1994 a mechanics model for sheet-metal stamping using flexible dies.pdf

8/18/2019 1994 A mechanics model for sheet-metal stamping using flexible dies.pdf

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ELSEVIER Journal of Materials Processing Technology 53 (1995) 798 810

Journal ofM a t e r i a l s

P r o c e s s i n gTe c h n o l o g y

A mechanics model for shee t -meta l s tamping us ingdeformable dies

L.C. Z h a n g

Department of Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia

Received 22 April 1994

Industrial Sum m ary

This paper studied the deformation mechanisms of sheet metals stamped by rigid punchesand deformable dies. Emphasis was placed on the development of elastic plastic zones in thesheet workpieces, on the functions of the die parameters and on their correlations withspring-back. It was found that the reduction of spring-back was due mainly to the interfaceinteraction offered by the dies. The paper provides new insight into processing techniquessubjected to plane-strain conditions.

Nomenclature

a half cont act- leng th between the plate and the punch , see Fig. l and Eq. (19)b width of the platec half-width of the elastic stress/strain distri bution on a plate section,

see Figs. 5 and 6d distance between the mid-pla ne and the neutra l stress/strain plane of the plate,

see also Figs. 5 and 6E Young' s modulu s of the plate materialEf Young' s modulu s of the elastic foun datio nEp plastic hard eni ng mod ulu s of the plate material, see also Fig. 4e non-dime nsional parameter, defined as Ep/Eh thickness of the platek spring consta nt, see Fig. 3L half-length of the plate in co nt ac t with the die, see Figs. 1 3M bending mom ent on a plate sectionMe max imu m elastic bendi ng mome nt, defined by Eq. (3)m non -dim ens iona l bendi ng mome nt, see Eq. (2)N axial force on a plate sectionNe ma xi mu m elastic axial force, defined by Eq. (3)

0924-0136/95/ 09.50 ~+ 1995 Elsevier Science S.A. All rights reserved.SSDI 0924-0136(94101765-S


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L.C. Zhang / Journal of Materials Processing Technology 53 (1995) 798-810 799

n no n-d im en s ion a l ax ia l fo rce, def ined by Eq . (2 )p(x) normal fo rce on the p la te sur face , see F ig . 2q(x) t angent ia l fo rce on the p la te sur face , see F ig . 2

u ho r i z on t a l d i sp l acemen t a t t he end o f t he p l a te , s ee F ig . 3w def lec t ion of the p la tex h o r i z o n t a l c o o r d i n a t e7 , 6 non -d im ens ion a l pa r am e te r s , de f ined by Eq . (4)e s t ra in~v yield strain , --Crv/E

n o n - d i m e n s i o n a l c o o r d i n a t e , d e f i n ed a sx / Lt/ non -d im ens io na l de f l ec t ion , de f ineda s w/ (LZtce )O the spr ing bac k ra t io , def ined by Eq . (13)~c cu rv atu re a t x~ce m ax im um e las t ic cur va tu re a t x

propor t iona l func t ion for in te r face f r ic t ion , def ined by Eq . (1 )v Po i s son ' s r a t io o f t he p l a t e m a te r i a lv f Po i s son ' s r a t i o o f t he e l as t ic f ou nd a t ion

mater ia l parameter, see F ig . 4~rv yield s t ress of the pla te m ater ia l , see Fig. 4~b no n-d im en s ion a l curv a ture , def ined by Eq . (2 )~bR non -d im ens io na l r e s idua l cu rva tu re , de f ineda s / £ R / / ( e

Superscr ip t sc cen tral sect ion, i .e ., the pla te sect ion at x = 0U upp e r su r face o f t he p l a t eL low er sur face of the p la te

