Materials Processing Defects (Studies in Applied Mechanics)

STUDIES IN APPLIED MECHANICS 43 Materials Processing Defects STUDIES IN APPLIED MECHANICS Methods of Functional Anal...

Author: S. K. Ghosh

397 downloads 2943 Views 8MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

STUDIES IN APPLIED MECHANICS 43

Materials Processing Defects

STUDIES IN APPLIED MECHANICS Methods of Functional Analysis for Application in Solid Mechanics (Mason) 10. Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates (Kitahara) 11. Mechanics of Material Interfaces (Selvadurai and Voyiadjis, Editors) 12. Local Effects in the Analysis of Structures (Ladeveze, Editor) 13. Ordinary Differential Equations (Kurzweil) 14. Random Vibration- Status and Recent Developments. The Stephen Harry Crandall Festschrift (Elishakoff and Lyon, Editors) 15. Computational Methods for Predicting Material Processing Defects (Predeleanu, Editor) 16. Developments in Engineering Mechanics (Selvadurai, Editor) 17. The Mechanics of Vibrations of Cylindrical Shells (MarkuP) 18. Theory of Plasticity and Limit Design of Plates (Sobotka) 19. Buckling of Structures-Theory and Experiment. The Josef Singer Anniversary Volume (Elishakoff, Babcock, Arbocz and Libai, Editors) 20. Micromechanics of Granular Materials (Satake and Jenkins, Editors) 21. Plasticity. Theory and Engineering Applications (Kaliszky) 22. Stability in the Dynamics of Metal Cutting (Chiriacescu) 23. Stress Analysis by Boundary Element Methods (Bala~, SIcSdekand Sladek) 24. Advances in the Theory of Plates and Shells (Voyiadjis and Karamanlidis, Editors) 25. Convex Models of Uncertainty in Applied Mechanics (Ben-Haim and Elishakoff) 26. Strength of Structural Elements (Zyczkowski, Editor) 27. Mechanics (Skalmierski) 28. Foundations of Mechanics (Zorski, Editor) 29. Mechanics of Composite Materials- A Unified Micromechanical Approach (Aboudi) 30. Vibrations and Waves (Kaliski) 31. Advances in Micromechanics of Granular Materials (Shen, Satake, Mehrabadi, Chang and Campbell, Editors) 32. New Advances in Computational Structural Mechanics (Ladeveze and Zienkiewicz, Editors) 33. Numerical Methods for Problems in Infinite Domains (Givoli) 34. Damage in Composite Materials (Voyiadjis, Editor) 35. Mechanics of Materials and Structures (Voyiadjis, Bank and Jacobs, Editors) 36. Advanced Theories of Hypoid Gears (Wang and Ghosh) 37A. Constitutive Equations for Engineering Materials Volume 1: Elasticity and Modeling (Chen and Saleeb) 37B. Constitutive Equations for Engineering Materials Volume 2: Plasticity and Modeling (Chen) 38. Problems of Technological Plasticity (Druyanov and Nepershin) 39. Probabilistic and Convex Modelling of Acoustically Excited Structures (Elishakoff, Lin and Zhu) 40. Stability of Structures by Finite Element Methods (Waszczyszyn, Cicho~ and Radwahska) 41. Inelasticity and Micromechanics of Metal Matrix Composites (Voyiadjis and Ju, Editors) 42. Mechanics of Geomaterial Interfaces (Selvadurai and Boulon, Editors) 43. Materials Processing Defects (Ghosh and Predeleanu, Editors) .

General Advisory Editor to this Series: Professor Isaac Elishakoff, Center for Applied Stochastics Research, Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, U.S.A.

STUDIES IN APPLIED MECHANICS 43

Materials Defects

Processing

Edited by

S.K. Ghosh GKN International College of Engineering Lohmar, Germany

M. Predeleanu LMT University of Paris V/ Cachan, France

L~~

l

1995 ELSEVIER Amsterdam- Lausanne- New York- Oxford- Shannon-Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

pages 17-58 reprinted from Journal of Materials Processing Technology, Vol. 32, nos. 1-2 (1992) ISBN 0-444-81706-9 91995 Elsevier Science B.V. All rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science BV, unless otherwise specified. No responsibility is assumed bythe publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. pp. 373-386: Copyright not transferred. This book is printed on acid-free paper. Printed in The Netherlands

Preface The Second International Conference on Materials Processing Defects (Proceedings of which was published as a special issue of the Journal of Materials Processing Technology, Vol. 32, nos. 1-2, 1992, 530 pages) was held GKN Automotive AG during 1 -3 July 1992. We believe that this technological field of defects, and more appropriately, avoidance of them, is very current in perhaps all sectors of the manufacturing industry. This is particularly important to reduce/minimize waste everywhere to address lean production procedures. The recent advances in finite plasticity and visioplasticity, damage modelling, instability theories, fracture modelling, computer numerical techniques and process simulation etc. offer new approaches and tools for defect prediction, analyses and guidelines for designing components to be manufactured by traditional and emerging process technologies. The present volume includes contributions from well known researchers and experts in this field, most of whom were also involved with the aforementioned Conference. The main aim of the contributions has been to extend and generalize somewhat their past contribution in the form a self-contained chapter on individual title topics such that a volume would be possible. We do hope that it matches to a large extent with the stated objectives considering the vast field of research into defects in all kinds of materials processing and associated topics: - Micro- and Macro-scale observation of defects Localization and instability analysis Damage modelling and fracture criteria - F ormability characterization - Defect prediction methods - Design considerations to avoid defects - Practical process/material considerations. -

-

We are very indebted to the authors and referees of this volume for their generous and very kind support. We would also thank Mrs. Heike Erlenkamp for all her help towards this volume. Finally, the team at Elsevier, Amsterdam: Mrs. Mary McAdam, Ms. Wilma van Wezenbeek, Dr. Bas van der Hoek and Dr. van der Hoop deserve special thanks for their patience and care for professional preparation and presentation of the book.

S K Ghosh Lohmar, Germany

M Predeleanu Paris, France 3 November 1994

vii

Dedicated to

Professor Frank W Travis DSc Professor Vellore C Venkatesh DSc

on the occasion of their 60th birthdays

The proceedings of the Conference, based on which this present volume has been initiated, was dedicated in 1992 to p r o f e s s o r William J o h n s o n F R S F E n g - T h i s book on M a t e r i a l s processing D e f e c t s is dedicated to my earlier teachers, p r o f e s s o r s Travis and Venkatesh. I do this with great pleasure on behalf of all their students, research assistants, research fellows, post doctoral researchers, colleagues and all others who have had contact with them throughout their professional careers. I t is very fitting indeed that this dedication appears in a volume published by Elsevier since both of them have been contributing through this publisher to the scientific community of materials processing worldwide for many years. M y dedication to p r o f e s s o r s Travis and Venkatesh will be incomplete without reference to their wives, M r s . J e a n T r a v i s and M r s . Gita Venkatesh concerning their generous support, help and kind hospitality always and everywhere, and especially, that extended to those, so-called 'foreign students/researchers abroad'. I t gives me great pleasure on all your behalf in wishing them continued good health and happiness.

I~rofeee)or Swaclhin K u m a r Ghoe)h Dirsc~0r GKN I n ~ c ~ r n a t i o n a l College of: Engineering Lohmar, G e r m a n y

This Page Intentionally Left Blank

CONTENTS

Preface oo VII

Dedication Some Comments on the Structure of Technology of Plasticity in R&D and Production K. Lange James Nasmyth (1808-1890): The Steam Hammer and the Mechanics of Vee-anvil Forging W. Johnson

17

Modeling Dynamic Strain ~ i z a t i o n M. Predeleanu

59

in Inelastic Solids

Void Growth under Triaxial Stress State and its Influence on Sheet Metal Forming Limits R. C. Chaturvedi

75

The Prediction of Necking and Wrinkles in Deep Drawing Processes Using the FEM E. Doege, T. El-Dsoki and D. Seibert

91

Constitutive Models for Microvoid Nucleation, Growth and Coalescence in Elastoplasticity, Finite Element Reference Modelling J. Oudin, B. Bennani and P. Picart

107

Theoretical and Numerical Modelling of Isotropic or Anisotropic Ductile Damage in Metal Forming Processes J. C. Gelin

123

Research on Forging Processes for Production a + /3 Titanium Alloy TCll Disks Sencan Chen, Yu Xinlu, Zongshi Hu and Shaolin Wang

141

Modelling of Fracture Initiation in Metalforming Processes Y.Y. Zhu, S. Cescotto and A.M. Habraken

155

Formability Determination for Production Control J.A. Schey

171

Design of Experiments, a Statistical Method to Analyse Sheet Metal Forming Defects Effectively D. Bauer and R. Leidolf

187

Formability, Damage and Corrosion Resistance of Coated Steel Sheets J.Z. Gronostajski and Z.J. Gronostajski

203

Model of Metal Fracture in Cold Deformation and Ductivity Restoration by Annealing V.L. Kolmogorov

219

Prediction of Necking in 3-D Sheet Metal Forming Processes with Finite Element Simulation M. Brunet

235

Deformability versus Fracture Limit Diagrams A. G. Atkins

251

Prediction of Geometrical Defects in Sheet Metal Forming Processes by Semi-Implicit FEM A. Makinouchi and M. Kawka

265

Evolution of Structural Anisotropy in Metal Forming Processes J. Tirosh

283

Computer Aided Design of Optimised Forgings S. Tichkiewitch

297

Defects in Thermal Sprayed and Vapour Deposited Thick and Thin Hard-wearing Coatings M.S.J. Hashmi

311

A Study of Workability Criteria in Bulk Forming Processes A.S. Wifi, N. El-Abbasi and A. Abdel-Hamid

333

Degradation of Metal Matrix Composite under Plastic Straining N. Kanetake and T. Choh

359

Crack Prevention and Increase of Workability of Brittle Materials by Cold Extrusion H. W. Wagener and J. Haats

373

A Database for some Physical Defects in Metal Forming Processes M.M. Al-Mousawi, A.M. Daragheh and S.K. Ghosh

387

Split Ends and Central Burst Defects in Rolling S. Turczyn and Z. Malinowski

401

Form-Filling in Forging and Section-Rolling P.F. Thomson, C.-J. Chong and T. Ramakrishnan

417

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

Some Comments on the Structure of Technology of Plasticity in R&D and Production

o. Prof. em. Dr.-Ing. Dr.h.c. Kurt Lange Institut fQr Umformtechnik, Universit~t Stuttgart, Stuttgart, Germany

The complexity of the increasingly expanding field of production techniques and technologies led to various attempts to develop structures for the systematic treatment of problems. This is also true for the technology of plasticity. In 1956 O. Kienzle, University of Hannover, formulated seven points which must be or should be considered for the solution of metal forming problems. Only a few years later, around 1965, W. Backofen, MIT Cambridge/Boston, developed a system comprising four zones for his systematic approach to metal forming with emphasis on the behaviour of materials during deformation processes, including especially damage. The four zones were:

.

2. 3. 4.

Deformation zone with plastic material behaviour Friction and lubrication between work material and tool Material properties before deformation Material properties after deformation

The tool itself was not of so much concern for him being a materials scientist. Kienzle, however, had already included in his seven points the machine tool and the factory besides the tool as a key to successful production of formed parts. Based on the list of seven points by Kienzle (fig. 1, items 1-5,7,8) and Backofen (Fig.l, items 1-4) a general system for the investigation and development of metal forming processes was designed by the author taking the original scetch-flatrolling by Backofen as the first process example, but demonstrating also, that the same numbering is also valid for other processes (Fig.l). The idea was to demonstrate the close interconnnection of items 1-5, the sub-system workpiece-tool, and to push it into production by machine tool and factory, symbolized by the two surrouunding rings. Also introduced into the system was item 6, taking into account possible exchange between workpiece and surrounding atmosphere, following suggestions by Gebhardt, Stuttgart and Schey, Chicago, which might cause e.g. surface contamination.

For better understanding some short and incomplete descriptions of the meaning of the eight points-or areas-and their interaction are given in the following:

9A r e a 1, the plastic zone, concerns the determination of the material behaviour in the plastic state. Using plasticity theory and initially assuming an idealized, isotropic material, the stresses, strains, and material flow may be determined. Based on these, the temperature distributions may be found at different locations and for different points in time. Metallurgy allows a description of the behaviour of the material on a microscopic scale (anisotropy, textures etc.). 9 A r e a 2 deals with the characteristics of the workpiece before deformation.

These affect more or less the behaviour of the material in the deformation zone and the characteristics of the resulting workpiece. Besides the chemical composition, mechanical properties play an important role here as well as the crystal structure, texture and microstructure (such as grain size, amount and type of second-phase particles). Apart from the chemical composition, all of the properties mentioned can be changed to a greater or lesser degree by heat treatment.

Further, the

surface properties and the surface treatment prior to the forming process are also of significance. 9A r e a 3 concerns the woz'l,'piece characteristics. These are primarily the mechanical properties, surface properties, and workpiece accuracy after deformation. The workpiece characteristics after the forming process largely determine how the component will behave in service (e.g. work hardening in fastener production). 9A r e a 4 considers the boundary area between the partly elastic (rigid), partly plastic workpiece and the ela.~tic tool (= gap) and concerns all the questions

connected with friction, lubrication, and wear. The interaction of the workpiece and the tool materials plays an important role here. Further, in this area considerable changes in the original workpiece surface may sometimes occur.

9A metal-forming operation cannot be regarded in isolation from the forming tool. For this reason, area 5 deals with the many-sided problems connected with tool layout and tool materials. Appropriate design (e.g. to achieve the required stiffness of the machine and for guiding moving tool parts with respect to each other) and manufacture directly influences workpiece accuracy (area 31. 9In area 6, which is outside the zone of tool - workpiece interaction, surface reactions can take place between the workpiece and the surrounding atmosphere,

such as formation of oxides during hot forming or gas absorption when forming exotic metals. On the one hand, these operations may considerably affect the resulting surface properties, and on the other hand they may also influence the workpiece characteristics in the same area, for example, with exotic metals through the absorption of small amounts of gases. 9The t o o l - workpiece system (problem areas i to 6) is always incorporated in a machine tool (e.g. forging hammer, mechanical or hydraulic press, or rolling

mill). The machine tool is symbolized by the inner circle, area 7. It must provide the necessary energy and forces for the different operations at each stage of the process and assure sufficiently accurate guiding of the different parts of the tool with respect to each other. This calls for appropriate dimensioning of the tooling and, where necessary, of the workpiece-handling equipment for the application concerned (e.g. bulk forming or sheet forming). Finally, important factors affecting productivity are the stroke rate, setup time, and so on. 9Beyond this, area 8 is concerned with the integration of the metal-forming process itself into the production system as a whole. It covers all auxiliary equipment and functions (e.g. heat treating, cleaning, handling, and automation) on the shop floor as well as factory organisation (e.g. work preparation, production control, and cost estimation).

The systematic approach to the solution of problems in metal forming processes has proved to be realistic as a base for R&D especially also in connection with the systematic grouping of forming processes in the German DIN-standards 8582 to 8587[2-7]. This is based mainly on the important differences in effective stresses. However, simple descriptions of stress states are not possible since, depending on the kind of operation, different stress states may occur simultaneously or they may change during the course of a forming operation. Therefore the predominant stress states were chosen as classification criteria, resulting in five groups of metal forming processes:

1. C o m p r e s s i v e f o r m i n g (forming under compressive stresses). German standard DIN 8583 covers the deformation of a solid body in which the plastic state is achieved mainly by uni- or multiaxial compressive loading. 2. C o m b i n e d tensile and c o m p r e s s i v e f o r m i n g (forming under combined tensile and compressive stresses). DIN 8584 covers the deformation of a solid body in which the plastic state is achieved mainly by combined uni- or multiaxial tensile and compressive loading. 3. Tensile f o r m i n g (forming under tensile stresses). DIN 8585 covers the deformation of a solid body in which the plastic state is achieved mainly through unior m ultiaxial tensile stresses. 4. F o r m i n g b y b e n d i n g (forming by means of bending stresses). DIN 8586 covers the deformation of a solid body in which the plastic state is achieved mainly by means of a bending load. 5. F o r m i n g b y s h e a r i n g (forming under shearing stresses). DIN 8587 covers the deformation of a solid body in which the plastic state is achieved mainly by means of a shearing load. The examples in Figs.2 to 9 represent some of the more than 200 processes defined in the standards. They contain the same basic interaction between tool and workpiece, i.e. the relevant geometw features and the basic kinematic. It is obvious that the points or areas 1 to 8 as given above (see Fig.l) can be easily defined, so that in consequence the process scetches in DIN 8582 to 8587 will fit in our general system for the investigation of metal forming processes. Together they might also serve vew well as a basic introduction of students to forming technologies (see also [8]).

Development of Technology of Plasticity after 1965 For the time being, one should not forget, however, that the above presented systematic approach with eight areas was designed between 1960 and 1969 approximately. Since then, the scientific technical fundaments of the technology of plasticity have been and still are widened and deepened considerably with accelerating velocity. This development is driven by the need for more economic use of energy and materials on the one hand and by the intention to keep the prices of products for a rapidly growing number of consumers as low as possible. Especially metal forming processes have the potential to meet these challenges including the development of near-net or net-shape processes, e.g. in hot, warm and cold forging. The aim is to replace machining for metal removal by forming with material saving and high-dimensional accuracy as well as high surface quality as far as possible. During the last 20 to 25 years, the rapidly increasing introduction of permanently more powerful computers with appropriate software has not only contributed significantly to the above mentioned development but has also pushed it ahead. Process analysis and simulation with FEM-programs are improving the understanding of forming processes considerably and hence are offering the key to unforeseen possibilities of process optimization. Modern metrology technology supported by computer data processing has also contributed to this development which comprises also triboIogy phenomena, but there will be still very much to be clone to develop and improve theories and to produce data - both for the application in sound process layout. Finally, the influence of the computer on the functions and control of the modern forming machine tools should be not forgotten as well as its impetus on flexible process automation including material handling and tool exchange. Last not least the very important modern development of the die and tool technology - design increasingly by CAD and aided by FEM -, materials, surface coating and treatment must be mentioned here, as the functioning, reliable and economic tool is the key to economic production by forming.

It should be mentioned here that all of these developments and changes have been backed up by rapidly growing stimulating exchange between scientists and technologists, e.g. by books and other publications, by conferences and by cooperative work

in organisations.

To sum it up: Metal forming seems to be on the move to new

standards and capabilities in order to be permanently prepared for new challenges in modern industrial production.

New Systematic Structure of Metal Forming Processes The changes and partly rapid developments since approximately 1965 were reason enough to reflect on the systematic approach of 1968. It was the special concern of the author to demonstrate more clearly the interdependence of influencing factors on the process, and, hence, the product. All the more, the strong influence of the modern computer technologies and of the tools on the process should be made more transparent. This goal was finally reached by defining a "process core" or heart just by geometry and kinematic as shown in Fig.2 to 9, and to place it in the system centre, surrounded by the influencing areas = process components as satellites. The process core is interconnected with the six satellites Material, Tribology, Tool (design, manufacture) Machine-Tool + Automation, Production and Theory of Plasticity + C A - Techniques which is the usual starting area for a process development or investigation. All six areas are interacting with the process core and with one another. These interconnections between the satellites and/or the process core may be partly described mathematically or by data flow structures. Only the process core plus the six satellites or components together will represent a process completely. This is integrated into the general technical and economical conditions and will be influenced by the location and the market - the latter being determined more and more by global aspects. Consequently further process development

must be directed towards increased product quality, improved flexibility,

productivity and economy.

For fundamental scientific studies the interconnection of the process core with only one or a few of the six satellites, e.g. material, theory of plasticity may be sufficient - many publications just deal with the analysis or simulation of a "process" by combining geometry and kinematic with assumed material and friction laws. Frequently also the interconnection theory-process core-tool is being used effectively for the FEM-assisted design of tool geometries optimized for e.g. material flow with limited strain gradients. But for a real production process, the whole system must be considered.

This may be underlined by some closer looks to the contents and structure of the system components=satellites.

There is no doubt or at least there should not be about the

extraordinary significance of materials for metal forming.

Material and optimized forming process together will lead to a product meeting the customers needs in a very global meaning. It is expected that after the end of the long-lasting cold war the free exchange of products, ideas and research results will create a new era of materials characterized by the relationship between materials and manufacturing solutions to follow the new challenges for economy, low weight/high strength, environmental friendliness, recycability and minimum energy consumption for material generation. From the materials engineers point of view the satellite "Material" might be represented in more detail as shown in Fig. 1 1 [1 1]. Important is the interconnection: Materials design, all materials system, manufacturing- the latter including forming - which corresponds perfectly to the system in Fig. 10. Consequently the all material system could be easily redesigned and could serve as level 2, if the presentation in Fig. 10 is considered to be level 1. In another level 3, information on specific material properties e.g. flow stress vs. strain (Fig. 1 2) or vs. strain rate (Fig. 13) might be contained in various manners, while in more other levels relation describing equations, data storages etc. could be presented. Figure 13 emphasizes the interaction between "Material" and "Process core", the kinematic of which determines - besides the machine t o o l - the strain rate.

These ideas will have to be worked out in more detail, but t w o other systems- for tribology and tools- seem to confirm the general applicability of the forming system proposed in Figure 10. The tribological system acc. to Figure 14 as a generally accepted standard might be used as a first step into tribology on level 2. It demonstrates already on this level t w o important interactions; with "Materials" (i.e. workpiece) and with "Tool". From the point of view of the satellite "Tool", as shown under the aspect of tool life in Figure 15, various interactions are obvious, thus demonstrating the feasibility of the system in Figure 10 again: Double link with "Material" - workpiece and tool material -, interactions with "Tribology" (lubrication, coating), "Theory" (tool design, tool geometry) and the "Process core" (geometry) [13].

It might have been shown already by these few examples that the expansion of the new system for metal forming processes to a broad and deep, multilevel information system will be possible and of great advantage. Compared with the system in Figure 1 the process structure core, components as satellites - allows easy improved approach to the individual items and their interconnections and interactions. In modern R&D as well as in production the improved understanding and data backed-up knowledge of these interactions will be more then up to now the sound base for development of reliable processes and economic manufacture of highquality formed parts.

Literature

1. Lange, K." The Investigation of Metal Forming Processes as Part of a Technical System. Proc. 10th International M.T.D.R. Conference, Manchester, September 1969. Pergamon Press: Oxford and New York- 1970. 2. DIN 8582: Fertigungsverfahren Umformen (Manufacturing Process" Forming)" 1st ed. Berlin, KSIn" Beuth 1971. 3. DIN 8583" Fertigungsverfahren Druckumformen (Manufacturing Process: Compressive Forming)" parts 1 and 6, 2 to 5 1st. ed. Berlin, KSIn" Beuth 1969, 1970. 4. DIN 8584" Fertigungsverfahren Zugdruckumformen (Manufacturing Process: Combined Tensile and Compressive Forming)" 1st. ed. Berlin, KSIn: Beuth 1970. 5. DIN 8585" Fertigungsverfahren Zugumformen (Manufacturing Process: Tensile Forming)" 1st. ed. Berlin, K61n: Beuth 1970. 6. DIN 8586: Fertigungsverfahren Biegeumformen (Manufacturing Process: Forming by Bending)" 1st ed. Berlin, K61n" Beuth 1970. 7. DIN 8587" Fertigungsverfahren Schubumformen (Manufacturing Process" Forming by Shearing)" 1st. ed. Berlin, K61n" Beuth 1969. 8. Lange, K. (editor)" Handbook of Metal Forming. New York etc." McGraw-Hill 1985. ISBN 007-036285-8. 2nd printing Dearborn: SME 1994. ISBN 0-87263-457-4. 9. Lange, K." Dohmen, H.G. (editors)" Pr~izisionsumformtechnik

(Precision Metal Forming

Technology). Results of the program "Precision metal forming technology" of the German Research Foundation (DFG). Berlin etc.: Springer 1990. ISBN 3-540-51943-2. 10. Lange, K. (editor)" Umformtechnik-Handbuch f~r Industrie und Wissenschaft (Metal Forming Technology-Handbook for industry and science) Vol.4. Berlin etc." Springer 1993. ISBN 3-54055939-6. 11. Bridenbaugh, P.R." Commercial Transportation" The next, best engine for advanced materials systems. ASM News (April 1993) 4 + 5. 12. DIN 50320 Verschleil3. Begriffe, Systemanalyse von Verschleil~vorg~ngen, Gliederung des Verschleil~gebietes (Wear Terms, Systems analysis of wear problems, Breakdown of wear region). 13. Lange, K." Cs~r, L." Geiger, M." Kals, J.A.G." Tool life and tool quality in bulk metal forming. Annals of the CIRP Vo1.41/2/1992, 667-675.

2

1 plastic zone 2 materia! properti es before Tormlng

4

3 material properties after Torming

9

4 contact zone 5 tool

6 work~iece and surrounoing atmosphere 7 forming maschine 8 factory

~ 2

a.

4

I

/

4

6

6

5

/,

2

1~-3

b.

Figure 1. A general system for the investigation of metal forming processes. (a) System structure. (b) Examples for various metal forming processes.

Threaded roll

Roll

Final shape

Initial shape

~' ~ , ~ Pressure Master . ~ , \ ~ - roller (a)

.or,,.e

e

(b)

(c)

Figure 2. Compressive forming. Examples of lateral rolling. (a) Thread rolling by the run-through method. (b) Lateral rolling of spheres. (c) Flow turning. (After [3].)

10

@

(a)

(b)

(c)

(d)

(e)

Figure 3. Compressive forming. Examples of die-forming processes. (a) Fullering. (b) Heading in a die. (c) Closed die forging without flash. (d) Closed impression die forging (with flash). (e) Upsetting in a die. (After [3].)

Drawing die

Drawing die

i ~~~~ . nrclning die

(plug)

~>"' ~ ~\~~I

(a)

~Roll

~

~ ~~_~kpiece

u

Roll , Workpiece be)

Floating mandrel (b) Figure 4. Combined tensile and compressive forming. Basic drawing processes. (a) Drawing through a die (rod drawing, drawing over a fixed mandrel, ironing). (b) Drawing through rolls (wire drawing, drawing over a floating mandrel). (After [4].)

1!

f

(b)

(a)

(c)

Figure 5. Combined tensile and compressive forming. Basic deep-drawing processes with rigid tools. (a) First draw with blankholder. (b) Redraw without blankholder. (c) Reverse drawing. (After [4].)

"

-

~

Spinning mandrel Tailstock

~.~/.. :.-.~,~~ ~>,~ .....

Die

Workpiece

~~-~"~~'~'~

Workpiece

Pressure roll

(b)

(a)

__~ G/_ ( ~ ~ . ~ (c)

Pressure roll ---Workpiece Mandrel

L~w

Pressure roll

orkpiece

(d)

Figure 6. Combined tensile and compressive forming. Spinning processes. (a) Spinning of hollow bodies starting from a blank. (b) Expanding by spinning. (c) Necking by spinning. (d) Thread forming by spinning. (After [4].)

12

!

Punch ~~~-Wor

k

~=~ p

i

Die

e

c

~

~-----!--

[

~9

Coil 9

(a)

Compressedair

..~ Workpiece

Workpiece

~~- Lower die

~.'./,~i ""'.../""',~{--.~- IDie

(b)

(c)

Figure 7. Tensile forming. Shallow and deep recessing processes. (a) Recessing with a rigid tool (e.g., stretch drawing, embossing). (b) Recessing by means of a pressure medium (static action). (c) Recessing by means of energy activation (e.g., electromagnetic field). (After [5].)

(a)

(b)

(c)

(d)

-'--'3

(e)

(f)

(g)

Figure 8. Forming by bending. Examples of formation by bending with linear tool motion. (a) Free or air bending. (b) Free round bending. (c) Die bending. (d) Die round bending. (e) Draw bending. (f) Edge rolling. (g) Bending by buckling. (After [6].)

13

Initial shape Upper die _~ . ~ / ) i n a l shape

__.-~

__o _.L.

Welding boss (a)

(b)

(c)

Figure 9. Forming by shearing. Shearing deformation processes. (a) Lateral displacement. (b) Embossing. (c) Twisting. (After [7].)

,/

//

/ Mater,al )

/ Ir,Do,ogy )

~'\\

,,

i

Market Technological Progress Location-costs (Labor costs Energy costs)

Product Properties

'i

/

/

/

Figure 10. New structure model of metal forming processes. (After [9,10].)

Productivity Flexibility Economy

14

Materials Systems for Manufacturing Solutions f

Paradigmfor Materials Competitiveness

/

Materials

,..~ Social, Legall ~~ M //Test,/ Environmental anufacturing/ /Evaluation and / Acceptance //CCharacterization ......... ~/, Figure 1 1. Materials systems for manufacturing solutions. (After [1 1].)

Z "%E ' 1000 =

1450 I

800

~,

1!60

b"- 6O0

E

870

~

"~

580

40o

200

~ N~

2901

0

0

l

"

I

i

04

~

~--~ . . . .

(A~S'1015) ,

OR Stroln

1000 o,,I

E E

8O0

-

Z

~- 600 -

-~

29

16

h~

1450

._ u~ ,=.

6

87O

.~ 40o - ~

580

u~

i

1160

I//

'

P Soft-onneolea !

o

'7

200 -

.._o L~.

,

~

290

o

"

i ,

04

~A 'S 10 35) 08 StroJn

"^

16

,~

Figure 12. Flow curves of t w o c o m m o n steels with influence of kind of heat treatment. (After [8].)

15

oa

E E

z

;. 2 0 0

b ~ 4-

.~ 3 _o

72.5

500-

100 50

-

= 29.0

-

~

14.5

-

,7

7.25

\

i

........ I -------

0 0.1

0

r

T= 1000oc(1832~

t

....

-~- -- ~L

', 1

_ ,.-

i

_ ~- ~ ]

-i~-+---~_.~--

J

"1

~--I T : 1200~ (2139~ 1 T = 1100~ (2012~ / 1

I

10 1O0 Averagestrain rate ~m, sl

J

1000

Figure 13. Flow stress versus strain rate for C15 at various temperatures, strain ~p = 0.5. (After [8].)

Collective load system 1

i.~ Structure of the Tribological System

2 t

1 ~'~'~'~~'~~~~ ~. 9 .

.~

,/

Sureface changes (Wear types) ~1

/-4 i i .

",,, i

1Basicbody(Tool) 2 COunterbOdy(wOrkpiece) 3 Intermediatelayer 4 Surroundingmedium

] Material loss I (Wear measure)

Wear characteristics Ii

Figure 14. Tribological system. (After DIN 50320112].)

16

Tool Manufacturing

Tool/Workpiece-lnterface Lubrication Tool-Material

Workpiece

9Wear 9Hardness 9Resistance 9Fracture Toughness

9Tolerances 9Surface Roughness Ra

TOOL-LIFE

~-d

Heat Treatment

Workpiece Material

Wear/Fracture

k,? "

Tool Design

I Tool Geometry

9Active Elements 9Prestressing

i 9Die Angles I 9Fillets, Corners

~'~"l'i~

I 9Deformation 9Surface Quality

tllll

9Coating

t

Figure 15. Different aspects of workpiece and forming process determining tool life by affecting wear and/or fracture. (After[13].)

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 1995 Elsevier Science B.V.

JA)~

17

I~ASI~/TH ( 1 8 0 8 - 1 8 9 0 ) "

T I l E S]'EAM ~ 1 ~

AID THE ~CHMIIC;~3 OF VEE--~VIL FORGING - byu

Johnsont

Summary An outline

biography

oi the 6:cotsman,

credit with the invention

James Nasmyth,

whom the British

ol the Steam Hammer and which the French deny,

is

~iven first oi all

C3

~

The enterprising in

hammer

Responding

and

production

that

to challenge,

iarge

diameter

paddle

also

to

introduced

anticipate

have that

talented he

Nasmyth

was

he overcame

ship

it would

able

the

successiui

to

late

retire

in

his

40s.

'gaggin 8' in !ilt hammers when iorging

crank--shaits

with

Vee-anvii,

promote

was so commercially

internal

his

his

s~eam hammer'

intuition

soundness

he seems

leading

in iorging

him

which

to the

flat anvil certainly did no~. At length we consider

the plasticity mechanics of these anvils and see

in w~at manner Nae,myth's expectation

was corroborated.

As well we consider some of the associated defects which arise.

Nasmyth's Steam-hammer .

.

.

.

.

.

.

.

.

.

.

*Emeritus Pro~essor of Mechanics, Visiting Professor in Mechanical

University ol Cambridge and Engineering

Science and Technology in U.M, I.S.T.

and in the History of

18 INTRODUCTION

Yasmyth

is a well known British

whose name is widely many

innovations

or

new

too.

portrait

and landscape

Nasmyl.h the

famous while

very at

Professor

young

ase

Stephenson

a

19,

in

1827

what

reputation

objective

was

(see

of

the

works.

2)

However,

to show to Maudslay

that he was taken on as Mr. apprenticeship

Maudslay's

business

in

up

up

~inancial pros

be

in

the

such

done'.

Maudslay

Arts

He

a

that Nas]myth was a

Note <-

l)and

' an to

project to

2.

- a road

because in

20,

for

see Fig.

George

process

aware

05

it became Nasmyth's

of the

chief

for its ~k~chine - making tools

His

father place

was

him

unable

to

he

instruments, models

made

the

manufactured

built

and

age

and

well

instrument

be

required that proved

know

-) in 1826,

caused

Coming

factory.

1821-1826

to

came to naught a

so renowned

from

came

see

of London,

Maudslay's

in

partner,

1831

as

an

his capability,

private

Nasmyth

told

to

pay

apprentice and

t.he at

mechanical

the result

assistant,

of

in a

it

interred

being

the need for any

Dale

happened

carried too n~ny tools and material

Woolwich

(London)

for a Mr.

that

he

intended

to

He was offered an apprentice

at

from the father

Suffice that

the

cost

el

because

annually,

to say that Nasmyth's iirst

iioor

premium

to him when

Manchester ZSO

begin

of a young man who

It was helpful

Piccadilly, a

Joshua

person'.

employer

Street,

property,

in

took to working

and this he took.

at a rate of 3%.

However

his

in his workshop,

business

backin~

was

- he was now 26.

in engineering offer

and

'a most systematic

Nasmyth

for 12 months

had ambitions

- and

on

loss t.o all.

1831

for himself

of ~50, set

died

a great

Late

in

being disregarded.

Maudsiay churchyard,

of

expansomet.er

dea.igned

to

Note

School

This ultimately

of Henry

organisation

Maudslay's

Field,

of

to visit this company,

!ar,~e premium

took

This demonstrates

(!Z66-1832,

working

waitin~

enterprisin~

family'.

and solid substances'

he

(I?@l-1848)was

'showing

drawings

type

8.

was

He produced

his father being predominantly

mathematics.

Leslie

new

see Pig.

national

he

i.

Edinburgh and

in bulk all metals

Steam-carriage,

the

the

subjects

and

engineer,

but talented to the extent that he desisned

see Fig.

attended

usual

Edinburgh

measuring

and

painter,

bridges',

designs

was Scottish,

of an 'artistic and mechanical

studied

7hen

engineering

His ancestry

'bow-and-string

century mechanical

linked to the origin el the steam hammer.

business

member

mid-19th

of

his

and

he got

business workshops

so that when a machine accidentally

19

Figure 1

Figure 2

20

v

I

9

The Road Steam-Carriage. By James NasmyLh. Figure 3

Bridgewater Foundry, PaLricrofl. From a Painting by Alexander Nasmyth. Figure 4

21 crashed

through

business

below he was asked to find new premises.

Rapidly,

the

floor

Nasmyth

side oi Manchester) §

transport

located

per

were

premises 1836,

The

see

design,

said

per the and

twenty

engine

and

S.S.

New

what

would

Nas~]th tools except

much

should

anywhere, dilemma,

the not

so

be

The problem Nasmyth , were

by

diameter o~

the

Francis

just

the

expected di splay. )

was

wrought

trunk

iron

been

found

to

'compass,

No

taken

up

'Nasmyth's O.W.R's

between Bristol

determined S.S.

was

paddle

shaft

build

Britain.

about

was such that by

O.%;.R.

Humphries, 'call'

diameter.

for

to approval

approached

he might

the

those

done

iorging.

ol 30 in.

to

Orear

especially

All

most

steamship

in

a

for cast

]he existing

of range and fall as well as want of power of versions though

a blow of adequate

was

by

to

company

the

that contemporary

enlarged

%o

fulfilled

engineer,

and

il he thought

a shaft

Nasmyth's

ladle.

The

the

their

engines.

undertake

was to forge

world,

intermediate

and asked

on

directors

yard

forged.

August.

invitation and

history oi travel

Humphries, ship

his. new

in

that they won a ZIO0 premium

ol

a

tilt hammers, 'smith's

'tilted' intensity

by

the

work

hand

see Figs.5(a) hammer'

and

and thus

to its full height was still - and indeed

of a forgin 8 was the more it became Sa@6~ed; hammer

time

accepted

led

in the

e~llent~ ,

He called

its

history'

company

west

bal~ed and the entire

Company's

was

connection

the

vertical

observed

to supply

He was called on

production

of

Railway

engines

Steamship

even the largest, forge hammer, unable

ahead

personal

that

before

hammers had not the (b)

This

wrote to Nasmyth

blow'.

Western

(his)

equip

large

the huge

could

much

81azier's

bed of brick clay

ground'.

started

proved so successful

largest

consulted

had

it

Western had a successful

so

was

that

ol the

a

7he . .site. had 'A fine

it was du~ out,

out

travelling

...

which

like

iron.

in

be the

constructing the

the Great

large

Oreat

York,

(,anal.

rental .

and

oi

at Patricro~t

and water).

ieature,

tender.

event

site

roo~

C]~allen~ and_l{espons~

The locomotives

important

the

and use in his foundry of a Salety Foundry

In the mid-]gsos Nasmyth.

risen

Foundry

A novel

The_bS_Z~m Hammer:

(rail

to have

Fi~.4.

for

a six acre

of the ground'

Bridgewater

introduction

tender

through

~.-~~,,~ jvard as

facilities

!ay below the surface buildings

thence

at the edge of the Bridgewater

~DaY 1 ~/4~ pence

close,

and

itself.

that many European museums o~ technology

the

larger

the

space for clear fall

(It may conlidently

be

will have tilt hammers on

22

This copy of the well-known painting by Joseph Wright of Derby, dated 1772, portrays in a very vivid m a n n e r the inside of an iron forge, showing a water-powered helve- or tilt-hammer at work. beating a large lump of white-hot iron. held by pincers on the anvil. Repeated heating (in 'finery' and 'chafery' hearths) and hammering were necessary to reduce the carbon content of pig iron and convert it into wrought (or malleable or bar) iron. until the 'puddling and rolling' process was developed in the later eighteenth century. The h a m m e r was lifted, as shown, by knobs or blocks of wood fitted into a circular wooden drum. fLxed on the axle of the water-wheel and thereby rotated: the weight of the hammer-fall was increased by the springy timber board fLxed above the hammer. Water-powered h a m m e r s of this type were introduced in the late fifteenth century, relieving the smith of the most laborious part of his work. See (b) for the 'oliver', an earlier treadle-operated tilt-hammer, [29]. Figure 5(a)

23

For heavy work in forge or smithy, manual labour was partly replaced by mechanical power in the form of the 'oliver', a treadle-operated tilt-hammer. The hammer, with its shaft attached to an axle, was operated by depressing the treadle; when the smith removed his foot the h a m m e r was raised automatically by the springy action of the bent pole turning the axle. This device was in use from the late Middle Ages onwards. Cf the pole-operated lathe, [29]. Figure 5(b)

24

First D r a w i n g of S t e a m H a m m e r , 24th Nov. 1839. Figure 6

25 Nasmyth telis us that he was able rapidly to sketch out a Steam Hammer with all

its. executant

details

in about

bali

an hour,

in his

Scheme

Book.

His primary aim was to lift a heavy iron mass above the ~orging and to then let

it

tall

hammer

was

~ig.6.

vertically entered

Note

sits,

the

its

hammer hammers, that,

to which

Steam

block

through

see

'Every

arrangement certainly

such

the

Scheme

as

hammer

the

head

beneath

forging. Book,

massive and

was attached

admitted

a human attendant

upon

personal

elements,

a piston

hammer-head.

guided,

his

blow-strikinz

containing block:

in

but

24

anvii

a piston

the

[2]. ) Nasmyth,

writing

nearly

detail

drawing

retains

which

i gave

to

one of the most

it,

43

iamous

years

the

steam

see

whoie

cylinder it and the

raise

the

hammer

the rise and fall oi the

<~or details

the

on which

could

drop-

1839,

rod between

piston

valve.

o~

steam

November,

T.he inverted

would be able to control

a slide

This

o!

hall

a

to

the

century

this

ago'.

'steam side'

His

later,

day

the

drawing,

in the hist.ory oI mechanical

of

wrote

form Fig.

and

6,

is

engineering

technology. 7he proposed

design was communicated

~rom his engineering

superiors-

to put the job of construction

which

to Humphries

included Chief

out to tender.

and he ~ot approval Engineer

However,

I. Brunel

-

the paddle shaft was

never forged and our story at this point proceeds in a different direction. .~ontroversies

I/ns!ish-Co~tinent.a.1 Priority

in discovery

in the minds o~ men, astonishing

example

distinguished

or invention

otherwise

mathematician,

to him and

day,

but even the time o~ day

in

research

he

for

many

with

are three which

quarrel: work.

years Oi

claim first

such

have

experiences

one

that

some

learned something

much

later

inlringed

these

nation

lamiiiarity.

claims

7here

and iluents, the

Royal

was

the

to

priorities

claiming

disputes,

visited

a great.

.just. as to %he a

assume

a priority

it is all a matter ol national

using the terms 11uxions

Leibniz

or

oi

of

international

discovered

memoir oi a

mattered

not

and

Perhaps most of us who have been

some

country

biographical

questions

[$3,

course,

An extreme

of discovery

...',

such distinctive

] can

argue

strong passions

modesty.

in r,he

'Priority

to

in the beliei

Newlon

in the i660s, his

read,

known

dimensions,

one of its 'sons' ihere

was

somewhere.

international

oi great

! recently encountered

deal

priority

raises exceedingly

seemingly

for

honour.

oI some magnitude the

Newton-Leibniz

dif~erentiai

calculus

but he did not pub!isn Society

in

London

and

ol it, either in conversation or he saw a copied paper oi

26 Newton's

work.

On

work and published ethics

of

this

resounded A.R.

returning

to continental

it - first!,

matter

is

of

long into the lgth.

Hall's

see [4]. no

further

century.

PhiJosophers

book,

st

Europe

he completed

his

own

]he quarrel about the details and concern

but

its

reverberations

This is examined in great detail

War~

Belonging

to

this

period

in

too,

there was the continuous assertion by Hooke that Newton had stolen the idea of gravitation

from him,

A personal the

[4 and 5 (p. lg)], .

experience

Ballistic

Pendulum.

Englishman,

Benjamin

!orty

or so earlier

years

of claim to priority All

Robins,

1750,

yet

another

experiments Robins

that

and

cannon balls.

Robins

century

such

be!ore

clearly with

Magnus

respect

it was

though

to

the some

not with the

Magnus

Eifect

conducted

simple

table-top

artificial to the

circumstances

shooting

of

it

if

we

attached

Truesdell

seems

was

state

about the

what

framed

as

what

large,

heavy

and certainly claims might

have well

examples

made

called

of it

Of promises.

later

o~

in

all.

Of

research

writing

it behoves us to course,

suggests

o~

names

prior

Boyle's

known

about

length

is proportional

Hooke's

law,

are

claims.

in Young's

for a

Maxwell!,

Tresca

Cardano

criterion

of

line,

The

he

first

remind

o~

outright

(1501-1570)

of solution

pursued

ol one !orm o~

- under pledge of s e c r e c y -

theft

or

saw

Nicolo

at

it

We might

and that best

Tartag!ia

cubic equation

<1904)

that English

who first reported or perceived it), cases

that

- nothin$

Huber's

adiabatic

analysts

the

ior some about

modulus.)

properly

as

namely

we may recall

[8].

since

(This will

'his'

seems

With wry amusement Clerk

are

definition

yield

Law

to load

(1944),

there

Gira!mo

to him his method discovered

well

change

in the 18@Os.

course

at

To to-day's gathering we need go no further

[7].

the Mises

Zener and Holloman,

names

though

not Hencky's!

be

use

when done

law,

is already

been

is deplorable

this

elastic

that

effects

[6].

exemplify

stress and strain. reflect

to

genuinely,

unsatisfactoriness

well

to important

simple ignorance of previous work;

intend

to

Towneiey-Power-Boyle than

of names

and through

right

sometimes

by

The mid to-late 19th century saw the name of Magnus attached

The attaching get

it

the

to this phenomenon - about the time when it interested Lord Rayleigh carelessly

of

used

In the same vein,

identiiied

in

inventor

attribute

by Robins.

clearly

the

a pendulum

and others,

as it was

lapsed

described

writers

fact,

Cassini

had

demonstrated

had already

in

by J.D.

same intention or effectiveness is. incontestable

English

but

surrounds

heat

line

described it is not [9]. of

broken

to reveal

which

he had

only later to lind it published

in

27 Cardano's

volume,

Algebra),

in 1548.

citing

Artis

oi Cardan's

only

of Hooke's

masnae

Ever since

The

third in

major

We shall

complete

this

can indeed

joint

o~

and

occasions of

section,

noting

of

much

is

in

... The

zhe U.K.

continent

knows

the same

item

both men!

probable

this

, we

confrontations

this

the

Nasmyth

that

Hurope

its

It

Of

similarly,

on the

Art

solution!

misappropriation,

the

story.

design

But

oi

before

unpleasantness

hope

arise.

a

good I

in

steam

doing

so we

in these

matters

of

some

is reminded

manner

how

of

these

comments

birth

understanding

concerning

the

example made

more

One

were treated

mechanisms,

hope

for

ol

about

how

the

the study ol the rational,

matters

-

or

his

1992 is the year that sees the olflcial

for priority

and

Great

I

well-

should

trust

be

helpful

-

[ i0].

It many,

which

men.

with.

elsewhere,

James

Since

claims

evolution

intentioned dealt

that

by

describe

economic

on

Darwin-Wallace fact

by

somewhat

whilst

is

presently

section

political

the

case

be avoided.

....

but in fact it ante-dates

recollection,

hammer.

Alsebra

name reminds me that,

is known as Cardan's joint, confusion

. ..

it has been known as Cardan's

may

finally

above,

varied

be

remarked

- and several

and important

that

the

not touched

as to warrant

issues

touched

upon

in

upon yet closely r e l a t e d a systematic

treatment

this

are so

in a longer

paper or book. The Visit by le~. Nasmyth Schneider

owner

Patricroft They

were

that

Mon.

details that

Schneider

whilst ol

plant shown

with

foreign

his

'took

see

his hosts

over

p.236,

extraordinary

Creuzot

later

and

the

French works -

single

smoothness

elsewhere,

in

France,

manager, of

the

Mon. his:

Bourdon.

to Nasmyth

constructive

Nasmyth

in his A u t o b i o g r a p h y

'a copy

of the

plan

had visit

Mon.

it was related

sketches

[ ], held

and details

ol my

6.

1842 when he was requested

the

engine

[i].

at

and

notes

had taken

on a visit

engineering

book

carelul

with the remark

marine

works

mechanical

that is, Fig.

with

foundry

iron

Scheme

visitors

Only in April, connection

iron

great

of the hammer',

his

to

the

and Bourdon from his

Nasmyth's

Bourdon

steam hammer',

also

away

Royal at

Dockyards

Creuzot

'the crank

and truth

did

large

eye

. . . had ...

to make a visit to France

and he ... been

thus learn,

had

the

after

', and in response

in

opportunity

complimenting

of a large wrought and

drifted

with

to the question,

28

A Portrait of J a m e s Nasmyth. Figure 7(a)

Steam Hammer in Full Work. From a Painting by J a m e s Nasmyth. Figure 7(b)

29

In 1839 J a m e s N a s m y t h invented his famous s t e a m h a m m e r (patented 1842), which soon displaced the old helve- or t i l t - h a m m e r s in forges. This h a m m e r was simply an inverted s t e a m - e n g i n e cylinder, fLxed on top of a massive iron frame, with a heavy iron head fitted on the end of the piston-rod. Much larger forgings could now be made, firstly b e c a u s e there was a bigger gap between the raised h a m m e r a n d the anvil, a n d secondly b e c a u s e blows of far greater force could be given, since the direct action of s t e a m w a s a d d e d to the weight of the h a m m e r - h e a d . In this early photograph, t a k e n in 1855, we see N a s m y t h himself working one of his h a m m e r s at his Bridgewater Foundry, Patricroft, n e a r Manchester, [29]. Figure 7(c)

30 'How

(wasll that crank

hammer'.

Further

.+.he result

it

urged

would

France,

revealed

of the piston

that. their

foundry.

steam hammer

'like

cartilage

packing

between

for instance

with the hammer

under the piston

bones'.

was

He saw and inspected

rod at its junction

Bourdon to put elastic

act

'It was forged by your steam

was told,

said,

He learned that they had had a few mishaps,

breaking

Nasmyth

he

of their visit to the Bridgewater

the steam hammer. frequent

... forged?',

enquiry,

On

rod,

returning

block. so that

home

from

1~asmyth lost no time in taking out a patent on his steam hamme.r and

indeed it was secured in less than two months. 'To constructed fall'.

demonstrate

one

with

a

Hand-working

piston

to

the

of

later

pieces.

performing

this

inventions between

(See

patent

for

of

armour

I

4 ft.

range

Of

applied

speed as

novel

at

the

above

the

impact

were

development

of a

3?

method

useful,

of Nasmyth

incidentally,

that

of

the

of

successfully

unpatented

Mechanical

are listed and given away

the

development

of the

pile-

admission

out

cast

iron

experiment

in May

1863

exterior

description

in June

- earned

to

contributions enjoyed

before

the

was just beaten as regards

chilled

was

shot

able

by Major

ballistics

for

to

of Nasmyth's

priority shell

for

be conducted,

Pa!liser

field.

and whose

But

technical

we

a

Palliser shall

not

accomplishments

many and various.

He was married retirement,

use

in

with

which remained

Nasmyth

the

taken

known

here

on which plate'

it was

is well

continue

for

steam

hammer,

in [ i].

to add,

advocacy

penetrating

o~n

Some

with

my

stamping for welding together- hot

Nasmyth's

Contrivances

a clear

hammer

as well

of

was intimately connected with that of the steam hammer.

the

shell

for

and

and

the

operation

operation. )

Another- matter was

controls

and the process of 3

capabilities

block

increase

improved

and Technical

We ought

and

hammer

steam

Note

pp. 38'7-431,

driver

powers

cwt.

and

die and stamp system, iron

30

accelerate

refinements

the

travel

lunar

abroad,

recreation,

a great

painting, to

1840 and retired

see

deal Yig.

science,

after

of money. F(a),

being

to

engaged

1853,

7(b)

(which

is very

- by his

He devoted

himself,

astronomy,

makin S

in

some

living to the good age oI 82.

see 5ig.

having

~ine

after several

archeology

Nasmyth

in colour),

and

he

painted

well

whilst

Fig.

F(c~ shows him 'on the job' . It

is

very

notable

-

and

very

elected to the Royal Society of London.

surprising

- that

Nasmyth

was

He was never a member of the

not.

31 Institution

of Mechanical

Engineers

knew one of its founders,

(founded in 1847,

George Stephenson,

of Civil

Engineers,

national

recognition or Royal honours.

founded

in 181F.

though

he obviously

well) or the older Institution

~urthermore

he did not receive

any

Charles Babbage wrote in his books

that these sorts of failure to recognise talent and the enterprise on which national prosperity rested, 19th century, Much edited

were far too common in the England of the mid-

[Ii, 12].

of the material

(though

but

related

lightly)

Nasmyth's biography,

by

above

Samuel

is from Nasmyth's

Smiles.

Smiles

AutsbioHraphy ,

relused

to

write

but for what reason is not known.

_fibre about the whole Story it is well known that several

scientists and engineers had long been

interested in achieving what l~asmyth proposed

in developing,

marketing and

selling his steam hammer and particularly that James Watt had envisaged for the same

purpose,

the attachment

the piston or piston rod of years before Nasmyth.

of a 'hammer or stamper

(an) engine';

... directly

to

this he made known in IF84,

55

A patent of 1806 described Nasmyth's design of 1842

remarkably closely having been taken out by W. Deverell. added

that

probably be.

Robert

The

critical

'steam hammer, efficiently

1864

on,

the by

paper was, was,

between

for

the was

reported

said

to

and

his

slide

consumption

points

Steam

Hammer,

intended

be

to

to a paper to the set

in many magazines.

Patricroft, it turned

the

was

out to

addition

of

the steam valves'

adapted

[13]

him

to

use

as

precisely

Cantreil notes on this point, the

hammer between

only

of

time

magnitude

of

the

found

of

which

was

using

at

valves of

etc.,

sLeam

read

for

as

in

in December

Patricro~t record

renewed

and them;

but

of

contact

interest

iis

Mechanics

right.

This

Wilson's original

system

workpi ece nature

pressure when

of

delicately

ali ected of heat the

between

considering

and

as

of

occasion

how the length oi time of contact

this transler the

a

to be

again adapted to place the hammer under the carelul control

transferred not

History

manager

the success

for enabling

admission

Cantreli

Wilson,

hammer-man

demanded.

sales

apparatus

continuous

further,

later,

for

later

the hammer

- effectively,

locomotives.

Institute

the

addition

self-acting the

Nasmyth's

for making

operated

facilitating railway

Wilson,

responsible

It remains to be

interfacing

them. the

el.me~.t~ ~ tubes in the middle o~ this century.

the

amount

is of course This

eificiency

of

suriaces

and

is a subject oi

heat

a •

nuclear

the that iuel

32

Figure 8

Figure 10

Figure 1 1

33 F O U ~ D A T ! OHS FOR HAKP[EI~S, [ 14 ] We length

intend

the

to

by these anvils

foundations

shock

and

possibly

designed

being

cam.

very

being da

helve-

with

hammers

height work

thick

were

billets

which

at

considerable

comparing

is

ground

relatively

conducive

t.o pres

but

and in

delivered

their to

metal

materia i

is in part transmitted

and

structures,

distant

an

locations

what

was

soon became

(or handle~

needed

accuracy, problems

a

they

in

could that

an earthoutside

oi

these

of

at

the

as 1440

shows

six

cam.

These

old

by

forging

by

that

for accurate bounce"

hammer,

impact,

the

di~ficuit

saying

As weii

types

thors

and

which in

for free

iorce

against,

became

described

the

hand-

As early

achieve

'gagged'

ordinary

hammer

[15],

it soon

to allow

For

the

water-driven

well

because

iorge

to fail.

p.!'70

and

to

raising

or allowed

is observed it

in effect,

anvil

for

drawing,

'feed'

in

it

were,

kinds,

a

them.

to

having

released gave

limited

no

hammers

different

iorgings

the helve

not

~-~=med

to

automatic:

lift

ham3~ers for large forging,

large

of

Vinci

were

their

of

blow

earliest

tripped,

A.TD., ieonardo helve

The

certainly

somewhat

to surrounding

the

though

ha~ers

dept.h and

during the forging process

conducted

that

hammers,

then

in

Vee-anvii,

itself.

it is held

latter

the

thence

being

the forging plant

a

below

and

character~~s~ics. *

Drocessin~ supported

to discuss

>'fat anvil

to the

die

also they foundations

was

re!ativeiy

iow. !~ig. 8, lb.'

is a sketch

of a drop hammer

two men were required

to an up-position'

the

height

in the plate seen conspicuously A steam Fi~.. 9.

hammer

was

]'he heavy anvil

on a concrete

~oundation

The patents

Nasmyth's

!0

hammers

s,~u~.ture. . . . . .

at the Iront and

sits, on timber

claimed of

made

i843.

be

Note

a

sketch

the

Woo]wic. h Arsenal

oI wood, concern the

ld,30, see

themselves

res.tin S

or shoulder.

themselves

they were meant of

in

with

the

to allay.

!irs+. order

for

of the anvil

and super-

Nasmyth'<_

one

cnmpanv_.. supplied

o~ to

a hammer with a tup weighing 20 tons.

We may remark hote, out. ane specify

Gave

cast pedestal

independence

. Just . . a{ter . . . his . . retir-~m~r:~, .

by using the ratchets

by Francois

did not

or the problems to

,50

the hammer

of the hammer.

baulks

with an inte[raily

o~ %'att and Deveril!

is

wi~h a ~up of weight

o! fall was controlled

patented

laying of hammer foundations Fig.

oi i550

at *..he end of the rope for bringing

foundation

that

it was apparently

details

Nasmyth's

for each forgin 5 machine

practice

to draw

design as a

34

Figure 12

Figure 14

Figure 13

35 whole.

He always

had the anvil

and hammer

on independent

foundations

and

he liked t.o ieei that everythin.$ was; 'firm'. The will

subject,

find

great

is now

left

interest

anvils and their

in

at this

point

inspecting

foundations

ol,

hammer bounce and to minimise

but

it is thought

~'igs. Ii to

14

'

or with damping materials,

the communication

the reader

note

how

heavy

aim to prevent

of forging blows to distant

i ocat ions.

The FrenchnEn~lish Controversy about th~ Invention of the ~ t e a j _ ~ r The

patent

for

taken out in 1843.

Nasmyth's

steam

hammer

expired

in

1856

Nearly 500 had by then been manufactured

i/4 million pounds-worth

sold,

customers being distributed

having

been

and more than

world-wide.

A strong dispute broke out between the French and English as to which first

invented

Cantrell

conversation writes

the

[13], at

Creuzot

Cantre!l.

about, which

The

Bourdon

impractical' A sketch

steam

hammer.

Bourdon'~ in

subject

was. a

at

and very different 1839

is

Nasmyth's

treated

alleted

Patricroft,

that

on the

from what the Creuzot Nasmyth hammer

in

depth

recollection

gross; distortion

engineers

notes

of the November

of

1842,

Creuzot

made

The

record

the

ol

the

machine

in

the

facts:,

Nasmyth

visit

by

of

device

1839,

'was.

later became_.

is seen top right

in Fig.

6,

and the bottom left sketch is said to be close to the 1842 patent..

Some of

the

LooKing

opinions

at the

i859

and

does

it

later,

now expressed device,

are different

we are

indeed

said to see

neglect

the

only

solution

~or example that the hammer block

oottom.

from those

of Cantrell.

a sketch

of

certain

of his general

idea

ieatures

came

which

is guided by slides at its top and

Could it have then been known to require ~uidance the whole length

of the

block?

small.

The

impact, load inserting

The piston

rigid

fitting

is; a

weakness

somethin~

rod,

one senses,

together

of

- though

elastic,

(on

the

there the

has a cross-sectional block

to the rod under

is a note

left).

area

on the

Nasmyth

a iar~e

diagram

implied

too

about

that

he

recogni~. =4_~.~ this: need ~or elastic cushioning and thus -'-oo~"~sted- packing when he visited ~,o surround

Creuzot the

in 1842.

guides

We believe

beneath

the

downwards again as much to protect its upward bounce. .just below Cantreil

the

writes?

(or see that)

steam

cylinder

15

the

accelerate

sprinz~ the

iup

i% as absorb any shock it might give on

]here is indeed a note too o~,

cylinder.

to

he intended

hammer

block

This is not clear to the writer.

'? spring' speciiied

on the right, at

3

We see the anvil

tons

as

block

36 surrounded nothing

by cork

to

'work

is

standard

ground

appears

vibration

steam

engine

'

...

observations

foundation suggest

and

parlance.

for use as a seal. that

- which

is a good point

in the later sketch.

exhaustively/expansively?'

probably about

to reduce

in this respect

'cut-off

There

And at the

There

at

2/3

is mention

foot

stroke'

too

of

of the right

from

(the?)

Block'.

there

is more

useful

- which

'leather'

column,

These comment

but

is a reference -

a note

few

incomplete

to the

top Figure

than the bottom~ It is not

clear

that

a chain

was

here

intended

to exist

between

a

piston rod and the piston to raise and lower them. There answers

need

are

a

to

be

~asmyth's half hour

number

of

technical

available

to

questions

assess

the

outstandin S

degree

of

for

which

originality

of

'brain-storm'.

DIES KWD KNKLYSIS:

S L I P L i l l e FIELD THEORY ( s . 1. f . t . ) :

PLANE STRAIN INDENTATIO~ AND FORGING

It hammers Russia

is pertinent

and

perhaps

to

vee-dies

similar

or comparable

helpful

historical

for

it

hitherto

ones,

This

scientific

1.

Sliu

work

to

be

is

paid

a an

evaluation

line

of p l a n e

with

that

described,

to the writer

been

field

s h o w i n g how t o

terms

Germany,

invention

central

for science/engineering

to

done thus

and

development

Europe,

the

of

U.S.A.

and

If there are claims from

pioneers who

in Britain

have performed

and France,

contributing

to

it would

a more

accurate

strain,

in various the

which

books [16,

load to cause

o f a mass o f i s o t r o p i c , Results

~or

shear

an orthogonal

to a given die geometry, and velocity

matter

of has

scientific

vee-dies only

as

been

attention

against

flat

susceptible

of

theory

o~ k, the yield

establishing

insufficient

introduction

in the last generation.

calculate

s.l.f.t..

%hat quite

the

important

Lengthy expositions

using

the

record.

It appears has

in

about

in the second quarter of the 19th century.

within these countries be

inquire

pressures

stress net

of

have l o n g been a v a i l a b l e

indentation

under

conditions

rigid-perfectly-plastic and

loads

of the material. lines

with sufficient,

(metal movement)

the

17]

of yield

are

S.l.f.t.

shear

materials

usually stress

given

in

consists

in

appropriate

(though perhaps implicit),

boundary conditions.

stress

37 for metals originated with Prandtl

S,l,f,i:..

later

with

Nadai,

in

Germany,

though

the

in the 1920s and slightly

notion

of

slip

planes

in

soil

plasticity long ante-dated that,

even back to the time of Coulomb [18].

full

of plasticity

interrogation

of

the

kind

situation

raised

below

becomes possible following 1950, after Hill had published his book,

2.

Indentin~

with

a flat

(i) A flat rigid conditions restin 8

of

on

plane

a

deformation

rigid

die, strain,

refers

to

only

[16].

die

width 2a,

is set to indent normally

a block

frictionless

The

of

material

foundation,

see

two-dimensional

of

Fig.

uniform

depth

15(a).

plastic

and under h

flow,

the

,

strain

flow

being

independent of the third dimension.) The Fig.

15(a)

indicates

an orthogonal

net

of slip lines stretching

between the indenting die, AB , and the foundation shear stress lines 45 ~ .

Using

point

left in

and

the

written

as,

between

two

pair

s.l.f,

of Hencky

die

of

can

amount

frictionless

on

the

BE,

p,

drawn

of

technical able

to

the

are

~

the yield

foundation at

used

h/a

hydrostatic

is the change

at

line

the

in connection

be

,~

and

of the plastically

the

can

stress as

the pressure

pressure

die p

upwards

to

or displacement deforming

its sides

as

+o cause this is given by

this

on Fig.

18

leads

Fig.

16
Fig.

15(a) when

h/a ~

us to conclude 8.7.

that

sideways so found

ol individual

material"

this

particles

is represented

see Fig.

15(b).

(ii) Indentation with a flat die can cause the expulsion ol material the

a

The principles of the method make it

in a velocity diagram commonly known as a hodograph,

below

in

After

integration, cause

be

change

points.)

will

to the foundation; ratio.

is

numerical which

at each

equations

in hydrostatic

respective

calculated

pressure

Hencky

and

two

manipulation

to ascertain the velocity

at all points

which

(The two

slip

parallel

is speciiic to the given possible

and

same

is

expulsion

equations

be calculated.

tangents

pressure

at

see [!6 and 17] and knowing that there is no force of AE

Z~p + 2 k , ~ p = O'

inclination certain

right

points

;

- the slip lines - meet the friction!ess

the

with specific s.l.f.i., to the

CD

indicated p/2k = that

in Fig.

1 +

the mode

~

16(a). 2.57.

o~

The

from die

Entering

indentation

here proposed prevails over that implicit

<see in

bJ O0

..j

Q

~I

(D

0

,,,,

-~

-ol

I

i~~~~3~~ i

A~

o \

o

\

\

'

\

-i

I- 0 m

]>

E-u

< II

<

~

39

Fig.

!7(a)

isosceles Plastic the

is

triangle

adaptation

of

been

triangle

off

the

indented strip was examined,

and

work-piece

itself

evidence

indented

of

the

where

smooth foundation;

it

had

at

the

a small

the side strips as they move

in the

placed

15(a),

causes

see

clear

Fig.

having

indentation

material

an

tip

of

foundation.

the

right-angled s.l.f,

outwards

If the

bottom of

h/a is in the appropriate range,

been

material

previously

seeming in

at. E.

to r~ise

to

contact

be

the

we would folded

with

the

it would seem to be a fairly clear concentrated

or

flat, defect

in the lower strip surlace. The hodograph for the s.l.f, the

large

slightly above

range

of

smaller

h/a,

die

and hence

this

pressure

plotted

of

Fig.

17(a) is seen in Fig.

particular or

on Fig.

load 18,

mode

than

would

of

does

deformation the

constitute

mode

of

17(b).

In

requires Fig.

a

15(a)

yet a third mode

of

deformation. This last mode was propounded by ~. Hill,
There

work piece,

is yet

a fourth

mode

prompted by the results

but later examined using a s.i.f, large values about

which

of

h/a

.

see [p. 14Y,

involves

ol

the

indented

oi a crude upper bound due to Dugdale,

by Dewhurst,

We shall

bending

19].

not

[19];

include

it just prevails for

here any further

discussion

t,his fourth mode.

5. For~ing betwee~ flat~_e~ded e q u _ a l _ ~ t h

~i9_~

Consider the indentation cutting or ~orging ol a thick slab as in Fig. !9. line,

Because it

the top and

follows

section 2 ( i )

that

situation

19

a s.l.f,

give the necessary

formally

slab,

stretching Evidently

shown in Fig. 20 applied

by

the

diecussed immediately before,

that

of

M K

and

M u

to move outwards,

from one die to the other - which Fig.

I~.

is able to be used to

[~QI

be

with

is to have the dies approach

6. For~in~ w i t h f l a t - - e n d e d d i e s o~ ~t~equal w i d t h : to

the centre

that

h/a.

The s.i.i,

about

identical

for any given value ol

required

p/2k

symmetrical

15.

deform this

suggests.

are

is

and for the two portions

we need to construct is what Fig.

the

above or u s i n g Fig.

To plastically each other,

bottom halves

will provide an upper bound to the load dies

to eflect

the non-identical

lorging. s.l.1's

From

the

cases

from AB

and

C.i~,

40

/

,oo,~

:

' ~'-3-7

I A PAIR OF INTERSECTING VELOCITY DISCONTINUITIES

0 2

I

Outwardly Displaced RigidBlock

T

\

Slip-line Field

CI O21D

Hill's slip-line field for indenting with two dies.

Figure 20

Figure 19

(a) f

(b)

9

)

\

]

(a) Reproduction of Nasmyth's sketch of a flat-anvil forging. (b) Reproduction of Nasmyth's sketch of a V-anvil forging. Figure 22

Cutting with opposed wedge-shaped indenters. Figure 21

41 meeting at

K

would seem to be able to provide a solution.

certainly

to be met are,

pressures

o~ course different)

right

of

BKD

calculations tY]~ . 2k

AKC

of forces

and

should

hydrostatic

be

the s.l.f,

against

pressure

from

zero.

hodosraph

Then is

above or below

K .

a

at /a

and

CD

completing

the

(normal

the

numerical

same

and i approached

whether

The ratio of die pressure

to

ratios is given in [20]

equations for angles

can be draw-n,

AB

~orces to the le~t and

provides specific angles ~or ~ K

H/al~or various

Once having solved

on

(ii) the total

for these two conditions,

The

through

and

(i) equality

The conditions

and

,

an accompanying

It is given in form for a stationary

and a top die moving down,wards in [20]'

simple

upper

bottom die

bounds for this same

situation are also given in [20]. Cutting ~rire with pliers, Fig. approaching

21

[21]

shows

one

the

another

and

conlirms

that

section

wire is substituted.

those

which

causes the

thus

the mechanisms can

prevail

compression

latter

s.l.f,

of course

a

cutting

for

the

the

cutting

reduces

case

sides

of of

pair

wide,

The situations

in

whilst

for

of

thick

the compressive

plate.

dies,

Experiment

are much the same

i~ circular

are evidently very similar to

forging. the

wedge-shaped

Approach

wedge

cause

of

the

lateral

~orce necessary

dies

tension;

to effect

the

cutting action and at the same time the tension tends to cause fracture. More realistically, this is addressed

8.

in [21,

The T e n d e n c y t o

When forging split

or crack

centre

line

closed when a certain the

small

triangle

form

of

hodograph. void

top

is

half

can

as they

open

of move

19, the effect relatively

it up,

as

is reached.

the

opposed

such

is zero.

appropriate which

that

Fig.

of a short

initial

magnitude>

on the

to

takes

and

maintaining specific

the

17(a)

s.l.f,

of the two interfaces apart,

small

If this latter

be re-constructed

on the crack,

When separation

is formed

to

h/a ratio

s.l.f,

standing

the

as in Fig.

than a certain

slab

and

cracks

or reducing

the

dies as pliers are flat-ended

22 and 23].

ovenine-uD

(of more

of

is exceeded

the wedge-shaped

stress

ratio in the

shows then the its

accompanying

at the crack

place

it

occurs

a

after

the

dies

have

all

the

solutions

approached each other by some small amount. It concerning

has

not

been

emphasised

the s.l.fs above,

in

connection

with

that every case is one which is only

II~IPROVEMENTS

IN

FORGING

AND

IN

WELDING

IRON.

tO

MR. N^S.~YTH described to the British Association, at Edinburgh, an improvement of his, which tends to increase the certainty of the production of perfectly sound and solid cylindrical forgings, especially those of large size, such as shafts, axles, and the like. In the common method, by which the metal is placed upon a fiat anvil, the effect of the hammer is to spread the mctal out in one direction, and this must be corrected by t u r n i n g it round, so that each successive blow may correct ~he spreading caused by the prcvious one. T h i s causes a

AN:.N:~..:L.

THE

SCIENTIFIC DISCOVERY:

fretting or mincing of the centre part of the metal of the shaft or other forging, resulting in a separation of the metal t h r o u g h o u t the entire

0 R,

90

ANNL'AI,

()1." , ~ ' I I . : N ' i ' I I . ' I C

I)IYCOVI.lilY.

centre, fr('qu('utlv t~, .,-uch :m ('xh.t~t as to l)eraLit tim i)assagc of air or w:tWr from c,,I t,, (',,d. 'l'l,is (,vii Mr. N. (',a'ru('ts by using a wedgesh:,l)mi, ,)r V :ulxll, lwtw~ t.n lit,. j:tw.s of wl~it'h tile u'~rk to be hamm('red is I)lace,l. In tl,is ('as(', i,~sw:ul ()t" a ten,h.ncy to spread, so as to render the c(,ntr:tl i,,)rti(,zl ,)f tl~(' metal le.,-s compact ami solid, we havc exactly the (q)l)().stl,: cli;:ct, I)esi,h.s wl~icl, tim arti(.le is m,)r(; easily kept under the hammer, and the scales or impurities which fall from the hot iron fall (low, into the apex of the V out of the way, tlauu renmvin~ th(; bh.mi..,IL :~n,I r(,u,.zlm,'ss xvl~i(.h is caused l)y the scah's collecting on the line of t}~(; :mE II and b(.iug bea1,,n into the surface of the met:t]. "]'lie (.,:Jlll~r(..,,mnl..r (.P.i.('t of ll,t: I~}oxvs, i,m, is so mu(.h enhauc('d that ;ts mu('ll can l,(' Ii:,,n;nt'r(',[ ,)lit at ,me " h(;:tt " I)y the new anvil as in tl,r(.e '" hr " hv th,. (',~,,,m,,,, ,m,'. "l'lw angle ,,f sO ~ is me,st ~em.rally sui,:~l~l, , for ilz~' iu('lin:,li,m ,,f 11,(' s~,l,,s of 111(; V" 1he edges should be well rou,,(Is,1 ,)ff and the .~urth('u ,~f tlw V sides curved in the direction of tile. a,rts ,,f the work, to the extent of one eighth of an inch in twelve incht's, s() ::s to !,~, " l)r()ud " in the centre, and ~hus facilitate the cx~ensi,m (axis-ways) of the work. One anvil will accommodate forffings of :ill ,li:~,m.u'rs, which :,re not so ]ar~'e as to rest on the UlTcr c~,rm,v.~, m,r s,, s,nall as to touch the apex. T h e s e anvils have be(;~, v('rv ~'x~,',~...iw'lv i,,~r,Hl~,'('d. Mr. N:;s~J,)')l, ul..-,, ,l,.~.,,I,,.,I ::~, i,,,i,r~,vm,w+,t i,~ wc.lding, l i e commenced by showiu,_: t!vtt Ill{' l'r~'~i,~('!~ld~.li'('t it~ w<'l,lin,M :tri,~es from the interposing (,t'.~cur[:.,: ,,r " ('in,h.r "' b:'~twuuu tht' wehlcd surfaces, which i)revent tile two ...urliv.,.s I'mm I,,.i,~u I)r[,,t,.,l~t in contact at all points. he " slabs " prodtLc'u~l uu~d~.r lh,' :wti,)a ut" a tbr~u-hammer and anvil usually have some l),,vti,us el' flu.it surlitc('s sli~:htly concave, and the concavity is (,rditmvily such tl~:Lt ll~e i~nrts w h w h come inh) contact first are the e x t e r i . r cd,~.w.~. Tim Idows (,f the hanm~er weld the parts in natural coutacl, :m(l in a tin'arty or less degree expel the seoriab which will esc;,lW as i,)ug as th,.rc is :~ 1)assagc (rot, but if, as has been said is generally the ca..s,:, ll;c t, xterior l)ortiou of the surf:aces of the slabs is the first wchled, th[. scoria; nmst remain, and no amount of h a m m e r i n g can remove th('m, :,.t,I thus ~ c have :m unsound welding. T h e remedy fi)r this great evil is a very simple one. It i~ only ncccssary so to form the surfaces to be wch]e(I that a fi'ce exit may be prescrved to thc last fi~r the scori~., and l!,is is (hme by m a k i n g one of the surfaces sligl~lly convex, s,) tits, the ~v,,hling l)e,,.,ins at the centre a n d proceeds outwards, thus fi,r('in~ out :ill tl~(' s('m'ia', and a l l o w i n g com1)h,le c o n t a c t . - (Sril ]':/~g~, r ,m,I Ar, h m , t ' s Jour~zal, ~%1,l.

YEAR-B00K OF FACTS IN SCIENCE AND AR '~'

MOST

IMPORTANT

DISCOVERIES

MECHANICS, USEFUL A R ' [ I $ , NATURAL Plill r CHEMISTRY, TOGETHER

Wlf

s

A CLA$SII~J~ IN

LIST

OF

V.MT OF

SCIENTIFIC

AND IMPROVEMENTS

ASTRONOMY, METEOROLOGY, ZOtJlX)GY BOTANY,

MZN ;

RECItNT

8CIS[NTIFIC

P A T E N T $ ; OBITtTARIES

s

SCIENTIFIC

INDEX

OF

JOURNALS~

PUJL/Cs

OF

IMPORTANT REPORTSj

IMINENT

Ps

ETC.

U D I T I t D BY

DAVID OF T H B

LAWRBNCE

A. W E L L S , SCIENTIFIC

A. M.,

8CHOOL~

AND

GEORGE

BLISS,

JR.

BOSTON: GOULD

AND 59 WASHINGTON

1851. Figure

23

LINCOLN, STREET.

IN

MINERALOGY, GEOLOGY, GEOGRAPHY, ANTIQUITIES,

CAMBRIDOE!

$

43 momentarily geometry

applicable.

After

any

small

is changed and a new s.l.f,

degree

should,

of

deformation,

field

be developed

which

strictly,

accords with the new geometry. NASNYTH AND TH~ V--AJVVIL

Nasmyth,

[i, p.420],

introduced the V-anvil

in 1845.

He gave as his

reason the fact that this form of block avoided "unsoundness" at the centre of the forging. list

of

Before quoting the remarks taken from his chronological

Inventions

recall

Hill's

dies;

this we have

cleaving

and

solution

action

Contrivances to the

problem

shown in Fig.

at

the

centre

19

ol

A

how

are

they

and

A'

is

worth

of forging block

oppositely moving zones

separated M&

and

with

and

after M L

relevant

a single

where

the

to

pair

slip-lines

intersect will be noted.

in the inset to Fig.

rapidly

while

of

and in particular the shearing and

the

tangential velocity discontinuities) particles

it

19 indicate

crossing

the

(or

Two adjacent

diagrammatically slip-lines

into

.

Nasmyth wrote: "In connection with my Steam Hammer,

great cylindrical shafts,

when employed in forging

I introduced what 1 termed my V-anvil.

Its employment has most importantly contributed to secure perfect soundness in such class of forgings. forging cylindrical shafts, anvil.

In the old system of

the bar was placed upon a flat-faced

The effect of each blow of the hammer upon the work was

to knock the shaft into an oval form, as in Fig. 22(a), and the inevitable result of a succession of such blows was destruction of the soundness of the centre or axis of the shaft.. to remedy this ~rave defect, flat-faced anvil,

In order

arising from the employment of a

I introduced my V-anvil face, the effect of

which was that the dispersive action of the blow of the hammer was changed into a converging action,

which ensured the perfect

soundness of the work", see Fig. 22(b). (Report

of Nasmy%h's

instanced in Fig. 23.)

idea abou%the

V-anvil

in popular

terms as of 1851

is

44

Radially Outward Moving R~gid Block

C_entr.a!$tationarl ~. Rllld Blockwhich ~ is notD~storted

i ]

[ /

Shp.|ineField

DieMovesInwards withUmtSpeed

~ (a)

FinalSpeedof. Outwardly DisplacedBlock

\

Unit Speed

(runch or Die)

Hodograph corresponding to the slipline field for the portion above DOD in (a); hodograph for other two portions is of similar shape.

(b) Figure 24

45 However,

given

this

observation

corresponding to that of Fig. 22(b) author to be as in Fi 5. 24(a), having

an

appropriate

how

must appear.

[24],

for three

a/r.

slip-line

field

This was proposed by the indenting or forging dies,

stress field has a 3-fold symmetry as has the velocity field of Fig.

24
peripheral

The

the

whole

the

of

ask

the

Specifically,

ratio

we

regions

predicted

over

occurs leave a central rigid core of material dictated

principally

actually

has an angle

drawn in Fig, 24(a) only

a trivial

agreement

by the geometry in the

lower

adaptation

some

which,

of

of the

p/2k

~iven

s.l.f,

versus

it is emphasised,

to

r/2a

are

bring

requires

about

admirable

but for ~our dies were used are

given

at length,

in

Fig.

8

of

[25].

by Yang [25] using the

it is not necessary to use it to obtain results.

the writer's opinion the traditional the mathematical

deformation

judgment and intuition.)

24(a)

This case has been investigated recently, matrix method though

plastic

of the diagram not

with Nasmyth's practical knowledge, results

of

to treat of this latter angle

S.l.fs similar to those in Fig. and

which

of the situation. 'third'

, but of 80~

form

In

approach is sufficiently satisfactory:

demands made on engineering students to attain familiarity

with the matrix technique are made much too high. The set of results given in the Fig. what we described earlier above, of

h/a

8, o~ Yang,

show,

interestingly,

namely that for sufficiently large values

, tensile stresses that aid cracking,

can be developed

in what

is

predominantly a compressive process. A large volume on Industrial l;or~in~ (in French), is

that

by

Codron

English-speaking of

machines

[26] ;

it

countries.

and

test

is

behaviour

engineer of the 1920s.

V-dies

the

and

manner

However,

that

seems

in

which

any

modern

in

terms

Particularly, they

act

so

recognisable

(2nd edition i923),

virtually

It contains many results

piece

professional compression.

one

unknown

in

and much discussion appropriate

to

the

some mention is made of as

to

promote

central

plastic analysis oi this

situaion is not found there. A simple

elementary

analysis

I have not seen carried

could be made to calculate the pressure along dies in Fig, il

it

25(a). were

a

AB

out but which

etc.,

is suggested

We treat sector I of the diagram of the physical set-up as case

of

extruding

through

a

60 ~ wedge-shaped

die.

The

elements supposed uniformly stressed through their whole thickness between

46

Figure 25

47 one pair oi neighbouring a Irictional

stress

dies,

~p

on the

equilibrium

equation

p

and ~ ,

with a necessary

is

easy

find

to

This approach Fig.

circumstance CC'

as

a

and

between

parallel

line;

there

centre

line. )

curvilinear

in

hill'

the

that

If.

the

that ~

pressure

ol

material

is

there

course

be

note

that

zones,

no

triangular

internally

sound core of the iorging.

'unminced'

move

the a

of

that

in the

withotlt

on the cenlre hill

triangular

just considered

marked

in

compression

iriction

straight-sided the

dies.

or to be,

outwards

case

in

(see

the

it

approach breaks down a~

real

cusp

by

note

have to appear

ol the two cases

a central

We

o3 course

the

up an

between

along AB',

straight

cannot (In

and

As indicated

Figure.

unacceptable.

of

/~

as either

the

= 0 applied

is

p

By setting

a yield criterion

oi this elementary

We

triangular

condition

ior

dies the gap would

can

the element.

the element

sector

because gap

ol

the better the smaller

for the utilisation

C'C"

or

creating

'Iriction

apply a normal pressure

introducing

boundary

we may consider

circular

ends

for an element,

will apply

25(a)

part,

a

AB and A'B',

portion

in

on or

the the

constiZute 9ig.

23),

or

Other geometr ies_
26,

the

and less so in

latter

workpiece contact

ior cases

case

a dead

interface; there

is

Fig.

when

the r/2a

ratio

is small,

27, drawn ior i/6 o5 the whole si[uation,

metal

zone extends

across

the whole

where there is, relatively,

movement

oi a block

of

the

metal

an even greater

outwards

in

oi the hammer-

relative

to

length o! the

~-c.,,~.~

die. The whole

six

and

discontinuities what

Nasmyth

discontinuities out

to

radius

which away

content of

in

not

the

tending did

to

not

destroy

single

six

meeting

cavities

as

between simple parallel

its

iorsing

on

is

for

oi

their

zone

has

the

What

has

zone

oi

intersection> been

should

dies!

in

Fig.

daylight

integral

shiited

six to

or

veiociT.y

happened

an~

centrai

be regions

perhaps

be interested

to the British Association which

remain

disintegrating

soundness.

The reader

reviewed

through

of

'mincing',

points

'mincing'

Irom the centre.

region

by

comment

(and the the

central

penetrated

of l~asmyth's report

Science

'minced'

parts

are

bali

to note

a the

for the Advancement

23.

It

contains

could

be

seen

in

relerence iorgings

to made

48

"Dead" Zone Attached ~, A

"..

" ,o~o,,or,~:~T ,: Lre Face~ /

Sco,,o,

A slip-line field (one-sixth of field, there being a six-fold symmetry), for a very rough indenter, w h e n r is m u c h less t h a n 2a, a n d the d e a d m e t a l zone e x t e n d s only over part of the die face.

o. f ~ , e. r , al

~~vf

Figure 26

r r =-

A slip-line field of the s a m e kind as t h a t in Figure 6, b u t r not m u c h less t h a n 2a.

Figure 27

49 (ii) the

In Fig.

three

faces

periphery

of

solution the

28

there are of

the

of

forging.

block.

applying

zone

the

The

because

enforced

six separate There

plastic

(iii)

Fig.

29

down

through

V-block

suggests

(sector

half-right

the

delormation

II

of

Fig.

and half-left.

The mechanics

forging

the

around

standard

the

Prandtl

not

penetrate

through

the

core elastic.

the

and

is

two to each ol

is so large that when forging,

in which

between

25(a)),

local

does

internal

a circumstance opening

regions,

presented ratio

r/2a

whole of the block but leaves the

metal

is just

solution

the

s.l.f,

two

only

there lower

is no

expulsion

inclined

upwards

and

ol

dies of the

outwards

to

the

o~ ~low in this case remains to be

explored. SINPLE This subject be a method much

is dealt with at some length in [17].

easily

comprehended

(though nearly all s.l.f, Alternative simple

little

as

s.l.fs,

bounds

in excess

The

standard

an

energy

In

Fig.

used

than

can

of those

method,

and

easily

provided

foundation}

method

It can be said to

for plane strain which is is

the

theory

of

s. 1. fs,

only upper bounds.)

one can make calculations

which

in theoretical

(i)

and

solutions are themselves

to using

upper

i0%

requirement using

BOUNDS

of treating plastic metal deformation

more

using

UPPER

give

results

(after

for die loads accurate

to

much

greater

effort

upper

bound

consists

as and

by s.l.f.t.

for

finding

one

form

a 2d

of

it

is

exemplified

in

immediately

below. enforced 1/6

of

by the the

consists material BDA'',

by

cross

tup

approprimate

or die

section,

sector

solid

block

arrow

(south-eastwards);

or shears

quantities

an

the

crosses

performed

show

one

CBD,

being

AB

pattern

of

plastic

on the workpiece.

Ill,

we

suppose

compelled it

by

slides

For just

that CB

along

flow

the

flow

to

move

as

CD.

As

the

BD , it is sheared to move as a portion of exiting

in the direction

slides

we

frictionless

whole

in

indicated

25(a)

on

of

~.

CD and

block

Work is done in shearing as the metal (ii) on crossing

to provide an over-estimate

The calculation

of the tup load on

P.u = k(CD.v c~.) +

are easily measured ofl,

BD.

DB 'v Df~ );

CB,

from the diagrams Figs.

(i) to be

P , is

25(a)

nd

(b).

50

l

1

l S(otionary

' ",:.

,'X

X

~.~

4"....~x-,'..

<.~

9

" , 4 "4-

........~..

". 2.~ : : 22".. .[4.%~. "

9

v

'..:',:~.:<"

(b) (a) Showing i n s t a n t a n e o u s slip-line field for forging with a smooth die, consisting of circular sectors a n d 45 deg. isosceles triangles. (b) H o d o g r a p h corresponding to rationalised field, a straight line "3" being s u b s t i t u t e d for the arc. Figure 28

Unit

Speed

1

IlPll 2

J~l

(b)

3

(~

(a) Slip-line field showing one indenter a n d two perfectly smooth supports. Dotted lines define the blocks of the rationalised stress field. (b) Hodograph of the rationalised field for one-sixth of the whole situation. Figure 29

51 We

~ind ~or the example shown, P

Point

D

makes

=

that if

can be chosen to be any on

p

CD = 3 units then

k(3.3~6 + 2. 4.7)/2 5 =

a minimum;

D

here

8'ik,

C~

or p/2k = i-6.

but that one is best chosen which

is not

necessarily

the

best

point

in the

example. The hexagonal the unruptured,

internal mass of which triangle

un-minced cohesive central

In some of the Figures,

OCD

is 1/6, represents

body.

dotted straight

lines could be used to provide

an upper bound in place of the curved s. ls.
An

upper

equilibrium flow

bound

force

field;

result

can

also

be

simply

obtained

by

diagram for each and every chosen polygon

these

latter

fields

should

accompany

an

drawing

an

in the plastic

acceptable

velocity

field. Fig.

25(c)

is the

equilibrium

forgi~ 8 block for the same circumstance yield shear load the

starting

is

line

BD

for

Calorimetric remaining

10'7, is retained

there

s.l.f,

is

boundary)~ but continuity heat in

dull

across

have

i/6

o2

the

whole

The normal N, and

jump

has

to

be

observed

that

of the freely emerging metal.

LI]IES

shown

that

metal

the

there

in

about

90%

reappears

slip

lines

strain.

iorgin~s.

is usually

shear

of

as

in the

mass

there

kind of slip

as bright lines

These

heat

one

set

strain~(frequently

Assuming

away from the first hot

it

the

energy

heat.

in the deformed met.aliurgica! structure. )

in ~orgin Z especially) red

which

HEAT

deforming

solutions

a finite

in deformation

transfer

end faces

studies

plastically

In most which

for

(i) above.

to the centre-line ADIABATIC

in

as in

iie!d

S , on each oi the three lines o2 triangle CBD are clear'

point

there is no force parallel

dissipated

iorce

lines

of

the

at

lines

across

the

plastic

field

there

little

time

for

they can show up

(on

is also

line,

of

is.

exist

unseen

in the

body

of

deforming hot metal. Conspicuous "9*I the

late Dr.

heat R,A.C.

lines Sister

- designated - can

'Limes oY Theri~l Discontinuity'

be seen

in [27],

the temperature

being evident because the imposed shear strain is rapid and finite.

jump

The

52

9

~

_____

The last 250 ton white hot ingot to be forged into a naval gun being s h u n t e d over Ashton Old Road in 1927 from Sir W.G. Armstrong Whitworth's North Street steel works to the Whitworth Bessemer Works forge. Sketched by F r a n k Wightman when an apprentice at Saxon's engineering works opposite. Note the traffic men (Whitworth's), the two locomotives and the heat shields. (courtesy F. Wightman) [28]. Figure 30

Figure 31

53 history

ol this topic up to the late 1960s was described

Association an

easy

of ~ngineers

matter

incidentally, These

to

Sister

estimate

were first

many

years

heat

It is

maximum

have,

these

reported

having

been

seem

only

they

mathematicians

lines

have

and scientists

the hard phase,

and dangerous

metal

become

being

infinite 25(a); were

see [27].

transfer

in the short

working

the

object

martensite)

in steel

oI report

research ol

~or

engineers

attention

by

[93.

<especially can they leave

thus constituting

a source

ol unwanted

imperlection. forged

temperatures

magnitude. there

shown

heat

in the last two decades,

Where there is sliding of metal block

using a pyrometer,

to

Heat lines can cause phase changes behind

jumps, which

in the 1860s the subject

known

to

temperature

(if we neglect

by Tresca well

to the Manchester

in a paper of the latter name.

been verilied approximately

adiabatic

term) though

by

Such

six such

lines

(of ~inite amount)

be extremely

situations

are of course

to be likely

can

over metal arise

large

along

- theoretically

typical

in the Figure.

of defect

by the

inside a

line

These

CD

latter

late Prolessor

o~

in Fig. lines

Tomlenov

oi

Moscow in the late 1950s. Strain enormously bands

and

of

horizontal

rates

as implied

can

approach

white

lines

o~

in the various infinity

near

martensite

s.l.ls

to

die

inclined

at

above,

or

tup

corners.

a

few

degrees

vary The

to

the

can often be found near to die corners.

These two issues of heat lines of nominally and

suggested

regions

discussions

of

infinite

of s.l.f.t.,

strain

rate

have

because

there

infinite temperature

never

are no

been

properly

accompanying

level

faced

in

thermodynamics

of s.l.fs. DEFECTS

i.

We

have

first

discontinuities

in

using the V-anvil

seen

flat-lorging

some

the

avoidance

was

in

of

Nasmyth's

intersecting mind

when

in order to avoid their disrupting ellects.

the latter secures central is relocated

that

distance

soundness,

he

velocity turned

to

Whilst doing

it is our belief that the unsoundness

from a block

centre

such as

and that there could be in this case as many as six sites.

D

in Fig.

24(a)

54 2. We have pointed to the possibility being

able

across slag

pull

voids.

small

apart

the

(non-cohesive)

inclusions

causing o~

to

the

which

of developing

interiaces

~aces

have

o5

o~

otherwise

'stringers'

been

extended

of

inclinations

and

-

in

It should not be too difficult

cracks

tension

closed

unwanted

the other

cracks

(or

metallurgical

course

to examine

locations

in iorging and

of

roilin~

-

the consequences

than

the

symmetrical

and horizontal. 3.

We

have

indicated

that

metallurgical

inside forgings as a consequence

de~ects

may

occur

oI heat lines. CONCLUSIONS

We have infancy

to

Ultimately Fig. it

a

the

'li~e

massive

and

unit

death' ~or

the steam hammer was superseded 31

show because how

followed

becoming shows

a remarkable

it indicates

became

of the forging

steam very

hammer

heavy

•

its

workpieces.

by other kinds ol press.

drawing

made by an apprentice

which

how massive 1orgings had become after W.W.I.

necessary

in

some

circumstances

to

separate

we and

casting

and

forging shops. We have seen some s.l.f.t, predict die

of the results

and its adaptations fairly

~orging

investigation techniques, plasticity.

accurately
- loads,

like) as

a

simplilicaions

pressures,

processes.

and elaboration, Plasticene

which modern plasticity

or engineering

There

theoretical model

~low patterns is

scope

and de~ects ~or

and experimental,

material

<using

mechanics

-

- can begin to

grids>

much

in

more

using etchin~ and

photo-

55

NOTES

I. (Sir)John LESLIE (i~66-1832) Leslie was a Scottish mathematician and natural Andrews

and

s 1804

He

Enquiry into and

was

published and

Edinburgh. made

various

arithmetic

a

which

notice

in

medallist

1809,

led

to

in

!806.

18i3 and

his

181g

James Hall

The expansometer,

Among

of ~%.

for

other

on geometry,

appointment

Philosophy at ~dinburgh University in 1819. Playiair,

philosopher

particularly

his

the Nature and Properties off Heat published

Rumford

books

received

to

the

things

in he

trigonometry

Chair

in

Natural

He was a colleague of John(?~

Alexander Nasmyth and David Brewster.

see Fig. 2, was made for display at Lesiie's lectures.

James Nasmyth,

p,109,

[i], acknowledged with pleasure Leslie's private

explanations to him ol matters in Dynamics and Mechanics.

2. The PremiumApprentice: or

than

trade',

a century

connnenced

his

pro~essional time

later

However,

and

journeymen.

sought

provoked

o! the ~actory', by

an

than

engineer

still

-

'a fee

was 1840s work

for

known

in the

engineering at

paid

a still

days

practice

who

had

instruction phenomenon when

and

entered

in

even

the

writer

many

older

industry

by

Maudslay had ceased to take on such apprentices by to

join him as many

ill

feeling

with

irregularly

of them had caused the

regular

him much

apprentices

and came wearing "gloves"'

and their

... they were very disturbing elements in the work

p. 121, [i]. army

the

with

were

'They attended

attendance was irregular

(premium S.O.E.D.)

association

Nasmyth

annoyance

1765,

engineers

virtue of it. the

Way_we were

premium apprentice

The profession more

The

colonel

working

at

an

ordnance

~actory

in the

middle of W.W. II ~or not presenting himself carrying gloves and a cane~

He

was a newly commnissioned Electrical and Mechanical Engineer.) The reader interested in socially useless conduct will find a lot of interesting 'classic'

discussion

and

absorbing

about book,

such

practices

in

the

Theory af the Leisure

now

forgotten,

Class;

once

An Ecanomic

Study aY Institutions by the American academic renegade Thorstein Veblen, !irst printed in 1899 but available in Mentor Books,

1953.

56 $. A_~_Improved Method of Y e l d i ~

Iron

Such is the title of a two-page entry, 1845

in his List

ol

Inventions

and

pp. 418-20,

Contrivances.

He

[i],

by Nasmyth in

finds

it necessary

first of all to carefully instruct his reader about the meaning ol the term clearly a new term for many.

weldin~-

in Fig.

81 he then

illustrates

the

With the aid o~ the ~our diagrams

wrong

final and perfect completion of welding'. malleable

iron possessing

hollow part, white

hot

firmly will

contain

effect

the right

way to secure

'a

(a) and

(b), first shuts up in the

scoriae or molten oxide of iron that clings tenaciously to the

metal

dislodge.

c o n c a v e form,

and

Hammering together two pieces oi

of

iron

the

surlaces'

impurity,

If now the uniting

two

from

which

pieces the

continued further

hanLmering

prepared

convex

outwards

and

are

centre

expulsion of any surface impurities,

hammering

see

(c)and

would

proceed

would

in form,

with

it

the

fail

to to

hammering sideways

(d).

This procedure Nasmyth gave to the Lords of the Admiralty when serving on a Co~smittee investigating the causes of failure in the deiecive welding of Anchors and Chain cables.

ACK]fOWLEDGJtENT The

author

wishes

to

thank

several drafts of this manuscript.

Heather

Johnson

for

typing

the

usual

57 REFEI~ENCES 1

S. Smiles

.fames Nasmyth, Engineer, an Autobiography, John Murray, London, 1885.

2

G.F. Charnock

Mechanical Technol ogy, Constable, 1916.

(Editor)

Biographical Memoirs of Fellows of the Royal Society, 1990, p. 9, Vol. 36, The Royal Society, London. 4

D. Ojertson

The Newton Handbook, Routledge and Kegan Paul, London,

5

S. Timoshenko

History of the Strength of Materials, McGraw Hill Publishing Co., 1953.

W. J ohnson

The Magnus Effect: Early Investigation and a Question of Pri ori ty, I.J,M.S., Ill, pp. 859-8?2, 1986.

