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... Calibration and Simulation Results 78 5.1 Quasi- static Tests at Room Temperature 78 5.2 Quasi- static Tests at High Temperature 83 5.3 Experiments for Deformation at High Strain. . .DESCRIBING LARGE DEFORMATION OF POLYMERS AT QUASI- STATIC AND HIGH STRAIN RATES HABIB POURIAYEVALI (M.SC., AMIRKABIR UNIVERSITY, IRAN) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY... Analysis and modelling of the quasi- static and dynamic behaviour of polymers are essential and will facilitate the use of computer simulation for designing products that incorporate polymer padding and
DESCRIBING LARGE DEFORMATION OF POLYMERS AT QUASI-STATIC AND HIGH STRAIN RATES HABIB POURIAYEVALI NATIONAL UNIVERSITY OF SINGAPORE 2013 DESCRIBING LARGE DEFORMATION OF POLYMERS AT QUASI-STATIC AND HIGH STRAIN RATES HABIB POURIAYEVALI (M.SC., AMIRKABIR UNIVERSITY, IRAN) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2013 Declaration I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Habib Pouriayevali I Acknowledgement First and foremost, I would like to express my heartfelt thanks to my supervisor, Professor Victor P.W. Shim. I have been really impressed by his attitude and his efforts in providing a positive and encouraging environment in his laboratory. I sincerely appreciate his patience and confidence in his students, and from our interaction, I gained profound understandings, knowledge and awareness, helping me to shape my future. I would also like to thank the laboratory officers, Mr. Joe Low Chee Wah and Mr. Alvin Goh Tiong Lai, for their pleasant support and technical guidance for undertaking the experiments. My sincerest thanks to my colleagues and friends in Impact Mechanics Laboratory; Dr. Kianoosh Marandi, Dr. Long bin Tan, Saeid Arabnejad, Nader Hamzavi, Chen Yang, Bharath Narayanan, Liu Jun, Jia Shu, Xu Juan, with whom I share unforgettable memories and I appreciate their valuable advice and contributions in my research work. I would also like to express my heartfelt gratitude to my parents and family for their patience, encouraging support and conveying strength to me. Last but not the least, I am grateful to Singapore and NUS for providing me with the honour to build and grow up my academic life in an extremely developed and convenient environment. II List of Contents Declaration .................................................................................................................... I Acknowledgement ....................................................................................................... II Summary .................................................................................................................... VI List of Tables ...........................................................................................................VIII List of Figures ............................................................................................................ IX List of Symbols ........................................................................................................XIII 1. Introduction .................................................................................................... 1 1.1 Polymers; Definition and Morphology .......................................................... 1 1.2 Objective and Scope ...................................................................................... 4 2. Modelling of Elastomers ................................................................................ 6 2.1 Literature Review .......................................................................................... 8 2.1.1 Hyperelastic Material.............................................................................. 8 2.1.2 Viscoelastic Material .............................................................................. 9 2.1.3 Elastic and Viscous Components of a Linear Viscoelastic Model ....... 10 2.1.4 Maxwell Model..................................................................................... 10 2.1.5 Kelvin-Voigt Model.............................................................................. 11 2.1.6 Generalized Maxwell Model ................................................................ 12 2.1.7 Relaxation Model.................................................................................. 13 2.1.8 Visco-hyperelastic Constitutive Model ................................................ 14 2.2 Proposed Visco-hyperelastic Model ............................................................ 16 2.2.1 Derivation of Second Piola-Kirchhoff and Cauchy Stresses – A Hyperelastic Model for Element A ....................................................... 17 2.2.2 Derivation of Cauchy Stresses for Element B ...................................... 20 2.2.3 Combination of Cauchy Stresses of Elements A and B ........................ 21 2.2.4 Relaxation Process in Elastomers ......................................................... 22 2.2.5 Hyperelastic Model Used for Uniaxial Loading of Element A............. 25 2.2.6 Visco-hyperelastic Model for Uniaxial Loading .................................. 26 2.3 High Strain Rate Experiments on Elastomers.............................................. 27 2.4 Application of Model and Discussion ......................................................... 27 2.5 Summary and Conclusion ............................................................................ 35 3. Quasi-static and Dynamic Response of a Semi-crystalline Polymer ........... 36 3.1 Quasi-static Experiments at Room Temperature ......................................... 36 3.1.1 Quasi-static Compression ..................................................................... 36 3.1.2 Quasi-static Tension ............................................................................. 39 III 3.2 Dynamic Mechanical Analysis (DMA) Testing .......................................... 43 3.3 Quasi-static Experiments at Higher Temperatures ...................................... 44 3.4 High Strain Rate Tests ................................................................................. 46 3.4.1 Split Hopkinson Bar ............................................................................. 46 3.4.2 Dynamic Compressive Loading ........................................................... 47 3.4.3 Dynamic Tensile Loading..................................................................... 52 3.5 4. Summary and Conclusion ............................................................................ 54 Modelling of Semi-crystalline Polymers ..................................................... 55 4.1 Literature Review ........................................................................................ 55 4.2 Constitutive Equation .................................................................................. 57 4.3 One-dimensional Form of the proposed Elastic-Viscoelastic-Viscoplastic Framework ................................................................................................... 58 4.4 Kinematic Considerations ............................................................................ 60 4.5 Thermodynamic Considerations .................................................................. 62 4.6 Simplification of Chain Rule for Time Derivative of Free Energy ............. 63 4.7 Second Piola-Kirchhoff and Cauchy Stresses ............................................. 66 4.8 Dissipation Inequality .................................................................................. 67 4.9 Evolution of Temperature Variation ............................................................ 68 4.10 Inelastic Flow Rule ...................................................................................... 69 4.11 Initiation of Plastic Deformation ................................................................. 71 4.12 Helmholtz Free Energy Density for the Proposed Model............................ 72 4.13 Summary ...................................................................................................... 76 5. Model Calibration and Simulation Results .................................................. 78 5.1 Quasi-static Tests at Room Temperature ..................................................... 78 5.2 Quasi-static Tests at High Temperature....................................................... 83 5.3 Experiments for Deformation at High Strain Rates ..................................... 85 5.4 Multi-Element FEM Model of Complex-Shaped Specimens ...................... 91 5.5 Short Thick Walled Tube............................................................................. 91 5.6 Plate with Two Semi-Circular Cut-Outs ...................................................... 93 5.7 Summary and Conclusions .......................................................................... 95 6. Conclusion and Recommendation for Future Work .................................... 97 6.1 Recommendations for Future Work ............................................................ 99 References ................................................................................................................ 100 Appendixes .............................................................................................................. 106 A. Split Hopkinson Bar Device ...................................................................... 106 A.1. Governing Equations ................................................................................. 106 IV A.2. Split Hopkinson Pressure Bar Device (SHPB) .......................................... 108 A.3. Split Hopkinson Tension Bar Device (SHTB) .......................................... 109 B. Visco-hyperelastic model employed for uniaxial tests of elastomers........ 111 C. One-dimensional Model for the Proposed Elastic-viscoelastic-viscoplastic model ......................................................................................................... 114 D. Time Integration Procedure of Writing a User-defined VUMAT Code for The Proposed Elastic-Viscoelastic-Viscoplastic Model ............................ 119 V Summary Polymeric materials are widely used, and their dynamic mechanical properties are of considerable interest and attention because many products and components are subjected to impacts and shocks. Analysis and modelling of the quasi-static and dynamic behaviour of polymers are essential and will facilitate the use of computer simulation for designing products that incorporate polymer padding and components. This thesis comprises two segments and focuses on two categories of polymeric materials – elastomers and semi-crystalline polymers. The quasi-static and dynamic behaviour of these materials were studied and described by two different constitutive models. An accompanying objective is to formulate these macro-scale models with a minimum number of material parameters to avoid the complexity of delving into the details of polymer microstructure. The first part focuses on elastomers which are nonlinear viscoelastic materials with a low elastic modulus, and exhibit rate sensitivity when subjected to dynamic loading. A visco-hyperelastic constitutive equation in integral form is proposed to describe the quasi-static and high rate, large deformation response of these approximately incompressible materials. The proposed model is based on a macromechanics-level approach and comprises two components: the first corresponds to hyperelasticity based on a strain energy function expressed as a polynomial, to characterise the quasistatic nonlinear response, while the second captures the rate-dependent response, and is an integral form of the first, based on the concept of fading-memory; i.e. the stress at a material point is a function of recent deformation gradients that occurred within a small neighbourhood of that point. The proposed model incorporates a relaxation-time function to capture rate sensitivity and strain history dependence; instead of a constant relaxation-time, a deformation-dependent function is proposed. The proposed threedimensional constitutive model was implemented in MATLAB to predict the uniaxial response of six types of elastomer with different hardnesses, namely U50, U70 polyurethane rubber, SHA40, SHA60 and SHA80 rubber, as well as EthylenePropylene-Diene-Monomer (EPDM) rubber, which have been subjected to quasistatic and dynamic tension and compression by other researchers. The second part of this study describes the behaviour of the largest group of commercially used polymers, which are semi-crystalline. From a micro-scale viewpoint, these materials can be considered a composite with rigid crystallites VI suspended within an amorphous phase. This is desirable, because it combines the strength of the crystalline phase with the flexibility of its amorphous counterpart. Nylon 6 is the semi-crystalline polymer studied; it is notably rate-dependent and exhibits a temperature increase under high rate deformation, whereby the temperature crosses the glass transition temperature and increases the compliance of the polymer. Material samples of were subjected to quasi-static compression and tension at room and higher temperatures using an Instron universal testing machine. High strain rate compressive and tensile deformation were also applied to material specimens using Split Hopkinson Bar devices. The specimen deformation and temperature change during high rate deformation were captured using a high speed optical and an infrared camera respectively. A three-dimensional thermo-mechanical large-deformation constitutive model based on thermodynamic consideration was developed for a homogenized isotropic description of material consisting of crystalline and amorphous phases. The constitutive description is formulated using a macromechanics approach and employs the hyperelastic model proposed for elastomers. The model describes elasticviscoelastic-viscoplastic behaviour, coupled with post-yield hardening, and is able to predict the response of the approximately incompressible semi-crystalline material, Nylon 6, subjected to compressive and tensile quasi-static and high rate deformation. The proposed model was implemented in an FEM software (ABAQUS) via a user-defined material subroutine (VUMAT). The model was calibrated and validated by compressive and tensile tests conducted at different temperatures and deformation rates. Material parameters, such as the stiffness coefficients, viscosity and hardening, were cast as functions of temperature, as well as the degree and rate of deformation. Simplicity of these material parameters is sought to preclude involvement in the details of the molecular structures of polymers. VII List of Tables Table 1.1 Nylon 6 properties. ........................................................................................ 3 Table 2.1. Material coefficients and relaxation time functions for dynamic tension and compression of SHA40, SHA60, SHA80, U50, U70 and EPDM rubber. .. 32 Table 3.1. Compression test specimens with three lengths.......................................... 48 Table 4.1. The material parameters and model coefficients/functions which are needed to be calibrated. ........................................................................................... 77 Table 5.1. Model parameters and material coefficients. .............................................. 90 VIII List of Figures Fig. 1.1. Molecular chain types: (a) straight, (b) branched, (c) cross-linked. ................ 1 Fig. 1.2. Mechanical components made of elastomers .................................................. 2 Fig. 1.3. Schematic representation of molecular chains in a semi-crystalline polymer . 3 Fig. 1.4. Mechanical components made of Nylon. ........................................................ 4 Fig. 2.1. "Spaghetti and meatball" schematic representation of molecular structure of elastomers ........................................................................................................ 6 Fig. 2.2. Experimental stress-strain curves (Shim et al., 2004) for SHA40, SHA60 and SHA80 rubber: (a) tension; (b) compression. .................................................. 7 Fig. 2.3. Experimental data for uniaxial compressive loading to various final strains at a strain rate of -0.01/s, followed by unloading (Bergstrom and Boyce, 1998). ......................................................................................................................... 8 Fig. 2.4. Maxwell model (Tschoegl, 1989). ................................................................. 11 Fig. 2.5. Kelvin–Voigt model (Tschoegl, 1989). ......................................................... 11 Fig. 2.6. Generalized Maxwell Model (Tschoegl, 1989). ............................................ 12 Fig. 2.7. Stress relaxation under constant strain .......................................................... 13 Fig. 2.8. Physical relaxation, reversible motion in a trapped entanglement under stress. The point of reference marked moves with time. When the stress is released, entropic forces return the chains to near their original positions (Sperling, 2006). ............................................................................................................. 14 Fig. 2.9. Chemical relaxation; chain portions change partners, causing a release of stress. (Bond interchange in polyesters and polysiloxanes) (Sperling, 2006). ....................................................................................................................... 14 Fig. 2.10. Parallel mechanical elements A and B. ........................................................ 16 Fig. 2.11. Schematic representation of deformation of elastomer molecular chains. Spheres indicate globules of network chains (a molecular chain is illustrated in the globule at the top). Thick lines represent connections between network chains. (a) Before deformation, all globules are spherical; (b) after deformation, only some of the globules are elongated – the darker ones (Tosaka et al., 2004). ..................................................................................... 