viscoplastic model

8/12/2019 Viscoplastic Model

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Aeta Mechanica 36, 213--230 (1980) C T M E C H N I C| by Springer-Verlag 1980

Viscoplas t ic M ater ia l M odel and I t s pp l ica t ion to Cyclic Loa ding

By

D. Kujawski and Z. Mr6z, Warsaw, Poland

With 14 Figures

Received October 15, 197 8)

S u m m a r y Z u s a m m e n f a s s u n g

A Vi s e o p l a s t i e M a t e r i a l M o d e l a n d I t s A p p l i c a t io n t o C y c l ic L o a d i n g . Akinematichardening model is generalized by introducing plastic and viscous residual back s t r e s sesa, ~ that govern the translation of the yield surface. The evolution equations for ec andare proposed and the material functions are identified for a construction steel by carryingout tension-compression tests at different strain rates. The cycli c tests with changingstrain amplitudes and frequencies are next carried out and model predictions are comparedwith experimental results.

E i n v i s k o p l a s t i s eh e s S t o f f m o d e l l u n d s e i n e A n w e n d u n g e n b e i z y k l i s e h e r B e l a s t u n g .Ein Modell mit kinetmatischer Verfestigung wird durch die Einfiihrung plastischer undviskoser bleibender ,,tt intergrundspannungen e~, ~ verallgemeinert, die die Bewegungender Flie$fl~che steuern. Die Wachstumsgleichungen fiir a und ~ werden aufgestellt undd ie Materialfunktionen fiir einen Baustahl aus Zug-Druckversuchen mit verschiedenerDehnungsrate bestimmt. Die zyklischen Versuche mit sieh ~ndernden Dehnungsamplitudenund Fequenzen werden als n~ehstes durchgeffihrt und die Modcllaussagen mit den experi-mcntellen Ergebnissen verglichen.

1. I n t r o d u c t i o n

The present paper constitutes an extension o previous work on modellingof cyclic behaviour of metals at moderate temperatures [1]--[4]. A viscoplasticmaterial model is proposed in order to simulate plastic hardening, softening,rat cheting and viscous creep or relaxation. Identific ation of material functionsis carried out for a uniaxial cyclic stress loading an d verifi cation of model pre-dictions is performed for several loading programs with varying strain amplitudeand freque ncy. However, a further verification for a multiaxial stress sta teshould be carried out in order to assess the range of applicability of this model.Our development will be based on the assumption that the residual stress statecan be decomposed into , plastic an d viscou s parts. Whereas the plasticresidual or ba ck stress depen ds only on the deforma tion history, the viscouspart depends also on the rate of deformation and represents macroscopicallythe multiplication and annihilation of dislocation mechanisms. In describingplastic hardenin g or softening, besides the contin uously varying state paramet ers,we introduce also discrete memory parameters that will record maximal prestress

0001-5970/80/0036/0213/$03.60


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214 D. Ku jawski and Z. Mr6z:

e v e n t s f r o m t h e p a s t h i s t o ry. I n t h i s w a y, t h e m o s t e s s en t ia l f e a t u r e s o f v is co -p l a s t i c d e f o r m a t i o n c a n b e i n c o r p o r a t e d i n t o t h e m o d e l .

I n t h e n e x t s e c t io n , w e s h a ll d is c u ss t h e b a s i c a s s u m p t i o n s a n d r e l a t i o n s

o f o u r m o d e l a n d i n s e c t io n 3 t h e i d e n t i f ic a t i o n p r o b l e m w i ll b e c o n s id e r e d ,w h e r e a s i n s e c ti o n 4 t h e v e r i f ic a t i o n t e s t s a r e d e s c r ib e d . T h o u g h t h e c o n s t i t u t i v er e l a t i o n s a r e d e r i v e d f o r a g e n e r a l s t r e s s s t a te , b o t h i d e n t if i c a ~ ib n a n d v e r i f i c a t i o nt e s t s w e r e c a r r i e d o u t f o r t h e u n i a x ia l t e n s i o n - c o m p r e s s i o n t es t s.

