Model for uniaxial compaction of ceramic powders C.-H. Park, S.-J. Park, H. N. Han, K. H. Oh, and K. W. Song

A phenomenological yield criterion has becn pro­posed for modelling the uniaxial compaction pro­cesses of various ceramic powder compacts on the basis of continuum mechanics. It includes three para­meters to cha racterisc the geometric hardening of thc powder compact and the strength of the base matcria l. The model was applied to uniaxial cO !11 paction of three ceramic powders which possess diffcrent pa r­ticle size distributio n, particle morphology, and base material property. Thc valucs of parameters in the yield cri terion werc determined through the uniax ia l compaction experiment. Using the yield criterion, thc elastoplastic fi nite element calculation was carricd o ut to analyse the compaction of the three ccramic powders. PMj0827

Dr C.-H. Park, Dr S.-J. Park, and Professor Oh are in the Division of κ1atcrials Science and Engineering, College of Engineering, Seoul National University, Seoul 151 - 742, Korea, Dr Han is part of thc Sheet Products and Process Research Team, Technical Research Laboratories, Pohang l ron and Steel Co. Ltd, Pohang 790- 785, Korea, and Dr Song is in the Department of Advanccd LWR Fuel Developmcnt, Korea Atomic Encrgy Research Institute, PO Box 105, Daeduk-Danji, Ta멍on 305- 600, Korea. Manuscrip t received 8 December 1998; acccptcd 24 M arch 1999.

~ 1999 10M Communications Ltd

INTRODUCTION T he d ic compaction of powders has been used in ma nufac­turing components for a broad range of applications. [n cngineering applications, the grccn compact of uniform density is a fundamental requi rement for the production of a good quality and high strength part. Density inhomogen­eity can be caused by friction force owing to interparticle movement and rela tive s[i p between the powder particles and the die wall. Also, the die geometry and the scquence of punch movements results in a lack of density homogen­eity for a compact of complex shape. Therefore, creating the right tooling design is very important for the success of powder compaction processes.

Generally, the design can be achieved by an empirical approach or a computer aided approach. A computer aided approach offers the designer a computa tional tool to reduce time and cost fo r process deve[opment, using an appropriate mathematical model to sim ula te a nd investigate thc com­paction pro야ss without actually constructing the system. For the computer aided approach, several mathematical models for porous material have been proposed.I-5 Lee a nd Kim4 modified a yield criterion for porous material which was suggested by Doraivelu et al.3 and could incorporate one empirical parameter that can be estimated from the yield stress v. initial relative density data. Using

ISSN 0032- 5899

the yicld critcrion, Ha n et al.5 for l1l ulated an elastoplastic finite element code and analyscd thc deformation of sintered porous metals in sil1lple upsetting,5 indenting,5 ring com­pression,6 and hot forging.7 Also, Han et al.8 calculatcd thc forging ’imit curves of sintered porous l1letals with thc Lce- Kuhn initial il1lperfection model.9

In thc case of powder compaction at room temperature, the manner of densifìcation differs from tha t of sintered porous material. ln cold compaction, the densifìcation of a powder cO l1lpact can be classificd into two stages. IO

-12 Tn

the fìrst stagc, where the arra ngement of powdεr particlcs changes, powder particles rearrange by sliding a nd local plastic dcformation or fracture at surface irregularitics. 12

Tn the sccond slage where the rclative motion among powder parlicles is small or negligible, the manner of densifìcation becol1les similar to that of sintered po rous l1la lerial.

