AC/~ mafer. Vol. 45, No. 10. pp. 10014018, 1997
0 1997 Acta Metallurgica Inc.
Published by Elsevier Science Ltd. A.1rights reserved
Pergamon
PII: PII: s1359-6454(97)00104-3
OPTIMIZING
Printed in Great Britain
1359-6454107
$17.00+ 0.00
THE CONSOLIDATION
OF TITANIUM
MATRIX COMPOSITES
RAVI VANCHEESWARAN,
DAVID G. MEYER?
and HAYDN N. G. WADLEYS
Dept. of Materials Science and Engineering, School of Engineering and Applied Science, University of
Virginia, Charlottesville, VA 22901, U.S.A.
(Received 29 December
1996; accepted 25 February 1997)
Abstract-The high temperature consolidation of fiber reinforced metal matrix composite preforms seeks
to eliminate matrix porosity (i.e. to increase the relative density) while simultaneously minimizing fiber
microbending/fracture and the growth of reaction products at the fiber-matrix interface. By combining
model predictive control concepts with previously developed time dependent consolidation models [18],
a method is presented for the design of near optimal process schedules that evolve performance defning
microstructural parameters (relative density, fiber fracture density and fiber-matrix reaction thickness) to
prechosen microstructural goal states that result in composites of acceptable mechanical performance. The
method is illustrated by exploring the design of process schedules for two titanium matrix composites with
very different creep properties. Feasible process schedules resulting in acceptable composite mechanical
performance are identified for a Ti-6Al--W/SCS-6 system. However, the method reveals the absence of
such schedules for the more creep resistant Ti-24Al-llNb/SCS-6
intermetallic matrix composite The
optimization methodology is then used to explore modifications to the process environment, the
composites component material properties (e.g. fiber strength or diffusion inhibiting coating), and the
preforms initial microstructure that together enable the successful consolidation of the intermetallic matrix
composite system. 0 1997 Acta Metallurgica Inc.
1. INTRODUCTION
Silicon carbide monofilament
reinforced titanium
and nickel matrix composites are attractive materials
for elevated temperature
unidirectionally
loaded
structures because of their high specific stiffness [l],
strength [224] and creep rupture life [6, 71 in the fiber
direction. Composites based either upon conventional, or intermetallic titanium alloys have attracted
particular attention because they enable the radical
redesign of aircraft gas turbine engines [ 10, 111.Many
laboratory scale manufacturing
processes are now
available for the manufacture of these materials. All
involve a two-step process sequence that seeks to
sidestep the aggressive chemical reaction that occurs
between liquid titanium or nickel alloys and silicon
carbide fibers. First, a composite monotape is
synthesized by either plasma spray methods [12, 131,
slurry casting [14] or vapor deposition [15-171. In the
second step, the unidirectionally reinforced tapes are
cut to shape, stacked to create a preform with an
appropriate fiber architecture and then consolidated
to theoretical density by hot isostatic pressing (HIP),
Fig. 1 [18].
Consolidation ideally results in a near net shape
composite component containing no residual matrix
tPresent address: Department of Electrical and Computer
Engineering. University of Colorado, Boulder, CO 803090425, U.S.A.
fTo whom all correspondence should be addressed.
porosity (to avoid premature failure oi the matrix
under static or fatigue loading), mmimal fiber
microbendinglfracture
(to avoid degradation of the
composites strength and creep/fatigue endurance
[2, 6]), a limited thickness of reaction product at the
fiber-matrix interface (to avoid undesired increases in
interfacial sliding stress or even a loss of the fiber’s
strength [30]) and the lowest achievable thermal
residual stress. Unless a “goal state” combination of
these microstructural attributes (i.e. relative density,
degree of fiber microbending stress/fracture, reaction
product thickness and residual stress) is achieved at
the completion of the consolidation process, the full
potential of the composite system fails to be realized,
and its competitive advantage over other materials
can be lost.
Extensive experimentation
has been needed to
deduce consolidation
schedules that result in
reasonable final microstructural states, Some apply
temperature and pressure simultaneously (to reduce
the ramp-up time), others delay the onset of pressure
until temperature ramping is complete in order to
soften the matrix, and hopefully limit fiber fracture
[18]. While, this empirical approach has led to the
identification
of successful schedules for a few
“highly processable”
material systems such as
Ti6Al4V
reinforced with SCS-6 SK monofilaments, successful processes have eluded discovery for
many other systems of potential interest, and it
remains unclear if they even exist,
400 1
4002
VANCHEESWARAN
et al.:
OPTIMIZING
An alternative approach to consolidation process
design seeks to develop time dependent micromechanics-based models that predict the evolution of
the microstructural
attributes given (1) a process
schedule, (2) the monotape’s geometry and (3) basic
properties of the matrix and fiber [18]. These
predictive consolidation models essentially. map a
process schedule to a trajectory in the microstructural
state space [18]. For example, Fig. 2 shows the
calculated evolution of relative density, fiber fracture
density and reaction product thickness for a
Ti4A14V/SCS-6
monotape layup subject to a
representative process schedule using models developed in [ 181.(See [ 181for the monotape geometry and
matrix/fiber thermo-physical properties used for the
calculation.) The models are seen to nonlinearly map
the consolidation process schedule to an evolving
microstructural
state and therefore transform the
layup from a defined initial, to a deduced final
microstructure state. We note that for the example
shown in Fig. 2, the final deduced state would be
unacceptable because of the large number of fiber
breaks per meter of fiber.
Analysis of models reveals the microstructural
states to have conflicting dependencies upon the
variables of the consolidation process, i.e. temperature and pressure as functions of time. For example,
densification
is most rapidly accomplished
by
consolidating at the highest available temperature
THE CONSOLIDATION
and pressure. However, fiber microbending/fracture
is minimized by consolidating at high temperature
while applying pressure very slowly (i.e. by using long
duration high temperature schedules). This unfortunately results in extensive chemical reaction at the
fiber-matrix
interface. The amount of chemical
reaction can only be reduced by either shortening the
high temperature exposure time or by reducing the
temperature. Both unfortunately increase the probability of fiber bending/fracture. Thus, the modelling
approach also provides a fundamental understanding
of why one strategy (for instance applying pressure
only after high temperatures are reached) sometimes
works while another does not.
By substituting a validated model for the real
material, trial and error methods can more economically seek processes that result in acceptable
outcomes. The trade-off between pressure and
temperature has been found to vary with time and is
also a sensitive function of the material system [18].
It is therefore difficult to be sure that one has deduced
the “best” process. This trial and error approach to
process schedule design does not fully exploit the
predictive power of models, the optimal schedules
may not have been identified, and for some material
systems it is possible to incorrectly label them as
“unprocessable”.
