OPTIMIZING THE CONSOLIDATION OF TITANIUM MATRIX · PDF fileAbstract-The high temperature consolidation of fiber reinforced metal matrix composite preforms seeks to eliminate matrix

Pergamon AC/~ mafer. Vol. 45, No. 10. pp. 10014018, 1997

0 1997 Acta Metallurgica Inc. Published by Elsevier Science Ltd. A.1 rights reserved

Printed in Great Britain PII: PII: s1359-6454(97)00104-3 1359-6454107 $17.00 + 0.00

OPTIMIZING 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 80309- 0425, 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 monofila- ments, 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 THE CONSOLIDATION

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 [ 181 for 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

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 economi- cally 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



Fig. 1. A schematic diagram illustrating the consolidation of a plasma sprayed composite monotape layup.

VANCHEESWARAN et al.: OPTIMIZING THE CONSOLIDATION 4003

Direct Simulation for Ti-6AI-4V / SCS-6

(b) Temperature Schedule (c) Pressure Schedule

$$F=J[~~

0 50 103 150 200 0 50 ICO 150 200 Time (min) 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.

depends 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

Process :

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;.

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).


(a) Direct Simulation for 7724AI- 11 Nb / SCS-6

FinatState,xf=[%]=[;“k]


cl 50 100 150 2w 0 50 100 150 x0 Time (mln) 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 irrevers- ible. Thus, the consequences of even a small “wrong” decision by the planner early in the process can result in the microstructure unrecover- ably missing the desired goal, making it look unreachable. Thus the planner needs to be “con- servative” 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 microelec- tronic components.

2. THE CONSOLIDATION PROCESS 2.1. Definitions

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 INb / SCS-6

FinalState,xf=[!]=[ ;“E]

Time (mln)

0 50 100 150 2w 0 100 150 2w Time (min) 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.


Table 1. Critical fiber deflections for each unit cell that result in a failure probability of 10%

19.1 17.3 15.5 13.7 11.9 10.1 8.3 6.5 4.7 2.9 159.0 131.0 105.0 82.0 62.0 45.0 30.0 18.5 9.7 3.7

1.14 0.94 0.75 0.59 0.44 0.32 0.22 0.13 o.c7 0.026

(dynamic) temperature within the sample. A machine model attempts to capture this dynamic behavior.

Abstractly, one can think of a HIP machine as a “box” with a (vector) input, a (vector) output, and a (vector) state, Fig. 4. The input vector, u(t), consists of commanded time varying pressure and temperature slew rates:

Tme(t) u(r) = Pm(t) [ 1 .

The state vector, .x,(t). then consists of the actual temperature and pressure within the sample:

x,(t) = T(r) [ 1 P(t)

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 =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 Pmp and Pmd are 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).

Thus, the lumped dynamic model of an ideal machine is

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


from U. The vector state, xh, of the microstructural model for a representative unit cell is thus D(t) xl(t) =

[ 1 u(t) (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

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

j&h, ?) = r Xh,celll 1

Xh,dl2 xh= . I : I ,_ Xh,celllO 1

i

j, [l--k(xh)(pLck(xh, 7) + DIF&h, f’f))l

AD, rl) - Mu, 7’) 1 (6) m@, T) which is the microstructural state for each unit cell in Y cells “stacked” together. The unit cell equations (5) are also stacked to get

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

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.

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.

An example of the simulated response for a Ti-6A14V/SCS-6 silicon carbide fiber TMC under- going 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

In the entire monotape layup, the fiber deflections are statistically distributed because of variations in the distance between the tops of asperities. This

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.


Table 2. Goal states for the two composite systems

Composite system $I rcnl (pm) D I’” r,

Ti-6A1-4V/SCS-6 0.1 0.6 0.999 0.47 0.8 Ti-24AI-1 lNb/SCS-6 0.1 1.0 0.999 0.5 0.8

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.

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

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 pressure trajectories (satisfied by (2)). If the 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))

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 machines

HIP Type

I 2 3 4

T,,. T-.x Pnu. Pm., T”n” P”,,x (YYjmin) (Yjmin) (MPa/min) (MPa/min) (‘C) (MPd

-20.0 20.0 -1.0 1.0 825.0 100.0 -20.0 20.0 - 2.0 2.0 825.0 100.0 -30.0 30.0 -1.0 1.0 825.0 100.0 -30.0 30.0 - 2.0 2.0 825.0 100.0


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.

2.

3.

4. 5.

The .Y, partition the state space and X, E ZI?i if and only if i = j. The range of plant dynamics is covered by the Xi. 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,. It is easy to detect which SYi the plant state is in. On each !Z”i a tractable path-planning scheme, S,, using the approximate plant is available.

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 /-Do n 1 - Da

T

n = 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, very mildly affected by changes in pressure.


