Analysis and Performance of Fiber Composites

BHAGWAN D. AGARWAL, PHD, is a former Vice President of engineering services at Bodycote Polymer—Broutman Laboratory, and Professor of Mechanical Engineering and Dean of Research and Development at the Indian Institute of Technology, Kanpur.

LAWRENCE J. BROUTMAN, PHD, is an independent consultant and founder of L.J. Broutman & Associates.

K. CHANDRASHEKHARA, PHD, is Professor of Mechanical and Aerospace Engineering and Director of the Composite Manufacturing Laboratory at the Missouri University of Science and Technology.

Preface xv

About the Companion Website xvii

1 Introduction 1

1.1 Definition 1

1.2 Classification 2

1.3 Particulate Composites 2

1.4 Fiber-Reinforced Composites 5

1.5 Applications of Fiber-Reinforced Polymer Composites 7

Exercise Problems 15

References 16

2 Fibers, Matrices, and Fabrication of Composites 17

2.1 Reinforcing Fibers 17

2.1.1 Glass Fibers 19

2.1.2 Carbon and Graphite Fibers 25

2.1.3 Aramid Fibers 29

2.1.4 Boron Fibers 30

2.1.5 Other Fibers 31

2.2 Matrix Materials 33

2.2.1 Polymers 33

2.2.2 Metals 44

2.3 Fabrication of Fiber Composite Products 45

2.3.1 Fabrication with Thermosetting Resin Matrices 45

2.3.2 Fabrication with Thermoplastic Resin Matrices 59

2.3.3 Sandwich Composites 61

2.3.4 Fabrication with Metal Matrices 63

2.3.5 Fabrication with Ceramic Matrices 64

Suggested Reading 65

3 Micromechanics of Unidirectional Composites 67

3.1 Introduction 67

3.1.1 Nomenclature 68

3.1.2 Volume and Weight Fractions 68

3.2 Longitudinal Loading: Deformation, Modulus, and Strength 70

3.2.1 Model 70

3.2.2 Deformation under Small Loads 71

3.2.3 Load Sharing 74

3.2.4 Behavior beyond Initial Deformation 76

3.2.5 Failure Mechanism and Longitudinal Strength 78

3.2.6 Factors Influencing Longitudinal Strength and Stiffness 80

3.3 Transverse Loading: Modulus and Strength 83

3.3.1 Model 83

3.3.2 Elasticity Methods of Stiffness Prediction 85

3.3.3 Halpin–Tsai Equations for Transverse Modulus 86

3.3.4 Transverse Strength 89

3.4 Shear Modulus 92

3.5 Poisson’s Ratios 96

3.6 Expansion Coefficients and Transport Properties 97

3.6.1 Thermal Expansion Coefficients 97

3.6.2 Moisture Absorption and Expansion Coefficients 99

3.6.3 Transport Properties 100

3.7 Failure of Unidirectional Composites 105

3.7.1 Microscopic Failure Events 105

3.7.2 Failure under Longitudinal Tensile Loads 108

3.7.3 Failure under Longitudinal Compressive Loads 111

3.7.4 Failure under Transverse Tensile Loads 115

3.7.5 Failure under Transverse Compressive Loads 116

3.7.6 Failure under In-Plane Shear Loads 120

3.8 Typical Properties of Unidirectional Fiber Composites 120

Exercise Problems 121

References 126

4 Short-Fiber Composites 129

4.1 Introduction 129

4.2 Load Transfer to Fibers 130

4.2.1 Simplified Analysis of Stress Transfer 130

4.2.2 Stress Distributions from Finite-Element Analysis 134

4.3 Predicting Modulus and Strength of Short-Fiber Composites 136

4.3.1 Average Fiber Stress 136

4.3.2 Longitudinal and Transverse Modulus of Aligned Short-Fiber Composites 137

4.3.3 Modulus of Randomly Oriented Short-Fiber Composites 138

4.3.4 Longitudinal Strength of Aligned Short-FiberComposites 142

4.3.5 Strength of Randomly Oriented Short-Fiber Composites 143

4.4 Influence of Matrix Ductility on Properties 144

Exercise Problems 148

References 149

5 Macromechanics Analysis of an Orthotropic Lamina 151

