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 G1, G2 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
Relaxation ........................................ 310
11.3.4 Applications of the Eyring Equation to Yield ...... 312
11.4 Multi-axial Deformation: Three-Dimensional Non-linear
Viscoelasticity .......................................... 313
References ............................................... 315
Further Reading .......................................... 318
12 Yielding and Instability in Polymers ..................... 319
12.1 Discussion of the Load-Elongation Curves in Tensile
Testing .................................................. 320
12.1.1 Necking and the Ultimate Stress ................... 321
12.1.2 Necking and Cold-Drawing: A Phenomenological
Discussion ........................................ 323
12.1.3 Use of the Considere Construction ................. 325
12.1.4 Definition of Yield Stress ........................ 326
12.2 Ideal Plastic Behaviour .................................. 327
12.2.1 The Yield Criterion: General Considerations ....... 327
12.2.2 The Tresca Yield Criterion ........................ 327
12.2.3 The Coulomb Yield Criterion ....................... 328
12.2.4 The von Mises Yield Criterion ..................... 329
12.2.5 Geometrical Representations of the Tresca, von
Mises and Coulomb Yield Criteria .................. 331
12.2.6 Combined Stress States ............................ 331
12.2.7 Yield Criteria for Anisotropic Materials .......... 333
12.2.8 The Plastic Potential ............................. 334
12.3 Historical Development of Understanding of the Yield
Process .................................................. 335
12.3.1 Adiabatic Heating ................................. 336
12.3.2 The Isothermal Yield Process: The Nature of the
Load Drop ......................................... 337
12.4 Experimental Evidence for Yield Criteria in Polymers ..... 338
12.4.1 Application of Coulomb Yield Criterion to Yield
Behaviour ......................................... 339
12.4.2 Direct Evidence for the Influence of Hydrostatic
Pressure on Yield Behaviour ....................... 339
12.5 The Molecular Interpretations of Yield ................... 342
12.5.1 Yield as an Activated Rate Process ................ 343
12.5.2 Yield Considered to Relate to the Movement of
Dislocations or Disclinations ..................... 351
12.6 Cold-Drawing, Strain Hardening and the True Stress-
Strain Curve ............................................. 359
12.6.1 General Considerations ............................ 359
12.6.2 Cold-Drawing and the Natural Draw Ratio ........... 359
12.6.3 The Concept of the True Stress-True Strain Curve
and the Network Draw Ratio ........................ 361
12.6.4 Strain Hardening and Strain Rate Sensitivity ...... 363
12.6.5 Process Flow Stress Paths ......................... 364
12.6.6 Neck Profiles ..................................... 365
12.6.7 Crystalline Polymers .............................. 366
12.7 Shear Bands .............................................. 366
12.8 Physical Considerations behind Viscoplastic Modelling .... 369
12.8.1 The Bauschinger Effect ........................... 370
12.9 Shape Memory Polymers .................................... 371
References ............................................... 372
Further Reading .......................................... 378
13 Breaking Phenomena ....................................... 379
13.1 Definition of Tough and Brittle Behaviour in Polymers .... 379
13.2 Principles of Brittle Fracture of Polymers ............... 380
13.2.1 Griffith Fracture Theory .......................... 380
13.2.2 The Irwin Model ................................... 381
13.2.3 The Strain Energy Release Rate .................... 382
13.3 Controlled Fracture in Brittle Polymers .................. 385
13.4 Crazing in Glassy Polymers ............................... 386
13.5 The Structure and Formation of Crazes .................... 391
13.5.1 The Structure of Crazes ........................... 392
13.5.2 Craze Initiation and Growth ....................... 395
13.5.3 Crazing in the Presence of Fluids and Gases:
Environmental Crazing ............................. 397
13.6 Controlled Fracture in Tough Polymers .................... 400
13.6.1 The/-Integral ..................................... 401
13.6.2 Essential Work of Fracture ........................ 404
13.6.3 Crack Opening Displacement ........................ 407
13.7 The Molecular Approach ................................... 413
13.8 Factors Influencing Brittle-Ductile Behaviour: Brittle-
Ductile Transitions ...................................... 414
13.8.1 The Ludwig-Davidenkov-Orowan Hypothesis ........... 414
13.8.2 Notch Sensitivity and Vincent's стд-ау Diagram .... 416
13.8.3 A Theory of Brittle-Ductile Transitions
Consistent with Fracture Mechanics: Fracture
Transitions ....................................... 419
13.9 The Impact Strength of Polymers .......................... 422
13.9.1 Flexed-Beam Impact ................................ 422
13.9.2 Falling-Weight Impact ............................. 426
13.9.3 Toughened Polymers: High-Impact Polyblends ........ 427
13.9.4 Crazing and Stress Whitening ...................... 429
13.9.5 Dilatation Bands .................................. 429
13.10 The Tensile Strength and Tearing of Polymers in the
Rubbery State ............................................ 430
13.10.1 The Tearing of Rubbers: Extension of Griffith
Theory ........................................... 430
13.10.2 Molecular Theories of the Tensile Strength of
Rubbers .......................................... 431
13.11 Effect of Strain Rate and Temperature ................... 432
13.12 Fatigue in Polymers ..................................... 434
References ............................................... 439
Further Reading .......................................... 447
Index ......................................................... 449
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