23 Fig. 2.12. Readjustment and relaxation of a free polymer chain loop located in a network (Bergstrom and Boyce, 1998). ......................................................... 23 Fig. 2.13. Comparison between experimental tension and compression data (Shim et al., 2004) with proposed visco-hyperelastic model for SHA40 rubber. ........ 29 Fig. 2.14. Comparison between experimental tension and compression data (Shim et al., 2004) with proposed visco-hyperelastic model for SHA60 rubber. ........ 30 Fig. 2.15. Comparison between experimental tension and compression data (Shim et al., 2004) with proposed visco-hyperelastic model for SHA80 rubber. ........ 31 Fig. 2.16. Comparison between experimental compression data (Song and Chen, 2004a) and proposed visco-hyperelastic model for EPDM rubber................ 32 Fig. 2.17. Comparison between experimental compression data (Doman et al., 2006) and proposed visco-hyperelastic model for U50 & U70 rubbers. ................. 33 IX Fig. 2.18. Comparison between relaxation time data obtained by fitting to dynamic experimental response: (a) U50, U70 rubber under compression; (b) SHA40, SHA60 and SHA80 rubber under tension...................................................... 34 Fig. 3.1. Compression test on a specimen fabricated according to ASTM D 695-08 standards; lateral slippage is obvious............................................................. 37 Fig. 3.2. Nylon 6 specimens before and after quasi-static compression. ..................... 37 Fig. 3.3. Response of three specimens to quasi-static compression. ........................... 38 Fig. 3.4. Quasi-static compressive linear elastic response of four specimens. ............ 38 Fig. 3.5. ASTM D638-Type V specimen dimension. .................................................. 39 Fig. 3.6. Specimen before and after quasi-static tension. ............................................ 40 Fig. 3.7. Quasi-static linear tensile elastic response of three specimens. .................... 40 Fig. 3.8. TEMA software (TrackEye Motion Analysis 3.5). ....................................... 41 Fig. 3.9. Response of three specimens to quasi-static tension. .................................... 41 Fig. 3.10. Experimental quasi-static response of Nylon 6 showing asymmetry in tension and compression ( = ). ..................................................... 42 Fig. 3.11. Vulcanized natural rubber under tension; (a) before deformation, (b) crystallization and lamellae generation (rectangular boxes) during tension (Tosaka et al., 2004). ..................................................................................... 42 Fig. 3.12. Nylon6 specimen subjected to dual cantilever bending DMA test using a TA Q800 DMA tester. ................................................................................... 43 Fig. 3.13. DMA results to identify the glass transition temperature for the Nylon6 studied. ........................................................................................................... 44 Fig. 3.14. Experimental compressive response of Nylon6 at various temperatures ( = ) .................................................................................................. 45 Fig. 3.15. Experimental tensile response of Nylon6 at various temperatures ( = ) ..................................................................................................... 45 Fig. 3.16. Schematic arrangement of SHB bars and specimen. ................................... 47 Fig. 3.17. High speed photographic images of dynamic compressive deformation of Nylon 6 specimen; (a) before deformation; (b) compressive engineering strain at after start of loading at a strain rate of . ............... 48 Fig. 3.18. Response of Nylon6 under dynamic compression....................................... 48 Fig. 3.19. Strain rates imposed by SHPB. .................................................................... 49 Fig. 3.20. Response of two specimens with different lengths ( mm and mm) under dynamic compression. ................................................................................... 50 Fig. 3.21. Response of two specimens with different lengths ( and ) under dynamic compression. ................................................................................... 50 Fig. 3.22. Comparison of forces on input bar interface and output bar interface for a mm thick specimen, during dynamic compression at a strain rate of . .................................................................................................................... 51 Fig. 3.23. Comparison of forces on input bar interface and output bar interface for a 4 mm thick specimen, during dynamic compression at a strain rate of . .................................................................................................................... 51 X Fig. 3.24. Infrared images showing temperature increase for high rate deformation: (a) before compression; (b) 5 K increase after engineering strain at . ....................................................................................................... 52 Fig. 3.25. (a) Nylon 6 specimens dynamic tension. (b) Specimen connection to input/output bars. High speed photographic images of dynamic deformation: (c) before loading; (d) tensile engineering strain at after commencement of loading at a strain rate of . .................................... 53 Fig. 3.26. Response of material under dynamic tension at high strain rates. ............... 53 Fig. 3.27. Constant strain rates imposed by the SHTB. ............................................... 54 Fig. 3.28. Infrared images showing temperature increase for high rate tension: (a) before loading; (b) 1.2 K increase after engineering strain at . .. 54 Fig. 4.1. Schematic diagram of proposed elastic-viscoelastic-viscoplastic model; p, e, v, ve denote the plastic, elastic, viscous and viscoelastic components. ......... 58 Fig. 4.2. Schematic diagram of a one-dimensional elastic-viscoelastic-viscoplastic model. ............................................................................................................ 59 Fig. 4.3. Response of various elements in the one-dimensional model for different strain rates. ..................................................................................................... 60 Fig. 4.4. Schematic representation of the right and left polar decompositions of (Wikipedia, Sep 2012). .................................................................................. 61 Fig. 5.1. Compressive and tensile responses of Nylon 6 at low and high strain rates. 80 Fig. 5.2. Identification of yield stress marking onset of plastic deformation, for compression and tension. ............................................................................... 81 Fig. 5.3. Comparison between test data and fit of proposed model for compression. . 82 Fig. 5.4. Comparison between test data and fit of proposed model for tension. .......... 83 Fig. 5.5. Comparison between experimental tension and compression data with fit of proposed model for a strain rate of and different temperatures. ... 84 Fig. 5.6. Comparison between tension and compression test data for different strain rates with proposed model. ............................................................................ 86 Fig. 5.7. Proposed values for to describe compressive loading at (a) constant temperature; =298 K; (b) different temperatures at low strain rates; (c) different temperatures at high strain rates...................................................... 87 Fig. 5.8. Comparison between temperature variations predicted by the single-element model and experimental data. ........................................................................ 88 Fig. 5.9. Comparison between temperature profile predicted by ABAQUS (multielement model) and thermo-graphic infrared camera images, (a) temperature increase at compressive engineering strain for a strain rate of , (b) temperature increase after tensile engineering strain for a strain rate of ........................................................................................................ 89 Fig. 5.10. (a) Nylon 6 specimen with a complex geometry; (b) ABAQUS model; (c) Quasi-static compression of specimen; (d) and (e): comparison between specimen geometry and model at compressive engineering strain. ...... 92 XI Fig. 5.11. Temperature profile corresponding to mm compressive deformation at a strain rate of (Fig. 5.12) (a) ABAQUS model; (b) infrared image. ....................................................................................................................... 92 Fig. 5.12. Comparison between experimental force-displacement data and FEM model for compression of thick walled tube with a transverse hole. ........................ 93 Fig. 5.13. (a) Nylon 6 strips with cut-outs; (b) ABAQUS model; (c) and (d): comparison between specimen geometry and model at tensile engineering strain. .......................................................................................... 94 Fig. 5.14. Comparison between experimental force-displacement data and FEM model for the tensile loading. ........................................................................ 95 Fig. A.1. Schematic arrangement of SHB bars and specimen. .................................. 106 Fig. A.2. Schematic diagram of a SHPB set up. ........................................................ 108 Fig. A.3. Typical strain gauge signals captured by the oscilloscope. ........................ 109 Fig. A.4. Pulse shaper used in SHPB device. ............................................................ 109 Fig. A.5. Schematic diagram of a Tensile Split Hopkinson device. .......................... 110 Fig. A.6. Pulse shaper used in SHTB device. ............................................................ 110 XII List of Symbols Bold Symbols denote tensorial variables , Position vectors in the reference and deformed configurations , Deformation gradient tensor and its determinant , , Right and Left stretch tensors, Rigid rotation tensor , Right and Left Cauchy-Green deformation tensors Principal invariants of Cauchy-Green deformation ( ) , Second and Fourth order identity tensors , , Second Piola-Kirchhoff and Cauchy stresses, Driving force tensor , Velocity gradient, Stretch and Spin rate tensors , , Magnitude and Direction of stretch rate tensor , Helmholtz free energy densities per unit reference volume , Internal energy and Entropy density per unit reference volume , , , , , ρ, Material constants Material constants , Density, Thermal expansion coefficient, Specific heat capacity , , , Elastic, Bulk and Shear moduli, Poisson's ratio , Hardening functions , Almansi plastic strain and Green-Lagrange strain tensors Inelastic work fraction generating the heat , Fluidity and Viscosity terms , , One-dimensional Stress and Strain, Stress in friction slider , , , Temperature, Temperature-dependent scalar functions Equivalent strain, Stretch term Undetermined pressure and Frame-independent matrix function Deformation-dependent relaxation time function Time variables Super and subscripts Elastic, Viscous, Viscoelastic, Plastic, Yield XIII Deviatoric, Volumetric , , , , , 0, o Tension, Compression, Instantaneous Tensor transposition, Total, Symmetry, Initial Framework elements XIV CHAPTER 1 1.Introduction 1.1 Polymers; Definition and Morphology The word "polymer" is derived from the ancient Greek word polumeres, which means 'consisting of many parts'. Polymeric materials result from the linking of many monomers and molecular chains, which are covalently bonded, by polymerization processes. Polymerization can occur naturally or synthetically and produce respectively natural polymers, such as wool and amber, and synthetic polymers such as synthetic rubber, nylon and PVC (poly vinyl chloride). Polymer morphology describes the arrangement of molecular chains and can be defined according to three categories – cross-linked, straight and branched molecular chains (Fig. 1.1). (a) (b) (c) Fig. 1.1. Molecular chain types: (a) straight, (b) branched, (c) cross-linked. 1 Cross-linked polymers consist of molecular chains linked together by bonds; these bonds can pull the chain configuration back to its original shape after unloading from a large amount of stretching. Elastomers such as rubbers are well-known cross-linked polymers which are widely used as elastic materials in industry (Fig. 1.2). Branched chain polymers are characterized by branches along the polymer chains, which prevent them from being stacked and packed in a regular sequence. The amount and type of branching affect polymer properties, such as density and viscosity. Straight chain polymers consist of long molecular chains which are arranged in two different forms – crystalline and amorphous. The crystalline phase is defined by threedimensional regions associated with the ordered folding and/or stacking of adjacent chains. Crystallinity makes a polymer rigid, but reduces ductility (Dusunceli and Colak, 2008). Conversely, the amorphous phase is made up of randomly coiled and entangled chains; these regions are softer and more deformable (Messler, 2011). When an amorphous region is heated, the molecular chains start to wiggle and vibrate rapidly at a particular temperature; this temperature is called the glass temperature , and is a property only of the amorphous region. Below , the amorphous region is in a glassy state – chains are frozen and hardly move. Above the glass transition temperature, the material becomes more rubbery and flexible, with a lower stiffness, and the chains are able to wiggle easily. The glass transition is a second order phase transition (continuous phase transition), whereby the thermodynamic and dynamic properties of the amorphous region, such as the energy, volume and viscosity, change continuously (Blundell and Blundell, 2010). Fig. 1.2. Mechanical components made of elastomers 2 The largest group of commercially used polymers are semi-crystalline. A semicrystalline polymer can be viewed as a composite with rigid crystallites suspended within an amorphous phase (Fig. 1.3); this is desirable, because it combines the strength of the crystalline phase with the flexibility of its amorphous counterpart. Crystallinity in polymers is characterized by its degree and expressed in terms of a weight or volume fraction, which ranges typically between 10% to 80% (Ehrenstein and Immergut, 2001). Amorphous phase Crystalline phase Fig. 1.3. Schematic representation of molecular chains in a semi-crystalline polymer Nylon is a generic designation for semi-crystalline polymers known as polyamides, initially produced by Wallace Carothers at DuPont's research facility at the DuPont Experimental Station (February 28, 1935). It was first used commercially in toothbrushes (1938), followed by women's stockings (1940). Nylon was the first commercially successful synthetic polymer (IdeaFinder, Sep-2010), and widespread applications of nylon are in carpet fibre, apparel, airbags, tires, ropes, conveyor belts, and hoses. Engineering-grade nylon is processed by extrusion, casting, and injection molding. Type 6.6 Nylon is the most common commercial grade of nylon, and Nylon 6 is the most common commercial grade of molded nylon (Table 1.1). Solid nylon is used for mechanical parts such as machine screws, gears and other low to mediumstress components previously cast in metal (Fig. 1.4). Table 1.1 Nylon 6 properties. Density 1145 kg/m3 Amorphous density at 1070 kg/m3 Crystalline density at 1240 kg/m3 Glass transition temperature Melting temperature 3 Fig. 1.4. Mechanical components made of Nylon. 1.2 Objective and Scope Polymeric materials are widely used because of many favourable characteristics, such as ease of forming, durability, recyclability, and relatively lower cost and weight. Polymers are able to accommodate large compressive and tensile deformation and possess damping characteristics, making them suitable for employment in the dissipation of kinetic energy associated with impacts and shocks. The dynamic mechanical properties of polymers are of considerable interest and attention, because many products and components are subjected to impacts and shocks and need to accommodate these and the energies involved. Effective application of polymers requires a good understanding of their thermo-mechanical response over a wide range of deformation, loading rates and temperature. Therefore, analysis and modelling of the quasi-static and dynamic behaviour of polymers are essential, and will facilitate the use of computer simulation for designing products that incorporate polymeric padding and components. The present research effort undertaken is described according to the following segments: Chapter 2 focuses on common synthetic elastomers. In general, elastomers are nonlinear elastic material which exhibit rate sensitivity when subjected to dynamic loading (Chen et al., 2002; Tsai and Huang, 2006). Therefore, a visco-hyperelastic constitutive equation in integral form is proposed to describe the high rate, large deformation response of these approximately incompressible materials. This equation comprises two components: the first corresponds to hyperelasticity based on a strain energy function expressed as a polynomial, to characterise the quasi-static nonlinear response, while the second is an integral form of the first, based on the concept of 4 fading-memory, and incorporates a relaxation-time function to capture rate sensitivity and strain history dependence. Instead of a constant relaxation-time, a deformationdependent function is proposed for the relaxation-time. The proposed threedimensional constitutive model is implemented in MATLAB to predict the uniaxial response of six types of elastomer with different hardnesses, namely U50, U70 polyurethane rubber, SHA40, SHA60 and SHA80 rubber, as well as EthylenePropylene-Diene-Monomer (EPDM) rubber, which has been subjected to quasi-static and dynamic tension and compression by other researchers, using universal testing machines and Split Hopkinson Bar devices. In Chapter 3, the mechanical response of semi-crystalline polymer, Nylon 6, is studied. Material samples are subjected to quasi-static compression and tension at room and higher temperatures using an Instron universal testing machine. High strain rate compressive and tensile deformation are also applied to material specimens using Split Hopkinson Bar devices. Specimen deformation and temperature change during high rate deformation are captured using a high speed and an infrared camera respectively. The experimental results obtained are used to calibrate and validate the proposed constitutive description developed in Chapter 4. Chapter 4 presents the establishment of a thermo-mechanical constitutive model that employs the hyperelastic model proposed in Chapter 2. The model is formulated from a thermodynamics basis using a macromechanics approach, and defines elasticviscoelastic-viscoplastic behaviour, coupled with post-yield hardening. It is able to predict the response of the incompressible semi-crystalline material, Nylon 6, subjected to compressive and tensile quasi-static and high rate deformation. Chapter 5 describes the implementation of the three-dimensional constitutive model in an FEM software (ABAQUS) via a user-defined material subroutine (VUMAT). The model is calibrated and validated by compressive and tensile tests conducted at different temperatures and deformation rates (Chapter 3). Material parameters, such as the stiffness coefficients, viscosity and hardening, are cast as functions of temperature, as well as degree and rate of deformation. Simplicity of these material parameters is sought to preclude involvement in the details of the molecular structures of polymers. 