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An upper bound solution of tube drawing
Joumalof
Materials Processing Technology ELSEVIER
Jourrnal of Materials Processing Technology 63 (1997) 43-48
An upper bound solution of tube drawing Kyung-Keun Urn, Dong Nyung Lee Division ofMaterial Science and Engineering, and Centerfor Advanced Materials Research, Seoul National University, Seoul 151-742, Korea
Abstract An upper bound solution of tube drawing process using a fixed tapered plug has been obtained, which reduces to a solution for tube sinking by setting friction factor between tube and plug at zero. Effects of various process parameters have been discussed based on the solution. Keywords: upper bound solution, tube drawing, tube sinking, fixed tapered plug, semi-die angle
1. Introduction R o and,
The tube drawing is a process in which a tube is pulled through a conical converging die, reducing both the diameter and the tube thickness. The various types of plugs used in the tube drawing are fixed tapered plugs, floating plugs, moving plugs and cylindrical plugs. In the tube drawing with a fixed tapered plug, a stationary tapered plug is used to reduce the outer and inner radii as well as a reduction in wall thickness. The drawing stress, which is the longitudinal stress in the tube wall at the exit of the die and plug, is determined by experimental analysis or numerical FEM methods. The experimental methods are very costly and time consuming. The analytical methods, such as slab analysis, slip line field solution, and upper bound solution, are still useful tools for drawing stress and for understanding effects of process variables, even though FEM has been used to simulate deformation processing last twenty years. A slab analysis of tube drawing was obtained by Hoffman and Sachs[l]. Guo et al.[2] dealt with tube sinking by an upper bound approach. Collins and Williams[3] obtained slip line fields for axisymmetric tube drawing. FEM solutions for tube drawing have been attempted by many authors [4-7]. The purpose of this study is to obtain an upper bound solution of tube drawing process using a fixed tapered plug.
To and to the final outer diameter and inner radii, and T f , using a die of semi-die angle at and a plug of semi-plug angle f3, a constant shear stress 'ria is assumed to prevail at the interface between the die and tube, while a constant shear stress -rf is assumed to prevail between the tube and plug. Figure lea) shows the flow lines and Fig.l(b), the hodograph. The paths of all particles are horizontal before crossing AA' and after exiting across BB'. The planes AA' and BB' are assumed to be vertical, which is a good approximation, for sinking, according to a FEM grid distortion in Fig.2. Between the inlet and outlet boundaries, the flow lines are straight and converging. All particles on a common vertical plane such as CC' have a common horizontal velocity component, Vx , but such velocities constantly increase from AA' to BB'. Upon crossing AA', a particle suffers a velocity discontinuity, v' A' that is proportional to its distance y from
Rf
the center line. At the outer surface, where y = R o , the velocity of a particle abruptly changes from v0 to
For the drawing of a tube from the initial outer and inner radii, 0924-0136/97/$15.00 © 1997 Elsevier Science SA All rights reserved PII S0924-0136(96)02597-6
due to the
velocity discontinuity V"A(S) = Vo tan a . At the inner surface, where, the velocity of a particle abruptly changes from v 0 to VQ( A) due to the velocity discontinuity V"A(Q) = V
2. Upper bound model
V s( A)
line, V"A(y)
otanp. For a particle at a distance y from the center
the =
velocity
(;o}otana .
discontinuity
along
AA'
IS
Kyung-Keun Urn, Dong Nyung Lee / Journal of Materials Processing Technology 63 (1997) 43-48
44
Q
-r-t-+--rF'-+--"~
lA'
_-_lir_-L-_:~_LLI' Lr: (b)
(a) Fig.l. (a) Flow lines and (b) a partial hodograph.
where vf is the horizontal velocity of a particle at the exit. Therefore, the rate of energy dissipation over the entire area at the exit is :
IIIIIIIII~
~~cl~I\~\mm~I~\
.. (4)
Fig.2. FEM mesh distortion after tube drawing.