5.1 Introduction 151

5.1.1 Orthotropic Materials 151

5.2 Stress–Strain Relations for Unidirectional Composites 153

5.2.1 Engineering Constants in Longitudinal and Transverse Directions 153

5.2.2 Off-Axis Engineering Constants 156

5.2.3 Transformation of Engineering Constants 158

5.3 Hooke’s Law and Stiffness and Compliance Matrices 167

5.3.1 General Anisotropic Material 167

5.3.2 Transformation of Stress, Strain, and Elasticity Constants 169

5.3.3 Stress–Strain Relations for Orthotropic Materials 169

5.3.4 Transversely Isotropic Material 170

5.3.5 Isotropic Material 171

5.3.6 Orthotropic Material under Plane Stress 172

5.3.7 Compliance Tensor and Compliance Matrix 173

5.3.8 Relations between Engineering Constants and Elements of Stiffness and Compliance Matrices 174

5.3.9 Restrictions on Elastic Constants 177

5.3.10 Transformation of Stiffness and Compliance Matrices 178

5.3.11 Invariant Forms of Stiffness and Compliance Matrices 182

5.4 Strengths of an Orthotropic Lamina 185

5.4.1 Maximum-Stress Theory 186

5.4.2 Maximum-Strain Theory 188

5.4.3 Maximum-Work Theory 190

5.4.4 Importance of Sign on Off-Axis Strength of Composites 193

Exercise Problems 196

References 200

6 Analysis of Laminated Composites 202

6.1 Classical Lamination Theory 202

6.1.1 Introduction 202

6.1.2 Laminate Displacements and Strains 202

6.1.3 Laminate Stresses 205

6.1.4 Resultant Forces and Moments 206

6.1.5 Laminate Constitutive Relations 207

6.2 Laminate Description System 213

6.3 Design, Construction, and Properties of Laminates 215

6.3.1 Symmetric Laminates 215

6.3.2 Unidirectional, Cross-Ply, and Angle-Ply Laminates 215

6.3.3 Quasi-isotropic Laminates 216

6.4 Failure of Laminates 224

6.4.1 Initial Failure 224

6.4.2 Laminate Analysis after Initial Failure 228

6.5 Hygrothermal Stresses in Laminates 238

6.5.1 Concepts of Thermal Stresses 238

6.5.2 Hygrothermal Stress Calculations 240

6.6 Laminate Analysis through Computers 251

Exercise Problems 255

References 259

7 Analysis of Laminated Plates and Beams 260

7.1 Introduction 260

7.2 Governing Equations for Plates 261

7.2.1 Equilibrium Equations 261

7.2.2 Equilibrium Equations in Terms of Displacements 264

7.3 Application of Plate Theory 266

7.3.1 Bending of Specially Orthotropic Laminates 266

7.3.2 Buckling 276

7.3.3 Free Vibrations 281

7.4 Deformations Due to Transverse Shear 286

7.4.1 First-Order Shear Deformation Theory 287

7.4.2 Higher-Order Shear Deformation Theory 290

7.5 Analysis of Laminated Beams 293

7.5.1 Governing Equations for Laminated Beams 293

7.5.2 Application of Beam Theory 295

Exercise Problems 299

References 301

8 Advanced Topics in Fiber Composites 302

8.1 Interlaminar Stresses and Free-Edge Effects 302

8.1.1 Concepts of Interlaminar Stresses 302

8.1.2 Determination of Interlaminar Stresses 304

8.1.3 Effect of Stacking Sequence on Interlaminar Stresses 306

8.1.4 Approximate Solutions for Interlaminar Stresses 308

8.1.5 Summary 312

8.2 Fracture Mechanics of Fiber Composites 313

8.2.1 Introduction 313

8.2.2 Fracture Mechanics Concepts and Measures of Fracture Toughness 315

8.2.3 Fracture Toughness of Composite Laminates 323

8.2.4 Whitney–Nuismer Failure Criteria for Notched Composites 327

8.3 Joints for Composite Structures 332

8.3.1 Adhesively Bonded Joints 333

8.3.2 Mechanically Fastened Joints 337

8.3.3 Bonded-Fastened Joints 339

