[K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)

5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)

1

Berichte aus

dem

Institut fur Umformtechnik

der

Universitiit Stuttgart

Herausgeber: Prof. Dr.-Ing. K. Lange

85


2

Simulation

of

Metal Forming Processes

by the Finite Element Method

(SIMOP-I)

Proceedings of the I. International Workshop

Stuttgart, June 3, 1985

Springer-Verlag

Berlin Heidelberg New York Tokyo 1986


3

Dr.-Ing. Kurt

Lange

o. Professor an

der

Universitiit Stuttgart

Institut

fOr

Umformtech nik

ISBN-13:978-3-540-16592-7

001: 10.1007/978-3-642-82810-2

e-ISBN-13:978-3-642-8281 0-2

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4

PREFACE

The production-costs of

formed

workpieces are in an increasing extent fixed

through the costs for designing

and

manufacturing the

tools.

Nowadays,

i t

is

possi b1e

to

reduce these redundant

tool-costs by

app

lyi ng modern numeri

ca 1

simulation techniques such as the finite element type procedures.

In

thi s context, the basic

ojecti

ve of the

workshop

SUtoP ( ~ i m u l ation of

: etal F ~ r m i n g

~ r o c e s s e s

by the

Finite

Element

Method) was

to determine

and -

especially

-

to

discuss the level of finite-element-simulations of

metal-forming processes with regard to technological utilization.

On this purpose,

eight presentations have

been selected to focus the

discussions onto the prime aspects such as:

- technological aspects (bulk metal forming versus sheet metal forming),

- constitutive laws (rigid-plastic

versus elastic-plastic versus visko-plas-

tic material laws),

- coupled analysis (thermo-mechanical coupling),

- kinematical description (Eulerian versus Lagrangian formulations,

co-rota-

tional formulations etc.),

- numerical problems

(incompressibility, integration

of

constitutive

equa-

tions, iterative and incremental schemas,

etc.),

as

well

as

- contact problems (friction, heat-transfer,

etc.).

In

order to promote

discussions,

the audience of the

workshop

was

limited to

50 participants. Due to this

fact,

we

had to refuse unfortunately

many

app

1

cat ions. However, we hope that

these proceedi

ngs

-

whi

ch also inc 1

ude

the discussions in an almost complete extent - will be a compensation for

those

who

could not attend the workshop

SIMOP-I.

The

proceedings contain the

eight

written manuscripts, the discussions after

each sub-session as well as the closing

discussions,

the "FORUM", at the

end

of the workshop.

Finally, as the organizers

we wish

to thank very deeply the Stiftung

Volkswagenwerk, Hannover,

for

the

financial

support of

this

workshop.

August 1985

Kurt Lange, A.Erman

Tekkaya


5

THE

SIMOP-PARTICIPANTS

(Numbers

in

the figure correspond

to the names

in

the l ist of

part icipants)


6

- 7 -

LIST OF PARTICIPANTS

(Names

of the authors

who

presented

the

papers are underlined)

1. Altan, T.,

Dr.

2. Argyris, J.H., em.Prof .Dr.Dr.h.c.mult.

3. Braun-Angott, P., Dr.-lng.

4. Dannenmann, E., Dipl.-lng.

5.

Doege,

E.,

Prof. Dr.-lng.

6. Dohmann, F., Prof. Dr.-lng.

7. Doltsinis, J.St., Dr.-lng.

S. Du, G.,

SSe.

(Eng.)

9.

Dung,

N.L., Dr.-lng.

Battelle

Columbus Laboratories

Engineering

and

Manufact. Techn.

Department

505 King Avenue

Columbus, Ohio 43201

/

USA

lnstitut

f. Computer-Anwendungen

Universitat Stuttgart

Pfaffenwaldring

27

7000 Stuttgart 80

Betriebsforschungsinstitut

des VDEh

Sohnstr.

65

4000

DUsseldorf

lnstitut

fUr Umformtechnik


Holzgartenstr. 17

7000 Stuttgart 1

lnstitut

fUr

Umformtechnik

und

Umformmaschinen

(lfUM)

Universitat Hannover

Welfengarten lA

3000

Hannover

1

Univ.-Gesamthochschule-Paderborn

Fachbereich 10 -Maschinentechnik

l

Umformende Fertigungsverfahren

Pohlweg

47-49, Postfach 1621

4790

Paderborn

Institut fUr

Computer-Anwendungen

Universitat

Stuttgart

Pfaffenwaldring 27

7000

Stuttgart SO

Shanghai Tiao Tang University

Shanghai /

PR

China

Presently at:

lnstitut

fUr Umformteehnik


Holzgartenstr. 17

7000

Stuttgart 1

Arbeitsbereich Meeresteehnik

- Strukturmeehanik -

Teehn.Univ. Hamburg-Harburg

EiBendorfer Str. 38, Postfaeh 901403

2100

Hamburg

90


7

- 8 -

10.EL-Magd,

E., Prof. Dr.-Ing.

11.Forschner, A., Dr.

12. Fugger, B.,

Dr.

- Ing.

13.Gerhardt, J. , Dipl.-Ing.

14.Grieger, I . , Dr.-Ing.

15. Hansen, R., Dipl.-Ing.

16. Hart 1ey, P., Dr.

17. Herrmann, M., Dipl.-Ing.

18. Hirt, G., Dipl.-Ing.

19.Hopf,

S., Dipl.-Ing.

20.Horlacher, U., Dipl.-Ing.

Lehrgebiet

fUr

Werkstoffkunde

RWTH

Aachen

Augustinerbach 4

5100 Aachen

Stiftung

Volkswagenwerk

Postfach

81

05 09

3000

Hannover

81

Daimler-Benz

AG

Werk Sindelfingen, Abt. WZE

Postfach

226

7032

Sindelfingen

Institut fUr Umformtechnik

Universitat

Stuttgart

Holzgartenstr.

17

7000

Stuttgart 1

Institut f. Statik und Dynamik

(ISD)


Pfaffenwaldring 27

7000

Stuttgart 80

AUDI AG, PKP

Postfach 220

8070

Ingolstadt

Dept. of Mechanical Engineering

The

University of

Birmingham

South West Campus, P.O. Box 363

Birmingham B15

2TT

/

GREAT BRITAIN

Institut

fUr

Umformtechnik


Holzgartenstr.

17

7000 Stuttgart 1

Institut f. Bildsame

Formgebung

RWTH Aachen

Intzestr. 10

5100

Aachen

Daimler-Benz AG

Werk Sindelfingen,CAD/CAM-Entwicklung

Postfach

226

7032

Sindelfingen

Institut fUr

Umformtechnik


Holzgartenstr.

17

7000

Stuttgart

1


8/

- 9 -

21.Jucker,

J. , Dr.-lng.

22.Jung, G.,

Dipl.-lng.

23.Kanetake, N., Dr.

24.Keck, P. (Student)

25.Konig, W.,

Dipl.-lng.

26.Lange, K., Prof.

Dr.-lng.

27.L ipowsky, H.-J., Dipl.-lng.

28.Luginsland, J.,

Dipl.-lng.

29.Mahrenholtz, 0.,

Prof.Dr.-Ing.

30.Mareczek, G., Dr.-lng.

31.Marten, J.,

Dipl.-lng.

Daimler-Benz

AG

Werk

Sindelfingen

Postfach 226

7032

Sindelfingen

Daimler-Benz AG

Abt. Verfahrensentwicklung

Mercedesstr.

136

7000 Stgt.

60 - UntertUrkheim

Nagoya University

Nagoya / JAPAN

Presently at:

lnstitut

fUr Umformtechnik

Universitat

Stuttgart

Holzgartenstr.

17

7000

Stuttgart 1

Universitat

Stuttgart

Lehrstuhl f. Fertigungstechnologie

Friedrich-Alexander-Universitat

Erlangen-NUrnberg

Egerlandstr. 11, Postfach

3429

8520 Erlangen

lnstitut

fUr Umformtechnik

Universitat

Stuttgart

Holzgartenstr. 17

7000

Stuttgart

1

AUDl AG, EGA

Postfach 220

8070

lngolstadt

lnstitut fUr Computer-Anwendungen

Universitat

Stuttgart

Pfaffenwaldring

27

7000

Stuttgart

80

Arbeitsbereich Meerestechnik

- Strukturmechanik -

TU Hamburg-Harburg

EiBendorfer Str. 38, Postfach 901403

2100

Hamburg

gO

lnstitut fUr Umformtechnik

und

Umformverfahren

(lfUM)

Universitat

Hannover

Welfengarten lA

3000 Hannover

1

Institut fUr Mechanik

Universitat

Hannover

Appe

1

str.

