describing realistic deep drawing processes. Also potential improvements concerning the choice of die radius of curvature and the blank holding force are ...
Int. J. Mech. ScL Vol. 36, No. 2, pp. 121-135, 1994 Printed in Great Britain.
0020-7403/94 $6.00 + .00 ~) 1993 Pergamon Press Ltd
THE ROLE OF DIE CURVATURE IN THE PERFORMANCE OF DEEP DRAWING (HYDRO-MECHANICAL) PROCESSES A. SHIRIZLY, S. YOSSIFON a n d J. TIROSH Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel (Received 5 January 1993; and in revised form 6 May 1993)
A k t m c t - - A hydromechanicai deep drawing process (which replaces the conventional rigid blank-holder tool with a hydrostatic fluid pressure) is utilized to study the roles played by die curvature, interfacial friction, material hardening, etc. in deep drawing performance. The analytical study is based on limit analysis in plasticity (applying both the upper and the lower bounds simultaneously) with a special emphasis on the geometry of the die profile. The resulting relationships between the various parameters obtained through the bounds are backed by an independent numerical solution using Woo's finite difference scheme. The associated experiments, with which the limit analysis is compared, were conducted with aluminium blanks at various die radii and with various holding fluid pressures. The relatively close proximity of the above solutions, in describing the observed behaviour of the process, enables one to draw a few general conclusions about the strength of the limit analysis in describing realistic deep drawing processes. Also potential improvements concerning the choice of die radius of curvature and the blank holding force are indicated.
NOTATION E
Young's modulus
f shear traction H J n
P,P, r, 0, 2' re
R Rp R, Rd Re t
U
~7
current travelling distance of the punch upper bound on energy consumption, applied power strain-hardening exponent fluid pressure polar coordinates a point on the blank at initial position of the process ratio of width strain to thickness strain in uniaxial tension punch radius instantaneous outer radius of the flange die opening radius outer radius of the blank blank thickness (assumed constant) velocity energy rate of deformation energy rate of interracial shear energy rate of applied tractions energy rate of discontinuity in velocities
Greek letters or, p, 0 torodial coordinates It the current angle by which the blank wraps the die and the punch curvatures 0 expression defined in Eqn (29) Pd die radius of curvature Pp punch radius of curvature effective strain and effective stress ¢~0, ~0 material constants [in the flow stress o --- ao(~o + ~)~] radial, tangential and normal strain in zone I ~, to, Ep radial, tangential and normal strain in zone II radial, tangential and normal stress in zone I ~==, 0"80, ~7a0 radial, tangential and normal stress in zone II #t Coulomb friction coefficient between the blank and the die ~2 Coulomb friction coefficient between the blank and the rubber diaphragm dot denotes time derivative •* asterisk denotes that [] is kinematically admissible D r F denotes as surface of discontinuity
121
122
A. SH1RIZLY, S. YOSmFON and J. TmOSH I. I N T R O D U C T I O N
In order to regulate the blank holding force, which is considered as a major issue in the performance of the deep drawing process [1], a technical modification to t.he classical process was recently suggested [2]. It is based on replacing the rigid blank-holder tool by a controllable hydrostatic fluid pressure (as shown in Fig. 1). The modification is called a "deep drawing process with fluid assisted blank-holder" (or shortly, FAB). This FAB includes some features in common with the hydroforming process. For example, the reduction in frictional resistance to the drawing ([see Refs 3, 4]). On the other hand, the FAB avoids some features which seem to be disadvantages. For example, in a hydroforming process, the curvature of the blank at the lip is self-adjusted by the current fluid pressure and thus highly prone to wrinkling [3], whereas in the FAB process this redundant flexibility does not exist. Instead, the curvature is beneficially imposed by the rigid die, as in the classical process, and may essentially be optimized with respect to the limit drawing ratio (or other criteria to maximize the drawing, performance). The main goal here is to employ the limit analysis of plasticity [5, 6] (upper and the lower bounds) to study the interrelationships of a wide range of material and geometrical variables acting during the deep drawing process, but in particular the die radius of curvature, not previously considered. Experimental study of the deep drawing process with the new FAB machine provided a limited (though essential) background for the analysis. As a complementary verification to the prediction capability of the dual bounds, a pure numerical solution was obtained, using Woo's finite difference scheme [7]. The geometrical complexity of the die (torus) and the need to trace the evolution of the contact area of the blank with the die during the progression of the process, leads naturally to lengthy algebra. For brevity, the main body of the paper will be focussed only on the essentials. 2. L O W E R B O U N D
(a) Preliminaries A lower bound solution may be reached by employing a trial stress field, to be denoted as tr~°~ i t , which satisfies: (a) the equilibrium equation: eto~ = 0; ij, j (b) the yield condition:
j~2o)= .~Sij(o)Sit(o) ~
(1)
k2;
(2)
(c) the boundary conditions: aijt°~..t" = Ti on
sr
(where tractions T/are prescribed).