1. In t roduc t ion

The t echn ique o f e l a s t i c -p l a s ti c shee t s t amp ing has been em p loy ed ex t ens ive ly i nthe fo rming o f a va r i e ty o f eng inee r ing comp onen t s . I n add i t i on t o t he con ven t ion a lp roces ses , w i th un i t a ry r i g id d ie s, a t e chn ique o f shee t s t amp in g by de fo rm ab lefo rming t oo l s (SSDFT) has been e s t ab l i shed and des igned i n to a w ide r ange o fp re s swo rk ing [1 ] . Co m par ed w i th con ven t io na l p roces s , SS D FT posse s se s t he di s-t inc t adv anta ges of low too l in g cos ts , fl ex ib il i ty o f p rodu c ib le shap es , the capa bi l i ty fo rp rov id ing s c r a t ch - fr ee su r face s wh ich is mo s t su it ab le fo r pa r t s w i th coa t ed o rpo l i shed su rf ace s , and a r em arkab l e r educ t i on o f sp r ing -back a nd wr ink l ing wh ich isex t r eme ly imp or t a n t f o r ha rd - to - fo rm ing shee t ma te r i a ls . Unfo r tuna t e ly, t he p roces shas n o t b een s tud i ed ca r e fu ll y so t ha t p rop e r gu ide l ine s a re n o t ava i lab l e fo r p rac t i c alm a n u f a c t u r i n g d e si gn .

The p re sen t pa pe r a ims t o i nves ti ga te , w i th t he a id o f mec han i c s mode l l i ng , t hed e f o r m a t i o n m e c h a n i s m s o f sh e et m e t a l s s ta m p e d b y r ig i d p u n c h e s a n d d e f o r m a b l ed ie s. The dev e lopm en t o f e l a s t i c -p la s t i c zones i n t he shee t workp i eces, t he func t i ons o fd i e pa rame te r s and t he i r co r r e l a t i ons w i th sp r ing -back a r e add re s sed i n de t a i l .


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800 L.C. Zhang /Journal of Materials Processing Technology 53 1995) 798-810

T h e s t u d y r e v e a ls t h a t t h e r e d u c t i o n o f s p r i n g - b a c k is d u e m a i n l y t o t h e i n t e rf a c ei n t e r a c t i o n o f f e r e d b y t h e d ie s. T h e p a p e r p r o v i d e s n e w i n s i g h t i n t o p r o c e s s i n gt e c h n i q u e s s u b j e c t e d t o p l a n e - s t r a i n c o n d i t i o n s .

2. Problem charac te r i sa t ion

2.1. Stamping o f wide plates by deformable dies

C o n s i d e r a w i d e t h i n p l a t e r e s ti n g o n a d e f o r m a b l e d ie s t a m p e d b y a ri g idc y l i n d r i c a l p u n c h , s e e F ig . 1. T h e d e f o r m a t i o n o f th e p l a t e is s y m m e t r i c a l a n d t h eb e n d i n g i s c y l i n d r ic a l . O n t h e u p p e r s u r f a c e o f t h e p l a te , t h e r e a re a n o r m a l c o n t a c tf o r c e p U(x ) a n d a t a n g e n t i a l f o r c e qU (x ) d i s t r i b u t e d o v e r a s m a l l c o n t a c t a r c ,AA .O nt h e l o w e r s u r f a c e, th e c o r r e s p o n d i n g f o r c e s a r epL(x)a n d qe(x),respe ct ively , see Fig . 2 ,e x e r t e d o n t h e c o n t a c t l e n g t h , 2 L , w h i c h i s l es s t h a n t h e w h o l e l e n g t h o f th e p l a t eb e c a u s e o f t h e e x i s t e n c e o f l if te d p a r t s a t t h e e n d s . O w i n g t o t h e s y m m e t r y o fd e f o r m a t i o n , p U (x ) a n dpL(x)m u s t b e s y m m e t r i c a l b u tqV(x)a n d qL(x)a n t i - s y m m e t r i -ca l , wi th respec t to x .

