Materials Science and Engineering A244 (1998) 58 – 66
Model-based optimization of consolidation processing
Ravi Vancheeswaran, Haydn N.G. Wadley *
Intelligent Processing of Materials Laboratory, Department of Materials Science and Engineering, Uni6ersity of Virginia, Charlottes6ille,
VA 22901 -2442, USA
Abstract
The performance of fiber reinforced titanium matrix composites (TMC) made by consolidation of spray deposited monotapes
is strongly influenced by the processing conditions used. This high temperature consolidation step must simultaneously increase
the relative density while minimizing fiber microbending/fracture and the interfacial reaction product layers at the fiber–matrix
interface. These three microstructural variables have conflicting dependencies upon the consolidation process variables (temperature, pressure and time), and it has been difficult to experimentally identify process pathways that lead to composites of acceptable
quality (where the fiber damage and the reaction layer thickness are kept below some bounds, while matrix porosity is eliminated).
Here, models for predicting the microstructure’s dependence upon process conditions (i.e. the time varying temperature and
pressure) are combined with consolidation equipment dynamics to simulate the microstructure evolution. We then introduce the
idea of process failure surfaces and show how a model predictive control algorithm, is able to design ‘locally’ optimal process
cycles that minimize fiber damage, reaction product layer thickness and porosity. As an example, we explore the planning of
process schedules that process failure surfaces for several TMC systems. © 1998 Elsevier Science S.A. All rights reserved.
Keywords: Model-based optimization; Consolidation processing; Titanium matrix composites
1. Introduction
Metal matrix composites are attractive alternative
materials to monolithic titanium and nickel base alloys
in gas turbine engines because of their higher specific
stiffness, strength, crack growth resistance, creep rupture life and fatigue endurance [1 – 6]. Those based upon
conventional or intermetallic titanium alloys have attracted considerable attention because they allow a
radical redesign of high performance aircraft engines
[7,8]. Many laboratory scale manufacturing processes
exist for the manufacture of these materials; however,
they all involve a two step sequence that seeks to reduce
the aggressive chemical reaction between all liquid titanium alloys and most candidate fibers. First a monotape is manufactured by plasma spray deposition,
slurry casting or physical vapor deposition coating of
colinear fiber arrays [9 – 14]. The tapes are then cut to
shape, stacked to create an appropriate fiber architecture and consolidated by either vacuum hot pressing
(VHP) or hot isostatic pressing (HIP) [15]. Ideally, this
* Corresponding author. Tel.: +1 804 9825670; fax: + 1 804
9825677; e-mail: [email protected]
0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved.
PII S 0 9 2 1 - 5 0 9 3 ( 9 7 ) 0 0 8 2 6 - 5
results in a near-net-shape composite component with
no residual porosity (to avoid premature matrix failure
during static or fatigue loading), minimal fiber microbending/fracture (to avoid degrading composite
strength and creep/fatigue endurance) and a limited
thickness of reaction product at the fiber–matrix interface (to avoid increases in interfacial sliding stress or
loss of fiber strength). Unless a ‘goal state’ combination
of these microstructural attributes (i.e. pore concentration or equivalently, relative density, fiber microbending stress/fracture and reaction product layer
thickness), the competitive advantage of these composites over other materials will not be realized.
A dynamic model has been developed that simulates
the evolution of these three microstructural parameters
for any hypothetical process cycle (i.e. temperature and
pressure schedules) for a lay-up of plasma sprayed
monotapes [15]. It reveals that the goals for each microstructural parameter have conflicting dependencies
upon the variables of the process (temperature and
pressure). For example, densification (the elimination of
matrix porosity) of plasma sprayed monotape preforms
is most quickly accomplished by consolidating at the
highest temperature and pressure. But fiber microbend-
R. Vancheeswaran, H.N.G. Wadley / Materials Science and Engineering A244 (1998) 58–66
59
Fig. 1. Simulation of the evolution of the microstructural states using the process schedule shown in (b).
ing/fracture can only be minimized by consolidating at
high temperatures while applying the pressure very
slowly (i.e. using long duration cycles), and reaction
product layer thickness can only be reduced by shortening the high temperature exposure time (i.e. by using
short, cold consolidation cycles).
