1569905126

Peter K. Kennedy Rong Zheng

Flow Analysis of Injection Molds

2nd Edition

Kennedy, Zheng Flow Analysis of Injection Molds

Peter Kennedy Rong Zheng

Flow Analysis of Injection Molds 2nd Edition

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 6915 Valley Avenue, Cincinnati, Ohio 45244-3029, USA Fax: (513) 527-8801 Phone: (513) 527-8977 www.hanserpublications.com Distributed in all other countries by Carl Hanser Verlag Postfach 86 04 20, 81631 München, Germany Fax: +49 (89) 98 48 09 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 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

To my Father Professor Zhi-Zhong Zheng 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.

— 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), 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 influence 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 Universiteit 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 sponsored 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 providing an excellent working environment and constant support to both of us during 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 encouragement that sustained us during our confrontation with an important, but difficult, industrial problem.

Preface

Injection molding is an ideal process for fabricating large numbers of geometrically complex 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. Consequentially, 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, various 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, materials 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 computer 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 discussion, 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 modeling and assumptions made. Modeling 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, graduate 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 possible 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 semicrystalline 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 molding process. Injection molding 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.

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

Contents

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

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

4.9 Runners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Contents

XIII

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

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .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 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167 12.1 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 . . . . . . . . . . . . . . . . . . .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 Viscoelastic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

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

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

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

Contents

XX

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

Notation

Symbols which have more than one meaning are listed with a semicolon dividing the meanings. 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/(ρc p )

ai j

second-order fiber orientation tensor

a i j kl

fourth-order fiber orientation tensor

ar

fiber aspect ratio

aT

time-temperature shift factor

A

area

b

FENE-P model parameter

A (A i 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

g g C1 , C2 C 10 , C 20

g

g

WLF universal constants (C 1 = 17.44, C 2 = 51.6 K) WLF constants

CB

stress-optical coefficient

CI

interaction coefficient in Folgar-Tucker model

Ct

stress-thermal coefficient

C (C i j )

interaction coefficient tensor in anisotropic rotary diffusion model

C (C i j kl )

stiffness tensor (also called elasticity tensor)

d

diameter; distance function

D (r ) D (D i j )

diffusion coefficient (isotropic) ³ ´ ∂v j ∂v rate-of-deformation tensor 21 ∂x i + ∂x

D(r ) (D i(rj ) )

diffusion coefficient tensor

Ea

activation energy

E (E i j kl )

Eshelby tensor

f

function

△F f

flow-induced free energy change

j

i

XXII

Notation

△F q

difference of free energies between melt and crystalline phases under quiescent condition

F (F i )

force vector

g (g i )

acceleration vector due to gravity

G

radial growth rate of spherulite; shear modulus

GN

melt plateau modulus

H

cavity half-thickness

H (t ) Hˆ

Heaviside unit step function

△Hc

latent heat of crystallization for perfect crystals

I (δi j )

unit tensor (also called the Kronecker tensor)

I (I i 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 (k i j )

thermal conductivity tensor

l

length

L (L i j )

velocity gradient tensor ∂v i /∂x j

Mn

number-average molecular weight

Mw

weight-average molecular weight

n

power-law exponent

n0

number of molecules per unit volume

specific enthalpy

n (n i )

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 (p i )

orientation unit vector

q (q i )

heat flux vector; orientation vector q = |q|p

Q

heat

R

radius

Rg

gas constant (8.3143 J / mol·K)

S1

one-dimensional fluidity for 2.5D runner approximation

Notation

S2

two-dimensional fluidity for 2.5D cavity approximation

S3

three-dimensional fluidity for pseudo 3D approximation

S

∥

shrinkage parallel to flow direction

S⊥ Sˆ

shrinkage perpendicular to flow direction specific entropy

S (S i j kl ) t, t

XXIII

′

elastic compliance tensor time

t 1/2

half-crystallization time

t (t i )

traction vector (also called stress vector)

T

temperature; as superscript denotes transpose of a tensor

Tg

glass transition temperature

o Tm

equilibrium melting temperature

u (u i ) Uˆ

velocity vector; unit orientation vector; displacement vector

U∗

activation energy

v (v i )

velocity vector

V Vˆ

volume

Wi

Weissenberg number ³ ´ ∂v j ∂v vorticity tensor 21 ∂x i − ∂x

W (Wi j )

internal specific energy

specific volume

j

i

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˙ i j ) γ

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/(a r2 + 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 (L i j )

effective velocity gradient L i j − ξD i j

Operator symbols D/D t

material derivative ∂/∂t + v k ∂x∂

△/△t

upper upper convected derivative defined as △( )i j △t

=

∂( )i j ∂t

+ vk

k

∂( )i j ∂x k

− L i 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 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 C I for a r = 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 transition 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 thickness, 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 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.1 Direct simulation results of the fiber configuration at different strains (reproduced from Fan et al. [103] with permission of Elsevier) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8.1 Reduced effective moduli scaled by E m vs. fiber volume fraction for a R = 20. Predicted using the Mori-Tanaka model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.2 Effective Poisson’s ratios vs. fiber volume fraction for a R = 20. Predicted using the Mori-Tanaka model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.3 Effective coefficients of thermal expansion vs. fiber volume fraction for a R = 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. Predicted using the Mori-Tanaka model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8.5 Effective Poisson’s ratios vs. fiber aspect ratio for φ = 0.20. Predicted using the Mori-Tanaka model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.6 Effective coefficients of thermal expansion vs. fiber aspect ratio for φ = 0.20. Predicted 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 structure (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

List of Figures

XXVII

10.3 Isothermal crystallization curve of the Borealis iPP sample at 132◦ C. Inset: Variation 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-Weeks method, for Borealis iPP. Melting point data measured on samples isothermally crystallized at different Tc s were used to determine the equilibrium melting temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 10.6 Isothermal crystal growth for Borealis iPP sample at 132◦ C under quiescent condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 10.7 Two-dimensional SAXS image patterns at different distances from the skin surface to the mid-surface 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% 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 12.4 Half-crystallization time vs. crystallization temperature of three samples: virgin 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 iPP (Zheng et al. [425]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

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. 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 A i , 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

I

The Current Status of Simulation

1

Introduction

Automotive and consumer electronics are two disparate industries that rely heavily on injection 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 parallelization of tasks from inception to manufacture. Regardless of the degree to which concurrent engineering is practiced, there is no doubt that simulation is a valuable aid in linking design 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 industry. 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, pressure 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 cavity. 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

4

1 Introduction

5. Plastication and screw back 6. Ejection Most effort in computer simulation has been devoted to phases 2–4. There have been significant 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 simulation 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 isotactic polypropylene at 200◦ C, was still evident after a period of 30 min with the sample maintained 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 industry 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 concept 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 properties of the melt fed into the cavities 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 1.1 A simple two-cavity mold

6

1 Introduction

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 Beaumont et al. [31] provide background for the interested reader. Moreover, they provide information 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 simulation and discussed in this book.

1.5 Why Simulate Injection Molding?

1.4.1

7

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 mechanisms, is transient in nature, and involves a phase change and time-varying boundary conditions at the frozen layer in filling, packing, and during cooling. While these considerations 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 processing 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

8

1 Introduction

of molding was developed, and it is interesting to note that CAE has been much more successful 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 inexpensive 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 injection 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 information on the issues of material data and assumptions 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 requires 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 area δA on the surface S. Define a normal, n, at some point of δA 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 force δF on the other part situated on the negative side of S. The force δF depends on the orientation of the normal n, and the area δA 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 on δA 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 called σzx , σz y , and σzz in the x, y, and z directions, respectively. The components in the x and y directions and tangential to the x-y plane (σzx and σz y ) are called shear stresses while the z-component σzz , 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

2.1 Stress in Fluids

13

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 secondorder tensor called the stress tensor and denoted by σ. It is convenient to replace subscripts x, y, z by 1, 2, 3, so that σx y , for instance, is written as σ12 . Each component may be identified by the symbol σi j , where i ∈ {1, 2, 3} and j ∈ {1, 2, 3}. The stress tensor may also be written as a matrix of its components:   σ11 σ12 σ13 σ = σ21 σ22 σ23  . (2.2) σ31 σ32 σ33

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 = σ j 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 tensor σ 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), t i = σi j n j .

2.1.2

The Extra Stress Tensor

For a fluid at rest, the stress is equal to the thermodynamic pressure p t : σ = −p t I .

(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 = −p t I + τ .

(2.5)

In compressible flows, the pressure p t is a function of the volumetric strain, while τ is independent 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 =

∂v i , ∂x j

(2.7)

2.2 Newtonian and Non-Newtonian Fluids

15

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

∂v j ∂x i

,

(2.8)

and the superscript “T” indicates the transpose operation. It should be noted that this definition 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¡ ∇v + (∇v)T , 2

(2.9)

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

¢ 1¡ (∇v)T − ∇v . 2

(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 tensor τ 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 µ ¶ 2 τ = 2µD − µ − µd (∇ · v) I , 3

(2.11)

where µ is the viscosity and µd is the dilatational viscosity. The dilatational viscosity is zero for ideal simple gases [35]. Moreover, in the case of an incompressible 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: 2 τ = 2µD − µ (∇ · v) I . 3

(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 p m I. So Equation 2.12 may be written: τ = 2µD − p m I .

(2.13)

16

2 Stress and Strain in Fluid Mechanics

Substituting this into Equation 2.5 we obtain σ = −(p t + p m )I + 2µD = −pI + 2µD ,

(2.14)

where p = p t + p m is some average pressure that involves both the equation of state and the motion of the fluid. For Newtonian fluids, the viscosity, while constant at a particular temperature, may vary with temperature. Fluids for which Equation 2.14 does not hold are called non-Newtonian fluids. Polymer melts show a variation of viscosity with temperature and also deformation rate. Consequently, they are non-Newtonian, and some modification to Equation 2.14 is necessary. One of the simplest is to allow the viscosity µ to be a function of the rate of deformation of the fluid.

2.3

The Generalized Newtonian Fluid

In the previous section, we noted the need to allow the viscosity to be a function of the deformation rate of the fluid. Let η denote a viscosity function. We want to define some relationship that allows η to depend on the rate of deformation. We cannot make η depend directly on the rate of deformation tensor D, as the components of the tensor change in different coordinate systems, while η is a scalar, and must remain unchanged under a change of frame. Every second-order tensor has three independent scalar invariants, so named because their values are independent of coordinate rotations. Consequently, the viscosity function should be based on the invariants of the rate of deformation tensor. For the rate of deformation tensor, the first invariant is identically zero for incompressible fluid flows; and for shearing-type flows or any plane flows, the third invariant is also zero [35]. Since injection molding is a shear dominated process, we are left with the second invariant, denoted I I D . To define the second invariant, we first need to introduce the double dot product or scalar product of a tensor. For two tensors, A and B, the double dot product is denoted A : B and defined by: A : B = Ai j B j i .

(2.15)

The second invariant of a tensor A is defined as: I IA = A : A = Ai j A j i . It is convenient to define, for all flows, a generalized strain rate by p γ˙ = 2I I D p = 2D : D q ¡ ¢ = 2tr D2 ,

(2.16)

(2.17)

and this may be used to define a viscosity function. In shear flows, γ˙ is the magnitude of the shear rate.

2.3 The Generalized Newtonian Fluid

17

When the flow behavior is dominated mainly by the viscosity, we can define the generalized Newtonian fluid (GNF), for which the viscosity is allowed to depend on the generalized strain rate: ˙ , τ = 2η(γ)D

(2.18)

˙ is called the viscosity function. where η(γ) The GNF is non-Newtonian but has neither memory nor elasticity. It is useful for modeling polymeric flows dominated by shear forces [35], such as in injection molding. Substituting Equation 2.18 in Equation 2.14 we obtain the following relationship valid for a compressible, generalized Newtonian fluid: ˙ . σ = −pI + 2η(γ)D ˙ to simply η in the sequel. For brevity, we will abbreviate η(γ)

(2.19)

3

Material Properties of Polymers

The physical properties of polymers depend to a large extent on their molecular structure. To simulate injection molding, we need to measure properties for their use in the simulation; however, the process itself affects the properties. The relationship between processing and properties is still an area of research. Indeed, this lack of understanding is a major deficiency of current simulation software. Some ideas on how to advance this area are given in Part II of this book. In this chapter, we identify the basic polymer types and discuss their properties. Our aim is to define the properties required for simulation. We also indicate some difficulties in property measurement that cause errors in simulation. As a prelude to Part II, we discuss errors introduced in simulation due to ignoring the effects of processing on the material itself.

3.1

Types of Polymers

The molecular structure of polymers used in injection molding directly influences their physical properties. For our purposes, a polymer molecule may be regarded as a long chain of covalently bonded units with additional chemical groups forming branches along the primary chain. This structure leads to two basic types of polymeric materials: thermosets and thermoplastics. Thermosetting materials become fluid on heating, but as the temperature increases, a reaction known as cross-linking causes molecules to bond at the branch groups along the chain. This process leads to the formation of a rigid three-dimensional lattice. Once cross-linking occurs, the material will not remelt on heating. Molecules in a thermoplastic material do not cross-link. These materials can be made fluid by heating and then solidify when cooled. Moreover, unlike thermosets, the cycle of becoming fluid and solidification upon heating and cooling, may be repeated indefinitely, providing the material does not undergo any permanent chemical change due to the applied heat. Because both thermosets and thermoplastics soften when heated, both can be used in injection molding. However, there is an important difference in processing. For thermosets, the mold is at a higher temperature than the injected material, while for thermoplastics, the mold is cooler than the injected material. This book focuses on thermoplastics. There is a further division of polymers that is useful; namely, amorphous and semi-crystalline materials. These polymer types have very different characteristics. Accurate simulation must take this into account. Unfortunately, this is easier said than done.

20

3 Material Properties of Polymers

3.2

Amorphous Polymers

For polymers, the bonding between atoms along the main molecular chain is flexible, in the sense that segments of the chain can rotate about the bonds. In the melt state, molecules have increased mobility and tend to adopt an entangled, disordered configuration. If the temperature is lowered, and in the absence of external forces, the molecular configuration remains entangled, the material is said to be amorphous. As an amorphous polymer in the melt state is cooled, it becomes rubbery. Further cooling will cause the material to change from a rubbery state to a hard glass-like material. The temperature at which this transition occurs is called the glass transition temperature, T g . While the existence of a transition temperature may imply a sudden change of properties, properties of amorphous polymers change gradually as temperature is changed. In the absence of external forces, molecular entanglement is preserved in an amorphous polymer as the melt cools. However, in injection molding, the final properties depend on the orientation of molecules. This depends on the processing history, the rate of cooling, and the relaxation characteristics of the polymer. Sometimes the molecular orientation in an injection molded part is referred to as the “frozen in” orientation. Engels [96] investigated the mechanical performance of injection molded amorphous materials. Using a model for yield stress that accounted for processing history, he found that the processing history determined the yield stress of the molded material. Govaerts et al. [131] linked a mechanical model to a molding simulation code to predict yield stress [131] according to the temperature/time history of the material. These are major accomplishments in the endeavor to determine ultimate properties after processing. Unfortunately, such success is still lacking for semi-crystalline materials.

3.3

Semi-Crystalline Polymers

While for amorphous polymers, molecular orientation is preserved as the melt cools, it is possible that molecules align themselves in a regular lattice shape. Such materials are said to be crystallizable. These materials in the cooled, solid state may possess both amorphous and crystalline regions, and are called semi-crystalline. It should be noted that the lattice structure here is not due to chemical bonding as in the case of cross-linking of thermosets. Indeed, the crystalline regions may be removed and restored with repeated heating and cooling cycles, provided there is no chemical change to the material. In the absence of external forces, when cooled from the melt to the solid state, semi-crystalline materials will form randomly oriented amorphous and crystalline regions. Properties differ in each phase. Consequently, these materials are composite in nature. Their properties are determined by the amount and orientation of the crystalline and amorphous phases in the material. The degree of crystalline content is affected by the rate of cooling. Generally, slow cooling results in higher degrees of crystallinity. High cooling rates, such as those near the mold wall in injection molding, tend to decrease the level of crystallinity.

3.4 Overview of Material Properties for Simulation

21

Over the past two decades, a lot of investigation into the effect of flow on crystallization has occurred. We will consider this in later chapters. For now, it suffices to note that flow enhances crystallization. This is important in injection molding. As we will see, the molding process itself affects crystallization of the material, and hence, the material properties. This raises a problem for molding simulation. For simulation, you need material properties; however, these properties are typically measured under different conditions to those in processing. Since the process affects the properties, there is an intrinsic error in the use of simply-measured properties. The solution is to have models that can simulate the process as well as the changes in material properties. This is the future for simulation science. At the time of this writing, no simulation software can do this. Part II of this book contains chapters that give some ideas to overcome this shortcoming for semi-crystalline polymers.

3.4

Overview of Material Properties for Simulation

As we saw in Chapter 1, injection molding is a complex process. We need to deal with flow of a compressible non-Newtonian fluid, variable temperature, and phase change. For simulation, we need to know some properties of the material. These material properties change during the process. Consequently, determination of material properties is not trivial. Laboratory tests to measure properties are frequently made under conditions that are very different to those encountered in the molding process. Using these data in simulation leads to intrinsic errors. In this section, we define the required material properties for simulation of the process, discuss some limitations of the assumptions made, and indicate some areas where improvements can be made. Material properties required to simulate injection molding depend on the simulation to be undertaken. For flow analysis (filling, packing, and cooling) the following are required: ■

viscosity

■

specific heat capacity

■

thermal conductivity

■

no-flow or transition temperature

■

PVT data

Should the user of simulation software want to simulate shrinkage and warpage, additional properties are required, such as: ■

elastic modulus

■

shear modulus

■

coefficient of thermal expansion

Some simulation software may also require specialized data for particular models that have been developed individually by the company supplying the software. Below we discuss these material properties which are used in currently available simulation software.

22

3 Material Properties of Polymers

3.5

Viscosity

An important property for simulation is the viscosity of the material. To define this property, imagine a fluid at a fixed temperature confined between two plates that are a distance, H , apart. Suppose the bottom plate is stationary and the top plate moves with a constant speed, v x , in the x-direction, due to the application of a force, F x , as illustrated in Figure 3.1. Such a

Figure 3.1 Steady simple shear flow

flow is called a steady (or simple) shear flow and is important in injection molding where the flow is approximated by flow in a narrow gap. The shear stress, τzx , acting on the top plate is given by τzx =

Fx , A

where A is the area of the plate. In 1687, Isaac Newton proposed that for such flows, “The resistance which arises from the lack of slipperiness of the parts of the liquid, other things being equal, is proportional to the velocity with which the parts of the liquid are separated from one another.” Viscosity is this “lack of slipperiness” [362]. Denoting the viscosity by η, this means vx H d vx =η dz = ηγ˙ zx ,

τzx = η

(3.1)

where γ˙ zx is called the shear rate. For more complex flows, it is necessary to rely on tensor quantities, and so ˙. τ = ηγ

(3.2)

Fluids for which η is constant are called Newtonian fluids. It is possible that η depends on pressure or temperature, however. Otherwise, these fluids are non-Newtonian. Molten polymers generally exhibit shear thinning whereby the viscosity reduces as the material is sheared. Consequently, polymers are non-Newtonian. Moreover, they are viscoelastic, meaning they exhibit both viscous and elastic properties. In particular, viscoelastic materials exhibit a time dependent strain when stressed. Viscoelasticity is important for calculation of warpage and the properties of molded material. We return to this in Section 5.9.1.2 and Chapter 13.

3.6 Modeling Viscosity

3.6

23

Modeling Viscosity

In this section, we discuss aspects of viscosity as they relate to simulation. In particular, we look at the effects of temperature, shear rate, and pressure on viscosity. Due to shear thinning effects, a modification to Equation 3.2 is made to allow viscosity to depend on the shear rate: ˙γ ˙, τ = η(γ)

(3.3)

˙ is called the viscosity function, and γ˙ is the generalized strain rate defined by Equawhere η(γ) tion 2.17. Equation 3.3 defines the generalized Newtonian fluid introduced in Section 2.3.

3.6.1

The Viscosity Function

A number of viscosity functions are available for injection molding simulation. Some of the most popular are: the power law model, the Carreau model, and the Cross model.

3.6.2

The Power Law Model

The power law model was developed in the 1920s [362] and is also called the Ostwald-de Waele model. It sets the viscosity proportional to a power of the shear rate and has the form η = m γ˙ n−1 ,

(3.4)

where m and n are constants. When m = µ and n = 1, we obtain the relationship for a Newtonian fluid; that is, Equation 2.14. Polymer melts exhibit shear thinning behavior. That is, the viscosity decreases with increasing shear rate. Consequently, for polymer melts, the constant n ≤ 1. Taking logarithms of both sides of Equation 3.4, we obtain log η = (n − 1) log γ˙ + log m .

(3.5)

A plot of log η against log γ˙ shows this relationship to be linear. At high shear rates, the constants m and n may be easily determined from experiments, and the power law model can represent the behavior of polymer melts in the high shear rate region. Injection molding is often thought of as a high shear rate process. However, it is important to consider the phase of the process under consideration. In the filling phase, there are often high shear rates, but in the packing phase, shear rates may be low. What’s more, it is possible to have low shear rates in the filling phase when the filling pattern is unbalanced. Figure 3.2 gives such an example.

3.6.3

The Carreau Model

The Carreau model (see Bird et al. [35]) is h i η − η∞ ˙2 = 1 + (λγ) η0 − η∞

n−1 2

,

(3.6)

24

3 Material Properties of Polymers

Figure 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

where η 0 is the zero shear rate viscosity, η ∞ is the upper limiting Newtonian viscosity, λ is a time constant, and n is the power law index. The model has been successful in correlating experimental viscosity data. When η ∞ = 0 and at high shear rates, the model reduces to the power law model with m = η 0 λn−1 .

3.6.4

The Cross Model

The Cross model [72] has the form η=

η0 , ˙ ∗ )1−n 1 + (η 0 γ/τ

(3.7)

which combines a Newtonian region and a power law shear thinning region. When γ˙ → 0, it predicts the zero shear rate viscosity η 0 , while at high shear rate, it predicts power law behavior. In Equation 3.7, τ∗ is a constant related to the shear stress at the transition between Newtonian and power law behavior, and n is the power law index—a measure of degree of the shear thinning behavior. In a comparison of the Carreau and Cross models, Hieber and Chiang [153] found that the Cross model provided a better fit for the shear rate dependence. Consequently, the Cross model has been widely adopted in commercial simulation.

3.6.5

Incorporation of Temperature Effects

The viscosity functions discussed above assume a melt at a particular temperature. In injection molding, the temperature varies from the melt temperature to that of the mold wall. Consequently, for simulation we require a viscosity function that includes temperature. Polymers are viscoelastic materials, and depend on both temperature and time. Although most commercial simulations do not consider viscoelastic effects explicitly, they do adopt temperature effects from viscoelastic theory. The basic idea is called time-temperature superposition.

3.6 Modeling Viscosity

25

For many polymers, the logarithmic plot of viscosity at temperature T may be obtained from that at temperature T0 by shifting the viscosity along the logarithmic time axis by the amount log a T (T ). This assumes thermo-rheological simplicity, meaning that all molecular mechanisms involved in relaxation mechanisms have the same temperature dependencies. The time-temperature shift factor, a T (T ), has been fitted by the WLF (Williams-Landel-Ferry) equation (Williams et al. [119, 402]): g

log a T = g

−C 1 (T − T g ) g

C 2 + T − Tg

,

(3.8)

g

where C 1 and C 2 are constants and T g is the glass transition temperature. Although empirical in nature, the WLF equation has a theoretical justification based on the free volume idea as g g described by Ferry [119]. At first, the constants C 1 and C 2 were thought to be universal for all g g polymers with values C 1 = 17.44, C 2 = 51.6 K when T g is chosen as the reference temperature. In simulations, the “universal values” should only be used in the absence of experimental data. If experimental data for a T are available, one may use an arbitrary reference temperature, T0 , to replace T g , and rewrite Equation 3.8 as log a T =

−C 10 (T − T0 ) C 20 + T − T0

.

(3.9)

The WLF equation is widely applicable to amorphous polymers in the temperature range T g < T < T g + 100 (◦ C or K). For higher temperatures, or where T g is irrelevant, the Arrhenius equation may be more appropriate [356]. It has the form log a T =

Ea , Rg

(3.10)

where E a is the activation energy and R g is the gas constant. Readers should note that in recent literature, the Arrhenius equation is also referred to as the Andrade equation. A commonly used model for viscosity in simulations is the Cross-WLF model. It has the form η=

η0 , ˙ ∗ )1−n 1 + (η 0 γ/τ

(3.11)

where ³ −C 0 (T − T0 ) ´ 1 η 0 = D 1 exp , C 20 + T − T0

(3.12)

and D 1 is a constant.

3.6.6

The Solidification Problem

The temperature effects discussed in the previous section are relevant when the material is at a high temperature and molten. In simulation of injection molding and most other forming

3 Material Properties of Polymers

26

processes, there is a need to consider the viscosity of the material as it solidifies. Unfortunately this is not easy. Viscosity is usually determined experimentally in capillary viscometers. In these experiments, as the melt temperature is reduced, there is an increase in viscosity. However, as the viscosity reduces, the temperature of the melt rises due to viscous dissipation. Consequently, though the instrument may indicate a particular temperature, the actual melt temperature may be higher due to viscous dissipation. For amorphous materials, the temperature sensitivity of viscosity depends on the difference between the glass transition temperature and the processing temperature and may be temperature sensitive. However, semi-crystalline materials undergo a very sharp increase in viscosity as temperature is reduced. This sudden increase is not captured by any of the temperaturedependent models described above. We discuss this further in Section 5.3.5.

3.7

Thermal Properties

Heat transfer in the injection molding process requires attention to three phenomena: ■

convection due to the incoming melt

■

conduction out of the melt, through the mold, and to the cooling channels

■

shear heating caused by the deformation of the melt as it flows into the mold cavity

For convection into the mold, the key property is the specific heat of the melt. This is a measure of the heat content of a volume of melt. Conduction of the melt to the cooling lines requires the conductivity of the melt as well as that of the mold. Shear heating of the material is a function of the melt’s viscosity and the rate of deformation.

3.7.1

Specific Heat Capacity

Specific heat capacity of a material is a measure of how much energy is required to raise the temperature of the material, and conversely, how much energy is contained in a material at a given temperature. For convection into the mold, the key property governing the heat into the mold is the specific heat capacity. To define this, suppose we have some mass of material to which we add some heat, Q, to bring about an increase in the temperature of ∆T . The mean heat capacity denoted, C¯ , of the body is defined to be the ratio of the change in heat to the change in temperature: C¯ =

Q . ∆T

(3.13)

The heat capacity of the body, C , at a particular temperature, is then defined as the limit approached by C¯ as the temperature difference tends to zero: C = lim

∆T →0

Q . ∆T

(3.14)

The specific heat capacity is defined to be the heat capacity per unit mass of material. The unit of specific heat is Joule per kilogram degree Kelvin (J/kg.K). Specific heat capacity may be measured under conditions of constant volume or pressure, and is denoted c v or c p , respectively.

3.7 Thermal Properties

27

Table 3.1 Specific Heat of Some Polymers and Metals Material

Specific Heat (J/kg.K)

ABS

1300

PA66

1700

Polycarbonate

1300

Polyethylene

2300

Polypropylene

1900

Polystyrene

1300

Steel (AISI 1020)

460

Steel (AISI P20)

460

Due to the large stresses exerted on the containing vessel when heating a sample for constant volume measurements, the use of c p is more common. Specific heat is used, in simulation, to calculate the heat lost from the melt to the mold cooling system during filling, packing, and cooling. It is also used to calculate the heat generated by viscous dissipation as the melt flows through the runner system and cavity. When the material solidifies there may be additional heat (latent heat) created, especially for semi-crystalline materials. This is frequently ignored by simulation programs as it requires explicit calculation of crystallization. Generally, the specific heat of polymers is much higher than that for metals. Table 3.1 shows some values for common polymers [311] and steels used for injection molding [247]. While the specific heat of metals is lower that that for polymers, the amount of heat to be removed from the mold is defined by the product of specific heat and density, ρc p . Steels have a density around one order of magnitude above polymers, however.

3.7.2

Thermal Conductivity

Thermal conductivity is a measure of how conductive to heat a material is. Materials with a high thermal conductivity are used to draw heat away from a heat source. Materials with low thermal conductivity are insulators. In simulation of molding, the thermal conductivities of the mold and polymer are required. More formally, imagine a slab of material with thickness ∆x, for which one side is at a temperature, T , and the other at a higher temperature, T +∆T , as illustrated in Figure 3.3. Let ∆Q be the heat flow across the cross-section of area A, in time ∆t . The average heat flow across the crosssection is then ∆Q/∆t . Experimentally, it has been determined that the average rate of heat flow is proportional to the area and the temperature difference, and inversely proportional to the thickness of the slab: ∆T ∆Q ∝A . ∆t ∆x

(3.15)

Introducing the positive constant of proportionality k, we have ∆T ∆Q = −k A , ∆t ∆x

(3.16)

28

3 Material Properties of Polymers

Figure 3.3 Definition of thermal conductivity Table 3.2 Thermal Conductivity of Polymers and Metals Material

Conductivity (W.mK)

ABS

0.25

PA 66

0.3

Polypropylene

0.60

Polystyrene

0.14

Aluminium

250

Copper

401

Steel (AISI 1020)

51.9

Steel (AISI P20)

51.9

and in the limit, as ∆x → 0 and ∆t → 0, we obtain: dQ dT = −k A . dt dx

(3.17)

Equation 3.17 defines the thermal conductivity, denoted k, of the material. The units of thermal conductivity are Joules per meter second Kelvin (J/m.s.K) or Watts per meter Kelvin (W/m.K). Being complex materials, the thermal conductivity of polymers varies with temperature, degree of crystallization and molecular orientation. Polymers generally have low thermal conductivities whereas metals are much more conductive and have high thermal conductivities. Some examples are given in Table 3.2. Experimental error becomes a problem as conductivity reduces. Due to their low values, thermal conductivity is a difficult property to measure for polymers. The most important aspect of thermal conduction, as it relates to simulation, is Fourier’s law, which can be deduced from Equation 3.17 by dividing by A and generalizing the derivative to get: q = −k∇T ,

(3.18)

3.8 Thermodynamic Relationships

29

where q is the local heat flux vector. Most simulation software will assume k is a scalar. This is discussed later in Sections 5.3.4 and 11.4.

3.8

Thermodynamic Relationships

Several thermodynamic properties are required for molding simulation. These are obtained from the equation of state of the material. This is often referred to as the PVT data. An equation of state relates these three variables, pressure p, specific volume Vˆ , and temperature T [326]. For any material, we can write the equation of state in the form f (p, Vˆ , T ) = 0 .

3.8.1

(3.19)

Expansivity and Compressibility

Given any two variables, the third may be obtained from the equation of state. In particular, we can write Vˆ = g (p, T ) ,

(3.20)

where g is some function. Graphing the function g , we obtain a PVT surface as in Figure 3.4. Imagine that the material at a temperature, T a , undergoes a change in temperature while the pressure is held constant. What is the subsequent change in volume? This can be answered by Figure 3.4, where this is illustrated as a change from point a to b. A change in temperature from T a to T a +∆T causes a change in volume equal to g (p a , T a +∆T )− g (p a , T a ). The average change in the volume over the temperature change is therefore ∆Vˆ g (p a , T a + ∆T ) − g (p a , T a ) = ∆T ∆T Vˆ (p a , T a + ∆T ) − Vˆ (p a , T a ) = . ∆T

(3.21)

In the limit as ∆T → 0, we obtain the instantaneous change in volume for the material which we denote by ³ ∂Vˆ ´ ∂T

p

(3.22)

where the subscript p indicates that the pressure is constant. The coefficient of volume expansion, β, of the material, is defined as β=

1 ³ ∂Vˆ ´ , Vˆ ∂T p

(3.23)

and has units of reciprocal Kelvin (K−1 ). The coefficient of volume expansion is also called the expansivity of the material.

30

3 Material Properties of Polymers

Figure 3.4 PVT surface

Now consider the change in volume due to a change in pressure while keeping the temperature constant. This is illustrated by moving from point b to c in Figure 3.4. The average change in volume due to a change in pressure is given by ∆Vˆ g (p a + ∆p, Tb ) − g (p a , Tb ) = ∆p ∆p Vˆ (p a + ∆p, Tb ) − Vˆ (p a , Tb ) = . ∆p

(3.24)

Once again, letting ∆p → 0, we obtain the instantaneous change in volume for the material that we denote by ³ ∂Vˆ ´ ∂p

T

.

(3.25)

The isothermal compressibility coefficient κ, is defined as κ=−

1 ³ ∂Vˆ ´ , Vˆ ∂p T

(3.26)

where the negative sign indicates that the volume decreases with increasing pressure. The isothermal compressibility coefficient has units of one square meter per Newton (m2 N−1 ).

3.9 Pressure-Volume-Temperature (PVT) Data

3.9

31

Pressure-Volume-Temperature (PVT) Data

For thermoplastic polymers, the equation of state is of the type Vˆ = Vˆ (p, T ) and is usually provided as a PVT diagram that gives the specific volume as a function of p and T . A commonly used equation is the Tait equation (van Krevelen [380], Zoller et al. [432], Zoller and Fakhreddine [433]) h ³ p ´¤ + Vˆt (p, T ) , (3.27) Vˆ (p, T ) = Vˆ0 (T ) 1 −C ln 1 + B (T ) where C = 0.0894 is a constant, and considered to be universal. Vˆ0 (T ) is given by Vˆ0 (T ) =

b 1(s) + b 2(s) (T − b 5 ), b 1(m) + b 2(m) (T − b 5 ),

½

where the superscripts tively. B (T ) is given by ½ B (T ) =

(s)

and

(m)

if T ≤ Ttrans if T > Ttrans ,

(3.28)

represent the solid and melt state of the polymer, respec-

¡ ¤ b 3(s) exp [−b 4(s) (T − b 5 ) , £ ¤ b 3(m) exp −b 4(m) (T − b 5 ) ,

if T ≤ Ttrans if T > Ttrans .

(3.29)

For amorphous polymers Vˆt (p, T ) = 0, while for semi-crystalline polymers, Vˆt (p, T ) =

½

£ ¤ b 7 exp b 8 (T − b 5 ) − b 9 p , 0

if T ≤ Ttrans if T > Ttrans ,

(3.30)

where the transition temperature, Ttrans , is assumed to be a linear function of pressure, that is, Ttrans = b 5 + b 6 p .

(3.31)

Given the PVT data for Vˆ = Vˆ (p, T ), it may be differentiated to yield the expansion and compressibility coefficients discussed in Section 3.8.1. As density is the inverse of specific volume, the PVT diagram also provides density at any pressure and temperature.

3.10

Fiber Orientation

Fiber-orientation analysis seeks to determine the orientation of fibers blended into the polymer. Models for this are discussed later in Chapters 4 and 7. The simplest models require the amount of fiber in the polymer. This is expressed as a weight fraction or a volume fraction. The aspect ratio of fibers may also be required. The fiber weight fraction, W f , is defined by Wf =

wf wc

=

ρf vf ρc vc

,

(3.32)

32

3 Material Properties of Polymers

where w f is the weight of the fiber, w c is the weight of the composite, ρ f is the density of the fiber, ρ c is the density of the composite, v f is the volume of fiber, and v c is the volume of composite. Usually for industrial materials, the weight fraction is given as a percentage. The fiber volume fraction φ, is defined by φ=

ρc Wf . ρf

(3.33)

The aspect ratio, a R , is defined as the length of the fiber divided by its diameter.

3.11

Shrinkage and Warpage

For shrinkage and warpage we need: ■

linear coefficients of expansion in directions 1 and 2

■

elastic moduli in directions 1 and 2

■

Poisson’s ratios

■

shear modulus

These properties are dealt with in basic texts on solid mechanics such as Shames [330]. We discuss them briefly for completeness and in preparation for later chapters where these properties will be modeled using the effects of processing. We have already discussed the coefficient of volume expansion, Equation 3.23, which tells us how the volume changes with a change in temperature. This may be sufficient for very simplified analysis of shrinkage and warpage. In such cases the linear coefficients of expansion are roughly 1/3 of the coefficient of volume expansion. Generally, though, molded parts exhibit anisotropic effects, meaning that the linear coefficients of expansion vary in three orthogonal directions. As injection molded parts are frequently thin-walled, if a coordinate system is defined such that the 3-ordinate is in the thickness direction, the required linear coefficients of expansion are those in the 1 and 2 directions. The linear coefficient of expansion in direction i is defined by αi =

1 d xi , where i ∈ {1, 2} , xi d T

(3.34)

and, despite the repeated index i , no summation is intended. Similar comments may be made for elastic modulus. Equation 3.26 provides the change in volume for a material subjected to uniform loading. But injection molded materials exhibit anisotropy in modulus as well. Adopting a similar coordinate system to that above, we can speak of modulus in three directions. Depending on the type of analysis, values of E 1 and E 2 may suffice. The elastic modulus, often called Young’s modulus or tensile modulus, is defined by the ratio of stress to strain in the direction of interest Ei =

σi , where i ∈ {1, 2} , εi

and despite the repeated index i , no summation is intended.

(3.35)

3.11 Shrinkage and Warpage

33

Poisson’s ratio is associated with the observed effect that when you stretch a material sample, by applying a tensile force in a given direction, there is a contraction in the width of the sample in a direction (or directions) perpendicular to the tensile force. Conversely, if you compress a sample of material, there is an increase in width in directions perpendicular to the compressive force. Shear modulus or modulus of rigidity, G, is a measure of how a material will behave in shear. It is defined as the ratio of shear stress to shear strain. Assuming three orthogonal directions as above, we have: Gi j =

τi j γi j

.

Again, despite the repeated indices, no summation is intended here.

(3.36)

4 4.1

Governing Equations

Introduction

In this chapter, the equations governing the flow of a compressible, viscous fluid are derived. These equations are applicable to the flow of a polymer melt and are obtained using the conservation principles of mass, momentum, and energy. In deriving the governing equations for a fluid, we assume that the fluid is a continuum. That is, we ignore the molecular structure of the material and assume it is possible to define physical properties such as velocity and density at a point in the fluid. More importantly we assume that these properties vary smoothly over space and time so that differentiation is permissible. Such an assumption greatly simplifies the modeling and allows us to use standard mathematical techniques. That said, it is worth noting that there are statistical approaches that may be used for simulation. Indeed there is an extensive literature on simple models of polymers with a statistical bias [36]. These are in some ways more directly applicable to polymers which are statistical materials, meaning commercial polymers comprise some distribution of molecular lengths. Interested readers may find an introduction to these techniques in Öttinger [280]. It is likely these techniques will become more common as computer technology improves, with regard to speed, and injection-molding simulation strives for better performance. At the time of writing however, all commercial codes are based on a continuum approach.

4.2

Mathematical Preliminaries

In this section, we introduce some mathematical concepts that will be used in later chapters. These are the concepts of material derivative, Gauss’s divergence theorem, and Reynolds transport theorem. Our discussion here will be brief. Additional detail may be found in specialist books such as Aris [10] and Morrison [260].

4.2.1

The Material Derivative

Some fluid properties are functions of both position and time. When analyzing fluid motion, it is necessary to take derivatives, with respect to time, of these properties. For example the density ρ, of a fluid, may depend on both position and time. Using Cartesian coordinates we can write ρ = ρ(x 1 , x 2 , x 3 , t ) ,

36

4 Governing Equations

where x 1 , x 2 , and x 3 are the spatial coordinates and t is time. Using the chain rule, we find the derivative of the density, with respect to time, is given by dρ ∂ρ ∂x 1 ∂ρ ∂x 2 ∂ρ ∂x 3 ∂ρ = + + + dt ∂x 1 ∂t ∂x 2 ∂t ∂x 3 ∂t ∂t ∂ρ ∂ρ ∂ρ ∂ρ = x˙1 + x˙2 + x˙3 + ∂x 1 ∂x 2 ∂x 3 ∂t ∂ρ = v · ∇ρ + , ∂t where v = (x˙1 , x˙2 , x˙3 ) is the velocity at a point in the fluid. More generally, for any scalar function of position and time f = f (x 1 , x 2 , x 3 , t ), we have df ∂f = v·∇f + , dt ∂t and denote this differentiation by ∂f Df = v·∇f + . Dt ∂t

(4.1)

We call D f /D t the material derivative of f . Some texts use “substantial derivative” or “differentiation following the fluid” instead of material derivative. Physically, the material derivative takes into account both the motion of the fluid and the changing value of the fluid particle with time. Although we have considered scalar functions above, we can take the material derivative of any scalar, vector, or tensor quantity that varies with position and time. For example, the material derivative of the vector c is given by Dc ∂c = v · ∇c + , Dt ∂t

(4.2)

while for a tensor, A, we have DA ∂A = v · ∇A + . Dt ∂t

4.2.2

(4.3)

The Gauss Divergence Theorem

This important result is called the Gauss Divergence Theorem, also known as the GaussOstrogradskii divergence theorem, which relates volume and surface integrals. More particularly, it relates the divergence of a vector function f within a volume to the flux of f through the surface enclosing enclosing the volume. It may be expressed as Z Z ∇ · f dV = f · n d S , (4.4) V

S

where V is a region in space with a closed boundary S and n is a unit outward normal to S. In physical terms, the theorem states that the accumulation of a quantity within the volume V , is given by the net flow, into or out of the region, through its bounding surface.

4.2 Mathematical Preliminaries

37

A similar result holds in lower dimension. Letting S be an area in the plane with closed boundary Γ, then Z Z ∇·fds = f·nd Γ. (4.5) Γ

S

The reader should note that there are technical requirements on the regions V and S but we will not worry about them here. Details may be found in Marsden and Tromba [239].

4.2.3

Reynolds Transport Theorem

To derive the governing equations of fluid motion, we will need to consider integrals of any function of position and time f (x, t ) over a volume of fluid. This volume will move with the fluid but consists of the same fluid particles. Such a volume is called a material volume and is denoted V (t ). The expression Z F (t ) = f (x, t ) dV (4.6) V (t )

defines a function of t . The Reynolds transport theorem [10] tells us how to calculate the derivative of F (t ) with respect to time. Note that because V (t ) varies with time and moves with the fluid, it is not possible to simply take the differentiation through the integral sign; it is necessary to use the material derivative of F (t ). The result is also known as the Leibnitz formula [260] and may be written ½ ¾ Z Z ∂f d f (x, t ) dV = + ∇ · f v dV , (4.7) d t V (t ) V (t ) ∂t where v is the velocity of a fluid particle. Applying the divergence theorem to the right-hand side of Equation 4.7, we obtain the following equivalent form of the transport theorem Z Z Z ∂f d f (x, t ) dV = dV + f v · n dS , (4.8) d t V (t ) V (t ) ∂t S(t ) where S(t ) is the surface of V (t ), and n is the outwardly directed unit normal on S(t ). Physically, Equation 4.8 says that the rate of change of the integral of f (x, t ) is equal to the integral of the rate of change of f (x, t ) over a fixed region V (t ), plus the resultant flow of f (x, t ) across the surface S(t ). The result may be applied to any scalar, vector, or tensor function f (x, t ).

4.2.4

Integration by Parts

Many books on calculus provide information on integration by parts. The formula takes the form: Z x2 ¯x=x2 Z x2 ′ ′ ¯ MN dx = MN¯ − M Ndx , (4.9) x1

x=x 1

x1

38

4 Governing Equations

where M and N are continuous and differentiable and ′ represents differentiation. We will have occasion to use it when simplifying the governing equations for injection molding. The ′ ′ essential idea is to choose M and N such that the integral of N and the derivative of M result ′ in the integral of M N being simpler than the original integral.

4.3

Conservation of Mass

If V (t ) is a material volume of fluid, the principle of mass conservation says that the mass of fluid contained in V (t ) does not change. Letting ρ(x, t ) denote the density of fluid at x at time t , the mass m within V (t ) is given by Z m= ρ(x, t ) d v . (4.10) V (t )

Consider now the derivative of m with respect to time, Z dm d = ρ(x, t ) d v dt d t V (t ) ½ ¾ Z ∂f = + ∇ · ρv dV , V (t ) ∂t

(4.11)

where we have used Reynolds transport theorem, Equation 4.7. According to the conservation of mass principle, the rate of change of m with respect to time is zero. That is ½ ¾ Z ∂ρ 0= + ∇ · ρv dV. (4.12) V (t ) ∂t But the region V (t ) is arbitrary. Consequently, the integrand must be identically zero and so ∂ρ + ∇ · ρv = 0 . ∂t Equation 4.13 is called the continuity equation.

(4.13)

Sometimes it is useful to express the continuity equation in terms of the material derivative. Expanding out Equation 4.13 we get ∂ρ + ρ∇ · v + v · ∇ρ = 0 . ∂t Using the definition of material derivative, we may write Dρ = −ρ∇ · v . Dt

4.4

(4.14)

(4.15)

Conservation of Momentum

Conservation of momentum requires that the time rate of change of fluid particle momentum in a material volume V (t ), be equal to the sum of external forces acting on V (t ). That is, Z X d ρv dV = Fext . (4.16) d t V (t )

4.4 Conservation of Momentum

39

The external forces acting on V (t ) include both body forces due to gravity and tractions. The total body force Fb , is given by Z Fb = ρg dV , (4.17) V (t )

where g is the total body force per unit mass. Usually g is due to gravitational effects. In injection molding, these forces are relevant in very thick moldings that are usually associated with variants of the process, such as gas injection molding discussed in Section 14.3.2. The traction force acting on an element d S of the boundary surface of V (t ) is given by td S, where t is the stress vector defined in Section 2.1. The total traction force Ft is then Z Ft = t dS S(t ) Z = σ · n dS , (4.18) S(t )

where S(t ) is the boundary surface of V (t ), σ is the stress tensor (see Section 2.2), and n is the unit outward normal to S(t ). Applying the divergence theorem, (Equation 4.4), we obtain, Z Ft = ∇ · σ dV . (4.19) V (t )

The total external force is given by the sum of the body forces and the tractions. Using Equations 4.17 and 4.19 we have X Fext = Fb + Ft Z Z = ρg dV + ∇ · σ dV V (t ) V (t ) Z = (ρg + ∇ · σ) dV . (4.20) V (t )

Substituting this result in Equation 4.16 gives Z Z d ρv dV = (ρg + ∇ · σ) dV . d t V (t ) V (t )

(4.21)

Applying the transport theorem, (Equation 4.7), to the left-hand side of the equation above we have, Z Z ³∂ ´ ¡ ¢ (ρv) + ∇ · (ρvv) dV = ρg + ∇ · σ dV . (4.22) V (t ) ∂t V (t ) However, the volume V (t ) is arbitrary. Consequently, after rearrangement, we have ∂ (ρv) = ρg + ∇ · σ − ∇ · (ρvv) . ∂t

(4.23)

This equation is called the momentum equation. The momentum equation can be simplified using the continuity Equation 4.14 [260]. To this end, consider the last term in Equation 4.23. Using the product rule for differentiation: ∇ · (ρvv) = ρ∇ · vv + ∇ρ · vv £ ¤ = ρ v · ∇v + (∇ · v)v + ∇ρ · vv

(4.24)

40

4 Governing Equations

where we have used Equation B.52. Consider now the left-hand side of Equation 4.23. Again using the product rule of differentiation: ∂¡ ¢ ∂v ∂ρ ρv = ρ + v. ∂t ∂t ∂t

(4.25)

Substituting Equations 4.24 and 4.25 into Equation 4.23 gives ρ

£ ¤ ∂v ∂ρ + v = −ρ v · ∇v + (∇ · v)v − ∇ρ · vv + ρg + ∇ · σ ∂t ∂t

which may be rearranged as: h ∂ρ i h ∂v i =− + ρ(∇ · v) + ∇ρ · v v + ρg + ∇ · σ ρ v · ∇v + ∂t ∂t h ∂ρ i =− + ∇ · ρv + ρg + ∇ · σ . ∂t

(4.26)

To conclude, we note that the term in the square brackets on the left-hand side is the material derivative of v defined in Section 4.2.1. Moreover, the terms in the square brackets on the righthand side are identically zero due to the continuity Equation 4.14. Hence, the momentum equation may be written: ρ

Dv = ρg + ∇ · σ . Dt

4.5

(4.27)

Conservation of Energy

The total energy of fluid in a material volume V (t ) is given by the sum of its kinetic and internal energies. Letting Uˆ be the internal energy per unit volume, the total energy in a material volume is Z Z Z 1 1 2 ρ( v 2 + Uˆ ) dV = ρv dV + ρUˆ dV , (4.28) 2 V (t ) V (t ) 2 V (t ) where v 2 = v · v . The increase in total energy of a material volume V (t ) with boundary S(t ) is equal to the work done on the volume less the heat lost through S(t ). The work done on the material volume is due to both traction and body forces. The rate of work done by traction forces is given by Z Z t · v dS = (σ · n) · v d S S(t ) S(t ) Z = (σ · v) · n d S S(t ) Z = ∇ · (σ · v) dV , (4.29) V (t )

where we have used the definition of the stress vector t, (Equations 2.1 and 2.3), and the divergence theorem (Section 4.2.2) on the right-hand side.

4.5 Conservation of Energy

41

We will consider body forces due to gravity only. These will have a negligible effect for injection molding except in gas injection molding as mentioned in the previous section and Section 14.3.2. Denoting the acceleration due to gravity by g, the rate of work done on a material volume by gravitational forces is given by Z ρ(g · v) dV . (4.30) V (t )

We now consider the rate of heat flux across the surface S(t ). Let q denote the heat flux vector at a point on S(t ). Letting n be the unit outward normal to S(t ), the rate of heat loss from S(t ) is given by Z Z − q · n dS = − ∇ · q dV , (4.31) S(t )

V (t )

where we have used the divergence theorem to transform the surface integral to a volume integral. We now equate the rate of change of the right-hand side of Equation 4.28 to the work due to tractions given by Equation 4.29, work due to gravitational forces (Equation 4.30), and heat loss (Equation 4.31) to get d dt |

1 2 d ρv dV + dt V (t ) 2 {z } |

Z

Z

Z ρUˆ dV = ∇ · (σ · v) dV V (t ) V (t ) {z } | {z }

1

2

Z +

V (t )

|

3

Z

∇ · q dV . ρg · v dV − V (t ) {z } | {z } 4

(4.32)

5

It is useful to consider the physical meaning of Equation 4.32. Each term has been numbered and their meanings are as follows: 1. The rate of change of kinetic energy of the material volume 2. The rate of change of internal energy of the material volume 3. The rate at which work is done on the fluid within the material volume by viscous forces 4. The rate of work done on the fluid within the material volume by gravitational forces 5. The heat loss from the fluid within the material volume due to conduction across S(t ) Now we seek an expression for the kinetic energy which may be substituted in Equation 4.32 to obtain a simplified expression for the rate of change of internal energy. Stewart [344] noted that this may be done by taking the scalar product of the momentum equation, Equation 4.23, with the velocity vector v. Taking the dot product: ∂ (ρv) · v = ρg · v + (∇ · σ) · v − ∇ · ρvv · v . ∂t

(4.33)

Note that, by the product rule of differentiation, ∂ ∂ ∂ (ρv · v) = (ρv) · v + v · (ρv) ∂t ∂t ∂t ∂ = 2 (ρv) · v , ∂t

(4.34)

42

4 Governing Equations

and so, ∂ 1 ∂ (ρv) · v = (ρv · v) ∂t 2 ∂t 1 ∂ = (ρv 2 ) , 2 ∂t

(4.35)

where v 2 = v · v. Using Equation B.56, we can write the second term of Equation 4.33 as (∇ · σ) · v = ∇ · (σ · v) − σ : ∇v .

(4.36)

Substituting Equations 4.35 and 4.36 into Equation 4.33 gives: 1 ∂ (ρv 2 ) = ρg · v + ∇ · (σ · v) − σ : ∇v − ∇ · ρvv · v . 2 ∂t

(4.37)

The above equation gives the kinetic energy for a unit volume of material. For the kinetic energy in a material volume, we must integrate Equation 4.37 over a material volume V (t ) to get 1 ∂ (ρv 2 ) dV = 2 ∂t

Z V (t )

Z V (t )

ρg · v dV +

Z −

V (t )

Z V (t )

σ : ∇v dV −

∇ · (σ · v) dV

Z V (t )

∇ · ρvv · v dV .

(4.38)

Setting f = 12 ρv 2 in Reynolds transport theorem, Equation 4.7, we have: d dt

Z V (t )

³1 2

Z ´ ρv 2 dV =

V (t )

Z =

V (t )

Z ∂ ³ 1 2´ 1 ρv dV + ∇ · ρv 2 v dV ∂t 2 V (t ) 2 Z ∂ ³ 1 2´ ρv dV + ∇ · ρvv · v dV . ∂t 2 V (t )

(4.39)

Substituting the above in Equation 4.38 we obtain d dt

Z V (t )

1 2 ρv dV = 2

Z V (t )

ρg · v dV −

Z V (t )

σ : (∇v) dV +

Z V (t )

∇ · (σ · v) dV ;

(4.40)

the required expression for the kinetic energy of a material volume. Substituting Equation 4.40 in Equation 4.32 we get d dt

Z V (t )

ρUˆ dV =

Z V (t )

σ : (∇v) dV −

Z V (t )

∇ · q dV .

(4.41)

Applying the transport theorem, Equation 4.7, to the left-hand side, we obtain, Z Z ³∂ ´ (ρUˆ ) + (∇ · ρUˆ v) dV = σ : (∇v) dV − ∇ · q dV . V (t ) ∂t V (t ) V (t )

Z

(4.42)

However, the region V (t ) is arbitrary, and so: ∂ (ρUˆ ) + ∇ · (ρUˆ v) = σ : (∇v) − ∇ · q . ∂t

(4.43)

4.5 Conservation of Energy

43

Expanding the left-hand side, and using the definition of the material derivative (Section 4.2.1), we have: ∂ ∂ (ρUˆ ) + ∇ · (ρUˆ v) = (ρUˆ ) + ρUˆ ∇ · v + v · ∇ρUˆ ∂t ∂t D = (ρUˆ ) + ρUˆ ∇ · v Dt Dρ DUˆ + Uˆ + ρUˆ ∇ · v =ρ Dt Dt DUˆ =ρ , Dt

(4.44)

where we have used the continuity equation, Equation 4.13, to get the last line. Substituting Equation 4.44 into Equation 4.43 we obtain: ρ

4.5.1

DUˆ = σ : (∇v) − ∇ · q . Dt

(4.45)

Relating Specific Energy to Temperature

Because the specific energy, Uˆ , cannot be measured directly, it is necessary to establish a relationship between specific anergy and a measurable quantity, like temperature. Consequently, in this section, we write Equation 4.45 in terms of temperature T , rather than specific energy Uˆ . This section requires some knowledge of thermodynamics, which is not required to understand the rest of the book. Nevertheless it is important, and is presented for completeness. Users of simulation software, unfamiliar with thermodynamical concepts, may prefer to skip this subsection and consider the final results in Section 4.5.2. Let Hˆ = Uˆ + p Vˆ

(4.46)

denote the specific enthalpy where p is the pressure and Vˆ the specific volume. A combination of the first and second laws of thermodynamics may be written in the following form [326], ´ 1³ d Hˆ − Vˆ d p , (4.47) d Sˆ = T where Sˆ is the specific entropy. We will suppose that Hˆ is a function of both pressure and temperature and so, d Hˆ =

³ ∂ Hˆ ´ ∂p

T

dp +

³ ∂ Hˆ ´ ∂T

p

dT .

(4.48)

Substituting this expression in Equation 4.47 we get d Sˆ =

i 1 h³ ∂ Hˆ ´ 1 ³ ∂ Hˆ ´ − Vˆ d p + dT . T ∂p T T ∂T p

(4.49)

But Sˆ is also a function of p and T , therefore d Sˆ =

³ ∂Sˆ ´ ∂p

T

dp +

³ ∂Sˆ ´ ∂T

p

dT .

(4.50)

44

4 Governing Equations

Equating coefficients of Equations 4.49 and 4.50 we obtain ³ ∂Sˆ ´ ∂p

T

=

i 1 h³ ∂ Hˆ ´ − Vˆ T ∂p T

(4.51)

=

1 ³ ∂ Hˆ ´ . T ∂T p

(4.52)

and ³ ∂Sˆ ´ ∂T

p

We now differentiate Equation 4.51, with respect to T , to get h ∂ ³ ∂Sˆ ´ i ∂2 Sˆ = ∂T ∂p ∂T ∂p T p i 1 µh ∂ ³ ∂ Hˆ ´ i ³ ∂Vˆ ´ ¶ 1 h³ ∂ Hˆ ´ =− 2 − Vˆ + − T ∂p T T ∂T ∂p T p ∂T p µ ¶ h³ ´ i ³ ´ 2 1 1 ∂ Hˆ ∂ Hˆ ∂Vˆ =− 2 − Vˆ + − . T ∂p T T ∂T ∂p ∂T p

(4.53)

Differentiating Equation 4.52 with respect to p we get, h ∂ ³ ∂Sˆ ´ i ∂2 Sˆ = ∂p∂T ∂p ∂T p T 1 h ∂ ³ ∂ Hˆ ´ i = T ∂p ∂T p T 1 ∂2 Hˆ . = T ∂T ∂p

(4.54)

Now, ∂2 Sˆ ∂2 Sˆ = , ∂T ∂p ∂p∂T and so we may equate Equations 4.53 and 4.54 to get ³ ∂ Hˆ ´ ∂p

T

= Vˆ − T

³ ∂Vˆ ´ ∂T

p

.

(4.55)

Substituting this result in Equation 4.48 we obtain ³ ³ ∂Vˆ ´ ´ ³ ∂ Hˆ ´ dp + dT . d Hˆ = Vˆ − T ∂T p ∂T p

(4.56)

The above equation may be simplified by noting that the specific heat of a material c p , and the coefficient of volume expansion β, are related to the partial derivatives as follows: cp =

³ ∂ Hˆ ´ ∂T

p

and β =

1 ³ ∂Vˆ ´ . Vˆ ∂T p

Hence Equation 4.56 may be written as d Hˆ = (1 − βT )Vˆ d p + c p d T .

(4.57)

4.5 Conservation of Energy

45

From Equation 4.46 we have d Hˆ = d Uˆ + Vˆ d p + p dV . Substituting this result into Equation 4.57 and rearranging for d Uˆ we obtain, d Uˆ = (1 − βT )Vˆ d p + c p d T − p d Vˆ − Vˆ d p = c p d T − βT Vˆ d p − p d Vˆ .

(4.58)

We can differentiate the above expression with respect to time to get d Uˆ dT dp d Vˆ = cp − βT Vˆ −p . dt dt dt dt

(4.59)

The above expression also holds true for material derivatives and so, DT Dp D Vˆ DUˆ = cp − βT Vˆ −p . Dt Dt Dt Dt

(4.60)

Consider now the term D Vˆ /D t . Because Vˆ = 1/ρ we have, D Vˆ ∂Vˆ = + v · ∇Vˆ Dt ∂t ³1´ ∂ ³1´ = +v·∇ ∂t ρ ρ h ∂ ³1´ i ∂ ³ 1 ´ ∂ρ = +v· ∇p ∂ρ ρ ∂t ∂ρ ρ ³ ´ 1 ∂ρ + v · ∇ρ =− 2 ρ ∂t 1 Dρ =− 2 ρ Dt 1 = ∇·v, ρ

(4.61)

where in the last line, we have used Equation 4.15. Substitution of Equation 4.61 in Equation 4.60 and multiplying both sides by the density gives ρ

DT Dp DUˆ = ρc p − βT − p∇ · v . Dt Dt Dt

(4.62)

This equation relates the rate of change in specific energy to the rate of change of measurable quantities such as temperature, pressure and volume.

4.5.2

The Energy Equation in Terms of Temperature

We are now in a position to express the energy equation in terms of temperature. Substituting Equation 4.62 in Equation 4.45 gives ρc p

Dp DT = βT + p∇ · v + σ : ∇v − ∇ · q . Dt Dt

(4.63)

46

4 Governing Equations

Expanding the material derivatives we obtain: ρc p

³ ∂T

´ ³ ∂p ´ + v · ∇T = βT + v · ∇p + p∇ · v + σ : ∇v − ∇ · q . ∂t ∂t

(4.64)

Our final step is to relate the heat flux vector q to temperature. This can be done using Fourier’s law of heat conduction [37], which states that q = −k∇T ,

(4.65)

where k is the thermal conductivity of the material. Substituting this result in Equation 4.64 gives the final form of the energy equation: ρc p

³ ∂T

´ ³ ∂p ´ + v · ∇T = βT + v · ∇p + p∇ · v + σ : ∇v + ∇ · (k∇T ) . ∂t ∂t

(4.66)

While this is quite a general equation, there are some physics that can be added to it to better deal with injection molding. In particular, there is the heat of reaction. This may be due to a chemical process, such as in a thermosetting material or a foaming process. More fundamentally, it may be due to the latent heat of fusion which occurs when a semi-crystalline polymer solidifies. Adopting a general approach, we add a heat source term to Equation 4.66. The energy equation then becomes ρc p

³ ∂T

´ ³ ∂p ´ + v · ∇T = βT + v · ∇p + p∇ · v + σ : ∇v + ∇ · (k∇T ) + Q˙ , ∂t ∂t

(4.67)

where Q˙ represents the heat source.

4.6

Boundary Conditions

The conservation equations of mass, momentum, and energy derived in the preceding sections of this chapter are very general and may be applied to a wide range of fluid flows. Our focus is injection molding however. In later chapters we will see there is scope to simplify the governing equations for simulation of injection molding. Nevertheless there are boundary conditions that are quite specific to injection molding and apply to both the general conservation equations and simplified equations. To conclude this chapter we consider these boundary conditions. Figure 4.1 illustrates a simple mold cavity for which we will discuss the required boundary conditions. There are several surfaces on which boundary conditions need to be described: ■

Σinj , the surface through which melt enters the cavity

■

Σem , the edge of the mold

■

ΣW+ , the top surface of the mold

■

ΣW− , the bottom surface of the mold

■

■

Σins , the surface defining any insert in the mold. While only one is shown in figure 4.1, it is possible to have any number in a real mold Σmf , the surface defining the melt front. Depending on the geometry of the mold, and the number of inserts, there may be any number of these

4.6 Boundary Conditions

47

Figure 4.1 Boundary conditions for simulation of filling, packing, and cooling

4.6.1

Pressure and Flow Rate Boundary Conditions

For general flow problems, boundary conditions at a surface are usually given in terms of either the tractions or velocities, or a combination of the two. For thin-cavity flows, however, it is convenient to use pressure and flow rate boundary conditions. Boundary conditions relating to pressure are: ■

At any impermeable boundary, the pressure gradient in the normal direction to the boundary is zero. The impermeable boundaries are the mold edges, mold walls, and any mold inserts. Therefore, ∂p = 0 on Σem , ΣW+ , ΣW− , Σins . ∂n

(4.68)

Physically this means that material cannot flow through the mold walls, edges, or inserts. ■

The melt flow rate q, or the pressure p, is specified on the surface where the melt enters the cavity. That is, q = q in j or p = p inj on Σinj .

(4.69)

Flow analysis software usually uses a specified flow rate in the filling phase. This is obtained by dividing the volume of the cavity to be filled by the user specified fill time. In recent years, there has been some improvements in this. For example, if there are constraints due to the machine, shear rate, or some other constraint, it may be that the flow rate is reduced or even increased. The packing phase boundary condition usually specifies a pressure at the point(s) of injection. In basic software, this may be constant. However, in order to ensure a more uniform density, pressure may be varied. Michaeli and Lauterbach [249] discussed the idea of using PVT data to vary the packing pressure such that an optimal packing pressure profile could be used to minimize shrinkage variation. ■

Assuming the pressure datum is atmospheric, the pressure is zero at the melt front. That is p = 0 on Σmf .

(4.70)

4 Governing Equations

48

While this is a reasonable assumption, because most molds are vented to allow air to escape, there are some processes that may require this to be changed. These include the counterpressure process in which the unfilled part of the mold is held at a pressure above atmospheric pressure to affect the movement of the flow front.

4.6.2

Temperature Boundary Conditions

The boundary conditions relating to temperature are: ■

The temperature profile through the cavity thickness, T (z), is prescribed for the surface through which the melt is injected. That is T (z) = Tinj (z)

(4.71)

Most flow analysis software assumes that the temperature is uniform at the point(s) of injection. In practice this is not too critical since the melt is rapidly convected into the cavity. An area where this may be critical occurs when a runner system is used to convey melt into several cavities. Due to shear heating in the runner and the flow of melt around corners, it is possible that viscosity differences occur at the gates of the various cavities and hence to filling imbalances. We discuss this later when we consider 3D analysis in Chapter 6. ■

The temperature, T , is prescribed on all mold boundaries. However, it is usual to allow different temperatures to be prescribed on each boundary such as, T = Tem on Σem , T = TW + on ΣW+ , T = TW − on ΣW− , T = Tins on Σins .

Alternatively, heat flux boundary conditions may be used. More generally, a combination of heat flux and temperature boundary conditions are prescribed on the corresponding surfaces. Heat flux boundary conditions are usually specified on the mold walls, ΣW+ and ΣW− . The heat fluxes may be calculated by coupling the filling/packing analysis with a mold cooling analysis. In this way the effect of the mold cooling system may be accounted for.

4.6.3

Mold Deformation Boundary Conditions

All of the above assumes that the mold does not deform. In practice, mold deformation may be important but we need to discuss the meaning of this term. There are two cases: 1. thin cavities 2. long cores and mold inserts

4.6.3.1

Thin Cavities

Since the first version of this book in 1995, the molding process has changed significantly. New material formulations and high pressure molding machines allow very thin cavities, such as for

4.7 Fiber-Filled Materials

49

laptop computer cases and personal electronic components, to be molded. These applications require much higher injection pressures. Specialized machines can deliver up to 250 MPa. Under such pressures, it is possible that mold compliance should be considered in calculations. This is not a new idea. Baaijens [21] included mold deformation for a thin-wall approximation to injection molding. More recently Delaunay et al. [77] have shown that for cavities less than 1 mm thick, mold compliance may be important.

4.6.3.2

Long Cores and Mold Inserts

Today there are a number of truly three-dimensional (3D), mold filling analyses available. We will look in detail at these in Chapter 6. For now, we note that 3D modeling permits explicit inclusion of cores and inserts. It is possible that with high injection pressure, cores and inserts may deflect and so alter the flow channel. This can lead to errors in flow front calculation and shrinkage and warpage if not accounted for. The boundary conditions in this section are very general. In the next chapter we consider simplification of governing equations, special cases for injection molding, and the necessary refinement of the boundary conditions.

4.7

Fiber-Filled Materials

It is common to include fibers in the polymer resin to improve mechanical properties. The amount of fiber added may vary from 10 to 60% by weight. Fibers may be of two types: short or long. Short fibers typically have a length of around 0.5 mm [388] and most commonly are made of glass, aramid, or carbon. However, stainless steel may be used for electrical shielding applications. Materials reinforced with short fibers are referred to as short fiber reinforced thermoplastics or SFRTPs. The addition of short fibers increases stiffness (modulus) but has little effect on strength of the material. It should be noted that the actual length of fiber found in moldings will depend on how they are processed. Plastication speed and back pressure affect the breakage of fibers and hence the ultimate length. Similarly gate design may have an affect on molded fiber length. Long fiber reinforced thermoplastics (LFRTPs) have fibers made of the same materials as SFRTPs. The length of the fiber is typically from 6–11 mm and is determined by the length of plastic pellet. As in the case of short fibers, the actual fiber length in a molding depends on processing and gate design. LFRTPs exhibit improved modulus over the base resin and higher impact resistance and strength. In this section we will consider only SFRTPs. We return to the challenges of simulating LFRTPs in Chapter 9 of this book. The simulation challenge for SFRTPs is to determine the location and orientation of fibers after molding. Given this information, it is possible to determine variation of thermo-mechanical properties throughout the molding. These properties can then be used for shrinkage, warpage, and structural analysis.

50

4.7.1

4 Governing Equations

Fiber Concentration

Fiber-filled materials are characterized by the aspect ratio of the fibers and the volume fraction. We will refer to the molten SFRTP as a fiber suspension. The fiber aspect ratio a R is defined as the length of the fiber divided by its diameter, d . The volume fraction φ is defined as the total volume of fibers in a unit volume of SFRTP. We define the number density of fibers, n, as the number of fibers per unit volume of SFRTP. Assuming a cylindrical shape, the volume of a single fiber is πd 2 l /4. Hence φ = nπd 2 l /4 ≈ nd 2 l . While the melt is filling the cavity, the rheological properties of the suspension and the ultimate fiber orientation distribution, depend on processing conditions and interaction among fibers in the suspension. The fiber interactions depend not only on the number of fibers in a certain volume but also on their length. Parameters such as nl 3 and nd l 2 are commonly used to classify concentration regimes of fiber suspensions. Since nl 3 = nd 2 l × (l /d )2 ≈ φa R2 and nd l 2 = nd 2 l × (l /d ) ≈ φa R , the two parameters are equivalent to φa R2 and φa R , respectively. The concentration of fiber suspensions is classified into three regimes: dilute, semiconcentrated (or semi-dilute), and concentrated. A dilute suspension has, on average, less than one fiber in a volume of v = l 3 . Each fiber may freely rotate about its three-rotational axes without any hindrance from other fibers. This leads to nl 3 < 1 or φa R2 < 1. In a semi-dilute suspension, each fiber is confined by nl 3 > 1 and nd l 2 < 1. In other words, 1 < φa R2 < a R where the average distance between fibers is greater than a fiber diameter but less than a fiber length. Consequently, the fiber is free to rotate about only two of its axes. Finally, we have the concentrated suspension in which φa R > 1. Here, the average distance between fibers is less than a fiber diameter so fibers cannot rotate about any axis other that their axis of symmetry. In such suspensions, fiber motion involves movement of surrounding fibers and fiber-fiber contacts are dominant.

