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Mechanics Research Communications 50 (2013) 8 11
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Mechanics Research Communications
jo ur nal homep age : www.elsev ier .com/ lo
Modeli ducylindr
Jos A. A . GeSan Diego State
a r t i c l
Article history:Received 12 AReceived in re14 November Available
onlin
Keywords:SinteringGravity-inducPorosityShape
distortiDensication
ng thntinuibes catioly det of thring
1. Introdu
In many sintering processes, gravity is an important
factoraffecting microstructure development and causing shape
distort-ions. In the last two decades, there has been growing
interestin uncovering the principles that govern the dynamics of
masstransport mity. Extensithe results experiment(Xue et al., 2003;
Johns
These efgravity on sgravitationaworks, diffephase sinteand
Germaduring sintaccompanyan analyticcal porous set al. (200under
solidnite elem
CorresponE-mail add
y. Likalso used by German and coworkers (Raman and German,
1995;Ganesan et al., 1998; Binet et al., 2004; Park et al., 2006)
to studydifferent distortion phenomena taking place due to
capillary andgravitational forces during liquid-phase sintering.
More recently,Nikoliv and Shinagawa (2012) introduced the concept
of three-
0093-6413/$ http://dx.doi.oechanisms during sintering with and
without grav-ve reviews of the research studies in this area
describeof on ground- and space-based microgravity sinterings
conducted by German and co-workers, and others1996; Naser et al.,
1998; Johnson et al., 1998; Ye et al.,on et al., 2009).forts have
been coupled with the modeling aspects ofintering, including the
interplay of microstructure andl forces following a continuum
formulation. In theserent constitutive formulations for solid- and
liquid-ring were proposed. Early numerical work (Olevskyn, 2000;
Olevsky et al., 2000) on the effects of gravityering focused on the
non-isotropic shape changes that
densication. Olevsky and German (2000) proposedal approach for
predicting shape changes of cylindri-pecimens in terms of aspect
and radius ratios. Olevsky0) investigated the gravity-induced shape
distortion- and liquid-phase sintering using an axisymmetricent
approach based on an average dissipation rate
ding author.ress: [email protected] (E.A. Olevsky).
dimensional skeleton units to investigate numerically how
gravityinduces an equilibrium conguration of solid phase domains
duringliquid phase sintering. Even though the existing models aid
in theinterpretation of experimental observations, many issues
regardingthe driving forces and processes affecting densication and
dis-tortion under the presence or absence of gravity still need to
beexplored.
One of the under-investigated areas relates to assessment of
thedegrees of the structure (density) heterogeneity imposed by
gravityas well as of shape distortions (deviations from
cylindricity for con-ventional cylindrical specimens) caused by
gravity forces. Despitesome analytical approximations describing
these phenomena sug-gested in the past (Olevsky and German, 2000),
their corroborationbased on comparison with more robust numerical
(e.g., nite ele-ment) solutions is missing. Indeed, some of the
nite elementcalculations of sintering considering gravity inuence
were focusedon the description of sintering of specic components
(Doege et al.,2001; Kraft and Riedel, 2004; Blaine et al., 2005;
Schoenberg et al.,2006) without comparing to analytical solutions
in a normalized(master-sintering like) form.
This paper presents a three-dimensional nite elementapproach for
modeling sintering with an embedded gravityinuence. The paper
assesses the impact of the densication
see front matter 2013 Elsevier Ltd. All rights
reserved.rg/10.1016/j.mechrescom.2013.02.007ng of gravity-induced
shape distortionsical specimens
lvarado-Contreras, Eugene A. Olevsky , Randall M University, San
Diego, CA 92182, USA
e i n f o
ugust 2012vised form2012e xxx
ed
on
a b s t r a c t
A nite element approach for modelitutive equations are based on
the comaterial behavior. The model descrinhomogeneity. Simulations
of densiresults are compared with a previousgravity. First time a
direct assessmeninduced shape distortion during sintecurve
approach.