2.2. Mechanics modelling

To e x p l o r e c l e a r l y t h e d e f o r m a t i o n m e c h a n i s m o f t h e p l a te , it is e ss e n t ia l t o d i s ti l

a m e c h a n i c s m o d e l f o r t h e a b o v e e n g i n e e r i n g p r o c e s s, t h a t c a n r e f l e ct t h e m a i nc h a r a c t e r i s t i c s o f t h e o r i g i n a l b u t a l s o c a n s i m p l if y t h e p r o b l e m t o a g r e a t e x t e n t s u c ht h a t a n a n a l y t i c a l s o l u t i o n c a n b e g e n e r a t e d . 1

I t is r e a s o n a b l e t o a s s u m e , b y n o t i n g t h e s m a l l p l a te t h i c k n e s s , t h a tqU(x)a n d qL(x)c o n t r i b u t e t o t h e t o t a l a x i a l f o r c eN(x)only, whi l s t pU(x) and pC(x) con t r ibu te to theb e n d in g m o m e n tM(x) o n l y. T h e m e c h a n i c s m o d e l is s h o w n i n F ig . 3. A t h i n p l a t e o ft h i c k n e s s h , w i d t h b a n d l e n g t h 2 L ( h ~ 2 L ) s u b j e c t e d t o a s e t o f t r a n s v e r s e l o a d s( d i s t r i b u t e d u n i f o r m l y a l o n g t h e w i d t h b ) is s i m p l y s u p p o r t e d b u t w i t h v a r i a b l e ax i a lr e s t r a i n t s ( s p ri n g s) a t t h e t w o e n d s . T h e s p r i n g s p r o v i d e a u n i f o r m a x i a l f o r c e ( asa f u n c t i o n o f t h e p l a t e d e f l e c t i o n to s i m u l a t e p o s s i b l e e x t e r n a l t e n s io n ) . A n o t h e r p a r to f N is a s s u m e d t o b e p r o p o r t i o n a l t o p L(X ) t o r e p r e s e n t t h e f r i c ti o n o n t h e i n t er f a c e .C o n s e q u e n t l y :

L

N x) = b f U ( 4 ) p L ( ~ ) d 4 + k u ,x

(1)

wh ere p (x) i s a p r op or t io na l fun c t ion , k is the s t i ffness o f the sp r ings an d u is theh o r i z o n t a l d i s p l a c e m e n t a t t h e p l a t e e n d s ( F i g . 3 ) .

l Alth oug h numerica l met hod s such as the finite-element and finite-difference met hod s could be applied totake into account more details, they cannot replace a modelling approach that reveals thoroughly thedeformation mechanisms.


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L.C. Zhang / Journal o f Materials Processing Technology 53 (1995) 798-810

~P Rigidpunch

, z

Fig. 1. Stamping by a deformable die.

801

h - -t1~

x

/ i,plate 2 ~ A ' pt~x\

~ x )

Fig. 2. Interface forces on the plate.

after deforn~tionbeforedeformation /

2L

Fig. 3. A mechanics model for plate stamping.

3. Analysis

3 .1 . L o a d - c u r v a t u r e r e la t i o n s h i p s

A c o m p l e t e r e l a t i o n s h i p b e t w e e n i n t e r n a l l o a d s a n d d e f o r m a t i o n a t a n y p l a t es e c t i o n s h o u l d b e e s t a b l i s h e d f ir st . T h e u n i a x i a l s tr e ss s t r a i n b e h a v i o u r o f m o s t s h e e t


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802 L.C. Zhan g / Journ al of Ma terials Processing Technologt , 53 (1995) 798 810

J

~ p

F i g. 4 . T h e s t r e s s s t r a i n c u r v e o f t h e p l a t e m a t e r i a l .

h/~

~c

h/2 - d - ~c :

_h /2 + d - ~c

t/2

F i g. 5 . S t r a i n d i s t r i b u t i o n a c r o s s t h e p la t e t h i c k n e s s .

m e t a l s c a n b e c h a r a c t e r i s e d b y th e m o d e l s h o w n i n F ig . 4 . W h e n t h e l e n g t h o f t h ey ie ld p la t fo rm a pp ro ac he s ze ro (i.e ., ~ ~ 0 ), i t r epre sen t s an e las t i c l inea r ly p las ti ch a r d e n i n g m a t e r i a l , b u t w h e n t h e h a r d e n i n g m o d u l u s E p i s z e r o , i t b e c o m e s a ne l a st i c p e r f e c t l y p l a s ti c m o d e l . A s s u m i n g t h a t p l a n e s e c t i o n s o f t h e p l a t e r e m a i n p l a n et h r o u g h o u t t h e d e f o r m a t i o n , t h e s t r a i n a c r o s s t h e p l a t e t h i c k n e s s is t h e n a l w a y s l i n e a r,s ee F ig . 5 . C o r r e s p o n d i n g l y, t h e s t re s s d i s t r i b u t i o n a c r o s s a p l a t e s e c t i o n a t c o o r d i n a t ex u n d e r d i f f e r e n t c o m b i n a t i o n s o f M a n d N m u s t b e o n e o f t h e f o ll o w i n g six ty p e s (s ee

