Strain Rate Potential for Metals and Its Application to Minimum Work Path Calculations

8/14/2019 Strain Rate Potential for Metals and Its Application to Minimum Work Path Calculations

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Internattonal Journal ofPlasttclty, VoI 9, pp 51-6 3, 1993 0749-6419/93 $6 00 + 00Pnnted m the USA Copyright 1993 Pergamon Press Ltd

S T R A I N R A T E P O T E N T I A L F O R M E T A L S A N DI T S A P P L I C A T I O N T O M I N I M U M P L A S T I CW O R K P A T H C A L C U L A T I O N S

F . B A R L A T , K . C H U N G , a n d O . R I C H M O N DA l c o a T e c h n i c al C e n t e r

( C o m m u n i c a t e d b y G e o r g e W e n g , R u t g e r s U n i v e r s i t y )

A b s t r a c t - - I n t h i s w o r k , a d e f l m U o n o f t h e s t r a t a r a t e p o t e n t i a l f o r p l a s ti c a l ly d e f o r m i n g m e t -a l s i s p r o p o s e d T h i s p o t e n t i a l i s d e f i n e d m s i x - d i m e n s i o n a l d e v i a t o n c s t r a i n r a t e s p a c e , a n d R sg r a c h e n t p r o v i d e s d e v a a t o n c s t r e s s e s i n t h e f l o w i n g m a t e r i a l . F o r i s o t r o p l c F C C m e t a l s , i t is s h o w nt h a t t h e p l a s t ic b e h a v i o r p r e d i c t e d w i t h t h i s p r o p o s e d p h e n o m e n o l o g l c a l d e s c r i p t i o n i s i d e n t i -c a l t o t h e b e h a w o r p r e d i c t e d w i t h t h e T a y l o r / B i s h o p a n d H i l l p o ly c r y s t a l p l a s U c it y m o d e l F o ro r t h o t r o p i c F C C m e t a l s , s i x m a t e r i a l c o e f f i ci e n t s c h a r a c t e ri z e t h e a m s o t r o p y . T t u s p o t e n t i a l p r o -r i d e s a d e f i m t i o n o f t h e e f f e c t iv e s t r a in r a t e . T o g e t h e r w i t h a w o r k - h a r d e n i n g c u r v e , t h i s e q u a -t i o n c o m p l e t e l y d e s c r ib e s t h e p l a s ti c b e h a v i o r o f i s o t r o p i c a l ly h a r d e n i n g m e t a l s . T h i s d e f i n i t i o ni s u s e f u l f o r t h e c a l c u l a t i o n o f w o r k a l o n g m i n i m u m p l a s t i c w o r k p a t h s , a s i s i l l u s t r a t e d f o r a nl s o t r o p i c F C C m e t a l a n d a s t r o n g l y t e x t u r e d a l u m i n u m a l lo y , s u b j e c t e d b o t h t o p u r e s h e a r a n ds i m p l e s h e a r d e f o r m a t i o n m o d e s_

1 . I N T R O D U C T I O NC a l c u l a t io n s o f m i n i m u m p l as ti c w o r k p a t h s p r o v i d e a u s e f u l ba s is f o r b o t h d e s ig n a n da n a l ys i s o f m e t a l f o r m i n g p r o c e s s e s ( C H u N 6 e t a l . [ 19 8 9 ]) . S u c h c a l c u l a t i o n s c a n b e s i m -p l i f ie d c o n s i d e r a b l y w h e n a n e x p l i c i t e x p r e s s i o n o f t h e e f f e c t i v e s t r a i n r a t e is k n o w n f r o mt h e c o n s t i t u t i v e e q u a t i o n s . P o l y c r y s t a l m o d e l s c a n b e u s e d t o d e s c r ib e t h e p l a s ti c b e h a v -i o r o f m e t a ls , b u t t h e y d o n o t p r o v i d e a n e x p li c it d e f i n i ti o n o f t h e e f f e c ti v e s t r ai n r a t e .R e c e n t l y , B A RL ATa n d L tA N [ 1 98 9 ] a n d B A R L A T , LEGE,a n d B RE M [ 19 9 1] p r o p o s e d p h e -n o m e n o l o g i c a l y i e l d f u n c t i o n s t h a t g i v e a n a n a l y t i c a l d e s c r i p t i o n o f t h e y i e ld s u r f a c e o ft e x t u r e d p o l y c r y s t a l s . T h e s e p h e n o m e n o l o g i c a l f u n c t i o n s w e r e s h o w n t o r e p r e s e n t t h ep l a s ti c b e h a v i o r o f 2 0 0 8- T 4 a n d 2 0 9 0 - T 3 a l u m i n u m a l lo y s a d e q u a t e l y (LEGE, BARLAT,& BREM [1989] , BAgLAT,LEGE,B R E M, & WAR RE N [ 1 9 9 1 ]) . T h e s e f u n c t i o n s p r o v id e a d e f -i n i t io n o f t h e e f f e c t i v e s tr e ss , b u t t h e i r m a t h e m a t i c a l f o r m s a r e n o t s i m p le e n o u g h t og i v e a n e x p l i c i t e x p r e s s i o n o f t h e e f f e c t i v e s t r a i n r a t e .Z IE G LE R [1 97 7] a n d H IL L [1 98 7] h a v e s h o w n , b a s e d o n t h e w o r k e q u i v a l e n t p r i n c i p l e ,t h a t a m e a n i n g f u l s t r a in r a t e p o t e n t i a l c a n b e a s s o c i a te d w i t h a n y c o n v e x y ie l d f u n c t i o n .T h e d e r i v a t i v e s o f s u c h a p o t e n t i a l w i t h r e s p e c t t o t h e s t r a i n r a t e l e a d t o t h e s t r e s sc o m p o n e n t s . FO RT UN IER [1 98 9] h a s i n t r o d u c e d t h e " f l o w p o l y h e d r o n " a s t h e d u a l p o -t e n t i a l o f a n F C C s i n g l e c r y s t a l . A R M IN JO N a n d B A CR OIX [ 19 9 1 ] h a v e g i v e n a n a n a l y t i -c a l d e s c r ip t i o n o f t h e s t r a i n ra t e p o t e n t i a l o f B C C p o l y c r y s t a l s w i t h q u a d r a t i c f u n c t i o n s .S i n c e a n a n a l y t i c a l d e s c r i p t i o n o f th e e f f e c t i v e s t r a i n r a t e is n o t p r o v i d e d b y t h e a n a -l y t i c al y i e ld f u n c t i o n s p r o p o s e d b y B ARLA Tan d LIAN [1989] an d BARLAT, LEGE an d BREM[ 19 9 1] , s u c h a n e x p l i c i t e q u a t i o n is p r o p o s e d i n t h i s w o r k t o d e s c r i b e t h e p l a st i c b e h a v -i o r o f F C C p o l y c r y s ta l s . W h e n c o m b i n e d w i t h a w o r k h a r d e n i n g c u r v e , t h is e q u a t i o ns e r v es as a d e f i n i t i o n o f t h e c o n s t i t u t i v e b e h a v i o r o f p l a s t i c a l l y fl o w i n g m a t e r i a l s . T h e

