Mechanical Properties of Solid Polymers

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 xiii1 Structure of Polymers 11.1 Chemical Composition 11.1.1 Polymerisation 11.1.2 Cross-Linking and Chain-Branching 31.1.3 Average Molecular Mass and Molecular Mass Distribution 41.1.4 Chemical and Steric Isomerism and Stereoregularity 51.1.5 Liquid Crystalline Polymers 71.1.6 Blends, Grafts and Copolymers 81.2 Physical Structure 91.2.1 Rotational Isomerism 91.2.2 Orientation and Crystallinity 10References 16Further Reading 172 The Mechanical Properties of Polymers: General Considerations 192.1 Objectives 192.2 The Different Types of Mechanical Behaviour 192.3 The Elastic Solid and the Behaviour of Polymers 212.4 Stress and Strain 222.4.1 The State of Stress 222.4.2 The State of Strain 232.5 The Generalised Hooke’s Law 26References 293 The Behaviour in the Rubber-Like State: Finite Strain Elasticity 313.1 The Generalised Definition of Strain 313.1.1 The Cauchy–Green Strain Measure 323.1.2 Principal Strains 343.1.3 Transformation of Strain 363.1.4 Examples of Elementary Strain Fields 383.1.5 Relationship of Engineering Strains to General Strains 413.1.6 Logarithmic Strain 423.2 The Stress Tensor 433.3 The Stress–Strain Relationships 443.4 The Use of a Strain Energy Function 473.4.1 Thermodynamic Considerations 473.4.2 The Form of the Strain Energy Function 513.4.3 The Strain Invariants 513.4.4 Application of the Invariant Approach 523.4.5 Application of the Principal Stretch Approach 54References 584 Rubber-Like Elasticity 614.1 General Features of Rubber-Like Behaviour 614.2 The Thermodynamics of Deformation 624.2.1 The Thermoelastic Inversion Effect 644.3 The Statistical Theory 654.3.1 Simplifying Assumptions 654.3.2 Average Length of a Molecule between Cross-Links 664.3.3 The Entropy of a Single Chain 674.3.4 The Elasticity of a Molecular Network 694.4 Modifications of Simple Molecular Theory 724.4.1 The Phantom Network Model 734.4.2 The Constrained Junction Model 734.4.3 The Slip Link Model 734.4.4 The Inverse Langevin Approximation 754.4.5 The Conformational Exhaustion Model 794.4.6 The Effect of Strain-Induced Crystallisation 804.5 The Internal Energy Contribution to Rubber Elasticity 804.6 Conclusions 83References 83Further Reading 855 Linear Viscoelastic Behaviour 875.1 Viscoelasticity as a Phenomenon 875.1.1 Linear Viscoelastic Behaviour 885.1.2 Creep 895.1.3 Stress Relaxation 915.2 Mathematical Representation of Linear Viscoelasticity 925.2.1 The Boltzmann Superposition Principle 935.2.2 The Stress Relaxation Modulus 965.2.3 The Formal Relationship between Creep and Stress Relaxation 965.2.4 Mechanical Models, Relaxation and Retardation Time Spectra 975.2.5 The Kelvin or Voigt Model 985.2.6 The Maxwell Model 995.2.7 The Standard Linear Solid 1005.2.8 Relaxation Time Spectra and Retardation Time Spectra 1015.3 Dynamical Mechanical Measurements: The Complex Modulus and Complex Compliance 1035.3.1 Experimental Patterns for G 1 , G 2 and so on as a Function of Frequency 1055.4 The Relationships between the Complex Moduli and the Stress Relaxation Modulus 1095.4.1 Formal Representations of the Stress Relaxation Modulus and the Complex Modulus 1115.4.2 Formal Representations of the Creep Compliance and the Complex Compliance 1135.4.3 The Formal Structure of Linear Viscoelasticity 1135.5 The Relaxation Strength 114References 116Further Reading 1176 The Measurement of Viscoelastic Behaviour 1196.1 Creep and Stress Relaxation 1196.1.1 Creep Conditioning 1196.1.2 Specimen Characterisation 1206.1.3 Experimental Precautions 1206.2 Dynamic Mechanical Measurements 1236.2.1 The Torsion Pendulum 1246.2.2 Forced Vibration Methods 1266.2.3 Dynamic Mechanical Thermal Analysis (DMTA) 1266.3 Wave-Propagation Methods 1276.3.1 The Kilohertz Frequency Range 1286.3.2 The Megahertz Frequency Range: Ultrasonic Methods 1296.3.3 The Hypersonic Frequency Range: Brillouin Spectroscopy 131References 131Further Reading 1337 Experimental Studies of Linear Viscoelastic Behaviour