Acta metall, mater. Vol. 39, No. 5, pp. 979-986, 1991
0956-7151/91 $3.00 + 0.00
Pergamon Press plc
Printed in Great Britain
SENSING A N D MODELING OF THE HOT ISOSTATIC
PRESSING OF COPPER PRESSING
H. N. G. W A D L E Y l, R. J. S C H A E F E R z, A. H. K A H N z,
M. F. A S H B Y ~, R. B. C L O U G H ' , Y. G E F F E N 4
and J. J. W L A S S I C I ~
IDepartment of Materials and Science, University of Virginia, Charlottesville, Virginia, U.S.A.
2MetaUurgy Division, National Institute of Standards and Technology, Galthersburg, Maryland, U.S.A.
3Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, England
4Nuclear Research Center, Negev, Beer Sheva, Israel
5BDM International, Inc., Rosslyn, Virginia, U.S.A.
(Received 29 May 1990)
Abstract--A detailed experimental evaluation of mathematical models for densification during hot
isostatic pressing (HIP) has been conducted using high purity copper powder as a model system. Using
a new eddy current sensor, the density of cylindrical compacts has been measured/n situ and compared
with model predictions for the HIP process. Pressure shielding by the can has been found to influence
the densifieation, and a simple plastic analysis of a thin-walled pressure vessel was used to account for
its effects in the models. The existence of a low temperature creep mechanism during consolidation has
been found and a formulation to account for its contribution to densification has been developed and
implemented in the models. Other effects, believed to be associated with transient creep and the
temperature dependence of power law creep parameters, have also been observed in the experiments and
suggest the need for further model refinement.
R~mm&---On a effectu6 une ~valuation exp~rimentale d6taillee des moddles math6matiques relatifs/L la
densification au cours de la compression hydrostatique ~ chaud (CHC), en utilisant une poudre de cuivre
de haute puret6 comme syst6me modele. En utilisant un nouveau eapteur d courant de Foucauli, on mesure
in situ la densit6 de compact6s cylindriques et on compare avec le mod61e pr~vu par le processus de CHC.
On trouve que l'effet protecteur de la bolte sur la pression influence la densification, et on utilise un modele
plastique simple de vase d paroi mince sous pression pour tenir compte de cet effet sur les mod61es. Il existe
un m~canisme de fluage a basse temperature pendant la consolidation; de plus on d6veloppe et on applique
aux moddles une formulation pour tenir compte de la contribution de ce m ~ n i s m e d la densification.
Dans les experiences on a anssi observe d'autres effets, que l'on pense 6tre associ6s au fluage transitoire
et d la d6pendance de la temp6rature vis ~i vis des paramdtres de la loi en puissance du fluage; on pense
que ces effets rendent n6cessaire une am61ioration du mod61e.
Zmammenf~An
hochreinem Kupferpulver als Modellsystem werden die mathematischen Modelle
ffir die Verdichtung w~rend des heiBisostatischen Pressens ("Hippens') experimentell fiberprfift. Mittels
eines neuen Sensors gird die Dichte des zylindrischen PreBlings in situ gemessen und mit den Voraussagen
der Modelle ffir den Hip-ProzeB vergiichen. Die Druckabschirmung durch das Gef'dfl beeinfluBt die
Verdichtung; mit einer einfachen plastischen Analyse eines d/innwandigen Druckgef~sses gird dieser
Effekt in den Modellen berficksichtigt. Gefunden gird ein Niedertemperatur-Kriechmechanismus, dessen
Einflul3 auf die Verdichtung mit einer entsprechenden Formulierung beschrieben und in die Modelle
eingebaut gird. Andere Einitdsse, yon denen angenommen gird, dab sie mit dem Obergangskriechen und
der Temperaturabh~ngigkeit der Parameter des Potenzgesetzkriechens zusammenh~ingen, werden auch in
den Experimenten beobachtet; sie legen eine weitere Verfeinerung der ModeUe nahe.
INTRODUCTION
desired density and microstructure. Here, we consider
the use o f new sensors and consolidation models to
measure and predict density during the H I P cycle, as
part of a larger study that seeks to apply Intelligent
Processing of Materials (IPM) concepts [1] to the H I P
o f alloy powders.