C. Truesdell

Essays in the History of Mechanics, Springer, N.Y., 1968.

7.

1986.

8. W. Johnson

Some Neglected Researchers in the Field of Metal Forki oK, CAD/CAM and Y.E.M. in Metal Forming: Keynote Address: pp. 1-24, April 1987, Pergamon Press.

9. W. Johnson

Henri Tresca as the Originator of Adiabatic Heat Lines, I.J.M.S., 29, pp. 301-310, 1987.

i0. W. Johnson

Benjamin Robins: 18th Century Founder of Scientific Ballistics: Some European Dimensions, lot. European Solid Mechanics Conference, Munich, Sept., 1991, Plenary Lecture. 1 . J . I m p a c t Eng. ( I n Press)

ii. C. Babbage

Reflections on the Decline of Science and Some Causes (in England), 1830.

12. A. Hyman

Charl es Babbage, Princeton University Press,

13. J.A. Cantrell

James Nasmyth and the Steam Hammer, The Newcomen Society, pp. 133-138, Vol. 5F, 1985/6.

1982.

]4. W.C. Andrews Large Hammers and their Foundation, and J.H.A. Crockett The Structural Engineer, Oct. 1945, pp. 453-492. 15. L. Reti

(Ed.)

The Unknown Leonardo. Hutchinson, London, 1974.

16. R. Hill

Mathematical Theory of Plasticity, O.U.P., 1950, pp. 355.

17. W. Johnson and P.B. Mellor

Engi neeri ng Pl ast i cit 7, Ellis Horwood and Co., pp. 646, 1973.

58 18. J. Heyman

Coulomb's Memoir on Statics, C.U.P. , 1972.

19. W. Johnson, R. Sowerby and R.D. Venter

Bibliography o f Plane Sg.rain Slip Line Fields for Metal Deformation Processes, A Source Book and Bibliography, Pergamon Press, pp. 364.

20. W. Johnson

Overestimates of Load for some 2-d Forging Operations, Proc. 3rd. U.S. Cong. of Applied Mechanics, pp,571-579, 1958.

2!, W. Johnson

The Cutting of Round ~ire with Knife-edge and F1 a t-face Tool s, App. Sci. Rec. Series A, pp. 65-87, 1957.

22. W. Johnson and H. Kudo

Cutting of Metal Strips Between Partly RoughKnife Tools, I.J.M.S., ~, pp. 224-230, 1961.

23. W. Johnson and F.U. Mahtab

Mechanics of Cutting Strip with Knife and Flatface Die~ l. JoMachine Tool Des. & Res., ~, pp. 335-356, 1962.

24. V. Johnson

Indentation and Forging and iVasmyth's Hammer, The Engineer, 205, pp. 348-350, 1958.

25. S. Yang

Research into the GFM forgin 8 process, Jnl. Matls. Proc. Tech., 28, 1991, pp. 307-319.

26. C. Codron

Procedes Forgeage dams L 'Industri e, Michel, Paris, 1923, pp. 566, 2nd, edit.

27. G.L. Baraya, R.A.C. Slater and %4. Johnson

On Heat Lines or Lines ol Thermal Discontinuity, I.J.M.S., ~t, pp. 409-414, i964.

28. E. France

A History of Oorton and Openshaw Logoprint, Manchester, 1989.

29. H. Chaloner and A.E. Musson

industry and Technology, Vista Books, 1963.

(Manchester),

1982,

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

59

Modeling dynamic strain localization in inelastic solids M. Predeleanu

Laboratoire de M6canique et Technologie, ENS de C a c h a n / C N R S / U n i v e r s i t 6 Paris 6, 61 avenue du Pr6sident Wilson, 94230 Cachan, France

Abstract

The paper reviews the main results and underlines certain fundamental aspects of dynamic strain localization modeling for inelastic solids, including the thermo-mechanical coupling. Some important points requiring further study are mentioned.

1 INTRODUCTION

Modeling localization p h e n o m e n a represents a challenge not only for mechanics of deformable media but more recently for other branches of physics concerned with the propagation of waves which could be trapped by the presence of a disorder. Indeed, Anderson's experience in 1958 on the dynamics of quantic electrons [1] has given way to experimental and theoretical investigations in optics, acoustics and electromagnetics. Firstly, this phenomenon has been observed in manufacturing operations. Under severe loading conditions the deformation of a solid localizes in certain zones, often in the form of narrow bands, where the thermal energy also accumulates increasing the temperature eventually to the point of phase change in the material. Henry Tresca [2] reported in 1878 the apparition of heat lines in the form of a white cross during hot forging in a platinum bar and later in 1922 and independently by Massey [3] (see for additional references Johnson's paper [4]). Zener and Hollomon [5] observed such white bands during punching operations in a steel plate and proposed a simplified analysis. Commonly, these strain localization bands are termed shear bands, although whithin them not only shearing but a more complex deformation regime is present. When it is assumed that the material is non-conductor, then the shear

60 b a n d s are called adiabatic shear bands (see review paper by Rogers [6]). Generally, the shear bands are planar and their width can only reach a few microns. This intensive concentration of deformation and heat induces high gradients of certain kinematical and dynamical fields. That has led to model the shear bands as singular surfaces across which these fields suffer jump discontinuities. The localization zones can also have a spherical aspect as reported by Pr6mmer [7] for the explosive compaction operation of tungsten powder. It has been observed that the converging shock waves concentrate the deformation in a very small point wise region (0,12 mm diameter) where the temperature reaches the melting point of the material (-- 3000 ~ C). The localization phenomena can occur inside the body but also at the boundary, which in general induces more complex effects. Considering the loading conditions, we can distinguish: i) Quasi-static localization which appears during slow processes for which inertia effects may be neglected. In certain cases, the strain-induced thermal effects may also be ignored and consequently the material behaviour is defined in isothermal regime. ii) Dynamic localization occurs during high-energy rate processes such as explosive forming operations, high-speed machining, punching, ballistic impact and power laser. In this case, phase-changes can appear in material structure whithin the localization zones called transformed zones or transformed bands. Otherwise, it is a mere deformation band. In both cases, within the localization zone, the material suffers important micro-structural modifications generally leading to failure. Consequently two deformation regimes take up within the body, one within the localized zone and the other outside it. This fact requires the eventual use of two different constitutive models for material behaviour, which obviously induce an additional difficulty in analysis. The experimental studies indicate that the nucleation and evolution of the localized zones present a great variety of patterns, depending on the loading conditions, material behaviour and body geometry. So, the localized zones once initiated can develop without traversing the material (non-propagating or stationary bands) or they propagate inside the solid (propagating or PortevinLe Chatelier's type bands). Geometrical and kinematical characteristics of the shear bands such as width, spating and propagating speed can be measured. Modeling localization phenomena to predict all the aspects mentioned above represents a formidable if not impossible task at this point in time. Most studies published up to now introduce simplifying assumptions such as : quasi-static approximation by ignoring inertia effects, adiabacity hypothesis for heat transfer, one-dimensional analysis and simplified constitutive models for material behaviour. Two kinds of approaches have been followed in localization modeling: One, which aims to characterize the material sensitivity to the formation of localization zones, by using field equations inside the body but ignoring boundary conditions (material type analysis). The purpose of such analysis is to find the necessary a n d / o r sufficient conditions (criteria) for the existence of shear bands.

61

The second approach aims to capture the localized zones by solving a mixed initial-boundary value problem. The parametric studies of the analytical solutions (when these can be obtained !) or numerical simulations also permit to discuss the constitutive sensitivity to localization effects. The present paper reviews the main results and discusses certain fondamental aspects of dynamic strain localization modeling for inelastic solids, including the thermo-mechanical coupling. The governing equations are presented in Section 1 for inelastic materials with internal state variables, by using a Lagrangian formulation. The well-posedness conditions for a mixed initial-boundary value problem are defined in Section 2 and some pathological cases are cited such as : non-existence of a global solution, blow-up of a local solution, hyperbolicity loss of the governing equations. Section 4 is concerned with the use of singular surfaces in modeling localized bands, either as stationary discontinuity solutions introduced and developed by Hadamard [8], Hill [9], Mandel [10], or as possible bifurcated solutions in respect to the homogeneous ones, as proposed by Rice [11]. This approach has been extensively employed for the last decade, for the case of quasi-static strain localization in hyperelasticity and time-independent plasticity. Necessary and sufficient conditions have been formulated for strain localization inside the body [12] and at the boundary [13]. Mesh-dependence of the numerical solutions in the presence of a strain localization in such materials has multiplied the studies in this field. New specific regularization procedures have been proposed to avoid these difficulties by i n t r o d u c i n g non-local constittttive models [14], [15], second gradient deformation theory [16] - [18] or viscoplastic models [19], [20]. The study of localization problems by examining the stability of the solutions in Liapunov's sense is presented in Section 5. Currently, the linear perturbation method is used for the sake of simplicity in the case of applications. That way, the linear or infinitesimal stability is established, which obviously does not imply general Liapunov's stability. Other limitations and critical aspects of the method are also mentioned. In Section 6 some numerical results are reviewed : they are intended to capture the dynamical strain localization including thermal effects. However, a lot of work remains to be done in this field to establish the most convenient numerical strategies and elucidate the sort of mesh dependence in terms of constitutive models, material characteristic length and stress level.

2 - GOVERNING

EQUATIONS

Let us consider a deformable body B, in which a material point M occupies the position X in a reference configuration (30 in the three-dimensional Euclidian

62 space E and the position x in the configuration Ct at time t. po and p respectively denote the mass densities of the body in the configuration Go and Ct. The motion of the body with respect to Go is described by a mapping X, defined by x = % (X, t) on Go (B) x I, where I is a time interval. The deformation gradient relative to X is noted by F = 3 X / 3 X with det F ~ 0 and the particle velocity is noted by v = )~- x where a superposed dot denotes the material derivative with respect to the current time holding X fixed. The thermo-mechanical behaviour of the body can be generally defined, in terms of F, temperature field 8, temperature gradient g = 38/3X and a n u m b e r of internal state variables ot, by means of the constitutive relations w

= ~ (F, O, g, or) m

n = n (F, 0, g, ~) P = P (F, 0, g, 0t)

(1)

q =q (F, 0, g, o0 ot = ot (F, 0, g, or) where ~ is the specific free energy, q the entropy, P first Piola-Kirchhoff's stress tensor and q the heat flux. From Clausius-Duhem's inequality 9

1

-~,-n 6 + - - p / : - ~

1

PO Po 0 we can deduce [21] that

gq>0

(2)

~ = ~ (F, 0,~) q = -3V ~ (F, 0, ~)

P = Po ~

(3)

(F, 0, cz)

i.e. the material behaviour is defined by k n o w i n g the free energy ~ , the constitutive relation for heat flux q and the evolution equations for the internal state variables or. The internal variables, which can be scalar, vector or tensorial functions are introduced to describe the inelastic properties of the solid (plasticity, viscoplasticity, hardening, softening, etc...) and eventually thermal properties. The balance equations in the Lagrangian formulation are

63

p det F = go (mass balance equation) div P + Po b = Po 4, ( m o m e n t u m balance equation)

(4)

Po ~ + Po (f~ T1 + 0 rl) - P f: + div q = Po r (energy equation) w h e r e b denotes the body forces per unit mass and r the heat sources per unit mass. The E u l e r i a n f o r m u l a t i o n of the d y n a m i c c o u p l e d t h e r m o - m e c h a n i c a l p r o b l e m can be obtained from the equations (1) - (4), by using the mechanical and thermal fields defined on the current configuration as for instance the rate strain tensor D = sym (1h F -1) and Cauchy's stress tensor ~ = 1 / d e t F PF T. To the equations (1) - (4) one must add the initial conditions at t = to and the b o u n d a r y conditions on the boundary of the body. Two kinds of problems can be posed : i) Initial value problems (or Cauchy's problem) for which the solution is required at t e I w h e n initial data at to < t are known, for X e E and regularity conditions at oo are imposed. ii) Mixed i n i t i a l - b o u n d a r y value p r o b l e m , for w h i c h the solution verifies : a) initial conditions at t = to and b) b o u n d a r y conditions on the b o u n d a r y 3B e x p r e s s e d in terms of the d i s p l a c e m e n t s or velocities on one part 3Bu of the b o u n d a r y and in terms of the tractions on 3 B o : the t e m p e r a t u r e on 3B 0 and the heat flux on 3Bq with 3BuU3B o = 3B and 0BoU3Bc~ = 0B. It is w o r t h n o t i n g that for certain c o n s t i t u t i v e relations, the s y s t e m of e q u a t i o n s (1) - (4) is a first order quasilinear s y s t e m of partial differential equations, written in the matrix form A1 (W, X, t) Wt + A2 (W, X, t) Wx + A3 (W, X, t) = 0

(5)

where W (X,t) is a column vector grouping all u n k n o w n variables of the problem and Wt and Wx are respectively the temporal and spatial derivatives of W.

3 - WELL-POSEDNESS

OF THE PROBLEM

A mixed initial b o u n d a r y value problem is well-posed in H a d a m a r d ' s sense [22], if the data of the problem are such t h a t : i) A solution exists and is unique and ii) The solution continuously depends on the data. Othervise the problem is said to be improperly-posed. At this time, the question of the existence and unicity of the solution of the d y n a m i c coupled thermo-mechanical problem in three-dimensional cases is still

64 open, even for particular classes of inelastic material models. For the first order quasilinear system in one space dimension it has been proved that Cauchy's problem has a unique solt~tion if the system is hyperbolic and regularity conditions of Lipschitz's type are verified by the matrix coefficients of the system and the initial data [23], [24]. Ne\'ertheless, these conditions do not ensure a global solution. The following pathological cases will be cited : a) Recent studies [25], [26] have s h o w n that for r a t e - d e p e n d e n t materials exhibiting thermal softening, the solution of the dynamic shearing problem can be local and in this case it considered in [25] and

blor~,s-llp

in finite time. c~ = g (0, 7) ~' has been

b = G ~ + h (0, ~)

(6)

in [26], where c~ is the shear stress, 7 the shear strain, ;[ shear rate strain, G , elastic shear modulus, H and h material functions. b) The change of type of go\'erning partial differential equations can lead to a i m p r o p e r l y - p o s e d problem and that is associated with the occurrence of a localized m o d e of the soltttion. Thus,in [27], [28], considering the onedimensional constituti\'e relation (s = F (7), the solution is analysed when the tangent m o d u l u s F' (7) goes to zero. In this case the speed of propagation of a disturbance vanishes, and that is associated with a trat~pi~ly, of the wave, which accumulates deformation in localized zones. For elastic-plastic solids with nonassociative flow rules it has been sho~vn [29] that a loss of hyperbolicity of the governing equations system could occur. Also for isothermal regime, in [30] it is shown that for rate-dependent solids characterized by g = al (E, S) IZ + a2 (E,S) where S = F-ip is the second Piola-Kirchhoft's stress tensor, E = 1/2 (FTF-1) is the L a g r a n g i a n strain tensor, and al, a2 are material fonctions, the g o v e r n i n g equation system may loss hyperbolicity for large strains and stresses. Exemples can also be given for quasi-static problems when the governing system may lose ellipticity and the problem becomes improperly-posed.

4 - SINGULAR

SURFACES

9W A V E S

AND

STATIONARY

DISCONTINUITIES

The solution of an initial-boundary' value problem for a solid subjected to dynamic loading or rapid structural modifications (apparition and growth of cavities and cracks) may present singular surfaces S, across which certain field variables a n d / o r their derivatives ha\,e jump discontinuities. Let f (X, t) = 0 be the Lagrangian representation of the regular surface S and U = - i : / I g r a d f I denotes the speed of propagation of ~.

65 A singular surface which traverses the material is n a m e d wave. A nonpropagating singular surface, always consisting of the same particles is called a material surface or a stationary discontinuity. For a material singular surface, represented by f (X) = 0, the speed of propagation is zero. The existence of stationary discontinuity was associated which the m e c h a n i s m of the strain localization phenomena. A singular surface S is of the first order with respect to the motion Z (X, t),if Z is continuous across S but at least one of their first derivatives F and v and the t e m p e r a t u r e 0 suffer jump discontinuities across S. The propagating singular surfaces of the first order are shock waves, across which the normal velocity is d i s c o n t i n u o u s and vortex sheets, across which the tangential velocity is discontinuous. Across a material surface of the first order, the velocity v is continuous, but at least one c o m p o n e n t of the d e f o r m a t i o n g r a d i e n t F is discontinuous. A singular surface S is of the second order with respect to the motion Z (X, t), if v, F, 0, o~ are c o n t i n u o u s while their first d e r i v a t i v e s m a y exibit j u m p discontinuities across S. The propagating singular surfaces of the second order are called acceleration waves. On a material singular surface of the second order, the acceleration Z and the velocity gradient are continuous, but grad F is discontinuous. When f} and grad 0 are continuous the acceleration wave is called homothernlal [31], [32]. The existence and nature of the singular surfaces for a particular problem are d e d u c e d by using the geometric and kinematic compatibility conditions [31] [x]=-Ua, [ F ] = a | [x]=U2a, [1~ ] = - U a | 1 7 4 1 7 4

(7)

[0] = - U b, [grad 0] = bn where [~] denotes the jump of ~ across S, a and b are respectively the mechanical and thermal amplitudes of the wave, U is the speed of propagation and n the unit normal vector to S. Similar relations are added for internal parameters. The dynamic compatibility relations are deduced from the balance equations (4) using the relations (7). Using compatibility relations we deduce that the amplitude and the speed of propagation of the acceleration waves are completely determined by solving a homogeneous linear system of algebric equations of the form A8=O (8) where A is a linear map and 5 is a N - d i m e n s i o n a l vector g r o u p i n g the mechanical, thermal and eventually internal variable a m p l i t u d e s of the discontinuities. The necessary propagation condition is given by the following equation det A = G (U) = 0. (9)

66 It is to be noted that the map A depends on the propagation speed U as well as on the material properties and direction of propagation. The corresponding conditions for the existence of a material singular surface (stationary discontinuity) are obtained from (9) by putting U = 0, that is G ( U = 0 ) =0 (10) From a physical point of view only waves with real amplitude and real speed of propagation are significant. The acceleration waves in elastic-plastic materials for isothermal regimes have been studied in [33] - [39], [9], [11]. In this case the linear map A is expressed by A = Q - U 2 pl (11) where the second-order tensor Q is called the elasto-plastic acoustic tensor. For the rate-independent plastic material defined by 6 = H grad v, where H is the elasto-plastic m o d u l u s tensor we have Q = n H n . The analysis of the wave propagation is reduced in this case to a spectral problem : to determine the proper values and proper vectors of Q. The major symmetric conditions for H ensure that the cubic equation (9) has three real roots [9]. More generally the strong ellipticity condition for Q, i.e Q k,, )~k ~m > 0 for arbitrary non-vanishing vectors ~, and n, provides, for any given propagation direction n, positive squared speeds c o r r e s p o n d i n g to each real acoustical direction. The ellipticity condition is equivalent to the condition according to which the symmetric part of the acoustic tensor is positive-definite for every propagation direction [32]. The condition for possible material surfaces (stationary discontinuities) is obtained from (11) putting U = 0, i.e detQ=0 (12) It is w o r t h noticing that the above condition formally coincides with that for quasi-static strain localization, where the inertia terms are dropped. However, in this case, it is an acceleration wave (in quasi-static regim) for which the speed of propagation may not vanish. The study of the acceleration waves in inelastic solids with internal variables, under thermo-mechanical loadings has been considered in [40] -[47]. The results obtained show that the wave pattern is strongly influenced by the mechanical model considered but also depends strongly on the assumptions concerning the heat conduction. The simplest constitutive law for the thermal flux (Fourier's law) leads to infinite speed of propagation of the thermal disturbances. The analysis of the possible coupled acceleration waves shows that [f~] = [grad O] = 0. An alternative constitutive law for the heat conduction has been proposed in order to obtain a finite speed of propagation by Maxwell [48] and Cataneo [49], [50] in the form ~Cl +]3q =-k grad O (13) Generalized heat conduction relations in differential or integral forms have been considered in [51] - [54]. Also in [45], [55], thermal internal variables have been used to describe the heat conduction properties. In general, for the inelastic solids defined by constitutive relations of the form (1) an algebraic equation of the eighth degree has been obtained [45], [55] for the

67 p r o p a g a t i o n condition. It leads to non-symmetric acceleration waves mainly caused by thermal effects. In [45] it is shown that, for the solids characterized by (1) under the following hypothesis, all acceleration waves are real: m

m

1. The fonctions ~ and 0~ verify an inequality of elliptic type in Bowen's sense [56] in the neighbourhood of a strong equilibrium state [57] 2. The acceleration waves are symmetric 3. ~3g 1 (F, 0, g, o~) = 0

(14)

4.0t (F, 0, g, 0t) = bl (F, 0, 0t) g + b2 (F, 0, v.) (15) where bl et b2 are material functions. Additional conditions on the material characteristics are indicated to obtain four positive squared speeds of propagation. Necessary and sufficient conditions are given in [47] for the existence of real propagation speeds of acceleration waves in viscoplastic materials for Cataneo's heat conduction law. In quasi-static cases, conditions for the existence of a singular surface of the second order across which the velocity gradient exhibits j u m p discontinuities have been given for r a t e - i n d e p e n d e n t materials and Fourier's conduction law in [58] for finite strains. Such conditions were deduced also in [79] for more complex constitutives law for elasto-plastic materials, including kinematic hardening effects and the softening effect generated by d a m a g e process. Possible localization phenomena at the b o u n d a r y of the body and the thermo-mechanical coupling effects are discussed in [59] for small strains. It is worth noting that an alternative approach of the singular surfaces of the second order may be adopted by using the characteristic manifold theory [60]. It is known that the acceleration waves is a subset of the characteristics family.

5- INSTABILITY AND LOCALIZATION

The instability p h e n o m e n o n which can occur at different length scales of the material structure during a deformation process are considered to be the principal factor for the onset of strain localization. Among various definitions of stability, Liapunov's concept seems to be the most adapted to localization problems. It permits to appreciate the b o u n d n e s s in time of a p e r t u r b e d or bifurcation solution, or to estimate the tendency an imperfection has to localize the deformation and the heat energy. Let W (X, t) be all field variables which define an evolution of the body over time interval I and W* (X,t) a perturbed solution. It is said that the solution W is stable in Liapunov's sense [61] if, for each positive number e, there exists a positive number 8 such that

d o [W (x, t0)- W*(X, to)] < 8 for any W*, implies

(16)

68

d [W (X, t) - W* (X, t)] < E, (17) where d,, denotes a measure of the distnricr between initial disturbances and d a measure which defines the departure o f W' from W over the entire time interval I. The use of two metrics d,, and d has been introduced by Movchan [62] but currently in applications d,, is considered equal to d . An evolution which is not stable is called instable. If the time interval I is semi-infinite, then W is asymptotically stable if there exists 6 > 0 such that d [W (X, t) - W* ( X , t)] -+ 0 as t 4 00 for each W' which satisfies d,, [W (X, to) - W*(X, t,,)] < 6. I t is worth noticing that to characterize an evolution W as stable, it is necessary to verify the relations (16) - (17) for n n y departure W'. On the contrary , to characterize an evolution as instable it is sufficient to show that there exists l i t /cost oiic disturbance which does not verify the stability criterion. Secondly, the diagnosis of dynamic stability or instability depends o n the choise of the measures d,, and d. Schield and Green [63] and Koiter [64] have given various exemples of evolutions which are stable with respect to certain measures but instable with respect to others. The application of the above stability criterion to non-linear problems is not easy.Instead, the linear perturbation method is used, by linearizing the governing equations for small perturbations about a basic solution (which may be for instance, a space homogeneous solution). I t is obvious that the linear or infinitesimal stability as defined do not imply the general stability concerning the non-linear evolutions. The studies devoted to localizatiotl phenomena by using a linear stability approach are concerned with : a ) The analysis of the sensitivity o f certain inelastic materials, defined by their constitutive relations and balance equations in exhibiting stability or intability phenomena. The boundness of small perturbations of the progressive plane wave type W' = W,, exy [i 5 x.11 + ct], W,, = const, W,, n = 0 (18) is examined determining the necessary conditions on the material characteristics for the onset of shear bands. The boundary conditions are ignored. In 1651 viscoplastic materials are considered exhibiting strain and strain-rate hardening, pressure hardening and thermal softening. The analysis is concerned with two and three-dimensional flows, taking thermo-mechanical couplings into account. b) The analysis of linear stability of the solutions and the localized modes for the mixed initial-boundary value problems. Most studies are concerned with one and bidimensional analysis, i n p a r t i c u l a r ivith the simple shearing problem for materials governed by the cnnstitiiti\,e relation : 0 = p (0) ym 'y" [66] - [721 (see also for reference the review paper b y 8ai [73]).I t has been deduced, by using a linear perturbation method, that instability does not always imply localization. The relative linear perturbation analysis used in [69], permits to establish more edifying relations concerning both phenomena.

69 A complete analysis of the dynamic shearing problem for elasto-viscoplastic solids, defined by (6) and including thermo-mechanical coupling is given in [26]. The existence, uniqueness and continuous d e p e n d e n c e of the solution are proved. The stability analysis is performed by studying the behaviour of the perturbed disturbances represented by Fourier series. In order to point out the mechanism of strain and thermal localization, the following stability parameter is introduced

P(t) = c~~ H0 (O~ c~~ + [H (O~(t), c~~ "c~ Hc~ (O~ c~~ - 4H0/O~ c;~ (O~ c~~ where H(O, c~) is the plastic component of the shear rate, c~~, O~ the homogeneous solution Ho, Hc~ are the deri\'ati\'e of tlne fttnction H in respect to O and c~. The comparison of P(t) with a qt~antit\' d, depending of material constants, the thickness of the layer and the \'elocit)' of the upper edge of the layer, permits to deduce the following conclusions regarding the beha\'iour of the perturbation" 1. If P(t) ___d for all t ~ [T1, T2], the n o n - h o m o g e n e o u s p e r t u r b a t i o n of the h o m o g e n e o u s solution will not increase on [TI, T2] and strain localiztion does not occur during the time inter\'al [TI, T2]. 2. If there exists n >_ 1 such that n2d < P(t) < (n + 1)2d for all t e [T1, T2] then we can expect that the strain xx'ill be localized in the points where ]cos krrx[ takes its maximum value, that is, at the point Xi.k = i/ k \vith 0 < i _< k and 1 < k <_ n. If n is small then we can expect that the strain will be localized in some special zones. For instance for n = 1 the strain can be localized at the boundary and for n = 2 at the b o u n d a r y or in the middle. If n is large then there exists no special zones where the strain will be localized and the localization of the strain will depend on the shape of the perturbation at t = T1. The above analysis performed on the time interval [T1, T2] is significant for the behaviour of the solution on [0, T] if the [T1, T2] is sufficiently large. Numerical simulations are in agreement with qualitative results obtained by linear stability analysis.

6- NUMERICAL STUDIES Many basic aspects concerning the apparition and evolution of localization p h e n o m e n a eventually till failure are expected from c o m p u t e r numerical simulations. These will permit to abandon the simplifications as adopted in most approaches, such as 9one-dimensional analysis, adiabacity hypothesis, quasi-static a p p r o x i m a t i o n , simplified constitutive models, simplified failure criteria u n c o u p l e d with constitutive relations via damage theory. Obviously, more complete numerical studies require theoretical developments concerning the existence, unicity and stability of the solution but also computational strategies well a d a p t e d for capturing narrow localized zones which eliminate mesh dependence effects.

70 There are very few numerical solutions to the fully coupled two or threedimensional thermo-mechanical problems including inertia forces. Recently Batra and Liu [74], [75] have studied the dynamic plane deformation problem for a block subjected to shearing or compressive loadings on the top and bottom faces. The material behaviour is described by a thermally softening viscoplastic model defined by c~ = - p (p) t + 21.1D, 2, = ~]3i; (1- v0)(1 + r

(19)

1 ID =~ tr D '2, p(p)= K(p--- 1)

where o 0 is the yield in simple tension or compression, v is the coefficient of thermal softening, r and m are the strain rate sensitivity parameters, ~c represents the volumetric behaviour modulus, D' is the deviator of rate strain deformation tensor D, 6 Cauchy's tensor. Fourier's law of heat conduction is used but the solution is obtained only for adiabatic regime. The finite element solution is obtained by using the updated Lagrangian formulation, which permits to solve the severe distorsion of the mesh within the strain localization zone more easily. The material inhomogeneity is modeled by introducing a p e r t u r b a t i o n of temperature at the center of the block or assuming a weak material characterized by a reduced flow stress. The numerical simulations show that a shear band initiates from the region of the defect and propagates in the direction of maximum shearing stress. In [76] the same problem is considered for a bimetallic body containing an elliptical void. The sites where the shear bands initiate are described along with their propagation directions. Though the inertia effects are ignored, the papers by Lemonds and Needelman [77] and Anand et al [78] present interesting results concerning the influence of various features of the material behaviour on the initiation and growth of the strain localization zones. There also appears in [77] that heat conduction delays the apparition of shear band localization and influences the shear band thickness, in terms of isotropic and kinematic hardening characteristics. To avoid the pathological aspects in numerical t r e a t e m e n t w h e n rateindependent elastic-plastic solids are considered (ill-posedness of initial- value problems, mesh-sensitivity), Loret et Prevost [20], have proposed a visco-plasticity regularisation procedure for one-dimensional problems, by introducing a new parameter (relaxation time) into constitutive model. In [79], these procedure is extended to three-dimensional problems and the authors can capture the shear bands during the dynamic deformation of a elastic-plastic regular parallelipiped.

7 - CONCLUDING REMARKS The present paper has underlined some features in the phenomenological m o d e l i n g of the thermo-mechanical localization u n d e r high-rate loading

71 conditions. The review of the main results in this field shows that important advances have been brought in by the use of the discontinuity surface theory, linear stability analysis or numerical simulations. However, most studies are based on simplifying assumptions for the sake of simplicity that lead to partial or non clarifying conclusions. Secondly, many questions remain open, either from a mathematical point of view, or concerning the physical background used in modeling localization phenomena. Here are some of them : a. Generally, strain localization is associated with a softening of the material, in particular thermal softening, which, included in the constitutive model, can lead to a improperly posed problem. In this case, examples have indicated a possible non-existence of a global solution, a blow-up of a local solution and a bifurcation of solutions in localized modes which can be instable. In non-linear d y n a m i c elasticity and in r a t e - i n d e p e n d e n t plasticity, general equivalence theoremes establish connections for isothermal regimes between the existence of real waves, the uniqueness and the stability of the solutions. Such m a t h e m a t i c a l results do not exist for general t h e r m o - m e c h a n i c a l problems defined in section 2. b. As it has been shown, the use of second order singular surfaces has permitted to characterize the sensitivity of the material model to localization and also to mathematically characterize the governing equation system. The first order singular surface theory has been less used until now. But it could bring new advances concerning the jump discontinuities of velocity and thermal fields (for the case of generalized conduction heat models). c. In localization problem analysis, the stability has been approached essentially by using the linear perturbation method. But linear or infinitesimal stability does not i m p l y the general (non-linear) stability. The latter d e m a n d s more complicated computational methods in applying general criteria of stability based either on a kinematic concept (of Liapunov's type) or on an energetic one. d. Previous analytical solutions and a few the influence of the various aspects of the and d e v e l o p m e n t of localization zones hardening, thermal softening, smoothness sensitivity. On the other hand, there remain to point c o n d u c t i o n models, tensile stress states, loadings.

numerical simulations have stressed material behaviour on the occurence such as : isotropic and kinematic of the yield surface and rate strain out the role of inertial forces, heat hydrostatic stress and multi-axial

e. A more complete analysis of tile late stage of the localization process requires the following i n g r e d i e n t s i) A constitutive characterization of tile material, differing wether it refers to the inside or the outside of tile localization region.

72

ii) A modeling of the e\ entiial phase transition in the material. iii) A failure criterion, defined, for Instance, via coupled d a m a g e modeling.

8 - REFERENCES

1 P.W. Anderson,Phys. Re\,., 109 (1958)1492. 2 H. Tresca, Proc. Inst. Mech. Engrs 30 (1878) 301. 3 H.F. Massey, Proc. Manchester Assoc. Engrs (1921) 21. 4 W. Johnson, Int. J. Mech. Sci. 29 (1987) no 5 301. 5 C. Zener arid J.H.Hollonwn, J. Applied Physics 15 (1944)22. 6 H.C. Rogers,Ann. Rc\.. X1atc.r. Scl. 9 (1979) 283. 7 R. Priinimer, In. Esplosi\.r IVtJlding, Forming a n d Compaction, T.Z. Blazynski (ed), Applied Science I'iiblishers, London, (1983) 369. 8 J. Hadamard, Leqons sur Li propagaticm des ondes et les equations d e I ' h y d r o d yn a m iq i ie , Pa L'i 5, 1O(13. 9 R. Hill, J. Mech. Phys. Solids, 10 [1%2)1. 10 J. Mandel, I n Rheology a n d Soil h4echanics, J . Kravtchenko a n d P.M. Sirieys(eds), Springer (1966) 58 11 J.R. Rice, In Theoretical a n d Applied Mechanics, W.T. Koiter (ed) NorthHolland, Amsterdam, (1976) 207. 12 G. Borre a n d G. Maier, hleccanica 24 (1989) 36. 13 A. Benallal, R.Rillardon et G. Gt~!,nionat,C.R. Acad. Sci. Paris 310 (1990), serie I1 679. 14 Z.P. Ehzant, T.R. F;elyt~chkaa n d T.P. Chang, J . E I I ~ I I h4e~h. ~. ASCE 110 (1984) 1666. 15 G. Pijattdier-Cabot and Z.1'. I h a n t , J. Engng. Mech. ASCE 113 ( 1987)15 12. 16 H.L. Schreyer a n d Z . Chen, I n ConntitLiti\,e Equations: Macro a n d Computational Aspects, K.J. I\Jilliam (ed) ASME, New-York, (1984) 193. 17 N. Triantafyllidis and E.C. Aifantis, J. Elasticity 16 (1986) 225. 18 D. Lasry and T. Belytschko, Int. J . Solids Structures 24 (1988) 581. 19 A. Needleman, Comp. hleth. Appl. Mrch. Eng. 67 (1988) 69. 20 B. Loret a n d J.H. Prevost, Comp. Meth. Appl. Eng. 83 (1990) 247. 21 B.D. Coleman a n d M.E. Gurtin, J. Chem.Phys. 47 (1967) 597. 22 J. Hadamard, Lectures on Cauchy's problem in linear partial differential equations, IVover, New-York 1952. 23 P.D. Lax, Corn. Pure Appl. k t h . 6 (1953) 231. 24 A. Jeffrey, @uasilinear hy~wrL7oIicsystems a n d waves, Pitman Publishing, London 1976. 25 A.E. Tzavaras, Arch. Rational Mech. a n d Anal. 99 (1987) 349. 26 I.R. Ionescu and M. Predeleanii, Quart. J. Mech. Appl. Math. 46 (1993), 437 27 D.C. Erlich, D.R. Curran a n d L.. Seaman, Further development of a Cornpiitationall Shear Rand hlodel SRI R e p . AMMRC TR 80-3 (1980). 28 F.H. W u a n d L.B. Freuncl, J. blech. Phys. Solids 32 (1984) 119.

73

29 B. Loret, J.H. Prevost and 0.Harireche, Eur. J . Mech. A/Solids 9 (1990) 225. 30 I. Suliciu, Meccanica 19 (1984) 38. 31 C. Truesdell and R.E. Toupin, The classical field theories Handbuch der Physit III/l, Springer Verlag, Berlin, 1960. 32 C. Truesdell and W. Noll, The nonlinear field theories of mechanics, Handbuch der Physik 111/3, Springer Verlag, Berlin, 1965. 33 T.Y. Thomas, J. Rat. Mech. Anal. 7 (1958) 893. 34 J.W. Craggs, In Progress in Solid Mechanics vol. 1I.N. Sneddon and R. Hill (eds), North-Holland Publ. Comp.,Amsterdam, (1961) 143. 35 J. Mandel, J. Mecanique, 1 (1962)3. 36 T. Tokuoka, Mem. Fac. Engng, Kyoto Univ. 33 (1971) 193. 37 T. Tokuoka, Int. J. Non-Linear Mech. 13(1978) 199. 38 M.M. Balaban, A.E. Green and P.M. Naghdi, Int. J. Engng Sci. 8 (1970) 315. 39 J. Mandel, In Mechanical Waves in Solids, J. Mandel, L. Brim (eds) Springer-Verlag, Berlin (1976) 40 W. Kosinski and P.Perzyna, Arch. Mech. 24 (1972)629 41 Y. Tokouka, J. Engng. Math. 8 (1974) 9. 42 M. Piau, J. MPcanique 14 (1975)l. 43 I. Suliciu, Arch. Mech. 27 (1975)841. 44 W. Kosinski, Arch. Mech. 27 (1975) 733. 45 M. Mihailescu and I. Suliciu, Int. J. Solids Structures 12 (1976)559. 46 V. Kukudzanov, Arch. Mech. 29 (1977) 325. 47 K. Woloszynska, Arch. Mech. 33 (1981) 261, 451. 48 J.C. Max\vell, Phil. Trans. Roy. Soc. London 157 (1867) 49. 49 C. Cattaneo, Atti. del Sem. Mntem. Fisico, Univ. di Modena 3 (1948) 83. 50 C. Cattaneo, C. R. Acad. Sc. Paris, 247 (1958) 431. 51 M.E. Gurtin and A.C. Pipkin, Arch. Rat. Mech. Anal. 31 (1968) 113. 52 N. Fox, Int. J. Engng Sci. 7 (1969) 437. 53 D.B. Bogy and P.M. Naghdi, J. Math. Phys. 11 (1970) 917 54 I. Muller, Arch. Rat. Mech. Anal. 40 (1971) 55 W. Kosinski, Field Singularities and Wave Analysis in Continuum Mechanics, PWN, Warszawa (1986). 56 R.M. Bowen, J. Chem. Phys. 49 (1968) 1625. 57 C. Truesdel, Rational Thermod~~iiarnics Mc Graw-Hill, New-York, 1969. 58 M.K. Duszek and P. Perzyna, Int. J. Solids Structures 27 (1991) 1419. 59 A. Benallal, C. R. Acad. Sci. Paris, 312 (1991) 117. 60 R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience Publishers, New-York, 1966. 61 A.M. Liap~inov,ProblPine g6116ral de la stabilite d u mouvement, Memoires de 1'Acad. Imp. Sci. St Petersburg VIII, ser. 22, n"5. 62 A.A. Movchan, Appl. Math. Mech. 24(1960) 988. 63 R.T. Shield and A.E. Green, Arch. Rat. Mech. Anal. 12 (1963) 354. 64 W.T. Koiter, Over de stabilitrit \'an het elastish evenivicht. Thesis, Delft, 1945. 65 L. Anand, K.H. Kim a n d T.G. Sha\vki, J. Mech. Phys. Solids 35 (1987)407.

74

66 R.J. Clifton, Report to the NRC Committee on Material Responses to Ultrasonic Loading Rates, 1978. 67 Y.L. Bai, J. Mech. Phys. Solids 30 (1982) 195. 68 T.J. Burns and T.G. Trucano, Mech. Mat. 1 (1982) 195. 69 C. Fressengeas and A. Molinari, J. Mech. Phys. Solids 35 (1987) 185. 70 T.W. Wright and J.M. Walter, J. Mech. Phys. Solids 35 (1987) 701. 71 C. Fressengeas, J. Physique 49 (1988) C3-278. 72 H.TZ. Chen, A.S. Douglas and R. Malek-Madani, Quart. App1. Math. XLVII (1989) 247. 73 Y. Bai, Res. Mechanica 31 (1990) 133. 74 R.C. Batra and De-Shin Liu, J. Appl. Mech. 56 (1989) 527. 75 R.C. Batra and De-Shin Liu, Int. J. Plasticity 6 (1990) 231. 76 R.C. Batra and De-Shin Liu, Int. J. Solids Structures 27 (1991) 1829. 77 J. Lemonds and A. Needleman, Mech. Mat. 5 (1986) 339, 363. 78 L. Anand,A.M. Lush and K.H. Kim, In Thermal Aspects in Manufacturing, M.H. Attia, L. Knops (eds), PED- vol. 30, A.S.M.E., NewYork, 1988. 79 M.K. Duszek, P. Perzyna and E. Stein, Int. J. of Plasticity 8 (1992), 361. 80 O. Harireche and B. Loret, Eur. J. Mech., A./Solids 11 (1992), 733.

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

75

VOID GROWTH UNDER TRIAXIAL STRESS STATE AND ITS INFLUENCE ON SHEET METAL FORMING LIMITS R.

C.

CHATURVEDI

DEPARTMENT OF M E C H A N I C A L E N G I N E E R I N G INDIAN INSTITUTE OF T E C H N O L O G Y , P O k l A I ,

BOMBAY

-

4.00076,

INDIA

ABSTRACT The presence of voids in sheet meta 1 has been e x p e r i m e n t a l ly c o n f i r m e d [2]. Increase in s i z e of t h e s e v o i d s with plastic strains has been e x p e r i m e n t a l ly confirmed by Parmar and Me[ for [2] and modeled by R i c e and Tracey [12]. Bridgman [14] h a s s h o w n t h a t the s t r e s s s t a t e in the n e c k of a b i a x i a l ly s t r e t c h e d sheet becomes triaxial and analyzed the stress state. R a o [17] u s e d the I i n e a r i s e d v e r s i o n of R i c e a n d Tracey's void growth equation to develop void growth and coalescence model and incorporated the triaxial stress state for p r e d i c t i n g its i n f l u e n c e on f o r m i n g limits. Padwal et al. [22,23] further extended the analysis to cover both the I inearised and non-linearised versions and gave an analysis for

plastic

instability

as

wel I

as

the

prediction

of

forming

I imits. It is observed that the use of non-I inearised model predicts a much sharper growth of void fraction especially when the magnitude of relative hydrostatic stress component is large. The forming limits predicted at t h e e q u i b i a x i a l strain end are appreciably lower compared to t h o s e f r o m M.K. models. It is a l s o o b s e r v e d t h a t an i n c r e a s e in v o i d f r a c t i o n , as well as i n c r e a s e in v o i d s i z e at c o n s t a n t void fraction reduces the forming I imits. This model also predicts a dependence of the forming I i m i t s on s h e e t thickness and this is m o r e p r o n o u n c e d compared to t h a t f r o m the m o d e l of R a o a n d C h a t u r v e d i [10].

1. NOTATION a

C

- Half - Stress

1

-

D1 C

vo

,

C

v

the

sheet

concentration

Volume g r o w t h

-

thickness

Initial

and

at

the

center

of

the

factor.

rate.

current

volume

fraction

of

voids.

neck.

76

-

f

Equivalent Harciniak inhomogeneity index.

- Marciniak inhomogeneity index due to voids.

f V

- Characteristic width

‘0

(initial) of inhomogeneity.

m

-

Strain rate sensitivity.

n

-

Strain hardening exponent.

R

-

Radius of curvature at the neck.

r3

- Thickness dimension of the void.

Ri

- Radial velocity

to* t

-

Initial and current thickness of the sheet.

vo v

-

Initial and current volume of the voids.

2

- Distance of the element from the center of the neck.

a

-

Stress ratio ( a / u

-

Relative increase of a void volume.

I

of void surface in the i direction.

2

1

).

-

Principal strains.

_E.E

-

Equivalent strain and strain rate.

E.

-

Strain rate field far from the void. - Pre-strain.

E

0

c1

-

Constant

P

-

Strain ratio ( % / e l ) .

ul,u2,u3

-

Principal stresses.

0 , am

-

Equivalent and mean stress.

U

-

Superimposed hydrostatic stress.

-

h

-

U

T

0

(9 ) (v)

in the expression for Dl.

Variable tensile stress across the neck.

-

Yield stress in shear.

-

Subscript denoting values in the necking band.

- Superscript denoting values related to voids.

77

2. INTRODUCTION A major contribution to the understanding of the onset of instability and prediction of FLD was made by Marciniak and (M-K) Cll. They introduced the concept of Kuczynski inhomogeneity factor and proposed a theory for predicting sheet This theory has been most widely metal forming I imits. accepted theory for predicting sheet metal forming limits. The inhomogeneity factor in M-K theory was taken to be in the form of a localized thickness reduction. However, inhomogeneity in actual material is very much different from the assumed thickness inhomogeneity. It is well known that the sheet in unstrained condition already contains some randomly located voids. The amount of voids or the so called void fraction (ratio of void voll~me to total volume under consideration) can be measured by scanning Parmar and Mellor 121 provided electron microscope ( S E M ) . experimental proof of void growth during plastic straining by measuring the relative changes in density of a sheet specimen subjected to plastic strain. They also provide an analysis of the phenomenon of coalescence of neighbouring voids in the shape of parallel long cylindrical holes under biaxial stretching. According to M-K analysis, when the material deforms, the localization of strain and formation of neck starts at the particular band, which has the lowest effective thickness due to the largest Concentration of voids. Since the void growth depends on strain, the rates of growth of void fraction inside and outside the band become different due to different strain states in these two regions. Due to this differential void growth, the effective inhomogeneity gets enhanced and gets superimposed o n the thickness inhomogeneity caused by the necking process. The localization of strain in the neck results in the continued growth of voids leading to their coalescence and finally to fracture. A number of attempts C3-101 have been made to express the assumed thickness inhomogeneity in M - K model in terms of Al 1 inhomogeneity caused by variation in void concentration. these studies showed that the overestimation of I imit strains near the equibiaxial tension state, using M - K analysis C l l , is significantly reduced in analyses considering the void growth. The growth of a void under biaxial stresses has been modeled previously by McClintock C111, and Rice and Tracey 1123. McClintock studied the growth of a cylindrical hole in a hardening plastic matrix, while, Rice and Tracey studied the growth of a spherical void in a non-hardening plastic matrix. Since voids inside a material are more likely to have a shape nearer the spherical than the cylindrical, the Rice and Tracey model is nearer the correct state than the McClintock model. Schmitt and Jalinier 1 1 3 1 proposed linearization of the Rice and Tracey model and obtained a simple analytical expres6ion for the change in volume fraction due to the growth of a void under biaxial stress as,

C

V

= C

vo

exp (1.92 ( 1 + p)

el)

(1)

Rao and Chaturvedi 1 1 0 1 used this void growth equation and proposed the simple statistical analysis of voId distribution to define the M-K material inhomogeneity as, fV

-

= 1

3

11’2

(2)

~ t ( 1 - c ~ ~ ) ~

[ r3 cvo

with the initial void fraction distribut.ion of voids as,

in

the

neck

due

to

random

Once the process of straining starts, the volume fraction of voids, both inside and outside the necking band, increases A t this stage the according to Eqns ( 1 ) and (3) respectively. inhomogeneity is contributed both by physical thickness inhomogeneity due to necking process as we1 I as thickness inhomogeneity due to the differential void fraction levels inside and outside the necking band, and is as given below, f =

g/t C(1

- CVB 1

/

(1

-

Cv)1

(4)

Schmitt and Jalfnier 1131 had assumed a biaxial stress state in the neck while proposing Eqn ( 1 ) f o r the change i n volume fraction due to the growth of a void. However, the stress state in the neck which is the critical region for C l 4 1 and that is an important instability is triaxial Bridgman C141 Shortcoming of the analyses so far discussed. has analyzed the triaxial stress state in the neck of a sheet undergoing plastic deformation in plane strain and shown that the stresses vary across the neck (Fig. t . ) following the equation : ox =

v1

[

1

+

log C 1 + (a/2R).( 1 - 2’

/

a2 f

1

]

(5)

This implies the presence of stresses along the thickness direction, and the stress state bacomes triaxial. This triaxial stress state in the neck can be determined b y swperimposing a hydrostatic tension component on the applied G i a k i a l stress field. Tt,is component. varies over the thickness of the neck, i t is ze113at the surface of the sheet ( 2 . a ) and attains its highest v a l ~ ~of, e Ch = C

1

log

c 1

t

a

/

2R

I

( 6 )

at. the center of the neck ( Z = O ) . The growth r a t e of voids wil I be influenced by the presence o f this hydrostatic stress component which -varies a c r n s 5 t h e thickness o t sheet. A s explained e a r i ier, the hydrostatic tensile stress, E q n ( 5 ) , varies a c r o s s the n e c k (Fig. 11. The average value of this tensile stress can b e calculated by Cl41,

79

I' Figure 1. Cross section of the inhomogeneity in the sheet.

a

av

= l/a S

a

dz

x

0

(7)

which gives, U

x av

= el Cf

where C

is a correction factor, given by.

f

C f = ( 1 + 2 R / a )'I2

log Cl+a/R + ( 2 a / R

(lta/2R )ll2]-

1

(8)

and o,is the major principal stress i n the neck, calculated disregarding the presence of three dimensional stress state. Thus during the necking process stress in the neck gets enhanced by this correction factor, due to the presence of three dimensional stress state. Based on the experimental observation of Bridgman 1141 and Hecker 1151, Ran and Chaturvedi C161 proposed the following simple relation for calculating the function a/R in Eq. ( 8 1 , a/R = K

-

1

(

tB/t

They also proposed that the factor K could constant and calculated using the relation,

K = 2

(

t

0

/

lo

1

2

(9)

be

considered

a

(101

This implies that K depends on the thickness t o of the sheet and width of inhomogeneity 1, ( F i g . 1). The value l o can be

80 considered a characteristic of the material dependent method of production but independent of the thickness. Thus i t is clear from the above discussion that the of voids in the neck considering the effect of triaxial state will b e different than that assuming plane condition in the neck.

on

the

growth stress stress

3. MODELING OF THE VOID GROWTH The change in volume and shape of the cavity initial radius R o e is given b y Rice and Tracey 1133 :

Ri

/

Ro =

E.

C1

+

D1

with

-

an

(11)

E

C 4 E i ia a deviatoric term ass?ciated w i t h the change in shape D,E is a spherical term associated with no volumetric growth.

D,

with the isotropic volume expansion. fol lowing expression C 1 2 1 :

D l = 0 . 5 5 8 sh

+

0.008 y ch

is

given

by

the

[$*]

o = ( a + o + o )/3 m 1 2 3

where, and

=

y

-35/(E1-E3)’

with

the

rule

f

1

> = E 21= € 3

Considering the presence of hydrostatic tensile stress ( o h ) superimposed on a biaxial stress state the new values of principal stresses become.

and a; =

(13)

0

h

Hence, a

m

= a /3C1+at3(ffh/0)l 1 1

Using Von

-

-

(14)

Mises yield function we get,

1/2 a = o [ 1 - a + a 2 1 1

(15)

81 and 7

2

0

112

(16)

= a C ( l - a + a ) / 3 1 1

Substituting a,,,and becomes, D 1 = 0.558 sh Cpl

t

in equation

T~

r

0,008

( 1 2 ) the expression

f o r D1

(17)

ch Cpl

where,

5

P =

c 1 - a + a2 Ill2

substituting

a = ' e w e 2 +P

get,

and -3p

(18)

for - 0 . 5 < = p < = 1

y = 2+p

In the expression for D, i f the second part which is negligible is not considered then we get,

Dl = 0.558 sh Cp1

(19)

Schmitt and Jalinier 1131 have linearieed the expression f o r Dq and arrived at the following relationship, D1 = 0 . 6 4 Cpl

(20)

In this paper both the linearised and

non-linearised forms of

D 1 are used. The radial deformation o f the void is given by, E"

-

(211

= C1 E. + D1 E

Substituting the value of €"

=

c1

€. t 1

0.64 C 1

t

p

and using

( 7 ) and ( 9 ) we get,

+ ( 2 t p ) (ah 1 0 1 1

The relative increase of void volume is, a

v

=

"ro=

exp (eV t 1

Using ( 1 1 ) we get,

5

t

1

E

1

122)

82 a

= exp

[ 1.92

I1

t

p +

( 2 t p ) (uh /

CY

1

11

E

(23)

1 1

When o ~ / # I = 0, the above expression reduces to the equation derived by Schmitt and Jalinier 1131 for the growth of a void in a plastic matrix under biaxial stress. Rao 1171 used the linearised form of D,, Eqn (20) and derived Eqn (23). For non-linsarised form of D,, Eqn (23) becomes,

a

= exp

[ 1.674

sh C ( 1 + p , ( 2 + p ) ( ~ ~ / # l ) ) /* A A1

where, A = C ( l + p t p2 )/311/2* 2 The above expression indicates growth is strain path dependent.

that

*

the

(24)

el]

rate

of

void

4. PLASTIC INSTABILITY The growth of a void in the neck of a sheet under superimposed hydrostatic tension, Eqn (241, is used to study its influence on the sheet metal forming limits using the M - K analysis incorporating the correction for triaxial stress state in the neck, Eq. ( 8 1 , and void growth equivalent of Marciniak inhomogeneity index, Eq. (4). I f the number of voids are considered to be constant i.e. the nucleation of new voids is ignored, the growth of a void fraction, as derived b y Schmitt and Jalinier C131, using non-linear form of Rice and Tracey C121 equation is, C v = C v o exp C1.674 sh C(l+p)/Al

*

A

*

ell

(251

where, A = C(l+pp2)/311/2 * 2 The growth of voids outside the groove will follow this rule. However, because of the superimposed hydrostatic tension due to triaxial stress state in the neck, the void growth in the neck will follow,

as derived earlier, Eqn (24). The equivalent flarciniak inhomogeneity index, given by Eqn (41, has been incorporated in the M-K model C11, instead of conventional physical thickness inhomogeneity and the equation describing the process of groove formation is derived as,

where, B = 314 Cp2/C1+ptpzll assuming isotropic material. The solution 118.191 of Eqn ( 2 7 ) enables determination of the limiting strain, when the function d G d z B a p p r o a c h e s zero.

83

5. RESULTS AND DISCUSSIot(( The changes in relative void volume with strains are given in Figs 2 and 3 for the equibiaxial and uniaxial tension path respectively, as a function of the relative hydrostatic stress, for both linearised ( Eqn 23) and complete (Eqn 2 4 ) forms of Rice and Tracey's equation. The curve for a value of u h / u , = O also represents the Schmi t t and Jal inier I131 void growth it can be seen from these figures that the equation (Eqn I). relative void volume and their growth rate increases with higher values of superimposed relative hydrostatic tension. The differences are more pronounced in equibiaxial tension than in uniaxial tension. The linearised model predicts a slower growth and growth rate, compared to the actual model and the difference is more for larger magnitudes of superimposed relative hydrostatic tension. The void growth is faster in equibiaxial tension compared to uniaxial tension. The effect of hydrostatic tension in the neck at constant inhomogeneity factor is to enhance its load carrying capacity. The growth of voids reduces the magnitude of the factor and has limits. The analysis an opposite effect on the forming presented above is therefore essential to get even a qualitative comprehension of the influence of the triaxial s t r e s s state. Fig. 4 shows t h e predictions o f this analysis i n curve C. It can be observed that the predictions are higher in plane strain region and lower in equibiaxial region than those by the M-K analysis (curve BI. The M - K analysis (without the consideration of void growth), using the influence of triaxial stress state only (curve A ) C l l , predicts values which are higher than usual M-K model throughout the range. The contribution of the analysis, based on triaxial stress state alone (curve A ) , is to predict a variation of forming limits with sheet thickness as observed experimentally by Haberfield and Boyles C201 but not predicted by conventional M - K analysis. However, as the values predicted are higher than those predicted b y M-K analysis, throughout the range, it does fl-K analysis, vir. not remove the other limitations of predicting higher forming limits i n the equibiaxial region C18,211. The analysis, considering void growth under triaxial stress state, corrects for this trend too. The effect of initial void fraction on l i m i t strains is Lower levels of initial void fraction shown in Fig. 5 . increase the l i m i t strains significantly. The effect of initial void size, at constant void fraction, on forming limits is shown in Fig. 6. This shows that the forming limits are higher when void size is smaller. Fig. 7 shows the e f f e c t o f increase in sheet thickness on l i m i t strains for various strain ratios. I t can be seen that the l i m i t strains increase significantly, with increase in the I t can also be observed that the thickness of the sheet. predictions based on the present analysis show relatively faster increase i n forming limits, with higher sheet thickness, as compared to the void growth model of Rao and Chaturvedi

84

8.0

&

i,i d

0 b,

i

,

,, r

tm

6.0

C~ O ~rhlo"I

I~I > 4.0

0.00 0.25 o 0.50 Linearized model Non-linearized model 9

9

I,i n,"

N

IZ

2.0

0.o

_ I . . . .

0.2

o.o

i

i

0.4

I

0.6

I

o~8

MAJOR TRUE STRAIN

Figure 2. Influence of superimposed hydrostatic tension on the change in volume of a void ( ~ =1.0).

8.0

a'h/~

W

9

9 [3

d

0

b,

6.0

o >

0.00 0.25 0.50 Lineerized model Non-lineorized model

,1:3

Ld > 4.0 I--

laJ r

.,.,.,J..13

2.0

. ~ ~ ~ : ~ 0.0 0.0

O.2

-~~''-~'4~-~'~

0.4

,

0.6

0.8

MAJOR TRUE STRAIN Fiqure 3. Influence of superimposed i n v o l u m e of a void ((~ = - 0 . 5 ) .

hydrostatic tension on the change

85 1.0

Z os o9

0.8 /

Ld 0.6 OC F-Qf O 0.4

/

n=0.22 R=I.0 ~o=0.006 /

/

0.2

/

/ O0

0.0

I,

0.4

MINOR TRUE STRAIN

Figure 4. Comparison of different models. A. M-K model c o n s i d e r i n g B.

C.

(f=0.99,

M-K

M-K

model

toll o = 0 . 8 ) .

0.8

forming

Iimits

triaxial

stresses

predicted

using

[ 16]

(f=O. 99).

model considering void growth under s t r e s s e s (Cvo=O. O l , r 3 / t = O . O 0 1 2 5 , t o / l c = 0 . 8 ) .

triaxial

[lO]. This rate of increase in f o r m i n g limits, with thickness, p r e d i c t e d by the p r e s e n t a n a l y s i s appears to b e t t e r a g r e e m e n t w i t h the e x p e r i m e n t a l observation (Fig. H a b e r f i e l d and B o y l e s [20].

sheet be in 8) of

6. CONCLUSI ONS The consideration of void growth with the presence of triaxial s t r e s s s t a t e p r o v i d e s a t h e o r y w h i c h can e x p l a i n the i n c r e a s e in f o r m i n g l i m i t s w i t h i n c r e a s e d s h e e t t h i c k n e s s w h i c h is not d o n e by the c o n v e n t i o n a l M-K a n a l y s i s . The void model also provides a phenomenological basis for the inhomogeneity factor

in

M-K

analysis,

The

I inearised

version

of

Rice

and

Tracey's equation, though accurate in the absence of hydrostatic stress component, is not a d e q u a t e in its p r e s e n c e at a p p r e c i a b l e levels. T h e a n a l y s i s a l s o p r o v i d e s a good b a s i s for p r e d i c t i n g forming I imits e v e n t h o u g h in the c u r r e n t form it i s o n l y for an i s o t r o p i c m a t e r i a l .

86

o.5 | Cvo-O.O02 z n.,i-u)

0.4

/ o=0.01

0.3 I--

o

,/

/

/ ,/

0.2 /t

0.1

Cvo=O.05

/

/

OOooF1

I

I

0.4 0.2 0.3 MINOR TRUE STRAIN

0.1

0.5

igure 5. Effect of initial void fraction on forming limits. n=O.22,R= 1 ,f= 1 , C o = 0 . 0 0 6 , r 3 = 0 . 0 0 1 , t o / I o = 0 . 6 )

o.~ / Z

J

r3=0.0002 0.4

/

t-U')

ILl 0 . 3

/ 0- - ) 0.2

005

./ ,/ o.1

I-

o.o ~ - / 0.0

[

/ 0.1

0.2

!

0.3

I

MINOR TRUE STRAIN

0.4

igure 6. Effect of void size on forming limits. n - O . 2 2 , R = 1 ,f= 1 ,Co=O.O06,Cvo=O.O 1 , t o / I o = 0 . 6 )

0.5

8?

__Z < 1.2

PRESENT MODEL VOC MODEL (10) 1.0

I'-" (/) o3 o3 LIJ

z

....-.,,o'-"

.....

.

1.0

o3

0.5

0.4

0.5 0.0

o T l-ILl :3 ~: I--.-

0.31

0.0 0.0

F ;

i--T

0.4

;~176 1.2

0.8

to/Io

Figure 7. Effect of sheet thickness on limit strain for various strain ratios (n=O.22,R= 1 ,f= 1,Co=0.006,r3=0.001 ,Cvo=O.O02).

Z

1.2

1.0

I-.o3

(/) o3 ILl Z Y (O T" I--LLI Z3 eI--

0.61 0.47 03

0.28 0.4

0.0

0.0

0.4

1

o18

SHEET THICKNESS

I

~.2 (ram)

Figure 8. Effect of sheet thickness on limit strain for various strain ratios (Experimental data for EDD stabilized steel (20)).

88

REFERENCES 1.

Marciniak, Z., Kuczynski, K. and Pokora, T. : 'Influence limit of plastic properties of a material on the forming diagram for sheet metal in tension', Int. J . of Mechanical Sciences, Vol. 1 5 , pp. 7 8 9 - 8 0 5 , ( 1 9 7 3 ) .

2.

Parmar, A. and Mellor, P. B. : 'Growth of voids in biaxial Int. J. Mech. Sci. V o l . 2 2 , pp. 1 3 3 - 1 5 0 , stress fields', (1980).

3.

Needleman, A. and Triantafyllidia, N. : 'Void growth and local necking in biaxially stretched sheets', Trans. ASME. J. of Eng. Materials and Technology, Vol. 100, pp. 164- 169,

(

1978).

4.

Chu, C.C.and Needlewan, A. : 'Void nucleation effects i n of Eng. Trans. ASME. J . biaxially stretched sheets', Materials and Technology, Vol. 1 0 2 , pp. 2 4 9 , ( 1 9 8 0 ) .

5.

Gurson, A.L. : *Continuum theory of ductile rupture by void nucleation and growth - Part I - Yield criteria and flow rules for porous ductile materials', Trans. ASME J. of Eng. Materials and Technology, Vo1.99, p p . 2 - 1 5 , ( 1 9 7 7 ) .

6.

Jalinier, J.H. and Schmitt, J.H. : forming-11-Plastic instability", Vol. 30, pp. 1 7 9 9 - 1 8 0 9 . 1982).

7.

Kim, K.H. and Kim, D.W. : 'The effect of void growth on Int. J. of Mechanical the l i m i t strains of steel sheets', Sciences, Vol. 25, pp. 2 9 3 , ( 1 9 8 3 ) .

8.

Barata Da Rocha, A., Barlat. F. and Jalinier, J . M . : "Predictions of the forming l i m i t diagrams of anisotropic National Sci. sheets i n linear and non-linear loading', Eng., France, ( 1 9 8 4 ) .

9.

Tai, W.H. : 'Prediction of l i m i t strains in sheet metal Int. J . of Mechanical using a plastic damage model', Sciences, Vol. 30, No. 2 , pp. 1 1 9 - 1 2 6 , ( 1 9 8 8 ) .

10.

Rao, U . S . and Chaturvedi, R.C. : 'Sheet metal forming limits under complex strain paths using void growth and coalescence model', Trans. ASME, J. Eng. Materials Tech., V o l . 108, pp. 2 4 0 - 2 4 4 , ( 1 9 8 6 ) .

11.

McClintock, F.A. : 'A criterion for ductile ASME. J. the growth of holes', Trans. Mechan ics, pp. 3 6 3 - 3 7 1 , ( 1968 )

.

12.

damage i n sheet metal Acta Metallurgica,

fracture by of Applied

Rice, J . R . and Tracey, D.M. : 'On the ductile enlargement J. of Mechanics and of voids in triaxial stress fields', Physics of Solids, Vol. 1 7 , pp. 2 0 1 - 2 1 7 , ( 1 9 6 9 ) .

89

13.

Jalinier, J.H. and Schmitt. J.H. : 'Damage in sheet metal forming - I - Physical behauiour', Acta Metal lurgica, V O I .30, pp. 1789- 1798, ( 1982I .

14.

Bridgman, P.U. : Studies in large fracture, McGraw Hill. pp. 32, (19521.

15.

Hecker, S. S. : 'Experimental studies of sheet stretch-ability', Formability analysis - Modeling and experimentation, Proceedings of Symposium held in Chicago, Illinois, pp. 150, (19771.

16.

Rao, U . S . and Chaturvedi, R.C. : 'A new model for predicting forming limits for strain rate sensitive materials', Manufacturing Simulation and Processes, ASME, pp, 119-127, (19861.

17.

Rao, U.S. : 'Sheet metal forming limits under simple and complex strain paths", Ph.D. Thesis, 1. I.T., Bombay, ( 1985 I .

18.

Padwal, S.B. and Chaturvedi, R.C. : 'Prediction of sheet metal forming limits", Proceedings The 2nd International Conference on Automation Technology, Taipei, Taiwan, July 1992.

19.

Padwal, S.B. and Chaturvedi, R.C. : .Computer aided International determination of forming l i m i t diagram., Conference on CADICAM, Robotics, & Autonomous Factories, 1. I.T. New Delhi, India, pp. 527-538, (19931.

20.

Haberf ield, A.B. and Boyles, M.U. : 'Laboratory determined forming I imi t diagrams', Sheet Metal Industries, v o l . 50, pp. 400, (19731.

21.

Padwal, S.B. and Chaturvedi, R.C. : .Prediction of forming limits using Hosford's modified yield criterion', International Journal of Mechanical Sciences, V o l . 34, No. 7, pp. 541-547, (1992).

22.

Padwal, S.B., Chaturvedi, R.C., and Rao, U . S . : .Influence of superimposed hydrostatic tension on void growth in the neck of a metal sheet in biaxial stress fields. Part - I Modelling', Journal of Materials processing Technology, Val. 32, N O S . 1-2, pp. 91-98, (19921.

23.

Padwal, S.B., Chaturvedi, R.C., and Ran, U.S. : 'Influence of superimposed hydrostatic tension on void growth in the neck of a metal sheet in biaxial stress fields. Part - 1 1 - Plastic Instability", Journal of Materials processing Technology, Vol. 32, Nos. 1-2, pp. 99-107, (1992).

plastic

flow

and

This Page Intentionally Left Blank

Materials Processing Dcfects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

91

The Prediction of Necking and Wrinkles in Deep Drawing Processes Using the FEM DOEGE, E.; EL-DSOKI, T. and

SEIBERT,

D.

Institute for Metal Forming and Metal Forming Machine Tools, University of Hannover, Welfengarten 1A, D-30167 Hannover, Germany Abstract

Starting out from elementary analytical approaches, the authors discuss the main factors affecting failure by necking and wrinkles. To discuss necking, a large number of macroscopic criteria is evaluated in the light of recent results obtained with the Finite Element Method (FEM). The section on the prediction of necking closes with an evaluation of damage mechanics as a means to analyze failure. Parameters that influence wrinkling such as the blank holder force are discussed. Wrinkling in sheetmetal forming operation are considered either by an implicit FE-Code or an explicit FE-Code.

Introduction One of the main reasons for the FEM increasingly to attract the interest of the sheet metal working industry is that this numerical tool can indeed help to reduce the number of try-outs needed for die design. However, this requires criteria when analyzing FE-plots which allow to predict whether a deep drawing operation is feasible or not, necking and wrinkling representing the most important failure types.

2 2.1

Failure by Necking Analytical Approach

Generally spoken, failure by necking is said to take place when 9 the deep drawing ratio, i.e. ratio of blank diameter to punch diameter, is too large 9 the radii of the die are too small *The authors wish to express their appreciation to the "Deutsche Forschungsgemeinschaft (DFG)" for their financial support of the projects Do/75-2 and SFB 300/B5. Greatfully acknowledged are further the provision of the FE program ABAQUS from Hibbitt, Karlsson and Sorensen, Inc. and the successfull cooperation with the German agency ABACOM as well as the "Regionale Rechenzentrum fiir Niedersachsen (RRZN)"

92 9 the blankholder force is too high 9 lubrication is insufficient 9 the deep drawing gap, i.e. the gap between die and punch, is too small A simple equation first proposed by SIEBEL and PANKNIN [1] may help to understand this: Consider an axisymmetric cup, with the bottom already formed. The punch force Ft, which is in equilibrium with all forces acting on the cup, must be transmitted through its wall. If the punch force is larger than the transmittable force, then rupture will take place. Refection will show that for equilibrium conditions, the punch force is given by Ft -- Fid "~- Fbend"~- tPfric,die "~" Ffric,bh

,

(1)

Fid representing the ideal forming force, Fb~nd the bending force, Ffric,die the accumulated friction force between die radius and blank and Ffric,bh the friction force between blankholder and sheet. From the geometry and the yield behaviour of the cup one can deduce the load carrying capacity of its wall Fbt, the force at which bottom tearing will occur, and one can readily see that F, < Fbt

(2)

must hold as to avoid rupture. The research activities on the prediction of rupture following the analytical approach aim at improving the description of the terms in equation 1, extending them to general geometries and implementing them in fast PC runnable programs [2]. The main advantage of the analytical technique is its quickness in delivering results, while its main drawback is the lack of accuracy and the poor local resolution- for general part geometries, the method is not able to give stress and strain distributions in a sheet.

2.2

General R e m a r k s on the P r e d i c t i o n of N e c k i n g U s i n g the FEM

At the present stage, advantages and disadvantages of the FEM can be judged as opposite to the analytical approach: It is a slow method, yet though both hardware and software ~re becoming increasingly efficient, but it offers a good local resolution, giving realistic strain and stress distributions in the sheet. The accuracy of the FEM relies heavily on the knowledge of the boundary conditions one ha~-to prescribe. In particular, this involves the description of friction and yield behaviour which are both difficult to measure. Friction is a highly local phenomenon, depending on the lubrication conditions [3], the evolution of surface asperities during the forming operation and correct contact search, which in turn requires a shell element formulation which is able to incorporate the thickness in the contact algorithm. The yield behaviour is both history and stress state dependant - measuring the flowcurve for example by hydraulic bulging corresponding to a biaxial stress state will result in values 1 5 - 20% higher than those obtained by a uniaxial test [4]. Strictly spoken, one would

93 have to measure the full yield surface taking path dependancy into account - a very timeconsuming task. For the experimental determination of the boundary conditions, the approach chosen in this work is to measure the yield stress in a hydraulic bulging test and to perform simulations of the bulging test, where no friction develops. The friction is determined as the unknown quantity when simulating for example an Erichsen test and is adapted such as to give optimum agreement of punch force and strain distribution in both experiment and simulation. The friction parameter found thus is then also used for other geometries, when experimental data is not available. 2.3

Macroscopic

Fracture

Criteria

The term "macroscopic fracture criteria" was proposed by GROCHE [4, 5, 6] and implies criteria consisting of products, integrals and sums of macroscopic stresses and strains. To determine the value of this criteria at the onset of failure, both experiments and FE-simulations of hydraulic deep drawing processes, simple stretch and deep drawing operations were conducted. In the simulations, standard LEvY-MISES-plasticity was used, anisotropy effects taken into account through a quasi-isotropic flowcurve after SEYDEL

[7]. After determining characteristic values of the different criteria, their accuracy in predicting the critical punch stroke at which rupture would take place was investigated. It was found that the main factor affecting the accuracy is the mode in which failure takes place, whether under deep drawing or under stretching conditions. The deep drawing condition is characterized by a halt of the flange draw-in in spite of an increasing punch stroke, while deep drawing condition can be recognized by the monotonic flange draw-in punch stroke curve. The results are summarized in figures 1 and 2, indicating the deviation of the predicted punchstroke from the value determined experimentally. These results are confined to deep drawing cracks, which reveals a severe drawback of these criteria: One must know beforehand what type of crack will take place, i.e. whether failure will occur under deep drawing or stretch drawing conditions, [4, 6]. The equivalent MISES stress was judged best for the prediction of both deep drawing and stretch drawing cracks. It turned out, however, that the locus of maximum equivalent MISES stress does not necessarily coincide with the locus of failure in the sheet [4, 6]. At this stage, some remarks on implicit and explicit FE integration schemes seem appropriate. The results above mentioned were obtained using the implicit Finite Element Method. In industrial applications involving large models however, the explicit integration scheme is becoming increasingly important [8], as long as elastic springback prediction is not involved. In the explicit integration scheme, dynamic effects may superpose the solution and will be very noticeable especially in the stress distribution plots. Thus, the thickness strain and the sheet thickness distribution are currently the most widely spread variables used when evaluating a FE-simulation of a sheet metal forming process. In spite of its popularity, however, this kinematic criterion also has several shortcomings: There is no material-dependant critical sheet thickness reduction, since this parameter is operation-dependant. As an example, the reader may refer to the results of -

94

Figure 1" Errors in the prediction of the critical punch stroke using diverse instantaneous macromechanical fracture criteria, after GROCItE [5]

Figure 2: Errors in the prediction of the critical punch stroke using diverse integral macromechanical fracture criteria, after GROCIIE [5]

95 the INPRO group [9], where major strains of over 180% were obtained in the actual multi stage forming and simulation of an oil pan out of mild steel. Moreover, the thickness distribution may also indicate the wrong locus of failure, [6]. For two processes A and B, figures 3 and 4 show the equivalent plastic strain and thickness distribution, respectively. Both processes lead to fracture, process A under deep drawing conditions, process B under stretching conditions. From the diagrams 3 and 4, however, one would presume that only operation B is not feasible, whereas operation A is, which is not confirmed by the experimental findings. Moreover, knowing that process A leads to failure, one would erroneously deduce failure to take place at about 40ram from the center, which is near the die radius instead of the punch edge radius. Therefore, sheet thickness distribution and equivalent plastic strain must also be interpreted with great care and experience when attempting to predict failure.

EP

1.0 0.8 0.6 0.4

\

0.2

process A 0

20

40

60 blank diameter [mm]

Figure 3: Distribution of the equivalent plastic strain in an axisymmetric cup, [6]

2.4

Microscopic

Fracture

Criteria

The drawbacks of the macroscopic fracture criteria gave rise to the idea of applying the concepts of damage mechanics to sheet metal forming. Describing the evolution of an initially flawless material to a microcrack, damage mechanics bridges the fields of continuum mechanics dedicated to the study of perfectly homogeneous deformable bodies, and fracture mechanics, the focus of which is crack propagation [10]. This is done by describing the microscopic processes that precede ductile failure, which is generally attributed to the growth and coalescence of voids nucleating at rigid second phase particles [11]. Some micrographs taken with a light optical and scanning electron microscope can be seen in the figures 5 and 6. They show void formation in the necking area close to the rupture surface. As one can see, outside the necking area hardly any voids can be found. For a more detailed discussion, the reader may refer to [13]. One plasticity model to account for interior damage is the GURSON model [12], which was derived in an attempt to model a plastic material containing randomly dispersed

96 0.9 sheet thickness [mm] 0.6 0.4

f

process A =..-~

"\

process B

0.2

20

40

r / [mml

60

Figure 4: Sheet thickness at initial failure, [6]

Figure 5: Micrograph of a ruptured X5 Cr Ni 18 10 sheet (light optical microscope)

97

Figure 6: Micrographs of a ruptured X5 Cr Ni 18 10 sheet (scanning electron microscope) voids. Studying a unit cell large enough to be statistically representative and applying admissible velocity fields, the yield surface was derived as

q)~ + 2qlf cosh(F = (-~I

) - (1 + q3f 2)

(3)

In equation 3, q is the root of the second stress deviator, p is the hydrostatic pressure, k/ is the yield stress and f is the void volume fraction. When interpreting f geometrically as a fraction of void volume to matrix volume, one can say that for sheet metal forming, the damage variable f is small [13]. When f is equal to zero, the GURSON model abridges to standard LEVY-MISES plasticity. A suggestion how to extend the Gurson model to anisotropic matrix behaviour so that it is suitable for simulating sheet metal forming is sketched in [14]. To implement this constitutive model in a commercial FE package, an integration algorithm due to ARAVAS [15] was utilized. Documentation of uniaxial and hydrostatic tests performed on an eight-node brick element is presented in [14]. When applying the algorithm to shell elements that use the plane stress assumption, modifications of the method are needed since the out-of-plane component is not defined kinematically. These modifications are briefly outlined in [15]. Further modifications are needed when applying the algorithm to explicit FE schemes. When the elastic predictor is very large, i.e. 3q2p/(2kt) > 30, difficulties may arise with calculating the cosh term. As a modification, the authors chose a subincrementation following OWEN and HINTON [16] in order to avoid premature abortion of the iteration process of the Backward Euler algorithm. Figures 7 and 8 shows contour plots of the MISES equivalent stress and the damage variable of a large rectangular cup. Though the calculations were performed at a very high punch speed, the damage variable distribution is still very reasonable, the maximum indicating well the locus of necking, while the MIsEs equivalent stress distribution leaves ample room for speculation. Ergo, the damage variable works successfully as a pointer to the endangered area. Whether the damage variable will also work as a failure criterion, has to be analyzed in future work.

98

Figure 7: MISESequivalent stress distribution in a rectangular cup. For symmetry reasons, only one quarter of the cup was modelled

Figure 8: Damage variable distribution in a rectangular cup

99

3

Failure

by

Wrinkling

Apart from cracks, wrinkling represents another important kind of failure in the area of sheet metal forming. Two different types of wrinkles are known: 9 wrinkles of first order in the flange (figure 9) 9 wrinkles of second order in the free forming zone between the punch radius and the die radius While wrinkles in the flange can be suppressed by the blank holder force, this is not possible for the secondary wrinkles.

Drawing Conditions: 'drawing ratio' Blankholder Force Punch Geometry Punch Stroke

= 1.77 = 81 kN = 220 mm * 110 mm = 70 mm

Figure 9: Undeformed and deformed mesh for a rectangular box

3.1

General R e m a r k s on the A p p e a r i n g of Wrinkles

When using thin sheets for drawing a cup, the flange may start to wrinkle. This tendency can be explained by considering an axisymmetrical cup. Concentric circles move inward and attain smaller radii. This movement results in a pressure stress in circumferencial direction and a tension stress in radial direction. The sheet starts to wrinkle for a critical ratio of both stresses. Pressure due to the blank-holder can help suppress the wrinkles somewhat, but if the force increases too much, wrinkles may be replaced by necking.

3.2

T h e Blank-Holder-Force

As above mentioned the primary wrinkles can be suppressed by using a blank-holder during the deep-drawing process. SIEBEL [20, 21] was the first one who analyzed the connection between the occuring of wrinkles and the blank-holder-force on a theoretical

100 base.

Nearly the same investigation was made by GELEJI [22] in a more simple way. More complex mathematical relations were done by SENIOR [23], Yu and JOHNSON [24] as well as M E I E R a n d R E I S S N E R [25]. For the calculation of the blank-holder-force SIEBEL [20] suggested for rotational parts: 0.5Do] (~o - 1)2 + 100so

Pbh,Siebel "= (2...3) 10-3R~

P~ ~0 Do so

(4)

tensile strength forming limit ratio blank diameter initial blank thickness

While GELEJI [22] gave the relation Pbh,Geleji -" 0.02Rv0.2

/~.2 dp u Do

dp + 2u ] Do + dp + 2u

(5)

yield strength punch diameter gap between punch and die blank diameter

Both equations give nearly the same results. However practical investigations with a rigid blank-holder have shown, that wrinkles appear even if the upper limit of the force, calculated with one of the equations mentioned above, acts during the deep drawing process. The experience shows, that the force for suppressing wrinkles can be calculated by Pbh,exp -~ 1.5pbh,Geleji

(6)

For rectangular parts, SOMMER [26] suggests to calculate the needed force by Pbh,rec. -- k

k m

Ao/Ast

m

(ao) Ast-

1

Rm

(7)

parameter considering the thickness distribution in the flange parameter taking into account the workpiece geometry blank area/projected punch area

To which extent the blank-holder-force influences the success of the deep-drawing operation is illustrated in figure 10. The abscissa stands for the reduction ratio and the ordinate for the blank-holder-force. In the diagram there are three regions

101 9 region where wrinkling is expected 9 region where a successful draw is expected 9 region where necking is expected For a given reduction ratio there are two critical points. The first one is when wrinkling is eliminated and a successfull draw is expected. The second one is when necking is expected [27, 28]. The second region increases if either the friction between blank and die decreases or the friction between punch and blank increases. For a reduction ratio greater than the maximum ratio wrinkling and/or necking always occurs.

Figure 10: The domains of wrinkling and necking

3.3

Aspects

of

Stability

The wrinkling represents a so-called stability problem. The specimen under force deforms so that the new geometry is from the mathematical point of view a stable state of equilibrium [17]. This is characteristical for this kind of problems. By continous increase of the force the state of equilibrium is formally maintained, but at a certain time it becomes instable. At this critical point, even the smallest disturbance such as a non-centered point of application of force, inaccuracy due to manufacturing etc., will lead to instability. This holds for buckling of a bar as well as for wrinkling of sheet metals. The state of equilibrium is stable. The engineer's duty is to avoid a switching over to the stable equilibrium, since a drawing piece with such a geometry can not fulfill the requirements of the design nor its original function. 3.4

EULER's Formula

The wrinkling during sheet metal forming processes is similar to the mechanism of the buckling of a bar, as it was described by EULER when deriving his formula. This comparison is similar to the one of SIEBEL.

102 This process was simulated using the FE-package ABAQUS/Standard and ABAQUS/Explicit (figure 11). In order to reduce the needed CP-time, a plain strain condition was assumed. Another advantage of this assumption is that the discretisation of the model would not influence the results in a wrong way. The model in figure 11 was

Figure 11: Undeformed and deformed mesh for the buckling problem discretised using 8"100 linear elements. After a displacement of u = 21ram every code gives a different result: 9 for the implicit code the process will resemble an upsetting of the specimen, as it is well known from the forging process. 9 the explicit code shows the buckling of the model. For the engineer's point of view it suffices to know that wrinkling or buckling appears. The question of the quantity and the quality of the wrinkles is of a theoretical and academical nature. However it is possible to explain both results by the mathematical formulation of the used integration scheme [19, 29, 30]. For this reason it is also possible to gain the same results using an implicit code. Therefore imperfections have to be considered in the model: 9 geometrical imperfections, i.e. nonuniform sheet thickness 9 physical imperfections, i.e. nonuniform u

4

modulus, nonuniform yield stress

Summary

Failure by necking and wrinkling are two important types of failure in deep drawing which can be predicted using the Finite Element Method. After a brief survey on analytical methods, a large number of macroscopic failure criteria are reviewed in the section devoted to the study of necking. In the framework of continuum mechanics, the highest accuracy in predicting the critical punch stroke is attained with the equivalent MISES stress, which

103 however falls short of indicating the locus of necking. The section on necking closes with an evaluation of damage mechanics. Focussing particularly on the GURSON model, the void volume fraction is prooved to work successfully as a pointer to the endangered area, regardless of geometry and type of operation. Wrinkles in the flange can be suppressed by an adequately chosen blank holder force. The friction behaviour at punch/sheet and die/sheet as well as the sheet thickness influence the succeeding of the deep-drawing operation. In order to produce very thin cups, a subsequent and separate ironing operation usually follows. Wrinkles can be simulated by either an implicit FF_,-Code or an explicit FE-Code.

References [1] SIEBEL, E. and PANKNIN, W.: Ziehverfahren der Blechbearbeitung. Metallkunde 47 (1956) 4, pp. 207-212

[2]

DOEGE, E. and SCHULTE,E.: Design of Deep Drawn Components with Elementary Calculation Methods. In: PIETRZYK, M. and KUSIAK, M. (Eds.): Proc. of the 4th Int. Conf. on Metal Forming, Krak6w, Poland, Sept. 20-24, 1992. Journal of Materials Processing Technology, Vol. 34, pp. 439-448 (1992)

[3]

BOCHMANN, E. and DOEGE, E.: Friction as a Critical Phenomenon in the Simulation of Sheet Metal Forming. In: CHENOT, J.-L.; WOOD, R.D. and ZIENKIEWlCZ, O.C. (Eds.): Proc. 4th Int. Conf. on Numerical Methods in Industrial Forming Processes- NUMIFORM '92, pp. 415-420, A.A. Balkema/Rotterdam/Brookfield (1992)

[4] GROCHE, P.: Bruchkriterien fSr die Blechumformung. Dissertation, University of Hanover, Fortschritt-Berichte VDI, Reihe 2: Fertigungstechnik, Nr. 229, VDI Verlag Dfisseldorf (1991)

[5]

EL-DSOKI, T.; DOEGE, E. and GROCHE, P.: Prediction of Cracks in Sheet Metal Forming with FEM Simulations. Proc. of the Int. Conf. FE-Simulation of 3-D Sheet Metal Forming Processes in Automotive Industry, Zfirich. VDI-Berichte 894, VDIVerlag, Dfisseldorf ( 1991)

[6]

DOEGE, E. and EL-DSOKI, T.: Deep-Drawing Cracks - Stretching Cracks: Two Different Types of Cracks in Deep-Drawing Processes. In: GHOSH, S.K. and PREDELEANU, M. (Eds.): Proc. of the 2nd Int. Conf. on Material Processing Defects, Siegburg, Germany, July 1 - 3, 1992, special issue of Journal of Materials Processing Technology, Vol. 32, Nos. 1-2 (1992)

[7] SEYDEL,

M.: Numerische Simulation der Blechumformung unter besonderer Berficksichtigung der Anisotropie. Fortschritt-Berichte VDI, Reihe 2: Fertigungstechnik, Nr. 182, VDI Verlag Dfisseldorf (1989)

IS] TAYLOR, L.; CAO, J.; KARAFILLIS,A.P.

and BOYCE, M." Numerical Investigations of Sheet Metal Forming Processes. In: MAKINOUCHI, E.; NAKAMACHI,E.;

104

OI~ATE, E. and WAGONER,R.H. (Eds.): Proc. of the 2nd Int. Conf. NUMISHEET '93, Tokyo, Japan, pp. 161-172 (1993) [9] VON SItONING, K.-V.; SiJNKEL, R.; HILLMANN, M.; BLiJMEL, K.W. and WOLFING, A.: Mathematical Modelling Bridges the Gap between Material and Tooling. Proc. NUMISHEET '93, ibid, pp. 321 ft. (1993) [10] CHABOCIIE, J.L. and LEMAITRE, J.: Mechanics of Solid Materials. Cambridge University Press (1990) [11] TItOMASON, P.F.: Ductile Fracture of Metals. Pergamon Press (1990) [12] GURSON, A.L.: Plastic Flow and Fracture Behaviour of Ductile Metals Incorporating Void Nucleation, Growth and Interaction. Dissertation, Brown University (1975) [13] DOEGE, E. and Seibert, D.: On a Failure Criterion for Sheet Metal Forming in the Framework of Continuum Damage Mechanics. Int. J. of Damage Mechanics, in preparation [14] DOEGE, E.; EL-DSOKI, T. and SEIBERT, D.: Prediction of Necking and Wrinkles in Sheet Metal Forming. NUMISHEET '93, ibid, pp. 187-197 (1993) [15] ARAVAS, N.: On the Integration of a Certain Class of Pressure Dependant Plasticity Models. Int. J. of Numerical Methods in Engineering, Vol. 24, pp. 1395-1416 (1987) [16] OWEN, D.R.J and HINTON, E.: Finite Elements in Plasticity, Theory and Practice. Pinderidge Press Ltd., Swansea, UK, 2nd reprint, p. 253 (1986) [17] MOTZ, H.-D.: Ingenieur-Mechanik. VDI-Verlag Dfisseldorf (1991)

[18] SIMON,H.: RechnerunterstStzte Ziehteilauslegung mit elementaren Berechnungsmethoden. Fortschritt- Berichte VDI, Reihe 2: Fertigungstechnik, Nr. 188, VDI Verlag, Dfisseldorf (1990)

[19] NAGTEGAAL, J. C. and TAYLOR, L. M.: Comparision of implicit and explicit finite element methods for analysis of sheet metal forming problems. Proc. of the Int. Conf. FE-Simulation of 3-D Sheet Metal Forming Processes in Automotive Industry, Zfirich, ibid (1991)

[20] SIEBEL, E.: Der Niederhalterdruck beim Tiefziehen. Stahl und Eisen 74, pp. 155-158 (1954) [21] SIEBEL, E. and BEISSWANGER, H.: Tiefziehen. Carl Hanser Verlag, Mfinchen (1955) [22] GELEJI, A.: Bildsame Formung der Metalle in Rechnung und Versuch. Berlin: Akademie (1960) [23] SENIOR, B. W.: Flange Wrinkling in Deep-Drawing-Operations. J. Mechanics and Physics of Solids 4, pp. 235-246, (1956)

105

[24]

Yu, T. X. and JOHNSON, W.: The Buckling of Annular Plates in Relation to the Deep Drawing Process. Int. J. Mech. Sci. 3, pp. 175-188 (1982)

[251

MEIER, M. and REISSNER, J.: Instability of the Annular Ring as Deep-Drawn Flange under Real Conditions. Annals of the CIRP, Vol. 32/1, pp. 187-190 (1983)

[26] SOMMER,N.:

Niederhalterdruck und Gestaltung des Niederhalters beim Tiefziehen yon Feinblechen. Fortschritt- Berichte VDI, Reihe 2: Fertigungstechnik, Nr. 115, VDI Verlag, Dfisseldorf (1986)

[27] SCHEY,J.

A.: Tribology in Metalworking, Friction, Lubrication and Wear. In: American Society for Metals (1983)

[28] AVITZUR, B.: Handbook of Metal-Forming Processes. A Wiley-Interscience Publication (1983) [29] TEODOSIU, C. et.al.: Implicit versus Explicit Methods in the Simulation of Sheet Metal Forming. Proc. of the Int. Conf. FF_,-Simulation of 3-D Sheet Metal Forming Processes in Automotive Industry, Zfirich, ibid (1991) [30] MATTIASSON, K. et. al.: On the Use of Explicit Time Integration in Finite Element Simulation of Industrial Sheet Forming Processes. Proc. of the Int. Conf. FESimulation of 3-D Sheet Metal Forming Processes in Automotive Industry, Zfirich, ibid (1991)

This Page Intentionally Left Blank

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

107

C o n s t i t u t i v e m o d e l s for m i c r o v o i d n u c l e a t i o n , g r o w t h a n d c o a l e s c e n c e in elastoplasticity, finite e l e m e n t reference m o d e l l i n g J. Oudin, B. Bennani and P. Picart Laboratoire de GEnie Mrcanique, Unite de Recherche AssociEe au CNRS, Universit6 de Valenciennes et du Hainaut CambrEsis, B.P. 311, 59304 Valenciennes Cedex, France. 1. I N T R O D U C T I O N To enhance design, development and optimization of secure and efficient new modern metal forming processes, know-how, empirical rules and expensive experiments are not suited to industrial requirements. The main interest of numerical methods using accurate mechanical models, either in elastic, elasto-plastic or visco-plastic problems in a finite element framework, is to make easier the reliable design of new modern mechanical parts and structures. For those imperative reasons, more and more problems require to take into account material microstructure variables such as microvoid volume fractions in the different material matrixes involved. The basic aim is now to get the most efficient solution scheme for such problems in relation with non linear large strain finite element framework. Typically, from the most recent microscopic observations, damage occurrence involves four phases more or less linked. The first one is an accommodation phase of the material matrix in which high stress and strain gradients appear around second phase particles, inclusions and precipitates. The second one is a new microvoid nucleation phase, either due to rupture of second phase particles, inclusions or matrix or to decohesion of inclusion-matrix interface. In an obvious way, this nucleation phase depends on particles and inclusions shapes, stresses and their hydrostatic part [ 1]. The increase of microvoid volume fraction during nucleation has been related to effective plastic strain rate in the matrix, effective yield stress and macroscopic hydrostatic stress [2,3]. The third phase begins with the growth of previous nucleated microvoids, the corresponding variation of microvoid volume fraction can be observed by density, modulus of elasticity or microhardeness measurements. The modifications of the mechanical properties have been described in using state variable damage parameter for isotropic material and damage tensor for anisotropic one. The increase of microvoids and the corresponding loss of load capacity is clearly linked to triaxiality of the stress field [4]. The triaxiality rate and microvoid volume fraction are introduced into a specific yield function for porous material [5,6]. The fourth and ultimate phase is obviously the most critical phase, occurring coalescence of nucleated-extended microvoids and finally ductile fracture of material. This phase has been predicted either from critical dimension of microvoids {7], critical dilatency, critical energy [8] or intrinsic limit function [9]. The present paper describes a solution schema well fitted for finite element framework in large strain elasto-plastic problems with porous material. The constitutive model is based on an isotropic elasto-plastic potential with three material parameters [ 10,11 ] and the main phases of damage evolution, microvoid nucleation, growth and coalescence, are taken into account. Microvoid nucleation is related to effective plastic strain rate, microvoid growth to material strain rate and associated elasto-plastic potential and microvoid coalescence to effective plastic strain rate. As reference, this model has been implemented in ASTRID non linear farge strain finite element code. The related algorithms and useful program are described in detail to permit

108 implementation in any finite element framework and three levels of computations are achieved forward: patchwork test of three node elements, collar test and pipe bulging to check its good implementation and to enhance its interest. 2. MODEL AND COMPUTING ASPECTS 2.1. C o n s t i t u t i v e model The constitutive model is based on an isotropic elasto-plastic potential with three material parameters. According to the irreversible character of ductile plastic damage, the isotropic elasto-plastic potential D.ep is defined as follows Oef + 2 ql f cosh f~ep = o'--5-

q 2. O m. OM

.kIIl+q3f2) .

0 withOm>0 (la)

and 2 Oef

f2eP = ~ I + 2 ql f - "(1 + q3 f21~ = 0 with O m <- 0

(lb)

in which f is the state variable available for microvoided material, called microvoid volume fraction and defined by f = VA - VM VA

(2)

with VA elementary apparent volume of material and V M the corresponding matrix one. In relations (la, lb), oef={3/2 s:s} 1/2 is the effective stress, s is the deviatoric Cauchy stress tensor, O M is the effective matrix yield stress of the matrix material, ql, q2, q3 are the material parameters and O m is the hydrostatic stress related to macroscopic Cauchy stress tensor a. The constitutive model takes into account the three main phases of damage evolution, so, microvoid volume fraction rate is given by

/. = i~n + i~g + ~

(3)

where fn, i~g and fc are respectively the microvoid volume fraction rates due to nucleation, growth and coalescence.

109 Considering a random distribution of second phase particles, microvoid volume fraction rate due to nucleation is expressed by

fn = fN S N ~

exp

(4)

2SNZ

where fN is the nucleated microvoid volume fraction consistent in regard to inclusions volume fraction, Ss is the Gaussian standard deviation, EM is the effective plastic strain and eN is the mean effective plastic strain at incipient nucleation. For ferritic steel matrix containing Fe-Fe3 and Cu-SiO2 inclusions, the usual values are fN=0.04, SN--O.1 and EN=0.2. Microvoid volume fraction rate due to growth is expressed by the apparent volume change, ?g = VM ~/A = (1 -- f) tr D p VA

(5)

where tr DP is the first invariant of macroscopic plastic strain rate tensor. Finally, microvoid volume fraction rate due to coalescence depends on effective plastic strain rate and is expressed by t'c = f u - for kM = fE kM AI~r

with f > for

(6a)

and fc = 0

(6b)

with f < for-

In these last relations, fu is the ultimate microvoid volume fraction at ductile rupture, fer is the critical fraction at incipient coalescence, Aer is the additional strain increment leading to ductile rupture and f is the current microvoid volume fraction. Solution of the microvoided material problem is achieved as follows: at each load increment, the consistency condition ~ bf~ /gf~ b f ~ i" = 0 f~ep + ~o :(~ + Oo M (~m + Of

= ~ep = 0 leads to solve (7a)

110 Then, the plastic multiplier is expressed by f~q, 8~ep O: ,, 8fin, 80 3f2eP (1- f) i9o :I + (A N + fE)(1 _ f)OM ~f

o: 8f2q, ~"~ep :C e . 3~"~eP C)~C'~ePh 00 ~0" ~0" ~O M (1 -- f)o M

(7b) where C e is the material elastic constitutive tensor, OM is the effective matrix yield stress of the material, h is the strain derivative of matrix stress-plastic strain function g(EM), f is the current microvoid volume fraction and I is the second order identity tensor. The effective plastic strain rate is then calculated from dissipated power expressions in the matrix and the material, given by

EM --

o:D p (1- f) O M

(8)

with the plastic strain rate tensor defined by (9)

D p = ~, ~ n e P .

80 Finally, the material elasto-plastic constitutive tensor is e 8f2ep)T.c ~ 3 ~ c

ceP = C e _ 8f~p :C ~ 8f~p 9 8o 8o

~'p

-7oo c ~p

: ~~ 8t2q, c: ~o 9 _..___.Sleep(1- f) 8f2n ' :I +(A N + fE h ( 1 f )o M Of 3 0 ) ( i ' f )o M 8OM (10)

2.2. P r e d i c t o r / c o r r e c t o r

algorithm

The previous formulations are introduced into a large strain finite element framework whose flowchart, related to the authors code ASTRID, is shown in Figure 1. The specific elastoplastic algorithm, corresponding to an elastic prediction of Cauchy stress tensor and a correction of this tensor on the yield surface is described in detail in Figure 2.

