Table of Contents |
1. | Introduction | 1 |
1.1. | Background | 1 |
1.2. | Outline of Following Chapters | 2 |
1.3. | Closure | 3 |
2. | Fibre Geometry and Fibre Mechanics | 5 |
2.1. | Introduction | 5 |
2.2. | Fibre Dimensionality | 6 |
2.3. | Fibre Properties | 9 |
2.3.1. | Fibre Size | 10 |
2.3.2. | Fibre Weight | 10 |
2.3.3. | Fibre Material Stress---Strain Behaviour | 10 |
2.4. | Primary Deformation, Extension | 13 |
2.4.1. | Polynomial Approximations | 14 |
2.4.2. | Strain Energy | 15 |
2.5. | Anelasticity | 16 |
2.5.1. | Viscoelasticity | 16 |
2.5.2. | Linear Viscoelasticity | 17 |
2.5.3. | Nonlinear Viscoelasticity | 20 |
2.5.4. | Large Strain Rate Modelling for Polymer Fibres | 23 |
2.5.5. | Hysteresis | 24 |
2.5.6. | Creep and Relaxation | 27 |
2.6. | Secondary Deformations | 30 |
2.6.1. | Fibre Torsion | 31 |
2.6.2. | Fibre Flexure | 32 |
2.7. | Strain Energy or the Energy of Deformation | 35 |
| References | 36 |
3. | Component Path Geometries | 37 |
3.1. | Curves in Space | 37 |
3.1.1. | General Path Geometry | 37 |
3.1.2. | Geometry Preserving Structures | 39 |
3.2. | The Helical Structure | 40 |
3.2.1. | The Helix Geometry | 40 |
3.2.2. | Extensional and Twist Deformations | 41 |
3.2.3. | Flexure in Helical Components | 42 |
3.2.4. | The Partitioning of Strain Energy | 43 |
| Reference | 46 |
4. | Transversely Continuous Structures | 47 |
4.1. | Geometrically Preserving Structures | 48 |
4.1.1. | The Assembly of Transversely Continuous Structures | 48 |
4.1.2. | The Straining of Transversely Continuous Structures | 50 |
4.1.3. | The Loading of Transversely Continuous Structures | 53 |
4.1.4. | An Analytic Result for Strain Energy | 53 |
4.1.5. | The Principle of Virtual Work for Transversely Continuous Structures | 54 |
4.1.6. | The Progressive Failure of Transversely Continuous Structures | 55 |
4.2. | Parallel Structures, Variability | 57 |
| References | 59 |
5. | Hierarchical Structures | 61 |
5.1. | Component Terminology | 61 |
5.2. | Normalisation of Hierarchical Structures | 64 |
| References | 65 |
6. | Transversely Discrete Structures | 67 |
6.1. | Numerous Small Component Structures | 67 |
6.1.1. | The Hierarchical Tree | 68 |
6.1.2. | The Assembly of Transversely Discrete Structures | 69 |
6.1.3. | The Straining of Transversely Discrete Structures | 70 |
6.1.4. | The Principle of Virtual Work for Transversely | |
| Discrete Structures | 71 |
6.2. | Component Geometries | 72 |
6.2.1. | The Assembly of Large Component Structures | 73 |
6.2.2. | The Loading of Large Component Structures | 77 |
6.3. | Tubular Woven Geometries: Braiding and Plaits | 78 |
6.3.1. | The Geometry of Braids and Plaits | 80 |
6.4. | Bending of Helical Structures | 83 |
6.4.1. | General Kinematics Associated with Bent Structures | 83 |
6.4.2. | No Slip (Friction) Bending | 84 |
6.4.3. | Geodesic (No Friction) Bending | 85 |
| References | 89 |
7. | Contact Force and Friction | 91 |
7.1. | Contact Forces and Pressures | 92 |
7.2. | Friction and Relative Slip: Inter Modes | 94 |
7.2.1. | Mode 1: Axial Slip | 95 |
7.2.2. | Mode 2: Component Twist | 98 |
7.2.3. | Mode 3: Scissoring | 100 |
7.2.4. | Mode 4: Sawing | 103 |
7.2.5. | Transition Between the Sliding Modes | 105 |
7.3. | Component Transverse Deformation and Set: Intra Modes | 106 |
7.3.1. | Dilation Measure, Ξ and Packing Factor Pf | 107 |
7.3.2. | Distortion Measure, Ω | 108 |
7.3.3. | The Set in Distortion and Dilation | 108 |
7.3.4. | The Energy of Dilation and Distortion | 111 |
7.3.5. | The Continuum Model | 112 |
| References | 113 |
8. | Component Wear, Life and Heating | 115 |
8.1. | The Work Done by Relative Motion Between Components | 115 |
8.2. | The Friction Parameters | 116 |
8.3. | Wear | 117 |
8.4. | Steady State or Equilibrium Temperature, Surface Heat Convection | 117 |
| References | 119 |
| About the Author | 121 |
| Appendix | 123 |