Modelling of powder die compaction

Engineering Materials and Processes

Series Editor Professor Brian Derby, Professor of Materials Science Manchester Materials Science Centre, Grosvenor Street, Manchester, M1 7HS, UK Other titles published in this series Fusion Bonding of Polymer Composites C. Ageorges and L. Ye Composite Materials D.D.L. Chung Titanium G. Lütjering and J.C. Williams Corrosion of Metals H. Kaesche Corrosion and Protection E. Bardal Intelligent Macromolecules for Smart Devices L. Dai Microstructure of Steels and Cast Irons M. Durand-Charre Phase Diagrams and Heterogeneous Equilibria B. Predel, M. Hoch and M. Pool Computational Mechanics of Composite Materials M. Kamiński Gallium Nitride Processing for Electronics, Sensors and Spintronics S.J. Pearton, C.R. Abernathy and F. Ren Materials for Information Technology E. Zschech, C. Whelan and T. Mikolajick Fuel Cell Technology N. Sammes Casting: An Analytical Approach A. Reikher and M.R. Barkhudarov Computational Quantum Mechanics for Materials Engineers L. Vitos

Peter R. Brewin • Olivier Coube • Pierre Doremus and James H. Tweed Editors

Modelling of Powder Die Compaction

123

P.R. Brewin European Powder Metallurgy Association (EPMA) 2nd Floor, Talbot House Market Street Shrewsbury, SY1 1LG UK

O. Coube, PhD European Powder Metallurgy Association (EPMA) 2nd Floor, Talbot House Market Street Shrewsbury, SY1 1LG UK

P. Doremus, PhD Laboratoire GPM2 ENSPG PO Box 46 Saint Martin d'Heres 38402 France

J.H. Tweed, PhD AEA Technology Gemini Building Harwell, Didcot Oxfordshire, OX11 0QR UK

ISBN 978-1-84628-098-6

e-ISBN 978-1-84628-099-3

Engineering Materials and Processes ISSN 1619-0181 British Library Cataloguing in Publication Data Modelling of powder die compaction. - (Engineering materials and processes) 1. Compacting 2. Compacting - Mathematical models I. Brewin, Peter R. 620.4'3 ISBN-13: 9781846280986 Library of Congress Control Number: 2007932623 © 2008 Springer-Verlag London Limited ABAQUS, ABAQUS/Standard, ABAQUS/Explicit and ABAQUS/CAE are trademarks or registered trademarks of ABAQUS, Inc., Rising Sun Mills, 166 Valley Street, Providence, RI 02909-2499, USA; http://www.abaqus.com/ Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Foreword

Die compaction of powders that develop green strength on compaction is the absolutely dominating forming technology for powdered materials. Areas of application are structural parts, hardmetal and ceramic indexable inserts, pharmaceutical tablets, electrical contacts, filters, hard magnets, soft magnetic composites, friction materials and many others. In particular, multi cross-sectional net-shape geometries have been gaining importance continuously, because the ability to deliver complex shapes with higher and higher productivity has contributed to competitive advantages over alternative forming techniques. Since the raw material usage is better than 90 % in die compaction, even in areas that could be served by competing manufacturing technologies, die compaction of powders is often the most economic solution. The industries applying this technique have seen tremendous innovation especially in shape capability, reproducibility and productivity over the last 15 years that has resulted in high added value, astonishing growth rates and increased employment also in high labour cost countries. In order to stay successful in an environment of competing shaping technologies, the manufacturers must also in the future improve every single processing step, which implies to reduce the compaction press downtime caused by tool-setting iterations or tool readjustments. This requires more sophisticated production planning and quantitative predictions of tool and press deformation, crack formation and prevention during compaction and ejection, density development during compaction and final density distribution in the compact as well as the high-density phenomenon of delaminations. The goal of modelling diecompaction processes is, therefore, to get a very detailed picture of the powder density distribution in the die cavity before, during and after finishing the compaction stroke to provide the precise local stress distribution within the compact and the tool together with the associated deformations as the key to precise and crack-free parts. Much progress has been made in this direction during recent years especially through the work of the EPMA-coordinated and EU-supported Modnet and Dienet groups. Modelling is on the brink of becoming a usable tool also in die compaction of powders. Many details require further intensive development work, because the

vi

Foreword

mechanical behaviour of compressible powders is more complex to deal with than that of incompressible solids. This book is a record of our present knowledge and the newest achievements in this field. It gives an overview of the present status and views and shows the existing application possibilities of compaction modelling in developing new tooling and optimising compaction sequences. Also, the remaining deficiencies and areas of future progress are mentioned. The book resembles a balance sheet of what compaction modelling is currently capable of performing for designers and practitioners. It includes rich databases of information useful for future work in the field. It is highly welcomed as the starting point to continue ongoing developments for reliable and user-friendly widespread application. Paul Beiss RWTH Aachen

Acknowledgements

The editors would like to thank all those who contributed to the text of this book. Additionally the editors would like to thank two individuals who originally laid the foundation for the Thematic Networks, Dr David Whittaker of DW Associates and Mr. Bernard Williams, former Executive Director, EPMA. The editors would also like to thank all those who participated in the Thematic Networks and the associated independently funded research projects, without whose enthusiasm and involvement little would have been achieved. Finally the editors would like to acknowledge the support of the European Union for this work. Peter Brewin EPMA Olivier Coube EPMA Pierre Doremus Institut National Polytechnique de Grenoble James Tweed AEA Technology

Figure Acknowledgements

The editors acknowlege permission to publish illustrations that have previously been published elsewhere. In particular, thanks are due to: •

• • •

AEA Technology for Figures 9.1, 9.2, 9.12, 9.13, 12.10 and 12.11. This work was undertaken as part of the project on “Minimising density vartiations in powder compacts” funded by the UK Department of Trade and Industry. Elsevier Ltd for Figures 9.3, 9.14, 9.15, 14.1 and 14.9. EPMA for Figures 9.8-9.11, 11.4, 11.10, 12.2, 12.8, 12.12 and 12.13, Appendix 2, figures on pages 299-307, 310-311and 315-317. John Wiley and Sons Ltd. for Figure 14.10.

Contents

Lists of Contributors and Project Partners......................................................xvii 1 Introduction....................................................................................................... 1 P.Brewin, O.Coube, P.Doremus and J.H.Tweed 1.1 Treatment of Main Subjects in Compaction Modelling...................................... 2 1.2 Summaries of Individual Chapters.. ................................................................... 2 2 Modelling and Part Manufacture.................................................................... 7 P.Brewin, O.Coube, J.A.Calero, H.Hodgson, R.Maassen and M.Satur 2.1 Introduction ........................................................................................................ 7 2.2. Requirements for Improving the PM Production Process ................................. 8 2.2.1 Introduction................................................................................................ 8 2.2.2 Selection of Powder Blends ..................................................................... 10 2.2.3 Tooling Design ........................................................................................ 14 2.2.4 Press Selection ......................................................................................... 17 2.2.5 Production and Quality Control ............................................................... 18 2.2.6 Sintering and Infiltration.......................................................................... 18 2.3 Requirements for Compaction Modelling ........................................................ 19 2.3.1 Input-Data Generation ............................................................................. 19 2.3.2 Modelling and Part Manufacture: Requirements of the Hardmetal Industry (“HM”) .......................................................... 21 2.3.3 Modelling and Parts Manufacture: Requirements of Ferrous Structural (“FS”) Parts Industry ............................................ 24 2.3.4 Validation ................................................................................................ 26 References…… ...................................................................................................... 28

xii

Contents

3 Mechanics of Powder Compaction ................................................................ 31 A.C.F.Cocks 3.1 Introduction ...................................................................................................... 31 3.2 Uniaxial Deformation ....................................................................................... 32 3.3 Deformation under Multiaxial States of Stress ................................................. 34 References........ ...................................................................................................... 41 4 Compaction Models ........................................................................................ 43 A.C.F.Cocks, D.T.Gethin, H.-Å. Häggblad, T.Kraft and O.Coube 4.1 Micromechanical Compaction Models ............................................................. 43 4.1.1 Stage 0 Models ........................................................................................ 44 4.1.2 Stage 1 Models ........................................................................................ 46 4.1.3 Stage 2 Models ........................................................................................ 54 4.2 Phenomenological Compaction Models ........................................................... 55 4.2.1 Introduction.............................................................................................. 55 4.2.2 Cap Model ............................................................................................... 57 4.2.3 Cam-Clay Model...................................................................................... 59 4.3 Closure…... ...................................................................................................... 62 References…… ...................................................................................................... 62 5 Model Input Data – Elastic Properties.......................................................... 65 M.D.Riera, J.M.Prado and P.Doremus 5.1 Introduction ...................................................................................................... 65 5.2 Elastic Model .................................................................................................. 65 5.3 Experimental Techniques ................................................................................. 68 5.3.1 Characterisation of Elastic Properties of Green Compacted Samples ..... 68 5.3.2 Characterisation of Elastic Properties at High Stresses ........................... 73 5.4 Conclusions ...................................................................................................... 76 References…… ...................................................................................................... 76 6 Model Input Data – Plastic Properties.......................................................... 77 P.Doremus 6.1 Introduction ...................................................................................................... 77 6.2 Closed-Die Compaction Test............................................................................ 77 6.2.1 Discussion of Assumptions A1,A2,A3 .................................................... 83 6.2.2 Influence of the Sample Aspect Ratio on Experimental Results.............. 85 6.3 Powder Characterisation from Triaxial Test..................................................... 88 6.4 Concluding Comments ..................................................................................... 92 References…… ...................................................................................................... 93 7 Model Input Data – Failure ........................................................................... 95 P.Doremus 7.1 Introduction ...................................................................................................... 95 7.2 Tensile Test ...................................................................................................... 96 7.3 Diametral Compression Test............................................................................. 98 7.4 Simple Compression Test ............................................................................... 100

Contents

xiii

7.5 Concluding Comments ................................................................................... 103 References………………………………………………………………………..103 8 Friction and its Measurement in Powder-Compaction Processes ............ 105 D.T.Gethin, N.Solimanjad, P.Doremus and D.Korachkin 8.1 Introduction .................................................................................................... 105 8.2 Friction Measurement by an Instrumented Die............................................... 107 8.3 Friction Measurement by a Shear Plate .......................................................... 111 8.4 Example Measurements.................................................................................. 112 8.4.1 Instrumented-Die Measurements ........................................................... 112 8.4.2 Shear-Plate Experiments........................................................................ 115 8.5 Factors that Affect Friction Behaviour ........................................................... 118 8.5.1 Surface Properties .................................................................................. 118 8.5.2 Compact and Process Influences ........................................................... 122 8.6 Other Friction Measurement Methods ............................................................ 125 8.7 Relevant Bibliography .................................................................................... 127 8.8 Concluding Comments ................................................................................... 127 References........ .................................................................................................... 128 Chapter Bibliography ........................................................................................... 128 9 Die Fill and Powder Transfer ...................................................................... 131 S.F.Burch, A.C.F.Cocks, J.M.Prado and J.H.Tweed 9.1 Introduction .................................................................................................... 131 9.2 Potential Sigificance of Die Fill Density Distribution .................................... 132 9.3 Die-Filling Rig................................................................................................ 133 9.4 The Flow Behaviour of Powder into Dies Containing Step-like Features ...... 136 9.5 Metallographic Techniques for Determining Density Variations ................... 139 9.6 Measurement of Die Fill Density Distribution by X-Ray Computerised Tomography............................................................ 140 9.6.1 Hardware Components Needed for X-Ray CT ...................................... 141 9.6.2 Technique for the Quantitative Measurement of Density Variations..... 142 9.6.3 Results for Die fill Density Distribution Using X-Ray Computerised Tomography ............................................. 143 9.7 Modelling of Die Filling ................................................................................. 146 9.8 Concluding Comments ................................................................................... 149 References ............................................................................................................ 149 10 Calibration of Compaction Models ............................................................. 151 P.Doremus 10.1 Introduction .................................................................................................. 151 10.2 Calibration of the Drucker-Prager Cap Model ............................................. 151 10.2.1 Elasticity .............................................................................................. 151 10.2.2 Calibration of Yield Stress Surface and Plastic Strain. Method Based on Simple Tests........................................................... 152 10.3 Calibration of the Cam-Clay Model ............................................................. 158 10.4 Calibration of the Drucker-Prager Cap Model from Triaxial Data............... 159 10.5 Comparison of the Two Calibrations............................................................ 161

xiv

Contents

10.6 Concluding Comments ................................................................................. 163 References ............................................................................................................ 163 11 Production of Case-Study Components....................................................... 165 T.Kranz, W.Markeli and J.H.Tweed 11.1 Introduction .................................................................................................. 165 11.2 Press Instrumentation for Force and Displacement ...................................... 166 11.2.1 Punch Force ......................................................................................... 166 11.2.2 Punch Travel ........................................................................................ 169 11.3 Side Effects of Load Buildup – Press and Punch Deflections ...................... 171 11.3.1 Press Deflections.................................................................................. 171 11.3.2 Punch Deflections ................................................................................ 173 11.3.3 Implications for the Acquired Position Values .................................... 176 11.3.4 Concluding Comments on System Deflections.................................... 177 11.4 Case-Study Components .............................................................................. 177 References ............................................................................................................ 178 12 Assessing Powder Compacts......................................................................... 179 S.F.Burch, J.A.Calero, M.Eriksson, B.Hoffman, A.Leuprecht, R.Maassen, F.M.M.Snijkers, W.Vandermeulen and J.H.Tweed 12.1 Introduction .................................................................................................. 179 12.2 Density Distribution by the Archimedes Method ......................................... 180 12.2.1 Hardmetals ........................................................................................... 180 12.2.2 Zirconia and Sm-Co Samples .............................................................. 181 12.3 Density Determination by Machining .......................................................... 185 12.4 Density Distribution Determined by SEM-EDS Line Scan of Polished Cross-Sections ........................................................................... 187 12.4.1 General Considerations on Density Measurements of Green Samples.......................................................................................... 187 12.4.2 Samples................................................................................................ 188 12.4.3 Experimental Procedure....................................................................... 188 12.4.4 SEM-EDS Method to Determine Density Distribution........................ 189 12.4.5 Results.................................................................................................. 189 12.4.6 Discussion and Conclusion .................................................................. 190 12.5 Density Determination by X-ray Computerised Tomography...................... 190 12.6 Comparison of Result of Density Distribution Measurement Techniques ... 192 12.7 Determination of Defect Distribution........................................................... 193 12.8 Concluding Comments ................................................................................. 195 References ............................................................................................................ 195 13 Case Studies: Discussion and Guidelines..................................................... 197 O.Coube and P.Jonsén 13.1 Introduction .................................................................................................. 197 13.2 Constitutive Parameters Sensitivity.............................................................. 198 13.2.1 Case Study 3 ........................................................................................ 198 13.2.2 Constitutive Model used for the Parameter Study ............................... 199 13.2.3 Influence of the Constitutive Parameter R ........................................... 200

Contents

xv

13.2.4 Influence of the Hardening for High Density Values .......................... 202 13.2.5 Looking for Good Agreement with Experimental Values ................... 205 13.2.6 Discussion............................................................................................ 207 13.3 Framework for the Numerical Simulation .................................................... 207 13.4 Influence of Meshing.................................................................................... 211 13.5 Influence of the Initial and Process Data ...................................................... 212 13.5.1 Reference Case .................................................................................... 213 13.5.2 Influence of the Fill Density Distribution ............................................ 214 13.5.3 Influence of the Punch Kinematics ...................................................... 217 13.5.4 Discussion............................................................................................ 220 13.6 Conclusions and Guideline........................................................................... 221 References ........................................................................................................... 222 14 Modelling Die Compaction in the Pharmaceutical Industry ..................... 223 I.C.Sinka and A.C.F.Cocks 14.1 Introduction .................................................................................................. 223 14.2 Pharmaceutical Formulations and Processes ................................................ 225 14.3 Rotary Tablet Press Production Cycle.......................................................... 226 14.3.1 Die Fill on Rotary Presses.................................................................... 227 14.3.2 Compression and Ejection ................................................................... 229 14.4 Tablet Compaction Modelling...................................................................... 230 14.4.1 Material Characterisation for Model Input .......................................... 230 14.4.2 Friction................................................................................................. 233 14.5 Case Studies ................................................................................................. 234 14.5.1 Case Study 1: The Density Distribution in Curved-Faced Tablets ...... 235 14.5.2 Case Study 2: The Density Distribution in Bi-layer Tablets................ 237 14.5.3 Case Study 3: The Density Distribution in Compression Coated Tablets .......................................................... 239 14.6 Summary and Conclusions ........................................................................... 240 14.7 Acknowledgements ...................................................................................... 241 References ............................................................................................................ 241 15 Applications in Industry ............................................................................... 243 P.Brewin, O.Coube, D.T.Gethin, H.Hodgson and S.Rolland 15.1 Numerical Simulation of Die Compaction and Sintering of Hardmetal Drill Tips.......... ........................................................ 243 15.1.1 Summary.............................................................................................. 243 15.1.2 Introduction.......................................................................................... 244 15.1.3 Numerical Simulation of Die Compaction........................................... 244 15.1.4 Numerical Simulation of Sintering ...................................................... 248 15.1.5 Discussion and Conclusions ................................................................ 251 15.2 Ceramic Case Studies ................................................................................... 252 15.3 Concluding Comments ................................................................................. 258 References ........ .................................................................................................... 258

xvi

Contents

A.1 Appendix 1 – Compaction Model Input Data for Powders...................... 259 A.1.1 Distaloy AE Powder ................................................................................... 259 A.1.2 WC-Co Powder........................................................................................... 266 A.1.3 Zirconia Powder. Low- and High-Pressure Closed Die Compaction ......... 275 A.1.4 Samarium Cobalt Powder Low- and High- Pressure Closed Die Compaction .............................................................................. 286 References........ .................................................................................................... 292 A.2 Appendix 2 – Case Study Components ..................................................... 295 A.2.1 Introduction ................................................................................................ 295 A.2.2 Data for Case Study Components............................................................... 296 References........ .................................................................................................... 318 Glossary........... .................................................................................................... 319 Index .................................................................................................................... 325

Lists of Contributors and Project Partners

P. Brewin European Powder Metallurgy Association, 2nd Floor, Talbot House, Market Street, Shrewsbury SY1 1LG, UK. [email protected] S.F. Burch ESR Technology Ltd, 16 North Central 127, Milton Park, Abingdon, Oxfordshire, OX14 4SA, UK. [email protected] J.A. Calero Ames S.A., Ctra.Nac., 340 Km 1.242 Pol.Ind. Les Fallulles, 08620 Sant Vicenc dels Horts, Barcelona, Spain. [email protected] A.C.F. Cocks Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK. [email protected]

O. Coube PLANSEE SE, 6600 Reutte, Austria. Now with: European Powder Metallurgy Association, 2nd Floor, Talbot House, Market Street, Shrewsbury SY1 1LG, UK. [email protected] P. Doremus Laboratoire GPM2, ENSPG, PO Box 46, Saint Martin d'Heres 38402 France. [email protected] M. Eriksson IVF Research and Development Corporation/ Swedish Ceramic Institute, Argongatan 30, SE-431 53 Mölndal, Sweden. [email protected] D.T. Gethin, University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, UK.

D.T. [email protected]

xviii List of Contributors and Project Partners

H.-Å. Häggblad, Lulea University of Technology, S 97 187 Lulea, Sweden. [email protected] H. Hodgson Dynamic Ceramic Limited, Crewe Hall Enterprise Park, Weston Road, Crewe, Cheshire CW1 6UA, UK. [email protected] B. Hoffman GKN Sinter Metals Engineering GmbH, D-42477 Radevormwald, Germany. bettina.hofmann@ gknsintermetals.com P. Jonsén Luleå University of Technology, Luleå SE-971 87, Sweden. [email protected] D. Korachkin University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, UK. T. Kraft Fraunhofer IWM, LB 4.1 Pulvertechnologie, Woehlerstr 11, 79 108 Freiburg, Germany. [email protected]

T. Kranz Komage Gellner GmbH, Kell am See, Germany. [email protected] A. Leuprecht Plansee SE, A-6600 Reutte, Tirol, Austria. [email protected]

R. Maassen GKN Sinter Metals Engineering GmbH, D-42477 Radevormwald, Germany. robert.maassen@ gknsintermetals.com W. Markeli Komage Gellner GmbH, Kell am See, Germany. [email protected] J.M. Prado Universidad Polytechica de Catalunya, Diagonal 687, 08028 Barcelona, Spain. [email protected] M.D. Riera CTM Technological Centre, Department Materials Science, Universidad Polytechica de Catalunya, Diagonal 687, 08028 Barcelona, Spain. [email protected] S. Rolland University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, UK. [email protected] M. Satur Swift Levick Limited, High Hazels Road, Barlborough Links, Barlborough, Derbys S43 4TZ, UK. [email protected] I.C. Sinka Merck Sharp and Dohme Ltd, Hoddesdon, Herts. EN11 9BU, UK. Now with: Department of Engineering, University of Leicester, University Road, Leicester LE1 7RH, UK. [email protected]

List of Contributors and Project Partners

F.M.M. Snijkers VITO, Boeretang 200, B-2400 MOL, Belgium. [email protected] J.H. Tweed AEA Technology, Gemini Building, Harwell, Didcot, Oxfordshire, OX11 0QR, UK. [email protected]

W. Vandermeulen VITO, Boeretang 200, B-2400 MOL, Belgium.

xix

xx

List of Contributors and Project Partners

Thematic Network Partners: 1. Dienet Thematic Network in Die Compaction Modelling “Dienet” EU Contract No. G5RT-CT-2001-05020 Partners at Project Completion February 2005 Partner Name

Country

European Powder Metallurgy Association

UK

AEA Technology plc

UK

Commissariat a l’Energie Atomique

F

CS Systemes d’Information

F

Centro Sviluppo Materials SpA

I

Fraunhofer Institut Angewandte Materialforschung

D

Fraunhofer-Institut fur Werkstoffmechanik

D

Institut National Polytechnique de Grenoble

F

University of Leicester

UK

Lulea University of Technology

S

Swedish Ceramic Institute

S

University of Wales Swansea

UK

Universitat Politecnica de Catalunya

E

Vlaamse Instellung voor Technologisch Onderzoeck NV

B

Dynamic Ceramic Limited

UK

Eurotungstene

F

GKN Sinter Materials Gmbh & Co KG

D

Hoeganaes AB

S

Komage Gellner Gmbh & Co

D

Makin Metal Powders Limited

UK

Tecsinter S.P.A.

I

Plansee AG

A

QMP Metal Powders GmbH

D

Swift Levick Magnets Limited

UK

Centro Internacional de Metodos Numericos en Ingenieria

E

Institute for Problems of Materials Sciences of National Academy of Sciences of Ukraine

UA

Aleaciones de Metales Sinterizados SA

E

List of Contributors and Project Partners

Thematic Network Partners: 2. PM Modnet Thematic Network in Powder Metallurgy Process Modelling “PM Dienet” EU Contract No. BRRT-CT97-5021 Partners at Project Completion September 2000 Partner Name

Country

European Powder Metallurgy Association

UK

GKN Sinter Metals

D

Leicester University

UK

MIBA Sintermetall

A

Monroe Belgium METC

B

Sintermetal S.A.

E

Hilti Corporation

FL

Sinterstahl Gmbh

D

University of Nottingham

UK

University of Wales

UK

Universidad Politecnica de Catalunya

E

Lulea University of Technology

S

Institut Nationale Polytechnique de Grenoble

F

CEA/CEREM

F

Katholieke Universitat Leuven

B

Vlaamse Instelling voor Technologisch Onderzoek

B

Fraunhofer-Institut fur Werkstoffmechanik

D

VTT Manufacturing Technology

FIN

AEA Technology

UK

Delft University of Technology

NL

Dorst Maschinen und Anlagenbau

D

Hoganas AB

S

xxi

1 Introduction P. Brewin1, O. Coube2, P. Doremus3 and J.H. Tweed4 1

European Powder Metallurgy Association, Talbot House, Market Street, Shrewsbury SY1 1LG, UK. 2 PLANSEE SE, 6600 Reutte, Austria. Currently as 1. 3 Institut National Polytechnique de Grenoble, France. 4 AEA Technology, Gemini Building, Harwell, Didcot, Oxfordshire, OX11 0QR, UK.

Computer modelling (“CM”) of powder die compaction has a reputation for being limited to density predictions on simple shapes, slow to perform and an academic subject of little practical relevance. Much has, however, happened in recent years to challenge this negative image: the modern desktop PC is fast and capable; CM can now be operated by nonspecialists; case studies have shown that providing the model input data is of sufficient accuracy, these models can provide tool designers with accurate quantitative information on stresses and press functions that previously could only be estimated. This textbook details research and validation work carried out in several European and industrially funded projects over the period 1990 - 2005 by leading European academic and industrial centres. Much of this work underpinned the various case studies carried out on several powder materials within the European funded Thematic Network PM DIENET (2001-5). In addition to comparing computer predictions with industrially produced components, this network was able to take advantage of new research on die filling and powder deformation at low pressures. The Dienet case studies showed that the accuracy of computer predictions could be affected strongly by inaccurate input data (die fill, press elasticity, powder plastic data). In general, where the pressed components were to be sintered at high temperature to “full” density, CMs were invaluable in assisting the manufacturer to design powder tooling and press kinematics to achieve optimum pressed density. Pressed components sintered at lower temperature with minimal size change often involved more complex geometries; in this case avoidance of shear cracking was more important than density uniformity. While important additional work needs to be done in the areas of die filling, powder transfer and crack prediction, this textbook aims to detail the state-of-theart and to provide the basis for new work.

2

P. Brewin, O. Coube, P. Doremus and J.H. Tweed

1.1 Treatment of Main Subjects in Compaction Modelling The table shows the treatment of the main subjects by chapters Main subject

Sub topic

Main chapter

Other chapters

Input data

Elasticity

5

2,14

Plasticity

6

2, 10

Shear failure

7

2,14

Die fill

9

2,11,14

Friction

8

2,6, 14

Modelling

3

Simulation

Plasticity models

4,10

3,7,14

Validation

Industrial techniques

11,12

2, 9

Case studies

Component production

11

14

Comparisons

13

14,15

Practical implementation of modelling in industry

Blend Tool design, press selection and tooling kinematics

2, 14 2

Sinter modelling Instrumented die Applications

14 2

6

2

14, 15

2

1.2 Summaries of Individual Chapters Chapter 2 “Modelling and Part Manufacture” This chapter considers the requirements of industry for: • how compaction models (“CM”) should improve the PM production process • how the operational aspects of CMs should integrate with industrial existing design and manufacturing methods • methods of validating computer predictions. The first part covers blend selection, tooling design, press selection, production and quality control, sintering and infiltration. The second part covers the need for practical methods of generating the necessary input data for the CMs, and includes statements from both hardmetal and ferrous part manufacturers. The third part is a critical review of validation techniques.

Introduction

3

Chapter 3 “Mechanics of Powder Compaction” This chapter introduces the mathematical treatment of die compaction starting from simple uniaxial stress-strain, introducing those elastic and plastic factors that convert the applied stresses to a multiaxial stress system. Different models of yield surface are reviewed, including Shima, Cam-Clay and the Drucker-Prager-Cap model. Chapter 4 “Compaction Models” This chapter describes both macro and microscopic models (also called phenomenological and micromechanical models). It points out that the different phenomenological models are largely differentiated by the mathematical treatment of the powder yield stage of compaction, the two models most widely used being the Cap and Cam-Clay. Chapter 5 “Model Input Data - Elastic Properties” This chapter gives a brief overview of the present knowledge of the elastic behaviour of granular materials, including a description of the tests commonly used to determine stresses in unloading and ejection. It covers contacts between loose particles as well as the elastic properties of pressings at different stages of compaction, Poisson's ratio and Young’s modulus being especially relevant in the establishment of radial stresses. Chapter 6 “Model Input Data – Plastic Properties” This chapter discusses methods of determining key compaction parameters at low cost using different types of instrumented dies. It discusses the effects of aspect ratio, of die-wall friction, and the variation of friction with both density and aspect ratio. Chapter 7 “Model Input Data - Failure” This chapter considers failure arising from excessive shear during the compaction stroke. Methods of measuring compact cohesion are presented and discussed. Chapter 8 - “Friction and its Measurement in Powder Compaction Processes” This chapter discusses the role of friction on powder compaction and on part ejection. Two main methods of measuring friction coefficient are described, instrumented die and shear plate. Factors affecting friction behaviour are discussed. Some other less common friction measurement methods are reviewed. Inverse modelling is introduced as a useful method for deriving friction data from compaction experiments. Chapter 9 “Die Fill and Powder Transfer” This chapter presents the results of sensitivity studies showing the effect of non-uniform die fill on the evolution of punch loads and pressed densities for different starting fill geometries. Techniques for measuring fill density distributions on laboratory diesets are described. The chapter ends with a discussion of die fill modelling using discrete element techniques.

4

P. Brewin, O. Coube, P. Doremus and J.H. Tweed

Chapter 10 “Calibration of Compaction Models” The DPC model has two sections, the failure line and the cap. The position of the failure line is determined by powder cohesion and by slope. The cap is fixed by several parameters: the stress needed to achieve a given density, the intersection of the cap with the failure line, and the cap eccentricity, which on some materials can be density dependent. This chapter compares inelastic (hardmetal) and ductile (ferrous) powders. It then compares cap models drawn from data produced in instrumented dies with those generated on triaxial test rigs. Chapter 11 “Production of Case Study Components” Case studies have confirmed the importance to accurate computer predictions of being able to specify the starting conditions in terms of punch position and fill density, and the motion of the press tooling during the compaction process. Determining these factors accurately on a modern production press usually requires additional instrumentation to measure deflections in press and tooling. Although small dimensionally, these can have a large effect on loads. Techniques for measuring load, tooling displacement and press deflection are discussed. The chapter then reviews components produced for the Modnet and Dienet Case Studies. Chapter 12 “Assessing Powder Compacts” This chapter reviews techniques for measuring density distributions and detecting cracks in pressed components using industrial samples. Chapter 13 “Case Studies: Discussion and Guidelines” This chapter reviews several case studies in which parts from different materials were first produced under industrial conditions then “virtually” reproduced by several different computing centres using numerical simulation. The “actual” and “virtual” parts are compared. Sensitivity studies show the effect on predicted density and punch forces of cap eccentricity, powder hardening, mesh size and starting fill density distribution. The discussion confirms the importance of knowing punch deflection accurately, and difficulties arising from friction between pressing tools. The chapter ends with suggested conclusions and guidelines Chapter 14 “Modelling Die Compaction in the Pharmaceutical industry”. Modelling is used in the pharmaceutical industry to ensure tablets are of uniform density, free from cracks or chips notwithstanding embossed lettering and break lines. The special features of pharmaceutical powder blends and presses are discussed; techniques for blend characterisation and different modelling approaches are reviewed.

This page intentionally blank

Introduction

5

Chapter 15 “Applications in Industry” This chapter gives examples of the successful use of CM by industry. It includes a discussion of numerical simulations of die compaction and sintering of hardmetal drill tips. Appendix This is in two parts. The first provides the input data used for four materials studied, together with information on the test techniques used. The second includes data on case studies carried out by the Modnet and Dienet Thematic Networks. These data include tool fill, kinematics and final compacted positions, punch loads and pressed part density distributions.

2 Modelling and Part Manufacture P.Brewin1, O. Coube2, J.A. Calero3, H. Hodgson4, R. Maassen5 and M. Satur6 1

European Powder Metall Association, Talbot House, Market St., Shrewsbury SY1 1LG, UK. 2 PLANSEE SE, Technologiezentrum (TZIK), A-6600 Reutte, Tirol, Austria. 3 AMES S.A, Ctra.Nac. 340 Km 1.242 Pol.Ind."Les Fallulles", 08620 Sant Vicenc dels Horts, Barcelona, Spain. 4 Dynamic-Ceramic Ltd., Crewe Hall Enterprise Park, Weston Road, Crewe, Cheshire CW1 6UA, UK. 5 GKN Sintermetals Service, Krebsoge 10, D-42477 Radevormwald, Germany. 6 Swift Levick Magnets Ltd., High Hazels Rd., Barlborough Links, Barlborough, Derbys S43 4TZ, UK.

2.1 Introduction Compaction is the central stage of the Powder Metall shaping process. Powder blends must balance free flow with high green strength and good compressibility (of special interest to powder makers); tooling must be of sufficient strength to withstand the stresses of production, and must incorporate design features that take into account not only finished-part geometry but also die fill, powder-transfer stages, press kinematics and ejection to achieve uniform pressed density and to avoid generation of shear or tensile cracks as a result of the compaction process (of special interest to tool designers and press makers). Where parts are to be sintered to full density, sintered-part accuracy will substantially be determined by uniformity of pressed density; where final-part properties are to be achieved without dimensional change by sintering at lower temperatures, parts can be more complex in shape (of special interest to component producers). Where parts are to be marketed in the as-pressed condition (such as in pharmaceuticals) they must have sufficient strength to withstand post-compaction operations and delivery to the customer. In this process the pressed part may incorporate necessary geometrical features (such as embossed letters) which weaken the component (see Chapter 14). Developments in compaction modelling (“CM”) offer the component manufacturer an advanced tool to calculate stresses and pressed part qualities for different possible tooling designs. Such a design tool can additionally be used in the selection of powder blends presses and tooling by testing alternative designs virtually and by sensitivity methods.

P. Brewin, O. Coube, J.A. Calero, H. Hodgson, R. Maassen and M. Satur

8

In addition to accurate and effective compaction modelling (“CM”) industry requires • •

cost-effective methods of generating the input data required for the compaction models, this to include powder blend constitutive data, friction and die fill characteristics. methods of validating the predictions on experimentally produced parts including predictions of pressed density, tooling loads and cracks.

This chapter considers the requirements of industry for both: • •

how CMs should improve the PM production process how the operational aspects of CMs should integrate with industry’s existing design and manufacturing methods.

It is intended that this discussion should assist researchers in prioritising their efforts in the development of compaction modelling techniques for industrial use, while further encouraging industry to implement them.

2.2 Requirements for Improving the PM Production Process 2.2.1 Introduction Because of its inherent flexibility, Powder Metall often offers several different processing routes to solve a single design problem. The increasing success of computers to model the compaction process is also a measure of our improved understanding of the process itself; for example, our understanding of the mechanisms controlling the filling of dies has greatly improved in recent years as a direct result of CM studies. CM also offers the ability to carry out computer studies to determine the sensitivity of a key parameter such as part density distribution to an independent variable such as die wall friction and tooling motion [1]. The results of such studies can be used better to focus engineering effort in terms of raw material selection, engineering design or processing route. Recent developments in CM have been greatly assisted by the continuing and fast improving power and processing speed of the desktop PC, these have made it possible to carry out simulations in a few hours that previously would have taken several days, and have opened up the possibility of using discrete element models. It is significant that many major industries converting particulates to components by die compaction have no direct experience of CM - indeed it is only recently that CM has started to be used in pharmaceuticals, one of the largest industries employing powder compaction. Industry requirements for CMs are as numerous as the number of in house functions involved [2]. Overall, however, industry wishes to use CM in all inhouse functions for calculating density distribution, crack prediction, tooling loading and press movements.

Modelling and Part Manufacture

9

Apart from the obvious uses of CM in research and development, full implementation will be in association with the design office and to some extent by production for press setting. Achieving this will require: • • • • •

material databases robust FE software integration of the software in an automatic or semi automatic optimisation tool good interfaces with other computer applications (both component design and press software) user-friendly interfaces for non specialists.

Additionally it should be possible to generate adequate input data in-house; externally developed software should provide good quality updates and support. Table 2.1 lists the uses and user friendly requirements by company function: Table 2.1. Uses of compaction modelling by company function Company function

Uses

User-Friendly requirements

R&D

Feasibility studies

Interfaces to FE and CAD software input data generation with in-house capabilities (instrumented die, tensile and compression tests)

New materials Process development

Tooling design office

Powder development

Interface between FE and optimisation tool

Feasibility studies for different tool solutions (materials, shapes, kinematics)

Interface between FE and CAD software

Die deformation

Production press setup

Interface between FE and optimisation tool

Springback of the green compact

Material database

Press setup

Interface to press software

Database of presses to be used

Interface to less-controlled mechanical presses Material database Robust software Speed Visual displays of output

Until low-cost software packages are commercially available, smaller companies will likely prefer to purchase simulations from commercial agencies.

P. Brewin, O. Coube, J.A. Calero, H. Hodgson, R. Maassen and M. Satur

10

2.2.2 Selection of Powder Blends 2.2.2.1 Introduction Industry can look to CM not only to predict pressed part density distributions and loads, but also as a tool to develop powder properties. Thus, powder makers can use CM to optimise powder properties for different component shapes; parts makers with in-house powder production can similarly use CM to optimise powder properties; other parts makers can choose the most appropriate combination of raw material and processing parameters to suit the component and the production equipment available. Granulation and other blend modifications can be used to optimise powder blends. (Where powders require magnetic alignment following die fill but before the application of pressure, granulation may not be possible.) 2.2.2.2 Granulation Several industries including ceramic, hardmetal, pharmaceutical and magnets use powders that are too fine to flow freely. Granulation using organic binders is used to provide relatively coarse rounded agglomerates that flow freely into dies. A variety of different granulation techniques is used [3], each producing different granule structures. The best granulation techniques • •

produce granules in which the binder is distributed uniformly allow granules to deform during compaction to fill all voids uniformly.

In hardmetal manufacture granulation can be used to reduce the tap:apparent density ratio to as low as 1.05 (unlubricated ferrous powders are typically 1.25) thereby improving die fill uniformity. Because the hardmetal particles are incompressible, the binder provides the compact green strength. 2.2.2.3 Fill and Flow Recent experimental work and numerical studies have done much to improve our understanding of die filling [4]. Not least, these studies have highlighted those areas where we lack understanding of the different mechanisms, particularly the role of entrapped air. Numerical studies are only of value if they can be validated; this has been done by: • • •

high speed photography of transparent dies using both monochrome powders and powders of varying colours and sizes sectioning and X-ray of filled dies after light sintering metallography of the finish sintered part.

(See Chapter 9 for further discussion.) Discrete element modelling enables us to map the evolving powder density distributions during die filling and thereby better to optimise the die design and fill kinematics. In this respect it is clear that fill-shoe speed must be controlled to match the evolution of air from the die cavity. Failure to optimise die filling can

Modelling and Part Manufacture

11

lead to large variations in fill density through the die cavity and also to incomplete filling at the die surface. Some key issues are as follows: •

•

•

•

• •

The fluidity of powder blends has a large influence on die filling, powder transfer and the early stages of compaction [5]. Fluidity is normally measured by Hall flow, although more recent work shows that a variableaperture flowmeter (“VAF”) more closely reproduces flow into closed dies ([6] p.77). The speed of the filling shoe must also be controlled: above a critical velocity dies will only fill partially ([6] Fig 6.) Recent modelling studies have demonstrated that on more complex parts, high-integrity compacts can only be achieved if the powder fill is uniform. High throughput compaction presses need to operate at fastest speeds consistent with good quality product, and therefore prefer free-flowing powders. On the other hand, free-flowing powders tend to give lower pressed-part green strength and greater tendency to ejection cracks. While to some extent this can more easily be achieved by optimising tool design and press kinematics, the most important factor is achieving a satisfactory compromise between powder flow and green strength. Filling studies [4] show the high degree of turbulence that powder blends can experience during die filling, largely as a consequence of exhausting the air entrapped during the filling process. This is particularly important on fine irregular powders that present considerable resistance to air flowing out of the die during the filling operation ([6] p.82). A comparison of critical velocities measured in air and vacuum give an indication of the tendency of certain powders to elutriate during filling ([6] p.84). Where two or more different powders are to be blended, the ease of mixing is greatest where powders are similar in particle size and material density. Thus, it is difficult to mix powders of widely differing particle size and density. Unfortunately many PM production processes require the addition of sub-micrometre powders (graphite, lubricants) to coarser powders. Such blends can segregate owing to air turbulence during the filling operation, resulting in undesirable variations in chemistry through the sintered part. Where this could lead to problems in final product performance, it will be advisable to stabilise the powder blend using appropriate binders. Powder blends may also be granulated where binders used are chemically compatible with the powder, and where high compacted densities are not required [5]. Measures such as suction fill can be used to reduce countercurrent air flow during die fill by withdrawing lower punches during powder filling. The abrasive action of the blending operation increases the surface activity of powders; therefore powders should always be compacted as soon as practicable after blending; where this is not practicable care must be taken to exclude moisture in storage.

12

P. Brewin, O. Coube, J.A. Calero, H. Hodgson, R. Maassen and M. Satur

2.2.2.4 Deformation Characteristics during Compaction In cold isostatic compaction powders are placed in flexible bags. After sealing, these bags are submerged in hydraulic fluids and then pressurised to the required pressure. Each powder particle sees the same pressure in all directions, and the compact shrinks approximately isotropically during compaction. In die compaction this is not so; axial pressure only is used to densify the powder, radial dimensions being constant and confined by the die wall. Powder particles are subjected to different pressures radially from axially, typical radial pressures being half the axial pressures. Compaction modellers have developed several basic methods of establishing the deformation characteristics of powders subjected to these non isotropic more complex stress systems (for more details see Chapters 3 and 4). To measure the densification and hardening of the powder during compaction three laboratory rigs are commonly used: 1. High-pressure Triaxial Stress Rigs Depending upon the design, these either measure the compaction characteristics of loose powders, or of a previously compacted part. These can work at pressures up to 1200 MPa axial and 700 MPa radial. 2. Low-pressure Triaxial Stress Rigs Similar in design to high-pressure rigs compacting loose powders, these measure the deformation characteristics below 1 MPa. Both the above are capital-intensive items unlikely to be purchased by industry for in-house use. 3. Instrumented Die A plain cylindrical die is used to compact the powder; sensors in the die wall measure radial pressures during compaction at different heights. To measure the green strength achieved by the powder during compaction, cylindrical or beam samples are first die compacted in order to reach several strategic densities and are then tested using one of the following methods: • Compression test a cylinder is compressed again in the axial direction (the original pressing direction). Fracture stress is measured at different densities. • Brazilian disc test a cylinder is compressed in the radial direction. Fracture stress is measured at different densities. • Four-point bend test a beam sample undergoes a four-point bend test. Fracture stress is measured at different densities. Our improved understanding of the material processes underlying compaction now enables us better to select powder blends for the more demanding applications. In the early stages of compaction, powders need to transfer more readily to the geometrical shape of the final part. This reflects in large axial strains

Modelling and Part Manufacture

13

for a given pressure (in [7] slide 23 the lubricant addition causes deterioration in low-pressure response). Finer spherical powders can transfer in the early stages of compaction by a series of collapsing bridges, explaining why good compacts can only be achieved at compaction speeds of 50% or less of normal ([7] slide 24). While granulates can exhibit superior powder-transfer characteristics, granulation has a significant effect on low-pressure compaction response. Thus, granulates deform and break down at lower pressures than non granulates giving rise to greater strains at a given pressure and a more complex low-pressure response ([7] slide 25). Computer simulators fit the raw plasticity data to relatively simple empirical models (Cam-Clay, Drucker-Prager-Cap etc). Such models are also useful in powder selection as follows: ([7] slides 27 and 28). • •

Powder densities are determined by a combination of mean and shear stress; different blends can be compared by comparing the respective DP failure lines, which shows the stress levels at which failure can occur. Powders with high cohesion* are less likely to form cracks.

*as measured by the cohesion and the cohesion angle on Drucker-Prager-Cap models ([7] and [8]). This is further discussed in Chapters 4 and 10. 2.2.2.5 Other The pressed density selected for a component depends on several factors: • • •

•

•

Where ferrous parts are to be sintered, the pressed density converts roughly to the sintered density and therefore largely determines the finished part mechanical strength. Where hardmetal, ceramic or refractory parts are to be sintered, the minimum pressed density is used consistent with a) good green-part handleability b) adequate sinterability. Die-wall friction gradients increase with increasing pressed density, as do internal stresses locked up in the pressed part. Both effects mean that it is often more difficult to achieve tight dimensional tolerances on components pressed to high densities. While elastic recovery of compaction punches can be calculated, elastic recovery of punch holders and tool and press frames is difficult to estimate, especially in the axial direction (see Section 2.3.4). They are directly measured on modern press machines. These effects become more important at higher compaction pressures; Components pressed at high pressure have a greater tendency to form tensile cracks on ejection.

Powders produced in prealloyed (rather than elementally blended) form have the advantage of known chemical composition unaffected by segregation caused by air turbulence during die filling. The main disadvantage of prealloying, however, is the inevitable loss of compressibility, although small additions of certain elements can sometimes be made that have a strongly beneficial effect on finished-part

14

P. Brewin, O. Coube, J.A. Calero, H. Hodgson, R. Maassen and M. Satur

mechanical properties without adversely affecting compressibility. (An example of this is the Fe-0.6Mo powder series.) Powder cost tends to be significant on larger components; on components below (typically) 10-20 g powder content is usually masked by processing costs. Components subjected to high stresses or other hostile environments should always be produced from high-purity powders with low surface oxygen. Inclusions should be minimised by taking care when manufacturing both the powder and the component, including use of magnetic separation (where possible). On materials sintered subsolidus maximum allowable inclusion size will correspond to maximum pore size; on materials sintered supersolidus maximum allowable inclusion size will generally be much finer, depending on stress levels and critical defect size. 2.2.3 Tooling Design 2.2.3.1 Introduction Tooling design is arguably the most important function in sintered component manufacture, and the function that is most closely guarded in a competitive world. One of the uses of CM is to make it easier for the production engineer to select the cheapest tool and simplest press consistent with a pressed component of adequate quality. In general, the part designer will strive to avoid the need for postsinter machining; however, this may be necessary on components incorporating features such as transverse holes or where a complex tool cannot be justified on economic grounds. Post-sinter machining may also be needed where sintered dimensional tolerances are insufficiently accurate. Examples of post-sinter machining are where supersolidus sintering is used on pressed components of nonuniform pressed density, or subsolidus sintering on components pressed at high pressure. Where distortion after sintering can be expected (uneven pressed density, slumping through gravity) it may be possible to incorporate the inverse of the anticipated distortion in the compaction tooling, to neutralise this. However, this is not usually the best practice. The tooling designer needs accurate information on tooling stresses if he is to minimise tool breakage, to avoid over-design and to be able to use the correct figures for elastic recovery. Without CM the designer has to rely on previous experience on similar parts. The fact that tooling costs are often amongst the largest items in the maintenance budget shows the potential for improvement here. Much of the above has to be decided at the quotation stage; CMs offer the potential to improve and accelerate this function with great benefit to the parts maker and his customer. Tooling for use at higher compaction pressures (over 500 MPa) is usually composite, incorporating a shrink-fitted hard-wearing insert inside a steel bolster. Because hardmetal has a significantly higher elastic modulus than steel (hardmetals ~550 GPa versus steel 200 GPa), hardmetal tooling exhibits less springback during part ejection.

Modelling and Part Manufacture

15

The principles of the composite steel/hardmetal design are that: • • • •

the insert remains in compression at all times the interference fit between the insert and the die is sufficiently strong to withstand ejection of the pressed part the thermal processes used in fitting have no adverse metallurgical effects the pre-stressing is within the strength limits of the materials.

Such calculations require accurate knowledge of radial stresses during both compaction and ejection, and must take into account pressed-part height [9]. 2.2.3.2 Powder Fill and Powder Transfer Sensitivity studies using CM are powerful tools for showing the potential effect of powder fill distribution on pressed-part integrity. One study [10] compared the effect of nonuniform powder fill on pressed density distributions on 2 different shapes (see also Chapter 9). The results showed that on a plain cylinder nonuniform fill density had little effect, whereas on a more complex geometry nonuniform fill resulted both in nonuniform pressed density distributions and large inaccuracies in predicting tool forces. From the production standpoint this implies that poor flow blends and high press speeds (underfilled tooling) may be tolerated on simple parts, but to obtain high-quality pressings on complex shapes every effort must be made to obtain uniform die fill by optimising powder-fill characteristics and press kinematics. In the case of pharmaceuticals a further requirement can be for accurate control of part weight using multitooled rotary presses (See Chapter 14). Before pressure is applied, powders may be moved freely within the die cavity; the ideal tool design will ensure that powder particles are transferred to their final relative positions before pressure is applied [11]. Where this is not practicable (many presses will lack this capability; some geometrical features can only be imparted by compaction) it is important that powder transfer takes place below a certain threshold density (typically 4.0 - 4.5 g/cm3 on a ferrous part); shear cracks can otherwise form. CM has considerable potential for optimising these early stages of compaction, including the ability to choose press kinematics to correspond to the powder fill characteristics and to minimise powder transfer above the advisable threshold densities. For a given final component shape there are usually several different ways to design the compaction tooling. While the final pressed shape is determined by the finished component design, the starting cavity is determined by a combination of powder fill density, the ability of the powder to transfer laterally, the capabilities of the compaction press and the economics of the tooling. Thus, large-batch production on a high added value component can usually justify using a complex tool on a complex press (which often operates at lower speeds). In contrast, it may be difficult to justify an expensive tool for small batch production on a low added value component; such parts may have to be compacted on simple presses. Such presses may not have the capability to provide the best fill profile of the part, and may relay on significant powder transfer during the early part of the compaction stroke. CM is a tool for the parts producer to decide the level of complexity that

P. Brewin, O. Coube, J.A. Calero, H. Hodgson, R. Maassen and M. Satur

16

can be imparted to the sintered component without introducing defects that cannot be machined out; additionally it enables him to optimise press selection. CM therefore offers the possibilities of designing tooling to: • • •

enable simple presses to produce high-integrity complex parts design sintered components requiring minimal machining enable complex presses to produce high-integrity complex parts incorporating geometrical features that would otherwise lead to internal shear cracks.

2.2.3.3 Friction It is known that friction between powder and die walls is one of the main contributors to density gradients in powder compacts [12]. Early simulations assumed that the powder:die-wall friction coefficient µ remained constant as pressed density increased. While this may indeed be a reasonable approximation for spherical powders, it is clear that irregular powders in ductile materials (such as atomised iron powders) will exhibit fast reducing µ in the early stages of compaction, as particles are forced under pressure to conform to the die wall ([12] Fig. 17). Factors to be considered include: • • •

die finish including grinding direction relative hardnesses of die and powders the role of admixed lubricant (quantity and type).

Depending upon these factors, on irregular powders in ductile materials at the end of the compaction stroke µ may reduce to 75 or even 50% of its value at the start of the stroke. Sensitivity studies using CM may be used in the selection of the best die material and finish; it will be advisable to characterise µ by a linear or algebraic expression to reflect its variation during the compaction stroke. 2.2.3.4 Tooling Stresses and Deflection Early CM studies validated tool-load predictions against measurements on production presses. It soon became clear, however, that data produced experimentally needed to take into account the elasticity not only of the punches but also of the entire press frame, especially where higher compaction pressures were used; where split punches were used, interpunch friction was by no means negligible; load data calculated from hydraulic pressures was often quite inaccurate and was affected by such factors as the compressibility of the hydraulic fluid at higher pressures. It was important therefore to generate “spring constant” data by compacting solid metal blocks before introducing powders for validation trials. Such effects were more important when modelling ferrous parts (compaction pressures up to 900 MPa) than ceramics or hardmetal (pressures of 90 and 200 MPa, respectively).

Modelling and Part Manufacture

17

2.2.3.5 Other In addition to compaction, tooling design must consider part ejection. This has two aspects: • •

the ability of the pressed compact to hold together during ejection the design of the tool.

The Pressed Compact The mechanical properties underlying pressed part integrity are usually measured at final pressed part density: • shear strength usually measured either by uniaxial compression on shallow cylindrical specimens (height: diameter < 1.0, aspect ratio H/D ≈ 2) or by diametral compression of thin circular discs (thickness less than 25% of diameter; often termed Brazilian Disc). See also Section 2.2.3. • tensile strength usually measured using a simple tensile test. The above tests are reviewed in [13]. All are readily carried out in-house using standard laboratory equipment. In both cases powder blends with higher values will produce higher-quality compacts than those with lower values. Tooling Design for Ejection Apart from using low-friction die materials, simple geometrical features that aid ejection can be incorporated in the tooling. These measures allow gradual relaxation of pressed part stresses during the ejection process. These can include: • •

providing a gradual expansion or “draft” in the top region of the die cavity radiusing the transition between the die cavity top and die table.

Ejection cracks can also be reduced by withdrawing the die while maintaining moderate punch axial pressure (sometimes termed “top punch hold-down”). 2.2.4 Press Selection In its finally developed form CM will help the parts maker to make the best use of the compaction press, whether mechanically or hydraulically driven. At the initial enquiry stage CM will enable him to make a preliminary selection of press type and to include press output and operating cost in his price quotation. In general, the mechanical press will be used for high-volume simple shapes and the hydraulically driven press for lower-volume, more complex shapes. Hydraulic presses are usually used above 100 tonne punch loads, and can incorporate complex tooling kinematics within a single press cycle including rapid advance (closing of the die cavity), medium speed (initial compaction) and slow speed (final compaction). CM will further enable the parts producer to evaluate different routes to the finished component: a simple low-cost sintered component machined to final shape may be more attractive to the customer (cheaper tooling, shorter lead time to first

18

P. Brewin, O. Coube, J.A. Calero, H. Hodgson, R. Maassen and M. Satur

production part) than a more complex alternative. CM can also potentially evaluate the viability of producing complex shapes from simple presses, e.g. the degree to which powder transfer will take place on a 2-level part without resorting to 2-piece tooling. 2.2.5 Production and Quality Control Setting up a new press tool can be an important cost as this involves skilled operators, uses press time unproductively and is a common cause of tool damage. CM integrated with press-actuating software potentially reduces this time significantly. Where die cavities are deep and narrow, and powder blends have poor fill characteristics, special care must be taken to obtain the most uniform die fill possible (filling-shoe vibration, bottom punch withdrawal for suction fill, slower filling rates for air evolution, etc.) On complex shapes involving significant powder transfer it will be important to ensure that this occurs below the pressed density at which shear cracks can form; quite small inaccuracies in punch motion can be the difference between cracked and crack-free parts. Quality control is most effective online early in the production sequence. Sensitivity analyses using CM may be used to calculate which powder and compaction process variables are likely to have the greatest effect on the quality of the final component. These data can then be used to set the allowable variations for these critical in process variables. 2.2.6 Sintering and Infiltration The purpose of this book is to examine die-compaction modelling; clearly models capable of calculating sintering shrinkage, warpage and even metallurgical structures are also of great importance [14]. 2.2.6.1 Supersolidus Sintering Phenomenological sinter models can be used to calculate the required uniformity of pressed density to avoid corrective machining of the sintered part. In this case constitutive data is generated using dilatometers; other characteristics can be predicted using differential scanning calorimeters and differential thermal analysers. Discrete element sinter models offer the potential to predict the sintering of a press part from fundamental principles. 2.2.6.2 Subsolidus Sintering Subsolidus sintering of ferrous parts is a compromise between the need to form strong interparticle bonds and the need to control dimensions. Although an undesirable practice metallurgically, admixed elemental compositions can be adjusted to control size change through sintering, increased admixed nickel causing increased shrinkage, admixed copper causing growth [15].

Modelling and Part Manufacture

19

2.2.6.3 Infiltration In this process the pressed part is placed in contact with a low melting point material the volume of which is equivalent to the volume of interconnected porosity in the pressed part. During sintering the low melting point material fills the pores by capillary action. For infiltration to be successful all part porosity must be interconnected. CM may be used to predict the distribution of density within the pressed component to ensure that density levels are statistically unlikely to give rise to closed-off pores (typically below 85% of full theoretical density).

2.3 Requirements for Compaction Modelling 2.3.1 Input-Data Generation CM requires reliable accurate input data on powder properties, interaction between powder and tooling, press kinematics and green-part properties. Table 2.2 below lists the key data and how generated (see also Table 8.1 and Chapter 14 for discussion of pharmaceuticals). Powder properties: Fill/Flow: while a die-filling rig such as that described in [16] is relatively inexpensive, for industry purposes it will be sufficient to characterise powders on an experimental press using tooling geometrically similar to that being studied. Instrumented die: it is seen that much data can be generated using this lower cost method. Low-pressure compaction and powder transfer: there are currently no simpler alternative methods to the laboratory techniques Press-frame stiffness or spring constant is measured by pressing solid metal blocks (Section 2.3.4) but results have to be corrected for punch elasticity. This aspect is discussed more fully in Chapter 11.

20

P. Brewin, O. Coube, J.A. Calero, H. Hodgson, R. Maassen and M. Satur Table 2.2. Key input data and how generated

NA = not available Research

Industry

Key Ref

Chapter

Plastic deformation

Triaxial stress rig

Instrumented die

[7]

6

Fill/flow

Die-filling rig (critical shoe velocity)

Experimental press with generic tooling

[4], [5], [6], [16]

9

Fill density distribution

Die/shoefilling rig + Xray CT

Die/shoe filling rig + metallography

[17], [18], [10], [16]

9, 11, 12

Low-pressure compaction

Low-pressure triaxial stress rig

NA

[7]

6

Powder transfer

Powdertransfer rig

NA

[19,] [20], [4], [21]

9

Powder/tool ing interface

Friction

Shear plate

Shear plate

[22]

8

Instrumented die

Instrumented die

Green-part properties

Poisson’s ratio

Axial compression

Young’s modulus

Uniaxial compression or ultrasonics

Shear failure line

Powder properties

Press operation

8 [23]

5

Uniaxial compression

[23]

5

Diametral or axial compression

Uniaxial compression

[24], [13], [21]

7

Tensile strength

Uniaxial compression

Uniaxial compression

[24], [13], [21]

6

Punch and die kinematics

Control signal driving press

Control signal driving press

[25], [26]

11

Load

Load cell

Hydraulic pressure

[27]

11

"Spring constant" tests

[25]

11

Press deflection

Modelling and Part Manufacture

21

2.3.2 Modelling and Part Manufacture: Requirements of the Hardmetal Industry (“HM”) What is a successful model from the viewpoint of industry? The answer is productivity. Thus there is a long distance to travel starting from the mathematical formula and arriving at the net-shape or crack-free pressed and eventually sintered part. In this respect some requirements from the hardmetal (HM) industry are put forward below. Although this discussion is intended to be as general as possible, differences exist between companies depending upon the level of internal expertise as a matter of course. 2.3.2.1 Reliability and Robustness The first of these requirements is of course reliable and robust modelling – by modelling is meant in this case the mathematical model and its implementation in the numerical code. This is not the case for all industries. For HM, dimensional control of the net-shape sintered parts is the main priority and this is currently satisfactorily described and predicted by compaction (and sintering) modelling (CM). However, the tolerances required by the market are very tight and can sometimes be less than 1 % of the actual dimension. This is a difficult challenge for modelling that is used more to give accurate trends rather than definitive results. Nevertheless, accurate trends can also be very useful in the optimisation process. 2.3.2.2 Other Requirements Once the reliability and robustness requirements are fulfilled, some other issues still remain for a real industrial use of CM. The requirements are less rigorous if the use of CM is confined to the company R&D department rather than the design office or the production department. By R&D, design office and production we mean the following typical qualifications: • • •

R&D: People with a degree in science, with basic to very good knowledge of finite element codes, basic or no CAD software knowledge and basic to good knowledge of production. Design: People with a technical degree, with very good knowledge of CAD software, good knowledge of production conditions and no or basic knowledge of finite element code. Production: People with no or basic CAD knowledge but with very good knowledge of production.

