Polymer Processing E-Book

Jean-François Agassant

• Herausgeber: Carl Hanser Verlag GmbH & Co. KG
• Kategorie: Fachliteratur
• Sprache: Englisch
• Veröffentlichungsjahr: 2017

Seitenzahl: 1159

Beschreibung

Engineering of polymers is not an easy exercise: with evolving technology, it often involves complex concepts and processes. This book is intended to provide the theoretical essentials: understanding of processes, a basis for the use of design software, and much more.

The necessary physical concepts such as continuum mechanics, rheological behavior and measurement methods, and thermal science with its application to heating-cooling problems and implications for flow behavior are analyzed in detail. This knowledge is then applied to key processing methods, including single-screw extrusion and extrusion die flow, twin-screw extrusion and its applications, injection molding, calendering, and processes involving stretching.

With many exercises with solutions offered throughout the book to reinforce the concepts presented, and extensive illustrations, this is an essential guide for mastering the art of plastics processing. Practical and didactic, Polymer Processing: Principles and Modeling is intended for engineers and technicians of the profession, as well as for advanced students in Polymer Science and Plastics Engineering.

Leseprobe

Jean-François Agassant Pierre Avenas Pierre J. Carreau Bruno Vergnes Michel Vincent

Polymer Processing

Principles and Modeling

The authors:

Jean-François Agassant, Professor, MINES ParisTech, CEMEF, CS 10207, 06904 Sophia Antipolis Cedex, FrancePierre Avenas, Former director of CEMEF and R+D in chemical industry, 249 rue Saint-Jacques, 75005 Paris, FrancePierre J. Carreau, Professor Emeritus, Polytechnique Montreal, C.P. 6079 suc. Centre-Ville, Montreal, QC H3C 3A7, Canada E-mail: [email protected] Vergnes, Directeur de Recherches, MINES ParisTech, CEMEF, CS 10207, 06904 Sophia Antipolis Cedex, FranceMichel Vincent,, Directeur de Recherches au CNRS, MINES ParisTech, CEMEF, CS 10207, 06904 Sophia Antipolis Cedex, France

