Acta metall, mater. Vol. 42, No. 7, pp. 2247 2260, 1994
Pergamon
0956-7151(94)E0031-B
Copyright © 1994 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0956-7151/94 $7.00 + 0.00
DENSIFICATION OF POROUS MATERIALS BY
POWER-LAW CREEP
YONG-MEI LIU, H. N. G. WADLEY and J. M. DUVA
School of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22903, U.S.A.
(Received 26July 1993; in revisedform 10 January 1994)
Abstract--The densification of a porous intermetallic alloy (Ti-14wt%AI-21wt%Nb)during the final stage
of densification has been investigated under various states of stress and compared with the predictions
of current models for densification by power-law creep. The experiments generally confirm the model
predictions that the densification rate is a sensitive function of the stress state. Experimentally,
unconstrained uniaxial compression resulted in the largest densification rates, while constrained uniaxial
compression resulted in the lowest. Hydrostatic loading resulted in a densificationrate similar (but slightly
higher) than that of constrained compression. This ordering of the densification rates agreed well with
the model predictions. However, the magnitudes of the measured densification rates are found not to be
accurately predicted. A number of factors, including pore shape, pore spatial distribution and matrix
microstructure have been observed to affect the densification rate, and the signifcance of each of these
factors to predictive modelling of creep consolidation processes is assessed.
1. INTRODUCTION
parameters are defined by the uniaxial power-law
creep formula
During the final stage of metal (or alloy) powder
consolidation by processes such as hot isostatic
pressing (HIP) and vacuum hot pressing (VHP), void
collapse by power-law creep is often the dominant
densification mechanism [1]. A number of models
have been proposed to describe this process, and they
have been used to predict the dependence of the
densification rate on the relative density (solid
volume/total volume), the consolidation pressure,
and the material's (temperature dependent) creep
parameters [1-5]. All assume that once the final
stage of densification has been reached, the voids
can be treated as isolated, spherical, and homogeneously distributed in a matrix deforming by
power-law creep. The pores in a powder compact
undergoing HIP consolidation have been predicted
to become isolated at a relative density of 0.92 and
the models are generally assumed valid for densities
above this [6].
Under general loading, creep deformation of
fully dense materials is caused only by the deviatoric
components of the stress state [7]. The strain rate
dependence upon stress for fully dense solids undergoing power-law creep can be written
•
3. [~r~\"-lS 0
~i/= ~ E o / - - )
z
\ O-0
--
170
(1)
3
where ac = x~gSoS!j
is the effective stress, Sq is the
stress deviator (S~= tr~--3trkk6~)
J
and the other
i = i0 cr0
where n is the stress exponent, tr0 and ~0 are
a reference stress and a reference strain rate,
respectively. The first creep model for final stage
densification of a porous material was developed by
Wilkinson and Ashby [1]. They analyzed the creep
collapse of a representative volume element consisting of a thick walled spherical shell subject to
externally applied hydrostatic loading. Their model
has been widely used for predicting the final stage of
HIP densification [8, 9].
During HIP consolidation, the stress state within a
compact is often not perfectly hydrostatic [8-11],
while for other consolidation processes, such as
VHP, the stress state evolves with density from a state
of near to uniaxial compression to a state of (laterally) constrained compression. Thus, creep consolidation occurs under a range of stress states that
contain significant deviatoric stresses. Piehler and
Watkins [12] have reported significant increases in
densification rate when a deviatoric stress component
was introduced into the consolidation process. Thus,
it became necessary to broaden the Wilkinson and
Ashby result to general loading conditions. This
has been accomplished by deducing an effective
(strain rate) potential for a porous body from
which the strain rate components (and thus densification rates) can be obtained when the body is
2247
2248
YONG-MEI LIU et al.: DENSIFICATION OF POROUS MATERIALS
subjected to a prescribed combination of compressive loads.
Cocks [2] developed a bound for the creep potential
by applying a convexity constraint to the potential to
find the strain rate field of a compressible power-law
creeping material. Ponte [3] used a variational
method and also obtained bounds to the potential of
a porous body. Both approaches resulted in lower
bound solutions to the true potential. Closer
estimates (for more geometrically restricted
problems) have been obtained using numerical
methods. Duva and Crow [5] have proposed an
approximate potential based upon the work of [1-3].
Recently, Sofronis and McMeeking [4] have used a
finite element analysis of a cell to determine the strain
rate components under pure hydrostatic and perfectly
deviatoric loadings. They fitted a simple elliptical
interpolation function between the two limits to allow
computation of strain rates under intermediate
conditions of loading. It is important to note that
both the Duva and Crow [5] and Sofronis and
McMeeking [4] potentials were deliberately
developed to predict a densification rate in the hydrostatic limit that is identical to the result of Wilkinson
and Ashby [1]. The predictions of these models for a
variety of stress states are analyzed in detail later (see
Section 4.1).
The work reported here is a first systematic
experimental study of the validity of these creep
potentials for modelling final stage creep densification. Using HIP/sinter methods, samples of a porous
Ti3A1 + Nb intermetallic alloy have been produced.
These were subsequently consolidated under different
loading conditions, but always such that power-law
creep was the dominant densification mechanism.
The densification rates have been measured and
compared to the predictions of the models above. The
general trends in the relationship between densification rate and state of stress are found to be quite well
represented by the models, but several important
discrepancies have also been observed. The most
significant of these is a large model underestimate of
the measured densification rates. This is believed to
be due to the nonspherical shape of the pores in
actual samples and their non-uniform distribution.
Serendipitously, the experiments showed a second
significant problem with the practical application of
the models to densification, namely, a strong
dependence of the creep parameters (n, ¢r0, E0) upon
both the microstructure of the creeping material
(which evolves during consolidation) and upon the
magnitude of the applied stress and temperature.
1oo
v
80
60
.>_
,...,
40
E 20
o
1 O0
200
I
400
300
500
Powder Size (pm)
Fig. 1. Particle size distribution of as-received powders.
is shown in terms of the weight percent (determined
by a sieve analysis at Nuclear Metals, Inc.) in Fig. 1.
