Densification of porous bodies in a granular pressure-transmitting medium E.A. Olevsky a, * , J. Ma a , J.C. LaSalvia b , M.A. Meyers c a San Diego State University, Mechanical Engineering Department, 5500 Campanile Drive, San Diego, CA 92182-1323, USA b Materials Division, US Army Research Laboratory, APG, MD 21005-5069, USA c Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0416, USA Received 5 July 2006; accepted 25 September 2006 Available online 11 December 2006 Abstract Densification is a critical step in the manufacture of near-net-shaped components via powder processing. A non-isostatic stress state will in general result in shape distortion in addition to densification. In the quasi-isostatic pressing (QIP) process the green body is placed into a granular pressure-transmitting medium (i.e. PTM), which is itself contained in a rigid die. Upon the application of a uniaxial load, the PTM redistributes the tractions on the green body, thereby creating a stress state that is quasi-isostatic. The character of the defor- mation of the PTM is studied using model experiments on pressing of the PTM in a rigid die and a scanning electron microscopy analysis of the PTM powder. An important problem of the optimization of the PTM chemical composition enabling the maximum densification of a porous specimen with the minimum possible shape distortion is solved. The results of modeling agree satisfactorily with the exper- imental data on cold QIPing Ti and Ni powder samples and hot QIPing TiC–TiNi cermet composites. Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Quasi-isostatic pressing; Shape change; Deformation of porous bodies; Self-propagating high-temperature synthesis 1. Introduction Densification of porous materials by the application of compressive stress is an important mechanical processing method. Porous bodies undergo volume change in addition to shape change during mechanical treatment; this intro- duces additional considerations with regard to possible deformation modes, in comparison to fully dense bodies. Several methods are used to insure densification of por- ous materials (for more details, see German [1]). The prin- cipal ones are shown in Fig. 1. Stress states imposed by the different methods depend on the loading mode. In the terminology of mechanical treatment of porous bodies, the volume change relative to the shape change cor- responds to the ‘‘stiffness’’ of the deformation modes. Free up-setting has the lowest stiffness of the deformation modes shown in Fig. 1. Isostatic pressing has the highest stiffness. It is evident that the higher stiffness causes the larger den- sification degree, which is of significant importance for final mechanical properties. In this regard, isostatic press- ing has the highest potential for the production of full- dense articles. However, the high cost of equipment associated with isostatic pressing (e.g. CIP and HIP) of particulate and porous bodies lends impetus for other cost-effective technologies, which are technically simple while providing a sufficiently high degree of stiffness with respect to the deformation mode. Uniaxial pressing with a pressure-transmitting medium (PTM) (Fig. 1) has been attracting attention as one such possible alternative method. Known as the Ceracon [2–4] or quasi-isostatic pressing (QIP) process, it has been utilized industrially in manufacturing [1,5,6] and, in particular, in combination with self-propagating high- temperature synthesis (SHS) [7–20]. This technological 1359-6454/$30.00 Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.09.039 * Corresponding author. Tel.: +1 619 5946329; fax: +1 619 5943599. E-mail address: [email protected](E.A. Olevsky). www.actamat-journals.com Acta Materialia 55 (2007) 1351–1366
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Acta Materialia 55 (2007) 1351–1366
Densification of porous bodies in a granularpressure-transmitting medium
E.A. Olevsky a,*, J. Ma a, J.C. LaSalvia b, M.A. Meyers c
a San Diego State University, Mechanical Engineering Department, 5500 Campanile Drive, San Diego, CA 92182-1323, USAb Materials Division, US Army Research Laboratory, APG, MD 21005-5069, USA
c Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0416, USA
Received 5 July 2006; accepted 25 September 2006Available online 11 December 2006
Abstract
Densification is a critical step in the manufacture of near-net-shaped components via powder processing. A non-isostatic stress statewill in general result in shape distortion in addition to densification. In the quasi-isostatic pressing (QIP) process the green body is placedinto a granular pressure-transmitting medium (i.e. PTM), which is itself contained in a rigid die. Upon the application of a uniaxial load,the PTM redistributes the tractions on the green body, thereby creating a stress state that is quasi-isostatic. The character of the defor-mation of the PTM is studied using model experiments on pressing of the PTM in a rigid die and a scanning electron microscopy analysisof the PTM powder. An important problem of the optimization of the PTM chemical composition enabling the maximum densificationof a porous specimen with the minimum possible shape distortion is solved. The results of modeling agree satisfactorily with the exper-imental data on cold QIPing Ti and Ni powder samples and hot QIPing TiC–TiNi cermet composites.� 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Densification of porous materials by the application ofcompressive stress is an important mechanical processingmethod. Porous bodies undergo volume change in additionto shape change during mechanical treatment; this intro-duces additional considerations with regard to possibledeformation modes, in comparison to fully dense bodies.
Several methods are used to insure densification of por-ous materials (for more details, see German [1]). The prin-cipal ones are shown in Fig. 1. Stress states imposed by thedifferent methods depend on the loading mode.