2 . F u n d a m e n t a l R e l a t i o n s f o r a Vi s e op l as ti e M o d e l

F i g . 1 p r e s e n t s s c h e m a t i c a ll y t y p i c a l p h e n o m e n a o b s e r v e d i n u n i a x ia lt e s t s u n d e r a p p l i e d c y c l i c s t r a i n o r s t re s s. F o r p r e s c r i b e d s t r a i n a m p l i t u d e ,t h e a n n e a l e d m a t e r i a l h a r d e n s a n d t h e t r a n s i e n t s t a t e is c h a r a c t e r i z e d b y g r o w i n gs t re s s a m p l i t u d e b e f o r e t h e s t e a d y s t a t e i s r e a c h e d , F i g . 1 a . T h e m a t e r i a l p r e -v i o u s l y h a r d e n e d b y p l a s ti c s t r a i n o r h e a t t r e a t m e n t m a y e x h i b i t c y c li c s o f t e n i n ga n d t h e d e c r e a s e O f s t re s s a m p f i t u d e , F i g . l b . T h e i m p o s e d a s y m m e t r i c s t r a i na m p l i t u d e i n d u c e s c y c li c r e l a x a t i o n , F i g. l c , a n d t h e p r e s c r i b e d a s y m m e t r i cs t r e s s c y c l e m a y i n d u c e c y c li c c r e e p r a t c h e t i n g ) w i t h t h e a s s o c i a t e d p l a s t i cs t r a in a c c u m u l a t i o n , F i g. l d . T h e s e p h e n o m e n a m a y h a v e i n s t a n t a n e o u s o rt i m e - d e p e n d e n t c h a r a c t er. To d i s ti n g u i sh b e t w e e n p l a s ti c t im e i n d e p e n d e n ta n d v i s c o u s d e p e n d e n t o n n a t u r a l t i m e sc a le ) e f f e c t s , t h e c y c l i c t e s t s w i t hv a r y i n g f r e q u e n c y o r w i t h h o l d - p e ri o d s s h o u ld b e c a r r i e d o u t .

l v v v vCycl ic harden i r lg

Icl W W

~Cyc lic so f ten ing 5 ~

C ycl ic re texat ion

VV VV ~ ~

C yc lic c r ee p

Fig, 1. Ty pica l effects during cyclic loadings: a cyclic h ardenin g, b cyclic softening,c cyclic relaxatio n, d cyclic creep


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A Viscoplastic Material M odel 21 5

F i g . 2 a s h o w s t h e c y c l ic s t re s s - s t r a i n d i a g r a m f o r a c o n s t r u c t i o n s t e e l( St 3S ) o b t a i n e d d u r i n g u n i a x i a l l o a d i n g b y c y c l i c a l ly o sc i ll a ti n g f o r c e b e t w e e n0 a n d P m a x . I t i s s e e n t h a t t h e p e r m a n e n t s t r a in a c c u m u l a t e s a f t e r e a c h c y c l e

u p t o t h e f i n a l f a i l u r e ( m e a n r a t e o f a c c u m u l a t i o n (~sp,~ - - 0 . 5 6 5 p e r c y c l e ).A t t h e u n l o a d i n g p o i n t t h e s t r e s s -s t ra i n c u r v e d o e s n o t f o ll o w e l as t ic p a t h b u tb e n d s fo r w a r d , i n d i c a t in g t h e r e b y t h e e x i s t e n c e o f v i sc o u s s t r a in . ]F ig . 2 bp r e s e n t s t h e d e p e n d e n c e o f a c c u m u l a t e d p l a s ti c s t r a in o n t h e n u m b e r o f cy c le sf o r d i f f e r e n t m a x i m a l s t re s s l ev e ls ( m e a s u r e d b y t h e n u m b e r o f c y cl e s t o f a i l u re-IV/). I t i s s e e n t h a t t h e s e c u r v e s r e s e m b l e m u c h t y p i c a l c r e e p C u r ve s u n d e r c o n s t a n ts tr e ss , t h e f a c t a l r e a d y n o t e d b y n u m e r o u s r e se a r c h er s . T h e fi n a l v a l u e o f s t r a i na t f a i lu r e d o e s n o t d e p e n d s i g n i fi c a n t ly o n t h e m a x i m a l s t r es s l ev e l, t h o u g h

IP lO ~ [ ]S t e e l S t S

N ~ 2

10 .7 I ~ 5

2 2

Nf=67e

2 H f = 1 3 5 N f =8

i s ~ 1 2 N ~ 6

16

14

12

o

2 6 1 4 2 2 3 3 8 4 5 5 4 6 2 7 N

b

Fig. 2. Cyclic creep of stee l finder unii~xial tensio n: a Stress-strain response, b A ccum ulatedplas t ic s t ra in versus num ber of cycle s

1 5 A c t a M e c h . 36/3 4


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A Viscoplas~ie M aterial Model 217

S u c h d i r e c ti o n a l v a r i a t io n o f t h e h a r d e n i n g m o d u l u s w a s d e s c ri b e d p r e v i o u s l y b yus ing m ul t i s u r f ace [1 ], [2 ] o r two -su r fac e mo de l s [7], [8].