51


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Strata rate potentml for metals 53

0 . 5

0 . 3

0 .1

- 0 . 1

-0 .3

- 0 . 5 - 0 . 5

I I I I I I I I I

Z r " " 3

3 n p l a n e, , T - , ,-0 .3 -0 .1 0 .1 0 .3 1 .5Fig 1 Strata rate potential m 7r-plane calculated for an lsotroplc FCC material_ TBH model (diamonds) Stratarate potential (# = 4/3)

t h r o u g h o u t t h is p a p e r m a k e u s e o f # = 4 / 3 . F i g u r e 1 s h o w s t h e s tr a i n r a te p o t e n t i a l f o ree = I/ M w h e r e M is th e T a y l o r f a c t o r f o r a n i s o t r o p i c F C C p o l y c r y s t a l s u b j e c te d t op l a n e s t r a i n t e n s i o n ( M = 3 . 3 3 ), i . e . , t h e r a t i o o f t h e p l a n e s t r a i n y i e ld st re s s t o t h e c r it -i c a l r e s o l v e d s h e a r s t r e s s o n t h e s l ip s y s t e m s ( C R S S ) . I n F i g . 2 , t h e c o n j u g a t e p o t e n t i a l( y i e l d s u r f a c e ) is r e p r e s e n t e d . I n o r d e r t o g e n e r a t e t h is f i g u r e , a w a s s e l e c te d s o t h a t t h ep l a n e s t r a i n t e n s i o n y i e l d s t re s s e q u a l s M .G e n e r a l i z a t i o n o f t h e p h e n o m e n o l o g i c a l s t r a i n r a t e p o t e n t i a l f o r a n i s o t r o p i c m a t e r i a l se x h i b i ti n g o r t h o t r o p i c s y m m e t r y c a n b e o b t a i n e d f o l l o w i n g t h e p r o c e d u r e u s e d f o r e x -t e n d i n g a n i s o t r o p i c y i e l d f u n c t i o n (H E R Sr m V [ 1 95 4 ]; H O S FO ~ D [ 1 97 2 ]) t o a n a n i s o t r o p i cf u n c t i o n (B A RL AX , L E ~ E B R~.M [ 1 9 91 ] ). T h e f o l l o w i n g m a t r i x h a s t o b e c o n s i d e r e d :

L =

C 3 ( ~ l l - - ~ 2 2 ) - - C 2 ( ~ 3 3 - - ~ | l ) C 6 E l 2 C 5 ~ 3 13C I ( E 2 2 - - E 3 3 ) - - C 3 ( E I I - - E 2 2 )C61~12 3 C4~23

C 2 ( ~ 3 3 - - ~ l l ) - - C I ( ~ 2 2 - - ~ 3 3 )(25 ~31 C 4 ~23 3

(4 )

T h i s m a t r i x i s a f u n c t i o n o f t h e si x p l a s t ic s t r a i n r a t e c o m p o n e n t s e x p r e s s e d i n th e a x e so f o r t h o t r o p i c s y m m e t r y 1 , 2 , a n d 3 . T h e s i x m a t e r i a l c o e f f i c ie n t s c , c h a r a c t e r i z e a n is o t -r o p y . T h e e i g e n v a l u e s o f t h i s m a t r i x L c a n b e c a l c u l a t e d b y s o l v i n g t h e f o l l o w i n ge q u a t i o n :