In the terminology of mechanical treatment of porousbodies, the volume change relative to the shape change cor-responds to the ‘‘stiffness’’ of the deformation modes. Free
1359-6454/$30.00 � 2006 Acta Materialia Inc. Published by Elsevier Ltd. All
up-setting has the lowest stiffness of the deformation modesshown in Fig. 1. Isostatic pressing has the highest stiffness.It is evident that the higher stiffness causes the larger den-sification degree, which is of significant importance forfinal mechanical properties. In this regard, isostatic press-ing has the highest potential for the production of full-dense articles. However, the high cost of equipmentassociated with isostatic pressing (e.g. CIP and HIP) ofparticulate and porous bodies lends impetus for othercost-effective technologies, which are technically simplewhile providing a sufficiently high degree of stiffness withrespect to the deformation mode.
Uniaxial pressing with a pressure-transmitting medium(PTM) (Fig. 1) has been attracting attention as one suchpossible alternative method. Known as the Ceracon [2–4]or quasi-isostatic pressing (QIP) process, it has beenutilized industrially in manufacturing [1,5,6] and, inparticular, in combination with self-propagating high-temperature synthesis (SHS) [7–20]. This technological
sequence was pioneered by Merzhanov and co-workers [21]in Russia and implemented in the US by Raman et al. [2].When combined with SHS, QIP offers a relatively simpleprocessing method by which hundreds of industriallyuseful materials can be produced and shaped into engineer-ing components [3]. A granular PTM (alumina or aluminawith graphite powder) serves not only as a load transmitter,but also as a natural thermal insulator which preventssubstantial heat loss and minimizes temperature gradientsduring SHS. This leads to increased flexibility in thepressurization cycle, as well as microstructural uniformity(grain size, phase distribution, etc.).
In contrast to a conventional containerless isostaticpressing, a considerable shape change is obtained underQIP [1,4].
In light of the development of near-net-shape technol-ogies, the analysis of both shape and volume changesunder QIP is of considerable importance. The factorswhich influence the shape and the volume change of aporous body include the initial porosities of both thePTM and the porous body, and their respective constitu-tive properties. The objective of the present work is theinvestigation of the effect of these factors on shape changeduring QIP.
The paper is organized as follows: Section 2 provides adescription of a constitutive model for nonlinear-viscousdeformation of porous bodies. Section 3 comprises the the-oretical analysis of the quasi-isostatic pressing. In this sec-tion, the change in the aspect ratio for a cylindrical porousbody is analyzed for the conditions of free up-setting,pressing in a rigid die, isostatic pressing and QIP. Section4 includes the experimental data on the constitutive behav-ior of PTM. Section 5 includes the solution of an importantproblem of the optimization of the PTM chemical compo-sition (the concentrations of graphite and alumina pow-ders) enabling maximum densification with minimum
possible shape distortion. Section 6 presents the results ofcold QIP experiments on Ti and Ni porous bodies andhot QIP experiments on TiC–TiNi cermet composites. Acomparison of experimental and theoretical results is givenin Section 7.
2. Theory of nonlinear-viscous deformation of porous bodies
The mechanical response of a nonlinear-viscous porousbody to an externally applied pressure (which is signifi-cantly higher than the sintering-imposed stresses) can bedescribed [22–26] by a rheological (constitutive) relation-ship of a continuum theory of sintering connecting compo-nents of the stress tensor rij and the strain rate tensor _eij
and omitting the effective sintering stress:
rij ¼rðW Þ
Wu_eij þ w� 1
3u
� �_edij
� �ð1Þ
where W is the so-called equivalent strain rate, and r(W) isthe equivalent stress, u and w are the shear and bulk viscos-ity moduli, which depend on porosity h, dij is a Kroneckersymbol (dij = 1 if i = j and dij = 0 if i 6¼ j) and _e is the firstinvariant of the strain rate tensor, i.e. the sum of the tensordiagonal components: _e ¼ _e11 þ _e22 þ _e33.
The porosity h is defined as 1� qqT
, where q and qT rel-
ative and theoretical (corresponding to a fully dense state)densities, respectively. Physically, _e represents the volumechange rate of a porous body.
Equivalent strain rate W is connected with the currentporosity and with the invariants of the strain rate tensor:
W ¼ 1ffiffiffiffiffiffiffiffiffiffiffi1� hp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu _c2 þ w _e2
pð2Þ
(the origin of this equation is explained in Ref. [24]); _c is thesecond invariant of the strain rate tensor deviator and rep-resents, physically, the shape change rate of a porous body:
Fig. 2. Conditions of biaxial loading of a porous sample during QIP.
Parameter r(W) determines the constitutive behavior ofa porous material. If r(W) is described by the linearrelationship r(W) = 2g0W, where g0 is the shear viscosityof a fully dense material, one obtains the equationcorresponding to the behavior of a linear-viscous porousbody (used to describe hot deformation of amorphousmaterials):
rij ¼ 2g0 u_eij þ w� 1
3u
� �_edij
� �ð4Þ
If r(W) is a constant (r(W) = ry, ry being the yieldstress for a fully dense material), the equation correspond-ing to a rigid-plastic porous body (used to describe colddeformation processing) is obtained:
rij ¼ry
ffiffiffiffiffiffiffiffiffiffiffi1� hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu _c2 þ w _e2
p u_eij þ w� 1
3u
� �_edij
� �ð5Þ
In the general case, r(W) is described by a nonlinearrelationship. For example, for hot deformation of crystal-line materials, a power law is used [1] (r(W) = AWm whereA and m are the material constants, 0 6 m 6 1). In thiscase, we have:
rij ¼ A
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu _c2 þ w _e2
pffiffiffiffiffiffiffiffiffiffiffi1� hp
!m�1
u_eij þ w� 1
3u
� �_edij
� �ð6Þ
It can be noted, that, for m = 1, Eq. (6) is transformed intoEq. (4) (A = 2g0) and, for m = 0, Eq. (6) is transformedinto Eq. (5) (A = ry).