ction densitcate /mechrescom
ring sintering of
rman
e gravity-affected sintering process is presented. The consti-um
theory of sintering in the framework of a linear viscousthe gravity
inuence on porosity evolution and shrinkagen, shape distortion, and
porosity gradients are presented. Theveloped analytical model of
sintering under the inuence ofe impact of the densication
inhomogeneity on the gravity-is provided in a generic form similar
to the master sintering
2013 Elsevier Ltd. All rights reserved.
ewise, nite element, and boundary methods have been
-
J.A. Alvarado-Contreras et al. / Mechanics Research
Communications 50 (2013) 8 11 9
inhomogeneity on the gravity-induced distortion during
sinteringin a generic form similar to the master sintering curve
approach.Based on the continuum theory of sintering proposed by
Olevsky(1998), the model describes the expected shape distortion
andinhomogenmodel is usimens of dgravity intecal approxiOlevsky
and
For simwhere lowez. For instasor are writsummationthe short
fodisplacemethey appear
2. Constitu
2.1. Genera
As a partering (Olevstresses ij
ij = 20[
where 0 is are normstrain ratesthe sinterinbringing ab
Considerthe strain rui,j is the ve
Followineters; the shof porosity
= (1 )2
In sinterable represspecimens tributed in adopted sucresult of
thtering proceV + Vui,i = rate occurristrain rate g
The matmaterial is relationshiptaken as
= 1
then the povolumetric
= (1 )e
For calculation of the transient problem of solid-state or
liquidphase sintering, a standard mechanical approach based on the
solu-tion of the equilibrium equation is used, i.e.
i =
ij,j T thationrce in thorce ction
nite e
s seclemome
h0. T are er onore,
are is alse rigis astical mpesion vsky
0
(1
the ues os from4).een fhe cyed in
ults
put of th
shont alome o
to thcimeporoe thaial dnsic
in andavity
and
= (1ig. 2V)
nd d anaeous porosity distribution induced by gravity. Theed to
simulate isothermal sintering of cylindrical spec-ifferent initial
compact aspect ratios, porosities, andnsities. The results are
compared with the analyti-mations for the dimensional changes as
proposed by
German (2000).plicity, equations are written using index
notations,rcase indices label the coordinate directions x, y,
and
nce, the displacement vector and the stress state ten-ten as u
and ij, respectively. A repeated index implies
over the index range. The notations ui, ui,j and ui,i arerms for
the time derivative, gradient, and divergence ofnt vector ui,
respectively. Other notations are dened as
in the equations.
tive equations and modeling
l formulations
ticular case of the continuum conceptualization of sin-sky,
1998), a linear viscous relationship is used to relateand strain
rates ij in the form:
ij + ( 13)kkij]
+ PLij (1)
the shear modulus of the fully-dense material, andalized
viscosity parameters, and kk is the volumetric, respectively. The
model considers the contribution ofg stress in the form of an
isotropic driving stress PLijout the densication with time.ing an
innitesimal strain-displacement relationship,ate components are
written as ij = 12 (ui,j + uj,i) wherelocity gradient tensor.g
Olevsky (1998), the two normalized viscosity param-ear and bulk
viscosities, can be written as function:
and = (2/3)(1 )3
(2)
ing modeling, porosity is considered an internal vari-enting the
fraction of volume of voids over the totalvolume. It is assumed
that pores are uniformly dis-a continuous way. The associated
porosity evolution ish that the porosity evolution rate is assumed
to be the
e process of bulk volume reduction caused by the sin-ss. From
the conservation of mass, it is established that0, where V and V
are the volume and volume changeng due to sintering shrinkage, and
ui,i is the volumetriciven as ui,i = ii = e.erial transformation
from the powder to a densiedaccompanied by a variation of the local
density. If the
between porosity and the relative density is simply
(3)
rosity evolution law can be written in terms of thestrain rate
as
(4)
ij,j + f
whereforces,accelerbody fog acts body fthe fra
2.2. Fi
Thinite emen geheightplanesonly ovTherefplanesdition relativ
It itheorehigh-teexpresby Ole
PL =2r
wherethe valevolvein Eq. (
As ster of texpect
3. Res
Comuationand (c)gradiecic tirelatedthe speinitial indicatthe
radthe de
The(Rb/Rt)ent gr(h0/R0)
that R
men. F = (1/solid acal andTui (5)
denotes the divergence of stress tensor, fi the volumee
theoretical density, the relative density, and ui the. In the
model, the inuence of gravity is considered as a
that acts throughout the volume. If gravity acceleratione
negative z-direction according to Fig. 1(a), then thehas components
fx = fy = 0 and fz = Tg, where is
of Earth gravity acceleration g used in the simulations.
lement modeling
tion introduces the modeling using the commercialent software
COMSOL Multiphysics. The model speci-tries consist of circular
cylinders of initial radius R0 andwo different aspect ratios are
analyzed. Two symmetry-used in each of the cylindrical specimens,
proceedinge-fourth of the original geometry, as shown in Fig.