T h e m a t e r i a l f u n c t i o n s c = c (2 c) a n d d = d (2 ) c a n b e i d e n t i f i e d f r o m u n i a x i a l

l o a d i n g - r e v e r s e l o a d in g t e s t s ( se e F i g . 5 ) a n d t h e y a r e i n g e n e r a l d e c r e a s i n g f u n c -t i o n s o f t h e p l a s t i c s t r a in . A s s u m i n g o 0( 2) = c o n s t . , i t f o l lo w s f r o m (7 ) t h a t t h em a x i m a l v a l u e o f h a r d e n i n g m o d u l u s w il l g r a d u a l l y d e c r e a s e d u r i n g c y c l ic l o a d i n ga n d t h e h y s t e r e s i s lo o p s w il l f l a t t e n b e f o r e t h e s t e a d y s t a t e i s r e a c h e d . To a v o i dt h i s e f fe c t , u s u a l l y n o t o b s e r v e d e x p e r i m e n t a l l y, w e s h a ll i n t r o d u c e a n e w m e a s u r eo f m e m o r y o f p r e v i o u s d e f o r m a t i o n h i s t o r y b y i n t ro d u c i n g t h e m a x i m a l p la s ti cs t r a i n a s a c h a r a c t e r i s t i c r e f e r e n c e v a l u e .

C o n s i d e r a d e f o r m a t i o n p r o c e s s , s a y 0 B , a s s h o w n i n F ig . 4 a . F o r t h i s p r o c e s sb o t h 2, a n d t h e a b s o l u t e v a l u e o f p l a s t i c s t r a i n eap i n c r e a s e , t h u s

2 ep . ~p > O, ~op = ear = eP sp

A n e w l o a d i n g e v e n t i s c h a r a c t e r i z e d b y t h e c h a n g e o f si g n o f i v . T h u s , f o r t il ec o n t i n u a t io n o f t h e p a t hB C 5 w e h a v e

> 0 , ~ v < 0 , % v > 0 . (9)

L e t u s d e n o t e t h e v a l u e o f 2 co r r e sp o n d i n g t o m a x i m u m o feap a t B b y 2 m. A n e wp a r a m e t e r h e w i ll b e i n t r o d u c e d w h i c h is d e f i n e d a s f o ll o w s

~0(k) = ~m 2 - - 2(k -l) (10)w h e r e 2 ( k-~ ) d e n o t e s t h e v a l u e s o f 2 r e a c h e d a t t h e e n d o f k - - 1 l o a d i n g e v e n t .E a c h l o a d i n g e v e n t i s c h a r a c t e r i z e d b y t h e c h a n g e o f s ig n o f t h e r a t e o f di st a r, c ef u n c t i o n

6 = [ 2 (~ .P - -~sP) . (eP- -~sP ) ] 1/2 (11)

w h e r e e~v d e n o t e s t h e v a l u e o f p l a s t i c s t r a i n a t t h e e n d o f t h e p r e c e d i n g l o a d i n ge v e n t . R e f e r r i n g t o F i g . 4 a , i t is s e e n t h a t a l o n g t h e i n it ia l s t r e s s p a t h t h e r e isd0p = e~v, ~0p > 0 a n d t h i s p r o c e s s t e r m i n a t e s o n c e ~ 0p c h a n g e s i t s s ig n . A l o n g t h ep a t h B C w e h a v e

i 2 )2

~BP3Bp - -~ e p - - es p )~P > 0, 2 m = 2(B)

an d e~v = ~Bv, ),~ = )0. I f , how ev er, f ro m C the s t res s p a th rev ers es so th a t 3m~ < 0 ,a n e w d i s t a n c e d c p is c r e a t e d , s u c h t h a t

~c p- - [ 3 e p - - ~ sP ) . eP - - e sP )] 1]2 ~gp > O ,i 3 )

e ~P = c o p , 2 ~ = ~ B ) , , ~ k -1 ) = , ~ C ) ,

a n d 2 c = ~ t(B ) + 2 ( P ) - - 2 ( C . H e r e P d e n o t e s t h e c u r r e n t p o s i t i o n o f t h e s t r e s sp o i n t o n t h e l o a d i n g p a t h .

