Analysis of superplastic forming processes using a finite-element method

JJef

Materials

Processing Technology

;! Journal of Materials Processing Technology 62 (1996) 90-99

ELSEVIER

Analysis of superplastic forming processes using a finite-element method Yong H. Kim a,*, S.S. Hong b, J.S. Lee b, R.H. Wagoner c aDepartment of Mechanical Design Engineering, Chnngnam National University, Daejon, South Korea bAgency Jot Defense Development, Daejon, South Korea CDepartment oJ"Materials Science and Engineering, Ohio State University, Cohoabus, USA Received 15 March 1995

Industrial summary

An incremental rigid-viscoplastic finite-element method (FEM) has been developed in order to simulate superplastic forming (SPF) processes. The optimum pressure-time relationship for a target strain rate is calculated simply by a liner interpolation of the pressure values at the previous iteration step. The material is assumed to be isotropic and a modified Coulomb friction law is adopted. SPF processes of aluminum alloy with various geometries have been analyzed using axisymmetric and plane-strain line elements, based on the membrane approximation. Through the analysis, optimum pressure-time curves and deformation behavior are predicted. Experiments have been performed successfully using the calculated optimum pressure-time curves as a control value. Comparison between the prediction of the FEM and the results of experiment shows good agreement, the present study demonstrating the usefulness of the method in process simulation of superplastic forming. Also, the combined stretch-blow forming process is modeled to improve thickness distribution.

Keyword~:SuperplasticIbrmingprocesses:Finite-elementmethod

1. Introduction

The superplastic forming (SPF) process is one of the advanced manufacturing methods for producing very complex thin-sheet components, used especially in aerospace industry, Superplastic materials exhibit very high tensile elongation, typically 500% or more, in a particular range of temperature and strain rate [1]. Such superplasticity in a material depends highly on the strain rate and occurs only in a narrow range of strain rate with an optimum value that is unique to each material: usually very low, such as 10-3-10-5/s. Therefore, it is crucially important to determine the pressure loading cycle before the actual forming to maintain the maximum strain rate near the optimum value throughout the whole forming process. For this reason, the process desigp of SPF is less amenable to trial-and-error approaches than conventional manufac*Corresponding author. Fax: +82 42 823 2931; e-mail: yonghkim@hanbat.¢hungnam.ac.kr ElsevierScienceS,A.

SSD10924-0136(95)02223-6

turing processes. The determination of the forming parameters, such as the pressure loading cycle and final thickness distribution, is essential in not only reducing the forming time but also in achieving successful forming of complex parts. For this purpose, it is necessary to model and analyze the forming process. A number of theoretical and numerical analyses have been performed for modeling superplastic forming. Recently, the finite-element method (FEM) has been used extensively to model SPF processes with more accurate prediction of the deformation behavior [2-12]. For complex shapes, FEM can be a viable approach to model non-linear material behavior and contact-friction phenomena. In this paper, a rigid-viscoplastic finite-element analysis with membrane approximation has been performed to simulate numerically the superplastic forming of thin sheets. Two-node line elements for both axisymmetric and plane-strain cases are used with a Newton-Raphson non-linear solution scheme. The pressure-time relationship for a given optimum strain-rate and thickness distribution are calculated. Various geometries,

91

Y.H. Kim el al. / Journal of Materials Processing Teclmology 62 (1996) 90 99

computational cost in the modeling of SPF, where the sheet deforms predominantly by stretching rather than by bending or shear. The material is assumed to be isotropic, following the von Mises flow rule. A modified Coulomb friction law is used to model the interfacial friction between sheet and tools. Both axisymmetrie and plane-strain line elements are used. More details of the formulation can be found in Refs. [13,14]. No

(:El ,Zl )

(x,.=)

(x

[

__~

(~.=,)

(=,,Zo)

(~,z~) (~)

(b)

x(r)

0.4

Fig. I, Geometries of (a) line and (b) arc segments. including cylindrical-cup, cone and rectangular-box shapes are analyzed using the developed FEM program. Experiments are performed using the calculated pressure-time curve, the validity of the formulation being verified by comparison between analysis and experiment. Also, the combined stretch-blow forming process is modeled to improve the thickness distribution.