~ 3 _ 3 1 2 ~ - 2 / ~ - - o , ( 5 )


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5 4 F BA R LA T e t a l

- 1

- 2

- 3- 3

I I I I I2 ' , , ~ T B H

0 C ~ ----

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Fig 2_ Yield surface in 7r-plane calculated for an lsotrop~c FCC m ater ial TBH mo del (diamonds)_ Derivedfrom strata rate potentml (Iz = 4/3)

w h e r e t h e c o e f f i c i e n t s I2 a n d / 3 a r e d e f i n e d b e l o w . W h e n a l l t h e c o e f f ic i e n t s a re e q u a lt o l , m a t r i x L r e d u c e s t o t h e p l a s t i c s t r a i n r a t e d e v i a t o r . I n t h i s c a s e , s u b s t i t u t i n g t h ee i g e n v a l u e s o f L = g i n e q n s ( 1 ) , (2 ), a n d ( 3 ) l e a d s t o t h e e x p r e s s i o n o f t h e i s o t r o p i c s t r a i nr a t e p o t e n t i a l . S i n c e t he p o t e n n a l ~ i s c o n v e x w i t h r e s p e c t to t h e p r i n c i p a l s t r a in r a t ec o m p o n e n t s , I t i s a l s o c o n v e x w i t h r e s p e c t t o t h e s t r a i n r a t e c o m p o n e n t s ~,j (LIPPM .A .N N[ 19 7 0 ]) . W h e n t h e c o e f f i c i e n t s c , a r e n o t s i m u l t a n e o u s l y e q u a l t o o n e , t h e f o l l o w i n g s eto f e q u a t i o n s d e f i n e s t h e s t ra i n r a t e p o t e n t i a l f o r a n i s o t r o p i c m a t e r i a l s :

2 = [ C 3 ( 6 1 1 - - 6 2 2 ) - - C 2 ( ~ 3 3 - - ~ 1 1 ) ] 2 [ C I ( 1~ 22 - - E 33 ) - - C 3 ( E I I - - 1 ~2 2 )] 2+5 4 5 4[ C 2 ( ~ 3 3 - - ~ 1 1 ) - - C 1 ( ~ 2 2 - - ~ 3 3 ) ] 2 ( C 4 6 2 3 ) 2 + ( C 5 6 3 1 ) 2 + ( C 6 6 1 2 ) 2+ +5 4 3

( 6 )

& =[ C 3 ( ~ 1 1 - - ~ 2 2 ) - - C 2 ( 1~ 3 3 - - I ~ 11 ) ] [ C ! ( ~ 2 2 - - 6 3 3 ) - - C 3 ( 6 1 1 - - 1 ~2 2) ]

X [ C 2 ( ~ 3 3 - - 6 1 1 ) - - C 1 ( 6 2 2 - - ~ 3 3 ) ]5 4

[ C 3 ( ~ 1 1 - - 17.22 - - C 2 ( 6 3 3 - - E l l ) ] ( C 4 1 ~ 2 3 ) 26

[1 5.1 (1 ~22 - - 1 ~3 3 ) - - C3 (E l l - - 1 ~22)] (C5 F_ ,3 1 )26

[ C 2 ( ~ 3 3 - - I ~ l l ) - - C l ( ~ 2 2 - - E ' 3 3) ] ( C 6 E ; I 2 ) 26

(7 )

- - + C4 C5 C6 E 23 ~3 1 E 1 2


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Strata rate potentml for metals 55

o = cos-X (t2 ~/2 ), o_< o__ ~- (8)

41 = 2 x /7 2 2c o s 4 2 = 2 f f 2 c o s 4 3 = 2 x/ ~2 c o s ( - - - ~ ) (9 )

+ , + ,

= I ~ , l , + ] 4 2 1 " + 1 4 ~1 ~ = 2 ~ .(1o)

T h e e f f e c t i v e s t r a i n r a t e ~e c a n b e d e f i n e d a s t h e s t r a i n r a t e f o r p l a n e s t r a t a t e n s i o n md i r e c t i o n 1 (~ 2 = - E l , ~3 = 0 ) . A g a i n , t h is a m s o t r o p i c f o r m u l a t i o n r e d u c e s t o t h e i so -t r o p i c c a s e , e q n s ( 1 ) a n d ( 2) , w h e n a l l t h e c o e f f i c i e n t s c , a r e e q u a l t o 1 . A l s o , i n t h i sc a s e , t h e f u n c t i o n s o f t h e s tr a i n r a te c o m p o n e n t s - 3 2 a n d 213 r e d u c e t o t h e s e c o n da n d t h e t h i rd i n v a r i a n t s o f t h e p l a s ti c s tr a in r a t e d e v i a t o r . F o r p l a n a r i s o t r o p y ( a x i s y m -m e t r y a b o u t d i r e c t i o n 3 ), th e f o l l o w i n g r e l a t io n s h i p b e t w e e n t h e c o e f fi c i e n ts c , c a n b es h o w n :

C 1 =C 2

c 4 = c 5 ( 11 )

