MODELLING OF THE PLASTICITY IN COLD COMPACTION OF METAL POWDERS
M. D. Rie a, J. M. P ado
Cen e Tecnològic de Man esa, Uni e si a Poli ècnica de Ca alunya, Spain
KEYWORDS: Plas ici y, G anula Ma e ials, Compac ion o Me al Powde s, Modelling.
1. INTRODUCTION
Powde Me allu gy, as a me alwo king p ocess, uni es o i s undeniable echnological in e es ,
impo an sa ings o ma e ial and ene gy; his has p omo ed i s accep ance in many di e en
indus ial sec o s. Mo eo e , his a i es when he Socie y wo ies abou he Ecology and he
p o ec ion o he En i onmen . This echnique does no gene a e smoke no chemical con aminan s.
Mos o he de eloped coun ies a e conscious o he necessi y o ecycle he ma e ials; hey a e
exhaus ible, mo e and mo e expensi e esou ces and o di icul eco e y. Ne e heless, me allic
powde s a e e y equen ly manu ac u ed om sc aps; mo eo e , PM componen s equi e e y
ew, o no, ope a ions o machining (ne shape and nea -ne shape) and, he e o e, i gene a es
minimum losses o ma e .
Ne e heless, Powde Me allu gy p esen s es ic ions; some o hem a e ine i able, depend on he
own p ocess and a ec mainly he design o he piece; geome ic equi emen s and hose
conce ning he oughness limi he possibili ies o his echnology. Ano he ype o limi a ions, also
well s ablished, a e mos ly associa ed o he compac ion s age: ic ion be ween pa icles and ools
induces a non-uni o m densi y dis ibu ion in he compac . The he e ogeneous s a e o s ess
de eloped p oduces e y equen ly c acks in he p e o m, especially du ing i s ejec ion om he
die, and he ac u e o he ools.
All hese p oblems ha e been adi ionally sol ed by means o ial and e o me hods. Howe e ,
he cu en de elopmen o new and mo e e icien ools o calculus educes he cos o he design
and manu ac u e p ocesses and con ibu es o imp o e he quali y o he p oduc . In his espec , he
main objec i e o he nume ical simula ion is o de e mine he op imal means o p oduce pieces
wi hou de ec s. In Me alwo king, he design and he con ol need a deep knowledge o he
mechanics o he de o ma ion du ing he p ocess; he in luence o a iables such as geome y,
ic ion and p ope ies o he ma e ials mus be well es ablished o de ine p ope ly he ools and
hei kinema ics, he pa ame e s o he p ocess and o p edic and p e en he appea ance o de ec s.
Compu e simula ion has become one o he mos in e es ing echnique in he mode n me alwo king
Indus y; howe e , i s success depends mainly on he model applied o ep esen he beha iou o
he ma e ial. Di e en ly o mos o he con en ional ma e ials, he mechanical beha iou o hese
agg ega es o pa icles is no ye well known. No many ime ago he sin e ing s age o he PM
echnique seemed o be he one esponsible o all he p ope ies o he inal componen . Recen ly, a
ABSTRACT: Powde Me allu gical Indus y is e y in e es ed in sp eading i s
applica ions and imp o e he quali y o he PM p oduc s. The e o e, a de ailed
knowledge o hei p ocesses, in o de o con ol hem, is necessa y. Many di e en
ypes o models ha e been used o ep esen he mechanical beha iou o hese
ma e ials; howe e , nowadays, mos o he g oups wo king in his ield admi ha
me allic powde s and g een compac s ha e o be conside ed as g anula ma e ials.
Plas ici y models specially de ined o geological ma e ials a e being applied on
me allic agg ega es. Ne e heless, none o he known ones can ep esen adequa ely
hei mechanical beha iou , specially in s a es o ailu e.
pa o he in e es has been cen ed on he compac ion and me allic powde s a e al eady ea ed as
g anula ma e ials. Ne e heless he e is no a gene ally accep ed heo e ical body o explain he
mechanical beha iou o he pa icles du ing hei cold compac ion; he e is no a gene al ag eemen
abou he plas ic model and he elas ici y has been p ac ically igno ed.
In his pape he au ho s p e end o p esen he di e en ype o models cu en ly en isaged in his
ield and gi ing in o ma ion a di e en le els; all o hem a e use ul o unde s and he beha iou o
hese ma e ials.