111

Initial data

1

!

~"-if L~176 I

Compute the tangent stiffness matrix K T

...c oa'n,c- I

,l-.c ~

KT~Uin+l) =

B T C e B dV

f

KT~Uin+l

BT ~

dV

i 1 Solve the linear system to compute the nodal displacement increment Aun+ K T(Un+I) i i = AUn+l

g T ~n+l dV v

N T Fn + AFn+ 1 dS

!

Update the new nodal displacement vector i+l i i Un+1 = Un+1 + Aun+ 1 Update the loading force Wn+1 = Wn +/~Zn+ 1 -

!

Compute the strain increment A i A i ~+1 = B Un+1 !

Predictor/corrector algorithm I

Compute the material elasto-plastic constitutive tensor ( e~lf~p~T e~.ep

ODep Ce O~qep / ) % o : ~ --~--. " ~ -~V~- h(l_f) OM-

~[)ep a'-'~ (l-f) --j-d- "I + (AN + fe)(l_f) oM

I

Compute the residual n.ofil,_a.l force vector

,C ui.l"V

No

Figure 1. Finite element code ASTRID flowchart.

":S(3

112

Predictor/cormctoralgorithm ..... ,,,

I

Compute the Cauchy stress tensor after elastic stress predicu.'on O + 1 = O n + Ce: A~+I No, elastic loadin8 increment s, plastic loading increment

Compute plastic multiplier A~k+l

f2ep k ~ep

o b~p b f ~ C ~ igf~p ig~p hk : ;90 9 " " --k O0 ~0 ~o M (1-fk) OM ~.

ok t)flep

,bf~p 1-

:I+(A~fe)

fk)~

I Compute effective plastic strain and effective matrix yield stress

k+l k k k A~k+l eMn+ 1 = eMn+ 1 + Ae M = eMn+ 1 +

..k+l

"M = g ( ~ § !

o

k ~f2~p

/)a

k (1-f k) o M

l)

Compute the new microvoid volume fraction n+l = AN AEM + ( 1+fk)

I + fe AEM

3o ! Compute the Cauchy stress tensor k+ 1 k e ~,k +1 t)~')ep On+ 1 = On+ 1 - C A c}O"

No

Figure 2. Flowchart of the predictor/corrector algorithm.

:-~-o (i--'~')7M

113 3. S O M E

REFERENCE

RESULTS

The proposed predictor/corrector algorithm performances are enhanced through a validation example, a reference one and a typical one. Patch test, collar test and pipe bulging according to material and matrix properties are performed as defined in Table 1. _ ,

Table 1. Material and matrix properties. Continuous matrix Microvoided material Nucl~tion Growth Coalescence Modulus of elasticity E ql Poisson's ratio v fN q2 fu effective matrix yield stress CM SN q3 fcr I~N fi At;r 200,000 N.mm -2 15 0.3 0.04 1 1/ql 500(1 +eM) N.mm -2 0.1 2.25 0.2 0.2 0.1 0.25 200,000 N.mm -2 1.5 0.3 0.04 1 l/q1 500(1 +eM) N.mm -2 0.1 2.25 0.2 0.2 0.05 0.25 200,000 N.mm -2 1.5 0.3 0.04 1 l/q1 650(1 +CM~ N.mm -2 0.1 2.25 0.2 0.2 10 -4 0.25 ,,

.

Patch test _

.

.

.

.

.

.

.

,,

Collar test

,,,

Pipe bulging

3.1. P a t c h

,,,

test

Besides basic tests on each library element, it is interesting to look after patch of elements, such as the one shown on Figure 3.

Prescribed disolacements : Node 1 : Ur=13z--0 Node 2 : U z - - 0 Node 3 : U z = l mm Node 4 : Ur--0 and Uz=l mm

Figure 3. Patch test of three node elements - dimensions and boundary conditions. Three computations are achieved until 1 mm prescribed displacement, the first one only takes into account microvoid growth, the second one nucleation and growth, the third one nucleation growth and coalescence. Computed microvoid volume fractions are given on Figure 4 : monotonous variation of microvoid volume fraction versus effective plastic strain is, as expected, obtained from 0.1 to 0.198 (see curve (1)); the incidence of nucleation appears when the matrix strain reaches 0.2 mean nucleation strain and the microvoid volume fraction gradient increases from 0.195 to 0.262, the fraction obtained at 0.6 effective strain being now equal to 0.225 (see curve (2)).

114 When coalescence is considered, the previous nucleation-growth results are strongly modified when the fraction reaches 0.2 critical value, and consequently the fraction gradient becomes 1.49, the fraction at 0.6 effective plastic strain being equal to 0.349 (see curve (3)). An other attractive and engineering representation of this patch test is given on Figure 5, in term of axial load capacity : the basic modifications of the load-prescribed displacement curves are as visible as above, the axial load capacity decrease being well related to microvoid nucleation, growth, coalescence (see curves (1),(2),(3)); looking now to the results when considering a cavity free material, the loss of load capacity is exactly computed (see curve (4)).

Figure 4. Microvoid volume fraction versus effective plastic strain, ASTRID code computations: (1) growth, (2) nucleation and growth, (3) nucleation, growth and coalescence.

Figure 5. Axial loading versus prescribed axial displacement, ASTRID code computations: (1) growth, (2) nucleation and growth, (3) nucleation, growth and coalescence, (4) cavity free. 3.2. Collar test Collar test of material is one of the required experiments for the determination of intrinsic ductility curves. The numerical modelling of this problem is very useful especially if taking into account voided materials. The dimensions of the required specimen are often : for the cylinder height 21 mm and diameter 14 mm, for the collar thickness 3 mm and diameter 21 mm, for the cylinder-collar connecting surface radius of curvature 0.8 mm. The corresponding finite element mesh is built for a quarter of the specimen (Figure 6a) : 58 Q4 elements with 2x2 Gauss integration points for the main domain and 4 T2 elements with a single Gauss integration point, located at the upper right corner vicinity of the domain. The computation is achieved with 280 increments for a prescribed displacement of 8.4 mm which leads to a final

115 specimen height of 4.2 mm (Figure 6b). As expected, the maximum increases of microvoid volume fraction are predicted in the collar zone (Figure 7a) and a maximum of 0.289 located on the collar upper surface; this is related to path and positive values of the hydrostatic stress component and in the last increment, hydrostatic reaches in the maximum microvoid volume fraction zone 137 N.mm -2 compared to -1040 N.mm -2 in the close cylinder zone (Figure 7b).

B

Prescribed displacements upper face Ur---0 and Uz=8.4 mm

(a)

Figure 6. Collar test specimen, one fourth of the meridian plane" (a) initial mesh, (b) deformed mesh. The evolution of the microvoid volume fraction in the centre zone shows that microvoid nucleation and growth is not yet effective in element 1, microvoid nucleation and growth appears in element 2 located at the connecting zone at 40% height reduction (Figure 8). The coalescence is observed in element 3 located at the external part of the collar at 75% height reduction. A minimum

0 maximum

f

,A

------ A-

A B C N/ram 2 D E

- 1040 -483 -30 137 283

0 A

503 -2965

o

A B

(b)

? i CD ii,+D

0.066 0.117 0.193 0.236 0.263 0.278 0.289 0.050

BcD E

(a)

-----A

A B C D E F

r

m

_2 mm

Figure 7. Collar test, 80% height reduction, ASTRID code computations: (a) microvoid volume fraction, (b) hydrostatic stress.

116

Figure 8. Collar test, ASTRID code computations: (a) zoom of the centre zone, (b) microvoid volume fraction in 1, 2 and 3 elements. To complete the above analysis, it is useful to look after the collar dimensions all along the test: the thickness is going from 3 mm to 3.547 mm (+18%), the diameter from 21 mm to 32.8 mm (+56%). 12

]

Wi -

0.6

J

0.54'7/~

. ,..,q

~ ~ 6 0 ............................................................--/ 0

11.8]

I..... ! ~ 0 . 3 0~' - ................................................................................... --

4.0 8.0 8.4 0 (a) Uz prescribed axial displacement (mm)

4.0 (b)

8.0 8.4

Figure 9. Collar test, ASTRID code computations: (a) diameter increase of the collar versus prescribed axial displacement, (b) thickness increase of the collar versus prescribed axial displacement.

3.3. Pipe bulging Pipe bulging is one of the typical cold forging processes in which the material ductility limits are rather easily reached and therefore the development of new such cold forged parts has a strong handicap. The dimensions of the non-bulged pipe in the example are: length 470 mm, internal diameter 170 mm, external diameter 340 mm. The finite element mesh used here is made of 280 Q4 elements with 2x2 Gauss integration points (Figure 10a). The computations are performed on the above basis with a prescribed displacement of 70.5 mm on the upper and lower faces (Figure 10b), the pipe-punches contact being a sticking one, the pipe-shape die contact being a free-sliding one. The microvoid volume fraction reaches rather quickly 137x10.4 maximum value in the equatorial outer zone of the pipe (Figure 11a) coming from 10.4; the microvoid volume fraction r-gradient is also expressive, 4.3x10 -4 per mm. Hydrostatic stress map at the last step shows strong r and zgradients, the heterogeneity being increased by microvoid nucleation and growth whilst in the

117 main part of the pipe, the hydrostatic stress remains negative; the z-gradient is equal to 14-12 N.mm -1 in the upper and lower zones; the r-gradient in the centre zone is lower, from 6 N.mm: to 2.8 N.mm -1, going from the middle to the outer surface. On Figure 12, it is shown that microvoid volume fraction strongly increases in the centre zone as far as hydrostatic stress grows upper than zero and again increases when the effective plastic strain becomes greater than 0.2 the mean nucleation value.

Figure 10. Pipe bulging: (a) initial mesh, (b) 30% height reduction deformed mesh.

N.mm -2 A B C

A

0.0012 0.0046 0.0092

--

0.0137 0.0001

,

V.

0j

~-~0

t~ D"C -190 ''0 70 r 0 A,

.

(a)

A

.

(b)

-1180

-1507 137

minimum 50 mm

Figure 11. Pipe bulging, 30% height reduction, ASTRID code computations" (a) microvoid volume fraction, (b) hydrostatic stress.

118

Figure 12. Pipe bulging, centre zone outer surface vicinity, ASTRID code computations: (1) hydrostatic stress, (2) microvoid volume fraction versus effective plastic strain. CONCLUSIONS The proposed damage model is based on a kinematic model in which microvoid volume fraction evolution in the constitutive matrix is described. The main damage phases are well enhanced, microvoid nucleation is related to effective plastic strain rate, microvoid growth to material strain rate and associated elasto-plastic potential available for porous material, microvoid coalescence to effective plastic strain rate. This model has been implemented into a finite element framework, of which the related algorithms and computer programs are described. Throughout three levels of finite element computations, element patch test, collar test as reference problem for the determination of material properties, pipe bulging as tactical important problem for which microvoid volume fraction determination is essential, it is shown that (i) formulation of elasto-plastic potential leads to coherent results, (ii) strain-controlled approach for nucleation and coalescence seems to be available in cold forging and similar engineering applications, (iii) effective strain path, hydrostatic stress path principally are to be analysed for each problem. NOMENCLATURE f

fi fu fN f~

current microvoid volume fraction critical fraction at incipient coalescence initial microvoid volume fraction ultimate microvoid volume fraction at ductile rupture nucleated microvoid volume fraction consistent with inclusion volume fraction coalescence parameter microvoid volume fraction rate

fc

microvoid volume fraction coalescence rate

fg

microvoid volume fraction growth rate

119

g(eM) h i

k n

ql,qE,q3 s tr DP u

AN B

C~ Cep DP E F I

J1 J2

KT N R

S SN V

VA VM E

EM

eM EN Au AF Ae Aer v t~ O'ef (~m O"M ~"~ep ~'~ep--O

microvoid volume fraction nucleation rate stress-plastic strain function of the matrix material strain derivative of g(e M) iteration loop index local iteration loop index loading increment loop index material parameters deviatoric Cauchy stress tensor first invariant of macroscopic plastic strain rate tensor nodal displacement vector nucleation parameter strain displacement operator material elastic constitutive tensor material elasto-plastic constitutive tensor macroscopic plastic strain rate tensor modulus of elasticity loading force second order identity tensor first Cauchy stress invariant Jl=30rn second Cauchy stress invariant J2=1/2 s:s tangent stiffness operator interpolation function residual force vector limiting surface on which surfacics traction are prescribed Gaussian standard deviation plastic deformed volume of the material elementary apparent volume of material elementary volume of matrix material macroscopic Green strain tensor effective plastic strain effective plastic strain rate mean effective plastic strain at incipient nucleation nodal displacement increment loading force increment strain increment additional strain increment leading to ductile rupture elastoplastic multiplier Poisson's ratio macroscopic Cauchy stress tensor effective stress hydrostatic stress effective matrix yield stress of the matrix material microvoided material potential consistency condition

120 REFERENCES A. Needleman, J. of Applied Mechanics, N~ (1987) 525. E. Onate and E. Kleber, Proc. of NUMIFORM 86, (1986) 339. J.R. Rice and D.M. Tracey, J. Mech. Phys. Solids, N~ (1969) 201 C. Lamy, P. H~caj, J. O'udin, J.C. Gelin and Y. Ravalard, Int. J./Vlech. Sci., 33, 5 (1991) 339. A.L. Gurson, Proc. Int. Conf. Fract., D.M.R. TAPLIN Ed., N~ (1977) 357. A.L. Gurson, Proc. J. Eng. Matl. Tech., N~ (1977) 2. H. Sekigushi and K. Osakada, Bull. of J.S.M.E., N~ (1981) 534. M. Oyane, S. Shima and T. Tabata, J. Mech. Working Tech., N~ (1978) 325. J.C. Gelin, J. Oudin, Y. Ravalard and A. Moisan, Annals of the CIRP, N~ (1985) 151. B. Bennani, P. Picart and J. Oudin, Proc. 3rd Int. Conf. on Computational Plasticity, (1992) 1443. B. Bennani, P. Picart and J. Oudin, Int. J. of Damage Mech., 2, 2 (1993) 118.

10 11

APPENDIX FORTRAN COMPUTER PROGRAM FOR THE PREDICTOR/CORRECTOR ALGORITHM SUBROUTINE PRED CORR (EPST, FRVO, KGAUS, LPROP, NMATS, PROP, SIGM,SIGMA, STRAIN) C C Inout variables: C - E P S T 9array of effective plastic strains C - F R V O 9array o f microvoid volume fraction C - K G A U S 9Gauss integration point number C - L P R O P 9material element number C - N M A T S 9matrix material number C - PROP 9array o f matrix material C - SIGM 9array of Cauchy stress tensor C - S I G M A 9elasticaly predicted Cauchy stress tensor _

C - S T R A I N 9array of plastic strain increment A~ C IMPLICIT REAL*8 (A-FLO-Z) C O M M O N / N U C L E A T I O N / E P S N U , FN, SN C O M M O N / G R O W T H / E T O L , Q1, Q2, Q3 C O M M O N / C O A L E S C E N C E / D E , FCR C C C o m m o n variables 9

C c C C C C C

Calle~ subroutines" INVAM, S I G M A E Q G L O - I N V A M - calculation subroutine of deviatoric Cauchy stress tensor s (DEVIA) and two Cauchy stress invariants J1, J2 (VARJ1, V A R J 2 ) - S I G M A E Q G L O - calculation subroutine o f stress-

C plastic strain function o f matrix material o M C ( S I G M A M ) and its strain derivative h (SLOPE) C C Internal variables 9 C - AN 9 nucleation parameter, A N C - A V E C T 9normal vector at yield surface C - B E L A S 9bulk modulus of elasticity E/{ 3(1-2"v) } C - FEPS 9coalescence parameter, fe C - D E P S E F 9effective plastic strain increment,/Xe M C - D E V I A 9deviatoric C a u c h y stress tensor, sij C - D F " increment of microvoid volume fraction, Af C - D F C O A L E S C E N C E 9increment of microvoid

C - DE " additional strain increment leading to ductile C rupture, AEr

C volume fraction coalescence, Af c

C - E T O L " residual potential convergence tolerance C - FCR 9critical microvoid volume fraction, fcr

C fraction growth, Afg

C - F N " nucleated microvoid volume fraction, fN C - E P S N U 9mean effective plastic strain at incipient C nucleation, EN C - Q1, Q2, Q3 9material parameters q l , q2' q3 C - S N " Gaussian standard deviation, S N C D I M E N S I O N EPST(*),FRVO(*), SIGM(4,*) D I M E N S I O N PROP(NMATS,*), STRAIN(*) D I M E N S I O N AVEC'T(4), DEVIA(4), DSIGMA(4), D I M E N S I O N SIGMA(d),STRES(4)

C - D F G R O W T H 9increment of microvoid volume C - D F N U C L E A T I O N 9increment of microvoid C volume fraction nucleation, Af n C - D L A M B 9increment of plastic multiplier, AX C - D S I G M A 9increment o f Cauchy stress tensor AO C - EPSEF 9effective plastic strain, EM C - FU 9ultimate microvoid volume fraction at C ductile rupture., fu C - F 9microvoid volume fraction, f C - GELAS 9shear modulus, G=E/{2*(I+v)} C - lITER 9index of local iteration loop, k

121

C - SLOPE : g(EM) strain derivative, h C - POISSON : poisson's ratio, v C - POTEN : microvoided material potential, f2ep C - SIGMA : Cauchy stress tensor, oij C - SIGMAEF : effective plastic stress, Oef C C C C

- SIGMAM : effective yield stress, o M - STRES : elastic stress increment - VARJ1, VARJ2 : 1st and 2 nd Cauchy stress tensor invariants, J1 and J2

C - VELAS : V=E/{ 1-2*v} C - YOUNG : modulus of elasticity, E C IER=O PI=3.141592653589793 YOUNG=PROP(LPROP,1) POISSON=PROP(LPROP,2) GELAS=YOUNG/(2.D0*(1.DO+POISSON)) VELAS=YOUNG/(1.D0-2.D0*POISSON) BELAS=YOUNG/(3*(1.DO-2.D0*POISSON)) C C Initial values of state variables at previous C convergent loading increment C EPSEF=EPST(KGAUS) F=FRVO(KGAUS) CALL I N ' A M (DEVIA, LPROP, NMATS, PROP, SIGMA, VARJ2, VARJ1) C A L L S I G M A E Q G L O (EPSEF, LPROP, N M A T S , PROP, S I G M A M , SLOPE) C C Compute the elastic stress increment C TRAN=DEFA(1)+DEFA(2)+DEFA(4) D 11=STRAIN(1)-TRAN/3.D0 D22=STRAIN(2)-TRAN/3.D0 D12=STRAIN(3) D33=STRAIN(4)-TRAN/3.D0 STRES ( 1)=2.D0*G ELAS *D 11+B ELAS *TR AN STRES (2)=2.D0*G ELAS *D 11+BELAS *TR AN STRES(3)=2.D0*GELAS*D12 STRES (4)=2 .DO*G ELA S *D33 +B ELAS *TR AN C C Update the Cauchy stress tensor C DO I=1,4 SIGMA(I)=SIGM A(I)+STRES(I) ENDDO C C Check the loading type C C 1=Q2* V ARJ 1/(2.D0* S IG MAM) PHI=I.D0+Q3*F*F-2.D0*QI*F*DCOSH(C1) YIELD=(3.D0* VARJ2/(SIGMAM* S IGM AM)-PHI C C YIELD <=0 elastic loading C YIELD >0 plastic loading C IF(YIELD.LE.O) GOTO 20

C C Return to the yield surface, radial-return algorithm C ffI'ER=I 10 CONTINtYE C C Compute the nucleation parameter, A N C AI=EPSEF-EPSNU A2=SN* DSQRT(2.D0*PI) A3=FN*DEXP(0.SD0*AI*A1/(SN*SN)) AN=A3/A2 IF (EPSEF.LT.EPSNU) AN=0.DO C C Compute the coalescence parameter C FEPS=0 FU=I.D0/Q1 IF (F.GE.FCR) FEPS=(FU-FCR)/DE C C Compute the plastic multiplier increment A~. C C Tension state case C CI=Q2*VARJI/(2.D0*SIGMAM) C C Compression state case C IF (VARJ1.LT.0.DO) CI=0.D0 C C Square root radical sign test which validates C potential factorization C TEST=I.DO+Q3*F*F-2.DO*QI*F*DCOSH(C1) SIGN=I .DO IF (TEST.LT.0.D0) SIGN= -1.D0 C PHI=DSQRT(SIGN~ ALPHA=SIGN*QI*Q2*F*DSINH(C1)/2-DO SIGMAEF=DSQRT(3.DO*VARJ2) DL11=ALPHA/PHI DLI=3.DO*GELAS+3.DO*VELAS*DL11 9 *DL11 C C

/3 t'/cp

DL1 =

5o

Co

"

i3 D., p

9

5o

C

C

C C

DL2.I=PHI+((ALPHA* VARJ1 )/(PHI*SIGMAM)) DL22=DL21*DL21 DL2=S LOPE*DL22/(1 .D0-F) ~ i'2q, Of~q, o 9 DL2= ~ h i9o i9o a ( 1-f) o M DPHIFV=SIGMAM*SIGN*(QI*DCOSH(C1)9 Q3*F)/PHI DLNUCLEATION=AN*(DL21/(1-DO-F)) DLGROWTH=(1.DO-F)*3.DO*ALPHA~HI DLCOALESCENCE=FEPS*DL21/(1-DO-F)

122

C DO ISTRI=I,4 SIGMA(ISTR 1)--SIGMAflSTR 1)DLAMB*DSIGMA(ISTR 1) ENIXX)

DL3=DPHIFV*(DLNUCIXATION+ 9 DLGROWTH+DI.L-OALESCENCE) C /gt~ C C

DL2 =

n, Oc~M

h

~(I ( 1-f) c M

DLAMB=(SIGMAEF-SIGMAM*PHI) /(DL1 +DL2+DL3) C C Compute the effective plastic strain increment, kE M C DEPSEF=DLAMB*(SIGMAM*PHI*PHI+ 9 ALPHA*VARJ1)/((1.D0-F)*SIGMAM*PHI) C C Compute the microvoid volume fraction C increment due to nucleation Afn C DFNUCLEATION=AN*DEPS EF C C Compute the microvoid volume fraction C increment due to growth Afg C DFGROWTH=DLAMB*(1 .DO-F) *3 .DO*ALPHA/PHI C C Compute the microvoid volume fraction C increment due to coalescence Afc C DIR2OALESCENCE=FEPS*DEPSEF IF (DFNUCLEATION.LT.0.DO. OR.VARJ1.LT.0.DO) DFNUCLEATION=0.DO IF (DFGROWTH.LT.0.DO) DFGROWTH=0.DO IF (DFCOALESCENCE.LT.0) DFCOALESCENCE=O.D0 DF=DFNUCLEATION+DFGROWTH+ DFCX)ALESCENCE C C Update the effective plastic strain and microvoid C volume fraction C EPSEF=EPSEF+DEPSEF F=F+DF C C Compute the Cauchy stress tensor C SQJ2=DSQRT(VARJ2) AVEC~(1)=DSQRT(3)*DEVIA(1)/SQJ2 AVECT(2)=DSQRT(3)*DEVIA(2)/SQJ2 AVECI'(3)=DSQRT(3)*DEVIA(3)/SQJ2 AVECI'(4)=DSQRT(3)*DEVIA(4)/SQJ2 TRAVEL'q'=VELAS*(ALPHA/PHI) DSIGMA(1)=GELAS*AVECT(1)+TRAVECT DSIGMA(2)=GELAS*AVECT(2)+TRAVECI" DSIGMA(3)=2.DO*GELAS*AVECT(3) DSIGMA(4)=(3ELAS*AVF.L-q'(4)+TRAVEC~

C C Compute the new effective matrix yield stress and C 1st and 2nd Cauchy stress tensor invariants C CALL INVAM (DEVIA, LPROP, NMATS, PROP, SIGMA, VARJ2, VARJ1) CALL SIGMAEQGLO (EPSEF, LPROP, NMATS, PROP, SIGMAM, SLOPE) C C Compute the new microvoided plastic potential C C Tension state case C C 1=Q2* VARJ 1/(2. DO*S IGMAM) C C Compression state case C IF (VARJ1.LT.0.DO) CI=O.DO TEST= 1+Q3* F* F-2*Q 1*F* DCOS H(C 1) SIGN=I IF (TEST.LE.0.DO) SIGN=-I.D0 ALPHA=SIGN*QI*Q2*F*DSINH(C1) PHI=DSQRT(SIGN*TEST) SIGMAEF=DSQRT(3.D0*VARJ2) C POgEN=SIGMAEF-SIGMAM*PHI C IF (DAB S(POTEN).GT.ETOL.AND. IITER.LE. 100) THEN C C New local iteration C I1TER=I1TER+I GOTO 10 ENDIF C 20 CONTINUE C C State variables keeping C EPST(KGAUS)=EPSEF FRVO(KGAUS)=F DO ISTRI=I,4 SIGM(ISTR 1,KGAUS)=SIGMA(ISTR 1) ENDO9 C REFth~ END

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

123

Theoretical and numerical modelling of isotropic and anisotropic ductile damage in metal forming processes J.C. Gelin Laboratoire de M6canique Appliqu6e, URA CNRS 004, Universit~ de Franche-Comt~ 25030 BESANCON C~dex- France Abstract

The modelling of defects occurrence in metal forming is a real challenge for the engineer concerned with processing of new materials or new geometries. From an economical point of view it is important to be able to predict before setting the process if it leads or not to defects. As fracture in metal forming is mainly due to the development of ductile damage or plastic instabilities, a review of some models and methods is presented, applicable both in quasi-static as well as in transient dynamic situations, for cold or hot forming conditions. Numerical modelling of damage development in some industrial forming processes illustrates the possibilities of the proposed approach.

1. INTRODUCTION During metal forming, the main causes of fracture result either from plastic instabilities associated with localisation phenomena (necking in sheet metal forming, surface fracture observed in bulk metal forming .... ), or from the development of damage due to large straining of the material (central bursting in extrusion, internal fracturing in cold forging .... ). The plastic instabilities result from mechanical conditions associated with the loss of stability of the equilibrium problem and can be treated as a local problem with classical methods in that field as the singular perturbation technique [1 ], the linear stability analysis [2] or the Hill's stability theory [3]. The development of damage results from the apparition of microvoids or microcracks that growth under large straining conditions and lead to macroscopic discontinuities in the workpiece. From observations at the microscopic level, the main cause of internal damage in metal forming is the ductile fracture or intergranular fracture occurring in cristallographic materials [4]. Ductile fracture results from the decohesion of the matrix material around inclusions or second phase particles, then the microcavities or microvoids growth due to large straining and they tend to form macroscopic cavities or cracks leading to macroscopic fracture [4]. Such a mechanism mainly arrives in cold forming conditions. In hot forming conditions, i.e. when the deformation temperature is greater than about half of the melting temperature, the main cause of damage development is associated with intergranular damage that corresponds to the nucleation of cavities at grain boundaries, this phenomenon being mainly controlled by the viscoplastic strain rate. Although both mechanisms of internal damage in metal forming are different from the metallurgical point of view, we will show that they can be treated with approximately the same mechanical approaches.

124 On the other hand, forming processes are mainly controlled by boundary conditions (lubrication and friction), by the displacement and the average speed of the tools, and by the temperature deformation during the process. In modem metal forming industry, as example in rolling or wire drawing of steels or aluminium alloys, the average speed of the processes is such that the classical time independent plasticity is no longer applicable, so the need for better understanding the effects of high speed forming is necessary. Numerical modelling of metal forming has now gained the industrial stage, and it became possible to simulate metal deformation and to calculate stress and strain states for complex processes, see as example the increasing interest in the Numiform Conference in terms of papers and number of processes simulated [5]. Numerical modelling is mainly based on finite element models for bulk and sheet metal processes that are developed for steady or unsteady state conditions. The material behaviour considered is rigid-(visco)plastic, elasto-plastic or elastoviscoplastic [6]. In some cases the transient dynamic conditions are taken into account, sometimes for computational purposes, or in other cases to describe particular effects associated to dynamic loading [7][8]. The finite element simulation of defects occurrence in metal forming is a real challenge and there are two principal ways to achieve such an analysis. One way consists to carried out a conventional finite element simulation and, by postprocessing the finite element results, in using a ductile fracture failure criterion, to detect the zones where risks of fracture can occur. It is the approach employed in [9][ 10]. A somewhat modified form consists to take into account of damage accumulation in the flow stress curve of the material and to carried out a simulation with a flow stress that increases with hardening and decreases with damage, such a possibility have been applied to simulate localized necking in sheet metal forming. One of the major limitation of such an approach is that fracture criterion used have to be fitted with experimental results obtained with tensile, torsion or compression tests. In the case of complex metal forming processes, the process conditions influence significantly strain and stress paths and the material parameters obtained from simple tests and are no longer applicable. Another way consists to directly incorporate damage evolution in constitutive equations and to carried out the FEM simulation with such equations. First attempts have been carried out for extrusion where the Gurson model [ 11] have been employed. The results show that the finite element method can be successfully used to predict the formation of central bursts in extrusion [12][13]. In [14][15] an extension is given to treat the elastic-viscoplastic case and results obtained show that inclusion of viscoplastic effects generally tend to decrease the risk to obtain fracture as it is remarked in most forming processes. In that paper an isotropic model recently developed by the author and coworkers [ 16][ 17] that takes into account both degradation of elastic properties and modification of plastic flow when plastic straining increases is presented. Using elasto-(visco)plastic or rigid-(visco)viscoplastic finite element formulation, the model has proved to give satisfactory results in agreement with experimental ones. Then some basic ideas to do an extension of the model to take into account anisotropic damage are given[ 18]. As in many industrial cases the real conditions are a combination of dynamic loading and elasto-viscoplastic behaviour, with temperature and softening effects associated to adiabatic conditions, the present investigation describes the combined effects of dynamic loading, visco-

125 plasticity and temperature on the development of ductile fracture in metal forming processes. In section 2 of the paper we intend to underline some important points in the development of damage models. In section 3 and in the case of finite strain elasto-plasticity or elasto-viscoplasticity, an isotropic damage model is presented. The damage evolution is described in the framework of internal state variables. Of course the anisotropic damage presents some particular difficulties to define a damage tensor with significant constitutive properties and due to the fact that we work in finite strains. Some attempts in this case are mentioned. In section 4 of the paper some aspects of the numerical implementation of the proposed models are described and some difficulties inherent to finite element computations with damage constitutive equations are related. The last part of the paper concerns simulations of practical metal forming processes that have been carried out with the finite element method developed. The damage and ductile fracture development are analysed in extrusion for a very slow punch velocity and for a high punch velocity. Results obtained concerning the damaged zones in the extrudate clearly show that the damaged zones move from the centre of the extrudate to the external part when the punch velocity increases. The second example treated is a high speed upsetting of a cylinder along two generative lines. 2. MECHANICAL DESCRIPTION OF DUCTILE DAMAGE As previously mentioned the notion of macroscopic damage in forming operations is generally acknowledged by the observation of cracks were not present in the virgin material before loading/unloading process. Due to the microscopic origin of ductile damage it results that the classical theories such as elasticity or elasto-plasticity are not able to properly describe the behaviour of metals that exhibit ductile damage. It results that refined theories have been developed or are in progress to try to describe properly the behaviour of damaged materials, for a review see [ 19]. In order to model changes associated to damage some approaches were developed. Between these we can distinguish two major classes 91) the phenomenological approach starting with the Kachanov's model [20] initially used for describing damage in creep and generalized later for the ductile damage in small strains and to isotropic or anisotropic cases in finite strains; ii) a microscale-macroscale approach that intends to construct models at a macroscale using some considerations from the microscale and some additional assumptions. The basic ideas of the last approach goes back to the work of Hill [21 ] and were used substantially by Gurson [ 11] and are now an active domain of research. In his original one dimensional model, Kachanov [20] suggests that damage should be defined as the density of microvoids in any cross section of the material. The relative area of the voids in a cross section being represented with a scalar variable d(x) with 0 < d(x) < 1, where x stands for the current position. The damage variable d(x) used in [33][34][35] and in other author works is often interpreted as the effective area reduction caused by the distributed microscopic cracks and cavities due to material damage. It is clear that d(x) is well adapted to represent damage in one dimensional case, but the generalization to 3-dimensional cases suffer from serious limitations and numerous alternatives are proposed. It seems that an obvious choice for defining a damage variable representing nucleation, growth and coalescence of voids and microcavities is their volume density denoted by f(x) where x means for current position. For practical measurements, the

126 void volume fraction fix) is directly related to the density of the material through relation: (1)

p(x) = (l-f(x))Po(X)

where p(x) is the current density of the material, P0(X) is the initial one if we consider that the void volume fraction in the initial state is zero (f0(x) in other cases). Clearly the variable fix) is applicable in 3-dimensional cases but the remaining question here is how to take into account both the area of the cracks and their orientation. When damage is isotropic, it is tempting to try to relate d(x) to fix) introduced for the same purpose. As it is demonstrated in [22] it is not possible through purely mathematical arguments, the relation is an inequality relation expressed as: [1-(1-f(x))

2/3] < d ( x ) < [ f ( x ) 2/3]

(2)

The extension of the damage variables previously introduced in the anisotropic cases where the orientation of the cavities have to be taken into account can be formulated with the following arguments. Using remarks from Rabier [22], let d(n) the scalar function defined on the unit sphere centered at x position and that describes damage in direction n. Since the damage is anisotropic and since the function d(n) is even it follows that it can be developed in a generalized Fourier series by means of the spherical function. Thus the function d(n) can be expressed as : d (x, n) = D O(x) + gij (n) Dli j (x) + .......

(3)

where gij = (ni nj - ~5ij/3). Thus (D1)ij is a traceless and symmetric second order tensor. The six scalar variables D O and (D1)ij describe entirely that anisotropic damage state. Another arguments are given in [34] that consider two configurations of the same solid, a damaged one (B) and one corresponding to the undamaged fictitious configurations (Bud). If G denotes the fictitious deformation gradient from B to Bud it results from the Piola transformation that : n*dS* = jG-TndS

with

J = detG

(4)

where ndS is the area vector in B, whereas n dS is the area vector in Bud.The argument developed in [34] implies that the damage state of B can be described by the linear transformation JG T , and similary to the one directional case it is possible to introduce a damage tensor as:

1-D

= JG -T

(5)

It results that equation (4) can be written as follows: n*dS* = ( 1 - D ) n d S that generalizes to anisotropic cases the concept of net load carrying section.

(6)

127

3. D U C T I L E D A M A G E MODELS

3.1 The isotropie ease We consider in this paper a material containing microscopic voids or cracks. From the physical point of view it is possible in a first approximation to consider that the void volume fraction f represents a measure of the damage, figure 1.

Figure 1 - Schematic representation of a porous material containing micro-voids Taking into account that the deformation can be decomposed in an elastic part, in a pure dilatant part and in a plastic one that preserves the volume, the classical multiplicative decomposition can be extended as [ 16][ 17]: F = FeFdF p

(7)

where F e is the elastic deformation gradient, pO is the deformation gradient associated to the dilatation due to the growth of voids and FP is the plastic deformation gradient with detFP = 1. Finite strain decomposition (7) leads to express the relative change in volume during the deformation as a function of the determinant of the deformation gradient F and we have: detF = detFedetFddetFP =

p0 p

=

1 fo 1-f

(8)

with detF e = 0 e ; detF d = 0 a ; detF p = 1

(9)

where p0 is the initial density of the material, P the current density of the material, 0 e the volume change associated with elasticity and 0 d is the volume change associated with damage evolution (the plastic flow is assumed to be an isochoric one). The scalar measures 0 d or f can be indifferently used to characterize damage evolution, in our model we prefer to use 0 d because of its geometric signification. An important point to be remarked is that these scalar measures consider only volume change due to the growth of voids, but not their shape and orientation, clearly 0 d or f are isotropic damage variables. Let

Be

the Cauchy Green tensor associated to the elastic deformation gradient :

128

B e = FeF eT

(10)

and let ~e the Cauchy Green tensor associated to the isochoric elastic deformation as B e = (0e)

2/3B-e

(11)

with F e = (0 e) l/3~e

and

~e = ~e~e T

(12)

Let ~ the free energy defined on the current configuration. We suppose that elasticity, plasticity and damage are uncoupled, V is written in the following form: = ~,e (0e, ~e ) + ~p (0~) + ~1/d(Od)

(13)

For simplification we consider that the elasticity coefficients K and G (bulk modulus and shear modulus) are sufficient to describe the isotropic elastic behaviour and the term ~e is de fined as: v e ( ~ e , 0 e)

(14)

= 41 ( G t r ( ~ e ) + K U ( 0 e ) )

The Cauchy stress tensor c corresponding to V is given by the following relation: c = 2oB e ~ V _ 2pdev g *~V---~ + p ~ l ~)Be ~)ge 30 e

(15)

If we suppose that U (0 e ) = Ln (0 e ) 2 relation (15) can be written as: = 2K(Ln(0 e))l+Gdev(~e)

(16)

that coincides with the classical elasticity relation in the small strains limit. In order to establish the damage model let consider a plastic potential defined on the following form: ~)(G, 0d, o0(s p ) )

= ~/21devol + o 0 ( ~ p ) B ( 0 d ) g ( t r o ) - - ( I 0 ( I ~ P

)

(17)

where g(tro) is an unknown function of the first invariant of the stress tensor, o 0 is the flow strength of the undamaged material, devo is the deviatoric part of o and Idevol is the euclidian norm of devo. From the conservation of mass law and admitting that 0 e = 1 it results that: _ {~d

+ 0dtrD p

_.

0

(18)

Relation (18) is sufficient to define the form of the function g(tro) introduced as an unknown one in relation (17):

129 trc~ g (trc) - [3exp ( ~ )

(19)

where ~ is a constant that depends on the shape of voids (generally ]3=3/2) and c 0 is the flow strength of the undamaged material. Function u(0 d) can be chosen as u(0 d) = In 0 d accordingly with results associated to the growth of a spherical void in a spherical cell of material with a yon Mises flow rule [ 11]. It results that the plastic potential of the material containing voids is entirely defined as: r

0d, Oo ) = ~

d e v o ] - o o I 1 - 1 3 L n ( 0 d ) e x p ( ~trc~ ) 1

(20)

The plastic strain rate tensor is given by the normality relation as: devc I- Ln0d 2 13exp (2-~o) trcr 1 1 D p = ~ I~-32Idevc]

(21)

The plastic potential defined in relation (20) is very closed from the form obtained in [11] and used in [ 12][ 13] to predict defects occurrence in the axisymmetric extrusion process. A representation of the plastic potential defined in relation (20) is given in figure 2 in the space [Idevcl/c 0, tr(ff/2ff0)], the void volume fraction being considered as a parameter, we see that 0d =1 or f=0 corresponds to the well known von Mises case whereas f=l corresponds to the complete degradation of the material. In the physical reality f=l is never reach and the load carrying capacity of the material decreases considerably for small values of the void volume fraction (from f=0.05 to f---0.2). Relation (20) describes the plastic potential that can be used in cold forming considering the material as an elastic-plastic one. In hot forging cases we can consider that the elastic strains are small compared to the viscoplastic strains and then it becomes possible to use a rigidviscoplastic approach.

0

1

2 MEAN S T R E S S tr c~,'3.,\

3

Figure 2 - Effective reduced stress versus mean reduced stress as a function of void volume fraction

130 Let consider relation (20) as a viscoplastic potential [14], it results immediately that the viscoplastic strain rate is given by: d e v ~ + Ln0d 2 13exp ( 2t -r ~~ D p = ]?I~-32 IdevoJ

1

(22)

where ~" is in that case a time dependent parameter or a strain rate sensitivity parameter. To relate the spherical part of D p to the rate of void volume fraction the conservation of mass law is use, thus: tro = ( 1 - f) trD p = ( 1 - f) ~ 13Ln0dexp (~-~0)

(23)

The effective or the equivalent viscoplastic strain rate for the damaged material (sound material + voids) is given by: = q?

(24)

It is interesting to remark that for an undamaged material the effective viscoplastic strain rate is equal to q?. The factor q?can be expressed in assuming that the plastic power dissipated in the aggregate (sound material+voids) is equal to the plastic power dissipated in the sound material" O. D p = ( 1 - f) Oog5o

(25)

where o o and g~o are respectively the flow strength and the effective strain rate acting on the matrix (sound material). From relation (25) it results that ~, depends on devo, tro, f and. Generally the flow stress of the matrix (sound material) is expressed as a function of the temperature, effective viscoplastic strain rate and effective viscoplastic strain. Among the models that have been proposed, the most current is based on the observations by Sellars and Tegart [23] that steady state conditions for both creep and hot working are similar: o'* = 1 A r g s h I Z l t / n X

(26)

where: o* is the plateau or saturation stress, - or, A, n are temperature-independent constants for a specific microstructure, - Z, the Zener-Hollomon parameter defined as: -

z , xpE 1 with 9 -~ viscoplastic strain rate, - T absolute deformation temperature, Q activation energy. On the other hand, the work hardening regime (at small strains) is best represented by the fol-

-

131 lowing laws: (~0 = ~n ~m e XP I ~ T ]

(28)

or [24]: b c o o = (a+~+~-~)

in -m ~

(29)

Finally, to consider both the work hardening and the dynamic recovery, Voce's equation is usually used: o o - o* - (~* - oi) exp ( - N E p)

(30)

where 6 i is the flow stress of the matrix at a given cumulated plastic strain and temperature T. Although empirical, the above equations usually represent quite well the experimental data. More recently, the introduction of structural variables such as dislocation density and subgrain size for calculating the flow stress as a function of strain, strain rate and temperature leads to more physically based relationships [25][26]. Under adiabatic conditions the balance of energy gives: ~)T PCv-0--~ = k(~:D p

(31)

where p is the density in the deformed shape, C v is the heat capacity and k specifies the fraction of plastic energy transformed into heat, a typical value for metals is 0.9. To complete material description, the elastic behaviour has to include thermal sensitivity as: 6 = 2K (T) Ln (0 e) 1 + G (T) dev ( ~ e )

(32)

The model described does not consider a yield to define the plastic the viscoplastic flow. 3.2 The anisotropic case

We describe here a recent approach developed to take into account anisotropic ductile damage [18]. The damage variable introduced, following the arguments developed in [22] is a second order symmetric tensor denoted A. The macroscopic Cauchy stress tensor is decomposed into a part normal to A and a part parallel to A as follows: o = c

A

o:A + ~A

(33)

(above A 9B = Aij Bij and oA:A = 0). Then o A is the analog of the deviatoric part of o (denoted devo) while (o:A) / (A:A) is the analog of trff. Starting from this analogy it is postulated a yield condition in the form: A

f (~, A) = or" cr + CoO "~

A

2 Cl

A" A

(34)

132 where c o and Cl are material constants allowing to introduce qualitative constitutive aspects. It is easy to see that (34) can be written in the equivalent form: f(o,A)

= (1 + c 0 ) d e v o : d e v o + c 0 ( t r o ) 2rl(A) -Co 3

dev~:devA A:A

2

c1 A'A

(35)

(trA) 2 where 1"1(A) = 1 - 3A---7~ A and thus 11 = 0 if and only if A = a l . At this point at least two consequences are clear : firstly, the damage evolution changes qualitatively the yield condition, introducing plastic anisotropy. Secondly the form (35) which generalizes the von Mises yield condition can be also written for a yield function that includes the first invariant of the stress (its analog being (c:A) / (A:A)). It is shown in [18] that the yield function (35) has interesting properties concerning damage evolution. The damage tensor under loading conditions tends to become normal to the stress tensor. Other models based on the extension of the Kachanov model using a fourth order damage tensor are discussed in [27] and [28]. But the constitutive laws associated to damage variables have to be determined, the thermodynamical restrictions being not sufficient to formulate completely the model. 4. EQUILIBRIUM EQUATIONS AND VARIATIONAL F O R M IN THE DYNAMIC RANGE

4.1 Variational formulation and discretization The dynamic equilibrium problem consists to solve the following field equilibrium equations for all the points of the domain under consideration: ~)V

in

divo'+ f = p~-~ o.n=t u=fi

on on

f~ Ft

(36)

Fu

In equation (36) p is the material density, f is the body forces vector and t are the prescribed tractions imposed on a part of the boundary. The variational form associated to the equilibrium equations can be written in term of the total energy G(u) as:

G(u)

= fV( Fe,tx,0 d ) d V - f t u d S - f f u d S + J d ( u )

(37)

where g is the free energy per volume unit and Jd(u) is the energy associated to inertial loading. Solution u of the problem minimizes the total potential energy G(u). The minimum of quantity G(u) is obtained when the linearized derivative of equation (37) is equal to zero. The first variation of (37) relatively to the displacement field leads to: ~ F T : VSrldV - f t r l d s - f f r l d S + f p a . rldV DuG(U )1"1 = f~--F Ft

~

(38)

133 where 1"1is the homogeneous field associated to u and a is the acceleration field. Equation (38) is a non-linear one with respect to the displacement field u and the acceleration field a. To solve such an equation an iterative technique based on the Newton method is used: [DuG (u, 1"1) ]iAui + ~ + [G (u, rl ) ]i = 0

(39)

where u = [u, a] T, [.]i stands for quantity [.] at the ith iteration, and where DuG ( u ) 1"1 corresponds to the linearized derivative of G(u) in the 11 direction. A Newmark scheme for integration in time is used. Equation (39) is discretized at the element level in using a mixed formulation where the variational unknowns are the displacement field u(x, t), the acceleration field a(x,t), the dilatancy field 0(x,t) and the pressure field p(x,t) [29][30].

4.2 Integration of constitutive equations The numerical solution of elasto-plastic problems by the finite element method in displacement formulation leads to a sequence of displacements Un+l such that: Un + 1 = Un + Au n +

1

(40)

Displacement Un+1 corresponds to the (n+l)th loading increment, whereas displacement u n corresponds to the nth loading increment. In order to calculate the displacements, stresses and internal parameters corresponding to Un+ 1, it is necessary to define a procedure that can be explicit or implicit. The procedure that we have retain is an implicit one and is based on the following relation: Fn+l = ( l + V n U n + l ) F n

(41)

The approximation of the elastic gradient for the (n+ 1)th increment can be written as: e(o)1 = Fn + 1FnP-1Fnd-1 Fn+

(42)

Such an approximation supposes that the internal variables ot and 0 d are constant during the elastic prediction phase. From equation (42) it results that: Be(o) e(o) e(0) T n+l = Fn+l Fn+l

(43)

From the expression of the free energy potential, it results that the stress tensor is expressed as: ~e(O ) en(O) c)~ n+l = P ~ +l ~BeCo) n+l

(44)

The plastic multiplier and 0 d are calculated in using the expression of the constitutive functions and yield function. The algorithm used is an extension of the algorithm proposed by Simo and Ortiz [31 ], developed by Gelin [30] in the case of elasto-viscoplastic behaviour, and applied by Aravas [32] in the case of elastic-plastic behaviour with ductile damage.

134 5. N U M E R I C A L RESULTS 5.1 Extrusion of a cylindrical bar First example concerns the extrusion of a cylindrical bar (at low or high speed) represented in figure 3, extruded with a semi-cone die angle equal 5.7 ~ (respectively 15~ The initial dimensions of the bar are given in figure 3. The material properties used correspond to a mid-carbon steel with E = 210000 Mpa, v = 0.3 and o 0 = 500 ( 1 + 0 , 4 ~ '4) MPa, the operation is carried out under isothermal conditions. Figure 4a and figure 4b show the deformed mesh corresponding to a punch displacement L0 = 40mm with a punch velocity v0/L0 = 103 s -1.

Figure 3 - Geometry of the extrusion of a cylindrical bar through conical dies

zl

I

m

a

I I

zt I

mi m I

B i m i m m m m m n m m u m m n

im-,,u,m-,"' m m u m u m u m m

n n u m m m m n

~'"--""--"

m m m m m m m l

nmm,m n n m m n

.m.m.m.m.m.m.m., m

mmlmmmmmm

ram,

mmmmmmmm mmmmmmmmp-m

mmmm'amm m a m a m m Immmm,mami

_mmmm.,mm|m!m

m a m m m a m m

maw

l'-iiii~;

m~mmammmm

mlm

mm'mmmmmaml mmmmmmmmm

Immmmmmmm Rmmmmmmam

1 i

(a)

-- '--

mlumm,,,

i i

i

! i !

(b)

Figure 4 - Deformed mesh corresponding to (a) L0 = 40 mm, o~ = 5.7 ~ and v~ (b) L0 = 40 mm, cx = 15 ~ and v0/L0 = 103 s -1

- 103 s 1

Another computations have been carried out with a punch velocity equal 0.1 s -1. The contours of effective plastic strain are not sensibly influenced by the inertial terms. Figure 5a and 5b show the contours of void volume fraction for a punch displacement corresponding to 40mm

135

with v0FL0 = 103 s -1 whereas figure 6 shows the contours of void volume fraction for a punch displacement corresponding to 40mm with v0/L 0 = 10 -1 sl.It is clear that the inertial terms have an influence on the increasing of void volume fraction near the external surface of the extrudate. In that study the effects of inertial terms is to move the damaged zones from the central part of the extrudate to the external surface of the extrudate.

I'A iBC 4.08E-2 4.33E-2 4.59E-2 4.84E-2 5.09E-2 5.35E-2 i! 5.60E-2 5.86E-2 6.1 IE-2 6.36E-2

~

4.08E-2 4.90E-2 5.72E-2 6.54E-2 E 7.36E-2 8.18E-2 G 9.00E-2 H 9.8 IE-2 I 1.06E-1 J 1.15E-I

a Figure 5 - Contours of void volume fraction (a) L0 = 40 mm, o~ = 5,7 ~ and v0/L 0 = 103 s -1 (b) L0 = 40 mm, o~ = 15 ~ and v0FL 0 = 103 s "1

\ 5

/ ~

A B C D E F G H I

+2.053E-02 +2. 665E-02 +3. 276E-02 +3. 888E-02 +4.499E-02 +5.111E-02 +5. 722E-02 +6.334E-02 +6. 945E-02

Figure 6 - Contours of void volume fraction L0 = 40 ram, o~ = 15 ~ and v0/L0 = 10 -1 s -1 5.2 High speed upsetting test Computer simulations have been carried out that consider a simple upsetting of a steel cylindrical cylinder with an initial diameter DO= 20 mm, as it is represented in figure 7. The upsetting corresponds to a frictionless one and the die are supposed to be rigid. Two loading cases

136 have been simulated, one corresponding to a quasi-static case with v0/H 0 = 0.1 s -1 and the other corresponding to a dynamic loading case with v0/D0 =103s -1. The material properties used correspond to a steel with E = 210000Mpa, v = 0.3 and the flow strength is given by 0,4 c o = 500 ( 1 + (~:P) ) MPa. Isothermal conditions were assumed for the simulations. A previous study has revealed that in the initial deformation stage [7] the deformed mesh is strongly influenced by the inertial effects. Figure 8 shows the upsetting load and reveals that the load is influenced by the inertial terms. But the major result concerns the contours of void volume fraction plotted in figure 9 that shows an important void volume fraction increase at the free surface of the compressed cylinder, associated to an important increase of the axial stress. Our results are in agreement with results found in [36] using a viscoplastic finite element program to simulate upsetting tests.

llllt

___. l~I

L

m

IllI] 1 ,,, irr-~

Figure 7 - Initial and deformed mesh corresponding to the upsetting of a cylinder along generative lines a reduction in height equal 44% (D O = 20 mm, v0/H0=103s-]). 8O00 w

~

D

~ 9

uasi-static case

_ l - ~ .J

a

!

o

;

2 ; ; Punch Displacement (mm)

s

Figure 8 - Curves of force on the die versus die displacement corresponding to the quasi-static (v0/H 0 = 0.1 s -1) and dynamic loading case (v0/H 0 =103s -1)

137

rt A B C D E F G H

1.05E-5 8.46E-3 1.69E- 2 2.54E- 2 3.38E-2 4.23E-2 5.07E-2 5.92E-2

A B C D E F G H

1.11E-5 1.15E-2 2.31E-2 3.46E-2 4.62E-2 5.77E-2 6.92E-2 8.08E-2

A B C D E F GH

1.13E-5 1.43E-2 2.85E-2 4.28E-2 5.70E-2 7.12E-2 8.55E-2 9.97E-2

v

(b)Contours ot void volume tracUon (AD/D0=34%).

i<

(b)Contours of void volume fraction (AD/D0=40%)

u

Figure 9 - Contours of void volume fraction corresponding to the upsetting of a cylinder (DO= 20 mm, v0/H0=103s-l, AD/D0=44%) 6. CONCLUSIONS Material models to deal with the plastic flow of damaged metals that contain micro-voids or micro-cracks have been developed and used in situations of quasi-static and dynamic loading with strain rate and temperature effects. The constitutive models describe ductile damage in terms of a scalar or tensorial variable related to volume change. The strain rate effects, adiabatic heating and inertial terms are considered. The inertial effects are included in the equilibrium equations and a variational formulation have ben proposed that includes an accurate

138 treatment of the quasi-incompressibility conditions associated with the elasto-viscoplastic flow. The following results have been obtained for a stationary process: i) when loading velocity increases the inertial terms delay the central defects in the extrudate, ii) when inertial effects are taken into account the apparition of defects is moved in the peripheral part of the extrudate. In non-steady state process as the upsetting of a cylinder the inertial terms are important and lead to a concentration of damage on the external free surface of the cylinder. From our analyses, it is shown that in some cases of industrial relevance, it is necessary to take into account of inertial effects to be in agreement with experimental observations. REFERENCES

[ 1] J.R. RICE, The localization of plastic deformation, in Theoretical and Applied Mechanics, Ed. W.T. Koiter, North-Holland (1976). [2] D. DUDZINSKI, A. MOLINARI, Mod61isation et prrvision des instabilitrs plastiques en emboutissage, In Physique et m6canique de la mise en forme des m6taux, Presses du CNRS (1990) 444-460. [3] R. HILL, On discontinuous plastic states with special reference s to localized necking, J. Mech. Phys. Solids, 1 (1952) 19-36. [4] D. FRANCOIS, A. PINEAU, A. ZAOUI, Comportement Mrcanique des Matrriaux, Tome 2, Herm/~s, Pads (1993). [5] J.L. CHENOT, R.D. WOOD, O.C. ZIENKIEWICZ Eds., Numerical methods in industrial forming processes, A.A. Balkema, Rotterdam (1992). [6] J.C.GELIN, D. LOCHEHGNIES, Simulations num6riques en grandes deformations plastiques, In Physique et mrcanique de la mise en forme des mrtaux, Presses du CNRS (1990) 553-562. [7] J.C. GELIN, A finite element analysis of high speed metal forming processes, Annals of the CIRP, 40-1 (1991) 277-280. [8] J. TIROSH, Kinetic and dynamic effects on the upper bound in metal forming processes, J. ofAppl. Mech,43 (1976) 214-218. [9] L. CHEVALIER, P. LE NEVEZ, Calculs 61asto-plastiques en grandes drformations avec pr6vision d'endommagement ductile. Application ~ l'extrusion - Physique et Mrcanique de la Mise en Forme des Mrtaux. Presses du CNRS, Paris (1990) 553-560. [10] A. EL MENNI, M. DUBOIS, J.C. GELIN, Contribution ~ la simulation numErique du d6coupage des m6taux, Congr~,s STRUCOME 93, APCAS (1993) 567-579. [11] A.L. GURSON, Continuum theory of ductile fracture by void nucleation and growth: Part I- Yield criteria and flow rules for porous ductile media, J. Eng. Mat. Tech., 99 (1977) 225. [12] J.C. GELIN, J. OUDIN,Y. RAVALARD, An improved finite element method for the analysis of damage and ductile fracture in cold forming processes, Annals of the CIRP, 34-1 (1985) 209-213.

139 [13] N. ARAVAS, The analysis of void growth that leads to central bursts during extrusion, J. Mech. Phys. Solids, 34-1 (1986) 55-79. [14] J.C. GELIN, A viscoplastic model for void containing materials incorporating temperature effects, Proc. of the 2nd Int. Conf. on Computational Plasticity, Ed. by D.R.J. OWEN,E.HINTON and E. ONATE, Pineridge Press (1989) 515-526. [15] J.C. GELIN, Application of a thermo-viscoplastic model to the analysis of defects in warm forming conditions, Annals of the CIRP, 35/1 (1986) 157-160. [ 16] J.C. GELIN, M. PREDELEANU, On a finite strain elastic-plastic ductile damage model Model and computational aspects, 17th Int. Cong. Theoretical and Applied Mechanics, Grenoble, France (1988). -

[17] J.C. GELIN, M. PREDELEANU, Finite strain elasto-plasticity including damage Applications to metal forming problems, Proc. of NUMIFORM 89 Conf., Ed. by E.G. THOMPSON et al., A.A. Balkema (1989) 151-157. [18] J.C. GELIN, A. DANESCU, Constitutive model and computational strategies for finitestrain elasto-plasticity with isotropic or anisotropic ductile damage, In Proc. of 3rd Ht. Conf. Computational Plasticity, Ed by D.R.J. Owen et al, Pineridge Press (1992) 1413-1424. [19] M. PREDELEANU, Finite strain analysis of damage effects in metal forming processes, in Computational Methods for Predicting Materials Defects, Ed by M. PREDELEANU, Elsevier (1987) 295-308. [20] L.M. KACHANOV, Time of the rupture process under creep conditions, Isv. Akad. Nauk. SSR, 8 (1958) 26-31. [21] R. HILL, The essential structure of constitutive laws for metal composites and polycristals, J. Mech. Phys. Solids, 15 (1967) 79. [22] P. RABIER, Some remarks on damage theory, Int. J. Engrg. Sci., 27(1) (1989) 29-54. [23] C.M. SELLARS, W.J. Mc TEGART, La relation entre la rrsistance et la structure dans la d6formation ~ chaud, Mere. Sci. Rev. Met., 63 (1966) 731. [24] M. MOTUMURA et al., On relating the resistance to deformation of commercial pure aluminium and aluminium alloys to strain, strain rate and temperature, J. of Jap. Inst. of Light Metals, 9 (1976) 432. [25] O. RICHMOND, Microstructure-based modelling of deformation processes, J. of Metals,4 (1986) 16. [26] L.A. LALLI, A.J. DE ARDO, Experimental assessment of structure and property predictions during hot working, Met. Trans. A., 21 (1989) 301. [27] T.J. LU, C.L. CHOW, On constitutive equations of inelastic solids with anisotropic damage, Theoretical and Appl. Fract. Mech.,14 (1990) 187-218. [28] J. JU, On energy based elasto-plastic damage theories : constitutive modelling and computational aspects, Int. J. Solids Struct., 25 (1989) 803-833. [29] J.C. SIMO, On a fully three-dimensional finite strain viscoelastic damage model: Formulation and computational aspects- Comp. Meth. Appl. Mech Engrg, 60 (1987) 153-173.

140 [30] J. C. GELIN - Application of an implicit method for the analysis of damage with temperature effects in large strain plasticity problems, In Numerical Methods for Nonlinear problems, Ed. by C. Taylor et al., Pineridge Press (1986) 494-508. [31] J.C. SIMO, M. ORTIZ, A unified approach to finite deformation elasto-plastic analysis based on the use of hyperelastic constitutive equations, Comp. Meths. Appl. Mech. Eng. 49 (1985) 221-245. [32] N. ARAVAS, On the numerical integration of a class of pressure- dependent plasticity models - Int J. for Numer. Meth. in Engrg., Vol.24 (1987) 1395-1416. [33] J. LEMAITRE,J.L. CHABOCHE, Aspects ph6nom6nologiques de la rupture par endommagement, J. Mech. Appl., 2(3) (1978) 317-365. [34] S. MURAKAMI, Mechanical modelling of material damage, Trans. ASME, J. Appl. Mech., 55 (1988) 280-286. [35] D. KRAJCINOVIC, Constitutive equations for damaging materials, Trans. ASME, J. Appl. Mech, 50 (1984) 355-360. [36] N.L. DUNG, Prediction of the fracture initiation using a finite element method and various damage models, Proceedings of Plasticity 91 (1991) 607-611.

~

~

~

"

0~

~"0

oO ~

~

=

OO

~

~

~

~o

~

~

0

~

o

o

,..,.

0

in,.

,-... 9

~

~., ~-,

0

,._

o~=

0

~" ~ '

~~o

0

=

~

0

I~

9

9

~

~,

o ~ 0~ ~

~

. . . .

oC~

~ o

~

~

~

~ ~

~ 0~

~ ~~-

~

~ ~

~

-

~

-0~

~ ,.,.

~

- ~

~.~

~ ~D

~

~ ~ ~

~

~

~...~".

~

~~

~.~

,.

,-~ ~

~ 8

~

~D

~,-"

o~ ~~-

~.

~

0

~ .

~

o~.

~. 0

~

~ ~ ~

~ o o o ~

<

~ ~

~

"

~

~"

~ . ~ o ~ . g

-~--~

~ ~ ~ .

~" ~

~~o

o ~ ~ ~o

~.~y =

~.=~-

~

~" .

~ ~ ~-

~

8

~

~

....

~.

o

~ ~ 0~ ~

~~

~ ~ ~ ~~o ~ :~ ~ ~ ' ~

o =

o

~"

o~

9

.

~

--.

o.

0~

1.-,~

<

~

~

~.

~

~" ~

~

n~

~

~-'

~

o

0

0

-"

o

~--,.

~

o

,-"

:~

0

b~

'

O0

imJo

.~

"I

~,

":

~.~

;:r

~'o v

"

142 is worse, the conductance of most of which is about 1/15 that of aluminium alloy. The severe control condition is required during titanium alloy forging process because it is easy to produce overheated phenomenon and makes its structure and properties worse. The nominal composition of titanium alloy T C l l , which belongs to two phase titanium alloy,is Ti-6A1-3.5Mo-2Zr-0. 3Si. It has two kinds of microstructure under normal temperature,one is the acicular Weisz structure with good elevated temperature creep property obtained by heat treatment heating above the phase transition temperature (990~ another is the equiaxed ~-[-[3 structure with good elevated temperature fatigue property obtained by large deformation and heat treatment under phase transition temperature. The acicular t3 structure can be translated into the stable equiaxed a+[3 structure with the alloy's being heated and deformed under the phasetransus temperature. The property and extent of this transus is depend on heating temperature and strain. According to this, it is practicable to control the final structure and distribution by controlling blocking shape and forging parameters. In order to implement the computer-aided design and manufacturing ( C A D / C A M ) approach,detailed mathematical models of the workpiece material, the forming process, and the workpiece-tool interface conditions are required. However, these models did not exist in a sufficiently complete form. Nowadays, the finite element analysis of upsetting processes are subjected to uniform temperature field or added heat conduction which is not in conformity with practical produce. In fact,the initial state is a non-uniform temperature field,conversely,the heat produced during deformation effects the distribution of temperature field. However, there are no data for these models which could be referred. The objective of this paper is to establish and prove the model required and combines them to form a computer program of interactive mode used in forging design. The test of the deformation resistance in high temperature for this material is carried out as well. 2. THE M A T E R I A L M O D E L IN FORGING CONDITION

To establish a workpiece' s plastic flow equation or a constitutive equation the uniform compression tests under various temperature and strain rate are carried out.

143 2 . 1 Test process

1)Test samples

O 1_1

t

~6.4

f

__1

....

~8. o Figure. 1

I

--1

Size of sample

The initial structure which conforms with that of forging stock is 13 structure. The size of the sample is shown in Fig. 1. Lubricating pits have been caved on both ends. 2 ) T e s t temperature distribution The test temperature chosen should cover the entire heating range (T =800~-1050'C, ~= 1 0 - 3 ~ 2 . 5 s - ~ ) . In view of the active area of the dynamic-recrystallization and phase transition ( T = 9 0 0 ~ 1 0 0 0 ' C ) t h e points of test temperature are added fitly. Hence eight test temperatures are chosen: 8 0 0 C , 8 5 0 C , 900'C, 9 3 0 C , 960'C ,980"(2,1010"C,1050:C. The deformation of each strain rate point is above 50 percent reduction of basic height. All tests were conducted on a Gleeble-1500 material heat modeling test machine. The test temperatures and strain rates are keeped at constant . All test data are recorded automatically by computer system. 2.2 Experimental results and analysis The s t r e s s - s t r a i n curves by the amendment of data and temperatures are shown in Fig. 2. It can be seen that:

144

400

~,,,

250

(a)T=850C

~(b)T=

900C

2oo

300

[..,

200 U~

15o 100

m~

[-.,

[..., 100

5O |

0.2

200

0'.4

i

0.6

0.'8

TRUE STRAIN

0

(c)T=930 C

150

~

_

9

A

i

0.2 0.4 0.6 TRUE STRAIN

018

.(d)T=960 C

~'150

r~ 100

r~

[..., 100

&--10-,S-,

r.l.1 [-,

~---10-zS-i

50

~=10-3S-1 0.2

0.4 TRUE

0.6

t0.8

~-----lO-~S-I 0

0.2

STRAIN Figure2

0.4 TRUE

0.6

0.8

STRAIN

The stress strain curves

1) There is an extensive and obvious softening phenomenon in the alloy T C l l of forging condition. 2) Alloy T C l l is a kind of material sensitive to the strain rate. The more the strain rate, the more deformation resistance. 3) T C l l is also sensitive to temperature under phase transition point (990'(2). The temperature leads a large deformation resistance. 4) Alloy T C l l has a "stress peak" at the initial deformation, the strain at the peak stress point is nearly equal to the critical strain at the

145

beginning of dynamic-recrystallization r 1 6 2 is not c o n s t a n t , b u t changes as deformation temperature and strain rate vary. The constitutive equation for T C l l could be obtained by multivariate regression way for test data. Being sensitive to temperature, low conductivity and extensive deformation softening phenomenon,alloy T C l l gets non-uniform deformation during forging processes,which easily produces overheat structure,leads the workpiece properties worse and defects. 3. THE SIMULATION OF THERMO-VISCOPLASTIC FEM FOR TC11 WORKPIECE FORGING DEFORMATION

Some deformation field information which can not be obtained by physical experiment could be studied by computer simulation , such as deformation shape at any t i m e , s t r a i n , s t r a i n rate,temperature and so on at any points. According to the deformation field we could identify whether forging defect exists in an area, or according to the value of strain and strain rate, the structure change trend in an area could be identified. If the preforming shape and forging parameters could be changed, rational process parameters could be chosen conveniently and quickly on the computer and the necessary test could be reduced to the minimum. Because alloy T C l l is sensitive to strain rate, neglecting elastic deformation and using the rigid-viscoplastic models are more practicable. 3.1 The contitutive equation of deformation Processes The contitutive eqution of deformation processes is: 2(5.

~t

iJ ~

- - E: ij --3 - g.,

where o'~i is deviatoric stress t e n s o r ; 8 , ~ , i s reffers to equivalent stress and equivalent strain rate respectively. In general, the equivalent stress is the function of total strain,strain rate and temperature ,which will be given by: o=o(~,~,T) then for the incompressible rigid-viscoplastic material its energy funtional is taken as r~-- fvE(~j)dv-- I Sp F~,ds + I v k ~,j~,jdv

146 Here,plastic work E(e~i)can be shown as E(e~i)= [ ode 1 Where: ), express Lagrange's multiplier, and X = -~ o~, &i is Claisen tensor. The second item in the equation expresses external force work. General, forging is carried out under elevated temperature, the mechanics properties of forging workpiece,in especial titanium alloys, change obviously as the temperature is varified. During forging processes accompanied by deformation heat, if the die temperature is lower,relative large temperature gradient between the workpiece will be formed due to the heat loss created by heat conduction and radiation between the workpiece.the dies and the environment. Hence, the effect of temperature on forging processes must be considered. During elevated temperature deformation processes the plastic deformation work could lead local temperature high, phase transformation, and structure bad, conversely, which produces the change of deformation-resistance and mechanics properties and affect the plastic process itself again. According to this ,the calculation of deformation and temperature should be carried out by coupled way. Heat conduction energy balanced equation can be given as: KT,ii-qt-Jr--lo c "i"-- 0 Where:K T,ii refers to heat conductance;J r heat generation rate;p cT internal energy change rate. In general, the heat generation rate in the forgings mainly comes from plastic deformation work ~r --- ~Oij~ij

is usually O. 90,after the boundary condition is taken,the variational form of which can be found IvNT.iIT.,dv-~- I P C T I T d v - - I v vaii~ii~jTdv-

Is

q.iTdSq=0

Where:q, refers to boundary heat flow sagitta. The temperature of which is written into nodal temperature interpolating function,its matrix form is: C T+KT=Q Where.C refers to the matrix of heat capacity;K the matrix of heat conduction,Q the sagitta of heat flow. The following several factors are considered in the sagitta of heat flow:

--"

~

=

o

~

"

~-=

~

o~

""

~4 ~

~

~

--~

.

"

o =

~

,-,.

~.

~ o -~. . ~

~.

g

9

~,

~

~

. ~o ~

~

~'8 ~

9

~

~..

~

~D

~ ' ~

~

~"

~

~

~-~

-"

o

r~

~

F

~I

~

"~

~

,

.

~

~

~

ff

~

o

~

O~

~

~

~

~

9

~..~

-

:~--, 0~

~B.~

~

~

....

~

~ ~"

=~

~'

~m'~

~

~

~-~

~

.