Whether R&D and design exist as separate departments depends mainly on the size of the company. In HM the majority of manufacturing companies are large enough to have separate R&D and design offices. In the HM industry CM projects with the goal of optimising productivity are run by the R&D department in direct cooperation with production, the design office being partly involved. This may not be the case in other industry sectors where the design office often replaces R&D.

P. Brewin, O. Coube, J.A. Calero, H. Hodgson, R. Maassen and M. Satur

22

The following summarises the HM industry requirements for implementation: • • • • • • •

user friendliness computing power interface with CAD interface with press software selection of finite elements (codes) importing input data optimisation tools.

User friendliness enables CM to perform complex tasks more rapidly, with less manual work and possibly with less expertise than initially required. The degree of user friendliness may have to be developed differently depending on the current expertise in CM of the R&D and design department. It is also related to the optimisation tools since user friendliness eases the use of such tools. Regarding production, the whole process should be a “ready to use black box” for the optimisation of pressing schedules. Computing power is currently sufficient to carry out the calculation of a complex 3D model in a few hours to a few days. This is fast enough for development projects in R&D and design office but major improvements are needed if it is to be used in production. CAD is the main working tool of the design office, therefore Interface with CAD is a key issue. The main FE codes already allow R&D to import CAD files for numerical simulation. However, few CAD software packages in the design office are provided with FE interfaces with CM package. Interface with press software should be the final link in the chain of implementation of CM from research to production. Such an interface would allow, within a few minutes, the optimisation of the pressing schedule according to the final geometry, powder type. However, in the HM industry the optimisation of the pressing schedule is not as important as optimisation of the sintering cycle. Finite-element codes are nowadays mainly used by R&D. One can find even in commercial codes simple CM packages that can be updated. The design office should be able to use, through a suitable CAD interface, a simplified version of FE code committed to the prediction of powder pressing. Input Data for CM are well defined for R&D purpose with standard tests. Improved user friendliness could be, in this field, an advantage for implementation in design office. Optimisation tools combined with existing FE code should help the R&D department to provide fast and reliable solutions to design office and production problems. These could include integration possibilities (of different modules, e.g. pressing + sintering for HM), automation (of numerical prediction assessments, choice of alternative solutions and rerun of simulation) and user-friendly interfaces.

Modelling and Part Manufacture

23

These requirements are summarised in the following table with the legend: “Is currently sufficient”: ☺ (fairly) to ☺☺☺ (completely) “Must be improved”: X[1,2…] (slightly) to XXX[1,2…] (drastically) and depends on requirements No. 1,2… In Bold: Main requirements to be developed Underlined: Main CM tool for the department (current or to-be) Superscripts denote requirement numbers (e.g. ☺☺X[5] - refers to: finite elements (codes) Table 2.3. Requirements for the use of CM in HM industry No.

Requirement

R&D [5]

Design [5]

Production

1

user friendliness

☺☺X to XXX[5]

☺XX to XXX[5]

XXX

2

computing power

☺XX

☺XX

XXX

[5]

☺☺X

XXX

[1,5]

3

interface with CAD

4

interface with press software

-

-

XXX[1,2,3,5,7]

5

finite-element codes

☺☺X

XXX[1,3,7]

XXX

6

importing input data

☺☺X[1]

☺XX[1]

-

7

optimisation tools

XXX[1,5]

XXX[1]

XXX

☺XX

2.3.2.3 Discussion of Requirements In the table above the number of main requirements, X, can be totalled arithmetically as follows: R&D 3 to 6, design office 8 to 9, production 17. If we convert each mark by its equivalent in time and money, the first step of the implementation should be the R&D department. The main requirement is then the optimisation tools. Assuming that a FE code with CM and sintering package and CAD interface is available, numerical simulation of compaction and sintering of complex 3D geometries can be performed. The expertise required depends upon the level of user friendliness available. This is currently the most advanced status of CM in the HM industry. Manual optimisation tasks are then performed using parametric studies. Optimisation tools would decrease drastically development times and enable some difficult projects to be solved. Implementing CM directly in a design office requires additional improvements mostly in terms of user friendliness, interface with CAD software and optimisation tools. Since design engineers are not necessarily CM specialists, the principle of the black box must be more generalised in terms of model generation, importing input data, interface to FE code and optimisation.

24

P. Brewin, O. Coube, J.A. Calero, H. Hodgson, R. Maassen and M. Satur

Direct use of CM in production in HM industry is not conceivable for the moment, not least because the cost outweighs the benefits. Besides, onsite experience and improved press designs compete with CM implementation in optimising pressing schedules 2.3.2.4 Conclusion Once CM achieves a satisfactory level of reliability and robustness – which is not the case for all the PM sectors – additional requirements must also be fullfilled. CM could be successfully implemented as a first step in R&D with some improvements described above. It is after all one of the functions of the R&D department to introduce into the company innovative solutions coming from research. Implementation of CM in the design office could be of interest to SMEs but requires up to three times greater investments than for R&D. Production cannot currently use CM directly, as high productivity conditions require ready-to-use solutions that are not the case for CM at this level. Training should also be mentioned. CM training courses – at best directly in the company – could help to reduce the level of the requirements in R&D and furthermore in the design office for the implementation of CM in their working tool environment. 2.3.3 Modelling and Parts Manufacture: Requirements of Ferrous Structural (FS) Parts Industry As computer simulation techniques continue to improve, the question is repeatedly asked whether the time has come to implement these industrially. In the production of ferrous structural powder metal components, powder compaction is one of the core steps in the process. For modelling to be implemented requires that specific production problems can be solved efficiently. Along with sintering, axial die compaction is a major production step in ferrous-part production. Components are pressed near net shape to increasingly more complex geometries. Associated with this, press and press-tool design become more sophisticated. Controlling the pressing kinematics of a multilevel part using tooling incorporating several upper and lower punches and core rods needs skilled operators. Increasingly these are assisted by software tools. Small deviations from ideal pressing kinematics can easily cause defects in the pressed component. Determining the cause of such defects can be very difficult. Typical defects include: • inhomogeneous density distribution • shear cracks resulting from unfavourable transfers of powder in the die cavity • brittle cracks resulting from unloading and springback of the pressing tool • failure of the press tool itself owing to overloading.

Modelling and Part Manufacture

25

For the above reasons simulation software needs to predict both density distribution and crack formation. In ferrous-part production, size changes in sintering are relatively small and therefore part distortion through sintering is not important. 2.3.3.1 The Process Chain The process chain from part design to compaction usually starts with a 3D design model of the component. This is then used to design the compaction tooling. In turn this is then used to generate the press kinematics. Tool deflections can be predicted using loading data generated from compaction experiments. As far as possible proportional compaction is used in processing the powder from fill to final pressed density. The press setter approaches the final press kinematics carefully from the safe side, taking into account press-frame elasticity and filling effects. 2.3.3.2 Time Considerations Experience from structural mechanical FE analysis shows that it is not always possible to carry out a fully detailed 3D simulation in an acceptable time, even using powerful PCs. Small geometrical variations can increase the size of the FE model quickly up to one million degrees of freedom. For computing purposes, therefore, it is often necessary to simplify part geometry and to take out nonessential details. Considerable simplifications can be achieved by considering tooling to be stiff, and taking advantage of part symmetries. It may, for example, be sufficient to solve a smaller segment of part volume or even reduce the part from 3D to 2D especially in the case of axi symmetric parts. Such assumptions and simplifications can be successful but require good knowledge of FE, CAD and the interface between simulation and CAD software. For these reasons the wider application of simulation software is hindered by the shortage of FEM-experienced staff. Such skilled operators do not need well-developed user interfaces, a stable operating material routine implemented in a commercial FE code is sufficient. In contrast, the use of simulation as a “black box” by a typical designer or even on the shop floor would have to process nonsimplified FE models requiring comprehensive material database, robust computing and especially powerful computers. Preparing a detailed CAD model and carrying out the simulation can easily take one or two weeks. This is much too long a response time for solving typical production problems, which can be solved much faster by an experienced operator using trial and error methods. However, some problems are more difficult, and need to be solved without risking costly tool-design changes. In these cases simulation can be advantageous, providing the effective punch movements are known. The best numerically controlled presses provide accurate data on punch movements. In contrast, the wider use of simulation on older presses will be limited. Compaction simulation is also useful for basic feasibility studies and for the development of new tool-design concepts.

26

P. Brewin, O. Coube, J.A. Calero, H. Hodgson, R. Maassen and M. Satur

2.3.3.3 Cost Considerations The introduction of compaction simulation into a company can finally only be justified by a positive cost:benefit ratio. This includes the costs of hardware and software, operator training and modifications to presses to improve data logging. The benefits of using simulation are more difficult to quantify, since these only result from practical experience in solving some tough problems. 2.3.3.4 Conclusions It can be concluded that the first requirement for implementation of simulation is to satisfy the needs to predict density distribution, crack formation and tool loading. It is not currently clear whether the prediction of cracking using material models should best be carried out using continuum mechanics or using particle methods. On materials involving significant shrinkage and distortion through sintering the simulation of density distribution is an obvious advantage. In ferrous Powder Metall this is not so important. Simulation packages require good interfaces to CAD and press-control systems. The user interface is not very important currently, since uses of modelling are limited to FEM-experienced staff. The steady and fast development of both simulation techniques and computing power could ultimately lead to powder presses being controlled like modern CAM machining centres: after import of CAD data and powder information the pressing kinematics would be generated automatically. Until then considerable advances can be achieved by systematic guidelines and well-educated machine operators. 2.3.4 Validation Parts makers need to be able to validate CM predictions. Industry views on the validation techniques listed in Table 2.4 are given below (see also further detail on some of these techniques in Chapter 12). 2.3.4.1 Validation of Green-part-density Predictions Important decisions on part feasibility, tooling design and press settings will be taken in the light of green-part-density predictions. At the design stage the computer predictions will be compared with past experience on similar shapes; at the prototype stage techniques such as those listed in Chapter 12 will be used, with the emphasis on the simpler quicker methods. Where parts are to be sintered to full density evaluation of sintered dimension provides a simple additional check. 2.3.4.2 Validation of Internal-crack Predictions In the case of ferrous structural parts, internal porosity will be similar in size to the primary powder particles - typically up to 150 µm. For the purposes of defect detection, therefore, it will be necessary only to detect internal defects significantly larger than this figure - e.g. 0.3 mm and above. Where parts are to be sintered to full density, smaller defects may “heal”. However, in general this is not reliable, and similar efforts should be made to produce crack-free green parts as are made on ferrous components.

Modelling and Part Manufacture

27

Techniques for crack detection such as acoustic resonance are well proven for use on sintered parts; however, industry has a continuing high-priority requirement for online techniques for crack detection on green parts. Table 2.4. Validation: research and industry techniques Ref

Green-part density

[28, 17, 18]

Research

Industry tests Nondestructive

Destructive

XRT (see Section 11.5)

surface hardness

slice, weight and measurement

SEM-EDS (see Section 11.4)

bulk density by measurement and weight

quantitative metallography sintered dimension (lps only)

Internal shear cracks

[28]

acoustic resonance tests on sintered parts

microstructural

X-ray

X-ray (large section thicknesses)

ultrasound

Tensile cracks

[28]

visual, magnetic

etch

punch loads X-ray ultrasound eddy current

lps = materials subject to liquid phase sintering (e.g. Hardmetals)

28

P. Brewin, O. Coube, J.A. Calero, H. Hodgson, R. Maassen and M. Satur

References [1] Lavery N.P., et al, June 1998. Sensitivity study on powder compaction. AEA Technology Report AEAT-4035. [2] Maassen R., et al., October 2002. User Friendliness Aspects of Modelling Industry Standpoint. Presentation to Dienet Workshop No. 2, Nice October 2002. [3] Guyoncourt D.M.M., September 2004. Review of Granulate Performance. AEA Technology Report LD81000/02. [4] Coube O., et al., 2005. Experimental and numerical study of die filling, powder transfer and die compaction. Powder Metall 48/1 68ff. [5] Guyoncourt D.M.M., and Tweed J.H., 2003. Measurement of Powder Flow. Proceedings of Euro PM2003 Conference, EPMA. [6] Schneider L.C.R., et al., 2005. Comparison of filling behaviour of metallic, ceramic, hardmetal and magnetic powders. Powder Metall 2005 Vol 48/1 77ff. [7] Cocks A.C.F., November 2004. Characterising Powder Compaction. Presentation to MPM 5.2 Seminar. [8] Coube O, Nov 2005. Private communication by email [9] Armentani E., et al, 2003. Metal powder compacting dies: optimised design by analytical or numerical methods. Powder Metall 2003 Vol 46/4 349ff. [10] Korachkin D., and Gethin D.T., Nov 2004. An exploration of the effect of fill density variation in the compaction of ferrous, ceramic and hard metal powder system. AEA Technology Report o. LD81000/05. [11] Ernst E., 2003. Practical needs for simulation of powder compaction. Proceedings of Euro PM2003 Conference EPMA. [12] Cameron I.M., et al. 2002. Friction measurement in powder die compaction by shear plate technique. Powder Metall 2002 Vol 45/4 345ff. [13] Doremus P., May 2001. Simple tests standard procedure for the characterisation of green compacted powder. Proceedings of the NATO Advanced Research Workshop on Recent Developments in Computer Modelling of Powder Metallurgy Processes, Kiev Ukraine. Pub IOS Press, Amsterdam. [14] Leitner G., May 1998. Modelling of sintering. Presentation at EPMA Powder Metall Summer School, Meissen. [15] Metals Handbook Vol 7 Powder Metal Technologies and Applications. Published ASM 1998. ISBN 0-87170-387-4 [16] Wu C.Y., and Cocks A.C.F., 2004. Flow behaviour of powders during die filling. Powder Metall 2004 Vol 47/2 pp.127ff. [17] Burch S.F., et al., 2004. Measurement of density variations in compacted parts and filled dies using X-ray Computerised Tomography. Proceedings of Euro PM2004, EPMA. [18] Tweed J.H., et al. 2005. Validation data for modelling of powder compaction: Guidelines and an example from the European DIENET project. Proceedings of Euro PM2005 Conference, EPMA. [19] Cante J.C., et al., 2004. Numerical modelling of powder compaction processes: towards a virtual press. Proceedings of Euro PM2004 Conference, EPMA. [20] Cante J.C., et al., 2003. Powder Transfer modelling in powder compaction processes. Proceedings of Euro PM2003 Conference, EPMA. [21] Cante J.C., et al., 2005. On numerical simulation of powder compaction processes: powder transfer modelling and characterisation. Powder Metall Vol 48/1 85ff.

This page intentionally blank

Modelling and Part Manufacture

29

[22] PM Modnet Methods and Measurements Group, 2000. Measurement of friction for powder compaction modelling - comparison between laboratories. Powder Metall 2000 Vol 43/4 364f. [23] Frachon et al., 2001. Modelling of the springback of green compacts. Proceedings of PM2001 Conference, EPMA [24] Coube O., and Riedel H., 2000. Numerical simulation of metal powder die compaction with special consideration of cracking. Powder Metall 2000 Vol 43/2 123ff. [25] Coube O., 2005. Numerical simulation of die compaction: case studies and guidelines from the European Dienet Project. Proceedings of Euro PM2005, EPMA. [26] PM Modnet Research Group, 2002. Numerical simulation of powder compaction for two multilevel ferrous parts, including powder characterisation and experimental validation. Powder Metall 2002 Vol 45/4 335ff. [27] Tweed J.H., et al 2004. Validation data for modelling of powder compaction: guidelines and an example from the European DIENET project. Proceedings of Euro PM2004 Conference, EPMA. [28] Ernst E., and Donaldson I., 2004. The application of different NDT Processes for automotive PM components. Proceedings of Euro PM 2004 Conference, EPMA.

3 Mechanics of Powder Compaction A.C.F. Cocks1 1

Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK.

3.1 Introduction The purpose of this chapter is to cover the basic mechanics principles that underpin the development of the material and computational models described in this book. The chapter is aimed at the nonspecialist and covers the fundamental aspects of the mechanics of elastic and plastic deformation. During the compaction process a compact is loaded principally in compression. When evaluating the material response we therefore adopt the sign convention that compressive stresses and strains are positive. In practice, large-scale plastic straining of a body can occur during the compaction process. In this chapter we, however, express the material constitutive behaviour within the framework of a small-deformation theory of plasticity. This allows us to readily decompose the total strain into elastic and plastic components and to develop relatively simple expressions for the constitutive response. The general form of expressions we employ here is still applicable to large deformations, provided we define appropriate measures of stress, strain, stress rate and deformation rate for these problems. At this stage it is not necessary to precisely determine what these definitions are. All the information we need can be determined by making the simple assumption of small strains. We start by considering the simple case of uniaxial compression of a block of material and identify some elementary features of the material response and how we can present these in the form of a material constitutive model. We then generalise the results to describe the response of porous powder compacts subjected to general multiaxial stress histories. Throughout, we limit our attention to the development of models for isotropic materials. Further details of the underlying plasticity concepts can be found in the books by Calladine [1] and Khan and Huang [2]. A review of the application of these concepts to powder compaction is given by Trassoras et al. [3].

32

A.C.F. Cocks

3.2 Uniaxial Deformation Consider, initially, the situation where a cylindrical specimen of a material is subjected to a uniaxial stress σ and as a result experiences a strain ε . Figure 3.1 shows a typical material response. At low stress the response is elastic and there is a linear relationship between stress and strain:

ε=

σ

(3.1)

E

where E is Young’s modulus. As the stress is increased the material eventually yields and plastic strain accumulates. Figure 3.1 shows the situation where the specimen is loaded beyond the initial yield stress, σy, to a stress σ=s. The total strain at this point is εT. If the stress is now reduced to zero the specimen unloads elastically. The residual plastic strain accumulated as a result of this loading history is εp. If the stress is increased to a value σ<s, the subsequent response is elastic and the total strain experienced by the specimen is

εT = εe +ε p =

(a)

σ E

+ε p

(3.2)

(b)

Figure 3.1. Stress-strain curves for an elastoplastic material, showing the decomposition of the strain into elastic and plastic components

If the stress is increased to the point σ=s further plastic straining can occur. The magnitude of s is important since it gives the value of stress at which the response is no longer elastic, i.e. it is the instantaneous magnitude of the yield stress that results from the prior thermomechanical loading history. It is evident that s (the yield stress) depends on the accumulated plastic strain, i.e. s is a function of ε p , or mathematically:

Mechanics of Powder Compaction

( )

s=sεp

33

(3.3)

As the stress is increased and further plastic strain accumulates, s also increases, such that f =σ −s = 0

(3.4)

In practice, the state f = σ − s > 0 cannot be achieved, but as noted previously, if f = σ − s < 0 , the stress is less than the yield stress and the response is elastic (i.e, changing the stress only produces a change in the elastic strain). In the above construction we have implicitly assumed that Young’s modulus is a constant for a given material. This is a reasonable assumption for fully dense materials, but for porous materials the modulus can be a function of the state, i.e. s, or equivalently ε p , i.e.

( )

E = E (s ) = E ε p

(3.5)

In general, we are interested in complex stress histories. Equation 3.4 allows us to keep track of how the material responds, i.e. we only need to know the instantaneous value of s, not the details of the loading history that created this state. It is evident from Equations 3.3 and 3.4 that the function s ε p can readily be determined from a uniaxial test in which the stress is increased monotonically, such that after initial yield Equation 3.4 is satisfied throughout the loading history. The body can be unloaded periodically to determine the elastic response, i.e. how the modulus of Equation 3.5 depends on the state. When dealing with general stress histories it proves convenient to express the material response in incremental form. Consider the situation where, at a given instant, Equation 3.4 is satisfied. The stress is then increased by an amount dσ, as illustrated in Figure 3.1b. The resulting strain increment is given by (making use of Equations 3.2 to 3.5)

( )

dσ σ dE dσ + dε p − 2 E E ds  1  σ dE  dσ 1  =  1 − −  dσ = E ( ) E E ds h s     T

dε T = dε e + dε p =

(3.6)

ds and the term in the square bracket is the inverse of the tangent dε p modulus to the stress-strain curve, ET , as illustrated in Figure 3.1. Equation 3.6 is simply the incremental analogue of Equation 3.1 for the nonlinear stress-strain curve of Figure 3.1.

where h( s ) =

34

A.C.F. Cocks

3.3 Deformation under Multiaxial States of Stress We now extend the constitutive model of Section 3.1 to multiaxial states of stress. Under general multiaxial conditions we must consider 6 components of stress (3 normal components and 3 shear) and 6 corresponding components of strain. In this section we limit our consideration to the behaviour of isotropic materials. It then proves convenient to express the material response in terms of stress and strain invariants, i.e. measures of stress and strain that are independent of the axes used to define them. In 3D, there are three stress invariants, but in most situations it is only necessary to consider two of these: the hydrostatic pressure p and the von Mises equivalent stress q. Experimental studies are often conducted on compacts that are subjected to axisymmetric loading histories, such as that shown in Figure 3.2a, in which a cylindrical compact is subjected to axial and radial components of stress, σ a and σ r . It proves instructive to use this situation as an illustration of a representative multiaxial state of stress. The hydrostatic component of stress is the mean of the three principal stresses (there are two radial components) and for the axisymmetric stress state of Figure 3.2a is given by p=

1 3

(σ a + 2σ r )

(3.7)

The equivalent stress is related to the principal shear stresses and for the loading of Figure 3.2a is given by q = σa −σr

(a)

(3.8)

(b)

Figure 3.2. Axial and radial components of (a) stress and (b) strain on an axisymmetric powder compact

Mechanics of Powder Compaction

35

Under the axisymmetric loading of Figure 3.2a the compact experiences axial and radial strains, ε a and ε r , as shown in Figure 3.2b. When considering the elastic response, analogous to the consideration of stress, it proves convenient to define two strain invariants (as with stress there are three strain invariants, but it is only necessary to consider two of these): the volumetric strain, ε v , and the equivalent strain, ε e . For the axisymmetric conditions of Figure 3.2b:

ε v = ε a + 2ε r

ε e = 23 ε a − ε r

(3.9)

For an isotropic elastic material only two material constants are required to define the constitutive response (the relationship between stress and strain). Application of a pressure results in a volume change:

εv =

p K

(3.10)

and the effective stress leads to a shape change:

εe =

σe 3G

(3.11)

where K is the bulk modulus and G the shear modulus. The elastic response can alternatively be described in terms of the Young’s modulus of Equation 3.1 and Poisson’s ratio, ν. These are related directly to the bulk and shear moduli: G=

E E and K = 2(1 + ν ) 3(1 − 2ν )

(3.12)

When dealing with plastic deformation under multiaxial loading conditions, we need to introduce a number of additional concepts. When considering the uniaxial behaviour we introduced the concept of a yield stress, a stress below which the response is elastic. Under multiaxial loading, we can identify a yield surface, a convex surface in stress space; examples are given in Figures 3.3 to 3.5. For stress histories within the surface the response is elastic and a compact responds to changes in stress according to Equations 3.10 and 3.11. Plastic deformation can only occur if the stress state lies on the yield surface. We express the yield surface mathematically as: f = f (q, p, state) = 0

(3.13)

where f is a function of our two scalar stress measures, q and p and the current state of the material (which we defined in terms of s, or the accumulated plastic strain under uniaxial loading). As with uniaxial loading the response is elastic if f < 0 (i.e. the stress state lies within yield) and the state f > 0 is not achievable.

36

A.C.F. Cocks

Now we need to identify how to define the state of the material and determine how the compact deforms plastically at yield. Since plastic strain can accumulate at any point on the yield surface, it proves convenient to express the response in terms of increments of plastic strain: dε vp = dε ap + 2dε rp and dε p =

2 3

dε ap − dε rp

(3.14)

for the axisymmetric loading of Figure 3.2, where the superscript p refers to plastic components of strain. The effective strain increment, dε p , is always positive, thus effective strain will steadily accumulate: εp

∫

ε = dε p

p

(3.15)

0

An increment of volumetric strain can be related to the change in volume, V, of a sample: dε vp = −

dV V

(3.16)

Integrating this relationship from the initial volume Vo , when the strain is zero and the volume V, when the volumetric strain is ε vp , gives V = Vo exp− ε vp or

ρ = ρ o exp ε vp

(3.17)

where ρ o and ρ are the initial and current densities of the compact, with the second of these relationships determined from the fact that the mass of the compact remains constant. As with uniaxial loading it is convenient to describe the state of the material in terms of the plastic strain at a given instant, i.e. ε vp and ε p , or, since the density is directly related to the volumetric strain, by ρ and ε p . Then, the yield condition takes the form: f = f (q, p, ρ , ε vp ) = 0

(3.18)

It is generally assumed that the state can be adequately described in terms of the density. The yield condition can then be written in the form f = f ( q, p, ρ ) = 0

(3.19)

Mechanics of Powder Compaction

37

Different types of models and forms of yield surface for powder compacts are reviewed in Chapter 4. In this chapter we focus on some general features of these models. The detailed forms of the models are given in Chapter 4 and the calibration of these models is fully discussed in Chapter 10. The earliest plastic model of powder compaction was proposed by Green [4] and Shima and Oyane [5]. In this model the surface is an ellipse in p-q space centred on the origin, Figure 3.3, whose size and aspect ratio are a function of the density of the compact. Effectively, as the density increases the yield surface expands and higher stresses are required to deform the compact plastically. It proves convenient to express the yield function of Equation 3.19 in normalised form: 2

2

 q   p   +   − 1 = 0 f =   qo ( ρ )   p o ( ρ ) 

(3.20)

where qo (ρ ) and po (ρ ) are functions of density, and are equivalent to the state variable s introduced in Equation 3.3. They are simply the minor and major axes of the elliptic yield surface, as illustrated in Figure 3.3. The response is elastic for any loading history within the ellipse and plastic deformation can only occur if the stress state lies on the yield surface. The question now is: What are the relative magnitudes of the different strains?

Figure 3.3. Symmetric elliptical yield surface for the Green-Shima model

As the body deforms plastically there can be increments in the volumetric strain, dε vp and the effective strain, dε p . In our discussion of elastic behaviour, we noted that the pressure p, gives rise to volume change and the effective stress, q, gives rise to shape change, characterised by ε p . We can use this association to

38

A.C.F. Cocks

draw a vector at the current loading point on the yield surface, whose component parallel with the p-axis scales with dε vp , while the component parallel to the q-axis scales with dε p . Such a strain increment vector is shown in Figure 3.3, located at the point (p,q). Drucker [6] describes a stability postulate for the deformation of a material, in which the work done by additional stresses from any arbitrary starting point in stress space for any history of loading must be non-negative. Drucker [6, 13] demonstrates that this can only be satisfied if the plastic strain increment vector is normal to the yield surface, as drawn in Figure 3.3. Thus, if we know the shape of the yield surface, we can determine the relative magnitudes of the strain increments, i.e. the rule for deformation (or plastic flow) can be associated with the yield surface; this is generally referred to as an associated flow rule. We can express this mathematically as dε vp = λ

∂f ∂f and dε p = λ ∂p ∂q

(3.21)

where λ is a plastic multiplier, and is the length of the vector normal to the yield surface drawn in Figure 3.3. Within this model there are two unknown functions, qo (ρ ) and po (ρ ) . We

( )

saw in Section 3.1 that we could determine the equivalent function s ε p from a simple material test. A similar procedure can be followed when determining qo (ρ ) and po (ρ ) . Consider the situation where a powder is compacted in a rigid, cylindrical, frictionless die. If the die is instrumented, see Chapter 6, during the compaction process the axial and radial stresses can be monitored, allowing p and q to be determined from Equations 3.7 and 3.8. The instantaneous density can be determined from the position of the punch (i.e. the enclosed volume of the die occupied by the powder) and the mass of powder. For simplicity, we assume that the material is elastically rigid, i.e. only deforms plastically. Under this type of loading the radial strain experienced by the compact is zero. Increments of the effective and volumetric strains are then both proportional to the axial strain increment, dε a , Equations 3.14. Combining Equations 3.14 and 3.21 we find dε vp 3 ∂f / ∂p = = dε p 2 ∂f / ∂q

(3.22)

At a given instant, there are two unknowns ( qo (ρ ) and po (ρ ) ) and two Equations 3.20 and 3.22, which allows the two unknown quantities to be determined: po ( ρ ) =

p ( p + 23 q ) and qo (ρ ) = q( 32 p + q )

(3.23)

Mechanics of Powder Compaction

39

An analytical constitutive law can be obtained by plotting these calculated values of qo (ρ ) and po (ρ ) against ρ and fitting appropriate forms of relationship to the data. The model is now complete. For this model, compaction occurs if the hydrostatic stress, p, i.e. the volumetric component of the strain increment points to the right on Figure 3.3 (i.e. is positive) and the yield surface expands. Dilation occurs for tensile hydrostatic stress states, i.e. the volumetric component of the strain increment points to the left on Figure 3.3 (i.e. is negative) resulting in softening of the material and shrinkage of the yield surface. The yield surface is also symmetric about the q-axis, indicating that once a particular state is achieved, the same magnitude of tensile and compressive hydrostatic stress is required to yield the compact. This feature of the model is not supported by experimental observations of the plastic deformation of loose granular assemblies [7,8], although, provided the model is fit using the procedures described above and in the loading conditions of interest all the components of strain are compressive, it can provide good predictions of the behaviour of real components during compaction [9].

Figure 3.4. The Cam-Clay model

Following Gurson and McCabe [10], Brown and Abou-Chedid [8] proposed a modified form of Cam-Clay model, originally developed for soils [11,12] to model their experiments. The Cam-Clay model is described by a simple translation of the ellipse of Figure 3.3 until it passes through the origin, Figure 3.4. As with the Green-Shima model the plastic strain increment vector is normal to the yield surface. This model can be expressed mathematically in terms of the following yield function:

40

A.C.F. Cocks

 q   p − po (ρ )   +   − 1 = 0 f =   q o ( ρ )   po ( ρ )  2

2

(3.24)

with the plastic strain increments given by the associated flow rule of Equation 3.21. As with the Green-Shima model there are two unknown functions, qo (ρ ) and po (ρ ) , which have the same geometric definitions as before, Figure 3.4. These can be determined from a frictionless closed-die compaction experiment, using the same procedures as above, although the detailed form of relationships that result from manipulation of the data are different. With this model, dilation and softening of a component can now occur when the hydrostatic component of stress is compressive. It would be possible to generalise this type of model, by relaxing the requirement that the yield surface should pass through the origin, so that it lies somewhere between the Green-Shima and Cam-Clay surfaces, thus allowing the material to exhibit some cohesion.

Figure 3.5. The Drucker-Prager two-surface model

Based on their experimental observations, Watson and Wert [7] proposed the adoption of Drucker and Prager’s [13] two-part yield surface, Figure 3.5, which was also originally developed to model the behaviour of soils. In the form currently employed by researchers, it consists of: An elliptic compaction region, i.e. a section of the Green-Shima model of Figure 3.3, whose centre is displaced with respect to the origin, along which the normality condition is satisfied, i.e. the flow is associated; a shear failure line along which the flow is nonassociated (the strain increment vector is not normal to the surface); and a transition region, which simply smoothes the response between the compaction and shear-Clayfailure regimes, where normality of the strain increment vector is again assumed. In Figure 3.5 the shear failure line has been drawn such that a compact has a finite shear stress, i.e. there is some cohesion. A number of extensions of this model have been proposed [14] and schemes have been developed to experimentally

Mechanics of Powder Compaction

41

determine the parameters for the different regimes [15,16]. These are more complex than the simple procedure described above for the Green-Shima and CamClay models and are fully described in Chapter 4. Other models have been proposed in the literature, either based on micromechanical considerations [17] and/or taking into account the anisotropy in material response that can develop as the powder mass is compacted [18]. The general features of the models described here are retained, i.e. a yield surface can be identified and in the majority of situations associated flow is assumed. We noted in our description of the elastic response under uniaxial loading that the elastic properties can depend on the state of the material. This obviously also applies to multiaxial loading conditions. For an isotropic material we identified two elastic constants, e.g., the bulk modulus, K, and the shear modulus, G, Equation 3.12. These constants are a function of the state of the compact, which we have described in terms of the density, ρ . Therefore, K = K (ρ ) and G = G (ρ )

(3.25)

Experimental procedures for the determination of these functions and a description of the forms of relationship used to fit the data are presented in Chapter 10.

References [1]

Calladine CR. 2000, Plasticity for Engineers: Theory and Application, Horwood Publishing, Chichester, UK. [2] Khan AS and Huang S. 1995, Continuum Theory of Plasticity, John Wiley and Sons, New York. [3] Trasorras JRL, Parameswaran R, and Cocks ACF. 1998, Mechanical behaviour of metal powders and powder compaction modelling, ASM Handbook, Vol 7, 1998, 326-342. [4] Green RJ. 1972, A Plasticity Theory for Porous Solids, Int J. Mech. Sci., Vol 14, 215224. [5] Shima S and Oyane M. 1976, Plasticity Theory for Porous Metals, Int. J. Mech Sci , Vol 18, 1976, 285. [6] Drucker DC. 1959, A definition of a stable inelastic material, J. Appl. Mech. 26, 101106. [7] Watson TJ and Wert JA. 1993, On the development of constitutive relations for metallic powders, Metall. Trans. A, 24A, 2071-2081. [8] Brown S, and Abou-Chedid G. 1994, Yield behaviour of metal powder assemblages, J. Mech. Phys. Solids, 42, pp383. [9] Parameswaran R, Trasorras JRL and Cocks ACF. 2002, improvements in tool load vs. displacement predictions in the compaction of multilevel parts in a production press, Process Modelling in Powder Metallurgy and Particulate Materials, A Lawley and JE Smugeresky, MPIF, Princeton, NJ. [10] Gurson AL and McCabe TJ. 1992, Experimental determination of yield functions for compaction of blended powders, Proc. MPIF/APMI World Congress on Powder Metallurgy and Particulate Materials, San Francisco. [11] Schofield A and Wroth CP. 1968, Critical State Soil Mechanics, McGraw Hill.

42

A.C.F. Cocks

[12] Wood DM. 1990, Soil Behaviour and Critical State Soil Mechanics, Cambridge University Press. [13] Drucker DC and Prager W. Soil Mechanics and Plastic Analysis of Limit Design, Quaterly Appl. Math., Vol 10, 1952, 157-165. [14] DiMaggio FL and Sandler IS. Material Models for Granular Soils, J. Eng. Mech. Div . ASCE, Vol. 96, 1971, 935-950. [15] Doremus P, Toussaint F and Alvain O. Simple Tests and Standard Procedures for the Characterization of Green Compacted Powder, in Recent Developments in Computer Modeling of Powder Metallurgy Processes, A. Zavariangos and A. Laptev, IOS Press, Amsterdam, 2001. [16] Coube O and Riedel H. Numerical Simulation of Metal Powder Die Compaction with Special Consideration of Cracking, Powder Metall., Vol 43, 2000, 123-131. [17] Fleck NA. 1995, On the cold compaction of powders, J. Mech. Phys. Solids, Vol 43, 1409-1431. [18] Schneider LCR and Cocks ACF. 2002, Experimental investigation of yield behaviour of metal powders, Powder Metall., 45, 237-245.

4 Compaction Models A.C.F. Cocks1, D.T. Gethin2, H.-Å. Häggblad3, T. Kraft4 and O. Coube5 1

Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK. 2 School of Engineering, UW Swansea, Singleton Park, Swansea SA2 8PP, UK. 3 Luleå University of Technology, S-97 187 Lulea, Sweden. 4 Fraunhofer IWM, Wohlerstrasse 11, D 79108 Freiburg, Germany. 5 PLANSEE SE, 6600 Reutte, Austria.

Modelling correctly the behaviour of a powder during pressing is the first issue that needs to be resolved before thinking of reliable numerical simulation. For example, in the early 1950’s it was established that cracking was a consequence of both shearing and hydrostatic stresses. At this time, soils and their mechanics was the centre of interest for scientists and engineers. In the 1960’s and 1970’s models appeared with density-dependent hardening laws. It was noticed that soil behaviour depends on the achieved density and that in addition to cracking the state of stress could also influence densification and hardening. These phenomena were modelled – as for other “materials” – with yield surfaces governed by mathematical relations between stress and strain. As the PM industry grew, the scientific community started to apply the rules of soil mechanics to powders during their pressing stage. In the 1980’s and 1990’s, special approaches were also developed devoted to studying the interaction between grains during compaction at the micromechanical scale. Phenomenological models were adopted by the finite-element method specialists in the 1980’s and 1990’s in order to simulate the pressing of real parts. Due to increasing computer power and development of new codes based on particle interaction – e.g. discrete element methods, micromechanical models are emerging as more and more attractive for numerical simulation. These two approaches are described in this chapter. Due to the complexity of the subject we restrict ourselves to the plastic behaviour. Elasticity that is also included in the models and boundary conditions like friction are treated in Chapters 5 and 8, respectively.

4.1

Micromechanical Compaction Models

Analytical micromechanical models divide the material response into three major stages, which we will refer to as: stage 0, which is dominated by particle

44

A.C.F. Cocks, D.T. Gethin, H.-Å. Häggblad, T. Kraft and O. Coube

rearrangement; stage 1, in which plastic deformation is confined to the contact regions between particles; and stage 2, in which the entire particle deforms plastically. Models for simulating stage 0 have been developed in which the interaction between particles assumes that they are rigid and their kinematic behaviour is a consequence of forces that are generated between confining surfaces and between particles. These models are particularly appropriate for simulating powder flow [1,2]. Models to capture stage 1 have received the most attention in the literature. These models generally assume an initial dense random packing of monosized spherical particles [3]. Stage 2 also includes the plastic deformation of the particles and has received the least attention, principally because it is computationally very demanding and the computing power that is required to undertake such simulations is only now becoming available [4,5]. The following sections will present and explain the key issues associated with micromodels aimed at addressing each of these compaction stages. 4.1.1

Stage 0 Models

One of the first attempts to use a discrete approach to simulate compaction with application in PM was undertaken by Shima et al [6]. The method finds its origins in the work of Cundall and Strack [7] who were concerned with geotechnical applications. As an example, an assembly of large powder particles is shown in Figure 4.1.

Figure 4.1. Discrete particle assembly and a pair of interacting particles

Figure 4.1 also shows the interaction between a pair of particles. For the purpose of computing the force of interaction, each is surrounded by a thin halo and this halo is used to capture the forces of interaction. Normal forces will always exist at this interface, the existence of tangential forces depends on whether frictional mechanisms in the contact are taken into consideration. The forces of interaction between the two particles are shown in Figure 4.2. The interaction is based on a spring- and dashpot-type model, in which the spring captures the deformationbased force and the dashpot captures speed-related behaviour.

Compaction Models

F,a

F3

F=kx

T,α

r

m, I

F1

F=Cv

45

T1

T3

T2 F2

Figure 4.2. Forces and consequent accelerations on interacting particles

Computation of the force level depends on whether Stage 0, Stage 1 or Stage 2 models are being applied. For Stage 0 behaviour, a Hertzian contact model that is appropriate for rigid particles is often used. Contact laws that reflect plastic behaviour are used to model Stage 1 and this will be explained more fully in Section 4.1.2. Where complete deformation of the particle is accounted for, the force of interaction will be computed from a deformation analysis that is computed on each particle and this will be explained further in Section 4.1.3. The force balance on a particle imparts both linear and angular acceleration, also as shown in Figure 4.2. Particle kinematic response is achieved using a two-stage time-stepping algorithm. The resultant force F can be computed from the impact velocity v and penetration δ of overlapping particles and the prescribed halo interaction thickness ∆, viz.,

F =k

δ (∆ − δ )

+C

v

δ

(4.1)

The particle of mass (m) is given an acceleration (at) computed at time step (t)

at = k

δ m

−

Pv m

(4.2)

Pv is an external force. For the time increments ∆tt = t – tlast and ∆tnext = tnext – t, integration gives

vnext = vt + 0.5at ( ∆tt + ∆t next ) δ next = δ t + vnext ∆t next

(4.3) (4.4)

46

A.C.F. Cocks, D.T. Gethin, H.-Å. Häggblad, T. Kraft and O. Coube

The time step is chosen to ensure that the maximum movement and penetration of all particles are smaller than the halo thickness (∆). The final ingredient in this type of simulation comprises contact detection. This can be an extremely time consuming process because, in principle, any bodies within the simulation have the potential to interact. Practically this is never the case and so the search field for interactions can be narrowed significantly to make this part of the computation more efficient [8]. This approach to compaction simulation has been used to provide insight into the compaction process and to explore the impact of different mechanisms [9]. However, its most important application is in understanding powder flow, such as silo-emptying [2] or die-filling mechanisms [1]. The latter has the possibility to lead to a definition of density variation within a filled die, prior to compaction. 4.1.2

Stage 1 Models

In order to provide a useful structure for the macroscopic constitutive behaviour we further restrict our discussion to situations in which the matrix material exhibits a rigid perfectly plastic response. We now seek to identify the yield condition for a given internal structure, characterised by the internal porosity. Cocks [10] has demonstrated that for a rigid perfectly plastic matrix response it is possible to express the macroscopic yield condition in the form.

(

)

F = Σ Σ ij / σ y , S k − 1 = 0

(4.5)

In Equation 4.5, Σ is a homogeneous function of degree 1 in the normalised k stress, Σ ij / σ y for a given state S and σ y is the yield strength of the material. Cocks [10] further shows that a lower bound to Σ (which corresponds to an upper bound for the yield surface) can be obtained for a given state by assuming any c arbitrary internal strain-rate field ε&ij , which is compatible with the macroscopic c strain-rates E& ij . Then

Σ≥

Σ ij E& ijc V −1σ y ∫ ε&ec dV

= ΣL

(4.6)

Vm

In Equation 4.6, V is the volume of the macroscopic element and Vm is the volume occupied by the matrix material. It can further be shown that Σ L forms a c convex surface in stress space, with E& ij normal to the surface. In the remainder of

Compaction Models

47

this section we review the development of stage 1 micromechanical models in the context of the above bound.

Σ a , Ea

Σ r , Er

Σ r , Er

Σ a , Ea Figure 4.3. A cylindrical sample containing a random array of mono sized spherical particles subjected to an axisymmetric state of stress.

The situation we consider here is shown in Figure 4.3, which consists of a random array of mono sized spherical particles of radius R, which are free to slide with respect to each other. When applying Equation 4.6 to structures of this type it proves useful to first solve the problem of two contacting particles that deform under the action of a compressive force F. For a perfectly plastic contact, an analytical solution for the force required for the centres of the two particles to displace a distance, 2u towards each other [11,12] is given by

Fy = 6σ y πRu

(4.7)

when the contact radius is much less than R. When employing Equation 4.6 we assume a simple displacement pattern in the body by relating the velocity of approach of the grain centres to the macroscopic strain-rate:

u& c (ni ) = E& ijc ni n j R

(4.8)

48

A.C.F. Cocks, D.T. Gethin, H.-Å. Häggblad, T. Kraft and O. Coube

where ni is the normal to the contact. For a random array of particles the integral of Equation 4.6 can be converted to an integral over the surface of a representative particle, such that:

ΣL =

Σ ij E& ijc 3D 4πR 3

∫ z (ni ) Fy (ni ) u&

(4.9) c

(ni ) dS

S

where S is the surface area of the representative particle, F y ( ni ) is given by Equation 4.7 and depends on the contact normal and z(ni) is the probability of there being a contact with normal ni, which is simply the number of contacts per unit area with normal ni. The earliest stage 1 models were developed by Helle et al [13], who limited their consideration to hydrostatic stress states. Under these conditions the compact c experiences a pure volumetric strain-rate E& v and E& ij = 1 E& v δ ij , where δ ij is 3 c the Kroneker delta. The quantities z, Fy and u& are then the same for all contacts and Equation 4.9 becomes

ΣL =

3P DZF y R

(4.10)

2 where Z is the particle co-ordination number ( Z = 4πR z ) and P is the hydrostatic component of stress. Helle et al [13] assume that the coordination number and the relative displacement 2u of two neighbouring particles are only functions of the relative density of the compact

Z = 12 D

and

u=

1  D − Do   R 6  1 − Do 

(4.11)

where Do is the initial relative density (i.e. the compact density/the density of the constituent material). Combining these expressions with Equations 4.7 and 4.10 gives

ΣL =

P Po (D )

(4.12)

Compaction Models

 D − Do  1 − Do

2 where Po ( D ) = 3D 

49

 σ y . 

Thus, in this limit the material response can be expressed in terms of a single state variable, the relative density. This model was later generalised to multiaxial stress states by Fleck et al [14]. They retained the assumption that z and u are independent of the contact normal ni and are related to the relative density D through Equation 4.11. This combination of assumptions results in an isotropic model, where the state is once more described in terms of the relative density. For c the situation where all the principal components of strain-rate E& ij are compressive, we find that

ΣL =

Σ ij E& ijc 9 D 2  D − Do  & c   Eklσ y ∫ nk nl dS 4πR 2  1 − Do  S

=

PE& v + QE& e  D − Do  & σ y Ev 3D 2   1 − Do 

(4.13)

E& e is the equivalent strain-rate. Since the denominator does not depend on the effective strain-rate E& , this bound where Q is the von Mises effective stress and

e

can be optimised by making E& e as large as possible, subject to the constraint that all the principal strain components are compressive. This condition can be expressed in terms of the three strain-rate invariants, but the resulting expressions are cumbersome and it proves more instructive to consider a restrictive class of loading conditions. Following Fleck et al [14] we consider the axisymmetric stress state shown in Figure 4.3, where the body is subjected to an axial stress Σ a and a radial stress Σ r , resulting in strain-rates E& a and E& r . Under this loading condition the von Mises effective stress and hydrostatic stress are given by

Q = Σa − Σr

and

P=

1 3

(Σ a + 2Σ r )

The bound presented in Equation 4.13 is optimised when either E& a = 0 or

E& r = 0 , depending on the sign of Σ a − Σ r . E& e = 23 E& a = 23 E& v and Equation 4.13 becomes ΣL =

P + 23 Q Po ( D)

If

Σa ≥ Σr ,

E& r = 0 ,

(4.14)

50

A.C.F. Cocks, D.T. Gethin, H.-Å. Häggblad, T. Kraft and O. Coube

The yield condition represented by Equation 4.14 is plotted in Figure 4.4, together with the mechanism of plastic deformation that gives rise to this condition. Alternatively, if Σ r is the maximum compressive principal stress,

E& a = 0 ,

E& e = 23 E& r = 13 E& v , and Equation 4.14 becomes ΣL =

P + 13 Q Po ( D)

(4.15)

The difference between these expressions indicates the influence of the third stress invariant on the constitutive response. In order to make progress and avoid the complexity of considering the influence of the third invariant on the constitutive response, when generalising the results of the axisymmetric loading conditions to other stress states we use the results for Σ a ≥ Σ r . The full yield surface for the assumptions employed here has been determined by Fleck [15]. He obtained the yield condition 2 3 Q P  1  P    1 − 1 −  =0 F= − 1 − Po (D ) 2  Po (D )  4  Po (D )    

(4.16)

This expression is plotted in Figure 4.4, together with Equation 4.14 that forms a tangent to the curve at the vertex where the yield surface meets the hydrostaticpressure axis. If we examine this plot in the context of Equation 4.13, the mechanism gradually changes with increasing effective stress, such that the denominator of Equation 4.13 decreases and the magnitude of the effective strainrate increases above that used to determine Equation 4.14. As a result, the yield surface curves inside the tangent of Equation 4.14. The important feature of the yield surface in Figure 4.4 is that it contains a vertex for conditions of pure hydrostatic loading [14,15]. Also, if the specimen is loaded isostatically and the direction of stress is suddenly changed, then the resulting strain-rate immediately after this change will correspond to the mechanism illustrated in Figure 4.4. This result is confirmed by experimental studies on commercial iron powders [16,17], where it is observed that the radial strain-rate immediately after the axial load is applied in a consolidation test (i.e. a test in which a compact is initially loaded isostatically in a triaxial cell and at a particular density the cell pressure is kept constant, while an additional axial stress is applied to the specimen) is zero.

Compaction Models

51

1.2 1

Q/Q0(D)

0.8 Equation 4.14 Equation 4.16 Equation 4.18

0.6 0.4 0.2 0 0

0.5

1

1.5

P/P0(D) Figure 4.4. Yield surfaces for the situations where the the tensile strength of the contacts is the same as the compressive strength (Equation 4.16) and the tensile strength is zero (Equation 4.18), compared with the limiting behaviour of Equation 4.14, which forms a tangent to the two yield surfaces. The mechanism corresponding to this tangent is depicted in the figure and consists of a simple uniaxial straining of the compact.

The constitutive model of Equation 4.16 can be combined with the evolution law for the relative density, which follows from the associated flow law, the requirement that the stress remains on the yield surface during plastic flow [18] and the observation that the densification rate is related to the volumetric strainrate:

D& = − DE& kk

(4.17)

This allows the model to be incorporated in a conventional finite-element code. The presence of the vertex can, however, lead to numerical problems, since the direction of the strain increment vector is non unique for pure hydrostatic stress states. In practice, this non uniqueness can be bypassed by inserting a circular arc at the vertex to ensure a smooth continuous yield surface, see for example [19]. The analysis presented above effectively assumes that the contacts have the same strength Fy (Equation 4.7) in compression and tension. Fleck [15] subsequently relaxed this assumption. In many powder systems some interlocking of the particles occurs during compaction, but the effective tensile strength is much less than the compressive strength and can be taken as equal to zero. The resulting yield surface is given by:

52

A.C.F. Cocks, D.T. Gethin, H.-Å. Häggblad, T. Kraft and O. Coube 2 3 Q P   P    1 − 1 −  =0 F= − 1 − Po (D ) 2  Po (D )   Po (D )    

(4.18)

This surface is plotted in Figure 4.4, where it can be compared with the surface of Equation 4.16. Both surfaces share the common tangent of Equation 4.14 at the vertex where P = Po (D ) . As the hydrostatic pressure is reduced and the effective stress increases, the surface for zero tensile strength gradually curves away from the surface of Equation 4.16 towards the origin. If the contacts are unable to support a tensile stress the compact cannot carry any tensile hydrostatic loading. This type of behaviour is similar to that of the Cam-Clay and DruckerPrager-Cap models described later in this chapter. Fleck et al [14] also allowed for a distribution of contact patch sizes, which evolves as the compact deforms. A consequence of this assumption is that z and Fy in Equation 4.9 now depend on the contact normal ni and Fy only exists if u& is compressive. It is not now possible to solve the resulting Equations analytically. Fleck [15] demonstrated that the shape of the yield surface depends on the loading history and cannot be represented in terms of a single state variable such as the relative density. Also, apart from pure hydrostatic-stress histories, the response is anisotropic and the response can no longer be described in terms of stress invariants. In his model, a second-order tensor, which is related to the Green strain, is required to describe the state of the material. Fleck further demonstrates that for simple monotonically increasing loading histories there is a vertex on the yield surface coincident with the instantaneous stress state. A series of yield surfaces predicted by this model for a rigid plastic particle response is given in Figure 4.5 for a number of different initial loading paths. The general form of the yield surface predicted by this model has been verified by Akisanya et al [20] from a series of experiments on near-monosized spherical copper powders. The shape of the yield surface for an irregular steel powder is, however, not in such good agreement [21] with the model predictions.

Compaction Models

53

1.2 1 0.8

Deviatoric Stress

0.6 0.4 0.2 0 0.2 0.4 0.6

0

0.2

0.4

0.6

0.8

1

MeanStress Figure 4.5. Surface of constant complementary work (solid line) enveloping three yield surfaces for stress histories that terminate on the work surface. Each of the yield surfaces has a vertex where it touches the outer surface.

Fleck et al [22] further developed the anisotropic model using more complex contact laws derived from indentation studies and allowed for different particle sizes and properties [23]. In the process they developed a simplified model in which they relax some of the underlying assumptions, which do not significantly influence the response during the early stages of the process, such as the change in particle coordination number. As a result, they could obtain closed-form analytical expressions for the material behaviour for certain loading histories and particleproperty assumptions. Cocks and Sinka [24] further developed this class of micromechanical model by considering the material response along extremal paths in stress or strain space. Their model is based on an observation of Budiansky [25] that for situations where the actual yield surface contains vertices the material response is not too sensitive to loading path, i.e. a range of loading paths to the same terminal state in stress space produce the same final strain, provided these paths do not differ significantly from each other. They demonstrate that the material response corresponding to these paths can be determined from a surface of constant complementary work density that forms a convex surface in stress space. The reference state for the definition of this surface is the initial random packing of particles, which is taken to be macroscopically isotropic. The shape of this surface

54

A.C.F. Cocks, D.T. Gethin, H.-Å. Häggblad, T. Kraft and O. Coube

in stress space can therefore be expressed in terms of stress invariants. Cocks and Sinka [24] demonstrate that the shape of this surface is similar to that for the Drucker-Prager-Cap model described in section 4.2.2. They further demonstrate that a series of yield surfaces for loading paths which result in the same complementary work nest inside this surface, as illustrated in Figure 4.5. This feature of the material response has been verified experimental by Schneider [21] for a range of metallic powders. A number of researchers have studied the compaction of arrays of powders using the discrete element method (DEM), in which contact laws of the particles are specified and the motion of the individual particles is followed as the body is subjected to a macroscopic stress or strain history [26,27,28]. The continuum micromechanical models described in this section assume a random array of spherical particles with the deformation local to the contact related directly to the macroscopic strain. The appropriateness of this assumption of affine motion can be evaluated using DEM. Under isostatic loading, this assumption is reasonable and the results of the DEM studies are consistent with the micromechanical model [17,28]. But as the shear component of loading is increased the motion of the particles diverges increasingly away from an affine response and the difference between the micromechanical model and DEM simulations increases. As shown earlier the affine-motion assumption results in an upper bound to the yield condition for a given internal state. If the actual internal motion of the particles is different from that assumed in the analytical models the real yield surface lies inside the predictions presented earlier. Also, the DEM studies suggest that, if a micromechanical model is calibrated against the isostatic state, the predicted response in closed-die compaction is too stiff (i.e. the uniaxial load for a given density is too high). This result is confirmed by recent experimental studies on the triaxial response of a range of powders [24]. The general relationship between yield surfaces produced on different loading paths still holds, however, particularly the nesting character and general anisotropic response illustrated in Figure 4.5. The micromechanical models can then be employed to evaluate experimental data and guide the interpretation of this data to produce engineering constitutive laws [24]. 4.1.3

Stage 2 Models

Stage 2 models capture particle kinematics and deformation throughout the particle itself. This is achieved by combining discrete and finite-element analysis schemes as explained in general form by Munjiza [29]. Research in the context of powder forming has been applied in pharmaceutical tabletting, where the aim is to design a powder blend that has appropriate characteristics for compaction. In this instance, each particle is mapped by a finite-element mesh. The approach is the most computationally demanding since it undertakes a discrete simulation, see Section 4.1.1 to capture the kinematics of each particle and the deformation of each in response to load application is performed by a finite-element analysis carried out on each particle. An example compact is shown in Figure 4.6.

Compaction Models

55

Figure 4.6. Combined discrete and finite-element model of powder compaction

The method is likely to be most appropriate for providing insight into compaction phenomena and in predicting likely material response [4,5]. Through the emulation of characterisation experiments, it is possible to predict characteristics that can be used in continuum models. In this way the effect of mixture blends and particle surface characteristics can be explored, opening up the opportunity to design materials for compactability.

4.2

Phenomenological Compaction Models

As shown in Section 4.1, micromechanical models can give some interesting insight into the mechanisms occurring during compaction. They are conceptually convincing, and they were successful, e.g. in explaining the results of triaxial tests on metal powders with nearly spherical particles [20]. Excepting the combined discrete and finite-element simulation, for many commercial powders, however, the assumptions underlying the micromechanical models like spherical particles are apparently violated to an extent that the phenomenological models are more successful than the micromechanical counterparts [30,31,32]. Therefore, many of the numerous analyses published on compaction of more or less complex parts made of metal and ceramic powders are still based on phenomenological material models. There are a large number of constitutive models for simulation of compaction. Not all can be described here, the description below focuses on the two most widely used types of phenomenological compaction models, namely the Cap model and the Cam-Clay model. Some of the other models are mentioned in the introductory section below. 4.2.1

Introduction

The phenomenological compaction models, which were originally developed for soil mechanics, are usually incremental continuum plasticity models – sometimes also called critical-state models - characterised by a yield criterion, an isotropic or kinematic hardening function and a flow rule. The latter could be associated as often assumed for porous metals or non associated as for example in metal and ceramic powders [33,34]. Associated means the yield surface and the plastic

56

A.C.F. Cocks, D.T. Gethin, H.-Å. Häggblad, T. Kraft and O. Coube

potential surface are coincident and, as a consequence, the plastic strain increment is normal to the yield surface. Quite a lot of models for compaction of granular materials have been developed since the 1950s. Gens and Potts [35] give a short review about critical-state models used in geomechanics. Besides more complex models relatively simple approaches are still widely used in powder metallurgy. Secondi [36] gives some historical remarks about these kinds of models. In addition, he developed a pressure density law and found good correlation with experimental data for hard powders. In his simple treatment the influence of the deviatoric stress (or second deviatoric stress invariant) on particle and agglomerate rearrangement was neglected. However, this neglect could lead to erroneous predictions and, therefore, most models consider this invariant today. The influence of the third stress invariant is less understood. Bardet [37] states that neglecting the third stress invariant could lead to erroneous predictions in soils. Since experimental data concerning the effect of the third stress invariant are very scarce, this stress invariant is normally not included in the models. For a thorough study of the effect of the third stress invariant on the behaviour of powder compacts see Mosbah et al. [38]. Therefore, except for some rare cases, the flow behaviour is usually modelled in the stress space of the first and second invariants only. In addition, as observed by several groups (see, e.g. [20,35,39]) the compression response is path dependent, implying the development of anisotropy, which is, at least to our knowledge, not yet considered in any phenomenological model. The various phenomenological models developed differ by the functional form of the yield surface, which is often plotted in the p-q plane, where p is the hydrostatic pressure and q is the von Mises equivalent stress. For example, in the DiMaggio-Sandler model the failure surface is given by an exponential function approaching the yield stress of the fully dense material at high pressures [40-43]. A relatively simple model is that of Shima and Oyane [44], which is characterised by a single elliptic yield surface. In the Cam-Clay model both the failure surface and the cap are characterised by elliptic arcs with different eccentricity [45,46]. This model is already implemented in commercial finite-element codes like ABAQUS and used in several groups for compaction simulations [47-49]. The Cam-Clay model was recently extended by Schneider and Cocks [32] to better reproduce experimentally determined yield surfaces. These authors had also presented experimental data showing that particle shape and compact densification level have a strong effect on the yield surface. Kim et al. [50] have compared the Cam-Clay model with two other models, one is a new model developed by them and containing two empirical parameters describing the shape of the yield surface. For all models studied these authors found good agreement with experimental data for Si3N4. Since the parameters of the models are usually fitted to experimental data in the vicinity of the stress states occurring later, the good agreement is not surprising and the selection of a specific model is, therefore, not so important at least for predicting the densification behaviour. Modelling ejection will be more sensitive to the actual model due to the fact that the stress states occurring at different locations could be much more widespread. Also very important for simulation of ejection is correct modelling of the elastic behaviour, see Chapter 5.