The I P M concept, as it applies to HIPing, utilizes
advanced sensors to measure not only temperature
and pressure, but also quantities that determine a
component's fitness for its intended purpose (e.g.
density, microstructure, residual stress) and process
conditions, such as steep temperature gradients, that
Control of H o t Isostatic Pressing (HIP) is currently
achieved by sensing and feedback control o f temperature and pressure to achieve a predetermined temperature and pressure schedule. Schedules for both
variables are usually determined by trial and error
and seek to achieve a desired value of density.
Variations in powder properties, the method o f packing, unplanned upsets to the schedules, etc. can result
in uncontrolled c o m p o n e n t properties (such as
strength and ductility) due to failure to attain the
979
980
WADLEY et al.: SENSING AND MODELING OF HIP
Pressure (MPa)
0.1
1
10
100
g
1.0
(a)
8~
~' o.s
A
0.8
~-
0.7
6
0.6
........
,
-2
........
i
-1
, , ......
,
0
.......
5
Log (Normalized Pressure, P / Sin)
Temperature
1.0
300
'
400
'
//////'-'
,,
.o
600
~
//////
~, o.~ R
.'/1/L.
EXPERIMENTAL
i
PL - C R P 1
0.9
9
. c
/
.~
r,-
(°C)
500
~
v
0
Yield
I
0"~0,4
1
01.~)
a given pressure/temperature/time schedule, and
identifies the fields where single densification mechanisms are dominant. An eddy current sensor is used to
measure the change of diameter of simple cylindrical
samples during a HIP cycle. This change in diameter
is then converted to relative density and compared
with the density predicted by the models. By comparing the densities in this way it is possible to test the
validity of the modelling approach. While the overall
HIP response is found to be reasonable well modeled,
the experiments reveal several interesting phenomena,
such as can shielding, low temperature creep, etc.
that require further refinement of the modeling
approach.
I
016
0.7
Normalized Temperature T / Tm
Fig. I. The predicted HIP map for copper (a) shows the
effect of pressure at a fixed temperature (T = 550°C) upon
density (b) shows the effect of temperature at a fixed
pressure (P ffi 50 MPa). The particle size (a) is taken to be
75 ftm, the initial density (Do) is 0.62, and the gas pressure
between particles ( P p ) is Z e r o .
can cause undesirable distortions [2, 3]. Predictive
process models that relate density [4-6], microstructure [7], etc. to the process variables [temperature (T), pressure (P), and time (t)] are used to
determine the corrections to the T,P-schedule necessary to affect control of the component properties.
Non-linear constrained optimization methods can
then be used to resolve the sometimes conflicting
needs of an optimum schedule to achieve full densification and limited grain growth at minimum cost [3].
The opportunity to apply these control concepts to
HIP has emerged because of significant advances in
the modeling of the process [4-7] and the recent
development of an/n situ sensor that can measure the
diameter of cylindrical samples (and thus density) in
the HIP [8]. In this paper, we explore the use of this
sensor to experimentally validate the HIP models
developed by Ashby and coworkers. A model system,
high purity copper, has been selected as the material
for study since many of the plasticity, creep and
diffusion parameters used in the models are well
established for this material.
Figure 1 shows an example of the use of this model
to generate a HIP map for copper. The modeling
approach considers how plasticity, power-law creep
and diffusional flow at powder particle contacts lead
to densification. By summing the contribution of each
mechanism, it allows one to calculate the density for
The metal used in the current study, 99.99%
copper, was atomized in the National Institute of
Standards and Technology (NIST) inert gas atomization facility using argon for the atomization process.