Thus, linear-viscous and rigid-plastic behaviors are twolimiting cases for a nonlinear-viscous constitutive behavior.
3. Analysis of the densification processes
Based on the theory of nonlinear-viscous deformation ofporous bodies described in the previous section, it is possi-ble to assess both the densification kinetics and the shapeevolution of a specimen subjected to QIP.
It is assumed that the stresses are uniform within boththe PTM and the porous body. For simplicity, a cylindricalgeometry is assumed and a cylindrical coordinate system isused.
The volume–change rate _e and the shape–change rate _care given by:
_e ¼ _ezz þ 2_err ¼ 1þ 2_err
_ezz
� �� �_ezz ¼
_h1� h
ð7Þ
_c ¼ffiffiffi2
3
r_ezz � _errj j ¼
ffiffiffi2
3
r1� _err
_ezz
� ��������� _ezzj j ð8Þ
where _ezz, _err and h are the axial strain rate, radial strainrate and porosity, respectively. For a cylindrical speci-men, the axial and radial strain rates are given by (seeFig. 2):
_ezz ¼_H
H 0
; _err ¼_R
R0
ð9Þ
where H and R are the instantaneous cylinder height andradius. Substituting Eq. (3) into Eq. (8) gives the followingrelationship for the shape–change rate:
_c ¼ffiffiffi2
3
r_H
H�
_RR
�������� ð10Þ
This expression will be used to derive relationshipsbetween the height and the radius of the cylindrical speci-men and the porosity.
The radial rrr and the axial rzz stresses can be written as(see Eqs. (1) and (6)):
rrr ¼ AW m�1u_err
_ezz
� �� 1
31� 3
wu
� �� �1þ 2
_err
_ezz
� �� �� _ezz
ð11Þ
rzz ¼ AW m�1u 1� 1
31� 3
wu
� �� �1þ 2
_err
_ezz
� �� �� _ezz ð12Þ
where W is the equivalent strain rate, A and m (0 6 m 6 1)are material constants, and u and w are the normalizedshear and bulk viscosity moduli (see Section 2). The shearand bulk viscosity moduli depend upon porosity h and aregiven by [22–26]:
u ¼ ð1� hÞ2 ð13Þ
w ¼ 2
3
1� hð Þ3
hð14Þ
From Eqs. (13) and (14), the ratio of the bulk to shear vis-cosity moduli is obtained:
The equivalent strain rate W is defined, by substitutingEqs. (7), (8) and (13) into Eq. (2), as:
W ¼ u1� h
_c2 þ wu
� �_e2
� �� 12
¼ ð1� hÞ 2
31� _err
_ezz
� �� �2
þ wu
� �1þ 2
_err
_ezz
� �� �2( )" #1
2
_ezzj j
ð16ÞIf the specimen’s volume is much smaller than the vol-
ume of the PTM-containing rigid die, then the specimen’sdeformation is accompanied by a negligible deformationof the PTM material. Hence, the granular flow of PTM isminimal and it can be assumed that the PTM (i.e. PTM)behaves as a purely elastic porous body (see Section 4and Appendix). It is further assumed that the presence ofthe porous cylindrical body within the PTM provides anegligible effect on its state-of-stress as a result of theapplied axial load. This is equivalent to imagining the por-ous cylindrical body embedded in an infinitely extendedPTM with a far-field applied stress r1zz at its boundary.The PTM itself is assumed to be under the condition of auniaxial load with a lateral confinement (i.e. pressing in arigid die). Therefore, the sample can be considered underconditions of biaxial loading (Fig. 2).