To pursue a better design approach, we begin by
recognizing
that since mechanical
performance
Pressure, temperature
Pressure, temperature
Pressure, temperature
Fig. 1. A schematic
diagram
illustrating
the consolidation
of a plasma
sprayed
composite
monotape
layup.
et al.:
VANCHEESWARAN
Direct
Simulation
for Ti-6AI-4V
OPTIMIZING
THE CONSOLIDATION
/ SCS-6
Process
(b) Temperature Schedule
4003
:
(c) Pressure Schedule
Fig. 3. Pictoral
representation
of the microstructural
evolution in state space. An optimal process path can be
defined as the shortest one connecting
the initial state (X,)
with a user defined goal state A;.
$$F=J[~~
0
50
103
Time
150 200
(min)
0
50
ICO 150 200
Time
(min)
Fig. 2. Predicted microstructure
trajectory for the consolidation of a Ti-6AIdV/SCS-6
composite layup subjected to “a
ramp plus soak” process schedule. The final microstructural
state, Xt, is characterized
by a relative density, Dr, the
number of fiber breaks per meter of fiber, N,. and the
thickness of the reaction layer at the fiber matrix interface.
only on the microstructural
state achieved
at process completion,
it does not really matter if
either the pressure or the temperature
are applied
first, or if the full heating rate or temperature/pressure capacities of the equipment are utilized, all that
is important
is to find a trajectory
in the
microstructural
state space that transforms
the
monotape preform from a defined initial to a defined
goal state that is known to result in acceptable
mechanical performance,
Fig. 3. The solution to this
problem
is challenging
because
of the state’s
nonlinear
dependence
upon the process conditions
and the enormous
number
of possible
process
trajectories,
only a few of which might terminate at,
or acceptably close, to the goal. It is considerably
simplified by recognizing
that while many options
are available
for changing
future pressures
and
temperatures,
the irreversibility
of the microstructure’s evolution,
and the limited capacity of the
consolidation
equipment,
confine all these trajectories to a conical volume emanating
from the
current
state. Furthermore,
as the goal state is
approached,
fewer path alternatives exist, and these
eventually converge upon the goal state.
An optimal path-planning
problem can be defined
as the search for the process path (a pressure/temperature history) that reaches the microstructure
goal
in the shortest time. Here predictive consolidation
depends
models [ 181 that simulate the dynamics of microstructural evolution are coupled with a process optimization technique developed for control design [21, 221
to calculate optimal process paths to user defined
goal states. The technique
is motivated
by gain
scheduling methods
from feedback control theory
[23]. It uses a multistep
(microstructure)
state
predictor
and a receding horizon
philosophy
to
compute optimal perturbations
to the set of control
variables using only knowledge
about the current
state of the process and the consequences
of all
available changes to the process.
xg=
Fig. 4. A block diagram
of the path-planning
problem
showing the plant model (the HIP machine and monotape
consolidation
dynamics) and the model predictive planning
scheme used to find a process that ends at the goal state, X,.
The states of the HIP machine (X,) are the temperature
(T)
and pressure (P), while its inputs are the corresponding
slew
rates for temperature
(T,,,)
and pressure
(Pr,t.). The
evolving microstructural
states (Xh) are the relative density
(D). the fiber bend-cell deflections (r), and the reaction layer
thickness (r).
VANCHEESWARAN et al.: OPTIMIZING THE CONSOLIDATION
4004
(a) Direct Simulation for 7724AI- 11Nb / SCS-6
2. THE CONSOLIDATION PROCESS
2.1. Definitions
FinatState,xf=[%]=[;“k]
(b) Temperature Schedule
cl
50
100
150 2w
Time (mln)
(c) Pressure Schedule
0
50
100
150 x0
Time (min)
Fig. 5. Microstructure trajectory for the consolidation of an
intermetallic Ti-24Al-1 lNb/SCS-6 composite subjected to a
“ramp plus soak” process schedule.
The optimal path is found by minimization of
the distances between the goal point and each
intermediate point, x1, x2, etc., in a time series
representation
of the process trajectory, Fig. 3.
Although
the methodology
is computationally
intensive, such an approach is required because the
microstructural variables of interest are nonlinearly
dependent on process conditions and are irreversible. Thus, the consequences
of even a small
“wrong” decision by the planner early in the
process can result in the microstructure unrecoverably missing the desired goal, making it look
unreachable. Thus the planner needs to be “conservative” during the initial stages of path design
and the extensive use of predictions well into the
future via a multistep predictor/receding
horizon
approach greatly helps to avoid the chance of
overshoot.
The need for process path-planners of the type
developed here appears to be widespread
in
materials processing. For example, with relatively
minor modifications, the approach devised could
be combined with other process models and used
to design temperature histories that simultaneously
control precipitation
reactions and grain growth
during alloy heat treatment or oxide layer thickness and diffusion depth during the rapid thermal
processing step used in the synthesis of microelectronic components.
Process path optimization is a sub-field of control
theory. In control theory terminology, the entity to
which control is applied is commonly called the
“plant”, and a mathematical model of the plant is
called a “plant model”. In TMC consolidation, the
plant model has two easily distinguishable components: One is the consolidation machine (a hot
isostatic press (HIP) or vacuum hot press (VHP))
while the other is the composite layup inside it. Thus,
a plant model can be obtained by connecting a
mathematical
model of a HIP machine with a
mathematical model of the microstructural evolution
of the composite. The proper connection is a cascade
connection, making outputs of the HIP machine
model be the inputs to the microstructure model. The
proper cascade of the two models is shown pictorially
on the right hand side of Fig. 4. Both the machine and
the material microstructural models, and hence the
overall plant model, take the mathematical form of
coupled systems of nonlinear ordinary differential
equations [ 181.
2.2. The HIP machine model
A HIP machine has dynamic behavior. For
example, conduction, convection and radiation are
all in play when the working gas is being heated by
the furnace elements. Thus a pulse of current through
the elements eventually results in a time-varying
Direct
Simulation
for Ti-24AI-1
FinalState,xf=[!]=[
INb / SCS-6
;“E]
Time (mln)
0
50
100 150 2w
Time (min)
0
100 150 2w
rim.3(min)
Fig. 6. Time dependence of the microstructural states in a
Ti-24Al-1 lNb/SCS-6 composite subjected to a “ramp plus
soak” process schedule.
VANCHEESWARAN
et al.:
OPTIMIZING
1. Critical fiber deflections for each unit cell that result in a failure probability of 10%
Table
19.1
159.0
1.14
17.3
131.0
0.94
15.5
105.0
0.75
(dynamic)
temperature
within the
machine model attempts to capture
behavior.
Abstractly, one can think of a HIP
“box” with a (vector) input, a (vector)
(vector) state, Fig. 4. The input vector,
of commanded time varying pressure
ture slew rates:
13.7
82.0
0.59
11.9
62.0
0.44
sample. A
this dynamic
machine as a
output, and a
u(t), consists
and tempera-
1
Tme(t)
u(r)
=[ Pm(t)
The state vector,
actual
temperature
sample:
.x,(t).
and
x,(t) =
.
then consists of
within
pressure
T(r)
P(t)
[
the
the
1
The vector output of the machine model, q(t), is just
the temperature and pressure (thus, q = x,). A
complete time-history, q(t), is the process schedule
and a single reading of ;r?at an instant in time is a
process emironment
The above choice of input, state, and output is a
lumped approximation.
Very complicated mathematical models of a HIP machine with distributed
states are possible [9]. However a lumped model
can always be expressed in the simple symbolic
form:
dx,
dt
4005
THE CONSOLIDATION
=fmbl(t), u(t))
rl(t) = gm(.%n(r),
u(t))
Any real HIP machine has upper and lower limits on
the temperature and pressure and their respective
slew rates:
Pm d P G Prna
(2)
where T,,,,, and T,, are the maximum heating and
cooling rates respectively, while Pmpand Pmdare the
maximum pressurizing and depressurizing rates again
respectively. The meaning of T,,,, T,,,, P,,,, and P,,,
is obvious.
