## Preface to the Second Edition

The first edition of this book was published in 1997, and I am grateful for

the response and comments I have received about the book and the accompanying

PROMAL software. The changes in the book are mainly a result

of comments received from students who used this book in a course or as

a self-study.

In this edition, I have added a separate chapter on symmetric and unsymmetric

laminated beams. All the other chapters have been updated while

maintaining the flow of the content. Key terms and a summary have been

added at the end of each chapter. Multiple-choice questions to reinforce the

learning from each chapter have been added and are available at the textbook

Website: http://www.eng.usf.edu/~kaw/promal/book.html.

Specifically, in Chapter 1, new applications of composite materials have

been accommodated. With the ubiquitous presence of the Web, I have annotated

articles, videos, and Websites at the textbook Website. In Chapter 2,

we have added more examples and derivations have been added. The appendix

on matrix algebra has been extended because several engineering departments

no longer teach a separate course in matrix algebra. If the reader needs

more background knowledge of this subject, he or she can download a free

e-book on matrix algebra at http://numericalmethods.eng.usf.edu/ (click

on “matrix algebra”). In Chapter 3, derivations are given for the elasticity

model of finding the four elastic constants. Two more examples can be found

in Chapter 5: design of a pressure vessel and a drive shaft.

The PROMAL program has been updated to include elasticity models

in Chapter 3. PROMAL and the accompanying software are available to

the eligible buyers of the textbook only at the textbook Website (see the

“About the Software” section). The software and the manual will be continually

updated.

## Preface to the First Edition

Composites are becoming an essential part of today’s materials because they

offer advantages such as low weight, corrosion resistance, high fatigue

strength, faster assembly, etc. Composites are used as materials ranging from

making aircraft structures to golf clubs, electronic packaging to medical

equipment, and space vehicles to home building. Composites are generating

curiosity and interest in students all over the world. They are seeing everyday

applications of composite materials in the commercial market, and job

opportunities are also increasing in this field. The technology transfer initiative

of the U.S. government is opening new and large-scale opportunities

for use of advanced composite materials.

Many engineering colleges are offering courses in composite materials as

undergraduate technical electives and as graduate-level courses. In addition,

as part of their continuing education and retraining, many practicing engineers

are participating in workshops and taking short courses in composite

materials. The objective of this book is to introduce a senior undergraduateor

graduate-level student to the mechanical behavior of composites. Covering

all aspects of the mechanical behavior of composites is impossible to do

in one book; also, many aspects require knowledge of advanced graduate

study topics such as elasticity, fracture mechanics, and plates and shells

theory. Thus, this book emphasizes an overview of composites followed by

basic mechanical behavior of composites. Only then will a student form a

necessary foundation for further study of topics such as impact, fatigue,

fracture mechanics, creep, buckling and vibrations, etc. I think that these

topics are important and the interested student has many well-written texts

available to follow for that.

This book breaks some traditional rules followed in other textbooks on

composites. For example, in the first chapter, composites are introduced in

a question–answer format. These questions were raised through my own

thought process when I first took a course in composites and then by my

students at the University of South Florida, Tampa. Also, this is the first

textbook in its field that includes a professional software package. In addition,

the book has a format of successful undergraduate books, such as short

sections, adequate illustrations, exercise sets with objective questions and

numerical problems, reviews wherever necessary, simple language, and

many examples.