For the PTM, the axial and radial stresses are related tothe axial strain ezz by the Hooke’s law and are given by (seeEq. (A1) in Appendix):
rzz ¼1� m
ð1þ mÞð1� 2mÞ
� �Eezz ð17Þ
rrr ¼m
ð1þ mÞð1� 2mÞ
� �Eezz ð18Þ
where m and E are the Poisson’s ratio and Young’s modulusfor the PTM, respectively. These depend upon the PTMporosity hp and are given by (see Section 4 and Appendix):
where C is the concentration of graphite in the PTM. The ra-tio of the axial stress to the radial stress is therefore given by:
rzz
rrr¼ 1� m
v¼ 2
2� 3hp
¼ k ð21Þ
The ratio of the radial stress to the axial stress is alsoobtained from Eqs. (11) and (12) And it is given by:
rrr
rzz¼
errezz
�� 1
31� 3 w
u
�h i1þ 2 err
ezz
�h i1� 1
31� 3 w
u
�h i1þ 2 err
ezz
�h i ð22Þ
By substituting Eq. (15) into Eqs. (21) and (22), the fol-lowing expression for the axial/radial strain rate ratio isobtained:
_err
_ezz¼ h� hp
2hp þ ð1� 3hpÞhð23Þ
It is methodologically useful to compare Eq. (23) withthe following expressions for axial/radial strain rate ratioin conventional densification processes [28]:
_err
_ezz¼
1� 3 wu
�1þ 6 w
u
� ¼ � 2� 3h4� 3h
� �for free up-setting ðpressing
without lateral confinement; rrr ¼ 0Þ ð24Þ_err
_ezz¼ 0 for pressing in a rigid die ðpressing with lateral
If k ¼ 22�3h (which means that h = hp), then _err ¼ 0, and one
has conditions of pressing in a rigid die. If k = 1 (i.e. ifhp = 0, which means that PTM is an incompressible mate-rial), _err ¼ _ezz, and the conditions of isostatic pressing areachieved.
If k!1 (i.e. if hp = 2/3), which approximately corre-sponds to the density of packed isomeric spherical parti-cles, then _err ¼ 3h�2
4�3h_ezz, and one has the conditions of free
up-setting (see Eq. (24)).In Fig. 3, the curves corresponding to the relationships
between the radial/axial strain rate ratio and sample’sporosity (in accordance with Eq. (23)) for various PTMporosities are shown. For comparison, the curves corre-sponding to the conditions of free up-setting (Eq. (24)),
pressing in a rigid die _err_ezz¼ 0
�and isostatic pressing
_err_ezz¼ 1
�are also shown.
The data in Fig. 3 indicate that, for high PTM porosi-ties, the porous material deformation mode under QIPcan be closer to the conditions of free-upsetting rather thanto the isostatic pressing ones.
Mostly, the radial/axial strain rate ratio, for hp > 0.5, isin between the curves corresponding to pressing in a rigiddie and free up-setting. This fact testifies the intensiveshape change under QIP which is the distinctive featureof this process in comparison with the conventional con-tainerless isostatic pressing.
For QIP, the equivalent strain rate is given by:
W ¼ffiffiffi6p ð1� hpÞð1� hÞ
32
2hp þ ð1� 3hpÞh
" #hp
1� hp
� �2
þ h1� h
" #12
_ezzj j
ð27Þ
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.20.000 0.100 0.200 0.300 0.400 0.500 0.600
Porosity, θ
Rad
ial -
Axi
al s
trai
n r
ates
rat
io
Isostatic Pressing
Pressing in Rigid Die
QIP (PTM Porosity 10%)
50%
20%
30%
40%
60%
Free Up-Setting
67%
Fig. 3. Radial–axial strain rate ratio vs. porosity for different processes of treatment by pressure.
In deriving Eq. (35), it was assumed that hp is a constant(see Appendix). Eq. (35) indicates that the change in the as-pect ratio H/R does not depend upon the constitutivebehavior of either the PTM or densifying body, but de-pends only on the PTM’s porosity and the body’s initialdensity and dimensional parameters.
Eq. (35) can be compared with analogous expressionsfor conventional densification processes [28]:
When hp = 2/3, we obtain Eq. (36) derived for the condi-tions of free up-setting. If hp = 0, we have the conditionsof isostatic pressing (Eq. (38)) when H/R = const. Ifhp = h in the vicinity of h = h0, we obtain Eq. (37), corre-sponding to the aspect ratio evolution in the conditionsof pressing in a rigid die, as the first term of expansion ofexpression (35).
The results of the calculations in accordance with Eqs.(35)–(38) are shown in Fig. 5.
Change of the aspect ratio expressed by the relationshipHR =
H0
R0is represented as a function of the sample’s porosity.
The initial porosity of the sample is assumed to be h0 = 0.3.The calculation results indicate that, for sufficiently
dense PTM, having porosity hp < 0.2, the deformationstate under QIP is close to the isostatic one. However,for most cases, in the capacity of PTM, industrial sand(alumina) mixed with graphite powder in a loose state isused.
For this PTM kind, hp > 0.2. This means that the aspectratio evolution under QIP can be close to that one obtainedunder the conditions of pressing in a rigid die or free up-setting.
4. Constitutive behavior of PTM
In order to understand the shrinkage and shape distor-tion of porous bodies densified in a granular PTM, it isimportant to understand the mechanical behavior of thePTM itself under compression load. The major questionto be answered is the verification of the assumption madein Section 3. The model introduced in Section 3 assumedan elastic behavior of the PTM during compression. Ifthis hypothesis is valid, then the dependence of theYoung’s modulus of the PTM with respect to the PTM
Comparision of Ec/Er (50% C
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
50.0 55.0 60.0
Relative Density (%
Rat
io o
f E
c/E
r
Fig. 6. Normalized Young’s modulus vs. PTM relative density (the
composition and the PTM porosity should obey certainrules. A series of experiments have been conducted tounderstand the evolution of the PTM Young’s modulusduring densification.
