Rheology Processing 1

Rheology and Processingof Polymeric Materials

Volume 1

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RHEOLOGY AND PROCESSING

OF POLYMERIC MATERIALS

Volume 1Polymer Rheology

Chang Dae Han

Department of Polymer EngineeringThe University of Akron

2007

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Library of Congress Cataloging-in-Publication DataHan, Chang Dae.Rheology and processing of polymeric materials/Chang Dae Han.

v. cm.

Contents: v. 1 Polymer rheology; v. 2 Polymer processingIncludes bibliographical references and index.ISBN: 978-0-19-518782-3 (vol. 1); 978-0-19-518783-0 (vol. 2)

1. PolymersRheology. 1. Title.

QC189.5.H36 2006620.1920423dc22 2005036608

9 8 7 6 5 4 3 2 1Printed in the United States of Americaon acid-free paper

In Memory of My Parents

Preface

In the past, a number of textbooks and research monographs dealing with polymerrheology and polymer processing have been published. In the books that dealt withrheology, the authors, with a few exceptions, put emphasis on the continuum descrip-tion of homogeneous polymeric uids, while many industrially important polymericuids are heterogeneous, multicomponent, and/or multiphase in nature. The continuumtheory, though very useful in many instances, cannot describe the effects of molecularparameters on the rheological behavior of polymeric uids. On the other hand, thecurrently held molecular theory deals almost exclusively with homogeneous polymericuids, while there are many industrially important polymeric uids (e.g., block copoly-mers, liquid-crystalline polymers, and thermoplastic polyurethanes) that are composedof more than one component exhibiting complex morphologies during ow.

In the books that dealt with polymer processing, most of the authors placedemphasis on showing how to solve the equations of momentum and heat transportduring the ow of homogeneous thermoplastic polymers in a relatively simple owgeometry. In industrial polymer processing operations, more often than not, multi-component and/or multiphase heterogeneous polymeric materials are used. Suchmaterials include microphase-separated block copolymers, liquid-crystalline polymershaving mesophase, immiscible polymer blends, highly lled polymers, organoclaynanocomposites, and thermoplastic foams. Thus an understanding of the rheology ofhomogeneous (neat) thermoplastic polymers is of little help to control various process-ing operations of heterogeneous polymeric materials. For this, one must understand therheological behavior of each of those heterogeneous polymeric materials.

There is another very important class of polymeric materials, which are referredto thermosets. Such materials have been used for the past several decades for thefabrication of various products. Processing of thermosets requires an understandingof the rheological behavior during processing, during which low-molecular-weightoligomers (e.g., unsaturated polyester, urethanes, epoxy resins) having the molecular

viii PREFACE

weight of the order of a few thousands undergo chemical reactions ultimately givingrise to cross-linked networks. Thus, a better understanding of chemorheology is vitallyimportant to control the processing of thermosets. There are some books that dealt withthe chemorheology of thermosets, or processing of some thermosets. But, very few,if any, dealt with the processing of thermosets with chemorheology in a systematicfashion.

The preceding observations have motivated me to prepare this two-volume researchmonograph. Volume 1 aims to present the recent developments in polymer rheology,placing emphasis on the rheological behavior of structured polymeric uids. In sodoing, I rst present the fundamental principles of the rheology of polymeric uids:(1) the kinematics and stresses of deformable bodies, (2) the continuum theory for theviscoelasticity of exible homogeneous polymeric liquids, (3) the molecular theory forthe viscoelasticity of exible homogeneous polymeric liquids, and (4) experimentalmethods for measurement of the rheological properties of polymeric liquids. Thematerials presented are intended to set a stage for the subsequent chapters by intro-ducing the basic concepts and principles of rheology, from both phenomenological andmolecular perspectives, of structurally simple exible and homogeneous polymericliquids.

Next, I present the rheological behavior of various polymeric materials. Sincethere are so many polymeric materials, I had to make a conscious, though some-what arbitrary, decision on the selection of the polymeric materials to be covered inthis volume. Admittedly, the selection has been made on the basis of my researchactivities during the past three decades, since I am quite familiar with the subjects cov-ered. Specically, the various polymeric materials considered in Volume 1 range fromrheologically simple, exible thermoplastic homopolymers to rheologically complexpolymeric materials including (1) block copolymers, (2) liquid-crystalline polymers,(3) thermoplastic polyurethanes, (4) immiscible polymer blends, (5) particulate-lledpolymers, organoclay nanocomposites, and ber-reinforced thermoplastic composites,and (6) molten polymers with solubilized gaseous component. Also, chemorheology isincluded in Volume 1.

Volume 2 aims to present the fundamental principles related to polymer processingoperations. In presenting the materials in this volume, again, the objective was notto provide the recipes that necessarily guarantee better product quality. Rather, I putemphasis on presenting fundamental approach to effectively analyze processing prob-lems. Polymer processing operations require combined knowledge of polymer rheology,polymer solution thermodynamics, mass transfer, heat transfer, and equipment design.Specically, in Volume 2, I have presented the fundamental aspects of several pro-cessing operations (plasticating single-screw extrusion, wire coating extrusion, berspinning, tubular lm blowing, injection molding, coextrusion, and foam extrusion)of thermoplastic polymers and three processing operations (reaction injection molding,pultrusion, and compression molding) of thermosets. In presenting Volume 2, I haveused some materials presented in Volume 1.

In the preparation of this monograph, I have tried to present the fundamentalconcepts and/or principles associated with the rheology and processing of the variouspolymeric materials selected and I have tried to avoid presenting technological recipes.In so doing, I have pointed out an urgent need for further experimental and theoreticalinvestigations. I sincerely hope that the materials presented in this monograph will not

PREFACE ix

only encourage further experimental investigations but also stimulate future develop-ment of theory. I wish to point out that I have tried not to cite articles appearing inconference proceedings and patents unless absolutely essential, because they did notgo through rigorous peer review processes.

Much of the material presented in this monograph is based on my research activitieswith very capable graduate students at Polytechnic University from 1967 to 1992 and atthe University of Akron from 1993 to 2005. Without their participation and dedicationto the various research projects that I initiated, the completion of this monograph wouldnot have been possible. I would like to acknowledge with gratitude that Professor JinKon Kim at Pohang University of Science and Technology in Korea read the draft ofChapters 4, 6, 7, and 8 of Volume 1 and made very valuable comments and suggestionsfor improvement. Professor Ralph H. Colby at Pennsylvania State University read thedraft of Chapter 7 of Volume 1 and made helpful comments and suggestions, for whichI am very grateful. Professor Anthony J. McHugh at Lehigh University read the draftof Chapter 6 of Volume 2 and made many useful comments, for which I am verygrateful. It is my special privilege to acknowledge the wonderful collaboration I hadwith Professor Takeji Hashimoto at Kyoto University in Japan for the past 18 yearson phase transitions and phase behavior of block copolymers. The collaboration hasenabled me to add luster to Chapter 8 of Volume 1. The collaboration was very genuineand highly professional. Such a long collaboration was made possible by mutual respectand admiration.

Chang Dae HanThe University of Akron

Akron, OhioJune, 2005

Contents

Remarks on Volume 1, xix

1 Relationships Between Polymer Rheology and Polymer Processing, 3

1.1 What Is Polymer Rheology?, 31.2 How Does the Fluid Elasticity of Polymeric Liquids Manifest

Itself in Flow?, 41.3 Shear-Thinning Behavior of Viscosity of Polymeric Liquids, 71.4 Processing Characteristics of Polymeric Materials, 81.5 Application of Polymer Rheology for On-Line Control

of Polymerization Reactors, 10References, 11

Part I

Fundamental Principles of Polymer Rheology

2 Kinematics and Stresses of Deformable Bodies, 15

2.1 Introduction, 152.2 Description of Motion, 162.3 Some Representative Flow Fields, 18

2.3.1 Steady-State Shear Flow Field, 182.3.2 Steady-State Elongational Flow Field, 19

xii CONTENTS

2.4 Deformation Gradient Tensor, Strain Tensor, Velocity Gradient Tensorand Rate-of-Strain Tensor, 202.4.1 Deformation Gradient Tensor, 202.4.2 Strain Tensor, 222.4.3 Velocity Gradient Tensor and Rate-of-Strain Tensor, 25

2.5 Kinematics in Moving (Convected) Coordinates, 292.5.1 Convected Strain Tensor, 302.5.2 Time Derivative of Convected Coordinates, 32

2.6 The Description of Stress and Material Functions, 35Appendix 2A: Properties of Second-Order Tensors, 38

Invariants, 38Principal Values and Principal Directions, 39The Polar Decomposition Theorem, 40

Appendix 2B: Tensor Calculus, 41Curvilinear Coordinates and Metric Tensors, 41Time Derivatives of Second-Order Tensors, 42

Problems, 45Notes, 48References, 48

3 Continuum Theories for the Viscoelasticity of Flexible HomogeneousPolymeric Liquids, 50

3.1 Introduction, 503.2 Differential-Type Constitutive Equations for Viscoelastic Fluids, 51

3.2.1 Single-Mode Differential-Type Constitutive Equations, 513.2.2 Multimode Differential-Type Constitutive Equations, 58

3.3 Integral-Type Constitutive Equations for Viscoelastic Fluids, 603.4 Rate-Type Constitutive Equations for Viscoelastic Fluids, 643.5 Predicted Material Functions and Experimental Observations, 66

3.5.1 Material Functions for Steady-State Shear Flow, 663.5.2 Material Functions for Oscillatory Shear Flow, 723.5.3 Material Functions for Steady-State Elongational Flow, 76

3.6 Summary, 80Appendix 3A: Derivation of Equation (3.5), 81Appendix 3B: Derivation of Equation (3.16), 82Appendix 3C: Derivation of Equation (3.29), 83Appendix 3D: CayleyHamilton Theorem, 83Appendix 3E: Derivation of Equation (3.97), 84Appendix 3F: Derivation of Equation (3.103), 85Problems, 86Notes, 88References, 90

