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Stamping and springback of circular plates deformed in hemispherical dies
Int. J. Mech. Sci. Vol. 26, No. 2, pp. 131-148, 1984 Printed in Great Britain.
0020-7403/84 $3.00+ .00 © 1984 Pergamon Press Ltd.
STAMPING A N D SPRINGBACK OF CIRCULAR PLATES DEFORMED IN HEMISPHERICAL DIES T. X. Yu, W. JOHNSON and W. J. STRONGE Department of Engineering University of Cambridge, England
(Received 27 August 1983) Summary~Circular plates of mild steel with radius-to-thickness ratios between 30:1 and 60:1 were pressed or stamped into a hemispherical die by a matching punch. Measurements of the punch travel, the final curvature of the plates (i.e. after elastic springback) and a record of the wrinkling encountered during pressing are related to the punch force for a range of plate central deflection up to several plate thicknesses. A theoretical analysis, based on axisymmetric bending or stretching of rigid-perfectly plastic plates is presented. This analysis well accounts for the experimental variations in stamping force during pressing and the final curvature of the deformed plates.
a b c D E E0 Ep h
Alp No
N, No n P P* P0 P q R RD Rr
Re r, 0 S s AT zl U w w0 Y ct fl A AA Ap 3p ~c, x xe xr v p
NOTATION radius of a circular plate radius of a central portion of a plate arc length for the tensile region, see Fig. 12 flexural rigidity of a plate, Eh3/12(1 - v 2) Young's modulus buckling modulus strain-hardening modulus thickness of a plate fully plastic bending moment, Yh2/4 maximum plastic membrane force, Yh membrane force in r-direction membrane force in 0-direction wave number in wrinkling punch force the largest punch force applied initial collapse force, 2nMp non-dimensional punch force, P/Po distributed load caused by punch force average radius of punch and die ( R ° + Re)~2 radius of die final radius of curvature of a plate radius of punch polar coordinates area of a plate surface arc length the work done by membrane forces bending energy of a plate deflection elastic deflection at the plate centre yield stress non-dimensional parameter, a2/Rh an angle in Fig. 9, b/R deflection at the plate centre accompanying displacement punch travel non-dimensional punch travel, A/h buckling index, x / ( Y /Eo) a / h critical value of ( curvature maximum elastic curvature, 2Y(1 -v2)/Eh final curvature Poisson's ratio ratio, b/a an angle in Fig. 9 INTRODUCTION
Thin axisymmetric shells may be formed by stamping or pressing a plate into a die by a matching punch. Usually, the geometry of the shell is specified; the pressing force and the die shape that are required to form this shell are unknowns that depend on the geometry and material properties. 131
132
T. X, Yt3 et al.
In metal shells that are cold formed by pressing, the plate first elastically and then plastically deforms in the die. The deformation proceeds with an increasing radius of contact between the punch and plate as the punch travel is increased. In-plane forces dominate bending moments in determining the stress distribution over most of the plate when the central deflection is larger than the plate thickness. The deformed shape and uniformity of curvature in the final shell are strongly dependent on the pressing force. As the pressing force is removed, elastic springback substantially changes the final shape from that of the die. Previous experiments by Johnson and Singh[1] showed that springback in circular plates increased as the radius of the plate decreased. In these experiments on circular plates resting in a hemispherical die, a central gap was observed between the plate and the hemispherical punch at small deflections. This gap decreased and disappeared as the pressing force (and central deflection) increased. Similar separations have been observed on the contact surface of metal strips that are pressed by cylindrical punches [2]. The central separation for plates is much smaller than that for strips when other dimensions are comparable. To analyse these results, Johnson and Singh developed a theory based on plastic bending which neglected plate stretching; consequently, when the deflections are large, their theory is not useful for examining springback. DESCRIPTION OF EXPERIMENTS Circular plates were pressed into a spherical die by a matching punch. The plates initially rested horizontally in the die with the periphery in continuous contact as shown in Fig. 1. The spherical punch was pressed into the centre of the plate at a slow rate of 5 mm/min. The plates and dies had the following dimensions: radius o f the spherical die:
R ° = 288 mm
radius of the spherical punch:
R e = R ° - h = 286.4 mm
thickness of plate:
h = 1.6 mm (~ in.)
radii of plates:
a = 50, 75 and 100 mm.
No lubrication was used in these tests. Although the edge o f the plate is unrestrained, in-plane resultant forces are significant in pressing to large deformations. To expose this effect, a range o f plate radii were tested so 30 < a/h < 60. All plates were cut from rolled mild steel sheet and used in the as-received condition. Stress-strain curves from tensile tests of this material are shown in Fig. 2. EXPERIMENTAL RESULTS Punch force-travel curves Non-dimensionalised punch forces measured in these experiments are shown in Fig. 3 where 6p=Ap/h
(1)
p -- P/Co
(2)
and
with A. the punch travel and P the punch force. Punch force is non-dimensionalised by the initial plastic collapse force (or a rigid-perfectly plastic circular plate subjected to a concentrated force at the centre, P0. If Y is the yield stress of the material and Mp is the fully plastic bending moment, then[3], eo = 2nMp = nYh2/2.
(3)
In Fig. 3, the punch force to deform a 100 mm radius plate has a smaller rate of increase after 6p -~ 5. At this deflection, noticeable wrinkling first occurred at the edge o f the plate. Final curvature along a radius After the plates were removed from the press, the average final curvature was calculated from measurements o f the mean diameter of the periphery and the height o f the crown. These calculations assumed that the final curvature was a constant. The following observations concerning average final curvature ge can be made trom Fig. 4. (i) ff~increases with increasing punch force P*, up to P * / Y S ~ 3 x 10 -3. Thereafter, ~Fis almost a constant in this mild steel plate. Here, the surface area o f one side o f the plate is S = ~a 2 so that P * / Y S denotes the largest punch force divided by a critical average membrane stress for the material. (ii) fir increases with plate radius-to-thickness ratio a/h when the largest punch force is P * / Y S > 3 x 10 -3. Final profiles o f all plates were measured after springback by traversing with a Ferranti S U R F C O M along a diameter. The arc length and the distribution of final curvature were then calculated using a finite difference
Stamping and springback of circular plates
133
FIG. 1. Experimental setup.
(a)
(b)
FIG. 7. Photographs taken of a specimen of a = 100 mm and h = 1.6 mm, pressed to P* = 28 kN; the dark marks show the contact regions.
Flo. 17. Photographs showing the wrinkling shapes of plates after stamping.
Stamping and springback o f circular plates
135
method. Fig. 5 shows typical distributions of final curvature for a plate o f radius 75 mm, where s denotes the arc length measured along a radius from the centre, Ke denotes the final curvature and x ° = 3.472 m - ' is the curvature of the die. Comparison o f the three curves produced by different values o f P*/YS show that when P*/YS < 3 x 10 -3 the principal effect of increasing force is to increase the average curvature. When P*/YS > 3 x 10-3, increasing force principally makes the distribution o f final curvature more uniform. The distribution o f final curvature is more uniform in the central portion o f the plate than in the outer portion; spherical pressing is more effective in the central portion where the punch and die surfaces are almost perpendicular to the direction o f punch travel.
(w/..,)
~o.~
200t [
SPECIMENS' ORIENTATION
E/
I00
ROLL 1N~. DIRECTION
"4--"
OP THE SHEET
o
~
'
Ib
.
.
.
.
.
. 20 . .
.
3b
'
'
,~'o - 5
FIG. 2. Stress-strain curves for tensile specimens cut from rolled mild steel sheet of thickness 1.6 mm in three different orientations.
P
P
P
Po
2nMp
30-
E E II
n
E E o 0
20
10
/ Ap
1 0
a'l'= h . . . .
i
5
10
FIG. 3. Punch force-travel curves for circular plates o f radius a and thickness h = 1.6 mm stamped by a spherical punch and die o f R ° = 288 mm. MS Vol. 26, No. 2 - - D
136
T . X . Yu el aL
1.00.g ~ o
o
[]
0.7~
a (ram)
0.5"
TEST THEORY(42)
100 0.4-
0,2 0.5 i
0.,3
o.,s ~
75
o
50
,,
1,
,2 i
~
3
3
~
s ;o .
P~ .2~
( PTys)xl 0"
FIG. 4. Non-dimensional relationship between the average final curvature of plates and the largest punch force applied, h = 1.6 mm, R ~ = 288 mm.
2
(P*/yS)xl03
/k
1.6 I
3.2
I
6.s
/,,
-.-,,,,,./
~
,11
I
,
E z I
0'.5
'
U'
1.'0 s/ct
FI6. 5. Distribution of final curvature along a radius for a plate of a = 75 mm and h = 1.6mm.