4.7.2

Jeffery’s Equation

Most SFRTPs have fiber contents that put them into the concentrated suspension regime. Nevertheless, early attempts at modeling fibers in injection molding ignored this fact. The starting point of fiber suspension modeling is the work of Jeffery [182], who modeled the fiber as an inertialess rigid prolate spheroid suspended in a Newtonian fluid. Jeffery considered only a single particle that was free to rotate about all its axes. As a means of describing fiber motion, we introduce a unit vector p along the symmetrical axis of the prolate spheroid. Jeffery’s solution for the rate of orientation changes is: ˙ = W·p+ p

´ a r2 − 1 ³ D · p − (D : pp)p , 2 ar + 1

(4.72)

where W and D are, respectively, the vorticity and rate of strain tensors defined in Section 2.1.3 and a r is the ratio of the major to minor axes of the ellipsoid representing the particle. A derivation of Jeffrey’s equation is provided in Appendix C. When applying Equation 4.72 to cylindrical particles with length to diameter ratio a R , one may use the approximation, a r ≈ a R . Some authors have determined equivalent ellipsoidal ratios for cylinders however. Akczurowski and Mason [5] found that the equivalent ellipsoidal axis

4.7 Fiber-Filled Materials

51

ratio for a cylinder of a R = 0.86 is 1.0. They also found that for a R > 1.68, a r is smaller than a R and vice-versa. In shear flows of non-interacting fibers, the fibers exhibit a closed periodic rotation, known as a Jeffery’s orbit, with a period 2π ³ 1 ´ Tr = ar + . (4.73) γ˙ ar The Jeffery solution is suitable for dilute suspensions, but as mentioned earlier, commercial polymers are usually concentrated. Vincent [388] gives some typical figures. For aspect ratios of 10, the suspension can be considered semi-dilute for volume fractions from 0.01 to 0.10. Using typical values for densities of glass and polymer, these volume fractions correspond to weight fractions of 2.5% to 22%. Most SFRTPs for engineering application have weight fractions above 30%. Indeed in recent years, many suppliers have increased fiber content to 60% by weight. In order to realistically deal with these concentrated suspensions, the interaction between fibers must be considered.

4.7.3

A Statistical Approach

When dealing with a suspension of fibers, a probability density function (PDF) is a general way to describe the orientation state of the fibers. We define such a distribution, which we denote as ψ(p, t ), so that ψ(p, t )d p is the probability of finding a fiber oriented in the direction range from p to p + d p at time t . One of the properties of a probability density function is that it can only take values between 0 and 1, where 0 represents no possibility and 1 is a certainty. If we integrate the probability density function over all possible directions the value must be one. That is, Z ψ(p, t )d p = 1 . (4.74) Assuming the fibers are cylindrical and have no preferred end, the directions defined by p and −p should be the same. Consequently the PDF must be even. That is ψ(p, t ) = ψ(−p, t ). It can be shown [287] that ψ(p, t ) satisfies the Fokker-Planck equation: ´ i ∂ψ ∂ h (r ) ∂ψ ³ = · D (I − pp) · − L · p − (L : pp)p ψ , (4.75) ∂t ∂p ∂p where D (r ) is the diffusion coefficient and L is the effective velocity gradient, defined as L = L − ξD with ξ = 2/(a r2 + 1). Although the PDF provides a general description of the orientation state in the suspension, numerical solution of Equation 4.75 is computationally expensive. Indeed at the time of writing, we are unaware of any commercial code solving for the PDF. There are, however, several academic codes that may be used on small problems for research purposes. In the next chapter we will consider a popular method for predicting fiber orientation distributions in molded parts. There are some shortcomings associated with this and we reconsider the problem in Chapter 7 where some improved models are discussed. Moreover we have only dealt with short glass fibers. It is not uncommon to use long glass fibers for reinforcement and these may bend due to processing forces. These materials require alternative models. We discuss this further in Chapter 9.

4 Governing Equations

52

4.7.4

Mechanical Properties

One of the major incentives for predicting the fiber orientation of molded SFRTPs is to use that information to predict mechanical properties of the molded material. These properties may be used in structural analysis or as input to shrinkage and warpage calculations. For structural analysis, the main interest is in modulus prediction. For shrinkage and warpage, we need moduli and thermal expansion coefficients. There are a variety of models and methods to do this. Those used in commercial codes are often approximations and will be discussed in the next chapter. In Chapter 8 we consider some advanced methods that are yet to find implementation in commercial codes.

4.8

Shrinkage and Warpage

One of the major challenges of injection molding simulation is the prediction of shrinkage and warpage of injection molded materials. All polymers shrink and the degree of shrinkage depends on molding conditions. If the material shrinks uniformly, the molded article will simply be a faithful reproduction of the molded article albeit somewhat smaller. In this case, a mold maker merely needs to know the amount of shrinkage and can then make the mold appropriately bigger so that the molded article will conform to the desired dimensions. Geometric reduction in the dimensions of a molded part is referred to as mold-shrinkage, as-molded shrinkage, or simply shrinkage. Shrinkage may be defined according to standards such as the Standard Test Method of Measuring Shrinkage from Mold Dimensions of Thermoplastics (ASTM D955-08). In this test, shrinkage is measured 24 and 48 hours after ejection from the molding machine. The international standard, Plastics-Injection Moulding of Test Specimens-Part 4: Determination of Moulding Shrinkage (ISO 294-4), uses similar samples and methodology. However the reality is that shrinkage is a time dependent function. In practice, molded parts may be assembled well before (or after) the 24 or 48 hours specified in the standards. Worse still, molded parts may be subjected to changing environmental conditions after molding. An example is the painting of an automotive part. The painting process can involve exposure to higher than ambient temperatures, which can affect the amount of shrinkage and its anisotropy, and hence final shape (warpage). We refer to this type of shrinkage as post-molding shrinkage. Ideally simulation would predict shrinkage as a function of time. The above assumes shrinkage is uniform within the molding. Unfortunately this is rarely the case. Shrinkage of polymers depends on molding conditions—more particularly it depends on the temperature and pressure history of the material as it cools. These are local conditions and therefore shrinkage, and hence warpage may vary across the molding. The real challenge for injection-molding simulation is the accurate prediction of warpage of moldings. This entails accounting for local shrinkage as well as the effect of environmental factors after molding. What we really seek is a description of the deformation of the part immediately after molding and during its life history. This requires an appreciation of the mechanical properties of the material after molding. Apart from shrinkage and warpage, there is great value in predicting structural properties after processing. Some progress has been made in predicting properties such as yield stress for amorphous properties as a result of processing [96]. In Part II of this

4.9 Runners

53

book, we consider improvements in simulation that contribute to this ambitious goal for semicrystalline materials. We bring these ideas together in Chapter 13.

4.9

Runners

In practice there may be many mold cavities within a single mold. The cavities may not necessarily have the same geometry, but each cavity will need to be supplied with melt from a runner system. Figure 1.2 depicts a “cold runner” system, meaning the flow in the runner has a similar temperature boundary condition to the cavity; the polymer mold interface temperature is fixed or heat flux at the polymer mold interface is defined. On the other hand, for large molds, it may be necessary to introduce melt at various points of the molding to minimize the flow length and control the position of weldlines. In these cases, the runner system is heated with electrical elements so as to maintain a near constant temperature to the cavity and is called a hot runner. Whether runners are cold or hot, flow in the runner system, as for the cavity, is determined by the equations for conservation of mass, momentum, and energy. The difference is in the temperature boundary conditions at the plastic/mold interface, though it should be noted that the flow within a hot runner system may be complex. The type of runner used affects the geometry of the flow, and hence the shear deformation of the melt as it flows through the runner into the cavity. Moreover, since hot runners may be used on large moldings the flow rate may be quite high, resulting in significant shear heating. Pressures in hot runners may also be high and can affect viscosity. We will consider simplified mathematical modeling of runners in Section 5.10.

5 5.1

Approximations for Injection Molding

Introduction

The equations provided in Chapter 4 are quite general and comprise a very comprehensive model for injection molding. In this chapter we consider some approximations that may be made to allow simulation of the molding process. Many approximations are available and each will affect accuracy, computational speed and, choice of numerical method for solution. It is important to note that approximations arise from material modeling, the governing equations, and the numerical method for solution. Therefore in every simulation code, there are material models, a mathematical model (a set of equations to be solved with appropriate boundary conditions), and some numerical method for obtaining a solution. Whatever the approximations made, the fact is that both temperature and pressure need to be determined at each point in the cavity. This is at odds with many standard fluid mechanical and rheological problems where temperature is frequently ignored and stress, rather than pressure, is to be determined. At the factory level, the injection molding process depends on flow rate, temperature, and pressure. These are the settings that may be modified on an injection molding machine and commercial simulation codes must relate to these variables. Another important issue arises in the validation and verification of codes simulating the process [272, 310]. Put simply, validation concerns the applicability of a model to a process while verification concerns how well the calculations are implemented in a particular code. Both aspects may be difficult to assess. Validation usually involves a comparison between real results, obtained from experiment, and the results of simulation. For the end user of a commercial code only limited verification can be done, such as systematic mesh refinement testing; usually it can be very difficult to verify the code except in cases of blatant error—unless the developer allows access to parts of the code. Few commercial companies provide this. Consequently, many potential buyers of a commercial code can only conduct an experiment and run a simulation to determine agreement. In performing an experiment, the potential buyer assumes that the mathematical model used is appropriate and is correctly implemented. For injection molding under production conditions, the variables that can be measured are temperature, flow front position, and pressure. Specialist molds have been constructed to assess these aspects [76, 233]. While this provides basic information on the filling phase, some people will try to compare shrinkage and warpage of the part seen in experiment against the simulation. This is a difficult task, and people wanting to pursue this must have considerable experimental expertise and resources. Our point here is that it is not a trivial task to validate an injection molding code.

56

5 Approximations for Injection Molding

In this chapter we look at the approximations used in injection-molding simulation. These affect the ability of a simulation to accurately predict an experimental result. We make no claim that this is the whole story. Hopefully it is a snapshot of where we are at the time of writing. There are many aspects of simulation that remain to be improved. These will occur over time and we present some potential areas of improvement in Part II of this book.

5.2

Material Property Approximations

Material properties are usually simplified due to lack of knowledge and inability to model material properties. In this section we discuss common material assumptions in simulation software. It is convenient to discuss these properties depending on the analysis undertaken. Material properties required to simulate injection molding depend on the simulation to be undertaken. For flow analysis (filling, packing, and cooling) the following are required: ■

thermal sources

■

viscosity

■

specific heat capacity

■

thermal conductivity

■

no-flow or transition temperature

■

PVT data

Should the user want to simulate shrinkage and warpage, additional properties are required such as: ■

elastic modulus

■

shear modulus

■

coefficient of thermal expansion

■

simulation software may also require specialized data for particular models

We discuss these assumptions below.

5.3

Filling, Packing, and Cooling Analysis

The properties required are: ■

thermal sources

■

viscosity

■

specific heat capacity

■

thermal conductivity

■

no-flow or transition temperature

■

PVT data

5.3 Filling, Packing, and Cooling Analysis

5.3.1

57

The Thermal Source Term in the Energy Equation

For thermosetting materials, the heat of reaction is considerable and the heat source term is essential. Our focus is on thermoplastics and so heat of reaction is ignored here. Semi-crystalline materials give up some heat on solidification and this should be included. However, few simulations deal with crystallization though it is a current research area. For now we ignore it and return to it in Part II of the book. For now we assume Q˙ = 0 .

5.3.2

(5.1)

Viscosity Modeling

We will assume the melt is a generalized Newtonian fluid (GNF), see Section 2.3. That is ˙ . σ = −pI + 2η(γ)D

(5.2)

This alters the terms ∇ · σ and σ : ∇v in Equations 4.23 and 4.67. Using Equation 2.19 we have ∇ · σ = ∇ · (−pI + 2ηD) = −∇p + 2∇ · ηD .

(5.3)

Polymer melts are viscoelastic in nature. However for injection-molding simulation, the assumption of a GNF can provide useful results. To carry out a full viscoelastic analysis for an injection mold raises a number of issues. Firstly, there is the viscoelastic model itself. In the past 40 years there have been some major improvements in the development of these models. However any model must be evaluated in terms of what improvement it brings to injectionmolding simulation. Regardless of their worth, these models stretch our computational ability even in simple geometries. For complicated molds, they are out of the question today. This is not all bad. A reasonable approximation for prediction of molecular orientation, shrinkage, and warpage of amorphous injection molded products has been provided by Baaijens [21] and we discuss this later in this chapter. In many commercial codes, this approximation has been applied to semi-crystalline materials. However semi-crystalline materials are more complex and deserve a more detailed treatment. We leave this topic and return to it in Part II of this book. Now we consider the assumption that the material is a GNF. To this end, consider the term σ : ∇v: σ : ∇v = (−pI + 2ηD) : ∇v = −pI : ∇v + 2ηD : ∇v ´ ³ = −p∇ · v + η ∇v + (∇v)T : ∇v ¢ ¡ ¢ 1 ¡ = −p∇ · v + η ∇v + (∇v)T : ∇v + (∇v)T 2 = −p∇ · v + 2η D : D = −p∇ · v + ηγ˙ 2 , where, in the last step, we have used Equation 2.17.

(5.4)

58

5 Approximations for Injection Molding

5.3.3

Specific Heat Capacity

Specific heat capacity can be readily obtained from differential scanning calorimetry (DSC). The problem is that it is measured under relatively slow temperature change due to the construction of the machines used to measure it. In injection molding, the rate of temperature change (cooling) can be very high and some error is introduced. For amorphous polymers these errors may not be too important but for semi-crystalline materials, the error is significant. Some codes may allow for values of specific heat in the melt and solid regions. However there is a need to introduce the rate of cooling as well. In other words, for semi-crystalline polymers, the specific heat needs to be coupled with the crystallization kinetics of the material. We consider this problem in Section 11.3.

5.3.4

Thermal Conductivity

Polymers have a relatively low thermal conductivity as we saw in Section 3.7.2. The low conductivity makes experimental measurement of the property difficult. The line source method of measurement has become the norm [229] however there are some more sophisticated methods developed by Schieber et al. [322]. Both amorphous and semi-crystalline polymers show an increase in thermal conductivity with temperature. Pressure also affects thermal conductivity. Dawson et al. [75] have shown that both amorphous and semi-crystalline materials show an increase in conductivity with pressure. Many simulation codes use a single value for amorphous and semi-crystalline materials. Moreover, the values are assumed to be scalar quantities. We challenge these assumptions and provide some details in Part II to support our view and indicate how modeling may be improved. For now we briefly comment on shortcomings of these assumptions in three categories: 1. unfilled amorphous 2. unfilled semi-crystalline 3. filled polymers

5.3.4.1

Unfilled Amorphous

These materials have been considered the simplest and there is a great deal of literature devoted to their properties. However, properties measured in the laboratory using scientific equipment are determined under very different conditions to those in injection molding. The assumption that a single value of thermal conductivity for an amorphous material was perhaps valid a couple of decades ago. Since then, moldings have thinner walls. Consumer electronics—particularly personal electronics—such as mobile phones, laptops, and other devices has lead the move to thinner moldings. Consequently amorphous polymers are being subjected to extreme pressures and shear rates. In what may be a visionary paper, van den Brule [374] proposed that molecular orientation may induce anisotropy in the thermal conductivity of an amorphous polymer. In particular, his theory suggests improved conductivity along the polymer backbone and reduced conductivity transverse (in the thickness direction) to it. The molecular backbone orientation will be generally in the direction of shear. Hence his theory suggests that the thermal conductivity transverse to the flow direction may be reduced.

5.3 Filling, Packing, and Cooling Analysis

59

Some experimental evidence to support this concept has been supplied by Schieber et al. [322] and Venerus et al. [385].

5.3.4.2

Unfilled Semi-Crystalline

Most properties of semi-crystalline material will exhibit a change at the transition temperature. Measurements of thermal conductivity that cover the transition will exhibit an increase in conductivity as the material solidifies. However, measurement techniques fail to take into account the morphology which develops during molding and the subsequent variation of properties. Until a comprehensive understanding of property development is available, the best that can be done is to use two values of thermal conductivity for the melt and solid phases respectively—ideally these will be as a function of temperature.

5.3.4.3

Filled Materials

A common filler for both amorphous and semi-crystalline materials is short glass fibers. Fibers of S2 type glass have a conductivity of 1.10–1.40 W/(mK), which is at least three times that of polymers. It is not uncommon for glass loadings to be as high as 60% by weight for semicrystalline materials. Given the higher conductivity of the glass, we suggest that the orientation of the glass fibers could give rise to a tensorial thermal conductivity. We investigate this further in Part II of the book but note that the assumption that thermal conductivity is a scalar should be challenged.

5.3.5

No-Flow or Transition Temperature

In Section 3.6.6 we noted that no viscosity function with WLF or Arrhenius temperature dependence can reproduce the viscosity as the material starts to cool to a solid. In injection-molding simulation this difficulty has been overcome by introducing a no-flow temperature or a transition temperature. This is a temperature at which the material in the simulation is assumed not to flow. In other words, below the “transition temperature” the material has an extremely high viscosity and is assumed to be solid. The no-flow temperature was introduced many years ago by Moldflow and was measured using extrusion through a capillary with a given pressure whilst the temperature was reduced. A no-flow temperature was measured when the rate of extrusion fell below a defined rate. There may have been some merit in this technique since it was discovered later that both deformation and temperature play a role in crystallization and hence solidification of semi-crystalline materials. However, there was no link in the simulation between crystallization and solidification. Measurement of transition temperature was usually performed in a differential scanning calorimeter (DSC). It was argued by proponents of this method that is was more repeatable than the no-flow test. This may have been true, but the DSC test was static with no fluid motion. Consequently it is invalid for semi-crystalline materials though perhaps more repeatable for amorphous materials. It is not our intention to argue for either of these methods. Both are limited and lead to errors. Instead, we recommend a more comprehensive modeling of the material properties—for both amorphous and semi-crystalline materials.

60

5 Approximations for Injection Molding

In computer simulation, the no-flow or transition temperature is implemented by assigning an extremely high value to the viscosity of the material when its temperature falls below the transition temperature. Figure 5.1 illustrates this.

Figure 5.1 Need for a transition or no-flow temperature

For amorphous materials, the error created is less than that for semi-crystalline materials. However, for semi-crystalline materials, the use of a single transition temperature makes no sense. In reality, semi-crystalline materials exhibit a very rapid increase in viscosity under cooling conditions, but this is not the only complication. The rate of temperature decrease has an effect on the crystallization and hence the viscosity. This has been known for many years. If a semi-crystalline is cooled at a very fast rate, it may continue to flow below a measured transition temperature. This is the phenomenon known as supercooling. That is, all things being equal, a melt cooled at a particular rate of cooling will have a sudden increase in viscosity at a lower temperature than that exhibited by a melt cooled at a lower rate of cooling. An added complication is the effect of deformation. Over the past two decades, this has been studied extensively. There is general agreement that the historical deformation of the material affects the temperature at which viscosity suddenly increases. Put roughly, the higher the shear rate that the material is exposed to, the higher the temperature at which we see the viscosity suddenly increase. So we have at least two factors to consider for semi-crystalline materials. The first is the cooling rate. As the cooling rate is increased the crystallization temperature is decreased. The second is the deformation history of the material. As the deformation rate increases, so does the crystallization temperature. These two effects act contrary to each other. At the time of writing there are a number of theories of how the crystallization should be modeled. We will discuss some of these in Part II. For now however, we will (wrongly) assume that a single transition temperature will suffice. That said, the reader should not be dismayed if their particular injection-molding simulation software uses a single value for the crystallization temperature. It all depends on what you want to do with the software. In molds where the filling phase takes place above the crystallization temperature, the issue is probably irrelevant and flow patterns and pressures may be

5.3 Filling, Packing, and Cooling Analysis

61

accurately predicted. In molds with significant variations of thickness and imbalanced flow, there could be inaccuracies. More particularly, under these conditions, the prediction of residual stresses and hence warpage may be in error.

5.3.6

Pressure-Volume-Temperature (PVT) Data

Early codes assumed the melt was incompressible during filling and compressible during packing [196]. This seems reasonable but the reality is that as a mold fills, there may be parts that are full and so undergoing packing as the rest of the mold fills. Figure 5.2, though simple, illustrates this idea. Melt is introduced at the injection point and fills the left side of the part

Figure 5.2 Mold for which material is in packing and filling phases

while the right-hand side of the mold continues to be filled. Material on the left of the injection point is basically in the packing phase—with little flow and cooling at the pressure defined by the injection pressure. So contrary to Kennedy [196], we assume the melt is compressible in both the filling and packing phases of the process. As discussed in Section 3.9, the compressibility, expansivity, and density are obtained from a PVT relationship and so their accuracy depends on that of the PVT data. Determination of such data is called high pressure dilatometry. For polymers, this is a challenging field due to complex experimental problems. This is evidenced by the few PVT testing machines commercially available. Almost all of these measure the data under static conditions by which we mean without any controlled deformation of the sample. The GNOMIX device arose from work by Zoller et al. [432] and encapsulated the polymer sample in mercury which was then pressurized and the resultant change in volume measured. One of the advantages of the GNOMIX device is that the sample undergoes hydrostatic pressure loading. Another device developed by SWO Polymertechnik GmbH was the PVT 100. This device used a piston to apply pressure to the polymer sample. It was criticized on the basis that the pressure applied need not be hydrostatic particularly as the sample solidified and potentially shrunk away from the walls of the measurement chamber. The reality was that both these devices had difficulty in assessing the effect of cooling rate on the PVT behavior of semi-crystalline materials. This was due to their robust design, which was required to allow measurement of PVT data at high temperature and particularly high pressures. While both machines could measure data with decreasing temperature, the

62

5 Approximations for Injection Molding

thermal gradients within the machines were unknown and so it was difficult to assess the actual cooling rate of the polymer sample even if the temperature was well controlled. Luyé et al. [234] used a modified PVT 100 and analytical solutions for heat transfer to determine thermal gradients in the sample and so estimate the real temperatures in the polymer sample as temperature was decreased. This approach was successful and showed the effect of cooling rate on PVT data. A major advance in PVT measurement came with the work of Forstner et al. [123]. They set out to achieve PVT measurements at high pressures, high cooling rates, and at controlled shear rates. This was done by using an annular sample of 0.5 mm thickness and allowing the sample to be sheared at a controlled rate. Despite all this work, at the time of writing, there appears to be only one commercial dilatometer for polymers on the market. It is based on the work of Forstner et al. [123] and van der Beek et al. [376–378] and is produced by IME Technologies in the Netherlands, (http://www.imetechnologies.nl/Pirouette-n236m299). To summarize then, the PVT data generated by commercial machines such as the GNOMIX and PVT 100 are not sufficient for injection molding of semi-crystalline materials. As we will see in Chapter 10 the deformation of the polymer has a profound effect on the transition temperature as does the rate of cooling. For now, perhaps the best approach is to use PVT data at near equilibrium conditions for temperature and shear rate and then, using crystallization modeling, modify the data to account for the effects of processing. We return to these problems in Part II of this book.

5.3.7

Fiber Orientation, Shrinkage, and Warpage

Generally, fiber orientation analysis is used to determine the mechanical and thermal properties of the molded material. These predicted properties are then used for shrinkage and warpage analysis. However, having obtained these properties, they may be used in other analyses beyond molding such as structural and thermal analysis.

5.3.7.1

Fiber Orientation Analysis

Material properties and subsequent approximations for fiber orientation analysis depend on the mathematical model used to determine the fiber orientation. Most commercial codes use the Folgar-Tucker model, which we discuss later in this chapter. For now it should be noted that the Folgar-Tucker model uses the velocity gradients of the melt during filling and packing to determine the orientation of fibers. That means that all the previous material assumptions discussed in this chapter, which affect the viscosity and hence the velocity gradients, will have an effect on the fiber orientation prediction. In terms of material properties, the Folgar-Tucker model requires the concentration by volume of glass fibers in the material and the aspect ratio of the fibers. The model assumes that the fiber length is about an order of magnitude less than the width of the cavity. Fiber concentration does not present any problem in measurement and requires no assumption other than that the fibers are uniformly distributed in the polymer prior to and during molding. The latter may be debatable. It is known that fibers can flocculate at the entry to very thin sections of moldings but this depends on the length of the fibers.

5.4 Summary of Material Assumptions

63

The aspect ratio of fibers is a more difficult material property to obtain. When compounding fibers into a polymer matrix, the fibers will initially have lengths conforming to some distribution. During compounding, there may be fiber breakage and so the length distribution will be changed. Finally during processing on the molding machine, there may be further fiber breakage and a consequent reduction in fiber length. This will depend on the processing parameters and mold features such as gate size. One area that is prone to error is processing of long fiber-filled materials. it is not uncommon to see the Folgar-Tucker model used even when long fiber materials are involved. This violates the assumption regarding fiber length that is implicit in the Folgar-Tucker model. In addition, the actual final length of fibers achieved during processing is very dependent on processing conditions. Processing of long fiber materials is an ongoing area of research and we return to it in Chapter 9.

5.3.7.2

Shrinkage and Warpage Analysis

For shrinkage and warpage we need: ■

linear coefficient of expansion in directions 1 and 2

■

elastic modulus in directions 1 and 2

■

Poisson’s ratios

■

shear modulus

Of these, Poisson’s ratios and shear modulus (moduli) are usually determined with some approximation as they are difficult to measure experimentally. This is especially true for semicrystalline and fiber filled materials. We will consider equations for prediction of linear coefficients of expansion and elastic moduli later in this chapter. For now we note that their determination for unfilled materials depends on the morphology of the material which in turn depends on its processing history. This is still an ongoing area of research. Typically, for unfilled materials, commercial codes will use a single values for linear coefficient of expansion and elastic modulus. For filled materials, there is an implicit assumption that the anisotropy introduced by the fiber filler is greater than that due to morphology. So most codes will use a formula that uses a single value for properties of the matrix and filler and then modify this using the calculated fiber orientation to get the anisotropic properties of the composite.

5.4

Summary of Material Assumptions

From the foregoing we see that the essential problem in obtaining relevant material properties for molding simulation are twofold. Firstly, there is the experimental difficulty to determine the properties and secondly the fact that the process itself changes properties. It seems to us that that the best way forward is to measure material data under well known conditions, accept its limitations for use in simulation, and develop material modeling techniques that allow us to transform the measured data to data that is relevant in simulations. In other words, the concept of material data generated under lab conditions that is input to simulation

64

5 Approximations for Injection Molding

programs and used to generate results should be replaced with the concept of inputting data generated under lab conditions and modifying it as the simulation progresses. Much of Part II of this book is devoted to this idea. While any advance in testing is welcome, its true value may lie in validating material modeling.

5.5

Governing Equations

The governing equations were derived in Chapter 4 and are: Conservation of mass: ∂ρ + ∇ · ρv = 0 ∂t

(5.5)

Conservation of momentum: ∂ (ρv) = ρg + ∇ · σ − ∇ · (ρvv) ∂t

(5.6)

Conservation of energy: ρc p

´ ³ ∂p ´ + v · ∇T = βT + v · ∇p + p∇ · v + σ : ∇v + ∇ · (k∇T ) + Q˙ ∂t ∂t

³ ∂T

(5.7)

We now consider implications and approximations that may be made due to the material assumptions, discussed in Section 5.2, on these equations. The conservation of mass equation remains unchanged and is the same as in Equation 5.5. However the material assumptions affect both the conservation of momentum and energy equations. We deal with the conservation of momentum first. Substituting Equation 5.3 into Equation 5.6, we obtain ∂ (ρv) = ρg − ∇p + ∇ · ηD − ∇ · (ρvv) . ∂t

(5.8)

The left-hand side of Equation 5.8 may be expanded to give: ∂ρ ∂v ∂ (ρv) = v+ρ ∂t ∂t ∂t = −(∇ · ρv)v + ρ

∂v , ∂t

(5.9)

where we have used the conservation of mass equation, Equation 5.5. The last term of Equation 5.8 may be expanded as follows: ∇ · (ρvv) = ρv · ∇v + (∇ · ρv)v .

(5.10)

Substituting Equations 5.9 and 5.10 into Equation 5.8, we obtain −(∇ · ρv)v + ρ

∂v = ρg − ∇p + ∇ · ηD − ρv · ∇v − (∇ · ρv)v . ∂t

(5.11)

5.6 The 2.5D Approximation

65

That is, ρ

∂v = ρg − ∇p + ∇ · ηD − ρv · ∇v . ∂t

(5.12)

Now consider the energy equation, Equation 5.7. Assuming constant thermal conductivity, and substituting Equations 5.1 and 5.4 into Equation 5.7, we get: ρc p

³ ∂T

´ ³ ∂p ´ + v · ∇T = βT + v · ∇p + ηγ˙ 2 + k∇2 T . ∂t ∂t

(5.13)

Most commercial codes will be based upon the Equations 5.5, 5.12, and 5.13 for the conservation of mass, momentum, and energy, respectively. To improve this situation, which is far from satisfactory, there is a need to develop more detailed models of material properties. If this sounds negative, it is important to note that any simulation can bring some insight provided assumptions and limitations are understood. It is for this reason that many of the codes using the above assumptions provide users with information to improve their designs despite the scientific deficiencies. The challenge for code developers—commercial and academic, is to provide a more complete modeling of the injection molding process; particularly the modeling of material properties, that are themselves affected by the process. We consider this further, and supply some ideas in Part II of this book to address this problem. For the remainder of this chapter though, we focus on numerical schemes for the solution of the governing equations. These too entail assumptions and subsequent simplifications that any user of simulation software should be aware of.

5.6

The 2.5D Approximation

The 2.5D approximation makes use of the fact that injection molding cavities are frequently thin walled. This is due to the low thermal conductivity, desire for fast production times, and lack of internal voids. For a thin molding, there is a high temperature gradient in the thickness direction, yet the pressure gradient throughout the molding is relatively low. The 2.5D approximation has remained in the simulation industry from around 1986 to present. Its roots lie in the paper by Hieber and Shen [154]. This paper introduced the idea that the temperature in the thickness may be best addressed by finite differences and the pressure in the midplane of the molding by finite elements. This model is so important, historically and commercially, that we devote some time to its detailed derivation in this section. These derivations have been provided in the past by Boshouwers and van der Werf [42] for the filling phase. The filling and packing phases were considered by Chiang et al. [60] and Kennedy [196]. Temperature during molding varies from point to point and, in particular, there is a high gradient across the wall section of the part. While the temperature gradient is less in the direction of flow, it still varies significantly. For these reasons, temperature should be solved in 3D. Pressure on the other hand varies from point to point, but there is relatively little pressure variation across the wall section. For this reason, pressure may be solved in 2D at points on an imaginary midplane of the actual 3D part. This is where the “2.5D” terminology comes from.

66

5 Approximations for Injection Molding

It should be remembered that approximations in injection-molding simulations are made due to lack of knowledge about material properties, boundary conditions, and lack of compute power. To the current reader, the latter may appear odd. However codes for injection molding were developed from the 1960s and lack of computer speed and memory was a significant factor. With today’s capabilities it is less of a problem. However from the point of view of a designer for an injection molded product, it is desirable to run many analyses to optimize a design. Consequentially speed is important. It is for this reason that, while full 3D analysis is available (see Section 6.2), it may not be the best way to optimize designs. For these reasons the 2.5D approximation still enjoys popularity today. In order to derive the 2.5D approximation, we introduce a Cartesian coordinate system to describe the cavity. The axes are arranged so that at any point in the cavity the x-y plane coincides with the midplane of the part and the z-axis points in the thickness direction, that is, normal to the midplane. Figure 5.3 shows a simple cavity with the axes defined at two points.

Figure 5.3 Thin-walled cavity with coordinates systems defined at two points

5.6.1

Governing Equations in Cartesian Coordinates

In this section we express the governing equations in Cartesian coordinates. We will then use dimensional analysis to remove lower order terms and so determine a simplified set of equations for solution.

5.6.1.1

Conservation of Mass

In Cartesian coordinates, the conservation of mass equation becomes: ³ ∂v ∂v y ∂v z ´ ³ ∂ρ ∂ρ ∂ρ ∂ρ ´ x = −ρ + + − vx + vy + vz . ∂t ∂x ∂y ∂z ∂x ∂y ∂z

(5.14)

5.6 The 2.5D Approximation

67

Now the density ρ is a state variable and depends on pressure p and temperature T as defined by the PVT relationship. Using the chain rule of differentiation we obtain: ∂ρ ∂x ∂ρ ∂y ∂ρ ∂z ∂ρ ∂t

³ ∂ρ ´ ∂p ∂p T ∂x ³ ∂ρ ´ ∂p = ∂p T ∂y ³ ∂ρ ´ ∂p = ∂p T ∂z ³ ∂ρ ´ ∂p = ∂p T ∂t =

³ ∂ρ ´ ∂T ∂T p ∂x ³ ∂ρ ´ ∂T + ∂T p ∂y ³ ∂ρ ´ ∂T + ∂T p ∂z ³ ∂ρ ´ ∂T + ∂T p ∂t +

,

(5.15)

,

(5.16)

,

(5.17)

.

(5.18)

Substituting the above expressions into Equation 5.14 and rearranging gives: 0=

µ ¶ µ ¶ 1 ∂ρ ∂p ∂p ∂p ∂p + vx + vy + vz ρ ∂p T ∂t ∂x ∂y ∂z µ ¶ µ ¶ 1 ∂ρ ∂T ∂T ∂T ∂T + + vx + vy + vz ρ ∂T p ∂t ∂x ∂y ∂z µ ¶ ∂v x ∂v y ∂v z + + + . ∂x ∂y ∂z

(5.19)

Recall that density is the inverse of specific volume, Vˆ . Hence we can write µ ¶ µ ¶ 1 ∂ρ 1 ∂ ³1´ = ρ ∂p T ρ ∂p Vˆ T ¶ µ 1 ∂ ³ 1 ´ ∂Vˆ = ρ ∂Vˆ Vˆ ∂p T µ ¶ 1 ³ 1 ´ ∂Vˆ = − ρ Vˆ 2 ∂p T µ ¶ 1 ∂Vˆ =− Vˆ ∂p T = κ,

(5.20)

where κ is the isothermal compressibility coefficient of the material [326] defined in Section 3.8.1. Similarly, µ ¶ µ ¶ 1 ∂ρ 1 ∂ ³1´ = ρ ∂T p ρ ∂T Vˆ p µ ¶ 1 ∂ ³ 1 ´ ∂Vˆ = ρ ∂Vˆ Vˆ ∂T p µ ¶ 1 ³ 1 ´ ∂Vˆ = − ρ Vˆ 2 ∂T p µ ¶ 1 ∂Vˆ =− Vˆ ∂T p = −β ,

(5.21)

68

5 Approximations for Injection Molding

where β is the expansivity of the material [326] defined in Section 3.8.1. Substituting Equations 5.20 and 5.21 into Equation 5.19 we obtain ³ ∂T ∂p ∂p ∂p ´ ∂T ∂T ∂T ´ + vy + vz −β + vx + vy + vz ∂t ∂x ∂y ∂z ∂t ∂x ∂y ∂z ³ ∂v ´ ∂v ∂v y x z + + + . ∂x ∂y ∂z

0=κ

5.6.1.2

³ ∂p

+ vx

(5.22)

Conservation of Momentum

The momentum equation, Equation 5.12, when expressed in Cartesian coordinates, yields the following three equations for each of its components: ■

x-component ρ

■

(5.23)

y-component ρ

■

µ ¶ ∂p ∂ ³ ∂v x ´ ∂ ³ ∂v y ∂v x ´ ∂v x = ρg x − + 2η + η + ∂t ∂x ∂x ∂x ∂y ∂x ∂y µ ³ ¶ ´ ∂ ∂v x ∂v z + η + ∂z ∂z ∂x ³ ∂v ∂v x ∂v x ´ x + vy + vz − ρ vx ∂x ∂y ∂z

∂v y ∂t

µ ¶ ∂p ∂ ³ ∂v x ∂v y ´ η + + ∂y ∂x ∂y ∂x µ ¶ ∂ ³ ∂v y ´ ∂ ³ ∂v z ∂v y ´ η + 2η + + ∂y ∂y ∂z ∂y ∂z ³ ∂v y ∂v y ∂v y ´ + vy + vz − ρ vx ∂x ∂y ∂z

= ρg y −

(5.24)

z-component ρ

5.6.1.3

µ ¶ ∂v z ∂p ∂ ³ ∂v x ∂v z ´ = ρg z − + + η ∂t ∂z ∂x ∂z ∂x µ ¶ ∂ ³ ∂v y ∂v z ´ ∂ ³ ∂v z ´ + η + + 2η ∂y ∂z ∂y ∂z ∂z ³ ∂v ´ ∂v z ∂v z z − ρ vx + vy + vz ∂x ∂y ∂z

(5.25)

Conservation of Energy

The equation of energy, Equation 5.13, in Cartesian coordinates, is: ρc p

³ ∂T ∂t

+ vx

³ ∂p ∂T ∂T ∂T ´ ∂p ∂p ∂p ´ + vy + vz = βT + vx + vy + vz ∂x ∂y ∂z ∂t ∂x ∂y ∂z ³ ∂2 T ∂2 T ∂2 T ´ + ηγ˙ 2 + k + + . ∂x 2 ∂y 2 ∂z 2

(5.26)

5.6 The 2.5D Approximation

69

Recall from Equation 2.17 that ¡ ¢ γ˙ 2 = 2D : D = 2tr D2 . In Cartesian coordinates, ∂v x   ³ ∂x  1 ∂v y ∂v x ´ D=  2 ∂x + ∂y   1 ³ ∂v x ∂v z ´ + 2 ∂z ∂x 

1 ³ ∂v y ∂v x ´ + 2 ∂x ∂y ∂v y 1 ³ ∂v

∂y z

2 ∂y

+

∂v y ´

 1 ³ ∂v x ∂v z ´ + 2 ∂z ∂x   1 ³ ∂v y ∂v z ´ . + 2 ∂z ∂y    ∂v z

∂z

(5.27)

∂z

We will simplify the term γ˙ 2 later.

5.6.2

Estimation of Relevant Terms

In order to simplify the governing equations we first use dimensional analysis. The idea is to use characteristic values of variables, such as cavity length, thickness, and melt velocity to evaluate the relative magnitude of the terms. Terms that are relatively small are dropped and provide simplification of the governing equations. Characteristic values need not be exact; order of magnitude estimations are sufficient. We use the following characteristic values [42, 196] but note that these may vary for extreme variants of injection-molding processes, such as micro-injection molding or ultra high-speed injection molding: ■

Cavity thickness, H = 10−3 m

■

Cavity length, L = H /δ m where δ = H /L ≪ 1

■

Melt velocity, V = 10−1 m/s

■

Cavity pressure, p 0 = 107 N/m2

■

Melt viscosity, η 0 = 104 Ns/m2

■

Melt expansivity, β = 10−3 1/K

■

Melt thermal conductivity, k = 10−1 W/mK

■

Melt density, ρ = 103 kg/m3

■

Temperature difference between mold and melt, T0 = 102 K

■

Acceleration due to gravity, g 0 = 10m/s2

■

Specific heat of melt, c p 0 = 103 J/kg K

The key assumption here is that δ = H /L is small—of the order of 10−3 . This is sometimes called the thin gap or lubrication approximation. Further details on this are provided in Appendix D. Using these typical values, the relevant variables in the equations can be defined in terms of dimensionless variables as follows: ■

x-coordinate: x = Lx ∗ = H /δx ∗

■

y-coordinate: x = Ly ∗ = H /δy ∗

■

z-coordinate: x = H z ∗

70

5 Approximations for Injection Molding

■

Time: t = [L/V ]t ∗ = (H /δV )t ∗

■

x-component of melt velocity: v x = (L/t )v x∗ = V v x∗

■

y-component of melt velocity: v y = (L/t )v ∗y = V v ∗y

■

z-component of melt velocity: v z = (H /t )v z∗ = δV v x∗

■

Pressure: p = p 0 p ∗

■

Viscosity: η = η 0 η∗

■

Temperature difference: ∆T = T0 ∆T ∗

■

Acceleration due to gravity, g = g 0 g ∗

■

Specific heat, c p = c p 0 c p∗

where the starred quantities are dimensionless and of order one. Our approach will involve substituting dimensionless variables into the governing equations and then estimating the order of magnitude of each term. For example, ∂v x h V i ∂v x∗ . = ∂x L ∂x ∗

(5.28)

Because the starred quantities are of order one, the order of magnitude of ∂v x /∂x is equal to the order of magnitude of V /L. For brevity we write: h ∂v i hV i x O =O , ∂x L

(5.29)

where O[∗] is read as the order of magnitude of [∗]. Using the characteristic values for V and L we find: hV i h ∂v i x =O O ∂x L h δV i =O H h 10−1 δ i =O 10−3 £ 2 ¤ = O 10 δ .

(5.30)

In Appendix D we use the above methods to estimate the relative value of terms in Equations 5.22 to 5.26. The results are: the conservation of mass equation, Equation 5.22, remains unchanged and is ³ ∂T ∂p ∂p ∂p ´ ∂T ∂T ∂T ´ + vy + vz −β + vx + vy + vz ∂t ∂x ∂y ∂z ∂t ∂x ∂y ∂z ³ ∂v ´ ∂v ∂v y x z + + + . ∂x ∂y ∂z

0=κ

³ ∂p

+ vx

(5.31)

The momentum equations, Equations 5.23 to 5.25, are reduced to ■

x-component ∂ ³ ∂v x ´ ∂p = η ∂x ∂z ∂z

(5.32)

5.6 The 2.5D Approximation

■

y-component ∂p ∂ ³ ∂v y ´ = η ∂y ∂z ∂z

■

71

(5.33)

z-component ∂p =0 ∂z

(5.34)

The energy equation is reduced to (see Equation D.51): ρc p

³ ∂T ∂t

+ vx

∂T ∂T ∂T ´ ∂p ∂2 T + vy + vz = βT + ηγ˙ 2 + k 2 . ∂x ∂y ∂z ∂t ∂z

Using dimensional analysis, it may be shown that, in the 2.5D approximation, s µ ¶ µ ¶ ∂v y 2 ∂v x 2 γ˙ = + , ∂z ∂z

(5.35)

(5.36)

and it is common to call γ˙ the shear rate.

5.6.3

Velocity in the z Direction

We see from Equations 5.32 to 5.34 that the pressure, p, is a function of x and y only. However, there is an anomaly here. Both the conservation of mass and energy equations have terms involving v z yet there is, from Equation 5.34, no pressure gradient in the z-direction. To provide consistency between the conservation equations for the 2.5D approximation we make a further assumption that v z = 0. Denn [78] considers an approximation to v z when the thickness is known as a function of x or y. However, in injection molding, the transitions at gates and changes of thickness in the cavity are frequently very sudden and not amenable to Denn’s treatment. Accordingly, we set v z = 0 and adopt the following equations for the 2.5D approximation: Conservation of mass ³ ∂p ³ ∂T ∂p ∂p ´ ∂T ∂T ´ ³ ∂v x ∂v y ´ 0=κ + vx + vy −β + vx + vy + + . ∂t ∂x ∂y ∂t ∂x ∂y ∂x ∂y

(5.37)

The momentum equations, Equations 5.32-5.34, are unchanged and the energy equation is ρc p

³ ∂T ∂t

+ vx

∂T ∂T ´ ∂p ∂2 T + vy = βT + ηγ˙ 2 + k 2 . ∂x ∂y ∂t ∂z

(5.38)

Neglecting v z affects the temperature near the flow front where fountain flow has the effect of convecting melt from the middle of the cavity to the walls. Once the flow front has passed a particular point in a thin-walled mold, the flow is laminar and so convection in the z-direction is not so important. A significant error may be introduced at corners where the melt changes direction rapidly. In these areas, which include runner systems, the errors in temperature convection can lead to error in flow front prediction. Only a true 3D simulation can identify this accurately. We consider this later in Section 6.4.

72

5 Approximations for Injection Molding

5.6.4

Integration of the Momentum Equations

As mentioned in the previous section, the momentum equations show that pressure depends only on x and y. This suggests that it may be useful to integrate these equations over the thickness, z, to get expressions for the velocities v x and v y . Before doing this however, we note that in injection molding the material is in both solid and melt states. Indeed, the incoming melt is sandwiched between a layer of frozen material adhering to the cavity walls. Obviously some way of differentiating these phases is required. A common, though unscientific, term is used in the molding industry to describe this and is called the “frozen layer.” By definition, the velocities v x and v y are zero in this region. From the simulation point of view, there is the problem of defining the frozen layer—most importantly, the transition between melt and solid. Most codes using the 2.5D approximation adopt temperature as the criterion. As discussed earlier, there are various terms used such as “noflow temperature” or “transition temperature.” For now we note that most commercial software uses fixed values for a temperature that defines the material to be in the solid or melt phase. Consequently there will be a frozen layer determined by this temperature. Figure 5.4 defines our notation based on this premise. Cavity thickness will be 2H with the cen-

Figure 5.4 Definition of frozen layer thickness

terline at z = 0 and the location of the interface between solid and liquid at a point (x, y) is at z = h + (x, y) and z = h − (x, y). Hence the thickness through which there is flow at any point (x, y) is h + (x, y) − h − (x, y). Note we make no assumption of symmetry here. That is |h − (x, y)| ̸= |h + (x, y)|. Readers should be aware, however, that symmetry may be assumed in commercial software to reduce calculation time; as only the half cavity is analyzed. To begin, we integrate Equation 5.32 with respect to z from h − (x, y) to z to obtain: Z

z

h−

∂p dz = ∂x

Z

z

h−

µ ¶ ∂ ∂v x η dz , ∂z ∂z

· ¸ · ¸ ∂p ∂p ∂v x ∂v x z− z =η + η , ∂x ∂x z=h − ∂z ∂z z=h −

(5.39)

(5.40)

5.6 The 2.5D Approximation

73

∂p ∂v x z =η + A(x, y) , ∂x ∂z

(5.41)

∂p A(x, y) = z ∂x

(5.42)

where ·

¸

· ¸ ∂v x − η . ∂z z=h − z=h −

We divide both sides of Equation 5.41 by η and again integrate both sides from h − to z to get: Z Z z Z z ∂p z z ∂v x 1 dz = d z + A(x, y) dz ∂x h − η h − ∂z h− η Z z 1 = v x (z) − v x (h − ) + A(x, y) dz . (5.43) h− η At the interface formed by the frozen layer where z = h − or z = h + , we have v x = 0. This is called a no-slip condition. Applying this condition and rearranging Equation 5.43, we obtain: Z Z z ∂p z z 1 v x (z) = d z + A(x, y) dz . (5.44) − ∂x h − η η h We can apply the no-slip condition again to Equation 5.44 by substituting z = h + to obtain: 0=

∂p ∂x

Z

h+

h−

z d z + A(x, y) η

Z

h+

h−

1 dz , η

(5.45)

and so h+

z dz ∂p h − η A(x, y) = − Z h+ ∂x 1 dz − η h ∂p =− C (x, y) , ∂x Z

(5.46)

where Z C (x, y) = Z

h+ h− h+ h−

z dz η 1 dz η

.

(5.47)

Substituting Equations 5.46 and 5.47 into Equation 5.44 we obtain the following expression for velocity in the x direction: ·Z z ¸ Z z ∂p z 1 d z −C (x, y) dz . (5.48) v x (z) = ∂x h − η h− η Carrying out the same steps as above on the y-component of momentum, Equation 5.33, we find a similar expression for the y-component of velocity namely: ·Z z ¸ Z z ∂p z 1 v y (z) = d z −C (x, y) dz . (5.49) ∂y h − η h− η

74

5 Approximations for Injection Molding

The width of the flow channel is defined by h + − h − . An average x velocity across the flow channel width is defined by v¯x =

1 + h − h−

Z

h+

h−

v x (z) d z .

(5.50)

Substituting Equation 5.48 into Equation 5.50 we get: v¯x =

∂p 1 h + − h − ∂x

·Z

h+ Z z h−

z d z d z −C (x, y) h− η

Z

h+ Z z

h−

¸ 1 dz dz . h− η

(5.51)

Equation 5.51 may be simplified using integration by parts (see Section 4.2.4). The first term Z z z may be evaluated by setting M = z and N = d z, then h− η Z

h+ Z z

h−

z dz dz = h− η

h+

dM dz dz · ¸z=h + Z − = NM Z

N

h−

z=h −

h+

M N ′d z

h−

¸z=h + Z h+ 2 z z − dz dz η − h− h− η z=h Z h+ Z h+ 2 ¡ ¢ z z = h+ − h− dz − dz . h− η h− η · Z = z

h+

(5.52)

We can again use integration by parts to evaluate the second integral in Equation 5.51 by setZ z 1 ting M = z and N = dz : h− η Z

h+ Z z

h−

h+

dM dz dz · ¸z=h + Z = NM −

1 dz dz = h− η

Z

N

h−

z=h −

h+

M N ′d z

h−

¸z=h + Z h+ 1 z dz dz − h− η h− η z=h − Z h+ Z ¡ ¢ h+ 1 z dz − dz . = h+ − h− η η h− h−

· Z = z

h+

(5.53)

5.6 The 2.5D Approximation

75

Substituting Equations 5.52 and 5.53 into Equation 5.51, and using Equation 5.47, we obtain the following expression for the average velocity in the x direction: · ½ ¾¸ Z + Z h+ 2 Z + Z h+ ¢ h z ¡ + ¢ h 1 ∂p ¡ + z z 1 − − h − h d z − d z −C (x, y) h − h d z − d z h + − h − ∂x h− η h− η h− η h− η  ´2  ³Z h + z d z Z Z Z + + +  h z2 ¢ h z ¡ ¢ h z 1 ∂p  h− η ¡ +  = + dz − d z − h+ − h− dz + Z +  h − h−  − h − − −  h − h ∂x  η η η h h h 1 dz − η h  ³Z h + z ´2  dz  Z h+ 2 1 ∂p  z h− η   = + − d z +   Z + h 1  h − h − ∂x  h − η dz h− η −2S 2 ∂p = + , (5.54) h − h − ∂x

v¯x =

where we have defined 

³Z h + z ´2  d z Z +  2 1 h− η  h z  S2 =  dz − Z + . h −  2 h η 1 dz − η h

(5.55)

The quantity S 2 is called the fluidity. We use the subscript to denote that it is associated with flow in a gap which is essentially two-dimensional. An average velocity in the y direction may be similarly defined by: v¯ y =

1 h+ − h−

Z

h+

h−

v y (z) d z .

(5.56)

Upon substitution of Equation 5.49 into Equation 5.56 and carrying out the integrations as we did for the average v x velocity, we obtain: v¯ y =

5.6.5

−2S 2 ∂p . h + − h − ∂y

(5.57)

Integration of the Continuity Equation

Recall from Section 5.6.3 that after simplification due to dimensional considerations, and ignoring the velocity in the z direction, the conservation of energy equation has the form: ρc p

³ ∂T ∂t

+ vx

∂T ∂T ´ ∂p ∂2 T + vy = βT + ηγ˙ 2 + k 2 . ∂x ∂y ∂t ∂z

(5.58)

This may be rearranged to give ∂T ∂T ∂T 1 ³ ∂p ∂2 T ´ + vx + vy = βT + ηγ˙ 2 + k 2 . ∂t ∂x ∂y ρc p ∂t ∂z

(5.59)

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5 Approximations for Injection Molding

Recall that the continuity equation from Section 5.6.3 is: 0=κ

³ ∂p

+ vx

∂t

³ ∂T ∂p ∂p ´ ∂T ∂T ´ ³ ∂v x ∂v y ´ + vy −β + vx + vy + + . ∂x ∂y ∂t ∂x ∂y ∂x ∂y

(5.60)

Substituting Equation 5.59 into Equation 5.60 gives ∂p ∂p ∂p ´ β ³ ∂2 T ´ ∂v x ∂v y βT + vy − + ηγ˙ 2 + k 2 + + ∂t ∂x ∂y ρc p ∂t ∂z ∂x ∂y ³ ³ ´ ³ 2 ´ 2 ´ ∂ T ∂v x ∂v y ∂p ∂p β β T ∂p ηγ˙ 2 + k 2 + + κ vx + vy − + . = κ− ρc p ∂t ∂x ∂y ρc p ∂z ∂x ∂y

0=κ

³ ∂p

+ vx

(5.61)

We now integrate this equation with respect to z across the cavity thickness, that is from z = −H to z = +H , to get +H ³

Z 0=

−H

Z −

Z +H ³ β2 T ´ ∂p ∂p ∂p ´ dz + κ vx + vy dz ρc p ∂t ∂x ∂y −H Z +H Z +H ∂v y β ³ 2 ∂2 T ´ ∂v x ηγ˙ + k 2 d z + dz + dz . ρc p ∂z −H ∂x −H ∂y

κ−

+H

−H

Consider now the first term. We may take Z

+H ³

−H

κ−

∂p β2 T ´ ∂p dz = ρc p ∂t ∂t

Z

+H ³ −H

= a(x, y)

(5.62)

∂p outside the integral and so ∂t

κ−

β2 T ´ dz ρc p

∂p , ∂t

(5.63)

where Z a(x, y) =

+H ³ −H

κ−

β2 T ´ dz ρc p

(5.64)

is a constant with respect to z. We now consider the first part of the second term of Equation 5.62. From Equation 5.48 we know ·Z z ¸ Z z ∂p z 1 d z −C (x, y) dz , (5.65) v x (z) = ∂x h − η h− η and so Z

+H

−H

κv x

¶ ·Z z ¸ Z z ∂p 2 z 1 d z −C (x, y) dz dz ∂x −H h− η h− η µ ¶2 · Z +H ³ Z z ¶ ¸ Z +H µ Z z ´ ∂p z 1 = κ d z d z −C (x, y) κ dz dz ∂x −H h− η −H h− η ¶2 µ ∂p = d (x, y) , ∂x

∂p dz = ∂x

Z

+H

κ

µ

(5.66)

5.6 The 2.5D Approximation

77

where ·Z d (x, y) =

+H µ

−H

κ

Z

z

h−

¶ ¶ ¸ Z +H µ Z z z 1 d z d z −C (x, y) κ dz dz . η −H h− η

(5.67)

Similar consideration of the second integral of the second term in Equation 5.62 yields: Z

+H

−H

κv y

¶ ·Z z ¸ Z z ∂p 2 z 1 d z −C (x, y) dz dz ∂y −H h− η h− η Z +H ³ Z z ³ ∂p ´2 · Z +H ³ Z z z ´ ´ ¸ 1 = κ d z d z −C (x, y) κ dz dz ∂y −H h− η −H h− η ¶2 µ ∂p . = d (x, y) ∂y

∂p dz = ∂y

Z

+H

κ

µ

For the third term of Equation 5.62 we set Z +H β ³ 2 ∂2 T ´ b(x, y) = ηγ˙ + k 2 d z . ∂z −H ρc p

(5.68)

(5.69)

Finally, we consider the fourth and fifth terms of Equation 5.62. Using our assumption that the velocity components are zero in the frozen layer, we have: Z

+H

−H

∂v x dz + ∂x

Z

+H

−H

∂v y ∂y

Z dz =

h+

h−

∂v x dz + ∂x

Z

h+ h−

∂v y ∂y Z h+

dz

Z + ∂ ∂ h vx d z + vy dz ∂x h − ∂y h − ´ ∂ ³ ´ ∂ ³ + = (h − h − )v¯x + (h + − h − )v¯ y ∂x ∂y ∂ ³ ∂p ´ ∂ ³ ∂p ´ =− 2S 2 − 2S 2 , ∂x ∂x ∂y ∂y =

(5.70)

where, in the last two expressions, we have used the definition and values of average velocities given in Equations 5.50, 5.54, 5.56, and 5.57. Substitution of Equations 5.63, 5.66, 5.68, 5.69, and 5.70 into Equation 5.62 gives: µ³ ¶ ∂p ∂p ´2 ³ ∂p ´2 + d (x, y) + − b(x, y) 0 = a(x, y) ∂t ∂x ∂y ∂p ´ ∂ ³ ∂p ´ ∂ ³ 2S 2 − 2S 2 . − ∂x ∂x ∂y ∂y

5.6.5.1

(5.71)

Summary of the 2.5D Approximation

We appreciate that some readers are not inclined to go through the detailed derivations above. Consequently we summarize the approximations and the resulting equations of the 2.5D approximation. The steps to get to the 2.5D equations are: ■

Start with the conservation equations derived in Chapter 4, namely, conservation of mass, Equation 5.5, conservation of momentum, Equation 5.6, and conservation of energy, Equation 4.67.

78

5 Approximations for Injection Molding

■

Ignore the source term Q˙ in the energy equation, Equation 5.7.

■

Assume the melt may be modeled by a generalized Newtonian fluid.

■

■

■

■

■

■

Assume constant thermal conductivity of the melt. This is not necessary for the 2.5D approximation but is used in many codes. One can also use two values—a value of thermal conductivity in the melt and another in the solid state. The ultimate level of sophistication is to determine the thermal conductivity from laboratory measurements under certain condition and take into account the effect of processing. Adopt a Cartesian coordinate system in which the z axis is in the thickness direction of the part. Use dimensional analysis to remove low order terms under the assumption that the part is thin walled. Due to the thin wall assumption, neglect velocity in the thickness direction (z direction). Integrate the momentum equations arising from the thin-wall assumption to get velocities in x and y directions that depend on pressure gradient where pressure, p = p(x, y). Integrate the continuity (or conservation of mass) equation to get an equation for pressure, p(x, y), Equation 5.71.

As a result of these steps, the conservation equations, Equations 4.13, 4.23, and 5.7 are reduced to a single equation for pressure: µ³ ¶ ∂p ∂p ´2 ³ ∂p ´2 + d (x, y) + − b(x, y) ∂t ∂x ∂y ∂p ´ ∂ ³ ∂p ´ ∂ ³ 2S 2 − 2S 2 , − ∂x ∂x ∂y ∂y

0 = a(x, y)

(5.72)

and the simplified energy equation: ρc p

³ ∂T

5.7

∂t

+ vx

∂T ∂T ´ ∂p ∂2 T + vy = βT + ηγ˙ 2 + k 2 . ∂x ∂y ∂t ∂z

(5.73)

Mold Cooling Analysis

We have spoken of cooling analysis previously by which we mean simulation of the cooling of the polymer. Heat from the polymer is conducted into the mold and then into the mold cooling system. Since cooling of the polymer accounts for the majority of time in the molding cycle, there is value in simulating the mold cooling phase with a view to reducing the cooling time. While of great importance, in practice, the mold’s cooling system is often an afterthought. The first priority of the toolmaker is to form the shape of the part. Frequently this requires mold inserts and slides. The cooling system must coexist with these mechanisms. Mold cooling systems usually comprise circular drillings in the mold through which coolant is pumped. Most mold cooling simulation programs perform a hydraulic analysis on the cooling lines and account for temperature conduction through the mold to the cooling line. The hydraulic analysis can indicate problem areas where coolant is stagnant and so not efficiently removing heat.

5.7 Mold Cooling Analysis

79

Usually some form of heat transfer coefficient is defined between the mold and cooling channel. This sounds easy but in practice can be difficult. When water is used as a coolant, scale or rust may build up on cooling channels and therefore reduce their efficiency. As well as cooling lines, bubblers, and baffles [31, 333], are used to introduce cooling in mold cores or areas where it is not possible to use standard cooling lines. Most mold cooling analysis software has provision for these. Note also that when we speak of “coolant” it may be water, a mixture of glycol and water, or oil. The latter is often used when the mold temperature needs to be above 100◦ C, which is a requirement for some high performance polymers. Apart from the hydraulic analysis of the coolant, mold cooling analysis software is based on the conduction equation: ∂2 T ∂T =α 2 , ∂t ∂z

(5.74)

where α = k/ρc p is the thermal diffusivity of the material. This is a transient equation, and early versions of mold cooling analysis did not take this into account. These dealt only with the right-hand side of Equation 5.74. Kwon et al. [214] introduced a relatively simple solution to mold cooling. This was extended by Himasekhar et al. [157]. By considering a 1D conduction problem with a finite difference scheme in the mold and melt, and several different numerical methods, the authors concluded that a cycle averaged temperature was sufficiently accurate for mold design purposes. Himasekhar et al. [157] then implemented a 3D solution for which the temperature in the mold was determined using a boundary element method (BEM), similar to that proposed by Burton and Rezayat [49], and a finite difference method for heat transfer in the polymer. This became the most common approach to cooling simulation. That is, use a finite difference or semianalytical solution in the polymer and conduct a full 3D steady (or cycle averaged) heat transfer analysis in the mold using the BEM. The boundary element method was attractive because it required only that the external mold surface be meshed. It was also capable of dealing with materials of different conductivity within the mold such as Beryllium Copper (Be-Cu) inserts. More recently, transient mold cooling analysis software has been developed. While this sounds like an advance, its value depends on the question being asked. If the purpose of the mold cooling analysis is to establish boundary conditions for the cooling of the polymer, it is reasonable to use a steady state (or cycle averaged) method. However, one can ask another question: “How long does it take for my mold to reach a steady state?” This is an important question for those who make large molds that may take considerable time to reach steady state conditions. In this case, a transient analysis is necessary. The work of Karjalainen [192] was a noteworthy but little-known contribution to the field. Here a finite element solution in the plastic and the mold metal was employed, unlike the boundary element approach used by others. Moreover, interface elements were used to model heat transfer between mold blocks and inserts. This is a sophisticated approach, but early mesh generators would not have been able to automatically mesh the mold. It is an example of the need for commercial simulation to be in step with existing technology to minimize operator time and maximize the results of simulation. In conclusion, we mention a further problem with mold cooling analysis. Most software assumes that the plastic is in contact with the mold during cooling. Due to shrinkage, this may not be so. Indeed it is common for the material to shrink onto the core and pull away from the cavity side. This greatly changes the heat conduction on each side of the polymer. This

5 Approximations for Injection Molding

80

is a good reason to adopt a holistic approach to molding simulation in which the entire mold and polymer are incorporated into the mathematical model. Despite the move to “concurrent engineering,” in injection molding, this approach is at odds with industry practice where the part is often designed with little consideration of how it is to be manufactured.

5.8

Fiber Orientation

In Section 4.7.3 we found that accurate modeling of fiber-filled materials in the concentrated range requires a statistical approach. Letting ψ(p, t )d p represent the probability of finding a fiber oriented in the direction range from p to p + d p at time t , ψ(p, t ) satisfies the equation: ´ i ∂ h (r ) ∂ψ ³ ∂ψ = · D (I − pp) · − L · p − (L : pp)p ψ , ∂t ∂p ∂p

(5.75)

where D (r ) is the diffusion coefficient. This equation may be solved directly but is numerically expensive. For example, suppose we have a 2.5D model consisting of 1000 nodes defining triangles on the midplane of the part. Letting the cavity thickness be covered by 10 nodes above and below the midplane, we have 21 nodes across the thickness at each midplane node location. If we use 20 increments in θ to cover 180 degrees and 40 increments in φ to cover 360 degrees then each node has 800 dof. Since there are 21,000 nodes, the model has 21, 000 × 800 dof = 16, 800, 000 dof. A model with 1000 nodes on the midplane is small by injection molding standards. It is not uncommon to have hundreds of thousands of nodes. Consequently we seek a more efficient way to compute orientation.

5.8.1

Orientation Tensors

An approach to overcome computational effort was introduced by Advani and Tucker [2]. They defined the second- order orientation tensor a i j by Z a i j ≡ 〈p i p j 〉 =

p i p j ψ(p, t )d p ,

(5.76)

where 〈·〉 is called the ensemble average with respect to the PDF ψ(p, t ). Since the distribution function is normalized, (see Equation 4.74) and p is a unit vector, it follows that the trace of a i j = 1. Moreover, from Equation 5.76 we have a i j = a j i . Consequently the second-order tensor a i j has only five independent components. Figure 5.5 shows some examples of the second-order tensor for some extreme cases of orientation. Similarly one can define a fourth-order orientation tensor a i j kl by Z a i j kl ≡ 〈p i p j p k p l 〉 =

p i p j p k p l ψ(p, t )d p .

(5.77)

We now turn our attention to calculation of a i j . The first step is to obtain an evolution equation for a i j . This requires some mathematical manipulation which is provided in Appendix C. From

5.8 Fiber Orientation

81

Figure 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

Equation C.46 we have the following evolution equation for a i j D ai j Dt

= Wi k a k j − a i k Wk j +

a r2 − 1 ³ a r2 + 1

D i k a k j + D j k a ki − 2 D kl a i j kl

´

¡ ¢ + 2 D (r ) δi j − 3a i j ,

(5.78)

where D (r ) is called the rotational diffusivity and accounts for the fact that injection molded composites are concentrated solutions and there will be interactions among the fibers. This is in contrast to Jeffery’s equation, Equation 4.72, for a single fiber that we discussed in the previous chapter.

5.8.2

Folgar-Tucker Equation

A common method for calculating the fiber orientation distribution in an injection molding is by means of the so called Folgar-Tucker equation [122]. Folgar and Tucker assumed D r = C I γ˙ in Equation 5.78 to get D ai j Dt

= Wi k a k j − a i k Wk j +

´ a r2 − 1 ³ D i k a k j + D j k a ki − 2 D kl a i j kl 2 ar + 1

¡ ¢ + 2C I γ˙ δi j − 3a i j ,

(5.79)

where C I is a constant called the interaction coefficient. Equation 5.79 raises two immediate problems. The first is that it is not closed. That is, it involves the rate of change of a i j as a function of the higher order tensor a i j kl . The second is how to determine the interaction coefficient C I .

5.8.3

Closure Approximations

The first problem mentioned above is referred to as the closure problem. Basically, we need to approximate the fourth order tensor a i j kl as a function of the second order tensor a i j . This

82

5 Approximations for Injection Molding

is not a trivial problem and has lead to several proposals. Zheng et al. [424] have recently reviewed several available closures in the context of injection molding. We will mention only four here: 1. the linear closure 2. the quadratic closure 3. the hybrid closure 4. the orthotropic closure It should be noted that closure approximations may take different forms in planar and 3D flows.

5.8.3.1

Linear Closure

The linear closure is described by Hand [145] and has the 3D form: ¢ 1 ¡ δi j δkl + δi k δ j l + δi l δ j k 35 ¢ 1¡ + a i j δkl + a i l δ j k + a kl δi j + a j l δi k + a j k δi l . 7

a i j kl ≈ a il ijnkl = −

(5.80)

For planar orientation, the expression is similar but the coefficient 1/35 is changed to 1/24 and the coefficient 1/7 is changed to 1/6. For random orientation, the linear closure is exact but may be unstable when the fibers become more oriented [3].

5.8.3.2

Quadratic Closure

The quadratic closure has been employed by Doi [80]. It has the form: quad

a i j kl ≈ a i j kl = a i j a kl .

(5.81)

This is exact when the fibers are perfectly aligned but can have poor behavior in transient flows or when the interaction coefficient is nonzero [3].

5.8.3.3

Hybrid Closure

Advani and Tucker [3] proposed a mixture of the linear and quadratic closures. It takes the form: h ybr i d

a i j kl ≈ a i j kl

quad

= (1 − f ) a il ijnkl + f a i j kl ,

(5.82)

where f = 1 − 27 det(a i j ) for 3D orientation ,

(5.83)

f = 1 − 4 det(a i j ) for planar orientation .

(5.84)

and

By construction, the hybrid approximation is exact when fibers are random or perfectly aligned. However, it does tend to accelerate orientation in transient shearing flows.

5.8 Fiber Orientation

5.8.3.4

83

Orthotropic Closure

In this closure scheme, introduced by Cintra and Tucker [62], the fourth-order tensor is approximated as a polynomial function of the principal values of the second-order orientation tensor. The coefficients of this function are obtained by fitting to the exact solution of the orientation distribution function for some simple flows (shear, elongation, or simple combinations of both). For further details, see Cintra and Tucker [62].

5.8.3.5

The Interaction Coefficient

The interaction coefficient presents its own problem in that there is still a lack of fundamental theory that could dictate how to determine its value, either theoretically or experimentally. An approach was presented by Bay [29] based on experimental values of a 11 in steady simple shear flows for different concentrations. These experimental results were compared to the numerically calculated a 11 using the value of C I that best agreed with the experimental data. The polymer matrices used in the experiments were nylon, polycarbonate, and polybutylene terephthalate. The C I values were then plotted against φa r to give the following empirical relationship: C I = 0.0184 exp(−0.7184φa r ) .

(5.85)

In Bay’s experiments, the measured a 11 value increases as the fiber volume fraction increases, therefore the empirical equation shows C I decreasing with increasing φa r . This is opposed to the trend observed by Folgar and Tucker [122] for nylon fibers in silicon oil. However, Folgar and Tucker’s data were measured in the semi-concentrated regime, while Bay’s data were measured in the concentrated regime. Bay [29] and Tucker and Advani [370] conjectured that there could be a changeover in the fiber-fiber interaction from “disturbances” to “caging” at a certain concentration (around φa r = 1). It also needs to be mentioned that in Bay’s work, the hybrid closure was used to model the data. With this closure, the values of C I were around 0.01 to match the experimental a 11 values. However, if one uses the orthotropic closure, the data are matched with C I = O(0.001). Therefore, it is important to notice that, although the measured values of a 11 are independent of any closure scheme, the values of C I , and hence the coefficients of the empirical Equation 5.85, depend on the closure approximation used as noted by Zheng et al. [424]. Frequently this difficulty is unknown to users of simulation software. The key point is that any attempt to determine the interaction coefficient must be in keeping with the closure approximation used in the simulation software. Whereas Bay [29] used experimental data to determine a relationship between C I and φa r , Phan-Thien et al. [289] performed a direct simulation of fiber motions in shear. They used a boundary element method to calculate the fiber orientation and, when the suspension was in equilibrium, calculated the value of C I . They proposed that £ ¤ C I = 0.03 1 − exp(−0.224φa r ) .

(5.86)

The results of the direct simulation and the empirical Equation 5.86 are comparable with the data of Folgar and Tucker [122] as shown in Figure 5.6.

84

5 Approximations for Injection Molding

Figure 5.6 A comparison of the simulated C I for a r = 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)

We complete this section by noting that there is no reason to make the interaction coefficient isotropic, that is, scalar in value. Indeed, Phan-Thien et al. [289] assumed the interaction coefficient was a tensor and the results shown in Figure 5.6 used one third of the trace of the interaction coefficient tensor for the value of C I . We discuss other short comings and improvements of the Folgar-Tucker model in Part II.

5.9

Shrinkage and Warpage

We have seen in the previous sections that there are many approximations made to the governing equations to make them more tractable due to our lack of understanding of material properties and, to a lesser extent, compute time. The latter, though not really a hurdle in academic work, has a big bearing on commercial codes. So far we have dealt with prediction of pressure, temperature, and velocities in injection molding. From this information the skilled user can determine the number of gates, their location, location of weldlines, and the size of the injection molding machine required to make the part. Using heuristic ideas and results from simulation, users can try to minimize stresses in moldings and enhance quality [31, 333]. Nevertheless, the big prize in molding simulation is the accurate prediction of shrinkage and warpage. The reader should note that there are several time scales involved. The first is postinjection, where components may be assembled shortly after molding. A longer time scale is introduced when post-molding operations, such as painting, which involves exposure to elevated temperatures, are considered and yet a longer time scale is the performance of the part during its lifetime. So shrinkage and warpage are time dependent. While advances in polymers have reduced the time scale problem, many commodity polymers of great commercial importance do require detailed consideration of their performance over time.

