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FREAKLEY PAYNE E T a N n w G dH iI tNP h ERE E RA RO C U I NT B R GI C B EY E Ro f Theory and Practice This book is based on the well-known CONTENTS Rubber in Engineering Practice by Davey and Payne, which was published Part I: Properties of Elastomeric in 1965 and has become a major work of Engineering with Materials. on the subject. The book has been 1. Introduction and review of prop largely rewritten and with the per erties. spective of 12 years of hew advances 2. Static load-deformation properties. and applications it has been possible to 3. Dynamic force-deformation prop provide a considerably more substantial Rubber erties. link between theory and practice and to Part II: Theory and Practice of Design. deal in detail with the origins and 4. Analysis of the deformation of limitations of the analytical techniques rubber units under equilibrium used extensively in design. The book is loading conditions. divided into three sections, furnishing 5. Vibration isolation and trans- the reader with a logical progression missibility. from physical properties to the design 6. Isolation of shock and transient of load bearing rubber units and then vibration. to the use of rubber units in a range of 7. Practical design details and typical applications. The layout is arranged to forms of rubber mountings. provide the user with a means of rapid 8. Practical mounting arrangements reference to specific items and to supply using rubber springs. the practising engineer with concise Part III: Applications. information in a form easily related to 9. Rubber in packaging. design problems. 10. Flexible couplings Р. К. FREAKLEY 11. Bridge bearings. 12. Piers and fenders. 13. Rubber in buildings. and 14. Transport engineering. 15. Industrial shock and vibration isolation. A. R. PAYNE 16. Noise control and structural damping. 17. Rubber in fluid sealing. 18. Flexible composites. Appendices: I. Properties of vulcanised rubber. II. Note on expressions for damping. III. Note on the decibel system. IV. Hardness testing of rubbers. Index. APPLIED SCIENCE PUBLISHERS THEORY AND PRACTICE OF ENGINEERING WITH RUBBER P. K. FREAKLEY Institute of Polymer Technology, Loughborough University, UK and A. R. PAYNE Formerly Director, Shoe and Allied Trades Research Association, UK APPLIED SCIENCE PUBLISHERS LTD LONDON APPLIED SCIENCE PUBLISHERS LTD RIPPLE ROAD, BARKING, ESSEX, ENGLAND British Library Cataloguing in Publication Data Freakley, P К Theory and practice of engineering with rubber. 1. Rubber I. Title II. Payne, Arthur Robert 620. 1'94 TA455.R8 ISBN 0-85334-772-7 WITH 34 TABLES AND 443 ILLUSTRATIONS © APPLIED SCIENCE PUBLISHERS LTD 1978 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, photocopy ing, recording, or otherwise, without the prior written permission of the publishers, Applied Science Publishers Ltd, Ripple Road, Barking, Essex, England Typeset by The European Printing Corporation Limited Printed in Great Britain by Galliard (Printers) Ltd, Great Yarmouth Acknowledgements In the text the authors have been happy to acknowledge the contri butions made by many workers in the field of ‘engineering with rubber’ and are grateful for their kind permission and that of their publishers to reproduce various material. Without this it would not have been possible to contemplate producing a book of this form. It is proper to mention here the unique and sustained contribution made by the staff of the Malaysian Rubber Producers’ Research Association (MRPRA). The scope of this contribution has resulted in Chapters 2 and 4 being based largely upon MRPRA publications. The extensive work of Professor J. C. Snowdon in the field of vibration and shock isolation, resulting in numerous references in the text, must also be mentioned. In order to update the sections and chapters dealing with com mercially available products and their application to particular cases, the help of numerous manufacturers was sought and in most cases given, enthusiastically and unstintingly. The authors wish to ac knowledge, with many thanks, the assistance of the following persons and companies among the many who helped them in the preparation of this book: Mr E. T. Dawson and Dunlop, Polymer Engineering Division; Mr W. R. Brake and BTR-Silvertown Ltd; Mr D. O’Brien, Mr A. T. Smith and Cementation Muffelite; Mr R. P. Torr and the Andre Rubber Co Ltd; Mr D. Turner, Mr R. A. Dove and the Avon Rubber Co. Ltd; Dr D. Holland and BTR-Solarbridge; Mr D. A. Rayner, Mr M. K. Snow and J. H. Fenner Co. Ltd; Mr G. N. Wright and Lord Kinematics; Mr J. Fee and Firestone Industrial Products; Mr