4 Molecular Theories for the Viscoelasticity of Flexible HomogeneousPolymeric Liquids, 91

4.1 Introduction, 91

CONTENTS xiii

4.2 Static Properties of Macromolecules and Stochastic Processes in theMotion of Macromolecular Chains, 934.2.1 Static Properties of Macromolecules, 944.2.2 Stochastic Processes in the Motion of Macromolecular

Chains, 974.3 Molecular Theory for the Viscoelasticity of Dilute Polymer Solutions

and Unentangled Polymer Melts, 1024.3.1 The Rouse Model, 1034.3.2 The Zimm Model, 1064.3.3 Prediction of Rheological Properties, 109

4.4 Molecular Theory for the Viscoelasticity of Concentrated PolymerSolutions and Entangled Polymer Melts, 1124.4.1 Reptation Mechanism and the Tube Model, 1154.4.2 The Dynamics of a Primitive Chain, 1174.4.3 Contour Length Fluctuation and Constraint Release

Mechanism, 1204.4.4 Constitutive Equations of State, 1254.4.5 Comparison of Prediction with Experiment, 131

4.5 Summary, 142Appendix 4A: Derivation of Equation (4.6), 143Appendix 4B: Derivation of Equation (4.71), 145Problems, 146Notes, 147References, 151

5 Experimental Methods for Measurement of the RheologicalProperties of Polymeric Fluids, 153

5.1 Introduction, 1535.2 Cone-and-Plate Rheometry, 154

5.2.1 Steady-State Shear Flow Measurement, 1545.2.2 Oscillatory Shear Flow Measurement, 160

5.3 Capillary and Slit Rheometry, 1635.3.1 Plunger-Type Capillary Rheometry, 1635.3.2 Continuous-Flow Capillary Rheometry, 1665.3.3 Slit Rheometry, 1735.3.4 Critical Assessment of Capillary and Slit Rheometry, 1805.3.5 Viscous Shear Heating in a Cylindrical or Slit Die, 188

5.4 Elongational Rheometry, 1895.5 Summary, 193


xiv CONTENTS

Part II

Rheological Behavior of Polymeric Materials

6 Rheology of Flexible Homopolymers, 203

6.1 Introduction, 2036.2 Rheology of Linear Flexible Homopolymers, 204

6.2.1 Temperature Dependence of Steady-State Shear Viscosityof Linear Flexible Homopolymers, 204

6.2.2 Temperature Dependence of Relaxation Time and First NormalStress Difference in Steady-State Shear Flow of Linear FlexibleHomopolymers, 210

6.2.3 Temperature-Independent Correlations for the Linear DynamicViscoelastic Properties of Linear FlexibleHomopolymers, 213

6.2.4 Effects of Molecular Weight and Molecular Weight Distributionon the Rheological Behavior of Linear FlexibleHomopolymers, 219

6.3 Rheology of Flexible Homopolymers with Long-ChainBranching, 233

6.4 Summary, 241Problems, 241Notes, 243References, 244

7 Rheology of Miscible Polymer Blends, 247

7.1 Introduction, 2477.2 Phase Behavior of Polymer Blend Systems, 2487.3 Experimental Observations of the Rheological Behavior of Miscible

Polymer Blends, 2527.3.1 TimeTemperature Superposition in Miscible Polymer

Blends, 2527.3.2 Rheology of Polymer Blends Exhibiting UCST, 2617.3.3 Rheology of Polymer Blends Exhibiting LCST, 269

7.4 Molecular Theory for the Linear Viscoelasticity of Miscible PolymerBlends and Comparison with Experiment, 2737.4.1 Linear Viscoelasticity Theory for Miscible

Polymer Blends, 2747.4.2 Comparison of Theory with Experiment, 279

7.5 Plateau Modulus of Miscible Polymer Blends, 2867.6 Summary, 288


CONTENTS xv

8 Rheology of Block Copolymers, 296

8.1 Introduction, 2968.2 Oscillatory Shear Rheometry of Microphase-Separated Block

Copolymers Exhibiting Upper Critical OrderDisorder TransitionBehavior, 3018.2.1 Oscillatory Shear Rheometry of Symmetric or Nearly Symmetric

Block Copolymers, 3028.2.2 Oscillatory Shear Rheometry of Highly Asymmetric Block

Copolymers, 3068.2.3 Effect of Thermal History on the Oscillatory Shear Rheometry

of Block Copolymers, 3198.3 Oscillatory Shear Rheometry of Microphase-Separated Block

Copolymers Exhibiting Lower Critical DisorderOrderTransition Behavior, 327

8.4 Linear Viscoelasticity of Disordered Block Copolymers, 3318.4.1 Effect of Molecular Weight on the Zero-Shear Viscosity

of Disordered Diblock Copolymers, 3328.4.2 Effect of Block Length Ratio on the Linear Dynamic

Viscoelasticity of Disordered Block Copolymers, 3378.4.3 Molecular Theory for the Linear Viscoelasticity of Disordered

Block Copolymers, 3458.5 Stress Relaxation Modulus of Microphase-Separated Block Copolymer

Upon Application of Step-Shear Strain, 3558.6 Steady-State Shear Viscosity of Microphase-Separated Block

Copolymers, 3598.7 Summary, 363

Notes, 364References, 365

9 Rheology of Liquid-Crystalline Polymers, 369

9.1 Introduction, 3699.2 Theory for the Rheology of LCPs, 379

9.2.1 Theory for Rigid Rodlike Macromolecules withMonodomains, 379

9.2.2 Theory for Rigid Rodlike Macromolecules withPolydomains, 394

9.3 Rheological Behavior of Lyotropic LCPs, 4009.4 Rheological Behavior of Thermotropic Main-Chain LCPs, 406

9.4.1 Effect of Thermal History on the Rheological Behavior ofThermotropic Main-Chain LCPs, 406

9.4.2 Transient Shear Flow of Thermotropic Main-Chain LCPs, 4139.4.3 Flow Aligning Behavior of Thermotropic

Main-Chain LCPs, 4249.4.4 Intermittent Shear Flow of Thermotropic Main-Chain LCPs, 4269.4.5 Evolution of Dynamic Moduli of Thermotropic Main-Chain

LCPs Upon Cessation of Shear Flow, 428

xvi CONTENTS

9.4.6 Effect of Preshearing of Thermotropic Main-Chain LCPs onthe Rheological Behavior, 430

9.4.7 Reversal Flow of Thermotropic Main-Chain LCPs, 4339.4.8 Effect of Molecular Weight on the Rheological Behavior of

Thermotropic Main-Chain LCPs, 4359.4.9 Effect of Bulkiness of Pendent Side Groups on the Rheo-

Optical Behavior of Thermotropic Main-Chain LCPs, 4419.5 Rheological Behavior of Thermotropic Side-Chain LCPs, 4449.6 Summary, 451

Appendix 9A: Derivation of Equation (9.3), 454Appendix 9B: Derivation of Equation (9.11), 455Appendix 9C: Derivation of Equation (9.15), 457Appendix 9D: Derivation of Equation (9.23), 458Appendix 9E: Derivation of Equation (9.28), 460Appendix 9F: Derivation of Equation (9.30), 461Appendix 9G: Derivation of Equation (9.49), 462Appendix 9H: Derivation of Equation (9.50), 463Notes, 464References, 465

10 Rheology of Thermoplastic Polyurethanes, 470

10.1 Introduction, 47010.2 Effect of Thermal History on the Rheological Behavior

of TPUs, 47410.2.1 Time Evolution of Dynamic Moduli of TPU during Isothermal

Annealing, 47410.2.2 Thermal Transitions in TPU during Isothermal

Annealing, 47710.2.3 Hydrogen Bonding in TPU during IsothermalAnnealing, 479

10.3 Linear Dynamic Viscoelasticity of TPUs, 48410.3.1 Frequency Dependence of Dynamic Moduli of TPU

under Isothermal Conditions, 48410.3.2 Temperature Dependence of Dynamic Moduli of TPU during

Isochronal Dynamic Temperature Sweep Experiment, 48610.4 Steady-State Shear Viscosity of TPU, 48810.5 Summary, 490

References, 491

11 Rheology of Immiscible Polymer Blends, 493

11.1 Introduction, 49311.2 Experimental Observations of RheologyMorphology Relationships

in Immiscible Polymer Blends, 49511.2.1 Effect of Flow Geometry on the Steady-State Shear Viscosity

and Morphology of Immiscible Polymer Blends, 49511.2.2 Effect of Blend Composition on the Steady-State Shear Flow

Properties of Immiscible Polymer Blends, 504

CONTENTS xvii

11.2.3 Linear Dynamic Viscoelastic Properties of ImmisciblePolymer Blends, 511

11.2.4 Extrudate Swell of Immiscible Polymer Blends, 51211.3 Consideration of Large Drop Deformation and Bulk Rheological

Properties of Immiscible Polymer Blends in Pressure-DrivenFlow, 51911.3.1 Finite Element Analysis of Large Drop Deformation in the

Entrance Region of a Cylindrical Tube, 52411.3.2 Theoretical Approach to the Prediction of Rheology

MorphologyProcessing Relationships in Pressure-DrivenFlow of Immiscible Polymer Blends, 536


12 Rheology of Particulate-Filled Polymers, Nanocomposites, andFiber-Reinforced Thermoplastic Composites, 547

12.1 Introduction, 54712.2 Rheology of Particulate-Filled Polymers, 548

12.2.1 Rheology of Particulate-Filled Molten Thermoplasticsand Elastomers, 549

12.2.2 Rheology of Molten Thermoplastics with Chemically TreatedFillers, 559

12.2.3 Theoretical Consideration of the Rheology ofParticulate-Filled Polymers, 565

12.3 Rheology of Nanocomposites, 56912.3.1 Rheology of Organoclay Nanocomposites Based on

Thermoplastic Polymer, 57512.3.2 Rheology of Organoclay Nanocomposites Based on Block

Copolymer, 58312.3.3 Rheology of Organoclay Nanocomposites Based on

End-Functionalized Polymer, 59312.4 Rheology of Fiber-Reinforced Thermoplastic Composites, 603

