Design and Optimization of Laminated Composite Materials [1 ed.] 9780471252764

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Design and Optimization of Laminated Composite Materials. Zafer Gurdal Virginia Polytechnic Institute and State University

Raphael T. Haftka University of Florida

Prabhat Hajela Rensselaer Polytechnic Institute

@ A Wiley-lnterscience Publication

JOHN WILEY & SONS, INC. New York I Chichester I Weinheim I Brisbane I Singapore I Toronto

CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi This book is printed on acid-free paper.

§ 1

Introduction

1

Copyright© 1999 by John Wiley & Sons. All rights reserved.

Introduction to Composite Materials I 4 1.1.1 Classification of Composite Materials I 5 1.1.2 Fiber-Reinforced Composite Materials I 8 1.2 Properties of Laminated Composites I 9 1.2.1 Material Orthotropy I 9 1.2.2 Rule of Mixtures, Complementation, and Interaction I 11 1.2.3 Laminate Definition I 16 1.3 Design of Composite Laminates I 19 1.3.1 Historical Perspective I 19 1.3.2 Material-Related Design Issues I 21 1.4 Design Optimization I 24 1.4.1 Mathematical Optimization I 24 1.4.2 Stacking Sequence Optimization I 29 Exercises I 31 References I 32 1.1

Published simultaneously in Canada. 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 as permitted under Section 107 or 108 or the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: [email protected]. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with 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 person should be sought.

Library of Congress Cataloging-in-Publication Data: Giirdal, Zafer. Design and optimization of laminated composite materials/Zafer Giirdal, Raphael T. Haftka, Prabhat Hajela. p. em. Includes index. ISBN 0-471-25276-X (hardcover : alk. paper) I. Laminated materials. 2. Composite materials. 3. Structural optimization. I. Haftka, Raphael T. II. Hajela, Prabhat, 1956- . III. Title. TA418.9.L3G87 1998 620.1'18--dc21 98-22855 Printed in the United States of America 10 9 8 7 6 5 4 3 2 I

2

Mechanics of Laminated Composite Materials . . . . . . 33

Governing Equations for Elastic Medium I 34 2.1.1 Strain-Displacement Relations I 34 2.1.2 Stress-Strain Relations I 34 2.1.3 Equilibrium Equations I 36 2.2 In-Plane Response of Isotropic Layer(s) I 39 2.2.1 Plane Stress I 39 2.2.2 Single Isotropic Layer I 41

2.1

v

VI

CONTENTS

2.2.3 Symmetrically Laminated Layers I 43 2.3 Bending Deformations of Isotropic Layer(s) I 49 2.3.1 Bending Response of a Single Layer I 50 2.3.2 Bending Response of Symmetrically Laminated Layers I 52 2.3.3 Bending-Extension Coupling of Unsymmetrically Laminated Layers I 54 2.4 Orthotropic Layers I 61 2.4.1 Stress-Strain Relations for Orthotropic Layers I 62 2.4.2 Orthotropic Layers Oriented at an Angle I 64 2.4.3 Laminates of Orthotropic Plies I 69 2.4.4 Elastic Properties of Composite Laminates I 73 2.5 Properties of Laminates Made of Sublaminates I 83 Exercises I 87 References I 88

3

Hygrothermal Analysis of Laminated Composites . . .

4.1.2 Mathematical Optimization Formulation I 133 4.2 Graphical Solution Procedures I 138 4.2.1 Optimization of Orientations of Layers I 138 4.2.2 Graphical Design of Coefficients of Thermal Expansion I 153 4.2.3 Optimization of Stack Thicknesses I 155 4.3 Dealing with the Discreteness of the Design Problem I 160 Exercises I 166 References I 167

5 Integer Programming

3.1

4

Laminate In-Plane Stiffness Design . . . . . . . . . . . 127 4.1

Design Optimization Problem Formulation I 128 4.1.1 Design Formulation of In-Plane Stiffness Problem I 128

. . . . . . . . . 169

5.1 Integer Linear Programming I 170 5.2 In-plane Stiffness Design as a Linear Integer Programming Problem I 172 5.3 Solution of Integer linear Programming Problems I 177 5.3.1 Enumeration I 177 5.3.2 Branch-and-Bound Algorithm I 180 5.4 Genetic Algorithms I 184 5 .4.1 Design Coding I 185 5.4.2 Initial Population I 188 5.4.3 Selection and Fitness I 191 5.4.4 Crossover I 201 5.4.5 Mutation I 206 5.4.6 Permutation, Ply Addition, and Deletion I 208 5.4.7 Computational Cost and Reliability I 209 Exercises I 218 References I 219

