1569905126

Peter K. Kennedy

Rong Zheng

Flow Analysis of

Injection Molds

Flow Analysis of

Injection Molds

Rong Zheng

Hanser Publishers, Munich Hanser Publications, Cincinnati

The Authors: Dr. Peter Kennedy,

Helmet Investments, 141/99 Spring St., Melbourne, Victoria 3000, Australia Dr. Rong Zheng,

School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia

Distributed in North and South America by: Hanser Publications

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Phone: (513) 527-8977 www.hanserpublications.com Distributed in all other countries by Carl Hanser Verlag

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www.hanser.de

The use of general descriptive names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

While the advice and information in this book are believed to be true and accurate at the date of going to press, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

Library of Congress Cataloging-in-Publication Data Kennedy, Peter (Peter K.)

Flow analysis of injection molds / Peter Kennedy, Rong Zheng. -- 2nd edition.

pages cm

Includes bibliographical references and index.

ISBN 978-1-56990-512-8 (hardcover) -- ISBN 978-1-56990-522-7 (e-book) (print) 1. Injection molding of plastics. 2. Mathematical modeling. I. Zheng, Rong, 1947- II. Title.

TP1150.K45 2012 668.4’120685--dc23

2012025721 Bibliografische Information Der Deutschen Bibliothek

Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über <http://dnb.d-nb.de> abrufbar. ISBN 978-1-56990-512-8

E-Book-ISBN 978-1-56990-522-7

All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying or by any information storage and retrieval system, without permission in writing from the publisher.

© Carl Hanser Verlag, Munich 2013 Production Management: Steffen Jörg

Coverconcept: Marc Müller-Bremer, www.rebranding.de, München Coverdesign: Stephan Rönigk

Printed and bound by CPI buch bücher gmbh Printed in Germany

for his love and professional spirit that have guided my life.

— Rong Zheng

To my Children William and Anthony

for their support, understanding and love.

We wish to record our sincere thanks to Professors Roger I. Tanner (University of Sydney), H.E.H. Meijer (Technische Universiteit Eindhoven), Xi-Jun Fan (University of Sydney), and Nhan Phan-Thien (National University of Singapore—formerly of the University of Sydney). Many results and ideas presented in this book came from their works and from collaborative research work with them and their colleagues. From this book one may see their deep in-fluence on our work. Thanks are also due to Professor Charles Tucker (University of Illinois, Urbana-Champaign), Professors Gerrit Peters, and Patrick Anderson (both of Technische Uni-versiteit Eindhoven) for fruitful discussions and advice from which we benefited.

We also want to thank our former Moldflow colleagues in Melbourne, Australia and Ithaca, USA with whom we both used to work. Our interactions with them broadened our knowledge in several different aspects and lead to deep friendships. Their work can also be seen in this book.

Much of our early work was conducted with several consortiums located in France and spon-sored by Moldflow Corporation and some other industrial partners. In particular, we would like to thank Professors G. Regnier (formerly ENSAM Paris, now Arts et Métiers ParisTech), R. Fulchiron (Université de Lyon), D. Delaunay (Université de Nantes), and Dr. V. Leo (Solvay) for participation in several projects that showed how complex the injection molding process is but nevertheless produced some results that are of practical use.

We are indebted to the Australian Cooperative Research Center for Polymers for providing an opportunity of doing collaborative research with research teams from Monash and Sydney Universities. In particular, we were grateful to obtain access to the Australian Synchrotron. Special thanks go to the former Moldflow Corporation (now part of Autodesk Inc.) for provid-ing an excellent workprovid-ing environment and constant support to both of us durprovid-ing the period we were working there.

Professors H.E.H. Meijer, Nhan Phan-Thien, and Roger I. Tanner reviewed the draft of the whole book and made very valuable comments and suggestions for improvement; their help is gratefully acknowledged.

We also wish to thank the editors of this book’s publisher for their patience and professional assistance.

On the personal side, we want to thank our families for the love, understanding, and encour-agement that sustained us during our confrontation with an important, but difficult, industrial problem.

Injection molding is an ideal process for fabricating large numbers of geometrically com-plex parts. Many everyday items are injection molded: mobile phone housings, automobile bumpers, television cabinets, compact discs, and lunch boxes are all examples of injection molded parts. Parts produced by the process are also becoming commonplace in less obvious applications. For example, the relatively new area of micro-injection molding is providing new methods of drug delivery and optical couplers [195].

Variations of injection molding that have been developed over the years include co-injection or two-component molding, water injection, and gas-assisted injection molding (GAIM). All these processes provide additional scope for designers of plastic parts. Excellent examples are provided by Neerincx [267] and Neerincx et al. [268, 269]. Indeed it is possible to combine these variations with each other or injection molding to achieve other processes. In particular, Neerincx and Meijer combined GAIM and two-component molding [270] to produce a part with unique qualities.

An important characteristic of injection molding, including variations, is that it may not be possible to fix a part defect in production by simply varying process conditions. Frequently the mold must be modified to overcome a problem. This is expensive and costs valuable time. It is far better to avoid problems in the design phase than to fix them in production. Conse-quentially, simulation of injection molding is industrially valuable.

Not surprisingly, there are several commercial companies offering software for simulation of injection molding and its variants. Due to the complexity of the physics of the process, vari-ous assumptions are made to simplify the mathematical model used for simulation. Over the years many descriptions of modeling and simulation of injection molding have appeared in academic journals and books. While readily available to specialist readers, an understanding of principles used in simulation software is difficult for nonspecialists to obtain. This is due to the multi-disciplinary nature of simulation software. In particular, aspects of rheology, ma-terials science, and numerical methods are used. There are some excellent books on polymer processing that discuss injection molding. One of the original classics was by Tadmor and Gogos [351]. This was followed by Tucker’s book [368] which focused on modeling for com-puter simulation. More recently, Osswald and Hernández-Ortiz [279] provided an overview of modeling and simulation for polymer processing, while Kamal et al. [190] have produced a book focused on injection molding that discusses variations and other aspects of the injection molding process.

Given the importance of injection molding as a process, and the simulation industry that has grown to support it, we believe there is a need for a book that deals solely with modeling and simulation of injection molding. One of the authors wrote a book in 1995 [196] along these lines. It discussed filling and packing phase simulation, but is no longer in print. Moreover, there have been many developments in modeling and simulation since that time.

The current book is intended to address this need. It provides a comprehensive description of modeling and simulation of injection molding. While some parts of the book may be relevant

X Preface

to other polymer forming processes, we assume injection molding is the process under discus-sion, and so do not deal with variants.

The book is divided into two parts and a considerable number of appendices. Each appendix is meant to provide detailed information on the topics discussed in the main parts of the book. Hopefully, moving specialist and routine information into appendices makes the book more readable.

Part I is written for the user of simulation software who seeks an explanation of the basic mod-eling and assumptions made. Modmod-eling and simulation details of filling, packing, residual stress, shrinkage, and warpage of amorphous, semi-crystalline, and fiber filled materials are described. Additionally, it introduces numerical methods for solving mathematical models of the process. This part is intended to be self-contained but presumes knowledge of algebra and calculus at the level of a degree in physical sciences or engineering. Tensor concepts are given in Appendix B.