12.4.1 Theoretical Consideration of Fiber Orientation in Flow, 60312.4.2 Experimental Observations, 609

12.5 Summary, 614Appendix 12A: Derivation of Equation (12.19), 615Appendix 12B: Derivation of Three Material Functions for

Steady-State Shear Flow from Equation (12.30), 616Problems, 617Notes, 618References, 620

xviii CONTENTS

13 Rheology of Molten Polymers with Solubilized GaseousComponent, 623

13.1 Introduction, 62313.2 Rheological Behavior of Molten Polymers with Solubilized

Gaseous Component, 62413.2.1 Experimental Methods for Rheological Measurements of

Molten Polymers with Solubilized Gaseous Component, 62413.2.2 Experimental Observations of Reduction in Melt Viscosity by

Solubilized Gaseous Component, 62913.3 Theoretical Consideration of Reduction in Melt Viscosity

by Solubilized Gaseous Component, 63913.3.1 Depression of Glass Transition Temperature of Amorphous

Polymer by the Addition of Low-Molecular-Weight SolubleDiluent, 639

13.3.2 Depression of Melting Point of Semicrystalline Polymer bythe Addition of Low-Molecular-Weight Soluble Diluent, 641

13.3.3 Theoretical Interpretation of Reduction in Melt Viscosity bySolubilized Gaseous Component, 641


14 Chemorheology of Thermosets, 651

14.1 Introduction, 65114.2 Chemorheology of Unsaturated Polyester, 656

14.2.1 Viscosity Rise during Cure of Neat UnsaturatedPolyester, 658

14.2.2 Chemorheological Model for Neat UnsaturatedPolyester, 660

14.2.3 Cure Kinetics of Neat Unsaturated Polyester, 66414.2.4 Effects of Particulates on the Chemorheology of Unsaturated

Polyester, 67314.2.5 Effects of Low-Prole Additive on the Chemorheology

of Unsaturated Polyester, 67714.2.6 Oscillatory Shear Flow during Cure of Unsaturated

Polyester, 68214.3 Chemorheology of Epoxy Resin, 68314.4 Chemorheology of Thermosetting Polyurethane, 68814.5 Summary, 691


Author Index, 695

Subject Index, 704

Remarks on Volume 1

This volume consists of two parts. Part I describes the fundamental principles ofthe rheology of polymeric uids: (1) the kinematics and stresses of deformablebodies, (2) the continuum theories for the viscoelasticity of exible homogeneouspolymeric liquids, (3) the molecular theories for the viscoelasticity of exible homo-geneous polymeric liquids, and (4) experimental methods for measurement of therheological properties of polymeric liquids. Part I is intended to set a stage for thesubsequent chapters by introducing the basic concepts and principles of rheology,from both phenomenological and molecular perspectives, of structurally simple exi-ble and homogeneous polymeric liquids. Part II describes the rheology of variouspolymeric materials, ranging from exible ordinary thermoplastic homopolymers tothermosets, namely, (1) homopolymers, (2) miscible polymer blends, (3) block copoly-mers, (4) liquid-crystalline polymers, (5) thermoplastic polyurethanes, (6) immisciblepolymer blends, (7) particulate-lled polymers, organoclay nanocomposites, and ber-reinforced thermoplastic composites, (8) molten polymers with solubilized gaseouscomponent, and (9) thermosets. In presenting the materials in Part II, I have pointedout an urgent need for further experimental and theoretical investigations. I sincerelyhope that the materials presented in Part II will not only encourage further experimentalinvestigations, but also stimulate future development of theory.

C.D.H.

Rheology and Processingof Polymeric Materials

Volume 1

1Relationships Between PolymerRheology and Polymer Processing

Polymer products have long been used for a variety of applications in our daily lives,as well as for some more exotic applications, such as biomedical devices, super-high-speed airplanes, and outer-space vehicles. Other applications are too numerous tomention them all here. There are many steps involved in the production of polymerproducts, from the synthesis of raw materials to the manufacturing of the nished prod-ucts. Of the many steps involved, the fabrication (processing) step plays a pivotal rolein determining the quality of the nal products. Successful processing of polymericmaterials requires a good understanding of their rheological behavior (Han 1976, 1981).Thus, intimate relationships exist between polymer rheology and polymer processing.In this chapter we describe briey some of these close relationships between polymerrheology and polymer processing.

1.1 What Is Polymer Rheology?

Rheology is the science that deals with the deformation and ow of matter. Hence,polymer rheology is the science that deals with the deformation and ow of poly-meric materials. Since there are a variety of polymeric materials, we can classifypolymer rheology further into different categories, depending upon the nature of thepolymeric materials; for instance, (1) the rheology of homogeneous polymers, (2) therheology of miscible polymer blends, (3) the rheology of immiscible polymer blends,(4) the rheology of particulate-lled polymers, (5) the rheology of berglass-reinforcedpolymers, (6) the rheology of organoclay nanocomposites, (7) the rheology of poly-meric foams, (8) the rheology of thermosets, (9) the rheology of block copolymers,

3

4 RHEOLOGY AND PROCESSING OF POLYMERIC MATERIALS

and (10) the rheology of liquid-crystalline polymers. Each of these polymeric materialsexhibits its own unique rheological characteristics. Thus, different theories are neededto interpret the experimental results of the rheological behavior of different polymericmaterials. However, at present we do not have a comprehensive theory that can describethe rheological behavior of some polymeric materials and thus we must resort to empir-ical correlations to interpret the experimentally observed rheological behavior of thosematerials. It is then fair to state that a complete understanding of the rheologicalbehavior of all polymeric materials remains quite a challenge indeed.

Most of the polymeric materials of practical use exhibit viscoelastic behaviorduring ow, meaning that they exhibit not only viscous behavior but also elastic(rubberlike) behavior in the liquid state. There are several different ways of describingthe uid elasticity of polymeric materials, and this subject is dealt with in Chapters 3and 4 from a theoretical point of view and in Chapter 5 from an experimental pointof view. The viscosity of a polymer is proportional to its molecular weight (M) whenit is lower than a certain critical value (Mc), but the shear viscosity is proportional tothe 3.4-th power of its M when M Mc (Berry and Fox 1968). The polymer havingM < Mc is referred to as unentangled polymer, and the polymer having M Mc isreferred to as entangled polymer. It is well established that entangled polymers arehighly viscoelastic, while unentangled polymers are not (Ferry 1980).

The rheological properties of polymeric materials vary with their chemicalstructures. Therefore, it is highly desirable to be able to relate the rheological propertiesof polymeric materials to their chemical structures. For instance, one may ask: Whyare the rheological properties of polystyrene so different from the rheological proper-ties of polyethylene under an identical ow condition? Unfortunately, at present thereis no comprehensive molecular theory that can answer such a seemingly simple andfundamental question.

There are some molecular theories that can explain the effects of molecular weightand molecular weight distribution on the rheological properties of exible homopoly-mers (Rouse 1953; Doi and Edwards 1986). This subject is discussed in Chapter 4.However, some polymers are heterogeneous when they are polymerized (e.g., blockcopolymers, liquid-crystalline polymers), exhibiting two phases in the liquid state.Also, one often prepares heterogeneous polymeric materials by mixing a homogenouspolymer with other components (e.g., particulate llers, chemically modied clay,glass bers, and carbon black). The rheological behavior of such polymeric materialsis quite different from that of homogenous polymers. In several chapters of Volume 1,we discuss the rheological behavior of heterogeneous polymeric materials.

1.2 How Does the Fluid Elasticity of Polymeric Liquids ManifestItself in Flow?

There are several ways of demonstrating, experimentally, that polymeric uids exhibitelastic characteristics. One very well known experimental observation is the behaviorof liquid climb-up on a rotating rod in a polymer solution. Figure 1.1 demonstratesa dramatic difference in the behavior of liquid climb-up on a rotating rod between(a) 4 wt % aqueous solution of polyacrylamide and (b) glycerin. It is seen in Figure 1.1that the polyacrylamide solution climbs the rod rotating within it, whereas no climb-up

RELATIONSHIPS BETWEEN POLYMER RHEOLOGY AND POLYMER PROCESSING 5

Figure 1.1 Difference in liquid climb-up behavior on a rotating rod between (a) 4 wt % aqueoussolution of polyacrylamide (viscoelastic uid) and (b) glycerin (Newtonian uid).

of glycerin is seen on the rotating rod. The phenomenon of liquid climb-up is quitecontrary to what one would expect from the effect of centrifugal force (see Figure 1.1b);the faster the rod rotates, the higher the liquid climbs. The phenomenon was rstobserved by Garner and Nissan (1946) and later properly explained by Weissenberg(1947). The question is: What causes the liquid to climb up the rod? It is very importantto notice in Figure 1.1a that the direction of liquid climb-up is perpendicular to therotational ow direction of the liquid. That is, during the rotational ow of a liquidin the beaker, a force is generated in the direction perpendicular to the rotationaldirection. Apparently, such an experimental observation prompted Weissenberg (1949)to design, for the rst time, a cone-and-plate rheometer, which is known today as theWeissenberg rheogoniometer, enabling one to determine rst normal stress difference(N1) in steady-state shear ow of viscoelastic polymeric uids.

To illustrate the point, let us consider the schematic given in Figure 1.2, where auid is placed in the gap between the cone and the plate, and imagine the followingsimple experiment. Namely, place a uid in the cone-and-plate xture and then shearit by rotating the cone at a xed angular speed while the upper plate is held in itsoriginal position. Then, while the uid in the cone-and-plate xture is rotated, try todetermine, via a transducer mounted at the upper plate, if a force F is generated in thedirection perpendicular to the rotational direction of the cone. That is, the measurementof liquid heightL in the climb-up experiment is replaced by the measurement of force Fin the cone-and-plate ow experiment, the principles involved in both experiments

Figure 1.2 Schematic showingthe ow of a test uid placedin the cone-and-plate xture,where force F perpendicular tothe ow direction is measuredas a function of rotationalspeed .


Figure 1.3 Plots of rst normal stressdifference (N1) versus shear rate ( )for 4 wt% aqueous solution ofpolyacrylamide in steady-state shearow at 25 C.

being identical. In Chapter 5 we will elaborate quantitatively on the principle of shearow in the cone-and-plate xture.