~ ~

=

~

.

~~~

~*--, ~ . .

~.~.=

o

~

o

~

-.

;

~

0

..

T

I~

...

O0

0

p..,~

r

Cb

0

db

Oo

0

+

-].

0

-co

g

~"

"C~

"

o

~

~

II

o

r

.

-

o

o

=~.

~

~

r

~"

~

0

~

=

:=r

0

~

~

~

=

~

cr~

~

~"

e

'-" o~

~

O~

~o~

9

~ ] ~ ~

9

db

,.~

0

o

~

~

~

~)

|I~

~

9

~

"-" 0

O~

r

"~

0~-~

o'q

.

0

=-.o

>~

~

""

9 ~.

. ~ ~

148

die

1.14 ~~'~'~9"~0"79/ ~__~____~0Q97

X \ O.60\

1~.,:--2:1.32,1.14v \ ~" \ ~ 1

51 ~"

-

\i

i

(a)lsogram of effectivestrain (die)

494~486~ ~'--~--511--:518~-'~'~ \ ~1~16!'1173-~1185~~_..._.~197~ ~ 1220 ~1

]21 I ]209 i--~..:._.,!233 l'''''-'--'-'~-20/

~/~ 12~~

(b) The temperature of workpieceand die Figure. 3 Size of workpiece ,{~250X 300mm,the initial temperature of workpiece T--1203K.

I

die

|

.

475

9

~1.6Gz!-46,~._2~\ \ .. -

\

{a) Isogram of effective strain

___1[...._12~38~242~1244 ~

4

6

J \\\ \ \ X ~~X~ ~

(b)The temperatures of workpiece and die Figure. 4 Size of workpiece ~250 X 300mm,the initial temperature of workpieee T=1233K

149

_f\ (a)Isogram of effective strain die 472 171

'

1245

(rA

240

(b) The temperature of workpiece and die Figure. 5

Size of workpiece Zf290 • 150mm,the initial

temperature of workpiece T = 1233K. 3)According to the distribution of temperature,the temperature rises led by deformations are of frequent occurence. On the parts of the stock near the dies, the temperature distribution is mainly controlled by the boundary's heat exchange. While in the center and diagonal areas, the thermal conductivity for titanium alloy is relatively bad and the deformatoin rate is fast, the rising of temperatures created by deformation is not easy to release, so the temperature distribution is mainly controlled by the local deformation quantities.which belongs to the lifting temperature area,its form of distribution is basically similar to distribution of deformation. 4) While titanium alloy T C l l is deformed,there will be some isolated "island"on the temperature distribution. The"island" only appears when the initial forging temperature is relatively low (Figure 3 ) , a n d it will disappear as the initial temperature is risen (Figure 4). This indicates that there exists an extremely nonuniform deformation when forging under low temperature. On the shear zone near the "dead area", the

150

drastic shearing deformation will create a quick lift of temperature higher than the areas around,and form an isolated island distribution . This kind of distribution will deepen and augment as the deformation is enlarged. 5)During the entire deformation process ,at the part near the dies, the temperature of the workpiece continuously goes down , and by the friction’ s effect, the deformation is very small, the forged structure and initial structure should have no obvious difference; but at the center area ,especially a t the center shear zone, as the deformation increases, the temperature of workpiece will increase continuously. At some local area such as the “island”, the temperature will probably surpass the phase transition point ( 990.C 1. This will create a phase transition, greatly affect the forged structure ,grain size and the performance of the forgings ,and this is undesired. 4. PROVING THE “ ISLAND TEMPERATURE FIELD

(a)At the island area Figure. 6

”

PHENOMENON

ON

THE

(b) At the center

T h e sample microstructure

In theoretic analysis for the temperature field ,is it inevitable or a fault that the ” island” appears during calculation? To prove this, the following experiment was arranged :Use titanium alloy T C l l stocks 120 X 12Omm, the initial microstructure is a p. Given 70 percentage deformations to the stocks under two kinds of temperature: 1233K (9SO’C) and 1133K (860.C). T h e workpieces were forged into disks. Divided the disk in the center and observe the microstructure on the longitudinal direction. In the ” dead area” the microstructure is the

+

151

initial uniform c~ if- 13 . According to the elevated temperature microstructure observed, it is proved that near the island the temperature rose(shown in Fig. 6-a)and the microstructure at the center of the tested sample(Fig. 6-b) was equiaxial under the condition of low temperature (1133K). The experiment indicates undoubtedly that the heat area " island" exists, especially when the stock' s temperature is low. this will be a guide in theoretic analysising and process planning. 5. M O D E L I N G OF DEFORMATION

DYNAMIC

MATERIAL

BEHAVIOR

IN

HOT

A new method of modeling material behavior which accounts for the dynamic metallurgical processes occurring during hot deformation is presented. The approach in this method is to consider the workpiece as a dissipator of energy in the total processing system and to evaluate the dissipated energy co-content J =

jia.ao from the constitutive equation

relating the strain rate ( ~ ) to the flow stress ( a ) . The optimum processing conditions of temperature and strain rate are those corresponding to the maximum or peak in J. It is shown that J is related to the strain-rate sensitivity index ( m ) of the material and reaches a maximum value (Jma~)when m = 1. A typical constitutive relation for a simple dissipator is schematically represented in Figure 7 (a) in the form of the variation of flow stress with strain rate (flow)at constant temperature and strain. At any given strain r a t e , t h e power P (per unit volume) absorbed by the workpiece during plastic flow is given by

or

P=~247

J=Ii d~

In figure 7(a) total energe of dissipation is given by the rectangle,the area below the curve is G , t h e dissipator content,and the area above the curve is J, the dissipator co-content. The G term represents the energy dissipated by plastic work, most of which is converted into heat; the remaining small part is stored in the lattice defects. The dissipator cocontent J is related to the metallurgical mechanisms of occurred dynamic

152 heat dissipation.

J = COiCONTENT

FLOW STRESS

G+CONTENTI1 9, '

I

~-

1g

_

STPAIN RATE (a)

,-,~,~,~ , . ~ , ~ ,,,\x,, a~, \'\\5,

9

" '

'~':)~,~ " "

I

FLOw ~ ) ~ ~ f Y v m = 1II ST RESS Xk'~k~,~ ,,~z I I

STRAIN RATE (b) Figure 7 (a) Schematic representation of G content and J co-content

for workpiece having a constitutive

equation represented by curve a - - f ( e ) . ( b ) S c h e m a t i c representation showing Jm,x which occurs when strainrate sensitivity ( m ) of material is equal to one. ( F r o m

E4-1). The rate of the power dissipation ( J / J m a x ) through whole metallurgical processes is shown to be an index of the dynamic behavior of the material and is useful in obtaining a unique combination of processing temperature and strain rate and also in delineating the regions of internal fracture. Metallurgical processes such as dynamic recovery, dynamic recrystallization,internal fracture (void formation or wedge c r a c k i n g ) , separation or growth of particles or phases under

153 dynamic conditions,dynamic spheroidization of acicular s t r u c t u r e s , a n d deformation-induced phase transformation or precipitation under dynamic conditions contribute to the changes in the dissipated cocontent J. Let the rate of power dissipation be r/ J _ 2m 1 For processing of materials the most favorable conditions are those which provide the highest J dissipated in the most efficient fashion (highest r/) and lie within the "safe" regions.

Figure. 8

T C l l ( W ) stable working region ( E = 0 . 6)

The energy co-content J serves as the most useful index for characterizing dynamic material behavior in processing for the following reasons 9 1. It defines unique conbinations of T and ~ for processing (peak values of J and r/) and also distinguishes the regions which produce internal fracture. 2. It is a continuous parameter and can be integrated with the finiteelement analysis. From it ,an algorithm can be developed which can be incorporated into process defects control. T C l l is a a-b-t3 titanium alloy whose hot-working characteristics are

154 element analysis. From it ,an algorithm can be developed which can be incorporated into process defects control. TC11 is a a q-t3 titanium alloy whose hot-working characteristics are very sensitive to the initial microstructure and processing parameters. Fig 8 is an isograph of r/for T C l l (W) at 0. 6 strain. It shows TC11 (W) stable working region. 6. CONCLUSION Defects occurred in forging process of titanium alloy TC11 referred to complex energy field change and dynamics metallurgics behavior. 1. Coupled thermo-viscoplastic FEM simulation is the base for titanium alloy forging defects analysis. 2. Due to the deformation heat and the poor thermal conductivity for titanium alloy,the so--called isolated "island"exists in temperature field after forging process ,and it may induce local defect. 3. The dynamic metallurgy analysis offers useful referential judgement for optinum forging parameters choice and avoidance of defects. REFERENCES 1. G. D. Lahoti, T. Altan, Research to Develop Process Models for Producing a Dual Property Titanium Alloy Compressor Disk, AD/ Al12271 ,Interim Technical Report ,AFWAL-TR-81-4130,1"--7,19~21,52~'65,267~-324,Oct. 1981. 2. S. I. Oh, J. J. Park, S. Kobayashi, T. Altan, Application of FEM Modeling to Simulate Metal Flow in Forging a Titanium Alloy Engine Disk, Transactions of the ASME, November 198:], Vol. 105,251-258. 3. Chen Sencan, Hu Zongshi, Wang Shaolin, et al, Research on Forging Processes for Producing Two Phase Titanium Alloy T C l l Disks, Journal of Tsinghua University Vol. 32. No. $2 1992. 4. Y. V. R. K. Prasad, H. L. Gegel, et al, Modeling of Dynamic Material Behavior in Hot Deformation. Forging of Ti-6242, Metallurgical Transaction, Volume 15A, October 1984.

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

155

M o d e l l i n g of F r a c t u r e I n i t i a t i o n in M e t a l f o r m i n g P r o c e s s e s Y.Y. Zhu, S. Cescotto and A.M. Habraken M.S.M. Department, University of Liege, 6 (BELGIUM)

Quai Banning, B-4000

LIEGE

Abstract

In this paper, two kinds of approaches for modelling the fracture initiation in metalforming processes are reviewed. One is an uncoupled approach based on various published fracture criteria; another one is a coupled approach based on the continuous damage mechanics. Recent development of energy-based isotropic damage model with two damage variables is described in some details. A viscous regularization of the Duvaut-Lions type is also proposed to take into account effects of strain rate and mesh sensitivity. Both fracture criteria and damage model have been implemented in finite element code and compared with experimental work. It leads to the conclusion that the described damage model is a powerful tool for predicting material processing defects.

1. I N T R O D U C T I O N Ductile fracture of metals implies the appearance of damage processes which grow gradually. Many investigations [1] have shown that ductile fracture involves four successive damage processes which are the nucleation of void from inclusions, void growth, void coalescence (onset of a crack) and cracking propagation. From the viewpoint of application in metalforming processes, it is very important to define the fracture event, because the ultimate stage of the workpieces is preceded by or corresponds to crack initiation and propagation. When a material is formed by processes as forging, rolling, drawing, etc. it experiences large unrecoverable deformations. These deformations load to the development of zones of high strain concentration and, consequently, the onset of internal or surface cracks. The strain localization is the cause of many defects. For example, free-surface cracks occur in such processes as upsetting, bending and rolling; internal cracks in extrusion and drawing and in some forging processes. Although the appearance of a crack during the deformation process is, in most of the cases, undesirable, in some particular situations of deep drawing for example, the initiation and propagation of a crack is sometimes expected in order to soften the behaviour of the sheet in a zone that will be cut off at the end of

156 drawing operation. Furthermore, during metal cutting, the removal of chips is only possible because cracks have been created in the machined part of the cutting tool. Thus cracking is an inherent part of such processes. For occurrence of surface cracks, the fracture criterion may be constructed experimentally. However, for predicting internal fracturing, formulations of fracture criteria under general deformation are required. Since damage processes still remain difficult to define and proper mechanical models are not yet fully developed, recently, many methods have been investigated. [2-4]. There are two kinds of approaches, including uncoupled and coupled ones. In the uncoupled approach, the damage is computed from the stress and strain fields but does not modify these fields. The onset of fracture is determined according to the fracture criteria using the classical constitutive laws. By using the finite element method in conjunction with the fracture criteria, numerical predictions of the fracture event and its initiation sites are obtained. Maps of each cumulative fracture criterion value are computed and a crack occurs where one criterium reaches or exceeds its threshold value experimentally measured. This approach is well adapted to the cases where the redistribution of the stresses due to damage can be neglected, and is thus generally sufficient for most initiation fracture analyses. For example, Oh et al [5] employed a rigid plastic finite element technique to examine the use of the Cockroft and Latham's criterion [6] and a modified version of the McClintock's criterion [7] to predict fracture in axisymmetric extrusion and drawing. Sowerby et al [8, 9], Dung [1013] used a rigid plastic finite element model to examine the capability of Mc Clintock's void growth model, Crockcroi~ and Latham's criterion and Oyane's formulation [14] to predict damage accumulation in the upsetting of steel specimens. Their numerical results showed that the McClintock's model is appropriate to assess the forgeability of some steels. Clii~ et al [15-17], Pillinger et al [18] and Hartley et al [19] presented an investigation on the ability of elastic-plastic finite element simulations to predict the initiation of ductile fracture in bodies undergoing large plastic deformation. They found that the criterion based on generalized plastic work (Freudenthal's model [20]) was the most successful. In the coupled approach, the damage processes are incorporated into the constitutive relations. In this case, the redistribution of stresses or strains due to the damage accumulation is taken into account. Prior to achieving the critical amount of damage, the stress distribution based on the coupled approach is similar to that obtained with the uncoupled one. However, the coupled method gives a more accurate numerical simulation because the damage development and stress drop continue aider the onset of void coalescence. Therefore, after local fracture initiation, further damage can cause stress redistributions that will automatically induce fracture propagation as long as the coalescence criterion is exceeded and the large crack extensions can be simulated continuously. This approach implemented in a finite element code allows the prediction of defect occurrence. For examples : Aravas [21] studied the behaviour of microvoids nucleated at second phase particles during direct axisymmetric extrusion, using large deformation finite element analysis together with Gurson's constitutive

157 model. Onate [22, 23] found a formal analogy between the equation of pure plastic and viscoplastic flow for void-containing metals (Gurson's model [24]) and those of standard nonlinear elasticity. According to this approach, the effect of nucleation, growth and coalescence of voids could be treated by classical nonlinear elasticity, that is, to allow standard finite element formulations developed for elastic problems to be used for the analysis of complex metalforming processes including the effects of voids. Predeleanu et al [25], Gelin [26] proposed a finite strain elastoplastic model incorporating ductile damage mechanisms of Lemaitre's theory [27]. Their model included the strain softening of the material when ductile fracture occurs. Tirosh [28] suggested a computational procedure to couple the porosity of the material and the impact loading for solving explosive forming processes with materials which obey Gurson's yield criterion. A more detailed review of the applications of second approach to metalforming processes can be found in [2-4]. A satisfactory coupled constitutive relation should not only describe the initiation and propagation of fracture but also check the efficiency of the fracture criterion. Therefore, it is still necessary to implement several fracture criteria into the coupled constitutive law, on the one hand to define the new damage variables, on the other hand to determine the critical values when the material points fracture [29].

2. F R A C T U R E CRITERIA There are a lot of fracture criteria. It is obviously advantageous to keep the number of experimentally determined parameters to a minimum [3]. In this paper, only six previously published fracture criteria are chosen. In the following formulae, Cl, c6 are the critical material dependent values, at fracture they are denoted by the subscript f; A, K are material constants to be determined from experiments; ~1, c2, ~3 are the principal stresses; ~m is the hydrostatic stress; ~, e are the equivalent stress and strain. 2.1. F r e u d e n t h a r s m o d e l Freudenthal [20] proposed that the absorbed energy per unit volume is the critical parameter at fracture, that is 9

~o etude =Cl

(1)

This criterion does not consider the influence of hydrostatic stress and high tensile stress explicitely. 2.2. C o c k c r o f t - L a t h a m ' s m o d e l Cockroft and Latham [6] proposed that it is the principal tensile stress, rather than the equivalent stress, which is important in fracture initiation. They postulated that fracture occurs when the integral of the largest tensile principal stress component over the plastic strain path to fracture equals a critical value for the material, namely : m

= c:

<2)

158 This criterion does not take into account the influence of hydrostatic stress explicitely on fracture. 2.3. Brozzo's m o d e l Brozzo et al [30] proposed an empirical modification of Cockrot~ and Latham's model to consider the effect of hydrostatic stress explicitely. The modified equation is presented in the following form" .~o e

2~ 1

3(Ol-Om)

(3)

de =c3

2.4. G h o s h ' s m o d e l Ghosh [31] proposed a fracture criterion for plane strain state based on the statistical process of shear joining of voids in sheet metalforming. Here, we extend it to three dimensional state, that is 9 (~1 + ~2 + ~3)(~1 - c~3) = c4 (4) 2.5. O y a n e ' s m o d e l

Oyane et al [14] considered a void growth model and postulated that" (5) 2.6. M c C l i n t o c k ' s

model

McClintock et al [7] analyzed the expansion of long cylindrical cavities under a triaxial system of fixed orientation. They derived a closed form expression for the damage accumulation 9 I

~/3(1- n) sh

2

~

=

(6)

where n is the work hardening exponent of the material which obeys the following power hardening law 9 o=Ke n (7) There are two other equations for the coalescence of voids with their axes parallel to the other principal tensile stresses. m

3. A FULLY C O U P L E D I S O T R O P I C E L A S T O P L A S T I C DAMAGE M O D E L There exist many coupled constitutive models for local approach of ductile fracture, such as Gurson's models [24, 32], Perzyna's model [33], Lemaitre's model [27], Rousselie's model [1], Ladev~ze's model [34], Simo-Ju's model [35]. In the nonlinear finite element code LAGAMINE (developed in MSM Department, University of Liege), elastoplastic Gurson's law, damage Ladev~ze's law have been introduced. More recently, new fully coupled elasto-(visco)-plastic damage models for isotropic and anisotropic materials have been implemented [29]. Hereai~er, the isotropic damage model will be reviewed briefly.

159 3.1. D a m a g e v a r i a b l e s The damage variables, based on various equivalence hypothesis, represent the average material degradation which reflects the various types of damage at the microscale like nucleation and growth of void, microcracks and other microscopic defects. Ladev~ze [34] suggested a damage model with two scalar parameters d (deviatoric component) and 8 (volumetric component) by which not only the elastic modulus but also Poisson's ratio can vary with the damage growth. In this paper, these two damage variables are used. The true stress tensor g_can be transformed into the effective stress tensor G with the help of the damage variables d, 5, viz G' Gm G' = - -1-d'

Gm =

(8)

1-5

with G' the deviatoric stress tensor, Gin the hydrostatic stress. The coefficients (l-d) and (1-8) in (1) are reduction factors associated with the amount of damage in thematerial. 3.2. Equivalence

Hypotheses

For continuum damage models, various equivalence hypotheses have been proposed in order to transform the damage state into virgin state, such as strain equivalence [27], stress equivalence [35], elastic energy equivalence (Cordebois and Sidoroff [36]). From the viewpoint of energy conservation, the energy equivalence may be of more physical significance. In this paper, we propose an extension of this hypothesis, in the case of the two damage variables d, 8 model. Finally we can obtain the following relations between damaged material and virgin material.

e_. =e_ (1-d),em=em(1-8),ePq=ePq(1-d)

(9)

Here, e' is the deviatoric strain tensor, am the average strain, eePq the equivalent plastic strain. 3.3. P l a s t i c yield s u r f a c e The yield function Fp used in this paper is made of the energy-based Von Mises yield criterion with both isotropic and kinematic hardening, in the form : Fp

V

4--G~l:dfi- - R ~

R(a)

(10)

in which 2: is the deviatoric component of the shii~ stress tensor 2, Ro denotes initial plastic hardening threshold; R is plastic hardening threshold; a is accumulated plastic energy variable. Fig. 1 shows the corresponding yield surface for different values of the damage variables. We can see that with the growth of damage, the elastic region is reduced.

160 .o2/ay

!o o. l,,i

2.

O.

d=O-

~

.

"

.5

.o.s

i

.2

.2

/ -x'.5

d,-O.

9

~

allay q.

9

1./-(!

.///. -

l.s

z

Fig. 1 Plastic yield surface 3.4.

Damage

evolution

surface

In order to describe the growth of damage, several damage evolution criteria can be defined, such as, strain-based criterion [27, 37], stress-based criterion [38, 39], energy-based criterion (Ladev~ze [34]). In the present model, a modified energy-based damage evolution criterion is proposed: Fd = - Y d - < x > YS-Bo-B(I 3) G"G' < 1;>G2 (11) = 9 +__m _ Bo_B(~) 2G(I- d)3 9((I-8)3 With the definition, < ~ > = 8/d forGm >0; (12) 0 forGm_< 0 the difference of mechanical effects observed under tension and compression states can be described. Here B o denotes initialdamage strengthening; B is the damage strengthening threshold, [3is overall damage. The physical meaning of (11) is that the negative hydrostatic component does not contribute to damage evolution. Fig. 2 displays the evolution of the initial damage surface in stress space.

=o

~ _ _ ~ = 0 .

Fig. 2 Damage evolution surface

161 3.5. V i s c o u s r e g u l a r i z a t i o n of i n v i s c i d d a m a g e m o d e l s The local approach of ductile fracture based on the coupled constitutive theory is a useful tool to predict initiation in ductile fracture condition and enables the analysis of the propagation of completely damaged zones. However, developments are still needed especially in the case of very localized zones to handle the possible instabilities and bifurcation of the solution corresponding to local strain softening or loss of positive definiteness of the global stiffness matrix [40]. In rate independent materials, the localization corresponds to a bifurcation of the local behaviour of the material and to the occurrence of strain rate jumps through singular surfaces [37]. In some cases capturing the shear band of localization have the problem associated to the mesh dependency. Viscous regularization seems to be one of powerful approach to solve localization problem associated to material softening [35, 41, 42]. In fact, in viscous models, there are no plastic and damage consistency conditions, thus no strain rate jumping phenomenon [40]. On the other hand rate dependence naturally introduces a length-scale into the dynamic initial boundary value problem. In the present isotropic model, there exists non-smooth corner regions between the plastic yield surface and the damage evolution surface. The softening phenomenon can be captured with this model. As an extension of the proposal of Simo et al [44] for non-smooth multi-surface viscoplasticity and the suggestion of Loret and Prevost [42, 43] for softening elastoplasticity, we construct the viscous regularization of the present rate independent damage model by Duvaut-Lions form [45] as : ~(0) + At / * -n+l n ~n+l -~ = l+At /[t n -Tn(0) + l + At n /ktT* -n+l -Tn+l = l+At /kt n

qn+l=

qn +At n /l.tqn - +1 l+At /~

(13)

n ,

in which, qn + 1 is the vector of state variables, (~(0) (0) are the solutions n+l and Tn+l , * of the elastic predictor step, ~ n + l , T n + l are the inviscid solutions of isotropic elastoplastic damage models, ~ is the viscosity coefficient. More details on this model can be found in [29]. 4. N U M E R I C A L EXAMPLES AND D I S C U S S I O N S 4.1. D y n a m i c forging and fracture c r i t e r i a As an attractive example, let us consider a dynamic contact modelling of steel forging at 1150~ with the uncoupled approach. High strain rates and large variations of the contact area are effective in this example. The material

162 properties of the workpiece are assumed to be represented by an elasto-viscoplastic constitutive equation [46] in which all the parameters are determined according to the given temperature 9Young's elastic modulus E = 1.2 x 105 MPa; Poisson's ratio = 0.4; strain rate exponent n = 9.259; strain rate coefficient B = 0.034; initial yield limit K o = 50 MPa; the mass density p = 7800 kg/m 3. For definition of contact elements, the penalty coefficient on the contact pressure Kp = K~ = 5 x 1013 N/m 3 and the Coulomb's friction coefficient ~ = 0.3 are chosen.

discretiz~

a crack appears here

i

plane O!symmetrY ,

i i

before forging

after forging

Fig. 3 Dynamic forging This simulation corresponds to a practical case of metalforming in which a fracture was observed during the forming process. Although the actual piece was three dimensional, the region in which the crack developed could be adequately modelled as axisymmetric (fig. 3). Furthermore, due to the existence of an horizontal plane of symmetry, only one half of the piece is discretized. Since the strains are very large, the choice of an appropriate initial finite element mesh becomes a very important aspect. In fact, it is necessary to make sure that the simulation results are practically mesh independent. Therefore, three different meshes are used (fig. 4) 9'2~IESHI" 6-node firfite elements with constant mesh density; '2VIESH2" 6-node finite elements with variable mesh density; '2VIESH3" 8-node finite elements with variable mesh density. Fig. 4 shows the initial meshes (solid line) together with corresponding deformed configurations (dashed line) at time t=0.5 ms obtained by implicit dynamic scheme. At this time, fracture initiation sites of workpiece near the comer of the die can be observed experimentally. On fig. 4, we can observe that the elements near the right corner of the die are severely distorted. This means that the finite element mesh should be refined in the region of the flash and that the solution presented on fig. 4 does not model the flash with precision. However, we are more interested in the region near the lef~ re-entrant comer of the die, where the mesh is not too distorted. All the numerical simulatin show that the accumulative values of the six fracture criteria present a very sharp maximum near this corner at time t = 0.5 ms. These values are given in table 1. The location of these maxima are indicated by a cross on fig. 5 (the results are those given by implicit dynamic simulation with MESH3). On the same figures, other crosses appear, in the flash region. This means that the accumulative values at these Gauss integration

163 points are equal or larger than at the point near the left comer. Hence, in the flash, there are some points at which the critical value is larger than at the point near the lef~ corner of the die. However the flash will be cut off at the end of the forging process. Furthermore, it was pointed out that the solution in the flash is not reliable because of the excessive mesh distortion.

(a) mesh 1

(b) mesh 2

j.

(c) mesh 3

Fig. 4 Initial and deformed meshes

i Criterion Mesh 1 Implicit Mesh 2 Implicit Explicit Mesh 3 Implicit

I

Table 1. V ~ u e of fracture criteria 1 2 3 4 8.84 2.03 1.23 4.02 6.85 1.12 0.70 20.1 7.05 1.22 0.76 20.2 9.17 0.91 0.66 16.0

5 0.45 0.29 0.32 0.27

6 3.16 2.51 2.63 3.31

,.

Table 1 shows the broad agreement of accumulative values of each criterion given by different meshes and different time integration schemes. The differences may be due to the discretizations. Fig. 5 indicates that the fracture initiation locations based on each criterion are almost the same, and the damage accumulations are very local near the lef~ corner of the die which is confirmed by the experimental results. This may be due to the existence of high stress concentrations and localization of strains near this corner.

164

~

kp

(a) c r l t e r i o n - i

(b) c r l t e r l o n - 2

(c) c r i t e r i o n - 3

(d) crlterlon-4

(e) c r l t e r l o n - 5

(f) crlterlon-6

Fig. 5 Fracture criteria for mesh 3

165 4.2. C o l l a r tests, c o m p a r i s o n s b e t w e e n e x p e r i m e n t s , f r a c t u r e c r i t e r i a

and damage theory The upsetting of a circular cylinder is often used to assess the cold forgeability, but with ductile materials, the test can result in excessively high loads before surface cracking occurs. To overcome this difficulty, some alternative upsetting procedures are described in literature and so-called collar tests are recommended when studying the upsetting of ductile materials [8-11]. The collar tests often result in lower fracture strains in comparison with the upsetting of the circular cylinder [8]. In the present collar tests, two kinds of specimens are used : specimen with one flange as shown in fig. 6.a; specimen with three flanges as seen in fig. 6.b. The ratio of the height to the diameter must be low enough to prevent buckling but large enough to give sufficient deformation to induce fracture. All the tests were terminated when a surface crack could be detected with the naked eye.

COMPLEX UPSE'IqqNG (COLLAR TESTING)

J

l

t ee~

l

,

I

I

20

2O , -

t----

t__.

30

~

--|

3O

(b) with three flanges

(a) with one flange

Fig. 6 Sizes of collar specimens (a) with one flange (b) with three flanges.

~(MPa)

B+Bo (MPa)

//

~}0.,

3~).

200.

20.

I00.

I0.

~.

~b.

p(~)

:~. ~. ~(s)

(a) effective stress - strain c u r v e

(b) damage evolution

Fig. 7 Material properties of aluminium

166 The isotropic damage constitutive law for aluminium is determined with a uniaxial tensile test [29]. The resulting p a r a m e t e r s are : E = 7.47 x 1010 Pa, v = 0.333, ~ = 3.5; ~ = 10 -5 s; the effective stress-strain curve for virgin material and the damage B-~ curve are shown in fig. 7. For contact, the Coulomb's friction coefficient ~ = 0.17 is used.

Fig. 8 Final deformation at fracture (a) specimen with one flange, (b) specimen with three flanges Fig. 8 shows the experimental results of final deformation at fracture for bush specimens respectively. The cracks appear to propagate inwards to a depth of 2 to 3 m m and cover the full height of the collar. It is also found t h a t these cracks are at approximately 45 ~. It means t h a t the flange at the equatorial free surface is fracturing with a "shear-type" failure mode. At high height reduction, some other smaller cracks n e a r the contacting sites between the free surface of body and the flanges can be observed for the collar test with three flanges. The theoretical distributions of Von Mises stress, of deviatoric and volumetric damage variables, and of the six fracture criteria are shown in fig. 9. As expected, the m a x i m u m values of volumetric damage variable 5 and fracture criteria 2 to 4 are located in the collar. In general, the hydrostatic stress o m becomes larger at the equatorial free surface. Cracks are usually formed there owing to high tensile state where the Von Mises stress may not be too high. The distribution of fracture criterion 1 and criterion 6 are similar but they do not give ideal prediction of fracture for the present collar tests.

167

Fig. 9 Distribution of stress, damage and fracture criteria at 50 % height reduction ((a) for specimen with one flange)

168

Fig. 9 Distribution of stress, damage and fracture criteria at 50 % height reduction ((b) for specimen with three flanges)

169 5. C O N C L U S I O N S AND REMARKS In this paper, two approaches for modelling of fracture initiation in metalforming processes have been presented. The uncoupled approach based on various fracture criteria which is very easily introduced in any structural analysis code. The corresponding postprocessing induces very low extra computation costs. This approach is generally justified if the redistribution of stresses and strains due to damage can be neglected and is thus generally suited for most predictions of fracture initiation. However, it should be pointed out that until now, none of the criteria mentioned in this paper could adequately describe the observed behaviour for all types of experiments [29]. In reality, the plastic damage, loading ultimately to failure, can be caused by many different mechanisms, such as internal and external necking, large shear deformation, nucleation, growth and coalescence of voids and so on. Any one or more mechanisms can cause final rupture. Therefore in order to predict fracture initiation, several criteria should be implemented together, each of them describing more accurately the different mechanisms. The fully coupled approach based on the continuum damage theory is of course the most attractive one. In this paper, an energy-based isotropic damage model has been proposed to characterize microcrack initiation and growth in ductile materials. Rate-dependent effects are accommodated and the numerical problem of mesh dependency is improved by means of viscoplastic regularization of Duvaut-Lions' type. Therefore the proposed damage model is a useful tool for modelling of fracture initiation and propagation in metalforming processes. Our further research will be focused on the extension of the present model to a new one with non local damage framework in order to completely avoid the meshdependency problem.

REFERENCES G. Rousselier, Nuclear Engineering, Design, 105 (1987) 97. 2. M. Predeleanu, Computational Methods for Predicting Material Processing Defects, Elsevier, (1987), 295. Y.Y. Zhu and S. Cescotto, Programme mobilisateur multimat~riaux de la r~gion wallonne, n ~ 1, Contrat n ~ 1758 (1991). J.C. G~lin and M. Predeleanu, NUMIFORM'92, (1992) 214. 5. S.L. Oh, C.C. Chen and S. Kobayashi, Trans. of ASME, 101 (1979) 36. 6. M.G. Cockroft and D.J. Latham, J. Inst. Metals, 96 (1968) 33. 7. F.A. McClintock, J. Appl. Mech. 35 (1968) 363. 8. R. Sowerby et al., J. Engng Maters. Tech., 106 (1984) 101. 9. R. Sowerby et al, VDI-Forschung im Ingenieurween 5 -(1985) 51. 10. N.L. Dung, Forsh. Ing. - Wes., 50 -1984) 55. 11. N.L. Dung, NUMIFORM'86 (1986) 261. 12. N.L. Dung, PLASTICITY~89 (1989) 53. 13. N.L. Dung, PLASTICITY'91 (1991) 607. 14. M. Oyane, Bulletin of JSME, 15 (1972) 1507. 15. S.E. Clift et al., 25th MTDR, Birmingham (1985) 413. 16. S.E. Clift et al, Int. J. Mech. Sci., 32 (1990) 1. o

o

,

170 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

S.E. Clift, Numerical Modelling of Material Deformation Processes, Springer (1992) 406. I. Pillinger et al, Int. J. Num. Meth. Engng, 25 (1988) 87. P. Hartley et al, Res. Mech., 28 (1989) 269. A.M. Freudenthal, The inelastic behaviour of engineering materials and structures, Wiley, New-York (1950). N. Aravas, J. Mech. Phys. Solids, 34 (1986) 55. E. Onate and M. Kleiber, NUMIFORM'86, (1986) 339. E. Onate and M. Kleiber, Int. J. Num. Meth. Engng, 25 (1988) 27. A.L. Gurson, J. Engng Maters. Tech., 99 (1977) 2. M. Predeleanu et al, NUMIFORM'86, (1986) 277. J.C. G~lin, Ann. CIRP 35 (1986) 157. J. Lema~tre, J. Engng Maters. Tech., 107 (1985) 83. J. Tirosh and .D Iddan, J. Mater. Proc. Tech., 24 (1990) 203. Y.Y. Zhu, Doctoral Thesis, MSM, University of Liege, (1992). P. Brozzo et al, 7th Biennal Conf. (1972). A.K. Ghosh, Metal. Trans., 7A (1976) 523. V. Tvergrrad and A. Needleman, Acta Metall. 32 (1984) 157. P. Perzyna, Int. J. Solid Stru., 22 (1986) 797. P. Ladev~ze, Rapport interne n ~ 34, LMT, Cachan, 1984. J.C. Simo and J.W. Ju, Comp. Mech., 5 (1989) 375. J.P. Cordebois and F. Sidoroff, EUROMECH 115 (1979). A. Benallal et al, Cracking and Damage, Elsevier (1989) 295. C.L. Chow and J. Wang, Engng Frac. Mech., 30 (1988) 83. G.Z. Voyiadjis and P.I. Katton, Int. J. Engng. Sci. 28 (1990) 505. A. Benallal, R. Billardon and J. Lema~tre, Comp. Meth. Appl. Mech. Engng, 92 (1991), 141. O. Harireche, B. Loret, Eur. J. Mech. A/Vll (1992) 733. B. Loret and J.H. Prevost, Comp. Meth. Appl. Mech. Engng 83 (1990) 247. J.H. Prevost and B. Loret, Comp. Meth. Appl. Mech. Engng 83 (1990) 275. J.C. Simo et al, Int. J. Num. Meth. Engng, 26 (1988) 2161. G. Duvaut and J.L. Lions, Les in~quations en m~canique et en physique, Dunod, Paris (1977). S. Cescotto and R. Charlier, SMIRT'85 (1985).

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

171

Formability determination for production control John A. Schey Department of Mechanical Engineering, University of Waterloo Waterloo, Ontario, Canada N2L 3G1

ABSTRACT A critical feature of pressworking is that sheet contact with the die surface delays strain localization and subsequent failure. Therefore, basic tests such as the tension test and other inplane forming tests show relatively poor correlation with production performance. Only simulation tests--such as the limiting dome height (LDH), stretch-bend, and hole-expansion tests--that also involve tool/sheet contact have the potential for good correlation. This means, however, that variables relating to geometry, tribology, and other process parameters are introduced. If tests are to yield reproducible, meaningful information, the effect of these variables must be understood. Extensive round-robin tests will have to be conducted before any of the tests can be accepted as general standards.

1. I N T R O D U C T I O N Formability is a technological property and as such suffers from a vagueness of definition that results from the complexity of the reality to which it refers. Definitions can be deceptively simple: "Formability is the technical term used to describe the relative ease with which a metal can be shaped through plastic deformation" or "the ability of a material to undergo plastic deformation without fracture" [1]. Although some view formability as synonymous with workability, here we will limit the term to describe the ease of shaping in sheet metalworking, with workability more appropriately reserved for bulk deformation processes. Translation of the definition into practical terms is difficult, because formability is a system property. It may sometimes be strongly related to a material property, but more frequently it depends on unique combinations of several material properties and process variables. Complex forming processes usually combine several modes of forming, requiring different formability measures at various locations of the stamping and during different stages of the forming process. The main task is to find appropriate descriptors of formability and then develop tests that allow the determination of these properties. A simulative test for formability assessment must duplicate the critical strain state or forming mode found in the actual process. It should also be relatively be simple so that reproducible, unambiguous results can be obtained. Several valuable contributions to the subject have appeared in journals as well as books and conference volumes [2-10]; the present review surveys progress in the more recent time period, moving from applied to basic tests.

172 2. D R A W A B I L I T Y The drawing of deep cups, in which average sheet thickness remains essentially unchanged, represents a special case in that formability, now better termed drawability, is linked to the plastic anisotropy of the sheet, as expressed by the r value [11, 12]. The primary measure of drawability remains the limiting draw ratio (LDR) which can be readily and quite reproducibly determined in cupping tests [2, 13], provided that tooling configuration, blankholder pressure, press speed, and surface topography of the tooling are clearly specified. Friction is a powerful variable but can be neutralized by the use of oiled polyethylene sheet. Plastic anisotropy reflects the preferred orientation of crystals (texture) in the sheet. Texture also results in differences in elastic properties and hence in the speed of propagation of sound waves. This allows the ultrasonic determination of r values (see [ 14]). Commercial instruments are suitable for static measurements; recent efforts aim at techniques for measuring the r value on line, on sheet moving at 150 m/min, with a resolution of r to 0.05 or better [ 15].

3. F O R M I N G

LIMIT DIAGRAM

(FLD)

The most significant development of the last decades has been the introduction of the FLD (also known as the Keeler-Goodwin diagram) to describe forming limits under strain states ranging from balanced biaxial tension through plane strain to combined tension/compression. An entire volume [7] is devoted to the subject, and it will suffice here to look only at selected recent developments. Continuing progress is being made in the test technique itself. The use of strips of varying widths was introduced by Nakazima et al [16]. The technique now used was established by Hecker (see [17]): gridded specimens of varying widths are firmly clamped and stretched, under well-lubricated conditions, with a punch of 100 mm or 4 in (101.6 mm) diameter, until localized necking is observed or the maximum load is sensed. The strain ratio in the vicinity of the neck is obtained from the distorted circles of the grid, giving points on the forming limit curve (FLC). The standard technique was given in 1981 [ 18] together with the application of the grid technique to die development and troubleshooting. The tedium and uncertainties associated with manual measurement of circles are alleviated by computer-based image-analysis techniques [19, 20]. Surface strains have also been measured by taking two views of the surface [21, 22]. Yet faster results were obtained by the use of a camera to take photos of the grid during the deformation process itself, together with photos of a reference grid, so that strains could be immediately computed [23]. It remains difficult to decide what circle to take as the definitive one: the technique described by Bragard [24] uses parabolic interpolation but has, apparently, not been widely adopted. The shape and elevation of the FLC are a function of material properties, punch speed (strain rate), grid size, and even of the definition accepted for necking. Since substantial straining may be accommodated by diffuse necking, sheet thickness is a powerful factor by providing more material for deformation prior to the onset of local necking. For typical lowcarbon steels, Keeler and Brazier [25] found that the position of the plane-strain intercept (in percent) is defined by sheet thickness t (mm) and strain hardening exponent n: FLCo = a (23.3 + 14.1 t)

173 where a = n/0.21 (max. 1). Many recently introduced steels give a better fit if a = n/0.21 is used without limitation. It is then assumed that all steels have a FLC of the same, "standard" shape (Fig. I), hence it is sufficient for control purposes to determine only n for each coil. A safety zone of 10% can be added to account for process variability [ 18].

.

~\

80

STANDARD" FLC

"

111,,1

I--Or)ooC_)

<1~

\~ fl/-

-

x_10%,SAFETY

20-

ZONE

i

0

I , I I

-4O-20 MINOR

,

I

, I i I

0 2O 4O STRAIN , %

60

Fig. 1 Forming limit diagrams [18, 26]

The tacit assumption is that the limit strain is independent of strain-rate sensitivity (m value). However, it is clear--as discussed under Tension Test--that the m value has powerful influence on post-necking strain. Thus, it is reasonable to base the correlation on total (final) elongation in the tension test ef. Indeed, Raghavan et al [26] propose the relation FLCo = 2.78 + 3.24t + 0.892 ef where ef is the total elongation in percent of a transverse tensile specimen of 50.8 mm gage length. Furthermore, they use a "standard" shape that has gradual transition at plane strain (Fig. 1), similar to that found by Hecker [17]. The formula gives better agreement with experimental FLCo values than the one based on n, especially for the newer steels, including interstitial-free, bake-hardening, and coated steels. Whether two materials have the same formability is usually judged from a visual comparison of their FLCs. Buchar [27] argues that judgment should be based on statistical evaluation. It has long been recognized that the FLD is affected also by the strain path and that a production part formed along a non-linear path may fail either below or beyond the strain predicted by the standard FLC. Predictive work is difficult but beginnings have been made. Thus, Tse et al [28] developed a multilinear semi-empirical model to show the effect of a sequence of operations on the position of the FLC. They also performed three-dimensional grid analysis, evaluating strains from square arrays of grid points with a two-view system [21 ] and calculating the remaining ductility (taken as the residual n value) for each operation. The volume of metals does not change with plastic deformation. Thus, forming severity may be simply judged from thickness measured with a micrometer or ultrasonic gage [29]. Since the sum of natural (logarithmic) strains is zero, the maximum tensile strain can be calculated from

174 the thickness strain and minor strain. Minor strain is known to be zero at plane strain, hence here major strain equals through-thickness strain in absolute value. For the convenience of press-shop operators, a thickness strain diagram can be constructed which gives permissible thickness reductions, if desired, with a 10% safety zone. Away from plane strain the magnitude of minor strains must be detected by grid analysis; from this, permissible major strain is calculated for different parts of the pressing. Subsequent production control or die development efforts rely simply on measured thickness. Again, for the convenience of operators, diagrams or look-up tables can be constructed [30]. The FLC can be considered a property of the material, with the understanding that strain rates, strain paths, and in-plane vs. out-of-plane measurement can affect the result. The application of the FLC (or thickness strain analysis) is most powerful, however, as a tool for measuring the state and performance of the entire forming system. This can include the combined effects of the material, lubricant, stamping design (strain state and magnitude), tooling, stamping press, and even ambient conditions such as temperature and humidity [31]. Its application, however, is limited to existing processes; no information is available prior to designing and building the forming system. The results, while measuring the quality or state of the process, are nonspecific regarding appropriate measures for remedial steps or improvements to the process. Operator experience and, frequently, further testing are required to isolate a problem.

4. L I M I T I N G

DOME HEIGHT (LDH) TEST

The FLD provides a most powerful tool for production control but requires considerable effort. Furthermore, an FLC based of n or ef is often not sensitive enough to separate good coils from bad ones. There remains thus a need for tests that evaluate stretch formability with less effort. Such tests have been around for some eighty years. The Erichsen test has been widely used in Europe, the Olsen test in North America. Neither of them correlate with press performance for several reasons: some draw-in is always possible, the biaxiality of stress state is significant and difficult to control, and the small punch radius introduces a considerable bending component and makes results greatly dependent on sheet thickness [32, 33]. The Fukui test [2] gives a combined measure of stretchability and drawability. The newer dome tests are variants of the FLD determination method. Firmly clamped sheet specimens are deformed to fracture with a relatively large hemispherical punch, and the stress state is changed by varying the width of the specimen.

4.1.

Development of test method

In 1975 Ghosh [34] proposed that formability relevant to production conditions could be determined if gridded blanks of widths resulting in near-zero minor strain were tested. The critical minor strain measured next to the fracture site is plotted against specimen width. Similarly, the minor strain in the critical area of the press-formed part is determined. From the plot of minor strain vs. specimen width, the specimen width that reproduces the critical minor strain in the pressing can be chosen. The quality of sheets can then be routinely checked by performing dome tests at this width, with dome height as the critical value. A further simplification is introduced from the observation that some 85% of failures in stampings occur at or near plane strain [33]. One may dispense with the grid and plot dome

175 height vs. specimen width. The plot goes through a minimum, denoted as LDHo or simply LDH. Good correlation with press shop performance [33, 35] encouraged a significant cooperative development effort by the North American Deep Drawing Research Group (NADDRG), leading to a recommended referee practice [36]. Specimen widths are changed in 3-ram increments until two blank widths on each side of a minimum height value are obtained. The critical blank width is that which produces the minimum average dome height; five more tests are conducted at this width. The LDH value is then the average dome height of the eight blanks, reported to one decimal place in mm. It should be noted that there is a positive minor strain, as there would also be in a practical pressing. To find the intercept at plane strain, the specimen would have to be gridded or, at least, two parallel lines scribed at the expected fracture site [37]. At plane strain, however, discrimination is often reduced [37, 38]). Blank width, number of tests, and standard deviation of dome heights are reported. Test conditions appear to be well defined, and recent round-robin tests organized by the NADDRG [36] showed that reproducibility was acceptable within each participating laboratory, with standard deviations of 0.076 to 0.2 mm. However, the means ranged from 32.2 to 34.8 mm, a range wide enough to qualify the material as good or bad from a practical point of view. Evidently, test parameters were not adequately controlled. The LDH is a system property and, if reproducible results are to be obtained, it is essential that influencing factors be understood. In addition to laboratory investigations, substantial efforts were expended also in industrial applications [39 ].

4.2. Test geometry Test geometry is reasonably well fixed. The punch is a hemisphere (often a half ball-beating ball) of 100 mm or 101.6 mm (4 in) diameter. Smaller punches have been used, e.g., 75 mm [37], but this changes the ratio of sheet thickness to punch radius and increases the limit strain. Hence absolute values of LDH should not be compared for sheets of different thicknesses [35]. A small change in punch diameter from 100 to 101.6 mm, however, has no effect [40]. Greater variations exist in binder design. Cavity diameter and radius are well defined and small deviations from the specifications are of little significance. More important is lockbead configuration. The original design incorporated a triangular lock bead with a conforming female die. In the course of time the groove in the female die was changed to a square with little effect on the strain state or LDH. Yet later the binder surface was relieved by 0.075 mm from the bead inward [36]. This leads to more draw in, necessitating a slightly wider blank and increasing the LDH by some 0.5 mm [40]. Thus, results obtained with different binder designs are not comparable. Wear of tooling used in quality-control applications can also be sufficient to affect the results.

4.3. Binder (clamp) force. The test results critically depend on the absence of draw in, specified [36] as not more than 0.25 mm measured in the center of lock beads. Evidently, the clamp force required depends on bead design and friction on the binder surface. With the standard design and a smooth (ground) surface, Meuleman et al [41] found 200 kN sufficient for sheet of 0.84 mm thickness. Metals Handbook [2] indicates 250 kN. A systematic investigation confirmed that 250 kN is required to prevent draw in [42]. Below this, both mean dome height and scatter increase (Fig. 2); this is a potential problem with machines used in several quality-control laboratories [39].

176

131 m

34.0

130

E E

129 -iI-c3

E E

128 33.0

127 ~

I a

126

J

-125o

32.0

I I00

31.0 0

1 200 CLAMP

1 300

I 400

FORCE,

kN

i 500

Fig. 2 Effect of clamp force on LDH [42]

4.4. I n t e r f a c e conditions Friction determines, at a given blank width, the strain distribution in the course of stretching. It is imperative, therefore, that friction be controlled and made highly reproducible. Many factors, mutually interacting, determine friction, hence a strict procedure for surface preparation and lubrication must be followed. Sheets are usually supplied with mill oil of often unspecified or unknown composition on their surface. A logical step would be to remove this oil and test in the "dry" condition. Friction is high, forcing the fracture site away from the pole, and plane-strain conditions are readily attained. Dome height is low and sensitive to material properties such as the n value [32, 33, 35] but, in repeated testing of the same sheet, random variations are often observed [41 ]. 34.0

-J

_ 335 33.0

E "

32.5

O J

-

i

~ ~

~

~

~

o

z

t " ~" o', -

a,.

5

"-" ~ j e"i 5

w D

~

-~"'

u

~0~ ~ o

~

o

o_

~ o

w D

~

f

o_

-

~ w

32.0

-3

31.0

Fig. 3 Changes in LDH with surface preparation and lubrication [44]

177 The problem is common to all metalworking tests [43]. Polar cleaning fluids remove lubricant residues quite effectively but can form patchy EP films of their own, leading to higher LDH values (Fig. 3) [44]. Acetone, while popular, does not remove all oil [38, 44, 45]. If a dry surface is problematic, a slightly lubricated one may be acceptable. The simplest method is based on wiping the sheet surface, spreading out and removing excess mill oil. This has been adopted for quality control [39] but can give problems in comparisons of materials from different sources. Better reproducibility is obtained [44] by first removing mill oil residues with a non-polar solvent such as mineral spirits (paint thinner) and then applying a lowviscosity lubricant such as the so-called mineral seal oil of 4.6 cSt viscosity at 38 ~ (this is similar to the wash oil used by Meuleman et al [41 ]). Heavier oils increase dome height and, more importantly, the scatter of results (Fig. 3), no doubt because the strain ratio becomes more variable as fracture moves closer to the apex, as noted also by Ayres [38]. Reacted residues of mill oil can affect the results, as shown by much higher LDH found on sheets immersed in oleic acid or chlorinated paraffin for several weeks [44]. Reacted films can be removed by pickling, but surface topography could change with unknown consequences. LDH values stabilize only after the interface reaches equilibrium. An important part of the standard process is, therefore, conditioning of the tooling by drawing ten dummy blanks, the readings of which are ignored. Punch surface becomes even more important in the testing of coated (e.g., galvanized) sheet, since metal transfer (pick-up) would, if not removed, affect subsequent tests even on bare steel [37]. This can be particularly disturbing in industrial quality control where a large variety of sheet materials are tested in quick succession. It appears, however, that wiping the tooling with dry paper towel is adequate; any disturbing effects disappear after ten samples are tested [39].

4.5. Temperature Testing in a press shop environment introduces further variables such as temperature. Superimposed on changes in ambient temperature are localized changes, primarily from heating of the punch due to heating of the hydraulic oil used for operating most test equipment. Higher temperature lowers the viscosity of oil at the interface and reduces the LDH, signifying increased friction (Fig. 4). This leads to the conclusion--supported by observations of the deformed sheet surface--that the test operates in the mixed-film lubrication regime. Hence, increasing temperature should also reduce the LDH, and this has indeed been observed, with a decline of some 0.5 mm over a temperature range from 25 ~ to 45 ~ [45, 46]. The widely divergent results below 25 ~ indicate a further influencing factor. Tests conducted in the winter followed the trend expected from changes in lubricant viscosity. In contrast, those conducted in the summer showed a precipitous drop in LDH at 15 ~ where water condenses on the punch preventing the development of a continuous, effective oil film. These effects can be minimized by heating the tooling to 38 ~ [39].

4.6. Punch s p e e d Not all test facilities are capable of developing the recommended speed of 250 mm/min. The effect of speed is somewhat controversial [36, 47]. In tests with dry interfaces, Ayres [38] found with increasing speed a slight increase in LDH for aluminum-killed steel and a definite decrease for 2024-T4 aluminum. He attributed the difference, at least in part, to the positive and negative strain-rate sensitivities of the two materials, respectively.

178

[4s]

VISCOSITY, cSt

x

o 4.6

I o

33

[46]I

38~

x4.7ot

62

at

38~

38oc

~------~ 62 ot TEST TEMP.

32

_J 3O 29 10

I 15

I 20

1 25

l

1

30

1

35

,l 45

40

TEMPERATURE, ~

Fig. 4 Effect of tooling temperature on LDH [45, 46]

4.7. Interactions Uncertainties of interpretation are bound to remain if only a single variable is studied at a time. It is necessary to conduct tests with the main factors affecting film thickness--including viscosity, temperature, punch speed, and surface roughness--systematically varied within the ranges expected in testing practice [46]. A factor of known importance is roughness of punch and sheet. Specifications [36] simply state that punch roughness should be of 0.125 ~m (5 ~in) Ra maximum and are silent on roughness orientation. To reveal effects of temperature on formability, Fischer and Schey [46] conducted tests with punches of 0.033 and 0.144 ~m Ra circumferential or random finishes, at punch speeds (v) of 0.83 to 8.3 mm/s, temperatures of 25 and 50 ~ and with oils blended to give the same viscosity (q) at the two temperatures. Humidity was not controlled and changed according to the weather and heating season. 33.0 -

I

32.0

v

I

E E 31.0

-r a --.I

I

30.029.0 28.0

Z

ooooo Unknown Humidity ooooo Low Humidity ~0 03L~ High Humidity 0

~

~

i

I

i

25

t

i

i

~

i

50

i

i

i

i

i

i

i

i

i

i

75 100 "r/V ( m m 3 / s 2)

i

J

i

J

i

125

i

~

i

i

i

150

Fig. 5 Changes in LDH value with oil viscosity, punch velocity, and humidity [46]

179 LDH values were lowest at high (over 40%) relative humidity (RH)(Fig. 5) and increased with qv rising from 30 to 110, indicating decreasing friction, as expected for mixed-film lubrication. Dome heights obtained in a winter period characterized by very low (but not accurately known) RH were much higher and failed to give a clear trend. The rougher punch resulted in some 0.25 mm higher domes, suggesting that the deeper (but in absolute values still very fine) troughs helped lubricant retention and lowered friction. For a constant qv, increasing temperature decreased the LDH by 1 mm, indicating a true change in formability. This agrees with Ayres [48] and Stevenson [49] who found that m and thus total elongation dropped with increasing temperature in this low-temperature regime. In routine testing the punch is refinished in situ with an abrasive (not specified in the recommended practice, but usually crocus cloth, 600-grit SiC paper, or Scotchbrite). It is assumed but seldom checked that the finish is indeed random and unchanging, perhaps because earlier work showed the direction of finish to be unimportant [41 ]. Finishing the punch with 600-grit SiC paper until repeated refinishing caused no further change in roughness gave Ra - 0.04 pm. The LDH was now sensibly independent of humidity. It is not obvious why this should be so; humidity has unpredictable effects on lubricant performance (in [43], p. 54). Results indicate that the effect can vary with surface roughness, most likely through its effect on lubrication mechanism. Whatever the explanation may be, it is reassuring to know that a random finish is much less sensitive. Roughness must, nevertheless, be controlled, as shown by the some 1 mm higher LDH obtained with a rougher (Ra = 0.144 pm) random finish. Any factor that affects lubrication also affects friction. For this reason, otherwise identical sheets of different surface topographies give different LDH values. For example, production sheets of rougher finish gave lower LDH than laboratory-rolled sheets of finer finish, even at the same n values [50].

4.8. Test technique A very large source of potential error is the use of incorrect blank width. Determination of the critical width is tedious and there is a temptation to conduct tests on blanks of constant width (typically, 133 mm) for all materials. As seen in Fig. 6a, the dome height changes quite

@

(.9

/

,,4

2

o a 3 mm--~, ~_ I I I I I I I J BLANK WIDTH, mm

1

I_I

2rnm-,~ ~ I

I

I

I

~-~ I

BLANK WIDTH, mm

Fig. 6 Determination of LDH by two different methods

I

180 steeply away from the critical width, and replicate tests on blanks of incorrect width will simply yield an inaccurate LDH with greater precision. This would explain the tight dispersion yet wide range of LDH results obtained in round-robin tests. These problems can be avoided and the effort reduced if the test method is changed as follows: For a given material group, the approximate critical width is explored by testing blanks in 6-mm increments. Routine tests are then done on blanks cut at 2-mm increments, with the widths chosen to assure at least three points on each side of the critical width. Triplicate tests at, say, 7 to 9 widths again require 21 or 27 blanks, but this time they can all be cut at once. As shown by the typical plot in Fig. 6b, the curve of dome heights is well defined. It is then a matter of agreement whether to take as the LDH the minimum of the best-fit curve or the intercept of two best-fit straight lines. For all its proven benefits, the LDH test is time-consuming and subject to a relatively large scatter of results, attributable to a number of causes [reviewed in 51 ]. There are continuing efforts to improve the test technique. Kim and Park [52] retain the blankholder of the standard die set but replace the hemispherical punch with a cylindrical punch of 37.5 mm radius and 70 mm length, thereby ensuring fracture in plane strain. A standard specimen width of 126 mm was found suitable for a variety of steels. Height at fracture (limiting punch height, LPH) was very close to the LDH value but scatter, expressed as standard deviation, was greatly reduced. It appears that, by forcing fracture to take place in plane strain, sensitivity to variations in friction is also reduced, minimizing scatter and eliminating the need for finding an optimum blank width. Taking a different approach, Miles et al [5 l] replace the circular lock ring of the standard die set with straight lock beads and use a cylindrical punch with an elliptical profile to obtain fracture near plane strain. This "OSU Test" gave punch heights at fracture that correlated well with the LDH value, and the number of test specimens required was less than in the standard LDH test. The scatter of data was, however, no better than with a well-controlled LDH test. The value of any of these newer tests would have to be proven in strictly designed and controlled round-robin tests. Until such trials are made, a properly conducted LDH test still gives the most sensitive measure of formability in plane strain. Random variations, otten attributed to the poor reproducibility of the test, are then revealed as true changes in formability. For example, dayto-day variations in LDH on a coil-annealed material made no sense until it was realized that the variations were due to a true change in formability along the length of the coil [44]. The test lends itself to considerable automation. The load-displacement curve begins to drop when necking sets in; hence, with the advent of computer-based data acquisition and process control, it is easy to detect the maximum punch force and the corresponding dome height. The punch can be automatically stopped at, say, 1% drop in force. At this point, there is incipient fracture at the failure site. The LDH test is a technological test, in essence simulating a production part made with a hemispherical punch and a completely locked blank. As such, it is subject to all the variables that affect a pressing. This is a disadvantage in that other variables must be strictly controlled if a single variable, such as the formability of sheets, is to be evaluated. It is, however, also an advantage in that it highlights the importance of other variables in the system, alerting one to the sensitivity of the stretching process to a host of influences. In stamping plants where the test has been introduced, it has proven of great help in separating problems due to material quality from those due to process variables [39].

181

5. S T R E T C H - B E N D

TESTS

The LDH test and its variants fail to predict formability when bending strains dominate. Various stretch-bend tests have been proposed, although none of them have achieved the status of a standard. Demery [53] performed extensive testing with two configurations. The hemispherical stretch-bend test (HSBT) is similar to an LDH test but the radius of the hemispherical punch is reduced. The load reaches a maximum when the limiting height is attained. Narrower blanks stretched over larger-radius punches fracture outside the contact zone in a tensile mode. Lowest limit strains are obtained with wide (fully constrained) specimens, giving the most critical evaluation of formability. At a given radius, the limiting height increases with increasing sheet thickness because more material is available for distributing the necking strain. Similarly, for a constant thickness, limiting height increases with increasing punch radius because the larger punch distributes strain more uniformly prior to necking. For practical applications, presentation as a function of t/g (increasingly severe curvature) is most illuminating. The angular stretch-bend test (ASBT) is the plane-strain analog of the HSBT. Rectangular blanks are gripped on their ends and stretched by wedge-shaped punches of varying tip radii. The limiting height again increases with tip radius, but decreases with sheet thickness, presumably because of the dominance of thickness strain. A simplified model of the latter test [54] correctly accounts for several effects of friction. Lower friction, as produced by polyethylene sheet lubricated with oil, promotes sliding and thus thinning over the punch where fracture occurs. For a given friction, increasing t/R (tightening bend) promotes bending strain and thus fracture on the punch. Fracture strain can be regarded as a material constant. (It should be noted that the strain relevant to this situation is neither the uniform strain nor strain in the neck, rather some intermediate value which should correlate with elongation measured over a short gage length, in which localized strain in the neck receives greater weighting.) In the course of stretching, strain over the punch increases more rapidly under lubricated conditions, hence the fracture strain is reached at a lower height. Sliding over the punch makes more material available for the free-stretching ligaments, increasing limiting height when ligament fracture controls the event. Evidently, for the test to yield reproducible results, interface conditions have to be controlled as they are in the LDH test. For sharp radii (t/g < 0.85) friction effects are swamped by the dominance of through-thickness strain. However, fracture is then forced to occur over a very short distance, reducing sensitivity of the test to material variables that promote distribution of strain over a punch surface. Correlation studies with industrial press performance have not been published, but some companies do use the test routinely as an adjunct to the LDH test [55].

6. T E N S I O N T E S T There is a perhaps never-ending search for a measure of formability as an intrinsic material property. Some form of tensile testing would appear ideal to determine formability in the absence of the disturbing influences of friction, with well-controlled stress states free of the bending that is characteristic of all out-of-plane tests.

182

6.1

Standard tension test

The left-hand side of the FLD gives forming limits in a tensile/compressive strain state (i.e., at negative minor to major strain ratios). Ostensibly, the simplest measure is elongation in the standard tension test. Following the arguments and experiments of Ghosh [56] it is now well recognized that while uniform strain is more or less proportional to strain hardening (n), postnecking strain is largely governed by strain-rate sensitivity (m), thus, both contribute to elongation to fracture (final strain) ef. Finite-element analysis [57] shows that both n and m are important throughout the entire strain history. It is, therefore, reasonable that ef should correlate reasonably well with observed rankings of formability in press working, as recognized early in this century. Correlation is certainly much better than with n. The correlation is, however, far from perfect and greatly depends on the geometry of the tensile specimen on the one hand and on the stress state prevailing in production on the other. There is also a question of correlation with simulation tests. We have seen that for most steels, FLDo intercepts correlate better with ef than n. The same holds for LDH values [36]. Stevenson [49] found that among tensile properties only ef gave general correlation with LDH for a wide range of steels, aluminum alloys, and zinc. The equation LDH (mm) = 10.0 § 0.508 ef described the relationship with a standard error of 1.62 mm when ef was taken from ASTM specimens of 50.8-mm gage length. Since the error is larger than the 0.5 to 1.0 mm difference in LDH that separates a good lot from a bad one, Stevenson too suggested that ef can be regarded only as a gross indicator of formability. Story et al [50] found that, even for aluminum alloys which show very little post-necking strain, dome heights correlated best with an instantaneous strain hardening exponent combined with ef. A seldom recognized problem is the poor reproducibility of the tension test. This is surprising in view of the long history of this widely practiced test, yet the problem was evident in each round-robin test conducted by NADDRG [58].

6.2. Plane-strain tension Since most failures in pressworking occur near plane strain, it is reasonable to search for methods that eliminate transverse strain in a tensile specimen. For thick materials, transverse grooves can be machined so that gage length is reduced and the non-deforming shoulders restrain lateral contraction. With decreasing gage length to width ratio, the strain state moves closer to plane strain. For thinner sheet, Devenpeck and Richmond [59] achieved the same objective by sandwiching the sheet between end plates and defining the gage length by transverse electron-beam welds. In-plane strains were obtained from photos of grids, throughthickness deformation from holographic interferometry [60]. For the steel used, the limit strains were only slightly lower than those determined in a dome test. Other forms of planestrain testing have been proposed. For example, Graf and Hosford [61] achieve plane strain over a large portion of the width of simple rectangular specimens by wrapping the sheet around grips that allow free choice of the deforming length. With gridded sheets it should be possible to obtain the plane-strain intercept of an FLC. Ziaja [62] uses a gridded, side-notched tensile specimen and computes the left-hand side of the FLC from a modified McClintock [63] fracture criterion.

183 Sang and Nishikawa [64] completely eliminated lateral strain by attaching to the sheet specimen a spring-loaded clamp with four knife edges indenting the two sides of the sheet parallel to the loading direction. They defined the plane-strain limit strain as the point where the longitudinal thickness gradient reached 0.1; this coincided with the development of a visible neck. Fracture strain was calculated from thickness at fracture. No comparisons with FLDs were made, however, Timothy [65] found that, in stretching over an unlubricated punch of 50 mm diameter, necking strain was some 40% higher than in plane-strain tension. This effect was already noted by Ghosh and Hecker [66] who pointed out that friction over the punch and the gradual wrapping of sheet over the punch surface combine to postpone necking. Since the same effects operate in practical operations, the use of plane-strain tests is questionable, especially for materials showing negative strain-rate sensitivity and thus a tendency to flow localization. An apparently simpler test has been proposed by Richmond and Devenpeck [67] for assessing the material's ability to distribute strains and thus giving a measure of stretchability. A tapered specimen is produced by machining circular arcs in the width of the specimen. The extent of plastic zone is judged from grids or brittle lacquer coating. The test was successful in showing that a dual-phase steel has greater stretchability than a low-carbon DQSK steel, even though standard tensile test data (including n and ef) would indicate otherwise.

6.3. Biaxial tension In-plane biaxial tension can be generated by an ingenious method proposed by Marciniak and Kuczynski [68]. A circular specimen is clamped and stretched over a flat-face punch together with another specimen in which a central hole had been made. Several specimens are stretched to find the point at which necking begins. The technique is used also in conjunction with punch stretching to determine the FLC around plane strain (below + 10 % minor strain). Alternatively, the strain just before fracture is found by continuous observation of a grid [50]. No stress-strain information is, however, obtained since the load at necking and failure cannot be measured. Out-of-plane biaxial tension, such as produced by hydrostatic bulge testing, can provide stress/strain curves to large strains [69] but does not give a good measure of formability [70] since the crucial element of contact with the die is missing. Similarly, biaxial stretching over a punch has limited value since few pressings fail in this mode and correlation with plant performance is poor [70].

7. H O L E E X P A N S I O N A problem of different dimension is the edge cracking that sets a limit to the expansion of holes, whether expansion is into a straight flange (perpendicular to the plane of sheet) or offset (like the bottom of a cup formed by a flat-faced punch on a previously pierced sheet). An intrinsic edge formability can be determined by expanding holes of reamed (or otherwise burr-free) surfaces with a tapered, well-lubricated punch. Intrinsic edge formability is related not just to ductility as expressed, for example, by reduction of area in the tension test, but also to an absence of inclusions or, in general, of weak interfaces in a two-phase structure. Since a single stress raiser located at the edge can initiate fracture, limits of expansion are not directly predictable from tensile tests done in only one direction. Indeed, even plastic anisotropy (both

184 normal and planar) plays a role, since hole expansion requires in-plane deformation and is adversely affected by through-thickness deformation. Formability tests such as the LDH test are not relevant, because strain accommodation around an inclusion reduces harmful effects. For the same reason, flanging limit increases with increasing sheet thickness [71, 72]. Any stress raiser present will reduce flanging limits. Burrs resulting from punching are most harmful. Burrs of highly jagged configuration are more deleterious and are prone to form on materials of lower ductility. For a given material, degradation of expansion limits is usually a function of burr height to sheet thickness ratio [73]. Because burrs or other edge damage significantly reduce expanding limits, surface finish--and even residual stress distribution--in the wall of the hole must be strictly standardized, for example, by detailed specification of the preparation method.

8. C O N C L U S I O N In judging the value of formability tests for production purposes, it is worthwhile to reflect on the purpose of such tests. The primary goal is judgment of suitability for specific press forming applications in various contexts: matching the material to the demands of the process; modifying a process or design to suit material capabilities; troubleshooting and production control; and development of new materials. Ideally, one would measure some basic property and then, with the aid of appropriate process models, predict production performance. Despite enormous effort and undeniable progress, this goal has not yet been achieved. Until such time when total solutions become available, continuing efforts must be exerted to find simulation tests that take the full complexity of reality into account. Of fundamental importance is die contact which aids in delaying strain localization. This introduces interface friction as a complicating factor which, for testing purposes, needs to be neutralized. If, however, production conditions are to be strictly simulated, the effects of lubricants must be understood; this leads into the very large and complex area of tribology as it pertains to sheet metalworking. As the example of the LDH test shows, progress in these directions can be slow but measurable. Progress would be accelerated by conducting more round-robin tests, preferably in global cooperation. One can only agree with Levy [55] that no unique test is likely to exist and a battery of tests must be applied. For production control purposes the likely candidates are the LDH test or its variants, the angular stretch-bend test, and hole expansion. The tension test remains important as a quick check on overall quality. Truly relevant simulation tests, combined with statistical process control [74], would go a long way to put production on a firm footing.

Acknowledgments The author is grateful to the Natural Sciences and Engineering Research Council of Canada for support of research and to R.F. Fischer for helpful discussions. Special thanks are due to Dr. D. J. Meuleman who critically reviewed the manuscript and offered valuable suggestions.

185 REFERENCES

Metals Handbook, 9th ed., Vol. 14, Forging and Forming, ASM, Metals Park OH, 1988. B. Taylor, in [ 1], pp. 877-899. S.S. Hecker, A.K. Ghosh, and H.L. Gegel (eds.), Formability: Analysis, Modeling, and Experimentation, A/ME, New York, 1978. J.R. Newby and B.A. Niemeier (eds.), Formability of Metallic Materials -2000 A.D., ASTM STP 753, Philadelphia, PA, 1982. R.H. Wagoner (ed.), Novel Techniques in Metal Deformation Testing, TMS, Warrendale, PA, 1983. A.K. Sachdev and J.D. Embury (eds.), Formability and Metallurgical Structure, TMS, Warrendale, PA, 1987. R.H. Wagoner, K.S. Chan, and S.P. Keeler (eds.), Forming Limit Diagrams: Concepts, Methods, and Applications, TMS, Warrendale, PA, 1989. Autobody Stamping Technology Progress, SAE SP-865, 1991. 8 Autobody Stamping Applications and Analysis, SAE SP-897, 1992. 9 10 Sheet Metal and Stamping Symposium, SAE SP-944, 1993. 11 W.T. Lankford, S.C. Snyder, and J.A. Bauscher, Trans. ASM, 42 (1950) 1192-1235. 12 R.L. Whiteley, Trans. ASM, 52 (1960) 154-168. S.Y. Chung and H.W. Swim, Proc. Inst. Mech. Eng., 165 (1951) 199-223. 13 14 K. Sakata, D. Daniel, J.J. Jonas, and J.F. Bussiere, Met. Trans., 21A (1990) 697-706. 15 A.V. Clark, N. Izworski, M. Hirao, and Y. Cohen, in [8], pp. 1-13. 16 K. Nakazima, T. Kikuma, and K. Hasuka, Tech. Rep. 264, Yawata Iron & Steel Co., Sep. 1968, p. 141. 17 S.S. Hecker, in [3], pp. 150-182. 18 S. Dinda, K.F. James, S.P. Keeler, and P.A. Stine, How to Use Circle Grid Analysis for Die Tryout, ASM, Metals Park, OH, 1981. 19 R.A. Ayres, E.G. Brewer, and S.W. Holland, Trans. SAE, 88 (1979) 2630-2634. 20 D.N. Harvey, in Proc. 13th Bienn. Congr. IDDRG, Melbourne 1984, pp. 403-414. J.H. Vogel and D. Lee, J. Mater. Shaping Technol., 6 (1989) 205-216. 21 22 M.I. Kapij, J.J. Moleski, T. Ohwue, J.H. Vogel, and D. Lee, in [9], SAE Paper 920436. Y. Wang, A.E. Bayoumi, and H.M. Zbib, J. Mater. Shaping Technol., 9 (1991) 143-152. 23 24 A. Bragard, in [7], pp. 9-19. S.P. Keeler and W.G. Brazier, in Microalloying 75, New York, 1977, pp. 517-530. 25 26 K.S. Raghavan, R.C. van Kuren, and H. Darlington, in [9], SAE Paper 920437. 27 Z. Buchar, IDDRG Working Group Meeting, Linz, Austria, 1993. 28 W. Tse, T. Aboutour, J.S. Thomas, A. Assempoor, and M. Karima, in [9], SAE Paper 920437. 29 S.P. Keeler, SAE Paper 850278. 30 D.J. Hogarth, C.A. Gregoire, and S.L. Caswell, in [8], pp. 87-95. J.F. Siekirk, J. Appl. Metalwork, 4 (1986) 262-269. 31 S.S. Hecker, Met. Eng. Quarterly, 14 (1974 Nov.) 30-36. 32 R.A. Ayres, W.G. Brazier, and V.F. Sajewski, J. Appl. Metalwork., 1 (1979) 41-49. 33 A.K. Ghosh, Met. Eng. Quarterly, 15(3) (1975) 53-64. 34 J.M. Story, J. Appl. Metalwork., 2 (1982) 119-125. 35 36 R. Thompson and NADDRG, in [ 10], SAE Paper 930815.

186 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

H. Vegter and C.M. Dane, IDDRG Working Group Meeting,, Amsterdam, 1985. R.A. Ayres, in [5], pp. 47-64. A.F. Graf and N. Izworski, in [ 10], SAE Paper 930816. R.F. Fischer, M.A.Sc. Thesis, University of Waterloo, 1992. D.J. Meuleman, J. Siles, and J.J. Zoldak, SAE Paper 850005. J.A. Schey and R.F. Fischer, J. Mater. Shaping Technol., 9 (1991) 85-86. J.A. Schey, Tribology in Metalworking: Friction, Lubrication and Wear, ASM, Metals Park, OH, 1983, pp. 233-235. J.A. Schey, J. Mater. Shaping Technol., 6 (1988) 103-111. N. Izworski and A. Graf, in [8], SAE Paper 910512. R.F. Fischer and J.A. Schey, in [9], SAE Paper 920434. J.M. Story, J. Appl. Metalwork, 3 (1984) 292-300. R.A. Ayres, Met. Trans., 16A (1985) 37-43. R. Stevenson, J. Appl. Metalwork, 3 (1984) 272-280. J.M. Story, D.J. Lege, and W.H. Hunt, Jr., in [5], pp. 113-129. M.P. Miles, J.L. Siles, R.H. Wagoner, and K. Narasimhan, Met. Trans., 24A (1993) 1143-1151. Y. Kim and K. Park, to be published in Met. Trans., Ser. A M.Y. Demeri, J. Appl. Metalwork., 2 (1981) 3-10. O.S. Narayanaswamy and M.Y. Demeri, in [5], pp. 99-112. B.S. Levy, in Proc. 13th Bienn. Congr. IDDRG, Melbourne, 1984, pp. 592-605. A.K. Ghosh, in [3], pp. 14-28. K. Chung and R.H. Wagoner, in [6], pp. 261-277. B.A. Niemeier, in [4], pp. 296-314. M.L. Devenpeck and O. Richmond, in [5], pp. 79-88. M.L. Devenpeck, in [5], pp. 219-233. A.F. Graf and W.F. Hosford, in [ 10], SAE Paper 930812. G. Ziaja, in microCAD-Systems 93, University of Miskolc, 1993. F.A. McClintock, J. Appl. Mech. 90 (1968) 363-371. H. Sang and Y. Nishikawa, in [5], pp. 3-13. S.P. Timothy, in [7], pp. 21-36. A.K. Ghosh and S.S. Hecker, Met. Trans., 5 (1974) 2161, 6A (1975) 1065-1074. O. Richmond and M.L. Devenpeck, in [5], pp. 89-98. Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci., 9 (1967) 609-620. R.F. Young, J.E. Bird, and J.L. Duncan, J. Appl. Metalwork, 2 (1981) 11-18. J.R. Newby, in [4], pp. 60-83. M.Y. Demeri, in Proc. 15th Biennial Congress IDDRG, Dearborn, MI, ASM, 1988, pp. 231-236. L.D. Kenny and H. Sang, in [5], pp. 15-30. S.P. Keeler, Machinery, (1968 June) 98-104. S.P. Keeler, in [1], pp. 928-939.

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

187

Design o f Experiments, a Statistical M e t h o d to analyse Sheet Metal Forming Defects effectively D. Bauer a and R. Leidolf b a,b Institute of Production Engineering, University Siegen, P.O. Box 101240, 57068 Siegen, Germany

1. INTRODUCTION Because of constantly increasing shortage of raw material it is necessary to optimize material utilization in sheet metal forming processes. In order to determine wether a particular sheet metal can be deep drawn without defects the so-called cup test has been developed [ 1]. The tools of this test consist of the punch, the die, the blank holder and auxiliary equipement to center the blank. By means of these an axissymmetric deep drawing process can be performed by the cup test. This is done by placing a circular blank over the die and pressing the blank into die with the punch. Additionally, a force is required to press the blank holder against the blank to prevent wrinkling. Before deep drawing takes place the blank is coated, too, by an appropriate lubricant in order to reduce friction and wear. The ratio of the initial blank diameter d to the punch diameter dst is the well-known drawing ratio 13= d/dst. Its maximum value is the so-called limited drawing ratio [3max. This is generally dependent on a large number of drawing parameters, e.g. lubricant, die radius, blankholder force, etc.. The best economic material utilization can be obtained with a parameter combination that provides the maximum limited drawing ratio. In the following it will be shown how this aim can be achieved by the statistical method called design of experiments.

2. LIMITED DRAWING RATIO AND MAXIMUM DRAWING FORCE One method to determine 13max is given by keeping the punch diameter dst constant and drawing blanks with progressively enlarged diameters d, for example in steps of l mm. The largest

blank

diameter

dmax that

can be

drawn

without

failure

delivers

finally

188 13max

= dmax/dSt.This method is commonly preferred. But it is a laborious and an expensive

one. A more effective one is the method according to Schmidt-Kapfenberg. Starting point of this method published by M. Schmidt [2] is the analysis of the drawing force required to produce a cup. This force F z is measured in dependence of the punch travel during the cup test. Fig. 1 shows three representative results of such measurements carried out with constant punch diameter dst = 110mm and different blank diameters d 1 = 150mm, d 2 = 200mmm, d 3 = 240mm and corresponding drawing ratios 131 - 1.36, 132 = 1.82, 133 = 2.18. In case of d 1 and d 2 a cup can be drawn without fallout. But in case of d 3 failure occurs before the cup is formed completely. Usually, the failure occurs in a narrow band of material in the cup wall just above the radius of the punch which has undergone no radial drawing or bending but is subjected essentially to tensile straining and thinning. Thus, in this critical region necking defects, followed by tearing defects, will destroy the cup as soon as the stress state induced by the drawing force is getting equal or exceeds the strength of the wall material. At this critical instant the corresponding drawing force curve is interrupted, Fig. 1. I

78,1

'

d 3 = 2#Omm N

U.. (D O 0 q.13) c" "13

,

52.6

~

i

i

~_-------i ! d2 =! 200mm

[

I

35.8

17.5

F 8

6

13

19

punch displacement h -

2:6

m

32

~,,.-

Figure 1. Drawing force versus punch displacement. Sheet metal AIMg0.4Sil.2, thickness so = lmm, d 1 - 150mm, d 2 = 200mm, d 3 = 240mm; 131 = 1.36, 132 = 1.82 133=2.18.

M. Schmidt observed that there exists generally a linear relationship between the maximum of the drawing force Fmax and the increase of the blank diameter d, resp. the drawing ratio 13, if

189 Fmax is lower than the failure load FBr. Because of this M. Schmidt suggested to determine 13max only by three different runs of the cup test: two with different blank diameters smaller than dmax in order to fix the straight line in the subcritical area Fmax _< FBr, and one with a blank diameter larger than dmax to obtain the failure load FBr, Fig. 2. The point of intersection than gives the value of dmax resp. 13max. The failure load FBr is only dependent on the strenght and the thickness of the sheet material. For a given test arrangement it is a constant and cannot be varied. But against that Fmax in the subcritical area is dependent on the complete set of parameters as listed in Fig. 4. Thus, by systematic variation of these parameters the maximum values of the drawing force in the subcritical area can be minimized. Fig. 3 demonstrates this for two different parameter combinations of the cup test. The failure load gives for both combinations the same value. But in the subcritical area combination II is delivering lower Fmaxl and Fmax2 values than combination I. By reason of this, the point of intersection is shifted to the fight and 13max increases. This means, the limited drawing ratio is not a material property of sheet metal. On the contrary, it is not only depending on the strenght and thickness of sheet material but on lubricant, tool geometry, blankholder force, sheet roughness and all of other forming parameters, too. Thus, by an optimal combination of the forming parameters the stress state in the cup wall induced by the maximum drawing force in the subcritical area can be minimized, i.e. the corresponding limited drawing ratio can be improved and maximized. F~lax

hypercritical

area

A F~r

4

FmaxZ

F~ax1

subcritical

area

i

i ....

v

i ~I

~2

~max

~Br

Figure 2. Determination of the limited drawing ratio according to SCHMIDTKAPFENBERG.

190

max

~r

combination combination

l

/ /

Fmax2I Fmax2II FmaxlI FmaxllI

I II

--/ / / J/

'

lj fll

f12 flmaxI flmaxlI

Figure 3. Increase of the limited drawing ratio by reducing the maximum drawing forces Fmaxl and Fmax2.

3.

TRADITIONAL AND MODERN METHODS FOR DESIGN OF EXPERIMENTS

The traditional method to design and analyse experiments is the so-called one-factor-at-atime-method. One influential factor is set on different levels and the effect of this factor on a target value can be examined, while the remaining parameters of the process are on a constant level. There are a lot of studies like that regarding to the effect of different metal forming parameters upon limited drawing ratio. So, for example, W. Ziegler analysed the influence of the die radius upon 13max [3], while R. Woska investigated the controllable variable "tool coating" [4]. But the traditional method has two considerable disadvantages: 1. It is not possible to make out interactions between factors. So, it would be likely that the increasing of the die radius also increases the amount of 13max, if a special tool coating is used, while the same increasing of the die radius doesn't change 13max by the use of another tool coating. 2.

It requires a lot of temporal and financial experimental power input. This disadvantage today gains in importance because of global competition. Because apart from a very high quality standard it is possible to get advantages compared with competitives, if someone has brief, but efficiently development times and low development costs.

191 These are the reasons for a lot of industrial branches to favour the methods of statistical design of experiments (DOE), today. The number of required tests is fewer. In addition to the low power input it is a great advantage to determine the single influence of a factor as influences between two or more factors

(interactions).

(effect)

as well

Because of the special way to

analyse the experimental results, it is possible to get high certainty regarding to the effects of treatment factors. This is very important, because today any producer has to minimize the risks as a result of poor product quality. If necessary, he has to prove by data, that his products have no safety reducing quality defects. Further, it is hardly or even not possible to create mathematical models and without a model it is not possible to optimize the investigated process. For optimization it is not necessary to have a model, which is in accordance to natural laws. Very often a statistical model is sufficient. The design of experiment allways provides a statistical model,-which allows to suggest main effects and interactions. Also a correlation between controllable variables

(factors)

and a target value can be set up. To determine linear

relationships it is enough to investigate two levels of the selected factors. For many years the methods of DOE are used successfully in agricultural sciences [6], chemical engineering [7] and plastic material manufacturing [8]. In this paper it will be shown how to apply DOE on a metal forming problem. The deep drawing conditions are analysed with regard to maximum material economy. The single effect of different metal forming parameters on the limited drawing ratio was studied in great detail [9,10], but interactions are neglected until now. So, in this work full factorial design will be used, because the other well-known DOE-method TAGUCHI's experimental design- is a simplified form of factorial design. So, the TAGUCI-I/plans only supply meaningful results, if a lot of informations about the process and especially about the interactions are given [ 11 ].

4.

CONTROLLABLE VARIABLES UPON MAXIMUM DRAWING FORCE At the beginning of statistical design of experiments it is necessary to analyse which

controllable variables are probably important for the target value. The drawing force of a deep drawing process consists of different components [9]: -

ideal plastic deformation load Fid, needed for the plastic, lost-free tension-compressiondeformation,

-

-

flexural load FBie to bend over the die and punch radius, friction force F R in the areas of contact between tool and cup.

192 This drawing force components could be influenced by changes of the deep drawing system, which consists of forming and tribological parts. The forming part contents the loads and stresses, caused by metal forming. So, there are radial tensile stresses and tangential compressive stresses in the flange. Finally the flange is bent over the die and punch radius. Changes of the drawing force by one of this forming system parts can be realized by tool geometry (punch, die) and by increasing or decreasing blankholder force. Also the tribological system can be used to change the drawing force. The tribological system consists of all contact surfaces between tool and cup. The friction force in the four different areas (1: cup - die / 2: cup - blankholder / 3: cup - die radius / 4: cup - punch radius) is influenced by kind and quantity of lubrication and by mating of materials between tool and test sheet. So these parameters are also controllable factors regarding to the drawing force, because friction force is part of drawing force. Figure 4 describes all factors regarding to maximum drawing force for the used experimental arrangement.

Control[able variables (rneta[ forming parameter) .pmc~

= diameter

dst

= punch radius = material. - diameter

d,r,

= die racrmJs = materiat = L-[earance

uz

9blankl~der: = b(ankhotder force FN = material.

9test sheet:

= diameter

d

= gauge so = surface microslT'ucture = material. 9Lubricant:

- kind = quantity L

Figure 4. Controllable variables

:,~

_ I

(factors) influencing the maximum drawing force.

193 Four of these factors are chosen for a full factorial 24-plan. These n=4 factors are: blankholder force

FN

lubrication quantity S die radius

rz

blank diameter

d

The blank diameter has of course a great effect on maximum drawing force (compare Fig. 1). So, this factor is chosen to study, if the other investigated factors are independent from d. In this case the straight line in subcritical area would be shifted parallel analogous to Fig. 3. But if there are interactions between d and one or more than one factor, the straight line would be moved and turned, too. The next step of this work is how to investigate the effect on the limited drawing ratio. Further, a 23-plan is made for

133 =

2.18 (d = 240mm; hypercritical area).

This two plans are basic for the mathematical models to calculate the straight lines in sub- and hypercritical area. At least a relation between the calculated maximum drawing forces and the limited drawing ratio will be determined. All other metal forming parameters shown in Fig. 4 are kept on a constant level (table 1). Table 1 Test conditions. die material

AMPCO 25

blankholder material

AMPCO 25

punch material

1.2379

punch diameter

110 mm

,,

punch radius

3 mm

drawing gap

1.2 mm

lubricant

OEST Platinol BZK4

sheet material

AIMg0.4Sil.2

sheet surface microstructure

Isomill fein

gauge

0.8 mm

5.

FULL FACTORIAL DESIGN A full factorial 2n-experiment has n factors each at two levels. The range between the low

and high level should be as large as possible to calculate effects and interactions almost

194 certainly, if existing. Further, the uncertainty for the calculated relationship between factors and target value caused by random variance of testing results can be minimized [12]. The factor levels then are combined in such a way, that all possible combinations of the factor levels can be realized in one run. The notation of factors and runs is strongly systematic. The factors are marked with capital letters, started with "A". The name of a run indicates, which factor levels are realized in this run, because the name consists of all corresponding small letters of the factors, which are on high level, with exception of the test, where all factors are at their low level. This test is called "(1)". Table 2 shows the normal 24-experimental design. The four factors: A

= blankholder Force F N

B

= lubricant quantity S

C

= die radius r z

D

= blank diameter d or drawing ratio 13

comes to 24 = 16 runs.

Table 2 Normal 24-experimental design.

1

8

(1)

2

15

a

3

9

b

4

11

ab

5

14

c

6

1

ac

7

2

bc

8

10

abc

9

16

d

10

5

ad

11

4

bd

12

6

13 14

+

-

_

+ J

+

+

+

+

-

-

+

+

-

+

abd

+

-

+

13

cd

-

+

+

3

acd

15

12

bcd

16

7

abcd

+

+

+

-

+

+

+

+

+

+

+

+

195 Besides of the normal notation the table shows all levels which shall be realized in a run by a sign. (-) means the low level, (+) the high level. The first column contents the serial number, column two the testing number. This number is created by random. The experiments shall be realized in this random order to eliminate uncontrollable, systematic disruptive factors. It is important to prevent, that the effect of a disruptive factor is added to a controllable factor. By selecting the low and high level of blankholder force the low one has to prevent wrinkling and the high one should not create stretch forming processes. For the used experimental arrangement the values are fixed at F N = 35kN and 50kN, which corresponds to an onset value ofblankholder pressure of 1.31N/mm 2 and 1.87N/mm 2, resp.. The lubricant quantity levels S = 3g/m 2 and 6g/m 2 are selected according to the quantities usually used in practice. The die radius is fixed at r z = 3mm and 6mm. At least the chosen blank diameters are d = 150mm and 200mm, so both diameters are in the subcritical area.

6.

EXPERIMENTAL RESULTS To minimize the influence of random variation, each of the 16 combinations (table 2) is

replicated three times. In addition, by means of these replicates an analysis of variance can be conducted. This analysis of variance indicates certainly, if changes of the target value are influenced by the effect of a factor or is caused by stochastic errors in measuring. Table 3 contents the results of the 48 runs (column 2-4), and in addition, the statistical average ~Fma x (column 5). The other columns are explained later.

7. ANALYSIS OF VARIANCE It is not meaningful to list the experimental results without supplementary analysis. But because of the specific combination of parameters used in full factorial design, it is possible to calculate effects and interactions caused by the factors, too. This quantities are explaining the changes of the target value, in this case maximum drawing force, under the influence of a controllable factor. Then the analysis of variance

(ANOVA) helps to prove, if target value

changes are really influenced by increasing the level of one factor or if these changes are caused by experimental variations and disruptive factors. This ANOVA-procedure is explained very well in statistical literature [6, 7, 13, 14]. So, it will not be repeated here.

196 Table 3 Experimental results and analysis of variance (ANOVA). iname Fmax. 1st run /

kN

(1) a

28.093

i

c ac ,bc abc

l

d

a

26.256 26.354 ~

28.386

~

28.464

,25.905 1

,

25.983 62.125

ad

61.851

bd

1

cd

n

acd

kN

28.269 28.054

i

26.353 ' ,25.944 28.054

l

28.483

i

29.265

i

28.210

9

28.568

i

9

~ J

26.198 62.203

I

i

26.081 61.812

60.484

i

i

I

62.242 62.085

i

l

m

-0.110

1.769

no

-1.796

472.646

yes

I

62.476 62.359

l

62.046

i

60.549

ABC

I

62.099 62.125

no

0.037

0.197

no

-0.020

0.061

no

i -0.0951

1.328

no

|

i

BD

60.627 !

0.668

AC

D

i

l

0.068,

c

,61.838 mAD

60.523 i

26.087

1% /

A

,i25.976 B C

I

/

88.559

,25.866

9

signif.

Total

i26.159

60.874

bcd

28.353 i

kN

/

28.184

60.484

61.929

value

l

i28.034

i60.640

l

average!

28.054

m

60.523 61.577

effect- test-value

I

26.217 26.276 B I ~ ' L26.276 L26.191 A B

,62.203

1

•Fma x [ effect kN

28.347

,61.460

abd i

Fmax 3rd run

kN

28.444

b ab

Fmax 2nd run

CD 9

|

,

0.015 a

0.051

,

no i

23.393

!

yes !

-0.071

0.735

0.090

1.196

|

no yes

!

q

i

|

0.2E+06 0.035

0.400 i

ABD i

-0.019 34.098

|

no no i

ACD

-0.009

0.012

no

0.053

0.410

no

-0.1281

2.392 i

no

60.835

60.952

~60.640

60.809

BCD

abcd j61.030

60.210

60.366

60.536

ABCD

F(1, 32, 1%) = 7.500 factor A

blankholder force

min. 35kN

max. 50kN

factor B

die radius

min." 3mm

max. 6mm

factor C

lubricant quantity

min. 3g/m 2

max. 6g/m 2

factor D

drawing ratio

min. 1.36

max. 1.82

Table 3 (column 6-9) contains the results of the analysis of variance. The effects, usually named with the same capital letter as the corresponding factors (in this case A, B, C and D), are indicating the medium changes of target value, if one factor is kept at its low or high level. Effect B shows for example, that the maximum drawing force decreases for 1.796kN on average, if the die radius changes from 3 to 6mm, while effect C means an increasing maximum drawing force for 0.037kN by using 6g/m 2 lubricant instead of 3g/m 2. Also important is the interaction BD between die radius and blank diameter. This quantity indicates, that the

197 influence regarding to Fmax decreases by increasing of blank diameter. This fact is explained in Fig. 5. The schematic graph demonstrates, that it is possible to get a larger limited drawing ratio by using a greater die radius. But the effect of the high level of rz regarding to Fmaxreducing is not so great for the high level of 13 as for the low 13-level, so this is negative for the limited drawing ratio 13max. Fz

J

/

FZ, B r u c h

.7 J

~l-effect

J J

J

J

I

200

mm

"i Y

~J

B for d z =

T J

J

J

3 mm

rz = 6 mm 7 L'---

J

J J

rz=

/

effect

B for d 1 =

150 m m

i

J

J

i

straight

line

B low

straight

line

B high

Figure 5. Interaction BD between die radius and drawing ratio.

Changes of maximum drawing force are not only influenced by controllable variations of the investigated factors, but also by uncontrollable disruptive factors. The analysis of variance [ 13 ] allows to decide, if the maximum drawing force is really influenced by one or a series of investigated factors. Such a factor, which can be used for optimization of the target value, is called

significant. Said in simplified terms, the effects and interactions are significant, if this

values are greater as the natural variance of the experiment. To prove significance, a so-called F-test is used to compare experimental variance and effects. The experimental results, in this case the 48 values of Fmax, are used to calculate a test statistic for each effect or interaction. This test statistics are compared with the critical value of a fixed significance level, mostly 1% or 5%. In column 10 of table 3 the results of this test are listed by "yes" and "no". This means, that an effect is either ("yes") or not ("no") significant at the l%-risk of erroneous rejection. Comparing the test values (table 3) with the critical value F(1, 32, 1%) = 7.50 gives

198 significance for effect B (die radius), interaction BD (between die radius and drawing ratio), and naturally for effect D (drawing ratio). The confidence intervall for this result is 99%. Fig. 5 presents the results of the analysis of variance. This bar chart allows to compare the size of some selected effects and interactions. The most important part in this graph is the significance area, marked by a straight line. All factors, which are greater than this area, can be used for changes of maximum drawing force and so, for limited drawing ratio, too. So, in this case only r z can be used to optimize 13max.

l

8,5

significance area

-8.8

-1.4

-2.8

A

B

~

C

AC

BC

~C

Figure 6. Effects and interactions in comparision with significance area 1%.

8. MATHEMATICAL MODEL Started out from the analysis of variance a mathematical model can be established. This model describes the relationship between the target value (maximum drawing force Fmax) and the significant factors (die radius r z and drawing ratio 13) by means of a linear regression procedure [ 12] Fmax = f(rz, 13) .

199 This summarizing equation is for the presented 24-experiment:

/ ]

Fmax - [ 4 4 . 2 8 - 0.898. ] ~ - 3

+17.049.

+0.2. ~ - 3

.

0.227

0.227

-6.99

kN

(1)

By means of this equation any desired value of Fmax can be determined in the range 3mm _
and

1.3636 < 13 --- 1.8182 .

This model allows to calculate the maximum drawing force for all combinations of r z and 13 within range of validy. Because there are only two significant factors it is possible to present equ. (1) as an area in three-dimensional space (Fig. 7). According to this graph a minimum value of Fma x is given by the parameter combination r z = 6mm and 13 = 1.3636, this is Fmax = 26.133kN. The factors A = blankholder force and C = lubricant quantity could be fixed on any level between 35kN up to 50kN and 3g/m 2 up to 6g/m 2, resp.. In practice a level would be chosen, which is easier to create and/or of lower cost.

71

kN 61

L

/

51 41 E"o) E . c_

31

1 1.9 1.7"

E-o 9

4

5.5 die

radius rz

Figure 7. Diagram of model equation.

6 mm

1.3"

~x~

/

200 Equ. (1) describes the straight line in subcritical area, according to Fig. 2. In order to calculate the limited drawing ratio 13max, one point in hypercritical area is needed to determine the failure load FBr. According to the method by Schmidt-Kapfenberg this failure load is independent from the used metal forming parameters. So, one or a few experiments (to eleminate natural variance of measured values by calculating the mean) are enough. But to bear out this thesis, a 23-experiment with three replicates is carried out with blank diameter d 3 = 240mm resp. drawing ratio 133 = 2.18 and controllable factors A = F N, B = r z and C = S. The analysis of variance indicates, that none of this factors is significant regarding to Fmax. So, the calculated statistical average of all observed values of the 24 runs (each with three replicates) in the hypercritical area can be used as failure load. This calculation is delivering, here: FBr = 67.435kN

9.

(2)

O P T I M I Z A T I O N OF L I M I T E D D R A W I N G RATIO The summarizing equ. (1) for the 24-experiment describes the straight line in subcritical

area. This relationship together with failure load FBr (equ. 2) can be used to calculate the limited drawing ratio for any value of the significant factor r z. Therefore, equ. (1) is equated with equ. (2) and solved to 13. The solution is a summarizing equation for 13max: 135.59 + 1.532. rz / mm 13max(rz) = 72.367 + 0.587. rz / mm

(3)

By means of this equation any value of the limited drawing ratio can be determined for a given die radius in a range between 3mm and 6mm. For example, for a given value r z = 4mm a limited drawing ratio 13max = 1.896 can be realized. Optimum is found at 13max = 1.908 with r z = 6mm.

10. C O N C L U D I N G R E M A R K The intention of this work was the optimization of deep drawing conditions by reducing the maximum drawing force that is needed for metal forming and to increase the limited drawing ratio by means of designed experiments. A statistical design of experiment is a test, or a series of tests, in which the controllable input factors or process variables are varied

201 according to a particular pattern. The effects, which these variations have on the output response or target value are observed and evaluated statistically by an analysis of variance. This statistical technique enables to determine which factors and probable interactions between the factors have a significant influence on the target value. Finally, by means of a linear regression model a mathematical relationship between these significant factors and interactions on the one hand and the target value on the other hand is established. In the paper presented the designed experiment is a 2n-factorial one with three replicates in the blank diameter, die radius, lubricant quantity and blankholder force. By means of this the target value drawing force on the one hand and target value failure load on the other hand are investigated. Both investigations are delivering, finally, that the die radius is the only significant influential factor on the limited drawing ratio regarding the four input factors mentioned above. But, of course, other input factors can be studied in future, too, by the presented method, e.g. tool coating, kind of lubricant, etc..

REFERENCES

1. G.E. Dieter, Mechanical Metallurgy, Mc-Graw-Hill, Inc., New York 1986. 2. M. Schmidt, Archiv fur das Eisenhnttenwesen 3(1929)3, 213. 3. W. Ziegler, Industrie-Anzeiger 91 (1969)30, 19. 4. R. Woska, Werkstatt und Betrieb 114(1981)5, 342. 5. C. Schulze, QZ 36(1991)6, 334. 6. R.A. Fisher, The design of experiments, Oliver and Boyd, London, 1966. 7. O.L. Davies, The Design and Analysis of Industrial Experiments, Longman Group Ltd., London, New York, 1980. 8. J. Rabe, Kunststoffe 83(1993)4, 291. 9. K. Lange, Umformtechnik Band3: Blechbearbeitung, Springer-Verlag, Berlin, Heidelberg, New York, 1990. 10. E. Doege, et. al., VDI-Berichte 330(1978), 12. 11. H. Kuhn, VDI-Z 132(1990)12, 91. 12. B. Pegel, Empirische Modellbildung und Versuchsplanung, Akademie-Verlag, Berlin 1980. 13. K. D. C. Stoodley, Applied Statistical Techniques, Ellis Hoi'wood Ltd., England, 1980. 14. L. L. Lapin, Probabiltity and Statistics for modem engineering, 2nd. ed., PWS-Kent Publishing Company, Boston, 1990.

This Page Intentionally Left Blank

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

203

Formability, damage and corrosion resistance of coated steel sheets J.Z.Gronostajski and Z.].Gronostajski Institute of Mechanical Engineering and Automation, Technical University of Wroctaw, ul. Lukasiewicza 3/5, 50-371 Wroctaw, Poland

1. EN'TRODUCTION In recent years the demand for good corrosion protection in wide variety of finished products has resulted in the increasing use of various kinds of coated steel sheets [1-4] manufactured according to three processes: electrogalvanized coating - hot dip galvanized coating roller coating. Usually, the coatings are made of the following materials: pure zinc, alloyed zinc and plastic. A surface of coatings is essential not only for the protection, but also for the decoration, of a very wide range of products. The use of coated steel in press metal forming is becoming a quite common practice. The application, for example, of HSLA steels in the motor car industry enables a reduction in material thickness, in the weight of the car and in fuel consumption, but places higher demands on corrosion protection. The forming processes of coated steel sheets are limited by the similar phenomena as arise during the deformation of uncoated steel sheets, namely plastic instability, fracture, wrinkling caused by inappropriate mechanical properties, and additionally by the other phenomena like as: formation of cracks within the coating, lack of adherence between the base steel and the coating, powdering and abrasion, caused by the roughness of the surface layers. These phenomena are caused by the combination of a soft surface film and a hard base material and by differences in surface characteristic of coated steel sheets from those of ordinary cold rolled steel sheets. Special attention has to be paid to the influence of the deformation conditions on the performance of the coating, as regards protection against corrosion. The adhesion of the coatings- either metallic or organic - must remain good, and the surface should not be damaged during forming. When the coating beans to crack, the main property of coated steel sheet, i.e., its corrosion resistance, starts to deteriorate due to the exposure of the base steel to the atmosphere. Generally the application of coated steel sheets in press forming raises three main questions: the formability of the coated steel sheets, the adherence of the coating during the forming operations, and the effect of the forming on the behaviour of the coating. The paper gives the answer to above questions. -