For ease in interpretation of results, and to focus
on the microstructural
dynamics, the numerical
results presented later are for an “ideal” HIP machine
meaning that&(x,,
u) = u(t) and g_(xm, u) = x,(t).
10.1
45.0
0.32
8.3
30.0
0.22
Thus, the lumped
machine is
6.5
18.5
0.13
dynamic
4.7
9.7
o.c7
model
of an
2.9
3.7
0.026
ideal
drl
dt = l1
In other words, the ideal machine responds perfectly
to commanded
temperature
and pressure slews
provided the machine’s physical limitations (2) are
obeyed. The simple forms chosen for&,( ) and g,,,( )
are for clarifying the results only; it should be
recognized that the methodology presented can
readily handle more worldly choices.
2.3. The material model
Prior to consolidation, a typical spray deposited
TMC monotape layup contains 3545% internal
porosity. Most is located between the monotapes
due to their surface roughness, while the remainder
exists as isolated internal voids, Fig. 1 [18]. Upon
application
of an applied load, the laminate
densifies to a relative density, D, by inelastic
contact deformation of surface asperrties and by
matrix flow around isolated voids (see Fig. 1). The
asperity’s resistance to inelastic flow leads to large
localized contact stresses that cause fiber microbending displacements (u) which can lead to
fiber fracture. Any high temperature exposure also
results in the formation of a reaction product (of
thickness r), around the fiber. These microstructural quantities are the ones that cause a processing
dependence of the properties of the fully processed
composite [2-8, 36. 381, and it is their behavior that
must be captured mathematically in the microstructure model.
A model relating the time evolution of the relative
density, fiber deflection, fracture, and reaction
product thickness for a “unit cell” in response to a
process schedule, T(t) and P(t), has been assembled
from recent developments in contact mechanics
[24, 251, analyses of void collapse in power law
creeping solids [26,27], microbending/fiber
fracture
models [28, 291 and from experimental studies of the
kinetics of fiber-matrix interfacial reactions coupled
with push-out measurements of interfacial sliding
stress [30, 311. The surface asperity’s contribution to
densification (called the s-layer response) is calculated using a contact analysis for a randomly rough
surface.
In this microstructure model, the fiber fracture
density. N,, is related non-dynamically
to the fiber
deflection, v. through expressions that relate v to the
fiber stress CT,and the fiber stress to fiber failure
probability [ 181. Hence, a minimal material state can
exclude N,. since it can always be directly calculated
4006
VANCHEESWARAN
et al.:
xl(t)
=D(t)
u(t)
1
OPTIMIZING
from U. The vector state, xh, of the microstructural
model for a representative unit cell is thus
(4)
[ r(t)
Like the machine state (l), the state, xh in a lumped
microstructural model satisfies an ordinary differential equation
(5)
Notice that the time rate of change of the
microstructure depends on the applied temperature
and pressure (carried in rl) as one would naturally
expect.
The detailed form offh in (5) can be found in [18].
For our purpose, fh(xh, q) has the form
[l--k(xh)(pLck(xh,
AD,
i
results in a distribution of “beam lengths” in the
layupt which varies with density (and parametrically
with time) as a fiber span encounters additional
asperities randomly along its length. To capture the
effect of this statistical variation in the model, the
time evolution of xh is tracked for a representative set,
Yeells, of unit cells, each with a distinct fiber length.
A “coordination
number”,
N,(D, I) tracks
the
number of cells with a beam length 1 at a relative
density, D. Reasonable fidelity with experimental
data can be obtained by tracking xh with ten unit cells
whose beam lengths vary between about 2 and 20
fiber diameters [18]. The coordination numbers for
these beam lengths are taken to be static functions
(computed off-line by a Monte Carlo simulation) that
describe the variations of the beam length population
with densification. The complete material model is
thus an array of unit cell models and the material
microstructural state is given by
(6)
m@,
T) 1
j&h,
j,
THE CONSOLIDATION
?)
7)
rl)
-
r
=
+
Mu,
DIF&h,
f’f))l
7’)
In (6), the mechanisms for densification by power law
creep are denoted by PLC&,, q), and those for
diffusional flow by DIFk(xh, q). Their individual
contributions to the densification rate are additive.
Two distinct stages (1 and 2) of densification have
been analyzed. Stage 1 applies when D < 0.92, and
corresponds to the deformation of asperities. Stage 2
is operative where D 2 0.92 and treats the inelastic
collapse of isolated spherical voids.
The functional form for the densification rate due
to power law creep and diffusional flow differs
between Stage 1 and Stage 2, and gives rise to the “k”
subscripts. The r&h) are transitioning
functions
between stages that satisfy
I-1, l-2 2 0
r, + rz x i
r, z
1 in Stage 1, ~0 in Stage 2
rz z
1 in Stage 2, ~0 in Stage 1
The function, g(D, q), captures fiber bending due to
the load applied by the applied pressure, while
h(v, T), corresponds to fiber straightening due to
creep relaxation at the loaded asperities. The rate of
change of the reaction product thickness (m(r, T))
results strictly from a diffusional process. It is thus
only temperature and layer thickness dependent, and
is independent of all other variables including the
applied pressure.
In the entire monotape layup, the fiber deflections
are statistically distributed because of variations in
the distance between the tops of asperities. This
xh=
Xh,celll
1
Xh,dl2
.
I : 1I
,_Xh,celllO
which is the microstructural state for each unit cell in
Y cells“stacked” together. The unit cell equations (5)
are also stacked to get
Thus, the fiber fracture model consists of 10 vector
equations (5); each equation corresponding to a beam
of differing length. Since there is no beam length
effect on densification rate or reaction product
thickness, only the deflection variables in xh vary
from cell to cell and xh for the complete model has
12 components (relative density, reaction product
thickness, and 10 fiber deflections) and not 30.
An example of the simulated response for a
Ti-6A14V/SCS-6
silicon carbide fiber TMC undergoing a commonly used “ramp and soak” schedule is
shown in Fig. 2. In the figure, the x, axis is the fiber
fracture density for the monotape layup (not just one
unit cell). This was found by taking the deflection
variables in &, converting them into fiber fractures
and summing these to arrive at a cumulative fiber
fracture density. We note that this “ramp and soak”
schedule achieved full relative density, caused
approximately 9 fiber fractures per meter and resulted
in the growth of a 0.3 pm thick reaction product.
While this reaction product thickness is well below
the level where sliding would be adversely affected
(1 pm for many titanium matrices reinforced with
SCS-6 fibers) the particular schedule shown would
nevertheless result in a composite of poor mechanical
tThe fiber bending model is simple three-point
bending; a
fiber beam is defined by three-point
contact by three
asperities; since the distribution
of asperities is statistical,
the distribution
of beam lengths is also statistical. See [18]
for complete details.
VANCHEESWARAN
et al.:
OPTIMIZING
4007
THE CONSOLIDATION
Table 2. Goal states for the two composite systems
Composite system
$I
rcnl (pm)
D
Ti-6A1-4V/SCS-6
Ti-24AI-1 lNb/SCS-6
0.1
0.1
0.6
1.0
0.999
0.999
performance due to the large number of fiber
fractures [2].
When the Ti-6Al4V
matrix is replaced with a
more creep resistant intermetallic Ti-24Al-11Nb
matrix, a “ramp and soak” schedule type gives the
result shown in Fig. 5. For this material, the final
relative density was 0.999, but it was achieved at the
cost of 10.1 fiber breaks per meter and a 1.17 pm
reaction product thickness. Figure 6 shows the
dynamic response of the completed set of states.