At the entry, the rate of energy dissipation over a differential ring area, dA, at any arbitrary y is :
.
dWA = 21r Y dy k = 21r
It follows from Eqns.(2) and (4) that the total rate of energy dissipated along these two discontinuities is :
v' A(y)
k~tana y2 dy
(1)
Ro
where k is the shear yield stress of tube material. The rate of energy dissipation over the entire area at the entry is:
(2)
where Ao
= 1r(Ro 2 -
2
To )
is the initial cross sectional area of
the tube. At the exit, BB', the rate of energy dissipation over a differential ring area at any arbitrary y' is :
dW
B
= 21r
y'dy'k v'
•
Next, the frictional energy expended along the die-material interface and the plug-material interface is calculated. A ring slab whose outer and inner radii are R and T at some arbitrary position in the deformation zone is considered. The outer and inner contact areas of the slab are dr/sin a and dr / sin P, respectively, as shown in Fig.l(a).
•
The rate of energy dissipation dW a at the outer interface
f
is: dR a vS(C) sma
• a
B(y )
21r k v f
(5)
.2,
(3)
dWf = 21rR-.-Tj
(6)
---"-tan a y dy
Rf
where
.f
and
vS(C)
are the shear stress and velocity of a
Kyung-Keun Um, Dong Nyung Lee I Journal of Materials Processing Technology 63 (1997) 43-48
particle at the outer interface at position x, respectively. The velocity is related to the horizontal velocity V x :
45
In the homogeneous deformation, there is no shear strain. Therefore, the homogeneous deformation work per unit volume, dwH , is
(7)
(15) Constancy of volume requires: For tube drawing, Eq.(15) may be expresses as : (8) It follows from Eqns.(7) and (8) that:
(16) where
(9)
Therefore, the rate of energy dissipation at the outer interface is given by:
u] ,
u,
and
are the axial,
u ()
radial and
circumferential stresses, respectively, and &], &, and &0 are the axial, radial and circumferential strains, respectively. The stresses and strains are the principal quantities. The Levy-Mises equations may be expressed as :
(10)
(17) It follows from the geometry that: tanp r=--(R-Rf)+"f tan a
(11) The axial and circumferential strains may be expressed as :
The combination ofEqns.(10) and (11) give: (18) •
a
wa = 21fT; vo( R o
2
2
-"0 )
sin a cos a
f
JRO
(19)
RdR 2
2
2
Rf (1- t )R +2t(Rf t-rf)R + (2tR f "f - t R/ - r/) 2,'"A
=
I
V 0
0
•
[1
sin2P
It follows that:
Ro+Rot-Rft+rf --In-----'--"l+t Rf-rf
(12)
Ro+Rft-Rot-rf ] +---In--'---'-----=--~ I-t R f -rf 1
(20)
where t = tanp I tan a Similarly, the rate of energy dissipation at the inner interface is given by:
Jirp = 2Ti Aov• . sin2P
f
[2 1+t
ln
2J
3
(Rf-rf)t
(13)
1
R. +Rft- R.t- rf ] (Rf-"f)t
where u. is the uniaxial yield stress. The homogeneous strain may be expressed as :
is the shear stress at the inner interface.
Finally, the rate of energy of dissipation due to homogeneous deformation is calculated. The plastic deformation work increment per unit volume dw is given by :
It follows from Eqn.(17) that Eqn.(22) becomes: 2
dw = U11&11 + U22&22 + U33&33
+
2
2 2 =-dA.u.
+ R.t- Rft+ rf
I-t
If
2
1 [ (u]-u,) +(u,-u()) +(u()-u]) dwH=3dA.
(21) R.
+--In--~--=--''-
where
Substituting Eqn.(17) into Eqn.(16) and simplifying:
2(U12&12
+
U23&23
+
U3]&3])
2 ]/2
d&H=-(l+y+y)
(14)
J3
2
d&]=-dA. u 3 •
(23)
Kyung-Keun Urn. Dong Nyung Lee I Journal of Materials Processing Technology 63 (1997) 43-48
46
and it follows from Eqns.(2I) and (23) that: (24) (27)
Therefore:
(25)
where A o and AI are the initial and final cross sectional areas. For rod drawing,y= -1/2 and in turn
&H
= ]0"0 (1+y+y2)112 A v
Coo
v3
as
rr = mak
and
= &] = lnAol AI
and WH = 0" 0&]' which as well known relations. The rate of energy dissipation due to homogeneous deformation is given by :
wH
where rr and rf may be expressed rf = m Pk with m being the friction factor.
