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Pergamon

0020-7403(94) E0093-X

Int. J. Mech. Sci.Vol. 37, No. 8, pp. 883 897, 1995ElsevierScienceLtdPrinted in Great Britain.0020-7403/95 $9.50 + 0.00

F I N I T E E L E M E N T S I M U L A T I O N O F P O W D E R C O N S O L I D A T I O NI N T H E F O R M A T I O N O F F I B E R R E I N F O R C E D

C O M P O S I T E M A T E R I A L SJ. XU t and R. M. M c M E E K 1 N G ~

tConcurre nt Technologies Corpo ration 1450 Scalp Avenue, Johnstown, PA 15904, U.S.A. and ~ MechanicalEngineering Department, University of California, Santa Barbara, CA 93106, U.S.A.

(Received 15 October 1993; and in revised form 30 October 1994)Abstract--Finite element codes are developed to analyze powder consolidation due to time-inde-pende nt plasticity and time-dependent plasticity (power law creep and diffusional creep). F ortime-dependent plasticity, a new algorithm is propose d to obtain a prediction of the plastic strainincrement when solving for the nodal disp lacement increment. This eliminates the use of an ad hocNewton loop. A set of newly developed micromechanical models are adopted as the constitutivelaws for power deformatio n,where the influence of deviatoric stresses on densification is taken intoaccount. The densification maps are constructed for the powder consolidation in the formation offiber reinforced composite materials. The influence of fiber volume fraction, powder particle size,temperature and fiber arrangement is investigated,

1 . INTRODUCTIONOwing to its potential in manufacturing high performance components, hot isostaticpressing (HIP) has been motivating theoretical and experimental research. The densifica-t ion of a powd er com pact b y HI P is dependent on man y factors such as appl ied pressure ,temperature , po wd er particle size and the properties o f the material com posing the powder.A theoret ical foundat ion has been es tabl ished for the mechanisms contr ibut ing to pow derdensification and densification maps (mechanism diagrams) have been constructed I-1-5].In recent years, the fact that deviatoric stresses can influence powder consolidation hasmade i t the subject of a num ber of micromechanical analyses [6-15] . The resul ts of theseanalyses include the deviatoric stresses in the constitutive laws for the deformation ofa powder compact. With these micromechanical models, one can evaluate powder densifi-cation u nder n onun iform as well as nonh ydro static stress states. Som e of the above-mentioned models have been implemented in f in ite e lement programs to analyze problemssuch a s shap e cha nge, th e can effect and closed-d ie com pa ctio n 1-16, 17].

During densification, the powder compact experiences large geometric changes and theconstitutive laws are nonlinear. Th e impleme ntation of the above-m ention ed constitutivelaws in processing requires the use of a nonlinea r finite element analysis. The objective ofthis work is to develop a nonlinear finite element code to ev aluate pow der densification dueto time-independent and time-dependent plasticity. For the latter, a new algorithm ispro po sed to p rovide the first estimate of the nodal displacement increm ent an d a guess forthe plastic strain increment simultaneously. When the nonlinear problem is solved numer-ically, the accu racy and stability of the incremental solution is an impor tant issue. Indeed,this has long represented an interesting challenge in nonlinear finite element analysis.Several me thods have b een suggested for the integration of the elastoplastic constitutiveequatio ns o f time-indepe ndent and time-depe ndent plasticity 1-18-30]. The new algorithmformulated for the current paper exhibits excellent qualities in regard to accuracy andstability.Densif icat ion maps are constructed for powder consol idat ion during the formation offiber reinforced composite materials. Three mechanics, t ime-independent plasticity, powerlaw creep and diffusional creep, are considered. A set of newly develope d micromecha nicalmodels are adopted as the constitutive laws for powder deformation, where the influence of

883

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884 J. Xu and R. M. McMeekingdeviatoric stresses on the densification is taken into account. The transition of the consti tut-ive relations from the initial particle contact controlled stage to the final void controlleddensification stage is smoothed by a transitional interpolation stage between them. Theinfluence of fiber volume fraction, powder particle size and temperature is investigated.

