(
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`(
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`MECHANICS OF
`COMPOSITE MATERIALS
`Second Edition
`
`About the Book ...
`
`J
`
`About the Author ...
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`ISBN 1-56032-7i2-X
`
`9 0 0 0 0>
`
`ENG IN
`TA
`418. 9
`. C6J59
`1999
`
`. ,
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`ClearCorrect Exhibit 1045, Page 1 of 270
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`MECHANICS OF
`COMPOSITE MATERIALS
`
`SECOND EDITION
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`ClearCorrect Exhibit 1045, Page 2 of 270
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`MECHANICS OF
`COMPOSITE MATERIALS
`
`SECOND EDITION
`
`ROBERT M. JONES
`
`Professor of Engineering Science and Mechanics
`Virginia Polytechnic Institute and State University
`Blacksburg, Virginia 24061-0219
`
`)
`
`ClearCorrect Exhibit 1045, Page 3 of 270
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`(
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`USA
`
`Publishing Office:
`
`Distribution Center:
`
`UK
`
`Taylor & Francis, Inc.
`325 Chestnut Street
`Philadelphia, PA 19106
`Tel: (215) 625-8900
`Fax. (215) 625-2940
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`Taylor & Francis, Inc.
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`MECHANICS OF COMPOSITE MATERIALS, vr(
`
`Copyright © 1999 Taylor & Francis. All rights reserved. Printed in the United States of
`America. Except as permitted under the United States Copyright Act of 1976, no part of
`this publication may be reproduced or distributed in any form or by any means, or stored
`in a database or retrieval system, without the prior w1itten permission of the publisher.
`
`1234567890
`
`This book was produced in IBM Generalized Markup Language by Robert M. Jones and
`Karen S. Devens. Cover design by Michelle Fleitz.
`Printing by Edwards Brothers, Ann Arbor, Ml, I 998.
`
`A CIP catalog record for this book is available from the B,itish Library.
`© The paper in this publication meets the requirements of the ANSI Standard 239.48-
`1984 (Permanence of Paper)
`
`Library of Congress Cataloging-in-Publication Data
`
`(Robert Millard)
`Jones, Robert M.
`Mechanics of composite materials I Robert M. Jones --- 2nd ed.
`cm.
`p.
`Includes bibliographical references and index.
`ISBN 1-56032-712-X (hardcover : alk. paper)
`I. Composite materials --- Mechanical properties. 2. Laminated
`I. Title.
`materials --- Mechanical properties.
`TA418.9.C6J59 1999
`620.1 'I 892---dc2 I
`
`98-18290
`CIP
`
`ISBN: 1-56032-712-X (hardcover)
`
`CONTENTS
`
`PREFACE TO THE SECOND EDITION xiii
`
`PREFACE TO THE FIRST EDITION xv
`
`1
`1
`2
`
`1 INTRODUCTION TO COMPOSITE MATERIALS.......................................
`1.1 INTRODUCTION...................................................................................
`1.2 THE WHAT-WHAT IS A COMPOSITE MATERIAL?.....................
`1.2.1 Classification and Characteristics of Composite Materials 2
`1.2.1. 1 Fibrous Composite Materials 3
`1 .2.1.2 Laminated Composite Materials 6
`1.2.1.3 Particulate Composite Materials 8
`1.2.1.4 Combinations of Composite Materials 10
`1.2.2 Mechanical Behavior of Composite Materials 11
`1.2.3 Basic Terminology of
`Laminated Fiber-Reinforced Composite Materials 15
`1 .2.3.1 Laminae 15
`1.2.3.2 Laminates 17
`1.2.4 Manufacture of
`Laminated Fiber-Reinforced Composite Materials 18
`1.2.4.1 Initial Form of Constituent Materials 18
`1.2.4.2 Layup 19
`1.2.4.3 Curing 23
`1.3 THE WHY- CURRENT AND POTENTIAL ADVANTAGES
`OF FIBER-REINFORCED COMPOSITE MATERIALS....................... 26
`1.3.1 Strength and Stiffness Advantages 27
`1.3.2 Cost Advantages 31
