Model for compaction of metal powders -...

International Journal of Mechanical Sciences 41 (1999) 121—141

Model for compaction of metal powders

Seong-Jun Park!, Heung Nam Han", Kyu Hwan Oh!, Dong Nyung Lee!!Division of Materials Science and Engineering, Seoul National ºniversity, Seoul 151-742, Korea,

"Research Center for ¹hin Film Fabrication and Crystal Growing of Advanced Materials,Seoul National ºniversity, Seoul 151-742, Korea

Received 11 July 1997; and in revised form 5 January 1998

Abstract

A new yield criterion for metal powder compaction based on continuum mechanics has been proposed. It includesthree parameters to characterize the geometric hardening of powder compact and strain hardening of incompressiblemetal matrix. The elasto-plastic finite element method to describe compaction of metal powders has been formulatedusing the new yield criterion. The values of parameters in the yield criterion could be determined through cold isostaticpressing (CIP). The finite-element method was used to simulate compaction behaviour of copper powders of differentshape and mean particle size. ( 1998 Elsevier Science Ltd. All rights reserved.

Keywords: yield criterion; metal powder compaction; continuum mechanics

Notation

B0

bulk modulus of base materialBR

apparent bulk modulus of porous materialCE elastic stress—strain matrixCEP elasto-plastic stress—strain matrixE0

Young’s modulus of base materialER

apparent Young’s modulus of porous materialET

tangent modulus of base materialG

0shear modulus of base material

GR

apparent shear modulus of porous materialJ1

first invariant of stress tensorJ@2

second invariant of deviatoric stress componentK strength coefficientm geometric-hardening exponentn strain-hardening exponentR relative density of porous materialR

Ccritical relative density at which yield stress of sintered porous metal is zero

RPC

critical relative density at which yield stress of powder compact is zero

0020-7403/99/$ — see front matter ( 1998 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 0 - 7 4 0 3 ( 9 8 ) 0 0 0 2 2 - 8

RT

tap relative density of powder½

0yield stress of base material

½

Rapparent yield stress of porous material having relative density R

dij

Kronecker’s deltaeij

strain tensore60

effective plastic strain of base materiale6R

effective apparent plastic strain of porous materiale600

effective plastic strain rate of base materiale60R

effective apparent plastic strain rate of porous materiale60v

volumetric strain rate of porous materialg geometric hardening parameter ("½2

R/½2

0)

lR

apparent Poisson’s ratio of porous materialpij

stress tensorp@ij

deviatoric stress component

1. Introduction

The die compaction of metal powders has been used in manufacturing components for a broadrange of application. This manufacturing technique provides a method for near net shape partfabrication by which subsequent finishing operations are minimized or eliminated. In engineeringapplications, the green compact of uniform density and low defect is a fundamental requirement forthe production of a good quality and high strength finished part. The inhomogeneity in density ofa compact can be caused by friction force due to inter-particle movement and relative slip betweenthe powder particles and the die wall. Also, the die geometry and the sequence of punch movementsresult in a lack of homogeneity in density for a compact of complex shape.

Therefore, the success of the powder compaction process is largely dependent on the properdesign of the tooling as well as the mechanisms involved in the process itself. Generally, the properdesign can be achieved in two ways; by an empirical approach and a computer-aided approach.Although the empirical approach has been applied successfully to produce many shapes, it isiterative in nature, involving extensive development and validation programs which involvesignificant cost and time. Conversely, a computer-aided approach offers the designer a computa-tional tool to reduce time and cost for process development, using an appropriate mathematicalmodel to simulate and investigate the compaction process without actually constructing thesystem.

For the computer-aided approach, many mathematical models of metal powder compactionhave been proposed and can be classified broadly into two groups; micromechanical models andcontinuum models. The micromechanical models are based on the micromechanical structure ofeither regularly [1] or randomly [2, 3] close packed spherical particles. In these models, contactinteraction between particles is investigated and related to the macroscopic variables such as thenominal pressure and the overall density. Fischmeister et al. [4] investigated the contact geometryof plastically deformed particles and determined the number of contacts and contact regions atpressurized compaction of bronze powder. Arzt derived the relation between the size of localcontact regions and density of the compact and determined the global compliance [2], assumingthe random dense packing [5, 6] of spherical particle for perfect plasticity and power law creep.Fischmeister and Arzt [3] studied the influence of strain-hardening plasticity and considered the

122 S.-J. Park et al. / International Journal of Mechanical Sciences 41 (1999) 121—141

final stage of densification to obtain complete compaction formulae. Helle et al. [7] obtainedsimplified governing equations for the complete densification procedure. The resulting equationshave been generalized to apply to more general loadings by Fleck et al. [8] in the case of perfectlyplastic powder behaviour and by Kuhn and McMeeking [9], for power-law creep. Diffusionalcreep was also analyzed by McMeeking and Kuhn [10].