11

3000

Hannover

1


9/

32.Mattiasson, K.,

Dr.

33.Matzenmiller, A., MSc.(Eng.)

34 .Mayer, P., Di

P

1. -

I

ng

.

35.Meier, M.,

Dipl.-lng.

36.Dberlander, Th., Dipl.-lng.

37 .Dnate, E., Prof.

Dr.

38.Pillinger,

I . ,

Dr.

39.pohlandt, K., Dr.-lng.

habil.

40.Ramm, E., Prof. Dr.-Ing.

41

. Ro 11, K.,

Dr.

- Ing.

42.Rowe,

G.W.,

Prof.

Dr.

- 10 -

Dept. of

Structural Mechanics

Chalmers University of Technology

Sven

Hultins Gata 8

S-41296 Goteborg / SCHWEDEN

Institut fUr Baustatik


Pfaffenwaldring 7

7000 Stuttgart

80

Inst.f.Kernenergetik u.Energiesysteme


Pfaffenwaldring 31

7000

Stuttgart 80

lnstitut fUr Umformtechnik

ETH

ZUrich

Sonneggstr. 3

CH-8092

ZUrich

/ SCHWEIZ

lnstitut

fUr

Umformtechnik

Universitat

Stuttgart

Holzgartenstr. 17

7000

Stuttgart 1

Escola Tecnica Superior D'enginyers

de Camins,

Canals

I

Ports

Universitat Politecnica De Barcelona

Jordi Girona Salgado,

31

Barcelona - 34 / SPANlEN


The

University of

Birmingham

South West Campus, P.O. Box 363

Birmingham B15 2TT / GREAT BRITAIN

Institut

fUr Umformtechnik


Holzgartenstr. 17

7000 Stuttgart 1

Institut

fUr Baustatik

Universitat

Stuttgart

Pfaffenwaldring 7

7000 Stuttgart 80

Control Data GmbH.

Marienstr. 11-13

7000

Stuttgart

1


The University of Birmingham

South

West

Campus

P.O. Box 363

Birmingham

B15

2TT / GREAT BRITAIN


1

- 1 1 -

43.Sailer, C.,

Dipl.-Ing.

44.Schoch, F.-W., Dr.-Ing.

45.Schweizerhof, K., Dr.-Ing.

46.Stalmann, A.P., Dr.-Ing.

47.Steck, E., Prof. Dr.-Ing.

48.Sturgess, C.E.N., Dr.

49.Tang, S.C.,

Dr.

50.Tekkaya, A.E., MSc. (Eng.)

51.Traudt, Dr.-Ing.

52.Vu, T.C.,

Dipl.-Ing.

Lehrstuhl A fUr Mechanik

TU

MUnchen

Arcisstr. 21, Postfach

202420

8000 MUnchen 2

Staatliche

MaterialprUfungsanstalt

Universitat

Stuttgart

Pfaffenwaldring

32

7000 Stuttgart 80

Institut fUr Baustatik

Universitat

Stuttgart

Pfaffenwaldring 7

7000

Stuttgart

80

Institut

fUr

Umformtechnik und

Umformmaschinen

(IfUM)

Universitat Hannover

We lfengarten 1A

3000 Hannover 1

Institut f. Allgemeine Mechanik

und

Festigkeitslehre

(Mechanik B)

TU Braunschweig

GauBstr.

14

3300 Braunschweig


The

University of Birmingham

South

West

Campus

P.O. Box 363

Birmingham B15 2TT / GREAT BRITAIN

Ford Motor Company

Metallurgy Dept., S-2065

Scientific Research Labs.

2000 Rotunda

Drive

Dearborn,

Mich. 48121-2053/USA



Holzgartenstr.

17

7000 Stuttgart 1

Univ.-Gesamthochschule-Paderborn

Fachbereich

10

-Maschinentechnik

I

Umformende

Fertigungsverfahren

Pohlweg

47-49, Postfach

1621

4790 Paderborn


Universitat

Stuttgart

Holzgartenstr. 17

7000 Stuttgart

1


1

- l2 -

53.Wang,

N.-M.,

Dr.

54.Weimar, K.,

Dipl.-Ing.

55.Wilhelm,

M.

(Student)

56.WUstenberg, H., Dipl.-Ing.

Ford

Motor

Company

Metallurgy Dept., S-2047

Scientific

Research Labs.

2000

Rotunda

Drive

Dearborn,

Mich.

48121-2053

I

USA

Institut fUr

Baustatik


Pfaffenwaldring 7

7000 Stuttgart 80


Institut fUr Computer-Anwendungen

Universitat

Stuttgart

Pfaffenwaldring

27

7000

Stuttgart

80


1

CON

TEN T S

Opening

Address

K.

Lange

SESSION 1:

BULK METAL FORMING

Session lao Chairman:

E.

Steck

Thermomechanical Analysis of

Metal Forming

Processes

Through the

Combined Approach

FEM/FDM

O.

Mahrenholtz,

C.

Westerling, N.L.

Dung

Finite-Element-Simulation of

Metal Forming

Processes

Using Two

Different Material-Laws

A.E.

Tekkaya,

K.

Roll, J. Gerhardt,

M. Herrmann,

G. Du

Discussions (Session

la)

Session

lb.

Chairman: O. Mahrenholtz

Elastic-Plastic Three-Dimensional Finite-Element

Analysis of Bulk

Metal Forming

Processes

I. Pillinger,

P.

Hartley, C.E.N. Sturgess, G.W. Rowe

Three-Dimensional Thermomechanical Analysis

of

Metal Forming

Processes

J.H. Argyris,

J.St.

Doltsinis,

J.

Luginsland

Discussions (Session lb)

SESSION 2:

SHEET METAL FORMING

Session 2a. Chairman:

E. Ramm

Numerical Simulation of Stretch Forming Processes

K. Mattiasson, A. Melander

Page

15

19

50

86

91

125

161

170


1

- 14 -

Possibilities of the

Finite

Element Viscous Shell

Approach

for

Analysis of Thin Sheet

Metal Forming

Problems

E. Onate, R. Perez

Lama

Discussions (Session 2a)

Session 2b. Chairman: J.H. Argyris

Numerical Simulation of

the

Axisymmetric

Deep-Drawing Process

by

the

FEM

A.P. Stalmann

Applications of

the

Finite-Element-Method to

Sheet Metal Flanging Operations

N.-M. Wang,

S.C.

Tang

Discussions (Session 2b)

FOR U M

Page

214

254

261

279

307

309


1

-

15

-

OPENING

ADDRESS

K. Lange

It

is

my

very pleasure to welcome you in Stuttgart

and

to open this

workshop

on

"Simulation of

Metal

Forming

Processes

by the

Finite-E1ement-Method", or

briefly,

"SIMOP" - SIMOP-I, hoping that others may follow.

Let me

at first try

to explain the basic motivation

for

this workshop:

When I started to deal with meta1forming - this was about 35 years ago - my

colleagues

and

I were

really

proud

to predict

the forming load for a simple

axi symmetri c extrusion process withi n 20% 10% accuracy just in order

to

select the correct press. Although the fundamentals of the theory of

plasticity

were

given through in the

meantime

well-known -

at

that time

newly

pub 1 shed - book by Rodney

Hi

11, the uti 1 zati on of thi s theory was

diminishing1y small, because

by

applying

this

theory,

we ended up

with

impressive hyperbolic differential equations

which we

could not solve,

however, except for very crude assumptions. Therefore, the

theoretical

analysis of metal forming -

even

in the simplified version - was a job

for

highly skilled bright mathematicians but not for the engineer in any

production division of the industry, or even of a university. Hence, during

these years

nobody

could imagine that i t could be possible to

compute

strains and stresses, or

even

flow patterns in a workpiece during the course

of deformat ion, although

e1

ementary theori es such as the slab method had

already

lifted

metal forming technology

from

the blacksmith shop to the

drawing office level

and

hence contributed remarkably to

its

development.

Yet, this situation started to change in the mid-1960's with the

industrial

ut il

zati on

of e 1

ectri

ca 1

comput

i

n9 machi

nes, the soca 11 ed computers. Now,

the equi pment was

gi

ven to solve the di fferenti a1 equati ons without bei ng

necessarily

mathematically

skilled

or

bright. The keyword was

"numerical

methods". With these numerical methods,

which

could be

easily

handled by the

computer,

any differential

equation could be solved regardless of its

toughness.

From

this moment

on

the developments became drastic, in fact, I

would like to call

i t a revolution. This revolution

started

with the

first

attempts to computerize the

slip-line-field

solutions, went over to the


1

- 16 -

finite

difference

solutions and the weighted residual methods, and continued

with

the fi

ni

te

element methods. Pushed forward

by

a

non-stoppi ng

des

ire,

many

successfull

fi

nite-e1 ement-codes have been developed

in

the past

15

years

at universities and research

centers

which could serve as an

analysis

tool for metal forming processes.