1(a).
the displacements in the normal directions of theseconstrained
to zero. A similar motion constraint con-o used to restrain the
bottom of the cylinder against thed body motion. No friction is
considered.sumed that the powder material is copper with adensity T
= 8960 kg/m3 (fully dense material) andrature shear modulus 0 = 2.0
GPa s. In the model, thefor the sintering stress corresponds to the
one proposedand German (2000) as follows:
)2
(6)
average particle radius r0 and surface tension takef 44 m and
1.72 J/m2, respectively. In Eq. (6), porosity
an initial value 0 according to the evolution law given
rom Fig. 1(a), a relatively ner discretization at the
cen-linders bottom is used to capture the porosity gradient
that region during sintering.
and discussion
ations under different gravity forces allowed the eval-e
shrinkage inhomogeneity during sintering. Fig. 1(b)
ws the distribution of porosity and radial displacementng the
axis (ur,z) in the cylindrical specimen at the spe-f sintering of
1.5. The dimensionless sintering time ise real time t as s =
2t/r00. It is assumed that initiallyn is 7.5 mm in radius and 15.0
mm in height and has ansity of 0.4. The distributions shown in Fig.
1(b) and (c)t porosity is lower at the bottom of the cylinder;
andisplacement reveals a signicant inuence of gravity onation
response.uence of the densication on the bottom-to-top radii
height-to-radius (hR0/h0R) ratios under two differ- forces and
under two different initial aspect ratios
initial porosities 0 was investigated. It was assumed
/R) h0R dz and h is the current height of the speci-
shows the predicted ratios versus porosity , where
V dV is the porosity volume average. In this gure, the
ashed lines correspond to the results of the numeri-lytical
models, respectively. According to the analytical
-
10 J.A. Alvarado-Contreras et al. / Mechanics Research
Communications 50 (2013) 8 11
Fig. 1. (a) Computational domain of the circular cylindrical
specimen and distributions of (b) porosity and (c) radial
displacement gradient along the cylinders axis.
approach proposed by Olevsky and German (2000), the
expressionsfor the bottom-to-top radii and height-to-radius ratios
are given as
hR0h0R
=(
0
)2/(1+)and
RbRt
=[
1 01
(0
)2]2/(3(1+))(7)
where is a dimensionless parameter characterizing the
rela-tionship between surface tension and gravity-imposed
stresses(Olevsky and German, 2000) with the form
= PLohTg
= 12hTgr0
(8)
where h is the arithmetical mean of the initial and nal
heights.As pointed by German (2003), important differences on
dis-
tortion and densication have been found in Earth- and
in-spacesintered samples. Although dimensional precision is
inuenced bychanges in composition and sinter regime, gravity seems
to play
Fig. 2. A compradii and heighheight-to-radi
an important role on interparticle bonding and pore closure
kinet-ics. To study the impact of gravity on shape distortion, a
cylindricalsample with an initial aspect ratio of h0/R0 = 6.0 and
initial homo-geneous porosity of 0 = 0.4 is considered. As shown in
Fig. 2, threedifferent gravity accelerations (1.0, 0.5, and 0.1 g)
are considered.The model predicts that the shape distortion
increases more rapidlyas gravitational acceleration increases. For
these cases, when thespecimen reaches an average porosity of 0.01,
the model predictsfor the gravity accelerations of 1.0 g and 0.5 g
distortion differenceswith respect to the 0.1 g case of 7.7 and
2.1% for the bottom-to-topradii ratios and of 12.4 and 3.5% for
height-to-radius ratios, respec-tively. As expected for the same
initial aspect ratio, distortion isless pronounced as the factor
increases. Likewise, Fig. 2 showsthe shrinkage ratios as predicted
by analytical model (Olevskyand German, 2000). The most noteworthy
feature in the analyt-ical curves is that the distortion ratios are
less pronounced. Thisparticularity can be attributed to the fact
that the analytical modelarison of the numerical and analytical
results for the bottom-to-topt-to-radius ratios at different
gravity accelerations, specimen, initialus aspect ratios, and
porosities.
Fig. 3. A compgradients alongravity accelearison of the
numerical results for the average porosityand porosityg the
cylindrical specimens axis and bottom radius at three
differentrations.