The chemical composition of the powders is listed in
Table 1. Optical micrographs of the as-received powders are shown in Fig. 2. The majority of the particles
were spherical and fully dense. A few contained a
single (enclosed) pore. X-ray diffraction indicated
that the only phase present in the as-received powders
was the b.c.c, fl phase, a metastable phase formed as
a result of rapid quenching from above the fl transus
temperature [13]. This metastable phase has been
shown to transform to either a T%A! (h.c.p. DOI9 ~2
phase) +orthorhombic structure when heat treated
between 700 and 900°C or a Ti3AI (~t2) + fl structure
when heat treated above 900°C [13-16].
2.2. Sample preparation
Porous samples for studying power-law creep
densification were prepared by a two step process.
(a)
As polished
I
I
100 pm
2. E X P E R I M E N T A L
2.1. Materials
Intermetallic Ti-14A1-21Nb (wt%) powders were
used in this study. They were produced using a
Rotating Electrode Process (REP) by Nuclear
Metals, Inc. The - 3 5 mesh particle size distribution
Etched
i
i
50 pm
Fig. 2. Cross-sections of as-received Ti-I4A1-21Nb powder
particles.
YONG-MEI
L I U et al.:
DENSIFICATION
2249
OF POROUS MATERIALS
Table 1, Composition of Ti-14AI-21Nb powders (wt%)
AI
14.57
Nb
19.9
O
0.093
N
0.011
H
0,0021
First, the powders were packed in stainless steel tubes
(16-19 mm outside diameter with a wall thickness of
0.9 mm). These tubes were evacuated, sealed, and hot
isostatically pressed. To obtain samples with a range
of densities for later testing, the consolidation
pressure, temperature and hold time were varied. The
consolidation conditions and the resulting densities
for this first step are summarized in Table 2.
Examination of the partially densified HIPed
samples revealed cusp-shaped pores (see Section 3.1),
similar to those previously reported for other
materials [17, 18]. Since all the creep densification
models assume the pores to be spherical, most of the
samples were subjected to an additional high
temperature sintering treatment to spherodize the
pores. Samples for sintering were wrapped in
0.025 mm thick tantalum foil (to reduce oxidation),
sealed in quartz tubes under a partial high purity
argon atmosphere, and sintered at 1250°C (in the/3
phase field) for 48 h. They were then furnace cooled
to 100ff~C, held at that temperature for 2 h to stabilize
an ct2 +/3 microstructure, and finally air cooled to
room temperature. One sample was retained in the
as-HIPed condition in order to assess the significance
of cusp-shaped pores upon densification since this
shape may be more representative of practical consolidation. The fully dense samples shown in Table 2
(Nos 5-8) were used to measure the alloys creep
parameters in both the as-HIPed and HIPed plus
sintered conditions.
2.3. Creep tests
Compression creep tests were conducted in a
modified ATS model 2710 creep tester under vacuum
(0.01 Pa). The specimen length change was measured
with an extensometer and a linear variable capacitance transducer (LVCT). Specimen displacement
was measured with a precision of __+2 #m. A thermocouple measured the temperature near the test specimen. During all tests, the temperature was first
increased to the set point at a rate of 300°C/h and
held for 30 min so that the temperature inside the
chamber could stabilize. The load was then applied
quickly to the set load point at a rate of 0.7 MPa/s
Table 2. HIP conditions for Ti 14A1-21Nb samples
HIP run
No.
Pressure
(MPa)
Temperature
(°C)
Time
(h)
1
2
100
100 and
70
70
100
100
100
100
170
900
1000
1000
1000
1000
1000
1000
1000
1050
0.5
0.0
0.5
1.0
0.3
1.0
1.5
2.0
2,0
Density
Achieved
0.890
0.950
0,960
0,980
> 0.995
> 0.995
> 0.995
> 0,995
Cu
0.01
C
0.028
W
0,003
Fe
0,037
Ti
bal
and the length change monitored over time. During
the creep test, temperature was maintained constant
within +2°C of the set temperature. Load was
maintained constant within + 1% of the set point.
A special creep fixture (Fig. 3) was used for the hot
constrained/unconstrained compression tests. It
consisted of a single action die made from a high
temperature molybdenum alloy, TZM. The porous
specimens for constrained compression testing consisted of 14mm diameter cylinders with an initial
height (H0) to diameter (2R) ratio ranging from
about 1.5:1 to 0.4:1. Their areas of contact with the
fixture were lubricated with boron nitride to reduce
the effect of friction upon densification. As will be
discussed later, this was most successful for low
aspect ratio samples. The constrained compression
tests were conducted at a pressure of either 120 or
150 MPa and a temperature of either 980 or 1010°C.
Tests were also performed on porous samples without
constraint (i.e. uniaxial compression) at a pressure of
150MPa and at a temperature of 980°C. The
diameters of these samples were approximately
10 mm with an initial height to diameter ratio of 1.
Since we were unable to continuously measure the
samples radial expansion, we were unable to continuously monitor the densification during this compression; only the average densification rate could be
obtained by measuring the density before and after
creep densification.
Samples were also isostatically densified in a HIP
at a pressure of 150 MPa and at temperatures of
either 980 or 1010°C. The temperature was again first
increased to its set point before the pressure was
raised to the set point.
2.4. Density determination
The densities of samples before and after consolidation were determined using a hydrostatic weighing
method based on Archimedes' Principle. The sample
weight M was first measured in air, using a balance
with 10/~g precision. The sample was then immersed
in a liquid with a known density (high purity
Table 3. Test conditions for uniaxial constrained compression
experiments
Sample
Stress
(MPa)
T
(°C)
Time
(h)
Ho/2R
DO
D~nal
0.939
0.944
0.984
0.975
0.986
0.970
0.978
0.965
0.982
1A
2A
3A
4A
5A
6A
7A
8A
120
150
150
150
120
150
150
146
(a) HIPed + sintered condition
980
11
0.53
0.926
980
7.5
0.50
0.901
980
9
0.46
0.965
980
10.5
1.41
0.969
1010
10
0.58
0.975
1010
7
0.43
0,920
1010
8.7
1.43
0.973
1010
8
1.33
0.924
9A
150
(b) As-HIPed condition
980
6.0
0.40
0.920
2250
YONG-MEI LIU et al.: DENSIFICATION OF POROUS MATERIALS
[M alloy)
de
II
Ioy)
Fig. 3. Die for uniaxial compression creep tests.
methanol with density pL = 0.7914Mg/m3), and its
weight M ' again measured. From the loss of sample
weight in the liquid (M - M ' ) and the sample volume
V [ = ( M - M ' ) / p L ], the density of the sample, p,
( = M . P L / M -- M ') can be calculated.
