E icien and accu a e app oach o powde compac ion p oblems
A. Pe
´ ez-Fogue , A. Rod ı
´guez-Fe an, A. Hue a
Abs ac In his pape , a new app oach o powde cold
compac ion simula ions is p esen ed. A densi y-dependen
plas ic model wi hin he amewo k o ini e s ain mul i-
plica i e hype elas oplas ici y is used o desc ibe he
highly nonlinea ma e ial beha iou ; he Coulomb d y
ic ion model is used o cap u e ic ion e ec s a
die-powde con ac ; and an A bi a y Lag angian–Eule ian
(ALE) o mula ion is used o a oid he (usual) excessi e
dis o ion o Lag angian meshes caused by la ge mass
luxes. Se e al ep esen a i e examples, in ol ing s uc-
u ed and uns uc u ed meshes a e simula ed. The esul s
ob ained ag ee wi h he expe imen al da a and o he
nume ical esul s epo ed in he li e a u e. I is shown
ha , con a y o o he Lag angian and adap i e
h- emeshing app oaches ecen ly epo ed o his ype
o p oblems, he p esen app oach e i ies he mass
conse a ion p inciple wi h e y low ela i e e o s (less
han 1% in all ALE examples and exac ly in he pu e
Lag angian examples). Mo eo e , hanks o he use o an
ALE o mula ion and in con as wi h o he simula ions, he
p esen ed densi y dis ibu ions do no p esen spu ious
oscilla ions.
Keywo ds Powde compac ion, ini e s ain mul iplica i e
plas ici y, consis en angen moduli, A bi a y Lag an-
gian–Eule ian (ALE) o mula ion, densi y-dependen
models, nume ical di e en ia ion
1
In oduc ion
Cold compac ion p ocesses a e a key ing edien in powde
o ming p ocesses. They consis in he e ical compac ion
h ough he mo emen o a se o punches o a ine powde
ma e ial a oom empe a u e. The p ocess ans o ms he
loose powde in o a compac ed sample wi h a olume
educ ion (and he e o e a densi y inc ease) o abou 2–2.5
imes. The design o hese p ocesses includes he de ini ion
o he ini ial dimensions o he sample and he mo emen s
o he punches ha lead o compac ed samples wi h uni-
o m densi y dis ibu ions. In his con ex , e icien and
eliable nume ical simula ions can play an impo an ole
as a complemen o expe imen al es s.
Two ing edien s a e c ucial o he nume ical modelling
o powde compac ion p ocesses: he cons i u i e ela-
ionship and he kinema ic o mula ion o he p oblem.
Se e al cons i u i e models ha e been p oposed, including
mic oscopic models, low o mula ions and solid me-
chanics models, such as elas ic, plas ic o iscoplas ic
models; see Oli e e al. [1] and Lewis and Khoei [2] o a
gene al o e iew and e e ences o each ype o model.
One o he mos common app oaches is he use o elas-
oplas ic models based on po ous o ic ional ma e ials.
He e, plas ic models exp essed in e ms o he ela i e
densi y and he Ki chho s esses a e conside ed. The
cons i u i e ela ionship is o mula ed wi hin he ame-
wo k o iso opic ini e s ain mul iplica i e hype elas o-
plas ici y [3, 4]. This ype o models has al eady been
applied o powde compac ion p oblems wi h some sim-
pli ica ions de i ed om he assump ion o small elas ic
s ains [1]. In his wo k, la ge elas ic s ains a e included
in he o mula ion [5]. I is shown ha his does no
ep esen any d awback om a modelling poin o iew.
On he con a y, i allows o apply nume ical echniques
and ma e ial models de eloped o he gene al kinema ic
amewo k in a s aigh o wa d manne .
Up o da e, a common ea u e o powde compac ion
simula ions wi h solid mechanics cons i u i e models is
he use o a Lag angian kinema ic o mula ion. This
app oach has shown o be adequa e o p oblems ha do
no exhibi la ge mass luxes among di e en pa s o he
sample (i.e. homogeneous es s). Bu in p ac ical p ob-
lems, as hose which appea in ealis ic design p ocesses,
he Lag angian app oach leads o oo dis o ed meshes
[2, 6] o iola ions o he bounda y limi s [1]. In o de o
sol e hese p oblems, di e en h-adap i e p ocedu es ha e
been p esen ed ecen ly [6, 7]. Howe e , h- e inemen is
compu a ionally expensi e and in o ma ion mus be in-
e pola ed om he old mesh o he new mesh. Fo hese
easons an A bi a y Lag angian–Eule ian (ALE) app oach
is p e e ed in his wo k. ALE o mula ions we e i s
