mechanical, thermal and physical coupling methods in fe analysis of metal forming processes

8/6/2019 Mechanical, Thermal and Physical Coupling Methods in Fe Analysis of Metal Forming Processes

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JournalofMaterialsProcessingTechnologyE L S E V I E R Journal of Materials ProcessingTechnology60 (1996) 11-18

Mechanical, thermal and physical coupling m ethods in FE analysis o f me tal forming processesJ.-L. Chenot, Y. ChastelCEM EF, Ecole des Mines de Paris, URA CN RS 1374, BP 207, 06904 Sophia-Antipolis, France

A b s t r a c tThe thermal and mechanica l equations for large deformations occuring in metal forming processes are recalled. The finite elementapproaches for viscoplatic or for elastic viscoplastic materials are presented briefly. The coupling of the previous equations with thosedescribing the evo lution of physical internal parameters is analysed. A compact form of the physical and mechan ical equation is used toconsider any time integration method. The micro-macro approach is an alternate way to introduce physical phenomena Thismethodology is illustrated with the problem of texture evolution during forming of anisotropic materials. Finally the feasability of theinverse method for various types o f coupling is investigated, with application to the computat ion of parameters defin ing the mechanical,thermal and physical behav iour of the material.Keywords: Finite Element, Me cha nic al Ther mal Ph ysic al Texture evolution, Inverse meth od

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

While the mechanical engineer studies the macroscopic stressdistributions and the material flows, the metallurgist usuallyconcentrates on microstructural evolutions due to thethermomechanical treatments imposed to the metal as it isprocessed. But solving complex boundary value problemsinvolving large deformations oft en requires to couple the effectsof deformations on the metallurgy as well as that of themicrostructure on the mechanical response of the metal.Constitutive models are then needed to describe the mechanicalbehavior of metals as a function of microstructural statevariables. Comb inin g these layers of models increases thecomputat ional costs. Efficient treatments for full-sizerepresentations of forming operations can be obtained using finiteelement analys is on t oday' s computer architectures.We describe here a thermal and mechanical finite elementformulation meant for metal forming processes. Two approachesare then presented to take into account microstructural features ofthe metals within the finite element framework:when analytic expressions can be derived to relate statevariables and constitutive laws, the solution scheme for thecoupled problem with the finite element discretization consists inthe determination of the velovi ty field and the integration of themicrostructure kinetics [1-2];0924-0136/96/$15.00 1996 ElsevierScicace S.A. All rights reservedPI10924-0136 (96) 02302-3

for predicting the evolution of anisotropic properties throughthe deformation history, a constitutive framework is provided bythe polycrystalline plasticity theory. Such models can also beintroduced in the finite element analysis [3-4] to perform realisticsimulations.

2 . T h e r m a l a n d m e c h a n i c a l f i n it e e l e m e n t m o d e l l i n g o f metalf o r m i n g p r o c e s se s2.1 Finite element formulation fo r viscoplastic materialsThe general approach for viscoplastic materials is described indetails in [5]. When elasticity is neglected and the material isassumed incompressible, the isotropic constitutive equation willbe written: = c(e, ~, T) ~: with tr(~) = div(v) = 0 (1)where O" is the deviatoric stress tensor, ~ is the strain rate tensor,

the equivalen t strain and e the equivalent strain rate. If th ematerial is anisotropic, the scalar c can be replaced for exampleby a fourth rank tensor c. A viscoplastic friction law can beexpressed by:


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12 J.-L. Chenot, Y Chastel Journal of Materh~s Processing Technology 60 (1996) 11-18"~ = - c~(cn, [Av l, T) Av (2)w her e fin i s t he no r ma l s t r e s s, A v t he tangen t i a l ve l oc i t ydi f ference between the par t and the tool. For anisot ropic f r ic t ion,the scalar ~ is replaced by a tensor.

The virtual work principle is stated in the rate form, with a mixedformulat ion:

I f O ' : ~ * d V - I 3 g l c X . v * d S - I p d iv ( v* ) d V = 0

- f f 2 p*d i v ( v ) dV = 0

The heat equat ion i s wr i tt en:

pc T = div(kg rad(T )) + r s :1~

(3 )

(4 )where p i s the m ater ia l dens i ty , c i ts heat capac i ty , k the di f fus ioncoe f f i c i en t and r r ep r e sen t s t he amou n t o f v i s cop l a st i c ene r gywh ich is con ver te d into heat . Du e to thermal di la ta t ion wi thcoef f ic ient tx , the incompress ibi l i ty cont r ibut ion in the integra lformulat ion (3) i s replaced by:

w hi l e t he p r e s su r e f i e l d i s expr e s sed w i t h compa t i b l e shapefunct ions Mm:P = E P m M m ( 9 )

m

W ith these nota t ions (3) i s d i scre tized acco rding to , for any n andm :

. I ( 3 ' : B , * d V - f 3 ~ x N n d S ' f f p tr(B n) d V = O(lO)