Again, the composite produced by this schedule
would have poor mechanical performance because of
the large number of fiber breaks. Additionally, the
reaction product thickness is close to the thickness
where sliding stress has been experimentally observed
to rapidly increase [30]. The point here is that the
“obvious”
fix of increasing
the consolidation
temperature to lessen the fiber fracture density fails
because it results in too much fiber-matrix reaction.
It exemplifies the inherent competition between the
microstructural goals that makes a trial and error
approach to the design of process schedules tricky
and highly inefficient for processes of this complexity.
I’”
0.47
0.5
r,
0.8
0.8
process schedule, 9, that connects X, to .x8. Suppose
that at time t = tC the material is at x(t’) = xc, the
point labeled “current” in Fig. 3. Simple physical
reasoning indicates that when following the trajectory
away from the current point, one must always move
toward increasing relative density. fiber fracture
density, and reaction product thickness. The model
mirrors this physical reasoning because, the dynamic
equations satisfy
dD
dNf dr
dt.dt,z>O
when n#O
and so D(t),
N,(t),
and r(t) are monotone
non-decreasing functions of time. This means that all
physically feasible paths must lie in a conically
shaped region originating from xc. We call this the
infinite horizon (or infinite time) reachable set and
denote it by B(x”). a(xc) is therefore the set of
microstructure trajectories that can be generated for
all the admissible combinations of temperature and
(satisfied by (2)). If the
pressure trajectories
composite is at .xc and x,$W(x’) then it is impossible
to process the composite to achieve x,. Thus, the set
9(x,)
= {x1x&%(x))
3. PROCESS SCHEDULE DESIGN
3.1. Overview of path-planning
problem
The objective of path planning is the computation
of a process schedule that transforms a composite’s
microstructure to a predefined “goal” state, X,, at its
completion. At any moment during consolidation,
the material’s vector state, X,,, can be represented as
a point in a (twelve-dimensional)
relative density,
fiber deflection, reaction product thickness state
space. As explained in Section 2.3, Xi, can also be
projected into a (three-dimensional) relative density,
fiber fracture density, reaction product thickness
space by converting fiber deflections into fiber
fractures and summing over the unit cells using
NJ, D).
Consider Fig. 3 showing a trajectory which might
result from the application of a temperature and
pressure schedule. The material starts from an initial
condition X, in the lower left corner of the space at
time t = 0, and as time passes, it moves along the
trajectory shown. The problem is to find a realizable
is a forbidden zone for the goal xg and should be
avoided in path-planning.
One approach is to
determine if admissible processes exist for varied test
points x,,,,. At any given xc, computing :8(x’) exactly
is not tractable except for very simple dynamic
equations. However, a point x~~,~
will be in 9(x’) if it
can be reached at any future time. An “oracle” built
to decide the question would thus strictly have to
search over process schedules of arbitrarily long
duration. Of course, this problem can be removed in
practice by limiting the search to process schedules
whose duration are less than a suitable finite length
of time which can be chosen based on physical or
economic reasoning. A second, much more serious
impediment to computing B exactly is the path
dependence and nonlinearity of the dynamics which
makes the computational solution to this problem
NP-hard.
Since computing the exact reachable set from any
given current point, x’, is not practical, approximations to LJ?(x’)must be used in path-planning. The
Table 3. Physical limits for hwothetical
HIP
Type
I
2
3
4
T,,.
(YYjmin)
-20.0
-20.0
-30.0
-30.0
HIP machines
T-.x
(Yjmin)
Pnu.
(MPa/min)
Pm.,
(MPa/min)
T”n”
(‘C)
P”,,x
(MPd
20.0
20.0
30.0
30.0
-1.0
- 2.0
-1.0
- 2.0
1.0
2.0
1.0
2.0
825.0
825.0
825.0
825.0
100.0
100.0
100.0
100.0
4008
VANCHEESWARAN
et al.:
OPTIMIZING
method we explore constructs reachable set approximations from convex polytopes based upon a local
affine approximation of the dynamic equations. As
the approximation is local, the method takes many,
many small steps toward the goal state and
recomputes its approximation at each step. It is an
example of a receding horizon strategy.
A block diagram of such a path-planner is shown
in Fig. 4. The path-planner accepts as input a vector,
X,, for the material microstructural goal state. The
input X, is static and does not change with time. X,
gives desired values for the relative density (D*), and
the reaction product thickness (r*), and values (v&J
of deflections for each of the ten unit cells in the
model. The path-planner also accepts as input the
current material microstructural state, Xh(tC),and the
current process environment, q(tC). Both the Xh(tc)
and q(tC) inputs of course change with time. At
regular intervals in time the path-planning algorithm
then calculates and commands the HIP machine
model with appropriate temperature and pressure
slews, T,,, and P,,,, . A complete process schedule (i.e.
a temperature and pressure schedule), q(t), is then
constructed that steers Xh(t) to X,.
3.2. Path-planning approach
The model indicated by (5) is nonlinear, and so the
path-planning problem is computationally challenging. One possible approach is feedback linearization
which attempts to transform the state and input
coordinate (X,, and ?j) so that the transformed
equations become linear
Path-planning
the Xh states and q inputs is then
tractable because the nonlinearities
have been
removed. Once a suitable f(t) has been found, an
inverse transformation
is applied to get q(t).
Necessary and sufficient conditions under which such
a transformation
to linear equations
can be
accomplished are given by the theory of feedback
linearization [32]. Unfortunately,
the consolidation
model fails to satisfy the proper “relative degree”
conditions and so the approach fails.
Gain-scheduling
[19,20] motivates a second approach to path-planning. In this method, a set of N
points, Xi, and a set of N regions, xi, are chosen so
that:
1. The .Y, partition the state space and X, E ZI?iif
and only if i = j.
2. The range of plant dynamics is covered by the
Xi.
3. The dynamics of the plant do not change greatly
over each Zi and so a simpler approximate plant
can be used over each I,. A common choice is
a linear, time-invariant approximation about X,.
4. It is easy to detect which SYithe plant state is in.
5. On each !Z”ia tractable path-planning scheme,
S,, using the approximate plant is available.
THE CONSOLIDATION
The sets S?“i,and the points Xi are either chosen a
priori, or are generated as the process proceeds. As
the plant trajectory switches from xi to %,;, the
path-planner is switched from Si to Sj. Unfortunately,
the Hankel singular values of the linearized
consolidation
models vary over many orders of
magnitude during consolidation, and therefore the
method can make unrecoverable errors in designing
the schedule for the early stages of the process.
The method we present here combines aspects of
the gain-schedule motivated scheme described above
with a receding horizon philosophy [21]. A variant of
the method has been shown [23] to give good results
for consolidation of metal powders. It can be broadly
classified as model predictive [21,22, 33-351. The
method lets the Xi be the planned microstructural
progression at regular points in time, 0, At, 2At,
3At, . , and dynamically constructs each region, %,,
as an approximation to 9(X,). In our method, the
gain-scheduling properties 1 and 4 are not satisfied.
However this causes no difficulty; the gain-scheduling
property 3 is guaranteed by making At suitably small,
convex programming supplies the tractable scheme,
Si, required in property 5, and property 2 is satisfied
“implicitly” because the material dynamics change
rapidly when the temperature input is changed
rapidly? and, as we shall see, the planner gives a
schedule that uses the entire dynamic temperature
range of the HIP machine. The method is
implemented in a sequence of steps.
