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Backward can extrusion and materials behaviour

Citation for published version (APA):Sillekens, W. H. (1992). Backward can extrusion and materials behaviour. Technische Universiteit Eindhoven.https://doi.org/10.6100/IR374925

DOI:10.6100/IR374925

Document status and date:Published: 01/01/1992

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https://doi.org/10.6100/IR374925

https://doi.org/10.6100/IR374925

https://research.tue.nl/en/publications/backward-can-extrusion-and-materials-behaviour(107db83e-5dff-48ae-ab24-c67adeb0fcf3).html

W. H. Sillekens

Backward Can Extrusion

and

Materials Behaviour

CJP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Sillekens, Wilhelmus Hubertina

Backward can extrusion and materials behaviour 1 Wilhelmus Hubertina Sillekens.

- Eindhoven : Technische Universiteit Eindhoven. - 111.

Proefschrift Eindhoven. - Met lit. opg. - Met samenvatting in het Nederlands.

ISBN 90-386-0032·1

Trefw.: omvormtechniek.

Druk: drukkerij Creemers, Sint Odiliënberg

Dit proefschrift is goedgekeurd door de promotoren:

prof. ir. J.A.G. Kals

en

Prof. Dr.-Ing. R. Kopp

copromotor: dr. ir. J.H. Dautzenberg

Backward Can Extrusion

and

Materials Behaviour

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven,

op gezag van de Rector Magnificus, prof. dr .• I.H. van Lint, voor een commissie

aangewezen door het College van Dekanen in het openbaar te verdedigen op

vrijdag 5 juni 1992 om 14.00 uur

door

WILHELMUS HUBERTINA SILLEKENS geboren te Herten

vii

Summary

This thesis reports on a studyin the field of metal forming. Topic is the bulk-forming

process of backward can extrusion in relation to the plastic behaviour of the worked

material.

The issue of the workpiece material's behaviour during forming is an interesting

one: it is here where the fields of materials science and forming technology have a

common relevance. A better insight into these phenomena - tor the aim of better

process control - requires the inlegration of aspects trom both disciplines. lt is in this

context that the present contribution must be regarded.

Backward can extrusion is used in the mass production of discrete parts tor the

manufacture of can-shaped components. lndustrial productions cover a variety of

shapes, sizes and materials; these products are utilised in a diversity of consumer

goods.

Presently available methods of backward can extrusion modelling are reviewed.

These are classified in experimental, analytica!, and numerical methods. Special

emphasis is placed on the category of analytica! models. Same important upper-bound

models, proposed in the literature, are introduced. A comparison on the basis of the

upper-bound principle makes clear that each of the treated models has an outlined

validity range. From a combination of these models, it is possible to calculate the ram

pressure as a tunetion of the ram stroke tor the initia! and the final stage of the

process. In addition, information is acquired concerning the material flow.

A further account deals with the properties, which are important in the evaluation

of a (new) material tor forming applications. A number of characteristic quantities is

discussed; these reprasent distinct features of the tormability, like flow behaviour,

failure behaviour, and plastic anisotropy. These quantities, as they are obtained trom

basic material tests, can be used to formulate some directives for the application.

Aluminiuïn, redaimed trom scrap by rapid-solidification processing, is used as an

illustrative material.

viii Summary

Next, the attention concentratas on a particular aspect of the workpiece

material's properties: the flow behaviour. This flow behaviour can be represented by

means of a flow curve, which depiets the flow stress as a tunetion of the plastic strain.

Except tor the influence of strain rate and temperature, such a flow curve does depend

also on the strain path; this is of importance since different processes, in genera!, entail

different strain paths. The strain-path dependenee of flow curves is studied.

Corresponding results of compression, torsion and tension tests are compared to each

ether and to those trom tests, invalving an abrupt change in the deformation mode.

Obtained flow curves are described by means of an accommodated flow function.

Concerning the backgrounds to these phenomena, possible crigins are discussed.

In condusion, the behaviour of the workpiece material is interpreled with respect

to the backward can extrusion process. An application concerns the modelling of the

process trom a viewpoint of the workpiece material's ductility. An analytica! model tor

the calculation of local strains and stresses is introduced. Finite-element simuiatien

serves as a verification. The failure behaviour is determined experimentally, and is

represented by means of ductile-failure curves. From a confrontation of the analytica!

results with this experimental information, as well as with actual extrusion experiments,

an insight is gained into the process limit of material tracture.

i x

Samenvatting

Dit proefschrift beschrijft de resultaten van een studie op het gebied van de omvorm

techniek. Onderwerp is het massief-omvormproces achterwaartse hulsextrusie in relatie

tot het bewerkingsgedrag van het werkstukmateriaaL

Het gedrag van het werkstukmateriaal is van bijzondere interesse: de

kennisgebieden van materiaalkunde en omvormtechnologie hebben hier een raakvlak.

Inzicht in dit gedrag is van belang voor een betere procesbeheersing. Dit vereist een

integratie van kenniselementen uit beide disciplines; in deze context moet het

gepresenteerde onderzoek worden geplaatst.

Achterwaartse hulsextrusie (of slagextrusie) wordt toegepast in de massa

fabricage van busvormige metalen onderdelen. Industriële produkties omvatten een

scala aan vormen, afmetingen en materialen; deze produkten vinden toepassing in een

verscheidenheid aan gebruiksgoederen.

Beschikbare methoden van procesmodellering worden besproken. Deze zijn

ingedeeld naar de manier van aanpak in experimentele, analytische en numerieke

methoden. De belangstelling gaat vooral uit naar de klasse van bovengrensmodellen;

hiervan is een inventarisatie gemaakt- gebaseerd op de literatuur. Toepassing van het

bovengrensprincipe maakt duidelijk welk model het beste voldoet voor de verschillende

procescondities; elk van de modellen blijkt een eigen geldigheidsgebied te hebben.

Door combinatie van deze modellen wordt zowel het begin- als het eindstadium van

het proces analytisch beschreven, waarbij het verloop van de stempeldruk wordt

voorspeld. Bovendien wordt een indicatie verkregen betreffende de materiaalstroom.

Verdere aandacht gaat uit naar de materiaaleigenschappen, die van belang zijn

bij de beoordeling van een (nieuw) materiaal voor omvormende bewerkingen. Een

aantal karakteristieke grootheden wordt hierbij gehanteerd, die onderscheiden

kenmerken van de omvormbaarheid kwantificeren. Hierin zijn begrepen: het

vloeigedrag, de ductiliteit (deformatievermogen), de plastische anisotropie. Deze

grootheden zijn met behulp van materiaalproeven bepaald. Aan de hand hiervan

kunnen richtlijnen worden geformuleerd voor de bewerking. Als illustratief materiaal is

x Samenvatting

gerecycled aluminium gebruikt, verkregen uit het basismateriaal via een snelle

stollingstechniek.

Eén belangrijk aspect van het materiaalgedrag betreft het llloeigedrag. Dit

vloeigedrag kan worden weergegeven met behulp van een vloeikromme, die de

vloeispanning van het materiaal vastlegt als functie van de plastische rek. Behalve van

reksnelheid en temperatuur blijkt de vloeikromme ook afhankelijk te zijn van de rekweg;

dit is van belang omdat deze rekweg varieert per omvormproces. De rekweg

afhankelijkheid is experimenteel onderzocht, enerzijds aan de hand van stuik-, torsie

en trekproeven, anderzijds door gecombineerde proeven met abrupte overgangen in

de rekweg. Een aangepaste vloeifunctie is gebruikt om het gedrag te kwantificeren.

Mogelijke oorzaken van verschillen in vloeigedrag worden behandeld.

Tot slot wordt het gedrag van het werkstukmateriaal geëvalueerd met betrekking

tot achterwaartse hulsextrusie. Een toepassing betreft de modellering van het proces

uit oogpunt van de materiaalductiliteit. Geïntroduceerd wordt een analytisch model voor

de berekening van lokale rekken en spanningen. Als toetsing hiervan dient een eindige

elementensimulatie van het proces. Het duetiel falen van het werkstukmateriaal is

bepaald met behulp van proeven en is vastgelegd in zogenaamde faalkrommen. Vanuit

een confrontatie van het model met deze experimentele gegevens, aangevuld met

concrete hulsextrusieproeven, wordt een inzicht verkregen betreffende deze

procesgrens.

xi

Notation

In the following list the main symbols are compiled. The nomendature basically

confarms to the unified terminology, as recommended by the International lnstitution

for Production Engineering Research (CIRP). Reference: Armals of the CIRP 3.SL2 (1986): 575-577.

SI-di mension

b specimen width [LJ c characteristic stress [ML - 1T-2]

Cs Bridgman correction factor H D depth of penetratien [L]

d specimen diameter [L]

F force [MLT-2]

H chamber height [L]

h ram height [L]

L specimen length [L]

M moment [ML2T-2]

m plastic-friction factor [-]

n strain-hardening exponent [-]

PR average ram pressure [ML - 1T-2]

R extrusion ratio [-]

R specimen radius [L]

Re chamber radius [L]

RR ram radius [L]

extrusion reduction H anisotropy parameter [-]

s active-zone height [L]

xii

s

T

T

To

û

a

óx

e

p

q

q

Ut

am

ao T

8

c e F

Fr

m

max

min

0

opt

r, IJ, z

specimen thickness

tempersture

bottam thickness

billet height

velocity

semi-cone angle

experimental error in a quantity x

equivalent strain

indicator of reproducibility

profile radius

stress

effective stress

flow stress

hydrastatic stress component

constant flow stress

shear stress

twisting angle

Indices

compression

fin al

fracture

friction

initial

meen, average

maximum

minimum

original

optimal

polar coordinates

t

tot

0

tension

total

Nota ti on

[L)

[0]

[L)

[L]

[LT- 1]

[-] [mise]

[-]

(-]

[L)

[ML - 1T-2]

[ML - 1T-2J [ML -1T-2]

[ML - 1T-2]

[ML - 1T-2]

[ML -1T-2]

[-]

after pre-straining

xiii

Table of Contents

Summary vii

Samenvatting ix

Notation xi

1 Introduetion 1

REFERENCE LIST 8

2 Review of Backward Can Extrusion Models 9

2.1 EXPERIMENTAL TREATMENT 10

2.2 ANALYTICAL TREATMENT 11

2.2.1 Spherical model; 2.2.2 Two-zone model; 2.2.3 Three-zone

model; 2.2.4 Extended three-zone model; 2.2.5 Evalustion

2.3 NUMERICAL TREATMENT 26

2.4 CONCLUSIONS 27

REFERENCE LIST 28

3 Characteristic Quantities of Formability 31

3.1 RAPID-SOUDIFICATION PROCESSING OF ALUMINIUM SCRAP 32

3.1.1 Orientatîon; 3.1.2 Technique; 3.1.3 Processing parameters

3.2 FORMABILITY OF RECYCLED ALUMINIUM 37

3.2.1 Flow behaviour; 3.2.2 Failure behaviour; 3.2.3 Plastic

anisotropy; 3.2.4 Evaluatîon

3.3 CONCLUSIONS 49 REFERENCE LIST 50

xiv Table of Contents

4 Strain-Path Dependenee of Flow Curves 53

4.1 EXPERIMENTAL PROCEDURE 54

4.1.1 Testing techniques; 4.1.2 Experimental conditions and

materials

4.2 RESULTS 60

4.2.1 Standard flow curves; 4.2.2 Combined flow curves

4.3 DISCUSSION 68

4.3.1 Standard flow curves; 4.3.2 Combined flow curves;

4.3.3 Some practical aspects

4.4 CONCLUSIONS 75

REFERENCE LIST 76

5 On the Ductile-Failure Behaviour in Backward Can Extrusion 79

5.1 MODELLING ASPECTS 80

5.1.1 Slab method; 5.1.2 Power methad

5.2 TOOL LOADS 83

5.3 DUCTILITY 87

5.3.1 Ductile-failure curves; 5.3.2 Application; 5.3.3 Discussion

5.4 CONCLUSIONS 97

REFERENCE LIST 98

6 Conclusion 101

Appendices A Strain-Path Dependenee of Row Curves: Additional Results 103

B Error Analyses for the Standard Material Tests 111

C Backward Can Extrusion: Stress and Strain Analysis 119

1

Chapter 1

Introduetion

Backwerd can extrusion is an important metal-forming process for the production of

can-shaped parts. Other, more or less obsolete designations for this process are tube

piercing and impact extrusion. In German literature, the process is referred to by the

name of "Rückwärts-NapfflieBpressen".

Backward can extrusion is a typical process tor the mass production, where the

relatively high investment costs in tools and machinery can be spread over a large

number of products. The process is to be dassed among the category of cold-forging

operations; it is usually practised at room temperature, although the extrusion of

"difficult" products may require the process to be performed at elevated temperature.

Similar to other bulk-forming processes, the involved process torces are high. The

process can be used as a separate operation, but is applied also in combination with

other processas like hesding and ironing. For not too complicated products, such as

aluminium toothpaste tubes, a high dagree of production automation can be achieved;

production rates here may attain values of about 100-150 productsper minute [1.1).

The products which are produced by this process cover a variety of shapes,

sizes and materials. Customary workpiece matenals include non-terrous metals like

aluminium, copper and brass, as well as saveral grades of steel. The geometry aften

is axi-symmetric, but different shapes are produced also.

Parts, produced by means of the backwerd can extrusion process, are utitised

in a diversity of consumer goods. Examples of traditional products in the non-ferrous

sector are: collapsible tubes for toothpaste and other creams, beverage cans, battery

containers. Contemporary productions also cover such items as parts for video

equipment (electrotechnical industry), and housings for fuel pumps and filters

(automotive industry) [1.2). Steel parts find application where the strength requirements

are constderable. The automotive industry hereis a major customer, using high-quality

components in engines, transmissions, etcetera [1.3].

2 Chapter i

stripper

ram

ejector

Fjgure 1.1 Princlpal set-up for backward can axtrusion

In figure 1.1. the forming process ot backward can extrusion is drawn schematically.

The tooling consists of a ram (punch, mandrel) and a chamber (female die). First, !he

chamber is supplied with a billel (slug, disk), which fits the opening. By moving the ram

into the chamber, then, this workpiece material is enclosed and subsequently extruded

through the aperture between the ram side and the chamber wall. The escaping

material thus moves opposite to the forward direction of the ram, which explalns the

prefix "backward" in the name of the process. Lubricant, applied on the billet, prevents

direct metallic contact between tools and workpiece and reduces the loads. After

withdrawal of the ram, the product can be removed. For this purpose, the tools are

furnished with an ejector and a stripPer. To imprave !he stability of the process, the

ram nose otten is of a slighdy inclined conical shape. Further, the ram sha!t is designed

with a clearance; teaving only the ram land in contact with the intarior wall surface.

Introduetion 3

Of course, the previous description of the process is merely an outline. Detailed

intermation on the implementation has been issued by the International Gold Forging

Group (e.g., [1.4]); these data sheets and documents are concerned w~h various

practical aspects.

The extrusion ratio R or the extrusion reduction r are used as a global measure

of delermation in the extrusion process'l 11 the cross-sectionat area of the billet is

symbolised by Ao and the cross-sectionat area of the extruded material by A1, these

quantities are defined as:

Äo Ao-A, R=- and r=--.

A, Äo (1.1)

Restricting te the extrusion of cylindrical cans, these definitions can be converted into:

R (1.2)

where RR and Re denote the ram radius and the chamber radius respectively. For the

extrusion reduction, the whole extrusion range is comprised within distinct boundaries

(O<r<1); the range lor the extrusion ratio theoretically extends to infin~ (1 <R<~).

The extrudability aften is characterised by either of these quant~ies. Attainable

values in the appiication depend on the workpiece materiaL Being rather lim~ed lor the

(cold) extrusion of steels, these may extend well beyend R=10 (corresponding to

r=0.90) lor low-strength matenals (commerciatty·pure aluminium, lead, zinc).

In view of the long metal-forming history, cold-extrusion processas are of relatively

recent crigin [1.5]. Earlies1 implementations of the backward can extrusion process are

dated back to the beginning of the nineteenth century w~h the manufacture of

collapsible tubes trom lead and tin; large-scale production started about 1841 in the

Uniled Stales of Arnerica, and in 1879 in Germany. In 1915 a ~chover to pure

aluminium as a workpiece material followed.

Up to the thirties of this century, cold extrusion of steel was considered nat to

be possible. A. Uobergeld in 1934 succeeded in extruding a steel cartridge trom a pre-

1)English literature seems to preter the use of the extrusion ratio, whlle German sourees baslcally employ the extrusion reduction as a unit.

4 Chapter 1 ------------------------

formed can at the "Kabel- und Metallwerke Neumeyer· in Nümberg (Germany) (1.3,

1.5]. By subsequent experimentallons during 1935-37, the e>ctrusion of steel was

developed to industrial applicability; patents were obtained torsome extrusion variants,

including backward can extrusion. An important prerequis~e lor this development was

the application of a zinc phosphate layer on the biltets as a lubricant carrier - thus

preventing cold welding between the tools and the workpiece. This was an invention

by Singerat the same firm.

The use of the process in the produelion of consumer goods rapidty gained

significanee aHer 1945. Today, it is one of the established methods loran economical

produelion of can-shaped parts, in lhe terrous as wel! as !he non-terrous sector.

Process control is important tor a number of reasons. One of !he principal motives is

rendered by the necessity to keep up with the customer's constant demand lor a

better product quality - culminating even in a dietata lor 'zero defects". As lor the

increasing international competition, the condition of "producing at minimal costs'

remains particularly perceptible. This means lhat !he production must be efficient,

invalving a minimum of machine interruptions, tooi changes, etcetera. Present-day

developments in the environmental conte>ct urge tor a conscientious use of energy and

material resources, as well as a reduction (or even avoidance) of polluting residues

such as lubricants and detergents.

These conditlans have ledtosome distinguished trends in cold-Ierging practica

[1.6]. Among these is !he trend ol produel and process innovation, iocreasing the

variety ot extrudable materials and product geometries. Another ene is !he sustained

effortto imprave the dimensional accuracy and surface quality of the produel, pursumg

!he goal of "(near) net-shape" forming. Also, an e>ctension of the toollffe is sought tor

by using new tooi materials and surface coatings, as well as ether melhods of billet

treatrnent The implementation of computer-aided techniques (CAD and CAM) aims tor

an impravement of tooi-design and -manufaclure procedures.

The above mentioned trends demand lor a thorough understanding ol the

process mechanics. This insight is to be acquired by continued research, bath in the

fundamental and applied domain.

Introduetion 5

The process of backward can extrusion can be considered by means of the general

system, developed by Backofen, Gebhardt, Kienzle, Lange and Schey [1.7]. This

systematical approach has shown to be appropliate lor many different Iorming

processes. According to this system, the encountered problems can be divided into

eight areas.

Specified lor backwerd can extrusion, these areas are indicated in fiaure 1.2.

The central image shows a representation lor the beginning and the end of the

operation. The distinct problem areascan ba described as fellows:

Area 1 covers the plastic zone and is concerned with the determination of the

material's behaviour in tne plastic state.

Area 2 denales the oharacteristics of the workpleoe betore deformatlon,

such as the chemica! composilion, mechanica! properties, etcetera.

7

8

6

4

Figure 1.2 Systematic approach to metal-forming processes, CiamonstrataC! tor

backwarcJ can extrusion (atier Lange [1.7])

6 Chapter 1

Area 3 denotes !he charaoterlstics of the workplece alter delormatlon, in

particular !he mechanica! properties, surlace qualily, and product accuracy (!he

in-service propertles).

- Area 4 relales to !he boundary area between workplece and tooi and deals

with the toples of friction, lubrication, and wear.

Area 5 is concerned withall questions in conneetion with the Iorming tooi, such

as the tooi layout and tooi materials.

Area 6 relers to the area outSide the tool-workpiece interface and deals with any

surface reactlons which may occur between workpiece and surround1ng

atmosphere.

Area 7, visualised by the inner circle, symbolises the machinetooi and invoe.res

topics like tooi gulding and workpiece handling.

Area a is concemed with !he Inlegration lnto the production system as a whole; this lncludes diverse aspects, ranging from auxiliary equipment to factory

organisation.

These areas are basically the sarne lor all Iorming processes. Naturally, !hey are not

complete!y isolated from each ether, but are partially interrelated. For instance, the

phenomena in the plastic zone (area 1) inlluence the product properties (area 3), and

are in turn effected by !he workpiece properties befare deformation (area 2).

Such a systematic approach is particularly useful: experience obtained trom

individual processas can be categorised and hence may be transferred to other

processes, be it In a more or less modlfied form. Thus, one may hope to establish

scientlfic elements systematically - employing research and development efforts in an

optima! way.

The present contribution can be typified as a basic, rather than an applied study into

the backward can extrusion process with a special attention tor the malerial's

behaviour. ft principally deals with the phenomena in area 1 - being the core of !he

process in which many problems of process control find their source.

The determination of the workpiece behaviour during plastic werking, which is

the issue of area 1, entails aspectsof plasticity theory (stresses, strains, metalllow, ... )

and matenals science (textures, anisotropy, ... ). Traditionally, thesetopics are presenled

separately; that Is, studies in the field are concerned either with !he rnadelling of !he

Introduetion 7

process in a mathematica! interpretation, or with the behaviour of the material lrom a

metal-physical, often microscopie point of view. Aîm of this contribution is to give a

more embracing treatment of these matters, in order to gain a better insight into the

process limns: as a process characteristic and as a characteristic of the workpiece

materiaL 11 is believed that !he key to many technological problems is implied in such

an integlllll approach.

The arrangement in chapters is as !ollows. In chapter 2 the methods of

modelling lor the backwerd can extrusion process are reviewed, classified by the

adopted approach in experimental, analytica!, and numerical treatrnent. Four upper

bound models, obtained lrom various sourees in literature, are introduced and

evaluated vis-à-vis wtth respecttotheir individual validtty. These models apply to !he

important category of cylindrical products. The next two chapters are devoted to !he

plastic beheviour of materials in a more general sen se. In chapter 3 some characteristic

quantmes are discussed. reprasenting distinct features of the material's lormabiltty.

These characteristic quantnies can be used to assess the sunabiltty of new materlafs

lor Iorming applications. This is demonstraled lor a rather uncommon materie!:

recycled aluminium, reclaimed via rapid-solidification processing. The topic of the flow

behaviour, being one of the relevant features of !ormabiltty, is explored lurther in

chapter 4. Th is chapter is concerned with the ancmalies in flow behaviour, ruising trom

more or less abrupt changes in the strain path: many actual Iorming processes,

including backwerd can extrusion, invol\le such changes. Then, the attention shifts

back to the rnadelling of the extrusion process. Chapter 5 reporta on an application

which deals wnh the dUctiltty of the workpiece materlat This implementation combines

an analytica! model with a phenomenologically based representation of !he material's

behaviour, the so-called ductile-failure curve.