Dynamic models of this type enable a mapping from
the process state space (with temperature and pressure)
to a microstructure state space (in this case with axes of
relative density, fiber fracture density and reaction layer
thickness). An example is shown in Fig. 1 for the
Ti–6Al–4V/SCS-6 system (refer to [15] for the monotape geometry and matrix/fiber thermophysical properties used for the calculation). The successful
consolidation processing pathway is seen to be a nonlinear mechanism for transferring the microstructural
states from an initial configuration to a final ‘goal’
state. From this viewpoint, it no longer matters if
pressure or 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 process variable space (Fig. 2) that
takes the lay-up from its initial state to the goal state.
The dynamic simulations could be used as a trial and
error approach to find process pathways that accomplish this. For the more ‘processible’ composite systems
like Ti–6Al–4V/SCS-6, several potentially successful
cycles have been identified in this way [15]. However,
the processability of this system originates from the low
resistance to creep of the matrix at typical processing
temperatures. Efforts to extend the upper service temperature by using more creep resistant alloys rapidly
degrades the processability of the composite, and it
becomes increasingly difficult to find acceptable process
60
R. Vancheeswaran, H.N.G. Wadley / Materials Science and Engineering A244 (1998) 58–66
schedules, or to be sure at the outset of a materials
development campaign, that one even exists.
Here the models for microstructural evolution are
coupled with a model predictive control (MPC) algorithm to develop a methodology for calculating locally optimal process pathways. Control design
methods based on the MPC concept have been studied
and have found wide acceptance in industrial applications [17,18]. MPC techniques use a multistep predictor
and a receding horizon philosophy to predict the optimal set of control variables using the present knowledge of the process and candidate planner actions. This
is especially useful because the microstructural variables
like relative density, fiber fracture and reaction layer
thickness are irreversible processes. Therefore the planner needs to be ‘conservative’, since the consequences
of a ‘wrong’ decision by the planner can make the
microstructural states overshoot the desired goal
state thereby rendering the composite unusable. However, if extensive use is made of predictions well into
the plant rise time, disastrous overshoots can be
avoided.
The plant can be modeled as two distinct parts, the
machine model (i.e. HIP/VHP) and the material
(microstructural evolution) model [15]. The interconnection and interaction between these two models and
the automatic planner is shown in Fig. 3. Mathematical models capturing the dynamic behavior of the
machine and the material take the form of coupled
systems of nonlinear ordinary differential equations
[15,19–23].
Fig. 2. Schematic of the reachable set from a given point in state
space
Fig. 3. Block diagram of the control system.
2. The path planner
A simulated response of a matrix alloy Ti–6Al–4V/
SCS-6 silicon carbide fiber titanium matrix composite
(TMC) to a commonly used ‘ramp and soak’ temperature and pressure schedule is shown in Fig. 1. This
particular schedule fails to achieve full density, causes
about nine fiber fractures per meter of fiber and results
in the growth of :0.3 mm thick reaction layer at the
fiber–matrix interface. It would result in a composite of
poor mechanical performance [2] due to the large number of fractures.
The optimization problem that we pose is schematically illustrated in Fig. 2. The objective is to reach a
goal state at the completion of the process. At any
moment in the consolidation process, the material’s
state can be represented as a point in a relative density,
fiber deflection, reaction layer thickness space. It will
have reached this point along a path originating in the
lower left corner of the space. The consolidation process is ‘irreversible’1 hence all future feasible paths are
enclosed in a conically shaped region, called the reachable set, originating from the current point. If the goal
state is not inside this region, then it is impossible to
process the composite to achieve that goal. One seeks
to find the optimum process path in state space that
will take the current material state to the desired goal
state.
Ideally the indicated planning problem would be
tackled by computing the exact reachable set of microstructural states for a given set of process states.