-

204 2. E X P E R I M E N T A L PROCEDURE

The presented results were obtained on the base of investigation of fifth types of coated steel sheets [5-7]: zinc electrogalvanized steel sheet (ZEGS), zinc hot-dip galvanized steel sheet (ZHDGS), polyethylene coated zinc hot-dip galvanized steel sheet (PCZHDGS), acryl coated zinc hot-dip galvanized steel sheet (ACZHDGS) and PVC coated zinc hot-dip galvanized steel sheet (PVCCZHDGS). While the organic coating of the zinc layer was on one side only, the zinc coating was on both sides of the sheets. The thickness of the coatings and of the sheets is presented in Table 1, the chemical composition of the steels are given in Table 2. TABLE 1 The thickness of the coatinss and the total thickness of the coated steel sheets Thickness of Material Sheet Zinc Organic Intermetallic layer coating layer (mm) (~m) (~tm) (~m) ZEGS 1.03 10 ZHDGS 0.90 28 3.1 PCZHDGS 0.95 26 60 3.5 ACZHDGS 1.05 23 400 3.1 PVCCZHDGS 0.93 20 200 3.0 TABLE 2 Chemical composition of the steels investisated Material C Mn Si P weight % ZEGS 0.08 0.27 0.03 0.025 ZHDGS 0.08 0.35 0.01 0.025 PCZHDGS 0.09 0.37 0.01 0.028 ACZHDGS 0.08 0.36 0.02 0.035 PVCCZHDGS 0.09 0.44 0.01 0.035

S

AI

Ni

Cu

0.05 0.05 0.03 0.02 0.02

0.006 0.007 0.008 0.004 0.006

0.07 0.08 0.05 0.10 0.10

0.07 0.08 0.05 0.10 0.10

The forming limit diagrams were made by using specimens of varying width, to secure all the types of strain paths occurring in practice during a pressing operations. The FLDs were determined for simple strain paths (SSPs) and complex strain paths (CSPs), the latter consisting of two different linear stages of deformation. In the first stage the specimens were tensioned uniaxially to approximately e 1 = -0.06, or deformed equibiaxially to about e 1 = e 2 = 0.06, whilst in the second stage the strain paths were changed from uniaxial to nearly equibiaxial deformation. During the first stage of deformation in-plane conditions existed and in the second stage of deformation a hemispherical punch, 110 mm diameter, was used. The methods used for strain analysis and plotting of the FLDs have been described elsewhere [5]. The organic coated sheet steels were investigated in three various conditions: first, with the organic coatings, next with a zinc layer after the removal of the organic coatings, and finally, after the organic and zinc coatings had been removed, so that the base steels were in contact with the punch. The friction coefficients between various surface layers and the steel

205 tool were measured by using equipment similar to that one described elsewhere [8]. The study was performed without additional lubricant, but the soft zinc coating and the organic layers can be assumed to act as solid lubricant. The cracking resistance of the zinc coating was evaluated using the specimens that were tested to establish the FLDs. Several samples were cut from each of the specimens and examined under a scanning electron microscope. Scanning of the zinc coating was started at the fracture or necking band and was continued in the direction of major strain towards the less strained area away from the crack, until an area was reached in which no zinc cracking was noticed, the strain measured in this area indicating the cracking limit. The strain at which cracking began for a given path was marked on the same coordinates system as the FLDs. The cracking limit curve was drawn by connecting the points for the different paths, the flaking limit curve being evaluated in the same manner. As it was difficult, sometimes, to distinquish between cracking and flaking of the zinc layer using the scanning electron microscope, adhesive tape was employed to evaluate the flaking. SEM was also used to investigate the damages in the steel sheets such as: the type and size of cracks, the nucleation and growth of voids, the size, shape, quantity and chemical composition of the inclusions. The corrosion resistance was measured by the resistance to red rust in a corrosion test carried out according to American Standard C44-75. The corrosion test utilizes a one hour cycle that includes a 10 minutes period in a water solution of 3.5% NaCl followed by a 50 minutes period out of solution, dunng which the specimens are allowed to dry. This one cycle is continued 24 h/day for a total of 30 days at a temperature of 27_+ l~ and a humidity of 45_+6%. A general pictures of correlation between the strain level and corrosion resistance were made by drawing in the coordinate system g l-g2 the progress of corrosion for defined strain paths and degrees of deformation. 3. RESULTS AND DISCUSSION Comparison of the FLDs of the zinc hot-dip galvanized steel sheet and of the zinc electrogalvanized steel sheet for linear and complex strain paths can be seen in Fig. 1. Formability of zinc electrogalvanized steel sheet for all strain paths used in the study is much better than formability of zinc hot-dip galvanized steel sheets. The differences are mainly caused by presence of a brittle iron-zinc alloy layer between the zinc coating and the steel. Thickness of the layer is round about 3.1 ~tm (Table 1). The effect of the organic coatings and zinc coating obtained by hot-dip method on the FLDs related to the FLDs of base steel sheets is presented in Fig.2-4 [6]. Curves were determined for each of the above mentioned three conditions. Generally, the presence of coatings improves the formability of the sheets, but the improvement depends on the type of coating material. Comparing these curves, it can be noted that the PVC layer exerts a more distinct effect on the position of FLC than do polyethylene coatings, whilst the smallest influence is exerted by acryl. The effect of organic coatings on the ratio of the increment of effective strain caused by the coatings to the effective strain at the point of instability of the bare steel, determined along different linear strain paths, is shown in Fig5. From the figure it can be seen that the relative increase of formability caused by each of the organic coatings under study, is not dependent on the linear strain ratio. For the complex strain paths the differences in the FLCs between coated steel and bare steel are more distinct when the final stage involves biaxial stretching than when it involves uniaxial tension: this is because the area

206 of contact between the punch and the deformed sheet, as well as the contact pressure, are greater during biaxial than uniaxial deformation. As a consequence, the effect of friction due to different surface conditions becomes more significant.

E1

O,Z,O~

,

- ' - P~ ---zinc

o2.o

!

015

C.I0

.... sfeel

-0, 5-0J0-0 35 o o.os o.lo

9

-0,05

I

-o, s-o, o-ops o Qos o,lo

o,2o

Figure 1. FLDs of zinc hot-dip galvanized Figure 2. FLDs of polyethylene coated zinc and zinc electrogalvanized steel sheets for hot-dip galvanized steel sheet for simple and simple and complex strain paths complex strain paths Compared to steel, the coatings are soft and, consequently, their shear strength is low. Hence, they can be assumed to act as solid lubricants in the forming processes, with a large thickness as compared to the surface roughness of the bare metal. The measured coefficients of friction, ~t, are presented in Table 3, from which it can be noted that the lowest value was obtained for the PVC coating and the highest value for the bare steel. The decrease of the friction coefficient due to the presence of the zinc coating is in accordance with the findings of Ranta-Eskola [9] and M~kimattila [10], but according to Flossdorf [3] the effect of the zinc coating on the coefficient of friction can be varied, dependently on the sliding velocity, tool material, surface roughness and lubricant.The results of friction tests, as well as those of the analysis of the FLDs, establish that the position of the FLCs increases when the friction between the punch and the formed sheet decreases, which is in agreement with the results reported by Weidemann [11], Blanchard [12] and Gerdeen [13]. Weidemann investigated the drawability of coil coated steels and found that some of the coated steels, e.g., coated with PVC, can be drawn to a greater drawing ratio than uncoated steels, due to the reduction of friction between the tool and the sheet surface. Blanchard [12] concluded that the reduction in sheet-tool friction caused by different types of zinc coating made a noticeable contribution to the increase in the performance of the metal

207

E1

0/+0

~\%. '~x.~

~" 9

,,

_.<--

'

...... .... 9

~. ....

0115 "aJ"

acry1,0,10 ~ zinc steel 0,0~

, , -q15-o9o-qos

T I

l i ,

I

-

o o.o5 o,lo E2 q2

Figure 3. FLDs of an acryl coated zinc hotdip galvanized steel sheet for simple and complex strain paths

-.-

~v

.0~0

---

zinc steel

t3,05

i

i

-0;15-0,10-0,05

0 0 , OSO;lOl~ z 0,20

Figure 4. FLDs of the PVC coated zinc hot-dip galvanized steel sheet for simple and complex strain paths

TABLE 3 The coefficients of friction between the sheet and the tool Coefficient of friction Material Plastic coating Zinc-tool Steel-tool -tool ZEGS 0.18 0.20 ZHDGS 0.16 0.20 PCZHDGS 0.11 0.16 0.19 ACZHDGS 0.13 0.17 0.20 PVCCZHDGS 0.09 0.15 0.19 in press forming. Veerman [14] also reported different FLCs for different lubricants, showing that the FLCs decreased with increasing friction. The opinion is advanced in some reports that modification of the friction conditions between the tool and the sheet metal changes only the strain paths, and has no direct effect on the FLCs. In such analysis one additional factor must be taken into account, which is that the modification of the friction conditions changes not only the strain path but also the strain gradient in the necking area. It can be concluded that the one of the reasons for variation of formability with the type of surface conditions between the tool and the formed sheet is associated with the change of the strain gradient. The effect of the complex strain paths of the coated steel sheets on the FLDs is illustrated in Figs 1-4. The FLDs for the uniaxially prestrained specimens are displaced in the direction of negative minor strain, whilst those for equibiaxially prestrained specimens are displaced in the direction of positive minor strain. As shown by the plots of Figs 1-4, the forming limit

208

..... ....

PCV polyethylene ncryt

0,3

i)

0,55

FLC /r-'---

FI~C

(:U

,:,., L,O

0,2 "="

==.

===-

.=,~

.-=-

,===.

m l , = = , = , , ~ = , = = .

===,

~

I .

025"

' ~ ~

,0,20 -0#

-02

0

Q2

0,4

Figure 5. The effect of different coating on the ratio of Aee/e e for different simple strain paths

o Qos

E2 o2o

Figure 6. CLC, FkLC and FLC of the ZEGS for simple and complex strain paths

curves can be raised significantly in the area of deformation close to plane strain by using the CSPs if the primary forming sequence is close to uniaxial straining. This is of great importance to the press-forming industry, as it allows the formability of the material to be increased by controlling the strain paths, particularly since final states of deformation close to plane strain are dominant in pressed auto body panels. Comparing the FLCs for linear strain paths with those for primary forming in equibiaxial stretching, it becomes obvious that the FLCs for a CSPs are lower than those for the SSPs. The effect of the CSPs on the FLCs of coated steel sheets is in agreement with the data reported in the literature [ 15,16] for uncoated steel sheets. The effective work-hardening rates of the steel sheet under biaxial loading are greater than those under uniaxial loading: the strain softening which occurs due to the change in strain path from biaxial to uniaxial, therefore produces a rapid loss in the ability of the biaxially prestrained steel to undergo uniform deformation. The effect of changing of the strain path from uniaxial to equibiaxial on the limit strain can be explained similarly. The cracking (CLCs) and flaking limit curves (FkLCs) in comparison with the FLCs for zinc electrogalvanized, zinc hot-dip galvanized and polyethylene coated zinc hot dip galvanized steel sheets are shown in Figs 6-8, and for other sheets are given in papers [17]. From the Figs 6-8 one can notices, that the highest value of cracking strains has the electrogalvanized zinc coating. The coating has a constant chemical composition over the thickness of the coating without intermetallic layer, nearly the constant thickness and small grain size, what is a reason, which in negative minor strain region makes coating follows deformation of the steel core without cracking. In the positive minor strain region the cracking

209

limit curves of electrogalvanized zinc coating lie below the FLCs but much higher than CLCs of zinc hot-dip galvanized steel sheets. From Figs 6-8 it is evident that the cracking of the zinc layer depends on the strain ratio. For both linear and complex strain paths, the higher is the value of s2/e 1 ratio the lower is the value of the major cracking strain of ZHDGS sheets and the larger are discrepancies between FLCs and CLCs of ZEGS sheets. The cracking limit curves of the specimens that were prestrained equibiaxially are lower than those of the specimens that were prestrained uniaxially. By comparing the cracking limit curves for linear- and complex-strain paths (Figs 6-8), it can be noted that the cracking limit curves for uniaxially prestrained specimens lie at a higher major strain than the cracking limit curves for linearly deformed specimens, whilst the latter in turn lie at a higher major strain than the cracking limit curves for equibiaxially prestrained specimens. This shows that biaxial stretching is an unfavorable forming mode for hot-dip galvanized and electrogalvanized steel sheets and that uniaxial tension seems to be recommendable. From Figs 7-8 it can be noted also that for hot-dip galvanized steel sheets in the range of thickness of zinc layer investigated, the greater is the thickness of the layer (Table 1) the higher is the position of the cracking limit curves at the same SSP. The highest position is for the ZHDGS sheet with a zinc layer of 28 p.tm thickness (Fig.7) and the lowest is for the PVCCZHDGS sheet with a zinc layer of 20 lam thickness [ 17].

El 9,~

I . I

1

,

~,& ! / 7 1 u , ~ u i ~ ~ , 7 " ,

-~--..-.,r% i " ' , 3 6 - i

- /z

1

,7-,1'I-

Ix'4

!

i

CLC , , 1_

o,lo

1

..o.o5 I 1

-0 15-0,10-~5 0 ~5 0,10 ~2 0,20 Figure 7. Cracking limit curves, flaking limit curves and forming limit curves of the zinc hot-dip galvanized steel sheet for simple and complex strain paths

iL

.....

I

iI

0,0~

-0,15-0,10-0135 0 0,O5 0,10 s

02O

Figure 8. Cracking limit curves, flaking limit curves and forming limit curves of polyethylene coated zinc hot-dip galvanized steel sheet for simple and complex strain paths

Cracking of the zinc coating and the growth of the cracks during deformation was analysed in terms of the formability of the coating. The area fraction of the cracks 0rc) as a function of

210 the thickness strain (e3) for linear-and complex-strain paths is shown in Figs 9 and 10 respectively. The lines in each of these figures have been plotted using a simple model suggested by Makimattila and Ranta-Eskola [18]. In the model it is assumed that if the coating is rigid (i.e., does not deform plastically) the area fraction of the cracks is determined by the area increase due to deformation (solid lines). The formability index of zinc layer (k) was defined by the equation below.

(I)

fc = 1 - exp [(! - k)c~]

A value of k = 0 means that the coating does not deform plastically, whilst for k = 1 the coating deforms in the same manner as does the base steel (i.e., without cracking). The values of k at simple and complex strain paths for zinc electrogalvanized steel sheet are much higher than those for all kinds of hot-dip galvanized steel sheets (Table 4). Schedin et al [19] for zinc hot-dip galvanized steel sheet obtained the similar results of k = 0,8 as an average value for uniaxial tension, plane strain and equibiaxial stretching. The k values for all the steel sheets used in the study, are greater for uniaxially deformed or predeformed specimens than that for equibiaxially deformed or predeformed specimens. This means that the formability of the zinc coating for equibiaxial deformation is lower than that for uniaxial deformation, which is in agreement with the conclusions drawn from the cracking limit curves presented in Figs 6-8.

k..I

---catcut~afed -.-unbx~ot tension .... ploq,e simjn ;------:

,4---

.<E 0::: C)

z

0 I--L_J

---eQUIDIQ~IQ[

0,20

<

L_U C~

0,05 0

/~,

v

sfretch~ng/ I ck--0

~---eq9. ibioxio[ § , uniaxio( /I

(_.1

q20"- strain

Y

0,15

< 0,10

0:5 !1

] cotcutcd'~

q2s .-.-uniQx~l-,-equi-, biaxio[ strdin,~ f,L

I.I0

k-0,62,

i ! I i

i I

z 0,15 0 I--

I

tk:0?6 i

I -o,I -02 -0,3 -0,4 -0S THICKNESS STRAIN, E3

Figure 9. Area fraction of the cracks of the zinc layer as a function of the thickness strain of the PCZHDGS for simple strain paths

O,10

,,t

'k-0 !/ -.'""

u_ I_1_! r~

095

//

<1:

0

I" -0,I

I-

~" i

k-1

\ -02 -Q3 -o# -o5

TI-IICKNESS STRAIN, E3

Figure 10. Area fraction of the cracks of the zinc layer as a function of the thickness strain of the PCZHDGS for complex strain paths

The formability of the zinc coatings obtained in the hot dipping process to a certain degree is conditioned by thickness of intermetallic layer between zinc and steel. The differences of the

211 TABLE 4. The value of formability index of zinc layer Strain paths Material

ZEGS ZHDGS PCZHDGS ACZHDGS PVCCZHDGS

Uniax. strain 1.0 0.75 0.76 0.74 0.70

Simple Plane strain 0.95 0.65 0.67 0.65 0.60

Complex Equibiax. "Uniax.+equi- Equibiax.+unistrain biax. strain ax. strain 0.85 0.96 0.93 0.60 0.72 0.62 0.62 0.73 0.63 0.61 0.71 0.62 0.56 0.70 0.56

intermetallic layer thickness (Table 1) of various coated steel sheets are very small, and probably can be neglected in the analysis of the effect of the layer thickness on the cracking phenomena of zinc coating. The cracking usually is initiated within a brittle intermetallic layer, from where it propagates mostly to the surface. In this case no loss of adherence has occurred yet and the base steel has not cracked (Fig. 11). Early in the forming stage only intercrystalline cracks were formed for all forming strain paths, and for larger equibiaxial and plane deformation transcrystalline cracks were created also. According to Engberg at all [20] intercrystalline cracking in hot-dip galvanized zinc coating is associated to accumulation of oxygen due to grain boundary diffusion, but lead appearing as a thin film in interdendritic areas due to segregation during solidification is responsible for transcrystalline cracking.

Figure 11. Cross-section of the acryl coated Figure 12. Cracks in the zinc coating of an zinc hot-dip galvanized steel showing the acryl coated zinc hot-dip galvanized steel cracking of the zinc layer specimen uniaxially deformed to Ce=0.42 When the major strain was much higher than the minor strain, i.e., for uniaxial strain, the cracks were oriented mainly perpendicularly to the major strain (Fig. 12), but for nearly the

212 same values of major and minor strain as those of plane strain and equibiaxial stretching, the orientation of cracks is random (Fig. 13). Similar results were observed for complex strain paths. When uniaxially predeformed specimens are subjected to uniaxial or plane strain, the effect of prestrain dominates and the majority of the cracks are perpendicular to the major strain direction. The dominating effect of prestrain was not visible when in the second step the specimens were equibiaxially deformed: the orientation of the cracks is then random. When specimens were equibiaxially prestrained, the predeformation, regardless of the second strain paths, has a dominating effect on the orientation of the cracks, which are usually randomly oriented. However, in the case of specimens uniaxially and plane strained in the second stage, a somewhat greater number of cracks oriented perpendicularly to the major strain could be seen. The differences in the mechanical properties between the coating and the base steel are mainly responsible for the shear stress at the contact surface. When the shear stress being higher than the force of adhesion between the two layers the coating will be subjected to flaking at the interface with the steel (Fig. 14). Flaking of the zinc coating occurs at almost the same strain level as does the limit strain of the base steels (Fig.6-8), the small discrepancies between them growing significantly with increase in the value of the e2/el ratio, over the whole range of linear strain paths and in the second stage of deformation for complex strain paths. Good agreement of the FLCs and FkLCs means that the adhesive forces between the zinc layer and the steel are high.

Figure 13. Cracks in the zinc coating of an Figure 14. Separation between the zinc layer acryl coated zinc hot-dip galvanized steel and the base steel after plane deformation of an specimen equibiaxially deformed to ee-0.42 acryl coated zinc hot-dip galvanized steel The formability of coated steel sheets is connected with the occurrence of voids and their growth and coalescence, leading finally to the fracture. Voids growth can be described with various theoretical models, each of them is based on the some assumptions. Taking into account, that voids usually nucleate by decohesion at the interface between the inclusions and the matrix and those inclusions are stiff and behave in the

213 same way, the change of volume fraction f of voids during deformation can be expressed in the form similar to that one given by Melander [15] and Gladman et all [21]

f=fo

S

~-i-[exp(ee + eo)- 1]

(2)

where: fo- initial volume fraction of inclusions, s - strain concentration factor, r - width to length ratio of the inclusions, eo- fraction of voids in the as-received sheets and ce- effective strain. Metallographic examination of cross section of steel sheets by using micro-computer analyser revealed that fraction of voids eo is nearly the same in each steel sheet, and can be determined as 0.01, the inclusions fraction fo = 0.03 and the ratio of width to length of inclusions r = 0.6. According to results of Gladman et all [21] strain concentration factor was taken s = 2. The calculated growth of the voids during deformation described by equation (2) is presented in Figs 15 and 16. In the same figures the experimentally determined voids growth during deformation along different strain paths of zinc electrogalvanized and polyethylene coated zinc hot-dip galvanized steel sheets is illustrated. The similar informations about the behaviour of the other coated steel sheets are given in papers [22,23]. From the figures it can be seen, that the theoretical calculation gives lower values of volume fraction of voids than experimental results. This should be due to coalescence of voids, which is accelerated by closed spaced inclusions and voids. According to Brown and Emburry [24] voids coalescence when their length is equal to their spacing. The theory assumes a uniformity of voids and inclusions distribution, which is not usually observed in practice. This could be the main reason for observed discrepancies. The voids have a softening effect on the properties of the steel sheets, which accelerates necking, especially if a large density of voids and inclusions are nonuniformly distributed in the steel sheets The softening promotes strain localization and leads to sharp necking in the inclusion rich region of the steel sheets. From Figs 15 and 16 one can notices that the volume fraction of voids is dependent on the strain paths. The values of volume fraction of voids obtained at plane strain was higher, than those ones at the other strain paths. This may be one of the reason for low formability of steel sheets at plane deformation. More intensive growth of voids at plane strain can be attributed to the combined effect of the hydrostatic pressure and of preferable inclusion orientation against straining direction, both factors promote voids growth. To obtain a better agreement between calculated volume fraction of voids and experimental results, the strain concentration factor s should be changed according to rotation of the major strain against the orientation of inclusions, which take place during forming processes. Usually voids nucleate as a result of decohesion at the particle-matrix interface but sometimes as a result of fracture of inclusions (Fig. 17 and 18). There is the influence of the strain paths upon the type of the fracture surface of coated steel sheets. The fracture obtained at equibiaxial deformation is characterized by large and round shape dimples, plane strained specimens have fine and lit bit elongated dimples. Fracture surface of uniaxially deformed specimens is similar to scales of fishes. There are no distinct differences between the nucleation and the propagation of voids in the zinc coated steel sheets and in the uncoated steel sheets. The behaviour of zinc electrogalvanized steel sheet in the alternate immersion corrosion test after 720 h, for linear strain paths is illustrated in Fig. 19 and for polyethylene coated zinc

214

6

~o

I

f~o

- - - - - calcutafed

I

calcutafed

3

/ I

s"

/

/

2

/z~"

56-

"'yY>~" 1 o

o,2

0,4

EFFECTIVE STRAIN , Ee

.

1

0,6

Figure 15. Volume fraction of voidsf/fo as a function of effective strain at different strain paths (p = g2/~l) for the zinc electrogalvanized steel sheet

/ .,{//

-r

0,2 0,~ 0,6 EFFECTIVE STRAIN jEe Figure 16. Volume fraction of voids f/fo as a function of effective strain at different strain paths for the polyethylene coated zinc hot-dip galvanized steel sheet

Figure 17. Voids initiation due to Figure 18. fragmentation of inclusion and decohesion at fragmentation inclusion-matrix interface in ACZHDGS ACZHDGS

Voids initiation due of elongated inclusion

to in

215 hot-dip galvanized steel sheet in Fig.20, and for complex strain paths in Figs 21 and 22 respectively. The corrosion resistance of the other coated steel sheets described in Tables 1 and 2 is presented in paper [25]. In the figures, the forming limit curves of the bare steel sheets and the flaking-limit curves and cracking-limit curves of the coatings are shown also. The percentage of red rust varies, depending on the level of deformation and on the strain paths. The specimens deformed by uniaxial tension are characterized by a higher corrosion resistance as compared with the specimens strained by equibiaxial stretching (Fig.19 and 20). However, for plane strain the corrosion resistance is intermediate between that for uniaxial and that for equibiaxial deformation. Moreover, over the whole range of deformation, the amount of red rust increases with increase in the major strain. For the complex strain paths the corrosion resistance is lower in equibiaxially prestrained (positive minor strain region of Figs 21 and 22) than in uniaxially prestrained (negative minor strain region of Figs 21 and 22) specimens. For a given value of major strain, more red rust is noticed in the equibiaxially prestrained than in the uniaxially pre-strained specimens. Uniaxial linear deformation has a less distinct contribution to the deterioration of the corrosion resistance in zinc coated steel as compared with complex deformation in uniaxial followed by equibiaxial stretching or by plane stretching , which means that for uniaxially prestrained specimens, the maximum corrosion resistance occurs for uniaxial strain in both stages of deformation (Figs 21 and 22). Different results were obtained by Monford and Bragard [26] during investigation of zinc hot-dip coated and electrogalvanized steel sheets. They stated, that slight straining of hot-dip coated steels decreases the corrosion resistance, and further straining - up to the forming limit curve - has no additional effect, but for electrogalvanized zinc coated steels the straining has no significant effect on the corrosion resistance. It can be noted that the corrosion resistance depends on the type of material involved. Thus, the amount of red rust determined for the PVC coated zinc hot-dip galvanized steel is greater than that for the acryl coated zinc hot dip galvanized steel, that for the latter being greater than that for polyethylene coated zinc hot-dip galvanized steel [32]. This phenomenon should be attributed primarily to the slightly different thickness of the zinc layers (Table 1) and is due to the greater heterogeneity of the zinc layers (observed by SEM) in the PVC coated steel than that in the acryl and polyethylene coated steel sheets [34]. Analysis of the results obtained has shown that the corrosion resistance is influenced not only by different thickness and heterogeneity of the zinc layer but also by cracking and flaking of the zinc layer. When cracks are small steel sheets are protected by the cathode protection of zinc. 4. CONCLUSIONS According to the presented results of the investigation of fifth types of coated steel sheets, deformed along simple and complex strain paths one can come to the following conclusions. 1. The type of the coating influences the position and shape of the FLCs obtained for linear and complex strain paths. The greatest limit strain was obtained for zinc electrogalvanized steel sheet, and for such coating of hot-dip galvanized steel sheets that had the lowest coefficient of friction against the punch. 2. Deformation along complex strain paths with uniaxial tensile prestrain causes the growth of the FLCs of coated steel sheets whilst equibiaxial prestrain brings decrease of the FLCs, as

216

0,6 [~- [ii

I

red J

ASTIvl "

I

-1

FASrM ......

~

;

~ 1

r~ImSO

0,5 5~:la

0,4

'i

0,3

~FLC

i

0,2 0,1

-0,2

-0,1

0

0,1 Ez 02

-0,1

0

0,1 E2 0,2

Figure 19. The effect of simple path deformation Figure 20. The effect of simple paths on the corrosion resistance of the zinc deformation on the corrosion resistance of electrogalvanized steel sheet the PCZHDGS sheet compared with those obtained for linear strain paths. The effect of the complex strain path on the FLCs of coated steel sheets is in agreement with that for uncoated steel sheets. 3. The cracking limit curves of the zinc coating are influenced by the strain paths. For linear strain paths, the discrepancies between the CLCs and the FLCs reduce as the strain ratio e2/el decreases. For complex strain paths, uniaxially prestrained specimens have a higher cracking limit than those which were equibiaxially prestrained. 4. The FkLCs of the zinc coating are almost the same as the FLCs when over the whole range or in the second step of deformation the s2/e I ratio is changed from -1/2 to 0. For biaxiall deformation over the whole range or in the second step of deformation the discrepancies between the FkLC and the FLC reduce as the strain ratio s2/e 1 decreases. 5. The cracks in the zinc layer are nucleated mainly as intercrystalline and sometimes as transcrystalline. When the major principal strain is much greater than the minor principal strain, the cracks were oriented usually perpendicularly to the major strain direction. 6. The formability coefficient (k) of the zinc layer is affected by both SSPs and CSPs.

217

k lnAS71~f umber" ~E 1

e ~ 3 ~

i

'

i,

, C,LC/~

I

~5o

,~

~t~

10

/

Jd ~

~-

,

t 1

-0,2

.

-0,1

0

0,1

s

0,2

l

-0,1

0

0,1

_

E2 0,2

Figure 21. The effect of deformation along the Figure 22. The effect of deformation along complex strain paths on the corrosion resistance the complex strain paths on the corrosion of the zinc electrogalvanized steel sheet resistance of the polyethylene coated zinc hot dip galvanized steel sheet 7. The corrosion resistance is affected deteriously by deformation. For linear strain paths, the corrosion resistance decreases when the strain ratio e2/el increases. At complex strain paths, the equibiaxially prestrained zinc coated steels are more corroded than the uniaxially prestrained steels and the corrosion resistance decreases, additionally, when in the second stage of deformation the strain ratio e2/e 1 increases. 8. The corrosion resistance is a function of the thickness of the zinc layer and of its heterogeneity, and is correlated with the cracking and flaking of the zinc layer. REFERENCES 1. 2.

G.Arrigoni and M.Sarracino, Proc. Int. Deep Drawing Res. Group Meeting. Amsterdam, 1985, paper 12. D.Perry, A.Jussiaume and Y. Deleon, Proc. 15th Biennial Congress of Int. Deep Drawing Res. Group, Dearbon, 1988, 1

218

.

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

F.J.Flossdorf, Proc. Int. Deep Drawing Res. Group Meeting, Toronto, 1988, (Rapport of German Group). Y.Hishida and M.Yoshida, Proc. 16th Biennial Congress of Int. Deep Drawing Res. Group, Borlange, 1990, 173. J.Z.Gronostajski, W.J.AIi and M.S.Ghattas, Advanced Technology of Plasticity, Springer Verlag, 1 (1987), 423. J.Z.Gronostajski, W.J.Ali and M.S.Ghattas, J. Mater. Process. Technol., 22 (1990), 137. J.Z.Gronostajski, Z.J.Gronostajski and Z.Zimniak, Obrobka Plastyczna, 2 (1991),5.. J.Z.Gronostajski and M.Sulonen. Zesz. Nauk. AGH. Mechanika. 9, (1986), 269. A.J.Ranta-Eskola, Proc. Int. Deep Drawing Res. Group Meeting, Helsinki 1983. paper WGII/1. S.Makimattila, Proc. Int. Deep Drawing Res. Group Meeting, Helsinki, 1983, paper WGIII/9. Ch.Weidemann, Mem. Sci. Revue Metall, 77 (1980), 343. G.Blanchard, Proc. Int. Deep Drawing Res. Group Meeting, Amsterdam, 1985, paper 18 J.C.Gerdeen and B.A.EI-Jan, Proc. 16th Biennial Congress of Int. Deep Drawing Res. Group, Borlange, 1990, 131. C.C.Veerman, Sheet Metal Ind., 48 (1972), 351. A.Melander, E. Schedin and L.Gustavsson, Report Swed. Inst. Met. Res. 1983, 1 J.Z.Gronostajski, A.Dolny and T.Sobis, Proc. 12th Biennial Congress of Int. Deep Drawing Res. Group, S.Margherita, 1982, 39. J.Z.Gronostajski, W.J.AIi and M.S.Ghattas, J. Mater. Process. Technol., 23 (1990), 321. S.Makimattila and A.J.Ranta-Eskola, Proc. 13th Biennial Congress Int. Deep Drawing Res. Group, Melbourne 1984, 293. E.Schedin, S.Karlsson and A.Melander, Proc. 14th Biennial Congress of Int. Deep Drawing Res. Group., Koln 1986, 460. G.Engberg, A.Haglund and S.E.Hornstrom, Proc. 16th Biennial Congress Int. Deep Drawing Res. Group, Borlange, 1990, 131. T.Gladman, B.Holmes and I.D.McIvor, JISI (1976), 68 J.Z.Gronostajski, Proc. Int. Conf. on Computational Methods for Predicting Mat. Proc. Defect, Cachan 1987, 347. J.Z.Gronostajski, Proc.16th Biennial Congress of Int. Deep Drawing Res. Group, Borlange 1990, 331. L.A.Brown and J.D.Emburry, Proc. of 3rd Int. Conf. on Strength of Met. and Alloys, London, 1973, 164. J.Z.Gronostajski, W.J.Ali and M.S.Ghattas, J. Mater. Process. Technol., 23 (1990), 21. G.Monford and A.Bragard, Proc. Int. Deep Drawing Pres. Group Meeting, Amsterdam 1985, paper 13. J.Z.Gronostajski, Proc. 16th Biennial Congress of Int. Deep Drawing Res. Group, Borlange 1990, 233.

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

219

Model of metal fracture m cold deformation and ductility restoration by annealing V.L.Kolmogorov Institute of Engineering Science of the Russian Academy of Sciences (Ural Department), 91 Pervomaiskaya str., GSP-207, 620219 Ekaterinburg, Russia Cold plastic deformation of metal (rolling, stamping, etc.) may be accompanied by fracture of product in the process of its treatment with the appearance of macrodiscontinuities or breakage into fractions. The inadmissibility of that is evident. These macrodefects are easy to discover (external ones by visual inspection, internal ones by some methods of introscopy). Practical forming has evolved methods to prevent macrofracture. To avoid the phenomenon, the complete deformation is divided into stages and annealing is performed at the end of every cycle, in order to soften hardened metal and restorate met-,d ductility, i.e. ability to be deformed without fracture. It is known in the physics of metals that plastic deformations (especially large deformations in forming) are accompanied by progressive accumulation and development of microdiscontinuities. Microfracture process is hampered by selfdiffusion and recrystaUization that promote healing of these defects. As it seems to us, industry is not equipped with microfracture control devices. (By the way, the creation of such devices is urgent.) Therefore, products manufactured by plastic deforming may have microdefects. It is probable that microdefects appearing at the stage of manufacture of machine and construction elements under plastic deformation will influence the durability of these elements. The failure of metals (both on the macrolevel and on the microlevel) during processing is almost the main factor that dete~rmines the quality of metal products in their use and thereby the efficiency of technologies. That is why the creation of a theory (model) of metal failure under metal forming is topical. A phenomenological metal fracture model has been worked out under the guidance and with the participation of the author. The main features of the model will be briefly shown in a new light in this work. The reader can familiarize himself with the model in detail from Russian monographs [ 1-8]. 1. MODEL OF METAL FRACTURE IN COLD DEFORMATION It is generally recognized in the physics of metals that plastic deformation of metals is accompanied by the growth of microdiscontinuities ('damage'), both in number and extension, from the first moments of the start of deformation [9]. On this basis, the simplest description of material particle damage by a scalar value (that we mark here by

220 ql(t) as a function of time), is proposed in fracture mechanics [10-12]. This value is normalized: at the moment of fracture appearance qJ(tp)=l , at the initial instant when damages are absent ~(0)--0. Intermediate values of q~ indicate some level of damage by microdiscontinuities. The formulation of a kinetic differential equation

ddtv---f(w,t)

(1)

describing the evolution of microdiscontinuities into macrofailure on the basis of macroexperiments is the problem of fracture model creation considered in this work. Equation (1) must be integrated for the determination of qJ to answer the question whether the macrofracture of a material particle will come or not, and what level of microdamage it will be. The integration must be carried out for the specific material particle along the path of its motion. Working out the phenomenological fracture model of type (1) is achieved by the following scheme: creation of a model variant: its experimental verification, correction or new formulation of the model, new verification etc. We show the model in one of the latest modifications, and the simplest one (as one of many previous iterations). It is supposed hereinafter that the boundary-value problem of the specific technological process of plastic treatment of metal has been solved. It means that, at every instant in the volume of the body being deformed, not only the material particles' motion paths are determined, but also the tensor fields describing the stress-strain state and other belowmentioned parameters. The path of the particle motion should be divided into monotonic deformation stages. Deformation at a stage will be monotonic if all the components of the strain rate tensor in the accompanying (Lagrange's) system of coordinates will not change their sign during this stage. The border between the neighbotaing stages of monotonic deformation is such a moment of time (or a point on the material particle motion path) in which at least one component of the swain rate tensor becomes zero when changing its sign. Division of the motion path of a material particle subject to plastic deformation in the monotonic deformation stages is conditioned by physical reasons. The fact is that there is a difference in the mechanism of damage accumulation by both monotonic and nonmonotonic deformation (the latter having at least two stages of monotonic deformation and being characterized by change of the deformation increment direction on the boundaries of the monotonic deformation stages). Dislocations of one definite sign appear at the stage of monotonic deformation, developing in one direction. With the growth of deformation the number of such one-sign dislocations increases, and there appear aggregates of dislocations leading to the formation of microscopic pores and cracks. The change of the deformation direction that takes place on the boundary of the monotonic deformation stage leads to the appearance of dislocations with the different signs at the beginning of the following stage. They interact with the dislocations that have appeared at the previous stage of monotonic deformation. Annihilation (mutual destruction) occurs as a result. The process o1"fracture is thereby inhibited.

221 For the first stage of monotonic deformation equation (1) was proposed in the work [ 1] as follows: d_w__ H(t) dt -Ap(k t(t),k2(t)) ' ti-l
(2)

where H is the shear strain rates intensity, k I and k 2 are independent invariams of the stress tensor (all that is the result of solving the boundary-value problem specified for a certain material particle at a moment of lime t on the path of its motion); A.e is plasticity of metal as a function of characteristics of stress (this function is determined by experiments and called "constitutive relation"). The ability of a metal to be deformed without fracture (macrodiscontinuities) under monotonic deformation at constant k 1 and k 2 is called "metal plasticity" Ap=Ap(kl,k2). (It must not be confused here with the plasticity that is the ability of a materi',d to accumulate residual strains). A measure of plasticity is the shear deformation degree which is calculated with the use of the formula (the integral is taken for a specific material particle) 1

A=fHd~

.

(3)

o

So, A~=A in the moment of macrofracture when t=tp. Gene~rally speaking, temperature and strain rate may also be included as arguments of the function Ap=Ap(kt,k2). However, for the processes of cold deformation the indicated set of ~rariables characterizing only the stress state is quite sufficient. The stress state must be described by three basic (independent) invariants of the stress tensor that are always reduced to two nondimensional mvariams. It is expedient to select from all the possible combinations of k 1 and k 2 that are traditional, and the most essential in the function Ap considered. The essential influence of hydrostatic pressure p or mean normal stress (the first invariant of the stress tensor) o----p on plasticity was shown in the works of Karman and Bridgman [ 13, 14] done in the first half of the century. Such invariants as shear stress intensity T and the factor of the shape of the stress deviator (Lode factor) P.o are known in the theory of plasticity. Thus, it can be assumed that kl~,

(4)

k : ~ =2~~22-~a~-r~ 1, G 11-G33

qlz~+_O_~ where, as is known, o-= 3 ; ~11, o22, ~33 are principal normal stresses.

(5)

222 If a material particle is being deformed all the time only monotonically under plastic processing, then damage may be calculated with the use of equation (2) for any stage of forming at the moment of time t. t

HO:) . Ap(k! O:),k20:))dx"

(6)

0

If a material particle is being deformed nonmonotonicaUy (the solution of the boundary value problem will indicate that), then the damage by the moment of time t must be calculated in another way: n

qJ(t)=~q~ai , 1

(7)

;

where n is the number of monotonic deformation stages that the particle has overcome at time t; qJi is the value calculated for the i-th stage of monotonic deformation (i=l...n) in accordance with relation (6): ai is an exponent that is taken as a mean value for the conditions corresponding to the i-th monotonic deformation stage (it is found ai>l, thus formally reflects the damage development hampering effect upon the change of the deformation direction). It is found that ai=a(kl,k2), and it is the second (after Ap) constitutive relation in the fracture theory presented here. 2. CONSTITUTIVE REI~TIONS AND THE EQUIVALENCY OF THE MODEL Consider experiments to determine a dependence of A. and a on the stress state factors k I and k 2. The technique of such experiments is not smaple. It is complicated to select such test types (as a rule, it is impossible to restrict our tests to one type) when at the degree of deformation preceding macrofracture A=Ap and factors k I and k 2 could be determined at the fracture site, when the test proceeded under constant stress state factors, and the deformations were monotonic. It is not simple to foresee the site on the sample where the fracture could be expected first and fixed in time. A certain success was achieved overcoming all these difficulties, based on the technique of testing a material in a liquid under high pressure. The author can recommend the experience of his collaborators working at the Institute of Physics of Metals and at the Ural Polyteclmical Institute (Ekaterinburg), [5, 7, 15, 16]. Schematically we describe the testing of cylindrical specimens by torsion in one direction in a liquid under high pressure p (Fig. 1a). Before testing, a line is marked on the specimen's surface parallel to the specimen's axis. The deformation is monotonic in this case. It is known that a pure shear takes place in torsion. The principal normal stresses are ott=l:s-p and o33=-r (where ~s is yield strength or deformation resistance under shear) which are directed at an angle x/4 with respect to the axis of the

223 sample on its surface. (The value of ~s is determined by known techniques under atmospheric pressure, and Bridgman just showed that it does not depend on the pressure.) The stresses acting along the normal of the sample surface are o~2=-p. It is sufficient to calculate stress state factors on the sample surface that take place' during the testing. (Recall thai T=~s. ) kt=-~" k2=0 9

(8)

As deformations develop under torsion of the sample, the material will somehow change its resistance Zs (due to hardening or softening for unstable materials). To ensure k,=const, a test device can be equipped with a program control system (manual or automatic) adjusting p as it is necessary. Macrofracture begins on the sample surface (it is impossible to observe it visually because the specimen is in a thick-walled steel container). The moment of fracture initiation can be fixed upon reaching the maximum of the torsion moment or, simpler, after inspecting and measuriag the specimen when it is taken out of the container. Cracks will be seen on the specimen's surface, or the sample will break after testing. The angle qo (see Fig.la) which the printed line will form with the axis of the specimen is measured at the crack border or on the edge of a broken sample. It may be shown that Ap--tgqo.

(9)

Other values of Ap for a separate specimen may be found by the method just described at some value of k I (k2=0). The dependence

Ap=Ap(kl) [ k2=0

(I0)

may be acquired after carrying out a set of tests for different k i (for every specimen with its own value of kl) and after the approximation of the test resuhs. A diagram of dependence (10) is called "plasticity diagram" (see Fig.2, Ref. No 1). With the utilization of the same apparatus the dependence a=a(kl) at k2=0 can be obtained. For this purpose the test (alternating torsion) is carried to fracture in a highpressure liquid with p constant or slightly regulated to ensure k=const in the tested specimen during the test. In such test the length L of the sample (see Fig.la) should be small enough to prevent localization within the length. The value of torsion in one direction (the monotonic deformation stage) should be kepl constant during the test of a separate specimen. The number n of monotonic deformation stages accumulated by metal before failure is fixed by fracture of the specimen (M=0). Testing of a sel of samples is effected at constant k 1 changing the value of torsion in one direction (a swing of torsion or an increment of the deformation degree is designated as AA ) from one specimen to another. The empirical dependence of AA on n results from testing a sel of samples at some k I (recall that in torsion k 2 always equals zero).

224

/////,/

/./ ////

/ / /

/

/

-

,

///////////;'

Figure 1. Schemes of plasticity test in a liquid being under high pressure: a) torsion, b) tension.

"h, v. \

O~ ~~

o.

.....

6.0 ~.0 ~- ~ \ % \ . 0

2.0 v. !
o.

C

~ a.o

I~ I ,.0 q.o2.t; , 0 n -~.o o.fi o.s - ~ 5 - f . s - ~ 5 /'~,r = O/T

Figure 2. Diagrams of A.=A+.(kt) for some alloys (1-in t o r s i o n , k2=(~; 2-~n tension, k 1=-1): a is the 08XI 8H10T grade steel: b is the 12X1M grade steel: c is the A~-1 grade alloy.

3

3

2 t_

o

_

~a

f

-3

....

-2

I

n

.-1 R~, =ff,/T

0

Figure 3. Diagrams of a=a(kx) for some steels (1 is the steel grade of 08X18H 10T: 2 is the steel grade of C~r45: 3 is the steel grade of UIX-15) in torsion, k 2 = 0 .

The approximation of this empirical dependence was found to be successful within the fracture model of metal examined in section 1. If model (6) and (7) is true and if the macrofracture comes during the test of specimens under alternating torsion in accordance with the above-mentioned technique (at some kx), it can be put down as follows:

225 n

Zwa=I, or nwa=l ,

(11)

1

where n is the number of monotonic deformation stages in testing of a specimen to AA fracture by alternating torsion; a is a so-far unknown exponent; q ~ ~ , as within the tesl (even the test of a set when k i is constant) the integral is easily calculated; Ap(kl) is the already known diagram of plasticity. As a result, the relation (11) acquiressuch a form:

AAnl/a=Ap(kt)

(12)

The empirical dependence AAn~-c connecting the swing of deformation with the number of cycles before fracture is known in low-cycle fatigue. It coincides with the relation (12) within the designations. It is indicative of the equivalency of the fracture model (6) and (7). Below we shall return to it once more. The relation (12) has one unknown value a. One can determine it by the results of the set of tests with kx=const selecting the value of a with the use of the least squares method. The second and the last constitutive relation a-a(kx) for the fracture model (6) and (7) can be obtained by organizing several successive sets of tests by varying the value of k 1 from set to set. In Fig.3 some experimental data are shown for some steels. Unfortunately, at present there are no data concerning the influence of the factor k 2 o n a. However, as it will be shown below, the model has a sufficient engineering accuracy. Let us describe another test of plasticity (with another k2) which is tension of cylindrical specimens in a high-pressure liquid (Fig.l b). The tension test displays a phenomenon that essentially complicates the processing of the experimental data. It is the loss of stability of the homogeneous flow of the specimen upon reaching some deformation. The deformation becomes localized, a neck (a local narrowing) appears where further plastic flow develops. The stress state becomes heterogeneous and to a great extent depends on the parameters a and R of the neck (Fig.la). The macrofracture begins in the centre of the neck. There is the most unfavourable stress state there. The cavity originating in the centre of the neck propagates quickly to the periphery of the sample, and the latter breaks down. The process of propagation is rapid enough, and the diameter of the specimen's neck has no time for an essential change. It is ascertained also that strains are homogeneous all over the net section of the specimen. Bridgman [14] and Pugh [17] studied the kinetics of the neck formation. Two conclusions follow from their investigation: 1) the neck is formed and develops under tension differently in different materials, and a universal dependence between the relation a/R and A does not exist (even for steels of various grades): 2) the dependence between a/R and A for one and the same material will be identical both in tension under atmospheric pressure and in a compressed liquid. Plasticity and stress state factors for tensile testing of cylindrical specimens in a liquid under high pressure may be calculated in accordance with the following formulas:

226

Ap=2",,~l(~) 9 3a

(13)

(14)

Here the dimensions of the specimen before and after the breakage zre marked by indices 0 and p respectively: the factor k 1 was calculated with the use of Davidenkov's results [ 18]. The test of 9plasticity A P =A.(kt) under tension (k2=-1) is rea!iTed in the following way. v Some spectmens from the ones prepared for the test should be broken under atmospheric pressure to examine the kinetics of the change of the sample parameter dR depending on A and to obtain a hardening curve ~s=l:s(A). Then the testing of steel specimens is carried out in sets. Every set is tested with its own k 1. The breakage of each specimen should be made by controlling pressure p (manually or automatically) so as to keep up the given value of k 1. The results of determination of plasticity diagrams in tension at k2=-1 are shown for some alloys in Fig.2 by curve 2. Tension tests of cylindric-,d specimens with notches having different values of ao/Ro are applied when it is necessary to know the values of Ap.at k1>0.58. The notch produces high stress triaxiality near the net section of the spectmen and makes the stress state more stiff in comparison with the state of a smooth cylindrical sample. Information about Ap at k2= 1 may be obtained by the testing of a membrane bulged in a liquid under high pressure. At present a lot of data about plasticity diagrams are accumulated in the literature (but there is no data bank, unfortunately). Here are some general conclusions about plasticity diagrams by the example of Fig.2. (1). If one conceptually superposes the diagrams of plasticity on each other, they will cross. That is why it is impossible to speak in general about relative plasticity of different metals. A comparison may only be made at identical k 1 and k-,. (2). The dependence of A_ on k I with k2=constant always decreases and is well-approximated by an exponenti~relation which can essentially reduce the number of tests for the determination of a plasticity diagram. (3). The change of plasticity is not simple with k ranging from -1 to +1 when kl=COnSt, (it can both increase and decrease). Consider the problem of the equivalence of the model examined in this section. To begin with, we check whether any calculated value characterizes the level of microdamage. Photographs of the microslructure of the specimens made of CT3I(n steel with different levels of W (calculated) from 0 to 1 are shown in Fig.4. (Sharp notches, or stress concentrators, were machined on the initial specimen and on the specimens after their plastic deformation. The specimens were cooled down to 77oK, i.e. below the limit of cold brittleness, and subjected to breakage.) Big flat facets of cleavage with a peculiar fan-shaped pattern are well-seen on the photograph of the specimen microstructure with qJ-~. The whole microstructure surface represents pits of different sizes surrounded by crests of plastically deformed metal. It is obvious that the process of failure begins with the appearance of microcracks and pores that unite into larger defects as the deformation develops and are perceived in the photograph as pits divided by crests of metal cracked

227

Figure 4. Microstructure of specimens having ditTerenl microdamage (picture taken w ilh the aid of the JSM-V3 scanning electron microscope with 15000 X magnification)" a is ~=0.00; b is ~=0.45; c is ~=0.75; d is ~= 1.00.

228

at the last moment during the preparation of the specimens. It turns out that the appearance and development of microdamages under plastic deformation are connected with the changes in the density of the material. The results of measuring the density of steel of the CT3KrI grade against the calculated W are shown in Fig.5 by dots. It is remarkable that the variation of p with qJ depends linearly (unbroken

line). As mentioned above, the equivalence of the model under study is present at least at the level of the equivalence of the Manson-Coffin fracture model under low-cycle fatigue. How accurate is the model when there is especially small number of monotonic deformation stages (often the number of such stages is 0 to 3)? The results of experiments with two-stage deformation to fracture are shown by dots in Fig.6. Specimens underwent tension to some value of damage W1 and then torsion by the value of q~2 to failure. The calculation of qJl and qJ2 was made by formula (6), and lhe summation of qJl and qJ2 was made by formula (7). The continuous line is the result of using the model (6) and (7). It is described by the equation qja1+w%2=l. The model has a sufficient accuracy.

2,~/'3 i

~'~"~Oo',,.

7875

~ o 0 0'~_ n 0 0

~s 0.s

0,2 78~S l

o

I

O,2

I

,J

I

!

I

0,#

Figure 5. Change in steel density versus damage. The grade of steel is CT3KIL

Figure 6. Experimental results (dots) and the curve calculated in accordance with the model of twostage deformation to fracture.

Undoubtedly, the accuracy of the model can be increased. Here we mark some ways of perfecting the model having no claim on completeness. I.A.Kiiko, in accordance with A.A.Ilyushin's ideas, suggests describing damage by a tensor of a more high rank, but not a scalar value [ 19]. A.A.Bogatov offers to represent the expression under integral (6) by a nonlinear form depending on the plastic deformation accumulated [5]. B.A.Migachev and the author proposed a more general kinetic equation (2) [20]. V.A.Ogorodnikov considers that the expression under integral (6) must contain not only the values of invariants k 1 and k 2, but also their derivatives with respect of time [8].

229 3. MODEL OF DAMAGE IN ANNEALING Cold deformation of sheets, tubes, wire, etc. is often attained by a series of operations, each of which is accompanied by intermediate and final annealings. The processes of metal softening and restoration of plastic properties proceed under heat treatment by annealing. It should be noted that the nature of these processes is different. Restoration of plasticity is based on diffusive processes of transferring the substance into pores and microcracks and these discontinuities to the surface of the body. Apart from diffusion, the process of recrystallization is going on. To calculate the manufacture technology of cold-deformed products with annealing, it is necessary to give a mathematical description (within the above-mentioned model) of how the restoration of the plasticity margin (or the reduction of microdamage) proceeds in annealing. The following technique is developed for the solution of this problem. The tests are carried out with metal, the plasticity of which A~,=Ap(kl,k2) is already known. Specimens of the test batch undergo different amounts of plastic deformation, each specimen to different degrees of damage qJl (for example by tension not to fracture but to different shear deformation degree Al=2,~f31n(do/dt), where d o and dl are diameters of specimens before and after tension; v1-AI/Ap is determined by using the plasticity diagram for every, specimen, provided in mind that the process of tension is monotonic). Then all the specimens undergo annealing in accordance with the chosen regime (0 temperature and duration t of annealing). Plasticity restoration or damage decrease by the value AW takes place in annealing. All the specimens after annealing again undergo plastic deformation in the same direction, but this time to fracture, then A 2 is determined and calculated w2=A2/Ap. The second plastic deformation plays an auxiliary part for the determination of AqJ. Evidently, the total value for specimens is W=W1-AW+W2=I, since the specimens have been brought to fracture by the second deformation. This allows us to determine the unknown decrease in damage resulting from annealing in accordance with the chosen regime. A~=X~l+~2-1 9

(15)

The results of the determination of damage decrease by annealing of some steels and a titanium alloy are presented in Fig.7 as an example. The annealing regimes were: 0 for steels ranging from 500 to 750~ and t ranging from 5 to 300 minutes, and for the titanium alloy 0=680~ and t=60 min. The dependence presented here was found in other studies to apply also to various other metals and alloys, so we may make general conclusions [5]. The restoration of plastic properties, or decrease in microdamage Aq~, under annealing by recrystaUization depends on W1 to a great extent. Three intervals of qJl can be indicated, divided by two critical values of damage ql, and qJ** , within which the intensity of restoration will be different. If the damage resulting from deformation is 0
230

M,e*<~I<~**, then not all damage is completely removed by normal annealing. Even so, AqJ continues to increase with the growth of qJt but at a lower rate. When qJl>V** AqJ decreases and goes to zero when qj 1= 1 .

A

aq'

o

o~,,,ko.q

Csl

o,s N

Figure 7. Decrease in metal microdamage resulting from recrystallization annealing (for fixed time and fixed temperature) of steels 12X18H10T and CT3Kn and the BT1-0 titanium alloy damaged on plastic deformation ql 1.

to f

o

I

#0

i,.,,.,.-o - 0.2 It

80

t

!

fro

I

/

~"

l&O z.

Figure 8. Kinetics of damage change qJ in heat treatment (at 600~ The grade of steel is 20.

Given qJx, it is necessary to decide which regime of heat treatment (temperature and duration of treatment) is to be selected, in order to ensure complete restoration of plasticity or microdamage healing. For that purpose it is necessary to consider the change of ~Vwith time at different annealing temperatures with qJ=~x at t = 0. The above-described method allows to solve this problem. The dependence of damage on lime for the steel of grade 20 and the temperature of 600~ is shown in Fig.8 as an example. The lower curve at qlt=0 demonstrates that the annealing of a blank to be deformed of, for example, hot-rolled metal can lead to increased plasticity due to the healing of microdamages having appeared on the stage of hot rolling. The curves have three distinctive parts: AB is exponential rapid decrease in damage: BC is considerable deceleration (and even stopping) of the process of metal plasticity restoration; CD is further acceleration of the process. Assume that material damage qJl received as a Rsult of plastic deformation changes according to the exponent (in the example presented in Fig.8 within 0.5 h.), i. e. it may be written as follows:

,p (0

texp(- O.

(16)

231 Here 13>0 is the index of steepness of the exponent. Generally speaking, the exponent in this equation is a function of heat treatment and the third constitutive relation of the fracture theory considered here. 4. APPLICATION OF MODELS Consider some simplest examples that do not demand the solution of a complicated boundary-value problem. On the edges of cold-roUed sheets, unfavourable combinations of the stress-strain state occur. Small cracks (tears) can be observed there. The rest of the metal is rolled under more favourable conditions. How is the deformation to be determined when cracks appear on the edge of a sheet? Without solving the boundary-value problem it is possible to easily determine the factors of the stress state in this spot: k l=k2=0. Therefore, there is no need to make a complete plasticity diagram for determining Ap, and it is enough to carry out a torsion test and to receive A.... by formula (9). In the middle of [l~e height of the edge the metal is deformed by rolling under monotonic conditions at constant k 1 and k 2. Therefore, the damage will be calculated by formula (6). It follows therefrom that at the moment of tearing Ap/Apo=l. The strain of the sheet at which the tear appears will be

sinse under conditions of monotonic plane deformation A=21n(h0Pal). So, to avoid tearing, one must have h0/hl<exp(Apd'2 ). Another example concerns the estimation of the possibility of fracture in cold drawing of wire. It can be shown (by solving the boundary-value problem) that the deformation proceeds monotonically on the axis, as distinct from the periphery, where the deformation is nonmonotonic and one draw consists of two monotonic deformation stages. Here the most intensive fracture takes place and is confirmed by experiments [ 1]. The factors of the stress state in this location of the wire range approximately within k1=-1...+1, k2 =-1. For the determination of plasticity the breaking of specimens in a high-pressure liquid should be applied, at least at two levels (for one level it can be assumed that p=0). The damage in the case considered will be according to (6):

li 11

v=:s r

ti-1

.

(18)

232

where n is the number of draws (or passes) in drawing; (li_1, ti) is the duration of the iqh pass. If the calculation shows that ~ exceeds unity (or an other level of admissible damage) for the first time in the k-th pass, then the allowed number of passes without fracture in the axis zone is k-1.

O.g 0.6 0.4 0.~.

1 o.z

I o.~

o.6

o.e

~,,

Figure 9. Intensification of damage healing (the restoration of plasticity) on annealing. As it is mentioned above, the damage of wire on the periphery (in the subsurface volumes) will grow more slowly. The deformation will proceed nonmonotonically there. One should take into account formulas (6) and (7) in the calculation of ~. (As a result, each integral in (18) will be divided into two parts, every of which being raised to the power a> 1.) Consider examples of the application of the models of damage healing by heat treatment. Experimental data show (Fig. 6) that complete healing of damage .can be reached by conventional annealing when the damage after plastic deformation does not exceed 0.3. Naturally, that can create additional difficulties: to ensure the receipt of finished products at ~--0, the total deformation is to be divided into a greater number of cycles, and it is necessary to introduce additional annealing. The model described in Section 4 is applied to examine the possibilities of heat treatment intensification [21, 22]. It proves (Fig.9) that the use of thermocycling in recrystallization annealing intensifies the healing, making the value of admissible damage ~ , higher. Essential increase in lhe efficiency of recrystallization annealing can be obtained by hydrostatic compression in heat treatment if the pressure is commensurable with the creep stress. These measures, demanding, of course, great expenditures, can reduce the number of production cycles.

233 REFERENCES 1. V.L.Kolmogorov, Napryazheniya, deformatsii, razrushenie. M. Metallurgiya, 1970. 2. Plastichnost i razrushenie, pod red. bv V.L.Kolmogorov, M. Metallurgiya, 1977. 3. V.A.Parshin, E.G.Zudov, V.L.Kolmogorov, Deformiruemost i kachestvo. Metallurgiya, 1979. 4. B.A.Migachev, A.I.Potapov, Plastictmost instrumentalnykla stalei i splavov. Spravochnik. M. Metallurgiya, 1980. 5. A.A.Bogatov, O.I.Mizhin'tsky, S.V.Smirnov, Resurs plasticlmosti metallov pri obrabotke davleniem. M. Metallurgiya, 1984. 6. E.P.Unksov, W.Johnson, V.L.Kolmogorov i dr. Teoriya plasticheskikh deformatsyi, pod. red. E.P.Unksov, A.G.Ovchinnikov, M. Mashinostroyenie, 1983. 7. V.L.Kolmogorov, Mekhanika obrabotki metallov davleniem. M.Metallurgiya, 1986. 8. E.P.Unksov, W.Jotmson, V.L.Kolmogorov i dr. Teoriya kovki i shtampovki, pod redaktsiei E.P.Unksov, A.G.Ovchinnikov, 2-e izd. pererab, i dop. M. Mashinostroyenie, 1992. 9. V.I.Vladimirov, Fizicheskaya priroda razrusheniya metallov. M. Metallurgiya, 1984. 10. L.M.Kachanov, Osnovy meklaaniki razpusheniya. M. Nauka, 1974. 11. Yu.N.Rabotnov, Mekhanika deformiruemogo tverdogo tela. M. Nauka, 1979. 12. V.V.Bolotin, Prognozirovanie resursa maskin i konstruktsii. M. Mashinostroyenie, 1984. 13. T.Karman, Mitteilung Forschungsarbeit Vereinigte deutsche Ingenieur, 118, Berlin, 1913. 14. P.W.Bridgman, Studies in large plastic flow and fracture, McGraw-Hill, N.Y., 1952. 15. A.A.Bogatov, S.V.Smirnov, V.N.Bykov, A.B.Nesterenko, Av. svid. SU N 1422090 A1. Byulleten izobretenii i otkrylii N33 (1988). 16. A.A.Bogatov, S.V.Smirnov, V.N.Bykov, A.V. Nesterenko. Obrabotka metallov davleniem. Mezhvuzovskii sbornik. Sverdlovsk, izd. UPI, 1991, p 45. 17. H.L.D.Pugh, The mechanical properties and deformation characteristics of metals and alloys under pressure, NEL Report, N 142, March, 1964. 18. N.N.Davidenkov, N.I.Spiridonova, Zavodskaya laboratoriya, 1945, V. XI, N6, p.583. 19. I.A.Kiiko. Obrabotka metallov davleniem. Vyp.9. Mezhvuzovskii sbomik. Sverdlovsk, izd. UPI, 1982, p.27. 20. V.L.Kolmogorov, B.A.Migachev. Izvestiya AN SSSR, Metally, 3, 1991. p.124. 21. V.A.Belov, A.A.Bogatov, V.A.Golovin i dr. Fizika metallov i metaUovedenie, t.60, V.5, 1985, p. 1004. 22. V.L.Kolmogorov, A.A.Bogatov, S.V.Smirnov, O.I.Mizhin'tsky, Sb. Legkie i zharoprochnye splavy i ikh obrabotka. M. Nauka, 1986, p.7.

This Page Intentionally Left Blank

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

235

PREDICTION O F N E C K I N G IN 3-D. S H E E T M E T A L F O R M I N G WITH FINITE ELEMENT SIMULATION

PROCESSES

M. BRUNET Laboratoire de M6canique des Solides I.N.S.A. 304 20 Av. A. Einstein Villeurbanne 69621 F R A N C E Abstract:

The aim of this study is to propose a calculation method based on Hill's yield criterion and limit stress diagram to predict neclang in Finite Element simulation of 3-D. sheet metal .forming processes. The .forming limit diagrams of anisotropic sheets are first determined experimentally .[or various strain path shapes . From these diagrams, the limit stress states are calculated and plotted on a single curve . These limit stress values are then introduced as a set of additional data by. mean of subroutines in our Explicit Finite Elements codes where stresses and internal ,forces are calculated . Results from numerical studies of deep-drawing of anisotropic sheet metals are presented. I. E X P E R I M E N T A L

STUDY :

1.1 M a t e r i a l s c h a r a c t e r i s t i c s : Two types of materials are considered in this study, the ULC/Ti which is a killed mild-steel and the XD340 which is a high-strength steel . The elastic limits of these steels are determined along the rolling direction and the transverse direction by mean of mechanical extensometers to get enough precision : ULC/Ti Steel:

0-0 = 150 MPa E 0 = 207830 MPa

XD340 Steel :

0`0 = 357 MPa E 0 = 224280 MPa

o-90= 154 MPa E 9 0 = 218460 MPa o-90= 373 MPa E 9 0 = 233980 MPa

We also determined the strain-stress curve along the two directions . These curves are modelized according to the Swift's analytical formulation : 0`=k(c

+e)

ULC/ti Steel :

XD340Steel :

n

(1)

k = 510 MPa

c = 0.00352

n = 0.217

atO ~

k = 498 MPa

c = 0.00349

n = 0.208

at 9 0 ~

k = 729 MPa

c = 0.01345

n = 0.166

at 0 ~

k = 786 MPa

c = 0.02475

n = 0.201

at 90 ~

The anisotropy or L a n ~ o r d ' s parameter r is defined as the strain ratio of the two main directions without stress on a tensile test specimen . If we carry out a tensile

236 test along the direction 1 having an angle/3 with the rolling direction x, we have " r/3 = de 2 / de z

(2)

where de z is the strain along the normal direction to the sheet surface . In the orthotropic axis x,y it comes 9 r/3 = (dcxxSin2/3 + deyyCOS2/3 - 2 dcxySin~ cosg ) / de z

(3)

and the stresses in the orthotropic axis : O-xx= o-1 c~

O-yy-- o-1 sin2/3

O-xy - o-1 sir03 cos/3

(4)

The evolution of r 0 and r90 are studied during the straining. On each specimens , a square grid is deposed to measure large strain by optical measures and the specimens are strained by increments. After each, the tensile strength is relaxed and we measure the plastic strain increments along the x and y directions. The through-thickness strain is obtained by the plastic incompressibility. After a certain deformation level ( about 10%) , the r 0 and r90values stabilized such that 9 ULC/Ti Steel"

r0= 1.50

r90= 1.85

XD340

r0= 0.833

r90 = 1.075

Steel"

1.2 F o r m i n g L i m i t D i a g r a m s -

The F.L.D. curves for the steels used in this investigation are determined by means of a flat-headed punch (Marciniak's tool) and they are established in both rolling and transverse directions. Because of the anisotropic features of the materials, the F.L.D. are plotted along the physical axis which are the rolling direction x and the transverse direction y. The strain values at the onset of necking are obtained by the successive strain states analysis during the step by step deformation of the ~ i m e n using a video camera/1/. Concerning the F.L.D. with pre-strain by tension, the specimens are obtained from stripes pre-deformed by means of a large capability tensile machine . For the prestrain by equibiaxial stretching the flat-headed punch is used with different widths of the specimens - 40,60,80,100,120,140,160 mm and the lengths have been always kept equal to 160 ram. It is generally noticed that a pre-strain by tension leads to a h!gher F.L.D. in the stretching range while a pre-strain by equibiaxial stretching gwes a lower F.L.D. /2/,/3/. The figures (1) and (2) are relatives to the F.L.D. for the ULC/Ti and XD340 steels respectively with rectilinear strain paths and prestrain by stretching up to 12 %. In case of isotropic materials , these diagrams would be perfectly symmetric with respect of the first bisector. 2. T H E F O R M I N G LIMIT S T R E S S C U R V E "

2.1 Strain calculation for anisotropic material " If we neglect the transverse shear, the Hill's criterion of plasticity in the orthotropic axis of a thin sheet can be written 9 ~2

= H(o"x- O'y)2 + F(O-y- O-z)2+ G(o-z- O-x)2+ 2P O'xy 2

(5)

237

80,

EPS. X % R O L L I N G D I R E C T I O N ,

-. I

60

~

"-

".

.

.

, ~

.

Linear strain path

t

Linear strain path

~

.

.

',,~

~

~

--

.

, .

4o~ ULC/Ti STEEL

.

:

20

.

.

.

-

~

,

.

.

~

, !

0

i ~

t

+

I

1

Prestrain stretching

i '

\

] \

-

'

~

.

- a n..~-v

-- +

~

~

.

. :

-40

-20

EPS.

F i g u r e (1) U L C / T i

50

Y

%

Steel

--'--

\ .

,

~

TRANSVERSE

}

t

'

20

40

~

60

80

DIRECTION

F. L. D.

!

~

.

30--

XD340 STEEL

.

+

.--?

EPS. X % R O L L I N G D I R E C T I O N

, 40

,

!

i

i

0

Strain

r

~

1

-60

-60

;

---~

,

-t-~;

,

-20 --

!

'

I

.

.

.

20

Linear strain path Linear strain path

10

Prestrain stretching

4 t ......

i

-10

-20

-30 -30

i

- 2 0 -10 0 10 20 30 40 50 EPS. Y % T R A N S V E R S E D I R E C T I O N

Figure (2) X D 3 4 0

Steel

Strain

F. L. D.

238 where H, F, G, P are the dimensionless coefficients of Hill's criterion which take the values : H = F - G = 1/2 and P = 3/2 in the isotropic case. The associate flow rule for a material under plastic loading conditions can be expressed as: de.lj = d;~ oQ/oo'ij = 1 dh oO-2eq/ oo-.. .

2

(6)

11

where dcij are the plastic strain increment and dx is a proportionality factor which can be determined by the equivalence of plastic work dissipated during the plastic straining: dW = o'.. de.. = o" de lj q eq eq

(7)

The normality rule of the plastic flow (6) with (5) leads to : de x = dx [ G(o"x- O'z) + H(o"x- O'y) ] dcy = d,~[ F(O'y- O'z) + H(O'y- O'x) ] dc z = dx [ G(o"z- o"x) + F(o"z- O'y) ] de = d~ Po" xy xy

(8)

and with (7) we get the equivalent plastic strain increment: dceq = cix O-eq

(9)

In the general case, the anisotropy parameter r in the direction of angle 13 with the rolling direction x can be expressed with (2),(3),(4) and (8) by: r/3 = {(cos2/3- Hsin2/3) sin2/3 + [ (F+H)sin2/3- Hcos2/3 ] cos2/3- 2P(sint3cos/3) 2 } / {- Gcos2/3 - Fsin2/3 }

(10)

Now if we impose the equality between the equivalent stress o-eq and the effective stress corresponding to the rolling direction x then : (G+H) -

1

(11)

and x is the reference direction. On the other hand, if the transverse direction y is chosen as the reference direction we get : (F+H) = 1

(12)

Putting /3=0 ~ /3=45 ~ and /3=90 ~ and choosing the rolling direction x as reference direction we obtain : r 0 = H/(1-H) and inversely :

r90= H/F

r45 = - ( 1 - H + F - 2 P ) / I 2 ( F + 1 - H ) I

(13)

239

H = r0/(1 +r0)

F = r0/r90(1 + r 0)

P = (r90+r0) (2r45+ 1)/12r90(1 +r0)]

(14)

2 . 2 Determination o f the Forming Limit Stress Curve "

It is important to note that in this section, the fundamental assumption is made that the main solicitation directions are assumed to be overlayed with the ~ i m e n orthotropic axis of the specimen. In order to express the equivalent strain eeq in terms of plastic strain increments which are the only measurable values, it is readily shown that from (8) 9 F de x- G dey = da [ (o"x- Cry) ( F G + G H + H F ) I H de Z - F de X = d,~l (o"z-~r x) ( F G + G H + H F ) ] G dey- H de z = d~ [ (O'y-O'z) ( F G + G H + H F ) I (15) Using the relations (5),(9) and (15) , the effective plastic strain increment is given by" de2eq= I F(Gdey-Hdez )2 + G(Hdez-Fdex)2 + H ( F d e x - G d e y ) 2 1 / I F G + G H + H F I 2

(16)

Now we assume the plane stress state such that ~ = 0 and taking account the plastic '

Z

incompressibility 9 dez-- - (dex+ dey) , the stress state is given by " = ~ I (F+H)de + Hde I / i d e e q ( F G + G H + H F ) I x eq x y O-y= O-eq I Hde x + ( G + H ) d e y ] / I d e e q ( F G + G H + H F ) ]

(17)

The forming limit stress curves are then simply obtained in the orthotropic axis from experimental measured strain de and de by calculating the stresses o- and o- on each x y x y straight path assuming hardening of the form " O-eq= k (c + eeq)n where e

eq

is the equivalent plastic strain accumulated "

Ceq = Z deeq

(18)

The calculated stresses do not depend on the direction of reference chosen. These results are plotted on figure (3) for ULC/Ti steel and on figure (4) for XD340 steel. It may be noticed that the points for the rectilinear strain paths and for the prestrain by equibiaxial stretching are located on a same line. This result is the same as the one got with isotropic materials /5/,/6/. It shows that the forming limit stress curve is an intrinsic criterion for a given steel . It depends only on the final strains and on the strain hardening curve and Hills'parameters. However it is assumed that there is no induced anisotropy. The intrinsic aspect of this stress criterion cannot be due to the strain hardening curve flattening on the large strain range because, as it has been shown in /7/,/8/, the reverse calculation of strain from the limit stress curve would give a very large dispersion which is not the case. This result is also in agreement with the theoretical analysis given by Cordebois/9/.

240

700

S I G M A - X (MPA) R O L L I N G D I R E C T I O N ~ ~.

eoo!f

ULC/Ti STEEL

I

F.L. STRESS CURVES -J-

"

[

500

Linear strain paths Prestrain stretching

[]

Unlaxlal tests

X

Linear strain paths

'I

400 3 0 0 --

i

~

i

i

t

!1

I

!i .... ~

!

1

,

t

i/

300

400

200

//t /9

t 1

100

:

U.2

0

100

200

500

600

700

S I G M A - Y (MPA) T R A N S V E R S E D I R E C T I O N

Figure (3) ULC/Ti Steel Forming Limit Stress Curves

800

S I G M A - X (MPA) R O L L I N G D I R E C T I O N '

700

--'-- F.L. STRESS CURVES d-

Linear strain paths

-~

Prestrain stretching

[]

Unlaxlal tests

)<

Linear strain paths

500

; t

~ - - ~ - + -

600 XD340 STEEL

l

I

' i _

l '

r I

9~ - -

' l i

400

i

!

t

T ~

i

300 200

I

i i'

100

t

I

!

400

500

L

0

100

200

300

600

700

800

S I G M A - Y (MPA) T R A N S V E R S E D I R E C T I O N

Figure (4) XD340 Steel Forming Limit Stress Curves

241 3. MECHANICAL MODELLING

3.1 Explicit dynamic equations The rather complicated nature of industrial thin sheet metal forming, where large displacements and large strains, contact and large relative slips, friction and separation are present is posing a challenging test for the numerical methods. Explicit time integration seems to be the most efficient possibility to integrate the time variables in conjunction with finite element discretizations of time dependent problems in structural mechanics. In particular highly non-linear problems arising in thin sheet metal forming as large strain, contact, friction and multistage forming can be easily handled. In the Lagrangian mesh-Cauchy stress formulation the current configuration of the body V is used to establish equilibrium and the weak form of the differential equations is given by the principle of virtual work irrespective of material behavior as"

Yv p iii~ui dv + J'v c t~i~u i dv + J'v ~

dv =

Yv Fi~ui dv + J'st ti~ui dst+ ~ J'sci (fi ~gni + fit ~gti)dsci

(19)

~ui are the virtual displacements, ~cii are the associated virtual strains with the Cauchy stresses ~isJ ,F i are the body Jforces, t i are the surface tractions, o is the mass density, c the damping parameter and a dot refers to differentiation with respect to time. The last term in (19) includes the contribution of contact normal forces fn and friction tangential forces ft on the portion sc of the boundary S of volume V. Introducing a F.E. shell model, such as a three nodes triangular element /10/,/11/, we obtain from the spatial discretizations of (19) the following system of differential equations of second order in time 9 M i i + C 0 + fint = fext + fc

(20)

where M and C are the global mass and damping matrix respectively. At the element level, the internal nodal forces at a typical node I is given by 9 ~)int

_ yv Bj I ~

dv

(21)

with the gradient matrix B " Bjl : 0 N I l 0xj

(22)

where N I are the nodal shape functions of the element considered. In (22), the through thickness integration is performed using the 3 points Lobatto's rule desired to include the outermost points in the evaluation of stresses in the elasto-plastic material behavior.The most popular explicit method for time integration of equation (20) is based on the central difference formulas: v n + l / 2 = vn-l/2+ iin At

(23)

un+ 1

(24)

= u n + v n + 1/2 At

Provided the mass matrix M is rendered diagonal, displacements and velocities can be

242

updated directly without any need tbr solving systems of equations. It is well known that the explicit algorithm is conditionally stable and the main computational problem in the present context is the stress incrementation needed to update the internal forces and the evaluation of the contact and friction forces. 3 . 2 Stress i n c r e m e n t a t i o n

.Assuming that a local (x,y,z) orthonormal basis in each element coincides with the axis of orthotropy which rotates rigidly as the element deforms. In the following procedure, the strains and stresses are computed and stored in the local system and we employ the rate elasto-plastic constitutive equations in the implementation which facilitate maintaining zero normal-stress incrementally. With respect to a fixed coordinate system, the constitutive equations frequently used in large deformation take the following form with the Jaumarm's stress rate:

~ij : Cijkl ~ki

(25)

where the material time derivative of C a u c h y ' s stress is given as: (26)

6"ij = ~'ij + O'ik Wkj + O'jk Wki and the spin rate : wij =

(avj/axi- avi/axi)/2

(27)

We assume that the rate constitutive equations (25) are written with respect to the local coordinate system. The straightforward use of an expression like (26) preserves objectivity in an infinitesimal sense only. However, in explicit time integration the displacement and rotation increment are very small, thus incrementally objective algorithms are not needed /12/]/13/. Before updating, we rotate the stress o-i]l at time t n into the configuration at tn+ . *n+ 1

o'i j

n

n

n + 1/2

= o'ij + (O'ik tOkj

n

n + 1/2)

+ O'jk tOki

~t

(28)

The stress tensor can now be updated in the local system where Cijkl is the plane stress constitutive matrix in the local system: n+ 1

o-ij

*n+ 1

- crij

+ 1/2

+ Cijkl t ~ l

zxt

(29)

The first step in the elasto-plastic algorithm is to calculate the elastically induced trial stress with (29). The trial stress is then tested to see if it is inside or outside the yield surface. If it falls within or on the yield surface the process is an elastic one. If the trial stress is outside, the radial return algorithm is employed to generate a stress state on the updated yield surface in the plane O-z= 0. Usually fewer than 5 iterations are required for convergence /14/. 3 . 3 C o n t a c t and friction :

A discretized description of the geometry of the tools surface is convenient and it is done by the use of individual facet elements in a mesh on the contact surface. The numbering order of the facet nodal points defines the sign of the surface normal which is positive by convention when pointing into the rigid surface. The procedure to find the actual position of a material node k with that of a facet element is the same as the procedure described by K. Schweizerhof and J.O. Hallquist /13/ where particular

243 details can be found. The position of a material node k relative to the rigid surface is indicated by the normal distance to the closest facet element (Figure 5) :

Hotertol

)

,,

1

Figure (5) Die node m closest to material node k d n = n t Xmk

(30)

where Xmk denotes the vector from the closest facet node m to the material node k. Then at time t n + 1 of the time step d t , we use the non-penetration inequality 9 - d u n - dg n + d n

(31)

< 0

where du n is the normal displacement and dgn the normal motion of the surface. As long as ~ < 0 , the nodal point has not reached the tools and is left to move freely. If penetration is found or if the distance is within a certain accuracy limit, the contact conditions must be enforce. In order to retain the major advantage of the explicit methods that no solution of equation is needed, we apply the exterior penalty method: dt n = - k n (du n - dgn)

(32)

where k n is the positive penalty parameter. The node will stay in contact as long as the contact force fn updated by its material increment dfn remains negative. We assume that there exists a Coulomb's friction law so that no relative motion is observed if : (33)

Iftl -~ ~ Ifnl and the penalty formulation of the sticking condition is applied :

(34)

df t = _ k t (du t - dg t) The friction force at time tn+ 1 is updated : ~+l

= ~

+ Oft

(35)

and i f " I t ~ + l I > ~ If n] then the sliding condition gives 9 sign = ~ +1 / I ~ +11

and

~ +l

= sign # If n[

(36)

As an example, figure (6) shows a quarter model of deformed mesh with a flat-headed punch.

244

Figure (6) Deformed mesh with flat-headed punch

4. NECKING BY FINITE E L E M E N T SIMULATION "

4.1 Drawing by hemispherical punch : In order to simulate drawing and to predict necking of our experimental tests carried out in our laboratory on anisotropic steel sheets shown in the first section, the finite elements used are of triangular shell type. The previously determined limit stress curves are introduced as a set of additional data with the fitting ~y--f(~x ) in our Explicit Finite-Element Code "COQUE3" described in the preceding section where stresses are evaluated . In the experiment, the punch radius was 37.5 mm and the die profile radius was 6.25 mm. The ULC/Ti steel blank radius was 80 mm with a thickness of 0.7 mm and the sheet was modeled with 600 three nodes shells. Figure (7) shows the limit stress curves and the detected point by the finite element calculation which correspond to a punch travel of 36 mm where the experimental value observed is 38 mm. These small

245

700,

SIGMA X (MPA)

600

ULC/Ti -~

"~~ --~ -~

"I"---

500

Steel Rolling

~

direction

---F-

Transverse

-~

Calculated

[]

Detected

400

~--

300

~-;

'.

.

.

.

+

direction points point

200

t, i ....

1 0 0 ;-

-

t, '

"i

~-

;, . . . . . . .

,

,

~

~i

'

I= '

}~

, O

-T~

§

,i

0

100

200

300

400

SIGMA Y (MPA)

-

§ 500

Figure (7) Detected point on ULC/Ti steel Limit Stress Curves

-

600

246 differences may be due to the difficulties to estimate the value of the friction coefficient ( #=0.02 under the punch ) which has a strong influence and also the blank-holder pressure on the blank. The blanldaolder has been simulated by directional constraints. However, the calculated and measured true strains at the onset of necking are in good agreement as shown on figure (8) where el,e2,e 3 are respectively the radial , tangential and through-thickness strains . Moreover, the detected point on the transverse direction y of the limit stress curve corresponds to the radial crack observed on the experimental cup . LOG. STRAINS AT N E C K I N G

~176 I

4

~i

40%

i

~.

eli~xp.

' i

'~

e2 !exp.

20% 0% -20% -40%

1

,

-60% -80%

0

.

, 10

.

i

. 20

.

Ii i

,

l

J

30

40

50

60

70

80

I N I T I A L RADIUS (mm)

Figure (8) Strain distribution with hemispherical punch ( transverse direction y ) 4-2 Drawing b y f i a t - h e a d e d punch : In this case, the sheet metal ( XD340 Steel ), a quarter model consisting of 800 three node shells, was deep drawn from the binder region toward the die cavity (see fig.(6)) where a small coefficient of friction of 0.03 under the punch and 0.13 under the die and the blank-holder are assumed for the finite element calculation. The forming process was simulated by applying a punch velocity of 4 m/s which is faster than in reality. The increase in punch speed is identical to a reduction in the required numerical effort because the time integration is governed by the critical time step of the smallest element. It has been shown /13/ that global quantities as final form, displacements and forces are predicted rather close to results obtained at lower velocities if the masses have been scaled appropriately. Figure (9) shows the detected point on the XD340 steel limit stress curves which corresponds to a punch travel of 22.1 mm and the corresponding logarithmic strain distribution is shown in figure (10) where E 1, E 2, E 3 are respectively the radial,tangential and thickness strains . The localization of the strain peak ( About 30mm on initial radius ) corresponds to the necking observed on the blank which has occurred under the punch radius in this case. For a 22.1 mm punch travel, the outer edge of the blank has been drawn from its initial distance of 75 mm to about 69 mm.

247 SIGMA

800r

i

700

:

i

....-,

6001 XD340

X (MPA) . . . . . . . . . . . . .

I -~

:

I

!

i "

~ I ~,

~ '~,

i

"

i ~

Steel

Rolling direction Transverse

500

i

!

9

T

,

/i

}

~'

'

i ~

•~ ~

I

I

direction

-~

Calculated points

[]

Detected

300

point

.....

i

0

i

i

100 2 0 0 3 0 0

,

t

400

500

SIGMA

?

!

-~

i ! i ' i.. oi

4-

-

600

700 800

Y (MPA)

I

~

j

J

r~ Figure (9) Detected point on XD340 steel Limit Stress Curves

248

40

LOG. STRAINS AT NECKING (%) !

30 ........~ . . . . i i 20

'

i

i

/

-

Is: B

10

-10

-20

_30]

'

i t

1 ]- . . . . . . .

-40 ~' -50

10

0

20

30

40

50

60

1

:

70

80

INITIAL RADIUS (mm)

Figure (10) Strain distribution with flat-headed punch ( rolling direction x ) 5. C O N C L U S I O N

:

A determination method of predicting the onset of necking in finite element simulation of 3D. sheet metal forming process has been proposed . For sheet metal forming problems, it may be necessary to include the effects of the initial anisotropy, which has been done here by invoking the orthotropic Hill's criterion. This method is based on the representation of the forming limit curves on a stress diagram which is much more intrinsic than the usual F.L.D. strongly influenced by the strain paths . This stress criterion, easily determined from experimental points for rectilinear strain paths, can be introduced in finite element codes including sheet metal forming capability of simulation without difficulties. Very good agreements have been obtained between experimen~ results and fufite element simulations to predict the onset of necking because m thin-sheet metal forming shear deformations are less important and the forming limit stress curves can be established and used in the orthotropic axis. In the context of very complex strain paths, were are looking for refinements in the formulation of the criterion which need new experimental investigations. REFERENCES

"

1. Arrieux R., Le Gac H., Sevestre C., " A new experimental method for the determination of forming limit diagrams at necking . Proc. of the 14th IDDRG Congress Koln , (1986), 464-465 2. Gronostajski J. 9 Effect of strain path on the plastic instability. Proc. Symp. on Plastic Instability, Paris, (1985), 49-60 3. Arrieux R., ~ n C., Boivin M. 9 Determination of the strain path influence on the forming limit diagrams from the limit stress curves . CIRP Annals,34-1, (1985), 205-208

249 4. Hill R., Mathematical theory of plasticity , Clarendon Press Edts. Oxford. (1950) 5. Arrieux R., Bedrin C., Boivin M. : Determination of an intrinsic forming limit stress diagram for isotropic sheets. Proc. of the 12th IDDRG Congress, St. Margherita Ligure, (1982), 2, 61-71 6. Brunet M., Arrieux R., Boivin M. : D6termination par 616ments finis en grandes d6tbrmations des courbes limites de formage en contraintes. Proc.lnt. Symp. on Plastic Instability. Presses Ponts et Chauss6es Ed. Paris September, (1985), 227-233 7. Hage I. :Simulation de l'emboutissage des toles anisotropes par 616ments finis avec prediction des risques de striction Th~se de Doctorat, Institut National des Sciences Appliqu6es de Lyon, (1990), 1~i3 p. 8. Arrieux R.: D6termination th6orique et exp6rimentale des courbes limites de formage en contraintes . Th~se d'Etat, I.N.S.A. de Lyon et Univ. Lyon 1 , (1990), 235 p. 9. Cordebois J.P. : Crit~res d'instabilit6 plastique et endommagement ductile en grandes d6formations , Th~se d'Etat , Univ. Paris 6 , (1983), 209 p. 10. Brunet M., Sabourin F. : Explicit dynamic analysis with a simplified three nodes triangular shell element. Proc. Int. Conf. FEMCAD-CRASH 93, Technology Transfer Series, A. Niku-Lari ed. Paris, (1993), 14-20 11. Brunet M., Sabourin F. : A simplified triangular shell element with necking criterion for 3D. sheet forming analysis. Proc. of the 2nd. Int. Conf. Num. Simulation of 3D. sheet metal forming. NUMISHEET'93, 31 Aug.-2 Sept. ,(1993),Tokyo,Japan 12. HaUquist J.O. , Benson D.J., Goudreau G.L. : Implementation of a modified HughesLiu shell into a fully vectorized explicit finite element code. in F. E. Methods for Non-linear Problems. Ed.: Bergan, Bathe, Wunderlich. Springer Berlin. (1986) 465-479 13. Schweizerhof K., Hallquist J.O. : Explicit integration schemes and contact formulations for thin sheet metal tbrming, in FE-simulation of 3-D. sheet metal forming processes in automotive industry. VDI Berichte 894 (1991), 405-440 14. Brunet M.: Some computational aspects in three-dimensional and plane stress finite elastoplastic deformation problem. Engineering Analysis with boundary elements .(1989) Vol. 6 ,2, 78-83

This Page Intentionally Left Blank

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

251

"DEFORMABILITY VERSUS FRACTURE LIMIT DIAGRAMS" A.G. Atkins Department of Engineering, University of Reading Whiteknights, P.O. Box 225,Reading RG6 2AY, U.K.

1. Introduction

The fracture of sheet metals after prior plastic flow is a problem of concern to metalformers who attempt to make complicated pressings. They are concerned either with the choice of a suitable material whose mechanical properties will permit a given complicated shape to be manufactured, a n d / o r with the design of dies and tooling which will permit a given material to be employed to make a certain shape. The strain distribution in complicated pressings is inevitably non-uniform and can be highly non-uniform in certain regions of the pressing. Regions of high local strain, and high local strain gradient, are often areas where fractures can occur. Part of the skill of the tool and die maker is to avoid rapid changes of deformation pattern, and deformation history, during the pressing operation. Nowadays, some (but not all) industries are using large-deformation elastoplastic finite element codes, with even automatic remeshing facilities and thermomechanically-coupled consitutive relations, to predict metal flow patterns during sheet metal operations. The incorporation of elasticity, in place of merely rigid-plasticity, permits the important topic of elastic springback to be addressed and the important influence of combined bending, stretch forming and drawing on springback to be determined. Previously, the amount of "overbend" or "overstretch" required (which would recover to the desired size or shape product) was established empirically through the skill of the tool and die maker. Modern elastoplastic analyses also permit study of residual stresses in formed products: elastic unloading from non-uniform plastic flow fields inevitably produces a pattern of residual stresses, knowledge of which may be important for the behaviour of the pressed product in service. The deformation of sheets of ductile materials under conditions of biaxial loading is often complicated by the phenomenon of necking. Furthermore, in sheet forming, consideration has to be given to two types of neck, viz: (i) diffuse necking (similar to necking in axisymmetric tension of a round bar) and (ii) localised plane strain necking [1]. The former precedes the latter and, in many forming operations, the onset of necking is considered to be limiting. It is found

252 that u n d e r biaxial tensions, the localised plane strain neck forms perpendicularly to the major in-plane strain, i.e. along the minor in-plane strain direction. In the case of combined tensile and compressive strains, the localised neck forms at an angle to the principal minor in-plane strain, Figure 1, the angle becoming progressively greater the greater the magnitude of the compressive strain component; in simple tension of an isotropic sheet, which has compressive width and thickness strains of magnitude one-half the tensile strain, the angle has become about 35 ~ , i.e. some 55~ to the pulling axis in an isotropic sheet. 2. The Forming Limit Diagram (FLD) 2.1

Practical Considerations

It follows that metalformers have been interested for many years in the in-plane strain pairs at which necking occurs and the so-called "forming limit diagram" introduced by Keeler and Goodwin (see reference [2]) is the locus of such strains when diffuse and localised necking takes place, plotted on axes of major r and minor r in-plane stains. (The through-thickness true strain r is obtained from considerations of constancy of volume i.e. r = -(r + r Forming limit diagrams (FLD's) are constructed from experimental data obtained using sheets on which grids, circles or other coordinate points have been marked (by techniques such as etching, photoresist etc.) and which have been biaxially loaded to varying increasing degrees over the range of no necking through to necking and even fracture, Figure 2. Different strain biaxilities are traditionally achieved, in the tension-tension r r quadrant, by hydraulic bulging over elliptical orifices the circular bulge producing equibiaxial tension; in the tension-compression r r quadrant, by Hecker cup tests or Marciniak-Kuczynski 'driver plate' tests in which a parallel-sided strip of metal is clamped over a given span (usually across a circular hole) and deformed by a spherical punch (of the diameter of the hole). By varying the width of the strip, different tensile e I may be combined with compressive r Simple tension tests of course fall in the tension-compression quadrant with r =- r in an isotropic sheet. The tension-compression r162 quadrant is the region of sheet 'drawing' where the geometry of the starting materials is altered by extensive metal flow, e.g. when a flat sheet is drawn into the shape of a cup. We note that in most of these tests the deformation really combines bending with in-plane loading: rarely, if ever, are flat cruciform testpieces used for the purpose of constructing FLD's. Many authors have expressed concern over the uncertain effects of (i) friction in the Hecker and similar tests and (ii) absolute size of bulge testpieces (span-to-thickness considerations for which shell or membrane theory is expected to apply: many bulge testers seem too small and produce non-uniform stretching patterns of deformation between rim and pole).

253 Despite these reservations strains are determined, after deformation, from measurements of the deformed grid patterns, i.e. a circle of diameter d which is distorted into an ellipse of major axis 2a and minor axis 2b has been subjected to s

=ln(d)and

E2 = l n ( ~---b)

(1)

with the principal strain directions along the axes of the ellipse (these directions will vary from spot to spot in a complicated pressing). It is true to say that these procedures of measuring grid patterns in the presence of high strain gradients and localisations is both tedious and inaccurate. For a given load biaxiality, all experimental strain pairs are plotted in el e 2 space and careful distinction made between those for which necking did not occur, those where it definitely did occur, and those where in repeat experiments necking sometimes did and sometimes did not occur, Figure 2. In this way, over the whole range of biaxialities the necking locus is determined. Since the plastic strains at necking are not that small, it is preferable to use logarithmic strains as just indicated. This is certainly necessary if the associated stress components are later to be determined by plasticity theory from the effective strain and the flow rules. However, metallurgists perversely plot FLD's using engineering strains as percentages, and most FLD's in the literature and in handbooks will be found in that form. 2.2 Theoretical Considerations

While FLD's are produced by experiment, the question arises as to whether there is a relevant theoretical background. Briefly, it may be said that plasticity theory predicts the existence of both diffuse and localised necking in the tension-compression quadrant and predicts the angular orientation of the localised neck for both isotropic and anisotropic yield behaviour in terms of the biaxiality and the workhardening index n in c~ = ~o En where r~ and ~ are the von Mises effective stress and strain respectively. Experiments agree with the theory. However, while diffuse necking is predicted for biaxial tensile loading, conventional plasticity theory is paradoxically incapable of predicting localised necking in the tension-tension quadrant. It is true to say that rather artificial assumptions about developing corners on yield surfaces in biaxial tension can lead to the prediction of such localised necks, but many unanswered questions remain such as "if corners are necessary in the tension-tension quadrant, w h y not elsewhere?" and so on. There are difficulties in this area which bifurcation theory does not yet solve. In practical terms, the empirical PalmarMellor relation has been found for the locus of localised necking and the work hardening index n, which is r =n +r for annealed sheets.

254 While there is reasonable connexion between theory and the experimental data, in practice FLD's for alloys with various thermomechanical histories are invariably experimentally determined rather than theoretically predicted. A valuable review of strain localisation and fracture in metal sheets and thin-walled structures has recently been given by Ferron & Zeghloul [3].

2.3 Application of FLD's The way in which the metalformer uses the FLD is as follows [2]. Sheets photomarked with circles or marked in other ways are formed in either prototype or production tools. Local strains near failures or suspected trouble spots are measured and compared with the FLD. From this information, potential trouble spots can be identified and the severity of the strain pattern assessed even though failure did not occur. If the in-plane strain-pairs are near the necking locus, problems are likely to occur in production because of tool wear, variations in lubrication between tool/workpiece/die, tool alignment, and variations in material thickness and properties. Where the in-plane strain pairs lie in relation to the necking locus is important because the strains attainable before necking depend on the local stress and strain state, and it may be possible to alter that to advantage. That is, the lowest value of the principal in-plane strain at necking r occurs under plane strain conditions when the minor inplane strain r is zero, so if the deformation in a critical region is nearly plane strain, the tooling a n d / o r lubrication conditions could be altered in order to induce either more drawing (i.e. move into the + r162 quadrant) or more biaxial stretching (into the + E1, + r quadrant), in which regions the limiting r values are greater. Drawing is promoted by better lubrication and less constraint at the edges of the sheet; and vice-versa.

The Fracture Forming Limit Diagram (FFLD) 3.1 Practical Considerations Fracture of biaxially strained sheets most often occurs within the regions of previously-localised necking. (Sometimes in biaxial hydraulic bulging experiments fractures occur around the clamped rim rather than at the pole of the dome; this is often connected with superimposed excessive local bending in the clamped regions caused by too small a span-to-sheet thickness ratio). Depending on the condition of the material there may, or may not, be appreciable further straining before fracture occurs. Since all localised necks are plane strain necks, deformation after necking up to fracture consists of sheet thinning within the neck together with complementary tensile stretching perpendicular to the neck, and no straining along the neck. Since for all applied strain ratios in the tension-tension quadrant the 10calised plane strain necks usually form along the minor in plane strain r direction, it follows that there will always be a change of strain path in the deforming regions, in this quadrant,

255 from the given applied strain ratio de 1/dr 2 to the condition of d r 1 6 2

= ~' in the

neck. That is, all strain paths turn parallel to the r axis after necking, Figure 3. In the tension-compression quadrant, where the neck is inclined to the principal strain directions (which are the axes of the FLD), no change of strain ratio occurs on the FFLD plots in that quadrant, since the zero strain increment direction is along the neck which does not coincide with the axes. There are now T12 shear strains in the neck still refered to the 12 (non-principal axes). There has been some debate about this point and different authors have plotted the data in different ways [see Chapter 5 of reference 3]. When the fracture strain pairs are plotted on the FLD, the fracture forming limit diagram (FFLD) results. Results are shown in Figure 3. A practical matter on which there is, perhaps, some uncertainty is the effect of stiffness of the loading systems and its consequence on crack stability. Certainly hydraulic bulge testers are "soft" and there is no control over the extent of cracking during the dumping of fluid pressure which eventually arrests the test. Flat cruciform specimens in a screw-driven testing machine are better in that regard, but they are hardly ever used and there has been little, if any, systematic investigation on whether different failure strains result from different stiffness testing arrangements. What evidence there is, from cracks that have propagated different distances before the fluid pressure is released in otherwise identical specimens, is that the through-thickness failure strain is not markedly affected. In the tension-compression quadrant, the effect of stiffness is not known on data obtained using mechanical punches. Friction is likely to have a much more significant effect in these arrangements. Another aspect of fracture forming limit diagrams which has received little attention is whether neck profiles up to fracture are dependent on the biaxiality ratio. This could be important in theoretical predictions of failure strains (see Section 3.2) since hydrostatic stresses would be affected and the level of hydrostatic stress is important for cracking in materials failing by void growth and coalescence. Work is starting at Reading on this topic using flat cruciform specimens.

3.2 Theoretical Background It is clear that theories of crack initiation will be important in attempts to predict experimentally-determined fracture strains in biaxial fields. There is a whole host of empirical criteria for crack formation in ductile materials, a review of which is given in Chapter 5 of reference [4]. For commercial ductile materials containing particles of inclusions a n d / o r hard second phases, the mechanism of cracking will be void formation at the particles followed by void growth and coalescence. Under these circumstances, McClintock void-growth models will be important, in which an integrated "damage function" incorporating stress and strain attains at fracture a critical value dependent on the size and spacing of the

256 inclusions (modelled as holes). There is a strong dependency on stress triaxiality. Since the microstructures of practical materials do not consist of uniformlyspaced particles of uniform size spread through a matrix as assumed by the model, a quantitative connexion between the model and actual inclusion average spacing-to-size ratios is sometimes lacking. Even so, the correct functional dependence of the stress and strain variables is predicted. How fracture forming limit diagrams may be predicted using these models has been discussed extensively in Chapter 5 of reference [4]. Since, in sheets, necking precedes cracking and since, as explained in Section 3.1, there is a change in strain ratio at necking, the McClintock accumulated damage integration has to be performed in two parts, viz: before and after necking. There happens to be an unexpected simplification in these calculations, however, as a number of forms of the McClintock criterion are surprisingly path-independent. One leads to a prediction that coincides with a well-known empirical criterion for failure in sheets, viz: a constant through-thickness strain at fracture which gives a 45 ~ fracture locus in r r space, since r + r = r = constant. A factor to which insufficient attention has been given is that of neck profile and its effect on hydrostatic stress, and in particular the effect of biaxiality on neck profile. Linked to this sort of thing is the change in hydrostatic stress state with biaxiality and corresponding changes in hydrostatic stress once the loading paths change on the formation of a plane strain neck. For fracture after necking, anomalous behaviour is observed in the sense that conventional wisdom says that for monotonic loading increasing (tensile) hydrostatic stress results in smaller fracture strains, but if necking intervenes, the greatest fracture strain coincides with the greatest current hydrostatic stress [5]. Given that on the pre-neck portion of such loading paths the hydrostatic stress was least, the idea of some critical McClintock j'(oH/o)de) at fracture is reinforced. An alternative line of attack for the prediction of the fracture strain pairs in biaxially-loaded sheets appeals to the possible connexion between the stress and strain conditions in material ahead of a propagating crack into which the crack is running, and those which exist in crack-free material being loaded up prior to initiation. That is, propagation is viewed as a process of continuous reinitiation along the path of cracking. This seems a reasonable hypothesis for ductile materials failing, after excessive plasticity, by void coalescence and growth. The tensile fracture of thin ductile sheets into which starter cracks have been cut is characterised by necking down in the line of the crack; crack nucleation in nominally flaw-free sheets also occurs after necking as reported in Section 3.1. Although (again) possible differences in the actual geometry of the necked regions as affected by remote and local biaxiality are unknown, it seems reasonable to convert the specific work of fracture in the presence of large amounts of plastic flow, determined by Cotterell-Mai measurements on doubleedge notched sheets [5] into a plastic work per volume by dividing by the inplane dimension of the necked-down process zone, from which the effective strain at fracture may be derived, and converted into in-plane strain pairs, for given biaxialities.

257 The m e t h o d has been tried out and seems to work quite well [5].

4. Fracture Strains in Sheet Bending Empirical relations for limiting bend radii in sheets are of the form (Pi/to) = (1/2A r) - 1

(2)