`1.3.3 Weight Advantages 36
`1.4 THE HOW-APPLICATIONS OF COMPOSITE MATERIALS ......... 37
`1.4.1 Introduction 37
`1.4.2 Military Aircraft 38
`1.4.2.1 General Dynamics F-111 Wing-Pivot Fitting 38
`1.4.2.2 Vought A-7 Speedbrake 40
`1.4.2.3 Vought S-3A Spoiler 42
`1.4.2.4 Boeing F-18 43
`1.4.2.5 Boeing AV-BB Harrier 44
`1.4.2.6 Grumman X-29A 45
`1.4.2.7 Northrop Grumman B-2 45
`1.4.2.8 Lockheed Martin F-22 46
`1.4.3 Civil Aircraft 47
`1.4.3.1 Lockheed L-1011 Vertical Fin 47
`1.4.3.2 Rutan Voyager 48
`1.4.3.3 Boeing 777 49
`1.4.3.4 High-Speed Civil Transport 49
`1.4.4 Space Applications 50
`1.4.5 Automotive Applications 50
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`vi Content}
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`1.4.6 Commercial Applications 52
`1.5 SUMMARY............................................................................................ 52
`Problem Set 1 53
`
`REFERENCES 53
`
`2 MACROMECHANICAL BEHAVIOR OF A LAMINA .................................. 55
`2.1
`INTRODUCTION ................................................................................... 55
`2.2 STRESS-STRAIN RELATIONS FOR ANISOTROPIC MATERIALS.. 56
`2.3 STIFFNESSES, COMPLIANCES, AND
`ENGINEERING CONSTANTS FOR ORTHOTROPIC MATERIALS ... 63
`2.4 RESTRICTIONS ON ENGINEERING CONSTANTS ........................... 67
`2.4.1
`Isotropic Materials 67
`2.4.2 Orthotropic Materials 68
`Problem Set 2.4 70
`2.5 STRESS-STRAIN RELATIONS FOR PLANE STRESS
`IN AN ORTHOTROPIC MATERIAL..................................................... 70
`2.6 STRESS-STRAIN RELATIONS FOR
`A LAMINA OF ARBITRARY ORIENTATION ...................................... 74
`Problem Set 2.6 84
`2.7 INVARIANT PROPERTIES OF AN ORTHOTROPIC LAMINA ........... 85
`Problem Set 2.7 87
`2.8 STRENGTHS OF AN ORTHOTROPIC LAMINA................................. 88
`2.8.1 Strength Concepts 88
`2.8.2 Experimental Determination of Strength and Stiffness 91
`2.8.3 Summary of Mechanical Properties 100
`Problem Set 2.8 102
`2.9 BIAXIAL STRENGTH CRITERIA FOR AN ORTHOTROPIC LAMINA 102
`2.9.1 Maximum Stress Failure Criterion 106
`2.9.2 Maximum Strain Failure Criterion 107
`2.9.3 Tsai-Hill Failure Criterion 109
`2.9.4 Hoffman Failure Criterion 112
`2.9.5 Tsai-Wu Tensor Failure Criterion 114
`2.9.6 Summary of Failure Criteria 118
`Problem Set 2. 9 118
`2.10 SUMMARY .•......................•................................................................. 118
`REFERENCES 119
`
`3 MICROMECHANICAL BEHAVIOR OF A LAMINA .................................... 121
`3.1
`INTRODUCTION ................................................................................... 121
`3.2 MECHANICS OF MATERIALS APPROACH TO STIFFNESS ........... 126
`3.2.1 Determination of E1 127
`3.2.2 Determination of E2 129
`3.2.3 Determination of v12 132
`3.2.4 Determination of G12 133
`3.2.5 Summary Remarks 135
`
`Contents vii
`
`Problem Set 3.2 135
`3.3 ELASTICITY APPROACH TO STIFFNESS ........................................ 137
`3.3.1
`Introduction 137
`3.3.2 Bounding Techniques of Elasticity 137
`3.3.3 Exact Solutions 145
`3.3.4 Elasticity Solutions with Contiguity 147
`3.3.5 The Halpin-Tsai Equations 151
`3.3.6 Summary Remarks 157
`Problem Set 3.3 158
`3.4 COMPARISON OF APPROACHES TO STIFFNESS .......................... 158
`3.4.1 Particulate Composite Materials 158
`3.4.2 Fiber-Reinforced Composite Materials 160
`3.4.3 Summary Remarks 163
`3.5 MECHANICS OF MATERIALS APPROACH TO STRENGTH ........... 163
`3.5.1 Introduction 163
`3.5.2 Tensile Strength in the Fiber Direction 164
`3.5.3 Compressive Strength in the Fiber Direction 171
`Problem Set 3.5 184
`3.6 SUMMARY REMARKS ON MICROMECHANICS ............................... 184
`REFERENCES 185