2c3 + C 1C 6 -- 3S i m i l a r r e l a t i o n s c a n b e e s t a b l i s h e d w h e n t h e a x i s o f s y m m e t r y is 1 o r 2 . T h e d e v i a t o r i cs t r e s s e s a r e g i v e n b y t h e g r a d i e n t o f th e s t r a i n r a t e p o t e n t i a l :

a , a , ( a 4 , a l , a 4 , o / 3 / (12)T h e d e r i v a t i v e s o f 4 , w i t h r e s p e c t t o 12 a n d 1 3 a r e r e a d i l y o b t a i n e d b y d i f f e r e n t i a t i n ge q n ( 5) . T h e y a r e a l w a y s d e f i n e d e x c e p t f o r t h e c a s e s w h e r e 0 = 0 a n d 0 = 1 80 . T h ed e r i v a t i o n o f th e d e v i a t o n c s tr e ss c o m p o n e n t s f o r th e s e t w o c a se s c a n b e o b t a i n e d a st h e l i m i t o f t h e g e n e r a l c a l c u l a t i o n :

s u = cx ~ = c~ + - - qZ--_, fo r O = (13 )3120 a f t ' ( 1 012 1 a 1 3 ) f o r 0 = 1 80 . ( 14 )

T h i s n e w m o d e l c a n b e c o m p a r e d w i th t h e T B H a n a ly s is f o r th e c a s e o f a s tr o n g ly a n -i s o t r o p i c 2 0 9 0 - T 3 a l u m i n u m l i t h i u m s h e e t . F i g u r e 3 r e p r e s e n t s t h e ( 1 11 ) p o l e f i g u r eo f t hi s a ll o y r e su l ti n g f r o m 2 4 0 i n d i v id u a l g r a m o r i e n t a t io n s . T h e s e o r i e n t a t i o n s w e r em e a s u r e d b y m e a n s o f b a c k s c a t t e r e d K i k u c h~ d i f f r a c t i o n u s i n g a s c a n n i n g e l e c t r o n


6/13

56 F BARLATet al

q l.

(111) pole figure"1

, : ; . ' ,/ \

i ' , . " ". ' : : . + " ' I : " ~ " ~ " " " 'i It

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R DFig_ 3 (111) Pole figure for 2090-T3, 240 grams measured by EBSP technique

m i c r o s c o p e ( s e e B AR LA T, L I U , & D IC K EN S O N [1 99 1] f o r s i m i l a r m e a s u r e m e n t s ) . F i g u r e 3i n d i c a t e s t h a t t h e c r y s t a l l o g r a p h i c t e x t u r e i n t h i s m a t e r i a l i s v e r y s t r o n g . F i g u r e 4 r e p -r e s e n ts th e r - p l a n e s e c t i o n o f t h e T B H a n d t h e p h e n o m e n o l o g i c a l s t ra i n r a te p o t e n t i a lf o r e e = 1 / M ( M T a y l o r f a c t o r i n p la n e s t r a in t e n s i o n , LEE ~ - - e l l , o t h e r Ev = 0 ) . T h ec o e f f i c ie n t s c, w e r e c a l c u l a t e d u s i n g th e v a l u e o f t h e s t r a i n r a t e c o m p u t e d w i t h t h e T B Hm o d e l f o r si x d i f f e r e n t i m p o s e d s t r a i n s t at e s , a s s u m i n g t h e s a m e a m o u n t o f p la s t icw o r k e x p e n d e d a l o n g t h e s e d e f o r m a t i o n s t a t e s . T h e c u r v e s s h o w t h a t t h e p r o p o s e d s t r a i nr a te p o t e n t i a l is a g o o d a p p r o x i m a t i o n o f th e T B H m o d e l n o t o n l y f o r ls o t r o p lc b u t a l s of o r a n i s o t r o p i c m a t e r i a ls . F i g u r e 5 re p r e s e n t s t h e y i e ld s u r f a c e s a s s o c i a t e d w i th t h es t r a i n r a t e p o t e n t i a l o f F ig . 4 . A g a i n , a g o o d a g r e e m e n t i s o b t a i n e d . F i n a l ly , F i g . 6 r e p -r e s e n t s t h e d i r e c t io n a l i t y o f th e y i e l d s t re s s , w h e r e t h e h o r i z o n t a l a x i s d e n o t e s t h e a n g l eb e t w e e n t h e r o ll in g a n d t h e l o a d i n g d i r e c t io n s i n th e p l a n e o f th e s h e e t. T h e a g r e e m e n tb e t w e e n t h e t w o t h e o r i e s i s s a t i s f a c t o r y .

T h i s e x a m p l e s h o w s t h a t t h e s t r a in r a t e p o t e n t i a l p r o p o s e d i n t h is w o r k l e a d s to a d e -s c r i p t io n o f t h e p l a s ti c b e h a v i o r o f p o l y c r y s t a l s c o n s i s t e n t w i th t h e T B H m o d e l . H o w -e v e r , it is in t e r es t in g t o m e n t i o n t h a t a s li g h tl y b e t te r a g r e e m e n t c o u l d b e o b t a i n e d b yl o o k i n g a t th e o p t i m u m v a l u e o f t h e e x p o n e n t ix. M o r e o v e r , t h e c o n s t a n t s t h a t c h a r a c -t e ri z e a n l s o t r o p y c o u l d b e c a l c u l a te d u s i n g s o m e te s t r e su l ts . T h e n , t h e a d v a n t a g e s o ft h e s t ra i n r a t e p o t e n t i a l w o u l d b e c o n s i s t e n c y w i t h b o t h p o l y c r y s t a l c a l c u l a t i o n s a n d e x -p e r i m e n t a l d a t a , a n d n u m e r i c a l e f f i c i e n c y