The continuum models are based on continuum theory that treats powder compact as a porouscontinuous medium. The continuum models determine the yield condition of porous medium usingsecond deviatoric stress invariant and first stress invariant. Two well-known groups of continuummodels are those from soil plasticity and others from isotropic theory. Schwartz and Holland [11]carried out a high-pressure triaxial test to establish the relationship between hydrostatic stress andshear stress for an iron powder. For Mohr—Coulomb yield condition generally used in soilplasticity, they suggest that a cohesive stress is present at zero normal stress level and that thecohesive stress and internal friction angle are both density-dependent. Shima and Mimura [12]performed a triaxial test of ceramic powders, and suggested that there was a more significantcontribution from elastic behaviour for ceramic powders in comparison to iron powders whereplastic deformation was the most dominant mechanism. Recently, Gethin et al. [13] carried outa finite-element calculation employing a Mohr—Coulomb material model for the compaction ofiron, bronze, ceramic and carbon powders. The models of isotropic theory starts from the analysisof ductile fracture and deformation of sintered porous metal. Green [14] and Gurson [15]obtained yield criteria by combining Torre’s exact solution [16] for a volumetric response and theapproximation solutions for a deviatoric response. Kuhn and Downey [17], Shima and Oyane[18], and Kim et al. [19] have proposed empirical yield criteria. Lee and Kim [20] modified a yieldcriterion suggested by Doraivelu et al. [21] and it could incorporate one empirical parameter thatcan be estimated from the yield stress versus initial relative density data. Yield surfaces of thesemodels are elliptic and symmetric in deviatoric-hydrostatic stress space. These models are usedfrequently to simulate the deformation behaviour of sintered porous metal whose relative density issufficiently large such as over 0.8.

In case of powder compaction, the manner of densification varies as the shape of powder particlevaries. Generally, the powder compact of irregular particle shows low apparent relative density andis densified easily in initial stage of compaction, compared with the compact of spherical particle.The manner of densification also changes as the relative density of a powder compact increases. Incold compaction, the densification of powder can be classified into two stages [4, 22, 23]. In thefirst stage where the arrangement of powder particles fairly changes, powder particles rearrange bysliding and local plastic deformation at surface irregularities [23]. In the second stage where therelative motion among powder particles is small or negligible, plastic flow becomes homogeneousrather than confined in the vicinity of contact points. Progressive flattening makes the particlecenters closer to each other, producing overall densification.

The shape of powder particles is an important material characteristic at the first stage wheresliding and local plastic deformation play an important role. Thus, the effects of particle shape haveto be taken into account in the yield criterion to simulate overall powder compaction. Themechanical properties of metal powder can be regarded as different from those of the fully densifiedmaterial due to the effects of powder producing processes and the oxidized surface layer. Theseeffects of material properties on densification behaviour should also be taken into account in theyield criterion.

S.-J. Park et al. / International Journal of Mechanical Sciences 41 (1999) 121—141 123

In this paper, authors developed a new yield criterion that can take into account the effectsof morphological and mechanical characteristics of metal powders on densification behaviour.The densification behaviours of various copper powder compacts were analysed using the pro-posed model in cold isostatic pressing and uniaxial compaction. An elasto-plastic finite-elementcode was formulated incorporating the proposed model to simulate the compaction process ofmetal powders.