Circumferential wrinkling Measurements of the plate surface profile in the circumferential direction were also made with a Taylor-Hobson Talyrond 61 machine. In Fig. 6, the deviation of each profile from a datum circle (marked by a chain line) represents the variation in final height along a circle on the plate surface. It can be seen from Fig. 6 that the plates of radii 50 mm and 75 mm have two wrinkling waves or lobes, whilst plates of radius 100 mm have four deep lobes for small P* and eight faint lobes for large P*. Photographs of the deformed plates in Fig. 7 have dark marks at the contact regions between the tool and the plate of radius 100 mm. These marks show the radial distribution of these circumferential wrinkling lobes. ELASTIC D E F L E C T I O N OF PLATE In order to fully appreciate the behaviour of circular plates in the early stage of the pressing process and the onset of plasticity, the elastic deflection will be analysed.
(b)
(a)
(d)
(0
e)
'
FlG. 6. Measurement of specimens after pressing, showing the variation in the height along circles of latitude. (a) a = 5 0 m m , r = 4 5 m m , p*=3.5kN; (b) a = 5 0 m m , r = 4 5 m m , p * = 1 4 k N ; (c) a = 7 5 m m , r = 7 0 m m , p*=7.85kN; (d) a = 7 5 m m , r = 7 0 m m , p*=31.4kN; (e) a = 1 0 0 m m , r = ~( 95 50 m mm m -- -- ' p*=8.9kN; (0 a=100mm, = ~95 m m - r [50 mm__, p* = 89 kN.
-
(c)
/
.q
-.9..
o~
e~
0~
(ao
T . X . Yu et al.
138
First, examine the elastic solution of a simply-supported circular plate of radius a and thickness h subjected to a concentrated force P at its centre. In this case, the deflection of the plate is[4], (a2 - r:) + 2r 2 In ~ ]
w = 16riD /[ ~3 v+ v P L
= A (a ~"- r") + Br 2 In ~,
(4)
a
where D = Eh3/12(1 - v:) is the flexural rigidity of the plate, and A and B are coefficients independent of the radial coordinate r. The curvatures of the plate are, accordingly,
and
xo . . . . .
2A-B
r dr
(r) 21n
a
(5)
1 ,
so that G and x o tend to infinity at r = 0. This implies that this elastic solution is not strictly applicable to the circular plate loaded by a spherical punch. Even for a small force P, the centre contact point must develop into a small circle. Thus, consider a circular plate on which the load is uniformly distributed along a circle of radius b, as shown in Fig. 8. In this case, the deflection of the plate is[4], w = 8~tD ( b 2 + r : ) l n
+(a:-b2)(3+v)a2-(1-v)r21"-~+v-~ -.j
(6)
for r < b, and
w=8nPD{(a2-r2)[l+~(1-~)]+(b2+r2)ln
r}
(7)
for r > b. Thus the deflection at the centre of the plate is P [- 3 + v 2 Wo=~zbt~(a-b~)
b2
ln~]
(8)
and the deflection of a point on the loading circle is P
Wb
~L
F(3+v)a 2-(1-
~
+~
v)b2(a2-b2)-2b21n~]"
(9)
As a result, the relative deflection in the central portion is w
l_v
0-
L2(-i
(10)
)k
For the curvatures in the region r < b, (6) leads to P [ l-v [l_b2"~ a] ~=K°=~L2(I +v)\ a:]+ln~ .
(11)
Since the central portion of the plate will wrap around the punch, x, and x0 are equal to 1/R for r < b, where R is the average radius of the punch and die, i.e. R = (R D + Re)/2. Thus, from (11),
4nD /[- 1 - v
["
t' = W / L 2 ( w 7 7 6 ) [ l
b:\ -
a] + In g ] ,
(12,
and from (10),
wo - wb = b2/2R.
(13)
P(2Wb
I
1
FIG. 8. A circular plate subjected to the load uniformly distributed along a circle.
Stamping and springback of circular plates
139
The r.h.s, of (13) is equal to the displacement of the punch; thus expression (13) indicates that the centre of the plate and the punch pole have the same displacement; in other words, no separation occurs. By introducing non-dimensional parameters
tl - R/a and p =- b /a,
(14)
expressions (12) and (8) can be re-written in non-dimensional form as
--=--O ~
[2T~v)(1-p2)+ln
(15)
and
w°
a-
Pa/D[- 3 + v 8n [ 2 ( ~ v )
(I-p2)-p21n
~1
"
(16)
For a specified problem, v and ~/are given, so that the load-displacement relationship can be obtained from (15) and (1.6) by means of the parameter p which varies from 0 to 1. As a numerical example, by taking v = 0.3 and ~/= 1, i.e. R = a, the calculated results are as shown in Table 1. In this example, the load-displacement relationship remains nearly linear until the displacement w0 is as large as about 0.2a. The previous analysis is based on the small deflection theory so it is applicable only if the deflection of the plate is smaller than the thickness, i.e. w0 ~ h.
(17)
For a plate, the curvature at the initial yield state is (refer to[5]), 2Y(1 - v2)
re
Eh
(18)
Since the maximum curvature of the plate permitted in the investigation is 1/R, the above elastic solution is available only if I/R < r, or R E >- h 2Y(1 - v2)'
(19)
If (19) is violated, then the onset of plasticity will precede the wrapping of the plate around the punch pole. PLASTIC DEFORMATION
Joined spherical and conical deformation mode It was observed in the experiments that (i) the contact region between the punch and the plate developed from the centre outwards as the pressing proceeded, and (ii) the separation between the central portion of the plate and the punch pole was negligible when ratio a/h was large, say, > 30. Therefore, the following approximate deformation mode may be proposed. As shown in Fig. 9, the central portion of the deformed plate is assumed to be a spherical surface in which the radius matches the punch. In other words, this portion follows precisely the shape of the punch, whilst the outer portion of the deformed plate is assumed to form a conical surface having a smooth conjunction with the spherical portion. From the geometry shown in Fig. 9, the deflection at the centre of the plate relative to the outer drcumference is A = (a - b) sin/3 + R(1 - cos/~),
(20)
where R is the radius of the central spherical portion, b is the arc length from the centre to the edge of this portion.
TABLE I. THE ELASTIC BEHAVIOUR OF A CIRCULAR PLATE LOADED BY A SPHERICAL PUNCH, V = 0 . 3 AND R = a = bla
PaID
Wola
(Pa/D)(wo/a)
10 -6
0.8922
0.0451
19.80
10-5
1.0666
0.0539
19.80
10 -4
1.3256
0.0670
19.80
10-3
1.7509
0.0884
19.80
0.01
2.5780
0.1301
19.81
0.05
3.8496
0.1928
19.97
0.10
4.8914
0.2401
20.38
140
T. X. Yu et al.
DIE
I
F1G. 9. Joined spherical-conical deformation mode. and fl = b/R is the angle subtending the arc b. By taking sin fl ~ fl --- b/R and 1 - cos fl ~ fl2/2, (20) can be re-written in non-dimensional form as d ~ ctp (1 - P ) ,
(21)
where a2
ct - ~
b
and p - a
(22)
are non-dimensional parameters. To find the punch displacement dp, an accompanying displacement, d A, caused by the movement of the plate edge down along the die should be added to the deflection, A, that is dp=d
+A A
or
6p = J + t5A
(23)
where 6A = da/h is a non-dimensional accompanying displacement. It can be calculated as R ~ = ~ (cos 4~ - cos ¢0),
(24)
where ~b0 and tk are determined from sin 4>0 = a/R and sin q~ = s i n fl +(a/R)(1 - p ) cos ft.
(25)
It can be proved from (21), (24) and (25) that 6~/6 is of the same order as ab/R2. Hence, in most practical cases &A is negligible by comparison with 6, and thus from (23), 6, -~ 6.
(23)'
That is, the punch displacement can be regarded as the same as the deflection at the plate centre.
Load-carrying capacity estimated by plastic membrane theory The force required for continuing deformation of the plate has been calculated for large deflections by assuming in-plane stresses dominate the stresses due to flexure of the plate. Three different models of the punch force distribution (and the mode of deformation within the contact region) have been analysed. (i) Centralforce with joined spherical and conical deformation. Calladine[6] has proposed a simple method to minimise upper-bound solutions to load-carrying capacity and that is by considering the overall equilibrium of segments of the plate. For example, when a simply-supported circular plate carrying a concentrated load at its centre has a large deflection (A > h), a conical deformation mode may be assumed for an upper-bound calculation, and the rigid/plastic stress distribution will be as shown in Fig. 10(b). The position of the line I - I is determined by the 'equal areas' condition, which corresponds to overall equilibrium of forces acting on one half of the plate in the direction perpendicular to the diametral cut. By taking moments about I - I, the load is found to be
P =~
~[ dA
=4e0f ah2 Ja ~v[ dA,
(26)
where y is the vertical coordinate measured from the line I - L A is the area over the radial cross-section, and P0 is as defined by (3). For example, in a simply-supported plate with zl _> h, (26) yields p=6+~, where p = P/Po and 6 -= A/h.