J. E. Long and PSC-Equipment Ltd; Mr T. Rogers and Vauxhall Motors Ltd; Mr M. P. Hill and Rolls Royce Ltd; Mr P. Ward and Britten Norman Ltd; Mr J. Adam and Scottish Aviation Ltd. Finally, our thanks are due to Miss Linda Malins, who spent many hours at her typewriter in order to produce the final manuscript. V Foreword The use of rubbers in engineering applications has increased markedly in the last two decades. This increase, however, is small compared to the potential, and a major factor restricting growth is the low level of appreciation of the engineering properties of rubber and design procedures based on these. The research work necessary to underwrite the rational use of rubber in critical applications has, in large measure, been completed and it remains to inform and convince the engineer that design procedures are available and reliable. Pioneering work has been carried out in a few centres in the UK and the late Dr A. R. Payne was an early missionary, his efforts culminating in the book Rubber in Engineering Practice (co-author A. B. Davey). Dr A. R. Payne was an Industrial Professor at Lough borough University of Technology and it is fitting that this book, updating the earlier one and which was in preparation at the time of his death, should have been completed by the new co-author and member of the staff of the University, Philip Freakley, in the face of considerable difficulties resulting from the tragic event of 20 August, 1976. A. W. BlRLEY Director, Institute of Polymer Technology, Loughborough, UK vii Preface Work carried out in the field of engineering design with rubber has, in the past ten years, followed two main paths. The first is an extension and consolidation of the subject matter of the book by A. B. Davey and A. R. Payne, entitled Rubber in Engineering Practice. The second path is essentially computer oriented, using the modern digital computer’s capacity and speed to good advantage in solving complex problems. It was decided, at the conception of this book, to deal only with those methods which are capable of being handled using traditional mathematical methods or a pocket calculator (although the increasing sophistication of pocket calculators may blur this dividing line with the passage of time), thus producing an updated version of the earlier book. This decision was made on the basis that the majority of engineering problems are solved without recourse to a computer and that this text is designed to present the engineer, practising and potential, with a clear and precise review of the application of conventional analytical skills to the field of engineering with rubber. Indeed, the combination of analytical and computer methods in a single text dealing with a subject matter which is itself a departure from conventional engineering was thought to detract from the requirement for a logical progression from material properties to applications. The application of the digital computer to engineering with rubber has extended the capabilities of some of the available analytical methods detailed in the text. The problems best solved using computer methods are indicated and it is hoped that these will be the subject of a further text to be produced in the not too distant future. P. K. Freakley ix Contents Acknowledgements............................................................ v Foreword .......................................................................... vii Preface............................................................................... ix List of Symbols................................................................. xvii PART I: PROPERTIES OF ELASTOMERIC MATERIALS Chapter 1 Introduction and Review of Properties 1.1 Definition of ‘Rubber’.......................................... 3 1.2 Molecular Nature of Rubber................................ 3 1.3 Prediction of Stress-Strain Properties .... 6 1.4 Relations Among Time, Temperature and Rate of Deformation ........................................................ 8 1.5 Energy Absorption and Loss ............................ 9 1.6 Other Properties................................................... 10 1.7 Raw Elastomeric Materials ................................ 15 1.8 Vulcanisation and Compounding ....................... 18 1.9 Processing Techniques.......................................... 21 Chapter 2 Static Load-Deformation Properties 2.1 Introduction ........................................................ 24 2.2 Statistical or Kinetic Theory of Rubber Elasticity • 24 2.3 Phenomenological Theory of Rubber Elasticity . . 30 2.4 Effect of Carbon Black on Poisson’s Ratio and Rela tions Among E0 ,G and К (Bulk Modulus) . . . 