Mechanisms, Models, and Simulations of Metal-Coated Fiber
Consolidation
R. VANCHEESWARAN, J.M. KUNZE, D.M. ELZEY, and H.N.G. WADLEY
Recent experimental studies of the hot isostatic consolidation of Ti-6Al-4V–coated SiC fibers contained
in cylindrical canisters have revealed an unexpectedly high rate of creep densification. A creep
consolidation model has been developed to analyze its origin. The initial stage of consolidation has
been modeled using the results of contact analyses for perfectly plastic and power-law creeping
cylinders that contain an elastic ceramic core. Final stage densification was modeled using a creep
potential for a power-law material containing a dilute concentration of cusp-shaped voids with a shape
factor similar to that observed in the experiments. Creep rates were microstructure sensitive and so
the evolution of matrix grain size and the temperature dependence of the a/b-phase volume fractions
were introduced into the model using micromechanics-based creep constitutive relationships for the
matrix. To account for load shielding by the deformation resistant canister, the consolidation model
was combined with an analysis of the creep collapse of a fully dense pressure vessel. The predicted
densification rates were found to agree well with the experimental observations. The high densification
rate observed in experiments was the result of the small initial grain size of the vapor-deposited
matrix combined with retention of the cusp shape of the interfiber pores.
I. INTRODUCTION
CONTINUOUS fiber reinforced titanium matrix composites are being developed for a variety of applications in
gas turbine engines and other aerospace structures because
of their very high specific stiffness and strength in the fiber
direction.[1] For example, a Ti-6Al-4V/Sigma-1240 (SiC)
composite with a 44 pct SiC fiber volume fraction has an
axial Young’s modulus of more than 200 GPa, an ambient
temperature fracture strength of about 1800 MPa, and a 600
8C creep rupture life of several thousand hours at 1000
MPa.[2] One approach for manufacturing composites of this
type uses either a sputtering or electron-beam physical vapor
deposition (PVD) technique to evaporate and condense the
metal matrix on the fibers.[3,4] Composite components can
then be formed by aligning the coated fibers in a metal
canister that is subsequently evacuated, sealed, and consolidated at an elevated temperature by either hot isostatic pressing (HIP), vacuum hot pressing, or rolling.[4–7]
Recent HIP consolidation experiments using in situ density sensors reveal that densification of Ti-6Al-4V vaporcoated fibers occurs at a very high densification rate.[5] As
a result, complete densification can be accomplished at much
lower temperatures and pressures than is needed for similar
composition titanium alloy powders.[4,5] Two factors might
be responsible for this anomaly. First, recent studies of a
similar PVD Ti-6Al-4V alloy revealed a greatly enhanced
low pressure superplastic formability because of its fine (30
to 100 nm) grain size.[3] However, rapid grain coarsening
during elevated temperature consolidation can reduce the
R. VANCHEESWARAN, Graduate Student, is with the MBA Program,
Cornell University, Ithaca, NY 14853. J.M. KUNZE, Engineer, is with
Triton Systems Inc., Chelmsford, MA 01824. D.M. ELZEY, Research Assistant Professor, Department of Materials Science and Engineering, and
H.N.G. WADLEY, Edgar A. Starke Research Professor and Associate Dean
for Research, School of Engineering and Applied Science, are with the
University of Virginia, Charlottesville, VA 22903.
Manuscript submitted May 19, 1999.
METALLURGICAL AND MATERIALS TRANSACTIONS A
significance of this process. The second factor is the observation of large cusp-shaped pores that were created by the
metal-fiber contacts. Although these shrink in dimension,
they retain a cusp shape throughout the consolidation process.[6] Recent micromechanical analyses of the creep collapse of cusp-shaped pores has indicated that their collapse
rate can be as much as an order of magnitude greater than
that of their spherical pore counterparts.[8]
These observations are of significant consequence to the
design of composite fabrication processes. High performance can only be achieved with this class of composites
when complete densification of the coated fibers is accomplished while simultaneously avoiding significant bending
of fibers (a result of fiber-fiber crossovers usually present
in a random packing of aligned fibers[4]) and excess growth
of reaction product layers at the fiber-matrix interface.[9]
These three requirements have conflicting dependencies
upon the variables of the consolidation process (i.e., pressure, temperature, and time). For example, a lower temperature, high-pressure cycle designed to limit fiber-matrix
reactions can bend/fracture fibers, while a prolonged elevated temperature low-consolidation pressure cycle fully
densifies the composite without bending or fracturing the
fibers but provides ample opportunity for fiber-matrix reactions. If, however, the enhanced (fine grain size) superplastic
flow and rapid (cusp-shaped) void collapse were responsible
for accelerated densification in these systems, they might
provide an opportunity to design a process schedule that
shortened the time at temperature and reduced the pressure
for complete consolidation.
Here, we seek to unravel the complex interdependencies
between the microstructural states of importance and the
controllable variables of the process (T(t), P(t)) using a
model-based approach. Models for simulating microstructural evolution have already played an important role in
deducing the mechanisms and designing processes for densifying metal, alloy, and ceramic powders[10] as well as metal
VOLUME 31A, APRIL 2000—1271
matrix composite monotapes.[11–14] These approaches adopt
a micromechanics method for model development that facilitates the incorporation of process path dependent properties
of the constituents in the model. The models can then be
tested against experiments to evaluate the relative contributions of the various mechanisms to consolidation. Validated
process models of this type have been used to predict (and
optimize) the density-pressure-temperature-time relation for
consolidation of material systems for which the properties
of the constituent components are known (or can be
estimated).[11,16]
Such a micromechanics-based model is developed here
and is used to predict the time-dependent evolution of the
composite’s relative density given an applied pressure and
temperature schedule, the volume fraction of fiber, and the
initial matrix grain size. It incorporates the void’s cusp shape
and the time-temperature-dependent growth of both the
matrix grain size and the a/b-phase volume fraction. A
titanium alloy canister used to encapsulate the fibers was
found to support a significant fraction of the applied load.
This partially shielded the coated fibers from the pressure,
thereby retarding the consolidation rate.[17] The consolidation model has, therefore, been embedded in a pressure
vessel model to account for the shielding effect. The overall
model was compared with experimental data and confirmed
the earlier notion[5] that the accelerated rate of densification
resulted from the very fine grain size matrix and retention
of the cusp shape of the pores.
