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Loading...Fundamentals of Composite Materials
Vincent K. S. Choo
Department of Mechanical Engineering
New Mexico State University
New Mexico, U.s.a.contents
Chapter 1 Introduction
Chapter 2 Strain Tensor
2.1 Introduction
2.2 Infinitesimal Deformation
2.3 Strain Tensor Transformation
2.4 Strain and Displacement
2.5 the Equations of Compatibility
Exercises
References
Chapter 3 Stress Tensor
3.1 Introduction
3.2 Stress Tensor Transformation
3.3 Equilibrium Equations
Exercises
References
Chapter 4 Constitutive Equations
4.1 Introduction
4.2 Applications of the Laws of Thermodynamics
In Material Deformations
4.3 Transformation of the Material
Elastic Stiffness Tensor
Transformation of the Material
Compliance Tensor
The Constitutive Relations
4.5.1 Monoclinic Material
4.5.2 Orthotropic Material
4.5.3 Isotropic Material
Exercises
58x Contents
References 59
Chapter 5 Mechanics of a Lamina . . .
5.1 Introduction
5.2 Constitutive Relations of a Lamina
5.3 Transformation of Qijk,
And Sijti for a Lamina
5.4 Elastic Behavior of a Lamina
5.5 Deformation of a Constrained
Generally Orthotropic Lamina
Exercises
References
Chapter 6 Mechanics of a Laminate
6.2 Assumptions of the Classical Lamination
Theory
6.3 Classical Lamination Theory
6.4 Elastic Behavior of Laminated Plates
6.4.1 Specially Orthotropic Plates
Generally Orthotropic Plates
A Two-ply Laminated Plate
6.4.4 a Three-ply Laminated Plate
6.4^5 Balanced Laminates
Exercises
References
Chapter 7 Elastic Laminated Plate Theory
7.1 Introduction
7.2 Equilibrium Equations
Of a Laminated Plate
7.3 Bending of a Specially
Orthotropic Plate
7.4 Variational Method
Bending of a Specially Orthotropic Plate
With Clamped Edges
7.5 Governing Buckling Equations
7.6 Transverse Vibration of
A Laminated Plate
References
Chapter 8 Thermal and Edge Effects
8.1 Introduction
8.2 Thermal Residual Stresses
8.2.1 Microscopic Thermal Residual Stresses
Macroscopic Thermal Residual
Stresses in a Lamina
171CONTENTS Xt
8.2.3 Macroscopic Thermal Residual
Stresses in a Laminate
THERMAL GRADIENT
EDGE EFFECTS
8.4.1 Interlaminar Stresses in Angle-ply Laminates
8.4.2 Effect of Stacking Sequence on the
Interlaminar Stresses
EXERCISES 211
REFERENCES
CHAPTER 9 FAILURE CRITERIA AND
ANISOTROPIC PLASTICITY
9.1 INTRODUCTION
9.2 STRENGTHS OF UNIDIRECTIONAL LAMINAE
9.2.1
9.2.2
9.3 FAILURE CRITERIA
<11 A Uniaxial Loading 214 Biaxial Loading 217 220 9.3.1 Gol’denblat-Kopnov Strength Criterion for Brittle Anisotropic Materials 9.3.2 The Tsai-Wu Strength Tensor Polynomial Criterion 9.3.3 The Tsai-Wu Strain Tensor Polynomial Criterion 9.3.4 Maximum Stress Criterion 9.3.5 Maximum Strain Criterion 9.3.6 First Ply Failure Criterion 9.3.7 Last Ply Failure Criterion 9.3.8 Hashin Failure Criteria ANISOTROPIC PLASTICITY 9.4.1. Elastic Constitutive Relations 9.4.2 Plasticity Theory for an Elastoplastic Isotropic Solid 9.4.3 Plasticity Theory for an Elastoplastic Composite Material EXERCISES 260 REFERENCES CHAPTER 10 BEHAVIOR AT THE MICROLEVEL 10.1 INTRODUCTION 10.2 EXPERIMENTAL DETERMINATION OF THE EFFECTIVE ELASTIC STIFFNESS TENSOR … 10 3 THEORETICAL PREDICTION OF THE EFFECTIVE ELASTIC STIFFNESS TENSOR … 10.3.1 Variational Bounding Method 10.3.2 Strength of Material Approach 280Xil CONTENTS 10.4 STRENGTHS AND FRACTURE BEHAVIOR OF UNIDIRECTIONAL LAMINA 10.4.1 Transverse Tensile Failure 10.4.2 Transverse Compressive Failure 10.4.3 Tensile Strength along the Fiber Axis 10.4.4 Compressive Strength in the Fiber Direction 10.4.5 In-plane Shear Strength of a Unidirectional Lamina 10.5 MICRO-YIELDING IN COMPOSITE MATERIALS EXERCISES REFERENCES A DDPMniY T oon n.i x uii x Xd C/ Cf INDEX 310IndexINDEX 311 tensor 270 Effective strain increment 256 Effective stress 255 Eigenvalue problem 162 Elastic stiffness tensor 78 Elastic-perfectly plastic 257 material 257 Elasticity solution 193 Elastoplastic behavior 257 Engineering shear strain 237 Engineering shear strains 13 Engineering shear strains 46 Engineering strains 11 Equations of motion 163, 164, 165 Equivalent strain increment 256 Euler equation 154 Exact solutions 150, 152, 193 Example 5.1 74 Example 5.2 76 Example 5.3 80 Example 5.4 81 Example 5.5 85 Example 5.6 87 Example 5.7 97 Example 6.1 118 Example 6.2 122 Example 6.3 128 