- Email: [email protected]
Bulging analysis in the disk forging of porous metals
Journal of Mechanical Working Technology, 9 (1984) 193--200
193
Elsevier Science Publishers B.V., A m s t e r d a m - Printed in The Netherlands
BULGING ANALYSIS IN THE DISK FORGING OF POROUS METALS
HUNG KUK OH
Department of Mechanical Engineering, Ajou University, Seoul (South Korea) (Received August 27, 1982; accepted in revised form October 4, 1983)
Industrial summary The bulging shape and the volume change occurring during the simple compression of porous metal under the action of friction, which provide useful guidelines for preform design in powder forging, are quantified for various working conditions (different friction, initial density and specimen shape) by using plasticity theory for porous metal. The upperbound method is used to analyze the situation by assuming that the radial velocity distribution along the specimen axis is parabolic and that the necessary parameters are determined by the minimum energy condition. The fracture limit in porous metals is also discussed, in terms of the results o f this study.
Notations cylindrical coordinates in axi-symmetry. half-height of the specimen. outer radius of the specimen. components of velocity OR, Oo, O, eyy, ~RR~600, eRY strain-rate components. volume~change strain-rate. ~v equivalent strain-rate. relative density. P volume-change weighting factor. B bulging factor. b degree of influence of the hydrostatic stress component f on the onset of yielding. internal energy-rate of deformation. .wi friction loss energy-rate. Wf applied normal stress. ON effective yield stress.
R,y T Ro
Introduction The production of machine parts by powder forging appears to be an attractive process because it blends the timer, the material-, and the cost-saving
0378-3804/84/$03.00
© 1984 Elsevier Science Publishers B.V.
194
advantages of conventional (press and sinter) powder metallurgy with the high production rates and property enhancement of forging. However, it is still necessary to s t u d y the basic concepts and applications to specific processes for the utilization of the full potential of powder forging. The effects of the substantial volume fraction of voids in the material make the plastic deformation more complicated than that of conventional materials. Additionally, the voids are sites of weakness at which ductile fractures m a y initiate during forging. Careful selection of the process parameters can improve the fracture limit by reducing the triaxial tensile stress. To provide guidelines for the design of powder preforms for forging and to aid in parameter selection for the overall process, the bulging caused by friction in the simple compression test -- which is an elemental process of forging -- is studied for porous material. The bulging shape and the volume change are quantified for various working conditions (different friction, initial density and specimen shape) by using plasticity theory for porous material [1--3]. This kind o f study has been investigated qualitatively by Kuhn and D o w n e y [4] and by Bockstiegel and Olsen [5]. In the elementary solutions obtained for porous disks [ 5], the velocity was so assumed as to have the axis of the cylindrical coordinate system coincident with the principal axis. The strain-rate fields were free of shear components and solution was relatively easy. As a consequence of the previously assumed velocity field, the outer cylindrical surface of the disk remained cylindrical. In practice however, this cylinder barrels out or bulges, which bulging is the result of friction over the surface of the die. The velocity c o m p o n e n t 0 a at the surfaces in contact with the dies (y = +T in Fig. 1) is smaller than 0 R at the center of the disk (y = 0) and the reduced velocity c o m p o n e n t on the die surface leads to reduced friction losses. The velocity at the die surface does not drop to zero because the change in OR over the thickness of the disk introduces a shear c o m p o n e n t into the strain-rate field, with a corresponding increase in the internal power of deformation. As a result, bulging produces a slightly lower total power [6] and the actual shape is such that the required power is minimized. In the present analysis the axes were chosen as shown in Fig. 1, where the origin of the cylindrical coordinate system is at the center of the disk. The two platens move toward each other at the same absolute velocity 0/2. Y
~////////~ =-
R
Fig. 1. A x e s and dimensions used for the analysis.