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218 D. Ku jawski and Z . Mr6z :

F i g . 3 i l lu s t r a t e s t h e v a r i a t i o n o f t h e p a r a m e t e r s ) . a n d 2~ f o r a s p e c i f ie dv a r i a t i o n o f p l a s t ic s t r a i n . W h e r e a s ~ i n c r e a s e s m o n o t o n i c a l l y, t h e p a r a m e t e r ~ cm a y v a r y d i s c o n t i n u o u s l y, g r o w i n g f o r e a c h lo a d i n g e v e n t f r o m i ts i ni ti a l v a l u e

2 = ,~m c o r r e s p o n d i n g t o t h e m a x i m a l p l a s t ic s t r a i n e~P r e a c h e d d u r i n g t h e p a s td e f o r m a t i o n p r o c e s s. I n w h a t f o ll ow s , w e s h al l a s s u m e t h a tc ~ c ) ~ c )w h e r e a sd ~ d ~) , ~ 0 ~ ) = c o n s t ., t h a t i s w e n e g l e c t h a r d e n i n g a n d s o f t e n i n g d u e t o v a r i a t i o nof ao 2 ).

P L ~

X c

X

fl

~ ~ = 6 K = 5=6 K=7

Fig . 3 . Var ia t ion of harden ing param eters ~ and 2c wi th p las t ic s t ra in sp

L e t u s n o w d i s c u s s t h e v i s c o p l a s t ic m o d e l . I n s t e a d o f 1 ), l e t u s a s s u m e n o wt h a t

3 ( s - a - 8 ) ( s - a - 8 ) - ~ 0 ~ ( ~ ) = 0 ( 1 4 )s , a , 8 , : ~ , ; . c ) = ~

w h e r e ~ r e p r e s e n t s t h e v i s c o u s b a c k s t r e s s w h o s e r a t e o f v a r i a t i o n is e x p r e s s e d a sf o l l o w s

~ r = c ~ ~ 8 . ( 1 5 )63

T h e f i rs t t e r m o f 1 5) r e p r e s e n t s t h e h a r d e n i n g e f f e c t w h e r e a s t h e s e c o n d c o r re -s p o n d s t o v i s co u s a n n e a l i n g . W e a s s u m e f u r t h e r t h a t c2 is a m a t e r i a l c o n s t a n tw h e r e a s c3 m a y d e p e n d o n t h e s e c o n d i n v a r i a n t o f t h e p l a s ti c s t r a i n r a t e , t h u s

c2 = co ns t . , c3 ~- c3 ~) . 16)

L e t u s n o t e t h a t w h e n c3 = c o n s t ., t h e r e l a t io n 1 5) c a n b e i n t e g r a t e d t o g iv e

[3 0


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Viscoplast~c lV[aterial Mod el 21 9

a n d t h e r a t e o f fl c a n b e e x p r e s s e d a s f o ll o w s

( )c ~ ' - - c._A~ x p - - c 2 t , O ) + c 2 f ~ P e x p-~3 t d t , (18)% ~ % 0

H o w e v e r , a s i t w i l l b e s h o w n , t h e a s s u m p t i o n 1 6 ) t h a t c 3 d e p e n d s o n p l a s t i c

s t r a i n r a t e p r o v i d e s b e t t e r d e s c r i p t i o n o f v i s c o p l a s t i c m a t e r i a l r e s p o n s e . T h u s , t h e

t h r e e m a t e r i a l f u n c t i o n s c ~ c ~ ) , d - - d ) ~ ) , c a = c 3 ~ )a n d t h e c o n s t a n t p a r a m -

e t e r c s h o u l d b e i d e n t i f i e d f r o m e x p e r i m e n t s b e f o r e t h e p r o p o s e d m o d e l c a n b e