Example 6.4 129 Example 8.1 180 Example 9.1 231 Example 9.2 234 Example 10.1 278 Example 10.2 283 Example 10.3 287 Expansion 172 Extensional 113 stiffness tensor 113 143, 157 Approximate methods 155 Bending 115, 161 Bending moments 161 Bending stiffness tensor 113 116, 151 Biaxial loading 217 Boundary conditions 147 Boundary problem 295 Buckling 161, 162 Buckling equations 162 Buckling load 162 Calculus of variations 154 Cartesian tensor 16 Classical lamination theory 230 Closed failure envelope 230 Closed form solution 155 Compliance tensor 66, 71, 75 Compressive Strength 292 Compressive failure 292 Conservative system 35 n A! O O O v^uu»titunvc C4uanuu» *uo Constitutive relations 63 Constraint imposed 171 Coupled 147 Coupling 151 Coupling stiffness tensor 113 115, 116, 143, 147, 159 Covariant derivative 166 Curing stresses 171 Curing temperature 170 Determined exactlv 155 Deviatoric stress tensor 294 Direction cosines 23 Divergence theorem 28, 33 Fiber Direction 292 Finite element method 155 First law of thermodynamics 35 First variations 154 Edge effects 217 Edge phenomenon 202 Effective elastic stiffness 270312 INDBX Flow-rule 257 46 , 47, 53, 58, 63, 66 Fourth rank tensor 41, 43 Fracture behavior 285 Free edge 192 Matrix 61 Maximum Strain Criterion 244 Maximum Stress Criterion 239 Micromechanical stress 285 Microscopic thermal stresses 75 Microstresses 170 Midplane 121 Minimum Potential Energy 153 Moisture absorption 171 Monoclinic material 47, 50, 52 Generally orthotropic lamina 62 Geometric coordinate system 66 Global coordinate system 4, 66 Governing buckling equations 161 Hoop wound tubes 289 Octahedral shear stress 255 Off-axis angle 6 61 Orthotropic material 50, 51 Orthotropic medium 53 Inclination of failure 229 envelope 229 Index 19 Indices 19 Infinitesimal deformation 12 Initial strains method 295 Inplane shear strength 293 Inplane forces 161 Interaction 238 Interaction component ^1122 229 Interlaminar shear stress 195 Intralaminar shear stress 170 Invariants 249 Inverse material elastic stiffness tensor 43 Isotropic material 53, 57, 58 Partial differential equations 147 Path independent 141 Plane of material symmetry 53 Plastic shear strain increment 255 Poisson contraction 200 Poisson ratio 66 Positive definite 154 Prandtl-Reuss 257 Prebuckling state 161, 162 Principal material axis 193 Principal material coordinate system 4, 66 Principle of minimum complementary energy 277 potential energy 277 Proper minimum 153 Pure deformation 10 Kirchhoff condition 147 Kronecker delta 228 Laminae 61 Laminate 61 Rayleigh and Ritz method 155 Reduced stiffness matrix 66 , 69 Reduced stiffness tensor 115 Resin shrinkage 171 Rigid body 33 Rigid body motion 10 Macroscopic thermal stress 175 Material compliance tensor 43 Material elastic stiffness tensor 40 41 , 42, 44, 45INDEX 313 Second law of thermodynamics 38 Second rank tensor 16, 26 Second variations 154 Shear coupling 78, 79, 189 Shrinkage 171 Singular 193 Skew symmetric tensor 34 Skew-symmetric 9, 10, 40 Specially Orthotropic Plate 159, 160 Specially orthotropic 270 c* i.i_ _ A i .* z> 1
uruiowupit lauuudui
State of equilibrium 147
Stepwise linearization 295
Strain Tensor 236
Strain energy 153, 160
Strain magnification 285
Strain tensor 10, 16, 18
26, 49, 143, 157
Strength Criterion 220, 221
Strength tensor polynomial
criterion 229, 236
Qt.rpntrfli fPnanrc 99fi 991 95ft
W A A W W A v W J « ^ W \/
Strengths 284
Stress invariants 248
Stress tensor 23, 26, 34, 45, 49
Stresses 175
Stretching 115
Symmetric 10, 40
Symmetric tensor 34, 41, 113
Symmetrical 121
Symmetry properties 143
Tangent modulus method 295
Taylor series 17
Tensor transformation 173
Tensorial shear strains 11
Tensorial shear strains 46
Theorem 153
Thermal coefficient of
expansion 172
Thermal compressive stress 170
Thermal residual stresses 170
Thermal stress 169
Thermal tensile stress 170
Traction 23
Tractive forces 25
Transformation 10
Transformation rules 16, 41, 69
Transverse Compressive Failure 289
Transverse vibration 163
Transversely isotropic 270
Twist coupling 148
Upper bound 277
Variational bending moments 162
Variational displacements 162
Variational inplane loads 162
Variational methods 152, 153
Winding angle 6 61
Young’s modulus 66
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