The copper was maintained under inert (oxygen
free) conditions throughout subsequent sieving
( - 100 + 325 mesh), packing and sealing. Prior to the
sealing process, the powder was hydrogen treated at
350°C for 2 h to be sure that no copper oxide was
present on the particle surfaces. Nominally 22.22 mm
outside diameter and 152 mm long preannealed copper cans were used for HIPing, Fig. 2. Their diameters and length were measured prior to installation
in the HIP. The 0.2% yield stress of the can material
was measured and found to be 75 MPa. After a strain
of 15%, work hardening increased the flow stress to
-~200 MPa. An eddy current sensor was used to
measure the cross-sectional area (and hence sample
volume and density) during each HIP cycle. The
detailed construction and operating principle of the
sensor are described elsewhere [6, 7]. The sensor is
based upon a two coil technique in which a primary
Primaff Solenoid
SecondarySdeno~
c.,o~o~ Powd~
I
I
Fig. 2. A schematic section showing the eddy current sensor
with the copper test sample inserted.
WADLEY et al.: SENSING AND MODELING OF HIP
981
Table 1. H I P schedules for copper,
Sample
Test set
No.
Temperature
(°C)
Final pressure
(MPa)
P o w d e r size
(mesh)
A
16
14
13
18
450
500
550
600
50
50
50
50
--
100
100
100
100
+ 325
+ 325
+ 325
+ 325
B
21
13
20
550
550
550
25
50
75
- 100 + 325
-- 100 + 325
- 100 + 325
C
17
13
22
550
550
550
50
50
50
-325
- 100 + 325
- 1 0 0 + 120
~
0.85
i~
0.80
0.75
1
L
i
i
I
L
i
(a)
Time (hours)
coil induces a uniform electromagnetic field of vailable frequency and a secondary coil senses the perturbation to this field by the sample. At high frequencies,
because of the skin effect, the perturbations to the
field are controlled only by the geometry of the
component and not by its electrical conductivity, and
it is possible to measure a component's dimensions.
For a long cylindrical sample, in an axially uniform
field, the sample diameter is sensed, and typically
changes in diameter can be sensed with a resolution
of + 3 #m. To deduce the change in sample volume
and thus density, it has been assumed that the
fractional change in length is proportional to the
fraction change in measured diameter. Subsequently
(see below) interrupted HIP cycles have shown this to
be valid.
To investigate the effect of HIP temperature and
pressure, and the role of particle size, on the validity
of the models, a series of tests was performed according to Table 1.
For each test, the samples were pressurized to full
pressure without furnace heating (a small ~ 100°C
rise in temperature occurred due to adiabatic effects).
When full pressure has been achieved, the HIP
chamber was heated to 340°C at 25°C/min, with gas
being vented to maintain constant pressure. At
340°C, the temperature was then held for 30rain
prior to heating (again at 25°C/min) to the final
temperature indicated in Table 1. An additional series
of tests (not shown in the table) were conducted to
ascertain the relation between the diameter and
length changes. The sample was progressively HIPed,
with both the length and diameter intermittently
measured by removing the samples from the
chamber.
After each sample had been HIPed, the final
density and dimensions were measured and the
samples were examined metallographically. All
of the samples contained a few voids originating
from retained argon within the larger atomized
particles.
RESULTS
In Fig. 3(a) we show the temperature and pressure
schedules for sample 16, together with the simultaneously measured sample diameter corrected for
LO
.
.
I
2
.
.
.
.
.
.
6
7
!,
.
o
.~
O.S
IIC
00
O
3
4
5
Time Ihours)
i
$
i
9
L i
10 11
(b)
Fig. 3. (a) Temperature and pressure cycle for sample No.
16 together with the sensed sample diameter; (b) sample
density computed from the data shown in (a).
thermal expansion (i.e. the diameter is referenced to
that at room temperature). The shrinkage due to the
initial cold consolidation and subsequent HIP densification can be clearly seen. During the initial (cold)
pressurization, the measured diameter varied inversely with pressure as the powder particles deformed plastically. Heating to 340°C under constant
pressure resulted in further densification as the particle yield strength decreased. The shrinkage continued, at a low rate, during the hold at 340°C
(T/Tm = 0.45). Further heating resulted in increased
shrinkage due to the combined contributions of a
lowering of the yield strength and more rapid creep
deformation.
To convert the measured diameter to relative
density (in order to compare the data with model
predictions), it was necessary to determine the sample
volume using knowledge of its length, which also
changes during densification. The volume can most
simply be evaluated assuming the length change to be
linearly related to the diameter change with the
proportionality constant (a) determined after the
cycle by measuring the length of the consolidated
sample. Figure 4 shows this data for all of the tests
together with that accumulated from the interrupted
tests at various cold pressurization values, and during
subsequent heating. A typical value of a was 3.5 but
for low pressure tests it was near 5 and for high
pressures it was nearer to 2.5.
