Mechanics of Composite Materials By Arthur K kaw

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Mechanics of Composite Materials By Arthur K kaw

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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