Then a complete path can be found in a single shot
beginning at the starting conditions and ending at the
1
The relative density, fiber fracture density and reaction zone
thickness are non-decreasing functions of time.
R. Vancheeswaran, H.N.G. Wadley / Materials Science and Engineering A244 (1998) 58–66
goal state. However, solving the problem this way is
very difficult because the problem is computationally
intensive (NP Hard). Instead, as described below, a
predictive planning method is used where a series of
small steps are taken toward the goal state and at each
step a viable approximation to the exact reachable set is
computed.
A block diagram showing the planner is presented in
Fig. 3. The path planning system accepts as input a
vector signal, xg, for the goal state that contains desired
final values for the relative density, the fiber deflections
of a set of representative unit fiber cells and the reaction zone thickness. The planning system also accepts
as input the state vector, xh, giving measured or estimated values of the microstructural variables (density,
fiber deflection and reaction layer thickness), and the
process environment vector, h, giving the current machine temperature and pressure. The planner calculates
appropriate temperature and pressure slews and commands the HIP or VHP, thereby constructing a temperature and pressure schedule that will steer the current
material state to the desired goal state, or keep the
current material state as close to the goal state as
possible. Mathematically, therefore, the planner’s task
is to calculate u(t) such that xh (t) “ xg (see [16] for
details).
61
as a Taylor series and retaining only first order terms.
The discontinuity (plasticity) is not included in this
step. Letting x̃= xh − x ch and h̃= h− h c denote deviations from the current material state and machine environment, respectively, the linearization of the vector
field about x ch, h c can be written
dx̃h
= Ahx̃h + Bhh̃+ fh.
dt
(2)
3.3. Cascading
Once the linearization of the material model is calculated for a given operating point, x ch, h c, the overall
material-machine model can be written
dx̃
= Ax̃+ Bh̃ + f.
dt
(3)
Numerically integrating the system of differential Eq.
(3) by the forward Euler algorithm leads to the recursive relationship
x̃((k +1) t)=(I+ tA)x̃(kt)+ tBu(kt)+ tf
(4)
where t is the integration time step. Solving the difference Eq. (4) starting from x̃(0)= 0 (recall that x̃ represents deviations from the current operating point)
yields
n−1
x(nt)= % t(I+ tA)n − 1 − k(Bu(k)+ f ).
3. Path planner design
3.1. Normalization
To begin, a change of variables is made in the
material model so that each component of the states xh
and inputs h lie in the interval [0, 1]. This normalization
is done to provide good numerical conditioning in the
later stages of the path planning design where a convex
program is solved. Transformations of the density, the
unit cell deflections, the reaction zone growth, temperature and pressure are defined as follows
Dn =
=
D −D0 i
6i
r
T −Tmin
6 n = i rn =
T =
P
1− D0
6 crit
rcrit n Tmax −Tmin n
P − Pmin
Pmax − Pmin
(5)
k=0
(1)
where the n subscript denotes the normalized variable.
The parameters 6 icrit and rcrit are defined as the maximum (worst case) values that these states are allowed to
have. The critical value for the reaction zone is chosen
to be 0.6 mm and for the deflection states it is the value
that causes a failure probability of 0.1.
3.2. Linearization
A linear approximate model about the current operating point is computed by expanding the vector-field
Hence it is seen that at an operating point, [x ch, h c],
an approximation to the reachable set is given by the
range of the right hand side of Eq. (5) when subjected
to the machine constraints.
3.4. Con6ex sol6er
The relevant planning task is steering the material
state, xh, to the goal state xg, and so a logical choice of
convex objective is the Euclidean distance between
x(nt) and xg. This leads to a quadratic program which
can be efficiently solved.
The planner minimizes a weighted sum of the distance between the goal state and the projected future
material states (based on the linearization) given by
Nl
k
(x(kt)− xg )TW(x(kt)− xg )22
N
k=1
l
%
Nl
k
= %
k = 1 Nl
ÆW D × (Dg − Dl )k))2Ç
ÃW 6 × (6g − 6 il (k))2 Ã .
ÈW r × (rg − rl (k))2 É
(6)
The planner attempts to drive the relative density to
the desired final value while not fracturing fibers or
accruing a reaction product, and the weightings, WD,
62
R. Vancheeswaran, H.N.G. Wadley / Materials Science and Engineering A244 (1998) 58–66
Table 1
Goal states for the different material systems
Composite
8f
rcrit (mm)
D
6i
r
Ti– 6Al – 4V/SCS-6
Ti– 24Al – 11Nb/SCS-6
0.1
0.1
1.0
1.0
0.999
0.999
0.47 6crit
0.5 6crit
0.6 rcrit
2.0 rcrit
W6 and Wr in the objective function allow the trade-off
between these competing goals to be made quantitatively and precisely. Note that the method in fact allows
cross-weightings between the states because it only requires that W= W T \0 and not W to be a diagonal
matrix.
Once optimal values of temperature and pressure
slew rates are found, they are used as inputs to the full
machine and material (nonlinear) models (which now
includes the effects of plasticity) which are then integrated forward in time by a higher-order method (e.g.
Runge–Kutta) to give a new operating point which
includes the discontinuities of the vector field (plasticity). A linearization is then found about the new operating point and the entire planning design process is
repeated until either the goal state is reached or a final
time limit condition is reached.
4. Results
To explore the optimization approach, we choose a
goal state density of 0.999, fiber cell deflections that
result in a cumulative probability of fracture B 0.1
(thereby holding the fiber fracture density below a
reasonable bound of 1 fracture per meter), and a reaction layer thickness lie below 0.6 mm for a Ti–6Al–4V
matrix system and 1 mm for the Ti – 24Al – 11Nb system.
For weights, the selections, WD =10, W6 = 10, and
Wr = 10 were chosen. Hence, roughly, the algorithm
assumes an equal concern for densification, fiber/matrix
reaction and fiber microbending/fracture. A sampling
interval of 1 min was used (t= 60 in Eq. (5)), and a
look ahead horizon for the planner of 10 min (N= 10
in Eq. (5)). Therefore for each 1 min interval, the
planner solves a quadratic program of 20 variables and
about 40 constraints. This takes a very short time ( : 1
s) on a multi-user Sun SPARC-20™ station. A higher
weighting is given to states that are further away in
time from the operating point, thereby making the
planner aggressive in reducing process time to consolidate the matrix, since the process is completed either
when the density reaches 0.999 or the time elapsed is 10
h. An extra constraint was placed on the planner to
constrain the pressure rate at zero till the first fiber
deflection cell was created. The extra constraint caused
the planner to serially apply the temperature before
applying pressure. Since the critical fiber unit-cells of
interest are not created during the early stages of
densification, and therefore if this constraint is not
added, the planner makes the mistake of increasing the
pressure since it does not know of the existence of the
fiber cells. When the fiber cells are created, the planner
is not able to correct the pressure rate soon enough, to
reduce the deflection of the fiber cells beyond their
critical limit and this causes fracture (see Table 1).
4.1. The Ti– 6Al– 4V/SCS-6 system
A path for the microstructural variables to the given
goal state for the Ti–6Al–4V/SCS-6 composite monotape is computed. Fig. 4 shows a result where we
simulate this system with a nominal machine (machine
1) see Table 2.
In this case, the goal state density was reached and
the goal states for fiber deflections and reaction layer
thickness were held below their critical values. The
planner computed a very interesting and intuitive input
schedule. First the temperature is ramped at the maximum rate till the maximum machine temperature was
reached. At this time, the planner ramped the pressure
at the maximum rate till the bounds on the deflections
of the fiber cells were reached. Then the planner reduced the pressure rate so as to make sure that the
deflection bound is not violated while the densification
rate continued to increase. It was during this period
that the pressure profile underwent cycling or hunting.