where Pi is the inside bend radius and A r is the %-reduction in area to fracture in a tensile test; t o is the sheet thickness. Such relations are obtained by equating the true strain in bending to the true strain at fracture in the tension test cf which is related to A r by cf = In (1/(1 - Ar)). That is ln(1/(1 - Ar)) = In (1 + t o / 2 ( p i + to/2)) and so on. Figure 4 shows that experimental data follow the trend of the equation. The correlation is reasonable for materials with limited ductility (high Pi/to), but it is not accurate for sharp bends (low Pi/to) because the neutral axis shifts from the mid-plane, the amount or shift depending on applied tension and friction (over rollers of punches used in practice to effect the bend), Also, of course, strains are diffused into nominally "unbent" flat regions and simple "y/p" values are not reached at low bend angles.

An alternative approach for fracture strains is to use non-linear elastic fracture mechanics and, involving H e n c k y d e f o r m a t i o n plasticity, say that it applies to irreversible plastic flow. This is the usual approach behind use of the J-integral in yielding fracture mechanics. The sheet bends to increasing radii p (p here is the neutral axis radius, not the inside bend radius of the empirical relation) and eventually cracks on the tensile face. In general we consider a wide b e a m with a notch and, for simplicity, we first a s s u m e that the resistance to bending is provided by the whole beam but only by the material below the notch of the same size as the ligament i.e. by a whole beam of depth t. For a material following o = o o Cn, and taking a mean bending strain of E = (t/4 p), and also ignoring any change of neutral axis, we have for the bending moment

we2 /;Y

M =-~o0

(3)

where w is the width of the sheet. If the bending takes place over a span S = p0 where 0 is the rotation, it follows that

wt2

M =-~o0

(4)

258 The w o r k done in b e n d i n g up to rotation 0 is wt 2 r'xn = 4 c~O~,T~)

jMdO

On

(riu

(5)

In non-linear elasticity, this is recoverable elastic strain energy; in rigid-plasticity, it is dissipated work. For a notch of depth a, the crack area is A = wa = w (t o - t) w h e r e t o is the full thickness of the sheet. A f u n d a m e n t a l relationship of fracture mechanics is that

R = -~

3

(JMd0)

(6)

w h e r e R is the specific work of fracture, i.e. the fracture toughness [4]. Hence in this p r o b l e m

R

~

--

13 1 3 w ~- (~Md0) = +--w~- (JMd0)

G0 0 n+l (2+n) t l+n 4 l+n Sn(n+l) Thus

or

0n + l

(7)

crack =

41+n Sn (n+l) R o0(2+n) t l+n

(8)

crack =

S c~0 (2+n) t l+n 41+n (n+l) R

(9)

p n+l

using S = p0. For an uncracked sheet t = t o and instead of a critical rotation or radius for cracking we m a y express the relation as a critical surface b e n d i n g strain ecrac k = (to/2Pcrack). Hence

r

21+n(n+1) R

(10)

crack = S G0 (2+n)

A l t e r n a t i v e l y , if it is a r g u e d that the d e f o r m a t i o n is c o n c e n t r a t e d on the ligament, it m a y be s h o w n that [7]

259

0n+l

(n+l) R crack = 2 ClaO (2c2)n+lt0 l+n

(11)

in a full thickness sheet where the plastic volume being deformed is V = Cl W(tot) 2 ( ~

c 1 = n / 4 say

) and where the plastic strain is related to the rotation

by e = 2c10. It follows that e n + l f = (n+l)R 2Cla0t0

(12)

C o m p a r i s o n of the two expressions for ef says that e q u i v a l e n c e w o u l d be obtained w h e n

i.e.

21+n/s (2+n) = 1/2Clt0 S = 22+ncl to / (2+n)

For n = 0 (rigid-plasticity) S ~

2 c I t o = 1.6 t o using c I = n/4; for n = 0.5 (a highly

w o r k - h a r d e n i n g solid) S ---) 2.26 c I to = 1.8 t o. These n u m b e r s m a k e sense p h y s i c a l l y for d e f o r m a t i o n concentrated u n d e r a notch as o p p o s e d to being spread out around a radius p over the whole span S. To check the sense of the equations with the empirical relations for bend radius given at the beginning of this section, we use Equation (9) for Pcrack and write p = (Pi + to/2) since the empirical expressions are in terms of inside bend radius. We obtain for the full thickness sheet [S(~o (2+n)]l/(n+l) (~4) Pi + t0/2 = [ (n--+ii R- .]

or

pi/t0

1 FSc~0 (2+n)]1/(n+l) 1 = 4 [ (-n-+-D R J - ~

(13)

This seems sensible with the form of the data in Figure earlier. The connexion b e t w e e n A r and [ ( n + l ) R / S o o ( 2 + n ) ] l / n + l seems physically reasonable: big A r are o b t a i n e d from tough (high R) and ductile (low a o) materials.

Of interest

h o w e v e r is the i n v o l v e m e n t of the span S. If we use the observation that S (1--2) t o for "concentrated" bending, it suggests that p i / t o should be smaller for thinner materials. This is, in fact, borne out in Figure 4 and does not seem to have been explained as such before.

260 The question arises whether, in bending, these ef are reached before necking occurs. During bending, the outer fibres are supported by underlying material that suffers lower strains and, therefore, is less prone to neck. The extent to which necking of extreme fibres is suppressed by the materials closer to the neutral axis depends on section shape in general. For a sheet, necking may be completely suppressed in simple bending, so failure does occur by fracture. With thin-walled tubes on the other hand there is little support from the underlaying fibres, so failure by necking is likely [2]. Futhermore in thin-walled sections, buckling may occur on the compressive side of the bend. Buckling is controlled mainly by the slenderness ratio of the section (i.e. distance from neutral axis + wall thickness). In contrast the tensile bending strain depends on (distance from neutral axis + radius of bend).

References

~

o

.

W.A. Backofen "Deformation Processing" Reading, Mass: Addison-Wesley, 1972. W.F. Hosford & R.M. Caddell "Metal Forming- Mechanics and Metallurgy" Englewood Cliffs, N.J.: Prentice-Hall, 1983. G. Ferron & Z. Zeghloul. Chap.4 in "Structural Crashworthiness and Failure" (ed. Jones & Wierzbicki) London: Elsevier Applied Science, 1993. A.G. Atkins & Y.W. Mai, "Elastic & Plastic Fracture" Chichester Ellis Horwood/John Wiley, 1985 and 1988. A.G. Atkins, Metal Science, Feb, 1981, 81. A.G. Atkins & Y.W. Mai, Engr.Fract.Mech. 27, 1987, 291. A.G. Atkins, Proc. ECF-8, Turin 1990, 234. A.G. Atkins, "The Griffith Centenary Meeting- The Energetics of Fracture", London: Institute of Materials 1993, 158. S.S. Hecker, Sheet Metal Ind. 52, 1975, 671

261

( a nd '6, ~ )

"-~

I

,/

/L/

,,.,--o

\

Fig.1. Changing orientation of localised necks with different remote applied strain rations. Diagram shows arbitrarily anisotropic yield locus with principal stresses applied along reference axes x and y. (Taken from reference 1).

262

I

100

l~eplica of Frad'ur'ed Specirnen

O019

021

2O

22

18

...-... z rY

I

4

80

Q~ C)

=E

16

17

50~

~

012

60

I -24

I

10 13 14 15

1 -20

8

I

9

1 -16 MINOR STRAIN (%)

1

-12

Fig.2. Distortion of printed circles near a localized neck and a plot of the strains in the circles. Solid points are for grid circles through which failure occurred, open points are for grid circles removed from failure, and partially filled points are for grid circles very near failure. (Taken from reference 2; see reference 9).

263 (a) %% %

-

3.0

-

2.5

-

2.0

-

! 5

-

I0

E, I

% ~ %

~

,

,.

X ~

~

,

~

'~

,Y

Siml:::)leTension ~,

il~

~

',, ', Z', ~'~ l~ 11 ,, ',

LOCal Necl(~(; ~

~

I

I

',,", i "* ', ,, ',U "1,~, "" ,,,

Diffuse Necking

~ ~ 1 ~ I ",," $i

I

-1.4

I

11

- 2

I

I

-I 0

I

1

-0 8

I

[ -0

6

l

I

-0 4

I

I

-0.2

t

I

0

r

s"

B~xlal tension

sl

0

I Necking

t

012

1

I

0.4

E2

Fig.3. Orthogonal in-plane plastic strains measured on sheets of 5154 A1-Mg alloy. Diffuse and local necking shown along with fracture strain pairs. Note: In the tension-tension strain quadrant, kinks are shown in the strain paths after the onset of localised plane strain necking since de2=0, necks forming perpendicularly to the major in-plane strain whatever the tension strain ratio. In earlier illustrations of this figure, kinks were also shown in the tension-compression quadrant. However, there the necks are no longer perpendicular to the r but take up angles depending on the applied strain ratio. Hence dr = 0, rather than ds where the r is along the inclined neck. r and r are no longer principal directions and there are ~/12 strains which cannot be shown on this diagram (Taken from reference 8).

264 A 1/4 in magnesium B 1/8 in magnesium C 0.050 in AZ31B.H24 (Mg) D 0,032 in HK31A.H24 (Mg) E 1/4 in 2011 ST 6(AI) F 1/8 in 2011 ST 6(AI) G 1/4 in 70/30 brass (CR 10%) H 1/8 in 70/30 brass (CR 10%) I 1/4 in 1018 steel (CR 25%)

56

16 o

,-9

J 118 in 1018 steel(CR 25%) K 3/16 in RC 130 B(Ti) L 1/4 in RC 130 B(Ti) M 1/8 in B & S tool steel N 1/4 in 70/30 brass (50% CW) O 1/4 in B & S tool steel P 1/4 in 1100(A1) Q 1/8 in polystyrene R 9/64 in cast iron

12

.,=., Q.

-

8

-

4

_

E

C 9

A

o

" B

I 0

K D

10

I ..... 20

9

9

L 9 0

~ 30

~

0

J

I

M

........

50 P

40

60

70

80

Reduction in Area %

Fig.4. Correlation of limiting bend severity (Taken from reference 2).

Pi/to

with tensile ductility.

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

265

P r e d i c t i o n of g e o m e t r i c a l defects in s h e e t m e t a l f o r m i n g p r o c e s s e s by semi-implicit F E M A.Makinouchi and M.Kawka* The Institute of Physical and Chemical Research -RIKEN 2-1, Hirosawa, Wako, Saitama, 351-01 Japan A three dimensional finite element program - ITAS-3D has been developed to analyze sheet metal forming processes. The code is capable of simulating forming processes for automobile body panels of complex shapes in order to predict forming defects. The theoretical background and numerical implementation of the code are discussed. Industrial examples of calculations using ITAS-3D-D code are presented. 1. INTRODUCTION Most car body parts are made of sheet metal which is manufactured by press forming processes. Thus, sheet metal forming plays a very important role in the automotive industry. The design process of forming tools (die, punch, etc.) for manufacturing sheet metal parts is traditionally based on actual production experience. This situation leads to a time consuming and expensive tools tryout process through which tools are revised many times. Consequently, the design of tools for sheet metal forming is said to be an art. Since computers have been put into the practice, the finite element method has attracted engineers who want to use it in the design process of sheet metal forming tools. Although many commercial FEM codes that are designated for analysis of the sheet metal forming process are now offered, results from these codes do not fully satisfy engineers. Very long computation times, numerical instabilities (e.g. lack of convergence) and poor prediction of surface defects like wrinkles or dents are on the long list of difficulties encountered when using commercial FEM codes. The forming defects related to automobile body panels can be classified in four groups : breakage, springback, surface defects and wrinkling. Using simplified (e.g. geometrical) methods or existing databases only a small group of these forming defects can be analyzed. When the forming process for a part with complex shapes, like an automobile body panel, has to be examined, there is no alternative to analysis by means of FEM code. In this paper we introduce the ITAS-3D code widely used by the Japanese automotive industry to analyze sheet metal forming processes. This FEM package is capable of simulating complex industrial applications and gives accurate results. The code is being continuously developed by the authors through the support of the Sheet Forming Simulation Research Group in Japan. This paper contains up-todate information on the theoretical background and the numerical procedures implemented in the ITAS-3D code. Examples of industrial applications are also given. On leave from Warsaw University of Technology, Warsaw, Poland

266 2. F O R M U I ~ T I O N

There are two competing FEM approaches to sheet metal forming analysis : dynamic and static. The competition between them has received significant attention, but it is still not possible to indicate which approach is better [ 1-2]. ITAS-3D is a static semi-implicit FEM code. Precise theoretical formulation and improved numerical procedures are implemented in the ITAS-3D code, eliminating most of the problems characteristic of static implicit FEM codes. This section presents the theoretical background of ITAS-3D. 2.1 V a r i a t i o n a l p r i n c i p l e The updated Lagrangian rate formulation is the base of the incremental elastic-plastic finite element code- ITAS-3D. The principle of virtual velocity describes the rate form of the equilibrium equation and boundary conditions. The form of the principle is very similar to the form proposed by McMeeking and Rice [3]" (1) The one significant difference comes from the assumption t h a t body is incompressible i.e. det(Ox/0x o ) - 1. As a consequence of this assumption we have xJ - a J where ,~J is the d a u m a n n derivative of Kirchhoff stress and (~J is the daumann derivative of Cauchy stress. Here, by is the virtual velocity field satisfying the boundary condition by - 0 on St. V and S denote respectively, the region occupied by the body and its boundary at time t. St is a part of the boundary S, on which the rate of nominal traction f is prescribed. L is the gradient of the velocity field (L = 0v/0x), D and W are respectively the symmetric and antisymmetric parts of L. 2.2 C o n s t i t u t i v e e q u a t i o n

A small strain linear elasticity and the large deformation rate-independent work-hardening plasticity was assumed. In order to deal with anisotropy of metal sheets, Hill's quadratic yield function and the associative flow rule was used (for details see Cao and Teodosiu [4]). The constitutive equation can be written as in the form" o~J = C?~D~ P - C ~epL ~

(2)

ep where C~a is the tangent constitutive matrix. Introducing Eqa.(2) to Eqa.(1) the final form of the principle of virtual velocity is obtained"

fDijklLklSLijdW = f f i 6 v i d S V

St

1 where D ~ -- C ~ + X~ and Xi~ = -~'(o~5~ -o~5~ -o~5~ - o~5~).

(3)

267 3. F I N I T E E L E M E N T E Q U A T I O N S

The above stated equation is solved in the standard way : Eqa.(3) and Eqa.(2) are integrated from time t to t+At where At is a small time increment. The displacement increment, the J a u m a n n stress increment, and the increment of the displacement gradient are written as : Au = v- At, Ao J-- O J "At, AL = L- At, etc. and all rate quantities are simply replaced by incremental quantities, assuming t h at rates are kept constant within an incremental step. When performing a standard finite element discretization procedure, Eqa.(3) can be replaced by a system of algebraic equations : K. Au = AF + AC

(4)

where K is the elastic-plastic stiffness matrix. The terms AF and AC come from the right hand side of Eqa.(3) where the derivative f has been replaced by the expression f = f -e + f- e. The term AF - Af "e denotes the increment of the external force vector and the term AC = f . Ae expresses the rotation of the total force vector during the incremental step. In the ITAS-3D code a semi-implicit approach to the solution of Eqa.(4) is applied. A stiffness matrix K is described at time t and is regarded as constant within the time increment At. The so-called "r-min" method [4] is employed to impose a limitation on a size of the time step Some number of iteration are needed to solve Eqa.(4) due to the fact that the AC term can be calculated only if the final configuration is known (a change of the local coordinate base vector he has to be found). 4. E L E M E N T S

During stamping or deep drawing operations sheet metal is subjected to a very complex deformation history and very complex boundary conditions. For example, it is well known that some parts of the sheet deform under almost pure stretching, whereas other parts can develop significant bending strains. The deformation mode can be very complex in the parts of the sheet where severe bending or double side contact with tool exists. Consequently the theory needed to describe deformation in a sheet can vary from a two-dimensional (membrane assumption) to a fully three-dimensional. Three distinct classes of elements used in the simulation of sheet metal forming processes are known:membrane, shell and solid. Elements based on the membrane theory are the simplest ones. They are very efficient and cheap in simulation, but any results obtained by the means of membrane elements suffer from a lack of the bending effects. For example, when using only membrane elements, it is not possible to simulate springback after a stamping or deep-drawing process. From the practical point of view membrane elements have limitations in their use in simulation of sheet metal forming processes. It seems to be very natural to use elements based on the shell theory to simulate sheet metal forming processes. Unfortunately, not all shell theories are able to adequately represent the complex phenomena t h a t exist in deformed

268 sheets. Another problem is the computational difficulties associated with some of the shell theories. For example the Kirchhoff-Love theory needs continuously varying displacements and slopes over the shell surface (C 1 continuity). This condition is difficult to satisfy and sophisticated elements are needed to fulfill it. From the practical point of view elements based on the Kirchhoff-Love theory are not often used mainly due to computational disadvantages. Elements based on the Mindlin-Reissner theory which demands only C o continuity to dominate the shell analysis. The most popular elements among them is the family of degenerated shell elements. Despite some numerical drawbacks (like shear and membrane locking or spurious deformation modes), the degenerated shell element is quite efficient and simple. Solid (continuum) elements are used to simulate sheet forming only in situation where fully three dimensional theory is needed to describe the deformation process. Some of the solid elements properties make them difficult to employ in sheet forming analysis (e.g. far too stiff behavior in bending and problems with incompressibility condition). The remedy for these problems is to apply a special integration technique. 4.1 D e g e n e r a t e d shell e l e m e n t The idea of a degenerated shell element comes from the work of Ahmad et al. [5]. This type of element is based on the Mindlin-Reissner theory : deformation field is defined by means of the two independent variables - the displacement and the rotation of the mid surface nodes - Figure 1.

X A

y

Figure i. Degenerated shell element

The 4 node quadritarelal element with bilinear shape functions is preferred for its effectiveness and its simplicity of formulation. The disadvantage of this element is its severe, overstiff behavior in bending. This behavior stems from the fact that linear shape functions are not able to adequately describe the deformation field characteristics for the pure bending and spurious shear deformation that affects the solution. The kinematics of the degenerated shell element are defined in terms of the shape functions by invoking the isoparametric hypothesis. Thus the components of velocity field vi: v,(~, ~1,~) - N~(~, ~1)"v7 + N~(~, q)-~'~, h ~- 07 or

(5)

269 where v" and 0 a are respectively the velocity and the fiber rotation velocity for node "a". Symbols N~ and ~ mean the generalized shape function and the generalized fiber velocity V~= [v",0"]. Index "r" takes the value of 1 to 5 (i.e. V~ = [Vl, w, va ,0x ,O~] ). The generalized shape function N" is"

N~(~,~l,~) = { -0.5"~'N~(~,

n)'h='b~

r = 1,2,3 r=4 r-5

for

[0.5 "~'N~(~, ~l) "ha "bli

Using Eqa.(5), the velocity gradient L~(~, ~l,~) can be defined" L~(~j,~I,~)

=

OVi(~,]],~)/OXj

(6)

-- N ' n , j ( ~ , ' I ] , ~ ) ' V ~

where the partial differentiation operator 0/0x is designated be a comma "j". Introducing Eqa.(6) to Eqa.(3), the expression defining the elemental stiffness matrix K is obtained" K ~(~, 11,~) = ~"N~,j(~, ~q,~)" D~" N~,I(~, q, ~)dV

(7)

Fully integrated stiffness matrix (8 - integration points for a 4 - node shell element) performs very well for moderately thick shell situation. However, when working with thin shell structures, some parts of the stiffness matrix relate to membrane or shear deformation artificially dominate the total stiffness matrix ( membrane or shear locking). Consequently too stiff solution is obtained. Several ways to overcome this problems are known, for example : reduced integration - RI [6], selective reduced integration - SRI [7], assumed strain field for shear components ASF [8], and elements with stabilization matrix - SM [9-10].

4.2 The square cup deep d r a w i n g test The square cup deep drawing test Figure 2, is used by our group to verify the properties of the finite elements. The most interesting aspect of this test is the existence of buckles and wrinkles on a sheet surface - Figure 3. We find that some types of elements or some types of integration schemes make the elements resistant to the formation of wrinkles or wrinkling mode is different than obtained in experiments.

.:.:.:.:.:.-.:.-.:.:.~.:.

Iis,,Ti!iiiiii: 9

i

.......

147.5

. .__

9 ". ". 9. t . ".

J

150.5 -1 180(i00)

=-

Figure 2. Geometry of the square cup deep drawing test

270

Figure 3. The square cup drawing test (initial blank 80x80mm) a) experiment b) ASF element c) element SRI d) SM element with insufficient number of integration points

Figure 4. Comparison of CPU time for a one step in the square cup test. (700 elements, IBM RS6000, Model 550)

271

The blanks used in this test are square with an 80ram or a 100mm side length and a 0.7ram thickness. The test is performed until a 20ram of punch stroke, with or without the blank holder. The square cup test can also be used to compare the efficiency of the elements Figure 4. We have found from our experiments t h a t the cheapest and the most economical to calculate is the element with stabilization matrix - SM., However, the wrinkling mode obtained in calculation for this element is strongly affected by the number of integration points in the thickness direction - Figure 3d. The most reliable element to use is the one with the assumed strain field ASF, in which the results are physically correct in almost all cases.

5. FEM TECHNIQUES Substantial progress has been made in the last few years in the development of stable nonlinear 3D FEM codes. However, implicit codes are still said to be troubled by sever convergence problems. Most of the these problems are related to highly nonlinear contact-discontact phenomena, friction boundary conditions and physical instabilities like bifurcation points. To study and understand how these p h e n o m e n a influence the convergence, new stable F E M a l g o r i t h m s are continuously being developed. In this section some of these FEM algorithms which are used in the ITAS-3D code are presented. 5.1 C o n t a c t To keep all the boundary conditions unchanged within one incremental time step, r-min method [5] is employed to limit step size in such a way t h a t : (i) no free node comes into a contact, (ii) no contacting node starts to separate. The procedure for r-min calculation is as follows - Figure 5 : - for free nodes : at first the initial solution of Eqa.(4) is obtained, then condition (i) is used to calculate r-min in such a way that the node going into a contact with tool will be precisely on the tool's surface at the end of the incremental step; - for nodes contacting the tool's surface : if the normal component of a nodal force is positive - F3>0 at the beginning of the incremental step and the initial solution of Eqa.(4) gives F3+AF3<0 then r-min is calculated from relation F3+rmin*hF3=0. All the nodes are verified by the above procedure and the smallest r-min value is chosen to define the incremental step size.

\

F31 1

3

Et initial guess

update a

initial guess

update b

Figure 5. R-min method in contact analysis a - node goes into contact with tool b - node disontacts tool

Figure 6.Contact spring element En normal stiffness Et tangential stiffness

272

5.2 Friction Mathematical model of the friction implemented in ITAS-3D is assumed to have both an elastic or reversible part and a plastic or irreversible part [11-12]. The description is t h a t of non-associated plasticity. This formulation leads to the constitutive equation for contact behavior : o v - (E~ - a.P~ ).vj

(8)

where 9a is constant and equal 0 during sticking and equal 1 during sliding; o v is the co-rotational rate of the contact unite force (pressure) vector; Eij and Pij are respectively elastic and plastic constitutive tensors. Practical implementation of friction is done by using contact a spring element - Figure 6. This element is attached to each node that is in contact with the tool. We run a set of standard numerical tests to define values with stiffness parameters of Et and En. Combing Eqa.(8) and Eqa.(4) and integrating the contact spring elements we obtain" (K +

K fr )" A u =

K f'' Aut + AC

(9)

where K fr is the friction stiffness matrix- Eqa.(8), A n t is a tool's displacement vector. (given at the beginning of the incremental step). The friction matrix K fr ( size 3x3 ) exists only for degrees of freedom that correspond to the displacement of nodes that are in contact with the tool.

5.3 Springback Two different m e t h o d s are used in the ITAS-3D code to simulate the springback process - Figure 7. The first one, method A, is a simple continuation of s t a m p i n g process analysis : the tools' motion is reversed and calculation are carried out until no nodes are found in contact with tools. In the case of method B, geometrical contact with tool is replaced by the nodal force f equivalent to the reaction force, at first for all nodes,. Next, a new force boundary condition is prescribed for all these nodes : h f =- f . Then, calculations are continued until all nodal forces vanish. METHOD

stamping

A

springback

R stamping ~

METHOD

springback

B

Figure 7. Two methods of springback process calculation Method A - with tool approach; Method B - without tool approach.

273 There are almost no differences between the results of the calculations of method A and those of method B. However, the number of steps and calculation time needed for each method is very different. Method A is at a disadvantage in this regard, due to the contact analysis that needs to be performed for this method at each incremental step [13]. Trimming line

Removed elements

Unequilibrated forces

,-

I

|

F Force boundary L c~176 ~

I

F i

I,

f

l

~i

!

I~

l__L..-.a-~"

:"

d Figure 8. Simulation of trimming process a - group of elements if removed from mesh b - unequilibrated forces are calculated c - force boundary condition is imposed d - mesh free ofunequilibrated forces is obtained 5.4 T r i m m i n g To fulfill the quality requirements for automobile body panels such as body setup accuracy, all the forming process that affect the shape of final product have to be analyzed. Trimming is one of these processes. A new FEM technique of trimming process analysis is available for users of ITAS3D [14]. Simulation of trimming consists of three independent steps - Figure 8. First, the group of elements is removed from the mesh (trimming line is equivalent to the outer edge of removed group of elements). Next, unequlibrated forces are calculated based on the element's stress field for all nodes on trimming line. Then, in an approach similar to springback method B is applied : for each node on the trimming line an additional force boundary condition is imposed - Af =-f and calculations are continued until all nodal forces on the trimming line vanish. 6. EXAMPLES In the automobile industry, tool design usually requires an iterative procedure of trail and error in order to obtain a final component with good quality, t hat is without defects and keeping in mind the need for optimization of material and production. This process may take several months and is expensive in terms of man power and machining cost. Reductions in both time and money can be

274 achieved if designers have a good method for evaluating the formability of produced parts and to predict possible forming defects. FEM codes are examples of such widely used tools. In this section, three examples of ITAS-3D application for forming defects prediction are presented. 6.1 F o r m i n g of a truck frame m e m b e r A frame of a small size truck is presented in Figure 9a. The front part of the frame has alternation in its section geometry along the longitudinal direction which are responsible for several possible geometrical defects when part is ultimately formed- Figure 9c. FEM calculation was done in order to analyze the behavior of the frame member. The tested part is shown in Figure 9b, additionally experimental results have been compared with the simulation of this process [15]. The FEM model of the sheet was divided into 1355 shell elements. After the stamping process, the springback has then been simulated. Figures 10 and 11 present results of the simulation and of the experiments. Although the results of the simulations and the experiments do not agree in all the points, they present the same tendency and variation along the frame axis. 6.2 Multi-stage sheet metal forming processes In the case of multi-stage sheet forming processes, physical phenomena which appear in one process can affect sheet deformation in subsequent processes. This situation indicates that the history of sheet deformation must be taken into account if satisfactory results of simulation are expected. Simulation of a mini-van roof panel forming process is presented in this paper as an example of multi-stage FEM analysis. The real production process consists of three operations : stamping, trimming and flanging - Figure 12. The most frequent defect encountered on the production line for this panel is t h a t of difference between expected and obtained shape near the sun r o o f - Figure 13. This geometrical defect creates problems for assembling sun roofs parts. Profiles of a roof panel were measured after the three main stages of the forming process and geometrical defects accumulates during successive operations - Figure 14. The size of the roof panel (approximately 3 m length) and the complex local shapes made it impossible to analyze the entire panel (too many finite elements were needed and computers are not yet powerful enough). We decided to simplify the FEM model for the roof panel forming process, only part of a deformed sheet around the window was analyzed - Figure 15 ( bright marked area ). Additionally, displacement boundary conditions were imposed on the symmetric plane. In the first stage of simulations (stamping & springback) mesh consisted of 2820 shell elements, while 2357 elements were used in the second stage of simulations (flanging & springback). Figure 16 presents a flow chart of FEM calculations. In total 9 separate stages of calculations were needed to complete the whole simulation process. Although, the correlation between experimental and computational results is not satisfactory, Figure 14 and 17, the same tendency in sheet deformation is observed : contours of the roof panel become fiat after drawing and flanging. Differences are mainly due to boundary conditions. For example symmetric planes make sheet very rigid, contrary to very flexible big sheet panels.

275

..... i- liP"

Bending at change of section

Bending at offset section

lI'

Flange bending

Longitudinal twisting

Wrinkling and flange curvature at change of section

.,a.-----..--.ib.

--

F i g u r e 9. a) a t r u c k f r a m e ; b) t h e t e s t e d p a r t ; c) g e o m e t r i c a l d e f e c t s t h a t m a y o c c u r

vl!

276 0

design curve )~

L

_1

[

1t

measured curve

/

0.8 ~

simulation

0.6

./

experiment

0.4

....

02O ~9

0

\\

-

e.2

-

-e. 4

-

design curve

0 . 6 -0.8-

-1-w

!

100

I

I

,

140

,

,

180

'

'

,

- -

220

,

--

260

300

Measuring position on a frame axis [ram] Figure 10. Comparison of deviation from design curve between simulation and experiment _

18

et e

simulation

e. 8 0.6 ,.~

0.4

~

e.2

9

-0.2

~

-0.4 -0.6

/~~~

'

-'~__\ expemment

;

0

j

i

200

m

,

400

w

........ ,

600

Measuring position on a frame axis [mm] Figure 11. Comparison of twisting between simulation and experiment

277

_J

Figure 12. Three stages of a roof panel forming" sto_mping, trimming, flanging

A-A

Figure 13. Geometrical defects of a roof panel. Cross section A-A" continuos line - expected shape, dashed line - obtained shape.

278 2

ix drawing n

trimmirlg

x flanging --

expected shape

EASUREMENT AREA

250

300 350 400 450 distance from the symmetric surface [ram]

Figure 14. The position for contours measuements

Figure 15. The rael roof panel and the FEM model (bright marked area) 20

A cL~ ~ p . g []

o

+~I0 8

Lx zx ~

LXLX A

X

x

X

[]

[]

X

[]

[]

X

A

X

x z-.1

trimming

x

X

[]

X

r'l

o

r"l i::::z

X

8 ~q~O

'

I

.

I

.

I

.

I

45O 500 300 350 400 distance from the symmetric surface [mm]

Figure 17. Shapes of the roof panel after three main stages of calculation

500

279

al

Figure 18. A door assembly panels : door inner, door outer, inner reinforcement, outer reinforcement

a

b

c

Figure 19. Flange bending of the door inner reinforcement a) shapes before and after flange bending Comparison of the flanged part contours : experiment and simulation Two cases of calculation : without (b) and with friction (c)

280 Initial setting (stamping tools)

Dead-weight deformation

- •Stamping

"Springback" ~ ~ - ~

Trimming

~

Springback

Setting 9 (flangingtools) ~

Flanging

~-~

Springback

Figure 16. A flow chart of the roof panel forming process FEM simulation. 6.3 F o r m i n g of a door inner panel reinforcement A door inner panel and its reinforcement part are presented in Figure 18. Springback effect in the flanged part of reinforcement creates difficulties in the assembly process before welding of the door panel. This FEM simulation using ITAS-3D has been performed only for the last stage of the forming process flanging. Additionally, springback process has been analyzed to find the final shape of the flanged part. The FEM model consists of 950 shell elements. During calculations, friction has been modeled (~=0.1) by using a spring contact element described in section 5.2. Results of the experimental measurements have been compared to those of the simulations - Figure 19. A good agreement is observed between simulation and experiments. 7. CONCLUSION A numerically stable FEM code, ITAS-3D, simulating sheet metal forming processes has been tested and released to engineers in the automotive industry. This package is capable of predicting complex geometrical defects like springback and wrinkling. Presented results of calculations prove that FEM simulation can be a source of quick and valuable information for designers of metal sheet forming processes. ACKNOWLEDGMENT

Authors would like to thank Mr..Koichi.Kazama from Press Kogyo Co.,.Mr. Mikio Kimura and Mr. Yasuhiro Watanabe and from Fuji Heavy Industries Ltd.., Mr. Hideyuki Sunaga and Mr. Masato Takamura from Nissan Motor Co.,Ltd. for providing examples of calculation using ITAS-3D code. REFERENCES

1. Rebelo N. at al., Comparison of implicit and explicit finite element methods in the simulation of metal forming processes, Conference proceedings : Numerical methods in industrial forming processes - Numiform'92, Valbonne, France,, September 14-18, 1992, A.A. Balkema, Rotterdam, 1992, 99. 2. Teodosiu C. at al., Implicit versus explicit methods in the simulation of sheet metal forming, Conference proceedings : FE-simulation of 3-D sheet metal forming processes in automotive industry, Zurich, Switzerland, May 14-16, 1991, VDI Betrichte 894, VDI Verlag, Dusseldorf 1991, 601.

281 3. McMeeking R.M. and Rice J.R., Finite element formulation for problems of large elastic-plastic deformation, Int. J. Solids Structures, 11 (1975) 601. 4. Cao H.L. and Teodosiu C., Finite element calculation of springback effects and residual stress after 2D deep drawing, Conference proceedings : Computational Plasticity- fundamentals and applications, Barcelona, Spain, September 18-22 1989, Pineridge Press, Swansea,1989, 959. 5. Yamada Y., Yoshimura N. and Sakurai T., Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method, Int. J. Mech. Sci., 10 (1968) 343. 6. Ahmad S., Irons B.M. and Zienkiewicz O.C., Analysis of thick and thin shell structures by curved finite elements., Int. J. Num. Meth. Eng., 2 (1970) 418. 7. Huang H.C., Static and dynamic analysis of plates and shells., Berlin, Springer-Verlag, 1989. 8. Dvorkin E.M. and Bathe K.J., A continuum mechanics based four-node shell element for general non-linear analysis, Eng. Comp., 1 (1984) 77. 9. Liu W.K., Ong J.S.J. and Uras R.A., Finite element stabilization matrices - a unification approach, Comp. Methcds Appl. Mech. Eng., 53 (1985) 13. 10. Liu W.K., Law E.S., Lam D., Belytshko T., Resultant-stress degenerated shell element, Comp. Methcds Appl. Mech. Eng., 55 (1986) 259. 11. Michalowski R., Mroz Z., Associated and non-associated sliding rules in contact friction problems, Arch. Mech., 30 (1990), 259. 12. Wriggers P., Vu Van T., Stein E., Finite element formulation of large deformation impact-contact problems with friction, Comput. Struct., 32, 1989, 319. 13. Santos A. et al., Simulation of springback behavior of 3D sheet bending process by elastic-palstic FEM, Conference proceedings : Computational Plasticity fundamentals and applications, Barcelona, Spain, April 6-10, 1992, Pineridge Press, Swansea, 1992, 1334. 14. Kawka M. et al., Analysis of multi-operation automotive sheet metal forming processes, Conference proceedings : Advanced Technology of Plasticity 1993,, Beijing, China, September 5-9, 1993, International Academic Publishers, Beijing, 1811. 15. Kazama K., Nagai Y., Effect of die radius shape in a U-bending process, Conference proceeding : 1993 Japaneese spring conference for the technology of plasticity, May 25-27, 1993, Tsukuba, Japan

This Page Intentionally Left Blank

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

283

Evolution of Structural Anisotropy in Metal Forming Processes J. Tirosh Technion, Faculty of Mechanical Engineering, Haifa, Israel

Abstract The evolution of structural anisotropy during sheet forming processes of time-dependent materials is considered here in details. The eventual alignment of the elongated voids (pores, cavities, etc.) towards the principal tensile loading direction, constitutes the source for the anisotropy. Experimental observations on large plastic derformation (around 30 percent strain) during bulging of thin sheet metal with artificial voids, provided the background for this study. 1. I N T R O D U C T I O N In the area of sheet forming processes there is a need to predict the strain-induced anisotropic behavior of the final product. Such processes are usually performed at a relatively large plastic strain [of the order of 0(102) percent of equivalent strain]. It is this high level of strain which may bring an initial small void (pore, cavity, etc.) to a substantial size with a preferred orientation. The voids, as observed in the tests, are eventually stretched along the tensile loading direction and subsequently generate an anisotropy in the structural response. The present work is focused on the evolution of such a process and enlarges other works on the same subject [ 1,2]. It is assumed that the microvoids are evenly scattered in the material and their surface is stress-free, or nearly so (like micro-inclusions which are significantly softer than their surrounding matrix, e.g., sulfuric particles in steel [3]). As described in a previous work [4], each void has an elliptical shape, characterized by three variables (i.e., the two axes of the ellipse and the angle between the major axis and the loading direction). Such a solid resembles a porous material. Thus, the considered structure is modeled by a metal matrix with a periodic array of 2D voids, as described in Fig. 1. The matrix remains incompressible and obeys a certain time-dependent constitutive law. As observed in our bulging tests in Fig. 2 (using a double-layer of superplastic sheet metals [5]), each void undergoes a local distortion (expansion, extension and rotation). The induced rotation of the voids is accumulated in time and eventually causes a preferred alignment of their orientation. Consequently, the instantaneous compliance of the structure, in particular, the responses along and transverse the loading direction progressively differe one from the other.

This evolutionary process of anisotropy in cavitated metals is formulated here along with experimental data for comparison.

284

Fig. 1. The model of the porous material with the periodic distributed cavities. The shaded area points to a representative cell with a single cavity. The changes in the geometry of this representative cell (in particular, its rotation) during large plastic deformation generates the anisotropy in the material compliances.

285

,

o ] e, y e r s

pressure

.--mammmmnmumunuummumanuummmlnmommwmmm.. ,xlllIIlIllTll[lllIlllILIIlTlIIIJ~-~,d..r.,~ ~gommlmimQmaallsgmgguglmeaiHgamP ~ IIFIIIIll ~,,~-tl L L L I I I I T I I I I T I T ~ I I I I I I I I I | I I I I k ~ ltrll iI[lllllltilIllllI[lllll: 7 IF /1: F: I E . .I .i J. l.~. I.I.I I.I.I.I I.I.l .t l.I.l l.l .] I.I.I .I I.~. :,, hff ..... I,,I,: L[[II[tlIIIIILI~LIIIIIIII!II'~L.LL[]J_s ............. ~II iH!)lll11~llil: I ~ l l l l l l l i l l l l ~ ~,~,[L[III] LII II11IIIIIII11I: l IIIIIIIIIIII1._ ~I I I I -,,

Err

rH

[ tr-~-,,

--'lIlllI[[lllllIllIlllIll

.....

~....

,' "~ , ,", I

] ' I ' [ ~ "_', "

~/llllIllrlIIJ..d

~--mmmmmmmmmmmmmmmmmmm~ -om-mmmmmmmmmmmiLT,.~

lllIl~.

I1i~ :,II~ lliI

....

,,''~

V

--~:"

~r

L2

Figure 2. Schematic view of the bulge test. An hydrostatic fluid pressure causes a biaxial stretch of a double-layer thin sheet (Lead-Tin 50-50 percent) tightly held between rigid dies. The top layer is imbedded with artificial elliptical cut-outs whose configurational changes are monitored during the bulge process.

286 2. T H E C O N S T I T U T I V E

BEHAVIOR

Consider a dilatant solid of volume V which contains a uniform distribution of cavities, each of which has the same volume, denoted by V(c). The volume fraction of cavities f, is def'med as f _ ~ V (c)

V

where

V = V ( ~ + ~ V (r

(I)

and V0~) is the total volume occupied by the matrix. Let a self-equilibrating remote traction P

be prescribed on the outer boundary, A, such that

i

Pi = ~ ' ' n ~J J

on A

(2)

where n j are the components of the exterior unit normal on A. The average stress occupied in V is obtained by 1 r dV O'iJ = V JO'iJ

a ij

v Let the average strain-rate in the dilatant solid be defined similarly as

(3)

)

~ _ 1 J'd dV 'J"" V V ij , U i

1 I~ij = -2(ui , j + uj , i ) .

where

(4)

is the velocity field in the solid and the comma followed by an index denotes partial

differentiation with respect to the corresponding coordinate. It is now assumed ( i.e., [6]) that the average values of the stress and strain in (3) and (4) are composed of the weighed average of their constituents; the matrix (with superscript 'm') and the cavities (with superscript 'c'), namely

~.. = lj

o(m) + f_(c) ..

(I- f)-

ij (m)

-'.. E = (1 - f)~ .. ii

ij

lj (c)

+f~..

Ij

(5)

where, as defined before _(m) 1 I o 0.. (m) , ~ ij dV Ij V V _ (c)

11

1

(c) !,~ V v

dV

_(m) E..Ij

_(c) ij

1 v(m)

dV

~,fl~

V

1 ,y_,V (c)

ij

dV ~ I~ij (6)

287

In the considered problem, the cavities (before closure) are stress-free, so that ~(i~) = 0 With the aid of the divergence theorem, one gets from (4) the strain-rate induced by the motion of the cavities as _(c)

s

ij

-

I

I

(~ "~ (u in j + u . n )dA ~v(C) ~A"' j i

(7)

The matrix, unlike the whole bulk, is incompressible, ~ (m).. = 0 and obeys the following U power-law strain-rate equation: (m)

(m)

=

)n

min

0
(8)

where e rain and C are reference values (10 -5 l/sec and 5.39 Mpa, respectively, in the considered material) and n (=.35) is the rate sensitive exponent. In short, (8) reads o

(m)

=(E

(m))n

(8)a

such that, .3

or(m) = O(m)c

~.(m) ___t~Sij Sij )

, Sij

_

Oij

3 kk

(9)

and the equivalent strain-rate of (8)a (which satisfies the work-rate equivalence a~ = ~ ij~ ij ) yields (m]

2

(m) (m) I/2

--=( E ij Eij )

(10)

By averaging the above equivalent stress and strain-rate terms (o.(*)._.) ~(,))and substituting in (5), the

(~(m)...), ~(m))a n d

resulted 'rule of mixture' sets the (scalar)

instantaneous constitutive equation for the 'structure' (namely the matrix with the cavities). It reads -o

= (I - f ) ~ ( m ) + f-d(c)

g =(1-f)g

(In)

+tg

(c)

(5)a

288 Now, one has to define the evolution process of the variables in (5)a. It is possible to do it for the porosity rate of change, 1:, in terms of the surface motion of the cavity in the 'unit cell'. Both the cavity and the cell dimensions are changed during the creep. At any instance, the volume fraction of the cavity is changed according to ~uinids volume of unit cell where

ni

(5) b

is the unit normal outward to the cavity surface 's'. The solution to u. (details are 1

given in [4,7]) is modified from Mushkelishvili's fundamental elastic solution [8], and given in terms of the remote loading ( p 1' p 2) and the three geometrical variables of the elliptical cavity (a,b,and a ). It reads; 4

(u + iv ) = - ~ R [

(Pl + p 2 ) (Pl-P2) (~ _ m ) + 4 4 q

e 2i~

2-T--]

(5)c m

where ~ = e i~

0 < 0 < 2tr , m=( a-L"'a+ L'o)/~,u)

R=(a+b)/2

and r/(e) designate the plastic

modulus of the structure. In order to get the constitutive behavior of the structure (~

vs. ~) one needs to solve the five

equations [5a x2, 5b, and 5c x2] for the five unknowns ( ~ ,

E, f, u , v) provided the rate

variables (/~ and e) are related to their accumulated values, f and E, by a straight time integration. 3. I N S T A N T A N E O U S

COMPLIANCES

The constitutive equation of the matrix (8) can alternatively be written in a linearized tensorial form as; (m) (in) (in) s ~J'" = 2rl ~ij

(m) where

rl

1 .(m))n - 1 = -~(t;

(11)

and s ij is the stress deviator given in (9). The non-linear function 77(m) of (11) represents the i n s t a n t a n e o u s plastic m o d u l u s of the matrix. It varies (non-linearly) with the progression of the plastic flow. By substituting (7) into (5) and recalling that the voids in the solid are stress-free (and hence

S--ij= ( 1 - f)~m)) the change in the overall strain of the solid becomes the 'weighted sum' of the two contributions:

289 The strain-rate of the matrix (the first term on the right hand side of (12)) and the strain-rate attributed to the motion of the cavities (the last term on the fight hand side of (12)). It reads: S~176 ~~176

ij

2rl

~J 1 (m) + ' V

1

~) )~ (uin j + u . n i ) d A ~.,A'" J (12)

Let us designate the strain induced by the presence of the pores (the last term in (12)) as: _- 1 V HijldSkl

(~ ~.,'2 1 ( u i n j + u j ni)d A (13)

so that (12) can be rewritten in a compact form as:

ij = C ijld girl'

where

C ijld = (

1 _ + H ijkl ) 2rl (m)

(14/

The components C o~a are the instantaneous compliances of the plastic porous solid which we seek.

Fig. 3. Simulation of the configurational evolution of a single cavity in porous material. The initial shape (at time t--0) is the internal ellipse. The final shape (at time t=tmax) is the outer ellipse. The ellipses in between these two extremes are given at various time increments during the creeping process (while the hydroststic pressure stays constant). The gradual rotation of the ellipse major axis towards the loading direction is manifested.

290 4.

THE SOURCE OF INDUCED ANISOTROPY During plastic deformation the cavities tend to elongate in the direction of the principal

tensile load. Consequently, an initial isotropic porous solid becomes progressively anisotropic, namely, it responds differently in different directions. The net matrix is assumed to stay isotropic (i. e., no textural changes) while the overall structure (the matrix with the cavities) exhibits, on a macroscopic level, different compliance at different directions. This difference in the c o m p l i a n c e is our m e a s u r e to structural anisotropy, the evolution of which is formulated as follows: Consider a plate which consists of periodic, equally distributed micro-cavities subjected to in-plane load. In view of the well known similarity between an incompressibe linear elastic material 6.. = 21.re .. tJ ~J

(15)a

and a linear viscoplastic material (known as Rayleigh analogy) s ij = 2rll~ ij

(15)b

one can, in principle, apply elastic solutions to similar problems in viscoplastic continua with minor changes (o'# --~ s 0, e# ~ ~ , /,t ~ r/(~), v ~ 1/2). In the considered problem, the applied remote traction can be replaced by the effective stress deviator according to: (P r p 2' 1;12) ")(Sll' s22' s12 )

(16)a

By using Voigh notation we rename the following; (S'll' s'22' s'12) '') (Sl' s2' s6)

(ElF E22' 2E12) -")(~1' E2' E6) (16)b and similarly

Hij ~ ---~Hij ,

Con ~ C i j

(16)c

By substituting (16)a in (5)c and decomposing the outcome to the real and imaginary parts, the resulted velocity components along the border of a cavity (for plane stress condition) become:

291 R

Ul= 3~(m) ~[ (Sl + s2 )(1 - m)cos 0 + ( S l - s2 )(c~ 2or cos 0 + sin 2or sin 0) + 2g6(sin 2or cos 0 - cos 2or sin 0) ]

U2=

R

(m) n[ (Sl + s2 ) (1 + m)sin 0 + (Sl - s2 )(sin 2or cos 0 - cos 2or sin 0) 31"1 + 2g6(cos 2or cos 0 + sin 2cx sin 0) ] (17)

The normal to the unit circle along which the velocity of (17) is described is ni =(cos& sin0)

(18)

The integration of (13), using (17) and (18), leads to the following; HljS- =j =

"~1 A~(Ulnl)dA

~:RA V3rl (m) {[ (1 - m) + cos 2c~]g 1 + [ ( 1 - m) - cos 2cz]~ 2 + [2 sin 2oqg"6} (19)a

H2jsj= ~=

(u2n2)dA

nRA

V3rl(m ) {[ (1 + m) - cos 2oqg 1 + [ (1 + m) + cos 2ctlg 2 - [2 sin 2oqg6 } (19)b

H6jsj = V2 ~12- (uln2 + u 2 n 1)dA =

~RA

V3rl(m ) {[2 sin 2oqg 1 - [2 sin 2oqg 2 - [2 cos 2oqs 6} (19)c

After rearrangement of (19) along with (14), the matrix form of the instanteneous compliances (measured along the loading directions '1' and '2') becomes:

292

1

Cll-

2

2rl(m){l+~f"

(a + b)

2

a*b

[1-m+cos(2tz)]} (20)a

2 . (a + b)

2

1 {1+ --~-t. a , b c 22 - 21-1(m) 1

2

C12- 2rl(m){l+3f

(a + b)

[l+m+cos(2tz)]} (20)b 2

a*b

[1-m-cos(2tx)]} (20)c

1 {1+ 2 f (a ..... + b) = [ 1 + m - cos(2a) ] } 21/(m) 3 a*b 2 _ 1 4 (a + b) [sin (2ct)] } C =C 16 61 2rl(m) {1 + ~ f . a* b

C2 ~ =

C

26

C 66

=C

1

1

4

2rl(m){l+3f

(a + b) 2

a*b

(20)e

2

4 (a+b)

62 - 2 r l ( m ) { l + ~ f

(20)d

a*b

[-sin(2ct)]} (20)f 2

[-~cos(2tx)]}

(2o)g

It is seen that there is a complete symmetry in C ij if, and only if, the voids are circular (namely 'm'=0). Otherwise, the compliance tensor C ij is non-symmetric. For example, it appears that for any non zero value of 'm', it is seen that C12 ~ C21" A possible explanation is discussed in [9]. Most importantly, the anisotropy in the structure response is manifested in (20)a and (20)b by noting that for any ' m'~ 0 C ll ~ C 22 5.

(21)

DISCUSSION AND SUMMARY The configurational evolution of a single cavity during large deformation is simulated in

Fig. 3. The rotation of the cavity (beside its extension and expansion) is clearly manifested. These changes take place similarly throughout the cavitated material. As a result of these changes, a macroscopic anisotropy (formulated in the last paragraph) emerges. By bulge testing of thin sheets (see Fig.2) with large plastic deformation, we have observed the actual dimensional changes of the cavities. Their volume fraction 'f' (called 'porosity' ) is increased.

293 The analytical prediction of the changes in the porosity (given in Fig. 4a) is based on equation (5)b along with the solution of (5)c. In particular, we have recorded, as shown in Fig. 4b the gradual rotation of the two cavities (positioned initially at 43 and 48 degrees from the principal stress direction) towards their current position (8 and 12 degrees, respectively) after a certain creeping time (4500 seconds). The numerical simulation (the solid lines in Fig. 4) seems to follow (relatively) closely the observed rotation in spite of the underlying assumptions (e.g., the material is composed of a periodic array of cavities, there is a weak interaction between the cavities via the self consistent approximation, the matrix remains isotropic, etc.). For a particular case (the parameters of which match the bulge test), this evolution of anisotropy in the compliances C 11 and C22 is illustrated in Fig. 5. In general, as expected intuitively, the softening of the material is progressively increased during the creeping process due to the increase in the volume fraction of the cavities in the matrix. Due to the tendency of the cavities to orient their long axis towards the direction of the principal tensile stress, the softening seems to be more pronounced in the direction transverse to the principal loading direction, namely,

C 11 < C22

(22)

with quantitative values given in (20)a and (20)b.

o EXPERIMENT] ANALYSIS O,1

>.. I-r O IT O 13_

Jo oi.t

o

0.0

----L--l--I

0

o j

1

1000

1

D-""

o ~

Z

I

I

l

I

2000

,

l

l

l

1

3000

,

z

,

. .l . . 1 4000

i

1

l

l

!

i

l

j

5000

T I M E (s)

Fig. 4a. The evolution of the 'porosity' as measured in the bulge test (of Fig.2). The analysis is based on the expansion-rate of (5)b and the solution of (5)c.

294

50 I-

+

+ o ~ ~

o

~

- o --EXPE'RIMENT I : - L. ANALYSI..~~..__SI

--I 1

~-

4O "

o

o

30

:

o

2O

10

,

0

t

!

I

•

I

t

t

1000

,

l

,

t.

,

~

2000

t

,

,

3000

t

,

!

t

4000

!

,

,

!

,

5000

TIME (s)

Fig. 4b. The rotation of two elliptical cavities during bulge test ( shown in Fig. 2). The ratio of the two tensile loads which stretch the thin sheet depends on the aspect ratio of the elliptical profile of the die.

~d o

i~ O

C

1,5 84

2

22

~o

+

|.0

~'o

~o

z'o

'

T I M E (SEC XI00)

Fig. 5. The evolution of anisotropy in the compliances at a stretched sheet during the bulge test. The parameters for the simulation are those chosen for the bulge experiment of Fig. 4b.

295 ACKNOWLEDGEMENT This work was funded partially by the Naval Research Laboratory through the Office of Naval Research and the Center for Nondestructive Evaluation, GWC School of Engineering, The John Hopkins University. It was also partially funded by the Center of Robotics and Manufacturing Systems at the Faculty of Mechanical Engineering in the Technion, Haifa, ISRAEL. The author is highly indebted to Mr. Dan Harvey (NRL) for his help in carrying out the experiments. To Dr. Phillip Mast (NRL) for his role in setting up the experiments and his help with the Mathematica applications; to Prof. Robert E. Green Jr. (Center for NDE), Dr. Yapa Rajapakse (ONR), Dr. Robert Badaliance, Dr. John Michopoulos and Dr. Irvin Wolock (all of NRL) for their encouragement and fruitful discussions. REFERENCES

1. B. Budiansky, J.W. Hutchinson and S. Slutsky, "Void Growth and Collapse in Viscous Solids", Mechanics of Solids (ed. H.G. Hopkins and M.J. Sewell) Pergamon Press, (1982), 13. 2. G.J. Rodin and D.M. Parks, J. Mech. Phys. Solids. Vol. 36, No. 2 (1988), 237. 3. R. Becher, R.E. Smelser and O. Richmond. The Journal of Mechanics and Physics of

Solids, Vol. 37 (1989), 111. 4. J. Tirosh and A Miller, International Journal of Solids and Structures, Vol. 24, No. 6 (1988) 567. 5. J. Tirosh and A. Harvey, Proceedings of the 4th International Symposium on Plasticity and

its Applications. Baltimore, MD, July 19-23, (1993). 6. H. Horii and S. Nemat-Nasser, The Journal of Mechanics and Physics of Solids, Vol. 31 (1983) 155. 7. J. Tirosh, Proceedings of International Conference on Computational Methods for

Predicting Material Processing Defects, (Editor, M. Predeleanu, Elsivier Science Publishers) (1987) 331. 8. N.I. Muskhelishvili, "Some Basic Problems in Mathematical Theory of Elasticity", P. Noordhoff Ltd., Groningen, The Netherlands, 1963. 9. S. Vallipan, V. Murty and Z. Hohua, Eng. Frac. Mech., Vol. 35, No. 6 (1990) 1061.

This Page Intentionally Left Blank

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

297

Computer Aided Design of Optimised Forgings. S. Tichkiewitch Laboratoire Sols Solides Structures, I.N.P.G. / U.J.F. / URA C.N.R.S. 1511 Domaine Universitaire - BP 53X - 38()41 Grenoble Cedex - France

The design of a closed-die forging is a too diflicult problem in order to be let to a traditional designer. As it is not possible to always ask the forger to be present during the design activity, we propose here a computer aided design system in order to dress tooled parts in a self-acting mode or in an aided mode. This system plays tile part of an actor in integrated design context, as explain in a first chapter. The next chapter gives the models used lbr tile deformation modes in our system. The three last chapters arc concerned by tile dressing of parts, the third for revolution parts, tile fourth for plane sectit~ns and tile fifth for three dimensional parts. In conclusion, we give some perspectives lk~r such a system. 1. A C O N T E X T

OF INTEGRATED

DESIGN

The actual evolution of the company structures is based on the concurrent engineering notion [1-2], for which the Design office and the Method office are melt into an integrated design office. The cohabitation of the manufacturers and designers might allow the illSt to intervene as soon as possible into the definition of the forms of the part, using the rules of the art before it is too late. The goal is double, tc~ decrease tile time from the idea to market, aw~iding the go and return between services, and tt~ increase the final quality of the product simultaneously with the cost reduction [3]. However, in the most case, we cannot arrive to have sufficient exchanges between manufacturers and designers because they do not understand themselves. It is in such a context we decide to extract tile know-how of the forgers and to integrate it into traditional design tools. The design of a forged part is typically a case where parallelism must work and where integrated and concurrent design has to be competitive : 9first, the rough forged shape is absolutely necessary to the final tooled shape definition that has to be known in order t~ verify tile level t~l stresses during service, 9second, the laying out of the forged part is tecllnically to~ difficult to leave it to designer. The integration of this know-hc~w as an application module of the Euclid-IS | system (CADCAM system l'rt~m Matra Datavision), the C()PEST module, allows to be confront with the forgers. Starting witll a tr wished shape, the user has the ability to transform it into the minimal forged shape with a minimum of work. Wllen dressing a totaled part, C()PEST automatically adds specific forging features, or allows the user tt~ fix such and sucll attribute. These features concern the adding of machining allowances linked with the capability of the process, draught angles, ribs, fillet radii and flash.

298

Some of the corresponding attributes arc typically dependent on the final morphology of the part, or on the kn~)w-h()w of the manufacturers (value of the draught angles, depth and position of the ribs...), others attributes are more complex to control as comer radii. In fact the difficulty is coming from the radii, and with their links with the local deformation mode : upsetting or extrusion. This deformatic)n mode is itself linked to the global morphology of the part, to the following pressures when we close the dies and to the duration of these pressure existence. The pressure being the cc)nsequence of the corner values that we have to get, it is a loop system, and only iterative process allows to resolve these crossing links. Today, COPEST realises an analysis of the global morphology of the part and suggests the user the different deformation modes it determines in order to calculate the radii. 2. M O D E L S

FOR THE FLOWING

DEFORMATION

We have in a first time to introduce the notion of plasticity threshold. After we describe here the evolution of the shape of a billet during the upsetting test. Then we do the same for closed upsetting, and for extrusion. At this time, all the presented model are quasi static model, and with an isothermic evolution. 2.1 T h e p l a s t i c i t y

threshold The basic notion introduced intc) c)ur model is the notion of plasticity threshold, as it is defined into Chamouard's book [4]. This threshold, noted (Yo, is corresponding to the average equatorial pressure we have to carry on in order to obtain a flat cylinder with a H thickness and a D maximal diameter. It cmly gec)metrically depends of the K ratio defined by :

K = HID

(1)

Plasticity threshold is independent on the effective shape of the flat forged part, either this part has this cylindrical shape in the middle or completely being into a barrel form. This form of course has to be taken into account to value the volume of the matter we must introduce to obtain a part with such D diameter and H thickness. In other terms, the real shape has no influence on tim useful strength to do them. 8{){)

~-. 121)1~r l()(X),

Steel - I()()()~

6111) 0 ,=~

AU4G

- 450~

"~ 81111

oo

~" 4()1)

~., 6()1)

,.I:2 4==) o~=,(

~

9- 21)1 ()

.,..~

"~_.~

9

i

.

l

.

I

11,11 1),I 1),2 (),3

.

i

(),4

(a)

.

(),5

~

Chamouar~

4()() 21)1") ()

,,,

9

I

'

I

'

I

"

I

"

(b)

1),11 11,1 11,2 1),3 11,4 (),5

Figure 1 9plasticity threshold fo, steel at I()(I~I~ (a) and AU4G at 45()~ (b)

299

In counterpart, the plasticity threshold is dependent on the material and on the effective temperature used by its ~e elastic Mises threshold. ()ur actual formulation of the plasticity threshold is" O'o = (~e-[ 1, 44-1()-6.K-5 - 1,17.1()'4.K-4 + 4.1()-3 K -3 - 8,77-1()-2-K'2 + 1, 6 4 - 1 0 1 - K - 1 -9,()7 + 3 6 , 1 7 . K - 68,2.K 2 + 51,g9.K 3]

(2)

Tiffs formulation is an easy to use polynomial expression, with an asymptote for K upper than ().4. It is possible to find in [3] the origin for such a formulation, and also some values for the elasticity threshold Ge lbr different materials at different temperatures. The curves shown in Figure l.a and l.b respectively gives the Chamouard's experimental plasticity threshold and the threshold computed with our model (2) for steel at l()(X)~ and AU4G at 45()~ 2.2 T h e

upsetting

model

o f a billet b e t w e e n

two dies

2.2.1 Evolution of tile upsetting three Initial force allowing a cylindrical billet with Do diameter and HO height to be plastically deformed is deduced frorn the plasticity threshold by 9

Fo = O'O(Ho/Do).FI.Do2/4

{3)

For a starting billet with a rati~ K() upper than ().4, tile initial strain is made with the preservation of a cylinder part. The first step concerns the growing of the equatorial diameter, keeping constant tile Dt ct~ntact diameter with tile dies at the initial value Do. An increase of the force such as 9 F = F{} + AF

{4)

gives an evolution of the equatorial diameter in order to bring it to the diameter D, such as 9

(5)

F = ~0(0.4).1-1.D2/4 :

Pi

12.i/H

_...251-" -I ~II!M!!IIIlII!III!!!I!III!II!III!IIIIIII!IIIIlll!III~

D

d

D, 1

1l) ,..I

~i1111111111111111t1111111t111111t!111tlt!111t111t111111~ Figure 2. Characteristics for a billet during upsetting.

300

Height h0 is necessary to transform the contact diameter Dt to the equatorial diameter D, as shown into Figure 2. It is equal to ().2D, and this for any height of the cylindrical part. The billet with 2ho height is the thicker billet we can c>btain with the F fl+rce and the D diameter. We consider into our model that the curve which links the equatorial diameter D to the contact diameter Dt is a parabolic curve. So, twice ~I the volume generated by this curve is subtracted from the initial volume Vo in order to define the remaining w,+lume V of the cylinder part with D diameter. The increase of the Ii+rce F is possible with this scheme as long as the V volume exists. As the time of the V volume is cancelled, an evolution of the force always creates an increase of the equatorial diameter D. The cylinder part no longer exists. The height of the parabolic curve ho is now equal to HI2 and the correspr generate volume is equal to the half of the initial volume Vo. The equilibrium between the H height and the D diameter is got when the average pressure is equivalent to the plasticity threshc~ld, that is to say lkw the flwce : F = (Y0(H/D).FI.D2/4

(6)

During this second step, it is easiest to pilot the computational research starting from a decrease of the height H instead of starting with the increase of the force F as in the first step. 2.2.2 Dist!ibution of the pressure on the dies The applied force between the dies and the billet is in accordance with the average pressure (plasticity thresilold). The transmissit~n t~l this li~rce is nevertheless not done with a constant pressure on the contact area. As we ctmsidcr that the upsetting pressure gives an increase of the diameter of the billet, we can als~ consider as a dual effect that the internal stresses that resist to the increase of the diameter can result l'r~m a pressure in the same order as the retract effect, according to the Chamouard's theory [4]. So we consider a pressure distribution on the contact surfaces of the dies limited by the retracting curve determined with the diameter D and the height H. If i is the difference between the equatorial radius and the radius of a current point (see figure 2), the retract thrcsh~ld lt)r the p~int is given by Oo(2i/H)- For D/H upper than ().4, the curve of this threshold is a curve named a "'bell curve" as shown on figure 2. The relative position of the "bell curve" is given with the io value, the difference between the equatorial radius and the radius of the limited pressure pt~int. Upsetting pressure Pi in the position of the point with i index is so equal to the difference of the threshold in io and i. Pi = (YO(2io/H) - (YO(2i/tl)

(7)

The io value is defined in ~)rder t()obtain the upsetting force equal to the resulting force given by the integrati~m of the pressure sr evaluated. 2.2.3 Evolution of the cc~ntact diameter At the beginning of the upsetting, tile diameter of tile contact area between the billet and the die is equal to tile initial diameter DII. The ev~luti~n ~1"this diameter has to take into account two different factors 9the pressure distributi~n induced by the upsetting and the friction between die and billet.

301

We have just seen that it is possible t~)determine at any time the pressure distribution at the contact level. This distribution defines a maximal diameter Dp for the pressure such as 9 Dp = D - 2.io

(8)

It is clear that to effectively transmit a pressure, we must be in touch. So, in a first approximation, we consider in our model that the ccmtact diameter Dt remains identical to initial diameter D{I as so long as the latter is greater or equal to the diameter Dp. After this bound, the c{mtact diameter increases in ~rder to stay equal tc, the pressure diameter. Friction is introduced in our rn~del taking int~ accc~unt a Ifictit~n factor Cf which determines a fictive pressure diameter Dj-such as : Dr:

D - 2.i{). CI

(9)

The evolution of the c~ntact diameter is n~'~ l~ngcr done taking as reference the pressure diameter Dp but this fictive pressure diamctcr DI. A friction factor Cf equal to (I allows to keep a perfectly cylindrical billet with the Dt diameter always equal to D diameter. A friction factor equal to 1 keeps the pressure diamctcr Dp. Between II and 1, the frictic~n factor favours the increase ~,I the cc~ntact diameter, when greater than 1, it delays the effective gliding. This is in cc~rll'r with the ~hscrvatir we can d~ with a billet r which we have marked the external diameter ~}1"the plane surface in ct~ntact. The same crushed billet shows a greater external diameter tm this plane surface while the marked diameter is near from its initial value, as if the border ~I" the part unwinds on the dies tt~ adapt iLself. 2.2.4 Experimental c~,rnpmis~n 4,{}()e+6 Cf=0,7 3,{}()e+6

c - 14 M Pa c - 1(1MPa iment

2,{){}e+6

l,(){}e+6

{ },{ }{ }e+{ }

{},2

{},4

{},6

{},~

1,{}

1,2

1,4

1,6

1,8

2,{}

2,2

H/D Figure 3. Compalison between m~,del-expe~iment on a basic titanium billet from S N E C M A .

302

This upsetting rnc~del has bccn c~mparcd with a test done by S N E C M A (a French engine builder for aeronautic), with a billet on basic titanium material [5]. The dimensions of the initial billet are Do = 2()3 mm and Ho = 43() mm. The temperature at the beginning of the upsetting is 1()4()~ The elastic threshold of this material can be evaluated with the initial force we must reach in order to start the plastic deformation. This elastic threshold is 14 MPa. The figure 3 shows the curves of the force in function of the H/D ratio of the billet. The bold curve is the experimental curve. The upper thin curve is the result of our model for an 14 MPa elastic threshold, and a ().7 friction factor. This factor has not a big influence in our test, the variation of the force being only to 5% for a variation of the friction factor from ().25 to 2, as shown figure 4. 34()()-

7,

az

335()331)().

,.....

325(). 32{)()

9

I

131

"

1),5

I

"

I

1,11

"

I

1,5

"

2,{1

2,5

Frictio,.lactor CI Figure 4" Variation of tile final lt)rce versus tile friction factor The compariscm c~l"tile measured curve to tile proposed curve shows a growing difference with the decrease of tile H/D ratic~. In fact, tile measured curve can be enclosed between two curves (,14 and 1() MPa). This variation is much sensible. l(X)'Steel XC3X 9 Cham{mard's Illreshl~d versus tmnperature linear sm~c~lhing

v

8().

eb e = 223,1)6 - (), 17167.tin ~

40-

()

. 7()()

. . 81)()

.

. . 91)11

.

. i(X)()

1

SNECMA

~

1 11)()

12(X)

1 :~()()

temperatttre ~ Figure 5 9 Experimental variation ~I the elasticity threshold function of the temperature for steel and extrapolation for the S N E C M A material.

303 Explanation can be Ibund if we exa,nine the evolution of the elastic threshold function of the temperature of the material. Considering the same evolution of the threshold for basic titanium material as the evolution of the threshold liar steel, that we can obtain from Chamouard's experiments [3] and given figure 5, tile threslx~ld at 1() MPa is in accordance with a temperature of 1{)62~ This temperature can be effectively considered if we take into account that a part of the mechanical plastic strain work is transformed into heat. We are thinking about this thermal evolution in order to be able to take it int~ acc~mnt in our model. 2.3 The closed upsetting model of a billet into closed dies 2.3.1 Upsetting mode.1 o f a billet into a floating fine In a first case, let us imagine the billet upset between two dies as before. Outside the billet, we place a floating ring with the inner diameter greater than the diameter of the billet. The first upsetting is exactly as previously defined, as long as the external diameter D of the billet becomes equal to tile inner diameter of tile ring. At this time, the ring is centred by the billet. In these conditions, an increase of tile Ii~rce on the dies is traduced by an increase of the height of the cylinder part of the billet, D remaining constant. This upsetting is done without sliding between tile wall of tile ring and tile billet, st) without friction. Tile previous equations are always coherent because 9 9 the increase of the force gives all increase of the average pressure, so of the plasticity threshold. With a constant diameter, we must decrease the equivalent height of the cylinder in order to follow this evolution, s~ t~ decrease the height ho of tile parabolic .joining curve. 9 the increase of the li)rcc is done with tile increase of tile fictive diameter and so on the contact diameter, 9tile decrease of tile n~m cylindrical v{Hume on tile billet due to the decrease of the height h0 is more important than tile increase of the same volume due to tile increase of the contact diameter D(,. The cylindrical volume is increasing and its height too, in the same time as the global height is decreasing. In this closed upsetting, tile average pressure PM induces on the sides a pressure PR "

PR(h) = PM- O'0(2h/D) where h ranging between R and HI2. This pressure represent the reaction of the wall on the billet and its repartition is shown Figure 6. Tile upsetting pressure PM is directly related to the value of the height h(I. Indeed at the point A, the Chamouard's hypotheses implies that PR becomes null. Therefore" PM = O'0(2h0/D)

( 11 )

So we get a relation which links tile value of the radius to tile upsetting pressure. The upsetting problem is then completely defined. We have two relations and three unknowns (PM, PR, h()). By fixing one of thern, we determine tile others. In the following we will use the mean value of the pressure repartition PR distributed on tile inner side of the dies, so that : !112

PR,n,,y.H = 2.[PM .(HI2 -hI})! - 2.

j" O'll(h/D).dh

(12)

304

i

p

M

h Pp,(h) D

IIII!111!111i!11!111i111!111i 111!111 III!111!111!111!11li111!111i111i11 Figure 6. Pressure l'epartitioll for upsetting c~la billet into a floating ring 2.3.2 Closed-upsetting model of a billet intr a c~ntainer Consider now that tile ring is n~t ll~ating at the beginning of the upsetting, but that it is jointly and severally liable with tile Iciwer die. The external diaineter of tile upper die is exactly the same than the inner dialneter of tile ling, and can move down into the fixed lower die. When begging tile upsetting, we, always have cc~ntact at a time between the billet and the die, the lower one. At this time, each upsetting ~I the billet as to be done with a global translation of the material into the die. So, this upsetting is done with sliding between the wall of the die and the billet, with fliction. The fiiction is a true fiictioll, in tile sense of Coulomb, and we must take in order to take it into account a friction coefficient.f that is not the same as our friction factor Cf. PMh

htlh

h ()b

Ill Iill IIIII,I lli,II/H,I III,1I111111, 111, I1,111,111,111,111,111,111,111,111 111, I I I I I

PMb Figure 7. Pressure repartitic~ll for classed upsetting of a billet

305

By integration of the friction force along the wall of the lower die, we must have an average pressure growing from the lower face of the billet to the upper face, and the retract pressure is becoming as shown into figure 7. As the pressure is not the same at the lower face than at the upper face, we find different values for the height of the .joining parabolic curves (hob and hob) and for the contact diameters. The upsetting do not allow to have a symmetrical forged part. 2.3.3 Experimeqta.1 comparison 4000

~.

.

.

.

.

.

3000

-.---

c / = i.s

- - - - - - - Cl= o.5

ooo

~"

1000

,

9

l

"

l

'

l

!

'

,, I

9

V% 4 5 6 1 2 3 Figure 8. Evolution of the upsetting force in concordance with the friction factor

,

4000

9

9

H/D

= 2

H/D

= 1.5

H/D = 1

3000

~9 2 0 0 0 1 000

,

I

0

i

I

1

v~

2

Figure 9. Evolution of the upsetting force in concordance the initial ratio H/D We simulate tile experiment d~ne by Ihhadt~de and Dean [6] on the filling of the corners in precision stamping. The experiment consists of tile upsetting of a billet with initial diameter of 5(1 rnm, in commercially pure aluminium and at room temperature. This upsetting is made into a

container with a diameter 01' 63.55 mm. TWOi y p c ~01' I'iiction has heen tested. corresponding to good luhrication and a dry I'riciion. During cxpciiment, the authors have regularly record the upsetting force and have each time inca,sLircd the percentage of the unfilled volume V% in accordance to the volume of the hillct. On Figure 8 is shown the results of the force evolution for two different friction factors. and for the same initial ratio H/D. It is true that the friction factor has an inlluence on the evolution of the parabolic joining curve near the dies and SO on the evolution on the unfilled voluine. On Figure 9,we have the same test for the samc friction factor hut for different ratios H/D. This parameter has less influence on the result, hecause the height of' the cylindrical part of the hillet d o n o t affect the direct upsetting force. Only the friction on the wall o f the dies is here sensitive, hecause the fiiction surface is growing with the initial height. 2.4 The extrusion model o f a I)illet we consider now an extrusion cavity in an axial position, with an entry diameter Dth smaller than the initial Do diameter o f the hillst. The rcst ot' the dies is considered as closed upsetting dies with a D diameter greater thaii the Do diameter.

I n a first time, when decreasing ilic height H of the hillet, the matter in front of the extrusion cavity IS conSidcrcd as driven hy tlic dic. ;I.\ h w n i n l'igure 10. A spherical portion is created with a h~charactcristicsuch as :

The constant k is dependant of the matciial and of the temperature. This evolution is true as long as the ad angle is greater than aI.

Figure I 0 : first step o f extrusion

307 W h e n or d becomes equal tc~ o:t, the real extrusion begins. Each translation of a slice of metal into the extrusion cavity has to be done with a retract pressure in order to reduce the mean diameter of the slice, taking into account the draught angle of the cavity. As the strain is a plastic strain, the retract pressure has to be equal to O'()(d/hf), with d the diameter at the top of the extruded part. This pressure is equal to O'o{~.4} at the beginning of the extrusion. The extrusion is done with sliding on the die, so we also have to integrate the friction force along the extrusion part which inw~lves an evolution of the retract pressure as given into figure 11. The equilibrium of the force in the extruded part is realised with an entry pressure P f w h i c h participates in the upsetting of the billet and so is equal to the PM upsetting pressure.

d

-1

h

i

'~ .... ",.%'?:

I

P[

Figure 1 1 : mid-tcnn t~l the cxtrusitm

Figure 12 : ccmtact at the top of the cavity

When the extruded height is equal tt~ the extrusion cavity, contact is made at the top of the cavity. The contact area is grt~wing, such as is decreasing the free radius r (figure 12). The top pressure is equivalent tt~ a upsetting pressure lt~r a cylinder with Dlh diameter and 2.r height. 2.5 T o w a r d a fast simulation of forging d e f o r m a t i o n

system

With the presented mt~dels, we have build a simulation system for flwging allowing to the user help in order to quickly obtain the retest difficult areas of filling. This system is actually realised on a Silicon Graphics Indigo 41)l){I workstation, in C language. A view of the graphic interface of the system is given on figure 13. This interface has been built in order to truly ease the use of this tool and allt~w the user to test a lot of different hypothesis. A connection with the next C()PEST system is l~resee in order to test the proposed rough shape. 3. A S E L F - A C T I N G

THREE-DIMENSIONAL

DRESSING

Ill [7], one ~1 tile autht~rs sh~wcd the engaged w~rks l'~r a self-acting dressing of revolution parts. This w~rk has been enlarged in [~], with the ability to dress any plane section, this section being symmcuical ~r n~t. The li~ll~wing parts arc the result of C()PEST. The span

308

'

>'."~

i

Linboralolr~

I

LLiJ,,~

....

S-----IMULATIONgECRASEMENT

"I"B,r

_

I',~TF:ICE

LOPIN

F~SSE

l~il

i ~,.'"" r, I]~ ::ii::i ::i~ii::iii ::ii

(~ Ar.,cesso,res )

S'g~Mpa

1

Courbe de termeture

"

! ]~u,]

!

I_!~.o o

I

I',.'llrll

i

//"

/

P~,rllmell'~l

4

)

:".,eI a:).- [~' ~ela,-t- H.' DelaP- F.' _r~-~ . FQ--~-I

) ,

r,.t8:,,J

FF',OTTEMErgr

.

Ill................................................ Q

l

"~nne...~

~o11- $ o l l d r ~ - ~ I r u ~ ; l u r : ~

,~-

P....

;)

._

....

r~U._, ,. . . . . i,,,~.~ i

,[ 1:14~:I~S I

(Sracl ~

,v

.*.

. i,

~ tie41a H 9 [iN

.L:,

__. ~

I

Figure 13 9Flattening simulation environment.

figure 14.a is an example of revolution part whose rough and tooled cross-sections are given on figure 14.b. The clamp shown on ligurc 15 gives an example of three-dimensional dressed part.

Figure 14. Starnped span (a) with their rough and tooled cross-sections (b).

309

.. _ - ' *

Figure 15. Rough clamp, result of the C()PEST system. REFERENCES

1. A. Kusiak, Concurrent Engineeling, Wiley-Interscience, ISBN ()-471-55492-8 2. S. Tichkiewitch, Mod61isation pour la conception et processus d'intdgration, Int. Symp. : design in 2()()() and beyond, 24-27 May 1992, CNES, Strasbourg, 381-389 3. S. Tichkiewitch, J.F. Boujut, Designing to avoid potential defects, J. Mat. Proc. Tech., 32 (1992), 399-4()6 4. A. Chamouard, Estampage et forge, Dunod, Paris, vol 1 to 3 (1964-1966-1970) 5. J.F.Chevet, Action concertde de recherche "Moddlisation du forgeage", Groupe de travail n~ 3 nov 1992, CETIM, Saint-Etienne, France 6 A.O.A. Ibhadode, T.A. Dean, "Corner filling charactelistics in precision forging", Int. J. Machn. Tools. Manufact, V~I 2~, N~ pp 1()3-122, 1 9 ~ 7 S. Tichkiewitch, We have tt~ think stamping during design to become competitive, Proc. of XIIIth International Forging Congress, New Delhi, 23-27 Jan 199(), Vol 2-pp V.1-27 8 J.F. Boujut, S. Tichkiewitch, A step tc~ward automatic dressing of a three-dimensional stamped part, J. Mat. Proc. Tech., 34 (1992), 163-171

This Page Intentionally Left Blank

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

311

Defects in thermally sprayed and vapour deposited thick and thin hard-wearing coatings M.S.J. Hashmi School of Mechanical and Manufacturing Engineering Dublin City University, IRELAND

1. INTRODUCTION Surface engineering is the science and technology of changing or modification of the properties of a surface without influencing the bulk property of the component or product to suit a particular application. There are a large number of surface engineering techniques which are in use. Some are very old methods but still very effective and others are based on recent technological advancements which are still being perfected. Surface engineering techniques may be broadly classified into two main types. The first is whereby a layer of additional material is deposited on the original surface, the property of this additional layer being different and in certain ways superior to those of the substrate. The adhesion between this additional layer and the substrate may vary considerably depending on the type of application. Galvanising, electro-plating, painting, thick-and thinfilm coating etc are the processes which fall into this category. The second is whereby a layer of the actual substrate is modified to obtain the required surface characteristics of the component. Heat treatment, Nitriding, ion-implantation, etc. are examples of processes which belong to this category. Irrespective of the type of surface-engineering process, its success is dependent on a multitude of parameters such as the substrate type and shape, the surface preparation, the processes parameters and so on. For some of the processes even minor deviation from the ideal conditions of any of the important parameters would give rise to imperfect or defective results. Recent development, however, have resulted in reasonably reliable surface-coating techniques. The principal advantage of the coating is the ability to tailor coating properties to suit the application. A particular material may be deposited to form a hard or soft and porous or dense coating. This versatility presents difficulties when the actual properties of the applied coating are unknown. Proper characterisation techniques for the evaluation of coating properties and the response of these properties in actual application is not yet well established. Particularly, non-destructive testing for evaluating coatings still poses difficulties. In the absence of more appropriate testing methods, only qualitative test data and reasonable theoretical methods are available to provide qualitative information on a coating. In order to select and decide on a coating for a given application, knowledge of the effect of different parameters on the coating qualities is a prerequisite. The potential of the coating can be further improved by better understanding of its process parameters and the associated defects and, more importantly, the methods of avoiding these defects.

312 The conventional way of applying a coating is by wet processes, in which coating is applied in the form of a liquid or a solution. The more advanced techniques of applying a coating are dominated by the dry process, which means that the coating is deposited on to a substrate in the vapour (gaseous) or molten/semi-molten state. The term "deposition" signifies diffusion and overlay. Diffused coatings are applied by complete inter-diffusion of material applied to the substrate into the bulk of the substrate material (Figure 1). Examples of these are the diffusion of oxygen into metals to form various sub-oxide and oxide layers. An overlay coating is a add-on to the surface of the part. Depending upon the process parameters, an inter-diffusion layer between the substrate and the overlay coating may or may not be present. The physical dimension of the thickness of thick-and thin-film is not quite distinct. A thickness of 1 micron is often accepted as the boundary between thick and thin film [1]. A recent view point is that a film can be considered to be thick or thin depending on the application and discipline. This definition has been advanced with the idea that a coating used for surface properties is a thin film, whereas if it is used for bulk properties it is a thick film. 2. COATING/SUBSTRATE SYSTEM The performance of the coating applied to surface engineer a component does not depend solely upon the type of coating, rather it is very much influenced by the coating/substrate combination. The first consideration is that the substrate must be able to support the coating without yielding beyond the coating strain to failure. As such, the coating/substrate complex system acts together to deliver the desired performance. Figure 2 illustrates some of the inter-related properties of the complex system which may be controlled within specified limits to ensure that the overall engineering requirements of the system are fulfilled. There are a large number of process parameters, such as gas flow rates, gas composition, pressure, environment, substrate temperature and geometry, etc. which determine the quality of the coating. The application area is another factor affecting the process variables. Therefore, understanding the relationship between these process variables, to have an optimum coating/substrate composite system towards a definite application, is the key to assess the defects of coating. Figure 3 shows the relationship of coating process and application.

Figure 1.

Schematic illustration of a diffused coating and an overlay coating.

313

Figure 2. Inter-related properties of the coating/substrate system.

Figure 3. The inter-relationship of coating, substrate, process and application

314 3. COATING PROCESS A coating process can be divided into three steps: (a) synthesis or creation of depositing species; (b) transport of the species; and (c) accumulation or growth of the coating on the substrate. These steps can proceed completely separately from each other or can take place simultaneously, depending upon the process under consideration. The synthesis or creation and transport of the depositing species can be done in three distinct phases viz. Vapour (gaseous) phase, Liquid phase and molten or semi-molten phase. In vapour deposition processes the depositing species are created and transported in the vapour phase and deposited on the substrate atom by atom. In the thermal spray deposition process the synthesis or creation and transportation of the depositing species are in the molten or semi-molten state and deposited on the substrate particle by particle. Here we shall confine our discussion within vapour deposition and the thermal spray deposition process. A brief description of the processes are described below.

3.1 Vapour-Deposition Process In this process a vapour is generated by boiling or subliming a source material, the vapour then being transported from the source to the substrate where it condenses to a solid film. The vapour-deposition process has the ability to produce rather thin coatings with high purity, high adhesion, and unusual microstructure at high deposition rates. Most non-gassing substrate materials which can withstand the deposition temperature can be coated by this process. Coatings deposited by this method generally do not require post finishing. There are three classes of vapour deposition techniques: physical vapour deposition (PVD); chemical vapour deposition (CVD); and physical chemical deposition (P-CVD) [2-4]. 3.1.1. Physical Vapour Deposition (PVD) Physical vapour deposition is used to apply coatings by condensation of vapours in high v a c u u m (10 -6 to 10 Pa) atomistically at the substrate surface. This technology is very versatile, enabling one to deposit virtually every type of inorganic material: metals, alloys, compounds and mixtures there of as well as some organic materials. The deposition rates can be varied from 10 to 750,000 per minute, whilst the thickness of the deposits can vary from angstroms to millimetres. There are three physical deposition processes, namely evaporation, ion plating and sputtering. 3.1.1.1 Evaporation PVD Process In the evaporation process, vapour is produced from a material located in a source which is heated by direct resistance, radiation, eddy current, electron beam, leaser beam or an arc discharge. The process is usually carried out in vacuum so that the evaporated atoms undergo an essentially collisionless line-of-sight transport prior to condensation on the substrate. The substrate is usually a good potential i.e. not biased. Most pure metals, many alloys and compound that do not undergo dissociation can be directly evaporated in vacuum. In the more general sense, when a compound is evaporated or sputtered, the material is not transformed to the vapour state as a compound state but as fragments thereof. The fragments have to recombine on the substrate to reconstitute the compound. Satisfactory methods of preparing alloys and compounds with proper stoichiometric coatings include reactive evaporation, multiple-source evaporation, and flash evaporation. A plasma is sometimes included in the reactive evaporation to enhance the reaction between the reactants and to

315 cause the generation of ions and energetic neutrals: this process is known as activated reactive evaporation. The source material is normally in the form of powder, wire, or rod. Typically coating thickness ranges from 0.1 to 100 micron. Most substrate materials can be coated by this process.

3.1.1.2 Ion Plating The Ion plating process can be classifed into two broad categories: glow-discharge (plasma) ion plating performed in low vacuum (5 X 101 to 10 Pa) and ion beam ion plating performed in high vacuum (10 .5 to 10-2 Pa). In glow discharge ion plating processes, the material to be deposited is evaporated in a manner similar to ordinary evaporation, but it passes through a gaseous glow discharge on its way to the substrate, thus ionizing the evaporated atoms in the plasma. Condensation of the vapour takes place under the action of ions from either a carrier gas or the vapour itself. The impingement of ions on the substrate surface or the deposited coating is combined with a transfer of energy and momentum. The glow discharge is produced by biasing the substrate to a high negative potential and admitting an inert gas, usually argon, at a pressure into the chamber. The glow-discharge techniques can be classified based on the deposition system configuration, mode of production of vapour species, and method of enhancement of ionization of vapour species. In ion beam ion plating processes, the ion bombardment source is an external ionization source (gun). These guns utilize various ion beams; single or cluster ion beams. Ion beams can be of inert gas ions or ionized species of coating material and sputtering. By using a beam of desired ionized species, alloys or compounds can be formed. Ion beam plating is performed at high vacuum. A wide variety of metallic and non-metallic coatings have been applied onto metallic and non-metallic substrate by ion plating processes. These coatings are applied by various ionplating processes under various conditions of substrate biasing, evaporant species source and degree of ionization.

3.1.1.3 Sputtering Sputtering is a process whereby the coating material is dislodged and ejected from the solid surface due to the momentum exchange associated with surface bombardment by energetic particles. The sputtered material is ejected primarily in atomic form for the source of the coating material, called the target. The substrate is positioned in front of the target so as to intercept the flux of sputtered atoms. The atoms of coating material deposited on the substrate gives rise to a coating. This method is generally used for sputtering metals: an rf potential must be applied to the target when sputtering non-conducting materials. Since the coating material is passed into the vapour phase by a mechanical rather than a chemical or thermal process, virtually any material is a candidate for coating. The sputtering process can be classified based on the means of producing high energy ions: glow-discharge and ion beams from an ion source external to the sputtering system.

3.1.2 Chemical Vapour Deposition (CVD) Chemical vapour deposition (CVD) is a process in which a volatile component of coating material is thermally decomposed, or chemically reacts with other gases or vapours, to produce a non-volatile solid that deposits atomistically on a suitably placed hot substrate surface. The CVD reactions generally take place in the temperature range of 150~ to 2200~ at a pressure of 60 Pa to atmospheric pressure. The coating source material can be initially in solid, liquid, or vapour form and then converted to vapour phase by heating. The

316 coating quality depends on the substrate cleanliness, the compatibility of the coating and the substrate materials, and the thermodynamics and kinetics of the reaction involved. Since a large variety of chemical reactions are available, CVD is a versatile and flexible technique to produce a wide range of metallic and non-metallic coatings on any non-gassing substrate. Sometimes CVD processes are carried out at reduced pressure to make the process more thermodynamically favourable, this process being called Low-pressure CVD (LPCVD). With the decrease of pressure, the diffusivity of the gas increases, which increases the gas phase transfer of the reactants. The increased mean free path of the reactant gas molecules results in high throughput. In conventional or low-pressure CVD processes, sometimes it is required to limit the growth to a very small portion of the part, a laster beam then being used to heat the limited area of the substrate. This process is known as Laser-Induced CVD (LCVD).

3.1.3 Physical-Chemical Vapour Deposition Physical-Chemical vapour deposition (P-CVD) processes are hybrid processes which use glow discharge to activate CVD processes. These are broadly referred to as plasmaenhanced CVD (PECVD) or plasma-assisted CVD (PACVD) processes. These processes consist of the techniques of forming solid deposits by initiating chemical reactions in a gas with an electric discharge. Instead of requiring thermal energy as in CVD, the energetic electrons in the plasma can activate almost any chemical reaction. The reactions proceed at high reaction rate in a system at low processing temperature. Practically any gas or vapour including polymers can be used as a precursor material. Another method, known as the reactive pulsed plasma (RPP) deposition process, uses a high energy pulsed plasma to coat the substrate at room temperature.

3.2 Thermal Spray deposition process In this process finely divided metallic or non-metallic coating materials are sprayed in the molten or semi-molten state on a substrate without penetrating the substrate to form a spray deposit [5-7]. The spraying apparatus is designed to heat and spray the particles in the direction of the substrate and to condition the particles so that when they strike the substrate they are prone to adhere. Therefore the device must produce a hot zone containing so much enthalpy that the spraying particles melt in the hot zone. Moreover in the hot zone a gas flow must exist exerting such a drag that the particles arrive at the substrate having the required momentum. Depending on the heat source, thermal spraying can be divided into many types. 3.2.1 Plasma thermal spraying In plasma spraying the thermal energy of an electric arc and associated plasma is utilised for melting and projecting the coating materials. Coating materials in the form of powder is introduced into the plasma, melted and propelled onto the workpiece. The heat content, temperature and the velocity of the plasma jet are controlled by the nozzle type, the arc current, the mixture ratio of gases and the gas flow rate. Plasma spraying can be used to spray all materials on almost all substrates. Plasma spraying can be carried out in vacuum or in controlled atmosphere to maintain the desired coating qualities. In this process, the plasma gun and the workpiece are enclosed in a vacuum chamber. All operations (pretreatment, spraying, post heat-treatment, if required) are carried out in an inert low pressure (typically 7 kp or 50 torr) atmosphere. This development permits the application of a

317 complex coating system with proper stoichiometry and structure, high coating adhesion and low porosity. 3.2.2 Flame thermal spraying In the flame-spraying process, fine powder or wire is carried in a gas stream and is passed through an intense combustion flame, where it becomes molten. The gas stream, expanding rapidly because of the heating, then sprays the molten powder onto the substrate, where it solidifies. The flame-spraying process is widely used for corrosion resistance and the reclamation of worn or out-of-tolerance parts. To improve the coating qualities of the flamespraying process, a new spraying process was developed called the High Velocity Oxy-Fuel (HVOF) thermal spraying process [8-10]. There are tow types of HVOF process, the continuous-combustion HVOF process and the pulsed-combustion HVOF process. The continuous-combustion process uses an internal combustion (rocket) jet to generate superhypersonic gas velocity. During operation, the fuel and oxygen are mixed by means of coaxial jets and guided to the combustion zone where a pilot flame or external igniter initiates the "gas mixture" or "combustible mixture". The combustion products are allowed to expand in the nozzle where they are accelerated. The powder fed with nitrogen into the centre of the flame is heated while passing through the nozzle and is further accelerated. In the pulsed-combustion thermal spraying process a timed spark detonates the mixture of oxygen and fuel to produce heat and pressure waves which instantly heat the particles to a molten state and hurl them at supersonic velocity from the gun to the substrate surface. After the powder has exited the barrel, a pulse of Nitrogen gas purges the barrel. The cycle is repeated about 4 to 8 times per second. 3.2.3 Electric arc thermal spraying In electric arc thermal spraying, an electric arc is struck between two converging wires close to their intersection point. The high temperature arc melts the wire electrodes, which are formed into high-velocity molten particles by an atomizing gas flow. The wires are continuously fed to balance the loss. The molten particles are then deposited onto a substrate. By this process only relatively ductile and electrically conductive materials, which can be drawn in the wire form, can be deposited.

4. VARIATION OF THE PHYSICAL PROPERTIES OF THE COATING The physical properties of the coating vary widely, depending on the type of process and also depending on the process parameters. The large number of variables involved have limited the number of fundamental investigations of the process property relationship. The microstructure of the coating dictates many of the physical properties of the coating. Some of the important properties of the coating and their probable variation and probable defects are highlighted below. 4.1 Bonding and associated defects Adhesion and adhesive strength are macroscopic properties that depend on the bonding within the deposited particles or atoms and bonding across the interfacial region and also on the local stresses generated during deposition. The bonding and local stresses are determined by the environment, the chemical and thermal properties of the coating and the substrate materials, coating morphology, mechanical property, defected morphology of the interfacial

318 region and external stresses. In addition, adhesion failure may be time dependent. Defects of bonding depend on the processes and their mode of growth of the coating from the source material. These aspects are outlined below.

4.1.1 Bonding in Thermal spraying coating In thermal spraying coating, the bonding strength depends on the spray process, the associated fuel, raw material characteristics and the kinetics. The raw material characteristics are especially influential when the process uses powder materials. Particle shape, size, density and oxide content all contribute to the quality of the end product. 4.1.1.1 Defects arising during particle deposition in thermal spraying The basic criteria of thermal spraying is that the particle should melt completely without excessive vaporisation and should remain soft until it impinges onto the substrate. Vaporisation produces a sponge-like coating structure. A smaller particle size will ensure more complete melting but the velocity gradient restricts the particle to enter into the hot zone. Therefore some of the particles do not melt sufficiently and are transported on the periphery and deposited. Further, these particle cool more rapidly and therefore solidify before impact and give rise to defective coatings. The formation of oxide in the film is also controlled by the rate of cooling, the size and shape of the powder particle and the velocity of the particle and its location within the heat zone. During the spraying process, particles are heated and transported with different velocities and are flattened on the substrate or on previously deposited layers. The substrate surface or the previously deposited layer influences the formation of a metastable phase in the fluid particle, resulting from rapid solidification of the molten particle. Because of the high kinetic energy of the molten particles, considerable liquid flow and radial sliding occur upon particle impact. This spreading is, to some extent, dependent on the substrate surface topography and on the surface tension of the liquid particle. In Figure 4 taken from ref. 5 a schematic of the planner morphology of a solidifying particle is shown. At the central core zone where the particle first comes into contact with the substrate, heat is extracted through the substrate and the solid-liquid interface moves away from the substrate and a thin extended area with a peripheral rim is formed. This thin region may not be in full contact with the substrate. The proper splashing or wetting of the particle depends on the proper stand-off distance (Figure 5a). When the stand-off distance is too small the particle cannot become completely molten and the solid centre core can rebound from the surface, resulting in a void (Figure 5b and 5c). If the standoff distance is too large the molten particle may solidify before impact and may rebound or stock on the surface, resulting in less deposition efficiency or higher porosity (Figure 5d).

Figure 4. Schematic of the planar morphology of a solidified particle.

319

Figure 5. Splash forms of a sprayed metallic particle at different standoff distance. 4.1.1.2 Defects during coating formation Thespray pattern of any thermal spraying apparatus is conical with the spray concentrated in the central zone (Figure 6) and more sparse in the outer periphery. A circular or oval central zone exists, within which the densest and thickest coating is deposited. The particles in the periphery are widely spaced and adhere poorly to the substrate and are highly porous. The spray deposit density ratio is largely dependent on the particles within the peripheral zone. The final sprayed coating contains a mixture of material from each zone because the gun and the substrate are moved relative to each other during coating application. The extent of flattening depends on factors such as the degree of melting, the viscosity of the liquid and

320 wetting of the surface. The coating may contain voids arising from out-glassing, shrinkage or topographical effects (e.g. shadowing) and if it is a metal processed in air, probably due to the inclusion of oxide.

4.1.2 Bonding in vapour deposition coating In the vapour-deposition process where the coating is deposited atomistically, the nature and condition of the substrate surface determine many of the factors which control nucleation, interface formation and film growth. These in turn control the interfacial properties and defects. 4.1.2.1 Defect arising during nucleation When atoms impinge on a surface, they lose energy to the surface and finally condense by forming stable nuclei. A strong surface atom interaction will give a high density of nuclei and a weak interaction will result in widely spaced nuclei. It has been proposed the nuclei density and the nuclei growth mode determine the effective interfacial contact area and the development of voids in the interfacial region (Figure 7). Nuclei density and orientation formed during deposition can be affected by ion bombardment, electric fields, gaseous environment, contaminant layers, surface impurities, surface defects and deposition techniques. In addition to the effective contact area, the mode of growth of the nuclei will determine the defect morphology in the interracial region and the amount of diffusion and reaction between the depositing atoms and the substrate material. 4.1.2.2 Defect arising during interface formation Interfaces may be classified into different types viz mechanical, monolayer-to-monolayer (abrupt), compound, diffusion or pseudo-diffusion and combinations of these types. The formation of different types of interface depends on the substrate surface morphology, contamination, chemical interactions, the energy available during interface formation and the nucleation behaviour of the depositing atoms. The mechanical interface is characterised by mechanical interlocking and the strength of this interface depends on the mechanical properties of the materials and the surface roughness. The monolayer-to-monolayer type interface is characterised by an abrupt change of the coating material to the substrate material. This type of interface may be formed where there is no diffusion, lack of solubility between materials, little reaction energy available, or the presence of contaminant layers. Figure 8 shows a picture of the coating separated from the substrate due to a contaminant layer on the substrate. A compound interface may be formed either by an intermetallic compound or some other chemical compound such as an oxide. In this type of interface, there may be abrupt physical and chemical discontinuities associated with the abrupt phase boundaries. Often during compound dormation there is segregation of impurities at the phase boundaries and stress are generated due to lattice mismatching. Porosity may develop in the interracial region. In the diffusion type of interface there is gradual change in composition, intrinsic stress and lattice parameters across the interface region. Due to the diffusion rates, Kirkendell porosity may be formed. The diffusion process may be important in the defect structure of the interface region. A combination of several types of interracial regions is possible by controlling the environment or film composition during the initial stage of the film deposition, as well as by heating during and after film deposition.

321

Figure 6.

Schematic representation of the spray pattern and the distribution of the coating material in the spray and the resulting coating profile.

Figure 7. Schematic cross section of the model of a thin film on a substrate [14].

Figure 8. Separation of the coating from the substrate due to contamination.

322

4.1.2.3 Defect arising during film formation As the nuclei join together the film begins to form. The manner in which a film develops determines the properties of the coating. In the vapour deposited coating process a spit or small droplet of source material may be ejected with the vapour and land on the substrate and be incorporated with the coating. The composition of the droplet is different and therefore can be the initiator of corrosion. The spit may also fall out, leaving a pinhole behind which can be a stress raiser and the site for fatigue crack initiation. Spit and foreign particles can induce preferential growth of the deposit, termed flake, which can lead to crack formation or to the nucleation of corrosive attack. 5. MICROSTRUCTURE OF THE COATING AND THE ASSOCIATED DEFECTS A thermal spray coating consists of a lamellar structure, of which the boundaries are almost parallel to the substrate surface (Fig. 9). This structure is the result of the molten or semimolten powder particles. The impacting particles may split, with some small droplets branching out or separating from the central particle. As a result of rapid cooling, some coating may have no crystalline structure, and some have a thin amprphous layer next to the substrate followed by a crystalline structure. Thick sprayed deposits reveal a directionallyoriented columnar grain near to the interface where rapid cooling occurs through the substrate. The columnar orientation gradually decrease as the thickness of the coating increases. The change from columnar to random grain morphology is believed to be produced by the effective lowering of the cooling rate, because of the evolution of heat of fusion which gives time to reforln and change the structure. Another feature of the rapid solidification in thermal spraying is the formation of structural defects such as vacancy coagulation and dislocation. However, the structure depends heavily on the quench rate. In the vapour deposition process, the microstructure of the deposit is determined by nucleation and growth of films from the depositing atoms. Nucleation depends on the atom/substrate interaction and the surface mobility of the depositing atoms, the atomic arrangement on condensation and the subsequent rearrangement of the structure. Surface asperities or micro-projections on the substrate are shadowed depending on the angle of incidence of the vapour stream and cause localised rapid growth. The islands so formed encourage columnar defects and ultimately lead to a poorly bonded coating. Clearly, high roughness will lead to many defects and the effect is dependent on the substrate temperature during deposition, the angle of incidence of the vapour stream and the gas pressure. An elevated substrate temperature will affect the morphology of the coating by increasing surface mobility, enhancing bulk diffusion and allowing recycstallisation to occur. These effects have led to the structure zone model (Fig. 10) of deposited materials of Movchan and Demchishin. The model consists of the formation of three sones which depend on the ratio of the surface temperature (T) to the melting point of the deposited material (Tm). At low temperature the coating atoms have less mobility and gives rise to a columnar structure with taper out growths and longitudinal porosity (in zone 1). It also will have dislocations and a high level of residual stresses. With the increase of substrate temperature the mobility of the surface increases. As the substrate temperature increases above about 0.3Tm, the columns tend to be finer, with parallel boundaries normal to the substrate surface. In this zone adatom diffusion dominates and the columnar structure consists of less defects and large grain with a higher density boundary between the columns. The surface morphology of zone 2 material is more angular than that of zone 1 material and contains less

323 defects and no porosity. In zone 3 which is deposited at still higher substrate temperature, the structure shows equiaxed grain morphology, where bulk diffusion and recrystallisation dominates. These structures have little residual stresses, with high density grain boundary and large grains. Movchan and Demchishin studied deposits of pure metal, which was extended by Thorton [11]. He identified a zone T (Figure 11) which is not prominent in pure metals or singlephase alloy deposit, but becomes quite pronounced in deposits of complex alloys and compounds. In this zone the structure is identical to that generated in the open grains of zone 1 but the surface is sufficiently fiat because enough diffusion has occurred to overcome the main surface irregularities. Thorton's modification shows that the transition temperature may vary significantly and normally shifts towards higher temperatures as the gas pressure increases in the deposition process. It should be emphasized that all zones are not found in all deposits, for example, zone 3 is not seen very often in materials with high melting points.

=

Figure 9. Lamellar structure of thermally sprayed coating [5].

ZONE 1

ZONE 2

::-;...: .: .:..'~': . : . ' , , ,, , . . .

)'-:-. :'..-;.~

T1 SUBSTRATE ZONE 1

..-

..

ZONE 3

:.:

T2 TEMPERATURE ZONE 2

ZONE 3

METALS

<0.3T m

0 . 3 - 0.45 Tm