!1 I. M I N I M U M P L A S T I C W O R K A N D I TS A P P L I C A T I O NA d e f o r m a t i o n p a t h t h a t a c h ie v e s a de s ir e d h o m o g e n e o u s d e f o r m a t i o n w i th a m i n i-

m u m a m o u n t o f p l a s ti c w o r k h a s b e e n s t u d i e d b y N AD A I [1 96 31 a n d H IL L [1 9 86 ]. T h e


7/13

S t r a t a r a t e p o t e n t m l f o r m e t a l s 5 7

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, 3 , / . . . . . ,~ p l a n e- o . 1 o . ,

F i g . 4 S t r a i n r a t e p o t e n t i a l m r - p l a n e c a l c u l a t e d f o r 2 0 9 0 -T 3 T B H m o d e l ( d i a m o n d s ) . S t r a i n r a t e p o t e n -t i a l (~ . = 4 / 3 , c I = 1 _ 0 4 1 0 , c 2 = 0 8 4 5 3 , c a = 1 0 1 3 3 , c 4 = 0 7 9 9 4 , c 5 ---- 0 8 6 8 5 , c 6 = 0 7 7 6 9 )

m i n i m u m w o r k p a t h is a c h i e v e d w h e n t h e s et o f th r e e p r i n c i p a l a x e s o f s t r e tc h i n g i s f ix e dw i t h r e s p e c t to t h e m a t e r i a l a n d , a t th e s a m e t i m e , t h e r a t io s o f t r u e s t r a i n r a te s a l o n gt h o s e p r i n c i p a l m a t e r i a l l in e s a r e c o n s t a n t . T h e f ix e d m a t e r i a l h n e s m a y b e c h o s e n a r -b i t r a r il y f o r i s o t r o p i c m a t e r i a l s , b u t t h e y a r e m o r e c o n s t r a i n e d f o r a n i s o t r o p i c m a t e r i -

3

-1

- 2

- 3- 3

I I I I Io T B H

, 3 , o ~ p l a n e-2 .1 0 i 2 3

F i g . 5. Y i e l d s u r f a c e m 7 r -p la n e c a l c u l a t e d f o r 2 0 9 0 -T 3 T B H m o d e l ( d i a m o n d s ) . D e r i v e d f r o m s t r a i n r a t e p o -t e n U a l (/ ~ = 4 / 3 ) _


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58 F BARLATt u/

3 . 5 0

t/)t/)n-Ot/loL_qDom

3 . 3 0

3 . 1 0

2 . 9 0

2 . 7 0

2 . 5 0

T B H= 4 / 3 / '

/-x (

1 I I I0 1 5 3 0 4 5 6 0 7 5 9 0L o a d i n g d i r e c t i o n ( d e g . )

Fig. 6 Predictedyield stress directionality for 2090-T3, from TBH model (diamonds), and from strain ratepotential

als (CmSNG ~ RICHMOND [1992a]). The minimum effect ive strain on the min imum workpath can be easily calculated from the definition of the effective strain rate by simplyreplacing strain rate c omponent s with true s train com ponent s (CHuNG & RICHMOND[1991]). Therefore, the minimum effective strain for the strain rate potential proposedin eqn (10) becomes:

(15)

where ~, are the principal values of matrix L, whose strain rate components are re-placed with true strain components. Under such a circumstance, the following relation-ship is satisfied:

dffk~k-- dt (16)

This simple relationship between the minimum effective strain and the effective strainrate is possible for general anisotropic materials that harden isotropically. Here, Isotro-plc hardening implies that the strain rate potential defined for a material does not changealong the minimum work path.

For demonstration purposes, minimum effective strains are calculated for the 2090-T3 alloy, which undergoes simple shear deformation as shown in Fig. 7. Note that sim-ple shear is not the optimum deformation because the principal material lines are not


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S t r a i n r a t e p o t e n t i a l f o r m e t a l s 5 9

Y

/0 . 0 1 . 0

a)Y

~ X

F i g 7

0.0

R D

- X1 .0b)

( a) K i n e m a t i c s o f s h e a r d e f o r m a t m n ( b ) In i ti a l o ri e n t a t i o n o f t h e s p e c tm e n

