- Email: [email protected]
Deep Drawing of Shells Having Slopes in the Base
Deep Drawing of Shells Having Slopes in the Base Toshihiko Kuwabara, Takashi Jimma and lsao Matsuoka - Submitted by H. Kudo (1) Received on January 16,1989 ABSTRACT
A deep d r a w i n g process of s h e l l s having t w o couples o f s l o p e s i n t h e base
i s i n v e s t i g a t e d as a model f o r t h r e e dimensional sheet forming processes. The deformation b e h a v i o r o f r e c t a n g u l a r b l a n k s i s e x p e r i m e n t a l l y examined i n Furthermore, a simple, plane-strain deformation f i e l d (no d e t a i l by using f i v e punches having d i f f e r e n t shaped bases. change i n thickness) i n the flange and the combination o f simple shear and elongation i n the i n c l i n e d side w a l l s are assumed t o approximate t h e mode o f d e f o r m a t i o n o f t h e blanks. Based on t h i s model, t h e s t r a i n d i s t r i b u t i o n s o f t h e f i n i s h e d s h e l l s are predicted by the use o f an energy method, and the degree o f w r i n k l i n g i s evaluated by the amount o f t h e o v e r l a p p i n g m a t e r i a l a l o n g t h e r i d g e s o f t h e punches. The v a l i d i t y o f t h i s s i m u l a t i o n method i s d i s c u s s e d by comparing the experimental and calculated results. Key words: Three-dimensional sheet formina. Darts w i t h i n c l i n e d side walls, wrinkling, s t r a i n d i s t r i b u t i o n , beading, energy method, numerical simulation.
1. I n t r o d u c t i o n Deep drawing processes are complex, three-dimensional forming processes. i n which a blank i s stretched over the punch head and, a t t h e same time, drawn on t h e d i e surface. A t present, t h e FEM c a l c u l a t i o n seems t o be t h e most f f c t i v e a n a l y t l c a l t o o l f o r simulating such forming processesfi-2j. Oue t o a vast amount o f computation time, however, the FEM analysis i s s t i l l i n s u f f i c i e n t i n p r e d i c t i n g w r i n k l i n g and f r a c t u r i n g behavior o f sheet metal, p a r t i c u l a r l y when the p a r t has a deep, complex shaped bottom. On the other hand, several attempts t o simulate actual forming processes of autobcdy panels have been made by s i m p l i f y i n g the 3-D p r o c ss s i n t o two-dimensional, p l a n e - s t r a i n d e f o r m a t i o n ~ t o b l e m s ~ ) *Those ~ ~ . attempts have been successful i n p r e d i c t i n g f r a c t u r e s i t e s o f blanks: however, t h e y a r e o n l y a p p l i c a b l e t o t h e case where t h e shape o f t h e v e r t i c a l c r o s s - s e c t l o n o f t h e p a r t does n o t v a r y i n one d i r e c t i o n , so n o t a p p r o p r i a t e t o analyze general 3-0 problems. This study deals w i t h the deep drawing o f s h e l l s having slopes i n t h e i r basea, as a model of general 3-0 sheet f o r m i n g processes. F i r s t l y , t h e b e h a v i o r o f blanks d u r l n g t h e deep d r a w i n g processes i s e x p e r i m e n t a l l y examined by u s i n g f i v e d i f f e r e n t shaped punches. Then, i n order t o simulate the modes o f d e f o r m a t i o n of these p a r t s , a s i m p l e a n a l y t i c a l model based on t h e e n e r g y m e t h o d i s proposed. The s t r a i n d i s t r i b u t i o n s calculated by t h i s model are compared w i t h experimental results, and the v a l i d i t y o f the model proposed i s discussed.
2. Deformation analysis o f a blank based on energy method T h i s s t u d y d e a l s w i t h a t y p e of s h e l l s h a v i n g t w o c o u p l e s o f side w a l l s whose i n c l i n a t i o n angles are 8 1 on the longer side o f punch and 8 2 on the shorter side, as shown i n Fig.1. 2.1 Basic assumptions and procedures o f c a l c u l a t i o n Following assumptions are made: 1) the material i s i s o t r o p i c , r i g i d - p l a s t i c , and work-hardening. the equivalent stress-strain curve o f which being expressed by 0
eq'
Q
Ceqn
(1 )
2) Die c o r n e r r a d i u s and t h e p r o f i l e r a d i u s o f t h e d i e opening and the radius o f a l l ridges o f the punch are taken t o be zero. A r e c t a n g u l a r b l a n k i s supposed t o be used. The b l a n k i s d i v i d e d i n t o t e n r e g i o n s o f d e f o r m a t i o n , as shown i n Fig.1: due t o symmetry, only one quarter o f the flange and the corresponding p o r t i o n o f the d i e opening, dldpd3. need t o be considered. The xa x i s i s t a k e n t o be p a r a l l e l t o t h e s h o r t e r s i d e o f t h e d i e opening, d2d3. The number o f the parameters used t o express the deformation o f t h e b l a n k i s f o u r : t h e magnitude o f i n i t i a l d i s p l a c e m e n t o f
region 4, lal: the magnitude o f i n t t i a l displacement o f region 10. Ibl; and t h e magnitude o f d i s p l a c e m e n t o f r e g i o n 6,lcl. and i t s d i r e c t i o n + . With the values o f these f o u r parameters given, the ten regions i n Fig.1 i s supposed t o deform, according t o the mode o f deformation given i n section 2.2. t o make a s h e l l o f h e i g h t h. From t h i s deformation mode, the t o t a l energy dissipated during the deep drawing process, W t . i s determined as Wt=
wfl+
wsw
+ wf+
(2)
wb
where W f l i s the energy dissipated due t o p l a s t i c deformatlon i n the flange (region (4+5+6+8+10)), Wsw i s the one i n the i n c l i n e d side w a l l s (region (2+3+7+9)), Wf i s the work done by f r i c t i o n a l forces a t the blank-die and blank-blankholder interfaces, and Wb i s t h e energy d i s s i p a t e d due t o bending and unbending o f t h e m a t e r i a l passing over the edge o f the d i e opening. Consequently, the set o f the above four parameters t h a t minimize W t determines the mode o f deformation o f the blank. 2.2
Deformation analysis o f a blank
2.2.1 D e f o r m t i o n o f m a t e r i a l on punch head I n Fig.2, i t i s assumed t h a t t h e m a t e r i a l i n r e g i o n 1 i s subject t o uniform b i a x i a l s t r e t c h i n g on the punch head and t h a t t h e m a t e r i a l l i n e IJM reaches t h e edge o f t h e punch head, 1"'J"'M'. a t t h e end o f t h e deep d r a w i n g process. The n o m i n a l s t r a i n s o f the m a t e r i a l i n the d l r e c t i o n s Ox and Oy are given by epx=( 1"'J"'-IJ)/IJ
(3.1
epy-( J"'M'-JM)/JM
(3.2)
1
I n t h e p r e s e n t work. i n o r d e r t o s i m p l i f y t h e a n a l y s i s , t h e v a l u e s o f epx and e a r e g i v e n so t h a t t h e y m a t c h t h e experimentally m e a s u r J ones.