a p p l i e d t o s i m u l a t e i n e l a s t i c d e f o r m a t i o n p r o c e s s e s .L e t u s f ir s t d e r i v e t h e e x p r e s s io n f o r t h e h a r d e n i n g m o d u l u s . L e t u s n o t e t h a t

i n t r o d u c i n g t h e e f f e c t i v e s t r e s s s t a t e s ' = s - - ~ t h a t is t h e s tr e s s a c t i n g o n t h ep l a s t i c e l e m e n t , w e m a y a p p l y th e r e la t io n s ( 1 ) -- (7 ) a n d e x p r e s s b o t h t h e fl o w

r u l e a n d t h e h a r d e n i n g m o d u l u s i n t e r m s o f t h e e f f e c t i v e s t re s s , t h u s1 3

S P ~ - h--- n S . - - ( s - - a - - ~ ) [ ( s - - a - - { t ) . t i ] ( 1 9 )2h'~o2

a n d

h ' = c ( ~ ) - - d ( ~ ) ( s - ~ - , ~ ) . e + _ ~ 2 ~ 0 ' ( X ) . ( 2 0 )(Y0

H o w e v e r , i n o rd e r t o d e t e r m i n e t h e h a r d e n i n g m o d u l u s i n t e r m s o f t o t a l s tr es s,w e s h a l l s t a r t f r o m i ts d e f i n i t io n

h . n _ ~ ( 2 1 )( ~ . ~ v ) ~/ 2 ( ~ . ~ ) ~ I ~_ _

w h i c h l e a d s t o t h e r e l a t i o n

h = h ' -I- ~ (22)% ( s - a - 8 ) , S

1c 3 ( s - ~ - ~ )

a n d

h i 1 d - c ~ ( s - - r - - ~ ) ~ ] ~~ - ,~) d(] t) (s ~ a '3) eP d - c2 d - 2g- (23)

I t i s s e e n t h a t n o w t h e h a r d e n i n g m o d u l u s d e p e n d s o n ~ h e s t r e ss ra t e .A n a l t e r n a t i v e d e r i v a t i o n o f ( 2 3 ) c a n b e o b t a i n e d b y s t a r t i n g f r o m t h e f l o w

r u l e e x p r e s s e d i n t e r m s o f to t a l s t r e s s: t h u s

~v = A n = 1 n ~ . (24)h

w h e r e A i s a s y e t u n s p e c i f i e d m u l t i p l ie r. U s i n g t h e c o n s i s t e n c y c o n d i t i o n a n a l -ogo us to (6 ), we f ind

A = 1 (3 )~ /2 c_~( s - -_a - - ,~ ) -~+c~_ J ( s j - a__~_~)_ .~ (25)C30 0

(Y0

a n d t h e n c e w e o b t a i n t h e e x p r e s s i o n ( 23) fo r h . L e t u s n o t e t h a t n o w t h e a c t i v el o a d i n g c o n d i t i o n / 1 > 0 d o e s n o t r e q u i r e ~ > 0 . T h u s , i f w e p r e s e n t t h e f l o wr u l e in a u s u a l f o r m (3 ) o r (2 4) in t e r m s o f t o t a l r a t h e r t h a n e f f e c t i v e s tr e s s , t h e


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2 2 0 D. Ku ja w sk i and Z . Mrdz :

[o

r

E PB

B I

~

P P

C E C ~ E B

~e=c0n$[.

~ o =

Fig . 4. Trans la t ion of the y ie ld sur face for : a p las t ic a nd b v isco-p las tic mate r ia l

l o a d i n g - u n l o a d i n g c o n d i t io n s a r e r e s p e c t i v e l y m o d i f i e d t h o u g h t h e n o r m a l i t yp r o p e r t y is p r e s e r v e d . F i g . 4 p r e s e n t s t h e y i e l d s u r f a c e a f te r t h e l o a d i n g p a t hO B Cf o r p l a s ti c a n d v i sc o p la s ti c h a r d e n i n g m o d e l s . I t i s s e e n t h a t t h e h a r d e n i n g m o d u l ia t B a n d C a r e d i f f e r e n t a n d a s s u m i n g t h a t ( s - - a - - i~) 9 ~BP > 0 , (S - - a - - ~ ) . 8Cp< 0 i t f o l l o w s f r o m ( 20 ) a n d ( 23 ) t h a th ( C ) > h ( B ) .