`
`4 MACROMECHANICAL BEHAVIOR OF A LAMINATE .............................. 187
`4.1 INTRODUCTION ................................................................................... 187
`Problem Set 4.1 190
`4.2 CLASSICAL LAMINATION THEORY .................................................. 190
`4.2.1 Lamina Stress-Strain Behavior 191
`4.2.2 Stress and Strain Variation in a Laminate 191
`4.2.3 Resultant Laminate Forces and Moments 195
`4.2.4 Summary 199
`Problem Set 4.2 202
`4.3 SPECIAL CASES OF LAMINATE STIFFNESSES ............................. 203
`4.3.1 Single~Layered Configurations 203
`4.3.2 Symmetric Laminates 206
`4.3.3 Antisymmetric Laminates 214
`4.3.4 Unsymmetric Laminates 218
`4.3.5 Common Laminate Definitions 219
`4.3.6 Summary Remarks 221
`Problem Set 4.3 222
`4.4 THEORETICAL VERSUS MEASURED LAMINATE STIFFNESSES 222
`4.4.1 Inversion of Stiffness Equations 222
`4.4.2 Special Cross-Ply Laminate Stiffnesses 224
`4.4.3 Theoretical and Measured Cross-Ply Laminate Stiffnesses 229
`4.4.4 Special Angle-Ply Laminate Stiffnesses 232
`4.4.5 Theoretical and Measured Angle-Ply Laminate Stiffnesses 235
`4.4.6 Summary Remarks 237
`Problem Set 4.4 237
`4.5 STRENGTH OF LAMINATES .............................................................. 237
`4.5.1 Introduction 237
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`4.5.2 Laminate Strength-Analysis Procedure 240
`4.5.3 Thermal and Mechanical Stress Analysis 242
`4.5.4 Hygroscopic Stress Analysis 245
`4.5.5 Strength of Cross-Ply Laminates 246
`4.5.6 Strength of Angle-Ply Laminates 255
`4.5.7 Summary Remarks 258
`Problem Set 4.5 260
`4.6 INTERLAMINAR STRESSES ............................................................... 260
`4.6.1 Classical Lamination Theory 262
`4.6.2 Elasticity Formulation 264
`4.6.3 Elasticity Solution Results 267
`4.6.4 Experimental Confirmation of lnterlaminar Stresses 269
`4.6.5 lnterlaminar Stresses in Cross-Ply Laminates 271
`4.6.6 Implications of lnterlaminar Stresses 272
`4.6.7 Free-Edge Delamination-Suppression Concepts 274
`Problem Set 4.6 275
`REFERENCES 275
`
`5 BENDING, BUCKLING, AND VIBRATION OF LAMINATED PLATES •...• 277
`5.1 INTRODUCTION ................................................................................... 277
`5.2 GOVERNING EQUATIONS FOR BE~, BUCKLING, AND
`VIBRATION OF LAMINATED PLATES ............................................... 279
`5.2.1 Basic Restrictions, Assumptions, and Consequences 279
`5.2.2 Equilibrium Equations for Laminated Plates 282
`5.2.3 Buckling Equations for Laminated Plates 285
`5.2.4 Vibration Equations for Laminated Plates 288
`5.2.5 Solution Techniques 288
`5.3 DEFLECTION OF SIMPLY SUPPORTED LAMINATED PLATES
`UNDER DISTRIBUTED TRANSVERSE LOAD ................................... 289
`5.3.1 Specially Orthotropic Laminated Plates 290
`5.3.2 Symmetric Angle-Ply Laminated Plates 291
`5.3.3 Antisymmetric Cross-Ply Laminated Plates 295
`5.3.4 Antisymmetric Angle-Ply Laminated Plates 298
`Problem Set 5.3 301 -
`5.4 BUCKLING OF SIMPLY SUPPORTED LAMINATED PLATES
`UNDER IN-PLANE LOAD .................................................................... 301
`5.4.1 Specially Orthotropic Laminated Plates 303
`5.4.2 Symmetric Angle-Ply Laminated Plates 306
`5.4.3 Antisymmetric Cross-Ply Laminated Plates 307
`5.4.4 Antisymmetric Angle-Ply Laminated Plates 312
`Problem Set 5.4 315
`5.5 VIBRATION OF SIMPLY SUPPORTED LAMINATED PLATES ••.••••• ·315
`5.5.1 Specially Orthotropic Laminated Plates 315
`5.5.2 Symmetric Angle-Ply Laminated Plates 317
`5.5.3 Antisymmetric Cross-Ply Laminated Plates 318
`5.5.4 Antisymmetric Angle-Ply Laminated Plates 320
`Problem Set 5.5 322
`5.6 SUMMARY REMARKS ON EFFECTS OF STIFFNESSES ................ 323