1
forr>l,
(27)
Stamping and springback of circular plates
141
The rigid/plastic stress distribution shown in Fig. 10(b) leads to a distribution for the circumferential membrane force, No, as shown in Fig. 10(c), where N O denotes the maximum plastic membrane force which is equal to Yh. Between No/N o = ! and No/N o = - 1 there is a transition zone of width a/& When the deflection considered is very large (say, 6 = d / h > 5), Calladine's method can be further simplified• In this case, the second term on the r.h.s, of (27) is negligible in comparison with the first term, and the transition zone for the distribution of No~No vanishes. These results correspond to assuming that the radial cross-section can be condensed to its mid-line L - L, see Fig. 10(b), so that in effect the stress distribution through the thickness is neglected.Consequently, instead of SA [Yl dA, only a simple one-dimensional integration SL [Y[ dL along the mid-line L - L is required. To apply this simplified Calladine method to our deformation mode, Fig. 9, with the help of Fig. 11, the work done by P gives P p - ~ = ~k, ro
(28)
with ct = a2/Rh and p(1 - 2p2/3) k=
,
b 1 f o r P = a = < ~,
,
b 1 f o r p m--_>--.
1-p ½+p-p2/2 1 --p
(29)
a --2
This result is independent of the distribution of the punch load in the contact region r __
1 2
p h
4a
1 2
p
h • 4R'
• t a n ~b ~ -
(30)
where c is the arc length for the tensile region, see Fig. 12(b), and 4, can be calculated from equation (25). Since R ~> h, expression (30) indicates that c/a is only slightly smaller than ~ as long as p is not very large.
I~1.~a~LI
I
P/2~a
N0( 'N, a.r
-,
(a)
[
-1
(b)
(c)
FIG. 10. A simply-supported plate subjected to a concentrated force at its centre. (a) Equilibrium of one half of the plate. (b) Stress distribution when A > h. (c) Circumferential membrane force.
b
._
A
/ I
L b Y (a) (b) FIG. i 1. Mid-line L - L of the radial cross-section of a circular plate. (a) For p = b/a <~ 1/2. (b) For p = b/a >>.1/2.
P/2~R cos ~
(a)
~
~
(b)
F16. 12. Effect of the horizontal components of the die reaction. (a) Equilibrium of one half of the plate. (b) tensile and compressive regions in the radial cross-section.
142
T . X . Yu et al.
The distribution of circumferential membrane force is as shown in Fig. 13, i.e. No~No = ~ + 1, for 0 < r/a < c,/a, ( - 1 , for c/a < r/a < 1.
(31)
(ii) Uniformly distributed transverse force with conical deformation. Consider a simply-supported circular plate subjected to a uniformly distributed circular loading q within radius b as shown in Fig. 14. In the case where the boundary is free to move inward, Kondo and Pian[7] found the load-deflection relationship to be as follows,
.(,
1 1+5620,
when30< 1,
1 6o + ~ , .~oo
when3o>l,
P
(32)
where P = nb2q, Po = 2nMp and 60 is the non-dimensional deflection at the centre determined from a conical mode. If the punch force is assumed to be uniformly distributed over the central portion of the plate, then by using the previously defined non-dimensional parameters, it is found that 3o = ~p
(33)
and when 6o is large, P ~
~p 2p' 1 ----
(34)
3
where p - P/Po, ot - a2/Rh and p = b/a as defined before. Therefore, by considering a continuously increasing p, expression (34) provides an estimate to the load-carrying capacity of the circular plate loaded by a spherical punch. (iii) Uniformly distributed pressure with spherical central deformation. For a circularly loaded, simplysupported, circular plate, the previous analysis[7] also showed that in the central portion of the plate, iV, = No = No. In our problem this implies that the central portion under the punch is subjected to a uniform biaxial plastic tension; in other words, it becomes a plastic spherical membrane. From force equilibrium considerations, a uniform internal pressure is required to be applied on the central spherical portion of the plate, see Fig. 9, and its magnitude is found to be
2N0
q = --. R
(35)
Thus, the total punch-force is p = nb2q =
Dr Nob2
R
Nob2 = P0"--
(36)
M.R'
or in non-dimensional form, p = 4~p 2.
(37)
Since N o = No holds within 0 _-
Ne/No t
-I
i FIG. 13.
a
FIG. 14.
FIG. 13. Circumferential membrane force when the deflection of a plate is very large. FIG. 14. A simply-supported circular plate subjected to a uniformly distributed circular loading q.
Stamping and springback of circular plates
143
contact circle between the punch and the plate and found a punch force-travel relationship. Using our notation, it can be re-written as sin fl cos ~ -
cos
(4
(38)
fl)
-
and (39) where fl = b/R, and ~b is determined by (25). Expression (39) is equivalent to (21).
Comparison with experiments and discussion Four different approximate methods have been proposed. Accordingly, they estimate the punch force by expressions (28), (34), (37) and (38), respectively, whilst the punch travel, i.e. deflection at the centre of the plate, can be calculated by (21) for all cases. Fig. 15 shows the punch force-travel curves (i.e. the load-deflection curves for plates). The load-carrying capacities predicted by the simplified Calladine method, i.e. expression (28), and by employing a circular plate subjected to a continuously extended circular loading, i.e. expression (34), are in good agreement with the experimental data. The first three methods are essentially different from the last method proposed by Johnson and Singh. In the first three analyses the membrane forces are dominant and the punch load is assumed to be carried principally by the membrane forces developed in the plate. However, the method proposed by Johnson and Singh takes the radial bending moment M, as a dominant factor in the deformation process and it controls the travelling of the circular hinge. As a result, the method proposed by Johnson and Singh may provide a prediction to the p ~ ~ curve for the thick plates, say a/h < 20, since the deformation of thick plates is more closely represented by a stress distribution associated with flexure; but this method fails if the plate ratio a/h is large. In the case of thin plates, say a/h > 30, membrane forces are the dominant factor, and the methods based on membrane forces give much better prediction to the p ~ ~ curve of the plate. Previously, a small deflection analysis[8] has shown that the membrane forces are negligible only if a/h < 20. Beyond this range, the membrane forces have to be included. If a/h is very large, a pure membrane theory may be employed.
1.8o
THEORY
o -o
(26) (34) (37) (38)
~ =0.
I~jl
l!.
EXPERIMENT
l
•
O. --=100
mm
o
0,=
mm
75
7t/ I I II II II IIII ell I
1.0
I/7', ./ Vo ~ /
II
0.2 0
0.1
012
0.3
0 .4
0~5
FIG. 15. Comparison of the punch force-travel curves predicted by the different theories with experiments.
144
T . X . Yu et al.
SPRINGBACK Based on the distribution of membrane forces as associated with the simplified Calladine method, the springback of circular plates after pressing can also be approximately predicted as follows. Assume that at the end of pressing, the entire plate forms a partly spherical surface, see Fig. 16, i.e. p = h/a = I. At the circumference of the plate, i.e. point A in Fig. 16, assume that unloading leads to the entire release of the membrane force, No = -No = - Y h . Then an elastic circumferential strain of Y(I - v 2 ) / E must occur in a radius of R sin (a/R). Hence, point A should move outwards a distance Y(-!-- v2) R sin ( a ~ E \R]
(40)
Similarly, when the membrane force N,(=No in r < a/2) is released there is a corresponding shortening in the length of arc OA, which leads to a fall in the height of point ,4 of Y(I - v 2) 9 --• R. E 4
(41)
Hence, after unloading, point A moves to a new position A ', as shown in Fig. 16, and the coordinates of point ,4' can be determined by (40) and (41). By assuming OA' is a circular arc of radius R e, it can be shown that
XD - R r ~ 1
(42)
~
which is the ratio of the radii of the plate before and after springback. For comparison with our experimental results, take R = 288mm, v =0.28, E = 2 0 5 k N / m m 2 and Y = 302 N / m m 2 (i.e. choosing 10% higher than the initial yield stress of mild steel plate used, in order to allow somewhat for the strain-hardening which occurs during the pressing process), then for plates of radii a = 50, 75 and 100mm, the calculated values of xr/~c are 0.80, 0.91 and 0.95, respectively. The measured values in experiments are 0.71, 0.87 and 0.93 respectively, refer to Fig. 4. Expression (42) is based on assuming p = b/a = 1 and therefore it can be regarded as a limit ot xe/x when the punch load increases without limit. From this point of view, the agreement between the prediction by using (42) and the experiment, as shown in Fig. 4, is quite good. Two non-dimensional punch-force parameters P / Y S and plot have the following relation p :x
P Po~
P 2R YS h
(43)
As the experiments have shown, when P * / Y S > 3 × 10 -3, further increase in P* has little effect on the increase in Kr/~c°. Since R/h = 180 in these experiments, this is equivalent to stating that the increase in xF/x n is mainly brought about in the course of increasing p until p*/~ ~ 1. Alternatively, this refers to increasing p ( = b/a) until p = 0.5 ~ 0.6. As long as the largest punch-force applied is such that p/o~ > 1, i.e. P*>P#-
Ya 2h 2R '
(44)
the springback can be predicted by (42).
WRINKLING Theory To analyse technologically the plastic buckling or wrinkling of circular plates during spherical pressing, we adopt the well-known energy method and follow a procedure which was developed previously in[9], where the buckling of annular plates in relation to the deep-drawing process was considered. As observed by previous researchers, when a circular plate wrinkles during pressing, the wrinkling mainly takes place in its outer portion, see Fig. 17. Indeed, the plastic analysis above has indicated that the circumferential membrane force No is negative in the region of a/2 < r < a, and this compressive force must be the principal source of wrinkling.
