A cta me tall. mater. Vol. 41, No. 8, pp. 2297-2316, 1993
Printed in Great Britain. All fights reserved
0956-7151/93 $6.00+ 0.00
Copyright© 1993Pergamon Press Ltd
MODELING THE DENSIFICATION OF METAL MATRIX
COMPOSITE MONOTAPE
D. M. ELZEY and H. N. G. WADLEY
Department of Materials Science and Engineering, University of Virginia, Charlottesville,
VA 22903, U.S.A.
(Received 28 September 1992; in revised f o r m 9 February 1993)
Almraet--The consolidation of monotapes consisting of a layer of continuous aligned ceramic fibers
embedded in a plasma sprayed metal or intermetallic matrix is becoming a preferred approach for the
processing of high performance composite systems. We present a first model that enables prediction of
the density (and its time evolution) of a monotape lay-up subjected to a hot isostatic or vaceum hot
pressing consolidation cycle. Our approach has been to break down the complicated (and probabilistic)
consolidation problem into simple, analyzable parts and to combine them in a way that correctly
represents the statistical aspects of the problem, the change in the problems interior geometry and the
evolving contributions of the different deformation mechanisms. The model gives two types of output.
One is in the form of maps showing the relative density dependence upon pressure, temperature and time
for step function temperature and pressure cycles. They are useful for quickly determining the best place
to begin developing an optimized process. The second gives the evolution of density over time for any
(arbitrary) applied temperature and pressure cycle. This has promise for refining process cycles and
possibly for process control. Examples of the models application are given for Ti3A1+ Nb, ), TiAI,
Ti6AI4V and pure aluminum.
1. INTRODUCTION
Continuous fiber reinforced metal and intermetallic
matrix composites are attracting increasing interest
for high temperature aerospace applications because
of their attractive specific stiffness and high temperature strength. Many methods are being developed for their processing (for recent reviews see
Mehrabian [1] and others [2, 3]). In one increasingly
favored approach, foils (also called monotapes) consisting of uniformly spaced continuous fibers in a
porous alloy matrix are produced using plasma spray
methods [4]. Lay-ups of these porous, unidirectional
reinforced monotapes are then consolidated to form
near net shape composite components using processes such as hot isostatic pressing (HIP), vacuum hot pressing (VHP) and roll bonding. Here we
develop predictive models for the densification of
these lay-ups. We seek models that predict the effect
of process conditions (i.e. pressure, temperature
histories) and foil attributes (geometry, matrix properties . . . . ) upon densification. Our approach is
inspired by that of Arzt, Ashby and others, who
have developed a predictive densification model for
metal, alloy and ceramic powders undergoing hot
isostatic pressing (HIPing) [5].
The plasma spray process consists of winding
100-150tim diameter silicon carbide or aluminum
oxide fibers onto a mandrel which is then rotated
under a stream of (plasma melted) matrix alloy
droplets with diameters of 100-200/~m [4]. The
droplets, upon contact with the substrate, spread out
coating the fibers and filling the interstices between
with metal. Usually, the droplets freeze before fully
filling the interstices creating small unconnected
pores. Thus, at the completion of the spray process
one obtains a unidirectionally reinforced foil with
one relatively smooth surface (the one in contact with
the mandrel during deposition), one much rougher
surface (determined by the spreading of successive
droplets), and several percent of closed internal
porosity. The foils are then stacked to achieve a
desired lay-up and placed either in the die of a
vacuum hot press or suitably shaped HIP tooling
and subjected to heating and pressurizing cycles to
both fully densify the lay-up and obtain a near net
shape component.
During the consolidation process one seeks to
utilize the deformation mechanisms of plasticity,
power law creep and diffusional flow to close both
interfoil pores (i.e. those formed by contacts between
foils) and intrafoil (i.e. the closed) porosity. The
model we seek should predict the evolution of lay-up
density over time for any given (arbitrary) application of pressure and temperature, foil geometry
(foil thickness, surface roughness, internal porosity),
and matrix material properties. There are several
important reasons for wanting to do this. First, one
would like to find the most convenient process cycle
to densify a component without resorting to costly
and time consuming trial and error experiments.
Second, other phenomena during processing may
degrade the component's properties, i.e. fibers may be
damaged or broken, the fiber-matrix interface may
2297
2298
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
degrade, or the matrix may undergo phase transformations (because the foil matrix is often in a
metastable, rapidly solidified state [6]). Ideally, one
would use the model to design a process cycle that
reaches the target density without adversely impacting the component's properties. It is quite difficult
to do this purely by experiment because of the many
things that must be optimized and their complicated
dependence upon the process path. Third, one is
interested in determining the role of matrix properties on the feasibility of consolidation. For example,
we are interested in discovering which matrices are
most easily consolidated and how this is affected
by previous processing steps. Fourth, the model
should allow assessment of the influence of foil
surface roughness/internal porosity upon consolidation in order to identify those that will give the best
results. This can be used to help in defining a target
for the plasma spray process step. Finally, there is
increasing interest in using the model for feedback
control of consolidation processing [7]. Here, we
derive a first model that is capable of each of these
tasks and show its application to the densification
of several systems of current interest. Its application for understanding and controlling some of
the other phenomena during processing mentioned
above has been presented elsewhere [8, 9].
2. MODEL FORMULATION
Fig. 1. (a) Cross section of a plasma sprayed MMC monotape (fibers are 140#m diameter SCS-6 (SIC) within a
Ti-24Al-llNb alloy matrix). (b) Ti-24-11/SCS-6 monotapes following consolidation processing.
The cross-section of a typical continuous fiber
reinforced monotape produced by the Induction
Coupled Plasma Deposition (ICPD) process is shown
in Fig. l(a). One sees that the tapes produced by
this process are characterized by internal porosity,
one very rough surface (the other side in contact
with a mandrel during deposition remaining relatively smooth) and a certain fiber volume fraction.
Figure l(b) shows a fully consolidated lay-up of
TiaA1 + Nb/SiC monotapes (volume fraction of fiber
is 25%). Figure 2 illustrates schematically the physical situation encountered in a lay-up at the beginning of consolidation. When the monotapes are
stacked one on top of another, there is a large (30--40)
volume percent of porosity. Most is associated with
contacts. Typically, for a total initial void volume
fraction of 0.35, the surface roughness contributes
about 0.3 and internal porosity the remaining 0.05.
"~ Initial deformation
at contact points
Interlaminar porosity
due to surface roughness
Intermetallic Matrix
lib
I~B
Fiber (e.g. SiC, AI203)
HIP Cannist
[._j
v Intralaminar (closed) porosity
Fig. 2. Schematic view of the situation encountered during consolidation processing of a lay-up of plasma
sprayed monotapes.
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
This obviously is determined by statistical properties of the roughness and it is important that this
statistical representation be included in a realistic
model. A second important consequence of this
physical situation is that even in the absence of
reinforcing fibers and the application of hydrostatic
stress, the aggregate will deform anisotropically. This
is because deformation (at the contacts) occurs quite
readily perpendicular to the laminate plane, whereas
the compressive stresses in the two orthogonal directions are well supported by the relatively dense (and
thus more deformation resistant) monotapes. The
totaling used for HIP of planar components also
better supports the in-plane loads and encourages
only strains in the through thickness direction. The
presence of fibers also inhibits in-plane macroscopic
deformations with the result that almost any uniform
applied load combination leads to a state of constrained uniaxial compression during most of the
early stages of densification.