The estimated error was less than +0.5% and
Powder
was due to a combination of the weighing error
and the liquid density fluctuations with temperature.
The relative density was obtained by dividing the
measure density of the porous sample by the
measured density of fully densified material
(4.7 Mg/m3).
Interconnected
Interconnected
(a)
Sample I
Sample 1
Average Density = 0.89
I
I
100 prn
Average Density = 0.89
I
1~00 prn
(b)
Sample 3
Pore ~
S
Isolated
Pore
Average Density = 0.96
i
J
100 pm
Fig. 4. Micrographs of partially HIPed samples with
different densities: (a) HIPed sample 1 (D =0.89); (b)
HIPed sample 3 (D = 0.96).
Average Density = 0.96
i i
100 pm
Fig. 5. Micrographs of the two samples shown in Fig. 4
after sintering: (a) sample ! (Do = 0.89); (b) sample 3
(D o = 0.96).
2251
YONG-MEI LIU et al.: DENSIFICATION OF POROUS MATERIALS
3. RESULTS
constrained compression in a lubricated die using a
creep machine at constant load and fixed tempera3.1. Characteristics of the porous material
ture. The length change of the sample was measured
3.1. I. Pore geometry. The as-HIPed samples (see continuously during the test. The final length (Hr)
Table 2) were first examined using optical and final density (Dr), measured after unloading,
microscopy. Fig. 4 shows micrographs of the were subsequently used to deduce the density during
as-HIPed samples (numbers 1 and 3 in Table 2). the test using D = Dr × nf/H. Test conditions toFigure 4(a) shows that some pores are interconnected gether with densities before (D 0) and after ( D r ) testing
and therefore indicates that samples with densities of are given in Table 3. Figure 8 shows the variation of
less than ~0.9 had not reached the final stage of the measured relative density with time for the HIPed
densification (characterized by isolated pores). plus sintered sample listed in Table 3. It is apparent
Sample 3 with a higher density, Fig. 4(b), contained that the densification rate decreases with increase in
isolated pores. Figure 4 shows that in the as-HIPed density, and increases with consolidation pressure
condition the pores are cusp-shaped and not spherical and temperature.
as assumed by the models. The pores were randomly
Two additional aspects of the results are noteoriented and showed a marked variability in size, worthy. The first point to note is an experimental
especially at lower relative densities.
problem always associated with these types of
To obtain more spherical pores (for comparison experiments. It can be seen in Table 3, that the sample
with the theoretical predictions) the HIPed samples aspect ratio (Ho/2R) affected the observed densificawere sintered at 1250°C where diffusional processes tion. For instance, samples 3A and 4A had almost
(that round the pores) occur more rapidly. Micro- identical initial densities and were consolidated under
graphs of HIPed plus sintered samples are shown in identical conditions, but exhibited significantly
Fig. 5 Although the pores are more rounded, they different densification rates [Fig. 8 (a)]. A similar
are not completely spherical, especially in the lower observation was found for samples 6A and 8A
[Fig. 8 (b)]. In general, the smaller aspect ratio
density samples.
3.1.2. Microstructure. A microstructural evolution samples exhibited a higher densification rate. It was
was found to accompany both the HIP and sintering suspected that the reason for this was die-sample
treatments. During HIPing at 1000°C, the metastable friction, which would cause a decrease in the effective
phase fl (b.c.c.) of the as-received powder trans- pressure in the sample. Metallography was used to
formed to an equiaxed ct2 + fl structure, Fig. 6. The investigate the possibility.
microstructure consists of equiaxed ct2 grains ( ~ 4 # m
Figure 9 shows a cross section (in the axial plane)
in diameter) with the fl phase retained both at the ~2 of a high aspect ratio sample (Ho/D = 1.33). It can be
grain boundaries and within the ct2 grains. Sintering seen that inhomogeneous densification occurred in
resulted in a coarser microstructure consisting of this specimen. The end of the sample nearest the
aligned ~2 laths (width ~ 6/~m, and length ~ 75 pm)
moving punch (the top surface) underwent
with a thin fl phase layer retained at the ~2 lath significantly greater densification. In contrast, a lower
aspect ratio sample, Fig. 10 (Ho/2R = 0.43), exhibited
boundaries (Fig. 7).
homogeneous deformation. These microscopic
3.2. Densification
observations lend support to the view that in the high
3.2.1. Uniax&l constrained compression. HIPed aspect ratio tests, die wall friction significantly
and sintered samples were densified under uniaxial lowered the effective pressure and retarded the
I
J
10pm
Fig. 6. SEM micrographs of as-HIPed (100 MPa, 1000°C,
1 h) sample 5.
t
i
I O0 pm
Fig. 7. SEM micrographs of a HIPed plus sintered
sample.
2252
YONG-MEI LIU et al.: DENSIFICATION OF POROUS MATERIALS
1.00
I
i
i
I
I
(a) T= 9aO°C
I
ExperimentalData
(> 3A:Ho/2R=0.46
0.98
L•
150MPa
• 4A:H0/2R-1.41
150MPa
• 9A:H0/2R=0.40
150MPa
0.96
2A: H0 / 2R - 0.50 150 MPa
E3
O 1A:Ho/2R=0.53
120MPa
' ~ 0.94
rr
0.92
0.90
I
I
I
I
I
2
4
6
8
10
i
i
Time
1.00
(h)
i
I
'
(b) T= 1010°C
I
ExperimentalData
0.98
t(~
12
0.96
O 5A:Ho/2R=0.58
120MPa
O 7A:H0/2R=l.43
150MPa
A 6A:H0/2R.0.43
150MPa
• 8A:Ho/2R=1.33
146MPa
"~ 0.94
0.92
1"
0.90
0
error
I
I
I
I
I
2
4
6
8
10
Time
12
(h)
Fig. 8. D e n s i t y increase w i t h time u n d e r u n i a x i a l c o n s t r a i n e d c o m p r e s s i o n .
densification [18]. Thus, only the short aspect ratio
results are considered to be valid for latter comparisons with the model predictions. The measured
densification rate of higher aspect ratio samples
significantly underestimates the true rate of densification. They are presented only because, as discussed
later, they still exceed the predictions of certain of the
models.