~
R Rs'n R
.(,*o)
', ~F
~'
FIG. 16. Change in the shape of a plate during springback.
Stamping and springback of circular plates
145
Hence, it is reasonable to assume that the additional deflection due to wrinkling is
(a)
C r-~
sinn0,
for-a-
(45)
where C is a constant and C ~ 0 when the wrinkling occurs. Then
--
Or
a2w
= C sin nO,
= Cn r -
cos nO,
a2w = O, ~ = Cn cos nO
(46)
and
02~w=002- C n : ( r -
2 ) sin n0.
Although the punch load is carried by membrane forces, the additional deflection is assumed to be associated with the flexural rigidity when strain-hardening is considered. By referring to[4], the corresponding bending energy of the plate is
~fDfO2w 1 Ow
1 O2w'~2
O2w[1 Ow
1 O2w'~
- - D ( 1 - v)-~-Tr2t ; - ~ -r + ~ )
(1 O2w
+D(1 - v)t;ff~- 0
1 OwN2) rjff~)~rdrdO.
When applying (47) to a fully plastic plate, Poisson's ratio is
Eoh3
(47)
v
=
½
and the flexural rigidity is then
Eoh3
D = 12(1 -- v 2)
9 '
(48)
where E 0 is the buckling modulus determined by
Eo _
4EEp +
(49) 2
E is Young's modulus and Ep the tangent modulus of the material (see[10]). Substituting (46) and (48) into (47) leads to
u=- oc2~:,r°: {[l-,:( l-aq: ~)J + n't,~) / a '~2] J~T dr 2
C 2. ~ ' . 9
F(n),
(50)
where
( 5\ 4 ( 11\ F(,)-- tin 2 - ~). - t 2 In 2 - ~-)n:+ In 2.
(51)
The plastic stress distribution in the outer portion of the plate has been found using a rigid/plastic analysis, see[7], to be
Nr=No6[1
r
a (
1\ 2-] for~
1+~ ,
(52)
and
where N o = Yh, 6 = d/h and d is the deflection at the centre of the plate caused by pressing, as defined earlier. By referring to[l 1], the work done by the membrane force is 1 (" ('f ( ~ w \ 2
A T = --~ J J l N t ~ r J
/1 ~w'~ 2)
+ N o t r ~ ) ~rdrdO.
(53)
T . X . Yu et al.
146 Substituting (52) into (53) results in
AT=
7z
. (/2. Yha2.G(n, 6),
2
(54)
where
n21 1~-I 1 2 1 G(.,~)---7 l+ln2-(l+~)ln ( l+7)J-~(. +1)~8~n2 246~(.~-1).
(55)
By equating A U and A T, the critical condition for wrinkling is presumed to be obtained as
go h3 9
F(n) = Yha 2 G(n, 5).
(56)
If a non-dimensional buckling index for the plates is defined as -
.~,
(57)
the critical value of ( can be found from (56) as
,_ f,(,)
(<' = 3 ~1 G(n, ~)"
(58)
When ( > ~c,, wrinkling occurs; and when ( < ~c,, no wrinkling occurs. Expression (58) indicates that (<, depends on the wave number n as well as the deflection caused by pressing. The calculated results from (51), (55) and (58) are as shown in Fig. 18. This figure shows that: (i) (c, increases with the wave number n, and the smallest value o f (c, is given by the case of n = 2; and (ii) (c, falls when the deflection increases from 6 = A/h = 1. For 6 > 2, ~<, approaches a limiting value (~*,(n), for a particular n. Taking the limit of G in (55), as deflection increases without bound, it is found that 1
1
G*(n) - G(n, oo) = ~ (2 In 2 - 1)n 2 - ~,
Y
o
4-
BUCKLED
\ \
(~cr
r~= 8
(or
n =6
~,cr
n =A
(or
n=2
UNBUCKLED
0
j
~
~
~
~=e/h &
FIG. 18. Critical values of the buckling index ( = ( Y ~ o ) " (a/h) as a function of the deflection 5 = d / h and the wave number n.
(59)
Stamping and springback of circular plates
147
and accordingly,
*" , 1 / F(n)
(60)
Hence, the smallest buckling index for the wrinkling of plates is ln 2 - 1/2x~I/2 ~*,(2) = \ ~ ] ~ 1.68,
(61)
and a plate will remain unbuckled if its index satisfies ( -
.~ < 1.68.
(62)
The higher is the value of ( for a plate, the more waves will be created in the outer portion o f the plate during pressing. Indeed, if n is large enough, then *
1
/8]n2 - 5
~c,(n)~42~n
2
1 . n ~-0.4n.
(63)
Thus, for a very thin plate which will have a large (, the wave number can be easily estimated by n = 2.5(.
(64)
Comparison with experiments For the mild steel circular plates used in our experiments, E = 205 kN/mm 2, Y = 275 N/mm 2, and Ep ~ 0.1 E, so that n = 2.5 ( = 0.19 (a/h). When a plate of a = 100 mm and h = 1.6mm is examined, this equation gives n~12. Experimentally, we observed n = 4 ~ 6 when P * / Y S = l x 1 0 -3 and n = 8 ~ 1 2 when P*/YS = 10 x 10 3, refer to Fig. 6.
Doubling and ironing After the punch contacts the wrinkles in the outer portion o f plate, the development o f waves of large amplitude is prevented by the punch pressure, though the number o f waves is increased by a reverse buckling process, as shown diagrammatically in Fig. 19. This phenomenon is similar to the doubling of waves in a deep-drawing process when a blank-holder is applied. Johnson and Mellor[12] have described this phenomenon (first pointed out by B. Senior) and observed that the more numerous are the waves o f small amplitude so formed, the more easily are they ironed-out to give a good finish on products. CONCLUSIONS (1) To form a bowl from a thin metal plate, the punch force P required to achieve the full curvature from a hemispherical die set is
P
na2hy
Po
2R
Further increase in force may reduce wrinkles but it has almost no effect on elastic springback or the final curvature. At this punch force, the ratio o f final-to-die curvature will be
xe/x n -~ 1 -- 9Y(I - v2)R2/2Ea 2. (2) Wrinkling is a principal limitation on stamping to a large curvature. In this analysis of thin plates, wrinkling does not occur when
~/h < m.68e,/eo/r. "tRUESINE CURVE~
t
FIG. 19. The mechanism for the doubling in the number of waves (following [12]).
148
T.X. Yu et al.
These conclusions follow from an analysis of large plastic deformation where in-plane forces predominate during stamping and elastic springback occurs upon unloading. The forming history of metal plates has been completely followed using this analysis with a rigid-plastic material characterisation. Acknowledgements--One of the authors, T. X. Yu, gratefully acknowledges the support of the Educational Ministry of the The People's Republic of China and the Cambridge University Engineering Department. The authors wish to thank Mrs R. M. Orriss for typing their manuscript and Prof. S. R. Reid for his comments on the paper. REFERENCES I. W. JOHNSOH and A. N. SINGI-I,Springback in circular blanks. Metallurgia 275-280 (May 1980). 2. T. X. Yu and W. JOnNSOH, Cylindrical bending of metal strips. Metals Tech. 10, 439~,47 0983). 3. H. G. HOPKINSand W. PRAGER,The load carrying capacities of circular plates. J. Mech. Phys. Solids 2, 1-13 (1953). 4. S. TIMOSh'EHKOand S. WOIHOWSKY-KgmGER,Theory of Plates and Shells, 2nd Edn. McGraw-Hill, New York (1959). 5. T. X. Yu and W. JOHNSON,The large elastic-plastic deflection with springback of a circular plate subjected to circumferential moments. J. Appl. Mech. 49, 507-515 (1982). 6. C. R. CALLAD1NE,Simple ideas in the large-deflection plastic theory of plates and slabs. Engineering Plasticity (Edited by J. Heyman and F. A. Leckie), pp. 93--127, Cambridge University Press (1968). 7. K. KO~qDOand T. H. H. PIAH, Large deformations of rigid-plastic circular plates. Int. J. Solids Structures 17, 1043-1055 (1981). 8. W. JOHNSONand T. X. Yu, On the range of applicability of results for the springback of an elastic/perfectly plastic rectangular plate after subjecting it to biaxial pure bending-II. Int. J. Mech. Sci. 23, 631-637 (1981). 9. T. X. Yu and W. JOHNSON,The buckling of annular plates in relation to the deep-drawing process. Int. J. Mech. Sci. 24, 175-188 (1982). 10. J. W. GECKELER, Plastiche knicken der wandung von hohlzylindern und einige andern faltungserscheinungen. Z. Angewandte Mathematik und Mechanik 8, 341-352 (1928). I I. S. TIMOSh~NKOand J. M. GERE, Theory of Elastic Stability, 2rid Edn. McGraw-Hill, New York (1961). 12. W. JOHNSONand P. B. MELLOR,Engineering Plasticity, pp. 313-314. Van Nostrand Reinhold, London (1973).