¢=0.3

0.2

0.1

#=0.2 ,u, = 0 . 4

I

O.G 0

I

100 140 TIME, s

5'0

2. Rigid-viscop|asfic fin|te-element analysis

200

2,50

~a)

2.1. Finite-element Jbmmlation and solution procedure The membrane approximation is used and the effect of strain-rate hardening is taken into account. Use of the membrane element can be justified in terms of less

1.6

1.2

f[ =;

- LINE L ~MENT * * ° o ° TRIANGULAR ELEMENT

j

,u.

~4

°°

r~

P~ P~

r4

0.4

i

f

~ 5'0

0.0 0

1;o

. = 04

1;o

200

TIME, s (b)

(a)

0.4

- LINE ELEMENT ===o= TRIANGULAR ELEMENT

n~ 0"8

,u~O 2 r~

~=0,4 O1

0.0 fPlO0

'J

(b)

~84

'J (c)

Fig. 2. Geometries of the forming tools: (a) hemisphere; (b) cylindrical cup, and (el cone shape.

0

±

100

i

i

200 300 TIME, s

t

400

500

(c)

Fig. 3. Pressure-time curves for differentfriction coefficients for: ~a) hemisphere: (b) cylindrical cup, and (c) cone shape.

Y.H, Kim et ol. Journal ¢!/' Materials Processing Technology 62 (1996) 90-99

92

where #, 8(= j ~ d t ) , ~ and A~ are effective stress, effective strain, effective strain-rate and the increment of effective strain during At, respectively. Considering the external loading by a pressure P, the associate work functional All can be defined by: ~ + AV

t=90s

P dV

(2)

¢ o

where V0 is the cacrent volume enclosed by the sheet and AV is the increment of volume during the time interval. Now, let the sheet be represented by an assemblage of finite elements and AU be the nodal-displacement vector. Then application of the virtual work principle to Eq. (2) gives:

"xq

ANALYSIS

EXPERIMENT

Iv

All = A W -

ANALYSIS EXPERIMENT (a)

~AI-I [' ~3Ag .... =ho Ja,, | 0~c ~ ° d A ° cat./

c~AV P'~A-U=

F,~

(3)

80s

where F¢ is the external nodal force vector at time t + A t . Eq. (3) can be written in the form of force equilibrium as: ¥, EXPERIMENT

-

PIt

= F~

(4)

where,

ANALYSIS

f~

it)

c~Ag

# ~

F i = ho Fig. 4, Comparison of deformed shapes with experiments for different forming stages of: (a) homisphere; (b) cylindrical cup. and (el cone

dAD

|O

~AV dAU

shape.

H=--

evolution of the microscopic properties of material, such as cavity growth, grain growth, etc,, is considered. A rigid-viscoplastic FEM for simuhtting SPF processes was developed. The formulation is based on an incremental deformation theory, which assumes a minimum plastic-work path during a small time interwtl At [14]. Then, the internal work increment during the small time interval may be written as: /' [',~o+ a~ aw=/,o J.oJ~o # dgdAo (1)

The non-linear system of Eq. (4) is solved by the Newton-Rhapson scheme with linearization by: A U = AU* + ~U

(5)

where AU* is a trial solution and ~U is the correction vector. Then the iinearized form of Eq. (4) is: (K~ - PK2 - K3)6U = F~ - F~+ P H

(6)

where:

N+~,,NI~AvoAuj

K,=ho o L ~ + O.OlO

dAoiav= av, 6~2AV

K2 =

~ A--(~U

Av =

Av

~0,005

....

oOOOo

5b

: AIMEDV A L U E (0,00693/see)

16o

I~o

~6o

~o

TIME, see Fig. 5. Variation of the maximum effective strain rate with time for the forming of a hemisphere.

2.2. Treatment of contact

In order to consider the friction between tool and sheet, the following modified C o u l o m b friction law is used [i 5]:

zxu~ Ft =

-

,u IIF.II4,(AUJ-,,--~;-.. 1

lq

,, ~l~t~

I

{7)

Y.H, Khn et al. Jourmd (~! Materials Processing Technology 62 (1996)

90-99

93

(a) ,

ii

~<

~i

i~

i ~ i

~ ~iI~ ~,i

~,

~

,

(b)

(c) Fig. 6. Photographs of the formed parts: (a) hemisphere; (b) cup, al,,d (c) cone. where Ft and Fn are the tangential and normal components of the extenal force F~, respectively, and tt is the friction coefficient. 4)(AU,) in Eq. (7) is the following functic,n so that the friction force tends smoothly to

zero as the incremental displacement approaches zero [14]: q~(AUL)--- 1 if IIAU~II> d

Y.H. K~m et al./ Journal oJ'Materials Processing Technology 62 (1996) 90-99

94

¢(A/Jt) = ~

if

[lau, II ~ d

(8)

F,

This modification allows all nodes in contact to be fully slipping, even at very small displacements when Ft < pF. for equilibrium. The parameter d represents an accuracy tolerance for contact.