As the lay-up is raised to a high temperature and
pressure, densification is accomplished by a complex
sequence of deformations. Interfoil contacts, which
initially support the highest stresses, begin to deform
by plastic yielding and time-dependent creep. This
allows more contacts to form (and the areas of
previous ones to increase). As the applied stress rises,
the stresses around the closed pores within the foil
become sufficient to cause their collapse, again by a
combination of plasticity and creep. The addition
of these deformations results in a net shrinkage
(confined almost totally to the through thickness
direction) and densification of the laminate. The rate
of densification will depend upon process variables
(pressure, temperature and time), material properties (which determine the extent of deformation by
plasticity and creep processes) and the statistical
characteristics of the surface roughness and internal
porosity. All must be correctly incorporated in a
successful model.
Rather than attempting the potentially fruitless
task of seeking a single ad hoc constitutive law for
such a complex material behavior and using a finite
element model to solve our problem, we have sought
to model the micromechanical (plastic and creep)
behavior of representative (geometric) elements of the
problem and allow their contribution of the overall
densification to evolve during the process. Given the
representative volume element (RVE) or combination
of elements, (which is determined by the problem's
internal geometry), with the constitutive properties
of the macroscopic material, we model their response
to an arbitrary pressure/temperature cycle.
To ensure realistic predictions by this approach
one must select the RVE(s) such that:
1. The response of the element is truly like
that of the macroscopic body it replaces
(when subjected to identical boundary
conditions).
2299
2. The elements are small enough, relative to
the macroscopic body, that stresses and
strains can be defined as averages of such
field quantities determined within the volume element.
Once RVE's have been selected, and their micromechanical responses deduced, the effective (averaged)
properties of the RVE(s) can be used to represent
(at every point) the properties of the macroscopic
body, and its overall response to the process determined. The identification of a RVE which is a
sufficiently realistic and accurate representation of
the macroscopic body's heterogeneous behavior
while remaining simple enough to yield an analytical
solution is arguably the most critical aspect of the
model's development.
2.1. Representative volume element selection
We have visualized the porous lay-up as made
up of two alternating lamellar phases (see Fig. 3).
The r (reinforced) phase contains the fibers and
internal (closed) porosity formed by incomplete
infiltration. The s (surface contact) phase is highly
porous (because of the surface roughness), (but
contains no reinforcement. These are still complex
phases (i.e. they are heterogeneous). Their response
is evaluated independently by identifying an
appropriate RVE for each, deforming these under
conditions of perfect plasticity, power law creep
and diffusional flow and then recombining them to
simulate the overall lay-up response.
Both the r- and s-laminae are themselves made up
of two or more homogeneous phases; s contains
matrix and void while r is made up of fiber, matrix
and void. In Fig. 3, we identify representative volume
elements within the r and s laminae. It might at first
sight appear reasonable to model the r-lamina by
considering an elastic fiber within a cylindrical shell
of porous matrix material. Unfortunately, analyses
of such a 3-phase composite cylinders RVE for
arbitrary stress states and deformation mechanisms
have only been developed for the particular case of a
power law creeping matrix subjected to a hydrostatic
stress [10]. They show very little effect of a fiber upon
densification (other than reducing the available volume to deform and imposing anisotropic deformation). In view of this we use the well studied
concentric spheres RVE, to represent the porous
matrix, and consider only the particular case of plate
strain: no macroscopic strains are allowed to occur
in the fiber direction. The weak local effects due to
the fiber are thus neglected, while the macroscopic
effects of anisotropy induced by the fibers and the
reduced matrix volume fraction are preserved. For
the case of the r-laminae, we assume all voids to be
the same size and shape, to be uniformly distributed
and unconnected.
The s-lamina is made up of asperities whose heights
and radii can vary randomly. Therefore a volume
2300
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
Single Asperity (GRVE)
Z3
----~
1
j
X2
"
.e..:ig~ll.. e.'O.'. ¢2
I;
i s - lamina
,
i
',
#
:ca
,0,
x2
".: :.'" :.': .':4<D
~2(=xa)
7-,!
Concentric
spheres
RYE
Fig. 3. The complex deformation behavior of a monotape lay-up is treated by considering the response
of representative volume elements to simplified stress states.
element is chosen which is taken to be representative
of the geometry of all asperities. This "geometrically
representative volume element" (GRVE) consists of
a fully dense hemispherical cap enclosed within a
cell (see Fig. 3). The presence of the enclosing cell
permits definition of appropriate boundary conditions and the calculation of relative density. The
response of the GRVE is determined by approximate direct analysis methods for a perfectly plastic
matrix, power law creep and diffusional flow. Once
the behavior of a single asperity has been determined, the RVE response is obtained by modeling
the behavior of a statistical assemblage of asperity
unit cells representative of an actual surface.
A given RVE approximates the overall behavior
so long as the deformations imposed on it do not
significantly alter its geometry or distribution within
the macroscopic body. Otherwise the RYE ceases to
be representative of the actual material. In our case,
as density increases, the asperities making up the
s-lamina flatten and flow outwards toward each other
until only isolated, cusp-shaped voids remain. At this
point (occurring at a relative s-lamina density of
around 0.9) the s-lamina behavior is more accurately
represented by the concentric spheres model for
matrix containing isolated, spherical voids. We refer
to the range of densities below 0.9, during which the
rough surface RVE is applied, as stage I. For densities well above 0.9 (stage II), the s-lamina behavior
is modeled using the concentric spheres RVE. In the
vicinity of 0.9, the relative density is obtained as a
weighted average of the two predictions.
The respective RVE's are evaluated assuming that
the matrix deformation occurs by perfect plasticity,
power law creep (i.e. dislocation creep) and diffusional flow [through the lattice (Nabbro-Herring
creep) and along contact faces (Coble creep)]. Coble
creep by diffusional flux along grain boundaries
within the matrix has been neglected, implying that
the grain size is on the order of the asperity size and
pore spacing. The overall strain rate is taken simply
to be the sum of strain rates due to the individual
deformation mechanisms
where the superscripts refer to plastic, creep and
diffusive mechanisms. In addition, the s-lamina is
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
representative volume element making up the macroscopic body. Specifically, it assumed that a state of
plane strain prevails macroscopically, with no deformations occurring in the direction of the fibers. Both
volume and shape changes are allowed to occur.
However under these conditions shape changes are
limited to changes in the aspect ratio of the laminate.
The model is applicable to HIP and VHP of MMC
planar laminates. It may also provide a reasonable
approximation of the response of tubes and could
provide a starting point for understanding other
consolidation processes such as roll bonding.
Figure 4 summarizes the constitutive models
which are discussed in detail below. The macroscopic
considered in two stages, during stage I, the surface
roughness RVE is applied and during stage II, the
concentric spheres RVE. Thus a total of six constitutive models are needed. This approach follows
that developed with success earlier for powder
consolidation [5, 11].