-
J.A. Alvarado-Contreras et al. / Mechanics Research
Communications 50 (2013) 8 11 11
Fig. 4. Bottom
assumes anfor the cylinevolution foation of 1.0the same gwhen
comp
Fig. 3 preity gradienta selection an initial ascarried outmodel
predthe gravity inuence omodel, gravcylinders adients increfor the
axiaAfter those the materia
The gravand et are tsurface of talso in Fig. versus averis seen
thatcation is sspecimen. Hdensicatioing the denradial disto
4. Conclus
A nitesintering isshrinkage i
captured the inuence of porosity, initial specimens
height-to-radius aspect ratio, and gravity acceleration on the
evolution of thebottom-to-top radii, height-to-radius ratios, and
bottom-to-topvolume change rate ratios. First-of-a-kind assessment
of the inu-ence of the non-uniformity of the sintering shrinkage on
the shapedistortion induced by gravity is obtained in a general
material-independent form. The numerical results compare favorably
witha previously developed analytical model (Olevsky and
German,2000). This paper leaves the comparisons with experimental
datafor a future work.
wled
s respaceAV3Patto.
nces
., Lenc denssactio.C., Botional875.E., Bader p1., R., Gng
liq664.
, R.M., Scien, J.L., Csten hs. Met, J.L., Uon anograv866.
, Riedeicatio, Smithng liq573Z.A., Sced sputer, E.A.,
and E, E.A., ring
, E.A.,-to-top volume-change-rate ratio at different gravity
accelerations.
idealized uniform porosity and a truncated cone shapeder.
Similarly, Fig. 2 compares the predicted distortionr an initial
aspect ratio h0/R0 of 2.0 and gravity acceler-
g obtained from the numerical model. As expected, forravity
acceleration, distortion progresses more slowlyared to those cases
with larger initial aspect ratios.sents the evolution of the
average porosity and poros-s along the cylinders axis ,z and bottom
radius ,r forof three gravity accelerations (1.0, 0.5, and 0.1 g)
andpect ratio h0/R0 of 6.0. All calculation examples were
for the same initial porosity of 0.4. As expected, theicts
identical average porosity evolution, regardless ofintensity.
Nevertheless, the curves reveal a signicantf gravity on the
densication process. According to theity induces comparable
porosity gradients along the
xis and bottom radius. For all the three cases, these gra-ase as
time progresses, reaching their maximum valuesl and radial
directions at s = 1.1and 1.5, respectively.time periods, the
overall gradients decrease rapidly asl reaches fully dense
state.ity effect on densication can be seen also in Fig. 4. If ebhe
average volume change rates at the bottom and tophe cylinder,
gravity effect on densication can be seen4. This plot presents the
volume change rates (eb/ et)
Ackno
Thiand SNNX10James Center
Refere
Binet, CafterTran
Blaine, Dto ra867
Doege, pow576
Ganesanduri659
Germanrials
Johnsontungtion
Johnsontortimicr857
Kraft, T.appl
Naser, J.duri33, 5
Nikoliv, induCom
Olevskyence
Olevskysinte
Olevsky
age porosity at 1.0, 0.5, and 0.1 g. In the simulations, it
that at the beginning of the sintering process, densi-ignicantly
larger at the bottom than at the top of theowever, due to the
restraining forces at the bottom,
n at the top increases as porosity decreases overcom-sication of
its counterpart, which serves to increasertion.
ions
element formulation for modeling gravity-affected shown. The
model predicts the densication andnhomogeneity induced by gravity.
The simulations
sional cha1180.
Park, S.J., Chunof liquid p2745275
Raman, A., Gerduring liq653659.
Schoenberg, S2006. Stretion. Journ
Xue, Z., Noojitional Symvol. 37 , p.
Ye, S., He, Y., Ntion in miCommunigments
earch was supported by the National Aeronautics Administration,
Materials Science Program (Grant8G). The authors appreciate the
guidance provided byn Downey and Biliyar Bhat of the Marshall Space
Flight
oski, K.L., Heaney, D.F., German, R.M., 2004. Modeling of
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Modeling of gravity-induced shape distortions during sintering
of cylindrical specimens1 Introduction2 Constitutive equations and
modeling2.1 General formulations2.2 Finite element modeling
3 Results and discussion4
ConclusionsAcknowledgmentsReferences