Fig. 6): 2 ( i) the pu re e las t ic s ta te (S1); (i i) the s ingle-s ide per fect ly plas t ic s ta te ($2);( iii) the d ou ble - s ide pe r fec t ly p las t i c s t a te (S3(a ) ); (iv) the s ing le - side ha rde n in g p las t i cs ta te w i th the o th er s ide be ing e las t i c (S3(b) ); (v) the s ing le - s ide ha rde n in g p las t i c s t a tewi th the o the r s ide be ing per fe c t ly p las t i c ($4); and (v i) the dou ble - s ide ha rde n in gplas t ic s ta te ($5) .

T h e n o n - d i m e n s i o n a l b e n d i n g m o m e n t , a x i al f o rc e a n d c u r v a t u r e , a r e d ef i n ed a sfol lows:

M N /cIn - M e ' n - N e ' q5 = --'Ke (2)

2 I n t h e d e s c r i p t i o n o f F i g s . 5 a n d 6 it h a s b e e n a s s u m e d t h a t t h e n e u t r a l s t r a i n a n d n e u t r a l s t r e s s l a y e r s a r ciden t i ca l : r e f ined theor i e s can be found in [1 ] .


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L.C. Zhang / Journal o f Materials Processing Technology 53 (1995) 798 -810 803

Oy Oy$1 $2 ~ S3(a)

S3(b)

h/2 - d d i ~ C

12-c+d ~ ~h/2-c+d

~_ _ _t~ d

~ A C l ~ h / 2 - c -d- G y

$4 $5h/2~-- /2+ d ' ~ c ~ ~ h/2 + d'~ c

h/2 - d -~ct

Fig. 6. Possible stress states in the plate.

where

2 a vM e = u v b h 2, N e = u v b h , K e - h E (3)

a r e t he max imum e l a s t i c bend ing momen t ; t he max imum e l a s t i c ax i a l f o r ce ; and t hemaximum e las t ic curva ture ; respec t ive ly, in which Uv i s the y ie ld s t ress and E i sYo u n g ' s m o d u l u s . F u r t h e r m o r e , t h e n o n - d i m e n s i o n a l p a r a m e t e r s

c d6 - (4)

7 - ( h / 2 ) ' ( h / 2 )

a re i n t rodu ced t h ro ug ho u t t he ana ly s i s (c and d a re sho wn in F ig s . 5 & 6 ). Thel o a d - c u r v a t u r e r e la t io n s h i p s fo r t h e a b o v e s ix s ta te s a n d t h e ir c o r r e s p o n d i n g b o u n d -a ry cond i t i ons can be ob t a ined f rom F igs . 5 & 6 d i r ec t l y :

(S1) 1 + 6 - 7 ~ < 0 :6 1

n = - , m = - , 4 ' = m .7

($2) l + 6 - 7 > ~ O a n d 1 - 6 - 7 ~ < 0 :

1n = 1 ~7 (l -- t~ -}- 7) 2,

1m = 4 7 7 ( 1 - 6 + o / ) 2 1 - 3 - ( 1 - 6 + 7 ) ] ,

t m 12' = 4 ( 1 - n ) / 3 1-

(5)

(6)


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804 L.C. Zhang / Journal o f Materials Processing Technology 53 (1995) 798 -810

T h e r e a r e t w o p o s s i b i l it i e s f o r t h e f u r t h e r d e v e l o p m e n t o f t h e p r e s e n t s t r e ss s ta t e ,r e l y i n g o n t h e p a r a m e t e r o f t h e m a t e r i a l p r o p e r t y, 4. F o r e x a m p l e , i f ~ i s r e l a t i v e l yl a rge t he s t r e s s s t a t e w i l l evo lve i n to (S3 (a ) ) , o the rw i se i f i t i s ve ry c lo se t o u n i ty t he

fo l lowing s t a t e w i l l be (S3 (b ) ) .(S 3( a) ) 1 - 6 - ? > ~ O a n d 1 + ~ ~ / ~ < 0 :

n = 5, m = 3(1 --6 z )-- 172 , ~b = [3 (1 - n 2) - 2m ] - 1/2 (7)