195 Because of s y m m e t r y and t o facilitate computation, only the u p p e r half of the disk is considered• A velocity field for 0 ~< y ~< T is assumed such that [Jo = 0 (JR = -~ B R (a - b y 2 / T 2) = 0(Ry)
O,
-
2__~UT[ ( b +
(1) ~----) Y - 3
B is a weighting factor concerned with volume change (B = 1 for conventional solid metal and B < 1 for porous metal) whilst a and b are the constants of the parabolic equation which represents the bulged shape: the latter is based on physical observations of actual bulged cylinders. F r o m the boundary conditions (at y = 0, 0~ = 0 and at y = T, 6"~ = - 0 / 2 ) , a = b / 3 + 1/4
(2)
The strain-rate field becomes: eRR -
o ta-.b-'~:)
~R
= TB
~fR e°°
T
eY~
ay
eRR
T
• = -- RB(-by/T eR~ T
( a- b : ) 2)
o( :)
---2g ~-b~ (~-m
(4)
The equivalent strain rate is: 2
~2 = ~ ((~RR - ~00) 2 + (~00 - ~ ) 2
+ ( ~ . . _ ~RR)2 }
+ ~ .~
.2 + ~IR) + ( f e ~ ) : g (eRO + eOz
(5)
Substituting eqn. (3) into eqn. (5):
I_, where
(6)
196 a = + ~ (2 + B) 2 + 4 f 2 (1 - B ) :
(7)
= 4B2
(8)
1 f = 2.5x/T --p
(from Ref. 1)
(9)
The internal power of deformation, as applied to the strain-rote field, eqn. (3), becomes Wi =
f
~ e dV
(10)
V
where dV = 2u RdRdy. The effective strain-rate is simplified by approximation as follows, for easy integration, U [
(
= -T- ~
~y2b2R2/T4 ] +~ v~(a-by:/T2)J
(11)
--a
(12)
1
Y:
a-b-~)
The result is 2 R°
-b +V~
~ log 1-- bx/b-~/J
The minimum value of W i is obtained for b=0, and represents the ideal power of deformation. In the actual solution, b ¢ 0 and the internal power of deformation is higher than the ideal value, of which the difference is redundant power. The friction loss is
-W~ = fs rAVdS = 2 fs T
IvRI27rRdR
4 -- K~ -5 B ( a - b) nR~ ( R o / T ) ( /
(13)
3~/3
Finally, the average normalized normal forging pressure is aN
Wi+Wf
~[1+~ -
(..~)2 ( --
4 + ~ - ~ ~ B (a - b)
1 -b+-
1+~)] x/~log
--
(14)
To minimize (os/~), the o p t i m u m values of B and b must be chosen under the constraints b i> 0 and 0 ~ B ~<1 : these are sought by gridding the constrained domain in the computer. Eventually the volume change and the bulging shape are evolved from minimizing the total deformation power.
197
Results a n d discussion
Starting from the simpler case with negligible friction, ~ = 0 and b = 0; eqn. (14) is reduced to
oN/~
(15)
= v/~/2
Differentiating it with respect to B, 9f 2 - 2
B-
(16)
9f 2 +1 Alternatively,
iv
0 -
(1
2T
e y y - eRl~
=
-
B)
K . (1
+ B
(17)
Citing eqn. (9) from Ref. 3 e,y - eRR
~,
= l + B/2
1 -B
= __9 f2
(18)
2
Solving eqn. (18), the same result for B as eqn. (16) is obtained. It can be seen here that deformation proceeds towards minimum energy dissipation. For the general case with non-negligible friction, b y optimizing eqn. (14), the volume-change weighting factor (B) and the bulging factor (b) are sought under the several conditions that arise by varying the friction factor, the density and the shape factor. In Fig. 2, B is seen to decrease with increase in friction factor, where its gradient is steeper for flatter disks. Conversely, b increases with increase in friction factor, having a lesser gradient for higher shape factors. In Fig. 3, B is seen to increase approximately linearly with increase in density, whilst b is seen to decrease linearly. In Fig. 4, b o t h B and b are observed to decrease for higher shape factors. Of special concern is the densiW effect, as the porosity effect on the fracture limit in the simple compression test has been studied b y many researchers in this field [4, 5]. Because fracture is dependent on the local ductility of the material and also on the localized tensile stress at the barrelled outer surface, the preform porosity has very little influence on the strain-tofracture in compression, from the viewpoint that although increasing porosity decreases the local ductility of the material it also decreases the tensile stress generated on the barrelled outer surface. If it should be in keeping with the results of this study, the decreased tensile stress in porous material is probably due to the strain reduction being much more influenced by decreased B than by increased barrelling (b).