Part II deals with improved modeling. This part is aimed at interested users of software, gradu-ate students, and researchers who are interested in enhancing simulation. A knowledge of the history of simulation is useful for anyone so disposed. Appendix A provides some background on both academic and commercial developments in simulation to around 2008. Much of the material presented in Part II covers developments from 2000 to the present. At the time of writing, this information is not implemented in commercial simulation software, and is meant to be a starting point for improvement in modeling and simulation. It presents some models that incorporate more of the physics of the molding process. Although we present some pos-sible approaches, we do not cover all areas of improvement. We do, however, try to reference other approaches to the problems we consider. In particular, we focus on fiber-filled and semi-crystalline materials, but some ideas may be applied to amorphous materials. Hopefully it will be a source of ideas that lead to better simulations. Part II uses more advanced ideas of tensor calculus. Where these are not provided in the text, we prescribe external references.

We hope our readers enjoy the challenge of modeling and simulating the injection mold-ing process. Injection moldmold-ing is a technology that has been around for approximately 140 years [172]. However, it was only in the 1950s, with the development of the reciprocating screw method, that the process showed its true potential.

Despite the immaturity of computer technology, simulation of injection molding can be traced to 1960 [367]. Since then it has become a field of both academic and commercial interest. Moreover, the physics of injection molding are still being researched. It is this latter aspect that provides us with the hope that this book will inspire others to improve simulation by improved modeling and by taking advantage of the computational power available today and in the future.

Preface

IX

Notation

XXI

I

The Current Status of Simulation

1

1

Introduction

3

1.1 The Injection Molding Process . . . 3

1.2 Molding Terminology . . . 4

1.3 What is Simulation? . . . 5

1.4 The Challenges for Simulation. . . 6

1.4.1 Basic Physics of the Process . . . 7

1.5 Why Simulate Injection Molding? . . . 7

1.6 How Good is Simulation? . . . 8

2

Stress and Strain in Fluid Mechanics

11

2.1 Stress in Fluids . . . 11

2.1.1 The Stress Tensor . . . 11

2.1.2 The Extra Stress Tensor . . . 14

2.1.3 Rate of Strain Tensor . . . 14

2.2 Newtonian and Non-Newtonian Fluids . . . 15

2.3 The Generalized Newtonian Fluid . . . 16

3

Material Properties of Polymers

19

3.1 Types of Polymers . . . 19

3.2 Amorphous Polymers . . . 20

3.3 Semi-Crystalline Polymers . . . 20

3.4 Overview of Material Properties for Simulation . . . 21

3.5 Viscosity . . . 22

3.6 Modeling Viscosity . . . 23

XII Contents

3.6.2 The Power Law Model . . . 23

3.6.3 The Carreau Model . . . 23

3.6.4 The Cross Model . . . 24

3.6.5 Incorporation of Temperature Effects . . . 24

3.6.6 The Solidification Problem . . . 25

3.7 Thermal Properties . . . 26

3.7.1 Specific Heat Capacity . . . 26

3.7.2 Thermal Conductivity . . . 27

3.8 Thermodynamic Relationships . . . 29

3.8.1 Expansivity and Compressibility . . . 29

3.9 Pressure-Volume-Temperature (PVT) Data . . . 31

3.10 Fiber Orientation . . . 31

3.11 Shrinkage and Warpage. . . 32

4

Governing Equations

35

4.1 Introduction . . . 35

4.2 Mathematical Preliminaries . . . 35

4.2.1 The Material Derivative . . . 35

4.2.2 The Gauss Divergence Theorem . . . 36

4.2.3 Reynolds Transport Theorem . . . 37

4.2.4 Integration by Parts. . . 37

4.3 Conservation of Mass . . . 38

4.4 Conservation of Momentum . . . 38

4.5 Conservation of Energy . . . 40

4.5.1 Relating Specific Energy to Temperature . . . 43

4.5.2 The Energy Equation in Terms of Temperature . . . 45

4.6 Boundary Conditions . . . 46

4.6.1 Pressure and Flow Rate Boundary Conditions. . . 47

4.6.2 Temperature Boundary Conditions . . . 48

4.6.3 Mold Deformation Boundary Conditions . . . 48

4.6.3.1 Thin Cavities . . . 48

4.6.3.2 Long Cores and Mold Inserts . . . 49

4.7 Fiber-Filled Materials . . . 49

4.7.1 Fiber Concentration . . . 50

4.7.2 Jeffery’s Equation . . . 50

4.7.3 A Statistical Approach . . . 51

4.7.4 Mechanical Properties . . . 52

4.8 Shrinkage and Warpage. . . 52

5

Approximations for Injection Molding

55

5.1 Introduction . . . 55

5.2 Material Property Approximations . . . 56

5.3 Filling, Packing, and Cooling Analysis . . . 56

5.3.1 The Thermal Source Term in the Energy Equation . . . 57

5.3.2 Viscosity Modeling . . . 57

5.3.3 Specific Heat Capacity . . . 58

5.3.4 Thermal Conductivity . . . 58

5.3.4.1 Unfilled Amorphous . . . 58

5.3.4.2 Unfilled Semi-Crystalline . . . 59

5.3.4.3 Filled Materials . . . 59

5.3.5 No-Flow or Transition Temperature . . . 59

5.3.6 Pressure-Volume-Temperature (PVT) Data . . . 61

5.3.7 Fiber Orientation, Shrinkage, and Warpage. . . 62

5.3.7.1 Fiber Orientation Analysis . . . 62

5.3.7.2 Shrinkage and Warpage Analysis . . . 63

5.4 Summary of Material Assumptions . . . 63

5.5 Governing Equations. . . 64

5.6 The 2.5D Approximation . . . 65

5.6.1 Governing Equations in Cartesian Coordinates . . . 66

5.6.1.1 Conservation of Mass. . . 66

5.6.1.2 Conservation of Momentum . . . 68

5.6.1.3 Conservation of Energy . . . 68

5.6.2 Estimation of Relevant Terms . . . 69

5.6.3 Velocity in the z Direction . . . . 71

5.6.4 Integration of the Momentum Equations . . . 72

5.6.5 Integration of the Continuity Equation . . . 75

5.6.5.1 Summary of the 2.5D Approximation . . . 77

5.7 Mold Cooling Analysis . . . 78

5.8 Fiber Orientation . . . 80 5.8.1 Orientation Tensors . . . 80 5.8.2 Folgar-Tucker Equation . . . 81 5.8.3 Closure Approximations . . . 81 5.8.3.1 Linear Closure . . . 82 5.8.3.2 Quadratic Closure . . . 82 5.8.3.3 Hybrid Closure . . . 82 5.8.3.4 Orthotropic Closure . . . 83

XIV Contents

5.9 Shrinkage and Warpage. . . 84

5.9.1 Shrinkage Prediction . . . 85

5.9.1.1 Residual Strain Methods . . . 85

5.9.1.2 Residual Stress Models . . . 87

5.10 The 2.5D Approximation for Runners . . . 91

5.10.1 Conservation of Mass for Runners . . . 92

5.10.2 Conservation of Momentum for Runners . . . 93

5.10.3 Conservation of Energy for Runners . . . 93

5.10.4 Integration of the Momentum Equation for Runners . . . 94

5.10.5 Integration of the Continuity Equation for Runners . . . 96

6

Numerical Methods for Solution

99

6.1 Midplane Methods . . . 99

6.1.1 Extraction of a Midplane from a 3D Model . . . 100

6.1.2 Dual Domain Analysis for Flow . . . 101

6.1.3 Dual Domain Structural Analysis . . . 103

6.1.4 Warpage Analysis Using the Dual Domain FEM. . . 107

6.2 3D Analysis . . . 107

6.2.1 Finite Volume Methods . . . 107

6.2.2 A Pseudo-3D Approach . . . 108

6.3 Warpage and Shrinkage Analysis in 3D . . . 108

6.4 3D Analysis of Runner Systems . . . 109

II

Improving Molding Simulation

111

7

Improved Fiber Orientation Modeling

113

7.1 Introduction . . . 113 7.2 ARD Model . . . 114 7.2.1 Evolution Equation . . . 114 7.2.2 Direct Simulation . . . 116 7.2.3 Calculation of CI. . . 117 7.3 RSC Model . . . 117 7.4 Suspension Rheology . . . 118