5 CHAPTER 2 2. Modelling of Elastomers The molecular structure of elastomers can be visualized as a 'spaghetti and meatball' structure (Fig. 2.1), with the 'meatballs' denoting cross-links. Fig. 2.1. "Spaghetti and meatball" schematic representation of molecular structure of elastomers The stress-strain responses of elastomers exhibit nonlinear rate-dependent elastic behaviour associated with negligible residual strain after unloading from a large deformation (Bergstrom and Boyce, 1998; Yang et al., 2000) (Fig. 2.2 and Fig. 2.3). The observed rate-dependence corresponds to the readjustment of molecular chains, whereby the applied load is accommodated through various relaxation processes (e.g. rearrangement, reorientation, uncoiling, etc., of chains). When high rate deformation 6 is applied, the material does not have sufficient time for all relaxation processes to be completed, and the material response is thus affected by incomplete rearrangement of chains. (a) (b) Fig. 2.2. Experimental stress-strain curves (Shim et al., 2004) for SHA40, SHA60 and SHA80 rubber: (a) tension; (b) compression. 7 Fig. 2.3. Experimental data for uniaxial compressive loading to various final strains at a strain rate of -0.01/s, followed by unloading (Bergstrom and Boyce, 1998). Generally, an elastomer is viscoelastic, with a low elastic modulus and high yield strain compared with other polymers. In this Chapter, a hyperelastic constitutive model is proposed and the relationship between the strain experienced and the relaxation time is investigated. A strain-dependent relaxation time perspective is also defined for a visco-hyperelastic constitutive equation, to describe the large compressive and tensile deformation response of six types of incompressible elastomeric material with different stiffnesses at high strain rates. 2.1 Literature Review 2.1.1 Hyperelastic Material A Cauchy-elastic material is one in which the Cauchy stress at each material point is determined in the current state of deformation, and a hyperelastic material is a special case of a Cauchy-elastic material. For many materials, linear elastic models do 8 not fully describe the observed material behaviour accurately, and hyperelasticity provides a better description and model for the stress-strain response of such materials. Elastomers, biological tissues, rubbers, foams and polymers are often modelled using hyperelastic models (Hoo Fatt and Ouyang, 2007; Johnson et al., 1996; Shim et al., 2004; Song et al., 2004; Yang and Shim, 2004). A hyperelastic model defines the stress-strain relationship using a scalar function – a strain energy density function. Some common strain energy functions are (Arruda and Boyce, 1993; Ogden, 1984): o Neo-Hookean o Mooney-Rivlin o Ogden o Polynomial o Arruda-Boyce model 2.1.2 Viscoelastic Material Viscoelasticity is the property of a material that exhibits both viscous and elastic characteristics. Viscoelastic materials render the load-deformation relationship timedependent. Synthetic polymers, wood and human tissue, as well as metals at high temperatures, display significant viscoelastic behaviour. Some phenomena associated with viscoelastic materials are: o If the stress is held constant, the strain increases with time (creep). o If the strain is held constant, the stress decreases with time (relaxation). o The instantaneous stiffness depends on the rate of load application. o If cyclic loading is applied, hysteresis (a phase lag) occurs, leading to dissipation of mechanical energy. Specifically, viscoelasticity in polymeric materials corresponds to molecular chain rearrangements. When a constant stress is applied, parts of long polymer chains change position. This movement or rearrangement is called “creep”. When a constant deformation is applied, polymer chains are stretched to accommodate the deformation. This movement causes relaxation. Stress relaxation and the relaxation time describe how polymers relieve stress under a constant strain and how long the process takes. Polymers remain solid even when parts of their chains are rearranging to accommodate stress. Material creep or relaxation is described by the prefix "visco", 9 while full material recovery is associated with the suffix elasticity (McCrum et al., 1997). In the nineteenth century, physicists such as Maxwell, Boltzmann, and Kelvin studied and experimented with the creep and recovery of glasses, metals, and rubbers. Viscoelasticity was further examined in the late twentieth century, when synthetic polymers were engineered and used in a variety of applications. Viscoelasticity calculations depend heavily on the viscosity variable 2.1.3 (Meyers and Chawla, 1999). Elastic and Viscous Components of a Linear Viscoelastic Model A viscoelastic model has elastic and viscous characteristics, which are modelled respectively by linear combinations of springs and dashpots. Linear viscoelastic models differ in the arrangement of these elements. The elastic components are modelled as springs with a constant elastic modulus, (2.1) where is the stress, the elastic modulus of the material and the strain that accompanies a given stress, similar to Hooke's Law. Viscous response is modelled by dashpots, such that the stress-strain rate relationship is defined by, (2.2) where is the stress, the viscosity of the material and the time derivative of strain, or strain rate. The three most well-known linear viscoelastic models are the Maxwell, Kelvin-Voigt and Generalized Maxwell models (Tschoegl, 1989). 2.1.4 Maxwell Model The Maxwell model can be represented by a single viscous damper and a single elastic spring connected in series, as shown in Fig. 2.4 . This model is defined by the following relationship: 10 (2.3) Fig. 2.4. Maxwell model (Tschoegl, 1989). When a material is subjected to a constant stress, the strain has two components; an instantaneous elastic component corresponding to the spring, which relaxes immediately upon release of the stress. The second is a viscous component that grows with time as long as the stress is applied. The Maxwell model predicts that stress decays exponentially with time, which is appropriate for most polymers. One limitation of this model is that it does not predict creep accurately. The Maxwell model for creep or constant-stress conditions postulates that strain will increase linearly indefinitely with time. However, polymers show a strain rate that decreases with time (McCrum et al., 1997). 2.1.5 Kelvin-Voigt Model The Kelvin-Voigt model, also known as the Voigt model, consists of a Newtonian damper and a Hookean elastic spring connected in parallel, as shown in Fig. 2.5. It is used to explain creep in polymers, and the stress is described by. (2.4) Fig. 2.5. Kelvin–Voigt model (Tschoegl, 1989). Upon application of a constant stress, the material deforms at a decreasing rate, asymptotically approaching a steady-state strain. When the stress is released, the material gradually relaxes to its undeformed state. For constant stress (creep), the 11 model is quite realistic, as it predicts that the strain tends to with time. As with the Maxwell model, the Kelvin-Voigt model also has limitations. This model is extremely good for modelling creep in materials, but with regard to relaxation, the model is much less accurate. 2.1.6 Generalized Maxwell Model This model takes into account the fact that relaxation does not occur over a single characteristic time scale, but over several characteristic time periods. As molecular segments possess different lengths, with shorter ones contributing less to the deformation than longer ones, different relaxation times are proposed. This model describes such behaviour by having as many Maxwell spring-dashpot elements as are necessary to accurately represent the response. Fig. 2.6 is a schematic diagram of such a model. Fig. 2.6. Generalized Maxwell Model (Tschoegl, 1989). The relationship between stress and strain can be described specifically for particular strain rates. For high rates and short time periods, the time derivative components of the stress-strain relationship dominate. Dashpots resist changes in length, and they can be considered approximately rigid. Conversely, for low rates and longer time periods, the time derivative components are negligible and the dashpots can essentially be ignored. As a result, only the single spring connected in parallel to the dashpots accounts for the total strain in the system (Tschoegl, 1989). 12 2.1.7 Relaxation Model Stress relaxation describes how materials relieve stress under a constant strain and each relaxation process can be characterized by a relaxation time . The simplest theoretical description of relaxation as function of time is an exponential function. (2.5) where is the initial stress and is the time required for an exponential variable to of its initial value (Fig. 2.7). decrease to Fig. 2.7. Stress relaxation under constant strain The following non-material parameters affect stress relaxation in polymers (Junisbekov et al., 2003a) o Magnitude of initial loading o Speed of loading o Temperature (isothermal or non-isothermal conditions) o Loading medium o Friction and wear o Long-term storage In complex models, such as the Generalized Maxwell Model, a relaxation time spectrum is defined, since it is recognized that for a model to adequately predict the behaviour of viscoelastic materials, it cannot comprise a single relaxation time. Basically, stress relaxation is associated with either physical or chemical phenomena, as shown respectively in Fig. 2.8 and Fig. 2.9 (Sperling, 2006). 13 Fig. 2.8. Physical relaxation, reversible motion in a trapped entanglement under stress. The point of reference marked moves with time. When the stress is released, entropic forces return the chains to near their original positions (Sperling, 2006). INTERCHANGE ALONG DOTTED LINE Fig. 2.9. Chemical relaxation; chain portions change partners, causing a release of stress. (Bond interchange in polyesters and polysiloxanes) (Sperling, 2006). 2.1.8 Visco-hyperelastic Constitutive Model A visco-hyperelastic constitutive model is frequently employed to describe hyperelastic materials which exhibit time-dependent stress-strain behaviour. They are by definition, materials with fading memory; i.e. the stress at a material point is a function of recent deformation gradients which occurred within a very small neighbourhood of that point. Therefore, most of the early constitutive laws for visco14 hyperelastic materials were developed in the context of the theory of fading memory. Historically, the theory of nonlinear viscoelasticity was formulated about 50 years ago. The first constitutive models of simple materials with fading memory were proposed by Green and Rivlin, Coleman and Noll, and then by Pipkin and Wang (Coleman and Noll, 1961; Green and Rivlin, 1957; Pipkin, 1964; Wang, 1965). Later, books by Lockett, Findley and Carreau discussed constitutive theories and some of their applications, and these have received much attention by contemporary researchers (Carreau et al., 1997; Findley et al., 1976; Lockett, 1972). Recently, two comprehensive reviews on nonlinear viscoelastic behaviour have been presented by Wineman and Drapaca (Drapaca et al., 2007; Wineman, 2009). Wineman discussed several specific forms of constitutive equations proposed in literature, and attention was focused on constitutive equations that are phenomenological rather than molecular in origin. Drapaca also reviewed classical nonlinear viscoelastic models and provided a unifying framework using continuum mechanics. An overview of constitutive equations and models for the fracture and strength of nonlinear viscoelastic solids has been presented by Schapery (Schapery, 2000). Up to the present, there is no generally-accepted well-defined theory for nonlinear viscoelastic solids, as there are for linear viscoelastic materials. Several constitutive equations have been proposed for nonlinear viscoelastic solids. Pipkin and Rogers (Pipkin and Rogers, 1968) developed a constitutive theory for nonlinear viscoelastic solids based on a set of assumptions about their response to step strain histories. Fung (Fung, 1981) proposed Quasi-Linear Viscoelasticity; this constitutive description is used to represent the mechanical response of a variety of biological tissues. There is also the K-BKZ model, which is a nonlinear single integral constitutive equation in which the integrand is expressed in terms of finite strain tensors; this was proposed by Bernstein, Kearsley and Zapas (Bernstein et al., 1963). By considering the role of elastomer chains, Bergstorm (Bergstrom and Boyce, 1998) suggested a new micromechanism-inspired constitutive model based on the assumption that the behaviour can be captured by two networks acting in parallel — one describing the equilibrium state or quasi-static response of the material, and the second associated with rate-dependent response. The idea of decomposing the total stress into an elastic and a history-dependent component was proposed by Green and Tobolsky (Green and Tobolsky, 1946); their approach of modelling elastomers as two interacting networks 15 was used later by Johnson and co-workers (Johnson et al., 1995). A more recent description of viscoelastic materials was proposed by Yang (Yang et al., 2000). Their three-dimensional model comprises two parts, a hyperelastic and a viscoelastic component; this constitutive model is based on BKZ-type hereditary laws for high strain rate response. This has been followed by others studying elastomers (Hoo Fatt and Ouyang, 2007; Li and Lua, 2009; Shim et al., 2004; Yang and Shim, 2004). In essence, the present study aims to propose a three-dimensional constitutive description that is able to define the quasi-static and high rate, large deformation compressive and tensile mechanical responses of approximately incompressible elastomeric material. The model is based on a macromechanics–level approach, coupled with the novel proposition of a strain-dependent relaxation time. An accompanying objective is to formulate a model with a minimum number of material parameters and avoid the complexity of delving into the details of polymer structures. These prompt the development of a visco-hyperelastic description in integral form, based on the concept of fading-memory and comprising two components: the first corresponds to hyperelasticity based on a strain energy density function, expressed as a polynomial, to characterise the quasi-static nonlinear response; the second is an integral form of the first and incorporates a relaxation time function to capture rate sensitivity and deformation history dependence. Instead of employing a constant relaxation time, a novel approach of incorporating a deformation-dependent relaxation time function is adopted. 2.2 Proposed Visco-hyperelastic Model The behaviour of an elastomer is considered to be amenable to idealization by two parallel elements, A and B, representing mechanical responses, as depicted schematically in Fig. 2.10. B: rate-dependent A: rate-independent Fig. 2.10. Parallel mechanical elements A and B. 16 Element A defines the rate-independent quasi-static response and is modelled by a nonlinear spring that corresponds to incompressible hyperelasticity. Element B is associated with rate-dependent response, whereby molecular chains in the polymer encounter resistance to sudden stretching through entanglements with other long chains, constraints on chain mobility by cross-links, and interactions between molecular chains. Depending on the constraints imposed on the chains, they rearrange to a more relaxed configuration after being deformed, and this takes a specific period of time characterized by a relaxation time . Element B takes the idealized form of a nonlinear spring connected to a nonlinear dashpot. The total stress is therefore, 2.2.1 Derivation of Second Piola-Kirchhoff and Cauchy Stresses – A Hyperelastic Model for Element A Consider a generic particle in a body, identified by its position vector X in a reference configuration, and by x in the deformed configuration. The deformation gradient, its determinant, right and left Cauchy-Green deformation tensors are respectively, , , The Second Piola-Kirchhoff stress tensor and . for such a hyperelastic material is given by: (2.6) where is the Helmholtz free energy density and , and are the principal invariants of the right and left Cauchy-Green deformation tensors. (2.7) where is the identity tensor. 