198
(A)
{B).~_
0.8
/
: R o / T ----- 1
0.32
0.7
0.28
"-0
0.6
0.24
0.5
0.20
rn 0.4
0
:
V
:9=0.8
9=
X
:
0.9
9=0,7
X
0.16
0.3
×
012
=
~o/°7= 1
02
''
~x
0.08
---2
= 3
0.1
g=0.9 9=0.8
o
v 0
0.04
[
o.1
0.2
1
0
0,3
I
i
i
o.1
o. 2
0.3
Fig. 2. Showing the effect of friction factor on: (a) the volume-change weighting factor; (b) the bulging factor.
(A) Ro/T
V
o.9
=
1
---- 2
- -
m
---- 0.1
---
-~
= 0.2
----:
~
=
X
=
o\
022
.
O.lO
Ro/T ,,
= 1 =2
; O3 l
__×: ~
0.20
3
•
0.18
0.3
9.-,, x~%.\ \',~\
/V
0.8
0.16i
0.7
0.14
0.6
0.12
0.5
01C
OA'
\.
•
~
=
0.3
~o
\ "-4-.. \..o
rn
0.08
v//X/' X
0.3
0.06
02
0.04
0.1
0.02
0
0 : V :
(e,) 0
~s
I
I
I
0.7
0.8
0.9
0
s
I
I
I
0.7
0.8
0.9
Fig. 3. Showing the effect of density on: (a) the volume-change weighting factor; (b) the bulging factor.
199 (B)
(A)
0.30
0.8
0
: "fi'r
=
0.1
V
: ~
=
0.2
X
: i"i'i
=
0.3 0.7
- - : 9
" ~ 0.7
0
: ~ V : ~ X:Ni"
i" ~
= 0.1 ----0.2 =0.3
~
0.27
X 0.24
= 9
0.8 0.9
=
0.21 "" ~V
~'--.
0.18
toO.5
~ X
1.5
v\
\\.,..%.,
x
o,.
0.4
0.3
:9 --'--:
X
o ....
0.6
=
V --:9 --'--:
9 = 0.8 9 =09
0.06
- -X ~
•~.-~
0.03
0.2 0
0.09
I
1.0
[
2.0
Ro / 7
I
3.0
0
I
1.0
I
2.0
~ t
3.0
Ro/T
Fig. 4. Showing the effect o f shape factor on: (a) the volume-change weighting factor; (b) the bulging factor.
Conclusions (1) It is established from upper-bound analysis of the simpler case with negligible friction that the volume change in the plastic deformation of porous material proceeds towards minimum energy dissipation. (2) From determination of the volumechange weighting factor (B) and the bulging factor (b), using the condition of minimum deformation-energy dissipation, it is noted that the former decreases with friction factor and shape factor and increases approximately linearly with density, whilst the latter increases with friction factor and decreases with density and shape factor. (3) The decreased tensile stress in porous material at the barrelled outer surface is due to the strain reduction being much more influenced by decreased volume-change weighting factor (B) than by increased barrelling (b). References 1 S. Shima and M. Oyane, Plasticity theory for porous metals. Int. J. Mech. Sci., 18 (1976) 285---291. 2 M. Oyane and T. Tabata, Slip line field theory and upper bound theory for porous materials. Plasticity and Processing (Japan), 15(156) (1974) 43--51.
200 3 Hung Kuh Oh and Jae Ho Mun, A study on the ring compression of porous metal. Proceeding of the spring conference, 1982. Korean Society of Mechanical Engineering, Seoul, Korea. 4 H.A. Kuhn, C.L. Downey, Material behavior in powder preform forging. J. Eng. Mat. Tech., (Jan. 1973) 41--46. 5 G. Bockstiegel and H. Olsen, Processing parameters in the hot forming of powder preforms. Presented at the Joint Powder Metallurgy Group, Bristol, Nov. 1971. 6 B. Avitzur, Metal Forming. McGraw-Hill, New York, pp. 102--110.