It is worth mentioning that the density continued to
increase while the deflection of the fiber cells was kept
constant or decreased. In the latter stages of the process, the fiber deflection mechanism is not important
anymore (see [15]), so it then becomes essentially a race
to the finish between the densification and the reaction
layer kinetics. If the reaction layer reached the critical
state first, then the temperature was ramped down and
this stopped the densification of the composite. This
schedule is nearly time optimal since the planner kept a
constraint active at all time.
4.2. The Ti– 24Al – 11Nb/SCS-6 system
The Ti–6Al–4V matrix was then replaced with a
more creep resistant matrix, Ti–24Al–11Nb. A typical
industrial input schedule was simulated and compared
with a path planned simulation using HIP M/C 1. The
final states for these simulations have been tabulated in
Table 3.
R. Vancheeswaran, H.N.G. Wadley / Materials Science and Engineering A244 (1998) 58–66
63
Fig. 4. Simulation showing evolution of the microstructural states for the Ti – 6Al – 4V/SCS-6 system using a path-planned process schedule.
On comparison of the results for the direct simulation and the path-planned simulation, one finds that in
both cases the desired reaction layer thickness of 1 mm
is not met. The path-planned result provides better
performance for the fiber fracture density while trading
off the reaction layer thickness performance. In the
path-planned simulation, if the reaction layer thickness
requirement is made any smaller, the planner drops the
temperature to constrain the growth of the reaction
layer. An interesting effect of this problem is that when
the planner drops the temperature, applying high pressures does little to increase density and thus the planner
drops the pressure.
4.2.1. Machine limitations
The optimal planning approach can be used to determine what states are reachable. Fig. 5 shows the
boundary that separates those states for which optimal
process trajectories exist and those for which no trajectory exists. It shows that for a fixed relative density at
process completion, fiber fracture and reaction layer
thickness are inversely related. The best composite
would lie at the origin of Fig. 5 and we can explore how
the limitations affect this window. Four sets of numerical experiments have been performed using enhanced
temperature and pressure HIP machines (see Table 2)
to find what effect this has on the bounds of the
reachable set. It is interesting to note that regardless of
machine, as the goal state density is reduced, the performance of the reaction layer thickness and fiber fracture density are improved. As more stringent reaction
layer thickness goal states are chosen, the curves become asymptotic with the y-axis while the limitation of
how many fibers are broken is completely dependent on
the pressure ramp rate. One can think of reaction layer
thickness as a ‘fuel’ constraint and only a limited
thermal exposure (as monitored by the reaction layer
thickness) is allowed during consolidation. If this constraint is violated, the computed temperature trajectory
drops off, and severely limits further densification.
The nominal HIP was found to have the worst
performance. The performance of the planner was bet-
R. Vancheeswaran, H.N.G. Wadley / Materials Science and Engineering A244 (1998) 58–66
64
Table 2
Physical limits for the different HIP machines
HIP M/C
T: min (°C min−1)
T: max (°C min−1)
P: min (MPa min−1)
P: max (MPa min−1)
Tmax a (°C)
Pmax (MPa)
1
2
3
4
−20.0
−30.0
−20.0
−30.0
20.0
30.0
20.0
30.0
−1.0
−1.0
−2.0
−2.0
1.0
1.0
2.0
2.0
825.0
825.0
825.0
825.0
100.0
100.0
100.0
100.0
a
The maximum temperature for the Ti–24Al–11Nb/SCS-6 composite is 1000°C.
ter when the temperature rate was increased (M/C 2).
This was because if the temperature is increased at a
faster rate, the total (integrated) thermal exposure that
the composite was exposed to during consolidation was
reduced. The effect of increasing the pressure rate (M/C
3) had an even more significant effect on the consolidated final states of the composite. Pressures in the HIP
are ramped at a much slower rate than temperatures.