ln~ A
(26)
I
Equating the external work rate, O"dAlvl =O"dAovo, to the sum of Eqns.(5),(l2),(13) and (26), and solving for the drawing stress O"d :
3. Discussion 3.1. Drawing with plug
Fig.3 shows drawing stresses calculated using Eqn.(27), along with slab analysis results[8] and FEM results[9]. The drawing stresses increase linearly with friction factor. The drawing stress obtained using Eqn.(27) is the highest for a given friction factor, as expected. For given initial and final tube dimension and friction factor, the variation of drawing stress with semi-die angle a is shown in FigA. The optimum semi-die angle which makes the drawing stress minimum, is mainly determined by the internal shear and the friction. For given initial and final tube dimensions, optimum semi-die angle and the related drawing stress increase with increasing friction factor (Fig.5).
3
1.5
Contribution to ad
--Upper bound [eq.(27)] ------ Slab solution [8] •
FEM [9]
;
------ Homogeneous deformation ---- Internal shear - - Die-material friction .....-_ .. Plug-material frcition
2
bO - "C. 10 b
0.5
0.0 L-----'-_L...-..........._ L . -..........._L.---'----JL----'----J 0.00 0.02 0.04 0.06 0.08 0.10
o
5
15
20
m Fig.3. Tube drawing stress O"d calculated by various methods as a function of friction factor m. (Ro = 33.5mm, ro = 30mm , R r =27mm, ro =24mm, a=13° and m a =m P =m)
FigA. Tube drawing stress O"d as a function of semi-die angle for m=O.Ol. (R o = 33.5mm , ro =30mm, R I =27mm, ro = 24mm )
47
Kyung-Keun Urn, Dong Nyung Lee /Journal of Materials Processing Technology 63 (1997) 43-48
20.....---..---..--......--......--......--......--..., 1.0
16
-
1.20
0.8
1.10
-
0.6 ~
12
o
-.-
0
~ 8
Ro I Rr= 1.2 m =0.07
1.15
Cl
~12 ___ 9
0
en -;.. en
6 1.05
0.4 1.00
0.2
4
0.95
o ...J...-_...J...-_....L-_-'-_-L-_-'-_-L-_...J 0.0 0.00
0.02
0.01
0.03
m FigS Effect of friction factor on optimum die semi-angle a and drawing stress (T d' ( R o = 33.5mm , ro =30mm ,
Fig.6. Variation of wall thickness ratio, Sf/So, with semi-die angle a and initial tube dimension.
R f =27mm, ro =24mm)
3.2. Sinking 1.25
Since the inside of the tube is not supported in tube sinking, the sinking stress (T d( sink) can be obtained by setting = 0 in
-rf
1.20
Eqn.(27). It follows that:
--~~-----------
..._0'-
1.15
.=15'
en0
cif
(28)
1.10
..-
1.05 1.00
--_•.. _...._---
0.95
For given Ro/Rf and friction factor, the ratio of final and
0.90 0.00
0.05
initial wall thickness, Sf/So, which make the drawing stress minimum, can be obtained from Eqn.(28). The wall thickness ratio, Sf/So, increases with increasing Do/So, while it is insensitive to a, where Do = 2Ro (Fig.6). For given a and Ro/Rf values, Sf/So increases with friction factor and the increasing rate decreases with increasing semi-die angle (Fig.7). The results differ from those by Guo et al. [2], in which the wall thickness ratio decreased with increasing friction factor. This is due to the difference of velocity discontinuity planes in the two solutions. In this work, the velocity discontinuity planes are vertical, whereas Guo et al. used a little different discontinuity
0.10
0.15
0.20
m Fig.7. Variation of wall thickness ratio, Sf/So, with friction factor in tube sinking.
planes. FEM solution showed that the wall thickness ratio increased initially and then decreased with friction factor[8]. However, absolute values were not very different among the different methods. Therefore, this simpler solution is justified to be useful.