!ii

LL L

1,0

J

Fig. 8. Geometries of the forming tool for rectangular box forming (plane-strain case; dimensions in ram).

iI

0.8

35

Finally the external force is written by: ~0.6 ~0,4 t.-

Art Fo=F, + F. = (-~(AU,) ~ +,QIIFol I

'~51S

at contacting nodes

e

0.2

ooooo

:

~tAA: ooooo:

t

44

s

t =132

S

=

and:

t =~76 s '~'~'¢,* : t =220 s

0.0

z'0

60

4'0

DISTANCE

FROM

CENTER,

rnm

(a) 1.0 -PREDICTED(TRIANGULARELEMENT) • -- -- PREDICTEDtUNE ELEMENT) =~00¢1EXPERIMENT

0.8

~

Fe = 0

at non-contacting nodes.

The tool profile is represented by a number of digitized line and arc segments. The coordinates of the segments and their first and second derivatives with respect to the x-coordinate are represented by: (i) for a line segment (Fig. l(a)): Z ~ Z2 -- ZI X

X2 --

~0,6

~

D

O

~

0,4

,i

D

(9)

X I

S~ = z2 - zl X 2 -- X I

O

0,2

+ZlX2 - -- Z2XI x 2 -- X i S,,,, = 0

(I0)

(ii) for all arc segmem (Fig. l(b)): z = :o + ~

o.o o--

~% DISTANCE

;o FROM

e'o' THE

CENTER.

xj

X-- X o

8o

mm

-

& = ~:',/Ro 2_ -- ( x - - Xo)2

Sxx = -~ - [Ro2-- ( R°2 X - - Xo)2] 3/2

(b)

(11)

where (x,, z,) and (x2, z2) are the coordinates of the starting and ending points of the segment, respectively,

1,0

~0,8

PREDICTED{TRIANGULARELEMENT) - - PREDICTED(LINE ELEMENT) {:=O00 EXPERIMENT(t-140 s) ~^AA EXPERIMENT( t = ~

0,65

~0,6

--

E 0.60

Ro,4

{~

~ 0.2

Z

0,2

0,0

FEM

ooCO= EXPER~['[MENT

E ~0.55 ~n

,.vqo.5o 1;

DISTANCE

2b

FROM

3;

THE

4b

CENTER,

50

mm

(c)

Fig. 7. Comparison of thickness distributions at different forming times with experiments for: (a) hemisphere; (b) cylindrical cup, and (c) cone shape,

0.45

o.%.~

' '7.'o

141o ' ,~.o' ' ~6'.o '35.o

DISTANCE

FROM

THE

CENTER,

mm

Fig. 9. Comparison of the final thickness distribution with experimental data for rectangular box forming (plane-strain case).

Y.H. Kim et al. dourmdo! Materials Processing Technology 62 (1996) 90-99 ~

~

z× ;"

95

6° a

Z3o i

20

N

io

~.~

_~- c/2

06

20

40 X

A

-

AX2S,

60

60

mm

Fig. 12. Evolution of the deformed shapes for differemt forming times of the component with stiffeners.

A

c0 = :o + x/Ro: - ( x p - Xo) 2 e = IP-?tmn . Other die corner radii = 1.27m.m

(13)

with:

Fig. lO. Geometries of the forming tool for an aircraftcomponent wiih integralstiffeners(plane-straincase).