Now,

i t is not exceptional anymore that

people using these

codes speak of Almansi or Green-Lagrange

strains,

of

different sorts of

Piola-Kirchhoff-stresses,

of Jaumann rates etc. Even our

faithful true

stress changed

i ts name

and

became the

"Cauchy"

stress.

The euphoria slowed down, however. There are

three

reasons

for

this:

Fi rs t 1

y, the basi

c

theory

of p1ast city of

the

1950' s remai ned

the

same

although many

numerical procedures were

developed.

Being able

to

implement

all

details of

this theory, people started to see the limitations and

shortcomi ngs of

thi

s

theory.

Besi des,

the

uncertai

nti

es in

the

boundary

conditions

started

to

become

the

more delicate weakpoint of

the analysis.

Secondly,

there was no

diffi cul

ty to

grasp that

nature

has so

many

degrees

of freedom, in

fact

too

many

for a

conventional

computer

to

handle

economically.

Thirdly,

to

use a

ready-finite-element

code

in industry, specially trained

engineers were

s t i l l required.

Hence,

the application

of

finite-element-simulation

in metalforming

just

remained an academic exercise, and industrial engineers -

sti l l

utilizing

empirical heuristic

design procedures

- were happy

to know that there exist

some

guys

at the

universities who can predict a priori stresses,

strains

and

flow

patterns

in forming

processes.

In

the

past couple of

years

trends and

feeling

changed

again,

this time

stimulated through

the introduction

of the

new

computer

generation.

Having a

new archi tecture, such as

for the

array process i ng, and showi ng computa

tional

speeds around one giga-flops

(instead

of 10 to 100 mega-flops

for the

conventi

ona 1 computers), the handi cap of not bei ng economi c seems starting

to disappear.

The

speculations

about

intelligent

computers which

are

claimed

to

be in development

in

Japan and

the

States with 10 giga-flops or even

more, as well as the attempts to develop hardware ori ented numeri

cal


1

- 17 -

procedures - as for

instance,

the element by element procedures - are just

strengthening these

beliefs to be

at

the

end

economic.

However, there

will

be

another aspect

coming

up, once the accuracy of computing

strains,

stresses, forces will

be

improved

to errors

of

5%

or

even less:

the lack of

reliable,

broadscale material

data.

As long as

we

will not

know

flow stress

as

prec i se

as we can

compute, our results

wi 11 mai

ntai n a 1arger

scatter.

What we need

in my

opi ni

on

is

another round of determi nati on of p1

ast

i c

properties

of

materials

- metals - taking into account influences of

microstructure

and

microstructural phase transformations as well as of

process parameters such

as strai

ns,

strai

n

rates,

time, temperature.

The

goal

wi

11

be

materi al data banks

wi

th

comprehensi

ve

"constituti

ve

equations"

for

a large

variety

of metals. This

will,

however, become a time

and

money

consuming business but

i t must be

done.

Finally, this new situation has been

the basic motivation

for us to

organize

thi

s workshop. The

aims

herefore are

to di

scuss,

to

determi

ne

and,

even

maybe,

to

evaluate the present level and the trends of finite-element-simu

lations

of metal forming processes with a special emphasis onto the

technological

utilization.

This emphasis onto the technology

is

also the

reason

for

holding this meeting in a technological oriented research center

for

metal forming as this

is

the case

for

our institute.

Now,

for

thi s purpose,

we

wi 11

have

ei ght presentat ions today whi ch wi 11

focus the discussions onto the relevant aspects of the matter. I'm

especially

very glad that

all

of the

scientists we invited

as chairmen and

presenters have accepted our request - although some of them are under heavy

time pressure - so that I want to thank them here again very deeply. Also, I

want

to

thank

all

the participants,

from

whom I expect

that

they will give

valuable contributions through the discussions.

In

this context I

want to

inform you that

we

will record

all

the discussions

in order to pub 1

sh

them together with the written manuscri pts of the

presentations.

The

proceedings will be

available

within 3

to

4 months and

every

participant

will receive a copy. I hope that recording the questions

and

answers

wi 11

not prevent

you

or damp your enthusi

asm to participate

in

the discussions. I believe that the discussions -

especially

for our

workshop

today are at 1

east as

important

and

presentations

which have to

serve in

fact

- as

interesting

as the

said before -

for

stimulating the

discussions.

Furthermore, i t is one of our goals to bring

the contents of the discussions to those who are not oresent here.


1

-

18

-

The

multidisciplinar character

of the simulation of metal forming processes

is

also exhibited in the fields of interest of the

participants.

For

examp1e, there are bes ides academi c and i ndustri a 1 metal formi ng techno 10-

gists, representatives of computer-manufacturers, of pure and applied

mechanics, of material science and of

civil

engineering present. I expect

that this heterogeneous group will be able to discuss the rather complex

matter in nearly every aspect. Futhermore, I

hope that

the metal formi

ng

practicers

will give the pure

theoreticians some inspirations

but also

that

the

theoreti

ci

ans and

the

academi

c staff

can

show the practi cers the merits

of the numerical

analysis.

also

want to

express

my

special

gratitude to Frau Dr.

Forschner

representing the

Stiftung

Volkswagenwerk, Hannover,

who,

with their generous

financial support made this workshop possible.

That

is all that

I

have to

say. I

wish for all

of us a successful meeting.

Thank you

very

much.


1

-

19

-

Thermomechanical

Analysis

of Metal Forming Processes Through

t he Combined

Approach FEM/FDM

Oskar

Mahrenhol tz*, Claus

Wester l ing+ and Nguyen

L.

Dung*

*

s t ru c tu r a l Mechanics Divis ion ,

Technical Univers i ty of Hamburg

Harburg, Hamburg-Harburg, FR

Germany

+ I n s t i t u t e of

Mechanics, Univers i ty

of Hanover,

Hanover,

FR

Germany

Summary

During a

forming

process , the

temperature

of

t he formed pa r t

increases due to

t he

conversion of the

forming energy

and

t he

f r i c t i o n l osses

i n to

heat .

This

causes

the

thermomechanical

behaviour

of

the

mater ia l , i f

t he ma te r i a l

i s

tempera ture

sens i t ive .

The p l a s t i c deformation and the t empera ture

change

are coupled with

each

other ,

hence

it

i s

necessary to

develop

an ef fec t ive and economic method to

achieve

the

coupled

ana lys i s .

In

t h i s

paper ,

the

method,

based

on

the

f i n i t e

element

method

(FEM) for t he

p las t i c deformation and

t he

f i n i t e di f fe rence

method (FDM) for the hea t

t r a n s f e r ,

i s found to be sa t i s fac to ry

for the coupled ana ly s i s . This method

inc ludes

many s impl i f i ed

numerical procedures of

the FEM and the

FDM to

save computa

t i ona l

t ime.

Both

cold and

hot

forming processes

could be

c a l

cu la ted s t ep

by

s tep in

t h i s

way to

obta in the

r e levan t

data

for the des ign of

dies and

manufactur ing t echniques .

I

In t roduc t ion

Most

of the

forming

process so lu t ions a re

developed, for numeri

ca l

s impl ic i ty , with

an assumption

of

q u a s i - s t a t i c

and

i s o

thermal condi t ions . Such a method

i s

genera l ly sa t i s fac to ry

for the

ana lys i s of

s i t u a t i o n s in

which

t he ma te r i a l

i s not

t empera ture - sens i t ive

and

the cold forming processes are per

formed

slowly.

But , in many cases , the convers ion

of

forming

energy

i n to

heat

causes

a

high

t empera ture

grad ien t

dur ing

t he

process .

Then,

t he t empera ture

balancing in

t he workpiece and

t he

hea t

t r a n s f e r

to

t he

surrounding, due to

t he

t empera ture


1

- 20 -

grad ien t between

the

workpiece and t he surrounding , occur un

avoidably .

The

t emperature change i n f luences

the p l a s t i c

flow

of the mate r i a l . The

t emperature

e f f e c t s

must be

taken i n to

account

in

form

of

the

thermomechanical

behaviour of

the

mate

r i a l .

The

approach to

l a rge p l a s t i c deformat ion a t e leva ted tempera

t u r e cons i s t s

ge ne ra l l y of

so lu t ions for

p l a s t i c

deformat ion

and

for

heat t r a n s f e r in t he coupled

manner.

There have

been

many f i n i t e

element

methods

which

were employed for ca lcu la t ion

of

the

forming processes

under

cons ide ra t ion of t he temperature

i n f luence .