48
Kyung-Keun Um, Dong Nyung Lee / Journal of Materials Processing Technology 63 (1997) 43-48
4. Conclusions
References
1. The optimum semi-die angle which makes the drawing stress minimum, is mainly detennined by the internal shear and the friction.
[1] O.Hoffman and G.Shacs, Introduction to the Theory of Plasticity for Engineers, McGraw-Hill, (1953) pp.187-193 and pp.252-256. [2] N.C.Guo, Z.lLuo and G.Cui, Advanced Technology of plasticity - Proc. Fourth Int. Conf. on Technology ofPlasticity, (1993) 1023 [3] I.F.Collins and B.K.Williams, Int. J. Mech. Sci. 27 (1985) 225. [4] lRasty and D.Chapman, J. Mats. Eng. and Peiformance., 14 (1992) 547 [5] K.Sawarniphakdi, G.D.Lahoti and P.K.Kropp, J. Mats. Proc. Tech., 27 (1991) 179-190 [6] S.Urbanski, M.Packo, U.Stahllerg and H.Keife, J. Mats. Proc. Tech., 32 (1992) 531 [7] lM.Rigaut, lOudin and Y.Ravalard, Numiform 89, E.G.Thompson, R.D.Wood, O.C.Zienkiewicz and A.Samuclsson. (eds.) Balkema, Rotterdam (1989) 581 [8] K.-K.Um and D.N.Lee, unpublished results. [9] K.-K.Um and D.N.Lee, Proc. Fall meeting ofKorea Society for Technology ofPlasticity, Seou1,(1995) 76
2. The optimum semi-die angle and related the drawing stress increase with increasing friction factor. 3. For given Ro/Rf
and friction factor in tube sinking, the
wall thickness ratio, Sf / So, increases with increasing Do/So, while it is insensitive to a.
4. For given a and Ro/Rf values in tube sinking, Sf/So increases with increasing friction factor and the increasing rate decreases with increasing semi-die angle. Acknowledgement This study has been supported by KOSEF through RETCAM, Seoul National University.
Materials Processing Technology ELSEVIER
Jourrnal of Materials Processing Technology 63 (1997) 43-48
An upper bound solution of tube drawing Kyung-Keun Urn, Dong Nyung Lee Division ofMaterial Science and Engineering, and Centerfor Advanced Materials Research, Seoul National University, Seoul 151-742, Korea
Abstract An upper bound solution of tube drawing process using a fixed tapered plug has been obtained, which reduces to a solution for tube sinking by setting friction factor between tube and plug at zero. Effects of various process parameters have been discussed based on the solution. Keywords: upper bound solution, tube drawing, tube sinking, fixed tapered plug, semi-die angle
1. Introduction R o and,
The tube drawing is a process in which a tube is pulled through a conical converging die, reducing both the diameter and the tube thickness. The various types of plugs used in the tube drawing are fixed tapered plugs, floating plugs, moving plugs and cylindrical plugs. In the tube drawing with a fixed tapered plug, a stationary tapered plug is used to reduce the outer and inner radii as well as a reduction in wall thickness. The drawing stress, which is the longitudinal stress in the tube wall at the exit of the die and plug, is determined by experimental analysis or numerical FEM methods. The experimental methods are very costly and time consuming. The analytical methods, such as slab analysis, slip line field solution, and upper bound solution, are still useful tools for drawing stress and for understanding effects of process variables, even though FEM has been used to simulate deformation processing last twenty years. A slab analysis of tube drawing was obtained by Hoffman and Sachs[l]. Guo et al.[2] dealt with tube sinking by an upper bound approach. Collins and Williams[3] obtained slip line fields for axisymmetric tube drawing. FEM solutions for tube drawing have been attempted by many authors [4-7]. The purpose of this study is to obtain an upper bound solution of tube drawing process using a fixed tapered plug.