2. MECHANISM-BASED MICROMECHANICAL MODELSThe micromechanical models adopted in this paper are as follows. In particle contactcontrolled consolidation, known as Stage I, for time-independent plasticity, the model isdue to Fleck e t a l . [7]; for power law creep, Kuhn and McMeeking [9]; for diffusional creep,McMeeking and Kuhn [13]. In void controlled consolidation, known as Stage II, fortime-independent plasticity, the model is by Gurson [31]; for power law creep, Sofronis andMcMeeking [8]; for diffusional creep, Riedel [14]. According to the micromechanicalmodels, the deformation rate of the porous powder compact caused by different mechan-isms is governed by corresponding potentials. For time-independent plasticity, the macro-scopic strain rate ~ is determined by

= 2 ~ , (1)where (I) is the plastic potential, o is the macroscopic stress tensor and 2 is the plasticmultiplier [32]. For power law creep and diffusional creep,

8LF= O ~ ( 2 )

where W is the creep potential [27]. Here (I) and W are functions of the macroscopic stresstensor a, temperature 0, the relative density of the powder compact D, and materialparameters, i.e.

= ~(~, 0, D, material parameters) (3)andqJ = W(g, 0, D, material parameters). (4)

The analytical expressions for all the plastic and creep potentials are listed in the Appendix.The derivations of the micromechanical models were performed with geometries charac-

terizing different specific densification stages, either I or II. Since the topology of porosityexperiences a gradual but significant change from Stage I to Stage II, the model for theinitial stage of particle contact controlled deformation must be dispensed with completely infavor of the one for the final stage of void controlled consolidation above some transitionlevel of porosity. One way to simulate this shift of constitutive relations is to envisiona transitional stage (D1 < D < D2), where the plastic and creep potentials are obtained byinterpolation [7]. For example, let ~c be the plastic potential for Stage I due to Fleck e t a l .[7] and O~ that due to Gurson [31] for Stage II. In the transition range D1 < D < D2, theplastic potential is

D2 -- D D - Dj = Oc + O~. (5)D2 - D1 D2 - D1The choice of D 1 and D2 should best reflect the transition and depends on the stress state

and the deformation mechanism. For example, Ashby [4] suggests that D, = 0.8, andD2 -- 0.9 for hot isostatic pressing. In this analysis, we choose D1 = 0.75 and DE = 0.9.

3. FORMULATION OF THE BOUNDARY-VALUE PROBLEMThe formation of fiber reinforced composite materials through powder processing ismodeled by analyzing the densification of the powder compact around long rigid cylindricalinclusions. Due to the rigidity of the fiber, plane strain is assumed for the deformation of the

powder compact. The finite deformation of the powder compact is allowed for through the

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F i n i t e e l e m e n t s i m u l a t i o n o f p o w d e r c o n s o l i d a t i o n 8 8 5

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 [ 3 3 ] g o v e r n e d b y t h e v ir t u al w o r k e x p r e ss i o nfv [ ~ : r M + ~ 6 : 6 ( LX L - 2 M M )] d V = s 'r r v d S , (6 )

wh ere V i s t h e cu r ren t v o lu m e o f th e p l a in s t ra in p ro b l em, S is t h e b o u n d a ry o f V, 6 i s t h eCau ch y s tres s, ~ d en o t e s t h e Jau m an n ra t e o f K i rch h o f f s tre s s, L i s t h e v e lo c i t y g rad i en t , i .e .

~vL ~ x ' (7 )M i s t h e d e fo rm a t io n ra t e wh ich i s th e sy m met r i c p a r t o f L , T is th e su r face t rac ti o n o nS an d 6 d en o t e s an a rb i t ra ry v i r t u a l v a r i a ti o n , t r an d ~ a re re l a t ed b y