5.9 Shrinkage and Warpage

85

No codes, commercial or academic, exist today with the ability to consider time effects on molded products. This requires some explanation. By consideration of time effects, we mean calculating the performance of the molded part after ejection from the mold. While we can calculate shrinkage and warpage at ejection, a real molding may be subjected to a temperaturetime history after ejection and indeed, during its life. To account for this, we need to calculate properties after ejection and consider the effects of environment including mechanical interactions with other parts. Time dependence remains a major challenge for polymer science. We do not seek to blame simulation codes for their current deficiency. Instead we try to give an idea of the simplifications currently used that can create error. In Part II we give some initial suggestions to improve the science behind shrinkage and warpage prediction. For now though, we consider what is done in practice.

5.9.1

Shrinkage Prediction

One of the earliest attempts to deal with shrinkage calculation tried to correlate measured shrinkages on molded samples with injection molding conditions, such as injection time, holding pressure, and cooling time. Such an approach may be useful for a single mold and material combination but cannot be applied generally to any mold, even with a given material. A further development of this idea sought to lump injection molding conditions into functions that accounted for certain physically observed properties of injection moldings—namely anisotropic shrinkage, which is described in the next section.

5.9.1.1

Residual Strain Methods

The model introduced by Moldflow has the form [390]: S∥ =

iX =5

bi Mi ,

(5.87)

i =1

S⊥ =

iX =10

bi Mi ,

(5.88)

i =6

where S ∥ and S ⊥ are the calculated shrinkage strains parallel and perpendicular to the flow direction, b i are material constants, and the M i are measures of the effects of processing and were calculated using results from filling and packing analyses. M 1 = M 6 and is the volumetric shrinkage, M 2 = M 7 are measures of the level of crystallinity, M 3 ̸= M 8 are measures of molecular orientation, M 4 = M 9 are measures of the effect of relaxation and, M 5 = M 10 are constants and should be small if the model is valid for a given material. The material constants b i (1 = 1, . . . , 10) were obtained by the following procedure: ■

Twenty eight samples are molded with different process conditions or thicknesses. The samples were made in an instrumented mold and on an instrumented molding machine. Measured data were used to determine the processing conditions used for each sample.

86

■

■

■

5 Approximations for Injection Molding

The shrinkage of each sample was measured in directions parallel and perpendicular to the flow direction using a grid pattern etched on to the mold (see Figure 5.7). Four regions of the die are defined by the points 3254, 4589, 2165, and 6785. Measurements of shrinkage are made in each region. For example, in region 3254, the shrinkage is measured in the flow directions between Points 3 and 2 and between 4 and 5. Shrinkage across the flow direction is measured between Points 3 and 4 and between 2 and 5. Simulations were run at the same conditions used for molding to determine values of M i (i = 1, . . . , 10). Measured shrinkages and the calculated values of M i (i = 1, . . . , 10) were substituted in Equations 5.87 and 5.88, and the b i (1 = 1, . . . , 10) , were calculated by regression analysis.

Figure 5.7 Actual sample for shrinkage measurement

The material constants, b i (1 = 1, . . . , 10), are stored in a database. In use, the user runs an analysis to determine the M i (i = 1, . . . , 10), then Equations 5.87 and 5.88 are used to calculate strains parallel and perpendicular to the flow direction for every element in the mesh. The direction of flow angle was also calculated elementally based on flow analysis results. The calculated strains were then input to a structural analysis to determine the deformed shape. It is well known that any difference in the temperature of the mold halves can have a dramatic effect on warpage. Typically such differences are determined by cooling analysis. In the residual strain model, the effect of temperature differences between the mold halves was introduced by modifying the shrinkage strain on the top and bottom of the element so as to produce a bending moment. This technique was successfully applied to unfilled and short-fiber reinforced amorphous and semi-crystalline materials. The need to mold the samples to determine the shrinkage coefficients, b i , (i = 1, . . . , 10) was a disadvantage of the method but necessary to obtain reasonable accuracy. For short-fiber reinforced thermoplastics, one may assume that the level of crystallization, molecular orientation, and mold restraint are second-order effects compared to volumetric shrinkage and the resulting fiber orientation distribution. In 1997, a model for calculation of shrinkage strains specifically for reinforced materials was introduced by Zheng et al. [423]. This model did not

5.9 Shrinkage and Warpage

87

require the molding of samples and gave excellent predictions of deformed shape but tended to underestimate the actual deflection.

5.9.1.2

Residual Stress Models

Three types of residual stresses arise in injection molding: ■

■

■

Flow-induced stresses, arising from the effect of flow on the molecular configuration of the material; Pressure-induced stresses, arising due to a fluid core that exists within the frozen layers during the packing phase, and Thermal-residual stresses, which arise from the thermal contraction of the material as it solidifies.

Baaijens [21] has shown that the flow-induced stresses are an order of magnitude smaller than the pressure induced and thermal residual stresses. We note however that the flow-induced stresses have an important effect on the development of anisotropic material properties and so are important in warpage simulation [246]. Early work on the calculation of residual stresses was influenced by the literature on residual stresses in glass [219] and was concerned with the use of viscoelastic or elastic constitutive models [185, 320]. However, an important difference exists between glass cooling and molding—namely, the effect of packing pressure. The origin of stresses in a freely quenched material, as in glass making, is temperature change. The material cools from the outside, as in molding, and the resulting residual stress distribution is typically compressive at the surface and tensile in the core. In the case of injection molding, however, the residual stress distribution is determined by both the varying pressure history in the packing phase coupled with the frozen layer growth due to cooling, and the stresses can become tensile at the surface layer. Baaijens [21] noted this effect and developed a thermo-viscoelastic model that was isotropic and accounted for both thermal stress and the stress induced by pressure applied in the packing phase for an amorphous material. Residual stress models are generalizations of Hooke’s law which, for an elastic solid, has the form, σi j = c iej kl εkl ,

(5.89)

where σi j and εkl are, respectively, the stress and total strain tensors, and c iej kl is the tensor of elastic constants or stiffness tensor. The strain tensor is determined by differentiating the components of the displacement vector u and is defined to be εi j =

µ ¶ 1 ∂u i ∂u j + . 2 ∂x j ∂x i

(5.90)

Residual stress models are frequently formulated using a viscoelastic constitutive relationship [21, 308]. A general linear anisotropic thermoviscoelastic constitutive relationship may be written σi j =

Zt 0

µ ¶ ¡ ¢ ∂εkl ¡ ¢ ′ ∂T c i j kl ξ (t ) − ξ(t ′ ) − α ξ(t ) − ξ(t ) dt′ , kl ∂t ′ ∂t ′

(5.91)

88

5 Approximations for Injection Molding

where c i j kl is the viscoelastic relaxation modulus, t is time, T is temperature, αkl is the tensor of thermal coefficients of expansion, and ξ (t ) is a pseudo-time scale defined as ξ (t ) =

Z

t

0

1 dt′ , aT

(5.92)

where a T is the time temperature shift factor that accounts for the effect of temperature on material response. We encounter a problem when using Equation 5.91 for non-isothermal systems. First, Equation 5.91 assumes that the material is thermo-rheologically simple, by which we mean, that the change in linear viscoelastic behavior of the material, as a function of temperature, corresponds to a shift in logarithmic time scale [325]. Unfortunately, to obtain material data for relaxation functions satisfying the assumption of thermo-rheological simplicity is not always possible, since a large number of real materials are actually thermo-rheologically complex. Secondly, the relaxation functions used in Equation 5.91 may depend on the internal structures which themselves are in turn affected by processing conditions—particularly for those systems involving semi-crystalline materials and phase change. The exact relation between the internal structures and the relaxation functions is largely unknown (either theoretically or experimentally). Because of the above-mentioned complexity related to the viscoelastic data, it is common to further approximate the problem with a viscous-elastic calculation in which the material is assumed to sustain no stress above a certain temperature and is elastic below that temperature. But what is this temperature to be? We discussed a similar problem in Section 5.3.5. There, the problem was solved by introducing a transition or no-flow temperature. For warpage, it is common to adopt the same approach and assume the transition temperature T t is the temperature above which no stress is sustained in the material. Below this temperature the material is assumed elastic. Under this assumption we have  0 for T ≥ T t ,    Zt µ ¶ ¡ ′ ¢ ∂T ∂εkl σi j = e  c i j kl − αkl t d t ′ , for T < T t .   ∂t ′ ∂t ′

(5.93)

0

A more elaborate discussion of residual stress calculation, including the case of anisotropic materials such as short-fiber reinforced thermoplastics, is given by Zheng et al. [420]. Equation 5.93 is generally solved subject to the following assumptions: 1. With respect to the local coordinates in which the z direction is normal to the local midplane, the shear strains ϵ13 = ϵ23 = 0. 2. The normal stress σ33 is constant across the thickness. 3. As long as σ33 < 0 , the material sticks to the mold walls. 4. Before ejection, the part is fully constrained within the plane of the part such that the only nonzero component of strain is ϵ33 . 5. Mold elasticity is neglected. 6. The material behaves as an elastic solid after the part is ejected. In practice, the residual stresses are calculated by finite element analysis. The discretization involved means that the residual stress is calculated for each element at grid points through

5.9 Shrinkage and Warpage

the thickness. Hence, Equation 5.93 may be written    0 for T ≥ T t ,  ¶ µ Zt ¡ ′ ¢ ∂T ∂εkl σi j (e) (z i ) = e  − α t d t ′ , for T < T t , c kl  i j kl  ∂t ′ ∂t ′

89

(5.94)

0

where the subscript (e) refers to the element number and the stress is calculated at each gridpoint z i ∈ [−H , H ] . In order to obtain the shrinkage of the part, the calculated residual stresses are used as the loading condition in a structural analysis. This requires an additional set of boundary conditions to prevent rigid body motion of the geometry. We achieve this by selecting three nodes on the part that are not colinear. Denoting displacement degrees of freedom in the coordinate directions x, y, and z by u x , u y , and u z respectively, we define the boundary conditions to be: Node 1: u x = u y = u z = 0 ,

(5.95)

Node 2: u x = u y = 0 ,

(5.96)

Node 3: u z = 0 .

(5.97)

All rotational degrees of freedom are unconstrained. With these boundary conditions the part is free to shrink and deform, yet rigid body motion is prevented. Deformations may be calculated, and from these, warpage and shrinkage can be determined. To illustrate the performance of the viscous-elastic model described above, we provide an example. Luye [233] measured pressure and shrinkages in an ISO mold of dimension 60 mm × 60 mm × 3 mm for an iPP produced by Solvay (PHV 252). The processing conditions used for the molding is given in Table 5.1. Table 5.1 Molding Conditions Molding Parameter

Value

Melt temperature

220 ◦ C

Injection time

1s

Mold temperature

43 ◦ C

Coolant flow rate

7 l/min

Nozzle holding pressure

80 MPa

Holding time

15 s

Total cooling time (holding time + cooling time)

40 s

Figure 5.8 shows pressure traces from simulations run using the same conditions as Table 5.1, with fixed transition temperatures ranging from 118–160 ◦ C, while the measured trace is from [233]. These simulations involved no crystallization kinetics and were based on the 2.5D midplane theory previously presented in this chapter. The actual transition temperature for the material was measured to be 118◦ C using the DSC technique discussed earlier. Clearly the value of transition temperature chosen has a significant effect on the pressure decay in the packing phase. An obvious question is what effect does this have on shrinkage? In Figure 5.9 we calculate shrinkage in the flow direction using the transition temperatures of Figure 5.8 and the viscouselastic model described previously.

90

5 Approximations for Injection Molding 80 Measured

Pressure (MPa)

70

Temperature = 118 oC

60

Temperature = 136 oC

50

Temperature = 148 oC Temperature = 160 oC

40 30 20 10

0

10

20

30

40

50

Times (s)

Figure 5.8 Simulated and measured packing pressure versus time results for different transition temperatures

As expected, the effect is significant. Any attempt to improve shrinkage and warpage prediction for semi-crystalline materials will need to overcome the use of a fixed transition temperature. This will be further discussed in Chapters 10 and 11.

Figure 5.9 Calculated shrinkage in the parallel direction for different no-flow or transition temperatures. The measured value from Luye [233] is 0.8%

5.10 The 2.5D Approximation for Runners

91

Kennedy and Zheng [200] discussed the difficulty of obtaining material data for use in the theoretical models. To overcome this problem, they proposed a semi-empirical model that introduces some correction factors to tune the theoretical predicted results and used measured shrinkage data on simple geometrical molded specimens to determine the correction factors for different materials by the multi-variable regression technique. This approach is known as the CRIMS (Corrected Residual In-Mold Stress) method in commercial software.

5.10

The 2.5D Approximation for Runners

Runners are frequently trapezoidal in cross-section to ensure easy ejection from the mold. However most 2.5D mold-filling simulation software uses only runners of circular crosssection. While the user of a simulation program may choose various cross-sections, the software will generally reduce this to an equivalent circular cross-section. This may not be a satisfactory approximation, however, and can introduce error in shear heating calculations and convection of the melt around corners. Nevertheless, we ignore this for now and return to the problem in Section 6.4. Just as for the cavity, the flow in the runner system is determined by the conservation equations of mass, momentum, and energy. However, simplifications are obtained using the material approximations discussed in Section 5.2 and the assumption of symmetry about the runner axis. Figure 5.10 shows the geometry of the circular runner and its associated coordinate system. We

Figure 5.10 Geometry and coordinate system for runners

assume that the temperature and flow fields are symmetrical about the longitudinal axis of the runner. Consequently no quantity will depend on θ. In particular, it means that the interface between melt and frozen layer depends only on r and x. The most significant aspect in considering runner systems is the adoption of a cylindrical coordinate system for the conservation equations of mass, momentum and energy. We provide these here and then look at further approximations later in this chapter and Chapter 6.

92

5 Approximations for Injection Molding

5.10.1

Conservation of Mass for Runners

The conservation of mass equation, Equation 4.13, may be expressed in cylindrical coordinates [35, 260] as ´ ³1 ∂ ∂ρ v θ ∂ρ ∂ρ ∂ρ ∂x + v r +ρ + + vx = 0. ∂t r ∂v x ∂r r ∂θ ∂x

(5.98)

Due to symmetry about the axis of the runner, terms involving derivatives of θ are zero and Equation 5.98 becomes: ³1 ∂ ¡ ¢ ∂v x ´ ∂ρ ∂ρ ∂ρ +ρ r vr + + vr + vx = 0. ∂t r ∂r ∂x ∂r ∂x

(5.99)

The density ρ depends on both the temperature and pressure. Therefore, using the chain rule for differentiation, and the definitions of expansivity and compressibility given by Equations 3.23 and 3.26 we have: ∂ρ ³ ∂ρ ´ ∂p ³ ∂ρ ´ ∂T = + ∂t ∂p T ∂t ∂T p ∂t ∂p ∂T = ρκ − ρβ (5.100) ∂t ∂t ³ ´ ³ ´ ∂ρ ∂ρ ∂p ∂ρ ∂T = + ∂r ∂p T ∂r ∂T p ∂r ∂T ∂p − ρβ (5.101) = ρκ ∂r ∂r ³ ´ ³ ´ ∂ρ ∂ρ ∂p ∂ρ ∂T = + ∂x ∂p T ∂x ∂T p ∂x ∂p ∂T = ρκ − ρβ . (5.102) ∂x ∂x Substituting these expressions into Equation 5.99 and rearranging gives: κ

³ ∂p ∂t

+ vr

³ ∂T ´ ∂v ∂p ∂p ´ ∂T ∂T ´ 1 ∂ ³ x + vx −β + vr + vx + r vr + = 0. ∂r ∂x ∂t ∂r ∂x r ∂r ∂x

(5.103)

It is usual to ignore the pressure variation in the radial direction, as well as the pressure convection term and so, κ

³ ∂T ´ ∂v ∂p ∂T ∂T ´ 1 ∂ ³ x −β + vr + vx + r vr + = 0. ∂t ∂t ∂r ∂x r ∂r ∂x

(5.104)

Ignoring pressure dependence in the radial direction is analogous to ignoring the pressure dependence in the thickness direction of the cavity. It introduces errors at the flow front where convection of melt from the centerline is convected out to the runner wall (fountain flow). However, as in the cavity, these errors reduce after the flow front passes. The assumption of temperature symmetry about the runner axis, while apparently reasonable, introduces errors in convection of temperature fields when the runner feeds a number of similar cavities. These errors depend on processing conditions and runner configuration. If there is substantial shear heating in the runner system, the error becomes more prevalent and can lead to variations in filling times and properties of the molded components. We discuss this in more detail in Section 6.4.

5.10 The 2.5D Approximation for Runners

5.10.2

93

Conservation of Momentum for Runners

In cylindrical coordinates, the momentum equation, Equation 5.12, may be written as three scalar equations [35, 260]: ■

r -component ρ

■

µ ´¶ 2v ∂v r ∂p 2 ∂ ³ ∂v r ´ ∂ ³ ∂v r r = ρg r − + rη + η + ∂v x ∂r − 2 ∂t ∂r r ∂r ∂r ∂x ∂x r ³ ∂v v2 ∂v r ´ r − θ + vx − ρ vr ∂r r ∂x

θ-component µ ¶ ∂v θ 1 ∂p 1 ∂ 3 ∂ ³ vθ ´ ∂ ³ ∂v θ ´ ρ = ρg θ − + 2 r η + + η ∂t r ∂θ r ∂r ∂r r ∂x ∂x ³ ∂v ´ v v ∂v θ θ r θ −ρ v r − + vx ∂r r ∂x

■

(5.105)

(5.106)

x-component ρ

µ ³ ¶ ∂v x ∂p 1 ∂ ∂ ³ ∂v x ´ ∂v r ∂v x ´ = ρg x − + rη + +2 η ∂t ∂x r ∂r ∂x ∂r ∂x ∂x ³ ∂v vx ´ x − ρ vr + vx ∂r ∂x

(5.107)

Ignoring body forces, assuming symmetry about the center axis of the runner centerline and employing the dimensional analysis techniques of Section 5.6.2 to determine the magnitude of terms, the momentum equations may be reduced to ∂p =0 , ∂r ∂p =0 , ∂θ ∂p 1 ∂ ³ ∂v x ´ = rη . ∂x r ∂r ∂r

5.10.3

(5.108)

Conservation of Energy for Runners

Using the gradient and Laplacian operators in cylindrical coordinates [35, 260], the energy equation, Equation 5.13, may be written in cylindrical coordinates as: ρc p

³ ∂T ∂t

+ vr

³ ∂p ∂T v θ ∂T ∂T ´ ∂p v θ ∂T ∂T ´ + + vx = βT + vr + + vx ∂r r ∂θ ∂x ∂t ∂r r ∂θ ∂x ¶ µ 1 ∂ ³ ∂T ´ ∂2 T 2 r + + ηγ˙ + k , r ∂r ∂r ∂x 2

(5.109)

where γ˙ =

∂v x . ∂r

(5.110)

94

5 Approximations for Injection Molding

Assuming the runner is circular, and therefore the temperature and pressure fields are independent of θ, Equation 5.109 becomes: ρc p

µ

¶ µ ¶ ∂T ∂T ∂p ∂p ∂p ∂T + vr + vx = βT + vr + vx + ηγ˙ 2 ∂t ∂r ∂x ∂t ∂r ∂x ¶ µ 1 ∂ ³ ∂T ´ ∂2 T r + 2 . +k r ∂r ∂r ∂z

(5.111)

Further simplification is obtained by ignoring the pressure variation in the radial direction ∂p/∂r and the conduction term in the x direction ∂2 T /∂x 2 . The former is unjustified at the flow front and is analogous to ignoring the velocity v z in the cavity. Moreover, this assumption is unjustified at bends in the runner system. This can be important and is discussed in Section 6.4. Ignoring the temperature conduction along the x direction is reasonable and can be established by dimensional analysis. Finally we ignore the pressure convection term, v x ∂p/∂x. In view of these further assumptions, we arrive at the final form for the 2.5D approximation for the energy equation: ρc p

5.10.4

³ ∂T ∂t

+ vr

∂T ∂T ´ ∂p k ∂ ³ ∂T ´ + vx = βT + ηγ˙ 2 + r . ∂r ∂x ∂t r ∂r ∂r

(5.112)

Integration of the Momentum Equation for Runners

In Section 5.6.4 we integrated the momentum equations for the cavity to obtain expressions for the velocities v x and v y . A similar treatment may be given to the momentum equations for runners given by Equation 5.108. The nontrivial momentum equation, Equation 5.108, ∂p 1 ∂ ³ ∂v x ´ = rη , ∂x r ∂r ∂r

(5.113)

can be multiplied on both sides by r and then integrated with respect to r to obtain ∂p ∂x

Z

∂ ³ ′ ∂v x ´ ′ r η ′ dr . ′ ∂r 0 0 ∂r 1 ∂p 2 ∂v x r =rη . 2 ∂x ∂r r

r ′ dr ′ =

r

Z

(5.114)

Dividing both sides of Equation 5.114 by r η and integrating again with respect to r , 1 ∂p 2 ∂x

Z 0

r

r′ dr ′ = η

Z 0

r

∂v x dr ′ ∂r ′

= v x (r ) − v x (0) .

(5.115)

Let r + denote the value of r at the interface of the frozen layer and the melt as shown in Figure 5.10. By definition, the velocity v x is zero in the frozen layer, and so v x (r + ) = 0. Then from Equation 5.115, v x (0) = −

1 ∂p 2 ∂x

r+

Z 0

r′ dr ′ . η

(5.116)

5.10 The 2.5D Approximation for Runners

95

Consequently, Equation 5.115 becomes: v x (r ) =

1 ∂p 2 ∂x

r

½Z 0

r′ dr ′ − η

r+

Z 0

r′ dr ′ η

¾ (5.117)

An average velocity in the x direction v x , at a point x, may be defined as the total flow rate through the runner, at x, divided by the area A c , of the melt channel at x. That is, v¯x (x) =

1 Ac

Z v x (r ) d A .

Ac

(5.118)

From Figure 5.10, and since we have assumed a circular cross-section, we have A = πr 2 and 2 A c = 2πr + . So Equation 5.118 becomes, v¯x (x) = =

1

Z

πr + 2 Z 2 r +2

r+

0 r+

0

2πv x (r ) d r v x (r ) d r .

(5.119)

Equation 5.117 defines v x (r ) and may be substituted into Equation 5.119: v¯x (x) = = =

1 ∂p r + 2 ∂x

(Z

1 ∂p r + 2 ∂x

(Z

1 ∂p r + 2 ∂x

(Z

r+ ³Z r

0

0 r+

r

³Z

0

0 r+ ³Z r

0

0

r+ ³Z r+

) ´ r′ ′ dr r dr , η 0 0 ) Z r+ Z r+ ′ ´ r′ r ′ ′ dr r dr − dr , r dr η η 0 0 ) Z + ´ r′ 1 +2 r r ′ ′ ′ dr r dr − r dr . η 2 η 0 ´ r′ dr ′ r dr − η

Z

(5.120)

The double integral on the right-hand side of Equation 5.120 may be evaluated using integration by parts as discussed in Section 4.2.4. To this end, set r

Z M=

0

r′ dr ′ , η

(5.121)

and 1 N = r2. 2

(5.122)

Then the double integral on the right-hand side of Equation 5.120 may be written: Z 0

r+ ³Z r 0

´ r′ dr ′ r dr = η

Z

r+

MN′ dr

0

£ ¤r + = MN 0 − 1 2 = r+ 2

Z 0

r+

Z

r

M ′N dr

0

r′ 1 dr ′ − η 2

Z 0

r+

′3

′ r dr . η

96

5 Approximations for Injection Molding

Substituting this result into Equation 5.119 gives, ∂p 2 ∂x + 2r S 1 ∂p =− + , r ∂x 1

v¯x (x) = −

Z

r+

0

3

r′ dr ′ η (5.123)

where the fluidity S 1 , is defined by S1 =

1 2r +

Z 0

r+

3

r′ dr ′ . η

(5.124)

Note that the fluidity subscript of 1 is to distinguish it from the fluidity S 2 for the cavity in Section 5.6.4.

5.10.5

Integration of the Continuity Equation for Runners

The previous section developed an equation that related average velocity to the pressure drop along the runner. In order to combine the momentum and continuity equations, we integrate the continuity equation and use the average velocity to develop a single equation for pressure in the runner system. We rearrange the energy equation, Equation 5.112, to get ½ ¾ ∂T ∂T ∂T 1 ∂p k ∂ ³ ∂T ´ 2 ˙ + vr + vx = βT + ηγ + r . ∂t ∂r ∂x ρc p ∂t r ∂r ∂r

(5.125)

Substituting this into the continuity equation, Equation 5.104, we obtain, ³ ∂T ∂p ∂T ∂T ´ −β + vr + vx ∂t ∂t ∂r ∂x ¢ ∂v x 1 ∂ ¡ + r vr + r ∂r ∂x ½ ¾ ∂p β ∂p k ∂ ³ ∂T ´ =κ − βT + ηγ˙ 2 + r ∂t ρc p ∂t r ∂r ∂r ´ ∂v 1 ∂ ³ x + r vr + . r ∂r ∂x

0=κ

(5.126)

Using dimensional analysis, the quantity β2 T /ρc p is much smaller than κ and so may be ignored. Consequently, Equation 5.126 becomes ½ ¾ ´ ∂v ∂p β k ∂ ³ ∂T ´ 1 ∂ ³ x 0=κ − ηγ˙ 2 + r + r vr + . (5.127) ∂t ρc p r ∂r ∂r r ∂r ∂x Taking A c = πR 2 as the cross-sectional area of the runner, an average value of compressibility ¯ may be defined as: coefficient κ Z 1 ¯c = κ κd A A c Ac Z 2 R = 2 κr d r . (5.128) R 0

5.10 The 2.5D Approximation for Runners

97

Noting that d p/d t is independent of x, and using the average compressibility coefficient, Equation 5.127 may be integrated over the cross-section of the runner to get ¾ · ½ ∂p β k ∂ ³ ∂T ´ κ − r ηγ˙ 2 + ∂t ρc p r ∂r ∂r 0 ´ ∂v ¸ 1 ∂ ³ x + r vr + r dr r ∂r ∂x · ½ ¾ Z R Z R ∂p β k ∂ ³ ∂T ´ = 2π κr d r − ηγ˙ 2 + r r dr ∂t 0 r ∂r ∂r 0 ρc p ¾ ¸ Z R½ ¢ ∂v x 1 ∂ ¡ + r vr + r dr r ∂r ∂x 0 ½ ¾ · Z R ∂p β k ∂ ³ ∂T ´ 1 2 2 ¯cR − ηγ˙ + r r dr = 2π κ 2 ∂t r ∂r ∂r 0 ρc p # Z r+ ½ ´ ∂v ¾ 1 ∂ ³ x + r vr + r dr . r ∂r ∂x 0 Z

R

0 = 2π

(5.129)

Consider now the last term on the right-hand side of Equation 5.129,

Z 2π

r+ ½ 1

0

½h ¾ ir =r + ∂ Z r + ´ ∂v ¾ ∂ ³ x r d r = 2π r v r + v x (r )r d r r vr + r ∂r ∂x ∂x 0 r =0 ∂ ³ v¯x (x) r + ´ = 2π ∂x 2 ∂ ³ + ∂p ´ = −π r S1 , ∂x ∂x

(5.130)

where, in the last two steps, we have used the average velocity, Equation 5.123. Substituting Equation 5.130 into Equation 5.129 gives; ¯ c R2 0=κ

∂p −2 ∂t

Z 0

R

β n 2 k ∂ ³ ∂T ´o ∂ ³ + ∂p ´ ηγ˙ + r r dr − r S1 . ρc p r ∂r ∂r ∂x ∂x

This concludes the 2.5D treatment for runners.

(5.131)

6

Numerical Methods for Solution

Despite simplification of governing equations, the resulting equations are still such that a numerical approach to their solution is necessary. The key numerical methods are finite differences and finite elements. In this chapter, we discuss the application of these methods to the resulting equations. Some historical comments on numerical methods may be found in Appendix A. It should be noted that the numerical methods employed in commercial products were related to the available computer power at the time. So in looking at various numerical schemes, one should have some feeling for the CPU and memory limitations in the past. The finite difference and finite element methods are introduced in Appendices E and F. These appendices are intended to provide a basic understanding of these methods. A particular numerical technique for commercial use must be evaluated by its accuracy, the amount of effort to create the geometric model, the ability to obtain relevant material data, and the computational resources needed to implement it. Academic codes are generally more concerned with just one of these aspects if they are constrained at all. One measure of computational resources is the number of degrees of freedom (DOF) at a node. This refers to the number of variables that must be calculated by the simulation at each point of interest in the computational domain. We will discuss this measure with each of the methods described next.

6.1

Midplane Methods

A detailed derivation of the finite difference and finite element equations for the filling, packing, and cooling phases of the injection molding process is provided in Appendix G. This is usually referred to as the 2.5D approach, because the pressure is solved in two dimensions as defined by the midplane model in terms of x and y, while the temperature is determined in three dimensions. This concept, introduced by Hieber and Shen [154], was by far the preferred method in the period 1980–1997. The 2.5D approximation was adopted widely because it interfaced with available CAD systems, which were surface-based. Three-dimensional modeling was not the norm during this period. Wireframe and surface models were the norm, and the midplane approach suited using such CAD models. The advantage of the 2.5D approach was the use of finite element analysis to determine pressure, the use of finite differences to determine the temperature, and the simplified momentum equations that required just pressure gradient and the fluidity s 2 to be known at each node. At the time of its inception, it was a brilliant solution and remains so today. Pressure varies more gradually over the molding, while temperature has extreme changes over the thickness of the mold. Consequently, use of finite differences for the temperature field is ideal because once the finite element mesh is generated for the midplane, users can decide the level of detail across the thickness by choosing the number of nodes for the temperature solution.

100

6 Numerical Methods for Solution

Consequently, the DOF per node comprises pressure p, fluidity S 2 , and temperature T . An appropriate midplane model generally provided a good analysis result. It could be used for filling, packing, and cooling. After these analyses, residual stresses or strains could be calculated that could, in turn, be taken into a structural analysis that provided shrinkage and warpage results. The key factor here, from a commercial viewpoint, was that the same model could be used for all analyses. To today’s reader, this may seem obvious. But lack of standards and computing resources made the single model concept difficult to attain. The midplane method lagged behind the development of 3D CAD software. During the 1980s, there was a revolution in CAD. Led by Parametric Technology, the means of defining an object for manufacture was represented in a photo-realistic 3D representation. This created a divide between CAD models and their representation for use in injection molding analysis (or any other type of analysis). The CAD industry was centered on the concept of a single model for everything—tooling, assembly, analysis, and production. For injection molding analysis, which lagged during the CAD advancement, the main issue was the need to make a midplane representation from a true 3D model. This resulted in three approaches to overcome this difficulty: ■

Extraction of a midplane from a 3D model

■

Dual Domain analysis

■

Full 3D analysis

Of these, the first two are based on the 2.5D approximation. The last item, 3D analysis, is quite different. We discuss these methods next.

6.1.1

Extraction of a Midplane from a 3D Model

Automatic generation of a midplane from a 3D geometry is not a trivial task. Figure 6.1 shows the idea. The challenge is to automatically deduce the midplane mesh with thickness defined for all elements from the 3D geometry shown at left.

Figure 6.1 Generation of a midplane mesh

While the example given in Figure 6.1 is fairly simple, a more realistic part is shown in Figure 6.2. Much of the academic work in this area was focused on obtaining the medial axis of a 3D geometry, and was driven by needs in the fields of image analysis and hexahedral mesh generation.

6.1 Midplane Methods

101

Figure 6.2 3D representation of a complex injection molding

Many approaches were drawn from the image processing field, and attempted to find the medial axis of the object [82]. A drawback of these schemes was that the time required to compute the midplane was in the order of hours. A more important problem was that the midplane mesh was often not as planar as the original geometry. While this had little effect on the flow analysis, it looked disturbing to users. Worse still was that the structural characteristics of the part were not preserved, and so this approach prevented structural analysis; and hence, warpage analysis, on the part. Kennedy and Yu [198] presented an automatic midplane generator using a different approach. Here the original starting point was the mesh used for stereolithography, which was, at the time, a popular way of producing prototype components. The mesh consisted of planar triangles, often of very poor aspect ratio, that defined the exterior surfaces of the actual 3D component to be made. These were remeshed to improve aspect ratio, and then grouped into surfaces. Algorithms were developed to determine the direction in which a surface should be collapsed. During collapse, the thickness change from original location to the midplane was assigned to the resulting midplane elements. This approach was effective for planar thin parts, but had some difficulty with small features such as bosses. The generated mesh often had discontinuities which needed to be fixed manually. Manual correction is not commercially desirable as it involves user interaction, which is expensive. Commercially useful algorithms use computer rather than human resources.

6.1.2

Dual Domain Analysis for Flow

Instead of determining the midplane from a 3D geometry, another approach was to convert the 3D geometry to an equivalent 2.5D geometry. This gave rise to the Dual Domain method.

102

6 Numerical Methods for Solution

The starting point for this approach was an external mesh on a 3D geometry. This was frequently a stereolithography mesh, which was further refined as in the automatic midplane procedure mentioned in the previous section of this chapter. Figure 6.3 shows the basic idea. Consider a cross-section of a rectangular plate injected at its center as in Figures 6.3(a) and (b).

Figure 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)

If we prescribed a thickness for the surface elements, it would be possible to perform a 2.5D analysis on the surface mesh. However, such an analysis would not be physically consistent, as the material would flow over the top surface, around the edges, and then along the lower surface forming a weldline under the injection point in Figure 6.3(c). The solution is to link the top and bottom mesh at the injection point as shown in Figure 6.3(d). Material then flows simultaneously along the top and bottom surfaces as expected. This gives rise to the name Dual Domain Finite Element Analysis (DD/FEA). In fact, we perform two analyses, one on each side of the surface mesh. As we are in fact filling two domains, it is necessary to double the flow rate at the injection point to obtain the fill times calculated with a midplane mesh for a given geometry. Real parts are always more complex than simple plates. Consider the cross-section of a plate with two ribs, as shown in Figure 6.4. With the injection point linked to top and bottom surfaces, flow emanates and hits each rib, as in Figure 6.4(a). At the ribs, the flow continues only along one side, as in Figure 6.4(b). It is necessary to again form a link to the opposite surface so that flow goes up the rib, and continues past the rib, in a sensible way as shown in Figure 6.4(c). While the idea is simple, it is quite complex to implement. Nevertheless, a product based on this idea was released by Moldflow in 1997. Patents on the concept have been awarded in the United States [411] and Europe [410] and are pending in other jurisdictions. To summarize, Dual Domain flow analysis has three basic steps: ■

■

Generation of a surface mesh Establishing relationships between the elements on the top and bottom surfaces so that a thickness may be defined

6.1 Midplane Methods

103

Figure 6.4 Dual Domain flow analysis for a part with two ribs

■

Adding in connector elements to maintain physically realistic flow patterns

Full implementation details are disclosed in Yu and Thomas [411]. In terms of DOF per node, we have pressure p, fluidity S 2 , and temperature T , just as in the midplane case. However, as we have two meshes on which to solve—one on each side of the part—the total number of DOFs is approximately double that of the midplane case. Consequently, the Dual Domain approach does require more computational time. However, it is merely computer time and relatively inexpensive, compared to the cost of a human operator preparing a model for midplane analysis. This last point clarifies the distinction between academic and commercial goals. It is unlikely that the Dual Domain method would have ever been considered in an academic environment.

6.1.3

Dual Domain Structural Analysis

The Dual Domain concept was taken up very rapidly by industrial users, so much so that there was immediate pressure on software providers to extend the idea to mold cooling and warpage analysis. The mold cooling analysis was easily extended. The boundary element approach for the mold was retained, and coupled with a heat transfer analysis in the plastic material enclosed by the surface mesh. In order to permit warpage analysis, a structural analysis capability was first required. Much of the following description was published in [113].

104

6 Numerical Methods for Solution

The method is best introduced by means of an example. The essential idea is to model the structural performance, that is, the bending and membrane characteristics of a plate, using only a surface mesh defining the outer boundaries of the plate. Consider the plate shown in Figure 6.5.

Figure 6.5 A simple plate may be decomposed into two parts, each of half the original thickness, and perfectly bonded together

From the geometric point of view, the flat plate of thickness h can be seen as the perfect bonding of two plates each of thickness h/2. If we consider such an assembly, we can see that it could be modeled using two shells, each with their reference surface at the geometric center of the two plates. However, this is problematic, as the nodes defining the mid-surfaces are displaced from the outer surface. We want to use the mesh on the outer surface without modification. This is accomplished by using eccentric shell elements. As shown in Figure 6.6, a structural shell element may be defined such that its reference plane is located anywhere. The distance from which the reference plane is displaced from the midsurface is called the eccentricity. Returning to the problem, we can see in Figure 6.5 that the

Figure 6.6 Eccentric shell element for structural analysis

top plate can be modeled using eccentric shell elements with their top surfaces as reference surfaces. Similarly, the bottom plate can be modeled using eccentric shell elements with their bottom surfaces as reference surfaces. This solves the problem by allowing us to use existing nodes on the outer surfaces of the plate. In order to get the correct structural response, however, the two plates of thickness h/2 must be “bonded together” in some way. The bonding of the top and bottom plates involves imposing the Love-Kirchhoff assumption of classical

6.1 Midplane Methods

105

plate or shell theory [83] and requires that a normal to the plate or shell remains straight after deformation and be unchanged in length. This is accomplished by the use of multi-point constraints. In summary, for structural analysis, the Dual Domain method involves the following steps: ■

■

■

Meshing of the outer surface of the structure and establishing relationships between elements on the top and bottom surfaces to define local thickness Use of shell elements with their reference surfaces at the surfaces defining the outer boundary of the three-dimensional object The use of multi-point constraints to ensure that normals to the top and bottom surfaces remain straight after deformation

The constraints used depend on the type of element chosen. In the past, 3-node triangular elements involved constant strain, across the element and their performance was poor. Due to interest in optimization of structures, however, much improved element formulations were developed. One example is a plane triangular facet shell element with 18 degrees of freedom (six at each node—three displacements and three rotations). The element is constructed by superimposing the local membrane formulation due to Bergan and Felippa [34] with the bending formulation due to Batoz and Lardeur [28] and transforming the combined equations to the global coordinate system. The drilling rotation degree of freedom about a local reference surface normal is used in the membrane formulation, and is defined in the local element system by θz =

µ ¶ 1 ∂u y ∂u x − . 2 ∂x ∂y

(6.1)

To define the relationship between the degrees of freedoms of node n and those of its matching node p (see Figure 6.7), we require that the normals to the mid-surface before deformation remain straight after deformation.

Figure 6.7 Structural elements matched for Dual Domain analysis

Adopting the local coordinate system of the element, we denote the three displacement DOF and the three rotational DOF at node n by u xn , u y n , u zn and θxn , θ y n , θzn respectively. There are

106

6 Numerical Methods for Solution

then the following relationships between the degrees of freedom of node n and displacements and rotations of its matching point p: u xn = u x p − θ y p h ,

(6.2)

u y n = u y p + θx p h ,

(6.3)

u zn = u z p ,

(6.4)

θx n = θx p ,

(6.5)

θyn = θy p ,

(6.6)

and θz n = θz p +

· ¸ h ∂θx p ∂θ y p + , 2 ∂x ∂y

(6.7)

where h is the distance between node n and its matching point p. Note that the relationship in Equation 6.7 is obtained using Equation 6.1, and so is particular to the element type chosen. This system of constraints is imposed at all nodes on the bottom (or top) surface of the model with the exception of those at the edges. Elements forming the edge of the plate are assigned one-sixth of the thickness of the adjacent elements on the top and bottom surfaces. With these constraints, the structural performance of the composite structure is identical to the original plate. The composite model may now have appropriate boundary conditions and loading applied for structural analysis, and so be used for warpage analysis of the 3D geometry. In general, the mesh on the top surface is not coincident with the bottom mesh. Hence, a normal from a node n on the bottom surface will not usually coincide with a node on the top surface. Instead, it is more likely that the normal will intersect the top element at a point p (see Figure 6.8). In this case, we interpolate the required constraints using the three nodes defining the top element.

Figure 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

6.2 3D Analysis

6.1.4

107

Warpage Analysis Using the Dual Domain FEM

In order to use the Dual Domain approach for warpage analysis, we load the finite element model with shrinkage strains or membrane stresses derived from the analysis of the filling, packing, and cooling stages of the molding process, as discussed in Section 4.8. There is a slight computational overhead incurred in using the Dual Domain approach when compared to a midplane model. Considering the time required to create a midplane model, this increased solution time is inconsequential. Hence, the method is extremely efficient when judged by the total time taken to obtain a result. Patents for the Dual Domain structural analysis have been granted in the USA [114] and in several other jurisdictions. Details of the implementation of the method are provided in the patent.

6.2

3D Analysis

All of the above has been concerned with the 2.5D approximation. In particular, we have discussed the requirement to obtain a midplane model or, use the Dual Domain method. An alternative method is to perform a 3D analysis. This avoids the 2.5D approximation assumptions and, in principle, should be the ultimate method of simulation. In particular, 3D analysis conforms to the 3D CAD trend of modeling objects in 3D. Unlike the Dual Domain method, which uses both finite differences and finite elements, most 3D codes are based on a single numerical method. Finite element codes are the most common; we provide a description of the element formulation in Appendix H. Some finite differences and finite volume codes have their origin in metal-casting simulation. Hetû et al. [152] gave the earliest example of 3D injection molding analysis. This was followed by Pichelin and Coupez [298] and Talwar et al. [353]. Such analysis makes fewer assumptions; in particular, there is no restriction on the thickness of the part, but requires the computational domain to be meshed with tetrahedral or hexahedral elements. Tetrahedral meshing is generally preferred since it may be performed automatically. However, due to the low thermal conductivity of plastics, injection molded parts tend to be thin-walled. In the thickness direction, temperature gradients of the order of hundreds of degrees per millimeter necessitate the use of many elements across the part thickness. Consequently, in thin-walled parts, the number of elements for 3D analysis increases dramatically and leads to excessive compute times and resources. This has been addressed by the use of anisotropic meshes [334]. Not surprisingly, the computational requirements for 3D analysis are higher than any other method. At each node, we have pressure p, temperature T , and three components of velocity: v x , v y , and v z .

6.2.1

Finite Volume Methods

To date, our discussion has centered on finite difference and finite element methods. Chang and Yang [55] described a finite volume method for 3D simulation of injection molding. An

6 Numerical Methods for Solution

108

advantage of this approach is the relative ease of using different elemental shapes in the simulation. For example, tetrahedral elements may be used in the interior of the part, and rectangular prisms may be used near the walls to capture temperature changes there. So while the finite volume method provides some additional flexibility, the mesh generation becomes more complex, as discussed in Chang et al. [54].

6.2.2

A Pseudo-3D Approach

Nakano [265, 266] developed a scheme that is not truly 3D, but that permits direct analysis of the 3D geometry. Beginning with the continuity equation in Cartesian coordinates: ∂v x ∂v y ∂v z + + = 0, ∂x ∂y ∂z

(6.8)

Nakano makes a generalization of the thin wall approximation, which is valid in two dimensions, and sets v x = −S 3

∂p ∂p ∂p , v y = −S 3 , v z = −S 3 , ∂x ∂y ∂z

(6.9)

where S 3 is a 3D analogue of the 2D fluidity described in Equation 5.55. Substituting Equation 6.9 into 6.8 gives the following equation for pressure: S3

³ ∂2 p ∂x

+

∂2 p ∂2 p ´ + = 0. ∂y ∂z

(6.10)

Note that the right-hand side of the above equation is zero. However, if the material is assumed compressible, as indeed it is, the right-hand side is nonzero. Therefore, the S 3 should not be canceled out. The final part of Nakano’s approach is to determine the value of S 3 by solving the following equation, ∂2 S 3 ∂2 S 3 ∂2 S 3 1 + + =− , ∂x 2 ∂y 2 ∂z 2 η

(6.11)

where η is the viscosity of the material. The economy gained by the Nakano method is reflected in the degrees of freedom (DOF) to be determined at each node. Ignoring constraints, at each node Nakano requires a pressure, temperature, and fluidity, S 3 . The three velocity components are then determined from Equation 6.9. While temperature calculation is not discussed by Nakano [265, 266] directly, finite element methods could be employed.

6.3

Warpage and Shrinkage Analysis in 3D

Since the domain in 3D is meshed with 3D elements such as tetrahedra or hexagonal elements, to calculate shrinkage strain and warpage we need to be able to input stresses or strains into the structural analysis package. This is not a trivial task.

6.4 3D Analysis of Runner Systems

109

A crude approach is to take approximately one-third of the calculated volume shrinkage and apply it as a strain to each element and use this as the loading. For truly 3D parts, where there is little anisotropy in molded material properties, this may not be too bad. A significant problem is that, for a truly 3D part, it can be difficult to determine what parts of the simulation FEM model are rigidly constrained by the mold. While some regions of a part may be constrained, others may be free to pull away from the mold wall as the material cools and shrinks. Thus, the boundary conditions to be used are not trivial. If there is some concept of thickness in the model for analysis, such as when anisotropic meshing is used, some means of distributing the linear strains arising from constraint and flowinduced structure may be introduced. Of course, these must be consistent with the calculated volumetric shrinkage. For semi-crystalline materials, the problem is more difficult. True 3D parts rarely have uniform thickness, so there will be variable cooling rates and deformation of the material during processing. This will give rise to varying material properties throughout the molding, and hence, varying shrinkage. The end result will be a complex distribution of anisotropic shrinkage resulting in part deformation.

6.4

3D Analysis of Runner Systems

In this section, we will discuss the convection of heat around bends in runner systems, and show how it can lead to filling imbalances in multi-cavity molds. We will also show that 3D analysis is necessary to capture these effects. Recall from Section 5.10 that the 2.5D approximation for runners makes some major assumptions. The first is that the runner is assumed circular. Even if the cross-section of the runner is of non-circular shape, it is approximated by a circle according to some rule, often the hydraulic radius. This can introduce significant error in flow and temperature calculations in runners. Beaumont [30] has found that, in practice, convection of temperature in runners may create filling imbalances, and potentially different properties, of parts made in multi-cavity molds. Figure 6.9 illustrates how this can occur due to inaccurate calculation of thermal convection around corners in runner systems. As shown in Figure 6.9, assuming there is shear heating of the material in the runner system, convection around corners should result in melt of different temperatures entering the four cavities. As polymers will be less viscous at higher temperature, the higher temperature melt will preferentially flow into the outer cavities, resulting in a flow imbalance, and hence, a potential imbalance in weight of the molded parts and their properties. Since the 2.5D approximation for runners assumes temperature is the same radially, and makes no correction for corners, it is incapable of predicting this effect.

110

6 Numerical Methods for Solution

Figure 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 system

II

Improving Molding Simulation

7 7.1

Improved Fiber Orientation Modeling

Introduction to Limitations of the Folgar-Tucker Model

Many simulations for predicting orientation in short-fiber reinforced thermoplastics adopt the Folgar-Tucker model [122], which was described in Chapter 5. The Folgar-Tucker model adds into the Jeffery equation an isotropic rotary diffusion term to account for fiber-fiber interactions. The diffusivity is assumed to be proportional to the generalized shear rate, which is believed to be a valid assumption. However, the model delegates the strength of fiber-fiber interactions to just a single scalar constant, or the coefficient of interaction C I , which is used as a fitting parameter. Since there is no fundamental basis to determine the value of C I , Tucker and his co-workers find the value of C I by adjusting it to match the experimentally measured fiber orientation tensor component a 11 . There is, however, no guarantee for other components to match the experimental data at the same time. Since the fiber orientation is influenced by several factors, the diffusivity may be, on occasion, used inappropriately to correct errors due to other factors. Furthermore, as mentioned previously, closure approximations have to be used to relate the fourth-order tensor to the second-order one. The C I value has also been found to vary significantly with different closure approximation schemes (Cintra and Tucker [62]). Fan et al. [103] and Phan-Thien et al. [290] developed a direct simulation technique that does not require either the rotary diffusion assumption or the closure approximation. In fact, this fiber-level simulation technique provides insight into problems of the Folgar-Tucker model, and a way to find a constitutive fix for the model. The above authors realized that, with a single scalar coefficient of interaction, the Folgar-Tucker model is unable to predict the direct simulation results. Hence, one will expect the interaction coefficient to be anisotropic and in tensor form. They generalized the conventional Folgar-Tucker’s isotropic rotary diffusion model to an anisotropic rotary diffusion (ARD) model (Phan-Thien et al. [289]). It has also been found that the conventional Folgar-Tucker model predicts a much narrower core region compared to the experimental data in injection-molded end-gated strips, indicating that the kinetics of orientation change in the model is too quick, compared with the reality. The Jeffery terms in the Folgar-Tucker model are considered to be responsible for this limitation. Some models to slow the orientation change by modifying the Jeffery terms have been proposed by Tucker and his co-workers [296,372,393], Sepehr et al. [327], and Férec et al. [116]. There are also some other cases revealing the need for further enhancements to the FolgarTucker model. For example, experiments with dilute suspensions (∼5% by weight) showed remarkably different orientation behavior from concentrated suspensions, while the numerical predictions using the Folgar-Tucker model do not reflect these changes (As-Sultany [12]).

7 Improved Fiber Orientation Modeling

114

The theory produces better predictions for more concentrated systems (∼30% by weight) than for less concentrated systems ( 1. As noted in Chapter 5, Bay [29] and Tucker and Advani [370] have explained the reduction of C I by a “caging effect.” If this is true, the direct simulation could miss the effect, because the fiber diameter is not taken into account in the simulation. In addition, the simulated shear flow is in an unbounded space, while in Bay’s experiments a wall effect certainly exits.

7.3

Reduced-Strain Closure (RSC) Model

Experimental evidence has shown that the kinetics of orientation change in concentrated suspensions is much slower than predicted by the standard Folgar-Tucker model. Huynh [170] attributed this to the fact that the fibers move in clusters and experience a local strain which is smaller than the bulk strain. He introduced a “reduction factor” to the velocity gradients in the Folgar-Tucker equation, effectively reducing the rate of change of the orientation tensor. In the light of the non-affine motion, Sepehr et al. [327] modified the standard Folgar-Tucker model by introducing a slip deformation. This is essentially equivalent to the modification made by Huynh. In both cases, the modified Folgar-Tucker model no longer obeys the principle of material objectivity—the principle asserts that the response of a material to a given experience or history of motion must be independent of any change of reference frame [356]. Tucker and his co-workers (Tucker et al. [372] and Wang et al. [393]) proposed a new modified model that is objective. The approach is based on the spectral decomposition theorem,

7 Improved Fiber Orientation Modeling

118

which allows decomposition of the Folgar-Tucker equation into two isolated sets of deferential equations for the eigenvalues and eigenvectors of a i j , respectively. The expressions for the eigenvalue kinetics are then modified by reducing the velocity gradients by a factor of κ ≤ 1, while the expressions for the eigenvector kinetics remain unchanged. After the modification, the evolution equation for the material derivative D a i j /D t is then reconstructed. This new model is written as D ai j Dt

˙ i j − 3a i j ) = 0, = L i k a k j + L j k a ki − 2L kl Ai j kl + 2κC I γ(δ

(7.9)

where Ai j kl = a i j kl + (1 − κ)(Li j kl − Mi j mn a mnkl ) ,

(7.10)

with 3 X

L=

λi ei ei ei ei and M =

i =1

3 X

ei ei ei ei ,

(7.11)

i =1

where κ is a phenomenological parameter that is introduced to reduce the rate of fiber alignment. If κ = 1, the standard Folgar-Tucker equation is recovered. The vectors ei (i = 1, 2, 3) are the eigenvectors of the second-order orientation tensor, and λi (i = 1, 2, 3) are the corresponding eigenvalues. Note that Equation 7.11 does not use the summation convention; instead, all sums are indicated explicitly. Comparing Equation 7.9 to the standard Folgar-Tucker model, one can see that the fourthorder tensor a i j kl is replaced by Ai j kl , and C I is multiplied by κ. Since a i j kl is the term to be approximated by a closure approximation, this modification is equivalent to using a new closure approximation. This is why the model is called a “reduced strain closure” model. A similar model has also been independently proposed by Férec et al. [116]. The reduced-strain closure (RSC) model has been incorporated into the anisotropic rotary diffusion (ARD) model to form a unified evolution equation called the ARD-RSC model by Phelps and Tucker [296].

7.4

Suspension Rheology and Fiber-Flow Interaction

Suspension rheology is not the subject of Part I of this book, because fiber-flow interactions have been ignored and a decoupled approach is used in the simulation of injection molding flows. The decoupled approach calculates the flow field independently from the fibers, and then uses the obtained flow field for fiber orientation calculation. According to Tucker [369], whether or not such a simplification is allowed depends on the dimensionless group N p δ2 , where N p is the particle number varying with volume fraction and aspect ratio and can be estimated by Np ≈

φa r2 2(ln 2a r − 1.5)

,

(7.12)

7.4 Suspension Rheology

119

Table 7.1 Asymptotic Values of A i , i = 1 to 4 Cases

A1

A2

A3

A4

aR → ∞

a r2 2(ln 2a r −1.5)

6 ln 2a r −11 a r2

2

3a r2 ln 2a r −0.5

395 2 147 ε

15 395 2 14 ε − 588 ε

5 2 1 3 2 (1 − 7 ε + 3 ε )

9ε

10 3πa r

8 128 − 3πa + 1 − 9π 2 r

8 3πa r

12 − πa r

(rod-like) a r = 1 + ε, ε ≪ 1

(sphere-like) ar → 0

208 + 9π 2 −2

(disk-like)

while the parameter δ describes the out-of-plane fiber orientation, equal to the maximum of narrowness of the flow channel (characterized by the ratio of the typical gap hight to the typical in-plane dimension) and C I1/3 , where C I is the interaction coefficient. A decoupled approach is allowed when N p δ2 ≪ 1. Outside this regime, fiber-flow coupling needs to be considered. Lipscomb et al. [227] have also shown that, in complex flows, flow kinematics can be strongly affected by the fiber orientation so that decoupled calculations may be grossly in error. In order to take the fiber-flow interaction into account, it is necessary to consider the suspension rheology. In general, for a suspension of fibers in a Newtonian fluid, the constitutive equations have two parts contributing to the extra stress of the suspension: (p)

+ τi j , τi j = τ(s) ij

(7.13) (p)

is the viscous contribution of the suspending fluid, and τi j is the particlewhere τ(s) ij contributed stress. The difference among various constitutive models is the expression of the particle-contributed stresses. Although work on the rheology of fiber suspensions dates from the 1960s, the broader subject of the rheology of dispersions may be traced to the work of Einstein [94] in 1905–1906, who considered the case of dilute suspensions (φ < 0.03) of rigid spheres in Newtonian fluids. Einstein’s result for the effective extra stress of the suspension is given as ¡ ¢ τi j = 2η s 1 + 2.5φ D i j ,

(7.14)

where η s is the solvent viscosity. The fluid described by Einstein’s equation is isotropic. An anisotropic dilute suspension model, called the transversely isotropic fluid (TIF) model, was derived by Ericksen [97] and Hand [145]. The equation is written as ¡ ¢ (p) τi j = 2η s φ[A 1 D kl a i j kl + A 2 D i k a k j + a i k D k j + A 3 D i j + d r A 4 a i j ],

(7.15)

where d r is the rotational diffusivity due to the Brownian motion, and A 1 to A 4 are material constants depending on the aspect ratio a r . The asymptotic values of these constants are listed in Table 7.1. One can see that when a r ≫1, the first term in the right-hand side of Equation 7.15 is dominant, since A 1 = O(a r ), A 2 = O(a r−1 ), and the Brownian motion term is negligible at a ˙ 3 /k B T ), where η s is the solvent viscosity, γ˙ is the strain rate, l large Peclet number (Pe = O(η s γl is the length of the fibers, k B is the Boltzmann’s constant, and T is the absolute temperature).

120

7 Improved Fiber Orientation Modeling

The TIF model is only appropriate for dilute suspensions. Dinh and Armstrong [79] derived an equation for non-dilute fiber suspensions as follows: (p)

τi j = φη s

πl 3 N p3 6 ln(2H f /d )

L kl a i j kl ,

(7.16)

where L kl is the velocity gradient tensor, N p is the number of particles per unit volume, l and d are the fiber length and diameter, respectively, and H f is the average distance from a fiber to its nearest neighbor, given by ( Hf =

(N p l 2 )−1 (N p l )

−1/2

for random orientation , for fully aligned orientation .

(7.17)

Another modification to the TIF model for concentrated suspensions has been proposed by Phan-Thien and Graham [292]. The idea of their modification is to keep only the dominant term in the TIF model and to use a functional dependence on the volume fraction to replace the linear dependence of the volume fraction in the TIF model. In the Phan-Thien-Graham model, the particles-contributed stress is given by (p)

τi j = 2η s f (φ, a r )D kl a i j kl ,

(7.18)

where f is a function of the volume fraction and the aspect ratio given by ± φa r2 (2 − φ φm ) f (φ, a r ) = , ± 2 4(ln 2a r − 1.5)(1 − φ φm )

(7.19)

in which φm denotes the maximum volume packing. The parameter can be evaluated by a linear regression using the data of Kitano et al. [203], which leads to φm = 0.53 − 0.013a r ,

5 < a r < 30.

(7.20)

The Phan-Thien-Graham model has been further modified by Fan et al. [104] to include an extra term representing the contribution due to momentum transport caused by random motion of fibers (entropic contribution).

7.5

Brownian Dynamics Simulation

When solving the evolution equation in terms of the second-order orientation tensor as the primary variables, closure approximations are required, which introduce errors into the fiber orientation calculation. A different computational technology known as the Brownian dynamics simulation was adopted by Fan et al. [104]. In this method, the motion of a fiber is described by a stochastic process. The stochastic equation for the orientation vector of the n−th fiber, p i(n) , is written as p˙i(n) = L i j p (n) − L j k p i(n) p (n) p k(n) + (δi j − p i(n) p (n) )F j(n) (t ), j j j

(7.21)

7.5 Brownian Dynamics Simulation

121

where n = 1, ..., N with N being the total number of fibers, L i j is the effective velocity gradient tensor as defined previously, and F(b) (t ) is a random force, with properties D E F(b) (t ) = 0, (7.22) and D

E F(b) (t + s)F(b) (t ) = 2D (r ) δ(s)I.

(7.23)

In the above equations, D (r ) is the diffusivity, δ(s) denotes the Dirac delta function and I the unit tensor. The average result of the stochastic equations will be equivalent to the FolgarTucker equation if one chooses the following expression for D (r ) , ˙ D (r ) = C I γ,

(7.24)

where γ˙ is the generalized strain rate and C I is the interaction coefficient, which may be a function of φ and a r . The interaction coefficient can also be treated as a second-order tensor as in Phan-Thien et al. [291]. The random force can be expressed in terms of the white noise (Öttinger [280]): F i(b) (t ) =

p

2C I γ˙

d wi , dt

(7.25)

where w i is the Wiener process. After solving Equation 7.21 for the p i(n) , one can calculate the structure tensors using the ensemble average: ai j =

N N 1 X 1 X , a = p k(n) p l(n) , p i(n) p (n) p (n) p (n) i j kl j j N n=1 N n=1 i

(7.26)

where N is the total number of fibers in the ensemble. A technique, which combines the Brownian dynamics simulation method with the finite element method to solve the conservation equations, is termed the CONNFFESSIT (Calculation of Non-Newtonian Flow: Finite Elements and Stochastic Simulation Techniques) [280]. The technique makes it possible to simulate macroscopic flow fields directly from microscopic models, rather than from closedform constitutive equations. This is a kind of multi-scale simulation. For many flow problems of interest, the individual particle tracking involves a large number of degrees of freedom, and hence is expensive in computing time. Hulsen et al. [169] extended the CONNFFESSIT method to the so-called Brownian configuration field method for variance reduction. Fan et al. [104] employ this idea for simulation of non-dilute fiber suspension flows. They treat the random vector p(n) for fiber motion as a random vector field function depending on space x and time t , and then use the Euler time derivative to replace the Lagrange time derivative of p(n) . To solve Equation 7.21, a new configuration field q(n) (x, t ) (n = 1, ...N ) is introduced, which is parallel to p(n) (x, t ): q(n) (x, t ) = Q (n) p(n) (x, t ),

(7.27)

where Q (n) is the modulus of q(n) (x, t ). The equivalent stochastic field equations to Equation 7.21 is obtained as ∂ (n) q + u · ∇q(n) = L · q(n) +Q (n) F(b) (t ). ∂t

(7.28)

7 Improved Fiber Orientation Modeling

122

Expressing the random force in terms of the white noise and integrating Equation 7.28 from t to t + ∆t , we obtain its Euler scheme with first-order weak convergence (Öttinger [280]): q(n) (x,t + ∆t ) + u · ∇q(n) (x, t + ∆t ) = q(n) (x, t ) p ˙ (n) (x, t )∆w(t ), + L · q(n) (x, t )∆t + 2C I γQ

(7.29)

where ∆w(t ) is the increment of the Wiener process, which is uniform over space. The structure tensors can be determined approximately by averaging p over N configuration fields: ­

N µ q(n) ¶ µ q(n) ¶ ® 1 X p...p = ... (n) . N n=1 Q (n) Q

(7.30)

Still, either the Brownian dynamics simulation or the Brownian configuration field method is more expensive than calculating a i j directly from the evolution equation directly; but it does not require a closure approximation, so it is potentially more accurate. We can expect that the growth of parallel computing will increase the efficiency and practical usefulness of the technique. Application of the Brownian dynamics simulation to a simple injection molding problem has been attempted by Zheng et al. [419].

8 8.1

Improved Mechanical Property Modeling

Introduction

This chapter will be limited to the prediction of the elastic moduli and the coefficients of thermal expansion for fiber-reinforced composites. These properties are derived from calculated fiber orientation distributions during the molding process and used in warpage or structural analysis, as discussed in Section 5.3.7. These generally use short-term properties. In reality, these properties are time-dependent. Some progress in modeling efforts in properties like long-term failure of unfilled polymers can be found in Klompen et al. [205, 206]. In fiber-reinforced polymers, the geometry, concentration, thermo-mechanical properties, and orientation distribution of the fibers, as well as the thermo-mechanical properties of the polymer matrix, significantly affect the mechanical and thermal performances of the composite material. The idea of material modeling is to predict the macroscopic behavior of the injection-molded composite materials from the mechanics and physics on the microscopic level. An efficient way of predicting the influence of the micro-structure on the overall properties is the mean-field homogenization approach. Tucker and Liang [371] reviewed several mean-field homogenization based micro-mechanics models. They recommend the MoriTanaka model [258] as the best choice for estimating the stiffness of aligned short-fiber composites in injection molding. There are two main steps in the prediction procedures. The first step is to use a micromechanics model to predict unidirectional properties in fully aligned material. In the case of axi-symmetrical fibers, the properties of the unidirectional composite are transversely isotropic. The second step is to combine orientation information with the unidirectional properties of the fully aligned material to predict the properties of more complex systems, taking into account the effect of the orientation distribution. In what follows, we will use the superscript f to denote a quantity associated with the fibers, and the superscript m to denote a matrix quantity.

8 Improved Mechanical Property Modeling

124

8.2

8.2.1

Modeling Effective Properties of Unidirectional Composites

Effective Stiffness

Considering a two-phase composite comprising matrix of volume fraction φm and fibers of ¯ in the composite is given by volume fraction φ f (these satisfy φm +φ f = 1), the average stress σ ¯ = φf σ ¯ f + φm σ ¯ m, σ

(8.1)

¯ f and σ ¯ m are the volume-averaged stresses in fibers and matrix, respectively, given by where σ ¯f = σ

1

Z

Vf

Vf

¯m = σ(x)dV, σ

1 Vm

Z Vm

σ(x)dV,

(8.2)

where V f and V m are the volumes occupied by the fibers and the matrix, respectively. The average strain in the composite, ε¯ , is given by ε¯ = φ f ε¯ f + φm ε¯ m .

(8.3)

where ε¯ f and ε¯ m are the volume-averaged strains in fibers and matrix, respectively. Similarly, the stress-strain relations in the fiber and matrix phases are ¯ f = C f ε¯ f σ

(8.4)

¯ m = Cm ε¯ m , σ

(8.5)

and

respectively. The effective stiffness tensor of the composite is C (C i j kl ), which relates the average strain to the average stress as follows: ¯ = C¯ε. σ

(8.6)

Hill [156] defined a fourth-order tensor A called the strain concentration tensor, which is essentially the ratio between the average fiber strain and the overall average strain of the composite, thus, ε¯ f = A¯ε

f

(¯εi j = A i j kl ε¯ kl ).

(8.7)

Combining Equations 8.1–8.7, one obtains ¯ = C¯ε = φ f σ ¯ f + φm σ ¯m σ = φ f C f ε¯ f + φm Cm ε¯ m = φ f C f ε¯ f + Cm (¯ε − φ f ε¯ f ) = Cm ε¯ + φ f (C f − Cm )¯ε f = Cm ε¯ + φ f (C f − Cm )A¯ε,

(8.8)

8.2 Unidirectional Composites

125

and therefore, ³ ´ C = Cm + φ f C f − Cm A.

(8.9)

This is the required equation for the effective stiffness tensor C. Since C f , Cm , and φ f are all known, one only needs to find the strain-concentration tensor A. Once A is determined, the effective stiffness can be calculated. Different expressions of A represent different models. For example, if A = I, where I (I i j kl = (1/2)(δi k δ j l + δi l δ j k )) is the fourth-order unit tensor, Equation 8.9 reduces to C = φm Cm + φ f C f ,

(8.10)

which is the Voigt average. The Voigt average is an upper bound on the effective stiffness. A solution for the dilute case has been obtained by Eshelby [98]. For a single ellipsoidal particle embedded in an infinite matrix, the strain-concentration tensor, denoted by AEshelby , is given by A = AEshelby = [I + ESm (C f − Cm )]−1 ,

(8.11)

where E is the fourth-order Eshelby tensor [98], and Sm = (Cm )−1 is the elastic compliance tensor. The Eshelby model is only applicable to small concentrations where there is no particle interaction. Mori and Tanaka [258] have generalized the above model to the non-dilute case. Let us denote the strain-concentration tensor A by AMT . It is required that AMT satisfies the following conditions at the two extreme concentrations as AMT |φ f →0 = AEshelby ,

AMT |φ f →1 = I.

(8.12)

Clearly, the following equation satisfies the necessary conditions at both ends: A = AMT = [I + (1 − φ f )ESm (C f − Cm )]−1 . Substitution of Equation 8.13 into Equation 8.9 gives ³ ´ C = Cm + φ f C f − Cm [I + (1 − φ f )ESm (C f − Cm )]−1 ,

(8.13)

(8.14)

or, equivalently, C = Cm + φ f [(C f − Cm )−1 + (1 − φ f )ESm ]−1 .

(8.15)

This model was first derived by Mori and Tanaka [258], and hence is called the Mori-Tanaka model, which was later described by Benveniste [32] and Christensen [61] in a simpler, more direct way. The model gives the correct dilute solution, and also gives the correct composite property at high concentration. Useful closed-form expressions for the Eshelby tensor E for an isotropic matrix have been given by Tandon and Weng [354]. The general form of the Eshelby tensor is given in the book of Mura [262], and can be calculated numerically (Gavazzi and Langodas [129]). An equivalent form of Equation 8.15 is ³ ´ C = Cm + φ f C f − Cm AEshelby (φm I + φ f AEshelby )−1 , which is the equation presented in the paper of Benveniste [32].

(8.16)

126

8 Improved Mechanical Property Modeling

8.2.2

Effective Thermal Expansion Coefficients

The thermoelastic constitutive relation of the composite material can be given by ¯ + α△T, ε¯ = Sσ

(8.17)

where α is the effective thermal expansion coefficient tensor. Benveniste and Dvorak [33] have shown that α = α f + (S f − S)(Sm − S f )−1 (α f − αm ).

(8.18)

The elastic compliance tensor S = C−1 can be calculated from the Mori-Tanaka model. Equation 8.18 is identical to the following equation of Rosen and Hashin [314]: α = φ f α f + φm αm + (S − φ f S f − φm Sm )(S f − Sm )−1 (α f − αm ).

8.2.3

(8.19)

Effects of Fiber Concentration and Aspect Ratio

The behavior of the micro-mechanical models described in the previous sections will be illustrated by predictions of seven effective thermo-mechanical properties: elastic moduli E 11 , E 22 , G 12 , Poisson’s ratios ν12 and ν23 , and coefficients of thermal expansion (CTEs) α11 and α22 , for a full range of volume fractions 0 ≤ φ f ≤ 1, and varying aspect ratios. The results shown below are calculated for fibers aligned with the 1-direction. For the calculation, we assume a fiber/matrix system with properties given in Table 8.1. Table 8.1 Property Data for Components of a Short Glass Fiber-Reinforced Composite

8.2.3.1

Properties

Matrix

Fiber

Young’s modulus

E m = 1.57 GPa

E f = 72.5 GPa

m

Poisson’s ratio

ν = 0.335

ν f = 0.2

CTE

αm = 108.3 × 10−6 K−1

α f = 4.9 × 10−6 K−1

Effect of Fiber Concentration

The numerical results based on the Mori-Tanaka model for the five effective elastic constants E 11 , E 22 , G 12 (all scaled by E m ), ν12 , and ν23 for a fixed aspect ratio a r = 20 are plotted as a function of the fiber volume fraction φ(≡ φ f ) in Figures 8.1 and 8.2. It is noted that E 11 /E m and E 22 /E m converge to 1 for φ = 0, and to E f /E m for φ = 1 The variation of E 11 /E m with volume fraction is nearly linear, showing that for this aspect ratio the Mori-Tanaka predicted E 11 does not deviate much from the rule of mixtures. The values of transverse and shear moduli, E 22 /E m and G 12 /E m , are relatively low until they reach high volume fractions, indicating that the values of these moduli are dominated by the matrix. The effective thermal expansion coefficients (CTEs)are plotted against the fiber volume fraction φ in Figure 8.3. The slight increase in the transverse expansion coefficient at low volume fraction is attributed to the axial restraint due to fibers [321].

8.2 Unidirectional Composites

127

Figure 8.1 Reduced effective moduli scaled by E m vs. fiber volume fraction for a R = 20. Predicted using the Mori-Tanaka model

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

8.2.3.2

Effect of Fiber Aspect Ratio

Figures 8.4 and 8.5 show the predictions of the Mori-Tanaka model for the five effective elastic constants E 11 , E 22 , G 12 (all scaled by E m ), ν12 , and ν23 over a wide range of aspect ratio for volume fraction φ f = 0.2. Fiber aspect ratio has very little effect on E 22 /E m , G 12 /E m , and ν12 , but has a significant effect on the longitudinal Young’s modulus, E 11 /E m , and Poisson’s ratio ν23 . The greatest effect occurs at small aspect ratio. However, when the aspect ratio exceeded 50, E 11 /E m and ν23 become less sensitive to the aspect ratio and approach asymptotic values.