p oposed o luid p oblems wi h mo ing bounda ies [8, 9].
Nowadays, ALE o mula ions o luid p oblems a e widely
used in o ming p ocesses. On he o he hand, he ALE
o mula ion has been success ully employed in nonlinea
solid mechanics [10–14]. The ALE o mula ion o
Compu a ional Mechanics 30 (2003) 220–234 Sp inge -Ve lag 2003
DOI 10.1007/s00466-002-0381-4
220
Recei ed: 20 Ma ch 2002 / Accep ed: 15 Oc obe 2002
A. Pe
´ ez-Fogue (&), A. Rod ı
´guez-Fe an, A. Hue a
Depa . de Ma ema
` ica Aplicada III,
E.T.S. de Ingenie os de Caminos,
Canales y Pue os, Uni e si a Poli e
`cnica de Ca alunya,
Jo di Gi ona 1, E-08034 Ba celona, Spain
e-mail: agus i.[email p o ec ed]
The pa ial inancial suppo o he Minis e io de Ciencia y
Tecnologı
´a (g an numbe DPI 2001-2204) is g a e ully
acknowledged.
mul iplica i e hype elas oplas ici y ecen ly p esen ed by
Rod ı
´guez-Fa an e al. [14] is used he e.
Due o he high nonlinea i y o powde compac ion
simula ions, a common app oach is he explici ea men
o bo h he in eg a ion o he cons i u i e equa ions and
he solu ion o he equilib ium equa ion [2, 6, 15], al-
hough, implici schemes ha e also been success ully
applied [1, 7]. In ac , he de elopmen o obus , accu-
a e and e icien schemes, bo h implici and explici , is
s ill an ac i e esea ch opic, see o ins ance [16–19]
among many o he s. He e a mixed app oach is ollowed.
The in eg a ion o he ALE cons i u i e equa ions is done
by means o a ac ional-s ep me hod [3]: he ea men
o he Lag angian phase is implici and he scheme o
he con ec i e phase is explici . The equilib ium equa ion
is sol ed wi h an inc emen al-i e a i e app oach, whe e
only he Lag angian phase is pe o med wi hin he i e -
a ions o each load inc emen ( he pe u ba ion due o
he in eg a ion o he con ec i e phase is aken in o ac-
coun in he subsequen load inc emen ). A key poin o
he e icien implici solu ion o he equilib ium equa ion
is he consis en linea iza ion o he algo i hm, which is
gi en by he p ope consis en angen moduli [20]. The
ecen ly p esen ed exp ession o densi y-dependen
models by Pe
´ ez-Fogue e al. [5] is used he e. Speci ic
ools o wo o he main issues o implici app oaches
o complex elas oplas ic models a e also conside ed
wi hin his wo k: he con e gence o he plas ic co ec o
a Gauss-poin le el is gua an eed by he use o globally
con e gen New on–like schemes [21, 22], and nume ical
di e en ia ion schemes a e used o he compu a ion o
complex plas ic equa ions de i a i es ollowing e e ences
[18, 23].
An ou line o he pape ollows. The main ea u es o
he cons i u i e equa ions and he nume ical ime–in e-
g a ion scheme a e p esen ed in Sec . 2. The consis en
angen moduli is p esen ed in Sec . 3. A e ha , in Sec . 4,
he p oposed app oach is applied o se e al ep esen a i e
powde compac ion p oblems. The p esen esul s a e
compa ed wi h expe imen al da a and o he esul s o
nume ical simula ions p esen ed in he li e a u e. Sec ion 5
con ains some concluding ema ks.
2
P oblem s a emen
In his sec ion, he basis o he p oposed app oach is
b ie ly p esen ed. Fi s , he kinema ics o he ALE o -
mula ion a e e iewed. A e ha , he ini e s ain ellip ic
elas oplas ic model is desc ibed, ocusing in he yield
unc ion dependence on he densi y. Finally, he nume ical
ime-in eg a ion algo i hm is p esen ed.
2.1
Kinema ics
Le RXRndim (ndim ¼2;3) be he ma e ial con igu a ion
o a con inuum body wi h pa icles labelled by hei ini ial
posi ion ec o X2RX. In a Lag angian se ing, Xa e used
as he independen a iables in he desc ip ion o mo ion.
The mo ion o he body is desc ibed by he one-pa ame e
amily o mappings u :RX7! Rndim wi h 2½0;T.
Rx¼u ðRXÞis he spa ial con igu a ion o he body a
ime , and x¼u ðXÞ¼uðX; Þ2Rxis he cu en
posi ion o he ma e ial pa icle X.
The key ing edien o he ALE o mula ion is he e -
e en ial con igu a ion R , wi h g id (o e e ence) poin s
used as independen a iables o desc ibe body mo ion.
This e e en ial con igu a ion R is mapped in o he ma-
e ial and spa ial con igu a ions by Wand U espec i ely,
see igu e 1. The ini ial posi ion o a ma e ial pa icle is