2. Yield criterion for metal powder compaction

In compressible solid material, the apparent elastic strain energy per unit volume ¼ due to theapparent stresses may be expressed as

¼"

12

pijeij"

12 Ap@

ij#d

ij

pkk3 B A

1#lR

ER

p@ij!

lR

ER

dijpkkB

"

12 A

1#lR

ER

p@ijp@ij#

1!2lR

3ER

p2kkB , (1)

where pij, e

ijand p@

ijare stress components, strain components and deviatoric stress, respect-

ively. ER

and lR

are the apparent Young’s modulus and Poisson’s ratio, respectively. Assumingthat yielding occurs when the elastic strain energy reaches a critical value, it follows from Eqn (1)that

3(1#lR)p@

ijp@ij#(1!2l

R)p2

kk"C, (2)

where C is a constant. The yield condition at a uniaxial stress state (p11"½

R) gives

C"3½2R, (3)

where ½R

is the apparent yield stress or flow stress of metal powder compact. It follows from Eqn(3) that the yield condition (2) becomes

2(1#lR)J@

2#

1!2lR

3J21"½2

R"g½2

0. (4)

J@2

and J1

are the second deviatoric stress invariant and the first stress invariant, respectively. Forincompressible materials, l

R"0.5, Eqn (4) becomes the von Mises yield criterion and ½

Rbecomes

½0, which is the yield stress or flow stress of incompressible material.Zhdanovich [24] suggested the following relation between apparent Poisson’s ratio and relative

density, R, for the porous metals,

lR"0.5Ra. (5)

Kuhn [25] obtained a"2 from experimental results with sintered porous metals. Therefore, Eqn(4) for sintered porous metals becomes

(2#R2)J@2#

1!R2

3J21"½2

R. (6)


Table 1Expressions for A, B and g

Authors A(R) B(R) g (R)

Gurson [15]12

5#R

1!R

5!R

4R2

5!R

Shima et al. [18] 31

9a2(1!R)2mR2n

Doraivelu et al. [21] 2#R21!R2

3

R2!R2C

1!R2C

Lee et al. [20] 2#R21!R2

3 AR!R

C1!R

CB2

Present 2#R21!R2

3 AR!R

PC1!R

PCBm

Lee and Kim [20] noticed that the yield stress or flow stress of sintered porous metal, ½R, varies

linearly with the yield stress or flow stress of incompressible base metal, ½0, from experimental

data. They proposed an empirical yield criterion of a porous metal through a parameter, g, asfollows:

(2#R2)J@2#

1!R2

3J21"g½2

0"½2

R, (7)

g"½2

R½2

0

"AR!R

C1!R

CB2, (8)

where RC

is an experimental parameter that may be interpreted as the critical relative densitywhere the yield stress or flow stress of porous metal is zero; that is, ½

R"0 at R"R

C. The para-

meter g depends only on the relative density of porous metal and represents the square ofratio of the apparent yield stress, ½

R, to the yield stress of incompressible base metal, ½

0. A

number of authors proposed yield criteria of porous metal which can be generalized in the follow-ing form:

A(R)J@2#B(R)J2

1"g(R)½2

0"½2

R. (9)

Expressions for A, B and g of other authors in Eqn (9) are summarized in Table 1.In Eqn (8), the apparent yield stress of porous metal increases as the relative density increases

and finally equals to the yield stress of incompressible base metal when the relative density becomesunity. The hardening which is dependent only on relative density is termed ‘‘geometric hardening’’and it can be represented by parameter g in Eqn (7). The hardening which depends on the change ofmechanical properties of incompressible base metal could be termed ‘‘strain hardening’’ and isrepresented by parameter ½

0in Eqn (7). Thus, the total hardening of porous metals is affected by


‘‘geometric hardening’’ and ‘‘strain hardening’’ such as g½20

in Eqn (7). For non-porous metalswhose relative density R is unity, Eqn (7) becomes the von Mises yield criterion. Equations (7) and(8) could accurately describe the densification of sintered porous iron specimens with various initialporosities under hydrostatic pressure [26]. Han et al. [27] formulated a elasto-plastic finite-element code for analyzing deformation of sintered porous metals using Eqns (7) and (8), andpredicted the deformation behaviour of sintered porous metals in simple upsetting [27], indenting[27], ring compression [28] and hot forging [29]. Also, Han et al. [30] calculated the forging limitcurves of sintered porous metals using Eqns (7) and (8), with the Lee and Kuhn [31] initialimperfection model. However, during analysis of powder compaction, Eqns (7) and (8) requireda modification, because they do not reflect the effects of the morphological and mechanicalcharacteristics of powders on densification behaviour, especially at the first stage.