Quantitative experimental observation for 4 wt % aqueous solution of polyacry-lamide in the cone-and-plate xture is presented in Figure 1.3, in which the valuesof N1, which is proportional to the force F measured in the cone-and-plate xture(see Figure 1.2), are plotted against the values of shear rate ( ), which is proportionalto the rotational speed of the cone in the cone-and-plate xture. In Chapter 5 wewill present theoretical expressions that relate F to N1, and to . From Figure 1.3we can conclude that the faster the rotational speed of the cone, the larger are thevalues of force F generated during the rotation of the cone. Conversely, no measurableforce F can be detected when glycerin is placed in the cone-and-plate xture, which isconsistent with the absence of liquid climb-up of glycerin (see Figure 1.1b). In otherwords, the origin of the dramatic difference in the liquid climb-up behavior betweenthe 4 wt % aqueous solution of polyacrylamide and glycerin lies in the force F gener-ated for the 4 wt % aqueous solution of polyacrylamide in the direction perpendicular(normal) to the rotational direction of the liquid in the beaker. Such force is referredto as normal force. Today, it is well accepted that the normal force is related to uidelasticity. Thus, we can conclude that it is the elastic property of 4 wt % aqueoussolution of polyacrylamide that gave rise to liquid climb-up on a rotating rod shownin Figure 1.1a, and that glycerin does not exhibit uid elasticity.

Another well-known rheological experiment is illustrated in Figure 1.4, where(a) 4 wt % aqueous solution of polyacrylamide is jetting from a cylindrical tube and(b) glycerin is jetting from the same tube. It is clearly seen in Figure 1.4 that thediameter of liquid jet of 4 wt % aqueous solution of polyacrylamide swells, whereaslittle or no swell of the liquid jet from glycerin can be seen. The swell of 4 wt %aqueous solution of polyacrylamide, upon owing out of a cylindrical tube, is believedto arise from the recovery of the elastic energy that was stored in the liquid while itwas being sheared within the tube. Comparison of Figure 1.4 with Figure 1.1 showsvery clearly that for the same liquid there is a correlation between the swell of a liquidstream and the climb-up on a rotating rod. There are other phenomena (e.g., stressrelaxation, elastic recoil) observed experimentally that demonstrate the unique vis-coelastic characteristics of polymeric uids. This subject is discussed in other chaptersof this book.


Figure 1.4 Liquid jets of (a) 4 wt% aqueous solution of polyacrylamide (viscoelastic uid) and(b) glycerin (Newtonian uid) upon leaving a cylindrical tube.

1.3 Shear-Thinning Behavior of Viscosity of Polymeric Liquids

Polymeric liquids, like other types of liquids, possess viscosity, which is regardedas a measure of the resistance to ow. There is a unique rheological characteristicsof polymeric liquids, not seen in low-molecular-weight ordinary uids, during owin that the resistance to ow (viscosity) through a cylindrical tube decreases as theow rate is increased. This is illustrated in Figure 1.5, in which the viscosity () of4 wt % aqueous solution of polyacrylamide decreases with increasing shear rate ( ),

Figure 1.5 Plots of shear viscosity ()versus shear rate ( ) for () 4 wt%aqueous solution of polyacrylamide and ()glycerin in steady-state shear ow at 25 C.


which is proportional to ow rate, whereas the of glycerin is constant and inde-pendent of . The decreasing trend of with increasing , commonly referred to asshear-thinning behavior, is believed to arise from the stretching of an entangledstate of polymer chains to an oriented state when the applied shear rate is higherthan a certain critical value. Conversely, because glycerin is a small molecule, it can-not possibly have an entangled state and thus no shear-thinning behavior is expectedfrom glycerin. It should be mentioned that the shear-thinning behavior of observedin Figure 1.5 and the shear-rate dependence of N1 observed in Figure 1.3 are only twoof the unique rheological behavior of viscoelastic polymeric liquids. Other unique rhe-ological behavior of viscoelastic polymeric materials is discussed in several chaptersof this volume.

1.4 Processing Characteristics of Polymeric Materials

Figure 1.6 gives a schematic showing the interrelationships that exist between the manysteps that range from the production of a polymer to the physical/mechanical propertiesof the nal polymer products. One must control reactor variables to produce consis-tent quality in a polymer, and thus one needs a polymerization reactor simulator.The polymer produced from the reactor must be characterized in terms of its rheo-logical properties, and thus one needs a rheological property simulator. Since therheological properties of polymers depend on their molecular parameters, it is highlydesirable to relate the rheological properties of a polymer to its molecular parameters,and thus one must understand molecular viscoelasticity theory for polymeric materi-als. Since the rheological behavior of a polymer depends on temperature and pressure,and also on the geometry of a ow device, one needs a polymer processing simu-lator, which is intimately related to the rheological property simulator. It should be

Figure 1.6 Schematic describing intimate interrelationships that exist among the reactorvariables, rheological properties, processing variables, and physical/mechanical properties ofpolymer products.


pointed out that a polymer processing simulator must be based on the momentum andheat transfer equations at a minimum. Depending on the processes involved with thefabrication of a nal product, sometimes the polymer processing simulator requires, inaddition to momentum and heat transfer equations, mass transfer equations and/or reac-tion kinetic expression for reactive systems (including thermosets). Finally, one needsa property evaluation simulator, which evaluates the physical, mechanical and/oroptical properties of the fabricated products.

When the fabricated products do not meet the specications of physical, mechani-cal and/or optical properties, there are two routes that can be pursued further; namely,either modifying the chemical structure of the polymer or modifying (or optimiz-ing) processing conditions. A modication of the chemical structure of a polymerrequires the establishment of a new or revised rheological property simulator and thusa new or revised polymer processing simulator. There are many different fabrica-tion methods (processing techniques) for obtaining polymer products. Examples ofprocessing techniques that are currently used in industry include extrusion, pultrusion,injection molding, compression molding, reaction injection molding, tubular lm blow-ing, blow molding, thermoforming, ber spinning, calendering, and foaming. Needlessto say, each of these processing techniques requires a separate processing simulator.Once again, the rheological property simulator and polymer processing simulator areintimately related to each other.

The ultimate goal of the polymer fabrication industry is to manufacture productsthat meet the requirements for desired physical and/or mechanical properties. The endusers are not interested in knowing how the polymers were synthesized or fabricated.It is the responsibility of polymer scientists and polymer engineers to provide their cus-tomers with nal products that have the desired properties. It is worth mentioning thatthe mechanical/physical properties of a given polymer can vary in different fabricatedproducts, depending upon the processing conditions employed. This is because differentprocessing conditions (e.g., stretching rate in melt spinning or lm blowing, or coolingrate in melt spinning or injection molding) can greatly inuence the molecular orienta-tions, the rate of solidication, and the morphological state of the solidied products,thus affecting their mechanical/physical properties. For a given polymer, understandingthe relationships between processing variables and the mechanical/physical propertiesof fabricated products and relationships between processing variables and the mor-phology of fabricated products is highly desirable. However, at present the details ofsuch relationships are rarely available in the literature. In essence, one must developa criterion (or criteria) for processability for each processing operation; for example,ber spinnability, tubular lm blowability, injection moldability, blow moldability,and thermoformability. Processability criteria are needed to answer a fundamentalquestion: Why is a certain polymer suitable only for producing bers, while anotherpolymer is suitable only for producing bottles? Establishment of such processabilitycriteria is not a trivial task because many factors must be considered: material vari-ables, rheological properties, processing variables, and the morphology associated withthe physical/mechanical properties and the molecular orientation in the nal products.For instance, in melt spinning of a given polymer, bers of different tensile propertiescan be obtained by varying the rate of cooling or the rate of stretching.

After all, processing of polymeric materials requires ow through a shaping device.Thus, a rational design of a shaping device (e.g., die or mold) requires information on


the rheological properties of the polymer to be processed. And, for a given process thedetermination of an optimum processing condition (e.g., a minimum pressure drop)requires information on the temperature dependence of shear viscosity of a polymerto be processed. It is, then, fair to state that polymer rheology is an essential part ofpolymer processing operations.

1.5 Application of Polymer Rheology for On-Line Controlof Polymerization Reactors

In the manufacture of polymers, the control of reactor conditions is of utmost impor-tance for the production of a polymer with consistent quality. There are two methodsthat can be applied to control the reactor conditions. One method is to continu-ously monitor the weight-average molecular weight (Mw) or number-average molecularweight (Mn) and molecular weight distribution (Mw/Mn) of the polymer leaving thereactor and then use the measured quantities to adjust, via a feedback control strategy,reactor variables (e.g., monomer/catalyst ratio, feed rate to the reactor, or reactor tem-perature). However, at present, on-line measurements of Mw or Mn, and Mw/Mn(using gel permeation chromatograpy for instance) are not available. Another methodis to continuously monitor both the viscosity and elasticity of the polymer leaving thereactor and then use the measured quantities to adjust, via a feedback control strategy,reactor variables, as schematically shown in Figure 1.7. For this method, one mustdevelop a rheological property simulator that relates the viscosity and elasticity of agiven polymer to its molecular parameters (Mw or Mn, and Mw/Mn). A rheologicalproperty simulator can be constructed on the basis of empirical correlations when arigorous molecular viscoelasticity theory is not available. On-line measurement of theviscoelastic properties of a polymer can be realized using a capillary or slit rheometer,the principles of which are described in Chapter 5. On-line control of the rheologicalproperties of polymers is far more effective than on-line control of molecular weightand molecular weight distribution of polymers to control later polymer processingoperations. This is because the rheological properties of a polymer dictate the optimumprocessing conditions for a given piece of equipment.

In summary, in Volume 1 of this book we present the fundamental aspects of poly-mer rheology and the rheological behavior of different types of polymeric materials.In so doing, examples will be given that show relationships between the rheologicalproperties and the molecular parameters of specic polymeric materials. In Volume 2

Figure 1.7 Schematic describinghow on-line measurements of therheological properties of theefuent stream from thepolymerization reactor can beused to control the consistencyof polymer quality.


we present the unique processing characteristics of some polymeric materials. In sodoing, we choose several processing operations of thermoplastic polymers and threeprocessing operations of thermosets or thermoset composites. No attempt is made todescribe how to produce products that are better from a commercial point of view.Instead, emphasis is placed on presenting the fundamental concepts and principles, butnot recipes, for each polymer processing operation chosen.

