An approximate solution for the hydrostatic bulging of circular diaphragms with draw-in allowed

Journal o f Mechanical Working Technology, 13 (1986) 279--289 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

279

AN APPROXIMATE SOLUTION FOR THE HYDROSTATIC BULGING OF CIRCULAR DIAPHRAGMS WITH DRAW-IN ALLOWED

H.M. SHANG, F.S. CHAU, C.J. TAY and S.L. TOH Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge (Singapore 0511) (Received May 21, 1985; accepted January 27, 1986)

Industrial Summary The hydroforming of sheet metal into shells can be found in many engineering applications, including the manufacture of containers and bursting discs and the investigation of sheet-metal formability and basic plasticity laws. As such, there is vast amount of literature on both the experimental and analytical aspects of this forming process. In these previous investigations, draw-in of the flange during forming has always been neglected: in the production line, however, complete elimination of flange movement is not always ensured, as is evident from the presence of wrinkles at the flanges of some hydroformed shells. Based on previous experimental observations and an earlier discussion on the effect of die shoulder radius, an approximate solution is developed in this paper to include the influence of slight draw-in during the hydroforming of copper sheets into axisymmetrical shells. The solution is further compared with experimental results; the latter, in general, agreeing with reasonable accuracy. Some limitations of the present solution are also discussed in this paper.

Introduction M a n y s h e e t - m e t a l p r o d u c t s , including t h o s e in t h e f o r m o f surfaces o f rev o l u t i o n , are m a n u f a c t u r e d b y t h e h y d r o f o r m i n g process, in w h i c h the s h e e t m a t e r i a l is d e f o r m e d using h y d r a u l i c pressure. This m e t h o d o f d e f o r m i n g t h e b l a n k is also u s e d e x t e n s i v e l y f o r t h e a s s e s s m e n t o f s h e e t - m e t a l f o r m a b i l i t y a n d p l a s t i c i t y laws. H e n c e , n u m e r o u s investigations, b o t h e x p e r i m e n t a l [ 1 - 4] a n d a n a l y t i c a l [ 5 - - 1 0 ] , h a v e b e e n m a d e , especially f o r t h e s i t u a t i o n w h e r e t h e b l a n k is f o r m e d freely into a shell having r o t a t i o n a l s y m m e t r y . Past l i t e r a t u r e [1--3, 5--10] indicates t h a t little a t t e n t i o n has b e e n p a i d t o t h e e f f e c t o f draw-in d u r i n g h y d r o f o r m i n g . Using c o m m e r c i a l p u r i t y copp e r sheet, S h a n g et al. [4] r e c e n t l y r e p o r t e d a l o w e r i n g o f t h e f o r m i n g pressure w i t h a c o r r e s p o n d i n g r e d u c t i o n in t h e d e f o r m a t i o n w h e n a slight a m o u n t o f draw-in was p e r m i t t e d . A t a p o l a r height a p p r o a c h i n g failure, h o w e v e r , t h e f o r m i n g pressure w i t h draw-in p r e s e n t was higher t h a n t h a t w i t h draw-in absent. When draw-in was a b s e n t t h e overall bulge s h a p e was

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© 1986 Elsevier Science Publishers B.V.

280 observed to be close to spherical, b u t with draw-in present the overall shape became even closer to spherical form. The main objective of this paper is to develop an approximate solution for the hydroforming of commercial purity copper sheet into axisymmetrical shells where a slight a m o u n t of draw-in of the flange is permitted. It will be seen that this solution, which is based on previous experimental observations on the effect of draw-in [4] and an earlier paper describing the effect of die shoulder radius [10], agrees well with experimental results. Hydroforming with and w i t h o u t draw-in Figure 1 shows the meridional section and plan of an axisymmetrical shell h y d r o f o r m e d to polar height H by pressure P. If draw-in is prevented during forming, the various element particles in the clamped portion of the undeformed blank (denoted by the d o t t e d circles BB, CC and DD) will remain in their original positions on the deformed shell. If, however, draw-in is permitted, the d o t t e d circles become the solid circles (denoted b y B'B', C'C' and D'D') at polar height H. Thus, curves BB and B'B' represent, respectively, the rim of the undeformed blank and that of the deformed shell. At this polar height H, a material particle on curve CC on the flange of the undeformed blank has moved to the position where the die shoulder just begins (curve C'C'). Similarly, a material particle on curve DD, which is where the die shoulder (of radius b) just begins, has shifted to curve D'D' on the shell. The a m o u n t of draw-in present may be quantified adequately by the per-

de formed~

i

~

Ir I

undeformed

bLonk

B' |

\

C' D' l

|

\\

\\2

O, \ \ ~

r.