8

Improved Mechanical Property Modeling

123

8.1 Introduction . . . 123

8.2 Unidirectional Composites. . . 124

8.2.1 Effective Stiffness . . . 124

8.2.2 Effective Thermal Expansion Coefficients . . . 126

8.2.3 Effects of Fiber Concentration and Aspect Ratio . . . 126

8.2.3.1 Effect of Fiber Concentration. . . 126

8.2.3.2 Effect of Fiber Aspect Ratio . . . 127

8.3 Fiber Orientation Averaging. . . 130

9

Long Fiber-Filled Materials

131

9.1 Fiber Orientation Evolution Model . . . 131

9.2 Flow-Induced Fiber Migration Model . . . 132

9.3 Fiber Length Attrition Model. . . 134

9.4 Uniaxial Tensile Strength Model . . . 135

9.5 Flexible Fiber Modeling. . . 136

9.5.1 Direct Simulation Methods. . . 136

9.5.2 Continuum Modeling . . . 139

10

Crystallization

141

10.1 Quiescent Crystallization . . . 141

10.1.1 The Kolmogoroff-Avrami-Evans Model . . . 142

10.1.2 The Rate Equations of Schneider . . . 143

10.1.3 Quiescent Nuclei Number Density . . . 144

10.1.4 Growth Rate of Spherulites . . . 145

10.1.5 Material Characterization . . . 146

10.1.5.1 Half-Crystallization Time . . . 146

10.1.5.2 Equilibrium Melting Temperature . . . 146

10.1.5.3 Crystal Growth Rate. . . 148

10.2 Flow-Induced Crystallization . . . 149

10.2.1 Enhanced Nucleation . . . 150

10.2.2 Critical Parameters . . . 151

10.2.3 Shish-Kebab Structure . . . 152

10.2.4 Material Characterization . . . 152

11

Effects of Crystallization on Rheology and Thermal Properties

155

11.1 Effects of Crystallization on Rheology . . . 155

XVI Contents

11.1.2 Two-Phase Model . . . 157

11.2 Effect of Crystallization on PVT. . . 159

11.3 Effect of Crystallization on Specific Heat Capacity . . . 160

11.4 Effect of Crystallization on Thermal Conductivity . . . 161

11.4.1 Non-Fourier Thermal Conduction . . . 161

11.4.2 Van den Brule’s Law for Amorphous Polymers . . . 162

11.4.3 Extending the Van den Brule Approach to Semi-Crystalline Polymers. . . 162

11.5 Effect of Crystallization on Heat Transfer . . . 164

11.5.1 Stefan’s Solution . . . 164

11.5.2 Numerical Solution with Crystallization Kinetics . . . 165

11.6 Modification to the Hele-Shaw Equation . . . 166

12

Colorant Effects

167

13

Prediction of Post-Molding Shrinkage and Warpage

175

13.1 Introduction . . . 175

13.2 Governing Equations. . . 176

13.3 Constitutive Equations . . . 177

13.3.1 Viscoelastic Effect. . . 177

13.3.2 Thermal Expansion Effect . . . 178

14

Additional Issues of Injection-Molding Simulation

181

14.1 Weldlines . . . 181

14.2 Core Shift . . . 182

14.3 Non-Conventional Injection Molds . . . 182

14.3.1 Overmolding . . . 182

14.3.2 Gas-Assisted Injection Molding. . . 183

14.3.3 Microcellular Injection Foaming Molding . . . 186

14.3.4 Micro-Injection Molding . . . 188

14.4.1 Flow-Induced Residual Stress and Birefringence . . . 191

14.4.2 Viscoelastic Instability . . . 193

14.4.3 Viscoelastic Suspensions . . . 194

14.5 Other Numerical Methods . . . 196

14.5.1 Molecular Dynamics Simulation . . . 196

14.5.2 Meshless Methods . . . 197

15

Epilogue

201

Appendices

203

A

History of Injection-Molding Simulation

205

A.1 Early Academic Work on Simulation . . . 205

A.2 Early Commercial Simulation . . . 206

A.3 Simulation in the Eighties . . . 208

A.3.1 Academic Work in the Eighties. . . 209

A.3.1.1 Mold Filling . . . 209

A.3.1.2 Mold Cooling . . . 211

A.3.1.3 Warpage Analysis. . . 212

A.3.2 Commercial Simulation in the Eighties. . . 212

A.3.2.1 Codes Developed by Large Industrials and Not for Sale . . . 214

A.3.2.2 Codes Developed by Large Industrials for Sale in the Marketplace 214 A.3.2.3 Companies Devoted to Developing and Selling Simulation Codes 215 A.4 Simulation in the Nineties. . . 216