References

Berry GC, Fox RG (1968). Adv. Polym. Sci. 5:261.Doi M, Edwards SF (1986). The Theory of Polymer Dynamics, Oxford University Press,

Oxford.Ferry JD (1980). Viscoelastic Properties of Polymers, 3rd ed, John Wiley & Sons, New York.Garner FH, Nissan AH (1946). Nature (London), 158:634.Han CD (1976). Rheology in Polymer Processing, Academic Press, New York.Han CD (1981). Multiphase Flow in Polymer Processing, Academic Press, New York.Rouse PE (1953). J. Chem. Phys. 21:1272.Weissenberg K (1947). Nature (London), 159:310.Weissenberg K (1949). In Proceedings of the First International Congress on Rheology,

North-Holland, Amsterdam, Netherlands, p II-114.

Part I

Fundamental Principlesof Polymer Rheology

2Kinematics and Stresses of DeformableBodies

2.1 Introduction

The form of kinematics to be used for the description of a deformation process is largelydetermined by the kind of mechanical response that is being described. To describethe mechanical response of purely viscous uids it is convenient to use coordinates,which are xed in space, since purely viscous uids have no past memory and thereforeremain in the deformed state when loads are removed. In other words, the mechanicalresponse of purely viscous uids is determined solely by the instantaneous values ofthe time rate of deformation.

However, in order to describe the deformation of a viscoelastic uid it is necessaryto follow a given material element with time as it moves to dene a suitable measureof deformation that always refers to the same material element as time varies. Thereason is that when a material element undergoes a nite deformation the coordinatepositions of the given material element (with respect to a xed origin) will vary. Hence,any measure of deformation dened in terms of innitesimal deformation of xedcoordinate positions loses its physical signicance since it will not always be associatedwith the same material element.

In this chapter, we introduce some basic concepts of the kinematics and stressesof a deformable body from the point of view of continuum mechanics, and discussvarious representations of a deformation process in terms of the deformation (or strain)tensor and the rate-of-deformation (or rate-of-strain) tensor. In order to help the readersfollow the material in the text, the elementary properties of second-order tensors arepresented in Appendix 2A.

15

16 FUNDAMENTAL PRINCIPLES OF POLYMER RHEOLOGY

2.2 Description of Motion

In this section, we briey describe the motion of a body, which consists of a set ofparticles (or elements), sometimes called material points (or material elements)(Jaunzemis 1967). Let X(Xi ; i = 1, 2, 3) be the particles P of the body B in somereference conguration at time t = 0 (i.e., undeformed state) and then we have

X = (P) (2.1)

in which describes the shape of the body B in the undeformed state, which in generalis known to an observer. When the body B is deformed, the positions of the sameparticles P may be represented by (see Figure 2.1)

x(t) = (P, t) (2.2)

in which x (xi ; i = 1, 2, 3) are the positions of the particle at time t that have con-gurations . Because of the implicit assumption used that the body B is deformable, describes the shape of the body at time t . If we assume that one particle can occupyonly one position at a time, we can combine Eqs. (2.1) and (2.2) to give

x(t) = (X, t) (2.3)

Equation (2.3) states that the positions of the particles P in motion at any instant maybe determined from the information of the positions and conguration of the sameparticles in the undeformed state (t = 0), that is, in the reference conguration .Thus describes the shape of a body at time t in reference to the shape of the samebody in the undeformed state (i.e., in the reference conguration). The coordinatesXi are called the material coordinates, which describe the reference conguration

Figure 2.1 Deformation ofa material element.

KINEMATICS AND STRESSES OF DEFORMABLE BODIES 17

in Eq. (2.1), and the coordinates xi are called the spatial coordinates (Jaunzemis1967).

When dealing with motion, the present instant is usually singled out for specialattention and chosen as the reference conguration. This choice is of particular interestto the description of the motion of nonperfectly elastic materials (e.g., viscoelasticuids). The reason is that viscoelastic materials do not possess perfect memory, andtherefore such materials cannot return to their original (undeformed) state when externalforces are removed. It is then clear that the choice of the undeformed state as a referenceconguration is not convenient for the description of the motion of viscoelastic uids.

When the present conguration is chosen as the reference conguration, particlesare identied with the positions they occupy at time t, therefore from Eq. (2.2) we have

P = 1(x(t), t) (2.4)

Use of Eq. (2.4) in Eq. (2.2) gives

x(t ) = (1(x, t), t ) = t (x, t ) (2.5)where x(x1, x2, x3) are the positions of the particles at time t (< t) and t describesshapes of the body at time t relative to the shape at time t ; in other words, therelative conguration by means of which all other congurations are compared withthe present one. Frequently, one also uses the elapsed time s, dened by s = t t ,where 0 < s < and < t < t . Note further that Eq. (2.5) reduces to the trivialconsequence

x(t) = t (x, t) = x(t) (2.6)

for t = t .There is another way of describing the motion of a body consisting of particles,

which does not require knowledge of the paths of individual particles. In this descrip-tion, called the spatial description, the particle velocity v(t) at time t is consideredas a dependent variable:

v(t) = f(x, t) (2.7)

Note in Eq. (2.7) that x and t are independent variables; that is, in Eq. (2.7) x describesmerely a xed point in space.

The distinction between material and spatial descriptions is clear in that in theformer x(t) is the dependent variable and X and t are the independent variables, whereasin the latter v(t) is the dependent variable and x and t the independent variables.Frequently, the material coordinates are called Lagrangian, and the spatial coordinatesEulerian.

To illustrate the rules described previously, let us consider a motion described by

x1(t ) = X1(1 + t ), x2(t ) = X1t + X2, x3(t ) = X3 (2.8)


The spatial description of this motion may be obtained by rst substituting t = t intoEq. (2.8), yielding

x1(t) = X1(1 + t); x2(t) = X1t + X2; x3(t) = X3 (2.9)

which is of the form of Eq. (2.3), and then by eliminating X1, X2, and X3 withsubstitution of Eq. (2.9) into Eq. (2.8),

x1(t ) = (1 + t)x1(t)

1 + tx2(t ) = t

x1(t)1 + t + x

2(t) tx1(t)

1 + t (2.10)

x3(t ) = x3(t)

which is of the form of Eq. (2.5). Equation (2.10) describes the positions of particlesat time t (< t) relative to the positions of the same particles at present time t . It canbe easily shown that Eq. (2.10) reduces to the identity equations for t = t .

2.3 Some Representative Flow Fields

Here, we consider two important, frequently encountered ow elds: shear ow eldand elongational ow eld. They will be used throughout this chapter and in laterchapters.

2.3.1 Steady-State Shear Flow Field

A simple ow geometry of practical interest is schematically shown in Figure 2.2.It consists of two parallel plates forming a narrow gap whose distance h is very smallcompared with the width w of the plates (i.e., w h). Referring to Figure 2.2a, a uidis placed in the gap between the two parallel plates, and then the upper plate is forced

Figure 2.2 Schematic of shear ow eld for (a) uniform shear ow and (b) nonuniformshear ow.


to move along the z direction while the lower plate is kept stationary. Under suchsituations, the velocity prole vz is linear with respect to the y direction, giving riseto a constant velocity gradient, dvz/dy = constant. Such a ow eld is referred to asa uniform (or simple) shear ow eld. Referring to Figure 2.2b, a uid is forced toow through the gap between two stationary parallel plates. Under such situations, thevelocity prole vz varies with the y direction, giving rise to a parabolic velocity proleand a nonconstant velocity gradient, dvz/dy = f(y). Such a ow eld is referred to asa nonuniform shear ow eld. For steady-state shear ow, the velocity eld for anincompressible uid in Cartesian coordinates (x, y, z) can be expressed as

vz = y, vy = vx = 0 (2.11)

where = dvz/dy is the velocity gradient, commonly referred to as shear rate. Therate-of-strain tensor d for the steady-state shear ow eld can be described by

d =

0 /2 0 /2 0 00 0 0

(2.12)Note that appearing in Eq. (2.12) is constant for uniform shear ow and not constantfor nonuniform shear ow. In Chapter 5 we present experimental methods for thedetermination of the rheological properties of polymeric liquids in the uniform shearow eld using a cone-and-plate rheometer and in the nonuniform shear ow eldusing a capillary or slit rheometer.

2.3.2 Steady-State Elongational Flow Field

Another ow eld that is also of very practical importance is the elongational(or extensional) ow eld, which may be found in such polymer processing operationsas ber spinning, cast-lm extrusion, lm blowing, blow molding, and thermoforming.For uniaxial stretching, the velocity eld v(vx , vy , vz) of an incompressible uid inCartesian coordinates (x, y, z) is given by

dvz/dz = ; dvy/dy = dvx/dx (2.13)

where is the velocity gradient in the direction of stretching z (commonly referred toas the elongation rate), y is the direction perpendicular to the stretching, and x is theneutral direction. In order to satisfy the equation of continuity, we require that

dvzdz

+ dvydy

+ dvxdx

= 0 (2.14)


Using Eq. (2.13) in Eq. (2.14), we have the following expression for the rate-of-straintensor d in uniaxial elongational ow

d =

0 00 /2 00 0 /2

(2.15)Note in Eq. (2.15) that is constant for steady-state uniform, uniaxial elongational owand varies with the stretching direction z for nonuniform, uniaxial elongational ow.In Chapter 5 we present the rheological response of polymeric liquids in steady-stateuniform, uniaxial elongational ow, and in Chapter 6 of Volume 2 we present the rheo-logical response of polymeric liquids in steady-state nonuniform, uniaxial elongationalow that occurs in ber spinning.