exp essed on he e e ence domain as X¼Wð ; Þ, and he
cu en posi ion as x¼Uð ; Þ. The h ee mappings u,U
and Wa e ela ed by u¼UW1.
In an ALE se ing, di e en displacemen ields can be
de ined. Two o hem ha e special in e es , he pa icle
displacemen uand he mesh (spa ial) displacemen uU,
uðX; Þ¼xðX; ÞXand uUð ; Þ¼xð ; Þ :
ð1Þ
The pa icle eloci y and he mesh eloci y a e espec i ely
¼ox
o jXand mesh ¼ox
o j ;ð2Þ
whe e, ollowing s anda d no a ion, jmeans ‘‘holding 
ixed’’. The link be ween ma e ial and mesh mo ion is
p o ided by he con ec i e eloci y, c¼  mesh.
2.2
Cons i u i e model
Iso opic ini e s ain mul iplica i e plas ici y is assumed.
The pa icle de o ma ion g adien ,
FðX; Þ¼ou
oXðX; Þ;ð3Þ
is locally decomposed in o elas ic and plas ic pa s as
F¼FeFp. The he modynamic s a e is de ined by means o
he elas ic le Cauchy–G een enso be¼FeFeT, whe e he
supe sc ip T means anspose. As usually in densi y-
dependen plas ici y, no plas ic in e nal a iables a e
conside ed [1, 5]. The Ki chho s ess enso , s, is gi en
by he hype elas ic ela ionship
Fig. 1. Domains, mappings and de o ma ion g adien s in he
ALE desc ip ion
221
s¼2dWe
dbebe;ð4Þ
whe e Weis he ee-ene gy unc ion pe uni o unde-
o med olume. The Cauchy s ess enso is gi en by
¼s
de ðFÞ:ð5Þ
The plas ic esponse o he ma e ial is assumed iso opic
and densi y-dependen . The dependence on he densi y is
inco po a ed in he yield unc ion ðs;gÞand he low
di ec ion mðs;gÞ h ough he ela i e densi y gðX; Þo he
pa icle Xa ime . The ela i e densi y is equal o he eal
densi y o he ma e ial di ided by a e e ence alue, which
usually is he solid densi y o he compac ed ma e ial.
Associa i e plas ici y is cha ac e ized by
mðs;gÞ¼$s ðs;gÞ, wi h $equal o he g adien ope a o
wi h espec o he a iables .
The e olu ion o he elas ic le Cauchy–G een enso
(i.e. low ule) is de ined in he ALE se ing as [14]
obe
o j þc$xbelbebelT¼2_
ccmðs;gÞbe;ð6Þ
whe e l¼$x is he eloci y g adien enso and _
cc is he
plas ic mul iplie . Equa ion (6) is complemen ed wi h he
Kuhn-Tucke condi ions
_
cc 0; ðs;gÞ0 and _
cc ðs;gÞ¼0:ð7Þ
The e olu ion o he ela i e densi y gis gi en by he mass
conse a ion p inciple, which in ALE o mula ion eads
og
o j þc$xgþg$x ¼0:ð8Þ
2.3
Nume ical ime-in eg a ion: a ac ional-s ep me hod
In he ALE app oach, he wo undamen al unknowns in
e e y ime-s ep ½n ;nþ1 , wi h nþ1 ¼n þD , a e he in-
c emen o mesh displacemen s,
nþ1DuUð Þ¼nþ1xð Þnxð Þð9Þ
and he inc emen o pa icle displacemen s,
nþ1DuðnXð ÞÞ ¼ nþ1xðnXð ÞÞ  nxðnXð ÞÞ ð10Þ
which is e e ed o he pa icles nXassocia ed o g id
poin s a he beginning o he ime-s ep, nX¼Xð ;n Þ.
The mesh displacemen s a e ob ained om an ALE
emeshing algo i hm. The pa icle displacemen s a e
ound ia he inc emen al solu ion o he equilib ium
equa ion. The di e ence be ween he inc emen s o pa -
icle and mesh displacemen s is he so-called inc emen o
con ec i e displacemen s,
nþ1Ducon ð Þ¼nþ1Duð Þnþ1DuUð Þ;ð11Þ
which ep esen s he ela i e mo ion be ween pa icles and
g id poin s du ing he ime-s ep.
The quan i ies o in eg a e a e only beand g, see ema k
1 below. The nume ical ime-in eg a ion is done by means
o a ac ional-s ep me hod o e Equa ions (6) and (8).
E e y ime–s ep is di ided in o wo phases: he Lag angian
phase and he con ec ion phase.
Du ing he Lag angian phase con ec ion is neglec ed,
c¼0. In ha si ua ion, Eq. (6) simpli ies o he s anda d
spa ial exp ession o he low ule
L be¼2_
ccmðs;gÞbe;ð12Þ
whe e L be¼obe
o jxlbebelTis he Lie de i a i e wi h
espec o he pa icle eloci y; and Eq. (8) simpli ies o
gðX; Þ¼g0ðXÞ
de ðFÞ;ð13Þ
he ma e ial exp ession o he Lag angian ela i e densi y
e olu ion. The ini ial condi ions o equa ions (12) and
(13) a e he s a e a ime n (nx,nbeand ng). The p oblem is
s ain-d i en, hus, he Lag angian alues Lbeand Lga e
ob ained om nþ1Du. Recall ha nþ1Duis no equi ed o
be in equilib ium.
A e he Lag angian phase, an ALE emeshing algo-
i hm is employed o compu e he inc emen o mesh
displacemen s nþ1DuU. The algo i hm is de ined such as i
educes he elemen dis o ion o a pu e Lag angian ap-