Zienkiewicz

e t

a l

/ 1 / have

developed

a

coupled

ana lys i s of

thermomechanical problems

in ext rus ion . Rebelo and

Kobayashi

/ 4 / inco rpora ted

t emperature and

s t r a i n - r a t e

e f f ec t s

in to

a

v i scop la s t i c

t r ea tmen t of

an

axisymmetr ic problem,

while

P i l l i n g e r e t a l / 2 / made the f i r s t

s tage

in

the

development of

a thermomechanical f i n i t e element

ana lys i s for th ree-d imens ional

forming

proces ses .

The

previous

works

t r e a t e d

the

l a rge

p l a s t i c

deformat ion

a t

e leva ted t emperature

using

the f i n i t e

element

t echnique ex

c lus ive ly .

But , if

t he p l a s t i c deformat ion

and

t he

heat t r an s fe r

are ca lcu la ted separa te ly

using

t he f i n i t e di f fe rence

method

for

the hea t t r a n s f e r i ns t ead of FEM, the

computat ional

e f fo r t s

a re l e s s . Also,

l e s s computer core s torage i s necessary . Altan

and Kobayashi / 3 / have

model led

t he heat t r a n s f e r problem

with

cen t r a l d i f f e r ence method t o p red ic t

the

t emperature

d i s t r i

but ion

in

ext rus ion .

A

combined

approach

FEM +

FDM

has been

developed

by

the presen t authors / 7 , 8 / to

study

t empera tu re

e f f e c t s in wire drawing and r i ng compression. In

t he presen t work, t h i s combined approach i s

ex tended

to

the

thermomechanical

ana lys i s of hot forming processes with heated

dies .

Such

a metal forming

t echnique

al lows a lower ing of the

product ion

cos t s .

Thus,

the workpiece i s

not

cooled during the

process , the flow

behaviour

of mate r i a l and

t he compl icated

gap

f i l l i n g

a re promoted. A pre l imina ry comparison

between

the

coupled ana lys i s only wi th

the FEM

and

the coupled

ana lys i s

with

the

FEM

+

FDM i s a l so at tempted .


2

- 21 -

2 F in i t e Element Method

for

P l a s t i c Deformation

The

unsteady forming

processes

a r e ca l cu l a t ed by

means

of

t he

f i n i t e

element

method

us ing t he

r i g id -p l a s t i c

technique . The

FEM

i s

based

on

t he

modif ied

var ia t iona l pr inc ip le

1T =

D

J

y dV + J m ~ . . dv+ J ' [ Ivtl

dS

-

V V II SA

F

( 1 )

The i ncompr es s ib i l i t y

condi t ion

( ~ . . 0)

i s

mainta ined by

II

means

of

t he Lagrangian mu l t i p l i e r

am

which i d e n t i f i e s t he

mean s t r e s s .

The

f r i c t i on s t r e s s ' [ ac t s a n t i p a r a l l e l to

t he

t angen t ia l

ve loc i ty v

t

on t he i n t e r f a c i a l area S ~ between the

die

and

the

workpiece,

whi le

t he sur face

t r a c t i o n

po

ac t s

on

S ~

with ve loc i ty v

k

. The

f r i c t i o n l osses

are

L = j

T l v t l d S

Sa

F

(2 )

The so lu t ions

of

t he var ia t iona l problem (1) are the admiss ib le

ve loc i ty

f i e ld and t he f i e l d

of mean

s t r e s s . The

va r i a t iona l

problem i s then t ransformed

in to

t he f i n i t e element equat ions :

(3 )

o

where R

i s

the vec tor of the nodal

f r i c t i on

forces ; po the

vec tor of

the nodal forces . The

vec tor

~ I

and

Qm

inc lude

the

nodal

v e lo c i t i e s

and mean s t r e s ses .

The matr ix

0 has only zero

e lements .

The FEM

has

two

types

of l i n e a r e lements : t r i angu la r and quadr i

l a t e r a l . In t he t r i a n g u la r e lement , l i n e a r

func t ion

for ve lo

c i t i e s i s used;

t he

mean s t r e s s , yie ld

s t r e s s

Y and s t r a i n

r a t e s

E are cons tan t . The quadr i l a te ra l element

i s

a combination of

two

t r i a n g u la r

elements , where

t he

mean

s t r e s s

and

s t r a i n

ra tes

are a l so assumed to have

t he

same

value

in

each

pa i r

of

t r iangu

l a r

e lements .

The

hypermatr ices

~ o

and fa

are obta ined from t he

compat ib i l i ty

and incompress ib i l i ty

condi t ions .

Because

t he matr ix K

O

i s

a


2

- 22 -

func t ion

of the

ve loc i ty

f i e l d , t he nonl inea r se t of

equations

(3) should be computed

in

an i t e r a t i v e way.

The

unsteady

forming

processes

a r e

s imula ted

by

means

of

an

inc rementa l so lu t ion . The method

analyses t he

l a rge p las t i c

deformat ion by div id ing

it in to

many

quas i - s t a t iona ry small

deformat ion

s teps .

Therefore , the

ve loc i ty and

s t r e s s f i e ld

could be determined s tep by s tep . The mate r i a l flow i s displayed

by

the ve loc i ty

f i e l d

and

t he

displacement of

t he FE mesh

which

i s

updated

a f t e r each deformation

s tep .

The

t echn ica l

d e t a i l s

of

the

FEM

are

publ ished

in the

previous

works / 6 , 8 / and documented

in

t he u s e r ' s manual

of

the

programme

FARM / 5 / .

3 Fin i te Difference Method for Heat

Transfer

3.1 Coupled Analys i s

In

addi t i on

to the ca lcu la t ion

of t he

p l a s t i c deformation,

t he

heat genera t ion and heat t r a n s f e r a r e ana lysed

in

each t ime

increment

to

obta in the

temperature

d i s t r ibu t ion .

During

the

forming processes , the

tempera ture increases

within the

work

piece due to the hea t genera t ion from t he

forming energy

E

= Iv

Y

~ dV

( 4 )

In

the inhomogeneous cases , the f r i c t i o n

losses

(2) cause a

t empera ture grad ien t

on t he i n t e r f a c i a l area

add i t iona l ly .

The

heat genera t ion

in

the workpiece

and

the hea t t r a n s f e r in

work

piece

and

between

the

workpiece

and surrounding

occur

simul

t aneously .

The t empera ture changes

in

each deformation s tep of the

inc re

mental so lu t ion

i n f luence the mechanical behaviour

of the

mate r i a l . Then, t he

mater ia l behaviour

i s updated due

to

the

j u s t

pred ic t ed f i e l d of tempera ture . The

heat

ca l cu l a t i on

method i s based on

t he

FDM which i s modif ied

so

t h a t

it

i s

able to run compara t ive ly in the

programme FARM.


2

I

n

-

G

om

e

r

c

d

a

-

B

y

c

o

o

-

P

c

c

o

a

n

s

-

F

a

n

F

D

m

h

Q

u

-

s

a

o

y

t

r

e

m

e

n

o

p

o

(

F

E

M

)

r

i

-

F

m

s

r

u

u

a

s

f

n

m

a

r

x

J

c

-

_

r

F

m

o

d

m

a

r

x

c

o

d

n

f

c

o

I

T

.

S

v

h

s

e

o

e

q

o

.

S

u

o

V

e

o

y

a

n

m

n

s

r

e

f

e

d

j

C

c

u

a

e

s

r

e

a

n

s

r

a

n

f

e

d

o

d

C

k

b

y

c

o

o

(

c

o

a

p

o

e

m

I

n

L

S

v

a

p

o

m

a

o

I

y

~

1

D

-

L

o

d

o

m

a

o

s

e

p

I

E

N

r

1 1

I

y

C

c

u

a

o

o

e

m

p

a

u

e

I

f

e

d

n

w

o

k

e

o

w

i

h

F

D

M

.

F

m

n

e

n

g

F

c

o

o

e

H

g

n

a

o

hT

=

h

T

+

h

T

=

T

Y

l

,

)

I

_

H

r

a

n

e

T

=

F

o

o

T

z

o

g

o

m

r

y

a

h

t

T

I

T

m

n

e

m

e

n

s

L

I

n

1

I

I I

I

I

I I

I

I

t

-

1

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

-

-

I

~

F

g

1

F

ow

d

a

g

a

m

o

h

m

o

m

e

h

n

c

a

n

y

s

h

o

c

om

b

n

d

a

p

o

h

F

M

/

F

D

M

'

w


2

- 24 -

Fi r s t l y , a f t e r a t ime s tep ~ t

t he mechanical

quan t i t i e s l ike

s t r e s ses ,

forces , disp lacements , e tc . a re pred ic t ed on

the

f ixed

poin t s (nodes , e lements) by t he FEM. Subsequent ly, the

r e su l t ing heat genera t ion i s computed from the diss ipa t ive

forming and f r i c t i o n a l force

fo r the concerning

t ime s tep . The

r e su l t s

of t h i s computat ion show

an

inhomogeneous

temperature

f i e ld with a t ime dependent t empera ture

compensat ion.