89

Hygrothermal Behavior of Composite Laminates I 90 3.1.1 Temperature and Moisture Diffusion in Composite Laminates I 91 3.1.2 Hygrothermal Deformations I 94 3.1.3 Residual Stresses I 99 3.1.4 Hygrothermal Laminate Analysis and Hygrothermal Loads I 102 3.1.5 Coefficients of Hygrothermal Laminate Expansion I 105 3.2 Laminate Analysis for Combined Mechanical and HygrothermaL Loads I 108 3.3 Hygrothermal Design Considerations I 118 Exercises I 124 References I 125

vii

CONTENTS

6

Failure Criteria for Laminated Composites 6.1

. . . . . . 223

Failure Criteria for Isotropic Layers I 226 6.1.1 Maximum Normal Stress Criterion I 226 6.1.2 Maximum Strain Criterion I 227 6.1.3 Maximum Shear Stress (Tresca) Criterion I 229 6.1.4 Distortional Energy (von Mises) Criterion I 230

VIII

CONTENTS

Failure of Fiber-Reinforced Orthotropic Layers I 231 6.2.1 Maximum Stress and Maximum Strain Criteria I 233 6.2.2 Tsai-Hill Criterion I 237 6.2.3 Tsai-Wu Criterion I 241 6.3 Failure of Laminated Composites I 245 6.3.1 Failure under In-Plane Loads I 249 6.3.2 Failure under Bending Loads I 257 Exercises I 258 References I 259 6.2

7 Strength Design of Laminates .

7.1

. . . . . . . . . . . 261

Graphical Strength Design I 262

7 .I .1 Design for Specified Laminate Strain Limits I 262 7.1.2 Design of Laminates with Two-Fiber Orientations I 266 7.1.3. Design of Multiple-Ply Laminates with Discrete Fiber Orientations I 271 7.2 Numerical Strength Optimization Using continuous Variables I 274 7.2.1

Strength Design with Thickness Design Variables I 274 7 .2.2 Strength Design with Orientation Angle Design Variables I 286 7.3 Numerical Strength Optimization Using Discrete Variables I 288 7.3.1

Integer Linear Programming for Strength Design I 289 7.3.2 Genetic Algorithms for Strength Design I 292 Exercises I 294 References I 295 8

Laminate Design for Flexural and Combined Response . . . . . . . . . . . . . . . . . . . . . . . . . 297

8.1

Flexural response Equations I 298

8.2 Stiffness Design by Miki's Graphical Procedure I 301

CONTENTS

IX

8.2.1 Linear Problems I 30 l 8.2.2 Changes in Stacking Sequence I 306 8.2.3 Nonlinear Problems I 308 8.3 Flexural Stiffness Design by Integer Linear Programming I 310 8.3.1 Ply-Identity and Stack-Identity Design Variables I 310 8.3.2 Stiffness Design with Fixed Thickness I 313 8.3.3 Buckling Load Maximization with Stiffness and Strength Constraints I 317 8.3.4 Stiffness Design for Minimum Thickness I 322 Exercises I 328 References I 328 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

PREFACE

With rapid growth of the use of composite materials in many commercial products ranging from sports equipment to high-performance aircraft, literature on composite materials has proliferated. At the time of the publication of this book a simple search of a popular web site for books with the words "composite materials" in their title yielded more than 250 entries. Many of these titles are well-written textbooks on mechanics of composite materials and have been adopted by educational institutions for introductory courses. As the application of composites to commercial products has increased, so has the need for literature that focuses on the design aspects of these materials. However, the number of titles that focus on the mechanics of composites far outnumbers those dealing with design. In particular, books that focus on optimal design of composite materials virtually do not exist. It is the intent of this book to introduce readers to the emerging field of optimal design of laminated composite materials. The first and the foremost reason for writing this book was the desire to acquaint students with the latest techniques in the field. For many years, designers have treated optimization problems involving composite materials with continuous optimization techniques that were ill suited for these problems. The design of a composite laminate stacking sequence generally involves selecting discrete layer thickness and orientation angles-a discrete optimization problem. Researchers in this field have more recently focused on numerical and graphical methods useful for the solution of such problems; this book mirrors that focus. In particular, the book places emphasis on graphical design techniques developed by Professor Miki from Japan. These techniques allow representation of even the most complicated stacking sequences using two parameters bounded by a parabola and provide extremely valuable insight into the multiplicity of solutions available for laminate design problems. Another important motivation for the book was the need to provide condensed coverage that would be of use to the design engineer. Dexi

'"'