32 2.5 Creep and Stress Relaxation................................ 35 2.6 Rupture of Rubber.............................................. 50 2.7 Effect of Ozone................................................... 53 xi xii CONTENTS Chapter 3 Dynamic Force-Deformation Properties 3.1 Introduction ........................................................ 56 3.2 Linear Viscoelastic Behaviour............................ 56 3.3 Non-linear Viscoelastic Behaviour ................... 74 3.4 Energy Dissipation and Heat Build-up in Rubber Units...................................................................... 85 3.5 Wave Effects in Rubber ..................................... 88 3.6 Dependence of Ultimate Strength and Extension on Rate, Temperature and Hysteresis....................... 93 3.7 Failure of Rubber Under Conditions of Cyclic Load ing—Fatigue............................................................ 97 PART II: THEORY AND PRACTICE OF DESIGN Chapter 4 Analysis of the Deformation of Rubber Units Under Equilibrium Loading Conditions 4.1 Application on Deformation Theories................... 113 4.2 Compression of Rubber Blocks............................ 113 4.3 Shear of Rubber Blocks ....................... 119 4.4 Combined Compression and Shear ................... 120 4.5 Torsion Disk........................................................ 121 4.6 Annular Rubber Units in Axial Shear................... 125 4.7 Annular Rubber Units in Rotary Shear .... 129 4.8 Annular Rubber Units in Radial Compression . . 134 4.9 Annular Rubber Units in Transverse Loading . . 138 4.10 Elastic Stability and Buckling of Rubber Compression Units...................................................................... 140 4.11 Compression of Laterally Unrestrained Toroidal Rubber Rings........................................................ 146 4.12 Radial Compression of Long Hollow Circular Cylin ders ...................................................................... 148 4.13 Radial Deformation of Rubber-covered Rollers . . 154 4.14 Compression of a Rubber Sphere Between Plane Surfaces ............................................................ 156 4.15 Radial Compression of Half-Cylindrical and D-section Units...................................................................... 158 4.16 Hollow Cylinders in Axial Compression .... 162 4.17 Conclusions ........................................................ 164 CONTENTS xiii Chapter 5 Vibration Isolation and Transmissibility 5.1 Introduction ........................................................ 166 5.2 Vibration of a Simple Single-degree-of-freedom System................................................................. 166 5.3 Vibration of Systems Having Parallel Mountings 174 5.4 Vibration of Systems Mounted on Non-rigid Foun dations ................................................................. 179 5.5 Vibration of Systems Having Two Degrees of Freedom ............................................................ 183 5.6 Vibration of Systems Having Many Degrees of Freedom ............................................................ 187 5.7 Vibration of Systems Having a Distributed Mass . 189 5.8 Vibration of Systems Mounted on Non-linear Units 195 5.9 Vibration of Systems Having a Dynamic Absorber 198 5.10 Vibration Wavelength Effects on Rubber Mountings 203 Chapter 6 Isolation of Shock and Transient Vibration 6.1 Introduction ........................................................ 208 6.2 Transient Vibration of the Simple Linear System . 208 6.3 Transient Vibration of the Non-linear Simple System 217 6.4 Concepts of Mounting Performance...................... 223 6.5 Practical Considerations for Shock Isolation . . 230 Chapter 7 Practical Design Details and Typical Forms of Rubber Mountings 7.1 Introduction ........................................................ 236 7.2 Avoidance of Excessive Stress in the Rubber . . 236 7.3 Dimensional Tolerances and Clearances .... 243 7.4 Design for Stability.............................................. 248 7.5 Design for Manufacture ..................................... 255 7.6 Precautions to be Taken During Installation . . 258 7.7 Causes of Failure and Suitable Remedies . . . 260 7.8 Typical Forms of Rubber Mountings................... 264 Chapter 8 Practical Mounting Arrangements Using Rubber Springs 8.1 Parallel Mounting Arrangements ....................... 300 8.2 ‘Parallel Ring’ Type (Conventional) ................... 303 8.3 Convergent Arrangements ................................ 304 8.4 Other Arrangements .......................................... 310