The second point to note concerns the sample in
the as-HIPed condition (sample 9A). While most
porous samples were sintered before consolidation
tests, an as-HIPed sample (Ho/2R = 0.4) with cuspshaped pores was also densified under constrained
Table
Sample
1B
2B
4.
Uniaxial
compression at an axial stress of 150 MPa and a
temperature of 980°C. Its densification behavior is
shown in Fig. 8 (a) (sample 9A). By comparing with
the densification behavior of a HIPed plus sintered
sample tested under identical conditions (sample 2A),
it is clear that the combination of the cusp-shaped
pores and the finer scale microstructure of the asHIPed conditions resulted in a significantly greater
densification rate.
3.2.2. Unconstrained uniaxial compression. Two
porous samples in the HIPed plus sintered condition
were densified in uniaxial compression without
constraint from the die-wall. A stress of 150 MPa and
(unconstrained) compression results
HIPed + sintered condition
for
samples
in
Stress
(MPa)
T
(°C)
Time
(h)
DO
D~nat
Daveras
~
b x 103
(h- J )
150
150
150
150
980
980
980
980
0.30
0.98
1.27
0.67
0.906
0.918
0.950
0.972
0.918
0.943
0.962
0.976
0.912
0.931
0.956
0.974
41.0
26.0
9.7
6.1
YONG-MEI LIU et al.: DENSIFICATION OF POROUS MATERIALS
2253
V
H
I _ _ J
100 pm
Fig. 9. Cross-section along cylindrical axis of sample 8A after uniaxial constrained compression
(Ho/2R - 1.33).
a temperature of 980°C were applied until a strain of
about 10% had developed. Results obtained are given
in Table 4.
Metallography was performed to investigate the
effect of the consolidation upon the pore shape.
Figure 11 shows a cross-section along the cylindrical
axis of sample 1B after uniaxial unconstrained
compression. It clearly shows that many of the pores
changed shape and were significantly flattened by the
compressive stress. It is also noteworthy that pores
whose major axis was aligned with the direction of
loading have deformed less. This difference in pore
shape evolution is not treated by current models and,
as will be shown later, it is believed to be a significant
contribution to the discrepancies between model predictions and observations.
3.2.3. Hydrostatic pressing. Hot Isostatic Pressing
of HIPed plus sintered specimens were also
performed. Only an average density could be
obtained because of the difficulty of measuring
sample dimensions inside a HIP. The results are
shown in Table 5. For these tests, it took about 1.5 h
to increase the gas pressure to the present value. Since
the initial density was high, the densification accompanying the pressure transient period was small and
has been ignored.
3.3. Microstructure dependence of creep parameters
To compare the measured densification rate of the
porous samples tested with model predictions, the
parameters describing power-law creep (n, a0, ~0)
were carefully measured by creep compression testing
of fully dense material. The creep behavior was
investigated for the two microstructure states
involved: (1) the HIPed plus sintered condition and
(2) the as-HIPed condition.
Table 5. Hot isostatic pressing results for porous samples in the HIPed + sintered
condition
Sample
lC
2C
3C
4C
Stress
(MPa)
T
(°C)
Time
(h)
DO
Dfinal
Davera~
b × 103
(h ~)
150
150
150
150
980
980
980
1010
2.5
2.0
5.0
1.5
0.963
0.971
0.976
0.977
0.974
0.977
0.989
0.984
0.970
0.974
0.982
0.980
4.40
2.98
2.50
4.80
2254
YONG-MEI LIU et al.: DENSIFICATION OF POROUS MATERIALS
i
1
100 urn
Fig. 10. Cross-section along cylindrical axis of sample 6A after uniaxial constrained compression
(1to/2R = 0.43).
F o r all experiments, a steady state creep regime
was well established before the strain had exceeded
2%. The steady state creep rate (i) was determined
by calculating the slope of the linear portion o f
the strain-time creep-curves. Results are given in
Tables 6 and 7 for tests on the two material
conditions for different temperatures and stresses.
It is clear that the as-HIPed condition exhibited
2-10 times the strain rate of the HIP-ed + sintered
condition.
F o r materials which exhibit power-law creep, the
stress exponent n can be determined from the slope
of a log ~ against log a plot. Steady state creep rate
Table 6. Steady state creep rate for the
HIPed + sintered material
Sample T(°C) o (MPa) i (h -I)
1
980
50
0.0053
980
70
0.0123
2
980
100
0.0362
3
980
120
0.0816
4
980
150
0.2740
5
1010
50
0.0168
1010
70
0.0454
6
1010
100
0.1440
7
1010
120
0.3790
8
1010
150
1.2860
vs stress results are shown in Fig. 12. The creep data
are well represented by two single straight lines over
the test stress range. The stress exponent n, and the
material parameter go/a ~ were calculated for different
stress and temperature ranges and are summarized in
Table 8. The results indicate a strong effect of heat
treatment, stress and test temperature upon the creep
parameters.
4. DISCUSSION
The experiments described above show that the
stress state, pore shape and the microstructure each
affect the densification rate at a fixed pressure and
temperature. Each of these factors will be examined
in detail and their relationship to model predictions
evaluated.
Table 7. Steady state creep rate for as-HIPed
material
Sample T (°C) a (MPa) i (h-t )
l
980
50
0.069
980
70
0.109
2
980
1O0
O.158
3
980
130
0,352
4
980
150
0.574
YONG-MEI
L I U et al.:
DENSIFICATION
OF POROUS
Loading Direction
2
10 o
2255
MATERIALS
I
(a) H I # ~ + iInten~l
11- 14AI- 21Nb
4 - - ~ 9
~
2
nr"
10-~
.c:
4
IF
100 pm
4
2
10-3
100
55554355
i
(b)
i
i.8
C r o s s - s e c t i o n along cylindrical axis of sample IB
after uniaxJa] unconstrained compression.