2.2. Deformation and stress state
We restrict ourselves here to the analysis of
plane, unidirectional laminates subjected to homogeneous boundary conditions. In this case, one
can assume a macroscopically homogeneous stress
(strain rate) field to exist, which implies that the
overall deformation behavior is the same for every
Stage h
Macroscopic
rough surface
RVE
z~x E
°-
.,4
E = ~ J ' f (ph" (Pr" Fc (h' r' D' I~) dr dh
~~
Diffusional flow
Power law creep
Plastic yielding
o
Density, D
E ,~
'
~/I'lz
Fc
E = CDdiff~
o
'~Fc
' Zo(D olD 2)[~
Stage fl:
Macroscopic
porous
media
F C =
i
D O ~I ~ - n
, oc[r(h - Zo--5-)j
r2
kT Zo (Do/D2) ~
Fc = ~ gu(D) ac(D)'~fl (6Db+ 2pOv)
~
E [~1:':-"~:'. :" "" Z ~ : " • " ;t'o" • .'.'~<~
Power law creep
= ~0
, 3Em,
co, 1N I -,
•
0¢,
Eij = aT.~
Diffusional flow
0
T-,3 _
=0
i
"i Fc
FC =
E
Plastic yielding
,/%
~
, nI
Do
F c = 2~r(h - z o --5-)13o o
.o=
2301
n-1
~ij = 3B( "~o )
s~j%
~ 3 ~
= CDdiff ~T Sij
.!~
n+l
~c = ~n + l
la(D)2~2
+ b(D)2z~} 2
t
•
= kTR = ~ 6
=
x
v *
u
4
Fig. 4. Summary of local and macroscopic constitutive models for describing the dcnsification of MMC
monotapes. The overall response is the sum of surface contact deformation and shrinkage of internal voids
by plastic yielding, power law creep and diffusional flow.
A M 4118--D
mj
2302
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
uniaxial response (stress-density) of a rough surface
is expressed in terms of the statistical distribution of
asperity sizes (tph" tpr) and the behaviour of a single
asperity [F¢(h, r, D,/))] for plastic, creep and diffusional mechanisms. The plane strain response of the
r-lamina is expressed in terms of a potential, ~ for
each deformation mechanism.
2.3. S-Lamina
All engineering surfaces are rough on a sufficiently
microscopic scale. The behavior of rough surfaces in
contact plays an important role in problems involving
wear or the conduction of heat or electricity between
contacting bodies and a large volume of literature on
surface contact behavior exists [12-17]. The pressure
assisted diffusion bonding of surfaces is also well
studied. Models for diffusion bonding, such as those
by Guo and Ridley [18] and Hill and Wallach [19],
deal with the joining of prepared surfaces for which
the asperities are of uniform size and distribution.
The surface of a monotape is viewed as an assemblage
of asperities, all of which are hemispherical in shape,
but whose heights and radii are statistically distributed (Fig. 3). We construct the model as follows:
the force-displacement (alternatively, stress~lensity)
response of a single asperity is determined for a given
matrix material constitutive law (i.e. plasticity, power
law creep or diffusional flow). This effective response,
which is obtained as a function of the asperity radius
and height, is then used to obtain an approximation
for the response of the rough surface by considering
the statistical asperity height and radius distributions
(see Fig. 4).
2.3.1. Statistical model for rough surface RVE.
Consider a thin plate with one smooth side and one
rough side compressed between two smooth rigid
platens; initially, only a few widely spaced asperities make contact with the upper platen. These are
therefore subjected to high stresses and are easily
deformed and the total force resisting compaction is
small. As compaction proceeds, the number of contacting asperities, the contact area and hence the
resultant force to cause continued deformation all
monotonically increase. Taking strains in the plane of
the rough foil to be negligibly small relative to that
in the directon of compaction (i.e. a state of constrained uniaxial compression), conservation of mass
gives the relative density (D) as a function of the
spacing (z) between the platens
D ----z° " Do
(2)
divided by the total area of contacts. This depends on
the number of asperities making contact and their
size; both are governed by statistical parameters of
the rough surface.
Let the probability density function (PDF)
describing the distribution of asperity heights be
gOh(h). The cumulative fraction of asperities in contact after deforming a foil from z0 to z is given by
#h = S~° gOb(h)dh. Similarly, the distribution of asperity radii can be described by a PDF, tPr(r ), where
r is the asperity radius. Given the number of contacts and their size distribution, the beating area
can be determined as a function of z, and thus
density from (2).
Assuming the distribution of asperity radii to be
independent of asperity height, the probability of an
asperity of height, h and radius, r is
,~o(h, r)
=
~oh(h)" ~or(r).
(3)
Denoting the contact force required to cause further deformation of a single asperity as Fc(h, r, z, ~),
the (bivariate distribution of) forces, ~of required to
further deform asperities is given by
~f(h, r) = ~o. Fo.
(4)
Equation (4) associates a resistance force with each
asperity. The total force required to cause further
compaction of the surface is obtained by integrating (4) over all asperity radii, r and heights, h
encountered in compacting from zo to z
F(z, z) =
~f(h, r) dr dh.
(5)
The stress, I~, to cause compaction is obtained when
equation (5) is multiplied by the areal density of
asperities, l (number of asperities/unit area)
Z(z, ~) = I.
;°f0°
jr(h, r) dr dh.
(6)
Equation (6) is the constitutive relation we seek; it
relates the applied stress to the degree of compaction
(z) or relative density from equation (2).
It is simple to determine the areal density (l) and
distribution of asperity heights (~oh) and radii (Or)
from a profilometer measurement of a foil surface.
Our measurements of plasma sprayed surfaces indicate the heights are usually characterized by a normal
distribution:
Z
where z0 is the initial platen separation (determined
by the height of the highest asperity) and D o is the
initial relative density.
A relation between the applied stress (~z) and foil
thickness [from which the density can be obtained
using (2)], is next desired. The average applied stress
at the contacts is given by the applied force (known)
h(h) =
1
F
1 { h --
[[~qj
(7)
where h" is the mean height and ah the standard
deviation while the distribution of asperity radii is
approximately exponential
q,Rr)
= ~. e -~'.
(8)
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
Table 1. Typical statistical surface roughness data for Ti-24AI-11Nb
monotapes
Parameter (units)
Mean asperity height (/am)
Asperity height SD (/tm)
Asperity radii exponential factor (#m i)
Areal density of asperities (#m -2)
Symbol
Value
/~
ah
2
I
91.06
39.82
0.0178
5.0.10 -s
response
z = l-
~--
(h - z )
"exp[ -~(h-~2]I°~k-ah/JJo r . e x p ( - 2 r ) d r } d h .
Substitution of these functions into (6) gives the
constitutive relation for the RVE associated with
surface roughness
f~0f
l
E(z, 2) =I j~ i x / 2 ~ , ah
x
fo
.exp[ 1?-~Yl
L-~ \-~-) j
2.exp(-2r).Fc(h,r,z,~)dr
dh. (9)
The above relation is valid for any matrix constitutive behavior. It then only remains to calculate the
single contact resistance force, F~, which of course
depends on the mechanism of deformation.