PREFACE

sign of composite materials and structures requires both a thorough understanding of the mechanics of laminated composites and familiarity with optimization techniques that enable designers to find practical laminate configurations in an efficient manner. At present, a student wanting to learn about the application of optimization techniques to composite design will need to take a separate course in each subject. This is somewhat difficult within the constraints of an undergraduate curriculum. The book combines the study of the mechanics of composite laminates with optimization methods that are most useful for the design of such laminates. This book has been developed for senior-level undergraduate or early graduate courses in numerical design methods for laminated composite materials. Applications of composite materials have traditionally originated in weight-critical aerospace structures. More recently, these materials have become popular in civil engineering infrastructure applications (such as bridge and building construction) and mechanical engineering applications (such as mechanisms and lightweight robotic structures). Therefore, the book may be used in aerospace engineering, civil engineering, engineering science and mechanics, and mechanical engineering curricula. In addition, the book has technical material useful for practicing engineers in related fields. Researchers in composite materials are likely to benefit from state of the art methods introduced in the book. The first chapter reviews the types of composite materials in use and the terminology established for their description. The types of composites considered in this book are then identified and their properties are discussed within the context of mechanics. A brief review of the design issues relevant to composite materials is included. The chapter concludes with an introduction to the terminology and formulation of mathematical optimization problems, with special emphasis on laminate design problems. The second and third chapters introduce the basic equations and assumptions used in the analysis of laminated composites under mechanical and thermal loads. They emphasize the computation of elastic properties as functions of variables that can be changed during the design process and the effects of such changes on response quantities such as stresses and strains. Chapter 4 formulates the in-plane stiffness design as an optimization problem and introduces a simple graphical technique for its so-

PREFACE

XIII

lution. Also provided is a technique to handle the discrete nature of the thickness and orientation design variables. Two formal procedures, namely integer linear programming and genetic algorithms, suitable for handling discrete optimization problems specific to composite laminate design are introduced in Chapter 5. In particular, the formulation and solution of the in-plane stiffness design· problem is demonstrated. Chapters 6 and 7 address strength analysis and design, respectively. Commonly used failure criteria for laminated composite materials and their implementation for strength analyses are introduced in Chapter 6. Chapter 7 describes the implementation of strength constraints in design optimization based on graphical and mathematical optimization procedures. Finally, Chapter 8 introduces analysis and design for bending requirements. These include the transverse displacement of a simply supported laminate loaded by transverse loads, its natural vibration frequencies, and the buckling response of a simply supported laminate under in-plane loads. . We have used the material in our respective institutions for a combined senior-level undergraduate and first-year graduate course for several years. For a one-semester course with students who have no previous background in composites or optimization, we recommend that the course cover most of the material in Chapters 1, 2, 4, and 5, and parts of Chapter 6. In addition, it is probably possible to cover material from one more chapter. Depending on the emphasis of the course, Chapters 3, 7, or 8, which focus on thermomechanical, strength, or bending design characteristics, respectively, may be added. It is also possible to cover a combination of sections from the remaining chapters as the instructor sees fit. The authors wish to express their appreciation for many valuable suggestions from former students in courses that led to the development of this book. Thanks also go to the authors' respective universities for providing the opportunities to teach courses directly relevant to the book's content, and their fellow faculty members for providing valuable input and stimulating discussions. Special thanks are also due Professor Mitsunori Miki of the Doshisha University, Japan for introducing us to the graphical representation of laminate optimization problems, which is heavily used in the book. The authors also appreciate the input and suggestions provided by various individuals, in

'"" PREFACE

particular, Professor Valery V. Vasiliev of Moscow State University of Aviation Technology, Professor Ron Kander of Virginia Tech, Professor Giinay Anla~+ of Bogazic;i University, Dr. Walter Dauksher of Boeing, for reviewing some of the chapters of the book. The authors would also like to thank Dr. James H. Starnes, Jr., of NASA Langley Research Center. He has sponsored research leading to many of the results reported in the textbook, and his insight into design issues for composite materials enriched our appreciation of the subject. Zafer Gtirdal Raphael T. Haftka Prabhat Hajela

f,

I

1 INTRODUCTION

Structural designers seek the best possible design, be it a vehicle structure, a civil engineering structure, or a space structure, while using the least amount of resources. The measure of goodness of a design depends on the application, typically related to strength or stiffness, while resources are measured in terms of weight or cost. Therefore, the best design often means either the lowest weight (or cost) with limitations on the stiffness (or strength) properties or the maximum stiffness design with prescribed resources and strength limits. Traditionally, engineers have relied on experience to achieve such designs. For a given application, first a set of essential requirements are identified and designs that satisfy these requirements are obtained. Next, structural modifications that are likely to improve the performance or reduce the weight or the cost are implemented. The tasks, stated so simply in two sentences, are often extremely tedious and require a large number of iterations by the designer. For example, changes in the structural dimensions (cross-sectional areas, thicknesses, member lengths) implemented to improve the performance may yield designs that violate the strength or stiffness requirements. Sometimes these requirements are difficult to satisfy and may require many iterations by the designer. In some cases, it is so difficult to satisfy these requirements that the first design ever identified to satisfy them becomes the final design. 1