22.2
Deformation o f f l a n g e Deformation of the flange (region 4+5+6+8+10) i s assumed t o be .lade up of two successive deformation steps. I n the f i r s t step, i s assumed t h a t regions 4 and 10 move r i g i d l y , w h i l e regions 5
lp
Y
Y
X Fig.1 (a) D i v i s i o n o f a blank i n t o r e g i o n s o f deformation. (b) 0 , used f o r d e f o r m a t i o n The f o u r parameters, a. b. c. analysis o f the blank.
Annals of the CIRP Vol. 38/1/1989
X Fig. 2
Deformation analysis o f flange.
287
and 8 a r e s u b j e c t t o f i n i t e s i m p l e shear d e f o r m a t i o n as indicated by the dash-dotted l i n e s i n Fig.2. The shear s t r a i n s a t regions 5 and 8 are given by Y xy, 5-1 a1 /BC
(4.1)
Y xy. 81 ' b l /HL
(4.2)
I n the next deformation step, i t i s assumed t h a t region 6 moves from the y-axis by a distance IcI. w i t h p o i n t r i g i d l y a t angle G r e a c h i n g t h e d i e c o r n e r d2 and GC and GH becoming t a n g e n t i a l v e l o c i t y discontinuities, and i s assumed t h a t regions (4+5) and (8+10) are subject t o uniform compression i n the d i r e c t i o n Oy and Ox, respectively, w i t h no change i n thickness. Consequently, the f l a n g e deforms i n t o t h e shape as shown by t h l c k s o l i d l l n e s I n Fig.2. Thus, t h e e q u i v a l e n t p l a s t i c s t r a i n a t r e g i o n i (i-4.5.8,lO) i s calculated as
+
the f o l l o w i n g equation:
I (l+exp( u~/2))to/(4rd)~aotol(s)ds
wb'
(10)
where e i s taken as 8 1 f o r t h e s e c t i o n d i d 2 and as 8 2 f o r t h e section dpd3. to i s the i n i t i a l thickness o f the blank, r d i s the radius o f the d i e p r o f i l e , l(s) i s the magnitude o f displacement o f the material passing over the d i e edge a t p o s i t i o n s. The path o f i n t e g r a t i o n i s taken along d i e opening dldpd3. 2.2.3 Deformation o f i n c l i n e d s i d e walls The mode o f d e f o r m a t i o n o f i n c l i n e d s i d e w a l l , I"'J"'Gx'dpE', i s i l l u s t r a t e d s c h e m a t i c a l l y i n Fig.3. The d e f o r m a t i o n i s assumed t o be made up o f t h r e e successive d e f o r m a t i o n steps. F i r s t l y , accompanied w i t h t h e r i g i d d i s p l a c e m e n t o f r e g i o n 4. region 2 moves r i g i d l y i n t h e d i r e c t i o n Or), w h i l e r e g i o n 3 i s s u b j e c t t o f i n i t e s i m p l e shear deformation. Thus r e g i o n (2+3) becomes I'J'GE'. Here, the shear s t r a i n a t region 3 i s given by
n
Y xy.3'1
al/F'G
(11)
I n t h e n e x t d e f o r m a t i o n step. I'J'GE' deforms i n t o 1"J"Gx'E". accompanied w i t h t h e compressive d i s p l a c e m e n t o f the flange, u1, and w i t h the t e n s i l e displacement o f the materlal on t h e punch head, u2. F i n a l l y , 1"J"Gx'E" i s assumed t o be stretched i n the d i r e c t i o n O n , w i t h zero s t r a i n i n the d i r e c t i o n O f , t o become 1"'J"'Gx'E". I n the present analysis, the d e f o r m a t i o n mode o f r e g i o n ( 2 t 3 ) a f t e r I'J'GE' i s assumed t o be expressed by the f o l l o w i n g equations: (12.1) The energy dissipated a t GC and GH. due t o tangentlal v e l o c i t y d i s c o n t i n u i t y there, can be estimated as
Wvl-(
~o/m*GC(Gx'dp/2)
(7.1)
Wv2-( a0/fl)WGy'dp/2)
(7.2)
where a. i s t h e i n i t i a l y i e l d s t r e s s o f m a t e r i a l . Since the m a t e r i a l dldpGx'E'' i s equal i n volume t o t h a t d i s p l a c e d by region 6. Gx'dp-lclcos+.GC/dld S i m i l a r l y , Gy'd -lcl sin+.GH /dpd-j. Although the present mode? o f deformation alfows material t o o v e r l a p i n r e g i o n J"'Gy'd2Gx'. t h i s can n o t occur i n r e a l i t y , but a l o c a l in-plane compression o r out-of-plane d e f l e c t i o n o f material, such as wrinkling. i s deemed t o occur. Thus the amount o f w r i n k l i n g can be e v a l u a t e d by t h e sum o f Gx'd2 and Gy'dp. I n t h e p r e s e n t a n a l y s i s , t h e energy r e q u i r e d f o r t h o s e t y p e s o f deformation i s not considered. Thus, the t o t a l energy dissipated due t o the p l a s t i c deformation of t h e flange i s given by
where u and v a r e t h e displacements i n t h e d i r e c t i o n s O E and 0 II, respectively. As a result, the equivalent s t r a i n s a t regions. 2 and 3 are calculated by
.