(3)
The normality and convexity properties of the plastic yield function lead to the following maximum work principle;
I(~l °' -
t~u)~ijdv
~< O.
(4)
The inequality (4) states the lower bound theorem. It renders, after some mathematical manipulation:
f tr~°'ntu,ds~f~o~°'ds.
(6)
Performance of deep drawing processes
123
Force (F)
Fluid pressure (P)
er Diaphragm
FIG. 1. Description of the deep drawing process with fluid pressure assisted blank-holder (FAB). The rubber diaphragm is used here only to prevent fluid leakage. Otherwise, it does not interfere with the process.
124
A. SHIRIZLY, S. YOSSIFON and J. TIROSH
z
I ~° !~ooo~ ~IPll
//
±
N~NN~ "
' ~
_ ~ ~ ~
ZonelII I "xx ZoneW
~
Re
Pd
FIG. 2. The geometry of the workpiece and final products (produced with dies of various radii of curvatures as shown). For computational convenience the deformation along the blank is subdivided to various zones (as Zone |, Zone lI, etc.) each of which has its own flow pattern.
Performance of deep drawing processes
125
Equation (6) will have the final lower bound form (with superscript LB), to be used here as: FLn(U°) =
C ~°}ds"
(7)
dsiJ
(b) The trial stress solution in deep drawinff The plastic domain is partitioned, for convenience, into five distinct zones as described in Fig. 2. For each zone we will set the appropriate equilibrium equations to be satisfied by the trial solutions, the yield condition and the requirement of continuity of traction between the zones.
Zone I: stresses in the flanffe area. The equilibrium equations in the flange area are: ~0" r
~O'r=
c~-7-+ ~ ~O'rz
~-%-+
tr0
O"r - -
+
= O,
(8a)
-&-z + --r = o.
(8b)
~¢~z
r ¢rrz
The radial tensile stress, o, and the compressive hoop stress, no, are "nearly" principal stresses. They are exactly principal stresses when frictional shear across the thickness of the blank is neglected. This is not the case here, although the friction in the flange is relatively small in view of the fluid pressure acting on one of its face. Consequently the Tresca yield criterion is written approximately as: ~, - o0 = 2K.
(9)
For more generality, one may assume that strain hardening exists in the form of: 2K = tTo(g1 + Co)",
(10)
where g~is the current equivalent strain in zone I and eo is the initial strain. Using Eqn (10) in the lower bound analysis may be applicable only if the strain path is known a priori, which is very rare. Otherwise, the strain can be approximated, for example, via an admissible velocity field (as is done in the upper bound analysis). By doing so, however, the resulting solution is no longer considered as a strict lower bound, but merely as an approximation to it. The suggested general expressions (to be formulated next) can always be reduced to the strict lower bound by letting the strain hardening exponent [n in Eqn (10)] approach zero. In the above three equations (8a, b) and (9) there are four unknowns: ~,, no, tr~ and ~%. It is therefore adequate (in principle) to guess only one admissible stress component and solve for the other three. The suggested trial guess is a linear variation of the shear stress through the thickness of the blank. Thus, we take: o.(o)
z
,: = (#2 + #*)Pt-- # ' e '
(11)
and note that the shear stress conditions on both sides of the blank are satisfied, namely:
{
a~o)=_/zlp a~o~ ~2e
@ z=0, @ z=t.