II. PROCESS-STRUCTURE MODEL
The densification of metal-coated fibers with diameters
of 100 to 200 mm has many similarities to the consolidation
of alloy powders[18,19] and plasma-sprayed monotapes[12,13,14]
for which models describing the evolution of relative density
have been developed and experimentally verified.[8]
A schematic diagram of a coated fiber bundle contained
in a cylindrical canister is shown in Figure 1(a). Using the
approach for powder consolidation, several approximations
are used to simplify the model; it is assumed that (1) the
fibers do not rearrange their packing during densification,[20]
(2) no deformation occurs along the fiber axis (so deformation can be treated as plane strain) (3) all fibers are perfectly
aligned, and (4) the ceramic fibers remain elastic at the
temperatures and stresses used for consolidation. The transformation from a fiber-fiber contact to a pore collapse mode
of densification as the relative density increases is treated
by assuming densification to occur in two steps.[12] Stage I
involves localized deformations at interparticle contacts. A
recent contact analysis for coated fibers can be used to
obtain a density-dependent contact flow stress-displacement
relation for this problem (Figure 1(b)).[15,21] A simple law
for the evolution of the number of fiber contacts (i.e., the
coordination number) can then be used to link the contact
stress-displacement relation to the applied pressure.[5] At
high relative densities (D . 0.9), as many as six contacts
form around each fiber, and it is then reasonable to view
the body as a reinforced solid containing isolated cuspshaped voids. A stage II model addresses the collapse of
these voids using a creep potential for a power-law creeping
body containing a dilute distribution of two-dimensional
cusp-shaped voids,[22] as in Figure 1(c). An interpolation
1272—VOLUME 31A, APRIL 2000
Fig. 1—(a) A cross section of the HIP can containing an aligned randomly
packed coated fiber bundle. (b) Representative volume element for stage
I densification. (c) Stage II densification representative volume element.
The fibers were 150 mm in diameter.
function smoothly transitions from stage I to II behavior as
the relative density is increased from 0.8 to 0.9.
Warren et al. have examined and modeled the creep
response of vapor-deposited Ti-6Al-4V.[3] They found the
stress-strain rate response in the range of stresses encountered in consolidation to be highly dependent upon grain
size. They also found that it could be well modeled by the
METALLURGICAL AND MATERIALS TRANSACTIONS A
Ashby–Verrall relation, which combines diffusion-accommodated grain sliding (DAGS) and power-law creep (PLC)
deformation mechanisms.[23] The mechanical properties of
the nanocrystalline PVD matrix[3] and their effect on the
densification process can, therefore, be captured in the densification analysis by using the Ashby–Verrall model for
matrix creep. Because the grain size is a sensitive function
of the temperature-time path experienced during consolidation, a grain growth law is introduced into the formulation.[3]
A. Densification
Metal matrix composites are consolidated over a range of
temperatures, 0.4 # T/Tm # 0.8 (where T/Tm is the absolute
temperature normalized by the melting point of the matrix),
and applied pressures, 0 , P/sy # 0.8 (where P/sy is the
ratio of the applied pressure to the uniaxial yield strength
of the metallic matrix). Depending on the consolidation conditions used, several deformation mechanisms may contribute to the overall densification rate at any one time.[3] These
include time-independent plastic flow (yielding), PLC, and
DAGS. If these three mechanisms are incorporated into both
the stage I (contact deformation) and stage II models, a total
of six representative volume element models need to be
constructed.[3] Sintering mechanisms such as surface and
volume diffusion have not been included because their contributions at the temperatures and times used to consolidate
composites are anticipated to be relatively insignificant.[23]
The overall density can be obtained by summing the deformation mechanism contributions:
t
D(t) 5 D0 1 DDp (t) 1
e 1 o s (D˙ (t) 1 D˙ (t))2 dt
2
j
0
c
d
[1]
j51
where D0 is the starting relative density; DD
˙ p is the
˙ change
in relative density due to plastic yielding; Dc and Dd are the
densification rates due to PLC and DAGS, respectively; and
sj are smoothing functions used to interpolate between stages
I and II.[11]
1. Stage I: Interfiber contact deformation
Consider a random dense packing of perfectly aligned
fibers subjected to a hydrostatic pressure, P. The rigid fibers
fully support the load in the axial direction, and consolidation, therefore, occurs by the radial and circumferential displacement of matrix material at interparticle contacts. This
allows the particle centers to approach each other and densification to occur. A representative contacting particle pair
(Figure 1(b)) can be isolated and a local stress-strain relation
identified for it. A relation is then found between the applied,
or macroscopic stress, and the local average contact stress.
The center-to-center displacement of the contacting pair is
then related to the macroscopic strain or densification. The
assumptions made in this model are that (1) the deformation
geometry and loading of a single contact at the microscopic
level can be described by a square array of coated fibers
subjected to constrained uniaxial compression (Figure 1(b));
and (2) the macroscopic densification rate is the product
of this square array contact model and an experimentally
determined empirical relation for the evolution of the coordination number as a function of density (Eq. [7]).
The deformation response of a representative contact can
be described using two parameters, a flow coefficient, F,
METALLURGICAL AND MATERIALS TRANSACTIONS A
representing the contact’s resistance to flow, and an area
coefficient, c, describing the evolution of the contact’s
area.[15,21] The contact’s resistance to flow has been analyzed
using finite-element analysis and shown to be a function of
contact area, the volume fraction of fiber, and the matrix
constitutive behavior.[15,21] For a power-law creeping matrix,
strain rate
the average contact stress, sc , ˙and the effective
˙
at the contact (defined as «˙ c 5h/a, where h is the displacement rate of the contact and a is the contact semiwidth) can
be related by an expression of the form[15]
1/n
sc
«˙ c
5 F(a/r, vf , n) ˙
[2]
s0
«0
12
where F(a/r, vf , n) is the flow coefficient (expressed as a
function of the ratio of contact semiwidth to initial fiber
radius, a/r, the fiber volume fraction, vf , and the matrix creep
exponent, n); and s0 and «˙ 0 are the reference stress and
strain rate, respectively. The flow coefficient, F, has been
calculated as a function of contact strain (a/r) and fiber
volume fraction for a wide range of n values.[15] Note that
Eq. [2] describes plastic yielding as n approaches infinity.