8

REFERENCE LIST

[1.1] Sligte J.G.,

Spaanloos omvormen van metalen, deel 2,

Agon Elsevier, Amsterdam - Brussel {1974).

[1.2] lnformation brochure,

Advanced Metal Forming BV, Zwolle {1991).

[1.3] lnformat1on brochure,

Neumeyer-FiieBpressen GmbH. Nürnberg (1984).

[1.4] ICFG,

Chapter 1

General Recommendalions tor Design, Manutacture and Operational Aspects

of Gold Extrusion Tools tor Steel Components,

document no. 6/82 {1983).

[1.5] Feldmann H.-D., Liebergeld R.,

"Die historische Entwicklung des Kattmassivumformens van Stahl zum

wirtschaitlichen Fertigungsverfahren",

Draht 41 (1990): 431-437, 830-835.

[1.6] Geiger A.,

"State of the Art and Development Trends in Cold Forging Technology•,

Proceedings of the Secend International Conterenee on Technology of

Plasticity, Stuttgart (1987): 469-477.

[1.7] Lange K (editor),

Handbook of Met al Forming,

McGraw-Hill, New York (1985).

9

Chapter 2

Review of Backward Can Extrusion Models

Backwerd can extrusion is an instatïonary process~ which means that the pracess

variables depend on the penetration depth of the ram, and therefore are tîme

dependent. As a consequence the rnadelling is rather complicated, and generally nat

as advanced as lor related (quasi-)stationary processes such as forward extrusion.

Lrterature on the rnadelling of !he backwerd can extrusion process does almast

wrthout exception apply to a rotationally symmetrie geometry. The first onsets were

mainly experimental in !heir approach. Several of these early worl<s are based on large

numbers of extrusîon experiments, where the process farces were measured and

subsequently expressed by means of approximating formules. A further group involves

the category a! studies, in which the analytica! methods of plasticrty (slab method,

upper-bound method) are employed. The introduetion of powerlul computers induced

the development of new numerical techniques (such as the finite-element method) and

the application of these to plasticrty problems.

Presently availabte methods of rnadelling lor the backwerd can extrusion

process are reviewed in 1hiS chapter_ This survey is not complete, but is meant just to

picture the ·state of the art•. Special attention will be given to the category of analytica I

models. basedon the upper-bound approach: severat rnadeis trom lrterature are newty

evaruated from some different points of view.

10 Chapter 2

2.1 EXPERIMENTAL TREATMENT

Over the years, many empirica! formulas have been proposed lor the calculation of the

(maximum) ram force during extrusion, A dozen of these is compiled by Boes and

Pouw [2, 1], The application of such formulas olten is restricted to a limited range of

extrusion conditions andfor certain materials. Practical use of several of these further

is hampered because they include vague or impractical quantities, such as eHiciency

factors and alloy compositions. Unsatisfied with these previous formulas, Boes and

Pouw then introduced a general method which - in their view is practically useful and

easy to apply. They propose:

p = [a·ln(R) +b]·ub- (2, 1)

The symbol p represents a pressure, being defined as the maximum ram force divided

by the cross-sectionat area oi the billet The extrusion geometry is appointed by the

extrusion ratio R; ub denotes the ultimata tensile strength of the workpiee<l material.

ConstaniS a and b are determined experimentally (a=2.80 and b=0.75 tor the

extnusion of steel; lor non-terrous metals a=3.14 and b=0,80). The fonmula inthls form

applies to can extrusion, using a ram with a cone-shaped nose; if otherwise. the right

hand side of the equation must be muitiplied with a correction factor x (lor instance,

x= 1.2 lor a fiat ram).

Schmitt performed an extensive experimentel investigation info the backward can

extrusion of steel [2.2]. Saveral proe<lss fore<ls (ram force, chamber lorres, pull bacik

force, product ejection force) were manilored in dependenee on the workp.ace

material, billeLdimensions, and extrusion reduction. In the experiments a ram with a

cone-shaped nose was used. The experimental information is presented by means of

graphs, trom which the influence of the parameters can be interpreted. This work was

.an important step towards a better process controL To the aim of industrial use,

Schmitt's resuits later were incorporated in a nomogram (2.3]; this provides lor a

simple predietien of the maximum ram force within engineering accuracy (x 10 %).

The study of delermation pattems by means of visieplastic methods can also

be grouped among the experimental methods. As an example, _Kast reports on

bacikward can extnusion experiments using axially spl~ biliets [2.4J. This, technique

yields intermation on the iocation of dead zones and strain concentrations.

Review of Backward Can Extrusion Models 11

2.2 ANALYTICAL TREATMENT

The attention in this sectien will focus particularly on the upper-bound models of

backward can extrusion. Some ether studies (Dipper (2.5], Altan and Thomsen [2.6]),

based on the slab method, are just mentioned here. These models consicter !he

processas a double compression operation; trom the ana!ysis of stresses, a tormula

is derived lor the calculation of the ram pressure in dependenee on the ram stroke.

Upper-bound models are based on an analysis of the power consumption; a device

lor optimisation is given by the up per -bound theorem, which stat es - in general terms

- !hal !he total involved power strives to a minimum. Since the introduetion of !he

technique, saveral of these models tor the description of the backward can extrusion

process have been proposed. A separate category consists of these models, which

are developed lor the application to thin-walled cans; this geometry allows lor some

simplifying assumplions. The present review. however, directs itse" to the category of

general models, which apply to thiCk-walled as wen as thin-walled cans.

As a representation of the material, a constant flow stress a0 ("perfectly plastic

behaviour') is assumed in all models. Another aspect concerns the friction in the

interface of tools and workpiece; !he frictlonal shear stress 'Fr here is quantffied in

accordance with the von Mises model (or constant-friction model):

"o Tfr=m·- . ..[3

(2.2)

The value of the plastic-friction factor m depends on !he lubricating conditions and

ranges lrom m=O (na friction) to m= 1 (stiCking conditions).

By using the upper-bound technique, it is possible to derive an expressiontor

the total power consumption, required lor the process. 11 this resutt is divided by the

velocity of the ram and the ram's Irontal area, one arrives at an expression lor the

(average) ram pressure PR· A further normalisation on the workpiece material's ftow

stress yields an equation in dimensionless lorm. In this ferm, the resutts of the rnadeis

will be presented.

12 Chapter 2

2.2.1 Spherical model

This upper-bound model of the backward can extrusion process originally is proposed

by Avitzur et al. [2.7, 2.8]. The model is meant to describe the early stage of the

process; that is, tor small depths of ram penetration.

D

Figure 2. 1 Schematic of the spherical model (model 1)

In the analysis, the ram is assumed to be a flat one; by a minor adaptation,

however, it is also possible to consider a nosed ram of spherical shape. An image of

the model is shown in tigure 2.1. The workpiece body is divided into tour zones,

separated by surfaces of velocity discontinuity. Bath zone I and zone IV are dead-zone

regions; zone 111 represents the rigidly ascending wall. Only zone 11 experiences plastic

deformation. lt consists of a spherically shaped shell-section and is· appointed by its

semi-cone angle a. The velocity field tor this zone is derived on the analogy of a

previous analysis by Avitzur, concerning flow through conical converging dies. The


semi-cone angle serves as an optimisation parameter in minimising the total power

requirement, in accordance with the upper-bound theorem. This model is applicable

until the ram has penetrated to a depth, at which the top of the active zone 11 has

reached the bottam of the chamber.

The geometry of the process is represented by the ram radius AR, the chamber

radius Re, the ram height h, and the current depth of penetration D. lnternal

deformation, shearing along the surfaces of velocity discontinuity, friction, an externally

applied pressure on top of the wall, and inertia effects are calculated in the total power

consumption. Omitting the contributions of extemal pressure and inertia, the salution

tor the relative ram pressure reads:

PR =~· 1 ·{.fä·t(a}·ln(Re}+-0 --cota+(cota}·ln(Re} ao .[3 1-(RR/Rc)2 AR sin2a AR

+m·(RR ·[.E._ +te -1)·cota] +_!:!__)}, Re AR AR AR

where f(a) is the goniometrical function:

f(a} =-1-·[1-(cosa)·V1-(11/12}·sin2a

sin2a

+ 1 ·In( 1 +.ff1fï2 )] . J11·12 J11/12·cosa+V1-(11/12)·sin2a

(2.3)

{2.4)

These expressions state that the relativa ram pressure is a tunetion of the following

variables:

PR PA [Re D h l ao = ao AR' AR' RR,m,a .

(2.5)

The semi-cone angle a is a pseudo-independent parameter. lt is not possible

to derive an explicit expression for the optima! semi-cone angle aopt• which minimises

the required power; its value, however, can be obtained by numerical solution.1l This

1 lvet lt can be gathered from the ram-pressure function, that the optima! semi-cone angle does not

depend on the depth of panelration and thus remains constant throughout the process.

14 Chapter 2

optima! seml-cone angle and the accompanying ram pressure are considered to be the

actual ones, under the given set of conditions.

2.2.2 Two-zone model

The next model of backward can extrusion is designed tor the description of the end

of the stroke. This model, also, originates trom Avitzur and co-werkers [2.7, 2.9].

H

T

Fiaure 2.2 Schematic of the two-zone model (model 2)

Here, the workpiece body is divided into two zones, as is shown in figure 2.2.

Zone I - beneath the ram - is being plastically deformed; the assumed velocity field

corresponds with the one tor ideal compression. In zone 11, the material is moving

upward as a rigid body. The surface of velocity discontinuity, which ~eparates the two

zones, is shaped such that the respective velocity fields in the two zones are


compatible; this means that the velocity component perpendicular to the surface is

continuous while a discontinuity in velocity parallel to the surface is permitted. The

model in this form does not incorporate any optimisation parameters.

The geometry of the tools is represented by the ram radius RR, the chamber

radius Re, the ram height h, and the chamber height H. The current bottorn thickness

T appoints the position of the ram. lnternal deformation, shearing, friction, an ex:ternal

pressure on the wall, and inertia effects are included in the analysis. lf the contributions

of ex:ternal pressure and inertia are left out, the obtained expression tor the relativa ram

pressure is:

PR 2 Re 1 11 (Re/RR)3

-1 ( 4·(Rc/RR)-(T/RR)2

) = ·In(-)+-· -· · 1 +-----,--ao 1-(RR/Re)2 RR J3 3 T fRR ((Re/RR)2-1]2 (2.6)

+m·(.!· (Re/RR}3

+1 + 2·RR/Re ·[_!i_+_!:!_. Rel)}, 3 T /RR 1-(RR/Rc)2 RR RR RR

and resuming in terms of the independent variables:

PR PR[Re T H h l ao O"o RR' RR' RR' RR,m .

2.2.3 Three-zone model

(2.7)

This three-zone representation of the process is originally suggested by Kudo (2.1 0].

Later, it is explored by saveral others [2.9, 2.11, 2.12], including the implementation

into a nomogramfora practical determination of the ram pressure (2.13]. The model

is meant to describe the final stage of the process.

In figure 2.3 the schematic of this model is shown. In the plastic zone I, the

velocity field is assumed to be the one tor ideal compression. Zone 11 in the corner

area also is plastic; the flow direction here is altered towards zone 111, the rigidly

ascending wall. No optimisation parameter is incorporated.

16 Chapter 2

H

T

Figure 2.3 Schemaüc of the three-zone model (model 3)

Subsequentto the calculation of the power consumption by intemal deformation,

shearing and friction, an equation can be derived tor the relativa ram pressure. In tuil,

the salution is:

PA= 1 + 1 ·{In( ~(Re/RR)4 +3 +(Re/RR)2) +2 -~1 +3·(RR/Rc}4 O'o J3·[1-(RR/Re)2] 3

T 2 RR Re AR 1 AR AR 2 +-+-·-·---+-·-·(-)

RA 3 T RR T 3 T Re (2.8)

which can be summarised simply as:


(2.9)

2.2.4 Extended three-zone model

This model is a further development of the three-zone model. An extension is obtained

by adding a dead-zone region in the bottorn area of the workpiece [2.12, 2.14]. By

doing so, the application shifts from the final to the initia! stage of the process; this

extended model therefore is treated separately.

In this case the workpiece body is divided into four zones, as is indicated in

figure 2.4. The zones I, 11 and 111 correspond with these for the plain three-zone model;

zone IV is the newly introduced non-deforming region. At increasing ram penetration,

D

Figure 2.4 Schematic of the extended three-zone model (model 4)

18 Chapter 2

this zone's height decreasas untillhe activa zones touch lhe bottorn of the chamber;

the extended model then is no longer applicable.

Calculation of the power consumption due to internal deforrnation basically is

the same as lor the three-zone model; the corresponding contributions of shearing and

friction can also be obtained by re arrangement of these pnevious resu~s. The salution

lor !he relativa ram pressure then is:

PR -1 + 1 ·{In( J(Re/RR)4 +3 +(Rc/RRJ\ +2 -J1 +3·(RA/Rc)4

Uo /3·(1-(RR/RdJ 3

S 4 RR Re RA 2 RA AR 2 +- ·-·-·- -2·- +-·-·(-)

RR 3 S RR S 3 S Re

1 R R +-·~·[1-(~)2]

3 s Re

(2.10)

and recapi!ulating:

(2.11)

The lunetion can be minimised by op~mising !he pseudo-independent parameter

SfRR. For this optima! relativa zone height Sopt/RA an explicrt expression can be

derived, which is:

sopt-fiA-

4·(AcfRR) +(RR/Rd-5 +m·[1-(RR/RdJ

3·[1 •m·(RR/Ac)]

(2.12)

The accompanying relativa ram pressure is !he best attainable estimate lor this model.


2.2.5 Evaluation

Tile introduced models of backward can extrusion are dictaled by the l'pper-bound

theorem. Tilis principle states that the (klnematically admissible) velocity field, which

involves the lewest power consumption, prevails - lor the gillen set of conditions. For

the presumed constancy of the flow stress, this corresponds with the condition of

lowest ram pressure for the most appropnate choice. Essentially, the theorem does

apply nol only to the variety of fields within a particular model, blll also to distinctly

different velocity fields as contained in different models. Tilis means, in plain words,

that the best salution is given by the model which prediets the lewest ram pressure.

This instrument wil! be used in this werk for the evalustion of the present

models. First, the calculation of the ram pressure Is dlscussed. On the basis of this,

information is deduced about the flow pattem. Finally, some attention will be paid to

the use of the upper-bound technique tor the purpose of defect prediction.

• Ram pressYI§

The most direct use of the models concerns the calculation of ram pressures.

Two examples will be discussed: the first lor the extrusion of a thick-walled can

(extrusion ratio R=2), the second lor the extrusion of a thin-walled can (R=10).

Pract1cally, the high extrusion ratio in the latter example is attalnable merely in the

extrusion of soft materials, like some grades of aluminium, while the forst one may be

a typical example lor the extrusion of steel. lubrlcating conditloos are quantffied by the

plastic-lrietion factor; in these exarnples a value of m =0.1 is assumed.

Resuits are presenled in the figures 2.5 and Ul. Tile relativa ram pressure here

is plotled as a lunetion of the relabve ram stroke, which is defined as the current depth

of penetratien D normalised on the chamber height H. As lor the models 2 and 3, the

botlom thickness T is Iranslaled to !he penetratien depth by the assumption that the

billet's origlnal height equals the chamber height. Tilis implicates:

D=H-T. (2. 13)

Tile resuits lor the indivîdual rnadeis are shown as dashed curves.

The spherical as wel! as the extended three-zone model (models 1 and 4) do

not cover the complete range, but apply only to the quasi-stationary stage at the

beginning of the stroke. Ram pressures tor both models increase· linearly with ram

20 Chapter 2

8.-----.-----.-----.-----.-----, I Rc/RR~1.41 H/RR~2 h/RR~o 1 m=0.1

........ .L~---····

2 ..... ~- ........ .--1--'

Relotive ram stroke D /H [ -)

Figure 2.5 Prediction of ram pressures lor lhe extrusion ol a th/ek-wa/led can (R =2):

1 spher/cal model (aop1=42°), 2 two-zone model, 3 three-zone model,

4 extended three-zone model (Sop/RR=0.813)

penetration: this is conneeled wilt1 !he linear increase in frictional area. The slopes of

these lines equal each other and depend on !he value of the friction factor. The velocity

fields tor these modals do not change during !he stroke, which means that the

contributions of internar deformation and shearlng in !he total power consumption are

constant Optima! values tor !he pseudo-independent parameters are included in the

caption of !he figures.

The !Wo· zone model and !he three-zona model (models 2 and 3) are designed

to describe the instationary stage of !he process at !he end of the stroke. At smal!

dépths of ram panelration these models do not satisly: calculated ram pressures are

toa high. As a tunetion of the ram penetration, both rnadeis predict an initia! decrease

and a subsequent increase in ram pressure, which is linked up wilt1 !he changing

proportions of intemal delormation, shearlng and friction in !he total power


8.---~--------~--,.--~----,-----,

4 L

2+-"··········---~- -+············--+--···········1---~

Relotive ram stroke D/H [ -]

Figure 2.6 Prediction of ram pressures fortile extrusion of a thin-walled C81l (R = 10):

1 spherical model (aaptE 17°), 2 two-zone model, 3 liJree-zone model,

4 extended tilree-zone model (Sop,IRR=0.196)

consumption.

The actwal course of the ram pressure as a lunetion of the stroke is obtained

by combining the model predictions, sueh that !IJe towest pressure at each ram

position prevaj/s. The rasuiting course is the "enveloping lewest curve'. In the figures,

this curve is represented by a solid line. The results lor !he low 8l<tnusion ratio

(figure 2.5) show !hat the early stage of the processis described best by the spherical

model, whereas !he three-zone model governs !he end of !he stroke. For lhe high

extrusion ratio (figure 2.6), the situation is more 1nvoMng: !he preferenee hereis aftered

trom the extended !hree-zone model at the start, !hrough !he !hree-zone model, into

!he two-zone model at !he veryendof the stroke.

22

10 -- H/RR-2 m-0.1

I ~

c: 0 ·v;

5

·-·~-···

-···· .

~ 2 ... _ .... " w

,_ 1 0.0

i extended three-zone

sphericol model

I 1

~ !

0.2 0.4

Chapter 2

' mi• -·-

I L -·-· ·----·- ....... / !.. ....

model m/ /.. .. ···-·· I

(._~w~-:9-:

/ I

three-zone model '

0.6 0.8 1.0

Relotive rom stroke D /H [ -]

Figure 2. 7 Working spheres of fhe individuaJ models tor the exiTusion of high bil/ets

• Flow pattern

The combination of rnadeis results in a more detailed prediction of the ram

pressure as a tunetion of the stroke. Aiso, a deduction can be made concerning !he

flow of the materiaL This information is acquired by establishing the cond~ions, lor

which each of the models governs l!le process; !he accompanying veloc~ fields then

are indicative of the patterns as they actually develop.

Some illustrative resu~s are presenled in the f~gures 2. 7 and 2J!; these show the

prevalenee of the different models in dependenee on the penetratien depth and the

extrusion ratio. These results, which are gained numerically from the ram-pressure

!unctions, apply to relativa chamber heights of H/RA=2 and H/RA= 1 and correspond

to the extrusion of •high• and •tow" biliets respectlvely2 l As before, a friction factor

2)The lnfluence af the reiatlve ram heîghl on the ram pressure is ident!cal for all ·modets; this parameter lhen doesnotact upon the tmnsttions and therefore is of no consequence tothese tesWts.


10 H/RR-1 0.1 y ·-

m ···········~--· .. / /i _. ... ......

5

""

/ /i .. extended / / '1 three-zone model

' V V L=

.2 ë c 0

·v;

/ ( ~~zon~e~---spherical t;r

~ "' ~ ~ '"' w

2

1 0.0 0.2

three-zone model

0.4 0.6 08 1.0

Relelive ram stroke D/H [ -]

23

Figure 2.8 Working spheres of the individual models for the extrusion of low bil/ets

of m ~0.1 is assumed. Solid lines reprasent the transitions between !ha models. The

enclosed surfaces !hen delermine !he werking spheres of the individual models; each

of these models shows to have an outlined validity range.

AS an experimental oonfinnation, Kast's visioplastic study can be quoled [2.4].

Grid deformations here are examined lor !he quasi-stationary stage of the process -

when !ha bottorn area does nat yet perticipate in the deformation. Same of these

results are reproduced in f>gure 2.9. For tha situation which is shown in the lelt part of

the figure, a dead zone develops immediately beneath the ram while the actual

defonning region takes a spheroidal shape. Tha extrusion geometry hare corresponds

to an extrusion ra~o R ~ 1.46. No such dead zone exists lor !he situation shown in the

right part of !he figure; the acti\le zone beneath !he ram is shaped rather flat. In this

case tha extrusion geometry corresponds to R ~3.56. These observations agree with

the present results lor the inrilal stage, as can be interred trom figure 2. 7.

24 Chopter 2

Eiqure 2.9 Delarmat/on patterns, visu8lised by the dislortion of an or/g/nally square

grid (repn"nted trom Kast {2.4}; workplece male/ia/: Ma8 steel)

• Oefeet oredict!On

Avitzur and Luotailorad existing upper..!Jound modelsof backward can extrusion

lor the purpose of defect prediction by extending these wilh an additional optimisation

parameter [2.7, 2.14). Principally, this new parameter represems a change from sound

flow into a mode, which is connectad wilh the particular defect. By minimising the total

power requirement, then, the ccnditions are determined lor which the defect conneeled

flow type prevails over the one tor sound flow. In this manner, they studled some

distinct types Of delects (cavitetion, flshskin, indentation).

As an example, the model lor the study of the cavitation defect will be sketched.

This defect - which is known to occur at the end of the stroke - results when the

extruded material fails to flow into the corners of the die chamber. The model is based

on the introduced two-zone representation of the process (figure 2.2). An additional

parameter is dimensioned at !he irrtersection of chamber wal! and chamber bottom,

and represems !he altachment of the surface of velocity discontinuity" in this corner. By

varlation of the parameter, a release of this surface frorn the chamber-wall iS simulated,

which corresponds physioally with the developmerrt of a void. Friction arid extemal


pressure are considerad as soma means to prevent the occurrence of the defect. The

choice olthe two-zone model as a basis lor the study of this defect is an appropriate

one ~ this model governs the process lor the particular set of condilions (ligures 2. 7

and 2.8: end of the stroke).

