Professor Ian M. Ward is an internationally recognized and well respected authority on this subject. Chair in Physics at Leeds University since 1970, he has gained a reputation as an outstanding scientist. He is also a co-founder of the British Polymer Physics Group and the winner of several awards, including the Glazebrook medal of the Institute of Physics (2004) and the Netlon award (2004) both given for his work in polymer physics.

Professor John Sweeney holds a Personal Chair in Polymer Mechanics at the University of Bradford. He has researched in various areas of solid polymer behaviour, including viscoelasticity, fracture mechanics, shear banding, large deformations and nanocomposites. He is well known for his collaborations with Professor Ward and his association with the internationally recognized Polymer IRC (Interdisciplinary Research Centre).

Preface xiii

1 Structure of Polymers 1

1.1 Chemical Composition 1

1.1.1 Polymerisation 1

1.1.2 Cross-Linking and Chain-Branching 3

1.1.3 Average Molecular Mass and Molecular Mass Distribution 4

1.1.4 Chemical and Steric Isomerism and Stereoregularity 5

1.1.5 Liquid Crystalline Polymers 7

1.1.6 Blends, Grafts and Copolymers 8

1.2 Physical Structure 9

1.2.1 Rotational Isomerism 9

1.2.2 Orientation and Crystallinity 10

References 16

Further Reading 17

2 The Mechanical Properties of Polymers: General Considerations 19

2.1 Objectives 19

2.2 The Different Types of Mechanical Behaviour 19

2.3 The Elastic Solid and the Behaviour of Polymers 21

2.4 Stress and Strain 22

2.4.1 The State of Stress 22

2.4.2 The State of Strain 23

2.5 The Generalised Hooke's Law 26

References 29

3 The Behaviour in the Rubber-Like State: Finite Strain Elasticity 31

3.1 The Generalised Definition of Strain 31

3.1.1 The Cauchy-Green Strain Measure 32

3.1.2 Principal Strains 34

3.1.3 Transformation of Strain 36

3.1.4 Examples of Elementary Strain Fields 38

3.1.5 Relationship of Engineering Strains to General Strains 41

3.1.6 Logarithmic Strain 42

3.2 The Stress Tensor 43

3.3 The Stress-Strain Relationships 44

3.4 The Use of a Strain Energy Function 47

3.4.1 Thermodynamic Considerations 47

3.4.2 The Form of the Strain Energy Function 51

3.4.3 The Strain Invariants 51

3.4.4 Application of the Invariant Approach 52

3.4.5 Application of the Principal Stretch Approach 54

References 58

4 Rubber-Like Elasticity 61

4.1 General Features of Rubber-Like Behaviour 61

4.2 The Thermodynamics of Deformation 62

4.2.1 The Thermoelastic Inversion Effect 64

4.3 The Statistical Theory 65

4.3.1 Simplifying Assumptions 65

4.3.2 Average Length of a Molecule between Cross-Links 66

4.3.3 The Entropy of a Single Chain 67

4.3.4 The Elasticity of a Molecular Network 69

4.4 Modifications of Simple Molecular Theory 72

4.4.1 The Phantom Network Model 73

4.4.2 The Constrained Junction Model 73

4.4.3 The Slip Link Model 73

4.4.4 The Inverse Langevin Approximation 75

4.4.5 The Conformational Exhaustion Model 79

4.4.6 The Effect of Strain-Induced Crystallisation 80

4.5 The Internal Energy Contribution to Rubber Elasticity 80

4.6 Conclusions 83

References 83

Further Reading 85

5 Linear Viscoelastic Behaviour 87

5.1 Viscoelasticity as a Phenomenon 87

5.1.1 Linear Viscoelastic Behaviour 88

5.1.2 Creep 89

5.1.3 Stress Relaxation 91

5.2 Mathematical Representation of Linear Viscoelasticity 92

5.2.1 The Boltzmann Superposition Principle 93

5.2.2 The Stress Relaxation Modulus 96

5.2.3 The Formal Relationship between Creep and Stress Relaxation 96

5.2.4 Mechanical Models, Relaxation and Retardation Time Spectra 97

5.2.5 The Kelvin or Voigt Model 98

5.2.6 The Maxwell Model 99

5.2.7 The Standard Linear Solid 100

5.2.8 Relaxation Time Spectra and Retardation Time Spectra 101

5.3 Dynamical Mechanical Measurements: The Complex Modulus and Complex Compliance 103

5.3.1 Experimental Patterns for G 1 , G 2 and so on as a Function of Frequency 105

5.4 The Relationships between the Complex Moduli and the Stress Relaxation Modulus 109

5.4.1 Formal Representations of the Stress Relaxation Modulus and the Complex Modulus 111

5.4.2 Formal Representations of the Creep Compliance and the Complex Compliance 113

5.4.3 The Formal Structure of Linear Viscoelasticity 113

5.5 The Relaxation Strength 114

References 116

Further Reading 117

6 The Measurement of Viscoelastic Behaviour 119

6.1 Creep and Stress Relaxation 119

6.1.1 Creep Conditioning 119

6.1.2 Specimen Characterisation 120

6.1.3 Experimental Precautions 120

6.2 Dynamic Mechanical Measurements 123

6.2.1 The Torsion Pendulum 124

6.2.2 Forced Vibration Methods 126

6.2.3 Dynamic Mechanical Thermal Analysis (DMTA) 126

6.3 Wave-Propagation Methods 127

6.3.1 The Kilohertz Frequency Range 128

6.3.2 The Megahertz Frequency Range: Ultrasonic Methods 129

6.3.3 The Hypersonic Frequency Range: Brillouin Spectroscopy 131

References 131

Further Reading 133

7 Experimental Studies of Linear Viscoelastic Behaviour as a Function of Frequency and Temperature: Time-Temperature Equivalence 135

