1. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
1
Forging Analysis – 2
Cylindrical Forging
ver. 1
2. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
2
Overview
• Slab analysis
– frictionless
– with friction
– Rectangular
– Cylindrical
• Strain hardening and rate effects
• Flash
• Redundant work
3. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
3
Forging – cylindrical part
sliding region
h
p
4. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
4
Equilibrium in r direction
( ) ( ) q
s
s
q
s
q
µ
q
s
q d
h
dr
r
d
d
dr
h
dr
d
r
p
d
r
h
dF
r
r
r
r
×
×
+
×
+
+
×
×
×
×
-
×
×
×
×
×
-
×
×
×
-
=
=
å
2
2
2
0
neglecting HOTs
0
2 =
×
-
×
-
×
+
× r
r d
hr
dr
h
dr
h
dr
pr s
s
s
µ q
2
2
sin
q
q d
d
=
÷
ø
ö
ç
è
æ
N.B.
5. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
5
Axisymmetric flow and yield
For axisymmetric flow
By Tresca
flow
flow
r k
p t
s
s 2
2 =
=
=
+
dp
d r -
=
s p sr
sflow/2 = tflow
( )
q
q
q
s
s
e
e
p
p
p
e
e
=
=
=
-
+
=
=
r
r
r
r
dr
r
r
dr
r
r
dr
;
2
2
2
;
6. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
6
Stress in z direction
dp
hr
dr
pr ×
-
=
×
µ
2
0
2 =
×
+
×
-
×
+
× dp
hr
dr
h
dr
h
dr
pr r
r s
s
µ
substituting
or
rearranging
dr
h
p
dp µ
2
-
=
7. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
7
Forging pressure - sliding
dr
h
p
dp
R
r
p
flow
r
×
-
= ò
ò
µ
t
2
2
( )ú
û
ù
ê
ë
é
-
= r
R
h
p
flow
r µ
t
2
exp
2
for rk < r < R
! = #$%&';
Yield or flow stress, )$ = 2! = 2#$%&'
8. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
8
Average forging pressure –
sliding
( ) ( ) ( ) dr
r
r
R
h
r
R
dr
r
p
r
R
p
R
r
k
R
r flow
r
k
flow
ave
k
k
ò
ò ú
û
ù
ê
ë
é
-
-
=
×
×
-
=
µ
p
t
p
t
2
exp
2
2
2
1
2 2
2
2
2
( )
R
r
k
flow
ave
k
h
r
h
r
h
R
h
r
R
p
þ
ý
ü
î
í
ì
÷
ø
ö
ç
è
æ
-
-
×
÷
ø
ö
ç
è
æ -
÷
ø
ö
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
-
= 1
2
2
exp
2
exp
2
2
2
2
2
2
µ
µ
µ
µ
t
( ) þ
ý
ü
î
í
ì
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ
-
-
×
÷
ø
ö
ç
è
æ -
-
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ
-
-
×
÷
ø
ö
ç
è
æ -
÷
ø
ö
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
-
= 1
2
2
exp
1
2
2
exp
2
exp
2
2
2
2
2
2
h
r
h
r
h
R
h
R
h
R
h
r
R
p k
k
k
flow
ave µ
µ
µ
µ
µ
µ
t
9. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
9
Average forging pressure –
sliding
( )
( )
ú
û
ù
ê
ë
é
-
÷
ø
ö
ç
è
æ
-
÷
ø
ö
ç
è
æ
+
×
÷
ø
ö
ç
è
æ -
÷
÷
ø
ö
ç
ç
è
æ
×
-
= 1
2
1
2
2
exp
2
2
2
2
2
2
h
R
h
r
h
r
R
h
r
R
p k
k
k
flow
ave µ
µ
µ
µ
t
10. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
10
• Taking the first four terms of a Taylor’s
series expansion for the exponential
about 0 for
Average forging pressure –
all sliding approximation (rk = 0)
1
£
x
yields
( ) å
=
=
+
+
+
+
+
=
n
k
k
n
k
x
n
x
x
x
x
x
0
3
2
!
!
!
3
!
2
1
exp !