p oach.
Du ing he con ec ion phase, he con ec i e e m o
he e olu ion equa ions is aken in o accoun . The con-
ec i e eloci y is assumed cons an in he ime-s ep and
equal o c¼nþ1Ducon =D . The inal (con ec ed) alues
nþ1beand nþ1ga e compu ed in eg a ing he anspo
equa ions
obe
o j þc$xbe¼0and og
o j þc$xg¼0;ð14Þ
wi h he ini ial condi ions Lbeand Lg.
In his wo k, only he Lag angian phase is pe o med
du ing i e a ions o equilib ium wi hin each load inc e-
men . The emeshing and he con ec ion a e compu ed
wi h he con e ged esul s a he end o each ime-s ep.
This esul s in o a educed compu a ional cos o e head
wi h espec o he pu e Lag angian app oach. The equi-
lib ium pe u ba ion due o he emeshing and he con-
ec i e ime-in eg a ion o he con e ged esul s is aken
in o accoun in he subsequen load inc emen . The e o e,
he e a e no accumula i e e o s in he esul s. On he
o he hand, as he i e a i e scheme is pu e Lag angian, he
consis en ope a o is no in luenced by he emeshing
echnique no by he nume ical ime-in eg a ion scheme
used in he con ec i e phase. The o e all scheme is sum-
ma ized in able 1. The main cha ac e is ics o he nu-
me ical ime-in eg a ion o each phase a e p esen ed in he
ollowing wo subsec ions.
Rema k 1 In a gene al se ing, all quan i ies ela ed wi h
he pa icles should be con ec ed. This includes he ma-
e ial pa ame e s and he alue o de ðFÞ, in addi ion o be
and g(and he plas ic in e nal a iables i hey exis ).
Howe e , in he usual case ha he ma e ial su aces a e
acked by he ALE emeshing algo i hm and homoge-
neous ma e ials a e conside ed (as is he case in all he
examples p esen ed in his wo k), only he in eg a ion o
beand gis necessa y. The alue o de ðFÞcan be compu ed
om he ma e ial exp ession o he ela i e densi y
e olu ion, Eq. (13).
222
2.3.1
Lag angian phase
The esul s o he Lag angian phase, Lbeand Lg, a e
compu ed om nbe,ngand he inc emen al pa icle
de o ma ion g adien ,
L ¼Indim þ
nxnþ1Du;ð15Þ
which ela es he pa icle de o ma ion g adien s a ime n ,
nF, and a he end o he Lag angian phase, LF, h ough he
ela ionship L ¼LFðnFÞ1.Indim deno es a iden i y ma ix
o o de ndim.
The ela i e densi y, g, is in eg a ed exac ly (in he sense
ha no nume ical ime-in eg a ion scheme is used) be-
cause o he ma e ial exp ession o he mass conse a ion
p inciple, Eq. (13), which leads o
Lg¼g0
de ðLFÞ¼
ng
de ðL Þ:ð16Þ
The alue o Lbeis ob ained by means o he s anda d
elas ic p edic o -plas ic co ec o spli s a egy applied o
Eqs. (7) and (12). Rema kably, he dependence o he
cons i u i e equa ions on he densi y does no modi y he
algo i hm. The alue o Lgis gi en by Eq. (16), and
he e o e i plays he ole o a ixed pa ame e .
The esul o he elas ic p edic o s ep is he so-called
ial s a e. I is de ined by
be¼L nbeL T:ð17Þ
I he ial s a e is admissible, ðsð beÞ;LgÞ0, he alue
o Lbeis se equal o he ial s a e. I i is no , a plas ic
co ec o s ep is compu ed. Recall ha he ial s a e, Eq.
(17), does no depend on Lg, a key poin in he linea iza-
ion o he algo i hm p esen ed in Sec . 3.
The plas ic co ec ion s ep equi es he app oxima ion
o he low ule, Eq. (12). A s anda d app oxima ion
consis s in he use o he exponen ial map and he back-
wa d Eule in eg a ion scheme [3]. Unde he p e ious
iso opy assump ions, his app oach leads o a nonlinea
sys em o equa ions wi h he same s uc u e as ha o
in ini esimal elas oplas ici y. In o de o ob ain his non-
linea sys em o equa ions, h ee ec o s o Rndim a e de-
ined: ee,Leeand L
ss. The componen s o eeand Leea e
unc ions o he eigen alues o he enso s beand Lbe,
espec i ely, and he componen s o L
ss a e he eigen alues
o Ls:
*be¼X
ndim
i¼1expð½eeiÞ2ni
ni
and Ls¼X
ndim
i¼1
½L
ssini
ni
;
ð18Þ
whe e he supe sc ip * e e s o and L, ni
gi¼1;...;ndim a e
he eigen ec o s o he h ee enso s, and ½*igi¼1;...;ndim he
componen s o he h ee ec o s. The eigen ec o s a e he
same o he h ee enso s because o he iso opy as-
sump ions. Thus, hey a e ully speci ied by he ial s a e.
A e some manipula ions, he ollowing nonlinea
sys em o equa ions is ound:
LeeþDcmss 
ssðLeeÞ;Lg