\~-'u

D

C

B

I

I

i

iT

I

\i,",",,,"

Fig. 1. Hydroformed shell formed with and without draw-in.

281

centage area of blank material which is drawn into the die throat. Mathematically, this is expressed as AA AD

AC (~DD- 1 ) × 100%

(1)

where AC is the area of a circle with radius rc and AD is the area of a circle with radius r D. The

proposed approximate solution

The effect of the size of the die shoulder on the hydroforming of commercial purity copper sheet (having a linear hardening characteristic) has recently been discussed by Shang and Shim [10] using a simple model as well as existing models which have been modified to include the die shoulderradius effect. On comparison with experimental results, their studies showed that, whilst each of these models might have some advantage over the others, none could provide a complete description of the hydroforming process. For example, the approximate solution of Takahashi and Takeyama [7] gave good correlation with experiment on the effect of die shoulder-radius on pressure growth and polar strains but it could not predict strain distributions. Hill's solution [5], after suitable modification [10], enabled computation of strain distributions with reasonably accuracy but the effect of die shoulder-radius on the polar thickness-strain could not be analysed. Those equations which fit the experimental data well [10] are extracted from these models and summarised as follows: 2p - H Thickness strain, et' = 2 In [ ~ ] at the pole (r = 0) 2p (after Refs. [7,10] ) e t , = 2 In

R2

[R2 ] -+ Hl

forR~>r> 0 ( a f t e r R e f . [5])

(2)

where the radius of curvature (p) and vertical deflection (1) at r are given by the equations H2+R 2 p - - 2H

b

l= H - p + ¢,[p2_ r 2]

(3)

The physical meanings of the symbols used in eqns. (2) and (3) are illustrated in Fig. 1. It should be pointed out that both eqns. (2) are identical at the pole for a sharp-edged die (b = 0), and that the first of eqns. (2) was used by Takahashi and Takeyama [7] to show the effect of die shoulder-radius on the polar thickness-strain. The polar thickness-strain computed from the second of eqns. (2), however, is unaffected by the die shoulder-radius.

282 For linear-hardening materials, the following stress-strain equation has been used: )T = o0 (1 + K~)

(4)

where o and ~- denote, respectively, the effective stress and effective strain and where o0 is the initial yield stress in simple tension and K is a constant. Thus, from the condition of equilibrium, the forming pressure may be derived, i.e., P -

20o to p exp (- e0)

Keol

ll

~5}

where to and e0 denote, respectively, the undeformed sheet thickness and polar thickness-strain. Equations (2) to (5) were derived on the basis of a linear-hardening material which is h y d r o f o r m e d into a spherical shell w i t h o u t draw-in [10]. Without draw-in, the deformed shell was observed to be quite spherical, but when draw-in was allowed, the overall shape of the shell became even " m o r e spherical" and its surface area at a given polar height was reduced [4]. The utilisation of the in-feeding material during forming, derived from Figs. 8 and 10 of Ref. [4], is shown in Fig. 2. In this figure, the ordinate represents the dimensionless difference in surface area of two shells of equal polar heights, one formed with draw-in permitted (A¢ ~ o) and the other with draw-in prevented (A~ = 0). It is readily seen that the in-feeding material is practically unstretched since the a m o u n t of in-feeding material present is approximately equal to the difference in the surface areas. Figure 3, which is derived from Fig. 9 of Ref. [4], shows the difference (A) between the thicknessstrain distribution of shells formed with draw-in and that without draw-in. It is seen that at 24 mm polar height the difference is nearly constant. At 47 mm polar height, however, the difference is no longer constant because at this polar height deformation w i t h o u t draw-in is unstable whilst deformation with draw-in present is still in its stable regime. In developing the present ap-

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'

8t-

0

~i I

2

i ..... /,,;K. i,,Jx~" ~

~

6

8

10

'i -

12%

&A AD

Fig. 2. Utilisation of in-feeding material during forming.