0020-7403/84 $3.00+ .00 © 1984 Pergamon Press Ltd.
STAMPING A N D SPRINGBACK OF CIRCULAR PLATES DEFORMED IN HEMISPHERICAL DIES T. X. Yu, W. JOHNSON and W. J. STRONGE Department of Engineering University of Cambridge, England
(Received 27 August 1983) Summary~Circular plates of mild steel with radius-to-thickness ratios between 30:1 and 60:1 were pressed or stamped into a hemispherical die by a matching punch. Measurements of the punch travel, the final curvature of the plates (i.e. after elastic springback) and a record of the wrinkling encountered during pressing are related to the punch force for a range of plate central deflection up to several plate thicknesses. A theoretical analysis, based on axisymmetric bending or stretching of rigid-perfectly plastic plates is presented. This analysis well accounts for the experimental variations in stamping force during pressing and the final curvature of the deformed plates.
a b c D E E0 Ep h
Alp No
N, No n P P* P0 P q R RD Rr
Re r, 0 S s AT zl U w w0 Y ct fl A AA Ap 3p ~c, x xe xr v p
NOTATION radius of a circular plate radius of a central portion of a plate arc length for the tensile region, see Fig. 12 flexural rigidity of a plate, Eh3/12(1 - v 2) Young's modulus buckling modulus strain-hardening modulus thickness of a plate fully plastic bending moment, Yh2/4 maximum plastic membrane force, Yh membrane force in r-direction membrane force in 0-direction wave number in wrinkling punch force the largest punch force applied initial collapse force, 2nMp non-dimensional punch force, P/Po distributed load caused by punch force average radius of punch and die ( R ° + Re)~2 radius of die final radius of curvature of a plate radius of punch polar coordinates area of a plate surface arc length the work done by membrane forces bending energy of a plate deflection elastic deflection at the plate centre yield stress non-dimensional parameter, a2/Rh an angle in Fig. 9, b/R deflection at the plate centre accompanying displacement punch travel non-dimensional punch travel, A/h buckling index, x / ( Y /Eo) a / h critical value of ( curvature maximum elastic curvature, 2Y(1 -v2)/Eh final curvature Poisson's ratio ratio, b/a an angle in Fig. 9 INTRODUCTION
Thin axisymmetric shells may be formed by stamping or pressing a plate into a die by a matching punch. Usually, the geometry of the shell is specified; the pressing force and the die shape that are required to form this shell are unknowns that depend on the geometry and material properties. 131
132
T. X, Yt3 et al.
In metal shells that are cold formed by pressing, the plate first elastically and then plastically deforms in the die. The deformation proceeds with an increasing radius of contact between the punch and plate as the punch travel is increased. In-plane forces dominate bending moments in determining the stress distribution over most of the plate when the central deflection is larger than the plate thickness. The deformed shape and uniformity of curvature in the final shell are strongly dependent on the pressing force. As the pressing force is removed, elastic springback substantially changes the final shape from that of the die. Previous experiments by Johnson and Singh[1] showed that springback in circular plates increased as the radius of the plate decreased. In these experiments on circular plates resting in a hemispherical die, a central gap was observed between the plate and the hemispherical punch at small deflections. This gap decreased and disappeared as the pressing force (and central deflection) increased. Similar separations have been observed on the contact surface of metal strips that are pressed by cylindrical punches [2]. The central separation for plates is much smaller than that for strips when other dimensions are comparable. To analyse these results, Johnson and Singh developed a theory based on plastic bending which neglected plate stretching; consequently, when the deflections are large, their theory is not useful for examining springback. DESCRIPTION OF EXPERIMENTS Circular plates were pressed into a spherical die by a matching punch. The plates initially rested horizontally in the die with the periphery in continuous contact as shown in Fig. 1. The spherical punch was pressed into the centre of the plate at a slow rate of 5 mm/min. The plates and dies had the following dimensions: radius o f the spherical die:
R ° = 288 mm
radius of the spherical punch:
R e = R ° - h = 286.4 mm
thickness of plate:
h = 1.6 mm (~ in.)
radii of plates:
a = 50, 75 and 100 mm.
No lubrication was used in these tests. Although the edge o f the plate is unrestrained, in-plane resultant forces are significant in pressing to large deformations. To expose this effect, a range o f plate radii were tested so 30 < a/h < 60. All plates were cut from rolled mild steel sheet and used in the as-received condition. Stress-strain curves from tensile tests of this material are shown in Fig. 2. EXPERIMENTAL RESULTS Punch force-travel curves Non-dimensionalised punch forces measured in these experiments are shown in Fig. 3 where 6p=Ap/h
(1)
p -- P/Co
(2)
and
with A. the punch travel and P the punch force. Punch force is non-dimensionalised by the initial plastic collapse force (or a rigid-perfectly plastic circular plate subjected to a concentrated force at the centre, P0. If Y is the yield stress of the material and Mp is the fully plastic bending moment, then[3], eo = 2nMp = nYh2/2.
(3)
In Fig. 3, the punch force to deform a 100 mm radius plate has a smaller rate of increase after 6p -~ 5. At this deflection, noticeable wrinkling first occurred at the edge o f the plate. Final curvature along a radius After the plates were removed from the press, the average final curvature was calculated from measurements o f the mean diameter of the periphery and the height o f the crown. These calculations assumed that the final curvature was a constant. The following observations concerning average final curvature ge can be made trom Fig. 4. (i) ff~increases with increasing punch force P*, up to P * / Y S ~ 3 x 10 -3. Thereafter, ~Fis almost a constant in this mild steel plate. Here, the surface area o f one side o f the plate is S = ~a 2 so that P * / Y S denotes the largest punch force divided by a critical average membrane stress for the material. (ii) fir increases with plate radius-to-thickness ratio a/h when the largest punch force is P * / Y S > 3 x 10 -3. Final profiles o f all plates were measured after springback by traversing with a Ferranti S U R F C O M along a diameter. The arc length and the distribution of final curvature were then calculated using a finite difference
Stamping and springback of circular plates
133
FIG. 1. Experimental setup.
(a)
(b)
FIG. 7. Photographs taken of a specimen of a = 100 mm and h = 1.6 mm, pressed to P* = 28 kN; the dark marks show the contact regions.
Flo. 17. Photographs showing the wrinkling shapes of plates after stamping.
Stamping and springback o f circular plates
135
method. Fig. 5 shows typical distributions of final curvature for a plate o f radius 75 mm, where s denotes the arc length measured along a radius from the centre, Ke denotes the final curvature and x ° = 3.472 m - ' is the curvature of the die. Comparison o f the three curves produced by different values o f P*/YS show that when P*/YS < 3 x 10 -3 the principal effect of increasing force is to increase the average curvature. When P*/YS > 3 x 10-3, increasing force principally makes the distribution o f final curvature more uniform. The distribution o f final curvature is more uniform in the central portion o f the plate than in the outer portion; spherical pressing is more effective in the central portion where the punch and die surfaces are almost perpendicular to the direction o f punch travel.
(w/..,)
~o.~
200t [
SPECIMENS' ORIENTATION
E/
I00
ROLL 1N~. DIRECTION
"4--"
OP THE SHEET
o
~
'
Ib
.
.
.
.
.
. 20 . .
.
3b
'
'
,~'o - 5
FIG. 2. Stress-strain curves for tensile specimens cut from rolled mild steel sheet of thickness 1.6 mm in three different orientations.
P
P
P
Po
2nMp
30-
E E II
n
E E o 0
20
10
/ Ap
1 0
a'l'= h . . . .
i
5
10
FIG. 3. Punch force-travel curves for circular plates o f radius a and thickness h = 1.6 mm stamped by a spherical punch and die o f R ° = 288 mm. MS Vol. 26, No. 2 - - D
136
T . X . Yu el aL
1.00.g ~ o
o
[]
0.7~
a (ram)
0.5"
TEST THEORY(42)
100 0.4-
0,2 0.5 i
0.,3
o.,s ~
75
o
50
,,
1,
,2 i
~
3
3
~
s ;o .
P~ .2~
( PTys)xl 0"
FIG. 4. Non-dimensional relationship between the average final curvature of plates and the largest punch force applied, h = 1.6 mm, R ~ = 288 mm.
2
(P*/yS)xl03
/k
1.6 I
3.2
I
6.s
/,,
-.-,,,,,./
~
,11
I
,
E z I
0'.5
'
U'
1.'0 s/ct
FI6. 5. Distribution of final curvature along a radius for a plate of a = 75 mm and h = 1.6mm.
Circumferential wrinkling Measurements of the plate surface profile in the circumferential direction were also made with a Taylor-Hobson Talyrond 61 machine. In Fig. 6, the deviation of each profile from a datum circle (marked by a chain line) represents the variation in final height along a circle on the plate surface. It can be seen from Fig. 6 that the plates of radii 50 mm and 75 mm have two wrinkling waves or lobes, whilst plates of radius 100 mm have four deep lobes for small P* and eight faint lobes for large P*. Photographs of the deformed plates in Fig. 7 have dark marks at the contact regions between the tool and the plate of radius 100 mm. These marks show the radial distribution of these circumferential wrinkling lobes. ELASTIC D E F L E C T I O N OF PLATE In order to fully appreciate the behaviour of circular plates in the early stage of the pressing process and the onset of plasticity, the elastic deflection will be analysed.