Si
:i
aeq, i d o eq. i+ WVI+
Wv2
_F
Q n+l n+lsi( Ceq,i) + Wv1+ Wv2 (i45.8.10)
(8)
(13.1)
where
), En-ln(l+ 2L ),
c -ln(l+
an
E
YEn
-
au
ct--ce
- en
av
(thickness strain) (14)
+
S i m i l a r procedures o f c a l c u l a t i o n can be applied t o regions 7 and 9. Thus, the energy dissipated due t o the p l a s t i c deformation o f the side w a l l t s given by
1
where S i i s the area o f region i. The work done by f r i c t i o n a l forces a t the blank-die and blankblankholder interfaces, Wf. i s calculated by
Wf-
2uE P i I l(x.y)dxdy, i Si
(i-4.5,8,10)
(9)
where v i s t h e c o e f f i c i e n t o f f r i c t i o n , p i i s t h e b l a n k h o l d i n g f o r c e p e r u n i t area a c t i n g on r e g l o n i. and l(x.y) i s t h e magnitude o f displacement o f m a t e r i a l a t i n i t i a l c o o r d i n a t e (x.y), during the deep drawing process. The energy dissipated due t o bending and unbending o f material passing over the edge o f d i e opening, dldpd3. i s approximated by
3. Experimental setup and procedures Figure 4 shows the dimensions o f the f i v e punches used. The d i e used had a hollow c a v i t y w j t h a rectangular cross-section g i v i n g a clearance o f 0.6mm f o r a nominal blank thickness o f 0.5mm. The p r o f i l e r a d i u s a t t h e edge o f t h e d i e opening was taken as 5mm. The sheet m a t e r i a l used i n t h l s work was t i t a n l u n - k i l l e d lowcarbon steel. I t s mechanical properties are shown i n Table 1. It was c u t i n t o rectangular blanks o f several sizes.
/ \
Shorter side Fig.3
288
Oefornation analysis o f i n c l i n e d side wall.
Fig.4
P m A 15' 90' B 30' 90* c 4 F 90' D 4 5 60' E 45 45'
Longer side
Geometry o f punches.
Table 1 Mechanical properties o f material Orientation] ~ 0 . 2 1a t
fro'
rollins d i r e c t ion
0
90 45
]
I
~ 1 . n
[ W P ~ I [ M P ~ I [XI 163 173 168
321 319 312
44.2 47.7 47.7
Ir
-
Ir
value
value
value
0.26
1.43 2.21 1.76
1.94
0.26 0.26
I n the deep drawing t e s t s using the punches except punch C. the portions o f the blank i n contact w i t h the d i e and the blankholdcr were lubricated w i t h machine o i l . When using punch C, i n order t o prevent f r a c t u r e from occurring a t the punch p r o f i l e , a sheet o f v i n y l c h l o r i d e f i l m o f 0.05mm t h i c k (160mmx50mm) was inserted between the blank and the f l a t p o r t i o n o f the punch head: i n t h i s case, both sides o f the f i l m and the blank were l u b r i c a t e d w i t h machine o i l . Throughout the tests, blanks were set so t h a t t h e i r r o l l i n g d i r e c t i o n s c o i n c i d e w i t h t h e shorter side d i r e c t i o n o f t h e punch. The punch speed was about lmm/sec. The b l a n k h o l d i n g force was maintained constant during one drawing operation. The b l a n k h o l d i n g f o r c e p e r u n i t area a t r e g i o n i. p . (see Eq.(9)), was determined from the measurement w i t h PRESCALE] (FUJI PHOTO FILM CO.). From the measured d i s t r i b u t i o n o f pi, the t o t a l b l a n k h o l d i n g f o r c e F i n c a l c u l a t i o n was assumed t o d i s t r i b u t e ~ + ~ ~ ~ throughout t h e f l a n g e t o s a t i s f y ( p ~ and p6=O. where p4-p5, p8'p1@ and i s the area o f region i.
fi
~
)
~
(
~
~
~
~
+
~
~
~
~
4. Results and dlscussions 4.1 Results o f deep drawing t e s t s F o r t h e d r a w i n g w i t h punches A and B. t h e s h e l l s w e r e successfully drawn without f r a c t u r e o r wrinkling. Figure 5 shows series o f forming processes o f s h e l l s drawn w i t h punches E. D. and C. For punch E (Fig.5(a)). i t i s seen t h a t t h e b l a n k g r a d u a l l y conforms t h e punch, and t h e area o f c o n t a c t proceeds towards the lower p a r t o f the side walls. A t hdOmm the blank has completely conformed t o the punch. t h e b l a n k does n o t c o m p l e t e l y conform For punch 0 (Fig.5(b)). t o t h e punch even a t h-60mm, and w r i n k l e s remain a t t h e s i d e w a l l s and a t the l o w e r p o r t i o n o f t h e r i d g e s o f t h e punch. W i t h increasing the blankholding force up t o 71kN. breakage occurred a t the punch p r o f i l e . For punch C (Fig.5(c)). the degree o f w r i n k l i n g i s greater than when punch D i s used Moreover, the blankholding force t h a t does n o t cause f r a c t u r e a t the punch p r o f i l e lowers t o 43kN. I n order t o r e s t r a i n t h e w r i n k l i n g a t t h e s i d e w a l l s seen i n Fig.5(c). f o u r p i e c e s o f tape, l O m m wldth. 20mm long, O.lmm t h i c k , were sticked a t the edges o f the blank, i n f r o n t of the shorter sides o f t h e punch, as shown i n Fig.5(d). The d i s t a n c e between t h e tapes across t h e c e n t e r l i n e was 60mm. The tape l a y o u t o f Fig.5(d) was found t o make t h e amount o f w r i n k l i n g l e a s t : w i t h the distance between the tapes greater than 60mm. the amount o f w r i n k l i n g increases, and w i t h less than 60mm. f r a c t u r e occurred r i g h t beneath the punch p r o f i l e as shown i n Fig.5(e). These e x p e r l m e n t a l r e s u l t s suggest t h a t t h e c o n t r o l o f b l a n k h o l d i n g force alone i s not enough t o suppress w r i n k l i n g behavior o f sheet metal, but the layout o f beads i s a c r u c i a l factor. Another s i g n i f i c a n t e f f e c t o f u s i n g tapes i s t h a t t h e upper l i m i t o f b l a n k h o l d i n g f o r c e t h a t does n o t cause f r a c t u r e i s i n c r e a s e d and r e s u l t s i n t h e f u r t h e r r e s t r a i n i n g o f w r i n k l e s . The use o f tapes s t r e t c h e s t h e b l a n k and b r i n g s more m a t e r i a l i n t o c o n t a c t w i t h t h e punch than when they a r e n o t used. T h t s prevents the draw-in stress from concentrating on the m a t e r i a l a t the punch p r o f i l e and be e f f e c t i v e i n r e s t r a l n i n g fracture. 4.2
Strain d l s t r i b u t i o n s Figure 6 compares the s t r a i n d i s t r i b u t i o n s between experiment and c a l c u l a t i o n . a t t h e c e n t e r l i n e s o f f i n i s h e d s h e l l s . The calculated r e s u l t s o f strains, obtained f o r y 4 . 2 by the method o f c a l c u l a t i o n described i n section 2. are shown by three kinds o f t h i n l i n e s i n Fig.6 ( A n a l y s i s I). The c a l c u l a t i o n tends t o underestimate the s t r a i n s i n the d i r e c t i o n o f the center lines, e l , p a r t i c u l a r l y a t t h e s h o r t e r s i d e s o f punch C and D. T h i s d i s c r e p a n c y i s c o n s i d e r e d t o be caused by t h e assumption t h a t r e g i o n s 2 and 3 i s s u b j e c t t o t h e same amount o f s t r e t c h i n g i n t h e d i r e c t i o n On i n Fig.3 t o f o r m t h e s i d e w a l l 1"'J"'Gx'E": i n r e a l i t y , judging from the w r i n k l i n g shown i n Fig.5. region 3 i s not so stretched as region 2 is. Thus another c a l c u l a t l o n was done by v i r t u a l l y taking the values ofc,, i n Eq.(13.2) as zero a t r e g i o n s 3 and 7. The c a l c u l a t e d r e s u l t s o b t a i n e d f o r y-0.1 a r e shown b y t h r e e k i n d s o f t h i c k l i n e s i n Fig.6(b) and ( c ) (Analysis II ). E x p e r i m e n t a l r e s u l t s a r e between t h e t h i c k l i n e s (Analysis I ) and the t h i n l i n e s (AnalysisII ).