Consolida ion o a me allic powde begins by illing and ans e ing he powde in o he
compac ion die. This s age, cu en ly unde s udy, is cha ac e ised by he densi y dis ibu ion o he
mass o loose powde and depends no only on he na u e o he pa icles bu , also, on he me hods
o illing and ans e ence in o he die. Se e al models, based on di e en heo ies, a e being
published [1,2,3]. A e illing he mould, he powde is comp essed o p oduce a me allic
agg ega e. This p ocedu e has been quali a i ely desc ibed by Seeling and Wul [4] who de ined
h ee s ages du ing he comp ession. In he i s one, co esponding o he lowes le el o applied
p essu e, pa icles ea ange. The second pa in ol es elas ic and plas ic de o ma ion o pa icles
h oughou hei con ac a eas. These wo phenomena, sliding and de o ma ion, can occu
concu en ly om he beginning o he comp ession, as s a ed Fishmeis e , A z and Olsson [5].
Mo eo e , me allic pa icles s ain ha den; wo ypes o ha dening ac : ha co esponding o
pa icles and he geome ical one, due o he p og essi e inc ease o he con ac a eas be ween hese
pa icles. The e o e, in he las s age, o high s esses, he only mechanism go e ning he
densi ica ion is he elas ic de o ma ion. The powde compac is, he e o e, a
packing o de o med pa icles,
which a e able o mo e wi h ega d
o each o he ; i exhibi s he main
ea u es cha ac e ising a g anula
ma e ial: in e pa icle po osi y and
sliding o hese pa icles h oughou
hei con ac s. A ema kable ea u e
is i s dila a ion unde uniaxial
comp ession, which has been shown
o geological ma e ials [6] and,
mo e ecen ly, o g een powde
compac s [7, 8]. Figu e 1 p esen s he esul s co esponding o a simple comp ession es applied on
a g een e ous compac . Du ing he es , he e olu ion o he componen s o he s ain de ine h ee
pa s: a he beginning o he comp ession (I), he cu es show an ini ial oo co esponding o a
ansi o y plas ic beha iou which can be ela ed o some in e nal ic ion phenomena. Wi h u he
de o ma ion, he s ess g ows quickly wi h s ain (II), which has an impo an elas ic componen .
And a inal s age (III) whe e a sa u a ion s ess,
σ
max, is eached. The specimen ails du ing his las
egion. Ne e heless, be o e he ailu e and jus a he beginning o his hi d s age, he compac
s a s i s dila a ion (a
σ
SD
); he agg ega e expands, ins ead o densi ying as a po ous ma e ial (a
sin e ed compac , o ins ance).
A he p esen wo main ype o models a e being used o s udy and ep esen he mechanical
beha iou : Mic omechanical models and hose based on he Con inuum Mechanics.
2. MICROMECANICAL MODELLING
The in e es o de eloping mic os uc u al models is due o he ac ha desc ibing he beha iou o
he ma e ial a he mic os uc u al le el can lead o a be e unde s anding o he mechanisms ac ing
Fig. 1: E olu ion o he componen s o he s ain du ing an
uniaxial comp ession es , a e [9].
Axial s ess, MPa
Axial s ain
Radial s ain
Volume ic s ain
σ
max
σ
SD
du ing he compac ion. The in o ma ion abou he physical phenomena o he p ocess may be
c ucial o de ine a cons i u i e law. When analysing he compac ion p ocess om a mic omechanics
poin o iew, di e en app oaches can be dis inguished. The i s one consis s on s udying he po e
s uc u e and i s e olu ion du ing he applied loading pa h; he al e na i e analysis is based on a
pa icle a angemen .
The i s heo e ical model, in s ic sense, applied o s udy he comp ession o me allic pa icles
was de ined by To e [10]. He assumed ha he po es in he g een compac could be ep esen ed by
a single la ge po e in he cen e o a hollow igid, pe ec ly plas ic sphe e. By applying he T esca'
yielding c i e ion he calcula ed he po e olume as a unc ion o he hyd os a ic p essu e applied.
Heckel [11], Bocks iegel [12], Kla , Hewi , Wallace and de Malhe be [13] ha e also in es iga ed
his p ocess and de ined e isions o he To e' s model. G een [14] and Ga son [15] applied he
heo y o Plas ici y o a model also based on he change in he po e s uc u e.
Sunds öm and Fischmeis e [16] emphasised on he di icul y o ex end he one-po e model o a
sys em mul i-po e and s udied he comp ession o a conca e
po e de ined by ou pa icles; hey sol ed he beha iou o his
uni by means o he ini e elemen me hod (Fig. 2). This model
shows good ag eemen wi h expe imen al esul s a high
densi ies o he compac al hough i inco po a es he geome ical
ha dening o he sys em.
