1 Introduction to Composite Materials .......................... 1
1.1 Introduction ............................................ 1
1.2 Classification of Composite Materials ................... 1
1.2.1 Fibre Composites ................................. 2
1.2.2 Particle Composites .............................. 3
1.2.3 Laminated Composites ............................. 3
1.3 Characterisation of Unidirectional Fibre Composite
Materials ............................................... 4
1.3.1 Mechanics of Material Approach to the Moduli ..... 5
1.3.2 Elasticity Approach to the Moduli ................ 7
1.4 Three-dimensional Modelling of Laminated Composite
Materials ............................................... 9
1.4.1 Three-dimensional Analytical Modelling of
Laminated Composite Plates and Shells ........... 10
1.4.2 Three-dimensional Numerical Modelling of
Laminated Composite Plates and Shells ........... 11
2 Fundamental Equations of Three-dimensional Elasticity ....... 13
2.1 Introduction ........................................... 13
2.2 Equilibrium Equations of Stresses ...................... 14
2.3 Strain-Displacement Relations .......................... 19
2.4 Stress-Strain Relations ................................ 21
2.4.1 Hooke's Law of Elasticity ....................... 21
2.4.2 Engineering Constants for Stress-Strain
Relations of Orthotropic Materials .............. 24
2.4.3 Engineering Constants for Stress-Strain
Relations of a Lamina ........................... 26
3 State Variable Equation of a Continuous System .............. 29
3.1 Introduction ........................................... 29
3.2 Concept of State and State Variables ................... 30
3.3 Solutions for a Linear Time-invariant System ........... 32
3.3.1 The Eigenvalue Problem .......................... 35
3.3.2 Diagonalisation of Square Matrices .............. 39
3.3.3 Calculation of е|А|t by Series Expansions ....... 41
3.3.4 Calculation of e|А|t by Matrix Transformation ... 43
3.3.5 Calculation of e|А|t by the Cayley-Hamilton
Method .......................................... 46
3.3.6 Calculation of e|А|t by the Time Step
Integration Method .............................. 50
3.4 Solutions for a Time-varying System .................... 52
3.4.1 The Successive Approximation Method for
Homogeneous Equations ........................... 53
3.4.2 The Successive Approximation Method for
Non-homogeneous Equations ....................... 54
3.5 State Variable Equation of Elasticity .................. 57
4 Bending, Vibration and Buckling of Laminated Rectangular
Plates ...................................................... 59
4.1 Introduction ........................................... 59
4.2 Free Vibration of Cross-ply Laminated Plates ........... 60
4.2.1 Free Vibration of Simply Supported Plates ....... 63
4.2.1.1 Free Vibration of Single-layered
Plates ................................. 63
4.2.1.2 Free Vibration of Laminated Plates ..... 66
4.2.2 Free Vibration of Plates with Clamped Edges ..... 69
4.3 Buckling of Cross-ply Laminated Plates ................. 76
4.4 Bending of Cross-ply Laminated Plates .................. 77
4.4.1 Bending of Simply Supported Laminated Plates .... 78
4.4.2 Bending of Laminated Plates with Clamped
Edges ........................................... 83
4.4.3 Bending of Laminated Plates with Free Edges ..... 86
4.4.4 Other Topics on Plate Bending ................... 91
4.4.4.1 Bending of Laminated Continuous
Plates ................................. 91
4.4.4.2 Bending of Laminated Piezoelectric
Plates ................................. 91
4.4.5 Asymptotic Three-dimensional Solution ........... 97
5 Axisymmetric Vibration, Buckling and Bending of Laminated
Plates with Transversely Isotropic Layers ................... 99
5.1 Introduction ........................................... 99
5.2 Axisymmetric Free Vibration of Laminated Plates ........ 99
5.2.1 Axisymmetric Vibration of Laminated Circular
Plates ......................................... 102
5.2.2 Axisymmetric Vibration of Laminated Annular
Plates ......................................... 108
5.2.2.1 State Space Solution .................. 108
5.2.2.2 State Space with Hankel Transform
Method ................................ 112
5.3 Axisymmetric Buckling of Laminated Plates ............. 114
5.3.1 Axisymmetric Buckling of Laminated Circular
Plates ......................................... 115
5.3.2 Axisymmetric Buckling of Laminated Annular
Plates ......................................... 117
5.4 Axisymmetric Bending of Laminated Circular Plates ..... 119
5.4.1 Axisymmetric Bending of Circular Plates Under
Uniformly Distributed Transverse Pressure ...... 119
5.4.2 Axisymmetric Bending of Circular Plates on
Winkler's Foundation ........................... 123
6 Vibration, Buckling and Bending of Laminated Cylindrical