If O0(t) is the measured outer diameter of the can,
which varies with time (t) during the HIP cycle, and
WADLEY et aL: SENSING AND MODELING OF HIP
982
1° /
,
i
!
Final: 0.96~
1.000-I
°~
• Cok:iPressurizationStage
• CompletedRuns
o HeatingSt~e
L
o.s,~~
g6
0.606
o.8~0 I
0
Pre~e
J= 4 - -
--
~
•,
.
~,~
~
1.000~
®
o,2,- 1
>~
0.848 1
0.7?2
2
6
5
10
15
rr
20
Diameter Change (%)
Fig. 4. The change in length of a sample as a function of its
change of diameter. The circles correspond to a sample
whose dimensions were measured by interrupting the test.
I(t) is the can length, then the initial volume of the
cylindrical sample is ~tO0:(0)l(0)/4. During the HIP
cycle
~0(0)
--=
.
~
(a)
0.772
. . . .
o.6#s
0.620 I
0
,.oooJ
0.92,~
0.048 ~
0.772
0.620
,
SO
550 °C
25 MPa
,
,
,
~
~
,
,
,
120 180 240 300 360 420 480 540 600
f f
Final: 0.9994
/
f
,
60
(b)
55O ~C
50 MPa
,
,
,
,
,
,
,
,
120 180 240 300 360 420 480 540 600
Final: 1.0000
f
(c)
i
o
i
eo ~o ~so 24o see ~o 4=o
550 °C
75 MPa
i
i
480 540
Time (minutes)
.
(1)
Fig. 6. The measured (thick curve) and computed relative
density for samples at various pressures. The computation
used tuned power law creep parameters.
J
(2)
Substituting (2) into (3) and rearranging gives the
inner can diameter (i.e. powder compact diameter)
Upon rearranging this gives
_; (,
=
-~-~)
and the can volume becomes 7tdp2o(t)l(t)/4. This includes the volume of the can walls. If Oi(t) is the
diameter of the inside of the can, conservation of can
wall volume states that
l(0)[~bo~(0) -- ~0~(0)] = l(t)[d~(t)
1.000"I
- -
~
0.924"
O.84O0.772 -
~b~0(t)].
(3)
Final: 0,998S
0.6960.820
0
60
(a)
< 45 pm
550 °C
50 MPa
,
,
,
i
,
120 180 240 300 360 420 480 540 600
1000 "
I
f
60
45 - 150~m
550 °C
50 MPa
~
i
i
,
120 180 240 300 360 420 480 540 600
P
¢D
0.924"
o
o.e~-
>o
:,~
0.772-
rr'
O.SSS0.620
0
RnaJ:0.9994
(b)
~bi2(t) -- ~02(t) - A 1 - ~ [1 - ~bo(t)/~o(0) ]
(4)
where A = O~(t) - $~(t).
Since the mass of powder does not change, the
density, A(t), at any point in the cycle is given by the
product of the (easily measured) final density A(oo)
times the ratio of the final volume to instantaneous
volume
~bo:(oo)l(oo)
A(t) = A(oo) ¢,o2(t)l(t)
(5)
where A(oo) is measured after the HIP cycle is
completed.
This derived density, is plotted verus time in
Fig. 3(b). It can be seen that in this case the sample
was compacted to near full density. The curves for all
of the tests listed in Table 1 are shown in Figs 5, 6
and 7, together with the theoretically predicted behavior that is discussed below.
Flnil: 0.9993 I
1.000
°-,- l
/
DISCUSSION
0.772
-150gm I
°" t ~ , ~
0.gN
0.¢?.9 " ~ v ,
o
.
550 (°)
°C I
.
.
.
.,
,
SO Y P I I
eo 1=o lao =4o 3oo ~o 4=o =o 540 eoo
Time (minutes)
Fig. 5. The measured (thick curve) and computed relative
density for the samples with varying particle size. The
computation used tuned power law creep parameters.