2.3.2. Perfect plasticity. Upon application of
stress to an asperity at low temperature (T ~ 0.3Tin),
deformation occurs instantaneously by plastic yielding. We seek the stresses during plastic contact of a
sphere (or spherical cap) and a flat rigid plate. No
exact solution to the three dimensional problem has
even been obtained for general plasticity (i.e. yield
surface and incremental stress-strain relations). A
number of approximate solution methods are available based on either finite element analysis or slipline field theory [201. We are interested here in the
simplest process model providing an acceptable level
of accuracy. We have taken the criterion for flow
to be
Fo
--
=
tr~ I>
fl
-a 0
(10)
t/c
where trc is the contact stress, a~ the contact area,
a0 the uniaxial yield stress and fl, according to the
slip-line solution for a cylindrical punch indenting a
flat surface, has the value of about 3 [21]. Equation
(10) then states that yielding occurs when the average stress at a contact reaches about three times the
uniaxial yield strength of the matrix. While for
simplicity, we have assumed a constant value for fl,
detailed numerical work indicates that fl is more
generally a function of the relative density [22]. For
perfect plasticity, fl does not depend on the specific
material, only on the shape of the asperity.
From (10) the smallest force acting on a single
asperity to cause deformation is
Fc=ac'fl "cro~2Ztr(h --z)'fl "ao
(11)
where ac is approximated by 2nr(h- z) [20] and
(h - z ) represents the displacement of the asperity
peak to the current deformed height, z. Inserting the
constitutive relation for a single plastically deforming
asperity (11) into (9) gives the overall s-lamina plastic
2303
(12)
Equation (12) must be solved iteratively for the
height of compaction (z) from which the density
corresponding to the given applied stress is determined. Note that a 0 is usually a temperature and
microstructure dependent material constant. Using
the appropriate temperature dependence or yield
strength-structure relation allows equation (12) to be
evaluated under conditions of changing temperature
or evolving microstructure. (For the moment we
have ignored the latter, but it can be important in
systems where phase transformations or coarsening
occur during hot consolidation.)
2.3.3. Power law creeping asperity. At sufficiently
high temperatures (T > 0.3-0.4 Tm), the matrix can
continue to flow by time dependent creep. We
assume a power law relation for uniaxial creep of
the matrix
= B • a"
(13)
where B is a temperature dependent material parameter and n is the stress exponent (see Appendix).
(In general, B and n vary with stress and temperature, but may be assumed roughly constant if a
particular creep deformation mechanism predominates.) By a dimensional argument, which follows
that of Arzt et aL [5] for the creep deformation at the
contact between two spherical powder particles, we
can calculate the rate of change of asperity height in
terms of the contact stress. The physical problem is
likened to a cylindrical punch of cross-sectional area
ac = nx 2, indenting a problem law creeping solid. The
stresses, (f in the material below the punch then scale
with the contact stress, a c as
¢~ = Cl 'O"c.
(14)
The displacement rate of the punch must scale with
the strain rate ~ and with the radius of the contact x
giving £ = c2ix. Substituting (13) for ~ gives
=c2 "Bx(cl "at)".
05)
Writing B a s Eref/o'nef, where t=f and a~f are a reference
strain rate and stress, the constants c~ and c 2 are
determined by requiring 0 5 ) to reduce to the perfecfly plastic solution [see equation (10)] when n-~ oo
and O'r¢f is taken to be the uniaxial yield strength, and
to the elastic Hertz solution (for a sphere being
pressed into a flat plate) when n = l , i = E and
O'ref/Er¢f = E , where E is Young's modulus. Imposing
the limiting restrictions, c 1 = 1/fl, where fl is the
effective yield coefficient introduced earlier, and
2304
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
c2 = 1.36nil. From equation (15), the rate of change
of asperity height is [23]
= 1.36" nfl I -"xBa"~.
(16)
By recognizing that a c = nx 2"-" 2 n r ( h - z), we can
eliminate x in (16) to give the asperity force (Fig. 4)
Fc = \=B[r(h-- z)] '/z")
(17)
where at = 1.36(rcfl)t-"2 t/2-". Substituting (17) into
(9) gives the s-lamina creep response
e = otB '
Zn~
l
fZ°(h _z)l-t/2n
[ X / ~ . trh J ,
× expr
l
L-~\--~h ) _1 "a
x
r 1- 1/2.e x p ( - 2r) dr
.
(18)
where c is a geometry dependent constant, Ddiff is
the diffusion coefficient (lattice or grain boundary),
fl the atomic volume, k is Boltzmann's constant,
T is the absolute temperature and as before, a c is
the contact stress.
We assume the total flux of matter (m3/s) from the
contact to be the sum of grain boundary and lattice
diffusional contributions [11]
f~
f" =4~(aDb + 2pDv)-k---~" ac
(21)
where 6Db is the grain boundary diffusion coefficient
(i.e. along the contact interface) times the boundary
thickness, Dv is the lattice diffusion coefficient and p
is the radius of curvature of the neck which forms at
the perimeter of the contact.
Helle et al. [27] have considered the analogous
problem of diffusional sintering of two spheres.
The neck radius of curvature is (see Fig. 5),
p, = x2/[2(r- x)], where r is the contacting sphere
radius and x is again the contact radius. They
The densification rate is obtained by differentiating
(2) to give ~ =zo(Do/D2)D and substituting with
equation (18) for ~, whence:
D2
y,f
1
D = DoZo"
orB" (X/~"ah
x exp , ~ \
f:°(h-z),
~--Z/d
x fo°° rl-1/2, exp (--2r) dr }-" .
(19)
B and n are the only material parameters needed
to determine actual densification rates. They can be
complex functions of temperature, stress and microstructural state. For our work, we have used average
values representative of the conditions encountered
during consolidation. For more precise work one
could measure the creep constants and use a look-up
table for the most appropriate values of a particular
T, P combination.
2.3.4. Diffusional asperity deformation. In addition to dislocation creep, time dependent deformation can also occur (particularly at very high
temperatures) by stress-directed diffusion. The driving force is the reduction of chemical potential resulting from the removal of atoms from regions of high
compressive stress to regions of tensile stress (or
lower compressive stress). Thus material will be displaced from the asperity contacts to the traction free
surface of the asperity. We assume the uniaxial
stress-strain rate to be linear in the stress in accordance with the Nabarro-Herring [24, 25] and Coble
[26] creep mechanisms and to be of the form
= C " Ddilr • ~--~" Orc
(20)
:ering of two spheres
g of a sphere
plate
Fig. 5. The neck radius forming at the sintering contact
between a sphere and a fiat plate compared with that formed
at the contact of two spheres.
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
show that this can be well approximated as a
linear function of relative density, p, ~ - r ( D - Do).
