Mechanical Properties of Solid Polymers 3e

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 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 365 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 J - 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 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 13.11.1 Fatigue in Polymers 434 References 439 Further Reading 447 J-Integral 401sigmaB--sigmaY Diagram 416 Index