¼ ee

ssðLeeÞ;Lg

¼0;ð19Þ
whe e Dc¼D _
cc is he inc emen al plas ic mul iplie , mss is
he low ec o in he p incipal di ec ion space ( ha is,
ms¼Pndim
i¼1½mssini
ni
), he iso opic unc ions mss and
a e exp essed as unc ions o 
ss, and 
ss is gi en by he
hype elas ic ela ionship

ss ¼d
WWe
dee;ð20Þ
wi h 
WWeðeeÞde ined so ha Eq. (20) is equi alen o Eq.
(4). Equa ions (19) a e complemen ed wi h he es ic ion
Dc0.
This p oblem is ypically sol ed wi h he New on–
Raphson me hod. Rema kably, as Lgis ixed, he globally
con e gen ex ensions o he New on–Raphson me hod
p esen ed [22] o densi y-independen ini e s ain
models also apply o densi y-dependen plas ic p oblems.
Once Eqs. (19) a e sol ed, he Lag angian phase is
comple ed.
2.3.2
Con ec i e phase
Du ing he con ec i e phase, he anspo equa ions (14)
a e in eg a ed nume ically wi h a Goduno -like scheme
[13, 24]. The applica ion o his scheme o quasi-s a ic
p oblems is compa ed wi h o he al e na i es in [13]. This
scheme ci cum en s he compu a ion o g adien s wi h
espec o he spa ial coo dina es p esen in he anspo
equa ions, he main sou ce o ouble when disc e ized
p oblems a e conside ed. The only di e ence be ween he
p esen applica ion, wi h hype elas ic-plas ic models, and
ha p esen ed in [13, 24], wi h hypoelas ic-plas ic models,
is ha he e i is applied o he con ec ion o he elas ic le
Cauchy–G een enso ins ead o he Cauchy s esses. The
key poin s o he implemen a ion a e p esen ed in he
ollowing.
The anspo equa ions (14) con ains se en equi alen
scala equa ions, one o each componen o beand one o
g. The e o e, he scheme is de ised o a gene al scala
con ec i e equa ion
ou
o j þc$xu¼0;ð21Þ
Table 1. The o e all ALE scheme
Fo e e y ime-s ep [
n
,
nþ1
]:
Ma e ial phase
Neglec con ec i e e ms
Ad ance he solu ion i e a i ely in an upda ed Lag angian
ashion: compu e he inc emen o pa icle displacemen s
n+1
Du
and quan i ies
L
b
e
and
L
g(supe sc ip L deno es Lag angian)
Remeshing
Compu e he inc emen o mesh displacemen s
n+1
Du
F
and he
inc emen o con ec i e displacemen s
n+1
Du
con
by means o a
emeshing algo i hm ha educes elemen dis o ion.
Compu e he con ec i e eloci y c=
n+1
Du
con
/D
Con ec ion phase
Accoun o con ec i e e ms
Use he Goduno - ype echnique o con ec quan i ies
L
b
e
and
L
gin o
n+1
b
e
and
n+1
g
Compu e s esses
n+1
sand
n+1
223
wi h uð ;xÞ ep esen ing he di e en componen s o be
and g. The ini ial condi ion o Eq. (21) is he Lag angian
alue Lu.
The basic idea o he scheme is o di ide e e y ini e
elemen in a ious zones, each o hem co esponding o
he in luence domain o a Gauss poin . In wo-dimensional
p oblems, as hose p esen ed in his wo k, each zone has
an a ea Aand nedg edges. Then, a each Gauss poin , he
ollowing explici upda e equa ion is applied
nþ1u¼LuD
2AX
nedg
iedg¼1
Iiedg
Luc
iedg Lu

1signðIiedg Þ;
ð22Þ
whe e Luc
iedg is he alue o he a iable uin he con iguous
Gauss poin ac oss edge iedg and Iiedg is he lux o con-
ec i e eloci y cac oss edge iedg. This scheme leads o a
e y simple algo i hm. Mo eo e , as he space disc e iza-
ion and he con ec i e eloci y a e he same o he di -
e en scala anspo equa ions, he majo pa o he
compu a ions a e common o all o hem.
Once Eq. (22) is applied o he six componen s o beand
g he con ec i e phase is comple ed: he alues o nþ1be
and nþ1ga e de e mined. The s ess alues nþ1sand nþ1
a e compu ed wi h Eqs. (4) and (5).
3
Consis en angen ope a o
The consis en angen moduli a e needed o sol e he
equilib ium equa ion wi h quad a ic con e gence [20]. In
his wo k, as he con ec i e phase is no included wi hin
he i e a i e p ocess, hey a e he same needed in s anda d
Lag angian app oaches. Howe e , due o he densi y-
dependence o he plas ic equa ions, he well-known con-
sis en angen moduli o densi y-independen models o
mul iplica i e ini e s ain p oblems [3] esul s incomple e.
In he ollowing, he comple e exp ession o densi y-
dependen plas ic models is p esen ed.
The consis en angen moduli, c, a e he linea iza ion
o he Ki chho s esses ob ained om he Lag angian
phase, Ls, wi h espec o he g adien o he inc emen al
pa icle displacemen s, nxnþ1Du[3, 25]. They a e ound
by applying he chain ule o equa ion (18)2, which leads
o
c¼X
ndim
i¼1X
ndim
j¼1
½aijni
ni
nj
nj
þ2X
ndim
i¼1
½L
ssi^
cci
;
ð23Þ
wi h he i s e m co esponding o he linea iza ion o L
ss
and he second one o he linea iza ion o ni
gi¼1;...;ndim ,
and whe e he enso s ^
cci
gi¼1;...;ndim depend on
ni
gi¼1;...;ndim and ee[3], and ais a ma ix o o de ndim
de ined as
a¼dL
ss
d ee:ð24Þ
Equa ion (23) has he same exp ession o densi y-
dependen and densi y-indepeden plas ic models because
he densi y does no a ec he applica ion o he chain ule
no he ial s a e. In ac , he in luence o he densi y is
es ic ed o he alues o L
ss and ain plas ic s eps, i.e.
when he ial s a e is no admissible. In elas ic s eps, he
ma ix ais he Hessian o 
WWe,
a¼d2
WWe
dee2ee¼ ee
:ð25Þ
The alue o L
ss is ob ained, in bo h elas ic and plas ic
s eps, di ec ly om he hype elas ic ela ionship. The e-
o e, only he exp ession o a o plas ic s eps needs o be
de e mined.
In o de o do ha i is use ul o eph ase he depen-
dence o Lgon L , Eq. (16), as
Lg¼n^
gg exp  ð eeÞ;ð26Þ
wi h n^
gg ¼ngffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
de ðnbeÞ
pa known alue om he p e ious
ime s ep and whe e ð*Þmeans ace o ( ha is,
Pndim
i¼1½*i). Equa ion (26) is ound by applying he de e -
minan unc ion o bo h sides o Eq. (17) and subs i u ing
Eq. (18)1in o i .
A mo e con enien exp ession o ais ound om i s
de ini ion, Eq. (24), and he applica ion o he chain ule:
a¼dL
ss
dLee
dLee
d ee¼d2
WWe
dee2ee¼Lee
dLee
d ee:ð27Þ
This exp ession shows ha ais de e mined once he o al
in luence o eeon Lee;dLee
d ee, is ound. This in luence is
gi en by he he nonlinea sys em o Eq. (19) and he
ela ionship be ween Lgand ee, Eq. (26). Thus, i can be
compu ed by linea izing he nonlinea sys em o equa ions
LeeþDcm
ss 
ssðLeeÞ;Lg

