Peter K. KennedyRong Zheng

Flow Analysis ofInjection Molds

2nd Edition

Kennedy, Zheng Flow Analysis of Injection Molds

Flow Analysis ofInjection Molds

Peter KennedyRong Zheng

Hanser Publishers, Munich Hanser Publications, Cincinnati

2nd Edition

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 6915 Valley Avenue, Cincinnati, Ohio 45244-3029, USA Fax: (513) 527-8801 Phone: (513) 527-8977 www.hanserpublications.com

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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. -- 2ndedition. 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

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ISBN 978-1-56990-512-8E-Book-ISBN 978-1-56990-522-7

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To my Father

Professor Zhi-Zhong Zheng

for his love and professional spiritthat have guided my life.

— Rong Zheng

To my Children

William and Anthony

for their support,understanding and love.

— Peter K. Kennedy

Acknowledgements

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), andNhan 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 collaborativeresearch 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 knowledgein several different aspects and lead to deep friendships. Their work can also be seen in thisbook.

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 wouldlike 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 isbut nevertheless produced some results that are of practical use.

We are indebted to the Australian Cooperative Research Center for Polymers for providing anopportunity of doing collaborative research with research teams from Monash and SydneyUniversities. 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 working environment and constant support to both of us during the period wewere working there.

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

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

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, industrialproblem.

Preface

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

Variations of injection molding that have been developed over the years include co-injectionor two-component molding, water injection, and gas-assisted injection molding (GAIM). Allthese processes provide additional scope for designers of plastic parts. Excellent examples areprovided by Neerincx [267] and Neerincx et al. [268, 269]. Indeed it is possible to combinethese 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 partwith unique qualities.

An important characteristic of injection molding, including variations, is that it may not bepossible to fix a part defect in production by simply varying process conditions. Frequentlythe 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 ofinjection 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 theyears many descriptions of modeling and simulation of injection molding have appeared inacademic journals and books. While readily available to specialist readers, an understandingof principles used in simulation software is difficult for nonspecialists to obtain. This is dueto 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 polymerprocessing that discuss injection molding. One of the original classics was by Tadmor andGogos [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 overviewof modeling and simulation for polymer processing, while Kamal et al. [190] have produced abook focused on injection molding that discusses variations and other aspects of the injectionmolding process.

Given the importance of injection molding as a process, and the simulation industry that hasgrown to support it, we believe there is a need for a book that deals solely with modeling andsimulation of injection molding. One of the authors wrote a book in 1995 [196] along theselines. 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 ofmodeling 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 appendixis 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 morereadable.

Part I is written for the user of simulation software who seeks an explanation of the basic mod-eling and assumptions made. Modeling and simulation details of filling, packing, residualstress, shrinkage, and warpage of amorphous, semi-crystalline, and fiber filled materials aredescribed. Additionally, it introduces numerical methods for solving mathematical models ofthe process. This part is intended to be self-contained but presumes knowledge of algebra andcalculus at the level of a degree in physical sciences or engineering. Tensor concepts are givenin 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 thehistory of simulation is useful for anyone so disposed. Appendix A provides some backgroundon both academic and commercial developments in simulation to around 2008. Much of thematerial presented in Part II covers developments from 2000 to the present. At the time ofwriting, this information is not implemented in commercial simulation software, and is meantto be a starting point for improvement in modeling and simulation. It presents some modelsthat 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 referenceother 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 willbe a source of ideas that lead to better simulations. Part II uses more advanced ideas of tensorcalculus. 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 molding is a technology that has been around for approximately 140years [172]. However, it was only in the 1950s, with the development of the reciprocating screwmethod, that the process showed its true potential.

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

Peter K. Kennedy and Rong Zheng, Melbourne, Australia, 2013

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XXI

I The Current Status of Simulation 1

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 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

3.6.1 The Viscosity Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1 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

4.9 Runners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Contents XIII

5 Approximations for Injection Molding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.1 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

5.8.3.5 The Interaction Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.1 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1137.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.2 ARD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.2.1 Evolution Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.2.2 Direct Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.2.3 Calculation of C I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.3 RSC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.4 Suspension Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.5 Brownian Dynamics Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Contents XV

8 Improved Mechanical Property Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1238.1 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1319.1 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14110.1 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 . .15511.1 Effects of Crystallization on Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

11.1.1 Viscosity-Enhancement-Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

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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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16712.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

12.2 Material Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

12.2.1 Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

12.2.2 Specific Heat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

12.2.3 Half-Crystallization Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

12.2.3.1 Quiescent Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

12.2.3.2 Flow-Induced Crystallization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

12.3 Effect on Shrinkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

13 Prediction of Post-Molding Shrinkage and Warpage . . . . . . . . . . . . . . . . . . .17513.1 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 . . . . . . . . . . . . . . . . . . . . . . .18114.1 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 Viscoelastic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Contents XVII

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .205A.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

A.5.1.2 Timon .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227B.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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235C.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 . . . . . . . . . . . . . . . . . . . . . . . . . . .243D.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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253E.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

E.2.2 Implicit Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

Contents XIX

F The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261F.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 . . . . . . . . . . . . . . . . . . . . . . . . . .283G.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

G.6 Runners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

XX Contents

H Three-Dimensional FEM for Mold Filling Analysis . . . . . . . . . . . . . . . . . . . . . . .303H.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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .317J.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

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .345

Notation

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 thebook 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

C g1 , C g

2 WLF universal constants (C g1 = 17.44, C g

2 = 51.6 K)

C 01 , C 0

2 WLF constants

CB stress-optical coefficient

C I 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 tensor 12

(∂vi∂x j

+ ∂v j

∂xi

)D(r ) (D (r )

i j ) diffusion coefficient tensor

Ea activation energy

E (Ei j kl ) Eshelby tensor

f function

△F f flow-induced free energy change

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

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 × 10−23 J/K)

k (ki j ) thermal conductivity tensor

l length

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

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

N f 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|pQ heat

R radius

Rg gas constant (8.3143 J / mol·K)

S1 one-dimensional fluidity for 2.5D runner approximation

Notation XXIII

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

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 om equilibrium melting temperature

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

U internal specific energy

U∗ activation energy

v (vi ) velocity vector

V volume

V specific volume

W i Weissenberg number

W (Wi j ) vorticity tensor 12

(∂vi∂x j

− ∂v j

∂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

µd dilatational viscosity

XXIV Notation

ξ slip parameter 2/(a2r +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/Dt material derivative ∂/∂t + vk∂∂xk

△/△t upper upper convected derivative defined as

△( )i j

△t= ∂( )i j

∂t+ vk

∂( )i j

∂xk−Li k ( )k j −L j k ( )ki

List of Figures

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 phasewith very low shear rates and high pressures, while material near the flow front isat 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 inthe 1-direction; (b) random in the 1-2 plane; (c) random in 3D space . . . . . . . . . . . . . . . 81

5.6 A comparison of the simulated C I for ar = 10,16.9,20,30, and 31.9 [289] withexperimental data of Folgar and Tucker [122] (reproduced from Phan-Thien etal. [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 transitiontemperatures. 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 rectangularplate; (b) shows the flow in the cross-section of the plate; (c) shows the flow frontadvancement on the surface mesh, and (d) shows the use of a connector elementto 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, thenormal from node n of the bottom element, intersects the top element at somepoint p within the element. In this case, interpolation is required . . . . . . . . . . . . . . . . . . . 106

6.9 In 3D analysis, temperature is correctly convected around changes in directionin runners. In these cases, the temperature differences due to shear heating mayresult 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 E m 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 usingthe 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 E m vs. fiber aspect ratio for φ = 0.20. Pre-dicted using the Mori-Tanaka model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.5 Effective Poisson’s ratios vs. fiber aspect ratio for φ = 0.20. Predicted using theMori-Tanaka model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.6 Effective coefficients of thermal expansion vs. fiber aspect ratio for φ= 0.20. Pre-dicted using the Rosen-Hanshin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

9.1 Schematic representations of (a) short-fiber pellet and (b) long-fiber pellet usedfor 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 sampleof industrial iPP (reproduced from Koscher and Fulchiron [211] with permissionfrom Elsevier) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

List of Figures XXVII

10.3 Isothermal crystallization curve of the Borealis iPP sample at 132◦C. Inset: Varia-tion of half-crystallization time with crystallization temperature . . . . . . . . . . . . . . . . . . . . 147

10.4 Heat flow curves for Borealis iPP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

10.5 Determination of equilibrium melting temperature using Hoffman-Weeksmethod, for Borealis iPP. Melting point data measured on samples isothermallycrystallized at different Tc s 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-surface for an iPP (from Zhu and Edward [428], with permissionfrom American Chemical Society) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

11.1 PVT diagram for different cooling rates (from Luyé et al. [234], with permissionfrom John Wiley and Sons) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

11.2 Undisturbed equilibrium thermal conductivity against temperature forpolypropylene (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 permissionfrom Springer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

12.2 Morphologies of (a) virgin iPP at T = 132◦C, t = 180 s; (b) iPP mixed with UBcolorant 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% colorant by weight), denoted byPP, 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 etal. [425]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

12.5 Half-crystallization time vs. short-term shear rate for virgin iPP. Shearing time 1sec; temperatures: 132◦C and 136◦C. Symbols are experimental data, and solidand 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 experimentaldata, 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 experimentaldata, and solid and dotted lines are modeled results (Zheng et al. [425]) . . . . . . . . . . . . 172

12.8 Experimental and predicted parallel and perpendicular shrinkage for the virginiPP (Zheng et al. [425]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

XXVIII List of Figures

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

12.10 Experimental and predicted parallel and perpendicular shrinkage for the iPPwith 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 thefront (shown) and rear sides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

A.2 The “layflat” is created by unfolding the box to lie in a plane. Note though thatthe correct thickness for each surface of the box is retained. Dark lines representpossible 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.Huagang Yu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

List of Tables

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