4.3 Evaluation o f the amount o f u r l n k l l n g As has been mentioned i n section 2.2. the amount o f w r i n k l i n g can be evaluated by the amount o f the overlapping m a t e r i a l along Table 2 shows t h e a r i d g e o f the punch, Gx'dp+Gy'dz (Fig.2). calculated r e s u l t s o f them, the conditions o f c a l c u l a t i o n being the same as i n Fig.6. It i s clear, w i t h a look a t the photographs i n Fig.5. t h a t the amount o f w r i n k l i n g becomes greater w i t h the increase o f the values o f Gx'd2t Gy'd2: punch C>punch DIpunch E. This suggests t h a t the simulation method proposed i n t h i s study i s also useful i n p r e d i c t i n g the degree o f wrinkling.
5. conclusions 1) k e p drawing t e s t s were conducted by using f i v e punches having two couples o f side w a l l s whose i n c l i n a t i o n angles are 81 on the
l o n g e r s i d e o f punch and 8 2 on t h e s h o r t e r side. The o r d e r o f d i f f i c u l t y i n deep drawing was punch C(81-45'. 02=90'). D(49, 60'), E(45", 45"). B(30". go"), and A(15". 90"). Punch C caused t h e most conspicuous w r i n k l i n g and t h i n n i n g o f m a t e r i a l . An a p p r o p r i a t e l a y o u t o f tapes s t i c k e d t o t h e b l a n k was found t o r e s t r a i n t h e w r i n k l e s s u b s t a n t i a l l y and t o i n c r e a s e t h e upper l i m i t o f blankholding force t h a t does not cause fracture. 2) A s i m p l e method o f a n a l y s i s based on an energy method was proposed t o simulate the deformation behavior o f sheet metal i n t h e above deep d r a w i n g t e s t s . The b l a n k i s d i v i d e d i n t o t e n r e g i o n s whose b a s i c mode o f d e f o r m a t i o n i s g i v e n by f o u r parameters, and the set o f these f o u r parameters which minimizes Table 2 Calculated values o f Gx'd2 and Gy'd2
Punch
D 1.77
*Analysis I and II correspond t o the calculated r e s u l t s indicated by the t h i n l i n e s and the t h i c k l i n e s i n Fig.6, respec:tively.
289
~
~
)
30
20 n
N u
10
p c
e 5
-10
-20
-30 0
100
50
'
50
100
0
50
100
-50 0
I
1
I
I
30 20
20 CI
E
..-
5
10
L
s
x2 0
r
t
-10
f -10
-20
- 20
-30 0 30
-30 --
100
50
I 1
-30 0
Punch proflle
Shock line
I
50
Die Drofile
I
I
100
x coordinates i n undefoned blank [mm] Fig.6 S t r a i n d i s t r i b u t i o n s a t t h e c e n t e r l i n e s o f blanks. e l : nominal s t r a i n i n t h e d i r e c t i o n o f t h e c e n t e r l i n e o f t h e nominal t r a n s v e r s e s t r a i n : et: nominal t h i c k n e s s s t r a i n . r a w i'n g c o n d i t i o n s a r e t h e same as Fig.5 (a),(b), (c). respectively. I n the calculation, the c o e f f i c i e n t o f f r i c t i o n p i s t a k e n as 0.2 f o r t h i n l i n e s and 0.1 f o r t h i c k l i n e s . 0=407 * c026[Mpa] i s assumed.
s:
the t o t a l energy required t o deform the blank i n t o a given shape determines the f i n a l mode o f deformation o f the blank. The s t r a i n d i s t r i b u t i o n s p r e d i c t e d by t h i s method were i n f a i r agreement w i t h measured ones. This method i s also useful i n evaluating the amount o f w r i n k l i n g t o occur a t i n c l i n e d side w a l l s o f the parts. 3) These r e s u l t s v a l i d a t e the use o f an energy method t o p r e d i c t deformatlon behavior o f sheet metal i n general three-dimensional sheet forming processes. Ackmwledgrant T h i s work was supported by t h e Japan M i n i s t r y o f Education, Science and Culture under Grant-in-aid f o r encouragement o f young
290
o
c
r
f
10
c
Y)
y coordlnates I n u n d e f o r d blank [ m a ]
s c i e n t i s t ( 6 3 7 5 0 7 1 4 ) and p a r t l y by AMADA Foundation f o r M e t a l Work Technology. References 1) Tang,S.C.. llankamban,RL. and Ling,P.. 1988. A F i n i t e Element M o d e l i n g o f t h e Stretch-Draw Forming Process, Advances and Trends i n A u t o m o t i v e Sheet S t e e l Stamping, 101-107. The E n g i n e e r i n g S o c i e t y f o r Advancing M o b i l i t y Land Sea A i r and Space, Warrendale. 2) Nakamachi.E. and Wagoner.RK. ibid. 109-120. 3) Takahashi.4. Okamoto. I.. Hiramatsu.T. and Yamada.N.. 1985. E v a l u a t i o n M e t h o d s o f P r e s s F o r m i n g S e v e r i t y i n CAD A p p l i c a t i o n s , Computer M o d e l i n g o f Sheet M e t a l Forming P r o c e s s , Eds. Wang.N.-M. and Tang.S.C., 37-50, The Pennsylvania. M e t a l l u r g i c a l Society o f AIME. 4) Furu bayashi, T.. Uj ihara, S. and Sa kamoto,T.. 1988. S i mu 1a t ion of Forming S e v e r i t y on Autobody Panels U s i n g A CAD System A n a l y s i s of Drawbead and I t s Control-, IDDRG 1 5 t h B i e n n i a l Congress, 243-250. ASM.