4.1. Experiments on PTM pressing in a rigid die
In our study, a mixture of 99% pure fused irregularshaped alumina (Al2O3) with typical size of 200 lm and99% pure spherical shaped graphite (C) with typical sizeof 150 lm with different mass percentage was used asPTM. The alumina and graphite were mixed in a V-shapeplastic container fixed in a rotary mixing machine for atleast 8 h. The mixed PTMs were stored in sealed plasticcontainers for further use.
The PTM powders were placed in a rigid die with pol-ished walls and with the inner diameter of 0.5 in. ThePTM was pressed at a maximum load of 1.5 tons in anINSTRON� machine, which recorded the load and thedisplacement. Thereby the stress–strain dependence duringPTM compression has been determined for the composi-tion range from 0% graphite and 100% alumina to 100%graphite and 0% alumina with an increasing (decreasing)step of 20% of each of the components. Based on theseexperimental data, one can obtain the evolution of theYoung’s modulus of the PTM powders during compres-sion. At the same time, the relative density of the PTMpowders during compression can be determined based onmass conservation. Fig. 6 shows a set of the normalizedYoung’s modulus vs. relative density dependences for allthe tested PTM compositions. (The Young’s modulus forall the PTM compositions was normalized by the Young’smodulus Er (the reference Young’s modulus) of the compo-sition containing 50% Al2O3 and 50% C.)
+50% Al2O3)
65.0 70.0
)
100% C
80% C
60% C
50% C
40% C
25% C
20% C
15% C
10% C
7.5% C
5% C
2.5% C
0% C
reference modulus is obtained for a fixed PTM composition).
Fig. 7. SEM micrograph of the loose PTM powder: (a) 50% alumina and50% graphite; (b) cracks on surfaces of graphite particles; (c) graphiteparticles after compression.
The first set of tests indicated that there is a substantialchange in the densification behavior when the compositionof alumina is greater than 80%. Therefore, the additionaltests with graphite composition of 25%, 15%, 10%, 7.5%,5% and 2.5% were conducted.
The qualitative analysis of the PTM particles’ response tothe applied pressing load was carried out by scanning elec-tron microscopy (SEM). The results of the scanning electronmicroscopy analysis of the loose PTM powder are shown inFig. 7. It is evident (Fig. 7a and b) that graphite particlesexperience extensive damage at the first several cycles ofthe PTM usage (when ‘‘fresh’’ PTM is employed). Aluminaparticles, on the contrary, do not indicate any substantialevidence of cracking (Fig. 7a and c). In order to assess thedegree of the deformation of the PTM during quasi-isostaticpressing, the recycled PTM alumina powder has beenpressed with a polymeric binder (polychlorvinile) in a rigiddie under 100 MPa pressure. The pressed specimens wereremoved from the die and polished using standard metallo-graphic techniques. Examination of both the loose PTMpowder and the pressed specimens was conducted using aCambridge Stereoscan 360 SEM. The results of the SEManalysis of the pressed specimen are shown in Fig. 8.
No particle distortions or areas of a remarkable plasticflow are observed in Fig. 8. The results shown in Figs. 7cand 8 confirm an accepted hypothesis of a pure elasticdeformation of the recycled PTM powder. The applicabil-ity of an elastic model for the description of the ‘‘fresh’’PTM constitutive behavior is studied by the followingquantitative analysis of the obtained experimental dataon the PTM pressing in a rigid die.
4.2. Analysis of the experimental data
Fig. 5 indicates that the PTM’s normalized Young’smodulus (obtained as a slope on the correspondingstress–strain diagrams assuming the boundary conditionsof the deformation in a rigid die), does not experience asubstantial change when the relative density is greater than60%. Therefore, it is appropriate to make an assumptionthat the PTM Young’s modulus E(C,qP), which, in general,is a function of the PTM composition C and the PTM rel-ative density qP, can be represented as a product of twofunctions Ec(C) and EqP
ðqÞ, where Ec(C) is a function ofthe PTM composition C only and EqP
ðqÞ is a function ofthe PTM relative density q only. (Indeed, this functionalform justifies the independence of the concentration-normalized PTM Young’s modulus of the PTM relativedensity.)
According to our assumption, we have:
EðC; qÞ ¼ EC � EqPð39Þ
Since Ec does not depend on the PTM relative density, wecan use the effective Young’s modulus for a fully densePTM material for the purpose of finding a general expres-sion of EqP
as a function of the relative density qP. FromEq. (39) we have:
EqP¼ EðC; qPÞ=EC ðFully DenseÞ ð40Þ
Here EC (Fully Dense) is determined as a linear combina-tion of the corresponding elastic moduli for alumina andgraphite.
Fig. 8. SEM micrograph of the recycled alumina PTM powder pressed under 100 MPa with a polymeric binder in a rigid die.
Fig. 9 shows the experimental results for EqPfor different
PTM compositions. From this figure one can see that theobtained curves are very close to each other. This, in turn,indicates that EqP
does not depend considerably on thePTM composition. This fact agrees with the above-mentioned assumption.