For the sintering of a sphere to a flat plate, the
radius of curvature, p will be less than p, for a
given neck radius, as can be seen in Fig. 5. To a
first approximation therefore, p is taken to be
r (D -- DoI
P "" -4 \ 1 - Do ] "
(22)
The flux is related to the rate of uniaxial compaction by differentiating an expression for the volume of material displaced from the asperity contact
in deforming to cell height, z [28]
r,_d f2=Fl
dt [ 3
=) iF:
(r-
L
~
JL 2r - 2(r - z) JJ"
Carrying out the differentiation and making use of
(2) gives
r 3
(I = ~g.(D)£
(23)
2.4. R-Lamina
The r-lamina representative volume element has
the geometry of a spherical shell and there are
numerous papers with its analysis and application
[30-34]. The volume fraction of void in the actual
material is reflected in the ratio of the void and shell
radii• The concentric spheres (CS) RVE needs to be
evaluated for the case of plane strain (see Fig. 4), the
strain rate along the direction of the fiber being taken
to be zero• The results are the CS RVE model are
applied to both the r-lamina (throughout consolidation) and the stage II (D > 0.9) behavior of the
s-lamina•
We need the plane strain response of the CS RVE
for a rigid, perfectly plastic, a power law creeping and
a linear viscous (diffusion) creeping matrix. In each
case the relation beween stresses and strain rates has
been expressed in terms of a potential function cp such
that the macroscopic strain rates (for the RVE), may
be expressed as
Ocpm
/~U= A" a Z i j '
where
g~(D)=(1
D°\fD°/2
+
Do\ '
+~_2n2+Do/D'~
2 ' [ D - Do/
1 - - Do/D J ~- T
\~]•
Combining eqns (21) and (23), one obtains the
force-displacement relation (Fig. 4)
r 2
F¢ = ~
kT
"g~ac D (~Db + 2pD~) "
(24)
Substitution of (24) into (9) gives the stress-directed
diffusional response of the s-lamina RVE
•
x
O-h
e x p F A ? - ;YI
L 2 \ °~ / J
j,o~
g~(z)kT
}_l. (25)
r: e x p ( - ~.r) d r •
o
6D(rD b + 2pD~)
I
;o
The densification rate is given by
/)
D2 " Z '
ZoDo
x exp
.
l
x / ~ . oh
oY-2q(l-
)cosh(3
\a0/
(h - z/
(h-z)
[_I 2 (h\ ~h- ~121
dh " 2
] J
(27)
where A is a scalar factor which may depend on
the loading path, Z o are the macroscopic stresses
(i.e. stresses imposed at the RVE boundaries) and
g~m is the potential associated with deformation
mechanism, m.
2•4•1. Plasticity. We consider here the case where
the matrix is taken to be homogeneous and incompressible and deforms in a perfectly plastic manner. For this case a modified version of the plastic
potential developed by Gurson [32] can be used to
represent the yield surface of a porous body
+0 = (
=
x
2305
o/-,
=0
(28)
\2a0]
where Ye and Y'm are the macroscopic equivalent and
mean stresses, respectively, D is relative density and
a0 is the uniaxial tensile yield strength of the matrix.
The constant, q is taken to be ,.5 as indicated by
experimental and theoretical results [35]. The stress
state considered is axisymmetric with Y.2(=Y3) the
given applied stress and Z~ the as yet undetermined
stress along the fiber direction (Fig. 3). The condition of zero strain in the fiber direction yields a
second relation involving the unknown stress (Z1)
and density, D
a~PP
g, = ~ , ,
= 2
- Zs)
(x'-
O0
x
r 3exp(-- 2r) dr
6fl(t~Db + 2pD~))
. (26)
Evaluation of this expression requires estimates of
the diffusion constants which can be found for many
common materials in the standard handbooks• Frost
and Ashby [29] give simple rules for their estimation
in other cases.
(29)
\
za0
/
Simultaneous solution of eqns (28) and (29), which is
done numerically, gives the density as a function of
the applied stress as well as the ratio of stresses YI/Zs2.4.2. Power law creep. A lower bound estimate
for a power law creeping, porous material is given by
2306
ELZEY and WADLEY:
MODELING OF METAL MATRIX COMPOSITE MONOTAPE
Applied Stress (MPa)
1
10
100
1000
.....
oo
~ . . ~ - : - - Power
- - :- J ~- ' i / / /~'~
/ ~Z "7~
/I
Law / / ~
C-reep/ / / ; , ~ y , ~ /
o
-'~o"-=°0.8~
o)
._>
/
~@!21a
0.7
0.6 ~
I
e
a
n"
/ [ il
Id
'
~
~
s
i
h
' _j . . . . . . . . . . . . . . . . . .
'
L..,'
(8)
0.5
-2
-1
'
Co tours
'..
First = 0.25
Last = 16,0
Factor = 2.0
0
NormalizedPressure,P/ay
Applied Stress (MPa)
1.0
1
10
1 O0
1000
0.9
a
-.~ 0.8
a
~ 0.7
0.6
~n~u~
. . . . Initial Density -
First = 0.25
Last = 16.0
-~-I FaVor=2.0
0.5
-3
-2
-1
NormalizedPressure,P/ Gy
Fig. 6. (a, b) Caption
on facing page.
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
Applied Stress (MPa)
1
lO
lOO
'
' ' ' S
. . . .
o.1
1.o ..... ~
Diffusion ~
~.,/.~.
0.9
•~
~
f
0.8
e-
"~ 0.7
(c)
0.5-3
-2
-1
Normalized Pressure, P/Oy
Applied Stress (MPa)
o.1
1
O.Ol
1.0
.....
i
I
.
Diffusion
0.9
/
~ 0.7
-3
.~
- Pc ~er-
.
lO
.
~ ~
Creep.~
.
-- Initial D e n s i t y
0.5
.
-2
..........................
. = .
'~
-1
0
Normalized Pressure, P / Oy
Fig. 6. (a-d) Densification maps showing predicted MMC laminate density vs normalized applied stress
for Ti-24AI-11Nb (Ti3AI + Nb), TiAI (y), Ti-6A1-4V and A1 matrices reinforced with 25vo1.% SCS-6
(SIC) fiber.
2307
2308
ELZEY and WADLEY:
MODELING OF METAL MATRIX COMPOSITE MONOTAPE
Temperature (°C)
400
1.0
600
~
j
=
800
=
t
1000
,
1200
Law~
0.9
i
Power
Creep
.•0.8
e-
ID
>
'~ 0.7
p= IOMPa
n=2.5
0.6
,
.................
Initial Density . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contour=
,
First = 0.25 ""
Last = 16.0
Factor = 2.0
(8)
0.5
0.4
0.5
0.6
0.7
0.8
Homologous Temperature, T / T m
Temperature
600
.
1.0
.
.
.
(°C)
800
1000
1200
~
,
Diffusion
./
0.9
a
I
I
..~ 0.8 - C
a
~
Plastic Yield
o.7
TIAI
'
p - 50 MPa
ft.
n = 4.0
0.6
,
.....................
Initial Density . . . . . . . . . . . . . . . . . . . . . . . . .
(b)
0.5
0.5
I
Contour=
~i:stt :~.62.~ - - "
Factor = 2.0
0.6
0.7
0.8
Homologous
Temperature,
T/ Tm
Fig. 7. (a, b) Caption on facing page.