= J o , (8 )wh ere J i s t h e ra t i o o f v o lu m e in t h e cu r ren t s t a t e t o t h a t i n t h e re fe ren ce s t at e .A rep re sen t a t i v e sq u a re u n i t ce l l wi th sy mmet ry an d p e r io d i c i t y co n d i t i o n s o n t h ep e r ime te r i s u sed in t h e an a ly s i s a s sh o w n in F ig . 1 . A co m p ara t i v e s t u d y i s a l so p e r fo rme dwi th a he xago nal un i t ce ll (Fig . 2 ) to tes t the appr opr ia ten ess of the choice of un i t cel l andthe in f luence of f iber arran gem ent on the densi f ica t ion . In the process of densi f ica t ion , theperimeters ( four s ides for the square un i t ce l l and s ix s ides for the hexagonal un i t ce l l ) a reco n s t ra in ed t o rem a in s t ra ig h t. Becau se o f sy mm et ry , th e p e r ime te rs a re sh ea r s t res s free .A p e r fect b o n d in g b e tw een t h e p o w d er co m p ac t an d t h e i n c lu s io n is a s su med . Th e u n i t ce ll sa re eq u a l l y l o ad ed i n t h e x an d y d i rec t io n s . Us in g t h e sy mm et ry o f g eo m et ry an d l o ad in gcond i t ion , i t i s suff ic ien t to dea l wi th one-e igh th of the sq uare un i t ce ll and one-tw el f th of theh ex ag o n a l u n i t c e ll . Ho we v er , t h e f i ni te e l emen t co d e fo r t h e sq u a re u n i t ce ll is d ev e lo p e d fo ro n e - fo u r th o f th e u n i t ce ll , l e av in g l eeway fo r fu tu re i n v es t i g a t io n o f d en s i fi ca t io n u n d e ru n eq u a l l o ad in g . In t h is wo rk , th e v o lu m e f rac t io n s o f th e co n so l i d a t ed co m p o s i t e ma te r i a la r e f r o m 1 0 % t o 5 0 % .

P

- - - 0 , @ ,- . , - ,I I- ,-@ , @ ,. - - ,

t t t

p o w d e r c o m p a c tfiber

m

F i g . 1 . T h e u n i t c e l l f o r fi b e r s p a c k e d i n a s q u a r e a r r a y a n d a t y p i c a l m e s h u s e d i n t h e fi n i t e e l e m e n ta n a l y s i s .

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886 J. Xu and R. M. M c M e e k i n gpowder compact

_ .

t t t

I

I

iI

Fig. 2. T h e u n i t c e l l f o r f i b e r s p a c k e d i n h e x a g o n a l a r r a ye l e m e n t a n a l y s i s.

1

a n d a t y p i c a l m e s h u s e d i n t h e f i n i t e

In the f in ite e lem ent analys is , e ight-noded second-orde r isoparam etric e leme nts with fourstat ions for the integrat ion are used [3 4] . T he num ber of e lements and no des used in theanalysis for different unit cells and volume fractions are listed in Table 1.

In the con struct ion o f dens if icat ion maps , the co ntribut ion of t ime- independent p las tic ityis evaluated f irs t . To do th is , d isp lacement boundary condit ions are imposed at theperimeters of the unit cells. By use o f the resulting stresses in the po wd er com pac t, thenorm al surface tract ion on the perimeter of the unit cell can be obtained. T he average valueof the norm al surface tract ion is taken to be the applied pressure, wh ich , together w ith therelat ive dens ity of the powder compact , is the main output of the calculat ion . After theapplied pressure has reached a certain value, power law creep and diffusional creep start tocontrib ute to the densification. Th e total strain rate is the sum of the creep rates due to

T a b l e 1 . N u m b e r s o f e l e m e n t s a n d d e g r e e s o f f r e e d o m i n t h e f in i t ee l e m e n t a n a l y s i s

V o l u m e N u m b e r o ff r a ct io n N u m b e r o f d e g r ee s o f

(%) e l e m e n t s f r e e d o m

S q u a r e u n i t c e l l

64 450256 1666

30 72 50650 72 506

H e x a g o n a l u n i t c e ll10 50 36250 36 266

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Finite element simulation of powder consolidation 887

power l aw and d i f fu s iona l behav io r . A t t h i s s t age , t he ave rage no rma l su r face t rac t i on onthe pe r ime te r o f t he un i t cel l r ema ins cons t an t , t h oug h the l oca l no rm a l su r face trac t i onsm ay change ; t he ana ly s i s g ives t he re l a t ive dens i t y o f t he co m pac t a s a func t ion o f time. A l sothe con t r i bu t ions f rom power l aw c reep and d i f fu s iona l c reep a re compared t h rough thefo l l owing pa ram e te r , spa t i a l l y ave raged ove r t he dom ain o f the p rob l em:

I v PL ~PL d VR - " , (9)v ~ D F ~ D F d v

w h e r e ~ P L is t h e s t r a i n r a t e t e n s o r c a u s e d b y p o w e r l a w c r e e p a n d ~ D v b y d i f fu s i on a l c r e e p .T h e v a l u e o f R i s u s e d t o d e t e r m i n e t h e d o m i n a n t m e c h a n i s m b e t w e e n p o w e r l a w c r e e p a n dd i f f u s i o n al c r e e p . I f R > I , p o w e r l a w c r e e p d o m i n a t e s o n a v e r a g e i n V , w h e r e a s R < 1i n d i ca t e s t h a t d i f f u s i o n a l c r e e p d o m i n a t e s o n a v e r a g e .