(S 3(b )) 1 - 6 - 7 ~ 0 a n d 1 + 6 - 3 7 > ~ 0 :

1 (1 -- () -{- 7)2 q-e 1 + • 4? ) 2,= 1 - 4 5 ~ ? ( -

1 em = ~ (1 -- 6+ ?)2 [3 - - (1 -- c~ + 7)] + ~ (1 + 6 -- d?) 2 [3 - - (1 + 6 -- ~7)] , (8)

where e = E p / E .A c c o r d i n g t o E q . ( 8 a ) :

6 = ( 1 - - e ) - { l + e + ( 1 - - e ~ ) 7 - - ~ e [ ( ~ - - l ) 2 7 2 + 4 ( n - - ~ - -_,) }+ 4 .e(9)

T h e l o a d - c u r v a t u r e r e l a t i o n s h i p c a n b e o b t a i n e d e x p l i c i tl y i f E q . (9 ) i s s u b s t i t u t e d i n t oEq . (8b ) an d 7 i s r ep l ace d by ~b - 1. Pa r t i cu l a r ly, i f e app ro ac he s ze ro , (5 i n Eq . (9) w i llb e c o m e s t h a t o f ($ 2) . A n y f u r t h e r d e v e l o p m e n t o f s t re s s s t a te f r o m e i t h e r (S 3 (a )) o r(S3(b) ) wi l l l ead to ($4) .

($4) 1 + 6 - 3 7 ~ > 0 , 1 - 6 - 7 ~ > 0 a n d 1 - 6 - 3 7 ~ < 0 :

n = 6 + ~ ( 1 + 6 - - ~ ? ) 2 ,, [

1 0 )3 1 1 2 e 1m = ~ ( - - 6 2 ) - - ~ 7 + ~ / 7 ( + 5 - - ~ ?)2 [ 3 - - (1 + 6 - - ~ 7 ) ] .

A n e x p li c it l o a d - c u r v a t u r e r e l a t i o n s h i p c a n b e d e r i v e d w h e n ? = 4) 1 a n d

,5 = { 2 x /? [ 7 + e (1 - - ~ , + n ) ] e (1 - - 37) 27 } /e ( l l )

a r e sub s t i t u t ed i n to Eq . (10b ). S imi l a r ly, Eq . (1 l ) w i l l be i den t i ca l t o t h a t o f (S3 (a ))w h e n e a p p r o a c h e s z e ro .

( $ 5 ) 1 - 6 - ~ ; , ~ > 0 :

[ e 1= 1 + - ( 1 - - ¢ 7 ) 6,

7 (12)3 1 2 e

m = ~ (1 - - 6 2) - - ~ 7 + ~ [ (1- - ~ 7 ) 2 ( 2 + ~ 7 )+ 3 ~ ? ( ) 2 ] •


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806 L.C. Zhang /Journal of Materials Processing Technolo,Kv 53 1995) 798-810

3.4. Solution pro ced ure

The de t a i l ed exp l i c i t exp re s s ions de r ived above enab l e so lu t i ons t o be ob t a ined

eas ily. Th e fo l low ing proc edu res a re used for the num er ica l ca lcu la t ions in the presen tpaper :

(i) de term ine n with the value of r /* given;( i i ) compute m(( ) accord ing to the g iven loads pU and pL;(iii) calculate ~b(();( iv) integrate Eq. (17) to obtain t / (~) under the condi t ions specif ied by Eq. (18) ;(v) i f q~ ~ r /* , calcu late O an d s top, oth erw ise , con t inue ;(vi) let ~/* = t/c and return to step (i).