A deep d r a w i n g process of s h e l l s having t w o couples o f s l o p e s i n t h e base
i s i n v e s t i g a t e d as a model f o r t h r e e dimensional sheet forming processes. The deformation b e h a v i o r o f r e c t a n g u l a r b l a n k s i s e x p e r i m e n t a l l y examined i n Furthermore, a simple, plane-strain deformation f i e l d (no d e t a i l by using f i v e punches having d i f f e r e n t shaped bases. change i n thickness) i n the flange and the combination o f simple shear and elongation i n the i n c l i n e d side w a l l s are assumed t o approximate t h e mode o f d e f o r m a t i o n o f t h e blanks. Based on t h i s model, t h e s t r a i n d i s t r i b u t i o n s o f t h e f i n i s h e d s h e l l s are predicted by the use o f an energy method, and the degree o f w r i n k l i n g i s evaluated by the amount o f t h e o v e r l a p p i n g m a t e r i a l a l o n g t h e r i d g e s o f t h e punches. The v a l i d i t y o f t h i s s i m u l a t i o n method i s d i s c u s s e d by comparing the experimental and calculated results. Key words: Three-dimensional sheet formina. Darts w i t h i n c l i n e d side walls, wrinkling, s t r a i n d i s t r i b u t i o n , beading, energy method, numerical simulation.
1. I n t r o d u c t i o n Deep drawing processes are complex, three-dimensional forming processes. i n which a blank i s stretched over the punch head and, a t t h e same time, drawn on t h e d i e surface. A t present, t h e FEM c a l c u l a t i o n seems t o be t h e most f f c t i v e a n a l y t l c a l t o o l f o r simulating such forming processesfi-2j. Oue t o a vast amount o f computation time, however, the FEM analysis i s s t i l l i n s u f f i c i e n t i n p r e d i c t i n g w r i n k l i n g and f r a c t u r i n g behavior o f sheet metal, p a r t i c u l a r l y when the p a r t has a deep, complex shaped bottom. On the other hand, several attempts t o simulate actual forming processes of autobcdy panels have been made by s i m p l i f y i n g the 3-D p r o c ss s i n t o two-dimensional, p l a n e - s t r a i n d e f o r m a t i o n ~ t o b l e m s ~ ) *Those ~ ~ . attempts have been successful i n p r e d i c t i n g f r a c t u r e s i t e s o f blanks: however, t h e y a r e o n l y a p p l i c a b l e t o t h e case where t h e shape o f t h e v e r t i c a l c r o s s - s e c t l o n o f t h e p a r t does n o t v a r y i n one d i r e c t i o n , so n o t a p p r o p r i a t e t o analyze general 3-0 problems. This study deals w i t h the deep drawing o f s h e l l s having slopes i n t h e i r basea, as a model of general 3-0 sheet f o r m i n g processes. F i r s t l y , t h e b e h a v i o r o f blanks d u r l n g t h e deep d r a w i n g processes i s e x p e r i m e n t a l l y examined by u s i n g f i v e d i f f e r e n t shaped punches. Then, i n order t o simulate the modes o f d e f o r m a t i o n of these p a r t s , a s i m p l e a n a l y t i c a l model based on t h e e n e r g y m e t h o d i s proposed. The s t r a i n d i s t r i b u t i o n s calculated by t h i s model are compared w i t h experimental results, and the v a l i d i t y o f the model proposed i s discussed.
2. Deformation analysis o f a blank based on energy method T h i s s t u d y d e a l s w i t h a t y p e of s h e l l s h a v i n g t w o c o u p l e s o f side w a l l s whose i n c l i n a t i o n angles are 8 1 on the longer side o f punch and 8 2 on the shorter side, as shown i n Fig.1. 2.1 Basic assumptions and procedures o f c a l c u l a t i o n Following assumptions are made: 1) the material i s i s o t r o p i c , r i g i d - p l a s t i c , and work-hardening. the equivalent stress-strain curve o f which being expressed by 0
eq'
Q
Ceqn
(1 )
2) Die c o r n e r r a d i u s and t h e p r o f i l e r a d i u s o f t h e d i e opening and the radius o f a l l ridges o f the punch are taken t o be zero. A r e c t a n g u l a r b l a n k i s supposed t o be used. The b l a n k i s d i v i d e d i n t o t e n r e g i o n s o f d e f o r m a t i o n , as shown i n Fig.1: due t o symmetry, only one quarter o f the flange and the corresponding p o r t i o n o f the d i e opening, dldpd3. need t o be considered. The xa x i s i s t a k e n t o be p a r a l l e l t o t h e s h o r t e r s i d e o f t h e d i e opening, d2d3. The number o f the parameters used t o express the deformation o f t h e b l a n k i s f o u r : t h e magnitude o f i n i t i a l d i s p l a c e m e n t o f
region 4, lal: the magnitude o f i n t t i a l displacement o f region 10. Ibl; and t h e magnitude o f d i s p l a c e m e n t o f r e g i o n 6,lcl. and i t s d i r e c t i o n + . With the values o f these f o u r parameters given, the ten regions i n Fig.1 i s supposed t o deform, according t o the mode o f deformation given i n section 2.2. t o make a s h e l l o f h e i g h t h. From t h i s deformation mode, the t o t a l energy dissipated during the deep drawing process, W t . i s determined as Wt=
wfl+
wsw
+ wf+
(2)
wb
where W f l i s the energy dissipated due t o p l a s t i c deformatlon i n the flange (region (4+5+6+8+10)), Wsw i s the one i n the i n c l i n e d side w a l l s (region (2+3+7+9)), Wf i s the work done by f r i c t i o n a l forces a t the blank-die and blank-blankholder interfaces, and Wb i s t h e energy d i s s i p a t e d due t o bending and unbending o f t h e m a t e r i a l passing over the edge o f the d i e opening. Consequently, the set o f the above four parameters t h a t minimize W t determines the mode o f deformation o f the blank. 2.2
Deformation analysis o f a blank
2.2.1 D e f o r m t i o n o f m a t e r i a l on punch head I n Fig.2, i t i s assumed t h a t t h e m a t e r i a l i n r e g i o n 1 i s subject t o uniform b i a x i a l s t r e t c h i n g on the punch head and t h a t t h e m a t e r i a l l i n e IJM reaches t h e edge o f t h e punch head, 1"'J"'M'. a t t h e end o f t h e deep d r a w i n g process. The n o m i n a l s t r a i n s o f the m a t e r i a l i n the d l r e c t i o n s Ox and Oy are given by epx=( 1"'J"'-IJ)/IJ
(3.1
epy-( J"'M'-JM)/JM
(3.2)
1
I n t h e p r e s e n t work. i n o r d e r t o s i m p l i f y t h e a n a l y s i s , t h e v a l u e s o f epx and e a r e g i v e n so t h a t t h e y m a t c h t h e experimentally m e a s u r J ones.