Step 1. Normalization. First a change of variables is
made in the material model on each unit cell to get
states and inputs to lie in the interval [0, 11. This is
done to enhance numerical conditioning of a later
Step (6), where a convex program is solved.
Transformations of the density, the deflections of the
sample unit cells, the reaction zone growth, the
temperature, and the pressure are made as follows:
D
T
n
=
n
/-Do
1 - Da
T - Tmin
Max - Tmin
The n subscript therefore denotes the normalized
variable. The fiber deflection, v, and reaction product
thickness, r, are normalized using the parameters vCti,
and rcrit. To determine v,,~, we note that the fiber
failure probability, $r, in a unit cell is a one-to-one
tThe dynamics are, in comparison,
changes in pressure.
very mildly
affected
by
VANCHEESWARAN et al.: OPTIMIZING THE CONSOLIDATION
(a) Path Planned Simulation for TMAI-4 V / SCS-6
(b) Temperature Schedule
the examples that follow. For the other (Ti-24All
11Nb) matrix a critical value of 1.O pm is obtained
from experiments [30].
Step 2. Agglomeration.
The individual material
microstructural states, xh, for the 10 unit cells can be
combined with relative density and fiber-matrix layer
thickness to give an overall material microstructural
state;
(c) Pressure Schedule
The dynamics are then described by a system of
(twelve) ordinary differential equations
fgfffj$~/
!I
4009
(9)
2x3
Time
4w
603
0
(min)
xx)
Time
Nm
6cm
(mln)
Fig. 7.Path planned optimal process schedule ((b) & (c)) and
resulting microstructure trajectory (a) for the consolidation
of a Ti-6AMV/SCS-6 composite to the goal state X,. A
fiber deflection of 0.47 vcnr,results in 0.15 breaks per meter
of fiber.
function of the fiber deflection, v,
Note we assume a uniform temperature and pressure
across the sample so that each unit cell uses the
same process environment input, rl. This assumption
could be easily relaxed and the path-planning
method applied to samples containing temperature
gradients.
Step 3. Cascade connection. A cascade connection
of the material model (9) and the HIP machine
model (3) is performed to get the overall plant
model:
(8)
4
;i;=”
where I is the beam length in the unit cell, df is the
fiber diameter, Ef the fiber modulus, oref is the fiber
reference stress and m the Weibull modulus [18]. A
critical deflection, vent, corresponding to a failure
probability r&, of 0.1 is found by solving (8) for v.
If the condition v > v,,,~ were ever to occur, the
composite would be severely damaged and the
process schedule deemed worthless. For all reasonable goal states our choice for vccitensures that u,
varies on [0, 11. Recall that deflections for an entire
set of unit cells, each with a different beam length,
are evolved in time. Because the failure probability
depends on the (square of the) beam length, each
cell has a different vcrlt (Table 1).
The critical reaction product thickness, rent, has
been determined from fiber push-out experiments on
consolidated HIP specimens, For a typical SCS-6
fiber, it was found in [30] that sliding resistance
between fiber and matrix rapidly increases as r
increases above a critical value. The exact value
depends on the alloy system. We have chosen
rCrlt
system in one of
= 0.6 /*m for a Ti-6Al~V/SCS-6
which, letting
XII
X=
[I rl
and
F(X, U) =
[
Fdxh,
u9)
1
can be written simply as
z
= F(X, u)
(10)
Step 4. Local afine approximation. An affine
approximate model can be constructed around the
current material microstructural
state and current
process environment by taking a first-order Taylor
approximation to F(X, u) in (10).
4010
VANCHEESWARAN
et al.:
OPTIMIZING
If x” is the current plant state, we can denote
deviations from the current state by:
THE CONSOLIDATION
forward starting from f(O) = 0 (recall that 2
represents deviations from current) yields
(15)
N, - 1
&At) = c 6(Z+ 6A)Nf-‘-k(Bu(k)
k=O
D-D’
vcelll- VI,,,
&II2- l&Z
&elllO
-
+ C) (16)
where N, = At/a. Hence if at time tCthe model state
is x” and the HIP machine is commanded for a finite
horizon interval At with the slew rate control
u(t) = u(k)
&,,0
r - rC
T-T’
P - P”
for kS G t G (k + 1)6
k = 0, 1,2, . . . , $ - 1
(17)
we have the approximation
Then
d8
_
dt=‘4x+Z?u+c
(11)
X(t’+ At) x x’+
iv- I
c @I+ 6A)*-‘k=O
where
VW)
+
C)
(18)
for what the model state X will at the end of the
horizon.
Consider (18) as a mapping
and
@:{u(k)p=-,,’-+ X(t” + At)
,‘f = =(X9 u)
8X
B
=
mx>
u)
au
c
(12) taking commanded
X=X+=0
=
(13)
X=xC,U=O
F(F,
(14)
0)
gives an affine local approximation
to (10). The
derivatives required to evaluate the matrices A and B
and the vector C can be computed once, off-line,
using MATHEMATICATM or another similar symbolic manipulation package. Since x’ cannot be an
equilibrium point of the plant model unless T’ and P
are both identically zero, C appears additively in (11).
Since the ideal HIP machine model is linear, and
the affine approximation of a linear model is just
itself; there is no need for “-” to appear above the u
in equation (11) and thus the current HIP machine
input uc can be taken as identically zero. If a
non-ideal HIP machine model were used, then uc and
6 = u - uc would be used.
Step 5. Discretization in time and approximation of
&?‘(Fj. Discretizing (11) in time by Forward Euler?
gives the recursion
T((k + 1)6) = (I + &4)8(kS) + 6Bu(k6) + 6C
(15)
where 6 is the chosen sampling interval. Propagating
tAny other’continuous
to discrete transformation
could be used, for example Tustin (Bilinear).
HIP machine inputs to future
states. We denote by L& the range of @ when
subject1 to machine constraints (2). Thus, WA,is an
approximation to the set of Xs that can be reached
in a time At starting from x”. Since @ is an affine
map, and the constraints (2) are linear, Wa, is a convex
polytope with important consequences for optimization.
The values of v in WA,correspond, through (8), to
values of failure probability, &, in each cell and thus
through NJ/, D) to a value of N/ for the entire layup.
In this way the WA,set yields an approximation for the
conically shaded region shown in Fig. 3. As the
solution to a well-behaved differential equation is a
diffeomorphism from the space of initial states into
the motion space, the approximation
(18) is
asymptotically accurate as At and 6 go to zero.
Step ‘6. Local planning by convex optimization. The
approximately reachable set, 9&, , defined by (18) and
(2) is a convex polytope, so locally feasible controls
can be found by solving a convex feasibility problem.
Moreover, locally optimal controls can be found
by minimizing a convex objective function over
this convex polytope. The decision variables in
the program
are the values
of u(k)
for
k = 0, 1, 2, . . . , N, - 1. Since we seek to steer
X,,(t) + X,, a logical choice of convex objective is a
weighted Euclidean distance between X,,(t) and X,
which leads to a quadratic program which can be
efficiently solved.
method
$The constraints
are discretized
in time the obvious
way.
VANCHEESWARAN
The objective
solver is
r
presented
wD x
to
the
(8C.N+ DC-
et al.:
local
D,)>
OPTIMIZING
1
Trdk)
[ Pz&)
4011
4. IMPLEMENTATION
convex
which is a weighted sum of the distance between
the goal state and the projected
future material
states (based on the affine approximation).