~
i
i
i
I
]
i
7
i
a
i ]
9 10 a
As - HIPed
Ti -
Loading Direction
Fig. 11.
~
"~ 10-a
"R
14~
- 21Nb
2
n" 10_1
._=
CO
oC
4
4.1. Stress state comparisons
The constitutive relation for a porous body is
embodied in its strain rate potential
EOdO ( S
@(o.) = ~
L
6
10-1
Applied S t r e s s
~ n+l
\ ao /
(4)
where
(5)
s 2 = ao- ~ + ba 2m
is the mean stress, and o.~ = ~ 3
is the
effective stress• Here, S~ is the stress deviator
(S~= O'~--½Okk&O'), and the coefficients a and b are
functions of stress exponent n and current relative
density• Expressions proposed for a and b for each of
the densification models are listed in Table 9.
The strain rate components for a body subjected
to a multiaxial stress state can be found by differentiating
o. m = O ' k k / 3
_~f~:
gO S n _ l
(3a
).
~_S/j.+.~O.mrU
(6)
The densification rate is then obtained from
/) = - - D - ~ . , where D is the relative density and
~, = (~. + ~22+ ~33) is the dilatation rate.
Under
constrained
uniaxial
compression,
~H = ~22= 0 and only ~33~: 0. Thus, the densification
rate is given by
= -D
/):--D,33
• 1-
e°
o.g /
a(c-l)~+b
(1
9
Ia(l-c)+b(1
2~-~
+2c)]
~
.
io(
+2c)]o.g3
'11 ={722=O.
~
n-I
b\~/a
a +~)
I~0
n+l
b )~-
(a + ~
O-~3"
o-g\
(9)
Thus, the densification rate for a uniaxial stress state
is
D:-~D.a+~) t3)o-~3.
•
(
b ~/b
~ ,
(10)
Under pure hydrostatic stress, the three principal
strain rates are equal. The dilatation rate and the
densification rates are given by
E0
n+ 1
I?1v = ~kk = -~0 b ~-o- ",
(11)
/) = - D •~kk.
(12)
The result given by Wilkinson and Ashby [1] for a
porous creeping material under pure hydrostatic
pressure can be written (using the notation of [5]) as
-o.~
(7)
b),
t--~+~
E33~-
•
1
where c is the ratio of the lateral stress (o.l~ = o.22) to
the axial stress O-33' C = (a -- 2b)/(a -~ 4b), which c a n
be obtained by differentiating the potential • with
respect to o-H (or o-2:) to give ~,~ (or ~2~) and setting
this strain rate to zero.
Under uniaxial compression without lateral constraint, ~. = ~:: ~ ~33" The strain rate components are
given by
•
(MPa)
Fig. 12. Stress d e p e n d e n c e o f the s t e a d y s t a t e c r e e p o f
T i - 1 4 A I - 2 1 N b : (a): H I P e d + s i n t e r e d c o n d i t i o n (at 9 8 0 ° C
a n d 1010°C); (b) a s - H I P e d c o n d i t i o n (at 980°C).
2n
.a" (13)
which was derived by analyzing the collapse of a
spherical pore contained in a thick walled spherical
Table 8. Material dependent creep parameters for Ti-14Al-21Nb
Tests
HIPed+sintered
As-HIPed
0"33, (8)
[1 - - ~ _ ~ 1 / . ] ,
T(°C)
a (MPa)
n
i0/a~ (MPa "h i)
980
980
1011
1011
980
980
50-100
100-150
50-100
100-150
50-100
100-150
2.8
4.9
3.1
5.4
1.2
3.2
8.9 × 10 8
5.6 × 10 -~2
8.8 × 10 8
2.3 × l0 12
6.4 × 10 4
6.1 × 10 ~
2256
YONG-MEI LIU et al.: DENSIFICATION OF POROUS MATERIALS
Models
Table 9. Coefficients a and b for porous body creep potentials [5]
a
b
Cocks
1+ ( l - D )
9 2n I - D
4n+l 2-D
D 2./(. + i)
D 2./(.+ I)
l+~(l--O)
4~(1 - O )
Pont¢
D 2n/(n+ 1)
O 2s~/(.+ 1}
Duva and Crow
D 2"/("+')
McMeeking and Sofronis
[E l l - ( 1
[ 1 + ( - ~ - D ) ] 2"/"+ '
shell of power-law creeping material under hydrostatic pressure. By substituting for b in equation (11),
it can be shown that in the hydrostatic limit, the
predictions of Duva and Crow [5] and Sofronis and
McMeeking [4] are identical to equation (13).
By substituting expressions for a and b given in [5]
into equations (7), (10) and (12), and using the
measured creep parameters for a given temperature
and pressure (Table 8), it is possible to plot predicted
densification rate versus density for each of the three
states of stress and to compare this with experimental
results for the HIPed and sintered material tested
under each state of stress, Figs 13-15. Note that
under constrained compression, the predictions of
Sofronis and McMeeking [4] and Duva and Crow [5]
are too similar to distinguish and are represented
by the same curve in Fig. 13. Under hydrostatic
loading, the predictions of Duva and Crow [5],
Wilkinson and Ashby [1], and Sofronis and
MeMeeking [4] are identical and are again shown by
a single curve in Fig. 15.
Examination of Figs 13-15 reveals that all models
predict a slowing of the densification rate as full
density is approached and this trend agrees with
experiments. If we compare behaviors for
tr = 150 MPa and T = 980°C for the three states of
stress [Figs 13(b), 14 and 15], we see that the lower
bound Ponte potential consistently results in the
lowest predicted densification rate and only in the
case of unconstrained compression (Fig. 14) did it
more closely approximate the observed response. The
other models generate predictions that are roughly
similar and in better agreement with measurements.