3 . U n i a x i a l e a s e : I d e n t i f i c a t i o n o f M a t e r i a l F u n c t i o n s

L e t u s l i m i t o u r d i s c u s s i o n t o t h e u n i a x i a l s t r e s s s t a t e a l o n g t h e X l - d ir e c ti o na n d n e g l e c t t h e d e p e n d e n c e o f a0 o n X . W e s h al l d i sc u s s t h e l o a d i n g p r o g r a m s w i t hc o n s t a n t p l a s t i c s t r a i n r a t e , ~ = e P l = c o n s t . , c a k ) = c o n s t . S i n c e n o w s

1 1 1 .~2 2 = d ~ 33 = - - ~ 1 1 , ~ 2 2 = ~ 3 3 = - - ' ~ f l l l , ~ 2P 2= ~ P 3 = - - - 2 - e l i ,S q . (2 3) p r o v i d e s

( 1 + o-:~~/811,= ~ 1 7 6 _ 3 y d ( x ) ( ~ - ~ - ~ o ~ ) ~ + ~ (

a n d f r o m ( 1 7 ) i t f o l l o w s t h a t

fln = % ~ l [ 1 - - e x p ( - - t ) ] . (27)

S i n ce t h e f lo w r u le n o w t a k e s t h e f o r m

~Pl = ~ el i (28)


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A Viscoplastic M aterial Mo del 221

t h e h a r d e n i n g m o d u l u s i n ( 2 6 ) c a n b e e x p r e s s e d a s fo l lo w s

h = c , t c ) - 3 d , ~ ) 8 1 1 -anao-fi11)SlPl ~ c2 ex p ( - - ~-3c2 ). (29)

L e t u s n o t e t h a t f o r ePl = c o n s t . - > ~ , t h e l a s t t e r m i n ( 2 9 ) e q u a l s

l im c2 exp ( --c--~ t ) -~ c 2 (30)t-->O C3

an d fo r s f l = cons t . -> 0 ( s t a t i c case ) , the re is

c2 exp c~ t ) - -> O . (31)irat-~co \ ca ]

T h u s , f o r t h e s t a t i c c a s e t h e h a r d e n i n g m o d u l u s is e x p r e s s e d i d e n t i c a l l y a s fo r t h ep l a s t i c m o d e l

h = c(~c) -- 3 4(2) (sn -- ~n ) s~1 (32)2 a 0

a n d s i n c e3~ I811 - - ~ 11[ : 'frO (33)

t h e r e l a t i o n ( 3 2 ) p r o v i d e s t h e e x p r e s s i o n f o r t h e t a n g e n t m o d u l u s o f t h e s t r e s s -p las t i c s t ra in Curve

d a n 3 h ~ 3

K t = d~lP - - ~ - -~ [c()~c) :~: d(/~) elPi] (34 )w h e r e - - a n d - ~ s ig n s r e f e r t o l o a d in g a n d u n l o a d i n g p o r ti o n s. I n F i g . 5 s u c hl o a d i n g - u n lo a d i n g p o r t i o n s a r e s h o w n t o g e t h e r w i t h t h e c o r re s p o n d in g v a r i a t io no f t h e t a n g e n t m o d u l i , w h ic h c a n b e d e t e r m i n e d e x p e r i m e n t a l l y f r o m a n y t e n s io n -c o m p r e s s i o n t e s t . I n f a c t , a l o n g t h e l o a d i n g a n d r e v e r s e l o a d i n g c u r v e s t h e r e i s

3 3K L = -~- [c(),c) -- d(2) ~] , KR = ~ - [c(2c) + d(2) (2ec ~ -- Z)] (35)

since ep = 2 s c P - - ~a l o n g C D . F r o m (3 5) i t fo l lo w s t h a t

d ,~ ) = K R - - K L (36)eVP

E q s . ( 3 5 ) a n d ( 3 6 ) c a n b e u s e d t o d e t e r m i n e t h e f u n c t i o n s c(2 ~) a n dd )O .