4 . ALGORITHM FOR THE FINITE ELEMENT ANALYSISTo cons t ruc t t he dens i f i ca t i on maps , Eqn (6 ) i s so lved i nc remen ta l l y un t i l t he des i red

load ing and de fo rm a t ion l eve ls a re reached . Le t [Au] b e the a r ray o f noda l d i sp l acemen tinc remen t s fo r t he f i ni te e lemen t m ode l u sed t o rep resen t t he p rob l em. By choos ingapp rop r i a t e shape func t ions [A] and [B] , we have t he f i n i t e e l emen t equa t ions re su l t i ngf rom Eqn (6 )

[ K ] [ A u] = [AR] , (10 )w h e r e

[ K ] = f v [ B I T [ C ] [ B ] d V , ( i I )= f s [ A ] T i ~ d S A t ,A r l (12)

and [C ] i s ca l cu l a t ed f rom the l e f t-hand s ide o f Eqn (6 ) . No te t ha t [C ] depen ds on t hemate r i a l p rope r t i e s , t he s tre s s and t he a lgo r i thm used fo r t he i n t eg ra t i on o f t he cons t i t u t i veequa t ions .

The i nc rem en ta l s o lu t i on o f Eqn (6 ) i s com pl i ca t ed by t he geom et r i c non l inea r i t y a nd t hemate r i a l non l inea r i t y i n t he powder conso l ida t i on . In each i nc remen t , t he f i n i t e -e l emen tequa t ion s (10 ) a re a t f i r s t e s t ab l i shed based o n t he g eom et ry a t t he beg inn ing o f t ha ti nc remen t . When the noda l d i sp l acemen t i nc remen t i s so lved and the geomet ry i s changedacco rd ing ly , an unba l anced fo rce vec to r w i ll be gene ra t ed i n t he f i n i te -e l emen t ca l cu l a t i on[341 . O n the o the r hand , t he cons t i tu t i ve behav io r o f t he pow der com pac t depe nds on i t sre l a ti ve dens i t y wh ich a l so changes i n each i nc remen t . In t h is ana ly s is , t he backw ard Eu le rm e tho d i s chosen fo r t he num er i ca l i n t eg ra t i on o f t he e l a s top l a s t ic cons t i t u t i ve re la t ions .

N ew ton i t e ra t ion i s u sed fo r t he so lu t i on o f the no n l inea r f i n i te e l emen t equa t ions . Ino rde r t o sa t i s fy t he equ i l i b r ium equa t ions (10 ) a t t he end o f t he i nc remen t , a fu ll New tonscheme (ou t e r Newton loop ) i s u sed t o so lve t he co r re spond ing non l inea r a lgeb ra i c equa -t ions . In each i t e ra t ion , equa t ions (10 ) a re e s t ab l i shed base d on t he geom et ry , the re l a t ivedens i t y and t he s t re s s i n t he powder compac t upda t ed by t he l a s t i t e ra t i on o r i nc remen t .A co r rec t i on o f t he noda l d i sp l acem en t i nc remen t i s then so lved . The o u t e r N ew ton loo p i scon t inued un t i l the unba l anced noda l fo rce o r t he co r rec t i on o f t he noda l d i sp l acem en tincrement i s less than a prescribed to lerance .