`REFERENCES 329
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`ix
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`6 OTHER ANALYSIS AND BEHAVIOR TOPICS .......................................... 331
`6.1 INTRODUCTION ................................................................................... 331
`6.2 REVIEW OF CHAPTERS 1 THROUGH 5 ............•...••.......•..........•...... 332
`6.3 FATIGUE ...•......••..•••....•.•.......•.....................................•......................... 333
`6.4 HOLES IN LAMINATES ...•••...................................•..................•.......... 336
`6.5 FRACTURE MECHANICS ...............•..•.....•...•...•......•..•..............•.......... 339
`6.5.1 Basic Principles of Fracture Mechanics 340
`6.5.2 Application of Fracture Mechanics to Composite Materials 343
`6.6 TRANSVERSE SHEAR EFFECTS ••.....••.............•...••.....................•.... 345
`6.6.1 Exact Solutions for Cylindrical Bending 346
`6.6.2 Approximate Treatment of Transverse Shear Effects 350
`6.7 POSTCURING SHAPES OF UNSYMMETRIC LAMINATES .............. 356
`6.8 ENVIRONMENTAL EFFECTS ............................................................. 359
`6.9 SHELLS ........•..••.....•.•..••.......•........................•...............•.....•................ 361
`6.10 MISCELLANEOUS TOPICS ............................................................... 362
`REFERENCES 362
`
`7 INTRODUCTION TO DESIGN OF COMPOSITE STRUCTURES .............. 367
`7.1 INTRODUCTION .•..••.•••••.•....•........................••......•....•.....•......•....•........ 368
`7.1.1 Objectives 368
`7.1.2 Introduction to Structural Design 368
`7.1.3 New Uses of Composite Materials 368
`7.1.4 Manufacturing Processes 368
`7.1.5 Material Selection 369
`7 .1.6 Configuration Selection 369
`7.1.7 Joints· 369
`7 .1 .8 Design Requirements 370
`7.1.9 Optimization 370
`7.1.1 O Design Philosophy 371
`7.1.11 Summary 372
`7.2 INTRODUCTION TO STRUCTURAL DESIGN .................................... 372
`7.2.1 Introduction 372
`7.2.2 What Is Design? 372
`7.2.3 Elements of Design 376
`7.2.4 Steps in the Structural Design Process 380
`7.2.4.1 Structural Analysis 381
`7.2.4.2 Elements of Analysis in Design 381
`7.2.4.3 Failure Analysis 382
`7.2.4.4 Structural Reconfiguration 383
`7.2.4.5 Iterative Nature of Structural Design 384
`7.2.5 Design Objectives and Design Drivers 385
`7.2.6 Design-Analysis Stages 386
`7.2.6.1 Preliminary Design-Analysis 387
`7.2.6.2 Intermediate Design-Analysis 388
`7.2.6.3 Final Design-Analysis 388
`7.2.7 Summary 389
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`7.3 MATERIALS SELECTION .................................................................... 389
`7.3.1 Introduction 389
`7.3.2 Materials Selection Factors 390
`7.3.3 Fiber Selection Factors 391
`7.3.4 Matrix Selection Factors 392
`7.3.5 Importance of Constituents 393
`7.3.6 Space Truss Material Selection Example 394
`7.3.7 Summary 400
`7.4 CONFIGURATION SELECTION .......................................................... 400
`7.4.1 Introduction 400
`7.4.2 Stiffened Structures 400
`7.4.2.1 Advantages of Composite Materials in
`Stiffened Structures 401
`7.4.2.2 Types of Stiffeners 403
`7.4.2.3 Open- versus Closed-Section Stiffeners 405
`7.4.2.4 Stiffener Design 407
`7.4.2.5 Orthogrid 410
`7.4.3 Configuration in Design Cost 411
`7.4.4 Configuration versus Structure Size 413
`7.4.5 Reconfiguration of Composite Structures 414
`7.4.6 Summary 417
`7.5 LAMINATE JOINTS ................................ .:-:-....,_. ...................................... 417
`7.5.1 Introduction 417
`7.5.2 Bonded Joints 419
`7 .5.3 Bolted Joints 420
`7.5.4 Bonded-Bolted Joints 421
`7.5.5 Summary 422