17 A combination of Eqs. (2.6)–(2.7) yields the following expressions for the Second Piola-Kirchhoff stress and the Cauchy stress tensor . (2.8) The Helmholtz free energy density is assumed to consist of the deviatoric and volumetric terms of the stored energy. The isochoric term is defined by and and ; , which are the principal . The volumetric term is defined by , invariants of , which represents the volume change. Therefore, the strain energy is given by: (2.9) The Cauchy stress tensor which comprises deviatoric and volumetric components is explained as follows: (Anand and Ames, 2006; Simo et al., 1985). (2.10) , = , 18 Substitution of Eq. (2.10) into Eq. (2.8) defines the Cauchy stress as follows: (2.11) Following the analysis of Yang et al. (Yang et al., 2000) and Pouriayevali et al. (Pouriayevali et al., 2012), an isotropic incompressible hyperelastic material is described by , and , because . The Cauchy stress for element A is defined using Eq. (2.8) by the following: (2.12) where because is an undetermined pressure related to incompressibility, defines the bulk modulus (infinite value) for a fully incompressible material and for volume-conserving deformation. , and is a Helmholtz strain energy potential for element A and defined using a polynomial function of and as follows; (2.13) where, functions as a Lagrange multiplier to enforce the material incompressibility constraint. shear modulus are material constants and correspond to the isotropic as follows: (2.14) 19 2.2.2 Derivation of Cauchy Stresses for Element B With respect to a homogeneous, isotropic and incompressible viscoelastic material, its constitutive relationship can be expressed in the following form (Coleman and Noll, 1961; Wineman, 2009): (2.15) where is the Cauchy stress tensor, an undetermined pressure associated with incompressibility of the material, and is a frame-independent matrix function. Eq. (2.15) is used to describe element B and captures the fading-memory effect of recent deformation on the current stress state by the following: (2.16) where is the instantaneous stress in element B. Elastomeric materials are generally rate-dependent and become stiffer with strain rate; beyond a certain strain rate, they tend towards a limiting stress-strain behaviour because of insufficient time for relaxation to occur (Hoo Fatt and Ouyang, 2008); i.e. in Fig. 2.10, the dashpot essentially locks up. , is a function that describes the relaxation of the material and includes the relaxation time (Junisbekov et al., 2003b). In this study, the nonlinear stiffnesses of the springs in elements A and B are defined by and respectively (Fig. 2.10), and they are both functions of strain. Let the relationship between these two stiffnesses be defined by = , where is a strain-dependent function; therefore, the response of element B to an instantaneously applied load is (2.17) A combination of Eqs. (2.12) and (2.15)–(2.17) yields 20 (2.18) Eqs. (2.12) and (2.15) comprise two parts: and represent undetermined pressures corresponding to incompressibility of the material, and and define the stresses associated with the deformation gradient tensor that the undetermined pressure effect of , and . It is assumed in Eq. (2.18) defines only the fading-memory in Eq. (2.18) describes the fading-memory effect of recent stress states corresponding to . Therefore, is defined by the following: (2.19) A combination of Eqs. (2.15) and (2.19) yields, (2.20) Consequently, Eq. (2.20) defines a frame-independent constitutive equation in integral form to describe only the rate-dependent response of an incompressible isotropic elastomeric material. 2.2.3 Combination of Cauchy Stresses of Elements A and B A three-dimensional visco-hyperelastic constitutive equation is now established from a combination of two functions. One corresponds to the quasi-static response of the material defined by Eq. (2.12) and is related to element A. The second is linked to the rate-dependent response of the material; this behaviour is associated with element 21 B and modelled by Eq. (2.20), which is based on the quasi-static response coupled with fading-memory characteristics. Consequently, the visco-hyperelastic constitutive description is: (2.21) 2.2.4 Relaxation Process in Elastomers Tosaka et al. (Tosaka et al., 2004) modelled the deformation of molecular chains in an elastomer (natural rubber), based on wide-angle X-ray diffraction (WAXD) measurements. Their model is depicted schematically in Fig. 2.11. Spheres and thick lines represent globules of randomly coiled, oriented network chains, and the connections between them (Fig. 2.11.a). Long chains and convolutions can extend through more than one globule. When the sample is stretched to a small strain, the globules rearrange to accommodate macroscopic elongation. It was concluded that elongation of a small fraction of relatively short network chains may be sufficient to accommodate this deformation, and many globules can still remain in their original undeformed state, i.e. deformation is more localized (Fig. 2.11.b). When larger strains are applied, more molecular chains and globules, including long convolutions, are involved and stretched; i.e. there is a transition from localized to widespread deformation. 22 Fig. 2.11. Schematic representation of deformation of elastomer molecular chains. Spheres indicate globules of network chains (a molecular chain is illustrated in the globule at the top). Thick lines represent connections between network chains. (a) Before deformation, all globules are spherical; (b) after deformation, only some of the globules are elongated – the darker ones (Tosaka et al., 2004). Stress relaxation in elastomers is linked to various forms of molecular chain readjustments associated with different relaxation times. To illustrate these readjustments, Bergstrom and Boyce considered one loop chain within a network of chains, as shown in Fig. 2.12 (Bergstrom and Boyce, 1998). The network is deformed at a high rate, causing the polymer loop chain to be strained to conform to the network. Orientation of the chains in the direction of applied stress results in reduction of entropy of the network and increased resistance to deformation. If the deformation is maintained, the loop chain will gradually return to a more relaxed configuration associated with a lower stiffness. Fig. 2.12. Readjustment and relaxation of a free polymer chain loop located in a network (Bergstrom and Boyce, 1998). 23 An elastomer comprises a wide range of molecular network lengths subjected to various strains, and the molecular chains in the network are relaxed through various rearrangements. Within localized regions in the material, relaxation involves relatively rapid readjustments of short-length chains, while rearrangements of convolutions associated with long chains and excessive entanglements, require larger relaxation times (Hoo Fatt and Ouyang, 2007). In addition, strain-induced crystallites, as well as folded and stacked adjacent chains, appear in large strain deformation and generate additional resistance during the deformation and relaxation processes; thus, a larger relaxation time is expected (Van der Horst et al., 2006). Based on this, the present study on high rate deformation envisages that smaller applied strains are associated primarily with deformation of smaller localized regions and shorter relaxation times, while larger strains involve a greater proportion of chains and larger convolutions, and thus longer relaxation times. The use of several relaxation times to characterise elastomers has been proposed previously by other researchers (Hoo Fatt and Ouyang, 2007; Tosaka et al., 2006). They employed different relaxation times to describe low and high rate responses for different ranges of strain magnitudes. A larger number of relaxation times results in a better fit with experimental results (Hoo Fatt and Ouyang, 2007). In this study, instead of using a combination of different relaxation times, a single relaxation time function , in integral form, and which increases with strain, is proposed and defined by the following, (2.22) where is an equivalent strain and is the Green- Lagrange strain tensor. A combination of Eq. (2.21) and Eq. (2.22) yields (2.23) 24 2.2.5 Hyperelastic Model Used for Uniaxial Loading of Element A The Cauchy stress function defined for element A (Eq. (2.12)) is simplified for uniaxial loading along the 1-axis, where the stretch ; is the engineering strain in the loading direction. The principal stretches are thus , and and are defined by: (2.24) The stress corresponding to uniaxial loading is: (2.25) The hydrostatic pressure is determined from the condition that for uniaxial loading, , together with the fact that . (2.26) Consequently, the Cauchy stress-deformation relationship in the direction of loading is: (2.27) The Cauchy stress associated with the strain energy density function presented in Eq. (2.13) can be written in terms of the stretch and engineering strain , as follows: (2.28) 25 2.2.6 Visco-hyperelastic Model for Uniaxial Loading The Cauchy stress function defined in Eq. (2.23) is simplified for uniaxial loading along the 1-axis. In this study, the aim is to propose a constitutive equation that can be applied across a wide range of strains and strain rates, while keeping the number of material parameters to a minimum. Therefore, it is assumed that (Eq. (2.17)) can be defined by a constant, and that its value can be determined by fitting the proposed constitutive equation developed in this study (Eq.(2.33)) to experimental results. From Eqs. (2.12), (2.15) and (2.21), the stress for uniaxial loading along the 1-axis is given by (2.29) The hydrostatic pressure stresses is determined from the condition that the transverse . (2.30) Substitution of Eq. (2.26) ( Eq. (2.19) and noting that into Eq. (2.30) yields is constant, ; using is defined as follows: (2.31) Consequently, a combination of Eqs. (2.19), (2.25), (2.29)–(2.31) results in (2.32) 26 Eq. (2.32) describes the rate-dependent, stress-deformation response of an isotropic incompressible elastomer for uniaxial loading. Eq. (2.32) can be extended to the following equation by considering the engineering strain and strain rate in the direction of loading. (2.33) where 2.3 is defined by Eq. (2.28). High Strain Rate Experiments on Elastomers High strain rate experiments on elastomers (> 100/s), performed using SHB devices (see Appendix A) are quite challenging because of the relatively low mechanical impedance of rubber, whereby the stress propagated through such specimens into a metallic output bar is often too small to be captured, and modifications to SHB systems using metallic bars are necessary (Chen et al., 1999; Nie et al., 2009; Rao et al., 1997; Yang et al., 2000). In a study by Shim et al. (Shim et al., 2004), improvement in the output bar signal was attained by using polycarbonate bars instead of metallic ones, because of the smaller mechanical impedance difference between rubber and polycarbonate. They also investigated both the dynamic compressive and tensile responses of three types of rubber with different hardnesses; SHA40, SHA60, SHA80 rubber. These experimental results and the response of three types of elastomers – namely U50, U70 polyurethane rubber and Ethylene-PropyleneDiene-Monomer (EPDM) rubber, which have been reported in two previous works (Doman et al., 2006; Song and Chen, 2004a) – are used in this study to validate the proposed visco-hyperelastic constitutive model. 2.4 Application of Model and Discussion Figs 2.13–2.17 show quasi-static compressive and tensile uniaxial experimental data relating true stress to engineering strain at strain rates below ; these are from tests done by previous researchers on six types of rubber and it was assumed that at low strain rates, the material response is essentially rate-independent. Quasi-static 27 compression and tension data were used to fit Eq. (2.28) via a least squares approach. This was done by employing the curve-fitting routine available in MATLAB, which this yielded the constants (Eq. (2.13)). A good fit between the proposed hyperelastic model Eq. (2.28) and the experimental curves substantiates the validity of the model based on only three material parameters , and (Table 2.1). Dynamic compressive and tensile tests were undertaken by earlier researchers using Split Hopkinson Bar devices (Figs 2.13–2.17). Eq. (2.33), which incorporates was implemented in MATLAB (Appendix B), then the constants evaluated ( fitted to one set of dynamic compression test data each for U50, U70 and EPDM rubber (Figs 2.16–2.17) and one set of dynamic compression and one set of dynamic tension test data each for SHA40, SHA60 and SHA80 rubber (Figs 2.13–2.15) to determine and (Table 2.1). This was done by employing the approach of adopting the simplest function possible for the relaxation time by , and scaling it so that a good fit was obtained. Several iterations were required to achieve good correlation. This constitutive equation was then employed to predict one set of dynamic compressive responses for U50 and U70 rubber (Fig. 2.17), SHA40, SHA60 and SHA80 rubber (Figs 2.13–2.15), as well as two dynamic compressive responses, for EPDM rubber (Fig. 2.16). Figs 2.14–2.17 demonstrate favourable agreement between the proposed model and test data, indicating the potential of the model to describe the dynamic behaviour of elastomeric material over a range of strain rates. However, Fig. 2.13 indicates less than satisfactory correlation between the theoretical and experimental stress-strain responses for SHA40 rubber for a strain rate of . The reason for the difference between the model and the experimental results is unclear for this particular case, although the trend in behaviour with respect to strain rate is consistent. Finally, it is noted that the compressive and tensile behaviour of SHA40, SHA60 and SHA80 rubber can be fitted and modelled using a single relaxation time function with a common set of coefficients (Table 2.1). Fig. 2.18 shows (slightly) higher relaxation times for stiffer rubber for each category. Fig. 2.16 demonstrates the ability of the proposed model to predict the dynamic response of elastomers at strain rates higher and lower than those used for curve fitting; i.e. the model appears to do a reasonable job of interpolation, as well as extrapolation outside the range of strain rates used to derive the model parameter values. 28 Fig. 2.13. Comparison between experimental tension and compression data (Shim et al., 2004) with proposed visco-hyperelastic model for SHA40 rubber. 29 Fig. 2.14. Comparison between experimental tension and compression data (Shim et al., 2004) with proposed visco-hyperelastic model for SHA60 rubber. 30 Fig. 2.15. Comparison between experimental tension and compression data (Shim et al., 2004) with proposed visco-hyperelastic model for SHA80 rubber. 31 Fig. 2.16. Comparison between experimental compression data (Song and Chen, 2004a) and proposed visco-hyperelastic model for EPDM rubber. Table 2.1. Material coefficients and relaxation time functions for dynamic tension and compression of SHA40, SHA60, SHA80, U50, U70 and EPDM rubber. : Relaxation Time Material Loading (MPa) (MPa) (MPa) ( ) Equivalent strain SHA40 SHA60 SHA80 Tension and compression Tension and compression Tension and compression 0.1328 0.2853 0.01003 1.75 0. 14 (1+ 820 ) 0.164 0.5215 0.0304 1.8 0. 16 (1+ 870 ) -0.177 1.637 0.119 1.85 0. 2 (1+ 930 ) U50 compression 0.5149 -0.07167 0.00355 1.6 0.19 (1+ 1000 ) U70 compression 1.549 -0.384 0.02288 3 0.2 (1+ 1200 ) EPDM compression 192.68 -193.945 109.17 26 0.5(1+ 8 ) 32 U50 U70 Fig. 2.17. Comparison between experimental compression data (Doman et al., 2006) and proposed visco-hyperelastic model for U50 & U70 rubbers. 33 (a) (b) Fig. 2.18. Comparison between relaxation time data obtained by fitting to dynamic experimental response: (a) U50, U70 rubber under compression; (b) SHA40, SHA60 and SHA80 rubber under tension. 34 2.5 Summary and Conclusion A three-dimensional visco-hyperelastic constitutive equation that incorporates a deformation-dependent relaxation time has been proposed to describe the large deformation response of incompressible rubber under dynamic compression and tension. The model comprises two components: the first corresponds to hyperelasticity based on a strain energy density potential to characterize the quasistatic nonlinear response; the second is an integral of the first and incorporates a relaxation time function to capture the rate sensitivity and fading-memory characteristics of the material. The proposed model is applied to describe the high rate response of six types of rubber with different shore hardnesses and rate-sensitivity, and found to be able to capture their dynamic behaviour. It is noted that a three-term truncated polynomial series for the energy density function adequately characterises the hyperelastic character of the elastomers. The relaxation time function fitted indicates that relaxation time increases with equivalent strain. It is also noted that the compressive and tensile behaviour of elastomers can be defined by a single relaxation time function with a common set of material parameter values. The model has the potential to predict responses within and outside the range of strain rates used in deriving the material stiffness and viscosity parameters. 