- f~ Mm (div(v) - 3eq T) dV = 0

In the same w ay the thermal equat ion i s d i scre t ized w i th the helpof the Galerkin method b y replacing w = N n, in (6); we obta in:C . T + H . T + F = 0 ( 11 )where C i s the capaci ty mat r ix , H the heat conduct ivi ty mat r ixand F conta ins the other cont r ibut ions . Final ly (10) and (11) canbe rewrit ten sym bolic al ly as:

R ( X , V , P , T , ' l; ) = 0 w i t h d._X _X Vd t (12)

- f ~ p* ( d i v ( v ) - 3a , " i M )V = 0The integra l form o f the heat equat ion i s general ly wr i t ten:

(5 )Y (X , V , T , "F) = 0 wi th O__TT -i~d tT hese equa t i ons can be i n t eg r a t ed w i t h r e spec t t o t i me us i ngv a r i o u s n u m e r i c a l t e c h n i q u e s , t h e m o s t p o p u l a r f o r th emechan i ca l equa t i ons be i ng t he one s t ep exp l ic i t E u l e r s chemebut, in general, i t is not sat isfacto ry for the therm al equations.

I t ) ( PCd~ - - r s : ~ ) w dV + I kg r ad ( T ) .g r ad ( w ) dV

+IOD ~swdS = 0(6 )

where w i s a weight ing funct ion and ~s i s the imposed boundaryheat f lux.

2.2 Finite eleme nt formu lation fo r elastic viscoplastic materialsThe veloci ty approach for e las t ic vi scoplas t ic mater ia l s i s a l sorepor ted in [6]. Here w e s ti ll in t roduce the ve loci ty fie ld v as thema i n unknow n. T he add i ti ve s t ra i n ra t e decompo s i t ion i n to anelast ic (e) and a viscoplast ic (p) contribution is assumed:

T he ve l oc i t y f i e l d i s d i c r e t i zed us i ng i sopa r ame t r i c f i n i t ee l emen t s , de f i ned o n a m esh w i t h noda l coor d i na t e s X n , w i thshape funct ions N n and n odal vectors V n, so that :

~ = ~ + ~pthe e las tic p ar t obeying a ra te form of the Ho oke law:

(13)

V ---- E V n Nn w i t h x = ~ ' X . N n ( 7 )n n

and the B operator i s def ined by:= ~ B . v . ( 8 )

n

t ~ = ~ . 0e l + 2 B I ~ = D :I ~e ( 1 4 )

W hen mater ia l rota t ions cann ot be cons ide red as negl igible , theJaum an de r i va t i ve is i n t r oduced i n ( 14 ) . T he v i s co l a s t i ccontribu tion to the strain rate is such that:~P = 0 if f(ff) < 0 or (f(ff) < 0 and f(t~) < 0) (15)


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J.-L. Chenot, Y Chastel / Journal of Materials Processing Technology 60 (1996) 11 -18 13

~ P = ~ p O - ~ f f ) w i t h A 3 , p > 0 i f f (( ~ ( ~ ) = 0 a n d f ( f f ) > 0 ( 16 )0 1 1

w h e r e f i s t h e y i e l d c r i t e r i o n a n d ( p' i s t h e c o m p l e m e n t a r yv i s c o p l a s t i c p o t e n t i a l. T h e h e a t e q u a t i o n h a s a n a d d i t i o n a l t e r m(w hich i s o f t en neg l ec t ed ) :

The e vo lu t i on equa t i ons fo r t he se va r i ab l e s a r e :a ) T h e c o n s t i t u t i v e l a w o f t h e m a t e r i a l e x p r e s s e d e x p l i c i t l y o rim p l i c i t l y . I f~ i s the s t r a in r a te , w e w r i t e :

= h ( a , T , s ) ( 1 9 )

p c d T = d i v ( k g r a d ( T ) ) + o : ~ p + T ~--~--:1~d t O T (17)

A s im i l a r p rocedure a s t ha t u sed 2 . 1 can be fo l l ow ed t o ob t a in thed i s c r e t i z e d fo r m w h i c h i s a n a l o g o u s t o t h e p r e v i o u s o n e , e x c e p tf o r t h e s t r e ss t e n s o r w h i c h m u s t b e i n t r o d u c e d s o t h a t e q u a t i o n s( 1 2 ) m u s t b e r e p l a c e d b y :

b ) The equ a t i ons fo r t he evo lu t i on o f t he s t ruc tu r e va r i ab l e s s:= X ( o , T, s ) (20)

c ) T h e h e a t e q u a t i o n i s d e r i v e d f r o m t h e r m o d y n a m i c s o fcon t i nuum m ed ia ( s ee [7 ] ):

R( X , V , a , T ,q~) = 0 w ith d_._XX Vd tp c + = d iv (k g r a d ( T ) ) + o : e P - F s.s + T ( O : ~ e + / ) F ' .s ' ) ( 21 )3 T 3 T

Y ( X , V , i f , T , " i ' ) = 0 w i th d--T-T 'i ~d t

d___~_~ ~d t

(18)