In this work, we have investigated the geometric hardening and strain hardening of metalpowder compacts and suggested a new yield criterion by which the overall densification behavioursof various metal powder compacts can be described.

2.1. Geometric hardening of power compact

The geometric hardening can be described as the increase in flow stress caused by the increase ofrelative density of a porous medium. In case of sintered porous metal, the flow stress increaseslinearly as the relative density increases as can be seen in Eqn (8). However, the linear relationshipbetween relative density and flow stress is no more suitable for metal powder compacts, because thedensification mechanism of powder changes during compaction. When the relative density ofa powder compact is low, rearrangement of particles occurs through sliding and local plasticdeformation. Local intense deformation occurs at surface irregularities of powders of which theparticles are in non-spherical irregular shape [23]. These sliding and local plastic deformationmake the densification of powder compact relatively easy in the early stages of compaction. Inother words, the increase in flow stress of powder compact is very small with the occurrence ofsliding and local plastic deformation.

At the second stage, interlocking between particles prohibits the relative motion of sliding.Plastic deformation occurs more homogeneously in the particle, since the surface irregularitiesdeformed flat and work-hardened. As a result, the effects of sliding and local plastic deformation ondensification of powder compact vanish when the relative density of powder compact becomeshigher. At this state the powder compact loses the characteristics of granular material and starts tobehave like a sintered porous metal. This change would result in a nonlinear geometric hardeningmanner of metal powder compact. To describe this nonlinear geometric hardening, the authorsproposed a new empirical geometric hardening parameter, g, as follows:

g"½2

R½2

0

"AR!R

PC1!R

PCBm, (10)

where RPC

and m are experimental parameters varying with the characteristics of powder such asshape and size distribution. R

PCmay be interpreted as the critical relative density in which the flow

stress of metal powder compact becomes zero, i.e. ½R"0 at R"R

PC. In this work, we assume

RPC"R

T, where R

Tis the relative tap density of metal powder. The parameter m is incorporated to


Fig. 1. Normalized yield stress as a function of relative density at various geometrical hardening exponents, m.

reflect the effect of sliding and local plastic deformation on the geometric hardening and may becalled ‘‘nonlinear geometric-hardening exponent’’. Figure 1 shows the variation of geometricalhardening manner of powders, whose relative tap density is 0.45, at various values of m. Thevertical axis is the ratio of the apparent yield stress of powder compact to the yield stress ofincompressible base metal. This ratio is equivalent to the square root of g and becomes zero atrelative tap density and unity when the relative density is unity. The solid straight line in Fig. 1represents the case where m equals 2. In that case, Eqn (10) becomes Eqn (8), which means that thedensification behaviour of the metal powder compact is the same as that of sintered porous metal.That is, the flow stress of metal powder compact increases linearly with the relative density, i.e.linear geometric hardening. At m value larger than 2, the increase rate of yield stress ratio is small atlow relative density and high at high relative density.

It follows from Eqn (10) that the yield criterion of Eqn (7) becomes

(2#R2)J@2#

1!R2

3J21"g½2

0"A

R!RT

1!RTBm½2

0. (11)

2.2. Strain hardening of incompressible base metal

In order to obtain the flow curve that can describe the strain hardening of incompressible basemetal of powder compact, the following equation can be used:

½0"Ke6 n

0, (12)

where K is the strength coefficient, n is the strain hardening exponent and e60

is the accumulatedplastic strain in incompressible base metal. The mechanical properties of matrix of metal powdermay not be the same as that of bulk metal of the same kind due to different metallurgical historyand oxidation on the surface, etc. Thus, the parameters of m, K and n in Eqns (11) and (12) could beobtained by the best fitting of the relationship between hydrostatic pressure and relative density


measured during cold isostatic pressing (CIP). Since there is no friction between powder and diewall during CIP, the values of m, K and n determined from CIP are not affected by the frictionbetween powder compact and die.