For the three stress states investigated, experiments
indicated the largest densification rates accompany
unconstrained compression, followed by hydrostatic
pressing and then constrained compression. All four
models predict the same order of the densification
rate. In general, the Duva-Crow and SofronisMcMeeking predictions are in closest agreement with
the data. Note that the data and model comparisons
are only valid for material nonlinearities (i.e. n values)
in the 3-5 range. Other n values could result in better
or worse agreement.
l (3_y
•2n ,]
- O)'/"l" J
l (3_y
[ 1 - (1ZO)---~/"l" J
\2n]
The presence of the large deviatoric stress
component
associated
with
unconstrained
compression clearly results in accelerated creep rates
and shortened consolidation times for otherwise fixed
test conditions. This experimental trend is captured
by all the models and has two important practical
consequences. First, it may be more economical to
consolidate powder in equipment capable of exerting
large deviatoric stresses e.g. isothermal forging or
triaxial HIP [12]. Secondly, the complex tooling used
in HIP for some near net shapes can result in
significant deviatoric stresses within the powder that
will result in differential densification rates and
therefore shape distortions [8-11]. The D u v a ~ r o w
or Cocks potentials appear to be the best choices for
analyzing this behavior, at least for the range of
material nonlinearity explored here. The model
predictions still need to be used with care because the
simplifying assumptions necessary for their
derivation are not always achieved in reality. In
particular, it appears the shape of the pores can
significantly affect the rate of densification.
4.2. Pore shape effect
Examination of the results shown in Figs 13-15
reveals that while the trends are predicted, significant
differences exist between the magnitudes of the
modelled and measured rates of densification. A
central assumption of all the models has been the
spherical shape of the pores. However, experiments
have shown that the pores are cusp-shaped in the
as-HIPed condition and that even prolonged sintering results in more rounded, but still not completely
spherical pores. These pores are better approximated
as small aspect ratio (0.3-3) oblate or prolate ellipsoids of random alignment, Fig. 5(b).
4.2.1. Cusp-shaped pores. Experimentally, it has
been observed that the presence of cusp-shaped pores
(as opposed to ellipsoids) significantly increases
the measured densification rate [compare Fig. 16 to
Figl 13(b)]. At lower densities (where the cusp-shape
is more evident), the measured densification rate was
3 times greater than the model predictions, whereas
when ellipsoidal pores were present, only a ~ 2 0 %
YONG-MEI LIU et al.:
o = 120 MPa
0.006
7
•~
0.003
"~
0.002
t-
0.001
£3
0.000
"
o
O
O
O
Z= 4A: H 0 / 2 R = 1.41
O
McMeeldng - Sofronis
rr
O 2A: H0 / 2R. 0.53
r3 3A: HO/ 2R = 0.46
0.008
i~
¢O
:~
O
T = 980°C
Experimental Data
I
(b)
Tt,-
. . . . . . Cocks
- - - - - - Pome
o
0.004
¢O
I
0.010
Thc~ret~al PmdicCons
Duw - Crow I
0.005
2257
o = 150 MPa
T = 9800C
~ntlg
Data
O 1A: HO/ 2R - 0,53
=
(a)
DENSIFICATION OF POROUS MATERIALS
0,006
** 0
•
• ..
O
TheorelicalPredictions
Du~ - Crow /
McMeeking * Sofronis
O
O Oo
0.004
....
O
U) 0.002
r"
1~
I
I
0,92
0.91
I
l
0.000
0.91
~ "-I
0.93
0.93
0.95
0.94
0.97
Relative Density
o=150MPa
I o=120MPa
0.030
Ti v
S
Theoretical predic~ons
0.0020
Duva
i~
0.0015
tO
•~
0.0010
-
-
o
0.980
• .,
0.010
h
0.982
L. . . . . . . .
0.984
0.986
J...~
0.988
t-~
0.990
Cocks
------
Ponte
O
%.
o
%°%
"o..
~
o
0
0.000
0.91
0.92
0.93
Relative Density
0.94
0.95
0.96
0.97
0.98
Relative Density
o=146 MPa
0.020
I
T=I010°C
Experimental Data
=
(~')
T
O
......
0.015
u)
t-- 0,005
I" ":..:'--" "--' ~ ' ~ " . . . . ~¢,- . . . . . . .
O
• "*
¢"
O
0.0005
0 0000
O
•o
rr
Ponte
O
Eb
TheoreticalPredic~ons
Duva - Crow /
McMeeking Sofronis
Softonis
O
"f~-
I
•"¢: 0,025
0.020
......
I
/
- Crow
eeking
L
(d)
1"--1010°(7;
ER~edmentalData
O 6A: HO/ 2R = 0,43
Z~ 7A: H0/2R = 1,43
T=I010*C
Experimental Data
O 5A: H012R = 0.58
0.0025
0.99
Relative Density
O sintered sample
8A: H 0 / 2 R = 1.33
Theoretical Predictions
0.015
Duva - Crow /
McMeeking - Sofronis
rr
rO
•
0.010
• • •,
0
*••
Cocks
Ponle
•°%,
o
u~ 0.005
C
a
• .....
-- - - - -
°*,.•
"0"..
,Q
0.000
0.93
0.94
0.95
0.96
0.97
Relative Density
Fig. 13. A comparison between measured and predicted densification rate for creep consolidation
under constrained uniaxial compression: (a) a = I20MPa, T = 980°C; (b) ~ = 150 MPa, T = 980°C;
(c) a = 120 MPa, T = 1010°C; (d) a = 150MPa, T = 1010°C; (e) a = 146 MPa, T = I010°C.
increase above the model prediction was evident• No
theoretical studies have been reported for the creep
collapse of cusp-shaped pores. However, some
physical insight can be gained from the work of
Zimmerman [19] who theoretically examined the
elastic compressibility of two dimensional cavities of
various shapes including those with cusps. He showed
that the compressibility of a hole could be expressed
as the product of two terms, one dependent only
upon the elastic moduli of the surrounding matrix
and the other only upon the shape of the hole.
The shape dependent term was found to correlate
closely with a shape parameter defined as (perimeter)2/
(4~ × area)• Cusp-shaped pores, with their large
shape parameter, were shown to be more compliant
under loading than spherical cavities and presumably
also less resistant to creep collapse. It is interesting
to note that while the cusp-shaped pores may be
detrimental to efforts to model creep densification,
they are beneficial to the rapid densification of
powders, and consolidation cycles that to seek to
retain them (e.g. by avoiding excessive sintering) may
be of value.