By performing a numerical integration (with respect to time) of the Ashby et al. HIP model [4-7] it
is possible to calculate the relative density as a
function of time for any pressure-temperature cycle,
provided one knows the values of certain physical
parameters for the powder. The calculated density is
compared to that deduced from the HIP sensor
WADLEY et al.: SENSING AND MODELING OF HIP
1,000 -I
0.624
-
o.e4s
-
~---""
Table 2. Parameters for copper
Final: 0,~29
(a)
0.772 450 ~)C
50 MPa
0,e960 . 6 20
0
60
120
180
240
1,000 -]
300
360
420
r
480
540
600
Final: 0.9897
0.924 0.848 0.772 -
(b)
500 °C
g
50 MPa
0.~6-
o.62o
C)
o)
n-
o
60
12o
18o
240
300
360
420
480
540
600
Final: 0.9994
1.000
o'"- 1
0.606~
550 ~'C
~
60
~
120
,
180
-]
50 MPa
,
240
i
i
)
i
)
300
380
420
480
540
~...~
600
Flflal: 0.9999
0924 =
Y
0.848 0.7720.8960.620
0
60
120
(d)
600 "C
50 MPa
I
300
240
180
Time
I
I
I
i
360
420
480
S40
600
(~ = 2.5) in Fig. 8, using the generally accepted
material parameters for copper in Table 2.
We see that while there is general agreement in the
trends of both the experimental and calculated behavior, three primary problems are evident:
1. During cold pressurization, the simulation
predicts too highly a density.
2. At 340°C, the simulation fails to predict
continued densification (low temperature
creep).
3. During the hold at constant T and P (i.e.
t > 4 h ) the theoretically predicted high
temperature creep rate is less than was
measured.
T
•
9
rr n ~ ~- ~
-'-~
0
.
.
~- l~nmenW
.
.
.
.
SSO*C
.
3
¢
501MP"
S
Time
a
9
(hours)
Fig. 8. The measured (thick curve) and computed relative
density for samples 13. The computation used handbook
values for the power law creep parameters.
AM 39/~Q
MPa
m3
m2/s
ld/mol
m3/s
kJ/mol
m3/s
ld/mol
MPa
kJ/mol
kg/m 2
1360
1.72
145
0.540
55
0.540
1.18 x
6.20 x
207
5.12 x
I05
6.00 x
205
4.8
35
197
8960
10 -29
l0 -5
l0 - ts
l 0 - L0
The density during cold pressurization is controlled
by the extent of plastic deformation at particle to
particle contacts. This is determined by the yield
strength of the particles (cry), the contact area between the particles (controlled only by the value of
the current density, Ai), and the pressure (P) applied
to the particles [6]. It has the form shown in equation
(6) for Ai < 0.85.
A,
[ ( 1 - - ~ 0 ) p q_ A o ] 1/3
=
L
'
(6)
(minutes)
Fig. 7. T h e m e a s u r e d (thick c u r v e ) a n d c o m p u t e d relative
density of samples HIPed at various temperatures. The
computation used tuned power law creep parameters.
co~
K
J/m 2
GPa
CoM pressurization
(c)
I
0.620 i
0
1.000
J
~
Melting point
Surface energy
Young's modulus
T-dependence of modulus
Yield stress
T-dependence of yield stress
Atomic volume
Pre-exp., volume diffusion
Activ. energy, volume diffusion
Pre-exp., boundary diffusion
Activ. energy, boundary diffusion
Pre-~xp., surface diffusion
Activ. energy, surface diffusion
Power law creep exponent
Reference stress, P-L creep
Activ. energy for P-L creep
Theoretical density
Each effect is discussed more fully below.
j~-
°.°2,-1
983
where A0 is the initial relative density. The density
thus depends upon the ratio P/cry. We have used a
value for the room temperature yield stress cry of
55 MPa which is well supported by the literature [9].
Because the particle deformations are relatively
small, work hardening (neglected in the models) is
insufficient to account for the difference between
measurements and model predictions until the final
stage of densification (Ai>0.85) [10]. One possible
explanation is that the pressure that actually acts on
the powder is less than that applied by the HIP. One
way this could occur is for the can to be supporting
a fraction of the pressure---a possible explanation
also for the non-uniform shape change observed
during densification.