(12)
Inserting Eqns (11) and (12) in Eqn (8) and noting that on the top side of the flange the fluid pressure is prescribed (oz = - P ) and that at the outside rim (r = Re) the radial traction is prescribed by the fluid pressure (o, = - Pc), one can integrate Eqn (8) to get the other three admissible components as: (13)
A. SHIRIZLY, S. YOSSIFON and J. TIROSH
126
a~°) = -P(/~* +
t
r) +
aO(E I "~ r
,;(oo~= ~o)_ ~o(~,+
Co) dr - Pc,
(14)
(15)
~o)".
For a strict lower bound solution, one recalls that both the friction coefficient and the strain hardening exponent in the admissible stress solution (13)-(15) should be set equal to zero.
Zone l h stresses along the die curvature. A similar procedure is undertaken in the die curvature (zone II). The equilibrium equations are:
I(
Op + p %p - %"
+
aOpp.+ p ~ 06
+_l r [(ago -
%:)sin ct - %,cos ct] = 0,
(16)
) + -r1[(aoo -
a,,)cos at - %,sin 0t] = 0,
(17)
~ ] 2%,
where r = (Rd + Pd) -- psin 0t. All the expressions above were developed at p = Pd + ~- The Tresca yield criterion is written approximately as: (18)
O'~t, - - O'00 = 0"0(~II "4- EO) n.
In the above three equations (16), (17) and (18) there are four unknowns: ~,,, a00, app and %,. Here we can first evaluate (approximately) the normal stress, %o, acting on the curved die. Using Eqn (16) with the boundary condition of zero shear traction on the outer surface, and taking into account the relative thickness of the blank, 0%0/0 p ~ %o/t, it may be shown that: ¢r~
6o0 •
sin a(o)= P r PP 1 1 1 . -- -~. . . . sln~ t p r
(19)
For the case when p, r>>t, one can approximate the normal stress further to obtain:
o,o,,( op
~ %,-
%0sine
)
.
(20)
This expression coincides with the one obtained previously by Sitaraman, Kinzel and Altan [8]. Again, as in zone I, we guess a linear variation of an admissible shear stress component: a, = # l ( Ptd + 1 - - -Pt) - a oa (°), (0)
(21)
which satisfies:
6(0)=0 @ ,~(0) 0 vp~ lq~pp
%=
pd -F t, P pd.
(22)
=
Now we can solve for the other components by integration of the equilibrium equations while enforcing continuity of the normal stress with the solution from zone I, namely: (0) O'0t ~t ] ~ = 0 ~ 0(0) r, [r=Rd+Pd "
(23)
The results are:
,60) rOtOr
~-
f~--tD(r, ct, p)ao(g,, + %~'exp ( - - # , (Pd + t)ct)dct + a(°' . . . . R,+p, P
(24)
e x p ( - U , (Pd t +) C t ) p .(0)
,,.(0)
00 = ,,~ - ao(h, + Co)".