A.4.1 Academic Work in the Nineties . . . 217

A.4.2 Commercial Developments in the Nineties . . . 218

A.4.2.1 SDRC . . . 218

A.4.2.2 Moldflow . . . 219

A.4.2.3 AC Technology/C-MOLD . . . 220

A.4.2.4 Simcon. . . 220

A.4.2.5 Sigma Engineering . . . 220

A.4.2.6 Timon . . . 221

A.4.2.7 Transvalor . . . 221

A.4.2.8 CoreTech Systems . . . 221

A.5 Simulation Science Since 2000 . . . 221

A.5.1 Commercial Developments Since 2000. . . 223

A.5.1.1 Moldflow . . . 224

XVIII Contents

A.5.1.3 CoreTech Systems . . . 225

A.5.1.4 Autodesk. . . 225

A.5.2 Note for Students . . . 225

B

Tensor Notation

227

B.1 Index Notation . . . 227

B.2 Einstein Summation Convention . . . 228

B.3 Kronecker Delta. . . 229

B.4 Alternating Tensor . . . 229

B.5 Product Operations of Two Tensors . . . 230

B.6 Transpose Operation . . . 230

B.7 Transformation of Principal Axes . . . 231

B.8 Gradient of a Field . . . 233

B.9 Unit Vector p and Operator∂/∂p ... 233

B.10 Identities . . . 234

C

Derivation of Fiber Evolution Equations

235

C.1 The Langevin Equation . . . 235

C.2 Probability Density Function and Orientation Tensors . . . 237

C.3 Equations of Change for the Orientation Tensors. . . 238

C.3.1 Isotropic Rotary Diffusion Model (Folgar-Tucker Model) . . . 239

C.3.2 Anisotropic Rotary Diffusion Model . . . 241

D

Dimensional Analysis of Governing Equations

243

D.1 Conservation of Mass . . . 244

D.2 Conservation of Momentum . . . 245

D.3 The Energy Equation . . . 248

D.4 Summary . . . 250

D.4.1 Conservation of Mass . . . 250

D.4.2 Conservation of Momentum . . . 250

D.4.3 Energy Equation . . . 251

E

The Finite Difference Method

253

E.1 Introduction to the Finite Difference Method . . . 253

E.1.1 A Simple Example . . . 255

E.2 Application to Temperature Calculation . . . 257

E.2.1 Explicit Methods . . . 257

E.2.1.1 Stability Criteria for Explicit Methods . . . 258

F

The Finite Element Method

261

F.1 Basic Terminology . . . 261

F.2 The Finite Element Approach. . . 262

F.2.1 Geometric Modeling of the Solution Domain . . . 262

F.2.2 Meshing . . . 263

F.2.3 Derivation of Element Equations . . . 263

F.2.4 Assembly of Element Equations . . . 263

F.2.5 Application of Boundary Conditions . . . 264

F.2.6 Solution of the System Equations . . . 264

F.2.7 Display of Results . . . 264

F.3 The Nature of a Finite Element Solution . . . 265

F.4 Shape Functions . . . 267

F.5 Approximating Nodal Values. . . 267

F.5.1 Weighted Residual Methods. . . 268

F.6 Constraint Equations . . . 268

F.6.1 Special Case 1: Two Unknowns Equal . . . 271

F.6.2 Special Case 2: One Known Constraint . . . 272

F.7 A One-Dimensional Problem Solved Using the FEM . . . 273

F.7.1 Meshing . . . 273

F.7.2 Derivation of Element Equations . . . 274

F.7.3 Assembly . . . 278

F.7.4 Application of Boundary Conditions . . . 280

F.7.5 Solution of System Equations . . . 281

G

Numerical Methods for the 2.5D Approximation

283

G.1 Overview of Solution Process . . . 283

G.1.1 Numerical Methods . . . 284

G.2 Finite Element Formulation for the Pressure Field . . . 285

G.2.1 Interpolation Functions . . . 285

G.2.2 Area Coordinates . . . 286

G.3 Finite Element Derivation . . . 287

G.3.1 Assembly of Element Equations and Solution . . . 295

G.4 Solution of the Energy Equation. . . 296

G.4.1 Finite Difference Discretization . . . 296

G.4.2 Solution of the Conduction Problem . . . 297

G.4.3 Explicit Method . . . 297

G.5 Flow Front Advancement . . . 298

XX Contents

H

Three-Dimensional FEM for Mold Filling Analysis

303

H.1 Governing Equations. . . 303

H.2 Weak Formulations . . . 304

H.3 Finite Element Matrix Formulations . . . 305

H.4 Solution Procedures . . . 309

H.5 Flow-Front Advancement . . . 310

H.6 Numerical Solution For Temperature Field . . . 311

I

Level Set Method

313

J

Full Form of Mori-Tanaka Model

317

J.1 Eshelby Tensor Components. . . 317

J.1.1 Material with Isotropic Matrix and Inclusions. . . 317

J.1.2 General Anisotropic Materials . . . 318

J.2 Expanded Mori-Tanaka Equation . . . 319

J.2.1 Contracted Notation for Stiffness Tensor and Compliance Tensor . . . 319

J.2.2 Inverse of a Matrix . . . 319

J.2.3 Expanded Expression of the Mori-Tanaka Equation . . . 320

Bibliography

321

Symbols which have more than one meaning are listed with a semicolon dividing the mean-ings. To avoid being too lengthy and jumbled, not all symbols and their definitions used in the book are included in the notation list, but they are defined throughout the text.

Roman symbols

a thermal diffusivity k/(ρcp)

ai j second-order fiber orientation tensor

ai j kl fourth-order fiber orientation tensor

ar fiber aspect ratio

aT time-temperature shift factor

A area

b FENE-P model parameter

A (Ai j kl) strain-concentration tensor

c pseudo-concentration parameter

cp specific heat capacity under constant pressure

cv specific heat capacity under constant volume

C heat capacity

C₁g, C₂g WLF universal constants (C₁g= 17.44, C2g= 51.6 K)

C₁0, C₂0 WLF constants

CB stress-optical coefficient

CI interaction coefficient in Folgar-Tucker model

Ct stress-thermal coefficient

C (Ci j) interaction coefficient tensor in anisotropic rotary diffusion model

C (Ci j kl) stiffness tensor (also called elasticity tensor)

d diameter; distance function D(r ) diffusion coefficient (isotropic) D (Di j) rate-of-deformation tensor1₂ ³_∂v i ∂xj + ∂vj ∂xi ´

D(r )(D(r )_{i j}) diffusion coefficient tensor Ea activation energy

E (Ei j kl) Eshelby tensor

f function

XXII Notation

△Fq difference of free energies between melt and crystalline phases under

quies-cent condition F (Fi) force vector

g (gi) acceleration vector due to gravity

G radial growth rate of spherulite; shear modulus GN melt plateau modulus

H cavity half-thickness H (t ) Heaviside unit step function

ˆ

H specific enthalpy

△Hc latent heat of crystallization for perfect crystals

I (δi j) unit tensor (also called the Kronecker tensor)

I (Ii j kl) fourth-order unit tensor

J Jocobian of coordinate transformations

k thermal conductivity

kB Boltzmann’s constant (1.380658 × 10−23J/K)

k (ki j) thermal conductivity tensor

l length

L (Li j) velocity gradient tensor∂vi/∂xj

Mn number-average molecular weight

Mw weight-average molecular weight

n power-law exponent

n0 number of molecules per unit volume

n (ni) outward-pointing unit normal vector

N nuclei number density

Nf flow-induced nuclei number density

Np particle number

Nq quiescent nuclei number density

N0 constant nuclei number density

N (Ni) particle flux

O(A) mathematical symbol reading as the order of magnitude of A

p pressure

pt thermodynamic pressure

p (pi) orientation unit vector

q (qi) heat flux vector; orientation vector q = |q|p

Q heat

R radius

Rg gas constant (8.3143 J / mol·K)

S2 two-dimensional fluidity for 2.5D cavity approximation

S3 three-dimensional fluidity for pseudo 3D approximation

S∥ _{shrinkage parallel to flow direction}

S⊥ shrinkage perpendicular to flow direction ˆ

S specific entropy

S (Si j kl) elastic compliance tensor

t , t′ time

t1/2 half-crystallization time

t (ti) traction vector (also called stress vector)

T temperature; as superscript denotes transpose of a tensor Tg glass transition temperature

T_mo equilibrium melting temperature

u (ui) velocity vector; unit orientation vector; displacement vector

ˆ

U internal specific energy U∗ _{activation energy} v (vi) velocity vector V volume ˆ V specific volume W i Weissenberg number W (Wi j) vorticity tensor12 ³_∂v i ∂xj − ∂vj ∂xi ´ Greek symbols α relative crystallinity

α (αi j) linear thermal expansion coefficient tensor

β coefficient of volume expansion; empirical parameter of some equations

γ shear strain

˙

γ generalized strain rate ˙

γ ( ˙γi j) shear strain rate tensor

δ(t) Dirac delta function (also called impulse function) δi j Kronecker tensor (also called unit tensor)

ε (εi j) strain tensor

ζ dimensionless drag coefficient

η shear viscosity

η0 zero shear rate viscosity

λ time constant

µ viscosity

XXIV Notation

ξ slip parameter 2/(a2

r+ 1)

ξ(t) pseudo time

ρ mass density

σ surface tension coefficient, tensile strength σb tensile strength at perfectly bounded interface

σw tensile strength at weldline interface

σ (σi j) stress tensor

τ (τi j) extra stress tensor

φ volume fraction

χ absolute crystallinity

χ∞ ultimate absolute crystallinity

ψ probability density ω angular velocity; frequency

L (Li j) effective velocity gradient Li j− ξDi j

Operator symbols

D/D t material derivative∂/∂t + vk_∂x∂_k

△/△t upper upper convected derivative defined as △( )i j △t = ∂( )i j ∂t + vk ∂( )i j ∂xk − Li k ( )k j− Lj k( )ki