In these adapted rnadeis lor the aim of defect prediclion, anyhow, a relevant

characterisation of the workpiece material's behaviour is lacking. The use of this

technique theretore is restricted to the study al those defects, wnich originate from the

processing conditians (geometry, friction, ... ) rather than from the material itsen!

26 Chapter 2

2.3 NUMERICAL TREATMENT

Following the introduetion of dig~al computers in the 1960s, the rnadelling of Iorming

processas was enriched ~ha new category of techniques. The fin~·element methad

(FEM), which finds ~ crigin in structural mechanics, has established itseU during the

last decade as !he principal means of numerical rnadelling in the field of metal Iorming

[2.15]. Nowadays, a dive~ of user-oriented FEM software is available.

The concept of finite-element simulation is one of discretisation. Essentially, the

object of study is divided into an array of standardised e!ements, linked tagether in

their nadal points. The Iorming pracess is simuialed by imposing boundary conditlans

la !he structure; local flow lor the individual elements is described in terms of the nadal

displacements, settling lor the conditions of plaslicity {yielding, equilibrium). The

extensive ernauni of data processing is dea~ wilh by using powertul computers.

The technique is capable of obtaining detailed salutlans on !he process

mechanics, invalving many different quantities: velocities, stresses, strains, contact

pressure distributions, and so on. Also, complex behaviOur of the workpiece material

can be modelled; this includes strain hardening, strain-rate and tempersture effects,

as well as thermal coupling. The technique is. universa! in a sense, that a computer

code can be utilised lor a large variety of problems, just by changing the input data.

There are, however, also disadvantages. lnvestment casts in the hard- and software

are considerable. Specialised training is required. Each individual case must be

computed separately; computing times in general are long.

The development of the finite-element methad received a goad deal of scientific

attention in recent years; an overview is beyend the scope of this work. Conceming

~ current relevance, ft can be concluded that the technique has captured ijs position

among the tools of plasticity research, This add~ional methad of analysis is particularly

useful in the study of these problems, requiring delailed information on specilic

s~uations.


2.4 CONCLUSIONS

Several methods lor the study of the backward can extrusion process have been

reviewed in the previous sections. Each of these shows to have its merns and

demerits.

The use of empirically based formules and nomograms is a practical means lor

the calculation of the maximum loads. These may serve tor general calculatlons, such

as to estimate the required press capacny.

The application of up per -bOund models extends well beyend the calculation of

ram pressures. From a combination of these models, n is also posslble to obtain an

indication of the ftow pattems. Also, some models can be adapted tor the object of

defect prediction. For convenience, the (bulky) formulas may be treated numerically,

but this is not an essential requirement. An important disadvantage, anyhow, is the

incomplete representation of the workpiece material's behaviour (constant flow stress).

This, of course, can be by-passed to some extent but ~ must be realised that this

interferes wHh the upper-bOund presumptions. Nevertheless, ~ is believed that the

potentialities of this category of models lor such complicated processas as backward

can extrusion are not fully explo~ed yet. Further developments are to be expected in

a more detailed calculation of local quantities; an onset to this will be introduced in

chapter 5.

Numerical rnadelling techniques are an interesting new branch on the tree of

plastic~ methods. The fin~-element methad currently is the major exponent of these.

n incorporates the possibil~ of calculations on a locallevel, yielding an unprecedented

amount of inforrnation. The required elfort to obtein resuits, however, is large. At

present, a numerically supported analysis certainly can be beneficia! lor critica!

appllcations, but does not feature a standard salution to the problems of everyday

practica.

28 Chapter 2

REFERENCE UST

[2.1] Boes P.J.M., Pauw H.P.,

"A Practical Calculation Method tor Extrusion Pressures•,

Sheet Matsllndustries ~ (1966): 377-389.

(2.2] Schmilt G.,

Untersuchungoo über das Rûckwätts-NapfflieBpressen von Stahl bei

Raumtemperatur,

Berichte aus dem lnstitut für Umformtechnik, Univers~ät Stuttgart, Bericht 7,

Girardet, Essen (1966).

[2.3] Burgderf M., Müschenbom R.,

"Nomogramme zur Ermiltlung der Umformkraft beim AieBpressen·,

wt - Zeilschrift tür Industrie/Ie Fettigung 60 (1970): 503-506.

(2.4] Kast D.,

MocJel/gesetzmliB/gkeiten beim Rüc/t.wärtsllieBpressen geometrisch ähnlicher

Näpfe,

Berichte aus dem lnstitul für Umformtechnik, Universitàt Stuttgart, Bericht 13,


[2.5] Dipper M.,

"Das FlieBpressen von Hülsen in Rechnung und Versuch",

Archiv für das Eisenhüttoowesen ~ (1949): 275-286.

[2.6] Altan T., Thomsen E.G.,

"Pressures required tor Backward-Can Extrusions•,

Armals of the CIRP ~ (1966): 273-280.


[2.7] Avitzur B.,

Handbook of Metaftarming Processes,

John Wiley & Sans, New Vork (1983).

[2.8] Avitzur B., Bishop E.D., Hahn W.C. Jr.,

"Impact Extrusion: Upper Bound Analysis of the Early Stage",

Joumal of Engineering tor lndustry - Transactions of the ASME, series B

(1972): 1079-1086.

[2.9] Hahn W.C. Jr., Avitzur B., Bishop E.D.,

"Impact Extrusion: Upper Bound Analysis of the End of the Stroke",

Journat of Engineering tor lndustry- Transactions of the ASME, series B

(1973): 849-857.

[2.10] Kudo H.,

29

"Same Analytica! and Experimental Studies of Axisymmetric Cold Forging and

Extrusion - 1 ",

International Joumal of Mechanica/ Sciences 2 (1960): 102-127.

[2.11] Steek E.,

"Kraftberechnung bei Umformverfahren mit Hilfe der "oberen Schranke'"',

Werkstattstechnik 57 (1967): 273-279.

[2.12] Ramaakers J.A.H.,

Hulsextrusie 1: Berekeningen,

internal report WPT 0534, Eindhoven Univarsity of Technology (1982).

[2.13] Weber W.,

"Bestimmung der bezagenen Umformkraft beim Rückwärts-NapfflieBpressen",

Fertigungstechnik und Betrieb 21 (1971): 49-53.

30 Chapter 2

[2.14] Luo Z.J., Avitzur B.,

"Umitations of the Impact Extrusion Process",

International Journat of Machine Tooi Design and Research 2,2 (1982):

41-56.

[2.151 Kobayashi S., Oh S.-1., Altan T.,

Meta/ Forming and the Finite-Element Method,

Oxford Univarsity Press, New York- Oxford (1989).

31

Chapter 3

Characteristic Quantities of Formability

Advances in materials science largely conduced to the enormous increase in

industrially available materials within the last decades. Rapid-solidification processing

presently is one of the areas tor special attention. By using such a processing route

it is possible to alloy compositions which cannot be produced otherwise. Moreover,

very fine or even amorphous microstructures are obtained.

The performance of a (new) material often is evaluated from its in-service

properties, like hardness, yield strength, fatigue behaviour, etcetera. With respect to

its eventual application, however, the potentialities of a material to bring it in its final

shape also are important, if not crucial. From a manufacturing point of view thus it is

necessary to distinguish the characteristics which are relevant in conneetion with the

workability and quantify these as such. These characteristic quantities then can serve

as an intermediary between manufacturing technology on the one, and matenals

technology on the other hand.

Concentrating on forming technology, this means that it should be possible to

evaluate the suitability of a material to forming applications trom some characteristic

quantities, reprasenting distinct features of the formability. This is demonstraled in this

chapter, using material (semi-finished product) which is recycled trom aluminium scrap

by means of a rapid-solidification technique. For a proper understanding of the results,

the implemented technique will be outlined first.

32 Chapter 3

3.1 RAPID-SOLIDIFICA110N PROCESSING OF ALUMINIUM SCRAP

3.1.1 Orlentatlon

Recycling is becoming increasingly important because of energy- and material-saving

aspects and because it is an effective approach to the problem of environmental

pollution. The recycling of aluminium traditionally has been emphasised, mainly tor

economical reasons: production of aluminium from bauxite is very energy-intensive and

thus expensive. A high recycling rate is achieved for primary aluminium scrap, which

is scrap originating from the menutacture of aluminium (semi-finished) products and

therefore has a well-known chemica! composition. This material aften can be recycled

back into production directly. The recycling of aluminium from discarded products

(secondary scrap) is far more complicated. In genera!, this material isofan unknown

composition; besides it aften contains polluting residues. A conventional way of

recycling secondary scrap is by adding it into the production of casting alloys. The

aluminium then must be separated from compounds and has to be classified with

respect to its chemical composition.

At the Delft Univarsity of Technolog)' (the Netherlands) a new aluminium

recycling technique has been developed, in particuier for aluminium from car shredder

scrap [3.1 , 3.2, 3.3]. The essential part in this technique is the implementation of a

rapid-solidification process. By using such a process extremely high cooling rates are

obtained in the solidification of the melt, typically in the order of 105-106 °C/s. This

induces many effects to the material; as a result the properties are entirely different

from a conventionally solidified material.

Aluminium car shredder scrap in general consists of various alloys, mainly with

high levels of alloying elements (casting alloys, for instanee piston alloys}, and includes

many impurities (oil, remainders of balts, etcetera). H this scrap is recycled by

conventional casting techniques, microstructures with a high amount of coarse

(intermetallic} second phases are obtained. The material then is very brittle: it fractures

at zero plastic strain, soit cannot be used in forming operations. Recycling this scrap

by rapid-solidification processing leads to a refinement of the microstructure and a

homogeneaus distribution of second-phase particles. As a result the ductility is much

Characteristic Quantities of Formability 33

Figure 3. 1 Optica/ micrographs of (A) conventionally cast material and (B) rapidly

soliditled and conso/idated material

better; strains at fracture of about 40 %, at room temperature, are common values.

Ditterences in the microstructures are obvious from tigure 3.1.

The variations in concentratien of important elements in the aluminium scrap

showed to be quite moderatefora partJeular shredder [3.3]. To obtain reproducible

results, a representative specification is standardised (table 3.1). Si and Cu contents

are high, mainly due to the large fractîon of aluminium casting alloys in the scrap.

Tabla 3.1 Standard chemica/ composition of aluminium scrap (in wt. %)

Si Mg Zn Cu Fe Mn Al

5.5-6.5 0.5-1.0 1.5 2.8-3.8 0.7 <0.3 bal

34 Chapter 3

input ribbons flokes .. MELT SPINNING .. ~===c=un=I=NG====~~~ -.

chopper

*'' Figure 3.2 Rapld-sotidification processing: initia/ steps

3.1.2 Technlque

In this sub-section the new recyding technique for secondary aluminium scrap will be

described briefly. For a detailed treatment of the technique and its metallurgical

backgrounds the reader is referred to publications from the original investigators [3.1,

3.2, 3.3].

The distinct steps in rapid-solidification processing are reproduced schematically

in the figures 3.2 and 3.3. In the first step, ribbons are produced trom a melt by using

the malt-spinning process. In this process cooling rates of about 106 °C/s are reached

by casting the liquid metal upon a fast rotating copper wheel. To acquire such high

cooling rates it is necessary that the thickness of the produced ribbon is very small:

50-100 pm typically. In the present geometry the width is about 3 111m, the length is

esseritially unlimited.

Characteristic Quantmes of Formability 35

flokes biliets profiles

1 ..

I COLD COMPACTION .. HOT EXTRUSION

rom die

eh omber

Eigure 3.3 Rapld-solidification processing: final steps

Subsequent processing is needed to consolidate these ribbons. The ribbons are

cut by a chopper to produce flakes of some millimatres length. For easy handling,

these flakes are (cold) compacted to biliets in a preform. The actual consolidation

takes place by an extrusion process: direct extrusion at elevated temperature. In this

way profiles of various shapes can be produced. The extrusion process especially is

suitable tor the consolidation of the flakes, since it introduces large (shearing)

detormations to the material at high hydrastatic pressure. Both factors are important

for bonding the flakes: the {hydr-)oxide layers at the surfaces have to be stripped, the

flakes then can be "welded" together. The material obtained is of tuil density; without

pores, that is. Profiles produced by this technique can be used directly, tor instanee

for construction purposes, but they also may serve as an input material for forming

processes. The formability of the material then is of particuier interest.

36 ehapter 3

3.1.3 Processing parameters

In the technique as described above many processing parameters are involved.

Important parameters in the malt-spinning process are the wheel velocity (which

delermines the cooling rate) and the tempersture of the melt. The cold compaction

step introduces the billet density as a further parameter. eoncerning the hot extrusion

process, the extrusion temperature, the extrusion ratio and the pre-heating time before

extrusion are some variables.

lnvestigations were done into the influence of the next processing parameters:

wheel velocity, tempersture of the melt, time interval between melting and melt

spinning, pre-heating time befare extrusion, and extrusion tempersture [3.2]. lt was

concluded that the properties of the semi-tinishad product are mainly determined by

the extrusion conditions, provided that the cooling rate during melt spinning was

sufficiently high. A high extrusion tempersture leads to a devaluation in mechanica!

properties like hardness, yield strength, and ultimata tensile strength [3.2]. With respect

to the plastic properties of the extruded semi-finished product, however, a high

extrusion tempersture is rather profitable: flow-stress values are slightly lower, ductility

is better [3.4].

The processing parameters were optimised in conneetion with the recycling of

aluminium from car shredder scrap [3.3]. For melt spinning on a pilot scale unit the

standard conditions were: circumferential velocity of the wheel Vw=25 mfs,

tempersture of the melt TM= 750 oe. Extrusion was done on an industrial press at a

tempersture T e=450 oe, extruding the billets (0110 mm; pre-compaction of

approximately 60% of tuil density) with an extrusion ratio (or area reduction ratio)

R ... 30.


3.2 FORMABILITY OF RECVCLED ALUMINIUM

In this section the attention will focus on the formability of the semi-finished product,

redaimed from aluminium secondary scrap.

Recycling was done by rapid-solidification processing atthe standard conditions,

using scrap with a composition according to the standard. Semi-finished product was available in form of profiles with a dumb-bell shaped cross-section; from these profiles

both sheet material (thickness: 3.5 mm) and bar material (diameter: 12 mm) could be

extracted for experiments. Test pieces were annealed at 350 °C for 20 minutes in

ambient air. This heat trestment was done to reduce the intemal stresses caused by

machining, but it showed also to have a marked influence on the plastic properties.

The presented results in this section, anyhow, are restricted to the investigated

condition.

In order to acquire information about the formability of this material, a number

of tension and torsion tests was performed. The testing tempersture was used as a

parameter and ranged from room temperature to the annealing temperature. Two sets

of tensiontests ware carried out; test specimens, respectively, were taken longitudinally

(bar) and transversely (sheet) from the extruded profile. For comparison soma tension

specimens, taken longitudinally trom the sheet-section, were tested (room

temperature). The tension tests ware supplemented by a set of torsion tests on

specimens, taken trom the bar-sections of the profile. All tests ware performed in a

quasi-static tashion and in triplicate.

From the tests some distinct quantities, which are relevant in conneetion with the

formability of the tested material, can be derived. First, the flow behaviour can be

expressed in a flow curve and mathematically described by a flow function. Second,

the failure behaviour can be represented either by the strains at fracture from the

tension and torsion tests, or in a combined form by a ductile-failure curve. Further, the

plastic anisotropy tor sheet material can by quantified by means of the anisotropy

parameter.

Since the tests were done in triplicate, it is possible not only to present the

results as averages but also to obtain an impression of the reproducibility in these

results. As a maasure the range indicator I is used, which is defined as:

38

1 = Ymax-Ymin ,100 %, Ym

Chapter 3

(3.1)

Ymax and y- here denote !he maximum and minimum value respectivaly, and Ym the

arithmetical average of the trio.

The resu~s on the formability of recycled aluminium wil! be presenled in the

lollowing sub-seelions on !ha basis of !he above mentioned characteristics.

3.2.1 Flow behaviour

The flow behaviour often is represented by a flow curve (or strass-strain curve), which

depiets the relationship between !he equivalent plastic straln i: and the flow stress a1.

In Iorming processes, the knowiedge of !he flow stress and !he strain-hardening ability

of the workpiece material is of interest, lor instanee in conneetion with the tooi loads

and the workpiece material's susceptibility to instabie flow.

Calculation of !he flow curves trom !he lension tests required !he meesurement

of !he tensile force F1 and the specimen's cross-sectional area (bar: diameter d; sheet:

width b and lhickness s). The explored range was confined to unflorm straining.

Formules lor the calculation of equivalent strain and flow stress are:

- d Ft € = -2·1n(-) and u1=-- lor bar specimens,

do !.':-d2 (3.2)

4

and

i: • -ln(,É..)-In(~) and o1=_1'i lor sheet specimens. {3.3) b0 s0 b·s

The subscript o relers to !he original geometry. Nomina! dimensions of the specimens

were: d0 =4.0 mm {bar); and b0 =6.0 mm, s0

=3.5 mm (sheet).

Flow curves from !he torsion tests are darived using a calculstion methad

proposed by Pêhlandl et al. [3.5]. The nomina! specimen geometry vyas: gauge length

L = 24.0 mm and radius R =4.0 mm. Apart trom these, the twisting angle 8 and !he

twisting moment M ware measured during !he test. The von Mises yield .criterion is

used. Equivalent strain and flow stress are oblained trom:

Characteristic Quantmes of Formability 39

(3.4)

These formules apply 10 a so-called "critica! radius• ir1 the specimer1. The twisting ar~gle

is to be substituted in radians.

The Halloman flow lunetion (u1=C·ë"J showed to be a sultabla means for a

mathematica! description of the obtained flow curves. The results then are

characterised by two parameters: the characteristic stress C and the strain-hardening

exponent n. These were determined, using a PC programme lor fitting of flow lunetlans

in a least-squares sense [3.6].

lr1 figure 3.4 the resutts of the lension tests are reproduced. The accompanying

indicators of reproducibility average to I =6 % in the characteristic stress and I= 24 %

in the strain-hardening exponent. From the figure ~ can be seen that the dillerences

in flow behaviour between the longttudinal and transverse direction are onty marginal.

Further, the resutts on the sheet- and bar-sections in longttudinal direction show no

significant diflerences. Both the C- and n-values decrease with increasing temperature;

the characteristic stress in particuier exhibtts a strong dependance.

Figure 3.5 shows the results obtained trom the torslori tests; these are very

similar to those trom the lension tests. Here, the reproducibility is characterised by

average values of 1=8% lor the characteristic stress and 1=12% tor the strain

hardening exponenL

3.2.2 Fallure behaviour

An important limitation in the application of Iorming processas is constituted by fracture

of the workpiece materiaL Also, the occurrence of tracture obstructs lurther

experimentation in tensile and torsional testing.

For !he lension tests, the strains at tracture ~are deterrnined trom the neck of

the specimens Qocal values). Thus:

and

ëF o. -2-ln(~) lor bar specimens, do

(3.5)

40 Chapter 3

400 0.4 ~

lension ~

bor-section: N I E ~

E • longitudinolly

' c ..... sheet-seclion: z 0.3 ~ -c • longitudinolly .,

u c • lronsversely 0 0-

"' >< "' 200 " 0.2 mm•---+-•~••••'

..1:: "' "' c

.>! 'ë .,

-;;; 1" ·c 100 0 0.1 -.!!: "" (.) I l

"' c 'ë 0 ~

"" (Ï} '-' 0 0.0

0 100 200 300 400 0 100 200 300 400

Temperolure T [•c J Temparolure T [•c J

Fiqure 3 4 Flow-curve c/laracterisation of semi-tinisllec/ product as a tunetion of 1/le

temperature (tension tests)

F F sF = -ln(E._) -In(!_) lor sheet specimens.

bo So (3.6)

The superscript F denotes fracture.

These values are represented as a tunetion of the temperature In tigure 3.6. For

!he results in this figure an average indicator of reproducibility of I= 18 % is obtained.

At first ft is noticed that !he strains at fracture increase lor increasing temperature.

Further, there are significant dillerences between !he fractional strains in the two

directions. These dillerences are closely bound up with !he nature of the conso!idation

of the flakes: the extrusion process leadstoa pronounced orientati()n of the original

Hakes in longftudinal direction. When the material is strained '" a transverse direction,

!he bonding surfaces are loaded perpendicularly; this is unlavourable as compared to

straining in longrtudinal direction. The lension tests on sheet material in longitudinal


400 ' ,......., N

E E

............ z 300 ........,

u (/) (I)

200 (1.) .... -(I) u

:;::::; (/)

·;:::: 100 (1.) -

torsion

I~

\ ' ~

~--·· u 0 .._ 0

.J:: u

0 0 100 200 300 400

Temperature T [°C]

-c::: (1.) c::: g_

0.4 ,---!"'", -~-~......., I

. ................. J.---········i·········-·······~---··-·······

~ I I 0.3

.. -........ L ............. L ......... J ............... . .~ o.2 ! ! I ~ : i I

x (1.)

_g 0 1 ·-----···~---··- 1 ·-·······--+·-··-··--··· 1 ' i l

·ê I ûî i

0.0 +-......--+-..--;--..--+--,..........j 0 100 200

Temperature T

Figure 3.5 Flow-curve cha.racterisation of semi-tinisheel product as a tunetion of the

tempersture (torsion tests)

direction showed a strain at tracture intermediately to the two extremes. For practical

purposes, it is important to note that the material's ductility is restricted by the

behaviour in transverse direction.

In torsional testing, the strain is assumed to be distributed linearly over the

cross-section of the specimen trom zero strain on the axis to a maximum strain for the

outer radius. The strain at tracture ëF hare is determined by the strain for the outer

radius at tracture, and is calculated trom:

ëF =-1_._!!.9F. .J3L

(3.7)

Thus obtained values are shown as a function of the testing tempersture in

figure 3.7. The indicator of reproducibility hare averages to 1=22 %. From this tigure

a similar trend is observed as trom figure 3.6: the strain at tracture increases with an

42 3

2.0 bor-section:

• longitudinolly

1.5 sheet-seclion: "- t longitudinolly ,.., ., • tronsversely • ~

" û 1.0 .!": ö c 0.5 <5 ~

i/)

lension

0.0 0 100 200 300 400

Tempero\ure T r·c l

Figure 3.6 Strain at tracture of semi-linished product as a lunetion of the

tempera/ure (tension tests)

increase in temperature. Absolute values, nevertheless, are higher !han in tensile

testing. This illustrates !he influence of !he state of stress during deformation on the

feilure behaviour [3.7).