Thc shapc and initia l porosity of powder particles arc important materia l characteristics at the firs t stage, whcre sliding and local plastic deformat ion or fracture play an important role. Generally, a t the initial stage of compaction, the powder cO l1lpact of high porosity or irregular shapc shows low apparent relative density and can be densified more casily than the compact of low porosity or spherical shape. T hus, the eITects of particle shape or porosity have to be taken into account in thc yicld criterion to simulate overa ll powder cOl1lpaction. Park et alY modified the yield criterio n suggested by Lee and Kim,4 so tha t the modificd yield criterion could successflllly incorporate empirical parameters which reflect the characte ristics of coppcr powdcrs of diITerent particle shape during uniaxial compactlOn.

ln the case of ceramic powdcrs, it is not certain that the yield criterion proposed by Park et alY could be used in analysis of compaction or other densificat ion processcs, because ccramic powders have no plasticity. In the present papcr, thc densification behaviour of ccramic powders is a na lyscd lIsing the yield criterion sllggested by Pa rk et al.13

The relatio n between para l1leters in the yield criterion and morphological and mechanical characterist ics of ceramic powders is investigated. Also, the elastoplastic fìnite element calculation is carried out to simulate the uniaxial com­paction process of ceramic powders

YIELD C RITERION A numbcr of authors p roposcd yicld criteria of po ro us materia l which can be genera liscd in thc fo llowing form

AJ;+ βJr = “ õ = y~ . (1)

where '1 = yV yÕ, J1 = all + a22 + a33

Jí = Wall - a22)2 + (azz- σ33)2 + (a33 _ a ll )2]

+ ai2 + a~3 + a~1 Jí , J 1, and R are second deviatoric slress invariant, fìrSI stress invariant, and rclative densily respectively; A, ß, and '1 are functions of relative density; 1';‘ is the yield st rcss of porous malerial having relative density R, Yo is thc yicld

Powder Metallurgy 1999 Vol. 42 No. 3 269

270 p，ω‘k el 씨 Model for uniaxial compaction ofceramic powders

stress of base materia l. Expressions for A, B, and '1 of the proposed yield criteria are summarised in Table 1.

From experimental data, Lee and Kim noticed that the yield stress of sintered porous metal ι varies linearly with the yield stress of incompressible base metal YO (Ref. 4). They proposed an empirical yield criterion of a porous metal through a parameter‘ ”

l - R2 _

(2 + Rι)Ji + -5- Jf=ηY5 = Y~ . . .. (2)

'1 = 혈 = (혈y whcre Rc is an expcrimental parametcr that can be inter­preted as the critical relative density, where the yield stress of porous metal is zero; that is 짜 =0 at R=Rc. The parameter 끼 depends only on the relative density o f porous metal and represents the square of the ratio apparent yield stress/ yield stress of incompressible base metal (YR : YO) . lt can be seen in equation (3) that ι increases as the relative density increases and finally becomes YO when the relative density becomes unity.

The hardening, which is dependent only on relative density, is termed geometric hardening and could be represented by parameter '1 in equation (2) . The hardening, which depend s on the change of mechanical properties of base material, is termed strain hardening and could be represented by parameter YO in equation (2). Thus, the total hardening of porous material is affected by geometric hardening and strain hardening such as ’1 YÕ in equation (2) For non-porous material whose relative density is unity, equation (2) becomes the Von Mises yield criterion.

Park et al. proposed a new empirical geometric hardening parameter '1 fo r powder compacts13

Y~ (R - l셔\m '1 = ，，~ = 1-, -~-' I ....... . (4)

YÕ \ l - R,J where RT and m are experimental parameters varying with the characteristics o f the powder such as shape and size distribution. RT is the tap relative density of powder while parameter m is incorporated to reflect the effects of particle sliding and local plastic deformation or fracture in the first stage of densification on the geometric hardening and may be called the geometric hardening exponent. Applying equation (2) to metal powders, Park et al. used the fol lowing expression fo r a f1 0w curve to describe the strain hardening of incompressible base metalJ3

YO = Kε8 . ‘ ’ ‘ ’ . . . . (5)

where K is the strength coefficient, n is the strain hardening exponent, and [0 is the accumulated plastic strain in incompressible base metal. The mechanical properties of

Table I Expressions for A, B, a nd ."