4.2.2. Ellipsoidal pores. The majority of the
samples tested were sintered prior to testing, so that
the pores had prolate or oblate ellipsoidal shapes with
aspect ratios ranging from 1/3 to 3. The pore
orientations, with respect to the loading axis, were
random. Lee and Mear [20] have recently analyzed
the behavior of power-law solids containing dilute
concentrations of aligned ellipsoidal pores and
compared their dilatation rates to that of a power-law
material containing spherical pores. For a fixed void
volume fraction the densification rate is found to be
2258
YONG-MEI LIU et al.:
I o= 150MPa
0.06
,
i
f
0 . 0 5 - . ~..
• * ~.,~.
0.04
".
",~
0.03
O
0.02
....
"~" -,...
......
"~'.,,..
-
T=980"C
o = 150 MPa
Expedmental Data
O slnteredsamples
i
I
Q3
"~
fT'
C:
DENSIFICATION OF POROUS MATERIALS
------
The=re@c#PreDictions
McMeeking - Sofronis
Ouva- Crow
Cocks
Ponte
0.08
I
I
O
TheoreticalPredictions
Ouva - Crow /
McMeekJng - SolronFs
I~
"~
I'~
tO
O
0.91
".......
t
I
0.92
0.93
0.94
I
0.95
1~
I
I
0.96
0.97
0.98
Relative Density
0.00
0.92
Cocks
Ponta
o
~
I
......
-----
0.04
0.02
0.00
T = 980"C
Experimental Data
0 as - HIPed sample
HQ/2R = 0.4
I
0.06
0.01
D
I
"''",,.o
i
I
i
0.93
0.94
0.95
o
t - - ~ - - P- .L..,__.,.t..~
0.96
0.97
0.98
0.99
Relative Density
Fig. 14. A comparison between measured and predicted
densification rate for creep consolidation under unconstrained compression (tr = 150 MPa, T = 980°C).
Fig. 16. A comparison between measured (HIPed without
sintering) and predicted densification rate under uniaxial
constrained compression.
a complicated function of the mode of loading, the
nonlinearity of the power-law material, and the pore
shape. For uniaxial loading and n values in the 3-5
range, Lee and Mear find that prolate voids with an
aspect ratio of 3 give rise to a densitication rate that
is only 30% of that predicted for spherical voids,
while for oblate voids with an aspect ratio of 0.3, the
densification rate is about 3 times that predicted for
spherical voids.
These calculations can be used to understand the
response of a sample to uniaxial compression. The
initial densitication rate can be regarded as an
average of the densification rates of all voids (both
oblate and prolate shapes), and ought to be in rough
agreement with predictions based on the collapse rate
of spherical voids, since some of the void population
densities faster and some slower than a sphere. At
higher densities, when the oblate voids have become
small or collapsed completely, the overall response
will become dominated by the relatively slow densification rate of the prolate voids whose major axis is
aligned with the load. These trends are evident in the
densitication rate data of Fig. 14 and metallographically in Fig. 11.
Lee and Mear [ 2 0 ] have also performed
calculations for loadings that approach constrained
compression. For these higher triaxiality loads,
prolate voids densify at a rate equal to or even slightly
in excess of the rate for spherical voids. Oblate voids
densify at a rate that is 30% greater than the rate for
spherical voids (for n = 3 and an aspect ratio of 1/3).
Thus, samples containing both types of voids will
densify more quickly than the predictions of a model
based on spherical voids for all densities, and this is
what is observed in Figs 13 and 15.
4.2.3. Pore shape evolution. Budiansky et al. [21]
predicted that for a linear creeping material (n = 1)
under conditions of high load triaxiality (e.g.
hydrostatic and constrained compression), initially
spherical voids collapse to points whereas under pure
uniaxial loading they collapse to crack-like shapes.
This has been observed here. Figure 11, for example,
shows crack-like voids after creep consolidation
under uniaxial conditions. Budiansky et al. [21]
showed that the shape evolution was a complicated
function of the mode of loading and the matrix
nonlinearity. It has not been included in the models
of Duva-Crow and Sofronis-McMeeking which have
both assumed the voids to remain spherical in shape,
nor the work of Lee and Mear [20] who assume the
void's aspect ratio remains fixed. The Budiansky et al.
calculations indicate that the time to densify porous
bodies (in the linear creep regime at least) remains
significantly shorter under uniaxial ioadings than for
hydrostatic or constrained compression, but that the
magnitudes of the densitication rate are functions of
the instantaneous shape and matrix nonlinearity, as
one would expect in light of the recent Lee and Mear
calculation [20].
This suggests that in practice, shape evolution
under deviatoric loadings is a contributing factor in
determining the densification rate. Here, it appears
that ellipsoids with their major axes aligned with the
loading direction collapsed more slowly than those of
other orientations and this, we believe, has fortuitously resulted in reasonably good agreement with
spherical model predictions. This pore shape
evolution would seem to be an area for fruitful
further theoretical work on densification modelling.
a = 150 MPa
0.007
T
0.006
0
0.005
tt" 0.004
¢-.
.0
0.003
O
0.002
T=980°C
ExpenW~mta]Data
0 sintor~l samples
TheoretioMPredictions
Duva - Crow I
McMeeking - Solronis /
Ashby & Wilkim.on
......
Cocks
---Ponte
0.001
173 0 . 0 0 0
0.97
0.98
Relative Density
Fig. 15. A comparison between measured and predicted
densification rate for creep consolidation under hydrostatic
pressure (a = 150 MPa, T = 980°C).
YONG-MEI LIU et al.: DENSIFICATION OF POROUS MATERIALS
0
e
0
..........
o
y,X~ , °
0
~
ton
o
"'444/
0
4--..4
0
.
, " ' 4
4 4 -- - - "
4 4 ,""
Fig. 17. Pore volume fraction in partially HIPed (No. 3)
then sintered sample.