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Table of Contents

Title page

Copyright page

Table of Contents

Foreword to the English Edition

Preface to the Third French Edition

Acknowledgements

Introduction

1 Continuum Mechanics: Review of Principles

1.1  Strain and Rate-of-Strain Tensor

1.1.1  Strain Tensor

1.1.2  Rate-of-Strain Tensor

1.1.3  Continuity Equation

1.1.4  Problems

1.2  Stresses and Force Balances

1.2.1  Stress Tensor

1.2.2  Equation of Motion

1.2.3  Problems

1.3  General Equations of Mechanics

1.3.1  General Case

1.3.2  Incompressibility

1.3.3  Planar Flow

1.3.4  Problem: Stress Tensor in Simple Shear Flow

1.4  Appendices

1.4.1  Appendix 1: Basic Formulae

1.4.2  Appendix 2: Invariants of a Tensor

2 Rheological Behavior of Molten Polymers

2.1  Viscosity: Equations for Newtonian Fluids

2.1.1  Basic Experiment of Newtonian Behavior

2.1.2  Generalization to Three Dimensions

2.1.3  Magnitudes of the Forces Involved

2.1.4  Navier-Stokes Equations

2.1.5  Problems

2.2  Shear-Thinning Behavior

2.2.1  Phenomenological Description

2.2.2  Rheological Models in One Dimension

2.2.3  Physical Explanation of the Shear-Thinning Behavior of Polymers

2.2.4  Three-Dimensional Constitutive Equations

2.2.5  Applications of the Power Law to Simple Flows

2.2.6  Problems in Power-Law Fluids

2.3  Behavior of Filled Polymers

2.3.1  Rheological Behavior of Suspensions

2.3.2  Yield Stress Fluids

2.3.3  Problem: Pressure Flow of a Yield Stress Fluid in a Pipe

2.4  Viscoelastic Behavior

2.4.1  Physical Phenomena

2.4.2  Linear Viscoelasticity and the Maxwell Model

2.4.3  Normal Stress Difference in Simple Shear

2.4.4  Extrudate Swell

2.4.5  Convected Maxwell Model

2.4.6  Viscoelastic Dimensionless Numbers

2.4.7  Physical Interpretation of the Viscoelastic Behavior of Polymer Melts

2.4.8  Some Viscoelastic Constitutive Equations

2.4.9  Problems in the Convected Maxwell Model

2.5  Measurement of the Rheological Behavior of Polymer Melts

2.5.1  Capillary Rheometer: Viscosity Measurements

2.5.2  Slit Die Rheometer

2.5.3  Flow with a Wall Slip

2.5.4  Cone-and-Plate Rheometer

2.5.5  Parallel-Plate Rheometer

2.5.6  Elongational Rheometry

2.5.7  Notions of Rheo-optics

2.5.8  Perspective

2.6  Appendices

2.6.1  Appendix 1: Physics of Viscosity

2.6.2  Appendix 2: An Approach to Viscoelasticity: Elastic Dumbbell Model

2.6.3  Appendix 3: Material and Convected Derivatives

2.6.4  Appendix 4: Rabinowitsch Correction (Rabinowitsch, 1929)

2.6.5  Appendix 5: Flow of a Viscoelastic Fluid in a Cone-and-Plate Geometry

2.6.6  Appendix 6: Viscometric Flows

3 Energy and Heat Transfer in Polymer Processes

3.1  Basic Notions on Heat Transfer

3.1.1  First Law of Thermodynamics

3.1.2  Heat Received by the System

3.1.3  Power Generated by Internal Forces

3.1.4  Equation of Energy

3.1.5  Internal Energy

3.1.6  Boundary Conditions

3.1.7  Solutions of the Heat Transfer Equation

3.2  Cooling in Molds, in Air, and in Water

3.2.1  Context

3.2.2  Heat Transfer Equation

3.2.3  Heat Penetration Thickness

3.2.4  Interfacial Temperature

3.2.5  Heating (or Cooling) of a Plate

3.3  Polymer Flow and Heat Transfer

3.3.1  Importance of Viscous Heating: The Brinkman Number

3.3.2  Notion of a Thermal Regime

3.3.3  The Equations

3.3.4  Equilibrium Regime

3.3.5  Adiabatic Regime

3.3.6  Transition Regime for a Newtonian Fluid

3.3.7  Transition Regime with a Power-Law Fluid

3.3.8  Comparison with an Exact Solution

3.3.9  Other Flow Geometries

3.3.10  Application to Flat Die Extrusion

3.3.11  Conclusion

3.4  Appendices

3.4.1  Appendix 1: Convective Heat Transfer

3.4.2  Appendix 2: Radiation Heat Transfer

3.4.3  Appendix 3: Internal Energy for Compressible Materials

4 Approximations and Calculation Methods

4.1  Equations for Polymer Processing

4.2  Choice of a Relevant Rheological Constitutive Equation

4.3  Choice of Boundary Conditions

4.3.1  Kinematics Boundary Conditions

4.3.2  Heat Transfer Boundary Conditions

4.3.3  Inlet Conditions

4.3.4  Exit Conditions

4.4  Approximation Methods

4.4.1  Approximations Concerning the Geometry of the Flow

4.4.2  Kinematics Approximations

4.4.3  Approximations for the Temperature

4.4.4  Conclusion and Application Example

4.4.5  Problems

4.5  Pressure Buildup in Polymer Flows: Hydrodynamics Bearings

4.5.1  Introduction

4.5.2  Qualitative Analysis of Some Hydrodynamics Bearings

4.5.3  Pressure Generated by a Sudden Flow Restriction (Rayleigh Bearing)

4.5.4  Flow Calculation in a Bearing of Variable Gap: the Reynolds Equation

4.5.5  Problem: Reynolds Bearing

4.6  Calculation Methods

4.6.1  Calculation Methods as Functions of the Type of Flow

4.6.2  Solution of Unidirectional Flows: Slab Method (or Incremental Method)

4.6.3  Solution of the Hele-Shaw Equations

4.6.4  2D and 3D Viscous Flow Calculations with a Finite Elements Method

4.6.5  Isothermal Flow Viscoelastic Computations

4.7  Appendix

4.7.1  Appendix 1: Analysis of the Lubrication Approximations

5 Single-Screw Extrusion and Die Flows

5.1  Single-Screw Extrusion

5.1.1  Geometric and Kinematic Description

5.1.2  Feeding Zone

5.1.3  Melting Zone

5.1.4  Flow of the Molten Polymer

5.1.5  Overall Model of Single-Screw Extrusion

5.1.6  Extrusion Problems

5.2  Extrusion Dies

5.2.1  Introduction: Role of an Extrusion Die

5.2.2  Description of the Encountered Geometries

5.2.3  Assumptions and Calculation Methods Revisited

5.2.4  Examples of Results

5.2.5  Conclusion

5.2.6  Die Problems

5.3  Multilayer Flows

5.3.1  Interest of Multilayer Flows and Related Problems

5.3.2  Study of the Steady Flow of Two Viscous Fluids between Parallel Plates

5.3.3  Flat Die Coextrusion

5.3.4  Coextrusion Die Problems

5.4  Appendix

5.4.1  Appendix 1: Calculation of Solid Velocity in Single-Screw Extrusion

6 Twin-Screw Extrusion and Applications

6.1  General Description of Twin-Screw Extrusion Process

6.1.1  Different Types of Twin-Screw Extruders

6.1.2  Flow Types

6.1.3  Specific Features of Corotating Twin-Screw Extrusion

6.1.4  Geometry of Screws and Barrel

6.1.5  Classical Approximations

6.1.6  Different Modeling Approaches

6.1.7  Reference Extruder

6.2  Solid Conveying and Melting

6.2.1  Solid Conveying Zone

6.2.2  Melting Zone

6.3  Melt Flow

6.3.1  Right- and Left-Handed Screw Elements

6.3.2  Mixing Elements

6.4  Global Model of Twin-Screw Extrusion

6.4.1  General Description

6.4.2  Residence Time Distribution

6.4.3  Examples of Results

6.5  Application to the Production of Polymer Blends

6.5.1  Basic Mechanisms

6.5.2  Modeling along the Extruder and Examples of Results

6.6  Application to Compounding Operations

6.6.1  Different Types of Mixing

6.6.2  Distributive Mixing

6.6.3  Dispersive Mixing: Application to the Production of Nanocomposites

6.7  Application to Reactive Extrusion

6.8  Optimization and Scale-Up

6.9  Conclusion

6.10  Problem: Simplified Model of the Flow around a Kneading Disk

7 Injection Molding

7.1  Description

7.2  Filling Stage

7.2.1  Peculiarities of the Filling Phase

7.2.2  Main Hypotheses and Governing Equations

7.2.3  Unidirectional Flows

7.2.4  Thin Flow or Hele-Shaw Models

7.2.5  3D Computations

7.3  Packing and Holding Phase

7.3.1  Introduction

7.3.2  Simplified Calculations of the Packing Phase

7.3.3  Physical Data for the Packing‒Holding Calculations

7.3.4  Calculations

7.3.5  Conclusions

7.4  Residual Stresses and Deformations

7.4.1  Introduction

7.4.2  Main Physical Phenomena Involved

7.4.3  Measurement of Residual Stresses

7.4.4  Calculations of Residual Stresses

7.5  Nonstandard Injection-Molding Techniques

7.5.1  Gas-Assisted Injection Molding (GAIM)

7.5.2  Water-Assisted Injection Molding (WAIM)

7.5.3  Multicomponent Injection Molding

7.6  Injection of Short Fiber Reinforced Polymers

7.7  Conclusion

7.8  Problems

7.8.1  Filling of a Center-Gated Disk

7.8.2  Balancing of a Multicavity Mold

8 Calendering

8.1  Introduction

8.2  Rigid Film Calendering Process

8.2.1  Presentation

8.2.2  Calendering Problems

8.2.3  Aim of Calendering Process Modeling

8.2.4  Kinematics of Calendering

8.2.5  Isothermal Newtonian Model Based on Lubrication Approximations

8.2.6  More General Newtonian Models

8.2.7  Shear-Thinning Calendering Model

8.2.8  Thermal Effects in Calendering

8.2.9  Viscoelastic Models

8.2.10  Use of Calendering Models

8.3  Postextrusion Calendering Process

8.3.1  Presentation

8.3.2  Process Modeling

8.4  Appendix

8.4.1  Appendix 1: Calculations of Two-Dimensional Flow in the Calender Bank by a Finite Element Method

9 Polymer Stretching Processes

9.1  Introduction

9.2  Fiber Spinning

9.2.1  Different Fiber Spinning Situations

9.2.2  Isothermal Melt Spinning of a Newtonian Fluid

9.2.3  Isothermal Melt Spinning of a Viscoelastic Fluid

9.2.4  Drawing of a Viscous Fluid in Nonisothermal Conditions

9.2.5  More General Models of Fiber Spinning

9.3  Biaxial Drawing

9.3.1  Introduction

9.3.2  Biaxial Stretching of a Newtonian Liquid

9.4  Cast-Film Process

9.4.1  Presentation

9.4.2  Different Kinematics Approaches

9.4.3  One-Dimensional Newtonian Model

9.4.4  One-Dimensional Membrane Model

9.4.5  Two-Dimensional Membrane Model

9.4.6  Conclusions

9.4.7  Problems

9.5  Film-Blowing Process

9.5.1  Process Description

9.5.2  Film Geometry

9.5.3  Equations of the Film-Blowing Process

9.5.4  Nonisothermal Newtonian Model

9.5.5  Nonisothermal Viscoelastic Model

9.5.6  A Semiempirical Model of the Blown-Film Process

9.5.7  Conclusions

9.6  Manufacture of Hollow Plastic Bodies

9.6.1  Various Blow-Molding Processes

9.6.2  Modeling of Extrusion Blow Molding

9.6.3  Stretch Blow-Molding Process

9.6.4  Conclusions

9.6.5  Problems

9.7  Appendices

9.7.1  Appendix 1: Solution of the Isothermal Cast-Film Equations

9.7.2  Appendix 2: Cooling of Films in Air or Water