G e n e r a l l y t h e c o m p u t a t i o n o f t ~ c a n b e d o n e a t e l e m e n t l e v e l a n dr e q u i r e s a g l o b a l r e s o l u t i o n p r o c e d u r e o n l y w h e n a m i x e dfo rm ula t i on i s u sed .

w h e r e F s i s t h e t h e r m o d y n a m i c f o r c e d e f i n e d f r o m t h e f r e eene rgy func t i on ~ b y :

Fs = O 0 _ V~ s (22 )

A s an ex am p le , t he cons t i t u t ive r e l a t i onsh ip can be w r i t t en , u s inga pow er l aw w i th a s ca l a r m ic ros t ruc tu r a l va r i ab l e s :

G~q = ~o e e P~ RT /" (23 )

3 . P h y s i c a l c o u p l i n g w i t h i n t e r n a l v a r i a b l e s3 . 1 G e n e r a l f o r m u l a t i o nA s e t o f v a r i a b l e s h a s t o b e i d e n t i f i e d t o d e s c r i b e t h em i c r o s t r u c t r a l s t a t e a n d t h e m e c h a n i c a l b e h a v i o r o f t h e m e t a l .E a c h v a r i a b l e n e e d s t o b e a t r u e s t a t e v a r i a b l e w h i c h c a n b ee x p e r i m e n t a l l y m e a s u r e d o r e s t im a t e d u s i n g i n v e r se m e t h o d s .T h e a c t u a l s tr u c t u re o f m e t a l s i s v e r y c o m p l e x . T h e d e f o r m a t i o nm e c h a n i s m s d e p e n d a p r i o r i o n p h y s i c a l f e a t u r e s s u c h a s t h eg ra in o r d i s l oca t i on ce l l s i ze and shape , t he dens i t y o f c r i s ta l l i ned e f e c t s , t h e p o r o s i t y l e v e l , t h e n a t u r e a n d p r o p o r t i o n s o f t h ev a r i o u s p h a s e s , o r t h e c r y s t a l l o g r a p h i c t e x t u re s . T h e d o m i n a n te f f e c ts f o r a g i v e n i n v e s t i g a ti o n m u s t b e i d e n t i f ie d a n d d e s c r i b e di n t e r m s o f t h e r e l e v a n t m i c r o s t r u c t u r a l v a r i a b l e s . I n a f i r s ta p p r o a c h , t h e w e l l - k n o w n c o r r e l a t i o n b e t w e e n t h e d i s l o c a t i o nd e n s i t y o r t h e g r a i n s i z e a n d t h e f l o w s t r e s s c a n b e u s e d t o m o d e lm ech an i ca l e f f e t c s u s ing s ca l a r va r iab l e s .I n a g e n e r a l m a n n e r , o n e c a n c h o o s e t h e f o l l o w i n g s t a te v a r i a b l e s

t~ t he C au chy s t r e s s t enso r ,T t he t em pera tu r e ,s t h e s e t o f v a r i a b l e s c h o s e n t o d e s c r i b e a n d p a r a m e t r i z e th em ic ros t ruc tu r e .

and t he evo lu t i on o f t he s t ruc tu r e va r i ab l e can b e g iven a s [8 ] :

s = ho (1 - -~- I ~ (24 )S /

w h e r e s * i s a f u n c t i o n o f t h e Z e n e r - H o l l o m o n p a r a m e t e r Z :

s ' = h 3 Z q w it h Z = e . e x p ( Q I~ R T/ (25 )

A p o p u l a r e x a m p l e o f m i c r o s t m c t u r a l p a r a m e t e r is t h e g r a i n s i z eD . I f E l i s a p o w e r l a w o f D , w e o b t a in :

w i th t he c l a s s i ca l i n t e rp re ta t i on :m s = ( ~ / a n d m = m s + p ~/~ ln D --

31n ~ ]D.T ~31n e 13. ( 2 7 )


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14 J.-L. Chenot, Y. Chastel dournal of Materials Processing Technology 60 (1996) 11-183.2 FE discretization and time integrationT h e m i c r o s t r u c t u r e v e c t o r i s d i s c r e t i z e d i n t h e u s u a l w a y s o t h a t Sw i l l r e p r e s e n t t h e n o d a l v a l u e s o f s. F o r a v i s c o p l a s t i c m a t e r i a l ,e q u a t i o n s ( 1 2 ) a re r e p l a c e d b y :

R ( X , V , P , T , 'F , S ) = 0 w i t h d X = V (28a)d tY ( X , V , T , ' i ' , S ) = 0 w i t h d _ _ T T ~i" (28b)d t

T o w h i c h w e a d d t h e d i sc r e t iz e d f o r m o f e q u a t i o n w h i c h c a n b er e w r i t t e n a s :

9 9 9 . ~ . t B~ '