1.1 A simple two-cavity mold . . . 5 1.2 A simple two-cavity mold showing runners and gates. . . 6 2.1 Definition of the stress vector . . . 12 2.2 Resolution of stress vector . . . 13 2.3 Stress components . . . 13 3.1 Steady simple shear flow . . . 22 3.2 In unbalanced flow, some regions of the molding may be in the packing phase

with very low shear rates and high pressures, while material near the flow front is at low pressure but high shear rate . . . 24 3.3 Definition of thermal conductivity . . . 28 3.4 PVT surface . . . 30 4.1 Boundary conditions for simulation of filling, packing, and cooling . . . 47 5.1 Need for a transition or no-flow temperature . . . 60 5.2 Mold for which material is in packing and filling phases . . . 61 5.3 Thin-walled cavity with coordinates systems defined at two points. . . 66 5.4 Definition of frozen layer thickness . . . 72 5.5 Schematic representations of fiber orientation distributions (a) fully aligned in

the 1-direction; (b) random in the 1-2 plane; (c) random in 3D space . . . 81 5.6 A comparison of the simulated CI for ar = 10, 16.9, 20, 30, and 31.9 [289] with

experimental data of Folgar and Tucker [122] (reproduced from Phan-Thien et al. [289] with permission from Elsevier) . . . 84 5.7 Actual sample for shrinkage measurement . . . 86 5.8 Simulated and measured packing pressure versus time results for different

tran-sition temperatures . . . 90 5.9 Calculated shrinkage in the parallel direction for different no-flow or transition

temperatures. The measured value from Luye [233] is 0.8% . . . 90 5.10 Geometry and coordinate system for runners . . . 91 6.1 Generation of a midplane mesh . . . 100 6.2 3D representation of a complex injection molding . . . 101

XXVI List of Figures

6.3 Dual Domain flow analysis; (a) depicts injection into the center of a rectangular plate; (b) shows the flow in the cross-section of the plate; (c) shows the flow front advancement on the surface mesh, and (d) shows the use of a connector element to ensure physical agreement with the true flow shown in (b) . . . 102 6.4 Dual Domain flow analysis for a part with two ribs . . . 103 6.5 A simple plate may be decomposed into two parts, each of half the original

thick-ness, and perfectly bonded together . . . 104 6.6 Eccentric shell element for structural analysis. . . 104 6.7 Structural elements matched for Dual Domain analysis . . . 105 6.8 Elements on top and bottom surfaces are generally not coincident; that is, the

normal from node n of the bottom element, intersects the top element at some point p within the element. In this case, interpolation is required . . . 106 6.9 In 3D analysis, temperature is correctly convected around changes in direction

in runners. In these cases, the temperature differences due to shear heating may result in an imbalanced filling of cavities despite the naturally balanced feed sys-tem . . . 110 7.1 Direct simulation results of the fiber configuration at different strains

(repro-duced from Fan et al. [103] with permission of Elsevier) . . . 116 8.1 Reduced effective moduli scaled by Em vs. fiber volume fraction for aR = 20.

Predicted using the Mori-Tanaka model . . . 127 8.2 Effective Poisson’s ratios vs. fiber volume fraction for aR= 20. Predicted using

the Mori-Tanaka model . . . 127 8.3 Effective coefficients of thermal expansion vs. fiber volume fraction for aR= 20.

Predicted using the Rosen-Hanshin model . . . 128 8.4 Reduced effective moduli scaled by Emvs. fiber aspect ratio forφ = 0.20.

Pre-dicted using the Mori-Tanaka model . . . 128 8.5 Effective Poisson’s ratios vs. fiber aspect ratio forφ = 0.20. Predicted using the

Mori-Tanaka model . . . 129 8.6 Effective coefficients of thermal expansion vs. fiber aspect ratio forφ = 0.20.

Pre-dicted using the Rosen-Hanshin model . . . 129 9.1 Schematic representations of (a) short-fiber pellet and (b) long-fiber pellet used

for injection molding . . . 132 9.2 Schematic representations of flexible fiber models . . . 138 9.3 Bead-rod model of Strautins and Latz [346] . . . 139 10.1 Schematic representations of (a) spherulite structure, and (b) shish-kebab

struc-ture (from Zhao et al. [417] with permission from Cambridge University Press) . . . 142 10.2 Nuclei number density as a function of supercooling temperature for a sample

of industrial iPP (reproduced from Koscher and Fulchiron [211] with permission from Elsevier) . . . 144

10.3 Isothermal crystallization curve of the Borealis iPP sample at 132◦_{C. Inset:}

Varia-tion of half-crystallizaVaria-tion time with crystallizaVaria-tion temperature . . . 147 10.4 Heat flow curves for Borealis iPP. . . 147 10.5 Determination of equilibrium melting temperature using Hoffman-Weeks

method, for Borealis iPP. Melting point data measured on samples isothermally crystallized at different Tcs were used to determine the equilibrium melting

tem-perature . . . 148 10.6 Isothermal crystal growth for Borealis iPP sample at 132◦_{C under quiescent}

con-dition . . . 149 10.7 Two-dimensional SAXS image patterns at different distances from the skin

sur-face to the mid-sursur-face for an iPP (from Zhu and Edward [428], with permission from American Chemical Society) . . . 153 11.1 PVT diagram for different cooling rates (from Luyé et al. [234], with permission

from John Wiley and Sons) . . . 160 11.2 Undisturbed equilibrium thermal conductivity against temperature for

polypropylene (from Speight et al. [339]) . . . 163 11.3 Temperature evolution at the core region of an injection molded part . . . 165 12.1 The molecular structures of two types of blue pigments: (a) the UB-colorant; (b)

the CuPc-colorant (reproduced from Lee Wo and Tanner [404], with permission from Springer) . . . 168 12.2 Morphologies of (a) virgin iPP at T = 132◦_{C, t = 180 s; (b) iPP mixed with UB}

colorant at T = 132◦_{C, t = 180 s, and (c) iPP mixed with CuPc colorant at T =}

140◦_{C, t = 150 s, during quiescent crystallization ... 169}

12.3 Specific heat capacities of three samples: virgin iPP, UB-colored iPP (0.8% col-orant by weight), and CuPc-colored iPP (0.8% colcol-orant by weight), denoted by PP, PP+08U, and PP+08P, respectively (Zheng et al. [425]). . . 170 12.4 Half-crystallization time vs. crystallization temperature of three samples:

vir-gin iPP, UB-colored iPP (0.8% colorant by weight), and CuPc-colored iPP (0.8% colorant by weight), denoted by PP, PP+08U and PP+08P, respectively (Zheng et al. [425]) . . . 170 12.5 Half-crystallization time vs. short-term shear rate for virgin iPP. Shearing time 1

sec; temperatures: 132◦C and 136◦C. Symbols are experimental data, and solid and dotted lines are from modeled results (Zheng et al. [425]) . . . 171 12.6 Half-crystallization time vs. short-term shear rate for 0.8% UB-colored iPP.

Shearing time 1 sec; temperatures: 132◦_{C and 136}◦_{C. Symbols are experimental}

data, and the solid and dotted lines are modeled results (Zheng et al. [425]) . . . 172 12.7 Half-crystallization time vs. short-term shear rate for 0.8% CuPc-colored iPP.