0.9
ELZEY and WADLEY:
MODELING OF METAL MATRIX COMPOSITE MONOTAPE
Temperature (°C)
600
800
1 0O0
1.0
0.9 ~ - ~ P ~ V w w e r
/
D
._~ 0.8
r-
®
>
' ~ 0.7 m
!
0.6 . . . . . .
PlasticYield
Initial Density . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I
l
; pn = 4.0
10MPa
=
'
II
(C)
Cm~rm
' ..
First = 0.25
Last = 16.0
Factor = 2.0
0.5
0.4
0.5
0.6
0.7
Homologous Temperature, T / Tm
Temperature (°C)
100
400
300
20O
1.0
Creep ~
z-~ 0.9
g
a
2Z-
0.8
PlasbcY=eld
R r s t . 0.25
L a s t . 16.0
Factor = 2.0
(d)
0.7
0.4
0.5
0.6
0.7
0.8
HomologousTemperature, T / Tm
Fig. 7. (a--d) Predicted MMC laminate density vs homologous temperature for Ti-24Al-I1Nb, TiAI,
Ti-6AI-4V and AI matrices with 25vo1.% SCS-6 (SIC) fiber.
2309
2310
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
the strain rate potential originally developed by Ponte
[36] and modified by Duva and Crow [37]
q~c =
B
n+l
s" + l
(30)
where B and n are the familiar constants in the
Norton creep law, ~ = B a " and s 2 = a 2 ( D ) X 2 +
b2(D)Y~. The parameters a and b are functions of
the relative density only
a2(D) =
[1 + ~(1 -- D)]
Coble [26] creep and to Ashby's result for powder
densification [39].
Proceeding as for creep, the zero strain rate along
the fiber gives/~l = c3~d/c35Zl= 0, which can be solved
for the unknown stress component, 5.1
~"~'1~" (d" ~']3 where
(34)
with
c= 6
bZ(D) = [n(1 - D)" [1 - (1 - D)t/"]] 2/~"+1)
2c - ~d
(d -- 2C + ~
kTR 3
f~
d = 2" k - ~ "
tTDb+
RDv
(6Db + ~rpDv)"
The non-zero strain rates are given by
Rearrangement of the equation, /~1 = O~c/os.~ = 0
gives an explicit expression for the unknown stress
component
Y-1=(¢(D)'E3
where
(c(D)
a 2-~b 2
a2+~b2
(31)
E2 ~- E3 ~ (x. 5.3
(35)
with x = 2c(1 - ~d) + 4d( 1 + 2~d)- The densification
rate is obtained by substituting (35) into /5 = D/~3
/)
= 2D~
• 5.3.
(36)
and Z2 = X3. The non-zero strain rates are then
-/~2= ~"3 ~--ct(D)- BS.~
(32)
where
~(D) = a~(1 -~o)~+-¢-(1 +2~0) ~
b2
x (a2(l - ~ ) + ~- (1 + 2(~)).
The densification rate is obtained from /9 = - D .
2.4.3. Diffusion. The mean stress component, Z m
leads to a flow of matter from within the compressed
body (e.g. from grain boundary sources) to pore
surfaces (traction-free), resulting in pore shrinkage
and therefore densification. The deviatoric stress, 5.e
leads to fluxes from grain boundaries under high
compressive stress to those with lower stress thus
effecting shape changes. Both diffusive fluxes from
grain boundary giving shape change are considered,
whereby the driving forces are the mean stress and
deviatoric stress. Accordingly, the diffusion potential
is taken to be [381
f2
/5~ / D
2
\
' 5.2 + 2(tTDb + irpDv)52m
(33)
where R is grain size and rp, the pore radius, is taken
tobe
R ( 1 -- D ~ 1/3
rp ~-- \ 2D ] "
All other symbols have the same meanings as for the
s-lamina diffusion model. In the limiting cases of pure
shear and hydrostatic compression, (34) reduces to
the known limits of Nabarro-Herring [24, 25] and
3. COMPOSITE DENSIFICATION MAPS
Using the RVE models, the strain rate components
(and hence the densification rates) are determined for
the r- and s-laminae. Once the densities, Dr and Ds
have been determined [using equation (1)], the composite density is obtained by the rule-of-mixtures:
D =vrD~+vsDs, where v~ and vs are the volume
fractions of the r and s phases. (The volume fractions
are not constant, but depend on the relative rates at
which r and s densify. Thus the volume fractions must
be redetermined following each numerical integration
step.) When the s-lamina reaches a density of 0.92,
further increments in density are determined using
only the concentric spheres RVE.
The predictive relative density of the composite is
conveniently illustrated by means of densification
maps; plots of density vs normalized stress at constant temperature and of density vs temperature for
a given constant stress. We employ the same form for
the maps as that introduced for powder compaction
in Arzt et al. [5]. The maps are generated by numerical integration, implemented on an IBM-compatible
personal computer.
Figure 6(a) shows a map for Ti-24at.%Al1lat.%Nb (~2 + fl) matrix reinforced with SiC (SCS-6)
fibers like those shown in Fig. l for consolidation at
a constant temperature of 975°C. Material parameters used in the model are given in the Appendix.
The lowermost contour indicates the density achieved
at a given stress due only to plastic yielding, which is
assumed to occur instantaneously upon load application. Higher densities are achieved by allowing time
for thermally activated mechanisms (power law creep
and diffusion) to contribute. Thus the second contour
(at t = 0.25 h) indicates the density reached by holding the stress constant for a quarter of an hour.
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
Figure 7(a) shows the densification behavior as a
function of temperature (consolidation pressure is
constant at 10 MPa). As before, the lowermost contour represents t = 0 (densification due to plastic
yielding), while the remaining contour lines indicate
the increase in density due to creep and diffusion. The
regions of the map are labeled according to which
densification mechanism dominates; plastic yield,
power law creep or diffusion. The border between the
creep and diffusion fields represents values of applied
stress and density at which the two mechanisms
contribute equally to densification.
Figures 6(b--d) and 7(b-d) show the predicted
densification behavior for composites with TiAI (7),
Ti-6AI--4V (a + fl) and pure aluminum matrices. The
stress-density maps [Fig. 6(a--d)] have been constructed at roughly the same homologous temperature (T/Tm~0.68), while the temperature-density
maps [Fig. 7(a-d)] all illustrate behavior for a relatively low applied stress. Material parameters used in
constructing the maps are given in the Appendix.
Initial densities of s and r were taken to be 0.35 and
0.9 (excluding fiber), respectively, with initial volume
fractions of vs = 0.47 and v, = 0.53.
4. DISCUSSION
Our primary objective in developing this MMC
monotape densification model was to allow one to
predict how the density depends upon the applied
stress, temperature and time of a consolidation cycle
and to understand how this relates to the attributes
of the monotape (e.g. its physical/mechanical properties and geometry). This is important because HIP
cycles must attain full density in components, otherwise residual porosity severely reduces the high temperature structural integrity of finished components.
However a number of other processes that adversely
impact component properties also accompany the
HIP cycle, e.g. phase transformations, grain growth
[39] and fiber damage and fracture [40] and interracial
reactions. Densification maps such as those presented
above provide a convenient means of determining the
conditions that lead to full density, and these can be
I
I
100
evaluated for their impact upon other attributes of
the composite (e.g. the number of broken fibers [8]).