Compaction Models

57

In reality, it is difficult to define the exact location of the yield surface for many powder materials at least at low densities, because there is no distinct transition from elastic to elastic-plastic behaviour ([51], see also Appendix 1). To overcome this difficulty, more advanced compaction models have been developed. For example, Khoei and Bakhshiani [52] have adapted the endochronic theory to powder compaction, and, beside a short review, demonstrate the abilities by implementing it into a finite-element code and simulating several case studies. Another interesting model to overcome this difficulty is the multi-surface theory, see Häggblad [42] and Häggblad and Oldenburg [53]. 4.2.2

Cap Model

One of the most often used types of constitutive model in recent finite-element simulations is the so-called cap model. The various cap models developed differ by the functional form of the yield surface but they all have some kind of cap describing the hardening behaviour. Described here is the Drucker-Prager-Cap (DPC) model. The yield surface of this model consists of the Drucker-Prager failure line or surface Fs [54] and the elliptic cap surface Fc [40], which provide a combined convex yield surface in the plane of the first and second stress invariants (p-q plane) as shown in Figure 4.7 and characterised by the following Equations: Fc = (p − p a ) 2 + (Rq ) 2 − R (d + p a tan β) = 0

(4.19)

Fs = q − p tan β − d = 0

(4.20)

where p = hydrostatic pressure (i.e. negative mean stress), q = von Mises equivalent stress. The parameters R = cap eccentricity, d = cohesive strength, β = cohesion angle are constant in the original version of the model, and pa is a hardening function depending on the density. In Figure 4.7 two extensions of the original model are also included: the tension cutoff T, which characterizes the tension strength of the powder compact, and the von Mises yield strength σy of the dense material. Inside the yield surface the powder behaves elastically. If the stress state reaches the yield surface, the powder deforms plastically. The density increases, if the stress state is on the cap, whereas it decreases (dilatation), when the stress state reaches the failure line. Dilatation implies softening, so that strain localization and cracking may occur. Hofstetter et al. [43] propose a formulation of the cap model yield functions in order to ensure a good numerical stability of the model. This work was recently improved by Chtourou et al. [55] for ductile powders. In some general purpose finite-element packages like ABAQUS this DPC model in its basic form is already implemented. In the implementation in ABAQUS, for example, only the hardening variable pb depends on the volumetric strain (which is equivalent to the density), whereas the cap eccentricity R, the cohesion strength d, and the cohesion angle β are constants. In reality, however, the green strength increases for increasing density [56-58], see also Chapter 7.

58

A.C.F. Cocks, D.T. Gethin, H.-Å. Häggblad, T. Kraft and O. Coube

Figure 4.7. Modified Drucker-Prager-Cap model in the p-q plane (p=hydrostatic pressure, q=von Mises stress) with tension and von Mises cut offs

To describe the powder behaviour more realistically - especially with respect to crack formation during pressing, unloading or ejection - the Drucker-Prager-Cap model was modified by Coube and Riedel [56]. It is plausible that not only the hardening function pa, but also the cohesion parameters d, β and T as well as the shape of the cap R should depend on the density. In the following relations, the density ρ and the volumetric plastic strain εpvol are alternatively used. They are related by εpvol = ln(ρ / ρ0 )

(4.21)

where ρ0 is the fill density. As is common in soil mechanics, the volumetric strain is defined positive during compaction. The hardening relation, the cap eccentricity and the cohesion parameters are initially described by the following empirical expressions [56,60]:

(

(

ε pvol = W 1 − exp − c1 p a

c2

))

(4.22)

R = R1 + R2 exp( R3 ρ)

(4.23)

d = d1 exp( d 2εpvol )

(4.24)

tan β = b 1 − b 2 ε pvol

(4.25)

Compaction Models

59

The parameters W, c1, c2, R1, R2, R3, d1, d2, b1 and b2 are determined by experiments. For details about the measurement techniques see Chapters 6, 7 and 10. Numerical values for an iron-base powder were given by Coube and Riedel [56] and for an alumina powder by Riedel and Kraft [59]. Instead of the given Equations 4.22 - 4.25 alternative functional forms have also been proposed to describe the observed dependencies (see, e.g [47,60]). 4.2.3

Cam-Clay Model

The purpose of this section is to provide information concerning the application of a modified Cam-Clay material model1 to describe the yielding behaviour of powder as it is compacted in a die. It represents a “single surface” yield model that has the advantage of making use of a simpler material characterisation procedure based on the use of an instrumented-die measurement equipment (see e.g.[45,46]). The yield surface of a powder needs to capture the mechanism of densification that makes the powder more difficult to compact. The modified Cam-Clay model describes the yield surface by means of an ellipse function and a typical form is shown in Figure 4.8. 250

Q (MPa)

200

150

ρ5

100

ρ4

Stress path

50

0 0

ρ3

ρ2

ρ1 50

100

150

200

250

300

P (MPa)

Figure 4.8. Modified Cam-Clay yield model

The yield surface is expressed in terms of hydrostatic stress (P) and deviatoric stress (Q). For a plain cylindrical part that is typically used in an instrumented-die test, these are given by P=

σ z + 2σ r 3

(4.26)

1 The basic Cam-Clay model has the major and minor axes coincident with the origin of the hydrostatic and deviatoric stress plane. The modified form has the ellipse major axis offset such that all elliptical surfaces pass through the origin of deviatoric and hydrostatic stress – see Figure 4.8.

60

A.C.F. Cocks, D.T. Gethin, H.-Å. Häggblad, T. Kraft and O. Coube

Q = σz − σr

(4.27)

Figure 4.8 also includes the stress path that is mapped in an instrumented-die experiment and the properties of this path can be used to establish the yield model that is used within the simulation. The surfaces shown in Figure 4.8 are all ellipses and they are presented at different densification levels, thus capturing the increased resistance to compaction. Because the yield surfaces shown in the figure pass through the origin, they exclude cohesive behaviour since the powder will not sustain any shear (deviatoric) stress at zero hydrostatic stress. The surfaces can also be offset along the hydrostatic stress axis to account for this mechanism. Using the basic Equation to represent an ellipse, the yield Equation in its general form is written as:

f =

( P(σ ij ) - P0 ) 2 P02

+

Q 2 (σ ij ) Q02

=1

(4.28)

P0 and Q0 are, respectively, the half-lengths of the major and minor axes of the ellipse. P0 also represents the extent to which the ellipse is offset along the hydrostatic stress axis. These material parameters are assumed, as shown above, to vary with density to capture the hardening behaviour of the powder and this variation needs to be captured through appropriate Equation fits. Two Equations are required to determine the parameters P0 and Q0. The first Equation is derived by inserting Equation 4.26 and Equation 4.27 into Equation 4.28 2

σd + 2σdr − P0 ) ( z (σdz − σdr )2 3 + −1 = 0 f= P02 Q02

(4.29)

where σ z and σ r are, respectively, the axial and radial stresses that are generated within a compact, obtained typically from an instrumented-die test in which a cylindrical sample is formed. In the absence of further information it is common to assume that the model is associated. This is appropriate in the case of powder forming since the particles are generally small and are approximately uniform in size. This provides the second Equation that may be used to determine P0 and Q0. In this instance, the plastic strain-rate tensor is expressed as d

d

∂f ε& ijp = λ& ∂σ ij If the die is perfectly rigid, there is no radial displacement during die pressing and hence the plastic radial strain is zero, which implies

Compaction Models

 ∂f   ∂σ ij 

61

  =0   i = j= r

Application to Equation 4.29 gives

2 ∂f = ∂σ ij 3

(

σ zd + 2σ rd 3

− P0 )

P02

−

(σ zd − σ rd ) =0 Q02

(4.30)

From Equations 4.29 and 4.30, the functions P0 and Q0 are obtained as 2

P0 =

3P d + 2P d Q d 6 P d + 2Q d

(4.31)

( )

3 d 2 d P Q Q0 = Q + 2 2 2P d + Q d 3 d2

(4.32)

The material yield model as defined by the variation of P0 and Q0 with density and appropriate functional choices need to be made. To do this, appropriate Equations must be defined and the following relationships have been utilised. These Equations are not prescriptive and alternatives may be used.   ρ − ρ0 P0 = K 1  ln1 −    ρ − ρ max

    

K P Q 0 = Q max tanh 3 0  Q max

   

K2

(4.33) (4.34)

The terms K1 to K3 are curve-fit constants, ρ0 and ρmax are the fill and maximum theoretical density for the powder and Qmax is the maximum deviatoric stress that the fully dense powder can sustain. The functional form of Equations 4.33 and 4.34 has been chosen to obtain the best fit with experimental data and, further, Equation 4.34 ensures that the material behaviour is asymptotic to that of the fully dense powder. The parameters are determined by experiments. For details about the measurement techniques see Chapters 6, 7 and 10.

62

4.3

A.C.F. Cocks, D.T. Gethin, H.-Å. Häggblad, T. Kraft and O. Coube

Closure

This chapter has set out the essential details concerning modelling of the compaction stage of the powder-pressing cycle. Starting from a micromechanical framework, this has illustrated how the approach can be used to underpin the understanding of continuum-scale models. Two continuum material models have also been introduced, comprising either a single surface (Cam-Clay) or two surfaces (Drucker-Prager-Cap). Procedures for characterising both of these models are well documented and explained in other chapters of this book. It is also worth noting that other material models may be used, extending the framework set out in this chapter to cover aspects such as crack formation, ejection and powder transfer. However, these are not so mature at this time and are the subject of current research activity in this field.

References [1] [2] [3] [4] [5] [6] [7] [8]

[9] [10] [11] [12] [13] [14] [15] [16]

Wu CY, Cocks AFC and Gillia OT. 2002. Experimental and Numerical Investigationof Die Filling and Powder Transfer; Adv Powder Metal. Partic. Mater., 4, 258-272. Cleary PW and Sawley ML. 2002; DEM Modelling of Industrial Granular Flows: 3D Case Studies and the Effect of Particle Shape on Hopper Discharge; Applied Mathematical Modelling, 26, p89-111. Coube O, Heinrich B, Moseler M and Riedel H. 2004. Modelling and Numerical Simulation of Powder Die Compaction with a Particle Code; PM World Congress. Lewis RW, Gethin DT, Yang XS and Rowe RC. 2005. A Combined Finite-Discrete Element Method for Simulating Pharmaceutical Powder Tabletting; IJNME, 62, p853869. Procopio AT and Zavaliangos A. 2005. Simulation of Multiaxial Compaction of Granular Media from Loose to High Relative Densities; Journal of the Mechanics and Physics of Solids, 53, p1523-1551. Shima S, Kotera H and Ujie Y. 1995. A study of constitutive behaviour of powder assembly by particulate modeling; Materials Science Research International, 1, p163168. Cundall PA and Strack ODL. 1979. A Discrete Numerical Model for Granular Assemblies; Geotechnique, 29, p47-65. Yang XS, Lewis RW, Gethin DT, Ransing RS and Rowe R. 2002. Discrete-Finiteelement modelling of pharmaceutical powder compaction, in: Discrete Element Methods: Numerical Modelling of Discontinua, BK Cook and RP Jensen (eds.), ASCE Geotechnical Special Publication, p74-78. Hassanpour A and Ghadiri M. 2004. Distinct Element Analysis and Experimental Evaluation of the Heckel Analysis of Bulk Powder Compression; Powder Technology, 141, p251-261. Cocks ACF., 1989. J. Mech. Phys. Solids 37 693. Ashby MF, 2984. Adv Appl. Mech. 23: 117. Ashby MF, 1990. Background Reading HIP 6.0, Univ. of Cambridge. Helle AS, Easterling KE and Ashby MF, 1985. Acta Metall. 33: 2163. Fleck NA, Kuhn LT and McMeeking RM, 1992. J. Mech. Phys. Solids 40, 1139. Fleck NA, 1995. J. Mech Phys. Solids 43: 1409. Pavier E and Doremus P, 1999. Powder Met. 42: 345.

Compaction Models

63

[17] Sinka IC and Cocks ACF. to appear. [18] Trasorras JRL, Parameswaran R and Cocks ACF. 1998. Mechanical Behavior of Metal Powders and Powder Compaction Modeling; ASM Handbook, 7, pp 326-342. [19] Govindarajan RM and Aravas N. 1994. Deformation processing of metal powders: Part 1 - Cold isostatic pressing; Int. J. Mech. Sci., 36, p343-357. [20] Akisanya AR, Cocks ACF and Fleck NA. 1997. The Yield Behaviour of Metal Powders; Int. J. Mech. Sci. 39, p1315-1324. [21] Schneider LCR, 2004. PhD thesis University of Leicester. [22] Fleck NA, Storakers B and McMeeking RM. 1997 In: Fleck NA and Cocks ACF. ed Mechanics of Granular and Porous Materials. Kluwer, Dordrecht, 1. [23] Storakers B, Fleck NA and McMeeking RM. 1999. J. Mech Phys. Solids 47: 785. [24] Cocks ACF and Sinka IC. to appear. [25] Budiansky BJ. Appl. Mech. 1959:,26: 2. [26] Redanz P and Fleck NA. 2001. The compaction of a random distribution of metal cylinders by the discrete element method; Acta Mater. 49, p4325-35. [27] Martin CL and Bouvard D. 2003. Study of the cold compaction of composite powders by the discrete element method; Acta Mater. 51, p373-86. [28] Skrinjar O and Larsson P-L. 2004. Cold compaction of composite powders with size ratio, Acta Mater. 52, p1871-84. [29] Munjiza A. 2004. The Combined Finite-Discrete Element Method, John Wiley and Son, Chichester. [30] Sridhar I and Fleck NA. 2000. Yield behaviour of cold compacted composite powders; Acta Mater. 48, p3341-3352. [31] Rottmann G, Coube O and Riedel H. 2001. Comparison Between Triaxial Results and Models Prediction with Special Consideration of the Anisotropy, in: European Congress on Powder Metallurgy, 2001, Vol. 3, EPMA, Shrewsbury 29-37. [32] Schneider LCR and Cocks ACF. 2002. Experimental investigation of yield behaviour of metal powder compacts; Powder Metallurgy, 45, p237-245. [33] Bortzmeyer D. 1992. Modelling Ceramic Powder Compaction; Powder Tech. 70, p131-139. [34] Pavier E and Dorémus P. 1997. Int. Workshop on Modelling of Metal Powder Forming Processes, INPG, Université Joseph Fourier, CNRS, Grenoble, 1. [35] Gens A and Potts DM. 1988. Critical State Models in Computional Geomechanics; Eng. Comput, 5, p178-197. [36] Secondi J. 2002. Modelling powder compaction.From a pressure-density law to continuum mechanics, Powder Metallurgy, 45, p213-217. [37] Bardet JP. 1990. Lode Dependences for Isotropic Pressure-Sensitive Elastoplastic Materials, Trans ASME, 57, p498-506. [38] Mosbah P, Kojima J, Shima S and Kotera H. 1997. Int. Workshop on Modelling of Metal Powder Forming Processes, INPG, Université Joseph Fourier, CNRS, Grenoble, 19 [39] Brown BS and Abou-Chedid G. 1994. Yield Behaviour of Metal Powder Assemblages; J. Mech. Phys. Solids, 42, p383-399. [40] DiMaggio FL and Sandler IS. 1971. Material Model for Granular Soils; J Eng Mech Div 97, p935-950. [41] Sandler IS, DiMaggio FL, Baladi GY and Asce M. 1976. Generalized cap model for geological materials; J Geotech Engng Div 102, p683-699. [42] Häggblad HÅ. 1991. Constitutive Models for Powder Materials; Powder Tech. 67, p127-136. [43] Hofstetter G, Simo JC and Taylor RL. 1993. A Modified Cap Model: Closest Point Solution Algorithms; Comp Struct, 46, p203-214.

64

A.C.F. Cocks, D.T. Gethin, H.-Å. Häggblad, T. Kraft and O. Coube

[44] Shima S and Oyane M. 1976. Plasticity for porous metals; Int J Mech Sci 18, p285291. [45] Schofield A and Wroth CP. 1968. Critical State Soil Mechanics, McGraw-Hill, London. [46] Roscoe KH and Burland JB. 1968. On the generalised stress-strain behaviour of “wet” clay; Eng Plast, p535-609. [47] PM Modnet Computer Modelling Group, 1999. Comparison of computer models representing powder compaction process, Powder Metall 42, 301-311. [48] PM Modnet Research Group, 2002. Numerical simulation of powder compaction for two multilevel ferrous parts, including powder characterisation and experimental validation, Powder Metallurgy, 45, p335-353. [49] Favrot N, Besson J, Colin C and Delannay F. 1999. Cold Compaction and Solid-State Sintering of WC-Co-Based Structures:Experiments and Modeling; J Am Ceram Soc, 82, p1153-1161. [50] Kim HS, Oh ST and Lee JS. 2002. Constitutive model for cold compaction of ceramic powder; J Am Ceram Soc, 85, p2137-2138. [51] Perez-Foguet A, Rodriguez-ferran A and Huerta A. 2001. Consistent tangent matrices for density-dependend finite plasticity models; Int J Numer Anal Methods Geomech, 25, p1045-1075. [52] Khoei AR and Bakhshiani A. 2004. A hypoelasto-plastic finite strain simulation of powder compaction processes with density dependent endochronic model; Int J Solids Structures, 41, p6081-6110. [53] Häggblad HÅ and Oldenburg M. 1994. Modelling and simulation of metal powder die pressing with use of explicit time integration; Modelling Simul. Mater. Sci. Eng. (2), p893-911. [54] Drucker DC and Prager W. 1952. Soils Mechanics and Plastic Analysis of Limit Design; Quaterly Appl Math, 10, p157-164. [55] Chtourou H, Guillot M and Gakwaya A. 2002. Modeling of the metal powder compaction process using the cap model. Part II. Numerical implementation and practical applications; Int J Solids Structures, 39, p1077-1096. [56] Coube O and Riedel H. 2000. Numerical Simulation of Metal Powder Die Compaction with Special Consideration of Cracking; Powder Metallurgy, 43, p123131. [57] Bortzmeyer D, Langguth G and Orange G. 1993. Fracture Mechanics of Green Products; J Europ Ceram Soc, 11, p9-16. [58] Dorémus P, Toussaint F and Alvin O. 2001. Simple Tests Standard Procedure for the Characterisation of Green Compacted Powder. Recent Developments in Computer Modelling of Powder Metallurgy Processes, NATO Advanced Research Workshop, Series III: Computer and Systems Science vol 176 Zavaliangos A, Laptev A (eds), IOS Press, Amsterdam, p29-41. [59] Riedel H and Kraft T. 2004. Simulations in Powder Technology, Continuum Scale Simulation of Engineering Materials: Fundamentals – Microstructures – Process Applications, eds D Raabe, F Roters, F Barlat, LQ Chen, Wiley-VCH, Berlin, p641658. [60] Coube O and Riedel H. 2002. Modeling of Metal Powder Behavior under Low and High Pressure, Advances in Powder Metallurgy & Particulate 2002, Part 9, Arnhold V, Chu C-L, Jandeska WF, Jr. and Sanderow HI (eds). Metal Powder Industries Federation, Princeton, NJ, 199-208.

5 Model Input Data – Elastic Properties M.D. Riera1, J.M. Prado1 and P. Doremus2 1

CTM Technological Centre, Department Materials Science, UPC, Spain Institut National Polytechnique de Grenoble, France

2

5.1 Introduction It might seem that elastic behaviour plays a secondary role during powder compaction. This is true only during the initial densification of the powder in the die in which very inelastic mechanisms, i.e. particle sliding and rearrangement, are acting. However, as densification proceeds plastic deformation of particles, mainly in metals, becomes the prevailing compacting mechanism. The aggregate becomes mechanically coherent and is able to transmit elastic stresses, but it is during the final ejection stage when the elastic behaviour is most important. Final-part dimensions and, consequently, the achievement of the desired dimensional tolerances are highly dependent on the elastic springback that takes place during part ejection. Another effect deriving from elastic stresses is the cracking that can occasionally develop in this stage. Traditionally, experienced die designers using expensive trial and error methods have solved these two problems. Modern computer simulation techniques are powerful tools in the design of forming dies when reliable mechanical models of the behaviour of the materials involved are available. Unfortunately, the elastic behaviour of granular materials has not received enough research attention with the consequence that the necessary constitutive equations are not yet well established. This chapter gives a brief overview of the present knowledge of elastic behaviour of granular materials together with a description of the tests commonly used to determine elastic constants and existing data for some frequently used metal alloys powders.

5.2 Elastic Model Two different approaches can be found in the literature when dealing with the elastic behaviour of granular materials [1,2]. The micromechanical one considers the “hertzian” deformation of the contacts between neighbouring particles subjected to an applied strain field.

66

M.D. Riera, J.M. Prado and P. Doremus

The elastic behaviour predicted is non-linear and follows a potential law of the type [1,3]

σ = k εn

(5.1)

where σ and ε are the true stress and strain respectively, k is a proportionality constant which depends on the density of the compact and the exponent n takes the value of 3/2. In uniaxial tests Equation 5.1 refers to the axial stress and strain, meanwhile in a three dimensional state of stress σ and ε correspond to the hydrostatic and volumetric values. Depending on the state of stress the derivative dσ/dε gives either the instantaneous Young’s (E) or Bulk modulus (Kv). E = dσax/dεax = (3/2) kax εax1/2 = (3/2) kax2/3 σax1/3 Kv = dσv/dεv = (3/2) kv εv1/2 = (3/2) kv2/3 σv1/3

(5.2)

Both E and Kv are functions of density and applied stress. A theoretical expression for the Bulk modulus obtained by applying the micromechanical approach is given by 1  3φ 2 Z 2σ v   K v =  6  π 4 B 2 

1

3

(5.3)

where Z is the coordination number, φ the relative density and B an elastic parameter given by B=

1 1 1    + 4π  µ λ + µ 

(5.4)

where λ and µ denote the Lamé moduli of the bulk material. The coordination number changes with density and can be found in the literature [4]. However, authors working in continuum mechanics tend to consider the elastic moduli as linear depending only on part density. This is a much simpler approach and sufficiently satisfactory when density and applied stresses are high enough. It is also very convenient in numerical simulation as most of the commercial finite element codes only include linear elasticity. A model taking into account both approaches considers that the total elastic deformation has contributions from the local hertzian deformation of the contacts and also the whole linear elastic deformation of the metallic skeleton. At low stresses and densities, when the contact necks among particles are small, the nonlinear contribution of the elasticity will prevail. At high densities and applied stresses the pores become round and the elasticity is more linear in character.

Model Input Data – Elastic Properties

67

According to this criterion the total elastic deformation, εte, is the addition of the non-linear and linear contributions

εte = εne + εle ε

e t =

a (ρ,σ) (σ/k (ρ))

2/3

+ (1-a) (σ/E(ρ))

(5.5) (5.6)

where a is a parameter that weights the contribution of the non-linear elasticity. It is a function of density ρ and the state of stress. Density is related to the contact area between particles and consequently to the capacity to transmit force through them. At low densities necks between particles are only incipient and involve high local stresses, however, they remain small in the rest of the material.

(a)

(b)

Figure 5.1. Schematic representation of the deformation of a pore under a hydrostatic state of stress (a), or pure shear (b)

The elastic behaviour under these circumstances is mainly nonlinear. At high densities necks between particles are well developed and stresses are more homogeneously distributed throughout the bulk material, favouring linear elasticity. The state of stress also influences the type of elastic behaviour encountered because, as is schematically shown in Figure 5.1, hydrostatic stresses are more effective in closing pores than deviatoric ones. Deviatoric stresses mainly change pore shape. Hence, in closed-die compaction with a high hydrostatic stress component the elastic behaviour will tend to be predominantly linear from the early stages. This will explain the apparently contradictory results found by Riera et al. [5], which reported measurements of the elastic modulus in cyclic compression uniaxial tests, and those of Pavier and Doremus [2] obtained by loadings and unloadings in an instrumented compacting die. The strongly deviatoric character of the uniaxial compression tests makes them more useful in determining the nonlinear elastic behaviour.

68

M.D. Riera, J.M. Prado and P. Doremus

Another aspect to take into consideration when dealing with the elasticity of metal or ceramic powder compacts is their anisotropic character. This behaviour is because the magnitude of axial stresses, during compaction in rigid dies, is greater than the radial ones and, consequently, the size of the contact necks oriented perpendicularly to the axial stresses are also greater than those normal to the radial stresses. This complex elastic behaviour of metal and ceramic powders during compaction makes numerical simulation difficult. The compaction inside the die can be adequately simulated by taking into account only the linear elastic part, but for the correct simulation of the springback during ejection the nonlinear contribution is also necessary.

5.3 Experimental Techniques 5.3.1 Characterisation of Elastic Properties of Green Compacted Samples As previously described, the elastic parameters can be determined using either uniaxial compression tests carried out on samples previously compacted to a certain density or by loadings and unloadings during compaction in a rigid die. Another method that can also be used is to measure the velocity of ultrasound waves propagating through the compacted sample. In the uniaxial compression test samples are normally cylindrical with a relation between height H and diameter D H/D ∼ 1.5

(5.7)

The dimensions are a compromise between the need to have sufficient height to be able to accommodate a diametral extensometer and that of maintaining a homogeneous density. The main difference from a conventional compression test on bulk materials is the necessity of measuring the radial true strain that enables the determination, besides the Young’s modulus E, of both the volumetric elastic modulus and the Poisson´s coefficient ν. The volumetric strain is related to the axial and radial strains by the expression

εv = εax + 2 εr

(5.8)

The tests were carried out on a Distaloy AE powder compacted to a density of 6.75 Mg/m3. The chemical composition of the powder is given in Table 5.1. Table 5.1. Chemical composition of Distaloy AE, wt% Nickel

Copper

Molybdenum

Iron

4%

1.5 %

0.5 %

94 %

Axial strain, εax, can be measured by monitoring the displacement of the movable crosshead of the testing machine with a linear variable displacement

Model Input Data – Elastic Properties

69

transducer (LVDT), whereas the measurement of the radial strain, εr, needs the use of a diametral extensometer. A possible experimental setup is shown in Figure.5.2.

σax LOAD CELL

RADIAL EXTENSOMETER LVDT

Figure 5.2. Schematic experimental arrangement for a uniaxial compression test

The volumetric strain, εv, can be calculated by using Equation 5.8. During cycling the axial load is increased by a fixed amount (500 N) in each cycle; in this way the unloading part of a cycle can be considered as elastic; however, in the consequent reloading the sample will behave elastically only up to the load level reached in the previous cycle.

Distaloy AE Density 6.75AE Mg/m3 Distalloy

D = 6.75

Figure 5.3. Axial true strain during cyclic compression

70

M.D. Riera, J.M. Prado and P. Doremus

Distaloy Distalloy AEAE Density 6.75 Mg/m3 D = 6.75

Figure 5.4. Volumetric true strain during cyclic compression

Typical curves obtained when the applied axial true stress, σ, is represented as a function of εax and εv are shown in Figures 5.3 and 5.4 for the case of a sample compacted to a density of 6.75 Mg/m3. An important feature of the elastic loadingunloading cycles that can be observed in the above figures is the non-linear dependence between the applied true stress and the axial and volumetric true strains. When triaxial compression tests, such as loadings and unloadings in a rigid die are used, elastic parameters can no longer be deduced from Equations 5.1 and 5.2 but the classical elastic equations (Equations 5.9) relating stresses and strains have to be solved

1 [σ ax − ν (σ r + σ θ )] E 1 ε r = [σ r − ν (σ ax + σ θ )] E 1 ε θ = [σ θ − ν (σ ax + σ r )] E

ε ax =

(5.9)

The geometrical symmetry involved in a cylindrical die simplifies the experimental needs because εr = εθ and consequently σr = σθ. The knowledge of

Model Input Data – Elastic Properties

71

σax, σr, εax and εr will permit the determination of E and ν. To find these parameters two different techniques can be used: • •

Direct measurement of the axial and radial stresses by means of force transducers indirect measurement of the radial stress by means of strain gauges placed on the external surface of the die.

In the first case, the wall thickness of the die can be large enough to consider it as rigid and consequently εr = 0. Now, with the assumption of axial symmetry and a rigid die, Equations 5.9 simplify to

1 [σ ax −ν 2σ r ] E ν = σr /(σax+σr)

ε ax =

(5.10) (5.11)

In these conditions the determination of ν and E is of great simplicity because only εax, σax and σr have to be found experimentally. An experimental setup like the one shown in Figure 5.5 can be employed. Problems arise when the value of the radial stress is high. In this case it is recommended to avoid direct contact between the sensor and the powder, then, an internal ceramic or hardmetal sleeve must be used. Load cell

Cross-head Upper punch

LVDT

Die

Pressure sensor

Base plate Figure 5.5. Experimental setup in a cylindrical rigid die

When the radial force is measured by monitoring the deformation of the die wall it is necessary to start by calibrating the strain gauges. Calibration is normally done by means of a pressurized fluid that applies a uniform pressure on the internal surface of the die. A FEM simulation of this process can highlight the different problems involved during the calibration of a compacting die. A computer

72

M.D. Riera, J.M. Prado and P. Doremus

simulation has been carried out applying two kinds of radial stresses: a constant radial stress of 60 MPa, corresponding to the calibration stage, and another decreasing linearly along the height of the die wall similarly to the situation encountered during powder compaction. Three different die configurations with wall thicknesses of 5, 10 and 20 mm each with height of 15 mm were studied. Computer simulation (Figure 5.6) shows that the thickness of the die walls should be small enough to allow for a sufficient circumferential deformation, even under low radial stresses, which can be measured accurately. For this reason, dies should have a wall thickness less than 10 mm. However, a barrelling of the die always takes place and it is more marked for thinner die walls. This effect prevents direct measurement of the radial stress distribution along the die wall. Nevertheless, it has been shown by Mosbah [6] that the circumferential deformation values at the centre of the die wall are the same for both constant and linear stress distributions when they result in an equivalent total force on the wall. This fact enables the calibration of the strain gauge to obtain an average radial stress if a linear radial stress distribution is assumed.

200

Stress distribution: Constant Linear

5

180 160

Micro loop strain, εθθ

140 120 100

10

80 60 40

20

20 0 -1

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17

Height Z, (mm)

Figure 5.6. FEM simulation of the circumferential strain along the die-wall height for different wall thickness and radial stress distributions

The ultrasonic determination of the elasticity modulus is based on the fact that sound wave propagation depends on the modulus, E and the density ρ; the longitudinal wave speed v is given by v = (E/ρ )1/2

(5.12)

Model Input Data – Elastic Properties

73

To obtain the dependence of E on density and applied stress, it is necessary to measure the velocity of sound v for each density at different applied stresses. The problem with green compacted samples is the strong signal attenuation produced by porosity and unwelded particle contacts that limit the height of the sample. The measurement of the longitudinal and transverse sound wave propagation enables the study of the anisotropy of the elastic modulus [7]. 5.3.2 Characterisation of Elastic Properties at High Stresses As previously stated elastic properties depend on the density as well as the state of stress. However for high stresses, elasticity tends to depend only on density and becomes linear, which means that parameter a in Equation 5.6 tends to zero. To a first approximation die compaction generates radial stresses that are half the axial stresses. This corresponds to a mean pressure within the powder of about 2/3 σax. A compacting stress of 800 MPa leads to a mean pressure of 530 MPa and a deviatoric stress of 400 MPa. When simple compression is used for determining the elastic properties a maximum of 200 MPa is applied that corresponds to a mean stress of 70 MPa with a deviatoric stress of 200 MPa. The state of stress is two times greater in a die than in simple compression and the mean pressure is always higher than the deviatoric stress. For evaluating the elastic parameters of a powder under a high stress state one can use a triaxial cell. The test is similar to simple compression (Figure 5.2) except that the radial pressure is no longer zero but equal to the imposed confining pressure. Figure 5.7 shows the evolution of the axial stress for a constant radial stress of 400 MPa. While plasticity is observed during the increase of the axial stress, reversible cycles are achieved when a specimen is unloaded then reloaded, which is a characteristic of an elastic behaviour. It is, however, difficult to obtain the same accuracy as for simple compression. It is possible to examine the variation of the elastic properties as a function of the density, the mean stress P or the deviatoric stress Q. Figures 5.8, 5.9 and 5.10 illustrate results obtained with Distaloy AE. In the range of density and mean pressure tested the elastic modulus E and bulk elastic modulus K are not sensitive to mean stress P (Figure 5.8).

74

M.D. Riera, J.M. Prado and P. Doremus

Figure 5.7. Evolution of the axial stress and axial strain. Cycles can be described by reversible straight lines, the slope of which corresponds to the elastic modulus.

Figure 5.8. Representation of the elastic modulus E in space E(P,ρ) showing projection of experimental data on plane E(ρ)

Model Input Data – Elastic Properties

75

The influence of the density is shown in Figures 5.9 and 5.10. Experimental data can be fitted with analytical expressions as follows: Young’s modulus: E (Mpa) = (-28000 + 10120ρ) [ exp (ρ /6.8 )6 ] Bulk modulus: K (Mpa) = (-10500 + 3750ρ) [ exp (ρ /6.55 )6 ] with ρ in Mgm-3.

Figure 5.9. Evolution of the elastic modulus E as a function of density. The plain line corresponds to the fit given from the above expression.

Figure 5.10. Evolution of the bulk elastic modulus K as a function of density. The plain line corresponds to the fit given from the above expression.

76

M.D. Riera, J.M. Prado and P. Doremus

5.4 Conclusions Granular materials show a complex elastic behaviour highly dependent on density and the state of stress. In uniaxial compression tests elasticity is nonlinear and follows a potential-type law. At high densities and an triaxial state of stress with high values of mean stress the elasticity approaches a linear behaviour similar to that of bulk material. In this case triaxial cells are a convenient method of determining elastic constants.

References [1] Walton K. J. 1987. Mech. Phys. Solids. Vol. 35, 213-226. [2] Pavier E and Doremus P. 1996. Mechanical behaviour of a lubricated iron powder. PM´96. Advances in Powder Metallurgy & Particulate Materials, Vol.2, Part 6, pp. 2740. [3] Prado JM and Riera MD. 2001. NATO Science Series III: Computer and Science Systems. Vol.176, 63-70. [4] Arzt E. 1982. Acta Metall. Vol. 30, 1883- 1894. [5] Riera MD and Prado JM. 1998. Uniaxial compression tests on powder metallurgical compacts. Powder Metallurgy World Congress and Exhibition. CD-Document No.615, EPMA, UK. [6] Mosbah P. 1995. PhD Thesis, Université Joseph Fourier- Grenoble I. p 187. [7] Coube O. 1998. PhD Thesis, Université Pierre et Marie Curie, Paris VI

6 Model Input Data – Plastic Properties P. Doremus1 1

Institut National Polytechnique de Grenoble, France.

6.1 Introduction When powder is pressed densification occurs. However, powder density depends on the pressing method (hydrostatic pressing, die pressing, etc,) that is to say the state of stresses applied. During pressing plastic properties prevail. This chapter deals with the experimental methods that are commonly used for measuring and analysing plastic properties. Characterisation equipment must be instrumented sufficiently to link powder density to the measured applied stress. All types of powder can be tested using the two common techniques that are described below: instrumented die and triaxial cell. Each technique has its own inherent advantages. The die test requires certain assumptions in order to be useful. The method used for deriving plastic parameters is presented and assumptions are discussed. Triaxial tests are also presented. This sophisticated high-performance device is more suitable for finer-powder plastic behaviour analysis.

6.2 Closed-die Compaction Test Die compaction is certainly the most representative test for studying powder densification phenomena or compressibility [1,2]. Friction between die surface and powder compact can also be analysed by using a fully instrumented die [3,4,5]. A fully instrumented die is also required for the determination of material parameters when fitting constitutive models [6]. This test has several advantages: experimental equipments are not very complex and also not expensive compared to more sophisticated equipment such as a triaxial press. Tests are quickly performed and powder densification is achieved in a similar manner to industrial production. However, die compaction also has disadvantages. The most important one is due to die-wall friction that generates density variations leading to an inhomogeneous test. Despite this, it is possible to get information on powder/tool friction when die compaction is performed using single pressing action equipment. To be of interest, die equipment must be instrumented so that upper, lower and radial stresses are all measured during compaction. Generally, such equipment has a die with a constant

78

P. Doremus

cross section. Therefore the evolution of the mean density can be deduced by measuring compact height. Figure 6.1 represents die equipment with the different transducers that can be used. The fixing system of the intermediate plateau allows powder to be compacted in either a fixed die or floating die.

Figure 6.1. Die coMPaction rig showing details of the complete design

The upper and lower mean stresses are easily measured thanks to force transducers and the diameter D of the upper and lower punches (Figure 6.2).

Figure 6.2. Schematic representation of die equipment

Model Input Data – Plastic Properties

79

However, measuring compact height, that is to say the distance between the two pressing surfaces of the upper and lower punches, is not straightforward. Most often, laboratory devices or industrial presses are fitted with displacement transducers located as close as possible to the compact. Depending on press design, this measure requires assumptions to be made on at least punch stiffness or mechanical assembly stiffness. Therefore, calibration is needed to determine compact height prior to ejection. Concerning radial stress measurement different technologies have been developed. Figure 6.3 illustrates the most widely used. Each equipment has its advantages. The main differences are: •

Equipment A: the range of measurement depends on the inner layer thickness and force transducers. It is possible to get information on radial stress distribution along the compact height but potential inaccuracies or perturbations are introduced by the inner layer. Calibration is therefore needed due to the inner layer

•

Equipment B: the range of measurement depends on force transducers. At high density, transducer stiffness can affect density measurement. Information on radial stress distribution along the compact height is theoretically possible.

•

Equipment C: the range of measurement depends on the die-wall thickness. Calibration is needed and only the mean radial stress is accessible.

•

Equipment D: cubic specimen. The range of measurement depends on force transducers. There is no need for calibration but only the mean radial stress is measured.

σr

σr

Small pins in contact Small pin in contact with a thin inner layer with powder and connected to connected to force force transducers transducers

σr

Strain gauges stuck on the outer wall of the die

σr

Force transducer in contact with a moving part of the die

Cross sections of the different dies

Figure 6.3. Details of die equipment showing the different radial stress measurement methods. From left to right equipment A, B, C and D.

80

P. Doremus

When required, calibration is usually achieved on materials such as grease, modelling clay or elastic rubber. All these materials transmit a radial stress equal to the applied axial stress as they are quite incompressible. As previously said, the closed-die test is not homogeneous due to friction between the powder and the die wall. It is therefore necessary to make assumptions to derive intrinsic information characterizing the powder itself. The slab equilibrium theory is the most widely used for doing this. This theory is based on the following assumptions (A1) Radial stress σr is proportional to axial stress σz :

σr(r=R) = α σz(r=R)

(6.1)

Where α is the stress transmission coefficient, which is supposed independent of the axial coordinate z. (A2) Radial stress gradient is negligible. The previous expression can then be written: σr(r=R) = α σz (A3) Tangential stress τz is proportional to the radial stress on die wall:

τz = µ σr(r=R)

(6.2)

Where µ is the powder/tool friction coefficient supposed independent of the axial coordinate z. These assumptions will be discussed later. Neglecting gravity the equilibrium of a powder slice (Figure 6.4) of diameter D and thickness dz leads to: dσz πD2/4 = πDτz dz

(6.3)

Introducing the definitions of α and µ gives: dσz /σz = 4µαdz /D

(6.4) σz +dσz

σr

τz

dz

τz

σr

σz Figure 6.4. Stresses applied to a powder slice

Model Input Data – Plastic Properties

81

As the stress transmission coefficient and the friction coefficient do not depend on the axial coordinate z, integration of the previous equation leads to:

σz = σz lop exp(4µαz /D)

(6.5)

which gives for z = h (coMPact height)

σz up= σz lop exp(4µαh /D)

(6.6)

σz up is the stress applied on the upper punch and σz lop on the lower one. The axial stress at z = h/2 is given from the expression:

σz z=h/2 = (σz up σz lop)1/2

(6.7)

The mean radial stress σrm is measured thanks to strain gauges or force transducers and is expressed as:

σrm = 1/h

∫

h 0

σr(r=R) dz

(6.8)

Introducing Equation 6.5 in the integral 6.8 gives:

σrm = σz lop D[exp(hµα /4D)-1]/4µh

(6.9)

An illustration of the evolution of the axial and radial stress calculated from expressions 6.7 and 6.9 is shown in Figure 6.5 Axial and radial stresses as function of density

Stress

axial stress radial stress

ρo

Density

ρt

Figure 6.5. Evolution of the axial and radial stresses during compaction. ρ0 is the loose powder density and ρt the theoretical density (without pores)

Therefore, combining Equations 6.6, 6.7 and 6.9 leads to the stress transmission coefficient α. Its evolution is shown in Figure 6.6.

82

P. Doremus

α = σr m Log(σz up / σz lop)/(σz up - σz lop)

(6.10 )

Pressure transmission ratio α as function of density 0.7

α

ductile powder 0.5

hard powder

0.3

ρo

Density

ρt

Figure 6.6. Evolution of the stress transmission ratio, σr/σz during coMPaction as a function of density

In this illustration, the stress transmission ratio is largely independent of the density. More generally, α can increase slightly and ranges between 0.4 and 0.6. The friction coefficient (ratio of the tangential force to the normal force) can be deduced from the values of the different stresses:

µ = D(σz up - σz lop)/ 4hσrm

(6.11)

Friction coefficient µ as function of density 0.20

µ

ductile powder

0.15

hard powder 0.10 0.05

ρo

Density

ρt

Figure 6.7. Evolution of the mean friction coefficient as a function of density

Figure 6.7 shows the evolution of the friction coefficient. It can be observed that µ decreases with density. This is generally the case for all powders, metallic, ceramic, etc.

Model Input Data – Plastic Properties

83

Data scattering for low density is due to reduced transducer sensitivity at lower stresses. Axial and radial stresses can be expressed as mean stress p and deviatoric stress q as follows: P = -σz z=h/2 (1 + 2α)/3

and

Q = -σz z=h/2 (1 - α )

Typical results are illustrated in Figure 6.8 for two types of powder. Q : Deviatoric stress

ductile powder hard powder

P : Mean stress Figure 6.8. Mean and deviatoric stress loading path during die compaction. A straight line is typical of hard materials.

6.2.1 Discussion of Assumptions A1,A2,A3 The slab equilibrium theory developed in the previous section requires that the friction coefficient and the stress ratio are independent of the axial coordinate and considers that there is no radial stress gradient. These assumptions are more or less satisfied depending on the aspect ratio H/R of the compact. When the aspect ratio decreases the radial stress gradient becomes more and more important compared to the axial one. At the opposite when H/R increases, axial stress variation due to powder/tool friction induces density variation along compact height. The dependence on density of friction coefficient and stress ratio (Figures 6.6 and 6.7) is similar to a dependence on the axial coordinate. Therefore the assumption that µ and α are independent of z is a first approximation especially in that case when H/R>>1. Let us try to express the axial density gradient to estimate the error introduced by these assumptions. Considering no radial stress gradient (A1) expression (6.4) can be rewritten as: dz = (R/2µ α)dσz/σz

(6.12)

The mean density ρm corresponds to the following expression: ρm = 1 H

∫

H 0

ρz dz

(6.13)

84

P. Doremus

or

ρm = 1 H

∫

σzms dσz ρz dz = 1 ρz R 0 H σzmi 2µα σ z H

∫

(6.14)

As the friction coefficient and the stress ratio are independent of the axial coordinate z, one can write:

σzms dσzm σ ρ (2Hρm µ α) / R = ρm Log zms = σzmi σzmi z σzm

∫

(6.15)

Differentiation of this expression leads to the following expression: Log (σzs/σzi) dρm = (ρs - ρm) dσzs/σzs + (ρm - ρi) dσzi/σzi

(6.16)

with ρs = ρ(z=H) and ρi = ρ(z=0) Assuming a constant density gradient, a = (ρs - ρi)/H along the compact height [5], this expression gives: Log (σzs/σzi) = (aH/2) (1/σzs dσzs/dρm + 1/σzi dσzi/dρm)

(6.17)

The density gradient can therefore be deduced from this expression provided that the upper and lower stresses and the compact height are measured during densification. The density gradient can be plotted as a function of the mean density from Equation 6.17. Density gradient as function of density

Density gradient

0

ρo

Density

ρt

Figure 6.9. Evolution of the density gradient during compaction as a function of the mean density

At low density the gradient starts from zero and, depending upon the type of powder, increases to a maximum (Figure 6.9). The gradient decreases as the density tends towards full density. For filling density, gravity induces very weak stresses and friction forces. Therefore, density variations or density gradients are

Model Input Data – Plastic Properties

85

small. At high stresses the density gradient decreases and must certainly be zero when full density is reached within the whole sample. This result allows local density along the compact height to be calculated, and the evolution to be predicted of the axial stresses in three cross sections: the upper section, the lower section and middle-height section (Figure 6.10). These three curves overlap very well.

Stress

Axial stresses as function of local density

upper axial stress lower axial stress middle height axial stress

ρo

Local density

ρt

Figure 6.10. Upper, lower and middle-height axial stress have the same evolution when plotted against local density

With ductile powders, pressure transmission ratio and friction coefficient both depend on density. Therefore, for these powders the assumptions above cannot be used to predict density gradients. 6.2.2 Influence of the Sample Aspect Ratio on Experimental Results When performing tests to generate material data, it is safe to assume that sample geometry should not affect intrinsic material parameters. By way of example we can look at varying the aspect ratio H/R. When H/R decreases the radial stress gradient certainly becomes more and more important compared to the axial one. In reverse, when H/R increases the friction coefficient and stress transmission ratio should depend on the axial coordinate or density. Let us examine the two cases: • High H/R This is the compaction of a cylinder. Obviously, calculation from the slab equilibrium holds: dσz/σz = (2µ α/R)dz

(6.18)

As before, µ is in this case the die/powder friction coefficient that will be denoted hereafter µdie and α=σr(r=R)/σz(r=R) will be denoted αcyl to reflect the case of the compaction of a cylinder. This leads to the expression:

86

P. Doremus

Log(σzs/σzi)= 2 µdie αcyl Η/R

(6.19)

• Low H/R This is the case of the compaction of a disc (Figure 6.11). Assuming the existence of a radial stress gradient (but no axial stress gradient), and also of friction between powder and the two punches (with a friction coefficient µ punch) the equilibrium of an annulus of thickness dr gives: dσz(r)/σz(r) = -2µ punch dr /α H

(6.20)

Figure 6.11. Stresses applied to an annulus of thickness dr

In this expression α = σr(r)/σz(r). Considering µ punch and α to be independent of the radial coordinate, this expression gives after integration:

σz(r) = σz(R) exp[2µ punch (R-r)/α H]

(6.21)

As σz(r) depends on “r” it is experimentally difficult to measure σz(R) and therefore to deduce α = σr(R)/σz(R). A force transducer fitted to the upper punch enables us to calculate αdisc = σr(R)/σzm where σzm is the mean axial stress that is deduced from the integral of σz over the punch surface. Of course, if α = σr(R)/σz(R) can be considered as an intrinsic parameter of the powder, this is not the case for αdisc , which is deduced from the mean axial stress. The ratio of the mean stress to the axial stress P/σz and the ratio of the deviatoric stress to the axial stress Q/σz are easily expressed with α as: P/σz =(2α+1)/3

and Q/σz = 1-α

As σz(R)<σz(r) (see expression 6.21) therefore σz(R)<σzm from which is deduced:

αdisc = σr(R)/σzm < α = σr(R)/σz(R) and (1+ 2αdisc)/3 < (1+ 2α)/3

1- αdisc > 1- α

This result, illustrated in Figure 6.12, shows that the way α is measured can explain the difference of the loading path between a cylinder and a disc, see Figure 6.13.

Model Input Data – Plastic Properties

87

Figure 6.12. Location of the state of stress for the compaction of a cylinder with friction acting on the die wall and compaction of a disc with friction acting on the two punch surfaces

Figure 6.13. Loading stress path for die compaction of different initial powder heights

To conclude, it is recommended to compact a cylinder (H/R>1) for measuring the stress transmission coefficient.

88

P. Doremus

6.3 Powder Characterisation from Triaxial Test Triaxial equipment is commonly used to study the mechanical behaviour of granular material. Generally, oil under pressure is used to apply radial stress to a cylindrical specimen consisting of powder enclosed in a rubber container. Axial stress is applied separately using an hydraulic press (Figure 6.14).

Figure 6.14. Triaxial cell showing two types of compaction, hydrostatic and triaxial compression. Such a rig is not truly triaxial since only two stresses are independent, the axial and radial stresses.

Commonly used facilities differ in the way in which the radial strain is measured and in the maximum pressure. The first method consists of deducing the radial strain from the sample volumetric change [7, 8]. The volumetric variation of the oil pressuring the powder is measured. The measure of the volume of the specimen in addition to its height allows its diameter to be calculated, making the assumption that the sample remains cylindrical during the test. The second technique consists of measuring the variation of the sample diameter using transducers placed inside the cell and in contact with the specimen [9]. These two techniques allow powder characterisation from low to high density along many different loading paths in the stress space, as illustrated in Figure 6.15.

Model Input Data – Plastic Properties

89

Figure 6.15. Different conditions of pressing in the mean and deviatoric stress space

In contrast to the rigid die test, powder-tool friction has no influence. The state of strain and stress are more homogeneous, avoiding the need for assumptions necessary in calculating from closed-die tests. The possibility of applying radial and axial stresses separately and independently makes triaxial equipment an outstanding facility more dedicated to exploring powder behaviour than calibrating a preselected constitutive model. On the other hand, powder characterisation from triaxial equipment is expensive and tests are time consuming compared to the closed-die method. The information that can be derived from triaxial tests includes powder critical state, plastic flow direction and dependence of yield surface on loading path. Critical state is reached after sufficient deformation so that the powder flows with a constant density and state of stress. The Cap-Model and the Cam-Clay model that have an associated flow rule describe this phenomena. The critical state of stress corresponds to the top of the yield surface of each model

Figure 6.16. Location of the critical-state point for Cam-Clay and Cap-Models having an associated plastic-flow rule

90

P. Doremus

An illustration of this phenomenon is represented in Figure 6.17 that shows the critical state of an iron powder tested under a constant radial pressure of 100 MPa and 200 MPa [10]. The critical deviatoric stress is 300 MPa and 340 MPa corresponding to a density of 7.1 g cm-3 and of 7.2 g cm-3 .

Figure 6.17. Evolution of density and axial stress as a function of the axial strain during triaxial compaction of an iron powder. When a 0.4 axial strain is reached, density and axial stress remains constant. The powder has reached its critical state.

The critical state of stress and the corresponding density are represented for an iron powder in Figure 6.18. One more advantage of triaxial equipment is the possibility of checking whether the powder plastic flow rule is associated or not. Figure 6.19 shows the case for an iron powder of yield surface from consolidated tests and the related isopotential curves. Stress space is superimposed on strain space so that the volumetric axis corresponds to the mean stress axis and the deviatoric strain to the deviatoric stress. In this representation the direction of the plastic flow is perpendicular to the isopotential curves. When isopotential and yield surfaces are the same curves the material has an associated flow rule or the material is said to be standard. In the illustration, in the grey zone, one can consider behaviour to be associated. By contrast, near the mean stress axis powder flow is nonassociated.

Model Input Data – Plastic Properties

91

Figure 6.18. Critical deviatoric stress and density against mean pressure for an iron powder

Figure 6.19. Yield surface and isopotential curves. In the grey zone the material can be considered as standard with an associated flow rule.

92

P. Doremus

Another phenomenon observed is the dependence of the form of the yield surface on powder-pressing conditions. As previously stated, the triaxial cell allows powders to be pressed following different paths (Figure 6.15) such as consolidated, over-consolidated loading, loading similar to die pressing (no radial strain), etc. [11, 12]. Figure 6.20 represents the yield surfaces for different particle shapes and for loading paths similar to closed-die compaction. Obviously they differ from yield surfaces deduced from consolidated tests (Figure 6.19 compared to Figure 6.20)

Figure 6.20. Yield stress surfaces for a triaxial die compaction (no radial strain) of four different types of powder

This page intentionally blank

Model Input Data – Plastic Properties

93

6.4 Concluding Comments This chapter has presented the closed-die test, which is certainly the test that is most widely used to analyse powder compaction and to determine plastic parameters for given constitutive models. The instrumented die is required to measure radial and axial stress. It is then also possible to measure the friction coefficient between the powder and die surface. Triaxial equipment is more complex and more expensive. Such equipment is used to analyse powder plasticity and elasticity, yield surface, state of strain, critical state, dependence of plastic curves on loading stress conditions, etc. It is convenient to use triaxial installations for comparing different constitutive models or in the development of a new model.

References [1] Heckel RW. 1961. An analysis of powder compaction phenomena, Trans. Of the Metallurgical Society of AIME, Vol.221, 1001-1008. [2] Kawakita K and Lûdde KH. 1970/71. Some considerations on powder compression equations, Powder technology, Vol.4, 61-68. [3] Ernst E, Thummler F, Beiss P, Wahling R and Arnhold V. 1991. Friction measurements during powder compaction, Powder Metallurgy International, Vol.23, N°2, 77-84. [4] Gethin DT, Ariffin AK, Tran DV and Lewis RW. 1994. Compaction and ejection of green powder coMPacts, Powder Met., Vol.37, N°1, 42-52. [5] Mosbah P, Bouvard D, Ouedraogo E and Stutz P. 1997. Experimental techniques for analysis of die pressing and ejection of metal powder, Powder Met., 1997, Vol.40, N°4, 269-277. [6] Doremus P, Toussaint F and Alvain O. Simple tests and standard procedure for the characterisation of green compacted powder in ‘Recent developments in computer modelling of powder metallurgy processes’, NATO science series, series III Computer and systems sciences, Vol.176, ISSN 1387-6694, 29-41. [7] Koerner RM. 1971. Triaxial Compaction of Metal Powders, Powder Metallurgy International, Vol. 3, No. 4, S. 186-188. [8] Doremus P, Geindreau C, Martin A, Debove L, Lecot R and Dao M. 1995. Powder Met., 38, (4), 284-287. [9] Sinka IC, Cocks ACF, Morrison CJ and Lightfoot A. 2000. Powder Metall., 43, (3), 253-262. [10] Pavier E and Doremus P. 1999. Triaxial characterisation of iron powder behaviour, Powder Met., Vol. 42, No. 4, 345-352. [11] Schneider LCR and Cocks ACF. 2002. Experimental investigation of yield behaviour of metal powder coMPacts, Powder Met, Vol.45, No. 3, 237-244. [12] Rottmann G, Coube O and Riedel H. 2001. Comparison between triaxial results and models prediction with special consideration of the anisotropy, EURO PM 2001 Nice, EPMA.

7 Model Input Data – Failure P. Doremus1 1

Institut National Polytechnique de Grenoble, France.

7.1 Introduction In general, green compacted components are fragile. Particles have weak mechanical links or are just agglomerated by a binder. Therefore, failure cannot be considered as a rare phenomenon, especially when new part geometries are being prepared for production. Of course, failure occurs very often during the unloading and the ejection stage of the process. Failure can also occur during compaction, after a certain level of densification or powder cohesion. Generally, such a technical problem is more difficult to solve. Failure is generally encountered for a state of stress corresponding to a high deviatoric stress but a moderate mean stress. Under such stresses the powder fractures without plastic strain. This behaviour is considered by the DruckerPrager-Cap model, more exactly by the failure line that is only one part of the yield stress surface (Figure 7.1).

Figure 7.1. Representation of the Drucker-Prager-Cap showing position of the failure line in the mean stress (P) and deviatoric stress (Q) plane. The failure line is defined by its angle β and cohesion d.

96

P. Doremus

The position of the failure line depends on the density of the compact, more especially on its cohesion d. Failure analysis can be performed using several different tests. These tests require specimens compacted with a certain level of cohesion in order to handle and fix them to the testing machine. Specimens are often made from powders compacted in a floating die. Three tests are well known: •

tensile test

•

diametral compression

•

simple compression.

Techniques for these three tests are described below as well as material parameters that can be derived.

7.2 Tensile Test The tensile test consists in pulling on a cylindrical sample with two opposite and coaxial forces (Figure 7.2). A very simple way of performing this test is to glue cylindrical compacts in holders using epoxy resin. Stress is applied using two cables to ensure forces are truly coaxial.

F

F Figure 7.2. Schematic of tensile test

Figure 7.3 represents the evolution of the fracture stress as a function of the relative density for two different diameters of specimen, 8 mm and 12 mm. Fracture must occur near the middle of the specimen to avoid any influence of the holders. Specimens compacted in a floating die are then more suitable. Stress levels are very low and do not exceed a few MPa whatever the powder is. Results are shown to be slightly dependent on the diameter. However, due to the quite large data scatter (15 %) compared to the other tests of this chapter, it is difficult to give any specific recommendation.

Model Input Data – Failure

97

Figure 7.3. Tensile fracture stress as a function of relative density showing the influence of specimen diameter

Denoting F the tensile force, σt is the failure stress that is derived from the expression: σt=F/(πD2/4). In the P-Q plane the state of stress is: P = -σt /3

Q = σt

In the plane of the mean stress and deviatoric stress the loading path is within the region of negative mean stress (Figure 7.4). Deviatoric stress

Loading path of tensile test

Mean stress

Figure 7.4. Stress locus of the tensile test in the mean stress and deviatoric plane

98

P. Doremus

7.3 Diametral Compression Test The diametral compression test is particularly used for comparing different materials. However the success of this test, due to its simplicity, makes it also useful for determining material parameters. Tests are carried out by applying two diametrically opposed forces on circular discs previously compacted in a die with different densities (Figure 7.5). Considering behaviour to be isotropic and linear elastic, a uniform tensile stress σd is developed along the loaded diameter [1, 2]:

σd = 2F/πDt

(7.1)

(where F is the applied load, D and t are respectively the diameter and the thickness of the sample). However, the compressive stresses acting on this diameter are not uniform, (minimum at the centre of the disc σo = -3 σd) inducing a non-homogeneous state of stress.