9.7.3  Appendix 3: Solving the Film Blowing Equations

10 Flow Instabilities

10.1  Extrusion Defects

10.1.1  Description of the Various Defects Observed in Capillary Rheometry

10.1.2  Extrusion Defects of Linear Polymers

10.1.3  Extrusion Defects of Branched Polymers

10.1.4  Summary and Outlook

10.2  Coextrusion Defects

10.2.1  Investigation of Coextrusion Instabilities

10.2.2  Modeling Coextrusion Instabilities

10.2.3  Conclusions

10.3  Calendering Defects

10.3.1  Different Types of Defects

10.3.2  Analysis of the Matteness Defect

10.3.3  Analysis of the V-Shaped Defect

10.3.4  Analysis of the Rocket Defect

10.3.5  Conclusions

10.4  Drawing Instabilities

10.4.1  Description of Drawing Instabilities

10.4.2  Modeling Fiber Spinning Instability

10.4.3  Modeling Cast-Film Instability

10.4.4  Modeling Film-Blowing Instabilities

10.4.5  Conclusion

11 Notations

12 Color Supplement

4  Approximations and Calculation Methods

5  Single-Screw Extrusion and Die Flows

6  Twin-Screw Extrusion and Applications

7  Injection Molding

8  Calendering

9  Polymer Stretching Processes

10  Flow Instabilities

Foreword to the English Edition

It was with great enthusiasm that I agreed to compose this foreword for the second edition of Polymer Processing: Principles and Modeling (P3M-2). In 1994, when I arrived at the Mechanical Engineering Department of the University of Wisconsin ‒ Madison, it was Professor Tim Osswald who introduced me to teaching from the first edition of this book (P3M-1). I then taught the introductory course on polymer processing from P3M-1, twice a year, for years to come. My senior elective course classroom was well populated by students from the departments of Mechanical Engineering, Chemical Engineering, and Materials Science and Engineering. P3M-1 was a student favourite for its readability and its expert use of terms with plain meaning, wherever possible. I used this first edition until, disappointingly, it went out of print.

P3M-2 expands on P3M-1 from 6 chapters to 10, and P3M-2 is reorganized, now opting to cover rheology in one consolidated second chapter rather than postponing viscoelasticity until Chapter 6. This expansion and reorganization are clever improvements. I am pleased to report that Chapter 2 retains a clear explanation of the Jaumann derivative, making Chapter 2 a gem. I see that the writing style still employs terms with plain meaning, wherever possible. Undergraduate students, the hardest to please, will enjoy this book.

Each chapter is designed pedagogically to sets students free to solve a broad class of relevant problems, as it should. For instance, Chapter 7 on injection molding equips students to solve time-unsteady processing problems, Chapter 5 on single-screw extrusion enables students to attack problems with non-obvious coordinates systems, and Chapter 8 on calendering teaches students how an apparently complicated process geometry, cleverly chosen, may yield process working equations of remarkable simplicity. In Chapter 6 on twin-screw extrusion, new to P3M-2, we enjoy Vergnes’ special touch, the foremost authority on extrusion, and Chapter 8 on calendering, bears Agassant's signature, who for decades has been the foremost authority on this process. P3M-2 is a translation from the recent French fourth edition [Mise en forme des polymères (2014)] and, as was the case for P3M-1, P3M-2 has the readability of English first language authorship.

Our world's polymer processing industry continues to grow steadily, to employ and to govern our prosperity and quality of life. Creative polymer chemists and product designers continue to challenge plastics engineers with novel combinations of material and shape. Our need to arrive at solutions to the ensuing manufacturing problems, in a hurry, confidently, and inexpensively, more than ever, requires our plastics engineering community to be well versed in the fundamentals of plastics processing. P3M-2 addresses this need expertly by empowering plastics engineers to create knowledge about plastics processing, and thus, to fill knowledge gaps, as they arise, in our quickly evolving world of plastics manufacturing.

A. Jeffrey Giacomin, PhD, PEng, PE

Tier 1 Canada Research Chair in Rheology

Queen's University at Kingston, Canada

Preface to the Third French Edition

The viscoelastic properties of long chain molecules are quite extraordinary. Even in a highly diluted solution (100 parts per million), polyethylene oxide drastically reduces the turbulent losses of water. It also allows tubeless siphons to function, as discovered by James in Toronto, which are fascinating objects. The same for molten polymers: in very slow flows, they behave like liquids. In more rapid motions, they behave like rubber and, in flow near walls, they exhibit astonishing slip properties that we are beginning to examine at Collège de France using rather sophisticated optical techniques. All that I briefly described here has major practical implications, in particular for the processing of plastic materials. In injection molding, extrusion, or more sophisticated processes, consistently one has to force the liquid polymer to rapidly adopt preset shapes—which it does not like. Hence the many defects in the final product, such as sharkskin, which is a disaster for the manufacturer of extruded products. Plastics engineering is, therefore, a difficult art, and the authors describe here the basic notions based on extensive experiences, working directly with many manufacturers. Their approach is based mainly on principles of mechanics, but they have incorporated in their first chapters (and a few other places) a useful introduction to the physical underlying phenomena. Of course, this introduction is no substitute for basic textbooks such as that of John Ferry on viscoelasticity, or that of S. Edwards and M. Doi on the behavior of entangled chains. The first edition of this book has already been proven to be quite useful: chemical engineering communities in France and Canada have heavily relied on it. This new version, which is significantly expanded, should be of great service; I wish it great success.

P.G. de Gennes, Nobel Prize in Physics 1991

December 1995

Translated by P.J. Carreau

August 2016

Acknowledgements

Four authors of this book are or have been associated with the Centre de Mise en Forme des Matériaux (Materials Forming Center, CEMEF) of Ecole des Mines de Paris (now MINES-ParisTech).

This research center was established in 1974, and it was one of the first institutions to be established in the Sophia-Antipolis Techno-park (Alpes-Maritimes, France) in 1976. It has been associated with the Centre National de la Recherche Scientifique (CNRS) since 1979 (joint research Unit 7635). It now has nearly one hundred fifty people: professors, researchers, PhD students, advanced-master and master students, engineers, technicians, and administrative staff.

The role of CEMEF is twofold:

Training, in the field of engineering materials and processing, of engineers, master, and PhD students. Since the beginning, nearly 450 doctoral degrees and more than 350 advanced-master's degrees were supported by the center. These graduates are now working in many industrial companies with which the center is related.

Contribution to solving scientific and technical problems in the field of processing and forming of materials (particularly metals and polymers). The center maintains relations with the major French and European companies in the development, implementation, and use of materials.

Jean-François Agassant is an engineer from Ecole des Mines de Paris, Doctor of Science, and professor at the Ecole des Mines de Paris. He was deputy director of CEMEF (1981-2007) and director of the joint unit between MINES-ParisTech and CNRS (1989-2001). He is now responsible for the "Mechanical and Material Engineering" department and the head of MINES-ParisTech on the Sophia-Antipolis site.

Pierre Avenas is an alumnus of Ecole Polytechnique (Paris) and engineer "corps of Mines." He initiated research on polymers at the Ecole des Mines de Paris and helped create CEMEF, of which he was director from 1974 to late 1978. After heading the industrial research department at the Ministry of Industry of France (1979-1981), he held several positions in the chemical industry, including Director of R & D chemistry of Total group until 2004.

Bruno Vergnes is an engineer from ENSTA (École nationale supérieure de techniques avancées), Doctor-engineer from Ecole des Mines de Paris, and Doctor of Science. He worked from 1981 to 2008 at CEMEF, in the research group "Viscoelastic Flows", with J.F. Agassant and M. Vincent. He is currently director of research at MINES-ParisTech and responsible for continuous processes and rheological problems in the research unit "Polymers and Composites" at CEMEF.

Michel Vincent is an engineer from Ecole des Mines of Saint-Etienne and Doctor of engineering from Ecole des Mines de Paris. He is currently director of research at CNRS, and he is responsible within the research unit "Polymers and Composites" of CEMEF for the injection molding and reinforced polymers.

The fifth author, Pierre Carreau, was responsible for the translation and adaptation of the original French book into English. He graduated in chemical engineering from Ecole Polytechnique of Montreal. He obtained his PhD from the University of Wisconsin (Madison, USA). He is now professor emeritus of Ecole Polytechnique of Montreal. He was the founder of the Center on Applied Polymer Research and, more recently, of the Research Center for High Performance Polymer and Composite Systems (CREPEC). CREPEC is an interuniversity research center joining 50 of Quebec's scientists specialized in the development of new high performance polymers and composites and their transformation and implementation process.