Exercise Problems 339

References 340

9 Performance of Fiber Composites: Fatigue, Impact, and Environmental Effects 345

9.1 Fatigue 345

9.1.1 Introduction 345

9.1.2 Fatigue Damage 346

9.1.3 Factors Influencing Fatigue Behavior 354

9.1.4 Empirical Relations for Fatigue Damage and Fatigue Life 361

9.1.5 Fatigue of High-Modulus Fiber-Reinforced Composites 362

9.1.6 Fatigue of Short-Fiber Composites 366

9.2 Impact 371

9.2.1 Introduction and Fracture Process 371

9.2.2 Energy-Absorbing Mechanisms and Failure Models 373

9.2.3 Effect of Materials and Testing Variables on Impact Properties 377

9.2.4 Hybrid Composites and Their Impact Strength 383

9.2.5 Damage Due to Low-Velocity Impact 387

9.3 Environmental-Interaction Effects 391

9.3.1 Fiber Strength 391

9.3.2 Matrix Effects 397

Exercise Problems 405

References 406

10 Experimental Characterization of Composites 414

10.1 Introduction 414

10.2 Measurement of Physical Properties 415

10.2.1 Density 415

10.2.2 Constituent Weight and Volume Fractions 415

10.2.3 Void Volume Fraction 416

10.2.4 Thermal Expansion Coefficients 417

10.2.5 Moisture Absorption and Diffusivity 417

10.2.6 Moisture Expansion Coefficients 418

10.3 Measurement of Mechanical Properties 419

10.3.1 Properties in Tension 419

10.3.2 Properties in Compression 423

10.3.3 In-Plane Shear Properties 425

10.3.4 Flexural Properties 433

10.3.5 Interlaminar Shear Strength and Fracture Toughness 438

10.3.6 In-Plane Fracture Toughness Tests 442

10.3.7 Impact Tests 450

10.3.8 Tests for Aerospace Applications 455

10.4 Damage Identification Using Nondestructive Evaluation Techniques 457

10.4.1 Ultrasonics 457

10.4.2 Acoustic Emission 460

10.4.3 X-Radiography 461

10.4.4 Thermography 463

10.4.5 Laser Shearography 464

10.5 General Remarks on Characterization 464

Exercise Problems 468

References 470

11 Emerging Composite Materials 475

11.1 Nanocomposites 475

11.2 Carbon–Carbon Composites 477

11.3 Biocomposites 478

11.3.1 Biofibers 478

11.3.2 Wood–Plastic Composites (WPCs) 480

11.3.3 Biopolymers 481

11.4 Composites in “Smart” Structures 482

11.5 Further Emerging Trends 483

Suggested Reading 484

Appendix 1 Matrices and Tensors 488

A1.1 Matrix Definitions 488

A1.2 Matrix Operations 493

A1.3 Tensors 498

References 509

Appendix 2 Equations of Theory of Elasticity 510

A2.1 Analysis of Strain 510

A2.2 Analysis of Stress 514

A2.3 Stress–Strain Relations for Isotropic Materials 518

References 520

Appendix 3 Laminate Orientation Code 521

A3.1 Standard Code Elements 521

A3.2 Positive and Negative Angles 522

A3.3 Symmetric Laminates 524

A3.4 Sets 524

A3.5 Hybrid Laminates 525

Appendix 4 Properties of Fiber Composites 527

Appendix 5 Computer Programs for Laminate Analysis 532

Appendix 6 Introduction to MATLAB 534

A6.1 Introduction: Getting Started 534

A6.2 Vectors and Matrices 537

A6.2.1 Defining Matrices 537

A6.2.2 Basic Matrix Functions 537

A6.2.3 Extracting Parts of Matrices 539

A6.2.4 Basic Matrix Operations 539

A6.3 Programming in MATLAB 540

A6.3.1 Logical and Relational Operators 540

A6.3.2 Loop and Logical Statements 540

A6.3.3 MATLAB Functions: Saving Programs 540

A6.3.4 Input/Output Functions 541

A6.3.5 Controlling the Appearance of Floating Point Number 541

A6.4 Plotting Tools 542

A6.4.1 Basic Plot Commands 542

A6.4.2 Line Styles and Colors 543

Index 545

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