An attractive way to represent the ductility of a material as a tunetion of !he state

of stress is by means of a failure curve [3.8). This is a curve of !he equivalent strain ëF

versus the stress tri-axiality (a mful at ductile failure and can be determined by some

standardised tests. The (stress) tri-axiality is defined as the ratio of the hydrostatic

stress campenent am;(a1 +a2 +a3)/3 and !he effective flow stress u1.

lt is possible to estimate these ductile-failure curves trom the resu~s of the

lension and torsion tests.

Except tor !he strains at tracture, lhis requires !he determination of the

accompanying stress tri-axialities. For tension, the resu~ trom the tests in transverse

direction are used. The state of stress intension implies a tri-axiality value of 1/3 in

Characteristic Ouant~ies of Formability

4

~

I ~

3 ~

'"' ~

"' û 2 ~ 0 c ----

" û1 ~

0 0

' '

~ /

/ ····-.:1=·· ~

100 200

Temperature T [•c]

torsion

300

43

400

F•gure 3. 7 Strain at lracture of semi-fin/siled product as a lunetion of tlle

temperature (tors/on tests)

case of uniform straining; since the specimens showed soma necking at fracture,

however, a Bridgman correction is applied [3.9]. This correction accounts lor !he

necking of sheet specimens in lhickness direction. The procedure necessitates !he

addilional maasurement of the profile redlus p,F at fracture in this direction. For !he

centre plane, being normative, ~ holds:

Urn F 1 SF (~) -+ln(1+--).

"r 3 4·p/ (3.8)

For torsion, !he state of stress implies a tri-axiality value which equals zero throughout

testing, up to fracture:

44 ehapter 3

(3.9)

The faiture curve lor a particular temperature is obtained by the ccmbination of

!he corresponding results lor lension and torsion. For simplicity, the intermediale

course is interpolated linearly. These failure curves are shown in figure 3.8; !he

temperature is used as a parameter, which is a new aspect in 1he application of these

curves. The average indicator of reproducib1lity lor the stress tri-axiality is rather small

(I =3 %, tor tension) as compared to those lor the strains at fracture. From the ligure

it can be seen, that the ductility of the material approximately duplicates between 20

(room temperature) and 200 oe; the most significant improvements, however, manifest

between 200 and 300 oe. From 300 oe on no further increase in ductility is observed.

0.5

... 0.4 ~

'-....

.:1 "'

0.3 ~

::> ü .f 0.2 ~

"' g 0.1 ö ·;;

0 I

·;:; >- 0.0

~--

0 2

Stroin ot frocture i' [-J

temperature: o 20 •c + 100 •c o 1so •c • 2oo•c _ .6. 2so •c

3

• 30o •c o 350 •c -

4

Flgure 3.8 Duetlie-fai/ure cu/Yes of semi-tinishad product, using lhe temperature as

a parameter


3.2.3 Plastic antsotropy

Anisatrapie plastic behaviour is an important feature, especially in sheet metal torming.

Here, it is a major criterion tor the evaluation of materials, tor instanee in deep-drawing

operations.

As a maasure of plastic anisotrapy commonly the anisotropy parameterris used

[3.10]. This parameter can be calculated trom lhe resu~s of a lension test on sheet

material; it is defined as the ratio of lhe natura! strains in the width and lhickness

direction. Thus:

(3.10)

The anisotropy parameter (or plastic-strain ratio) may depend on the direction in the

material and on the longitudinal strain.

For lhe material under consideration, the plastic anisotropy was established in

terms of the r-parameter trom the resu~ of !he lension tests on sheet material [3.11].

As a tunetion of strain, the anisotropy parameter showed a stabilisation to a constant

value; the data tor each test therefore were averaged (lor &>0.04). Accompanying

standard deviations averaged to about 0.06.

A comprising resutt is obtained by averaging the r-values tor the trio of tests at

every testing temperature. This average anisotrapy parameter is reproduced as a

lunetion of !he temperature in ngure 3.9. The overall indicator of reproducibility tor

these resu~ is 1 = 12 %. In the transverse direction trom lhe extruded sheet, wihich is

!he so-ca!led 90"-direction, the r-parameter shows nodependenee on the temperature

and can be averaged toa value of r90(T)=0.64. This is visualised by the included line

in the figure. The result of the lension tests on sheet material in longitudinal direction

(!he 0°-direction) is also included.1l Represented values difler significantly trom the

neutra!, isotropie value (r= 1).

1)! .. ~~!!Sîle testing of lhe bars ~ongltudinal directlon) no significant anisalropy was observed; that Is, thc circular shape of the cross·section was preserved during the tests. This can be explained ftom

the rotatiorta! symmetry în the extrusion of the bar-section, this In contrast with the planar symmetry In

extruslon of the sheet·secUort

46 3

1.5 ,.-----;-----;-----;--r-------, sheet-section:

• longitudinally • tronsversely

..... _ ... ·······-·--- ·········+ ···-···-···--·-~"""-

j

• • • .......... .J_ ........... ~ ···········t-- ············-····

0.0 +----+----t---~--i------1 0 100 200 300 400

Temperature T ( 'C]

Figure 3.9 Anisotropy parameter of semi-fin/sfled product as a tunetion of the

temperature

The directional dependenee of !he plastic anisotropy is assessed trom !he

anisotropy parameter in three distinct directlans: !he 0 o-, 45 °- and 90 ° -direction. For

room temperature, the already obtained results (r0 •0.56 and r90 •0.63) were

compiemenled by a trio of lension tests in the 45°-direction trom which an average

value of r45 =1,09 was derived. These values can be combined in two charaderistic

quantities:

- The nonna! anisotropy paramelerr-(r0 +r90 +2f45)/4•0.84

- The planar anisotropy parameter Ar=(r0 +r90 -2<.s)/2= -0.50

The normal anisotropy parameter represents the average plastic anisotropy in the

distinct directions, whereas !he planar anisotropy parameter expressas the directional

variatien of plastic anisotropy. The investigated sheet material obViously deforms

preferably in thickness direction (i'< 1); !he directional varlation is large (Ar•OJ.


3.2.4 Evaluation

In the previous sub-sections a number of characteristic quantities is discussed, which

could be derived in a relatively simple mannar from the results of some basic tests

(tension and torsion)2>. These quantities can be used for a first evaluation of the

formability and, subsequently, to formulate some directives tor the application of the

investigated material to concrete torming processes.

Concerning the flow behaviour, it is notleed in tensile testing that there are no

significant ditterences between longitudinal and transverse direction. Also, the results

of tension and torsion are very similar. The experimental data, as expressed in the

characteristic stress G and the strain-hardening exponent n, reveal that a higher

werking temperature is connected both with a lower flow stress and a lower strain

hardening ability of the workpiece material (figures 3:4 and 3.5). Lower flow stresses

are favourable with respect to the toolloads. A low strain-hardening ability, however,

promotas the occurrence of strain concentration (instable flow) and thus may lead to

material failure or other deficiencies in the product.

Results on the failure behaviour are presented by some different means. The

strains at fracture ëF in longitudinal and transverse direction, determined from tension

tests, show significant ditterences (figure 3.6). The transverse direction here is critica!.

By using the concept of the failure curve - which setties tor the influence of the state

of stress - it is possible to reprasent the ductility in dependenee on the temperature

(figure 3.8). The ductility is better at elevated temperature, as could be expected;

considerable improvements manifest in the range trom 200 to 300 °G. Figure 3.8 can

be used as a guide-line in determining the proper conditions tor a forming process

from a viewpoint of the failure behaviour. This means that if the strains and stresses

can be assessed tor a particular process, it is feasible to estimate the required

minimum tempersture to obtain a sound product. An application of this tor the process

of backwerd can extrusion will be introduced in chapter 5.

The plastic anisotropy of the sheet material is represented by means of the

anisotropy parameter r. For the transverse direction in the material, this parameter

2>11 is recognised ;hat compressive testing could provide for some addltionallnformatlon, especially wlth respect to the flow-stress data. Conceming the talure behaviour, anyhow, the use of compresslon tests is limited slnce 1t is difficult here to derive accurate results.

48 Chapter 3

does notdepend on the tempersture (figure 3.9). Same supplementary tests enabled

the determination ofthe normal and planar anisotropy parameter, rand l:ir respectively

(room temperature). These quantities aften are used as an indication of the material's

suitability to deep-drawing operations: the first in conneetion with the maximum deep

drawing ratio, the second in conneetion with "earing". A suitable material requires a

normal anisotropy parameter r>1 and a planar anisotropy parameter l:ir"'O.

Gonsidaring the obtained values, it bacomes apparent that deep-drawing is not an

appropriate application for this material.

In practica, the working temperature can be used to adapt the material's

behaviour to the specific requirements of the process. Forming at elevated tempersture

combines the advanteges of lowerflow stress and better ductility. Consequently, higher

deformations can be obtained. Especially the range trom, roughly, 200 to 300 oe is of

interest since bath quantities here exhibit a strong dependenee on the temperature. A

complication in working at elevated temperatures is that the tools and the workpiece

have to be heated. Furthermore, other lubricants have to be used.

For this particuier material, bulk forming will offer soma advantages as compared

to sheet forming. First, there is the unfavourable plastic anisotropy for sheet-torming

operations. A second argument in tavour of bulk-forming applications is that these

praeeed generally in a compressive fashion, which impraves the workpiece material's

ductility (negative tri-axiality values!). Working at elevated temperature can be realised

by incorporating the heating equipment in the tools.


3.3 CONCLUSIONS

Rapid-solidification processing is a promising development in matenals science. This

technique is successfully employed at the Delft Univarsity of Technology for the

recycling of secondary aluminium scrap. The rapid-solidification step leads to an

extremely fine and homogeneaus microstructure as compared to the conventionally

solidified material; as a result the machanical properties are entirely different. The

material is in a semi-finished shape and has to be worked to obtain a final product.

From a number of basic experiments it was possible to quantify some important

features of formability; these include the flow behaviour, the failure behaviour, and the

plastic anisotropy. Thus, an impression is gained into the possibilities and limitations

of this new material for forming applications. lt is concluded that further processing of

the material into final products preferably may be done by using bulk-formlng

operations at elevated temperature; optima! results are to be expected at about

300 °C.

50

REFERENCE LIST

[3.1] Kool W.H., Kievits F.J.,

"Terugwinning van aluminium door flitsgieten•,

Metaal & kunststof 19 (1988): 62-65.

[3.2] Mallnowski A., Kool W.H., Kievits F.J.,

Recycling of Aluminium trom Metal Scrap by Rapid Solidification,

internal EEC-report, Delft Univarsity of Technology (1986).

[3.3] ZWart J.H., Kool W.H.,

"Recycling of Aluminium using Rapid Solidification Processing",

Jaarboek KNCV sectie Milieuchemie (1989): 119-126.

[3.4] Sillekens W.H., Oautzenberg J.H., Kals J.A.G.,

Chapter 3

"Formability of Recycled Aluminium: Advantages of a Rapiel Solldification

Process",

Anna/sof the CIRP 39/1 (1990): 287-290.

[3.5] Pöhlandt K., Tekkaya A.E., Lach E.,

"Prüfung des plastisohen Verhaltens metallischar Werkstofte in

Torsionsversuchen",

Zeitschrift für Werkstofftechnik 14 (1983): 181-189.

[3.6] Uempd J.H. van, Dautzenberg J.H., Kals J.A.G.,

"Een dataverwerker voor vloeiparameters•,

MB Produktietechniek ~ (1988): 374-379.

[3. 7] Stenger H.,

"Bedeutung des Formänderungsvermögens für die Umformung",

Bänder Bleche Rohre a (1967): 599-605.

Charaderistic Quantmes of Formability

(3.8] Bolt P.J.,

Prediction of Duet/Ie Failure,

Ph.D. thesis, Eindhoven Univarsity of Technology (1989).

[3.9] Bridgman P.W.,

Studies In Large Plastic Flow and Fracture,

McGraw-Hill, New Vork- London (1952).

[3.10] Atkinson M.,

"Assessing Normal Anisotropic Plasticity of Sheet Metals",

Sheet Meta/Industries {1967): 167-178.

[3.11] Yang J.H.,

Plastic Anisotropy of Recycled Aluminium,

intemal report WPA 1010, Bndhoven Univarsity of Technology (1991}.

51

52~---------------------------------------------~C~h~ap=t~er-=3

53

Chapter 4

Strain-Path Dependenee of Flow Curves

Row curves are an imponant link between materials science and Iorming technology.

Such a flow curve ar stress.strain curve - depiets the relationship between !he

equivalent plasticstrain ë and the flow stress o1 to maintain plastic deformation. The

flow curve of a metal depends not only on the temperature and the strain rate, but also

on !he strain path. A well-known expression ot this strain..path dependenee exhibits at

a total reversal instraining direction. This is called !he "Bauschinger effect" aftar the first

researcher who repor1ed a yield-stress drop at a reverse in straining direction, more

than a century ago [4.1]. The influence of the strair. path on the flow behaviour,

however, is nol llmited to this single illustration but is more comprislng.

The study of these phenomena is not merely theoretical: many Iorming

processes, more ar less, involve deflections trom a plain strain path. Backwerd can

extrusion in particular is such a process. Changes in strain path which lead to a flow

stress drop or a decrease in strain hardening are unfavourable since they may ceuse

unstable flow, strain concentration, and even material !allure. Several investigations

have been done Mheno, either from a theoretical-metallurgical [4.2] or trom a more

practical point of view [4.3, 4.4]. Vet a quantitative description for use in metal-torming

calculations is. to the author's knowfedge. not available.

This chapter deals with the flow-curve changes, resu~ing trom abrupt changes

in the strain path. These abrupt changes were actualised by subjecting specimens

successively to two different standerd material tests. In this way three types of

combined tests were investigated" The equivalent strain at the transition (pre-strain &0)

was varied. 11 will be shown that the resu~ can be expressed as parameter changes

in a conventional flow function. Preceding to this, the experimental procedure is

dlscussed.

4.1 EXPERIMENTAL PROCEDURE

An investigation into the strain-palh dependenee of flow curves must be supported by

an appropriate experimental set-up. This is discussed here. Moreover, information will

be given on the experimental condi!ions, as well as on lhe investigated materials.

4.1.1 Testlng te<:hnlquBS

Basic idea behind the present set-up is, that lhe strain palh can be allered abruptly by

subjecting one and the samespecimen successively !o two different toading silualtons.

Utilising the standard compression, torsion and tension test, lhis provides lor saveral

options, of which lhe next three we re adopted because of !heir practicalleasibili!y: the

tension-compression test, the lensiOn-torsion test, and the torsion-lension test. In

addition to these, the standard tests were carried out.

• Tension:eomoression test

The aim of these tests was to investigate the flow behaviour at a total reversal

in straining direction ("Bauschinger effect•). In this work, not lhe initia! yielding but the

behaviour at considerable reverse strains is emphasised.

Pre-straining was done by subjecting cylindrical test bars (in~ial diameter

d;=14 mm) toa tensite laad. Equivalent pre-straining value:

(4.1)

in which d0 is the diameter of the bar alter (uniform) pre-straining.

Next, compression specimens were machined trom the pre-strained sections.

These we re designed as Rastegaev specimens Onilial diameter d0 = 12 mm), as is

illustrated in fiQure 4.1, The two recessed face surfaces serve as lubrication pools;

lanolin was used as a Jubricant. Equivalent strain and ftow stress are calculated trom

the measured quantities, compressive force F c and specimen diameter d, as lo!lows:

Strain·Path Dependenee of Flow Curves 55

u

fitJure 4. 1 Experimental seH;p tor compression test aiter tensile pre-straining

(lett: tension bar; right: Rastegaev compression specimen, showing a lvbricaüon chamber}

- d - Fe e =2·1n(-) +e0 and u,"--· do ~-d2

(4.2)

4

The Rastegaev geometry in particular is suijable to maintain a uni-rudal stress

state and uniform deformation during testing. In !act, the shape of the specimen's rim

(dimensions u and t) has been optimised experimentally w~h regard to this [4.5).

• Tension-torsiqn test

For every test, two cylindrica test bars were used. These were identica, except

tor the diameter which was slightly different Onitial diameters of d11 =6.4 mm and

d12 =8.0 mm). Bot., specimens were strained in lension to the same uniform pre-strain,

according to:

56----------------------------------------------~C~ha~p~t~er __ 4

(4.3)

d01 and d02 are the respective diameters alter pre-straining. The gauge length of bath

spec:mens thus increased trom L; = 50 mm to Lu· depending on the actual pre-straining

value.

These bars were used toperfarm a differential torsion test (4.6], as is elucidated

in figure 4.2. The equivalent strain is carculated tor the mean radius in the layer,

constituted by "subtraction of the two specimens" (difference of the specimen radii R02

and Ro1 respective/y), The shear stress is considered to be constant in this incrementa/

layer, Flow-stress value& are obtained, using the van Mises yield criterion, from the

dilterenee in required moments (M2 and M 1) to twist the raspeelive specimens. The

relevant farmu/as are:

(4,4)

Twisting ongle 0

Figure 4,2 Experimenral set-up foT faTSion test alter tensite pre-straining

Strain-Path Dependenee of Flow C1.:1rves 57

lJ here stands tor the twisting angle, and is to be substituted in radians. Aow curves

thus are derived trom the recorded (M, 9)-graphs.

The advantage of a differentiel approach in torsional testing is that it is not

necessary to presuppose a flow function. Essentially, the methad is best suited to

specimens with a smal! radius difference, but this enhances the inaccuracy due to

experimental errors.

• Tersion-tension test

Forthese tests a similar approach as in the tension-torsion tests was adopted.

A pair of test bars of different diameter (d11 =6.4 mm and d12 =8.0 mm) was twisted to

a corresponding level oftorsional pre-strain (same twisting angle 80). The pre-straining

value for the radius intermediately to the two specimen radii (R11 and R12) is estimated

trom:

(4.5)

in which the gauge length of both specimens L1 = 50 mm.

Subsequently, the bars were used to perfarm a differentlal tension test. In

tigure 4.3 the experimental set-up is visualised. The flow curve is deduced trom the

imaginary tube, obtained by "subtraction of the two specimens". Respective tensile

torces (Fn and F12) and specimen diameters (d1 and d2} were measured. Equivalent

strain and flow stress are obtained trom:

respectively. The range is confined to uniform straining.

A small radius dîfference of the specimens is favourable to minimise the varlation

of torsional pre-strain in the considered "tube", but this affects the experimental

accuracy. The present geometry implies a variatien in the pre-straining value of ± 10 %.

• Standard tests

To evaluate the results of the combined tests, the corresponding standerd tests

(compression, torsion, and tension tests) were carried out also. Essentially, these ware

performed as combined tests without pre-strain.

58 Chapter 4

T ensile stro in ê1

Figure 4.3 Experimentalset-up tor tensiontest alter torsional pre-straining

Standard compression tests were done using Rastegaev specimens with an

original diameter d0 =14 mm.

Results for standerd torsion were obtained from differential torsion tests, in each

of which two specimens of different diameter (d01 =6.4 mm, d02 =8.0 mm) and equal

gauge length (L0 =50 mm) were involved.

For the standard tension tests, solid bar specimens (original diameter d0 = 7 mm)

were used - different from the combined tests. The employed strain range here was

extended to the necking range, utilising the Bridgman correction [4.7]. Except for the

tensile force Ft and the specimen diameter d at the smallest cross-section, this required

an additional maasurement of the profile radius p in the neck of the specimen. The

formulas for the calculation of equivalent strain and flow stress are: •

ë=-2·1n(~) and Ut=_!i_·C8 , in which Ce= 1 . do !:·d2 (1+ 4'P)·In(1+~)

4 d 4·p

(4.7}

Strain-Path Dependenee of Flow Curves 59

4.1.2 Experimental condHions and materials

The range of equivalent strains which was explored was confined globally to 0 < ë < 1.

Testing was done at room tempersture and in a quasi-static fashion; the equivalent

strain rate could be estimated to 1 0-3-1 0 - 2 s - 1, depending on the type of test.

Standard compression, torsion and tension tests were carried out in triplicate to

obtain an indication of the reproducibility.

Three materials were used tor the experiments: e22 steel, Armco steel, and

euZn37 brass. Relevant data of these matenals are listed in table 4.1. For each

material, the specimens were machined from the same rod {020 mm) and were

annealed to obtain a homogeneaus structure. As tor the heat trestment of CuZn37

brass, each specimen was enclosed in an individual quartz glass container {vacuum),

to counteract composition changes in the surface layer due to sublimation of Zn. The

homogeneity of the structures was checked both by hardness measurements and

microscopie examination in longitudinal and transverse direction: no significant

ditterences revealed.

Tabla 4. 1 Specification of the investigated materials

C22 steel Armcosteel CUZn37 brass

DIN indication 1.0402 - 2.0321

e 0.18-0.25 e 0.02

chemical Mn 0.3-0.6 Mn 0.08

composition Si 0.15-0.35 Cu 63 p <0.045 p <0.02 Zn bal [wt. %) s <0.045 s <0.015 Fe bal Fe bal

lattice bcc bcc fee

annealing 650 oe, 1 h, 650 oe, 0.5 h, 550 oe, 1 h, treatrnent vacuum vacuum vacuum

hardness HV 10 1360 940 630 [N/mm2]

60 Chapter 4

4.2 RESULTS

Results of the standard and combined tests will be presenled by means of examples

and by a suitable characterisation of the obtained flow curves. Tostart with, the results

of the standard tests are introduced.

4.2.1 Standard flow curves

The flow curves obtained trom the compression, torsion. and tension tests are

reproduced for the distinct matenals in the figures 4.4, ~. and 4.6. All three metenals

shcw discrepancies between !he standard flow curves. C22 steel and Armee steel

exhibit a similar behaviour; compression transcends bo!h lension and torsion. For

600

è 400

lt 200 0

c;:

............... _____ ........ , • compression

o lension

o torsion

(3 tests) (3 tests) (3 tests)

O+-~--~-----+--~--r-----~--~4 0.0 0.2 0.4 0.6 0.8 1.0

Equivalent s\roin i [ -]

Fiaure 4. 4 Standard flow curves ol C22 steel

Strain-Path of Flow Curves 61

800.-----~----~----~----~-----,

~mco 1 I .............. --+- ''''''''""""""""""""""""~-- ...................... t .. ... .. .

i j , *•'* ~ • • a i ! , • ., • ~~~<o ;; 0 o o I 1 ,.. "" ~1 8 G>

0 1° ! ~ • 1 ! cf'iflBII'E!I'Ji!§U i <00 "X'''""'-"'-'i'-'-- -:

200 1 ., ------·~: --- • compression ( 3 tests) -

o lension (3 tests)

600

o torsion (3 tests)

0+-----+-----+-----+-----+---~ 0.0 0.2 OA 0.6 0.8 LO

Equivalent stroin i: [ -]

Figure 4. 5 Standard flow curves o/ Armco steel

CuZn37 brass, a different resu~ is obtained; compression here coincides with tension,

torsion strongly deflects from these.