`7.6 DESIGN REQUIREMENTS AND DESIGN FAILURE CRITERIA ....... 422
`7.6.1 Introduction 422
`7.6.2 Design Requirements 422
`7.6.3 Design Load Definitions 424
`7 .6.4 Summary 425
`7.7 OPTIMIZATION CONCEPTS ............................................................... 425
`7.7.1 Introduction 425
`7.7.2 Fundamentals of Optimization 426
`7. 7.2.1 Structural Optimization 426
`7.7.2.2 Mathematics of Optimization 429
`7.7.2.3 Optimization of a Composite Laminate 431
`7.7.2.4 Strength Optimization Programs 435
`7.7.3 Invariant Laminate Stiffness Concepts 440
`7.7.3.1 Invariant Laminate Stiffnesses 440
`7.7.3.2 Special Results for Invariant Laminate Stiffnesses 443
`7.7.3.3 Use of Invariant Laminate Stiffnesses in Design 446
`Problem Set 7.7.3 447
`7.7.4 Design of Laminates 447
`7.7.5 Summary 453
`7.8 DESIGN ANALYSIS PHILOSOPHY FOR
`COMPOSITE STRUCTURES ...................•........................................... 453
`7.8.1 Introduction 453
`7.8.2 Problem Areas 454
`7.8.3 Design Philosophy 455
`
`Contents xi
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`7.8.4 'Anisotropic' Analysis 455
`7.8.5 Bending-Extension Coupling 456
`7.8.6 Micromechanics 457
`7.8.7 Nonlinear Behavior 458
`7.8.8 lnterlaminar Stresses 459
`7.8.9 Transverse Shearing Effects 460
`7.8.10 Laminate Optimization 461
`7.8.11 Summary 462
`7 .9 SUMMARY ...............................................•............................................ 463
`REFERENCES 465
`
`APPENDIX A: MATRICES AND TENSORS .................................................. 467
`A.1 MATRIX ALGEBRA .............................................................................. 467
`A.1.1 Matrix Definitions 467
`A.1.2 Matrix Operations 470
`A.2 TENSORS ...........•................................................................................. 472
`A.2.1 Transformation of Coordinates 473
`A.2.2 Definition of Various Tensor Orders 474
`A.2.3 Contracted Notation 475
`A.2.4 Matrix Form of Tensor Transformations 476
`REFERENCE 477
`
`APPENDIX B: MAXIMA AND MINIMA OF
`FUNCTIONS OF A SINGLE VARIABLE ....................•.......... 479
`REFERENCE 483
`
`APPENDIX C: TYPICAL STRESS-STRAIN CURVES .................................. 485
`C.1 FIBERGLASS-EPOXY STRESS-STRAIN CURVES ....•...................... 485
`C.2 BORON-EPOXY STRESS-STRAIN CURVES ..................................... 485
`C.3 GRAPHITE-EPOXY STRESS-STRAIN CURVES ..•••........•........•......... 485
`REFERENCES 494
`
`APPENDIX D: GOVERNING EQUATIONS FOR BEAM EQUILIBRIUM AND
`PLATE EQUILIBRIUM, BUCKLING, AND VIBRATION ....... 495
`D.1 INTRODUCTION ................................................................................... 495
`D.2 DERIVATION OF BEAM EQUILIBRIUM EQUATIONS ........•...........•.. 495
`D.3 DERIVATION OF PLATE EQUILIBRIUM EQUATIONS ..................... 498
`D.4 PLATE BUCKLING EQUATIONS ........................................................ 505
`D.5 PLATE VIBRATION EQUATIONS ........•.•..........•.•........•............•......... 506
`REFERENCES 506
`
`INDEX ............................................................................................................... 507
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`

`PREFACE TO THE SECOND EDITION
`
`More than two decades have passed since the first edition of this
`book appeared in 1975. During that time, composite materials have
`progressed from almost an engineering curiosity to a widely used mate(cid:173)
`rial in aerospace applications, as well as many other applications in ev(cid:173)
`eryday life. Accordingly, the contents of the first edition, although in most
`respects timeless fundamental mechanical behavior and mechanics an(cid:173)
`alyses, must be expanded and updated.