Shearing time 1 sec; temperatures: 144◦C and 148◦C. Symbols are experimental data, and solid and dotted lines are modeled results (Zheng et al. [425]) . . . 172 12.8 Experimental and predicted parallel and perpendicular shrinkage for the virgin

XXVIII List of Figures

12.9 Experimental and predicted parallel and perpendicular shrinkage for the iPP with UB colorant (Zheng et al. [425]) . . . 173 12.10 Experimental and predicted parallel and perpendicular shrinkage for the iPP

with CuPc colorant (Zheng et al. [425]) . . . 174 14.1 Pressure development during filling in (a) conventional injection molding and

(b) gas-injection molding (adapted from Turng [373]) . . . 184 14.2 Dynamic contact angle. . . 190 14.3 Unstable fountain flow (reproduced from Grillet et al. [132]) . . . 194 A.1 Flow progresses faster in the thick rim of the box and creates an air trap on the

front (shown) and rear sides . . . 207 A.2 The “layflat” is created by unfolding the box to lie in a plane. Note though that

the correct thickness for each surface of the box is retained. Dark lines represent possible flow paths for analysis . . . 207 A.3 An automotive component and its associated layflat model . . . 208 E.1 A finite difference mesh in the x-y plane . . . 253 E.2 A simple mesh for a one-dimensional finite difference solution . . . 255 F.1 Approximation of a simple curve . . . 265 F.2 Finite element solution of a two-dimensional problem . . . 266 F.3 Exact solution to 1D FEM sample problem . . . 273 F.4 Mesh for sample problem. . . 274 F.5 Finite element solution for sample problem using linear interpolation . . . 275 F.6 Comparison of exact and approximate FEM solution for the sample problem . . . 282 G.1 Area (barycentric) coordinates for triangular elements . . . 286 G.2 Geometry of triangular element for the pressure field solution . . . 291 H.1 MINI finite element with linear interpolation and bubble enrichment for velocity,

and linear interpolation for pressure . . . 308 I.1 Evolution of a free surface simulated by the level set method (provided by Dr.

3.1 Specific Heat of Some Polymers and Metals . . . 27 3.2 Thermal Conductivity of Polymers and Metals . . . 28 5.1 Molding Conditions . . . 89 7.1 Asymptotic Values of Ai, i = 1 to 4 ... 119

8.1 Property Data for Components of a Short Glass Fiber-Reinforced Composite . . . 126 J.1 Relation Between Indices in Contracted and Tensor Notations . . . 319

1

Introduction

Automotive and consumer electronics are two disparate industries that rely heavily on injec-tion molding. Moreover, they both involve updating of products on a regular basis. Both these industries have been leaders in the development of concurrent engineering, meaning the par-allelization of tasks from inception to manufacture. Regardless of the degree to which concur-rent engineering is practiced, there is no doubt that simulation is a valuable aid in linking de-sign to manufacture. For injection molding, the benefit of simulation is based on the fact that it is cheaper and faster to avoid problems in the design phase than to fix them in production. Simulation of injection molding, particularly flow analysis, has had a major impact on indus-try. Indeed, the editors of Plastics Technology magazine, a leading industrial journal, proposed a list of the fifty most important innovations in the plastic industry [264]. Number one was the reciprocating screw injection molding machine, while simulation of injection molding was listed nineteenth. Whether one agrees with the editor’s ranking or not, simulation of injection molding has been an outstanding aid to industry.

In this chapter, we review the molding process, terminology, and simulation so as to provide a background for the remainder of the book.

1.1

The Injection Molding Process

Injection molding is a cyclic process. Initially, the mold is closed to form the cavity into which the material is injected. The screw then moves forward as a piston, forcing molten material ahead of it into the cavity. This is the injection or filling phase. When filling is complete, pres-sure is maintained on the melt and the packing phase begins. The purpose of the packing phase is to add further material to compensate for shrinkage of material as it cools in the cav-ity. At some time during packing, the gate freezes and the cavity is effectively isolated from the pressure applied by the melt in the barrel. This marks the beginning of the cooling phase in which the material continues to cool until the component has sufficient mechanical stiffness to be ejected from the mold. During cooling, the screw starts to rotate and moves back. The rotation assists plastication of the material and a new charge of melt is created at the head of the screw. When the molded part is sufficiently solid, the mold opens and the part is ejected. The mold then closes and the cycle begins again.

In summary, the injection molding process is characterized by the following phases: 1. Mold closing

2. Injection 3. Packing 4. Cooling

5. Plastication and screw back 6. Ejection

Most effort in computer simulation has been devoted to phases 2–4. There have been signifi-cant advances in modeling plastication [162, 163, 173, 245, 261, 352, 387, 408] but generally, for molding simulation, it is assumed that the melt enters the cavity with a prescribed flow rate or pressure and a uniform temperature. While this may be reasonable, the ultimate goal of sim-ulation is to predict the properties of the molded material, both during and after the molding process. This requires a deep understanding of crystallization for semi-crystalline materials. Vleeshouwers and Meijer [389] reported that the effect of shear on crystallization of isotac-tic polypropylene at 200◦C, was still evident after a period of 30 min with the sample main-tained at 200◦C . Hence the plastication stage may be ultimately very important. Simulation of the ejection phase requires accurate shrinkage analysis and complex boundary conditions for the frictional resistance of the part on the core. Again, advances have been made in these areas [118], but today no simulation combines all these effects.

Readers of this book should know the basic terminology of the molding and simulation indus-try as explained in [31, 333]. For completeness we provide a very brief overview of some key concepts below.

1.2

Molding Terminology

In order to understand the molding process, it is important to define some basic terms. When we speak of a “mold” we are referring to a complex electric/mechanical assembly. There may be electrical heating elements within the mold, for some materials. There will certainly be some temperature regulation system consisting of a network of fluid channels. Fluids may be water, glycol/water, or oil. These may be used for both cooling the melt after injection or increasing the temperature of the mold. There may also be inserts of varying conductivity or even mechanically actuated parts of the mold that create holes or special features.

In the simplest case, the mold comprises two halves. One of these is called the fixed side or cavity side and is held fast to the injection molding machine. The other is the moving side and moves in one direction to form the mold cavity and in the opposite direction to allow ejection of the part.

These ideas are presented in Figure 1.1 where we depict a simple two-cavity mold. By “two cavity” we mean that two moldings will be produced each cycle—in this case two hemispheres. In this case the moldings are of the same shape. However that is not necessarily the case in reality. It may be that each molding is of different shape. Moreover, while we have shown just two cavities, it is possible that there may be many more cavities.

Despite its simplicity, Figure 1.1 illustrates some important terminology. In particular, the con-cept of mold core and cavity. The mold cavity is fixed to the molding machine whereas the mold core is able to move back as the mold opens. Usually, the molding will shrink onto the mold core, and so will be attached to the core, as the mold opens. Ejection of the part is done by mechanical or pneumatic means.

Note also the sprue. This is the channel by which melt from the injection machine flows into the mold. Figure 1.2 illustrates some further terms, namely runners and gates. Runners

1.3 What is Simulation? 5

transfer the melt from the sprue to the cavities. At the entry to each cavity is an area called the gate. The gate is generally much smaller in diameter that the runner. It allows the molded part to be removed easily from the runner system. It is important to realize that the proper-ties of the melt fed into the caviproper-ties will depend on the flow rate through the sprue, runners, and gates. This in turn will affect the properties of the material in the cavities during and after molding. Hence a comprehensive simulation should incorporate the entire system; within and from the molding machine, into the sprue and runners, through the gates, and finally into the cavity. While some commercial codes claim to do this, their analysis is subject to conditions that overlook some of these aspects.