If the can has a yield strength comparable to or
greater than that of the powder contained within, it
is able to support a fraction of the applied pressure.
For a cylindrical can, the principle stresses are unequal, which may also result in a non-uniform shape
change. Hill [9] give the principle stress in the radial
(err), tangential (crt) and axial (cr,) directions for a
cylindrical tube with thick walls, closed ends and
internal pressurization. When the wall thickness (t) is
small, and we note that for the radial distance r such
that (~i ~ 2r ~<~0
cr,
~02 - 4r 2
(7)
Thus, for a thin wall the radial stresses tend to zero,
and the situation becomes one of plain stress. The
984
WADLEY et aL: SENSING AND MODELING OF HIP
pressure (Pc) supported by the can will be the difference between the applied presure (P) and that supported by the powder (P0) i.e. Pc= P - P o . We
assume that the can behaves as though filled with a
compressible fluid. If the force exerted on the ends
of the can is (~02/4)Pc, and the area of the walls
supporting the axial force is 7r~bot, the axial stress is
¢0Pc
aa =
4t
(s)
Similarly, the force per unit length in the tangential
direction is q~0Pcand this is supported by a wall area
of 2t so the tangential stress is
~,0Pc
at=
2t
(9)
and at/a t = 2.
To calculate the plastic strains due to these stresses
we use a yon Mises yield condition
a~ - a,a, + a~ = ~
(10)
where a~ is the yield strength of the can. The strain
increment is normal to the yield surface so the
associated flow rule gives the plastic strains
dEt = d).[2a, - at]
(11)
d, t -- d~.[2a t - at]
(12)
d ~ ffi dA [at - at ]
(13)
where dA is a proportionality constant that depends
upon the stress and strain (for materials exhibiting
work hardening).
For a can containing powder that behaves as a
compressible fluid, we have at = 2*t and therefore the
axial strain is zero, which agrees with the observed
initial behavior for the interrupted test in Fig. 4. The
inverse slope, 1/:t, in Fig. 4 is given by
l/ct ffi d~---2= 2 -____~y
d~t 2y - 1
(14)
where y = a j a r . For a compressible fluid, y = 2 and
,,--,oo (i.e. zero axial strain). For isotropic densification, y = 1 and therefore ,, = 1. We see (in Fig. 4)
that the final value of • (at the end of densification)
suggests dcnsification to be progressing in an almost
isotropic manner.
Let el, a,, and a 3 be the principle stresses in the
powder. Then by balancing forces we get
al = P - (4t/O0)at
(15)
a2 = P -- (2t/~0)at
(16)
03 = e - (2t/~b0)at.
(17)
Note that initially, when y = 2, a I = a 2 = a3, i.e. the
stress on the powder is hydrostatic, and has the
average value
P* = (el + or2+ o'3)/3 -- P -- (4t/O0)o',.
(18)
For values of 7 # 2, the state of stress in the
powder is no longer purely hydrostatic as treated in
the models. Usually the deviatoric stress components
are small ( < 10% of the hydrostatic stress) so their
effects can be ignored.
We can relate the axial stress in the can to its yield
strength. Using the von Mises criterion for plastic
deformation of the can, equation (10) we have
at = --aye [y2--y + 1]-t/2 .
(19)
P* = P + (4t/O0)a~[y' - y + I]-I/2.
(20)
Thus
For
4t/¢~o= I/8 and with ~,-- 2
P* ~- P + 0.070"y¢
(21)
where P is a negative compressive stress. For
%, ~ 75 MPa, P* - P is ~5.2 M P a and therefore the
pressure used in equation (6) to estimate the density
is too high and predicts too high a density, as found
experimentally (Fig. 8). In Figs 5, 6 and 7 we have
corrected the model to account for this, and much
better agreement for cold pressurization is achieved.