`The specific revisions include more thorough explanation of many
`concepts, enhanced comparisons between theory and experiment, more
`reader-friendly figures, figures that are more visually obvious in portrayal
`of fibers and deformations, description of experimental measurements
`of properties, expanded coverage of lamina failure criteria including an
`evaluation of how failure criteria are obtained, and more comprehensive
`description of laminated plate deflection, buckling, and vibration prob(cid:173)
`lems. Moreover, laminate design is introduced as part of the structural
`design process.
`The 'latest' research results are deliberately not included. That is,
`this book is a fundamental teaching text, not a monograph on contem(cid:173)
`porary composite materials and structures topics. Thus, topics are cho(cid:173)
`sen for their importance to the basic philosophy which includes simplicity
`of presentation and 'absorbability' by newcomers to composite materials
`and structures. More advanced topics as well as the nuances of covered
`topics can be addressed after this book is digested.
`I have come to expect my students to interpret or transform the
`sometimes highly abbreviated, and thus relatively uninformative, problem
`set statements at the end of each section such as 'derive Equation (3.86)'
`to the more formal, descriptive, and revealing form:
`
`Given:
`Required:
`
`A composite material is to be designed.
`Find the critical fiber-volume fraction necessary to ensure
`that the composite material strength exceeds the matrix
`strength, i.e., derive Equation (3.86).
`
`Moreover, I expect students to explain on a physical basis where they
`start and what objectives they're trying to meet. In doing so, they should
`carefully explain the nature of the problem as well as its solution. I want
`students to gain some perspective on the problem to more fully under-
`
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`xiv Preface
`
`stand the text. That is, I want them to focus on The Why of each problem
`so they will develop a feeling for the behavior of composite materials and
`structures.
`I also expect use of appropriate figures that are well dis(cid:173)
`cussed. Figures that have not been fully interpreted for the reader are
`of questionable value and certainly leave room for misinterpretation.
`I expect students to explain and describe each step in the
`Also,
`problem-solving process with physically based reasons and explanations.
`Moreover, I expect observations, comments, and conclusions about what
`they learned at the end of each problem.
`I feel such requirements are
`good training for survival in today's and tomorrow's more competitive
`world.
`Completion of the problems will often require thoughtful analysis
`of the conditions and search for the correct solution. Thus, the problems
`are often not trivial or straightforward. The required mathematics are
`senior level except for the elasticity solutions in the micromechanics
`chapter where obviously the level must be higher (but the elasticity
`sections can be skipped in lower-level classes).
`I am delighted to express my appreciation to the attendees of more
`than 80 short courses from 1971 throu911. 1995 at government laborato(cid:173)
`ries, companies, and open locations.
`.' l'\ey helped shape this second
`edition by their questions and comments, as did the more than twenty
`university classes I taught over the years.
`I thank those who offered suggestions and corrections from their
`experience with the first edition. I am also delighted to express my ap(cid:173)
`preciation to those who contributed to both editions: Patrick Barr (now
`M.D.!) for illustrations in the first edition, some of which are used in the
`second edition; Ann Hardell for Adobe Illustrator illustrations in the sec(cid:173)
`ond edition; my daughter, Karen Devens, for IBM Script and GML text
`production; and my secretary, Norma Guynn, for miscellaneous typing.
`
`Blacksburg, Virginia
`April 1998
`
`PREFACE TO THE FIRST EDITION
`
`Composite materials are ideal for structural applications where high
`strength-to-weight and stiffness-to-weight ratios are required. Aircraft
`and spacecraft are typical weight-sensitive structures in which composite
`materials are cost-effective. When the full advantages of composite
`materials are utilized, both aircraft and spacecraft will be designed in a
`manner much different from the present.
`The study of composite materials actually involves many topics,
`such as, for example, manufacturing processes, anisotropic elasticity,
`strength of anisotropic materials, and micromechanics. Truly, no one
`individual can claim a complete understanding of all these areas. Any
`practitioner will be likely to limit his attention to one or two subareas of
`the broad possibilities of analysis versus design, micromechanics versus
`macromechanics, etc.
`The objective of this book is to introduce the student to the basic
`concepts of the mechanical behavior of composite materials. Actually,
`only an overview of this vast set of topics is offered. The balance of
`subject areas is intended to give a fundamental knowledge of the broad
`scope of composite materials. The mechanics of laminated fiber(cid:173)
`reinforced composite materials are developed as a continuing example.