.4=~(;-Zo)~ 2 [(x~ - xo)J + 1

and (Xo, Zo) and Ro are the coordinates of the center and radius of curvature of the arc. For each trial solution, the nodes may penetrate the tools (punch or die). The penetration conditions for every contact node are determined by checking the position of the node relative to the tool points. Then, all penetrated nodes are repositioned (projected) onto the nearest point of the tool surface. The coordinates of the projected position, (xp, z o) are computed directly as follows: (i) for a line segment:

zo}-xo

B=(z"-z°)Iz°xc-zcx°~ X (x~ o -- t

x~ - X o

C=xo-'+ \

"ZoX~-ZcX° Co - R o 2 Xc -- Xo

where (x~, :~) are the trial coordinates. A node located on the tool surface is constrained to move in the tangential direction of the tool surface. The constraint condition for contacting nodes can be written [14]: (14)

6U~ = S ~ U ~ Xp

(X,--X,)2Xc--(Z2--Z,)(z,x,-- Z2XI)+Zc(Z2--Z,)(X,--X,) ( z 2 - :,) 2 + (x2 _Z2--Z___.___~I

Zp x2_xtXo

-

x,) 2

.]ZIX2--Z2Xl

(12)

x2-x,

T3 (F~ - e n o - T~ (F~ - PH,) = 0

(15)

where T = (T,, T3), Fi = (F~, EA and H = (H~, 11,). After linearization by the Newton-Raphson scheme, Eq. (15) is:

(it) for a n arc s e g m e n t :

- s +_ , / h ~-- A C Xp =

where 6 U = {6U,, 6U:} i~ the solution of the stiffness Eq. (6). Considering the force equilibrium at a contacting node:

A

(16)

K ' 6 U = F'

7.0

6.0 /z=0.4 ¢~50

/I

~4.0

where,

~(F~-PHO g' = t 3

-

~-AU

(F~

-

O(F:enj T,

PHi) ~-..,1,~. U L~ ¢./ [

~AU

dT~ + ( F ~ - eHx) c~AU

=,w'

~ 3.0

~'2.0 ,o

1.0

0.0 0

1 0 0 0 2000 3000 4000 5000 6000 TIME. s

Fig. II. Pressure-time curve for the aircraftcomponent with integral

stiffeners.

t._,~

p

L

t.----a0-----~ Fig. 13. Tool geometry for the formingof the two-step pan and the punch (dimensionsin mm).

Y,H, Kim et al. Joltrnal ~!/" Materials Processing T'ecltmdogy 62 (1996) 90- 99

96

2.5

0,100

2.0 0.075

E

STRETCH FORMING

1.5 :~

~LOW FORMING

m

0,050

tn u~

iu >

0.

1.0

0,025

~

0,000

~

rE

0.5

I

"

"

D~ . . . . I000

I , 2000

,

,

.

. . aooo

.

.

. . . 4000

~

0.0 5000

TIME, S

Fig. 14, Optimal punch velocity-time and pressure-time curves for the SPF o f a two-step pan,

F'

=

T](F~ - PHi) - T.~(F~- e n o l a v = ~u'

This constraint condition will be imposed on the stiffness matrix after assembly. 2.3. Strain.rate control

One of the main goals of the numerical simulation of SPF is to predict the optimum loading curve for maintaining the superplastic properties of the material and for minimizing the forming time. Various approaches for determining the optimum load-time relatio,aship have been proposed [8-12]. In punch forming, the punch velocity is controlled so that the maximum strain rate is kept near to the optimum value. Since the strain rate is proportional to punch velocity, time increment and the punch velocity are modified to: - ~~ . ~ tAt~ ItADIlleW~ ~ /old t~opt

V. . . . .

(At)oia

unless the maximum strain rate, ~,~, is within a given error hound. Similarly, the pressure should be controlled so that the maximum strain rate is maintained near to the target value. An automatic pressure computation is done for maintaining the optimum strain rate [10]. After finding the displacement field with pressure p~ ~t each time step, the maximum strain rate, St, is computed. I~" ~ is not within a given error bound to the target value, an arbitrary pressure P2( = ~P~, • = 1.01) is chosen and the solution procedure is repeated. If the difference is not still within the error bound, a final pressure value is calculated using:

where ~t and 52 are the maxilaum effective strain rate at pressures p~ and P2, respectively, and ~op, is the target strain rate. Z 4. Computational procedure

The computation procedures for analyzing SPF processes by the previously-described method are as follows. I. At the beginning of each time step, update the equilibrium and contact condition by pressure. 2. Assuming trial displacement A U ~, check the position of the nodes. If a node has penetrated into the tool, that node is projected onto the tool surface by the method described in Section 2.2. 3. Compute the stiffness equation, Eq. (6). 4. Compute F, at all contacting nodes and release nodes if F, > 0. Impose the constraint condition, Eq. (15). 5. Solve the linearized stiffness equation and check the convergence. If the solution does not converge, repeat 2-5 until convergence is obtained. 6. If the convergence is obtained, check whether the maximum strain rate is within the given error bound with the target value. If the computed value is not within the error bound, modify the punch velocity or pressure value, Eq. (18) or (19), then repeat 2-6.