To and to the final outer diameter and inner radii, and T f , using a die of semi-die angle at and a plug of semi-plug angle f3, a constant shear stress 'ria is assumed to prevail at the interface between the die and tube, while a constant shear stress -rf is assumed to prevail between the tube and plug. Figure lea) shows the flow lines and Fig.l(b), the hodograph. The paths of all particles are horizontal before crossing AA' and after exiting across BB'. The planes AA' and BB' are assumed to be vertical, which is a good approximation, for sinking, according to a FEM grid distortion in Fig.2. Between the inlet and outlet boundaries, the flow lines are straight and converging. All particles on a common vertical plane such as CC' have a common horizontal velocity component, Vx , but such velocities constantly increase from AA' to BB'. Upon crossing AA', a particle suffers a velocity discontinuity, v' A' that is proportional to its distance y from
Rf
the center line. At the outer surface, where y = R o , the velocity of a particle abruptly changes from v0 to
For the drawing of a tube from the initial outer and inner radii, 0924-0136/97/$15.00 © 1997 Elsevier Science SA All rights reserved PII S0924-0136(96)02597-6
due to the
velocity discontinuity V"A(S) = Vo tan a . At the inner surface, where, the velocity of a particle abruptly changes from v 0 to VQ( A) due to the velocity discontinuity V"A(Q) = V
2. Upper bound model
V s( A)
line, V"A(y)
otanp. For a particle at a distance y from the center
the =
velocity
(;o}otana .
discontinuity
along
AA'
IS
Kyung-Keun Urn, Dong Nyung Lee / Journal of Materials Processing Technology 63 (1997) 43-48
44
Q
-r-t-+--rF'-+--"~
lA'
_-_lir_-L-_:~_LLI' Lr: (b)
(a) Fig.l. (a) Flow lines and (b) a partial hodograph.
where vf is the horizontal velocity of a particle at the exit. Therefore, the rate of energy dissipation over the entire area at the exit is :
IIIIIIIII~
~~cl~I\~\mm~I~\
.. (4)
Fig.2. FEM mesh distortion after tube drawing.
At the entry, the rate of energy dissipation over a differential ring area, dA, at any arbitrary y is :
.
dWA = 21r Y dy k = 21r
It follows from Eqns.(2) and (4) that the total rate of energy dissipated along these two discontinuities is :
v' A(y)
k~tana y2 dy
(1)
Ro
where k is the shear yield stress of tube material. The rate of energy dissipation over the entire area at the entry is:
(2)
where Ao
= 1r(Ro 2 -
2
To )
is the initial cross sectional area of
the tube. At the exit, BB', the rate of energy dissipation over a differential ring area at any arbitrary y' is :
dW
B
= 21r
y'dy'k v'
•
Next, the frictional energy expended along the die-material interface and the plug-material interface is calculated. A ring slab whose outer and inner radii are R and T at some arbitrary position in the deformation zone is considered. The outer and inner contact areas of the slab are dr/sin a and dr / sin P, respectively, as shown in Fig.l(a).
•
The rate of energy dissipation dW a at the outer interface
f
is: dR a vS(C) sma
• a
B(y )
21r k v f
(5)
.2,
(3)
dWf = 21rR-.-Tj
(6)
---"-tan a y dy
Rf
where
.f
and
vS(C)
are the shear stress and velocity of a
Kyung-Keun Um, Dong Nyung Lee I Journal of Materials Processing Technology 63 (1997) 43-48
particle at the outer interface at position x, respectively. The velocity is related to the horizontal velocity V x :
45
In the homogeneous deformation, there is no shear strain. Therefore, the homogeneous deformation work per unit volume, dwH , is
(7)
(15) Constancy of volume requires: For tube drawing, Eq.(15) may be expresses as : (8) It follows from Eqns.(7) and (8) that:
(16) where
(9)
Therefore, the rate of energy dissipation at the outer interface is given by:
u] ,
u,
and
are the axial,
u ()
radial and
circumferential stresses, respectively, and &], &, and &0 are the axial, radial and circumferential strains, respectively. The stresses and strains are the principal quantities. The Levy-Mises equations may be expressed as :
(10)
(17) It follows from the geometry that: tanp r=--(R-Rf)+"f tan a
(11) The axial and circumferential strains may be expressed as :
The combination ofEqns.(10) and (11) give: (18) •
a
wa = 21fT; vo( R o
2
2
-"0 )
sin a cos a
f
JRO
(19)
RdR 2
2
2
Rf (1- t )R +2t(Rf t-rf)R + (2tR f "f - t R/ - r/) 2,'"A
=
I
V 0
0
•
[1
sin2P
It follows that:
Ro+Rot-Rft+rf --In-----'--"l+t Rf-rf
(12)
Ro+Rft-Rot-rf ] +---In--'---'-----=--~ I-t R f -rf 1
(20)
where t = tanp I tan a Similarly, the rate of energy dissipation at the inner interface is given by:
Jirp = 2Ti Aov• . sin2P
f
[2 1+t
ln
2J
3
(Rf-rf)t
(13)
1
R. +Rft- R.t- rf ] (Rf-"f)t
where u. is the uniaxial yield stress. The homogeneous strain may be expressed as :
is the shear stress at the inner interface.