F

F Figure 7.5. Schematic drawing of diametral compression test

Generally, a disc diameter five times larger than disc thickness is needed to ensure plane stresses. Previous work [3] has shown that the type of failure and thus the failure stress depends on the shape of the surface used for loading the specimens. When the load is applied on a small surface using blotting paper for example or a large-diameter sample, compressive stresses are not too concentrated. This makes the test more homogeneous and prevents the compact from breaking under compressive stresses. Under such conditions a tensile failure initiated at the centre of the disc propagates rapidly along the loaded diameter (Figure 7.8). Figure 7.6 and Figure 7.7 represent failure stress as a function of relative density for various sample diameters and thicknesses. When the sample thickness is increased, the state of stress is no longer plane, affecting the failure stress by about 30 %. Finally, one can note that the tensile stress in a diametral compression test is perpendicular to the previous diecompaction direction applied to the powder. If the compact is anisotropic (induced by the die compression), the tensile stress could be different from the one obtained from a true tensile test.

Model Input Data – Failure

14

99

Failure stress (MPa)

12

Diameter: 16mm Diameter: 12mm Diameter: 8mm

10 8 6 4 2 0 0.65

0.7

0.75

0.8 0.85 Relative density

0.9

0.95

Figure 7.6. Failure stress for diametral compression showing the influence of specimen diameter

16

Failure stress (MPa)

14 12 10

thickness: 3mm thickness: 12mm thickness: 18mm

8 6 4 2 0 0.65

0.75

0.85

0.95

Relative density Figure 7.7. Failure stress for diametral compression showing the influence of specimen thickness

In conclusion, the diametral compression test must be carried out with sample aspect ratio t/D lower than or equal to 0.25. Moreover, it is important to ensure that failure arises from tensile fractures initiating at the centre of the specimen. Under such conditions material parameters can be determined using the two stresses:

100

P. Doremus

σd = 2P/πDt and σo = -3 σd The state of stress in the P-Q plane is: P = 2σd /3

Q=

13 σd

The loading path is represented in Figure 7.8. Deviatoric stress Loading path of diametral compression

Mean stress Figure 7.8. Stress locus of the diametral compression in the mean stress and deviatoric plane

7.4 Simple Compression Test The simple compression test is also extensively used for the determination of material parameters. It consists in applying two opposite and coaxial compressive forces (Figure 7.9). Spherical bearings are often used on one side of the specimen for controlling the applied forces. However it is important to take into account the specimen aspect ratio H/D (H being the specimen height and D its diameter) and the lubrication of the punches in determining the failure stress.

Figure 7.9. Schematic drawing of simple compression test

Model Input Data – Failure

101

Figure 7.10 shows failure stress as a function of mean relative density for different aspect ratios H/D. In each test a graphite sheet is placed between the sample and punches. 250

Failure stress (MPa)

200 150

H/D=1 H/D=1.5 H/D=2 H/D=2.5

100 50 0 0.65

0.7

0.75

0.8 0.85 Relative density

0.9

0.95

Figure 7.10. Failure stress as a function of mean relative density for different aspect ratios H/D

One can note that failure stress increases inversely to the aspect ratio H/D. During the test, compressive stresses applied on the specimen also induce shear stresses acting in the same way as a ring fitting the two ends of the specimen (Figure 7.11). This results in a nonhomogeneous state of stress increasing the failure stress level when the failure surface intersects the lower or upper cross section of the sample (H/D =1 or 1.5 see Figure 7.12). compressive stress Shear stress Failure line

Figure 7.11. Stress system acting on the two ends of the specimen during a simple compression

102

P. Doremus

Figure 7.12. Depending on aspect ratio, failure can intersect the ends of the specimen

When compacting samples of different heights to the same mean density in a floating die, the density in the middle cross section decreases as H increases due to die-wall friction. This can explain why the failure stress still decreases when H/D increases. The influence of the lubrication of the loading surfaces has been investigated with samples of aspect ratio H/D = 2 using graphite sheet, teflon, grease and no lubricant. Figure 7.13 represents the curves of failure stress as a function of relative density for various lubrication modes. The lubrication mode obviously does not have any influence on the values of failure stress even if the sample is directly in contact with the punches. In conclusion simple compression tests have to be carried out so that the failure does not cross the two ends of the sample. Such a condition is achieved with an aspect ratio H/D = 2 whatever the punch lubrication is. 250

Failure stress (MPa)

200 150

Without lubrication Teflon Grease Graphite sheet

100 50 0 0.65

0.75 0.85 Relative density

0.95

Figure 7.13. Failure stress in uniaxial compression for various modes of lubrication of the bearing surfaces

This page intentionally blank

Model Input Data – Failure

103

Denoting F the compression force, σc is the failure stress which is derived from the expression: σc =F/(πD2/4). In the P-Q plane the state of stress illustrated in Figure 7.14 is: P = σc /3

Q = σc

Deviatoric Loading path stress compression

of

diametral

Mean stress

Figure 7.14. Stress locus of the simple compression test in the mean stress and deviatoric plane

7.5 Concluding Comments This chapter has presented the main tests used for determining failure resistance of green compacts. Samples of different geometries are easily made and tested. Results of tensile tests and simple compression tests can be interpreted without assumptions. This is not the case for the diametral compression test that needs some assumptions (e.g. samples have an isotropic linear elastic behaviour). As previously mentioned in Chapter 5, compact elasticity is neither linear nor isotropic. However, the simplicity of this test gives its advantages over tests for other properties of powder compacts.

References [1] Timoshenko SP and Goodier JN. 1970. Theory of elasticity, McGraw-hill, New York. [2] Frocht MM. 1947. Photoelasticity, John Wiley and Sons, New York, 107-111. [3] Fell JT and Newton JM. 1970. Determination of tablet strength by the diametral compression test, J. Of Pharmaceut Sci, 39, N°5, 688-691.

8 Friction and its Measurement in Powder-Compaction Processes D.T. Gethin1, N. Solimanjad2, P. Doremus3 and D. Korachkin1 1

School of Engineering, UW Swansea, Singleton Park, Swansea SA2 8PP, UK. Hoganas AB, S-263 83 Hoganas, Sweden. 3 Laboratoire Sols, Solides, Structures, BP 53X, Domaine Universitaire, 38041 Grenoble Cedex, France. 2

8.1 Introduction Friction is present between all surfaces in mechanical contact and its importance depends on the application. For example, friction is essential in many powertransmission applications, but conversely needs to be minimised in power generation. In powder-compaction processes, friction is present between particles in contact and between the powder and tool surfaces. Relevant to process simulation within a continuum framework, interparticle friction is included in the measurement of powder-yielding behaviour and this is discussed fully in Chapter 6. The current chapter focuses on friction that is present between the powder mass tool set and the following is intended to highlight its importance. The presence of friction between the powder and tool surfaces is a critical consideration throughout the powder-compaction cycle. During the compression stage, it has a major impact on the generation of density gradients within the compact. For example, Figure 8.1 shows a plain cylinder that is compacted under conditions of low and high friction between the tool surfaces. For this shape, there is little difference in the form of the density variation, but differences exist in the density range and between top and bottom punch forces. Friction is also important during the unloading and ejection stages within the cycle. During the unloading stage, the punch loads are relaxed and elastic recovery of the press and tool components takes place. The compact itself also experiences some recovery, however, this will be resisted by the presence of friction between the powder and tool set surface. Friction is present due to the existence of residual stresses within the compact that have a component that is normal to the tool surface. At the end of the unloading phase, there will be a plane within the compact, above which the friction force between the powder and tool surfaces will equal the recovery force within the compact. For example, for the cylindrical compact shown in Figure 8.2, only a section towards the top of the compact springs

106

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

back since the recovery force (A) up to some plane within the compact exceeds the friction force (S) that is present over the zone above this plane [1].

72.57kN

77.58kN

68.36kN µ = 0.12

72.50kN µ = 0.001

Figure 8.1. The effect of friction on the compaction of a plain cylinder

S

Compacted section

A

Recovered section

Unrecovered section

Figure 8.2. Compact recovery on relaxation of compression loads

During ejection, the residual stresses normal to the die-wall surface interact with the friction that is also present, leading to a force that resists the ejection of the part. For simple cylindrical parts, friction can lead to further densification during the ejection of parts having a high aspect ratio [2]. For multilevel parts, consideration of friction is important in defining the ejection sequence because as tools are progressively “stripped” from the compact, the residual stresses are relaxed and the forces of ejection attenuated accordingly. Figure 8.3 shows an

Friction and its Measurement in Powder-Compaction Processes

107

example “stripping” sequence. An inappropriate sequence selection can lead to compact failure through cracking and in extreme cases, delamination. Clearly, the effect of friction between the powder and tool set surface is of critical importance and the current chapter will focus on ways of measuring it. There are a number of techniques that may be used. The main methods comprise instrumented-die and shear-plate techniques. The current chapter will present and discuss these approaches. Alternative and less common techniques will also be presented.

Compaction

Unload

Strip die

Ejected Part

Strip lower punches Figure 8.3. Example ejection sequence

8.2 Friction Measurement by an Instrumented-die The instrumented-die may be used to measure both yield and frictional behaviour of a powder. The principles of yield surface determination are set out in Chapter 6,

108

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

the current section focuses on derivation of friction information from this test. The experiment makes use of the difference that exists between top and bottom punch stresses (reflected as measured forces) that exists in a simple uniaxial compression test. The stresses that exist within the compact and on a disc within the compact are shown in Figure 8.4.

z

σua σr

σr

σla

σr

τar

σua

τar

σr

σla Figure 8.4. Stress balance on a plain cylindrical compact

Friction is derived from the application of a force balance on the disc that is integrated over the height of the compact [3], leading to Equation 8.1.

µ=

D (ln σ zt − ln σ zb ) 4kL

(8.1)

where: µ – coefficient of friction between powder and tool surface D – diameter of powder compact L – length of the compact σzt – axial stress on top σzb – axial stress on bottom

k=

σr σz

A similar approach may be applied to a flat disc in which the pressure on the top and bottom faces varies over the radius, leading to:

µ=

kH (ln po − ln p ) 2r

(8.2)

Friction and its Measurement in Powder-Compaction Processes

109

where: µ – the coefficient of friction H – the height of the compact po – axial pressure acting at the centre of the compacting area (i.e. at r=0) p - axial pressure acting on a certain radius r r - radius There are practical difficulties associated with investigating friction using these methods. It is necessary to know both the radial and axial stresses or the relationship between them. This is more practical for long cylinders, since experiments can facilitate radial stress measurement. This is less feasible for an experiment with a flat disc since ideally it is necessary to know or be able to measure the radial stress variation. The surface-finishes are either fixed or not easy to change. This makes it difficult to investigate parameters such as material hardness and die surface-finish. Thus a crucial requirement of the instrumented-die is the possibility to measure the radial stress on the compact. A number of designs may be proposed for doing this and three variants are shown schematically in Figure 8.5. The first is a simple cylindrical die that is instrumented with strain gauges to detect hoop stress on the outer surface of the die. The second comprises a thick cylinder that supports a liner [4]. The radial stress is measured through instrumented pins that are preloaded against the liner. The third method uses a pin that passes through the die-wall and is maintained flush against the wall by means of an appropriate closed-loop control circuit [5]. The latter is a particularly sensitive system that has been used principally for the characterization of powder at low pressure. Where possible, it is usual that a number of such transducers are fitted over the die height. For the former two methods, the choice of wall/liner thickness depends on the powder that is being measured and the consequent likely excursions in radial pressure. The sensitivity of the instrument is determined by the die-wall thickness and this must be balanced against the likely dilation of the die. Die-wall thickness may be selected based on a thick-cylinder-type calculation, or through a more complex finite-element analysis of the die.

110

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

LINER

Upper punch Powder

DIE L. V. D. T.

Strain gauges Strain gauges

DIE pin with strain gauge

Die Lower punch

11

[[4] ] die air vent

transducer air chamber

measured gap bracket

brass cylinder

pin

[5] Figure 8.5. Methods for radial-pressure measurement

A key practical requirement is the calibration of the radial-stress measurement within the die. As the powder is compressed, the difference between top and bottom axial stress also infers a radial stress gradient over the compact height [3]. Also, the radial-pressure measurement must account for the change in compact height and the consequent impact on the signals that are detected by either the strain gauges or pins that are located at fixed positions. The most common method of performing a calibration is to compact an elastomer plug that corresponds to the die diameter. Because of its incompressibility, it behaves hydrostatically and thus enables a direct measure of radial pressure on the die-wall. In performing this calibration, the elastomer may be lubricated to minimize friction effects at the diewall and it is usual to check calibration quality through measuring top and bottom punch loads. Under satisfactory calibration conditions, these loads are virtually

Friction and its Measurement in Powder-Compaction Processes

111

identical. To account for height changes that take place during compaction, elastomer plugs of different height are used to perform the calibration. The final calibration (σr) is then a function of hoop strain (εθθ) (or pin load) as measured by all transducers as a function of height (h), i.e.

σr = f (h ) εθ θ

(8.3)

8.3 Friction Measurement by a Shear-plate The shear-plate type apparatus provides a more direct measure of friction and also allows wider exploration of factors that affect friction. There are a number of possible configurations that include, for example, pin on disc, caliper and shearplate designs. The design of a typical shear-plate equipment is shown in Figure 8.6 [6]. Essentially, it comprises a die that is closed at one end by the target surface and powder is compacted to the required density by a top punch. Subsequently, the target surface that is mounted within a trolley running on linear bearings, is moved under the compact and the lateral force required to move it is recorded. The design of such equipment requires that only a very small clearance (typically 0.01 mm) exists between the die support and target plate, otherwise powder will be extruded through the gap. Force transducers should be designed to reflect the anticipated loads so that apparatus sensitivity is maximized. Normal force

Top punch

Die Powder compact Load cell

Die support Target surface

Shear force

Linear bearings

Base plate

Figure 8.6. Shear-plate apparatus

112

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

Inherent to the design shown in Figure 8.6 is the loss in pressure between the top and bottom of the compact due to friction between the compact and die-wall. For an accurate evaluation of friction, it is essential that this is taken into account when calculating the normal force at the target surface. This can be dealt with through the application of Equation 8.1, but this requires knowledge of the radial to axial stress ratio for the powder and this is derived typically from an instrumenteddie experiment. This will be expanded through illustration in Section 8.4.2. The problem associated with frictional losses within the die can be overcome through the use of a floating die within the shear-plate equipment [7]. A typical design is shown in Figure 8.7. In this instance, the die is supported on a split ring for the compaction stage of the process. The ring is then removed and the compact loaded against the target surface at a normal load that corresponds to the top-punch force. For both instrumented-die and shear-plate equipment, data is normally acquired through computational means and imported into spreadsheet software to facilitate data reduction. A typical experiment can comprise several thousand data points in order to capture the behaviour details during the experiment. Normal force

Top punch

Supporting ring Powder compact

Floating die Linear bearing Load cell

Die support Target surface

Shear force

Linear bearings

Base plate Figure 8.7. Floating-die shear-plate equipment

8.4 Example Measurements 8.4.1 Instrumented-Die Experiments Calibration is a key process in the application of an instrumented-die. Figure 8.8 shows a typical calibration curve performed for five elastomer lengths where the

Friction and its Measurement in Powder-Compaction Processes

calibration parameter represents

σr εθ θ

113

, Equation 8.3. In this instance, the radial

stress was recorded using two pairs of active strain gauges that formed part of a full Wheatstone bridge configuration. Both active gauges recorded hoop strain and a pair of gauges were used to maximize signal sensitivity. In this instance, five pairs of active gauges were used. There is not a requirement for using five elastomer calibration samples for five gauge sets, the technique will also work with fewer measurements of hoop strain as well as measurement using a single strain gauge at each location. The choice depends on resolution and sensitivity requirements. In conducting an instrumented-die experiment it is appropriate to check the calibration from time to time to assure test stability and hence results quality.

Figure 8.8. Example calibration curve for radial stress measurement using five elastomer plugs

Figure 8.9 illustrates the record of the axial and averaged radial stress states within the powder sample during an instrumented-die test. The top and bottom axial stress differ due to die-wall friction throughout the compression from 60 to 33 mm. During compression from 60 to 48 mm, the bottom and radial stress are nearly identical. On further compaction, the radial stress lies below the bottom axial stress for this powder type. In computing the ratio of axial to radial stress as required by Equation 8.1, it is usual that the top and bottom axial stresses are averaged. This is appropriate only if the variation over the compact height is linear, however, if the difference between these is not too significant, which is typical of lubricated powders, then this is an acceptable simplification.

114

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

Figure 8.9. Typical axial and radial stress evolution during an instrumented-die test

Figure 8.10 shows the variation of die-wall friction coefficient as a function of radial stress. The data may also be presented as a function of sample average density, The choice depends on the requirements of simulation software that may use such correlations. The variation shows a characteristic form. Many powders exhibit a high level of friction at the early stages of compaction (corresponding to low radial stress), reducing to lower levels as the compaction process proceeds. The levels exhibited often depend on the powder type, lubricant content and target surface properties, such as hardness, material and lubrication.

Figure 8.10. A typical friction characteristic from a typical powder

Friction and its Measurement in Powder-Compaction Processes

115

8.4.2 Shear-Plate Experiments The shear-plate experiment yields data that exhibit generic characteristics and thus initially, it is appropriate to present and explain typical output from a single experiment [6]. For this purpose, alumina compacted using a 12.5 kN load against a hardened D2 tool steel target surface of 0.02 µm Ra has been selected. In performing these experiments, it is usual to repeat each a number of time to establish repeatability and furthermore to “lubricate” the target surface. Starting from a clean target, experience shows that typically three runs are required to stabilize the results through “contaminating” the surface with lubricant extruded from the powder. Figure 8.11 shows the punch force and displacement history during the compaction and shearing process and Figure 8.12 shows the corresponding shear force and carriage-displacement histories. Figure 8.11 confirms that compaction force and punch position remained constant over the shear stage duration, whereas Figure 8.12 confirms a constant sliding velocity of about 0.2 mm/s and the existence of both static and dynamic friction mechanisms. The shear friction force may also be plotted as a function of carriage displacement and a typical representation is shown in Figure 8.13. The latter depicts the static friction clearly and dynamic value that increases slightly with displacement. Experience has shown that many powders behave in this way. 0:00 0:30 1:00 1:30 2:00 2:30 3:00 3:30 4:00 4:30 5:00

-5

15 Punch Force

12

-10

9

-15

6

-20

Punch Displacement

-25

Force (kN)

Displacement (mm)

0

3 0

Time (mm:ss)

Figure 8.11. Punch-force and punch-displacement history (time axis is in minutes and seconds)

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

10

2 Displacement

Displacement (mm)

8 6

1.6 1.2

Shear Force

4

0.8

2

0.4

Shear Force (kN)

116

0 0 0:00 0:30 1:00 1:30 2:00 2:30 3:00 3:30 4:00 4:30 5:00 Time (mm:ss)

Figure 8.12. Shear-plate force and displacement history

The conditions and powder used in this experiment have led to repeatability within ±2 %. However, in detail, some systems may not exhibit such consistent behaviour and, furthermore, the dynamic friction can show a strong increase with displacement. This can lead to difficulties in the identification of an appropriate dynamic value. This may be overcome for example by recording dynamic friction at a set displacement and comparing values at this displacement [6]. Friction variation over large displacements will be discussed further in Section 8.5.2.

Shear Force (kN)

2 1.5 1 0.5 0 0

2

4

6

8

10

H o ri z o n ta l D i sp l a c e m e n t (m m )

Figure 8.13. Shear force against displacement at 12.5 kN normal force

The static and dynamic shear forces can be determined for a range of normal forces, as measured at the top punch and an example graph is shown in Figure 8.14. In this instance, the relationship is clearly linear and therefore the friction coefficient is constant.

Friction and its Measurement in Powder-Compaction Processes

117

Shear Force (kN)

2.5 y = 0.1446x

2 1.5

Static Shear Dynamic Shear

1 y = 0.0768x

0.5 0 0

5

10

15

20

Normal Force (kN)

Figure 8.14. Normal force against shear force (hardened D2 tool steel 0.1 Ra)

As explained above, when reducing the data from this experiment it is also necessary to account for the friction effects at the die-wall. As a crude indicator and in the absence of data from an instrumented-die test, it is possible to assume that the powder behaves elastically, then Equation 8.1 can be manipulated to give

Fb = Ft e

−4 µ ν L D (1−ν )

(8.4)

where Fb – axial force on the bottom Ft – axial force on the top Use of Equation 8.4 now requires knowledge of Poisson’s ratio that may, for convenience, be based on elastic property data. A more rigorous approach is to use information from an instrumented-die test in conjunction with Equation 8.1. The following illustration assumes a Poisson’s ratio, ν, of 0.3 and this leads to a stress transmission ratio (

σr σz

) of 0.43. The friction coefficient (µ) chosen is

appropriate for a die steel material and is 0.1. Figure 8.15 shows graphs of shear force against normal force using both corrected and uncorrected normal forces. The difference in gradients between the two sets of data is the difference between the calculated coefficients of friction. Crucially, the differences are significant, underlining the importance of including this in the analysis of data from this type of experiment.

118

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

Shear Force (kN)

2.5 2 Corrected Static

1.5

Uncorrected Static Corrected Dynamic

1

Uncorrected Dynamic

0.5 0 0

5

10

15

20

Normal Force (kN)

Figure 8.15. Comparison between corrected and uncorrected forces

A number of benchmarking trials have been carried out in which shear-plate and instrumented-die equipments have been compared for identical powder systems [8]. The conclusion from this work is that the shear-plate equipments provide close agreement, whereas there is considerably more spread between the data from instrumented-die tests, attributed principally to instrument sensitivity and the methods that are used for data reduction.

8.5 Factors that Affect Friction Behaviour There are several factors that can affect the level of friction that exists between the powder and die surface. As well as the combination of powders and target surfaces, it will be affected by Surface-finish quality and its orientation, the hardness of the surface, the presence and level of lubricant addition and the distance over which sliding takes place [9]. It is important to look at both static and dynamic effects since the dynamic friction forces are important during compaction, and the static friction forces are particularly important during the ejection phase, since generally the compaction will halt before the part is ejected from the die. The trends that will be presented in the following paragraphs are relevant for a typical lubricated alumina powder [6]. 8.5.1 Surface Properties 8.5.1.1 Surface-finish Surface-finish effects may be illustrated through the trends that are present when a tool steel hardened to 64 Rockwell is used as the target surface. Figure 8.16 shows graphs of comparisons between static and dynamic forces for two series of tests, capturing the influence of roughness and its orientation.

Friction and its Measurement in Powder-Compaction Processes

119

For this powder, the variations are mapped by linear functions with a reasonable level of accuracy and the gradient gives a direct measure of the friction coefficient that is constant over the range of applicable force levels that are used in compaction. In the top two graphs the roughest surfaces were aligned perpendicular to the direction of sliding, in the bottom two graphs they were aligned parallel to the direction of sliding. Figure 8.17 shows images of the ends of the alumina compacts after testing against the four different target surfaces. The images shown were taken using an optical microscope at 50 times magnification.

Figure 8.16. Graphs of results of series of tests on alumina powder against all target surfaces. From top left to bottom right, static friction roughest surfaces perpendicular, dynamic friction roughest surfaces perpendicular, static friction roughest surfaces parallel, dynamic friction roughest surfaces parallel.

The tests against the smoother surfaces show evidence of the particles sliding over the surface, and retaining the same open structure. The tests against the rougher surfaces show evidence of particle shearing, which effectively seals the compact, and the contact area between the powder and die is again present over the complete surface. Similar results were reported in [10].

120

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

Figure 8.17. Alumina compacts tested against hardened tool steel at 12.5 kN (magnified 50 times). Top left Ra=0.02 µm, top right Ra=0.1 µm, bottom left Ra=0.5 µm perpendicular, bottom right Ra=1.0 µm perpendicular.

8.5.1.2 Material Hardness Die surface hardness effects may be illustrated through the use of a range of surfaces from soft (mild steel), untreated tool steel, tool steel heat treated to 64 Rockwell C and tungsten carbide. For this purpose, the target surfaces were finished to have a roughness average of 0.02 µm.

Figure 8.18. Static (left) and dynamic (right) shear force for target surface materials of varying hardness (Ra = 0.02 µm) for alumina compacts

The results for both static and dynamic friction are shown in Figure 8.18. For the hardest surface, the tungsten carbide, there is not a large difference between the static and dynamic friction forces. The difference between these forces for the next hardest surface, hardened tool steel, is greater, and greater still for the untreated tool steel. Whilst the difference between the static and dynamic friction forces was quite large for the two tool steel surfaces; these are material characteristics rather

Friction and its Measurement in Powder-Compaction Processes

121

than a mechanism of particle shearing as discussed in connection with surfacefinish effects [10]. This is supported by the observation that for the complete test series, these surfaces showed no evidence of degradation due to particle abrasion. Figure 8.19 shows the static and dynamic coefficients of friction against the hardness of the target surface material. Both static and dynamic coefficients reduce with increasing hardness.

Coefficient Of Friction

0.35 0.3 0.25 0.2

Static Dynamic

0.15 0.1 0.05 0 0

500

1000

1500

2000

Vickers Hardness

Figure 8.19. Coefficient of friction against hardness for alumina powder

The trends in Figure 8.19 confirm the benefit of increasing surface hardness and its impact on the friction coefficient between the powder and die-wall. It has the most impact on the static value and therefore will be most important at the commencement of ejection. The close agreement between static and dynamic levels of the tungsten carbide surface also suggests that this will eliminate the static and dynamic effects from the compaction process and clearly leads to the case of a minimum level of friction that will promote the uniformity of density throughout the compact. 8.5.1.3 Surface Treatments Surface treatments may be applied to the tool surfaces in the form of coatings or through lubrication. Coatings may be used to enhance tool life through wear reduction and may have additional benefits with respect to friction. Lubricant application has a direct impact on friction reduction and also provides the opportunity to reduce (or remove) the lubricant from within the powder mixture. However consistent die-wall lubricant application presents many practical difficulties, especially for complex part shapes comprising many levels. This concerns coverage completeness and the small quantities that need to be applied consistently. Example results that show the effect of surface treatment will be presented in a following section. Figure 8.20 shows the combined effect of admixed and die-wall lubricant on the static and dynamic friction coefficient for a ferrous powder [7]. The friction coefficient has been averaged from a number of tests conducted at different loads

122

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

and therefore summarises the impact of different levels of admixed lubricant and the benefit that may be gained through the application of direct die-wall lubrication. Clearly die-wall lubrication has significant benefits with regard to friction reduction, its consistent application in a tool set remains as a processdesign challenge.

Average Friction Coefficient

0.2

0.15 No Die Wall Lubricant, Static Friction No Die Wall Lubricant, Dynamic Friction

0.1

Die Wall Lubricant, Static Friction Die Wall Lubricant, Dynamic Friction

0.05

0

Figure 8.20. Combined effect of admixed and die-wall lubricant

8.5.2 Compact and Process Influences For given surface properties friction can also be influenced by compact density or process parameters as normal force, sliding distance, displacement speed, temperature, etc. Figure 8.21 shows a linear shear-plate device that is capable of measuring friction over an extended sliding distance and over a range of speeds. The purpose of exploring frictional behaviour over extended sliding distances is associated with the ejection stage of the cycle. The equipment also facilitates friction measurement at speeds that approach pressing conditions since the extended displacement facilitates acceleration of the target surface under the compact sample. The application of this device has led to a number of types of friction evolution as a function of sliding distance and these are shown schematically in Figure 8.22. The characteristics displayed in Figure 8.22b and Figure 8.22c are most applicable in powder-compaction. Preceding results have shown the existence of a high static coefficient, followed by a dynamic value that is either close to, or lower than the static level.

Friction and its Measurement in Powder-Compaction Processes

123

Figure 8.21. Extended shear-plate friction measurement device [11]

Figure 8.22. Generic form of friction evolution [9]

The influence of powder density and normal stress can be investigated independently using the shear-plate equipment. This is not the case with the instrumented-die test since density and stress are linked through the compressibility property of the powder. Figures 8.23a and b show these influences for an iron-based powder sliding on a slab made of tungsten carbide.

124

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

Figure 8.23. Influence of density (a) and normal stress (b) on the friction coefficient (a) constant normal stress 150 Mpa, (b) constant density 7.3 g/cm3

The friction coefficient decreases as density and normal stress increase. This is in agreement with the data of Figure 8.10 that corresponds to closed-die test measurement. Temperature and sliding speed influences are illustrated in Figures 8.24 a and b for an iron powder containing 1 % wax as lubricant. The powder and slab are heated to the same temperature ranging from 20 °C to 80 °C. At the upper temperature it is expected that softening of the lubricant will start to occur and that it will start to become more effective. Under industrial production, tool-surface heating occurs due to friction effects during the compaction and ejection stages. When considering the industrial speed of compaction or ejection (30 mm/s) as well as powder and tooling temperature (70 °C), these are likely to raise the powder temperature leading to further melting of the wax and ultimately to a reduction in the friction coefficient.

Friction and its Measurement in Powder-Compaction Processes

125

Figure 8.24. Influence of temperature (a) and sliding speed (b) on the friction coefficient. (a) normal stress 150 MPa, sliding speed 10 mm/s, density 7.3 g/cm3. (b) density 7.02 g/cm3, normal stress 150 MPa, temperature 20 °C.

8.6 Other Friction Measurement Methods Figure 8.25 shows a frictional measurement device that relies on a rotational principle in that friction between the powder and die surface is obtained from a torque measurement[12]. It is possible to carry out two kinds of experiments using this equipment. In the first one, the pressing of the powder and rotation of the die can occur simultaneously. In this type of experiment the powder particles are exposed to shearing against the punch surface (a solid surface corresponding to the die wall), which is similar to what happens in the real compaction process, where powder particles are forced to shear along the die wall. The measurement of the normal force, friction force and punch position occurs continuously during this process. In the second type of experiment the powder is compacted to a desired density before the rotation of the die. This kind of experiments is carried out at a

126

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

constant density and normal load that is similar to the ejection process. Parameters such as sliding distance (for which there is no limit), sliding velocity, etc. can be investigated easily. To the hydraulic

Hollow shaft motor

Punch Strain gauges Powder Die Core Hole for wires Thermocouple

Figure 8.25. Friction measurement using a rotational principle [12]

Due to the die confinement, the equipment is capable of measuring friction over a range of densities and Figure 8.26 shows some example results in terms of torque applied to the powder sample and time [13]. The high initial torque represents the static friction and the lower constant value its dynamic counterpart. The variation represents the characteristic form that has been presented previously.

Friction and its Measurement in Powder-Compaction Processes

127

160 b)

140

6.2 g/cc 5.9 g/cc 5.7 g/cc 5.6 g/cc 5.3 g/cc 5.2 g/cc 5.0 g/cc 4.8 g/cc 4.6 g/cc 4.4 g/cc 4.3 g/cc 4.1 g/cc 4.0 g/cc

Torque [Nm]

120 100 6.2 g/cm

80

3

5.9 g/cm3 60 40 20 0 1000

2000

3000

4000

Testing time [ms], (Sliding distance)

Figure 8.26. Static and dynamic friction from a rotational shear test [13]

8.7 Relevant Bibliography Due to its high relevance to the powder-compaction process, there have been a number of published works describing scientific investigations in this field. This work has been done over an extended period, one of the earliest systematic studies being reported in 1972 [14]. A full review of all aspects relating to friction in powder-compaction is a significant challenge and, furthermore, the interest in doing so will be influenced by the powder-processing sector. However, a selected number of more recent publications are set out in the bibliography at the end of this chapter.

8.8 Concluding Comments The chapter has described and set out the principles of a range of techniques for measuring friction between powder and a tool surface. Within the two thematic networks, Modnet and Dienet there have been a number of studies to draw the information from a number of trials together as well as characterisation experiments on a number of typical powders drawn from different powder types. The results form these studies and experiments are set out in Appendix 1.

128

D.T. Gethin, N. Solimanjad, P. Doremus and D. Korachkin

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

Gethin DT, Lewis RW and Ariffin AK. 1995. Modelling Compaction and Ejection Processes in the Generation of Green Powder Compacts; AMD-Vol 216, Nett Shape Processing of Powder Materials, ASME. Mair G. Correlation between Axial Density and Friction in the Compaction and Ejection of PM Shapes; Metal Powder Report, 46, p60-64. Bocchini GF. 1995. Friction Effects In Metal Powder-compaction Part One Theoretical Aspects, Adv Powder Metall Partic Mater, Vol. 1, No. 2. Guyoncourt DMM, Tweed JH, Gough A, Dawson J and Pater L. 2001. Constitutive Data and Friction Measurements of Powders Using Instrumented-die; Powder Metallurgy, 44, 25-33. Cocks ACF. Private communication. Cameron IM, Gethin DT and Tweed JH. 2002. Friction Measurement in Powder Die Compaction by a Shear-plate Technique; Powder Metallurgy, 45, 345-353. Korachkin D, Gethin DT and Tweed JH. 2005. Friction Measurement using a Floating Die Measurement Equipment; European PM Congress, Prague. PM Modnet Research Group, 2002. Numerical Simulation of Powder-compaction for two Multilevel Ferrous Parts, Including Powder Characterisation and Experimental Validation; Powder Metallurgy, 45, 335-344. Bonnefoy V, Doremus P and Puente G. 2003. Investigations on friction behaviour of treated and coated tools with poorly lubricated powder mixes; Powder Metallurgy, 46, 224-228. Strijbos S. 1976. Friction Between A Powder Compact and A Metal Wall, Science of Ceramics, Vol. 8, 415 – 427. Doremus P, Toussaint F and Pavier E. 2001. Investigation of Iron Powder Friction on a Tungsten Carbide Tool Wall, Powder Metallurgy, 44, No 3, 243-247. Solimanjad N. 2003. A New Method for Measuring and Characterisation of Friction at a Wide Range of Densities in Metal Powder-compaction, Powder Metallurgy, vol. 46, No.1, 49-54. Solimanjad N and Larsson R. 2003. Die-wall Friction and the Influence of some Process Parameters on Friction in Iron Powder-compaction, Materials Science and Technology vol. 19, issue 9, 49-54. Mallender RF, Dangerfield CJ and Coleman DS. 1972. Friction Coefficients between Iron Powder Compacts and Die-wall during Ejection Using Various Admixed Zinc Stearate Lubricants; Powder Metallurgy, 15, 130-152.

Chapter Bibliography The following are some key articles that describe studies on friction and friction measurement in powder-compaction. The list is not exhaustive, but includes publications, principally in the journals Powder Metallurgy and Powder Technology. [1] [2]

Mallender RF, Dangerfield CJ and Coleman DS. 1974. The Variation of Coefficient of Friction with Temperature and Compaction Variables for Iron Powder Stearate Lubricated System; Powder Metallurgy, 17, 288. Ward M and Billington B. 1979. Effect of Zinc Stearate on Apparent Density, Mixing and Compaction/Ejection of Iron Powder Compacts; Powder Metallurgy, 22, 201-208.

This page intentionally blank

Friction and its Measurement in Powder-Compaction Processes [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

129

Tabata T, Masaki S and Kamata K. 1980. Determination of the Coefficient of Friction Between Metal Powder and Die-wall in Compaction; Journal of Plasticity, 21, 773776. Tabata T, Masaki S and Kamata K. 1981. Coefficient of Friction Between a Metal Powder and Die-wall During Compaction, Powder Metallurgy International, 13, 179182. James BA. 1987. Die-wall Lubrication for Powder Compacting: A Feasible Solution; Powder Metallurgy, 30, 273. Amin KE. 1987. Friction in Metal Powders; International Journal of Powder Metallurgy, 23, 83. Ernst E, Thummler F, Beiss P, Wahling R and Arnhold V. 1991. Friction Measurements During Powder-compaction, Powder Metallurgy International, 23, 77– 84. Li Y, Liu H and Rockabrand A. 1996. Wall Friction and Lubrication During Compaction of Coal Logs; Powder Technology, 87, 259-267. Pavier E and Doremus P. 1997. Friction Behaviour of an Iron Powder Investigated with Two Different Apparatus, International Workshop on Modelling of Metal Powder Forming Processes, Grenoble, 345 - 350. Doremus P and Pavier E. 1998, Friction: Experimental Equipment and Measuring; Proceedings of World Congress of Powder Metallurgy 1998. Briscoe BJ and Rough SL. 1998. The Effect of Wall Friction on the Ejection of Pressed Ceramic Parts; Powder Technology, 99, 228-233. Turenne S, Godère C, Thomas Y and Mongeon P-É. 1999. Evaluation of Friction Conditions in Powder-compaction for Admixed and Die-wall Lubrication; Powder Metallurgy, 42, 263-268. Iacocca RG and German RM. 1999. The Experimental Evaluation of Die Compaction Lubricants Using Deterministic Chaos Theory; Powder Technology, 102, 253-265. Wikman B, Solimannezhad H, Larsson R, Oldenburg M and Häggblad H-Å. 2000. Wall Friction Coefficient Estimation Through Modelling of Powder Die Pressing Experiment; Powder Metallurgy, 43, 132-138. Turenne S, Godère C and Thomas Y. 2000. Effect of Temperature on the Behaviour of Lubricants During Powder-compaction; Powder Metallurgy, 43, 139-142. Lefebvre LP and Mongeon PÉ. 2003. Effect of Tool Coatings on Ejection Characteristics of Iron Powder Compacts; Powder Metallurgy, 46, 43-48. Sinka IC, Cunningham JC and Zavaliangos A. 2003. The Effect of Wall Friction in the Compaction of Pharmaceutical Tablets with Curved Faces: a Validation Study of the Drucker–Prager Cap Model; Powder Technology, 133, 33-43.

9 Die Fill and Powder Transfer S.F. Burch1, A.C.F. Cocks2, J.M. Prado3 and J.H. Tweed4 1

ESR Technology Ltd, 16 North Central 127, Milton Park, Abingdon, Oxfordshire, OX14 4SA, UK. 2 University of Oxford, Department of Engineering, Oxford, UK. 3 Universidad Polytechica de Catalunya, Diagonal 687, 08028 Barcelona, Spain. 4 AEA Technology, Gemini Building, Harwell, Didcot, Oxfordshire, OX11 0QR, UK.

9.1 Introduction Initial work on compaction modelling assumed that the density of the powder bed in a die at the start of compaction is uniform. This has been challenged from two directions. Firstly, measurements of the density distribution in compacted components did not always agree with predictions and nonuniform fill-density has been used as one explanatory factor [1]. Secondly, recent work on measuring the density distributions of powder beds in filled dies has suggested that variations of the order of 10 % to 15 % in local powder density can be developed [this chapter]. The potential significance of such variations has been assessed virtually by Korachkin and Gethin [2], using a compaction model. It was assumed for axisymmetric parts that two regions with densities differing by 10 % may exist at the start of compaction. For given compaction kinematics, punch loads and density distributions in compacted parts were calculated and compared with the case of uniform density and the same mass of powder. For simple parts with one column of powder, final loads and densities did not depend on the initial density distribution. However, for stepped parts, realistic, a non-uniform initial density led to much larger variations in the density distribution and higher punch loads for given compaction kinematics. In this chapter measurement of density distribution in a filled die by two techniques is illustrated. Both are based on a laboratory rig that simulates the major features of industrial die-filling systems. In the first case, the filled die is examined by X-ray computerised tomography to provide fine-scale detail of the effects of filling practice on density distributions in filled dies. In the second case, the powder bed in the die is lightly sintered, to facilitate sectioning and metallographic preparation. Density distributions are then estimated by metallographic techniques.

132

S.F. Burch, A.C.F. Cocks, J.M. Prado and J.H. Tweed

Modelling of die-filling is at an early stage of development. Some initial work is illustrated and an approach based on discrete element modelling shows good agreement with experimental observations. Approaches to further development are discussed.

9.2 Potential Sigificance of Die Fill-density Distribution In this section we examine the importance of the initial density distribution through the description of a computational study based on the compaction models described in Chapter 4. Computational simulations have been undertaken by Korachkin and Gethin [2] to evaluate the sensitivity of the final density distribution and punch loads on the initial density distribution. This study was motivated by experimentally observed initial density distributions determined using the techniques described in the main body of this chapter. This sensitivity study was performed for Distaloy AE, tungsten carbide and zirconia powders. A material characterisation process was performed to obtain model parameters for the three powders. A two-level part was used as the main shape for the study. For Distaloy AE powder a second two-level part and hollowcylinder geometries were also considered. In each case the following density distributions were considered: • •

uniform fill-density two fill regions differing in density by 10 %, but with the same total mass as for the uniform fill case.

The second case is based on measured fill-density variations for a two-level part (Figure 9.1). Two idealizations of the distribution shown in this figure were evaluated as illustrated in Figure 9.2, with the boundary between the low- and high-density regions either separated by a line at 45° or 0° to the horizontal, emanating from the corner of the transition. +9% relative to mean

-8% relative to mean Figure 9.1. X-ray CT density variations for a powder filled into a stepped ring-shaped die

The study showed that for simple shapes, such as hollow cylinders, there is little variation in final density distribution even for different vertical fill-density distributions with the same total mass of powder. It is still likely, however, that, where there is fill-density variation around a ring, this will be retained in the pressed part. For more complex stepped-die geometries the final density range is a lot higher, as a result of nonuniform fill-density distribution. An increase in tool forces was

Die Fill and Powder Transfer

133

also observed for nonuniform fill compacts for the conditions considered in this study (Figure 9.2). This study illustrates the importance of understanding the initial density distribution in a die and its effect on subsequent stages of the manufacturing process. In the remainder of this chapter we focus on the filling process and describe a range of techniques to characterize the density distribution in a filled die. Fill-density

1.1*ρ1

1.1*ρ2

3.15 g/cm3

ρ1

a)

ρ2

b)

c)

Press density

Load (kN)

129

166

175

Figure 9.2. Predictions from compaction model of variations in compact density and top punch load arising from variations in fill-density, assuming the same punch movements in each case. The neutral axis is at the change in section.

9.3 Die-filling Rig In this section we describe how a model die-shoe filling rig [3] can be used to investigate the flow behaviour of typical industrial metal, ceramic, hardmetal and magnetic powders. At present, flowability is evaluated in various ways in industry, but correlation of results generated by the different techniques is difficult to achieve. Comparison of different techniques shows that ranking the materials in order of their flowability often leads to different nonconclusive results. For example, zirconia has excellent flow characteristics when tested using a Hall flowmeter [4], however, it can be regarded as a poor-flowing powder when the model filling system employed in this programme is used, as shown later. It is evident that different features of the flow behaviour are measured in these different types of experiment. A flow-measurement system that mimics the major features of the industrial process of interest is therefore better suited to model the flow

134

S.F. Burch, A.C.F. Cocks, J.M. Prado and J.H. Tweed

behaviour. This chapter investigates how results generated using a die-shoe filling system can be used to provide information that can be employed to aid the design of die-filling systems. The flow properties of a powder depend on a combination of powder characteristics and operating conditions. Typical powder characteristics are particle size, composition, size distribution, shape and the loose and dense random packing densities. The operating conditions influence the way in which the particles interact with each other, with the shoe and with the die. Relevant environmental and operating parameters include relative humidity or moisture content, temperature, static charge, aeration and internal forces arising from gravity, air pressure, external loading, vibrations and the constraint imposed by the containers in which flow takes place. These conditions dictate the behaviour of the powder when it flows through a hopper orifice or during delivery into the die of a compaction press. A large number of experimental techniques have been developed to determine the flow behaviour of particulate materials. Physical measures such as the angle of repose and flow rate [5], critical aperture [4, 6], Hausner ratio [7] or Carr index [6] have been developed to characterise the flow properties. Other flow measures can be determined from avalanche studies, low-pressure triaxial cells [8], low-pressure instrumented dies [9] or shear cells [10]. Recently, Wu et al [3], have proposed the use of the concept of a critical velocity, i.e. the velocity above which incomplete filling of a standard die occurs in one pass of the shoe over the die, as a means of determining the flowability of a powder. A comprehensive overview of flowmeasurement techniques is given by Wu and Cocks [11]. The flow-measurement methods listed above can be used to characterise and classify the flow properties of powders. It was pointed out earlier, however, that these methods may give inconsistent classifications for a given material. It is therefore important that the flow characterisation is carried out using a device that captures the physical phenomena involved in the process under consideration. Hopper flow has been investigated extensively and many of the flowcharacterisation methods described above are employed to assist hopper design. As mentioned before, flow into closed dies has, however, received less attention and it is most likely that measures of flowability that are based on a die-shoe filling system are better suited to this situation compared to the other flow-measurement techniques, since it more closely resembles the industrial die-filling process. In order to use the results generated from a filling rig to aid the design of an industrial process, it is important to understand the flow mechanisms that operate and how these can change with changes in the die and shoe geometries, volume of powder handled and the kinematic operating conditions. Any numerical results generated from a rig must therefore be supplemented with information about the flow mechanisms that dominate under the experimental conditions employed. Within this study a model die-shoe filling system originally developed by Wu et al. [3] has been employed to study the flowability of powders under conditions that mimic those experienced during an industrial powder-delivery process. Use of transparent dies and shoes allows the details of the flow process to be observed using high-speed video. The original rig developed by Wu et al [3] is shown in Figure 9.3. The shoe is attached to a linear pneumatic actuator and the kinematics

Die Fill and Powder Transfer

135

of its motion are prescribed using a digital controller. A wide range of different types of motion can be programmed. In most experiments, it proves convenient to accelerate the shoe up to a steady-state velocity, which is maintained as the shoe passes over the die opening. Multiple passes can also be programmed and a series of shakes can be superimposed on the basic motion. Initial tests using this system demonstrated the importance of airflow on the filling process. If the bottom punch is in the lower position before filling commences, the die cavity is initially full of air. As powder flows into the die, air must escape. If there is no easy path for the air to escape, the air pressure can build up in the die, opposing the inward flow of powder, thus resulting in a slower net rate of filling. The system was constructed so that it could be placed inside a vacuum chamber. Comparison of tests in air and vacuum reveals the effect of air on the delivery process. A second rig has also been constructed. This is much larger than the original experimental rig and allows the use of shoes and dies that are representative of industrial systems. The system is also robust and contains temperature and humidity sensors that automatically log the environmental conditions during a test. These rigs were employed in each of the test programmes described in this chapter. We concentrate on two major aspects of the delivery process: the effect of features within a die on the filling process and the resulting density distribution; and the influence of multiple passes on the variation of density in the die.

control unit

pneumatic drive unit

shoe

die Figure 9.3. Die-filling rig, showing transparent shoe and die. The shoe is moved using a linear pneumatic actuator that is controlled using a digital programmable controller.

136

S.F. Burch, A.C.F. Cocks, J.M. Prado and J.H. Tweed

9.4 The Flow Behaviour of Powder into Dies Containing Steplike Features The kinematics of the flow process are sensitive to the microscopic properties of the powder and the kinematics of the shoe motion. The flow patterns also depend on the location of any features in the die with respect to the direction of shoe motion, and whether the experiments are conducted in air or vacuum. A series of experiments were conducted using stepped-dies of the type illustrated in Figure 9.4. The top section of the cavity is 30 mm wide, while the introduction of the step creates a narrower section of width 5 mm. A full description of the flow behaviour of various powders and how this depends on the location of the step and the environment (air or vacuum) is published elsewhere [3].

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9.4. Flow of DAE into a stepped-die in vacuum with the narrow section on the right at a shoe speed of 200 mm/s. The arrow in (c) indicates a forward concave shrink zone.

Figure 9.4 shows a sequence of images for the flow of a ferrous powder Distaloy AE (DAE) - into a stepped-die with the narrow section on the right. The experiment was conducted in vacuum at a shoe speed of 200 mm/s. At this particular shoe velocity, the flow pattern is not very sensitive to the location of the step. As the shoe reaches the die, powder particles are projected onto the ledge of the step creating a forward concave shrink zone adjacent to the nearside wall of the die, Figure 9.4. As the pile formed on the step grows, powder is fed into the narrow section as particles cascade over its surface (Figure 9.4b). Eventually, as the pile builds up further, the powder mass eventually bridges across the narrow section (Figures 9.4 c to e). Powder is now fed into the narrow section by detaching from the bottom of the bridge. This results in a slower flow rate into the narrow section than observed initially. The bridge can translate upwards as material detaches from it. Eventually, the narrow section becomes completely full as the upper surface of the deposited material merges with the mass of powder in the wider channel above

Die Fill and Powder Transfer

137

it (Figure 9.4f). Preliminary computational studies using the discrete element method reveal that the density is generally lower at the interface between these merged flow regimes [12]. As the narrow section fills with powder the upper right corner of the die fills by particles cascading backwards along the surface of the pile created on the ledge, which is exaggerated by the flow of powder into the narrow section. The initial stages of experiments conducted in air at a shoe velocity of 200 mm/s are similar to those described above, but as soon as the pile formed on the step spreads across the narrow opening, air becomes trapped in this section and the pressure rapidly builds up. The details of the subsequent flow process now depend upon the location of the step with respect to the direction of shoe motion – see Figures 9.5 and 9.6.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9.5. Flow of DAE into a stepped-die in air with the narrow section on the right at a shoe speed of 200 mm/s. The arrow in (c) indicates a forward concave shrink zone.

We initially consider the situation where the narrow section is upstream of the direction of shoe motion, i.e. it is on the right (see Figure 9.5). The powder initially builds up near the corner of the step (Figure 9.5a). As a heap forms in the corner, powder particles cascade over the surface back towards the edge of the step and into the narrow section (Figure 9.5b). The opening is quickly bridged (Figure 9.5c). Air initially breaks through the thin bridge formed and a chimney effect is observed. The chimney is quickly suppressed, however, by the incoming powder stream and an air bubble is formed that moves upwards by particles detaching from

138

S.F. Burch, A.C.F. Cocks, J.M. Prado and J.H. Tweed

the top and falling to the bottom surface (Figures 9.5 d to f). This continues until the powder bed above it is thin enough for the air to escape by pushing the powder away. A further variation on the flow pattern is observed when the narrow section is located on the left side of the die (see Figure 9.6). The powder first builds up as a heap on the ledge and then cascades down the surface towards the edge of the step as the shoe moves forward, where it falls into and then bridges over the narrow section. When the bridge first forms there is less powder in the narrow section of the die than when the narrow section is on the right. The intensity of the forward cascading motion over the surface of the heap into the leading corner increases, and a stronger chimney forms (Figures 9.6c and d) since more air is entrapped in the narrow section and a stronger air pressure builds up in a short period of time. Eventually, material is deposited directly above the chimney creating a bridge towards the top of the die that is sustained by the air pressure built up beneath it (Figure 9.6e). Powder is delivered gently into the void created as material detaches and falls from the surface of the bridge.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9.6. Flow of DAE into a stepped-die in air with the narrow section on the left at a shoe speed of 200 mm/s. The arrow in (c) indicates a forward concave shrink zone.

Die Fill and Powder Transfer

139

9.5 Metallographic Techniques for Determining Density Variations The same powder used in the study described above was used to fill metallic dies in which inserts were placed to create two narrower sections on the left and right of the die of widths 6 mm and 3 mm, which we refer to as the bottom wide and narrow sections, respectively, Figure 9.7. The die was cut in half along a plane parallel to the plane of the image in Figure 9.7 and the two parts were glued together, so that they could be readily split after sintering. The die was filled at a specified shoe speed with powder using the model shoe system shown in Figure 9.3. The filled die was then placed in an electrically heated furnace to lightly sinter the sample. The transfer process was carefully controlled to ensure that there was no settlement of the powder as the die was moved from the delivery system to the furnace. The furnace temperature was raised from atmospheric temperature to 900 o C at a rate of 5 oC/min. The temperature was then held at 900 oC for one hour, after which, it was reduced to room temperature at a rate of 5 oC/min. To avoid oxidation, nitrogen gas was pumped into the furnace at a flowrate of 173 cm3/s for half a hour before the furnace was switched on. The purpose of this operation is to expel the air from inside the furnace. Throughout sintering, nitrogen gas was continuously pumped into the furnace at a flowrate of 40 cm3/s. This light sintering process allows sufficient diffusion to occur for the particles to bond together at their points of contact, but there is minimal diffusional rearrangement of material. As a result, there is no change of density and the volume occupied by the powder remains constant throughout the sintering process. After sintering, each die was split into two halves with a small blade. The sectioned samples were then impregnated with resin in vacuum to hold the particles in place so that they could be prepared using standard metallographic techniques. Care was taken when polishing the samples to ensure that there was minimal pullout of particles from the surface. An optical microscope and a continuous-grab camera were used to capture the images of the polished sections, which showed the detailed arrangement of the particles. Preliminary tests revealed that a magnification of 100 produces consistent results with negligible variations. With such a magnification, the actual region captured in each image is 0.84 x 0.62 mm, which contains ca. 350 particles. This magnification was used to grab images for the results reported in this chapter. For each cross section, images were grabbed at 1 mm x 1mm intervals. The grabbed images were digitalised and analysed using an image analysis package (ImagePro Plus). For each image, the relative density, which is defined as the solid fraction of particles, was determined and a density map was produced. After creating the map the surface was ground back by about 1 mm and then 2 mm, with the above procedure followed after each grinding operation to create 3 density maps. The results presented here were determined by averaging over these three maps. Figure 9.7 shows the density distribution for the situation where the shoe moves from left to right at a speed of 200 mm/s. The different contours correspond to different relative densities; the darker the shading, the lower the density. It is clear that a low density is observed in two areas: at the trailing top edge of the cavity; and at the corner of the step adjacent to the bottom narrow section. The

140

S.F. Burch, A.C.F. Cocks, J.M. Prado and J.H. Tweed

former corresponds to the location of the concave shrink zone formed during diefilling, Figure 9.5, which is filled by the gentle backward flow of powder during the final stages of the filling process. The latter corresponds to the location of the interface between two different types of flow as powder deposited in the narrow section merges with the mass of powder above it, as observed in Figures. 9.5 and 9.6. A fuller evaluation of these results is given by Wu et al [13]. Direction of shoe motion

Figure 9.7. Density map for a die containing an insert that creates two lower narrow sections. The map is for the situation where the shoe moves from left to right.

Figure 9.1 shows the density variation using X-ray computerized tomography for an axisymmetric profile with a central core rod. This image shows a similar density profile to the map of Figure 9.7 with a lower density in the narrow section than in the larger body of the component.

9.6 Measurement of Die Fill-density Distribution by X-Ray Computerised Tomography X-ray computerised tomography (CT) is a nondestructive inspection technique that provides cross-sectional images in planes through a component [14]. The principle of third-generation CT imaging, as used in the industrial context, is illustrated in Figure 9.8. The component is placed on a precision turntable in the divergent beam of X-rays generated by an industrial X-ray source. A detector array (line or area array) is used to measure the intensities of the X-ray beam transmitted through the component, as the component is rotated in the beam. A mathematical algorithm

Die Fill and Powder Transfer

141

[15] is then used to generate (or “reconstruct”) the CT images from the measured transmitted intensities. For 2D area detectors, 3D CT data can be obtained from a single rotation of the component. Cone beam reconstruction algorithms [16] are then needed to compute the 3D CT dataset. The resultant CT images are true cross-sectional images, and show the geometry of the component in the plane of the cross-section. If an X-ray source with a very small size (microfocus source) is used, then the spatial resolution achievable can be very high (ca. 10 µm for mm-sized components). The CT image values (grey levels) provide information on the material X-ray attenuation coefficient at each point in the image. There is considerable current interest [17] in the correction for a number of effects, including especially “beam hardening”, which would allow the CT grey levels to be converted to values that are directly proportional to the local material density. Density measurements in powder compacts have been carried out since the early 1900s [18] and include techniques based on differential machining, hardness tests and X-ray shadows of lead grids placed in the compact. More modern techniques available to characterise compact microstructure were summarised in [19] and include: X-ray CT, acoustic-wave velocity measurements and nuclear magnetic resonance imaging. X-ray CT has been applied to characterise density distributions in powder compacts in various fields [19 to25].

R o tatio n

X - ray so urce

Co m po nent D etecto r

Figure 9.8. Principle of third-generation industrial X-ray computerised tomography (CT)

9.6.1 Hardware Components Needed for X-Ray CT Cabinet-based systems for real-time radiography contain suitable X-ray sources and detectors for industrial X-ray CT, and can be upgraded to provide a CT capability by addition of a precision turntable and a PC with appropriate image

142

S.F. Burch, A.C.F. Cocks, J.M. Prado and J.H. Tweed

acquisition cards, motor control capabilities and software for image reconstruction and display, etc. For the majority of work described in this chapter, the hardware comprised a 160 kV or 225 kV microfocus X-ray cabinet system, which included a detector consisting of an image intensifier optically coupled to a CCD camera generating real-time digital radiography images (typically 1024 x 1024 pixels resolution, with 12-bit dynamic range). The CT capabilities were provided by a TOMOHAWK CT upgrade system [22], which included a precision turntable and digital image acquisition card. 9.6.2 Technique for the Quantitative Measurement of Density Variations For the quantitative measurement of density variations within industrial components using X-ray CT, a number of specialised extensions to the standard technique were developed as follows. Firstly, in order to minimise the effects of scattered radiation, collimators close to the X-ray source and the detector can be used to obtain a narrow fan-beam of X-rays. However, this then restricts the vertical field of view and prevents 3D data from being acquired during a single component rotation. Alternative methods, including a single pinhole source collimator can then be used, although residual scatter levels are then higher than with the full dual-slit collimator technique. It is also necessary to measure and then correct for the non-linear effects of beam hardening. For each material, calibration discs were manufactured at constant density from the same material, and the X-ray attenuation was then measured for different thicknesses of the material. This allowed a beam-hardening calibration curve to be drawn up, as illustrated in Figure 9.9 for steel powders. A mathematical function was then fitted to the measured calibration points, which allowed the attenuations from the test object to be “linearised” [14], and hence corrected for beam hardening effects.

Figure 9.9. Measurements of beam hardening and fitted function that allows correction of the data by “linearisation”

Die Fill and Powder Transfer

143

Quantitative density maps can be obtained by applying corrections for the nonlinear effects described above. The X-ray CT technique is applicable to components predominantly composed of one material, in which the local density varies with location within the component, such as powder compact materials. Xray CT has been validated against quantitative density maps obtained using nuclear magnetic resonance imaging for pharmaceutical materials [26]. 9.6.3 Results for Die Fill Density Distribution Using X-Ray Computerised Tomography A series of CT measurements of filled dies, containing loose powder, prior to compaction was carried out. Initially, a rectangular die made of low X-ray attenuation material (perspex) was constructed, and filled with ferrous Distaloy AE powder. The effect of varying the orientation of the long axis of the die with respect to the direction of the fill-shoe motion was examined. In addition, experiments were made using different numbers of passes of the fill shoe over the die. During the rotational CT scanning it was ensured that any settling of the loose powder was negligible. Vertical CT slices obtained from 3D CT data are shown for four different cases in Figure 9.10, which include both die directions parallel and orthogonal to the shoe-fill direction and fills achieved with 1 and 20 passes of fill shoe. In Figure 9.10, it can be seen that there is a clear increase in powder density, by about 10 %, near the top of the die for 20 pass cases, but for the 1-pass cases the effects are much less. This density increase for the multipass cases can be attributed to friction and shear effects that act over a short distance below the shoe/die interface (the fill shoe was immediately above the top of the die). Validation of the above results was achieved by comparing the overall measured powder density for the filled dies, with the average material density derived from the CT data. The correlation achieved is given in Figure 9.11, which shows that the root mean square (rms) deviation of the CT densities from the best fit straight line was only 0.6 %.