4.3. The influence o f pore volume fraction variation
The cell model calculations that underlie the
potentials of Duva-Crow and Sofronis-McMeeking
are based on the assumption that the behavior of the
porous body is adequately approximated by the
behavior of a cell containing a single pore sized so
that the pore volume fraction in the cell is equal to
the average pore volume fraction in the body. The
shape of the cell is usually idealized to a sphere to
make the analysis easier. Figure 17 schematically
shows the pores in a typical partially compacted body
[cf. Fig. 5(b)]. We have shown the idealized spherical
cells that each enclose a pore are assumes by the
model approach. Clearly, the cells are not all of the
same void fractions, but cover a range of pore volume
fractions. We would expect the variation in the pore
volume fraction of the cells (independent of the pore
shape effects discussed above) to also influence the
densification rate of the body. The cell model
implicitly computes the average of the pore volume
fraction and the densification rate of this average cell.
1.5 - -
i
1.4
~
1.3
~
n
f
e
d
-
6av.____o_9_ 1.2
6 (5)
1.1
1.Of ~
~
b,
Unconstrained
--
0.9
0.8
0.00
i
0.01
i
0.02
0.03
AD
Fig. 18. The effect of cell size distribution on densification
rate: /5 = 0.97, stress exponent n = 4.9.
2259
However, a different result is obtained if we compute
the densification rate for each member in a distribution of cell pore volume fractions and then average
these rates (because the densification rate is a nonlinear function of pore volume fraction).
To illustrate this point, suppose there are N cells of
different void fractions, with a (volume) average
density of 0.97. We compute the densification rate for
each of the cells [D (i)] and the average densification
rate [ l ) , v ¢ = E { ) ( i ) / N , i = l , N ] ,
and compare the
result with that of the single cell of average density
calculation. Figure 18 shows that for a given stress
condition, the wider the cell size density distribution
(AD), the higher will be the predicted densification
rate for consolidation under hydrostatic or
constrained loading. Interestingly, the unconstrained
compression shows almost no effect of pore fraction
variability on densification rate.
For high stress triaxiality conditions (hot isostatic pressing and constrained compression), the
predictions based on a single cell analysis probably
underestimate the true densification rate. We believe
this, combined with the acceleration of creep consolidation for nonspherical pores, accounts for the
discrepancies between measurements and predictions
in Figs 13 and 15. These observations suggest that cell
void distribution evolution should also be considered
in future, more accurate efforts at modelling the creep
consolidation of nonlinear porous materials.
4.4.
Stress,
temperature
dependence o f creep
and
microstructure
The modelling of creep consolidation processes
usually assumes that creep parameters (such as stress
exponent n and ~o/tro) are material constants when
power-law creep (i.e. dislocation creep) is the
dominant densification process. For the Ti3AI + Nb
material studied here, both test conditions (pressure
and temperature) and sample microstructure have
been found to influence the material parameters that
characterize the creep rate, Fig. 12. For the material
investigated here at least, the underlying assumption
of constant material properties for a consolidation
cycle in which the pressure, temperature and microstructure vary, can lead to potentially serious predictive inaccuracies.
The original intent of the work reported here was
not to investigate the origin of these effects, but the
results illuminate some aspects of the phenomenon,
point to its underlying cause and connect it with other
work in this area. In agreement with results (Fig. 12),
previous studies of creep in Ti-14AI-21Nb [22-24]
have shown a strong dependence of the stress
exponent n on stress and test temperature. This effect
has been linked to the activation of competing
(diffusional flow/dislocation) mechanisms of creep as
the test conditions and microstructure are changed.
At low stresses (a < 100 MPa) for HIPed condition
material, the stress exponent approaches a value of
unity which is associated with (boundary) diffusional
2260
YONG-MEI L1U et al.: DENSIFICATION OF POROUS MATERIALS
creep. At higher stresses, n increases to values
normally associated with dislocation creep. However,
there is a strong microstructure effect superimposed
upon this. The coarse lath ct2 structure of the HIPed
plus sintered condition is more resistant to the
diffusional creep than is the equiaxed, fine grain size
as-HIPed structure. In the high stress region
(a > 100 MPa) dislocation creep dominants. It has
been suggested that the mean free path for slip in the
lath ~2 structure controls the extent of the dislocation
creep contribution to deformation [23]. This is
considerably lower than for the equiaxied ~2 structure
of the as-HIPed condition (the grain size was 4 pm)
compared to the HIPed plus sintered state (lath width
~ 6 # m , length ~75/~m). Incorporation of microstructure evolution, and thus creep parameter
changes into the models used to simulate consolidation processes would clearly be a desirable next step
in efforts to more realistically predict consolidation
behavior.
5. CONCLUSIONS
An experimental evaluation of final stage densification models for a power-law creeping material
(here a Ti3AI + N b intermetallic alloy) has been
conducted for a range of stress states, pore shapes,
and material nonlinearity parameters. It has been
found that:
1. All the current creep potentials correctly predict
observed trends of densification rate with density and
state of stress. Highest densification rates are associated with unconstrained compression which has the
largest deviatoric component of deformation. This
was followed by hydrostatic pressure and constrained
compression.
2. Quantitative
comparisons
between
the
experiments and the models have revealed a pore
shape effect. The most serious error introduced was
associated with cusp-shape pores in as-HIPed
samples. This led to a much greater (3 times) densification rate than predicted. Ellipsoidal pores also
result in elevated densification rates ( ~ 20% increase)
under conditions of high load triaxiality.
3. The models predict that only a 3% nonuniformity in pore volume fraction distribution is capable
of causing up to a 50% increase in the densification
rate for hydrostatic and constrained compression
loading, but it has almost no effect upon samples
densified by unconstrained compression.
4. The material parameters that characterize the
creep of this alloy system are shown to be quite
sensitive functions of loading conditions, temperature
and matrix microstructure.
The study has shown that the next level of
predictive accuracy will require the development of
creep consolidation models that capture the evolution
of pore shape, pore spatial distribution and
microstructure.
Acknowledgements--The authors would like to thank Dr
D. M. Elzey for valuable discussion and helpful comments,
Mr J. F. Groves for designing the creep fixture, Mr F. T.
Eanes for assistance during creep tests, and to Dr L. Hsiung
and P. E. Cantonwine for assisting with the SEM studies.
The financial support of the Advanced Research Projects
Agency (Program manager, W. Barker), the National
Aeronautics and Space Administration (Grant number
NAWG 1692) and the General Electric Company (through
the Office of Naval Research and ARPA) is also gratefully
acknowledged.
REFERENCES
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