The contact area can be described by a coefficient, c(n,
vf), which relates the contact displacement, h, to the contact
semiwidth, a:[15]
a
5 c(n, vf)
r0
!2 hr
[3]
A contact area coefficient of unity corresponds to a contact
width equal to the length of a chord created by truncating
a circle (representing the cross section of a coated fiber) a
distance, h, from the chord to the outer perimeter of the
circle. The equation c(n, vf) . 1 is indicative of material
piling up at the periphery of a contact, whereas c(n, vf) , 1
indicates material sinking in at the contact.[15] Finite-element
analyses have shown that for metal-coated fibers (MCFs)
with a fiber volume fraction of 50 pct, c has a value of about
1.42 for all n. The value of the contact area coefficient
has been shown to be relatively insensitive to the normal
displacement and, therefore, a/r0 (i.e., the size of the
contact).[15]
The contact pressure sc in Eq. [2] is the ratio of the force
supported by a contact ( fc) and the contact area (ac):
sc 5
fc
ac
[4]
The externally applied pressure is propagated in a random
packing of coated fibers as a network of forces acting through
interparticle contacts. An expression for the average force
acting on a contact for a random isotropic packing of coated
fibers can be related to the applied pressure by
fc
2p
5
P
ro L ZD
[5]
where ro is the initial coated fiber radius; P is the externally
applied hydrostatic stress; Z is the number of contacts per
particle at a given density, D; and L is the length of the fiber
bundle. This equation evokes a simple interpretation; the
average contact force is the compaction pressure times the
average contact area, which is approximated by the cylinder
surface divided by ZD. The force balance given by Eq. [5]
is analogous to a result obtained by Molerus[24] for the static
VOLUME 31A, APRIL 2000—1273
loading of a three-dimensional array of spherical powder
particles. Using Eq. [5] to eliminate fc in Eq. [4] and substituting Eq. [2] for sc , gives a macroscopic relationship between
density, strain rate, and the normal applied stress provided
the contact area, ac , is known.
n
2n
1
ac
«˙ 2p
«˙ c 5 0n
Pn
[6]
n
s 0 ZD
rL
a
F , vf , n
r
1 2
1
2
1 2
An expression for the average contact area (ac ,) is developed next.
The total contact area per fiber increases by the creation
of new contacts and the deformation of existing contacts as
the coated fiber bundle shrinks. The creation of contacts
can be characterized using a contact coordination number.
Experiments have shown that the coordination number (the
number of contacts per fiber) increases approximately linearly with relative density, D:[5]
Z 5 Z 8D 1 C1
[7]
where C1 5 25.3, and Z 8 5 10.8. Experiments have shown
that the average initial number of contacts per fiber is 2.5.[5]
A relation has been developed that relates the relative density
to the contact displacement. The growth in the area of the
contacts is related to the contact displacement, h, by Eq. [3],
which in turn depends on the relative density, D. Applying
conservation of mass to the single contact unit cell enables
an approximate relation between the relative density D and
the uniaxial displacement (h) to be obtained:
h
D
512 0
r0
D
[8]
where, as before, r0 is the initial coated fiber radius, and D0
is the initial relative density. The total contact area per fiber
Ac(D) is found by combining Eq. [8] with Eqs. [3] and [7]:
F
Ac(D)
5 (!2c(n)) Z0
r0 L
!
D
12 0
D
e!1 2 DD dDG
D
1 Z8
[9]
0
D0
The average contact area, ac , at any density is then
ac 5
Ac(D)
Z(D)
[10]
When Eq. [10] is substituted in Eq. [6], one can find an
expression that relates the applied pressure to the strain rate
«˙ c. What remains is˙ to relate the strain rate «˙ c to the overall
densification rate D. Differentiating Eq. [8] with respect to
time gives a relation for
˙ the densification rate in terms of
the displacement rate, h:
˙
˙
D
h
D5
[11]
r0
h
12
r0
˙
During consolidation, h and h are determined by the applied
pressure (which must be supported by the contacts) and by
the type of inelastic deformation mechanisms active at the
contacts. The densification rate can be written as a function
1
1274—VOLUME 31A, APRIL 2000
2
FG
of the radial strain rate using Eq. [11] and the average contact
radius a/r0, as follows:
1 2F
G
1
D2 a ˙
(«c)
[12]
c (n, vf) D0 r0
The overall radial strain rate («˙ c) is taken to be the sum of
the strain rates due to the different individual mechanisms
such as DAGS and PLC, which are discussed in Section 3.
˙
D5
2
2. Stage II densification
When the relative density exceeds about 0.9, the behavior
of the compact is more accurately described as a continuum
containing isolated, cusp-shaped voids that collapse by
matrix plasticity and/or PLC. When the voids are small
compared to the rigid fibers, a representative volume element
containing a single cusp-shaped void can be used to analyze
the densification rate (Figure 1(c)) Qian et al. have developed
a strain-rate potential for a creeping body containing aligned
isolated cusp-shaped voids subjected to in-plane states of
stress.[22] They have shown that bodies containing cuspshaped voids densify at much higher rates than similar density bodies containing an equal volume fraction of the spherical voids that have typically been used to analyze powder
and monotape densification.[25]
The Qian et al. potential is based on an analysis of a twodimensional cusp-shaped void in a monolithic matrix. For
a transversely isotropic, power-law creeping body containing
cylindrical voids, the plane strain* dilatational strain rate,
*The strain rate in the direction of the cylindrical voids is taken to be
zero; «˙ 33 5 0.
«˙ kk, is related to the mean stress, sm , by
1/n
«˙ kk
sm
3
5
b2(n11)/2n ˙
s
2
«
0
12
1/n
1 2
[13]
0
where for isostatic loading, sm 5 P (the applied pressure),
and l and b are density dependent functions given by
b5
3l
1 2 Dr
D(12n)/(11n), l 5
1 2 sl
s 1 Dr
[14]
The coefficients r and s depend on the pore shape.[22] Assuming four-pointed, cusp-shaped voids with a curvature of
1/3, r and s have been found to be 5.38 and 1.69, respectively.
Once the macroscopic strain rate, «˙ kk, has been determined
using the strain rate
˙ potential, the densification rate is simply
computed from D 5 2 «˙ kkD.
The stage II densification response for a matrix that
deforms plastically can again be obtained by taking the limit
as n approaches infinity. Thus, for the perfectly plastic case,
Eq. [13] simplifies to
sm 5
1
s
bp1/2 y
[15]
where sy is the uniaxial yield strength of a rigid, perfectly
3l
1
, and l, r, and s are
plastic material, bp(D) 5
1 2 sl D
defined previously. Because bp(D) is the only density-dependent variable, the densification due to plasticity may be
obtained by directly solving Eq. [15] for the density.
This stage II analysis only considers the presence of voids,
it does not account for the enhanced resistance of the matrix
F
G
METALLURGICAL AND MATERIALS TRANSACTIONS A
Table I. Values of the Fiber Restraint Factor, R90
Creep Exponent, n
Volume Fraction of
Fiber
1
2
5
10
25 pct
50 pct
1.6
8.0
1.5
3.5
1.2
1.8
1.1
1.4
to flow due to the presence of the fibers. At the start of
stage II densification, the voids are large and the strain-rate
fields of the matrix have been shown to interact strongly
with the fiber especially when the creep exponent is low
(n → 1).[15] However, the effect of the fiber diminishes as
the relative density approaches unity.