Both CEMEF and CREPEC have been associated for many years. Initially, under the France-Quebec collaboration program, a few joint research projects have been initiated. The first English book, published in 1991, and this revised and expanded version are major outcomes of this collaboration.

The initial French book was first published in 1982 and updated in 1986, 1996, and 2014. The second edition in 1986 was adapted and translated into English by Pierre Carreau; it was published by Hanser in 1991. The present translated version of the latest French edition is completely redesigned, both in the presentation and scope of the topics. It presents a synthesis of research and teaching approaches developed over more than thirty years in the field of processing of polymers at CEMEF.

We would like to mention all researchers, colleagues, doctoral and master's graduates, who were or are still at CEMEF and at Polytechnique Montreal, whose work has contributed to the realization of this book: H. Alles, J.M. André, B. Arpin, G. Ausias, Ph. Barq, C. Barrès, S. Batkam, P. Beaufils, M. Bellet, N. Bennani, C. Beraudo, F. Berzin, R. Blanc, F. Boitout, R. Bouamra, C. Champin, M. Coevoet, C. Combeaud, D. Cotto, T. Coupez, L. Delamare, Y. Demay, F. Démé, O. Denizart, E. Devilers, F. Dimier, T. Domenech, J.L. Dournaux, C. Dubrocq-Baritaud, R. Ducloux, V. Durand, A. Durin, M. Espy, E. Foudrinier, E. Gamache, J.F. Gobeau, S. d'Halewyn, J.M. Haudin, I. Hénaut, C. Hoareau, S. Karam, D. Kay, M. Koscher, P. Lafleur, P. Laure, M. Leboeuf, D. Le Roux, W. Lertwimolnun, O. Mahdaoui, H. Maders, R. Magnier, B. Magnin, J. Mauffrey, M. Mouazen, Ph. Mourniac, B. Neyret, I. Noé, H. Nouatin, L. Parent, C. Peiti, S. Philipon, A. Philippe, A. Piana, E. Pichelin, A. Poitou, A. Poulesquen, S. Mighry, L. Robert, A. Rodriguez-Villa, P. Saillard, G. Schlatter, F. Schmidt, D. Silagy, L. Silva, C. Sollogoub, G. Sornberger, B. Souloumiac, J. Tayeb, J. Teixeira-Pirès, R. Valette, C. Venet, E. Wey, and J.L. Willien. Our thanks go to them and to all those with whom we had the opportunity to work, in both French and foreign universities and in industry, on topics of rheology and polymer processing.

Finally, we thank Ms. Corinne Matarasso who improved the quality of many figures.

Introduction

This book is intended for engineers and technicians involved in various sectors of the plastics industry (processing, manufacture, and formulation of polymers; more specifically, the design of plastic objects and equipment), as well as students and researchers in physics and chemistry of polymers, continuum mechanics, rheology, modeling, and numerical simulation.

The rapid development of synthetic polymers since the 1960s is due to both the excellent usage value of these materials and the considerable progress that has been made in their implementation.

Effective and Innovative Materials

Contrary to popular belief, we find that the life cycle analysis (LCA) or eco-balance of synthetic polymers is better than most competing materials for the same application. This can be illustrated by several examples:

A glass container (e.g., a bottle) is much heavier than its plastic equivalent: it follows that, for a given volume of content, less energy is necessary for the manufacturing of plastic containers; furthermore, less energy consumption is required for their transportation and distribution, including recycling;

A mailing envelope made of plastic film has a lower energy cost than its equivalent in paper, due to its lower weight and the lower water consumption required for its manufacturing;

A PVC profile for construction is more durable than wood and provides more thermal insulation than aluminum.

The advantage of polymers is basically due to their lightness. It is logical that their systematic replacement by other materials would result in increases in energy consumption and, hence, larger CO2 emissions and a worsening of the situation of global warming.

Polymer processing techniques contribute to good environmental performances of these materials in two ways:

Low consumption of

energy, in particular because the processing temperature (generally below 250 °C) is moderate with respect to temperatures used for other materials, of the order of 1000 °C for glass and metals;

water, relative to paper manufacturing.

Their ability to integrate functions that often allow us to make a given object in one step processing, whereas it takes several steps for other materials, thus eliminating constraints and assembly costs.

However, complaints are regularly raised against synthetic polymers concerning long-term environmental issues (20 years, 50 years, or more).

Long-Term Concerns

The issues are twofold:

The consumption of fossil fuels (oil and gas), although plastics represent today's energy content equivalent to only 4 % of oil consumption.

The presence and low degradability of plastic wastes in the environment, particularly in the marine environment.

On the first point, there are already resources other than conventional oil and gas that are used, for example, sea or geological salts for PVC production and vegetal raw materials for some polymers such as polyamides, cellulose, and its derivatives. The progressive development of the use of biomass for producing bio-based polymers is possible provided there is no impact on the agricultural food chain. In the long term, beyond the reserves of oil and natural gas that can be used in a larger proportion than today for the production of plastics, a return to carbon chemistry, but with clean technologies, is quite conceivable. It can, therefore, be assumed that the issue of raw materials, which in any case affects all human activities, will be gradually solved based on economic considerations that are difficult to forecast.

On the second point, the only difficult question about the very long term is that of waste in the oceans. Controlling the spilling of these plastic wastes requires binding international decisions, which are still lacking today. Whether on land, in rivers and lakes, and even on the coasts, it will be possible to strengthen the control of spilling and repair the damage over time. However, this control needs to be combined, of course, with the development of plastic recycling and efficient energy production. An alternative is the use, in very special cases, of biodegradable polymers. However, in spite of real public expectations and of many studies, biodegradable polymers cannot be used in all applications.

After these considerations on the long and very long terms, let us get to the actual subject of this book: the transformation of polymers from their native conditions such as powder or pellets into semifinished or finished parts.

Issues of Polymer Processing

At the processing level, the problems consist in designing equipment and setting processing parameters in order to obtain satisfactory products at the lowest price. Satisfactory products mean:

Desired dimensions and surface finish, taking into account swellings, drawing ratios, shrinkages, and so on.

A physical structure adapted to the desired properties, which, depending on the case, requires a certain molecular orientation, low residual stresses in the part, a specific crystalline structure (size of the crystallites and spherulites), and a specific morphology in the case of blends of polymers or filled products.

At the lowest price means:

In general, to produce a minimum of waste or substandard parts and thus to rapidly reach a steady-state operation.

Maximum throughput in extrusion or calendering, and a minimum cycle time in injection or blow molding. In any case, vis-à-vis these objectives, there is an optimum to be found in equipment design (screws, dies, molds) and in the settings of the processing parameters, such as temperature, screw speed, and cooling rate.

At the production level of polymer parts, the challenge is to adapt the product to a given transformation process and desired physical criteria, such as mechanical properties, appearance, and so on. For this, one can examine the effects of:

Molecular structure (molecular weight distribution, copolymerization, length and type of branching…).

Formulation (blending of compatible or incompatible polymers, lubricating agents, processing aids, fillers…).

Globally, one has to master the relationships between the formulation, rheological behavior, and processing conditions. All of these subjects are parts of the mechanics, thermodynamics and rheology, that is to say the science of flow. These fields are quite vast and only the notions related to polymer processing will be presented in this book. The concepts and models presented are directly applicable to the processing of thermoplastics.