The tem

pera ture compensation i s ana lysed for

the

same t ime

s t ep

~ t by

t he

FDM. With the

help

of

t he

y ie ld

s t r e s s Y

=

Y

(E,

I , T), the

coupl ing

on

the

mechanical

behaviour

i s

ensued.

The

procedures

of

the

ca lcu la t ion of p las t i c deformation and

heat t r ans fe r

and

t he

coupled

ana lys i s

are

shown

in

Fig .

1.

For the hea t c a l c u l a t i o n , i t i s necessary to ass ign the element

t empera tures to d i sc r e t e re ference

poin ts .

These re ference

poin t s are the

middle

poin t s of t he f i n i t e e lements . To ca lcu

l a t e t he

temperature a t the

boundary

of

t he

workpiece,

more

addi t iona l t empera ture po in t s are needed a t the bound-

ary: All

boundary elements

have two and

t he

corner

elements

have

th ree

re fe rence

poin t s .

The boundary re fe rence

poin t s a re

placed on the center

of the boundary s ides of t he f i n i t e e lements . For ca l cu l a t i on

of

heat genera t ion and hea t t r a n s f e r , imaginary volumes

are

ass igned

to the

temperature poin t s a t

the boundary. All othe r

t empera ture poin t s ly ing

in

the cen te r

of t he

elements are

a l l o t t e d to the rea l element

volumes.

Fig . 2 shows the meshes

of

FEM

and

FDM

in

case

of

cy l inder upse t t ing .

3.2 The Basic Equat ions

of the

Heat Calcu la t ion

The

t o t a l t empera ture i nc rease ~ T o f each element or

reference

poin t

i s

obta ined from the heat

balance:

~ Q

-

G

~ Q o

~ Q

with

~ Q G

~

+

~ Q u

and

6Q

c ~

6T

(5 )

~ t


2

z

i

L

T i

T

r

T

- 25 -

~

J}

=

I

, ~ / ~

Die

t - - -Workpiece

FE -

mesh

FO- mesh

to

calculate

the

temperatu re fiel

r

Fig. 2: FE

mesh

and

mesh

of

t he

FDM

on workpiece

and die

llTu(81

IH

u

(12)

+ll TR(1

+ll T

R

(


2

- 26 -

lIQ

R

and lIOu a r e hea t f lows due

to

the d i s s i p a t e d f r i c t i o n

and

forming energy ; lIOo t he hea t t r a n s f e r e d

over

the element ; c

t he s p e c i f i c hea t capac i ty ; p t h e dens i ty and V t he volume of

t he

examined

e lement .

The s o l u t i on fo r

h e a t conduct ion

problems shown

in

t he l i t e r a

t u r e

i s ,

in c o n t r a s t to the here r e p r e s e n t e d method, formulated

fo r l o c a l l y f i xed

f i n i t e

d i f f e r e nc e

mesh.

The FD mesh

o f

t he

developed

method i s changed wi th

the f i n i t e

e lement

mesh.

3 .3 Calcu la t ion of Heat Gen era t i o n

The f r i c t i o n

lo s ses

and

the

forming energy

a r e

t rans fo rmed

i n t o

h e a t .

The

tempera ture inc rease

dur ing

a t ime increment i s

c a l

cu la ted as fo l lowing :

The t empe ra tu re

i n c r e a s e

llTR due

to

f r i c t i o n between workpiece

and too l can be given

wi th

and

T

as

FR lis = T llA ~ l i t

l i t

my/ 3

l i s / l i t

=

v

2 m Y llA v l i t

(6

)

fo r

an

area

llA. V means t he volume o f t he f r i c t i o n element

d iv ided

i n t o

two

equal ha lves

on t he workp iece

and

t he

t o o l

s i d e ;

c

w

' c

t

and Pw' P

t

are t he s p e c i f i c hea t c a pa c i t i e s

and

dens i t i e s of t he workp iece and t oo l m a t e r i a l r e s pe c t i ve l y . The

va lue

v

i s

the

s l i d i n g

ve l oc i t y

o f

the

node

a t

the

i n t e r f ace .

The tempera ture i n c r e a s e llTU due to

t he

d i s s i p a t e d forming

energy

i s

c a l c u l a t e d wi th

lIQ

U

= lIWU = n Y

I

lIV

liT = Y l i t

U

n

C

w

P

w

as

(7 )

The f a c t o r n (0 .85 :;; n :;;0.95) i s the thermal e f f i c i e nc y ; I i s t he

e q u i v a l e n t s t r a i n r a t e .


2

- 27 -

The determined

tempera ture

increase

i s ass igned

to the re ference

poin t of the e lement , Fig.

3.

At the i n t e r f ac i a l area , it i s

t he

sum

of 6T

R

and ~ T U .

3.4 Equat ions of Heat Transfer

As

it has been mentioned

ea r l i e r ,

t he developed

equations

of

hea t

t r a n s f e r a r e based on the temporar i ly changing f i n i t e e l e

ments . The

de r iva t ion

of the equat ion of hea t

t r a n s f e r

was

done

not

as usua l by compensat ing t he corresponding

di f fe rences

in

t he d i f f e r e n t i a l equat ion

of

the hea t conduc t ion ,

but

by

es t ab l i sh ing hea t

balances

t o the f i n i t e elements .

The hea t

balance

i s

es t ab l i s hed f o r each element . At

t he

addi

t iona l

r e fe rence po in t s

on t he boundary,

the

hea t

ba lances

es tab l i shed

cons ide r t he hea t conduc t ion and hea t convec t ion

as shown

in Fig. 4.

The s t a r t i n g

poin t

of t he fol lowing

cons idera t ion i s

the hea t

ba lance

equat ion :

By t he use of the

forward

d i f f e r ences ~ T O =

Tgt

- TO ' the

e x p l i c i t equat ion

of hea t ba lance

y ie ld s

from

equat ion (8):

(8 )

(9 )

The

tempera ture

T ~ t

a f t e r

the

t ime

increment 6 t

i s

ca l cu l a t ed

o

due to the hea t t r ans fe r to t he ne ighbour ing elements j . KOj

i s

the fac to r

of hea t t r a n s f e r and

i s given

as

AOJ

k

LOj

in case

of

heat conduct ion

between

i nne r e lements ;

(10

)

(11 )

in case of hea t t r a n s f e r

a t

flow QOj'

it becomes

boundary.

With a

presc r ibed

hea t


2

- 28 -

interior

element

boundary

element

Die l

l ~ ~

I-t-+----i

Air

~

Free

Surface

corner element

-

heat conduction

heat convection

Fig .

4: Scheme of a coupled ana lys i s through

combined

approach

FEM+FDM

~ Fluid/Air

Fig .

5: Contact

sur face

with

t he t empera ture

path

due

to t he hea t t r a n s f e r


2

-

29

-

(12)

Here,

k

i s

hea t conduc t iv i ty

c o e f f i c i e n t ;

a the average hea t

t r a n s f e r

c o e f f i c i e n t on

the

sur face

AOj ; AOj the average su r

face perpendicular

to t he di rec t ion of

hea t flow; and

LOj

the

d i s t ance

between

two tempera ture l eve l s a t 0

and

j .

The

Eq. (9) can be

wri t t en as

T ~ t

o

(13

)

Due to

t he s t a b i l i t y and

convergence condi t ion , the t ime s t ep

~ t

of the

so lu t ion method

has

to

f u l f i l l the fo l lowing equat ion

( 14 )

3.5 Boundary Condi t ions

The

hea t t r a n s f e r due to the convec t ion appears

mainly on

the

f ree boundar ies of the workpiece and t oo l . The hea t f low through

these boundary sur faces

i s

given by

(15

)

The

heat f low q

depends

on the d i f f e rence

between t he

surround

ing tempera ture

Ta

and

t he tempera ture

TR

of

the

boundary

su r

face and

on t he

average

hea t t r a n s f e r c o e f f i c i e n t

na

of the

surrounding

medium

( a i r ) .

Some problems a r e

appear ing

on f ix ing the boundary condi t ion

for the

hea t

t r a n s f e r i n to the d ie , Fig. 5. At the

i n t e r f a c i a l

area , the

l u b r i c a t i o n ,

the con tac t p res sure

and t he ox ida t ion

l ayer a f f e c t the hea t t r a n s f e r in add i t i on

to t he in f luences

o f

tempera ture

and

t he sur face f i n i sh .

The hea t

f low

across

the

area A a t t he i n t e r f a c e between

workpiece

and d ie

i s


2

- 30 -

(16

)

whereby

Tt

and

Tw

a re the su r face t emperatures

of

the d ie

and

workpiece ; ~ the

contac t

conduc tance d a t a

/ 10 / . The

con tac t

conduc tance da ta

i s

chosen under the

assumpt ion

of

an idea l

m a t e r i a l

c on t a c t .

The h e a t t r a n s f e r a t the contac t su r face i s t r e a t e d l i ke the

hea t convec t ion a t

t h e

f ree sur face . Heat r ad i a t i on i s neg lec t ed

in the s o l u t i on method.

The

con tac t conductance

da ta

a

K

i s

a

func t ion of

t h e

c o n t a c t pressure ,

t emperature

and su r face

roughness , but , fo r s i m p l i f i c a t i o n , the va lues

a

K

and

aa a re

assumed to be cons tan t in the c a l c u l a t i o n .

During an uns teady forming proces s , the change of the boundary

cond i t ions due to

the

contac t problem i s

a l s o

checked.

Since

t he re

a re some

nodes

on

the

f r ee su r face

o f

the

workpiece

touch

t h e

d i e

as t h i s

su r face i s bu lg ing

so

much. This causes an

inc rease

of

the i n t e r f a c e

between

d i e and workpiece . such a

con tac t

problem (normal and

f r i c t i o n a l

c on t a c t problem) i s

cons ide red

in

the

s o l u t i on

methods

fo r p l a s t i c deformat ion

and

hea t t r a n s f e r . The

i n c r e a s e

of t he

i n t e r f a c i a l

area

a l s o

means t h e i n c r e a s e

of

the f r i c t i o n lo s ses and

of

the hea t

t r a n s f e r

between

dice and

workpiece .

3.6 Model of FDM for Heat Trans fe r

(Supplement)

The

e lement

fo r hea t ca l cu l a t i on

with

the FDM i s developed as

fo l lowing:

For

elements in t h e i n t e r i o r of the workpiece and the t oo l , t h e

equa t ion

(9)

for hea t conduc t ion changes with

KOj

k

AOj

AOj SOj

"TSOj" Vo AO

11

TAO

11

LOj

to

TLlt

LIt

(

SOj

ITT ,11

(T

j - TO

+

TO

(17 )

a

AO

LOj

J J

The va lue

a

i s

the

the rmal

conduc t iv i ty

(

k/cp

)

.


3

- 31 -

The s t a b i l i t y and convergence condi t ion

becomes

lit

:;;

1

AO

a

l: SOi

liT

.11

i

LOi

J

(18)

In these

equa t ions ,

AO

and

Vo

i nd ica t e the sur face

and

volume

of t he

e lement ;

SOj i s t he s ide

where the hea t

flow goes

through; LOj means the dis tance between two

temperature l eve ls

a t

0 and j ; "TSOj" and "TAO" are

t he

"depths" of the cen te r of

t he edge

s ide

and

of

t he sur face of

an element .

"T "

SOj

{

"1"

2nr

SOj

_ E l ~ ~ e ____

_

axisymmetr ic

as :

itT II

J

"T

"

SOj

"TAO"

The

geometr ica l

dimensions of the preceeding equat ions are

shown

in

Fig. 6.

( 1 9

)

(20 )

(

21)

For boundary elements ,

one

has

in

add i t i on t he

hea t convec t ion:

wi th

the tempera ture T

of

surrounding medium

and

t he

assoc ia ted

depths ,

t he

equa t ion

(9 )

can

be wri t t en

as :

TLlt

l i t

4

SRj

(k l:

"TRj"

(T

j

- T

R

)

R

PRCRA

IR

j=2

L

Rj

(22 )

for

the

tempera ture

T ~ t a t

boundary

poin t

no.

1.


3

-

32

-

FDM

Point - - + - - ~ ' - ?

YIzI =-ro

c - r snO. j . - - - - -1

x/r

FEM: Four-Node

Element

Fig .

6: Geometr ical d imensions for t he hea t

conduct ion between two

ne ighbor ing

poin t s

4

2

Air/Die

3

'''L

/r


3

- 33 -

The assoc ia ted c r i t e r i o n of

convergence i s

given by

1

(k

4

E

j=2

"T "

Rj

In the

above

given

equa t ions , one

has

"T "

Rj

{

~ r

I ~ R ~ - ' p ~ a E ~ - - -- -

axisymmetr ic

(23)

(24)

The geometr ica l dimensions used a r e

i l l u s t r a t ed in

Fig . 7 . The

developed

element of

the

FDM

i s

r e l a t e d

to

t he quadr i l a te ra l

l i n e a r

element o f the FEM

/ 6 / .

The imaginary sur face AIR

i s

se lec ted to

be

a ha l f o f

the

sur face Ao of

the

observed

bound

ary e lement . The

surrounding medium (c ,

p,

a,

T)

i s

e i t h e r a i r

(c

a

'

P

a

' a a '

Ta)

o r too l (c

t

'

P

t

,

at'

T

t

)

i f

t he boundary

poin t

1

of the workpiece

i s

concerned.

The

corner

e lement , wi th s ides

no. 1

and 2 belonging

to t he

f r i c t i on i n t e r f ac i a l area ,

i s shown

in Fig.

8.

In

t h i s case ,

the sur face

and volume of

t he

element are given as :

(25 )

wl

"th { ' : ~ I ___

I 2 . . I ~ . . n ~

___ _

"T

SR

"= 2nr

SR

axisymmetr ic

to i n s e r t

in

Eq. (6) .

3.7 Convergence Condi t ion

The cons ide rab le i n f luence of t he

convergence

c r i t e r i a ,

given

in

Eqs. (9) &

(14) ,

could

be

exp la ined

in

Fig .

9

for

a t e s t

ca lcu la t ion . In a

simple conf igura t ion

of a volume

element ,

the tempera ture of the cen t r a l element

i s

determined

with


3

u

o

90

~

70

E

'"

Jj

-

34

-

E

r-""*,.,ml----,--...}i

' - - - - ' I - - - ' - . . L ~

j:7900

kg/rJ

c :0.477 kJ/kg

\ k:36.0 W/moC

~ S 0 t - - - - t ~ ~ ~ ~ ~ ~ ~ = = - - - - - - - - - - -

"U

"U

~

'0 30

: l

~

'"

10

a.

Convergence Criterion

~ t < 0 . 0 4 1

\

\

\

\

\

O + - ~ ~ ~ ~ ~ ~ - + ~ + - - + ~ ~ - + ~ \ _ + ~ + _ ~

f- 0,05

0.10 0.15

,0.20

-10

t [s] \

Fig.

9: Convergence condi t ion of a t e s t ca l cu l a t i on

us ing the

developed

FDM

1-----90---1

Thermoelement

-j ~ 1 ~ , ~ :

~ . . . , . :

: 7 : : ~ :.,..,.: ~ l

o

1--+----

_--

-----

1

1 - - - - - - - -200 - - - - - - - -1

Geometry:

Mater ia l :

60 x 60 x 200

rnm

C22 Stee l

o

w

; J \

1- - -+ - -+ - ' -

60

Fig . 10: Geometry

of

the t e s t piece

and

t he

pos i t ions

of

t he thermoelements

a t

the c r os s - s ec t i on


3

- 35 -

var ious t ime increment

~ t .

A worse r e s u l t i s obta ined for a

l a rge r t ime inc rement . And,

the re i s a n

o s c i l l a t o r y t empera ture

path pred ic ted , as t he maximal al lowable t ime increment i s

exceeded.

The maximal al lowable t ime

increment i s

r e l a t e d

with

t he

f i n e ~

ness

of t he

FO mesh and

t he

FE

mesh

r e spec t ive ly . It

i s

propor

t i o n a l

to

t he s i z e of

the element , i . e .

a f ine mesh renders

smal l t ime inc rement . The increas ing d i s t o r t i o n

of t he

mesh

could lead to a convergence problem. With regard to the g re a t e s t

poss ib le t ime inc rement , we pre fe r to use the equ i la te ra l tri

angles

and

quadrangles .

By

ca lcu la t ing

the

temperature

d i s t r i

but ion , t he maximal

al lowable

t ime

increment

6 t

i s

es t imated

by the

Eq. (23)

for

t he

smal les t i n t e r i o r element .

The t ime

increment

6 t of the hea t

ca lcu la t ion could

be d i f f e r e n t

to t he

t ime

s t ep of t he p las t i c deformat ion . To i nc rease the

accuracy

of t he

hea t

ca lcu la t ion , a smal le r t ime

s tep i s

chosen

for t h i s ca lcu la t ion . That means the t ime s t ep

of

p las t i c

deformation

i s

normal ly

div ided

in to many

t ime

s teps

in

order

t o p red ic t the tempera ture

f i e l d .

4 Numerical

Resul t s

4.1

Tes t Calcu la t ion

As t e s t example for the developed FD method, the process of

convect ive

cool ing of a q u a s i - i n f i n i t e l y long

rod i s analysed .

The i n i t i a l

tempera ture of the rod i s a t 102S

o

C.

The exper imen

t a l

r e su l t s

of

t h i s

t e s t

are

obta ined

a t

t he

RWTH

Aachen

/11 / .

The

geometry

of

t he

c ros s - sec t ion and

t he pos i t i ons

of the

thermoelements are

given

in Fig . 10. For t he

purpose

of com

par ing, the

tempera tures

a t the survey poin t s 1 and 5 are

determined t heore t i ca l ly with both FEM and FOM. Therefore , the

top

r i g h t quar te r of

the c ros s - sec t iona l area i s

div ided

i n to

36 l inea r

four-node

elements (49 nodes) in the FEM. Accordingly ,

the FO mesh i s

composed of

60

represen ta t ive

nodes (36 middle

nodes

of

t he

f i n i t e

elements

and 24

add i t i ona l

nodes

a t

the

boundar ies) .


3

I

F

S

I

I

~

,

P

n

N

1

,

~

~

,

E

m

e

a

,

F

l

a

u

~

.

a

9

~

c E

~

e

7

2

,

B

1

1

1

I

S

O

B

2

T

m

e

[

s

F

g

1

T

p

e

d

c

e

d

a

n

e

x

m

e

n

a

y

d

e

m

i

n

d

e

m

p

a

u

e

p

h

fo

a

s

h

m

e

T

O

[

9

E

m

e

a

9

F

M

F

N

E

e

m

n

/

1

I

n

e

g

m

t

o

n

P

n

-

9

S

v

P

n

N

.

5

5

6

7

I

I

,

P

n

N

O

.

5

,

1

\

l

O

'

9

,

U

e

B

,

~

'

\

0

.

t

.

'

-

F

D

M

-

.

I

~

.

3

e

~

6

E

~

5

I

-

3

t

s

~

I

2

4

.

-S

8

I

F

6

T

m

e

[

s

F

g

1

T

p

e

d

c

e

d

a

n

e

x

m

e

n

t

a

y

d

e

m

i

n

d

e

m

p

a

u

e

p

h

a

s

u

v

y

p

n

n

S

f

o

a

l

o

m

e

b

h

v

o

,

-

F

g

1

C

m

p

s

o

b

w

e

n

h

e

x

m

e

n

a

a

n

t

h

o

e

c

r

e

u

s

f

o

h

t

h

m

o

e

m

e

n

n

S

r

O

w

(


3

- 37 -

The

ca lcu la t ion

of

hea t t r a n s f e r with the FDM

y ie ld s

a

s t a b i l

and

nonosc i l l a to ry

temperature path if

t he t ime increment i s

se lec ted to be 0.5

sec .

The

FEM

r equ i res a

t ime

increment

t l t

=

0.467

sec .

The

exper imen ta l ly

pred ic ted

and

with

FDM

de

termined

tempera ture

paths a r e plo t ted in Fig. 11 (a t poin t

no.

1) and Fig .

12 (a t

poin t no. 5) for a shor t t ime behaviour.

They show

an

exce l l en t

agreement between t he experiments

and

the FD so lu t ions . But , if

t he t ime

per iod

observed

i s

longer ,

there wi l l be a cons ide rab le depa r ture between

both

r e s u l t s ,

Fig . 12. From

800

0

e,

t he t h e o re t i c a l l y

pred ic ted t empera tures

could

not

be

compared

with t he exper imenta l data , s ince the

physica l cons t an t s

of

t he ma te r i a l

are

assumed

to

be

t he

same

as those

a t

1025

0

e dur ing

t he cool ing

process . Whereas the FD

so lu t ion

i s smooth

and always l e s s

than

t he

exac t

so lu t ion for

a l l nodes

a t a l l t imes .

In

Fig . 13

it

can be shown t h a t t he FEM

y ie ld s

t he upper

bound

for the

tempera ture

path.

Although

the eigenvalues

of

the FEM

obta ined from t he r e su l t i n g d i f f e r ence equat ions a re usua l ly

somewhat

c loser

to

the

t rue

values

than

those

of

t he

FDM,

t he

FEM

i s

prone

to

t he problem of

tempera ture overshoots

for a

shor t t ime

behaviour /14 ,15 / .

The er ro r of t he FEM i s always

maximum a t

t he nodal

poin t neares t to

t he boundary, such as

a t

t he survey

poin t

no.

5. But

t hen ,

a good

agreement

between

t he

exper imenta l r e s u l t s and the FEM/FDM so lu t ions i s ensured

a t

t he

survey poin t s near the cen t r e

of t he

cross -sec t ion (points

I , 2,7) .

4.2 Forging an

Engine

Disk

The metal flow in forg ing

a Titanium a l loy

engine

disk,

Fig . 14,

i s

s imula ted

with the combined

approach FEM

+ FDM.

Such

a pro

cess

has been ana lysed before by

Oh

e t

a l

/12 /

using

the r i g id

v i scop la s t i c f i n i t e element

technique .

In

t he

observed uns teady

process , the preform a t the

i n i t i a l

temperature Tow

= gOOOe

i s

forged

between

curved

symmetric

dies with a

cons t an t

die

ve lo

c i t y

1.27

mm/s.


3

- 38 -

Upper Die V

I"

152.4 ~ . p ( 6 . 0 i n . l - - - - - t ,

rLtLtCLLL1.CLLL1.""'"lrr.---r

12

.7

mm

(0.5)

fl.l..WUJ.=UI.J

1 /

'---132mm(5.2 in.) --=-=-----I

zt

Preform - "I r

f - - - - 1 5 8 . 8 m m ( 6 . 2 5 ) - - - - o l

Air 20C

Fig .

14:

Schematic

drawing

of

di sk

forging die

and preform / 1 2 /

------ Equation

Experimental

150

0

n..

::E

"'

100

"'

II

"-

if)

'

::I

40

.

t-

20

0.0

0.2

0.4

0.6

True Plastic

Strain

Fig . 15:

Flow

s t r e s se s o f

Titanium a l l oy / 13 /


3

- 39 -

Two cases of forg ing processes

are

analysed: i so thermal forg ing

and ho t - d i e forging.

In

the i so thermal ana ly s i s , t he i n i t i a l

temperature of the

preform and of

the

dies a re the same and,

dur ing

t he

process ,

t he

temperature

dependence of t he flow

s t r e s s

i s

accounted

fo r .

The ho t - d i e

forging

process

i s per

formed with

t he

i n i t i a l tempera ture

Tot

= 371

0

C

and

t he

a i r

temperature a t

20

o

C.

The FE mesh

cons i s t s

60 l i n e a r

quadr i la

t e r a l

elements .

The f r i c t i o n f ac to r m =

0.3

i s

chosen, f r i c t i o n

s t r e s s

T

=

my/l3.

The values of t he phys i ca l cons t an t s fo r hea t

ca lcu la t ion are

taken from t he

papers

/ 7 , 1 2 / . But t he flow

s t r e s s

da ta

of Ti-6Al-2Sn-4Zr-2Mo-0. lSi a r e given by Dadras and

Thomas / 1 3 / ,

Fig.

15.

In

Fig . 16, t he

tempera ture

d i s t r i b u t i o n wi th in

the workpiece

dur ing t he

i so thermal forg ing process

i s shown

v i s - a -v i s those

of

the ho t - d i e forging process . The reduc t ion

in

he igh t

i s 70%

a t t h i s

i n t e rmedia t e s tage .

A

severe

temperature grad ien t can

be seen

in

t he

workpiece

in

t he ho t - d i e forg ing . The heat

t r a n s f e r , due to

t he

temperature

grad ien t

between t he workpiece

and

the

dies ,

i s

a lso

i n t ens ive

and

it

cool s t he

workpiece

p a r t i a l l y .

Logica l ly , t he forg ing load

in

t he

ho t - d i e forging

should be higher

than

the forg ing load requi red

in

t he

i so th e r

mal forg ing process

in

which

the tempera ture

grad ien t

within

the workpiece i s obvious ly unimportant . Simi l a r t empera ture

d i s t r i b u t i o n s

can be

found

in

t he

paper

of

Oh e t

a l / 1 2 / . But

d i r e c t comparison i s not a t tempted for lack of exac t

in format ion

on

mater ia l

proper t i e s .

The mater ia l f lows

a t

some i n t e rmedia t e s teps

are p lo t t ed in

Fig .

17

for t he ho t - d i e

forging process . The bulge of

t he

outer

sur face can be observed s tep by s tep . I t i nd ica t e s the

contac t

problem s ince the

sur face

folding

i s

l imi ted

to

t he

top and

bottom dies . The p a r t i c u l a r l y compl ica ted

die

p r o f i l e d ic t a t e s

t he

l a rge

number of t he

deformation s teps to be chosen.

The

computer

s imula t ions wi l l enable

t he process des igner

to

modify

the

die

and

manufac tur ing

technique

for

t he

purpose

of

yie ld ing

the des i red mater ia l

f low.


3

a

s

h

m

a

F

g

n

T

D

e

9

9

I

9

b

H

D

i

e

F

g

n

I

7

T

D

i

e

3

o

F

g

1

T

m

p

a

u

e

d

s

r

b

o

[

O

a

7

%

r

e

d

o

n

h

g

n

s

o

h

m

a

a

n

h

-

d

e

f

o

g

n

F

g

1

G

r

d

d

s

o

o

a

n

v

o

y

f

e

d

a

s

o

m

e

i

n

e

m

e

d

a

e

s

a

g

n

h

-

d

e

f

o

g

n

a

A

t

h

B

e

g

n

n

1

1

/

=

I

b

A

t

3

4

%

H

e

g

R

e

d

o

I

c

A

t

7

%

H

e

g

R

e

d

o

I

I

+

>

o


4

- 41 -

4.3 Closed-Die Forging

In

t he

l a s t

numerical example, the

technica l and

economical

as

pects of

t he

d i f f e r e n t manufacturing

techniques

are

discussed

by means

of

the

forg ing

process

in

Fig .

21.

The

i so thermal

fo r

gings

of

axis

ymmetrica

1 disk a re ana lysed for purpose of

com

par ing

with the

of t - u s ed

cold

forg ings .

The

condi t ions

of

four

forging

processes concerned are :

l.

Cold

forg ing process

with

T

ow

T

o t

20

0

C

and f r i c t i o n

fac to r

m = 1 . 0,

2.

Cold

forg ing

process wi th

T

T

o t

20

0

C

ow

and m

=

0 .3 ,

3 .

Iso thermal

forg ing process

with

T =T

=900

o C=constow

o t

.

and m

=

1 .

0,

4.

Iso thermal

forg ing

process

with

T

ow

Tot

900

0

C

and m

=

1 . 0,

S.

Iso thermal

forg ing process with T

T

ow

o t

900

0

C

and

m =

0 .3 .

In

a l l processes the

formed

pa r t

(Tow) and

t he dies

(Tot)

have

the

same

tempera ture

a t

the

beginning .

The

tempera ture

e f f e c t s

are

considered through the combined

approach FEM + FDM,

except

the case no. 3 (cons t an t

temperature

assumed

dur ing

the process) .

The

t empera ture - sens i t ive mater ia l of

t he

formed pa r t i s CIS

s t e e l

with i t s f low s t r e s s curves shown

in

Fig. 18 for

cold

forgings

and in Fig . 19 for i so thermal forg ings /17 / . The heat

t r a n s f e r

prope r t i e s

are

i l l u s t r a t ed

in

Fig.

20.

Fig .

22

shows

t he

temperature

d i s t r ibu t ions

pred ic ted

a t

40%

reduct ion

in

he igh t

in

four

forg ing

processes .

For

the

f r i c t i o n

f ac tor m =

1.0

(Figs . 22a,d) the

temperature

grad ien t in the

formed

pa r t and dies

are s l i g h t l y higher

than

those with

m = 0.3

(Figs . 22b,c)

due

to the in tens ive deformation of the mater ia l

in the formed p a r t

and

the high

d i s s ipa ted f r i c t i on

energy. In

both

cases

of

t he f r i c t i o n condi t ions , the m ater ia l , in the

zone

near the round corner

of

t he bottom

d i e , i s shear ing to

flow

toward

the

two

openings and press s t rongly

on t he bottom

die

(Fig. 22f) . Temperature peak can a l so be seen

in

t h i s zone. It

means t h a t the

die

corner

i s

a p a r t sub jec t

to

wear . At the


4

E

E

1

Z

8

0

0

>

:

~

6

0

0

-

0

Q

>

4

0

0

2

0

0

E

:

=

1

6

5

2

C

3

C

[

?

t

2

C

1

1

e

1

/

~

V

I

-

4

e

/

~

-

r

5

C

I

I

-

t

-

6

C

7

C

/

8

e

-

e

{

1

C

;

1

e

-

-

I

1

1

e

0

2

0

4

0

6

0

8

S

t

r

a

n

E

1

0

1

1 8

~

6

E

~

5

z >

4

U

U

~

3

V

0

2

Q

>

2

1

1

1

F

g

1

F

o

w

s

r

e

e

o

C

S

e

1

f

o

c

c

u

a

o

o

c

o

d

f

o

g

n

=

5

2

e

~

-

-

-

-

,

-

I

-

-

6

-

-

-

-

-

-

-

-

-

r

I

-

-

-

-

7

-

-

-

-

:

8

I

-

k

9

=

-

-

-

1

-

-

-

-

-

-

-

-

-

1

V

-

-

-

-

-

-

v

/

1

.

-

~

-

-

V

/

/

1

6

2

5

4

6

3

1

1

2

4

6

1

S

t

r

a

n

R

a

e

F

g

1

F

o

w

s

r

e

e

o

C

S

e

1

f

o

c

c

u

a

o

o

h

f

o

g

n

.

>

N


4

M

d

u

m

C

m

a

o

p

a

m

e

e

S

m

F

g

e

u

1

-

-

-

-

-

-

-

-

_

_

D

m

n

o

H

B

8

2

n

m

-

D

n

y

P

w

7

5

(

7

7

k

e

m

-

-

-

W

o

k

e

S

f

c

h

c

p

y

c

0

6

(

0

4

k

k

g

d

w

-

H

c

o

v

y

c

o

-

k

2

(5

W

/

m

g

d

c

e

n

w

1

-

T

m

e

c

e

n

y

1

0

8

-

r

~

-

I

n

a

e

m

p

a

u

e

T

9

(

2

M

-

T

c

k

S

R

0

0

n

m

C

a

a

y

C

a

c

o

a

n

d

a

a

K

1

k

W

/

m

g

d

-

-

-

D

e

n

y

P

0

0

k

e

m

-

-

-

A

i

r

S

f

c

h

c

p

y

c

1

1

(

1

0

k

k

g

d

a

H

r

a

n

e

c

o

c

e

n

a

a

0

0

k

w

m

g

d

D

e

n

y

P

7

7

(

7

7

k

e

m

S

f

c

h

c

p

y

c

t

0

5

(

0

4

k

k

g

d

T

H

c

o

v

y

c

o

-

k

2

(1

W

/

m

g

d

c

e

n

V

e

o

y

o

o

d

e

v

1

I

1

s

F

g

2

P

c

c

o

an

s

f

o

c

c

u

a

o

o

e

m

p

a

u

e

f

e

d

V

a

u

(

a

2

C

i

I

-

(

o


4

- 44 -

/

):/

280= B

v

"I Top Oil'

/

/

///

//

/ /

/

//////////

m

"+r

I

co

11

m

/

/

/ /

///////

-15

/

. .

H-h

Air

20C I

65

J Bottom Oil'

Height Reduction =

-H -

i , f

(v.tcitY',O)

/

Fig .

21: Schematic drawing of c losed-d ie

forg ings

b)

Tow

= 20

C

m =

0.3

@

25

40

SO

50

40

50

25

Fig . 22:

Temperature d i s t r i b u t i o n s

rOc]

a t 40%

reduct ion

in

he igh t

dur ing

var ious

forg ing processes


4

c

T

=

9

C

=

c

o

m

=

1

0

w

f

)

@

M

a

e

a

F

o

w

a

4

%

H

g

R

d

o

9

C

d

T

w

=

g

O

O

O

e

;

m

=

1

0

J

e

T

w

=

9

C

m

=

0

3

.

>

U

9

9

9

C

-

A

r

2

9

=

9

'

9

9

9

9

-

9

z

F

g

2

(

c

o

n

d

r

O


4

- 46 -

oute r opening,

the

r e l a t i v e movement of

Documents

[K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)