3. Experiment of forming into dies In order to evaluate the validity of the analysis, corresponding experiments were performed using alu-

Y.H. Kim et al./ Journal qf'MateriaA" Procesxing Techmdogy 62 (1996) 90 99 Table I L e a s t - s q u a r e s s u m o f t h i c k n e s s differences, E,"''~'* ( l . - t,) ~- f o r the c o m b i n e d stretch

b l o w f o r m i n g o f a l w o - s t e p p a n It,, = 0.411 m m J

t+= 0.2

Stretch-blow forming

t l = 15 mm H = 18 mm

/+= 0.4

Rp=3mm

Rp=4mm

R,,=Smm

R,=8 mm

Rp=4mm

Rv=Smm

0.024 0+017

0.027 0.018

0.030 0.020

0.047 0.029

0.089 0.063

0,150 0.110

Blow forming

0.t~S9

minum SUPRAL 150 sheet with an initial thickness of 0.88 mm. To obtain the strain-rate sensitivity of the material, tensile tests were performed separately, the following constitutive relationship being found: # =

')7

(19)

K~"

where K = 105.9 MPa s'" and the strain-rate sensitivity index m = 0.4. For this material, the optimum strain rate during the whole forming process is chosen as gopt=6.93 × 10--3/s, where m has its maximum value. The sheet is clamped over the die and heated to 470 °C for the forming test. Then the sheet was formed into the 0.8 /~ = 04 E 0.6

E

0270

die by compressed argon gas and the gas pressure was controlled so that it follows the computed pressuretime curve. Axisymmetric hemisphere-, cup- and coneshapes were formed. The geometries of the forming tools are shown in Fig. 2. Boron nitride is used for securing high interface lubrication, the friction coefficient being chosen as 0.4 [16,17].

4. Results and dlscussion The algorithm described above has been implemented ht the tinite-element code to simulate superplastic sheet forming processes. The simulation provides the pressure cycle as well as the deformed shape, the thickness distribution, the equivalent strain and the strain rate at each time step. Experiments were carried out for axisymmetric cases with the optimum pressure-time curve, as predicted by FEA.

M

I

0.2 .... +

0.0

~

*

m

5]~

~}5/+JLOt, LI~U',Sr~

~I = 2 0 i { ;1 : ~ I q :

"

'i; [ n w )

!

10

20 30 40 50 60 70 80 DISTANCE FROM THE CENTER, r n m

0.8 /,t, = 0 4 SO.6

,.~ 0.4

0.2 --

0.0

4.1. Amdysis o/" axisvnunetric hlow liwming

Ot c¢+

BLOW STRETCH/BLOW(PUNC-~ W=20+RP=4.H= I 5mm 5TRETCH/BLOW~ t PUNCH W=20,RP=4,W=1Brnr~

t'o 2'o a'o 4'o ~o do 7'o ~'o DISTANCE F R O M THE CENTER, rnrn Cb~

Fig. 15. Final thickness distributions for: (a) different punch corner radii, and (b) different punch travel (It = 0.4).

As an application to axisytnmetric modeling, a hemisphere, a cup and a cone shape, shown in Fig. 2, were simulated. Due to symmetry, only one half of the shapes was modeled using 2-noded line elements. The predicted optimum pressure-time curves for different friction coefficients are given in Fig. 3. As shown in this figure, the maximum pressure is increased with lower friction, whilst the forming time is reduced by a small amount for all three cases [16]. It should be mentioned that the experiments were performed by following the pressure cycle predicted by FEM with It =0.4. For comparison, results by triangular elements are given also [18] showing that there is almost no difference between the triangular and the line element. For hemisphere- and cone- forming, where the sheet contacts the tool at an early stage of the forming, the pressure reaches its maximum value at an early stage, after which it decreases with thinning of the sheet at the pole. In cup forming, however, there is a steep increase in the pressure once bottom contact occurs. Deformed shapes at different stages are shown in Fig. 4 along with experimental results, good agreement being found for the three cases. The variation of maxi-