To sa t i sfy the y ie ld corld i t ion = 0 ( 1 3 )

the f low ru le , Eqn (1) for t ime-independent p las t ic i ty , Eqn (2) for t ime-dependent p las t ic i ty ,a t t he end o f each i nc remen t , an i nne r N ew ton loo p is adop ted t o each in t eg ra t i on p o in t anda t each i t e ra t ion o f t he ou t e r Ne wto n loop . In t he i nne r New ton loop , the geom et ry is fi xed.HS 37-8-6

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888 J. Xu and R. M. McM eekingT h e p l a s t ic s t r a i n i n c r e m e n t i s c o r r e c t e d t o m a k e t h e r e s u l t i n g r e la t i v e d e n s i t y a n d t h e s t r e ssa t e a c h i n t e g r a t i o n p o i n t s a t is f y t h e a b o v e - m e n t i o n e d c o n s t i t u t i v e r e la t i o n s . T h e i n n e rN e w t o n l o o p i s r e p e a t e d u n t i l t h e p l a s t i c s t r a i n i n c r e m e n t c o n v e r g e s w i t h i n a p r e s c r i b e dt o l e r a n c e . D e t a i l s a b o u t t h i s it e r a t iv e l o o p c a n b e f o u n d e l s e w h e r e [ 3 5 ].T i m e - i n d e p e n d e n t p l a s t i c i t yT h e c a l c u l a t i o n o f t h e l i n e a r i z a t i o n m o d u l i ( m a t e r i a l s t i f f n e s s m a t r i x ) i s b a s e d o n t h ed e r i v a t i o n o f H i b b i t t [ 2 4 ]. I n t h i s w o r k , t h e n o n s y m m e t r i c s ti ff n e ss m a t r i x o b t a i n e d d i r e c t l yf r o m E q n s (6 ) a n d ( 1 1) i s u se d . C o m p a r e d w i t h t h e u s e o f it s s y m m e t r i c p a r t , t h i s in c r e a s est h e r a t e o f c o n v e r g e n c e o f t h e N e w t o n l o o p s . T h e i n n e r N e w t o n l o o p i s i n i t i a t e d w i t ha g u e s s o f t h e p l a s t ic s t r a i n i n c r e m e n t e s t i m a t e d b y t h e s u b s p a c e p r o j e c t i o n m e t h o d . ( N o t et h a t t h e r e i s a n o t h e r N e w t o n l o o p f o r t h is p u r po s e .) T h e c o n v e r g e n ce o f th e i n n e r N e w t o nl o o p i s in f l u e n c e d b y t h e r a t i o o f t h e y i e l d st re s s t o t h e Y o u n g ' s m o d u l u s (try~E). W h e n a y / Ei s 1 0 - 3, i t t a k e s n o m o r e t h a n t h r e e i t e r a t i o n s f o r t h e fu l l s t r e s s - s p a c e p r o b l e m t o c o n v e r g ew i t h i n a t o l e r a n c e o f 10 -6. L o w e r v a l u e s o f th e r a t i o i n c r e a s e t h e r a t e o f c o n v e r g e n c e . T h ec o n v e r g e n c e o f t h e o u t e r N e w t o n l o o p d e p e n d s o n t h e i n c r e m e n t , t h e r e la t i v e d e n s i t y a n dt h e f i b e r v o l u m e f r a c t i o n . I t i s e x p e d i e n t t o u s e s m a l l e r i n c r e m e n t s a t l o w e r r e l a ti v e d e n s i t ya n d t h e n i n c r e a s e t h e i n c r e m e n t s i ze w h e n t h e r e l a ti v e d e n s i t y in c r e as e s . B y p r o p e r c h o i c e o ft h e i n c r e m e n t , d e n s i f i c a t io n b y t i m e - i n d e p e n d e n t p l a s t ic i t y f r o m r e l a ti v e d e n s i t y 6 5 /'0 t ofu l ly dense ca n be acc om pl i she d in 3000 to 4000 s teps . Fob a square un i t ce ll , a typ ica l runw i t h a v o l u m e f r a c ti o n o f 10 % a n d a r / E = 1 0 - 3 r e q u i r e s 3 00 m i n u t e s o n a n I B M R S / 6 0 0 0M o d e l 3 2 0. S l i g h t ly lo n g e r t i m e i s n e e d e d f o r h i g h e r v o l u m e f r a c t i o n s a n d h i g h e r r a t i o s o fa r / E .T i m e - d e p e n d e n t p l a s t i c i t y