7.1 General Introduction 135

7.1.1 Amorphous Polymers 135

7.1.2 Temperature Dependence of Viscoelastic Behaviour 138

7.1.3 Crystallinity and Inclusions 138

7.2 Time-Temperature Equivalence and Superposition 140

7.3 Transition State Theories 143

7.3.1 The Site Model Theory 145

7.4 The Time-Temperature Equivalence of the Glass Transition Viscoelastic Behaviour in Amorphous Polymers and the Williams, Landel and Ferry (WLF) Equation 147

7.4.1 The Williams, Landel and Ferry Equation, the Free Volume Theory and Other Related Theories 153

7.4.2 The Free Volume Theory of Cohen and Turnbull 154

7.4.3 The Statistical Thermodynamic Theory of Adam and Gibbs 154

7.4.4 An Objection to Free Volume Theories 155

7.5 Normal Mode Theories Based on Motion of Isolated Flexible Chains 156

7.6 The Dynamics of Highly Entangled Polymers 160

References 163

8 Anisotropic Mechanical Behaviour 167

8.1 The Description of Anisotropic Mechanical Behaviour 167

8.2 Mechanical Anisotropy in Polymers 168

8.2.1 The Elastic Constants for Specimens Possessing Fibre Symmetry 168

8.2.2 The Elastic Constants for Specimens Possessing Orthorhombic Symmetry 170

8.3 Measurement of Elastic Constants 171

8.3.1 Measurements on Films or Sheets 171

8.3.2 Measurements on Fibres and Monofilaments 181

8.4 Experimental Studies of Mechanical Anisotropy in Oriented Polymers 185

8.4.1 Sheets of Low-Density Polyethylene 186

8.4.2 Filaments Tested at Room Temperature 186

8.5 Interpretation of Mechanical Anisotropy: General Considerations 192

8.5.1 Theoretical Calculation of Elastic Constants 192

8.5.2 Orientation and Morphology 197

8.6 Experimental Studies of Anisotropic Mechanical Behaviour and Their Interpretation 198

8.6.1 The Aggregate Model and Mechanical Anisotropy 198

8.6.2 Correlation of the Elastic Constants of an Oriented Polymer with Those of an Isotropic Polymer: The Aggregate Model 198

8.6.3 The Development of Mechanical Anisotropy with Molecular Orientation 201

8.6.4 The Sonic Velocity 206

8.6.5 Amorphous Polymers 208

8.6.6 Oriented Polyethylene Terephthalate Sheet with Orthorhombic Symmetry 209

8.7 The Aggregate Model for Chain-Extended Polyethylene and Liquid Crystalline Polymers 212

8.8 Auxetic Materials: Negative Poisson's Ratio 216

References 220

9 Polymer Composites: Macroscale and Microscale 227

9.1 Composites: A General Introduction 227

9.2 Mechanical Anisotropy of Polymer Composites 228

9.2.1 Mechanical Anisotropy of Lamellar Structures 228

9.2.2 Elastic Constants of Highly Aligned Fibre Composites 230

9.2.3 Mechanical Anisotropy and Strength of Uniaxially Aligned Fibre Composites 233

9.3 Short Fibre Composites 233

9.3.1 The Influence of Fibre Length: Shear Lag Theory 234

9.3.2 Debonding and Pull-Out 236

9.3.3 Partially Oriented Fibre Composites 236

9.4 Nanocomposites 238

9.5 Takayanagi Models for Semi-Crystalline Polymers 241

9.5.1 The Simple Takayanagi Model 242

9.5.2 Takayanagi Models for Dispersed Phases 242

9.5.3 Modelling Polymers with a Single-Crystal Texture 245

9.6 Ultra-High-Modulus Polyethylene 250

9.6.1 The Crystalline Fibril Model 250

9.6.2 The Crystalline Bridge Model 252

9.7 Conclusions 255

References 256

Further Reading 259

10 Relaxation Transitions: Experimental Behaviour and Molecular Interpretation 261

10.1 Amorphous Polymers: An Introduction 261

10.2 Factors Affecting the Glass Transition in Amorphous Polymers 263

10.2.1 Effect of Chemical Structure 263

10.2.2 Effect of Molecular Mass and Cross-Linking 265

10.2.3 Blends, Grafts and Copolymers 266

10.2.4 Effects of Plasticisers 267

10.3 Relaxation Transitions in Crystalline Polymers 269

10.3.1 General Introduction 269

10.3.2 Relaxation in Low-Crystallinity Polymers 270

10.3.3 Relaxation Processes in Polyethylene 272

10.3.4 Relaxation Processes in Liquid Crystalline Polymers 278

10.4 Conclusions 282

References 282

11 Non-linear Viscoelastic Behaviour 285

11.1 The Engineering Approach 286

11.1.1 Isochronous Stress-Strain Curves 286

11.1.2 Power Laws 287

11.2 The Rheological Approach 289

11.2.1 Historical Introduction to Non-linear Viscoelasticity Theory 289

11.2.2 Adaptations of Linear Theory - Differential Models 294

11.2.3 Adaptations of Linear Theory - Integral Models 299

11.2.4 More Complicated Single-Integral Representations 303

11.2.5 Comparison of Single-Integral Models 306

11.3 Creep and Stress Relaxation as Thermally Activated Processes 306

11.3.1 The Eyring Equation 307

11.3.2 Applications of the Eyring Equation to Creep 308

11.3.3 Applications of the Eyring Equation to Stress...