283 35%

30

-

I

25

--

A

--

•

I

20

10 H = 24ram

5 i

I i

0 0

10

20

30

40 Current

50 Rodius,

60

70

80mm

r

Fig. 3. Effect of draw-in on thickness-strain distribution.

proximate solution, it is therefore reasonable to assume that the deformed shell is spherical and that the additional blank material drawn into the die throat for attaining a polar height H is unstretched. Furthermore, this infeeding material is assumed to be uniformly distributed over the entire shell surface by an a m o u n t tu given by A C - AD tu = - - to A

(6)

where (Ac - AD) denotes the additional blank area due to draw-in; to denotes the original blank thickness; and A denotes the surface area of the shell (of projected radius rD) at polar height H and given by ref. [9] as A = 2•

[bR[3 +

(1 - Cosfi)(p 2 - b2)]

R

(7)

where fi = sin -1 [ p - ~ J The final thickness of the shell at polar height H is assumed to be the algebraic sum of the equivalent thickness (tu) and the shell thickness if draw-in had been absent. Hence, combining eqns. (2) and (6), the theoretical thickness-strain distribution in the shell formed with draw-in permitted is given by the equation

284 tu

et = In [~0 + exp (et')l

{8)

The present solution, which involves superposing the two thickness-strain distributions (thinning due to the absence of draw-in during forming and uniform thickening due to the in-feeding material), bears some similarity with the approximate solution b y G a m b y and Lampi [9]. In their solution, which neglects the die shoulder and draw-in effects, deformation during hydroforming is considered to be the superposition of a main large displacement field for spherical configuration onto an additional small displacement field which is necessary to pass from the spherical configuration to the actual one. Results and discussion In the experimental investigation b y Shang et al. [4,10], the material used was 1.215 mm thick commercial copper sheet having a linear work-hardening characteristic given by o = 255.6 (1 + 1.055-e) MPa

i9)

The bulge size (R) and die shoulder-radius (b) were, respectively, 88.9 mm and 7 mm. The a m o u n t of draw-in present in their tests [4] is illustrated in Fig. 4. Using the above data, the forming pressure and the polar effectivestrain for various polar heights were c o m p u t e d using the derived solution. It is readily seen in Fig. 5 that the present solution describes the actual behaviour well.

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i :'( :

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Pressuresu~e/

,01-

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' /

,/oi

{

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,

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10

20

30

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t

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40 50 Pol, or H e i g h t

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Fig. 4. Draw-in characteristic during hydroforming. Fig. 5. Theoretical forming pressure and polar effective-strain with draw-in present.

285

Pc = 3 2 0 6

-

-

-

kN / • -

-

AA

AD S

0

_~_.._o_..T._, o 0

10

20

,

o/S,e~tsec

30 ~.0 Point Height, H

Omm

Fig. 6. Variation of draw-in with blank-holding load. ? HPa

'

I

P = 320 kN 6

'

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5

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Serrated

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,

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10

20

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, 50 50 Height, H

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Fig. 7. Forming pressure for various draw-in conditions. To further demonstrate the adequacy of this approximate solution, additional tests were performed using the same sheet material and the same die sets, but with blank holding loads of 320 kN and 380 kN to produce different draw-in characteristics (Fig. 6). Figure 7 shows that the predicted forming pressure for various draw-in conditions agrees well with experiment (approximately 6% maximum error). When draw-in is prevented, the predict-

286 ed instability pressure is higher than the actual value. When draw-in is allowed, the instability pressure cannot be predicted for lack of experimental data since excessive shifting of the pole was observed after a polar height of 45 mm had been attained. The trend of the experimental pressure-growth curves in Fig. 7 shows that at a polar height of less than approximately 27 mm, draw-in lowers the forming pressure for a given polar height. However, for polar heights greater than 27 mm, draw-in results in a higher forming pressure. The present approximate solution also shows this trend, except that the predicted cross-over point occurs at a b o u t 15 mm polar height. Below this polar height, the effect of draw-in is negligibly small. For example, at 10 mm polar height, the maxim u m pressure variation is only 0.1 kPa. Nevertheless, the difference between theory and experiment is estimated to be -+ 0.1 MPa at this polar height. amounting to an error of approximately 6%. At greater polar heights (eg 45 mm polar height), theory gives a spread in pressure of approximately 0.3 MPa owing to draw-in whilst experiment shows a spread of approximately 0.5 MPa. The m a x i m u m error incurred is thus estimated to be 5%. Figures 8 to 10 show that the predicted thickness-strain distributions for various draw-in conditions agree well with experiment. The maximum error in the polar thickness-strain is approximately 22% which occurs at 45 mm polar height when a clamping load of 320 kN is used (Fig. 10). This magnitude of error, which also occurs at 15 mm polar height when clamping loads of 380 kN (Fig. 9) and 320 kN {Fig. 10) are used, is due to the small de-55% -SO

-45 ~-~o 35 -30 _~ - 2 5 2C

-1{ -5 C

0

10

20

30

40

SO [urrenf

50

70

Rodius,

BO

90ram

r

Fig. 8. Thickness-strain distributions for zero draw-in.