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ
+
=
h
R
p
flow
ave
3
2
1
2
µ
t
11. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
11
Forging force –
all sliding approximation
2
3
2
1
2 R
h
R
F flow
forging p
µ
t ×
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ
+
×
=
2
R
p
A
p
F ave
ave
forging ×
×
=
×
= p
12. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
12
Transition sticking / sliding
• Set tflow = µp and solve for rk
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ -
=
h
r
R
p k
flow
µ
t
2
exp
2
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ -
=
× h
r
R
p
p k
µ
µ
2
exp
2
÷
ø
ö
ç
è
æ -
=
÷
÷
ø
ö
ç
ç
è
æ
h
r
R k
µ
µ
2
2
1
ln ÷
÷
ø
ö
ç
ç
è
æ
-
=
µ
µ 2
1
ln
2
h
R
rk
13. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
13
Forging pressure - sticking
region
• Use the same method as for sliding
• Substitute µp = tflow,
• Assume Tresca yield criterion
dr
h
dp
flow
t
2
-
=
dp
hr
dr
pr ×
-
=
×
µ
2
dp
hr
dr
r
flow ×
-
=
×
t
2
14. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
14
Forging pressure - sticking
region
dr
h
dp
r
r
flow
p
p k
r
k
r
ò
ò -
=
t
2
( )
k
flow
r
r r
r
h
p
p k
-
-
=
-
t
2
( )
h
r
r
p
p k
flow
r
r k
-
=
-
t
2
15. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
15
Forging pressure - sticking
region
( ) ( )
h
r
r
r
R
h
p k
k
flow
r -
+
ú
û
ù
ê
ë
é
-
=
µ
t
2
exp
2
for 0 < r < rk
prk
determined from sliding equation
( )ú
û
ù
ê
ë
é
-
= k
flow
r
r
R
h
p k
µ
t
2
exp
2
16. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
16
Average forging pressure -
sticking
( ) rdr
h
r
r
r
R
h
r
dr
r
p
r
p k
k r
k
k
k
r
r
k
flow
ave
×
÷
ø
ö
ç
è
æ -
+
ú
û
ù
ê
ë
é
-
=
×
×
= ò
ò 0
2
0
2
2
exp
2
2
1
2
µ
p
p
t
( ) dr
r
h
h
r
r
r
R
h
r
r
p k
r
k
k
k
flow
ave
×
÷
ø
ö
ç
è
æ
-
×
+
ú
û
ù
ê
ë
é
-
×
= ò
0
2
2
1
2
exp
2
2
µ
t
( )
k
r
k
k
k
flow
ave
h
r
h
r
r
r
R
h
r
r
p
0
3
2
2
2
3
2
2
exp
2
2
2 ÷
÷
ø
ö
ç
ç
è
æ
-
×
+
ú
û
ù
ê
ë
é
-
×
=
µ
t
17. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
17
Average forging pressure - sticking
( ) ÷
÷
ø
ö
ç
ç
è
æ
-
+
ú
û
ù
ê
ë
é
-
×
=
h
r
h
r
r
R
h
r
r
p k
k
k
k
k
flow
ave
3
2
2
exp
2
2
2
3
3
2
2
µ
t
( ) ÷
ø
ö
ç
è
æ
+
ú
û
ù
ê
ë
é
-
=
h
r
r
R
h
p k
k
flow
ave
3
2
exp
2
µ
t
( )
k
r
k
k
k
flow
ave
h
r
h
r
r
r
R
h
r
r
p
0
3
2
2
2
3
2
2
exp
2
2
2 ÷
÷
ø
ö
ç
ç
è
æ
-
×
+
ú
û
ù
ê
ë
é
-
×
=
µ
t
18. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
18
Forging force – sticking region
2
k
ave
ave
forging r
p
A
p
F ×
×
=
×
= p
( ) 2
3
2
exp
2 k
k
k
flow
forging r
h
r
r
R
h
F ×
×
÷
ø
ö
ç
è
æ
+
ú
û
ù
ê
ë
é
-
×
= p
µ
t
19. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
19
Sticking and sliding
• If you have both sticking and sliding, and you
can’t approximate by one or the other,
• Then you need to include both in your
pressure and average pressure calculations.
( ) ( )sticking
ave
sliding
ave
forging A
p
A
p
F ×
+
×
=
sticking
sliding
forging F
F
F +
=
20. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
20
Strain hardening
(cold - below recrystallization
point)
n
flow K
Y e
t =
=
2
Tresca
21. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
21
Strain rate effect
(hot – above recrystallization point)
Tresca
( )m
flow C
Y e
t !
=
=
2
height
ous
instantane
velocity
platen
h
v
dt
dh
h
1
ε =
=
=
!
22. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
22
Flash for closed die forging
(plane strain)
• Say we have a typical flash with
thickness h/20 and length w/4
w
w/4
h/20
h
23. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
23
Average forging pressure
• in forging (Tresca)
• in flash (Tresca)
÷
ø
ö
ç
è
æ
+
×
=
h
w
p flow
ave
2
1
2
µ
t
÷
ø
ö
ç
è
æ
+
×
=
h
w
p flow
ave
µ
t
5
1
2
24. Prof. Ramesh Singh, Notes by Dr.
Singh/ Dr. Colton
24
Flash
• Flash’s high deformation resistance results
in filled mold
• Process wouldn’t work without friction