W e also note that as the temperature increases, ay~
decreases, and the effect of the canister becomes
smaller. At 550°C, we estimate the yield stress(aye)to
be reduced by an order of magnitude, and the
shielding of pressure to be less than I MPa. Under
these conditions, almost isotropic deformation of the
can would be expected. It is also interesting to note
that once affects for the can wall are accounted for,
equation (6) provides a method for calculating the
yield strength of the powder from a measured change
of density. This may be important for many alloy
powders because their small size,and refined microstructures make it very difficult to estimate or
measure directly their yield strength.
Low temperature creep
The modelling approach for densification due to
dislocation creep, is based upon a power law approximation to the creep behavior. For uniaxial deformation it takes the form
~c = Aane-Qa~r
(22)
where d, is the uniaxial strain rate caused by the
uniaxial stress o at temperature T, Q¢ is the creep
activation energy, R is the gas constant, A is a
constant (whose value varies from 1 to 10tT), and n
is the creep exponent. To eliminate the difficult to
determine A constant, one can define a stress, am
whose application results in a strain rate of 10 -6 s -i
at T = Tin/2 (a typical value during HIP densification). Then
,o-.r-:
To,<p-r,:.,,
This leads to an expression for the densification rate
for the initial stage (I), where A < 0.85, (i.e. when
WADLEY et al.: SENSING AND MODELING OF HIP
pores are still interconnected), due to power law creep
of the form [6]
3.1
rC (P - p0)]"
L
30-refA2
and for stage II
/~ = 1.5DLrcA(I - A)
(24)
J
985
x
1.5(P-P,)[
(n +2)17~f
1
~+2
1 --(1 --A) I/n+2
$2 (32)
where
where
l-A0
C -- - A-Ao
(25)
r Q" (..-~-2)]
De = 10 -6 exp - LRTm
Ao =
A=
P =
P0 =
DLTC= 10-texp
(26,
initial relative density
relative density
externally applied pressure (MPa)
initial pore pressure (MPa).
For A ~ 0 . 9 5 , when pores are
(referred to as stage II densification)
unconnected
A = I.SDeA(I - A )
X
-
-
17,~f
1 -- (1
-
A) l:~
(27)
where
P = Pi =
~ P0
(28)
and Ac is the relative density at which the pores close.
This formulation applies to regular (high temperature) creep with T > T~d2. At lower temperatures,
creep occurs faster than predicted by equation (22).
Detailed studies [9] show that at low temperatures the
data fit an equation like equation (20) but with a
smaller activation energy and a higher stress power
dependence (typically n + 2)
~LTC = ALTC 17" + 2 exp - QLT..._CC
RT
(29)
were typically QLrc=0.6Qc. Physically, the enhanced creep results from core-diffusion controlled
dislocation creep, the coefficients for which are
difficult to obtain. By again defining a reference stress
to be that at which ~ = 10 -6 s -~ at C~ Tm we can write
/ 17 \.+2
iLc-r = 10-6 tO--~f) e x p - - [L C2Q¢
R T . (-~- - ~ 1 ) ]
(30,
with the most probably values for C~ and C2 being 0.5
and 0.6. The parameter C~ then determines the transition from low temperature to high temperature
creep (which can be read from the deformation
mechanism map for copper [9]. The second parameter
C2 is QLrc/Q~ and it controls the temperature dependence of the low temperature creep process. Since the
low temperature and high temperature creep may
occur simultaneously, we obtain the total densification rate for creep by adding to equations (24) and
(27) the contributions from low temperature creep.
For stage I
3.1
[-C(P- Po)-~ +2
_[c2o,
ro 1]
L RTm T (1
(33)
where St and $2 are smoothing functions for the stage
I and stage II transition (i.e. S~ tends to zero and $2
tends to one as A tends from 0.85 to 0.95).
Thus, we would expect that with this additional
contribution to the densification, we may be able to
account for the finite rate of densification at 340°C.
In Figs 5, 6 and 7 we show predicted densification
behavior when corrections for the load supported by
the can and for low temperature creep have been
made. We see that the agreement between experiment
and prediction is now better for cold pressurization
and during the hold at 340°C, where the low temperature creep occurred. The revised model results in
a non zero densification rate at 340°C but the
rate remains a little low suggesting the model may
still underestimate slightly the contributions to
densification of low temperature creep.