Fu he p og ess ega ding de ailed mechanisms is mainly based
on he e olu ion o a ays o pa icles in con ac . A e he i s
wo ks, in ol ing only wo sphe es (Mo ison and Richmond
[17], Kaka and Chaklade [18], ...), he mo e in e es ing ones
suppose egula o andom a angemen o monosized pa icles
submi ed o di e en s a es o s ess. The assump ion o
egula ly packed pa icles simpli ies he analysis. In p ac ical
si ua ions, howe e , he andom a angemen is mos likely o
occu , as demons a ed Sco [19] and Mason [20]. Fischmeis e ,
A z and Olsson [21] s a ed ha he ela i e mo emen among
pa icles is limi ed o an ini ial s age o he densi ica ion p ocess;
as he applied p essu e inc eases, duc ile pa icles de o m and
wo k ha dens; he con ac s be ween hem a y, in in ensi y
and in numbe . These au ho s ound expe imen ally ha he
a e age numbe o con ac s (coo dina ion numbe ) pe
pa icle inc eased almos linea ly wi h he compac ion
p essu e.
A z [22] sugges ed he use o an a e age Vo onoi cell. As
depic ed in Fig. 3, i consis s on a polyhed on con aining one
powde pa icle whe e he numbe o aces o adjacen cells is
de e mined by he numbe o nea es neighbou s o he
pa icle in ques ion. Du ing he compac ion, he pa icles g ow
in an imagina y way a ound ixed cen es wi h cons an
olume o he p e o m e ained. A z cha ac e ise he shape o
his cell and i s e olu ion by means o he Radial densi y
unc ion (RDF) gi en by Sco [19] and Mason [20]. He de i ed ela ionships be ween he shape o
he con ac s and he densi y o he agg ega e. Howe e his s udy is alid only o pe ec ly plas ic
ma e ials. A pos e io i, Fischmeis e and A z [23] inco po a ed he e ec o he s ain and he
geome ical ha dening. An scheme o his model is shown in Fig. 4.
Fig. 2: Bidimensional model
s udied by Sunds öm and
Fischmeis e [16]. Change in
he po e s uc u e o di e en
le el o po
osi y: A=13.7%,
B=7.3%, C=1.6%.
Fig. 3: Bidemensional scheme o
he Vo onoi cell, a e [21].
O he esea ches adop ed his ype o ea men .
Among hem, he wo k done by Helle, Eas e ling
and Ashby [24] should be men ioned. They
de ined a mac oscopic yielding unc ion
h oughou a simple ma hema ical ea men ,
alid o pu ely hyd os a ic s a es o s ess.
McMeeking [25], Xu and McMeeking [26] and
Fleck, Kuhn and McMeeking [27] ound a mo e
gene al unc ion which also ake in o accoun
shea s esses. Fig. 5 ep esen s he model
analysed by hese au ho s and a de ail o he
s esses in he con ac be ween wo sphe ical
pa icles.
Ogbonna and Fleck [28], s a ed ha he yielding beha iou o a pa icle agg ega e depends no only
on i s densi y, bu also on he applied loading pa h. Fig. 6 shows he yield loci co esponding o wo
di e en s a es desc ibed by hese au ho s.
This ield con inues gi ing e y in e es ing in o ma ion abou he compac ion o a angemen o
pa icles. As Mole us s a ed in a e y in e es ing pape abou cohesi e powde s, "F om he
p ac ical as well as om he scien i ic poin o iew, i should be use ul o sea ch o a connec ion
be ween he con inuum mechanics ap oach and he pa icles app oach, in o de o a i e a a
deepe insigh in o he beha iou o cohesi e powde s" [29]. The au ho s o he p esen pape
pa icipa e o his idea.
3. CONTINUUM MODELLING
Cold compac ion o me allic powde s has been s udied om a mo e heo e ical poin o iew: he
consolida ion, as a ime independen plas ic low p ocess, has been ep esen ed by means o g oups
o yield su aces dependen on he ela i e densi y, R, and on he in a ian s o he s ess. The
yielding o no ully dense ma e ials is mo e complica ed han ha p esen ed by he bulk ma e ial;
in he o me case he mechanical beha iou usually in ol es changes o olume; in his si ua ion
he yielding phenomenon is de e mined no only by he de ia o ic componen o he s ess, bu ,
also, by he hyd os a ic p essu e. The e o e, many esea ches de ined models o Plas ici y based on
modi ying he on Mises' yielding c i e ion in o de o include he e ec o he hyd os a ic
componen [30].