The general behavior of the four composite
material systems for which maps have been included
(Figs 6 and 7) is similar. The behavior indicated by
the maps is also similar in form to that found for
the compaction of powders [27]. Comparison of the
stress-density map for Ti-24AI-11Nb [Fig. 6(a)]
with that of TiAI [Fig. 6(b)] indicates the relative
importance of creep deformation in achieving high
density components; diffusion plays a relatively
minor role for the Ti3AI + Nb (Ti-24AI-11Nb) composite while for the more creep resistant TiAI (7)
matrix composite, diffusional mechanisms are much
more important for equivalent homologous temperatures. This is especially so at the low stresses required
to avoid fiber fracture. As observed for powder
consolidation [5], densification at higher stresses
(tr > 10 MPa) is principally due to creep where/)~ttr",
with only the last few percent achieved by diffusion.
Although the maps have the disadvantage of viewing the consolidation process cycle as one in which
both applied stress and temperature are applied as
step functions (i.e. they are suddenly switched 'ON'
to certain values, then held constant for a certain
length of time and then switched 'OFF'), densification maps are still useful for their intended application. For more detailed work the transient response
for a general consolidation cycle can be evaluated by
integrating the densification rates along an arbitrary
process path, simulating the response to specific
pressure-temperature-time cycles. Figure 8 is an
example. This more sophisticated output of the modeling approach has been investiated for model-based
process control [9]. It has been used to find the
optimum process cycle [i.e. p(t), T(t)] that results in
a desired density. Departures from the intended D(t)
can then be sensed and, using the model, feedback
corrections to the remaining p(t), T(t) schedules
computed.
The transient response model output can also be
used to investigate the error incurred when using a
map to predict densification. Maps overestimate the
density at any given time since they neglect the time
required to increase the stress and temperature to
1200
IOOMPa
~°°° ~11
800
850"C
2311
1.0
~
t
I
0.9
e-
600 .,-,
~
2O
- 200
0
l
500
Time
I
1000
(rain)
0
1500
F-
t'~
(D
0.8
._>
0.7
n"
0.6
0.5~
0
I
I
500
1000
Time (min)
Fig. 8. Simulated densification for a typical pressure-temperature-time cycle.
1~ 0
2312
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
their respective soak values. The error is typically
greatest for short consolidation timest, falling to
within a few percent at densities above 0.9.
The models describing the constitutive response of
the plasma sprayed surface [equations (12, 19, 26)]
require as input the mean asperity height (h'), and
radius (1/2), the standard deviation of heights (ah),
the areal density of asperities (1) and the initial
relative density of the surface layer (D~). Since these
parameters are determined by the thermal spray
conditions (e.g. powder size, superheat, droplet velocity, etc.), the models allow the influence of spray
conditions on consolidation behavior to be investigated. For example decreasing the standard deviation
of asperity heights (keeping all else constant) can be
shown to decrease the density achieved for any given
level of applied stress since the number of additional
contacts per increment of compaction is increased.
However at the same time, decreasing ah decreases
the nominal thickness of the s layer (taken to be
h" + 3ah) and so also the volume fraction of s; thus the
decreased densification rates are to some extent compensated for by the higher initial composite density.
While there may be no substantial advantage in
reducing surface roughness to achieve efficient consolidation, (for the reason cited above and because
the applied stresses and temperatures are required
anyway to collapse internal voids), minimizing surface roughness has been shown to be essential to
avoid fiber fracture during consolidation [8, 40].
This first approach to a very complex process
has perforce required the use of many simplifying
approximations and these are worthy of further
consideration. The modeling approach used here is
analogous to that of Arzt et al. [5] for powder
consolidation; representative composite volume
elements are identified whose stress-strain response
is then obtained either directly (i.e. by solution of
the appropriate boundary value problem based on
equilibrium, compatibility, etc.) or by energy
methods. Equally significant is the treatment of large
deformations by applying separate volume element
analyses to various density regimes ("stages")~thus
the distinction between behavior at lower densities,
which is dominated by deformation at points where
foil surface asperities contact adjacent foils, and
higher densities, where the shrinkage of isolated
internal voids more accurately represents the macroscopic behavior. The apparent success of this method
in simulating the deformation behavior of powders is
an indication of its promise in modeling the deformation processing of more complex (heterogeneous,
anisotropic) materials. However it might be added
that the alternative to this, namely the derivation of
a unified model able to handle the transition from low
tAt t = O, the map predicts a density given by the initial
density (Do) plus the densification due to plastic yielding
(ADp) whereas the actual density is still Do; thus the
initial error is just ADP.
density (where the void forms a continuous phase)
to high density (where the metal is the continuous, or
matrix phase), poses very serious challenges.
Approximating the influence of continuous fibers
by evaluating the continuum models for the case of
plane strain, (with no deformations occurring along
the fiber direction), has the consequences of underestimating the actual densification rate under an
applied hydrostatic stress and of neglecting any
effects on densification rates near the fibers due to
stress concentrations (esp. enhanced shear stresses).
Theoretical investigations of porous, fiber reinforced
materials so far indicate these effects are of second
order [10, 41] and we feel justified in making this
greatly simplifying assumption. More serious in our
view is the assumption of spherical voids. In reality,
the voids in plasma sprayed foils are much more
compact in one direction, i.e. pancake-shaped (see
Fig. 1), than our spherical idealization. Consequently
they collapse more easily [42]. Further experimental
work is needed to address the significance of this.
We have viewed thin laminates as tending to
densify primarily by uniaxial deformations occurring perpendicular to the plane of the laminate even
when the applied stress is hydrostatic. This point
is worthy of some expansion. There are several
factors that we believe make this so:
(i) Because about 90% of the porosity
initially present in the composite laminate
is concentrated in the surface roughness
layers, the composite is much like a stack
of nearly rigid solid sheets alternating with
sheets of foam. Such a body is much easier
to compress perpendicular to the plane of
the laminate, i.e. to compress the foam
layers, than to deform in-plane.
(ii) The fibers shield the matrix from densifying stresses in the plane of the laminate
and along the fiber direction.
(iii) The tooling used in HIPing of thin sheets
tends to preferentially shield the composite from in-plane loads.
As a consequence, the spray-deposited foil laminate densities predominantly uniaxially, regardless of
whether the stress is applied isostatically or uniaxially and this has been found to be so in experiments
[40]. As the density increases, a unidirectional layup becomes more nearly transversely isotropic and
as a result, one might expect different densification
rates for HIP and VHP. However as full density is
approached, the macroscopic stress state during
VHP (constrained uniaxial compression) approaches
hydrostatic, neglecting frictional effects. Large deviatoric stresses caused by friction, which are generally
significant during VHP consolidation, are of less
concern when consolidating thin laminates. These
arguments lead us to speculate that the densification
response of thin laminates should be very similar
ELZEY and WADLEY:
MODELING OF METAL MATRIX COMPOSITE MONOTAPE
in practice for both H I P and V H P consolidation
processes.