¼ ee

ssðLeeÞ;Lg

¼0
Lg¼n^
gg exp  ð eeÞ:
ð28Þ
Once dLee=d eeis de e mined, he exp ession o ais ound
subs i u ing i in o Eq. (27). The esul can be ea anged
as
a¼agindep þag;ð29Þ
wi h agindep equal o he s anda d consis en moduli o
densi y-independen plas ic models,
agindep ¼~
GG 
~
GGm
ss 
ss T~
GG

ss T~
GGmss
;ð30Þ
whe e
~
GG ¼d2
WWe
dee21
þDc 
ssm
ss1
;ð31Þ
and
ag¼gDc~
GG 
~
GGm
ss 
ss T~
GG

ss T~
GGm
ss om
ss
og11;ndim
þg
o
og
~
GGm
ss

ss T~
GGm
ss
11;ndim ;ð32Þ
a e m ha akes in o accoun he in luence o he densi y.
All quan i ies in ol ed in he compu a ion o a, Eqs.
(30–32), a e e alua ed a he end o he Lag angian phase.
224

In he densi y-independen case, symme ic angen
moduli a e ob ained o associa i e ma e ial models. On
he con a y, unsymme ic moduli a e ound wi h all
densi y-dependen ma e ial models because agis, in gen-
e al, unsymme ic. Fo his eason, in densi y-dependen
plas ici y, unsymme ic linea sol e s ha e o be used in
o de o keep he cha ac e is ic quad a ic con e gence o
he New on–Raphson me hod.
On he o he hand, i is impo an o ema k ha he
exp ession o agcan be compu ed wi h jus a ew mo e
ma ix– ec o p oduc s han he s anda d agindep, com-
pa e Eqs. (32) and (30), and, as expec ed, he addi ional
in o ma ion
om
ss
ogand o
og:ð33Þ
3.1
Nume ical di e en ia ion
A key poin in he compu a ion o consis en angen
moduli in densi y-independen plas ici y is he compu a-
ion o low ec o and ha dening law de i a i es. Re e -
ence [18] p esen s an analysis o di e en nume ical
di e en ia ion schemes o compu e hese de i a i es and
e e ences [23, 26] show he applica ion o di e en non-
i ial elas oplas ic models and ime-in eg a ion ules. The
main conclusion o hese wo ks is ha simple i s and
second o de di e ence schemes do no dis u b he qua-
d a ic con e gence o he New on–Raphson me hod p o-
ided ha he s epsize is ixed in a ela i e way. Mo eo e ,
in ha e e ences i is shown ha nume ical di e en ia ion
is an e icien al e na i e o analy ical de i a i es e en i
a e eadily a ailable.
These schemes a e di ec ly applicable o densi y-
dependen plas ic models. In his case, i can also be applied
o he compu a ion o he de i a i es o low ec o and he
yield unc ion wi h espec o he ela i e densi y, see Eq.
(33), a si ua ion o special in e es when ealis ic cons i u-
i e laws a e conside ed. In his wo k, a i s o de di e ence
scheme wi h a ela i e s epsize equal o 105has been used
o app oxima e hese de i a i es.
4
Nume ical simula ions
In his sec ion, he p oposed app oach is applied o he
simula ion o se e al powde compac ion p oblems. In he
i s wo cases he a en ion is ocused in he compa ison
o he p esen nume ical esul s wi h nume ical and
expe imen al ones epo ed in he li e a u e. The i s one
is sol ed wi h a pu e Lag angian app oach, and he second
one using he ALE o mula ion p esen ed in Sec . 2.
Speci ic analysis ela ed wi h he applica ion o he ALE
o mula ion, he de ini ion o he globally con e gen
schemes o he plas ic co ec o s ep, and he compu a ion
o he consis en angen moduli ia nume ical di e en ia-
ion schemes can be ound in e e ences [14, 18, 22, 23].
The powde ma e ial is modelled wi h he Hencky’s
hype elas ic law, which leads o a linea ela ionship be-
ween 
ss and ee, and associa e plas ici y wi h he ollowing
ellip ic yield unc ion [1, 27]
ellipðs;gÞ¼2J2ðsÞþa1ðgÞI1ðsÞ
3