(25)
Performanceof deep drawing processes
127
where: @(r, ot, p) = Pcos~ - Pt r
sinct --
+ - r
sin2~t
- (-Prsin ct - 2 ) ( ~ + ; - P ) s i n ~tI • The solution given above, when reduced to rigid-perfectly plastic materials, reads: Zone I:
(Re - r) + aoln Rc _ Pc, t r
air O) = - - e ( / ~ l
+/~2)
a~o) ---- _
+ #2)(R¢
p(p,
- - -t
r)
+ aoln Rc r
(26)
PC -- ao,
(27)
°")=PI(/~1+/a2)t(:-l)+#l~(l-~)-11"2 r
(28)
Zone II:
f 2 - - O ( r , ~ , p ) a o e x p ( - - # t (Po+ + t)g)d~ p
a(O)
a~=
o(o) O0 -~" .Ao) - -
....a,+p,
a(o)
((pd+t)a) exp - Pl P
,
(29)
(30)
ao,
#(o)= -pt ( t r(°' ~ - -Pra(0° 'sin~) . ~PP
(31)
(c) The punch load The normal stress a~l , at the end of the curvature (zone II) is "carried" (by the traction continuity) through the tensile stress of zone III into the bottom of the cup (shown in Fig. 1). It means that:
a(O)
• a,v ,=#
= 040) Rd ~" a=aRp
psin ft. pp ~ psinff
~- Pd - -
(32a)
Therefore, the total lower bound load, •LB, provided by the punch is: ~LB
F . . . . 2~Rpteo
a(o) _ • ~,vI~- p sin fl ao
(32b)
and the solution is completed.
3. UPPER BOUND ANALYSIS (1) Admissible velocity fields In the following analysis, an attempt has been focussed on generating an admissible velocity field which will include, beside the flange area, the curvature of the blank dictated by the die curvature. Presently, the curvature of the product idictated by the curvature of the punch) is also accounted for, but in the experiments which follow it remains fixed. The velocity field throughout the blank is entirely smooth (namely, without velocity discontinuities), provided that the thickness of the blank is assumed to remain unaltered. This assumption is permissible, since it does not contradict the upper bound rule. In practice, the wall thickness of the product was found to be changed between - 5 % and up to 10% (including the residual flange) at drawing ratio of 1.8 (see pp. 69 and 70 in Ref. [9]).
128
A. SHIRIZLY, S. YOSSIFON and J. TIROSH
Because of the non-steady nature of the process, the computation of the total strains needs a continuous tracing of the current geometrical changes with respect to the initial configuration. For instance, the hoop strain at a particular material point along the blank at distance r depends on the original position of the same material point ro at the very beginning of the process. Therefore, admissible strain components [denoted with superscript (*)] can be written (for proportional loading path) in a logarithmic form as:
eo
= ln( 2nr'~ \2--~ro/"
(33)
If we assume that the thickness of the blank remains constant throughout the process then the radial strain is obtained from the incompressibility condition as: e , = --e0.
(34)
The expression for ro in Eqn (31) can be expressed in terms of a single geometrical variable, say fl (the current angle by which the blank warps the curvatures). Since the operating speed Uo is held constant, ro = ro(~) varies during the process by the rate of change of ft. The strain rate is then:
d
r
e'=d(time)lnr-~
=
1 Oro(fl) dfl . ro(fl) Off d(time)'
(35)
t~ro(fl)/t~fl and dfl/d(time) are given in the Appendix. The velocity field is then derived from the line integration of the appropriate strain rate as:
U, = J'~,dr.
(36)
This procedure is repeated for all the different zones and continuity of the velocity throughout is assumed via the constant of integration. Having the strain rate distribution in the current configuration, one can compute the work rate in the usual way, as follows. The upper bound on energy consumption is partitioned conveniently into: J ~ l~i~ + l~f* + I~ly - ff'~,
(37)
where the actual force F applied during the process is:
F = J/Uo.
(38)
The work rate of deformation is expressed as [10]:
a'7=ff, *dv= fv *dv,
{
(39)
# = ao(~ + Co)", _.,
/2T1-
) r,
(40)
The work rate over the boundary between the tool and the material is given as:
I ~ = ~ f l A U ,dsf,
(41)
where, the shear traction in Eqn (41) is taken as f = laP (P is the prescribed fluid pressure for holding the flange; see Fig. 1). Since the velocity field in the present work is continuous (AUr = 0), the work rate along lines of velocity discontinuity IS'* = S~rOAUrdsr is zero.