f i x e d . F o r t h i s c a l c u l a ti o n , t h e m a t e r m l is a s s u m e d t o r o t a t e a l o n g w i t h t h e s p in t e n s o r( the an t i - sym m et r i c pa r t o f t he spa t ia l g r ad i en t o f t he ve loc i ty ) . I n F ig . 7 (a ), t he p a ram -e t e r d is a t r ave l d i s t ance o f a po in t wh ich is o r ig ina l ly loca t e d a t (x ,y ) = (0 ,1 ) an d t r av-e l in g w i t h t h e c o n s t a n t v e l o c i t y , v ( t h e r e f o r e , d = v t , w h e r e t d e n o t e s t i m e ) . T h e i n i ti a lp o s i t ro n o f t h is a m s o t r o p i c m a t e r i a l is d e s i g n a t e d b y t h e a n g l e , O o , b e t w e e n t h e r o l li n gdi rect ion (or , 1-axis) and the x-axis in F ig. 7(b) .T h e m m i m u m e f f e c ti v e s t ra i n w h i c h is t h e e f f e c ti v e s t r a ta o f p u r e s h e a r l e a d in g t o t h es a m e s h a p e c h a n g e a s s i m p l e sh e a r i s d e n o t e d e ~ n. T h is m i n i m u m e f f e c t w e s t r a ta i s o b -t a i n e d f r o m e q n ( 15 ) w h e n t h e s t r a i n r a t e c o m p o n e n t s in e q n ( 4 ) a r e r e p l a c e d w i t h th ef o ll o w i n g t ru e s t ra i n c o m p o n e n t s b e t w e e n t h e i n it ia l a n d f in a l c o n f i g u r a t m n s :

[ E q ] =

~rc o s O - s i n O 0 [ [ l n A I 0l !s i n O c o s O 0 0 1 J [ 00 l nA 20

i ] c os 0- s i n 00

s i n OCOS 19

0(17)

w h e r ed + x / - d 5 + 4 12 , ~2 ~1 0 = ~- - 0 0l - - and Op = a rc t an (A l ) . ( 18)


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60 F BARLATet al

I n e q n ( 17 ) , e v a r e t h e t r u e s t r a i n c o m p o n e n t s r e p r e s e n t e d w i t h r e s p e c t t o t h e C a r t e s i a nc o o r d i n a t e s y s t e m a l i g n e d w i t h t h e r o l l i n g (1 - a x is ) a n d t r a n s v e r s e ( 2 -a x i s ) d i r e c t i o n s . I ne q n ( 1 8 ), A , ' s a n d O p r e p r e s e n t t h e p r i n c i p a l v a l u e s a n d d i r e c t i o n s , r e s p e c t i v e l y

T h e s t r a i n r a t e c o m p o n e n t s n e e d e d t o c a lc u l a t e th e e f f ec t i v e s t r a in o f s im p l e s h e a r ,s i reI~ e , are "

[ ~ v ] =c o s O s i n O

- s i n O c o s O0 0 i ] / 20 . 2 o O

c o c o _ s in .si n 0 COS 0 ,

0 0( 1 9 )

w h e r e

v - f0 = O o - - - ( 2 0 )2U n d e r t h e s i m p l e sh e a r d e f o r m a t i o n , t h e m a t e r i a l r o t a t e s c o n t i n u o u s l y s o t h a t t h e a n -g l e, /9 , b e t w e e n t h e r o l l i n g d i r e c t i o n i n t h e m a t e r i a l a n d t h e x - a x i s , c h a n g e s a s s h o w ni n e q n ( 20 ) . T h e e f f e c t i v e s t r a i n o f s i m p l e s h e a r , e ~ ,m IS o b t a i n e d b y n u m e r i c a l l y i n t e -g r a t i n g e q n ( 1 0) a f te r t h e s t r a i n r a t e c o m p o n e n t s i n e q n ( 19 ) a r e s u b s t i t u t e d i n t o e q n ( 4) .

T h e r e s u l t i n g e f f e c t i v e s t r a i n s , e~ ' " a n d e eslm, a r e d e p e n d e n t o n t h e a m o u n t o f d e f o r -m a t i o n , d , a n d t h e i n i t i a l p o s i t i o n o f t h e m a t e r i a l , O 0. T h e y a r e p l o t t e d a s c u r v e d s u r -f a c e s i n F i g . 8 . D e p e n d e n c e o f t h e e f fe c t i v e s t ra i n s o n t h e i n i t i a l p o s i t i o n a r e p l o t t e d i nF i g . 9 f o r d = 0 . 5 u p t o d = 5 . 0 a t e v e r y 0 .5 i n t e r v a l . T h e i n i t i a l p o s i t i o n s o f t h e m a t e -r ia l , w h i c h c o m p l y w i t h t h e m i n i m u m v a l u e s o f e m 'n f o r g i v e n d v a l u e s , a r e t h e o p t i -m u m m a t e r i a l p o s i t i o n s n e e d e d fo r u l t i m a t e o p t i m u m d e f o r m a t i o n s o f th e a n i s o t r o p i c

Fig 8plot)_


11/13

Strain rate potential for metals 61

3 . 0

2 . 5

2 . 0

E e 1 . 51 . 0

- - - S i m p l e s h e a rP u r e s h e a r

0 . 5

0 . 09 0 7 5 6 0 4 5 3 0 1 5 0

O 0Fig. 9. Effectwe strain as a function of shear displacement and initial spe ome n orientation for 2090-T3 (crosssection)

m a t e r i a l . T h i s f i g u r e c o n f i r m s t h a t th e f i x e d m a t e r i a l l m e s n e e d e d f o r o p t i m u m d e f o r -m a t i o n s a r e r e s t r i c t e d f o r a n i s o t r o p i c m a t e r i a l s .F o r i s o t r o p i c m a t e r i a l s , t h e e f f e c t i v e s t r a i n s , E e m a n d e~ m , a r e d e p e n d e n t o n d , b u tn o t o n t h e i n i t i a l p o s i t i o n . T h e s e v a l u e s f o r c , = 1 b e c o m e