2
2
3a2ðgÞð yÞ2;
ð34Þ
wi h I1ðsÞequal o he i s in a ian o s;J2ðsÞequal o
he second in a ian o he de ia o ic pa o s, and he
densi y–dependen pa ame e s
a1ðgÞ¼
1g2
2þg2

n1
g<1
0g1
(and
a2ðgÞ¼
0:02g0
10:98g0

n2
gg0
g0:98g0
10:98g0

n2
g>g0
8
>
<
>
:
ð35Þ
The dependence o a1and a2on g o he ma e ial pa-
ame e s p esen ed in Table 2 is depic ed in Fig. 2. The
ace o he yield unc ion on he me idian plane psqs,
wi h ps¼I1ðsÞ
3and qs¼ffiffiffiffiffiffiffiffiffiffiffiffi
3J2ðsÞ
p, o di e en ela i e
densi ies a e depic ed in Fig. 3. No e ha ellip becomes he
on Mises yield unc ion o g1.
4.1
Compac ion o a plain bush componen
The i s example is he uniaxial compac ion o a plain
bush componen . Expe imen al da a [28] and nume ical
esul s [2, 28, 29] a e a ailable o his example. Bo h a e
used he e o compa a i e pu poses. The es is pe o med
wi h he powde ma e ial pa ame e s p esen ed in Table 2.
These ma e ial pa ame e s a e calib a ed in [29] by com-
pa ing he nume ical esul s wi h hose p esen ed in [28].
The componen is modelled by an axisymme ic ep-
esen a ion as illus a ed in Fig. 4 [2]. A 2D s uc u ed
mesh o 200 bilinea elemen s is used. The die wall ic ion
is simula ed wi h a Coulomb ic ion coe icien l¼0:15
Table 2. Ma e ial pa ame e s
E50 000. [MPa]
m0.37
y
12. [MPa]
g
0
0.41
g
1
0.5
g
2
2.2
Fig. 2. Dependence o pa ame e s a1ðgÞand a2ðgÞon he ela i e
densi y, g
225
[28, 29] ac ing in he inne and ou e walls o he sample,
segmen s BC and DA o Fig. 4. In [29] he ela i e adial
mo emen o op and bo om su aces wi h espec o he
punches, segmen s AB and CD, is allowed, wi h a Coulomb
ic ion coe icien equal o ha o he la e al su aces.
He e, ollowing [2], he adial displacemen o he op and
he bo om o he sample is es ained. The e ical dis-
placemen o segmen AB is also se equal o ze o, and a
e ical displacemen o 11.5 mm is imposed o segmen
CD, simula ing he op punch mo emen .
The ela i e densi y p o ile a adius equal o 10.5 mm
o a op punch displacemen o 10 mm is shown in Fig. 5.
The nume ical esul s o [28, 29] and he expe imen al
da a o [28] a e included in he same igu e. The esul s o
he p esen nume ical simula ion a e in ag eemen wi h all
o hem. Howe e , as expec ed because o he simila i ies
o he ma e ial model, he bes ag eemen is ound wi h he
esul s o [29]. The lowe ela i e densi y a he op pa o
he sample ound by [29], wi h espec o he p esen one,
and he highe alues in he lowe pa a e di ec ly ela ed
wi h he di e en ea men o he ic ion e ec s o op
and bo om punches.
The e olu ion o he e ical eac ion o he punch wi h
espec o i s e ical displacemen is depic ed in Fig. 6.
The nume ical esul s o [2] and he expe imen al da a o
[28] a e also included. In e e ence [2] he esul s ob ained
wi h di e en o mula ions a e shown. Those depic ed in
Fig. 6 a e he mos simila o he expe imen al da a. The
ag eemen be ween he esul s o he p esen simula ion
and he expe imen al ones is good.
Finally, he e olu ion o he ela i e densi y dis ibu ion
o e he sample is depic ed in Fig. 7. Resul s o di e en
e ical op punch displacemen s ( om 4 o 11 mm) a e
shown. Two di e en ela i e densi y scales a e used (one
in each ow) in o de o show be e he non-homogeneous
dis ibu ion o he densi y. The shape o he ela i e
densi y dis ibu ions a e in gene al ag eemen wi h hose
Fig. 3. T ace o he ellip ic yield unc ion on he me idian plane
qsps o di e en ela i e densi ies, g
Fig. 4. Plain bush componen . P oblem de ini ion (a e Lewis
and Khoei 1998) and compu a ional mesh
Fig. 5. Plain bush componen . Rela i e densi y p o ile a adius
equal o 10.5 mm
Fig. 6. Plain bush componen . Rela ionship be ween op punch
e ical eac ion and i s e ical displacemen
226
p esen ed by [2], al hough he quan i a i e alues a e a
li le bi highe .
In o de o check he p esen esul s, he e olu ion o
he sample mass du ing he simula ion has been com-
pu ed. A cons an alue equal o he ini ial one has been
ound, up o six signi ican digi s. The e o e, he mass
conse a ion p inciple is e i ied. This implies, o in-
s ance, a mean alue o he ela i e densi y equal o 2g0 o
a heigh educ ion o 50% (i.e. gmean ¼0:82 o a op
displacemen o 10 mm, see Figs. 6 and 7).
4.2
Compac ion o a o a ional langed componen
The second example co espond o he compac ion o a
langed componen which is modelled by an axisymme ic
ep esen a ion, as illus a ed in Fig. 8. The example is used