Performance of deep drawing processes
129
The work rate supplied by the surface tractions I~'* =SsT~ U~dsr is caused by the applied hydrostatic fluid pressure acting along the thickness of the blank's outer rim, though it remains relatively small.
The admissible strain and velocity fields. Zone I: Rc i> r/> R d + Pal,
co,
A~
d(time)
r 2 _ (R d + pd) 2
+I]
fl' ~=0
~*r[
=
--
~,*,=
~*,,
O,
(42)
Uo*= O, ~f2
u,*, =
u,*,,i::o
°
t a n - ' (x/f~ : (Rd +- pd)2(r = R-*)
+
~x f 2 __ (R d + pd)2 + rR
I r > / 0 ,
= 0,
] (46)
Ug =O, U~* = - Uo.
4. E X P E R I M E N T A L
STUDY
A series of experiments were conducted on aluminium blanks (AI 1100) of thickness t = 0.47 m m having the following properties: Rolling direction
R value
ao [ M P a ]
n
0° 45 ° 90 ° Average properties
0.93 0.72 0.82 0.80
130 129 130 130
0.217 0.211 0.218 0.215
The average normal anisotropy: R =
Ro + 2R4~ + R 9 o ~ 0.80. 4
The planar anisotropy (not accounted for in this work) is: AR =
Ro + 2R45 + R 9 o - 0.155. 2
The constitutive equation is therefore: tr = 130(e + 3.87 x 10-4) °215 [ M N m -2] and the (normalized) Young's modulus is E/ao = 538. The diameter of the punch was 2Rp = 50 mm, the punch radius of curvature was pd = 5 m m (pd/t = 10.63), and the die radii of curvature were pp = 2, 5, 6.5 and 8 mm. The blanks in the tests had an initial diameter of 90 m m (hence the drawing ratios for the tests were 1.8). The blanks were imprinted with circles and squares by photo-chemical plating to visually observe the strain magnitudes and directions. The oil lubricant was commercial Molykote (a paste of oil with MoS2). Each test was conducted under a prescribed (constant) fluid pressure. The fluid pressure varied incrementally from test to test (at various die radii of curvature) in order to reveal (a) the buckling locus and (b) the rupture locus for any given radius of die. A series of experiments were conducted under the same constant pressure in order to inspect: (a) the variation of thickness of the drawing cups and (b) the punch force vs punch stroke for various die radii.
Performance of deep drawing processes
131
5. R E S U L T S
The predictions of the loading paths by the rigorous dual bounds [upper bound solution with maximal friction coefficient, and lower bound with minimal (i.e. zero) friction coefficient] are given in Fig. 3. It is seen that the experiments conducted under the laboratory conditions (where friction is approximated as/~ = 0.15) are scattered along a path which is contained between the bounds. In addition, an independent numerical solution (using a classical finite difference scheme [7]) closely follows the experiments. The large distance between the upper and the lower bounds, although consistent with the notion of bounds, is not too practical for designers, except perhaps, as a "first iteration" for getting the order of magnitude of the required load. One may, however, approximate the solutions by replacing the extreme values of the friction factors in the solution by the value characteristic of the laboratory conditions. In this case, the friction coefficient was set to be 0.15 and the strain hardening exponent was set to the measured value n = 0.215 (rather than n = 0). This brings the predicted lines, shown in Fig. 4, substantially closer to each other. They still embrace the experimental data and can be used, when needed, to predict the behaviour of the process. The role of the die curvature anticipated by the approximate bounds is given in 1.2
g.
i
e~ eq
i
~
t
t
i
i / i ~ i U p p e r BOund ................... i......../...............}............ ~ . i .............. Lower Bound i"7 ............~....' ~ ...... Numerical
\
0.8
. . . . . . . . . . . . . . . . / ~ ! + ~ " :
0.6
•
..................