(b)
(a)
(d)
(0
e)
'
FlG. 6. Measurement of specimens after pressing, showing the variation in the height along circles of latitude. (a) a = 5 0 m m , r = 4 5 m m , p*=3.5kN; (b) a = 5 0 m m , r = 4 5 m m , p * = 1 4 k N ; (c) a = 7 5 m m , r = 7 0 m m , p*=7.85kN; (d) a = 7 5 m m , r = 7 0 m m , p*=31.4kN; (e) a = 1 0 0 m m , r = ~( 95 50 m mm m -- -- ' p*=8.9kN; (0 a=100mm, = ~95 m m - r [50 mm__, p* = 89 kN.
-
(c)
/
.q
-.9..
o~
e~
0~
(ao
T . X . Yu et al.
138
First, examine the elastic solution of a simply-supported circular plate of radius a and thickness h subjected to a concentrated force P at its centre. In this case, the deflection of the plate is[4], (a2 - r:) + 2r 2 In ~ ]
w = 16riD /[ ~3 v+ v P L
= A (a ~"- r") + Br 2 In ~,
(4)
a
where D = Eh3/12(1 - v:) is the flexural rigidity of the plate, and A and B are coefficients independent of the radial coordinate r. The curvatures of the plate are, accordingly,
and
xo . . . . .
2A-B
r dr
(r) 21n
a
(5)
1 ,
so that G and x o tend to infinity at r = 0. This implies that this elastic solution is not strictly applicable to the circular plate loaded by a spherical punch. Even for a small force P, the centre contact point must develop into a small circle. Thus, consider a circular plate on which the load is uniformly distributed along a circle of radius b, as shown in Fig. 8. In this case, the deflection of the plate is[4], w = 8~tD ( b 2 + r : ) l n
+(a:-b2)(3+v)a2-(1-v)r21"-~+v-~ -.j
(6)
for r < b, and
w=8nPD{(a2-r2)[l+~(1-~)]+(b2+r2)ln
r}
(7)
for r > b. Thus the deflection at the centre of the plate is P [- 3 + v 2 Wo=~zbt~(a-b~)
b2
ln~]
(8)
and the deflection of a point on the loading circle is P
Wb
~L
F(3+v)a 2-(1-
~
+~
v)b2(a2-b2)-2b21n~]"
(9)
As a result, the relative deflection in the central portion is w
l_v
0-
L2(-i
(10)
)k
For the curvatures in the region r < b, (6) leads to P [ l-v [l_b2"~ a] ~=K°=~L2(I +v)\ a:]+ln~ .
(11)
Since the central portion of the plate will wrap around the punch, x, and x0 are equal to 1/R for r < b, where R is the average radius of the punch and die, i.e. R = (R D + Re)/2. Thus, from (11),
4nD /[- 1 - v
["
t' = W / L 2 ( w 7 7 6 ) [ l
b:\ -
a] + In g ] ,
(12,
and from (10),
wo - wb = b2/2R.
(13)
P(2Wb
I
1
FIG. 8. A circular plate subjected to the load uniformly distributed along a circle.
Stamping and springback of circular plates
139
The r.h.s, of (13) is equal to the displacement of the punch; thus expression (13) indicates that the centre of the plate and the punch pole have the same displacement; in other words, no separation occurs. By introducing non-dimensional parameters
tl - R/a and p =- b /a,
(14)
expressions (12) and (8) can be re-written in non-dimensional form as
--=--O ~
[2T~v)(1-p2)+ln
(15)
and
w°
a-
Pa/D[- 3 + v 8n [ 2 ( ~ v )
(I-p2)-p21n
~1
"
(16)
For a specified problem, v and ~/are given, so that the load-displacement relationship can be obtained from (15) and (1.6) by means of the parameter p which varies from 0 to 1. As a numerical example, by taking v = 0.3 and ~/= 1, i.e. R = a, the calculated results are as shown in Table 1. In this example, the load-displacement relationship remains nearly linear until the displacement w0 is as large as about 0.2a. The previous analysis is based on the small deflection theory so it is applicable only if the deflection of the plate is smaller than the thickness, i.e. w0 ~ h.
(17)
For a plate, the curvature at the initial yield state is (refer to[5]), 2Y(1 - v2)
re
Eh
(18)
Since the maximum curvature of the plate permitted in the investigation is 1/R, the above elastic solution is available only if I/R < r, or R E >- h 2Y(1 - v2)'
(19)
If (19) is violated, then the onset of plasticity will precede the wrapping of the plate around the punch pole. PLASTIC DEFORMATION
Joined spherical and conical deformation mode It was observed in the experiments that (i) the contact region between the punch and the plate developed from the centre outwards as the pressing proceeded, and (ii) the separation between the central portion of the plate and the punch pole was negligible when ratio a/h was large, say, > 30. Therefore, the following approximate deformation mode may be proposed. As shown in Fig. 9, the central portion of the deformed plate is assumed to be a spherical surface in which the radius matches the punch. In other words, this portion follows precisely the shape of the punch, whilst the outer portion of the deformed plate is assumed to form a conical surface having a smooth conjunction with the spherical portion. From the geometry shown in Fig. 9, the deflection at the centre of the plate relative to the outer drcumference is A = (a - b) sin/3 + R(1 - cos/~),
(20)
where R is the radius of the central spherical portion, b is the arc length from the centre to the edge of this portion.
TABLE I. THE ELASTIC BEHAVIOUR OF A CIRCULAR PLATE LOADED BY A SPHERICAL PUNCH, V = 0 . 3 AND R = a = bla
PaID
Wola
(Pa/D)(wo/a)
10 -6
0.8922
0.0451
19.80
10-5
1.0666
0.0539
19.80
10 -4
1.3256
0.0670
19.80
10-3
1.7509
0.0884
19.80
0.01
2.5780
0.1301
19.81
0.05
3.8496
0.1928
19.97
0.10
4.8914
0.2401
20.38
140
T. X. Yu et al.
DIE
I
F1G. 9. Joined spherical-conical deformation mode. and fl = b/R is the angle subtending the arc b. By taking sin fl ~ fl --- b/R and 1 - cos fl ~ fl2/2, (20) can be re-written in non-dimensional form as d ~ ctp (1 - P ) ,
(21)
where a2
ct - ~
b
and p - a
(22)
are non-dimensional parameters. To find the punch displacement dp, an accompanying displacement, d A, caused by the movement of the plate edge down along the die should be added to the deflection, A, that is dp=d
+A A
or
6p = J + t5A
(23)
where 6A = da/h is a non-dimensional accompanying displacement. It can be calculated as R ~ = ~ (cos 4~ - cos ¢0),
(24)
where ~b0 and tk are determined from sin 4>0 = a/R and sin q~ = s i n fl +(a/R)(1 - p ) cos ft.
(25)
It can be proved from (21), (24) and (25) that 6~/6 is of the same order as ab/R2. Hence, in most practical cases &A is negligible by comparison with 6, and thus from (23), 6, -~ 6.
(23)'
That is, the punch displacement can be regarded as the same as the deflection at the plate centre.
Load-carrying capacity estimated by plastic membrane theory The force required for continuing deformation of the plate has been calculated for large deflections by assuming in-plane stresses dominate the stresses due to flexure of the plate. Three different models of the punch force distribution (and the mode of deformation within the contact region) have been analysed. (i) Centralforce with joined spherical and conical deformation. Calladine[6] has proposed a simple method to minimise upper-bound solutions to load-carrying capacity and that is by considering the overall equilibrium of segments of the plate. For example, when a simply-supported circular plate carrying a concentrated load at its centre has a large deflection (A > h), a conical deformation mode may be assumed for an upper-bound calculation, and the rigid/plastic stress distribution will be as shown in Fig. 10(b). The position of the line I - I is determined by the 'equal areas' condition, which corresponds to overall equilibrium of forces acting on one half of the plate in the direction perpendicular to the diametral cut. By taking moments about I - I, the load is found to be
P =~
~[ dA
=4e0f ah2 Ja ~v[ dA,
(26)
where y is the vertical coordinate measured from the line I - L A is the area over the radial cross-section, and P0 is as defined by (3). For example, in a simply-supported plate with zl _> h, (26) yields p=6+~, where p = P/Po and 6 -= A/h.