E ~ ' n = l n ( d + v r ~ + 4 ) 2 (21)a n d

dE ~ m = - - . (22)2T h e y a r e p l o t t e d i n F i g . 1 0 . N o t e t h a t t h e r e s u l t s m e q n s ( 2 1) a n d ( 22 ) a r e v a l i d f o r a n y# ; t h e r e f o r e , t h e r e s u l t s a r e g e n e r a l f o r a l l i s o t r o p i c m a t e r i a l s p r o p o s e d m e q n ( 2 ) i n -c l u d i n g M i s e s a n d T r e s c a m a t e r i a l s .I n a p p l i c a t i o n , t h e m i n i m u m p l a s t i c w o r k p r o v i d e s a u s e f u l b a s i s f o r d e s i g n a s w e l la s f o r a n a l y s is o f m e t a l f o r m i n g p r o c e s se s , a s s u m m a r i z e d b y C H U N C e t a l . [1989]. Ford e s i g n p u r p o s e s , t h e s o - c a l l e d i d e a l f o r m i n g t h e o r y h a s b e e n d e v e l o p e d b y C H U N ~ a n dRICI-II~OND [199 2a,b] ba se d o n th e ea r l ier w or k b y RICI-IMONI~ [1968]. In thi s d esi gn th e-o r y , m a t e r m l is a s s u m e d t o d e f o r m m m i n i m u m w o r k p a t h s. T h e n , t h e o p t i m u m s t r a ind i s t r i b u t i o n s a r e o b t a i n e d f r o m t h e f o l l o w i n g r e l a t io n :

d W f N F .,~nd u - tr e ~ d V = 0, (23)


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62 F BARLATet al

Fig. 10

3 . 0

2 . 5

2 . 0

E e 1 . 5

1 . 0

0 . 5

0.0

m i nE e ,t/- - ' E s i m /

e ///

/

/ /

i i i i0 1 2 3 4 5dEffecuve strain as a function of shear displacement for an lsotropic materml

w h e r e W ~s t h e p l a s ti c w o r k w h e n m a t e r i a ls d e f o r m i n m i n i m u m w o r k p a t h s ; o e , u , a n dV a r e t h e e f f e c t i v e s t re s s , d i s p l a c e m e n t , a n d v o l u m e , r e s p e c t i v e l y .

F o r a n a l y si s p u r p o s e s , a w h o l e f o r m i n g p r o c e s s ~s d l sc r et ~z e d i n t o a f i n i t e n u m b e r o fs m a ll f o r m i n g i n c re m e n t s f o r u p d a t e d L a g r a n g i a n f o r m u l a t i o n s . T h e n , a d e f o r m a t i o np a t h i s a s s u m e d d u r m g d i s cr e ti z ed in c r e m e n t s. E m p l o y i n g t h e m i n i m u m w o r k p a t h d u r -i n g t h e d l s c r e t ~ z e d s t e p i s o n e o f m a n y p o s s i b l e w a y s t o a s s u m e t h e d e f o r m a t i o n p a t h .U n d e r s u c h c i r c u m s t a n c e s , t h e f o ll o w i n g e q u i h b n u m c o n d i t i o n i s a p p h e d f o r e a c h d ls -c r e t i z e d s t e p f o r r i g i d - p l a s ti c m a t e r i a l s :

d ( A W ) _ f d ( A C emln)d ( A u ) ae d ( A u ) d V = F , (24)w h e r e F is t h e e x t e r n a l f o r c e , a n d t h e q u a n t i t i e s w ~ th A d e n o t e t h e i n c r e m e n t a l q u a n t i -t ie s . C o n s e q u e n t l y , t h e p r o p o s e d a n i s o tr o p ~ c e f f e c t w e s t ra i n r a t e a n d i ts m i n i m u m e f -f e c ti v e s t r ai n p r o v i d e a u s e fu l m a t h e m a t i c a l b as is f o r m i m m u m - w o r k b a s e d f o r m u l a t i o n su s e d b o t h i n d es ig n a n d a n a ly s is o f d e f o r m a t i o n p r o c e s se s f o r a n i s o t r o p i c m a t e ri a ls .

IV. CONCLUSIONS

I n th ~s w o r k , t h e d e f i n i t i o n o f a p la s t i c p o t e n t m l e x p r e s s e d i n s t r a i n r a t e s p a c e , t h es t r a i n r a t e p o t e n t i a l , h a s b e e n p r o p o s e d f o r t e x t u r e d m e t a l s . T h i s d e f i n i t i o n le a d s t oa d e s c r i p t i o n o f th e p l a s t i c f l o w o f m a t e r i a l s i n g o o d a g r e e m e n t w ~ th t h e p l a s ti c b e h a v i o rp r e d i c t e d w i t h t h e T a y l o r / B i s h o p a n d H x ll p o l y c r y s t a l m o d e l . I n a d d i t i o n , t h i s p h e n o m -e n o lo g i ca l f o r m u l a t i o n c a n l e ad t o n u m e r i c a ll y ef f ic i en t c o m p u t a t i o n m m a t h e m a u c a ls i m u l a t i o n s o f f o r m i n g p r o c e s s e s , p a r t i c u l a r l y f o r c o d e s b a s e d o n m i n i m u m w o r k p a t h s .