The
factor j/N, in the summation
of (19) places higher
weight on plant states farther in time from the
current
state x’; and makes the planner
more
aggressive
in reducing
the total duration
of the
process schedule. The planner attempts to densify
the layup while not deflecting/fracturing
fibers
and
not
accruing
unacceptable
amounts
of
reaction
product.
The weightings
Wo, W,,, and
W, in the objective function could allow a tradeoff between
these competing
goals to be made
quantitatively.
The method
even allows
crossweightings
between
the states
because
it only
W = W > 0 and not that
W is
requires
that
diagonal.
Necessary and sufficient conditions
satisfied by a
minimizer of (19) subject to (18) and (2) are given
by the Kuhn-Tucker
equations,
and the solution
can be found using the constrained
optimization
routine
in the Optimization
Toolbox
in MATLABTM or any number
of other commercially
available convex programming
packages. We mention that sequential quadratic
programming
(SQP)
methods
are applicable
and they give superlinear
convergence
by using a quasi-Newton
updating
procedure
for accumulating
second
order information.
The method
described
is thus suitably
fast for on-line implementation
[37].
Step 7. Local control application,
evolution of
state. repetition. Once optimal values for
u(k) =
THE CONSOLIDATION
To illustrate
the method’s
applicauon,
we will
show path planning analyses to goal states for two
different TMCs: Tip6A14V-matrix/SCS-6-fiber
and
the Ti-24Al-1 lNb-matrix/SCS-6-fiber.
The desired
goal for each material system is given in Table 2. The
goal fiber deflection and reaction product thickness
are stated in the normalized variables defined in (7).
It should be kept in mind that, because v,,,, differs
among unit cells (refer to Table l), the single table
entry for ~1” represents
different
goal deflection
distances for each unit cell in the model.
To simplify interpretation
of the results,
the
objective weights in (19) were selected to be equal
(W, = 10, W, = 10, and
W, = 10). Hence,
the
planner
has equal concern
about
densification,
fiber/matrix reaction, and fiber fracture. A sampling
interval of 1 min with a look ahead horizon 10 min
(6 = 60 and At = 600 in equation (16)) was used for
all the calculations.
Hence, N, = 10, and so in each
Step 6 performed,
the planner solves a quadratic
program
consisting
of 20 variables with 40 constraints. This could be computed in about 1 s even on
a multi-user Sun SPARC20TM station. The pressure
rate was additionally constrained to be zero until the
relative density increased enough to give a non-zero
coordination
number for at least one unit cell. This
Path Planned
Simulation
for Ti-6AI-4V / SCS-6
xg=[‘j]+$g
1
are found, the control (17) is applied for the interval
6t to the full nonlinear plant model (10). This is then
integrated
forward in time by a suitable numerical
method (e.g. Runge-Kutta)
to give a new current
state x” and we then return to Step 4.
The stopping criterion is a suitably small optimum
objective value in Step 6, indicating nearness to the
goal X,, or passage into the forbidden zone 9(X,)
indicating X, cannot be reached.
0
200
Time
400
(min)
SW
0
xi)
Time
400
600
(min)
Fig. 8. The path planned optimal process schedule and
microstructural
state
evolution
for Tik 6A14V/SCS-6
composite consolidation
to a goal state, X, = [0.999,0.15/m,
0.48 pm].
VANCHEESWARAN et al.: OPTIMIZING THE CONSOLIDATION
4012
(a) Path Planned Simulation for F24AI- 1 1Nb/ SCS6
Path Planned
Simulation
for Ti-24AI-11Nb
/ SCS-6
xg=~]+J
(b) Temperature Schedule
Time
(c) Pressure Schedule
Time
(mln)
Time
(min)
Time
(min)
Time
(min)
Time
(min)
(mln)
Fig. 9. Path planned optimal process schedule and
microstructure trajectory for a Ti-24Al-1 lNb/SCS-6
composite.
constraint forces the planner to begin ramping the
temperature before beginning the pressure ramp.
The optimal process schedule is governed by the
HIP machine dynamics (l), the HIP machine
limitations (2), and the material dynamics (9). To
illustrate the approach to process path optimization,
we present detailed numerical results for path-planning (i.e. steering X” to X,) for the Ti-6A1_4V/SCS-6
and Ti-24Al-1 lNb/SCS-6 matrix composite systems
consolidated
using one of four different HIP
machines with the physical limits given in Table 3.
Fig. 10. Path-planned optimal process schedule and
microstructural state evolution for a Ti-24Al-1 lNb/SCS-6
composite.
rapidly approach their goals, the planner actively
adjusted the pressure. Because of the differing
dynamics of each cell, some cells can slightly
overshoot their deflection goals while many others
barely touch their goals. In trying to keep all the
deflections near their goals while still pursuing
densification, the planner “hunts” in pressure as it
searches for a near optimal strategy. As Stage 2
densification is reached, the stiffness of the cells
rapidly increase, fiber fracture becomes less important, and a “race” then develops between “densification to theoretical density” and “reaction product
5. PLANNED PATHS FOR THE TWO COMPOSITES
5.1. Ti-6Al-4V/SCS-6
System
The Ti_6A1_4V/SCS-6 system is readily processable. Here we seek to find the best, among the many
possible process schedules. Placing this material
system in a “nominal” HIP machine (Type 1 in
Table 3) and applying the path planning method
gives the result shown in Figs 7 and 8. For this case,
the desired density was reached while the goal states
for fiber deflections and reaction product thickness
were also held below their critical values. Examination of Fig. 8 indicates the planner computed a very
interesting process schedule. First the temperature
was ramped at the maximum (machine limited) rate
until the soak temperature was reached. The planner
then began ramping the pressure at the maximum
rate, However, as the fiber deflections began to
UNREACHABLE
0.0
0.1
0.2
0.3
Reaction layer thickness
0.4
0.5
(pm)
Fig. 11. Projections of the reachable states on the fiber
fracture density-reaction layer thickness plane for a
Ti-6Al_4V/SCS-6 composite.
VANCHEESWARAN
et al.:
OPTIMIZING
THE CONSOLIDATION
4013
Ti-24Al-11 Nbl SCS-6
\ \
25 -
Enhanced l’ HIP (1.5i)
20 -
15-
10 -
5-
OL
0.0
Reaction layer thickness
(pm)
Fig. 12. Projections of the reachable states on the fiber
fracture density-reaction
layer thickness plane for a
Ti-24Al-1 lNb/SCS-6 composite.
layer growth to the critical value”. Since density
depends on pressure while reaction layer thickness
does not, the planner ramps the pressure at the
maximum
rate. It is plausible
that the process
schedule constructed
is nearly time optimal since the
planner kept a constraint
active at all times.
5.2. Ti-24Al-1
lNb/SCS-6
50
.s?
E
40
$
$
30
5
!I
20
f
1.0
1.5
Reaction layer thickness
2.0
(km)
Fig. 14. Projections
of the reachable
states on the fiber
fracture
density-reaction
layer thickness
plane
for a
Ti-24Al-1 lNb/SCS-6
composite
consolidated
for (hypothetical) HIP machines 1 through 4.
to constrain the growth of reaction product. As the
planner dropped the temperature, the authority of the
pressure input over density decreased while authority
over the fiber fracture increased. Thus, the planner
also dropped the pressure, and it became impossible
to reach a better goal state.