35 CHAPTER 3 3. Quasi-static and Dynamic Response of a Semi-crystalline Polymer Nylon 6 is selected as the semi-crystalline polymer to be studied (Table 1.1). Compression and tension samples of Nylon 6 are fabricated from a thin sheet and a long rod respectively, which are provided by a commercial supplier . Samples are subjected to quasi-static compression and tension at room and higher temperatures using an Instron universal testing machine. High strain rate compressive and tensile deformation are also applied to material specimens using Split Hopkinson Bar devices. The specimen deformation and temperature change during high rate loading are captured using a high speed and an infrared camera respectively. The experimental results obtained are used to calibrate and validate the proposed constitutive description of semi-crystalline polymer. 3.1 Quasi-static Experiments at Room Temperature 3.1.1 Quasi-static Compression Compression test specimens were fabricated in the form of cylinders with a diameter of mm and a length of mm. (These dimensions do not correspond to ASTM D 695-08 standards, which specify that the specimen height should be twice that of the diameter). Fig. 3.1 shows a specimen fabricated according to ASTM D 36 695-08 specifications, subjected to compression; it exhibits obvious lateral slippage. Consequently, specimens of a smaller aspect ratio were used to prevent lateral slippage of the specimen during compression. A small amount of grease was applied to the top and bottom surfaces to reduce friction with the loading platens. Quasi-static compression tests were carried out using an Instron universal testing machine at a constant crosshead speed of of mm/min; this corresponds to a nominal strain rate . Fig. 3.1. Compression test on a specimen fabricated according to ASTM D 695-08 standards; lateral slippage is obvious. Fig. 3.2 shows a sequence of three images of quasi-static compression of a specimen; the resulting stress-strain curve is presented in Fig. 3.3. Fig. 3.2 also shows two vertical lines drawn on the specimen surface and these remained parallel throughout deformation, confirming that deformation was indeed uniaxial. A strain gauge designed for use with polymers was attached to the mid-point of some specimens to measure the axial strain up to , enabling calculation of the compressive modulus (Fig. 3.1). The average compressive elastic modulus of this material, , was determined from the results of four tests, as shown in Fig. 3.4. Fig. 3.2. Nylon 6 specimens before and after quasi-static compression. 37 Fig. 3.3. Response of three specimens to quasi-static compression. Fig. 3.4. Quasi-static compressive linear elastic response of four specimens. 38 3.1.2 Quasi-static Tension Specimens for quasi-static tension tests were fabricated in accordance with ASTM D638-08 (Type V) standards (Fig. 3.5). W OW G L T D R OL W- Width of narrow section 3.18 mm OW- Overall width 9.53 mm L- Length of narrow section 9.53 mm OL- Overall Length 80 mm G- Gage Length 7.62 mm D- Distance between grips 25.4 mm R- Radius of fillet 12.7 mm T-Thickness 5 mm Fig. 3.5. ASTM D638-Type V specimen dimension. For tensile tests, a universal testing machine fitted with special grips to clamp specimens was used, and a strain rate of was imposed (Fig. 3.6). A strain gauge was attached to specimens to measure tensile strains up to 1%, from which the average elastic modulus was calculated to be (Fig. 3.7). 39 Fig. 3.6. Specimen before and after quasi-static tension. Fig. 3.7. Quasi-static linear tensile elastic response of three specimens. An optical technique was used to measure the extension of specimens. Three points within the gauge section of each specimen were marked by ink dots. Visual images of the specimen deformation were captured by a video camera and the images processed using TEMA software (TrackEye Motion Analysis 3.5). This software tracks the position of points marked on the surface of a specimen by analysing visual images. Each point is associated with several image pixels and TEMA identifies the points by recognizing the contrast of the pixels compared to that of surrounding points. Therefore, points marked should be as small as possible with a distinct colour relative 40 to that of the specimen surface (Fig. 3.8). Fig. 3.9 shows the response of three specimens under quasi-static tension, whereby the strains were obtained using the TEMA software. Time ms) Marked Points position of points during tracking Fig. 3.8. TEMA software (TrackEye Motion Analysis 3.5). Fig. 3.9. Response of three specimens to quasi-static tension. 41 Fig. 3.10 shows a comparison of the material response for quasi-static tension and compression. The results display an asymmetry in tension and compression; this is attributed to possible polymer crystallization — a process associated with partial alignment of polymer chains during tensile loading (Fig. 3.11). The rearrangement of parallel chains form an ordered region called lamellae, which makes a polymer stiffer under tension (Mandelkern, 2002). Fig. 3.10. Experimental quasi-static response of Nylon 6 showing asymmetry in tension and compression ( = ). Fig. 3.11. Vulcanized natural rubber under tension; (a) before deformation, (b) crystallization and lamellae generation (rectangular boxes) during tension (Tosaka et al., 2004). 42 3.2 Dynamic Mechanical Analysis (DMA) Testing Nylon 6 is a semi-crystalline polymer with a glass transition temperature varies within the range of that (Gordon, 1971). The glass transition temperature is reduced by the amount of plasticizer added, which comprises small molecules to create gaps between polymer chains for greater mobility and reduced inter-chain interactions, resulting in greater toughness for industrial usage. Humidified Nylon 6 exhibits a reduced , because humidity plasticizes the material (Sauer and Lim, 1977). Samples of Nylon6 are tested using Dynamic Mechanical Analysis, DMA, which is a technique to characterize viscoelastic material and allows identification of the glass transition temperature. Specimens of Nylon6 in the form of rectangular strips measuring were subjected to cyclic finite-strain dual-cantilever bending at a frequency of 1 Hz, and a range of temperatures that varied from to . Each sample was anchored at both ends using fixed clamps and displacement was applied to the mid-point by a shaft linked to a rotary motor (Fig. 3.12). DMA tests elicit the storage modulus, which represents the stiffness of a viscoelastic material and its value drops dramatically at . Fig. 3.13 shows the DMA results and the intersection of tangent lines which are used to identify for the Nylon6 studied. Fig. 3.12. Nylon6 specimen subjected to dual cantilever bending DMA test using a TA Q800 DMA tester. 43 Fig. 3.13. DMA results to identify the glass transition temperature for the Nylon6 studied. 3.3 Quasi-static Experiments at Higher Temperatures Nylon6 specimens ware subjected to quasi-static large-deformation compression and tension at temperatures that cross to observe the material response. Fig. 3.14 and Fig. 3.15 show more compliant responses with a temperature increase; these experimental results are employed in Section 5.2 for calibration of the proposed constitutive model for high strain rate deformation, whereby significant temperature increases are observed. 44 Fig. 3.14. Experimental compressive response of Nylon6 at various temperatures ( = ) Fig. 3.15. Experimental tensile response of Nylon6 at various temperatures ( = ) 45 3.4 High Strain Rate Tests High strain rate loading is imposed on material specimens using Split Hopkinson Bar devices (SHB) — a Split Hopkinson Pressure Bar (SHPB) and a Split Hopkinson Tension Bar (SHTB) respectively, for compressive and tensile loading. The specimen deformation and temperature change during high rate loading are also captured using a high speed optical camera and an infrared camera respectively. 3.4.1 Split Hopkinson Bar A Split Hopkinson Bar is an apparatus for eliciting the dynamic stress-strain response of materials at various strain rates, and a schematic diagram of the device is shown in Fig. 3.16. The device comprises two long bars, commonly known as the input (incident) and output (transmission) bar. A striker, propelled by pressurized gas, impacts the free end of the input bar and generates a stress wave which propagates at the speed of sound through the input bar into the specimen mounted between the input and output bars. A portion of the incident stress wave is transmitted into the specimen and output bar, while the remaining portion is reflected back into the input bar. The transmitted stress wave is able to load the specimen at an approximately constant strain rate that can range from to (Gray, 2000). Based on one-dimensional wave propagation theory, the stress and strain in the specimen can be determined from the incident and reflected strains transmitted strain and , induced in the input bar, and the , in the output bar; these are measured by the strain gauges mounted on the input and output bars (Fig. 3.16). The formulae used to calculate the specimen stress and strain, and details of the SHPB and SHTB arrangement are presented in Appendix A. Application of the SHB technique to testing compliant low-strength, low impedance materials encounters significant problems, and modifications must be made to overcome them. Only a small portion of the incident pulse is transmitted through the low impedance specimen into the output bar; most of the incident pulse is reflected back into the input bar, so the transmitted pulse has a very small amplitude. Moreover, nonhomogenous deformation of the specimen results a nonequilibrium stress state – i.e. the specimen deforms significantly near the input bar interface, while deformation remains small near the output bar end. Thus, the loading pulse has to propagate back and forth within the specimen over several cycles before a uniform stress state is attained. A more constant strain rate can be obtained by using a pulse 46 shaper, in the form of an additional thin material layer inserted at the interface between the striker and the input bar. This material deforms plastically upon impact, slowing the rise of the incident pulse in the input bar. This facilitates capture of information on the specimen response at desired strain rates. (Fig. A.4 and Fig. A.6) (Chen et al., 1999; Fábio Benassi, 2006; Yang and Shim, 2005). Striker Input (Incident) bar Strain gauge Specimen Output (Transmission) bar Strain gauge Fig. 3.16. Schematic arrangement of SHB bars and specimen. 3.4.2 Dynamic Compressive Loading A SHPB was used to obtain the response of the material under dynamic compression. Specimens were mounted between two steel bars of diameter. Fig. 3.17 shows two high-speed images of a specimen during dynamic compression. The first is of the initial state of the specimen; the second corresponds to the end of compression, 163 µs after commencement of loading. The strain rate imposed on a specimen by a SHPB depends on the velocity of the striker and the specimen thickness. Therefore, specimens of three different thicknesses were fabricated to obtain a wider range of strain rates, given the limited range of striker velocities that could be generated (Table 3.1). The contact surfaces of specimens were smoothened and made parallel; a thin layer of grease was used to facilitate mounting of the specimen between the input and output bars and to reduce friction during compression. Dynamic compression was applied to specimens over a range of strain rates from to . Fig. 3.18 shows the rate-dependent stress-strain response of the material in this strain rate range, and Fig. 3.19 shows the strain rates induced by the SHPB. 47 (a) (b) Fig. 3.17. High speed photographic images of dynamic compressive deformation of Nylon 6 specimen; (a) before deformation; (b) engineering strain at compressive after start of loading at a strain rate of . Table 3.1. Compression test specimens with three lengths. Lengths (mm) Diameter (mm) Specimen type 1 4 6.4 Specimen type 2 3 6.4 Specimen type 3 2 6.4 Fig. 3.18. Response of Nylon6 under dynamic compression. 48 Fig. 3.19. Strain rates imposed by SHPB. Fig. 3.20 and Fig. 3.21 depict the stress-strain responses of three specimens at two strain rates. It can be seen that a difference in specimen length does not have affect the stress-strain response at these strain rates. Fig. 3.22 and Fig. 3.23 correspond to Fig. 3.20 and show the forces at the output bar and input bar interfaces during specimen compression. It can be concluded that a reduction in the specimen length facilitates earlier achievement of dynamic force equilibrium at both sides of the specimen. Basically, a long specimen results in nonhomogenous deformation and delays attainment of dynamic force equilibrium. On the other hand, there are some limitations to the reduction of specimen length, such as deviation from a uniaxial stress state for very small slenderness ratios (Song and Chen, 2004b). 49 Fig. 3.20. Response of two specimens with different lengths ( mm and mm) under dynamic compression. Fig. 3.21. Response of two specimens with different lengths ( and ) under dynamic compression. 50 Fig. 3.22. Comparison of forces on input bar interface and output bar interface for a mm thick specimen, during dynamic compression at a strain rate of . Fig. 3.23. Comparison of forces on input bar interface and output bar interface for a 4 mm thick specimen, during dynamic compression at a strain rate of . 51 An infrared camera operating at a framing rate of was used to capture the temperature increase for high rate compressive deformation (Fig. 3.24). (a) (b) Fig. 3.24. Infrared images showing temperature increase for high rate deformation: (a) before compression; (b) 5 K increase after engineering strain at . 3.4.3 Dynamic Tensile Loading A SHTB was used to obtain the material response under dynamic tension. Specimens were mounted between diameter aluminium alloy input and output bars (Fig. 3.25). The specimens were fabricated such that they could be screwed into the bars. This provides a close-fitting connection and facilitates better wave transmission through the specimen. Fig. 3.25 shows two sequential images of a specimen during dynamic tension. The first depicts the initial state of the specimen and the second corresponds to the end of tension, 365 µs after load commencement. Dynamic tension was applied to specimens over a range of strain rates from to Fig. 3.26 shows the rate-dependent stress-strain responses of the material for this strain rate range and Fig. 3.27 shows the strain rates induced by the SHTB. An infrared camera operating at a framing rate of was used to capture the temperature increase for high rate tensile deformation (Fig. 3.28). 52 (a) (b) (c) (d) Fig. 3.25. (a) Nylon 6 specimens dynamic tension. (b) Specimen connection to input/output bars. High speed photographic images deformation: (c) before loading; (d) of dynamic tensile engineering strain at after commencement of loading at a strain rate of . Fig. 3.26. Response of material under dynamic tension at high strain rates. 53 Fig. 3.27. Constant strain rates imposed by the SHTB. (a) (b) Fig. 3.28. Infrared images showing temperature increase for high rate tension: (a) before loading; (b) 1.2 K increase after 3.5 engineering strain at . Summary and Conclusion Experimental compressive and tensile quasi-static and dynamic results show that the semi-crystalline polymer studied is a rate and temperature dependent material that exhibits post-yield hardening. The material response becomes more compliant with an increase in temperature beyond . A significant temperature increase can be observed during high rate deformation and this causes softening of the polymer. Both quasi-static and high strain rate behaviour display asymmetry between tension and compression. 54 CHAPTER 4 4. Modelling of Semi-crystalline Polymers Based on the experimental results presented in Chapter 3, the quasi-static and dynamic compressive and tensile behaviour of semi-crystalline polymer, Nylon 6, can be described by a three-dimensional thermo-mechanical constitutive model which incorporates a yield stress, post-yield hardening, as well as rate and temperature dependence. 