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Strain rate potential for metals 63

19381951a1951b195419581963196819701972197219771986198719891989

1989198919891991199119911991

19911992a1992b

R E F E R E N C E STA~ort, G. I., "Plast ic Strain m Metals," J Inst_ Metals, 62, 307.BIsHoP, J.W.E, and HrtL, R , "A Theory o f the Plastic Distortion of a Polycrystalhne Aggregate Un-der Combined Stresses," Phil. M ag , 42, 414.BisHoP, J.W E, and HILL, R., "A Theoretical Derivation of the Plastic Properties of a Polycrystal-line Face-Centred Metal ," Phil. Mag,, 42, 1298.HERSHEY, A. V, "The Plasticity of an Isotroplc Aggregate of Amsotroplc Face Centred Cubic Crys-tals," J. Appl Mech., 21, 241.EOG~STON, H G., "General Properties of Convex Functions," in Convexaty (Ch Ill), Umverslty Press,Cambridge, pp. 45-58.NAD~a, A., Theory o f Flow and Fracture of Sohds, McGraw-Hall, New YorkRICHMOND, O, "Theory of Streamlined Dies for Drawing and Extrusion," in Mechamcs of Sohd State,in Rn~moTr, F.P.J, and SCrIW~GHOFE~t, J (eds), University of Toronto Press, Toronto, pp 154-167LIPPMANN, H , "Matr ixunglelchungen und die Konvexttat der Fliessflache," Zelt. Angew_ Mech., 50,134.HOSFORD, W H., "A Generalized Anlsotropic Y~eld Crite rion ," J Appl Mech., 39, 607ROCKAr'EU.AR, R. T, "Convex Functions," in Convex Analysis, (Section 4), Princeton University Press,Princeton, pp 23-31.ZIECLER, H., An Introduction to Thermomechanlcs, North-Holland Puhhshlng Company, Am-sterdamHILL, R_, "External Paths of Plastic Work and Defo rmat ion ," J Mech Phys. Solids, 34, 511HILL, R., "Consti tutive Dual Potent ials in Classical Plastic ity," J. Mech Phys Solids, 35, 22BARLAT, F, and LL~N, J., "Plastic Behavior and Stretchablh ty of Sheet Metals. Part h A Yield Func-tion for Orthotropic Sheets under Plane Stress Conditions," Int. J_ Plasticity, 5, 51CHUNC, K , RICHMOt,rO, O-, GERM~aN, Y, and WAC,ONEa, R H , "An Incremental Appr oach to Plas-ticity and Its Application to Finite Element Modeling," in TrIOMrSON, E_G et al (eds), NUMI-FORM'89, Belkema, Rotte rdam, pp 129-134FORXtmmR, R , "Dual Potentials and Extremum Work Principles in Single Crystal Plastloty," J. Mech.Phys Solids, 37, 779.GERMAIN, Y, CrIUNG, K , and WAGOr~ER, R_H., "A Rlgid-Vlscoplastlc Finite Element Program forSheet Metal Forming Analys is," Int. J. Mech So , 31, lLECE, D.J ., BARLAT, F, and BREM, J.C., "Characterization and Modehng of the Mechanical Behav-mr and Formabill ty of a 2008-T4 Sheet Sample," lnt J. Mech Scl., 31, 549.BArtLAT, F, Lru, J , and DICKENSON, R C. , "Prediction of Plastic Proper ties of Rods From Individ-ual Grain Orientation Measurements," Textures and MIcrostructures, 14-18, 1179ARMINJON, M., and BACROIX, B , "On Plastic Potential s for Anlsot ropic Metals and Their Deriva-tion from the Texture Function," Acta Mechamca, 8 8 , 219BARLAT, F., LEGE, D.J , and BREM, J C , "A Six-Component Yield Function for Anlsotropic Mate-rials," Int. J. Plasticity, 7, 693.BARLAT, F., LEGE, D. J, BREM, J.C , and WARREN, C.J , "Constitut ive Behavior for Amsotroplc Ma-terials and Application to a 2090-T3 AI-LI Alloy," in LowE, T_, ROLLETT, A.D. , FOLLANSnEE, P S ,and Da~EHN, G S (eds) Modeling the Defor mation of Crystalline Solids Physical Theory, Apph-cation, and Experimental Comparison , TMS, Warrendale, PA, pp 189-203_CHUNG, K., and RICHMOND, O , "A Deformation Theory o f Plasticity Based on Mlmmum WorkPaths" (submitted to lnt J of Plasticity).CHUNG, K., and RICHMOND, O , "Ideal Forming, Part l Homogeneous Deformation With MinimumPlastic Wor k," Int J_ Mech Scl., 34, 575CHUNG, K , and RICHMOND, O., "Ideal Forming, Part II Sheet Forming With Most Uni form De-formati on" Int J Mech So. , 34, 617

Aluminum Company of AmericaAlcoa Technical Center100 Techmcal DriveAlcoa Center, PA 15069-0001, USA(Received 20 January 1992, in final revised form 25 July 1992)

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Strain Rate Potential for Metals and Its Application to Minimum Work Path Calculations