Mechanics of Aeronautical Composite Materials Christophe Bouvet First published 2017 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 John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2017 The rights of Christophe Bouvet 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: 2017941155 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-114-7 Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Chapter 1. Presentation of an Aeronautical Unidirectional Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Carbon/epoxy composite T300/914 . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3. Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chapter 2. Characteristics of UD Ply . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1. State of stress of UD ply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2. Tensile test in the l-direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3. Tensile test along the t-direction . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4. Shear test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5. General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Chapter 3. Characteristics of a UD Ply in a Given Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1. Off-axis tensile test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Chapter 4. Fracture of a Laminated Composite . . . . . . . . . . . . . . . . . . 37 4.1. Fracture of a UD ply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1.1. Longitudinal tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1.2. Longitudinal compression . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 vi Mechanics of Aeronautical Composite Materials 4.1.3. Transverse tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1.4. Transverse compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1.5. In-plane shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2. Fracture of a laminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Chapter 5. Fracture Criteria of a UD Ply . . . . . . . . . . . . . . . . . . . . . . . 49 5.1. Maximum stress fracture criterion . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2. Maximum strain fracture criterion . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.3. Hill’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.4. Tsai–Wu criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.5. Yamada–Sun criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Chapter 6. Membrane Behavior of a Laminated Composite Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.1. Generalities and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.2. Membrane behavior, bending behavior and mirror symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.3. Resultant forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.4. Displacement field, stress field and strain field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.5. Tension / shear coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Chapter 7. Bending Behavior of a Laminated Composite Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.1. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2. Resultant moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.3. Displacement field, stress field and strain field . . . . . . . . . . . . . . . . . . 99 7.4. Bending/twisting coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Chapter 8. The Fracture Criterion of a Laminate . . . . . . . . . . . . . . . . . 115 8.1. The sizing criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.2. Test on a composite structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.3. Sizing principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.4. Sizing a given structure for a given loading . . . . . . . . . . . . . . . . . . . . 119 8.5. Optimal structure for a given load . . . . . . . . . . . . . . . . . . . . . . . . . 131 Chapter 9. Damage Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.1. The principle of damage tolerance . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.2. Damage during impact and compression after impact . . . . . . . . . . . . . . 144 9.3. Sizing for impact damage tolerance . . . . . . . . . . . . . . . . . . . . . . . . 148 Contents vii Chapter 10. Interlaminar and Out-of-Plane Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 10.1. Tension of a cross-ply laminate [0,90] . . . . . . . . . . . . . . . . . . . . . 151 S 10.2. Tension of a cross-ply laminate [45,–45] . . . . . . . . . . . . . . . . . . . . 153 S 10.3. Out-of-plane shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Chapter 11. Holed and Bolted Plates . . . . . . . . . . . . . . . . . . . . . . . . 157 11.1. Calculating holed composite plates . . . . . . . . . . . . . . . . . . . . . . . . 157 11.2. Calculating the multi-bolt composite joints . . . . . . . . . . . . . . . . . . . 167 Chapter 12. Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 12.1. Reminder surrounding beam buckling . . . . . . . . . . . . . . . . . . . . . . 179 12.2. Buckling of plates under compression . . . . . . . . . . . . . . . . . . . . . . 180 12.3. Plate buckling under shear loading . . . . . . . . . . . . . . . . . . . . . . . . 186 Chapter 13. Miscellaneous Rules for Stacking . . . . . . . . . . . . . . . . . . 189 Chapter 14. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 14.1. Experimental determination of the characteristics of a UD material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 14.2. Fracture of a laminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 14.3. Shear modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 14.4. Optimization of stacking sequence . . . . . . . . . . . . . . . . . . . . . . . . 195 14.5. Composite tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 14.6. Laminate calculation without calculation . . . . . . . . . . . . . . . . . . . . 196 14.7. Sandwich beam under bending . . . . . . . . . . . . . . . . . . . . . . . . . . 