~o
i
0.4
.,.,
i
........... P d / t = l O . 6 3 --
"7.'--.
~°T*°.°
....
i
.............. i.........................
"~..
i
.................... ~:-*-~ ............ i............................ ~.......................... i.............. ~Ji ........................
I
0.2
~*
:
~
.o'',
i
?'- g = O 15
i
!
O
Z
0
0
0.2
0.4 0.6 0.8 Normalized Stroke [I-bRp]
I
1.2
FIG. 3. Comparison between experiments, limit analysis (upper and lower bounds) and numerical solution (by finite difference method). The limit analysis solutions are drawn here using rigorous values of friction (namely, maximum friction resistance along the interface with the tools in the u p p e r bound solution and zero friction resistance in the lower bound solution). Hence, the gap between them is relatively wide.
"~ 0.8
..................................................................................................
0.6
.................... U p p e r b o u n d .........................i ............................i........... g = 0 . 1 5 .................. i ~ ~ . . iI n=0.215
0.4
.........................÷. . . . . . . . . . . . . . . . . . . . . . .
eq
~o
Analysis Finite Difference - - ~ - Experimental - - - * - .... !
i Po/t=10.63
--~
0.2
Z
0 0
.......................
.................................
0.2
0.4 Normalized
0.6 Stroke
0.8 [H/Rp]
1
1.2
( Drawingratio 1.8., blankholdingpressure20 bar, lubricant: MoS2) F r o . 4. Comparison between experiments, approximate limit analysis and numerical solution (by finite difference method). The approximate nature of the upper and the lower bounds solutions are obtained using the specific value for the friction coefficient, taken from the experiments, instead of the limiting values used in F i g . 3.
A. SHIRIZLY, S. YOSSIFON a n d J. TIROSH
132
~
0.7
~, 0.6 e~ 0.5 0.4 ~, 0.3 .~ 0.2 N ~ 0.1 Z 0
........... /........i.;,::.:...::. ........... i .......................... i............................ i........." .....~
0
0.2
0.4 0.6 0.8 Normalized Punch Stroke [I-I/Rp]
.
_
l
1.2
(P= 20 bar, g = 0.15, n=0.215, &awing ratio 1.8.)
FIG. 5. The effect of die curvature on the drawing force via an upper bound analysis. It is seen that the greater the curvature, the lower the force needed for the drawing process.
0.5 '
!
i
I
,
'
~
~ 0.4 ~'0.3
.... ~ .....
~
'
i
i 10.6i 13.82 .17i02 ! ,.25 ~ ; . ~ ~ ~
i
2~3
ii ................... ~.......................
~.....i..":.i'-..:..-.-..-.i.....................
~
i--~
~
.
0.2
..........
,i~"
~V
i
~
~ "~
..........i].........................
~ V .................................................... ~'~*~ i........................ IH
~ 0.1 o
Z
" ~ i
¢
0
0.2
i,
0.4
,
0.6
i
0.8
1
1.2
Normalized Punch Stroke [H/Rp] (P= 20 bar, g = 0.15, n---0.215,&awing ratio 1.83
FIG. 6. The effect of die curvature on the drawing force via a lower bound analysis. It is seen that increasing the die curvature causes only minor effects on the driving force. It shifts its peak value "downstream".
0.7 1 0.6
I
! ....................... ;.......................... T.................. .....
................. ~ ; r , ~ ~ - ? - t o : ~ - ~
0.5 .......................+.--../-----./.iT-;----~÷:~: ........... ~
...................... i .........................
0,4
i/,"
i
0.2 .........
o/'"
IR,
!~_
i
0.2
i
,
i . - ...................i
',1
i
0.4 0.6 0.8 1 Normalized Punch Stroke [H/Rp]
1.2
(P=20 bar, tt = 0.15, n=0.215, drawing ratio 1.8)
FIG. 7. The effect of die curvature on the drawing force via numerical solution (by finite difference scheme). The effect is shown to be somewhat similar to the limit analysis solution of Fig. 4.