Combination of Eqns (11) and (12) gives the final form of the yield criterion for metal powdercompact as follows:

(2#R2)J@2#

1!R2

3J21"A

R!RT

1!RTBm(Ke6 n

0)2"g½2

0. (13)

2.3. Analysis of cold isostatic pressing

For cold isostatic pressing, the yield criterion of Eqn (13) can be expressed as

3(1!R2)P2"AR!R

T1!R

TBm(Ke6 n

0)2, (14)

where P is the hydrostatic pressure. The relationship between the apparent plastic deformationenergy rate and that of base metal is given by

½Re60R"R½

0e600, (15)

where e60R

is the effective apparent plastic strain rate. From Eqns (10) and (15), the relationshipsbetween the apparent plastic strain and strain rate of powder compact, and those of base metal aregiven by

e600"

JgR

e60R

and e60"P

JgR

de6R

(16)

Combining of Eqns (14) and (16) yields the following equation:

3(1!R2)P2"AR!R

T1!R

TBm

GKCP AR!R

T1!R

TBm@2 1

Rde6

RDn

H2. (17)

From the yield criterion of Eqn (13) and the associated flow rule, the apparent plastic strain rate isexpressed by

(e60R)2"

23(2#R2)

[(e601!e60

2)2#(e60

2!e60

3)2#(e60

3!e60

1)2]#

13(1!R2)

(e60v)2, (18)

where e601, e60

2and e60

3are the principal strain rates and e60

vis the volumetric strain rate. In hydrostatic

pressing, e601"e60

2"e60

3, and the relationship between relative density and volumetric strain rate,

e60v"!RQ /R gives the following equation:

(e60R)2"

13(1!R2) A

RQRB

2. (19)

From Eqns (17) and (19), the following equation can be obtained:

P"C1

3(1!R2)D1@2

AR!R

T1!R

TBm@2

GKCPAR!R

T1!R

TBm@2 1

M3(1!R2)N1@2R2dRD

n

H. (20)


Fig. 2. Coordinate system for uniaxial compaction.

In the yield criterion for metal powder compaction of Eqn (13), parameters K, n and m, can beobtained from Eqn (20) and CIP data using a nonlinear optimization method [32].

2.4. Analysis of uniaxial die compaction

For uniaxial die compaction, the relation between the load and relative density can be derivedfrom Eqn (13). Figure 2 shows a powder compact and the coordinate system. Assuming that there isno friction between powder compact and die wall, i.e. no shear stress, the stress state and strain rateof powder compact during uniaxial die compaction become as follows:

p33"s, p

11"p

22"t and p

12"p

23"p

31"0

e511

"e522"e5

12"e5

23"e5

31"0 (21)

Applying the associated flow rule to the yield criterion, Eqn (13), strain rate, e5ij

can be expressed as

e5ij"

e60R

2½R

[(2#R2)pij!R2p

kkdij], (22)

where

e60 2R"

12#R2 G

23

[(e511!e5

22)2#(e5

22!e5

33)2#(e5

33!e5

11)2]#(c5 2

12#c5 2

23#c5 2

31)H

#

13(1!R2)

(e511#e5

22#e5

33)2.

Combining Eqns (21) and (22), the relation between s and t can be obtained as follows:

t"R2

2!R2s. (23)

As the shear strain rate components are zero under no friction condition, the pressure applied ontop surface of compact, s, is derived from Eqns (22) and (23) as follows:

s"SAR!R

T1!R

TBm (2!R2)

(1!R2)(2#R2)½

0, (24)

where ½0"Ken

0.


Using Eqn (23), the first stress invariant, J1, and the second deviatoric stress invariant. J@

2, can be

expressed as follows:

J1"

2#R2

2!R2s and J@

2"

43 A

1!R2

2!R2B2s2. (25)

Using Eqns (24) and (25), the stress path during uniaxial die compaction can be calculated. Thesecalculated stress path on the J

1and J@

2space can be compared with stress path from FEM

calculation and experimental measurement.