Chapter 1 introduces basic ideas about composites including why composites

are becoming important in today’s market. Other topics in Chapter

1 include types of fibers and matrices, manufacturing, applications, recycling,

and basic definitions used in the mechanics of composites. In Chapter

© 2006 by Taylor & Francis Grwoupw, LLwC.Techbooksyard.com

2, I start with a review of basic topics of stress, strain, elastic moduli, and

strain energy. Then I discuss the mechanical behavior of a single lamina,

including concepts about stress–strain relationship for a lamina, stiffness and

strength of a lamina, and the stress–strain response due to temperature and

moisture change. In Chapter 3, I develop equations for mechanical properties

of a lamina such as stiffness, strength, and coefficients of thermal and moisture

expansion from individual properties of the constituents (long continuous

fibers and matrix) of composites. I introduce experimental

characterization of the mechanical properties of a lamina at appropriate

places in Chapter 3. Chapter 4 is an extension of Chapter 2, in which the

macromechanics of a single lamina are extended to the macromechanics of

a laminate. I develop stress–strain equations for a laminate based on individual

properties of the laminae that make it. I also discuss stiffness and

strength of a laminate and effects of temperature and moisture on residual

stresses in a laminate. In Chapter 5, special cases of laminates used in the

market are introduced. I develop procedures for analyzing the failure and

design of laminated composites. Other mechanical design issues, such as

fatigue, environmental effects, and impact, are introduced.

A separate chapter for using the user-friendly software PROMAL is

included for supplementing the understanding of Chapter 2 through Chapter

5. Students using PROMAL can instantly conduct pragmatic parametric

studies, compare failure theories, and have the information available in

tables and graphs instantaneously.

The availability of computer laboratories across the nation allows the

instructor to use PROMAL as a teaching tool. Many questions asked by the

student can be answered instantly. PROMAL is more than a black box

because it shows intermediate results as well. At the end of the course, it

will allow students to design laminated composite structures in the classroom.

The computer program still maintains the student’s need to think

about the various inputs to the program to get an optimum design.