By taking a linear approximation of the average of allthe curves in Fig. 9, a general expression for Eq can bedetermined:
Fig. 9. Normalized Young’s modulus vs. PTM relative density
EqP¼ 0:0188 � qP � 0:8764 ð41Þ
Following a similar process, from Eq. (39) we have:
EC ¼ E C; qPð Þ=EqPð42Þ
From Fig. 6, it follows that Ec/Er does not change sub-stantially for relative densities greater than 60%. Thisallows an approximation (as shown in Fig. 10) for Ec as:
Ec ¼ 384:55 � ðC þ 1Þ�0:22475 ð43Þ
(the reference modulus is obtained for a fully dense PTM).
Fig. 10. Approximation of the concentration dependence of the PTM Young’s modulus.
Combining Eqs. (41) and (43), we obtain the generalexpression for the PTM Young’s modulus as a functionof the PTM composition and the PTM relative density(used earlier as Eq. (20)):
where C is the mass percentage of graphite in PTM and qp
is the PTM relative density during the compression process.
5. Optimization of PTM composition
Having determined the functional form of the elasticmodulus, one can consider the influence of the PTM com-position on the distortion of a porous body subjected toQIP. The elastic behavior of the PTM is determined byHook’s law. Therefore, the axial stress in the PTM canbe represented as:
rzp ¼E
1þ mezz þ
m1� 2m
ðezz þ 2errÞh i
ð45Þ
where E and m are the PTM’s Young’s modulus and Pois-son’s ratio, respectively; ezz and err are the PTM axialand radial deformations, respectively. For a rigid die,err = 0, therefore:
rzp ¼Eð1� mÞ
ð1� 2mÞð1þ mÞ ezz ð46Þ
The Poisson’s ratio for a compressible PTM is acceptedas [29]:
m ¼ 2� 3hp
4� 3hp
ð47Þ
where hP = 1 � qP is the relative porosity of the PTMs.From Eqs. (6) and (9), Eq. (8) can be written as:
In Section 3, for the engineering assessment of the shapechange during QIP (see Eq. (35)), the constancy of thePTM porosity hP has been assumed. Here, for the rigorousoptimization analysis, we consider a more general case ofevolving hP. Both solutions (with constant and changeablePTM porosity) are compared with the experimental dataon cold and hot QIPing of cylindrical specimens in Section 7.
Substituting the expression for the axial strain (based onmass conservation),
where parameters A and m are material constants. From ourformer study [30] on the testing by indentation of combustionsynthesized cermet specimens, the values of A = 180 MPa0.2
and m = 0.2 were used in this work (these values were ob-tained for a TiC–NiTi cermet composite with the 30 vol.%NiTi). Equilibrium in the system PTM–specimen requires:
rzp ¼ rzz ð52ÞFor the purpose of the generalization of the optimiza-
tion results, a normalized time of s ¼ AE0
�1mt is introduced,
where E0 is the effective Young’s modulus for a fully densePTM material and t is the physical time of the process.Therefore, from Eqs. (12)–(14), we obtain:
Based on Eqs. (8), (10) and (23), the following kineticdifferential equations are valid for the normalized dimen-sions H/H0 and R/R0, where H and H0 are the currentand the original heights of the specimen, respectively, and
dðhP=hP0Þ
ds¼
qSpe
qPTM2 R
R0
HH0
dðR=R0Þds ð1�hÞþ R
R0
�2dðH=H0Þ
ds ð1�hÞ� RR0
�2HH0
dhds
� �� R0
d
R0
�2_#
H0
AE0
�1mþ2 R
R0
HH0
d R=R0ð Þds þ R
R0
�2d H=H0ð Þ
ds
� �ð1�hPÞ
� R0
d
R0
�2 H0d� _# A=E0ð Þ
1ms
H0
� �� R
R0
�2HH0
� � ð59Þ
R and R0 are the current and the initial radii of the speci-men, respectively:
dðH=H 0Þds
¼ H3H 0
2hP þ ð1� 3hPÞhhð1� hÞð1� hPÞ
� �dhds
ð54Þ
dðR=R0Þds
¼ R3R0
h� hP
hð1� hÞð1� hPÞ
� �dhds
ð55Þ
Based on the mass conservation of the whole system,including both a PTM and a porous cylindrical specimen,the following is correct:
dMdt¼ qSpe
dV Spe
dtþ qPTM
dV PTM
dt¼ 0 ð56Þ
where qSpe is the density of the fully dense specimen, VSpe isthe skeleton volume (substance volume excluding the vol-ume of pores) of the specimen, qPTM is the density of the
fully dense PTM and VPTM is the skeleton volume of thePTM.
The following geometric relationships are valid:
V Spe ¼ pR2Hð1� hÞ
V PTM ¼ pðR02
d H d � R2HÞð1� hPÞð57Þ
where R0d is the radius of the rigid die used to contain the
whole composite cell (composite cell includes both PTMand densified specimen) and Hd is the current height ofthe whole composite cell, which is equal to H 0
d � _#tðH 0d is
the initial height of the composite cell and _# is the speedof punch).