Processing Stages

The processing of thermoplastics may involve three distinct thermo-mechanical stages:

The melting stage in which the polymer goes from a solid state as a powder or pellets to a sufficiently homogeneous and fluid liquid state. The term melting covers many different physical phenomena, depending if the polymer is semicrystalline or amorphous. However, in order to simplify the nomenclature in the text, the terms melting and molten state will be used for all polymers.

A second stage, where a specified shape is imposed to the molten polymer by bringing it to flow under pressure into molds or dies.

A third stage, where the final shape or conformation of the finish product is achieved under cooling and eventually drawing, biaxial stretching, or blowing.

General polymer processing stages

The properties of the final part depend not only on the chosen polymer, but also on the thermo-mechanical history of this polymer during processing.

Logic for Polymer Process Design

The specific physical characteristics of polymers have led to process designs adapted to these materials.

Thermal Insulating Character of Polymer Melts

The thermal conductivity of the polymers is of the order of 0.2 W·m−1·°C−1, which is a thousand times lower than that of copper. Under these conditions, it takes about:

17 minutes for heating or cooling of a sheet to reach a depth of 1 cm;

10 s for a depth of 1 mm;

0.1 s for a depth of 0.1 mm.

This shows that melting of a polymer by unique conduction from walls of equipment would lead to unacceptably long residence times and low flow rates. That is why all melting processes require a significant input of mechanical energy, often much larger than thermal energy transferred by conduction. The low thermal conductivity also implies that the majority of plastic products are less than a few millimeters thick to minimize the cooling times and favor high production rates.

Very High Viscosity of Molten Polymers

Molten polymers have viscosities of about 103 Pa·s under typical processing temperatures; this is a million times larger than that of water. This property has three practical consequences:

Firstly, the Reynolds number that characterizes the weight of inertia terms with respect to the viscosity terms is low; therefore, there will never be any turbulent flow of molten polymer and the inertial terms are often neglected in comparison with the terms of viscosity.

Secondly, the heating of polymers by viscous dissipation is easy to achieve, and it is operative during melting. Under most common extrusion operating conditions, the energy introduced into the polymer by heating and by viscous dissipation are of the same order of magnitude.

Finally, the high viscosities mean high pressures to ensure the flow of polymer in dies and molds at high flow rates, typically:

100 MPa or more for injection pressures;

5 to 50 MPa in extruders to ensure the flow in dies.

This pressure is obtained by two main methods: In injection, the screw, after its rotation during the melting phase is stopped, acts as a piston and moves in the barrel during the phases of filling and post-filling of the mold. In extrusion, the screw-barrel system acts as a pump; the pressure generated in extruders results from the equilibrium of pressure and drag flows in the channel of the screw on one hand and the pressure flow in the die on the other hand.

Combined Insulating Character and High Viscosity

The heating by viscous dissipation is often localized, and the low thermal conductivity of the polymer then creates large thermal gradients. Because of the risks of thermal degradation of the polymer, the amount of energy that can be provided by viscous dissipation needs to be restricted or imposed flow rates are to be limited. On the other hand, the high viscosity of polymers coupled with their low conductivity, allows us to design processes in ambient air where extrudates remain in the molten state before going through shaping devices or drawing systems. These properties have led to the development of many downstream extrusion processes:

Spinning, that is to say drawing polymer filaments for the manufacture of textile fibers;

Blowing, batch-wise, bottles and containers;

Blowing, continuous ducts;

Stretching and biaxial stretching of films.

Some of these processes are derived from glass-making techniques.

Sensitivity of Viscosity to Shear Rate and Temperature

Typically, the viscosity of a molten polymer may decrease by a factor of five when the shear rate is increased by a factor of ten, and it may drop by a factor of two when the temperature is increased by 20 °C. If the viscosity of the polymer was constant, doubling the flow rate would result in doubling the pressure required for the flow. The high sensitivity of the viscosity to the shear rate and temperature allows wide variations of the operating parameters while keeping the die head pressure or injection pressure at the same order of magnitude.

Viscoelasticity

Viscoelasticity plays a subtler role as it stabilizes the drawing or biaxial stretching of polymer melts. It is what makes possible the spinning of polymers at a high strain rate and blow molding and thermoforming with homogenized thicknesses of the parts. In contrast, shear flows are limited by the onset of instabilities, sometimes of viscoelastic origin, which result in defects such as volume defects in extrusion or coextrusion interface instabilities. Finally, viscoelasticity is responsible for the swelling at the outlet of dies, which complicates the design of the equipment.

Outline of the Book

Most processes for commodity thermoplastics of economic importance are mentioned in the figure below. The basic principles of these processes are the result of four essential properties of molten polymers: low thermal conductivity, high viscosity, viscosity-dependence on the rate of strain and temperature, and viscoelasticity.

Major commodity polymers and main applications

The reader will find in the first chapters of this book the details of the physical concepts briefly outlined above:

Basics of continuum mechanics in Chapter 1.

Rheological behavior and measurement methods in Chapter 2.

Basics of heat transfer and their applications to problems of heating and cooling in polymer processing in Chapter 3.

Chapter 4 exposes the reader how to implement these mechanical and physical concepts to understand and model processes.

The following chapters are devoted to specific processes:

Single-screw extrusion and flows in dies in Chapter 5.

Twin-screw extrusion and applications in Chapter 6.

Injection molding in Chapter 7.

Calendering processes in Chapter 8.

Processes involving stretching (spinning, blowing, cast film extrusion) in Chapter 9.

Chapter 10 is devoted to instabilities and defects that limit certain processes.

Problems and their solutions are proposed through the book for helping the reader in mastering the concepts.

1Continuum Mechanics: Review of Principles

1.1 Strain and Rate-of-Strain Tensor1.1.1 Strain Tensor1.1.1.1 Phenomenological Definitions

Phenomenological definitions of strain are first presented in the following examples.

1.1.1.1.1 Extension (or Compression)

In extension, a volume element of length l is elongated by Δl in the x direction, as illustrated by Figure 1.1. The strain can be defined, from a phenomenological point of view, as εl /l.

Figure 1.1 Strain in extension

For a homogeneous deformation of the volume element, the displacement U on the x-axis is , and . Hence another definition of the strain is .

1.1.1.1.2 Pure Shear

A volume element of square section h × h in the x-y plane is sheared by a value a in the x-direction, as shown in Figure 1.2. Intuitively, the strain may be defined as γa /h. For a homogeneous deformation of the volume element, the displacement (U , V) of point M(x , y) is

(1.1)

Hence, another possible definition of the strain is .

Figure 1.2 Strain in pure shear

1.1.1.2 Displacement Gradient

More generally, any strain in a continuous medium is defined through a field of the displacement vector U(x , y , z) with coordinates

The intuitive definitions of strain make use of the derivatives of U , V , and W with respect to x , y , and z , that is, of their gradients. For a three-dimensional flow, the material can be deformed in nine different ways: three in extension (or compression) and six in shear. Therefore, it is natural to introduce the nine components of the displacement gradient tensor ∇U:

(1.2)

This notion of displacement gradient applied to the two previous deformations presented in Section 1.1.1.1 leads to the following expressions:

Extension deformation:

(1.3)

Shear deformation:

(1.4)

If this notion is applied to a volume element that has rotated θ degrees without being deformed, as shown in Figure 1.3, the displacement vector can be written as

(1.5)

Figure 1.3 Rigid rotation

(1.6)

(1.7)

It is obvious from this result that ∇U cannot physically describe the strain of the material since it is not equal to zero when the material is under rigid rotation without being deformed.