193
Elsevier Science Publishers B.V., A m s t e r d a m - Printed in The Netherlands
BULGING ANALYSIS IN THE DISK FORGING OF POROUS METALS
HUNG KUK OH
Department of Mechanical Engineering, Ajou University, Seoul (South Korea) (Received August 27, 1982; accepted in revised form October 4, 1983)
Industrial summary The bulging shape and the volume change occurring during the simple compression of porous metal under the action of friction, which provide useful guidelines for preform design in powder forging, are quantified for various working conditions (different friction, initial density and specimen shape) by using plasticity theory for porous metal. The upperbound method is used to analyze the situation by assuming that the radial velocity distribution along the specimen axis is parabolic and that the necessary parameters are determined by the minimum energy condition. The fracture limit in porous metals is also discussed, in terms of the results o f this study.
Notations cylindrical coordinates in axi-symmetry. half-height of the specimen. outer radius of the specimen. components of velocity OR, Oo, O, eyy, ~RR~600, eRY strain-rate components. volume~change strain-rate. ~v equivalent strain-rate. relative density. P volume-change weighting factor. B bulging factor. b degree of influence of the hydrostatic stress component f on the onset of yielding. internal energy-rate of deformation. .wi friction loss energy-rate. Wf applied normal stress. ON effective yield stress.
R,y T Ro
Introduction The production of machine parts by powder forging appears to be an attractive process because it blends the timer, the material-, and the cost-saving
0378-3804/84/$03.00
© 1984 Elsevier Science Publishers B.V.
194
advantages of conventional (press and sinter) powder metallurgy with the high production rates and property enhancement of forging. However, it is still necessary to s t u d y the basic concepts and applications to specific processes for the utilization of the full potential of powder forging. The effects of the substantial volume fraction of voids in the material make the plastic deformation more complicated than that of conventional materials. Additionally, the voids are sites of weakness at which ductile fractures m a y initiate during forging. Careful selection of the process parameters can improve the fracture limit by reducing the triaxial tensile stress. To provide guidelines for the design of powder preforms for forging and to aid in parameter selection for the overall process, the bulging caused by friction in the simple compression test -- which is an elemental process of forging -- is studied for porous material. The bulging shape and the volume change are quantified for various working conditions (different friction, initial density and specimen shape) by using plasticity theory for porous material [1--3]. This kind o f study has been investigated qualitatively by Kuhn and D o w n e y [4] and by Bockstiegel and Olsen [5]. In the elementary solutions obtained for porous disks [ 5], the velocity was so assumed as to have the axis of the cylindrical coordinate system coincident with the principal axis. The strain-rate fields were free of shear components and solution was relatively easy. As a consequence of the previously assumed velocity field, the outer cylindrical surface of the disk remained cylindrical. In practice however, this cylinder barrels out or bulges, which bulging is the result of friction over the surface of the die. The velocity c o m p o n e n t 0 a at the surfaces in contact with the dies (y = +T in Fig. 1) is smaller than 0 R at the center of the disk (y = 0) and the reduced velocity c o m p o n e n t on the die surface leads to reduced friction losses. The velocity at the die surface does not drop to zero because the change in OR over the thickness of the disk introduces a shear c o m p o n e n t into the strain-rate field, with a corresponding increase in the internal power of deformation. As a result, bulging produces a slightly lower total power [6] and the actual shape is such that the required power is minimized. In the present analysis the axes were chosen as shown in Fig. 1, where the origin of the cylindrical coordinate system is at the center of the disk. The two platens move toward each other at the same absolute velocity 0/2. Y
~////////~ =-
R
Fig. 1. A x e s and dimensions used for the analysis.