Performance of deep drawing processes
133
0.7 ~o 0.6
[ P_-0
P'40ba
e--
" p=p
¢:h
-i
T
~
i
...... "-'-'~, "......... i.................. i................ i .................
0.5 ~
0.4
H
0.3
oN
0.2
c~
O.1 Q
Z
0 0
0.2 0.4 0.6 0.8 Normalized P u n c h Stroke [H/Rp] (Die radius 5 ram, Friction coefficient 0.15)
FIG. 8. The effect of the blank holding fluid pressure on the punch force path. In addition, the dotted lines indicate what would have been the punch force path if the fluid pressure traction on the rim had been accounted for. 0.8
6' "~ t'q
0.7 0.6
~, 0.5
!iiiiiiiiiii...................'. . . !!!i!!!!!!!i
0.4 O
0.3 "d
0.2
O
0.1
R,
Z
"'"
0
0
i
I
................i ..............................
0.2 0.4 0.6 0.8 Normalized P u n c h Stroke [H/Rp] (P= 20 bar, Ix = 0.15, Pd/t=10.63, &awingratio 1.8)
FIG. 9. The effect of the strain-hardening exponent n on the punch force "path. it is seen that the higher the n value, the lower is the peak load path of the punch.
2 1.6 Upper B o u n d ~
i
.2
~_. 0.8 "N
0.4
O
Z
Oo
I
0.2
i
0.4 0.6 0.8 Normalized stroke [H/Rp]
(drawing ratio 1.8.blankholdingr
1
pressure 20 bar,friction coefficient
1.2 0.15.)
FIo. 10. A solution to the loading path of the punch of a deep drawing process conducted with a sharp die corner. Comparison between a strain-hardening material and rigid-perfectly plastic material. The fact that the loading path in the n = 0 case is a monotonically decreasing function (as opposed to a "bell-shaped" configuration) is attributed to the sharp die corner.
134
A. SHIR1ZLY,S. YOSSlFONand J. TIROSH
Figs 5 and 6 along with the numerical solution shown in Fig. 7. It is shown that, in general, the greater the curvature, the lower is the punch load with some shift in the peak along the loading path. A unique characteristic of the F A B process is the presence of a fluid pressure acting on the flange area and along the perimeter of the rim simultaneously. It is seen that by increasing the fluid pressure level by a factor of 2 (from 20 to 40 bar as in Fig. 8), it causes a relatively small increase in the load (about 20% in this case). This is a useful advantage as c o m p a r e d with the hydroforming process suggested earlier [3, 4] where the punch load under the same conditions is substantially higher (by more than 100%). In addition, Fig. 8 indicates that the pressure acting on the rim has only a minute effect on the load. The incorporation of the die radius of curvature into the analysis sheds light (see Figs 9 and 10) on the difference between a classical model of deep drawing (which is d o n e with a sharp die corner) and the present model (with a finite die radius). The presence of the die radius in the limit analysis (with perfectly plastic material) gives the flow resistance a "bell-shaped" feature, which is closer to reality, as shown in all the previous figures. A sharp corner solution, however, leads to an entirely different path (a monotonically decreasing function, shown in Fig. l0 for n = 0). Such a description has not yet been validated experimentally, although it is consistent with independent analytical solutions (for example, Budiansky and W a n g [l 1], C h a n g and K o b a y a s h i [12], Tirosh and Sayir [13]). So, the incorporation of die radius of curvature (no matter how small) in sheet-forming analysis seems to be essential in describing the drawing process more realistically.
Acknowledoement--The authors are indebted to the J. W. UIIman Center for Manufacturing Systems and
Robotics Research (CMSR) (in the Faculty of Mechanical Engineering at Technion, Haifa) for continuing support. The authors wish to thank Professor A. Ber and Dr L. Rubinski for their kind academic help at various stages of the work.