To approximate the fiber strengthening during the beginning of stage II, and to force continuity in the matrix flow
resistance predicted using the finite element analysis and
the Qian et al. (stage II) potential (at a transition density,
D 5 0.9), a density and volume fraction–dependent fiber
restraint parameter, RD(vf), is used. The flow resistance of
the two models was computed at a transition relative density
(D 5 0.9) for different values of creep exponent (n) and
volume fraction (vf). A fiber restraint parameter R90(vf)
(which is the ratio of the flow resistance in the stage II
model to the flow resistance in the stage I model) was then
computed and is presented in Table I. It is assumed that the
influence of the fiber on densification reduces as consolidation progresses and disappears in the limit as D → 1 to
recover Qian et al.’s original model result.
The exponentially decaying smoothing function was chosen to retard the flow resistance of the matrix as the density
increased and so has the form
R(D, vf) 5 R90(vf) 1 1.8 3 10235e80D[1 2 R90(vf)] [16]
The effect of fiber restraint was incorporated into the stage
II densification model by multiplying the right-hand side of
Eq. [13] by R(D,vf). A plot showing the matrix flow resistance per contact as a function of relative density for different
volume fractions of fiber is shown in Figure 2. The curves
in this figure show the approximate relation of flow resistance of the matrix as a function of relative density as determined from the stage I model, Eq. [2], and the stage II
Fig. 2—A plot of the matrix flow resistance per contact as a function of
relative density showing the effect of the fiber volume fraction. The Qian
et al.[22] model result has been multiplied by an exponentially decreasing
fiber restraint factor R(D, nf), which incorporates the fiber’s effect on matrix
flow as the sample densities.
model, Eq. [13], multiplied by the restraint factor for fiber
volume fractions of 25 and 50 pct, respectively. As the
density is increased, the fiber restraint disappears and Qian
et al.’s original result is recovered at full relative density.
3. Matrix deformation mechanisms
At a low temperature (T ,500 8C), the dominant densification mechanism of Ti-6Al-4V alloys is plastic yielding. The
criterion for plastic deformation of contacts (Eq. [2]) can
be written as
sc $ Fp
1r , v 2s
a
f
y
[17]
0
where sc is the average contact stress; sy is the uniaxial
yield strength of the matrix; and Fp is the plastic flow coefficient, which is dependent on both the contact size (as given
by the semiwidth of the contact to the fiber radius a/r0) and
the volume fraction (vf) of the reinforcing phase. This flow
Table II. Material Properties of PVD Ti-6Al-4V[3]
Material Parameter
Burger’s vector, b (m)
Atomic volume, V (m3)
Grain boundary width, d (m)
Grain boundary energy, G (J/m2)
Shear modulus at room temperature, mo (MPa)
Lattice diffusion pre-exponential factor, Dov (m2/s)
Lattice diffusion activation energy, Qv (KJ/mol)
Grain boundary diffusion pre-exponential factor, Dogb (m2/s)
Grain boundary diffusion activation energy, Qgb (KJ/mol)
PLC exponent, n
PLC constant, A
Bulk tensile yield strength at room temperature, sy (MPa)*
Temperature-dependent yield strength, sy (MPa)
Grain boundary activation energy, Qm (KJ/mol)
Grain boundary mobility pre-exponential factor, Dom (m2/s)
a Phase
3 3 10210
1.7 3 10229
6 3 10210
0.35
4.35 3 10210
6.6 3 1029
169
1.3 3 1028
101
4.85
3.6 3 109
884
884 to 0.92T
122
1.2 3 10228
b Phase
3 3 10210
1.7 3 10229
6 3 10210
0.35
2.05 3 10210
4.5 3 1028
131
1.3 3 1027
77
3.78
1.2 3 106
884
884 to 0.92T
122
1.2 3 10228
* Values shown for bulk Ti-6Al-4V.
METALLURGICAL AND MATERIALS TRANSACTIONS A
VOLUME 31A, APRIL 2000—1275
coefficient and its dependence on the normalized contact
semiwidth and the fiber volume fraction has been determined
using finite-element analysis.[21]
To implement the plasticity model, one can rewrite the
contact stress using Eq. [4] in terms of a contact force ( fc)
and a contact area (ac). By substituting expressions for the
contact force and contact area using Eqs. [5] and [10], one
can obtain an equation that relates the applied pressure to
the relative density:
P
5
sy
where Fp
DFp
1r , v 2A (D)
a
f
c
0
[18]
2p
1r , v 2 is found from finite-element analysis for
a
f
0
a work hardening matrix,[21] and Ac(D) is an expression for
the evolution of the total contact area and is presented in
Eq. [9]. A Gauss–Newton root-finding procedure is used to
find the value of D that satisfies Eq. [18]. If this density is
greater than the current integrated density (due to PLC and
DAGS), then plasticity is assumed to have occurred, and
the solution to Eq. [18] is used to obtain the new density.
As the temperature is increased above ,500 8C, PLC
increasingly dominates the deformation response. Under uniaxial loading, the observed behavior is best described by a
power-law relationship of the form[23]
n
mb s
«˙ PLC 5 A
Dv
kT m
12
[19]
where m is the temperature-dependent shear modulus, b is
the Burger’s vector, A is the Dorn constant, and n is the
creep exponent. Between 720 8C and 880 8C, superplastic
flow occurs, and the superplastic behavior of PVD Ti-6Al4V is well described by a combination of DAGS and PLC.[3]
The uniaxial stress-strain rate relation for deformation due
to grain sliding is[26]
3.38dDgb
100VDv
0.72G
«˙ DAGS 5
s2
11
2
kTd
d
dDv
1
21
2
[20]
where T is the temperature; d is the grain size; G is the grain
boundary energy; d is the grain boundary width; V is the
atomic volume; Dv and Dgb are the volume and grain boundary diffusivities, respectively; and k is the Boltzmann’s constant. Note that the strain rate (and, consequently, the
densification rate) associated with the DAGS mechanism is
inversely proportional to the square of the grain size. As
0.72G
long as the s 2
term in Eq. [1] remains positive,
d
the influence of grain size on densification rate will be
strong. Matrix materials with a small grain size can, therefore, deform at a very high strain rate with significant consequences for the overall densification rate.
Material parameters in the creep rate expressions are different for the a and b phases of Ti-6Al-4V, and the overall
creep response, therefore, also depends upon their volume
fractions. The volume fractions of the constituent phases in
the Ti-6Al-4V matrix (a 1 b) are temperature dependent,
and this dependence has been quantified in Reference 3.
The effective response of the two-phase (a 1 b) Ti-6Al4V matrix was, therefore, estimated (for a given constitutive
1
2
1276—VOLUME 31A, APRIL 2000
behavior) using an isostrain-rate assumption[3] for the DAGS
and PLC mechanisms:
«˙ 5 «˙ 5 «˙ 5 («˙ 1 «˙ ) 5 («˙ 1 «˙ )
[21]
a
b
d
c a
c b
d
This isostrain rate condition was combined with a ruleof-mixtures expression to compute the stress supported by
each phase during densification. The preceding constitutive
responses were then used to obtain the strain rate in each
phase. These strain rates were then substituted in Eq. [12]
to yield the densification rate. Substituting Eq. [12] and the
solution for Eq. [18] into Eq. [1] gives an expression for
the temporal evolution of relative density.