3 2

INTRODUCTION

Over the past two decades, mathematical optimization, which deals with either the maximization or minimization of an objective function subject to constraint functions, has emerged as a powerful tool for structural design. The use of mathematical optimization for design relieves the designer of the burden of repeated iterations and transforms the design process into a systematic well-organized activity. Within the context of optimization, the weight or the performance becomes the objective function and the variables controlling the structural dimensions (such as thicknesses or cross-sectional areas of members) are design variables that are sized to achieve the best configuration. Formulation of the engineering design optimization, structural optimization, and the methods of solution are topics covered in extensive detail in a number of textbooks;, see, for example, Kirsch (1993), Vanderplaats (1998), Arora (1989), and Haftka and Gtirdal (1993). However, none of these books covers in detail a subset of the field of structural optimization, namely optimization of structures made of composite materials, with the exception of the last one, which has a chapter devoted to optimization of laminated composites. Use of composite materials in structural design has gained popularity over the past three decades because of several advantages that these materials offer compared to traditional structural materials, such as steel, aluminum, and various alloys. One of the primary reasons for their popularity is their weight advantage. Composite materials such as Graphite/Epoxy and Glass/Epoxy have smaller weight densities compared to metallic materials. For example, the weight densities of high-strength Graphite/Epoxy and Glass/Epoxy are 0.056 lb/in3 and 0.065 lblin 3 , respectively, compared to the weight density of Aluminum which is 0.10 lb/in 3 • In addition to their weight advantage per unit volume, some composites provide better stiffness and strength properties compared to metals as well. That is, structural members made out of composite materials may undergo smaller deformations, and carry larger static loads than their metallic counterparts. Stiffness of high strength Graphite/Epoxy, for example, is around 22 x 106 lb/in 2 compared to Aluminum's stiffness of 10 x 106 lb/in 2 . The weight advantage may also be reflected in the performance properties described above by introducing specific stiffness and specific strength, defined by dividing the respective property with the material density. Therefore, even if the stiffness and/or strength performance of a composite material is comparable to that of a conventional alloy, the advantages

INTRODUCTION

of high specific stiffness and/or specific strength make composites more attractive than alloys. Composite materials are also known to perform better under cyclic loads than metallic materials because of their fatigue resistance. It was these advantages that initially stimulated widespread use of composites in structural design, especially for weight-critical applica~ tions such as design of spacecraft and aircraft; see, for example Niu (1992) and Zagainov and Lozino-Lozinski (1996). Beyond their weight advantage, the use of composite materials in structural design had significant implications in terms of the design problem formulation, particularly in terms of optimal design. As will be explained in · this chapter and in a subsequent chapter, with the introduction of composite materials into a design problem formulation, designers obtained a new flexibility through the use of variables that directly change the properties of the material, and therefore optimal design of structures has acquired a new meaning. In order to improve structural performance and meet the requirements of a specific design situation, it is now possible to tailor the properties of the structural material in addition to structural dimensions. Design of a material, however, is not a task that can be taken lightly. Many of the properties of composite materials are unfamiliar to many design engineers who have no formal training in composite materials. There are also various composite material response mechanisms that are not fully understood and are still topics for continuing research. Therefore, most designers limit themselves to aspects of material response that are rather well defined for engineering applications. Another area which is not fully investigated is the integration of the design and manufacturing aspects of composite structures. There are a number of different ways to manufacture a composite part or a structure, but not all configurations designed to tailor the material properties are manufacturable. Therefore, most designers limit themselves to configurations known to be manufacturable, thereby limiting the performance gains that can be achieved through the use of composites. However, the fast pace of advances in composite material manufacturing technology is forging ahead of the present state of established design practices. Therefore, there is a continued need for developing tools and methodologies for design of composite materials and structures to achieve maximal performance gains.

4

INTRODUCTION

The application of the methods of mathematical optimization to the design of structures made of composite materials attracted the attention of many researchers. However, the approach has not been fully accepted by practicing engineers, primarily because of the impracticality of many designs obtained via the optimization process. Early design optimization research studies employed mainly extensions of the approaches used for designing structures made of traditional materials and lacked the ability to handle features that are unique to composite materials. For example, composite material design calls for use of design variables that assume values only from a discrete set of prescribed values. Use of traditional optimization techniques, which treat design variables as continuous valued, produces designs with limited practical value. Only recently has research on the design of composite structures started to introduce features that are important to .attain practical designs. This book attempts to address the unique design features associated with a limited class of composite materials, called laminated composite materials, via the application of the methods of mathematical optimization. In subsequent sections of this chapter, we first provide an introduction to the types of composite materials in use, with the intent of familiarizing the reader with general terminology in the materials field. The types of composites t.hat are specifically considered in this book are then identified, and the properties of such composites are discussed within the context of mechanics. A very brief and probably incomplete review of composite material design issues is included to warn designers about the pitfalls associated with tailoring the properties of a composite material. The chapter concludes with a brief discussion of the terminology and formulation of mathematical optimization problems, with a special emphasis on laminate design problems.