> 0 . 4 5 Tm

OXIDES

O.26-0-45T m

>'0-45 Trn

Figure 10. Movchan/Demchishin diagram, showing the structural zones in a vapour-deposited coating [4].

324 6. EFFECT OF SUBSTRATE BIAS ON MICROSTRUCTURE The substrate bias during the deposition of a coating has an effect upon the growth and resultant microstructure of a PVD coating. Substrate bias is an alternative to deposition temperature to increase the surface mobility. Messier et al [12] have suggested improvements to Thorton's model which account for the evaluation of morphology with increasing film thickness and the effects of both thermal and bombardment induced mobility. In this model all the distinct levels of physical structure column/void sizes are considered and assigned to subgroups 1A, 1B, 1C, 1D and 1E as shown in Figure 12. This model shows that ion bombardment promotes a dense structure of zone T. During ion bombardment it has been shown that the void-filling process involves a combination of forward sputtering of coating atoms into the void region, impact induced surface diffusion and local heating such that recrystallization takes place during the growth and the underlying crystal structure is adopted. These changes in properties are expected in going from zone 1 to zone T.

7. COMPOSITION OF THE COATING AND IMPURITIES INCORPORATION The composition of the coating may vary from the composition of the sprayed material in thermal spraying due to the reaction of the molten particle with the gas environment. Particularly, the extent of their oxidation is very important to the properties of the coating. In a study of the effect of oxidation on aluminium and bronze, it was found that even minor oxidation during deposition is detrimental to compressional strength both parallel and perpendicular to the surface. However discrete oxide particles on the other hand not only strengthen the coating but also add wear resistance. The loss of carbon from Tungsten carbide coating through oxidation has been reported. Metallic or cermet coating may also react with air, forming oxide scales on the particle and dissolving the gases in the molten droplet. The extent of these reactions varies with the process parameters. As a result of rapid quenching a non equilibrium phase may be present. In alumina coating a slightly superheated particle on impact on a high thermally-conductive substrate gives delta and theta phase in addition to gamma with alpha suppressed. Due to super heating, selective evaporation of one component in an alloy or decomposition to gas or reaction with the atmosphere may happen. The loss will be more rapid from a fine powder than from a coarse powder. The composition and the stoichiometry of the compound of the deposit in the vapourdeposition process is controlled by the deposition parameter. For some alloy and composites sometimes the constituents segregate out in patches. In other systems more volatile components of the film may be sputtered out by ion bombardment affecting the film stoichiometry. In the vapour-deposition process solute impurity atoms in a deposit may react with the host material to form compounds or stay in solution either in an interstitial or substitutional position or segregate to the surface or to interfaces. Segregation and concentration of an impurity at grain boundaries may drastically affect the fracture mode of the material, generally resulting in embrittlement. Segregation to an internal surface will also inhibit recrystallisation and growth, as does the formation of second-phase material.

325 TRANSITION STRUCTURE CONSISTING OF DENSELY PACKED FIBROUS GRAINS COLUMNAR GRAINS POROUS STRUCTURE CONSISTING OF TAPERED CRYSTALLITES

%-~. :rflk~-.

\

,-'~--'-%. '.,2-'-.

~ . : : : %- ~

~c:

.."

:-~ ~ -

,,,-.-~_ " ~

~

"

~

GRAIN STRUCTURE

; a ~ ;'4,.

-~'

-~;-~~!iii -.:-"

--~~- NI ~~~1111fff17II{lll ~ ;.:.! ~ ~ l ~ i ~ ; I

I/-1-~ ~"

1

0.9

II ! J.~o.~~

I I i 11~o.7

0

, ~ "-

4

%

-o,

,,~

Figure 11. Influence of the substrate temperature and the deposition pressure on the microstructure of sputtered films.

E D C B A

--- 100 eV

~~, BombardmentInduced Mobility

0

<~ ThermalInduced Mobility

Characler,sllc s,ze A = 1lo3nm B = 5 Io 20 nm C = 20Io40nm D = 501o200nm E = 2001o400nm

Figure 12. Revised structure zone model showing the effect of substrate bias.

9

,~:

,.~

~"

326 Impurities may arise from a number of sources. In vacuum processes, Oxygen, Nitrogen, Hydrogen and Carbon are common from residual gases, vacuum leaks and out-gassing. In sputter deposition, the implantation of several atom of the sputtering gas is not at all unusual, particularly when a substrate bias is used. In chemical vapour deposition reaction products, carrier gases and unreacted or partially reacted source gases are commonly incorporated. 8. RESIDUAL STRESSES In thermal spraying, residual stress occurs as a result of the cooling individual powder particles on splat from above their melting point to room temperature. The magnitude of the residual stress is a function of the spray gun parameters, the deposition rate, the thermal properties of both the coating and substrate materials and the amount of auxiliary cooling used. Use of finer powder also leads to higher residual stress. In thick sprayed coating, the residual stress increases linearly with coating thickness above some minimum initial thickness Residual stress can cause cracking and spalling (Figure 13). Deposited coatings almost always contain residual stress, which appear at the interfacial region in a very complex manner due to geometric effects, variation in the physical properties of the material and the usually non-homogeneous nature of the film and interfacial material. The presence of pores and voids in the interfacial region will give stress concentration and alter the value of the tensile and shear components of the interfacial stress (Figure 14). The total stresses are composed of a thermal stress due to the difference in the coefficient of thermal expansion of the coating and substrate materials and an intrinsic stress arising from the accumulation effect of crystallographic flaws which are incorporated in to the film during deposition. The intrinsic stress is a function of the deposition process. It may be affected by a number of processing and growth parameters: film deposition rate, angle of incidence, the presence of residual gas, deposition temperature and gas incorporation. For low melting point materials, the deposition condition will generally involve high values of T/TIn so that the intrinsic stresses are significantly reduced by recovery during the coating growth. Thermal stresses are therefore of primary importance for such materials. High melting point materials are generally deposited at a sufficiently low value of T/TIn (0.25) so that the intrinsic stress dominates the thermal stresses. The residual stress in evaporated metal films is tensile and the residual stress in the sputtered metal films can be either tensile or compressive depending on the deposition rates, with lower gas pressure and higher negative substrate bias resulting in higher values of compressive stress. 9. INTERFACIAL FRACTURE Fracture of a coating is initiated from a flaw which allows stress concentration and weakening of bond strength. Propagation of fracture will occur by repeated bond breaking. Bonds may be broken by physical straining (Figure 15), by chemical effects or by a combination of both. Pores and cracks in the interfacial region initiate crack and stress concentration and thus help to propagate cracks. In stress corrosion cracking or static fatigue, a foreign species at the crack tip will strain or weaken the bond, allowing the fracture to propagate.

327

Figure 13. Residual stress causing cracking and spalling of a thermally sprayed coating.

Figure 14.

Showing the presence of pores and voids in the interracial region of a thermally-sprayed coating.

Figure 15. Propagation of fracture by repeated bond breaking in a magnetron sputtering coating.

328 10. TIME-DEPENDENT INTERFACIAL CHANGES Stress relief in the interfacial region is time dependent and increases the bond strength of the coating. However, thermal and mechanical cycling may lead to time-dependent failure by fatigue. Crack in the coating not only acts as the point of fatigue failure of the coating but may also act as the point of fatigue failure of the component when the adhesive bond strength is very high. Treatment at high temperature may improve adhesion by forming a desirable type of interface or may decrease adhesion by void formation or by diffusion of reactive species away from the interracial region. Corrosion may give a long-term loss of adhesion. Corrosion not only depends on the environment and the materials involved, but also on the availability of the corrosive media at the interface, i.e. the presence of porosity and pin-holes in the coating (Figure 16 is a picture showing pin-holes) [13]. 11. DEFECTS ARISING FROM SURFACE PREPARATION PROCESS Proper substrate preparation prior to coating application is the most critical step influencing the bond strength and the addition of the coating to the substrate. Roughening is an important step as cleaning for substrate preparation. Normally it is done by grit blasting the substrate. During impacting the surface the grit particles cut small platelets from the surface which may adhere to the surface causing problems by mechanical adherence. In some cases sharp cutting edges of the grit may embed in soft substrates such as sluminium. Grit blasting sometimes may create stress in the substrate. Surface cleaning may lead to an undesirable surface composition, which is the most critical step in surface preparation processes. It may affect the morphology of the surface by selective etching and thermal faceting. This change in morphology may affect subsequent processing. The cleaning process may leave undesirable residues on the surface, for instance, some electropolishing solution leaves inorganic compounds on the surface. 12. EFFECTS ON HARDNESS OF COATING Thermally sprayed coatings posses a heterogenous structure consisting of the coating material, oxide and voids. As a result, the macrohardness values are less than those of the equivalent material in either a cast or a wrought form. The hardness is usually reduced for a given material if the coating is applied in an inert atmosphere as compared to spraying in air. Oxidation, although it may increase the hardness of the coating, will weaken its internal strength and thus may be detrimental to the coating performance. Hardness in thermally sprayed coating, normally measured on a test specimen, may differ from that in actual parts due to difference in the angle of deposition and stand off distance and in some cases due to difference in residual stress. In the vapour-deposition process, process parameters such as the deposition rate, pressure and ion bombardment, can cause considerable change in the microstructure, resulting in low hardness. An increase in the substrate temperature commonly increases the coating hardness for refractory compounds (Figure 17) in contrast to the behaviour of metal films. Other factors affecting the hardness value are the interatomic forces, the stress level, the adhesion of the coating, the impurity content and the film texture.

329

Figure 16.

Showing pin-holes.

----- TiN 5000

TiC Ti Bz

om

A

E

4OOO

TiBz Bulk --~--

tt~ C "10

o 3000 -r"

-

TiCt.o Bulk

TiNl.o Bulk ---~--

2000

_

~

t

t

t.

I

500

~

t

t

L

L

9

t

L

lO00

Substrate Temperoture,Ts

,

1

1500

(~

Figure 17. Film hardness as a function of substrate temperature for some refractory compounds[ 14].

330 13. ANISOTROPY IN COATING Successive particles in thermal spraying acquire the same lenticular shape over the material already deposited, so that the coating develops an anisotropic lamellar structure parallel to the interface. The mechanical as well as other properties of a thermally sprayed coating are anisotropic for its structure and directional solidification. This anisotrophy is probably more pronounced for cermets and metallic coating. In the microstructural model (Figure 10) zone 1, zone T and zone 2 have some preferred orientation of the growing crystallites. It has been reported that coatings are significantly harder when tested parallel to their growth direction compared with the values derived for the other two orthogonal directions. The thermal and elastic properties parallel to the coating plane and perpendicular to the coating plane are also different. In contrast, coatings that exhibit a fine equiaxed grain structure (zone 3 in Figure 10) do not show a hardness anisotropy effect [15]. 14. OTHER EFFECTS The mechanical properties of thermal sprayed coatings are sensitive to the deposition parameters used. As such, any general tabulation of properties would be misleading. The ductility of metallic thermal spray coating is slightly better than that of cast iron but is less than those of most cast metals. The ductility of a sprayed coating rarely exceeds 2%. The surface finish of a coated surface is extremely important. The smooth finish that can be obtained on a given coating is a function of its composition and the method of deposition. Normally, vapour deposition coatings do not need any surface finishing but most of the coatings deposited by thermal spray need surface finishing. The corrosion resistance of a deposited coating is different than that of its cast or sintered composition. For example, alpha alumina is very corrosive resistant, but when it is sprayed alumina is a mixture of phases, all of which are not corrosion resistant. Most thermally sprayed coatings have varying degrees of inter-connected porosity that allow attack of the substrate in corrosive environments. Galvanic corrosion can occur in some environments when improper selection of coating composition is made. 15. CONCLUSIONS The mechanical properties of the coating are dominatingly influenced by factors such as (1) the microstructure; (2) incorporated impurities; and (3) internal stresses. All of these are dependent on the deposition variables. Whatsoever be the variables, a coating material has unique properties, which are far different from those of materials fabricated by melting and solidification or by sintering processes. These differences are attributed to the microstructure, the phase compositions, impurity incorporation and/or composition or phase gradients. The physical and chemical compatibility of the mating surfaces are, of course, important in selecting a coating. Laboratory tests can be an excellent guide to selection, as long as the other considerations such as load, environment, temperature etc. are similar to those in service. It is quite obvious that coating applications need individual consideration of every coating and substrate combination and every fine points of the surface demands careful attention.

331 REFERENCES

.

~

,

.

,

~

.

,,

i0. 11. 12. 13. 14. 15.

R.F. Bunshah et al. (eds.), Deposition Technologies for Films and Coatings: Developments and Applications, Noyls Publications, Park Ridge, New Jersey, U.S.A. 1982. J.L. Vossen and W. Kern (eds.), Thin Film Processes, Academic Press. Inc. , San Diego, California, 1978. J.L. Vossen and W. Kern (eds.), Thin Film Processes II, Academic Press. Inc., Sand Diego, California, 1991. B. Bhusan and B.K. Gupta (eds.), Handbook of Tribology, McGraw Hill, New York, 1991. Anonymous, Thermal Spraying: Practice, Theory and Application, American Welding Society, Inc., Miami, Florida, 1985. H.J. Zaat, A Quarter of a Century of Plasma Spraying, Annual Review of Material Science, 13, (1987), 9. D.R. Rickerby and A. Matthews (eds.), Advanced Surface Coating: A Handbook of Surface Engineering, Chapman and Hall, New York, 1991. R. Kaufold, et al., Deposition of Coatings using a New High Velocity Combustion Spray Gun, Proc. 3rd NTSC, Long Beach, CA, (1990). E.J. Kubet, Thermal Spraying Technology: From Art to Science, Advanced Materials and Processing, 12, (1987). M.L. Thrope and H.S. Richter, Pragmatic Analysis and Comparison of the HVOF Process, Proc. ITSC, Florida, U.S.A. (1992). J.A. Thorton, J. Vacuum Science and Technology, Vol. II, No. 4, (1974), 666. R. Messier, A.P. Giri and R.A. Roy, J. Vacuum Science and Technology, A2, (1984), 500. P.V. Kola, S. Daniels, D.C. Cameron and M.S.J. Hashmi, Proc. AMPT'93, Vol. II, Dublin, Ireland, (1993), 1323. J.E. Sundgren and H.T.G. Hentzell, J. Vacuum Science and Technology, A4(5), (1986), 2259. H. Nakahira, K. Tani, K. Miyajima and Y. Harada, Anisotrophy of Thermally Sprayed Coatings, Proc. ITSC, Orlando, Florida, U.S.A., (1992), 1011.

This Page Intentionally Left Blank

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

A STUDY OF WORKABILITY CRITERIA PROCESSES

333

IN B U L K F O R M I N G

A.S. Wifi l, N. EI-Abbasi 2 and A. Abdel-Hamid 3

1 Professor, Department of Mechanical Design and Production, Cairo University, Egypt. 2 Graduate student, American University in Cairo, Egypt. 3 Schlumberger Professor, and Dean of Sciences and Engineering, American University in Cairo, Egypt. ABSTRACT This study presents a critical review, and comparison of workability criteria in bulk forming processes. Various well established criteria, are classified into appropriate groups, and a simple comparison between them is performed based on theoretical basis. An elastic-plastic large strain finite element technique is also used to predict workability limits using some of these criteria, with special emphasis on upsetting operations. The workability criteria are investigated based on their ability to predict fracture initiation sites and critical level of deformation up to fracture. NOTATIONS Ci D Dc Ff H I1 J2 K N R WD

WDC k m n cx 13

Constants used in workability criteria in Table 1 Diameter of disk Diameter of collar Friction force Height First invariant of the stress tensor Second invariant of the stress deviator Strength coefficient Normal compressive force Void radius Initial void radius Ductile fracture parameter Critical value of ductile fracture parameter Shear strength Shear friction factor Strain hardening exponent Strain ratio Stress formability index

334 -/

Relative fracture index Principal strain component i

6m~x

Maximum tensile strain

c

Effective strain

m

ci

Effective fracture strain

Co

Fracture strain in uniaxial loading

oe*

~t

Critical value of 82 at plane strain Coefficient of Coulomb friction

c%,

Average principal tensile stress normal to the axis of mechanical texturing

crH oi

Hydrostatic or mean normal stress Principal stress component i

Crr,~x

Maximum tensile stress

(7"

Effective stress Critical shear stress Frictional shear stress

rc Xf

1. I N T R O D U C T I O N Workability may be defined as the extent to which a material can be deformed without failure. It is not a unique material property since it is affected by both process and material variables. Material factors that affect workability are the local state of stress, strain, strain rate and temperature, whereas the main process variables are the geometries of the workpiece, punch and die, as well as the friction between them. Ductile fracture is usually the limiting factor that determines the workability of cold bulk forming processes. These processes are characterized by the application of compressive force to the material to impart a change in its shape and dimensions. The resulting deformation however can induce secondary stress and strain states which can be tensile. Here fracture usually follows one of two modes, either free surface fracture, or center bursting type fracture. Free surface fracture occurs on the surface of the workpiece where the stress normal to the surface is zero, and they are more common to upsetting, rolling, and heading processes. Center bursting fracture occurs inside the workpiece, mainly in extrusion and forging operations [1]. Many criteria have been developed to predict ductile fracture. Table 1 summarizes some of the criteria commonly used in the literature. Some of them are based purely on experimental data i.e. empirical, while others are developed from theoretical foundations. It is obvious that there are many criteria, all describing the same phenomenon, that is workability, yet each criterion is based on a different hypothesis. The situation is further complicated due to the fact that some of these criteria are only applicable to one of the common failure modes, while others claim to be applicable to more than one mode (surface fracture and center bursting fracture). The major aim of this study is to present a critical review of some of the common workability criteria for cold bulk forming processes with special emphasis on upsetting

335

Reference

Workability

criteria

43

McClintock [2]

n2(1- - - - ~ sinh( Rice and Tracey [3]

. 3 4 (l-n) o,+o,)+_ 2

a

4

a 2_ - a I ] d? cr

(1)

= C,-~ exp(C2crn /a) i(l + an /C3 ~r)d-e

(2)

Oyane et al. [4] Cockroft- Latham [5]

I O-maxd~ 9

(4/

In(R/Ro)

2 am,, a-C

Brozzo et al. [6]

(3)

(5)

f 3(O'm~ -CYn) ac

Norris et al. [7]

(6)

(1-Ca.) Atkins [8]

j (l +_~a)d-c/(l _Csan )

Cliff et al [9]

fc~

Tie-Jun [ 10]

f g(ff~-~)h(-~p)d-cp

(7)

(8) = W~c

(9)

O"

~,

Kuhn et al. [ 11-13 ]

Hoffmanner [ 14]

(lo)

In ~'y = C'7( c r y ) + ln(Cs)

(11)

O" Shabaik-Vujovic [ 15-16]

fl 3or. = _

=

Ii

,

~

a"I - f ( f l )

(12)

Sowerby et al [17-18]

~c=O~/2

(13)

Maximum tensile stress

crm~"

(14)

Maximum tensile strain

Cm~

Integral of hydrostatic stress

~a,d-E

(lS) (16)

Table 1- Common workability criteria. (For upsetting o t and el are axial stress and strain respectively, and 0 2 and ~2 are circumferential stress and strain respectively).

336 operations. Further, an elastic plastic large strain finite element code is used to predict workability limits, fracture initiation sites and the influence of changing the strain path on workability. The finite element results are then used in a relative comparative study to assess various workability criteria. 2. WORKABILITY CRITERIA BASED ON THEORETICAL FOUNDATIONS These criteria are developed by careful consideration of the nature of fracture both at the microscopic and macroscopic levels. Void growth which is an accepted cause of ductile fracture has been used at the microscopic level, by McClintock and others [2-4,19], to formulate fracture criteria. Localized thinning due to inhomogeneity, which is quite evident in sheet metal forming, has been used by Kuhn [11], to develop a fracture criterion for bulk forming operations. Other criteria based on integrals of stress and strain functions have been developed from macroscopic considerations by Cockrofi and Latham and others [5-9]. Continuum damage mechanics techniques have also been used to predict ductile fracture [10,20]. 2.1. Criteria based on void growth and coalescence

This is the most common approach, relating fracture to the formation (nucleation), growth and coalescence of voids The voids are usually initiated at inclusions or second phase particles, however, they could also start at interfaces or at slip band intersections. Factors such as panicle size, stress state, interfacial cohesion, and volume fraction of inclusions affect nucleation, and therefore affect workability [21]. This stage is then followed by the expansion of the voids in volume and shape until they finally link together leading to fracture. There is no clear cut distinction between the different stages, and while one type of voids is in its growing stage, other types can be growing at different rates, or even still in the nucleation stage due to their stronger bond with the matrix material, or their smaller size [22]. McClintock [2] modeled void growth by assuming three mutually perpendicular sets of cylindrical holes of circular or elliptical cross-section with axis parallel to the principal direction of the applied stress. Fracture is considered to occur when the holes grow to touch adjacent holes [2]. This criterion (Eq. 1 in Table 1) included some inherent simplifying assumptions. It is assumed that the principal components of stress do not rotate relative to the material, and the interaction between voids is limited. In fact the voids are assumed to be isolated holes in an infinite media. Two solutions are then obtained, one for a perfectly plastic material, and the other for an ideal plastic material model. The solution for power hardening materials is then interpolated between the two functions. A similar analysis was performed by McClintock, Kaplan and Berg [19] to account for fracture due to void growth in shear bands, and by Rice and Tracey [3], but for a spherical void using a rigid perfectly plastic material model (Eq. 2 in Table 1). The same hypothesis, limitations, and assumptions, were also adopted by Oyane et al. [4] to formulate a ductile fracture criterion (see Eq. 3 in Table 1). Most of these criteria adopt an integral of stress function to predict fracture that is applicable for both surface cracks and center bursting. 2.2. Criterion based on localized thinning

Marciniak and Kuczynski [23] devised a method to analyze the behavior of metal sheets containing inhomogeneity. This analysis turned out to be invaluable for predicting necking in

337

sheet metal forming. A similar analysis but for bulk forming processes has been attempted by Lee and Kuhn [11]. In the latter study, two geometric elements were considered (Fig. 1), each containing a groove representing an inhomogeneity in one direction. In one of them the groove was in the radial direction (R-model), and in the other it was in the axial direction (Zmodel) [11]. These specimens were subjected to combined axial compression and lateral tension, to simulate the conditions that they would usually be subjected to in an actual upsetting operation. Under the effect of the load the groove deformed faster than the rest of the specimen. The analysis was continued up to a limiting condition that represents final fracture. This condition was taken as 0 groove thickness at which fracture strains were calculated and compared with experimental data. The results in Fig. 2 show that the R-model predicts the empirical Kuhn type fracture line of slope - 1/2 (see Section 3. ! ) for relatively large strains while the Z-model predicts fracture behavior at low values of strain, with a slope of-1. The results obtained are highly dependent on the ratio of the initial thickness of the groove, to the size of the specimen, which limits the use of this criterion. The final thickness at fracture, contrary to the assumption made above, is usually not zero, which again gives uncertainty to the fracture strains predicted using this method. This model is only applicable for surface fracture which is evident since the grooves are only on the surface of the specimens.

I 1 11

/t

1

f,1

I !_k+ 1 ,l

' 1- 1'I-

Fig. 1 - Maleriai elcnlenls used by Lee and Kuhn for localized thinning analysis: (a) is R-model (b) is Z-model I111 i.0

i

0.$

.~-nnodel 1

"~176176 ~ 0

0~,

i

Of, 04 Compressive slfaln

.

Z~

Fig. 2 - Limiting slrains for R and Z models I I 1] 2.3. Criteria based on integral fnnctions of stress and strain These workability criteria are based on macro mechanical considerations, arid claim that fracture occurs when a certain integral of a stress function over the effective strain field reaches a critical value [22]

338

The most widely used criterion of this class is that of Cockrofi and Latham [5], which states that fracture depends on the principal tensile stresses. For a given material, temperature and strain rate, the criterion suggests that fracture occurs when the tensile strain energy reaches a critical value (Eq. 4 in Table 1). Brozzo et al. [6] proposed a modification where the stress function is explicitly dependent on the hydrostatic stress (Eq. 5 in Table 1). Norris et al. [7] proposed a similar criterion based on a different stress function (Eq. 6 in Table 1). Atkins [8] later modified Norris' criterion to account for the effect of the strain ratio (Eq. 7 in Table 1) [9]. Cliff et al. [9] suggested a different workability criterion that uses a critical value of the generalized plastic work per unit volume (Eq. 8 in Table 1). After using 3 dimensional elasticplastic finite element analysis program to solve the problems of simple upsetting, axisymmetric extrusion, and strip compression and tension, they were able to evaluate 9 criteria, to predict fracture initiation sites. They reported that only their criterion was successful in accurately predicting fracture sites in all the experiments that they investigated. It was also reported that their criterion was not as successful in determining the amount of deformation before fracture [9]. All of these workability criteria may be applicable to both surface fracture and center bursting, due to their dependence on the state of stress, as well as strain. It is worth noting that some of these criteria, are not theoretical in a true sense, yet they are all based on macro mechanical considerations. 2.4 Criteria based on continuum damage mechanics

Traditionally fracture mechanics techniques have only been used to describe macroscopic fracture for brittle materials under elastic loading [24], using the linear elastic fracture mechanics approach. This method was not suitable for predicting fracture in ductile materials, where extensive plastic deformation precedes and accompanies the formation and growth of cracks [24]. This led to the development of ductile fracture mechanics techniques that are capable of handling large scale plastic yielding, blunting of crack edges, and the high energy which is absorbed during ductile, compared to brittle fracture. Instead of a critical value of the stress intensity factor, as in linear elastic fracture mechanics, a J-integral was introduced by Rice [:25] and Cherepanov [26], which is defined for any contour around the crack tip. An alternative formulation involved measuring the Crack Tip Opening Displacement, 8t-CTOD [24]. These techniques have been used by various investigators, to predict ductile fracture [10,20,27]. Tie-Jun [10], formulated a local criterion for ductile fracture, based on the local ductile fracture parameter WD, which should reach a constant, material dependent value WDC, before fracture (Eq. 9 in Table 1). In this relation g(x) is a function of stress and h(ep) is a function of the plastic strain state. A similar criterion was also formulated by Lemaitre [20]. These criteria were not used to predict fracture, in any bulk manufacturing process. They have been compared however, to fracture strains estimated using some of the above mentioned criteria. Tie-Jun for example included in his discussion a comparison of the predictions of his criterion, with that of the growth of voids, formulated by Rice and Tracey [3]. A similar comparison was performed by Lemaitre, between the continuum damage failure criterion, which he formulated, and those related to void growth, mainly due to McClintock [2], and Rice and

339 Tracey [3]. He demonstrated that the two criteria are quite similar (Fig. 3), over a certain range of hydrostatic stress. Such good comparisons suggest that there is a need for further investigations regarding the applicability of continuum damage mechanics techniques to bulk forming. E

'i ~

Domain covered by void growth model Domain covered by Lemaitre model

q5,

0

n

1

2

3

O"

Fig. 3. Comparison between a void growth criterion, and continuum damage mechanics criterion of Lernaitre [201.

3. E M P I R I C A L W O R K A B I L I T Y CRITERIA 3.1. Strain based criteria An empirical workability criterion was suggested by Kuhn et al [11-13], which states that the axial and circumferential strains at fracture provide a measure of workability, since they fall on a straight line for a given temperature, strain rate, and material's microstructure. This conclusion is in agreement with the results previously reached by Kudo and Aoi [28], and those of Kobayashi [29]. Figure 4 shows the strain paths for the various tests, while Fig. 5 shows the corresponding fracture line for steel 1045. Upsetting tests were used to determine various points on the fracture line, by varying the specimen height to diameter ratio, and the lubrication conditions at the die/workpiece interface. The strains were measured using 3 mm grids that were placed on the cylindrical surface of the workpiece by electro-mechanical preetching of the specimens. The results show that all the strains at fracture fall on a straight line relating axial strains to circumferential strains. This line is nearly parallel to the homogeneous (frictionless) deformation line, with a slope o f - l / 2 (Eq. l0 in Table 1). Where ~1 and ~2 are the axial and circumferential strains at fracture, and C 6 is the fracture strain at plane-strain conditions. Due to its popularity, and simplicity this criterion is often used as a benchmark against which other workability criteria are compared and evaluated. This approach will also be adopted in the present study.

340

0.8 I

!-

Z

. UPSET ~~LUBRICATED 0.8~- TESTS~"//~MOQ TM \ / ~ DIES ~,X_cL j

[ L/D: I. 5: LUBRI CATED UPSET / 75.' ~'~~H ~"LID :0.

rr BEN ING I-o') /PLANE j_j 0.4

ss

0.2

.

,~ . . . . .

<~ rr 0.6 I.I

DIES ~

O0

I

0.2

0.4

I

0.6

I

~.illo~,

~

ROLLING

BEND,NG z uJ ....I~PLANE -S,,,~O0 I- u.LI-- STRAIN ,- ~'0 G~'" i

I

\

~,o~

9

I

0.8

1.0

VO

COMPRESS IVE STRAIN

0 2

0 4

0 6

0.8

1.0

COMPRESSIVE STRAIN

Fig. 4. Strain paths for some deformation processes [ 121

Fig. 5. Fracture locus for steel 1045 [121

In a consequent study Erman and Kuhn devised novel specimens in order to verify that the fracture line is applicable close to the plane strain axis. These specimens give large tensile stress for a small axial compression [30]. Kuhn later argued that a dual slope fracture line exists for some materials such as steel 1045 (Fig. 6), and Aluminum 2024-T351, with a slope o f- 1 close to the plane strain axis, followed by the usual slope of-l/2. Apart from the dual sloped line, this criterion is otherwise identical to the original strain based criterion in the procedure used to evaluate the fracture locus and to access workability.

04

9

!E:o

--.

/

1.0

0!8

- O'

- 0.4

- 02

0

Compressive s l r a m

Fig. 6. Dual fracture locus for AISI 1045 cold drawn steel [31 ]

Evaluation of workability limits using Kuhn type strain based fracture criterion, may b e carried out as follows 9 9 Determine the fracture line or lines for the material under consideration (either experimentally o r f r o m available literature).

341 9 Determine the strain path at various locations on the deformed workpiece (using experimental, theoretical, or numerical techniques). 9 Plot the different strain paths against the fracture line, and if any of them intersects the line, the workpiece is most liable to fracture at that strain level. In case of fracture, modification has to be made either by changing the material or the processing conditions [31]. It should be emphasized that the strain based criteria are only applicable to free surface fracture, where the stress normal to the surface is zero. Fracture occurring at the center of the workpiece or at the die/workpiece interface experience a triaxial state of stress and therefore cannot be anticipated using this workability criterion. 3.2. Stress based criteria Realizing that plastic instability and ductile fracture are much influenced by the stress field, several investigators attempted to develop stress based fracture criteria. One of the earliest attempts was due to Hoffmanner [ 14], who used the tensile stress induced in the workpiece in the direction normal to the axis of mechanical texturing. In the relation he proposed (Eq. 11 in Table 1), C 7 is dependent on process variables such as the strain rate, temperature and heat treatment, and C 8 accounts for the effects of mechanical texturing and should not change for a certain material [14]. It was demonstrated however that this criterion gives results that are totally different to those of other fracture criteria [32]. Due to the significance of the hydrostatic component of stress on fracture, Shabaik and Vujovic [15,16] used a parameter 13 which they defined as shown in Equation 12 in Table 1. They argued that damage or failure would occur when the effective strain at any point in the deformation region reaches a critical value, eft that is a function of the stress parameter 13. To evaluate ef various mechanical tests were performed including: Simple tension, compression, torsion and upsetting tests as well as thin walled tube subjected to internal pressures. These tests were chosen due to their simplicity and the range of [3 values that they cover,-2 < 13 < 2 (see Table 2).

Applied stress . . . . .

Equibiaxial tension Thin walled pressurized tube Simple tension Pure shear Simple compression Equibiaxial compression

o'n

9

o I c~ o cy c~ -o -or

o? o o/2 0 -o 0 -o

.

o3 0 0 0 0 0 0

.

.

.

o.

2o /3 cy / 2 o /3 0 -o / 3 -2 ~ / 3

.

.

.

[3 .

.

o 43 o / 2 o ~/3 c cr o

2

,/3 1 0 -1 -2 ....

,,

Table 2- Stress parameter 13 evaluated for different simple mechanical tests [ 16]. Sowerby et al. [17,18] used experimental results based on the upsetting of three different specimen geometries; a plain disk, a collar and a punch. Results of their investigation demonstrated that the fracture strains do not satisfy a linear fracture line. They argued however, that using the results of only the disk compression test, a Kuhn type fracture line can be obtained, but there was no guarantee that the straight line behavior would be valid in the domain close to plane strain [17]. Investigations of the collar and punch test, enforced their

342 claim that the straight line behavior is not valid in the plane strain region. They also evaluated the stresses and found that they do not satisfy the Cockrof or the Oyane criteria explained previously in this review. They concluded that for the materials considered (AISI 1045, 1146, 1541 and 4340), a critical value of shear stress, equal to half the circumferential stress, seems to be a more plausible workability criterion for free surface ductile fracture (Eq. 13 in Table 1). They supported their hypothesis by experimental evidence and some simple calculations [18], but without a theoretical basis. Other workability criteria have been quoted by Atkins [22] and Cliff [9]. These include critical maximum stress (Eq. 14 in Table 1) or maximum tensile strain (Eq. 15 in Table 1), and a critical value for the integral of hydrostatic stress (Eq. 16 in Table 1). However, these criteria seem to have very limited applicability. 4. T H E O R E T I C A L ANALYSIS OF W O R K A B I L I T Y C R I T E R I A It is evident from the previous review that the various workability criteria do not have a common basis, which makes their relative comparison, a rather difficult task. In the present work, various criteria are compared based on simplified theoretical considerations. The critical strains are determined from the various functions or integrals, and are used to plot a fracture locus relating axial to circumferential strain, in a manner similar to the Kuhn type fracture line. In this way the relative agreement between the criteria can be assessed for both free surface fractures, and fracture under general loading conditions. 4.1 Theoretical analysis of free surface ductile fracture A material following the power hardening law is considered,

~ = K~"

(17)

where K and n are the strength coefficient and the strain hardening exponent respectively. If the strain path is assumed to be linear, then a constant strain ratio, or, can be defined as: a . d~2 . . . ~'2 de~ ~'~

(18)

Using the deformation theory of plasticity, one can express the integrals or functions used in the various workability criteria, in terms of the axial and circumferential strains, as well as the material constants K, and n (Appendix). To accomplish this task, the different constants in the Appendix have to be evaluated for a given material. Equations involving one constant, need only one data point for their evaluation. In this study, the plane strain fracture point (e I = 0, e 2 = e ) was used since it is . easy to verify experimentally. Here the critical value, e , ~s taken to be the intercept of the Kuhn type fracture line with the ordinate. The criteria of Sowerby and Chandrasekaran, is an exception however, since it is supposed to predict a decrease in the fracture strains near plane strain conditions [18]. If the criteria involved another constant in the integral expression, it was selected in such a way to give an initial slope of about -0.5 near the plane strain axis.

343 A set of values of critical strains is then calculated for each criterion, based on the previously defined constants for 303 Stainless Steel, and it is used to plot a fracture strain locus. The results in Fig. 7, demonstrate that some of the criteria, predict a fracture strain locus, close to a straight line behavior with a slope close to -0.5, at least over some interval of strains. The workability criterion of Clit~ et al. [9] however, predict an elliptical fracture strain locus. It is obvious from the results that the various workability criteria behave differently. At this stage it is difficult to comment on their relative accuracy, due to inherent approximations, and the lack of supporting experimental evidence. However, a finite element relative comparison is attempted in section 6. 4.2. Effect of hydrostatic stress. The experimental straight line behavior predicted by Kuhn is only valid for free surface fracture. It cannot be used to predict fracture due to central bursting, nor due to tool-die interface. Such fracture conditions, are characterized by an additional hydrostatic stress component which is imposed on the free surface conditions [31 ]. To evaluate workability in such circumstances, some of the workability criteria that are applicable under a general state of stress, are used, and the theoretical analysis is modified to account for the additional hydrostatic stress component, which is assumed to be constant (See Appendix). The results for the Cockrofl, Brozzo, McClintock and the Oyane fracture criteria, (Fig. 8) are in accordance with experimental observations that a compressive hydrostatic stress improves the workability of metals, whereas an externally applied tensile stress decreases the predicted strains to fracture. This explains the high strains reached in extrusion, where fracture occurs at the contact between the die and the workpiece where a normal compressive stress is superimposed on the free surface conditions, permitting higher strains to fracture. The results support the idea of a movable fracture line, suggested by Kuhn [31], whose intercept with the ordinate, e*, is dependent on the state of stress, as well as its natural dependence on the material of the workpiece. It is also evident from Fig. 8, that there is less agreement among the different criteria stated above, regarding the values of the fracture strains, for the same value of external hydrostatic stress. Cliff's criterion however, does not predict any change in the fracture strains due an externally applied hydrostatic stress, since it is only dependent on the effective stress and strain which are insensitive to hydrostatic stress. Other workability criteria are not applicable except for free surface fractures and were therefore not tested. It is worth noting that the effect of hydrostatic stress on workability is being carefully considered by the authors, and finite element results, on this issue will be shown elsewhere.

5. ESSENTIAL FEATURES OF A GOOD W O R K A B I L I T Y CRITERION For a criterion to be successful in predicting workability limits in a bulk forming process, it should be capable of determining the amount of deformation before fracture, as well as the fracture initiation site, and accomplish these tasks with least sensitivity to calculation errors (minimum uncertainty). Estimating the amount of deformation up to fracture with reasonable accuracy is an important requirement for workability analysis. Being able to predict the limit to deformation,

344

Clift prediction

McClintock prediction r"

0.8 0.6

......

,

0

-0.5

0.5

.

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

.

.

1

.

.

.

i

"9

0 5,

(p}

-0.5

.

1.5

-0._0.5~~ 5

2

~f z

1.2 i

0.5

1

E

0

=

0

-0.5

1.5

. . . . . . . 0.5

i

prediction

1.5

2

Cockroft prediction

1.5T t..

E~

0.5

to oL..

t't v

~ 0

0.5

1

~

,

,

1

2

~

-1

1.5 - ( Axial strain )

- ( A x i a l strain )

I

1

( Axial strain )

I

1.2 1 0.8 0.6 0.4 0.2 -0.5

0

'

- ( A x i a l strain )

Oyane

1

--

0'0, 0

0.75

= ~

Z .....

0. -0.5

0.5

Brozzo prediction

prediction

1.5 1

s

- ( Axial strain )

- ( A x i a l strain ) Sowerby

j J J

Linear fracture line

!

~ = KT" = 1655 n =0.26

K

Fig. 7. Theoretical fracture strain loci based on different workability criteria.

MPa

345

Oyane

Brozzo 2.5

3.5 3

~ o

1.5

o o

2

o.s

o

~

1.5

1

-0.5

~

2.5

~

~

~

.

o

0.5 0

,

i

t

0.5

1

1.5

- ( Axial

cockroft

0

~

0

-0.5

0.5

- ( Axial

strain )

1.5

1

strain )

McClintock

and Latham

2

1.5

I"

~

1

:3 r

.'//

9149

. o

1.5

s ~176 ~ ~'~'~

. . . .

,

f

,

~149

, -

-0.5 strain

f ~

~'

1

9 1 7 6 ~.

~

-

_11 - ( Axial

o ~

0.5,

j

9~ ~149176~.~

0

0.5 - ( Axial

)

1 strain

1.5 )

Clift 1

. . . . . .

0.8

$3 = 0

0.6 0.4 J

-O.5

0.5 -

. . . . .

$3 =-500

........

$3 =- 1000

MPa MPa

1

~

- ( Axial

Fig. 8 -

$3 = +300MPa

strain

)

Effect of hydrostatic stress, $3, on fracture locus as predicted by various workability criteria

346 and the proximity of the workpiece to that limit, gives useful insight for the designer regarding the feasibility of his design. Determining the correct fracture initiation site, is also an essential requirement. The selected workability criterion, must at least be capable of predicting the most critical locations, where fracture could occur. A simple example to demonstrate this, is the upsetting of disks, where fracture occurs at the outer surface. Predicting otherwise would render the criterion unusable for this type of processes. Traditionally the selection of a workability criterion was restricted since most of them either depend on parameters that are difficult to evaluate empirically, like stress components, or require the evaluation of integrals. Such integrals, need continuous monitoring of stress and strain components, which is usually achieved by dividing the forming operation into several incremental steps, that requires laborious experimental effort. Here the finite element technique lends itself for use as a valuable tool, since all the relevant process and material parameters can be evaluated at all locations, and at any time during the forming process. 6. FINITE ELEMENT ANALYSIS OF WORKABILITY LIMITS Prior to the development of the finite element method, the slab method, slip line field method, the visioplasticity method and the upper bound method were more popular analysis tools [34]. However, these methods are usually limited in application when complicated geometry and complex material behavior are considered. The finite element technique, has recently become the most popular tool available for analyzing metal forming operations with tremendous capabilities of handling three dimensional contact problems, and material behavior. It is therefore possible to accurately simulate a forming process, by starting with the correct geometric, material, and frictional models and monitoring the variation of all material and process variables throughout the deformation process. There are certain basic features which must be present in a finite element program, if it is to be used for the successful analysis of bulk forming processes. These are : 9 Capability of handling large deformations, rotations, and large strains. This can be achieved through the use of the updated Lagrangian elastic plastic formulation, which uses the current state as the reference state. Another method that has been used with great success for forming operations, is the flow formulation [35], using either a rigid plastic or a rigid viscoplastic material model. This method has been used extensively by Kobayashi and others [36-37]. The flow formulation is much faster, but the updated Lagrangian elastic plastic solution, is capable of handling elasticity effects, which the flow formulations cannot do. This additional capability is necessary in some cold forming operations. It is therefore advantageous for a finite element program to handle both formulations, giving the user the freedom to choose the most suitable, based on the nature of the problem at hand. 9 Capability of handling contact boundary conditions. In most bulk forming processes, plastic deformation is induced by the motion of a die, or punch relative to a workpiece. To handle these situations, a finite element program must be capable of handling multiple contact surfaces, of various geometries, which are used as restraining surfaces to govern the flow of material. Furthermore, the surfaces of the die or punch are not ideally rigid, but they undergo some deformation as they contact the workpiece material. It is therefore an advantage if the program can handle deformable contacting surfaces.

347 Capability of handling friction. Friction in bulk forming processes is a very complex phenomenon, that depends on the die and workpiece materials, lubrication, if any, and the nature of the forming process. Empirically the frictional force is usually expressed in one of two forms, either using a shear friction factor (m) which relates the shear frictional stress, "of, to the shear strength of the material (k) as :

(19)

"~f= mk

or using coulomb friction (It), which relates the frictional force, Ff, to the normal applied compressive load (N) a s Ff= p.tN

(20)

A suitable finite element program must be capable of modeling both types of friction, and preferably any other user defined friction model, where friction is a function of any number of material and process variables. Both sticking and sliding conditions should also be correctly modeled. Three dimensional capabilities. This is necessary, since many forming operations, cannot adequately be described as a plane stress, plane strain, or axisymmetric problem. In the present work, MARC K5.2 general purpose finite element program [38], which is one of the well known non-linear analysis program, was adopted as the finite element tool since it has all the basic capabilities mentioned above. In order to predict workability limits, strain paths and/or stress paths should be determined. This is accomplished using an axisymmetric updated Lagrangian finite element formulation with shear contact friction for 303 Stainless Steel. Two basic problems are considered namely, the upsetting of disks and collars. The relevant geometric and process variables, as well as the finite element mesh for each model are shown in Fig. 9, and table 3. Model

Disk upsetting

H/ D= 1

Collar upsetting H / D = 1, D c / D - 1.2 Collar upsetting H / D = 1, D c / D - 1.1 Table 3. Finite element models used.

Friction factor (m)

Model No.

0.1

(1)

0.2 03 04 0.5 l 0.~ o.1

(2) (3) (4) (5) (6) (7) (8)

Figure 10 shows the strain paths for the disk and collar tests as predicted by finite elements. The trends in these curves, agree with the experimental data reported in the literature [37,39] for disk upsetting, where an increase in friction leads to a steeper strain path. The results of

348

I,,

D

DC

II

.I

I

I

"1

U,I .......~.......:::::::::::::::::::::::::::::::: .........~.........!...........~:~::~.......!::.......~:~:~:~:~:~::"

•-::•:•:•:•:•:•:•:•:•:•:•:•:•::•:::•:•:•:•:.::•:•:•:•:•:•:•::•:•:•:•:•:•:•:•:.:•:•:•:•:•:•:•:::•::•:•:•:•:•:

I/I

---

-

_

_

---j--

1 1' ,

:::.i:~:i:.:~:.:~:i:i:i:i:i:::::::::::::::::::::::::::~:i:i:i:~i:i:::i:i:)~:~:::.~:~:}:i:i:i:i:.:.:i:i:~:i:~:i:}:.:):i::::~ --

I

.

=

i=~=

zK I

.......

:tL

:t

1. . . . . .

.

.

=

.

,,i ,1 . . . .

9 ,

I

x

l, ItI

I

I

I

I

II

i1 11 X ' I I ~ I I ~ I I

I,=l=

=

= s,=~=~l!I

i

=

9149

=

9

I

9I

1

1

1

I'11'

1~

I

I

I

I

1

I

l'

II

I

I

I!1'

li

1'

[

9

9 i, 9

.

.

.

.

.

.

4 . ,

.

,9

. 4

. 4 - - 4

.

. 4

.

.

--4

.

.

..

.

.

=

,,,

.

,,

.

.

,

,

=

9

x

=

.

.

.

=

=

=

1,

=

=

=

x

.

.

=

x

x s x 4 x

x~

x

x

=

9

9

9

9

9

9

'

.

=

x

x t = 4 x

x

x

.

x

.

x

=

x

x

x

o

=

.

=4

,,4

.

..

-

x

l

1'

I'

l!

I

1

I'

I'

li

I

II

9

i.

J.

.

.

.

.

.

.4

4 .

4

.l

.

4

.

4

.

4

.

I

q

4

.

--

-

4

~

--

.

Undeformed shape

t- f, t. t D " t-.~.~S~_~L-

L"!. /-" .'_%r-./-r-

i-

I-

I - i - i - f - I - r

I,-.I" - ~ - l

F.i.-7.%~_- f -5"-',, --~ ~+ -'+ _'+ z g - J

~

~

"

! - 1 " ] ' ] ' I

-'J

" #

--

I

- #

--1

- #

--

1

- # - #

--

t

"

I

- ~

-

d

~

I

: ~

"'-1- ' ~

-

t

---t[

i

-

--

'JE ii

-

4

-4

t

i.

'

Deformed shape

Die

14s ~

Disk

~' -

" ~ ~z4

81 /

I

158

~

\

~

I

I

55

"

l '

9

~6o 6

3: /" 15

16

Fig. 9. Finite element modeling of disk and collar upsetting problems. Node numbers shown indicate nodal points considered in the analysis for possible fracture locations.

349 the collar test also confirm the data supplied by Sowerby et al [18], where the strain follows a steep, nearly linear path. 0.90 0.80

-

0.70 "~

/

/

/~

0.60

"

Disk, m=0.2 Disk, m=0.3

"

--

"~- 0.50

"~

o

Disk, m=0.4

- - o - - Disk, m=0.5

0.40

i

Disk, m=0.1

Disk, m = 1.0

0.30

---"-- Collar 1, m=0.1

0.20

~

Collar 2, m=0.1

0.10

~

Fracture line

0.00 0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

- ( Axial strain )

Fig. 10. Strain paths for different disk and collar upsetting operations. The finite element results were then fed into a simple program to evaluate the functions or integrals of some of the workability criteria given in Table 1 and to determine the level of deformation at the onset of fracture. Various nodes were considered as potential locations to determine the fracture initiation site.

6.1. Relative comparison among various workability criteria In this section, relative comparison is made among various workability criteria, as an extension to a previous work done by the authors [33] The aim being to check the validity of the inherent assumption that a damage function or integral has a constant critical value for a given material irrespective of the process parameters. Knowing the constants used in the damage functions or integrals, as discussed in section 4, enables one to evaluate the workability limit using the finite element predicted stress and strain paths directly. However, this may be laborious in view of the fact that various potential fracture locations should be examined. Such analysis is under investigation and is to be shown elsewhere. Alternatively, since the Kuhn type fracture line is essentially an experimental one, it will be used here to get the workability limit (~':), based on the finite element determined strain path as explained in section 3.1. This limit is introduced into the various fracture criteria and the value of the damage functions or integrals are accordingly evaluated for the various models under consideration. To avoid dimensional conflict, a relative fracture index, y is introduced. This is calculated by normalizing the values of the functions or integrals with respect to the critical values theoretically predicted at plane strain conditions. Table 4 shows the variation of "t for various models, as well as the average and standard deviation for the predictions of each criterion. The McClintock, Oyane and Brozzo workability criteria seem to give less fluctuations in the results, with a maximum deviation of about 10% percent, which suggests that the critical

350 values of their damage functions are nearly constant for the processes under investigation. The workability criteria of Cockrofl, Atkins and Norris however, show a wider variation in the predicted values of V, while ClitYs criterion seems to give the maximum standard deviation. One may then conclude that the latter group of criteria do not necessarily agree with that of Kuhn regarding fracture strains. An interesting point is depicted by Fig. 11, which shows the variation of the relative fracture index of different criteria with the friction factor, m. For high values of the friction factor, and thus highly non-linear strain paths, various criteria seem to asymptotically converge to a constant value of y. Workability criteria

Model

McClintock

Norris

Atkins

Cliff

0.859

Relative fracture index, T 0.963 1.135 1.243

0.495

3.390

1.057

0.506

2.892

0.536

2.061

Oyane Cockroft Brozzo

Finite elements 0 759

2 3

0757

0.843

1 195

0.925

4

0 732

0.806

1 072

0.835

0.963

5

0.788

0.922

0 886

0.798

0.902

1.009

0.977

6

0.780

0.864

0913

0.769

0.813

0.788

1.094

7

0.765

0.954

1 447

1.094

1.517

0.465

5.715

8

0.754

0.879

1 301

1.004

1.255

0.477

4.159

Average

0.7621

0.8753

1.1511

0.9128

1.0918

0.6110

2.8982

Minimum

0.732

0.806

0.886

0.769

0.813

0.465

0.977

Maximum Standard deviation

0.788

0.954

1.447

1.094

1.517

1.009

5.715

0.0168

0.0458

0.1902

0.1096

0.2205

0.1924

1.5772

,,

Table 4.

3.5

Comparison between relative fracture index, Y, during upsetting of disks (models 26) and collars (7-8).

r "-

Cockroft

r

2.5

Brozzo 9

2 = 1.5

Atkins Oyane

1 0.5

N orris

~ : ~ ~

r

---i

i

l

i

l

,

0.2

0.4

0.6

0.8

1

Clilt McClintock

F r i c t i o n fa c t o r

Fig. 1 1. Variation of relative fracture index with friction factor for upsetting of disk

351 6.2. Prediction of fracture initiation sites

The ability of some of the workability criteria to correctly predict the fracture initiation site was tested, by evaluating the functions or integrals of the different criteria at various nodes in the finite element mesh, to locate the most critical section. From experimental observations, fracture should initiate at the center of the outer surface in simple disk upsetting and at the outermost diameter in the collar test. Figure 12 shows the use of the empirical Kuhn, and Shabaik-Vujovic fracture criteria in predicting fracture initiation site for surface nodes, for model 3 (upsetting of disk with 0.3 friction factor). The results show that the center of the outer surface is most liable to fracture. Figure 13 shows the variation of the fracture integrals of various workability criteria throughout the deformation process, for the same model. The location of the selected nodes in the finite element mesh is shown in Fig. 9. The node that has the highest value of the integral is considered the site of fracture initiation. It is worth noting that most of the workability criteria correctly predict the fracture sites, at the center of the outer surface of the disk, some exceptions being those of CliR and Norris.

Kuhn

Shabaik-Vujovic

0.9 .=, O.S 0.7

1.5 1.3

-~9 0.6

0.9 0.

.,.,

"~ 0.5 0.4

,nw

/

0.3

._~ 0.2 r.) o.I o

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-

Nodel6

---*--Node48

-0.5

-2.5

- ( Axial strain )

1.5

Formability index (Beta)

~

Node 96

~Node144

Fracture locus

Fig. 12. Prediction of fracture initiation site for some empirical workability criteria 7. CONCLUDING R E M A R K S In this study, various workability criteria have been analyzed from a theoretical standpoint, as well as using the finite element technique. They were tested for predicting fracture initiation sites and the level of deformation up to fracture (workability limits). Based on theoretical analysis, it has been shown that McClintock, Brozzo, CockroIt and Kuhn workability criteria predict similar straight line fracture loci when relating axial to

352

Oyane

Norris

1.5 "~

0.5

""(

0

~

-o.5 -1

,~

-1.5

N ~

-2.5 -3 -3.5 -4

2 1.8

._..5 :'~_. -

~

I/I/I/x/I

1.6

o

1.4 1.2

~

l

80

',~ 0.8 0.6 0.4 ~. 0.2 o

,

Percent deformation

.

~

2

,, . . . . - = x . = ~ ~

\=\\

~-3

40

i

~ = ~-~

-2

~

Cockroft & Latham

2

!

8o

~ ,

-4

-5 i

Percent deformation

Percent deformation

Cliff

Brozzo

6

2

9,-(

!

x .xt~t.-- - x

4

/

l;~

x/

3 2

~

.~.~J

/ /

"~

.

Percent deformation

McClintock

i~

.

o

J/-4)

9

=

. .+.

,x B" ) . m'~ 8+ I t-/-+ x/" B-" Ir + ...~.- ~>---2~_• ...-A, ~z ~ .-'-~.~-<-'~J ~ - ~ ~ ~,4 ~x._~,~,-~'= x--~ _---

]

/ l 1

"l i

.~ -3

~

-\~--~,,~..~_~_~

i

-d 9 -4 i o

4o

80

Percent deformation =

Node 1 Node 96 Node 145

+

N o d e 16 -"

~

-5 i

Percent deformation *

Node 48

Node 144

--o--

N o d e 15

Node 158

--'--

Node 160

Node 73 --•

Node 32

Fig. 13. Predicting fracture initiation site using various workability criteria

-"

N o d e 81

---+-- Node 55

80

353 circumferential strain, and thus one expects comparable workability limits for a given material and working conditions. The workability criteria of Sowerby and Oyane, however, predict slightly curved fracture strain loci. An elliptical shape has been obtained when the Cliff criterion was analyzed, which might not be compatible with experimental observations in upsetting problems. The theoretical analysis has also indicated that applying hydrostatic pressure increases workability limits, in all the criteria considered except for that of Cliff, since it is inherently insensitive to hydrostatic stress. Different criteria, however, predict different changes in fracture strains for the same hydrostatic stress. A relative fracture index, ~,, has been introduced, which is defined for each criterion, as the value of the function or integral normalized with respect to its theoretically determined fracture value at plane strain conditions. This index enables one to perform a relative comparison among workability criteria while avoiding dimensional incompatibility. In this study, ), was evaluated for the upsetting of disks and collars that were analyzed by the finite element method. The criteria of McClintock, Oyane and Brozzo have shown a low standard deviation in the value of 7, indicating that the critical value of their integral or damage function is nearly constant. This was not the case for the Atkins, as well as the Cliff criterion. All criteria with the exception of those of Cliff and Norris have also been found to successfully predict fracture locations at the outer surface of disks, under upsetting conditions. It should be emphasized that the question of the assessment of various fracture criteria is really complicated and there is a need to determine the range of applicability best suitable for each criterion. Reaching this goal requires extensive theoretical and finite element analysis, supported by comprehensive experimental work. The analysis of the present study may be of help in enhancing the understanding of different workability criteria, regarding their applicability to upsetting processes. Despite being based on different foundations, there seems to be some agreement among the majority of the workability criteria in predicting workability limits, as well as fracture initiation sites. REFERENCES [1]

G.E. Dieter, Evaluation of Workability - Introduction, Metals Handbook, vol. 14. 1988, pp. 363-372. [2] F.A. McClintock, A Criterion for Ductile Fracture by the Growth of Holes, J. of Appl. Mech., ASME, 1968, June 1968, pp. 363-371 [3] J.R. Rice and D.M. Tracey, On the Ductile Enlargement of Voids in Triaxial Stress Fields, J. Mech. Phys. Solids, vol. 17, 1969, pp. 201-217. [4] M. Oyane, T. Sato, K. Okimoto and S. Shima, Criteria for Ductile Fracture and Their Applications, J. Mech. Work. Tech., vol. 4, 1980, pp. 65-81. [5] M.G. Cockroff and D.J. Latham, Ductility and workability of Metals, J. Instit. of Metals, Volume 96, 1968, pp. 33-39. [6] P. Brozzo, B. DeLuca and R. Rendina, A New Method for the Prediction of Formability Limits in Metal Sheets. Sheet Metal Forming and Formability; Proceedings of the 7th Biennial Conference of the International Deep Drawing Research Group, 1972. [7] D.M. Norris, J.E. Reaugh, B. Moran and D.F. Quinnones, A Plastic Strain Mean-Stress Criterion for Ductile Fracture, J. Eng. Mat. Tech., vol. 100, 1978, pp. 279-286.

354

[8] A.G. Atkins, Possible Explanation for the Unexpected Departure in Hydrostatic Tension-Fracture Strain Relations. Metals Sci., Feb. 1981. S.E. CIiR, P. Hartley, C.E.N. Sturgess and G.W. Row, Fracture Prediction in Plastic Deformation Processes, Int. J. Mech. Sci., vol. 32, 1990, pp. 1-17. [10] W. Tie-Jun, Unified CDM Model and Local Criterion for Ductile Fracture - II. Ductile Fracture Local Criterion Based on the CDM Model, Eng. Frac. Mech., vol. 42, No. 1, 1992, pp. 185-193. [11] P.W. Lee and H.A. Kuhn, Fracture in Cold Upset Forging- A Criterion and Model, Metall. Trans. 1973, vol. 4, Apr. 1973, pp. 969-974. [12] H.A. Kuhn, P.W. Lee and T. Erturck, A Fracture Criterion for Cold Forming, ASME J. Eng. Mat. Tech., Oct. 1973, pp. 213-218. [13] J.J. Shah and H.A. Kuhn, An Empirical Formula for Workability Limits in Cold Upsetting and Bold Heading, J. of Appl. Metalworking, Volume 4, No. 3, July 1986, pp. 255-261 [14] A.L. Hoffmanner, The Use of Workability Test Results to Predict Processing Limits, Metal forming Interrelation between Theory and practice, Proceedings of a Symposium on the Relation Between Theory and Practice of Metal Forming, Plenum Press, 1971, pp. 349-391. [15] V. Vujovic and A.H. Shabaik, A New Workability Criterion for Ductile Metals, J. of Eng. Mat. Tech., Volume 108, July 1986, pp. 245-249. [16] A.H. Shabaik and V. Vujovic, On the Stress Formability Index, Int. Conf., on Mech. Design and Production, Cairo, 1988. [17] R. Sowerby, I. O'Reilly, N. Chandrasekaran, N.L. Dung, Materials Tested for Cold Upsetting, ASME J. Eng. Mat. Tech., vol. 106, Jan. 84, pp. 101-106. [18] R. Sowerby and N. Chandrasekaran, The Upsetting and Free Surface Ductility of Some Commercial Steels, J. Applied Metalworking, vol. 3, No 3, July 1984, pp. 257-263. [191 F.A. McClintock, S.M. Kaplan and C.A. Berg, Ductile Fracture by Hole Growth in Shear Bands. Int. J. Mech. Sci. vol. 2, 1966, pp. 614. [201 J. Lemaitre, A Continuum Damage Mechanics Model for Ductile Fracture, ASME J. Eng. Mat. Tech., vol. 107, Jan. 1985, pp. 83-89. [21] W.M. Garrison Jr. and N.R. Moody, Ductile fracture, J. Phys. Chem. Solids, vol. 48, No. 11, 1987, pp. 1035-1074. [22] A.G. Atkins, Fracture in Forming, Proc. of Int. Conf. on Adv. in Mat. and Proc. Tech., Dublin, AMPT 93. [23] Z. Marciniak and K. Kuczynski, A Model of Localized Thinning in Sheet Metalforming, Int. J. Mech. Sci., vol. 9, 1967, pp. 609. [24] R.O. Ritchie, Why Ductile Fracture Mechanics?, ASME J. Eng. Mat. Tech., Vol. 105, 1983. [25] J.R. Rice, ASME J. Appl. Mech., vol. 35, 1968, pp. 379-385. [26] G.P. Cherepanov, Appl. Math. Mech., vol. 31, 1967, pp. 476. [27] W.H. Tai, An Improvement of CTOD Criterion for Ductile Fracture, Eng. Fract. Mech., Vol. 41, 1992, pp. 153-157. [281 H. Kudo and K. Aoi, Effect of Compression Test Conditions Upon Fracturing of a Medium Carbon Steel - Study on Cold Forgeability Test: Part II, J. of Japan Soc. of Tech. in Plasticity, vol. 8, 1967, pp. 17-27

[9]

355 [29] S. Kobayashi, Deformation Characteristics and Ductile Fracture of 1040 Steel in Simple Upsetting of Solid Cylinders and Rings, J. Eng. Ind., Trans. ASME 1970, pp. 391-399. [30] E. Erman and H.A. Kuhn, Novel Test Specimens, for Workability Measurements, in Compression Testing of Homogeneous Materials and Composites, STP 808, ASTM, 1983, pp. 279-290. [31] H.A. Kuhn, Workability Theory and Application in Bulk Forming Processes, Metals Handbook, vol. 14, 1988, pp. 388-404. [32] H.A. Kuhn, Forming Limit Criteria -Bulk Deformation Processes, Advances in Deformation Processes, Plenum Press, J.J. Burke & V. Weiss (eds.), 1978, pp. 159-186. [33] A.S. Wifi, A. Abdel-Hamid, H. El-Monayri and N. EI-Abbasi, Finite Element Determination of Workability Limits for Disks and Rings under Different Upsetting Conditions, Proc. of Int. Conf. on Adv. in Mat. and Proc. Tech., Dublin, AMPT 93. [34] E.M. Mienik, Metalworking Science and Engineering, McGraw Hill, New York, 1991. [35] S. Kobayashi, S.I. Oh and T. Altan, Metal Forming and the Finite Element Method, Oxford Series on Advanced Manufacturing, Oxford University Press, 1989. [36] C.C. Chen and S. Kobayashi, Rigid Plastic Finite Element Analysis of Ring Compression, Application of Numerical Methods to Forming Processes, ASME, AMD, vol. 28, pp. 163. [37] J.L. Frater and B.R. Penza, Predicting Fracture in Cold Upset Forging by Finite Element Methods, J. Mat. Shap. Tech., 1989,57-62. [38] MARC, Analysis Research Corporation, Palo Alto, California, 94306, USA. [39] S. Oh and S. Kobayashi., Workability of Aluminum Alloy 7075-T6 in Upsetting and Rolling, ASME J. of Eng. Ind., Aug. 1976, pp. 800-806. Appendix

Based on the assumption of proportionality of loading, and using the deformation theory of plasticity, the following relations, can be derived for free surface conditions: (A1)

El -[- ~2 "[- e3 ~" 0

-

2

cr~ . cr

,

2 ~2 + 2 g~ . . 3

o-z -

2

~

=

o"

-

3

~

"

x/l+a+a

a +2 ~3(1 + a +

.

2 a'~ + 2& = -

2

2 x

a: )

2a +1 =

c

~'

s x~ le',]

o

-

(A3)

~'~

..

x ~

43( l + c t + a 2 )

(A4)

I<1

0-3 =0 o-H

(A2)

(AS) 2 3

E I d-~2 -

~ =

+1 a, 2

x ~

~

(A6)

356 based on these relations, the integrals and functions of the workability criteria in Table 1, can be expressed as a function, of the strain ratio, or, as well as the material constants, K and n, as shown in Table A1.

Workability criteria

McClintock

Oyane

Integrals or functions used in criteria

" ~ ( a - 1 ) J~x~C' C=[-2[ ~-~ sinh(3(l-n) l+a (1-.) ------7 ~/1 + cr + a 2 ) + 4 4 1 + a + a z ]c,I

( 1+ 1

C-

1+

(A7)

(A8)

(~:6 43(1+ a + cr2)x

Cockrofi - Latham

1+ 2 a C-

K -(n.I) c

(A9)

~1

- - x ~

~ / 3 ( l + a + a 2) /l+l

g,I (A10)

Brozzo C-

Norris

2(1+2a)_ _ o r 3 a

;

l+a 4 3 ( 1 + a + a 2)

~

Atkins

-

gl

Kgnx Ic]]

(1 + 2 a ) d E 1+ a Ka'"x ez

C= [

(A12)

I ,1

/3(1 + a +

~ Cli~

(All)

de

X~.("+'~

(A13)

n+l Kuhn

(A14) C-

Sowerby

C=

a+

~

- ~ 1+ 2 a

K -c" x ~'l

(A15)

243(1 + a + ct" ) Table A I. Integrals or functions used in various workability criteria expressed in terms of axial and circumferential strains.

357

On the basis of the same assumptions, an additional constant external hydrostatic stress can be superimposed on the free surface conditions to reach the following relations: eI +e z +e 3 = 0 o-1 o-

2 e 2 + 2e", 3 e

(A16) 0"3 o-

(A17)

o2_ = -----=----+2 e, + 2e: ~ o3 e o 4~..o-3=

0 +

(A18)

~-

(A19)

O-

o-. _ 2 e, +_e2 +_-=-~ o3 e o-

(A20)

using these relations it is possible to derive new expressions for the different workability criteria, in terms of the axial and circumferential strain, as well as the material parameters, K and n. For instance when o 3 is negative, one can get the following expressions for some of the workability criteria under consideration For Cockrofl 9 --(n+l)

C

l+2a 3(l+a+

Ke ) n+l

e,

-

(A21)

For Oyane 9 ( 1 1+ ct f!'~1 o"3 -(~-,,) C = k 1+ x e4 - - e (76 4 3 ( 1 + ct+ ct 2) [e,[;' (7.6 X ( 1 - n )

(A22)

For Brozzo 9 C

2(1+2a)~.+ 3 a

2o- 3 f f l + a + a -~'3K a

-(l-n)

2 e 1-n

e, [e,I

(A23)

lIP.

Materials Processing Dcfects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

Degradation of m e t a l

matrix

359

composite under plastic straining

N. Kanetake and T. Choh Department of Materials Processing Engineering, School of Engineering, Nagoya University, Frocho, Chikusaku, Nagoya 464-01, Japan

The m a c r o - and microscopic degradations under plastic deformation of aluminum matrix composites are investigated and discussed in the present work. The change in strength brought about by forging and subsequent heat treating is investigated from the macroscopic viewpoint. The strengths at room and elevated temperatures after upsetting and heat treating are measured using small tensile specimens machined from upset materials. The strength of the composite is decreased by forging, and it can not be recovered by post upsetting heat treating. Furthermore the in situ tensile test in the SEM is carried out to observe directly the microscopic degradation during plastic straining. Some microcracks are initiated at small plastic strain after yielding far before maximum loading point. The microcracks do not propagate in the matrix because of its ductility, but a number of new microcracks are developed with increasing strain. Most microcracks are occurred at the interface between the matrix and the particle, though some particle fractures are also observed. The decrease in strength by the forging can be explained by taking the degradation in the microstructure during plastic deformation into consideration. 1. I N T R O D U C T I O N Metal matrix composites (MMCs) reinforced with particle or short fiber dispersions are new structural materials which have high specific strength and may take the place of steel structures. However the MMCs have not yet been put to practical use, though some trial products have been made in various industrial fields. As compared to the development of new fabricating processes for the MMCs, the manufacturing technologies for complex shape products of the MMCs have not been investigated extensively. To apply the MMCs much more to the mass-products in the automobile or machine industries, further development of their secondary processing is very necessary. The MMCs, especially particle reinforced MMCs, are generally fabricated by ei-

360 ther liquid phase methods, those are melt stirring, rheocasting, compocasting and spray deposition methods, or solid phase methods, mainly the powder process like a powder extruding, a hot pressing, a HIP process. The former is more economical processes, while the latter has an advantage in being able to disperse fine particles uniformly regardless of the wettability between matrix and particle materials. In any fabricating processes the economical process which is employed for conventional materials is desired for the use of MMCs in the mass-products. In that case their intermediate products like a bar or a plate are fabricated by the above mentioned methods. Then the intermediate MMC products are processed subsequently to the final products by secondary processing like deformation processing or machining. The M M C s can be deformed like conventional metals. In applying deformation p r o cessing to MMCs, there is an advantage in being able to use the existing forming techniques without large modifications. From the above viewpoint the authors are now focusing on the deformation processing of MMCs. The upsettability of particle dispersed aluminum matrix composites [1-3 ] and their flow stresses [4-6] have been studied experimentally and theoretically. The particle dispersed MMC consists of ductile matrix metal, hard particles and their interfaces. The MMC will be subjected to various loading during the deformation processing as well as in using them as structural parts. It is predictable, in this case, that the internal stress and strain distributions under the external loading are much complicated as compared with conventional metals. Therefore microstructural degradation such as microcrack can be occurred due to the stress localization. It would influence the mechanical properties of deformed composites. Recently, Lloyd [7] reported that the Young's modulus of SIC/6061 aluminum composite is reduced by plastic straining and that the reduction is owing to particle damage during straining. Mochida et al [8] followed the phenomena by an analytical model which can explain the particle failure mode observed in the experiment. The authors [9] also reported that the strength of forged SIC/6061 aluminum composite is decreased as compared with that before forging. Further the authors [10,11 ] have tried to observe directly the change in microstructure under plastic loading by the in-situ tensile test in a scanning electron microscope (SEM) equipped with a deformation stage. To apply the deformation processing to MMCs, the influences of various factors of matrix, particles and their interfaces which depends on their fabricating processes on the m a c r o - and microscopic degradation during plastic straining must be cleared. And it is very necessary to investigate the optimum processing conditions for MMCs as well as their deformability.In this report some of our work related with the degradation in properties and microstructure of particle dispersed aluminum matrix c o m posites under plastic straining are summarized and discussed. The composites used in the works were prepared by the powder extruding method in order to get uniform dispersion of particles and to prevent some complicated reactions between the matrix

361 Table 1. Composition of matrix aluminum alloy powder. Mg

Si

Fe

Cu

Cr

Zn

Mn

1.11

0.60

0.42

0.34

0.19

0.01

0.01

Ti

A1 bal.

and particles. 2. S T R E N G T H OF F O R G E D M M C 2.1 Tensile test after upsetting The SiC particle dispersed 6061 aluminum alloy composite was prepared for the test. The matrix aluminum is 6061 alloy powder whose composition is shown in Table 1 and the reinforcement is SiC particle whose mean diameter and volume fraction are 4.6~tm and 20% respectively. The matrix aluminum alloy powder and SiC particles were mixed by use of a V-type mixer and the mixed powder was directly hot extruded at 773K in the rod with diameter of 35mm. For the comparison the 6061 aluminum alloy rod was also prepared by the same processing from the 6061 matrix powder, which is herein referred to as the matrix material. The upsettability of the composite was investigated in the other work [3]. At higher temperature than 300~ the upsettability is very well as conventional aluminum alloys, and it is possible to produce complex shape parts of MMC by the forging process. However the localized deformation in the material is predicted in forging at higher temperature than 300~ because no strain hardening can be seen in its flow stress at that temperature. From these results the hot forging process followed by slight cold forging can be considered as one of useful forging processes for the MMCs. The process is also useful for improving the precision and the strength of forged parts. In order to investigate the change of mechanical properties brought about by d e formation processing and subsequent heat treatment, tensile tests were carried out using specimens machined from the upset materials. In all, following six types of upsetting and heat treating processing conditions including before upsetting were examined for both materials.

Process 1: before upsetting (as extruded material). Process 2: upsetting by 50% reduction in height at 300~ Process 3: upsetting by 25% reduction at 300~ followed by 25% reduction at room temperature. Processes 4, 5 and 6: processes 1, 2 and 3 followed by heat treating respectively,

362

to=2

5R

r--..

/11

10 Figure 1. Specimen for tensile test after upsetting.

that is water quenched after solid solution treatment at 530~ hours and this was followed by aging at 175~ for 8 hours.

for 2

The extruded cylindrical billet of r was upset in the extruded direction and heat treated as mentioned above. Then, small tensile specimens were machined in the longitudinal (upsetting ; L) and radial (T) directions from the upset materials as shown in Figure 1. In order to assess the effect of the specimen shape on measured values, the strength of the material before upsetting was compared with that measured by using a conventional parallel rod tensile specimen. The comparison showed that the strength measured using the specimen shown in Figure 1 is hardly different from that using the conventional rod specimen. Tensile tests were carried out at room, 200 ~ and 300~ temperatures.

2.2 Strength after upsetting Figure 2 shows the tensile strengths at room temperature of the matrix and c o m posite materials after processes 1, 2 and 3. For the both materials no clear difference in strength in longitudinal and radial directions was observed either before or after upsetting. The strength of the matrix material upset at 300~ is not different from that as extruded (before upsetting), and it increases after upsetting followed by it at room temperature due to strain hardening. This is the same result as that in general metals. On the other hand, the strength of the composite material upset at 300~ is clearly lower than that as extruded. Although the strength is increased by upsetting at room temperature as well as the matrix material, the strength after the process 3 is almost the same as before upsetting. The difference in strengths of the matrix and composite materials, namely strengthening in the composite, is about 100MPa before upsetting, but about 50MPa after upsetting. This means that the strengthening effect

363

of dispersed SiC particles is decreased by upsetting. Figure 3 shows the strength in the longitudinal direction at various temperatures

~ ]

L direction , '

As

~

I

E

T direction '

I

] L direction , '

'

As

extruded

~

!

T direction '

I

extruded

300 ~ 50%

300 ~ 50%

3000C, 25%

300~

25 %

R.T., 25%

R.T., 25%

300 100 200 Tensile strength o u/MPa

0

100 200 300 Tensile strength Ou/MPa

(b) Composite material

(a) Matrix material

Figure 2. Tensile strength of matrix and composite materials at room temperature before and after upsetting.

500

'

I

" .........

I

'

500

I

----O-- As extruded 400

--~-

300~

---D--

300~

'

I

'

-----O---

I'

As

'

I

extruded

--~-

300~

---O--

300~

400

%---RT,25% 300

300 e,o

I.i

I--4

200

200 o

.,..,

r

["

lOO

t--

-

,

0

I

100

~

I

,

200

Temperature /~ (a) Matrix material

I

300

~m

100

A

0

I

100

,

I

,

200

I

300

Temperature /~ (b) Composite material

Figure 3. Tensile strength of matrix and composite materials at various temperatures before and after upsetting.

364 500

r

'

I

'

I

'

500 -

I

400

400

300

~= 300 ,= ed)

e~0

,

i

'

u

'

i

= ~D

O

200

200 O

O----

.,...4 r~

---~---

O

[" 100

---D--

I

100

O----

~

1 O0

300~

,

I

,

'200

Temperature /~ (a) Matrix material

I

300

300~

---D--

300~

j

0

As extruded

--~-

300~ ["

,

0

As extruded

I

100

,

..

I

,

200

I

300

Temperature /~ Co) Composite material

Figure 4. Tensile strength of matrix and composite materials at various temperatures after upsetting followed by T6 heat treatment.

after the processes 1, 2 and 3. The effect of temperature on the strength is similar in both matrix and composite materials. Although the decrease in strength is small at 200~ it is considerable at 300~ The above mentioned phenomenon, that is d e crease in strength at room temperature after upsetting at 300~ is also seen in the strengths at elevated temperatures. The decreasing rate is also similar at various temperature. The increase in strength by strain hardening in upsetting at room t e m perature is not disappeared even in the tensile test at 300~ because the holding time at elevated temperature in testing is very short, about 5 minutes. Figure 4 shows the strength after post upsetting heat treating, that is after processes 4, 5 and 6. No difference in strength between processes can be seen with the matrix material, because strain hardening disappears by heat treating. However the decrease in strength of the composite material after upsetting is maintained after subsequent heat treating. The strengths after processes 5 and 6 are the same, because the strain hardening disappears by heat treating as well as the matrix material. This suggests that only decrease in strength by upsetting maintains even after heat treating. It is predictable from the result that the decrease in strength by upsetting is owing to m i crostructural damages which can not be recovered in subsequent heat treating.

365 3. M I C R O S T R U C T U R E AF'I'ER P L A S T I C S T R A I N I N G 3.1 In situ tensile test in S E M

The A1203 particle dispersed 6061 aluminum alloy composite was prepared for the test by the same powder extruding method. The matrix aluminum alloy powder is the same as above experiment and the reinforcement is A1203 spherical particulate whose mean diameter is 20ktm and volume fraction is 10%. The spherical shape and relatively large mean diameter were employed for an easy in situ observation of the m i crostructure. The composite material was extruded in the plate of 15x2mm. In situ observation of microstructure under tensile loading was carried out using a scanning electron microscope (SEM) equipped with a deformation stage. Both t e n sile and compressive loads can be applied on the deformation stage according to a preliminary arranged loading path. The continuous observation of microstructural change under the loading is possible in the SEM. A load (stress) - displacement (strain) curve can be also simultaneously measured by using a data acquisition system with a personal computer. The surface of the specimen whose shape is discussed in detail later is polished, and it is mounted in the deformation stage. Then the specimen is loaded under constant displacement rate 0.1 mm/min, and its microstructure is observed simultaneously in the SEM. The loading is stopped at appropriate points to be photographed. The other high resolution scanning electron microscope was also used to observe microcracks in detail. The static tensile properties were also e x a m ined using a tensile testing machine. 3.2 S p e c i m e n for in situ tensile test

In order to observe easily the change of microstructure under loading in the SEM, it's desired that the site to be observed on the specimen is specified in a limited area. In that case, however, the microcracks can be occurred due to not only the m i c r o -

R10

50 Figure 5. Specimen used for in situ tensile test in SEM.

366 structural degradation but also stress localization owing to the specimen shape. The specimen should be deformed under uniform stress state to prevent the microcracks due to the stress localization. In order to satisfy the both conditions which are c o n trary to each other, the shape of specimen shown in Figure 5 which has a narrow part in the center was employed for the in situ tensile test in the SEM. The stress and strain distributions in the center area of the specimen under external tensile loading was predicted by an elasto-plastic FEM calculation. In the calculation the specimen was assumed as a homogeneous material which has the stress-strain property corresponding to the used composite material. Figure 6 shows the calculated distributions of tensile stress and equivalent strain in the specimen. From the c a l c u lated results it can be confirmed that no considerable localization but relative h o m o geneity of their distributions would be generated in the center area of the specimen. Though the stress state in the center part is not pure uniaxial tension but slightly biaxial tension, the stress value in the tensile direction is around the same as that given by dividing the external load by the cross section area of the center. Therefore it can be expected that most of all microcracks observed in the specimen could be related

100

200

(a) Tensile stress (MPa)

(b) Equivalent strain

Figure 6. Calculated distributions of tensile stress and equivalent strain in the center area of the specimen for in situ tensile test.

367 with the microstructural degradation under external loading.

300

I

I

3.3 M i c r o s t r u c t u r e under plastic 200

straining Figure 7 shows a measured stressstrain curve of the used composite. Micrographs observed at two loading p o i n t s A and B in the f i g u r e are shown in Figure 8. Few microcracks initiation can be found at the interface between the particle and matrix and in the particle at the point A in small p l a s t i c strain after yielding. The stress value at the loading point A is clearly lower than that at a m a x i m u m loading point. It means that the m i crostructural degradation is occurred

o0

B

100

00

0.05

0.1

0.15

Strain Figure 7. Nominal stress-strain curve of used composite.

under relatively lower external load, namely at smaller plastic strain, though the local internal stress at the site of the microcracks may be relatively higher than that in other areas. The microcracks can be seen not only near side edge of the specimen but also in the center area. It supports that the microcracks are occurred due to not the stress localization owing to the specimen shape but the microstructural weakness based on the above calculated stress and strain distributions. In the micrograph observed at the loading point B near the m a x i m u m loading point, a number of new microcracks can be seen at the interfaces and in the particles. But primary initiated microcracks do not propagate in the matrix because of its ductility, though the microcracks are expanded. Many sharing are also seen in the matrix, which means the considerable plastic straining of the matrix. The same behavior was observed until just before fracturing. From these observations it is found that the microstructural degradations like debonding of interface and particle fracture are d e veloped with increasing external strain from small strain. 3.4 F r a c t u r e surface Figure 9 shows the typical microcracks occurred during tensile loading. And m i crographs of the particles observed in the fracture surface are shown in Figure 10 with a received particle. In the case of debonding at the interface many dimples of matrix aluminum alloy to prove ductile fracture can be seen on the surface of the particle, though a part of as received surface can be also seen as it is. It means that the bonding between the matrix and particle is relatively strong, but the interface debonding is

368

Figure 8. Micrographs observed during tensile loading.

369 occurred owing to a part of incomplete bonding. When the bonding of them would be complete all over the particle, no interface debonding but the particle fracture would occur as shown in Figure 10(c). In the other work [11] in which the composite fabricated by a melt stirring method was used, very brittle surface on the particle was observed in the fracture surface. And most of microcracks were occurred at the interfaces during plastic straining. From comparison with the work it can be found that the bonding strength at the interface of the composite fabricated by powder extruding method is relatively strong. 4. D I S C U S S I O N In the in situ tensile test in SEM, first few microcracks are initiated at small external loading, namely at small deformation. The microcracks do not propagate in m a trix at subsequent plastic deformation, but a number of new microcracks are occurred with increasing deformation. On the other hand the decrease in strength brought about by upsetting is not disappeared by post upsetting heat treatment. Therefore the decrease in strength will be owing to no metallurgical defects but microstructural damages which can not be recovered in subsequent heat treating. It is very difficult to observe such microcracks in the upset specimen, because the microcracks are filled with removed materials during polishing. Although such m i crocracks were not observed in the specimen after upsetting, its occurrence is quite possible, because the tensile stress is applied in radial and tangential directions during upsetting. In the case of upsetting by 50% reduction, 25% tensile strain is applied in radial and tangential directions, and 12.5% tensile strain even for 25% reduction. A

Figure 9. Typical microcracks occurred during tensile loading.

370

Figure 10. Micrographs of particles as recieved and observed in fracture surface.

debonded interface and a fractured particle will not be rebonded during subsequent heat treatment. Once the debonding and particle fracture are occurred, strengthening mechanisms of the particle dispersed composite, those are increasing in load carrying capacity and increasing in resistance for dislocation moving by dispersed particles, are invalidated. Then the matrix metal in the composite can be deformed without local restriction by the dispersed particles. Therefore the strength of the composite would be decreased after upsetting. 5. C O N C L U S I O N The change in strength after upsetting and post upsetting heat treatment were investigated for the particle reinforced aluminum matrix composite. Also the change in

371

microstructure of the composite under tensile loading was observed in the scanning electron microscope equipped with a deformation stage. From those experiments some phenomena were observed as follows. 1. The strength at room and elevated temperatures after upsetting at 300~ is lower than that before upsetting. 2. The decrease in strength is maintained after post upsetting heat treatment. 3. In the process of the upsetting at 300~ followed by it at room temperature, the strength is higher than upsetting at 300~ alone because of strain hardening of the matrix, although the strain hardening disappears by post upsetting heat treatment. 4. When the composite is subjected to tensile loading, some microcrack initiations are found at the interface between the particle and the matrix and in the particle at small external loading point. 5. The microcracks do not propagate in the matrix, but a number of new microcracks are occurred with increasing deformation. 6. The decrease in strength after upsetting can be explained by taking the occurrence of microcracks during upsetting into consideration. REFERENCES 1. N. Kanetake, N. Nakamura and S. Terada, J. Japan Inst. Light Metals, 40 (1990) 271. 2. N. Kanetake, Advanced Technology of Plasticity 1990, Japan Society for Technology of Plasticity, 1 (1990)53. 3. N. Kanetake, T. Choh and M. Ozaki, Sci. and Eng. of Light Metals, Japan Inst. Light Metals, (1991) 513. 4. N. Kanetake and H. Ohira, J. Mater. Pro. Tech., 24 (1990) 281. 5. N. Kanetake and N. Nakamura, J. Japan Soc. Technology of Plasticity, 31 (1990) 536. 6. N. Kanetake, M. Ozaki and T. Choh, J. Japan Soc. Technology of Plasticity, (1993) accepted. 7. D.J. Lloyd, Acta metall, mater., 39 (1991) 59. 8.. T. Mochida, M. Taya and D. J. Lloyd, Mater. Trans. JIM, 32(10), 931(1991). 9. N. Kanetake and T. Choh, Proc. 3rd Int. SAMPE Metals Conf., (1992) M414. 10. N. Kanetake, T. Choh and M. Nomura, Proc. 2nd Japan Int. SAMPE Sympo., (1991) 778. 11. N. Kanetake and T. Choh, Proc. 9th Int. Conf. Composite Materials, (1993) 634.

This Page Intentionally Left Blank

Materials Processing Defects S.K. Ghosh and M. Predcleanu (Editors) 1995 Elsevier Science B.V.

CRACK PREVENTION AND INCREASE OF BRITTLE MATERIALS BY COLD EXTRUSION

373

WORKABILITY

OF

H. W. Wagener and J. Haats Metal Forming Laboratory, University of Kassel, Germany

1. SUMMARY A titanium alloy, metal-matrix composites (MMCs) on aluminium base and free-machining (FM) steels were cold extruded and the main forming data (force, pressure) of the extrusion processes were determined. It was possible to avoid the generation of cracks during and after the extrusion process of these materials by increasing the hydrostatic pressure in the forming zone. Due to this measure both the ductility of these materials was increased and the plastic deformation of the workpieces was more homogeneous. The magnitude of counter pressure to be applied to the extruded part of the workpiece can be approximated by fundamentals of theory of plasticity.

2. INTRODUCTION

The application of titanium and titanium alloys as structural materials seems promising because of their excellent strength-density ratio and the corrosive resistance to chemical agents. Because of the high producer's price for these materials, it is necessary to employ production processes without any waste of material, i. e. cold bulk forming. The tendency to decrease the weight of structural components makes it imperative to use MMCs of aluminium alloy base. By variation of fibre and particle content, one can design the desired strength and ductility properties. In many cases, the final dimensions, the final accuracy, and surface finish of cold extruded workpieces are achieved by machining. To avoid the problems connected with the machining of cold forged low-carbon steels, one could make use of FM-steels which have an improved machinability even after cold forging.

374 3. WORKABILITY

The workability of a metal is defined as the degree of deformation that can be achieved in a particular metal working process without creating an undesirable condition, e. g. cracking. The workability is a function of the ductility of the material and of the stress and strain imposed by the forming process (Ref. 1). In the case of the (tx+/3)-titanium alloy TiA16V4, which is considered the most frequently applied titanium alloy, the high strength combined with the small ductility makes cold forming problematic. With regard to the FM-steels the MnS-inclusions act like non-deforming particles in a ductile matrix and also the lead inclusions intersect the metallic bonds of the matrix; the workability of FM-steels is considerably reduced. The same problem exists for MMCs. The fibrous and particle strengthening act like material separations in the matrix. During cold compression test a limit strain of ~ = 1.0 was found for this brittle type of metal. Due to these facts, generation and propagation of cracks must be expected during cold extrusion operations. To determine the main parameters of the cold extrusion of these brittle metals, experiments were carded out utilizing the following standard extrusion processes (Ref. 2): 1. 2. 3. 4.

Forward Bar Extrusion (FBE) Forward Tube Extrusion (Hooker, FTE) Backward Can Extrusion (Cupping, BCE) Forward Can Extrusion (FCE)

4. EXTRUSION EQUIPMENT AND MATERIAL DESCRIFFION

The extrusion tests were performed on a 1000 kN hydraulic press and on an eccentric press of a nominal force of 2500 kN. The punches and the dies were made of high speed steel. The dies were reinforced by double shrink rings. The inner diameter of the dies was 25.1 mm. Table 1 shows the lubrication systems of the tested materials. In the ease of MMCs the slugs for FBE and FTE are tapered, with half the die angle t~/2 = 45* on one face. Due to this method a reduction of the maximum value of extrusion force can be achieved. Figs. 1 and 2 show the metaUographic properties of the titanium alloy. The stretching of the tx-granules in the fl-matdx due to the extrusion process can be distinctly observed. This phenomenon must be taken into consideration during the interpretation of the experimental results.

375 Material FM-steels TiA16V4 MMC

Table 1:

Lubrication systems Zinc-phospate/Bonderlube Titaniumoxide/MoS2 Copper/MoS2 combined with MoS2-paste

Fig. 1: Micrographs of titanium alloy TiA16V4 in longitudinal axis (400:1)

Lubrication systems for the tested materials (Refs. 3 and 4)

Fig. 2: Micrographs of titanium alloy TiA16V4 in transversal axis (400:1)

The rods of the examined MMCs (type A) were made by the hot extrusion operation of an aluminium alloy produced by a powder-metallurgical process. The MMCs of type B were produced by mixing A1203 particles into the liquid aluminium alloy, followed by a hot extrusion process after solidification.

Fig. 3: Micrographs of the MMC A1MgSil Fig. 4: Micrographs of the MMC A1MgSil (15 Vol. % SiC) in longitudinal axis (400:1) (15 Vol. % SiC) in transversal axis (400:1) Figs. 3 and 4 show the metallographic properties of the MMCs as exemplified by the A1MgSil (15 Vol. % SIC). A longitudinal orientation of the SiC-particles due to the extrusion process can be observed.

376 Furthermore, there are areas both without any particles and with particle accumulations. In these areas there is low adhesion between matrix and particles, so that cracks inside the material occur.

Figs. 5 to 8: Micrographs of free-machining steels as-received in longitudinal axis (75:1) Figs. 5 to 8 show the metallographic properties of the four FM-steels under test. Inherent in steel 35S20 (AISI 1139) are slightly larger and more concentrated inclusions than in steel 45S20 (AISI 1146), a fact which is detrimental to the extrudability of 35S20.

Material

Mech. Properties

Chemical Composition (Weight %)

R,, 1~,o.2 As C

Si

Mn

P

S

Ni

Cr

Cu

[MPal

[%1

0.02

796 573

8.5

1060 989

16

45S20 (AISI 1146)

0.44 0.27 1.02 0.07 0.23 0.04 0.03

TiAI6V4

0.13 0.02 0.18 0.01 0.01 6.44 4.0 Balance

Fe

Cu

C

Fe

O

Si

N

Mg

H

Mn

A1

Zn

V

Cr

Ti

AI

A1MgSi 1

0.06 0.27 0.86 0.64 0.62 0.04 0.15 Balance

203

AISil2

0.08 0.27 12.4 0.01 0.10 0.09

-

Balance

198 105 17.5

A1MglSiCu

0.20 0.70 0.60 0.90 0.15

-

Balance

182 100 20.0

-

138 18.5

Table 2: Chemical Composition (Weight %) and Mechanical Properties of TiA16V4, 45S20 and MMCs (matrix material).

377 Produced by

Material

R. [MPa]

R~,2 [MPa]

A5 [%]

A

A1MgSil / 15% SiCp

209

162

7.4

A

AIMgSil / 25 % SiCp

204

161

3.0

A

AISil2/ 15% SiCp

229

140

5.4

A

AISil2 / 25 % SiCp

194

135

2.1

B

A1MglSiCu / 15 % Al~O3

216

113

16.5

B

A1MglSiCu / 20% A1203

234

95

9.0

A: Dr. Kainer, University of Clausthal, Germany; B: AMAG, Ranshofen, Austria

Table 3: Mechanical properties of the tested MMC materials. The composition and the mechanical properties of the materials are described in table 2. Table 3 shows the mechanical properties of the aluminium matrix materials after reinforcement. The flow curves of the tested materials are plotted in Figs. 9 to 11.

~'400 o. " " 360

"~'

T " 20-C

I.L

_.~.~..

I

l

_ :c e.2,ss t -

L~.~---r---,~-~, tn 320 ~ / 03 uJ tr 280 //r/--- a: AIMgSilll 5 Vol. % SiC I-~/ b: AIMgSi1/25 Vol.% SiC 03 c: AISi12/15 Vol.% SiC N 240 d: AISi12/25 Vol.% SiC O, 200 ' I l I 1 LL 0 014 0.8 1.2

r

TRUE STRAIN Ch ["]

Fig. 9: Flow curves of MMCs (type A).

6OO

=.500

Composite: Matrix: AIMglSiCu, A12Oa.Partide.Reinforcement: 1" 0 Vol. % 2:15 Vol.% I 2

1~400 03 300 - t 200 100~ LL

0

0

,

I

.

i ! |

I-II1'

UNI-AXIAL ISOTHERMAL COMPRESSIONTEST T-20"C; ~=4$'t LUBRICATION: PTFE

o.a

1.2

TRUE STRAIN E h [']

Fig. 10: Flow curves of MMCs (type B).

It is noticeable that the limit strain of the brittle MMC has an unexpectedly high value. Certainly this phenomenon is the result of the heterogeneous behaviour of this material. This high limit strain cannot be reached in the case of another distribution of strengthening particles.

378 5. EXTRUSION FORCE-PUNCH TRAVEL DIAGRAMS

The main characteristics and forming data of cold extrusion processes can be studied by means of the extrusion-force vs. punch-travel diagrams. In Fig. 12 such a diagram for FBE is plotted for the titanium alloy TiA16V4. At the beginning of the extrusion operation, there is a steep increase of the extrusion force, then the period of steady-state flow can be observed. For larger values of strain ~^, a reduction of the force takes place after the maximum, due to decreasing frictional forces and increasing temperature of the material in the forming zone.

~' 1500

1~'-12~176 / /

~

900 / ~ ~ -

I---

600 V ~

~:

300 ~

q

0

u.

0

~ Ti'AI6VJl I I I ~ ..45S20

~ ~~

~_~.,

~-------~~1

I ~

-.

E - 0,9 S

~

, 1 T = 20 ~C

,

0.4

,

35s20

-~10S20 I

9SMnPb28 ~,

I

MMC

0.8

TRUE STRAIN Eh [']

Z

1000

~.

8oo

=..._,,

1632

W

~ 600

1224

o

4oo

816

200

4O8

u• Z

I,

1.2

2040 ix.

0 U.l

13.

co

03 UJ n" n

o

o 5 ~o ~s 2o 2s

PUNCH TRAVEL

Su [mm]

Fig. 11: Flow curves of the tested materials. Fig. 12: Force-travel diagrams for FBE of TiA16V4, do = 25 ram. In the case of small to medium values of strain there is an inhomogeneous distribution of strain over the cross-section of the workpiece (Refs. 4 and 5), so that the actual strain in the outer zone of the workpiece is of high magnitude. Due to this fact the formation of a concave face form of the extruded part of the component can be observed. Fig. 13 shows the distribution of residual stresses in a workpiece of a low carbon steel for FBE with E^ = 0.7 (Ref. 8). Additionally, the values of actual strains over the cross-section of the extruded piece can be observed. In the outer zone of the workpiece the actual strain is ev --- 1.2. This is nearly twice the value of the actual strain in the inner area of the workpiece. The cracks due to the extrusion process of TiA16V4 appeared a few days after the process. Upon consideration of the minimal creep-inclination of this material (Ref. 6), the retarded occurrence of cracks is caused by an embrittlement due to the strain-aging process. The diagram in Fig. 14 shows the respective force-travel characteristics as exemplified by the free-machining steel 35S20.

379 Because of the high brittleness of the MMC, the workpiece in the first instance of the experiment shows cracks comparable to those of the extruded free-machining steels (Refs. 2 and 7). At the end of the extrusion process the workpieces crumble totally due to the combination of low workability and high gradient of actual strain over the cross-section of the extruded piece.

......

O

WORKPIECE

[rc"

i

7.1 na/a F

~ lr~- 4.4 mini e ira= 0.7 mml

RES,DUA' STRESSES ---1

800

~-- LOAOSrRSSSSS

AXIAL

STRESS

O z [MPa]

-800

TANGENTIAL

0

~""

lit'

STRESS

ACTUAL 1.o STRAIN [-I

~

l

,.

.=~J

--I

--

.....

~/

I ---'--T

~

'----"

~"

=

I

I~%,

|

rlia rb! ~r ~C" c

_/:_. ~_._--..!

0

10

20

30

-

I '%-~1

..!.......:!,

AXIAL DISTANCE

'f

~

o.,,.,,:-J- j --r lrL 4-0

.

1 ~" ~

____.______L__-t__~_, .....

"

(It [MPa]-1000

Ev

_ ~

I n

r~ rl~

z

40

[ram]

50

Fig. 13: Residual stresses and actual strain Ev for FBE of a low carbon steel (Ref. 8).

6. CRACK PREVENTION To increase the workability, i. e. to avoid cracks resulting from cold extrusion, special methods are explored which improve the ductility of the material and the state of stress during and after extrusion (Ref. 7). In the case of the brittle materials under test, the most promising measure is the employment of a counter force during extrusion, because the workability is improved if the state of stress is more to the compressive side (Fig. 15).

380 Nearly all extrusion tests with the four types of FM-steels produced cracks if the material was extruded in the as-received condition (not annealed) and if the normal cylindrical slugs were used. If the hydrostatic pressure acting in the forming zone is increased by the application of a counter force by means of a counter punch, the workability is also increased (Fig. 15).

Z

+" CRACKS

IX,

-=-12501 MATE-R,AL"

[2550

1000 | AS.RECEIVED

n." 750 1 ,,'t"-~,

+

~

L.E^'I-2. I 1 5 3 0 ~

5001 I//I EA'0"7~ :05 rr"

I IP..Z tI 25O~ri o

~_t'" islo -i~ -~,,F---'-

,~r NORMAL DIE EAT

0

5

AFi'ER STENGER

o=-(o~+ o=)12

10 15

PUNCH TRAVEL

0.3

20

L0

25

Su [mm]

Fig. 14: Force-travel diagrams for FBE of 35S20 in the as-received condition, do = 25 mm.

tO~tl

/ o,-o=>~~~ ,

COMPRESSION I \

TEST 1 \

IT:

0

uJ

0.33

o~/a

Fig. 15: Dependence of the extent of deformation on the specific mean stress (Ref. 9).

Fig. 16 shows the force-travel diagrams of the FTE-process when a counter force is applied. The FM-steel can be cold extruded crack-free due to the employment of the counter-pressure during extrusion (Ref. 7).

z 1000

CYUNDRICAL BORE DIE

Fg: COUNTERFORCE a: WITHFg u. 800 b: WITHOUTFg

~u 3.2

z

iii

n" 600 O u. 400

O

~

2.4

,x emum,r

. Z(o~--)-l.ls ~..

TIAI6V4

5s20 (Aim1139)

(::> (D 2 o o - 0

I

'

Z

n."

4. 0 a: WITHOUT COUNTER PR. b: COUNTER PR. (200 MPa)

0

5

10

15

PUNCH TRAVEL

20

25

Su [mm]

Fig. 16: Force-travel diagrams for FTE of 35S20 with counter pressure, do = 25 mm.

0.0

.

-2.0

Om/~

.

.

-1.0

.

0.0

['1

Fig. 17: Momentary strain vs. relative mean stress diagrams for FBE of TiA16V4.

381 Due to this fact, the employment of a hydrostatic counter force for crack prevention of brittle materials seems most suitable. Therefore, this type of crack prevention is applied during the forming processes of TiA16V4 and MMCs. The magnitude of the counter pressure to be applied to the extruded part of the workpiece for the prevention of cracks can be approximated by the following method (Fig. 17): If the function of workability (strain to fracture) versus the (r=/; -ratio (ratio of hydrostatic pressure to flow stress) is considered to be linear, according to (Ref. 9) the values of the gradients of the linear functions (straight lines) can be assumed to be between 1 and 3. The larger value belongs to steels which show a good ductility even under uni-axial compressive stress, e. g. for St 37 (AISI 1015) the gradient is 3.0 and for X8Crl7 (AISI 430) it is 2.4 (Ref. 9). Of course, the value of the counter pressure cannot be increased deliberately due to limitation

.--. 4. 0 a.: WITHOUT COUNTER PR.

b: COUNTER PR. (650 MPa)

>3.2 Z

_.1

,< :3

,<

I

2.4 1.6

A E~c-r

X(o ) "~

0.8 AIMgSill

15 Vol.% SiC

O. O -

-P_O

b

w

Z

n-' I-.

J

9

,-., 4. 0 a: WlTI~OUTcOUNTER PFI. 1 ,,j..., b: COUNTER PR. (650 MPa) I >

._I

<

a -1.0

am/O Fig. 18: Limit strain vs. relative mean principal stress for FBE of A1MgSil/15 Vol. % SiC.

I--

0.0 [-]

3.2

2.4

,:-c~

I

ZONJE~I

1.6 0.8

0.0

bi

g' i

I lsv~

-2,0

" a -1.0

1 0.0

[-]

Fig. 19: Limit strain vs. relative mean principal stress for FCE of A1MgSi1/15 Vol. % SiC.

of permissable punch and die stresses. Considering that the dependence of the workability on the aJ~-ratio is approximately linear, the determination of the critical limits of the materials by using two different upsetting operations is more accurate. The ftrst upsetting process is an uni-axial one. The resulting a,./'~-rafio is -0.33. The second point of the critical line can be found by determination of the workability in a multi-axial upsetting test. The a=/~-ratio belonging to this determined workability can be calculated by considering the actual workpiece dimensions. In Figs. 17 and 18 the limit-lines of the examined materials are plotted for FBE. The determination of the maximum actual strain is the result of the relation of hardness after extrusion to flow stress of the material (Ref. 10). By consideration of this relation one can find the values of actual strain over the cross-section of an extruded component.

382 In the case of TiA16V4 (Fig. 17) the maximum actual strain in the outer zone of the extruded workpiece is determined as ev = 0.9 for FBE with eA = 0.7. The minimum counter pressure to avoid crack generation due to the concave face form of the extruded part of the component has been calculated as 200 M~a. Because of the crack formation due to FBE of MMC with e^ = 0.7 and the knowledge of the critical line of this material, the maximum actual strain must be approximately ev = 1.2 (Fig. 17). The limit curve of AMgSil/15 Vol. % SiC for FBE is shown in Figure 18 and for FCE in Figure 19. The limit curve (line) is determined by compression tests of the A1MgSil/15 Vol. % SiC composite. The negative slope of the limit line for this MMC has the value 0.44. In the case of FBE (Fig. 18) for conventional extrusion (without counter pressure, a), the value of a=/~ (relative mean stress = mean stress/flow stress) is - 0.57; for an actual strain at the outer radius of ev = 1.1, cracks must be expected. If a counter pressure of CP = 650 MPa is applied for the FBE operations of this MIVIC (b), the value of the relative mean stress is reduced to -1.2.

_

,

,

,

Z__~1000 ] MATERIAl_ .I.iAI6V4

,,=

8 o o t T - 2ooc, , A -

0.7

2o4o =-

i

1632

~

'--'Zl000 ~

EA= 0"7

800

2040 ~' 1632~

600 /[~---t t \l~ul 1224~:::) ~ 400 /-Fu:EXTR.FORCE - 816 U) i rFg: COUNTERFORCE ~ ff COUNTERPRESSURE: 1 I I/Fg tr" 0 ' ' , , " 0 ~ 0 5 10 15 20 25 LU PUNCH TRAVEL Su [turn]

r! ~600 11 1 IFORCE'= 1224 I /Fu: EXTR. U. 4001--/Fg:COUNTERFORCE 816 I / COUNTERPAESSURE: ~ ~ 650MPa 408 ~ ' Fg cc 0 I 0 ~ ~ 0 5 10 15 20 25 UJ PUNCH TRAVEL Su [ram]

Fig. 20: Force-travel diagrams for FBE of TiA16V4 with counter pressure, do = 25 mm.

Fig. 21: Force-travel diagrams for FBE of A1MgSil / 15 Vol. % SiC with counter pressure, do = 25 mm.

ootyl i

For the same value of the actual strain (ev = 1.1), a certain safety margin (approx. A~ = 0.44) can be observed to produce crack-free components. This diagram proves that, due to the application of a counter pressure, the extrusion process is far off the limit-line, so that failure of the material can be avoided. For the FCE operation (Fig. 19), the limit curve is identical, since specific for one material. In cupping, very much greater values of strain are applied than in FBE. The maximum value of strain is at the bottom edge (bottom/wall) of the inner surface of the cup. Here the actual strain is approx. Ev = 2.1. It is very often observed that also at these areas the main cracks occur if high carbon steels, for example, are extruded by cup extrusion.

383

If the cupping is performed without counter pressure (conventional extrusion), the relative mean stress is -0.78 (a); this indicates that under this state of stress in the forming zone the material must crack because the actual stress is very much above the limit curve. If a counter pressure CP = 270 MPa is applied to the ring-like upper surface of the cup, the relative mean stress is reduced to -1.9 (b). In this case the actual strain is just below the limit curve. Hence, it may be assumed that components without cracks can be extruded. In Figures 20 and 21 the force-travel diagrams for FBE with r = 0.7 are shown. Additionally, the employed counter force can be obtained from the diagrams. In all examined cases it was possible to produce crack-free workpieces of these brittle materials.

INFLUENCE OF MATERIAL C O M I ~ S I T I O N

It is obvious that the different types of matrix material and also the percentage of reinforcement material will have an influence on the forming parameters and on the strength and hardness of the cold forged components. ,Z. . . , --.91000 "~ U.. LU 800 "

600

CYUNDRICAL SLUGS: BORE DIE TAPERED ~ " 2050 5 6 ~ ~ 1640 ~ , looo '