4 . D i s c u s s i o n

To ob t a in a de t a i l ed d i ag ram fo r t he co r r e l a t i on o f t he de fo rm a t ion o f t he p l at e andthe d ie , a conven i en t and na tu ra l s t a r t i ng po in t i s t o a s sum e tha t b o thpL(x)a n d pU(x)are Hertzian, i .e. :

pU(x ) = / )U (1 - X 2~1/2 (1 X 2~1 /2a~/I , pL (x ) = fiL -- ~2 y (19)

wh ere a i s the h a l f l eng th o f the co ntac t a rcA A be tween t he punch and p l a t e , s eeF ig. 1 , and /5 L and /5 U a r e t he m ax im um p re s su re va lues o f pL(x ) andpU(x) at x = 0re la ted by force equi l ib r ium in the z -d i rec t ion ( i t i s assumed tha t the spr ings do no tco ntr ibu te in this di rect ion) . This gives r ise to/~L =afiU/L.T h e b e n d i n g m o m e n t i n t h eplate is therefore:

aLpU 2 ~ _ ( 2 ) 3 _ ( ~m ( ~ )= h20.~ - --~ a2 / j

-- ~ 1 - - ~ + a r c s l n T - ~ . - -

f o r O ~ ( ~ ,(2o1

t(()----- h2- -vfo r (a ~< (~ < 1,

where (a =a l L . As wi l l be d i scussed be low, G i s an impor tan t parameter fo r theSS D FT tha t r ep re sen t s l a rge ly the r e l a ti ve mechan ica l p rope r t i e s o f t he p l a te a nd d iema ter ia l s , an d which in f luences the de for m at io n of the p la te s ign i f ican t ly.


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L.C. Zhang / Journal o f Materials Processing Technology 53 (1995) 79 8-81 0 8 0 7

0

0 . 7

o . s g = 0 . 0 2

I x = 0 . 10 . 3

0 .2 i i0 . 0 1 0 . 0 2 0 . 0 3

0

°° , ,g = 0 . 5

I i ~ i I I I I

0 . 0 5 0 . 1 0 . 2 0 . 3 0 . 5

F i g . 7 . E f f e c t o f # o n t h e s p r i n g b a c k r a t i o , f o r ~a = 0 . 8 a n d k = 1 .5 .

A r e l e v a n t s t u d y c o n c e r n i n g t h e r e c e d i n g c o n t a c t b e t w e e n a p l a t e a n d a n e l a st icf o u n d a t i o n s h o w e d t h a t t h e v a r i a t i o n o f ~a is n o n - l i n e a r e v e n w h e n t h e p l a t e is el a s ti c[ 3 ] . N e v e r t h e l e s s , w h e n a a p p r o a c h e s h , t h e v a l u e o f L w i ll b e c o m e c o n s t a n t [ 4 ] ,g o v e r n e d b y t h e m a t e r i a l s c o n s t a n t s o f t h e p la t e a n d f o u n d a t i o n :

L=h(l 'g45E(1E~i--~)--2) ]l/3J (21)

w h e r e Ef a n d vf a r e Yo u n g ' s m o d u l u s a n d P o i s s o n ' s r a t i o o f t h e f o u n d a t i o n , r e s pe c -t iv e ly. A l t h o u g h t h e c o n d i t i o n f o r o b t a i n i n g E q . (2 1) d i ff e r s t o a c e r t a i n e x t e n t f r o m t h ep r e s e n t s t a m p i n g p r o c e s s e s , i t c a n b e u s e d t o d e m o n s t r a t e q u a l i t a t i v e l y t h e i m p o r t -a n c e o f t h e r a t i o o f E t o E e i n a s t a m p i n g p r o c e s s i f t h e f o u n d a t i o n is t h o u g h t o f a s

a d ie . I t ind ica tes tha t fo r a g iven p la te m ate r ia l (E g iven) , a so f te r d ie ( smal le r Ef ) wi llb r i n g a b o u t a l a rg e r v a l u e o f L a n d v i ce v e r sa .

T he ax ia l fo rce in the p la te is, acc ord ing to E q . (1 ),

alt~ U fro ) kL 2n = 2avh \ 2 - ar c sin ~ - ~ ~ + E b ~ t /c, (22)

w h e r e / t h a s b e e n a s s u m e d t o b e i n d e p e n d e n t o f x a n d t h e c e n t ra l d e f le c t io n o f t h ep l a t e , w o ( it s d i m e n s i o n l e s s q u a n t i t y is t/ c) , is c o n s i d e r e d t o b e m u c h s m a l l e r t h a n L .

