Materials Processing Defects (Studies in Applied Mechanics)

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 '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'

E-u

< II


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 ~

~

""

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:

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_)

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

)
0 at the beginning of the incremental step and the initial solution of Eqa.(4) gives F3+AF3~~~ ,

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

,