1.1.1.3 Deformation or Strain Tensor ε

To obtain a tensor that physically represents the local deformation, we must make the tensor ∇U symmetrical, as follows:

Write the transposed tensor (symmetry with respect to the principal diagonal); the transposed deformation tensor is

(1.8)

Write the half sum of the two tensors, each transposed with respect to the other:

(1.9)

(1.10)

where Ui stands for U , V , or W and xi for x , y , or z.

Let us now reexamine the three previous cases:

In extension (or compression):

(1.11)

The deformation tensor ε is equal to the displacement gradient tensor ∇U.

In pure shear:

(1.12)

The tensor ε is symmetric, whereas ∇U is not. We see that pure shear is physically imposed in a nonsymmetrical manner with respect to x and y; however, the strain experienced by the material is symmetrical.

In rigid rotation:

(1.13)

The definition of ε is such that the deformation is nil in rigid rotation; it is physically satisfactory, whereas the use of ∇U for the deformation is not correct.

As a general result, the tensor ε is always symmetrical; that is, it contains only six independent components:

three in extension or compression: εxx , εyy , εzz

three in shear: εxyεyx , εyzεzy , εzxεxz

Important Remarks

(a) The definition of the tensor ε used here is a simplified one. One can show rigorously that the strain tensor in a material is mathematically described by the tensor Δ (Salençon, 1988):

(1.14)

This definition of the tensor ε is valid only if the terms ∂Ui /∂xj are small. So the expressions for the tensor written above are usable only if ε, γ, θ , and so on are small (typically less than 5 %). This condition is not generally satisfied for the flow of polymer melts. As will be shown, in those cases, we will use the rate-of-strain tensor .

(b) The deformation can also be described by following the homogeneous deformation of a continuum media with time. The Cauchy tensor is then used, defined by

(1.15)

where xi are the coordinates at time t of a point initially at Xi , and Ft is the transpose of F. The inverse tensor, called the Finger tensor, will be used in Chapter 2:

(1.16)

1.1.1.4 Volume Variation During Deformation

Only in extension or compression the strain may result in a variation of the volume. If lx , ly , lz are the dimensions along the three axes, the volume, , is then

(1.17)

1.1.2 Rate-of-Strain Tensor

For a velocity field u(x , y , z), the rate-of-strain tensor is defined as the limit:

(1.18)

where is the deformation tensor between times t and t + dt. However, in this time interval the displacement vector is dUudt. Hence,

(1.19)

where uiu , v , w) are the components of the velocity vector. The components of the rate-of-strain tensor become

(1.20)

As in the case of ε, this tensor is symmetrical:

(1.21)

The diagonal terms are elongational rates; the other terms are shear rates. They are often denoted and , respectively.

Remark: Equation (1.20) is the general expression for the components of the rate-of-strain tensor, but its derivation from the expression (1.18) for the strain tensor is correct only if the deformations and the displacements are infinitely small (as in the case of a high-modulus elastic body). For a liquid material, it is not possible, in general, to make use of expression (1.19). Indeed, a liquid experiences very large deformations for which the tensor ε has no physical meaning. Tensors Δ, C, or C−1 are used instead.

1.1.3 Continuity Equation1.1.3.1 Mass Balance

Let us consider a volume element of fluid dx dy dz (Figure 1.4). The fluid density is ρ(x , y , z , t).

Figure 1.4 Mass balance on a cubic volume element

The variation of mass in the volume element with respect to time is . This variation is due to a balance of mass fluxes across the faces of the volume element:

In the x direction:

In the y direction:

In the z direction:

Hence, dividing by dx dy dz and taking the limits, we get

(1.22)

which can be written through the definition of the divergence as

(1.23)

This is the continuity equation.

Remark: This equation can be written using the material derivative , leading to .

1.1.3.2 Incompressible Materials

For incompressible materials, ρ is a constant, and the continuity equation reduces to

(1.24)

This result can be obtained from the expression for the volume variation in small deformations:

(1.25)

(1.26)

(1.27)

1.1.4 Problems1.1.4.1 Analysis of Simple Shear Flow

Simple shear flow is representative of the rate of deformation experienced in many practical situations. Homogeneous, simple planar shear flow is defined by the following velocity field:

where Ox is the direction of the velocity, Oxy is the shear plane, and planes parallel to Oxz are sheared surfaces; is the shear rate. Write down the expression for the tensor for this simple planar shear flow.

Figure 1.5 Flow between parallel plates

Solution

(1.28)

1.1.4.2 Study of Several Simple Shear Flows

One can assume that any flow situation is locally simple shear if, at that given point, the rate-of-strain tensor is given by the above expression (Eq. (1.28)). Then show that all the following flows, encountered in practical situations, are locally simple shear flows. Obtain in each case the directions 1, 2, 3 (equivalent to x , y , z for planar shear) and the expression of the shear rate (use the expressions of in cylindrical and spherical coordinates given in Appendix 1, see Section 1.4.1).

1.1.4.2.1 Flow between Parallel Plates (Figure 1.6)

The velocity vector components are u(y), vw

Figure 1.6 Flow between parallel plates

Solution

(1.29)

1.1.4.2.2 Flow in a Circular Tube (Figure 1.7)

The components of the velocity vector u(r, θ, z) in a cylindrical frame are uvww(r).

Figure 1.7 Flow in a circular tube

Solution

(1.30)

Directions 1, 2, and 3 are respectively z , r , and θ. The shear rate is .

1.1.4.2.3 Flow between Two Parallel Disks

The upper disk is rotating at an angular velocity Ω0, and the lower one is fixed (Figure 1.8). The velocity field in cylindrical coordinates has the following expression: u(r, θ, z):uv(r, z),w

Figure 1.8 Flow between parallel disks

(a) Show that the tensor does not have the form defined in Section 1.1.4.1.

(b) The sheared surfaces are now assumed to be parallel to the disks and rotate at an angular velocity Ω(z). Calculate v(r , z) and show that the tensor is a simple shear one.

Solution

(a)

(1.31)

(b) If v(r , z) = r Ω (z), then and is a simple shear tensor. The shear rate is and directions 1, 2, and 3 are θ , z , and r , respectively.

1.1.4.2.4 Flow between a Cone and a Plate

A cone of half angle θ0 rotates with the angular velocity Ω0. The apex of the cone is on the disk, which is fixed (Figure 1.9). The sheared surfaces are assumed to be cones with the same axis and apex as the cone-and-plate system; they rotate at an angular velocity Ω(θ).

Figure 1.9 Flow in a cone-and-plate system

Solution

In spherical coordinates (r , θ , φ), the velocity vector components are uvwr sinθ Ω(θ).

(1.32)

The shear rate is , and directions 1, 2, and 3 are φ, θ , and r , respectively.

1.1.4.2.5 Couette Flow

A fluid is sheared between the inner cylinder of radius R1 rotating at the angular velocity Ω0 and the outer fixed cylinder of radius R2 (Figure 1.10). The components of the velocity vector u(r , θ , z) in cylindrical coordinates are uv(r), and w

Figure 1.10 Couette flow

Solution

(1.33)

The shear rate is , and directions 1, 2, and 3 are θ, r , and z , respectively.

1.1.4.3 Pure Elongational Flow

A flow is purely elongational or extensional at a given point if the rate-of-strain tensor at this point has only nonzero components on the diagonal.

1.1.4.3.1 Simple Elongation

An incompressible parallelepiped specimen of square section is stretched in direction x (Figure 1.11). Then is called the elongation rate in the x-direction. Write down the expression of .