144

S.F. Burch, A.C.F. Cocks, J.M. Prado and J.H. Tweed

Fill from 1 shoe pass, die long axis parallel to shoe motion

Fill from 1 shoe pass, die axis orthogonal

Fill from 20 shoe passes, die axis parallel

Fill from 20 shoe passes, die axis orthogonal

Figure 9.10. Vertical CT sections from a rectangular die filled with Distaloy AE powder under four conditions (lighter colours denote higher densities)

The work with rectangular dies was extended to a ring die with 25 mm outer diameter and 2 mm wall thickness. X-ray CT density information was averaged through the wall thickness of the die. Positions around the die surface were then converted to polar coordinates, with 0 degree being the portion of the die which is first filled by the fill shoe on its first pass, and 180 degree being the portion that the fill shoe crosses last on its first pass. Polar density plots are presented in Figure 9.12. The fill-shoe speed of 90 mm/s fills the die in one pass, whilst a speed of 200 mm/s requires more than one pass for filling. All the plots show an axis of symmetry in the direction of shoe motion.

Die Fill and Powder Transfer

145

Figure 9.11. Mean powder density plotted against the mean CT derived density, derived from the 3D CT data from the whole die. The line shows the best least squares fit through the origin.

a) One pass at 90 mm/s

b) Two passes at 200 mm/s

c) Six passes at 200 mm/s

d) Ten passes at 200 mm/s

Figure 9.12. Polar CT sections from a ring die filled with Distaloy AE powder under four conditions (lighter colours denote higher densities)

With increasing numbers of passes, the density of the powder at the top of the die increases significantly, presumably due to the shearing action of the powder remaining in the fill shoe. The magnitude and depth of this effect is greater at 90 degree and 270 degree locations where the length of interaction between powder in the die and fill shoe is greatest. The three tests with a shoe speed of 200 mm/s all show increased density at the 0 degree position at about half the die height. This corresponds with the powder surface after the first shoe pass, as illustrated in Figure 9.13 – this shows powder surfaces for four tests after one shoe pass at 200 mm/s.

146

S.F. Burch, A.C.F. Cocks, J.M. Prado and J.H. Tweed

0

180

360

0

180

360

Test No. 2

Test No. 1 0

180

360

Test No. 3

0

180

360

Test No. 4

Figure 9.13. Polar plots of powder height in a ring die after one shoe pass at 200 mm/s. The shoe first crosses the die at 0/360 degree and last crosses the die at 1800 degree.

A technique combining the use of a laboratory filling system and computerised tomography was used to provide an initial die fill-density distribution for one of the case studies reported in Chapter 13 (see, for instance, Figure 13.14).

9.7 Modelling of Die-filling The experiments and characterisations described above provide insights into the relationship between the details of the filling process and the initial fill-density distribution in the die. Eventually, it will be important to be able to develop predictive models of the die-filling process so that a much wider range of geometric shapes and flow conditions can be evaluated. Here, we describe a preliminary attempt to do this based on the discrete element method. In the discrete element method (DEM), the particles are treated as individual entities with Newtonian motion in a gravitational field. The fundamental equations governing the motion of an individual particle in 2D are mi

Ii

d 2xi dt 2 d 2θ i dt 2

= mi g + f c + f a = Mi

(9.1) (9.2)

where mi and I i are the mass and the moment of inertia of particle i, x i and

θi

are the position of the centroid of the particle and its angular orientation, g is the gravitational acceleration and t is the time; f c , f a and M i are the contact force, the force due to air pressure and the moment acting on particle i. It is obvious that

Die Fill and Powder Transfer

147

f c , f a and M i have to be determined prior to a complete solution for particle i being obtained. The stage in which f c , f a and M i are determined is referred to as contact modelling, which can be subdivided into two phases: (1) contact detection and (2) modelling of the interaction between two particles. The task of the former is to detect any potential contacting pairs of particles, while that of the latter is to determine the contact force between a pair of contacting particles. Both stages become more complicated when irregular-shaped particles (e.g., polygons) are considered, compared to the modelling of circular or spherical particles. The procedures for determining f c and M i are outlined by Gillia and Cocks [27] and a simple procedure for taking into the account of air pressure is described by Wu et al [12]. Integration of the equations of motion given by Equations 9.1 and 9.2 to determine the projectile of the particles is also discussed by Gillia and Cocks [27]. In this section we present simulations of the filling of the stepped-dies illustrated in Section 9.4. We concentrate on the situation where a die is filled in vacuum. Simulations that include the effect of air-pressure buildup are described by Wu et al [12]. From the DEM simulations, the total mass fed into the cavity and the density distributions inside a cavity can be determined, since the trajectory of each particle can be traced. The powder flow patterns for the situation considered in Figure 9.7 are shown in Figure 9.14.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9.14. DEM simulation of powder flow into a stepped-die at a shoe speed of 200 mm/s. The black arrows in (c) and (d) indicates a forward concave shrink zone. The white arrow in (c) indicates a merge surface.

148

S.F. Burch, A.C.F. Cocks, J.M. Prado and J.H. Tweed

For these simulations, 2000 particles were used and the particle-wall and interparticle friction coefficients were set to 0.15 and 0.5, respectively. It can be seen from Figures 9.14 that the flow into the wider section is smoother than that into the narrower section. Arching over the narrow section is observed in both cases (see Figure 9.14c). In addition, a forward concave shrink zone adjacent to the top trailing side is apparent. This zone appears to be filled mainly by an arching and breaking process as a result of the large particle size. This indicates that arching is more likely to occur around this zone. This is in broad agreement with the experimental observations described above. Figure 9.15 shows the corresponding relative density distribution after diefilling, in which the relative density is determined by dividing the die into a grid and calculating the area fraction of solid particles in each cell. The distribution is smoothed to create contour plots of the relative density inside the die.

Figure 9.15. Density variation in the filled die of Figure 9.14f

It can be seen from Figure 9.15 that the high relative densities are generally produced in the top wide region, although significant density gradients are observed near the top surface due to boundary effects. It is interesting to find that even in the top wide region, the relative density in the forward concave shrink zone is relatively low. This is because the formation of arching during die-filling resists the packing of powder into these zones. This phenomenon is also observed in laboratory experiments described in Section 9.4 in which the filling density was quantified using metallographic techniques. The formation of arching generally results in a low-density zone being developed underneath it because it slows down the flow of powder. Arching is also responsible for the lower density produced in the bottom narrow regions. On the other hand, if arching significantly reduces the flow intensity, a particle that detaches from the arch can fall and roll into an equilibrium position before other falling particles have the opportunity to interact with it. Under these conditions a dense packing can also be achieved. This effect is demonstrated in Figure 9.15, which shows that a higher-density zone is developed in the bottom narrow region. Nevertheless, arching at the top of the narrow section also leads to a merge plane being developed. Where the two flow regions merge a lower density is obtained as observed experimentally in Section 9.5. This is potentially detrimental as this lower-density zone may persist during subsequent

Die Fill and Powder Transfer

149

stages of the manufacturing process. Consequently, since the low-density zone develops adjacent to a corner of the finished component, defects such as cracks are more likely to develop in this region [28]. A more detailed comparison of the DEM studies with the experimental results is given by Wu and Cocks [12]. They also evaluate the effect of particle shape and die-wall friction on density variations within a die.

9.8 Concluding Comments In this chapter we have described two experimental techniques for the determination of density distributions within filled dies; metallographic techniques and X-ray computerised tomography. These have been used to determine density distributions in dies with a range of geometric profiles using a model die-filling system. The system also allows the delivery process to be observed using highspeed video, thus allowing the detailed flow mechanisms to be identified as well as the relationship between the observed mechanisms and the density distribution profiles observed at the end of the delivery process. Preliminary studies have been presented of DEM simulations of die-filling. The flow mechanisms and density distributions replicate those that are observed in the experimental studies. The DEM approach offers significant potential in modelling the filling process with the resulting density distributions used as input into finite-element studies of the compaction process. This modelling approach is described more fully by Coube et al [28].

References [1]

Kergadallan J et al. 1997. Compression of an axisymmetric part with an instrumented press, Proc. Int. Workshop on Modelling of metal powder forming processes, Grenoble, July 1997, 277–285. [2] Korachkin D and Gethin DT. 2004. An exploration of the effect of fill-density variation in the compaction of ferrous, ceramic and hard metal powder systems, AEAT/LD81000/05, November 2004. [3] Wu CY, Dihoru L, Cocks ACF. 2003: The flow of powder into simple and steppeddies. Powder Technol, 134, 24-39. [4] Guyoncourt DMM and Tweed JH. 2003. Measurement of powder flow, Proc PM2003, Vol 3 23-28, EPMA. [5] Carr RL. 1965. Evaluating the flow properties of solids. Chem. Eng., 72, 163-168. [6] German RM. 1994. Powder metallurgy science. Metal Powder Industries Federation, Princeton New Jersey. [7] Hausner HH. 1967. Friction conditions in a mass of metal powder. Int. J. Powder Metallurgy, 3, 7-13. [8] Li and Puri VM. 1996. Measurement of anisotropic behaviour of dry cohesive and cohesionless powders using a cubical triaxial tester. Powder Technology, 89, 197-207. [9] Schneider LCR and Cocks ACF. 2005. Development and test results of a low pressure instrumented die. To appear in Powder Metallurgy. [10] Jenike AW. 1964. Storage and flow of solids. Bulletin 123, Engineering and Experiment Station, University of Utah, USA.

150

S.F. Burch, A.C.F. Cocks, J.M. Prado and J.H. Tweed

[11] Wu CY and Cocks ACF. 2004. Flow behaviour of powders during die-filling, Powder Metall., 47, 127-135. [12] Wu CY and Cocks ACF. 2006. Numerical and experimental investigations of the flow of powder into a confined space, Mechanics of Materials, 38, 304-324. [13] Wu CY, Dihoru L and Cocks ACF. An experimental investigation of the variation of packing density of powder particles in filled dies, J. Mater. Proc. Tech., to appear. [14] Kak AC. 1979. Computerized Tomography with X-ray, Emission and Ultrasound Sources, Proceedings of the IEEE, Vol 67, No 9, pp 1245-1272. [15] Mersereau RM. 1974. Digital Reconstruction of Multidimensional Signals from Their Projections, Proceedings of the IEEE, Vol 62, No 10, pp 1319-1338. [16] Feldkamp LA, Davis LC and Kress JW. 1984. J. Opt. Soc. Am., 1 612. [17] Phillips DH and Lannutti JJ. 1997. Measuring physical density with X-ray computed tomography, NDT&E International, Vol 30, No 6, 339-350 [18] Train D. 1957: Transmission of Forces through a Powder Mass during the Process of Pelleting; Trans. Instn. Chem. Engrs., Vol 35, pp.258-266. [19] Lannutti JJ. 1997: Characterisation and control of compact microstructure, MRS Bulletin, Vo. 22, No. 12, pp3/8-44. [20] Lin CL and Miller JD. 2000: Pore structure and network analysis of filter cake; Chemical Engineering Journal, Vol. 80, pp.221-231. [21] Kong CM and Lannutti JJ. 2000: Localised Densification during the compaction of Alumina granules: the stage I-II transition; J. Am. Ceram. Soc., Vol. 83, No. 4, pp.685-690. [22] Burch SF. 2001. Measurement of density variations in compacted parts using X-ray computed tomography; Proc. EuroPM2001, October 22-24, Nice, France, pp.398-404. [23] Li W, Nam J and Lannutti JJ. 2002: Density gradients formed during compaction of bronze powders: the origins of part-to-part variations; Metallurgical and Materials Transactions A, Vol. 33A, January, pp165-170. [24] Nielsen SF, Poulsen HF, Beckmann F, Thorning C and Wert JA. 2003: Measurements of plastic displacement gradient components in three dimensions using marker particles and synchrotron X-ray absorption microtomography; Acta Materialia, Vol. 51, pp.2407-2415. [25] Sinka IC, Burch SF, Tweed JH and Cunningham JC. 2004: Measurement of density variations in tablets using X-ray computed tomography; International Journal of Pharmaceutics, Volume 271, Issue 1-2, pp. 215-224. [26] Sinka IC, Djemai A, Burch SF, and Tweed JH. Characterisation of density distribution in tablets using X-ray CT and NMR imaging, to be submitted to European Journal of Pharmaceutical Sciences. [27] Gillia OT and Cocks ACF. Modelling die-filling, submitted [28] Coube C, Cocks ACF and Wu C-Y. 2005. Experimental and numerical study of diefilling, powder transfer and die compaction, Powder Metall., 48, 68-76.

10 Calibration of Compaction Models P. Doremus1 1

Institut National Polytechnique de Grenoble, France.

10.1 Introduction Several constitutive equations are available for modelling powder densification. They have been presented in Chapter 4. Phenomenological constitutive equations are widely used to simulate forming processes. They are calibrated to predict as well as possible the mechanical aspects of the densification at a macroscopic level whilst ignoring physical aspects at a microscopic level. This chapter deals with the calibration of the Drucker-Prager-Cap model and the Cam-Clay model (which are the two models that are most frequently integrated in finite-element codes). Several methods are available for determining the values of the parameters incorporated in these models. One way consists of fitting models from the simplest and minimum characterisation tests. This method is based on tests described Chapters 5, 6 and 7. Another possible method, supported by triaxial tests is also described. A comparison of methods is made at the end of this chapter.

10.2 Calibration of the Drucker-Prager-Cap model The Drucker-Prager-Cap is an elastic-plastic model, the hardening parameters being the volumetric plastic strain that can be replaced by the density when elastic strains are negligible. 10.2.1 Elasticity Due to experimental facilities and because generally only isotropic linear elastic models are integrated into finite-element codes, the complete determination of the elastic behaviour needs two parameters, Young’s modulus E and Poisson’s ratio ν. As has been said in Chapter 5 - input data elasticity - the true elastic behaviour is nonlinear and nonisotropic. Moreover, elastic parameters depend on the density and state of stress. Examples of calibration are given in Appendix 1.

152

P. Doremus

10.2.2 Calibration of Yield Stress Surface and Plastic Strain. Method Based on Simple Tests 10.2.2.1 Determination of the Failure Line According to the Cap model, powder fails when the state of stresses is located on a straight line called the failure line, Figure 10.1.

Figure 10.1. Representation of the Cap model in the P-Q plane

This line is completely determined from two parameters, the cohesion d which is the intersection with the deviatoric axis and β the slope. In Figure 10.2 which shows results of three tests (diametral compression, simple compression and tensile test), it can be observed that for a given density, data are not exactly aligned on the same straight line.

Figure 10.2. Loading path of the three tests used for determining the failure line

Calibration of Compaction Models

153

Following the Cap model, only two experimental data are required for a complete determination of the failure line. Results of two tests from the four described in Chapter 7 are sufficient. Since they are easily performed, diametral and simple compression are often chosen. In the P-Q plane, the slope of the straight line corresponding to simple compression is 71.6°. When the failure line is determined by the two previous tests, the failure stress from simple compression is generally much higher than that from diametral compression. This means that diametral failure data are located near the origin and simple compression failure data are more or less far from the origin. This is why the failure line angle is often near 70° (Figure 10.3). This highlights that this value is linked to the test used and is not necessarily an intrinsic value of the material.

Figure 10.3. Failure line for different densities deduced from the diametral and simple compression tests

In the P-Q plane the equation of the failure line is the following: Q = P tanβ + d

(10.1)

The evolution of the cohesion d (Figure 10.4) and angle β (Figure 10.5) can easily be deduced from data from diametral and simple compression tests. The cohesion always increases with density, weakly for hardmetal powder and more strongly for ductile powders.

154

P. Doremus

Figure 10.4. Evolution of the cohesion function with density for hardmetal

As an example, the cohesion d can be fitted with the expression: d (MPa) = A(( ρ /ρ0)B) -1) where A and B are constants. Such an expression assumes a zero cohesion for the filled density ρ0.

Figure 10.5. Evolution of the cohesion as a function of density

As previously mentioned, the slope of the failure line can be considered constant to a first approximation. In this case: β = 68°.

Calibration of Compaction Models

155

10.2.2.2 Identification of the Cap The cap represents the yield surface in the densification region of the P-Q plane. The cap depends on the volumetric plastic strain or the density (neglecting the elastic strain) and is described by an elliptical shape the expression of which is: Q = [((Pb-Pa)² - (P-Pa)²) /R²]1/2

(10.2)

This expression depends on three parameters Pb, (point of intersection of the Cap and the mean stress axis) Pa (mean stress coordinate of the top of the Cap) and R (eccentricity of the ellipse representing the Cap), all functions of density. These are defined in Figure 10.1. Consequently three expressions are necessary for a complete determination. Results from an instrumented die test follow the evolution of the state of stress (P0,Q0) with density during compaction. Obviously (P0,Q0) is located on the yield surface of the corresponding density. This gives the first equation for determining the three parameters: Q0 = [((Pb-Pa)² - (P0-Pa)²) /R²]1/2

(10.3)

At a given density, the top of the cap is also on the failure line. Since the evolution of the two parameters of the failure line (cohesion d and angle β) against density has been previously known, the second equation needed can be expressed as : Pb = Pa + R (Pa . tanβ + d)

(10.4)

The Cap model is associated. This means that the direction of the plastic flow vector dεpl is perpendicular to the yield surface at the corresponding state of stress P0,Q0 (isopotential and yield surface are identical). This will give the third equation. The two components of the plastic flow vector can be noted dεpl=[dεp, dεq] in the plane (εp,εq) when superimposed to the P-Q plane: dεp = dεz + 2 dεr dεq = (2/3).[dεz - dεr]

(10.5) (10.6)

Considering a rigid die, no radial strain occurs during compaction. This means that dεr =0. One can express from Equations 10.5 and 10.6: dεq / dεp = 2/3

(10.7)

At stresses (P0,Q0), the slope of the perpendicular direction to the ellipse is 2/3 (Figure 10.6).

156

P. Doremus

d + Pa.tanβ

Q

β

Slope = 2/3

(P0,Q0)

d

P

R (d + Pa.tanβ)

Pa

Pb

Figure 10.6. Direction of the plastic flow strain at stresses P0,Q0 on the die loading path

The slope of the tangent to the Cap at stresses P0,Q0 is : dQ/dP(P0,Q0) = (Pa-P0) / (Q0R²) = -3/2

(10.8)

The eccentricity R is then deduced : R² = (2/3) (P0-Pa)/Q0

(10.9)

The three Equations 10.3, 10.4 and 10.9 lead to the following equation: A. Pa² + B. Pa + C =0 With

A = 2 tan²β

B = 3Q0 + 4d.tanβ

(10.10) C = 2d²-3P0Q0-2Q0²

Let us note ∆ = B² - 4AC, then : Pa = (-B + ∆1/2) / 2A The eccentricity R is deduced from Equation 10.9 and Pb from Equation 10.4. Using this approach it is therefore possible to determine R and Pb with densification of the powder. Pa is not interesting since it is known through Pb, R, d and β. Figures 10.7 and 10.8 represent the evolution of R and Pb as a function of density.

Calibration of Compaction Models

157

Figure 10.7. Evolution of the eccentricity R as function of density

To a first approximation it is possible to consider the eccentricity as constant: R = 0.70.

Figure 10.8. Evolution of Pb during densification

The evolution of Pb can be expressed as: Pb (MPa) = a[(ρ /ρ0)b-1]

(10.11)

with a = 0.0211 MPa and b = 9.84. Figure 10.9 gives an idea of the shape of the yield surface in the P-Q plane.

158

P. Doremus

Figure 10.9. Representation of the Cap model deduced from die test data and for hardmetal powder

10.3 Calibration of the Cam-Clay Model Calibration of the Cam-Clay model is simpler than the calibration of the DruckerPrager-Cap model since yield surfaces are represented by complete ellipses (Figure 10.10). Considering an associated model, yield surfaces are fully determined when Pb, Pa and R = (Pb – Pa)/ Qa are known as a function of the volumetric plastic strain or density. The expression for the yield surface is: Q = [((Pb-Pa)² - (P-Pa)²) /R²]1/2

(10.12)

As previously seen, an instrumented-die test provides the following two equations Q0 = [((Pb-Pa)² - (P0-Pa)²) /R²]1/2 indicating that the stress point P0 and Q0 belongs to the yield stress and dQ/dP(P0,Q0) = (Pa-P0) / (Q0R²) = -3/2

(10.13)

indicating the slope of the tangent of the yield stress at the stress point P0, Q0.

Calibration of Compaction Models

159

Figure 10.10. Representation of yield-stress surface of the Cam-Clay model

The third equation needs a complementary test. This one can be chosen from simple compression, diametral compression or tensile tests. For example, if diametral compression is used this leads to the third equation. Qd = [((Pb-Pa)² - (Pd-Pa)²) /R²]1/2

(10.14)

10.4 Calibration of the Drucker-Prager-Cap model from Triaxial Data In the previous method the calibration of the Drucker-Prager-Cap model is based on the exact number of tests needed for determining all the parameters. The calibration is unique and therefore it is impossible to get an idea of the ability of this model to predict the mechanical behaviour of the powder. Yield-stress surfaces are represented (isodensity curves) for a ductile powder (Figure 10.11) and for a hardmetal powder (Figure 10.12) from consolidated tests [1]. The fact that a Cap model is chosen for representing powder behaviour imposes an elliptical surface on the isodensity curves. It seems reasonable to make this approximation whatever the type of powder is. Moreover, the Cap model stipulates that the location of the top of the ellipse (critical state) belongs to the failure line previously defined. According to the first method, which has been described earlier in this chapter, failure lines are generally represented by a narrow set of straight lines with approximately an orientation of 70°. The second method consists in using triaxial data and more especially the critical state line (the location of the top of the ellipses) and one of the tests already mentioned for determining the failure line: simple compression, diametral compression, tensile test. As the critical curve is not a straight line for ductile powder this method leads to a set of failure lines having different angles (Figure 10.11).

160

P. Doremus

Figure 10.11. Yield-stress surfaces obtained from triaxial data for ductile powder

For a hard powder a straight line is more appropriate to represent the criticalstate curve (Figure 10.12). However, this method obviously leads to a failure line with an orientation of about 50°.

Figure 10.12. Yield-stress surface from triaxial data of hardmetal powder

Calibration of Compaction Models

161

A failure line with an angle less than 70° for a given density amounts to saying that simple compression of a specimen of this density is practically impossible. Such an inconsistent deduction is based on the choice of a straight line to represent fracture behaviour. Anyway, the calibration of the Cap model can be achieved as follows. For a given density, the failure line passes through points of the same density belonging to the critical state curve on one hand and the loading path of one test as described before on the other hand (simple compression, diametral compression, tensile test). The evolutions of the cohesion d and angle β are then deduced from the set of failure lines. The Cap is determined as the best portion of an ellipse, the top of which belongs to the critical state curve. The way to choose the best ellipse can be deduced using only a hydrostatic test (Figure 10.13) or from any other method taking into account all data obtained from triaxial tests.

Figure 10.13. Evolution of the density as a function of mean pressure for ductile powder and hardmetal powder

10.5 Comparison of the Two Calibrations Fits emerging from the two methods differ. For whatever method is chosen for triaxial data analysis and the only analysis possible from thedie test, the difference is easily noticeable. Eccentricities from triaxial data are about two times greater for small stresses (Figure 10.14) and tend to become similar as the density increases.

162

P. Doremus

Figure 10.14. Eccentricity of the Drucker-Prager-Cap model deduced from die test and triaxial test

The yield surfaces are therefore also different (Figure 10.15). However, it is difficult to say which one is better for predicting the die-compaction process when running numerical simulations [2, 3].

Figure 10.15. Comparison of the yield-stress surfaces deduced from die test and triaxial test

This page intentionally blank

Calibration of Compaction Models

163

10.6 Concluding Comments The way a constitutive model is calibrated depends on which powder material database is available. A method has been presented in this chapter that minimises the number of data required, that is to say minimises the number of tests that must be performed for model calibration. Following this procedure, the closed die test is the most important since powder is compacted nearly as in industrial processes. When model calibration is achieved based on a minimum number of tests, the method generally consists in finding the analytical expression of the curve that best fits the experimental data.

References [1] Bonnefoy V and Doremus P. 2004. Guidelines for modelling cold compaction behaviour of various powders, Powder Met. 47, N°3, 285-290. [2] Alvain O, Doremus P and Bouvard D. 2002. Numerical simulation of die compaction and sintering of cemented carbide, PM 2002, Orlando, 16-21 June, USA, Vol.9, 158171. [3] Kim HG, Gillia O, Doremus P and Bouvard D. 2002. Near net shape processing of a sintered alumina component: adjustment of pressing parameters through finite-element simulation, Int J Mech Sci, 14, n° 12, 2523-2539.

11 Production of Case-study Components T. Kranz1, W. Markeli1 and J.H. Tweed2 1

Komage Gellner GmbH, Kell am See, Germany. AEA Technology, Gemini Building, Harwell, Didcot, Oxfordshire, OX11 0QR, UK.

2

11.1 Introduction In order to model the compaction of powder components, we need to be able to specify the starting conditions in terms of punch positions and powder fill density, the powder properties during compaction and the motions of the press tooling during the compaction process. A model will then provide predictions of the density distribution in the compacted part and the loads on the tooling. A summary of the input data required for compaction modelling, together with the output data that may be obtained is given in Table 11.1. This also indicates where each topic is covered in this book. Table 11.1.

Input and output data for a powder-compaction model

Input data

Origin

Output data

Validated by

Initial geometry

Design

Press loads

Load-cell measurements

Fill density (distribution)

Measure mass and infer from fill volume. Or Xray CT or other techniques (Chapter 9)

Component dimensions

Measurement

Control kinematics

Control signal driving press

Crack locations

Inspection, metallography (Chapter 12)

Achieved kinematics

Measure tool movements and characterise tool deflections

Density distribution

Section and Archimedes. Or X-ray CT or other techniques (Chapter 12)

Powder characteristics

Range of tests (Chapters 5 to 8)

(Chapters 9, 12)

166

T. Kranz, W. Markeli and J.H. Tweed

When compaction modelling is to be applied to parts that have been produced, then it is important to provide accurate measures of the tooling movements during compaction as input to the model and to take accurate measurements of punch loads to test the model predictions. The approach to instrumentation used in the Dienet project is outlined. This involved recording platen displacements and punch loads on a servo hydraulic press. During compaction, there may be minor deflections in the press and tooling. Means of measuring and estimating these are discussed in order to provide accurate data for punch movements in compaction models. A summary of case-study components used in this book is given.

11.2 Press Instrumentation for Force and Displacement Practical trials of powder compaction are essential to test the theoretical models of compaction processes. These give both data for input to models, such as punch displacements and also data for testing models, such as punch loads or forces. A combination of practical trials and model predictions leads to an iterative improvement in both. Trials assist in the application of models to practical compactions. The models give insight into the relative importance of parameters controlling compaction in practice. In order to minimize the differences between the model and the practical process, it is necessary to have an adequate knowledge of the pressing parameters. The dominant parameters are the force (F) and displacement (S) for each punch. 11.2.1 Punch Force Punch force can be measured in a variety of ways. Sensor technology [1] and positioning are important for measurement accuracy. The easiest method of determining the press force in hydraulic presses – the hydraulic pressure measured only on the piston side – is hardly used any longer. Due to a variety of frictional forces in the loading train, it insufficiently reflects the actual prevailing conditions in the press tooling (see upper half of Figure 11.1 – pressure p1). It is advantageous to use a differential pressure measurement as the effect of the counter force on the piston ring surface is then accounted for (p2 in Figure 11.1, leading to force measurement F1). For production presses, this is the current state-of-the-art. However, this measurement still includes a number of frictional forces as well as the punch force (top half of Figure 11.1). A better representation of the punch stress is given by an alternative measurement strategy (bottom half of Figure 11.1) as the parasitic friction from the guides and seals does not flow into the force analysis. For this purpose, suitable sensors – such as wire resistance gauges, force-measuring rings, etc., depending on force to be measured and space available – are placed directly under the tool punch. However, the influence of friction between punch, powder and die cannot be eliminated even in this case. Depending on the compacted material and compact geometry it can be the dominant element of the punch force, e.g. for some ceramic materials. Measurement results in Figures 11.2 and 11.3 show the difference in force

Production of Case-study Components

167

transmission between the top and bottom punches for an iron and a ceramic powder. In this case, pressing was from the top only and the force difference gives a visualisation of the top-punch force transmitted by friction to the die. An example of loads measured for a ferrous part during the Dienet project is given in Figure 11.4.

F error Fp

FR seal

FR die

Y=f(x)

F1

FG

F compact FG

F error

p2

FR seal

FR guidings

F measured

F measured

Fp p1

FR seal FR seal FR guidings

Fp

F compact

FR die

FR die

F compact

FR die

F compact

F2

FR guidings

FR guidings FR seal FR seal FG

FG FR seal FR seal Fp

Fp : F1,F2 : F measured F error : FG: F compact : p1,p2 : FR seal : FR guidings : F R die :

resulting pressing force measured press force error ratio of measured force weight force pressing force pressure result - friction of guide and density result - friction in guide elements die friction Figure 11.1. Overview of forces in a press system

The measurements shown in Figures 11.2 and 11.3 were determined through comparison measurements on tooling with simple geometry. In order to simplify the interpretation of the results, the bottom punch remains in position during the measuring cycle and only the top punch submerges into the die (one-sided pressing from above). The resulting friction forces depend on the material being pressed, the die material and the tooling geometry and motion (see, e.g. [2]).

168

T. Kranz, W. Markeli and J.H. Tweed

Figure 11.2. Force transmission – ceramic material – note that the top-punch load is much greater than the bottom-punch load

Figure 11.3. Force transmission – iron powder – note that the top-punch load is only slightly greater than the bottom-punch load

Production of Case-study Components

169

1200

1000

1

2 3

4

5

6

800

Load, kN

1 Fill position, UP closes die 2 UP, LIP move to 20% of displacement 3 LIP only moves to 60 % of displacement 4 UP, LIP move to final displacement 5 Hold at maximum displacement 6 Ejection

Upper punch UP

600

400

Lower inner punch, LIP

200

Lower outer punch, LOP

0 0

5000

10000

15000

20000

25000

-200 Time, ms

Figure 11.4. Loads measured for a ferrous component manufactured as a case-study during the Dienet project (Case-study 1 Ferrous kinematics A – see also Appendix 2)

Many processes influence the pressing force. Selection of a press and load sensor will depend on the requirements for the part to be pressed, taking into consideration the complex tribology between the powder and tooling and also between tooling components. For materials that cause high friction forces within the die, the expenditure for a high-precision sensor technology seems questionable because the error rate can be considerable during pressing and will also vary with density. However, provided pressing conditions remain similar, accurate measurement of press forces is a valuable quality tool. 11.2.2 Punch Travel The radial geometry of the compact is mostly determined by the die. Tool punches determine the axial geometry limitations and offer, within the limits of travel on each punch, great freedom in determining the press process. This is very important, particularly when processing complex shapes to tight tolerances, as details of punch travel as well as the start and end points may be critical for stable production of crack-free products. In order to achieve this, precise position monitoring of the punches at any time in the pressing process is necessary. For most applications, an absolute accuracy of <+0.01 mm is sufficient. This is the current state-of-the-art. In order to reach this, the applied measuring systems must have a significantly higher resolution (<5 µm), since all participating links of the measuring chain for the signal processing contribute to the total error rate of the position data [3]. For electronic position monitoring, various physical principles apply. These fall into two broad categories: incremental and absolute value systems. Generally, the difference lies in the way the position value is determined - in the incremental

170

T. Kranz, W. Markeli and J.H. Tweed

method by the counting of steps with a defined step size, and in the absolute value method by comparing positions within a maximum length interval. The physical principles behind these techniques are discussed by Hoffmann [4], and illustrated in Figures 11.5–11.7. The incrementally operating systems can claim slight advantages regarding the resolution - for example: with optical systems resolution is of the order of the wavelength of the applied light [5]. However, if position information is lost (e.g. through loss of voltage) the requirement to travel to a reference position and the associated risks of tool damage is a disadvantage Light source Collimator lens Shutter Rotating code disc

Illuminated fields (sensor positions) Axle

Figure 11.5. Principle of an incremental measurement system – in this case an angular sensor

Figure 11.6. Code disc of an angular sensor, showing graduations

Production of Case-study Components

Mechanischer Resulting mechanical torsion impulse Torsionsimpuls

171

Magnetic field of Magnetfeld permanent magnet Permanentmagnet

Magnet field of Magnetfeld current

Stromimpuls Current pulse

Moving magnet Beweglicher (position)

Stromimpuls

Positionsmagnet Transducer for torsion Torsionsimpuls impulse

Wandlersystem

Magnetostrictive Magnetostriktives measuring element Messelement (waveguide)

(Wellenleiter)

Figure 11.7. Principle of absolute value position measurement using a magnetostricitve process

The optomechanical system also requires a high production accuracy because the mechanical quality has a big influence on the durability of the function (e.g. the quality of mechanical construction (Figure 11.5) or of the graduation lines on the code disc (Figure 11.6)). Based on these facts, contactless, absolute-value measuring systems are especially advisable for application in press systems, since the functional reliability due to the simple mechanical construction guarantees the necessary dependability in rough everyday operation. Figure 11.7 shows the physical principle of an elegant device for absolute value measurement based on magnetostriction. The position of the permanent magnet is converted without contact into a proportional measuring value. Position information is available on start-up, thus assisting press operation.

11.3 Side Effects of Load Buildup – Press and Punch Deflections 11.3.1 Press Deflections As press forces build up during powder compaction and based on the equality of action and reaction and the equilibrium of forces on each element ΣFY = 0

(11.1)

172

T. Kranz, W. Markeli and J.H. Tweed

there will be force transfer from the tooling to the surrounding press structure (see Figure 11.8). This expansion is calculated in the elastic area using the following relations [6]:

σ=

F A

(11.2)

with σ = normal tension; F = force (suction or pressure); A = working surface of the force

σ = E *ε

(11.3)

L0

Pressing force

∆L

with ε = expansion (suction); E = Elasticity modulus (material constant)

Tensile force 2 Area 2

Figure 11.8. Schematic diagram of expansion in a loaded-press system

ε=

∆L L0

with ∆L = change in length; L0 = original length From Equations 11.2 to 11.4

(11.4)

Production of Case-study Components

σ=

Thus, ∆L =

∆L F = E *ε = E * L0 A

173

(11.5)

F *L

(11.6)

A *E

The press supplier has the possibility, with this simple relationship, to design the frame of the press to minimise expansion values and to correct the absolute position software-technically at the same time. Of course, this requires a precise force measurement. However, the effect of this expansion can be avoided through an elegant displacement of the measuring axis into the middle of the press. This is illustrated in Figure 11.9. Pressing

Die closing

1 2 3 4 5 6 7

Compensation of Elongation

Deviation

1 2 3 4 5 6 7

1 2 3 4 5 6 7

X

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

Set value of top punch = 4, set value of bottom punch = 2 to reach part height of X

K

kt

Figure 11.9. Design of the measuring system to compensate for press expansion (used on e.g. Komage hydraulic press)

Through this design referenced to the press centre, the distances to the sole plate are individually evaluated and the positions are automatically corrected. The effect of expansion in the press frame is significantly reduced 11.3.2 Punch Deflections By analogy to the effects in the press frame, all elements in the force stream of the punch setup are subject to compressive forces leading to compression or buckling. This includes, for example for hydraulic presses the piston and piston rods, tool adaptor, pressure plates and tool punches. Figure 11.10 shows the construction of press tooling and instrumentation for trials for the Dienet programme.

174

T. Kranz, W. Markeli and J.H. Tweed

Fd

∆L

L0

Figure 11.10. Tool construction for the Dienet project using a Komage hydraulic press S300-5E-CNC

L0

∆L

Fd

C o m p re ssio n Figure 11.11. Schematic diagram of punch compression

Figure 11.11 illustrates the principle of compression of punches. By analogy with the calculation of the expansion (Equation 11.6) the formula for calculation of compression is:

∆L =

Fd * L0 A* E

where Fd = pressure force

(11.7)

Production of Case-study Components

175

This calculation formula can also be used for software corrections of the actual punch position. However, if all forces are not measured, additional inputs are required from the user because the force distribution varies with the manufactured product (tool geometry). The influence of the buckling (Euler) is considered in the initial design by the tool constructor since a tendency to buckle can reduce tool life. If the centres of the force buildup of pressing force and reaction force are not along the same line of action, the buckling of force-transmitting components becomes a further side effect for the position determination. Figure 11.12 shows the principal effects on the structure. With increasing complexity of tool design this unfavourable strain is difficult to avoid and its effects must be taken into consideration by the press manufacturer and in production or computer modeling of the pressing process. Pressing force F

Adapter plate

Punch (lower)

F/2

Deflection curve F/2

Figure 11.12. Schematic diagram of bending behavior of an adapter plate during strain

The amount of bending of single components can also be predetermined according to the equations from the science of material strength [7] and the dependency on pressing force corrected using software.

176

T. Kranz, W. Markeli and J.H. Tweed

11.3.3 Implications for the Acquired Position Values All the above physical influence factors, each and in summation, lead to a difference between the true- and the indicated value of the punch position. By suitable installation of measuring systems in the angle of the bending line or a respective axis (Figure 11.12), the bending influence is decreased. However, the mathematical process requires an accurate determination of the boundary conditions in order to achieve a precise conformity with the reality. In practice, the detailed system deflections and their effects are frequently difficult to determine by the user. This can lead to operational errors of the correction system. A practical method of correction was applied in the trials for the Dienet programme. Here, the participating punches move to a force-sensored measuring block. If the press force values are increased gradually and the received position data is outlined, a chain of pair variates develop that offers a good reflection of the static behaviour of the machine under strain. A function can be determined (see Figure 11.13) that describes with the desired accuracy the behaviour of the position values with force. This deviation can then be used as an offset to the control circuit, thus giving the actual punch position. Extension X1/Y1 Lower outer punch (LOP)

740kN 2.00mm

2 1.8 1.6

Position, mm

1.4 1.2 1 Extrapolation

0.8 0.6 0.4 0.2 0 0

100

200

300

400

500

600

700

800

900

1000

Power, kN

Figure 11.13. Correction of tool design for Dienet on a hydraulic press by KOMAGE S1505level CNC. Data measured to 200 kN

11.3.4 Concluding Comments on System Deflections Many different physical influences affect the agreement of the indicated positions values with the actual punch position. Mathematic corrections based on materials science can be determined. A much simpler approach is the calibration method described above. This could be applied in operational practice as well. Depending

Production of Case-study Components

177

on the tool design (for example: with facets), it may not be possible to use this calibration method. In these cases an option for calculation of the compression becomes necessary. 11.3.4.1 Signification for the Dienet – Simulation Project Data determined for the Dienet project have been corrected to allow for presssystem deflections using methods based on those in the preceding sections. These corrected displacements are presented in Appendix 2 for a range of case studies.

11.4 Case-study Components Four case-study components are considered in detail in this book. These were produced in the European Modnet and Dienet programmes that are outlined in Chapter 1. Geometries of the case-study components are in Figure 11.14 and the materials and numbers of kinematics used for each case-study are given in Table 11.2. Details of some of the case studies are presented in Appendix 2, so that they can be used to test compaction models. Modnet

78

13.4

29 12.6 68

Dienet Case-study 1

47.8

H1 H2

25.8 37.9

3.2

R 1

50.0 10.0 25.3

Case-study 2

30.0

Figure 11.14. Generic geometry for case-study components. All dimensions are in mm.

178

T. Kranz, W. Markeli and J.H. Tweed

Figure 11.14 (Continued). Generic geometry for case-study components. All dimensions are in mm.

Table 11.2. Case-study components produced (details in Appendix 2) Programme

Case-study

Material

Kinematics

MODNET

CS1

Ferrous

1

DIENET

CS1

Hardmetal

2

Ferrous

2

Ceramic

2

CS2

Ferrous

1

CS3

Hardmetal

1

Ceramic

2

Magnet

2

References [1] [2] [3] [4] [5] [6] [7]

Kuratle R. Motor measuring technology, 1st edn, S77 ff, Vogel Publishing House, Prague. TU Dresden; http://tu-dresden.de/mwiww/ansfw/matrizenwerkstoff01.pdf. Hering, Martin and Stohrer. 1995. Physik für Ingenieure; 5th. edn 1995; VDIpublishing, 11 ff. Hoffmann J. Taschenbuch der Messtechnik; 1998; Carl Hanser Verlag, Munchen; e.a. 230 ff. Hering, Martin and, Stohrer. 1995. Physik für Ingenieure; 5th. edn 1995; VDI publishing 459 ff Gieck K. 1995. 30th. edition, Gieck Publishing, Munchen. 1 ff. Ibid. 9 ff.,

12 Assessing Powder Compacts S.F. Burch1, J.A. Calero2, M. Eriksson3, B. Hoffman4, A. Leuprecht5, R. Maassen4, F.M.M. Snijkers6, W. Vandermeulen6 and J.H. Tweed7 1

ESR Technology Ltd, 16 North Central 127, Milton Park, Abingdon, Oxfordshire, OX14 4SA, UK. 2 Ames S.A., Ctra.Nac. 340 Km 1.242 Pol.Ind. “Les Fallulles”, 08620 Sant Vicenc dels Horts, Barcelona, Spain. 3 IVF Research and Development Corporation/Swedish Ceramic Institute, Argongatan 30, SE-431 53 Mölndal, Sweden. 4 GKN Sinter Metals Engineering GmbH, D-42477 Radevormwald, Germany. 5 Plansee SE, A-6600 Reutte, Tirol, Austria. 6 VITO, Boeretang 200, B-2400 MOL, Belgium. 7 AEA Technology, Gemini Building, Harwell, Didcot, Oxfordshire, OX11 0QR, UK.

12.1 Introduction Powder-compaction models predict density distributions in green powder compacts and also, potentially, the location of cracks. In order to test model predictions, it is important to be able to assess the density distribution in powder compacts and also the location of cracks. A number of methods for assessing density distributions are presented. Machining and Archimedes methods allow the density of significant portions of a part to be determined. However, they can not give fine detail in density variations. These can be assessed using metallography and X-ray computerised tomography. Methods of assessing crack locations and extent are presented. The results of the techniques considered are compared and it is suggested that combination of a coarse-scale with a fine-scale technique gives an effective basis for testing model predictions.

180

S.F. Burch et al

12.2 Density Distribution by the Archimedes Method 12.2.1 Hardmetals Density determinations are frequently performed by means of Archimedes’ principle (buoyancy method). Archimedes’ principle states that a body immersed in a fluid apparently loses weight by an amount equal to the weight of the fluid it displaces. In a first step, the weight of the sample is determined by weighing the sample in air. In a second weighing, the volume of the sample is determined indirectly from the weight the sample displaces in the liquid of known density. From these two weighings, the density of the sample can be calculated as follows:

ρSample =

mAir VSample

VSample =

mAir − mWater ρWater

ρSample =

mAir mAir − mWater ρ Water

where: VSample Volume of the sample ρSample Density of the sample mAir Mass in air mWater Mass in water ρWater Density of the water 12.2.1.1 Sample Preparation Each part was cut into at least six regions as schematically shown in Figure 12.1 and these were filled up with hot paraffin wax to close the pores, so that no water could penetrate the sample during the weighing in water.

Figure 12.1. Regions for sample preparation

Assessing Powder Compacts

181

12.2.1.2 Results As an example, the density distribution of a hardmetal sample (No. HM20 – Kinematic 1) is shown in Figure 12.2

Figure 12.2. Results for sample HM20 in g/cm³

12.2.2 Zirconia and Sm-Co Samples The density that is being considered in this section is the so-called apparent density, i.e. the mass divided by the apparent (outer) volume of the sample. Determining the apparent density distribution of green components using direct physical methods generally involves cutting these samples into smaller parts that are assumed to have a homogeneous density. From the density of these different parts and their position in the component the density distribution can be assessed. However, on applying this method one can encounter problems for obvious reasons: a powder compact can have an unfavourable shape or it can be very fragile. This makes it difficult to divide it into pieces. In addition, as the binder may dissolve if a water-immersion method is used, the individual pieces may fall apart. The above problems can be overcome if a number of precautions are taken. Sealing of the surfaces with wax (the amount of wax should be accounted for) is one possibility. Care must be taken not to damage the fragile surface when applying the wax. Another possibility is to give a minimum strength to the green body such that handling of the body becomes possible without the risk of disturbing its volume and shape. This can be done by applying a presintering treatment to the green body. In the following section it is described how the mercury-pycnometer method is applied to determine the density distribution of compacted bodies using presintering. The compacts considered in this study are pressed ceramic (zirconia) and magnet (Sm-Co) powder from Dienet case study 3 (CS3). 12.2.2.1 Sample Preparation The ceramic components (ZrO2) were received as-pressed from Dynamic Ceramic Ltd. The magnet components (Sm-Co) were provided by Swift Levick Ltd. in the presintered condition. No information on the presintering conditions of the latter is available.

182

S.F. Burch et al

In order to allow the green body to be divided into individual parts by cutting, the green ceramic samples were presintered at 1100°C for 1 hr. Since presintering causes some shrinkage the diameter and the central height of the samples were measured before and after the treatment. In the axial direction (height = pressing direction) presintering caused on average a shrinkage of about 1.1 %; the shrinkage of the diameter was about 2.0 %. Cutting used the following procedure: First, a slice was cut parallel to the mark indicating the filling direction. This slice was cut in two symmetrical halves named A and B, A being the position of the mark. From the two remaining parts of the component slices C and D were cut perpendicularly to the AB direction. These slices are of course shorter by half the thickness of the AB slice. The slice thickness was in all cases 4.0 mm. 12.2.2.2 Density Measurement Since the shape of the pieces is too irregular to calculate the volume from the dimensions the volume has been measured by means of a mercury porosimeter used as a pycnometer (Autoscan 33 porosimeter, Quantachrome, Florida, USA). A normal pycnometer therefore would of course also have been fit for the purpose. The principle of this method is as follows. The sample is loaded in a glass vessel with accurately known weight and volume. The vessel is then filled with mercury and weighed. From the total weight, the volume and the mercury density the sample volume can be calculated. With the equipment used the mercury pressure was limited to 0.165 MPa (1240 mm Hg) in order to avoid intrusion of the mercury in the sample. A picture of the glass vessel containing the specimen is shown in Figure 12.3.

Figure 12.3. Glass container with sample used as pycnometer

The apparent density is then calculated using: Density = sample mass / apparent volume The precision of this method was checked by measuring the volume of two ZrO2 samples that were carefully machined to a cylindrical shape with a diameter of 5.01 ± 0.01 mm and a height of 12.01 ± 0.01 mm. The difference between the

Assessing Powder Compacts

183

calculated volume and the volume measured with the mercury method was found to be less than 1.5 %. In order to check for any unwanted mercury uptake by the samples, these were weighed before and after measurement. It was found that the zirconia samples showed no weight increase. The magnet samples showed a weight increase of less than 1.2 %. This means that a small amount of mercury has entered the sample, or adheres to the sample or has reacted with the sample. In the case of weight increase by intrusion this corresponds to an underestimation of the sample volume by a maximum of 0.5 %. The corresponding density is therefore overestimated by maximum 0.5 % of its value. If the weight increase is due to adherence to the sample, this does not affect the result. For comparison with calculated density values from modelling the observed effect can be neglected. On comparing the results obtained with the presintering technique with results obtained on the as-pressed sample two factors should be accounted for: shrinkage due to presintering as mentioned above and binder removal due to presintering. These will be briefly discussed below. The volume shrinkage can be estimated to be twice the diametral shrinkage plus the axial shrinkage. For the zirconia samples this amounts to 5.1 %. This means that for the presintering cycle used, the density is 5.1 % higher relative to the as-pressed condition. From the weight change of the samples after presintering it was found that the binder content of the zirconia was 1.6 %. Relative to the as-pressed condition this leads to a density decrease of 1.6 %. Combining both factors, it follows that the aspressed density can be obtained by decreasing the presintered density by 3.5 % (multiplication with a factor 0.965). It should be emphasized that this change is not related to measurement errors but merely accounts for the difference between the as-pressed and the presintered state of the sample. 12.2.2.3 Results As an example, the results obtained on components 19 (ceramic) and 76 (magnet) are shown in Figures 12.4 and 12.5, respectively, in a schematic representation of the slices and their subdivision. The dimensions are given in mm, the density in g/cm³. It can be seen that for both materials the density variation in the component is rather small. The density values shown in the figures correspond to the presintered condition. Conversion to the as-pressed condition can easily be done by multiplication with the factor 0.965 for the ceramic samples. For the magnet samples this factor could not be determined because the as-pressed dimensions were not available. However, as an alternative for determining the correction factor from the shrinkage and binder burn out, this factor might also be obtained by taking the ratio of the average density of the component in the as-pressed condition (measured, e.g., geometrically) to the weighed average of the density measured by the present method. The weighing factors should take into account the sample volume represented by the different pieces. It should indeed be remembered that the pieces on the outer side of the component represent a larger volume than those close to the axis.

184

S.F. Burch et al

Some pieces showed an unexpected density taking into account their position in the sample. These “outliers” may be explained by possible uneven filling of the die. B

A

2.72

2.71

2.77

2.79

2.74

2.69

2.81

2.80

2.74

2.77

4.9

D

2.78

5.7

C

2.72

2.75

2.71

2.77

2.73

2.79

2.80

2.72

2.77

Figure 12.4. Schematic representation of the density distribution in ceramic component 19. Data refer to the presintered condition. The dimensions are given in mm, the density in g/cm³.

B

4.60

A

4.37

4.35

4.42

4.41

4.32

4.36

4.36

4.32

4.52

9.5

D

4.52

8.4

C

4.42

4.39

4.38

4.42

4.34

4.42

4.35

4.31

4.54

Figure 12.5. Schematic representation of the density distribution in magnet component 76. Data refer to the presintered condition. The dimensions are given in mm, the density in g/cm³.

Assessing Powder Compacts

185

12.3 Density Determination by Machining Two methods based on mechanical removal of material are described below. One approach to a density distribution map is based exclusively on measurements of geometry and weight (Figure 12.6).

Figure 12.6. Machining ferrous component to infer density distribution

The procedure consists of a first step of measurement of the dimensions and weight of the part. The second step is to presinter the part at 850 ºC, minimising the dissolution of carbon. This step is required to produce a part with sufficient strength to allow it to withstand machining. This mechanical step is done by a lathe removing material from the top to the bottom of the part. Each removed portion is obtained by weighing and measuring the final part and comparing the initial values recorded. This gives us the opportunity to calculate the volume and weight loss, which is the density of the removed disc. The height and depth of the removed disc depends on the precision of the tools used. For dimensional measurements it is good practice to make more than five measurements along the final diameter of the part and apply statistical tools to get the values. An analytical balance is required to obtain a measurement of weight with sufficient precision. This method is extremely labour intensive if accurate results are to be obtained. There is another procedure combining the weight loss and Archimedes methods, which offers the advantages of both. Instead of removing all the material, it is suggested to remove a small piece and to measure the density of the remaining by the Archimedes method. This method requires less time for validation and permits drawing of a qualitative map of the density distribution over the pseudo green part.

186

S.F. Burch et al

The method shares some of the steps with the first mentioned. Essentially it consists of presintering the axisymmetrical green part at 850 ºC, minimising carbon combination in the metal matrix. The part is then weighed and measured. The Archimedes method is used to determine the global density. A degreasing stage is then used to remove the soaking product (paraffin) from the surface of the part. By using a precision cutting tool the part is cut into different slices. The densities of the rings are measured by the Archimedes method (followed by degreasing) and an outer layer of material is machined away using a lathe. The process is then repeated to build up a three-dimensional map of densities.

Figure 12.7 Machining ring from ferrous component to infer density distribution

Each resulting ring is then measured and immersed in paraffin to calculate its density by Archimedes. Applying the formulae

ρ ring =

mrest

M ring

ρ rest

1 mremoved +

M ring

ρ removed

Where ρring is the full ring density recorded, Mring the mass of the full ring; mrest and ρrest the mass and the density of ring obtained after the machining step, respectively, and mremoved and ρremoved, the mass and the density of the removal ring, respectively. The unknown parameter is ρremoved and it is obtained from the preceding formulae.

Assessing Powder Compacts

187

This method is applied for each machining step to obtain the density distribution map of each ring and relies on the high precision of the analytical balance as the unique experimental measurement to carry out (see for example Figure 12.8). Dienet Case Study 1, Ferrous Kinematics A (CS1 Fer A)

Dienet Case Study 1, Ferrous Kinematics B (CS1 Fer B)

Fer32 detail

Fer44 detail

Fer32 summary

2.98

3.00

2.50

2.57

4.53

7.09

7.01

7.12

7.04

4.40

7.09

7.03

7.10

4.25

6.92

7.05

4.97

6.99

7.01

4.01

6.83

6.93

4.37

6.87

4.42 4.48 6.05

Fer44 summary

3.01

3.03

2.46

2.55

4.55

7.03

7.09

7.12

7.08

7.03

4.35

7.15

6.96

7.12

7.05

7.10

7.01

4.44

6.95

7.15

7.03

7.06

7.10

7.00

5.06

6.86

7.17

7.03

7.04

4.45

6.94

6.93

6.96

4.51

6.90

6.98

6.91

7.01

4.79

6.99

6.97

6.97

7.04

4.65

7.01

7.03

7.05

7.02

4.71

6.98

7.06

7.05 7.07

7.00 7.06

6.92

7.05 7.09

7.04 7.04

6.95

25.83 φ

25.82 φ

37.89 φ

37.89 φ

47.92 φ

47.92 φ

Dimensions, mm Densities, g/cc

7.02

Part, 7.02

Dimensions, mm Densities, g/cc

7.02

Part, 7.03

Figure 12.8. Density distribution for parts made with Dienet ferrous Case Study 1, Kinematics A and B

12.4 Density Distribution Determined by SEM-EDS Line Scan of Polished Cross Sections This section describes a method to determine the density distribution in a green (unsintered) powder compact, in this case a pressed zirconia powder compact (CS3 Ceramic). The method uses a scanning electron microscope with an energy dispersive spectrometry detector (SEM-EDS), to perform a line scan on an epoxy immersed, cut and polished cross section of the sample in question. The method could also be used on materials other than zirconia. 12.4.1 General Considerations on Density Measurements of Green Samples When the green density distribution is to be determined in a powder compact, it is not always easy to do direct physical measurements. The powder compact can have an unfavourable shape. The powder compact can also be very fragile, making it difficult to divide it into pieces to measure the individual pieces without damaging corners, etc. Also, the pieces taken out could not be measured with water intrusion

188

S.F. Burch et al

(they would dissolve) or helium pycnometry (because the pieces do not have a dense surface). The surfaces could be sealed with wax and measured but even that is tricky with fragile and small pieces. The amount of wax should also be accounted for, and the fragile surface should not be affected when applying the wax. Another method that could be used is volumetric density measuring, where the sample volume is measured by immersing the samples in a chamber with small rolling balls, but again the samples may be too fragile and this method is not so accurate when you are assessing density variations around 1 unit of percentage. The method described below (SEM-EDS line scan) was not found to be accurate to get a value of the absolute density, but the density gradients throughout the sample could be measured with a good level of accuracy. 12.4.2 Samples Three axially pressed zirconia CS3 (case study 3) parts were measured in the green state. The samples were named 34, 36 and 38. The samples were provided from Dynamic Ceramic Ltd. A mark on each sample denoted the side of the compact away from the fill-shoe position. 12.4.3 Experimental Procedure The samples were immersed in epoxy under vacuum. The epoxy could not penetrate the porous structure of the samples completely, but the immersion fixed the entire sample for further processing. The samples were cut vertically by a thin diamond wheel under slight water cooling. The cut was performed offset and parallel to the mark denoting the fill direction by about 1 mm in order to have some material to lose during polishing yet still end up with a polished cross sectional surface representing the middle of the sample. A vertical slice a few mm thick was cut from the central part. The slice was dried under vacuum for 24 h, and then immersed again in epoxy under vacuum. This was performed to impregnate the central parts of the sample that were not fully impregnated under the first immersion. The epoxy penetrated about 1.5 mm deep into the surface of the samples. The penetrated side that was first cut 1 mm from the mark was then ground and polished using a standard scheme of finer and finer diamond sizes, until about 1.0 mm had been removed by grinding and polishing. The polished surface obtained was a vertical cross section of the sample straight through the mark. The polished surfaces were given a thin layer of carbon for conducting purposes. Initially the polished surfaces were analysed with image analysis using backscattered electons from a scanning electron microscope – electron probe micro-analyzer (SEM-EPMA). The light intensity of the material was translated into density. It was found, however, that this technique did not work very well for this application, because the added carbon layer interfered with the measurement. The carbon-layering technique usually yields a smooth variation in carbon thickness depending on how large the samples are. Larger samples, as in this case, more easily get a larger carbon thickness gradient, with a thickness around approximately 0.005 – 0.01 µm. Since the detection depth of the technique was only about 0.02 µm, the carbon layer represents a substantial part of the “density

Assessing Powder Compacts

189

gradient” observed. Therefore, this technique was abandoned and is not recommended for this kind of light-intensity density measurements. Instead of using SEM-EPMA, the samples were measured with a line scan using an EDS (energy dispersive detector). The zirconium (Zr) contents were detected by X-ray photons excited by the electron beam. The detection depth with this technique was about 1 µm, which made the influence of the much thinner carbon layer insignificant.

Figure 12.9. Cross section showing the shape of the zirconia powder compact and the approximate location of the six plus six areas with the measuring locations

12.4.4 SEM-EDS Method to Determine Density Distribution Each of the three samples was analysed in six areas on the half-side of the cross section containing the mark and at six corresponding areas on the other half of the sample (Figure 12.9). Each area was measured at three locations on a horizontal line. Each measuring location was a cross sectional area of 0.5 x 0.75 mm, compared to each of the six areas that was roughly cm-sized, in which the average surface brightness between black (epoxy) and white (100 % dense sintered zirconia) was measured. The porous green samples gave values in between these extreme values, with higher values representing a higher green density. Each surface brightness value was then translated to density as a percentage of theoretical density. A total of 36 measuring points per sample were taken (six plus six areas, three points per area). 12.4.5 Results The density gradient results of the parts 34, 36 and 38 are presented below (Table 12.1). The numbers represent “% of theoretical density” and are average values of the three measuring points wihin each of the six plus six areas.