## Contents

Introduction to Composite Materials ……………………………………. 1

Chapter Objectives……………………………………………………………………………………..1

1.1 Introduction ……………………………………………………………………………………….1

1.2 Classification…………………………………………………………………………………….16

1.2.1 Polymer Matrix Composites………………………………………………….19

1.2.2 Metal Matrix Composites ……………………………………………………..40

1.2.3 Ceramic Matrix Composites………………………………………………….45

1.2.4 Carbon–Carbon Composites …………………………………………………46

1.3 Recycling Fiber-Reinforced Composites …………………………………………..50

1.4 Mechanics Terminology……………………………………………………………………51

1.5 Summary………………………………………………………………………………………….54

Key Terms ………………………………………………………………………………………………..54

Exercise Set ………………………………………………………………………………………………55

References…………………………………………………………………………………………………57

General References …………………………………………………………………………………..58

Video References………………………………………………………………………………………59

2

Macromechanical Analysis of a Lamina ……………………………… 61

Chapter Objectives……………………………………………………………………………………61

2.1 Introduction ……………………………………………………………………………………..61

2.2 Review of Definitions……………………………………………………………………….65

2.2.1 Stress……………………………………………………………………………………..65

2.2.2 Strain …………………………………………………………………………………….68

2.2.3 Elastic Moduli……………………………………………………………………….75

2.2.4 Strain Energy…………………………………………………………………………77

2.3 Hooke’s Law for Different Types of Materials …………………………………79

2.3.1 Anisotropic Material……………………………………………………………..81

2.3.2 Monoclinic Material………………………………………………………………82

2.3.3 Orthotropic Material (Orthogonally Anisotropic)/Specially

Orthotropic ……………………………………………………………………………84

2.3.4 Transversely Isotropic Material …………………………………………….87

2.3.5 Isotropic Material ………………………………………………………………….88

2.4 Hooke’s Law for a Two-Dimensional Unidirectional Lamina………….99

2.4.1 Plane Stress Assumption……………………………………………………….99

2.4.2 Reduction of Hooke’s Law in Three Dimensions to Two

Dimensions………………………………………………………………………….100

2.4.3 Relationship of Compliance and Stiffness Matrix to

Engineering Elastic Constants of a Lamina…………………………101

2.5 Hooke’s Law for a Two-Dimensional Angle Lamina……………………..109

© 2006 by Taylor & Francis Grwoupw, LLwC.Techbooksyard.com

2.6 Engineering Constants of an Angle Lamina…………………………………..121

2.7 Invariant Form of Stiffness and Compliance Matrices for an

Angle Lamina …………………………………………………………………………………132

2.8 Strength Failure Theories of an Angle Lamina ………………………………137

2.8.1 Maximum Stress Failure Theory …………………………………………139

2.8.2 Strength Ratio ……………………………………………………………………..143

2.8.3 Failure Envelopes………………………………………………………………..144

2.8.4 Maximum Strain Failure Theory …………………………………………146

2.8.5 Tsai–Hill Failure Theory………………………………………………………149

2.8.6 Tsai–Wu Failure Theory ………………………………………………………153

2.8.7 Comparison of Experimental Results with Failure

Theories……………………………………………………………………………….158

2.9 Hygrothermal Stresses and Strains in a Lamina…………………………….160

2.9.1 Hygrothermal Stress–Strain Relationships for a

Unidirectional Lamina…………………………………………………………163

2.9.2 Hygrothermal Stress–Strain Relationships for an

Angle Lamina ……………………………………………………………………..164

2.10 Summary………………………………………………………………………………………..167

Key Terms ………………………………………………………………………………………………167

Exercise Set …………………………………………………………………………………………….168

References……………………………………………………………………………………………….174

Appendix A: Matrix Algebra ………………………………………………………………….175

Key Terms ………………………………………………………………………………………………195

Appendix B: Transformation of Stresses and Strains ……………………………..197

B.1 Transformation of Stress ……………………………………………………..197

B.2 Transformation of Strains ……………………………………………………199

Key Terms ………………………………………………………………………………………………202

3

Micromechanical Analysis of a Lamina …………………………….. 203

Chapter Objectives………………………………………………………………………………….203

3.1 Introduction ……………………………………………………………………………………203

3.2 Volume and Mass Fractions, Density, and Void Content ……………….204

3.2.1 Volume Fractions…………………………………………………………………204

3.2.2 Mass Fractions …………………………………………………………………….205

3.2.3 Density ………………………………………………………………………………..207

3.2.4 Void Content ………………………………………………………………………. 211

3.3 Evaluation of the Four Elastic Moduli……………………………………………215

3.3.1 Strength of Materials Approach ………………………………………….216

3.3.1.1 Longitudinal Young’s Modulus………………………………218

3.3.1.2 Transverse Young’s Modulus………………………………….221

3.3.1.3 Major Poisson’s Ratio……………………………………………..227

3.3.1.4 In-Plane Shear Modulus …………………………………………229

3.3.2 Semi-Empirical Models ……………………………………………………….232

3.3.2.1 Longitudinal Young’s Modulus………………………………234