195 Because of s y m m e t r y and t o facilitate computation, only the u p p e r half of the disk is considered• A velocity field for 0 ~< y ~< T is assumed such that [Jo = 0 (JR = -~ B R (a - b y 2 / T 2) = 0(Ry)
O,
-
2__~UT[ ( b +
(1) ~----) Y - 3
B is a weighting factor concerned with volume change (B = 1 for conventional solid metal and B < 1 for porous metal) whilst a and b are the constants of the parabolic equation which represents the bulged shape: the latter is based on physical observations of actual bulged cylinders. F r o m the boundary conditions (at y = 0, 0~ = 0 and at y = T, 6"~ = - 0 / 2 ) , a = b / 3 + 1/4
(2)
The strain-rate field becomes: eRR -
o ta-.b-'~:)
~R
= TB
~fR e°°
T
eY~
ay
eRR
T
• = -- RB(-by/T eR~ T
( a- b : ) 2)
o( :)
---2g ~-b~ (~-m
(4)
The equivalent strain rate is: 2
~2 = ~ ((~RR - ~00) 2 + (~00 - ~ ) 2
+ ( ~ . . _ ~RR)2 }
+ ~ .~
.2 + ~IR) + ( f e ~ ) : g (eRO + eOz
(5)
Substituting eqn. (3) into eqn. (5):
I_, where
(6)
196 a = + ~ (2 + B) 2 + 4 f 2 (1 - B ) :
(7)
= 4B2
(8)
1 f = 2.5x/T --p
(from Ref. 1)
(9)
The internal power of deformation, as applied to the strain-rote field, eqn. (3), becomes Wi =
f
~ e dV
(10)
V
where dV = 2u RdRdy. The effective strain-rate is simplified by approximation as follows, for easy integration, U [
(
= -T- ~
~y2b2R2/T4 ] +~ v~(a-by:/T2)J
(11)
--a
(12)
1
Y:
a-b-~)
The result is 2 R°
-b +V~
~ log 1-- bx/b-~/J
The minimum value of W i is obtained for b=0, and represents the ideal power of deformation. In the actual solution, b ¢ 0 and the internal power of deformation is higher than the ideal value, of which the difference is redundant power. The friction loss is
-W~ = fs rAVdS = 2 fs T
IvRI27rRdR
4 -- K~ -5 B ( a - b) nR~ ( R o / T ) ( /
(13)
3~/3
Finally, the average normalized normal forging pressure is aN
Wi+Wf
~[1+~ -
(..~)2 ( --
4 + ~ - ~ ~ B (a - b)
1 -b+-
1+~)] x/~log
--
(14)
To minimize (os/~), the o p t i m u m values of B and b must be chosen under the constraints b i> 0 and 0 ~ B ~<1 : these are sought by gridding the constrained domain in the computer. Eventually the volume change and the bulging shape are evolved from minimizing the total deformation power.
197
Results a n d discussion
Starting from the simpler case with negligible friction, ~ = 0 and b = 0; eqn. (14) is reduced to
oN/~
(15)
= v/~/2
Differentiating it with respect to B, 9f 2 - 2
B-
(16)
9f 2 +1 Alternatively,
iv
0 -
(1
2T
e y y - eRl~
=
-
B)
K . (1
+ B
(17)
Citing eqn. (9) from Ref. 3 e,y - eRR
~,
= l + B/2
1 -B
= __9 f2
(18)
2
Solving eqn. (18), the same result for B as eqn. (16) is obtained. It can be seen here that deformation proceeds towards minimum energy dissipation. For the general case with non-negligible friction, b y optimizing eqn. (14), the volume-change weighting factor (B) and the bulging factor (b) are sought under the several conditions that arise by varying the friction factor, the density and the shape factor. In Fig. 2, B is seen to decrease with increase in friction factor, where its gradient is steeper for flatter disks. Conversely, b increases with increase in friction factor, having a lesser gradient for higher shape factors. In Fig. 3, B is seen to increase approximately linearly with increase in density, whilst b is seen to decrease linearly. In Fig. 4, b o t h B and b are observed to decrease for higher shape factors. Of special concern is the densiW effect, as the porosity effect on the fracture limit in the simple compression test has been studied b y many researchers in this field [4, 5]. Because fracture is dependent on the local ductility of the material and also on the localized tensile stress at the barrelled outer surface, the preform porosity has very little influence on the strain-tofracture in compression, from the viewpoint that although increasing porosity decreases the local ductility of the material it also decreases the tensile stress generated on the barrelled outer surface. If it should be in keeping with the results of this study, the decreased tensile stress in porous material is probably due to the strain reduction being much more influenced by decreased B than by increased barrelling (b).