190

S.F. Burch et al Table 12.1. Results of the SEM-EDS density gradient measurements (% density)

Part 34:

Mark

67.3 69.4

1: 67.9 68.7

69.1

2: 67.8

3: 66.4

68.4

69.2

5: 68.1

6: 65.7

Part 36:

Mark

69.8 69.5

1: 70.5 69.5

70.3

2: 69.4

3: 68.8

68.6

70.4

5: 70.1

6: 69.9

Part 38:

4: 68.8

Mark

67.9 69.7

4: 66.4

1: 70.3 69.6

69.7

2: 69.9

3: 69.5

69.1

70.0

5: 70.5

6: 69.5

4: 69.3

12.4.6 Discussion and Conclusion The absolute density values obtained using this method are too high compared to the physical measurements. The reason for this could be that information from zirconia grains just under the polished epoxy surface mixes with the exposed zirconia grains, which should result in higher values of zirconia density than in reality. This problem should be more pronounced if the powder was very fine grained, which it was in this case. However, the subject of study, the density gradient (difference from area to area), should be comparable and accurate with this method. The density-gradient values obtained could be scaled and compared with other density-measuring techniques.

12.5 Density Determination by X-ray Computerised Tomography Details of the X-ray computerised tomogaphy technique have been given in Chapter 9. This section presents a calibration study and one example of results using this technique. X-ray computerised tomography has been applied to a wide range of specimens. Application to a ceramic test sample is described here and application to pharmaceutical samples is described in Chapter 14.

Assessing Powder Compacts

191

As an example of the ability to measure density variations, a compound sample was made by pressing eight ceramic discs of 30 mm diameter, two each at 3, 4, 6 and 10 tonnes pressing load. These discs were then assembled into a stack (Figure 12.10a) and a CT reconstruction of a slice through the stack is in Figure 12.10b.

Figure 12.10. Compound specimen made from ceramic discs pressed to differing densities a Photograph b CT reconstruction of section through centre

A correlation between CT image level, after correction for “beam hardening,” and disc density is in Figure 12.11. Ceramic disc calibration sample

CT image value

22000

20000

18000

2

2.1

2.2

2.3

2.4

2.5

Density (g/cm3) Figure 12.11. Mean density of each ceramic disc plotted against the mean CT image value derived from a narrow strip across the diameter of each disc

A case study of the application of the X-ray computed tomography technique to density variations in pharmaceutical tablets is given in Chapter 14.

192

S.F. Burch et al

12.6 Comparison of Result of Density Distribution Measurement Techniques Two methods have been used to determine the density distribution for zirconia parts. The first is a variant of Archimedes method applied to lightly sintered parts (Section 12.2.2). Results for one case are shown schematically in Figure 12.12. Average A+B

Average A+B 2.66

2.63

2.71

2.64

2.63

0.8%

-0.4%

2.7%

-0.1%

-0.3%

Figure 12.12 . Schematic diagram of density and % density distribution for zirconia parts 35 and 39 of Dienet CS3 determined by a variant of Archimedes method. Each measurement is an average of four, two from sections parallel to the fill-shoe direction (A-B) for each of parts 35 and 39.

The second method uses SEM-EDS line scans on polished cross sections as described in Section 2.4. The signal brightness is determined as an average from three locations each 0.75 mm x 0.5 mm and scaled from 0 % (epoxy resin) to 100 % (fully sintered zirconia). Results from this method are high by about 60 % (range 57 % to 61 %) compared with the average density for the whole part inferred by the Archimedes method. This may reflect the detection depth of about 1 µm (in fully dense zirconia) for this method. As the zirconia powder particle size is under 1 µm, the technique may be sampling particles below the surface. It may thus, in effect, be registering macropores between granules but not the very fine pores within granules and between individual power particles. In order to compare with the results from the Archimedes method, the results from the EDS method have also been normalised to the measured part density (Figure 12.13) Despite the different basis of the two methods for determining the density variation of the zirconia parts, the indications are similar, with densities near the average towards the edge of the part and a peak towards the middle and the base of the part.

Assessing Powder Compacts

Average A+B

2.64

2.66

2.63

2.67

2.63

2.64

Average A+B

193

0.2%

0.8%

-0.1%

1.3%

-0.4%

0.0%

Figure 12.13. Schematic diagram of density and % density distribution for zirconia parts 34, 36 and 38 of Dienet CS3 determined by an SEM-EDS method. Each measurement is an average of six, two from sections parallel to the fill-shoe direction (A-B) for each of the three parts.

12.7 Determination of Defect Distribution The assessment of press defects in powder compacts according to their location, type and size is usually done by preparation of a metallographic cut and observation through a light microscope. The detection of a crack can be carried out on a green (nonsintered) or on a sintered sample. Due to the higher effort needed for the preparation in the green status the sintered sample is often preferred, particularly for ferrous parts where there is little shape change on sintering. To avoid damaging the sample, the preparation of a metallographic cut in the green status requires a careful and time-intensive procedure for cutting and embedding due to the low green strength. The powder compact is infiltrated under vacuum with epoxy resin. The hardening time is about 24 h. Only in the infiltrated and hardened state can the cutting of the sample in the plane where the crack is expected be performed. It might be necessary to infiltrate the cut again in case the infiltration is not complete. After successful infiltration the powder particles are sufficiently fixed to continue with the usual preparation steps like grinding and polishing. When the preparation is complete the sample can be observed with a light microscope. In the case of green samples the light is applied by interference contrast, in the case of a sintered sample the usual bright field is used. The sample preparation in the sintered state (at least presintered) allows simpler and faster processing of the sample mounting due to its much higher strength. Concerning the embedding, grinding and polishing no specific precautions need to be considered. The typical press defects in a powder compact are cracks. Two types of cracks can be distinguished, the shear crack and the tensile crack. A typical shear crack is shown in Figure 12.14 and a typical tensile crack is shown in Figure 12.15.

194

S.F. Burch et al

Figure 12.14. Shear crack

Figure 12.15. Tensile crack

A shear crack occurs when powder transfer during the pressing cycle takes place at too high a density. Particles at the crack boundary are deformed and the boundaries look smooth and relatively straight. In contrast to the shear crack, the tensile crack is irregular in its path and follows pores and particles.

Figure 12.16. Dead water zone

Another type of press defect can be zones of very low density. Due to the low density the number of pores as well as their size is increased. Figure 12.16 shows such a case. The powder flow does not follow the curvature of the radius, and this region is often termed a “dead water zone”. One more possible defect in green compacts is the so-called delamination. When pressing with high compaction pressures, the die is significantly radially deformed. On unloading the punches, the radial springback of the die can damage the powder compact. Planes perpendicular to the pressing direction are delaminated and cracks are formed. Usually these cracks do not appear at the diewall surface of the compact and are therefore not visible on the surface. Due to the very small crack width sample preparation needs to be done very carefully since cracks are easily closed with grinding. Because press defects do not heal with sintering, it is possible to detect them in a sintered sample for ferrous parts. Due to the danger of damaging the structure while preparing a green sample, again sintered samples are preferred.

This page intentionally blank

Assessing Powder Compacts

195

With the methods discussed for sample preparation the detection of crack size and type is limited to the location of the cutting plane. In an axi-symmetric case a radial cut will most probably show the relevant crack details. In other 3D cases it can be difficult to hit the crack with a cut. However, cracks can also be assessed by eye. In particular, when setting up a press for a good pressing schedule fast evaluation of the crack situation is needed. By breaking up the compact by hand along the crack, shear cracks show bright surfaces and can clearly be distinguished from tensile cracks. The defect assessment for the hardmetals was carried out visually. Before this the samples were sintered, because with the additive of 2 % PEG the cracks will be closed after cutting in the green state. The additive increases the mouldability of the powder and will be vapourised after the sinter process. After sintering, the samples could be examined without any difficulties to get a visual validation of the formation of cracks caused by the compaction process. In production, there may be an additional requirement for nondestructive detection of defects. This is not covered here, but a good introduction is given by Ernst and Donaldson [1].

12.8 Concluding Comments Comparison of compaction models is often most effectively done by comparing predictions for the density in significant regions of a component – see Chapter 13. To validate these predictions, only measurements of the same scale are required and the sectioning of large components into regions for density determination by machining, the Archimedes method or some equivalent method is appropriate. However it is known that densities can vary significantly over small regions - at changes in section of large components (e.g. Figure 12.15) or in small, complex components (e.g. Chapter 14). Thus, methods that give coarse-scale information should be complemented by methods such as metallography or X-ray tomography that can give this finer information. Crack detection by visual or metallographic means is sufficient for validation of model predictions.

References [1]

Ernst E and Donaldson I. 2004. The Application of Different NDT Processes for Automotive PM Components. Proceedings of Euro PM2004 , EPMA, 514-518.

13 Case Studies: Discussion and Guidelines O. Coube1 and P. Jonsén2 1

PLANSEE SE, 6600 Reutte, Austria. Now with European Powder Metallurgy Association, Talbot House, Market Street, Shrewsbury SY1 1LG, UK. 2 Luleå University of Technology, Luleå SE-971 87, Sweden.

This chapter presents a review of different case studies, i.e. practical examples of powder die compaction. Parts from four different powders – ferrous, hardmetal, ceramic and magnet – were first fabricated in an industrial environment and then reproduced virtually using numerical simulation. Results are given in terms of density distribution and loading. Parametric studies analyse the influence of different factors. The results obtained and their analyses serve as the basis for a guideline for a successful numerical simulation of powder compaction.

13.1 Introduction In order to analyse the difficulties of modelling and numerical simulation of powder compaction, three different case studies were designed. Geometries were produced in an industrial environment with different characterised powders – tungsten carbide, iron, ceramic and magnet – and experimental data collected. In the first case study (“CS1”) a tungsten carbide axisymmetrical flanged bushing geometry was designed. Parts were fabricated and numerical simulation of compaction was carried out. Case Study 1 is not described in this chapter, for more details on Case Study 1 see Appendix 2. In the second case – Case Study 2 (“CS2”) – the simulation of the compaction of a second axisymmetrical study on a ferrous powder part was performed. The third case study – Case Study 3 (“CS3”) – used a similar geometry to CS1 but with a curved upper punch and a larger inner corner to assess the influence of the powder transfer for the three nonferrous powders (tungsten carbide, ceramic and magnet). Seven Computing Centres (CEA/CEREM, CIMNE, Fraunhofer IWM, INPG/3S, Luleå University of Technology, University of Wales-Swansea and the Institute for Problems of Materials Sciences-Kiev) performed the simulations. This chapter does not intend to describe in detail all the three case studies. For those who would like to reproduce the simulation of one of them, they can find their complete descriptions and their main results in the Appendices and in [3], [4]

198

O. Coube and P. Jonsén

and [5], respectively. This chapter rather points out the lessons coming from these case studies. A parameter-sensitivity study was first performed on CS3 with tungsten carbide powder and points out the influence of two constitutive parameters on the results. In a second part, assuming that the powder is perfectly characterised, an analysis of CS2 with ferrous powder shows the influence of the initial and boundary conditions on the results of the simulation.

13.2 Constitutive Parameters Sensitivity 13.2.1

Case Study 3

An important factor to achieve good simulation result is to have correct data for the constitutive model [1]. The influence of some characteristic constitutive data is studied here. This is done for Case Study 3, which is a two-level geometry with a dished top produced by use of a domed upper punch and two lower punches as shown in Figure 13.1. Hardmetal powder WC-Co from Eurotungstene is used in this study. The tools in the simulation are considered rigid. The goals of the simulation are to predict the gradient of density within the compacted part and the punch forces. An experimentally assessed constant coefficient of friction µ = 0.25 was applied during the process. Figure 13.2 shows the sectioning for the density distribution for CS3.

(a) Figure 13.1. Case Study 3 geometry (a): To the left the axisymmetric model with domed upper punch; (b): On the right side the final shape with target height H1 = 20.07 mm, H2 = 9.72 mm and R2 = 2.00 mm. The heights set for the hardmetal powder are H1 = 47.5 mm and H2 = 19.97 mm before pressing.

Case Studies: Discussion and Guidelines

199

Figure 13.2. Green density sections for Case Study 3 in (g/cm3)

Analysis of Case Study 3 initiated the following parameter study regarding constitutive data in order to understand how rheological powder data may influence density and load results. For more information regarding Case Study 3, see the Appendix. 13.2.2

Constitutive Model used for the Parameter Study

A commonly used constitutive model for powder compaction is the so-called Drucker-Prager-Cap model. It consists of a failure line and a hardening cap as shown in Figure 13.3. The equation representing the shear failure surface is: Fs= q – ptanβ – d = 0

(13.1)

where β is the failure line angle and d the cohesion stress. The cap yield surface, of elliptical shape is defined by: Fc =

(p − p a )2 + (Rq )2 − R (d + p a tan β) = 0

(13.2)

where R, the eccentricity parameter, controls the shape of the cap, pb is the yield stress in hydrostatic compression and pa is the pressure of the intersection between the cap and the failure line, see Figure 13.3. The flow rule is defined assuming a potential similar to the yield surface. The movement of the cap is controlled by the hardening function as explained in Section 13.2.4. The parameters control the material flow during compaction and will influence density distribution and punch forces. To check the sensitivity, parameter studies on R and the hardening function are performed. For more information about constitutive models for powder compaction, see Chapter 4.

200

O. Coube and P. Jonsén

Figure 13.3. Representation of the Cap model in the p-q plane

13.2.3

Influence of the Constitutive Parameter R

The constitutive parameter R represents the eccentricity in the model and its value controls the shape of the elliptic cap surface. Depending on the type of powder, R may be density dependent. Transverse flow of powder during compaction and according to a density-dependent R is investigated elsewhere [2]. Since R appeared to be constant according to experimental characterisation of the tungsten carbide powder used in the Case Study 3, the present study is limited to constant values of R. In the parameter study four different values of R (0.50, 0.60, 0.80 and 0.90) have been investigated and compared with the reference simulation where R = 0.70. A change in R results in a change of pa, which must be readjusted with respect to model consistency, but not pb. The value of pb remains invariant, it is an experimental result of an isostatic compaction test. Figure 13.4 illustrates how the parameter R affects the shape of the cap model.

Figure 13.4. The shape of the Cap is dependent on the value of R. A Cap model is shown with different values of R, pb remains invariant.

Case Studies: Discussion and Guidelines

201

In the case of a die compaction test, the hydrostatic stress p, calculated from the axial stress, becomes the reference value. The influence of R upon final density distribution and force balance is studied. For all cases, punch force and six areas of mean densities are calculated, as shown in Figure 13.2. The major results from the study are presented in Figure 13.5, Table 13.1 and Table 13.2. Table 13.1. Experimental and simulated green densities. The reference simulation is marked with * Section

Experiment 3

R = 0.50 3

R = 0.60 3

R = 0.70* 3

R = 0.80 3

R = 0.90

[g/cm ]

[g/cm ]

[g/cm ]

[g/cm ]

[g/cm ]

[g/cm3]

1

8.088

8.258

8.286

8.310

8.331

8.343

2

8.018

8.220

8.201

8.188

8.177

8.167

3

8.068

8.236

8.227

8.217

8.208

8.200

4

8.284

8.305

8.294

8.283

8.272

8.263

5

8.036

8.016

8.028

8.044

8.056

8.067

6

7.912

7.926

7.944

7.965

7.984

8.002

Figure 13.5 represents the densification of each section according to R. 8.35 Section 1

8.30

Section 4

8.25 Section 3

ρ, [g/cm³]

8.20 8.15

Section 2

8.10

Section 5

8.05 8.00 Section 6

7.95 7.90

0.5

0.6

0.7

0.8

0.9

R Figure 13.5. Final density in each section as a function of cap eccentricity, R

The result shows that a larger value of R results in a better transverse flow. As expected, the compaction of a powder with “poor” transverse flow will assist the densification of the core of the part – i.e, Sections 2-4 – whereas the extremities –

202

O. Coube and P. Jonsén

Sections 1, 5 and 6 – benefit from large R-values. Notice also that the overall density gradient slightly decreases with increasing values of R, reducing from 0.38 to 0.34 g/cm³ for R=0.5 and 0.9, respectively. Table 13.2 reviews the loading balance between the punches and the ratio between the reacting forces of the lower outer – LOP – and inner punch – LIP. Table 13.2. Experimental and simulated punch forces. The reference simulation is marked with *. Values are given in their positive form, “UP” sign being the opposite to “LOP” and “LIP” signs. Punch

Experiment

R = 0.50

R = 0.60

R = 0.70*

R = 0.80

R = 0.90

[kN]

[kN]

[kN]

[kN]

[kN]

[kN]

UP

570

528

515

508

506

506

LOP

348

258

240

225

213

203

LIP

262

238

236

238

242

246

LOP/LIP

1.33

1.08

1.02

0.95

0.88

0.83

The results above shows that the constitutive parameter R has an influence on the force balance between the punches. Higher transverse flow capacity in the powder leads to a decrease of the forces necessary to compact this geometry. Furthermore, R influences the ratio between the reacting forces of the two lower punches. That is, the LOP/LIP ratio becomes smaller than one for R-values larger than 0.63 approximately. This result can be explained by the densification. As shown before, a lower value of R assists the densification of the core and particularly in Section 4 due to lower powder flow around the inner corner and lower densification of Section 1. Thus, material consolidates close to LOP, which increases the reacting force on LOP. These results are based on a supposed well-known hardening of the powder, which may also be an issue, as discussed in the following section. 13.2.4

Influence of the Hardening for High Density Values

The influence of the hardening upon density distribution and force is studied. In all simulations a value of R = 0.70 is used, except in Section 0. Figure 13.6 presents the experimental data for the hardening of the tungsten carbide powder. The data is fitted with a function of mean stress against density by: pb= a [(ρ/ρ0)b – 1]

(13.3)

where a and b are material constants and ρ0 the initial density. The density range of the measurements is between 5.00 g/cm3 and 8.25 g/cm3. In the simulation of Case Study 3, the highest local values for the density are around 8.6 g/cm3. Thus, a

Case Studies: Discussion and Guidelines

203

parameter study of the hardening function is carried out at high density values. To this end, the slope of the hardening function is altered for densities above 7.50 g/cm3. In this study, the same reference simulation as above is used, see Figure 13.6.

Figure 13.6. Experimental data for the hardening behaviour and the fitted function Equation 12.3 for the reference case; a = 0.0211 MPa and b = 9.84 with ρ0=3.20 g/cm³

In order to vary the hardening of the powder, the simulation of compaction is carried out with three different functions denoted H1, H2 and H3, as represented in Figure 13.7 together with the reference hardening function. H1 and H2 have a steeper slope for high density values than the reference hardening function, while H3 shows a flatter slope.

204

O. Coube and P. Jonsén

Figure 13.7. The different hardening functions and experinental data for Case Study 3

Table 13.3 shows the simulated green densities in the six sections in Case Study 3. For all the sections, change of density caused by hardening is negligible and density distributions are close to reference simulation values. Changes in hardening slope at high densities do not affect the transverse flow during compaction. Loading on the punches is summarised in Table 13.4, which shows that not only is there an influence on the total value for the punch forces, but also on the balance between the punch force of LIP and LOP. Both H1 and H2 show a larger punch Table 13.3. Experimental and simulated green density for different hardening functions Section

Experiment 3

H1

H2 3

Reference 3

3

H3

[g/cm ]

[g/cm ]

[g/cm ]

[g/cm ]

[g/cm3]

1

8.088

8.290

8.296

8.310

8.294

2

8.018

8.169

8.174

8.188

8.184

3

8.068

8.214

8.214

8.217

8.221

4

8.284

8.280

8.279

8.283

8.267

5

8.036

8.064

8.062

8.044

8.041

6

7.912

7.979

7.979

7.965

7.965

Case Studies: Discussion and Guidelines

205

Table 13.4. Experimental and simulated punch forces for different hardening functions Punch

Experiment

H1

H2

Reference

H3

[kN]

[kN]

[kN]

[kN]

[kN]

UP

570

560

518

508

455

LOP

348

277

245

225

203

LIP

262

242

230

238

212

LOP/LIP

1.33

1.14

1.07

0.95

0.96

force on LOP than on LIP, which is in agreement with the experimental results. H1 agrees well for the force on UP and LIP compared to experimental measurements, but the forces on LOP are still too low. H3 shows low forces for the punches and a poor balance between the lower punches. 13.2.5

Looking for Good Agreement with Experimental Values

In order to achieve a result closer to experiment, a simulation with R = 0.5 and hardening function of H2 is performed. This combination is interesting as the simulation from H2 gives better force balance between the lower punches and a better level on the forces. The hardening function of H2 is also close to the experimental data in Figure 13.7, the deviation of the slope starts close to ρ = 8.25. According to Table 13.1 and Table 13.2 the parameter R = 0.5 gives the best results compared to the experiment. Results from the combined simulation are shown in Table 13.5 and Table 13.6. The largest difference in green density is 2.3 % and found in Section 2, where reliability of experimental measurements is in the same range or higher due to the complexity of the geometry - see Chapter 12. However, this simulation does not improve significantly the prediction of the density distribution compared to the reference simulation. Table 13.5. Experimental and simulated density for the combined simulation Section

Experiment 3

Comb sim

Diff

[g/cm ]

3

[g/cm ]

[%]

1

8.088

8.259

2.1

2

8.018

8.200

2.3

3

8.068

8.235

2.1

4

8.284

8.305

0.3

5

8.036

8.028

0.1

6

7.912

7.934

0.3

206

O. Coube and P. Jonsén

Table 13.6. Experimental and simulated punch forces for the combined simulation. Positive values in column Diff indicate that measured forces are larger than predicted forces. Punch

Experiment

Comb sim

Diff

[kN]

[kN]

[%]

UP

570

541

5.1

LOP

348

290

16.7

LIP

262

225

14.1

LOP/LIP

1.33

1.29

3.0

The real improvement is, however, in term of force prediction. Precision for UP and LOP is improved by a factor of 2 compared to the reference simulation, see Figure 13.8, and the total pressure level is increased. The balance between LOP and LIP is improved compared to experimental measurements. Precision for LIP decreases slightly but remains under a 20 % threshold that is considered acceptable. The residual difference in forces is probably due to an interaction between the die and the punches or between the lower punches themselves, since the equilibrium in axial forces between the upper punch and the two lower punches gives 40 kN for

Diff in % between Exp. and Sim.

Figure 13.8 represents the improvements of the force predictions compared to the experimental measurement for different combinations of hardenings and eccentricity parameters R. 36 32 28 24 20

H2 / R=0.5 H2 / R=0.7 Href / R=0.5 Href (R=0.7)

16 12 8 4 0 UP

LOP

LIP

Punch Figure 13.8. Difference in percentage between measured and predicted forces for different hardening and R combinations, H ref being the reference hardening. Positive values indicate that measured forces are larger than predicted.

Case Studies: Discussion and Guidelines

207

the experiment and between -32 and -57 kN for the simulations. This difference of sign between measurement and simulation results from the interaction – friction – between the pressing tools during the real process. This cannot be taken into account in the simulation due to the rigid modelling of the tools. Another example of this problem is given in Section 13.5.3. 13.2.6

Discussion

This example points out the influence of the eccentricity parameter R and the hardening curve upon the density distribution and force predictions. The parameter R has a direct influence on the transverse flow, i.e, the density distribution, and then indirectly on the force predictions. The influence of R varies according to the geometry. A multilevel geometry with large inner corners and curved punches, like Case Study 3, is more subject to transverse flow than a “one block part” and therefore the results are more R dependent. The main issue in terms of force prediction remains the hardening curve. Fitting a curve with a potential function leads to possible errors of the same order. Therefore, extrapolation after the measured density range should be avoided. Otherwise, forces and stresses predictions can only be qualitative and sometimes misleading like for the ratio LOP/LIP of the forces in Case Study 3.

13.3 Framework for the Numerical Simulation In addition to the material data of the powder, a detailed framework for the numerical simulation of powder compaction must be established prior to any calculation. This framework should contain specific initial and process data explained as shown in Table 13.7. Table 13.7. Framework for the numerical simulation of powder die compaction: Initial and process data Initial data 1

Geometry of the pressing tools – punches, die, core – [mm]

Initial data 2

Initial position of the pressing tools [mm] prior to compaction. Initial height(s) of the powder [mm] prior to compaction

Initial data 3

Fill density [g/cm³] or mass of the powder [g] Optional: Fill density distribution [g/cm³]

Process data 1

Friction coefficient between powder and pressing tools

Process data 2

Kinematics of the pressing tools at the powder surface [mm] and/or Loading history [N/mm²] (should be avoided, see comment below)

For 3D simulation CAD data should be available for Initial Data 1. Starting from the press tooling surfaces finite-element preprocessors reconstruct the powder at its

208

O. Coube and P. Jonsén

fill state. This step requires, however, a certain expertise in the preprocessor software. In an industrial context Initial Data 3 is generally the mass of the green – i.e. compacted – part, which gives, together with the filled powder volume – Initial Data 2 – the fill density. The fill or starting density is then the volume of the poured powder divided by the mass of the green part. Except for fill density distribution, the Initial Data can be easily obtained from production and design departments. The Process Data 1 describes the friction between powder and tooling; it can be assessed by experiment or found in the literature. Simulation gernerally uses a constant Coulomb coefficient that reflects the average ratio between normal and tangential forces on the tooling walls during die compaction. The Process Data 2 is the so-called boundary conditions in numerical simulation. It must be defined on the whole boundary with a combination of displacements and forces, where available, in order to ensure a solution of the problem. It is the most problematic data necessary for simulation. First, it is difficult to ascertain the kinematics – especially in mechanical press apparatus – and second elastic tool deflections have to be taken into account in order to reproduce the real punch movement at the powder surface. An important point must be highlighted here: The pressing tools are often modelled as rigid bodies in the numerical simulation and therefore only those surfaces in contact with the powder require to be meshed. Loading history could be an alternative but in this case, the exponential form of the powder hardening is a handicap for the correct prediction of the density distribution and dimension, as shown in Section 13.2.4. Once the material characterisation is performed and implemented in the simulation, the initial and process data become well defined, and this assures that a numerical simulation can be performed in a proper manner, as will be demonstrated in the following example. In this example, the compaction of a two-level geometry as shown in Figure 13.9 – “Case Study 2” – is studied. The powder used for this study is an iron-based powder – Distaloy AE – provided by Hoganas AB.

Case Studies: Discussion and Guidelines

209

Figure 13.9. Case Study 2 geometry. The fixed diameters are: D1=10 mm, D2=30 mm, D3=50 mm. The heights set for the Distaloy AE powder are HF2=6 mm and HILP= 56 mm after filling. Left: Sections used for the assessement of the density distribution after compaction according to the numerical simulation and the experimental measurements. The heights of Sections 3-7 are more or less equal.

One of the goals of the current simulations is to predict the gradient of density within the compacted part. For this purpose, seven simulation Centres used different finite-element-codes (ABAQUS/Explicit, ABAQUS/Standard, PREcad, LS-DYNA, Merlin and MRCP/FEA) but similar versions of the modified DruckerPrager-Cap models, except the Code Merlin that uses the Cam-Clay Model. All the Centres received the same material data for implementation into the models. The die compaction of the metal powder was numerically simulated using the same initial and process data. The density was then calculated in seven sections along with the resulting forces, as shown in Figure 13.10.

210

O. Coube and P. Jonsén

Figure 13.10. Simulated density per sections (in g/cm³) and forces applied by the punches (in kN) according to the different FE codes

Numerical predictions show about 1 % difference in pressed density between the different computing centres and about 14 % in forces applied by the punches. The discrepancy in forces may be a result of the different mesh sizes used by the computing centres but is more likely due to the different mathematical fits used by the centres for the experimentally assessed hardening data. For practical reasons each centre was free to use its own mathematical interpolation. Since the relation between volumetric strain and pressure is exponential, a difference of a few per cent in the strain prediction may lead to a ten times larger difference in the stress prediction. This example shows, however, that with a good common databank on powder properties and the same initial and process data, the type of FE codes should not influence the results.

Case Studies: Discussion and Guidelines

211

13.4 Influence of Meshing Finite element codes use a mesh of elements made of nodes. Displacements of nodes reproduce the deformation of the mesh, while the results are interpolated in the elements in terms of stress, deformation, state variable, etc. Thus, a discrete structure is used to predict the behaviour of a continuous environment. The question is then not, does the mesh size matter – the answer is yes – but to what extent? In order to study the influence of mesh size the numerical simulation of compaction of Case Study 2 is performed again with different mesh sizes. Meshes using 416, 1027 and 2480 elements respectively were used for the simulation. The corresponding density distributions are shown in Figure 13.11. The results are similar for the three meshes, except in the upper part where the coarse meshing predicts a slightly lower density (less than 1 % difference). The critical part is the inner corner, where the element size influences the assessment of the density. This is because the mesh “penetrates” the lower outer punch (“OLP” in Figure 13.9) in the corner area because of the geometry and the pressing schedule. It happens for all three meshes. It could be avoided by slightly rounding the 90° corner as it certainly is in practice; a fine mesh must then be applied fully to reproduce the rounding. An alternative method may be a remeshing procedure that exists in an automatic form in some finite-element codes but not in all. This defect is, however, negligible for the finer mesh but is not for the coarse one and leads to a lower densification in the corner area.

Figure 13.11. Simulated density distributions in g/cm³ according to the mesh size[OC1]

Densities in different sections are shown in Figure 13.12, including tool forces. The forces on the tooling are similarly influenced by the mesh size with a maximum difference of about 5 % between the coarsest and the finest mesh.

212

O. Coube and P. Jonsén

Figure 13.12. Simulated density per section (in g/cm³) and forces applied by the punches (in kN) according to the mesh size

Another study [6] has shown that modelling of cracking can be strongly influenced by the size of mesh. Since cracking can be considered as a geometrical singularity, the mesh should be in the same order of size as the potential cracking zone.

13.5 Influence of the Initial and Process Data In an industrial environment, the goal of the numerical simulation of powder compaction is to improve the productivity by feasibility study, comparison study, tool-geometry optimisation, etc. The precision required for such tasks is then equal to the required tolerances of the final parts that are in the PM industry generally high. Therefore, it is essential to have a framework that is as precise as possible, which means reliable material, initial and process data. In a laboratory, these conditions can be well defined but in an industrial environment uncertainties cannot be avoided. The case studies analysed in this work were performed on an industrial press machine operated by PM companies. The measurements of Initial Data 1 and 2 were easily obtained. Initial Data 3 was given in terms of mass of the part. The influence of a possible fill density distribution will also be studied later. The Process Data 1 was obtained from measurements. The friction coefficient is actually a complex parameter that depends on density, pressure and even on velocity. However, the compaction usually occurs at a constant velocity and pressure relates to density. Thus, the friction coefficient can be considered in a first approximation as a function of solely the density. Usually, a mean value over the whole density range of the compaction is used for the numerical simulation.

Case Studies: Discussion and Guidelines

213

Process Data 2 was obtained from the kinematics of the pressing tools but with elastic deflection that could not be measured directly on the press machine. Current press machines can correct the elastic deflection of the apparatus only until the punches. There then remains the deflection of the punches, which must be taken into account. In the case studies, the machine deflections were assessed from the final dimension of the part and from the forces applied on the punches. Therefore, punch deflections could only be estimated roughly. Numerical simulations of die compaction of Case Study 2 were performed, taking into consideration these uncertainties. A discussion of these issues will follow. 13.5.1

Reference Case

A simulation is first performed according to the defined material, initial and process data in order to reproduce the experimental compaction of a part. Initial Data 3 is in a first step given by the mass of the part. Figure 13.13 shows the density per section and the forces applied by the punches for the numerical simulation of the reference case and for the experiment.

Figure 13.13. Simulated density per section (in g/cm³) and forces applied by the punches (in kN) for the reference case compared to the experiment

The simulation that is conducted under the given framework did not show a good agreement with the experimental results. The prediction of densities and forces showed differences of up to 8 % and 134 %, respectively. Assuming that the powder had been characterised with a suitable level of accuracy, the framework, i.e, the initial and process data, must then be modified and improved. Two weak points of this simulation framework are that the fill density is assumed constant and that the kinematics are given with only approximate estimates for deflections. The following sections analyse the effect of these weaknesses on the results.

214

O. Coube and P. Jonsén

13.5.2

Influence of the Fill Density Distribution

An important initial condition in simulation of powder compaction is the fill density. Instead of a constant fill density, a distributed density can be applied. The influence of fill density distribution upon green – pressed – density is investigated in Case Study 2. The mass of the studied part is 145 g, which corresponds to an overall fill density of 3.39 g/cm3. The fill geometry was divided into seven sections in order to reproduce a filling density distribution. The densities in each section were experimentally assessed. Numerical simulation of the compaction was conducted with and without fill density distribution that is the reference case. In both cases, the overall the fill density distribution remains at 3.39 g/cm3. According to the experimental assessment, the starting density is higher in the upper regions 1 and 2, which exhibit better fill conditions probably due to the small height to fill and to the contact with the fill shoe and in the lower regions 6 and 7 – probably due to self-densification by gravity – as shown in Figure 13.14.

Figure 13.14. Fill density distribution in the simulation (values in g/cm³) prior to compaction

Figure 13.15 shows the density per section and the forces applied by the punches for the numerical simulation of the distributed fill density variant (dfdv) and for the experiment.

Case Studies: Discussion and Guidelines

215

Figure 13.15. Simulated density per section (in g/cm³) and forces applied by the punches (in kN) for the distributed fill density variant (dfdv) compared to the experiment

The influence of the fill density distribution upon the green part in the numerical simulation of compaction is shown in Figure 13.16. The increase or decrease of fill and green densities, respectively, between the constant (cfdv) and the distributed fill density variant (dfdv) are represented in per cent for each section. Thus, the flange (Section 1) starts the dfdv compaction process with ca. 1 % more fill density than the average value and keeps it at the end due to a proportional pressing schedule. On the other hand, the leg (Sections 2-7) starts the compaction for the dfdv with strong density variations as in the average fill density, i.e. from -2.5 % to +2.5 % depending on the section. However, after compaction it does not show such variation but rather a constant slight decrease in the density of between -0.13 and -0.21 % – compared to the cfdv. In this case, the column or leg had dominated the sections and asserted its overall fill density variation of -0.19 % compared to the average density.

216

O. Coube and P. Jonsén

Differrence of density in %

3

Increase or decrease of the fill density vs the cfdv (3.39 g/cc)

2 1

Section

0 1

2

3

4

5

6

7

-1 Increase or decrease of the green density -2 vs the cfdv Difference in fill density Difference in green density

Figure 13.16. Difference of density in each section between the constant and the distributed fill density variants – dfdv – prior and after the compaction. Positive value means an increase of the density in the section for the dfdv compare to the cfdv.

Predicted densities in the seven sections are compared to experimental green density measured by means of Archimedes’ principle, i.e. the buoyancy method. Figure 13.17 shows the difference in density found between the measurement and the prediction for the constant – cfdv – and the distributed fill density variant (dfdv). In both simulations, the lower Sections (3-7) are slightly denser than in the experiment. The upper Sections 1 and 2 show, however, predicted lower densities as measured. The dfdv improves slightly the density prediction for all sections except Section 2.

Differrence of density in %

2 0 -2

1

2

3

4

5

6

7

Section

-4 -6 -8

Difference between experiment and dfdv Difference between experiment and cfdv

Figure 13.17. Difference of density in each section between the experiment and both variants dfdv and cfdv. Positive values means an overestimation of simulation compared to experiment.

Case Studies: Discussion and Guidelines

217

No major improvements have been achieved with the distributed fill density variant – dfdv. The density and force prediction still do not show a good agreement with the measurement. This variant helps us however to put into perspective the influence of the fill density distribution upon the green density distribution. As shown in Figure 13.16 the density distribution in the axial direction is homogenised by the compaction and only the mass difference in the columns influences the result. Two additional comments must be made. First, the kinematics or the form of the punches can influence this result. A transfer phase prior to or during compaction or a curved punch may increase the radial flow of the powder. Secondly, filling is not an ideal axisymmetrical process as several studies have shown in [7] and Chapter 9. Therefore, a numerical simulation in 3D of the influence of the filling density on the green one should be carried out for more accuracy. The attempt to improve the results of the simulation in comparison to the experiment was not successful. The next section analyses the influence of Process Data 2: The punch kinematics in this case. 13.5.3

Influence of the Punch Kinematics

Looking at the density distributions and applied forces, we can see large discrepancies between the numerical prediction and experimental measurement in the two upper sections and on the upper (UP) and lower outer punches (LOP) for the densities and the forces, respectively. In fact, they both are underestimated by the numerical simulation, which indicates an error in the punch kinematics of the lower outer punch. Knowing that the punch deflections were not measured but assessed from applied forces on the punches and from final part dimensions, the estimates may not be entirely accurate. The deflections for each punch were estimated to be of the order of 1 mm each. Estimation based of a correct green density of section 1 leads to a subtraction of 0.2 mm from the deflection of the lower outer punch. Another numerical simulation noted cdvd-defl – constant fill density variant with new deflection – was performed with the corrected deflection. Figure 13.18 shows the results in term of density per section and forces.

218

O. Coube and P. Jonsén

Figure 13.18. Simulated density per section (in g/cm³) and forces applied by the punches (in kN) for the constant fill density variant (dfdv) compared to the experiment

When the deflection is corrected by 0.2 mm, Section 1 increases its density by 5 % and the forces on the upper and lower outer punch by 20 and 21 %, respectively. This corresponds more closely to the experimental measurements of density and punch load, respectively. Figure 13.19 shows the difference in density found between the different variants and the measurement. It includes a third variant denoted dfdv-defl, which also takes into account the fill density distribution in addition to the corrected deflection.

Differrence of density in %

2

cfdv-defl dfdv-defl

0

1

2

3

4

5

6

7

Section

-2

Difference between experiment and cfdv Difference between experiment and dfdv Difference between experiment and cfdv-defl Difference between experiment and dfdv-defl

-4 -6 dfdv

-8

cfdv

Figure 13.19. Difference of density in each section between the experiment and both variants dfdv and cfdv with and without deflection correction. cfdv: Constant fill density variant; dfdv: Distributed fill density variant; -defl: Variant with deflection correction. Positive value means an overestimation of simulation compared to experiment.

Case Studies: Discussion and Guidelines

219

The lower regions 3 to 7 are slightly more densified with the variants with deflection compared to the one without deflection. However, the prediction remains within acceptable levels as the difference with the measurement was below a 2 % threshold. Similar results are obtained in Sections 1 and 2 with the corrected deflection. Taking into account the fill density distribution in addition to the deflection, the same improvement as before the kinematics correction is achieved but with a greater benefit for Section 1. The same comparison is carried out in Figure 13.20 for the applied forces by the punches. Again, the deflection variants improve the prediction such that UP and LIP punches show a discrepancy of below 10 and 5 %, respectively. However, in the case of the LOP, although the prediction is improved by a factor higher than 1.5, the simulations with corrected deflection still shows a difference of about 33 ± 3 % depending on the fill density option.

Difference of force in %

20

0

Difference between experiment and dfdv Difference between experiment and cfdv Difference between experiment and cfdv-defl Difference between experiment and dfdv-defl

UP

Punch LIP

LOP

-20

-40

-60

dfdv-defl cfdv-defl

dfdv cfdv

Figure 13.20. Difference of forces applied by the punches between experiment and both variants dfdv and cfdv with and without the deflection correction. cfdv: Constant fill density variant; dfdv: Distributed fill density variant; -defl: Variant with deflection correction. Negative values mean an underestimation in absolute value of simulation compared to experiment.

A closer look at the forces measured experimentally shows that the equilibrium is achieved with a contribution of -143 kN, which is usually provided by the friction between the powder and the die and the core. In this case, the value is excessively high and pointing in the other direction to expectation. According to numerical simulation, the contribution to the axial forces due to friction is 20 kN. This means that the die and core must react with -163 kN against one of both lower punches. In all probability, the lower outer punch can be said to interact with the die and must

220

O. Coube and P. Jonsén

develop an additional force to the one needed for the powder compaction. The latter is then of the order of 587-163 = 424 kN. This gives, finally, for LOP a discrepancy of 12 % and 5.7 % with the numerical prediction, which improves the first result with a factor 4.75 and 10 for the cfdv-defl and dfdv-defl variants, respectively. 13.5.4

Discussion

This case study is a very good example of the difficulties that may be encountered in the numerical simulation for predicting a real pressing process to achieve high relative densities in the compacted part The fill density distribution is a minor factor in this study improving the maximal deviations with experiment from 8 to 7 % and from 54 to 57 % for the density and the forces, respectively. On the other hand, a correction of 0.2 mm in the kinematics of one punch – representing less than 1% from the total displacement of the punch by pressing – can increase accuracy in predicting the density and forces from 8 to 2 % and 57 to 36 %, respectively. Furthermore, it turns out that the remaining error of 35 % is mainly due to friction between pressing tools. The Tables 13.8 and 13.9 show the results obtained with the different variants in term of precision in the prediction of density and load distribution, respectively. Based on the results of this work, the following conclusions could be drawn. At high levels of pressure and density, it is recommended to model punches and Table 13.8. Difference in % between measured and simulated density for the different variants. Positive values means an overestimation of simulation compared to experiment. Variant

Section 1

Section 2

Section 3

Section 4

Section 5

Section 6

Section 7

cfdv

-8.25

-1.79

0.27

0.95

1.59

1.24

1.15

dfdv

-7.17

-1.94

0.06

0.79

1.43

1.11

1.02

cfdv-defl

-1.91

-1.28

0.49

1.14

1.74

1.38

1.28

dfdv-defl

-1.01

-1.55

0.14

0.84

1.48

1.15

1.06

Table 13.9. Difference in % between measured and simulated forces for the different variants. Negative values means an underestimation in absolute value of simulation compared to experiment. Variant

Upper punch

Lower Inner Punch

Lower Outer Punch

cfdv

-27.06

-2.70

-57.3

dfdv

-25.30

-4.17

-54.66

cfdv-defl

-7.88

-2.46

-36.44

dfdv-defl

-4.37

-3.91

-31.87

Case Studies: Discussion and Guidelines

221

contingently die and cores because punch deflection and tool interaction may influence the result in term of densities, loads and eventually stress states. All these data are, for example, essential for the study of cracking during die compaction, which is one of the main concerns for the PM industry.

13.6 Conclusions and Guideline The engineer or scientist who starts a project of simulation of powder compaction should keep in mind two possible main sources of error. One comes from imprecise definitions of the framework data – Table 13.7 – the second may result from imperfections in the powder characterisation. Therefore, the following summary guidelines are proposed. These may not be exhaustive. •

•

•

• • •

•

•

If the powder characterisation for some variables or a framework data seems to be deficient, a parameter study should be undertaken to assess the possible range of error. Possible parameter studies are R, hardening, punch deflection, friction coefficient, etc. The main issue in terms of force prediction remains the hardening curve. Fitting a curve with a potential function leads to possible errors of the same order. Therefore, extrapolation after the measured density range should be avoided. The hardening experimental data point of the powder should cover the highest density obtained in the simulation and preferably a few per cent higher if the experimental apparatus allows it. Although the eccentricity parameter R has a direct influence on the powder transverse flow during compaction, small variations of R lead to changes in density that remain acceptable but can indirectly influence loading prediction. If it is possible to assess the fill density distribution for Initial Data 3 of the framework, columns values are sufficient in a first order since they dominate sections values – see Figure 13.16. Filling density distribution is, under normal process conditions, asymmetric and should lead to 3D simulation even for an axisymmetric geometry. Axial force balance should be compared between simulation and process. Too large a difference may result either from an incorrect friction coefficient used for the simulation or from an interaction between the pressing tools in the process. If precise loading and stress state must be precisely predicted – e.g., for a cracking study – punches and contingently die and cores should be modelled as deformable bodies instead of as rigid bodies. This would, however, increase the computing time and the possibility of divergence of the simulation. Coarse meshing may influence results in term of punch loading – 5 % deviation in the example of Section 13.4 – local density distribution – corner and other geometrical singularities – and even cracking that can be seen as a “singularity”. Use finer meshing around these singularities and avoid 90° corners that usally exhibit rounding in the real process.

222

O. Coube and P. Jonsén

The different case studies have shown the following best order of precision when compared with experiment: 1-2 % for density distribution in sections compared with Archimedes-method measurements, 10-20 % for punches loading in comparison with direct measurements on the machine. This should remind us that modelling using numerical simulation remains an instrument that gives an approximation of the reality. However, when the assessment reaches precision of order 2 % or below for the density distribution, the engineer and scientist can trust the instrument. Especially when he keeps in mind that the 10-20 % force discrepancy are due to known factors like modelling of hardening or interactions between tooling elements.

References [1] [2] [3]

[4]

[5]

[6] [7]

PM Modnet Computer Modelling Group. 2000. Sensitivity of numerical simulation to input data, Proceedings of 2000 Powder Metallurgy World Congress, JPMA, Kyoto, Japan. Coube O and Riedel H. 2002. Modelling of Metal Powder Behaviour under Low and High Pressures, Proceedings of the 2002 World Congress on Powder Metallurgy & Particulate Materials, MPIF, Orlando, Florida. USA. Coube O, Federzoni L, Cante J, Oldenburg M, Chen Y, Imbault D, Dorémus P, Tweed J, Leuprecht A, Markeli W and Brewin P. 2003. A Complete Study of the Die Compaction of a Flanged Bushing – From the Powder Characterisation to the Validation of the Numerical Simulation, European Powder Metallurgy Conference on Meeting the Challenges of a Changing Market Place, EPMA, Shrewsbury, UK. Vol. 3, 71–76. Coube O, Chen Y, Imbault D, Dorémus P, Maassen R, Federzoni L, Jonsén P, Tweed J, Gethin D, Rolland S, Maydanyuk A and Shtern M. 2004. Computer Simulation of Die Compaction: Guidelines and An Example from the European Dienet Project, Euro PM2004, Powder Metallurgy World Congress & Exhibition, EPMA, Shrewsbury, UK. Vol. 5, 209–214. Coube O, Jonsén P, Kraft T, Chen Y, Imbault D, Dorémus P, Gethin D, Rolland S, Federzoni L, Maydanyuk A, Shtern M, Cocks A, Tweed J and Markeli W. 2005. Numerical Simulation of Die Compaction: Case Studies and Guidelines from the European Dienet Project, Euro PM2005, Powder Metallurgy Congress & Exhibition, EPMA, Shrewsbury, UK. Vol. 3, 313–320. Coube O. 1998. Modelling and Numerical Simulation of Powder Die Compaction with Consideration of Cracking, Dissertation, Thèse de l’Université Pierre et Marie Curie Paris 6, Paris. Hjortsberg E and, Bergquist B. 2005. Powder Metall., 45 (2), 146-153.

14 Modelling Die Compaction in the Pharmaceutical Industry I.C. Sinka1 and A.C.F. Cocks2 1

Merck Sharp and Dohme Ltd, Hoddesdon, Herts. EN11 9BU, UK. Now with Department of Engineering, University of Leicester, University Road, Leicester LE1 7RH, UK. 2 Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK.

This chapter presents the current state of die compaction modelling in the pharmaceutical industry. The topic is introduced with a review of the particular features of pharmaceutical powder formulations. The operation of rotary production presses is described together with experimental procedures developed to characterise die-fill and high-speed compression. The experimental techniques described in Chapters 4-8 are employed to characterise the powder– die-wall friction interaction and to calibrate a modified Drucker-Prager-Cap model for microcrystalline cellulose, where the material parameters are expressed as functions of relative density. Three case studies are presented on the density distribution in curved faced tablets, bilayer tablets and compression coated tablet systems. The model predictions are validated using experimental data. The effect of tablet microstructure on strength and breakage behaviour is illustrated. The application of modelling to formulation design, process development, tablet image and tool design is discussed.

14.1 Introduction Approximately 80 % of all medication is administered in solid dosage form, primarily as tablets. Tablets are elegant, easy to use and lend themselves to highvolume production. A modern rotary tablet press can compress in excess of 500,000 units/hour in normal operation. Tablets are complex powder systems containing one or more active ingredients and a range of excipients to achieve the desired mechanical strength and bioavailability (disintegration or dissolution) profiles. Mechanical strength is necessary to maintain the integrity of tablets during post-compaction operations, such as coating, handling, packaging and transport. More common defects include cracks, laminations, edge chipping, erosion, etc. In broad terms these problems originate from insufficient material strength or flaws

224

I.C. Sinka and A.C.F. Cocks

induced during manufacture and can be traced to the porosity and microstructure of the tablet. These characteristics also influence the behaviour of the tablets in disintegration or dissolution media. The internal structure of powder compacts has been studied since the early 1900s. Train [1] describes techniques based on differential machining, hardness testing or the examination of X-ray shadows of lead grids placed in the compact. In the 1970s Macleod and Marshall [2] performed X-ray autoradiography experiments of radioactive ceramic powders and related the density distributions to die-wall friction. In recent years, nondestructive tomographic techniques (X-ray and acoustic waves) and nuclear magnetic resonance imaging (NMRI) have been developed, as summarised by Lannutti [3]. The techniques that are most widely used today are summarised in Chapter 12. X-ray computerised tomography (CT) has been employed for characterising a wide variety of powder compacts. The internal density distribution of pharmaceutical tablets has been studied by Sinka et al. [4]. Wu et al. [5] used Xray CT to detect cracks within a tablet and explored the influence of powder composition and processing parameters on the propensity to form cracks. Other studies have employed the metal tracer method [6] and NMRI [7] to examine the internal structure of pharmaceutical tablets with curved faces. Figure 14.1 illustrates the density distribution profiles determined using Xray CT in the vertical (YZ) plane of two pharmaceutical tablets made of microcrystalline cellulose. In these tomographs, the darker the colour the higher the density, as indicated in the legend. The two tablets have approximately the same weight and average relative density and were compressed in identical dies using punches having different curvatures to highlight the effect of tablet shape on density distribution. The tablets contain a notch (break line) at the top to aid splitting and letters are embossed on the lower surface to identify the tablet and manufacturer. Significant variations in density occur at these discontinuities, which may induce local differences in material properties. Low-density regions, such as between the

y z x

2 mm

Figure 14.1. Density distribution in tablets using X-ray CT (after Sinka et al. [4])

Modelling Die Compaction in the Pharmaceutical Industry

225

letters embossed on the lower face of the tablet, or along the flanks of the break line, may lead to erosion during bulk handling of the tablets or during coating, which affects the elegance of the product. High-density regions around the outer edge may be either beneficial in reducing chipping and breakage at the edges, or detrimental; for example, the high-density region just below the break line (present in the top tablet only) may require higher forces to break the tablet into two parts, or may result in halves having weights outside specification. It is therefore important to understand and control the microstructure and properties of the tablet. The importance of process models in tablet design has been recognized since the 1980s [8], however, compaction modelling was initially driven by the metal and ceramic powder processing industries. It is only in recent years that compaction models have been applied in the pharmaceutical industry [5, 9-12].

14.2 Pharmaceutical Formulations and Processes Typically, tablets contain active pharmaceutical ingredients (API) and excipients that have a wide range of functions, for example: fillers (to add bulk); binders (to aid compactibility); disintegrants (usually polymeric materials that swell in contact with fluids); lubricants (to reduce die-wall friction); glidants (to aid powder flow); antioxidants; colouring; flavouring agents, etc. Mixing of these components together is a key operation, with the objective of producing a uniform powder blend. One of the simplest production routes is direct compression, as it involves only mixing, followed by lubrication (when a lubricant is admixed to the powder blend) and compaction. In many cases, however, an agglomeration step is necessary to achieve a more uniform blend and improve flowability and compactibility. There are two main methods used in granulation of pharmaceutical powders: wet and dry. Wet granulation involves adding a granulating fluid to form liquid bridges between particles. This fluid is removed during a subsequent drying step. Milling is also generally necessary to achieve the desired size distribution of the granules. Dry granulation includes roller compaction and slugging. Roller compaction involves pressing the powder between rollers to produce a strip, which is then milled down to granules. Slugging involves pressing larger compacts to low density, which are also milled. Pharmaceutical materials are relatively light, having a true density in the range 1.5-2.5 Mg m-3. The relative density of most formulations and bulk excipients is in the range 0.3-0.4, which is significantly lower than for metal and ceramic powders. As a result of these low densities, pharmaceutical formulations generally have poor flow properties. Detailed descriptions of pharmaceutical tablet formulations and processing can be found in specialised texts and monographs [13-16]. A succinct presentation of the special features of pharmaceutical tableting, compared with other powder systems and compaction processes presented in this book, is given below. Pharmaceutical tablets are relatively simple and the tooling consists of a die, a single upper punch and a single lower punch. However, the tablet geometry can be complex, including a series of obvious or more subtle features, as illustrated

226

I.C. Sinka and A.C.F. Cocks

in Figure 14.1. The tableting operation consists of die-filling, compaction and ejection. Powder transfer, as employed in the production of complex multilevel parts, is absent as a distinct stage. The density and strength achieved upon compaction is final, in the sense that compaction is not followed by sintering. Tableting presses can be divided into two broad categories: 1. single-station (or single punch, or eccentric) presses are used in development and low-volume production, and are similar to presses used for other powder systems 2. multi-station or rotary presses, which are used in high-volume tablet manufacturing. In addition to the characteristics of the powder or granules, the choice of process parameters during tablet compaction influences the properties of the tablets. The vast majority of tablets are produced using rotary presses and we concentrate on this type of press in the remainder of this chapter. The operation of a typical rotary tablet press is outlined below with emphasis on the process parameters that are used as input data for numerical modelling. In order to illustrate how pharmaceutical powders perform during different stages of the manufacturing process, we base any discussion and models on the commonly used pharmaceutical excipient, microcrystalline cellulose. Microcrystalline cellulose is added as an adsorbent or disintegrant to tablet formulations. It occurs as a white crystalline powder composed of irregular porous particles that result from spray drying, as illustrated in Figure 14.2. It is commercially available in different particle sizes and moisture grades that have different properties and applications. In this work we use Avicel PH102 (manufactured by FMC Biopolymer) that has an average particle size of 100 µm. The bulk density of the powder is around 300 kg m-3 and the full Figure 14.2. Microcrystalline cellulose density is 1520 kg m-3.

14.3 Rotary Tablet Press Production Cycle Rotary presses are used for high-volume production, of the order of 100,000 500,000 tablets per hour. The central component of a rotary press is a round die table with a number of stations, which consist of upper-punch – die – lower-punch assemblies. The operating cycle of a rotary tablet press is illustrated in Figure 14.3,

Modelling Die Compaction in the Pharmaceutical Industry

227

which is based in broad terms on the layout of a Fette 1000 rotary press (Fette GMBH, Schwarzenbeck, Germany). As the die table (1) rotates as indicated in Figure 14.3a, each tooling station passes successively through the die-fill mechanism (4), compression rollers (6 and 7) and ejection cam mechanism (8). The specific characteristics of die-fill, compression and ejection are discussed in turn. 14.3.1 Die-fill on Rotary Presses The importance of die fill has been highlighted in Chapter 9, with focus on the mechanisms specific to single-station presses that employ sliding-shoe delivery systems. The dominant driving force for this configuration is gravity. On rotary presses, the die-fill system consists of a mass flow hopper connected to a feed frame. The feed frame consists of a gravity hopper and can include motor-driven powder-transfer mechanisms depending on the size of the press and flow properties of the material that is being compressed. For a typical production press the feed frame consists of a number of paddle wheels as illustrated in Figure 14.3a. The paddle wheels (3) and (5), which are in the immediate vicinity of the die table, are referred to as feeding and metering wheels, respectively. As described elsewhere in more detail [17] the die-fill mechanisms in addition to gravity fill also include: • force feed: the feeding wheel has profiled paddles that stir and transfer the powder towards the die opening • suction fill: the lower punch is moved downwards using a fill cam to create the die cavity while the top of the die is exposed to powder • weight-control mechanism: after the die is filled, the lower punch is moved upwards and part of the powder is ejected (weigh uniformity is critical for pharmaceutical tablets) • for larger rotary presses, such as that illustrated in Figure 14.3a, weight uniformity is assisted further by the presence of a second paddle wheel (metering wheel) • additional effects include: centrifugal forces and vibration of the system during the operation of the press. The contributing factors can be investigated experimentally using a model dieshoe system [18]. The mass of powder deposited in the die is dependent on the shoe velocity. The use of the concept of a critical velocity (i.e. the shoe velocity above which incomplete filling occurs) to describe the flowability of a powder has been developed by Wu et al. [18]. The flow behaviour of seven pharmaceutical powders and formulations has been characterized using the model die-shoe system [19], which allowed evaluation of the flow behaviour under the effect of gravity and gravity assisted by suction filling. The critical velocity for microcrystalline cellulose is presented in Figure 14.4. Dimensional analysis was employed to provide a framework for interpretation of the experimental results and to guide the extrapolation procedures to other die and shoe geometries [19]. When scaling the experimental results from a model die shoe system using gravity feed only to the rotary tablet press it was found that the die would be less than half full at conventional operating speeds. This result highlights the importance of the suction effect.

228

I.C. Sinka and A.C.F. Cocks

7

8

7

4 6

6

1

1

9 10

2

11 3

5

6

4

8

7

(a)

2

(b)

Figure 14.3. Rotary press production cycle a) top view, b) unfolded view; 1 – die table, 2 – fill cam, 3 – feed wheel, 4 – die-fill area, 5 – metering wheel, 6 – precompression roller, 7 – main compression roller, 8 – ejection cam, 9 – upper punch, 10 – die, 11 – lower punch.

Figure 14.4 indicates that suction filling improves the critical velocity by a factor of 2.5. Use of this higher velocity gives a more accurate prediction of the die-fill behaviour of the powder in the rotary press. Also, under suction filling the fill ratio drops less steeply at velocities in excess of the critical velocity and over 80 % of the die is filled at the highest shoe velocities used in the experiment. This observation can be employed to aid the design of the metering process to maximize the operational speed of the press. These initial fundamental studies illustrate how a fundamental understanding of the flow process can be used to guide the design of the manufacturing process. However, more detailed studies are necessary to scale 1.2

Fill ratio

.

1 suction

0.8 0.6 0.4 0.2

gravity

0 0

200

400

600

800

Shoe velocity, mm/s

Figure 14.4. The fill ratio (the fraction of the die-filled after a single pass of the shoe) plotted aginst the shoe velocity for a microcrystalline cellulose powder. The critical velocity is the velocity above which incomplete filling occurs.