Figure 1.11 Deformation of a specimen in elongation

Solution

Assuming a homogeneous deformation, the velocity vector is and

(1.34)

The sample section remains square during the deformation, so . Incompressibility implies . Therefore, and

(1.35)

1.1.4.3.2 Biaxial Stretching: Bubble Inflation

The inflation of a bubble of radius R and thickness e small compared to R is considered in Figure 1.12.

Write the rate-of-strain components in the r,θ,φ directions.

Write the continuity equation for an incompressible material and integrate it.

Show the equivalence between the continuity equation and the volume conservation.

Figure 1.12 Bubble inflation

Solution

(a) The bubble is assumed to remain spherical and to deform homogeneously so that the shear components are zero. The rate-of-strain components are as follows:

In the thickness (r) direction:

In the θ-direction:

In the φ-direction:

(b) For an incompressible material, , which can be integrated to obtain R2e

(c) This is equivalent to the global volume conservation: .

1.2 Stresses and Force Balances1.2.1 Stress Tensor1.2.1.1 Phenomenological Definitions1.2.1.1.1 Extension (or Compression) (Figure 1.13)

An extension force applied on a cylinder of section S induces a normal stress σnF /S.

Figure 1.13 Stress in extension

1.2.1.1.2 Simple Shear (Figure 1.14)

A force tangentially applied to a surface S yields a shear stress τF /S.

The units of the stresses are those of pressure: pascals (Pa).

Figure 1.14 Stress in simple shear

1.2.1.2 Stress Vector

Let us consider, in a more general situation, a surface element dS in a continuum. The part of the continuum located on one side of dS exerts on the other part a force dF. As the interactions between both parts of the continuum are at small distances, the stress vector T at a point O on this surface is defined as the limit:

(1.36)

At point O, the normal to the surface is defined by the unit vector, n, in the outward direction, as illustrated in Figure 1.15.

Figure 1.15 Stress applied to a surface element

The stress components can be obtained from projections of the stress vector:

Projection on n: σnT·n

where σn is the normal stress (in extension, σn > 0; in compression, σn < 0).

Projection on the surface: τ is the shear stress.

1.2.1.3 Stress Tensor

The stress vector cannot characterize the state of stresses at a given point since it is a function of the orientation of the surface element, that is, of n. Thus, a tensile force induces a stress on a surface element perpendicular to the orientation of the force, but it induces no stress on a parallel surface element (Figure 1.16).

Figure 1.16 Stress vector and surface orientation

The state of stresses is in fact characterized by the relation between T and n and, as we will see, this relation is tensorial. Let us consider an elementary tetrahedron OABC along the axes Oxyz (Figure 1.17): the x , y , and z components of the unit normal vector to the ABC plane are the ratios of the surfaces OAB , OBC , and OCA to ABC:

Figure 1.17 Stresses exerted on an elementary tetrahedron

Let us define the components of the stress tensor in the following table:

Projection on

of the stress vector exerted on the face normal to

Ox

Oy

Oz

Ox

σxx

σxy

σxz

Oy

σyx

σyy

σyz

Oz

σzx

σzy

σzz

The net surface forces acting along the three directions of the axes are as follows:

with OA , OB , OC being of the order of d; the surfaces OAB , OBC , and OCA are of the order of d2; and the volume OABC is of the order of d3. The surface forces are of the order of Td2 and the volume forces of the order of Fd3 (e. g., Fρg for the gravitational force per unit volume).

When the dimension d of the tetrahedron tends to zero, the volume forces become negligible compared with the surface forces, and the net forces, as expressed above, are equal to zero. Hence, in terms of the components of n:

(1.37)

This result can be written in tensorial notation as

(1.38)

where σ is the stress tensor, which contains three normal components and six shear components defined for the three axes. As in the case of the strain, the state of the stresses is described by a tensor.

1.2.1.4 Isotropic Stress or Hydrostatic Pressure

The hydrostatic pressure translates into a stress vector that is in the direction of n for any orientation of the surface:

(1.39)

The corresponding tensor is proportional to the unit tensor I:

(1.40)

1.2.1.5 Deviatoric Stress Tensor

For any general state of stresses, the pressure can be defined in terms of the trace of the stress tensor as

(1.41)

The pressure is independent of the axes since the trace of the stress tensor is an invariant (see Appendix 2, see Section 1.4.2). It could be positive (compressive state) or relatively negative (extensive state, possibly leading to cavitation problems in a liquid).

The stress tensor can be written as a sum of two terms, the pressure term and a traceless stress term, called the deviatoric stress tensor σ′:

(1.42)

Examples

Uniaxial extension (or compression):

(1.43)

Simple shear under a hydrostatic pressure p:

(1.44)

More generally, we will see that the stress tensor can be decomposed into an isotropic arbitrary part denoted as p′I, and a tensor called the extra-stress tensor σ′. The expressions of the constitutive equations in Chapter 2 will use either the deviatoric part of the stress tensor σ′ for viscous behaviors or the extra-stress tensor σ′ for viscoelastic behaviors (in this case, σ′ is no longer a deviator, and p′ is not the hydrostatic pressure).

1.2.2 Equation of Motion1.2.2.1 Force Balances

Considering an elementary volume of material with a characteristic dimension d:

The surface forces are of the order of d2, but the definition of the stress tensor is such that their contribution to a force balance is nil.

The volume forces (gravity, inertia) are of the order of d3, and they must balance the derivatives of the surface forces, which are also of the order of d3.

We will write that the resultant force is nil (Figure 1.18).

Figure 1.18 Balance of forces exerted on a volume element

The forces acting on a volume element dx dy dz are the following:

The mass force (generally gravity): Fdx dy dz

The inertial force: ργdx dy dzρ (du/dt) dx dy dz

The net surface force exerted by the surroundings in the x-direction:

and similar terms for the y and z-directions.

Dividing by dx dy dz and taking the limits, we obtain for the x , y , and z components:

(1.45)

The derivatives of σij are the components of a vector, which is the divergence of the tensor σ. Equation (1.45) may be written as

(1.46)

This is the equation of motion, also called the dynamic equilibrium. It is often convenient to express the stress tensor as the sum of the pressure and the deviatoric stress:

(1.47)

1.2.2.2 Torque Balances

Let us consider a small volume element of linear dimension d; the mass forces of the order of d3 induce torques of the order of d4. There is no mass torque, which would result in torques of the order of d3 (as in the case of a magnetic medium). Finally, the surface forces of the order of d2 induce torques of the order of d3, so only the net torque resulting from these forces must be equal to zero.

If we consider the moments about the z-axis (Figure 1.19), only the shear stresses σxy and σyx on the upper (U) and lateral (L) surfaces of the element dx dy dz lead to torques. They are obtained by taking the following vector products:

(1.48)

(1.49)

Figure 1.19 Torque balance on a volume element

A torque balance, in the absence of a mass torque, yields σxyσyx. In a similar way, σyzσzy and σzxσxz. The absence of a volume torque then implies the symmetry of the stress tensor. Therefore, as for the strain tensor ε, the stress tensor has only six independent components (three normal and three shear components).

1.2.3 Problems1.2.3.1 Shear Stress at the Surface of a Tube

A tube of length L and radius R is filled with a homogeneous material. A pressure difference Δp is applied between the end sections. The material necessarily exerts a shear stress on the wall of the tube. If the shear stress τ is uniform, show that

(1.50)

Solution

The shear stress τ applies on the lateral surface of the tube, 2πRL , inducing a force τ2πRL. This force is balanced by the pressure applied to the section of the tube, πR2: τ2πRLpπR2.

Therefore .

1.2.3.2 Stresses in a Shell

Consider a thin spherical shell of radius R and thickness e small compared to R. The internal pressure inside the sphere is p.