1.1

INTRODUCTION TO COMPOSITE MATERIALS

The most general definition of a composite material is very closely related to the dictionary definition of the word composite, meaning made up of different parts or materials. Composite materials are constructed of two or more materials, commonly referred to as constituents, and have characteristics derived from the individual constituents. Depending on the manner in which the constituents are put together,

1.1

INTRODUCTION TO COMPOSITE MATERIALS

5

the resulting composite materials may have the combined characteristics of the constituents or have substantially different properties than the individual constituents. As explained later, sometimes the properties of the composite even exceed those of the constituents. The following classification is given by Schwartz ( 1984).

1.1.1

Classification of Composite Materials

One approach to the general classification of composite materials is based on the nature of the constituent materials. The constituents may be either organic or inorganic. The designation of organic refers to materials originating from plants or animals or materials of hydrocarbon origin (natural or synthetic) such as carbon. Inorganic materials are those that cannot be classified as organic matter, for example, metals and minerals. Historically, perhaps the most common structural material of organic nature has been wood. Wood is a composite material because it is composed of two distinct constituents: stiff and strong fibers surrounded by a supporting structure of softer cells. In fact, most modern composite materials imitate wood in that they consist of strong fibers embedded in softer supporting material. The supporting material in such an arrangement is commonly referred to as matrix material. Before getting into a more complete description of the modern composite materials, we continue with general descriptions. For most highly loaded single-material structures, the use of purely organic materials is avoided because of their generally low stiffness and strength properties. They are also sensitive to environmental effects such as temperature and moisture and have low resistance to chemicals and solvents. Despite their disadvantages, organic materials such as polymers are commonly included in composites to bring in certain characteristics that may not be achievable by using only inorganic constituents. For inorganic composites, the constituents may be all metallic, all nonmetallic, or a combination of the two. The most common inorganic materials used for highly loaded structures are metals. Despite their good stiffness and strength properties, metals have high specific weight-a disadvantage in weight-critical applications, such as aircraft and spacecraft. Therefore, metallic composites are used mostly when there is need for specific properties, such as resistance to high temperatures. As a matter of fact, in contrast to single

8

INTRODUCTION

r !

1.1

material structures, which are mostly metallic, most frequently used composite materials are made up of all nonmetallic constituents, which include polymeric constituents as well as inorganic glasslike substances. Polymer matrix materials are made of long chain of organic molecules, generally classified as thermoset or thermoplastic. The two categories have substantially different characteristics as matrix materials. Fibers preimpregnated with thermoset matrix (thermoset prepregs) are soft and malleable. They are tacky and can be draped easily, hence allowing fabrication of t:omplex shapes with ease. They require low processing temperatures but long curing times. They undergo a chemical change (cross-linking of polymer chains) during curing, and the process is irreversible. Thermoplastics, on the other hand, do not require curing and they can be reshaped and reused. However, thermoplastic prepregs are hard and boardlike and lack drape and tack; making them harder to handle. They also require very high processing temperatures to shape them, although processing time can be very short. Thermoplastics also offer high resistance to moisture and can operate at higher temperatures of up to 500°F (260°C) compared to thermoset operating temperatures of about 250°F (121 °C). Typical thermoset matrices include Epoxy, Polyimide, Polyester, and Phenolic materials. Among popular thermoplastics are Polyethylene, Polystyrene, and Polyetheretherketone (PEEK) materials. A more complete discussion of advantages and disadvantages of thermosets and thermoplastics is provided by Niu (1992). A more traditional classification than the one based on the nature of the constituents is derived from their form. Several examples of composite materials with different forms are presented in Fig. 1.1. Particulate composites are generally made up of a randomly dispersed hard particle constituent in a softer matrix. Examples of particulate composites are metal particles in metallic, plastic, or ceramic matrices. A widely used particulate composite is concrete in which gravel is mixed in cement. Flake composites, as the name suggests, are formed by adding thin flakes to the matrix material. Although flake dispersion in the matrix is generally random, the flakes may be made to align with one another, forming a more orderly structure compared to particulate composites. This alignment provides for more uniform properties in the plane of the flakes than with particulate composites. Typical examples of flake

7

INTRODUCTION TO COMPOSITE MATERIALS

Particulate composite

Fiber reinforced composite

Flake composite

Laminated composite

Figure 1.1. Composite materials with different forms of constituents.