as a Function of Frequency and Temperature: Time–Temperature Equivalence 1357.1 General Introduction 1357.1.1 Amorphous Polymers 1357.1.2 Temperature Dependence of Viscoelastic Behaviour 1387.1.3 Crystallinity and Inclusions 1387.2 Time–Temperature Equivalence and Superposition 1407.3 Transition State Theories 1437.3.1 The Site Model Theory 1457.4 The Time–Temperature Equivalence of the Glass Transition Viscoelastic Behaviour in Amorphous Polymers and the Williams, Landel and Ferry (WLF) Equation 1477.4.1 The Williams, Landel and Ferry Equation, the Free Volume Theory and Other Related Theories 1537.4.2 The Free Volume Theory of Cohen and Turnbull 1547.4.3 The Statistical Thermodynamic Theory of Adam and Gibbs 1547.4.4 An Objection to Free Volume Theories 1557.5 Normal Mode Theories Based on Motion of Isolated Flexible Chains 1567.6 The Dynamics of Highly Entangled Polymers 160References 1638 Anisotropic Mechanical Behaviour 1678.1 The Description of Anisotropic Mechanical Behaviour 1678.2 Mechanical Anisotropy in Polymers 1688.2.1 The Elastic Constants for Specimens Possessing Fibre Symmetry 1688.2.2 The Elastic Constants for Specimens Possessing Orthorhombic Symmetry 1708.3 Measurement of Elastic Constants 1718.3.1 Measurements on Films or Sheets 1718.3.2 Measurements on Fibres and Monofilaments 1818.4 Experimental Studies of Mechanical Anisotropy in Oriented Polymers 1858.4.1 Sheets of Low-Density Polyethylene 1868.4.2 Filaments Tested at Room Temperature 1868.5 Interpretation of Mechanical Anisotropy: General Considerations 1928.5.1 Theoretical Calculation of Elastic Constants 1928.5.2 Orientation and Morphology 1978.6 Experimental Studies of Anisotropic Mechanical Behaviour and Their Interpretation 1988.6.1 The Aggregate Model and Mechanical Anisotropy 1988.6.2 Correlation of the Elastic Constants of an Oriented Polymer with Those of an Isotropic Polymer: The Aggregate Model 1988.6.3 The Development of Mechanical Anisotropy with Molecular Orientation 2018.6.4 The Sonic Velocity 2068.6.5 Amorphous Polymers 2088.6.6 Oriented Polyethylene Terephthalate Sheet with Orthorhombic Symmetry 2098.7 The Aggregate Model for Chain-Extended Polyethylene and Liquid Crystalline Polymers 2128.8 Auxetic Materials: Negative Poisson’s Ratio 216References 2209 Polymer Composites: Macroscale and Microscale 2279.1 Composites: A General Introduction 2279.2 Mechanical Anisotropy of Polymer Composites 2289.2.1 Mechanical Anisotropy of Lamellar Structures 2289.2.2 Elastic Constants of Highly Aligned Fibre Composites 2309.2.3 Mechanical Anisotropy and Strength of Uniaxially Aligned Fibre Composites 2339.3 Short Fibre Composites 2339.3.1 The Influence of Fibre Length: Shear Lag Theory 2349.3.2 Debonding and Pull-Out 2369.3.3 Partially Oriented Fibre Composites 2369.4 Nanocomposites 2389.5 Takayanagi Models for Semi-Crystalline Polymers 2419.5.1 The Simple Takayanagi Model 2429.5.2 Takayanagi Models for Dispersed Phases 2429.5.3 Modelling Polymers with a Single-Crystal Texture 2459.6 Ultra-High-Modulus Polyethylene 2509.6.1 The Crystalline Fibril Model 2509.6.2 The Crystalline Bridge Model 2529.7 Conclusions 255References 256Further Reading 25910 Relaxation Transitions: Experimental Behaviour and Molecular Interpretation 26110.1 Amorphous Polymers: An Introduction 26110.2 Factors Affecting the Glass Transition in Amorphous Polymers 26310.2.1 Effect of Chemical Structure 26310.2.2 Effect of Molecular Mass and Cross-Linking 26510.2.3 Blends, Grafts and Copolymers 26610.2.4 Effects of Plasticisers 26710.3 Relaxation Transitions in Crystalline Polymers 26910.3.1 General Introduction 26910.3.2 Relaxation in Low-Crystallinity Polymers 27010.3.3 Relaxation Processes in Polyethylene 27210.3.4 Relaxation Processes in Liquid Crystalline Polymers 27810.4 Conclusions 282References 28211 Non-linear Viscoelastic Behaviour 28511.1 The Engineering Approach 28611.1.1 Isochronous Stress–Strain Curves 28611.1.2 Power Laws 28711.2 The Rheological Approach 28911.2.1 Historical Introduction