Our depictions in Figures 1.1 and 1.2 show what are known as cold sprues/runners. In practice, since the molding is the item of value, the sprue and runner system become scrap material. This material may be recycled however. An alternative method used in large molds, or molds with many cavities, it to use hot runners.

Hot runners use heating elements to maintain the melt at an optimum temperature prior to injection into the mold. They are frequently used in large molds, such as large automotive components like bumpers or large panels, or multi-cavity molds, such as bottle tops. Cold or hot runners obey the same physical laws from a simulation viewpoint. The difference is in the boundary conditions of the governing equations.

1.3

What is Simulation?

Simulation of injection molding involves using a computer to solve a set of equations, and their associated boundary conditions, that constitute a mathematical model of the molding

Figure 1.2 A simple two-cavity mold showing runners and gates

process. Generally speaking, today’s simulations lead to a huge amount of calculated data that are frequently displayed as colored contour plots of some particular variable of interest, such as:

■ fill patterns

■ pressure distributions ■ shrinkages

■ warpage of the component under consideration

In this book we do not attempt to interpret or discuss these results. Shoemaker [333] and Beau-mont et al. [31] provide background for the interested reader. Moreover, they provide informa-tion on the molding process and industry practice.

1.4

The Challenges for Simulation

While the description of the process in the previous section appears straightforward there are complications, namely:

■ the nature of injection molding, in particular the basic physics of the process ■ the properties of the material

■ the geometric complexity of the mold

We now briefly introduce each of these as background to the problems associated with simu-lation and discussed in this book.

1.5 Why Simulate Injection Molding? 7

1.4.1

Basic Physics of the Process

The filling phase is characterized by high flow rates and hence high shear rate. During mold filling, the molten material enters the mold and convection of the melt is the dominant heat transfer mechanism. Due to the rapid speed of injection, heat may also be generated by viscous dissipation. Viscous dissipation depends on both the viscosity and deformation rate of the material.

Viscous heating may be most apparent in the runner system and gates where flow rates are highest, however, it can also occur in the cavity if flow rates are sufficiently high or the material is very viscous.

In addition to forming the shape of the part to be made, the mold causes solidification of the material. Heat is removed from the melt by conduction through the mold wall and out to the cooling system. As a result of this heat loss, a thin layer of solidified material is formed as the melt contacts the mold wall. Depending on the local flow rate of the melt, this “frozen layer” may rapidly reach equilibrium thickness or continue to grow thereby restricting the flow of the incoming melt. This has a significant bearing on the pressure required to fill the mold and an important role in shrinkage and warpage prediction. When the cavity is volumetrically filled, the filling phase is complete but pressure is maintained by the molding machine. This is the start of the packing or holding phase.

Since the cavity is now full, mass flow rate into the cavity is much smaller than during injection. Indeed, further flow is due to shrinkage of the material and consequently both convection and viscous dissipation are minor effects—though they can be important locally such as at the gate or in a thin region that feeds a thicker region. During packing, conduction becomes the major heat transfer mechanism and the frozen layer continues to increase in thickness. At some time, the gate will freeze, thereby isolating the cavity from the applied pressure. Conduction is still the dominant heat transfer mechanism as the material solidifies and shrinks in the mold. It is possible that the material will pull away from the mold wall during this time [43, 76]; a condition that greatly complicates the calculation of the temperature of the material whilst in the mold. Finally, when the part is sufficiently solidified, it is ejected from the mold.

To summarize then, we see the injection molding process involves several heat transfer mech-anisms, is transient in nature, and involves a phase change and time-varying boundary condi-tions at the frozen layer in filling, packing, and during cooling. While these consideracondi-tions are substantive, simulation of the process is further complicated by material properties and the geometry of the part.

1.5

Why Simulate Injection Molding?

The previous section provides some feeling for the complexity of the molding process. It is no surprise that part quality is related to processing conditions. Indeed, the notion that process-ing has a dramatic effect on the properties of the manufactured article has been known since plastic processing began. In practice, the relationship between process variables and article quality is extremely complex. It is very difficult to gain an understanding of the relationship between processing and part quality by experience alone. It is for this reason that simulation

of molding was developed, and it is interesting to note that CAE has been much more success-ful in injection molding than in other areas of polymer processing.

The last point requires some explanation. Many polymer forming processes are continuous and, although the process physics may be complex, the die is generally quite simple and in-expensive to make. Moreover, there is considerable flexibility in changing process conditions. For blow-molding and thermoforming, the cost of tooling is relatively inexpensive. In fact the cost of a blow-molding mold can be as low as one-tenth that of an injection mold for a similar article [127]. Moreover, blow-molding machines provide the operator with enormous control so problems can often be solved on the factory floor.

By contrast, in injection molding, problems experienced in production may not be fixed by varying process conditions as with other processes. While there is scope to adjust process conditions to solve one problem, often the change introduces another. For example, increasing the melt temperature, and so decreasing the viscosity of the melt, may cure a mold that is difficult to fill and that is flashing slightly. The increase in temperature may, however, cause gassing or degradation of the material that leads to visual imperfections on the product. The fix may be to increase the number of gates or mold the part on a larger machine. Both of these are economically unfavorable. The first, involving significant retooling, is also costly in terms of time, and the second will erode profit margins as quotes for molding were based on the original machine, which would be cheaper to operate. On the other hand, simulation can be performed relatively cheaply in the early stages of part and mold design and offers the ability to evaluate different options in terms of part design, material, and mold design.

1.6

How Good is Simulation?

Frequently people ask the question, “How good is molding simulation?” A simple question, but it belies the complexity, both scientific and in human terms, of simulation. Two common criteria used to assess the success or otherwise of a simulation are:

1. Did the simulation lead to an improved design of the mold or part? 2. Did the simulation agree with what we saw in the molding plant?

Both these criteria require some explanation. The first depends on the fidelity of the results and the competence of the user. This book is devoted to the former. However, the competence of the user also requires an understanding of the assumptions made in simulation as well as industrial experience in injection molding. Users of injection molding software will find details of assumptions, and shortfalls in mathematical modeling, in the first part of this book. The second criterion is often used by companies as a test prior to buying simulation software. While it appears reasonable, it is an extremely complex area. One major problem is that injec-tion molding machines are not laboratory instruments. Settings by an operator on the control panel of a molding machine, and what happens in reality, may not correlate. Consequently, inputs to the software from machine settings may give rise to errors in simulation.

Generally, we can say that the results of simulation will only be as good as the data given to the simulation and the assumptions made by the software. This book attempts to provide informa-tion on the issues of material data and assumpinforma-tions made in software. The issue of comparing

1.6 How Good is Simulation? 9

simulation results to those seen in a molding plant, or even a well-equipped laboratory, are not discussed here. Comparison of computer simulation results with experiments is a huge topic and a relatively new scientific discipline. It is often referred to as validation and verification of software. Interested readers are referred to the works of Roache [310] and Oberkampf and Roy [272].

2

Stress and Strain

in Fluid Mechanics

In this chapter, some necessary terms are defined. In particular, the concept of stress in a fluid is introduced. We also define the rate of strain tensor for a fluid and introduce the generalized Newtonian fluid. These concepts are used in subsequent chapters when discussing material properties and the governing equations for simulation of injection molding.

2.1

Stress in Fluids

The flow of melt in injection molding involves the deformation of the material due to forces applied by both the molding machine and the mold. Any attempt to determine the flow re-quires a description of how these forces are transmitted to and within the melt. The concept of stress allows us to consider the affect of these forces.