128

8 Improved Mechanical Property Modeling

Figure 8.3 Effective coefficients of thermal expansion vs. fiber volume fraction for a R = 20. Predicted using the Rosen-Hanshin model

Figure 8.6 shows the modeling results of coefficients of thermal expansion versus the fiber aspect ratio for φ = 0.2, predicted by the Rosen and Hashin [314] model.

Figure 8.4 Reduced effective moduli scaled by E m vs. fiber aspect ratio for φ = 0.20. Predicted using the Mori-Tanaka model

8.2 Unidirectional Composites

129

Figure 8.5 Effective Poisson’s ratios vs. fiber aspect ratio for φ = 0.20. Predicted using the MoriTanaka model

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

130

8 Improved Mechanical Property Modeling

8.3

Fiber Orientation Averaging

In practical situations, the short fibers are rarely fully aligned in the composite. Of particular interest here is to predict how the fiber orientation distribution affects the thermo-mechanical properties. The basic approach involves an averaging process over a large number of subunits, which may be regarded as comprising one fiber and the surrounding matrix material. The overall property is then simply the average of the properties of the subunits over all directions, weighted by the number of subunits. Such a procedure is termed the orientation average. As has been described previously, fiber orientation tensors are defined as the dyadic products of the fiber unit orientation vectors p multiplied by the distribution function and integrated over all direction. The fiber orientation tensors may be used to do the orientation averaging (Advani and Tucker [2]). The fourth order stiffness tensor, C i j kl , of a transversely isotropic subunit can be averaged to give the composite stiffness, 〈C i j kl 〉, as follows [2]: 〈C i j kl 〉 = B 1 a i j kl + B 2 (a i j δkl + a kl δi j ) + B 3 (a i k δ j l + a i l δ j k + a j l δi k + a j k δi l ) + B 4 (δi j δkl ) + B 5 (δi k δ j l + δi l δ j k ).

(8.20)

The constants Bi are invariants of the tensor C i j kl , given by B 1 = C 1111 +C 2222 − 2C 1122 − 4C 1212 ,

(8.21)

B 2 = C 1122 −C 2233 ,

(8.22)

B 3 = C 1212 + 0.5 (C 2233 − 2C 2222 ) ,

(8.23)

B 4 = C 2233 ,

(8.24)

B 5 = 0.5 (C 2222 −C 2233 ) .

(8.25)

In general, 〈C i j kl 〉 is no longer transversely isotropic. Equation 8.20 suggests that the composite stiffness depends on both the second order orientation tensor a i j and the fourth order orientation tensor a i j kl . In fact, it is only necessary to know the fourth order tensor. Noting that p k p k = 1, one can show that a i j kk = 〈p i p j p k p k 〉 = 〈p i p j 〉 = a i j .

(8.26)

This means that the second order tensor is incorporated in the fourth order tensor, for the case of transversely isotropic subunits. The orientation averaging method can also be applied to predict average thermal expansion coefficients. The result is 〈αkl 〉 = 〈C i j kl 〉−1 〈C i j mn αmn 〉,

(8.27)

where the average of the fourth order stiffness tensor is given in Equation 8.20. The other orientation average involving a second order tensor is given by Bay [29]: 〈C i j mn αmn 〉 = [(C 1111 −C 1122 )α11 + (2C 1122 −C 2222 −C 2233 )α22 ] a i j + [C 2211 α11 + (C 2222 +C 2233 )α22 ] δi j .

(8.28)

Knowing the average values of 〈C i j kl 〉 and 〈αi j 〉 allows us to predict the structural performance of the component in warpage or structural analysis. Other approaches that are more sophisticated than Equation 8.20 are also available. See, for example, Eduljee et al. [93] and Lin et al. [226].

9

Long Fiber-Filled Materials

Traditional short fiber-reinforced injection molding compounds, available since the 1960s, have been widely used in industry, because they not only produce improved strength properties, but also have the ability to flow readily in injection molds. Short fiber thermoplastic composite pellets used for injection molding are manufactured by extrusion, where the polymer and chopped strands of fibers are mixed in single or twin screw extruders, extruded, and pelletized. As a result of fiber length attrition during extrusion compounding, this technique produces short fibers, which are typically 0.2–0.4 mm long with aspect ratios of 20–50. The fibers are encapsulated in the polymer matrix to form the pellets, inside which the fibers are randomly oriented; as shown in Figure 9.1(a). Since the mid-1980s, injection moldable long fiber-reinforced thermoplastic (LFRTP) composites have been commercially available (Cattanach et al. [52]). Long fibers are still discontinuous, however. The long fiber thermoplastic composite pellets are made by a pultrusion process, in which continuous fibers with a polymer matrix surrounding them are pulled through a circular die. As the resulting fiber-filled strand solidifies, it is sliced into individual pellets. These pellets are typically 10–13 mm long, with equally long fibers aligned along the pellet length, see Figure 9.1(b). The aspect ratios of long fibers can be more than 100. For example, McClelland and Gibson [243] reported an injection molded nylon 66 composite containing long glass fibers with an aspect ratio of 364, (see Lafranche et al. [216]). Long-fiber-reinforced thermoplastic composites offer better mechanical properties than their short fiber counterparts, and still preserve the ability to flow in injection molds. Many issues arise in the modeling of injection molding of long-fiber composites. These include modeling of fiber orientation, fiber migration, fiber breakage, fiber bending during processing, and estimation of the composite strength.

9.1

Fiber Orientation Evolution Model

It has been found (Bailey and Rzepka [23] and Toll and Andersson [366]) that the long fibers aligned themselves in a “sandwich” structure in injection-molded edge-gated plaques. There are two skin layers of fibers aligned parallel to the mold filling direction and a core region of fibers aligned perpendicular to the mold filling direction, which is a typical structure often observed in traditional short fiber-reinforced moldings. However, the increased fiber aspect ratio results in increased relative core thickness. The orientation distribution in both the skin and core layers was found to be more random for the long-fiber composites. This could be explained as a consequence of enhanced interaction between the fibers, which inhibits the motion and reorientation of longer fibers.

132

9 Long Fiber-Filled Materials

Figure 9.1 Schematic representations of (a) short-fiber pellet and (b) long-fiber pellet used for injection molding

One may anticipate that the existing short-fiber orientation models could be used to represent orientation in long-fibers. However, Phelps and Tucker [296] have found that neither the traditional Folgar-Tucker nor the reduced-strain closure models (Sections 5.8, 7.3) can predict experimentally observed long-fiber orientation distributions accurately. However, a combination of the anisotropic rotary diffusion model (Section 7.2) with the reduced-strain closure model, that is, the ARD-RSC model, does well in predicting orientation of long fibers. This indicates that for long fibers, the anisotropic character of the fiber-fiber interaction is important and must be taken into account in the simulation.

9.2

Flow-Induced Fiber Migration Model

Fiber migration during flow is another issue in long fiber-reinforced composites processing. Spahr and Friedrich [338] have presented a comparative study of the properties of short (aspect ratio = 70) and long (aspect ratio = 320) fiber-reinforced polypropylene moldings. The longfiber samples were found to have a considerably higher concentration of fibers in the core region relative to the skin regions. Flow-induced migration of particles in suspensions has been an active research topic in suspension rheology during the last three decades. Most investigators dealt with spherical particles, except Mondy et al. [256], who reported some experimental data on the migration of rods of aspect ratios ranging from 2 to 18 at volume fractions of either 0.3 or 0.4. They observed that the fibers in the flowing suspension move away from the region of higher shear rate toward the region of lower shear rate; this migration rate increases with increasing rod volume, but does

9.2 Flow-Induced Fiber Migration Model

133

not depend much on the fiber aspect ratio. The final extent of migration is also relatively independent of the aspect ratio, except in regions near the walls. Their observations indicate that fiber migration has a similar mechanism to the migration of spheres. Therefore, for modeling purposes, a similar approach to that for spherical particles can be employed as the first step. Phillips et al. [297] proposed a diffusive flux model to describe the migration of spheres. This model has been modified by Fan et al. [105] for short fiber suspensions, and numerical results obtained from this modified model have been validated by the experimental data of Mondy et al. [256]. Both experimental and numerical data indicate that migration is a slow process. The time to reach a steady-state concentration profile is mainly dependent on the volume of the particles. The smaller the volume, the slower the migration process will be. For small-size particles such as short fibers, the time required to reach the steady-state concentration profile is much longer than the typical filling time in the injection molding processes. Therefore, short-fiber migration during injection molding processing may not be remarkable, except for the gate region. This may not be true for long fibers, as indicated by Spahr and Friedrich’s experiments [338]. No theoretical and numerical work has been done on long-fiber migration to date. Nevertheless, the conservation law of the solid phase is the same for all bodies. If we assume that the long fibers have the same migration mechanism as the short fibers, we may also employ the model of Phillips et al. [297] to simulate the long-fiber migration phenomenon. In the model, the conservation law for solid particles is written in an Eulerian reference frame as Dφ ∂Ni = , Dt ∂x i

(9.1)

where φ is the volume fraction of fibers, and Ni is the particle flux, expressed as Ni = k c a 2 φ

∂ ln η r ∂ ˙ + k η a 2 γφ ˙ 2 (γφ) , ∂x i ∂x i

(9.2)

where k c and k η are two empirical parameters determined from the experimental data, η r is the relative viscosity defined by the ratio of suspension viscosity to the solvent viscosity (η r = η/η s ), a is the dimensionless equivalent radius of particles based on the volume of a fiber, p and γ˙ = 2D : D is the generalized strain rate, see Equation 2.17. The particle flux consists of a contribution from the collisions between fibers (the first term on the right-hand side of Equation 9.2) and a contribution from the gradient of viscosity (the second term on the right-hand side). In the Phillips model, η r is usually expressed by using Krieger’s empirical form [212] for spherical particles. For fiber suspensions, it is advisable to use the Phan-Thien-Graham model (Equation 7.18), in which the relative viscosity η r is a strong function of the fiber aspect ratio. This allows for the flux N to depend on fiber aspect ratio. However, the empirical equation of Kitano et al. [203] (Equation 7.20) for φm may need to be re-correlated with a different range of aspect ratios for long fibers. Equation 9.1 is subject to the no-flux boundary condition N · n = 0 at the solid boundary, where n is the outward normal unit vector. A uniform volume fraction is also needed to be specified as the initial condition. The level of the average volume fraction over the whole domain will be a constant during the process. A drawback of the above model is that it predicts a cusp-like concentration profile in cavity flows. This arises from the fact that, in the plane of symmetry, γ˙ and its gradient are zero and so Ni = 0. In reality, however, particles in the plane of symmetry would not rest absolutely due to perturbation from the bulk flow. Fan et al. [105] therefore modified Equation 9.2 to i ∂ ln η r ∂ h ˙ 2 (γ˙ + γ˙ 0 )φ + k η a 2 γφ , (9.3) Ni = k c a 2 φ ∂x i ∂x i

134

9 Long Fiber-Filled Materials

where γ˙ 0 is a parameter characterizing the perturbation of the bulk flow.

9.3

Fiber Length Attrition Model

Fiber length degradation during processing is a problem in long fibers. Phelps [295] developed a fiber length attrition model based on the buckling force in a hydrodynamically loaded fiber to predict fiber length distribution during injection molding processing. In this model, a breakage criterion is proposed. The criterion states that a fiber of length l (i ) , diameter d , and orientation state 〈pp〉 will break if ¡ ¢ Γ − 2D : 〈pp〉 > 1 ,

(9.4)

where D is the rate of deformation tensor, and Γ is given by Γ=

£ ¤4 4ζη s l (i ) π3 E f d 4

.

(9.5)

Here, η s is the resin viscosity, E f is the fiber elastic modulus, d is the fiber diameter, and ζ is a dimensionless drag coefficient given by Dinh and Armstrong [79] as ζ=

2π , ln(H f /d )

(9.6)

where H f is the average lateral spacing from a given fiber to its nearest neighbor:

Hf =

h ¡ (i ) ¢2 i−1    n l

for random orientation

h i−1/2    nl (i )

for fully aligned orientation,

(9.7)

where n is the number of particles per unit volume. In practical applications, ζ can be treated as a fitting parameter. Phelps [295] expresses the probability that a fiber of length l (i ) will break during a time increment ∆t as P (i ) ∆t , where P (i ) is given by £ ¤ P (i ) = C B γ˙ max 0, 1 − exp(1 − Γ) ,

(9.8)

where C B is a fitting parameter that scales the breakage rate. Although the length of fibers may be reduced as a result of fiber attrition, the total fiber length under a given fiber length distribution does not change. Let M denote the total number of discretizations of the fiber length distribution and set ∆l = l max /M , where l max is the maximum fiber length in the fiber length distribution. If N (i ) (i = 1 to M ) is the number of fibers of length i ∆l (∆l = l max /M ), then it satisfies the following conservation equation: M X DN (i ) = −P (i ) N (i ) + R (i k) N (k) , Dt k|k≥i

(9.9)

9.4 Uniaxial Tensile Strength Model

135

where R (i k) is a probability density function defined such that the probability of finding a fiber of length l (k) breaking to form a fiber of length l (i ) (where l (i ) < l (k) ) in time t is R (i k) ∆t . The function is assumed to be a normal probability density function given by " # (l (i ) − 0.5l (k) )2 1 (i k) , (9.10) R = p exp − 2(Sl (k) )2 Sl (k) 2π where Sl (k) is the standard deviation, and the variable S is a dimensionless fitting parameter which controls the shape of the Gaussian breakage profile. A smaller value of S corresponds to a higher probability of breakage occurring at the midpoint of the fiber, while a larger S value corresponds to more evenly distributed probable breakage points along the fiber length. Equations 9.4, 9.8, and 9.9 provide a fiber length attrition model (Phelps [295]) to solve for the fiber length distribution during injection molding. Some implementation details can be found in Nguyen et al. [271].

9.4

Uniaxial Tensile Strength Model

In this section, we describe a uniaxial tensile strength model, which provides a general understanding of the composite strength. Let us consider a single fiber in the polymer matrix, oriented in the loading direction. The applied load is transferred via constant shear stresses to the fiber at the fiber/matrix interface. The fiber, which is stretched by the shear force acting at the interface, is assumed to be in tension only. If τ denotes the shear stress at the interface, and l and d are the fiber length and diameter, respectively, the tensile stress in the fiber at a distance x from the end of the fiber (where x = 0 refers to one end of the fiber and extends to x = l /2 at the middle of the fiber) is given by σ( f ) =

4xτ . d

(9.11)

For a discontinuous fiber to achieve a reinforcement efficiency of a continuous fiber, the stress over the length must be fully developed, and a critical fiber aspect ratio exists for which the peak stress at the middle of the fiber just reaches the ultimate fiber strength σ( f ,ul t ) (Kelly and Tyson [194]; Guell and Benard [135]). The critical aspect ratio is given by µ

¶ lc σ( f ,ul t ) = , d 2τ y

(9.12)

where τ y is the shear strength of the interface, identified with the yield shear stress of the matrix. We call l c /d the critical fiber aspect ratio and l c the critical fiber length. If we assume the matrix is at the onset of yield, then the tensile stress within the fiber at a distance x from the end of the fiber is written as  2x ( f ,ul t )   σ when x < l2c (f ) σ = lc (9.13)   ( f ,ul t ) σ when x ≥ l2c .

9 Long Fiber-Filled Materials

136

If the fiber is longer than l c , the surface shear stresses build up sufficient tensile stress within the fiber to fracture the fiber. If the fiber is shorter than l c , the tensile stress in the fiber never reaches its ultimate value, and the fiber will pull out from the matrix when the composite breaks. Provided that the medium is statistically homogeneous, the tensile strength of a composite is given by ¯ ( f ) + (1 − φ)σ ¯ (m) , σ(c,ul t ) = φσ

(9.14)

¯ ( f ) and σ ¯ (m) are average stresses, at composite failure, in the fiber and matrix, respecwhere σ ¯ (m) = σ(m,ul t ) , and then σ ¯ ( f ) is tively. We can set the matrix stress equal to the matrix strength σ given by  l ( f ,ul t )   σ when l < l c Z l /2  2 2l c ¯ (f ) = ¶ σ( f ) d x = µ (9.15) σ  l l 0   1 − c σ( f ,ul t ) when l ≥ l c . 2l The simple model described above is useful for determining the load-bearing capacity of a given fiber-reinforced composite.

9.5 9.5.1

Flexible Fiber Modeling

Direct Simulation Methods

Long fibers are not perfectly rigid, and therefore may exhibit deformation (bending and/or twisting) during processing. Many authors use direct simulations to model the flexible fiber motion at the particle level, where the individual flexible fiber model consists of multiple rigid parts connected with each other. The bending and twisting deformation is reflected by allowing movement of the rigid parts relative to each other. Several physical models (Yamamota and Matsuoka [405–407]; Ross and Klingenberg [316]; Joung et al. [183, 184]; Tang and Advani [355]; Schmid et al. [323]; Wang et al. [391]; Kittipoomwong and Jabbarzadeh [204]) have been employed to represent flexible fibers (Figure 9.2), from which mathematical models can be constructed based on force balances. We use the model developed by Joung et al. [183, 184] as an example, where the flexible fiber suspension is modeled by a chain of spherical beads in viscous flow. The beads are linked with each other by inextensible connectors. Only the beads interact with the fluid. The connectors do not interact with the fluid, but serve to transmit internal forces and maintain the configuration of the fiber. Within each bead is a joint linking two connectors. Each joint allows limited bending and torsion, and produces a restoring moment acting to straighten the fibers when they are deflected from equilibrium. Given the Young’s and shear moduli, one can calculate the internal bending moment, the torsion moment, and the bend and twist angles for each joint. Joint deflections are caused by fiber interaction forces. There are two types of forces acting on each sphere: external forces and internal forces. External forces arise from viscous drag (including long-range velocity disturbances from other

9.5 Flexible Fiber Modeling

137

beads), and short-range lubrication forces between beads when they are in close proximity. The internal force is the connector tension. Since the connectors are rigid and inextensible, the “tension” is simply the reactive force that exactly opposes external forces acting in the direction of the connector. Once the external forces are determined, one can also calculate the moments acting at each joint. The internal moment at a joint is equal to the moment balance generated by forces on beads to either side of the joint. The external drag forces can be calculated by µ

˜ µ )]. Fdrag = −ζ[˙rµ − (uµ + u

(9.16)

This equation describes the drag force acting on bead µ located at position r µ . In Equation 9.16, ζ is the Stokes drag coefficient given by ζ = 6πη s a, where η s is the solvent viscosity and a is the radius of the spherical bead. According to the expression of Equation 9.16, the force is ˜ µ) proportional to the difference between the bead velocity r˙ µ and the solvent velocity (u µ + u µ at bead µ. The velocity u is determined assuming a homogeneous velocity gradient L tensor: ˜ µ is the cumulative perturbation velocity at bead µ from all other u µ = L · r µ . The velocity u spheres ν in the suspension, described by ˜µ= u

X ν̸=µ

ν , Ω µν · Fdrag

(9.17)

ν where Fdrag is the drag force acting on all other suspension spheres ν, and Ωµν is called the Oseen tensor and is given by

Ω

µν

µ ¶ r µν r µν 1 I + µν µν , = 8πη s |r µν | |r r |

(9.18)

where r µν = rν − r µ is the position vector from bead µ to bead ν. Short-range lubrication forces are calculated using the following equation: µν

Flub = −3πη s

· µν ¸ a2 r µν r ν µ · (u − u ) . |rµν | − 2a |r µν | |r µν |

(9.19)

Since all interactions are between beads only, and interactions between beads are governed by the same laws regardless of whether they belong to the same fiber or different fibers, one does not need to distinguish beads of different fibers while determining external forces. For a rigid fiber, the orientation of the fiber is described by a single unit direction vector along the major axis of the fiber. Similarly, the orientation of a flexible fiber can be characterized by a normalized end-to-end vector p of the bead-chain model. In simulations, once the fiber rotation and translation rates are known, their new position and orientation can be calculated for the current time step. The vector p is overlaid by the configuration of chain-of-beads. The interaction forces are calculated using Equations 9.16 and 9.19. Then an additional calculation is performed to determine the deflection of the bead-chain along the fiber length. Fiber deformation may lead to a further small change in the end-to-end orientation vector p. The orientation results can be used to predict rheological properties of the suspension. Joung et al. [184] have shown that there is a distinct increase in bulk viscosity as fiber curvature is increased.

138

9 Long Fiber-Filled Materials

Figure 9.2 Schematic representations of flexible fiber models

9.5 Flexible Fiber Modeling

139

Figure 9.3 Bead-rod model of Strautins and Latz [346]

9.5.2

Continuum Modeling

Direct simulations are useful to explore the particle-particle interactions, such as long and short-range hydrodynamic effects. However, they are currently not efficient for modeling the entire molding process. A continuum model that accounts for the orientation evolution of semi-flexible fibers is that proposed by Strautins and Latz [346], which is derived from the socalled bead-rod model. In this model, a semi-flexible fiber is modeled as two connected rods of orientation p and q, each of length l rod that may flex about a central pivot point, as shown in Figure 9.3. The moments of the p and q vectors, with respect to the orientation distribution function ψ(p, q, t ), are defined as Z Z a(m,n) = pm qn ψ(p, q, t )d pd q, (9.20) where pm qn is the (m + n)-th rank tensor. Only the following 4 cases are important: Z Z a i(2,0) = p i p j ψ(p, q, t )d pd q , j Z Z a i(1,1) p i q j ψ(p, q, t )d pd q , = j Z Z a i(1,0) = p i ψ(p, q, t )d pd q , Z Z a i(4,0) = p i p j p k p l ψ(p, q, t )d pd q . j kl

(9.21) (9.22) (9.23) (9.24)

One can then obtain the equation for orientation tensor of the long fibers as follows (see Ortman et al. [274] for details): D a i(2,0) j Dt

(2,0) − L i k a k(2,0) − L j k a ki + 2 L kl a i(4,0) j j kl

=

´ l rod ³ (1,0) (1,0) (2,0) a i m j + m i a (1,0) − 2m a a k j ij k 2 ³ ´ ¡ ¢ (1,1) (1,1) (2,0) − 2κ a i j − a kk a i j + 2C I γ˙ δi j − 3a i(2,0) , j

(9.25)

140

9 Long Fiber-Filled Materials

D a i(1,1) j Dt

(1,1) − L i k a k(1,1) − L j k a ki + 2 L kl a l(2,0) a i(1,1) j j k

=

´ l rod ³ (1,0) a i m j + m i a (1,0) − 2m k a k(1,0) a i(1,1) j j 2 ³ ´

(1,1) (1,1) , − 2κ a i(2,0) − a kk ai j − 4C I γ˙ a i(1,1) j j

D a i(1,0) Dt

(2,0) − L i k a k(1,0) + a kl L l k a i(1,0) =

´ l rod ³ m i − a i(1,0) m j a (1,0) j 2 ³ ´ (1,1) − κa i(1,0) 1 − a kk − 2C I γ˙ a i(1,0) ,

(9.26)

(9.27)

with mi =

∂2 u i a (2,0) , ∂x j ∂x k j k

(9.28)

where u i is the velocity vector. Within these equations, C I is the interaction coefficient as defined in the Folgar-Tucker model, and κ is the resistive bending potential coefficient. When the value of κ decreases, the fiber becomes more flexible. Conversely, as the value of κ increases, the model behaves more like a rigid fiber, and when κ → ∞, the solution of the model converges towards the Folgar-Tucker model for high aspect ratio rigid fibers. Ortman et al. [274] proposed a stress law for the bead-rod model as follows: ¢ ¡ :D σbead−rod = −pI + 2η m D + f 1 φD + f 2 a(4,0) 4 ¡ ¢ 3 + 2 η m κφa r tr(r) a(2,0) − R , 4l rod

(9.29)

where p is the isotropic pressure, η m is the matrix viscosity, D is the deformation rate tensor, f 1 and f 2 are empirical parameters to be determined by fitting rheological data of long-fiber suspensions, φ is the volume fraction of fibers, r is a second order, dimensional end-to-end orientation tensor, defined as the second moment of the end-to-end vector, l rod (p − q), with respect to ψ: Z Z 2 2 r≡ l rod (p − q)(p − q)ψ(p, q, t )d pd q = 2l rod (a(2,0) − a(1,1) ), (9.30) and R is defined as a normalized and dimensionless end-to-end orientation tensor: R≡

a(2,0) − a(1,1) r = . tr(r) 1 − tr(a(1,1) )

(9.31)

The last term in Equation 9.29 is the bending stress term. For a perfectly straight fiber, p = −q, we have a(2,0) = R, so that the bending stress term vanishes and the equation reduces to the constitutive model for rigid fiber suspensions (Lipscomb et al. [227]).

10

Crystallization

An essential requirement for improving injection molding simulation is the incorporation of crystallization effects. In Section 5.3.5, we remarked that the use of a single no-flow or transition temperature is unjustified for semi-crystalline materials. It can lead to errors in fill pattern prediction and calculation of shrinkage and warpage. Moreover, if we are to be able to predict properties of semi-crystalline materials after molding, and when subjected to variable environments, we need to incorporate crystallization calculations explicitly. A major research trend in the simulation of injection molding is to predict properties of the final product through the prediction of the processing-induced material morphology. For amorphous polymers, the term morphology refers to molecular orientation and deformation. For semi-crystalline polymers, morphology also includes the degree of crystallinity, the shape, the sizes, and the orientation of the crystalline structures. This chapter focuses on prediction of morphology for semi-crystalline polymers in injection molding.

10.1

Crystallization Kinetics in Quiescent Melts

Crystallization of a polymer consists of two stages: the nucleation and the subsequent growth of nuclei. The nucleation is the formation, within the liquid phase, of active nuclei, from which crystals emerge. The nuclei can appear instantaneously at the beginning of the process with a fixed nuclei number density, or can appear continuously with a finite nucleation rate. The former is called the instantaneous nucleation, and the latter is called the sporadic nucleation. If the nuclei are sufficiently strained under flow, they may grow into threads. Otherwise, especially under a quiescent condition, they will grow radially in space and form spherical structures, called spherulites, see Figure 10.1(a), with the size of radii varying from one micron to several millimeters. A spherulite is a semi-crystalline morphological entity, consisting of radiating crystallites separated by amorphous phase material. The thread-like nuclei may lead to Keller and Kolnaar’s model of shish-kebabs [193]. As shown in Figure 10.1(b), the shish-kebab consists of extended-chain crystals (shish) and fold-chain crystals (kebab)that grow perpendicularly to the shish. We will consider the flow-induced thread structures in the next section. In this section, we consider the formation of spherulites from a quiescent semi-crystalline polymeric melt which 0 is supercooled down from a temperature above the equilibrium melting point (Tm ) to a tem0 perature below the Tm . The equilibrium melt point is defined as the temperature at which the equilibrium condition exists between the crystal and liquid, as both phases have the same value of the Gibbs free energy. Below the equilibrium melting point, the crystals are stable. Above the equilibrium melting point, the melt is more stable.

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142

Figure 10.1 Schematic representations of (a) spherulite structure, and (b) shish-kebab structure (from Zhao et al. [417] with permission from Cambridge University Press)

10.1.1

The Kolmogoroff-Avrami-Evans Model

The growth of a spherulite can be described as a sphere with a growing radius. Let G(t ) denote the growth rate of the sphere radius as a function of time. The volume of a sphere, which grows £R t ¤3 unrestrictedly during a time interval between s and t, is given by υ(s, t ) = (4π/3) s G(u)d u . Let N q (t ) be the nuclei number per unit volume at time t (noting that N q (t ) is not necessarily an integer). If the growth of the spherulite volume is assumed to be unrestricted and without impingement, the volume fraction φ is then given by φ(t ) =

4π 3

t

Z 0

N˙ q (s)

¸3

t

·Z

G(u)d u

ds ,

(10.1)

s

where N˙ q is the rate of nuclei number per unit volume. The calculated volume fraction is also called the fictive volume fraction. The assumption of unrestricted growth is physically unrealistic, but the fictive volume fraction can be related to the actual volume fraction of the spherulites, denoted by α. Avrami [16–18] proposed that, in the case when the nuclei are uniformly distributed throughout the entire volume during the growth process, the actual volume fraction increases by a quantity equal to the (1 − α) fraction of the fictitious increase, d α = (1 − α)d φ,

(10.2)

which can be easily integrated to give α = 1 − exp(−φ).

(10.3)

A few years before Avrami, Kolmogoroff [209] obtained the same equation using a more general statistical approach. Later than Avrami, Evans [99] also independently developed the theory, using an analogy between crystallization and the expanding circular waves created by raindrops on a pond. Therefore, the equation is called the Kolmogoroff-Avrami-Evans model.

10.1 Quiescent Crystallization

143

In the instantaneous nucleation cases, the nuclei number density would be of the form N q (t ) = N0 H (t ), where H (t ) is the Heaviside unit step function, zero for negative t , and unity for t Ê 0. One then has N˙ q (t ) = N0 δ(t ), where δ(t ) is the Dirac delta function concentrated on t = 0. We assume that the spherulite growth rate is constant at a fixed crystallization temperature. The Kolmogoroff-Avrami-Evans equation reduces to ¶ µ 4π α = 1 − exp − N0G 3 t 3 . 3

(10.4)

The power index of time is called the Avrami index, which equals 3 for spherulitic growth and instantaneous nucleation. If N˙ q is constant, one has ³ π ´ α = 1 − exp − N˙ q G 3 t 4 . 3

(10.5)

Thus, in sporadic nucleation cases, the Avrami index becomes 4. In fact, the volume fraction of the spherulites, α, can also be interpreted as the ratio of crystallized volume to the total crystallizable volume, and be called the relative crystallinity, which ranges from 0 to 1. One can also describe the crystallization degree by absolute crystallinity, χ, defined as the ratio between the crystalline volume and the total volume. The value of χ never reaches 1, but has the following relation with α: χ = χ∞ α ,

(10.6)

where χ∞ represents the ultimate absolute crystallinity.

10.1.2

The Rate Equations of Schneider

To consider the non-isothermal crystallization, Schneider et al. [324] transformed the integral appearing in Equation 10.1 to a system of rate equations comprising four first-order differential equations. By differentiating Equation 10.1 four times with respect to time t , one has φ0 = φ =

4π 3

Z t ·Z

¸3

t

G(u)d u 0

s

d N q (s),

¸2 Z t ·Z t 1 φ˙0 = 4π G(u)d u d N q (s), G 0 s ¸ Z t ·Z t 1 φ2 = φ˙1 = 8π G(u)d u d N q (s), G 0 s 1 φ3 = φ˙2 = 8πN q (t ), G φ4 = φ˙3 = 8πN˙ q (t ). φ1 =

(10.7) (10.8) (10.9) (10.10) (10.11)

10 Crystallization

144

Hence, a system of rate equations can be obtained: φ˙3 = 8πN˙ q (t ), φ˙2 = Gφ3 ,

(10.12)

φ˙1 = Gφ2 , φ˙0 = Gφ1 ,

(10.14)

φ0 = − ln(1 − α) .

(10.16)

(10.13) (10.15)

The functions φi can be related to the crystalline morphology for unrestricted growth of spherulites: φ0 is the total volume of the undisturbed spherulites per unit volume, φ1 is the total surface area of the undisturbed spherulites per unit volume, φ2 is 8π times the sum of the radii of the undisturbed spherulites per unit volume, and φ3 is 8π times the number of the undisturbed spherulites per unit volume. If N˙q and G depend on the instantaneous, local temperature T (t ) only, the above Schneider’s rate equations can be applied to non-isothermal crystallization of spherulites.

10.1.3

Quiescent Nuclei Number Density

Under the quiescent condition, the nuclei number density has been found to increase with 0 0 increasing degree of supercooling, ∆T = Tm − T , where Tm is the equilibrium melting temperature. For instantaneous nucleation, the dependence of the nuclei number density on ∆T can be described by the following equation (Koscher and Fulchiron [211]) : N q = N0 δ(t ), ln N0 = a N ∆T + b N ,

(10.17)

where aN and bN are constants to be determined by fitting the equation to experimental data of N 0 (T ). An example showing the linear variation of ln N0 with ∆T is given in Figure 10.2 [211].

Figure 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)

10.1 Quiescent Crystallization

145

Ziabicki [431] proposed an equation that expressed the nucleation rate as a function of temperature. The equation, in the form as used by Coppola et al. [67] and Chen et al. [58], is given by µ ¶ µ ¶ Ea Kn N˙q = C 0 k B T ∆F q exp − exp − , kB T T (∆F q )n

(10.18)

where C 0 includes energetic and geometrical constants, k B is the Boltzmann constant, T is the absolute temperature, ∆F q represents the difference of Gibbs free energy per unit volume between melt and crystalline phase under quiescent conditions, E a is the activation energy of the supercooled liquid-nucleus interface, and Kn is a constant containing energetic and geometrical factors of crystalline nuclei. The power index n, also appearing as a subscript for K, accounts for the temperature region where the homogeneous nucleation occurs, and generally assumes the value 2 for the formation of primary nuclei in the homogeneous melt, or 1 for the secondary nuclei [218]. For small degrees of supercooling, ∆F q can be expressed as ∆F q = ∆Hc

∆T 0 Tm

,

(10.19)

0 where ∆T is the supercooling, Tm is the equilibrium melting temperature, and ∆Hc is the latent heat of crystallization. The latent heat is the amount of heat per unit volume (in J/m3 ) of a substance absolved or emitted during a phase transition. It is called “latent” because the temperature of the material does not change during the phase change, despite the input or release of heat.

10.1.4

Growth Rate of Spherulites

For a spherulite, the radial growth rate G can be calculated by applying the Hoffman-Lauritzen theory [218]: " ¡ ¢# · ¸ 0 K g T + Tm U∗ G(T ) = G 0 exp − exp − , (10.20) R g (T − T∞ ) 2T 2 ∆T where U ∗ is the activation energy of motion, often taking a generic value 6270 J/mol (Koscher and Fulchiron [211]), R g is the gas constant, T∞ = T g − 30 with T g being the glass transition temperature. The equilibrium melting temperature may depend on pressure. Fulchiron et al. [126] express the pressure dependence as a polynomial function 0 0 Tm (p) = Tm (0) + a 1 p + a 2 p 2 ,

(10.21)

where p is the pressure, and a 1 and a 2 are constants that can be determined from the PressureVolume-Temperature diagram. The parameters G 0 and K g in Equation 10.20 are determined 0 by plotting lnG +U ∗ /R g (T −T∞ ) against (T +Tm )/2T 2 ∆T . The experimental method for measurement of G(T ) will be described later. There are also some simpler empirical or semi-empirical relations describing the temperature dependence of the growth rate. The following empirical equation is adopted by Eder et al. [91] and Zuidema et al. [435]: G(T ) = G max exp[−cG (T − Tref )2 ],

(10.22)

146

10 Crystallization

where G max , cG , and Tref are fitting parameters. Van Krevelen [380] derived a semi-empirical equation for the growth rate of spherulites based on literature data obtained from a variety of typical semi-crystalline polymers. The equation reads à ! 0 Tm T0 50 + , (10.23) logG(T ) = logG 0 − 2.3 m 0 0 T Tm − T g Tm −T with G 0 = 7.5 × 108 µm/s.

10.1.5

Material Characterization for Quiescent Crystallization Kinetics

One of the major problems in using any kinetic model is the availability of enough experimental data for fully describing the kinetics. In industrial applications, and especially for simulation, we need to collect the data efficiently. The experiments described below were mainly performed using differential scanning calorimetry (DSC), except for the crystal growth rate measurements, for which the shearing hot-stage Linkam CSS450 is employed. The Linkam cell was used in conjunction with an optical microscope and a computer-operated camera for image production at times of interest. The DSC experiments involve a switch from the cooling mode to the isothermal mode. If the crystallization starts too soon before reaching the isothermal stage, the DSC cannot be reliable when applied to isothermal crystallization. Therefore, the experiments can only be performed in a limited temperature range, depending on the material to be tested. As an example to describe the experimental procedures, we choose an industrial isotactic polypropylene (iPP), grade HD601CF, produced by Borealis, as the sample material. The iPP has M w = 367 kg/mol, M n = 74 kg/mol, and M w /M n = 4.96. The sample size in the DSC is approximately 5 mg, while for the Linkam cell 4 − 5 pellets are required for each run. The following experimental methods have been presented by Hadinata et al. [141].

10.1.5.1

Half-Crystallization Time

The half-crystallization time t 1/2 is defined as the time required to reach a relative crystallinity of 0.5. The sample is annealed first at 210◦ C for 5 minutes and then cooled to a desired crystallization temperature Tc with a cooling rate of 30◦ C/min. Quiescent, isothermal crystallization occurs at Tc and is shown by the DSC as an exothermic peak. Figure 10.3 gives an example for the iPP sample at 132◦ C. The time-dependent relative crystallinity is calculated by integrating the area inside the DSC crystallization curve. Various crystallization temperatures can be tested. The typical temperature range for iPP is 128–140◦ C. The range is chosen so that the crystallization times are not too long or too short, ensuring that the DSC signal can still be read clearly and no crystallization occur during cooling. The half-crystallization times at different Tc s can then be obtained, which is also shown in Figure 10.3. The area under the peak is calculated to give the latent heat of crystallization.

10.1.5.2

Equilibrium Melting Temperature

The state of the semi-crystalline polymers in most cases is far from equilibrium, and therefore melting points (Tm ) directly measured from DSC curves are almost always non-equilibrium

10.1 Quiescent Crystallization

147

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

Figure 10.4 Heat flow curves for Borealis iPP

0 ones. The equilibrium melting temperature (Tm ) can be determined by extrapolations. A widely used method is the Hoffman-Weeks extrapolation method [158], which is described next, using the Borealis iPP sample as an example again. First, the sample is melted and annealed at 210◦ C for 5 minutes. Then the sample is cooled to the desired crystallization tem-

148

10 Crystallization

Figure 10.5 Determination of equilibrium melting temperature using Hoffman-Weeks method, for Borealis iPP. Melting point data measured on samples isothermally crystallized at different Tc s were used to determine the equilibrium melting temperature

perature (ranging between 128–140◦ C) with a cooling rate of 30◦ C/min. At this temperature, the sample is held until sufficiently crystallized. The sample is heated up again to 210◦ C to obtain the nominal melting point Tm . The above experiment is done for several crystallization temperatures. Tm varies with Tc , as shown in Figure 10.4. The melting temperature Tm is plotted as a function of the crystallization temperature, Tc . A linear relationship between Tm and Tc is observed. The Tm = f (Tc ) curve is extrapolated up to its intersection with the 0 (Figure 10.5). The Hoffman-Weeks Tm = Tc straight line. The intersection gives the value of Tm extrapolation method, however, is not very accurate. Improved experimental methods have also been discussed by Marand et al. [238] and Al-Hussein and Strobl [6]. For many crystalline polymers, the ratio T g /Tm approximately equals 2/3 [380]. This empirical rule is useful in the absence of data for T g .

10.1.5.3

Crystal Growth Rate

For the crystal growth rate experiment, the Linkam cell is first heated up to the annealing temperature. Then, the sample (a few pellets) is put in, melted, and pressed to a thickness of 150 µm. After placing the Linkam cell under the microscope, the temperature is cooled down to the desired crystallization temperature with cooling rate 30◦ C/min. The instant the Tc is reached is defined as t = 0. The development of structures is then monitored with the microscope, and photographs are taken at regular time intervals using the camera and the computer. By measuring the radius of the spherulites at different times, the growth rate is known. Several

10.2 Flow-Induced Crystallization

149

Tc s are tested to obtain the function G(T ). Figure 10.6 show the iPP spherulite growth at 132◦ C.

Figure 10.6 Isothermal crystal growth for Borealis iPP sample at 132◦ C under quiescent condition

For some industrial polymers, especially those having nucleating agents, the crystallization occurs too quickly for an observation of the spherulite growth under the microscope at the interesting temperature ranges. In these cases, one may use the van Krevelen equation (Equation 10.23) to estimate the thermal dependence of G(T ). After t 1/2 (T ) and G(T ) have been obtained, the nuclei number density can be evaluated by ± 3 N0 = 3 ln 2 (4πG 3 t 1/2 ). Here, we assume the nucleation is instantaneous in the quiescent crystallization.

10.2

Flow-Induced Crystallization

It is now well known that the deformation of polymers during the flow has dramatic effects on the crystallization and subsequent solidification of polymers in two aspects: enhancement of crystallization rate and formation of thread-like nuclei that grow into shish-kebab structures if the deformation of the fluid is sufficiently strong (Zhu and Edward [428]). The two phenomena are known as the flow-induced crystallization. Early investigations in this field can be traced back to around the early 1970s (Hass and Maxwell [147]; Mackley and Keller [236]). In a vast literature, for useful results towards the prediction of structure formation during injection molding, one should especially note the work of the polymer research group at University of Linz (Eder and Janeschitz-Kriegl [91]; Janeschitz-Kriegl [181]). In this book we mainly consider the post-shear crystallization of polymers, that is, crystallization occurring after the material experienced a short-term shearing. The short-term shearing

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150

treatment is relevant to the injection-molding process. The reader who is interested in crystallization under continuous shearing is referred to the work of Hadinata et al. [140].

10.2.1

Enhanced Nucleation

Eder et al. [92] have suggested the following first-order differential equation for the flowenhanced nucleation: N˙f +

1 Nf = f , λN

(10.24)

where N f is the flow-induced nuclei number density, λN is a temperature-dependent relaxation time that usually takes a large number, and f is a function depending on flow variables and temperature. Eder et al. consider the shear rate as a driving force for the flow-enhanced nucleation, and write f in the following form: µ f = gn

γ˙ γ˙n

¶2 ,

(10.25)

where g n is a factor with the dimension s −1 m −3 , and γ˙n is a shear rate of activation. The quadratic dependence on γ˙ 2 was introduced originally based on the argument that the nucleation process should not depend on the shearing direction. Janeschitz-Kriegl [181] later interpreted this term as the specific work. Several other assumptions on the driving force have been adopted by different authors. Zuidema et al. [435] replaced the γ˙ 2 term by the second invariant of the deviatoric part of the recoverable strain, which is considered to be a representative measure for the molecular orientation. The underlying idea is that, not the flow kinematics, but the behavior of molecules influences the speed of nucleation. Koscher and Fulchiron [211] considered the first normal stress difference as a driving force for nucleation. Coppola et al. [67] modeled the flow-enhanced nucleation based on free energy considerations. They took the right-hand side of Equation 10.18 to construct the function f and modified it by adding an extra term ∆F f onto ∆F q in the equation, where ∆F f is the flow-induced free energy change, drawn from Doi-Edwards model. Zheng and Kennedy [418] also considered the function f as a function of the free energy, and they used the FENE-P dumbbell model to calculate ∆F f . Both the DoiEdwards free energy and the FENE-P dumbbell free energy exhibit the following limiting behavior in shear flows [67, 359]. For low shear rates, one has ∆F f n0 kB T

˙ 2, ∝ (λa γ)

(10.26)

and for high shear rates, one has ∆F f n0 kB T

˙ ∝ ln(λa γ),

(10.27)

where n 0 is the number density of the molecules, k B is the Boltzmann’s constant, λa is the relaxation time of the polymer melt, and γ˙ is the generalized strain rate. The quantity n 0 k B T is normally related to the shear modulus that can be estimated or measured.

10.2 Flow-Induced Crystallization

151

Tanner and Qi [359] also proposed a strain and strain-rate formulation for f : f = a γ˙ p γ,

(10.28)

˙ s is the strain with t s the shearing time. The where a and p are empirical constants, and γ = γt equation gives a very good fit to experimental results of Wassner and Maier [396] in simple shearing and small-train oscillation.

10.2.2

Critical Parameters for Shish-Kebab Structure Formation

As mentioned above, the effect of flow can also introduce a morphological change from isotropic spherulites to oriented crystalline shish-kebab structures. Eder and Janeschitz-Kriegl [91] assume that the influence of flow on this type of morphology is due to the formation of thread-like precursor from which lamellae grow perpendicularly. Experimental observation of Mykhaylyk et al. [263] shows that two parameters are responsible for the formation of shish-kebab morphology by flow-induced crystallization: a critical shear rate γ˙ c below which the oriented structure is unlikely to be formed, and a critical amount of a specific work, w c , which is required to create oriented nuclei to form shish-kebab structures for shear rates above γ˙ c . The critical shear rate is estimated by γ˙ c ≈ 1/λR , where λR is the longest Rouse relaxation time of molecules [81, 312], proportional to the square of molecular weight. The specific work, w, is given by the integral of the product of the viscosity, η, and the square of the strain rate over the total shearing time, t s : Z w=

ts

0

˙ )]γ˙ 2 (t )d t . η[γ(t

(10.29)

Therefore, the specific work contains the effect of shearing time in addition to the strain rate. The governing criterion to signify the formation of shish-kebab structure is thus given by ˙ R > 1 and γλ

Z 0

ts

˙ )]γ˙ 2 (t )d t > w c . η[γ(t

(10.30)

Mykhaylyk et al. [263] found that the critical specific work is independent of the shear rate, but decreases with increasing concentration of long chains. These observations are further supported by the experimental work of Housmans et al. [161]. Based on these results, Steenbakkers and Peters [341] and Steenbakkers [340] modeled the flow-enhanced nucleation rate (N˙ f ) using a fourth-order dependence on the molecular stretch of the slowest relaxation mode:

N˙ f =

·    g n (Λ4 − 1) 1 −  

0

Nf N f ,max

¸ if

˙ γ>0

if

γ˙ = 0

,

(10.31)

where g n is a scaling parameter that depends on temperature only, N f ,max is the saturated number density, and Λ is the stretch of the high molecular weight chains that can be determined from the chosen viscoelastic constitutive equation. Based on the experimental evidence

10 Crystallization

152

that a critical specific work needs to be exceeded to change from the isotropic growth regime to the oriented regime, a critical molecular stretch Λcrit is also introduced that acts as a threshold to generate oriented crystalline morphologies once Λ > Λcrit . Mykhaylyk et al. [263] also showed that the critical shear rate and the critical work can be measured experimentally. The plate-plate geometry can be used to produce a radial distribution of shear rates across a sheared sample (where the shear rate at the radial position r is given by γ˙ = ωr /d where ω is the angular velocity of the rotating plate and d is the gap between the plates). Structure-related methods such as small-angle X-ray scattering (SAXS) and birefringence can be used to identify the boundary between isotropic and oriented structures.

10.2.3

Kinetics Equation of Shish-Kebab Structure

For the purpose of modeling, the geometry of a shish-kebab can be represented by a cylinder. The fictive volume fraction of the shish-kebabs is then given by φ = πL total

·Z

¸2

t

G(u)d u

,

(10.32)

0

where L total is the total length of threads per unit volume, which can be calculated by Z L total =

0

ts

N˙ f (s)l (t s − s)d s,

(10.33)

where t s is the shearing time and l (t s − s) is the length of the thread-like nucleus at time t s , activated at time s. For more details and further references, see Janeschitz-Kriegl [181]. If we denote the fictive volume fractions of the spherulites and the shish-kebabs by φspher es and φr od s , respectively, we can calculate the relative crystallinity by £ ¤ α = 1 − exp −(φspher es + φr od s ) .

10.2.4

(10.34)

Material Characterization for Flow-Induced Crystallization Kinetics

To measure crystallinity and the half-crystallization time for flow-induced crystallization, one cannot use conventional DSC, since it does not allow one to impose a flow on the sample. One may use the Linkam shearing hot stage combined with a microscope and a light intensity reader (Koscher and Fulchiron [211]; Hadinata et al. [142]) . In the experiment, the light intensity transmitted through the sample between crossed polarizers is recorded with time. The experiment begins with an initial light intensity I 0 , and is finished when the intensity reaches a stable value I ∞ , indicating that crystallization is completed. The relative crystallinity α is estimated as follows: α(t ) =

I (t ) − I 0 , I∞ − I0

(10.35)

10.2 Flow-Induced Crystallization

153

and the half-crystallization time is estimated as the time giving the intensity I 1/2 (t 1/2 ) =

I∞ − I0 . 2

(10.36)

The variation of morphology through the thickness direction of an injection-molded part can be investigated using synchrotron small-angle and/or wide-angle X-ray techniques. Figure 10.7 shows an example of small-angle X-ray scattering (SAXS) experimental results reported by Zhu and Edward [428] for a sample of iPP. The figure shows 2D SAXS image patterns at different distances from the wall surface to the mid-surface through the sample thickness. At 100 and 200 µ, we can see two distinct maxima, one along the equatorial direction and the other in the meridional direction, indicating the formation of oriented shish-kebab structure. The equatorial and the meridional maximums are attributed to the shish structure parallel to the flow direction and the kebab structure perpendicular to the flow direction, respectively. At 300 µ away from the wall, the equatorial maximum decreases significantly, while the meridional maximum shows little change. At 500 µ, both the equatorial and meridional maxima become very weak. With the distance further increasing toward the mid-surface, the image patterns indicate isotropic structures. Detailed descriptions on characterization of flow-induced crystallization can be found in van Erp [379].

Figure 10.7 Two-dimensional SAXS image patterns at different distances from the skin surface to the mid-surface for an iPP (from Zhu and Edward [428], with permission from American Chemical Society)

11 11.1 11.1.1

Effects of Crystallization on Rheology and Thermal Properties

Effects of Crystallization on Rheology

Viscosity-Enhancement-Factor Model

Injection molding simulation needs to consider both molten flow and solidification of the material. To achieve accurate results in a flow analysis, it is important to predict the solidification layer thickness accurately. There are two ways in which the solidification layer thickness can be determined: 1. Using the concept of a no-flow temperature, which indicates the temperature at which the liquid-solid phase transition takes place. 2. Using a solidification criterion based on viscosity increase: the region where the viscosity increases by one order of magnitude will determine the solidification layer. For amorphous polymers, the no-flow temperature is related to the glass transition temperature T g . The T g is not very dependent on cooling rate and flow. This fact allows the use of a single temperature as a criterion to determine the melt and solid phases. Therefore, the concept of the no-flow temperature provides a useful simplification in the simulation of injection molding for amorphous materials (Section 5.3.5). Solidification behavior of semi-crystalline polymers is more difficult to model. The polymers solidify due to crystallization above the glass transition temperature. The crystallization rate is influenced by both thermal history and flow. It is well known that a semi-crystalline polymer under higher cooling rates solidifies at lower temperatures. It is obvious that, for semicrystalline polymers, there is no single value of a “no-flow temperature” that can be identified. To simulate the solidification behavior, it would be better to use the viscosity increase criterion. Thus, detailed knowledge of material rheology during crystallization is important. To consider the effect of crystallization kinetics on rheology, one needs to build a relationship between the viscosity and the relative crystallinity of the material. The simplest method to construct such a relationship is to introduce an enhancement factor f η (α) into existing constitutive equations. With this idea, Pantani et al. [283] adopted the following modified Cross model: η=

η0 f η (α), ˙ ∗ )1−n 1 + (η 0 γ/τ

(11.1)

156

11 Effects of Crystallization on Rheology and Thermal Properties

where η 0 is the zero-shear-rate viscosity of the amorphous phase, τ∗ is a constant related to the shear stress at the transition between Newtonian and power-law behavior, and n is the power-law index, a measure of the degree of the shear-thinning behavior. The enhancement factor, f η (α) is assumed to have the following empirical form (Titomanlio et al. [365]): ¶ µ β1 f η = 1 + β exp − β , α 2

(11.2)

where β, β1 , and β2 are fitting parameters. Tanner [357] considered the rheology of semi-crystalline polymers in the linear viscoelastic strain regime, based on suspension theory. He proposed using two separate models for low and high concentrations, respectively, and used an interpolation between solutions of the two models to ensure a continuous transition at intermediate volume fractions (crystallinities). If the complex shear moduli at low and high concentrations are G 0∗ and G 1∗ , respectively, the overall complex shear modulus G ∗ is G ∗ (ω, α) = fG (α)G 0∗ (ω) + hG (α)G 1∗ (ω) ,

(11.3)

where ω is the frequency. Tanner [357] determines fG (α) and hG (α) directly by fitting them to the oscillatory shear data of Boutahar et al. [44] for a polypropylene melt containing different volume fractions of spherulites. For flows in the non-linear viscoelastic regime, Tanner and his co-workers [357, 359] proposed using a Phan-Thien-Tanner (PTT) model [286, 293] in the following modified form: λa

¶ µ ∆τ tr τ = 2 f η (α)η 0 D , + τ exp λa ε ∆t f η (α)η 0

(11.4)

where ∆τ/∆t represents the upper convected derivative of τ, the extra stress tensor, defined as △τi j △t

=

∂τi j ∂t

+ uk

∂τi j ∂x k

− L i k τk j − L j k τki ,

(11.5)

ε is a parameter, tr τ is the trace of τ, η 0 is the zero-shear viscosity of the amorphous phase, and D is the rate of deformation tensor. The function f η is expressed as follows, based on the suspension theory of Metzner [248]: fη =

1 , (1 − α/A)2

(11.6)

where A is a parameter representing the geometrical effect. For smooth spheres, A ≈ 0.68. For rough compact crystals, A ≈ 0.44. By varying A, Equation 11.6 can also be applied to nonspherical shapes. Pantani et al. [281] provide a review of more models with several different enhancement factors. Generally, these models predict a sharp upturn of viscosity with increasing relative crystallinity. Some models, such as Equation 11.6, show that viscosity turns up and goes to infinity at reasonably low crystallinity, while others, such as Equation 11.2, show that the viscosity turns up first and then levels off, reaching a finite value.

11.1 Effects of Crystallization on Rheology

11.1.2

157

Two-Phase Model

Doufas et al. [87, 88] proposed a two-phase model in which the semi-crystalline phase is modeled as a rigid dumbbell, and the amorphous phase is modeled using a modified Giesekus constitutive equation. The total extra stress tensor of the system is given by an additive rule: τ = τa + τsc ,

(11.7)

where the subscript “a” stands for the amorphous matrix, and “sc” for the partially crystalline, partially amorphous material inside the spherulites, namely, the semi-crystalline phase. The amorphous extra stress tensor τa is calculated from a modified Giesekus model, with the relaxation time depending on the relative crystallinity as λa (α, T ) = λa (0, T )(1 − α)2 .

(11.8)

Here, the degree of crystallinity is built into the amorphous constitutive equation to account for the loss of chain segments due to crystallization. The semi-crystalline contributed extra stress tensor τsc is calculated from a rigid dumbbell model, given by ¡ ¢ τsc = µ 3〈uu〉 − I + 6λsc D : 〈uuuu〉 (11.9) where µ = n 0 k B T is the melt shear modulus, with n 0 being the number density of the molecules, k B being the Boltzmann’s constant, and T the absolute temperature. λsc is the relaxation time of the semi-crystalline phase, D is the deformation rate tensor, and u is the unit vector of orientation. The angular brackets denote the average with respect to the distribution function of the semi-crystalline phase, and 〈uu〉 and 〈uuuu〉 are the second and fourth orientation tensors, respectively. The changes in the orientation tensor are governed by an equation of the form ´ ³ D〈uu〉 1 λsc − L · 〈uu〉 − 〈uu〉 · LT + 2D : 〈uuuu〉 + 〈uu〉 = I , (11.10) Dt 3 which is exactly the expression of the rigid dumbbell model given by Bird et al. [36]. Zheng and Kennedy [418] followed the approach of the two-phase model. In their work, the semi-crystalline phase is also modeled by the rigid dumbbell model, but the amorphous phase is modeled by a FENE-P model for simplicity. The FENE-P model is a non-linear elastic dumbbell model, in which the dumbbell is constrained to a maximum allowable length. The model is relatively simple, but it captures most important non-linear rheological properties of polymer solutions such as memory effects and shear thinning. Its rheological properties are well known [399]. In Zheng and Kennedy’s approach [418], the effect of crystallinity is not built into the amorphous parameters. Instead, the system is viewed as a suspension of semi-crystalline phase (spherulites or shish-kebabs modeled as rigid rods) in a matrix of amorphous material. The physical properties of the amorphous phase, such as the viscosity η a and the relaxation time λa , are independent of the relative crystallinity, while the properties of the whole system is a function of the relative crystallinity. Let us consider the response of the model when u has a near-equilibrium isotropic distribution. In this case, the leading term of the distribution function that satisfies the Fokker-Planck equation (Equations 4.75 and C.27) in the u space is (4π)−1 . Using the integral theorem described by Brenner [45], one obtains D : 〈uuuu〉 =

2 D, 15

(11.11)

158

11 Effects of Crystallization on Rheology and Thermal Properties

and 2 L · 〈uu〉 + 〈uu〉 · LT = D . 3

(11.12)

From Equation 11.10, assuming steady state (D〈uu〉/D t = 0) and using Equations 11.11 and 11.12, we then obtain 6 3〈uu〉 − I = λsc D . 5

(11.13)

Substituting Equations 11.11 and 11.13 into Equation 11.9 gives τsc = 2µλsc D,

(11.14)

where µλsc should be a function of α. At the limit α → 0, the amorphous phase dominates, and µλsc → 0. At high values of α, the semi-crystalline phase dominates the response. The dependence of µλsc on the relative crystallinity is approximated as follows [418]: µλsc =

η a (α/A)β1 (1 − α/A)β

,

(11.15)

where A is the same as defined in Equation 11.6 and β and β1 are empirical constants. We then write Equation 11.14 as τsc =

2η a (α/A)β1 (1 − α/A)β

(11.16)

D.

In a simple shear flow, Equation 11.16 reduces to τ(sc) 13 =

(α/A)β1 (1 − α/A)β

˙ η a γ,

(11.17)

while the amorphous phase contributed shear stress is given by ˙ τ(a) 13 = η a γ .

(11.18)

The total shear stress is then " (sc) τ13 = τ(a) 13 + τ13

= 1+

(α/A)β1 (1 − α/A)β

# η a γ˙ .

Thus, we can write the shear viscosity function as # " τ13 (α/A)β1 η= = 1+ ηa , γ˙ (1 − α/A)β

(11.19)

(11.20)

which behaves similarly to the suspension model (Equation 11.6). Clearly, all of the above equations apply only for α < A. When α → A, jamming and cessation of flow take place, and the viscosity increases and tends to infinity. However, as pointed out by Tanner [357], the stress law of Equation 11.7 oversimplifies the real picture of structure. The additive rule envisages parallel components of amorphous and crystalline phase at each point. Photographs of crystallizing polypropylene such as those shown by Koscher and Fulchiron [211] do not support this assumption.

11.2 Effect of Crystallization on PVT

11.2

159

Effect of Crystallization on PVT

Consider a representative volume V containing crystalline and amorphous phases. The volume occupied by the crystalline phase is Vc , and the volume occupied by the amorphous phase is Va . Since we only consider a two-phase system, we have V = Vc + V a ,

(11.21)

ρV = ρ c Vc + ρ a Va ,

(11.22)

and

where ρ is the density of the representative volume and ρ c and ρ a are the densities of the crystalline phase and the amorphous phase, respectively. The absolute crystallinity is simply χ = Vc /V , and the volume fraction of the amorphous phase is 1 − χ = Va /V . Therefore, from Equation 11.22 we obtain ρ = ρ c χ + ρ a (1 − χ) .

(11.23)

We may assume that the crystallinity of the solid polymer has reached the ultimate value χ∞ . Then the density of the solid is ρ sc = ρ c χ∞ + ρ a (1 − χ∞ ) .

(11.24)

Recalling that χ = αχ∞ , Equation 11.23 can be recast in the form ρ = ρ sc α + ρ a (1 − α) ,

(11.25)

which can also be re-written in terms of the specific volume: α 1−α 1 = + , Vˆ Vˆsc Vˆa

(11.26)

where Vˆsc ≡ 1/ρ sc , Vˆa ≡ 1/ρ a , and Vˆ ≡ 1/ρ are the specific volumes for the semi-crystalline phase, the amorphous phase, and the total system, respectively. The dependence of Vˆsc and Vˆa on temperature and pressure is given by the Tait equation: · µ ¶¸ p , i = sc, a , (11.27) Vˆi (T, P ) = Vˆ0(i ) (T ) 1 − 0.0894 ln 1 + B i (T ) in which Vˆ0(i ) = b 1(i ) + b 2(i ) (T − b 5 ) , h i B i = b 3(i ) exp −b 4(i ) (T − b 5 ) ,

(11.28) (11.29)

and b 1(i ) to b 4(i ) and b 5 are model constants to be determined by fitting the equation to PVT measurement data. Equation 11.26 takes into account the variation of the PVT transition region with the cooling rate and the material deformation, because of the strong dependence of the crystallization kinetics on these variables. The key point of using Equation 11.26 in simulation is to permit the PVT diagram, which is usually measured under almost steady-state conditions, to vary with the processing.

160

11 Effects of Crystallization on Rheology and Thermal Properties

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

Figure 11.1 displays experimental and modeled results reported by Luyé et al. [234] that show the shift of the PVT transition zone due to the effect of cooling rate. The specific volume equation in the paper of Luyé et al. reads Vˆ = αVˆsc + (1 − α)Vˆa , which apparently differs from Equation 11.26. However, one should notice that the symbol α in the Luyé et al. paper denotes the relative mass crystallinity, and in fact the two expressions are equivalent. For more discussions on this subject, see Zuidema [434] and Forstner et al. [123].

11.3

Effect of Crystallization on Specific Heat Capacity

As described in Chapter 3, the specific heat capacity indicates how much heat is needed to raise the temperature of a unit mass of material by 1◦ C. To simulate the effect of crystallinity on the specific heat capacity, Le Bot [43] and Luyé et al. [234] proposed the following approximation: c p−1 = α−1 c p(s) + (1 − α)−1 c p(a) , with c p(s) = a 1 + a 2 T, c p(a) = b 1 + b 2 T,

(11.30)

11.4 Effect of Crystallization on Thermal Conductivity

161

where c p(s) is the specific heat of the solid phase, and c p(a) is the specific heat of the liquid phase. a 1 , a 2 , b 1 , and b 2 are fitting parameters. In [43, 234], the corresponding equation is written as c p−1 = αc p(s) + (1 − α)c p(a) because the symbol α in those papers is defined as the relative mass crystallinity.

11.4

11.4.1

Effect of Crystallization on Thermal Conductivity

Non-Fourier Thermal Conduction

In the cooling phase of the injection molding process, heat travels from the hotter polymer melt to the colder metal mold, where the polymer melt and the metal mold are in physical contact. This form of heat transfer is known as heat conduction. Heat conduction occurs by diffusion, as the atoms or molecules give over part of their kinetic energy to their neighbors. The rate of heat transfer by conduction per unit length required to alter the temperature of the material by 1◦ C is called the thermal conductivity. A mathematical definition of the thermal conductivity has been given by Equation 3.17. In the classical Fourier law, the flux vector is proportional to the temperature gradient with the thermal conductivity being the coefficient, q = k∇T,

(11.31)

where q is the heat flux vector and k is the thermal conductivity. In Fourier’s classical theory, the thermal conductivity is a scalar that can vary with temperature and pressure. It is obvious that the classical law assumes that the heat flux vector is always parallel to the temperature gradient. The use of thermal conductivity for material thermal calculations in injection molding flow analysis is essential, since it influences the solidification behavior and the development of initial stresses in the material. However, in injection molding processing, polymer melts are highly oriented and show observable physical anisotropy properties due to the oriented structures. The classical form of Fourier’s law is inadequate to describe heat conduction in deforming polymer liquids. Van den Brule [374] suggests that heat transport mechanisms along the backbone of a polymer chain should be more efficient than those between neighboring chains, so that the flow-induced anisotropic orientation of polymer chain segments leads to anisotropic thermal conductivity. To describe anisotropic thermal conduction, Equation 11.31 must be generalized as follows: q = k · ∇T,

(11.32)

where k is the thermal conductivity tensor. If the conductivity coefficient k = k 0 I, with the scalar k 0 only depending on the temperature and/or the pressure, and I being the unit tensor, then the heat flow is said to obey the Fourier law of heat conduction. Any deviation from this will be termed as non-Fourier heat conduction [168]. Thus, the pressure and temperature dependence of thermal conductivity discussed recently by Speight et al. [339] still falls into the

11 Effects of Crystallization on Rheology and Thermal Properties

162

category of the classical Fourier law. On the other hand, the work of van den Brule [374, 375] leads to non-Fourier effects because of the dependence of the conductivity tensor on the flowinduced molecular orientation. Similarly, fiber-filled thermoplastics have anisotropic thermal conductivities, so Equation 11.32 should be used [57, 421].

11.4.2

Van den Brule’s Law for Amorphous Polymers

On the basis of a network theory, van den Brule [374] proposed a linear relationship between the thermal conductivity tensor (k) and the stress tensor (σ) for amorphous polymers: ¸ · 1 1 (11.33) k − tr(k)I = k 0C t σ − tr(σ)I , 3 3 where tr denotes the trace of a tensor, C t is the stress-thermal coefficient, and k 0 is the undisturbed equilibrium scalar conductivity. This stress-thermal rule is analogous to the well known stress-optic rule (Janeschitz-Kriegl [180]). Venerus et al. [383, 384] found that several stressed polymers (polyisobutylene and polysiloxane rubber) gave a nearly universal value for the dimensionless stress-thermal coefficient, C t G N ≈ 0.03, where G N is the melt plateau modulus. The values of C t for these materials are about 1.2 ∼ 1.9 × 10−7 Pa−1 .

11.4.3

Extending the Van den Brule Approach to Semi-Crystalline Polymers

Dai and Tanner [74] measured two components of the thermal diffusivity tensor (parallel and perpendicular to the shear direction, denoted by k ∥ and k ⊥ , respectively) for isotactic polypropylene (iPP) specimens. They had two groups of specimens. The first group were injection molded specimens deformed at high shear rates, and the second group were sheared specimens prepared on the Bohlin-VOR rheometer at low shear rates, and then quenched. The thermal conductivities parallel and perpendicular to the shear direction were measured using modulated differential scanning calorimetry in accordance with ASTM E1952-01. Their experimental results showed that the thermal conductivity of the sheared polymer was anisotropic with an increase in the shear direction; k ∥/k ⊥ > 1. The ratio of k ∥/k ⊥ varied with the applied deformation and stresses. By correlating the changes of thermal conductivity with stress using the stress-thermal rule, Dai and Tanner found that the coefficient C t is O(10−5 ) Pa−1 , about 5 times larger than the value reported by Venerus et al. [383, 384] for amorphous polymers. Dai and Tanner interpret the results as the overall effects of the “van den Brule effect,” and the effect due to structural changes. From the viewpoint of simulation of polymer processing for semi-crystalline polymers, they suggest using the same stress-thermal rule with an enhanced value of C t . Zheng and co-workers [421, 425] have applied this idea in simulation of injection molding. In their simulations, the equilibrium conductivity k 0 is assumed to depend on the relative crystallinity, and can be roughly approximated by the “slab model” as follows: 1 α 1−α = + , k 0 (α, T ) k a (T ) k s (T )

(11.34)

11.4 Effect of Crystallization on Thermal Conductivity

163

where k a (T ) and k s (T ) are the temperature-dependent equilibrium thermal conductivity in melt state and in solid state, respectively. They can be obtained from experiments. Figure 11.2 shows experimental data of thermal conductivity against temperature in both the solid stage and the melt stage for a typical polypropylene. The different curves indicate the experimental repeatability. Equation 11.34 is based on the following assumptions:

Figure 11.2 Undisturbed equilibrium thermal conductivity against temperature for polypropylene (from Speight et al. [339])

1. The amorphous (liquid) component and the solid (semi-crystalline) component lie in series with each other. 2. The heat fluxes through the two constituents are equal: k 0 ∇T = k a ∇T a = k s ∇T s . 3. The temperature gradients in the two constituents are related to the overall average value by ∇T = α∇T s + (1 − α)∇T a . These assumptions are crude. Equation 11.34 is accurate at two extremes, and it predicts the lower bound of the equilibrium thermal conductivity values in the transition zone.

11 Effects of Crystallization on Rheology and Thermal Properties

164

11.5

11.5.1

Effect of Crystallization on Heat Transfer

Stefan’s Solution

Transient heat transfer problems involving a liquid-solid phase change are usually referred to as the “Stefan problem,” raised by Stefan in his early work in 1891 on ice formation in the polar oceans [342]. Here, for injection molding application, we consider a simplified system of two semi-infinite bodies in perfect contact. The first body is the metal mold. The second body is a polymer medium with an initial temperature T initial and crystallization temperature Tc . The interface temperature at the polymer-metal contact surface is denoted by T w , which can be proved to be independent of time. If T w > Tc , the polymer medium will remain liquid, otherwise the polymer medium will solidify. We suppose that there is no difference between the solid and liquid density of the polymer medium. In addition, no convective effects are considered in the liquid phase of the polymer. If the x-axis starts at the mold-polymer contact plane and points to the polymer medium, one has the analytical solution for the temperature of the solid phase in the polymer medium as follows [231]: p T solid (x, t ) − T w erf[x/(2 a solid t )] = , Tc − T w erf(ξ)

(11.35)

and the temperature of the liquid phase is given by T liquid (x, t ) − T initial Tc − T initial

p erfc[x/(2 a liquid t )] , = p erfc(ξ a solid /a liquid )

(11.36)

where a solid and a liquid are the polymer’s thermal diffusivities for the solid phase and the liquid phase, respectively. A thermal diffusivity is defined as a = k/ρc p , where k is the thermal conductivity, ρ is the density, and c p is the specific heat. The parameter ξ is determined from the solution of the following transcendental equation: exp(−ξ2 ) k liquid + solid erf(ξ) k

s

p a liquid Tc − T initial exp(−ξ2 a solid /a liquid ) ξ△Hc π = p a solid Tc − T w erfc(ξ a solid /a liquid ) c psolid (Tc − T w )

(11.37)

where k solid and k solid are thermal conductivities of the polymer for the solid phase and the liquid phase, respectively, △Hc is the latent heat of crystallization, and c psolid is the specific heat of the solid polymer. For more details, see Loulou and Delaunay [231]. The location of the solid-liquid interface in the polymer medium is given by x interface (t ) = 2ξ

p

a solid t .

(11.38)

A fundamental assumption in the Stefan solution is that the solidification occurs at a single temperature. For semi-crystalline polymers, however, the phase change is always associated with crystallization development, and hence the material solidifies over a temperature range, rather than at a single temperature. There are also other important assumptions:

11.5 Effect of Crystallization on Heat Transfer

165

1. None of the media in practical cases is of infinite extent. 2. The effects of convection in injection molding flows may not be negligible. In the next section, we consider the numerical solution of the three-dimensional heat transfer problems.

11.5.2

Numerical Solution with Crystallization Kinetics

By using a heat of crystallization term as the heat source term in Equation 4.67, the equation of heat transfer can be written as µ ¶ Dp ∂q i Dα DT T ∂ρ + = τi j D i j − + ρ△Hc χ∞ , (11.39) ρc p Dt ρ ∂T P D t ∂x i Dt where p is the pressure, τi j D i j is the heat due to viscous dissipation with τi j being the extra stress tensor and D i j being the deformation rate tensor, q i is the heat flux vector, △Hc is the latent heat of crystallization (in J/kg) for perfect crystals, Dα /Dt is the rate of relative crystallization given by the crystallization kinetics described above, and χ∞ is the ultimate absolute crystallinity.

Figure 11.3 Temperature evolution at the core region of an injection molded part

A numerically calculated temperature evolution at the core region of an injection molded iPP part is presented in Figure 11.3. One can notice that a plateau-like transition zone appears on the temperature curve at 14–16 s, where the corresponding temperature is about 120◦ C. The existence of the plateau is caused by the release of the crystallization latent heat, which compensates for the heat taken away by the cold mold. The plateau temperature is the crystallization temperature at which the phase transformation from melt to solid takes place.

166

11 Effects of Crystallization on Rheology and Thermal Properties

11.6

Modification to the Hele-Shaw Equation

Chapter 5 showed, in detail, how to derive a 2D pressure equation (Equation 5.71, known as the Hele-Shaw equation) for a generalized Newtonian fluid flow in a thin cavity of arbitrary geometry. In order to incorporate the evolution of the relevant crystallinity in the Hele-Shaw equation, we should note that the fluid density is not only a function of pressure and temperature, but also a function of the relative crystallinity α, so that ¶ ¶ ¶ µ µ µ Dp DT Dα ∂ρ ∂ρ ∂ρ Dρ = + + . Dt ∂p T,α D t ∂T p,α D t ∂α T,p D t

(11.40)

Following the same derivation procedure as described in Chapter 5, we obtain ∂ ³ ∂p ´ ∂ ³ ∂p ´ ∂p 2S 2 + 2S 2 = a(x, y) − b(x, y) ∂x ∂x ∂y ∂y ∂t ¶ µ³ Z H ∂p ´2 ³ ∂p ´2 1 ∂ρ Dα + + dz , + d (x, y) ∂x ∂y ρ ∂α D t −H

(11.41)

where S 2 , a(x, y), b(x, y), and d (x, y) are the same as used in Equation 5.71 in Chapter 5, but the last term on the right-hand side of the above equation is an extra term not appearing in Equation 5.71. If we use ρ = ρ sc α + ρ a (1 − α), namely Equation 11.25, to approximate the density ρ, we have ∂ρ = ρ sc − ρ a . ∂α The term Dα/D t is determined by the crystallization kinetics.

(11.42)

12 12.1

Colorant Effects on Crystallization and Shrinkage

Introduction

In injection molding, various colorants are typically added to the virgin polymer to produce colorful products, ranging from toys and mobile phones to computer cases and car parts. Colorants can be either soluble organic dyes or insoluble particulate pigments. While the former are often used to color textiles, the latter are of more importance in the plastics industry. The primary purpose of colorants added to polymers is to enhance their cosmetic appeal and appearance. However, they also have other effects. These tiny pigment particles may act as nucleating agents that change the crystallization rate and morphology of the polymer, and thus further affect the rheological and thermal properties of the melt, as well as mechanical properties and dimensional stability of the finished product. Pigment particles come in a variety of forms, with primary particle sizes ranging from 0.01 µm to 1 µm in diameter, and with varying degrees of combination into aggregates or agglomerates where the tightly packed aggregates reduce the pigment surface area for a given loading, whereas the loosely packed agglomerates do not affect the surface area exposed when compared to that of the primary particles. The shape and size of pigment particles play a major role in determining both the melt properties and crystallization processes in a polymer. Despite the fact that various colors are typically added to mold a part, numerical analysis of injection molding often inputs data obtained for virgin material. This leads to the simulation being of poor accuracy for colored products. In order to develop theoretical models to predict the properties and crystallization kinetics of polymers in the presence of colorants, several recent investigations have been devoted to the subject. In the studies of Hadinata et al. [142], Zhu et al. [429], Lee Wo and Tanner [404], and Zheng et al. [425], the virgin polymer used is an isotactic polypropylene (grade HD601CF supplied by Borealis), which is compounded with an inorganic blue pigment and an organic blue pigment, respectively. The inorganic colorant is Sodium Alumino Sulpho Silicate (commercial name: Ultramarine Blue), referred to as UB, which is a three-dimensional aluminosilicate lattice with entrapped sodium ions and ionic sulfur groups, having a light blue color and a spherical surface full of holes of a uniform size as shown in Figure 12.1(a). The organic colorant is Cu-Phthalocyanine (commercial name: PV Fast Blue), referred to as CuPc, a macrocyclic compound with an alternative nitrogen atom-carbon atom ring structure, as shown in Figure 12.1(b), having a dark blue color and a flat surface full of narrow cracks of nonuniform sizes. The pigments were mixed into the virgin iPP with various concentrations of 0.2%, 0.5%, and 0.8% by weight. Ma et al. [235] have also considered a similar system (the iPP with 0.2% of CuPc by weight).

12 Colorant Effects

168

Figure 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)

It is suggested [425] that the basic kinetic models (the Kolmogoroff-Avrami-Evans model, the Hoffman-Lauritzen model, and the nucleation model) as described in previous chapters could be applied to the colored system as well, if the model parameters have been determined for each combination of the additive and the base polymer.

12.2 12.2.1

Material Characterization

Morphology

Figures 12.2 (a)–(c) show the morphologies of the three samples: the virgin iPP, the UB-colored iPP and the CuPc-colored iPP, respectively. For the virgin iPP and the UB-colored iPP, images were captured at temperature T = 135◦ C at time t = 180 s. However, for the CuPc-colored iPP at the same temperature, the crystallization occurred too fast for the growth of nuclei to be captured with optical microscopy, and therefore a higher temperature (T = 140◦ C) had to be used. While the UB pigment increased the number of spherulites per unit volume only slightly, the CuPc pigment significantly increased the nuclei density, causing reduction in the spherulite size. Lee Wo and Tanner [404] also reported observations of non-spherical crystallites in the CuPc-colored iPP that were not found in either the virgin iPP or the UB-colored iPP. The images of crystalline structures obtained from the microscopy are important for measurements of growth rate and nuclei number density. However, this direct measurement may become difficult because the crystallization occurs too quickly, and/or the colorants or the large number of nuclei reduce the visibility of the images.

12.2 Material Characterization

169

Figure 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

There is a workaround if the growth rate is independent of the nuclei number density, even when the nuclei number density is influenced by the colorant and by flow [235]. In this case, the growth rate can be estimated from the experimental results of the virgin polymer sample, or from the empirical equation of van Krevelen (Equation 10.23). Then, the nuclei number density can be calculated from the Kolmogoroff-Avrami-Evans equation, given the data of the relative crystallinity, or the half-crystallization time. For further detailed microstructural analysis using the Wide Angle X-ray Scattering (WAXS) technique for the CuPc-colored iPP, see Zhu et al. [430].

12.2.2

Specific Heat

Figure 12.3 shows the DSC traces of the specific heat of samples of the virgin iPP, UB-colored iPP, and CuPc-colored iPP [425]. The concentrations of the colorants are 0.8% by weight in both cases. The figure shows that the temperature points of the peaks of the virgin iPP and the UB-colored iPP are 113◦ C and 118◦ C, respectively, varying by 5◦ C, while the CuPc-colored iPP has a peak at 129◦ C, which is 16◦ C above that of the virgin iPP.

12.2.3

Half-Crystallization Time

12.2.3.1

Quiescent Crystallization

The quiescent half-crystallization times of the colored polymer samples are compared with that of the virgin polymer sample in Figure 12.4. The UB colorant shows very slight influence on the crystallization speed, but the CuPc colorant dramatically reduces the half crystallization time by one order of magnitude.

12.2.3.2

Flow-Induced Crystallization

Figures 12.5, 12.6, and 12.7 display the short-term shearing experimental results for the change in the half-crystallization times with the shear rates for the three samples of the virgin iPP, the

170

12 Colorant Effects

Figure 12.3 Specific heat capacities of three samples: virgin 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])

Figure 12.4 Half-crystallization time vs. crystallization temperature of three samples: virgin iPP, UBcolored 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])

UB-colored iPP and the CuPc-colored iPP, in each case, for two temperatures. The experiments were performed on a Linkam shearing hot stage with shearing time 1 sec. The CuPc-colored iPP was observed to crystallize so fast that higher temperatures had to be chosen for the tests

12.3 Effect on Shrinkage

171

to obtain accurate half-crystallization times. For the virgin iPP and the UB-colored iPP, there was no significant difference within the low shear rate region. At the high shear rate region, the flow effect was slightly enhanced by the UB pigment.

Figure 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])

12.3

Effect on Shrinkage

Experience shows that colorants may influence the dimensional instability of injection molded articles. It is not uncommon for colored plastic articles that can be assembled well with a specific colorant, fail to be assembled just because another colorant is used, even if the molding conditions are the same. As far as the semi-crystalline polymers are concerned, the possible causes of the influence are: ■

■

■

Different types of colorants have different effects on crystallization speeds, causing different solidification behavior and leading to different degrees of shrinkages throughout the part. As seen in Figures 12.2 (a)–(c), colorant type also influences the coarseness of the spherulites. Coarser spherulites appear less homogeneous than the fine-spherulite structure. This affects the uniformity of the shrinkage distribution. Some types of colorants encourage the formation of an oriented crystalline structure, while others do not. Oriented morphology will cause anisotropic shrinkage.

If a careful set of crystallization kinetic parameters were available for a specific colored polymer, it would be possible to predict the influence on shrinkage by colorants for this mate-

172

12 Colorant Effects

Figure 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])

Figure 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])

rial. The predicted shrinkage results (parallel and perpendicular to flow) obtained by Zheng et al. [425] are shown in Figures 12.8, 12.9, and 12.10, respectively, for the virgin iPP, the UBcolored iPP, and the CuPc-colored iPP. Experimentally measured data are also plotted for comparison. The simulated experimental samples are injection molded rectangular plates, 200 mm long and 40 mm wide, with thicknesses 1.7 mm, 2 mm, 3 mm, and 5 mm. Typically,

12.3 Effect on Shrinkage

173

Figure 12.8 Experimental and predicted parallel and perpendicular shrinkage for the virgin iPP (Zheng et al. [425])

28 sets of processing conditions are used to mold the samples, as mentioned previously in Part 1. Processing conditions are designed by varying sample thickness, injection flow rate, packing pressure, hold time, and cooling time. Refer to Kennedy and Zheng [200] for more details. The undulation of the data in the figures reflects the influence of the processing condition sets. Comparisons between the predictions and experiments are promising. The overall trends

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

174

12 Colorant Effects

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

are well predicted, although the agreement in magnitude is not entirely satisfactory for a few processing conditions. Comparing the three figures, the CuPc-colored iPP shows a noticeably higher degree of anisotropy, indicated by the different levels of the parallel shrinkage and the perpendicular shrinkage. Costa et al. [68] have used the same method to study the anisotropic shrinkage behavior of injection molded polybutylene terephthalate (PBT) parts.

13 13.1

Toward the Prediction of Post-Molding Shrinkage and Warpage

Introduction

Successful manufacture of injection molded products requires not only meeting the designed dimensional tolerances, but also having a long-term dimensional stability under conditions at which the product is subjected to post-molding thermal treatments. It would be useful to clarify the term “shrinkage,” since the same term may have different meanings when used in different situations. In this case, we have two types of shrinkage. The first is the geometric reduction in size of the fresh part compared to the dimensions of the mold cavity. This type of product dimension change is referred to as “mold shrinkage,” “processing shrinkage,” “as-molded shrinkage,” or simply “shrinkage,” often measured 24–48 hours after ejection of the part (ASTM D99508) [284], as defined in Section 4.8. In fact, dimensions of the products may continue to change slowly during storage due to stress relaxation, physical aging, and/or recrystallization. Although in principle the creep model as discussed by Caspers [51] has enabled the prediction of the transient dimensional change for amorphous materials, the effects of recrystallization of semi-crystalline materials are yet to be investigated. The second type of shrinkage is the dimensional change of a product after thermal treatments. A practical example is the baking process for painted automotive plastic moldings, in which the injection molded parts of known dimensions are placed in an oven under programmed temperature conditions for a certain period of time. In this process, shrinkage is measured by comparing the dimensions of a product before and after the thermal treatment. We call this type of shrinkage the post-molding shrinkage. The post-molding shrinkage, usually not uniform, may cause post-molding warpage as well. There is still a lack of fundamental understanding about the mechanisms of dimensional instabilities occurring during the post-molding thermal treatment. As a starting point for further investigations, Fan et al. [108, 109] have considered the following two possible causes: ■

■

The stress relaxation and creep recovery of the polymer during thermal treatments due to the viscoelasticity of the polymer. The hysteresis loop behavior of thermal expansion due to microstructure changes in semicrystalline polymers during thermal treatments.

The simulation method proposed by Fan et al. [108, 109] that is based on the above assumptions will be described in the following sections.

176

13 Prediction of Post-Molding Shrinkage and Warpage

13.2

Governing Equations

To simulate the post-molding shrinkage and warpage, one needs to solve a problem of stress analysis with heat transfer. The stress and deformation are described by the force balance equation ∇ · σ = 0,

(13.1)

where σ is the Cauchy stress tensor, which is usually a nonlinear function of the mechanical and thermal history of the material. Here we neglect body forces. The heat transfer is governed by the heat conduction equation ρc P

∂T = ∇ · (k∇T ), ∂t

(13.2)

where ρ, c p , and k are the density, specific heat, and thermal conductivity of the material, respectively. If we consider here only the thin walled parts and choose to use shell elements, Equation 13.1 can be simplified to the equivalent 2D ones: ∂N xx ∂N x y + =0 ∂x ∂y

(13.3)

and ∂N x y ∂x

+

∂N y y ∂y

= 0,

(13.4)

where x and y are the local in-plane coordinates in the midplane of the thin structure, with the z direction along the normal of the midplane (the thickness direction). Ni j are the resultant force components per unit length, given by h

Z Ni j =

−h

σi j d z,

(13.5)

where h is the half thickness of the part. The out-of-plane response is governed by the following equation: ∂2 M x y ∂2 M y y ∂2 M xx +2 + = 0, 2 ∂x ∂x∂y ∂y 2

(13.6)

where M i j are resultant bending moment components per unit length, defined as Z Mi j =

h

−h

σi j zd z.

(13.7)

For thin parts, the heat conduction across the thickness is much larger than the conduction along the larger dimensions of the part. Hence, the heat conduction equation reduces to µ ¶ ∂T ∂ ∂T ρc p = k . ∂t ∂z ∂z

(13.8)

13.3 Constitutive Equations

13.3 13.3.1

177

Constitutive Equations

Viscoelastic Effect

For the sake of simplicity, we assume that, during the thermal treatment, the material behaves as a thermo-rheologically simple solid, and the strain increment would be sufficiently small, so that the a linear viscoelastic constitutive equation is applicable. Initially, the injection molded part in equilibrium is usually in a state of stress. Thus, the constitutive equation can be written as Z t i ∂ h th (x, t ′ ) d t ′ (13.9) σi j (x, t ) − σ0i j (x) = C i j kl (ξ(x, t ) − ξ(x, t ′ )) ′ εkl (x, t ′ ) − εkl ∂t 0 where σi j is the total stress tensor, σ0i j is the initial stress, εi j and εit hj are the total strain and the thermal strain, respectively, and ξ is a pseudo-time scale. We have already shown in Section 5.9.1.2 that the use of pseudo-time is a useful approach to account for the effect of temperature on material response, and it is related to the actual time as follows: ξ=

t

Z 0

dt′ dξ 1 , = , a T (T (x, t ′ )) d t a T (T (x, t ))

(13.10)

where a T is the time-temperature shift function, often characterized by the WLF equation or by the Andrade equation, as discussed in Chapter 3. If only isotropic materials are considered, C i j kl can be expressed by µ C i j kl =

¶ 2ν δi j δkl + δi k δ j l + δi l δ j k G(t ), 1 − 2ν

(13.11)

where δi j is the Kronecker delta, ν is the Poisson’s ratio, and G(t ) is the shear relaxation function given in terms of the Prony series: G(t ) = G 0 ϕ(t ),

(13.12)

with " ϕ(t ) = ϕ g ∞ +

m X

# g i exp(−t /λi ) ,

i =1

where G 0 is the shear modulus, related to the Young’s modulus through E /[2(1+ν)], g ∞ are the long-term dimensionless modulus, λi are the relaxation times, and g i is the relative dimensionless modulus satisfying g∞ +

m X

g i = 1.

(13.13)

i =1

For isotropic materials we also have 1 th εkl = εtvh δkl , 3

(13.14)

13 Prediction of Post-Molding Shrinkage and Warpage

178

where εtvh is called the volumetric thermal strain. By substituting Equations 13.11 and 13.14 into Equation 13.9, we obtain Z t i ∂ h σi j (x, t ) − σ0i j (x) = δi j K (ξ(x, t ) − ξ(x, t ′ )) ′ εkk (x, t ′ ) − εtvh (x, t ′ ) d t ′ ∂t 0 Z t ∂εdij (x, t ′ ) +2 G(ξ(x, t ) − ξ(x, t ′ )) dt′ , (13.15) ∂t ′ 0 where εdij is the component of the deviatoric strain tensor defined as 1 εdij = εi j − εkk δi j , 3

(13.16)

and K (t ) is given by K (t ) = K 0 ϕ(t ) ,

(13.17)

where K 0 = E /[3(1−2ν)] is the bulk modulus, and we have assumed that K (t ) and G(t ) depend on the same relaxation function ϕ(t ). The thermo-rheological simplicity assumption demands that all the molecular mechanisms involved in the relaxation process have the same temperature dependence. Only for limited materials this assumption is valid. As mentioned previously, a large number of real materials are actually thermo-rheologically complex. To deal with thermo-rheologically complex materials, see Dutta and Edward [90].

13.3.2

Thermal Expansion Effect

From Equations 13.9 and 13.15 we see that the stress depends on the thermal strain history. Fan et al. [108, 109] have shown that a thermal expansion hysteresis loop occurs, so that the curves of specific volume variation in heating and cooling processes can be different. They measured the specific volume of samples made from injection-molded polypropylene using a DMTA (Dynamic Mechanical-Thermo Analysis) device. Controlled heating and cooling of the samples were performed. The samples were heated from room temperature (20◦ C) to a maximum temperature of 69◦ C, 79◦ C, 89◦ C, and 109◦ C, respectively. Then, the temperature was held at the maximum value for 90 min before cooling the samples from the maximum temperature down to room temperature. It was noted that the cooling curves of the specific volume vs. temperature deviated from the heating curves. The difference of the specific volumes between heating and cooling at the same temperature increases with the maximum temperature. It was also found that, when the maximum temperature was lower than a critical value, the difference between heating and cooling curves becomes negligible. As a first attempt to take into consideration the thermal behavior in post-molding shrinkage and warpage simulations, Fan et al. [108, 109] attributed this phenomenon to microstructural changes due to the secondary crystallization (Huang [165]) during the post-molding thermal treatment, and used a semi-empirical approach to describe the thermal expansion behavior. First, the crystallinity due to the secondary crystallization was evaluated by ´o n h i ³ T −T g 0.05(T −T )2 χ0 1 +C 1 − (T −T )k2 log 1 + τ1˜ + ia˜τ˜ g k h i ³ ´ , (13.18) χ= T −T g 0.05(Ti −Tk )2 1 +C 1 − (T −T log 1 + ia˜τ˜ )2 g

k

13.3 Constitutive Equations

179

where χ0 is the initial crystallinity, Ti and T g are the initial and glass transition temperatures, τ˜ is a constant with the dimension of time, a˜ is a virtual rate of temperature change, which is chosen to be the initial temperature rate of the thermal treatment, and Tk = (T g + Tm )/2, with Tm being the melt temperature. The specific volume is then calculated by −1 v −1 = χv s−1 + (1 − χ)v m ,

(13.19)

where v s and v m are the specific volume of solid and melt of the polymer, respectively. Finally, the volumetric thermal strain in Equation 13.15 at time t may be given by εtvh =

v − v0 , v0

where v 0 is the initial specific volume of the material.

(13.20)

14 14.1

Additional Issues of Injection-Molding Simulation

Weldlines

The formation of weldlines is undesirable in injection molding, since it results in poor appearance and poor mechanical strength of injection-molded products. A weldline is formed due to the impingement of two melt fronts. This occurs when the cavity has multi-gates, or when there are inserts in the cavity. When two melt fronts meet, initially the molecular chains at the melt fronts are parallel to the interface induced by the fountain flow, so that no molecular chains bridge the interface. Then diffusion occurs and causes molecular chains to move across the interface, gradually randomizing the molecular orientation and bridging the interface with long-chained molecules. For homogeneous polymers, the bonding strength at the weldline depends on whether the polymer chains have enough time to diffuse across the interface to form a strong bond. If the melt solidifies before a sufficient number of molecular chains cross the interface, the poor intermolecular diffusion results in a certain degree of incomplete bonding. The weakness of the weldline also arises from the existence of V-notches at the surface around the weldline, caused by the entrapped air which is forced to the wall. Weldlines cannot be eliminated,therefore it is desirable to predict the location and strength of the weldline in the flow analysis. The mold, particularly gate locations, can then be redesigned to position weldlines in the least sensitive regions, considering both aesthetic and structural demands. The locations of weldlines can be predicted in the filling analysis by first finding the initial meeting point of two flow fronts and then expanding from the initial meeting point to obtain the entire weldline (Zhou and Li [426]). Weldline location prediction would be sensitive to mesh density. There have been attempts to quantify the weldline strength for homogeneous polymers. Let A 0 be the initial cross-sectional area of the interface, and A(t ) be the unbounded area where there is no molecular chain across the interface at time t . So the difference (A 0 − A(t )) represents the bonded area at time t . According to Kim and Suh [202], the degree of bonding is given by σw A 0 − A(t ) = , σb A0

(14.1)

at the instant when the melt temperature drops below T g . In the expression, σw and σb are the tensile strength with and without weldline, respectively. To obtain the value of A, Kim and Suh further proposed that the time variation rate of the non-bounded area, d A/d t , can be calculated by Fick’s first law of diffusion.

14 Additional Issues of Injection-Molding Simulation

182

This diffusion model is not adequate for fiber reinforced polymers because the flow-induced fiber orientation at the weldline interface does not relax, and usually remains parallel to the weldline, leading to a serious reduction in the weldline strength. The loss of strength is found to be related to the aspect ratio of the fibers. It has also been found in experiments that values of the weldline strength of fiber reinforced composites are close to the matrix strength (Meddad and Fesa [244]).

14.2

Core Shift

A core is the part of a mold that shapes the inside of a molded part. In Chapter 4, we mentioned the issue of long cores. Molds having long cantilevered cores are used to manufacture slender hollow parts with a closed end. When the pressure distribution over the surface of the core during the filling and packing phases is nonuniform, such a core may deflect and misshape the moldings. The spatial deviation of the position of the core is called core shift. The simulation of the core shift phenomenon requires solving a liquid-solid interaction problem. A structural analysis of the core is coupled with the plastic flow analysis (Bakharev et al. [24]). This is a two-way coupling. The fluid pressure exerts a net lateral force on the solid core surface, causing the core to deflect; and, in turn, the displacement of the solid core has an impact on the fluid flow. On the interface of the melt and the core, one applies both the no-penetration condition (which prevents fluid particles from penetrating the solid object) and the no-slip boundary conditions. If we denote the fluid velocity field by v and the core displacement field by u, the ˙ = v. no-penetration and the no-slip boundary conditions simply require u To simulate the geometric change of the cavity due to the deformation of the core, remeshing is required during the simulation. The simplest way is to update only the positions of the nodes at the fluid-core interface based on the calculated nodal displacement. For better numerical performance, one may use the arbitrary Lagrangian-Eulerian (ALE) method (Hughes et al. [166]).

14.3 14.3.1

Non-Conventional Injection Molds

Overmolding

Overmolding is a multi-component consequential molding process in which one polymer material is molded (or partially molded) onto a previously molded and solidified part of a different polymer material. This process extends the capability of conventional injection molding processes to produce multi-functional parts with a variety of colors. Here, we only consider the case of two components. The two-component injection molding process can be divided into three stages: (i) filling and post-filling of the first component (substrate), (ii) switchover of the

14.3 Non-Conventional Injection Molds

183

core to create a new cavity for the second component (overmold), (iii) filling and post-filling of the overmold. Accordingly, the numerical analysis procedure consists of two steps. First, the filling and postfilling analysis is performed on the first component. This step is in fact the same as the conventional injection molding flow analysis. In the second step, the filling and post-filling analysis is performed for the second component. Because part of the cavity wall is made up by the substrate for the second shot, the second-step analysis has to take into account the boundary conditions on the interface between the polymer melt of the second shot and the solidified polymer substrate. The initial temperature of the substrate usually is not uniform and can be determined by the first-step analysis. For the sake of simplicity, one also needs to made some assumptions. For example, Ray and Costa [306] assumed that, during the overmolding stage, the first component will not remelt (although the approach described earlier by Caspers [51] could allow remelting), will be rigid, will have a perfect adhesion with the second component, and will keep a good thermal contact with the mold. In the second-step analysis, the heat exchanges between the first component and the mold, and between the first component and the second polymer melt, need to be evaluated. In numerical simulations, the first component and the cavity for the second shot are often meshed separately, so that the nodes at the cavity and the substrate interface may not be matched. The temperature of the cavity and the substrate should be solved iteratively, together with mapping both temperatures and fluxes across the interface, until the temperature differences at both sides of the interface are smaller than a required converging criterion. Caspers [51] found that the thermally and pressure-induced residual stresses, computed for two-shot injection molded parts, are highly anisotropic, and this would often lead to severe warpage. For warpage analysis of over-molded parts, a special issue is how to establish relationships between degrees of freedom at the nodes in both sides of the interface of the first and second components. In the approach developed by Fan et al. [110], typical multi-point constraint (MPC) equations are written to establish the relationships, and then an elimination method based on the Lagrange multiplier formulations is used for dealing with MPC equations.

14.3.2

Gas-Assisted Injection Molding

Gas-assisted injection molding (GAIM) is derived from the conventional injection molding process to produce plastic parts with a hollow core. The GAIM process begins with a partial injection of polymer melt into the mold cavity. Compressed gas, usually nitrogen, is then injected under pressure into the core of the polymer melt to complete the filling and packing of the mold. The gas typically penetrates along the path of least flow resistance. Once the polymer melt has solidified, the pressurized gas is released just before opening the mold and ejecting the part. The GAIM can produce lower and more uniformly distributed cavity pressure, compared to the conventional injection molding process (Figure 14.1). This contributes to several advantages over the technology of conventional injection molding such as reduced clamping force, lower residual stresses, and less tendency to warp. Other advantages include reduced cycle time, reduced part weight, improved surface smoothness, lack of sink marks, and flexibility in

184

14 Additional Issues of Injection-Molding Simulation

Figure 14.1 Pressure development during filling in (a) conventional injection molding and (b) gasinjection molding (adapted from Turng [373])

part design, such as the consolidation with both thick and thin sections. Compared with conventional injection molding, the GAIM involves additional process parameters associated with gas, such as gas injection delay time, gas pressure, and gas hold time. As for the conventional injection molding process, modeling of the GAIM process can be done either using 2.5D HeleShaw approximation (Sherbelis [332], Turng [373], and Chen et al. [59]) or 3D models (Haagh et al. [138, 139]). The 3D simulation is able to capture the kinematics of the flow front and the form of the final cross-section, which the 2.5D approach is unable to do. We shall only review the 3D approach below. For 3D modeling, the pseudo-concentration approach proposed by Haagh et al. [139] allows us to solve the flow problem on a fixed finite element mesh that covers the entire cavity, without the need for re-meshing during the simulation. Both the injected gas and the air downstream of the polymer front are represented by a fictitious fluid that does not contribute to the pressure drop in the mold. The viscosity of fictitious fluid is set to be of order 10−3 of the polymer viscosity. This value is higher than the real gas viscosity to keep the Reynolds number small, while it is still low enough for the gas pressure drop to be negligibly small compared to the pressure drop in an equivalent molten polymer core. Consequently, the pressure can be considered constant throughout the gas core. The fictitious fluid has the same density as the gas.

14.3 Non-Conventional Injection Molds

185

As a result, the flow problem can be described by the Stokes equation for the entire computation domain including both polymer and the fictitious fluid ∇ · σ = 0,

(14.2)

where σ is the total stress. Here, the inertia effect and the body forces have been neglected. The conservation of mass and energy equations are also applied. Boundary conditions are prescribed at the mold entrance, the mold wall, and the air vents. At the mold entrance, either the injection flow rate or the injection pressure is given. At the mold wall, if it is covered with polymer, a no-slip condition is prescribed, while a no-stick condition is imposed at the air-wall interface downstream of the flow front. At the air vent boundary, the traction normal to the wall is set to zero. Material properties are functions of the type of material. The distinction between polymer and gas is made by labeling fluid particles with the pseudo-concentration parameter c, where c = 1 for the polymer and c = 0 for gas. Local density and viscosity are calculated as ρ = cρ polymer + (1 − c)ρ fictitious ,

(14.3)

η = cη polymer + (1 − c)η fictitious ,

(14.4)

or, they can be defined as discontinuous functions of the concentration: ( ρ=

ρ polymer

if

c ≥0.5

ρ fictitious

if

c 0, the disturbance will grow in time, indicating an unstable flow. Thus, the case Re(σ)= 0 signals the onset of instability. A measure of the viscoelasticity of the melt is the Weissenberg number defined as [356] Wi =λ

U , H

(14.27)

where λ is the characteristic time of the fluid, U is the characteristic velocity, usually taken as the average velocity of the melt, and H is the characteristic length scale, usually taken as the half gap width of the cavity in injection molding simulations. Bogaerds et al. [40] investigated the linear stability of a viscoelastic fluid modeled by an extended Pom-Pom constitutive equation [358,386] for various model parameters and Weissenberg numbers, and predicted the critical conditions in terms of the Weissenberg number beyond which the flow becomes unstable.

14.4.3

Viscoelastic Suspensions

In previous chapters, we mentioned suspension theory and the importance of the suspension theory on studies of flows of fiber-filled polymers and crystallizing polymers in the injection molding process. A suspension is a heterogeneous mixture made up of solid particles suspended in a liquid. The size of the particles is several orders of magnitudes smaller than the typical size of the apparatus. Most suspension theories applied in injection molding have considered Newtonian suspensions, or suspensions of particles in a Newtonian liquid, although in practice particles are usually in viscoelastic matrices. Hwang et al. [171] simulated particle suspensions in simple shear flows of Oldroyd-B fluids. They predicted the common experimental observation of the scaling of the first normal stress to the shear stress. Furthermore, they showed that both the shear viscosity and the first normal stress coefficient increase with the generalized shear rate, as well as with the volume fraction of particles. However, the model is not sufficient to give the correct behavior for viscoelastic

14.4 Viscoelastic Effects

195

suspensions in some aspects. For instance, the experimental study of Mall-Gleissle et al. [237] showed that the first normal stress difference is positive in magnitude but decreases with increasing volume fraction of the particles, while the second normal stress difference is negative with a magnitude that increases with increasing volume fraction of the particles. These were not predicted by the model of Hwang et al., because the Oldroyd-B fluid does not predict a second normal stress difference in planar shear flow. Using a different approach, Tanner and Qi [360] developed a phenomenological nonlinear viscoelastic suspension model. The stress tensor in the model consists of two modes. One is described by a PTT model with volume fraction dependence of the relaxation time: λ(φ)

∆τv + F (trτv )τv = 2η p D, ∆t

(14.28)

where λ is a time constant as a function of the volume fraction φ, ∆/∆t is the upper convected time derivative, τv is the extra-stress term, with the suffix v denoting “viscoelastic” behavior, term η P is a polymer viscosity, F is a function of trτv and D is the rate of deformation tensor. The other is described by a Reiner-Rivlin model with a volume fraction-dependent viscosity: µ ¶ β∗(φ) D τN = 2η 0 (µr − 1) I − ·D (14.29) γ˙ where τN is the extra-stress term, with the suffix N denoting “Newtonian” behavior, η 0 is the Newtonian solvent viscosity, µr is the relative viscosity that could be described by µr = (1 − φ/φm )−2 , with φm being the maximum value of allowable volume fraction, β∗ is a volume fraction dependent parameter, and γ˙ the generalized shear rate. The extra stress of the suspension is proposed to be the simple addition of the two kinds of stress terms: τ = τN + τv .

(14.30)

The complete response of this model shows the following features: (i) a positive first normal stress difference, (ii) a negative second normal stress difference, (iii) the dimensionless first normal stress difference strongly dependent on the shear rate and decreasing with volume fraction, (iv) the dimensionless second normal stress difference (in magnitude) nearly independent of the shear rate and increasing with the volume fraction. Computed results are in good agreement with experimental data [237]. Housiadas and Tanner [159] used a perturbation analysis to obtain the analytical solution for the pressure and velocity of a dilute suspension of rigid spheres in a weakly viscoelastic fluid. They also show that one can extend the dilute suspension results to concentrated regimes by using the Roscoe procedure [313] as follows: 1. Find the response of a dilute suspension. 2. Assume that a small amount of additional particles are added to the existing suspension, which will enhance the viscosity of the suspension. Replace d φ by the effective increase of concentration using a crowding function, such as d φ/(1 − φ/φm ). 3. By integration, one then finds the properties for finite concentrations. Phan-Thien and Fan [288] and Fan et al. [102] considered the slender particles in viscoelastic fluids. Their work is relevant to the fiber orientation problem. First, they used a boundary element method to simulate a single slender particle motion in a weakly viscoelastic shear flow,

14 Additional Issues of Injection-Molding Simulation

196

and showed how the particle rotates and leaves Jeffery’s orbit due to the viscoelastic effect. Based on the numerical information, they then introduced a modified effective velocity gradient into Jeffery’s equation to take into account the effect of weak elasticity in flow. The modified model was solved by a Brownian dynamics simulation method, and tested against the boundary element result. Furthermore, the modified model is extended to include the interactions between particles by adding a random force into the equation. The resulting evolution equation for the orientation tensor 〈pp〉 is D ˜ · 〈pp〉 + 〈pp〉 · L ˜ − 2L ˜ : 〈pppp〉 + 2C I γ(I ˙ − 3〈pp〉) 〈pp〉 = L Dt

(14.31)

˜ is given by where the modified effective velocity gradient L s

" ˜ = L− 1− L

1−

4 − β2 a r2 (a r + a r−1 )2

# D−

4cD · D γ˙

(14.32)

where L is the velocity gradient and β and c are parameters depending on viscoelasticity contribution. When β = 0 and c = 0, the model reduces to the standard Folgar-Tucker equation. Other relevant studies can be found in References [4, 20, 101, 160, 361].

14.5

Other Numerical Methods

In addition to the traditional grid-based numerical methods such as finite difference methods, finite element methods, and finite volume methods, other methods have also been used in the studies of injection molding problems, including molecular dynamics (MD) simulation and meshless methods. They should be considered to be complements of the traditional methods for some special purposes, rather than replacements of the traditional methods.

14.5.1

Molecular Dynamics Simulation

We have known that the processability and ultimate properties of materials used for injection molding depend on the nano/micro structure developed during processing. Molecular level simulations may help to understand the molecular origin of the phenomena and the relationship between the processing and the microstructure and the macro-properties. Molecular dynamics (MD) simulation is among the molecular-scale simulation methods that integrate Newton’s equations of motion for a set of molecules (Allen and Tildesley [7]). The thermal energy in an MD simulation is just the average kinetic energy of the atoms. In the MD simulation, the equations are deterministic, and the Brownian motion of the molecules is produced by a direct simulation of a huge number of intermolecular collisions, very similar to the case in real fluids. It is this feature that distinguishes MD from other molecular simulation methods such as the Brownian dynamics and the Monte Carlo methods. Jabbarzadeh and Tanner [179] have used the MD to study the effects of shear rate and strain on flow-induced crystallization of polymers.

14.5 Other Numerical Methods

197

In the work of Jabbarzadeh and Tanner [179], linear polyethylene C162 H326 was modeled using a united atom model. The groups of CH2 and of CH3 were treated as single interaction sites. Intra-molecular architecture was used in the model, including bond stretching, angle bending, and dihedral (torsional) potentials. Then, the molecular positions, velocities, and trajectories were calculated from a set of equations as the following: P(i )

˙ 3(i ) + ϵ˙r(i ) , + e1 γr m (i ) ˙ (i ) = F(i ) − e1 P (i ) γ˙ − ζP(i ) − ϵ˙P(i ) , P 3 V˙ = 3˙ϵV , r˙(i ) =

(i )

(i )

(14.33) (14.34) (14.35)

(i )

where r , P , and m are position, peculiar momenta, and mass of atom i, respectively; e1 is the unit vector in the flow direction; γ˙ is the shear rate; F(i ) is the total force applied by all other atoms in the system on atom i ; and V is the volume of the simulation box, dynamics of which are governed by the dilation rate ϵ˙. Periodic boundary conditions in all three directions were applied for quiescent simulations. Lees-Edwards sliding brick periodic boundary conditions [223] were applied for planar shear simulations. Finally, the desired macroscopic properties such as temperature, pressure, density, coefficients of thermal expansion, crystallinity, and stresses were calculated from the microscopic information. The simulation is able to monitor the morphological transition from the amorphous structure to the semi-crystalline structure under controlled temperature, pressure, cooling rate, and flow conditions for desired polymer system, and has helped to unravel the effect of shearing or pre-shearing on the enhancement of crystallization speed. Research and development of the MD simulation method is still progressing. Currently, MD simulations are computationally restricted to very tiny time scales (about tens of nanoseconds) and very large shear rates (≥ 107 s−1 ). The cost of computer time increases dramatically with increasing number of atoms in the model of molecular chains. The growth of computational power and the development of parallel algorithms will help researchers to simulate crystallization for large systems.

14.5.2

Meshless Methods

Some efforts have been made to explore the applicability of meshless methods to solve injection molding problems, in an attempt to eliminate the requirement for mesh generation. This section is devoted to a review of the Smoothed Particle Hydrodynamics (SPH) method, which is a meshless and full Lagrangian method. The SPH method was first introduced by Gingold and Monaghan [130] and Lucy [232] in simulations of non-axisymmetric phenomena in astrophysics. The book of Liu and Liu [228] gives a detailed description of SPH theory, including numerical programs and some application examples. The early implementation of the SPH method was appropriate for an inviscid fluid in gas dynamics. It has been further developed for wider applications for problems of Newtonian, generalized Newtonian (Shao and Lo [331], Rafiee [302]), viscoelastic (Ellero and Tanner [95], ´ Fang et al. [115], Rafiee et al. [303], Vazquez-Quesada and Ellero [381]), and Bingham-like fluid flows (Zhu et al. [427]). Cleary et al. [63] and Prakash et al. [301] used SPH to simulate the metal flow in the casting and furnace emptying with some success. The first attempt to use the SPH method to simulate injection molding of polymers was tried by Fan et al. [106, 107].

198

14 Additional Issues of Injection-Molding Simulation

In SPH, the fluid is discretized into a finite number of moving points, or “particles,” in which any physical quantity f (x) associated with the particle at the position x is interpolated using function values at neighboring particles within a small local support domain of the position x. Such an interpolated value of the function at x is approximated by its kernel estimate: f (x) ≈

N m X ¡ ¢ b f (xb )W |x − xb |, h , b=1 ρ b

(14.36)

where f (x) is known only at N discrete particles at xb (b = 1, 2, ...N ), and the particle b has mass m b and density ρ b , W is a kernel function, and h is the smoothing length representing the effective width of the kernel. Several commonly used kernel functions can be found in the book of Liu and Liu [228]. The SPH discretized governing equations are as follows. The mass conservation equation is ρ(x a ) =

N X

m b Wab ,

(14.37)

b=1

and the momentum equations are given by " Ã ! Ã !# N p a pb d ua X τ a τb m b − 2 + 2 I + 2 + 2 · ∇a Wab + b, = dt ρ a ρb ρ a ρb b=1

(14.38)

where u is the velocity vector, p is the pressure, τ is the extra stress tensor, and b the field force. The subscripts “a” and “b” denote particles a and b, respectively. The extra stress tensor depends on the constitutive equations used. For a specific formulation for generalized Newtonian fluids, see Fan et al. [106, 107]. The energy conservation equation (Cleary and Monaghan [64]) is N 4m k a kb (xa − xb ) · ∇a Wab d ea X b , = (T a − Tb ) 2 dt r ab + ξ2 b=1 ρ a ρ b (k a + k b )

(14.39)

where e a is the internal energy corresponding to particle a; k a and k b are the thermal conductivities corresponding to particles a and b, respectively; T a and Tb are the temperatures conductivities corresponding to particles a and b, respectively; r ab is the distance between particles a and b; and ξ is a small parameter to avoid the singularity when r ab → 0. The equation has neglected the viscous dissipation and the heat source terms. To simulate confined fluid flow such as mold filling, one usually models the solid boundaries by “solid particles.” These particles are fixed or moving with the velocity of the moving boundaries, and interact with fluid particles. Several approaches have been proposed to handle the solid boundary conditions. Most common approaches are the repulsive boundary condition method proposed by Monaghan [253–255] and the ghost particle method described by Morris et al. [259] and Ferrari et al. [117]. Fan et al. [106, 107] have applied the SPH method to mold-filling simulation. They have obtained stable solution for a power-law fluid flow with the highest zero shear viscosity 1.22 × 104 Pa·s and a lowest power-law exponent 0.294, and the highest pressure in the range of O(108 ) to O(1010 ) Pa, at a low Reynolds number of O(10−5 −10−4 ). It is shown that, in addition to the elimination of mesh generation, another advantage of SPH over the mesh methods is that

14.5 Other Numerical Methods

199

the moving free surfaces can be automatically caught. This makes it much easier to update complex free surfaces during the simulation. However, the SPH has also shown disadvantages that frustrate its application. The SPH method for high viscosity fluid flows is highly sensitive to instability. To obtain a stable solution using the standard explicit SPH, the time step must be reduced to a very small value according to the diffusive stability constraint. Hence, in the current early stage of development, the SPH method is not yet competitive with traditional grid based methods in polymer injection molding simulations.

15

Epilogue

It has been 17 years since the first edition of Flow Analysis of Injection Molds [196] was published in 1995. Over the years, there has been significant progress and changes of focus in the research areas relevant to injection-molding simulation. The classical 2.5D Hele-Shaw flow approximation is still the most efficient and useful method in the simulation of injection molding of thin-walled parts, and therefore the method is still a main topic in the new edition (Chapter 5 and Appendix G). However, the scope has been largely expanded, and many important topics not touched on in the earlier edition have now been covered, which can be summarized as follows: 1. Considerable additions to material properties have been made in Chapter 3. 2. The modeling of flow-induced fiber orientation in injection molded composite materials has been discussed, including several new issues such as anisotropic fiber-fiber interactions, strain reduction between fibers, and simulation of long fibers (Chapters 7 and 9, and Appendix C). 3. Micromechanics model to predict the stiffness and thermal expansion of fiber-reinforced polymers has been described in Chapters 8 and Appendix J. The model, together with a fiber orientation averaging method, links the predicted fiber orientation distribution to the shrinkage and warpage analysis. 4. There is an active worldwide effort focused on crystallization and its effects on injection molding processes. Chapters 10–12 have been devoted to this important topic. Our objective is not only to simulate crystallization during the processing, but also to predict how the established microstructure affects the flow behavior and the final properties of the product. Since there is no generally accepted fundamental theory linking the crystalline microstructure to the macroscopic properties of the material, we reviewed some different models currently available in the literature. 5. Predictive methods for shrinkage and warpage have been presented (Sections 5.3.7.2 and 12.3), in order to reflect the increasing demand on the dimensional precision of injection molded products. A new subject of the post-molding shrinkage and warpage has been highlighted in Chapter 13. 6. While the pioneering work in the development of 2.5D midplane finite element/finite difference methods for injection molding flow analysis has been successfully completed, research and development of alternative numerical techniques continues, either for overcoming mesh generation difficulties or for certain applications. The dual-domain approach, the full 3D simulation, and the SPH method are some examples that we have reviewed in this book (Chapters 6 and 14, and Appendix H). 7. Many additional and up-to-date references have been provided. We believe that some specific subjects discussed in this book will be investigated further. In particular, developments of commercially feasible methods of material characterization,

202

15 Epilogue

especially for semi-crystalline materials, are needed to advance further the current numerical methods for injection-molding simulation. A long-term goal would be to predict the material microstructures and their macroscopic properties after molding. While some efforts, either successful or failed, have been made toward this end, much still remains to be done. We hope that our readers find this book useful.

Appendices

A

History of Injection-Molding Simulation

This appendix provides a historical background to injection-molding simulation. We have tried to present a commercial and academic view of this history because this book is aimed at both users of commercial packages and those interested in improving simulation. With the benefit of hindsight, people may ridicule past methods—particularly those of commercial companies. A historical perspective can give insight into why something happened when it did and can be useful in a field that embraces science and engineering such as molding simulation. Indeed the broader field of polymer melt processing has attracted attention from some keen scientific minds as well as engineers trying to do the best they can. The scientific view, though vitally important, often dismisses the constraints and pressures on commercial enterprise. Koen [208] defined the engineering method as “the use of heuristics to cause the best change in a poorly understood situation within the available resources.” This is perhaps the appropriate attitude to molding simulation at any given time. However, as the understanding via science improves, it is essential that simulation incorporates it.

A.1

Early Academic Work on Simulation

Injection molding was practiced a long time prior to the advent of simulation. While the observation that part quality was affected by processing was well known, due to the complex interplay of the factors involved, injection molding was something of an art. Experience was the only means of dealing with problems encountered in the process. An overview of this approach is given by Rubin [317]. The bibliography of this book cites hundreds of empirical studies, each contributing to the relationship between processing and part quality. Early work on simulation began in the late 1950s with the work of Toor et al. [367], where the authors introduced a scheme to calculate the average velocity of a polymer melt filling a cold rectangular cavity and so obtain the maximum flow length of the polymer. These results could then be used to deduce the time to fill a cavity of given length. Their calculations accounted for conductional heat loss and used experimentally determined parameters for the effect of temperature and shear rate on viscosity. No viscous dissipation effects were accounted for, and the pressure equations solved were obtained by a force balance. It is interesting to note that the equations were solved on an IBM 702 computer with an average run time per simulation of 20 hours! Demand for increased quality of molded parts in the 1970s saw an increased interest in mathematical modeling of the injection-molding process. During this time many pioneering studies were published. In the early seventies there was some interest by mathematicians in Hele-

206

A History of Injection-Molding Simulation

Shaw flow [309]; however, these works focused on mathematical issues and did not consider application to injection molding. In 1971 Barrie [26] gave an analysis of the pressure drop in both delivery system and a disk cavity. He avoided the need for temperature calculations by assuming the frozen layer had a uniform thickness that is proportional to the cube root of the filling time. Interestingly, Barrie remarked that a tensile (extensional) viscosity may be required for prediction of cavity pressure in the region near the sprue due to the extension rate there. It is a sobering thought that to this day no commercial package includes such terms in the cavity, although pressure losses at sudden contractions such as gates are often included. The work of Kamal and Kenig [191] was especially noteworthy as they considered filling, packing, and cooling phases in their analysis. They used finite differences to solve for the pressure and temperature fields. Williams and Lord [401] analyzed the runner system using the finite difference method. This was extended to analysis of the cavity in the filling phase again using finite differences [230]. In Germany, the Institut für Kunstoffverarbeitung (IKV), formed at the University of Aachen in 1950, produced a method for simulation called the Füllbildmethode. This was based on simple flow paths (similar to the layflat method described in Section A.2) and assumed the melt was isothermal [318]. The formation of the Cornell Injection Molding Program (CIMP) at Cornell University in 1974 saw a focus on the scientific principles of injection molding. Early work focused on the filling stage [343]. This consortium had a significant effect on injection-molding simulation. All of the above work focused on rather simple geometries and, while of academic interest, offered little assistance to engineers involved with injection molding. Nevertheless, these studies provided the scientific base for commercial simulation tools.

A.2

Early Commercial Simulation

Development of commercial software for injection-molding simulation relied on the scientific understanding of the process as well as the state of the CAD and computer industries. The first company devoted to simulation of injection molding was founded in Australia by Colin Austin in 1978. In explaining the greatest influences on his early thinking [15], Austin named the works of Kamal and Kenig [191], Lord and Williams [230, 401] and Barrie [26]. Austin named his company Moldflow and, due to a purchase by Autodesk in 2008, it continues under the name Autodesk/Moldflow today. As computers were extremely costly in the early 1980s, Moldflow’s first products were distributed primarily by time-share services whereby users could buy access to the programs via satellite links to central computers. Consequently users around the world were granted access to the software. An important part of the Moldflow product at that time was the Moldflow Design Principles [14]. These were a set of guidelines for improving the design of plastic parts and runner systems. The Design Principles defined what people should do, while the software gave a quantitative indication of how closely they achieved these goals. Moldflow Design Principles are still valuable and were reprinted, with additional information, by Shoemaker in 2006 [333].

A.2 Early Commercial Simulation

207

Early Moldflow software used the “layflat” approach developed by Austin [13]. The layflat was a representation of the part under consideration that reduced the problem of flow in a threedimensional thin-walled geometry to flow in a plane. For example, consider an open box with a thickened lip at the open end. If the box is to be injected at the center of its base, a potential problem could arise from polymer flowing around the rim of the box and forming an air trap as shown in Figure A.1.

Figure A.1 Flow progresses faster in the thick rim of the box and creates an air trap on the front (shown) and rear sides

The layflat of the box is shown in Figure A.2. As can be seen, the box has been “folded out” to form the layflat.

Figure 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

Analysis could be performed on the various flow paths on the layflat (the dark lines). The analysis was essentially one-dimensional with regard to pressure drop, although temperature variation through the thickness and along the flow path was accounted for. While the box seems simple, considerable skill was required to produce the layflat for more complex parts. Figure A.3 shows the layflat for an early automotive component. It can be seen that unfolding the part and determining the flow paths is not straightforward.

208

A History of Injection-Molding Simulation

Figure A.3 An automotive component and its associated layflat model

Thermal calculations used either a method similar to that proposed by Barrie [26] for calculation of frozen layer thickness, or a finite difference scheme with grid points through the thickness and along the flow path. A constant mold temperature was assumed at the plastic mold interface. While the melt temperature at injection points was assumed constant, viscous dissipation, convection of heat due to incoming melt, and conduction to the mold were accounted for. Viscosity of the melt was modeled with a power law or second-order model as used by Williams and Lord [230] and included shear thinning and temperature effects. Pressure drop was calculated using analytic functions for flow in simple geometries—parallel plates or round tubes. Results from the analysis were displayed in tabular form for each of the analyzed flow paths. Due to the relatively simple assumptions made, the analysis was sufficiently fast to allow users to interactively modify thicknesses to achieve their design goal. By determining the pressures and times to fill along each flow path, the user could increase (or decrease) the thickness of the component so as to balance the fill time along each flow path and eliminate the air trap. While this type of analysis was undoubtedly of benefit, it required the user to analyze an abstraction (the layflat) of the real geometry. For complex parts, the determination of the layflat required considerable skill. However a solution to this problem was not far away. Giorgio Bertacchi formed Plastics & Computer, an Italian company devoted to molding software, in 1978. Products from Plastics & Computer were also distributed by time-share systems. These products were aimed at all aspects of injection molding. While there was some simulation capability, the software also dealt with costing estimation.

A.3

Simulation in the Eighties

Apart from research on molding simulation, the eighties saw the introduction of CAD as a mainstream part of product design. CAD systems of the period were predominantly surface

A.3 Simulation in the Eighties

209

or wireframe based. This meant that geometry was represented as surfaces with no thickness displayed. However, when meshed the local thickness information was assigned to elements. It was thus a time ripe for the introduction of finite element methods using 2.5D or Hele-Shaw approximation. Indeed the 2.5D approximation enabled the advancement of both academic and commercial software. Such is its importance that we provide a detailed derivation of the equations in Chapter 5. This decade saw a rapid evolution of computer hardware. In the early eighties, large mainframe systems and time-share distribution of software were common. In the mid-eighties the hardware moved to the super mini, while at the end of the decade the UNIX workstation was introduced. The latter provided vastly improved graphics and higher computational speed.

A.3.1

Academic Work in the Eighties

In the eighties, academic and commercial interest extended to other aspects of the process. Certainly there were further advances in simulating the filling phase, but interest shifted to other phases of the process. Consequently we find a broadening of simulation to the packing and cooling phases. Another feature of this period is the formation of several centers focusing on the injection-molding process. Each center was based around a university department, and each produced its own computer code to further research on simulation of molding.

A.3.1.1

Mold Filling

McGill University had a team led by Musa Kamal. As well as academic work on foundations, the McGill group developed the McKam software for molding simulation [187]. McKam used the finite difference method for numerical calculations and utilized the most advanced algorithms available. Analysis of both filling and packing phases was possible, and the program focused on the long-term goal of determining product properties such as birefringence and tensile modulus. In 1986 Lafleur and Kamal [215] presented an analysis of injection molding that included the filling, packing, and cooling phases with a viscoelastic material model. The Cornell Injection Molding Program (CIMP) led by K.K. Wang was also very active in the eighties. The work of Hieber and Shen in 1980 [154] was arguably the most influential work from the Cornell Injection Molding Program. Assuming an incompressible material, a symmetric flow field about the cavity center line, and adopting the Hele-Shaw approximation, the pressure equation solved was µ ¶ µ ¶ ∂p ∂ ∂p ∂ S + S = 0. (A.1) ∂x ∂x ∂y ∂y A more general form of this equation, Equation 5.72, is derived in Chapter 5. The above equation results by setting the functions a(x, y), b(x, y), and d (x, y) of Equation 5.72 to zero in accord with the assumption of incompressibility and other simplifications. The important point is that the pressure field is two-dimensional—there is no pressure variation in the thickness (local z) direction. Under the same assumptions, the energy equation was

ρc p

µ ¶ DT ∂ ∂T = ηγ˙ 2 + k . Dt ∂z ∂z

(A.2)

210

A History of Injection-Molding Simulation

See Equation 5.73 in Chapter 5 for details. A finite element scheme using quadratic elements was used to solve the pressure equation. As there was no need to calculate pressure in the local z direction, the mesh required was located on a local x-y plane. Finite differences were used to solve the temperature field, which was assumed to be symmetric about the cavity center line. After solving for the pressure field, the velocities in the x and y directions could be obtained using  H  Z ′ ZH ¡ ¢ ∂p  z d z′  ′ d z −C x, y , v x (x, y, z) = ∂x η η

(A.3)

 H  Z ′ ZH ′ ¡ ¢ ∂p  z d z . v y (x, y, z) = d z ′ −C x, y ∂y η η

(A.4)

−H

−H

−H

−H

For details, see Equations 5.48 and 5.49 in Chapter 5. With the velocity field known, the total flow into a nodal control volume could be determined. In this way the flow front could be propagated at each time step until the part was filled. Most importantly, this paper introduced the idea of analyzing the thin-walled geometry as a set of shell elements, with the required model being much closer to the original geometry than the one-dimensional flow paths analyzed in the layflat method and finite difference codes using simplified geometry. Figure 6.1 shows an example of a 3D component and a midplane shell element mesh. The use of finite elements and finite differences lead to this approach being described as a “hybrid” approach. Moreover, as the pressure field was two-dimensional and the velocity and temperature field three-dimensional, this method of molding simulation was often referred to as 2.5D analysis. We will use the term “2.5D midplane” analysis, by which we mean the use of a finite element solution of a 2D pressure solution using a mesh at the centerline of the product and the 3D solution of the temperature equation using finite differences with a control volume approach to the propagation of the flow front. The CIMP produced several software codes based on this work. In 1980 the code TM-2 was completed. It used the 2.5D midplane method of Hieber and Shen [154] and was limited to single-gated analysis of one cavity. TM-2 was extended to TM-7 in 1986. This code allowed analysis of the runner system and a variable thickness cavity, using beam and triangular elements respectively. Viscosity was modeled with temperature, shear rate, and pressure dependence. The final code from the CIMP in this period was distributed in 1989 and was called TM-10-C. This software offered 2.5D midplane analysis in the filling and packing stage of injection molding. It was applicable to thin-walled geometry with variation in thickness and used a compressible fluid model. In Germany, the Institut für Kunstoffverarbeitung (IKV) at Aachen, continued to research all aspects of plastic processing—not just injection molding. However, they too developed an injection-molding code called CADMOULD. This was also a 2.5D midplane analysis code. A group led by J. Vlachopoulos at McMaster University in Canada formed the Centre for Advanced Polymer Processing and Design (CAPPA-D). Generally CAPPA-D dealt with processes

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other than injection molding, but they did fundamental work on the so-called fountain flow that takes place in injection molding [241, 242]. The importance of this work was that all prior work had assumed the melt flow in injection molding is two-dimensional. That is, there was no pressure variation through the thickness of the part and hence no velocity gradient in the thickness direction. Consequently, when considering the energy equation, there was no explicit convection calculation of temperature in the thickness direction. Researchers openly stated that these assumptions were invalid at the flow front, but it was generally accepted that the assumption was appropriate once the flow front passed a given point. Interestingly it has only recently been demonstrated that the stability of the flow front has an important role in the surface defect known as “tiger stripes” [39]. The late eighties also saw the Eindhoven group, led by H.E.H Meijer and F.P.T. Baaijens, begin work on molding simulation. In conjunction with the Philips Centre for Fabrication Technology (Philips CFT), they introduced a series of codes called Inject-1 and Inject-2. Sitters [336] and Boshouwers and van der Werf [42] introduced simulations using the 2.5D midplane approach, which led to Inject-3, the first 2.5D midplane analysis from the collaboration. This code dealt rigorously with the filling phase for amorphous materials.

A.3.1.2

Mold Cooling

People familiar with the injection-molding process realized the cooling phase accounted for the majority of time in any given cycle. Naturally, there was an interest in optimizing the cooling system so as to reduce cooling time and increase productivity. Industrial interest in cooling was also motivated by the effect of cooling on warpage of injection-molded parts. Several groups contributed to development of mold-cooling software. Interestingly, development of cooling simulation was led by commercial companies rather than academia. This was perhaps due to the lack of new science involved in cooling simulation. Despite this, the CIMP did make some major contributions. Kwon et al. [214] introduced a relatively simple solution to mold cooling. This was extended by Himasekhar et al. [157]. By considering a 1D conduction problem with a finite difference scheme in the mold and melt and several different numerical methods, the authors concluded that a cycle-averaged temperature was sufficiently accurate for mold design purposes. Himasekhar et al. [157] then implemented a 3D solution for which the temperature in the mold was determined using a boundary element method (BEM), similar to that proposed by Burton and Rezayat [49], and a finite difference method for heat transfer in the polymer. This became the most common approach to cooling simulation. That is, use a finite difference or semi-analytical solution in the polymer and conduct a full 3D heat transfer analysis in the mold using the BEM. The work of Karjalainen [192] was a noteworthy but little-known contribution to the field. Here a finite element solution in the plastic and the mold metal was employed, unlike the boundary element approach used by others. Moreover, interface elements were used to model heat transfer between mold blocks and inserts. It is worth remarking that the trend to use the BEM method for the 3D mold cooling problem was due to the need to mesh only the outer surface of the mold. It is unlikely that the mesh generators of the day would have been able to produce a 3D mesh of the mold. It is for this reason that the BEM method became the standard solution.

212

A.3.1.3

A History of Injection-Molding Simulation

Warpage Analysis

One of the great problems in injection molding is part warpage. Some major steps toward simulation of this phenomenon were made in this period. We make no attempt to detail all of this work here but note the work of greatest impact on simulation. To understand the development of warpage simulation, it is important to appreciate that warpage results from inhomogeneous polymer shrinkage. While all polymers shrink on cooling from the melt to solid phase, processing causes variation in shrinkage, and it is this variation that results in part deformation. One can break the problem into two parts—prediction of isotropic shrinkage and prediction of anisotropic effects. The former is influenced greatly by the pressure and temperature history of the part. Consequently the packing phase is important. Development of anisotropic shrinkage effects is related to structure development of the material as it solidifies. For an amorphous polymer, molecular orientation is important. The problem is more difficult for semi-crystalline materials. It follows then that warpage simulation rests on our ability to model the filling, packing, and cooling phases of the molding process. There is little work in the literature on the prediction of warpage specifically. Instead, research focused on understanding the residual stress in injection-molded products. The early work in this area was influenced by the literature on residual stress in glass [219]. While this accounted for residual stresses due to cooling, it neglected the effect of the pressure applied in the packing phase. Isayev et al. [176] considered the residual stress in an amorphous polymer. They showed that the flow-induced stresses tended to be tensile and of maximum value at the surface of a molded strip. On the other hand, purely thermal stresses are compressive at the surface. An excellent review of this and the work of others is provided in Isayev [174]. The link between packing pressures and the development of residual stresses was investigated by Titomanlio et al. [363]. A simple model for residual stress was used to calculate stress distributions in rectangular plate moldings of polystyrene. Results compared reasonably with experimental data.

A.3.2

Commercial Simulation in the Eighties

The eighties saw a big increase in commercially available programs for simulation of injection molding. In the early eighties, the only commercial companies involved in molding simulation were Moldflow and Plastics & Computer. By the end of the decade there were simulation codes from ■

General Electric

■

Philips/Technical University of Eindhoven

■

Graftek Inc.

■

Structural Dynamics Research Corporation (SDRC)

■

AC Technology

■

Moldflow Pty. Ltd.

■

Simcon GmbH

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These offerings may be grouped into three categories: 1. Codes developed by large industrials and used for internal advantage but not for sale in the open marketplace; 2. Codes developed by large industrials and available for sale in the open marketplace; 3. Codes developed by companies devoted to developing and selling simulation codes. We consider each later in this section. Some general remarks are in order regarding the development of commercial software in this period. By far the most important development was the general acceptance of the 2.5D midplane solution for flow analysis. The use of finite elements immediately provided the advantage of displaying results on something that resembled the actual geometry. This was a great advantage over layflats and tabular results. Generally, triangular-shaped elements were used to model the cavity due to ease of mesh generation. Runners were modeled with beam elements of circular cross-section. While software was chiefly distributed by time-share in the early eighties, some large companies purchased mainframe systems. These included the VAX VMS, Control Data CDC Cyber, and IBM machines running the VM operating system. These systems started to decrease in popularity as the decade wore on due to inroads by PCs and UNIX workstations. Aside from any technological advance in the modeling of injection molding, the power of computers increased dramatically. Gordon Moore [257] published a paper in 1965, suggesting that the number of transistors per chip would double every 18 months or so. Nobody at the time could have imagined that this prediction would remain in force for so long, particularly when you look at the scant data on which it was based. As well as the predicted changes in fundamental semiconductor technology, there were enormous changes in commercially available hardware—approximately a doubling of speed every 18 months. An important change in hardware was heralded by IBM’s introduction of a personal computer in 1981. An improved model, the AT, was introduced in 1983. For the first time a computer with worldwide support was available at a reasonable price and lead to simulation software being distributed on media such as floppy disks. These machines had 16-bit processors and required extenders to increase addressable memory for molding simulation. However, it was not until 1985 when Intel introduced the 386 chip—a 32-bit processor—that the PC showed its true potential. Toward the end of the decade, a new hardware platform was introduced—the UNIX workstation. Manufacturers such as Apollo, Silicon Graphics International (SGI), Hewlett-Packard (HP), SUN, and Digital Electronics Corporation (DEC) introduced machines running variants of UNIX. These machines had 32-bit operating systems, were aimed at the scientific computing industry, were faster than PCs and offered very high graphics performance. The latter was a major factor in the acceptance of 2.5D midplane analysis as the standard for molding simulation. Importantly, the graphics capability of these machines enabled the development of photorealistic rendering of parts in the CAD systems of the day. The major CAD systems of the eighties were Computervision (CADDS), Intergraph (IGDS and Interact), McDonnell-Douglas (Unigraphics), GE/CALMA, and IBM/Dassault (CADAM and CATIA). All of these systems offered wireframe and surface modeling. Consequently they were ideally suited to production of the model required for the 2.5D midplane analysis employed in commercial simulation software.

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A History of Injection-Molding Simulation

Commercial simulation was extended to the packing and cooling phases of the molding process. A significant impediment to introduction of this software was the lack of PVT data. In the eighties the largest source of data was a German work published by VDMA—an industrial association of German plastic processing machine makers [382]. Fortunately, two commercial machines were introduced later in the decade. One was developed by Paul Zoller [432] and sold under the name Gnomix. This machine immersed the sample in a confining fluid—silicon oil or mercury. Pressure was applied to the confining fluid, which in turn applied a uniform pressure to the sample. The temperature and pressure of the fluid and the change in volume were measured, from which the PVT characteristics were derived. The other machine was developed by SWO Polymertechnik GmbH. Rather than a confining fluid, the sample was compressed by a piston. Measurement of the pressure and temperature of the melt and the volume change allowed calculation of the PVT behavior of the material. Wiegmann and Oehmke [400] describe each method and the associated advantages and disadvantages. Against this background we now discuss the available simulation codes.

A.3.2.1

Codes Developed by Large Industrials and Not for Sale

A.3.2.1.1 General Electric As mentioned in a previous section, cooling phase simulation was considered commercially important. Singh [335] described a system developed within General Electric in the early eighties that used one-dimensional heat transfer theory to optimize the design of cooling circuits. This code was known as POLYCOOL. It was further developed within GE and then commercialized by SDRC (see below). GE also developed some flow analysis software. Named FEMAP, this code used the 2.5D midplane approach [392]. Unlike the work of Hieber and Shen [154], linear finite elements were used for pressure calculation. Post-processing of results was done in the SDRC environment. A.3.2.1.2 Philips/Technical University of Eindhoven The Inject-3 code mentioned earlier [42] was used within Philips for simulation. It dealt with the filling phase using the hybrid 2.5D midplane approach. Its academic roots meant it was capable of detailed analysis when used by experts. While developed for internal use, there was an attempt to commercialize it. This failed when another Philips division adopted C-Flow (see section entitled “Companies Devoted to Developing and Selling Simulation Codes” later in this appendix).

A.3.2.2

Codes Developed by Large Industrials for Sale in the Marketplace

A.3.2.2.1 SDRC Structural Dynamics Research Corporation (SDRC) was an early pioneer of finite element dynamics analysis and a major CAD company. They became involved in CAE for injection molding when they commercialized the POLYCOOL code from GE. In 1982, SDRC offered its first product called POLYCOOL 1. This was a two-dimensional quasi-static thermal analysis of mold cooling. The program used shape factors to describe the mold geometry [335]. In 1984 SDRC embarked on a new development for a cooling analysis code that did not have the drawbacks of POLYCOOL 1, namely the lack of a true 3D description of the mold, part, and cooling

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line system. This culminated in the limited release of POLYCOOL 2.0 in late November 1984 and a wider release of POLYCOOL 2.1 in 1985. This code used a one-dimensional transient finite difference method for heat conduction in the plastic coupled with a 3D boundary element method for heat transfer in the mold [49]. Heat transfer from the mold to the cooling circuits was steady state. At that time, SRDC also distributed flow analysis software produced by Moldflow. SDRC developed interfaces so that POLYCOOL 2.1 could accept the initial temperature distribution of the plastic in the mold calculated from Moldflow flow analysis and then commence cooling analysis. POLYCOOL 2.1 was the state of the art in cooling analysis software at the time. Indeed the approach used, or an approximation to it, became the standard method for mold cooling analysis. A.3.2.2.2 GRAFTEK GRAFTEK was formed in 1980 and believed the plastics injection-molding market would be best served by an integrated CAD/CAM and CAE system. The company sold a turnkey system for 3D mechanical design and numerical control machining. Its first filling simulation product was called SIMUFLOW. This was a finite difference branching flow program not unlike that offered by Moldflow. A 2.5D midplane analysis called SIMUFLOW 3D, which was based on code developed by the CIMP, was offered later [50]. GRAFTEK also supplied SIMUCOOL for mold cooling analysis. In 1984 GRAFTEK was acquired by the Burroughs Corporation. It underwent further changes of ownership and disappeared in the 1990s. SIMUFLOW 3D has recently reappeared in the marketplace due to reinvestment in the technology by another company.

A.3.2.3

Companies Devoted to Developing and Selling Simulation Codes

A.3.2.3.1 AC Technology The Cornell Injection Molding Program gave rise to AC Technology—an incorporated company formed in 1986. AC Technology marketed the C-Flow filling code in 1986 [395]. Based on the work of Hieber and Shen [154], this code sought to make the 2.5D midplane analysis more tractable on the computer systems of the day by using linear finite elements for the pressure equation. It also incorporated a high-level graphical user interface (GUI) to facilitate use by people who were not expert in the field of analysis. The original C-MOLD product performed only filling analysis but did not assume symmetry about the cavity center line and so used a finite difference grid for temperature calculation over the entire thickness. Analysis of the packing phase [394] and mold cooling analysis were introduced in 1988. The cooling analysis used the BEM in the mold [157]. Heat fluxes calculated by the cooling analysis were used as boundary conditions for the filling and packing analyses, thereby coupling the flow and cooling phases. A.3.2.3.2 Moldflow Moldflow developed a finite element flow analysis program in the early eighties. While it used linear elements for pressure, it differed from the approach used by other companies in that it did not use the finite difference method for temperature calculations. Instead it used a proprietary scheme based on the ideas of Barrie [26] for frozen layer thickness and a semianalytic method for temperature. From a commercial viewpoint the big problem was the lack of suitable mesh generation and graphical display of results. This was overcome in 1982 when

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A History of Injection-Molding Simulation

the first finite element software with meshing and graphical display software was released by Moldflow [13]. Moldflow began development of a 3D cooling analysis in the early eighties. The development was completed in 1986 [350]. This code was similar to the SDRC POLYCOOL 2 development. However, it was never released. Unlike finite element methods, the boundary element method required solution of a full matrix. The company decided solution of such matrices was too compute intensive for the computers available at the time time. Consequently, Moldflow developed a near-node boundary element method. Rather than calculate the effect of each plastic element on every other plastic element and each cooling line element, this technique considered only those elements near the element under consideration. This resulted in much smaller matrices and reduced computer requirements greatly. It was released for sale in 1985. In 1987, Moldflow started an industrial consortium with the acronym SWIS: Shrinkage Warpage Interface to Stress. It was aimed at predicting the warpage of injection-molded parts. For this project, Moldflow adopted a 2.5D midplane analysis for flow analysis that was similar to that used in C-MOLD—that is, linear finite elements for pressure and finite differences for temperatures. A packing analysis was also introduced. In order to reduce the necessary computer requirements, the filling and packing analyses assumed the flow field was symmetric about the cavity center line. The near-node boundary element mold cooling analysis was extended to give asymmetric temperatures, and these were averaged to interface to the filling and packing analysis. Shrinkage calculations used the results from flow and cooling analysis to determine shrinkage strains calculated from an equation [252] (see Equations 5.87 and 5.88 in Chapter 5). These strains were calculated on the top and bottom of each element in the model, thereby accounting for differential temperature effects, and in directions parallel and transverse to flow. The deformation of the part was then determined by converting these strains to thermal strains and inputting them to commercial structural analysis solvers such as ABAQUS, ANSYS, NASTRAN, and ADINA. It should be noted that this approach does not involve calculation of residual stress; rather, residual strains are calculated, and these are then “corrected” with measured values of shrinkage. Section 4.8 provides more detail on this procedure. A.3.2.3.3 Simcon Kunststofftechnische Software GmbH Simcon was founded in 1988. Located in Aachen, the company maintained a close relationship with IKV and commercialized the Cadmould program that was developed within IKV. Simcon’s products used 2.5D midplane analysis with their own pre- and post-processing. They allowed analysis of the filling and cooling phases of the molding process.

A.4

Simulation in the Nineties

Toward the end of the 1980s, the UNIX workstation became the machine of choice for simulation. However, PC development continued. In 1990, the Windows operating system was introduced. This enabled better user interfaces and improved graphics performance. Apart from the hardware advances, another important factor was the development of computer-aided drafting (CAD) software. Many users of design software saw immediate

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benefits in CAD systems rather than simulation. The rationale was that CAD systems offered a single design environment in which product development could occur. The notion that designs were captured on disk and available for change quickly displaced many drafting boards with computer screens. In the eighties CAD was restricted by computer power and industrial needs. In 1985 the formation of Parametric Technology Corporation lead to the introduction of 3D modeling with parametric constraints on geometry. Suddenly the entire CAD landscape changed. Surface and wire-frame modeling were no longer the state of the art. Threedimensional modeling became the norm in the CAD world. However, injection-molding simulation was focused on the Hele-Shaw approximation and required midplane representations of the 3D geometry. The notion of performing full 3D analysis on 3D geometry in the early nineties was not viable with the available computer resources. Nevertheless, the nineties can be described as the period in which 3D geometry started to dominate the injection-molding simulation industry. An important side effect of this move to 3D was the general trend of all CAE companies to introduce analysis products that could be used by non-specialists. These were targeted at product designers rather than specialist analysts. Development of these “design” products was fueled by the recognition that analysis was more beneficial when used early in product development. In regard to molding simulation, this trend signaled a change in emphasis from troubleshooting to preliminary analysis of initial designs. This was reflected in the products for molding simulation developed in this decade.

A.4.1

Academic Work in the Nineties

For the first five years of the decade, there was little recognition from academia on the fundamental change that occurred in the CAD industry, namely, the move to 3D modeling systems. Instead the focus was on calculating the effects of processing on residual stress and properties—both necessary for improved shrinkage and warpage prediction. The CIMP published several early papers on warpage of molded parts. Santhanam and Wang [320] considered the warpage due to temperature differences across the mold halves. Using both thermo-elastic and thermo-viscoelastic models, their study showed that both models could calculate similar deflections. The effect of packing pressure was not considered however. Chiang et al. developed models for the packing phase in the early nineties [60]. Around the same time Hieber et al. [155] showed the effect of packing on warpage of a center-gated disk. All work from CIMP during this period used the 2.5D midplane analysis method. Significant contributions from the Technical University of Eindhoven also emerged at this time. They always had a focus on properties and offered sophisticated simulations of residual stress as the first step to prediction of properties. Douven [89] simulated the development of residual stresses using viscoelastic models for an amorphous polymer. Using analysis of the filling and packing phases, Douven used a compressible Leonov model to determine the residual stress in a molded part. Two methods of implementing the viscoelastic model were investigated. The first, known as decoupled, used a generalized Newtonian fluid model to determine the kinematics of the flow to drive the viscoelastic stress model, the assumption being that the flow-induced stress does not affect the rheology of the material. A fully coupled scheme was also used. Douven showed that the results from the decoupled solution were comparable to

A History of Injection-Molding Simulation

218

the fully coupled approach for a simple flow. Consequently the decoupled approach, requiring far less computer resources, was adopted by several authors. Similar findings were reported by Baaijens [21]. Caspers [51] used the decoupled approach to compute shrinkage, warpage, and the elastic recovery of a molded amorphous material. He used an ageing term in the PVT model to determine the density as a function of time. As for the CIMP, all work at Eindhoven utilized the 2.5D midplane analysis method and resulted in the code VIp (6p), which was an abbreviation for Polymer Processing and Product Properties Prediction Program. Contributions to simulation were also made by the group of Titomanlio in Italy. Using a Williams and Lord approach [230, 401], that was extended to the packing phase, they studied the decay of pressure during the packing phase using a crystallization model [364] and concluded that it was necessary to link crystallization to flow. In Titomanlio et al. [365] the theory was further developed to allow the crystallization kinetics to be a function of the shear stress. Moreover the viscosity was related to the degree of crystallization. In Canada the Industrial Materials Institute of the National Research Council of Canada (CNRC IMI) also developed 2.5D software for injection molding simulation. This was also based on the 2.5D midplane approach. More importantly, they undertook development of a true 3D code for filling analysis. The first of its type, the IMI code was first described by Hétu et al. [152]. Using a finite element solution for pressure and temperature, the code used tetrahedral discretization of the mold geometry and solved a Navier-Stokes equation for pressure and three velocity components at each node. Instead of the control volume approaches used to propagate the flow front in 2.5D midplane analysis, they used a pseudo-fluid method. This involved solution of a further equation DF ∂F = + v·∇F , Dt ∂t

(A.5)

where F ∈ [0, 1] and represents the concentration of polymer. When F = 0 the cavity is unfilled, whereas F = 1 corresponds to a filled region. Of course for numerical implementation some intermediate value must be chosen for partially filled regions. The Centre de Mise en Forme des Matériaux (CEMEF) at the Ecole Nationale Supérieure des Mines des Paris was also active during this period. Boitout et al. [41] used a simple thermoelastic constitutive model for development of residual stresses in a simple geometry and considered the effect of mold deformation. Like the CNRC IMI, CEMEF recognized the importance of a true three-dimensional approach and developed a 3D code [298]. This code solved the non-isothermal Stokes equations with Equation A.5 for flow front advancement. It also used a tetrahedral discretization.

A.4.2 A.4.2.1

Commercial Developments in the Nineties SDRC

In the 1980s SDRC distributed Moldflow flow analysis code and developed its own mold cooling analysis. Early in the nineties, however, the company decided to offer its own flow analysis and warpage analysis products. The flow analysis was based on 2.5D midplane analysis, and they included residual stress calculations for warpage prediction [308].

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In 1994, SDRC decided to stop the development of their proprietary molding simulation software. Moldflow and SDRC entered into an agreement in which Moldflow solvers for flow, cooling, and warpage were embedded in the SDRC CAD environment. This product was then marketed and sold by SDRC and its distributors.

A.4.2.2

Moldflow

Moldflow’s SWIS consortium resulted in a commercial warpage product in 1990. This used a 2.5D midplane analysis of the filling and packing phase with linear finite elements for pressure prediction. However, to save time and memory, it assumed the flow and temperature fields were symmetric about the midplane. The symmetry assumption was only possible because of the approach to warpage prediction used. While all other commercial codes used the residual stress method, Moldflow used a strain-based approach [252] to calculate shrinkage in each finite element in directions parallel and perpendicular to flow. Bending moments, due to temperature differences on the mold halves, were introduced by modifying the parallel and perpendicular shrinkages on the top and bottom of each element according to the temperature calculated by cooling analysis. Details of this approach are provided in Section 4.8. The trend to 3D solid modeling was taken very seriously by Moldflow and led to developments on three fronts: ■

Automatic midplane generation

■

Dual Domain Finite Element Analysis

■

Full 3D analysis

Moldflow released an automatic midplane generator in 1995. Kennedy and Yu [198] described the system, in which a representation of the geometry was input in stereolithography format (STL), remeshed and converted to a mesh of triangular elements with assigned thicknesses, located at the midplane of the geometry. The method was useful for many parts but could lead to a model that required some manual cleanup from the operator. The decision to use the STL format was to facilitate integration with CAD systems. Whereas mesh generators were frequently an expensive add-on to a CAD system, almost all CAD systems could output STL to facilitate rapid prototyping. In 1997, Moldflow introduced Dual Domain Finite Element Analysis (DDFEA) technology for filling analysis. Here the idea was to use a mesh on the exterior of the 3D geometry for the flow analysis. This method again used STL input of solid geometry. The exterior skin of the part was then meshed with triangles. Each triangle was assigned a local thickness when it could be matched to another parallel triangle on the other side of the mesh. Special boundary conditions were applied to ensure that the flow on each side of the part was synchronized. The essential idea is to introduce connections at strategic points such that the flows on each surface mesh remain synchronized. This method is discussed in detail in Section 6.1.2. The Dual Domain approach allowed a simple means of providing flow analysis on a solid geometry. DDFEA was extended to the Moldflow advanced products for filling, packing, and cooling analysis. The popularity of the method pushed Moldflow to extend the Dual Domain approach to shrinkage and warpage analysis. To achieve this, it was necessary to develop a structural analysis that used the mesh on the exterior of the 3D geometry. We discuss this in detail in Section 6.1.3. Moldflow undertook a large development effort in the nineties to develop its 3D analysis software [124]. Full 3D filling analysis was introduced to the marketplace by Moldflow in 1998 [304]

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A History of Injection-Molding Simulation

and was extended to packing in 1999 [353]. However, as noted earlier, the first report of true 3D filling analysis was reported by workers at the Industrial Materials Institute, National Research Council Canada (Institut des Matériaux Industriels Conseil National de Recherches Canada)in Hétu et al. [152].

A.4.2.3

AC Technology/C-MOLD

During the nineties AC Technology adopted a more commercial stance to the marketplace and renamed the company C-MOLD, thereby emphasizing its primary business. Having developed filling packing and cooling analysis, C-MOLD introduced a residual stress calculation module called C-PACK/W in 1991. This was based on the CIMP work [60, 155, 319, 320] and performed a viscous-elastic residual stress analysis using packing and cooling analysis results. Calculated residual stresses were then used as input to the structural analysis package ABAQUSr , which calculated the deformed shape of the component after ejection from the mold using linear or nonlinear geometric analysis. In 1992, C-MOLD released C-STRESS, a linear structural analysis program for calculation of the warpage from the residual stresses computed in C-PACK/W. This was later modified to permit nonlinear geometric analysis. In response to the growing movement to promote CAE at the design stage, C-MOLD developed some special products. The first was Quickfill, a 2.5D midplane analysis tool with a fast solver. Although it had a limited range of results, the product was intended to be used by nonspecialists. A later version called 3D Quickfill, released in 1998, used the dual domain technique introduced by Moldflow.

A.4.2.4

Simcon

Simcon continued to develop their 2.5D software to encompass warpage. In 1998 Simcon introduced a product called Rapid Mesh that, like the Moldflow dual domain method discussed above, utilized an exterior mesh on a 3D geometry. Designed for quick evaluation of mold designs, Rapid Mesh had a limited set of results and was a competitor to similar products from Moldflow and C-MOLD. This technique, which was called, the Simcon Surface Model Method, was then introduced to the main 2.5D product line and marketed as Cadmould Pro.

A.4.2.5

Sigma Engineering

Sigma Engineering was a joint venture of IKV Aachen, Simcon, and MAGMA GmbH. MAGMA had developed a 3D code for simulation of casting called MAGMASOFT. Initiated in 1998, Sigma produced a 3D injection molding simulation called SIGMASOFT. This was based on the code MAGMASOFT and provided a full 3D analysis using voxel meshing. Voxel meshing is a structured mesh generation technique in which the 3D part geometry is divided into a series of smaller and smaller hexahedra (voxels). Meshing is stopped when it is deemed that there are sufficient voxels to permit accurate analysis. Coming from the casting industry, SIGMASOFT solved the Navier-Stokes equations using finite differences. SIGMASOFT incorporated inertia, gravity and, a flow front propagation scheme that could predict jetting. That is, regions of the mold that were initially filled could be unfilled at a later time. Unlike most commercial plastics CAE companies, Sigma Engineering does not offer a 2.5D midplane analysis.

A.5 Simulation Science Since 2000

A.4.2.6

221

Timon

Timon started in business in 1986 but did not move into injection molding until the midnineties. A subsidiary of the Toray Corporation, Timon recognized the need to interface to 3D geometry and produced a pseudo-3D simulation in 1996 called 3D TIMON. Unlike other companies that solved the Navier-Stokes equations for their 3D simulation, Timon extended the Hele-Shaw approximation to 3D dimensions [265]. Given a pressure distribution p, they assumed the three velocity components could be obtained as v x = −κ

∂p ∂p ∂p , v y = −κ , v z = −κ ∂x ∂y ∂z

(A.6)

where κ solves the equation 1 ∇2 κ = − , η

(A.7)

and η is the viscosity. Hence, for an incompressible fluid, instead of solving the Navier-Stokes equations for p at each node, it is only necessary to calculate a pressure field from the Laplace equation ∇2 p = 0 ,

(A.8)

and the fluid conductance from Equation A.7. The velocities are then obtained from Equation A.6. Compared to a Navier-Stokes solution, the number of unknowns at each node is reduced from four to two, and computational effort is reduced considerably.

A.4.2.7

Transvalor

Transvalor is the commercial arm of the CEMEF. They commercialize, sell, and support software that is developed at the CEMEF. In the late 1990s they distributed a product called REM3D that is designed for 3D analysis of injection molding. Like Sigma Engineering, Transvalor did not develop and market any products using 2.5D midplane analysis. Their injection-molding simulation focuses on 3D only.

A.4.2.8

CoreTech Systems

Research funded by the Taiwanese government and carried out at the National Tsing-Hua University (NTHU) in Taiwan investigated injection molding from 1989 to 1999. This lead to the creation of a commercial entity called CoreTech System Co. Ltd. in 1995. Marketed under the name Moldex, their original products were based on 2.5D midplane technology and offered analysis of all phases of injection molding.

A.5

Simulation Science Since 2000

While most academic work utilized finite differences or finite element methods for solution of the governing equations, the finite volume method was widely used in some fields of Newtonian and non-Newtonian fluid mechanics. A finite volume formulation for simulation of injection molding was given by Chang and Wang [55] in 2001.

222

A History of Injection-Molding Simulation

Despite this contribution, the numerical methods for injection molding simulation are relatively mature, at least for the case of generalized Newtonian fluids and the indirect use of viscoelastic models as proposed by Douven [89] and Baaijens [21]. Consequently, academic simulation has focused more on linking properties to processing. Central to this work is the development of morphology in semi-crystalline materials. Much of this effort has been influenced by Janeschitz-Kriegl and Eder at the University of Linz [91, 92]. While much previous work had been done on crystallization in injection molding, it focused on thermal effects only. Eder et al. [91, 92] demonstrated the profound effect of shear on both the crystallization kinetics and the resulting morphology. It is now well established that a short, high shear treatment greatly increases the number of nuclei and hence the crystallization rate, whereas the total shear experienced affects the morphology [211]. For low shear, spherulitic structures are formed, while higher shear leads to oriented structures. Eder proposed the following equation for the effect of flow on the nucleation rate N˙ f ; N˙ f +

1 Nf = f , λN

(A.9)

where λN is a relaxation time that, according to Eder and Janeschitz-Kriegl [91], has a large value and varies with temperature, and f is a function that takes into account the effect of flow. Eder set the right-hand side to be a function of the shear rate squared, γ˙ 2 . Zuidema [434] produced the first simulation to predict morphology in injection molding. He used the Schneider equations [324] to explicitly determine the distribution of oriented and spherulitic structures. To account for flow-induced crystallization, Zuidema [434] and Zuidema et al. [435] used the recoverable strain in the right-hand side of Equation A.9, rather than γ˙ 2 as originally proposed by Eder et al. [92]. Kennedy and Zheng [199] presented an alternative method. The right-hand side of Equation A.9 is set equal to the change in free energy of the melt. Instead of using differential equations, they used the following integral equation for crystallinity α f ,

αf = gm

Zt 0



Zt

m

N˙ (s)  G (u) d u  d s ,

(A.10)

s

where g m is a constant that depends on m, with m = 4 − 3 〈uu〉 : 〈uu〉 . Here u is a unit vector in the direction of the c-axis of the crystalline structure and is calculated according to the flow field. Hence, m will vary from 1 for linear structures to 3 for spherical semi-crystalline regions, thereby providing information on the morphology. This approach has the advantage that micromechanics theories can be used for determining properties. Further details are provided by Zheng and Kennedy [418] and Kennedy [197]. Further developments on this work are provided in Chapters 10 and 11 of this book. The development of morphology calculation is under rapid development. Pantani et al. [281] give a review of progress thus far. A major challenge is determining material properties given a calculated morphology. Explicit calculation of the crystallization kinetics and the resulting morphology has enabled some progress to be made in the area of solidification. We discuss this further in Chapters 11 and 12. Despite these attempts, there is no generally accepted model for solidification, and it remains an area of research.

A.5 Simulation Science Since 2000

223

We conclude this section with some comments on PVT property determination. In an earlier section, we noted the availability of commercial apparatus from Gnomix and SWO. These devices are designed to measure PVT characteristics at high temperatures and pressures. Due to their robust design, they have high thermal inertia and so cannot make measurements at the high cooling rates seen in injection molding. This is a problem when trying to understand the crystallization process as it relates to injection molding, where cooling rates are very high. At the center of a molding that is 3 mm thick, the cooling rate is of the order of ten degrees per second, whereas closer to the mold wall it will be higher. Han and Wang [143] presented a method to adjust PVT data obtained under slow cooling rates for use at high cooling rates using a crystallization model. They showed an improvement in linear shrinkage prediction of molded samples using the transformed data for both PA66 and fiber-filled PBT. However their crystallization model did not include any flow-induced effects. Brucato et al. [46] presented an apparatus that permitted study of the density of solidified polymer under high pressure and high cooling rates. They concluded that, at high cooling rates, pressure effects tend to be insignificant. This is contrary to observations made with low cooling rate equipment. Most recently a new PVT device has been designed in the Netherlands. It permits both high cooling rate, high pressure, and shear effects. Using this apparatus and Wide Angle X-ray Diffraction (WAXD), Van der Beek et al. [378] studied the effect of shear and temperature on the specific volume and morphology of two isotactic polypropylene (iPP) samples. They concluded that flow effects on specific volume evolution increased with increased shear rate, pressure, and average molecular weight. On the other hand, the sensitivity of specific volume to flow effects decreases with the temperature at which shear is applied. The authors further surmised that crystallization models that consider only one phase, for example the β-crystalline phase in iPP, may not be able to fully describe the crystallization kinetics due to flow.

A.5.1

Commercial Developments Since 2000

An important industrial trend is in-mold assembly. Essentially the idea is to create as much of a system in the mold as possible. This has lead to development of insert molding and overmolding. The latter has become popular with soft-touch thermoplastic polyurethanes (TPUs) for improved grip on handheld appliances. Three-dimensional analysis is advantageous for these processes as the components have complex geometry that often cannot be represented by shell models. Most of the commercial suppliers of simulation software now offer a 3D code, with varying levels of support for these features. Another trend is the move to thinner-walled moldings. This has been fueled in part by the personal electronics industry, particularly laptop computers and mobile phones. In an effort to reduce weight and cost, wall thicknesses in these devices are often under 1 mm. The desire to pack more functions into laptops and phones has lead to miniaturization of electrical connectors. Wall thicknesses in these parts can be as low as 0.14 mm. Successful analysis of such moldings requires accurate thermal boundary conditions at the part-mold interface. Unfortunately there has been little fundamental work in this area. In-situ measurements of thermal contact resistance (TCR) and mold wall temperatures under molding conditions were described by Delaunay et al. [76]. This provided thermal contact resistance as a function of time

224

A History of Injection-Molding Simulation

for an iPP resin (Solvay PHV 252) in contact with 40CMD8 mold steel. The TCR varies considerably with time and differs from one side of the mold to the other. Other investigators [412] have noted that the default heat transfer coefficient values used in commercial simulation packages may have to be reduced in order to agree with experimental data on molding with very thin micro-features. The current decade has seen major changes in the companies offering simulation codes. Indeed there has been a consolidation of many CAD/CAM/CAE companies since 2000. One significant commercial development was the initial public offering by Moldflow. In March 2000, Moldflow Corporation successfully listed on the NASDAQ stock exchange. It was the first plastic CAE company to reach this milestone. A further milestone occurred in April 2000 when Moldflow acquired AC-Technology and the C-MOLD range of products. After the acquisition, Moldflow was by far the largest company involved in simulation of injection molding. Further consolidation in the industry occurred with the acquisition of SDRC by EDS in 2001. Plastics & Computer was acquired in 2005 by the VI Group, a publicly listed CAD/CAM supplier to the injection-molding industry.

A.5.1.1

Moldflow

Post-acquisition, Moldflow retained the technical staff from C-MOLD and built new premises in Ithaca, New York, to house the existing staff, and a new laboratory to facilitate growth. The first product from the post-acquisition company was MPI 3. It was a combination of the best technologies from Moldflow and C-MOLD. Since then there have been a regular series of product releases—mostly aimed at improving analysis capabilities for midplane, dual domain and true 3D solvers. The major advance in 3D simulation was the introduction of warpage analysis in 3D. Fan et al. [112] introduced a 3D warpage analysis that included calculation of mechanical properties from the calculated fiber orientation distribution. Following the industrial trend toward in-mold assembly, Moldflow 3D and dual domain analysis products have been extended to over-molding and insert analysis. It is now possible to consider the effect of an insert or an over-molded part on the flow of the encapsulating materials [111]. In the case of plastic over-molding, the temperature of the first shot, after packing and cooling, is used as a boundary condition for the injection of the second material. For inserts, an initial temperature is specified and heat transfer within the insert is calculated as a function of time. Generally, simulation has considered that the mold cavity dimensions do not change during processing. Baaijens [21] considered the effect of mold elasticity on pressure calculation by adding a term to the pressure equation. Such an approach is valid if the mold deformation is uniform. However, due to pressure gradients inside the mold, this is rarely the case. Mold elasticity effects are especially significant in thin-wall moldings where pressures are high. It is not unusual to see injection pressures of 250 to 300 MPa. Leo and Cuvelliez [224] showed the effect of mold elasticity on the pressure decay during packing—an important factor for shrinkage and warpage prediction. Further investigations including mold deformation were undertaken by Delaunay et al. [77]. In view of this work and the above-mentioned industrial trend to thinwalled molding, mold filling analysis has been extended to include mold deformation—either

A.5 Simulation Science Since 2000

225

due to the structure of the mold and its elastic properties or, in particular, core shift. Such an analysis requires coupling of the flow and structural analyses [24]. For a given time step, the calculated pressure distribution is used to determine the mold deflection. The deflection is used to update the flow domain by deforming the mesh (for 3D analysis), or changing the thickness (for 2.5D midplane analysis). The flow in the next time step is then calculated using the updated domain, and the process continues until the end of fill and packing. Using this technique it is possible to more accurately calculate the residual stress in the part and hence its deformation. Moreover, it is possible to determine the residual stresses in the mold during the molding cycle. These results can be interfaced to metal fatigue analysis so as to assess mold life—an important factor in mass-produced items.

A.5.1.2

Timon

Timon continued to develop its range of 3D products. Using the method discussed in the previous section, Timon developed a hybrid analysis [266] in which areas that were meshed with three or fewer elements across the thickness were solved using 2.5D analysis, whereas those with more than four elements were analyzed using the generalized Hele-Shaw scheme described earlier. In 2003, Timon introduced a product for designing optical lenses using a prediction of birefringence. This was the first commercial program to predict optical properties of injection-molded products.

A.5.1.3

CoreTech Systems

Using the finite volume method proposed by Chang and Yang [55], CoreTech produced a 3D analysis in 2001 called Moldex3D/Solid. This product uses a variety of element shapes for analysis. For example, it is possible to create a mesh with several wedge elements near the mold wall, and tetrahedral elements in the center. The idea is to improve the heat transfer calculations near the mold wall. While the approach may be effective for the filling phase, the location of the frozen layer during packing may not be predicted accurately. Moldex3D/Solid has since been extended to include filling, packing, cooling, and warpage.

A.5.1.4

Autodesk

Autodesk is a well-known CAD/CAM software supplier and has been actively acquiring other businesses to achieve its vision of virtual manufacturing. In 2008, Autodesk acquired Moldflow Corporation for US$297 million, less the amount in Moldflow’s cash balance and proceeds from options exercises. It is likely that such acquisitions will assist the development of simulation in the manufacturing environment whereby the actual mold can be modeled and used to provide boundary conditions for the simulation.

A.5.2

Note for Students

While there have been many contributions from industrial and academic organizations to injection-molding simulation, any student of the subject should avail themselves of the literature available from the Technische Universiteit Eindhoven. Over many years, under the guidance of Professors Meijer and Baaijens, this institution has researched many aspects

226

A History of Injection-Molding Simulation

of injection-molding simulation. Most recently, there has been substantial progress in the prediction of properties of injection-molded materials. Much of this information is contained in Ph.D theses and their corresponding publications. Information may be found at http://www.mate.tue.nl/mate/.

B

Tensor Notation

In this appendix we provide some basic information on tensor notation and algebra. This should be sufficient for reading the book but there are many good references available.

B.1

Index Notation

As a way of introducing index notation, we begin with scalar variables. These simply have a magnitude and are represented as numbers. Examples are temperature, speed, pressure, and distance between two points. Scalar variables do not depend on direction. Vectors, on the other hand, have both a magnitude and direction associated with them. For example, to define a point traveling with a certain velocity, it is necessary to know the speed (a scalar quantity—and often called the magnitude of the velocity), and the direction to define where it is going. Tensors are next. These are quantities that require a definition of a plane and a vector. For example, if a block of material is influenced by a force, the stress in the block would require us to know the plane in which the force acts, as well as the direction and magnitude of the force. Both vectors and tensors exist independently of any coordinate system. Yet they may be specified in a particular coordinate system by a certain set of components. In a three-dimensional Euclidean space, the number of components of a tensor is 3n , where n is the order of the tensor. A tensor of order zero is specified in the space by one component. Therefore zero-order tensors are scalars. First-order tensors have three components in the space and are known as vectors. Second-order tensors have nine components in the space. Several important quantities, such as the stress, are represented by second-order tensors. Higher order tensors, such as fourth-order tensors, often appear in the description of anisotropic mechanical properties. A vector v can be expressed in matrix form   v1    v =  v2  v3

(B.1)

or v = v 1 e1 + v 2 e2 + v 3 e3 ,

(B.2)

where v 1 , v 2 , and v 3 are the components of the vector v, and e1 , e2 , and e3 are orthogonal unit basis vectors in the Cartesian coordinate system. The vector v is also expressed for brevity’s sake in the index notation as v i , i = 1, 2, 3, where the subscript i is referred to as index or suffix.

228

B Tensor Notation

A second-order tensor A can also be expressed in matrix form by giving the nine components in a square array as   A 11 A 12 A 13    A= (B.3)  A 21 A 22 A 23  A 31 A 32 A 33 or A =A 11 e1 e1 + A 12 e1 e2 + A 13 e1 e3 A 21 e2 e1 + A 22 e2 e2 + A 13 e2 e3 A 31 e3 e1 + A 32 e3 e2 + A 33 e3 e3 .

(B.4)

A shorter notation for the second-order tensor is the index notation with two unrepeated indices, such as A i j , i , j = 1, 2, 3. Unrepeated indices are known as free indices. In the rectangular Cartesian coordinate system, a free index is to take on the values 1, 2, or 3 (in a 2D problem, it takes values 1 and 2 as specified). The index notation clearly and concisely represents the components of a tensor of any order and indeed the tensor itself. The number of the free index determines the order of a tensor. For example, the following expressions are second-order tensors because they all have and only have two free indices: A i j kk , A i j B j k , A mn u k v k . The following expressions are all vectors since each of them has only one free index: A i kk , A i j v j , εi j k u j v k . The expression A i i is a scalar (a zero-order tensor) because it has no free index. Similarly, the third tensor has three free indices such as B i j k , and the fourth-order tensor has four free indices such as C i j kl , and so on.

B.2

Einstein Summation Convention

If an index appears twice in a term, the repeated index is called a dummy index, which implies a summation of that term over the range of the index. For example, A i i = A 11 + A 22 + A 33 , uk

∂ρ ∂ρ ∂ρ ∂ρ = u1 + u2 + u3 . ∂x k ∂x 1 ∂x 2 ∂x 3

(B.5) (B.6)

The following system of equations a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 , a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 , a 31 x 1 + a 32 x 2 + a 33 x 3 = b 3 ,

(B.7)

B.3 Kronecker Delta

229

can be written in compact form as ai j x j = bi .

(B.8)

A pair of dummy indices can be replaced by another letter, for example, A i j b j ≡ A i k b k ≡ A i m b m , as long as the letter is not the same as the one used for the free index in the same term. The summation convention requires that one must never allow a dummy index to appear more than twice in a term. Because of this rule, it is sometimes necessary to replace a pair of dummy indices by other letters to avoid having three or more repeated indices occurring in the same term.

B.3

Kronecker Delta

The Kronecker delta is defined as ( 0 if i ̸= j δi j = , 1 if i = j

(B.9)

which is also called the unit tensor, denoted by I in the boldface notation. Using the above definition and the summation convention one obtains δi i = 3 ,

(B.10)

δi j δ i j = 3 ,

(B.11)

δi j δi k δ j k = 3.

(B.12)

and

We also have u i δi j = u j .

(B.13)

Here the subscript index i in u i is replaced by j, hence the Kronecker delta is also called the substitution tensor. Similarly, for a tensor A i j , one has Ai j δ j k = Ai k .

B.4

(B.14)

Alternating Tensor

The alternating tensor—also called the permutation symbol—is defined as   + 1 if i j k = 123, 231 or 312   εi j k = − 1 if i j k = 321, 132 or 213   0 if any two indices are alike ,

(B.15)

which satisfies the following identities: δi j εi j k = 0 ,

(B.16)

εi mn ε j mn = 2δi j ,

(B.17)

εi j k εmnk = δi m δ j n − δi n δ j m .

(B.18)

230

B Tensor Notation

The cross product of two vectors u (u i ) and v (v i ), w = u × v, can consequently be written as w k = εi j k u i v j .

B.5

(B.19)

Product Operations of Two Tensors

Given two second-order tensors A (A i j ) and B (B i j ), we have different types of products: the dyadic product, the single dot product, and the double dot product. — The dyadic product is a fourth-order tensor, written as C i j kl = A i j B kl (C = A ⊗ B).

(B.20)

— The single dot product is a second-order tensor, written as C i j = A i k B k j (C = A · B).

(B.21)

— The double dot product is a scalar, written as C = A i j B j i (C = A : B).

(B.22)

We also define A2 = A · A. It can be shown that trA2 = A : A = Aij Aji .

B.6

(B.23)

Transpose Operation

A tensor A i j is called symmetric if A i j = A j i , or A = AT .

(B.24)

In the case A i j = −A j i , or A = −AT ,

(B.25)

we call it anti-symmetric. Obviously an anti-symmetric tensor must have zero diagonal terms. Any second-order tensor can be decomposed into symmetric and anti-symmetric parts: 1 1 A i j = (A i j + A j i ) + (A i j − A j i ) . 2 2

(B.26)

B.7 Transformation of Principal Axes

B.7

231

Transformation of Principal Axes

For a second-order tensor A, there is a transformation law from axes x to x′ so that we can write A′ = R · A · RT (A ′i j = R i k A kl R j l ),

(B.27)

where R is called the rotation matrix, which is an orthogonal matrix, that is, its inverse is also its transpose. For a symmetric tensor A, there is always an orthogonal transformation R such that after the transformation the tensor has a diagonal form 

λ1

 A′ =  0 0



0

0

λ2

 0  . λ3

0

(B.28)

The elements λ1 , λ2 , and λ3 are called the eigenvalues of the tensor A. Eigenvalue can also be defined as follows. A scalar λ is an eigenvalue of a second-order tensor A i j , if there exists a direction (defined by a nonzero unit vector n i ) and a scalar λ , satisfying A i j n j = λn i ,

(B.29)

where the vector n i is called the eigenvector. Noting that n i = δi j n j , the above equation can be written as (A i j − λδi j )n j = 0,

(B.30)

which is a system of three linear equations for n j (i.e., n 1 , n 2 , and n 3 ). Obviously any multiple of an eigenvector n i will also be an eigenvector, but we will not consider such multiples as being distinct eigenvectors. In order that there be a nontrivial solution for n j , it is necessary that det(A i j − λδi j ) = 0 ,

(B.31)

where det stands for determinant. If expanded out, this gives a cubic characteristic equation for λ: λ3 − I 1 λ2 + I 2 λ − I 3 = 0 ,

(B.32)

where the coefficients are given by I1 = Ai i , ¢ 1¡ Ai i A j j − Ai j A j i , I2 = 2 I 3 = det A i j .

(B.33) (B.34) (B.35)

Here, the determinant of A i j (written as det A i j ) can also be expressed in the form εi j k A 1i A 2 j A 3k . The quantities I 1 , I 2 and I 3 are called the first, second, and third invariants of the tensor A i j .

232

B Tensor Notation

As an example, we take an exercise problem provided in Tanner’s book [356] as follows. Consider a symmetric tensor A:   3 −1 0   0 (B.36) A= .  −1 3 0 0 1 To find the eigenvalues, we solve det(A i j − λδi j ) = 0, that is, ¯ ¯3−λ ¯ ¯ ¯ −1 ¯ ¯ ¯0

−1 3−λ 0

¯ ¯ ¯ ¯ ¯ = 0, 0 ¯ ¯ 1 − λ¯ 0

(B.37)

or (1 − λ)[(3 − λ)2 − 1] = 0,

(B.38)

which gives λ1 = 4,

(B.39)

λ2 = 2,

(B.40)

λ3 = 1.

(B.41)

To find sets of eigenvectors, we solve    (3 − λ)n 1 − n 2 = 0 ,  − n 1 + (3 − λ)n 2 = 0 ,    (1 − λ)n = 0 .

(B.42)

3

p p For λ = λ1 = 4, we find n(λ1 ) = [−1/ 2, 1/ 2, 0]T . p p For λ = λ2 = 2, we find n(λ2 ) = [1/ 2, 1/ 2, 0]T . For λ = λ3 = 1, we find n(λ3 ) = [0, 0, 1]T . The rotation matrix formed from the eigenvalues is p p   − 1/ 2 1/ 2 0  p  p R= 1/ 2 0  1/ 2 , 0 0 1 which transforms A to the principal direction:   4 0 0    A′ = R · A · RT =   0 2 0 . 0 0 1

(B.43)

(B.44)

B.8 Gradient of a Field

233

Invariants of A are I 1 = λ1 + λ2 + λ3 = 7 ,

(B.45)

I 1 = λ1 λ2 + λ2 λ3 + λ3 λ1 = 14 ,

(B.46)

I 3 = λ1 λ2 λ3 = 8.

(B.47)

B.8

Gradient of a Field

A function of the position vector x is called a field. We can have a scalar field, a vector field, or a tensor field. Derivatives with respect to position vectors are performed using the vector differential operator ∇, know as the del operator. It is written as ∂/∂x i in the Cartesian tensor notation. The operator can be treated as a vector but it cannot stand alone. It must operate on a scalar, a vector, or a tensor. Gradient of a scalar The gradient of a scalar field ρ is a vector defined by ∇ρ, or ∂ρ/∂x i . Gradient of a vector The gradient of a vector field u is defined by ∇u. The corresponding Cartesian tensor notation is ∂u j /∂i . Divergence of a vector The divergence of a vector field u is a scalar defined by ∇ · u = ∂u i /∂x i (≡ ∂u 1 /∂x 1 + ∂u 2 /∂x 2 + ∂u 3 /∂x 3 ). Curl of a vector The curl of a vector u is a vector defined by ∇ × u, or εi j k ∂u k /∂x j . Divergence of a tensor The divergence of a tensor field A is a vector, defined by ∇ · A, or ∂A i j /∂x i .

B.9

Unit Vector p and Operator ∂/∂p

∂ denotes the gradient operator on the surface ∂p of a unit sphere. If we denote the base vector of the spherical coordinate system by er , eθ and eφ , the operator is defined as: If p is a unit vector, |p| = 1, then the operator

∂ ∂ 1 ∂ = eθ + eφ . ∂p ∂θ sin θ ∂θ Consider a vector Q i = Qp i (Q = Qp), where Q ≡ |Q| is the norm (length) of Q. We have µ ¶ ∂p i ∂ Qi =Q ∂p j ∂Q j Q ¶ µ 1 ∂Q i Q i ∂Q =Q − 2 Q ∂Q j Q ∂Q j = δi j −

Qi Q j Q2

= δi j − p i p j ,

(B.48)

B Tensor Notation

234

or, ∂p = I − pp . ∂p

(B.49)

We also have ∂(p i p j ) ∂p k

= pi

∂p j ∂p k

+

∂p i pj ∂p k

= p i (δ j k − p j p k ) + (δi k − p i p k )p j = p i δ j k + p j δi k − 2p i p j p k , or, ∂(pp) = pI + Ip − 2ppp. ∂p

B.10

(B.50)

Identities

If s is a scalar, a, b, and v are vectors, and σ is a second-order tensor. We have the following identities: ∇ · (sv) = s∇ · v + v · ∇v ,

(B.51)

∇ · (ab) = a · ∇b + (∇ · a)b ,

(B.52)

σ : ∇v = ∇ · (σ · v) − v · (∇ · σ) .

(B.53)

As an example of the use of indicial notation we prove Equation B.53 as follows: ¢ ∂σi j ∂v j ∂ ¡ σi j v j = v j + σi j , ∂x i ∂x i ∂x i

(B.54)

so that σi j

∂v j ∂x i

=

¢ ∂σi j ∂ ¡ σi j v j − v j , ∂x i ∂x i

(B.55)

which is identical to Equation B.53. Note that the last term of Equation B.53 is the dot product of two vectors, v and ∇ · σ. The dot product for two vectors is commutative, and so v · (∇ · σ) = (∇ · σ) · v. Consequently, we may write Equation B.53 as σ : ∇v = ∇ · (σ · v) − (∇ · σ) · v .

(B.56)

C

Derivation of Fiber Evolution Equations

In this appendix we show the derivation of the equations that govern the fiber orientation of dilute and concentrated solutions. In particular we derive the well-known Folgar-Tucker equation and the anisotropic rotary diffusion model.

C.1

The Langevin Equation

Consider a vector R = R p,

(C.1)

where R is the length of R, and p is a unit vector that represents the orientation direction of a fiber in a suspension flow. We assume that the local flow field is homogenous with a velocity gradient L (= ∇uT ) and also assume an affine motion of R at this stage, by which we mean that the macroscopic deformation of the fluid can also be applied to the microscopic motion. Then the relative velocity between the two ends of the vector R is ˙ = L · R. R

(C.2)

If the motion of this fiber is affected by other fibers, we can model the interaction effect by adding a random motion component on the right-hand side of the equation: ˙ = L · R + R F(b) , R

(C.3)

where F(b) are the random forces, with the following properties 〈F(b) (t )〉 = 0,

(C.4)

〈F(b) (t )F(b) (t ′ )〉 = 2D(r ) δ(t − t ′ ),

(C.5)

and

where D(r ) is the rotational diffusion tensor, δ(t − t ′ ) is the delta or impulse function, and the angular bracket denotes the ensemble average with respect to the probability density function of the process, which will be described in more detail later. Equations C.4 and C.5 come from the concept that the random force is a “rapidly varying, highly irregular function” [128]. If we assume the rotational diffusion tensor is isotropic, D(r ) = D (r ) I, then Equation C.5 becomes 〈F(b) (t )F(b) (t ′ )〉 = 2D (r ) δ(t − t ′ )I.

(C.6)

236

C Derivation of Fiber Evolution Equations

For application to short-fiber suspensions, we can also take the effect of aspect ratio a r into account by introducing a non-affine parameter ξ = 2/(a r2 +1), and replacing the velocity gradient Li j by L = L − ξD,

(C.7)

where L is called the effective velocity gradient, and D is the deformation-rate tensor. Thus we rewrite Equation C.3 as ˙ = L · R + RF(b) R = RL · p + R F(b) .

(C.8)

Differentiating Equation C.1 with respect to time t , we also have ˙ = R˙ p + R p ˙, R

(C.9)

˙ − R˙ p . ˙ =R Rp

(C.10)

so that

Since p is a unit vector, p·p = 1.

(C.11)

˙ ·p+p·p ˙ = 0, and so Differentiating Equation C.11 with respect to time t gives p ˙ = 0. p·p

(C.12)

Multiplying Equation C.9 by p we obtain ˙ · p = R˙ p · p + R p ˙ ·p, R

(C.13)

which may be simplified using Equations C.11 and C.12 to give: ˙ ·p. R˙ = R

(C.14)

Substitution of Equations C.1 and C.8 into C.14 gives R˙ = RL : (pp) + R p · F(b) .

(C.15)

Substitution of Equations C.8 and C.15 into C.10 gives ˙ = RL · p + R F(b) − RL : (pp)p + Rpp · F(b) , Rp

(C.16)

and, after division by R, we obtain: ˙ = L · p − L : (pp)p + (I − pp) · F(b) , p

(C.17)

or, in the Cartesian tensor notation form, p˙i = L i k p k − L kl p k p l p i + (δi k − p i p k )F k(b) .

(C.18)

C.2 Probability Density Function and Orientation Tensors

237

The factor (I−pp) in front of F(b) ensures that the random force is always acting perpendicular to p, so that only rotational random motion is allowed. Equation C.17 or C.18 is a stochastic equation known as the Langevin equation. If there is only a single fiber, or if the suspension is dilute, then F(b) = 0 and Equation C.17 reduces to ˙ = L · p − L : ppp, p

(C.19)

which is equivalent to ˙ = W·p+ p

a r2 − 1 ¡ a r2 + 1

¢ D · p − D : ppp ,

(C.20)

where W = (∇uT − ∇u)/2 is the vorticity tensor, and D = (∇uT + ∇u)/2 is the deformation-rate tensor. Equation C.20 is known as Jeffery’s equation. Jeffery’s equation is only suitable for a single fiber or dilute suspensions. For concentrated solutions, we use the Langevin equation as our starting point to derive evolution equations.

C.2

Probability Density Function and Orientation Tensors

To complete the description of fiber orientation state, the probability density function must be introduced, which is a function denoted ψ(p, t ) and defined such that ψ(p, t )d p is the probability of finding a fiber orientated in the range between p and p + d p at time t . Such a function satisfies the Fokker-Planck equation in the form (Chandrasekhar [53], see also Huilgol and Phan-Thien [167]): · ¸ ∂ψ ∂ △p△p ∂ψ △p = · 〈 〉· −〈 〉ψ . (C.21) ∂t ∂p 2△t ∂p △t From the Langevin equation (Equation C.17) we have 〈

△p 〉 = L · p − L : ppp △t

(C.22)

and Z 〈△p△p〉 =

0

△t

Z 0

△t

(I − pp) · 〈F(b) (t ′ )F(b) (t ′′ )〉 · (I − pp)d t ′ d t ′′ + O(△t 2 )

= 2△t (I − pp) · D(r ) · (I − pp) + O(△t 2 )

(C.23)

where △t is small compared to the relaxation times of the process, so that 〈

△p△p 〉 = (I − pp) · D(r ) · (I − pp), 2△t

(C.24)

which, in the case of isotropic diffusion, reduces to 〈

△p△p 〉 = D (r ) (I − pp). 2△t

(C.25)

C Derivation of Fiber Evolution Equations

238

The diffusion equation for isotropic rotary diffusion then becomes ¢ i ∂ψ ∂ h (r ) ∂ψ ¡ = · D (I − pp) · − L · p − L : ppp ψ ∂t ∂p ∂p

(C.26)

and, for general anisotropic rotary diffusion cases, it is · ¸ ¡ ¢ ¡ ¢ ∂ψ ¡ ¢ ∂ψ ∂ = · I − pp · D(r ) · I − pp · − L · p − L : ppp ψ . ∂t ∂p ∂p

(C.27)

Numerical solution of the Fokker-Planck equation is computationally expensive. A more compact and efficient approach is to use orientation tensors, which were introduced by Advani and Tucker [2]. Orientation tensors are defined in terms of the ensemble average of the dyadic products of the unit vector p, that is, Z a i j ≡ 〈p i p j 〉 = p i p j ψd p , (C.28) Z a i j kl ≡ 〈p i p j p k p l 〉 = p i p j p k p l ψd p , (C.29) R where the ensemble average is defined as 〈∗〉 = ∗ψ(p, t )d p. Note that ψ is an even function, therefore all the odd-order orientation tensors are zero. There are different approaches to deriving an evolution equation in terms of the orientation tensors. In most published work, the evolution equation is derived from the Fokker-Planck equation. First, the Fokker-Planck equation is multiplied by a tensor A i j = p i p j , and then the equation is integrated over the surface of a unit sphere. The use of a non-Cartesian coordinate system makes the integration complex (see Bird et al. [36] for details). In the next section we shall follow an alternative approach similar to the one described by Phan-Thien and Zheng [294] and Huilgol and Phan-Thien [167], without having to use the Fokker-Planck equation.

C.3

Equations of Change for the Orientation Tensors

We begin with the Langevin equation (Equation C.18) and write it as p˙i = Ai + Bi k F k(b) ,

(C.30)

Ai = L i k p k − L kl p k p l p i ,

(C.31)

B i k = δi k − p i p k .

(C.32)

where

and

C.3 Equations of Change for the Orientation Tensors

239

We note that ¢ D ¡ p i p j = p˙i p j + p i p˙ j Dt ¡ ¢ ¡ ¢ = Ai + Bi k F k(b) p j + p i A j + B j k F k(b) = Ai p j + p i A j + Bi k F k(b) p j + p i B j k F k(b) . ¡ ¢ ¡ ¢ = L i k p k − L kl p k p l p i p j + p i L j k p k − L kl p k p l p i p j ¡ ¢ ¡ ¢ + δi k − p i p k F k(b) p j + p i δ j k − p j p k F k(b) = L i k p k p j + L j k p k p i − 2L kl p i p j p k p l + F i(b) p j + p i F j(b) − 2p i p j p k F k(b) .

(C.33)

Thus, taking the ensemble average, we have D 〈p i p j 〉 = L i k 〈p k p j 〉 + L j k 〈p k p i 〉 − 2L kl 〈p i p j p k p l 〉 Dt + 〈F i(b) p j 〉 + 〈p i F j(b) 〉 − 2 〈p i p j p k F k(b) 〉.

(C.34)

We now need to estimate the terms 〈F i(b) p j 〉, 〈p i F j(b) 〉 and 〈p i p j p k F k(b) 〉. For isotropic and anisotropic diffusions, the results will be different. The isotropic case leads to the conventional Folgar-Tucker equation, and the anisotropic case leads to the anisotropic rotary diffusion model developed by Fan et al. [103] and Phan-Thien et al. [289].

C.3.1

Isotropic Rotary Diffusion Model (Folgar-Tucker Model)

To estimate 〈F i(b) p j 〉, one should keep in mind the fast time scale of the random force, and hence 〈F i(b) p j 〉 = 〈F i(b) (t + ∆t )p j (t + ∆t )〉 .

(C.35)

We use Equation C.30 to obtain the following expression for p j (t + ∆t ): p j (t + ∆t ) = p j (t ) + A j ∆t +

t +∆t

Z t

B j k F k(b) (t ′ )d t ′ ,

(C.36)

so that 〈F i(b) p j 〉 = 〈F i(b) (t + ∆t )p j (t + ∆t )〉 Z t +∆t = 〈F i(b) (p j + A j ∆t )〉 + 〈B j k F kb (t ′ )F i(b) (t + ∆t )〉d t ′ . | {z } t

(C.37)

drift terms

Note that the drift terms are linear in the random force. According to Equation C.4, the terms will vanish in the average process. By applying Equation C.6, we have Z t +∆t ¢ 1¡ 〈F jb (t ′ )F i(b) (t + ∆t )〉d t ′ = 2D (r ) δi j = D (r ) δi j (C.38) 2 t

240

C Derivation of Fiber Evolution Equations

where the factor (1/2) arises because the delta correlation function overlaps the upper limit of the integral. Then we obtain 〈F i(b) p j 〉 = 〈B j k 〉D (r ) δki = D (r ) (δi j − 〈p i p j 〉).

(C.39)

Similarly, ¡ ¢ 〈p i F j(b) 〉 = D (r ) δi j − 〈p i p j 〉

(C.40)

〈p i p j p k F k(b) 〉 = 2D (r ) 〈p i p j 〉.

(C.41)

and

By substituting Equations C.39, C.40, and C.41 into Equation C.34, we obtain D 〈p i p j 〉 − L i k 〈p k p j 〉 − L j k 〈p k p i 〉 + 2 L kl 〈p i p j p k p l 〉 Dt ¢ ¡ = 2 D (r ) δi j − 3p i p j

(C.42)

or D ai j Dt

¢ ¡ − L i k a k j − L j k a ki + 2 L kl a i j kl = 2 D (r ) δi j − 3a i j .

(C.43)

When L i j is identified with L i j , Equation C.42 or C.43 is exactly the rigid dumbbell model given by Bird et al. [36]. Folgar and Tucker [122] assume that the rotational diffusivity is isotropic and proportional to the generalized shear rate γ˙ = (2D : D)1/2 : D (r ) = C I γ˙

(C.44)

where the scalar parameter C I is known as the interaction coefficient, and Equation C.43 becomes D ai j Dt

¡ ¢ − L i k a k j − L j k a ki + 2 L kl a i j kl = 2C I γ˙ δi j − 3a i j .

(C.45)

This equation is known as the Folgar-Tucker model. In much of the polymer processing literature it is not expressed in terms of the effective velocity gradient tensor L but rather in terms of the rate-of-deformation tensor D and the vorticity tensor W. Consequently it is frequently written in the form: D ai j Dt

= Wi k a k j − a i k Wk j + ¡ ¢ + 2C I γ˙ δi j − 3a i j .

´ a r2 − 1 ³ D i k a k j + D j k a ki − 2 D kl a i j kl 2 ar + 1 (C.46)

C.3 Equations of Change for the Orientation Tensors

C.3.2

241

Anisotropic Rotary Diffusion Model

To derive an evolution equation for the anisotropic rotary diffusion, we begin with the estimation of the terms 〈F i(b) p j 〉, 〈p i F j(b) 〉, and 〈p i p j p k F k(b) 〉 in Equation C.34. Again, we take the average 〈F i(b) p j 〉 = 〈F i(b) (t + ∆t )p j (t + ∆t )〉 Z t +∆t = 〈F i(b) (p j + A j ∆t )〉 + 〈B j k F k(b) (t ′ )F i(b) (t + ∆t )〉d t ′ , {z } t |

(C.47)

drift terms

which contains the drift terms that will not survive in the average process and terms which are quadratic in the random force. The latter will contribute to the average, since t +∆t

Z t

〈F jb (t ′ )F i(b) (t + ∆t )〉d t ′ =

1 ³ (r ) ´ 2D i j = D i(rj ) . 2

(C.48)

Here we have used Equation C.5. We then obtain (r ) (r ) 〈F i(b) p j 〉 = 〈B j k 〉D ki = D i(rj ) − 〈p j p k 〉D ki .

(C.49)

Similarly, 〈p i F j(b) 〉 = D i(rj ) − 〈p i p k 〉D k(rj) .

(C.50)

To evaluate the term 〈p i p j p k F k(b) 〉 we follow a procedure similar to the above. First, we find the time derivative of P i j k = p i p j p k , that is, P˙ i j k = p˙i p j p k +p i p˙ j p k +p i p j p˙k , and substitute Equation C.30 to the time-derivative terms. We obtain

(b) (b) P˙ i j k = (Ai + Bi m F m )p j p k + p i (A j + B j m F m )p k (b) ) + p i p j (Ak + Bkm F m

= Ai p j p k + A j p i p k + Ak p i p j | {z }

drift terms (b) (b) + Bi m F m p j p k + B j m F m pi pk

(b) pi p j . + Bkm F m

(C.51)

Next, we use the above equation to obtain P i j k (t + ∆t ) and then take the average 〈p i p j p k F k(b) 〉 = 〈P i j k (t + ∆t )F k (t + ∆t )〉. We obtain 〈p i p j p k F k(b) 〉 =

t +∆t

Z t

(〈Bi m p j p k + B j m p i p k

b ′ (b) + Bkm p i p j 〉)〈F m (t )F k (t + ∆t )〉d t ′ (r ) = 〈Bi m p j p k + B j m p i p k + Bkm p i p j 〉D mk ) (r ) (r ) = D i(rk) 〈p k p j 〉 + D (r 〈p k p i 〉 + D kk 〈p i p j 〉 − 3D km 〈p k p m p i p j 〉. jk

(C.52)

C Derivation of Fiber Evolution Equations

242

By substituting Equations C.50, C.49, and C.52 into Equation C.34, we obtain D 〈p i p j 〉 − L i k 〈p k p j 〉 − L j k 〈p k p i 〉 + 2L kl 〈p i p j p k p l 〉 Dt (r ) ) = 2(D i(rj ) − D kk 〈p i p j 〉) − 3(D i(rk) 〈p k p j 〉 + D (r 〈p k p i 〉 jk (r ) − 2D kl 〈p i p j p k p l 〉) ,

(C.53)

or D ai j Dt

− L i k a k j − L j k a ki + 2L kl a i j kl (r ) ) (r ) = 2(D i(rj ) − D kk a i j ) − 3(D i(rk) a k j + D (r a − 2D kl a i j kl ). j k ki

(C.54)

Assuming ˙ D i(rj ) = C i j γ,

(C.55)

we have D ai j Dt

− L i k a k j − L j k a ki + 2L kl a i j kl ˙ = γ[2(C i j −C kk a i j ) − 3(C i k a k j +C j k a ki − 2C kl a i j kl )] .

This is the anisotropic rotary diffusion model proposed by Phan-Thien et al. [289].

(C.56)

D

Dimensional Analysis of Governing Equations

In this appendix we estimate the order of magnitude of terms in the governing equation for the 2.5D method discussed in Chapter 5. In order to simplify the governing equations we first use dimensional analysis. The idea is to use characteristic values of variables such as cavity length, thickness, and melt velocity to evaluate the relative magnitude of the terms. Terms that are relatively small are dropped, providing simplification of the governing equations. Characteristic values need not be exact. Order of magnitude estimation is sufficient. We use the following characteristic values [42, 196]: ■

Cavity thickness, H = 10−3 m

■

Cavity length, L = H /δ m = 10−3 /δ where δ = H /L ≪ 1

■

Melt velocity, v 0 = 10−1 m/s

■

Cavity pressure, p 0 = 107 N/m2

■

Melt viscosity, η 0 = 104 Ns/m2

■

Melt expansivity, β0 = 10−3 1/K

■

Melt compressibility, κ0 = 10−7 m2 N−1

■

Melt thermal conductivity, k 0 = 10−1 W/mK

■

Melt density, ρ 0 = 103 kg/m3

■

Temperature difference between mold and melt, T0 = 102 K

■

Acceleration due to gravity, g 0 = 10 m/s2

■

Specific heat of melt, c p 0 = 103 J/kg K

Using these typical values, the relevant variables in the equations can be defined in terms of dimensionless variables as follows: ■

x-coordinate: x = Lx ∗ = H /δ x ∗ = (10−3 /δ)x ∗

■

y-coordinate: y = Ly ∗ = H /δ y ∗ = (10−3 /δ)y ∗

■

z-coordinate: z = H z ∗ = 10−3 z ∗

■

Time: t = [L/V ]t ∗ = (H /δv 0 )t ∗ = (10−3 /10−1 δ)t ∗ = (10−2 /δ)t ∗

■

x component of melt velocity: v x = (L/t )v x∗ = v 0 v x∗

■

y component of melt velocity: v y = (L/t )v ∗y = v 0 v ∗y

■

z component of melt velocity: v z = (H /t )v z∗ = δv 0 v x∗

■

Pressure: p = p 0 p ∗

■

Viscosity: η = η 0 η∗

■

Melt expansivity: β = β0 β∗

244

D Dimensional Analysis of Governing Equations

■

Melt compressibility: κ = κ0 κ∗

■

Temperature difference: ∆T = T0 ∆T ∗

■

Acceleration due to gravity: g = g 0 g ∗

■

Specific heat: c p = c p 0 c p∗

where the starred quantities are dimensionless and of order one.

D.1

Conservation of Mass

Recall the conservation of mass equation in Cartesian coordinates, Equation 5.22, is 0=κ

³ ∂p ³ ∂T ∂p ∂T ∂p ∂p ´ ∂T ∂T ´ + vx + vx + vy + vz −β + vy + vz ∂t | {z∂x} ∂y ∂z ∂t | {z∂x} ∂y ∂z |{z} |{z} | {z } | {z } | {z } | {z } (i )

(i i )

(i i i )

(i v)

(v)

(vi )

(vi i )

(vi i i )

∂v y ∂v z ´ x + + + ∂x ∂y |{z} ∂z |{z} |{z} ³ ∂v

(i x)

(x)

(D.1)

(xi )

We now estimate each term. Term (i): h ∂p i h h h i δv 0 i 10−1 δ i O κ = O κ0 p 0 = O 10−7 107 = O 102 δ −3 ∂t H 10

(D.2)

Term (ii): h h h h i δ i ∂p i p0 i O κv x = O κ0 v 0 −3 δ = O 10−7 10−1 107 −3 = O 102 δ ∂x 10 10

(D.3)

Term (iii): h h h i h p0 i δ i ∂p i = O κ0 v 0 −3 δ = O 10−7 10−1 107 −3 = O 102 δ O κv y ∂y 10 10

(D.4)

Term (iv): h h h h i ∂p i p0 i 10−1 i O κv z = O κ0 v 0 δ −3 = O 10−7 107 −3 δ = O 102 δ ∂z 10 10

(D.5)

Term (v): h ∂T i h T δi h h i 102 δ i 0 O β = O β0 −2 = O 10−3 −2 = O 10δ ∂t 10 10

(D.6)

Term (vi): h h h h i ∂T i 102 δ i 102 δ i O βv x = O β0 v 0 −3 = O 10−3 10−1 −3 = O 10δ ∂x 10 10

(D.7)

D.2 Conservation of Momentum

245

Term (vii): h h h h i ∂T i 102 δ i 102 δ i O βv y = O β0 v 0 −3 = O 10−3 10−1 −3 = O 10δ ∂y 10 10

(D.8)

Term (viii): h h h i h i ∂T i 102 i O βv z = O β0 δv 0 −3 = O 10−3 10−1 105 δ = O 10δ ∂z 10

(D.9)

Term (ix): O

h ∂v i x

∂x

=O

hv i 0

L

=O

h 10−1 i h i 2 δ = O 10 δ 10−3

(D.10)

h i h 10−1 i δ = O 102 δ −3 10

(D.11)

Term (x): O

h ∂v y i ∂y

=O

hv i 0

L

=O

Term (xi): O

h ∂v i z

∂z

=O

h i h δv i h 10−1 i 0 2 δ = O 10 δ = O 10−3 10−3

(D.12)

From the above we can see that various levels of approximation may be made. The largest terms are Equations D.2 to D.5 and Equations D.10 to D.12. These must be retained. However there is just a factor of 10 separating the magnitude of these terms from Equations D.6 to D.9. Given the variation in injection-molding conditions, we believe all terms should be retained. Hence the thin-walled approximation leads to no simplification of the conservation of mass equation.

D.2

Conservation of Momentum

We repeat below Equation 5.23, which gives the x-component of the momentum equation, when expressed in Cartesian coordinates: ρ

µ ¶ ∂v x ∂p ∂ ³ ∂v x ´ ∂ ³ ∂v y ∂v x ´ = ρg x − + 2η + η + ∂t ∂x ∂x ∂x ∂y ∂x ∂y µ ³ ¶ ´ ∂v x ∂v z ∂ + η + ∂z ∂z ∂x ³ ∂v ∂v x ∂v x ´ x − ρ vx + vy + vz . ∂x ∂y ∂z

Expanding out the right-hand side terms and identifying each we have:

(D.13)

D Dimensional Analysis of Governing Equations

246

∂p ∂ ³ ∂v x ´ ∂ ³ ∂v y ´ ∂v x = ρg x − + 2η + η ρ ∂t } |{z} |{z} ∂x |∂x {z ∂x } ∂y ∂x | {z | {z } (i )

(i i )

(i i i )

(i v)

(v)

µ ¶ µ ¶ ∂ ³ ∂v x ´ ∂ ∂v x ∂ ∂v z + η + η + η ∂y ∂y ∂z ∂z ∂z ∂x | {z } | {z } | {z } (vi i )

(vi )

(vi i i )

∂v x ∂v x ∂v x − ρv y − ρv z − ρv x . ∂x ∂y | {z } | {z } | {z∂z} (i x)

(x)

(D.14)

(xi )

We now evaluate the relative magnitudes of the terms in the equation. Term (i): h h i h ∂v i h δ i δ i x O ρ = O ρ 0 v 0 −3 = O 103 10−1 −2 = O 104 δ ∂t 10 10

(D.15)

Term (ii): h i h i h i h i O ρg x = O ρ 0 g 0 = O 103 10 = O 104

(D.16)

Term (iii): i h 107 i h i h ∂p i h p 0 O δ =O δ = O 1010 δ =O −3 −3 ∂x 10 10

(D.17)

Term (iv): h ∂ ³ ∂v ´i h δ ³ v δ ´i x 0 O 2η =O η 0 −3 ∂x ∂x 10−3 10 h i h δ ³ −1 ´i 4 10 δ 9 2 10 = O 10 δ =O 10−3 10−3

(D.18)

Term (v): h ∂ ³ ∂v y ´i h δ ³ v δ ´i 0 O η =O η 0 −3 ∂y ∂x 10−3 10 i h δ ³ h 10−1 δ ´i =O 104 = O 109 δ2 −3 −3 10 10

(D.19)

Term (vi): h ∂ ³ ∂v ´i h δ ³ v δ ´i x 0 O η =O η 0 −3 ∂y ∂y 10−3 10 h δ ³ h i 10−1 δ ´i =O 104 = O 109 δ2 −3 −3 10 10

(D.20)

D.2 Conservation of Momentum

247

Term (vii): O

h ∂ ³ ∂v ´i h 1 ³ v 0 ´i x η =O η 0 −3 −3 ∂z ∂z 10 10 h 1 ³ −1 ´i 4 10 10 =O 10−3 10−3 h i 9 = O 10

(D.21)

Term (viii): O

h 1 ³ v δ2 ´i h ∂ ³ ∂v ´i z 0 η =O η 0 −3 ∂z ∂x 10−3 10 h 1 ³ 10−1 ´i 104 −3 δ2 =O −3 10 10 h i 9 2 = O 10 δ

(D.22)

Term (ix): h h i h h 10−1 i ∂v x i v0δ i O ρv x = O ρ 0 v 0 −3 = O 103 10−1 −3 δ = O 104 δ ∂x 10 10

(D.23)

Term (x): h h h h i ∂v x i v0δ i 10−1 i O ρv y = O ρ 0 v 0 −3 = O 103 10−1 −3 δ = O 104 δ ∂y 10 10

(D.24)

Term (xi): h i h h h ∂v x i v0δ i 10−1 i O ρv z = O ρ 0 v 0 −3 = O 103 10−1 −3 δ = O 104 δ ∂z 10 10

(D.25)

Looking at the relative sizes of terms we conclude that the main terms to be considered are Equations D.17 and D.21 . That is, the x-component of the momentum equation may be reduced to ∂p ∂ ³ ∂v x ´ = η . ∂x ∂z ∂z

(D.26)

Using the same procedure as above for the y-component of momentum, Equation 5.24, we conclude that the y-component of the momentum equation may be reduced to ∂p ∂ ³ ∂v y ´ = η . ∂y ∂z ∂z

(D.27)

Finally, the same procedure applied to Equation 5.25 provides the following simplification: ∂p = 0. ∂z

(D.28)

D Dimensional Analysis of Governing Equations

248

D.3

The Energy Equation

The energy equation in Cartesian coordinates is from Equation 5.26: ρc p

³ ∂p ∂T ∂T ∂p ∂p ∂T ´ ∂p ´ + vx + vy = βT + vx + vy + vz + vz ∂t | {z∂x} ∂y ∂z ∂t | {z∂x} ∂y ∂z |{z} |{z} | {z } | {z } | {z } | {z }

³ ∂T (i )

(i i )

(i i i )

(v)

(i v)

(vi )

(vi i )

(vi i i )

1 ∂ T ∂2 T ´ + ηγ˙ 2 +k + 2+ 2 2 ∂z |2 {z } |∂x {z } |∂y {z } | {z } ³ ∂2 T

(i x)

(x)

2

(xi )

(D.29)

(xi i )

Evaluating the magnitude of each term we get: Term (i): h ³ ∂T ´i h h h i 102 δ i T0 δ i O ρc p = O ρ 0 c p 0 −2 = O 103 103 −2 = O 1010 δ ∂t 10 10

(D.30)

Term (ii): h ³ ∂T ´i h T0 δ i O ρc p v x = O ρ 0 v 0 c p 0 −3 ∂x 10 h i h 102 δ i = O 103 10−1 103 −3 = O 1010 δ 10

(D.31)

Term (iii): h ³ ∂T ´i h T0 δ i O ρc p v y = O ρ 0 v 0 c p 0 −3 ∂y 10 h h i 102 δ i = O 103 10−1 103 −3 = O 1010 δ 10

(D.32)

Term (iv): ³ ∂T ´i h h T0 i O ρc p v z = O ρ 0 c p 0 δv 0 −3 ∂z 10 h i h i = O 103 103 10−1 105 δ = O 1010 δ

(D.33)

Term (v): h ³ ∂p ´i h p0δ i O βT = O β0 T0 −2 ∂t 10 h i h 107 δ i = O 10−3 102 −2 = O 108 δ 10

(D.34)

Term (vi): h ³ ∂p ´i h p0δ i O βT v x = O β0 T0 v 0 −3 ∂x 10 h h i 107 δ i = O 10−3 102 10−1 −3 = O 108 δ 10

(D.35)

D.3 The Energy Equation

249

Term (vii): h ³ ∂p ´i h p0δ i O βT v y = O β0 T0 v 0 −3 ∂y 10 h h i 107 δ i = O 10−3 102 10−1 −3 = O 108 δ 10

(D.36)

Term (viii): h h ³ ∂p ´i p0δ i = O β0 T0 v 0 δ −3 O βT v z ∂z 10 h h i 7 i −3 2 −1 10 δ = O 10 10 10 = O 108 δ −3 10

(D.37)

Term (ix): From Equation 2.17 we have: ˙ 2 = 2D : D = 2tr(D2 ) . γ In Cartesian coordinates, ∂v x   ³ ∂x  1 ∂v y ∂v x ´ D=  2 ∂x + ∂y   1 ³ ∂v z ∂v x ´ + 2 ∂x ∂z 

1 ³ ∂v x ∂v y ´ + 2 ∂y ∂x ∂v y ∂y

1 ³ ∂v z ∂v y ´ + 2 ∂y ∂z

 1 ³ ∂v x ∂v z ´ + 2 ∂z ∂x   1 ³ ∂v y ∂v z ´ . + 2 ∂z ∂y    ∂v z

(D.38)

∂z

Hence O[D] is given by the order of its components. That is, O[102 δ] O[102 δ + 102 δ] 2 2 O[102 δ] O[D] = O[10 δ + 10 δ] 2 2 2 O[10 δ + 10 ] O[102 δ2 + 102 ]   O[102 δ] O[102 δ] O[102 ] 2 2 2 = O[10 δ] O[10 δ] O[10 ]  . O[102 ] O[102 ] O[102 δ] 

 O[102 + 102 δ2 ] 2 2 2 O[10 + 10 δ O[102 δ]

(D.39)

(D.40)

Consequently, O[tr(D2 )] = O[γ˙ 2 ] = O[104 ]

(D.41)

and so: O[ηγ˙ 2 ] = O[η 0 104 ] = O[108 ] .

(D.42)

Term (x): h ∂2 T i h T δ2 i h h i 102 δ2 i 0 O k 2 = O k 0 −6 = O 10−1 = O 107 δ2 −6 ∂x 10 10

(D.43)

Term (xi): h ∂2 T i h T δ2 i h h i 102 δ2 i 0 7 2 O k 2 = O k 0 −6 = O 10−1 = O 10 δ ∂y 10 10−6

(D.44)

D Dimensional Analysis of Governing Equations

250

Term (xii): h ∂2 T i h h h i T0 i 102 i O k 2 = O k 0 −6 = O 10−1 −6 = O 107 ∂z 10 10

(D.45)

Looking at the magnitude of terms we see that the energy equation for thin-walled parts may be reduced to ρc p

³ ∂T ∂t

D.4

+ vx

∂T ∂T ∂T ´ ∂p ∂2 T + vy + vz = βT + ηγ˙ 2 + k 2 . ∂x ∂y ∂z ∂t ∂z

(D.46)

Summary

The assumption that the mold is thin walled, by which we mean the thickness is three orders of magnitude less than the length of the mold, enables some simplification of the governing equations for momentum and energy; however, the conservation of mass equation is unchanged. The resulting equations are:

D.4.1

Conservation of Mass ³ ∂p

³ ∂T ∂p ∂p ∂p ´ ∂T ∂T ∂T ´ + vy + vz −β + vx + vy + vz ∂t ∂x ∂y ∂z ∂t ∂x ∂y ∂z ³ ∂v ´ ∂v y ∂v z x + + + ∂x ∂y ∂z

0=κ

D.4.2 ■

+ vx

Conservation of Momentum

x-component ∂p ∂ ³ ∂v x ´ = η ∂x ∂z ∂z

■

(D.48)

y-component ∂p ∂ ³ ∂v y ´ = η ∂y ∂z ∂z

■

(D.47)

(D.49)

z-component ∂p =0 ∂z

(D.50)

D.4 Summary

D.4.3

251

Energy Equation ρc p

³ ∂T ∂t

+ vx

∂T ∂T ∂T ´ ∂p ∂2 T + vy + vz = βT + ηγ˙ 2 + k 2 ∂x ∂y ∂z ∂t ∂z

(D.51)

We note that we have already removed the Q˙ term from the energy equation. However it may be incorporated in the 2.5D approximation when considering thermosetting materials or crystallization kinetics. While we do not consider the former, we consider crystallization in Part II of this book.

252

D Dimensional Analysis of Governing Equations

E

The Finite Difference Method

The finite difference method (FDM) is one of the oldest methods for obtaining approximate solutions to real physical problems. In this appendix we briefly review some basic concepts associated with the FDM. We do not attempt a complete presentation here. We describe only what is necessary for injection-molding simulation. There are many books on the FDM, (see for example [8, 121, 337]). An excellent introduction, relevant to polymer processing, is given by Güçeri [134].

E.1

Introduction to the Finite Difference Method

The FDM discretizes a differential equation into a system of linear algebraic equations. The simplest approach to this process is via a Taylor series expansion. As for the finite element method, the FDM requires a mesh. Generally the mesh comprises a set of points, called nodes, that cover the domain of interest. Figure E.1 shows a section of mesh in the x-y plane.

Figure E.1 A finite difference mesh in the x - y plane

Suppose u(x, y) is some function of x and y. We denote the value of u(x, y) at the node (x i , y j ) by u i j . If u(x, y) is continuous, its value u i +1, j at the point (x i +1 , y j ) may be written in terms of

254

E The Finite Difference Method

its value u i , j at the point (x i , y j ) by means of a Taylor expansion as follows: u i +1, j = u i , j +

∂2 u ¯¯ (∆x)2 ∂3 u ¯¯ (∆x)3 ∂u ¯¯ + 3¯ +... ¯ (∆x) + 2 ¯ ∂x i , j ∂x i , j 2! ∂x i , j 3!

(E.1)

where ∆x = x i +1 − x i . We have assumed here that ∆x = x i +1 − x i is constant, but that is not necessary. Equation E.1 may be rearranged to make the first derivative of u i , j the subject as follows: u i +1, j − u i , j ∂2 u ¯¯ (∆x) ∂3 u ¯¯ (∆x)2 ∂u ¯¯ − 2¯ − 3¯ −... ¯ = ∂x i , j ∆x ∂x i , j 2! ∂x i , j 3! u i +1, j − u i , j = + O(∆x) , ∆x

(E.2)

where O(∆x) indicates the higher order remaining terms. An estimate of the first derivative is obtained by truncating the series and ignoring O(∆x). That is u i +1, j − u i , j ∂u ¯¯ . ¯ ≈ ∂x i , j ∆x

(E.3)

Equation E.3 is called a first order, forward difference approximation for the first derivative because it ignores terms of order two and higher and is written in terms of the value of u at a forward node, u i +1, j . If we choose ∆x = x i − x i −1 < 0, the value of u i −1, j may be obtained in terms of u i , j using the following Taylor series: u i −1, j = u i , j +

∂u ¯¯ ∂2 u ¯¯ (∆x)2 ∂3 u ¯¯ (∆x)3 + 3¯ +... ¯ (∆x) + 2 ¯ ∂x i , j ∂x i , j 2! ∂x i , j 3!

(E.4)

Rearranging this equation and ignoring higher order terms we obtain the first order, backward difference approximation to the first derivative: u i , j − u i −1, j ∂u ¯¯ . ¯ ≈ ∂x i , j ∆x To get more accurate approximations we subtract Equation E.3 from E.1 to get u i +1, j − u i −1, j = 2

∂u ¯¯ ∂3 u ¯¯ (∆x)3 ... ¯ (∆x) + 2 3 ¯ ∂x i , j ∂x i , j 3!

(E.5)

from which we obtain: u i +1, j − u i −1, j ∂3 u ¯¯ (∆x)2 ∂u ¯¯ + 3¯ +... ¯ = ∂x i , j 2(∆x) ∂x i , j 3! u i +1, j − u i −1, j = + O(∆x)2 . 2(∆x)

(E.6)

This leads to a second order, central difference approximation to the first derivative: u i +1, j − u i −1, j ∂u ¯¯ . ¯ ≈ ∂x i , j 2(∆x)

(E.7)

E.1 Introduction to the Finite Difference Method

255

It is also possible to approximate second derivatives. Adding Equations E.1 and E.4 we get: u i +1, j + u i −1, j = 2u i , j + 2

∂2 u ¯¯ (∆x)2 ∂4 u ¯¯ (∆x)4 + 2 +... ¯ ¯ ∂x 2 i , j 2! ∂x 4 i , j 4!

(E.8)

Rearranging we get ³ ´ u i +1, j − 2u i , j + u i −1, j ∂2 u ¯¯ 4 = + O (∆x) ) , ¯ ∂x 2 i , j (∆x)2

(E.9)

from which we obtain a second order, central difference approximation for the second derivative: u i +1, j − 2u i , j + u i −1, j ∂2 u ¯¯ . ¯ ≈ ∂x 2 i , j (∆x)2

(E.10)

Similar expressions may be found for first and second derivatives with respect to y.

E.1.1

A Simple Example

As an illustration of the FDM, we solve the one-dimensional problem solved in Section F.7 of Appendix F. That is, find the solution u(x) such that d 2u − 2 = 0, d x2

for 1 ≤ x ≤ 4 ,

(E.11)

subject to the boundary conditions; du = 3, at x = 1 , dx u(x) = 20, at x = 4 .

(E.12) (E.13)

Recall from Section F.7 that this equation may be solved exactly, and the solution is: u(x) = x 2 + x .

(E.14)

Just as for the finite element method, the FDM requires a mesh. For our example, the mesh is illustrated in Figure E.2. It comprises four nodes equally spaced along the x-axis at x = 1, 2, 3 and 4.

Figure E.2 A simple mesh for a one-dimensional finite difference solution

Using a second order, central difference approximation to the second derivative, as in Equation E.10, Equation E.11 becomes u i +1 − 2u i + u i −1 −2 = 0 (∆x)2

(E.15)

E The Finite Difference Method

256

or, after rearrangement, u i +1 = 2(∆x)2 + 2u i − u i −1 .

(E.16)

In this example the inter-nodal spacing (∆x) = 1 and so u i +1 = 2 + 2u i − u i −1 .

(E.17)

Consider now the first of the boundary conditions in Equation E.12. Using a first order, forward difference approximation as in Equation E.3 we get: u2 − u1 = 3. (∆x)

(E.18)

u2 = 3 + u1 .

(E.19)

That is,

Setting i = 2 in Equation E.17 and substituting the result into Equation E.19 we obtain: u 3 = 2 + 2(3 + u 1 ) − u 1 = 8 + u1 .

(E.20)

Now set i = 3 in Equation E.17 to get: u 4 = 2 + 2u 3 − u 2 .

(E.21)

Substituting Equations E.19 and E.20 into Equation E.21 we have: u 4 = 2 + 2(8 + u 1 ) − (3 + u 1 ) = 15 + u 1 .

(E.22)

But from the second of the boundary conditions in Equation E.12, u 4 = 20 and so, from Equation E.22, u1 = 5 .

(E.23)

The values for u 2 and u 3 may be found by substituting u 1 = 5 in Equations E.19 and E.20, respectively, to get u 2 = 8 and u 3 = 13. Our final solution is therefore: u 1 = 5 (2) , u 2 = 8 (6) , u 3 = 13 (12) , u 4 = 20 (20) , where the exact solution is given in parentheses. At first sight the approximate solution may look poor but we used a very coarse mesh. If we refine the mesh, the approximations will improve. In this section we have barely touched the FDM. Interested readers are referred to the cited references at the start of this appendix for further details and methods.

E.2 Application to Temperature Calculation

E.2

257

Application to Temperature Calculation

For the so-called 2.5D approximation we are required to determine the temperature as a function of time due to the conduction of heat from the polymer into the mold. The energy equation for the 2.5D approximation is given by Equation 5.73: ρc p

³ ∂T ∂t |{z}

time derivative

∂T ∂T ´ + vx + vy = ∂x ∂y | {z } convection

∂p βT | {z∂t}

compressive heating

+

ηγ˙ 2 |{z}

viscous dissipation

+

∂2 T k 2 . | ∂z {z }

conduction to mold

(E.24) For our purposes here we only need to consider the time derivative and the conduction terms, that is: ρc p

∂T ∂2 T =k 2 . ∂t ∂z

(E.25)

This is commonly written as: ∂T ∂2 T =α 2 . ∂t ∂z

(E.26)

where α = k/ρc p is called the diffusivity of the material. We will consider the solution of Equation E.26 in this section. In doing this, we introduce the concepts of explicit and implicit finite difference schemes.

E.2.1

Explicit Methods

An explicit finite difference scheme arises when the spatial derivatives are evaluated at a previous time step. Before considering this in more detail, we modify our notation slightly. In previous sections, z was considered a continuous variable through the thickness, and T (z, t ) was taken to be the temperature at z at time t . Since we are going to discretize both the spatial variable z at a finite number of points (grid points or nodes) through the thickness and at discrete values of t , we use subscripts for the spatial node position and superscripts for the temporal discretization. Thus the value of T (z, t ) at node i in the z direction and time step n will be denoted Tin . To develop an explicit finite difference scheme for Equation E.26, the temporal derivative is replaced with a first order finite difference approximation: T n+1 − Tin ∂T ≈ i . ∂t ∆t

(E.27)

The spatial derivative is approximated with a second order, central difference scheme: Tin+1 − 2Tin + Tin−1 ∂2 T ≈ . ∂z 2 (∆z)2

(E.28)

E The Finite Difference Method

258

Substituting the approximate Equations E.27 and E.28 into Equation E.26 gives: Tin+1 − Tin ∆T

³ T n − 2T n + T n ´ i i −1 = α i +1 , (∆z)2

(E.29)

which may be rearranged to give the value of T at the new time n + 1 as: ¢ α ∆t ¡ n T − 2Tin + Tin−1 + Tin (∆z)2 i +1 ¢ ¡ = M Tin+1 − 2Tin + Tin−1 + Tin

Tin+1 =

(E.30)

where M = α∆t /(∆z)2 . The explicit nature of Equation E.30 arises from the fact that temperature values at time step n + 1 are determined explicitly from values at the old time step n.

E.2.1.1

Stability Criteria for Explicit Methods

Explicit methods are computationally fast but require the time step ∆t to be sufficiently small. The step size depends on the spatial discretization and is computable. It is known as the Courant-Friedrichs-Lewy (CFL) criterion. This was published in German in 1928 [69] and reprinted in English in 1967 [70]. It should be noted that the CFL condition is necessary but not sufficient for a converged solution. For stable solution of Equation E.30, a necessary condition for stability is [8]: M=

α∆t 1 ≤ . 2 (∆z) 2

(E.31)

Consequently the software developer and possibly the user must be careful in determining explicit solutions using the FDM.

E.2.2

Implicit Methods

In an implicit finite difference scheme, the time derivative is approximated in the same way as for the explicit method but the spatial derivative is approximated by a second order, central difference formula at the new time. Hence instead of Equation E.28 we use: Tin+1 − 2Tin+1 + Tin+1 ∂2 T +1 −1 ≈ . ∂z 2 (∆z)2

(E.32)

Substitution of Equations E.27 and E.32 into Equation E.26 gives: Tin+1 − Tin ∆T

³ T n+1 − 2T n+1 + T n+1 ´ i i −1 = α i +1 . (∆z)2

After rearrangement we obtain: ¡ ¢ n+1 n 1 + 2M Tin+1 − M Tin+1 +1 − M Ti −1 = Ti .

(E.33)

(E.34)

When written for each node (gridpoint) in the z direction, Equation E.34 yields a set of linear equations that may be expressed in matrix form as follows: [ A ]{ T }n+1 = { T }n .

(E.35)

E.2 Application to Temperature Calculation

259

Here the matrix [ A ] is a square matrix of known terms involving M and constants, { T }n+1 is a matrix of temperatures at the new time, and { T }n is a matrix of temperatures at the old time. By inverting [ A ] and premultiplying both sides of Equation E.35 by the inverse, we can obtain the new temperatures { T }n+1 . Implicit methods do not suffer from the instability problem in Equation E.31. They remain stable for any step size. However, if the time step is too large they can converge to the wrong solution, albeit in a stable manner. The choice of explicit or implicit method depends on the stage of the solution that is under consideration. In filling, it may be better to use implicit methods, while in packing where pressure can change rapidly, explicit methods may be better, though slower. The choice of these is generally not given to the user. It has been determined by the developer of the software code being used.

260

E The Finite Difference Method

F

The Finite Element Method

In Chapter 5 we discussed the approximation that reduced injection-molding simulation to a 2D pressure problem and 3D temperature problem. In this appendix we will provide an introduction to the finite element method. The concepts will be valid for the so-called “midplane” analysis, which is also relevant for the dual domain method described in Chapter 6. Moreover, the concepts are also valid for the 3D analysis discussed in Chapter 6. Detailed derivations of the finite element method for the 2.5D approximation and full 3D analysis are provided in Appendices G and H respectively. Generally speaking, the equations governing the flow of a polymer melt cannot be solved in any but the simplest of geometries. Given the complexity of injection molds a general method of solution is required. The finite element method is well suited to this type of problem. Indeed the finite element method (FEM) is a general method for solving engineering problems. Historically it was developed for structural analysis. Soon, however, it was realized the method could be applied to a very wide range of problems. When first introduced the method had a dubious theoretical background. In 1973 a rigorous basis was supplied by Strang and Fix [345].

F.1

Basic Terminology

Before looking at the FEM we need some terminology. A system is an object or process under investigation. Typically the system has a physical meaning, and you want to know the effect of outside influences on the system. In our case the system is the injection-molding process. A mathematical model is a set of governing equations with appropriate boundary conditions that describes the actual system—or some approximation to it. Frequently, a mathematical model that describes the system exactly does not exist or is too complicated to be of use. The latter can involve computation time, though this is becoming less of a problem with the astounding improvement in computer power. In many cases, particularly injection-molding, it is a lack of material data and suitable boundary conditions for the process. In either case, simplifications are made so that an approximate mathematical model may be used. In injectionmolding analysis, we have material models for viscosity as a function of temperature and shear rate. In Chapter 4 we derived the governing equations and discussed boundary conditions. We then simplified these in Chapter 5. These equations form the mathematical model we will use. For a given set of equations, there can be many solutions. Boundary conditions restrict the number of possible solutions and ensure a unique solution to the problem under consideration. For example, in filling analysis, the flow rate of the melt, the temperature of the melt and the mold wall, and the pressure at the flow front constitute a set of boundary conditions for the filling phase.

F The Finite Element Method

262

A solution domain is a region of space and time throughout which the mathematical model is defined and solutions are sought. In injection-molding simulation, the solution domain comprises the physical space through which the polymer melt flows and the time during which filling, packing, and cooling take place. In practice we normally have two components of the solution domain. The first is a description of the physical space in which a solution is sought. We refer to this as the geometric domain. The other is the time over which the solution is sought. In Section 14.2 and Chapter 13, the solution domain is extended, with an appropriate mathematical model and boundary conditions, to account for deformation of the part after molding and when exposed to environmental factors over some time period.

F.2

The Finite Element Approach

A finite element analysis may be broken into the following steps: ■

geometric modeling of the solution domain

■

meshing of the solution domain

■

derivation of element equations

■

assembly of element equations

■

application of boundary conditions

■

solution of system equations

■

display of results

Geometric modeling and meshing are often called preprocessing. This may also entail application of boundary conditions. For example, in filling analysis, users may be able to designate parts of the mold that have defined temperatures. Other basic boundary conditions for filling are the temperature of the melt, the flow rate of the melt, (which may be defined as an injection time; the flow rate is then calculated from the volume of the part), and the points of injection. Derivation of element equations, assembly of element equations, application of boundary conditions, and solution of the system equation are all performed by the flow analysis software. Results of finite element analysis are usually displayed graphically as color coded contours of the variable of interest.

F.2.1

Geometric Modeling of the Solution Domain

Geometric modeling is the process in which the user creates a representation of the domain in which a solution is sought. Sometimes this representation is called a model, more explicitly a model for analysis. This is quite different to the mathematical model we referred to above. It is restricted to the spatial domain of the solution. The concept of a “model” has changed markedly over the history of injection-molding simulation. Chapter 6 outlines some of the strategies for solution, but each of these requires a particular model for the solution domain. Some historical background to the different models used is provided in Appendix A.

F.2 The Finite Element Approach

F.2.2

263

Meshing

Meshing is the process in which the geometric domain is discretized into smaller regions called elements. Elements are defined by designating nodes. In midplane and Dual Domain analysis the elements are usually of triangular shape, and each requires three nodes for definition. It is however possible to define quadrilateral elements, each defined by four nodes. This is rare in commercial codes, as triangular mesh generation has reached a stage where it can be automatically accomplished. For 3D analysis, the elements may be tetrahedral or hexahedral and require four or eight nodes, respectively. Tetrahedral meshing is today quite automatic. However, automatic generation of a hexahedral mesh is not common, especially for complex geometries such as encountered in injection molding. Flow analysis must also consider the runner system. In early codes, runners were basically pipes. Each element was defined by two nodes, and the temperature and flow field were considered symmetric about the central axis. True 3D analysis has shown the error this can introduce. This is discussed in Section 6.4.

F.2.3

Derivation of Element Equations

Injection-molded parts often have complex shapes, and direct solution of the mathematical model is impossible. However, by breaking up the geometric domain into a number of smaller parts (elements), progress can be made. Due to the relatively simple shape and small size of the elements, algebraic equations that represent the mathematical model can be derived for each element. This entails making some approximations. Nevertheless these approximations are reasonable given that the element is a small region of the geometric spatial domain over which a solution is required. Algebraically the equations are identical for a given type of element. For example, all triangular elements in the mesh will have the same algebraic equations. Only the values of variables will change from element to element. This will usually reflect the change in material properties. Fortunately, element equations are handled by the software you use. You only need to specify the type of element you wish to use. The necessary mathematical analysis has been done by the software developer.

F.2.4

Assembly of Element Equations

The element equations are only valid for small regions of the geometric solution domain. To find a solution for the entire domain, the element equations are assembled to form system equations. This is performed by the software. The system equations comprise a large set of equations that are solved to provide a solution for the entire solution domain. System equations are represented in matrix form. For filling analysis the system equations may take the form {Q} = [K f ]{P } ,

(F.1)

264

F The Finite Element Method

where {Q} is the nodal flow rate vector, that is the flow rate into each node in the mesh. [K f ] is called the stiffness matrix and {P } is the nodal pressure vector. For another application such as structural analysis, the system equations may be written {F } = [K s ]{U } ,

(F.2)

where {F } is the nodal force vector, [K s ] is the stiffness matrix, and {U } is the nodal displacement vector. Note that Equations F.1 and F.2 have identical structures. This highlights the versatility of the FEM. All solutions using the FEM involve equations of this form.

F.2.5

Application of Boundary Conditions

The boundary conditions specify physical effects that cause the system to change. They are introduced by modifying the system equations. This aspect of analysis is performed automatically by the software, although the user gives the program some help. For example, in flow analysis, the user specifies the mold and melt temperature and the injection time. Nodes are convenient places to define boundary conditions. In flow analysis, the user may define an injection point at a node. The program can calculate the volume of the part, and using the user-specified value of injection time, calculate the required flow rate at the node.

F.2.6

Solution of the System Equations

System equations are solved using numerical techniques. There are two basic solution types: direct and iterative. Generally iterative solvers are used for flow analysis software. In practice the software user does not have to worry about solution of the equations. It is done by the chosen software. However users may be able to set tolerances on the accuracy of solution. Such a facility may be used to increase or decrease accuracy of the solution. While the latter may seem odd, decreased accuracy often leads to faster solutions. Consequently in a design situation, where you may want to try several gating positions, you could reduce tolerances to get a faster solution and so be able to assess more options.

F.2.7

Display of Results

Finite element analysis of complex parts that are modeled with many elements produces huge amounts of data. Typically this is stored in results files. Some results are nodal and others elemental. Nodal results are usually calculated at nodes and include scalar quantities such as pressure and temperature. Elemental results include velocity and shear rate. Note that many programs average elemental results to create a nodal result because this gives rise to “prettier” displays. However, the averaging can hide maximum and minimum values.

F.3 The Nature of a Finite Element Solution

F.3

265

The Nature of a Finite Element Solution

The previous section may appear abstract to the reader. To provide further relevance to the terms introduced earlier we now consider some specific examples in one and two dimensions. The first is the problem of approximating a simple curve by line segments. An example is shown in Figure F.1 where the exact solution is the dotted line and the solid line segments are ˜ i ) are approximations of the exthe approximation. The points defining the line segments u(x act solution at the points x i , i ∈ {1, . . . , 6}. It is apparent that the accuracy of the approximation

Figure F.1 Approximation of a simple curve

depends on ˜ i ), i ∈ {1, . . . , 6} approximate the exact solution; 1. the accuracy with which the points u(x ˜ i ) used to approximate the exact solution; 2. the number of points u(x 3. the shape of the line segments. In finite element terms, this approximation would be described as follows. Finite element solution of the problem will provide the approximate values of the exact solution at points that we ˜ i ), i ∈ {1, . . . , 6}. This is done by ensuring that for a given numcall nodes. These are denoted u(x ber of elements the approximate solution is “best” in some sense. We will try to sharpen this ˜ i ), i ∈ {1, . . . , 6} vague concept later in the appendix. The quality of the approximate values of u(x depends on the mathematical model, including boundary conditions, and how faithfully the model describes the physical process. There is also a need to have sufficient elements to cover the solution domain so that it makes sense to obtain approximate solutions. This is usually referred to as mesh density. All things being equal, a finer mesh, that is more elements, or a higher mesh density should improve the approximated solution.

266

F The Finite Element Method

The solution domain {x : x 1 ≤ x ≤ x 6 } is divided into five subdomains: {x : x 1 ≤ x ≤ x 2 }, {x : x 2 ≤ x ≤ x 3 }, {x : x 3 ≤ x ≤ x 4 }, {x : x 4 ≤ x ≤ x 5 }, and {x : x 5 ≤ x ≤ x 6 }. Each of these subdomains is called an element and is defined by the x-ordinates x i ∈ {1, . . . , 6} called nodes. Within the subdomain, we assume some variation of the variable. In our case we have used linear segments and thus assume a linear variation. This is called linear interpolation. Linear interpolation is the simplest, but it is possible to use higher order interpolation such as quadratic to achieve a better solution. For example, in our example if we used quadratic interpolation, each element may be able to provide a better approximation to the solution as it will be curved. Alternatively we may use smaller linear elements, that is, increase the mesh density. This area is still under research and goes under the topic of h − p approximation. Creating a finer mesh is called the h-approach whereas increasing the interpolation order is called the p-approach. Indeed there are some general purpose FEM codes that use both approaches but this is not yet used in molding analysis. Our first example was one dimensional in that the elements were line segments defined by two nodes. It is essential to use at least two-dimensional finite elements for molding analysis. The simplest type of element has a triangular shape. This is most common in molding simulation, as a grid of triangles may be generated automatically over a midplane geometry of an injection mold. As an example, Figure F.2 shows a finite element solution to a two-dimensional problem in which the elements are planar triangles. The planarity arises due to the assumption of linear interpolation of the solution throughout the element. However it is possible to increase the interpolation order, in which case the solution would be curved rather than having planar surfaces. Just as in the one-dimensional case we discussed earlier, the quality of approximated

Figure F.2 Finite element solution of a two-dimensional problem

solution is determined by the underlying mathematical model and boundary conditions and the number of elements used to discretize the domain.

F.4 Shape Functions

F.4

267

Shape Functions

As pointed out in the previous section, the way we interpolate the approximate solution between nodes (linear, quadratic, or higher order) affects the accuracy of the FEM solution. As in previous sections, when dealing with a solution to a problem with exact solution u(x), we ˜ will denote the approximate solution by u(x). The nodal values of the approximate solution will be denoted u i , where i is the node number. Consider Element 1 in the example depicted in Figure F.1. We will denote the approximate solution in Element 1 by u˜ (1) (x). This is a straight line and may be defined in terms of the nodal values u 1 and u 2 as follows: ³ x −x ´ ³ x −x ´ 1 2 u1 + u 2 , for x 1 ≤ x ≤ x 2 , u˜ (1) (x) = x2 − x1 x2 − x1 = N1(1) u 1 + N2(1) u 2 ,

(F.3)

where ³ x −x ´ 2 x2 − x1 ³ x −x ´ 1 . N2(1) = x2 − x1 N1(1) =

(F.4) (F.5)

The functions N1(1) and N2(1) are called the shape functions for element one. They define the type of interpolation, between nodal values, used within the element. As discussed earlier, it is possible to use higher order interpolation such as quadratic or cubic. We will consider only linear shape functions, but the principles outlined in this section are valid for higher order interpolation. Equation F.3 may be generalized for element number k defined by nodes numbered i and j with x-coordinates x i and x j , respectively, as follows: ³ xj − x ´ ³ x −x ´ i u˜ (k) (x) = ui + u 2 , for x 1 ≤ x ≤ x 2 x j − xi x j − xi = Ni(k) u i + N j(k) u j .

(F.6)

Interpolation provides the means of writing an approximate solution within an element in terms of the approximate nodal values u˜ i and u˜ j of the problem. However, u˜ i and u˜ j are unknown. They are merely an approximation to the real solution at nodes u i and u j (which of course are also unknown). In the following section we discuss a method of determining the “best” approximation for u˜ i and u˜ j .

F.5

Approximating Nodal Values

In Section F.3 when discussing the approximation of the curve shown in Figure F.1 we noted that the quality of approximation depends on several factors such as the number of points (nodes) used (mesh density), the shape of elements (and, particularly, their order), and the ˜ i ), i ∈ {1, . . . , 6} approximate the exact solution. This section accuracy with which the points u(x is devoted to the latter.

268

F The Finite Element Method

F.5.1

Weighted Residual Methods

Let D define a differential operator. By this we mean D(u) represents an expression involving the function u and its derivatives. For example we may have D(u) =

∂2 u ∂u − 3u 2 + 5u . ∂x 2 ∂x

(F.7)

A general partial differential equation can then be defined as D(u) − f = 0 ,

(F.8)

where f is some known function of the independent variables. Now suppose we substitute ˜ which is not an exact solution into Equation F.8. Then the right-hand side of some function u, ˜ so Equation F.8 will be nonzero. We call this a residual and denote this nonzero value by R(u) Equation F.8 becomes ˜ − f = R(u). ˜ D(u)

(F.9)

˜ to the actual solution u we seek to minimize the residual To find the “best” approximation (u) ˜ There are several methods that can be used; they are generally referred to as weighted R(u). residual methods. Burnett [48] provides details on the following: ■

collocation method

■

subdomain method

■

least squares method

■

Galerkin method

We note that the least squares method has become popular in recent years, particularly with regard to so-called “meshless” methods. Further information on these is provided in Section 14.5.2. Our focus here will be on the Galerkin method (sometimes called the Bubnov-Galerkin method). For the Galerkin method we enforce the following condition: Z ˜ i(k) d E = 0 , R(u)N

(F.10)

E

where the Ni(k) are the element shape functions defined in Section F.4 and E is the domain of the element. At this stage we say no more about this. However we will return to the Galerkin method in Section F.7 to derive element equations.

F.6

Constraint Equations

This section describes the way constraints are used to alter systems of equations. Constraints are implemented in the FEM during assembly of equations, and application of boundary conditions. We introduce the idea of constraint equations by example on a set of three equations.

F.6 Constraint Equations

269

Suppose we need to solve the following three equations: k 11 u 1 + k 12 u 2 + k 13 u 3 = F 1 k 21 u 1 + k 22 u 2 + k 23 u 3 = F 2

(F.11)

k 31 u 1 + k 32 u 2 + k 33 u 3 = F 3 , for the u i , where the k i j and F i are known. These equations may be written in matrix notation as follows:      k 11 k 12 k 13 u 1 F1 k 21 k 22 k 23  u 2  = F 2  . (F.12) k 31

k 32

k 33

u3

F3

We will refer to the square matrix with terms k i j as the stiffness matrix. A constraint equation for this set of equations describes a relationship between the unknowns u 1 , u 2 , and u 3 . A general constraint equation is c1 u1 + c2 u2 + c3 u3 = b ,

(F.13)

where c 1 , c 2 , c 3 , and b are known constants. If a constraint equation exists, u 1 , u 2 , and u 3 are no longer independent, and any one may be written in terms of the other two. This greatly simplifies the solution process. As an example we will write u 1 in terms of u 2 and u 3 . So we get u1 =

c3 b c2 − u2 − u3 . c1 c1 c1

(F.14)

Now substitute Equation F.14 in Equation F.11 and so eliminate u 1 to get k 11 k 21 k 31

³b c1 ³b c1 ³b c1

c2 c3 ´ u 2 − u 3 + k 12 u 2 + k 13 u 3 = F 1 , c1 c1 c2 c3 ´ − u 2 − u 3 + k 22 u 2 + k 23 u 3 = F 2 , c1 c1 c2 c3 ´ − u 2 − u 3 + k 32 u 2 + k 33 u 3 = F 3 . c1 c1 −

Rearranging we get ´ ³ ´ c2 c3 b k 11 u 2 + k 13 − k 11 u 3 = F 1 − k 11 c1 c1 c1 ³ ´ ³ ´ c2 c3 b k 22 − k 21 u 2 + k 23 − k 21 u 3 = F 2 − k 21 c1 c1 c1 ³ ´ ³ ´ c2 c3 b k 32 − k 31 u 2 + k 33 − k 31 u 3 = F 3 − k 31 c1 c1 c1 ³

k 12 −

which may be written in matrix form as     k 12 − (c 2 /c 1 )k 11 k 13 − (c 3 /c 1 )k 11 µ ¶ F 1 − (b/c 1 )k 11 u 2 k 22 − (c 2 /c 1 )k 21 k 23 − (c 3 /c 1 )k 21  = F 2 − (b/c 1 )k 21  . u3 k 31 − (c 2 /c 1 )k 31 k 33 − (c 3 /c 1 )k 31 F 3 − (b/c 1 )k 31

(F.15) (F.16) (F.17)

(F.18)

Comparing Equations F.12 and F.18, the procedure used to derive Equation F.18 may be considered as a set of column operations on Equation F.12, namely:

270

F The Finite Element Method

1. Multiply the first column of the stiffness matrix by −c 2 /c 1 and add to the second column. 2. Multiply the first column of the stiffness matrix by −c 3 /c 1 and add to the third column. 3. Multiply the first column of the stiffness matrix by b/c 1 and subtract from the right-hand-side matrix. 4. Delete the first column of the stiffness matrix and u 1 from the second matrix on the left-hand side. A similar set of row operations can then be performed on the rows of Equation F.18: 1. Multiply the first column of the new stiffness matrix in Equation F.18 and the right-hand-side matrix by −c 2 /c 1 and add to the second row. 2. Multiply the first column of the new stiffness matrix and the right-hand-side matrix by −c 3 /c 1 and add to the third row. 3. Delete the first row of the new stiffness matrix and the first row of the right-hand-side matrix. Applying these row operations to Equation F.18 gives  ¢ ¢   c2 c2 ¡ c3 c2 ¡ c2 c3 u2 k 22 − k 21 − k 12 − k 11 k 23 − k 21 − k 13 − k 11    c1 c1 c1 c1 c1 c1      =   ¢ ¢   c2 c2 ¡ c2 c3 c3 ¡ c3 k 32 − k 31 − k 12 − k 11 k 33 − k 31 − k 13 − k 11 u3 c1 c1 c1 c1 c1 c1   ¢ ¡ b c2 b F 2 − c k 21 − c F 1 − c k 11  1 1 1    .    ¢ b c3 ¡ b F 3 − k 31 − F 1 − k 11 c1 c1 c1

(F.19)

The original problem has now been reduced to two equations in two unknowns. It may be solved to find u 2 and u 3 whereupon u 1 may be determined from the constraint Equation F.14. Note that while this manipulation appears tedious when done manually, it is very easy to implement in a computer program. Note also that we chose to eliminate u 1 in this example. In principle it makes no difference which unknown we eliminate; however, for numerical stability, when using a computer to solve the equations, the u i with the largest coefficient is often the best choice. To conclude this section we consider two special cases of constraint equations. They are: 1. When two of the constraints are equal. That is, the constraint equation has the form u i = u j , i ̸= j . 2. When one of the constraints is a constant. That is, the constraint equation has the form u i = c. We will use both these types of constraint equation in later sections when discussing assembly of element equations and setting boundary conditions.

F.6 Constraint Equations

F.6.1

271

Special Case 1: Two Unknowns Equal

In this case we have the constraint equation u i = u j . This type of constraint is important when assembling the element equations to determine the system equations. We will consider the equations  (1) k 11 k (1)  21   0 0

(1) k 12 (1) k 22 0 0

0 0 (2) k 11 (2) k 21

    0 u1 F1     0   u 2  F 2  (2)    =    u 3  F 3  k 12 (2) u4 F4 k 22

(F.20)

subject to the constraint u2 = u3 .

(F.21)

Although the notation for the entries in the matrix on the left-hand side of Equation F.20 may appear strange, it will be explained later. For now we ask the reader to take the entries to be unique numbers. Writing the constraint in the general form of Equation F.13, c1 u1 + c2 u2 + c3 u3 + c4 u4 = b

(F.22)

with b = c 1 = c 4 = 0, c 2 = 1, and c 3 = −1 The column operations become 1. Multiply the second column of the stiffness matrix in Equation F.20 by 0 and add to the first column. 2. Multiply the second column of the stiffness matrix in Equation F.20 by 1 and add to the third column. 3. Multiply the second column of the stiffness matrix in Equation F.20 by 0 and add to the right-hand-side matrix. 4. Delete the second column of the stiffness matrix in Equation F.20 and u 2 . Of these, only Items 2 and 4 cause any change. When applied to Equation F.20 we get (1) k 11 k (1)  21   0 0



(1) k 12 (1) k 22 (2) k 11 (2) k 21

   0   F1 u 1   0     F 2  (2)  u 3 =   .  F 3  k 12 u4 (2) F4 k 22

(F.23)

The row operations become 1. Multiply the second row of the stiffness matrix and right-hand-side matrix in Equation F.23 by 0 and add to the first row. 2. Multiply the second row of the stiffness matrix and right-hand-side matrix in Equation F.23 by 1 and add to the third row. 3. Delete the second row of the stiffness matrix in Equation F.23 and the second row of the right-hand-side matrix.

272

F The Finite Element Method

Here, only list Items 2 and 3 have an effect. Applying these operations to Equation F.23 gives  (1)     (1) k 11 k 21 0 u1 F1  (1) (1) (2) (2)     (F.24) k 21 k 22 + k 11 k 12  u 3 = F 2 + F 3  . (2) (2) u4 F4 0 k 21 k 22 This may then be solved for u 1 , u 3 and u 4 . The value of u 2 then follows from Equation F.21.

F.6.2

Special Case 2: One Known Constraint

In this case the constraint equation is u i = constant. This type of constraint arises when applying boundary conditions. As an example we consider the following equations  k 11 k 21 k 31

k 12 k 22 k 32

    k 13 u 1 F1 k 23  u 2  = F 2  k 33 u 3 F3

(F.25)

with the constraint equation u 2 = c where c is a constant. Writing the constraint in the form of Equation F.14 we get u 2 = c + 0u 1 + 0u 3 .

(F.26)

Steps 1 and 2 of the column operations have no effect. Only Steps 3 and 4 are relevant and become 1. Multiply column two of the stiffness matrix in Equation F.25 by c and subtract from the right-hand-side matrix. 2. Delete the second column of stiffness matrix and u 2 . Applying these operations to Equation F.25 gives  k 11 k 21 k 31

   k 13 µ ¶ F 1 − ck 12 u 1 k 23  = F 2 − ck 22  . u3 k 33 F 3 − ck 32

(F.27)

Steps 1 and 2 of the row operations have no effect. Only Step 3 need be done, and it becomes: delete the second row of the stiffness matrix and the right-hand-side matrix. Hence Equation F.27 becomes · ¸µ ¶ µ ¶ k 11 k 13 u 1 F 1 − ck 12 = . (F.28) k 31 k 33 u 3 F 3 − ck 32 This may be solved for u 1 and u 3 . Note that if c = 0, that is u 2 = 0, then Equation F.28 becomes · k 11 k 31

k 13 k 33

¸µ

¶ µ ¶ u1 F1 = u3 F3

(F.29)

Comparing this with the original Equation F.25 we see that the constraint u 2 = 0 may be applied by deleting the second row and second column from each of the matrices in Equation F.25. In general, the constraint u i = 0 is applied by deleting the i t h rows and columns from the relevant matrices.

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

F.7

273

A One-Dimensional Problem Solved Using the FEM

In this section we illustrate many of the important concepts of finite element analysis in a simple example. We will solve the following equation d 2u − 2 = 0, d x2

for

1≤x ≤4

(F.30)

subject to the boundary conditions du = 3, at x = 1 dx u(x) = 20, at x = 4 .

(F.31) (F.32)

This equation may be solved exactly by integrating twice and applying the boundary conditions. The solution is u(x) = x 2 + x

(F.33)

and is shown graphically in Figure F.3. We will use the FEM to derive an approximate solution and compare it to the exact solution.

Figure F.3 Exact solution to 1D FEM sample problem

F.7.1

Meshing

The domain for our problem is {x : 1 ≤ x ≤ 4}. For the FEM we will break this up into three elements, each of which is defined by two nodes as shown in Figure F.4. A better solution can be found by using more elements (a finer mesh), but this will make the hand calculations too tedious. In total then we have three elements and four nodes. We need to approximate these four nodal values.

274

F The Finite Element Method

F.7.2

Derivation of Element Equations

We will use linear shape functions in this example. That is we seek a solution of the type shown in Figure F.5. Let u˜ 1 , u˜ 2 , u˜ 3 , and u˜ 4 be the approximate solution to the problem at Nodes 1, 2, 3 and 4, respectively. These are the unknowns that we will determine with the FEM. Let u˜ (1) , u˜ (2) and u˜ (3) represent the approximate solution in Elements 1, 2, and 3, respectively. The first step is to represent the approximate solution for each element in terms of the unknown nodal values u˜ 1 , u˜ 2 , u˜ 3 , and u˜ 4 . For Element 1 we have u˜ (1) (x) = (2 − x)u˜ 1 + (x − 1)u˜ 2 = N1(1) u˜ 1 + N2(1) u˜ 2 ,

(F.34)

where N1(1) = (2 − x)

(F.35)

N2(1)

(F.36)

= (x − 1)

are the shape functions for Element 1 and defined in Section F.4. Similarly for Element 2 we write u˜ (2) (x) = (3 − x)u˜ 2 + (x − 2)u˜ 3 = N1(2) u˜ 2 + N2(2) u˜ 3 ,

(F.37)

where N1(2) = (3 − x)

(F.38)

N2(2) = (x − 2) .

(F.39)

For Element 3 we have u˜ (3) (x) = (4 − x)u˜ 3 + (x − 3)u˜ 4 = N1(3) u˜ 3 + N2(3) u˜ 4 ,

(F.40)

where N1(3) = (4 − x)

(F.41)

N2(3)

(F.42)

= (x − 3) .

The residual (see Section F.5.1) for each element is given by ³ ´ d 2 ³ (e) ´ u˜ (x) − 2 R u˜ (e) (x) = d x2 ³ d d u˜ (e) ´ − 2, where e = 1, 2, 3. = dx dx

Figure F.4 Mesh for sample problem

(F.43)

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

275

Figure F.5 Finite element solution for sample problem using linear interpolation

Galerkin’s method (see Section F.5.1) requires that for each element Z xb ³ ´ R u˜ (e) (x) Ni(e) d x = 0, where i = 1, 2

(F.44)

xa

where x a and x b are the x-coordinates for the nodes defining element number e. Note that for each element, Equation F.44 gives two equations. Substituting Equation F.43 into Equation F.44 we get Z 0=

xa

Z =

i d ³ d u˜ (e) ´ − 2 Ni(e) d x dx dx Z xb d ³ d u˜ (e) ´ (e) Ni d x − 2Ni(e) d x . dx dx xa

xb h

xb xa

(F.45)

The first term on the right-hand side may be integrated by parts to give Z

xb xa

h d u˜ (e) ix=xb d ³ d u˜ (e) ´ (e) − Ni d x = Ni(e) dx dx dx x=x a

Z

xb xa

(e) d u˜ (e) d Ni dx . dx dx

(F.46)

Substituting Equation F.46 into Equation F.45 then gives 0=

Z xb h d u˜ (e) ix=xb Z xb d u˜ (e) d N (e) i Ni(e) − dx − 2Ni(e) d x . dx dx dx x=x a xa xa

(F.47)

Rearranging we have Z

xb xa

(e) h d u˜ (e) ix=xb d u˜ (e) d Ni dx = Ni(e) − dx dx dx x=x a

Z

xb xa

2Ni(e) d x .

(F.48)

F The Finite Element Method

276

Consider now the left-hand side of the above equation. Substituting u˜ (1) from Equation F.34 and changing the integration limits gives Z

2

1

(1) d u˜ (1) d Ni dx = dx dx

(1)

´dN d ³ (1) i N1 u˜ 1 + N2(1) u˜ 2 dx dx 1 dx µZ 2 ¶ (1) d N1(1) d Ni = d x u˜ 1 dx dx 1 µZ 2 ¶ (1) (1) d N2 d Ni + d x u˜ 2 . dx dx 1 Z

2

(F.49)

Substituting the above equation into Equation F.48 we find the following two equations for Element 1, corresponding to i = 1 and i = 2 respectively: µZ

2

d N1(1) d N1(1) dx

1

dx

µZ ¶ d x u˜ 1 +

2

d N2(1) d N1(1) dx

1

dx ¸x=2

d u˜ (1) (1) = N1 dx ·

µZ

2

d N1(1) d N2(1)

1

dx

dx

µZ ¶ d x u˜ 1 +

2

dx

x=1

2

1

2N1(1) d x

(F.50)

2N2(1) d x

(F.51)

¶ d x u˜ 2

dx ¸x=2

d u˜ (1) (1) = N2 dx ·

Z −

d N2(1) d N2(1)

1

¶ d x u˜ 2

Z −

x=1

1

2

Equations F.50 and F.51 may be written in matrix form as follows: Ã ! " # (1) (1) µ ¶ u˜ 1 F 1(1) k 11 k 12 = (1) (1) (1) u˜ 2 F2 k 21 k 22 where (1) k 11 = (1) = k 12 (1) k 21 = (1) k 22 =

F 1(1) =

Z

2

dx

1

Z

2

2

Z 1

2

dx

d N1(1) d N2(1) dx

1

dx

d N2(1) d N1(1) dx

1

Z

d N1(1) d N1(1)

dx

d N2(1) d N2(1) dx

h d u˜ (1)

N1(1)

dx ix=2

dx = 1, d x = −1 , d x = −1 , dx = 1, Z

2

2N1(1) d x dx x=1 1 ³ d u˜ (1) ´ = −1, d x x=1 h d u˜ (1) ix=2 Z 2 F 2(1) = N2(1) − 2N2(1) d x dx x=1 1 ³ d u˜ (1) ´ = −1. d x x=2 −

(F.52)

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

277

The terms k i(1) and F i(1) above were evaluated by first substituting for u˜ (1) and Ni(1) using Equaj tions F.34 and F.35 respectively, and then carrying out the integrations. Equations F.52 are called the element equations for Element 1. The square matrix on the lefthand side of Equations F.52 is called the stiffness matrix for Element 1. An important property of the stiffness matrix is its symmetry, that is k i(1) = k (1) . j ji Using the same steps used to produce Equations F.52, it is possible to write similar equations for Elements 2 and 3. In fact the only changes required are 1. Replace the superscript (1) by (2) for Element 2 and (3) for Element 3. 2. Change the limits of integration for each element according to the mesh we defined in Section F.7.1. 3. Change the subscripts on the nodal unknowns accordingly. For Element 2 we get "

(2) k 11 (2) k 21

(2) k 12 (2) k 22

! #µ ¶ Ã u˜ 2 F 1(2) = (2) u˜ 3 F2

(F.53)

where (2) k 11 = (2) k 12 = (2) k 21 = (2) k 22 =

3

Z

d N1(2) d N1(2) dx

2 3

Z

d N1(2) d N2(2) dx

2 3

Z

dx 3

dx

d N2(2) d N1(2)

2

Z

dx

dx

d N2(2) d N2(2)

2

dx

h d u˜ (2)

d x = −1 , d x = −1 , dx = 1, Z

3

2N1(2) d x dx x=2 2 ³ d u˜ (2) ´ = −1, d x x=2 h d u˜ (2) ix=3 Z 3 F 2(2) = N2(2) − 2N2(2) d x dx x=2 2 ³ d u˜ (2) ´ = −1. d x x=3

F 1(2) =

N1(2)

dx ix=3

dx = 1,

−

Finally, for Element 3, we get "

(3) k 11 (3) k 21

(3) k 12 (3) k 22

#µ ¶ Ã ! u˜ 3 F 1(3) = (3) u˜ 4 F2

(F.54)

278

F The Finite Element Method

where (3) k 11 = (3) k 12 = (3) k 21 = (3) k 22 =

4

Z

d N1(3) d N1(3) dx

3 4

Z

d N1(3) d N2(3) dx

3 4

Z

dx 4

dx

d N2(3) d N1(3)

3

Z

dx

dx

d N2(3) d N2(3)

3

dx

h d u˜ (3)

dx ix=4

dx = 1, d x = −1 , d x = −1 , dx = 1, Z

4

2N1(3) d x dx x=3 3 ³ d u˜ (3) ´ = −1, d x x=3 h d u˜ (3) ix=4 Z 4 F 2(3) = N2(3) − 2N2(3) d x dx x=3 3 ³ d u˜ (3) ´ = −1. d x x=4

F 1(3) =

N1(3)

−

This completes the derivation of the element equations. Though tedious by hand, the procedure is readily implemented on a computer. Before considering the assembly of the element equations, we review the main points so far. First we chose linear interpolation for the value of u˜ between two nodes (or within an element). This allowed us to define the shape functions for each element. The Galerkin method was then applied to each element to minimize the residual (and the error in the approximation). This resulted in two algebraic equations for each element, which may be expressed in matrix form, namely, Equations F.52, F.53 and F.54. This last point is crucial. In effect the FEM procedure reduced a differential equation to a set of algebraic equations for the unknown u˜ i .

F.7.3

Assembly

For the assembly description, we introduce some new notation to associate the nodal unknowns to the element(s) they are associated with. Generally, u˜ i(e) is the value of the approximate solution at Node i that is associated with element number e. Using this notation, the element Equations F.52, F.53, and F.54 can be written as a single matrix equation:  (1) k 11  (1) k 21   0   0    0 0

(1) k 12 (1) k 22 0 0 0 0

0 0 (2) k 11 (2) k 21 0 0

0 0 (2) k 12 (2) k 22 0 0

0 0 0 0 (3) k 11 (3) k 21

  (1)   (1)  u˜ 1 F1 0     0  u˜ 2(1)  F 2(1)    (2)   (2)      0   u˜ 2  = F 1  . u˜ (2)  F (2)  0   3   2  (3)   (3)   (3)   u˜ 3  F 1  k 12 (3) u˜ 4(3) F 2(3) k 22

(F.55)

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

279

Now consider the terms u˜ 2(1) and u˜ 2(2) . These are the values of the approximate solution at x = 2 for Elements 1 and 2 respectively. Substituting x = 2 in Equations F.34 and F.37 gives u˜ 2(1) (2) = (2 − 1)u˜ 2 = u˜ 2 u˜ 2(2) (2) = (3 − 2)u˜ 2 = u˜ 2 . Therefore u˜ 2(1) (2) = u˜ 2(2) (2) = u˜ 2 .

(F.56)

That this is so is no great surprise. Intuitively any “reasonable” approximation should be continuous across the node at x = 2. Indeed we assumed it in Figure F.5. This requirement is called an inter-element boundary condition (IBC). Assembly of the element equations is the process of enforcing the IBCs. In a similar way we find from Equations F.37 and F.40 that u˜ 2(3) (3) = u˜ 3(3) (3) = u˜ 3 .

(F.57)

Equations F.56 and F.57 are constraint equations on the system of equations Equation F.55 of the kind discussed in Section F.6.1. Applying the row and column operations of Section F.6.1 to Equations F.55 we enforce the IBCs and so assemble the element equations to obtain the following system equations:  (1) k 11 k (1)  21   0 0

(1) k 12 (1) (2) k 22 + k 11 (2) k 21 0

    F 1(1) 0 u˜ 1    (1) (2)  0   u˜ 2  F 2 + F 1  (3)  . (3)    =  (2) k 12  u˜ 3  F 2 + F 1  (3) u˜ 4 F 2(3) k 22

0 (2) k 12 (2) (3) k 22 + k 11 (3) k 21

(F.58)

These equations represent the FEM solution to our problem over the whole solution domain, namely, {x : 1 ≤ x ≤ 4}. In the next section we discuss the application of boundary conditions. Before this we substitute values for the terms in Equation F.58. These were given in Section F.7.2. Upon substitution, Equation F.58 becomes  ³ d u˜ 1 ´ − − 1     ³ (1) ´ d x ³ x=1(2) ´   d u˜  0 u˜ 1 ˜ d u     − − 2  0  u˜ 2   .    = ³ d x(2) ´x=2 ³ d x(3) ´x=2  ˜ ˜ d u d u −1 u˜ 3    − − 2   0 u˜ 4 d x x=3  d x x=3  ³ d u˜ (3) ´   −1 d x x=4 



1 −1   0 0

−1 2 −1 0

0 −1 2 −1

(F.59)

280

F The Finite Element Method

F.7.4

Application of Boundary Conditions

The boundary conditions were given in Section F.7 and are as follows: du = 3, at x = 1 dx u(x) = 20, at x = 4 .

(F.60) (F.61)

The first of these may be applied by substitution in Equation F.59 and gives 



1 −1   0 0

−1 2 −1 0

0 −1 2 −1

 −4 ³ d u˜ (2) ´ ³ d u˜ (1) ´   u˜ 1 0 − − 2       d x x=2 d x x=2 0  u˜ 2   ³ ³ ´ ´  . (3) (2) =   d u˜ d u˜  −1 u˜ 3   − − 2   d x x=3  d x x=3 u˜ 4 0 ³ d u˜ (3) ´   −1 d x x=4 



(F.62)

The second boundary condition u(4) = 20 is a constraint equation of the type dealt with in Section F.6.2 and is applied using the row and column operations of Section F.6.2. That is, we multiply Column 4 of the stiffness equation in Equation F.62 by 20, subtract this from the righthand-side matrix, delete Column 4 of the stiffness matrix, delete u 4 , and finally delete Row 4 of the stiffness matrix and the right-hand-side matrix. This gives

1 −1 0 

−1 2 −1

  −4   ³ d u˜ (2) ´  ³ d u˜ (1) ´  u˜ 1 0   − −2  . −1 u˜ 2  =   d x x=2  d x x=2 ³ d u˜ (3) ´  ³ d u˜ (2) ´ u˜ 3 2 − + 18 d x x=3 d x x=3

(F.63)

We now consider the right-hand-side terms: ³ d u˜ (1) ´

³ d u˜ (2) ´

−2 d x x=2 d x x=2 ³ d u˜ (3) ´ ³ d u˜ (2) ´ − + 18 . d x x=3 d x x=3 −

(F.64) (F.65)

Since our problem involves a second derivative, the first derivative d u/d x must be continuous over the solution domain. In our example, the first derivatives of the approximate solution d u˜ (i )/d x will not be continuous across the nodes due to our use of linear shape functions. Despite this fact, we enforce continuity of first derivatives on our approximate solution. That is, we set ³ d u˜ (1) ´

=

³ d u˜ (2) ´

d x x=2 d x x=2 ³ d u˜ (2) ´ ³ d u˜ (3) ´ = . d x x=3 d x x=3

(F.66) (F.67)

Equations F.66 are referred to as natural boundary conditions. They arise “naturally” from the finite element formulation. Substituting Equation F.66 into the right-hand side of Equation

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

281

F.63 gives 1 −1 0 

−1 2 −1

    0 u˜ 1 −4 −1 u˜ 2  = −2 . 2 u˜ 3 18

(F.68)

Equation F.68 is the final system of equations for the FEM solution of our problem. All that remains to do is to solve it.

F.7.5

Solution of System Equations

Our example has led to a simple set of equations that may be solved using elementary methods. Writing the equations separately we have u˜ 1 − u˜ 2

=

−4

−u˜ 1 + 2u˜ 2 − u˜ 3

=

−2

−u˜ 2 + 2u˜ 3

=

18 .

(F.69)

Adding these equations we get u˜ 3 = 12. Substituting this in the last of Equations F.69 gives u˜ 2 = 6. Finally substitution of this in the first equation gives u˜ 1 = 2. From the boundary condition given in Section F.7 we know u˜ 4 = u 4 = 20. It should be noted that real finite element problems can involve the solution of hundreds of thousands of linear equations. In these cases elementary methods are insufficient and a numerical method is required. Our approximate solution over the entire domain is determined by substituting the approximated nodal values into the element equations namely, Equations F.34, F.37, and F.40. We obtain u (1) (x) = 4x − 2 for 1 ≤ x ≤ 2 , u (2) (x) = 6x − 6 for 2 ≤ x ≤ 3 , u

(3)

(F.70)

(x) = 8x − 12 for 3 ≤ x ≤ 4 .

The exact solution and the approximation of Equation F.70 are shown in Figure F.6. The reader should note that our example of solving the simple one-dimensional problem using FEM is like driving in a thumbtack with a sledge hammer. The work involved in carrying out an FEM solution cannot be justified for simple problems. However, when no exact solutions are available for a problem, especially a large scale engineering problem, the FEM is a wonderful tool. While the hand calculations above are tedious, they may be readily implemented in computer code. It is this capability that makes the FEM so attractive for large scale engineering problems.

282

F The Finite Element Method

Figure F.6 Comparison of exact and approximate FEM solution for the sample problem

G

Numerical Methods for the 2.5D Approximation

In this appendix we consider the numerical solution of the 2.5D approximation described in Section 5.6. In particular, we will use the concepts of the previous appendix to derive a finite element approach for the 2.5D approximation. We also use the concepts of the finite difference method, introduced in Appendix E, to solve the energy equation for the 2.5D approximation.

G.1

Overview of Solution Process

Recall from Section 5.6.5.1 that the 2.5D approximation results in two equations—one for pressure and another for temperature—which may be written, respectively, as µ³ ¶ ∂p ´2 ³ ∂p ´2 ∂p + d (x, y) + − b(x, y) ∂t ∂x ∂y ∂ ³ ∂p ´ ∂ ³ ∂p ´ − 2S 2 − 2S 2 , ∂x ∂x ∂y ∂y

0 = a(x, y)

(G.1)

and the simplified energy equation ρc p

³ ∂T ∂t

+ vx

∂T ∂T ´ ∂p ∂2 T + vy = βT + ηγ˙ 2 + k 2 . ∂x ∂y ∂t ∂z

(G.2)

These equations are to be solved subject to the boundary conditions discussed in Section 4.6. These are briefly repeated below. Boundary conditions relating to pressure are: ■

At any impermeable boundary, the pressure gradient in the normal direction to the boundary is zero. That is, ∂p = 0 on Σem , ΣW+ , ΣW− , Σins . ∂n

■

(G.3)

The melt flow rate q, or the pressure p, is specified on the surface where the melt enters the cavity. That is, q = q inj or p = p inj on Σinj .

(G.4)

The packing phase boundary condition usually specifies a pressure at the point(s) of injection. In basic software, this may be constant. However in order to ensure a more uniform

284

G Numerical Methods for the 2.5D Approximation

density, pressure may be varied during the packing phase. This usually consists of a reduction in packing pressure as the material solidifies. This idea has been used as the basis of some molding machine control systems. An early example was given by Michaeli and Lauterbach [249]. ■

Assuming the pressure datum is atmospheric, the pressure is zero at the melt front. That is, p = 0 on Σmf .

(G.5)

The boundary conditions relating to temperature are: ■

The temperature profile through the cavity thickness, T (z), is prescribed for the surface through which the melt is injected. That is, T (z) = Tinj (z) .

■

(G.6)

The temperature, T , is prescribed on all mold boundaries however, it is usual to allow different temperatures to be prescribed on each boundary, such as T = Tem on Σem , T = TW + on ΣW+ , T = TW − on ΣW− , T = Tins on Σins . Alternatively, instead of temperature, or in conjunction with temperature, the heat flux across a boundary may be defined. This approach is often used when a cooling analysis is used with a fill and pack analysis.

Since the fluidity, S 2 , depends on viscosity, which depends on temperature and shear rate, Equations G.1 and G.2 must be solved simultaneously. For numerical solution, however, the equations are decoupled by using small time increments. This means that for a short time, temperature is assumed constant while pressure is calculated using the viscosity at that temperature. Providing the time increments are sufficiently small, this method gives satisfactory results.

G.1.1

Numerical Methods

Early schemes for simulation of injection molding frequently further simplified the pressure equation by reducing it to a one-dimensional equation. Often these programs used finite difference methods to solve equations for temperature and pressure. The problem here is mesh generation in complex geometries. While excellent for developing theories and results on simple geometries, they lacked “industrial strength.” Their use has been restricted to academic studies while commercial injection-molding simulation adopted other methods that were better suited to complex geometry and the development of computer-aided design (CAD) systems. However, it should be noted that some of the early academic finite difference codes introduced advanced material properties such as crystallization and viscoelasticity. However, the problem was that they could be used only in the simplest mold geometries such as circles

G.2 Finite Element Formulation for the Pressure Field

285

and rectangles. Further information is presented in Appendix A, which discusses the history of molding simulation. Today all commercial codes use a hybrid combination of finite elements and finite differences when using the 2.5D approximation. We focus on this in this appendix. More specifically the pressure equation, Equation G.1, is solved using the finite element method and the energy equation, Equation G.2, is solved using the finite difference method. Brief overviews of these methods are given in Appendices E and F . Readers unfamiliar with numerical methods may like to read these before going to the next sections of this appendix. There are some commercial codes that solve the full 3D set of equations. These do not make the thin wall assumption that gives rise to the 2.5D approximation. At first sight this may seem a more accurate approach and mathematically it is. However there are implementation issues that could result in 3D codes being less accurate than those using the 2.5D approximation. An overview of these methods and their practical complications is provided in Chapter 6.

G.2

Finite Element Formulation for the Pressure Field

To define a solution mesh, the midplane of the part (where z = 0) is covered with a net of triangular elements. Although other shapes, such as quadrilaterals, may be used, triangles can “fill” any shape and in particular, the complex shapes of injection-molded parts. Moreover, it is possible to automatically generate this net of triangular elements. The latter point is important as manual mesh generation requires user interaction and consequently is expensive. So the mesh for the pressure field comprises triangles, each of which is defined by three points. Runners are modeled with cylindrical elements. These are defined by two nodes defining the length of the runner element.

G.2.1

Interpolation Functions

As the cavity is meshed with triangular elements we need to choose appropriate interpolation functions for pressure over the element. The original paper of Hieber and Shen [154], which introduced the hybrid FEM/FDM approach, used quadratic interpolation. However, commercial software generally uses linear interpolation, and so the pressure at any point (x, y) within element number e is given by: p˜ (e) (x, y) =

3 X

Ni p˜i

(G.7)

i =1

where Ni are the linear shape functions, and the p˜i are the approximate nodal pressures at Node i , which are to be determined by the FEM. The shape functions take a particularly simple form if we work in area coordinates [66]. Sometimes these are called barycentric coordinates.

286

G.2.2

G Numerical Methods for the 2.5D Approximation

Area Coordinates

Consider the triangular element shown in Figure G.1. Area coordinates are based on the fact

Figure G.1 Area (barycentric) coordinates for triangular elements

that lines from the triangle’s vertices to a point P (x, y) in the triangle divide the triangle into three areas. Area coordinates ξ1 , ξ2 , and ξ3 are defined as follows: A1 A2 A3 , ξ2 = , ξ3 = , A A A where A is the area of the triangular element. Note that ξ1 =

(G.8)

ξ1 + ξ2 + ξ3 = 1 .

(G.9)

The relationship between the global x-y coordinate system and the area coordinates is given by      1 1 1 1 ξ1  x  =  x 1 x 2 x 3  ξ2  , (G.10) y y 1 y 2 y 3 ξ3 and this may be inverted to give    x 2 y 3 − x 3 y 2 y 23 ξ1 1 x 3 y 1 − x 1 y 3 y 31 ξ2  = 2A x 1 y 2 − x 2 y 1 y 12 ξ3

  x 32 1 x 13  x  . x 21 y

(G.11)

Using area coordinates the Ni of Equation G.7 take the simple form N 1 = ξ1 , N 2 = ξ2 , N 3 = ξ3 .

(G.12)

G.3 Finite Element Derivation

287

Hence Equation G.7 may be written in area coordinates as p˜ (e) = (ξ1 p˜1 + ξ2 p˜2 + ξ3 p˜3 ) .

G.3

(G.13)

Finite Element Derivation

For convenience, we now introduce a more concise notation. We write ∂ ³ ∂p ´ ∂ ³ ∂p ´ 2S 2 + 2S 2 = 2∇ · (S 2 ∇p) , ∂x ∂x ∂y ∂y

(G.14)

where ∇ denotes the two-dimensional divergence and gradient operator. Hence the pressure equation, Equation G.1, may be written as 0 = a(x, y)

µ³ ¶ ∂p ∂p ´2 ³ ∂p ´2 + d (x, y) + − b(x, y) − 2∇ · (S 2 ∇p) . ∂t ∂x ∂y

(G.15)

Now returning to the finite element derivation, in Appendix F we defined weighted residual methods for a one-dimensional problem. Equation G.15 is two-dimensional but the process is the same. We define a differential operator D as follows: µ³ ¶ ∂p ∂p ´2 ³ ∂p ´2 D(p) = a(x, y) − 2∇ · (S 2 ∇p) + d (x, y) + . (G.16) ∂t ∂x ∂y So Equation G.15 may be written: D(p) − b(x, y) = 0 .

(G.17)

˜ y) into Equation G.17 we Now, as in Appendix F if we substitute an approximate solution p(x, ˜ That is, Equation G.17 becomes will get some error R(p). ˜ − b(x, y) = R(p) ˜ . D(p)

(G.18)

˜ in the sense The Galerkin method, introduced in Appendix F, seeks to minimize the error R(p) that Z ˜ i(e) d A = 0 for i ∈ {1, 2, 3} , R(p)N (G.19) A

where the Ni(e) are the element shape functions for element e, and A is the domain of the triangular element. For our problem, Equation G.19 becomes Z · 0=

a(x, y) A

µ³ ¶ ∂p˜ ∂p˜ ´2 ³ ∂p˜ ´2 ˜ + d (x, y) − 2∇ · (S 2 ∇p) + ∂t ∂x ∂y

¸ − b(x, y) Ni d A , for i ∈ {1, 2, 3}.

(G.20)

288

G Numerical Methods for the 2.5D Approximation

Consider now the first term: Z ∂p˜ Ni d A . a(x, y) ∂t A

(G.21)

From Equation 5.64 we have Z a(x, y) =

+H ³ −H

κ−

β2 T ´ dz . ρc p

(G.22)

Assuming no variation of κ, β, ρ, and c p over the element, and substituting for p˜ and Ni from Equations G.7 and G.12 respectively, Equation G.21 becomes Z Z ∂p˜ ∂p˜ a(x, y) Ni d A = a(x, y) Ni d A ∂t A A ∂t Z ¢ ∂¡ ξ1 p˜1 + ξ2 p˜2 + ξ3 p˜3 Ni d A = a(x, y) ∂t A Z ³ ∂p˜1 ∂p˜2 ∂p˜3 ´ = a(x, y) ξ1 ξi + ξ2 ξi + ξ3 ξi d A ∂t ∂t ∂t A Z Z n ∂p˜ ∂p˜2 1 ξ1 ξ i d A + ξ2 ξi d A = a(x, y) ∂t A ∂t A Z o ∂p˜3 + ξ3 ξi d A . (G.23) ∂t A For i = 1, 2, and 3, Equation G.23 gives the following three equations: Z Z n ∂p˜ Z ∂p˜2 ∂p˜ 1 ξ1 ξ1 d A + ξ2 ξ1 d A a(x, y) N1 d A = a(x, y) ∂t A ∂t A A ∂t Z o ∂p˜3 + ξ3 ξ1 d A , ∂t A Z a(x, y) A

Z a(x, y) A

Z n ∂p˜ Z ∂p˜ ∂p˜2 1 N2 d A = a(x, y) ξ1 ξ2 d A + ξ2 ξ2 d A ∂t ∂t A ∂t A Z o ∂p˜3 + ξ3 ξ2 d A , ∂t A Z n ∂p˜ Z ∂p˜ ∂p˜2 1 N3 d A = a(x, y) ξ1 ξ3 d A + ξ2 ξ3 d A ∂t ∂t A ∂t A Z o ∂p˜3 + ξ3 ξ3 d A . ∂t A

(G.24)

(G.25)

(G.26)

It is possible to evaluate the integrals on the right-hand side (RHS) of Equations G.24 to G.26 using the formula [66]: Z k! l ! m! ξk1 ξl2 ξm , (G.27) 3 d A = 2A (2 + k + l + m)! A where A is the area of the element. Hence Equations G.24 to G.26 become Z ∂p˜ a(x, y)A n ∂p˜1 ∂p˜2 ∂p˜3 o a(x, y) N1 d A = 2 + + , 12 ∂t ∂t ∂t A ∂t

(G.28)

G.3 Finite Element Derivation

289

(G.29)

A

∂p˜ a(x, y)A n ∂p˜1 ∂p˜2 ∂p˜3 o N2 d A = +2 + , ∂t 12 ∂t ∂t ∂t

(G.30)

A

a(x, y)A n ∂p˜1 ∂p˜2 ∂p˜3 o ∂p˜ N3 d A = + +2 . ∂t 12 ∂t ∂t ∂t

Z a(x, y)

Z a(x, y)

Equations G.28 to G.30 may be written in matrix form as 

Z a(x, y)

 ZA   a(x, y)  ZA  a(x, y) A

 ∂p˜ N1 d A   ∂t 2 1  A ∂p˜  a(x, y) 1 2 N2 d A  =  12 ∂t  1 1 ∂p˜ N3 d A ∂t A = a(x, y)[C ]e p˙ , 12

  1 p˙1 1 p˙2  2 p˙3

(G.31)

where  2 [C ] = 1 1 e

1 2 1

 1 1 , 2

(G.32)

and is called the “compliance matrix” for the element “e.” Consider now the second term of Equation G.20, namely Z Z ˜ i d A = 2 ∇ · (S 2 ∇p)N ˜ i dA. 2∇ · (S 2 ∇p)N A

(G.33)

A

The divergence of the product of a scalar a and vector b is given by Equation B.51. Setting s = Ni and applying this identity to Equation G.33 gives: Z Z Z ˜ i d A = 2 ∇ · (Ni S 2 ∇p) ˜ d A − 2 S 2 ∇p˜ · ∇Ni d A . 2 ∇ · (S 2 ∇p)N (G.34) A

A

A

Applying the divergence theorem (see Section 4.2.2) to the first term on the RHS we obtain Z Z Z ˜ i d A = 2 (Ni S 2 ∇p) ˜ · n d Σ − 2 S 2 ∇p˜ · ∇Ni d A , 2 ∇ · (S 2 ∇p)N (G.35) A

Σ

A

where Σ is the boundary of the element. In order to use Equation G.35 we need expressions for ∇p˜ and ∇Ni . To this end, note that from Equation G.13 we can write ∂ ∂p˜ = (ξ1 p 1 + ξ2 p 2 + ξ3 p 3 ) , ∂x ∂x

(G.36)

∂p˜ ∂ = (ξ1 p 1 + ξ2 p 2 + ξ3 p 3 ) . ∂y ∂y

(G.37)

290

G Numerical Methods for the 2.5D Approximation

Differentiating Equation G.11 we obtain ∂ξ1 y 23 = ∂x 2A

∂ξ1 x 32 = ∂y 2A

(G.38)

∂ξ2 y 31 = ∂x 2A

∂ξ2 x 13 = ∂y 2A

(G.39)

∂ξ3 y 12 = ∂x 2A

∂ξ3 x 21 = , ∂y 2A

(G.40)

and so we have ´ ´ ∂ ³ ∂ ³ ξ1 p˜1 + ξ2 p˜2 + ξ3 p˜3 eˆ1 + ξ1 p˜1 + ξ2 p˜2 + ξ3 p˜3 eˆ2 ∂x ∂y ´ ´ 1 ³ 1 ³ = y 23 p˜1 + y 31 p˜2 + y 12 p˜3 eˆ1 + x 32 p˜1 + x 13 p˜2 + x 21 p˜3 eˆ2 . 2A 2A

∇p˜ =

(G.41)

We can use Equation G.12 to get the following expression for ∇Ni : ∇Ni = ∇ξi .