Modelling Die Compaction in the Pharmaceutical Industry

229

the kinematics of suction feed and to include the force feed effect for scaling calculations. 14.3.2 Compression and Ejection A typical rotary press tool set assembly (Figure 14.3b) consists of an upper punch (9), a die (10) and a lower punch (11). The punches are guided so that only vertical motion with respect to the die is allowed. The vertical position is determined by the cam mechanisms that guide the punches around the die table between the filling station, compression rollers and ejection cam. During compression, the punch head makes contact with the compression rollers forming a cam-follower system, which determines the details of the compression schedule. The compression sequence is biaxial, in the sense that both the upper and lower punches are moved in tandem. The punch displacement during compression is given by the vertical position of the rollers, which are adjusted automatically to achieve the same maximum compression force tablet after tablet. More modern systems allow a wider range of settings for the position of the cams and rollers. Compression takes place in two stages, under the precompression and main compression rollers (Figure 14. 3b). The compaction event itself is very short, from a few milliseconds to tens of milliseconds. Precompression is usually necessary to eliminate defects like cracks or laminations that occur during high-speed tableting. The cause of these defects can often be traced to the effect of air that is compressed in the reducing pore space, given that the initial porosity is often of the order of 70 %. This porosity is reduced according to the level of precompression applied. At this stage the porosity is open, and the dwell between pre- and main compression allows some air to escape and relieve the pressure within the pores. In the main compression stage the powder is compacted to its final dimensions. In general, for any given rotary press, the geometry and relative position of the components, as well as the selection of the processing conditions determine the details of the compression sequence that affects the strength of the resulting tablets accordingly [20]. The punch-displacement profiles can be determined analytically or measured using specialised instrumentation [21]. In order to mimic the operation cycle of any press in a research and development environment, compaction simulators are used. A compaction simulator is a high-speed servo-hydraulic press that can be programmed to execute a prescribed displacement profile. Figure 14.5 shows a set of force-displacement data for microcrystalline cellulose obtained using a compaction simulator. The top-punch displacement was ramped linearly. The lower punch was maintained stationary in order to facilitate data analysis for model calibration as described in the following section. Finally, ejection takes place using the ejection cam (8), Figure 14.3. The tablet is then subject to dedusting, metal checks, bulk handling, coating, packaging, storage, transport, etc., processes during which it must retain its prescribed integrity and bioavailability behaviour. In this chapter we discuss only the mechanical aspects of the tablet-compression problem.

I.C. Sinka and A.C.F. Cocks

upper and lower force

10

16

upper

14

Displacement, mm

5

lower

0

12 10 8

-5

6

-10

Force, kN

230

4 -15

2

-20

0 0

5

10

15

20

25

30

35

40

45

Time, s

ejection

compression unloading

Figure 14.5. Compression schedule (compaction simulator data). The reference for punch displacement is the top of the die table.

14.4 Tablet-Compaction Modelling As described in the introduction, it is only in recent years that compaction models have been used to model the tabletting process in the pharmaceutical industry [5, 912]. In general, modelling tablet compaction requires a knowledge of the following main factors: • constitutive model for the powder material • friction interaction between the powder and tooling • geometry of die and punches, which determines the tablet image • sequence of punch motions • initial conditions that relate to the state of the powder after the die was filled. Similar to other industrial sectors the incremental plasticity framework has been adopted for pharmaceutical die compaction modelling at present. This approach is employed in the next section to illustrate the effect of the internal structure of the tablet on tablet properties. 14.4.1 Material Characterisation for Model Input The general methodologies for determining the elastic, plastic and failure behaviour of powders and compacts and characterising die-wall friction are described in detail in Chapters 4-8. In the following sections we present constitutive data for microcrystalline cellulose that are used as input parameters for modelling.

Modelling Die Compaction in the Pharmaceutical Industry

231

We adopt a modified Drucker-Prager-Cap model where the elastic and plastic model parameters are expressed as functions of relative density. Density is used almost exclusively as a state variable for modelling the compaction behaviour of all classes of powder materials. A more detailed description of the experimental procedures is given elsewhere [12].

(a)

10 9 8 7 6 5 4 3 2 1 0

(b)

0.35 0.3 Poisson's R atio

Young’s Modulus, Young's modulus, M P GPa

14.4.1.1 Elastic Properties Isotropic linear elasticity is assumed. Young’s modulus and Poisson’s ratio can be determined from the unloading curve of a die-compaction experiment. The

0.25 0.2 0.15 0.1 0.05 0

0.2

0.4

0.6

0.8

1

0.2

0.4

10 9 8 7 6 5 4 3 2 1 0

80

(c)

0.8

1

(d)

70 60 50 40 30 20 10 0

0.2

0.4

0.6

0.8

0.2

1

0.4

Relative Density

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

(e)

0.2

0.4

0.6

0.6

0.8

1

Relative Density Hydrostatic Yield Stress , M P

Cap Excentricity Paramete

0.6

Relative Density

Friction Angle, Degree

Co hesio n, MPa

Relative Density

0.8

Relative Density

1

160

(f)

140 120 100 80 60 40 20 0 0

0.3

0.6

0.9

1.2

Volumetric Plastic Strain

Figure 14.6. Constitutive model data for microcrystalline cellulose, using relative density as state variable. Elastic properties: (a) Young’s modulus, (b) Poisson’s ratio. Shear-failure line parameters: (c) cohesion, (d) internal friction angle. Parameters of cap surface: (e) shape parameter (cap eccentricity), (f) hydrostatic yield stress.

232

I.C. Sinka and A.C.F. Cocks

material is compressed to a given density in a prelubricated die to minimize friction. As knowledge of the radial stress is necessary, a die instrumented with radial stress sensors was used. Young’s modulus and Poisson’s ratio are determined from the radial and axial stresses and axial strain during unloading. The evolution of Young’s modulus and Poisson’s ratio as a function of relative density are given in Figures 14.6a and b. 14.4.1.2 Drucker-Prager-Cap Model Parameters The shear-failure line is characterised by cohesion and the internal friction angle and can be determined using a number of simple tests, such as tension, compression, etc. A series of compacts was pressed in a prelubricated die to different relative densities. Die-wall lubrication is used to reduce the density variations due to friction and generate compacts under more uniform stress states for further material characterization. The shear-failure line was determined using the following simple experiments: • diametrical compression of tablets, which is used extensively in the pharmaceutical industry to characterize strength [20] • uniaxial compression of cylindrical compacts. The resulting cohesion and the internal friction angle are presented in Figures 14.6c and d as functions of relative density. The cap surface is described by an ellipse, which is also characterized by two parameters, the size and shape, which are presented in Figures 14.6d and e. These parameters are obtained by compacting the powder in a die instrumented with radial stress measurement sensors. The stress state (axial and radial stresses) and the strain increment direction (zero radial strain) at the loading point are known. The parameters of the cap surface are determined using the normality rule, as described in Chapter 3. Families of Drucker-Prager-Cap surfaces with increasing relative density are presented in Figure 14.7, illustrating the evolution of the size and shape of yield surfaces during densification. The labels indicate relative density. The stress path corresponding to closed-die compaction is also indicated in Figure 14.7. For more detailed characterization of the cap surface, triaxial testing can be employed [22].

Modelling Die Compaction in the Pharmaceutical Industry

233

150

Effective Stress, MPa

125 100 0.93

75 0.90 50

0.80 0.75

0.85

25 0 -25

0

25

50

75

100

125

150

Hydrostatic Stress, MPa

Figure 14.7. Drucker-Prager yield surfaces for microcrystalline cellulose

14.4.2 Friction The friction is measured using a die instrumented with radial stress sensors and the friction coefficient is calculated using the Janssen-Walker method of differential slices using a relationship derived elsewhere [9]. The analysis is based on one of the punches being stationary. The compression schedule illustrated in Figure 14.5 provides the maximum amount of information from a single compaction simulator experiment when a limited amount of powder is available. The friction coefficient is dependent on contact pressure. It is highest during the initial stages of compaction and asymptotes to a lower value as the contact pressure (and density) is increased. Figure 14.8 illustrates the variation of friction coefficient for two extreme cases: • clean die-wall condition, where the die-wall is degreased prior to compaction, and • lubricated die-wall condition, where a tablet of pure lubricant powder (magnesium stearate) is compressed prior to the experiment. The clean and lubricated die-wall conditions result in asymptotic values of the friction coefficient of 0.48 and 0.08, respectively, as illustrated in Figure 14.8.

234

I.C. Sinka and A.C.F. Cocks

1

Coefficient of Friction

0.9 0.8 0.7 0.6

clean die

0.5 0.4 0.3 0.2

lubricated die

0.1 0 0

15

30

45

60

75

90

Contact Pressure, MPa

Figure 14.8. Coefficient of friction between powder and die-wall

14.5 Case Studies Three cases are presented below. Case Study 1 illustrates the effect of friction between powder and tooling on the density distribution in round, curved-faced tablets. The effect of density distribution on tablet strength and breakage is illustrated. Case studies 2 and 3 are concerned with more complex tablet systems developed to: co-administer two different drugs in the same dosage form; release the same drug at different rates (i.e. immediate release for rapid onset of action followed by extended release over a longer period); modify the release property of a formulation, etc. Case Study 2 examines the compaction of bilayer tablets. Bilayer tablets are manufactured on special rotary presses described in detail elsewhere [23] and summarised as follows. The first layer is partially compressed but it is not ejected from the die. Then the second powder is fed into the die, followed by compaction and ejection. This Case Study focuses on the density distributions in the powder at various stages of the compaction process. Case Study 3 examines the compression coating process, which is also used to combine two different drug formulations. It essentially involves enclosing a compressed tablet (formulation 1) into a powder bed (formulation 2). After compression tablet 1 is completely enclosed in tablet 2. This Case Study is also concerned with the density distribution in the powder system. Complex tablet systems can be designed using variations of layered and compression coated systems. Trilayer tablets and three nesting tablet systems have been marketed for a number of years. The case studies presented in this chapter highlight the use of numerical modelling for formulation design, process development, tablet image and tool design.

Modelling Die Compaction in the Pharmaceutical Industry

235

14.5.1 Case Study 1: The Density Distribution in Curved-Faced Tablets Curved-faced tablets were compressed using a 25 mm diameter round die and concave punches having a radius of 19.82 mm. The friction interaction between powder and tooling was controlled and two tablets were compressed to the same average relative density (i.e. same weight and thickness) using clean (degreased) and prelubricated tooling, respectively resulting in high and low friction coefficients. The bottom punch was maintained stationary during the compression. The die was filled by hand with attention to obtaining a uniform initial packing in the powder bed. The relative density distribution in vertical cross-sections of the tablets was characterized using the indentation-hardness mapping technique [11]. The relative density distribution is presented in Figures 14.9 a and b for high and low friction, respectively. Numerical analysis was carried out using input factors as described in the previous section. The Drucker-Prager-Cap model, where all material properties were expressed as a function of relative density (Figure 14.6) was used. The low friction (prelubricated die) case was modeled using variable friction data as presented in Figure 14.8. For the high-friction case the friction coefficient was taken as 0.5 (constant). The relative density distributions predicted by the model for the high- and low-friction cases are presented in Figures 14.9c and d, which are in good agreement with the experimental density maps. The only difference between the two cases is the friction interaction between powder and tooling. High friction hinders the relative sliding of the powder into the top punch cavity during compression (initially the top surface of the powder is flat), which results in high-density regions around the edge of the tablet. Although the powder movement is relatively complex, the high-friction case can be described as pressing columns of powders to different heights, developing the density pattern illustrated in Figure 14.9c. If relative sliding is facilitated (low friction coefficient) then a more complex radial movement of the powder takes place, which results in a density distribution that is the inverse of that for high friction, i.e. the density is highest in the middle of the tablet. As a consequence, two identical tablets in terms of shape, weight and material have different microstructures, which affect their strength, friability, disintegration and failure behaviour. The break force during diametrical compression experiments for series of tablets compacted to different relative densities in clean and lubricated dies is presented in Figure 14.10a, and it can be observed that the strength is not unique.

236

I.C. Sinka and A.C.F. Cocks

(a)

(b)

0.6 0.57 0.55 0.62 0.5560.544 0.5440.539 0.5620.556 0.64 0.5660.582 0.66 0.575 0.5620.562 0.575 0.575 0.604 0.601 0.7 0.643 0.5970.601 0.5790.566 0.572 0.625 0.562 0.575 0.6330.6160.597 0.712 0.689 0.5720.559 0.586 0.579 0.5720.572 0.647 0.579 0.616 0.667 0.5560.566 0.556 0.569 0.604 0.5750.569 0.5690.547 0.544 0.6040.566 0.530 0.484?0.5150.472?0.474?0.482?

0.5590.589

0.647 0.657 0.689 0.662 0.700 0.620 0.694 0.5970.612 0.662 0.620 0.575 0.586 0.556 0.5930.586 0.5970.608

0.64 0.65 0.66 0.612 0.643 0.6200.647 0.625 0.638 0.643 0.66 0.65 0.652 0.643 0.638 0.64 0.638 0.6380.657 0.652 0.6520.643 0.647 0.550 0.643 0.53 0.6 0.657 0.620 0.643 0.616 0.629 0.53 0.633 0.6290.638 0.633 0.652 0.6470.657 0.672 0.662 0.667 0.643 0.525 0.647 0.638 0.647 0.547 0.629 0.638 0.657 0.6520.625 0.652 0.6520.633 0.647 0.652 0.643 0.662 0.647 0.539 0.608 0.652 0.643 0.652 0.633 0.6470.643 0.647 0.6470.643 0.643 0.652 0.633 0.633 0.6290.638 0.638 0.633 0.633 0.6250.616 0.625 0.629 0.61

0.643

5 mm

(c)

Clean (unlubricated) die and punches

0.63

0.64

5 mm

(d)

Lubricated die and punches

Figure 14.9. Relative density distribution in curved faced tablets. Experimental data for tablet compressed using a) clean and b) lubricated tooling. Numerical results for c) high and d) low friction (from Sinka et al., [11]).

Figure 14.10b presents the failure modes of the tablet during diametrical compression experiments. Tablets compacted in clean dies fail across the diameter in a consistent way with the stress field induced in thin linear elastic disks subject to point loading (the “Brazilian” test), while tablets compressed in prelubricated dies delaminate. It is important to note that this failure mode cannot be predicted using the isotropic Drucker-Prager-Cap model. Predicting tablet strength is discussed in more detail elsewhere [20]. The numerical analysis of tablet compaction can aid formulation design and selection of process parameters. By taking the behaviour of the powder into consideration it is possible to establish more realistic normal and shear stress distributions on the tooling, which can aid tool design, estimation of maximum compression forces and assessment of tool life. This can represent a considerable improvement over the methodology currently used in tool design, where stress analysis is carried out under the assumption that the load applied to the punch face is a uniform pressure that is given by the compaction force divided by the crosssectional area of the die.

Modelling Die Compaction in the Pharmaceutical Industry

clean

237

lubricated

unlubric ated die

200 180 Break Force, N

160

lubricated die

140 120 100 80 60 40 20 0 0.35

0.4

0.45

0.5

0.55

0.6

0.65

Relative Density (Average)

(a)

(b)

Figure 14.10. Effect of microstructure on tablet behaviour. (a) break force in diametral compression experiments, (b) failure mode for tablet compacted in clean and prelubricated dies (from Sinka et al. [12]).

14.5.2 Case Study 2: The Density Distribution in Bilayer Tablets

Displacement, mm

Numerical analysis has been carried out for compressing a bilayer tablet in a round, 20 mm diameter die using flat punches. For both layers microcrystalline cellulose was used, starting from an initial relative density of 0.3. The weights of the first (bottom) and second (top) layers were 2.00 g and 2.25 g, respectively. The first layer was compressed to 10 kN and the second layer to 40 kN, as indicated in Figure 14.11. The bottom punch was maintained stationary during the entire compression schedule. 20

upper punch

10 0 -10 lower punch

-20 -30 50

Force, kN

Reference: top of die table

40 30 20 10

fill height fill height 1st layer 2nd layer

compression 2nd layer

compression 1st layer ejection

0

Figure 14.11. Compression schedule for the bilayer tablet

238

I.C. Sinka and A.C.F. Cocks

Figure 14.12a illustrates the density distribution at the end of the compression stage for the first layer. This type of density distribution is specific to single-layer tablets. For round, flat tablets the density variations arise from the frictional interaction between powder and die-wall. Detailed parametric studies of the effect of friction are presented in [9]. In the present Case Study, in order to accentuate frictional effects, the friction coefficient was set to 0.5 (constant), which is a relatively high value specific to clean (unlubricated) die-wall conditions. After the compression of the first layer, the powder for the second layer is delivered into the die, Figure 14.12a. The initial density of the second layer is uniform (this assumption is used throughout this chapter). Figure 14.12b is a snapshot of the density distribution during bilayer compaction. At this stage, densification occurs in the second (top) layer only and the density distribution in the first layer has not yet changed. Concurrent densification of both layers occurs only after the second layer reaches a similar state as the first layer, which occurs approximately when the compaction force reaches the value used to compress the first layer (10 kN in this case). The density distribution in the second layer assumes a similar pattern as in the first layer because of the friction between powder and die-wall. However, at the interface region, especially at the die-wall the density in the two layers is different. As densification progresses, both layers first reach the same density around the centre of the interface. The mismatch at the die-wall gradually disappears towards the end of the bilayer compaction (Figure 14.12c). It is also interesting to note that the interface between the two layers becomes distorted. The top surface of the first layer becomes concave and the model predicts a variation in height of 0.5 mm for the input data used in the simulation. The prediction of face curvature can be verified experimentally by performing a tensile test, which results in separation at RD

RD

(a)

RD

(b)

(c)

Figure 14.12. Density distribution in bilayer tablets during compaction, (a) end of first layer compaction after the second layer was filled, (b) snapshot during bilayer compression, (c) end of bilayer compression

Modelling Die Compaction in the Pharmaceutical Industry

239

the interface. If the friction coefficient is lowered or the compression force of the first layer is increased, the depth of the concavity is reduced. The effect is also reduced in bilayer tablets with a larger overall height to diameter aspect ratio. In the above analysis the interaction between the two layers was described as frictional sliding, using a high friction coefficient (0.5). In practice, an adhesive normal interaction develops between the layers. If the first layer is compressed to a high density, then interlocking with the second layer becomes difficult and it is not possible to produce intact bilayer tablets. The interface properties are of considerable practical importance as they affect the strength and properties of the bilayer tablet. A comprehensive analysis of the stress states during compression, unloading and ejection requires more detailed experimental characterisation of the interlayer strength as well as numerical implementation. 14.5.3 Case Study 3: The Density Distribution in Compression Coated Tablets Compression coating is also referred to as core coating or dry-powder coating and involves enclosing a smaller tablet (core) into a larger one (coating). In this Case Study we examine the compression coating process for flat-faced geometries to illustrate a number of generic trends. While the density variations in the core can be examined in detail (the core is a single-layer tablet), in this Case Study it is assumed that the core is initially uniform. Microcrystalline cellulose is considered for both core and coating. The initial relative densities were set to 0.6955 and 0.3536, for core and coating, respectively, as indicated in Figure 14.13a. The material is compressed to near full density. Figure 14.13 illustrates that the RD

RD

(a)

RD

(b)

(c)

Figure 14.13. Density distribution in a compression coated tablet system, (a) initial relative density 0.6955 and 0.3536 in core and coating, respectively, (b) density distribution in the coating after compression, (c) density distribution in core after compression

240

I.C. Sinka and A.C.F. Cocks

geometry chosen is not optimal: the material in the core is almost fully densified together with the material in the area between the core and the punches. The material around the vertical edges of the coating, on the other hand only has a relative density of the order of 0.75-0.8. Lower density implies lower strength, which makes fractures in these regions possible. A first-order design should ensure that columns of powder are compressed proportionally. The compression coated tablet analysis is further complicated by the compressible (partially densified) core. Marketed compression coated tablets usually have curved faces, which adds to the complexity of the problem. At the same time, a rounded core facilitates powder movement as compared to a sharp corner. Detailed analysis of all stages of the problem can identify the conditions that are favourable for defect formation during compression, unloading, ejection, or subsequent handling. The geometry of the core as well as the material properties can be optimised from a mechanical point of view. The process parameters, particularly the initial conditions for the core are important in determining the mechanical response at the interface between the two materials, which has an effect on the final strength and mechanical properties.

14.6 Summary and Conclusions Tablet-compaction models can be employed in industry to assist decision making at various stage of product development. The structure of a tablet depends on the contribution of all contributing factors: powder behaviour (constitutive model), friction, tablet geometry, pressing sequence, and initial conditions of the powder after die-fill. By performing numerical studies it is possible to optimize parameters related to each of these factors. The powder behaviour during compaction can be controlled through its composition (active pharmaceutical ingredients, excipients, lubricant, glidants, disintegrants, etc.) and the selection of process parameters for use in manufacture, such as mixing, wet or dry granulation, etc. The friction between powder and die-wall is determined by lubricating the powder (admixing a lubricant to the powder blend) or by lubricating the die-wall before die-fill. A robust model allows the structure and properties to be predicted for any tablet shape. The effect of the sequence of punch motions can be evaluated numerically and used as input for setting up high-speed rotary presses. The initial density distribution in the powder after die-fill is determined by the feed system and choice of process parameters. The effect of die-fill propagates through the compaction process. The five factors described above are interrelated, therefore it is difficult to generalise the conclusions. Each case should be examined individually. Compaction models today offer insight, guidance and solutions to practical problems in an industrial environment and are used to optimize formulation design, process development, tablet image and tool design. Tablets having simple or more complicated geometry can be analysed, as well as complex solid dosage forms such as multilayer tablets or compression coated tablet systems. However, currently the models suffer from robustness when used for detailed modelling of the subtle geometric features. In addition, compaction models to date do not normally take into account the effect of loading rates on the

Modelling Die Compaction in the Pharmaceutical Industry

241

material, anisotropy or the effect of air pressure buildup in high-speed compaction, although rate effects that affect friction can be readily implemented. This chapter centred on consideration of the density distribution in solid dosage form. Mechanical integrity, however, is only one of the many product quality criteria. The role of a tablet is to deliver medication to a patient, thus tablets must satisfy chemical stability and bioavailability requirements, which are also affected by composition and manufacturing route.

14.7 Acknowledgements Drs. Steve Burch and James Tweed are acknowledged for X-ray CT characterisation of tablets (Figure 14.1), carried out as part of the project “Minimising density variations in powder compacts MPM5.2” funded by the Department of Trade and Industry (UK) and led by AEA Technology. Dr. Ludwig Schneider is acknowledged for powder flow and die-fill experiments carried out at the University of Leicester as part of a project funded by Merck and Co., Inc. Ms. Sarah Jackson is acknowledged for examining the effect of suction fill (Figure 14.4). Dr. Adam Procopio of Merck and Co., Inc. is acknowledged for producing the electron microscope image of microcrystalline cellulose (Figure 14.2). Professor Antonios Zavaliangos of Drexel University, Philadelphia, PA, USA is acknowledged for guiding the calibration and implementation of the constitutive model (Figures 14.6 and 14.7) part of GOALI Grant NSF/CMS0100063. Mr. J.C. Cunningham of Merck and Co. Inc. (currently with Johnson & Johnson Pharmaceutical Research & Development, LLC) is acknowledged for supporting this project. Dr. Kendal Pitt of Merck Sharp and Dohme Ltd. and Ms. Sharon Inman from Imperial College London are acknowledged for discussions related to practical aspects of bilayer tableting.

References [1] [2] [3] [4] [5]

Train D. 1957: Transmission of forces through a powder mass during the process of pelleting; Trans. Instn. Chem. Eng., vol 35, 258-266. Macleod HM and Marshall K. 1977: The determination of density distributions in ceramic compacts using autoradiography, Powder Technol. vol. 16, pp. 107-122. Lannutti JJ. 1997: Characterisation and control of compact microstructure; MRS Bulletin, vol. 22, no. 12, pp. 38-44. Sinka IC, Burch SF, Tweed JH and Cunningham JC. 2004: Measurement of density variations in tablets using X-ray computed tomography; International Journal of Pharmaceutics, vol. 271, issue 1-2, pp. 215-224. Wu C-Y, Ruddy OM, Bentham AC, Hancock BC, Best SM and Elliott JA. 2005: Modelling the mechanical behaviour of pharmaceutical powders during compaction. Powder Technology, vol. 152, issues 1-3, 29 April 2005, pp. 107-117.

242

I.C. Sinka and A.C.F. Cocks

[6]

Eiliazadeh B, Pitt K and Briscoe B. 2004. Effects of punch geometry on powder movement during pharmaceutical tabletting processes. International Journal of Solids and Structures, vol. 41, no. 21, pp.5967-5977. Djemai A, and Sinka IC. NMR imaging of density distributions in tablets. International Journal of Pharmaceutics. (To appear). Khattat IM and Al-Hassani ST. 1987: Towards a computer aided analysis and design of tablet compaction. Chemical Engineering Science, vol. 42, no.4, pp.702-712. Sinka IC, Cunningham JC and Zavaliangos A. 2001: Experimental characterization and numerical simulation of die-wall friction in pharmaceutical powder compaction. Proc. PM2TEC 2001 International Conference on Powder Metallurgy & Particulate Materials, 13-17 May, New Orleans, Louisiana, USA, ISBN: 1-878954-82-2, part 1, pp.46-60. Michrafy A, Ringenbacher D and Tchoreloff P. 2002: Modelling the compaction behaviour of powders: application to pharmaceutical powders, Powder Technology, vol. 127, pp.257– 266. Sinka IC, Cunningham JC and Zavaliangos A. 2003: The effect of wall friction in the compaction of pharmaceutical tablets with curved faces: a validation study of the Drucker-Prager-Cap model. Powder Technology, vol. 133, issue 1-3, pp.33-43. Sinka IC, Cunningham JC and Zavaliangos A. 2004: Analysis of tablet compaction. Part 2 – Finite element analysis of density distribution in convex tablets. Journal of Pharmaceutical Sciences. vol. 93, no. 8, pp.2040-2052. Lieberman HA, Lachman L and Schwartz JB. (eds.), 1990: Pharmaceutical dosage forms: tablets. vol. 3. 2nd edn, Marcel Dekker Inc., New York, Basel. Swarbrick J and Boylan JC. (eds.) 1992: Encyclopedia of Pharmaceutical Technology. Marcel Dekker Inc., New York, Basel, Hong Kong. Chulia D, Deleuil M and Pourcelot Y. (eds.) 1994: Powder Technology and Pharmaceutical Processes. Handbook of Powder Technology, volume 9. Elsevier Amsterdam, London, New York, Tokyo. Banker GS and Rhodes CT. (eds.) 1996: Modern pharmaceutics. Drugs and Pharmaceutical Sciences, vol. 72. 3rd edn., Marcel Dekker New York, Basel. Sinka IC, Schneider LCR and Cocks ACF. 2004: Measurement of the flow properties of powders with special reference to die-fill. International Journal of Pharmaceutics, vol. 280, issue 1-2, pp.27-38. Wu C-Y, Dihoru L, Cocks ACF. 2003. The flow of powder into simple and stepped dies. Powder Technology, vol. 134, pp.24-39. Schneider LCR, Sinka IC and Cocks ACF. Characterisation of the flow behaviour of pharmaceutical powders using a model die-shoe filling system. (to appear). Sinka IC, Pitt KG and Cocks ACF. The strength of pharmaceutical tablets. In: Particle Breakage. (Eds.) A.D. Salman, M.J. Hounslow and M. Ghadiri. Elsevier, (to appear). Ridgeway Watt P. 1988: Tablet press instrumentation in pharmaceutics: principles and practice. Ellis Horwood Ltd., Chichester, UK. Sinka, IC, Cocks ACF and Tweed JH. 2001: Constitutive Data for Powder Compaction Modelling. Journal of Engineering Materials and Technology, Transactions of the ASME, vol. 123, no. 2, pp.176-183. Pitt KG and Sinka IC. 2006: Tableting. In: Granulation (Handbook of Powder Technology, Volume 11). )Eds.) A Salman, M Hounslow and JPK Seville. Elsevier Science, Netherlands.

[7] [8] [9]

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

15 Applications in Industry P. Brewin1, O. Coube2, D.T. Gethin3, H. Hodgson4 and S. Rolland3 1

European Powder Metallurgy Association, Talbot House, Market St., Shrewsbury SY1 1LG, UK. 2 PLANSEE SE, 6600 Reutte, Austria. Now with European Powder Metallurgy Association, Talbot House, Market Street, Shrewsbury SY1 1LG, UK. 3 School of Engineering, University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, UK. 4 Dynamic-Ceramic Ltd., Crewe Hall Enterprise Park, Weston Road, Crewe, Cheshire CW1 6UA, UK.

This chapter gives examples of the successful use of compaction modelling (CM) by industry. The examples complement those for the parmaceutical industry in Chapter 14 and have been selected to illustrate the use of CM in solving common industry problems.

15.1 Numerical Simulation of Die Compaction and Sintering of Hardmetal Drill Tips 15.1.1 Summary Die compaction and sintering processes of tungsten carbide drill tips are studied numerically to assess the influence of process parameters on green density distribution and sintered final distortions. The corresponding 3D computer-aided design (CAD) models provide geometries for tools, filled powder, and green body. Simulations of the pressing and sintering processes are carried out with both the explicit and implicit versions of the finite-element (FE) package ABAQUS® specifically by adopting user-defined material subroutines for powder compaction and sintering. Besides other macroscale results such as pressing forces and microscale results such as stress and strain measures the predicted density distribution after pressing in the green body is the main focus of attention. The latter serves as an initial condition in the simulation of sintering, since distortions of the sintered body are mainly governed by inhomogeneous density distribution in the green body. Based on this procedure a parameter-sensitivity study is carried out

244

P. Brewin, O. Coube, D.T. Gethin, H. Hodgson and S. Rolland

in terms of geometry and process kinematics variants in order to extract the functional dependencies between geometrical design parameters, materials properties, and processing parameters, respectively. 15.1.2 Introduction The main production route for hardmetal parts as well as for some other materials like sinter metals or ceramics is pressing of powder followed by sintering. The two processes permit the fabrication of complex geometries at relatively low cost or even enable it at all for some materials. Nevertheless, die compaction of such geometries leads to a nonhomogeneous density distribution, which is the principal cause of distortions during sintering and that could finally lead to rejects due to tight tolerances for certain components. Production lines are still using trial and error methods to solve problems related to inhomogeneous density distribution. These methods are usually time consuming and show limits in the efficiency. On the other hand, the precision of the simulation tools is very sensitive to the complexity of the material behaviour, which means that characterisation of the powder behaviour during compaction at low and high densities needs permanent improvement. Distortions can also be affected by the imprecision of process parameters relative to the demanding tolerance of the part dimensions. Elastic tool deflections during pressing, e.g., are sometimes an order of magnitude higher than the required final-part tolerances. These conditions for a successful simulation are difficult to obtain, especially in an industrial environment and may lead to discrepancies between prediction and practice [1]. Nevertheless, simulation used with some knowledge and experience, is a useful tool for process and design optimisation, where the influence of every parameter can be assessed in comparison with a calibrated reference case [2]. Concerning the hardmetal component of this study, numerical prediction of the powder behaviour is carried out during two fabrication steps, pressing and sintering. If gravity and friction during sintering are neglected and if the part reaches full or nearly full density after sintering, the latter can be seen as a straight forward process, where the actual constitutive model used to describe sintering has only an insignificant influence on the predicted shape distortions [3]. The principal parameter will then be the green (i.e. after pressing) density distribution. Therefore, the principal challenge is a reliable simulation of die compaction. 15.1.3 Numerical Simulation of Die Compaction In the following section, numerical simulation of the die compaction of a hardmetal powder will be presented. Die compaction is the first fabrication step of a hardmetal percussion drill tip for stone working shown in Figure 15.1.

Applications in Industry

245

Figure 15.1. Hardmetal percussion drill tip (Type 50051) for stone working produced by CERATIZIT.

To model the behaviour of the powder during compaction a modified DruckerPrager-Cap Model [4,5] is implemented as a user subroutine (VUMAT) with the FE code ABAQUS/Explicit®. The parameters of the model were characterised for a WC-Co powder. An essential aspect is the relation between the pressure and the density. Figure 15.2 shows, for example, a typical hardening curve for WC-Co powders [6] (and, furthermore, for every powder at least qualitatively). Whereas the powder can be pressed very easily at low densities, the hardening increases exponentially at high densities, which leads to a requirement for high pressure for the compaction. Further details about the modelling and the parameters characterisation are given by Coube and Riedel [7]. Friction between the powder and tooling is modelled with a constant Coulomb friction coefficient of 0.2 for the whole compaction process. This is an average value observed experimentally for WC-Co powder. The pressing tools are imported from CAD files. Their surfaces, which are in contact with the powder, are retained for the simulation. The powder starting volume is then reconstituted by using the volume of the die cavity and the lower punch surface. The powder and the pressing tools are shown in Figure 15.3. To create the finite-element meshes the mesh-generation program ABAQUS CAE® was used. The generated mesh of the loose powder contains about 44,000 brick elements. Meshing of 3D geometries for the FE simulation of powder compaction may become an issue for the success of such projects. Complex 3D parts are often difficult and sometimes impossible to mesh with brick elements.

246

P. Brewin, O. Coube, D.T. Gethin, H. Hodgson and S. Rolland

Axiale Pressure in MPa

300 250 200 150 100 50 0

4

5

6

7

8

9

Density in g/cm³ Figure 15.2. Typical hardening curve for WC-Co powders [6]

The alternative is then to use tetrahedral elements, which allow flexible automatic meshing. However tetrahedral meshes are less optimal from a computational point of view. In this study the authors carried out the simulation with both kinds of elements in ABAQUS/Explicit and noticed that the simulation with a tetrahedral mesh (with element type C3D4) did not run until the end of the process due to an excessive distortion of the elements. This was provoked in particular by the special kinematics of the pressing tool (immersion of the upper punch) and the special geometry of the upper punch. Simulation with brick elements (C3D8R) run under the same conditions showed enough stability to complete the process. The compaction process in the simulation follows the kinematics shown in Figure 15.4. The time t of the compaction is normalised to the arbitrary value of 2. From t=0 to 1 the pressing tools do not move except the upper punch, which immerses in the powder up to 9 mm. In practice, after t=1, the die follows the movement of the upper punch. The powder is then die compacted according to the pressing schedule with a compaction ratio of 2.2 (ratio from fill height to final height). Figure 15.5 shows the mesh in its fill and final positions.

Applications in Industry

Figure 15.3. Powder and pressing tools imported from CAD

Displacement in mm

6

Lower Punch

4 2 0

Die

-2 -4 -6 -8 -10 -12 -14 -16 -18 0.0

Upper Punch

0.5

1.0

1.5

Normalised Time Figure 15.4. Pressing kinematics for the drill tip

2.0

247

248

P. Brewin, O. Coube, D.T. Gethin, H. Hodgson and S. Rolland

Figure 15.5. Position of the tools and the powder after filling (left) and pressing (right)

The die pressing of the powder leads to inhomogeneities in the density distribution of the green part. Starting with a homogeneous fill density of 3.5 g/cm³, the density distribution after compaction is in a range from 6.1 to 9.3 g/cm³ with an average of 8.2 g/cm³. In particular, the upper volume is not densified as much as the rest of the component as shown in Figure 15.6. This is due to the form of the upper punch, which allows different pressing heights and therefore different compaction ratios. 15.1.4 Numerical Simulation of Sintering After virtual pressing, a numerical simulation of sintering is carried out. The predicted density distribution and the deformed shape are used as input data and starting shape for the sintering, respectively. The modelling of sintering is based on the model developed by Riedel and coworkers [8, 9, 10]. Although both solid- and liquid-phase sintering occurs for WC-Co, the prediction of shrinkage and shape distortion for a full-density sintering process can be carried out solely with a solidstate sintering since the predicted shape distortions are not very sensitive to the actual constitutive model used to describe sintering, at least for cases with uniform

Applications in Industry

249

6.3 g/cm³

8 g/cm³

9 g/cm³

Figure 15.6. Density distribution (SDV9) in g/cm³ in the part after pressing. View from the side and from the top.

temperature [2,3,11]. The solid-state sintering model is implemented as a user subroutine in ABAQUS/Standard®. The deformed mesh after pressing is used as the starting shape for the sintering simulation. An alternative method would be to generate a new mesh with the same dimensions as the pressed one and to transfer the density-distribution data. This is useful when large distortions of elements occur during pressing, which may lead to deceptive results or to program aborts. This was not the case for this example and the pressed mesh could be used. The green part is sintered to a final density of 14.4 g/cm³ and underwent shrinkage of about 17 % of its initial dimensions (see Figure 15.7). The burnout of the organic component during sintering is taken into account in the ratio between the apparent and the theoretical density, which calibrates the green density distribution for the sintering simulation.

250

P. Brewin, O. Coube, D.T. Gethin, H. Hodgson and S. Rolland

Green shape

Sintered Shape

Bottom Figure 15.7. Green and final shape of the drill tip after sintering. View from two sides and from the bottom.

Since direct pressed hardmetal parts must respect very tight tolerances, one of the main post-processing actions after the simulation is to assess the distortion caused by the shrinkage. With finite-element meshing, the new dimensions of the part can be calculated at each node. Edges, profiles or angles can be numerically predicted. Figure 15.8 shows an example of numerical control of tolerances. Two axial profiles right and left of the part are compared between the green and the sintered state. Maximal unevenness and deviation angle, respectively 5 µm and 0.11°, are calculated and can be used to control the quality of the part with the current fabrication process. This geometrical prediction can now be used to analyse the relative and absolute influence of different process or material parameters on final-part geometry. Doing this analysis in this virtual environment, a detailed process understanding can be obtained, which cannot be achieved in this systematic form by real experiments.

Applications in Industry

251

Figure 15.8. Assessment of the part distortion after sintering

15.1.5. Discussion and Conclusions Two fabrication steps, pressing and sintering, were modelled and implemented in the finite-element code ABAQUS to predict first the density distribution after pressing and secondly shrinkage and associated geometrical distortion after sintering. To demonstrate the feasibility, numerical simulation of pressing and sintering of a complex 3D geometry, a hardmetal percussion drill tip from CERATIZIT was performed. Although the main results of the pressing and sintering study are the final dimensions and distortions of the sintered part, the main parameter in this study is the density distribution obtained after die compaction. Inhomogeneities in the density distribution lead to an unbalanced shrinkage process. In the case of sintering to full or nearly full density under uniform temperature condition, the latter can be considered as a straightforward process, which sanctions the inhomogeneous green density distribution in term of warpage. Such studies are, however, confronted with numerous difficulties and sources of inaccuracy: • Conversion of CAD files to usable finite-element meshes (brick often better than tetrahedral element) • Data transfer of the industrial fabrication process (fill density, kinematics…) to the simulation

252

P. Brewin, O. Coube, D.T. Gethin, H. Hodgson and S. Rolland

•

Characterisation of the powder in terms of compressibility, flow, friction and contingently green strength, at low, medium and high density/pressure ranges • Simulation of 3D forming processes with large deformations and contact (computing time, convergence issues) • Transfer of results from pressing to sintering and combination of two numerical simulations for a final result • Hypothesis of uniform temperature field and full density during and after sintering, respectively. Therefore, this numerical tool must be used carefully for optimisation purposes. If it is available, a well-known reference case should first be numerically reproduced and numerical predictions could be calibrated to experimentally measured values. In a second stage, different versions can be numerically assessed with respect to the reference case. More generally for comparisons between different alternatives and parameter studies, numerical simulation of pressing and sintering is a powerful tool, which once directly implemented in the industrial environment, may save time and provide support to the production teams by visualising and quantifying the influences of single process parameters on the final part quality.

15.2 Ceramic Case Studies The objective of the case study shown in Figure 15.9 was to define the best way to minimise heterogeneity and forces in a top-end compaction of an axisymmetric component shown below. The strategic purpose of the case study was to explore the density variation through the part when compacted uniaxially and to compare this with practical experience of isostatic pressing. This comparison may be used to guide process selection and, furthermore, decisions on process investment.

Figure 15.9. Nozzle part for the first ceramic case study (not to scale)

Applications in Industry

253

The approach to this particular case study was to consider four possible ways to manufacture the component, simulate them and carry out observations on the results. The possible manufacturing methods were: with and without the core rod (requiring more machining) and with the contoured tool at the top or at the bottom; combining these yields four potential pressing schemes. For each scheme, the figure below illustrates the resulting density distribution.

Figure 15.10. Density contours in the nozzle part; the boxed area will be machined off

254

P. Brewin, O. Coube, D.T. Gethin, H. Hodgson and S. Rolland

A homogeneity criterion was devised to measure the quality of the density distribution based on the standard deviation weighted with the radius. The compaction scheme 2 with the core rod and the profiled tool at the top yields the best result. The second ceramic case study is the compaction of a simple ring. The aim in this case study is to provide an insight to the compaction of zirconia powder by observing the compaction of a simple part. The results of the simulation on this component showed that the friction has a great impact on the compaction of this powder. This is made evident through the two zones that comprise high and low density levels. Unloaded density :

relative time :

0.0

0.1 0.2

0.3

0.4

0.5 0.6 0.7

0.8 0.9

1.0

Figure 15.11. Compaction of a zirconia ring

For comparison, and to explore the issue of high and low density zones, the same numerical experiment was reproduced using a ferrous powder: Unloaded density :

relative time :

0.0 0.1 0.2 0.3

0.4 0.5 0.6 0.7 0.8 0.9

1.0

Figure 15.12. Compaction of a ferrous ring

The formation of a compacted front in the zirconia powder shows that friction is opposing the homogenous compaction during the process, which does not occur during the compaction of a ferrous component. The possibility of a double-ended compaction was considered to try to minimise the effects of friction.

Applications in Industry

255

Figure 15.13. Comparison of single- and double-ended compaction for a zirconia ring

In the case of the single-ended compaction, the range of density is from 2.25 g/cm3 to 2.75 g/cm3, and in a double ended compaction it is from 2.31 g/cm3 to 2.70 g/cm3. The double-ended compaction reduces heterogeneity and it also considerably reduces the compaction forces from 80.5 kN to 73 kN. From this modelling task, it is clear that friction plays a major role in the generation of the compaction forces, but it is impossible to know whether the friction forces are mainly due to the importance of the pressure transmission coefficient or to a high friction coefficient. To determine the significance of these two factors, the friction coefficient was reduced from 0.204 to 0.05. The original value is the one normally used as defined experimentally. The results are shown in Figure 15.14 and Figure 15.15. The force required from the upper punch (Tool 1-Y) drops from 81 kN to 68 kN, which represents 15.5 %. This is not negligible and highlights the importance of lubrication. Another important observation must be made: the force exerted on the side walls (Tool 2-X) is very high, and the compacted front is still present in both cases. This combines with the fact that half of the compaction force is only reached at 93 % of the compaction stroke to show that the zirconia powder rearranges and maybe breaks down its granular structure very easily, before entering a plastic deformation phase, in which the compaction force rises sharply. The layered compaction phenomenon appears to be attributable to a high pressure transmission coefficient in the powder and this combines with the high friction coefficient to give high compaction forces.

256

P. Brewin, O. Coube, D.T. Gethin, H. Hodgson and S. Rolland

Evolution of tool forces 100000 Tool 1-X

Tool 1-Y

Tool 2-X

Tool 2-Y

Tool 3-X

Tool 3-Y

Tool 4-X

Tool 4-Y

80000 60000

forces(N)

40000 20000 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-20000 -40000 -60000 -80000 relative time

Figure 15.14. Evolution of the tool forces and density during compaction; µ=0.204

1

Applications in Industry

257

Evolution of tool forces 80000 Tool 1-X

Tool 1-Y

Tool 2-X

Tool 2-Y

Tool 3-X

Tool 3-Y

Tool 4-X

Tool 4-Y

60000 40000

forces(N)

20000 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-20000

-40000 -60000

-80000 relative time

Figure 15.15. Evolution of the tool forces and density during compaction; µ=0.05

The case studies on compaction of zirconia have quantified the density variation and punch-force levels that occur in the compaction of relatively simple shapes. The simulations have contrasted different processing configurations and confirmed the benefits of different compaction kinematics. Strategically, the results from uniaxial compaction may be compared with experience based on the isostatic process. The components may be manufactured by either route. Potentially, the uniaxial pressing appears to be more effective, but more uniform density may be achieved through isostatic pressing, but with substantially increased machining. As in the case study on the drill bit, shrinkage following sintering determines the final part shape, geometric tolerance and final grind requirements and the extent of local shrinkage is determined by the density variation within the pressed part. Simulation, such as that carried out above, quantifies the potential for using uniaxial pressing and may be used to guide process and equipment selection.

258

P. Brewin, O. Coube, D.T. Gethin, H. Hodgson and S. Rolland

15.3 Concluding Comments The above examples show the benefits to industry of the practical use of compaction modelling in reducing costs, calculating tooling stresses and predicting part quality. In total, these confirm the usefulness of this design tool and its potential impact on the bottom line

References [1]

Coube O, Federzoni L, Cante J, Oldenburg M, Chen Y, Imbault D, Dorémus P, Tweed J, Leuprecht A, Markeli W and Brewin P. 2003. European Powder Metallurgy Conference, EPMA, Shrewsbury, UK. Vol. 3, 71 - 76. [2] Kraft T. 2003. Modelling Simul. Mater. Sci. Eng, 11 381-400 [3] McHugh PE and H. Riedel H. 1997 Acta Mater. 45 2995 [4] Drucker DC and Prager W. Appl. Math. 1952, 10 157-175. [5] Sandler S, Dimaggio FL and Baladi GY. 1976. J. Geotech. Eng Div. ASCE, 102, 683699. [6] Doremus P. Private Communication, Input Data INPG L3S for Dienet Network. [7] Coube O and Riedel H. 2000. Powder Metall. 43, 123-131. [8] Svoboda J, Riedel H and Zipse H. 1994 Acta Mater. 42 435. [9] Svoboda J, Riedel H and Gaebel R. 1996 Acta Metall. Mater. 44 3215. [10] Riedel H and Blug B. 2001. Multiscale Deformation and Fracture in Materials and Structures, the James R. Rice 60th Anniversary Volume, T-J Chuang and JW Rudnicki, Eds., Kluwer Academic Publishers, Netherlands. 49-70. [11] Riedel H, Coube O, Gaebel R and Sun D-Z. 1997. Werkstoffwoche ’96, Band 8, Hrsg. J. Hirsch, DGM Informationsgesellschaft-Verlag, Frankfurt S. 217-224.

A.1 Appendix 1 – Compaction Model Input Data for Powders

This appendix presents experimental results and analytical expressions for four powders assessed in the Modnet and Dienet projects. These are used to generate parameters used as input for compaction models.

A.1.1 Distaloy AE Powder Composition: C-UF4: 0.5 %, Kenolube: 0.6 %, balance Distaloy AE (from Hoganas AB) Apparent density: Full density of green compact:

3.10 g/cm3 7.48 g/cm3

A.1.1.1 Elasticity The expressions of the analytical curves shown in Figure A.1.1 are: Young’s modulus: with ρ in g/cm3

E (MPa) = (-28000 + 10120ρ) [ exp (ρ /6.8 )6 ]

(A.1.1)

Bulk modulus: with ρ in g/cm3

K (MPa) = (-10500 + 3750ρ) [ exp (ρ /6.55 )6 ]

(A.1.2)

260

Appendix 1 – Compaction Model Input Data for Powders

Figure A.1.1. Young’s and bulk modulus as function of density for Distaloy AE

Poisson's ratio is deduced from the expression:

ν = 0.5 – E/(6K) Figure A.1.2 shows the evolution of ν as function of the density.

Figure A.1.2. Poisson’s ratio as a function of density for Distaloy AE

(A.1.3)

Appendix 1 – Compaction Model Input Data for Powders

261

A.1.1.2 Plasticity A.1.1.2.1. Diametral Compression (Figure A.1.3) and Simple Compression (Figure A.1.4) as Function of Density

Figure A.1.3. Diametral compression failureσd stress as a function of density for Distaloy AE

Figure A.1.4. Simple compression failure stress σs as a function of density for Distaloy AE

A.1.1. 2.2. Determination of the Failure Line Failure lines were fitted using only the simple and diametral compression tests (Figure A.1.5).

262

Appendix 1 – Compaction Model Input Data for Powders

Figure A.1.5. Failure line for different densities deduced from the diametral and simple compression test

In the P-Q plane the equation of the failure line is: Q = P tanβ + d

(A.1.4)

The evolution of the cohesion d (Figure A.1.6) and angle β (Figure A.1.7) are deduced from data of diametral and simple compression tests.

Figure A.1.6. Evolution of the cohesion function of density for Distaloy AE

Appendix 1 – Compaction Model Input Data for Powders

263

Cohesion d can be fitted with the expression : d (MPa) = A(( ρ /ρ0)B) -1)

(A.1.5)

with A = 0.0054 MPa; B =10.3; ρ0 = 3.1 g/cm3

Figure A.1.7. Evolution of the failure line angle as a function of density for Distaloy AE

The slope of the failure line can be considered as constant in first approximation: β = 70°. A.1.1.2.2. Determination of the Cap

Figure A.1.8. Loading path of die compaction in the mean P-Q stress plane

264

Appendix 1 – Compaction Model Input Data for Powders

Figure A.1.9. Evolution of the eccentricity R as a function of density

The evolution of R can be expressed as: R = 0.0039 [(ρ /ρ0)5.6 -1] + 0.55

Figure A.1.10. Evolution of Pb during densification

(A.1.6)

Appendix 1 – Compaction Model Input Data for Powders

265

Figure A.1.11. Representation of the yield surface in the P-Q plane

A.1.1.3. Friction Friction coefficient is constant, µ = 0.1 or to be more accurate:

µ = - 0.0049 ρ + 0.115 with ρ in g/cm3.

(A.1.7)

266

Appendix 1 – Compaction Model Input Data for Powders

A.1.1.4. Suggestion for analytical input data for Distaloy AE Table A.1.1. Suggestion for analytical input data DISTALOY AE Parameters of the model Elasticity

Young’s modulus Bulk modulus

Expression for fitting the evolutions Constants, units

ρ in g/cm3

E (MPa) = (-28000 + 10120ρ) [ exp (ρ /6.8 ) ] 6

ρ in g/cm3

K (MPa) = (-10500 + 3750ρ) [ exp (ρ /6.55 ) ] 6

Poisson’s ratio Failure line

d cohesion

ν = 0.5 - E/6K d (MPa) = A(( ρ /ρ0)B) -1)

A = 0.0054 MPa B = 10.3

ρ0 = 3.1 g/cm3 ρ in g/cm3 Angle β Cap

R eccentricity Pb isostatic pressure

β = 70° R = 0.0039 [(ρ /ρ0)5,6 -1] + 0.55 Pb (MPa) = a[(ρ /ρ0)b-1]

ρ in g/cm3 a = 3.2 MPa b = 5.8

ρ in g/cm3

A.1.2 WC-Co Powder Composition: 10 % Co, 2 % PEG, WC balance (from Eurotungstene) Apparent density: Full density of green compact: Full density after sintering:

3.20 g/cm3 11.83 g/cm3 14.44 g/cm3

Appendix 1 – Compaction Model Input Data for Powders

267

A.1.2.1. Elasticity

Figure A.1.12.Young’s modulus as function of density for WC-Co

The expression of the analytical curve shown in Figure A.1.12 is : E (MPa) = 63.74 (exp((ρ -3.2)/0.8)-1)

(A.1.8)

with ρ in g/cm3.

Figure A.1.13. Bulk modulus as a function of density for WC-Co

The expression of the analytical curve shown in Figure A.1.13 is: K(MPa) = 20(exp((ρ -3.2)/0.73)-1)

(A.1.9)

268

Appendix 1 – Compaction Model Input Data for Powders

with ρ in g/cm3. Poisson’s ratio (Figure A.1.14) is deduced from E and K using the relationship:

ν = 0.5-E/6K

(A.1.10)

E and K being given from Equations (A.1.8) and (A.1.9).

Figure A.1.14. Poisson’s ratio as a function of density for WC-Co

A.1.2.2. Plasticity A.1.2.2.1. Failure Line Tensile Test The failure stress is defined by: σt =F/(πD2/4). In the P-Q plane the state of stress is: P = -σt /3; Q = σt

(A.1.11)

Appendix 1 – Compaction Model Input Data for Powders

269

Figure A.1.15. Tensile failure stress σt as function of density for WC-Co

Diametral compression test The failure stress is defined by: σd = 2F/πDt. P = 2σd /3; Q =

13 σd

(A.1.12)

Figure A.1.16. Diametral compression failure σd stress as function of density for WC-Co

270

Appendix 1 – Compaction Model Input Data for Powders

Simple compression test The failure stress is defined by: σs=F/(πD2/4). P = σs /3; Q = σs

(A.1.13)

Figure A.1.17. Simple compression failure stress σs as function of density for WC-Co

Determination of the failure line

Figure A.1.18. Loading path of the three tests used for determining the failure line

Failure lines have been fitted using only the simple and diametral compression tests (Figure A.1.19).

Appendix 1 – Compaction Model Input Data for Powders

271

Figure A.1.19. Failure line for different densities deduced from the diametral and simple compression test

Figure A.1.20. Evolution of the cohesion function of density for WC-Co

Cohesion d can be fitted with the expression: d (MPa) = A(( ρ /ρ0)B) -1) with A = 0.00216 MPa; B = 8.455; ρ0 = 3.2 g/cm3.

(A.1.14)

272

Appendix 1 – Compaction Model Input Data for Powders

Figure A.1.21. Evolution of the failure line angle as a function of density for WC-Co

The slope of the failure line can be considered as constant in first approximation: β = 68°. A.1.2.2.2 Identification of the Cap

Figure A.1.22. Evolution of the eccentricity R as a function of density

To a first approximation it is possible to consider the eccentricity as constant: R = 0.70.

Appendix 1 – Compaction Model Input Data for Powders

273

Figure A.1.23. Evolution of Pb during densification

The evolution of Pb can be expressed as : Pb (MPa) = a[(ρ /ρ0)b-1]

(A.1.15)

with a = 0.0211 MPa and b = 9.84. Figure A.1.24 gives an idea of the shape of the yield surface in the P-Q plane.

Figure A.1.24. Representation of the yield surface in the P-Q plane

274

Appendix 1 – Compaction Model Input Data for Powders

A.1.2.3 Friction The evolution of µ with the density of the compact is represented Figure A.1.25.

Figure A.1.25. Evolution of the friction coefficient µ as function of density

The evolution of the stress transmission coefficient α is shown Figure A.1.26 :

α = σr m Log(σz up / σz lop)/(σz up - σz lop)

(A.1.16)

Figure A.1.26. Evolution of the stress transmission coefficient α as a function of density

Appendix 1 – Compaction Model Input Data for Powders

275

A.1.2.4 Suggestion for analytical input data WC-Co Table A.1.2 . Suggestion for analytical input data WC-Co Parameters of the Cap model Elasticity Young’s modulus Bulk modulus

Expressions E (MPa) =

Constants, Units ρ in g/cm3

63.74(exp((ρ-3.2)/0.8)-1) K(MPa) =

ρ in g/cm3

20(exp((ρ -3.2)/0.73)-1) Poisson’s ratio Failure line

d cohesion

ν = 0.5-E/6K d (MPa) = A(( ρ /ρ0)B) -1)

A = 0.00216 MPa B = 8.455

ρ0 = 3.2 g/cm3 Angle β

β = 68°

R eccentricity

R = 0.70

Cap Pb isostatic pressure

Pb (MPa) = a[(ρ /ρ0)b-1]

a = 0.0211 MPa b = 9.84

A.1.3 Zirconia Powder. Low- and High-Pressure Closed-die Compaction Tests carried out for Dienet by Ludwig Schneider, David Thomson and Alan Cocks, University of Leicester, Department of Engineering, University Road, Leicester LE1 7RH, UK A.1.3.1 General Low- and high-pressure closed-die compaction tests were carried out for “Dienet” on zirconia (ZrO2) powder. The material consists of small particles with 100 nm size that are granulated to a granule size of 30 to 100 µm. The zirconia powder was delivered by Dynamic-Ceramic Ltd.

276

Appendix 1 – Compaction Model Input Data for Powders

A.1.3.2. Low-pressure Compaction A.1.3.2.1. Results and Discussion This section shows typical results and plots that can be obtained using the lowpressure facility. All tests on ZrO2 were carried out using a glass die. Depending on the material investigated there can be significant scatter in experimental results mainly due to the different initial state of the material [1]. A number of tests on ZrO2 were carried out. Reproducibility of the test results is quantified in a plot of top axial stress versus density (Figure A.1.27a). Densities achieved for a given stress were within a range smaller than ±1 %. For clarity, only one representative test is shown in all the following plots. A.1.3.2.2. Compaction and Friction Response at Low-pressure 120 interfacial shear stress /kPa

1200

top axial stress /kPa

1000 800 600 400 200 0 1.40

100 80 60 40 20 0 1.40

1.60 density /(Mg/m^3)

1.80

1.60

1.80

density /(Mg/m^3)

Figure A.1.27. (a) Top axial stress and (b) interfacial shear stress as a function of density

Figure A.1.28 shows that the stress ratio (radial / axial) of approximately 0.46 and the friction coefficient of 0.25 are independent of density within the range investigated.

Appendix 1 – Compaction Model Input Data for Powders

1

0.5

0.9

0.45 coefficient of friction

stress ratio (ra/ax)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

277

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0 1.40

1.60

0 1.40

1.80

1.60

density /(Mg/m^3)

1.80

density /(Mg/m^3)

Figure A.1.28. Evolution of (a) stress ratio and (b) friction coefficient with density

A.1.3.2.4. Cam-Clay Surfaces at Low-pressures t ta ls a c iti line cr

Σ

softening

Q0

e

hardening

elastic material response

P0 / 2

P0

Σm

Figure A.1.29. Cam-Clay model 700 1.81 Mg/(m^3)

(a)

700

strain direction

600 1.75

500 400

1.62

tangent

300 200

1.49 loading path

100 0 0

100 200 300 400 500 600 700 mean stress /kPa

ellipse semi-axes /kPa

deviatoric stress /kPa

600

Qo 500 400 300 200

Po/2

100 0 1.4

1.6 1.8 density /(Mgm-3)

2

Figure A.1.30. (a) Cam-Clay surfaces determined from a low-pressure closed-die test at four densification levels. (b) Ellipse semiaxes Po/2 and Qo as a function of density

278

Appendix 1 – Compaction Model Input Data for Powders

A.1.3.2.5. Drucker-Prage- Cap Surfaces at Low-pressures Alternatively to the Cam-Clay model the Drucker-Prager-cap material model [2] is widely used. The model consists of a shear failure line and a spherical cap as shown in Figure A.1.31.

Σ

cap yield surface Fc

yie ar e h s

ace urf s ld

c

φ

Fc

ha

g nin e rd

elastic response R(c + pa tan φ )

Σm

pb

pa Figure A.1.31. Drucker-Prager model

600

CamClay

2

1.75

500 400 1.62

300 200

2.5

(b)

1.81 Mg/(m^3)

Pb

500

1.5

400

R

1

300 0.5

Pa

200

1.49

0

100

100

0

0

-0.5

1.4

0

100 200 300 400 500 600 700

R

600 deviatoric stress /kPa

700

(a)

Pa and Pb /kPa

700

1.6

1.8

density /(Mgm-3)

mean stress /kPa

Figure A.1.32. Drucker-Prager surfaces determined from a low-pressure closed-die test at four densification levels. (b) Model parameters as a function of density.