7464~

= Z ( V , T , S ) a n d d S = S ( 2 8 c )d t

A t t h i s s t a g e , i t i s p o s s i b l e t o c o n s i d e r a n y t i m e i n t e g r a t i o ns c h e m e t o s o l v e e q u a t i o n s ( 2 5 a , b , c ) s i m u l t a n e o u s l y o r i n as e q u e n c e . I t i s i n t e r e s t i n g to n o t e t h a t i n g e n e r a l th e r e i s n od i f f u s i o n c o n t r i b u t i o n t o ( 2 5 c ) s o t h a t t h e r e i s n o n e e d t o s o l v e i tg l o b a l l y , e v e n w i t h a n i m p l i c i t s c h em e .T h e r e m e s h i n g p r o c e d u r e , w h i c h i s c o m p u l s o r y f o r m a n yf o r m i n g p r o c e s s e s s i m u l a t i o n , c a n b e d o n e i n t h e s t a n d a r d w a y .A n A L E f o r m u l a t i o n c a n a l s o b e i n t r o d u c e d i n o r d e r t o d e l a yl a r g e m e s h d i s t o r s i o n s , t h e s t r u c t u r a l p a r a m e t e r s S c a n b e t r e a t e das i t i s descr ibed in [9 ] .A v e r y s i m i l a r f o r m c a n b e o b t a i n e d w i t h t h e e l a s t ic v i s c o p l a s t i cm a t e r i a l a n d a d i s p l a c e m e n t fo r m u l a t i o n , d i ff e r e n c e .

3.3 ApplicationW e a p p l y t h e g e n e r a l f o r m u l a t i o n w i t h i n t e r n a l v a r i a b l e s to t h ep r e d i c t i o n o f m i c r o s t r u c t u r e s a f t e r e x t r u s i o n [ 8 ]. T h e g o a l i s t oi n v e s t i g a t e th e g r a i n s i z e e v o l u t i o n d u r i n g t h e f o r m i n g p r o c e s s.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 t h e m a t e r i a l to b e e x t r u d e d i sd e s c r i b e d b y a v i s c o p l a s t i c N o r t o n - H o f f l a w a n d t h e e v o l u t i o nl a w f o r t h e g r a i n s i z e i s g i v e n b y :

l ~ = h o Z ~ ( 1 ___D_D) w i t h h 0 = h l e x p ( h 2 Z m )D * (29)D * r e p r e s e n t s a s a t u r a t i o n v a l u e f o r t h e g r a i n s i z e a n d i s a l s o af u n c t i o n o f Z , th e Z e n e r - H o l l o m o n p ar a m e te r W h e n i n t r o d u c e di n a th r e e - d i m e n s i o n a l f in i t e e l e m e n t f o r m u l a t i o n s u c h a s F o r g e 3o n e c a n m o d e l t h e e x t r u s i o n o f a r e c t a n g u l a r b i ll e t t h r o u g h ac y l i n d r i c a l d i e .

F i g u r e 1 : G r a i n s i ze e v o l u t i o n d u r i n g 3 - D e x t r u s i o n

D u e t o t h e f r i c t i o n w i t h t h e t o o l , l a r g e s t r a i n ra t e s a r e r e a c h e d i nt h e o u t e r l a y e r s a n d , g i v e n t h e e v o l u t i o n l a w , t h i s l e a d s t o m o r ep r o n o u n c e d c h a n g e s i n t h e g r a i n s i z e a t t h e s u r f a c e o f t h e b i l l e t(F igure 1 ) .

4 . P h y si c a l c o u p l in g w i t h m i c r o - m a c r o d e s c r i p t i o n s

4.1 Prediction of texture with classical polycrystall ine models4.1.1 Polycristalline models

I n t h e l a s t d e c a d e , p o l y c r y s t a l l i n e p l a s t ic i t y h a s b e e n a f i e l d o fi n t e n s i v e re s e a r c h a n d d e v e l o p m e n t . I t a ll o w s t o t a k e i n t o a c c o u n te x p l i c i t l y t h e p o l y c r y s t a l l i n e n a t u r e o f t h e m e t a l s . T h e g o a l i s t or e l a t e t h e m o d e s o f p l a s t i c d e f o r m a t i o n a t t h e s l i p s y s t e m l e v e lw i t h i n th e g r a i n s t o t h e m a t e r ia l f l o w o b s e r v e d a n d m e a s u r e d a tt h e m a c r o s c o p i c l e v e l. T h e o r i e n t a t i o n o f a g r a i n , g c , c a n b ed e s c r i b e d b y t h r e e s c a l a r v a r i a b l e s s u c h a s E u l e r a n g l e s .E x p e r i m e n t a l l y th e s e o r i e n t a t i o n s a r e m e a s u r e d g l o b a l l y o n as a m p l e b y X - r a y d i f f r a c t io n , o r g r a i n b y g r a i n t h a n k s t o E l e c t r o nB a c k S c a t t e re d P a t t e r n s a n a l y s i s . T h e v o l u m e r a t i o o f e a c ho r i e n t a t i o n i n t h e m a t e r i a l i s s t a t i s t i c a ll y q u a n t i f i e d b y a f f e c t i n g aw e i g h t w c t o e a c h o r i e n t a t i o n g c . T h i s r e l a t i v e w e i g h t i s t o b eu s e d a s a w e i g h i n g f a c to r f o r c a l c u l a t i n g t h e a v e r a g e m e c h a n i c a lp r o p e r t i e s o f t h e p o l y c r y s t a l l i n e e n s e m b l e .F o r m a n y o p e r a t i o n s o n l y g l i d e o n s l i p s y s t e m s o r t w i n n i n ga c c o m o d a t e t h e d e f o r m a t i o n a n d t h e o t h e r d e f o r m a t i o n m o d e ss u c h a s g r a i n b o u n d a r y s l i d i n g o r d i f fu s i o n m e c h a n i s m s c a n b en e g l e c t e d .T h e c o n s t i t u t i v e l a w i s t h e n d e f i n e d a t t h e s l i p s y s t e ml e v e l. F o r a s li p s y s te m ~ t h e v i s c o p l a s ti c b e h a v i o r c a n b ed e s c r i b e d a s a p o w e r - l a w r e l a t i n g t h e a p p l i e d e f f o r t o n ac r y s t a l l o g r a p h i c p l a n e ( t h e r e s o l v e d s h e a r s t r e s s xa ) a n d t h e s h e a r