B. Grain Growth
The superplastic constitutive response of the matrix
depends strongly on grain size (l/d 2). The grain size of PVD
Ti-6Al-4V has been shown to change by a factor of 103
during a typical consolidation cycle.[3] It is, therefore,
important to include the effects of grain growth into the
constitutive response. Experimental measurements of singlephase materials indicate that grain size dependence upon
time are well represented by a parabolic law,[27] whereas
the temperature dependence is exponential as shown in the
following equation:
1
d(t)2 2 d 20 5 tC exp 2
2
Qm
RT
[22]
where d(t) is the average grain diameter after annealing at
a temperature T for time t, d0 is the initial grain size, Qm is
the effective activation energy for grain boundary mobility,
and R is the gas constant. In two-phase systems such as that
encountered in Ti-6Al-4V below the b transus, the “other
phase” slows grain growth resulting in a grain-growth exponent that is less than the single-phase value of 0.5.[28] Warren
et al.[3] measured the grain size of PVD Ti-6Al-4V after
various annealing treatments and found that the grain diameter-time relation had an exponent of 0.2, i.e., d } t0.2. We,
therefore, model grain growth using
d 5 d0 1 (KMb(T )t) p
where p 5 0.2, K 5 1,
mobility[29] given by
Mb(T ) 5
[28]
[23]
and Mb(T ) is a grain boundary
F G
V2/3 Dom
Q
exp 2 m
kT
RT
[24]
In this expression, V is the atomic volume, D0m is a preexponential grain-boundary mobility, and k is the Boltzmann
constant. Differentiating Eq. [23] with respect to t gives the
grain growth rate as
d5
F G2
Q
K V2/3 Dom
exp 2 m
5d 4
kT
RT
1
[25]
This phenomenological model compares well with Warren
et al.’s experimentally measured grain-growth kinetics.[3]
This model can be integrated forward using a Runge–Kutta–
Fehlberg (RKF) integration scheme in conjunction with the
densification rate mechanisms. The time evolving grain size
is then used as an input in the DAGS contribution to
densification.
METALLURGICAL AND MATERIALS TRANSACTIONS A
C. Can Shielding
During consolidation, the MCFs were contained in a cylindrical, thin-walled, coarse-grained pure CP titanium canister.[5] At low pressures and temperatures, this canister acts
as a pressure vessel and is able to support a significant
fraction of the applied pressure, thereby reducing that acting
on the fibers.[17,30,31,32] The largest shielding occurs in the
axial direction, but densification cannot occur in this direction due to the presence of rigid fibers. Significant radial
pressure shielding also occurs under some consolidation conditions. As a result, the effective in-plane radial stress
sMCF
actually applied to the fibers will be less than the
rr
applied pressure (P). The calculation of shielding is complicated because it depends on the compliance of the consolidating material.[32]
We note that the effective in-plane radial stress sMCF
for
rr
the two-dimensional problem shown in Figure 1(a) can be
deduced by first noting that the inner surface of the pressure
vessel is supported by an unknown force per unit length
times the inner circumference of the vessel.
given by sMCF
rr
Assuming the can and its contents are in equilibrium, one
can find an expression for the hoop stress in the can in terms
of its inner radial stress and the applied pressure:
Fe
p
1
s 5
2d
C
uu
G
˙
(P 2 sMCF
rr ) (R sin u) du
0
(P 2 sMCF
rr ) R
d
[27]
1 2
where «˙ 0 and s0 are the reference strain rate and stress, n
is Norton’s creep exponent, and se is the effective stress
3
S S , where Sij is the stress deviator, Sij 5
(i.e., se 5
2 ij ij
1
sij 2 skk dij). The hoop strain rate in the can is obtained
3
by differentiation of Eq. [28] with respect to the hoop stress:
!
n21
1 2
Suu
s0
[29]
After substituting expressions for se 5 s2rr 1 s2uu 2
srrsuu, Suu and Eq. [29] in Eq. [27], the hoop strain rate of
the container is obtained as
METALLURGICAL AND MATERIALS TRANSACTIONS A
Value
15.9
1.6
4.3
7.7 3 104
242
1.3 3 1023
1933
2.95 3 10210
F
1
R R2
C
5 «˙ MCF
5 B 12 1 2
«˙ uu
uu
2
d d
1
G
2
[30]
R
n
2 2 1 (P 2 sMCF
rr )
d
where B 5 («˙ 0/sn0). Expressions for the strain rate of the
MCFs are given in Eq. [2] and Eq. [13], and because the
can is axially symmetric, equating the strain rate of the
MCFs to Eq. [30] will give a nonlinear equation that can
be iteratively solved for sMCF
rr .
III. SIMULATION METHOD
If the can is well bonded to the composite, the hoop strain
C
5
rate of the can and the MCFs must be equal, i.e., «˙ uu
˙«MCF, and this compatibility condition can be used to find
uu
the radial stress transmitted to the MCFs, sMCF
rr .
The vessel is assumed to deform by creep, which is
described using the multiaxial stress potential for a powerlaw creeping material:
n11
«˙ s se
[28]
F(s) 5 0 0
n 1 1 s0
s
­F
3
C
5
5 «˙ 0 e
«˙ uu
­suu 2
s0
Geometric or Material Parameter
Initial outer diameter of the can (mm)
Initial wall thickness of the can (mm)
PLC exponent, n
PLC constant, B
Creep activation energy, Qcr (KJ/mol)
Creep pre-exponential factor, Dcr (m2/s)
Melting point (8C)
Burger’s vector, b (m)
[26]
where R and d are the radius and thickness of the vessel,
respectively; and sMCF
is the radial stress. This radial stress
rr
is the stress that is actually applied to the MCFs. Assuming
the can wall to be thin (i.e., d /R , 1/5), the radial stress
can be taken as constant across the thickness of the can (i.e.,
suu5suu (r)). Integration of Eq. [26] then gives
C
s uu
5
Table III. Material and Geometric Properties of the CPTi Can[3]
A simulation algorithm was programmed on a SUN
SPARC-20* using MATLAB.** The simulation modeled
*SUN SPARC-20 is a trademark of Sun Microsystems, Inc.