4.1 Literature Review A semi-crystalline polymer can be viewed as a composite with rigid crystallites suspended within an amorphous phase; this is desirable, because it combines the strength of the crystalline phase with the flexibility of its amorphous counterpart. When a semi-crystalline polymer is subjected to large deformation, deformation is first accommodated by rearrangement of chains in the amorphous region; this occurs for any degree of strain. When deformation becomes sufficiently large, intra-crystal sliding, twinning, martensitic transformation and fragmentation of crystalline blocks are induced, and these result in considerable energy dissipation (Bartczak and Galeski, 2010; Drozdov and Christiansen, 2004; G'Sell and Dahoun, 1994; Hiss et al., 1999; Shojaei and Li). It is well-known that the large strain response of semicrystalline polymers displays rate and temperature dependence linked to irreversible 55 deformation that follows yield, and this is accompanied by strain hardening. (Brusselle-Dupend et al., 2003; Drozdov et al., 2004; Greco and Nicolais, 1976; Holmes et al., 2006; Khan and Farrokh, 2006; Lin and Argon, 1994; Nitta and Takayanagi, 1999; Peric and Dettmer, 2003; Schrauwen et al., 2004; van Dommelen et al., 2003). A comprehensive review of constitutive descriptions of semi-crystalline material has been presented by Holmes and his co-workers (Holmes et al., 2006); it is noted that material behavior is commonly described via two general approaches – micromechanics and macromechanics. At the micro level, material is modeled by two structural phases – crystalline and amorphous. An anisotropic arrangement is assumed for molecular chains and their rearrangement is activated by thermal energy. Drozdov and his co-workers have done considerable work on the micromechanics of semicrystalline polymers, particularly on the kinetics of chain rearrangement, when an activated chain is separated from a junction and merged with another one. (Drozdov, 2007; Drozdov and Christiansen, 2007; Drozdov et al., 2009). At this scale, the crystalline and amorphous phases are arranged in two different ways – an upper bound approach (stiffness and flow of amorphous and crystalline phases occur in parallel) and a lower bound approach (stiffness and flow of the amorphous and crystalline phases are considered in series with each other) (Ahzi et al., 2003; Dusunceli and Colak, 2008), and the proposed models employ a rule of mixtures to take into account the influence of degree of crystallinity (Ayoub et al., 2010; Regrain et al., 2009; van Dommelen et al., 2003). Generally, micro-scale modeling is accompanied by considerable difficulty in associating the mechanical response of polymers with complex micro-scale mechanisms. In macro-scale modelling, it is assumed that the deformation of semi-crystalline polymers can be modelled by considering a homogenized isotropic material consisting of crystalline and amorphous phases (Bergstrom et al., 2002). A wide variety of rheological frameworks, which are combinations of elastic, viscoelastic and viscoplastic components are proposed to describe this. Elastic springs, viscous dashpots, friction sliders, etc, are idealized elements employed to capture a wide spectrum of responses. Findely et al., Kitagawa et al. and Schapery et al. all made early contributions towards validating elasto-viscoelastic-viscoplastic frameworks for application to a wide range of semicrystalline polymers. Later, the proposed models 56 were developed and extended by other researchers to incorporate a yield stress and post-yield behaviour, such as kinematic and isotropic hardenings. (Brusselle-Dupend et al., 2003; Dusunceli and Colak, 2008; Findley et al., 1976; Holmes et al., 2006; Khan et al., 2006; Kitagawa and Takagi, 1990; Nedjar, 2002; Peric and Dettmer, 2003; Reese and Govindjee, 1998; Schapery, 1997). Anand et al. (Anand and Ames, 2006; Anand and Gurtin, 2003) extended a widely-used model formed by Parks, Argon, Boyce, Arruda, and their co-workers (Arruda and Boyce, 1993; Boyce et al., 1988) to describe the behaviour of an amorphous polymer. This well-defined model comprises a resistance component assocciated with molecular network interaction in parallel with several clusters of micro-mechanisms describing intermolecular interaction. Srivastava et al. employed Anand’s theory to propose a thermomechanically-coupled large-deformation model for amorphous polymers in a temperature range which crosses the glass transition (Srivastava et al., 2010a). It is noted that there are few macro-scale models that characterize comprehensively the three-dimensional thermo-mechanical large-deformation response of polymers and incorporate rate and temperature dependence, as well as a yield criterion and postyield hardening (Anand and Ames, 2006; Anand et al., 2009; Anand and Gurtin, 2003; Boyce et al., 1988; Srivastava et al., 2010b); existing formulations generally consist of descriptions of quasi-static behaviour. Consequently, this study aims to propose a three-dimensional constitutive model to predict the response of an incompressible semi-crystalline material, Nylon 6, under compressive and tensile quasi-static and dynamic deformation. The proposed model is formulated on a thermodynamics basis, using a macromechanics approach, and defines elastic-viscoelastic-viscoplastic behaviour, coupled with post-yield hardening. Simple forms for material parameters are sought, to preclude involvement in the complexity of polymer molecular structures. The constitutive model can be implemented in an FEM software (ABAQUS) via a user-defined material subroutine (VUMAT). 4.2 Constitutive Equation A three-dimensional constitutive model is formulated to describe the quasi-static and dynamic response of semi-crystalline polymers. It is schematically depicted in Fig. 4.1, and captures large strain deformation associated with elastic-viscoelastic57 viscoplastic behavior. The proposed model comprises two groups of idealized mechanical components connected in series; the first captures rate-dependent reversible behavior and is modeled by a hyperelastic element A, which acts in parallel with a visco-hyperelastic component that consists of a hyperelastic element B and a viscous element defined by a viscosity coefficient . The second cluster describes irreversible rate-dependent response and is activated when the stress in the material exceeds the yield value . It has two parallel elements – a friction slider that defines and a viscous element defined by the coefficient . Elastic-viscoelastic Viscoplastic Hyperelastic element A Friction slider ( ) Viscous dashpot ( ) Viscous dashpot ( ) Hyperelastic element B Fig. 4.1. Schematic diagram of proposed elastic-viscoelastic-viscoplastic model; p, e, v, ve denote the plastic, elastic, viscous and viscoelastic components. 4.3 One-dimensional Form of the proposed Elastic-Viscoelastic-Viscoplastic Framework A one-dimensional form of the elastic-viscoelastic-viscoplastic model (Fig. 4.1) is described in Fig. 4.2. Motion of the friction slider is activated when the total stress exceeds a threshold value , . The stress and strain components in the various elements of the model are governed by the relationships in Fig. 4.2. 58 = = = + + ve Fig. 4.2. Schematic diagram of a one-dimensional elastic-viscoelastic-viscoplastic model. (4.1) where is the yield stress and a material constant, is a function of the plastic strain and describes hardening. In addition to this threshold stress for plastic deformation, there is an additional component that is associated with strain rate, which is captured by the viscoplastic dashpot. (4.2) where, is defined by the following. (4.3) Assumption of a Newtonian dashpot for the viscoplastic component yields the following expression for the plastic strain rate. (4.4) 59 is a viscosity coefficient. The one-dimensional model (Fig. 4.2) is where formulated and implemented in MATLAB (Appendix C). Fig. 4.3 shows schematically the response of the various elements in the one-dimensional model, for different strain rates; . Fig. 4.3. Response of various elements in the one-dimensional model for different strain rates. 4.4 Kinematic Considerations Consider a deformable body comprising material points called particles. A typical particle is identified or labelled by its position vector X in the reference configuration. After deformation, the particle moves to a new position defined by the vector x in the current configuration. The deformation gradient relates these quantities and is defined by; 60 (4.5) The right and left polar decompositions of are given by the following relationship and schematically represented in Fig. 4.4. (4.6) Reference configuration R U V R Deformed configuration Fig. 4.4. Schematic representation of the right and left polar decompositions of (Wikipedia, Sep 2012). The proper orthogonal tensor R characterizes rigid body rotation, while the positive definite symmetric tensors and are respectively the right and left stretch tensors and describe local deformation in the material. Since and are usually not easily computed, it is convenient to use the following (positive definite symmetric) tensors: , (4.7) 61 and are respectively the right and left Cauchy-Green deformation tensors. Multiplicative decomposition of the deformation gradient facilitates analysis, and the total deformation is composed of elastic, viscoelastic and plastic components (Gurtin and Anand, 2005; Kroner, 1960; Lee, 1969). In line with the schematic diagram in Fig. 4.1, decomposition of the deformation gradient deformation tensor and the velocity gradient , the right Cauchy-Green are effected as follows: (4.8) , = = where, , , = and are the respective deformation gradient tensors associated with deformation of the hyperelastic element A, hyperelastic element B, viscous dashpot connected to element B and friction slider in the viscoplastic cluster. 4.5 Thermodynamic Considerations The continuum mechanics formulation adopted is based on a scalar quantity known as the free energy density , which is a function of frame-invariant variables, commonly defined using parameters that describe elastic deformation ( (4.8)), the absolute temperature , ; Eq. as well as internal variables affecting the free energy stored. Generally, the total free energy can be defined as the sum of free energies associated with the different rheological components (Anand et al., 2009; Holmes et al., 2006; Peric and Dettmer, 2003; Reese and Govindjee, 1998). In this study, the proposed elastic–viscoelastic–viscoplastic constitutive model (Fig. 4.1) is described by the following free energy function. (4.9) The constitutive equation must satisfy fundamental thermodynamic considerations, and it is assumed that quasi-static deformation corresponds to a constant temperature 62 process and that high rate deformation is adiabatic. Therefore, in the absence of heat flux and a heat supply, the balance of energy and Second Law of Thermodynamics (inequality) per unit reference volume are expressed as (Anand, 1985; Anand et al., 2009; Thamburaja and Ekambaram, 2007). (4.10) (4.11) where is the Second Piola-Kirchhoff stress, applied load per unit reference volume, unit reference volume, the work-rate of an externally the internal energy density per the time derivative of the internal energy, free energy density per unit reference volume, and entropy density per unit reference volume. the Helmholtz (Eq. (4.25)) is the is obtained by applying the chain rule to the time derivative of Eq. (4.9). (4.12) 4.6 Simplification of Chain Rule for Time Derivative of Free Energy Eq. (4.8) relates the deformation gradient deformation tensor and the right Cauchy-Green to their elastic and plastic components; . The chain rule for the time derivative of the free energy , (Eq. (4.12)) is simplified as follows: (4.13) The first term on the right-hand side of Eq. (4.13) is simplified by: 63 (4.14) where the fourth order identity tensor ( Using the notation is given by: kronecker delta) , , (4.15) and application of the double tensor contraction, , and to Eq. (4.14) results in (4.16) The second term on the right-hand side of Eq. (4.13) is simplified by (4.17) Differentiation of the inverse of a second order tensor with respect to itself is expressed by the following fourth order tensor: (4.18) Substitution of the following relationships and Eq. (4.18) into Eq. (4.17) yields Eq. (4.20). 64 (4.19) (4.20) A combination of Eqs. (4.13), (4.14), (4.16) and (4.20) yields the simplified time derivative of the free energy (Eq. (4.12)) as follows: (4.21) As with simplification of the chain rule for the time derivative of the free energy , the simplified time derivative of the free energy using Eq. (4.21) and the decomposition of the deformation gradient Cauchy-Green deformation tensor ; , is expressed and right = (Eq. (4.8)) as follows: (4.22) 65 Substitution of Eqs. (4.21), (4.22) and (4.12) in Eq. (4.11) yields: (4.23) 4.7 Second Piola-Kirchhoff and Cauchy Stresses A well-accepted argument in the derivation of constitutive equations is applied to Eq. (4.23) to obtain the Second Piola-Kirchhoff stress and the entropy expressed respectively by Eqs. (4.24) and (4.25) (Anand et al., 2009). (4.24) (4.25) The Second Law inequality presented in Eq. (4.23) must hold for all values of , , , , , and . Therefore, in the absence of inelastic deformation ( ) and temperature variation (isothermal conditions; ), only the first term of the inequality remains. Therefore, the relationship defined in Eq. (4.24) is a possible expression for the Second Piola-Kirchhoff stress which satisfies the inequality for any rate of deformation . Eq. (4.25) is an expression for the entropy , which satisfies the inequality for purely thermal deformation ( (4.24) and (4.25) are possible solutions and . Eqs. “….we content ourselves with constitutive equations that are only sufficient… by Anand” (Anand et al., 2009). 66 The relationship between the Second Piola-Kirchhoff stress and the Cauchy stress in the deformed configuration, is (4.26) where, , and are the determinants of the deformation gradient tensors , and (Eq. (4.8)), and are related by; (4.27) It is assumed that inelastic deformation is isochoric (Section 4.10); thus and 4.8 . Dissipation Inequality The last two terms of the inequality defining the Second Law of Thermodynamics (Eq. (4.23)) are expressed in terms of the plastic and viscoelastic work associated with inelastic deformation rates – , (Eq. (4.8)). In Eq. (4.23), and are symmetric tensors (Second Piola-Kirchhoff stresses, Eq. (4.24)); thus, Eq. (4.23) can be expressed as: (4.28) The inequality expressed in Eq. (4.28) is simplified by the tensor double contraction as follows: 67 (4.29) Substitution of the Cauchy stresses , tensors , and (Eq. (4.26)) and velocity gradient (Eq. (4.8)) into Eq. (4.29) results in following relationship: (4.30) where , 4.9 are defined as driving forces which are power conjugates of and . Evolution of Temperature Variation The entropy is related to the inelastic work dissipated using a combination of Eqs. (4.11), (4.23), (4.24), (4.25), (4.28) and (4.30), via the following: 68 (4.31) This relationship is employed to capture the temperature variation and heat generated by the inelastic work dissipated (Eq. (4.54)). 4.10 Inelastic Flow Rule The viscoelastic and plastic deformation are described respectively by the velocity gradients ( , and (Eq. (4.8)), and are generally decomposed into inelastic stretch ) and spin rate tensor , ) components, which are defined by: (4.32) It is commonly assumed that inelastic flow is irrotational and incompressible (Anand et al., 2009; Gurtin and Anand, 2005; Weber et al., 1990), therefore (irrotational flow) (4.33) (incompressibility) Substitution of Eqs. (4.32)–(4.33) into Eq. (4.30) results in the inelastic work dissipated associated with irrotational and incompressible inelastic flow as follows: (4.34) where the deviatoric part of the symmetric driving force tensors are denoted by the following: 69 (4.35) Assuming that the material is strongly dissipative, the inequality is satisfied by the following: (4.36) , The stretch rate tensors are related to the corresponding driving forces as follows: (4.37) where, and represent the magnitudes of the stretch rate tensors, and define their directions. , and are the fluidities, or reciprocal of viscosities, which are expressed in terms of deviatoric and volumetric components as follows (Holmes et al., 2006; Holzapfel, 1996; Peric and Dettmer, 2003; Reese and Govindjee, 1998): (4.38) where , are the deviatoric and , the volumetric components of the viscosities. Substitution of Eq. (4.38) into Eq. (4.37) defines the inelastic stretch rate tensors and satisfies the dissipation inequalities (Eq. (4.36)) as follows: 70 (4.39) (4.40) In the following section, is modified in order to incorporate a yield criterion and post-yield hardening. 4.11 Initiation of Plastic Deformation In the proposed model, motion of the friction slider element (Fig. 4.1) defines plastic flow when the driving force exceeds a threshold. This threshold is made to increase with plastic deformation, to capture the hardening observed in experiments, as well as results reported in previous work on semicrystalline materials (G'Sell and Dahoun, 1994; Holmes et al., 2006). The details on the formulation of a onedimensional description of plastic flow based on the proposed model is described in the Section 4.3. A von Mises criterion, accompanied by a hardening function , is proposed as the yield threshold for the three-dimensional constitutive description, and which defines the stress beyond the yield threshold, is expressed using Eq. (4.3) as follows: (4.41) where is the yield stress (a material constant) (Section 5.1), and is a frame- invariant scalar that increases with plastic deformation. The Almansi plastic strain , which is work-conjugate with the Cauchy stress, is employed to define via the simplest function that exhibits good agreement with experiments (Section 5.1). (4.42) 71 where and are material constants. The plastic flow defined in Eq. (4.39), is modified to incorporate a yield criterion and post-yield hardening behavior, by substitution of Eqs. (4.41) into Eq. (4.39); it satisfies the dissipation inequality (Eq. (4.40)) as follows: (4.43) 4.12 Helmholtz Free Energy Density for the Proposed Model The Helmholtz free energy density is defined in general form to describe the hyperelastic elements A, B in Fig. 4.1. The energy density per unit reference volume is basically a function of frame-invariant parameters, and the following separable form is proposed: (4.44) where, denotes the energy stored by pure elastic deformation; the purely thermal portion of the free energy and elastic portion of the free energy. represents represents the thermo- consists of the stored energy of the deviatoric and volumetric deformations. Volume-conservation is captured by , , which are the principal invariants of part of the strain energy is defined using change (Srivastava et al., 2010b). Therefore, , and and . The volumetric , which represents the volume is expressed in the following form: (4.45) 72 A polynomial form of is proposed and described by: (4.46) where, and bulk modulus are material constants and correspond to the shear modulus and as follows: (4.47) is given by (Pan et al., 2007; Srivastava et al., 2010a): = where (4.48) is the reference temperature and the specific heat capacity per unit reference volume, which is governed by (4.49) where is the internal energy density per unit reference volume and is the entropy density per unit reference volume. is described by (Anand and Ames, 2006; Anand et al., 2009; Pan et al., 2007). 