U..Z 400 ,'1 2 3 ~3 820 O ~ / 2 5 Vol.% SiC 200 . ~ ) 1 to 3: AIMgSil 410 ec 4 to 6:AISi12 o

,#

, .

0 4 8 12 PUNCH TRAVEL

,

o

16 20 Su [mm]

Fig. 22: Force-travel diagrams for FBE of two different matrix materials, do = 25 mm.

,---, ~ .~ ~"

CYLINDRICAL BORE DIE SLUGS:TAPERED 1000 ~ ~ ~ :~ i--7 5 6 O. U. 800 LU W == o 6oo w~ nn ~ tu

o

u. 400 z ////~ O r 200 D o

u.I

,i

i

~" IX. 2050 :~ 1640 nlil

1 820

AISi12 + SiC: 410 1 to 3:15 Vol.% 4 to 6:25 Vol.% 0 o

o"' =

tu tr n

,.

0 4 .. 1 ' - - . . 20 PUNCH TRAVEL Su [ram]

Fig. 23: Force-travel diagrams for FBE for two different percentages of SiC, do = 25ram.

The expected brittle behaviour of the matrix material A1Sil2 is verified by the force-travel diagrams. Figure 22 represents the comparison between A1MgSil and A1Sil2 for different values of strain. Both materials are reinforced by 25 Vol. % SiC. For the extrusion of A1Sil2 (curve 6) with the strain of e^ = 0.7 a maximum value of extrusion pressure (1505 MPa) is required while the corresponding value for the alloy A1MgSil is 1416 MPa only.

384 The difference is approx. 6 %. For lower values of strain (curves 2, 3, and 4), the differences are not so clearly defined. For the MMCs matrix A1Sil2, Figure 23 indicates that for a strain of e^ -- 0.7 the extrusion pressure p has to be increased from approx. 1440 MPa to 1505 MPa if the percentage of SiC is increased from 15 % to 25 %.

CYLINDRICALBOREDIE SLUGS:TAPERED "~ 1000 CP - 650 MPa 2050 ~: ::3 .l~ PVD-COPPER " 8001640 ~ u.I 1 :~3 0" 0n- 600~ 1230~ ~"

u. 0 :~ Ix:

1,~

~"-' "~ 1000

CYLINDRICAL BORE DIE SLUGS: TAPERED '~" 2500 ... ,.

''= 800 2000 Ill o 600 igl . . - r - 1500tu /%0.; posite: L_Ell /1 PM-AISi12 + O 820 to ' II 25 Vol.% SiCp 1000~176 o3 ,,z 400 oo 400 i LU UJ ,-1.25/j cP- i P, 1" 0 Vol.% AI203 410 a. rr ~O 500 0CO. 200 2:15 Vol.% ,&J203 2oo 1,/_ I l,Fgl 3:20 Vol,%/~J203 0 0 n~0 o ! |i 0 4 8 12 16 20 ..~ 0 4 8 12 16 20 PUNCH TRAVEL Su [mm] PUNCH TRAVEL Su [mm]

Fig. 24: Force-travel diagrams for FBE of MMCs (type B), do = 25 mm.

Figure 24 represents the force travel diagrams for the B-type MMCs which demonstrates that, with increasing value of A1203 content, the extrusion pressure increases, too. The values of maximum extrusion force and maximum extrusion pressure are nearly the same as those of A12Sil2, 15 Vol.% SiC, Figure 23, curve 5. It is assumed that forward tube extrusion is a more difficult cold forging process than FBE, because of the additional frictional forces acting on the mandrel and because of the phenomenon of fish-skin surface on the inner bore. Figure 25 shows the force-travel diagram of an A-type MMC for FTE for a strain e^ = 1.25 and with a counter pressure of CP = 400 MPa. The characteristic of the diagram corresponds to the diagram of conventional materials used for cold extrusion, e.g. low carbon steel.

Fig. 25: Force-travel diagrams for FTE of A1Si12 / 25 Vol. % SiC with counter pressure, do = 25 mm.

CYLINDRICALBOREDIE SLUGS:TAPERED 1000 ~ ! MATRIX:AIMgSil 2050 u. = 800 L--[-I~iL1:15 Vol.% SiC ! ~| 2:25 Vol.% SIC " 1640 I~. ~"

o" ' 600 0 e^zU" 400 0 nLu

I

200 0

..... 0 4 812 PUNCH TRAVEL

I

1230~

/1

820 410 m n

0 16 20 Su [mm]

Fig. 26: Force-travel diagrams for FCE of matrix A1MgSi 1 for two values of SiC percentage, do = 25 mm.

n=

385 The surface of the inner bore shows no sign of the development of fish-skin. The inner and outer surface of the componentes were free of cracks and the surface quality corresponds to that value which can be expected in general in cold extrusion. If, similar to the FBE and FTE tests, for Forward Cup Extrusion (FCE) a counter pressure of 270 MPa is applied by means of a ring-shaped counter punch, the generation of cracks can be avoided. This phenomenon is all the more amazing as cupping requires the highest value of extrusion pressure and maximum expansion of the components surface takes place, thereby putting maximum load on the tribological system. Fig. 26 shows the force-travel diagrams for A1MgSil with 15 Vol. % and 25 Vol. % SiC. As expected, the forces increase corresponding to the increased value of particle content. In general, the characteristic of the diagrams is similar to those of normal homogeneous materials, like aluminium, low carbon steel or titanium (Ref. 3).

CONCLUSIONS

Brittle metals or alloys can be successfully subjected to cold extrusion in the same way as ordinary low carbon steel. Due to the employment of a counter force during extrusion processes, an increase of hydrostatic pressure in the forming zone can be observed. By this measure, the workability is improved because of the displacement of the mean stress more to the compressive side. Additionally, due to the inhomogeneous strain distribution in the cross-section, the residual tensile stresses in the outer zone of the extruded component are clearly decreased by the application of a counter pressure by means of a counter punch. Because of the combination of increasing the workability and decreasing the gradient of actual value of strain over the cross-section of extruded workpieces, crack formation can be avoided by the employment of counter pressure during cold extrusion processes.

REFERENCES

1 G . E . Dieter, Introduction in Evaluation of Workability, Metals Handbook 9th Edition, Vol. 14, Forming and Forging, ASM International, Ohio, 1988, p. 364/372. 2 H.-W. Wagener, J. Haats, Cold Extrusion and Machinability of Free-Machining Steels, Journal of Materials Processing Technology, 24 (1990), S. 235/244. 3 H.-W. Wagener, K.-H. Tampe, Beitrag zum KaltflielIpressen von Titan, Reihe 2, Betriebstechnik, Nr. 101, VDI-Verlag. 4 K. Lange, Umformtechnik- Handbuch ffir Industrie und Wissenschaft. Band 1: Grundlagen, Springer-Verlag, Berlin, Heidelberg 1984. 5 K.l.ange, Umformtechnik- Handbuch ffir Industrie und Wissenschaft. Band 2: Massivumformung, Springer-Verlag Berlin, Heidelberg 1988.

386 6 U. Zwicker, Titan und Titanlegierungen, Springer-Verlag Berlin, Heidelberg, New York 1974. 7 H.-W. Wagener, H.-J. Engel, J. Haats, Kaltfliel3pressen von Automatenstahl- preBfertige Teile ohne Risse, Industrie-Anzeiger 112 (1990), Heft 28, S. 38/40, Heft 33, S. 44/48, Heft 39, S. 38/42 und Heft 41, S. 70/74. 8 A.E. Tekkaya, Ermittlung von Eigenspannungen in der Kaltmassivumformung, Berichte aus dem IFU Stuttgart, Nr. 83, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo 1986. 9 H. Stenger, Influence of the State of Stress on the Ductility of Metals (in German) Dr.-Ing. Thesis, Tech. Univ. Aachen 1965. 10 H. Wilhelm, Untersuchungen fiber den Zusammenhang zwischen VickershS.rte und VergleichsformS.nderung bei Kaltumformvorg~ngen, Berichte aus dem IFU TH Stuttgart, Nr. 9, Girardet 1969.

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

387

A data base for some physical defects in metal forming processes M . M . A I - M o u s a w i , , A . M . D a r a g h e h , and S . K . G h o s h b. 9School of Engineering, Staffordshire University, Beaconside, Stafford, Staffordshire, ST18 0AD, England. b International College of Engineering, GKN Automotive Postfach 1951 Alte Lohmarer Strasse 59, 5200 Siegburg, Germany. Abstract Some physical defects in metal forming processes such as rolling and forging, which are also known as bulk metal forming processes, are examined to determine the reasons for their occurrence and to suggest remedies. A database is being developed for access to the information about each physical defect in a particular forming process. 1. INTRODUCTION There are a wide range of physical defects which occur during metal forming processes. These defects, which may occur on the surface or be internal, are undesirable not only because of the surface appearance, but because they may adversely affect the strength, formability and other manufacturing characteristics of the material. A method of dealing with these defects is of considerable interest because the cost to industry in terms of lost time and material is essentially proportional to percentage of material being rejected. Some physical defects in metal forming processes such as rolling and forging, which are also known as bulk metal forming processes, are examined in this work to determine the reasons for their occurrence and to suggest remedies and also establish some common criteria leading to a more successful forming process. For instance a careful balance of such factors as the control of temperatures, intermediate annealing, control of scaling, lubrication, the condition of roll surfaces, angular speed and mill stiffness and finally soundness of ingot would achieve a more successful rolling process. The greatest advantages in combating rolling defects have been made by improving the melting and casting techniques. It is important that the workpiece be free of weakening features on the surface and along the central plane and possess some basic ductility. The causes for the occurrence of each defect will be discussed. Limited ductility, variation of stresses across the width of the rolled material and uneven deformation at the edges are some of the main reasons for such defects. Some forging defects are dealt with in the same manner. Defects in forging reduces its strength. They occur for a number of reasons including faults in original metal, incorrect die design, improper heating and lubrication. One of the defects associated with forging is centre burst which is a rupture in the centre of the billets and sometimes occurs when temperature of metal increases significantly as a result of large, rapid reduction. Also during forging

388

operations, substantial tensile stresses are produced in addition to the applied compressive stress. Where the material is weak, possibly from pipe, porosity, segregation or inclusion, the tensile stresses can be sufficiently high to tear the material apart internally, particularly if the forging temperature is too high. Professors W. Johnson, S. K. Ghosh and A. G. Mamalis [ 14] co-authored several papers on defects in the processes of metals and composites and assembled their results in tabular form. A database is being developed for access to the information about each physical defect in a particular forming process. No such database currently exists and the information about these defects are not easily accessible. It is hoped that by creating a database most of the information can be rapidly obtained and this would be of great interest to metal forming engineers and workers, saving time and reducing costs considerably. The database could also be useful for the academic teaching of the subject of material processing mechanics. 2. DEFECTS IN ROLLING Defects may be present on the surfaces of rolled plates and sheets or they may be internal and are undesirable not only because of surface appearance but because they may adversely effect the strength, formability and other manufacturing characteristics. The greatest advantages in combatting rolling defects have been made by improving the melting and casting techniques. It is important that the workpiece be free of weakening features on the surface and along the central plane and posseses some minimum, although qualitatively as yet undefined, basic ductility. Successful rolling practice requires a careful balance of such factors as the control of temperature, intermediate annealing, soundness of ingot, control of scaling, lubrications, the condition of roll surfaces and angular speed and mill stiffness. 2.1. Edge cracking Fracturing is observed in the rolling of slabs at the edges where longitudinal tension may develop under certain rolling conditions. In hot rolling, steel is deformed principally in plane strain compression, except near the edges where a rounded profile develops unless the mill is equipped with edge rolls. Thus, in a multi pass schedule when the edges of the slab or plate are free to spread, the edges are not compressed by the rolls but are forced to elongate with the bulk of the steel to maintain continuity. In materials with low ductility the secondary tensile stresses are sufficient to form edge crack. Causes:- Edge cracking was examined by Schey [5] and three causes were outlined for its occurrence: (i) Limited ductility:- Latham and Cockcroft [6] demonstrated the effect of material ductility determined by a tensile test on the onset of edge cracking in their experiments on a large number of alloys with a wide range of ductilities. Not unexpectedly, as the ductility increased, the cracking reduction increased. A material of almost infinite ductility, such as gold, will not show edge cracking under practically any rolling condition. (ii) Variation of stresses across the width of the rolled material:- While essentially plane strain conditions exist in the central portions of the rolled slab or sheet, lateral flow (spread) is bound to develop near the edges. This results in a gradual drop of interface pressure towards the edges. Therefore, the edges of the strip deform longitudinally only because they are bodily attached to the bulk of the strip and yielding is initiated by a combination of compressive and

389 secondary tensile stresses. In other words, tensile stresses are again responsible for edge deformation and eventual edge cracking. When barrelling occurs due to inhomogeneity of deformation in the thickness direction, stresses become much higher. Very often edge cracking develops at the later stages of hot rolling when the temperature is lower. When the h/l ratio is below unity, the side surface develops single barrels. Chilling of the comer may transfer the crack location to the comers, (h=slab thickness and l=length of contact). (iii) Uneven deformation at the edges:- When rolling a relatively thick slab with light reductions, the deformed zone penetrates to a fraction of total thickness. The surface of the slab deforms in all directions including laterally, while the centre remains at an unchanged width. In further passes, the "overhanging" material is not directly compressed but it is forced to elongate by bulk deformation of the neighbouring material. Thus, it is subjected to high tensile stresses which lead to cracking in a material of limited ductility. Early cross-rolling with light passes aggravates the situation as shown by Kasz and Varley [7] and may cause edge cracking even in relatively ductile materials such as aluminum. Also when the slab is relatively thin and the pass reduction heavy, the edges barrel and are forced to elongate by the dragging effect of the bulk of the material, again causing the edge cracking of materials with limited ductility. Dodd and Boddington [8] suggested that the cause of edge cracking in cold rolling for a given material are best categorized in terms of the initial width/thickness ratio of the work material. In wide strip rolling, where the strips have a width-thickness ratio often greatly in excess of 50, edge cracking is caused by using incorrectly camber~ rolls. This is essentially a problem of overcoming the mismatch between the ingoing strip and roll gap profiles. For narrow stock with a width-thickness ratio of less than about 8, both the lateral spread and the initial edge shape determine the induced longitudinal tensile stresses at the edges and therefore determine the onset of cracking. Thomson and Burman [9] examined an edge cracked sample from a large number of hot-rolled A1-Mg alloy ingots, as shown in figurel, and concluded that most edge cracking which occurs during the industrial hot rolling of AI-Mg alloys can be attributed to the presence of a segregation band at the edge of the ingot. This segregation band contains many inclusions and precipitation particles which initiate and assist the propagation of the edge cracks under the influence of the stresses produced in the edges during rolling. The effect of lubrication on the onset of edge cracking was shown by Latham and Cockcroft [6] to be of secondary importance. This result has been conf'tmaed by Oh and Kobayashi [ 10] in their multi pass rolling experiments on rectangular bars of 7075 AI alloys. Remedies:- Edge cracking can be combatted by either improving the workability of the material or changing the process itself. The first method can be extremely effective as shown by the success of the aluminium industry in improving the quality and homogeneity of strong alloys such as A1-Cu and A1-Mg. Similar successes have been achieved with vacuum-melted tool steels and vacuum-arc or electroslag remelted super alloys. Since the most severe secondary tensile stresses are due to double or single barrelling, the choice of the M ratio is critical. Continuous casting yields thinner slabs and this allows rolling with more favourable h/1 ratios. Highly inhomogeneous deformation is, however, unavoidable in the early passes of rolling thick slabs and the resultant double barrelling leads to large trimming losses and edge cracking. The obvious remedy is casting the slabs with a V-shaped edge, or with a moderate curvature (R=l.3 has been suggested by Salt), so that double barrelling super imposed on the starting shape results in a straight edge. If the material is of very limited ductility, there is of course danger of cracking in the cast ingot before deformation can be equalized. The worse

390 possible condition is given by a cast structure with concave edges. Conversely, problems are to be expected when the cast ingot has an octagonal shape because the side portions of the ingot will be elongated purely by secondary tensile stresses and are bound to crack. Rolling in closed passes of blooming mills also limits the total spread to some extent. However, possibilities are sharply limited by the danger of pinching the material at the pass line. As a consideration of three dimensional configuration will reveal, the roll pass is truly closed only in the plane through the centre line of rolls and the pass opens up towards the entry side. Therefore, spread must be allowed in the design of pass, otherwise material will be pinched and the workpiece will be wedged into the roll grooves and it may even wind around the rolls. Schey [5] suggested that spread in the roll gap can be prevented only by special devices. In the so called edge-restraint process rigid bars are f'mnly guided in the grooves of the rolls so that they move together with the deforming workpiece. A lateral compressive stress is induced, spread is restricted and the edges are maintained square. By providing deeper grooves, the rolls can be closed down in the customary fashion, and the slab may be thinned down in a succession of passes. Since no spread is allowed, secondary tensile stresses cannot develop and materials of poor workability can be successfully rolled. This was shown for an A1-8%Mg alloy, for titanium alloys and nickel base super alloys. Thomson and Burman [9] concluded that edge cracking can be controlled by optimizing both the ductility of the material being rolled and the deformation conditions to which it is subjected. High material ductility is obtained by the removal or depletion of possible embrittling agents by degassing or reaction combined with the application of the correct heat treatment and rolling temperatures. They also suggested that the initial shape of the ingot and the rolling schedules must be altered by trial and error to maintain an approximately rectangular slab cross section.

2.2. Alligatoring A fracture situation of practical importance, although very rare, is the rolling defect of alligatoring. In the rolling of slabs, particularly of aluminium alloys, the work piece may split along a horizontal plane on exit, with the top and bottom parts following the rotation of their respective rolls. Figure 2 illustrates alligatoring in an A1-8%Mg alloy slab. Causes:- Alligatoring occurs only in materials of limited ductility, such as AI-Mg alloys of higher Mg content and some of the Zn and Cu base alloys. It develops at some stage of hot rolling, in the production example quoted by Kasz and Varley [7] at h/l= 1.33 (slab thickness to length of contact), by Meadows and Pearson [11] at h/l= 1. 55 and in the hot rolling experiment of Schey [5] on A1-8Mg billets at h/1---0.5-0.7. Alligatoring at high h/l ratios (say above 1) is difficult to explain. Perhaps, as suggested by Kasz and Varley [7], inhomogeneous deformation leaves the centre layers weakened by porosity in the brittle as cast condition. Once a crack is initiated, it propagates very rapidly because the two halves of the billet now curve around the rolls, as though each half were deformed between the rolls and a central roll of infinite diameter. From consideration of material flow, it is much more likely that alligatoring should occur when deformation is more severe in the centre of the billet, that is, at low h/1 ratios. Meadows and Pearson [11 ] pointed out that the extrusion effect would result in tension normal to the rolling plane opening up the slab nose. Polakowski [ 12] suggested that in the initial stages of rolling (at high h/1 ratios) existing defects are enlarged and perhaps cracks are initiated in the centre section of the as-cast structure. On further rolling, when the central plane of the slab is more heavily deformed, the tensile stresses generated on the surface and compressive stresses produced in the centre would form a torque couple that

391 opens up the slab along its already weakened centre. The position of the pass line has a marked and often decisive influence on the occurrence of alligatoring. When the slab is entered into the roll gap with its centre plane above or below the centre line of the roll gap, the issueing material curls. In addition to the stresses imposed by curling, a shear stress develops also along the centre plane and this may contribute to opening up of the nose. Zhu and Avitzur [13] suggested that alligatoring occurs usually after severe successive reduction without anneal. Remedies:- Alligatoring initiates at the leading end of the slab, and can be prevented even under otherwise damaging h/l conditions. If splitting of the nose is to be prevented, one of the practical remedies is to taper the end of the slab so that the critical reduction is reached some distance away from the leading edge. The undeformed leading edge then acts as a clamp and in the course of further rolling the damaging stress pattern disappears. In the rolling of AI-Mg alloys the better centre quality of semi-continuously cast slab brought great improvement relative to chill cast ingots. The cross rolling of slabs cast with a pronounced V-edge is also effective as it neutralizes the tendency to the development of fish tail and eliminates the resulting stress concentration.

2.3. Fish-tail Unwanted end shapes can be developed particularly in the rolling of slabs and blooms and can have important consequences for production cost. The term encountered at the front end of an initially square ended ingot or slab after rolling is termed overhang, and that at the rear when well developed combines overlap and fish-tail. Causes:- Overlap is due to folding over of the ingots head and tail end in the direction of its thickness and fish-tail is a result of folding over of the ingots head and tail end in the direction of the width. These ends are cut off, constituting a crop-loss, and may account for about 5% of a total through put of bloom or slab weight. Even a small average percentage reduction in crop loss of the annual volume of production of a mill leads to substantial material and monetary savings. Remedies:- There are several factors that effect the slab yield, but particularly important among those are such defects as overlap and fish-tail. Matuzald et al [ 14] mentioned a new method of slab rolling developed by Kawasaki Steel Corporation developed for the purpose of improving the yield of slabs by preventing growth of fishtail during slabbing. From the result of observation of metal flow during slabbing, the new method provides a step of forming transverse recesses at one end portion of a steel ingot by use of rolling rolls. The recesses thus formed absorb metal flow during rolling, preventing metal flow towards the crop end. By this method a crop loss may be greatly diminished, resulting in 96% yield of capped steel. Ikushima et al [15] used plasticine models to examine the effects of pass schedules, type of rolling mills and shape of ingots on the crop loss which occurs during slabbing. He concluded that: Overlap decreases with heavier reduction and that generally, less fishtails develop when the edging is carried out at a later stage of rolling and crop loss gets less in the order of high-lift mill, high lift universal mill and universal mill. Also, fish-tail loss can be reduced with smaller radius on longer side of ingot cross section,and overlaps can also be reduced when ingots have an appropriate convex shape on their bottom ends.

392

2.4. Ridges.spouty material These are characterized by the presence of a series of bulges which are due to greater elongation in an area as against that in adjacent zones. The ridges may appear in the centre of the strip, as shown in figure 3, or roughly equidistant on each side of the centre line, on either or both edges of strip or in any combination at any point across the strip width. Causes:- The incident of ridging in cold reduced sheet and tin plates created such a situation that considerable thought, time and money were being devoted to tracking down and eliminating causes of ridge defect. It was determined that burnish marks in hot rolled strip will lead to ridge in cold reduction. Other reasons for the occurrence of ridging include badly fitted wiper boards in cold reduction mills, rolling too much tonnage of any particular width on any set of rolls in hot strip mill, ill-conceived contours on work rolls and back up rolls, careless setting of edger rolls and entry guides, substandard rolls, unevenly reheated slabs, cambered and uneven slab dimensions. Any single one or any combination of these conditions could lead to ridges in subsequent cold rolling of light gauges, and most of them could result in irregular hot rolled strip cross-section profiles. Remedies:- Burnish marks in hot rolled strip will lead to ridge in cold reduction. It is quite often difficult to track down the cause of burnish marking, particularly when the burnish is more pronounced on the underside of the strip. However, the hot strip cross-section profile contains two 'high spots' which burnish by the finishing mill delivery table rolls, the coiler pinch rolls and wrapper rolls.

2.5. Other forms of rolling defects Transverse cracks may propagate from the edges of a strip if too large a reduction is attempted without interstage annealing. The fibbing effect is the appearance of parallel alternating transverse dark strips on the surface of cold-rolled sheets used for fabrication of automobile bodywork parts. It occurs due to the surface finish of the rolls. Figure 4 illustrates a sinusoidal fracture as another type of fracture which occurs during the cold rolling of high-purity ferritic iron-nickel alloy (0.0125C, 18.8%Ni, Fe balance). The crack appears to initiate after the leading edge emerges from between the rolls and is propagated at approximately the speed of emergence. The crack takes the form of damped wave, the amplitude and length of which is similar to those of sinusoidial fracture sometimes seen in gas pipelines. When working conditions in cross rolling stepped shafts with two rolls are not correct, the products often have central cavities or cracks. They are presumed to have been caused by the residual radial tensile stress and the shear deformation induced repeatedly at the central zone of billets during the process. Such stresses and strains are generated when a pair of forming tools are used in a similar way to rotary piercing by the mannesmann method. It seems to be desirable to form stepped shafts by cross rolling with three rolls in order to avoid the tendency for central cavity formation. 3. DEFECTS IN FORGING Defects reduce the strength and the life of a forging. Therefore, it is essential to determine their causes and reduce the expensive cost associated with their occurrence. There is always a reason for a defective forging. Mistakes can and do occur in both material manufacture and in

393 forging operations. Faults in the original metal, incorrect die design, improper heating or improper heating operations are some of the reasons for forging defects. Defects can also occur due to machining-induced flaws which give rise to surface cracks when proper machining operation procedures are not observed [ 16]. 3.1. Hot tears and tears Hot tears are surface defects that occur when metals rupture during forging. An example of hot tears during upset forging is shown in figure 5. Causes:- The presence of segregation, seams or low melting or brittle second phases promotes hot tearing at the surface of a forging. The internal discontinuities in forgings caused by faulty forging techniques which appear as these cracks or tears, may also be the result of either forging with too light a hammer or from continuing forging after the metal has cooled down below a safe forging temperature. 3.2. Centre Burst Centre bursts are ruptures which occur in the centre of billets and are most frequently encountered when side forging. Figure 6 shows a centre burst defect. Causes:- Centre bursts sometimes occur when the temperature of the metal increases significantly as a result of large, rapid reductions and incipient melting occurs. Nickel base and magnesium base alloys are particularly sensitive to centre bursts from this cause. Also, during forging operations, substantial tensile stresses are produced in addition to the applied compressive stress. Where the material is weak the tensile stresses can be sufficiently high to tear the material apart internally, particularly if the forging temperature is too high. Similarly, if the metal contains low-melting phases resulting from segregation, these phases may rupture during forging. Remedies:- Internal cracks developed during upsetting of a cylinder or a round as a result of circumferential tensile stresses may be minimized by proper die design.To combat centre bursting in open-die forging, flat dies may be replaced by curved or so called swaging dies which introduce favourable lateral compression. Centre burst is prevalent in close-die forging because lateral compressive stresses are developed by the reaction of the work with the die wall. 3.3. Cracks due to tangential velocity discontinuities and thermal cracks Cracks due to tangential velocity discontinuities (t.v.d.) may arise in material which is insufficiently ductile. The fracture characteristics of cylindrical rods of aluminium alloy in plane strain side-pressing has been reported. The localised deformation zone revealed by etching suggested that fracturing occurred along a line of t.v.d. Cracks caused by stresses resulting from non-uniform temperatures within a metal are called "thermal cracks". Causes:- In light sections of high-hardenability steels, thermal cracks occur when the forging is allowed to cool too rapidly. It originates at the surface and extends into the body of the forging. Thermal cracks can also develop when forgings are heated too rapidly. The internal ruptures form because the hotter surface layers expand more than the cooler metal near the centre. Since the tensile stresses developed at the centre depend on the temperature gradient, such cracks are more likely to be encountered in metals with poor thermal diffusivity. A typical example of thermal cracks in large forgins due to rapid heating is illustrated in figure7. Higher coefficients of thermal expansion are also unfavourable. Cracks of this type occur

394 more frequently in forgings having section sizes larger than about 3 inches as was discussed by Sabroff et al[ 17]. Remedies:. Thermal cracks resulting from rapid cooling down of the forging can obviously be avoided by cooling in insulating material or in a furnace. Those that are due to forging being heated too rapidly may be avoided by either of the following measures: i) After forging, reheat the part in the forge furnace to remove most of cold-work stresses, then air cool. ii) If the forgings are cooled to room temperature, after forging, reheat slowly through the range of 1000 to 1500 F before solution heat treating. The second method is the usual choice for many materials because of possible problems of grain-size control with the f'trst. 3.4. Folds or laps A common surface defect in closed-die forging is folds or laps. It occurs as a result of a protrusion of hot metal being folded over and forged into the surface. The oxide present on the internal surface of the lap or fold prevents the metal in the crevice from joining. A discontinuity with a sharp root is thus created, resulting in stress concentration. It occurs mostly in the die forging of sections where vertical and horizontal contours intersect. Causes:- This defect occurs when metal flows past part of the die cavity that has already been filled or that is only partly filled because the metal failed to fill in due to sharp comers. Wu [18] studied the metal flow and defect formation in a rib-web type forging. Four cases of different die shapes and preform geometries were simulated with emphasis on investigating the occurrence of laps or folds during forging. He determined that for a Oven preform shape, the fillet radii and flash design are the most important parameters which effect the formation of laps. Metal flowing non-uniformly in vertical cavities may also form a lap when the metal finally fills the cavity. This is a particular problem when the vertical section of a forging varies significantly in volume requirements. Folds at the edge of hammer products originate from too small a reduction per pass when local deformation occurs near the surface layer. Remedies:- In his study of occurrence of laps in a rib-web type forging, Wu et al [18] suggested that to obtain a sound product three design approaches may be taken. i) Enlarge the fillet radius which leads to an increase of weight. ii) Add a blocker operation before fmish forging which leads to an extra operation and cost. iii) Change the parting line from the base to the top of the rib which complicates the die design and may cause the forging load to increase. 3.5. Flash or fins The position and size of the flash, which serves as an outlet for superfluous metal and also as a source of back pressure to force the stock into the thinner sections of forging, are factors that cause the incident of flash cracking. Causes:- Cracking may occur where the flash is thin in relation to the thickness of forging from which it emerges, either because in undergoing a high reduction the transverse strength has decreased or because of marked temperature drop in the flash. Such cracks may propagate from the flash into the body of forging. Remedies:- To avoid flash cracking the flash should be as thick as possible and preferably not less than about 1/4 of the original thickness of the metal. Where such an approach is not

395 successful or is impracticable, the flash sometimes may be transferred to a less objectionable position.

3.6. Orange peel Forging billets containing coarse grains, whether as cast or wrought, will develop wrinkles during forging. These wrinkles are comn~nly known as orange peel. Figure 8 shows an upset forged billet with wrinkles caused by coarse grains. Causes:- When billets with coarse grain are forged in closed dies, the wrinkles often fold in to cause a series of small laps. Although they are seldom very deep, these laps produce a poor surface appearance that often necessitates considerable grinding and restrike forging. 4. DATABASE DESIGN Databases are becoming an important source of information for engineers involving four distinct functions namely analysis, design, coding and testing, and with advanced computer technology are able to facilitate rapid and efficient processing. The first databases were designed as data files with software and hardware constraints in mind rather than the need of end users. Nowadays, the concept of the database as a single central store of data with multiple uses prevails. An entity-relationship model is used in this work as opposed to the hierarchical relationship model. It is a conceptual scheme and is built by gradually integrating diagrams and entries in the data dictionary for individual functions. The principal philosophy follows the six features for rigorous database design as expounded by Fidel [19]. These cover 'Intrinsicality' which stipulates that each component of the conceptual scheme should relate to some function of the database and 'Representability' which provides for the identification and determination of each individual entry, relationship, or attribute. 'Reliability' requires that information for each component in the real domain is on the fight level of specificity. Further, 'Continuance' dictates that all domains are likely to be stable over time and 'Resolution' requires that each component of the scheme be clearly distinguished from other components. Finally 'Consistency' ensures that the definition of each component be consistent with the definitions of other components. Once the database is operative, it will be evaluated by testing quality features that are critical for the database usefulness. These have been identified as 'Flexibility' which assesses the degree to which the conceptual schema can accommodate changes in the work it represents and 'Clarity' which designates the applications of functions and rules in an unambiguous and understandable way. 'Efficiency' leads to an examination of the number of components with relation to the functions of the database and 'Schematic integrity' assesses the capability of a database to support the development of meaningful inferences. 'Completeness' estimates the degree to which all the entity types, relationship types and attribute types are included in the conceptual schema and finally 'Specificity' is a measure of the degree to which the database can answer questions and express various characteristics of an entity. There is a trade-off amongst the features just described. For instance, between 'Flexibility' and 'Clarity' where the database design has to take into account the relative importance of each feature and consequently the database can not achieve maximum quality for all purposes. A more detailed discussion is presented by Oxborrow [20].

396 The database design is currently in its initial stages and, once completed, it should provide a valuable source of information for academic researchers and 'material' formers. The database will be flexible and can be updated to include new information and data on defects. The importance of such a pictorial dictionary was recently emphasised by Johnson [21 ]. 5. CONCLUSIONS Defects can be peculiar to some materials or associated with some forming processes. It is a topic that, despite its importance, seems to attract little attention from academic researchers who deal mostly with 'ideal' materials and processes. The design of a database cataloguing and describing defects in metal forming processes should help engineers and scientists in their search for new creative processes avoiding those defects. 6. REFERENCES

1 W. Johnson and A. G. Marnalis, Proc. 17th. Int. M.T.D.R. Conf., Macmillan, London, 1977 2 S. K. Ghosh, Int. J. Mech. Sci., 23 (1981) 195. 3 W. Johnson and A. G. Mamalis, Plasticity today: Modelling, Methods and al~'.',:ations, A. Sawczuk and G. Bianchi (eds.), Elsevier, Amsterdam, 1984. 4 A. G. Mamalis and W. Johnson, Computational methods for predicting material processing defects, M. Predeleanu (edt.), Elsevier, Amsterdam, 1987. 5 J. A. Schey, J. App. Met. Work., 1 (1980) 48. 6 D. J. Latham and M. G. Cockcroft, N.E.L. Reprt No. 216 (1966). 7 F. Kasz and P. C. Varley, J. Inst. Met., 76 (1950) 423. 8 B. Dodd and P. Boddington, J. Mech. Work. Tech., 3 (1980) 239. 9 P. F. Thomson and N. M. Burman, Mat. Sci. Eng., 45 (1980) 95. 10 S. I. Oh and S. Kobayashi, J. Eng. Ind., ASME Trans., 98 (1976) 800. 11 Meadows and Pearson, J. Inst. Met., 92 (1964) 254. 12 Polakowski, J. Inst. Met., 76 (1950) 754. 13 Y.d.Zhu ;B. Avitzur, J.Eng. Ind.,ASME Trans.,110 (1988) 162. 14 M.Matsuzaki et al,Int.Conf. Man. Eng., Melbourne.Inst.Eng.Aust.Barton.(1980) 111. 15 H. II~shima et al, Nippon Kokan Technical Report Overseas No. 20 (1975). 16 G A Honeyman, Nondestructive Testing, Inst. Mat., London 1989. 17 A. M. Sabroff et al, Forging materials and practices, Brit. Lib. Reinhold Book Corp. 18 W.T.Wu et al. 12th NAMRC North American Manufacturing res. conf. Manufacturing Eng. transactions. (1984) 159. 19 R. Fidel, Database Design for Information Retrieval, Wiley and Sons,London,1987. 20 E. Oxborrow, Databases and Database Systems: Concepts and Issues, Chartwell Bratt, London, 1989. 21 W. Johnson, J. Mat. Proc. Tech., 26 (1991) 97.

397

Figure 1" Edge cracking on an AI-8% Mg alloy workpieee

Figure 2: Alligatoring in an AI-8% Mg alloy workpieee

398

Figure 3: Central ridges appearing as bulges on the strip

Figure 4: Sinusoidal fracture in the form of a damped wave

399

Figure 5" Hot tears on billets upset

Figure 6: Centre burst due to upset forging

400

Figure 7: Thermal cracks due to rapid heating

Figure 8: "Orange peel" Wrinkles due to the preset,- e of coarse grains

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

401

Split ends and central burst defects in rolling Stanistaw Turczyn, Zbigniew Malinowski Faculty of Metallurgy and Materials Eng., Mickiewicza 30, 30-059 Krak6w, Poland 1.

University of Mining and Metallurgy,

INTRODUCTION

The occurrence of some defects in rolling processes is a crutial problem for industry, especially for mass production like slab or plate rolling. So that the prevention of particular defects which can arise during rolling plays important role and is a mean for improving the quality and increasing quantity of products. To resolve this problem, two research activities are known: one deals with the improvement of metal formability by controlling the inclusion content and predicting the microstructure. The other one interests in improving the rolling process itself, through better roll pass design, deformation-zone geometry control, temperature and friction selection. Defects prediction in rolling processes is a complex problem because of the superposition of multiple parameters influences in a context of complicated geometries and boundary conditions. The resolution of these problems requires better understanding of the basic phenomena, involving the prediction of stress and strain state, flow- and deformation induced fracture that can occur in rolling. N

o

m

- rate of deformation tensor, - Young's modulus, h I, ho - strip thickness, H' - elastoplastic hardening modulus of the rolled material, j - total power per unit width, j. - total relative power, J~, - relative shear power, k - yield stress in shear, ld - length of the projection of the arc of contact, m - friction factor, Ra - distance, (see Fig. 3), roll radius, Ro Rr, - discontinuity surface radius, Dij

e

n

c

l

a

t

u

r

e

sij

E

,

-

t vi v l, vo

vr V X~, Y~ Yo

ac .7 A

- deviator of the Cauchy stress tensor, time, kinematically admissible velocity field, - velocity of the workpiece, - roll surface velocity - control volume, - x, y coordinate of point A, - yield stress, - contact angle, - constant equals 1 for loading and -1 for unloading, increment, - relative reduction, -

-

-

402

Rolling defects such as split ends or 'crocodiling' [5,9] and central bursts [3,11] take place during various rolling processes. The consequences of these defects are not limited only to yield loss or rolling disturbances but these defects can also cause damage to the rolls and mill accessories. The phenomenon of split ends is a defect which initiates as a crack, forming along the center plane of the deformed material. The crack can vary in severity from a slight separation of the upper and lower halves of the material rolled to complete separation or encasement of the rolls. Sometimes the splitting can also be formed at any position along the length of the rolled material. Typical split ends defects arisen during rolling are presented in Fig. la and lb. The phenomenon of internal voids or central burst is a defect which can occur during rolling of flat products. This defect, found in wire or strip drawing is reported by Rogers and Coffin [11] and a v i t z u r [2]. Early work by Avitzur, Van Tyne and Turczyn [3] has provided some criteria for the prevention of central burst during rolling. An example of this defect formed during steel beams rolling is shown in Fig. 2. One of the main aims of the present work is to update the study on the split ends and central burst defects formation in the flat rolling process. An energetic criteria for avoiding defects forming in the centre plane of the rolled material is discussed. Using an upper bound method with assumption of a rigid body uni-triangular velocity field for deformation zone the model of these defects is proposed. The power solutions obtained for analysed flow patterns allowed to classify the rolling parameters into safe and defect ranges. The shape coefficient of the deformation zone [4] plays an important role in this study [12,13]. Finally, criteria for split ends and central burst avoiding are sought so that preventive steps can be taken during the rolling process to eliminate these defects. In order to investigate closer the mechanical conditions under which the rolling defects are formed the stress analysis is performed. The elastoplastic finite element approach developed by Malinowski and Lenard is employed to compute the stress field in the deformed material. Nomenclature C

-

gij

-

lJ

II

-

-

m

o" o'ij o'ij O'ra or n O'p

oi~ (7"x , O'y

-

-

effective strain rate, strain r a t e t e n s o r , Poisson's ratio of the workpiece, functional of power, effective stress, Cauchy stress tensor, J a u m a n n rate of stress, mean stress, normal stress, flow stress, stress specified on St, components of stress,

(cont.)

r r,

-

shear stress, shear stress specified on S, error norm, ~1, r angles (see Fig. 3), a;,wo rotational velocity of zone II and the roll, ,~ij - spin tensor, [] A ][ - velocity discontinuity on S, i

- serial, i = 1,2,3 - optimal, - split ends, sound flow.

S u b s c r i p t s

opt se, s f

403

Figure 1: Split ends defects arisen during rolling of: a - aluminium alloy, b - steel bars. Two types of the workpiece are analysed. Firstly, the material is assumed to have no defects which results in a one type of the stress field. Secondly, the stress fields are computed in the notched specimens subjected to rolling. The results of the stress analysis are compared with the upper bound predictions and they confirm an important role of the shape coefficient of the deformation zone. Finally, the results of experimental rolling of copper and aluminium alloy are presented. It has been found that split ends and central bursts are more likely to occur in thick plates and sheets when small reductions are applied. 2.

UPPER

BOUND

MODEL

OF

DEFECTS

FORMATION

To model failure modes during rolling, an upper bound method of limit analysis has been applied in a classical formulation. According to Prager and Hodge [10] a statement of the upper bound theorem is that "among all kinematically admissible strain rate fields the actual one minimizes the following expression"

(1) F

T

404 The upper bound theorem states that the externally supplied power J" is less than or equal to the sum of the internal power, frictional power losses, power losses due to the surfaces of velocity discontinuity and losses or gains due to external tractions. For a rigid body motion and absence of front and back tractions the deformation energy is assumed to be dissipated along the surfaces of the velocity discontinuity. The upper bound analysis for a process where rigid body motion is separated by surfaces of the velocity discontinuity has been described by Johnson and Mellor [6] and then by Avitzur and Pachla [1].

Figure 2: Central burst during steel beams rolling. An essential step in the application of the upper-bound theorem is the determination of the strain rate field which is usually derived from the kinematically admissible velocity field. The number of admissible flow patterns, called velocity fields, is unlimited but it is presumed that in order to get higher accuracy of solutions velocity fields should be as closely as possible to actual ones. In this analysis three kinds of a triangular rotational velocity field are applied on purpose to model defects formation during rolling. These patterns are illustrated in Fig. 3. The workpiece of thickness ho enters the roll gap with a velocity vo (zone I) and exits from it with a thickness hf and a velocity vs (zone III). The rolls of radius Ro rotate at an angular velocity coo, thus peripheral velocity of the roll attends vr = cooRo. In general case, Fig. 3a, the deformation region (zone II) is separated by three surfaces of velocity discontinuity F1, F2 and Fa and rotates as a rigid body with angular velocity co. The inaterial is assumed to be an isotropic strain-hardening body. Surfaces F1 and F2 separate a linear body motion from the one with rotational motion. The velocity field contains two pseudo-independent parameters X~ and t~. These parameters are used to position the apex A of the triangle within the deformation zone. The values of X~ and Y~ are determined from the principle of minimum energy. The parameter Y= is the half distance of the apex A of tile deformation zone from the center line of workpiece, Fig. 3a, is used to permit the formation of an internal voids if it is energetically favorable. This flow pattern is used to model central burst generation during the rolling process. An assumption of Y~ = 0 leads to another flow pattern which describes sound flow with no defects formation, Fig. b. When point A of the deformation zone is located at

405

b /

........

W"~0 " -

"

a

t

fol

-~

'.

-o~-,v-

--:7.~~

70NET/--.x~,~v-.._ONE~..._ -

. . . . . . .

-"-'" .....

I ,~h~p,.o

A, < ~..-'<>"

'----I - ~

I

'

;

"\ ~.._~,.0.'

,

Figure 3" Assumed flow patterns: a - central bursts arising, b - sound flow, c - split ends formation.

.( ..., c~'-k ~

/

/

"...

\ \

/

J

the exit plane, the value of Y~ and X~ can be set equal to zero, furthermore, the surface of the velocity discontinuity F2 can be removed. The triangle (zone II) would continue to rotate around the roll axis and split ends would develop. This flow pattern, shown in Fig. 3c, is used to simulate phenomena of split ends formation during rolling of flat products. The computation of upper-bound on power for each flow pattern enables to predict which scheme is prevailing during deformation. The domain where the lowest power required is for a field that describes the flow with defects formation indicates the process parameters under which the rolling failures are expected. This results in establishing proper criteria in order to eliminate occurrence of such defects during flat rolling. 3.

P O W E R

EQUATIONS

FOR

SPECIFIC

FLOW

PATTERN

Assuming the plane strain and the unit width of the workpiece, the incompressibility condition for a general flow pattern, Fig. 3a, can be written as follows

406 (ho-

Z o ) V o = (R~o - R : o ) . , = (h~ - Y o ) ~

(2)

From Eq. (2), the kinematically admissible velocity field is deriued in the form 0.)

Lt_ _ Y__._

Vf

~

Ro

v~ " Ro (~)2-

Wo

Ro

(3)

1

The relative power losses (divided by by following equations" - surface F1 and F2

2koRovr) for a strain hardening material are given

ki w (Rr,)2 J~" = 2r

(4)

-~o

- surface F3 k w j ~ = 0.5rnc~ ~ l - -

1I

(5)

COo

- associated with tensions 9

j; =

_

~

~ (1 + - a - , a -

7--

a)~

Ro

Ro

Ro

Ro

~-~

(6)

where R~, Rr,, ld, ~i and ac can be simply calculated from the geometry of the deformation zone as a function of four parameters, namely ho/h], ho/Ro, Xa/Ro and Y~/Ro. Consequently, the total relative power for a unit width of the material rolled is obtained by summing up above mentioned power losses along three surfaces Fi and power losses or gains due to tensions, if applied. For a general flow pattern it results in j* =

J*

.,

.,

,

2koRovr = 2(j~. + Jr2 + 3r3) - Jt

(7)

Substitution of Eqs. (4-6) into Eq. (7) and optimizing the resulting formulae with respect to pseudo-independent parameters cZ/wo, Xa/Ro and Y,/Ro leads to a final equation which can be presented symbolically as the function of process independent parameters

Jo*~t- f(k/ko, c, ho/Ro, G, ~f)

(8)

P o w e r for s o u n d flow p a t t e r n . As stated earlier in cases where sound flow occurs the value of Y= is equal to zero and the apex A of the deformation zone is located on the center line of the workpiece. Using an assumption Y~/Ro = 0 in Eqs. (4-6) and optimizing Eqs. (4-6) with respect to pseudo-independent parameters leads to the power equation for a sound flow kl RF,)2 k2 RF2)2 RF2 hi (Js*ilopt - 2(I)l ~o ( - ~ o ~- 2(I)2 ~o (-~o -~- (1 + -~o ) Roo (4b -- ~f)

(9)

407

P o w e r for split ends p a t t e r n . Unlike the sound flow case, point A in the velocity field assumed for split ends coincides with the exit plane, thus Xa = 0. Moreover, there is no the velocity discontinuity surface F2 at the exit and therefore no shear loss J~2 occurs in the power equation for split ends. By eliminating the shear loss J~2, substituting for (I)1 a proper term and assuming no external tensions, the power equation for a sound flow (Eq. 9) transforms into that for the split ends pattern ka RF1)2sin_a [ ~ R o i ( ~ d ~

= 2 Vo

4.

PREVENTION

CRITERIA

+

1 ho ] FOR

ROLLING

(10)

DEFECTS

To obtain criteria for the prevention of central b u r s t s in rolling requires determination of an optimal value of the parameter Ya/Ro. As stated earlier when the optimal value of Ya/Ro is greater than zero the model indicates that the flow pattern with a central burst is energetically more favorable. The critical point is the condition where the optimal ~/Ro ratio becomes zero. Such points identified for a wide range of process parameters give the central bursts criteria for flat rolling, shown in Fig. 4. This figure shows the range of independent process parameters where sound flow is expected (labeled as "safe zone" in Fig. 4) and the range where a central burst is energetically more favorable. Besides that, Fig. 4 indicates that external tractions promote central bursting and that the relative back stress has a greater effect on it as compared to the relative front stress. The criterion of split ends defect is developed through the following procedure. If for a given set of process parameters, the power consumption for a sound flow is higher than that for a split end pattern, the split ends defect is likely to occur. The criterion of split ends defect will be established when the whole range of the independent process parameters e and ho/Ro has been covered. The power consumption for a sound flow pattern can be acquired by Eq. (9) through a numerical optimization procedure with respect to pseudo-independent parameter X~/Ro , while the power requirement for the split ends flow pattern can be directly calculated from Eq. (10). The obtained solution can be applied both for perfectly plastic and strain-hardening materials. For the first one the substitution k/ko = 1 is made in computations. For the strain-hardening materials, the yield stress variation along the deformation zone is calculated from the relation applicable to the plane strain conditions

2 2 ho 2k = ~ Yo[1 + --~ B In (__)]~

(11)

rt

where Yo, B and n are the material parameters. These parameters used in calculations for annealed aluminium alloy 6061-T6 are: Yo = 43 MPa, B = 50, n = 0.204 and for the hardened one: Yo = 252 MPa, B = 42, n = 0.149. The criteria obtained for the perfectly plastic material and for the aluminium alloy are presented in Fig. 5 and for the SAE-1020 steel in Fig. 6. To the right and below the line split ends defects are expected while to the left and above the line a sound flow prevails. Fig. 7 shows the joint criteria for both failures: central burst and split ends. It can be

408

0.5

0.5

"

erfectly ~ 06 I-T6

plastic hardened 6061-T6 annealed I

0.4

1

I

I

I::1

0

0

om,,I

o

0.3

0.3

Q)

~D

I>

i>

safe zone I

-5/

~D

0.2 ' ~ c

::c.c.:

0.1

0.0

i

0.0

,~

plastic materml = 0.00 ,~e = 0 . 0 0 = 0 . 0 0 ,if = 0 . 2 5 = 0 . 0 0 lit = 0.50 = = = =

0.1

0.2

0.3

0.25 0.25 0.50 0.50

0.4

,if ,if = ,if ':t =

0.00 0.501 0.00 0.50

0.5

Figure 4: Central bursts criteria in the rolling process with external tractions.

/, ' ~

0.1

(

[

"// ,2"

i

0.6

Relative thickness ho/Ro

,',,j

0.2

split ends e x p e c t e d 0.0

0.0

0.1

i

i

i

i

i

0.2

i

i

i

i

i

0.3

!

i

I

i

!

0.4

i

i

i

i

i

0.5

i

i

i

i

0.6

Realtive thickness ho/Ro

Figure 5" Split ends criteria for aluminium alloy rolling.

observed that the range of process parameters where a central burst is expected is much wider than that for a split ends. The lines in Fig. 7 showing criteria for the perfectly plastic material can be approximated by the following expressions: - for central burst defects ho

g

R--~ = 0.55 1~-~

(12)

- for split ends defects ho

g

Ro = 1.81 1 - e

(13)

The central bursts or the split ends are expected when the relative thickness is higher than the left side of the Eqs. (12) and (13), respectively. For strain-hardening materials (cold rolling processes) the split ends zone is slightly larger and this negative effect depends on the strain-hardening characteristic of the deformed metals, Figs. (5-6). The developed criteria indicate that for a given material the central bursts and split ends defects tend to be promoted by small roll radius Ro, large initial thickness of the workpiece ho and small thickness reductions e. Therefore, these defects can occur when

409 0.5

p e r f e c t l y plastic SAE- 1020 SAE-1020 ,

.

0.4

perfectly plastic material " ~b = ~ = 0

hardened annealed

, I

I

I

I

0.4

I

safe

~

0

0

9a,,,4

~ 0.3

o 0.3 j4 f J

GJ

@//

J

.p,,q

.,a 0 . 2 Q)

0.1-

"

j~

"

/

0.2

p

4)

,y

split ends expected 0.0

0.1

,

,,

I,,

0.2

Realtive Figure 6: rolling.

,,

I

0.3

, , , , l , , , ,

0.4

thickness

u~,'Tx

0.5

0.6

ho/Ro

.

.

.

cU~ ~ (co] Jl

/

-

0.00.0

central bursts and split

ends

expected .

0.1

0.2

Realtive

Split ends criteria for steel

.

i I

S ,

.

expected

/

/

0.1

/

L!,.l

zone

,.o j ~

0,0

/

0.5 -,

i ....

0.3

_

0.4

thickness

0.5

0.6

ho/Ro

Figure 7: Joint criteria for central burst and split ends defects.

shape coefficient of the deformation zone/k achieves a relatively high value [11,12]. Such conditions exists during rolling of slabs, heavy gauge plates or in the roughing groups of hot strip mills. 5. S T R E S S SAMPLES

DISTRIBUTION

IN

NOTCHED

AND

SOLID

In order to investigate closer the mechanical conditions under which split ends and central burst defects are formed the stress analysis has been performed. The elastoplastic finite element approach developed by Malinowski and Lenard [7] is employed to compute the stress field in the deformed material. Two types of the workpiece are analysed. Firstly, the material is assumed to have no defects which results in one type of the stress fields. Secondly, the stress analysis is carried out for the notched specimens subjected to rolling. The problem is formulated in the Eulerian reference frame. The material's constitutive relation is taken to be governed by a modified Prandtl-Reuss equation [8] associated with the Huber-Mises yield criterion. The actual velocity field is calculated from the minimum condition of the following functional:

410

n (v,)

=

fv [7 ~ i + ~1 X t / 4 ' ( i ) ' ] dV (14)

Minimization of the function (14) gives a kinematically admissible velocity field describing the elastoplastic flow of the material during the process. The multiplier 7 equals 1 for loading and -1 for unloading. According to the Huber-Mises yield criterion in the plastic zone g = crp while for elastic material the effective stress varies from 0 to ~rp. The first term in Eq. (14) represents the deviatoric part of the deformation power, the second one gives the power dissipated due to elastic volume changes and the third introduces the friction power. The friction stress ts on the strip/roll interface is modeled using the friction factor [8]. Having the velocity field the Cauchy stress field is computed solving the stress rate equation in the Eulerian reference frame. It can be accomplished by minimizing the error norm r

=

fv( ~176

-'~xk Vk -- Crll - - (adlk O'k 1 "q- Orlkadkl

)~

dV

+

- ~ z k v k -- &12 -- c01k crk2 + crlk ~k2

dV

+

-~zkvk

dV

-- &22 -- ~2k crk2 + cr2k wk2

(15)

under the constraints Ooij

Ozk = O i n V

er,~=0

and

3 sij sij

and 2(rrp)2 r=0

on Sf

1 < 0 in V

(16) (17)

with the initial condition o"0 = eri~ on S,

(18)

where St is that part of the strip surface on which the stresses are specified. To solve the problem an iterative finite element procedure involving updating the specimen and roll geometry, the strain and stress fields is employed. The geometry of the rolled specimen is discretized using 4-node linear elements. The velocity and stress field are approximated by quadrilateral elements with parabolic and Hermitian shape functions, respectively. The computations are carried out for cold rolling of 6061-T6 aluminium alloy. Specimens 6.35 mm in hight. The roll radius of 21 mm and the speed of 50 rpm are assumed

411 200

MATERIAL: A1 6 0 6 1 - T 6 h a r d e n e d plane _

. . . . .

- - - -

_

.

.

.

.

.

.

.

.

.

.

.

.

.

r

-200~ .~a

-

-400_ c=_--=o =====

-600~

ho/Ro

-8oo~~

i i ! i i !

2.

~b i

w i i i i i i i i | ! ii

9

6.0

i i i i

g'd

v = n =

i i i i i i i 1 | ! 1 1 1 1 1 1 1 1

9

Sample

10.0

|111111

12.0

stress stress =

cr= ~

0,302

0,175

50

1

min-

~'~~9 ~'8'.d ....~'8. 0 |

! i i i

length, x, m m

Figure 8: Variatiori of crx and cru stresses at the centre plane of the solid sample during rolling.

400 CMATERIAL: A1 6 0 6 1 - T 6

-

hardened

200 -

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

o-_: -2oo-

1

-400"

-600:q

2.0

I

........

4'~

~"d

Sample

J ~

~[ ~ ~-~- = o I~

~~

I ho/Ro

~'~'~....~'~' l e n g t h , x, m m

stress stress

~ ~,

= 0, 3 0 2

iiii~'8.o

Figure 9: Variation of crx and cry stresses at the surface of the solid sample during rolling.

412

400 MATERIAL: A1 6 0 6 1 - T 6 Notched specimen I -

hardened centre plane

200 -

J 0-

+a

-200-

,,

~r

,

[

c .~ __. _.- o s t r e s s o-xi c =. = = = s t r e s s o'y] h~/Ro = 0,302 [

-4005 -

r n

9

i ! i i i i i

.

! i i I i i i i i i i i ! i i I

8.0

Sample

i ! i i i i

I

length,

.

i ! ! i ! i

= =

0,10 50

I

i i i i i

.

x,

,I

min-

I I

i i ! i i

.0

m m

Figure 10: Variation of cr. and cru stresses at the centre plane of the notched sample during rolling.

800'

MATERIAL: A1 6 0 6 1 - T 6 Notched specimen-

hardened centre plane

400

r

0

U~

~%-

-400:

c===o stress ~ = = = =' s t r e s s

I

o-= %

/Ro = 0.302

s = 0,175 n - 5 0 r a i n -1 -BOO-

. . . . . . . . . . . . . . . . . . . . . . . . . 2.0

~'.lo

6l I~

8

, I~

Sample

~.'0

.... O 9

length,

,

....

1,'2.0

, ~1l~'. 0

x,

.......

l 16.0

..... ~tB

mm

Figure 11" Variation of cr~ and ay stresses at the centre plane of the notched sample during rolling.

413

200__

MATERIAL: A1 6 0 6 1 " - T 6

hardened

0

-200~

-400~

-600~_

cvvvo~ ho/Ro = n = 50

:

: -8oo-,oo" 9

.....

6 ' .5 . . . . . . .

'i.'6

Distance

.......

from

i .

the

stress stress 0,30 rain

" ......

~ffi ~y

o ~

2 . 0' . . . . . . . .

centre

v-

0.100

Q ~ v= 9 ~ v=

0.175 0.300

2 .5 . . . . . .

plane,

3 .0

mm

Figure 12: Variation of cr= and ay stresses across the solid sample near the exit plane.

in calculations. The distribution of stress components are presented in Fig. 8 and 9 for solid specimen and in Fig. 10 and 12 for notched specimens. The stress patterns in the specimen centre are mainly compressive. Tensile axial stresses are only noted at entrance to the deformation zone. Patterns of axial stresses at the sample surface shown in Fig. 9 indicate that at entrance and exit significant tensile stresses occur. As far as formation of defects is concerned, the tensile axial stresses at the specimen surface can cause surface defects. The central burst defects or split ends formation more likely can be explained by presence of tensile stresses acting along the strip thickness. The computation, however, gave no evidence of such stresses in the case of solid specimens rolling, Fig. 12. Significantly different results have been obtained for rolling of notched specimens. Typical stress patterns in the center of the sample are presented in Fig. 10 and 11. High tensile stresses acting along the specimen thickness at the tip of notch are noted. Thus, near inclusions on tensile stresses can develop on rolling leading to macro defects such as central bursts or split ends. 6.

EXPERIMENTAL

RESULTS

The analytically obtained split ends criteria have been compared with the results of experimental rolling. Two different specimens shapes have been used. Firstly, copper samples, folded in the half of the length, were rolled. Four series of rolling experiments using constant specimens height 2, 3, 4 and 5 mm, respectively, were done. The rolls of 25 mm in diameter were used. During rolling both part of the folded specimens were bent in opposite direction with various curvature, Fig. 13. The curvature of the split ends

414

Figure 13: Example of series of the folded copper specimens of 3 mm in height after rolling (increasing reduction from 3.4% for sample No. 1 to 14.9% for sample No. 8).

Figure 14: Example of the A series of the solid aluminium specimens of 6.5 mm in height after rolling (increasing reductions with increasing sample No.).

415 0.5

AI

6061-T6 perfectly plastic

0.4

annealed

r

9 .

0 ~162 0

.

-

split

o o o

-

sound

I

0.3

I

ends flow i

I

/ / a

safe

zone

/ /

, /

CD

/

/

/ /

,,, P,.4 ~ 0.2

=

~IU

0.1

9

A/

9 9

, :-ze

9

//-

9 =

/

~ ~

"

"

//t,

= ,,/

9

~

=

/=/. =

z

s

,0

e 9

GFZ 0.0 0.0

0.1

Realtive

0.2

0.3

s

expected B

0.4

thickness

C

0.5

0.6

0.7

ho/Ro

Figure 15: Comparison of experimental results with developed criteria. obtained after rolling shows tendency to splitting. For the second part of the experiment solid specimens has been prepared. They were cut from 6061-T6 aluminium alloy rectangular bars 6.35; 7.94; 9.53 and 12.70 mm of height and 25.4 mm of width. Two pairs of work rolls of 21 and 67 mm in diameter were used for the experiment, therefore, eight values of parameter ho/Ro, ranging from 0.095 to 0.605 were obtained The results obtained during rolling are shown in Fig. 15 with comparison to the developed criteria. It can be seen from Fig. 15 that almost all split ends defects which occur during experimental rolling are in the area below the line where 'crocodiling' is expected. 7. S U M M A R Y

AND

CONCLUSIONS

The two-dimensional model of the limit analysis is presented for the simulation of central burst and split ends formation during flat rolling. The determination of the total relative power for three flow patterns have allowed to establish the proper criteria that classify process parameters into safe and central burst or split ends zones, which are illustrated in Figs. 4, 5, 6 and 7 of the paper. The main conclusion of the study is that for a given material central bursting or splitting of the ends in rolling tends to be promoted by the following independent parameters of the process: large initial thickness of the sample ho, small thickness reductions e and small work roll radius Ro.

416 Moreover, it has been found that internal burst defects are more likely to occur than split ends; both failures are expected in thick plates and sheets when small reduction are applied. By using these criteria in rolling practice, it has become possible to predict necessary rolling conditions in order to avoid split ends and internal burst defects. Comparison of analytically developed criteria for split ends defects with data obtained in experimental rolling of aluminium specimens show generally good agreement. The stress fields computed for solid samples did not reveal essential tensile stresses which could lead to internal defects. The results obtained for notched specimens, however, showed high tensile stresses acting in the direction perpendicular to the sample centre plane. Such stresses can lead to internal bursts or split ends formation. The comparison of the stress fields obtained for the two type of specimens confirmed the important role of inclusions or other material discontinuities in the internal defects creation. By using these criteria in rolling practice, it has become possible to predict necessary rolling conditions in order to avoid split ends and internal burst defects. A c k n o w l e d g m e n t . The financial assistance of the Polish Scientific Research Committee (Grant No. 7.0510.91.01) is gratefully acknowledged. REFERENCES 1. B. Avitzur and W. Pachla, ASME, J. Eng. Ind., ASME Trans.,108 (1986) 295. 2. B. Avitzur and J. C. Choi, J. Eng. Ind., ASME Trans., 108 (1986). 3. B. Avitzur, C. J. Van Tyne and S. Turczyn, J. Eng. Ind., ASME Trans., 110 (1988) 173. 4. W . A . Backofen, Deformation Processing, Addison-Wesley Publishing Co., Reading, Massachusetts, 1972. 5. K.L. Barlow, P. R. Lancaster and R. T. Maddison, Metals Technology 11 (1984) 14. 6. W. Johnson and P. B. Mellor, Engineering Plasticity, Van Nostrand Reinhold Co. Ltd, New York, 1973. 7. Z. Malinowski and J. G. Lenard, Comput. Meths Appl. Mech. Eng., 104 (1993) 1. 8. Z. Malinowski, Metallurgy and Foundry Eng., 19 (1993) 323. 9. M.M. A1-Mousawi, A.M. Daragheh, S.K. Ghosh and D.K. Harrison, J. Mat. Proc. Tech., 32 (1992) 461. 10. W. Prager and P. G. Hodge Jr., Theory of Perfectly Plastic Solids, Chapman and Hall, Ltd, London, 1951. 11. H.C. Rogers and L. F. Coffin Jr., Proc. Manufact. Tech., Univ. of Michigan, 1967, 1137. 12. S. Turczyn, Steel Research 63 (1992) 69. 13. S. Turczyn and M. Pietrzyk, J. Mat. Proc. Tech. 32 (1992) 509.

Materials Processing Defects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

F o r m - f i l l i n g in f o r g i n g a n d s e c t i o n - r o l l i n g P.F. Thomson*, C.-J. Chong** and T. Ramakrishnan # *

INTRODUCTION The object of the present paper is to investigate the factors governing the filling of tools in forging and section rolling and to evaluate the use of forging to study causes of underfill in open calibre rolling of beam sections. The analogy might be expected to be between closed die forging and calibre rolling on one hand and between open die forging and universal rolling on the other. It is always to be expected that some systematic differences exist between rolling and forging(i) (ii) (iii)

(iv)

In rolling without tension, friction provides the force to draw the material into the deformation zone. In rolling there are components of both rolling and sliding in the relative motion. In rolling, only the head and tail ends are deformed under non-steady state conditions, whereas forging is inherently a non-steady state process in which geometry changes throughout the process. As a corollary to the above, material passes through the deformation zone in the rolling process, whereas in forging the material essentially remains within the deformation zone as geometry changes.

1.1 D e f e c t s in c l o s e d - d i e forging and beam rolling Some of the defects more commonly observed in beam section rolling [1] are: a. Underfill b. Overfill and resultant formation of fins c. Malformation of sections due to scraping of the guides d. Twisting arising from inhomogeneous deformation e. Seams caused by wrinkling at the free surface during large reduction Chitkara and Hardy [2] found that overfill was promoted by a small ratio, w o / h o , of original width to thickness of the work piece. The more common defect, underfill, has been found to occur most notably on the inside

*

Department of Materials Engineering, Monash university, Clayton, VIC 3168 Australia ** BHP Research Melbourne Laboratories, 245-273 Wellington Road, Mulgrave, VIC 3170 # Present address: MMC Oil & Gas Engineering Sdn. Bhd., P.O. Box 10936, 50730 Kuala Lumpur, Malaysia

417

418 face of the flange near the root radius [2,3,4]. Specimens with larger shape factor (wo/ho) showed evidence of underfill, while specimens with smaller shape factor over-filled at large drafts and spread excessively. Overfill of the roll groove also tends to promote the formation of a fin which leads in turn to the occurrence of laps. This can be ameliorated by designing the groove passes with convex bottoms and using large radii at the opening of the pass [1]. A typical break-down rolling schedule with open calibre rolls prior to the universal rolling of I- and H-shapes is shown in Fig. 1 [5]. ,,

.

~ S

OF DEFORMATION

RELATION B ~ N CALIBER AND MATERIAL

I. Centeringroiling 2.

Rolling both web and flange with different reduction

3. Ed~gtoning 4.

Enlarging web height

%. ~ i n g

flange width w.1

6.

Rolling only web part

7.

Rollingonly flangepart

v

Figure 1. Schematic outline of pass design for production of beam blanks by open calibre rolling - after Kusaba and Sasaki [5] It is a schedule of such passes which is the subject of the present investigation. In a schedule of some twenty eight passes alternating between bull head (fiat)and calibre (profile) passes, for rolling an I-beam, Westrope and Glover [4] observed that the underfill (which occurred on the inside flank of the flange adjacent to the root fillet and on the crest) frequently developed between the twentieth and twenty second passes, although underfill up to 3 mm occurred on the inner flank in all passes after Pass 12. A similar observation was made by Bodnar, Bamfitt and Ronemus [3] who reported pull-out of the flange in their experiments. Westrope and Glover suggested that underfill on the inner flank of the flange resulted primarily from a tensile stress applied to the flange by the larger reduction in the web. It would seem desirable, then, to select relative web and flange areas and reductions to minimise this effect. However draft, elongation and spread are

419 interdependent. Elongation and spread are affected by reduction, rolling temperature, friction and yield strength of the work piece. The cause of defects found in the mill production of blanks for the I-beam designated 760 UB was investigated by Westrope and Glover [4] in experiments in which Ibeam blanks were rolled at one-sixth full size using 'Plasticine' (held at 20oc) as a model material. The straight sides of the ingot prepared for the model were rolled to form the outside of the flange while the concave sides became faces of the web and the inside of the flange. The rolls used to produce the 760 UB beam blank were designed with bull head and profile grooves (Fig. 2) in which the bull head groove were used to control the overall height of the flanges and overall width, while the profile groove were used to reduce thickness of the web and to develop the flange.

7v.

s ! i

|

I i

/-C cF L F

I

Figure 2. Profile and bullhead passes form for simulated 760 UB beam blank rolling- after Westrope and Glover [4]. To study the effect of changes of geometry of the roll profiles on underfill, Westrope and Glover [4] successively increased the initial thickness of the web by 40%, changed the profile of the inside flank to decrease the slope and effectively to increase the root radius (Fig. 3) and they increased the width of the crest of the flange by some 50% at the expense of the width of the web.

420 I

t

. . . . . . . . . . .

~.._.

I ROLL PROFILE MODIFICATION 1

ROLL PROFILE MODIFICATION 2

Figure 3. Modifications 1 and 2 to the roll profile for producing the 760 UB beam blank- after Westrope and Glover [4]. As the inner fillet radius and the flank angle increased, the thickness of the web at its junction with the flange increased, decreasing the percentage reduction of the web locally and promoting greater transfer of material from web to flange, with the expectation that this would decrease elongation of the web and stretching of the flange. An increase in the width of the flange, increasing the amount of material in it, was expected to decrease elongation of the flange by the web during rolling. These modifications were recorded as having made "little difference" and underfill (defined in Fig. 4 as the horizontal distance between the roll surface and the traced section of the pass) was found to reach about 3 mm in the 'Plasticine' scale model and 18 mm in steel, suggesting effective modelling. Commenting on the establishment of the profile in early passes, they observed that the grooves were not completely filled until after Pass 14 and, even then, some end defects were still observed.

~

FLANGE

/

UNDERFILL lgmm on steel samples 3ram on plasticine samples. Scale 1:6

_ ~

/;

~ ~

ROLL PROHLE

~ _ ~ / /

STEEL PROHLE (actual traced shape)

Y ~ . / P L A S T I C I N E PROHLE Inner flange / fillet profile - recedes from the roll profile with increased number of passes in the roll profile groove WEB

Figure 4. Shape of inner flange and fillet developed during rolling of the 760UB beam blank - Westrope and Glover [4].

421 It is not proposed here to review methods of roll pass design. Although numerical analysis, notably by the finite element method, is developing rapidly, the c.p.u, time consumed by a three-dimensional analysis with many degrees of freedom has meant that relatively few solutions to three-dimensional problems in rolling exist and finite element (or finite difference) analysis with many degrees of freedom is not routinely used as part of the roll pass design procedure, but some of the numerical analyses have clearly been directed to process or tool design. Analysis of shape or section rolling by the finite element method include [6-10] and those of closed-die forging include [11-13]. Work in which finite element analysis has been directed to process or tool design in rolling or forging include [11-16]. The most extensive and significant work on preform design is undoubtedly that of Kobayashi, whose paper [13] describes a method of backward tracing to optimise design; specific mention is made of design for complete die filling, and recommended preform designs, related to the geometry (ratio of breath to height) of the flange, are very similar to the more empirically-based designs recommended by Lange (Fig. 5) whom he quotes. Another paper by the same author [17] in which an upper bound technique is applied is part of a series of that author's papers on preform design in metal forming. Among recent work on optimising process design is that of Joun and Hwang [19,20]. A somewhat different approach to the analysis and hence possible optimisation of shape rolling is taken by several Russian workers [21-24] who express the velocity field in the deformation zone in terms of an analytical function of the surface profile and optimise the resulting expression for work-rate by a variational technique. The profiling of (frictionless) dies for minimum work in steady state processes provided by Richmond and his co-workers [25] also deserves mention. 1%------~. UPSET STOCK

PREFORM

NOhO

-ltr FINISH 9

Figure 5. Recommended preforms for manufacture of steel forgings of H-Section [18]

....

i

Figure 6. Definition of preform ratio (after Biswas and Knight [30])

422 Raghupathi and Altan [26] reviewed the German technical literature at the time and found that the programs developed for design related essentially to special cases. They noted that a general approach to mathematical simulation of shape rolling was still to be developed. They referred to the work of Geleji [27]. This author reviewed a number of techniques for rollpass design in use. The majority depended on defining the geometry of a complex shape by dividing it into simple modules, the interaction of which could then be considered by elementary means. It is significant to the present investigation that this approach was used by six authors to develop the eight formulae for elongation quoted by Wusatowski [26], each of which is derived by considering the elongation of the separate modules. Neumann (quoted by Raghupathi and Altan [26]) concluded that the elongation of the flange of an I-beam should be smaller than the average elongation except in the first four passes, in which the reverse should be the case. Discussing the principles of design appropriate to I-beams, Krueger (quoted by Raghupathi and Altan), who divided the web and the flange into various modules, concluded that the reduction of each should be uniform. Different authors proposed different modes of division. Neuschutz and Thies [28] investigated the importance of uniform deformation of web and flange in universal rolling, indicating that uniformity could be produced by an increase or a decrease in the width of the flange, stretching of the web by the flange, transverse flow from flange to web or vice versa as appropriate. Transverse flow at early stages (in thick sections) to secure uniform elongation was acceptable, but high friction hindered such flow in thin sections, making early elongation of the web and flow from the flanges desirable, to preclude the need for later transverse flow from web to flange. Nakagawa, Hira, Abe et al. [29] found experimentally that the transfer of metal between the web and the flange increased linearly with the difference, (rf-rw), of the reductions in the flange and the web. When the reduction in the web was large, axial stresses developed in the web, leading to buckling. As the ratio of the cross-sectional areas of the web and flange increased, spread of the flange increased. Biswas and Knight [30] found that there was a minimum preform ratio w=(R2-R1)/(h-b/2), defined in Fig. 6, below which the preform did not completely fill the final die during forging. The use of preforms minimised die wear and assisted complete filling. In their studies of groove filling, Subramaniam, Venkateshwar, Lahoti et al., [31] found that filling near the root of the cavity improved as the shoulder or root radius increased. Laps may arise in forging when the metal flows away from the corners and back on itself. This was attributed by Watkins [32] to inadequate root radii. In his simulation of the forging of an H-section, Abebe [33] observed underfill on the inner flank of the flange, followed by overlap during subsequent filling. Higher friction reduced overlap but resulted in a greater forging load. If the web of the preform is too thin, laps may form in the web [34]. A sharp root radius in the flange may also cause laps and underfill [34].

423 Some evidence of the effect, and possible benefit of, differential lubrication was obtained by Aung and Thomson [35] in their investigation of upsetting cylindrical billets of microcrystalline wax into square and rectangular dies as a model of the hot die forging of steel. With coefficients of friction on platens and die walls respectively of 0.4 and 0.02, die filling was better and deformation more stable (showing less tendency to laps) than with unlubricated tooling or with all tool surfaces lubricated. Summarising, then, it appears that underfill may arise early in the process as a result of poor distribution of material, i.e. poor development of the modular structure of the section, usually as a result of requiring excessive transverse flow, or later in the sequence of passes. In the latter case, suggested mechanisms of underfill are principally:(a) transverse flow- from web to flange - coupled with an inadequate inside root fillet radius, promoting flow away from the fillet, and (b) greater elongation of web than flange (also resulting from greater reduction in the web than the flange) causing tensile deformation in the flange and contraction on its inner flank. Buckling of these sections as a result of differential flow or partial contact in deep grooves may also contribute. If the first of these mechanisms is significant, underfill will increase with the initial area of the web cross section even if the reduction in the flange and the web are equal, and if it is of overwhelming importance minimal evidence of a gradient in the longitudinal component of velocity across the section may be expected. If the second is more important, evidence of a strong transverse gradient in the longitudinal component of velocity is expected and underfiU would be eliminated if equal reduction in height of web and flange could be achieved in the absence of transverse flow. Relative thickness of web and flange, rather than relative area, may dominate. Other geometrical factors include notably ones which will affect ease of transverse flow, namely - taper of the web, radius of root and crest fillets on the web and angle of the inner flank of the flange. 2. 2.1

SIMULATION Conditions to be Simulated The focus of the present work then rested on the cause of underfiU in later passes in the production of a 760 UB beam blank of C-Mn steel; in particular on the role of differential reduction of web and flange leading to diferential elongation, the effect of relative a r e a of web and flange and the significance of transverse flow, resulting in the transfer of metal between modules.

424 2.1.1

Friction Work by Ekelund, quoted by Male [36] indicates that the effective coefficient of friction between 700-1100oc decreases with temperature (a function of the mechanical properties of the oxide film), whereas that of bare steel surfaces increases with temperature. This agrees qualitatively with the work of Male and Cockroft [37] who suggested a value between 0.25 and 0.4. Values of the coefficient of friction are specific to the governing mode of deformation [38], so that friction assessed in one-dimensional flow is unlikely to give results appropriate to t h r e e - dimensional flow. The coefficient of friction varied from 0.32-0.57 in typical ring tests conducted by Licka and Wozniak [39] in which, however, the lubrication was not specified. Theocaris, Stassinakis and Mamalis [40] found that the coefficient of friction in hot rolling varied from 0.5 to zero through the roll gap, being highest at entry, decreasing to zero at the neutral point and increasing parabolically toward exit. A mean coefficient of friction therefore has little significance. Chijiiwa, Hatamura and Hasagawa [41] found a coefficient of friction ~=0.3-0.4 in both forging and rolling.

2.1.2

Material Properties The flow stress appropriate to hot rolling of steels over a range of carbon contents and temperatures was estimated from expressions due to Licka and Wozniak [39] and Hodgson, Szalla and Campbell [42], the latter being for typical C-Mn steels. In the former case, the strain hardening exponent was calculated according to given expressions and in the latter, n=0.13 was used; a strain rate exponent of 0.24 [42] and a mean strain rate of =2 [43] were assumed. Values of yield strength calculated from the two equations agreed well (to within 0.5% at 1100oc) and a yield stress of 85 MPa at that temperature was adopted in the calculations. E=100 GPa was taken as representative of C-Mn steels at 1100oc, which may be compared with a value of 107 GPa for pure iron calculated from the equation given by Fields and Ashby [44]. Simulation of Forging as an Analogue to Rolling Numerical Model To simplify the contact boundary conditions and to provide an initialvalue problem with fewer degrees of freedom, forging was adopted as indicative of groove filling in the rolling of a 760 UB beam blank in profile and bullhead passes (Fig. 3). The dimension of the model in the rolling direction (the "axial

2.2 2.1.1

direction") was taken as

Lp -~/R~h, where

R (=184 mm) is the radius of a

typical roll used for beam blank rolling and ~h is the typical reduction in height of a web of thickness 50 mm. With 10% reduction in thickness, Lp=30mm. The tool/work-piece interface was modelled by gap friction

425 elements. Normal separation of tool and work-piece was arbitrarily chosen as a measure of underfill because this was calculated in the finite element package, ABAQUS Version 4.5, in contact problems, although the volume between separated surfaces or the area of lost contact may have been significant characteristics. It was found by trial that an incremental closure of about 1% in the gap between faces of the web was the maximum at which convergence could be achieved. In the finite element simulation, one quarter of the cross-section was modelled (Fig. 7) and a deformation zone with two layers of elements in the "axial" or rolling direction was analysed. The angle of bite was ignored and the deformation was represented by deformation between opposed dies. Because the approach was that of investigating the effect of the geometrical and process parameters on underfill rather than on empirical simulation of flow, it was considered acceptable to include only elements within the dies (roll gap) and to apply nodal constraints on the "front" and "back" faces representing respectively the effect of the feed and the product on the deformation zone. The simulation commenced at a web thickness of 68 mm and continuing to a web thickness of 23 mm; the ratios of areas A w / A f, reductions in height r w / r f and reductions in area R w / R f of web and flange are shown in Table 1. This corresponds to the rolling process from Pass 12 to Pass 24 in Westrope and Glover's experiments where underfiU was observed. Table 1 shows that a large reduction in the web of a die of given geometry therefore produced relatively little change in the parameters of interest. Table 1" Geometry of web and flange modules of standard beam blank profile in Passes 12-24 of rolling schedule. Pass No 12 24

Initial Thickness (mm) Web Flange 68 108 23.7 63.7

Aw/Af 1.94 1.3

rw/rf " 1.59 2.69

Rw/Rf 1.38 2.05

To provide a basis for relating the effects of differential elongation and of transverse flow at various geometries on underfill, the loss of contact on the inside face of the idealised H-section shown in Fig. 8, resulting from differential stretching under frictionless compression and from transverse flow under frictionless upsetting in plane strain, was calculated. The assumptions of no flow between modules (no transverse flow) and transverse flow only (no longitudinal extension), respectively, were made leading to the estimates of normal separation (underfiU) shown in Table 2. The models on which these calculations were based can only provide indicative values. In particular that based on transverse flow does not apply to a groove which is

426 completely filled initially. However, on the basis of this analysis, it might be suggested that underfill as a result of transverse flow is more likely to occur when the web is wide and that underfiU resulting from differential stretching may be promoted when web and flange modules are of approximately equal section and, to some extent, when the web is thin. Table 2: Notional effect of H-beam geometry on underfill resulting from differential stretching of modules (U1) and from transverse flow from web to flange (U2). (Parameters as defined in Fig. 8). (All dimensions in mm). hw 23 23 23 23 23 70 70 70 70 70

hf 63 63 63 63 63 110 110 110 110 110

bw 60 57 42.5 35 28 60 57 42.5 35 28

bf 25 28 42.5 50 57 25 28 42.5 50 57

ont

d Section

U1 0.36 0.37 0.35 0.31 0.265 0.26 0.27 0.28 0.26 3.5

I L

[

U2 6.0 5.7 4.25 3.5 2.8 6.0 5.7 4.25 0.23 2.8

i,i

bw

I [

Face

Figure 7. Three-dimensional model of deformation zone for simulating the forging of a 760 UB beam blank showing subdivision into elements.

Figure 8. Idealised model of beam blank used for elementary investigation of differential extension of web and flange.

427 3. 3.1

RESULTS OF COMPUTATION The effects of constraint, including frictional constraint As expected, in computational models without friction, restraint of axial flow on the entry and exit planes of the deformation zone were found to promote separation at the inside root fillet of the flange, while removal of restraint at exit, indicative of unconstrained non-equilibrium flow at the head of the section, promoted separation at the inside root fillet, at the crest of the flange and at the exterior fillet on the crest of the flange. Transverse restraint, with freedom of axial displacement, promoted separation on the outside flank of the flange. When computation was repeated, with a coefficient of friction increasing to 0.3 [37,45], with and without constraint at exit, the loss of contact at the inside root fillet and at the crest of the flange both decreased, separation being 3,2 and 0.2 mm respectively with coefficients of friction of 0, 0.1 and 0.3, but the computed difference in strain in the axial direction increased from approximately 5 mm to 8 mm so that an effect of friction was to promote differential elongation. It should be noted that no constraints were applied to nodes in the exit plane, so the results were more indicative of conditions at the head end of a rolled section. It was found that differences in axial flow and flow into the flange almost balanced each other, so that the overall difference in transverse flow was very small. With high friction (~=0.3) in the web and zero friction in the flange, the difference in elongation of the web and the flange was greater than when the variation in friction was reversed. However, when the web was frictionless, differential elongation was minimised, irrespective of the level of friction in the flange. This did not correlate with the magnitude of separation at the inside root fillet, which was of the order of 3 mm at the inside root fillet when the flange was frictionless, but negligible when the coefficient of friction was 0.3, irrespective of friction in the web (Fig. 9). It appears the separation in this case was not controlled by differential elongation and consequent stretching of the flange, but perhaps by transverse flow from the web away from the fillet, promoted by lack of friction in the flange to provide a back pressure and hence the spread required for complete filling.

428 0.40

VI! I

0.35

o~

.•0.30

o~

7

O.25 ~ 0.20 r~

Z~ 0.15-

I\

,

In

lE

..l~osol lOO

ss

o.j

.~ 0.10 Ii 9]1oso] lOO 3s o.3 t i 0.8 "l 7S~176176190 0.3 ~',X f I "~ 0.6

"nllOSOlWO 9o 0.1 I'\t I I!',~ I

o

~ Z

0.4 0.2-

0.50

'IT

tf

0.10

0

0.12

A

~.s, rZ~ooe i

9

I

a

I

20 40 60 80 100 DistanceFromLongitudinalAxis (nun)

Figure 9. The (predicted) effect of differential lubrication of web and flange on underfiU (normal separation) in forging of a 760UB beam blank section.

0

.

0

1

20

1

I

40

I

~l

i

60

_

80

i

100

Distancefrom LongitudinaaAxis (nun)

Figure 10. Investigation of the effect of material properties on underfill (normal separation)

3.2

The Effect of Geometrical Factors The shape of web and flange and their geometry govern the possible interaction of axial deformation and transfer of metal between them during rolling. Radii of fillets and the angle of the flanks of the flanges may assist in controlling flow of metal to achieve good filling of the tool form. Subramaniam, Venkateshwar, Lahoti et al. [31] found that filling of the groove improved with an increase in the radius of the root fillet of the flange. In the present work, the length of the crest of the flange relative to that of the web in a frictionless three-dimensional model of the 760 UB beam blank was increased progressively, changing the ratio of volumes (before deformation) from 1.25 to 5.9 in four steps. The same pattern of underfill on the root fillet and on the inside face of the flange after 10% reduction in height was observed at exit from the deformation zone in each case, although the severity increased by some 40% as the width of the flange was increased. Tapering the web by 4 ~ in either sense (i.e. opening it toward the root fillet of the flange or the reverse) changed the elongation of the web by an amount which corresponded closely to the local change in reduction of the web. As usual, there was some loss of contact at the root radius of the flange, but this was unaffected by taper of the web. Chang and Choi [46] found that an increase in die temperature resulted in reduced underfill. In the present work, the effect of yield strength and work hardening were investigated through a constitutive equation of type

429 r =r

n

9

r > (r3r

in which yield strength was variously assumed to be 35, 85, 190 MPa (where (~y = 85 MPa had previously been assumed as typical of C-Mn steels at 1050oc) and the work hardening exponent was given values of 0.1 (a reasonable lower estimate from the expression due to Licka and Wozniak [39]) and 0.3 (an extreme or over-estimate). Young's modulus was E=100 GPa. A yield strength of 190 MPa was taken as representative of yield strength at a working temperature of about 750oc. Separation at the root radius of the flange seemed to be strongly associated with high yield strength of the work piece, although a very low yield strength produced apparently contradictory results (Fig. 10). This contradiction may be only apparent if the range over which tensile deformation is transferred into the flange depends on yield strength, so that loss of contact is transferred from the inside flank of the flange toward the root radius as yield strength decreases. Separation increased with work hardening. Underfill in Initial Passes Abebe [33] suggested that underfill in closed die forging from a rectangular block resulted from flow away from interior fillets in the initial stages, so that a thinner, unsupported flange may form, reaching the crest of the die before full contact on the flanks [Fig. 8]. Upsetting (transverse flow) is necessary in early, blocking, passes so that transfer between modules is likely. This does not necessarily conflict with the suggested later loss of contact in forging or rolling [3,4]. Analysis of forging the given profile (760 UB beam blank) from a rectangular block was performed in 2-D with NIKE2D [47] and ABAQUS Version 4.7 and in 3-D with ABAQUS. Although some difficulty was experienced with interpenetration between the tool and the work-piece with the latter program at very large strains, it appeared that any apparent loss of contact in either case was attributable to the relatively large mesh size used. However, Lapovok and Thomson [48], who investigated the effect of initial shape of the billet in the 2-D forging of an H-beam using the program LUSAS [49], found a small amount of underfiU at the root fillet of the flange when flow occurred from flange to web and at the crest of the flange when forging commenced from a rectangular preform. However, as might be expected, their results showed that the distribution of strain was strongly dependent on preform shape (flange-to-web thickness ratio) and on friction. For the particular section investigated - one in which the flange had an aspect ratio of about 3 - the recommended preform shape to minimise heterogeneity of deformation at a coefficient of friction, I~ = 0.15, was one with a web which thinned toward the flange, differing substantially from that suggested by Akgerman, Becker and Altan [18] for a similar product, but bearing a strong resemblance to that recommended by Kobayashi [13] and Lange [13, 18]. 3.3

430 4.

EXPERIMENT Initially-rectangular blocks of commercially-pure lead, lubricated with sheets of PTFE, were compressed between a roll, grooved with the 760UB profile, and a fiat base to simulate forging of a beam blank (taking advantage of one axis of symmetry). To correspond a p p r o x i m a t e l y with the conditions used in the calculations, the work-piece was restrained axially on one face outside the deformation zone. The overhang on that end was variously allowed to spread or was clamped to prevent spread. Overhang on the other end was allowed to move freely. When lead was used simulate hot steel, the roll groove filled completely. When 'Plasticine' was used, lubricated with calcium carbonate to give a coefficient of friction 0.3<~t<0.4 [41], underfill occurred as shown in Figs 11(a) and (b). Evidently, increased transverse restraint caused an increase in separation at the root fillet of the flange. However, it became evident that the result was extremely sensitive to the constraints imposed by the constitutive behaviour of the work material as well as by the boundary conditions.

ActualProfile of Plasticine . . . ~ ~ Profileof Template

ActualProfile of Pl~ticine Profileof Template

Figure 11. Loss of contact (underfill) when a 'Plasticine' billet was forged into the profile of 760 UB beam blank (a) With transverse restraint on "front face". (b) With axial restraint, only, on "front face".

5.

DISCUSSION

Study of the literature suggests strongly that underfiU of tooling can occur early in a multistage process as a consequence of a requirement of excessive initial shape change or poor transverse flow of material resulting in poor establishment of a modular configuration, or it may occur later in the sequence, most notably as a result of deficient interaction between "modules". In the former case, it is especially likely to occur in rolling under non-steady state conditions (at the head and tail of billets) because of the lack of material outside the deformation zone to provide constraint. In the latter case, underfill may also occur as a result of transverse flow (i.e. transfer of material between modules of the section) or as a result of differential longitudinal flow. In fact, it appears that the relative importance of these mechanisms

431 probably depends on the dimensions of the web and flange and on process variables, including initial thickness of web and flange and the level of friction. Buckling of a thin web or flange as a result of faulty preform or process design may also occur. It is concluded tentatively that the effect of differential elongation of the web and flange in causing stretching of the flange and underfiU on its inside flank by transverse contraction is more significant when the web is thin, i.e. the ratios of the areas of the corresponding modules are largest. The constraint on flow (transverse and axial) provided by friction tends to decrease underfiU, which is dependent on friction in the flange, but not on that in the web. "Secondary" geometrical factors such as root radius and inside angle of flank appeared to be of relatively small significance in determining die filling. Taper of the web was not found to have any effect, although increasing the relative width of the flange increased underfill, suggesting that a simple dependence on differential stretching to cause contraction away from the root of the inside fillet and flank of the flange is improbable. UnderfiU seems to be strongly dependent on material properties; it is promoted by work hardening and generally by high flow stress. These are essentially parameters high values of which would inhibit free transverse movement. Moderate freedom in the flange may assist complete filling of the root radius of the fillet and the inner flange, but friction above a critical level certainly results in incomplete filling of the crest [48]. That form filling is likely to be highly sensitive to constraints applied to the deformation zone - perhaps to a greater extent than to the shape of the groove itself or of the preform - is suggested by the analysis of Lapovok and Thomson [48] in their investigation of optimum preform shape for a given H-beam in plane-strain forging. Against this is the evidence of buckling in deep, thin, flanges leading to laps [33] and to the recommendations, related to flange geometry, made by Kobayashi [13] on number and shape of preforms, which agree closely with those of Lange [18]. It is noteworthy that the recommended shape, which coincides essentially with that for deep flanges obtained by Lapovok and Thomson [48] by a rational design procedure, differs from that suggested by Akgerman, Becker and Altan [18], but this may be because the recommendations of the latter seem to have pertained to an Hsection with a shorter web, the significance of which difference in geometry was recognised by the former authors. Although underfill was found in the present investigation to be relatively insensitive to geometry, it is noted that a very coarse mesh was used in the analysis and constraint due to material outside the deformation zone was not well simulated. A much finer mesh was used in the analysis of Lapovok and Thomson [48] and it extended sufficiently far outside the deformation zone to model constraint. Further, a much wider range of geometries was included; one large pass rather than a series of incremental passes was considered, so that some lack of complete filling in the

432 intermediate stage was accepted, and homogeneity of deformation was the prime consideration. Although the analyses performed cannot be applied directly to rolling, they provide some evidence relevant to groove filling in rolling and to filling of closed dies in forging, but the constraints applied in the computations are likely to have provided an indication of behaviour at the head end of a rolled section or close to the end of a forged section. The importance of constraint by material outside the deformation zone in determining the occurrence and extent of underfill is remarked by Westope and Glover [4] in their comments on observed end effects. 6.

CONCLUSIONS The results of the numerical analysis of closed die forging described in this paper cannot be interpreted directly to study the causes of underfiU in shape rolling. However, some probable conclusions can be drawn. Underfill may originate early in a sequence of passes as a result of inadequate transverse flow, or a requirement of excessive transverse flow (poor design of initial billet shape or poor preform design) to establish a modular form in which substantial redistribution of material is not required. UnderfiU, notably at the inside root fillet and on the internal flank of the beam blank investigated, originating later in the sequence of passes, was not a result of a simple contraction of the flange produced by differential elongation. A more potent cause seemed to be lack of back pressure in the flange to produce transverse deformation, a result, for instance, of insufficient friction in the flange. Material properties, notably high flow strength and high rate of work hardening which promoted underfiU, possibly also by affecting freedom of transverse spread, were significant. There seems to be some evidence that a degree of underfiU at critical locations, e.g. at the inside fillet and flank of the flange in the H-section, is more or less inevitable during production and it is essentially a matter of choosing a production schedule which will minimise it. It appears that geometrical factors such as taper of the web, angle of the inside flank of the flange, radius of the root fillet of the flange and even width of the flange are of no more than secondary importance. Constraints on the deformation zone, exerted for instance by material outside the deformation zone, are likely to act in somewhat the same manner as friction, providing a back stress and hence promoting complete filling. However, it may be concluded that a system of rational design with the capacity to include a range of geometries is needed, so that form-filling in grooves of any shape can be investigated individually and the preform best able to provide filling can be determined in each case.

433 7. ACKNOWLEDGEMENTS The work was largely funded by the BHP Co. Ltd. and thanks are due to the Company for permission to publish the work. One of the authors (P.F.T.) also wishes to acknowledge the support of the Australian Research Committee. .

1. 0

3. 0

~

6. 7. ~

9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21.

REFERENCES R.N. Parkins, Mechanical Treatment of Metals, Allen & Unwin, London, 1968. N.R. Chitkara and G.M. Hardy, Int. J. Mech. Sci., ~ 1977, pp 575-594. R.L. Bodnar, D.C. Ronemus, B.L. Bamfitt and D.C. Shah, Trans ISS, 1986, pp 35-45. A.R. Westrope and G. Glover, Study of Pass Filling in Beam Blank Rolling, BHP-MRL Report No. MRL/PS32/86/007, April, 1986. T. Kusaba and T. Sasaki, Kawasaki Steel Tech. Rep., 11 (5), 1979, pp 1-15. J.J. Park and S.I. Oh, J. Eng. Ind. (Trans ASME), 112 (1), 1990, pp 36-46. C. Bertrand-Corsini, C. David, P. Montmitonnet, J.L. Chenot and P. Buessler, Rev. Met., 85(10), 1988, pp 771-782. A. Faessel, Rev.Met., 85(10), 1988, pp 783-785. W. Dahl, M. Hagen, R. Kopp and M.M. Desouza, , Stahl u. Eisen, 109(20), 1990, pp36-46. K. Komori, K. Kato, T. Murota, J. Endo and Y. D o i , JSME Internat. J., 30(262), 1987, pp 574-580. N. Kim and S. Kobayashi, Int. J. Mach.Tools & Manuf., 3_.00(2),1990, pp 243-268. D.Y. Yang, N.K. Lee, J.H.Yoon, J.L. Chenot and N. Soyris, Proc. Instn Mech. Engrs B- J.Eng.Manuf., (205)(2), 1991, pp 111-122. S. Kobayashi, "Process design in metal forming by the finite element method", Advanced Technology of Plasticity, 1987, Proc. Second Internat. Conf. Technol. Plast., Stuttgart, 24-28 August 1987, K. Lange (ed.), Vol II, Springer-Verlag, Berlin, 1987, pp 1213-1219. C. Hung and S. Kobayashi, J. Eng. Ind. (Trans. ASME), 11___44(4),1992, pp 459-464. S.M. Hwang and O. Heimlund, Int. J. Mach. Tool Des. Res., 24(4), 1984, pp 253-266. S. Tjotta and O. Heimlund, J. Mat. Process. Technol., ~ 1992, pp 79-96. S. Kobayashi, Int. J. Mach. Tool Des. Res., 24(3), 1984, pp 215-224. N. Akgerman, J.R. Becker and T. Altan, Metallurgia and Metal Forming, 4_0.,01973,pp 135-138. M.S. Joun and S.M. Hwang, Int. J. Mach. Tools & Manuf., 33(1), 1993, pp 51-62. M.S. Joun and S.M. Hwang, Int. J. Mach. Tools & Manuf., 33(1), 1993, pp 103-108. B.A. Nikiforov, Izv. V.U.Z. Chem. Metall., (6), 1980, pp 37-40. [BISI Translation No. 19449].

434 20 3@

24. 5,

26.

27. 8.

29. 30. 31.

32. 33.

4@

35. 36. 37. 38. 39.

F.A. Pisarenko et al., Communication-I, Izv. V.U.Z. Chern. Metall., (10), 1978, pp 104-106. [BISI Translation No. 18123]. F.A. Pisarenko et al., Communication 2. Izv. V.U.Z. Chern. Metall., (2), 1979, pp 56-59. [BISI Translation No. 18468]. F.A. Pisarenko, A.V. Yakovchenko and A. Yu Litvinenko, Russian Metallurgy, (3), 1979, pp 92-99. [Translation of Izvestiya Akademii nauk SSR Metally]. F. Barlat, K. Chung and O. Richmond, Int. J. Plast., 9(1), 1993, pp 51-64. P.S. Raghupathi and T. Altan, "Roll Pass Design in Shape Rolling (A survey of the German Technical Literature)", Applications of Computer-Aided Design in Rolling of Strip, Shapes and Plates [Proc. Conf.], Columbus, Ohio, 30 October, 1979, Battelle-Columbus Laboratories, Columbus, Ohio, 1979, pp 42. A. Geleji, Arch. Eisenh/itt., ~ (2), 1967, pp 99-103, quoted by Raghupathi and Altan [25]. E. Neuschutz and H. Thies, Stahl u. Eisen, 99(2), 1979, pp 49-54. K. Nakagawa, T. Hira, H. Abe, S. Kanari and H. Hayashi, Kawasaki Tech. Rep., 11(4), 1979, pp 501-513. [BHP Co. Ltd. CRL/Translation No. 10805]. S.K. Biswas and W.A. Knight, Int. J. Mach.Tool Des. Res., 15(3), 1975, pp 179-193. T.L. Subramaniam, R. Venkateshwar, G.D. Lahoti and F.M. Lee, "Experimental and computer modelling of die cavity fill in radial forging of rifling", Process Modeling: Fundamentals and Applications to Metals, Materials and Processing Congresses, 1978-79 [Proc. Conf.], ASM, Metals Park, Ohio, 1979, pp 185-203. M.J. Watkins, "Metal Forming Forging and Related Processes", Engineering Design Guides, No. 11, OUP, London, 1975. M. Abebe, "Simulation of plane strain H-shape forging using non-FEM method, "Advanced Technology of Plasticity 1987, Proc. 2nd Internat. Conf. Technol. Plast., Stuttgart, 24-28 August 1987, K. Lange (ed.), Vol II, Springer-Verlag, Berlin, 1987, pp 1097-1104. S.L. Semiatin, "Workability in forging" in Workability Testing Techniques, G.E. Dieter, (ed.), A.S.M., Metals Park, Cleveland, Ohio, 1984, Ch. 8., quoting A.M. Sabroff, F.W. Boulger and H.J. Henning, Forging, Materials and Practices, Rheinhold, New York, 1968. Myoe Aung and P.F. Thomson, Indian J. Technol., 2_2,21984, pp 133-138. A.T. Male, Jnl Inst. Metals, ~ 1964-65, pp 489-494. A.T. Male and M.G. Cockroft, Jrd Inst. Metals, ~ 1964-65, pp 38-46. E.G. Thomson, C.J. Yang and S. Kobayashi, Plastic Deformation in Metal Processing, Macmillan, New York, N.Y. 1965, Ch 13. S. Licka and J. Wozniak, "A model of the development of structure and mechanical properties of steel in hot rolling", Kovove materially, 20(5), 1982, pp 562-576. [BHP Co. Ltd. CRL/Translation No. 11377].

435 0@ 41. 42.

3@ 4@

45. 46. 47.

8Q

49.

P.S. Theocaris, C.A. Stassinakis and A.G. Mamalis, Int. J. Mech. Sci., 25(11), 1983, pp 833-844. K. Chijiiwa, Y. Hatamura and N. Hasagawa, Trans. ISIJ, 2_!,11981, pp 178-186. P.D. Hodgson, J.A. Szalla and P.J. Campbell, "Modelling of thermomechanical and metallurgical processes in plate rolling", 4th Internat Steel Rolling Conf., Deauville, France, 1-3 June, 1987, pp C8.1C8.6. A. Gittins, R.H. Moiler and J.IL Everett, B.H.P. Tech. Bull., 18(1), 1974, pp 2-8. R.J. Fields and M.F. Ashby, "A Fracture Mechanism map for Pure Iron", Report CUED/C/MATS/TR.32, Engineering Department, University of Cambridge, p4. H. Takahashi and J.M. Alexander, Jnl Inst. Metals, 9_.0_,01961-62, pp 72-79. K.T. Chang and I.J. Choi, "Influence of some parameters on closed-die forging", CIRP Ann., 24(1), 1975, pp 163-165. J.O. Hallquist, NIKE2D- A Vectorized Implicit, Finite Deformation Finite Element Code for Analyzing the Static and Dynamic Response of 2-D Solids with Interactive Rezoning and Graphics, Lawrence Livermore National Laboratory, December 1986. R.Y. Lapovok and P.F. Thomson, "An approach to preform design", to be published. LUSAS, Version 10.1, Finite Element Analysis Ltd, Kingston on Thames, Surrey, United Kingdom, 1991.

This Page Intentionally Left Blank