. . ,{ ~tr a t e i n t h e s h p d l r e c t m n T :


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J.-L. Chenot, Y. Chastel / Jour nal of Materials ProcessLng Technology 60 (1996) 11 -18 15

a (30)

where x is a stress threshold and a is a rigidity factor.Work hardening or softening is given by the evolution of x , or ofan average hardness over the grain, with the deformation, thestrain rate and the temperature. For each grain, a set of four statevariables are needed, an average hardness over the grain andthree Euler angles. The relationship between the deviatoric stressand the deviatoric strain rate of a grain can be written as:o 'c = M cfic (31 a)

and can be introduced directly in the principle of virtual work andin the finite element formulation to capture the mechanicalresponse of the polycrystal contained in the element. Theconstitutive law defined at the slip system level beingviscoplastic, the matrix M is itsel f a funct ion of the local strainrate. The mechanical response of each finite eleme nt will bedescribed by the behavior of the polycrystalline aggregate itcontains. The finite elemnt formulation will then connect allpolycrystalline ensembles to solve the global bou ndary valueproblem. The integration in time of the microstructural andtexture evolution laws as the deformation proceeds willautomatically update the evolving anisotropy of thepolycrystalline material and allow for the simulation of largestrain forming processes.

with:.6

(31b)

where is the Schmid tensor descr ibing the geometry.The next step is to calculate the global mechanical response of apolycrystal knowing that of each of its cons tituent grains. Variouspolycrystalline models can be used to perform this length scaletransition. The bound theorems provide a range for the flowstress of the polycristal over a strain increment. The Taylorassumption gives an upper bound by assuming a uniform strainrate within the polycrysta l i.e. equal for all grains. A static modelwhich imposes a uniform stress field gives in turn a lower bound.Another approach starts from the equili brium and compatibilityequations and consider all grains in turn as ellipsoidal domainssurrounded by equival ent matrix. This forms the basis of theself-consistent class o f models.The mechanical responses of all grains are then averaged. Thequality of these various estimates depends on the structure of themetal itself (its crystallographic structure, the presence ofprecipitates, etc.) and on the thermomechanical parameters. In theTaylor hypothesis, the deformations are homogeneous and equalto the macroscopic deformation ~ applied to the polycrystal:~c = k (32)The deviatoric macroscopic stress, ~ ', can be written as:

4 .1 .3 A p p l i c a t i o nAlum inium profiles are typically produced by extrusion. Theseoperations and the subsequent heat treatments induce signi ficanttexture developments in the product. Due to its heterogeneouspolycrystalline nature and its pronouced crystallographic texturethe material behavior is anisotropic. The bend ing and shapingoperations of theses profiles involv e large local deformations. Tostudy these operations we apply the polycrystal line finite elementformulation extended to an elastic viscoplastic law [11]. Theinitial texture is measured throughout the part and is found to be astrong Cube texture. The effect of texture-induced anisotropy isfound critical to model the geometry of the final cross-section inthe area of large strains (Figure 2).

w , M ( xo o , , = E c . c =c c

(33) I I . I S E - 02 I

Figure 2 : Profile after bending

4.1.2 Introduction in a FE simulation codeThe stiffness matrix M for the polycrystalline volume is gi~,en bythe average over all the grains within the polycrystal:M = ~ wcMc (34)

4 .2 . F i n i t e e le m e n t m o d e l o f a n a g g r e g a t eIn classical polycrystalline plasticity the average equi valentmaterial behavior for an aggregate is calculated using varioushomogeneizing theories. Finite element methods have been usedto discretize the grains of an aggregate into finite elements and tosolve boundary value problems for the whole multicrystal