A R n

Shima and OyaneJ 3 R2n 9“

2(1 _ R)2m

12 I - R 4R2 Gurson2

5- R 5 - R 5+ R

1 - R2 R2 _ 1간 Doraivelu et (1 1.3 2 + R2

1 - R~ 3

2 + R2 1- R2

(짧~y Lcc and Kim4

3

2 + R2 1- R2

(환줬~r Park et al‘ 13 3

Powdcr Melallurgy 1999 Vol. 42 No. J

the matrix of metal powder may not be the same as that of the bulk metal of the same kind owing to dif-rerent metallurgical history a nd oxidation on the surface, etc. The parameters m, K , and n in equations (4) and (5) could be obtained by best fi tting to experimental data.

In the case of metal oxide or ceramic material, fracture occurs before plastic yielding. 1n the present work, it is assumed that ceram ic materials have constant yield st ress

YO = K . . . . . . ‘ . . . . . (6)

(3) where K is constant

A combination of equations (2), (4), and (6) gives the final fo rm of the yield criterion for a ceramic powder compact

1 - R2 _ ( R - RT \ m _

(2 + R')J~ + -~ - Jf = 1 -, - n- ' I Kι = '1 Y5 \ 1 - RT)

(7)

ANALYSIS OF U N IAXJAL OIE COM PACTION For uniaxial die compaction, the relation between the compaction pressure and relative density can be derived from equation (7). Assuming that there is no friction betwecn powder compact and die wall, i.e. no shear stress, the stress state and strain rate of the powder compact during uniaxial die compaction becomes

a33 = P, all = a22 = S, and a12 = a23 = σ3 1 = 0

ι 1 1 = 822 = ι 12 = ß23 = ι31 =0 (8)

where P and S are uniaxial compaction stress and radial stress respectively. Applying the associated flow rule to the yield criterion, strain rate 6ij can be expressed as

6 ij = 척 [(2 + R2 )aij - R2akk시J . . . . . (9) 2YR

where

1 (2 캡 = ~ ~~ [(8 11 - 822 )2 + (è22 - 깅33 )2 + (ε33 -깅 1 1fJ

2 + R" 13

+ (y션 }녕3 + 성I)}

+ _. _~ ， ( e ， ， + 8?? + κ" )2 3(1 - R‘) 、 “ “ 0 "

In combining equations (8) and (9), a relation bctwcen S and P can be obtained

R2

S =-, - .P ‘ • ‘ • . . • (1이 2 - R2

As the shear strain rate components are zero undcr no frict ion, the pressure applied on the top surface of compact P is derivcd from equations (9) and (10)

P = I(~二RT'f_(2 - R2

) 11/2

- I \ 1 - RT ) (1 - R2)(2 + R2) I ) I l (

It can be noted in eq uation (11) that as relative density approaches unity, uniaxial pressure approaches infinity, thus the effects of m and K cannot be distinguished. ln the usual compaction range, R ~ 0'8, the effects of m and K can be seen in Fig. 1. Figure la shows the variation of uniaxial compaction pressure P for d iITerent m values as a function of relative dens ity when Rr = 0.3 and K = 4000 in eq uation ( 11 ). At high values of m, the curve increases slowly at low relative density and increases rapidly at high l히ative density. Therefore, m characterises the shape of press ure- relative density curve of uniaxial compaction. Figure lb shows the variation of pressure for different K values as a function of relative density when RT = 0.3 a nd m. = 4 in equation ( 11). As K increases, pressure increases

271 Modcl fo r uniaxial compaction of ccramic powdcrs Park e! (11.

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Initial mesh for finite element analysis of single action uniaxial powder compaction: units in mm