Finally, the rate of energy of dissipation due to homogeneous deformation is calculated. The plastic deformation work increment per unit volume dw is given by :
It follows from Eqn.(17) that Eqn.(22) becomes: 2
dw = U11&11 + U22&22 + U33&33
+
2
2 2 =-dA.u.
+ R.t- Rft+ rf
I-t
If
2
1 [ (u]-u,) +(u,-u()) +(u()-u]) dwH=3dA.
(21) R.
+--In--~--=--''-
where
Substituting Eqn.(17) into Eqn.(16) and simplifying:
2(U12&12
+
U23&23
+
U3]&3])
2 ]/2
d&H=-(l+y+y)
(14)
J3
2
d&]=-dA. u 3 •
(23)
Kyung-Keun Urn. Dong Nyung Lee I Journal of Materials Processing Technology 63 (1997) 43-48
46
and it follows from Eqns.(2I) and (23) that: (24) (27)
Therefore:
(25)
where A o and AI are the initial and final cross sectional areas. For rod drawing,y= -1/2 and in turn
&H
= ]0"0 (1+y+y2)112 A v
Coo
v3
as
rr = mak
and
= &] = lnAol AI
and WH = 0" 0&]' which as well known relations. The rate of energy dissipation due to homogeneous deformation is given by :
wH
where rr and rf may be expressed rf = m Pk with m being the friction factor.
ln~ A
(26)
I
Equating the external work rate, O"dAlvl =O"dAovo, to the sum of Eqns.(5),(l2),(13) and (26), and solving for the drawing stress O"d :
3. Discussion 3.1. Drawing with plug
Fig.3 shows drawing stresses calculated using Eqn.(27), along with slab analysis results[8] and FEM results[9]. The drawing stresses increase linearly with friction factor. The drawing stress obtained using Eqn.(27) is the highest for a given friction factor, as expected. For given initial and final tube dimension and friction factor, the variation of drawing stress with semi-die angle a is shown in FigA. The optimum semi-die angle which makes the drawing stress minimum, is mainly determined by the internal shear and the friction. For given initial and final tube dimensions, optimum semi-die angle and the related drawing stress increase with increasing friction factor (Fig.5).
3
1.5
Contribution to ad
--Upper bound [eq.(27)] ------ Slab solution [8] •
FEM [9]
;
------ Homogeneous deformation ---- Internal shear - - Die-material friction .....-_ .. Plug-material frcition
2
bO - "C. 10 b
0.5
0.0 L-----'-_L...-..........._ L . -..........._L.---'----JL----'----J 0.00 0.02 0.04 0.06 0.08 0.10
o
5
15
20
m Fig.3. Tube drawing stress O"d calculated by various methods as a function of friction factor m. (Ro = 33.5mm, ro = 30mm , R r =27mm, ro =24mm, a=13° and m a =m P =m)
FigA. Tube drawing stress O"d as a function of semi-die angle for m=O.Ol. (R o = 33.5mm , ro =30mm, R I =27mm, ro = 24mm )
47
Kyung-Keun Urn, Dong Nyung Lee /Journal of Materials Processing Technology 63 (1997) 43-48
20.....---..---..--......--......--......--......--..., 1.0
16
-
1.20
0.8
1.10
-
0.6 ~
12
o
-.-
0
~ 8
Ro I Rr= 1.2 m =0.07
1.15
Cl
~12 ___ 9
0
en -;.. en
6 1.05
0.4 1.00
0.2
4
0.95
o ...J...-_...J...-_....L-_-'-_-L-_-'-_-L-_...J 0.0 0.00
0.02
0.01
0.03
m FigS Effect of friction factor on optimum die semi-angle a and drawing stress (T d' ( R o = 33.5mm , ro =30mm ,
Fig.6. Variation of wall thickness ratio, Sf/So, with semi-die angle a and initial tube dimension.