materials are glass, mica, metals, and carbon. The size, shape, and material of the flake to be used depend on the type of application, which also determines the amount of matrix material. The matrix material (either plastics, metals, or epoxy resins) used in a flake composite may make up the bulk of the composite or be in small amounts just sufficient to provide bonding of the flakes. For example, aluminum flakes are sometimes used in molded plastic parts to provide decorative effects. In such a composite, the plastic matrix makes up the bulk of the composite. Fiber-reinforced composites (or fibrous composites) are the most commonly used form of the constituent combinations. The fibers of such composites are generally strong and stiff and therefore serve as the primary load-carrying constituent. The matrix holds the fibers together and serves as an agent to redistribute the loads from a broken fiber to the adjacent fibers in the material when fibers start failing under excessive loads. This property of the matrix constituent contributes to one of the most important characteristics of the fibrous composites, namely, improved strength compared to the individual constituents. This is elaborated upon in Section 1.2.2, which deals with the different mechanisms by which the properties of composites are obtained from those of its constituents.

l

INTRODUCTION

The last form of composite materials is thin layers of material fully bonded together to form so-called composite laminates. The layers of a laminated composite material may be different single materials, such as clad metals that are bonded together or the same material, such as wood put together with different orientations. The layers may be composites themselves, such as fibrous composite layers placed so that different layers have different characteristics. This type of composite is the most commonly encountered laminated composite material used in design of high-performance structures and is the primary subject of this book. discussed next.The various forms of laminated composite materials are

9

1.2 PROPERTIES OF LAMINATED COMPOSITES

.~~~/ Random short fibers

Oriented short fibers

Continuous fibers

.7

Plain Fibrous Layers

1.1.2 Fiber-Reinforced Composite Materials An individual layer of a laminated composite material may assume a number of different forms, depending on the arrangement of the fiber constituent (see Fig. 1.2). The layers may be composed of short fibers embedded in a matrix. The short fibers may be distributed at random orientations, or they may be aligned in some manner forming oriented short-fiber composites. A typical example of a random short-fiber composite is fiberglass. Continuous fiber-reinforced composite layers are made up of bundles of small-diameter circular fibers. Typically, the radii of these fibers are in the order of 0.0002 in (0.005 mm), such as the radius of carbon fibers. The largest diameter fibers, such as boron fibers, are of the order of 0.002 in (0.05 mm). Continuous fiber-reinforced composite materials are commercially available in the form of unidirectional tape, with fibers aligned along the length of the tape. The fibers of the tape are preimpregnated with the matrix material, and for this reason the tape is sometimes referred to as a prepreg. As discussed in Section 1.2, this arrangement of aligning the fibers in a given direction ·provides the unique feature of the material properties of fiberreinforced composites. Another form of continuous fiber-reinforced composite layers is the woven fabric type composite, where fiber tows, which are large bundles of fibers (generally 10,000 or more fibers), are woven in two or more directions (see Fig. 1.2). Various fabric types with different weave features producing different material characteristics are available (see, for example, Middleton, 1990).

1.2 PROPERTIES OF LAMINATED COMPOSITES 1.2.1

Material Orthotropy

Composite layer properties (such as strength, stiffness, thermal and moisture conductivity, wear and environmental resistance) strongly depend on the form of the reinforcement in the laminate. The directional nature of the fibers in a fiber-reinforced laminate introduces directional dependence to most of those properties. Materials whose properties are independent of direction are called isotropic materials. Conversely, those materials with different properties in different directions are called anisotropic. A special case of anisotropy is the existence of two mutually perpendicular planes of symmetry in material properties. Such materials are referred to as orthotropic. Fibrous composites with either oriented short fibers or continuous fibers are orthotropic in nature. In such composites, properties are defined in

IV

INTRODUCTION

the plane of the layer in two directions-the direction along the fibers and the direction perpendicular to the fiber orientation. A typical example of an orthotropic material property of unidirectional fiber-reinforced composites is the stiffness. The matrix portion of the composite, which holds the fibers together, is generally isotropic. For all practical purposes, the fibers, which usually have much higher stiffness than the matrix material, are also isotropic. However, when combined together, the properties of the composite are no longer isotropic. Consider, for example, the fiber-reinforced material shown in Fig. 1.3 loaded along either the fiber direction, xl' or transverse to the fiber direction, x2 . When the material is loaded along the fiber direction, the deformation is smaller compared to the deformation under a load of the same magnitude applied in a direction perpendicular to the fiber. Since the amount of deformation under specified load reflects the stiffness of a material, the unidirectional composite has different stiffness properties along these two mutually perpendicular directions. The stiffness of the composite in the fiber direction is much closer to that of the fiber stiffness, E1 , and the stiffness perpendicular to the fiber direction is governed mostly by the properties of the matrix material, Em, Gm, and vm. Directions that are along the fiber and perpendicular to the fiber directions are commonly referred to as the principal material directions or principal axes of the material. Four elastic stiffness properties, commonly referred to as the engineering constants, fully describe the mechanical properties of an orthotropic layer in its plane (more detailed discussion of the stiffness properties is given in Section 2.4). These are two Young's moduli, 1 E and E2, along the fiber and transverse to the fiber directions, respectively, and the shear modulus G 12 and Poisson's ratio v in the 12

~ ·g

Table 1.1.