3. Elasto-plasticity of powder compact

3.1. Elasto-plastic constitutive equations of metal powders in finite-element method

For the yield condition of Eqn (13), under isotropic hardening, the yield function F at time t#*tcan be expressed as

F"(2#R2)J@2/3#(1!R2)J2

1/9!g½2

0/3, (26)

where R, ½0

and g are the state variables which are dependent on the plastic strains, ePij. The yield

function, F, is used to calculate the plastic strain increments, dePij, and the matrix notation of the

stress increments, dp6, as follows:

dePij"j

LFLp

ij

"

qTCEdepTq#qTCEq

LFLp

ij

(27)

dr"CE(de!deP)"CEPde"CCE!qTCE(qTCE)TpTq#qTCEqDde (28)

Here, de is the matrix notation of total strain increments, j is the positive scalar and CE and CEP arethe relative-density-dependent elastic stress—strain matrix and elastoplastic stress—strain matrix,respectively. Vectors, pT and qT, are defined by

pT"[p11

p22

p33

p12

p23

p31

] (29)

qT"[q11

q22

q33

2q12

2q23

2q31

] (30)

pij"!

LFLeP

ij

and qij"

LFLp

ij

, (31)

where pT and qT are transposes of p and q, respectively.The relationship between the apparent plastic deformation energy increment per unit volume of

powder compact, dwP, and that of matrix, dwP0, is given by

dwP"RdwP0

or pijdeP

ij"R½

0deP

0, (32)

where deP0

is the effective plastic strain increment of the matrix. Since the yield stress is a function ofthe plastic work per unit volume, we can evaluate p

ijand q

ijusing the following equations:

pij"

23

gRC

E0E

TE

0!E

TDp

ij, (33)


qij"

2#R2

3p@ij#

29

(1!R2)J1dij, (34)

where E0, E

Tand p@

ijare the elastic modulus, tangent modulus of the non-porous base metal and

the deviatoric stresses, respectively.

3.2. Elastic properties of metal powders

In this work, the elastic responses of the metal powder compact are assumed to be isotropic, andthe ‘‘self-consistent’’ estimate by Budiansky [33] was employed to evaluate elastic properties ofpowder compact which is dependent on the relative density and that of the non-porous base metal,Budiansky’s method estimates the properties of composite material which consists of a randommixture of N isotropic constituents. The metal powder compact is assumed to be a random mixtureof void and non-porous base metal.

Since the bulk modulus and shear modulus of the void in metal powder compact is zero, itfollows from Budiansky’s estimates that bulk modulus, B

R, and shear modulus, G

R, of the metal

powder compact become

1!R1!a

#

R1!a#a (B

0/B

R)"1 and

1!R1!b

#

R1!b#b(G

0/G

R)"1, (35)

where B0

and G0

are the bulk and shear moduli of non-porous base metal, and

a"13 A

1#lR

1!lRB and b"

215A

4!5lR

1!lRB . (36)

Here lR

is Poisson’s ratio of the metal powder compact given by

lR"

3BR!2G

R6B

R#2G

R

. (37)

4. Experimental procedure

The experiments were conducted using commercially available electrolytic and water-atomizedcopper powders, characteristics of which are given in Table 2. As shown in Table 2, the apparentand tap densities of electrolytic copper powders are lower than those of water-atomized copperpowders, due to the dendritic shape of electrolytic powders. Atomized powder shows more or lessround shape. Figure 3 shows shape of the electrolytic and water-atomized powder.

The cold isostatic pressing experiments were carried out using a wet-bag tooling in the range ofhydrostatic pressure from 30 to 170 MPa. After the cold isostatic pressing, the density of greencompact was measured by Archimedes’ principle. The hydrostatic pressure versus relative densitydata obtained from CIP were best fitted to Eqn (20) to yield the values of K, n and m.

The uniaxial die compaction of copper powders was conducted in a 15 mm diameter die uptoa load of 5.5 tn using an Instron tension and compression testing system. The crosshead speed was1 mm/min. The powder was filled in the die by tapping. The initial relative density of powder


Table 2Characteristics of copper powders

Fine Coarse Fine Coarsewater-atomized water-atomized electrolytic electrolyticcopper powder copper powder copper powder copper powder

Apparent relative density 0.395 0.356 0.203 0.161

Tap relative density 0.449 0.427 0.239 0.240

Particle size distribution*!140 1.0!140 to #200 1.4 20.8 0.7!200 to #325 23.3 34.1 1.4 32.2!325 75.3 44.1 98.6 67.1

*US standard sieve, wt%.

compact was obtained by measuring the height of the powder compact and weight of greencompact. During uniaxial die compaction, the compaction stroke and the load were measured andcompared with the result calculated by finite-element method.