R f =27mm, ro =24mm)
3.2. Sinking 1.25
Since the inside of the tube is not supported in tube sinking, the sinking stress (T d( sink) can be obtained by setting = 0 in
-rf
1.20
Eqn.(27). It follows that:
--~~-----------
..._0'-
1.15
.=15'
en0
cif
(28)
1.10
..-
1.05 1.00
--_•.. _...._---
0.95
For given Ro/Rf and friction factor, the ratio of final and
0.90 0.00
0.05
initial wall thickness, Sf/So, which make the drawing stress minimum, can be obtained from Eqn.(28). The wall thickness ratio, Sf/So, increases with increasing Do/So, while it is insensitive to a, where Do = 2Ro (Fig.6). For given a and Ro/Rf values, Sf/So increases with friction factor and the increasing rate decreases with increasing semi-die angle (Fig.7). The results differ from those by Guo et al. [2], in which the wall thickness ratio decreased with increasing friction factor. This is due to the difference of velocity discontinuity planes in the two solutions. In this work, the velocity discontinuity planes are vertical, whereas Guo et al. used a little different discontinuity
0.10
0.15
0.20
m Fig.7. Variation of wall thickness ratio, Sf/So, with friction factor in tube sinking.
planes. FEM solution showed that the wall thickness ratio increased initially and then decreased with friction factor[8]. However, absolute values were not very different among the different methods. Therefore, this simpler solution is justified to be useful.
48
Kyung-Keun Um, Dong Nyung Lee / Journal of Materials Processing Technology 63 (1997) 43-48
4. Conclusions
References
1. The optimum semi-die angle which makes the drawing stress minimum, is mainly detennined by the internal shear and the friction.
[1] O.Hoffman and G.Shacs, Introduction to the Theory of Plasticity for Engineers, McGraw-Hill, (1953) pp.187-193 and pp.252-256. [2] N.C.Guo, Z.lLuo and G.Cui, Advanced Technology of plasticity - Proc. Fourth Int. Conf. on Technology ofPlasticity, (1993) 1023 [3] I.F.Collins and B.K.Williams, Int. J. Mech. Sci. 27 (1985) 225. [4] lRasty and D.Chapman, J. Mats. Eng. and Peiformance., 14 (1992) 547 [5] K.Sawarniphakdi, G.D.Lahoti and P.K.Kropp, J. Mats. Proc. Tech., 27 (1991) 179-190 [6] S.Urbanski, M.Packo, U.Stahllerg and H.Keife, J. Mats. Proc. Tech., 32 (1992) 531 [7] lM.Rigaut, lOudin and Y.Ravalard, Numiform 89, E.G.Thompson, R.D.Wood, O.C.Zienkiewicz and A.Samuclsson. (eds.) Balkema, Rotterdam (1989) 581 [8] K.-K.Um and D.N.Lee, unpublished results. [9] K.-K.Um and D.N.Lee, Proc. Fall meeting ofKorea Society for Technology ofPlasticity, Seou1,(1995) 76
2. The optimum semi-die angle and related the drawing stress increase with increasing friction factor. 3. For given Ro/Rf
and friction factor in tube sinking, the
wall thickness ratio, Sf / So, increases with increasing Do/So, while it is insensitive to a.
4. For given a and Ro/Rf values in tube sinking, Sf/So increases with increasing friction factor and the increasing rate decreases with increasing semi-die angle. Acknowledgement This study has been supported by KOSEF through RETCAM, Seoul National University.