For equal biaxial stretching, the rate-of-strain tensor d can be expressed as

d =

B 0 00 B 00 0 2B

(2.16)where B is the elongation rate in equal biaxial stretching and is dened as

B = dvz/dz = dvy/dy (2.17)

Note that Eqs. (2.14) and (2.17) are used to obtain Eq. (2.16).For unequal biaxial stretching, the rate-of-strain tensor d is expressed as

d =

a 0 00 b 00 0 (a + b)

(2.18)where a and b are the elongation rates in unequal biaxial stretching and are dened as

a = dvz/dz; b = dvy/dy (2.19)

In Chapter 7 of Volume 2, we present the rheological response of polymeric liquids insteady-state biaxial elongational ow.

2.4 Deformation Gradient Tensor, Strain Tensor, VelocityGradient Tensor and Rate-of-Strain Tensor

2.4.1 Deformation Gradient Tensor

For the description of motion given by Eq. (2.3), consider two particles in the referenceconguration (at t = 0) that are a distance dX apart. Then in the conguration


(at some other time t) these same two particles are a distance dx apart, given by(Jaunzemis 1967)

dx(t) = (X + dX, t) (X, t) (2.20)

Using Taylors theorem we may approximate (X + dX, t) by

(X + dX, t) = (X, t) + (/X) dX (2.21)

as the magnitude |dX| of dX approaches zero. Use of Eq. (2.21) in Eq. (2.20) gives

dx(t) = F(t) dX (2.22)

where F is called the deformation gradient tensor represented given by

F(t) =x(X, t)X

=

x1/X1 x1/X2 x1/X3

x2/X1 x2/X2 x2/X3

x3/X1 x3/X2 x3/X3

(2.23)One may also interpret F as a linear operator, which maps the neighborhood of theparticles X in the reference conguration into the conguration .

Using the relative conguration t dened by Eq. (2.5), we can also dene

dx(t ) = Ft (t ) dx(t) (2.24)

where Ft (t ) is called the relative deformation gradient tensor. It is seen in Eq. (2.24)that at t = t we have

Ft (t) = I (2.25)

where I is the unit second-order tensor.For a motion described by Eq. (2.9), for instance, we have

F(t) =

1 + t 0 0t 1 00 0 1

(2.26)and

Ft(t , t) =

(1 + t ) / (1 + t) 0 0(t t) / (1 + t) 1 0

0 0 1

(2.27)


2.4.2 Strain Tensor

We can dene other deformation tensors, also, in terms of the deformation gradienttensor F. According to the polar decomposition theorem of the second-order tensor(see Appendix 2A), the deformation gradient tensor F, which is an asymmetric tensorand is assumed to be nonsingular (i.e., det F = 0), can be expressed as a product ofa positive symmetric tensor with an orthogonal tensor (Jaunzemis 1967):

F(t) = R(t)U(t) (2.28)

where U is a positive symmetric tensor and R is an orthogonal tensor. A geometricalinterpretation of Eq. (2.28) may best be illustrated in Figure 2.3. That is, the deformationF = RU may be said to occur rst by the stretches U in the principal direction, followedby the rotation R. From Eq. (2.28) one has

C(t) = FT(t)F(t) = U(t)2 (2.29)

where C is called the CauchyGreen (deformation) tensor. Note that FT in Eq. (2.29)is the transpose of F and the orthogonality property of R (i.e., RRT = RR1 = I) hasbeen used.

The practical signicance of Eqs. (2.28) and (2.29) lies in that, because of thepositive deniteness of the symmetric tensor C, once F is known one can determinethe stretches U from

U(t) = (C(t))1/2 = (FT(t)F(t))1/2 (2.30)and the rotation R from

R(t) = F(t)U1(t) (2.31)

Figure 2.3 Geometrical interpretation of the polar decomposition of the deformation process,where (a) denotes undeformed state, (b) denotes the deformed state by stretchesU, and (c) denotesthe deformed state after rotation R following stretches U.


In terms of the relative deformation gradient tensor Ft (t ), Eq. (2.29) may be written as

C(x, t, t ) = Ct (t ) = FTt (t )Ft (t ) (2.32)

where Ct (t) is called the relative CauchyGreen (deformation) tensor, which describesthe change in shape of a small material element between time t and t . Note that att = t we have

Ct (t) = I (2.33)

In terms of the components of the relative deformation tensors, we can write Eq. (2.32)with the aid of Eq. (2.24) as

Cij (x, t, t) = (xm/xi)(xn/xj )gmn(x) (2.34)

where x(x1, x2, x3) are the spatial coordinates of the place occupied by materialelements at time t (< t), and x(x1, x2, x3) are the spatial coordinates of the placeoccupied by the same material elements at present time t . Note that gmn (x) is the metrictensor (seeAppendix 2B) referred to the spatial coordinates x for curvilinear coordinatesystems (Hawkins 1963; Jeffreys 1961). For rectangular Cartesian coordinates we have

gmn(x) = mn(x) (2.35)

where mn is the Kronecker delta and is a second-order tensor. At t = t (hence x = x),Eq. (2.34) reduces to

Cij (x) = gij (x) (2.36)

Using Cij one can dene the quantity Eij (Eringen 1962; Jaunzemis 1967)

Eij (x, t, t) = gij (x) Cij (x, t, t ) (2.37)

which may be interpreted as strains that a material element located at x at time t (< t)has experienced during the time period t t . Eij (x, t , t ) are the covariant componentsof the nite strain tensor E(x, t , t ). Similarly, one can also dene the contravariantcomponents Eij (x, t , t ) of the nite strain tensor E(x, t , t ) by

Eij (x, t, t ) = [C1(x, t, t )]ij gij (x) (2.38)where (C1)ij are the contravariant components of the Finger deformation tensorC1(x, t, t ) dened by

[C1(x, t, t )

]ij = (xi/xm)(xj /xn)gmn(x) (2.39)


Note that the CauchyGreen and Finger tensors are related by

Cij (x)[C1(x)

]jk = ki (2.40)Let us now consider steady-state simple shear ow, for which we have the velocity

eld of the form

v1 = x2, v2 = v3 = 0 (2.41)

where is the shear rate. Now the relative deformation function x(t) can be foundby solving the differential equations

dx1dt

= x2; dx2

dt = 0; dx

3dt

= 0 (2.42)

with the initial conditions

x(t )t =t = x (2.43)

giving rise to

x1(t ) = x1 + (t t) x2; x2(t ) = x2; x3(t ) = x3 (2.44)

The components of the relative deformation gradient tensor Ft (t ) may be obtained byuse of Eq. (2.44) in Eq. (2.24)

Ft (t , t) =

1 (t t) 00 1 00 0 1

(2.45)The components of the relative CauchyGreen deformation tensor Ct (t ) may beobtained by use of Eq. (2.45) in Eq. (2.32)

Ct (t , t) =

1 (t t) 0(t t) 1 + (t t)2 2 0

0 0 1

(2.46)and the components of the relative Finger deformation tensor C1t (t ) by

C1t (t , t) =

1 + (t t)2 2 (t t) 0(t t) 1 0

0 0 1

(2.47)


Therefore, the covariant components Eij (t , t) of the nite strain tensor E(t , t) aregiven by

E(t , t) =

0 (t t) 0(t t) (t t)2 2 0

0 0 0

(2.48)and the contravariant components Eij (t , t) of the nite strain tensor E(t , t) aregiven by

E(t , t) =

(t t)2 2 (t t) 0(t t) 0 0

0 0 0

(2.49)We have shown here that the CauchyGreen and Finger tensors are not equivalentmeasures of nite strain, which is a very important fact to remember in the formulationof constitutive equations, as is discussed in Chapter 3.

2.4.3 Velocity Gradient Tensor and Rate-of-Strain Tensor

We may take the time derivative of the deformation gradient tensor F (see Eq. (2.23))as (Jaunzemis 1967)

F(t) = F(t)t

= t

(x(X, t)

X

)(2.50)

But since, in the material description, X and t are independent variables, the order ofdifferentiation with respect to X and t can be interchanged:

t

(xi(X, t)

Xj

)=

Xj

(xi(X, t)

t

)= v

i(X, t)Xj

(2.51)

It is seen on the right side of Eq. (2.51) that the gradient of the instantaneous velocitywith respect to the coordinates Xj in the reference conguration is not a rate tensor.But the gradient of velocity with respect to the present coordinates xj does constitutea rate tensor. We can accomplish this by using the chain rule

vi/Xj =(vi/xm

) (xm/Xj

)(2.52)

or

F(t) = L(t)F(t) (2.53)

in which L is the velocity gradient tensor dened as

Lij (t) = vi(x, t)/xj (2.54)


It is seen in Eq. (2.53) that the velocity gradient tensor L(t) can be determined fromthe rate of deformation gradient tensor F(t) and the inverse of the deformation gradienttensor F1(t), that is,

L(t) = F(t)F1(t) (2.55)

The velocity gradient tensor L(t) can be determined from the relative deformationgradient tensor Ft (t ) also, since we have

Ft (t) = F(t )F1 (t) (2.56)

Therefore

Ft (t) =Ft (t , t)

t

t =t

= F(t)F1(t) = L(t) (2.57)

and any higher-order derivative of Ft (t) may be dened as (Rivlin and Ericksen 1955)

F(n)t (t) =nFt (t , t)

t n

t =t

= L(n)(t) (2.58)

where L(n) is the nth acceleration gradient tensor.We can now show further that L may be decomposed into symmetric and

asymmetric parts

L = d + (2.59)

in which d and are dened as

d = 12 (L + LT) or dij = 12(vi

xj+ v

j

xi

)(2.60)

and

= 12 (L LT) or ij = 12(vi

xj v

j

xi

)(2.61)

respectively. Note that dij = dji and ij = ji . The physical interpretations of and d are as follows. is called the vorticity tensor, which is the asymmetricpart of L, and it is the material derivative of the nite rotation tensor R taken withrespect to the present conguration (i.e., = Rt (t)). d is called the rate-of-straintensor (or rate-of-deformation tensor), which is the symmetric part of L, and it is thematerial derivative of the positive symmetric tensors U taken with respect to the presentconguration (Ut (t)).