Here VR is Poisson’s ratio of the powder compact given by

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EXPERJMENTAL PROCEDURE Commercial Al20 3 and Zr02 powders and U02 powdcr were used_ The particle shape of the powders was observed by scanning electron microscope. Particle size distributions based on volllme percentage were obtained lIsing a Malvern laser particle size analyser. Tap relative density of powders was measllred by tapping and measuring volume and weigh t. The measured tap relative density RT of each powder is given in Table 2. The single action uniaxial compaction experiments were carried out using an J nst ron tension and compression testing system in the presSllre range 0- 450 MPa. The diameter of the steel die was 15 mm and the crosshead speed of the lnstron was 2 mm min - 1. After lIniaxial compaction, the average density of the green compact was obtained by measuring the dimensions and weight. During uniaxial die compaction, compaction strokc and load wcre measllred_ Figure 2 shows the drawing and finite element mesh for analysing lIniaxial compaction behaviollr of powders

0.9 0.8 O 0.3

“ RT = 0-3, K = 4000; b RT = 0,3, m = 4

Pressure as function of relative density

proportionally_ Therefore, K characterises the level of lIniaxial compaction pressure

RESULTS AND DISCUSSION Figure 3a and b shows scanning electron micrographs of Al20 3 powder consisting of small constitllent particles_ Tt can be seen in Fig. 3b that the constitllent particle size of A120 3 powder is abollt 1- 5 μm_ The constitllent particles are anglllar and show a smooth sllrface_ They form porolls agglomerates whose size is abollt 20- 40 μm as shown in Fig_ 3a. This agglomeration is a result of the manufactur­ing process. Figure 3c and d shows SEM images of Zr02 powders, showing rOllnd and irreglllar shaped parlicles

ELASTIC PROPERTY OF POWDER COMPACT ln this work, the elastic responses of the powder compact are assumed to be isotropic, and the self-consistent estimate by Budiansky 14 was employed to evalllate the elastic properties of a powder compact which is dependent on relative density and thal of the non-porolls base material. BlIdiansky’s method estimates the propert ies of a composite material which consists of a random mixture of N isot ropic constituents. The powder compact is asslImed to be a random mixture of void and non-porolls base material.

Since the bulk modlllus and shear modllllls of the void in the powder compact is zero, it follows from Budiansky’s estimates that the blllk modlllus BR and shear modlllus GR of the powder compact become

1- R R 1 - a + l - Q + a(Ro/ BR) I

and

l - R R

I - b + 1 - b + b(Go/ GR) l

where Bo and Go are the blllk and shear modllli of non­porous base material, and Values of measured RT and fitted K and 11 for

powders Table 2

”--

u‘

+ --

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’l -1)

= U02

-R m K

0.235 5‘ 19

4417

Zr02

0.371 3.80

3820

AI20 3

0’ 261 10-4

58300

and

b= 융(慧)

NO. 3 Yol. 42 Powdcr Mctallurgy 1999

272 Park et al. Modcl fol' uniaxial compaclion of ceramic powders

a

c

c

a, b A120 3; C, d Zr0 2; e,f U0 2 3 Scanning clcctron micrographs of studied powders

with a rough surface. Figure 3d and e shows U02 powde r with rOll nd particles and a smooth surface.

Figurc 4 shows the measured size distriblltions of all three powders. T he Al20 3 powder shows a narrow size distribution with an avcragc size of 52.06 μm. T he Zr02 powder shows a bimodal distribution with an average particlc sizc of 56.25 μm. The bimodal distribution of the Zr02 powder might resu lt in the highest tap relative density, as given in Ta ble 2. The U02 powder shows a uni form particle size distribution with an averagc sizc of 21.09μm.

The singlc action uniaxial die compaction was conducted for the thrcc powders. Figure 5 shows the measured compaction presSll re as a function of relative density during uniaxial compactio n for al l the powders. The compaction pressure of the AI20 3 powder increases very slowly at low relativc dcnsity up to a relative density of aboll t 0-4 and rapidly al high relalive density as rel“tive density increases The Al20 3 powders a re agglomerates of high porosity and resu lt in easy compaction a t low relative density. Thc compaction pressllre of thc Zr02 powder starts to increasc a t higher relat ivc dcnsity because of its high tap rclalivc dcnsity, and thcn increases most steadily as relative density

Powder Melallurgy 1999 Vo l. 42 NO. 3

increases. The compact ion curve of thc U02 powder shows intermediate behaviour betwcen that of AI20 3 and Zr02.