22.2
Deformation o f f l a n g e Deformation of the flange (region 4+5+6+8+10) i s assumed t o be .lade up of two successive deformation steps. I n the f i r s t step, i s assumed t h a t regions 4 and 10 move r i g i d l y , w h i l e regions 5
lp
Y
Y
X Fig.1 (a) D i v i s i o n o f a blank i n t o r e g i o n s o f deformation. (b) 0 , used f o r d e f o r m a t i o n The f o u r parameters, a. b. c. analysis o f the blank.
Annals of the CIRP Vol. 38/1/1989
X Fig. 2
Deformation analysis o f flange.
287
and 8 a r e s u b j e c t t o f i n i t e s i m p l e shear d e f o r m a t i o n as indicated by the dash-dotted l i n e s i n Fig.2. The shear s t r a i n s a t regions 5 and 8 are given by Y xy, 5-1 a1 /BC
(4.1)
Y xy. 81 ' b l /HL
(4.2)
I n the next deformation step, i t i s assumed t h a t region 6 moves from the y-axis by a distance IcI. w i t h p o i n t r i g i d l y a t angle G r e a c h i n g t h e d i e c o r n e r d2 and GC and GH becoming t a n g e n t i a l v e l o c i t y discontinuities, and i s assumed t h a t regions (4+5) and (8+10) are subject t o uniform compression i n the d i r e c t i o n Oy and Ox, respectively, w i t h no change i n thickness. Consequently, the f l a n g e deforms i n t o t h e shape as shown by t h l c k s o l i d l l n e s I n Fig.2. Thus, t h e e q u i v a l e n t p l a s t i c s t r a i n a t r e g i o n i (i-4.5.8,lO) i s calculated as
+
the f o l l o w i n g equation:
I (l+exp( u~/2))to/(4rd)~aotol(s)ds
wb'
(10)
where e i s taken as 8 1 f o r t h e s e c t i o n d i d 2 and as 8 2 f o r t h e section dpd3. to i s the i n i t i a l thickness o f the blank, r d i s the radius o f the d i e p r o f i l e , l(s) i s the magnitude o f displacement o f the material passing over the d i e edge a t p o s i t i o n s. The path o f i n t e g r a t i o n i s taken along d i e opening dldpd3. 2.2.3 Deformation o f i n c l i n e d s i d e walls The mode o f d e f o r m a t i o n o f i n c l i n e d s i d e w a l l , I"'J"'Gx'dpE', i s i l l u s t r a t e d s c h e m a t i c a l l y i n Fig.3. The d e f o r m a t i o n i s assumed t o be made up o f t h r e e successive d e f o r m a t i o n steps. F i r s t l y , accompanied w i t h t h e r i g i d d i s p l a c e m e n t o f r e g i o n 4. region 2 moves r i g i d l y i n t h e d i r e c t i o n Or), w h i l e r e g i o n 3 i s s u b j e c t t o f i n i t e s i m p l e shear deformation. Thus r e g i o n (2+3) becomes I'J'GE'. Here, the shear s t r a i n a t region 3 i s given by
n
Y xy.3'1
al/F'G
(11)
I n t h e n e x t d e f o r m a t i o n step. I'J'GE' deforms i n t o 1"J"Gx'E". accompanied w i t h t h e compressive d i s p l a c e m e n t o f the flange, u1, and w i t h the t e n s i l e displacement o f the materlal on t h e punch head, u2. F i n a l l y , 1"J"Gx'E" i s assumed t o be stretched i n the d i r e c t i o n O n , w i t h zero s t r a i n i n the d i r e c t i o n O f , t o become 1"'J"'Gx'E". I n the present analysis, the d e f o r m a t i o n mode o f r e g i o n ( 2 t 3 ) a f t e r I'J'GE' i s assumed t o be expressed by the f o l l o w i n g equations: (12.1) The energy dissipated a t GC and GH. due t o tangentlal v e l o c i t y d i s c o n t i n u i t y there, can be estimated as
Wvl-(
~o/m*GC(Gx'dp/2)
(7.1)
Wv2-( a0/fl)WGy'dp/2)
(7.2)
where a. i s t h e i n i t i a l y i e l d s t r e s s o f m a t e r i a l . Since the m a t e r i a l dldpGx'E'' i s equal i n volume t o t h a t d i s p l a c e d by region 6. Gx'dp-lclcos+.GC/dld S i m i l a r l y , Gy'd -lcl sin+.GH /dpd-j. Although the present mode? o f deformation alfows material t o o v e r l a p i n r e g i o n J"'Gy'd2Gx'. t h i s can n o t occur i n r e a l i t y , but a l o c a l in-plane compression o r out-of-plane d e f l e c t i o n o f material, such as wrinkling. i s deemed t o occur. Thus the amount o f w r i n k l i n g can be e v a l u a t e d by t h e sum o f Gx'd2 and Gy'dp. I n t h e p r e s e n t a n a l y s i s , t h e energy r e q u i r e d f o r t h o s e t y p e s o f deformation i s not considered. Thus, the t o t a l energy dissipated due t o the p l a s t i c deformation of t h e flange i s given by
where u and v a r e t h e displacements i n t h e d i r e c t i o n s O E and 0 II, respectively. As a result, the equivalent s t r a i n s a t regions. 2 and 3 are calculated by
.
Si
:i
aeq, i d o eq. i+ WVI+
Wv2
_F
Q n+l n+lsi( Ceq,i) + Wv1+ Wv2 (i45.8.10)
(8)
(13.1)
where
), En-ln(l+ 2L ),
c -ln(l+
an
E
YEn
-
au
ct--ce
- en
av
(thickness strain) (14)
+
S i m i l a r procedures o f c a l c u l a t i o n can be applied t o regions 7 and 9. Thus, the energy dissipated due t o the p l a s t i c deformation o f the side w a l l t s given by
1
where S i i s the area o f region i. The work done by f r i c t i o n a l forces a t the blank-die and blankblankholder interfaces, Wf. i s calculated by
Wf-
2uE P i I l(x.y)dxdy, i Si
(i-4.5,8,10)
(9)
where v i s t h e c o e f f i c i e n t o f f r i c t i o n , p i i s t h e b l a n k h o l d i n g f o r c e p e r u n i t area a c t i n g on r e g l o n i. and l(x.y) i s t h e magnitude o f displacement o f m a t e r i a l a t i n i t i a l c o o r d i n a t e (x.y), during the deep drawing process. The energy dissipated due t o bending and unbending o f material passing over the edge o f d i e opening, dldpd3. i s approximated by
3. Experimental setup and procedures Figure 4 shows the dimensions o f the f i v e punches used. The d i e used had a hollow c a v i t y w j t h a rectangular cross-section g i v i n g a clearance o f 0.6mm f o r a nominal blank thickness o f 0.5mm. The p r o f i l e r a d i u s a t t h e edge o f t h e d i e opening was taken as 5mm. The sheet m a t e r i a l used i n t h l s work was t i t a n l u n - k i l l e d lowcarbon steel. I t s mechanical properties are shown i n Table 1. It was c u t i n t o rectangular blanks o f several sizes.