A.1.3.3. High-Pressure Compaction A high-pressure triaxial cell was used to perform simulated closed-die compaction tests where the material is compacted by increasing the axial strain while the cell oil pressure is controlled to keep the radial strain zero. The uniaxial straining is frictionless and therefore the compact density is uniform along its axial length.

Appendix 1 – Compaction Model Input Data for Powders

279

The high-pressure testing facility as well as the specimen preparation is described in detail in [3]. Similar tests on different powders can be found in [1, 3-5]. A.1.3.3.1. Results and Discussion Three closed-die compaction tests up to 500 MPa radial stress were carried out for the ZrO2 material. Typical plots are shown in the following subsections. The results show good repeatability of the testing procedure up to 300 MPa radial stress. For clarity, only one representative test is shown in all the following plots. Compaction Response at High-Pressure 800

1

(a)

700

0.9 0.8 stress ratio (ra/ax)

600 stresses /MPa

(b)

axial stress

500 400

radial stress

300

A

200

0.7 0.6 0.5 0.4 0.3 0.2

100

0.1

0 1.66

2.16

2.66

0 1.66

3.16 -3

2.16

2.66

3.16 -3

density /(Mgm )

density /(Mgm )

Figure A.1.33. Evolution of (a) stresses and (b) stress ratio with density. Figure (a) shows the comparison between triaxial data (lines) and instrumented die results from AEA.

Cam-Clay Surfaces 700

400

(a)

350 3.37 Mg/(m^3)

500 400 3.25

300 3.06

200

0 0

(b)

300 250 Qo

200 150 100 50

2.77

100

ellipse semi-axes

deviatoric stress /MPa

600

100 200 300 400 500 600 700 mean stress /MPa

0 1.66

Po/2 2.16

2.66

3.16

density /(Mgm-3)

Figure A.1.34. (a) Cam-Clay surfaces determined from a high-pressure simulated closed-die test at four densification levels. (b) Ellipse semiaxes Po/2 and Qo as a function of density.

280

Appendix 1 – Compaction Model Input Data for Powders

Drucker-Prager Surfaces

600

Cam-Clay

3.37 Mg/(m^3)

500

Pa and Pb /MPa

deviatoric stress /MPa

700

(a)

400 3.25

300 3.06

200 2.77

100

600

Pb

500

R

1.5 1

400 300

0.5

Pa

200

0

100

0 0

2

(b)

R

700

0 1.66

100 200 300 400 500 600 700

-0.5 2.16

2.66

3.16

density /(Mgm-3)

mean stress /MPa

Figure A.1.35. Drucker-Prager surfaces determined from a high-pressure closed-die test at four densification levels. (b) Model parameters as function of density.

A.1.3.4. Regression to Low- and High-Pressure Compaction Response Fitting an analytical expression to the experimental data to predict how the CamClay or the Drucker-Prager surfaces evolve with density is advantageous for describing the material response within a computer model. When fitting an expression to this data, it is important to note that in a practical design situation a powder will be compacted from its initial loose-packed state (characterised by the apparent density) to a high-density state, similar to that achieved in the triaxial cell. It is apparent from the plot, e.g. of P0/2 against density that the low-pressure response blends into the high-pressure response. When fitting this data we ignore the early stages of high-pressure compaction, which is sensitive to the initial (dense random packed) state of the powder and concentrate on the regime where the stresses are coMParable with or exceed those experienced in the later stages of the low-pressure compaction tests. We use a common form of expression to represent how P0/2, Q0 (Cam-Clay) and Pb, R (Drucker-Prager) evolve with density from the initial density up to the density achieved with the high-pressure compaction. Each of these quantities is assumed to evolve according to a relationship of the form Bi

 ρ − ρ0   ρ − ρ0   + Di  + Ci  V = Ai   ρ0   ρ0 

(A.1.17)

where V is the quantity ( P0 / 2 , Q0 , Pb , R ) that is being fitted, Ai , Bi , Ci and Di are fitting parameters, and

ρ

and

ρ0

are the current and the initial

Appendix 1 – Compaction Model Input Data for Powders

281

density, respectively. The initial density of 1.40 Mgm-3 is determined in the lowpressure experiments. At this density the system is sensitive enough to measure the top axial stress. Initially stresses are very low for large axial strains. This is the reason why the initial density can be slightly higher coMPared to the apparent density determined by other methods. The fitting parameters Ai , Bi , Ci and Di to achieve a good fit in the low- and the high-pressure regions are given in Table A.1.3 for an initial density of 1.40 Mg m-3. The annexe to Section A.1.3 contains the table in a similar format but Ai , Bi , Ci and Di were determined for three different initial densities where the 1.28 Mg m-3 is the apparent density for this material. Table A.1.3. Calibrated model parameters using expression (A.1.17). coefficient of friction [7]

0.15

function

 ρ − ρ0 V = A i   ρ0

   

Bi

 ρ − ρ0 + C i   ρ0

Cam-Clay model value

V

ρ0

/Mg m-3

Ai

(MPa)

P0 / 2

(MPa)

  + Di   Drucker-Prager model

Q0

(MPa)

Pb

(MPa)

R

1.40

1.40

1.40

1.40

56.97

86.30

113.78

0

Bi

4.96

4.29

4.97

0

Ci (MPa)

1.000

1.400

3.140

0.284

Di (MPa)

0

0

0

0.648

For the Cam-Clay model only the semiaxis P0/2 is used to show how the expression (A.1.17) fits the experimental data. The fitting to the semi-axis Q0 is of similar quality. Figure A.1.36 shows that the fit is satisfactory in the low- (Figure A.1.36a) and high-density regimes (Figure A.1.36b). The fitting quality can also be evaluated from Figure A.1.37, which is a double-logarithm plot of the semiaxis P0/2 versus density.

Appendix 1 – Compaction Model Input Data for Powders

2 1.8 1.6 1.4 Po/2 /MPa

350

(a)

1.2 1

(b) 300 high pressure data

250 high pressure data

Po/2 /MPa

282

200

fitted expression

150

0.8 0.6

fitted expression

100

0.4 low pressure data

0.2 0 1.20

1.40

1.60

1.80

2.00

low pressure data

50 2.20

-3

density /(Mgm )

0 1.20

2.20

3.20

density /(Mgm-3)

Figure A.1.36. Comparison of data with fitted analytical expression. Evolution of semiaxis P0/2 with density in the (a) low- and (b) high-pressure region. 3 2

log(P0/2/MPa)

1 0 0 -1 -2

0.2

0.4

0.6

high pressure data fitted expression

-3 -4

low pressure data

-5 log(density/(Mgm-3))

Figure A.1.37. Double-logarithm plot of semiaxis Po/2 versus density to show the material response in the low-pressure region more clearly.

For the Drucker-Prager model the fit to the data of the eccentricity (Figure A.1.38) and the isostatic yield stress (Figure A.1.39 and Figure A.1.40) are satisfactory in the low- and high-pressure compaction regimes. The hardening parameter Pa can be expressed as Pa = Pb /(3R + 1) .

Appendix 1 – Compaction Model Input Data for Powders

283

1.4 fitted expression

1.2

R

1 0.8 0.6

high pressure data

0.4 0.2

low pressure data

0 1.4

1.9

2.4

2.9

density /(Mgm-3)

Figure A.1.38. Comparison of data with fitted analytical expression. Evolution of eccentricity with density from low- and high-pressure compaction.

12 10

600 fitted expression

500

8 6 4

400

high pressure data

300

low pressure data

100

1.60

1.80

fitted expression

high pressure data

200

2 0 1.40

(b)

Pb /MPa

Pb /MPa

700

(a)

2.00

density /(Mgm-3)

2.20

0 1.40

low pressure data

1.90

2.40

2.90

3.40

density /(Mgm-3)

Figure A.1.39. Comparison of data with fitted analytical expression. Evolution of isostatic yield stress with density in the (a) low- and (b) high-pressure region.

284

Appendix 1 – Compaction Model Input Data for Powders

3.5 3 2.5

fitted expression

log(Pb /MPa)

2 1.5 1

high pressure data

0.5 0 -0.5 0

0.2

-1

0.4

0.6

low pressure data

-1.5 -2

log(density/(Mgm-3)) Figure A.1.40. Double-logarithm plot of isostatic yield stress versus density to show the material response in the low-pressure region more clearly.

A.1.3.5. Friction Measurements [6] Using a shear box method Vandermeulen and Hendrix [6] determined the static and dynamic coefficients of friction of powder compacts against shear plates made of hardmetal and zirconia. In this test the shear plate represents the die wall. It was found that lubrication and surface finish have a strong influence on the evolution of the friction coefficient. The compact was generated in a rigid closeddie using axial stresses between 0.7 and 31 MPa, which corresponds to densities of 1.73 to 2.69. The die, punch and shear plate were lubricated with zinc stearate. For stresses of 7 and 31 MPa, i.e. densities of 2.03 and 2.69 Mg m-3 the dynamic friction coefficients on lubricated hardmetal surfaces (Ra=0.06 µm) were 0.15. It can be expected that this value does not change significantly for compacts of higher densities, i.e. higher stresses.

Appendix 1 – Compaction Model Input Data for Powders

A.1.3 Annexe Table A.1.4. All calibrated model parameters for three different initial densities. Coefficient of friction

0.15

function

 ρ − ρ0 V = A i   ρ0

   

Bi

 ρ − ρ0 + C i   ρ0

Cam-Clay model value

V

P0 / 2

(MPa)

  + Di   Drucker-Prager model

Q0 (MPa)

Pb

(MPa)

R

ρ0 Mgm

1.20

1.20

1.20

1.20

Ai (MPa)

12.25

22.74

24.79

0

Bi

5.51

4.78

5.50

0

Ci (MPa)

0.196

0.300

0.364

0.244

Di (MPa)

0

0

0

0.609

-3

Cam-Clay model value

V

P0 / 2

(MPa)

Drucker-Prager model

Q0

(MPa)

Pb

(MPa)

R

ρ0 (Mgm-3)

1.28

1.28

1.28

1.28

Ai (MPa)

23.86

40.73

48.21

0

Bi

5.29

4.58

5.28

0

Ci (MPa)

0.595

0.500

1.163

0.261

Di (MPa)

0

0

0

0.625

Cam-Clay model value

V

P0 / 2

(MPa)

Drucker-Prager model

Q0

(MPa)

Pb

(MPa)

R

ρ0 (Mgm-3)

1.40

1.40

1.40

1.40

Ai

56.97

86.30

113.78

0

4.96

4.29

4.97

0

(MPa)

Bi Ci

(MPa)

1.000

1.400

3.140

0.284

Di

(MPa)

0

0

0

0.648

285

286

Appendix 1 – Compaction Model Input Data for Powders

A.1.4 Samarium Cobalt Powder Low- and High-Pressure Closeddie Compaction Tests carried out for Dienet by Ludwig Schneider, David Thomson and Alan Cocks University of Leicester, Department of Engineering, University Road, Leicester LE1 7RH, UK A.1.4.1. General Low- and High-Pressure closed-die compaction tests were carried out for “Dienet” on samarium cobalt (SmCo) powder. It was delivered by Swift Levick Magnets Ltd. The material consists of small magnets with a very small particle size in the range from 1 to 5 µm. This results in very poor flow and packing properties and therefore a very small initial density. Blending powder before testing was not possible as it is very reactive and flammable mainly due to its small particle size. A.1.4.2. Low-pressure Compaction A.1.4.2.1. Results and Discussion This section shows typical results and plots that can be obtained using the lowpressure facility. All tests on the SmCo were carried out using a glass die. Depending on the material investigated there can be significant scatter in experimental results mainly due to the different initial state of the material [1]. A number of tests on SmCo were carried out. The initial density varied between 2.12 and 2.24 Mg/m3. Reproducibility of the tests results is quantified in a plot of top axial stress versus density (Figure A.1.40a). Densities achieved for a given top axial stress above 1 kPa were within a range smaller than ±2 %. For clarity, only one representative test is shown in all the following plots. Compaction and Friction Response 120

1400 (a)

interfacial shear stress /kPa

(b)

top axial stress /kPa

1200 1000 800 600 400 200 0 2.12

2.52

2.92

3.32

density /(Mg/m^3)

100 80 60 40 20 0 2.12

2.52

2.92

3.32

density /(Mg/m^3)

Figure A.1.41. (a) Top axial stress and (b) interfacial shear stress as a function of density.

Appendix 1 – Compaction Model Input Data for Powders

287

Stress ratio and coefficient of friction were determined by the Janssen-Walker method of differential slices [7]. Figure 1.41 shows that measurement of stress ratio and friction cannot be carried out from the initial density onwards because in the very early stages the material is very soft and densification starts without any measurable axial and radial stress. Figure A.1.42 shows that the stress ratio (radial / axial) of approximately 0.5 is independent of density within the density range investigated. The friction coefficient, however, decreases from 0.27 to 0.20 with increasing density.

0.5

1 (a)

0.9

0.45 coefficient of friction

stress ratio (ra/ax)

0.8 0.7 0.6 0.5 0.4 0.3 0.2

(b)

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.1 0 2.12

2.52

2.92

0 2.12

3.32

2.52

2.92

3.32

density /(Mg/m^3)

density /(Mg/m^3)

Figure A.1.42. Evolution of (a) stress ratio and (b) friction coefficient with density

Cam-Clay Surfaces

3.57 Mg/(m^3)

600 3.51

400 3.42 tangent

200

3.31

0 200

400

(b)

600 Qo

500 400 300 200 100

loading path

0

700

strain direction

ellipse semi-axes

800 deviatoric stress /kPa

800

(a)

600

mean stress /kPa

800

0 2.12

Po/2 2.62

3.12

3.62

4.12

density /(Mg/m^3)

Figure A.1.43. (a) Cam-Clay surfaces determined from a low-pressure closed-die test at four densification levels. (b) Ellipse semiaxes Po/2 and Qo as a function of density.

288

Appendix 1 – Compaction Model Input Data for Powders

Drucker-Prager Cap Surfaces

800

CamClay

1.8 1.6

700

600

3.51

400 3.42

200

2

(b)

1.4

600

R

500

1

400

0.8

300 Pb

200

3.31

1.2

0.6

Pa

0.4

100 0 0

200

400

600

0.2

0 2.12

800

R

3.57 Mgm

Pa and Pb /kPa

deviatoric stress /kPa

800

900

-3

(a)

mean stress /kPa

2.62

3.12

0 3.62

-3

density /(Mgm )

Figure A.1.44. (a) Drucker-Prager surfaces determined from a low-pressure closed-die test at four densification levels. (b) Model parameters as function of density.

A.1.4.3. High-Pressure Compaction A.1.4.3.1 Results and Discussion Compaction Response 700

1

(a)

0.9

(b)

0.8

500

stress ratio (ra/ax)

stresses /MPa

600 axial stress

400 300 200

radial stress

100 0 3.31

4.31 density /(Mg/m^3)

5.31

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 3.31

4.31 density /(Mg/m^3)

Figure A.1.45. Evolution of (a) stresses and (b) stress ratio with density

5.31

Appendix 1 – Compaction Model Input Data for Powders

289

Cam-Clay Surfaces 400

(a)

(b)

350

500 400

ellipse semi-axes

deviatoric stress Q /MPa

600

5.33 Mg/(m^3)

300 5.08

200

4.76 4.28

100

300 Qo

250 200 150 100 50

0 0

100

200

300

400

500

Po/2

0 3.31

600

mean stress P /MPa

4.31

5.31

density /(Mg/m^3)

Figure A.1.46. (a) Cam-Clay surfaces determined from a high-pressure simulated closed-die test at four densification levels. (b) Ellipse semiaxis with P0/2 and Q0 as a function of density.

Drucker-Prager Surfaces 600

500

2

(b)

1.8

500

400

5.33 Mg/(m^3)

CamClay

300 5.08

200

4.76 4.28

100

1.6

Pb

400

1.4 1.2

300 200

1

0.8

Pa

R

R

(a)

Pa and Pb /MPa

deviatoric stress Q /MPa

600

0.6 0.4

100

0.2

0 0

100

200

300

400

500

mean stress P /MPa

600

0 3.31

0

4.31

5.31 -3

density /(Mgm )

Figure A.1.47. Drucker-Prager surfaces determined from a high-pressure closed-die test at four densification levels. (b) Model parameters as function of density.

290

Appendix 1 – Compaction Model Input Data for Powders

A.1.4.4. Regression to Low- and High-Pressure Compaction Response Table A.1.5. Calibrated model parameters using expression (A.1.17) Cam-Clay model

Drucker-Prager model

low-pressure (LP)

high-pressure (HP)

LP

HP

LP and HP

parameters

P0 / 2

Q0

P0 / 2

Q0

Pb

Pb

R

A (MPa)

11.8

20.1

1294

1213

24.3

1746

0

B

9.02

9.02

2.97

2.48

9.07

2.47

0

C (MPa)

0.020

0.025

0

0

0.043

0

0.050

D (MPa)

0

0

0

0

0

0

0.850

These separate fits are then used to define a transition regime, where the semiaxis Wtransition is valid within a density range from ρ L to ρ H Wtransition = Wlow

ρH − ρ ρ − ρL + Whigh ρH − ρL ρH − ρL

(A.1.18)

ρ L =3.50 to ρ H =3.60 Q0 the transition density range lies between ρ L =3.40 to

The transition density range for P0 / 2 and Pb ranges from Mgm-3. For the semiaxis

ρ H =3.50 Mgm-3. 350

(a)

(b)

300

1.6

semi-axis P0/2 in MPa

semi-axis P0/2 in MPa

2

250

1.2

high pressure data

0.8

fitted expression

high pressure data

200

fitted expression

150

low pressure data

100

0.4

low pressure data

50

0

0 2

2.5

3

3.5 -3

density /(Mgm )

4

2

3

4

5

6

-3

density /(Mgm )

Figure A.1.48. Comparison of data with fitted analytical expression. Evolution of semiaxis

P0 / 2

with density in the (a) low- and (b) high-pressure region.

Appendix 1 – Compaction Model Input Data for Powders

291

3 2

log(Po/2/MPa)

1 0 -1

0

0.2

0.4

0.6

0.8

high pressure data

-2 -3

fitted expression

-4

low pressure data

-5

-3

log(density/(Mgm ))

Figure A.1.49. Double-logarithm plot of semiaxis Po/2 versus density to show the material response in the low-pressure region more clearly

For the Drucker-Prager model the fit to the data of the eccentricity and the isostatic yield are satisfactory in the low- and high-pressure compaction regime. The hardening parameter Pa can be expressed as Pa = Pb /(3R + 1) .

1.4

high pressure data

1.2

R

1 0.8 0.6 0.4 0.2

low pressure data fitted expression

0 2.12

3.12

4.12

5.12

6.12

density /(Mgm-3)

Figure A.1.50. Comparison of data with fitted analytical expression. eccentricity with density from low- and high-pressure compaction.

Evolution of

292

Appendix 1 – Compaction Model Input Data for Powders

2

600

(a)

1.8

(b) 500

1.6

400

high pressure data

1.2

high pressure data

Pb /MPa

Pb /MPa

1.4

300

1 0.8

fitted expression

0.6 0.4

200

low pressure data

0.2

fitted expression low pressure data

100 0

0 2

2.5

3

3.5

2

4

3

4

5

6

-3

density /(Mgm )

-3

density /(Mgm )

Figure A.1.51. Comparison of data with fitted analytical expression. Evolution of isostatic yield stress with density in the (a) low- and (b) high-pressure region. 3

log(Pb /MPa)

2 1

fitted expression

0 0.4 -1

0.5

0.6

0.7

0.8

high pressure data

-2 low pressure data

-3 -3

log(density/(Mgm ))

Figure A.1.52 Double-logarithm plot of isostatic yield stress versus density to show the material response in the low-pressure region more clearly.

References [1] Schneider LCR. 2003. Compaction And Yield Behaviour of Particulate Materials. PhD Thesis, in progress. [2] Drucker DC and Prager W. 1952. Soil Mechanics and Plastic Analysis or Limit Design. Quarterly of Applied Mathematics, vol 10, no 2, p 157-175. [3] Sinka IC, Cocks ACF, Morrison CJ and Lightfoot A. 2000. High-Pressure Triaxial Facility for Powder Compaction. Powder Metallurgy, vol 43, no 3, p 253-261. [4] Sinka IC, Cocks ACF and Tweed JH. 2001. Constitutive Data for Powder Compaction Modelling. Journal of Engineering Materials and Technology, Transactions of the ASME; Series H, vol 123, no 2, p 176-183. [5] Schneider LCR and Cocks ACF. 2002. An Experimental Investigation of the yield behaviour of metal powder compacts. Powder Metallurgy, vol 45, no 3, p 237-245.

Appendix 1 – Compaction Model Input Data for Powders

293

[6] Vandermeulen W and Hendrix W. 2002. Determination of The Friction Coefficient of Compacted ZrO2, WC-Co and SmCo Powders Against WC and ZrO2 Die Materials. Work reported in the framework of the TN DIENET EC Contract G5RT-CT-200105020, Sept. 2002. [7] German RM. 1994. Powder Metallurgy Science. Metal Powder Industries Federation, Princeton New Jersey.

A.2 Appendix 2 – Case-study Components

A.2.1 Introduction As mentioned in Chapter 11, several case-study components have been produced and studied during the Modnet and Dienet programmes. This set of case studies potentally provides a valuable resource against which to test current and newly developed compaction models, and hence details of a range of these case studies is drawn together in this appendix. All have been produced with the materials for whom input data parameters are given in Appendix 1. The case studies produced are listed in Table A.2.1. This also indicates those for which detailed information is provided in this appendix as well as those for whom additional information has already been published. The next section introduces the data and presents the case studies. Table A.2.1.

Case-study components produced

Programme

Case-study

Material

Kinematics

This appendix

Modnet

CS1

Ferrous

1

Reference [1]

Dienet

CS1

Hardmetal

2

2 (Also [2])

Ferrous

2

2 (Also [3])

Ceramic

2

CS2

Ferrous

1

1

CS3

Hardmetal

3

1

Ceramic

2

2

Magnet

2

2

296

Appendix 2 – Case-study Components

A.2.2 Data for Case-study Components All the case studies that follow have been produced using the techniques presented in Chapter 11 and evaluated using techniques in Chapter 12. Key points are: Materials All case studies have been produced with one of the four materials for which input data is in Appendix 1. Average fill density This is from the part mass and filling volume. In one case (CS2), a fill density distribution is given. This used a technique mentioned in Chapter 9, and the results have been used in Chapter 13. Punch positions These are recorded as in Chapter 11. The displacements at maximum load have been corrected for punch deflection. Part geometries after pressing and sintering These have generally used measuring calipers as the geometries are simple. For CS3, where one surface is dished, some measurements used a coordinate measuring machine. Loads These are measured as in Chapter 11. Full load evolutions were measured as in Figure 11.4 and for some cases additional information is given in [2] and [3]. Pressed-part density This is by weighing and measuring as the part geometries are simple. Density distribution This was determined using one or more of the techniques described in Chapter 12. Defects Again these were assessed using metallograpic techniques as described in Chapter 12.

Appendix 2 – Case-study Components

297

A.2.2.1 Dienet Case-study 1, Hardmetal, Kinematics A (Dienet CS1 HM A) Model input data Filling

Geometry

Part reference

Dienet CS1 HM16

Powder

Dienet hardmetal

H1, filling, mm

25

H2, filling, mm

49.98

Part mass, g

195.11

Average fill density, Mg m-3

3.14

47.8

H1

R 1

25.8

H2

37.9

Punch positions Measured punch positions, mm, with respect to die top surface (down is negative) Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

1233

0.48

-74.98

-25.00

Upper punch closes die

2230

-2.95

-68.97

-25.00

20 % move on UP, LIP

3213

-3.00

-56.82

-25.00

LIP only moves to 60 %

5260

-15.08

-45.59

-25.66

UP, LIP move to 100 %

6130

-15.31

-45.62

-25.66

Hold at maximum

Validation data for compaction model predictions Pressed-part geometry (CS1 HM20) Dimension

As pressed, mm

H1

Springback, % (if measured)

As-sintered, mm

Shrinkage, % (if measured)

10.40

8.40

23.8

H2

20.00

15.85

26.2

Diameter, stub

38.00

-0.1

30.40

25.0

Diameter, flange

48.00

0.2

38.92

23.3

298

Appendix 2 – Case-study Components

Punch loads Measured punch loads, kN Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

1233

0

0

0

Upper punch closes die

2230

1

0

0

20 % move on UP, LIP

3213

1

3

-1

LIP only moves to 60 %

5260

201

83

80

UP, LIP move to 100 %

6130

245

78

111

Average part density

Hold at maximum -3

7.72 Mg m (Part 16)

Density distribution Part CS1 HM16

Sintered density Defects CS1 Part HM19

Part CS1 HM18

Appendix 2 – Case-study Components

Shape change on sintering (Part CS1 HM20) As-pressed

Sintered

Change on sintering

299

300

Appendix 2 – Case-study Components

A.2.2.2 Dienet Case-study 3, Hardmetal, Kinematics B (Dienet CS3 HM B) Model input data Filling

Geometry

Part reference

Dienet CS1 HM5

Powder

Dienet hardmetal

H1, filling, mm

25

H2, filling, mm

49.98

Part mass, g

194.96

Average fill density, Mg m-3

3.14

47.8

H1

R 1

25.8

H2

37.9

Punch positions Measured punch positions, mm, with respect to die top surface (down is negative) Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

1612

0.545

-74.98

-25

Upper punch closes die

2559

-1.435

-71.97

-25

10 % move on UP, LIP

3357

-1.51

-59.665

-25

LIP only moves to 50 %

5312

-15.12

-45.75

-25.68

UP, LIP move to 100 %

6129

-15.31

-45.78

-25.68

Hold at maximum

Validation data for compaction model predictions Pressed-part geometry Dimension

Aspressed, mm

H1

10.35

H2

20.10

Diameter, stub

37.87

-0.1

Diameter, flange

47.90

0.2

Springback, % (if measured)

As-sintered, mm

Shrinkage, % (if measured)

Appendix 2 – Case-study Components Punch loads Measured punch loads, kN Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

1612

0

0

0

Upper punch closes die

2559

0

0

0

10 % move on UP, LIP

3357

1

3

-1

LIP only moves to 50 %

5312

220

167

96

UP, LIP move to 100 %

6129

236

161

112

Hold at maximum -3

Average part density

7.69 Mg m (Part HM5)

Density distribution CS1 HM4

CS1 HM6

Sintered density

Not determined

Defects CS1 HM4

a

b a

b

301

302

Appendix 2 – Case-study Components

A.2.2.3

Dienet Case-study 1, Ferrous, Kinematics A (Dienet CS3 Fer A)

Model input data Filling

Geometry

Part reference

Dienet CS1 Fer32

Powder

Dienet ferrous

H1, filling, mm

40.31

H2, filling, mm

68.68

Part mass, g

287.63

Average fill density, Mg m-3

3.10

47.8

H1

R 1

25.8

H2

37.9

Punch positions Measured punch positions, mm, with respect to die top surface (down is negative) Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

1554

0.01

-107.97

-40.31

Upper punch closes die

2004

-3.78

-100.41

-40.3

20 % move on UP, LIP

3751

-4.05

-83.24

-40.3

LIP only moves to 60 %

6712

-21.22

-67.08

6712

UP, LIP move to 100 %

12078

-21.35

-67.08

12078

Hold at maximum

Validation data for compaction model predictions Pressed-part geometry Dimension

As pressed, mm

H1

19.77

H2

25.96

Diameter, stub

37.88

-0.1

Diameter, flange

47.91

0.2

Springback, % (if measured)

As-sintered, mm

Shrinkage, % (if measured)

Appendix 2 – Case-study Components Punch loads Measured punch loads, kN Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

1554

0

0

0

Upper punch closes die

2004

1

0

0

20 % move on UP, LIP

3751

4

14

-4

LIP only moves to 60 %

6712

781

407

192

UP, LIP move to 100 %

12078

772

392

200

Hold at maximum -3

Average part density

7.02 Mg m (CS1 Fer32)

Density distribution CS1 Fer 32 (detail)

CS1 Fer 32 (summary)

2.98

3.00

2.50

2.57

4.53

7.09

7.01

7.12

7.04

4.40

7.09

7.03

7.10

7.03

4.25

6.92

7.05

7.10

7.01

4.97

6.99

7.01

7.10

7.00

4.01

6.83

6.93

4.37

6.87

6.96

4.42

6.91

7.01

4.48

6.97

7.04

6.05

7.05

7.02

7.05 7.07

7.00 7.06

6.92

25.82 φ 37.89 φ 47.92 φ

7.02

Dimensions, mm Densities, g/cc

Sintered density Defects CS1 Fer 27

Not determined

303

304

Appendix 2 – Case-study Components

A.2.2.4

Dienet Case-study 1, Ferrous, Kinematics B (Dienet CS1 Fer B)

Model input data Filling

Geometry

Part reference

Dienet CS1 Fer44

Powder

Dienet ferrous

H1, filling, mm

40.32

H2, filling, mm

68.66

Part mass, g

287.97

Average fill density, Mg m-3

3.10

47.8

H1

R 1

25.8

H2

37.9

Punch positions Measured punch positions, mm, with respect to die top surface (down is negative) Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

1744

0

-108.09

-40.31

Upper punch closes die

2700

-7.92

-91.88

-40.3

40 % move on UP, LIP

4500

-8.1

-74.73

-40.3

LIP only moves to 80 %

6000

-21.22

-67.08

-41.18

UP, LIP move to 100 %

8155

-21.36

-67.08

-41.19

Hold at maximum

Validation data for compaction model predictions Pressed-part geometry Dimension

As pressed, mm

H1

19.83

H2

25.89

Diameter, stub

37.88

-0.1

Diameter, flange

47.91

0.2

Springback, % (if measured)

As-sintered, mm

Shrinkage, % (if measured)

Appendix 2 – Case-study Components Punch loads Measured punch loads, kN Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

1744

0

0

0

Upper punch closes die

2700

11

7

0

40 % move on UP, LIP

4500

27

60

-8

LIP only moves to 80 %

6000

794

391

215

UP, LIP move to 100 %

8155

790

382

230

Hold at maximum -3

Average part density

7.02 Mg m (CS1 Fer44)

Density distribution CS1 Fer 44 (detail)

CS1 Fer 44 (summary)

3.01

3.03

2.46

2.55

4.55

7.03

7.09

7.12

7.08

4.35

7.15

6.96

7.12

7.05

4.44

6.95

7.15

7.03

7.06

5.06

6.86

7.17

7.03

7.04

4.45

6.94

6.93

4.51

6.90

6.98

4.79

6.99

6.97

4.65

7.01

7.03

4.71

6.98

7.06

7.05 7.09

7.04 7.04

6.95

25.83 φ 37.89 φ 47.92 φ

7.02

Dimensions, mm Densities, g/cc

Sintered density Defects CS1 Fer 39

Not determined

305

306

Appendix 2 – Case-study Components

A.2.2.5

Dienet Case-study 2, Ferrous, (Dienet CS2)

Model input data Filling

Geometry

Part reference

CS2/P1

Powder

Dienet ferrous

H1, filling, mm

6.0

3.2

50.0 10.0

H2, filling, mm

50.0

Part mass, g

144.7

Average fill density, Mg m-3

3.39

25.3 30.0

Punch positions Measured punch positions, mm, with respect to LIP (down is negative) Step

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Die

Comments

1

100

0

50

56

Filling

2

56

0

50

56.5

Closure

3

27.5

-1

24.3

34.5

Compaction

4

100

0

25.3

24.5

Ejection (1)

Validation data for compaction model predictions Pressed-part geometry Dimension

As pressed, mm

Springback, % (if measured)

As-sintered, mm

Shrinkage, % (if measured)

(Av of 2)

H1

3.15

3.15

0

H2

28.30

28.25

0.2

Diameter, stub

30.04

0.1

30.03

0

Diameter, flange

50.09

0.2

50.09

0

Appendix 2 – Case-study Components Punch loads Measured punch loads, kN Step

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

1

Filling

2

Closure

3

659

215

587

Compaction

4

Ejection (1) -3

Average part density

6.63 Mg m (CS2 P1)

Density distribution CS2 P1

Die fill density distribution Average fill density 3.39 g/cc Part mass = 144.8g

3

6,86

1

6

6,66

2

6,57

3

5

6,56

4

5

6,55

5

6,60

6

5

6,63

7

5

1 3.43 10

10

25

50

10

5

3 3.38

10

4 3.31

10

5 3.32

10

6 3.40

10

7 3.45

10

10

Sintered density Defects CS2

Region number Density, g/cc

2 3.48

6.61 Mg m-3 (CS2 S1)

5

307

308

Appendix 2 – Case-study Components

A.2.2.6 Dienet Case-study 3, Hardmetal, Kinematics A (Dienet CS3 HM A) Model input data Filling

Geometry

Part reference

CS3 HM 20

Powder

Dienet Hardmetal

H1, filling, mm

47.50

H2, filling, mm

19.97

Part mass, g

351.90

Average fill density, Mg m-3

3.27

Punch positions Measured punch positions, mm, with respect to die top surface (down is negative) Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

5560

81.53

-67.47

-47.5

Punch filling position, before underfill by 11 mm

6600

0

-78.48

-58.49

UP at 0

6940

-6.63

-78.48

-58.49

UP contacts powder

7020

-8.2

-78.48

-58.5

LIP, LOP start

12140

-32.24

-62.03

-52.31

Max load, punch hold

Validation data for compaction model predictions Pressed-part geometry Dimension

As-pressed, mm

Springback, % (if measured)

H1

20.07

H2

9.72

Diameter, stub

37.92

0.3

Diameter, flange

47.95

0.3

As-sintered, mm

Shrinkage, % (if measured)

See next page

See next page

Appendix 2 – Case-study Components Punch loads Measured punch loads, kN Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

5560

0

0

0

Punch filling position, before underfill by 11 mm

6600

0

0

0

UP at 0

6940

0

0

0

UP contacts powder

7020

0

0

0

LIP, LOP start

12140

570

262

348

Average part density

Comments

Max load, punch hold

-3

8.09 Mg m

Density distribution Part CS3 HM20

Sintered density, Mg m-3

Defects

No defects could be detected

Shape change on sintering Part CS3 HM21 line

green sintered shrinkage

ay

25,36

20,26

20,11 %

bx

24,01

19,46

18,95 %

by

29,72

23,92

19,52 %

cx

24,00

19,38

19,25 %

cy

9,68

7,74

20,04 %

dx

18,97

15,20

19,87%

309

310

Appendix 2 – Case-study Components

A.2.2.7

Dienet Case-study 3, Ceramic, Kinematics A (Dienet CS3 Cer A)

Model input data Filling

Geometry

Part reference

CS3 Cer19

Powder

Dienet zirconia

H1, filling, mm

15.99

H2, filling, mm

8.00

Part mass, g

52.347

Average fill density, Mg m-3

1.39

Punch positions Measured punch positions, mm, with respect to die top surface (down is negative) (Cer25) Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

9 230

81.54

-23.99

-15.99

Punch filling position, before underfill by 11 mm

10 230

0

-34.99

-27

UP at 0

10 600

-6.63

-34.99

-26.97

UP contacts powder

10 680

-7.6

-34.99

-26.99

LIP, LOP start

14 500

-16.21

-31.20

-25.97

Max load, punch hold

Validation data for compaction model predictions Pressed-part geometry Dimension

As-pressed, mm

Springback, % (if measured)

As-sintered, mm (Cer20)

Shrinkage, % (if measured)

H1

14.99

H2

5.26

Diameter, stub

38.08

0.7

36.55

24.4

Diameter, flange

48.21

0.8

28.80

24.2

Diameter, dish(1)

67.51

52.89

21.7

Diameter, dish(2)

67.63

52.81

21.9

Appendix 2 – Case-study Components Punch loads Measured punch loads, kN (Cer25) Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

0

0

0

Punch filling position, before underfill by 11 mm

10 230

0

0

0

UP at 0

10 600

0

0

0

UP contacts powder

10 680

0

0

0

LIP, LOP start

14 500

150

92

50

9 230

Max load, punch hold -3

2.63 Mg m

Average part density Density distribution Part CS3 (Cer 19)

H1 2

3*

5

6

4* H2

R2 Approximately equal

Average Part 19 Average A,B,C,D 2.64 2.68

2.62 2.60

2.64

Archimedes method Average from sections parallel and orthogonal to filling direction Sintered density, Mg m-3 Defects Not assessed

6.01 (Cer15), 6.02 (Cer20)

311

312

Appendix 2 – Case-study Components

A.2.2.8

Dienet Case-study 3, Ceramic, Kinematics B (Dienet CS3 Cer B)

Model input data Filling

Geometry

Part reference

CS3 Cer39

Powder

Dienet zirconia

H1, filling, mm

16.00

H2, filling, mm

8.00

Part mass, g

52.29

Average fill density, Mg m-3

1.39

Punch positions Measured punch positions, mm, with respect to die top surface (down is negative) (Cer40) Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

3380

81.52

-23.99

-15.995

Punch starting position, underfill by 11 mm

4390

0

-34.99

-26.99

UP at 0

4730

-6.63

-34.99

-26.99

UP contacts powder

4850

-8.3

-34.99

-26.99

LIP, LOP start

8590

-14.25

-29.12

-24.09

Max load, punch hold

Validation data for compaction model predictions Pressed-part geometry Dimension

As-pressed, mm

Springback, % (if measured)

As-sintered, mm (Cer37)

Shrinkage, % (if measured)

H1

9.84

H2

5.03

Diameter, stub

38.11

0.8

28.69

24.7

Diameter, flange

48.20

0.8

36.44

24.4

Diameter, dish(1)

67.62

54.29

19.7

Diameter, dish(2)

67.53

53.08

21.4

Appendix 2 – Case-study Components

313

Punch loads Measured punch loads, kN Part CS3 Cer40 Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

3 380

0

0

0

Punch starting position, underfill by 11 mm

4 390

0

0

0

UP at 0

4 730

0

0

0

UP contacts powder

4 850

0

0

0

LIP, LOP start

8 590

144

76

81

Max load, punch hold -3

2.64 Mg m

Average part density

Comments

(Range from 2.63 to 2.65)

Density distribution Parts CS3 Cer35 and Cer39

Parts CS3 Cer34, Cer36 and Cer38 1b

H1 2

3*

5

6

4* H2

R2 Approximately equal

Part 35 Average A+B 2.65 2.72 Part 39 Average A+B 2.67 2.71

Average parts 35, 39 Average A+B 2.66 2.71

R2

2

3

5

6

1a

H1

4 H2

Approximately equal

2.62 2.67

2.64 2.61

2.63 2.64

Archimedes method Average from two halves of section parallel to filling direction.

2.61

2.66

2.63

Part 34 Average A+B 2.67 2.68

2.63 2.61

Part 36 Average A+B 2.66 2.67

2.63 2.63

Part 38 Average A+B 2.65 2.67

2.64 2.63

Average Parts 34, 36 38 Average A+B 2.66 2.63 2.67 2.63

2.63 2.65

2.67 2.63

2.62 2.64

2.64 2.64

Metallography and EDS mapping Average from two halves of section parallel to filling direction.

Sintered density, Mg m-3 Defects Not assessed

6.02 (Cer33), 6.01 (Cer37)

314

Appendix 2 – Case-study Components

A.2.2.9

Dienet Case-study 3, Magnet, Kinematics A (Dienet CS3 Mag A)

Model input data Filling

Geometry

Part reference

CS3 Mag 76

Powder

Dienet Magnet

H1, filling, mm

30

H2, filling, mm

14.96

Part mass, g

126.60

Average fill density, Mg m-3

1.79

Punch positions Measured punch positions, mm, with respect to die top surface (down is negative) Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

1300

81.5

-44.96

-30

Punch filling position, before underfill by 11 mm

2320

0

-55.96

-40.99

UP at zero

2660

-6.63

-55.96

-41

UP contacts powder

3060

-12.82

-55.96

-41

LIP, LOP start

27960

-20.00

-42.42

-32.07

Max load, punch hold

Validation data for compaction model predictions Pressed-part geometry Dimension

As-pressed, mm

H1

12.07

H2

10.35

Diameter, stub

37.95

Diameter, flange

48.05

Springback, % (if measured)

As-sintered, mm

Shrinkage, % (if measured)

Appendix 2 – Case-study Components Punch loads Measured punch loads, kN Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

1300

0

0

0

Punch filling position, before underfill by 11 mm

2320

0

0

0

UP at zero

2660

0

0

0

UP contacts powder

3060

1

0

1

LIP, LOP start

27960

154

71

84 -3

4.26 Mg m

Average part density

Comments

Max load, punch hold

(Part CS3 Mag 77)

Density distribution Part CS3 Mag 76 H1 2

3*

5

6

4* H2

R2 Approximately equal

Part 76 Average A,B,C,D 4.23 4.25 4.22

4.17

Archimedes method Average from two halves of section parallel to filling direction. Sintered density, Mg m-3 Not determined

4.39

315

316

Appendix 2 – Case-study Components

A.2.2.10

Dienet Case-study 3, Magnet, Kinematics B (Dienet CS3 Mag B)

Model input data Filling

Geometry

Part reference

CS3 Mag 64

Powder

Dienet Magnet

H1, filling, mm

51.99

H2, filling, mm

6.46

Part mass, g

189.41

Average fill density, Mg m-3

1.88

Punch positions Measured punch positions, mm, with respect to die top surface (down is negative) Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

760

81.51

-58.45

-51.99

Punch filling position, before underfill by 11 mm

1844

0

-69.46

-63.01

UP at zero

2180

-6.63

-69.45

-62.99

UP contacts powder

2540

-12.93

-69.45

-62.99

LIP, LOP start

10300

-26.04

-54.68

-49.42

Max load, punch hold

Validation data for compaction model predictions Pressed-part geometry Dimension

As-pressed, mm

H1

23.38

H2

5.26

Diameter, stub

37.95

Diameter, flange

48.05

Springback, % (if measured)

As-sintered, mm

Shrinkage, % (if measured)

Appendix 2 – Case-study Components Punch loads Measured punch loads, kN Time, ms

Upper punch (UP)

Lower inner punch (LIP)

Lower outer punch (LOP)

Comments

0

0

0

Punch filling position, before underfill by 11 mm

1844

0

0

0

UP at zero

2180

1

0

0

UP contacts powder

2540

2

0

0

LIP, LOP start

10300

142

61

85

760

Max load, punch hold -3

4.26 Mg m

Average part density

(Part CS3 Mag 62)

Density distribution CS3 Mag 64 H1 2

3*

5

6

4* H2

R2 Approximately equal

Part 64 Average A+B

4.25 4.32

4.23 4.18

Archimedes method Average from two halves of section parallel to filling direction. Sintered density, Mg m-3 Not determined

4.32

317

318

Appendix 2 – Case-study Components

References [1] [2] [3]

Kergadallan J et al. 1997. Compression of an axisymmetric part with an instrumented press, Proc. Int. Workshop on Modelling of metal powder forming processes, Grenoble, July 1997, pp 277–285. Tweed JH et al. 2003. Production and characterisation of a hardmetal flanged bushing for a compaction modelling study Proc. Euro PM2003, Valencia, October 2003, Vol. 3, 57-63. Tweed JH et al., 2004. Validation data for modelling of powder compaction: Guidelines and an example from the European DIENET project Proc PM2004, Vienna, October 2004, Vol.5, 223-228.

Glossary

The following defines the most important compaction modelling terms used in this book Phrase

Explanation

“p” or “P” stress

hydrostatic stress, p= - 1/3 (σ11 + σ22 + σ33)

“q” or “Q” stress

von Mises equivalent stress, q= 3 / 2(s ij : s ij )

axisymmetric 2D parts

pressed parts the shape of which can be described by a simple 360 degree rotation of one section

2D parts

geometry that can be represented with accuracy in two dimensions for the numerical simulation.

3D parts

geometry that cannot be represented with accuracy in two dimensions for the numerical simulation.

API

active pharmaceutical ingredient

Archimedes method

method of measuring green part volume using water displacement

associated flow

materials where the plastic strain increment is normal to the yield surface

beam hardening

reduction in intensity of X-ray beam as it passes through the specimen

Brazilian disc test

see Diametral compression test

Bulk modulus

volumetric elastic modulus of the compact under isostatic pressure

CAD

computer-aided design

Cam-Clay model

mathematical model describing the plastic deformation of powders during die-compaction

320

Glossary

Phrase

Explanation

Cap models

mathematical models describing the hardening of the powder by a eccentricity in the p-q plane

case study

comparison between simulation and actual parts produced under production conditions

cohesion

shear strength of powder at zero hydrostatic pressure

cohesion angle

angle between failure line and x-axis on Cap model

compaction models

computer models for predicting key compact properties during and after die-compaction

compaction ratio

ratio of fill height to final pressed height

compression test

method of measuring compact green strength by stressing a cylindrical specimen along the die-compaction direction

computing power

microprocessor speed combined with RAM size

computing time

time required by the computer central processing unit to carry out one full simulation of the compaction cycle

constitutive data

data on the elastic and plastic properties of powder blends during die-compaction determining the relationship between stress and strain

constitutive parameter

see “eccentricity”

continuum models

see “macro model”

Coulomb friction

friction in which the transverse force is linearly proportional to the normal force

crack prediction

use of computers to predict incidence of shear or tensile cracks

critical velocity

shoe velocity in die filling above which incomplete filling occurs

dead water zone

a zone of very poor powder flow during compaction and thus of low density

deviatoric stress tensor s

volumetric 3D representation of the stress state in a body responsible for shearing. sij= σij + δijp.

deviatoric stress tensor component sij

δij is the Kronecker symbol: δij=1 if i=j, 0 otherwise

deviatoric stress

“q” or shear stress in Cam-Clay model

diametral compression test

method of measuring compact green strength by stressing across the diameter of a thin cylindrical specimen, sometimes called Brazilian disc test

Dienet

European-funded Thematic Network on Die-compaction Modelling of Several Different Materials

Glossary

321

Phrase

Explanation

discrete-element methods (DEM)

numerical simulation (here of powder compaction) computing the interaction between particles (here grains) to predict a solution at the macroscopic level.

Distaloy AE

a sponge iron powder to which Cu, Mo and Ni have been added in powder form by a diffusion process

Drucker–Prager Cap model

mathematical model describing the plastic deformation of powders during die-compaction. (This model includes an allowance for compact cohesion with the Drucker-Prager failure line.)

dynamic or kinetic friction

friction during sliding

eccentricity

ratio of Cap model ellipse minor and major axes

EDS

energy dispersive detector (in SEM)

excipients

substances admixed to medicines as vehicles for administration of the API (q.v.)

explicit FEM versions

straightforward resolution of the stress-strain equilibrium during a process calculation iteration followed by slight corrections in the next iteration. Convenient for highdeformation process.

failure line

the line above which shear failure occurs (see DruckerPrager Cap model)

FE software

computer packages using finite elements

fill ratio

the fraction of the die filled after a single pass of the filling shoe

fixed die

compaction toolset in which the die is fixed relative to the lower punch

floating die

compaction toolset in which the die is free to move to equalise forces of top and lower punches

flow rule

can be “associated” or “nonassociated”

friction

friction during compaction between compact and tool, and between powder particles. Coefficient is expressed as ratio of tangential force to normal force.

green part

the pressed component after ejection from the die

green strength

mechanical strength of the powder compact

hardening

see “powder hardening”

hardening cap

see “yield surface”

hardening function

density-dependent relationship between stress and strain

hardware

physical constituents of the simulation computer (screen, tower, keyboard, etc.)

322

Glossary

Phrase

Explanation

implicit FEM versions

consistent resolution of the stress-strain equilibrium for each iteration of process calculation. Convenient for fine stress analysis.

input data

data on the properties of powder blends required by computer models

instrumented die

plain cylindrical die incorporating sensors in the die wall for the measurement of radial stresses during compaction

interfaces

interfaces between simulation computer and operator or other software (e.g. CAD, press software)

inverse modelling

prediction of compaction starting conditions by calculating back from data on the finished part

kinematics

see “press kinematics”

macro model

model of the behaviour based on data derived from hardening and strength tests on the material

mean stress

hydrostatic stress or “p” on Cam-Clay model

mesh

mapping of the 2D or 3D geometry by nodes linked together to form a “mesh”; used in simulation by finite element to predict the deformation, stress and other variables

micromechanical model

see “microscopic model”

microscopic model

model of the compact behaviour based on data calculated from assumptions about individual powder particles

Modnet

European-funded Thematic Network on Die-compaction Modelling of Ferrous Materials

multilevel component

component incorporating 2 or more different thicknesses in the axial direction

nonassociated flow

materials that do not deform plastically normal to the yield surface

numerical simulation

use of computers to simulate manufacturing processes, component behaviour, etc

parametric studies

see “sensitivity studies”

particle methods

see “discrete element method”

phenomenological Model

see “macro model”

plasticity models

mathematical models describing the plastic deformation of powders during die-compaction

Poisson’s ratio

elastic movement of compact in radial direction divided by that in axial direction

Glossary

323

Phrase

Explanation

powder cohesion

ability of powder blends to withstand shear during the various stages of powder compaction

powder hardening

plastic deformation during compaction. This apparent hardening of the compact during the compaction stroke is caused by rearrangement, collapse and deformation of individual powder particles

powder transfer

sideways movement of powder during the early stages of powder compaction

p-q or P-Q plane

the hydrostatic von Mises stress plane used for the representation of the cap and failure line in most models

pre-sinter

method of improving the strength of a green part by heating to a moderate temperature without causing significant size change

press kinematics

relative motions of tooling during compaction and ejection

relative density

compact density as a fraction of theoretical density of pore-free bulk material

remeshing

recalculation of finite-element mesh after a certain distortion

SEM

scanning electron microscope

sensitivity studies

the use of computer simulations to determine the compaction parameters likely to have the greatest effect on compact quality or press loads

shear cracks

cracks occurring during compaction

shear failure

failure of powder blends during compaction owing to excessive shear stresses

shear modulus

elastic modulus of the compact under shear

shear plate

method of measuring friction coefficient between compact and die surface.

Shima model

mathematical model describing the plastic deformation of powders during die-compaction

singularity

theoretically infinite stresses are predicted for example as a result of sharp corner

software

mathematic and textual programs and data used to operate the simulation computers

static friction

friction at zero relative movement

strain tensor ε

volumetric 3D representation of the strain state in a body through the combination of three reference directions i,j.=1,2,3 (or x,y,z)

strain tensor component εij

324

Glossary

Phrase

Explanation

stress invariant

measure of stress independent of the axes used to define

stress tensor, σ

volumetric 3D representation of the stress state in a body through the combination of three reference directions i,j=1,2,3 (or x,y,z)

stress tensor component, σij suction fill

method of filling a die in which the bottom punch is withdrawn during the fill to “suck” powders into the die cavity

tensile cracks

cracks occurring on compact ejection

tool deflection

elastic deformation of tooling during the compaction stroke

triaxial stress rig

typically a high-pressure cold isostatic press capable of measuring the axial and radial deformation of loose powders or green compacts subjected to a combination of differing radial and axial pressures. Used chiefly to generate plasticity data for computer models.

user friendliness

extent to which computer models and interfaces can be operated by nonspecialist personnel

validation

techniques for comparing computer predictions of key compact properties including density and load with parts produced under industrial conditions

volumetric modulus

see “bulk modulus”

von Mises material

material exhibiting deformation behaviour of a fully dense solid

X-ray computerised tomography (CT)

method of measuring density distributions in pressed parts using X-ray methods

yield stress

the stress below which material behaviour is elastic

yield surface

a convex surface in stress space within which material behaviour is elastic

Young’s modulus

elastic modulus of the compact in tension

Index

alumina, 115, 118ff applications in industry ceramics, 252-257 hardmetal drill tips, 243-252 blend selection, powder cost, 14 fill and flow, 10 granulation, 10, 12 prealloyed versus elementally blended, 8, 13 Cam-Clay model calibration of, 158 Cap eccentricity 157, 162 identification of, 155 cap surface, 232 case studies 197ff comparisons, 197ff component production, 165ff Modnet and Dienet components diagrams, materials, kinematics, 177, 178 cellulose, microcrystalline, 223, 226, 237, 239 cohesion 57ff hardmetal powder, 154 compaction, 225 numerical simulation of hardmetals, 248

compaction modelling basic mechanics, 31 fill density distribution, influence of , 214 industry requirements: ferrous structural, 24 industry requirements: hardmetals, 21 punch kinematics, influence of, 217 sensitivity studies, use of, 8, 13, 221 tablets, 230 use in tool design, 9 See also phenomenological compaction models, micromechanical models compaction simulation, 229 compaction tooling, die, 225 die table, 226 punch, 225 component failure, cracking, 107 delamination, 107 compression test diametral, 98ff simple, 100ff uniaxial, 108, 232 constitutive model, 230 crack distribution – see defect distribution

326

Index

cracks shear, 193 tensile, 193 data framework needed initial, 207, 212 process, 207, 212 defect distribution, 193 nondestructive detection, 195 visual detection, 195 defects in tablets, cracks, 223, 229 edge chipping, 223, 225 erosion, 223, 225 laminations, 223, 229 deformation, multiaxial, 34 uniaxial, 32 delamination, 194 density, average fill density, 248 green, 66 density, distribution fill, 234 green, 235, 238, 239, 241, 248, 251-257 density-distribution measurement, die fill, metallography, 139-140 X-ray CT, 132, 140-146 density distribution, fill, sensitivity study, 132 density-distribution measurement, green acoustic tomography, 224 Archimedes method, 180-184, 186-192 buoyancy method, 180 comparison of methods, 192,193 indentation hardness mapping, 235 machining, 185-187 nuclear magnetic resonance imaging, 221 porosimeter, 182 pycnometer, 182

SEM-EDS line scan method, 187 X-ray techniques, 190, 224 density gradient, 105 deviatoric stress, 83, 90 diametral compression (Brazilian disc) test, 236 die-filling mechanisms, centrifugal forces, 227 force feed, 227, 229 metering wheel, 227 suction fill, 227-229 vibration, 227 weight control, 227 die-filling process, 225, 227 airflow, 135 die features, 136 modelling by DEM, 137, 146-148 multiple passes, 135 pharmaceutical tablets, 225, 227 powder bridging, 136 rotary presses, 227 die-filling rig, 133, 227 critical velocity, 134, 227 die liner, 109 DIENET thematic network, 1 Distaloy AE powder, 132ff Drucker-Prager Cap model calibration of 159 (see also Cap model) ejection, 105, 106, 107, 225 elastic properties, 231 measurement, 68ff at high stress, 73 strain, 65 elastic recovery, 105 failure line 95ff angle of, 95, 152, 232 cohesion, 95, 152, 232 determination, 152 diametral compression, 152 simple compression, 152 tensile test, 152 ferrous components, 185 ferrous powder, 121 flow, powder, 4, 10

Index

friction, 14, 18, 115ff axial stress, effect of, 109, 113 dynamic, 115, 116, 118, 120, 121, 122, 126 importance of, 16 interparticle, 105 radial stress, effect of, 109ff static, 115, 118, 120, 121, 122, 126 with tooling, 105 see also friction coefficient friction, coefficient, 108, 117 variation with density, 114 friction coefficient, powder-tooling, 80ff, 108, 117, 233, 235, 238 caliper, 111 comparison, 118 floating-die, 112 instrumented-die, 107-111, 112114, 233 pin on disc, 111 rotational principle, 125 shear-plate, 107,111,112, 115-118 variation with density, 114 friction measurement, shear plate technique, 111ff friction, powder and tooling, 230 friction, powder-tooling, affected by, compact density, 124 compaction speed, 122 normal stress, 124 powder admixed lubricant, 122 powders, 118 sliding distance, 123 temperature, 122 tooling lubrication, 122 tooling material, 118 tooling material hardness, 120 tooling surface coatings, 121 tooling surface finish, 118-120 granulation, tablets, 225 green properties, 20 cohesion parameters, 58 strength, 11, 17 Hall flowmeter, 133

327

hardening function, 58, 202 hardmetals, 180 infiltration, 19 input data, die fill, 131ff elasticity, 65ff friction, 105ff plasticity, 77ff requirements for, 19 shear failure, 95ff instrumented die, 232 elastomer plug, 110, 112 radial stress calibration, 110ff instrumented pins, 109 kinematics, 207, 217 lubricant, 114, 233 lubrication, tablets, 225 meshing, 211 microcystalline cellulose, 223, 226, 237, 239 micromechanical models: Cocks, 46 Cundall and Strack, 44 discrete element method, 54 Fleck, 49 Helle, 48 modelling, 31ff practical implementation in industry applications, 223ff, 243ff blend, 7ff, 223ff instrumented-die, 77ff press selection, 223ff sinter modelling, 7ff tool design, 7ff tooling kinematics, 223ff multilevel parts, 106 pharmaceutical industry, die-compaction modelling, 223ff phenomenological compaction models, 55 Cam-Clay, 39, 59

328

Index

cap eccentricity, 199 cap model, 57 Drucker, 38 Drucker-Prager, 40 Drucker-Prager Cap, 232ff Green and Shima, 37 plain cylinder, 105 Poisson’s ratio, 68, 117, 151, 231,232 porosity, 229 powder fill, 11, 15 critical velocity, 11 suction fill, 11, 18 uniformity, importance of, 15 powder fill density measurement for press operation, 165 powder transfer, 15, 225 threshold density, 15 press deflection, 171, 173 selection, 17 setting up, 18 press instrumentation, force differential measurement, 167 effect of friction, 166-169 force rings, 166 hydraulic pressure, 166 wire resistance gauges, 166 press kinematics achieved, 165 control signal, 165 punch deflections, 173-176 correction for, 176 radial pressure measurement, 110 residual stresses, compact, 105, 106 rotary production press, 223 samarium cobalt (Sm-Co), 181ff sensitivity studies, 8 constitutive parameters, 198 simulation, plasticity models, 53ff, 151ff sintering infiltration, 19 numerical simulation of hardmetals, 248

shrinkage, 249 subsolidus, 19 supersolidus, 18 springback, 105 stress axial, 80, 109, 113 deviatoric, 83, 90 hoop, 109 mean, 83, 91, 152 radial, 80 ratio, 80, 112, 113 stress path, 232 tablets, 223 active ingredients, 223, 225 bilayer, 234, 237-239 compression coating process, 234, 239,240 exipients, 223, 225 tableting presses, rotary, 225 single station, 225 tensile test, 96, 97 bilayer tablets, 237 tooling, 107 design, 9, 14, 15 hardness, 114 lubrication, 114 rotary press, 229 stresses, 16 surface finish, 109, 114 tooling material, mild steel, 120 tool steel, 115, 117, 118, 120 tungsten carbide, 120, 123 triaxial stress rig, 12, 19 triaxial testing, 232 tungsten carbide powder, 132 unloading, 105 validation, 26 industrial techniques, 165ff, 179ff

Index

X-ray computerised tomography (CT), beam hardening, 142 calibration, 142 die fill, 140-146 hardware required, 141

329

Young’s modulus, 66, 151, 231 measurement by ultrasonics, 72 zirconia powder, 132, 133, 187, 192