(a) (bl

Fig. 5. a loading-reverse loading curve, b variation of tan ge nt mo dulus


10/18

2 2 2 D . K u j a w s k i a n d Z , M r 6 z :

F o r a f i n i t e v a l u e o f if1 t h e t a n g e n t m o d u l u s c a n b e o b t a i n e d f r o m (2 9) ,n a m e l y

K ~ ~ =_ 2_ 3 c ( 2 ~ ) - - d 0 0 s ~ l + c 2 e x p - - t -----K L + - ~ - c ~ e x p - - ~ t (3 7)

a n d b o t h s t a t i c ( iP l - -> 0 ) a n d d y n a m i c ( i p = c o n s t. ) s t r e s s - s tr a i n c u r v e sl ia r e

a~ = [c0.~) - - d(Z) en ]d e n ,

0 3 s )

C2~ = ~ ~

a n d '

~-C3~113t) [ ( c2 ) ]~ --- - ~v - - o~ - -- - 1 - - e x p - - -~-3 t l (39)

w h e r e z ]~ d e n o t e s t h e d i f f e r e n c e b e t w e e n d y n a m i c a n d s t a t i c s t r e s s f o r t h e s a m ev a l u e o f p l a s t i c s t ra i n . D e n o t i n g b y v =-Ca/C2t h e r e l a x a t i o n t i m e f o r g i v e n i~1,i t i s s e e n t h a t f o r t ~ r t h e r e i s

e x p - + 0 , A ~ - + ~ -

a n d t h e m a t e r i a l f u n c t i o n c3 ---- c a(i~ l) c a n b e e x p r e s s e d i n t h e f o r m

3 Aoc a ( e P l ) - 2 ~ i ( 41 )

U s i n g ( 39 ) w i t h ~ =c3/c~, t -= soP/ i~l ,f o r t = T w e o b t a i n

A ~ = - A a t = ,= 3 ca ~ l ( 1 e_ i ) = O.632A(r. (42)Z

A5

A61

F i g . 6. D y n a m i c a n d s t a t i c s t r e s s d if f e r en c e A o v e r s u s p l a s t i c s t r a i n

F i g . 6 p r e s e n t s t h e d i a g r a m o f A a = / ( e ~ l ) f o r c o n s t a n t p l a s t i c s t r a i n r a t e .F r o m t h is d i a g r a m w e d e t e r m i n e A a~ a n d s p a n d n e x t t h e r e l a x a t i o n t i m e

= - e~P/~ . T h e m a t e r i a l p a r a m e t e r c2 c a n n o w b e c a l c u l a t e d f r o m t h e r e l a t i o n

c2 ---- c~(~p). (43 )T

C o n s i d e r , f o r i n s t a n c e , t h e c a s e o f t e n s i o n w i t h c o n s t a n t p l a s t i c s t r a i n r a t e ~P a n da s s u m e t h a t p r o c e s s t e r m i n a t e s a t t h e s t r a i n ep = s0p , F i g . 7 . W h e n c , d , c~ a n d c3


11/18

A Viscoplastic ~ at er ia l YiOdel 223

a r e c o n s t a n t , t h e s t r e s s - s tr a i n c u r v e c a n b e e x p l i c it l y e x p r e s s e d i n t e r m s o f p la s t ics tr a in . T h e t a n g e n t m o d u l u s n o w e q u a ls

3 [ , ( " c~c---2t)] (44)~ - - ~ c - - deP 4 - c2 ex p

a n d

= a o 4 - f K d e ~ = a o - ~ - ~ C eo '~ --d + c ~e P~ 1 - - e xp - - (45)0

w h e r e T -~ .eoP/~oa n d f o r p a r t i c u l a r s t r a i n ~ a t e s . t h e f i n a l t i m e T. e q u a l s r e s p e c-1t i v e l y -~ ~, v, 2v,an d ~ . F ig . 7 b shows the cor resp ond ing s t ress - s t ra in . curves .

la

P ~ . --- -

9 i

Fig. 7. a Constant plastic strain rate loading programmes and b respective stress-strainc u r v e s

C o n s i d e r n o w t h e c y c li c l o a d i n g p r og r a 'm f o r p l a s t ic s t r a i n v a r y i n g b e t w e e n 0

and sop w i t h t h e c o n s t a n t s t r a i n r a t e ~P. U s i n g (1 7), w e c a n e x p r e s s t h e v a r i a t i o nof f l( /c ) du r ing k - th semicyc le

fi(k ) = % (i P )~ P ex p ( - - ~ ) [ fl'(k-1)c~k--4 - e x p \ ~k / ( @ / - - 1 ]

w h e r e f l ' (k - -1) d e n o t e s t h e v a l u e o f f l ( k - 1) a t t h e e n d o f s e m i cy c l e k - 1a n d ~ ,v is a s s u m e d p o s i t iv e f o r t e n s i o n a n d n e g a t i v e f o r c o m p r e s s i o n ;t kd e n o t e st h e t i m e m e a s u r e d f r o m t h e b e g i n n i n g o f t h e k - t h s e m ic y cl e. T h e t a n g e n t m o d u l u sd u r i n g t h e k - t h s e m i c y c l e c a n b e c a l c u l a t e d f r o m (2 6), n a m e l y