73 (4.50) where, is the thermal expansion coefficient. The proposed model (Fig. 4.1) employs two different free energy densities – – which correspond respectively to the and hyperelastic elements A and B. The total free energy defined in Eq. (4.45) is redefined as follows: (4.51) , are defined using Eq. (4.46) in a form with the minimum number of material parameters that provide reasonable agreement with experiments (Section 5.1). (4.52) , and are expressed using Eqs. (4.48)–(4.50) as follows: (4.53) 74 In order to capture the temperature variation and heat generated by inelastic deformation, is defined using a combination of Eqs. (4.31) and Eqs. (4.51)–(4.53) as follows: (4.54) Thus, (4.55) where , and it is assumed that only a portion of the inelastic work generates heat (Table 5.1). The Cauchy stress in the proposed three-dimensional model (Fig. 4.1) consists of the hyperelastic stresses of elements A and B; is thus defined by substituting Eqs. (4.51)–(4.53) into Eq. (4.26) as follows: 75 (4.56) In order to keep the number of parameters to a minimum, is defined to be a constant that links the coefficients of element A with those of element B as follows: (4.57) where, is determined by fitting the proposed constitutive equation to experimental data (Section 5.1). 4.13 Summary A three-dimensional thermo-mechanical constitutive model that employs the hyperelastic description proposed in Chapter 2 has been developed to describe the quasi-static and high rate large-deformation response of polymers. This model is 76 based on an elastic-viscoelastic-viscoplastic framework coupled with post-yield hardening, to capture the temperature and rate-dependent response of an incompressible semi-crystalline polymer, Nylon 6. The material parameters and model coefficients/functions which require to be calibrated and specified, are listed in Table 4.1. Table 4.1. The material parameters and model coefficients/functions which are needed to be calibrated. Equation (4.39) (4.41) (4.41), (4.42) (4.47) , (4.53) (4.55) (4.52), (4.56) (4.57) 77 CHAPTER 5 5. Model Calibration and Simulation Results Nylon 6 is the semi-crystalline polymer studied. The experimental results in Chapter 3 are the quasi-static and dynamic responses of the material, which are employed to calibrate and validate the proposed constitutive model presented in Chapter 4. The model is first cast in one-dimensional form and implemented using MATLAB, and subsequently in three-dimensional form for implementation in ABAQUS via its VUMAT subroutine. The one-dimensional model and a singleelement of the three-dimensional finite element model are fitted to experimental data for calibration and also used to predict the response at low and high rates deformation. In Section 5.4, a multi-element three-dimensional model is used to simulate the deformation and force-displacement response of complex-shaped specimens; the results are compared with experiments. 5.1 Quasi-static Tests at Room Temperature Samples of Nylon 6 were subjected to quasi-static compression and tension at room temperature; (Section 3.1). It is generally acknowledged that Nylon 6 behaves as an approximately incompressible polymer; thus, a large value of the Poisson's ratio ϑ is proposed to calculate the shear modulus bulk modulus ϑ and the ϑ (Table 5.1). A value of ϑ = 0.495 results in less than 2% compressibility for polymers, which has been found from Monte Carlo 78 simulations (Chui and Boyce, 1999) and experiments (Buckley et al., 2004; Khan and Farrokh, 2006). Fig. 5.1 shows the compressive and tensile responses of the material at different strain rates; it is observed that the material is relatively rate-insensitive at low strain rates, but significantly rate-sensitive at high deformation rates. Therefore, in the proposed model (Fig. 4.1), it is assumed that the hyperelastic element B does not contribute to the stress at the lowest strain rate of ; i.e. the dashpot connected to element B provides no resistance. Fig. 5.2 shows the intersections of tangent lines which are taken to define the yield stresses that trigger plastic deformation and motion of the friction slider (Fig. 4.1) (Srivastava et al., 2010a). The response of the material before yield is considered to be nonlinearly elastic and modelled by the hyperelastic description proposed by Pouriayevali et al. (Pouriayevali et al., 2012), and details are provided in Chapter 2 (Eq. (2.28)). This model is implemented in MATLAB and fitted to the uniaxial tensile and compressive responses of the material at strain rates of material constants for the energy density function With the value of to obtain the (Eq. (4.52), Table 5.1). determined, the three-dimensional model presented in Chapter 4, implemented in ABAQUS via its VUMAT subroutine (Appendix D), is fitted to the large strain material response for low strain rates (Fig. 5.3 and Fig. 5.4) to obtain the parameters , , , , and (Table 4.1 and Table 5.1). 79 Fig. 5.1. Compressive and tensile responses of Nylon 6 at low and high strain rates. 80 Fig. 5.2. Identification of yield stress marking onset of plastic deformation, for compression and tension. 81 Fig. 5.3. Comparison between test data and fit of proposed model for compression. 82 Fig. 5.4. Comparison between test data and fit of proposed model for tension. 5.2 Quasi-static Tests at High Temperature Nylon 6 has temperature-dependent material properties and the material parameters vary when the temperature crosses (Sections 3.2 and 3.3). Fig. 5.5 shows the quasi- static response of the material at different temperatures. Two temperature-dependent 83 functions, and , are employed to fit the proposed model to experiments, where is defined by an exponential function commonly used to decrease material stiffness with temperature (Mahieux and Reifsnider, 2001; Srivastava et al., 2010a). The material constants and are multiplied by (Table 4.1, Table 5.1). Similarly, is defined by an exponential function derived empirically, and frequently employed to describe Newtonian viscosity (Arcos et al., 2011; Dantzig and Tucker, 2001; Dobbels and Mewis, 1977); and are multiplied by (Table 4.1, Table 5.1). Fig. 5.5. Comparison between experimental tension and compression data with fit of proposed model for a strain rate of and different temperatures. 84 5.3 Experiments for Deformation at High Strain Rates Samples of Nylon 6 are subjected to high strain rate compression and tension using compressive and tensile Split Hopkinson Bar devices (Fig. 3.17 and Fig. 3.25). The responses at high strain rates are predicted using the proposed constitutive model with parameter values obtained by fits with experimental data at low strain rates (Sections 5.1, 5.2). Fitting of the constitutive model to experimental test results for low strain rates of is presented in Fig. 5.3 and Fig. 5.4, and duplicated in Fig. 5.6. The model incorporating low strain rate tension and compression parameters obtained according to Sections 5.1 and 5.2 (Table 5.1) is fitted respectively to dynamic test results for tensile strain rates of was found that only and and , and compression at (Fig. 5.6). It are rate-dependent, and their values obtained for low strain rate experiments in Section 5.1, must be modified for high strain rates (Table 5.1). Fig. 5.7 shows the values of proposed for different temperatures and strain rates. The model with compression and tension parameters for high strain rates (Table 5.1) is able to predict the responses for compression at strain rates of , and tension at strain rates of and and (Fig. 5.6). 85 Fig. 5.6. Comparison between tension and compression test data for different strain rates with proposed model. There is a significant temperature increase during high rate deformation due to considerable viscoelastic and viscoplastic work (Eq. (4.55)); Fig. 5.6 shows softening in the material response at a strain rate of because of this. This behaviour is reasonably well described by the proposed model. Fig. 5.8 shows comparisons between the temperature variations predicted by the single-element model and the maximum temperature captured in experimental results (Fig. 3.24 and Fig. 3.28), 86 while Fig. 5.9 shows the temperature profile predicted by the multi-element model compared with experimental results (Fig. 3.24 and Fig. 3.28). (a) (b) (c) Fig. 5.7. Proposed values for temperature; to describe compressive loading at (a) constant =298 K; (b) different temperatures at low strain rates; (c) different temperatures at high strain rates. 87 Fig. 5.8. Comparison between temperature variations predicted by the single-element model and experimental data. 88 (a) (b) Fig. 5.9. Comparison between temperature profile predicted by ABAQUS (multielement model) and thermo-graphic infrared camera images, (a) temperature increase at rate of compressive engineering strain for a strain , (b) temperature increase after strain for a strain rate of tensile engineering . 89 Table 5.1. Model parameters and material coefficients. Strain rate # # * Compression Tension Low High Low High 2.0 e8 3.5 e4 7.0 e8 5.0 e4 2.0 e8 2.2 e4 9.0 e7 5.0 e4 * 280.9 -249.8 * -215 330 * 52.07 67.7 1.2e4 1.55e4 35.0 46.0 720, 240.8, 958, 320.4, 2.4e4 3.2e4 117.97 120.4 1.2e4 = 1.2e4 10.0 147.9 160.2 1.55e4 1.6e4 Unit 55.3 1700 7e-5 ρ 1145 0.45 ϑ 0.495 5 0.9 Parameters identified with #, * are temperature-dependent and modified by multiplication with and , respectively. 90 5.4 Multi-Element FEM Model of Complex-Shaped Specimens The quasi-static and high strain rate responses of two complex-shaped specimens are observed experimentally and simulated by FEM. Experimental results are compared with that from FEM to validate the potential of the proposed constitutive model in predicting the response of structures and components made of Nylon 6. 5.5 Short Thick Walled Tube Fig. 5.10 (a) shows the geometry of a short thick walled tube with a circular hole drilled across its diameter. A three-dimensional FEM model, employing the compression parameters (Table 5.1) and C3D4 elements, is used to predict the quasistatic and high strain rate response to axial compression. Fig. 5.10 compares the geometry of the specimen with the FEM model for quasi-static compression, while Fig. 5.11 depicts the temperature profile from the FEM model and that captured experimentally for high rate deformation. Experimental force-displacement data are also compared with values generated by the FEM model in Fig. 5.12. The experimental data shows a more compliant response at large strains compared to the FEM model. This could be attributed to either the specimen geometry or the type of FEM element employed. The complex geometry induces non-uniform shear deformation which is negligible in the compressive deformation of the simple solid cylindrical specimen used for calibrating the proposed model (Fig. 3.2, Section 5.1). Moreover, the C3D4 tetrahedral elements used are the only elements that can accommodate the large deformations induced in complex geometry specimens, since alternative C3D8R hexahedral elements become excessively distorted. Nevertheless, the comparison shows that the proposed constitutive model implemented in a threedimensional FEM is able to simulate the overall quasi-static and dynamic response of complex-shaped specimens. 91 (a) (b) (c) (e) (d) Fig. 5.10. (a) Nylon 6 specimen with a complex geometry; (b) ABAQUS model; (c) Quasi-static compression of specimen; (d) and (e): comparison between specimen geometry and model at (a) compressive engineering strain. (b) Fig. 5.11. Temperature profile corresponding to a strain rate of mm compressive deformation at (Fig. 5.12) (a) ABAQUS model; (b) infrared image. 92 Fig. 5.12. Comparison between experimental force-displacement data and FEM model for compression of thick walled tube with a transverse hole. 5.6 Plate with Two Semi-Circular Cut-Outs A Nylon 6 plate with two semi-circular cut-outs on opposite edges, depicted in Fig. 5.13, is subjected to quasi-static tension. The FEM model, incorporating low rate tension parameters (Table 5.1) and C3D4 elements, is used to simulate the response. Fig. 5.13 shows the deformed specimen geometry and that generated by the FEM model, while Fig. 5.14 compares the experimental force-displacement data with FEM results. Fig. 5.14 shows that the model matches the experimental data in terms of overall trends. However, the experimental response is again more compliant than the 93 FEM results. As with the model of the thick walled tube, the difference probably arises from the complex specimen geometry or the type of FEM element employed. (a) (b) (c) (d) Thickness Fig. 5.13. (a) Nylon 6 strips with cut-outs; (b) ABAQUS model; (c) and (d): comparison between specimen geometry and model at tensile engineering strain. 94 Fig. 5.14. Comparison between experimental force-displacement data and FEM model for the tensile loading. 5.7 Summary and Conclusions A three-dimensional thermo-mechanical proposed model was calibrated to predict the quasi-static and dynamic response of semi-crystalline; Nylon 6. It is noted that the model requires two sets of parameters to capture the material behavior, one for compression and another for tension. The yield stress, stiffness coefficients and viscosity parameters are temperature-dependent, and the viscosity parameters are also found to be rate-dependent. The C3D4 tetrahedral elements used are the only FEM elements that can accommodate the large deformations induced in complex geometry specimens, since C3D8R hexahedral elements become excessively distorted. The experimental data shows a more compliant response at large strains for complex95 shaped specimens compared to the FEM model. However, the results indicate that the proposed constitutive model has good potential for predicting the thermo-mechanical response of complex-shaped components made of semi-crystalline polymer. 96 CHAPTER 6 6. Conclusion and Recommendation for Future Work This study comprises two aspects and focuses on two categories of polymeric materials; elastomers and semi-crystalline polymers. The quasi-static and dynamic behaviour of these materials were studied and described by two constitutive models. The first part of the work focused on elastomers which are nonlinear viscoelastic materials with a low elastic modulus, and exhibit rate sensitivity when subjected to dynamic loading. A large-deformation three-dimensional visco-hyperelastic constitutive equation that incorporates a deformation-dependent relaxation time was proposed to describe the large deformation response of incompressible elastomers under quasi-static and dynamic compression and tension. The model comprises two components: the first corresponds to hyperelasticity based on a strain energy density potential to characterize the quasi-static nonlinear response; the second is an integral of the first and incorporates a relaxation time function to capture the rate sensitivity and fading-memory characteristics of the material. The proposed model was applied to describe the high rate response of six types of rubber with different shore hardnesses and rate-sensitivity, and found to be able to capture their dynamic behaviour. It is noted that a three-term truncated polynomial series for the energy 97 density function adequately characterises the hyperelastic character of the elastomers. The relaxation time function fitted indicates that the relaxation time increases with equivalent strain. It is also noted that the compressive and tensile behaviour of elastomers can be defined by a single relaxation time function with a common set of material parameter values. The model has the potential to predict responses within and outside the range of strain rates used in deriving the material stiffness and viscosity parameters. In the second part of the study, a semi-crystalline polymer, Nylon6, was investigated. Samples were subjected to quasi-static compression and tension at room and higher temperatures, using an Instron universal testing machine. High strain rate compressive and tensile deformation was also applied to material specimens using Split Hopkinson Bar devices. The specimen deformation and temperature change during high rate deformation were captured using a high speed optical and an infrared camera respectively. Experimental results show that the semi-crystalline polymer studied is rate and temperature dependent, and exhibits post-yield hardening. The response becomes more compliant with an increase in temperature beyond . A significant temperature increase can be observed during high rate deformation and this causes softening of the polymer. Both quasi-static and high strain rate behaviour display asymmetry between tension and compression. In terms of macro-scale modelling, the semi-crystalline polymer was considered a homogenized isotropic material consisting of crystalline and amorphous phases, and a thermo-mechanical constitutive model for large deformation was proposed and applied to describe quasi-static and high rate compressive and tensile deformation. The model was developed from an elastic-viscoelastic-viscoplastic perspective and a desire to minimize the number of model parameters. It was implemented in FEA software (ABAQUS) by the writing of a user-defined material subroutine using ABAQUS' VUMAT function. Predictions based on the model show good agreement with test results for material response at low and high rate deformation. It is noted that the model requires two sets of parameters to capture the material behavior, one for compression and another for tension. The yield stress, stiffness coefficients and viscosity parameters are temperature-dependent, and the viscosity parameters are also found to be rate-dependent. The proposed constitutive model shows good potential in 98 predicting the thermo-mechanical response of complex-shaped components made of semi-crystalline polymer. 6.1 Recommendations for Future Work With regard to the limitations of the present work and possible future efforts; The quasi-static experiments in this study correspond mostly performed to uniaxial loading. Different loading conditions, such as torsion, bending, indentation, as well as a combination of uniaxial loading with torsion will enhance understanding of the quasi-static response of the material examined. Short and long-term relaxation and creep of polymers and soft materials are of interest, and such tests can be also undertaken to see if the proposed constitutive model can be extended and calibrated to predict relaxation and creep responses. Polymers are significantly temperature-dependent and are used in a wide range of temperature environments in industry. Quasi-static and dynamic tests over a wide temperature range can be done to examine their responses. Behavior under loading-unloading conditions could be examined to determine permanent deformation after unloading from large-strains. The proposed model could then be calibrated for such conditions and employed to predict the final geometry and shape of components fabricated by processes such as deep drawing and extrusion. 