`Many important topics are ignored in order to restrict the coverage to a
`one-semester graduate course. However, the areas covered do provide
`a firm foundation for further study and research and are carefully selected
`to provide continuity and balance. Moreover, the subject matter is cho(cid:173)
`sen to exhibit a high degree of comparison between theory and exper(cid:173)
`iment in order to establish confidence in the derived theories.
`The whole gamut of topics from micromechanics and macrome(cid:173)
`chanics through lamination theory and examples of plate bending,
`buckling, and vibration problems is treated so that the physical signif(cid:173)
`icance of the concepts is made clear. A comprehensive introduction to
`composite materials and motivation for their use in current structural ap(cid:173)
`plications is given in Chapter 1. Stress-strain relations for a lamina are
`displayed with engineering material constants in Chapter 2. Strength
`In Chapter 3,
`theories are also compared with experimental results.
`micromechanics is introduced by both the mechanics of materials ap(cid:173)
`proach and the elasticity approach. Predicted moduli are compared with
`measured values. Lamination theory is presented in Chapter 4 with the
`aid of a new laminate classification scheme. Laminate stiffness pred(cid:173)
`ictions are compared with experimental results. Laminate strength con-
`
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`xvi Pref(
`
`cepts, as well as interlaminar stresses and design, are also discussed.
`In Chapter 5, bending, buckling, _and vibration of a simply supported plate
`with various lamination characteristics is examined to display the effects
`of coupling stiffnesses in a physically meaningful problem. Miscella(cid:173)
`neous topics such as fatigue, fracture mechanics, and transverse shear
`effects are addressed in Chapter 6. Appendices on matrices and
`tensors, maxima and minima of functions of a single variable, and typical
`stress-strain curves are provided.
`This book was written primarily as a graduate-level textbook, but is
`well suited as a guide for self-study of composite materials. Accordingly,
`the theories presented are simple and illustrate the basic concepts, al(cid:173)
`though they may not be the most accurate. Emphasis is placed on an(cid:173)
`alyses compared with experimental results, rather than on the most
`recent analysis for the material currently 'in vogue.' Accuracy may suffer,
`but educational objectives are better met. Many references are included
`to facilitate further study. The background of the reader should include
`an advanced mechanics of materials course or separate courses in which
`three-dimensional stress-strain relations and plate theory are introduced.
`In addition, knowledge of anisotropic elasticity is desirable, although not
`essential.
`Many people have been most generous in their support of this
`writing effort. I would like to especially thank Dr. Stephen W. Tsai, of the
`Air Force Materials Laboratory, for his inspiration by example over the
`past ten years and for his guidance throughout the past several years.
`I deeply appreciate Steve's efforts and those of Dr. R. Bryon Pipes of the
`University of Delaware and Dr. Thomas Cruse of Pratt and Whitney Air(cid:173)
`craft, who reviewed the manuscript and made many helpful comments.
`Still others contributed material for the book. My thanks to Marvin
`Howeth of General Dynamics, Forth Worth, Texas, for many photo(cid:173)
`graphs; to John Pimm of LTV Aerospace Corporation for the photograph
`in Section 4.7; to Dr. Nicholas Pagano of the Air Force Materials Labo(cid:173)
`ratory for many figures; to Dr. R. Byron Pipes of the University of
`Delaware for many photographs and figures in Section 4.6; and to Dr.
`B. Walter Rosen of Materials Sciences Corporation, Blue Bell,
`I also appreciate the per(cid:173)
`Pennsylvania, for the photo in Section 3.5.
`mission of the Technomic Publishing Company, Inc., of Westport,
`Connecticut, to reprint throughout the text many figures which have ap(cid:173)
`peared in the various Technomic books and in the Journal of Composite
`Materials over the past several years.
`I am very grateful for support by
`the Air Force Office of Scientific Research (Directorate of Aerospace
`Sciences) and the Office of Naval Research (Structural Mechanics Pro(cid:173)
`gram) of my research on laminated plates and shells discussed in
`Chapters 5 and 6. I am also indebted to several classes at the Southern
`Methodist Institute of Technology and the Naval Air Development Center,
`Warminister, Pennsylvania, for their patience and help during the devel(cid:173)
`opment of class notes that led to this book. Finally, I must single out
`Harold S. Morgan for his numerous contributions and Marty Kunkle for
`her manuscript typing (although I did some of the typing myself!).