**MATLAB is a trademark of The Math Works, Inc.
the temporal evolution of two microstructural states, the
density, and the grain size. An RKF adaptive step size integration routine was used to integrate the rate-dependent contributions of relative density (which are PLC and DAGS)
and grain size. Plasticity contributions to densification for
each stage were calculated at each time-step before the RKF
integrator was executed. To compute the densification contribution by plasticity (Eq. [18]), a Gauss–Newton zero-finding
routine was used.
Because Ti-6Al-4V is a two-phase material, and the volume fractions of the constituent phases have been found to
be temperature dependent,[3] an iso-strain-rate condition (Eq.
[21]) has been assumed for both phases. A rule of mixtures
was used to partition the applied stress between each phase.
To find the stress in each phase, a Gauss–Newton zerofinding routine was used to satisfy Eq. [21] during both
stages of densification. To compute the effect of can
shielding, the strain rate of the can and the coated fibers
was assumed to be equal (Eq. [30]), and, therefore, a Gauss–
Newton routine was used to compute the stress transmitted
to the fibers. Consequently, Eqs. [21] and [30] were solved
using a nonlinear least-squares algorithm, which solves a
vector of nonlinear equations simultaneously. The RKF routine adaptively varied the integration time-step for the densification and grain growth models to maintain the infinity
norm distance between the fourth- and fifth-order approximations below 1026.
The material properties of the matrix coating (PVD Ti6Al-4V) are presented in Table I. The material and geometric
properties of the can are given in Table III.
VOLUME 31A, APRIL 2000—1277
IV. COMPARISONS WITH EXPERIMENTS
Simulations of the five HIP experiments reported in Reference 5 have been conducted. The volume fractions of fiber
were determined by quantitative image analysis of metallographically prepared specimens (Table IV) and used as inputs
to the model. A linear interpolation routine was used to find
the flow coefficients (F and Fp), and the restraint factor
(R) for the fiber volume fractions and creep exponents for
situations that were not explicitly determined by the finiteelement analyses.[15] Simulations were stopped either when
a density of 0.999 was reached or when the elapsed time of
the experimental process cycle was reached.
The process schedule of experiment A used for the simulation raised the temperature to 900 8C with an initial pressure
of 2 MPa. The pressure was then ramped from 2 to 20 MPa
in 30 minutes, held at 20 MPa for 30 minutes, and then
finally ramped to a pressure of 100 MPa at a rate of 0.92
MPa/min (solid line), as in Figure 3(a). The dashed line in
this figure shows the pressure exerted by the inner wall of the
canister after can shielding was accounted for. The internal
pressure and the temperature paths governed the temporal
evolution of the density and grain size during consolidation.
It can be seen that during the initial temperature ramp-up
phase, a pressure of 2 MPa was applied, but because of the
can’s high flow resistance, it effectively transmitted a very
small stress to the fibers. As the pressure was subsequently
increased, the can began to collapse and the internal pressure
then approached the applied pressure. Figure 3(b) presents
a comparison between the experimentally measured density
profile (the dashed line) and the model prediction (the solid
line). The small initial grain size enabled the coated fiber
bundle to densify by Ashby–Verrall creep (DAGS) as the
temperature was ramped to 900 8C even though the internal
consolidation pressure was very low. At the start of the
pressure ramp, the grains had evolved in size to 1.3 mm,
which is still small enough that the predominant mechanism
for densification is Ashby–Verrall creep. Thus, during the
early stages of pressure ramping, a large increase in creep
densification rate occurred. The densification rate quickly
slowed even though the pressure was continuously increased.
This was because of the combined effects of an increase in
grain size (Figure 3(c)) and a decrease in contact stress due
to an increase in contact area and relative density. The model
indicates that exploitation of the grain-size-enhanced densification rates is possible if the integrated thermal exposure
Fig. 3—(a) through (c) A comparison of the simulated and experimentally
measured densification evolution for experiment A (900 8C, 20 MPa for
30 min followed by 100 MPa for 1 h) and the predicted grain size evolution.
of the composite is reduced by increasing the temperature
ramp-rate capability of the equipment.
The simulation slightly underpredicted the densification
rate during stage I densification, while during stage II densification, the simulated density profile did not harden as
rapidly as the experiment and so mildly overpredicted the
densification rate. During stage II, a four-point cusp-shaped
pore with a curvature of 1/3 was assumed to be the most
dominant void configuration. In reality, voids were present
that had three to seven points, and the assumption of the
average pore shape to have four points could have contributed to the discrepancy.
To investigate the accuracy of the process simulation for
a range of holding (plateau) temperatures, two simulations
were performed using the process schedules that were used
in experiment B (holding temperature 5 840 8C, Figure 4)
Table IV. Fiber Volume Fraction, Densities, and Process Schedules of Hipping Experiments[5]
Experiment
Variable
Material parameters
Fiber volume fraction, vf
Initial density, D0
Final density, Df
Initial grain size, d (mm)
Process inputs
Temperature, 8C
First pressure hold, MPa
Time at first pressure hold, min
Second pressure hold, MPa
Time at second pressure hold, min
1278—VOLUME 31A, APRIL 2000
A
B
C
D
E
0.449
0.725
1.000
0.1
0.494
0.699
1.000
0.1
0.483
0.716
0.999
0.1
0.468
0.711
0.931
0.1
0.468
0.697
0.893
0.1
900
20
30
100
60
840
20
30
100
60
760
20
30
100
60
840
10
60
—
—
840
5
360
—
—
METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 4—(a) through (c) A comparison of the simulated and experimentally
measured densification evolution for experiment B (840 8C, 20 MPa for
30 min followed by 100 MPa for 1 h) and the predicted grain size evolution.
and experiment C (holding temperature 5 760 8C, Figure
5). Given the error in the experimental data and the uncertainty in material property data, the simulation predictions
agreed quite well with the experimental data.
Fig. 5—(a) through (c) A comparison of the simulated and experimentally
measured densification evolution for experiment C (760 8C, 20 MPa for
30 min followed by 100 MPa for 1 h) and the predicted grain size evolution.
METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 6—(a) through (c) A comparison of the simulated and experimentally
measured densification evolution for experiment D (840 8C, 10 MPa for
1 h) and the predicted grain size evolution.
The effect of holding pressure on consolidation was investigated by examining two experiments conducted with a
holding temperature of 840 8C but with holding pressures
of 10 MPa (experiment D) and 5 MPa (experiment E),
respectively. Figure 6 compares the simulation predictions
with the results of experiment D. In this simulation, after
raising the temperature to its soak value, the pressure was
ramped to 10 MPa and held for 60 minutes. The simulation
predicted a slightly higher densification than measured during the temperature ramp. Excellent agreement was observed
as the pressure was ramped at a rate of 4.92 MPa/min.
During this pressure ramp, the very high densification rate
was observed because of the relatively low grain size (1.1
mm) and the high internal pressure. The experimental and
simulated density saturated at almost identical values for
this test.
For experiment E, the pressure was increased (at a high
rate of 4.92 MPa/min) until a pressure of 5 MPa was reached
and was then held constant for 6 hours (Figure 7). In this
case, the simulation overpredicted the densification rate during stage I and resulted in a predicted density consistently
higher than that experimentally observed. For this low stress
cycle, the canister supported a large fraction (about 25 pct)
of the stress. As the simulation proceeded, the two profiles
became parallel, indicating comparable densification rates
but at different values of relative density. Uncertainty in the
can’s constitutive response and the simplifying assumption
used in the analysis of the can’s collapse could have
accounted for the overprediction. In particular, the canshielding model neglects elastic deformation, thus, allowing
a higher internal pressure to be applied in this simulation.
The prominence of this elastic effect would be greatest at
low applied pressures.
VOLUME 31A, APRIL 2000—1279
(a)
(b)
Fig. 8—(a) Process cycle and (b) relative density evolution showing the
effect of pore shape.
Fig. 7—(a) through (c) A comparison of the simulated and experimentally
measured densification evolution for experiment E (840 8C, 5 MPa for 6
h) and the predicted grain size evolution.
V. DISCUSSION
The comparison of simulated results against those of
experiments conducted over a broad range of temperatures
and pressures indicates reasonably good agreement between
the model and experiments. It suggests that all the dominant
micromechanical and thermophysical effects have been
incorporated in the model. The simulation methodology
therefore can be used to investigate the dynamic response
of metal-coated fibers during their transient consolidation
and to explore the (often) complex inter-relationships
between the process path, the geometry of both the fibers
and the can, fiber/matrix system properties, and the resulting
evolution of the microstructural state of the composite. A
subset of these inter-relationships is investigated systematically by examining the effects of the pore shape, the effects
of initial grain size, and the prominence of can shielding on
the evolution of density.
A. Pore Shape Effect
The effect of the pore shape during the final stages of
densification (stage II) was first noticed experimentally by
Liu et al. when they compared the densification rates of
partially consolidated powder compacts that had been heat
treated to round out the pores with other compacts that
retained cusp-shaped pores.[8] Kunze et al. in their work
with the densification of metal-coated fibers also suggested
that the shape of the voids between the aligned metal-coated
fibers would increase the densification rate.[5] The effect of
pore shape has been modeled by Qian et al.[22] This model
reported results for a four-sided cusp-shaped pore where the
curvature of the cusps is a variable parameter that is used
1280—VOLUME 31A, APRIL 2000
to simulate the effect of the pore shape on the densification
response. To study the effect of pore shape on densification,
we performed two simulations: one with cusp-shaped pores,
which used a cusp curvature of 1/9, and the other with
circular pores. Figure 8 presents the macroscopic densification response of these two simulations. The densification
response of the material with circular pores is shown using
the solid line, and that for the cusp-shaped pores uses a
dashed line. The densification rate of the material with circular pores is much less than that of an identical material with
cusp-shaped pores. Clearly, the densification process can be
made more time efficient if one can retain the initial cuspshaped nature of the pores by reducing the integrated thermal
exposure and, therefore, pore spherodization by diffusion.
It should be noted that the large (,200 mm) diameter of the
coated fibers leads to large diameter pores that require very
long times at a high temperature to spherodize.
B. Grain Size Effect
The enhanced densification response of the physical
vapor-deposited Ti-6Al-4V–coated fibers was experimentally studied and reported by Kunze and Wadley.[5] Warren
et al. in their experimental study of a similar PVD alloy
observed enhanced superplastic deformation and attributed
it to the ultrafine grain size that increased the strain-rate
contribution of the DAGS mechanism of creep, which is
inversely proportional to d 2.[3] The initial grain size of vapordeposited materials is sensitive to the temperature and the
rate at which deposition occurs and to some extent can
be controlled. To investigate the effect of grain size on
densification, a set of three simulations with the same input
schedule but three different initial grain sizes (of 0.1, 1, and
2 mm) were simulated. Figure 9 shows that as the initial
grain size was increased from 0.1 to 1 mm, the time to
densify the composite increased by 15 minutes, and for an
increase in initial grain size of 1 to 2 mm, the time to
densify the composite increased by 80 minutes. Starting with
materials with smaller grain sizes will, therefore, enable
METALLURGICAL AND MATERIALS TRANSACTIONS A
(a)
(a)
(b)
(b)
(c)
Fig. 9—(a) Process cycle, (b) relative density evolution, and (c) the grain
size evolution for three different initial grain sizes.
reductions in the thickness of the fiber matrix reaction without a decrease in density. The use of a material with a small
initial grain size also enables a lower consolidation pressure
to be applied, which in turn reduces the likelihood of fiber
fracture. It is, therefore, desirable to use high deposition
rates and low fiber temperatures during vapor deposition to
create a coating with as small a grain size as possible and
to avoid grain growth during early stages of consolidation
by using a low consolidation temperature.
C. Can Shielding Effect
The canister used for the consolidation of the metal-coated
fibers can play an important role in optimization of the
consolidation cycle used for densifying materials. In a consolidation study of electron beam Ti-6Al-4V–coated fibers,
Ward-Close and Loader[34] found that about 2 hours at 150
MPa were required to fully densify a composite at 760 8C
when using a titanium tube canister with a wall thickness
of 10.5 mm. In the work of Kunze and Wadley[5] with a 1.6mm-thick canister, only an hour was required at a pressure
of 100 MPa to accomplish full consolidation. Our experimentally validated simulation shows that it is possible to
reach full density at much lower pressures and considerably
shorter periods of time (Figure 5). The observation is consistent with the increasing shielding of the applied pressure as
METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 10—(a) Process cycle and (b) relative density evolution showing the
effect of can shielding on the densification response.
the thickness of the can is increased. This shielding effect
can be particularly important during the early stages of densification. To evaluate the effect of can shielding, two simulations were performed, and their results are presented in
Figure 10. The first simulation was performed using a CPtitanium can like that in the Kunze and Wadley experiments[5]
with a radius of 15.9 mm and a thickness of 1.6 mm. Its
densification response is shown using a solid line. The second simulation assumed “no can” was present during consolidation, and the densification response is shown using a
dashed line. The same pressure and temperature schedules
were applied to both the simulations. The simulation results
show that the can shielding effect on the densification
response is most prominent as the temperature is being
ramped. During this stage, the no-can simulation (dashed
line) achieved a much higher density even before the onset
of the main pressure ramp because of the low grain size and
the high effective pressure. The simulation using a 1.6mm can only started to significantly densify as the pressure
started to ramp because much of the pressure that was applied
during temperature ramp (2 MPa) was supported by the can.