The experimental data were processed, using a PC programme tor fitting of flow

tunctions in a least-squares sense [4.8). The Hollornon flow tunetion (a1=C·ê")

showed good results lor the lhree materials and features a convenient characterisation,

with only two parameters. Accompanying coefficients of correlation R were lewest lor

lhe torsion tests, yet in every case R <!:0.9895.

Each flow curve thus is characterised by a characteristic stress C and a strain

hardening exponent n. Average values are !isted in table 4.2. Since the tests were

carned out in triplicate, it was possible to acquire some intermation on the

· reproducibility of these results. This is done by means of the range indicator I, which

is defined as the difference of maximum and minimum value, divided by the average

of the three values and expressed as a percentage. For the standerd tests, an overall

62

600

b 400

"' ~ .!:;;

"'

Chapter 4

~ 200 0

G:

* compression (3 tests) "

(3 tests)

{3 tests)

o lension

o torsion

0+-~--+-----~--~-r--~-+----~ 0.0 0.2 0.4 0.6 0.8 1.0

Equivalent stroin i [ -]

Figure 4.6 Standard flow CUf\188 of CuZn37 brass

average is obtained of I ~4 % in the characteristic stress C, and I ~6 % in the strain-

hardaning exponent n; !he reproducibility theretore is judged to be goed.

Tabla 4.2 Characterisation of standard flow curves

u,=C-ê" compresslon tenslon torsion

C22 steel C [Njmm2] 832 756 670 n r-f 0.197 0.232 0.172

Armee steel C [Njmm2) 595 560 471 n [-] 0.250 0.292 0.214

CuZn37 brass C [Njmm2] 750 759 477 n [-] 0.504 0.503 0.403


4.2.2 Combined flow curves

Some of the results of the combined tests will be introduced here. The reader is

referred to appendix A for a presentstion of the supplementary experimental data.

• Tension-compressjon flow curves

An example of a compression flow curve after pre-straining in tension is shown

in figure 4.7; corresponding standard compression and tensionflow curves arealso

included. At reverse straining in compression, the flow-stress level is lower as ·

compared to uni-directional compressive straining.

.---. "" E E

............ z ........

800~----~------~----~------~----~ i *i* i .c.0 !* •* l .o.!.:.o C22 l.c. A O!

······-------·1-:~:J. A ~ .:.o A I : __ : ________ L·-------------· * * i.c. i ! ~ . i * o<r i ;

* oOO ! j Î

600

b 400 ... o0

' 1 i i

?'~ VJ VJ <V .... ..... VJ

200 I A compression after tension

-·-----··-!-1·---··--·-................... + ... -.. ( ~0 = 0.203) o tension * compression

i i I 0+---~~--~--~--~--r-~---+--~~

0.0 0.2 0.4 0.6 0.8 1.0

Equivalent stro in i [-]

Figure 4. 7 Examples of flow curves tor C22 steel: tension-compression

lt is found that the compression branches of the combined curves can be

described by an accommodated form of the Hollornon flow function. Here, the

64 Chapter 4

charaderistic stress C and the strain-hardening exponent n are considered to be

interrelated with the pre-strain ë0• Or:

(4.8)

The symbol ë then denotes the accumulated equivalent strain (sum of the pre-strain

and the current strain). In this manner the results could be recorded satisfactorily; they

are represented tor C22 steel in figure 4.8. Matching values trom the standerd

compression tests are reproduced on the ordinates of the graphs. For this material,

there is a slight dependenee of the charaderistic stress on the pre-strain (about 1 0 %

decrease, over the explored range), whereas the strain-hardening exponent scatters

around a constant value.

A decrease in the parameter C can be interpreted as denoting a permanent

influence of the Bauschinger effect on the flow behaviour - an influence which shows

to be only marginal tor this particuier materiaL

,......, N

E E

........... z ..........

1/) 1/) Q) ..... -1/)

u :.;::;

1/) ·;::: Q) ....... u a .._ 0

..s::: ,ü

1000 .,.-----?"", --~--. ' '

' C22 ,_..,. __ ,._~'.!. • • i

t I : i ., 600 -·---+~-· ·t----1

800

400 ......... -............. t" ............ -...... t .... ___ ," ___ , .. ,, ! 1

200 _, ___ i + .. ·---· .. I 0 +----+---.-+-..,..-""'i

0.0 0.1 0.2 0.3

Pre-strain &0 [ -]

0.4 ......----r---.1--...., ,......, I ..........

.: 0.3 ------------r---····---·! .. ,_ c:::

i0.2~~~~ 0'1 • i i ·~ : ! ~ I I -T o. , --·-···---r .................... r--------· c I I '!5 - ! ~

Vi i ~ 0.0 +---.--1'--.--1-' -..--""'i

0.0 0.1 0.2 0.3

Pre-strain &0 [-]

Eigure 4.8 Characterisation of flow behaviour tor C22 steel: compression atter

tenslon


• Tension-torsion flow curves

In tigure 4.9, an example of a flow curve trom torsion aftar pre-straining in

tension is shown, accompanied by some of the corresponding standard flow curves.

The change in deformation mode at i 0 initially results in a significant flow-stress

increase. At continued straining this effect vanishes in reistion to standard torsion:

these curves merge at high strains.

,........ N

E E ' z .........

800~----~----~------~----~----~

r I ~! ..... l------

1 ~ l A 0 1. Î, : 0 0 'fO

/i t:. ;:j ~ /i A .:~oi à A '1:J (t l .:~. t:. 6 4 6 i 0 oooojccoo

600

b 400 -····------}-.................... ~:~ .. J..Cj..P.. .. P.._'?_~ .. P~ .............................. j .............. ___ .. .. l<èJCU o 0 i i i

cP!f i i i d9~0 i ! i i

oB i ! e:. torsion after tension "l.':r-----.... ··t·-------~......................... ( &

0= 0.202) 200

i ! o tension i ! i i c torsion : l

0+---~~--~--~--~--~~---+--~~

0.0 0.2 0.4 0.6 0.8 1.0


Figure 4.9 Examples of flow cuNes tor Armco steel: tension-torsion

The torsion branches of the obtained flow curves could be described by the

accommodated Hollornon flow tunetion in the way depicted above. The parameters c and n, thus determined tor Armco steel, are shown as a tunetion of the pre-strain in

tigure 4.10. Matching values tor standard torsion are included. These flow-curve

changes exhibit by a deeresse in the strain-hardening exponent; the characteristic

stress approximately remains constant, as compared to the values for simple torsion.

66

,........, N

E E

........... z ........

u V) V) Q) ..... -V)

(..) :.::; V)

'i:: Q) ....-(..)

c .... c

..r::. u

1000

800

600

........... ......... 400

200

0 0.0 0.1

~ 1 Armco

~

0.2 0.3

Pre-strain ë0 [-]

Chapter 4

0.4 ,...--~--.....,.~--..

-+--j_ 0.3 . . ~ I 1

î 0.2 ~-~~-!-+· 6 .. ! -T 0.1 _"""""_""", i ........... -1 .... _"._""""_ c j + ... 'ë i i

01 l I 0.0

,........, I .........

c

0.0 0.1 0.2 0.3

Pre-strain ë0 [-]

Figure 4.10 Characterisation of flow behaviour tor Armco steel: torsion alter tenslon

In the example (figure 4~9) it is shown that the abrupt increase in flow stress at

initial re-straining does nat persist, but it neutralised at continued straining; this results

in a declined strain-hardening ability and merging with the standerd torsion curve.

These observations now are quantified by the parameters n and C respectively.

• Torsion-tension flow curves

In the torsion-tension tests, the strain range was limited by the occurrence of

necking of the specimens. CuZn37 brass showed the largast employable range, which

is connected with its high strain-hardening ability.

An example of a tension flow curve aftar torsional pre-straining is shown in

tigure 4.11. tagether with some of the matching standerd flow curves. The transition

from torsion to tension tor this material results in a gradual adaptation. of the flow stress

to the new deformation mode.

Strain-Path Dependenee of Flow Curves

...--. N

E E

........... :z ..........

soo~----~!-1------.~----~------~----~

CuZn37 ! : :,! :.: 0 !.I i 0 0 ... i 600 --··-··-·-··---·.Î.l i ··----·--····-····-··-·•·- ·---·--

!I Î j 1,

i I . A i Î I èfJ AA~ i ; ; I 00; Ali ; ~ C

0

Î 0 .t.Aii Î p' ......... 0 O b 400 +---······· .. ··----+'' ......";;..0.~ ~ n o .--i 1-

0 1:> i 0 ° i i : 'V Ai:> : [l : :

PI A~ 0 i i l o i hA n i ' ;

200 ........ -e-~ö: .... 0! ~ 0

!__.................... 6 tension after torsion

, o,

1

i I : ::~:;~~·= 0.217)

0+---~-+~~--+------+--~--+------; 0.0 0.2 0.4 0.6 0.8 1.0

Equivalent strain Ë [-]

Figure 4. 11 Examples of flow curves tor CuZn37 brass: torsion-tension

67

For CuZn37 brass it was possible to fit the tension branches of the combined

curves to the flow function, as introduced before. This yielded the results shown in

figure 4.12. The values for standard tension (uniform straining range) are included.

Over the explored range, the characteristic stress depends linearly on the pre-strain

(up to 15% decrease), whereas the strain-hardening exponent doesnotshow a clear

trend but seems to scatter around its standard tension value.

In the example of tigure 4.11 it is shown that the combined flow curve deflects

trom the torsion flow curve towards the tension curve. This is connected with a similar

strain hardening and a persisting lower flow stress, as compared to standard tension.

In the proposed characterisation, these features are expressed by a constancy of n

and a decreasein G respectively.

68

{/') {/') <U ... -(f)

1000 .....------,.---~-.......

lcuZn37

·----. 1-----~r-.. ·----.. ·--i A iA ~ ~

800

600

400 ························i······ .. ··· .. ······· .. ··+--·--······ ..

I 200 +--.-......i----i--1 -0+--r--~---+--~~

0.0 0.2 0.4 0.6

Pre-strain t0 [-]

c: -c: <U c: g_ il1 0'1 .E

c::: <U

'E

Chapter4

0.8 ......----r---"!'"", ----.,

--~--+-·-i • ~· ~ 1 l

0.6

0.4 ---·---1-................ _ _J___ .......... .

I I ---·-·--··-··T ............... ____ r ....................... . _g 0.2 I

c::: "ê -(/')

i 1

i i 0.0 +---..--+-..---+------t 0.0 0.2 0.4 0.6

Pre-strain ~:0 [-]

Figure 4.12 Characterisation of flow behaviour tor CuZn37 brass: tensionaftar torsion

4.3 DISCUSSION

The results, presented in the previous section, demonstrata that the flow behaviour of

a material may signfficantly be influenced. Now, the backgrounds to and implications

of these phenomena will be considered.

4.3.1 Standard flow curves

The flow curves trom the compression, torsion, and tensiontests differ. Essentially, this

may originate trom a number of different causes, which can be dass!fied with respect

totheir backgrounds in eKperimenta/, theoretica/, and metallurgical causes.

Strain-Path Dependenee of Flow CuiVes 69

• Experimental consideratjons

The determination of accurate experimental information depends highly on a

solid experimental procedure.

First, this requires a proper experimental set-up. For compression, the

Rastegaev test is a suitable means. As a result of the excellent lubricating conditions,

the cylindrical shape of the specimen is preseNed (absence of barreling) up to high

dagrees of deformation. The uniformity of deformation in these tests is confirmed both

by the hardness distribution [4.5] and the grid pattem [4.9] after testing. Maasurement

of the specimen diameter, instead of height, further impraves the accuracy (4.10]. In ·

the tension test, the deformation is uniform by its nature (uni-axial stress state), at least

before necking occurs. Else, the Bridgman correction is an established methad to

extend the explored strain range; this will be discussed later. In torsion, the

complications arising trom the non-uniformity of strain are met by using the differential

method; this is to be preterred especially when a flow tunetion cannot be presumed.

Since the Hollornon tunetion could be used to describe the flow behaviour, it was

possible to check on the results afterwards. To this goal the differential results were

compared with those, gained trom the individual solid specimens by using Pöhlandt's

methad [4.11]. These results did mutually deviate less than 3% (in the characteristic

stress C).

A second requirement included in the experimental category concerns the use

of calibrated testing equipment. This expressas the importance of an accurate

meesurement of all quantities, involved in the calculation of the flow cuiVe (dirnensions,

loads, ... ). The extent to which the errors of maasurement affect the accuracy of the

flow cuiVe can be assessed trom an error analysis; the results of these calculations tor

the standard tests are presented in appendix B. From these calculations it is concluded

that one has to consider a general inaccuracy tor the distinct tests of about 2-4 %.

These, however, are conselValive estimations since the analyses assume the

(hypothetical) case that all errors enhance. The actual accuracy will be better.

The homogeneity of the materials is also important. Ta assure good

reproducibility within the sets of experiments, the specimens for each material were

taken trom the same rad. The annealing treatments resulted in equi-axial

microstructures and similar grain sizes in both longitudinal and transverse direction.

Further, the geometry of the specimens was checked after testing, trom which it was

70 Chapter 4

concluded that no significant plastic anisatrapie effects appeared (no deviations of a

circular cross-section).

Rnally, it is repeated that the tests were attuned with respect to the

experimental conditions (room temperature, quasi-statically).

• Theoretica! consjderations

Theoretica! backgrounds also play a role in the camparisen of flow curves,

derived trom the different tests.

Rrst, this category involves the assumption of a yield crlterlon. This is

necessary for the evaluation of the (effective) flow stress in torsion, and in case of

specimen necking in tension. In both applications the von Mises yield criterion is used.

Another theoretica! aspect concerns the utilisation of a correction tor necklng

in tension (here: the Bridgman method). Except for the already mentioned yield

criterion, this analysis of necking adds some other assumptions which may lead to

inaccuracies. For example, it is indicated that the assumption of uniform deformation

in the smallest cross-sectien does not hold [4.12]. In the explored strain range,

anyhow, the Bridgman correction amounts to maximally 10% of the flow stress;

inaccuracies in the flow-stress correction thus influence a very small traction in the final

result and therefore are of secondary importance.

• Metallurgical considerations

Metallurgical effectscan induce El different flow behaviour. This is supported by

previous investigations.

Ditterences between the flow curves trom uni-axial testing and torsion can be

explained trom the development of bath different textures and different

microstructures in the deforming material. lt is stated by Canova et al. [4.13] that the

development of different textures - which is connected with different modes of

deformation - leads to a departure of the initial isotropie behaviour and in itself

produces a divergence in the flow behaviour. Furthermore, they argue that the rates

of strain hardening (ar accumulations of dislocations) depend on the combination of

activa slip systems and therefore may be different tor uni-axial testing and torsion.

These percaptions are amplified by Witzei and HaeBner [4.14] on· the basis of the

Taylor equation. Here, the flow stress uf in a polycrystalline metal and the crhical shear

Strain-Path Dependenee of Row Curves 71

stress r crit in the individual slip systems of a crystal are linked by introduetion of the

Taylor factor Mp which is determined by the orientation distribution of the crystals.

Thus:

(4.9)

lt is indicated (tor fcc-metals) that the texture developments during deformation result

in a Taylor factor increase tor tension and compression, and in a decrease for torsion.

Moreover, it is suggested that the increase in the critica! shear stress by strain

hardening is larger for uni-axial deformation than for torsional deformation. These .

effects lead to flow curves which diverge with strain and trom which the torsion curve

is positioned lower.

Ditterences between the flow curves trom compression and tension can be

explained trom an influence of the hydrastatic stress. According to Lorrek [4.15], the

flow stress depends on a superimposed hydrastatic pressure p as:

(4.10)

The actual flow stress a/P) thus is composed of the flow stress a,(o) at atmospheric

pressure and a term, containing a proportionality factor w tor the material's sensitivity

to hydrastatic pressure (w > 0). Herbertz and Wiegels [4.16] elaborate on this and

assign the hydrastatic pressure to the hydrastatic stress component am in testing,

which differs for compression and tension. The flow behaviour here is described by

means of the ludwik tunetion (af=aro+C·ën); this corresponds with the Hollornon

function, including an initial yield stress ato. By doing so, the sensitivity factor w can

be estimated trom the results of compression and tension tests (uniform straining

range). In formula:

(4.11)

the subscripts c and t denote compression and tension respectively. The interpretation

is confirmed by the similarity between their experimentally obtained values for w and

the results of lorrek, obtained trom tension tests at superimposed hydrastatic

pressures. For the present results of the compression and tension tests, the sensitivity

factor is derived as an average w of three values at different strains. This corresponds

72 Chapter 4

with the approach of Herbertz and Wiegels; they obtained values of

w ~ 0.1-0.4 GPa _,, dopending on !he materiaL As compared to this, the resu~s ot C22

steel (w~0.7 GPa- 1) andArmco steel (w~o.s GPa _,)are more pronounoad, whereas

CuZn37 brass (w~o.o GPa- 1) raveals nodependenee at all. Metal-physically, a flow

stress increase due to a superimposed hydrastatic pressure can be attributed to a

number of dilterent causes, each acting more or less reversibly. Herbertz and Wiegels

list: an increase in elastic modulus, an influence on !he activatien energy, local

hardening effects, and hindering of delermation mechanisms which are conneeled w~h

an increase in specific volume. They conetude !hal the compression and tension flow

curves cannot be brought into coincidenoe for certain matenals - depending on w.

From this discussion it is daar that many factors are involved in !he delerminatien of

tha etandard flow curves. The requirements appointed by a proper experimental

procedure were attained to as much as possible in lhis wor1<. On theether hand, tt is

not teasible to exclude all uncertainties arising trom !he theoretica! assumptions.

Metallurgical effects, nevertheless, are considered to be decisive for the observed

dilterences in !he compression, tension, and torsion flow behaviour.

11 is 10 be realised that lhe individual metal-physical factors act simullaneously

on the flow behaviour. An inftuence of dillerenUy davaloping textures and

microstnuctures, however, does nol appear in~ially but wil! intensily with strain; !he

hydrastatic stress, on the other hand, will influence the overall flow behaviour. The

presenUy obtained resu~ may be interpreled as fellows.

For C22 steeland Armee steel the resu~ are very similar (flgures 4.4 and 4.5).

The hydrastatic stress has a significant infiuence on !he flow behaviour, which exhib~

particularly in ihe dillerences between compression and tension. ff ~was tor this sole

effect, !he torsion flow curve should be positioned in between; yet !he development of

dillerent textures andfor microstructures does lead to a deflection of the torsion curve

to a lower flow-stress level.

The resu~ obteined for CuZn37 brass (lig ure 4.6) are rather dillerent trom these

of bath steels. Here, !he hydrastatic stress has no obvious influence on the flow stress;

the flow curves trom compression and lension (including the necking range) coincide.

Dillerences between !he resu~ trom uni-axial testing and torsionai testing !hen are

attributed to !he influence of dillerenUy developing textures and/or microstructures:


these effects manifest increasingly at oontinued straining and resu~ in a pronounced

deftection of the torsion curve to a oonsiderably lower flow-stress level.

4.3.2 Comblned flow curves

The possible crigins of dillerent flow behaviour, as discussed lor the stal1dard flow

curves, are to be eerried in mind also in interpretlog the resuks of the combined tests.

On the metallurgical baCkgrounds, furtherrnore, some add~ional comments can be

made.

Phenomena, connected w~ a total reversal in stralnlng direction are relerred

to as 'Bauschinger effect'. About the principles causing this effect, a general

agreement exists (e.g., [4.2]). Here, a lower yield stress in areverse straining direction

basically Is explalned trom a larger motionallreedom of dislocations, as compared to

the forward straining direction.1l The magnitude of the Bauschinger effect aften is

expressed using the yield-stress drop as a characteristic [4.17]; the oorrasponding

lower yield stress may be representatlve lor constructive application of oold-formed

parts. For subsequent Iorming operations, hcwever, the flow behaviour at considerable

reverse strains is also of importance.

In an aarlier werk, an indication of these effectS was obtained by introduetion of

some characteristics, reprasenting the in~ial yield-stress drop as well as the

'permanent" influences [4.18]. The present characterisation by means of the

accommodated flow tunetion seems to be a more suitable means lor use in metal

forming calculations.

Metal-physically, both the lension-torsion tests and the Iorsion-tension tests

involva a transition to a different combination of activaled slip systems. This contrasis

~ the tension-to-compression transition, in which the same combination of slip

systems is app!ied - be ~ in a reverse sense. Ranta-Eskola perforrned simi!ar

experiments using sheet material (4. 19j. Trans~ions in the loading path were realised

t)l, î<: to be no~lced that the actual Bauschinger eHect may be obscured by ether effecta For iru.;tance, the reversa! can be accornpanied by a change In tha hydrostalic stress; the yleló-stress drop at a transition trom compresslon to tension theretore will be different trom tt\e drop at a transition trom tension to compresslon!

74 Chapter 4

by combining balanced bi-axial stretching and uni-axial lension tests in the experiment

Hitl's macroscopie theory of anisotropy could not explain the observed ancmalies in

flow behaviour; it was conduded that the dependenee of !he stress-strain relationships

on lhe loading path should be considered a material property.

4.3.3 Some practical aspects

A furlher interest in the present context concerns the practical implications.

In a first approach, the strain-peth dependenee ol the flow behaviour can be

considered by selecting a standard testto lil the application, lor which lhe information

is intended. In other words: lhe flow curve must be obtained from a test, resembling

the actual process. The material's behaviour tor an applicaüon like backward can

extrusion thus is best represented by the resu~s trom a compression test. Other

examples: the torsion test represents punching, the lension test represems stretdher

levelling. A further implication is, that any documentation on flow curves must be

provided with intermation on the type of test, trom whidh the results are obtained.

Specific catculations may require more detailed intermation on the ftow

behaviour. An example of this relales to a study by Crafoord, concerning plast•c sheet

bending (4.20]. In this process, a zone near to the centre layer of the bend is

subjeeled first to compression and lhen to tension. This expression of lhe Bausdhinger

effect has been taken into account in order to imprave the calculating accuracy

(bending moment, final sheet thickness).