Obtain the expression of the stress tensor in the wall of the sphere and justify the thin-shell approximation.

Answer the same questions for a tube.

Solution

Sphere: The shear stresses are negligible compared to the elongational stresses σθθ , σφφ , and σrr. The sphere is cut in two parts in an equatorial plane perpendicular to the z-axis (see Figure 1.12). The force 2πReσθθ is exerted on the cut surface. On the upper half-sphere, the pressure force projected along the z-axis is

(1.51)

So 2πReσθθpπR2, that is,

(1.52)

Cutting the sphere by a plane perpendicular to the previous one leads to

(1.53)

The stress σrr is undefined, but its value is between 0 and p. If R is large compared to e (thin-shell approximation), then σrr is small compared to σθθ and σφφ.

Cylinder: For a cylinder of length L , radius R , and thickness e , a force balance leads to

(1.54)

(1.55)

Hence, and . If R is large compared to e , σrr is small compared to σθθ and σzz.

1.3 General Equations of Mechanics1.3.1 General Case

A problem in mechanics involves 10 unknown functions of x , y , z , and t:

the density ρ ,

the vector u with three components u , v , w (from which are obtained the strain and the rate of strain), and

the stress tensor σ with six components.

Four equations are available to solve the problem:

The equation of continuity (Eq. (1.23)):

The three components of the equation of motion (Eq. (1.46)):

We need six other equations. These are the constitutive relations between the stresses and the strain or rate of strain in a material element. The elastic behavior is well known; the constitutive relation in terms of the Lamé and Clapeyron coefficients λ and μ is

(1.56)

where δij is the Kronecker delta. This relation contains six equations. Other constitutive relations are introduced in Chapter 2.

The constitutive equation is the mathematical expression of the rheological behavior of a material, determined from experiments conducted in simple flows with a viscometer, a rheometer, or a tension or torsion device.

1.3.2 Incompressibility

For incompressible materials, ρ is constant, and the equation of continuity is simplified, but it is still useful (Eq. (1.24)):

On the other hand, as shown by Mandel (1974), the constitutive relation is no longer written in terms of the stress tensor σ, but in terms of a tensor σ′ that characterizes the state of stresses within an isotropic term (hydrostatic pressure). This stress tensor is frequently taken as the deviatoric stress tensor. Under these conditions, the constitutive relation contains only five equations. A given problem now represents nine unknowns and may be solved theoretically with the help of nine equations:

1.3.3 Planar Flow

We are frequently interested in planar flow problems. The most important one is planar deformation flow, defined by the following. The z-component of the velocity vector is zero, and u and v are independent of z. For any rheological behavior (see the problem in Section 1.3.4), σxzσyzσzz is constant. The problem is then reduced to two dimensions with only five unknowns: u , v , and σxx , σxy , σyy.

Example

The flow in the gap between two rolls in the calendering process is not strictly speaking a planar flow, except in the plane of symmetry, since the thickness of the bank in the material decreases with the distance from the center. However, over a large portion of the width, the approximation of planar flow is quite acceptable.

1.3.4 Problem: Stress Tensor in Simple Shear Flow

A material is submitted to a homogeneous simple shear flow under steady-state conditions with a velocity in the x-direction and varying with the y-axis. The velocity vector has only one nonzero component u in direction Ox depending only on y. The stresses in this material are then steady and uniform. For steady and uniform conditions and for a given rheological behavior, there is a unique relation between the tensors and σ.

For these considerations and making an appropriate change of coordinates, show that the stresses developed in simple shear for any rheological behavior are such that σxzσyz

Show that it is also the case for any type of planar flow.

Solution

In the coordinates (x , y , z) the rate-of-strain tensor is

(1.57)

and the general expression of the stress tensor is

(1.58)

The force acting on the surface defined by the unit normal vector n (i.e., surface normal to n) is T = σ· n, or

(1.59)

Let us consider the change of coordinates (x , y , z) to (x , y , −z). The tensor is not modified, and therefore the tensor σ is also unmodified. The coordinates of n and T are respectively (nx , ny , −nz) and (Tx , Ty , −Tz), and the same relation T = σ· n leads to

(1.60)

The equality of Tx , Ty , and Tz in these two sets of results for any values of nx , ny , and nz implies that σxzσyzx , y).

1.4 Appendices1.4.1 Appendix 1: Basic Formulae1.4.1.1 Cylindrical Coordinates

Gradient of a scalar:

(1.61)

Material or substantial derivative of a scalar:

(1.62)

Laplacian of a scalar:

(1.63)

Velocity-gradient tensor:

(1.64)

Rate-of-strain tensor:

(1.65)

Dynamic equilibrium:

(1.66)

Navier‒Stokes equations:

r-component:

(1.67)

θ-component:

(1.68)

z-component:

(1.69)

Material or substantial derivative of a tensor a:

(1.70)

1.4.1.2 Spherical Coordinates

Gradient of a scalar:

(1.71)

Material or substantial derivative of a scalar:

(1.72)

Laplacian of a scalar:

(1.73)

Velocity-gradient tensor:

(1.74)

Rate-of-strain tensor:

(1.75)

Dynamic equilibrium:

(1.76)

Navier‒Stokes equations:

(1.77)

where

(1.78)

1.4.2 Appendix 2: Invariants of a Tensor1.4.2.1 Definitions

A second-order tensor can be represented as a matrix M in a frame of reference R. Defining P as the matrix of transformation from the frame of reference R to R′, the tensor in the frame R′ is then given by

(1.79)

For many tensors, for example all symmetric tensors, there is a unique frame of reference in which M is a diagonal matrix:

(1.80)

Here, λi are the eigenvalues or principal values, and the frame of reference is defined by the directions of the eigenvectors, usually called in fluid mechanics the principal directions. If Mε, then λi are the principal extensions; if Mσ, the λi are the principal stresses. By definition, the eigenvalues are characteristics of the tensor at a given point and are independent of the frame of reference (or choice of coordinate system). They are solutions of the characteristic equation

(1.81)

where

(1.82)

I1, I2, and I3 are the invariants of the tensor M. We can define other invariants:

(1.83)

These invariants are in fact combinations of the first three:

(1.84)

1.4.2.2 Invariants Used in Fluid Mechanics

The most frequently used invariants in fluid mechanics are as follows:

The trace:

Hydrostatic pressure:

Compressibility:

For incompressible fluids, tr , and both sets of invariants (I and J) of the rate-of-strain tensor have the same absolute value.

The second invariant J2:

If one considers M as a tensor of nine coordinates mij in a nine-dimensional space, J2 is proportional to the square of the length of the vector. Thus, is a measure of the length of the vector, that is, of the scalar intensity of the tensor. Following the theory of plasticity, one can use the Von Mises criterion to write

(1.85)

where σy is the yield stress.

A generalized rate of deformation can be defined by

(1.86)

However, as shear deformations play a larger role than elongational deformations in many polymer processes, it is more appropriate to define the following generalized deformation rate:

(1.87)

In simple shear flow (Eq. (1.12)):

Hence, the generalized deformation rate reduces to the shear rate: .

In uniaxial extensional flow (Eq. (1.35)):

Hence, .

The third invariant:

For a large class of flow situations, the third invariant J3 is equal to zero, for example in planar and viscometric flows. In elongational situations such as those presented in Section 1.1.4.3, J3 is not zero.

References

Mandel J (1974). Introduction à la Mécanique des Milieux Continus Déformables, Editions Scientifiques de Pologne, Varsovie.

Salençon, J (1988). Mécanique des Milieux Continus, Ellipses, Paris.