5. Results and discussion

5.1. Determination of parameters K, n from cold isostatic pressing

Figure 4 shows the relationships between the hydrostatic pressures and the relative densities offour kinds of copper powder compacts obtained from cold isostatic pressing. At the low value ofpressure, the relative densities of water-atomized powder compacts could not be measured due totheir poor strength of green compact. It is noteworthy that at pressures above about 90 MPa, orequivalently relative densities above about 0.6, the pressure versus density data from a narrowband regardless of powder characteristics, i.e., the compaction behaviours of the four differentkinds of powders are similar. At this stage, sliding and local plastic deformation at particleinterfaces seem to become negligible and densification behaviour become similar to that of sinteredporous metals in which geometric hardening exponent, m, is equal to 2, i.e. linear geometrichardening. Therefore, the parameters K and n of base metal of copper powder could be determinedby best fitting Eqn (20) in the range of the high relative density above 0.6. When m is equal to two,the critical relative density, R

T, of powder compact could be the critical relative density, R

C, of

sintered porous metal as in Eqns (7) and (8). The critical relative density of sintered porous copperis known to be 0.45 [27]. Figure 5(a) and (b) shows the hydrostatic pressure as a function of relativedensity from CIP and best fitted curves for the water-atomized copper powders and the electrolyticcopper powders, respectively. The constrained Rosenbrock optimization technique [32] was usedto obtain K and n in Eqn (20). The optimized values of K and n of the atomized powder compactsare 538.7 MPa and 0.1996, respectively. Those of electrolytic powder compact are 401.5 MPa and0.0731, respectively. The optimized strength coefficient, K, and strain-hardening exponent, n, ofwater-atomized copper powder are greater than those of electrolytic copper powder.


Fig. 3. Scanning electron micrographs of (a) fine water-atomized, (b) coarse water-atomized, (c) fine electrolytic and (d)coarse electrolytic copper powders.


.


Fig. 4. Hydrostatic pressure as a function of relative density in cold isostatic pressing of copper powders.

Fig. 5. Hydrostatic pressure as a function of relative density in cold isostatic pressing of (a) water-atomized and (b)electrolytic copper powders. Arrow indicates a boundary at 90 MPa.

From the calculated strain-hardening parameters of K and n, the geometric hardening exponent,m, can be obtained in the whole range of relative density using Eqn (20). Figure 6(a)—(d) show therelationships between the hydrostatic pressure and the relative density in the cold isostatic pressingof the four different copper powders. The curves represent the values calculated using Eqn (20). Them values in Eqn (20) were obtained by the best fitting of all the measured data including the taprelative densities, and are given in Table 3. The dashed curves in Fig 6(a), (b) and (c), (d) are the sameas the curves in Fig. 5(a) and (b), respectively. In Fig. 6(a) and (b), the solid and dashed curvesalmost coincide. The geometric-hardening parameters, m and R

T, of atomized copper powder

compacts are close to 2 and 0.45 which are those of sintered copper, respectively. This coincidenceshows that the geometric-hardening behaviour of water-atomized copper powder compact issimilar to that of sintered porous copper, which implies that hardening due to sliding and localplastic deformation after tapping is not major mechanism. On the other hand, the geometrichardening parameters, m and R

T, for electrolytic copper powders differ much from those for


Fig. 6. Hydrostatic pressure as a function of relative density in cold isostatic pressing of (a) fine water-atomized, (b)coarse water-atomized, (c) fine electrolytic and (d) coarse electrolytic copper powders.

Table 3The values of K, n, m, R

Cand R

Tfor four different copper powders

K n m RT

Fine water-atomized copper powder 538.7 0.1996 2.026 0.449Coarse water-atomized copper powder 538.7 0.1996 2.148 0.427Fine electrolytic copper powder 401.5 0.0731 3.255 0.239Coarse electrolytic copper powder 401.5 0.0731 3.464 0.240

sintered copper. The higher m values reflect the higher densification rate due to particle sliding andlocal plastic deformation or fracture of dendrites of electrolytic powder particles at the first stage ofcompaction. It can be noted that the different geometric-hardening behaviour of water-atomizedand electrolytic copper powder compacts are described well by the parameters, m and R

T. The

effect of mean particle size on the geometric hardening seems to be small because the m values offine and coarse powders are not quite different.