T h e d e p e n d e n c e o f t h e s p r i n g - b a c k r a t i o u p o n p a r a m e t e r s k, ~ a n d # a r e p r e s e n t e din F igs . 7 -9 wi th in the id ea l i sa t io n o f the p r esen t mo de l l in g (in a l l cases ~ = 1 .05,e = 0.2,ay/E = 1/3,h/L = 0 .1 and b = 1). G ene ra l ly , inc re as ing k and /~ d ecrea ses thes p r in g b a c k r a t i o a n d h e n c e i n d i ca t e s th a t b o t h a x i al t e n s i o n a n d i n t e r f a ce f r ic t io nb e t w e e n t h e p l a t e a n d d i e a r e p r i n c i p a l f a c t o r s i n t h e r e d u c t i o n o f O . I t is e a s y t o s e e,


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808 L.C. Zhang / Journal o f Ma terials Processing T echnology 53 (1995) 798 810

0

1

0 . 9

0.8

0 . 7

0 . 6

0.5 /k = 0 . 7 5

k = 0 . 1 5

[]U

O

U O[] ©

O©

O©

O©

0 . 4 ° <

k = 1 . 50 . 3 I I I , , , L I I I

0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 5 0 . 1 0 . 2 0 . 3 0 . 5

Fig. 8. E f f e c t o f k o n t h e s p r i n g b a c k r a t i o , f o r~a = 0.8 and/a = 0.1.

0.7

0 . 5

O 0 . 3

0.2

U

[]

~ a = 0 . 0 8

~e ~9~ G = 0.8

%G = 0 .4

0 . 1 L I I , , i 4 I I L , , i

0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 5 0 . 1 0 . 2 0 . 3 0 . 5 1

Fig. 9. E f f e c t o f ~ a o n t h e s p r i n g b a c k r a t i o , f o rk = 1.5 an d/ ~ = 0.1.

h o w e v e r , t h a t t h e c o n t r i b u t i o n o f / z i s m o r e l o c a l i s e d u n d e r t h e a s s u m p t i o n t h a t q L (x )i s p r o p o r t i o n a l t o p L ( x) a n d t h a t i t se l f i s i n d e p e n d e n t o f x . F u r t h e r m o r e , in t h e c a s eo f z e r o e x t e r n a l t e n s i o n , d i e s w i t h h i g h e r f r i c t i o n p r o p e r t i e s w o u l d b e b e t t er f o r t h es h a p e m a i n t a i n i n g o f w o r k p i e c e s a f te r u n l o a d i n g .

N o w c o n s i d e r t h e e ff e ct o f ( a i n F i g . 9. A s h a v e b e e n d i s c u s s e d , a la rg e r v a l u e o f (ac o u l d b e c o n s i d e r e d a s a s t a m p i n g a s s o c i a t e d w i t h a h a r d e r d ie ( a n d v i c e v e rs a ) w h e r e


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L.C. Zhang / Journal of Materials Processing Technology 53 1995) 798-810

I - - 1 Elast ic l l Plas tic

E I x. . . . . . . . . . ~a )

1 x

(bl

8 0 9

rc)

F i g . 1 0 . D e v e l o p m e n t o f p l a s t i c z o n e s f o r ( a = 0 . 8 , # = 0 . 5 a n d : ( a ) q c = 0 . 7 , k = 1 . 5; ( b ) ~/c = 1 . 2 , k = 0 . 5 ;

an d (c) r /c = 1.2, k = 1.5.

acco rd ing t o t he r e su l t s shown in F ig . 9 , t h i s c ause s l a rge r sp r ing -back r a t i o .Ho we ver, i f the d ie i s too sof t, a s i s the case of Ca = 0 .08 , the res idua l c urv a tur e o f thep l a t e w i ll be ve ry no n -u n i fo r m w h ich i s u sua l l y no t de s ir ab l e i n p r ac t i c a l s tamp ing .