198
(A)
{B).~_
0.8
/
: R o / T ----- 1
0.32
0.7
0.28
"-0
0.6
0.24
0.5
0.20
rn 0.4
0
:
V
:9=0.8
9=
X
:
0.9
9=0,7
X
0.16
0.3
×
012
=
~o/°7= 1
02
''
~x
0.08
---2
= 3
0.1
g=0.9 9=0.8
o
v 0
0.04
[
o.1
0.2
1
0
0,3
I
i
i
o.1
o. 2
0.3
Fig. 2. Showing the effect of friction factor on: (a) the volume-change weighting factor; (b) the bulging factor.
(A) Ro/T
V
o.9
=
1
---- 2
- -
m
---- 0.1
---
-~
= 0.2
----:
~
=
X
=
o\
022
.
O.lO
Ro/T ,,
= 1 =2
; O3 l
__×: ~
0.20
3
•
0.18
0.3
9.-,, x~%.\ \',~\
/V
0.8
0.16i
0.7
0.14
0.6
0.12
0.5
01C
OA'
\.
•
~
=
0.3
~o
\ "-4-.. \..o
rn
0.08
v//X/' X
0.3
0.06
02
0.04
0.1
0.02
0
0 : V :
(e,) 0
~s
I
I
I
0.7
0.8
0.9
0
s
I
I
I
0.7
0.8
0.9
Fig. 3. Showing the effect of density on: (a) the volume-change weighting factor; (b) the bulging factor.
199 (B)
(A)
0.30
0.8
0
: "fi'r
=
0.1
V
: ~
=
0.2
X
: i"i'i
=
0.3 0.7
- - : 9
" ~ 0.7
0
: ~ V : ~ X:Ni"
i" ~
= 0.1 ----0.2 =0.3
~
0.27
X 0.24
= 9
0.8 0.9
=
0.21 "" ~V
~'--.
0.18
toO.5
~ X
1.5
v\
\\.,..%.,
x
o,.
0.4
0.3
:9 --'--:
X
o ....
0.6
=
V --:9 --'--:
9 = 0.8 9 =09
0.06
- -X ~
•~.-~
0.03
0.2 0
0.09
I
1.0
[
2.0
Ro / 7
I
3.0
0
I
1.0
I
2.0
~ t
3.0
Ro/T
Fig. 4. Showing the effect o f shape factor on: (a) the volume-change weighting factor; (b) the bulging factor.
Conclusions (1) It is established from upper-bound analysis of the simpler case with negligible friction that the volume change in the plastic deformation of porous material proceeds towards minimum energy dissipation. (2) From determination of the volumechange weighting factor (B) and the bulging factor (b), using the condition of minimum deformation-energy dissipation, it is noted that the former decreases with friction factor and shape factor and increases approximately linearly with density, whilst the latter increases with friction factor and decreases with density and shape factor. (3) The decreased tensile stress in porous material at the barrelled outer surface is due to the strain reduction being much more influenced by decreased volume-change weighting factor (B) than by increased barrelling (b). References 1 S. Shima and M. Oyane, Plasticity theory for porous metals. Int. J. Mech. Sci., 18 (1976) 285---291. 2 M. Oyane and T. Tabata, Slip line field theory and upper bound theory for porous materials. Plasticity and Processing (Japan), 15(156) (1974) 43--51.
200 3 Hung Kuh Oh and Jae Ho Mun, A study on the ring compression of porous metal. Proceeding of the spring conference, 1982. Korean Society of Mechanical Engineering, Seoul, Korea. 4 H.A. Kuhn, C.L. Downey, Material behavior in powder preform forging. J. Eng. Mat. Tech., (Jan. 1973) 41--46. 5 G. Bockstiegel and H. Olsen, Processing parameters in the hot forming of powder preforms. Presented at the Joint Powder Metallurgy Group, Bristol, Nov. 1971. 6 B. Avitzur, Metal Forming. McGraw-Hill, New York, pp. 102--110.