Changes in the strain path, which lead to a decreased strain hardening are of

special importance. A pronounaad example is lhe tension-to-torsion transition tor

Armco steel (figure 4.9). Here, the resutting strain hardening tor the second branch

shows to be very low; this is particularly untavourable lor the stability of deformation.

The torsîan-to-tension transition lor CuZn37 brass, as an oppos~e example, resu~s in

an increased strain hardening (figure 4.11). These observations may have implications

lor processes, which consist of two - or more - subsequent Iorming operations:

untavourable flow behaviour tor a particular combination possibly can be avoided by

a~ering the werking sequence!


4.4 CONCLUSIONS

From the foregoing it is clear, that the flow behaviour depends on the strain path.

In the first place, this reveals trom the results of the standard material tests: the

compression, torsion and tension flow curves do not coincide. This expresses itself in

different values for the flow parameters (here: the characteristic stress and the strain

hardening exponent, according to the Hollemen flow function). Apparently, these

parameters are not to be treated as material constants! Backgrounds to this non

coincidence of the standard flow curves may be of various origin; the metallurgical

effects, nevertheless, are considered to be decisive.

A further expression of the strain-path dependenee is obvious from the results

of the combined tests: a change in deformation mode leads to changes in the flow

curve. These flow-curve changes are of an initial andfor permanent nature, in that they

appear directly after the transition andfor maintain up to higher degrees of

deformation. For the presented examples, the changes in the course of the flow curve

could be quantified by accommodation of the flow function.

General conclusion is that flow curves, in practice, are to be used with due care.

76 Chapter 4

REFERENCE LIST

[4.1] Bauschinger J.,

"Ueber die Veränderung der Elasticitätsgrenze und des Elasticitätsmoduls

verschiedener Metalle",

Civllingenieur 21.. (1881): 289-348.

[4.2] Sowerby R., Uko D.K., Tomita Y.,

"A Review of Certain Aspects of the Bauschinger Effect in Metals",

Materials Science and Engineering 41 (1979): 43-58.

[4.3] Huml P.,

"Utilization of Flow Stress in Metal Forming Calculations•,

Armals of the CIRP 33L1 (1984): 147-149.

[4.4] Thomsen E.G.,

"What Stress-strain Curve shall I use?",

Proceedings of the NAMR/.11 (1983): 1.57-161.

[4.5] Pöhlandt K.,

Werkstoffprüfung für die Umformtechnlk,

Springer-Verlag, Berlin Heidelberg (1986).

[4.6] Shvartsbart Ya.S., Stepanov V.P.,

"Differential Methad of Determining Shear Stresses in Hot-Torsion Tests",

lndustrial Labaratory 40 (1974): 901-904.

[4.7] Bridgman P.W.,

Studies in Large Plastic Flow and Fracture,

McGraw-Hill, New Vork- London (1952).

Strain-Path Dependenee of Row Curves

[4.8] Uempd J.H. van, Dautzenberg J.H., Kals J.A.G.,

"Een dataverwerker voor vloeiparameters",

MB Produktietechniek .M (1988}: 374-379.

[4.9] Herbertz R., Wiegels H.,

"Ein Verfahren zur Verwirklichung des reibungsfreien

Zylinderstauchversuches für die Ermittlung van FlieBkurven•,

Stahl und Eisen 101 (1981): 89-92.

[4.10] Rasmussen S.N., Nester W., Pöhlandt K.,

"Weiterentwicklung des Rastegaev-Stauchversuchs zur Aufnahme van

FlieBkurven",

wt- Zeilschrift fûr industrie/Ie Fertigung 74 (1984): 667-670.

[4.11] Pöhlandt K., Tekkaya A.E., Lach E.,

"Prüfung des plastischen Verhalteos metallischer Werkstofte in

Torsionsversuchen",

Zeilschrift für Werkstofftechnlk 14 (1983): 181-189.

[4.12] Bollen T.G.S.,

Ductile Fal/ure: Ver/tication of Methods tor the Determination of Local

Stresses and Strains in Axisymmetric Test Specimens,

internel report WPA 0874, Eindhoven Univarsity of Technology {1990).

[4.13] Canova G.R., Shrivastava S., Jonas J.J., G'Seli C.,

'The Use of Torsion Testing to Assess Material Formability",

Formabillty of Metallic Materials - 2000 AD., ASTM STP 753 (1982):

189-210.

[4.14] Witzei W., HaeBner F.,

"Zur Vergleichbarkeit von Werkstoffzuständen nach Dehnen, Stauchen und

-:-ordieren",

Zelfschrift für Metallkunde zs (1987): 316-323.

77

78

[4.15] Lorrek W.,

EinfluB von hydrostatisehem Druck auf FlieBspannung und

Formánderungsvermögen metall/scher Werkstoffe,

Ph.D. thesis, Clausthai Univarsity of Technology (1972).

[4.16] Herbertz R., Wiegels H.,

Chapter 4

"Der Unterschied zwischen Zug- und DruckflieBkurve, gedeutet durch den

hydrostatischen DruckeinfluB",

Archiv für das Eisenhüttenwesen .si (1980): 413-416.

[4.17] Kishi T., Gokyu 1.,

"A New Relationship Between Pre-strain and Yield Stress Drop Due to

Bauschinger Effect",

Metallurg/cal Transactions ~ (1973): 390-392.

[4.18] Sillekens W.H., Dautzenberg J.H., Kals JAG.,

"Flow Curves for C45 Steel at Abrupt Changes in the Strain Path",

Armals of the CIRP azL1 (1988): 213-216.

[4.19] Ranta-Eskola A.J.,

"Effect of Loading Path on Stress-Strain Relationships of Sheet Steel and

Brass",

Metals Technology (1980): 45-49.

[4.20] Grafoord R.,

Plastic Sheet Bending,

Ph.D. thesis, Chalmers Univarsity of Technology (1970).

Chapter 5

On the Ductile-Fallure Behaviour in Backward Can Extrusion

79

Ductility usually is defined in terms like: "An indication or maasure of the amount of

plastic deformation which a material will undergo without fracture, ... " [5.1]. The ductility

of a material is influenced by the environmental parameters during deformation, such

as the state of stress, the temperature, and the strain rate. Wlth raferenee to industrial

practica, the ductility of a workpiece material is of importance since it may determine

the process limits in the application of plastic processes. The ductility therefore is one

of the relevant aspects in the evaluation of a material for forming applications. Even,

ductility is one of the major issues in the development of new materials (for instance,

alloys produced by powder metallurgy, ceramics).

Processas which proceed in a compressive fashion are favourable with respect

to ductility. One of these is backward can extrusion. This process therefore is one of

the interesting options tor forming of moderately ductile materials.

An implementation of this idea will be discussed in this chapter. As a

representation of the workpiece material's behaviour, the so-called ductile-failure curve

is used. The application to backward can extrusion requires the calculation of local

quantites during the process: stresses and strains. For this purpose an analytica!

model is introduced, comprising a combination of slab and power analysis. This model

does provide also for a predietien of the tooi loads; a separate section will deal with

this preliminary application. Most attention, anyhow, goes to the study of ductility. The

interpretation is based on both analysis and experiment.

80 Chapter 5

5.1 MODELLING ASPECTS

The required intermation is gained trom an analytica! model. The model in detail is

treated in appendix C; here, only the main features will be summarised. This model

involves a combined approach, in a way that it consists of two separate analyses: one

is based on the slab method, the ether on the power method. Common model

assumptions:

Thin-walled, rotationally symmetrie cans are produced.

- No dead zone appears in the bottorn area of the workpiece.

The von Mises criterion is used as a condition tor plastic yielding:

Ut=U= ..!.-[(u1-u2}2 +(u2-u3}2 +(u3-u1}2] , 2

as expressed in the principal stress components.

The flow stress is assumed to be constant across the workpiece body:

(5.1}

(5.2}

- The frictional shear stress r Fr in the interface between tools and workpiece is

quantified according to the von Mises model:

(5.3}

which introduces the plastic-friction factor m as a variable. lts value ranges trom

m = 0 to m = 1 ; the latter corresponds to shearing.

The model is based on the three-zone representation as is shown in tigure 5.1. From

chapter 2 it is clear that this is an acceptable choice to describe the process tor the

extrusion of low billets. Thus, the workpiece body is divided into zones, trom which the

zones I and 11 experience plastic deformation while zone 111 is rigid. These zones are

separated by so-called surfaces of discontinuity r1 and r2. Polar coordinates (r, 8, z}

are used. The geometry of the process is represented by the ram radius RR, the

chamber radius Re, and the current bottorn thickness T as well as the billet height T 0

.

The symbol ü denotes the ram velocity. The (frictional} influence of the ram height is

neglected.

On the Ductile-Failure Behaviour in Backward Can Extrusion 81

T

~oo----Rc---1

Eigure 5. 1 Three-zone representation of the process

The model is derived with a limitation to thin-walled products (extrusion ratio

R>2-2.5). This is nat an essential requirement, but it keeps the deductions

manageable. Excluded from the analysis is the pressnee of a non-deforming .zone in

the bottam region, which means that the model is useful merely to lew-billet extrusion.

This, anyhow, is mostly the case for the production of thin-walled cans.

5.1.1 Slab method

The calculation of stresses is based on a slab analysis. This double compression

model originates from Dipper [5.2]; the version as it applies to the current set of

assumptions is borrowed from [5.3].

82 Chapter 5

lhe analysis departs trom !he condition of force equilibrium on strategically

defined 'slabs' in !he respective zones. Only the principal stress components are

considered. Plastic friction between tools and workpiece is laken into account as lar

as it concerns !he zones I and 11. Zone 111 is assumed to be !ree of stresses. For !he

bollom zone, !he solLilion is:

u, ue 2 1•m T 2·m RA-r (-)1=(-),=--·(1•-·--)--·-, and u0 u0 l3 4 Re-RA l3 T

(5.4)

For !he comer zone:

<T, 2 1+m T-z Ue 1 1+m T-z (-)u=----·--, (-)u=----·--, and

<To 13 13 Re-RA <To 13 13 Ac-AR (5.5)

IJ1 1+m T -z (-)u=--·-~. u0 l3 Ac-RA

5.1 .2 Power metlied

lhe next kinematically admissible velocity field is used as an approximation lor thin·

walled cans:

(5.6)

In zone 111, !he wall moves upward as a rigid body. On !he basis of this field, an

equation can be derived lor the total power consumption, comprising the contributions

of intemal delormation, shearing, and friction [5.4]. In add'rtion, flow flnes and strain

quantilies can be deduced trom !he velocity field (5.5]. A listing of these bulkyformulas

is given in appendix C.


5.2 TOOL LOADS

In the process of backward can extrusion, the tools are exposed to extremely high

mechanica! loads. This, in tact, is one of the important process lim~s which sets

boundaries lor many workpiece materials.

Bath the slab and the power methad provide lor a relatively simple formula to

calculate the ram pressure. Concerning the slab analysis, an expression lor the

average pressure PA on the ram is obtained by averaging inlegration of the axial stress

component (a z)1 over the Irontal area of the ram. In a dimensionless farm, this yields:

PA 2 1 +m T/RA 2·m RA (-)slab=1 +-+--· +--·-. a0 /3 2/3 (Rc/RA)-1 3,f3 T

(5.7)

An analogous resu~ lor the power analysis is deduced by dividing the power tunetion

by the ram's velocity and Irontal area. Expressed as a tunetion of non-dimensional

variables, the salution reads:

Bath solutions dilter only as lar as~ concerns a term, \vhich represents shearing along

r 2 and bottorn friction in zone IL

For the slab as well as the power method, the relativa average ram pressure is

plotled in dependenee on the relativa ram stroke in figure 5.2. The extrusion geometry

and the friction factor to which these resu~ apply are also indicated in the figure; Te

denotes the final bottorn thickness. Dillerences between bath solutions show merely

to be marginal. Relativa ram pressures decrease at proceeding ram penetration, yet

tend to increase again at the very end of the stroke.

The relative ram pressure in ~se~ is not a quant~ that can be checked by

experiment directly, but~ may serve as an input to estimate the ram force. Sincestrain

hardening is common to most materials, it. is necessary to include it in the conversion.

Roughly, this may be done by adopting a flow stress lor the workpiece body which

depends on some global maasure of deformation in the process. Dipper proposes as

a maasure of defonnation:

84

5 ,........,

I ........

.§' 4 ...........

c.:: 0..

Q) 3 ...... ;::, IJ) IJ) Q) ...... Cl.. 2 E 0 .... Q) > :.;:; 0

(i) a:::

0

Rc/RR= 1.2

.....

I -~--~--

I 0.0 0.2

Chapter 5

_J --~~--------~-------------·t>-·1···---·-·-----·

0.4

-j------··-r-

0.6

a power method

b slab method

0.8 1.0

Relative ram stroke (T 0 - T)/T 0 [-]

Figure 5.2 Predietien of average ram pressures

- To - To RR e1 =In(-) and e11 =ln(-}·[1 + ] ,

T T 4·(Rc-RR) (5.9)

for the zones I and 11 respectively [5.2]. To obtain one global measure, these can be

averaged according to Schmitt [5.6] as:

(5.10)

in which v1 and v11 denote the volume fractions of the involved zones. Th is global strain

then is linked to the flow stress of the workpiece material by means of a flow curve, or

its reprasenting flow function.

Same results are shown in figure 5.3. The ram force here is recorded as a

tunetion of the ram stroke; the geometry of the process is indicated in the figure.

Asterisk symbols mark the results of an extrusion experiment, using recyded aluminium

On the Ductile-Failure Behaviour in Backward Can Extrusion

~ (\.) u .._ 0 -E 0

0:::

80~------~------~------~------~ Rc=6mm RR=Smm T0 =4mm Te= 1mm

m=O.B . I 60 ....................... -----t .......................................... ~·-1 . :

40 ....

* 20

*. experiment (230 °C) 0+---~---r--~--~~--~--+---~--~

0 2 3 4

Ram stroke T 0 - T [ mm]

Figure 5.3 Comparison of predieteel and actual ram force

85

as a workpiece material. This experiment was performed at elevated temperature; the

reason of which will be explained in section 5.3. Corresponding flow behaviour is

represented by the Hollornon flow tunetion (u1=C·i';, where the parameters are

approximated by C=176 N/mm2 and n=0.13.1) Model predictions of the ram force

were obtained on the basis of these values, using the averaged ram pressure of slab

and power analysis as a further input. This yielded the results shown in the tigure as

solid lines. The influence of friction is demonstrated by including the plastic-friction

factor as a parameter. Apart from the initial stage, the experimental result agrees well

with the calculated curve for m=0.4. Considering the global approach which is

followed, anyhow, it seems ambiguous to conclude tor a definite value of the friction

parameter.

1)0btained from tenslon tests. These values are used in the absence of more appropriate data.

86 Chapter 5

The slab analysis can be used also to obtain an indication of the local stresses,

acting upon ram and chamber. These tooi stresses can be dec:luced from the stress

formules by calculating them tor the interface boundaries - after all, workpiece stresses

are conveyed via the toots.

As an example, figure 5.4 visualises the normal stress components on the tools;

the two halved representations show data for the initial and final position of the ram

during the operation. Of course, all these are compressive stresses. The chosen

geometry is the same as the one used previously.

Outoornes such as presented in this section have a clear practical significance.

Direct access to this kind of information for the benefrt of the process designer can be

realised by incorporating the formulas into an expert system [5.7]. Supplemented with

practical knowledge, these computer programmes provide tor a device with which the

process layout can be optimised on-line, leeving tedious calculations to the computer.

Rc/RR= 1.2 T0 /RR=0.8 T.JRR=0.2 m=0.2

~

I

3.83 I

az I ao I 3.65

crr i ....•.. /

ao .·······~ ...... ·.1

3.93 ~v... 1.84

az cro

3.83 3.65

Figure 5.4 Normal stress components on the tools tor the initia/ and flnal posft/on of

the ram (slab ana/ysis)


5.3 DUCllLITY

This study of the ductile-failure behaviour in backward can extrusion is backed up with

calculations, based on the analytica! analysis as well as on FEM simulation. On the

results of actual extrusion experiments will be reported also. To start with, a practical

methad to describe the ductility of a workpiece material is discussed.

5.3.1 Duetlie-fallure curves

A ductile-failure curve is a convenient means to quantify the ductility of a workpiece

material in dependenee on the state of stress. By using this concept, the global

definition of ductility is settled in a manageable form; that is, concrete intermation is

obtained which can be employed tor practical aims. lt is to be emphasised here, that

these failure curves feature a phenomenologically based method to describe the

ductility rather than an accurate representation of the metal-physical effects. This,

however, is the exact intention: to provide for a practically oriented approach towards

the implications of ductility in forming.

A ductile-failure curve - after Hancock and Màckenzie [5.8] - is a curve of the

equivalent strain ëF versus the stress tri-axiality (um/ül at ductile failure. The stress

tri-axiality is the non-dimensional representative of the state of stress, and is defined

as the ratio of the hydrastatic stress component um= (u1 +u2+u3)/3 and the effective

flow stress uf.

• Experimental determination

A ductile.failure curve can be obtained by means of some basic material tests.

Essentially, these tests involve different losding situations in order to vary the state of

stress. For each of the experiments, the equivalent strain and the accompanying stress

tri-axiality are assessed tor which fracture occurs; these determine a single point on the

taUure curve. By combining the results of the different tests, then, the course of the

failure curve is estimated.

88 Chapter 5

Wlth regard to this concept, Bolt explored the non-negative tri-axiality range tor

saveral terrous and non-terrous metals [5.9). Two dissimilar loading tests were

employed in this investigation: the torsion test and the tension test. A further variatien

in the stress tri-axiality was obtained by the use of pre-notched tensile specimens, in

addition to standerd cylindrical ones. Thus, the tri-axiality range could be extended up

to values of 1.5. The state of stress in the nee!< of these specimens was studied by

means of FEM simulation. For the investigated conditions, the failure curves allowed

tor a linear approximation. An application relates to the process of punching, in which

the occurrence of tracture is an essential feature; this application also was FEM

assisted.

On the basis of this work, a simplified procedure can be proposed tor a practical

determination of the failure curve [5.1 0]. This method utilises the standard torsion and

tension test. The concession of using a minimum number of different tests showed to

be acceptable in an application to sheet bending. Predietien of ductile failure for this

process entails an extrapolation of the failure curve towards higher tri-axiality values.

• Examples

Ductile-failure curves can be used to rank different matenals with respect to their

ductility. An illustration of this is shown in tigure 5.5.

The failure curves indicated by solid lines reprasent the behaviour of recycled

aluminium; the testing tempersture hereis used as a parameter. These results ware

introduced in chapter 3 as one of the important characteristics of this material with

respect to its formability. Experiments were performed quasi-statically. Tensile data

originate trom tests on sheet specimens, taken trom the transverse direction in the

material: this direction proved to be the critical one with regard to ductility. Feilure

curves of two other grades of aluminium, AleuBiPb (DIN indication 3.1655) and

AIMgSi1 (DIN indication 3.2315), are included in the figure as dashed lines. These are

obtained trom bar specimens; tests were performed at room tempersture and in a

quasi-static fashion.

The failure curve of AICuBiPb nearly coincides with the one of recycled

aluminium at 200 oe; the curves of recycled aluminium at 300 and 350 oe show

resembienee to the failure curve of AIMgSi1. These two conventional grades of

aluminium are widely used as workpiece materials tor cold-forming processes,


......... 0.6

I I

AleuBiPb I ' recycled Al: ........ • ' AIMgSi1 I ' 0 20 oe .... ... \ ......... 100 oe n:) .. i ' I +

............ ' 150 °C .J 0.4 \ i ' c r··-·--ï----- ----·•·····-·······-----...._.,

! '

" x 200 oe

<I> ' 250 oe .... À ::l

i - 300 oe u • 0 ,\ .... ~ ' ~ 350 oe - \ ...... _,..,., .................... 0

>. :t:: :2 x 0 I

•;::: ....... 0.0

0 2 3 4

Stroin ot frocture ëF [-]

Flgure 5.5 Examples of duetlie-fallure curves

induding backward can extrusion. Backward can extrusion, performed at temperatures

above 200 °C therefore is suggested to be a suitable oparation for recyded aluminium

from a ductility point of view.

5.3.2 Application

The application of the failure-curve concept to concrete forming processas requires the

calculation of local quantities; that is, the development of the stress tri-axiality as a

tunetion of the equivalent plastic strain. For an instationary process, as is the case with

backward can extrusion, this course will be different for each material point. The

procedure basically assumes that fracture will occur if any of these lines intersacts the

faiture curve.

90 Chapter 5

First, !he anafytical resufts from !he model are introduced. Corresponding

experimentaf results are obtained from actual extrusion experiments. Numerical results

from a finite-element simuiatien of !he process serve as an additional souree of

information.

·~ From the slab analysis, the derived tormulas tor !he principal stress components

- as introduced in sectien 5. t - can be combined to obtain the next expresslons tor

!he stress tri-axiality in the zones I and 11:

Om [ 1 2 1+m T 2·m AA-r] (-)1 ~- -+-·(!+--·--)+-·-- ,and o0 3 {3 4 Ac-AR /3 T

(5. t 1)

Zone 111 is !ree of stresses. lhe minus signs indicate that the process proceeds under

pressure. 1t is to be realised, that the stress tri-axiality is a momentary quantity: the

state of stress is maintained only as long as the ram perfarms its p!ercing action.

lhe flow of a material point of original coordinates (r 0

, z.,J is appointed by the

flow-line equations, obtained from !he power analysis. lhis means that !he path of each

material point can be tracked duling the process. By application of !he accompanying

strain formulas, the development olthe equivalent strain is assessed. lhis includes the

comributions of internat deformation in !he respective zones, and shearing deformation

along the surfaces of discontinuity.

For oonvenience, the formulas are incorporated in a PC calculation programma.

Same illustrative results are shown in figure 5.6. Calculated strains and stresses

- as lhey develop during the process - hereare visualised lor three material points,

symbolised by A, B, and C. The inset shows the original positions of these points in

the billet. These data apply toa set of conditions as is indicated in lhe figure. Passages

through the surfaces of discontinuity are represented as dashed segments: invotved

quantities are undetermined lor these. Further, the failure curves of recycled aluminium

at 200 and 300 oe are included; !he dotled branChes are !he linear e'!fensions into the

negative tli-axiality domain. In !he backward can extrusion of recycled aluminium at

these temperatures, obviously, point C is a critical point which is liable to fracture.