6/8

16 J.-L. Chenot, Y Chastel Journal of Materials Processing Technology 60 (1996) 11-18ensem ble (e .g . [4] , [12]) . The l ink betwee n the microsco pic scaleof the gra in and the m acrosc opic lengthscale of the polycrys ta l isthen di rect ly obta ined f rom the f ini te e lement analysi s. The m eshof each g r a i n can be r e f i ned t o a l l ow f o r i n tr ag r anu l a r s t ra i nhe t e r ogene i t ie s an d t he deve l op men t o f l oca l shear bands . I nparallel , loc al texture , stress and strain me asure me nt techn iqueshave i mpr oved and expe r i men t s can be pe r f o r med on pa r t icu l a rsamples to v al idate th is m ul t icrys ta l fin ite e lem ent mo del .

a V 3 qa Y a Va V 3 q

3 R 3 V 3 R 3 T 3 R a X 3 R O T d a X 3 V_ _ _ _ .+ - - . - - = - _ _ . . ; - -a T a q 3 X a q ~ T a q d t a q a q3 Y a T 3 Y 3 X 3 Y a T d a T a T3 T 3 q a X 3 q a T a q d t a q a q

E q u a t i o n s ( 3 8 )aT aT- - a n d - - .3 q a q

c a n b e s o l v e d n u m e r ic a l ly f o r - -

(38)aX 3Va q a q

5 . F e a s i b i l i t y o f t h e i n v e r s e m e t h o d

5.1 The inverse method applied to constitutive parametersT he p r ob l em o f de t e r mi n i ng t he cons t i t u t i ve pa r ame t e r s by t heinverse method , appl ied to the F. E . mo del l ing of a l aboratory tes tinvolv ing large or very large s tra ins , was ad ressed to by severa lauthors e .g . [13~ 14] . I t can be summ ar ized in the fol lowing wayfor a vi scoplas t ic mater ia l : l e t be q a se t of parameters descr ibingthe cons t i tu t ive equat ion. The n equat ions (12) mu st be rewr it tenas :

The c omp utat ion of sensi tiv ity fac tors i s more com pl icated in thecase of an e las t ic vi scoplas t ic mater ia l . W e must in t rodu ce t h eder ivativ e o f the stress field: ~._~o,which m ust be integrated wi th3 qrespect to t ime from:

d t ~ 3 q / 3 q (39)

R ( X , V , T , T ,q ) = 0 w ith d X = vd tY ( X , V , T , J ' , q ) = 0 w ith d T = ' i "d t

(35)

Where the pressure P was e l iminated for s impl ic i ty ( th i s can bedone a t l eas t theoret ica l ly , as there i s no memory ef fect in th i sca se ) . W e suppo se t ha t t he expe r i men t a l measu r emen t s a r es tored in a vector M ~x. T hey cor respon d to compu ted values M e ,w hi ch can be expr e s sed i n t e r ms o f t he ve l oc i t y V and t hecoor d i na t e s X ( and pos s i b l y t he t emper a t u r e T ) . T he meansquare di f ference between the exper imenta l and computed valuesis written:- - . . . . . IAM = ~L (M~,"- M~)2 (36)

k

and minim ized wi th respect to the mater ia l parameters q . For th ispur pose t he G aus s N ew t on m e t hod i s u sed w h i ch necess i ta t e t hecom put a t i on o f t he s ens i ti v i t y f ac t o r s . F r om our p r ev i oushypo theses these sens i t iv i ty fac tors are f i rs t obta ined by:

~M~, aM~ ~v ~M~ av aM~, T- - - - . - - + . - - + . ~ _ _3 q 3 X 3 q 3 V 3 q 3 T 3 q (37)

Equat ions (35) are di f ferent ia ted wi th respect to q , so that weobta in the sys tem:

5.2 Feasability of the inverse method for internal physicalparametersThe same approach can be eas i ly general ized to the case where qcon t a i ns a l so phys i ca l pa r ame t e r s r ep r e sen t i ng e i t he r t hei n f l uence o f phys i ca l pa r ame t e r s on t he cons t i t u t i ve equa t i on(17), or the law of evolut ion of the phys ical parameters (18). Thes e t o f e x p e r im e n t a l m e a s u r e m e n t s m u s t n o w e n c l o s echaracter iza t ion of the micros t ructure of the sample a t d i f ferentt ime intervals so that (37) is replaced by :

aM~ aM~,~v aM~ av aM~ aT aM~ aS- - - -. - I -. - - - -. . - - + - - - -3 q 3 X 3 q 3 V a q a T a q 3 S 3 q (40 )

No w the differentiat ion of (25) with respe ct to q give s:

3 R a V 3 R a T 3 R 3 X 3 R a T 3 R a S_ _ _ _ + _ _ . - -3 V a q 3 T a q 3 X a q ~ T a q a S a q

3Y 3V 3Y 3"i" 3Y 0X O Y 0T 3Y 0S3 V 0 q 0 T 3 q 3 X 3 q 3 T a q 3 S 0 q

~ S 3 ~ 0 X 3 ' ~ 0 T ~ X 3 S-- + +3 q 3 X 0 q 3 T 3 q 3 S 3 q

(41a)

(41b)

(41c)