In the following section, we present a definition of stress that is suitable for our purposes. Our treatment follows that of Tanner [356] and the reader is referred to this work for further details. Stress may be more rigourously defined as a result of Cauchy’s Theorem for the existence of stress [136]. This arises from the conservation of momentum—both linear and angular. Denn [78] also defines stress from this viewpoint and the interested reader is encouraged to explore these references.

2.1.1

The Stress Tensor

Stress is a measure of the forces transmitted when an external force acts on a body of material. A body of material may be a mass of any material, regardless of liquid or solid. To define stress in a quantitative way, consider a body of material, V , that contains a closed surface S of the material within it, as shown in Figure 2.1. We can expect that the part of material outside S and that in the interior exert a force distribution on each other across the surface S. We want to consider the interaction of the material within S with that outside.

There are two basic classes of interactions: body forces and surface forces (or contact forces). The surface force per unit area is called the surface traction .

Body forces act on the elements of mass within the body and have the units force per unit volume or force per unit of mass. A common example is gravity. Surface tractions act directly on the surface S and are given in units of force per unit area. An example is the force exerted on the skin of a balloon by the gas within. Commonly, this is called pressure and is directed

12 2 Stress and Strain in Fluid Mechanics

Figure 2.1 Definition of the stress vector

normal to the skin of the balloon. Surface tractions need not always act normal to the surface however. Friction, for example, is a surface traction that acts tangentially to a surface.

To further our definition of stress, we refer again to Figure 2.1. Consider a small areaδA on the surface S. Define a normal, n, at some point ofδA such that n points away from the interior of S. Denote the side to which the normal points as the positive side. Consider the portion of material situated on the positive side. The part exerts a forceδF on the other part situated on the negative side of S. The forceδF depends on the orientation of the normal n, and the areaδA and its location. If we assume as δA tends to zero, the ratio δF/δA tends to a definite limit, and that the moments of the force acting onδA tend to zero at any point within δA, we can write,

t = lim

δA→0

δF

δA. (2.1)

The vector t is called the stress vector (or the traction vector) and describes the force per unit area acting at a point on a surface within a body. The surface force tδA may be resolved into components. For example, suppose that the normal to the surface at the point where the stress vector is defined points in the z-direction. Then the surface force can be resolved along the x, y, z directions respectively into components. Then the components of force per unit area in these directions can be found. These components per unit area are calledσzx,σz y, and

σzzin the x, y, and z directions, respectively. The components in the x and y directions and

tangential to the x-y plane (σzxandσz y) are called shear stresses while the z-componentσzz,

is called the normal stress as shown in Figure 2.2.

Note that in the notation we have introduced, the first subscript indicates the direction of the normal to the area at the point where the stress vector is defined, and the second subscript gives the direction of resolution. Note also that we assume tensile stresses to be positive and hence pressure is negative. There is no standard convention and the reader is warned that other texts may adopt a different definition.

It can be shown [330] that if the stress is defined on three orthogonal planes passing through a point, stresses for any plane passing through the point may be obtained. Figure 2.3 shows

Figure 2.2 Resolution of stress vector

the stresses acting at a point O within a body. For clarity, the planes that should pass through O are displaced. Hence, the definition of stress requires the definition of a normal stress and

Figure 2.3 Stress components

two shear stresses for each of the three planes. That is nine stresses that are required to define the stress at an interior point of a body. These nine stresses form the components of a second-order tensor called the stress tensor and denoted byσ. It is convenient to replace subscripts x, y, z by 1, 2, 3, so thatσx y, for instance, is written asσ12. Each component may be identified

by the symbolσi j, where i ∈ {1,2,3} and j ∈ {1,2,3}. The stress tensor may also be written as a

matrix of its components:

σ =   σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33  . (2.2)

14 2 Stress and Strain in Fluid Mechanics

The stress tensor is symmetric in the absence of couples on the faces in Figure 2.3. That is, σi j= σj i. There is no a priori reason why the couples must vanish, so the symmetry of stress

tensor is just a constitutive assumption.

The relationship between the stress tensorσ at a point and the stress vector t at that point on a plane with outward normal n is given by

t = σ · n, (2.3)

that is, using the Einstein summation convention (see Appendix B), ti= σi jnj.

2.1.2

The Extra Stress Tensor

For a fluid at rest, the stress is equal to the thermodynamic pressure pt:

σ = −ptI . (2.4)

Note the negative sign on the right-hand side. As mentioned above, tensile stresses are taken to be positive, while compressive stresses are negative. The thermodynamic pressure is defined by the PVT relationship for the material. This defines an equation of state that relates the pressure to the specific volume V for the material, and its temperature, T . We discuss this further in Chapter 3. On the other hand, if the fluid is moving, there are additional stresses that must be considered. This requires us to add another term to the stress tensor to account for this:

σ = −ptI + τ. (2.5)

In compressible flows, the pressure ptis a function of the volumetric strain, whileτ is

indepen-dent of volume change. For incompressible material,τ is called the extra stress tensor, which can be computed from the constitutive equation when the motion is known, and the pressure must be found from the momentum balance and boundary conditions. We consider the form of this later.

2.1.3

Rate of Strain Tensor

General motion of a fluid involves translation, deformation, and rotation. The translation of a point in the fluid is defined by its velocity vector v. The deformation and rotation of the fluid at a point depends on its velocity gradient tensor. Adopting the convention of several other books such as [287] we define the velocity gradient tensor L as

L = (∇v)T, (2.6)

with components Li j=∂vi

∂xj

and where ∇v is a tensor called the gradient of the velocity, with components (∇v)i j=

∂vj

∂xi

, (2.8)

and the superscript “T” indicates the transpose operation. It should be noted that this defini-tion does vary in texts on rheology.

The velocity gradient tensor may be decomposed into a symmetric part called the rate of strain (or deformation) tensor D, defined as

D =1

2¡∇v + (∇v)

T_{¢ , (2.9)}

and an anti-symmetric part called the vorticity tensor W, defined as W =1

2¡(∇v)

T_{− ∇v¢ . (2.10)}

Later we will use the rate of strain tensor extensively in viscosity modeling, while the vorticity tensor plays an important part in fiber orientation prediction.

2.2

Newtonian and Non-Newtonian Fluids

In order to model the flow of fluids and, in particular, injection molding, we need a relationship between the extra stress tensorτ and the rate of strain tensor D. Such a relationship is called a constitutive equation.

Newtonian fluids have a particularly simple constitutive equation of the form

τ = 2µD −µ 2

3µ − µd ¶

(∇ · v)I , (2.11)

whereµ is the viscosity and µdis the dilatational viscosity.

The dilatational viscosity is zero for ideal simple gases [35]. Moreover, in the case of an incom-pressible flow, ∇·v = 0, the dilatational viscosity becomes irrelevant. Given that the dilatational viscosity has no effect in these extreme cases, we assume its effect to be negligible for a polymer melt and so ignore it in what follows. Equation 2.11 then becomes:

τ = 2µD −2

3µ(∇ · v)I . (2.12)

Equation 2.12 shows that, for a Newtonian fluid, the extra stress tensor is linearly related to the deviatoric rate of strain tensor. Furthermore, Equation 2.12 predicts trτ = 0, even for compressible flows, so that pure volumetric change trD does not affect the stress trτ. Note also that the last term in Equation 2.12is isotropic. We identify this with a pressure that arises from the motion of the fluid and its compressibility and denote it by pmI. So Equation 2.12

may be written: