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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
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
Das Werk ist urheberrechtlich geschotzt. Die dadurch begrOndeten Rechte, insbesondere
die der Obersetzung, des Nachdrucks,
der
Entnahme von Abbildungen, der Funksendung,
der Wiedergabe auf photomechanischem oder iihnlichem Wege und
der
Speicherung in
Datenverarbeitungsanlagen bleiben, auch bei nur auszugsweiser Verwendung, vorbehalten.
Die VergatungsansprOche des
54, Abs. 2 UrhG werden durch die "Verwertungsgesellschaft
Wort", MOnchen, wahrgenommen.
Springer-Verlag, Berlin, Heidelberg 1986.
Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in diesem
Werk berechtigt auch ohne besondere Kennzeichnung nicht zu der Annahme, daB solche
Namen
im
Sinne der Warenzeichen- und Markenschutz-Gesetzgebung als frei zu betrachten
wiiren und daher von jedermann benutzt werden dOrften.
Gesamtherstellung: Copydruck GmbH, Offsetdruckerei,lndustriestraBe 1-3,7258 Heimsheim
Telefon
07033/3825-26
2362/3020-543210
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
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5
THE
SIMOP-PARTICIPANTS
(Numbers
in
the figure correspond
to the names
in
the l ist of
part icipants)
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
Universitat Stuttgart
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
Universitat Stuttgart
Holzgartenstr. 17
7000
Stuttgart 1
Arbeitsbereich Meeresteehnik
- Strukturmeehanik -
Teehn.Univ. Hamburg-Harburg
EiBendorfer Str. 38, Postfaeh 901403
2100
Hamburg
90
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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)
Universitat Stuttgart
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
Universitat Stuttgart
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
Universitat Stuttgart
Holzgartenstr.
17
7000
Stuttgart
1
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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- 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
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
Universitat Stuttgart
Pfaffenwaldring 7
7000 Stuttgart
80
Inst.f.Kernenergetik u.Energiesysteme
Universitat Stuttgart
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
Dept. of Mechanical Engineering
The
University of
Birmingham
South West Campus, P.O. Box 363
Birmingham B15 2TT / GREAT BRITAIN
Institut
fUr Umformtechnik
Universitat Stuttgart
Holzgartenstr. 17
7000 Stuttgart 1
Institut
fUr Baustatik
Universitat
Stuttgart
Pfaffenwaldring 7
7000 Stuttgart 80
Control Data GmbH.
Marienstr. 11-13
7000
Stuttgart
1
Dept. of Mechanical Engineering
The University of Birmingham
South
West
Campus
P.O. Box 363
Birmingham
B15
2TT / GREAT BRITAIN
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
Dept. of Mechanical Engineering
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
Institut fUr Umformtechnik
Universitat Stuttgart
Holzgartenstr.
17
7000 Stuttgart 1
Univ.-Gesamthochschule-Paderborn
Fachbereich
10
-Maschinentechnik
I
Umformende
Fertigungsverfahren
Pohlweg
47-49, Postfach
1621
4790 Paderborn
Institut fUr Umformtechnik
Universitat
Stuttgart
Holzgartenstr. 17
7000 Stuttgart
1
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
Universitat Stuttgart
Pfaffenwaldring 7
7000 Stuttgart 80
Universitat Stuttgart
Institut fUr Computer-Anwendungen
Universitat
Stuttgart
Pfaffenwaldring
27
7000
Stuttgart
80
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
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- 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
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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.
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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.
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
1
-
19
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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
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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 .
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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
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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.
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
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r
a
n
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d
C
k
b
y
c
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(
c
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p
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m
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n
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v
a
p
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a
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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
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m
p
a
u
e
I
f
e
d
n
w
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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
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n
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n
1
I
I I
I
I
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t
-
1
_
_
_
_
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_
_
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_
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-
-
I
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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
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
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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
(
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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 .
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- 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
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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
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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
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- 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
)
.
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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.
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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
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- 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
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
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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) .
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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
(
-
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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.
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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 /
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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.
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
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-
d
e
f
o
g
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F
g
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G
r
d
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s
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a
g
n
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-
d
e
f
o
g
n
a
A
t
h
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g
n
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1
1
/
=
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A
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3
4
%
H
e
g
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A
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7
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H
e
g
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e
d
o
I
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o
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
4
E
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5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
4
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-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
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
-
5/20/2018 [K. Lange (Auth.), Prof. Dr.-ing. Kurt Lange (e(BookZZ.org)
4
- 46 -
oute r opening,
the
r e l a t i v e movement of