Using the relative CauchyGreen tensor Ct (t , t) one can dene other rate tensors,such as (Rivlin and Ericksen 1955)

A(n) =dnCt (t , t)

dt n

t =t

=n

k=0

(n

k

)LT(k)L(nk) (2.62)

in which use is made of Eqs. (2.32) and (2.58). A(n) is called the nth-orderRivlinEricksen tensor. It should be noted that RivlinEricksen tensors play animportant role in formulating constitutive equations, which is discussed in Chapter 3.

To illustrate the usefulness of the various forms of rate tensors we have introduced,let us consider steady-state simple shear ow whose velocity eld is given by Eq. (2.41)and whose motion is given by Eq. (2.44). Use of Eq. (2.45) in (2.57) gives

L =

0 00 0 00 0 0

(2.63)We now have the rate-of-strain tensor d given by Eq. (2.12) and the vorticity tensor from Eq. (2.61):

=

0 /2 0 /2 0 0

0 0 0

(2.64)Further, use of Eq. (2.46) in (2.62) gives

A(1) =

0 0 0 00 0 0

(2.65)

A(2) =

0 0 00 2 2 00 0 0

(2.66)and

A(n) = 0 for n 3 (2.67)

Note that

A(1) = 2d (2.68)


and

Ct (t, t) = I + (t t)A(1) + 12 (t t)2A(2) (2.69)

It is of interest to note that the relative CauchyGreen tensor Ct (t ) can beexpressed in terms of the RivlinEricksen tensors A(m) by (Coleman 1962; Rivlinand Ericksen 1955)

Ct (t s) = I +n1m=1

(1)m sm

m! A(m)(t) (2.70)

where s is the elapsed time dened as s = t t .For the steady-state uniaxial elongational ow, the relative deformation gradient

tensor Ft (t , t) can be written

Ft (t , t) =

e(tt) 0 00 e(t t)/2 00 0 e(t t)/2

(2.71)Use of Eq. (2.71) in (2.57) gives the rate-of-deformation tensor d dened by Eq. (2.15).Note that for the uniaxial elongational ow, from Eq. (2.59) we have d = L becausethe vorrticity tensor vanishes. Further, we have

Ct (t , t) =

e2(tt) 0 0

0 e(t t) 00 0 e(t t)

(2.72)

C1t (t , t) =

e2(t t) 0 00 e(t t) 00 0 e(t t)

(2.73)

Since E = C1t I, we have

E(t , t) =

e2(t t) 1 0 00 e(t t) 1 00 0 e(t t) 1

(2.74)


For steady-state equal biaxial elongational ow, the relative deformation gradienttensor Ft (t , t) can be written, with the aid of Eqs. (2.16) and (2.57), as

Ft (t , t) =

eB(tt) 0 0

0 eB(t t) 00 0 e2B(t t)

(2.75)

Further, we have

Ct (t , t) =

e2B(tt) 0 0

0 e2B(t t) 00 0 e4B(t t)

(2.76)C1t (t , t) =

e2B(t t) 0 0

0 e2B(t t) 00 0 e4B(t t)

(2.77)E(t , t) =

e2B(t t) 1 0 0

0 e2B(t t) 1 00 0 e4B(t t) 1

(2.78)

2.5 Kinematics in Moving (Convected) Coordinates

The primary thrust of this section is to prepare ourselves in order to be able to write thekinematic quantities dened in a moving coordinate system via the transformation rulesin terms of the Cartesian components in the xed coordinate system. This is necessary,as will be discussed in the next chapter, for transforming a constitutive equation, whichwas rst written in a moving coordinate system, into a xed coordinates so that it canbe used in conjunction with the equations of continuity, motion, and energy that arenormally written in the xed coordinates.

In describing the kinematics of a deformable body, instead of using a coordinatesystem xed in space, it is convenient to use a coordinate system embedded in themoving object. This is frequently referred to as a convected coordinate system, andwas rst introduced by Oldroyd (1950). Any measure of deformation (strain) denedrelative to such a coordinate system always refers to the same element of materials,and therefore should be independent of the local rate of translation or rotation. As willbe shown in this section, if they are going to be useful, all kinematic variables denedin terms of the convected coordinates must be transformed to a xed coordinate systemas all physical measurements are made relative to the xed coordinate system.


2.5.1 Convected Strain Tensor

Let a convected coordinate system be denoted by i(i = 1, 2, 3) and a xed coordinatesystem by xi(i = 1, 2, 3). Then, states of a material element may be described byfunctions

xi = f i(, t) (2.79)

which have a unique inverse

i = i(x, t) (2.80)

where t represents time.A deformation may be said to occur when the magnitude of the distance between

any two points in a material element changes. The square of the distance betweenthe two points in a space may then be used as a quantitative measure of deformation(strain). In terms of spatial coordinates, the distance between two material points maybe represented by

(ds)2 = dx dx = gmn(x) dxmdxn (2.81)

where gmn is the spatial metric tensor (see Appendix 2B). From Eq. (2.79) we have

dxk = (xk/ i) d i (2.82)

Therefore, in terms of convected coordinates, the distance between two material pointsmay be represented, by use of Eq. (2.82) in (2.81), as

(ds)2 = ij (, t) d i dj (2.83)

where ij is called the convected covariant metric, which is related to the spatial metricgij by

ij (, t) = (xm/ i)(xn/j )gmn(x) (2.84)

Similarly, the convected contravariant metric ij (, t) is related to the spatialmetric gij by

ij (, t) = ( i/xm)(j /xn) gmn(x) (2.85)

The change in the distance between the material points at two different times,t and t (> t ), may be used as a measure of the strain, and it may be written,


from Eq. (2.83), as (Eringen 1962)

ds2(t) ds2(t ) = ij d idj (2.86)

where ij are the components of the convected covariant strain tensor,

ij (, t, t) = ij (, t) ij (, t ) (2.87)

Similarly, the components of the convected contravariant strain tensor ij may bedened as

ij (, t, t ) = ij (, t ) ij (, t) (2.88)

The denition of the convected strain tensor involves the difference between two quan-tities associated with a given material point at different times, and it refers to the samematerial point in convected coordinates. Now, we must transform the quantities ij (, t)and ij (, t ) (also ij (, t ), and ij (, t)) in such a manner that they both refer to thesame point in a coordinate system xed in space, because physical quantities (kine-matic and dynamic variables) can only be measured relative to a frame of referencexed in space. This can be done by making use of the transformation relations betweentwo coordinate systems.

Remembering that the coordinate systems xi and i are arbitrary (except that xi arexed in space and i in the material), let us choose an arbitrary spatial coordinate systemxi and then choose a convected coordinate system i that coincides with the spatialcoordinate system at present time t. Note that, in this choice, the present congurationis a reference conguration, so that all other congurations at time t (< t) are comparedwith the present one. From Eq. (2.84) we then have

ij (, t)

=x= gij (x) (2.89)

and

ij (, t)

=x,t=t = (x

m/xi)(xn/xj )gmn(x) (2.90)

Hence use of Eqs. (2.89) and (2.90) in (2.87) gives

ij (, t, t)=x= gij (x) (x

m/xi)(xn/xj )gmn(x) (2.91)

Similarly, we can also obtain from Eq. (2.85)

ij (, t)

=x= g

ij (x) (2.92)

ij (, t)

=x,t=t = (x

i/xm)(xj /xn)gmn(x) (2.93)


and from Eq. (2.88)

ij (, t, t )=x = (x

i/xm)(xj /xn)gmn(x) gij (x) (2.94)

We have shown how the strain tensors in the spatial coordinates may be obtained fromthose in the convected coordinates,

ij (, t)t Cij (x, t, t ) (2.95)

ij (, t)t [C1(x, t, t )]ij (2.96)

ij (, t, t) t

Eij (x, t, t ) (2.97)

ij (, t, t ) t Eij (x, t, t ) (2.98)

2.5.2 Time Derivative of Convected Coordinates

Having dened strain tensors in convected coordinates, we now describe the rate-of-strain (or rate-of-deformation) tensor. This may be obtained by taking the derivativeof a strain tensor with time, with the convected coordinates held constant. Such aderivative is commonly referred to as the material derivative, which may be con-sidered as the time rate of change as seen by an observer in a convected coordinatesystem. Using the notation D/Dt for the substantial (material) time derivative, we havefrom Eq. (2.86)

DDt[ds2(t) ds2(t )] = D

Dt[ij (, t, t

) d i dj] (2.99)

Since every material point always has the same convected coordinate position at alltimes, regardless of the extent of deformation of the medium, the relative coordinatedisplacements between any two points must be constant, so that any change in theactual distance between the points must be reected by a change in the metric ij .That is, if the distance between two points ds changes with time, the convected metricij must change accordingly with time since, by denition, the convected coordinates i of a material point are independent of time. Therefore Eq. (2.99) may be rewrittenwith the aid of Eq. (2.86) as

DDt[ds2(t) ds2(t )] = Dij

Dtd i dj = Dij (, t)

Dtd i dj (2.100)

Similarly, for the contravariant convected strain tensor we have

DDt[ds2(t) ds2(t )] = D ij

Dtdi dj =

Dij (, t)Dt

di dj (2.101)


It should be remembered that the metric tensor ij and ij in a convected coordinatesystem are related to the metric tensors gij and gij in a spatial coordinate system byEqs. (2.84) and (2.85).

Therefore, we can write the following general rule of coordinate transformation ofa second-order tensor:

Amn(, t) = (xi/m)(xj /n)aij (x, t) (2.102)

where Amn(, t) and aij (x, t) are covariant components of a tensor of second order inconvected and xed coordinate systems, respectively. Then the material derivative ofAmn(, t) requires the material derivative of the right-hand side of Eq. (2.102), yielding(see Appendix 2B)

DAmnDt

=(

xi

mxj

n

)aijt

(2.103)

where

aijt

= aijt

+ vk aijxk

+ vk

xiakj +

vk

xjaik (2.104)

or in direct notation

a

t= Da

Dt+ (v) a + a (v)T (2.105)

Here, /t is called the convected derivative due to Oldroyd (1950), and it isthe xed coordinate equivalent of the material derivative of a second-order tensorreferred to in convected coordinates. The physical interpretation of the right-hand sideof Eq. (2.104) may be given as follows. The rst two terms represent the derivative oftensor aij with time, with the xed coordinate held constant (i.e., Daij /Dt), which maybe considered as the time rate of change as seen by an observer in a xed coordinatesystem. The third and fourth terms represent the stretching and rotational motions ofa material element referred to in a xed coordinate system. This is because the veloc-ity gradient vk/xi (or the velocity gradient tensor L dened by Eq. (2.59)) maybe considered as a sum of the rate of pure stretching and the material derivative of thenite rotation. For this reason, the convected derivative is sometimes referred to as thecodeformational derivative (Bird et al. 1987).