7/8

d.-L. Chenot, Y Chastel dournal of Materials Processing Technology 60 (1996) 11-18 17d ~X o~V d ~T q'~ .d ~S ~d t ~ q ~ q d t ~ q ~ q d t ~ q ~ q (41d)

t h a t a l lo w s t h e c o m p u t a t i o n o f t h e d e r i v a t i v e s w i t h r e s p e c t t o qw h i c h a r e n e c e s s a r y i n (4 0 ) . T h e r e l a t i v e s i m p l i c i t y o f th ed e r i v a t i v e o f t h e s t r u c t u r e e v o l u t i o n i s d u e t o t h e f a c t t h a t th eg e n e r a l f o r m w e h a v e c o n s i d e r e d s o f a r d o e s n o t t a k e i n t oaccoun t any d i f fu s ion .

m a c r o m o d e l s . T h e e x p r e s s i o n s o f s e n s i t iv i t y fa c t o r s a r e d e r i v e di n c l o s e f o r m f o r t h e s e d i f f e r e n t t y p e s o f m i c r o s t r u c t u r a l a n dt e x t u r e e v o l u t i o n l a w s . O n c e i n c o r p o r a t e d i n o p t i m i z a t i o nm o d u l e s a n d f i n i te e l e m e n t m o d e l s o f m e c h a n i c a l t e s t s t h e y w i l la l l o w f o r a u t o m a t i c i d e n t i f i c a ti o n o f t h e c o n s t i t u t iv e p a r a m e t e r sa s w e l l a s t h e p h y s i c a l p a r a m e t e r s o f t h e m i c r o s t r u c t u r e a n dt ex tu re m ode l s .

5.3 Feasability of the inverse method fo r a micro-macr approachW e s u p p o s e t h a t , in t h e m i c r o - m a c r o m o d e l , w e i n t r o d u c e d a s e tq o f p a r a m e t e r s , w h i c h d e s c r i b e s t h e m e c h a n i c a l b e h a v i o u r o fa n y g r a i n a n d p o s s i b l y t h e i r i n t e r a c t io n s . F o r e x a m p l e , i n th es i m p l e T a y l o r m o d e l s u m m a r i z e d b r i e f l y i n s e c t io n 4 . 1. 1 , w e c a nA tXcho ose t he pa ram e te r s : a , m , x,~ o r t hose o f a f unc t i on o f ' t ~ ,c o r r e s p o n d i n g t o e q u a t i o n ( 30 ) . I n t h is c a s e w e s e e i m m e d i a t e l ytha t t he s t r e s s de r i va t i ve w i th r e spec t t o t he pa ram e te r s i s e a s i l yc o m p u t e d b y :

3o" _ O M . :~ + M :3 I~ (42 )~)q o~q ~)qand us ing (34 ) w e can w r i t e :O M = ~ wc 3M----~ (4 3)0 q ~ 0 qi n w h i c h w e h a v e t o c o m p u t e f o r e a c h g r a in :

~ - U t )~ M c + ) ' 3- " M C :~ ' ~--~- Xa-) pt,p,~ :M c~ q X 3q ~ T ~ (pa p~ )

I f th e d e r i v a t i v e s o f t h e S c h m i d t e n s o r s e x p r e s s e d i n t h e f i x e dr e f e r e n c e f r a m e a r e n e g l e c t e d a t f i x e d s t a t e w e s e e t h a t f e wa d d i t i o n a l a r ra y s a r e n e e d e d i n t h e c o m p u t a t i o n o f t h e s e n s i ti v i t yf ac to r s .O n t h e c o n t r a r y i f t h e r o t a t i o n s o f t h e c r y s t a l s a r e a l s o e x p l i c i t l ye x p r e s s e d i n t e r m s o f th e q p a r a m e t e r s w e m u s t s t o r e t h e

0vn e c e s s a r y i n f o r m a t i o n a b o u t t h e d e r i v a t iv e s ~ f o r e a c h g r a i n .

6 . C o n c l u s i o n s

I n t h e c o n t e x t o f f i n i t e e l e m e n t a n a l y s i s f o r m e t a l f o r m i n gp r o c e s s e s t h e g e n e r a l f o r m u l a t i o n f o r i n c l u d i n g m e c h a n i c a l ,t h e r m a l a n d m i c r o s t r u c t u r a l c o u p l i n g a r e r e c a l l e d a n di m p l e m e n t e d t o s o l v e t h r e e - d i m e n s i o n a l p r o b l e m s i n v o l v i n g l a r g ed e f o r m a t i o n s o f t h e m e t a l . V a r i o u s m i c r o s t r u c tu r a l m o d e l s h a v eb e e n c o n s i d e r e d : c l a s s i c a l s t a t e v a r i a b l e m o d e l s a n d m i c r o -