D u r i n g t h e t i m e i n c r e m e n t f o r t i m e d e p e n d e n t p l a s t i c it y , t h e a v e r a g e n o r m a l t r a c t i o n o nt h e p e r i m e t e r o f t h e u n i t c e ll i s r e q u i r e d t o r e m a i n c o n s t a n t , t h o u g h t h e l o c a l t r a c t i o n m a yc h a n g e . T h i s m e a n s t h a t i n t h e u n i t c el l, s a y t h e s q u a r e o n e a s s h o w n i n F i g . 1 ,

f s T d S A t = 0, (14)iw h e r e S i d e n o t e d lm a n d m n. T h i s i s a c o m m o n c a s e i n p o w d e r c o n s o l i d a t i o n s i n c ed e n s i f i c a t i o n b y t i m e - d e p e n d e n t p l a s t i c i t y i s o f t e n a c c o m p l i s h e d u n d e r c o n s t a n t a p p l i e dp r e s su r e . C a r e f u l e x a m i n a t i o n o f E q n (6 ) s h o w s t h a t , f o r a f i x e d ti m e i n c r e m e n t , i n t h e f i r stg u e s s o f [ A u ] , T i s u n k n o w n . I n t h i s w o r k , f o r t h e f ir s t g u e s s o f [ A u ] , T i s a s s u m e d t o b ez e r o. A t t h e s a m e t i m e , a n a t t e m p t i s m a d e t o d e t e r m i n e t h e f i rs t g u e s s o f t h e n o d a ld i s p l a c e m e n t i n c r e m e n t a n d t h e e s t i m a t e o f t h e p l a st i c s tr a i n i n c r e m e n t s i m u l t a n e o u s l y . T h eder iva t ion i s as fo l lows .

A s s u m e a n a d d i t i v e s t r a i n r a t e d e c o m p o s i t i o n , t h e s t r e s s i n c r e m e n t d u e t o t h e J a u m a n nra te i s

Ao = Ce:A e e - Ce:A~ - Ce:AI; p (15)whe re C e i s the fo u r th -o rd er e las t i c i ty t enso r , Ae = M At , Aee an d AI~p a re to ta l , e l as t i c andp las t i c s t r a in inc rem en ts .

I n t h e b a c k w a r d E u l e r m e t h o d , t h e f l o w r u l e a n d t h e e v o l u t i o n e q u a t i o n f o r t h eh a r d e n i n g p a r a m e t e r a r e s a t i sf i e d a t t h e e n d o f e a c h i n c r e m e n t , i.e .

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Finite elemen t simulation of pow der consolidation 889I n t h e f i r st g u e s s f o r t h e n o d a l d i s p l a c e m e n t i n c r e m e n t , A c p a n d A D a r e e s t i m a t e d b y

(19 )O k [ t ) 1 /0 2 k ~ ' \ 1 / 0 2 ~ x

A D = A t h t + A t ~ o + A tE l i m i n a t i n g A D f r o m E q n s ( 19 ) a n d ( 20 ) p r o v i d e s

a n d

w h e r e

a n d

o r

A ~ p ----- A ~ + A t N : A 6 ,

(20 )

(21)

- - -O h J A t (22 )A~ = + O6OD I _ At_ ~ t

: O 2 k i _ _ 0 2 k I A t ~ . ( 23 )8 hN ~ + O 6 0 D 1 - A t ~ - ~ ,

S u b s t i t u t i n g E q n ( 2 1) i n t o ( 1 5) , w e h a v e

w h e r e

[ I + A t C ~ : N ] ' A 6 = C ~ : A e - C ~ ' A ~

A f t = C P : A 8 - - C P : A ~ ,

( 2 4 )

(25)

t "[ K ] = I , , [B]T [C] [B]d V, (29 )[ A R ] = J s [ A ] T [ T ] dSAt , (30)[ A p ] = f v [ B I T [ U P ] [ A ~ ] d V (3 1)

a n d [ C ] i s c a l c u l a t e d f r o m t h e l e f t - h a n d s i d e o f E q n (2 7) . T h e p l a s t i c s t r a i n i n c r e m e n t i se s t i m a t e d b y E q n ( 21 ) o r , i n a n e q u i v a l e n t w a y a s th i s w o r k d o e s , b y s u b s t i t u t in g E q n (1 5)i n t o ( 2 1 ) w h i c h l e a d s t o