,]/(]c) : (~ ) T : d(~) ekp 4- ca T~vc~\ca / ' 4- ex p -- ~ ctk' (47)a n d t h e - - s ig n r e f e r s t o c o m p r e s s i o n a n d 4 - t o t e n s i o n . F o r t h e f i r s t s e m i-cy cle, th e re is fl '(k ~--:1)--~ fl'(0) ---- 0.


12/18

224 D. Kujawski and Z. Mr6z:

4. Identification Tests

In the preceding section, we indicated how material functions can be identified

from simple uniaxial tests. Now, let us discuss this procedure for a constructionsteel (St3S, carbon content 0.16~o), understanding that it can be applied to anyother material as well. This identification of material functions constitutes a firststep in quantitative testing of our model. Whereas identification tests should beas simple as possible, we next apply our model to predict material behaviour formore complex loading programs. This verification step provides additional

,5~07[N,~244

4 0 -

t e e [ S t 3 S 3 ~ -32 -

2 B -

1 ~ 4 2 5 , 6 i ' ~ o

. . . . , 2 . . . . ~ . . . . 4 2 ~ 1 8 . . . . 4 8 -

8

-12

-16

-20

-24o - co~vJP ~tionQ [ur ~'-28 9 ernpiricGI urve

32

-36

-40

- 4 4

Fig. 8. Stress-strain curve for construction steel: neglect of the perfectly plastic segment

information which can be used in improving our identification step. In order todete rmine the material function s c = c(),c) and d ~ d(2) the "st at ic" tension-compression test s were carr ied out with the ra te ~P --- 6.96 9 10 -~ s -1. Since theactual stress-strain curve is characterized by a horizontal portion before strainhardening occurs, this perfectly-plastic segment was neglected and the hardeningportion was extrapolated backward, Fig. 8. ]in fact, during unloading and sub-sequent cycles this segment disappears and t he material exhibits a hardeningstress-strain curve. The corresponding variation of tange nt modulus on t heloading and reloading c urves is shown in Fig. 9. The func tion d = d(2) is deter-


13/18

A Viscoplastic Material Model 225

mined from the formula 36). The experimental data are approx imat ed by theanalytical expression

d 2) = A exp B2) 48)

where the values A = 26460, B =- - 36 .3 9 are obtained through the meansquare minimiz ation process. Similarly, c = c 2c) is app rox ima te d by a n expo-nentia l func tion 48) with par ame ter s A = 533.87, B = --16.39.

6 -

2

S~ee t S t3S

K ~

1

4 6 8 l; li 1=;~.~ z

Fig. 9. Variation of tangent moduli on loading and reverse loading portions

In order to determine the function c a = ca iP) and th e par am et er c2, the spec-imens were ini tiall y deformed byeo nd the perfe ctl y plastic s egme nt e~n ----- 0.016 7)and unloaded. The y were subsequently defor med with c onstant ra te of elongationand the plastic strain rate was calculated by accounting for the elastic straincomponent. The resulting curves are shown in Fig. 10. The material function wasapproximated by the formula

c3 = a ~p)~ 49)

and th e parame ter s a = 8.298, b = --0 .9475 were identifie d using a log-log plotof 49) repr esen ted b y a straight line. The param et er c2 was evalua ted asc2 = 516 kg/mm ~.

We have thus identified the material functions by using only monotonic orloading-reverse loading tests. In the next section, we shall discuss the materialresponse under cyclic loading and provide the verification of the pr esente d modelfor more complex loading programmes.


14/18

2 2 6 D . K u j a w s k i a n d Z . M r 6z :

.1C~iN/m ] S le e t S t 3 S _ . ~0--~-7_0- -~

5 5 ' s - = -5 -3 8 1 0 ~ : ~ 1 -

I o . . '

viscoplastic model

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