7 . R e f e r e n c e s

[ l ] R . E . S m e l s e r , O . R i c h m o n d & E . G . T h o m p s o n , An u m e r i c a l s t u d y o f t h e e f f e c t s o f d i e p r o f i l e o n e x t r u s i o n , P r o c .O f t h e N U M I F O R M C o n f . , e d b y K . M a t t i a s s o n e t a l , A . A .B a lkem a , R o t t e rdam , 305 (1986)[2 ] J . K us i ak , M . P i e t r zyk & J . -L . C heno t , D ie shape des ign ande v a l u a t i o n o f m i c r o s r t u c tu r e c o n t r o l i n t h e c l o s e d - d i ea x i s y m m e t r i c fo r g i n g b y u si n g F O R G E 2 p r o g r a m , I S I JInterna t ional , 34, n 9 , 755 (1994)

[ 3 ] P . R . D a w s o n , A . J . B e a u d o i n , a n d K . K . M a t h u r . F i n i t ee l e m e n t m o d e l i n g o f p o l y c r y s t a l l i n e so l i d s . N u m e r i c a lp r e d i c t i o n s o f d e f o r m a t i o n p r o c e s s e s a n d t h e b e h a v i o r o f r e a lm a te r i a l s , 15 th R i so In t e rna t i ona l Sym pos ium , 33 (1994)[ 4 ] A . J . B e a u d o i n , K . K . M a t h u r , P . R . D a w s o n , a n d G . C .J o h n s o n , T h r e e - d i m e n s i o n a l d e f o r m a t i o n p r o c e s s s i m u l a t i o n w i t he x p l i c i t u s e o f p o l y c r y s t a l l i n e p l a s t i c it y m o d e l s . I n t e r n a t i o n a lJou rna l o f P l a s t i c i t y , 833 , 9 (1993)[5 ] J . -L . C heno t & M . B eU e t, The v i s c op l a s t i c app ro ach fo r t hef i n i t e e l e m e n t m o d e l l i n g o f m e t a l f o r m i n g p r o c e s s e s , i nN u m e r i c a l M o d e l l i n g o f M a t e r i a l D e f o r m a t i o n p r o c e s s e s .R e s e a r c h , D e v e l o p m e n t s a n d A p p l i c a t i o n s , e d . b y P . H a r t l e y e ta l , Sp r inge r V er l ag , Lond on , 179 (1992).[ 6 ] J . - L . C h e n o t & M . B e l l e t , A v e l o c i t y a p p r o a c h toe l a s t o p l a s t i c a n d e l a s t o - v i s c o p l a s t i c c a l c u l a t i o n b y t h e f i n i t ee l e m e n t m e t h o d , J o u rn a l o f E n g i n e e r i n g f o r I n d u s t r y A S M E , V o l .112, n 2, 151 (199 0).1 .7 .]. Z i eg l e r , A n In t rodu c t i on t o The rm odyn am ic s , N or th H o l l andP u b l i s h in g C o m p a n y , , A m s t e r d a m ( 1 9 8 3 ).[ 8 ] L . A n a n d . C o n s t i t u t i v e e q u a t i o n s f o r t h e r a t e - d e p e n d e n td e f o r m a t i o n o f m e t a l s a t e l e v a t e d t e m p e r a t u r e . J . E n g . M a t .Tech. , 104, 12 (1982)[ 9 ] J . - L . C h e n o t a n d M . B e l l e t . T h e A L E m e t h o d f o r t h en u m e r i c a l s i m u l a t i o n o f m a t e r i a l f o r m i n g p r o c e s s e s . S i m u l a t i o no f m a te r i a l s p roces s ing , N um i fo rm 95 , 39 (1995)


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18 J .-L . Chenot , Y Chas te l / Journ al o f M ate r ia l s Proces s ing Technology 60 (1996) 11- 18

[ 10 ] Y . T r one l , C on t r i bu t i on t o t he numer i ca l mode l l i ng o fext rus ion: predic t ion of mater ia l f low and micros tructure, . DE Arepor t , CEMEF, ENSMP (1989) ( in f rench)[11] A. Hac quin, P. M ontm i tonne t , and J . -P. Gui l leraul t . Asteady s ta te the rmo-elas toviscop las t ic fin ite e lem ent mode l ofrol l ing wi th coupled thermo-elas t ic rol l deformat ion. This i ssueof Me ta l Fo r mi ng 96 p r oceed ings .[ 12 ] R . J . A sa r o and A . N eed l eman . T ex t u r e deve l opmen t and

s t r a i n ha r den i ng i n r a t e dependen t po l yc r ys t a l s . A c t a Me t a l l .Mater . , 33,923 (1985)[ 13] J .C . G e l i n and O . G hou a t i . A n i nve r se me t hod f o rdetermining vi scoplas t ic proper t ies of a luminium al loys , Journalof Mater ia l s Proc ess ing Tech nolog y, 45 ,43 5 (1994)[ 14] A . G avr us , E . Masson i , and J . -L . C heno t . C ons t i t u t i vepa r ame t e r i den t i f i ca t i on us i ng a C omput e r A i ded R heo l ogyapp r oac h . . S i mul a t ion o f ma t er i a ls p r oces s i ng , N um i f o r m 95 ,563 (1995)

mechanical, thermal and physical coupling methods in fe analysis of metal forming processes

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