REFERENCES 1. Z. MARCINIAKand J. DUNCAN,Mechanics of Sheet Metal Forming. Edward Arnold (1992) 2. S. YOSSIFONand J. TIROSI-I,Deep drawing with fluid pressure assisted blankholder. J. Enon0 Manuf B 206, 247 (1992). 3. S. YOSSlFONand J. TIROSH,Buckling prevention by lateral fluid pressure in deep drawing. Int. J. Mech. Sci. 27, 177 (1985). 4. S. YOSSIFON,J. TIROSHand E. KOCHAVl,On suppression of plastic buckling in hydroforming processes. Int. J. Mech. Sci. 26, 389 (1984). 5. S. YOSSIFONand J. TIROSH,Rupture instability in hydroforming deep-drawing processes. Int. J. Mech. Sci. 27, 559 (1985). 6. R. HILL, The Mathematical Theory of Plasticity, Chap. 12. Oxford Unversity Press, Oxford, U.K. (1950). 7. D. M. Woo, Analysis of the cup-drawing process. Int. J. Mech. Sci. 6, 116 (1964). 8. S. K. SITARAMEN,G. L. KINZELand T. ALTAN,Process sequence design for multistage forming of axisymmetric sheet metal parts. Report ERC/NSM-S-89-49, The Ohio State University (1989). 9. A. SHIRIZLV,Deep drawing with fluid pressure assisted blankholder. M.Sc. Thesis, Technion-lsrael Institute of Technology, Haifa (1992). 10. S. KOaAYASHI,S. O1-Iand T. ATLAS,Metal Forming and the Finite Element Method Chaps 3 and 11. Oxford University Press, Oxford, U.K. (1989). 11. B. BUD1ANSKYand N. M. WANG,On the swift cup test. J. Mech. Phys. Solids 14, 357 (1966). 12. D. C. CHIANGand S. KOaAVASHI,The effect of anisotropy and work-hardening characteristics on the stress and strain distribution in deep drawing. Trans. ASME, J. EnonO Ind. 443 (1966). 13. -J. TIROSHand M. SAVlR,High speed deep drawing of hardening and rate sensitive solids with small inteffacial friction. J. Mech. Phys. Solids 35, 479 (1987).
APPENDIX The height H of the product depends on the current angle by which the blank warps the die curvature ft. By some simple algebra, it finally reads: H(fl) = (pp +
Pd "l-
t)(l-cosfl) +
['R d - - R p -t- Pv -[- Pd - - (Pp -4= Pd +
t)sinfl]tanfl.
Performance of deep drawing processes
135
The current position of the blank is given by:
The original position of a material point at the very beginning of the process, to, and the change in its position due to the change in the current contact angle fl, are given at each zone. Zone V: t~r o - - = 0.
r 0 = r,
Zone IV: r0 =
(Rp -- pp)2 ..~ 2 pp -~- ~
( g p - pp)~ ~-
pp -[- ~ (1 - cos0t)
,
-~
= 0.
Zone Ill:
1
t~fl
R
Pp +
t
,sio [ (Rp--pp+
2roCOS2fl r 2 -
2
c~s~3'
p p + ~ sins
y]
.
Zone I1: r0
I
~ro
t
2
+t
2
1 sinfl --(pd+t . 2 t . 2 2roCOS:fl[(Rd+Pd ~)smfl) - - ( R p - - p p + ( p p + ~ ) s l n ~ ) ]
Zone I: ro=
r 2 - ( R ~ + p d ) 2+(r0~)2~0
t 1/2 63r0 ' ~/~
ro~ c~(ro,) ro ~/~
Since the punch has a constant speed Uo, the rate of change in time of the contact angle, fl, represents a time-like parameter in the analysis. It reads: -
Uoc°S2/~ [ R d - Rp + Pd "4- pp -- (Pd + Pp +
t)sinfl]"