Hence , t he s e lec t ion o f the d i e ma te r i a l sho u ld be m ade ca r e fu ll y by co ns ide r ing t hepa r t i cu l a r sh ap ing r equ i r em en t o f i nd iv idua l s t am p ing p rocesse s.The i nhe ren t m echa n i sm o f t he va r i a t i on o f O w i th t he change o f k, # a nd Ca is

i n d e e d d u e t o t h e c h a n g e o f t h e d e f o r m a t i o n m e c h a n i s m o f th e p l a te , a s sh o w nin F ig . 10. W i th co ns ta n t va lues o f k, /~ an d Ca, the upp er com press ive p las t iczone sh r inks when t he punch d i sp l acemen t i nc r ea se s ( r ep re sen t ed by t he i nc r ea seof r/c here , see F igs . 10(a) and (c )) . Tens i le p las t ic de for m at io n i s do m in an t in th i sca se . When k dec rea se s , howeve r, t he compres s ive p l a s t i c de fo rma t ion becomescons ide rab l e, wh ich i s why the va lue o f O is r ema rkab l e fo r sma l le r va lue s o f k(Fig. 8).

A com par i so n o f t he p r e sen t r e su lt s w i th those o f conv en t iona l s t amp ing b y r i g idd i es can a l so be m ade . An obv iou s d i ff e rence is tha t a SS D FT l eads to w orkp i eces w i tha mu ch sm oo the r d i s t r i bu t i on o f cu rva tu re , wh ich is no t d i f f icu l t t o u nde r s t and i f t hed i s t inc t l oad in g s tages (b l ank bend ing an d i ron ing ) and t he i r co r r e sp ond ing de fo rm a-t i on mechan i sms i n t he conven t iona l p roces se s [1 ] a r e r eca l l ed . Dur ing b l ankbend ing , t he con t ac t be tween t he p l a t e and t he punch su r f ace i s no t con t inuous ,caused by m an y loca li s ed cu rva tu re peaks . These peaks a r e u sua l l y ve ry d if f icu l t t o bei roned ou t i n t he fo ll owing i ron ing s t age . I n a SSD FT , how eve r, the r e is no b l ankbend ing , t he de fo rma b le d ie a lways tr y ing t o m ake t he p l a t e ma tch t he p unc h su r f aceth rough the i n t e r ac t i on fo r cep L ( x ) .

Resu l t s r ep re sen t i ng r ea l s t amp ing p roces se s can be ob t a ined by u s ing a r ea l i s t i cd i s t r i bu t i on o f i n t er f ace fo rce s de t e rm ined b y t he t h eo ry o f con t ac t m echan i c s o r b ye x p e r i m e n t a l m e a s u r e m e n t .


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810 L.C. Zhang /Jo ur na l o f Materials Processing Technology 53 (1995) 7 98-8 10

5. Conclusions

A mechanics model has been proposed to study the plate stamping process by

deformable dies subjected to plane-strain conditions. There are six possible stressstates for any plate section, depending on the combination of axial force and bendingmoment. It has been found that the spring-back ratio is affected strongly by theproperties of the deformable die for a given plate material. The friction properties andthe relative magnitude of the elastic constants of the dies are the key parameters inprocess design. All of these can be interpreted by the detailed analysis of the develop-ment of the plastic zone in the plate.

Acknowledgement

This work is supported by an ARC Small Grant,

References

[1] T.X. Yu and L.C. Zhang,Plastic Bendin9: Theory and Applications,Science Press, Beijing, 1992.[2] T.X. Yu and W. Johnson, Inf luence of ax ia l force on the e las t ic -p las t ic bending and spr ingback of

a beam, J. Mech. Work. Tech.,6 (1982) 5.[3] M. Ratwa ni and F. Erdogan, O n the p lane contac t problem for a f r ic t ionless e las tic layer,Int. J. Solids

Structs, 43 (1973) 921.[4] G.M.L. Gladwel l , On some unbounded contac t problems in p lane e las t ic i ty theory,Trans . ASME,

J. Appl. Mech.,43 (1976) 263.

1994 a mechanics model for sheet-metal stamping using flexible dies.pdf

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