Modelling of powder die compaction 9781846280986, 1846280982

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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

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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

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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

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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.

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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

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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.

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

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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.

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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.

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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

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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.

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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

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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

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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.

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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).

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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

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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].

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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.

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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

1

user friendliness

2

computing power

R&D

☺☺X to XXX[5]

[5]

☺XX

☺☺X

[5]

3

interface with CAD

4

interface with press software

5

finite-element codes

6

importing input data

☺☺X[1]

7

optimisation tools

XXX[1,5]

Design

☺XX to XXX[5]

[5]

☺XX

XXX

[1,5]

Production XXX

XXX

☺XX

-

-

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

☺☺X

XXX[1,3,7]

XXX

☺XX[1] XXX[1]

XXX

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.

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[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 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 =

where h( s ) =

(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.

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

2

(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 ∂q ∂p

(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

T3

m, I

F1

F=Cv

45

T2 T1

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

δ

v

(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

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 In Equation 4.6, V is the volume of the macroscopic element and Vm is the

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

P + 13 Q Po ( D)

E& a = 0 ,

(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 Q P  1  P   3  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 Q P   P   3  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

Fs = q − p tan β − d = 0

(4.19) (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 R = R1 + R2 exp( R3 ρ)

c2

))

(4.22) (4.23)

d = d1 exp( d 2εpvol )

(4.24)

tan β = b 1 − b 2 εpvol

(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

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

(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 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

(4.32)

P0 =

2

d2

( )

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.

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[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.

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[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.

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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.

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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

RADI AL 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

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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

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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 Die

LVDT 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

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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).

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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.

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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

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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.

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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

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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

∫

0

h

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

Stress

Axial and radial stresses as function of density

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.

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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

ρt

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

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

∫

ρz dz

H 0

(6.13)

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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 (2Hρm µ α) / R = ρm Log zms = ρ 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)

Log (σzs/σzi) = (aH/2) (1/σzs dσzs/dρm + 1/σzi dσzi/dρm)

(6.17)

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:

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:

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P. Doremus

Log(σzs/σzi)= 2 µdie αcyl Η/R

(6.19)

dσz(r)/σz(r) = -2µ punch dr /α H

(6.20)

• 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:

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)1) for measuring the stress transmission coefficient.

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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.

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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

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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.

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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

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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.

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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.

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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

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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:

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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

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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

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

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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

72.50kN µ = 0.001

68.36kN µ = 0.12

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

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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,

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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 σla

τar

σr

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)

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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.

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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

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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

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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 Floating die Supporting ring Powder compact

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.

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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

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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 15 Punch Force -5

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)

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

10

2 Displacement 1.6

6

1.2 Shear Force

4

0.8

2

0.4

Shear Force (kN)

Displacement (mm)

8

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.

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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.

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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].

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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

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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

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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.

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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.

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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.

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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

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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.

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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.

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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.

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.

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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

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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

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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

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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.

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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

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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

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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.

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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

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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

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[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

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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”

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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 %.

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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.

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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.

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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

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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.

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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

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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.

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[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.

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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

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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.

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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°.

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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).

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Q

d + Pa.tanβ

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.

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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.

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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.

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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).

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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

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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.

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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

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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)

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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

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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. Fp p1

FR seal FR seal FR guidings

F error Fp

p2

FR seal FR seal

FR die

Y=f(x)

F1

FG

FR guidings F measured

F compact FG

F measured

F error

F compact

FR die

F compact

FR die

F compact

Fp

FR die

F2

FR guidings

FR guidings FR seal FR seal

FG FR seal FR seal

FG 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]).

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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

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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