A p = [ I + A t N : C e ] - l : [ A ~ + A t N : C e : A ~ ] . (3 2)T h e a b o v e A 8 p i s t h e n u s e d a s t h e i n i ti a l g u e s s f o r th e i n n e r N e w t o n l o o p . W i t h E q n (3 2) ,t h e r e i s n o n e e d t o u s e t h e s u b s p a c e p r o j e c t i o n m e t h o d t o f i n d th e i n i ti a l g u e s s f o r p la s t i c it y

w h e r e

f r o m w h i c h w e o b t a i n t h e f in i te e l e m e n t e q u a t i o n s[ K ] [A u ] = E A R ] + [ A p ] , ( 2 8 )

(27)

C o = [ I + A t C : N ] - 1 : C ~. (2 6)F i n a l l y , c o m b i n i n g E q n s ( 6) , ( 8) a n d ( 24 ) g i v e s

fv [6 M :(C P :A ~ . + A Jo ) + 6 : A t 6 ( L T L - 2 M M ) ] d V = f s6 V A t .A T d S + f v 6 M : C P : A ~ d V

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8 9 0 J . X u a n d R . M . M c M e e k i n g

[35]. This eliminates the use of a corresponding ad hoc Newton loop. In general, twoiterations are required for the convergence of the stress correction when the ratio trr/Eis 10 -3 .

5 . F I N I T E E L E M E N T R E S U L T S A N D D I S C U S S I O NResults of pressurizationHere pressurization means that time-independent plasticity is the only mechanism fordensification. It is found that under pressurization, the average relative density of thepowder compact is not sensitive to the composite volume fraction but depends only on theapplied pressure. The average density of the powder compact is defined to be the volume ofpowder divided by the volume of space it occupies excluding the fiber. The result is observedfor both the square and the hexagonal unit cells. The fiber serves only to fill space and doesnot cause any constraint on the densification of the powder. The density of powder in thecompact is almost uniform during pressurization. The implication is that, as far as overallrelative density is concerned, the densification of a pure powder compact (zero volumefraction) under the same loading condition would give a very good approximation to thedensification of a fiber reinforced composite after the volume occupied by the fiber is takeninto account. In addition, since the densification of a pure powder compact is independentof the choice of the unit cell, the overall densification behavior obtained from the squareunit cell should be quite close to that of the hexagonal unit cell.

Figures 3 and 4 are the comparisons of the finite element results for 10% and 50% fibervolume fractions in copper powder for the hexagonal unit cell with predictions for the purecopper powder compact obtained from numerical integration. In the figures, the solid linedenotes the average normal traction on the outer perimeter of the unit cell normalized bythe yield stress of copper at 800 K, whereas the dashed line shows the average axial stress.Table 2 contains a list of material parameters used in the analysis. The circles and trianglesin Figs 3 and 4 are the results for pure powder. The agreement with the finite element resultsfor pressurization of the composite material is obvious. This confirms the result of a pre-vious analysis by McMeeing [36] in Stage I, which showed the presence of fibers has noeffect on the pressure required to consolidate the material. The same result holds forGurson's model when it is used to evaluate plastic deformation of powder compact aroundspherical inclusion [37] and cylindrical inclusion [38]. Hence a combination of these twomodels yields the same result. Figures 3 and 4 also show that in pressurization, the averageaxial stress Ez on the powder compact is very close to the average pressure P.

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1. 51

0. 50 0.65

C o p p e r p o w d e r w i t h 1 0 % r i g i d f ibers by volume fraction /P - app l ied p ressure ]~;~ - a v e r a g e a x i a l stress ]~" F in i te e lem ent resu l ts fo r / / / x_ - - the com pos i te mater ia l P l ay / ~ /

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0.7 0.75 0,8 0.85 0.9 0.95O V E R A L L R E L A T I V E D E N S I T Y

F i g . 3 . P l o t o f t h e n o r m a l i z e d a v e r a g e l a t e r a l a n d a x i a l s tr e s s e s o n a c o p p e r p o w d e r c o m p a c t v s t h eo v e r a ll r e l a t iv e d e n s i t y d u r i n g p r e s s u r i z at i o n . T h e f i b e r v o l u m e f r a c t i o n i s 1 0 % .

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Fin ite element simulation of pow der consolidation 8913.5

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