5.3. Achievable
states
Given the initial condition,
X0, for .I layup, it is
interesting to identify which microstructures
can be
reached with a given HIP machine and material
system. The planning method can be used to find this
reachable region by several methods. The simplest is
to just repeatedly run the planner for different X,.
Those which the planner succeeds in reaching are
placed in the reachable region. For clarity, we show
two-dimensional
projections of the process region by
varying two of the goals while holding the third
constant at its target value.
Figure 11 shows the reachable states for Ti-6Al4V/S CS-6 using HIP machine 1. Three regions are
shown corresponding
to three different density goals.
Figure 12 shows similar calculations
for a Ti-24Al
60
.E.
z
0.5
7
System
When the Ti-6A14V
matrix is replaced with a
more creep resistant Ti-24Al-1lNb
matrix, it is no
longer
possible
to successfully
consolidate
the
composite. Using HIP machine 1 (but with a larger
7’,,, of 1000°C to accommodate
the more creep
resistant matrix), the results of path-planning
are
shown in Figs 9 and 10. A comparison
of the path
planned process schedule with the nonplanned
“ramp
and soak” schedule in Fig. 2, reveals that both result
in a reaction product thickness in excess of the 1 /*m
goal. However. the path-planned
schedule is better in
that it achieves less fiber fracture for the same
reaction product thickness. When we attempted
to
decrease the reaction product thickness goal below
1 pm, the planner attempted to drop the temperature
‘i
STATES
EnhancedT8.P
10
0
0.1
0.2
0.3
Reaction layer thickness
0.4
(pm)
Fig. 13. Projections of the reachable states on the fiber
fracture
density-reaction
layer
thickness
plane
for a
Ti-6A1_4V/SCS-6
composite
consolidated
for (hypothetical) HIP machines 1 through 4.
Reaction layer thickness
Fig.
(pm
15. Projections
of the reachable
states
increasing reference strength.
I
for fibers
of
VANCHEESWARAN et al.: OPTIMIZING THE CONSOLIDATION
4014
-7 10’
.E.
.g loo
v)
5
lo-'
P,
9 IO-2
g
m 104
.k
8
g
10-4
LL
10-S
4
5
6
7
a
9
Fiber reference strength (GPa)
10
Fig. 16. The dependence of fiber fracture (incurred during
densification to a density of 0.999) upon fiber reference
strength for three reaction layer thicknesses. Increasing the
reference strength from 4.5 to about 7 GPa reduces the
fracture density from 60 to about 0.1 breaks/m for
r = 0.96 pm and D = 0.999.
1lNb system, again for the nominal HIP machine
(Type 1). The curves quantitatively illustrate the
tradeoff between reaction product thickness and fiber
fracture density at fixed densification, and show a
reduced reaction product thickness and fiber fracture
density can only be obtained at the expense of the
relative density.
A comparison of Figs 11 and 12 quantitatively
reveals how much more processable the Ti-6A14V/
SCS-6 is compared to the (more creep resistant)
Ti-24Al-11Nb matrix composite. If a relatively dense
composite (0.999) is desired with less than 1 pm of
reaction product and 1 fiber fracture per meter,
Ti-6A14V/SCS-6
is an acceptable material system
choice whereas Ti-24Al-1 lNb/SCS-6 is not if Type 1
HIP machine is the only one available.
5.4. HIP machine performance effects
The path-planning method can be used to explore
the effects of HIP machine limitations on the region
of states that can be reached. To illustrate, consider
the four HIP machines listed in Table 3. Machine 1
has temperature and pressure rate capabilities that
are similar to many current HIP machines. Machine
2 is an “enhanced pressure HIP” with slew rates for
temperatures similar to a conventional HIP, but with
a pressure slew rate increased by 100%. Machine 3 is
an “enhanced temperature HIP” which has the same
pressure rate capabilities as machine 1, but the
temperature slew rate is increased by 50%. Machine
4 has both enhanced temperature and pressure
capabilities. Each machine has the same maximum
pressure and temperature.
Figures 13 and 14 show reachable regions for each
machine. They show that the nominal HIP has the
worst performance. Increasing the temperature slew
rate slightly improves the reachable region, while
increasing the pressure slew rate gives a more
pronounced performance improvement. Machine 4,
with both enhanced temperature and pressure slew
capabilities, clearly results in the best performance.
As the heating and pressurization rates continue to be
increased, we find that the regions of reachable states
cease to expand, and the states that can be reached
are effectively material limited. Thus, the methodology reveals that no HIP machine will enlarge the
processable region of a Ti-24Al-1 lNb/SCS-6 composite enough to achieve a useful (i.e. r < 1 pm,
N, < l/m and G > 0.999) composite. To process a
Ti-24Al-llNb/SCS-6
composite to a useful goal
state microstructure,
it is necessary to solve the
system’s materials limitations either by decreasing the
effective rate (i.e. the kinetics) of the fiber-matrix
reaction (e.g. coating the fiber with a low diffusivity
material), by developing a stronger fiber or reducing
the monotapes surface roughness.
5.5. Fiber property effects
The path-planning method can be used to explore
how much of a change needs to be made to overcome
material limitations, i.e. it can be used to design
composite systems that will have acceptable microstructures after processing with available equipment.
To illustrate, we examine the role of fiber strength.
Figure 15 shows the dependence of fiber fracture on
the fiber’s reference strength for the intermetallic
composite system. As the strength increases, fracture
is rapidly lessened and the region of reachable states
expands. In the limit, as the strength tends to infinity,
fracture ceases to be an issue and the states that can
be reached almost completely fill the microstructure
space.
In practise, the developer of a fiber is more
interested in determining just how much extra
strength is needed to successfully process a composite
system. Figure 16 shows one way this can be
addressed. By plotting the fiber fracture density
against the reference strength for three differing
reaction layer thicknesses, we see that by increasing
the reference strength, Q, from 4.5 GPa to 7.0 GPa,
the fiber fracture density can be reduced from around
60 to about 0.1 breaks per meter while maintaining r < 1 pm and achieving complete densification.
Obviously, the procedure can be repeated for fibers
with reaction inhibiting (diffusion retarding) coatings. This will result in the design of a composite
system that can be successfully processed, and the
process cycle that accomplishes it.
6. DISCUSSION
By combining
micromechanics-based
process
models for composite consolidation with a model
predictive planning method, we have been able to
calculate optimal process trajectories that evolve
composite systems to goal states that result in a
chosen level of mechanical performance. Because the
approach uses micromechanical models, it can be
applied to any composite system (fiber/matrix type)
for which basic mechanical properties are known or
VANCHEESWARAN
et al.:
OPTIMIZING
can be estimated. We have illustrated the approach
by a detailed analysis of two matrix systems
(TidAl4V
and Ti-24Al-1 lNb), both reinforced
with the same SCS-6 fiber. Optimal process schedules
resulting in a goal state with acceptable mechanical
performance have been found for the Ti-6Al4V/
SCS-6 system. The same approach reveals that no
process will similarly transform the states of a
Ti-24Al-11 Nb intermetallic matrix composite with
the same fiber system. Thus, the optimization
approach has enabled the rapid determination of an
infeasible composite system, and this idea could be
used to filter the many potential fiber-matrix systems
during materials selection.
a) Ti - 6A/ - 4WSCS
THE CONSOLIDATION
4015
The approach also enables a systema tic evaluation
of the effect of the consolidation
equipment
performance upon the outcome of optimal processes.