High temperature creep
In Figs 5, 6 and 7 the comparison between experimental and predicted density beyond the low temperature creep region shows that for some tests a
rather poor quantitative agreement is achieved during
the high temperature creep regime. This region (see
Fig. 1) is dominated by power law creep. To make
detailed comparisons between the different tests, we
have tuned the creep parameters i.e. we have found
the values that best fit the data of sample 13. These
values are 17~f= 30 MPa and n = 4.5, values that are
encouragingly within the range expected for pure
copper.
Using these creep parameters we can observe how
well the model works as we vary particle size, HIP
pressure and HIP temperature. Turning first to Fig.
5, we see that varying the particle size for tests at the
same temperature and pressure as sample 13 had
relatively little effect upon the level of agreement,
except for the case of very coarse powders, Fig. 5(c),
where the measured density was less than that predicted. This may have been related to the presence of
argon gas retained in a small (10%) fraction of the
(larger) particles due to the atomization process.
Metallography of this sample indicated the presence
of (presumably gas filled) voids at the centers of some
particles after the HIP run. The pressure within the
enclosed gas-filled cavity would tend to oppose densification, and since the effect is not included in the
models, it may account for the observed discrepancy.
Variations of HIP pressure, while the temperature
and particle size are kept the same, are shown in
986
WADLEY et al.: SENSING AND MODELING OF HIP
Fig. 6. The power law creep parameters for the test
at 50 MPa [Fig. 6(b), sample No. 13], describe well
the behavior of the test performed at higher pressure.
However, the models predict too low a densification
rate at lower pressure. At the lower pressures transient creep, which has a higher strain rate and which
is not included in the models can make a greater
contribution to the total strain. If the pressures are
low enough, steady state creep may only be reached
near the end of the test or not at all, and since only
steady state creep (with its lower strain rate) is
included in the models, the disagreement shown in
Fig. 6(a) is, perhaps, not altogether surprising.
The effect of HIP temperatures is shown in Fig. 7.
The test used to tune the power law creep parameters
(sample No. 13) is again shown [in Fig. 7(c)]. It can
be seen that the parameters used at 550°C
(a~f= 30 MPa, n = 4.5) predict too high a densification rate at high temperature and too low a densification rate at low temperature. The test data suggest
that the value of the exponent n may thus not be
independent of temperature, and that at lower temperatures creep exponents. In fact, simulations indicate that at low temperatures e.g. 450 ° the creep rates
are characterized by higher exponents e.g. of n + 3,
while at temperatures above 550°C, smaller n values
(e.g. n - 3) fit the data better (here n is the value that
best fits the data at 550°C). This observation may also
help explain the discrepancy between predicted and
measured densification rates during low temperature
creep. Only n + 2 was used for low temperature
creep. The results found indicate that affects associated with the creep exponents temperature dependence are significant and that since the creep
exponent is a temperature dependent quantity [12],
there is a need to refine the HIP model treatment of
the power law creep process to account for this
phenomenon.
CONCLUSIONS
A detailed comparison between measured and
modelled HIP induced time dependent densification
of copper powder has been performed using a novel
in situ sensor to measure density during a HIP cycle.
The results show the importance of including the load
shielding of the container in the model, and when this
is done, the HIP models developed by Ashby et ai. for
plastic yield predict well the observed behavior at low
temperature. At intermediate temperatures ( ~ 0.4T m)
a new mechanism, low temperature creep, is found to
make significant contributions to the densification
rate. This mechanism has been incorporated into a
revised form of the model and it successfully accounts
for the experimental observations. At higher temperatures, where traditional power-law creep dominates densification, we have discovered systematic
effects of pressure and temperature upon densification that are not embodied in current models.
These effects may reflect the absence of transient
creep in the models, and a temperature dependence of
the power law creep exponent (n), with higher n
values more accurately representing the creep conditions at low temperature.
Acknowledgements--We are grateful to F. Biancaniello for
preparing the copper powder and cans used for this study
and to M. Mester for his help with the eddy current
measurements. This research was supported by the Advance
Research Projects Agency, DARPA order number 6340,
under the direction of W. Barker.
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