Substituting Eq. (57) into (56) and simplifying theresults, we have:
Applying the same normalization procedures as usedbefore in Eq. (53) for the time t, the specimen height H,the specimen radius R and the PTM porosity hP, Eq. (58)can be rewritten as:
Eqs. (53)–(55) and (59) represent a set of four first-orderdifferential equations with respect to the four unknownfunctions of the specific time s: specimen’s porosity h, spec-imen’s height H, specimen’s radius R and the PTM poros-ity hP. Fig. 11 shows the solution of the above-mentionedset of equations (using the fourth-order Runge–Kutta algo-rithm). This solution has been obtained for the PTM withalumina concentration of 50%. Similar solutions wereobtained for various compositions of the PTM. All of themindicate almost no porosity and distortion level change forthe time range exceeding 12 s. The shrinkage and distortionrates for this time point are comparatively assessed inFig. 12. This figure shows the rate of the distortion
dððH=H0Þ=ðR=R0ÞÞdt
�and the densification (dh/dt) rate for differ-
ent PTM compositions. From this figure one can see that atthe composition of 75% Al2O3 and 25% C the curve of den-sification rate crosses the curve of the distortion rate. Thispoint corresponds to the highest densification rate while
Fig. 11. Kinetics of shrinkage and distortion.
Fig. 12. Comparative analysis of densification and distortion rates.
the distortion rate is at its lowest possible level. Therefore,for the chosen values of the specimen’s constitutive param-eters (constants A and m), the PTM composition contain-ing 75% of alumina and 25% of graphite should be optimal.
6. QIP experiments
6.1. Cold QIP experiments
For the analysis of the aspect ratio evolution and theverification of the modeling results obtained in previous
sections, quasi-isostatic pressing of Ni and Ti powder sam-ples was carried out in a rigid cylindrical die at room tem-perature. A graphite powder transmitting medium wasused in the experiments. The velocity of pressing wasapproximately 0.001 m/s. The die dimensions were 0.06 mdiameter and 0.1 m height.
The die was designed to withstand 200 MPa of inter-nal pressures. A smaller, thinner can (approximately1 mm) was encased in a structural thick wall cylinder.The piston maintained pressure with a hardened steelseal plate.
Table 1Comparison of the theoretical and experimental results on QIPing of Ni and Ti porous samples
Nickel and titanium specimens were produced by thepreliminary cold pressing of loose powders. The cold press-ing was performed to produce three nickel and three tita-nium samples, the heights and diameters of which aregiven in Table 1.
Each cold pressed sample was heated in a furnace tolower its yield stress and to improve the integrity of thesamples by providing some degree of sintering; the nickelsamples were heated to 615 �C and the titanium samplesto 350 �C. Without pre-sintering of the samples, they couldpotentially have been crushed during the QIP process.Upon removal from the furnace each sample was immedi-ately placed in a thin-walled can (6 in diameter) and sur-rounded by the PTM. The can was introduced inside athick-walled pressure cylinder and the piston with the plun-ger assembly was placed on the top of the PTM. The sam-ples were then pressed to a variety of forces, ranging from2.5 to 10 kN.
The diameter, the height and the porosity of all the sam-ples were measured before and after QIP. The porositymeasurements were performed by the method of opticalmetallography. The results of experiments are representedin Table 1.
6.2. Hot QIP experiments
The hot QIP was conducted at the post-combustion(SHS) densification stage employed for the synthesis ofTiC–TiNi cermet composites (for more details, see ourprevious publications [13,15]). The first step was bakingof the elemental titanium, nickel and graphite powders(with 30 vol.% of TiNi – 0.76 mol fraction of TiC).The elemental powders were baked under a temperatureof 110 �C and under a less than 25 mm Hg pressure in avacuumed oven for at least 24 h. The baked powderswere loaded under argon protection into polyethylene
Table 2Comparison of the theoretical and experimental results on QIPing of TiC–TiN
jars and dry mixed with corundum grinding balls(96.3% Al2O3 and 2.75% SiO2) in a grinding ball/powdermass ratio of 4:1. Thereafter, the polyethylene jars werefixed onto a rotary blender machine and dry mixed forat least 24 h. After mixing, the mixture was put backinto the vacuum oven and again baked for at leastanother 24 h under 110 �C and less than 25 mm Hgpressure to remove the moisture absorbed in the mixture.After baking, mixtures of the powders were uniaxiallypressed into cylindrical specimens with dimension ofapproximately either diameter 1.25 in. and height 0.6 in.(weight approximately 25 g) or diameter 1.25 in. andheight 1.25 in. (weight approximately 50 g) under acompressing load of 2 tons in a rigid die. Right afterthe green sample was ready it was loaded into the centerof a pool of PTM (of the optimal compositiondetermined in Section 5 – 75% Al2O3 and 25% C) in a6-inch-diameter die. Thereafter, it was ignited andsequentially consolidated using PHI and ENERPACpresses.
