further reading 16 2 The Deformation of an Elastic Solid The state of stress The state of strain The engineering components of strain The generalized Hooke s law Finite strain elasticity: the behaviour of polymers in the rubber-like state The definition of components of stress The generalized definition of strain The strain energy function 28 References 30 Further reading 30 3 Rubber-Like Elasticity General features of rubber-like behaviour The thermodynamics of deformation The statistical theory Simplifying assumptions 35 An Introduction to the Mechanical Properties of Solid Polymers I. M. Ward and J. Sweeney # 2004 John Wiley & Sons, Ltd ISBN: (HB); X (PB)

8 vi CONTENTS The average length of a molecule between cross-links The entropy of a single chain The elasticity of a molecular network Modifications of the simple molecular network Recent developments in the molecular theory of rubber elasticity 46 References 51 Problems for Chapters 2 and Principles of Linear Viscoelasticity Viscoelasticity as a phenomenon Linear viscoelastic behaviour Creep Stress relaxation Mathematical representation of linear viscoelasticity The Boltzmann superposition principle The stress relaxation modulus Mechanical models, retardation and relaxation time spectra Dynamic mechanical measurements: the complex modulus and complex compliance Experimental patterns for G 1, G 2, etc. as a function of frequency The Alfrey approximation 73 References 76 Problems for Chapter The Measurement of Viscoelastic Behaviour Creep and stress relaxation Creep conditioning Specimen characterization Experimental precautions Dynamic mechanical measurements The torsion pendulum Forced vibration methods Dynamic mechanical thermal analysis (DMTA) Wave-propagation methods The kilohertz frequency range The megahertz frequency range: ultrasonic methods The hypersonic frequency range: Brillouin spectroscopy 92 References 92 6 Experimental Studies of Linear Viscoelastic Behaviour as a Function of Frequency and Temperature: Time Temperature Equivalence General introduction Amorphous polymers Temperature dependence of viscoelastic behaviour Crystallinity and inclusions 101

9 CONTENTS vii 6.2 Time temperature equivalence and superpostion Molecular interpretations of time temperature equivalence Molecular rate processes with a constant activation energy: the site model theory The Williams Landel Ferry (WLF) equation Flexible molecular chain models Normal mode theories The dynamics of highly entangled polymers 116 References Anisotropic Mechanical Behaviour Elastic constants and polymer symmetry Specimens possessing orthohormbic symmetry Specimens possessing uniaxial symmetry, often termed transverse isotropy Measuring elastic constants Measurements on films or sheets Measurements on filaments Experimental studies of mechanical anisotropy in oriented polymers Sheets of low-density polyethylene Filaments tested at room temperature Interpretation of mechanical anisotropy: general considerations Theoretical calculations of elastic constants Orientation and morphology Experimental studies of anisotropic mechanical behaviour and their interpretation The aggregate model and mechanical anisotropy Correlation between the elastic constants of a highly oriented and an isotropic polymer The development of mechanical anisotropy with molecular orientation The anisotropy of amorphous polymers Later applications of the aggregate model The aggregate model for chain-extended polyethylene and liquid crystalline polymers Auxetic materials: negative Poisson s ratio 157 References Polymer Composites: Macroscale and Microscale Composites: a general introduction Mechanical anisotropy of polymer composites Mechanical anisotropy of lamellar structures Elastic constants of highly aligned fibre composites Mechanical anisotropy and strength of uniaxially aligned fibre composites 169

10 viii CONTENTS 8.3 Short fibre composites The influence of fibre length: shear lag theory Debonding and pull-out Partially oriented fibre composites Takayanagi models for semicrystalline polymers The simple Takayanagi model Takayanagi models for dispersed phases Modelling polymers with a single-crystal texture Ultrahigh-modulus polyethylene The crystalline fibril model The crystalline bridge model Conclusions 190 References 190 Problems for Chapters 7 and Relaxation Transitions: Experimental Behaviour and Molecular Interpretation Amorphous polymers: an introduction Factors affecting the glass transition in amorphous polymers Effect of chemical structure Effect of molecular mass and cross-linking Blends, grafts and copolymers Effect of plasticizers Relaxation transitions in crystalline polymers General introduction Relaxation in low crystallinity polymers Relaxation processes in polyethylene Relaxation processes in liquid crystalline polymers Conclusions 216 References Creep, Stress Relaxation and Non-linear Viscoelasticity The engineering approach Isochronous stress strain curves The rheological approach Adaptations of linear theory differential models Adaptations of linear theory integral models More complicated single-integral representations Comparison of single-integral models Creep and stress relaxations as thermally activated processes The Eyring equation Applications of the Eyring equation to creep Applications of the Eyring equation to stress relaxation Applications of the Eyring equation to yield 238 References 239