/ \
Shorter side Fig.3
288
Oefornation analysis o f i n c l i n e d side wall.
Fig.4
P m A 15' 90' B 30' 90* c 4 F 90' D 4 5 60' E 45 45'
Longer side
Geometry o f punches.
Table 1 Mechanical properties o f material Orientation] ~ 0 . 2 1a t
fro'
rollins d i r e c t ion
0
90 45
]
I
~ 1 . n
[ W P ~ I [ M P ~ I [XI 163 173 168
321 319 312
44.2 47.7 47.7
Ir
-
Ir
value
value
value
0.26
1.43 2.21 1.76
1.94
0.26 0.26
I n the deep drawing t e s t s using the punches except punch C. the portions o f the blank i n contact w i t h the d i e and the blankholdcr were lubricated w i t h machine o i l . When using punch C, i n order t o prevent f r a c t u r e from occurring a t the punch p r o f i l e , a sheet o f v i n y l c h l o r i d e f i l m o f 0.05mm t h i c k (160mmx50mm) was inserted between the blank and the f l a t p o r t i o n o f the punch head: i n t h i s case, both sides o f the f i l m and the blank were l u b r i c a t e d w i t h machine o i l . Throughout the tests, blanks were set so t h a t t h e i r r o l l i n g d i r e c t i o n s c o i n c i d e w i t h t h e shorter side d i r e c t i o n o f t h e punch. The punch speed was about lmm/sec. The b l a n k h o l d i n g force was maintained constant during one drawing operation. The b l a n k h o l d i n g f o r c e p e r u n i t area a t r e g i o n i. p . (see Eq.(9)), was determined from the measurement w i t h PRESCALE] (FUJI PHOTO FILM CO.). From the measured d i s t r i b u t i o n o f pi, the t o t a l b l a n k h o l d i n g f o r c e F i n c a l c u l a t i o n was assumed t o d i s t r i b u t e ~ + ~ ~ ~ throughout t h e f l a n g e t o s a t i s f y ( p ~ and p6=O. where p4-p5, p8'p1@ and i s the area o f region i.
fi
~
)
~
(
~
~
~
~
+
~
~
~
~
4. Results and dlscussions 4.1 Results o f deep drawing t e s t s F o r t h e d r a w i n g w i t h punches A and B. t h e s h e l l s w e r e successfully drawn without f r a c t u r e o r wrinkling. Figure 5 shows series o f forming processes o f s h e l l s drawn w i t h punches E. D. and C. For punch E (Fig.5(a)). i t i s seen t h a t t h e b l a n k g r a d u a l l y conforms t h e punch, and t h e area o f c o n t a c t proceeds towards the lower p a r t o f the side walls. A t hdOmm the blank has completely conformed t o the punch. t h e b l a n k does n o t c o m p l e t e l y conform For punch 0 (Fig.5(b)). t o t h e punch even a t h-60mm, and w r i n k l e s remain a t t h e s i d e w a l l s and a t the l o w e r p o r t i o n o f t h e r i d g e s o f t h e punch. W i t h increasing the blankholding force up t o 71kN. breakage occurred a t the punch p r o f i l e . For punch C (Fig.5(c)). the degree o f w r i n k l i n g i s greater than when punch D i s used Moreover, the blankholding force t h a t does n o t cause f r a c t u r e a t the punch p r o f i l e lowers t o 43kN. I n order t o r e s t r a i n t h e w r i n k l i n g a t t h e s i d e w a l l s seen i n Fig.5(c). f o u r p i e c e s o f tape, l O m m wldth. 20mm long, O.lmm t h i c k , were sticked a t the edges o f the blank, i n f r o n t of the shorter sides o f t h e punch, as shown i n Fig.5(d). The d i s t a n c e between t h e tapes across t h e c e n t e r l i n e was 60mm. The tape l a y o u t o f Fig.5(d) was found t o make t h e amount o f w r i n k l i n g l e a s t : w i t h the distance between the tapes greater than 60mm. the amount o f w r i n k l i n g increases, and w i t h less than 60mm. f r a c t u r e occurred r i g h t beneath the punch p r o f i l e as shown i n Fig.5(e). These e x p e r l m e n t a l r e s u l t s suggest t h a t t h e c o n t r o l o f b l a n k h o l d i n g force alone i s not enough t o suppress w r i n k l i n g behavior o f sheet metal, but the layout o f beads i s a c r u c i a l factor. Another s i g n i f i c a n t e f f e c t o f u s i n g tapes i s t h a t t h e upper l i m i t o f b l a n k h o l d i n g f o r c e t h a t does n o t cause f r a c t u r e i s i n c r e a s e d and r e s u l t s i n t h e f u r t h e r r e s t r a i n i n g o f w r i n k l e s . The use o f tapes s t r e t c h e s t h e b l a n k and b r i n g s more m a t e r i a l i n t o c o n t a c t w i t h t h e punch than when they a r e n o t used. T h t s prevents the draw-in stress from concentrating on the m a t e r i a l a t the punch p r o f i l e and be e f f e c t i v e i n r e s t r a l n i n g fracture. 4.2
Strain d l s t r i b u t i o n s Figure 6 compares the s t r a i n d i s t r i b u t i o n s between experiment and c a l c u l a t i o n . a t t h e c e n t e r l i n e s o f f i n i s h e d s h e l l s . The calculated r e s u l t s o f strains, obtained f o r y 4 . 2 by the method o f c a l c u l a t i o n described i n section 2. are shown by three kinds o f t h i n l i n e s i n Fig.6 ( A n a l y s i s I). The c a l c u l a t i o n tends t o underestimate the s t r a i n s i n the d i r e c t i o n o f the center lines, e l , p a r t i c u l a r l y a t t h e s h o r t e r s i d e s o f punch C and D. T h i s d i s c r e p a n c y i s c o n s i d e r e d t o be caused by t h e assumption t h a t r e g i o n s 2 and 3 i s s u b j e c t t o t h e same amount o f s t r e t c h i n g i n t h e d i r e c t i o n On i n Fig.3 t o f o r m t h e s i d e w a l l 1"'J"'Gx'E": i n r e a l i t y , judging from the w r i n k l i n g shown i n Fig.5. region 3 i s not so stretched as region 2 is. Thus another c a l c u l a t l o n was done by v i r t u a l l y taking the values ofc,, i n Eq.(13.2) as zero a t r e g i o n s 3 and 7. The c a l c u l a t e d r e s u l t s o b t a i n e d f o r y-0.1 a r e shown b y t h r e e k i n d s o f t h i c k l i n e s i n Fig.6(b) and ( c ) (Analysis II ). E x p e r i m e n t a l r e s u l t s a r e between t h e t h i c k l i n e s (Analysis I ) and the t h i n l i n e s (AnalysisII ).