Fig. 7. Pressure as a function of relative density in single action uniaxial compaction of copper powders.

Fig. 8. Initial mesh for finite-element analysis of single-action uniaxial powder compaction.

5.2. Uniaxial compaction

The single action uniaxial die compaction was conducted for the four kinds of copper powders.Figure 7 shows the compaction load as a function of relative density in uniaxial compaction for allcopper powders. The difference among data of different shaped powders in low relative densityregion can be seen clearly in Fig. 7. In order to simulate these experimental results, the finiteelement calculation for the uniaxial compaction was carried out. Figure 8 shows the initial mesh forthe finite-element calculation. The Poisson’s ratio and the elastic modulus of base copper used in


Fig. 9. Pressure as a function of relative density in single action uniaxial compaction of (a) fine water-atomized, (b) coarsewater-atomized, (c) fine electrolytic and (d) coarse electrolytic copper powders.

FEM calculation are 0.34 and 123 GPa, respectively. Figure 9(a)—(d) shows the FEM results atdifferent friction coefficients along with the experimental results for each kind of powder. Thevalues of parameters included in the yield criterion of Eqn (13) for the finite-element calculation aregiven in Table 3. The solid and dashed curves in Fig. 9(a)—(d) are calculated at the frictioncoefficients of 0.0 and 0.1 between the die and the powder compact, respectively. It can be seen thatthe loads are not sensitive to friction conditions and the calculated results are in good agreementwith experimental data, indicating that the yield criterion of Eqn (13) can simulate well the wholerange of densification behaviour of copper powder compacts.

Figure 10(a)—(d) shows the yield surface obtained from Eqn (13) for each kind of copper powder.As the hydrostatic and deviatoric stresses are divided by the yield stress of incompressible coppermatrix, the expansion of yield surface shows only the geometric hardening manner of the powdercompact. The geometric hardening manner of sintered porous copper which was proposed by Lee


Fig. 10. Yield surface and stress paths of (a) fine water-atomized, (b) coarse water-atomized, (c) fine electrolytic and (d)

coarse electrolytic copper powders in J3J@2/½

0and J

1/½

0space.

and Kim [20] is obtained from Eqns (7) and (8) and compared with that of each copper powder inFig 10(a)—(d). It can be seen in Fig. 10(a) and (b) that the geometric hardening manner of water-atomized copper powders is similar to that of sintered porous copper due to little contribution ofparticle sliding and local plastic deformation at surface irregularities to densification. However,Fig. 10(c) and (d) show the geometric-hardening manner of electrolytic copper powders which isdifferent from that of sintered porous copper. The electrolytic copper powder starts geometric-hardening from low relative density due to its irregular particle shape. The yield surface ofelectrolytic copper shows more expanded than that of sintered porous copper in the range of low


relative density. The stress path in uniaxial die compaction can be calculated from Eqns (24) and(25) using associated flow rule and assuming the rigid die and no friction between powder compactand die wall. The analytically calculated stress path is compared with the stress paths which areobtained from measured load or by FEM calculation for each copper powders in Figs 10(a)—(d).The stress paths obtained by different ways show good agreement with each other.

6. Conclusions

The yield criterion which can describe the mechanical behaviour of powder compact wasproposed. Three parameters in the yield criterion for metal powder compaction could be deter-mined from the hydrostatic stress versus relative density data. The geometrical hardening expo-nent, m, for more or less spherical water-atomized copper powders was close to 2 which is the valuefor sintered porous metal, the geometric-hardening exponent, m, for electrolytic copper powderswas larger than that for water-atomized copper powders.

A elasto-plastic finite-element code for powder compaction has been formulated using the yieldcriterion. The calculated pressure versus relative density values in single action uniaxial compac-tion of copper powders were in good agreement with experimental data. The geometric-hardeningmanner of each powder was compared with that of sintered porous metal. Water-atomized copperpowders showed a geometric hardening manner similar to that of porous sintered copper.Electrolytic copper powders showed a different geometric-hardening manner from that of poroussintered copper due to their irregular particle shape.

Acknowledgements

This work has been supported by Korea Science and Engineering Foundation through ResearchCenter for Thin Film Fabrication and Crystal Growing of Advanced Materials, Seoul NationalUniversity.

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