287

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t

I

I

1 /15 /2s /3s

-~5 ~T

-~o

I /

22.~ / o / I lo.6/

o -35 -30 H=35mm

c Z=

-25 -20 -15 -10 -5 0

10

20

30

~-0

50 [urrenf

60

70

Radius,

80

90mm

r

Fig. 9. Thickness-strain distributions for a clamping load of 380 kN.

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-,oil-45

~

,

~

I

I

'

--,hoar, i

--

'

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,, a, ~=0 t

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--

-10".

~ I

-5 0 0

10

20

30

/~0

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60

70

Radius,

80

90 mm

r

Fig. 10. Thickness strain distributions for a clamping load o f 320 kN.

288

formation at the small polar height. The inset in Fig. 8 shows that, when draw-in is prevented, the percentage difference between theory and experim e n t is consistent with an average value of 9.3%. Figures 9 and 10 show that when draw-in is allowed, the error at 25 mm polar height is less than 8%; but for a polar height greater than 25 ram, the error generally increases with polar height. The increasing error in polar thickness-strain with polar height may be attributed to the assumption that the in-feeding material is unstretched throughout the forming process. Figure 11 shows that the in-feeding material is unstretched only at small polar height (up to approximately 27 mm) but at greater polar height, the in-feeding material is stretched. It will be noted that in computing the theoretical thickness-strain distributions, equations from two different models have been used because these equations have been shown (Fig. 5 and Ref. [10] ) to give reasonably good correlation with experiment. The first equation of eqns. (2) for polar thickness-strain is based on the approximate solution of Takahashi and Takeyama [7], whilst the second is based on the modified Hill's solution [10]. Although the present results show that the error in the polar thickness-strain might be reduced if the modified Hill's solution is used, it should be noted that in this case the polar thickness-strain would be shown to be unaffected by the die shoulder radius.

II

i',

~.o~---+ )

'°<'

:/

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~/

i

2#/'/~-

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!

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o.,W Lo- ......1-- +,-.......¢ .... r.......i oLh-c/T 2 ] 2 i i7 0

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3

4

6

%

7%

AA AD Fig. I I, Utilisation o f in-feeding material for various draw-in c o n d i t i o n s .

Although t h e o r y shows that the stress--strain relationship of the sheet material used in this investigation will not yield a spherical shell [5], experimental results [4,10] show that the overall shape of shells formed with and without draw-in may be approximated adequately by a spherical surface. With this assumption of spherical surfaces, the present solution is unable to describe the effect of draw-in on the polar radius of curvature and the surface area of the deformed shell.

289

Conclusion In this paper, an approximate solution is developed to describe the deformation of commercial copper sheets subjected to hydrostatic pressure. This solution accommodates the presence of draw-in during forming and the variation of the die shoulder radius of the forming dies. In spite of the assumptions used, reasonable correlation of the predicted and experimental results is observed: the present solution may thus be used as a working model for the hydroforming process.

Acknowledgements The authors gratefully acknowledge the assistance of the National University of Singapore in providing financial support for this investigation.

References 1 W.F. Brown and G. Sachs, Strength and failure characteristics of thin circular membranes, Trans. ASME, 70 (1948) 241--249. 2 P.B. Mellor, Stretch forming under fluid pressure, J. Mech. Phys. Solids, 5 (1956) 41--56. 3 H.M. Shang and T.C. Hsu, Deformation and curvatures in sheet metal in the bulge test, Trans. ASME, 101 (1979) 341--347. 4 H.M. Shang, F.S. Chau, C.J. Tay and S.L. Toh, Assessment of behaviour of sheet metal during hydroforming, Proc. 13th Biennial Congress of IDDRG, Melbourne, Australia, 1984, pp. 106--117. 5 R. Hill, A theory of the bulging of a metal diaphragm by lateral pressure. Philos. Mag., 41 (1950) 1133--1142. 6 N.M. Wang and M.M. Shammany, On the plastic bulging of a circular diaphragm by hydrostatic pressure, J. Mech. Phys. Solids, 17 (1969) 43--61. 7 H. Takahashi and H. Takeyama, The analysis of hydrostatic bulging of metal diaphragms, Technology Reports, Tohoku University, Japan, 1969, Vol. 34, pp. 205-245. 8 M.F. Ilahi, A. Parmar and P.B. Mellor, Hydrostatic bulging of a circular aluminium diaphragm, Int. J. Mech. Sci., 23 (1981) 221--227. 9 D. Gamby and L.H. Lampi, An approximate numerical method for circular membranes under lateral pressure, J. Strain Anal. 19 (1984) 261--267. 10 H.M. Shang and V.P.W. Shim, A model study of the effect of the size of the die shoulder in hydroforming, J. Mech. Work. Tech., 10 (1984) 307--323.