11 CONTENTS ix 11 Yielding and Instability in Polymers Discussion of load elongation curves in tensile testing Necking and the ultimate stress Necking and cold-drawing: a phenomenological discussion Use of the Considère construction Definition of yield stress Ideal plastic behaviour The yield criterion: general considerations The Tresca yield criterion The Coulomb yield criterion The von Mises yield criterion Geometrical representations of the Tresca, von Mises and Coulomb yield criteria Combined stress states Historical development of understanding of the yield process Adiabatic heating The isothermal yield process: the nature of the load drop Experimental evidence for yield criteria in polymers Application of Coulomb yield criterion to yield behaviour Direct evidence of the influence of hydrostatic pressure on yield behaviour The molecular interpretations of yield and cold-drawing Yield as an activated rate process: the Eyring equation Alternative models: nucleation-controlled mechanisms Pressure dependence and general states of stress Cold-drawing General considerations The natural draw ratio, maximum draw ratios and molecular networks Crystalline polymers 270 References Breaking Phenomena Definition of tough and brittle behaviour in polymers Principles of brittle fracture of polymers Griffith fracture theory The Irwin model The strain energy release rate Controlled fracture in brittle polymers Crazing in glassy polymers The structure and formation of crazes The structure of crazes Craze initiation and growth Crazing in the presence of fluids and gases: environmental crazing Controlled fracture in tough polymers The J-integral Essential work of fracture 302

12 x CONTENTS Crack opening displacement The molecular approach Factors influencing brittle ductile behaviour: brittle ductile transitions The Ludwig Davidenkov Orowan hypothesis Notch sensitivity and Vincent s ó B ó Y diagram The impact strength of polymers Flexed-beam impact Falling-weight impact Toughened polymers: high-impact polyblends Crazing and stress whitening Dilatation bands The tensile strength and tearing of polymers in the rubbery state The tearing of rubbers: extension of Griffith theory Molecular theories of the tensile strength of rubbers Effect of strain rate and temperature Fatigue in polymers 331 References 335 Problems for Chapters 11 and Appendix A1.1 Scalars, vectors and tensors 341 A1.2 Tensor components of stress 341 A1.3 Tensor components of strain 342 A1.4 Generalized Hooke s law 342 A1.5 Engineering strains and matrix notation 343 A1.6 The elastic moduli of isotropic materials 345 A1.7 Transformation of tensors from one set of coordinate axes to another 347 A1.8 The Mohr circle construction 350 References 351 Appendix A2.1 Rivlin, Mooney, Ogden 353 References 356 Answers to Problems 357 Index 377

13 Preface This book is the second edition of An Introduction to the Mechanical Properties of Solid Polymers. Its aim is to provide an introduction to the mechanical behaviour of solid polymers at a fairly elementary level for research workers in polymer science and for postgraduate students with first degrees in physics, chemistry, engineering or materials science. It follows the approach of the first edition in developing the mechanics of behaviour first and then discussing molecular and structural interpretations. The individual chapters are self-contained so that they can be read as reviews of progress in different areas. Since the publication of the first edition, in 1993, the subject has advanced and this has been dealt with in some instances by adding additional sections with the latest developments. In other cases, although the overall original format has been retained, there has been substantial rewriting and many additions to the text. We are very grateful to Margaret Ward for undertaking the majority of the initial typing of the new text. We also wish to thank Colin Morath and Jagan Mohanraj for their assistance with the preparation of new diagrams. I. M. Ward J. Sweeney An Introduction to the Mechanical Properties of Solid Polymers I. M. Ward and J. Sweeney # 2004 John Wiley & Sons, Ltd ISBN: (HB); X (PB)

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15 1 Structure of Polymers The mechanical properties that form the subject of this book are a consequence of the chemical composition of the polymer and also of its structure at the molecular and supermolecular levels. We shall therefore introduce a few elementary ideas concerning these aspects. 1.1 Chemical composition Polymerization Linear polymers consist of long molecular chains of covalently bonded atoms, each chain being a repetition of much smaller chemical units. One of the simplest polymers is polyethylene, which is an addition polymer made by polymerizing the monomer ethylene, CH 2 ¾CH 2, to form the polymer Note that the double bond is removed during the polymerization (Figure 1.1). The well-known vinyl polymers are made by polymerizing compounds of the form where X represents a chemical group; examples are as follows: An Introduction to the Mechanical Properties of Solid Polymers I. M. Ward and J. Sweeney # 2004 John Wiley & Sons, Ltd ISBN: (HB); X (PB)