RcfRR= 1.2 T./RR=O.B T,/RR=02 m=0.2

I 0 ' ' l ' ' ~

A '" -1 ········-··~::::::ih., ......_

.:1 ~ -2 0

·;;; 0 I zl [·A ~ -3 B· I I Ie ' r

-4 0 2 3 4 5 6

Equivalent strain i [ -1

Fiaure 5.6 Ductility diagram, showlng data tor backward can extrusion (analytjca/

resutts) and tai/ure cutves ot recycted aluminium

An additional indication of the critica! spots is obtainec:t from the calculated strain

distribution in the extruded product. For the present geometry, this dislribution is

shown in figure 5.7. Locations of equal equivalent strain are marked by solid lines and

labelled by the respective values. Dotted lines separate the areas of different straining

history. The final positions of the points A, B and C are also plotted in the figure. Spots

which have expetienced highest strains deserve special attention; these are located at

the exterior side of the can in the near vicinity of the corner. The maximum equivalent

strain is ëmax=5.4. The adoptec:t velocity field implicates uniform deformation In the

bottorn zone; here, the value of the equivalent strain corresponds to the one lor ldeal

compression. In the corner and walt of the produced can, the strain distribut!on is

distinctly inhomogeneous. Noteworthy, the equivalent strain in the outer par1 of the wall

varles strongly along the axial coordinate, whereas the gradient in the inner part is less

pronounced.

92 Chapter 5

Equivalent stro in t [-]

r---

Re ~ 0.2 RR = 1.2 \

I~ A

To 0.5

f\ if=o.a

~ R

Te 1.0 RR =0.2 2.2

' ·\- 2.0 2.1

B

V 3.0

2.2 ~ 4.0

2.5 ~c t-i ----,-.4-------!1~\- 4.0

i 2.0 3.0 2.5

I

Fiqure 5. 7 Deformation distribution in the longitudina/ section of the extruded can:

analytica/ results


• Experimental results

Actual experiments were done at elevated temperatures, using tlle recyded

aluminium [5.11]. The extrusion geometry matehad the one for the calculations:

RR = 5 mm, Re =6 mm, T 0 =4 mm, and Te= 1 mm. Tools (both ram and chamber) were

equipped with heating devices. The tempersture range was confined to 200--300 oe;

this is the range where the material's ductility is highly dependent (figure 5.5). Besides,

for lower temperatures the toolloads were expected to be too high to be carried by

the set-up. Molykote HTF was applied as a lubricant, suitable forthese temperatures. ·

The experiments revealed that sound products could be obtained, even at

200 oe. No fracture of the workpiece material was visually detectable; these

observations were affirmed by microscopie examination.

Anyhow, subsequent forming of the cans by means of an ironing process - in

order to reduce the wall thickness to 0. 75 mm -- showed not to be possible: the

extruded cans fractured in the corner. Similar cans, not extruded but machined trom

the semi-finished material, could be worked by this ironing process at these

temperatures without fracture.

• Numerical resu!ts

In order to check the results from the analytica! model, additional calculations

on the basis of finite-element simulation were performed [5.12]. Forthese computations

the ABAQUS programme (version 4.8, 1989) was used.

In the FEM model, the initia! mesh comprised 600 tour-node quadrilateral bi

linear elements. Row behaviour was represented by data for the recycled aluminium

-- oorrasponding to one of the experiments (230 oe). In the programma, classic

Coulomb friction is modelled; a friction coefficient of p = 0.1 was adopted. The analysis

was carried out until the bottorn thickness was reduced by 75 %, which involved about

60 rezoning steps.

For the available programma, unfortunately, it showed not to be practicabie to

simulate the course of stress tri-axiality versus strain in the ductility diagram, in analogy

to tigure 5.6. Being one of the relevant results, the equivalent-strain distribution in the

product is reproduced in tigure 5.8. The maximum equivalent strain hereis ëmax=2.8.

94 Chapter 5

A comparison of the analytically and numerically obtained contour plots

(figures 5. 7 and 5.8) raveals that there are eertsin discongruencies on a local level.

Specifically, the locations of maximum strain are different; the oorrasponding values

even show a discrepancy of a factor 2. Ón a less detailed scale, however, there are

also similarities. One of the common characteristics Is that of a different directionality

of the strain gradient for the inner and outer wall sections. This has implications for the

machanical properties of the produced can, such as the hardness distribution.

5.3.3 Discussion

The results on backward can extrusion in relation to the material's ductility are quite

ambiguous. Intersectien of the analytica! data ~ike point C in tigure 5.6) and the

extended failure curves suggests that ductile failure of the workpiece material will occur

during the operation. On first thought this contradiets the experiments, from which

visually sound products were obtained. Nevertheless, the impracticability to impose

additional deformation tothese cans by means of a subsequent ironing process fuels

the doubt that actual material damage may have occurred.

Anyhow, the comparison of the analytica! strain distribution with the distribution

computed by finite-element simuiatien shows considerable differences; in particular, the

analytica! model seems to overestimate the maximum strain. The assumed velocity field

in tact is a rather simpte one: no optimisation parameters, nor the inttuenee of plastic

friction are included. A further shortcoming is that the relevant quantmes cannot be

calculated tor the surfaces of discontinuity. The present model, however, was meant

to be a first onset. An impravement of this kind of model is certainly possible, yet will

lead to elaborate formulations. Another salution may be to implament the idea

altogather by means of numerical rnadelling techniques.

Also, as a further point of discussion, there are certain indications that the failure

curve may deviate from a straight line in the compression range. One of the most

prominent of these is from a study by Pugh and Green [5.13). They performed tension

tests at superimposed hydrastatic pressures; tor various materials, it Was observed that

fracture deferred more than proportionally at increasing pressure. For the determination

of a feilure curve, by implication, the use of supplementary tests may be cèlnsidered.

On the Ductile-Failure Behaviour in Backward Can Betrusion 95

Equivalent stroin i [-]

Jl. =0.1

1.0

1.5

1.5 2.0 1.0 1.0 1.0

2.0 1.0

1.5 1.0 0.5 2.5

Figure 5.8 Deformation distribution in the longitudlnal sectien of the extruded can:

numerical results

96 Chapter 5

Plain compression tests are a simpte means but their outromes can be obscured by

the occurrence of disturbing effects, such as buiging. Alternatives - as employed by

Brldgman [5.14] - may be tension or torsion tests performed under superimposed

pressure, yet these methods are quite invoMng and therefore of limited practical

significance.


5.4 CONCLUSIONS

The failure-curve concept is a useful approach towards the implications of ductility in

the field of metal forming.

A ductile-failure curve of a workpiece material can be determined efficiently by

means of a torsion test and a tension test. Amplification, anyhow, may be necessary

tor the compression range. Failure curves of different materials can be used to obtain

a ranking of their ductility, being an indirect criterion to select suitable forming

applications. Also, failure curves are apt to describe the ductility of a material in

dependenee on the temperature; these curves are useful in assigning the proper

working tempersture to a particuier forming process.

The strategy of combining the ductile-failure curve with detailed calculation on

concrete forming processas uses aspects trom different fields of study. Presented

results for the process of backward can extrusion show as yet not to be a ready-made

solution; this implementation, nevertheless, certainly is a lead for further research. Tie

effectiveness of the approach depends highly on an accurate predietien of local

quantities during the process. Although the practical accessibility benefits trom an

analytica! descriptlon, these calculations in principle may also be carried out by

numerical techniques.

98

REFERENCE LIST

[5.1] CIRP,

Dlctlonary of Production Engineering, volume 5,


[5.2] Dipper M.,

"Das FlieBpressen von Hülsen in Rechnung und Versuch",

Archiv für das Elsenhüttenwesen 2Q (1949): 275-286.

[5.3] Ramaakers J.A.H., Houtsckers L.J.A., Peeters P.B.G.,

Plastisch bewerken van metalen: Procesbeheersing in de

onderdelenfabrikage,

Wibro, Helmond (1990).

[5.4] Ramaakers J.A.H.,

Hulsextrus/a 1: Berekeningen,

Chapter 5

internal report WPT 0534, Eindhoven Univarsity of Technology (1982).

[5.5] Hocgenboom S.M.,

personal communication, Eindhoven Univarsity of Technology (1991).

[5.6] Schmitt G.,

Untersuchungen über das Rückwärts-NapfflieBpressen von Stahl bel

Raumtemperatur,

Berichteaus dem lnstitut für Umformtechnik, Universität Stuttgart, Bericht 7,

Girarclet, Essen (1968).

[5.7] Franse C.J.M.,

De ontwikkeling van een technologiebank voor achterwaartse hulsextrusie,

internal report WPA 0384, Eindhoven Univarsity of Technology (1987).

On the Ductile-Failure Behaviour in Backward Gan Extrusion

[5.8] Hancock J.W., Mackenzie A.C.,

"On the Mechanisms of Ductile Failure in High-Strength Steels Subjected to

Multi-Axial Stress-States•,

Joumal of the Mechanfcs and Physics of Sollds 24 (1976): 147-169.

[5.9] Bolt P.J.,

Prediction of Ductfle Fallure,

Ph.D. thesis, Eindhoven Univarsity of Technology (1989).

[5.10] Sillekens W.H., Dautzenberg J.H., Hocgenboom S.M. and Kals J.A.G.,

"Practical Verification of Ductile Failure Curves•,

accepted tor publication in: Annals of the CIRP !1,L1 (1992).

[5.11] Joosten L.A.J.,

De ductiliteit van materialen bij hulsextrusie en duntrekken,

internal report WPA 1182, Eindhoven Univarsity of Technology {1991).

[5.12] Yang J.H.,

A Finite Element Ana/ysis of Bacl<ward Can ~sion,

internal report WPA 1191, Eindhoven Univarsity of Technology {1991).

[5.13] Pugh H.U.D., Green 0.,

"The Effect of Hydrastatic Pressure on the Plastic Flow and Fracture of

Metals",

Proceedings of the lnstitutfon of Mechanica/ Engineers, part 1 m (1965):

415-437.

(5.14] Bridgman P.W.,

Studies In Large Plastic Flow and Fracture,

McGraw-Hill, New York - London {1952).

99

100 Chapter 5

101

Chapter 6

Conclusion

In the introductory chapter, the objective of this workis described as to obtain a better

insight into the behaviour of the workpiece material during backward can extrusion. A

dual approach on the basis of process and material is thereby put forward as a matter

of special interest.

The review of models on the basis of the upper-bound principle in chapter 2 reveals

that the overall description of the backward can extrusion process is rather

complicated: there is no such thing as the universa! model. The nature of the process

differs for the initia! and final stage, as well as tor the production of thick-walled and

thin-walled cans. Each of the presented models shows to have an outlined validity

range, thereby justifying its very existence. More specific, it follows that the various

applications require a different analytica! approach.

The number of industrially available materials has increased explosively within the last

decades, following the introduetion of new material-processing techniques (powder

metallurgy, thermo-mechanical treatment, etcetera). Traditional characterisations - such

as: chemica! composition, hardness, ultimata tensile strength - are absolutely

inadequate to assess the werking behaviour; that is, to evaluate the possibilities and

restrictions of a material in the manufacture of final products. For manufacturing

engineers in genera!, and forming engineers in particular, the use of some other, truly

characteristic quantities may help to overcoma this problem.

Characteristic quantities of formability which are discussed in chapter 3 include

the flow curve, the ductile-failure curve, and the plastic-anisotropy parameter. All these

can be derived from some standardised material tests: the tensiontest and the torsion

test. The use of these quantities is illustrated tor an unconventional material; it is shown

102 Chapter 6

that the directional dependenee in the material, and the temperature dependenee of

these quantities can be expressed.

As a future employment of these characteristic quantities, one may think of

recording them systematically for all relevant materials on data sheets - as a

supplement to the traditional information.

The issue of the flow behaviour is a complicated one, entailing many different aspects.

In addition to the temperature and the strain rate, the strain path shows also to be of

importanee.

This strain-path dependenee of flow curves is treated in chapter 4. Depending

on the material, the flow curves trom the standerd tests (compression, torsion, tension)

ditter more or less. These differences can be explained on the basis of metal-physical

influences. As for the investigated abrupt changes in the strain path, these result in

additional deviations. Anyhow, these rather elementary transitlens in the strain path are

merely indicative of the actual changes which occur in such complicated processas as

backwerd can extrusion. What these results do show is that not all problems in

conneetion with forming processes may be tackled, just by improving the rnadelling

techniques without due consideration for the representation of the material's behaviour.

Results on the ductility of the workpiece behaviour during backward can extrusion are

described in chapter 5. These are not yet operational in a technica! sense; that is, there

still remain some questions which demand clarification. The adopted strategy, however,

does certainly show that an approach which combines aspects of materials scienee

(the ductile-failure curve) and methods of plasticity (the calculation of stresses and

strains) is a beneficia! one, providing fora tresh look upon these phenomena.

As a practical prospect, one may imagine the following implementation. Suppose

that for any forming process the range of critica! strains and stress tri-rudalities in the

ductility diagram is known sufficiently accurate. A simple comparison of these results

with the ductile-failure curve of a workpieee material would then be adequate to deelde

tor the applicability of this material trom a ductility point of view. This perspective,

worthwhile, is to be investigated further.

103

Appendix A

Strain-Path Dependenee of Flow Curves: Additional Results

The complementary results of the combined tests from chapter 4 will be introduced in

this appendix.

For the tension-compression and tension-torsion tests, lt is possible to describe

the second branch of the combined flow curves by means of the accommodated flow

function:

(A.1)

Results are presented in the figures A. 1. A.2, M and M. In the upper part of these

figures some examples of obtained flow curves are shown; exemplary results of the

corresponding standerd tests are included. The results for the complete set of

experiments are summarised in the accompanying lower part of the figures, in which

the parameters C and n are represented graphically as a tunetion of the pre-strain 80•

Values of bath C and n from the standerd tests, corresponding wlth the second

deformation mode, are reproduced on the ordinates of these graphs. These can be

interpreted as resulting from a combined test with zero pre-strain and servlil as points

of reference.

The proposed flow function, unfortunately, was not apt to describe the results

of the torsion-tension tests, as obtained for C22 steel and Armco steel. This shows

trom the figures A.5 and A.S, in which some examples are reproduced. Both materials

exhiblt a similar behaviour: pre-straining in torsion leads to a significant harelening in

a subsequent tensile deformation mode.

104

(I) (I) (I) .... ..... (I)

V :.::; (I)

·;:: (I)

t) 0 ..... 0 .c (..)

Appendix A

aoo~----~1-1 ------~----~1 ------~-----,

Armco I: I ...-. N

E E

-......... z

600 ! I I ! '

···-·······----------t~·-····-·--·-t··----·-··················+-----:--~·J···t;-~ 11 i ,6 j.C.. * ' 0 ........, i Al .e. .'!i• ; 0 o qo

---.--tll~~ .. t-~ ...... i ...... : ...................... J .. ~--····--1. .................. ._ ... ._ * 1bo i i i

0o<!'1 i ; i * 000 j I j i f

~o0 j 1 i t:. compression otter tension ··0"···-·····-·-········n····························t-······ o tension ( Ëo = 0.219 )

Î I i jl l

t) 400 -

200

* compression 0+-~--~'~~--~' --~~--~--~----~ 0.0 0.2 0.4 0.6 0.8 1.0

Equivalent strain l [-]

0.4 ,.......... I ..........

1000 -r---"""""1"'"--,...---.

I Armco

c:: ..... 0.3 c:: (I) c:: 0 a. x (I) 0.2 o> r:: '2

. I ·······················-~························r·······················

~ ! ••• ~· i -··················l·················r-... -----·······

i i i ! i i

400 +---+- --+---1

800

600

(I) -o .... 0 0.1 .c I r:: ·o ....

200 I

Oi 0 I I 0.0 +---.---ir---.....--+--.---1 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3

Pre-strain fo [-] Pre-strain l 0 [-]

Elgure A. 1 Results of tension-compression tests tor Armco steel


~

"" E E

........... :z: ...........

u !/) !/) (!) ..... -!/)

u :.;::; !/)

·;::: (!) ....... u 0 ..... 0

..c: u

aoo~-------~!-,---------------~1 ---------------~----~1 ---------------~ Î I i .

,....... N

E E

........... ~

600

b 400

31: 200 0

c;:

CuZn37 i 11 •' , -~ l .. ç.* 0

.o.

··----··--..l.~·····--·-- ' ~ I : ·--ilf.···-··-.···--·~-········If ........ lS ...... 'Iê .............. Î .... -... -............... .. 'I (jllh.o. '

--1~4- i --+--....... _ o•~:~ i i i

..... ~·~·~ ... : ........ ~ .................... _ ! ...... A compren~o~ 0~~~e~) tension

1 o tension

* compression 0+---~~~~--~--~--~·--~--~----~

0.0 0.2

1000

800 ... IJio. • • • •

600

400

200

0.4 0.6 0.8 1.0

Equivalent strain Ë [-]

CuZn37

0.8 -.------.,.--.....,-----. ..........

I .......... ~

c i - 0.6 ····--··------·1--··---···-t-~·-······

i i c (I) c 0 c.. x (I)

.. : ' ... ~-~+ 0.4

I . ........................ i, .. ----······,;. .................... ..

I I : i

I I 0.0 -+--......--+---.-+-.......... --!

105

0 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6

Pre-strain ë0 [-] Pre-strain l 0 [-]

Figure A.2 Resu!ts of tension-compression tests tor CuZn37 brass

106

,......, "" E E

............. z ........

u l/) l/) Q} ... ..... l/)

(.) :;::; l/)

·;;:: Q} ....., (.) 0 ,_ 0

.t:: u

Appendix A

800~----~----~------~----~-----,

C22 ' ! ' 0 0 0 I 0 0 ...... Ë ~ z .........

600

~ 400

200

i j 0 i b. i b. A b. ' i I o ~> ~> i~> ~> ~> o oio o o o o c

---·~·······t······----;.···ïi:""dil:s·-~ïJï!f"C"~-e-.Q-O...O. .......... t················--··········· 1 4 6

0 o0,0 i i

0oofo 0 i i i

0<o1o ; 1 ; ;

-r-··~·······1...··--·····--·- ! ·-···t-------------t-t ! !

.: ............... LJ_·····--···-·-·-1·····-···············- A . torsion aft~r tension I i i O:o = 0. 145) I i i o tension

; 1 o torsion ! ~

0+---~~--~--~--~--+-~~~--~--~

0.0 0.2 0.4 0.6 0.8 1.0

Equivalent stro in 'i [-]

1000 0.4 ~

I ......... . I C22 I 800 c:: 0.3 -=···----·x~---~··=····r······-··········· 00

'0

'00000

'0

'0'''''''''10000000000 ______ ..,.. __ .... ~00HOOOOOO<OOOO

600

400

200

.... c: Q) c: 0 Q_ x Q) 0.2 0'1 c:

"2 Q)

1: 0 0.1 .t:: I c:

....................... J ............. -......... L ...................... . ~ i

+----ili-----+--- . . I

·---···--·+··········---+------

I I ...................... +--------1·--· .. ·············· • ! i

" i ••• i. I ~

"ë ..... U)

! 1 l l l ~

0+-~~·--~~·~~~ 0.0 +----+--.----+----f 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3

Pre-strain 'i0 [-] Pre-strain e0 [-]

Figure A3 Results of tension-torsion tests tor C22 steel

Strain-Path Dependenee of Aow Curves: Additional Results

.--. N

E E

.......... z .........

(.)

(I) (/) Cl.> .._ ...... (/)

(,) :.;:::; (/)

·.:::: Cl.> u 0 .._ 0

..r= (.)

soo~----~------~----~----~----~

CuZn37

1

I I ,....., .....

! ! 0 t ~ 0 i

0

E E

.......... z

600 --ril- -~ ~~-----...... ! oif ! AA<lAAA~

I ! 00 i A A A ~ A ! 0 0 [

···············---......L·--f-····-o-0·Ä····-n--~··········--o-fet······.O...--~··---·········--····· I ; o Î o 0 I I

b 400 I R A 0 I i I p :.().. 0 ! ; l

0 j c i

200 o t ei l t:. torsion ofter ten si on

·-~··9--·e~·····:····t······················---r·······--·········· ( i0 = o .16 7)

1000

800

600

400

200

P 0 1 1 1 o tension

1 o torsion 0+-~--~--~--~~--~----~--~~

0.0 0.2 0.4 0.6 0.8 1.0

Equivalent strain ë [-]

lcuZn37

·-----···-····-~··-----.. ·----~--------·····-

: ! ···--·····----··--i············ .. -----~·-·····················

~.. .... • ... 1 ... • I ~. i i ·······-·········--·-i·~··············· .. ····Î···--···---

! I i ~

f---····-···-·i-----······-······1 ....... --·-·-· i !

i 1 i :

0.8 ,..----.....--'""!! __ ..., ,....., I .........

c:

- 0.6 . rt-c: Cl.> c: 0 Q. >< Cl.> 0.4

·.... ... I ••t.. j ----t-···--·----r---·

I I 0+-~--~'~-r--~'--~~ ~ ! 0.0 +----.....--+-...---t---.----1

107

0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6

Pre-strain ë0 [-] Pre-strain i0 [-]

Figure A4 Results of tension-torsion tests for CuZn37 brass

108

,......., N

E E

.......... :z .........

Appendix A

800~----~----~------~----,-----~

I C22 I I 1

..... ~ ........ ___ J ..... 1 ........................ .J .... ~L- ! --······--··········· .. .1. ............................... . 600 I j I "'-~+ 0 0• 0 o ~ I i O.q.q.1:i· 1° l i . ~IJ . . .

I i ~ i T""" "I 0! 0 0 0

I 9/f(_f i I i i ..... l ........... ~~"-"·+····l---·····) ..... 1.. ..................... j i

: ~+ I : I : tensio~ after torsion: b 400 (/) (/)

~ -(/) :J: 0

G:

<J>0 j I ~ I + &0 =0.020 I ! I ! I A Ëo= 0.120

_,.L ..... -~-+-.l·---··------f---L-··-··-··--· X t = 0 220 I ; I i I 0 "

l 1 i 1 o torsion [ I i I

! I I I 0+-~~--~~---+~~--~--~--+---~~

o tension

200

0.0 0.1 0.2 0.3 0.4 0.5


Figure A.5 Examples of flow curves tor C22 steel: torsion-tension


........ N

E E

............. z: ..__.

800~------------~----~----~------, lt .