Following the same trend, M/C 4 performed better
than all the other HIPs. Even with these enhanced HIP
machines an acceptable goal state fracture and reaction
layer thickness were never reached. For this system,
fiber deflection and fracture becomes less of a problem
since the material creeps readily, but reaction zone
growth becomes critical for processability. The density
and reaction layer thickness requirements were met
using the enhanced pressure HIP and the enhanced
temperature and pressure HIP but at the expense of
breaking 10 fibers m − 1. To make this system processible, either a better interfacial coating (with slower
reaction kinetics) is needed, or a stronger fiber must be
developed. For example, if the fiber reference strength
were increased from 4.5 to 6.2 GPa it becomes possible
to reach the composite’s goal state with M/C 4.
fibers are easily deflected and damaged by the application of pressure. Therefore the planner ramps the temperature as fast as possible while holding the pressure
at 1 MPa until the matrix starts to creep rapidly. Once
the temperature reaches the maximum temperature allowable, the pressure is aggressively ramped till the
critical fiber deflections are in sight. At this time the
pressure rate is reduced to trade off the densification
achieved while not increasing the fiber deflections beyond the critical values. After the density reaches 0.92
(the critical transition density between stage 1 and stage
2 densification), it becomes a ‘race’ for the planner to
achieve the highest possible relative density without
overshooting the upper bound on the reaction zone.
This strategy is nearly time-optimal as the densification
rate is maximized while the fracture rate is held acceptably low.
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 Fig. 6, it is possible to plot surfaces of constant
reaction layer thickness and fiber fracture density in a
three-dimensional (Prate − T− t) process space. we have
chosen failure surfaces corresponding to 1 fracture m − 1
5. Discussion
The fundamental trade-off between the final composite density, the number of fibers fractured and the
reaction layer thickness is evident in all path-planned
runs. Evidently, consolidation can be naturally divided
into two distinct time intervals. The first interval is
roughly the time taken to densify the matrix to a
relative density of 0.92. Here the fiber deflections and
thus number of fractures of the composite are the
critical parameters. It has been shown previously [15]
that in the beginning while the temperature is low,
Table 3
Final states for the direct and path planned simulation
Simulation
D
Nf (m−1)
r (mm−1)
Direct
Path planned
0.999
0.999
10.1
0.15
1.17
1.9
Fig. 5. The bounds on the process window for the Ti –24Al–11Nb/
SCS-6 system for a densification goal state of D =0.995 using different HIP machines
R. Vancheeswaran, H.N.G. Wadley / Materials Science and Engineering A244 (1998) 58–66
65
Fig. 6. Microstructure failure surface in process space for a Ti – 6Al – 4V/SCS-6 composite.
and r =0.47 mm. When the optimal trajectory (shown
by the black line) is 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. We see that the trajectory is initially pressure rate limited by the fracture
surface. However, after : 200 min of consolidation,
the Prate bound disappears and is replaced by the
need to avoid a reaction layer failure. The relatively
low consolidation temperature and reactivity of the
Ti–6Al–4V matrix enables this to be accomplished.
Thus, the optimal process is 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.
6. Conclusions
A model predictive planner has been developed for
guiding the microstructural evolution in TMCs to a
predefined goal state during consolidation. The path
planning method uses constantly updated linearizations and constrained convex optimization to compute
planner actions. For long look-ahead times and short
integration time steps, sequentially local reachable
goals are very nearly global reachable goals. It identifies a viable process path that reaches the desired
goal state for a Ti–6Al–4V/SCS-6 composite system
while showing that no such path exists for a similar
composite with a Ti–24Al–11Nb matrix. Reachable
process windows have been constructed by this
method for each composite system. The path planning method can also be used to redesign the processing equipment, the fiber’s reaction inhibiting coating,
the fiber’s strength or the monotape surface roughness so that difficult material systems, such as the
Ti–24Al–11Nb matrix composite, can be successfully
processed. A simple graphical method is also used for
the estimation of near optimal process trajectories for
processes where the microstructural states have conflicting dependencies upon the process variables.
Acknowledgements
We are grateful to R. Kosut for helpful discussions
about this research and to D. Elzey for suggestions in
implementing the models. This work has been funded
by DARPA through a contract with Integrated Sys-
R. Vancheeswaran, H.N.G. Wadley / Materials Science and Engineering A244 (1998) 58–66
66
tems, Santa Clara, CA (Dr Anna Tsao, Program Manager).
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