Shells ..................................................... 127
6.1 Introduction .......................................... 127
6.2 Vibration of Laminated Cylindrical Shells ............. 128
6.2.1 Vibration of Simply Supported Laminated
Cylindrical Shells ............................. 133
6.2.1.1 Free Vibration of Simply Supported
Laminated Cylindrical shells .......... 133
6.2.1.2 Forced Vibration of Simply Supported
Laminated Cylindrical Shells .......... 141
6.2.2 Axisymmetric Vibration of Simply Supported
Laminated Cylinders Composed of Monoclinic
Material Layers ................................ 143
6.2.3 Free Vibration of Cross-ply Laminated
Cylindrical Shells with Clamped Edges .......... 151
6.2.4 Free Vibration of Cross-ply Laminated
Cylindrical Shells with Interior Point
Supports ....................................... 156
6.2.5 Axisymmetric Vibration of Cross-ply Laminated
Cylinders with Ring Stiffeners ................. 158
6.3 Buckling of Laminated Cross-ply Cylindrical Shells .... 159
6.3.1 Pre-buckling State in a Thin Orthotropic
Layer .......................................... 159
6.3.2 Perturbed State in a Thin Orthotropic Layer .... 162
6.3.3 Solution for Thick or Laminated Cylindrical
Shells ......................................... 164
6.4 Buckling of Multi-laminated Angle-ply Cylindrical
Shells ................................................ 168
6.5 Bending of Laminated Cylindrical Shells ............... 175
6.5.1 Bending of Simply Supported Cylindrical
Shells ......................................... 175
6.5.1.1 Bending of Cross-ply Laminated
Cylindrical Shells .................... 175
6.5.1.2 Axisymmetric Bending of Angle-ply
Laminated Cylinders ................... 178
6.5.2 Bending of Cross-ply Laminated Cylinders with
Clamped Edges .................................. 181
6.5.3 Bending of Cross-ply Laminated Cylinders with
Free Edges ..................................... 183
6.5.4 Bending of Laminated Continuous Cylinders ...... 184
6.5.5 Generalized Plane Strain, Torsion and Bending
of Laminated Cylinders ......................... 185
6.6 Stationary Thermo-elastic Bending of Laminated
Cylinders ............................................. 186
7 Analysis of Laminated Shells of Various Shapes ............. 191
7.1 Introduction .......................................... 191
7.2 Free Vibration of Transversely Isotropic Spheres ...... 191
7.2.1 Free Axisymmetric Vibration of Transversely
Isotropic Thin Spheres ......................... 194
7.2.2 Free Axisymmetric Vibration of a Laminated
Thick Sphere Composed of Transversely
Isotropic Layers ............................... 196
7.3 Static Analysis of Piezoelectric Hollow Spheres ....... 199
7.4 Free Vibration of Doubly-curved Shells ................ 200
7.5 Bending of Anisotropic Inhomogeneous, Non-circular
Cylinders ............................................. 204
8 Decay of Edge Effects in Laminated Composites .............. 207
8.1 Introduction .......................................... 207
8.2 Decay Rate of Stresses in Hollow Cylinders with
Cross-ply Lay-ups ..................................... 208
8.3 Decay Rate of Stresses in Hollow Cylinders with
Angle-ply Lay-ups ..................................... 217
8.4 Decay of Stresses in Laminated Strips ................. 224
9 Semi-analytical State Equation Method ....................... 227
9.1 Introduction .......................................... 227
9.2 Semi-analytical State Space Finite Element Method
for Laminated Plates .................................. 228
9.2.1 Matrix Representation of the Three-
dimensional Equations of Elasticity ............ 228
9.2.2 The Principle of Virtual Displacements ......... 230
9.2.3 The Principle of Virtual Forces ................ 232
9.2.4 Finite Element Approximation in the Plane of
a Plate ........................................ 233
9.2.5 State Equation of the Semi-analytical FE
Solution ....................................... 234
9.3 Semi-analytical Finite Difference State Equation
Method for Laminated Plates and Shells ................ 247
9.3.1 General Formulation ............................ 247
9.3.2 Finite Difference Approximation ................ 248
9.4 State Space Boundary Element Method ................... 250
9.4.1 Derivation of the Fundamental Solution ......... 250
9.4.2 The Boundary Element Method .................... 252
9.5 Semi-analytical Variational State Space Method for
Laminated Plates and Shells ........................... 254
9.5.1 Galerkin Weighting and State Space Method ...... 254
9.5.2 Weighted Form of State Equation ................ 255
References .................................................... 257
Index ......................................................... 269
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