600

I ~ I Armco d ;

I d I ,._.J, ___ " ........... ..IJ ............................. _-f.. ... l ............................ + ......................... --.f.. ........ __ ", ................ J

I :1 ~.;i I ll

I li !~ b 400 _"l.. ..................... A.-... - ............ ~ná<~<· ............. _. o o o o o ..

1 ~l .. ~!!,-E>'f<tt 0 terfsion otter torsion: lf) lf)

e -lf) I ;tjdé~-l - 0 020 I +++b'IQ j I + :o = · 1 ~tfa0 ·~ 1 ' ~ :o= 0.095 .... ~ .. --.. ~r ............................... r-7-----· x t

0 = o.219

1 ,j l 1 o torsion !Ï Ï I 1Î j 1 o tension

200

! i 0+-~~~~-----+~----~--~--~--~~

0.0 0.1 0.2 0.3 0.4 0.5

Equivalent strain ê [-]

Figure A6 Examples of flow curves tor Almco steel: torsion-fension

109

110 Appendix A

111

Appendix B

Error Analyses for the Standard Materlal Tests

An accurate determination of the flow curve trom a material test requires a precise

maasurement of the quantities, involved in the calculation of stress and strain. Topic ·

of this appendix is the extent to which the errors of maasurement influence the

accuracy of the flow curve. Error analyses will be presented for the standard

compression, torsion and tension test, as employed in chapter 4.

These calculations are based on a previous study by Wiegels and Herbertz,

concerning the measuring accuracy in the compression test [8.1]. In the present work,

a similar approach is adopted. This will be explained first.

PROCEDURE

Suppose, a variabief is a tunetion of saveral (experimentally determined) quantities ~:

(8.1)

The experimental errors óX; = óx1, óx

2, ... , óx", of a systematic nature, then determine

the compound error ó1 in the result as:

ót='Ë~·óx.=~·óx +~·óx + ... +~·óx • 1=1 axj I ax, 1 ax2 2 élXn n

(8.2)

according to the rule of error propagation. Gonsidaring the errors óX; not to be

interdependent, an upper error limit is obtained by replacing the individual contributions

by their absolute values.

For a particular material test, the errors in the determination of the equivalent

strain ë and the flow stress O't thus can be assessed by application of this methad to

the respective calculating formulas. Th is results in two distinct errors: one tor the strain,

the ether for the stress. These separate results are combined to obtain an overall error

112 Appendix B

in the flow curve. The error in the strain therefore is converted into an additional error

in the stress, in aceordenee with:

aaf Óuf(óë) = ai 'Ói. (8.3}

To quantify this expression, the strain-hardening behaviour is assumed to be the

Hollornon behaviour (a1=Ci'1, which yields for the additional error:

n ó (ó-) -a1• ·óO't e - e e· (8.4)

In this manner, the total error in the flow stress is composed of a direct contribution

trom the stress formula and an indirect contribution trom the strain formula.

The procedure now is the following. For each standard test, an expression is

derived for the total error limits in the flow curve. These depend on the experimental

quantities (loads, dimensions, ... ) and their accompanying errors of maasurement Next,

an assumed flow behaviour is used to simulate the experiment; that is, experimental

quantities are calculated, utilising reversed stress and strain formulas. This

•experimental" intermation then serves as an input for the error forrnula, together with

representative data on the accuracy of the testing equipment. Thus, it is possible to

estimate the error limits in the flow curve as a tunetion of the equivalent strain.

Results for the three standard tests are illustrated, using actual data on the flow

behaviour of Armco steel as presumed inforrnation for the simulations.

COMPRESSION

Measured quantities in the compression test are the compressive force F c and the

specimen diameter d. Forrnulas for the calculation of strain and stress:

- d Fe t =2·1n(-) and Ut=--,

do !!·d2 (8.5)

4

in which d0

denotes the original diameter of the specimen. From these, it is possible

to derive the total error limits in the result as:

(8.6)

Error Analyses lor the Standard Material Tests 113

Expertmental quantilies (d and F0) are simulated tor the assumed flow

behaviour, using the lormulas (6.5) reversely; the original diameter of the present

specimens is d0~14 mm.

The inaccuracy of the loadcell lor measuring the compressive force is

represented by lóF I =1000 N, corresponding with 0.1 %of !he maximum rated laad. ' Diameters were measured by photographical registration; a representative value lor the

maximum error is I.Sd I ~0.1 mm.

For this input, !he estimated error in !he flow curve is visualised in ligure B. 1,

The lelt graph in !he ligure shows !he assumed flow curve, tagether with its upper and

lower error limits. The other graph shows !he corresponding total relativa error, and !he

distinct proportions which cantribiJle to this. From this graph ~ is obvious, that the

accuracy of !he resutt impraves with strain. The "strain" • proportion dominatas at smal I

strains, which is conneeled with !he Sleep slope of the flow curve in this range.

800 10 simulotion 0 totol

~ C=595 [N/mm2J ~

8 !''O!>ortions: N ~ E n=0.25 ( -] 1 force E 600 .

..._., ~

2 diameter .!S 3 strain b

«> b 400 5 ~ ~

~ li; 4 "' .::,

.~ ., ., 200 0 0 .,

2 c;: Ct: 0 2 1

0 0 3 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Equivalent strain • [ -] Equivalent s\roin < [-]

Figure B. 1 Absolute and rela/Jve lnaccuracy In the oompress/on flow curve

114 Appendix B

TORSION

Differentlal torsional testing involves the maasurement of the twisting angle 8 and the

accompanying moments M1 and M2 tor the two specimens. Strain and stress are

calculated by an approximative method from:

- 1 A0 1 +Ro2 3./3 MrM1 e =--· ·8 and u1=--· ,

2{3 Lo 2rr Ro23-Ro13 (B.7)

8 here is expressed in radians. R01 and Ro2 stand tor !he respective specimen radii,

L0 denotes the gauge length. Using these forrnulas, the total error limits can be derived

to be:

(B.8)

The twisting angle is simuialed by reversing the strain torrnula in (8. 7);

accompanying moments are deduced tor the assumed flow behaviour by solving the

inlegral equation tor the shear stress over the spoomen's cross-sectionel area (B.2],

which yields:

M 211·C ( 8 )" 1 R n+3 12=--·-- ·--· o1o2 ' · {3 La·f3 n+3 '

(8.9)

as a mathemetically exact solution. Nomina! dimensions of the present specimens are:

La=SO mm, Ra1 =3.2 mm and R02 =4.0 mm ..

The inaccuracy of !he equipment lor torsional testing is represented by

lóM,I = lóM,I =200 Nmm lor the loadcell, and lóol =0.05·11 rad lor the angle gauge.

The estimated error in the flow curve, resutting from this input, is reproduced in

figure 8,2. At small strains the accuracy is poor, due to a considerable relativa

inaccuracy in both the twisting moment and the twisting angle. Except tor the

proportions of moment and twist, !he figure includes an add~ional proportion - !he

"formula"-proportion - which quantifies the error in !he applied forrnulas (8.7). This

arro~ originates from the assumption lhat lhe shear stress is constant in the differential

layer. As compared to the experimental inaccuracy, the error resutting from this

assumption shows to be inappreciable!

Error Analyses lor !he Standard Material Tests 115

800 10 ' simulotion 0 toto!

~

N

E 600 E

' 2E.,

.;- 400 "' "' ... ~

-:;;

"' 200 0

G::

C"4 70 {N/mm2]

" n=0.21 H -i !

I -V ~

0

~ 8

--€ 6 .;-

'0

t ~ ~ 4 "' ., > :g v 2 "'

prapartions: 1 moment

2 twist

3 lormulo

\ ,""-['--.. 0 '...,...,..._

I , ...

~> 0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Equivalent stro in ë [ -] Equivalent strain f [ -)

Flgure B.2 Absolute aná relativa inaccuracy in the torsion flow curve

TENSION

In tensile tesUng, the Bridgman methad is used. This requires the meas;;rement of the

tensile force F1 and the specimen diameter d at !he smallest cross"section, as well as

!he profile radius p in the neck of !he specimen (during necking). Formulas lor !he

calculation of strain and stress are:

- d t = -2·1n(-) and u1 do

F, C . wh' h C 1 --· 8 , 1n IC s==----,---'---,-::_.d2 (1 +-4·_p)'ln(1 +-d-l 4 d 4~

(8.10)

The angina! diameter of !he specimen is indicated by d0 . C9 is !he Bridgman correction

factor, which accounts lor !he deviation of !he uni-axial stress state in the necking

range. Derived trom these formulas, !he total error limits are:

116 B

(B. 11)

The si mulation of the three quantlties (F1, d, p) from the available formules (B. 1 0)

is possible only by introduetion of an additionallink. Therefore an empirica! reletienship

is utilised, alter [B.3]:

p= d ,fore>n(neckingrange). 2·/i-n

(B.12)

Though this is a rough approximation, n satisfies tor this purpose. The original diameter

of the present specimens is d0 = 7 mm.

Concerning the tensile force, the inaccurecy of the loadcell is represented by

1 óF 1 = 150 N. A representetive value for the inaccuracy in the diameter maasurement ' is [ódl =0.01 mm. Profile radii ware measured on a profile projector; the maximum

error hers is estimated to be 1 óp I =0, 15 p.

The resuk tor this input is shown in figure B.3. The distinct proportions add up

to a total relativa error in the flow curve, which deelinas as a lunetion of strain in the

un~orm range, up to a minimum, and subsequently increases during necking. This

increase is mainly caused by the additional contobution of the profile radius.

CONCLUDING REMAAKS

The total errors in the flow curves, as presenled in this appendix, are conservative

estimations tor the actual errors in the experiments. This basically originetas trom the

assumption that all individual errors enhance. ti Furthermore, the inaccuracy of the

measuring davlees is represented by an upper lim~. n is nacognised, on the other

hand, that the analyses do nottake erroneous measurements in the original specimen

geometry into account; these provide the resuH with a lurther inaccuracy.

1l1n particular, thls may not hold ft a quanUty Is ioduded in bOth the straln end the stress formula.

Error Analyses tor the Standard Material Tests 117

800 10 .----,-----.,-..,.--,-----,

600

simulation

C=560 [N/mmZ] n=0.29 [-]

15 2oo tr-t-+·······-+--+-·-·· 1 c;:

0 1~-+-~ ....... -+-~r--.--4 0.0 0.2 0.4 0.6 0.8 1.0

Equivalent strain i' [ -]

0 total

proportions: 1 force 2 diameter 3 straîn 4 radius

Equivalent strain i [ -]

F/aure 8.3 Absolute and relative inaccuracy in the rension flow CUNe

In the interpretation of tha flow curves lrom chapter 4, in conclusion, one has

to consider an inaCCIJracy of some percents - due 10 the errors of measurement.

Special reseJValion, anyhow, is to be made tor tha range of very small strains.

REFERENCE UST

[8.1] Wiegels H., Herbertz R.,

'EinfluB der MeBgenauigke~ von Kraft und Weg au! die Unsicherheit bei der

FlieBkurvenerrnittlung im Zylinderstauchversuch',

Sta/11 und esen 1QQ (1980): 1548-1552.

118 --····---------

(8.2] Ramaakers J.A.H., Houtackers LJ.A., Peeters P.8.G.,

PlastiSCh bewerken van metalen: Procesbeheersing in de

onderdelen fabrikage,


(8.3] EI-Magd E.,

'Ermitllung der Flie8kurve im Zugversucho,

Archiv fûr das Eisenhüttenwesen § (1974): 83-89.

Appendix 8

119

Appendix C

Backward Can Extrusion: Stress and Strain Analysis

In this appendix the reader will find the edditional information on !he model of

backward can extrusion, to reconstruot the results of chapter 5.

Although !he analyses of stress end strain are based on different analytical

methods, they use the same three-zone representation. This representation and the

further model assumptions are already introduced in chapter 5.

STRESS AHALYSIS

The calculation of stresses is based on a slab analysis. This so-called double

compression model - in a somewhat different version - originally is proposed by

Dipper [C.1]. The analysis in the current version is borrowed trom (C.2]. In figure C.1,

the situation is drawn schematically. Deductions can be summarised as follows.

• Zone 111 (the walll

In this representa~on, the ascending wallis assumed to betree of stresses; the

principal stress components equal zero:

(C.1)

• Zone 11 Bhe corner)

The restrietion to thin-walled produels implies that a plane-strain situation can

be presumed in the corner zone:

~e =0. (C.2)

Considering !he Levy-von Mises equations, lh1s means tor !he circumferential stress

component:

120

T T ~~'i ti.. .·A··

1----Rc----'

•zone m··

r.%i ~{me.:-

' n~

Flgure C.1 Allalysis of stresses tor the three-zone representation

1 "o =2·(u,•u,),

whieh transfarms the von Mises yield criterion into:

2 !u,-u,l =-·u0 •

13

Appendix C

(C.3)

(C.4)

The equilibrium of axiaJ lorces on a slab of incremental dimensions dz and d8

(figure C.1) is formularised by:

(u2 +du2)·r·d9·(Rc-RR) -u2·r·d8·(Rc-RR) (rmax•'Frl·r·d8·dz =0. (C.5)

Raarrangement simplifies this expression to:

dz du2 =(rmax+Tfrl·---.

Re-RA (C.6)

Substitution of the friction model, foliowed by inlegration {bouodary condition:

(u2}2 .T=O) yields as a solu1ion for the axial stress component:

Backward Can Extrusion: Stress and Strain Analysis

from which the ether stress components can ba derived to ba:

and

• Zone I (the botterol

2 (u,)u = -- ·uo ra

121

(C.7)

(C.S)

(C.9)

The botlom zone is considered to be subjeoted to pure compression. For the

strain components in radial and clrcumlerential direction, it means that these equal

each other:

er ""ee. (C.10)

Then ij fellows trom the Levy-von Mises equations:

(C.11)

which simpillies the yield criterion to:

iuz-Url ""o· (C.12)

The equilibrium of radial lorces on a slab of incremental dimensions dr and d9

(figure C.1) can be expressed as:

(u,+du,)·(r+dr)·d8·T -u,-r·d9·T -2·rFrr·d9·dr -2·u9 • d: ·dr·T =0. (C.13)

This can ba rearranged to oblain a first order differentiel equation lor the radial stress

component:

2·TFr du,=-r·dr. (C.14)

To solve this equation, the global equilibrium of radial lorces on the discontinuity

surfac~ ::a:ween 'he zones I and !I is considered. Departing trom !he known radial

stress in zone 11, the average stress on this surface is:

122 Appendix c

- 2 1 +m T (u,)r=R =--·(1 +--·---)·uo,

R .f3 4 Rc-RR (C.15)

which serves as a boundary condition tor the radial stress in zone I. The salution to the

differential equation then is:

2 1+m T 2·m RR-r (u,)1 = --·(1 + --·---)·u0 --·-- ·u0 .

/3 4 Rc-RR /3 T (C.16)

From this point, it is one fin al step to derive the ether stress components:

2 1+m T 2·m RR-r (uz)1 = -u0 --·(1 +--·---)·u0 --·--·u0 ,

/3 4 Rc-RR /3 T (C.17)

and

(C.18)

An expression tor the average ram pressure pA arrives trom the integration of the axial

stress component in zone I over the ram's frontal area, according to:

2" AR

PR"""RR2 =- r I (uz)l·r·dr·d9. tl=0 r~

The resuning formula reads:

STRAIN ANALYSIS

2 1 +m T 2·m RR PR = (1 +- + --·---+--·-)·uo.

/3 2/3 Rc-RR 3/3 T

(C.19)

(C.20)

The calculation of strains is based on a power analysis. This analysis departs trom the

assumption of a kinematically admissible velocity field to describe the flow of the

workpiece material. For the proposed three-zone representation, the next field is used

as an approximation tor thin-walled cans:

Backward Can Extrusion: Stress and Strain Analysis 123

(û,)u RA • Rc-r ·u (u ),,-o (u l RA . z ·u 2·(Rc-RRl T- ' 8 - ' z 11 2·(Rc-RR) T ' (C.21)

R and (û,)111 =0, (u8)111 = 0, (û2) 111 = A ·û,

2·(Rc-RR)

in whieh û representS !he velocity of !he ram. Departing lrom such a velocity field, tt is

an established procedure to derive !he total power consumption P required by !he

process (C.3]. This involves an inlegration of specific power terms over tine workpiece

body, reprasenting internel deformation, shearing, and friction. For this case, !he

u~imate resutt as an approximation tor thin-walled produels is (C.4]:

(C.22)

The assumed velocity field can be used also to derive a tormulation on the flow

lines and !he detormations. Since the tuil deductions are rather elaborate, this will be

presenled by an outline of the procedure foliowed by a summary of !he final resutts.

Flow lines trace the flow of a material point during tine process. These flow lines

can be obtained trom !he velocity field by inlegration of the velocity components over

!he time [C.5]. Thus, !he posrtion of a point of orlginal coordinates (r 0 , zo> can be

foliowed in dependenee on the current posrtion of the ram. Accompanying strains also

are determined by !he velocity field. Essentially, tinese are derived by inlegration of the

strain-rate formulas, using !he flow-line equations to describe !he course of !he point;

in addition, shearlng deformations are cataulaled tor the passages lhrough !he r 1- and

r 2-surface (C.5]. Depending on the flow line, a material point experiences a eertsin

combination of these individual strain componentS.

As tor the resulting strain distribution, the areas of different "history• in the

longrtudinal sectien of the workpiece are visualised in figure C.2. A material point starts

either in zone I (srtuation a) or in zone 11 (situation b). At a praeeed of the process, the

point moves through one or more different zones. Relevant results are summarlsed in

the next listing.

124 Appendix C

T

'----Re---<

Fiaure C.2 Aroos of different straining histoty for the three-zone representatlon

• snuation a: origine! posnion in zone 1 10 <r ~

The ram posmans at whieh the material point will reaeh !he r 1- and r2-surlace

are appointed by:

(C.23)

respectively. Depending on the actual ram posrtion, !he point finels nseK in zone I (case

a1), zone 11 (case a2), or zone 111 (case a3). The current position (r, z) and !he

accompanying equivalent strain ii which the point has experienced are obtained from

!he following algorithm.

11 T> T1 (case a1):

(C.24)

Backward Can Extrusion: stress and strain Analysis 125

(C.25)

lf T sT2 (case a3):

RR RR T2 RR r83 = Rc-(Rc-RAJ-<-)exp[--]'(-)exp[ ].

r 0 Re-AR T 0 2·(Rc-RR)

r0 RA T0 RA RA (C.26) z83 = Z0·(-)exp[--+2]·(-)exp[ ] + ·(T2 -T),

RA Re-RA T 2 2·(Rc-RA) 2·(Rc-AR)

• Situatian b· orîgtnal oostt1on in zone U fRR~

The ram posijion at which !he material point will reach !he r 2-surtace is

appoinled by:

(C.27)

In !he course of !he process. !he point is sijuated either in zone 11 (case b2) or in zone

111 (case b3). The next algorithm applies.

lf T> T 2 (case b2):

(C.28)

lf T ;,;T2 (case b3):

126

REFERENCE UST

[C.1 J Dipper M .•

"Das Fliel3pressen von Hülsen in Rechnung und Versuch",

Archiv für das Eisenhültenwesen &!) (1949): 275-286.

[C.2] Ramaakers J.A.H., Houteekers LJA, Peeters P.B.G.,

Plastisch bewerlren van metalen: Procesbeheersing In de

onderdelenfabrikage,


[C.3] Avitzur 8.,

Metal Forming: The Application oi Limit Analysis,

Marcel Dekker, New York - Base! (1980).

[C.4] Ramaakers J.A.H.,

Hulsextrus/a 1: Berekeningen,

Appendix C

(C.29)

internat report WPT 0534, Eindhoven Univarsity of Technology (1982).

[C.5] Hocgenboom S.M.,

unpublished notes, Eindhoven University of Technology (1991).

Curriculum Vitae

Wim Sillekens was born on April26, 1963, in Herten, the Netherlands. He attended the

Rijksscholengemeenschap in Roermond, the Netherlands, where he obtained his

Atheneum-S diploma in 1981.

After that he entered Eindhoven Univarsity of Technology, Eindhoven, the

Netherlands, to begin his study in Machanical Engineering. He received his Master's

dagree in December 1987. lmmediately afterwards he was engaged by the Faculty of

Mechanica! Engineering at Eindhoven Univarsity of Technology, where he started his

doctoral studies in the field of forming technology.

Stellingen

behorende bij het proefschrift

Backward Can Extrusion and Materials Behavlour van W.H. Sillekens

Bndhoven,5]uni 1992

Oe duetiele-faalkromme is een technisch bruikbaar concept om de ductiliteit

van een werkstukmateriaal afhankelijk van de bewerkingstemperatuur vast te

leggen.

- Hoofdstukken 3 en 5 van dit proefschrift

- Stelling 1 behorende bij hl3t proefschrift "Prediction of Ductile

Failure" van P.J. Bolt

2 Voor de toekomstig vereiste voorspeiUng~Jkracht van processimulaties mogen

rekweg-effecten op het spanning-rek-gedrag van metalen niet worden

verwaarloosd.

- Hoofdstuk 4 van dit proefschrift

3 De traditionele beoordeling door gebruikers van een materiaal aan de hand

van de chemische samenstelling loochent de recente ontwikkelingen in de

materiaalkunde en is derhalve aan te merken als anachronistisch.

4 Het gebruik van termen als "true stress" en ''true strain" in de

omvormtechniek is te suggestief en daardoor misleidend.

5 Het ontbreken van een natuurwetenSöhappelijke onderbouwing van het

bovengrenstheorema in studieboeken is didactisch onaanvaardbaar.

6 De analytische benadering van omvormproblemen is door de opkomst van

numerieke methoden niet overbodig geworden.

7 Bij "make or buy"-besfissingen wordt doorgaans niet verder gekeken dan de

neus lang is.

8 De beoordeling van wetenschappers aan de hand van een citatie-index is

onjuist omdat hierin de afzonderlijke aspecten van kwaliteit en omstredenheld

zijn l:lflgrepen.

9 De opvatting dat technologen ook sociaal geschoold moeten zijn blijkt van

een zekere eenzijdigheid als men het gapruik van technische

verworv(:!nheden door sociologen beziet.

1 0 In tegenstelling tot het klassieke gezegde •oe gustibus non ast disputandum•

mag èn moet er zelfs over smaak worden gedisputeerd.

11 Het ware gezicht van het Limburgse en Brabantse carnaval gaat voor de

Bovenmoerdijker verloren door het masker van de Hilversumse

carnavalskraker.

12 De beste stellingen zijn niet waar.

13 In afwijking van wat gedacht wordt, werken magnetrons vaak voornamelijk

aan het oppervlak.

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