to Non-linear Viscoelasticity Theory 28911.2.2 Adaptations of Linear Theory – Differential Models 29411.2.3 Adaptations of Linear Theory – Integral Models 29911.2.4 More Complicated Single-Integral Representations 30311.2.5 Comparison of Single-Integral Models 30611.3 Creep and Stress Relaxation as Thermally Activated Processes 30611.3.1 The Eyring Equation 30711.3.2 Applications of the Eyring Equation to Creep 30811.3.3 Applications of the Eyring Equation to Stress Relaxation 31011.3.4 Applications of the Eyring Equation to Yield 31211.4 Multi-axial Deformation: Three-Dimensional Non-linear Viscoelasticity 313References 315Further Reading 31812 Yielding and Instability in Polymers 31912.1 Discussion of the Load–Elongation Curves in Tensile Testing 32012.1.1 Necking and the Ultimate Stress 32112.1.2 Necking and Cold-Drawing: A Phenomenological Discussion 32312.1.3 Use of the Considère Construction 32512.1.4 Definition of Yield Stress 32612.2 Ideal Plastic Behaviour 32712.2.1 The Yield Criterion: General Considerations 32712.2.2 The Tresca Yield Criterion 32712.2.3 The Coulomb Yield Criterion 32812.2.4 The von Mises Yield Criterion 32912.2.5 Geometrical Representations of the Tresca, von Mises and Coulomb Yield Criteria 33112.2.6 Combined Stress States 33112.2.7 Yield Criteria for Anisotropic Materials 33312.2.8 The Plastic Potential 33412.3 Historical Development of Understanding of the Yield Process 33512.3.1 Adiabatic Heating 33612.3.2 The Isothermal Yield Process: The Nature of the Load Drop 33712.4 Experimental Evidence for Yield Criteria in Polymers 33812.4.1 Application of Coulomb Yield Criterion to Yield Behaviour 33912.4.2 Direct Evidence for the Influence of Hydrostatic Pressure on Yield Behaviour 33912.5 The Molecular Interpretations of Yield 34212.5.1 Yield as an Activated Rate Process 34312.5.2 Yield Considered to Relate to the Movement of Dislocations or Disclinations 35112.6 Cold-Drawing, Strain Hardening and the True Stress–Strain Curve 35912.6.1 General Considerations 35912.6.2 Cold-Drawing and the Natural Draw Ratio 35912.6.3 The Concept of the True Stress–True Strain Curve and the Network Draw Ratio 36112.6.4 Strain Hardening and Strain Rate Sensitivity 36312.6.5 Process Flow Stress Paths 36412.6.6 Neck Profiles 36512.6.7 Crystalline Polymers 36612.7 Shear Bands 36612.8 Physical Considerations behind Viscoplastic Modelling 36912.8.1 The Bauschinger Effect 37012.9 Shape Memory Polymers 371References 372Further Reading 37813 Breaking Phenomena 37913.1 Definition of Tough and Brittle Behaviour in Polymers 37913.2 Principles of Brittle Fracture of Polymers 38013.2.1 Griffith Fracture Theory 38013.2.2 The Irwin Model 38113.2.3 The Strain Energy Release Rate 38213.3 Controlled Fracture in Brittle Polymers 38513.4 Crazing in Glassy Polymers 38613.5 The Structure and Formation of Crazes 39113.5.1 The Structure of Crazes 39213.5.2 Craze Initiation and Growth 39513.5.3 Crazing in the Presence of Fluids and Gases: Environmental Crazing 39713.6 Controlled Fracture in Tough Polymers 40013.6.1 The J-Integral 40113.6.2 Essential Work of Fracture 40413.6.3 Crack Opening Displacement 40713.7 The Molecular Approach 41313.8 Factors Influencing Brittle–Ductile Behaviour: Brittle–Ductile Transitions 41413.8.1 The Ludwig–Davidenkov–Orowan Hypothesis 41413.8.2 Notch Sensitivity and Vincent’s σ B –σ Y Diagram 41613.8.3 A Theory of Brittle–Ductile Transitions Consistent with Fracture Mechanics: Fracture Transitions 41913.9 The Impact Strength of Polymers 42213.9.1 Flexed-Beam Impact 42213.9.2 Falling-Weight Impact 42613.9.3 Toughened Polymers: High-Impact Polyblends 42713.9.4 Crazing and Stress Whitening 42913.9.5 Dilatation Bands 42913.10 The Tensile Strength and Tearing of Polymers in the Rubbery State 43013.10.1 The Tearing of Rubbers: Extension of Griffith Theory 43013.10.2 Molecular Theories of the Tensile Strength of Rubbers 43113.11 Effect of Strain Rate and Temperature 43213.12 Fatigue in Polymers 434References 439Further Reading 447Index 449