in [2] o illus a e he applicabili y o a dynamic app oach
wi h a hypoelas oplas ic model. The same example is used
in [6] o show he u ili y o an h-adap i e emeshing
echnique o educe mesh dis o ion. He e he hype elas-
oplas ic model and he ALE o mula ion p esen ed p e-
iously a e used. Some expe imen al esul s [30] a e
a ailable.
The p esen esul s illus a e ha he p oposed ap-
p oach allows o simula e highly demanding powde
compac ion p ocesses wi hou mesh dis o ion and spu-
ious oscilla ions in he esul s. Mo eo e , i is shown ha
he mass conse a ion p inciple is e i ied wi h a low
ela i e e o .
Th ee di e en compac ion es s a e simula ed [2]: 1) a
e ical mo emen o he op punch (6.06 mm); 2) a e -
ical mo emen o he bo om punch (5.10 mm); and 3) a
simul aneous mo emen o bo h punches (6.06 mm he op
punch and 7.70 mm he bo om punch). The same s uc-
u ed mesh o 170 eigh -noded elemen s wi h educed
in eg a ion ( ou Gauss poin s pe elemen ) is used in he
Fig. 7. Plain bush componen . Rela i e
densi y dis ibu ion o di e en op
punch mo emen s. No e ha wo di e en
scales a e used
227
h ee es s. The die wall ic ion is simula ed wi h a Cou-
lomb ic ion coe icien l¼0:08 ac ing in he segmen s
BC, CD, DE and FA, see Fig. 8, and he adial displacemen
a he punches is es ained [2].
The analysis is pe o med wi h he powde ma e ial o
Table 2, calib a ed in [1] o he compac ion o he plain
bush componen . The ela ionships be ween dimension-
less e ical loads and punch displacemen s ob ained in
his wo k a e compa ed wi h hose o [2] in Fig. 9. The
ag eemen be ween he wo se s o cu es is e iden .
Howe e , he e e ence load is di e en : 392 kN o he
p esen esul s and 1550 kN o he esul s o [2] (in bo h
cases he inal eac ion o he op punch in he double-
punch compac ion es ). This di e ence can be ela ed
wi h he di e en modelling app oach (dynamic e sus
s a ic) and, especially, wi h he g ea di e ence be ween
he elas ic moduli used in bo h cases (40 MPa in [2], h ee
o de s o magni ude lowe han he one used he e). In he
ollowing, a de ailed analysis o he ela i e densi y dis-
ibu ion ob ained in each o he h ee es s is p esen ed.
In he i s es , op punch compac ion, he mesh egion
ABCG is Eule ian, and equal heigh elemen s a e p e-
sc ibed in he mesh egion GDEF. The a ia ion o he
mass o he sample du ing he simula ion is depic ed in
Fig. 10. A inal loss o 0.75% is ound. This small a ia ion
co esponds o he unca ion e o in he disc e iza ion o
he con ec i e e m o he ALE o mula ion (bo h
empo al and spa ial disc e iza ion).
The e olu ion o he ela i e densi y dis ibu ion is
summa ized in Fig. 11(a–d). The compac ion p ocess
leads o a clea ly non-homogenous densi y dis ibu ion. As
expec ed, highe alues a e ound in he ou e egion o he
sample and lowe ones close o he bo om su ace. A
smoo h ansi ion om highe o lowe densi ies is ound.
A dense zone is de ec ed in he co ne egion, jus o e he
poin C, du ing all he p ocess. Recall ha he e is no
mesh dis o ion because, al hough he lux o mass is
impo an , he mesh does no ollow he ma e ial pa icles
in he ALE o mula ion. The inal ela i e densi y p o ile a
1.88 mm om line GD is depic ed in Fig. 11(e). The
p esen esul s a e in gene al ag eemen wi h hose
p esen ed in [2]. Two zones wi h a quasi-uni o m ela i e
densi y a e ound in bo h cases. Howe e , he densi y
p o ile ob ained in [2] p esen s a big oscilla ion be ween
hese wo zones, and, on he con a y, he ansi ion
ob ained in his wo k is smoo h.
The second es consis s in a bo om punch compac ion.
In his case he mesh egion GDEF is Eule ian and equal
heigh elemen s a e p esc ibed in egion ABCG. The
a ia ion o he mass o he sample du ing he simula ion
is depic ed in Fig. 10. The mass gain is less han 1% a he
end o he simula ion. The e o is a bi la ge han o he
op punch es . This indica es ha he con ec i e e ec s
a e mo e impo an in his case.
Fig. 8. Flanged componen . P oblem de ini ion (a e Lewis and
Khoei 1998) and compu a ional mesh
Fig. 9. Flanged componen . Rela ionships be ween he e ical
eac ions o he punches and hei e ical mo emen s: op
eac ion o op punch compac ion, bo om eac ion o bo om
punch compac ion, and op and bo om eac ions o double-
punch compac ion
Fig. 10. Rela i e mass a ia ion du ing he h ee compac ion
p ocesses o he langed componen . The load le els a e e e ed
o he punch displacemen s imposed a he end o each es
228