3.3.2.2 Transverse Young’s Modulus………………………………….234

© 2006 by Taylor & Francis Grwoupw, LLwC.Techbooksyard.com

3.3.2.3 Major Poisson’s Ratio……………………………………………..236

3.3.2.4 In-Plane Shear Modulus …………………………………………237

3.3.3 Elasticity Approach……………………………………………………………..239

3.3.3.1 Longitudinal Young’s Modulus………………………………241

3.3.3.2 Major Poisson’s Ratio……………………………………………..249

3.3.3.3 Transverse Young’s Modulus………………………………….251

3.3.3.4 Axial Shear Modulus ……………………………………………..256

3.3.4 Elastic Moduli of Lamina with Transversely Isotropic

Fibers …………………………………………………………………………………..268

3.4 Ultimate Strengths of a Unidirectional Lamina ……………………………..271

3.4.1 Longitudinal Tensile Strength……………………………………………..271

3.4.2 Longitudinal Compressive Strength ……………………………………277

3.4.3 Transverse Tensile Strength …………………………………………………284

3.4.4 Transverse Compressive Strength ……………………………………….289

3.4.5 In-Plane Shear Strength……………………………………………………….291

3.5 Coefficients of Thermal Expansion…………………………………………………296

3.5.1 Longitudinal Thermal Expansion Coefficient ……………………..297

3.5.2 Transverse Thermal Expansion Coefficient …………………………298

3.6 Coefficients of Moisture Expansion………………………………………………..303

3.7 Summary………………………………………………………………………………………..307

Key Terms ………………………………………………………………………………………………308

Exercise Set …………………………………………………………………………………………….308

References………………………………………………………………………………………………. 311

4

Macromechanical Analysis of Laminates…………………………… 315

Chapter Objectives………………………………………………………………………………….315

4.1 Introduction ……………………………………………………………………………………315

4.2 Laminate Code ……………………………………………………………………………….316

4.3 Stress–Strain Relations for a Laminate …………………………………………..318

4.3.1 One–Dimensional Isotropic Beam Stress–Strain

Relation ……………………………………………………………………………….318

4.3.2 Strain-Displacement Equations……………………………………………320

4.3.3 Strain and Stress in a Laminate …………………………………………..325

4.3.4 Force and Moment Resultants Related to Midplane

Strains and Curvatures ………………………………………………………..326

4.4 In-Plane and Flexural Modulus of a Laminate ………………………………340

4.4.1 In-Plane Engineering Constants of a Laminate …………………..341

4.4.2 Flexural Engineering Constants of a Laminate……………………344

4.5 Hygrothermal Effects in a Laminate ………………………………………………350

4.5.1 Hygrothermal Stresses and Strains ……………………………………..350

4.5.2 Coefficients of Thermal and Moisture Expansion of

Laminates…………………………………………………………………………….358

4.5.3 Warpage of Laminates…………………………………………………………362

4.6 Summary………………………………………………………………………………………..363

Key Terms ………………………………………………………………………………………………364

© 2006 by Taylor & Francis Grwoupw, LLwC.Techbooksyard.com

Exercise Set …………………………………………………………………………………………….364

References……………………………………………………………………………………………….367

5

Failure, Analysis, and Design of Laminates ……………………… 369

Chapter Objectives………………………………………………………………………………….369

5.1 Introduction ……………………………………………………………………………………369

5.2 Special Cases of Laminates …………………………………………………………….370

5.2.1 Symmetric Laminates ………………………………………………………….370

5.2.2 Cross-Ply Laminates ……………………………………………………………371

5.2.3 Angle Ply Laminates …………………………………………………………..372

5.2.4 Antisymmetric Laminates……………………………………………………372

5.2.5 Balanced Laminate………………………………………………………………373

5.2.6 Quasi-Isotropic Laminates…………………………………………………..373

5.3 Failure Criterion for a Laminate …………………………………………………….380

5.4 Design of a Laminated Composite …………………………………………………393

5.5 Other Mechanical Design Issues…………………………………………………….419

5.5.1 Sandwich Composites …………………………………………………………419

5.5.2 Long-Term Environmental Effects……………………………………….420

5.5.3 Interlaminar Stresses……………………………………………………………421

5.5.4 Impact Resistance………………………………………………………………..422

5.5.5 Fracture Resistance ……………………………………………………………..423

5.5.6 Fatigue Resistance……………………………………………………………….424

5.6 Summary………………………………………………………………………………………..425

Key Terms ………………………………………………………………………………………………426

Exercise Set …………………………………………………………………………………………….426

References……………………………………………………………………………………………….430

6

Bending of Beams …………………………………………………………….. 431

Chapter Objectives………………………………………………………………………………….431

6.1 Introduction ……………………………………………………………………………………431

6.2 Symmetric Beams …………………………………………………………………………..433

6.3 Nonsymmetric Beams…………………………………………………………………….444

6.4 Summary………………………………………………………………………………………..455

Key Terms ………………………………………………………………………………………………455

Exercise Set …………………………………………………………………………………………….456

References……………………………………………………………………………………………….457

© 2006 by Taylor & Francis Grwoupw, LLwC.Techbooksyard.com