Fig. 4: Model o Fischmeis e and A z: a) he
pa icles g ow a ound ixed cen es; b) pa o
he ma e ial is ex uded o he oids; c) he
alue o he RDF
de e mines he cu en
numbe o coo dina ion. A e [23].
Fig. 5: Mac oscopic elemen co esponding o
he model by Fleck, Kuhn and McMeeking and
de ail o he con ac be ween pa icles. [27].
Fig. 6: Yield loci ob ained by means o isos a ic
comp ession and die comp ession. [28].
Mos o hese yield models gene a e ellip ical yield su aces which p edic he same beha iou in
ension and in comp ession (Fig. 7). The models p oposed by G een [14] and Gu son
[15] a e he mos impo an and well known,
specially he second one. In gene al, his ype o
Plas ici y laws a e use ul o model he mechanical
beha iou o po ous me als.
Roscoe and his co-wo ke s [32] showed ha a
g oup o s eel balls submi ed o di e en loading
pa hs, beha e like a g anula ma e ial, a
cohesionless ma e ial; hen, he ype o models
de ined speci ically o geological ma e ials can be
success ully applied o ep esen he mechanical
beha iou o agg ega es o me allic pa icles. They
de ined he concep o c i ical s a e and, based on
i , hese esea ches s a ed he CAM-clay Plas ici y,
de eloped o clays in he i ies and used ecen ly
o ep esen he cold comp ession o me al and
ce amic powde s. Fig. 8 shows he yield su aces in
he space o he hyd os a ic and de ia o ic s esses
co esponding o he modi ied CAM-clay model.
These models a e subs an ially di e en om he
quad a ic, symme ic yield su aces p e iously
de ined: hey ake in o accoun he limi ed cohesion
be ween pa icles; hence, he ensile s eng h o
hese ma e ials is much lowe han hei s eng h in
s a es o comp ession. Among his g oup o
Plas ici y models, he D ucke -P age /CAP one is
cu en ly he mos equen ly used o ep esen he
mechanics in he powde compac ion p ocess. I
seems o be app op ia e, especially in he
consolida ion egion. Ne e heless, he au ho s o
he p esen pape ha e some doub s abou i s
e ec i eness in ep esen ing i s pe o mance in
s a es o ailu e.
D ucke and P age [34] p oposed a ailu e c i e ion
ha consis s o a s aigh line, he ailu e line, he ailu e su ace (Fig. 9), in he space o he i s
and second in a ian s o he s ess and de ia o ic s ess enso s, J1 and J2D
, espec i ely. Bu hey
ealized ha mos o he ma e ials ha hey we e s udying showed plas ic s ain om he i s s ages
o he loading p ocess; he e o e, du ing he loading pa h, L-L' in Fig.9, he ma e ial is yielding
cons inuously up o he ailu e, o ul ima e s a e, which can be conside ed as he las yield su ace;
mo eo e , du ing he successi e yielding he ma e ial ha dens. This beha iou can be ep esen ed
by means o a se ies o yield su aces, he ha dening caps, p e ious o he ailu e. This idea was i s
p oposed by D ucke , Gibson and Henkel [35] and ep esen ed as i is shown in Fig. 10. Fo
simplici y hese au ho s supposed ha he caps we e o ci cula shape, bu i depends on he
ma e ial and i can be expe imen ally de e mined. The CAP model [36] ini ially used o simula ed
he plas ic beha iou o geological ma e ials, has also been shown as e y adequa ed in he case o
me al powde compac ion.
Fig. 7: Yield su aces calcula ed using he
ollowing equa ion: AJ2D + BJ12=YR2=
η
Yo2
(A, B and
η
a e pa ame e s dependen on he
densi y o he ma e ial; J2D and J1 a e he
second in a ian o he de ia o ic s ess and
he i s in a ian o he s ess enso ,
espec i ely; YR and Yo a e he yield s ess o
he ma e ial wi h a ela i e densi y o R and
he yield s ess o bulk ma e ial, espe i ely.
A e Lee and Kim [31].
Fig. 8: Modi ied CAM-clay Plas ici y
model. [33].