Using the uniaxial compaction data, m and K values in equation (11) for each powder could be determined by a no n-linca r best fiUing method with experimentally meas­ured RT • Figure 6 shows the measured and calclllated compaction pressure as a function of relalivc density for A120 3, Zr02' and U0 2 powders. T he dashed curves, using data from eqllation (11 ), in Fig. 6, show good agreement wilb experimental data. This indicates that the different ha rden ing behaviour of oxide powders could be described well by the three parameters m, K, and RT . Thc dctermined valucs of m and K are given in Table 2. As can bc seen in F igs. l a and 6a, the Al20 3 powder, whose compaction pressure increases s lowly a t low relative density and rapid ly a t high rela tive density, shows a high geometric hardening exponent m of 10-4. The high m valuc of thc AI20 3 powder reflects casy dcnsi fication owing to high initial porosi ly of agglomerates a t the first stage of compaction wherc particle sliding is an importan t mechanism of densifìcation. The Zr02 powder, which has the lowest inil ia l porosity or highest tap relati ve density, shows the lowest m valuc of

0.6

0.6

0.6

compaction could be determined from uniaxial compaction presslI re v. relative density data. The geometric hardening exponent m is related to the shape of the compaction curve. The strength coefficient of base material K determines uniaxial compaction pressure at a constant m. The densification behaviour of the A120 3, Zr02, and U02 powders were well described by the parameters m and K. An elastoplastic finite element code for ceramic powder compaction has been formulated using the new yield criterion. The calculated pressure v. relative density values in single action uniaxial compaction of ceramic powders were in good agreement with the experimental data

ACKNOWLEDGEMENT This proJκcαt was carried out under the N lIclear R&D Program by the Korean Ministry of Science and Technology.

Vo l. 42 NO.3

a A120 3; b Zr02; c U02 Pressure as function of relative density in uniaxial compaction of powders

273

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(a}

(b)

Model for uniaxial compaction of ccnunic powdcrs

*

0.5

0.5

1999

0.5

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Relative Density

Equation (11 ) (m=3.80 , K=3820) - - FEM

Equation (11) (m= 104. K=58300) -- FEM

. Equation (11) (m=5.194, K=4417) --FEM

Powder Metallurgy

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0 .4

0.3

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400

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300

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300

500

400

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3-8. The m value of U02 powder is 5.19 which is rather close to that of Zr02.

The strength coefficient K of AI20 3 powder is 58 300 which is more than 10 times the strength coefficients of Zr02 and U02. However, the hardness of A120 3 is only 1.5 times that of Zr02.15 Fracture stress of the A120 3 powder particle would be high owing to its smooth surface Therefore, the high fracture stress would result in high K values, because the compaction pressure wOllld be pro­portional to the fractllre stress of powder particles at the second stage of compaction. The K value of the Zr02 powder with a rough particle surface, has the lowest value of 3820 among the three powders

ln order to simulate these experimental results, finite element calculation for the uniaxial compaction was carried out. Determined m and K values and measured RT values (Table 2) were used for finite element calculation. The FEM results were presented as solid curves along with experimental results for each powder in Fig. 6. J n FEM calculation, the friction cocfficient between die and powder compact is set to zero because the detennined m and K values already included the friction etfects. lt can be seen that the calculated results are in good agreement with experimental data.

CONCLUSIONS The yield criterion, which can describe the densification behaviour of ceramic powder, was proposed. Two para­meters in the new yield criterion for ceramic powder

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274 Park et al. Model for uniaxial compaction of ccramic powders

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