Similarly, for contravariant componentsAmn(, t) and aij (x, t) of a tensor of secondorder in convected and xed coordinate systems, respectively, we have

DAmn

Dt=(m

xi

n

xj

)aij

t(2.106)


where

aij

t= a

ij

t+ vk a

ij

xk v

i

xkakj v

j

xkaik (2.107)

or in direct notation

a

t= Da

Dt (v)T a a (v) (2.108)

In Chapter 3, we show that the contravariant and covariant components, respectively,of the convected derivative of the stress tensor give rise to different expressions for thematerial functions in steady-state simple shear ow. When compared with experimentaldata, it turns out that the material functions predicted from the contravariant componentsof the convected derivative of the stress tensor give rise to a correct trend, while thematerial functions predicted from the covariant components of the convected derivativeof the stress tensor do not.

Now, we can apply the general rule of transformation to the material derivative ofa strain tensor in the convected coordinates, given by Eqs. (2.100) and (2.101). Forinstance, from Eq. (2.84) we have

Dij (, t)Dt

=(xm

i

xn

j

)gmnt

(2.109)

Since the spatial metric gmn(x) is independent of time, it can be easily shown that(Oldroyd 1950)

gmnt

= 2dmn (2.110)

where dmn are the components of the rate-of-strain tensor d dened by Eq. (2.60).It is important to note that there are other types of time derivatives which also

transform as a tensor from convected to xed coordinates. One particular time deriva-tive that has received particular attention by rheologists is the so-called Jaumannderivative, which was suggested rst by Zaremba (1903) and later reformulated byother investigators (DeWitt 1955; Fromm 1947). The Jaumann derivative /t of asecond-order tensor aij is dened as

aijt

= aijt

+ vk aijxk

ikajk jkaik (2.111)

or in direct notation1

a

t= Da

Dt ( a) ( a)T (2.112)


where is the vorticity tensor dened by Eq. (2.61). The physical interpretation of theright-hand side of Eq. (2.111) may be given as follows. The rst two terms representthe material derivative of aij , similar to the rst two terms on the right-hand side ofEq. (2.104). However, the third and fourth terms containing only the vorticity tensor represent the rotational motion of a material element referred to in a xed coordinatesystem. For this reason, the Jaumann derivative is sometimes referred to as the coro-tational derivative (Bird et al. 1987). In Chapter 3 we show that the contravariant andcovariant components, respectively, of the Jaumann derivative of the stress tensor giverise to identical expressions for the material functions in steady-state simple shear ow,predicting the same trend as that observed experimentally.

2.6 The Description of Stress and Material Functions

Let us consider now the stress tensor, which causes or arises from deformation.In order to give the reason why a second-order tensor is required to describe the stress,a development of Cauchys law of motion is needed. The physical signicance of thestress tensor may be illustrated best by considering the three forces acting on threefaces (one force on each face) of a small cube element of uid, as schematically shownin Figure 2.4. For instance, a force (which is the vector) acting on the face ABCDwith an arbitrary direction may be resolved in three component directions: the forceacting in the x1 direction is T11dx2dx3, the force acting in the x2 direction is T12dx2dx3,and the force acting in the x3 direction is T13dx2dx3. Similarly, the forces acting onface BCFE are T21dx1dx3 in the x1 direction, T22dx1dx3 in the x2 direction, T23dx1dx3

Figure 2.4 Stress components on a cube.


in the x3 direction. Likewise, the forces acting on face DCFG are T31dx1dx2 in thex1 direction, T32dx1dx2 in the x2 direction, and T33dx1dx2 in the x3 direction.

In dealing with the state of stresses of incompressible uids under deformation orin ow, the total stress tensor T is divided into two parts:

Tij =

T11 T12 T13T21 T22 T23T31 T32 T33

=

p 0 00 p 00 0 p

+

11 12 1321 22 2331 32 33

(2.113)where the component Tij of the stress tensor T is the force acting in the xi directionon unit area of a surface normal to the xi direction. The components T11, T22, and T33are called normal stresses since they act normally to surfaces, the mixed componentsT12, T13, and so on, are called shear stresses. In direct notation, Eq. (2.113), usingCartesian coordinates, can be expressed by

T = p+ (2.114)

where the is the unit tensor, is the deviatoric stress tensor (or the extra stress tensor)that vanishes in the absence of deformation or ow, and p is the isotropic pressure.Note in Eq. (2.113) or Eq. (2.114) that p has a negative sign since it acts in the directionopposite to a normal stress (T11, T22, T33), which by convention is chosen as pointingout of the cube (see Figure 2.4). It should be mentioned that in an incompressibleliquid, the state of stress is determined by the strain or strain history only to withinan additive isotropic constant, and thus p appearing in Eq. (2.113) or in Eq. (2.114)is the pressure that can be determined within the accuracy of an isotropic term. Asis shown in some later chapters (e.g., Chapter 5), only pressure gradient plays a rolein describing uid motion. Thus the isotropic term, p in Eq. (2.114) has no effecton uid motion, i.e., the addition of an isotropic term of arbitrary magnitude has noconsequence to the total stress tensor T when a uid is in motion.

Special types of states of stress are of particular importance. In a liquid that hasbeen at rest (i.e., there is no deformation of a uid) for a sufciently long time, thereis no tangential component of stress on any plane of a cube and the normal componentof stress is the same for all three planes, each perpendicular to the others. This is thesituation where only hydrostatic pressure, p, exists. In such a situation, Eq. (2.113)reduces to

Tij =

p 0 00 p 00 0 p

(2.115)From Eq. (2.115) we can now dene pressure as

p = 13 (T11 + T22 + T33) (2.116)

Note that Eq. (2.116) can also be obtained from Eq. (2.113) with the assumption,11 + 22 + 33 = 0. Since such an assumption is quite arbitrary, the denition of


pressure p given by Eq. (2.116) can be regarded as a somewhat arbitrary one. In fact,in general p is the thermodynamic pressure, which is related to the density and thetemperature through a thermodynamic equations of state, p = p(, T ); that is, thisis taken to be the same function as that used in thermal equilibrium (Bird et al. 1987).

If we now consider the state of stress in an isotropic material, by denition thematerial has no preferred directions. In simple shear ow, we have

T13 = T31 = 0; T23 = T32 = 0; T12 = T21 = 0 (2.117)in which the subscript 1 denotes the direction of ow, the subscript 2 denotes thedirection perpendicular to ow, and the subscript 3 denotes the remaining (neutral)direction. It follows therefore from Eq. (2.113) that the most general possible state ofstress for an isotropic material in simple shear ow may be represented by

T11 T12 0T12 T22 00 0 T33

=

p 0 00 p 00 0 p

+

11 12 012 22 00 0 33

(2.118)Note that one cannot measure p and the components of the extra stress tensor separately during ow of a liquid. Therefore, the absolute value of any one normalcomponent of stress is of no rheological signicance. The values of the differences ofnormal stress components are, however, not altered by the addition of any isotropicpressure (see Eq. (2.118)), and they presumably depend on the rheological propertiesof the material. It follows, therefore, that there are only three independent stress quan-tities of rheological signicance, namely, one shear component and two differences ofnormal components:

12; T11 T22 = 11 22; T22 T33 = 22 33 (2.119)Note that the normal stress difference 11 33 becomes redundant since we haveassumed 11 +22 +33 = 0 in dening p by Eq. (2.116). In the rheology community,N1 = 11 22 is referred to as the rst normal stress difference and N2 = 11 33as the second normal stress difference. It now remains to be discussed how the stressquantities may be related to strain or rate of strain to describe the rheological propertiesof materials, in particular polymeric materials.

For steady-state shear ow, the components of the stress tensor T may be expressedin terms of three independent functions:

12 = ( ) N1 = 1( ) 2 N2 = 2( ) 2 (2.120)where ( ) is referred to as the shear-rate dependent viscosity, 1( ) as the rstnormal stress difference coefcient, and 2( ) as the second normal stress differencecoefcient. Often, ( ), 1( ), and 2( ) are referred to as the material functionsin steady-state shear ow. Note that N1 and N2, or 1( ) and 2( ), describe the uidelasticity, which is elaborated on in Chapter 3.

In the past, numerous investigators have reported measurements of the rheologicalproperties of polymeric liquids. Until now, very few polymeric uids, if any, whichexhibit a constant value of shear viscosity (i.e., ( ) = 0) exhibit measurable values


of N1 and N2. In other words, almost all polymeric uids showing measurable values ofN1 and N2 have also been found to exhibit shear-rate dependent (i.e., non-Newtonian)viscosity ( ). Also, polymeric liquids showing measurable values of N1 and N2 havebeen found to exhibit unusual ow behavior, such as climbing-up a rotating rod (seeFigure 1.1) and extrudate swell (see Figure 1.4). It can then be understood why N1 andN2 are regarded as being the material functions that describes the elastic behavior ofpolymeric uids. In Chapter 3 we present how the material functions vary with shearrate on the basis of continuum viscoelasticity theory, in Chapter 4 we present how thematerial functions vary with shear rate on the basis of molecular viscoelasticity theory,and in Chapter 5 we present experimental methods to determine the material functions.

Appendix 2A: Properties of Second-Order Tensors

Invariants

If a is an arbitrary vector, we can nd a linear transformation Ta, where T is a linearoperator, such that Ta has the same direction as vector a itself, but the two vectors Taand a would differ in magnitude. That is,

Ta = a (2A.1)

where is a real scalar to be determined. Equation (2A.1) may be rewritten as

(T I)a = 0, or (Tij ij )aj = 0 (2A

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