Increasing the heating and pressurization rates of a
HIP enlarges the reachable states and enables better
performing composites to be synthesized. However,
even after increasing the heating rate by 50% and
doubling the maximum pressurization
rate, the
microstructural outcome for the best Ti--24Al-1 lNb/
SCS-6 schedules still result in either too much fiber
bending or too much fiber-matrix reaction. As the
performance of equipment is enhanced, we find the
set of reachable states becomes materials limited.
Using the modelling plus optimization strategy, we
- 6 COMPOSITE
N, = 1 fiber fracture per meter
= 0.45pm
W
Ti - 24Al-
11Nb / SCS - 6 COMPOSITE
fracture
E
a0
2
60
.
permeter
Z.
40
Q)
5 20
p!
1
O
$
=lpm
-20
Fig.
17. Microstructure
failure
surfaces
in process space for the Ti-6A1&4V/SCS-6
SCS-6 composite systems.
and Ti-24AlkI
lNb/
4016
VANCHEESWARAN et al.:
a) Ti - 6AI - 4WSCS
2
OPTIMIZING THE CONSOLIDATION
b) Ti-24Al-lINb/SCS-6
-6
80
.
$
60
z
40
a,
E 20
g!
3
O
z?
f
L”
p
0
200
400
600
800
1000
,II
0
2
400
600
800
1000
Temperature (“C)
Temperature (“C)
.
$
200
80
60
r
40
Q)
g
20
p!
I
0
n,
h -201
0
I
100
I
200
I
400
I
300
I
500
0
100
200
300
400
500
400
500
Time (mins)
Time (mins)
f i 0.45pm
loo0 -1
800
600
400
200
0
-0
Time (mins)
100
200
300
Time (mins)
Fig. 18. Projections of the failure surfaces in process space for the Ti-6A1_4V/SCS-6 and
Ti-24Al-1 lNb/SCS-6 composite systems. The grey zone shows the projection of the fiber fracture failure
surface corresponding to 1 fracture/meter onto the 2-D plane.
have then been able to systematically explore changes
to the properties of the composites components
that enhance the processability. As an example,
it was shown that an increase in fiber strength
from 4.5 to 7 GPa would be sufficient to achieve
a Ti-24ACllNb
matrix composite microstructural
state with good mechanical performance. The region
of reachable states is also enlarged by reducing the
surface roughness of the monotapes [18].
These results have revealed that the task of finding
a process schedule is fundamentally driven by the
trade-off between D, Nf,and r. The modelling and
optimization methodology provides a means for
dynamically making this trade-off, but at a considerable computational
expense. Examination
of the
optimal strategies for both the composite systems
reveals the optimal process schedules to have a
similar form, Figs 7 and 9. First, the temperature was
always raised before the pressure, the pressure was
then applied at the maximum rate until the density
was about 0.7, whereupon the pressure rate was
reduced for a prolonged period (until the density
exceeded 0.9) before being increased to the maximum
available value.
VANCHEESWARAN
et al.:
OPTIMIZING
A short cut to a near optimal process can be found
by viewing consolidation as a process in which fiber
fracture and fiber-matrix
reaction
are failure
mechanisms for the process. For processes of the type
shown in Figs 7 and 9, it is possible to plot surfaces
of constant reaction layer thickness and fiber fracture
density in a three dimensional (Prate- T - t) process
space. Figure 17 shows the result for both the
TidAl4V
and Ti-24Al-11Nb
matrices reinforced
with SCS-6 fibers. For the former system, the failure
surfaces correspond to about 1 fracture/m and
Y= 0.47 pm, while for the latter, r was increased to
1.0 pm. When optimal trajectories (shown by the
black line) are superimposed, the optimal process is
seen to have found a way to avoid the fracture and
reaction layer failure surfaces while still densifying
the composite completely. Where this is possible (the
Ti-6Al4V matrix case), we see that the trajectory is
initially pressure rate limited by the fracture surface.
However. after about 200 min of consolidation, the
P,,,, bound disappears (see Fig. 18(a)) and is replaced
by the need to avoid a reaction layer failure. The
relatively low consolidation temperature and reactivity of the Ti-6A14V
matrix enables this to be
accomplished. Although the failure surfaces have a
similar form in the Ti-24AllllNb
matrix system, the
higher temperature needed to accomplish densification causes a more rapid growth of reaction product,
Fig. 18(b). Thus, by the time the process turns the
corner of the fiber failure surface and the P,,,,
constraint is relaxed (again at about 200 min), the
reaction had progressed too far to allow complete
densification before intersection of the reaction layer
failure surface. Thus, optimal processes are therefore
seen to hug the first failure surface encountered and
to then jump to follow the second failure surface until
complete densification is accomplished. By mapping
the failure surfaces in a process trajectory space, a
convenient graphical method can be used to estimate
a near optimal process.
We note that once an appropriate model is
developed, the method could be extended to address
optimization of the residual stress that forms on
cooling (due to coefficient of thermal expansion
mismatch between the fibers, the matrix and the
tooling). This stress is roughly proportional to the
consolidation temperature and only weakly affected
by cooling rate. Since the maximum temperature
calculated above is constrained by the need to limit
growth of the reaction layer, we anticipate that the
optimal process trajectories are close to those that
also minimize
residual stress (consistent
with
achieving the other goal states).
This path optimization approach could be used to
estimate optimal trajectories for many other processes where the various microstructural states have
conflicting dependencies upon the process variables.
The failure surfaces so calculated
define the
fundamental
performance
limits for a process
providing a rational mechanism for process selection.
THE CONSOLIDATION
4017
7. CONCLUSIONS
A consolidation path planning method has been
developed to determine
near optimal
process
schedules that guide the microstructure of a fiber
reinforced composite to a pre-selected gcal state. The
path-planning method uses constantly updated local
linear approximations
to micromechanics
based
models of consolidation together with constrained
convex optimization in a receding horizon philosophy to compute the optimal control action time
series for any composite system with known
mechanical properties. The methodology naturally
constrains the planner’s control acti,ons by the
physical limitations of the consolidatior. equipment,
and has been applied to the consolidation of several
titanium matrix composites. It identifies a viable
process path that results in good mechanical
performance
for a Ti-6AlMV/SCS-6’ composite
system and shows that no such path exists for a
similar composite with a Ti-24Al-11Nb matrix. By
varying the goal state and repetitively performing
optimal path planning calculations, attainable regions in the microstructure space are constructed for
each composite system. This attainable space depends
upon the composites component mechanical properties, the monotape geometry and the physical
limitations of the process equipment. It is shown that
the path planning method can be used to redesign the
processing equipment the initial microstructural
state, the fiber’s reaction inhibiting coating, the fiber’s
strength or the surface roughness so that difficult
material systems, such as the Ti-24Al-i 1Nb matrix
composite, can be successfully processed. A simple
graphical method is proposed for the estimation of
near optimal process trajectories for processes where
the microstructural states have conflicting dependencies upon the process variables.
AcknoM,[edgements-Weare grateful to R. Kosut for helpful
discussions
about
this research
and to 11. Elzey for
suggestions in implementing
the models. This work has been
funded by DARPA
through
a contract
with Integrated
Systems Inc., Santa Clara, CA (Dr Anna Tsao. Program
Manager).
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