4.3 Evaluation o f the amount o f u r l n k l l n g As has been mentioned i n section 2.2. the amount o f w r i n k l i n g can be evaluated by the amount o f the overlapping m a t e r i a l along Table 2 shows t h e a r i d g e o f the punch, Gx'dp+Gy'dz (Fig.2). calculated r e s u l t s o f them, the conditions o f c a l c u l a t i o n being the same as i n Fig.6. It i s clear, w i t h a look a t the photographs i n Fig.5. t h a t the amount o f w r i n k l i n g becomes greater w i t h the increase o f the values o f Gx'd2t Gy'd2: punch C>punch DIpunch E. This suggests t h a t the simulation method proposed i n t h i s study i s also useful i n p r e d i c t i n g the degree o f wrinkling.
5. conclusions 1) k e p drawing t e s t s were conducted by using f i v e punches having two couples o f side w a l l s whose i n c l i n a t i o n angles are 81 on the
l o n g e r s i d e o f punch and 8 2 on t h e s h o r t e r side. The o r d e r o f d i f f i c u l t y i n deep drawing was punch C(81-45'. 02=90'). D(49, 60'), E(45", 45"). B(30". go"), and A(15". 90"). Punch C caused t h e most conspicuous w r i n k l i n g and t h i n n i n g o f m a t e r i a l . An a p p r o p r i a t e l a y o u t o f tapes s t i c k e d t o t h e b l a n k was found t o r e s t r a i n t h e w r i n k l e s s u b s t a n t i a l l y and t o i n c r e a s e t h e upper l i m i t o f blankholding force t h a t does not cause fracture. 2) A s i m p l e method o f a n a l y s i s based on an energy method was proposed t o simulate the deformation behavior o f sheet metal i n t h e above deep d r a w i n g t e s t s . The b l a n k i s d i v i d e d i n t o t e n r e g i o n s whose b a s i c mode o f d e f o r m a t i o n i s g i v e n by f o u r parameters, and the set o f these f o u r parameters which minimizes Table 2 Calculated values o f Gx'd2 and Gy'd2
Punch
D 1.77
*Analysis I and II correspond t o the calculated r e s u l t s indicated by the t h i n l i n e s and the t h i c k l i n e s i n Fig.6, respec:tively.
289
~
~
)
30
20 n
N u
10
p c
e 5
-10
-20
-30 0
100
50
'
50
100
0
50
100
-50 0
I
1
I
I
30 20
20 CI
E
..-
5
10
L
s
x2 0
r
t
-10
f -10
-20
- 20
-30 0 30
-30 --
100
50
I 1
-30 0
Punch proflle
Shock line
I
50
Die Drofile
I
I
100
x coordinates i n undefoned blank [mm] Fig.6 S t r a i n d i s t r i b u t i o n s a t t h e c e n t e r l i n e s o f blanks. e l : nominal s t r a i n i n t h e d i r e c t i o n o f t h e c e n t e r l i n e o f t h e nominal t r a n s v e r s e s t r a i n : et: nominal t h i c k n e s s s t r a i n . r a w i'n g c o n d i t i o n s a r e t h e same as Fig.5 (a),(b), (c). respectively. I n the calculation, the c o e f f i c i e n t o f f r i c t i o n p i s t a k e n as 0.2 f o r t h i n l i n e s and 0.1 f o r t h i c k l i n e s . 0=407 * c026[Mpa] i s assumed.
s:
the t o t a l energy required t o deform the blank i n t o a given shape determines the f i n a l mode o f deformation o f the blank. The s t r a i n d i s t r i b u t i o n s p r e d i c t e d by t h i s method were i n f a i r agreement w i t h measured ones. This method i s also useful i n evaluating the amount o f w r i n k l i n g t o occur a t i n c l i n e d side w a l l s o f the parts. 3) These r e s u l t s v a l i d a t e the use o f an energy method t o p r e d i c t deformatlon behavior o f sheet metal i n general three-dimensional sheet forming processes. Ackmwledgrant T h i s work was supported by t h e Japan M i n i s t r y o f Education, Science and Culture under Grant-in-aid f o r encouragement o f young
290
o
c
r
f
10
c
Y)
y coordlnates I n u n d e f o r d blank [ m a ]
s c i e n t i s t ( 6 3 7 5 0 7 1 4 ) and p a r t l y by AMADA Foundation f o r M e t a l Work Technology. References 1) Tang,S.C.. llankamban,RL. and Ling,P.. 1988. A F i n i t e Element M o d e l i n g o f t h e Stretch-Draw Forming Process, Advances and Trends i n A u t o m o t i v e Sheet S t e e l Stamping, 101-107. The E n g i n e e r i n g S o c i e t y f o r Advancing M o b i l i t y Land Sea A i r and Space, Warrendale. 2) Nakamachi.E. and Wagoner.RK. ibid. 109-120. 3) Takahashi.4. Okamoto. I.. Hiramatsu.T. and Yamada.N.. 1985. E v a l u a t i o n M e t h o d s o f P r e s s F o r m i n g S e v e r i t y i n CAD A p p l i c a t i o n s , Computer M o d e l i n g o f Sheet M e t a l Forming P r o c e s s , Eds. Wang.N.-M. and Tang.S.C., 37-50, The Pennsylvania. M e t a l l u r g i c a l Society o f AIME. 4) Furu bayashi, T.. Uj ihara, S. and Sa kamoto,T.. 1988. S i mu 1a t ion of Forming S e v e r i t y on Autobody Panels U s i n g A CAD System A n a l y s i s of Drawbead and I t s Control-, IDDRG 1 5 t h B i e n n i a l Congress, 243-250. ASM.