(a) Path Planned Simulation for TMAI-4 V / SCS-6


fgfffj$~/

!I 2x3 4w 603 0 xx) Nm 6cm Time (min) Time (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,

(8)

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 vccit ensures 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 r Crlt = 0.6 /*m for a Ti-6Al~V/SCS-6 system in one of

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;

The dynamics are then described by a system of (twelve) ordinary differential equations

(9)

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 model (3) is performed to get the model:

4 ;i;=”

which, letting

X= XII

[I rl

and

F(X, U) = Fdxh, 9) [ 1 u

can be written simply as

z = F(X, u)

HIP machine overall plant

(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).


If x” is the current plant state, we can denote deviations from the current state by:

Then

D-D’ - where N, = At/a. Hence if at time tC the model state vcelll - VI,,, is x” and the HIP machine is commanded for a finite &II2 - l&Z horizon interval At with the slew rate control

&elllO - &,,0

r - rC T-T’ P - P”

u(t) = u(k) for kS G t G (k + 1)6

k = 0, 1,2, . . . , $ - 1 (17)

d8 _ dt=‘4x+Z?u+c

where

and

,‘f = =(X9 u) 8X

X=X+=0

B = mx> u)

au X=xC,U=O

c = F(F, 0)

(11)

(12)

(13)

(14)

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 method

(15) forward starting from f(O) = 0 (recall that 2 represents deviations from current) yields

N, - 1

&At) = c 6(Z+ 6A)Nf-‘-k(Bu(k) + C) (16) k=O

we have the approximation

iv- I X(t’+ At) x x’+ c @I+ 6A)*-‘-

k=O

VW) + C) (18)

for what the model state X will at the end of the horizon.

Consider (18) as a mapping

@:{u(k)p=-,,’ -+ X(t” + At)

taking commanded 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.

could be used, for example Tustin (Bilinear). $The constraints are discretized in time the obvious way.


The objective presented to the local convex solver is

r wD x (8C.N + DC - D,)> 1

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 trade- off between these competing goals to be made quantitatively. The method even allows cross- weightings between the states because it only requires that W = W > 0 and not that W is 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 MAT- LABTM or any number of other commercially available convex programming packages. We men- tion that sequential quadratic programming (SQP) methods are applicable and they give superlinear convergence by using a quasi-Newton updating procedure for accumulating second order infor- mation. 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) = Trdk) [ 1 Pz&)

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.

4. IMPLEMENTATION

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

0 200 400 SW 0 xi) 400 600

Time (min) Time (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].


(a) Path Planned Simulation for F24AI- 1 1Nb / SCS6


Time (mln) Time (mln)

Path Planned Simulation for Ti-24AI-11Nb / SCS-6

xg=~]+J


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.

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. Examin- ation 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


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

UNREACHABLE

0.0 0.1 0.2 0.3 0.4 0.5

Reaction layer thickness (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.


Ti-24Al-11 Nbl SCS-6 \ \


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 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

60

.E. 50

.s? E 40

$ $ 30

5 !I 20

z f 10

0 0.1 0.2 0.3 0.4


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 (hypotheti-

cal) HIP machines 1 through 4.

25 -

20 -

15-

10 -

5-

Enhanced l’ HIP (1.5i)

STATES 7

OL 0.0 0.5 1.0 1.5 2.0

Reaction layer thickness (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 (hypo-

thetical) 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-6Al- 4V/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

EnhancedT8.P

Reaction layer thickness (pm I

Fig. 15. Projections of the reachable states for fibers of increasing reference strength.


-7 .E.

10’

.g loo v) 5 lo-'

P,

9 IO-2

g m .k 104

8 g 10-4 LL

10-S 4 5 6 7 a 9 10

Fiber reference strength (GPa)

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 maintain- ing r < 1 pm and achieving complete densification. Obviously, the procedure can be repeated for fibers with reaction inhibiting (diffusion retarding) coat- ings. 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


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.

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

a) Ti - 6A/ - 4WSCS - 6 COMPOSITE N, = 1 fiber fracture per meter

W Ti - 24Al- 11 Nb / SCS - 6 COMPOSITE

E a0 . 2 60

Z. 40

Q) 5 20 p! 1 O

$ -20

= 0.45pm

fracture per meter

=lpm

Fig. 17. Microstructure failure surfaces in process space for the Ti-6A1&4V/SCS-6 and Ti-24AlkI lNb/ SCS-6 composite systems.


a) Ti - 6AI - 4WSCS - 6 b) Ti-24Al-lINb/SCS-6

2 80 .

$ 60

z 40

a, E 20 g! 3 O z?

L”

p 0 200 400 600 800 1000

Temperature (“C)

2 80 . $ 60 r - 40 Q) g 20 p! I 0 n,

h -201 I I I I I 0 100 200 300 400 500

Time (mins) f i 0.45pm

loo0 -1 800

600

400

200

0

Time (mins)

f ,II 0 200 400 600 800 1000

Temperature (“C)

0 100 200 300 400 500

Time (mins)

-0 100 200 300 400 500

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 consider- able 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.


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 reactiv- ity 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.

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-We are 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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OPTIMIZING THE CONSOLIDATION OF TITANIUM MATRIX · PDF fileAbstract-The high temperature consolidation of fiber reinforced metal matrix composite preforms seeks to eliminate matrix