Ef' Gf' v1

Em, Gm, vm

Properties of various fiber-reinforced composite layers

E1 Constituents

Material T300/5208 AS4/3501 8(4)/5505 Kevlar49/Ep Scotchply1002 Source:

Graphite/Epoxy Graphite/Epoxy Boron/Epoxy Aramid/Epoxy Glass/Epoxy

E2

r;

, '"

106 psi (GPa) 106 psi (GPa) 106 psi (GPa) 26.3 (181) 20.0 (138) 29.6 (204) 11.0 (76) 5.60 (38.6)

1.49 (10.3) 1.30 (8.96) 2.68 (18.5) 0.80 (5.50) 1.20 (8.27)

1.04 (7.17) 1.03 (7.10) 0.81 (5.59) 0.33 (2.30) 0.60 (4.14)

1/12

0.28 0.30 0.23 0.34 0.26

vJ 0.7 0.66 . 0.5 0.6 0.45

Tsai and Hahn, 1980.

plane of the layer. As discussed in the next section, these properties are related to the amount of fiber present in the composite layer. A quantity, "f. referred to as the fiber volume fraction, is commonly used to measure the amount of fiber in a composite material. Typical stiffness properties of various unidirectional composite materials systems are given in Table 1.1 along with the fiber volume fraction of the composite. Some of the properties of a fiber-reinforced composite can be estimated from the properties of its constituents by simple calculations. Others have to be measured. There are three different mechanisms Jl~Jo._o_htain_properties of the composite from those of its constituents. As shown in the next section, properties of different natures are determined through different m:ecnanisms.

1.2.2

.. XI

11

1.2 PROPERTIES OF LAMINATED COMPOSITES

( \_..&:. ~ '-"-,p ~ Vf.-rj ML C+Urv.0 Rule of Mixtur-es,_ Complementation, and Interaction

One of the ways to estimate composite material properties is summation of the properties of the individual constituents based on their contribution to the overall material volume. This method is commonly referred to as the rule of mixtures. The rule of mixtures employs the volume fraction of the constituents to estimate the properties of the composite. For example, in the case of a continuous fiber-reinforced composite layer, we have a fiber volume fraction V1 and a matrix volume fraction Vm, which must satisfy

Figure 1.3. Elastic properties of a unidirectional fiber-reinforced composite layer.

VI+ Vm

= 1.

(1.2.1)

\

12 INTRODUCTION

Based on the rule of mixtures, a property p is estimated from the constituent properties, p and Pm' as

13

1.2 PROPERTIES OF LAMINATED COMPOSITES

matrix

fiber

matrix

1

P =P1~+ Pm Vm

=P1~+ Pm(l -

~).

(1.2.2)

For example, the longitudinal (fiber direction) stiffness property, EP of the composite may be calculated from the Young's moduli of the constituents E1 and Em, using this rule of mixtures as El =EI~+EmVm.

(1.2.3)

Equation (1.2.3) may be derived with analogy to the calculation of the overall stiffness of two springs connected in parallel; see Fig. 1.4. The fibers are assumed to be fully bonded to the matrix, and the total end-deformation o of the composite is identical in the fiber and the matrix, J- ~

.\

Figure 6.6. Axial strength of Boron/Epoxy layer based on Tsai-Hill criterion.

eluded in the figure as dashed lines are the stress limits from Fig. 6.4 for comparison. The strength predicted by Tsai-Hill is almost identical to the ones predicted by the maximum stress criteria used in Example 6.2.1 except in the neighborhood of the fiber-orientation angles 0, where two of the maximum stress criteria intersect. For example, the largest difference between the Tsai-Hill and a maximum stress criterion prediction is for the fiber-orientation angle of about 3°, where maximum stress criterion caused by fiber tensile failure and matrix shear failure is predicted at a tensile stress level of ax= 1264 MPa; see Fig. 6.6. For the same fiber orientation, Tsai-Hill predicts the failure to be at about ax= 893 MPa, an almost 30% lower strength than the maximum stress criterion. For cases of uniaxial tension or compression, the strengths predicted by the Tsai-Hill and maximum stress criteria are identical for the oo and 90° layers. In the case of biaxial loading, however, existence of transverse stresses can cause transverse matrix failure even for the 0° layer. For example, it was shown in Example 6.2.1 that for r =a/ax= 0.05, the failure of the layer is due to transverse matrix failure. Use of an interactive failure criterion, such as the Tsai-Hill, heightens the sensitivity of the failure load to the existence of biaxial stress state with small transverse stress. Failure predictions are plotted in Fig. 6.7a and

y

200f

\

~

1

d"

30 (a)

45

a, (deg)

75

60

~ 800 600

200

-

15

1000

400

~~\' / I /._~,{---"' Tsai-Hill

90

I1 I I

transverse,' matrix 1 failure 1

\\

.