1
forr>l,
(27)
Stamping and springback of circular plates
141
The rigid/plastic stress distribution shown in Fig. 10(b) leads to a distribution for the circumferential membrane force, No, as shown in Fig. 10(c), where N O denotes the maximum plastic membrane force which is equal to Yh. Between No/N o = ! and No/N o = - 1 there is a transition zone of width a/& When the deflection considered is very large (say, 6 = d / h > 5), Calladine's method can be further simplified• In this case, the second term on the r.h.s, of (27) is negligible in comparison with the first term, and the transition zone for the distribution of No~No vanishes. These results correspond to assuming that the radial cross-section can be condensed to its mid-line L - L, see Fig. 10(b), so that in effect the stress distribution through the thickness is neglected.Consequently, instead of SA [Yl dA, only a simple one-dimensional integration SL [Y[ dL along the mid-line L - L is required. To apply this simplified Calladine method to our deformation mode, Fig. 9, with the help of Fig. 11, the work done by P gives P p - ~ = ~k, ro
(28)
with ct = a2/Rh and p(1 - 2p2/3) k=
,
b 1 f o r P = a = < ~,
,
b 1 f o r p m--_>--.
1-p ½+p-p2/2 1 --p
(29)
a --2
This result is independent of the distribution of the punch load in the contact region r __
1 2
p h
4a
1 2
p
h • 4R'
• t a n ~b ~ -
(30)
where c is the arc length for the tensile region, see Fig. 12(b), and 4, can be calculated from equation (25). Since R ~> h, expression (30) indicates that c/a is only slightly smaller than ~ as long as p is not very large.
I~1.~a~LI
I
P/2~a
N0( 'N, a.r
-,
(a)
[
-1
(b)
(c)
FIG. 10. A simply-supported plate subjected to a concentrated force at its centre. (a) Equilibrium of one half of the plate. (b) Stress distribution when A > h. (c) Circumferential membrane force.
b
._
A
/ I
L b Y (a) (b) FIG. i 1. Mid-line L - L of the radial cross-section of a circular plate. (a) For p = b/a <~ 1/2. (b) For p = b/a >>.1/2.
P/2~R cos ~
(a)
~
~
(b)
F16. 12. Effect of the horizontal components of the die reaction. (a) Equilibrium of one half of the plate. (b) tensile and compressive regions in the radial cross-section.
142
T . X . Yu et al.
The distribution of circumferential membrane force is as shown in Fig. 13, i.e. No~No = ~ + 1, for 0 < r/a < c,/a, ( - 1 , for c/a < r/a < 1.
(31)
(ii) Uniformly distributed transverse force with conical deformation. Consider a simply-supported circular plate subjected to a uniformly distributed circular loading q within radius b as shown in Fig. 14. In the case where the boundary is free to move inward, Kondo and Pian[7] found the load-deflection relationship to be as follows,
.(,
1 1+5620,
when30< 1,
1 6o + ~ , .~oo
when3o>l,
P
(32)
where P = nb2q, Po = 2nMp and 60 is the non-dimensional deflection at the centre determined from a conical mode. If the punch force is assumed to be uniformly distributed over the central portion of the plate, then by using the previously defined non-dimensional parameters, it is found that 3o = ~p
(33)
and when 6o is large, P ~
~p 2p' 1 ----
(34)
3
where p - P/Po, ot - a2/Rh and p = b/a as defined before. Therefore, by considering a continuously increasing p, expression (34) provides an estimate to the load-carrying capacity of the circular plate loaded by a spherical punch. (iii) Uniformly distributed pressure with spherical central deformation. For a circularly loaded, simplysupported, circular plate, the previous analysis[7] also showed that in the central portion of the plate, iV, = No = No. In our problem this implies that the central portion under the punch is subjected to a uniform biaxial plastic tension; in other words, it becomes a plastic spherical membrane. From force equilibrium considerations, a uniform internal pressure is required to be applied on the central spherical portion of the plate, see Fig. 9, and its magnitude is found to be
2N0
q = --. R
(35)
Thus, the total punch-force is p = nb2q =
Dr Nob2
R
Nob2 = P0"--
(36)
M.R'
or in non-dimensional form, p = 4~p 2.
(37)
Since N o = No holds within 0 _-
Ne/No t
-I
i FIG. 13.
a
FIG. 14.
FIG. 13. Circumferential membrane force when the deflection of a plate is very large. FIG. 14. A simply-supported circular plate subjected to a uniformly distributed circular loading q.
Stamping and springback of circular plates
143
contact circle between the punch and the plate and found a punch force-travel relationship. Using our notation, it can be re-written as sin fl cos ~ -
cos
(4
(38)
fl)
-
and (39) where fl = b/R, and ~b is determined by (25). Expression (39) is equivalent to (21).
Comparison with experiments and discussion Four different approximate methods have been proposed. Accordingly, they estimate the punch force by expressions (28), (34), (37) and (38), respectively, whilst the punch travel, i.e. deflection at the centre of the plate, can be calculated by (21) for all cases. Fig. 15 shows the punch force-travel curves (i.e. the load-deflection curves for plates). The load-carrying capacities predicted by the simplified Calladine method, i.e. expression (28), and by employing a circular plate subjected to a continuously extended circular loading, i.e. expression (34), are in good agreement with the experimental data. The first three methods are essentially different from the last method proposed by Johnson and Singh. In the first three analyses the membrane forces are dominant and the punch load is assumed to be carried principally by the membrane forces developed in the plate. However, the method proposed by Johnson and Singh takes the radial bending moment M, as a dominant factor in the deformation process and it controls the travelling of the circular hinge. As a result, the method proposed by Johnson and Singh may provide a prediction to the p ~ ~ curve for the thick plates, say a/h < 20, since the deformation of thick plates is more closely represented by a stress distribution associated with flexure; but this method fails if the plate ratio a/h is large. In the case of thin plates, say a/h > 30, membrane forces are the dominant factor, and the methods based on membrane forces give much better prediction to the p ~ ~ curve of the plate. Previously, a small deflection analysis[8] has shown that the membrane forces are negligible only if a/h < 20. Beyond this range, the membrane forces have to be included. If a/h is very large, a pure membrane theory may be employed.
1.8o
THEORY
o -o
(26) (34) (37) (38)
~ =0.
I~jl
l!.
EXPERIMENT
l
•
O. --=100
mm
o
0,=
mm
75
7t/ I I II II II IIII ell I
1.0
I/7', ./ Vo ~ /
II
0.2 0
0.1
012
0.3
0 .4
0~5
FIG. 15. Comparison of the punch force-travel curves predicted by the different theories with experiments.
144
T . X . Yu et al.
SPRINGBACK Based on the distribution of membrane forces as associated with the simplified Calladine method, the springback of circular plates after pressing can also be approximately predicted as follows. Assume that at the end of pressing, the entire plate forms a partly spherical surface, see Fig. 16, i.e. p = h/a = I. At the circumference of the plate, i.e. point A in Fig. 16, assume that unloading leads to the entire release of the membrane force, No = -No = - Y h . Then an elastic circumferential strain of Y(I - v 2 ) / E must occur in a radius of R sin (a/R). Hence, point A should move outwards a distance Y(-!-- v2) R sin ( a ~ E \R]
(40)
Similarly, when the membrane force N,(=No in r < a/2) is released there is a corresponding shortening in the length of arc OA, which leads to a fall in the height of point ,4 of Y(I - v 2) 9 --• R. E 4
(41)
Hence, after unloading, point A moves to a new position A ', as shown in Fig. 16, and the coordinates of point ,4' can be determined by (40) and (41). By assuming OA' is a circular arc of radius R e, it can be shown that
XD - R r ~ 1
(42)
~
which is the ratio of the radii of the plate before and after springback. For comparison with our experimental results, take R = 288mm, v =0.28, E = 2 0 5 k N / m m 2 and Y = 302 N / m m 2 (i.e. choosing 10% higher than the initial yield stress of mild steel plate used, in order to allow somewhat for the strain-hardening which occurs during the pressing process), then for plates of radii a = 50, 75 and 100mm, the calculated values of xr/~c are 0.80, 0.91 and 0.95, respectively. The measured values in experiments are 0.71, 0.87 and 0.93 respectively, refer to Fig. 4. Expression (42) is based on assuming p = b/a = 1 and therefore it can be regarded as a limit ot xe/x when the punch load increases without limit. From this point of view, the agreement between the prediction by using (42) and the experiment, as shown in Fig. 4, is quite good. Two non-dimensional punch-force parameters P / Y S and plot have the following relation p :x
P Po~
P 2R YS h
(43)
As the experiments have shown, when P * / Y S > 3 × 10 -3, further increase in P* has little effect on the increase in Kr/~c°. Since R/h = 180 in these experiments, this is equivalent to stating that the increase in xF/x n is mainly brought about in the course of increasing p until p*/~ ~ 1. Alternatively, this refers to increasing p ( = b/a) until p = 0.5 ~ 0.6. As long as the largest punch-force applied is such that p/o~ > 1, i.e. P*>P#-
Ya 2h 2R '
(44)
the springback can be predicted by (42).
WRINKLING Theory To analyse technologically the plastic buckling or wrinkling of circular plates during spherical pressing, we adopt the well-known energy method and follow a procedure which was developed previously in[9], where the buckling of annular plates in relation to the deep-drawing process was considered. As observed by previous researchers, when a circular plate wrinkles during pressing, the wrinkling mainly takes place in its outer portion, see Fig. 17. Indeed, the plastic analysis above has indicated that the circumferential membrane force No is negative in the region of a/2 < r < a, and this compressive force must be the principal source of wrinkling.