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International Journal of Impact Engineering 24, 545-560. 105 Appendixes A. Split Hopkinson Bar Device A Split Hopkinson Bar setup (SHB) comprises two long bars, commonly known as the input (incident) and output (transmission) bar (Fig. A.1). A striker, propelled by pressurized gas, impacts the free end of the input bar, generating a stress wave that propagates down the input bar into the specimen. A portion of the incident wave is transmitted into the specimen and output bar, while the remaining portion of the wave is reflected back into the input bar. These waves are measured using strain gauges mounted on the input and output bars. The strain gauges on the input bar measure the incident and reflected waves, while the gauges on the output bar measure the transmitted wave (JiHui, 2010; Lindholm, 1964; Sasso et al., 2008). A.1. Governing Equations Input (Incident) bar Striker Output (Transmission) bar Specimen Strain gauge Strain gauge Fig. A.1. Schematic arrangement of SHB bars and specimen. SHB theory is based on several assumptions: o Waves propagating in the bars conform to one-dimensional elastic wave propagation. o The stress and strain distribution within the specimen is uniform. o Friction is negligible. Consider the interface between the input bar and specimen. The interface velocity is defined by; (A.1) where is the velocity at the interface between input bar and specimen, particle velocity of the incident wave, the the particle velocity of the reflected wave 106 and and SHTB; is the elastic wave velocity in the input bar; the reflected strain wave. is the incident strain wave is compressive for an SHPB and tensile for an is the reflected wave; tensile for an SHPB and compressive for an SHTB. The force at the interface between the input bar and specimen is defined by: (A.2) where, and is the cross-sectional area of the input bar, the stress at that interface, the elastic modulus of the input bar. The velocity at the interface between the output bar and the specimen is: (A.3) where is the particle velocity of the transmitted wave and the transmitted strain wave. The force at the output bar interface is defined by: (A.4) where is the stress at the interface between the output bar and specimen. The specimen strain rate is given by; (A.5) where is the specimen length. Assuming that the specimen does not accelerate, equilibrium implies that (A.6) Inserting Eq. (A.2) and Eq. (A.5) into Eq. (A.6) yields: (A.7) Substitution of Eq. (A.7) into Eq. (A.5) results in the strain rate : (A.8) The strain of the specimen is obtained by integrating Eq. (A.8): 107 (A.9) The stress in the specimen is and is given using Eq. (A.4); (A.10) where A.2. is the cross-sectional area of the specimen. Split Hopkinson Pressure Bar Device (SHPB) Fig. A.2 is a schematic diagram of an SHPB set up. In using an SHB, the length of the striker should be selected to ensure no superposition of the incident and reflected waves in the input bar. The strain gauge attached to the input bar should be sited to capture the incident and reflected waves without overlap between the two. Gas chamber Striker bar Support Input bar Specimen Output bar Strain gauge Wheatstone bridge Wheatstone bridge Dynamic Strain Meter Dynamic Strain Meter Meter Oscilloscope Fig. A.2. Schematic diagram of a SHPB set up. Fig. A.3 shows typical strain gauge signals captured by the oscilloscope. Copper disks of different diameters and thicknesses are employed as a pulse shapers to provide a more constant strain rate (Fig. A.4). From experience, the disks should be compressed once before use in actual tests. To achieve stress equilibrium in a specimen, the stress wave should generally travel back and forth within the specimen more than three times. Consequently, stress equilibrium is attained 12 µs after load commencement for the specimens studied. 108 Reflected ignal Incident Signal Transmitted Signal Fig. A.3. Typical strain gauge signals captured by the oscilloscope. Input Bar Pulse Shaper Striker Fig. A.4. Pulse shaper used in SHPB device. A.3. Split Hopkinson Tension Bar Device (SHTB) Tensile testing using a SHTB is more complex because there are various ways to attach specimens. A schematic diagram of an SHTB set up is shown in Fig. A.5. For this device, a commercially available ductile steel washer is used as a pulse shaper (Fig. A.6). Stress equilibrium in the specimen is attained 30 µs after commencement of loading on the specimen. 109 Barrel Stopper Input Bar Tubular striker Pressurized Gas chamber Screw attachment Support Specimen Output Bar Strain gauges connected to Wheatstone bridge Fig. A.5. Schematic diagram of a Tensile Split Hopkinson device. Input Bar Pulse Shaper Striker Fig. A.6. Pulse shaper used in SHTB device. 110 B. Visco-hyperelastic model employed for uniaxial tests of elastomers clc clear %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Visco-hyperelastic model employed for uniaxial tests of elastomers % EPDM Rubber % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% syms x a b c d e k n t ditime xdot % Maximum strain maxstrain=5 % Number of Strain points strainpoint=500 % defining Strain points for i=1:strainpoint X(i)=(i-1)*maxstrain/pointnumber end % Hyperelastic coefficients avalue=192.68; bvalue=-193.945; cvalue=109.17; dvalue=0 ; evalue=0; % Relationship between elastic and viscoelastic springs BETA kvalue=26; % Strain Rate xdotvalue=3500; % constant used in relaxation function nvalue=0.5e-6; coefficents=[avalue;bvalue;cvalue;dvalue;evalue;kvalue;nvalue]; % Model parameters landa=1+x; B11=landa^2; B11B11=landa^4; B22=landa^-1; B22B22=landa^-2; Ione=landa^2+2*landa^-1; Itwo=landa^-2+2*landa; 111 % Cauchy Hyperelastic Stress sigmahyper =a*(((2*(1+x)^2*(1-(1+x)^-3))))+b*(((2*(1+x)*(1-(1+x)^3))))+c*(((2*(1+x)*(1-(1+x)^-3)*(((1+x)^2+2*(1+x)^-1)3+(1+x)*((1+x)^-2+2*(1+x)-3)))))+d*(2*2*((((1+x)^2)+2*((1+x)^-1))3))*((1+x)^2-(1+x)^-1)+e*((2*2*(((1+x)^2)+2*((1+x)^-1))*((((1+x)^2)+2*((1+x)))-3))*((1+x)^2-(1+x)^-1)-2*2*((((1+x)^-2)+2*((1+x)))3)*((1+x)^4-(1+x)^-2)); sigmahyperdot=diff(sigmahyper,x); for pointnumber=1:(strainpoint-1) % time interval time(pointnumber)=X(pointnumber)/xdotvalue; time(pointnumber+1)=X(pointnumber+1)/xdotvalue; stressigmahyperdot(pointnumber)=subs(sigmahyperdot,{x,a,b,c,d,e,k, n,t,xdot},{X(pointnumber),coefficents(1),coefficents(2),coefficents(3 ),coefficents(4),coefficents(5),coefficents(6),coefficents(7),time(po intnumber),xdotvalue}); Localstress(pointnumber)=(stressigmahyperdot(pointnumber))*1*(X(po intnumber+1)-X(pointnumber)); stressigmahyper(pointnumber)=subs(sigmahyper,{x,a,b,c,d,e,k,n,t,xd ot},{X(pointnumber),coefficents(1),coefficents(2),coefficents(3),coef ficents(4),coefficents(5),coefficents(6),coefficents(7),time(pointnum ber),xdotvalue}); engineeingstressigmahyper(pointnumber)=stressigmahyper(pointnumber )/(1+X(pointnumber)); Stressrelaxationrate(pointnumber)=0; %Equivalent Strain Xequivalent(pointnumber)= sqrt(0.25*((((1+X(pointnumber))^21))^2+2*(((1+X(pointnumber))^(-1)-1)^2))); %Stress Relaxation for point=1:pointnumber %Relaxation Function exptime=-(X(pointnumber)-X(point))/xdotvalue/(nvalue*(1+8* Xequivalent(pointnumber))); Stressrelaxationrate(pointnumber)=Stressrelaxationrate(pointnumber )+Localstress(point)*exp(exptime); end Engineerintotalstress(pointnumber)=(stressigmahyper(pointnumber)+k value*Stressrelaxationrate(pointnumber))/(1+X(pointnumber)); 112 XX(pointnumber)=X(pointnumber); ttime(pointnumber)=-X(pointnumber)/xdotvalue; end figure (1) plot(XX,stressigmahyper,'.g','LineWidth',1) hold on figure (2) plot(XX,stressigmahyper+kvalue*Stressrelaxationrate,'.g','LineWidt h',1) hold on 113 One-dimensional Model for the Proposed Elastic-viscoelastic- C. viscoplastic model clc clear %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % One-dimensional Elastic-viscoelastic-viscoplastic model % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Maximum Engineering Strain maxstrain=0.9; % Strain Rate1 Srate1=0.05; % Time interval timedelta=maxstrain/Srate1/5000 ; % Strain Rate1 to Rate2 at the following strain jumpstrain12=0.3; % Strain Rate2 Srate2=0.05; % Strain Rate2 to Rate3 at the following strain jumpstrain23=0.6; % Strain Rate3 Srate3=.01; % Total time totaltime(1,1)=0; %Total engineering strain Epsilon(1,1)=0; Epsilondelta=0; % Total Caushy stress Totalstress(1,1)=0; Strain=0; i=1; j=1; % Hyperelastic Parameters a=-249.8D6 b=330.D6 c=67.7D6 % Strain in Elastic component 114 StrainE(1,1)=0; % Viscoelastic Parameters % Relationship between Elastic spring and viscoelastic spring springrelation=4; % Strain in Viscoelastic component StrainV(1,1)=0; % Strain in Elastic part of Viscoelastic component StrainVe(1,1)=0; % Strain in Viscous part of Viscoelastic component StrainVi(1,1)=0; StrainVidelta=0; % Viscous coefficient in Viscoelastic component miuV=3e8; % Viscoplastic Parameters % Yield Stress Yieldstress=46e6; % Post-yield isotropic hardening isoyieldstress(1,1)=yieldstress; Kiso=60e6; % Strain in Viscoplastic component StrainP(1,1)=0; StrainPdelta=0; % Viscous coefficient in Viscoplastic component miuP=1e8; while Strain = jumpstrain12) %First rate jump SrateNew(1,i)=Srate2; timedeltaNew=Srate1*timedelta/SrateNew(1,i); end if (Strain >= jumpstrain23) %Second rate jump SrateNew(1,i)=Srate3; timedeltaNew=Srate1*timedelta/SrateNew(1,i); end 115 totaltime(1,i)=totaltime(1,i-1)+timedeltaNew; Epsilondelta=SrateNew(1,i)*timedeltaNew; % Total strain Epsilon(1,i)=Epsilon(1,i-1)+Epsilondelta; if (StrainP(1,i-1)+StrainE(1,i-1))> Epsilon(1,i) StrainE(1,i)=StrainE(1,i-1); StrainP(1,i-1)= Epsilon(1,i)-StrainE(1,i); else StrainE(1,i)=Epsilon(1,i)-StrainP(1,i-1); end % Cauchy Stress in Elastic component sigmahyper(1,i) =a*(((2*(1+StrainE(1,i))^2*(1(1+StrainE(1,i))^-3))))+b*(((2*(1+StrainE(1,i))*(1-(1+StrainE(1,i))^3))))+c*(((2*(1+StrainE(1,i))*(1-(1+StrainE(1,i))^3)*(((1+StrainE(1,i))^2+2*(1+StrainE(1,i))^-1)3+(1+StrainE(1,i))*((1+StrainE(1,i))^-2+2*(1+StrainE(1,i))-3))))); % Cauchy Stress in viscoelastic component StrainV(1,i)=StrainE(1,i); StrainVe(1,i)=StrainE(1,i)-StrainVi(1,i-1); sigmaviscohyper(1,i) =springrelation*(a*(((2*(1+StrainVe(1,i))^2*(1-(1+StrainVe(1,i))^3))))+b*(((2*(1+StrainVe(1,i))*(1-(1+StrainVe(1,i))^3))))+c*(((2*(1+StrainVe(1,i))*(1-(1+StrainVe(1,i))^3)*(((1+StrainVe(1,i))^2+2*(1+StrainVe(1,i))^-1)3+(1+StrainVe(1,i))*((1+StrainVe(1,i))^-2+2*(1+StrainVe(1,i))3)))))); StrainVidelta(1,i)=sigmaviscohyper(1,i)/miuV*timedeltaNew; StrainVi(1,i)=StrainVi(1,i-1)+StrainVidelta(1,i); if StrainVi(1,i) > StrainE(1,i) StrainVi(1,i) = StrainE(1,i); end % Total Cauchy Stress Totalstress(1,i)=sigmaviscohyper(1,i)+sigmahyper(1,i); % Flow rule in Viscoplastic Component isoyieldstress(1,i)=yieldstress+Kiso*StrainP(1,i-1); 116 if Totalstress(1,i) >= isoyieldstress(1,i) StrainPdelta(1,i)=(Totalstress(1,i)isoyieldstress(1,i))/miuP*timedeltaNew; StrainP(1,i)=StrainP(1,i-1)+StrainPdelta(1,i); else StrainP(1,i)=StrainP(1,i-1); end Strain=Epsilon(1,i); end figure(1) subplot (2,2,1) plot( Epsilon, Totalstress, '.r') title ('Totalstress') hold on subplot (2,2,2) plot( Epsilon, sigmaviscohyper, '.r') title ('sigmaviscohyper') hold on subplot (2,2,2) plot( Epsilon, sigmahyper, '.k') title ('Totalstress,sigmahyper') subplot (2,2,3) plot( Epsilon, StrainVi, '.r') title ('StrainVi') hold on subplot (2,2,3) plot( Epsilon, StrainVe, '.r') title ('StrainVe') hold on subplot (2,2,3) plot( Epsilon, StrainE, '.b') title ('StrainVe,StrainVi,StrainE') hold on subplot (2,2,4) plot( Epsilon, StrainP, '.r') title ('StrainP') hold on subplot (2,2,4) 117 plot( Epsilon, Epsilon, '.k') title ('StrainP,Epsilon') hold on subplot (2,2,4) plot( Epsilon, StrainE, '.b') title ('StrainE,StrainP,Epsilon') hold on xlswrite('1dd.xls', StrainE','aaa','b4') xlswrite('1dd.xls', StrainVe','aaa','e4') xlswrite('1dd.xls', StrainVi','aaa','h4') xlswrite('1dd.xls', StrainP','aaa','k4') xlswrite('1dd.xls', Epsilon','aaa','n4') xlswrite('1dd.xls', sigmahyper','aaa','q4') xlswrite('1dd.xls', sigmaviscohyper','aaa','t4') xlswrite('1dd.xls', Totalstress','aaa','w4') xlswrite('1dd.xls', SrateNew','aaa','y4') xlswrite('1dd.xls', isoyieldstress','aaa','aa4') xlswrite('1dd.xls', totaltime','aaa','Ac4') 118 D. Time Integration Procedure of Writing a User-defined VUMAT Code for The Proposed Elastic-Viscoelastic-Viscoplastic Model Step 1: Importing the material parameters and model coefficients at room temperature , , ϑ, , ρ, , , , . , Step 2: At initial time step Step 3: At time Inputs by ABAQUS , Recalling the stored data from the last time step . , , , , , , Updating the material parameters and model coefficients at current temperature. , , , , Step 4: If movement of friction slider was not triggered at the last time step, go to Step 6. Unless Step 5: If i.e. overflow in viscous element of viscoplastic component 119 Correction of Correcting the hardening function and inelastic work Step 6: Stresses in Elastic and Viscoelastic components Stress data are saved to transfer to ABAQUS at the end of VUMAT code. , Eq. (4.56) Step 7: Flow in viscous element of viscoelastic component If i.e. overflow in viscous element of viscoplastic component 120 (getting back to Step 6 and recalculation is recommended) Step 8: Calculating the hardening function Step 9: Calculating the driving force Movement of friction slider in plastic component If movement of friction slider has not been triggered, go to Step 12. Unless Step 10: Flow in viscous element and friction slider of viscoplastic component 121 Step 11: Updating the hardening function and inelastic work Step 12: Sending data to ABAQUS, Total Cauchy stress The inelastic work data taken from the last time step 122 [...]... different strain rates 60 Fig 4.4 Schematic representation of the right and left polar decompositions of (Wikipedia, Sep 2012) 61 Fig 5.1 Compressive and tensile responses of Nylon 6 at low and high strain rates 80 Fig 5.2 Identification of yield stress marking onset of plastic deformation, for compression and tension 81 Fig 5.3 Comparison between test data and fit of. .. response of semi-crystalline polymer, Nylon 6, is studied Material samples are subjected to quasi- static compression and tension at room and higher temperatures using an Instron universal testing machine High strain rate compressive and tensile deformation are also applied to material specimens using Split Hopkinson Bar devices Specimen deformation and temperature change during high rate deformation are... engineering strain at 52 Fig 3.25 (a) Nylon 6 specimens dynamic tension (b) Specimen connection to input/output bars High speed photographic images of dynamic deformation: (c) before loading; (d) tensile engineering strain at after commencement of loading at a strain rate of 53 Fig 3.26 Response of material under dynamic tension at high strain rates 53 Fig 3.27 Constant strain rates. .. response of the incompressible semi-crystalline material, Nylon 6, subjected to compressive and tensile quasi- static and high rate deformation Chapter 5 describes the implementation of the three-dimensional constitutive model in an FEM software (ABAQUS) via a user-defined material subroutine (VUMAT) The model is calibrated and validated by compressive and tensile tests conducted at different temperatures and. .. stiffnesses at high strain rates 2.1 Literature Review 2.1.1 Hyperelastic Material A Cauchy-elastic material is one in which the Cauchy stress at each material point is determined in the current state of deformation, and a hyperelastic material is a special case of a Cauchy-elastic material For many materials, linear elastic models do 8 not fully describe the observed material behaviour accurately, and hyperelasticity... dissipation of kinetic energy associated with impacts and shocks The dynamic mechanical properties of polymers are of considerable interest and attention, because many products and components are subjected to impacts and shocks and need to accommodate these and the energies involved Effective application of polymers requires a good understanding of their thermo-mechanical response over a wide range of deformation, ... temperature; =298 K; (b) different temperatures at low strain rates; (c) different temperatures at high strain rates 87 Fig 5.8 Comparison between temperature variations predicted by the single-element model and experimental data 88 Fig 5.9 Comparison between temperature profile predicted by ABAQUS (multielement model) and thermo-graphic infrared camera images, (a) temperature increase at. .. understanding of their thermo-mechanical response over a wide range of deformation, loading rates and temperature Therefore, analysis and modelling of the quasi- static and dynamic behaviour of polymers are essential, and will facilitate the use of computer simulation for designing products that incorporate polymeric padding and components The present research effort undertaken is described according... temperatures and deformation rates (Chapter 3) Material parameters, such as the stiffness coefficients, viscosity and hardening, are cast as functions of temperature, as well as degree and rate of deformation Simplicity of these material parameters is sought to preclude involvement in the details of the molecular structures of polymers 5 CHAPTER 2 2 Modelling of Elastomers The molecular structure of elastomers... specimens before and after quasi- static compression 37 Fig 3.3 Response of three specimens to quasi- static compression 38 Fig 3.4 Quasi- static compressive linear elastic response of four specimens 38 Fig 3.5 ASTM D638-Type V specimen dimension 39 Fig 3.6 Specimen before and after quasi- static tension 40 Fig 3.7 Quasi- static linear tensile elastic response of three specimens