98

Y.H. Kim et al./ Jo,,a;al of Materials Processing Technology62 (1996) 90-99

mum strain rate with time for hemisphere forming is shown in Fig, 5. The maximum strian rate is maintained within an error bound ( + 5%) of the target value, showing the validity of the present pressure control method. Use of a small error bound (2%) results in almost no difference in the final thickness distribution whilst causing a small increase in CPU time. Fig. 6 presents photographs of formed parts. The calculat~ thickness distributions along a radius for the final shape, together with experimental results and those from three-dimensional analysis, are given in Fig. 7. FEM prediction shows good agreement with experimental measurements. The difference between FEM and experiment can be attributed either to peripheral ,~k:w-in during the experiment or to possible void growth in the material. 4.2. Plane-strain analysis

In order to verify the program developed, planestrain blow-forming processes are analyzed and the results are compared with those in the literature. No experiment was carried out for the plane-strain case. The first example is the forming of a simple rectangular box section, which can be represented by a planestrain strip. The tool geometry is given in Fig. 8, material chosen being Ti-6AI-4V with K = 460 MPa s" and m = 0.5. The target strain rate is 3 x 10-4/s and the friction coefficient chosen is 0.2. The thickness distribution in Fig, 9 shows good agreement with the result of Bellet and Chenot [3]. Another example is chosen from Ref. [I l] showing the capability of the current formulation to handle complex geometry. The tool geometry is shown in Fig. 10, the material being A17475 with K=2332.5 MPa s" and m = 0,6324. The target strain rate is 2 × 10='4/$ and the friction coefficient chosen is 0.4. In Ref. [11], sticking condition was assumed for the contact between the sheet and the tool. Fig. 11 presents the computed pressure-time curve, showing less forming time compared with those in the literature. Two steep rises in pressue occur when the sheet starts to fill the stiffener cavities. The evolution of the deformed shapes at the different stages are presented in Fig. 12, showing the sequence of contact. 4.3. Analysis o f combined stretch-blow forming

The influence of the final thickness distribution in SPF processes on the mechanical properties of the product becomes very crucial as the geometry of the product becomes more complicated. It is desirable to design the process to improve the uniformity of the final thickness distribution, even though some amount of deviation is unavoidable due to the sequence of contact of the sheet and the tool. Until now, most effort

has been placed on experimental studies to improve the thickness distribution, using the trial-and-error method [19-2i], Since the sheet on contact tends to resist further deformation, the thickness becomes smaller where the contact occurs later. It is known that a stretch-blow forming can contribute the improvement of the thickness distribution by suppressing the deformation at the region where contact occurs later in blow forming. In this study, a combined stretch-blow forming process was simulated numerically to study the effect of the corner radius of the punch and the amount of punch travel on the final thickness distribution in the forming of a two-step pan, shown in Fig. 13. The width of the punch is fixed at 20 ram. The material is assumed to be Ti-6AI-4V with an initial thickness of I ram. Fig. 14 shows the variation of punch velocity and pressure for the optimum strain rate. In the stretching stage, the punch speed must be decreased rapidly for keeping the maximum strain rate uniform. The pressure-time curve shows a simlar tendency to those for blow forming. To measure the uniformity of the thickness distribution quantitatively, the least-squares sum of the nodal thickness deviation, Zlotys ( t o - 0 2 is used, where to is the uniform thickness computed from volume constancy. Table 1 shows the effect of the punch geometry and the punch travel on the uniformity of the thickness distribution for two different friction coefficients. Final thickness distributions for stretchblow forming are presented in Fig. 15, showing the effect of the punch corner radius and the punch travel: a greater friction coefficient causes a greater thickness variation, whilst stretch-blow forming provides less thickness deviation compared with blow forming. The result shows that a better thickness distribution can be obtained by proper choice of the punch geometry and punch travel.

5. Conclusions

Superplastic forming of thin sheet has been simulated using FEM with membrane approximation. A simple iterative method was proposed for the calculation of the pressure-time curve for the optimization of the superplasticity of the material. Several examples with different geometries have been modeled and the corresponding experiments performed using the calculated pressure-time curve. Comparison between FEM and experimental results shows a good agreement in shape and thickness distribution. The examples show that the present method can successfully predict the optimum forming paths for SPF and the method can be a useful tool for the process design of SPF. Simulation of the combined stretch-blow forming process with a view to improving the thickness distribution has been carried out; the results show the possibility of useful process design in SPF by numerical simulation.

Y.H. Kun et al. Joarmd ~! Materials Pro~vssing Teehaalogy 62 (1996) 90-99

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