5. S U M M A R Y
A set of micromechanical models have been
used to predict the densification behavior of fiber
reinforced metal matrix composite monotapes during
consolidation processing. To simplify the complex
deformation geometry, the composite foil is broken
up into two simpler sub-laminae whose relative densities are evaluated separately; one of these contains
the fiber reinforcement, has all smooth sides and
initially contains less than 10% internal porosity,
while the other sub-lamina contains the surface
roughness. The effective constitutive behavior of the
sub-laminae was obtained by the identification and
analysis of representative volume elements within
each sub-lamina. Direct (approximate) and variational (bounding) methods were applied to the
R V E ' s assuming matrix constitutive behavior corresponding to plastic yielding, dislocation and diffusional creep. Predicted results of the densification
behavior of several fiber reinforced composite systems of current interest have been presented in the
form of densification maps. Applications of the
model include H I P and V H P processing of thin
M M C laminates, tubes, and cones and HIPing of
thicker components provided nearly homogeneous
stress and temperature fields exist within the
components.
Acknowledgements--The authors would like to thank M. F.
Ashby, R. M. McMeeking and J. M. Duva for many helpful discussions and advice. R. Vancheeswaran was instrumental in developing simulation software used to produce
Fig. 8. The financial support of the Defense Advanced
Research Projects Agency (Program manager, W. Barker),
the National Aeronautics and Space Administration and the
General Electric Company (through the Office of Naval
Research and DARPA~ is also acknowledged.
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APPENDIX
Material Data
T i - 2 4 A I - l l N b [43-51]
Elastic properties
Young's modulus as f(temp) [GPa]
--R.T.
--low temp range (T ~<5O0°C):
---high temp range (T > 500°C):
lO0.0
E(t) = lO0.0-0.04*T
E(t) = 140.0-0.12.T
Plasticity
Yield strength as f(temp) [MPa]
--R.T.
--low temp range (20 ~< T ~< 800°C):
--high temp range (T > 800°C):
539.9
try(T) = 539.94).32.T
%(T) = 1019.9--0.92.T
Creep t
Power law creep constant, A [11-1]
Stress exponent, n
Activation energy, Qc [kJ/mol]
6.0
Diffusion :~
Pre-exp boundary diffusion, t~Dbo [m3/s]
Activation energy, Qb [kJ/tool]
Pre-exp volume diffusion, D,~ [mZ/s]
Activation energy, Qv [kJ/mol]
Misc. properties
Melting point [K]
Density [g/em3]
Atomic volume [m3]
Surface energy, ), [J/m2]
× 1017
2.5
285
3 x 10 -12
202
5.9 x 10 -3
330
1875
4.6
1.0 x 10-29
1.8
Ti-Al [51-54]
Elastic properties
Young's modulus as f(temp) [GPa]
--R.T.
--E(temp)
Plasticity
Yield strength as f(temp) [MPa]
--R.T.
---low temp range (20 ~< T ~<6O0°C):
--high temp range (T > 6O0°C)
Creep
Power law creep constant, A [h -I]
Stress exponent, n
Activation energy, Qc [kJ/tool]
173.0
E(T) = 172.0-0.03.T
473.0
try(T) = 475.0-0.10* T
%(T) = 875.0-0.76.T
7.62 x 1022
4.0
3O0
tThe steady state creep rate is described by
o" n
~ = A " ( E l exp(--Qc/RTk)"
The values given for the creep parameters appear to represent the observed creep behavior well for stresses below about
170 MPa and temperatures up to 870°C, the highest test temperature at which creep data could be found.
~Values given are for TiAI.
ELZEY and WADLEY:
M O D E L I N G OF M E T A L M A T R I X COMPOSITE M O N O T A P E
2315
Diffusion t
Misc. properties
Melting point [K]
Density [g/cm 3]
Atomic volume [m 3]
Surface energy, ~ [J/m 2]
1725
3.8
1.63 × 10 -29
1.8
T i ~ A I - 4 V [55~0]
Elastic properties
Young's modulus as f(temp) [GPa]
--R.T.
- - l o w temp range (T ~< 500°C):
- - h i g h temp range (T > 500°C):
114.0
E(T) = 115.0-0.056.T
E(T) = 172.44).16.T
Plasticity
Yield strength as f(temp) [MPa]
--R.T.
--try (temp) ( T < 960°C)
970.0
ay (T) = 884.0-0.92. T
Creep
Power law creep constant, A [h -I]
Stress exponent, n
Activation energy, Qc [kJ/moll
8.4 x 1024
4.0
280
Diffusion
Pre-exp boundary, diffusion, 6Dbo[m3/s]
Activation energy, Qb [kJ/mol]
Pre-exp volume diffusion, D~0 [m2/s]
Activation energy, Qv [k J/moll
3.5 × 10 -5
125
9.5 x 10 -ts
152
Misc. properties
Melting point [K]
Density [g/cm 3]
Atomic volume [m 3]
Surface energy, 7 [J/m2]
1941
4.42
1.78 × 10 -29
1.7
Aluminum [29]
Elastic properties§
Shear modulus as f(temp) [GPa]
--R.T.
--temperature dependence [Gpa/C]
Poisson ratio, v
25.4
5.0 x 10 -4
0.31
Plasticity
Yield strength as f(temp) [MPa]
--R.T.
----try(temp) ( T < 460°C)
41.5
try(T) = 43.4-0.094. T
Creep¶
Dorn constant, B [h -l]
Stress exponent, n
Burger's vector, b [m]
3.4 × 106
4.4
2.86 x I0 -l°
Diffusion
Pre-exp core diffusion, D~0 [m4/s]
Activation energy, Qc [k J/moll
Pre-exp boundary diffusion, 6Dbo [m3/s]
Activation energy, Qb [kJ/moll
7.0 x 10 -25
82
5 × 10 -14
84
tSee Ti-24Al-11Nb for data.
:~Values obtained using simple rule-of-mixtures applied to data for ~t-Ti and fl-Ti [58] assuming equal volume fractions of
ct and ft.
§Shear modulus calculated as
# =,Uo' [1 + (T-- 300"~• td~p]
where T and Tm are in Kelvin and ta~v denotes the temperature dependence in [GPa/K]. Young's modulus is then
determined from E = 2/~(1 + v).
¶The steady state creep rate is described by
Eb [ 2 ( I + v ) ]
1OacDc ~ t r
i¢=A.kT
(x/~),+l . D v I + W .
-• -b
where A is the Dorn constant, T the absolute temperature v Poisson's ratio, a~b~ the core diffusion coefficient, b the
burgers vector, D v the lattice diffusion coefficient, n the stress exponent, tr the stress and E is Young's modulus.
'~-t
[
"~2] ( a "y'
D,, ~,x,/~p,,]2 ~,,x/~E.,]
2316
ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE
Pre-exp volume diffusion, D~0 [m2/s]
Activation energy, Qv [kJ/mol]
Misc. properties
Melting point [K]
Density [g/cm2]
Atomic volume [m3]
Surface energy 3' [J/m 2]
1.7 X 10 -4
142
933
2.70
1.66 x 10-29
1.7