(G.42)

Equations G.41 and G.42 may be substituted into Equation G.35 to get three equations, each corresponding to i = 1, 2, 3: Z Z Z ˜ i d A = 2 (Ni S 2 ∇p) ˜ · n d Σ − 2 S 2 ∇p˜ · ∇N1 d A . 2 ∇ · (S 2 ∇p)N (G.43) A

Σ

A

We now evaluate the terms on the right-hand side of Equation G.43. To begin, we assume that at any instant of time, S 2 is constant with respect to x and y over the element. This sounds like a big simplification, but remember that the material properties will be changing more rapidly in the z-direction where there are high temperature and shear rate gradients. In Figure G.2, we depict the element. The ni are unit outward normals to the element boundary which comprises three lines, Σ1 , Σ2 , and Σ3 . Because N1 = ξ1 is identically zero on Σ1 , the only contribution to the first term on the right-hand side of Equation G.43 is along sides Σ2 and Σ3 . With this in mind, and substituting Equations G.41 and G.42 into the first term of Equation G.43, we obtain Z Z ˜ · n d Σ = 2S 2 2 (N1 S 2 ∇p) ξ1 ∇p˜ · n d Σ Σ2 ∪Σ3

Σ

Z ∂p˜ ∂p˜ ξ1 d Σ + 2S 2 dΣ ∂n ∂n Σ2 Σ3 2 3 Z 1 Z 1 ∂p˜ ∂p˜ = 2S 2 |Σ2 | x d x + 2S 2 |Σ3 | (1 − x) d x ∂n 2 ∂n 3 0 0 ∂p˜ ∂p˜ = S2 |Σ2 | + S 2 |Σ3 | ∂n 2 ∂n 3 Z

= 2S 2

ξ1

(G.44)

G.3 Finite Element Derivation

291

Figure G.2 Geometry of triangular element for the pressure field solution

where |Σ2 | and |Σ3 | are the lengths of Sides 2 and 3 of the element, respectively. We denote ∂p˜ , ∂n 2 ∂p˜ q3 = S 3 . ∂n 3 q2 = S 2

(G.45) (G.46)

Quantities q 2 and q 3 may be identified as flow rates per unit length along Sides 2 and 3 of the element respectively. Hence we can define the total flow rate at Node 1, denoted Q 1 by Q 1 = |Σ2 |q 2 + |Σ3 |q 3 . Hence Equation G.44 becomes Z ˜ · n d Σ = Q1 . 2 (N1 S 2 ∇p) Σ

(G.47)

(G.48)

We can perform the same calculations as above on equation G.43 for i = 2, 3 to get two more equations: Z ˜ · n d Σ = Q2 , 2 (N2 S 2 ∇p) (G.49) Σ

Z 2

Σ

˜ · n d Σ = Q3 . (N3 S 2 ∇p)

(G.50)

Now consider the second term of Equation G.43. Assuming again that S 2 is constant over the element at a given time and hence temperature, using Equations G.41 and G.42 we obtain for

292

G Numerical Methods for the 2.5D Approximation

the second term on the right-hand side of Equation G.43: Z Z 2 S 2 ∇p˜ · ∇N1 d A = 2S 2 ∇p˜ · ∇N1 d A A A Z ´ ∂ξ S 2 n³ 1 = y 23 p˜1 + y 31 p˜2 + y 12 p˜3 A A ∂x ´ ∂ξ o ³ 1 + x 32 p˜1 + x 13 p˜2 + x 21 p˜3 dA ∂y Z n³ ´ ³ ´ S2 2 2 = y 23 p˜1 + y 31 y 23 + x 13 x 32 p˜2 + x 32 2 2A A ³ ´ o + y 12 y 23 + x 21 x 32 p˜3 d A ´ ³ ´ S 2 n³ 2 2 = y 23 + x 32 p˜1 + y 31 y 23 + x 13 x 32 p˜2 2A ³ ´ o + y 12 y 23 + x 21 x 32 p˜3 .

(G.51)

Substituting i = 2, 3 into Equation G.35 and repeating the steps above provides another two equations, namely:

Z 2S 2

A

∇p˜ · ∇N2 d A =

Z 2S 2

A

∇p˜ · ∇N3 d A =

´ ³ ´ S 2 n³ 2 2 y 23 y 31 + x 32 x 13 p˜1 + y 23 + x 32 p˜2 2A ³ ´ o + y 12 y 31 + x 21 x 13 p˜3 ,

(G.52)

´ ³ ´ S 2 n³ y 23 y 12 + x 32 x 21 p˜1 + y 31 y 12 + x 13 x 21 p˜2 2A ³ ´ o 2 2 p˜3 . + y 12 + x 21

(G.53)

Now we return to Equation G.20. The second term may be written as three algebraic equations corresponding to i = 1, 2, 3. For i = 1, Z Z ˜ · N1 d A = 2 ∇ · (N1 S 2 ∇p˜ d A 2∇ · (S 2 ∇p) A A Z − 2 S 2 ∇p˜ · ∇N1 d A using Equation G.34 Z A ˜ ·n d A = 2 (N1 S 2 ∇p) Σ Z − 2 S 2 ∇p˜ · ∇N1 d A using Equation G.35 A Z = Q 1 − 2 S 2 ∇p˜ · ∇N1 d A using Equation G.48 A

¢ ¡ ¢ S 2 n¡ 2 2 = Q1 − 2 y 23 + x 32 p˜1 + y 31 y 23 + x 13 x 32 p˜2 2A ¡ ¢ o + y 12 y 23 + x 21 x 32 p˜3 from Equation G.51 .

(G.54)

G.3 Finite Element Derivation

That is, for i = 1, the second term of Equation G.20 becomes Z ¢ S 2 n¡ 2 2 ˜ · N1 d A = Q 1 − 2 2∇ · (S 2 ∇p) y 23 + x 32 p˜1 2A A ¡ ¢ + y 31 y 23 + x 13 x 32 p˜2 ¡ ¢ o + y 12 y 23 + x 21 x 32 p˜3 .

293

(G.55)

Setting i = 2, and using Equations G.35, G.49, and G.52, the second term of Equation G.20 becomes Z ¢ S 2 n¡ ˜ · N2 d A = Q 2 − 2 y 23 y 31 + x 32 x 13 p˜1 2∇ · (S 2 ∇p) 2A A ¡ 2 ¢ 2 p˜2 + y 31 + x 13 ¡ ¢ o + y 12 y 31 + x 21 x 13 p˜3 . (G.56) Setting i = 3, and using Equations G.43, G.50, and G.53, the second term of Equation G.20 becomes Z ¢ S 2 n¡ ˜ · N3 d A = Q 3 − 2 2∇ · (S 2 ∇p) y 23 y 12 + x 32 x 21 p˜1 2A A ¡ ¢ + y 31 y 12 + x 13 x 21 p˜2 ¡ 2 ¢ o 2 + y 12 + x 21 p˜3 . (G.57) Equations G.55 to G.57 may be written in matrix form as follows:  Z ˜    Z A 2∇ · (S 2 ∇p 1 )N1 d A  Q  k 11 k 12 k 13 p˜1 1      S2   k 21 k 22 k 23  p˜2   2∇ · (S 2 ∇p˜2 )N2 d A  = Q 2 −  Z A 4A   Q3 k 31 k 32 k 33 p˜3 2∇ · (S 2 ∇p˜3 )N3 d A A

= {Q}e −

S2 ˜ , [K ]e {p} 4A

(G.58)

where [K ]e is called the elemental stiffness matrix as denoted by the superscript “e,” and 2 2 k 11 = (y 23 + x 32 ); k 12 = (y 31 y 32 + x 13 x 32 ); k 13 = (y 12 y 32 + x 21 x 32 ) 2 2 k 21 = (y 23 y 31 + x 32 x 13 ); k 22 = (y 31 + x 13 ); k 23 = (y 12 y 31 + x 21 x 13 ) 2 2 k 31 = (y 23 y 12 + x 32 x 21 ); k 32 = (y 31 y 12 + x 13 x 21 ); k 33 = (y 12 + x 21 ).

(G.59)

Consider now the third term in Equation G.20, namely, µ³ ¶ Z ∂p˜ ´2 ³ ∂p˜ ´2 d (x, y) + Ni d A ∂x ∂y A for i ∈ {1, 2, 3} , where h Z +H ³ Z d (x, y) = κ −H

z

h−

Z +H ³ Z z ´ ´ i z 1 d z− d z −C (x, y) κ dz dz , η −H h− η

(G.60)

294

G Numerical Methods for the 2.5D Approximation

and we will assume it does not vary over the element. From Equation G.41 we have ´ 1 ³ ∂p˜ = y 23 p˜1 + y 31 p˜2 + y 12 p˜3 ∂x 2A ´ 1 ³ ∂p˜ = x 32 p˜1 + x 13 p˜2 + x 21 p˜3 . ∂y 2A

(G.61) (G.62)

From this we find ³ ∂p˜ ´2 ∂x

+

³ ∂p˜ ´2 ∂y

=

1 ˜ p} ˜ T, [K ]e {p}{ 4A 2

(G.63)

where [K ]e was defined by Equations G.58 and G.59. Using Equation G.63, the third term of Equation G.20 may be written Z 1 e T ˜ ˜ d (x, y)[K ] { p}{ p} Ni d A , (G.64) 4A 2 A for i ∈ {1, 2, 3}. Since from Equation G.12, Ni = ξi , we can use Equation G.27 to evaluate the integrals involving the shape function in Equation G.64 to get Z Z 1 (G.65) Ni d A = ξi d A = A, ∀i ∈ {1, 2, 3} . 3 A A Subsequently, using Equations G.63 to G.65, the third term in Equation G.20 may be written in matrix form as follows:   h³ ´2 ³ ´2 i R ∂p˜ ∂p˜ N1 d A   A d (x, y) ∂x + ∂y R  h³ ´2 ³ ´2 i 1   ˜ p} ˜ T. d (x, y)[K ]e {p}{ (G.66)  A d (x, y) ∂∂xp˜ + ∂∂yp˜ N2 d A  =   12A h³ ´2 ³ ´2 i R  ∂p˜ ∂p˜ + ∂y N3 d A A d (x, y) ∂x Finally we consider the fourth term in Equation G.20: Z b(x, y)Ni d A , A

where b(x, y) was defined in Equation 5.69 as Z b(x, y) =

+H −H

∂2 T ´ β ³ 2 ηγ˙ + k 2 d z . ρc p ∂z

(G.67)

Assuming no variation of b(x, y) over the element and using Equation G.65, the fourth term of Equation G.20 may be written in matrix form as R    b(x, y)Ni d A b(x, y) RA  b(x, y)Ni d A  = A b(x, y) RA 3 b(x, y) A b(x, y)Ni d A =

A {B }e , 3

(G.68)

G.3 Finite Element Derivation

295

 b(x, y) where {B }e = b(x, y). b(x, y) 

Equations G.31, G.58, G.66, and G.68 provide matrix representations for the terms on the righthand side of Equation G.20. Substituting these into Equation G.20 we get: S2 A ˙ + ˜ − {Q}e a(x, y)[C ]e {p} [K ]e {p} 12 4A 1 A ˜ p} ˜ T − {B }e + d (x, y)[K ]e {p}{ 12A 3 ˙ + 3S 2 [K ]e {p} − 12A{Q}e = A 2 a(x, y)[C ]e {p}

0=

+ d (x, y)[K ]e {p}{p}T − 4A 2 {B }e .

(G.69)

Equation G.69 may be rearranged to give £ ¤e £ ¤e £ ¤e ˙ + K {p} + K c {p}{p}T = {Q}e − {B }e , C {p}

(G.70)

where the underscore indicates that the pre-multiplication constants have been taken inside the matrix, and the subscript “c” indicates that [K ]c arises from pressure convection.

G.3.1

Assembly of Element Equations and Solution

Equation G.70 gives the element equations for the pressure solution, that is, the relationship between pressure and flow rates for each individual element. In order to represent the entire domain, the element equation must be assembled as discussed in Section F.7.3. We denote the assembled equations as £ ¤ £ ¤ £ ¤ ˙ + K {p} + K c {p}{p}T = {Q} − {B} . C {p}

(G.71)

It should be noted that these matrices may be very large. Their size depends on the number of elements used to model the part. It is not uncommon to have over 1,000,000 elements in a model. In order to solve Equation G.71 we use a first-order finite difference scheme to approximate the pressure derivative. That is, [C]

³ {pn+1 } − {p ˙n} ´ ∆t

T

+ [K]{p}n+1 + [K]c {p}n {p}n = {Q} − {B} .

(G.72)

This may be rearranged to give ¡

¢ T [C] + ∆t [K] {p}n+1 + ∆t [K]c {p}n {p}n = {Q} − {B} .

(G.73)

After setting appropriate boundary conditions the matrices may be created and the system solved for {p}n+1 .

G Numerical Methods for the 2.5D Approximation

296

G.4

Solution of the Energy Equation

The energy equation for the 2.5D approximation is given by Equation 5.73: ρc p

³

∂T ∂t |{z}

rate of change

∂T ∂T ´ + vx + vy = ∂x ∂y | {z } convection

∂p βT | {z∂t}

compressive heating

+

ηγ˙ 2 |{z}

viscous dissipation

+

∂2 T k 2 {z } | ∂z

(G.74)

conduction to mold

where we have underlined the terms to represent their physical significance. Equation 5.73 is to be solved subject to the following boundary conditions: 1. The melt temperature is defined at the point of injection. 2. The temperature at the cavity wall or at some point in the mold is defined. Alternatively, a heat flux may be prescribed on the mold. These appear straightforward but are quite complex in practice. They are discussed in detail in Chapter 5. Solution of the energy equation emphasizes the link between pressure and temperature solutions. The energy equation is solved with a finite difference method (see Appendix E) but the convective terms use results determined from the finite element solution for pressure and hence velocity discussed earlier in this appendix.

G.4.1

Finite Difference Discretization

The finite difference scheme requires discretization in both time and the spatial variable z. Time discretization may be linked to the scheme used for advancement of the flow front— a topic we will discuss briefly in Section G.5. Spatial discretization must ensure there are enough nodes—sometimes called grid points in finite difference literature—to capture the rapidly changing temperature field over the relatively small thickness of the part. An interesting problem arises at the mold/polymer interface. Some codes extend the finite difference grid from the polymer into the mold metal. Others define a heat transfer coefficient at the mold polymer interface and calculate temperature only within the polymer. While it is common to have a regular grid spacing in the polymer, some older codes varied the spacing. The idea was to bunch grid points near the mold/polymer interface to capture the highest temperature gradients. While valid for the filling stage, such schemes lose accuracy during packing where a substantial frozen layer forms. Subsequently residual stress calculation is compromised in the packing phase, and thus warpage and shrinkage calculations will be in error. For finite difference solution of the energy equation in a given time step, the convection, compressive heating, and viscous dissipation terms are from the previous time step and treated as source terms. That is, Equation G.74 may be written ρc p

µh

¶ h ∂T in h ∂T ∂T in−1 ∂p in−1 h 2 in−1 h ∂2 T in + vx + vy = βT + ηγ˙ + k 2 . ∂t i ∂x ∂y i ∂t i ∂z i i

(G.75)

G.4 Solution of the Energy Equation

G.4.2

297

Solution of the Conduction Problem

Ignoring the the convection, compressive heating, and viscous dissipation terms for the moment, we are left with the conduction from the plastic into the mold (or mold cooling system). Here we assume there is some point in the mold at some reference temperature or a point at which a heat transfer coefficient (HTC) is defined. The reference temperature or HTC may change on various time steps. This is probable if the mold filling/packing analysis has been coupled with an analysis of the mold cooling system. In Appendix E.2, we discussed the use of explicit and implicit methods for solution of the equation ∂2 T ∂T =α 2 , ∂t ∂z

(G.76)

where α = k/ρc p is called the diffusivity of the material. We use the same ideas here and recall from Section E.2.1.1 that a necessary condition for stable solution of Equation G.76 is that M=

α∆t 1 ≤ . 2 (∆z) 2

(G.77)

We deal only with the explicit solution scheme in the following.

G.4.3

Explicit Method

From Equation E.30 in Section E.2.1 we found Equation G.76 may be written as the explicit finite difference scheme: ¡ ¢ Tin+1 = M Tin+1 − 2Tin + Tin−1 + Tin .

(G.78)

Adding in the convection, compressive heating, and viscous dissipation terms, we get ¡ ¢ Tin+1 = M Tin+1 − 2Tin + Tin−1 + Tin .

(G.79)

If we now include the convection, compressive heating and viscous dissipation terms we get ¡ ¢ Tin+1 = M Tin+1 − 2Tin + Tin−1 + Tin ¸ · ¸n · ¸n · ∂T n ∆t ∂T + vy + ∆T + ηγ˙ 2 , − ∆t v x ∂x ∂y i ρc p comp i

(G.80)

£ ¤n where ∆T comp is the temperature increase (or decrease) due to compression (or expansion). We will define this term more carefully soon. For now we note that Equation G.80 may be solved for Tin+1 . £ ¤n We now make some comments on the term ∆T comp . Note that we do not have the temperature change associated with a particular grid point as with the other terms. This is because in the 2.5D approximation there is no pressure variation in the z direction and hence no way of calculating compressive heat at a grid point located in the thickness direction. Somehow the temperature increase must be smeared over the grid points in the thickness. We do not consider this aspect of numerical implementation.

298

G Numerical Methods for the 2.5D Approximation

£ ¤n To determine ∆T comp we need to recall that PVT data is used to determine several polymer properties, namely, density ρ, isothermal compressibility coefficient κ, and the coefficient of expansivity β. With these quantities known, we can determine the temperature change of the material due to compression. This is given by ³ βT n ´ ∂p £ ¤n+1 ∆T comp = ∆t ρc p ∂t ³ βT n ´ p n+1 − p n = ∆t ρc p ∆t n ¡ ¢ βT = p n+1 − p n . ρc p

(G.81)

Using this temperature change, a new value of density ρ, isothermal compressibility coefficient κ and the coefficient of expansivity β can be determined form the PVT data. The derivatives ∂ρ/∂p and ∂ρ/∂t can be calculated from the PVT data and so ∂p/∂t may be found. The new £ values of compressibility are then used to update the matrices C ]e in Equation G.70 and hence [C] in the assembled Equation G.71. The increase in temperature from Equation G.81 is also used to update viscosity. However, viscosity is calculated at points through the thickness whereas the temperature change due to pressure change is smeared over the thickness in some way. The temperature used to calculate viscosity will also depend on the viscous dissipation, convection, and conduction terms. Moreover the shear rate corresponding to the new pressure p n+1 will affect the viscosity due to viscous dissipation. With the viscosity established, the fluidity S 2 can be determined. Consequently the matrices [C] and [K] may be updated in Equations G.71.

G.5

Flow Front Advancement

In the 2.5D approximation, the flow front is usually advanced using a control volume method. This requires that each node of the finite element mesh is assigned a volume called a control volume. The idea is described in Kennedy [196]. A more advanced method was given by Boshouwers and van der Werf [42]. Other more advanced methods have since been developed. In particular, level set methods are probably the best available. We discuss these in the 3D context in Appendix I, but note they may also be used in 2D problems.

G.6

Runners

In practice there may be many mold cavities with a single mold. The cavities may not necessarily have the same geometry, but each cavity will need to be supplied with melt from a runner system. On the other hand, for large molds it may be necessary to introduce melt at various points of the molding to minimize the flow length. Runners are frequently trapezoidal

G.6 Runners

299

in cross-section to ensure easy ejection from the mold. However most 2.5D mold-filling simulation software uses runners of circular cross-section. While the user may choose various cross-sections the software will generally reduce this to an equivalent circular cross-section. In Chapter 5 we discussed the governing equations for runners. For the 2.5D approximation, the final equation governing pressure was Equation 5.131. Here we give the finite element derivation of the runner elements. Recall Equation 5.131: Z R ∂p β n 2 k ∂ ³ ∂T ´o ¯ c R2 −2 ηγ˙ + r r dr 0=κ ∂t r ∂r ∂r 0 ρc p ∂ ³ + ∂p ´ − r S1 . (G.82) ∂x ∂x Setting Z b = −2

0

R

β n 2 k ∂ ³ ∂T ´o ηγ˙ + r r dr , ρc p r ∂r ∂r

Equation G.82 can be abbreviated to ∂p ∂ ³ + ∂p ´ ¯ c R2 0=κ +b − r S1 . ∂t ∂x ∂x

(G.83)

(G.84)

Using linear interpolation between the two nodes defining the runner element, the approximate solution p˜ takes the form ³ x −x ´ ³ x −x ´ 1 2 p˜1 + p˜2 p˜ = x2 − x1 x2 − x1 = N1 p˜1 + N2 p˜2 ,

(G.85)

where N1 =

¢ x2 − x 1¡ = x2 − x x2 − x1 L

(G.86)

N2 =

¢ x − x1 1¡ = x − x1 , x2 − x1 L

(G.87)

and L is the length of the element. A residual R(x) is defined by substituting Equation G.85 into Equation G.84: µ ¶ ∂p˜ ∂ + ∂p˜ ¯ c R2 R(x) = κ +b − r S1 . ∂t ∂x ∂x Galerkin’s method requires that Z x1 R(x)Ni d x = 0 , i = 1, 2 .

(G.88)

(G.89)

x0

Substituting Equation G.88 into Equation G.89 gives the following two equations, corresponding to i = 1 and i = 2, respectively: Z x2 Z x2 Z x2 ∂p˜ ¯ c R2 R(x)N1 d x = κ N1 d x + bN1 d x x1 x1 x 1 ∂t µ ¶ Z x2 ∂ + ∂p˜ − r S1 N1 d x , (G.90) ∂x x 1 ∂x

G Numerical Methods for the 2.5D Approximation

300

Z

x2 x1

¯ c R2 R(x)N2 d x = κ Z −

Z

x2 x1

Z x2 ∂p˜ N1 d x + bN2 d x x 1 ∂t x1 µ ¶ ∂ + ∂p˜ r S1 N2 d x . ∂x ∂x x2

(G.91)

We now try to simplify the above equations. Consider the first integral on the right-hand side of Equation G.90. Writing ∂p˜i /∂t = p˙˜i , where i = 1, 2, and using Equation G.85 we have: Z x2 ³ Z x2 ´ ∂p˜ ˜ 2 ˜ 2 N1 d x = κR N1 p˙˜1 + N2 p˙˜2 N1 d x κR x1 x 1 ∂t Z x2 o n Z x2 2 ˙ ˜ = κR p˜1 N1 2 d x + p˙˜2 N2 N1 d x . (G.92) x1

x1

Using Equations G.86 and G.87, the integrals on the right hand side of Equation G.92 may be evaluated: Z Z x2 ´2 1 x2 ³ x2 − x d x N1 2 d x = 2 L x1 x1 · ´ 31 ¸x=x2 1³ 1 = 2 − x2 − x L 3 x=x 1 ³ ´ 1 = 2 x2 − x1 3L L = , (G.93) 3 Z

x2 x1

N2 N1 d x =

1 L2

Z

x2 ³ x1

´³ ´ x2 − x x − x1 d x

·

1 1 1 1 x2 x 2 − x2 x1 x − x 3 + x 2 x1 L2 2 3 2 ¢3 1 ¡ = 2 x2 − x1 6L L = . 6

¸x=x2

=

x=x 1

(G.94)

Substituting these values in Equation G.90 we obtain ¯ c R2 κ

Z

x2 x1

´ ¯ c LR 2 ³ ∂p˜ κ N1 d x = 2p˙1 + p˙2 . ∂t 6

(G.95)

Applying the same procedure to the first integral on the right-hand side of Equation G.91 we get, ¯ c R2 κ

Z

x2 x1

´ ¯ c LR 2 ³ ∂p˜ κ N2 d x = p˙1 + 2p˙2 . ∂t 6

(G.96)

Consider now the last integral on the right-hand side of Equations G.90 and G.91. Integrating by parts we obtain Z x2 Z x2 ³ ∂ ³ + ∂p˜ ´ ∂p˜ ´ ¯¯x=x2 ∂p˜ ∂Ni r S1 Ni d x = r + S 1 Ni ¯ − r +S1 dx , (G.97) ∂x ∂x ∂x ∂x ∂x x=x x1 x1 1

G.6 Runners

for i = 1, 2. Setting i = 1 in Equation G.97, and substituting Equations G.86 and G.87, Z x2 Z x2 ³ ∂ ³ + ∂p˜ ´ ∂p˜ ∂N1 ∂p˜ ´ ¯¯ − r +S1 r S1 N1 d x = − r + S 1 dx ¯ ∂x ∂x x=x1 ∂x ∂x x1 x 1 ∂x Z ³ ∂p˜ ´ ¯¯ 1 x2 + ∂p˜ = − r +S1 + r S1 dx ¯ ∂x x=x1 L x1 ∂x Z ³ ´ 1 x2 + ∂p˜ ´ ¯¯ ∂ ³ + = − r +S1 r S1 N1 p 1 + N2 p 2 d x ¯ ∂x x=x1 L x1 ∂x Z x2 ¯ ³ ´ ´ ³ 1 ∂p˜ ¯ r +S1 p2 − p1 d x + 2 = − r +S1 ¯ ∂x x=x1 L x1 ³ ´ Z x2 ∂ p˜ ´ ¯¯ r +S1 ³ = − r +S1 + 2 p2 − p1 dx ¯ ∂x x=x1 L x1 ´ ³ r +S1 ³ ∂p˜ ´ ¯¯ + p2 − p1 . = − r +S1 ¯ ∂x x=x1 L

301

(G.98)

The first term on the right-hand side of Equation G.98 may be identified as the flow rate into the node at x = 1 and we denote it ³ ∂p˜ ´ ¯¯ q1 = − r + S 1 . (G.99) ¯ ∂x x=x1 Using Equations G.85, G.86, and G.87, Equation G.99 may be written ³ ¢´ ¯¯ ∂ ¡ N1 p 1 + N2 p 2 ¯ q1 = − r + S 1 ∂x x=x 1 ³ r + S ∂ ³¡ ¢ ¡ ¢ ´´ ¯¯ 1 =− x2 − x p 1 + x − x1 p 2 ¯ L ∂x x=x 1 ³ r +S ¡ ´¯ ¢ ¯ 1 = p1 − p2 ¯ . L x=x 1 Hence Equation G.98 becomes Z x2 ´ r +S1 ³ ∂ ³ + ∂p˜ ´ r S1 N1 d x = −q 1 + p2 − p1 . ∂x L x 1 ∂x

(G.100)

(G.101)

Setting i = 2 in Equation G.97, and following the procedure yielding Equations G.98 to G.101, described above, we obtain the flow rate out of Node 2 of the runner element ³ r +S ¡ ¢´¯¯ 1 q2 = p1 − p2 ¯ . (G.102) L x=x 2 Consequently Equation G.97 becomes Z x2 ´ r +S1 ³ ∂ ³ + ∂p˜ ´ r S1 N2 d x = −q 2 + p2 − p1 . ∂x L x 1 ∂x

(G.103)

We can now rewrite Equations G.90 and G.91. Substituting Equations G.95, G.83, and G.101 into Equation G.90 we obtain, Z x2 ´ Z x2 ´ ¯ c LR 2 ³ κ r +S1 ³ R(x)N1 d x = 2p˙1 + p˙2 + bN1 d x − q 1 + p1 − p2 6 L x1 x1 ´ ´ ¯ c LR 2 ³ κ r +S1 ³ = 2p˙1 + p˙2 + b 1 − q 1 + p1 − p2 , (G.104) 6 L

G Numerical Methods for the 2.5D Approximation

302

where Z b1 =

x2 x1

bN1 d x .

(G.105)

Substituting Equations G.95, G.83, and G.101 into Equation G.90, Z

´ Z x2 ´ ¯ c LR 2 ³ κ r +S1 ³ 2p˙1 + p˙2 + bN2 d x − q 2 − p1 − p2 6 L x1 ´ ³ ³ ´ + 2 ¯ c LR r S1 κ 2p˙1 + p˙2 + b 2 − q 2 + p1 − p2 , = 6 L

x2 x1

R(x)N2 d x =

(G.106)

where Z b2 =

x2 x1

bN2 d x .

(G.107)

Note that according to the Galerkin requirement, Equation G.89, the left-hand sides of Equations G.104 and G.106 are zero and may be combined into a single matrix equation, · ¯ c LR 2 2 κ 1 6

1 2

¸µ

¶ ¸µ ¶ · r + S 1 1 −1 p 1 p˙1 + p˙2 −1 1 p 2 L µ ¶ µ ¶ q1 b1 = − . q2 b2

(G.108)

This may be abbreviated to £ ¤e £ ¤e ˙ + K {p} = {Q}e − {B }e , C {p}

(G.109)

and is the element equation for a runner. Note that Equation G.109 has identical form to the element equation for the cavity, Equation G.70, except that we neglected the pressure convection term [K ]c in the runner formulation. Solution for the pressure in runners requires assembly of the runner element equations and their solution according to Section G.3.1. Thermal solution for the runner system is similar to that outlined in Section G.4; however, a cylindrical coordinate system is used.

H

Three-Dimensional Finite Element Method for Mold Filling Analysis

In Appendix F we introduced the basic concepts of finite element analysis, and gave a 1D example. In Appendix G we discussed the finite element formulation of a two dimensional problem, namely the 2.5D pressure approximation. In this appendix, we discuss the 3D finite element solution for injection-molding simulation. Accordingly, we introduce some new notation and concepts that the reader is likely to find in the relevant literature. These comprise function spaces and inner products. A useful reference is Reddy [307].

H.1

Governing Equations

In this appendix, we solve the mold flow problems for the velocity u and the pressure p fields using 3D finite element methods with low-order tetrahedral elements. We neglect inertia and gravity and assume that the fluid is incompressible during the mold filling. Further, we choose the generalized Newtonian fluid to be the constitutive model. The governing equations we want to solve are ∇·σ = 0,

in Ω ,

(H.1)

∇·u = 0,

in Ω ,

(H.2)

where Ω is the 3D global computational domain. For the generalized Newtonian fluid the total stress tensor σ is given by ¡ ¢ σ = −pI + η ∇u + ∇uT (H.3) where I is the unit tensor (δi j ), and η is the shear-rate- and temperature-dependent viscosity. On the boundary ∂Ω of the domain Ω, the boundary conditions can be either velocity or traction boundary conditions: u = u0 ,

∀x ∈ ∂Ωu ,

(H.4)

t = σ · n = t0 ,

∀x ∈ ∂Ωσ ,

(H.5)

where t is the traction vector (also called the total normal stress), n is the outward normal, and u0 and t0 are the prescribed velocity and traction, respectively. In the mold filling modeling, a zero velocity boundary condition is assumed at the mold/polymer interface, and a zero traction boundary condition is assumed at the flow-front boundary. At the inlet boundary, either the injection velocity or the traction conditions are imposed. Note that the boundary condition in terms of the traction is different from the pressure boundary conditions. In many

304

H Three-Dimensional FEM for Mold Filling Analysis

practical cases, in the inlet boundary, we require that the tangential component of traction vanishes there and only nonzero normal traction components are specified. In this case, from the constitutive equation (Equation H.3), one has tn = −p + 2η

∂u n , ∂n

(H.6)

± where un is the component of velocity normal to the boundary, and ∂ ∂n stands for the out± ward normal gradient at the boundary. If η∂u n ∂n is negligibly small, then the traction boundary condition is essentially the same as the pressure boundary condition used for the 2.5D formulation.

H.2

Weak Formulations

The finite element procedure begins with the division of the computational domain Ω into a set of nonoverlapping elements, for example, a collection of tetrahedral elements Ωe (Ω = ∪ Ωe ). We adopt an Eulerian approach, meaning the elements are fixed in space. Methods such as the VOF or the level set method, Appendix I, are used to determine the moving flow front. Let V and Q denote the function spaces of admissible functions for velocity and pressure. For test functions w ∈ V and q ∈ Q, we try to satisfy 〈∇ · σ, w〉Ω = 0 ,

(H.7)

〈∇ · u, q〉Ω = 0,

(H.8)

where 〈 , 〉 denotes the inner product. We shall use the following notation: Z 〈s 1 , s 2 〉Ω = s 1 s 2 d Ω , ZΩ 〈v1 , v2 〉Ω = v1 · v2 d Ω , ZΩ 〈v1 , v2 〉∂Ω = v1 · v2 d ∂Ω , Z∂Ω 〈T1 , T2 〉Ω = T1 : T2 d Ω, Ω

(H.9) (H.10) (H.11) (H.12)

where s1 and s2 are scalar functions, v1 and v2 are vector functions, and T1 and T2 are tensor functions. We integrate Equation H.7 by parts, with help from the Gauss-Green theorem, and obtain 〈σ, ∇w〉Ω − 〈σ · n, w〉∂Ω = 0,

(H.13)

which is rewritten as, according to Equation H.3, −〈pI, ∇w〉Ω + 〈η(∇u + ∇uT ), ∇w〉Ω = 〈σ · n, w〉∂Ω .

(H.14)

This is known as the weak formulation. Since the boundary condition for the traction (t = σ·n) is directly incorporated in the weak formulation, the traction boundary condition is sometimes called the natural boundary condition.

H.3 Finite Element Matrix Formulations

H.3

305

Finite Element Matrix Formulations

On each tetrahedral element Ωe , the velocity and pressure are approximated by simple piecewise varying functions, using a linear combination of simple interpolation functions (also called the shape functions), usually polynomials of less than cubic in their orders. Lower order shape functions result in smaller total number of degrees of freedom. It has been known that for incompressible or nearly incompressible fluid flow problems, the velocity shape function should be one order higher than the pressure shape function to satisfy the Babuska-Brezzi stability condition. In case that both the velocity and the pressure fields are approximated by linear shape functions, the Babuska-Brezzi condition is not satisfied. One way to avoid the difficulty is to use the so-called MINI element of Arnold et al. [11]. The method has been applied to 3D injection-molding simulations by Pichelin and Coupez [298] and Yu and Kennedy [409]. In this method, a finite element approximation is constructed as follows. In the iso-parametric element formulation the position x in each tetrahedron is expressed in terms of natural coordinates ξ = [ξ1 , ξ2 , ξ3 , ξ4 ] as x(ξ) =

4 X

Nn (ξ)x(e) n ,

(H.15)

n=1

where x(e) n is the position of the k node of the element Ωe . For a 4-node linear tetrahedron the shape functions Nn are given by N 1 = ξ1 , N 2 = ξ2 , N 3 = ξ3 , N 4 = ξ4 = 1 − ξ1 − ξ2 − ξ3 ,

(H.16)

where ξk =

ak + bk x + ck y + dk z 6V (e)

, k = 1, 2, 3, 4

(H.17)

with V (e) being the volume of tetrahedron given by 

1  1  6V (e) = det   1 1

x 1(e) x 2(e) x 3(e) x 4(e)

y 1(e) y 2(e) y 3(e) y 4(e)

The coefficients of a k , b k , c k , d k are   (e) x y 2(e) z 2(e)  2(e)  a 1 = det  x 3 y 3(e) z 3(e)  x 4(e) y 4(e) z 4(e) 

1  b 1 = − det  1 1

y 2(e) y 3(e) y 4(e)

 z 2(e)  z 3(e)  (e) z4

z 1(e) z 2(e) z 3(e) z 4(e)

   . 

(H.18)

(H.19)

(H.20)

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H Three-Dimensional FEM for Mold Filling Analysis

x (e)  2(e) c 1 = − det  x 3 x 4(e)

1 1 1

x (e)  2(e) d 1 = − det  x 3 x 4(e)

y 2(e) y 3(e) y 4(e)





 z 2(e)  z 3(e)  (e) z4  1  1  1

(H.21)

(H.22)

with the other coefficients defined by cyclic interchange of the subscripts in the order 1, 2, 3, 4. The Jacobian of coordinate transformations can be expressed as  (e)  x 1 − x 4(e) y 1(e) − y 4(e) z 1(e) − z 4(e) 4 ∂N (ξ) ∂x X   k J= = x(e) = x 2(e) − x 4(e) y 2(e) − y 4(e) z 2(e) − z 4(e)  . k ∂ξ k=1 ∂ξ (e) (e) (e) (e) (e) (e) x3 − x4 y3 − y4 z3 − z4

(H.23)

The Jacobian relates the global coordinate derivatives ∂/∂x to the local coordinate derivatives ∂/∂ξ in the following way: ∂ ∂ ∂ ∂ =J , = J−1 . ∂ξ ∂x ∂x ∂ξ

(H.24)

Therefore,    ∂Nk ∂Nk  ∂x   ∂ξ1       ∂N   ∂N  k   k ,   = J  ∂y   ∂ξ2       ∂Nk   ∂Nk  ∂ξ3 ∂z 

(H.25)

and    ∂Nk ∂Nk  ∂x   ∂ξ1       ∂N    k  −1  ∂Nk  = J    .  ∂y   ∂ξ2       ∂Nk   ∂Nk  

(H.26)

∂ξ3

∂z

The velocity and the pressure fields are discretized by u(ξ) =

4 X n=1

(e) Nn (ξ)u(e) n + Nbub (ξ)ub ,

(H.27)

Nk (ξ)p n(e) ,

(H.28)

and p(ξ) ≈

4 X n=1

H.3 Finite Element Matrix Formulations

307

(e) (e) where u(e) n and p n are the nodal unknowns at the vertices of the tetrahedral element, ub is the so-called “bubble” velocity at the internal node (Figure H.1), and Nbub is a nonlinear bubble shape function. One option for the shape function is

Nbub (ξ) = 44 ξ1 ξ2 ξ3 ξ4 ,

(H.29)

which is called the conforming bubble. The value of Nbub is equal to 1 at the center of the tetrahedral element and equal to 0 at the boundary of the element. The derivatives of the bubble shape function are given by ∂Nbub 44 = (b 1 ξ2 ξ3 ξ4 + b 2 ξ3 ξ4 ξ1 + b 3 ξ4 ξ1 ξ2 + b 4 ξ1 ξ2 ξ3 ) , ∂x 6V (e) 44 ∂Nbub = (c 1 ξ2 ξ3 ξ4 + c 2 ξ3 ξ4 ξ1 + c 3 ξ4 ξ1 ξ2 + c 4 ξ1 ξ2 ξ3 ) , ∂y 6V (e) 44 ∂Nbub = (d 1 ξ2 ξ3 ξ4 + d 2 ξ3 ξ4 ξ1 + d 3 ξ4 ξ1 ξ2 + d 4 ξ1 ξ2 ξ3 ) . ∂z 6V (e)

(H.30) (H.31) (H.32)

One can also choose to use nonconforming bubble functions for a better convergence, such as Nbub (ξ) = 2 − 4

4 X k=1

ξ2k ,

(H.33)

which is called the quadratic bubble (see, for example, Pierre [299]). The approximation for the velocity field (Equation H.27) is said to be linearly enriched, denoted by P 1+ − C 0 , and the approximation for the pressure is linear of order P 1 − C 0 . Here the notation C m is the degree of continuity of the shape functions at element interfaces. If the shape function is continuous but its first derivatives are not, we say that we have C 0 continuity. If, in addition, its first derivatives are continuous but second derivatives are not, we have C 1 continuity, and so on. The notation P m is referring to the order of the polynomials of the shape functions Nk . Using the linear interpolation functions to approximate both velocities (without the bubble) and pressure, and taking w to be Nm , from Equation H.14 we obtain the following equations for element Ωe : ¶ Z 4 ·µZ X ∂Nm ∂Nn ∂Nm ∂Nn η δi k dV + η δ j k dV (u k )(e) n ∂x j ∂x j ∂x j ∂x i Ωe Ωe n=1 ¸ Z µZ ¶ ∂Nm σi j n j Nm d Γ, (H.34) Nn dV p n(e) = − ∂Ωe Ωe ∂x i and from Equation H.8 we have ¸ ¶ 4 ·µZ X ∂Nn (e) Nm dV (u i )n = 0. ∂x i Ωe n=1

(H.35)

Similarly, by taking w to be Nbub , we obtain µZ ¶ Z ∂Nbub ∂Nbub ∂Nbub ∂Nbub η δi k dV + η δ j k dV (u k )(e) b ∂x j ∂x j ∂x j ∂x i Ωe Ωe ·µZ ¶ ¸ Z 4 X ∂Nbub − Nn dV p n(e) = σi j n j Nbub d Γ, Ωe ∂x i ∂Ωe n=1 (H.36)

H Three-Dimensional FEM for Mold Filling Analysis

308

Figure H.1 MINI finite element with linear interpolation and bubble enrichment for velocity, and linear interpolation for pressure

and 4 X

·µZ

n=1

Ωe

Nbub

¶ ¸ ∂Nn dV (u i )(e) = 0. n ∂x i

(H.37)

The above Equations H.34 to H.37 can be rearranged in the elemental matrix form 

K11

 0  K31









0

K13

(b) K22

      (b)   (b)  K(b) 23   V  =  R  ,

K(b) 32

0

V

P

R

(H.38)

0

where Ki j and K(b) form the elemental stiffness matrix, resulting from the integration terms in ij the round brackets in Equations H.34 to H.37; V is the nodal velocity vector, V(b) is the bubble velocity vector, and P is the nodal pressure vector; R and R(b) are the load vectors resulting from the right-hand sides of Equations H.34 to H.37. By a classical condensation technique, we can write the bubble velocity vector as −1 (b) V(b) = [K(b) − K(b) 23 P). 22 ] (R

(H.39)

Thus, the system of equations H.38 reduces to "

K11

K13

K31

(b) −1 (b) − K(b) 32 (K22 ) K23

#" # V P

" =

R (b) −1 (b) − K(b) 32 (K22 ) R

# .

(H.40)

Since the bubble enriched velocity is zero at the boundary of each element, the nodal values of V are in fact the required solution. The unknown V(b) does not need to be solved.

H.4 Solution Procedures

309

The major task in the formation of the elemental stiffness matrix is integration. Generally, the integrals can be evaluated numerically by the Gauss integration method as follows: Z Ωe

1

Z f (x)dV (x) = ≈

Z

1

Z

1

−1 −1 −1 NX GP N GP N GP X X i

j

f (ξ1 , ξ2 , ξ3 )|J|d ξ1 d ξ2 d ξ3 f (ξGP )|J(ξGP )|Wi W j Wk ,

(H.41)

k

where NGP is the number of the Gauss points, Wi are the weights, and ξGP are the Gauss points at which the integrand is to be evaluated. Monomials of any order can be integrated on the plane-sided tetrahedral element in the closure form. The following integration formula is useful for this purpose (compare with Equation G.27): Z Ωe

ξ1a ξb2 ξc3 ξd4 dV = 6Ve

a!b!c!d ! . (a + b + c + d + 3)!

(H.42)

After the formulation of the elemental stiffness equations, one then assembles the elemental stiffness equations over their common nodes to find a global system of algebraic equations. Then the boundary conditions can be imposed into the system of algebraic equations. The traction boundary conditions appear naturally in the surface integrals, which contribute to the right-hand side of the system of equations to be solved. The velocity boundary conditions are now set in, and redundant equations corresponding to the boundary nodes where the nodal velocities are known are removed from the system of equations. Finally, after some algebraic manipulations we obtain a system of equations for the unknown nodal velocity and pressure components arranged as a vector U: KU = F ,

(H.43)

where K is the global stiffness matrix and F is the right-hand-side vector.

H.4

Solution Procedures

Generally in Equation H.43, the matrix K is a function of the unknowns U due to the shearrate-dependent viscosity. So we may write the equation in the form K(U)U = F .

(H.44)

This makes the system of equations nonlinear, and therefore linearization and iteration are necessary for solving the nonlinear system of equations. Either a Picard iteration scheme (also called the fixed point scheme) or a Newton-Raphson iteration scheme can be used for this purpose (see, for example, Pichelin and Coupez [298]). In the first case, if an estimated solution U(k) is obtained at step k, then the (k + 1)th solution U(k+1) is found by solving the following linear system of equations: K(U(k) )U(k+1) = F.

(H.45)

310

H Three-Dimensional FEM for Mold Filling Analysis

With the Newton-Raphson scheme, we write the residual of Equation H.44 as Q(U) = K(U)U − F,

(H.46)

and hence the problem we wish to solve is to find U such that Q(U) = 0.

(H.47)

If an estimated solution U(k) at step k is available, then the Newton-Raphson iteration scheme produces an improved solution U(k+1) at step (k + 1) by expanding Equation H.47 about U(k) : ¯ ∂Q ¯¯ Q(Uk ) + (U(k+1) − U(k) ) = 0. (H.48) ∂U ¯U=U(k) Usually one solves the linear system (Equation H.48) for (U(k+1) − U(k) ) and obtains a better solution U(k+1) . Both the Picard and the Newton-Raphson schemes reduce the nonlinear problem to the solution of a sequence of linear systems of equations. There are two types of solvers for linear systems: direct solvers and iterative solvers. Direct solvers are usually considered to be better for solving small- and medium-sized problems (the size of the problem is characterized by the number of unknowns or the number of degrees of freedom in the finite element model). Iterative solvers are more effective for large-sized problems with a sparse matrix K. However, the choice between the two types of equation solvers depends not only on the number of degrees of freedom in the finite element model, but also on the hardware architecture. For example, iterative solvers are potentially more suitable for parallel computations. Practical three-dimensional injection-molding simulation problems are usually huge. Several computational techniques have been developed for computing economy and speed. One of the most popular techniques is the Algebraic Multigrid (AMG) method. The basic idea behind the AMG is to extend the classical ideas of geometric multigrid to large sparse linear algebraic systems of equations. In contrast to geometric multigrid methods, the AMG does not need to make use of any geometric information. It constructs a sequence of “grids” with different levels of coarseness based solely on algebraic information contained in the given matrix. However, predefined geometric coarse grids based on the actual analysis mesh may be preferred sometimes in order to reduce the set-up time of the AMG solver. The methods are said to be scalable because they can solve a linear system with N unknowns with only O(N ) work. Details of the AMG methods can be found in the literature (for example, Stüben [347]; Webster [397, 398]; Klöppel and Wall [207]).

H.5

Flow-Front Advancement

In the mold-filling simulation, we need to determine the location of the moving flow front (also called the melt front). The calculations can be based on either a Lagrangian method or Eulerian method. In the Lagrangian method, only the fluid domain is meshed, and the location of the fluid front is represented by the frontal boundary of the moving mesh. On the other hand, in the Eulerian method, the whole cavity is meshed, and the mesh is fixed. The

H.6 Numerical Solution For Temperature Field

311

free surface (or interface) is considered as a scalar field to be solved. The disadvantages of the Lagrangian method are the mesh distortion and the need for a time-consuming remeshing. In injection-molding simulations, usually the Eulerian method is favored. In the Eulerian techniques, we can distinguish different methods that have been reviewed in the book of Zheng et al. [424]. Among these methods, the level set method (LSM) offers the most accurate results. Further explanations about this method will be given in Appendix I.

H.6

Numerical Solution For Temperature Field

To solve the energy equation for the temperature field, finite element methods can also be used. The thermal conductivity of polymers is very low, and hence in the flow direction the dominant heat transport mechanism is convection. As emphasized before, the classical Galerkin techniques are often unstable in solving convection-dominated problems. An upwinding scheme is the most commonly used technique to avoid the stability difficulty. Moreover, the 3D heat transfer simulation has another problem in calculating the temperature field across the thickness, where the main heat transport mechanism is thermal conduction. It is difficult to capture the rapid spatial change of the temperature along the thickness direction, especially within the narrow region next to the mold-polymer interface. This may require the use of many elements across the part thickness. However, for typical thin-walled parts, isotropic mesh refinement in the thickness direction can lead to an enormous increase in the total number of elements and prohibitive computational cost. The problem is alleviated by Gruan and Coupez [133], who have developed a 3D tetrahedral, unstructured, and anisotropic mesh generator that can place enough meshes through the thickness direction without introducing too many nodes in other directions. A different technique was presented by Friedl et al. [125], who adapted the analytical solution for a one-dimensional semi-infinite heat conduction to approximate the gap-wise temperature profile, and hence they can retain a relatively coarse mesh across the thickness.

312

H Three-Dimensional FEM for Mold Filling Analysis

I

Level Set Method

In injection-molding simulations, the flow front, that is, the interface of the melt and the air, can be modeled as the embedded zero level set of an implicit time-dependent function, and hence the advancement of the flow front can be followed by tracking the zero level set of the implicit function. This method, known as the level set method, was proposed by Osher and Sethian [276], and it has been applied to a variety of interface evolution problems, such as multi-phase flow (e.g., Zhao et al. [415]), free surface flow of liquid film (e.g., Dou et al. [86]), and a moving flow front in a filling process (e.g., Dou et al. [85]). Books describing the level set method are available (e.g., Sethian [328]; Osher and Fedkiw [275]). Suppose the computational domain is Ω. We divide it into two subdomains: the interior domain Ωinterior and the external domain Ωexterior . The interior domain Ωinterior represents the liquid-filled region, while the external domain Ωexterior is the empty or gas-filled region. The interface between the interior and exterior regions is denoted by ∂Ω. A distance function d (x), x ∈ Ω, is defined as d (x) = min(|x − xl |) for all xl ∈ ∂Ω,

(I.1)

implying that d (x) = 0 on the interface. The basic idea of the level set method is to define a signed distance function Φ(x, t ) in the computational domain, from which the zero level isocontour of the function is the interface representing the position of the flow front. Initially, Φ equals the distance from any point x to the initial ∂Ω, negative inside the filled region and positive outside the filled region, where the initial ∂Ω represents the inlet boundary. That is, for all xl ∈ ∂Ω,   − min (|x − xl |) 0 Φ(x, t = 0) =  min (|x − xl |)

for x ∈ Ωinterior for x ∈ ∂Ω for x ∈ Ωexterior

(the liquid-filled regions) , (the liquid-air interface) , (the air-filled regions) ,

(I.2)

which is continuous over the whole computational domain. In addition, the signed distance function has the following property: |∇Φ| = 1.

(I.3)

Let x(t ) be a particle on the interface. Then we must have Φ(x(t ), t ) = 0.

(I.4)

By the chain rule we have ∂ Φ d x(t ) + · ∇Φ = 0. ∂t dt

(I.5)

314

I Level Set Method

In general, since d x/d t is equal to the fluid velocity u, the time evolution of any level set of Φ is given by ∂Φ + u · ∇Φ = 0, ∂t

(I.6)

for the initial value Φ(x, 0) = Φ0 (x), which is given at the beginning of calculation for the entire domain. This Eulerian formulation for the motion of Φ will move the zero level set, Φ = 0, exactly as the flow front advances. The level set Equation I.6 moves the interface along its normal vector field with the normal speed u · n, and any tangential component will have no effect on the position of the front. The normal unit vector on the interface, drawn from the liquid to the air, can be expressed as ¯ ∇Φ ¯¯ . (I.7) n= |∇Φ| ¯Φ = 0 Thus one can rewrite Equation I.6 as 1 ∂Φ + u · n = 0. |∇Φ| ∂t

(I.8)

Equation I.6 or I.8 can be solved by a time marching scheme. After each time step, the zero level set function should represent the new position of the interface. Since the equation is of the hyperbolic type, stabilization techniques such as upwinding are usually employed. Because of the numerical approximation, the level set function Φ(x, t ) may not remain a signed distance function at later time steps, in particular after a long simulation time. Therefore, after the level set function has been advected, it is necessary that the level set function Φ be reinitialized so that it remains a distance function. The reinitialization can be achieved by solving the following partial differential equation: ∂Φ = si g n(Φ)(1 − |∇Φ|), ∂τ

(I.9)

with initial conditions Φ(x, 0) = Φ0 (x). Here, si g n is the sign function as 1 when Φ > 0, −1 when Φ < 0, and 0 on the interface where we want Φ to remain zero, and τ is the artificial time. Solving the equation to steady state with the artificial time τ provides a new value for Φ that satisfies |∇Φ| = 1, since the steady state is reached when the right-hand side approaches zero. The procedure is to stop the level set calculation periodically and solve Equation I.9 until reaching a steady state (Sussman et al. [348, 349]; Sethian and Smereka [329]; Dou et al. [85]). An example of the evolution of a free surface simulated using the level set method is given in Figure I.1.

I Level Set Method

315

Figure I.1 Evolution of a free surface simulated by the level set method (provided by Dr. Huagang Yu)

316

I Level Set Method

J J.1 J.1.1

Full Form of Mori-Tanaka Model

Eshelby Tensor Components

Material with Isotropic Matrix and Inclusions

When the matrix is isotropic and the inclusion is spheroidal with the symmetric axis identified as x 1 , the nonvanishing components of the Eshelby tensor were derived by Tandon and Weng [354] as µ ¶ ¸ · 3a r2 3a r2 − 1 1 m m − 1 − 2ν + g , (J.1) E 1111 = 1 − 2ν + 2 (1 − νm ) a r2 − 1 a r2 − 1 ¸ · 3a r2 1 9 ¢+ ¡ E 2222 = g, (J.2) 1 − 2νm − 4 (1 − νm ) 8 (1 − νm ) a r2 − 1 4(a r2 − 1) · ¸ ¾ ½ a r2 1 3 m E 2233 = − 1 − 2ν + g , (J.3) 4(1 − νm ) 2(a r2 − 1) 4(a r2 − 1) · ¸ a r2 3a r2 1 E 2211 = − + − (1 − 2νm ) g , (J.4) 2(1 − νm )(a r2 − 1) 4(1 − νm ) a r2 − 1 ¸ ¸ · · 1 1 1 3 + g, (J.5) E 1122 = − 1 − 2νm + 2 1 − 2νm + 2(1 − νm ) 2(1 − νm ) ar − 1 2(a r2 − 1) ½ · ¸ ¾ a r2 1 3 m E 2323 = + 1 − 2ν − g , (J.6) 4(1 − νm ) 2(a r2 − 1) 4(a r2 − 1) ½ · ¸ ¾ a r2 + 1 1 3(a r2 + 1) 1 m m 1 − 2ν − 1 − 2ν − g , (J.7) − E 1212 = 4(1 − νm ) a r2 − 1 2 a r2 − 1 E 3333 = E 2222 , E 3322 = E 2233 , E 3311 = E 2211 , E 1133 = E 1122 , E 1313 = E 1212 ,

(J.8)

where νm is Poisson’s ratio of the matrix, a r is the aspect ratio of the fiber, and g is given by £ ¤ ar (J.9) g= 2 a r (a r2 − 1)1/2 − cosh−1 a r , for a r > 1; 3/2 (a r − 1) £ ¤ ar g= (J.10) cosh−1 a r − a r (1 − a r2 )1/2 , for a r < 1. 2 3/2 (1 − a r ) For a spherical inclusion, a r = 1, one has 7 − 5νm , 15(1 − νm ) 5νm − 1 E 1122 = E 2233 = E 3311 = , 15(1 − νm ) m 4 − 5ν E 1212 = E 2323 = E 3131 = . 15(1 − νm ) E 1111 = E 2222 = E 3333 =

(J.11) (J.12) (J.13)

318

J Full Form of Mori-Tanaka Model

J.1.2

General Anisotropic Materials

For an ellipsoidal inclusion imbedded in a general anisotropic material, the Eshelby tensor E is given in terms of a double integral. Consider an ellipsoid inclusion whose three principal axes are a 1 , a 2 , and a 3 , respectively, and a 1 is its major radius. A rectangular Cartesian coordinate system [x i (i = 1, 2, 3)] is introduced such that the origin coincides with the center of the ellipsoid and axes x i are aligned with the principal axes a i . For a generic anisotropic material the Eshelby tensor is given by (Mura [262]) E i j kl =

1 m C 8π pqkl

Z

1

−1

dζ¯1

Z

2π h

0

i ¯ +G j pi q (ξ) ¯ dθ, G i p j q (ξ)

(J.14)

in which ¯ ¯ with ξ¯k = ζ¯k /a k , ¯ = ξ¯k ξ¯l Ni j (ξ)/Ω( ξ), G i j kl (ξ)

(J.15)

where ζ¯1 is an integration variable in [–1, 1], ζ¯2

= (1 − ζ¯21 )1/2 cos θ ,

(J.16)

ζ¯3

=

(1 − ζ¯21 )1/2 sin θ ,

(J.17)

¯ Ω(ξ)

= εmnl K m2 K n3 K l 1 , 1 εi kl ε j pq K kp K l q , 2 = C imj kl ξ¯ j ξ¯l ,

(J.18)

¯ = Ni j (ξ)

(J.19)

Ki k

(J.20)

and εi j k is the permutation tensor (see Appendix B). The double integral in Equation J.14 can be calculated numerically (Gavazzi and Lagoudas [129]) using the Gaussian quadrature formula as follows: E i j kl =

M X N 1 X C m {G i p j q (θn , ζ1m ) +G j pi q (θn , ζ1m )}Wmn , 8π m=1 n=1 pqkl

(J.21)

where M and N refer to the points used for the integration over ζ1 and θ, respectively, and Wmn are the Gaussian weights. The Gaussian points M and N are selected according to the aspect ratios a 1 /a 2 and a 2 /a 3 of the inclusion, as well as the disparity in the mechanical properties between the inclusions and the matrix. For cylindrical inclusions (a 2 /a 3 = 1 and a 1 /a 2 ≫ 1) and typical glass fiber reinforced polymers, M = 10 and N = 32 will provide sufficiently accurate results. A high ratio of inclusion/matrix mechanical properties may reduce the accuracy of predicted effective mechanical properties, and hence a higher number of the Gaussian points is required in this case.

J.2 Expanded Mori-Tanaka Equation

J.2

319

Expanded Mori-Tanaka Equation

In this section, the expanded form of Equation 8.15 for the Mori-Tanaka model will be given in terms of the contracted notation shown in Table J.1. Table J.1 Relation Between Indices in Contracted and Tensor Notations

J.2.1

Contracted Notation

Tensor Notation

1

11

2

22

3

33

4

23

5

13

6

12

Contracted Notation for Stiffness Tensor and Compliance Tensor

Since σi j and εi j are symmetric tensors, each of them has 6 independent components. The fourth-order stiffness tensor C i j kl contains at most 36 independent constants, such that it can be displayed as a 6×6 matrix of components using contracted notation, C mn , where m, n = 1, 2, 3, 4, 5, 6. There is a unique correspondence between C mn and C i j kl . The index m is related to ij, and n is related to kl, as shown in Table J.1. For instance, C 1122 = C 12 , and C 1323 = C 54 . Since C mn = C nm , the number of independent constants is generally 21. For orthotropic materials, the number of independent constants further reduces to 9. When the fourth-order tensor is transformed to the principal axes, all C i j = 0, except for C 11 , C 22 , C 33 , C 12 , C 13 , C 23 , C 44 , C 55 , and C 66 . Similarly, the fourth-order elastic compliance tensor S i j kl can be expressed in a contracted form S mn .

J.2.2

Inverse of a Matrix

In mechanical property calculations of fiber-reinforced polymers, one often needs to calculate the inverse of a matrix P i j .      [P i j ] =    

P 11 P 21 P 31 0 0 0

P 12 P 22 P 32 0 0 0

P 13 P 23 P 33 0 0 0

0 0 0 P 44 0 0

0 0 0 0 P 55 0

0 0 0 0 0 P 66

     .   

(J.22)

320

J Full Form of Mori-Tanaka Model

If there is a matrix Q i j , such that P i k Q k j = δi j , then the matrix P i j is said to be invertible, and Q i j is called the inverse of P i j , written as Q i j = P i−1 . j The components of Q i j can be calculated by Q 11 = (P 22 P 33 − P 23 P 23 )/ det(P i j ) , Q 22 = (P 11 P 33 − P 13 B 13 )/ det(P i j ) , Q 33 = P 11 P 22 − P 12 P 12 )/ det(P i j ) , Q 23 = (P 12 P 13 − P 11 P 23 )/ det(P i j ) , Q 13 = (P 12 P 23 − P 13 P 22 )/ det(P i j ) , Q 12 = (P 13 P 23 − P 12 P 33 )/ det(P i j ) , Q 44 = 1/P 44 , Q 55 = 1/P 55 , Q 66 = 1/P 66 .

(J.23)

where det(P i j ) = P 11 (P 22 P 33 − P 23 P 23 ) + P 12 (P 13 P 23 − P 12 P 33 ) + P 13 (P 12 P 23 − P 13 P 22 ) .

(J.24)

For example, given the contracted stiffness matrix C i j , one can use the above equations to calculate the contracted compliance matrix S i j , or vice versa,

J.2.3

Expanded Expression of the Mori-Tanaka Equation

Using the contracted notation, we write Equation 8.15 as C i j = C imj + φ f B i−1 j .

(J.25)

For orthotropic properties, all C i j = 0, except for C 11 , C 22 , C 33 , C 23 = C 32 , C 13 = C 31 , C 12 = C 21 , C 44 , C 55 , and C 66 . The components of B i j are given by f

m m m m −1 ) , B 11 = φm (E 1111 S 11 + E 1122 S 12 + E 1133 S 13 ) + (C 11 −C 11 f

m m m m −1 ) , B 22 = φm (E 2211 S 12 + E 2222 S 22 + E 2233 S 23 ) + (C 22 −C 22 f

m m m m −1 B 33 = φm (E 3311 S 13 + E 3322 S 23 + E 3333 S 33 ) + (C 33 −C 33 ) , f

m m m m −1 ) , B 23 = φm (E 2211 S 13 + E 2222 S 23 + E 2233 S 33 ) + (C 23 −C 23 f

m −1 m m m B 13 = φm (E 1111 S 13 + E 1122 S 23 + E 1133 S 33 ) + (C 13 −C 13 ) , f

m m m m −1 B 12 = φm (E 1111 S 12 + E 1122 S 22 + E 1133 S 23 ) + (C 12 −C 12 ) , f

m m −1 B 44 = 2φm E 2323 S 44 + (C 44 −C 44 ) , f

m m −1 B 55 = 2φm E 1313 S 55 + (C 55 −C 55 ) , f

m m −1 B 66 = 2φm E 1212 S 66 + (C 66 −C 66 ) .

(J.26)

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Index

A AC Technology see C-Mold advancement of the flow front see flow front advancement affine motion 235 algebraic multigrid (AMG) method 310 amorphous polymers 20 Andrade equation see Arrhenius equation anisotropic rotary diffusion model 114, 115, 241, 242 ARD-RSC model 118 Arrhenius equation 25, 177 Autodesk 225 Avrami index 143

B Babuska-Brezzi stability condition 305 barycentric coordinates 285 body force 11, 39 boundary element method 79, 83, 211, 215, 216 Brownian configuration field method 121 Brownian dynamics simulation 120 bubble shape function 307, 308 – conforming bubble 307 Bubnov-Galerkin method see Galerkin method

– inorganic colorant 167 – organic colorant 167 concurrent engineering 3 CONNFFESSIT 121 conservation laws – conservation of energy 35, 40–43, 45, 46, 53, 68, 248 – conservation of mass 35, 38, 46, 53, 66, 244, 250, 251 – conservation of momentum 35, 38–40, 46, 53, 68, 245, 250 constitutive equation 15 – Newtonian fluids 15, 22 – non-Newtonian fluids 16, 22 continuity equation see conservation of mass Cornell Injection Molding Program (CIMP) 206, 209–211, 215, 217, 218, 220 CRIMS 91 critical fiber aspect ratio 135 critical fiber length 135 crystallinity – absolute crystallinity 143 – relative crystallinity 143, 152 – ultimate absolute crystallinity 159, 165 crystallization 4, 21, 27, 86, 141 – flow-induced crystallization 149, 151–153, 169 – quiescent crystallization 141, 146, 149, 169

D C C-MOLD 215, 216, 220, 224 closure approximation 82 – hybrid closure 82 – linear closure 82 – orthotropic closure 82, 83 – quadratic closure 82 colorant 167–171, 173, 174

differential scanning calorimetry (DSC) 58, 59, 146, 152, 169 Dinh and Armstrong model 120 direct simulation 113, 116, 136 dual domain finite element analysis (DD/FEA) – flow analysis 101–103 – structural analysis 103–106

346

Index

– eccentric shell elements 104

E effective stiffness tensor 124 elastic modulus 32, 177 engineering method 205 ensemble average 80 equation of state 29 equilibrium melting temperature 141, 144–148 Eshelby model 125 Eshelby tensor 125 expansivity 29 – coefficient of volume expansion 29 – linear coefficients of expansion 32

F FENE-P model 150 fiber aspect ratio 50 – equivalent ellipsoidal axis ratio 51 fiber concentration 50 – concentrated regime 50 – dilute regime 50 – semi-concentrated regime 50 fiber length attrition model 134 fiber migration 132 fiber orientation averaging 130 fiber suspension 50 fiber volume fraction 32, 50 fiber weight fraction 31 finite difference method 253, 255, 257 – explicit scheme 257 – implicit scheme 258 finite element method 261, 262, 303–305, 308, 311 – C k continuity 307 – 1D analysis 261 – 2.5D approximation 65, 66, 99, 100, 102, 109, 243, 251, 257, 283 – 3D analysis 100, 107, 303 – application of boundary conditions 262, 264, 268, 280 – assembly of element equations 262, 263, 268, 270, 271, 278, 279

– constraint equation 268, 269 – derivation of element equations 262, 263, 274 – display of results 264 – interpolation functions see shape function – natural boundary condition 304 – natural coordinate 305 – nodal displacement vector 264 – nodal flow rate vector 264 – nodal flow values 267 – nodal force vector 264 – nodal pressure vector 264 – nodal unknowns 277, 278 – nodal values 267 – shape functions 267, 268, 274, 278, 280, 305, 307 – solution of system equations 262, 281 – stiffness matrix 264, 269 – system 261 – system equations 263 – weak formulation 304 finite volume method 107 flexible fibers 136 flow front advancement 298, 304, 310, 313 Fokker-Planck equation 51, 237, 238 Folgar-Tucker model 113–115, 117, 118, 140, 239 fountain flow 193 free energy 141, 145, 150 frozen layer 7, 72, 73, 77, 87, 91, 94, 206, 225 – growth 87 – thickness 72, 208, 215

G Galerkin method 268, 275, 278, 311 gate 5, 6 generalized Newtonian fluids 16 – Carreau model 23 – Cross model 23, 24 – Cross-WLF model 25 – definition 17, 23 – power law model 23 generalized strain rate – definition 16 glass transition temperature 20, 25, 26, 145

Index

growth 141, 142, 168 – growth rate 143, 145, 148, 168, 169 – spherulitic growth 143, 149 – unrestricted 142

347

Linkam shearing hot stage 146, 148, 152, 170 long fibers 49, 131

M H half crystallization time 169–172 half-crystallization time 146, 152, 153 heat capacity 26 Hele-Shaw approximation 166 high pressure dilatometry 61 Hoffman-Lauritzen theory 145, 168 Hoffman-Weeks extrapolation method 147, 148 hydrodynamic interactions 116 – long range interaction 116 – short range interaction 116

I instantaneous nucleation 141 interaction coefficient 81, 83, 113, 114 – interaction coefficient tensor 114, 115, 117 isothermal compressibility coefficient 30

J Jeffery’s equation 50, 113, 237 Jeffery’s orbit 51, 114

material volume 37 mean heat capacity 26 micro-injection molding 188 microcellular injection foaming molding 186 midplane approach 99, 100 MINI element 305, 308 modulus of rigidity see shear modulus mold 4 – cavity side see fixed side – fixed side 4, 5 – moving side 4, 5 Moldex 221, 225 Moldflow 59, 85, 206, 207, 212, 215, 216, 218–220, 224, 225 – Moldflow design principles 206 Mori-Tanaka model 123, 125–127, 319, 320 morphology 141, 142

N Newton-Raphson iteration scheme 309 no-flow temperature 59, 60, 72, 88, 155 no-slip condition 73 non-affine motion 236 non-Fourier thermal conduction 161 number density of fibers 50

K O Kolmogoroff-Avrami-Evans model 142, 168, 169 Kronecker delta 177

L Langevin equation 235, 237, 238 latent heat 146 level set method 313 – reinitialization 314 – signed distance function 313

orientation tensor 114–118, 120, 130, 238 Oseen tensor 137

P Peclet number 119 Phan-Thien-Graham model 120, 133 Phan-Thien-Tanner (PTT) model 156 Picard iteration scheme 309 Poisson’s ratio 33

348

Index

polymers – acrylonitrile-butadiene-styrene (ABS) 27, 28 – isotactic polypropylene (iPP) 4, 27, 28, 144, 146, 162, 165, 167, 169–174 – polyamide 66 (PA 66) 27, 28 – polybutylene (PBT) 174 – polycarbonate (PC) 27, 28 – polyethylene (PE) 27, 28 – polystyrene (PS) 27, 28 post-molding shrinkage 175 post-molding warpage 175 probability density function 51, 235, 237 Prony series 177 pseudo-time 177 PVT data 21, 29–31, 61

R rate-of-deformation tensor – definition 15 reduced-strain closure model 117, 118 Reynolds transport theorem 42 Rosen-Hashin model 126 runner 6, 53, 91, 96, 97 – cold runner 53 – hot runner 5, 53 – imbalanced filling 109, 110

S SAXS 152 semi-crystalline polymers 20, 141 shear heating 26, 48, 53, 92, 109, 110 shear modulus 33, 177 shear rate – definition 22 shear thinning 22 shish-kebab 141, 142 short fibers 49, 131 shrinkage 52, 84, 171, 173–175 – parallel shrinkage 85, 173, 174 – perpendicular shrinkage 85, 173, 174 – residual strain model 85 – residual stress model 87 – viscous-elastic model 88

– viscoelastic model 87 specific heat capacity 26 spherulites 141, 142 sporadic nucleation 141 stochastic equation 121, 237 stochastic process 120 strain concentration tensor 124, 125 stress relaxation 175 stress tensor 13, 39 – components 13 – normal stresses 12 – shear stress 12, 22 stress vector (traction vector) 12, 13, 39 stress-thermal coefficient 162 surface force see surface traction surface traction 11, 12, 39

T Tait equation 31 Taylor series expansion 253 tensile modulus see elastic modulus tensor – del operator 233 – Cartesian 228 – components 228 – double dot product 230 – dummy index 228 – dyadic product 230 – eigenvalue 231 – eigenvector 231 – Einstein summation convention 228 – free index 228 – orthogonal matrix 231 – orthogonal transformation 231 – rotation matrix 231 – single dot product 230 thermal conductivity 161 thermo-rheological simplicity 25, 177, 178 thermodynamic pressure 14 thermoplastics 19 thermosets 19 time-temperature shift factor 25, 177 time-temperature superposition 24 Timon 221 transition temperature 59, 60, 72

Index

transversely isotropic fluid (TIF) model 119

U uniaxial tensile strength model 135 upper convected derivative 156

V validation 9, 55 van den Brule’s law 162 velocity gradient 14, 15 – effective velocity gradient 51, 121, 236, 240 verification 9, 55 viscoelasticity 22 viscosity 17 – dilatational viscosity 15 – function definition 23

349

Voigt average 125 volume of fluid method (VOF) 304 vorticity tensor – definition 15

W warpage 84, 175 WAXS 169 weighted residual method 268 Weissenberg number 194 weldline 102 Williams-Landel-Ferry (WLF) equation 25, 177

Y Young’s modulus see elastic modulus

Kennedy · Zheng

Flow Analysis of Injection Molds Given the importance of injection molding as a process as well as the simulation industry that supports it, there was a need for a book that deals solely with the modeling and simulation of injection molding. This book meets that need. The modeling and simulation details of filling, packing, residual stress, shrinkage, and warpage of amorphous, semi-crystalline, and fiber-filled materials are described. This book is essential for simulation software users, as well as for graduate students and researchers who are interested in enhancing simulation. And for the specialist, numerous appendices provide detailed information on the topics discussed in the chapters. Contents: Part 1 The Current State of Simulation: Introduction, Stress and Strain in Fluid Mechanics, Material Properties of Polymers, Governing Equations, Approximations for Injection Molding, Numerical Methods for Solution Part 2 Improving Molding Simulation: Improved Fiber Orientation Modeling, Improved Mechanical Property Modeling, Long Fiber-Filled Materials, Crystallization, Effects of Crystallizations on Rheology and Thermal Properties, Colorant Effects, Prediction of PostMolding Shrinkage and Warpage, Additional Issues of Injection-Molding Simulation, Epilogue Appendices: History of Injection-Molding Simulation, Tensor Notation, Derivation of Fiber Evolution Equations, Dimensional Analysis of Governing Equations, The Finite Difference Method, The Finite Element Method, Numerical Methods for the 2.5D Approximation, ThreeDimensional FEM for Mold Filling Analysis, Level Set Method, Full Form of Mori-Tanaka Model

www.hanserpublications.com Hanser Publications ISBN 978-1-56990-512-8