Interestingly, because the can shielding eventually disappears, the simulation using a can ultimately catches up with
the no-can simulation during the later stages of consolidation. From this comparison, it seems that reducing the thickness of the can might not necessarily have as much impact
on the time at which consolidation is accomplished.
VI. CONCLUSIONS
An analysis of the HIP densification of randomly packed
Ti-6Al-4V–coated SiC monofilaments encased in a cylindrical canister has been conducted. The analysis was based
upon micromechanical models for coated-fiber contact and
cusp-shaped void collapse. Those models were implemented
for a power-law creeping material and included the special
cases of plasticity and creep superplasticity (DAGS). The
effects of process path-dependent changes in the matrix grain
size and a/b-phase volume fractions were incorporated
VOLUME 31A, APRIL 2000—1281
together with the shielding effect of the canister. The simulation was then validated by comparing its predictions with
published experiments. The simulation predictions of density
evolution showed good agreement with experiments over a
wide range of processing conditions. Analysis of the modeled behavior indicated the following.
1. The initial grain size of PVD coatings has a very strong
effect on consolidation. Increasing the initial grain size
from 100 nm to 2 mm reduces the densification rate by
a factor of 2 during the initial stages of densification and
can more than double the time needed to fully densify
a sample.
2. The cusp shape of the pores can be retained during finalstage densification due to the pores’ size and the use of
a process cycle that minimizes surface diffusion. Cuspshaped pores are more easily densified and are responsible for significant acceleration of final stage
densification.
3. A can-shielding effect significantly retards the early stage
consolidation when the coated fiber array is most
compliant.
ACKNOWLEDGMENTS
We are grateful to R. Kosut for helpful discussions about
this research. This work has been funded by DARPA through
a contract with Integrated Systems Inc., Santa Clara, CA
(Dr. Anna Tsao, Program Manager).
REFERENCES
1. Z.X. Guo and B. Derby: Progr. Mater. Sci., 1995, vol. 39 (4), pp.
411-95.
2. C.H. Weber, Z.-Z. Du, and F.W. Zok: Acta Mater., 1996, vol. 44 (2),
pp. 683-95.
3. J. Warren, L.M. Hsiung, and H.N.G. Wadley: Acta Metall, 1995, vol.
43, p. 2773.
4. H.E. Deve, D.M. Elzey, J.M. Warren, and H.N.G. Wadley: Advanced
Structural Fiber Composites, Proc. 8th CIMTEC World Ceramic Congress and Forum on New Materials, P. Vincenzini, ed., Techna,
Florence, Italy, 1995, p. 313.
5. J.M. Kunze and H.N.G. Wadley: Acta Mater., 1997, vol. 45 (5), pp.
1851-65.
1282—VOLUME 31A, APRIL 2000
6. J.M. Kunze and H.N.G. Wadley: Mater. Sci. Eng., 1998, vol. A244,
pp. 138-44.
7. D.M. Elzey and H.N.G. Wadley: University of Virginia, Charlottesville, VA, private communication, 1999.
8. Y.-M. Liu, H.N.G. Wadley, and J.M. Duva: Acta Metall., 1994, vol.
42 (7), pp. 2247-60.
9. P.E. Cantonwine and H.N.G. Wadley: Composites Eng., 1994, vol. 4
(1), p. 210.
10. E. Arzt, M.G. Ashby, and K.E. Easterling: Metall. Trans. A, 1983,
vol. 14A, p. 211.
11. R. Vancheeswaran, D.M. Elzey, and H.N.G. Wadley: Acta Metall.
Mater., 1996, vol. 44 (6), pp. 2175-99.
12. D.M. Elzey and H.N.G. Wadley: Acta Metall. Mater., 1993, vol. 41
(8), p. 2297.
13. R. Gampala, D.M. Elzey, and H.N.G. Wadley: Acta Metall. Mater.,
1994, vol. 42 (9), p. 3209.
14. D.M. Elzey and H.N.G. Wadley: Acta Metall. Mater., 1994, vol. 42
(12), p. 3997.
15. D.M. Elzey, R. Gampala, and H.N.G. Wadley: Acta Mater., 1997, vol.
46 (1), pp. 193-205.
16. R. Vancheeswaran, D.G. Meyer, and H.N.G. Wadley: Acta Mater.,
1997, vol. 44 (6), pp. 2175-99.
17. H.N.G. Wadley, R.J. Schaefer, A.H. Kahn, M.F. Ashby, R.B. Clough,
Y. Geffen, and J.J. Wlassich: Acta Metall. Mater., 1991, vol. 39 (5),
pp. 979-86.
18. E. Arzt: Acta Metall., 1982, vol. 30, p. 1883.
19. B.W. Choi, Y.G. Deng, C. McCullough, B. Paden, and R. Mehrabian:
Acta Metall. Mater., 1990, vol. 38 (11), p. 2245.
20. H.F. Fischmeister and E. Arzt: Powder Metall., 1983, vol. 26 (2), pp.
82-88.
21. H.N.G. Wadley, T.S. Davison, and J.M. Kunze: Composites Eng.,
1997, vol. 45 (5), pp. 1851-65.
22. Z. Qian, J.M. Duva, and H.N.G. Wadley: Acta Mater., 1996, vol. 44
(12), pp. 4815-24.
23. H.J. Frost and M.F. Ashby: Deformation Mechanism Maps, Pergamon
Press, Oxford, United Kingdom, 1992, pp. 43-52.
24. O. Molerus: Powder Technol., 1975, vol. 12, p. 259.
25. J.M. Duva and P. Crow: Mech. Mater., 1994, vol. 17 (1), p. 25.
26. M.F. Ashby and R.A. Verall: Acta Metall., 1973, vol. 21, p. 149.
27. J.E. Burke and D. Turnbull: Progr. Met. Phys., 1952, vol. 3, p. 220.
28. M. Hillert: Acta Metall., 1965, vol. 13, pp. 227-38.
29. M.F. Ashby: Report No. CUED/C-MATS/TR.170, Cambridge University Cambridge, United Kingdom, 1989.
30. J.K. McCoy, L.E. Muttart, and R.R. Wills: Am. Ceram. Soc. Bull.,
1985, vol. 64, pp. 1240-44.
31. J.K. McCoy and R.R. Wills: Acta Metall., 1987, vol. 35, pp. 577-85.
32. J. Besson and M. Abouaf : Int. J. Solids Struct., 1991, vol. 28 (6), pp.
691-702.
33. P.P. Castenada: J. Mech. Phys. Solids, 1991, vol. 39, p. 45.
34. C.M. Ward-Close and C. Loader: in Recent Advances in Titanium
Metal Matrix Composites, F.H. Froes and J. Storer, eds., TMS, Warrendale, PA, 1995, p. 19.
METALLURGICAL AND MATERIALS TRANSACTIONS A