197 14.8. Laminate plate under compression . . . . . . . . . . . . . . . . . . . . . . . . 200 14.9. Tube under torsion/internal pressure . . . . . . . . . . . . . . . . . . . . . . . 203 14.10. Optimization of a fabric with a strain fracture criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 14.10.1. Part 1: preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 14.10.2. Part 2: quasi-isotropic stacking sequence . . . . . . . . . . . . . . . . . 205 14.10.3. Part 3: stacking sequence optimization . . . . . . . . . . . . . . . . . . . 206 14.10.4. Part 4: stacking sequence optimization under bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 14.11. Open hole tensile test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 14.12. Multi-bolt composite joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 viii Mechanics of Aeronautical Composite Materials Chapter 15. Solutions to the Exercises . . . . . . . . . . . . . . . . . . . . . . . 211 15.1. Experimental determination of the characteristics of a UD material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 15.2. Fracture of a laminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 15.3. Shear modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 15.4. Optimization of stacking sequence . . . . . . . . . . . . . . . . . . . . . . . . 229 15.5. Composite tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 15.6. Laminate calculation without calculation . . . . . . . . . . . . . . . . . . . . 240 15.7. Sandwich beam under bending . . . . . . . . . . . . . . . . . . . . . . . . . . 242 15.8. Laminate plate under compression . . . . . . . . . . . . . . . . . . . . . . . . 253 15.9. Tube under torsion/internal pressure . . . . . . . . . . . . . . . . . . . . . . . 263 15.10. Optimization of a fabric with a strain fracture criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 15.11. Open hole tensile test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 15.12. Multi-bolt composite joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Foreword When my young colleague gave me his manuscript to proofread and asked me to write a few words for a foreword, it is with great pleasure that I accepted, even though the preamble and title were quite sufficient to present this book. Many books have been written on composite materials. The originality of this one comes in part from the objective, which is targeted towards aircraft structures made of composite materials. To this end, the calculation of holed structures, and their subsequent multi-bolt joints, and the theme of damage tolerance are presented alongside the problems that they carry with them, as well as the calculation methods that are currently in use. These two axes are rarely presented in the literature, yet they are essential to perform an accurate and realistic sizing since they often govern the resistance of a given structure and thus its sizing. This book covers all classic notions surrounding composite structures. Theoretical formulae are presented concisely and without unnecessary developments allowing the focus to remain on the essentials. When studying criteria, another crucial point, the differences between first damage and final fracture are made abundantly clear, something that is essential for achieving optimal structures and performing realistic calculations. This is another essential aspect of composite calculations that is nonetheless quite absent from the literature. x Mechanics of Aeronautical Composite Materials Lastly, a number of exercises surrounding all studied themes are proposed with complete and detailed corrections. They will allow the readers to assimilate the presented themes more easily. This is without doubt one of this book’s strongest assets as there are few other titles that carry quite this many examples. I recommend reading this book as it uses concision and precision to provide essential bases for designing and sizing composite structures. Jean-Jacques BARRAU Former Professor at the University Paul Sabatier Toulouse, France Preface The objective of this lesson on composite structure sizing is to present the principles that allow the sizing of composite laminates widely used in composite structures. After a brief presentation of the primary material used in aircraft structures, the basic theory of laminated plates under membrane and bending loading as well as their associated fracture criteria is touched on. The fracture of a UD ply is then explained in detail, in order to demonstrate its inherent complexity and the limits of the criteria in use. Next, these criteria of the base ply are used to size a complete composite laminate. Lastly, two fundamental cases of structure calculations are presented: sizing holes and multi-bolt joints, as well as the study of buckling. The criteria that are specific to aviation, in particular the notion of limit loads and ultimate loads, are addressed. The notion of damage tolerance specific to aviation is then presented, and in particular the notion of impact damage tolerance. These notions are fundamental to understanding the specificities of sizing aircraft composite structures. Corrected exercises then allow curious readers to test their understanding of the different subjects. These corrected exercises are typical for sizing aircraft composite structures. Engineers will also find exercises that resemble their case studies. Lastly, an Excel spreadsheet allows the presentation of the calculations in the book in detail for review (available to download at www.iste.co.uk/bouvet/aeronautical2.zip). The originality of this particular title is that it places itself very clearly in the field of aviation, where the sizing criteria are very specific. Take the calculation of holed plates and multi-bolt joints that are typical for composite aircraft structures and yet are not often touched on in the literature. Nonetheless, the notions in this book remain valid for most industrial purposes.