To aid the initiation of the combustion and thereafter aplanar combustion wave from the top of the specimen,samples were placed beneath a layer of loose stoichiometrictitanium–graphite mixture. An electrochemical system con-sisting of a resistant Ni–Cr heating wire was wrappedaround a wooden matchstick and buried in the loose pow-der. A remote variable transformer was connected to theresistant heating wire and the combustion was initiatedby passing an electrical current through the resistant heat-ing wire and consecutive ignition of the matchstick. Fol-lowing the combustion synthesis, after a time delay of15 s, a load of 10 kN have been applied for 20 s. Fig. 14compares the microstructure of the as-reacted material(Fig. 14a) to the same material loaded to 10 kN after adelay time of 15 s (Fig. 14b). The final porosity is in therange of 1–3%.
i powder cermet composites
inalorosity
Change of aspect ratio(experimental)
Change of aspect ratio(theoretical)
Relativeerror
.032 0.385 0.365 0.054
.033 0.401 0.363 0.095
Fig. 13. Relative change of the PTM porosityhp0�hp
hp0vs. sample porosity for different sample skeleton strain rate sensitivities (m) and process rates ( _e).
Based on the experimental data on initial parameters(aspect ratio and porosity) for QIP of Ni and Ti powdersamples, the calculations were performed in conformityto Eq. (35). The final aspect ratio was determined withthe knowledge of the sample final porosity. PTM (graphitepowder) porosity was determined experimentally (usingmass conservation-based calculations): hp = 0.4. This valuewas used in the analysis.
The experimental and theoretical data are given in Table1. One can see that the experimental and calculated resultsagree well: for all the cases, the relative error in the valuesof the final aspect ratio is less than 6.5%. Thus, Eq. (35) canbe recommended for use as a practical tool for the quanti-tative prediction of the shape change during (cold) QIP.
It should be noted that Eq. (35) is not valid for a fastpressing accompanied by high-strain-rate modes (fasterthan 1 s�1; see Fig. 13). In this case, Eq. (35) in its differen-tial form should be solved with regard to Eq. (A6) (seeAppendix).
Fig. 14. Microstructures of (a) the as-reacted and (b) the densified(subjected to QIP) TiC–TiNi cermet.
The comparison of the experimental data on the shapeevolution during hot QIPing of combustion-synthesizedTiC–TiNi cermet composites with model predictions pro-vided by the solution of Eqs. ((53)–(55) and (59)) is givenin Table 2 for both size sets of specimens mentioned in Sec-tion 6. The initial PTM porosity assumed was hP = 50%.One can see that the relative error in the values of the finalaspect ratio is smaller than 10% (see Table 2).
8. Conclusions
The principal results obtained in the investigation can besummarized as follows:
1. A mathematical model of the quasi-isostatic pressing(QIP) is developed.
2. The model predicts an essential shape change under QIPfor large porosities of the pressure-transmitting medium(PTM).
3. It is shown that, for most cases, the QIP deformationmode has an intermediate position between the defor-mation modes of pressing in rigid dies and free up-setting.
4. The ratio between the sample and the PTM porositiesinfluences the evolution of the integral aspect ratio.
5. The assumption of the elastic character of the PTM con-stitutive behavior is analyzed. The dependence of thePTM elastic modulus on the PTM composition andthe PTM relative density is determined.
6. An algorithm for the determination of the optimal PTMcomposition is developed. The algorithm is aimed atfinding the PTM composition that enables the highestdensification of the porous specimen with the minimumpossible distortion. It is shown that, for TiC–TiNi cer-met composites, the optimal PTM composition corre-sponds to 75 mass% alumina and 25% graphite.
7. A comparison of the experimental and calculationresults on cold QIPing of Ni and Ti powders and hotQIPing of TiC–TiNi cermet composites demonstratesgood quantitative correspondence.
Acknowledgement
This work was partially supported by the NSF Divisionof Manufacturing and Industrial Innovation, Grant DMI-0354857.
Appendix. Evolution of PTM porosity during QIP
In the following, E and m are the PTM Young’s modulusand Poisson’s ratio, respectively; ezz and err are the PTMaxial and radial deformations, respectively.
The following expressions for E and m are valid (by anal-ogy ‘‘linear viscosity = elasticity’’: g0 $ E0
2ð1þm0Þfor fully
dense material, g0u$ E2ð1þmÞ for effective porous material;
ðA6ÞThe latter expression enables the analysis of the relativechange of the PTM porosity
hp0�hp
hp0corresponding to definite
values of the sample porosity, the strain rate sensitivity m
and the volume change rate _e. Note that, for mostmaterials, A
E0� 10�3 . . . 10�2 sm for the range of tempera-
tures corresponding to hot deformation (A depends onthe temperature) [27,31]
The results of the calculations in accordance with (A6)are represented in Fig. 13. Here, the initial PTM porosityhp0 is assumed to be 0.65, and A
E0is taken 10�2 sm.
The calculation data indicate that, for higher strainrates, influence of the sample material strain rate sensitivitym on the change of the PTM porosity increases. For higherstrain rates, the larger change of the PTM porosity occurswhen the sample properties are close to the linear-viscousones (m! 1). For smaller strain rates (slower pressing),
the higher change of the PTM porosity corresponds tothe ideal-plastic properties of the sample (m = 0).
The smaller the sample porosity, the higher the relativechange of the PTM porosity.
For usual uniaxial pressing, strain rates vary in therange 10�3 . . . 10�2 s�1. As it follows from Fig. 13, for sucha level of _e, the relative change of the PTM porosity isalways less than �6%. Therefore, for slow QIP processing,one can assume that the PTM porosity is unchangeable.
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