The o iginal model uses wo yield su ace segmen s: a
p essu e dependen D ucke -P age shea ailu e su ace and
a comp ession cap yield su ace. The ailu e su ace is
pe ec ly plas ic in he sense ha no mechanical s ain
ha dening occu s, bu plas ic low on his su ace p oduces
inelas ic olume inc ease; in o he wo ds, o s a es o s ess
on his su ace he sample dila es unde cons an applied
s ess. The equa ion desc ibing his ailu e su ace is he
ollowing:
1
= q-p an
β
-d = 0 (1)
q and p a e he de ia o ic and he hyd os a ic s esses,
espec i ely;
β
, is he angle o ic ion, and d, he cohesion
o he ma e ial. The shape o his su ace in he p-q plane
is, hen, a s aigh line
The cap yield su ace has an ellip ical shape in he p-q
plane and ha dens (expands) o so ens (con ac s) as a
unc ion o he olume ic plas ic s ain: olume ic plas ic
compac ion (yielding on he cap) causes ha dening, while
olume ic plas ic dila a ion (yielding on he shea ailu e
su ace) causes so ening. The equa ion desc ibing he cap
yield su ace is:
2= [(p-pa)2 + (Rq)2]1/2 - R(d+ pa
an
β
) (2)
po is an e olu ion pa ame e ela ed o he hyd os a ic comp ession yield s ess, pb
, which
ep esen s he olume ic ha dening o so ening, and R is a ma e ial pa ame e ha con ols he
shape o he cap. All hese ea u es a e shown in Fig. 11.
Ne e heless, he esul s p esen ed in Fig. 1 sugges ha a ce ain numbe o modi ica ions should
be in oduced in his la e model in o de o ep esen he plas ic beha iou o me al powde . The
successi e s a es o s ess exis ing in a compac du ing
an uniaxial comp ession es a e ep esen ed in he p-q
space by means o a s aigh line o slope 3 and passing
h ough he o igin. The ac ha du ing he comp ession
he sample dila es means ha he comp ession loading
pa h cu s he D ucke -P age line when he inc ease in
olume begins; bu , opposed o he beha iou o
geological ma e ials, dila a ion only con inues unde
p og essi ely g ea e applied s esses. The e o e, he
D ucke -P age line changes du ing his dila a ion
s age. This is e lec ed in equa ion (1) by a dec ease o
he ma e ial cohesion, d, and an inc ease in he ic ion
angle,
β
. The dila a ion o he sample akes place, as i
has been al eady s a ed, h ough an inc ease in he
dis ance be ween he cen es o neighbou ing pa icles; his gi es ise o a highe olume ac ion o
po osi y and a lowe ma e ial cohesion. The ha dening o he su aces o con ac be ween pa icles
inc eases he in e nal ic ion in he sample and, consequen ly, he ic ion angle. The uniaxial
comp ession es dos no gi e di ec in o ma ion abou he e olu ion o he cap yield su ace du ing
his ype o plas ic de o ma ion, bu he ac ha pa icles ha e ha dened should p oduce an
Fig. 9: D ucke -P age c i e ion.
d
ε
pij is an inc emen o he plas ic
s ain and d and
β
a e pa ame e s o
he ma e ial. The line L-L'
co esponds o a loading pa h. A e
[33].
Fig. 10: Mechanical beha iou o
s ain ha dening g anula ma e ials,
a e [33].
Fig. 11: Yield su aces o he D ucke -
P age /CAP model in he space o
hyd os a ic, p,-de ia o ic, q
, s esses,
wi h a ansi ion egion. A e [37].
expansion o he cap su ace. In o de o check he co ec ness o his la e sugges ion iaxial
comp ession es s on p e iously uniaxially comp essed
samples should be pe o med. Fig. 12 shows he expec ed
yield su aces o a sample be o e and a e an uniaxial
comp ession es , s opped be o e ailu e. Hence, he new
yield su ace has expanded, excep in he icini y o he
ensile p axis, in spi e o he ac ha he densi y o he
sample has dec eased; his is an impo an di e ence wi h
espec o he case o geological ma e ials. This pa icula
beha iou is he consequence o he mechanical ha dening
capaci y o he me al powde pa icles. In o de o e lec
he obse ed beha iou o me al powde compac s, he
pa ame e s pa and
β
in p e ious equa ions (1) and (2)
should be adequa ely modi ied. In geological ma e ials,
due o he ce amic na u e o hei pa icles, ha dening is
he esul only o olume dec ease; in me al powde s, he
plas ic de o ma ion o he pa icles has o be conside ed as
a new mechanism o sample ha dening. Hence, he
pa ame e s pa
and
β
should be dependen no only on
sample densi y bu , also, on pa icle ha dening.
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