~
R Rs'n R
.(,*o)
', ~F
~'
FIG. 16. Change in the shape of a plate during springback.
Stamping and springback of circular plates
145
Hence, it is reasonable to assume that the additional deflection due to wrinkling is
(a)
C r-~
sinn0,
for-a-
(45)
where C is a constant and C ~ 0 when the wrinkling occurs. Then
--
Or
a2w
= C sin nO,
= Cn r -
cos nO,
a2w = O, ~ = Cn cos nO
(46)
and
02~w=002- C n : ( r -
2 ) sin n0.
Although the punch load is carried by membrane forces, the additional deflection is assumed to be associated with the flexural rigidity when strain-hardening is considered. By referring to[4], the corresponding bending energy of the plate is
~fDfO2w 1 Ow
1 O2w'~2
O2w[1 Ow
1 O2w'~
- - D ( 1 - v)-~-Tr2t ; - ~ -r + ~ )
(1 O2w
+D(1 - v)t;ff~- 0
1 OwN2) rjff~)~rdrdO.
When applying (47) to a fully plastic plate, Poisson's ratio is
Eoh3
(47)
v
=
½
and the flexural rigidity is then
Eoh3
D = 12(1 -- v 2)
9 '
(48)
where E 0 is the buckling modulus determined by
Eo _
4EEp +
(49) 2
E is Young's modulus and Ep the tangent modulus of the material (see[10]). Substituting (46) and (48) into (47) leads to
u=- oc2~:,r°: {[l-,:( l-aq: ~)J + n't,~) / a '~2] J~T dr 2
C 2. ~ ' . 9
F(n),
(50)
where
( 5\ 4 ( 11\ F(,)-- tin 2 - ~). - t 2 In 2 - ~-)n:+ In 2.
(51)
The plastic stress distribution in the outer portion of the plate has been found using a rigid/plastic analysis, see[7], to be
Nr=No6[1
r
a (
1\ 2-] for~
1+~ ,
(52)
and
where N o = Yh, 6 = d/h and d is the deflection at the centre of the plate caused by pressing, as defined earlier. By referring to[l 1], the work done by the membrane force is 1 (" ('f ( ~ w \ 2
A T = --~ J J l N t ~ r J
/1 ~w'~ 2)
+ N o t r ~ ) ~rdrdO.
(53)
T . X . Yu et al.
146 Substituting (52) into (53) results in
AT=
7z
. (/2. Yha2.G(n, 6),
2
(54)
where
n21 1~-I 1 2 1 G(.,~)---7 l+ln2-(l+~)ln ( l+7)J-~(. +1)~8~n2 246~(.~-1).
(55)
By equating A U and A T, the critical condition for wrinkling is presumed to be obtained as
go h3 9
F(n) = Yha 2 G(n, 5).
(56)
If a non-dimensional buckling index for the plates is defined as -
.~,
(57)
the critical value of ( can be found from (56) as
,_ f,(,)
(<' = 3 ~1 G(n, ~)"
(58)
When ( > ~c,, wrinkling occurs; and when ( < ~c,, no wrinkling occurs. Expression (58) indicates that (<, depends on the wave number n as well as the deflection caused by pressing. The calculated results from (51), (55) and (58) are as shown in Fig. 18. This figure shows that: (i) (c, increases with the wave number n, and the smallest value o f (c, is given by the case of n = 2; and (ii) (c, falls when the deflection increases from 6 = A/h = 1. For 6 > 2, ~<, approaches a limiting value (~*,(n), for a particular n. Taking the limit of G in (55), as deflection increases without bound, it is found that 1
1
G*(n) - G(n, oo) = ~ (2 In 2 - 1)n 2 - ~,
Y
o
4-
BUCKLED
\ \
(~cr
r~= 8
(or
n =6
~,cr
n =A
(or
n=2
UNBUCKLED
0
j
~
~
~
~=e/h &
FIG. 18. Critical values of the buckling index ( = ( Y ~ o ) " (a/h) as a function of the deflection 5 = d / h and the wave number n.
(59)
Stamping and springback of circular plates
147
and accordingly,
*" , 1 / F(n)
(60)
Hence, the smallest buckling index for the wrinkling of plates is ln 2 - 1/2x~I/2 ~*,(2) = \ ~ ] ~ 1.68,
(61)
and a plate will remain unbuckled if its index satisfies ( -
.~ < 1.68.
(62)
The higher is the value of ( for a plate, the more waves will be created in the outer portion o f the plate during pressing. Indeed, if n is large enough, then *
1
/8]n2 - 5
~c,(n)~42~n
2
1 . n ~-0.4n.
(63)
Thus, for a very thin plate which will have a large (, the wave number can be easily estimated by n = 2.5(.
(64)
Comparison with experiments For the mild steel circular plates used in our experiments, E = 205 kN/mm 2, Y = 275 N/mm 2, and Ep ~ 0.1 E, so that n = 2.5 ( = 0.19 (a/h). When a plate of a = 100 mm and h = 1.6mm is examined, this equation gives n~12. Experimentally, we observed n = 4 ~ 6 when P * / Y S = l x 1 0 -3 and n = 8 ~ 1 2 when P*/YS = 10 x 10 3, refer to Fig. 6.
Doubling and ironing After the punch contacts the wrinkles in the outer portion o f plate, the development o f waves of large amplitude is prevented by the punch pressure, though the number o f waves is increased by a reverse buckling process, as shown diagrammatically in Fig. 19. This phenomenon is similar to the doubling of waves in a deep-drawing process when a blank-holder is applied. Johnson and Mellor[12] have described this phenomenon (first pointed out by B. Senior) and observed that the more numerous are the waves o f small amplitude so formed, the more easily are they ironed-out to give a good finish on products. CONCLUSIONS (1) To form a bowl from a thin metal plate, the punch force P required to achieve the full curvature from a hemispherical die set is
P
na2hy
Po
2R
Further increase in force may reduce wrinkles but it has almost no effect on elastic springback or the final curvature. At this punch force, the ratio o f final-to-die curvature will be
xe/x n -~ 1 -- 9Y(I - v2)R2/2Ea 2. (2) Wrinkling is a principal limitation on stamping to a large curvature. In this analysis of thin plates, wrinkling does not occur when
~/h < m.68e,/eo/r. "tRUESINE CURVE~
t
FIG. 19. The mechanism for the doubling in the number of waves (following [12]).
148
T.X. Yu et al.
These conclusions follow from an analysis of large plastic deformation where in-plane forces predominate during stamping and elastic springback occurs upon unloading. The forming history of metal plates has been completely followed using this analysis with a rigid-plastic material characterisation. Acknowledgements--One of the authors, T. X. Yu, gratefully acknowledges the support of the Educational Ministry of the The People's Republic of China and the Cambridge University Engineering Department. The authors wish to thank Mrs R. M. Orriss for typing their manuscript and Prof. S. R. Reid for his comments on the paper. REFERENCES I. W. JOHNSOH and A. N. SINGI-I,Springback in circular blanks. Metallurgia 275-280 (May 1980). 2. T. X. Yu and W. JOnNSOH, Cylindrical bending of metal strips. Metals Tech. 10, 439~,47 0983). 3. H. G. HOPKINSand W. PRAGER,The load carrying capacities of circular plates. J. Mech. Phys. Solids 2, 1-13 (1953). 4. S. TIMOSh'EHKOand S. WOIHOWSKY-KgmGER,Theory of Plates and Shells, 2nd Edn. McGraw-Hill, New York (1959). 5. T. X. Yu and W. JOHNSON,The large elastic-plastic deflection with springback of a circular plate subjected to circumferential moments. J. Appl. Mech. 49, 507-515 (1982). 6. C. R. CALLAD1NE,Simple ideas in the large-deflection plastic theory of plates and slabs. Engineering Plasticity (Edited by J. Heyman and F. A. Leckie), pp. 93--127, Cambridge University Press (1968). 7. K. KO~qDOand T. H. H. PIAH, Large deformations of rigid-plastic circular plates. Int. J. Solids Structures 17, 1043-1055 (1981). 8. W. JOHNSONand T. X. Yu, On the range of applicability of results for the springback of an elastic/perfectly plastic rectangular plate after subjecting it to biaxial pure bending-II. Int. J. Mech. Sci. 23, 631-637 (1981). 9. T. X. Yu and W. JOHNSON,The buckling of annular plates in relation to the deep-drawing process. Int. J. Mech. Sci. 24, 175-188 (1982). 10. J. W. GECKELER, Plastiche knicken der wandung von hohlzylindern und einige andern faltungserscheinungen. Z. Angewandte Mathematik und Mechanik 8, 341-352 (1928). I I. S. TIMOSh~NKOand J. M. GERE, Theory of Elastic Stability, 2rid Edn. McGraw-Hill, New York (1961). 12. W. JOHNSONand P. B. MELLOR,Engineering Plasticity, pp. 313-314. Van Nostrand Reinhold, London (1973).