`
`R.M.J.
`
`First Edition:
`
`Second Edition:
`
`To my neglected family:
`Donna, Mark, Karen, and Christopher
`
`To Christopher:
`He helped many others,
`but he couldn't help himself
`
`ClearCorrect Exhibit 1045, Page 10 of 270
`
`

`

`(
`
`Chapter 1
`
`INTRODUCTION TO
`COMPOSITE MATERIALS
`
`1.1 INTRODUCTION
`
`The objective of this chapter is to address the three basic questions
`of composite materials and structures in Figure 1-1: (1) What is a com(cid:173)
`posite material?
`(2) Why are composite materials used instead of
`metals? and (3) How are composite materials used in structures? As
`part of The What, the general set of composite materials will be defined,
`classified, and characterized. Then, our attention will be focused on
`laminated fiber-reinforced composite materials for this book. Finally, to
`help us understand the nature of the material we are trying to model with
`mechanics equations, we will briefly describe manufacturing of composite
`materials and structures. In The Why, we will investigate the advantages
`of composite materials over metals from the standpoints of strength,
`stiffness, weight, and cost among others. Finally, in The How, we will
`look into examples and short case histories of important structural appli(cid:173)
`cations of composite materials to see even more reasons why composite
`materials play an ever-expanding role in today's and tomorrow's struc(cid:173)
`tures.
`
`•THE WHAT
`WHAT IS A COMPOSITE MATERIAL?
`•THE WHY
`WHY ARE COMPOSITE MATERIALS USED INSTEAD OF METALS?
`•THE HOW
`HOW ARE COMPOSITE MATERIALS USED IN STRUCTURAL APPLICATIONS?
`
`Figure 1-1 Basic Questions of Composite Materials and Structures
`
`ClearCorrect Exhibit 1045, Page 11 of 270
`
`

`

`2 Mechanics u1 Composite Materials
`
`1.2 THE WHAT-WHAT IS A COMPOSITE MATERIAL?
`
`The word composite in the term composite material signifies that
`two or more materials are combined on a macroscopic scale to form a
`useful third material. The key is the macroscopic examination of a ma(cid:173)
`terial wherein the components can be identified by the naked eye. Dif(cid:173)
`ferent materials can be combined on a microscopic scale, such as in
`alloying of metals, but the resulting material is, for all practical purposes,
`macroscopically homogeneous, i.e., the components cannot be distin(cid:173)
`guished by the naked eye and essentially act together. The advantage
`of composite materials is that, if well designed, they usually exhibit the
`best qualities of their components or constituents and often some quali(cid:173)
`ties that neither constituent possesses. Some of the properties that can
`be improved by forming a composite material are
`
`• strength
`• stiffness
`• corrosion resistance
`• wear resistaRc-e
`• attractiveness
`• weight
`
`• fatigue life
`• temperature-dependent behavior
`• thermal insulation
`• thermal c-enducttv#y
`• acoustical insulation
`
`Naturally, not all of these properties are improved at the same time nor
`is there usually any requirement to do so. In fact, some of the properties
`are in conflict with one another, e.g., thermal insulation versus thermal
`conductivity. The objective is merely to create a material that has only
`the characteristics needed to perform the design task.
`Composite materials have a long history of usage. Their precise
`beginnings are unknown, but all recorded history contains references to
`some form of composite material. For example, straw was used by the
`Israelites to strengthen mud bricks. Plywood was used by the ancient
`Egyptians when they realized that wood could be rearranged to achieve
`superior strength and resistance to thermal expansion as well as to
`swelling caused by the absorption of moisture. Medieval swords and
`armor were constructed with layers of different metals. More recently,
`fiber-reinforced, resin-matrix composite materials that have high strength(cid:173)
`to-weight and stiffness-to-weight ratios have become important in weight(cid:173)
`sensitive applications such as aircraft and space vehicles.
`
`1.2.1 Classification and Characteristics of Composite Materials
`
`Four commonly accepted types of composite materials are:
`
`(1) Fibrous composite materials that consist of fibers in a matrix
`(2) Laminated composite materials that consist of layers of various
`materials
`(3) Particulate composite materials that are composed of particles
`in a matrix
`(4) Combinations of some or all of the first three types
`
`(
`Introduction to Composite Materials 3
`
`These types of composite materials are described and discussed in the
`following subsections. I am indebted to Professor A. G. H. Dietz (1-1] f