I

45 60 75 9, (deg) (b) compressive loading

shear failure

\

I

1

30

I I I I 1 1

1

11

I ---=----"'

1400 I / 1-11._ ""-... jibe r 1200 1 1 failure

,.

1200t-l .... "'"'-.fiber \ failure

I

failure

1

~~ I

1400r 1

I I I

I

\

I

75

I

241

6.2 FAILURE OF FII!R·RI!INFORCED ORTHOTROPIC LAYERS

I I II

I I I I I

~-r

I/

failure

1

transverse matrix I

I I I

-~~'-- ?> __-__

90

r = O"/O"x = 0.05

..::::...,._-___,-/-;..,

15

30 (b)

45

60

a. (deg) r =O"/O"x =0.15

75

90

Figure 6.7. Biaxial tensile strength of Boron/Epoxy layer based on Tsai-Hill criterion.

6.7b for two levels of stress ratio, r = 0.05 and r = 0.15, considered in the previous example. For a 0° layer, the failure stress predicted by the Tsai-Hill criterion is substantially lower than the one predicted by the maximum stress criterion based on shear failure.

6.2.3 Tsai-Wu Criterion A more general form of the failure criterion for orthotropic materials under plane stress assumption is expressed as

F 11 af + F22 a~ + F66 -cf2 + 2F 1p 1a 2 + 2F16a 1-c 12

+ 2Fz6az'tu + F,al + Fzaz + F6-c12 < 1.

(6.2.14)

Evaluation of the strength coefficients F 11 , F 22 , Fw F 12 , F 16 , F2e Fl' F2, and F6 requires using the results of experiments on unidirectional fiber-reinforced specimens under simple load conditions. For example, if the failure stresses of a specimen loaded along the fiber direction in tension and compression are X1 and Xc, respectively, then we get F11 X~ + F 1 X1 = 1,

(6.2.15)

242

FAILURE CRITERIA FOR LAMINATED COMPOSITES

F 11 X~- F 1Xc= 1.

(6.2.16)

6.2

and, since the value of scomb is independent of the shear direction 2

F11r S~omb + F66S~omb- 2F,6rS~omb +FirS comb= 1.

Equations (6.2.15) and (6.2.16) are a system of two equations in two unknowns, solution of which yields F =__!_X x1 1

and

c

1

1

Fll

= Xtxc·

(6.2.17)

Another set of equations similar to Eqs. (6.2.15) and (6.2.16) is obtained by considering tension and compression tests in a direction transverse to the fiber direction. These equations are F 22 Y; + F 2 Y1 = 1,

(6.2.18)

= 1,

(6.2.19)

F22 Y~- F 2 Yc

243

FAILURE OF FIBER·REINFORCED ORTHOTROPIC LAYERS

(6.2.24)

The last two equations are almost identical except for the sign change in front of the F 16 term. Subtracting Eq. (6.2.24) from Eq. (6.2.23) yields (6.2.25)

F 16 =0.

A similar reasoning for an experiment that involves a combination of a 2 and 't 12 leads to the determination of

Fz6 = 0.

(6.2.26)

The only coefficient left to be determined, F 12, is the one that reflects the effect of interaction of the two normal stresses on failure:

and their solution is 1 1 Fz =y--yc

and

t

1 Fzz = ytyc·

(6.2.20)

and

F 66S 2 - F6 S = 1.

(6.2.21)

Solution of these equations leads to the determination of two more coefficients, which are given as F 6 =0

and

1 F66= S2'

(6.2.22)

In order to determine the coefficients that reflect the interaction of the normal and shearing stresses, namely F 16 and F26 , we need experiments that will generate combined stress states involving a 1 and -r 12 and a 2 and 't 12. For example, consider an experiment with l-r 121> 0 and a 1 > 0 with a 2 = 0 and a/'t 12 = ±r. If, for the experiment under consideration, the specimen fails at 't 12 =±Scomb' then from Eq. (6.2.14) we have

Fli?S~omb + F66S~omb + 2F,6rS~omb + F,rScomb = 1,

t

c

t

c

t

t

(6.2.27)

We next consider pure positive and negative shear tests where the specimens fail at a stress level of 't 12 = ±S. Therefore, F 66S 2 + F 6S = 1

X-X1) a,+ [1 y--y-1) az