A FIBER FRACTURE MODEL FOR METAL MATRIX COMPOSITE · PDF fileELZEY and WADLEY: FIBER FRACTURE...

Acta metall, mater. Vol. 42, No. 12, pp. 39974013, 1994 Copyright © 1994 Elsevier Science Ltd

Pergamon 0956-7151(94)00191-X Printed in Great Britain. All rights reserved 0956-7151/94 $7.00 + 0.00

A FIBER F R A C T U R E M O D E L FOR METAL MATRIX COMPOSITE MONOTAPE CONSOLIDATION

D. M. ELZEY and H. N. G. WADLEY Department of Materials Science and Engineering, University of Virginia, Charlottesville,

VA 22903, U.S.A.

(Received 9 March 1994)

Abstract--The consolidation of plasma sprayed monotapes is emerging as a promising route for producing metal and intermetallic matrix composites reinforced with continuous ceramic fibers. Significant fiber fracture has been reported to accompany the consolidation of some fiber/matrix systems, particularly those with creep resistant matrices. Groves et al. [Acta metall, mater. 42, 2089 (1994)] determined the predominant mechanism to be bending at monotape surface asperities and showed a strong dependence of damage upon process conditions. Here, a previous model for the densification of monotapes [Elzey and Wadley, Acta metall, mater. 41, 2297 (1993)] has been used with a stochastic model of the fiber failure process to predict the evolution of fiber fracture during either hot isostatic or vacuum hot pressing. Using surface profilometer measured roughness data for the monotapes and handbook values for the mechanical properties of different matrices and fibers, this new model is used to elucidate the damage dependence on process conditions, monotape surface roughness, and the mechanical properties both of the fiber and matrix. The model is used to investigate the "processibility" of several currently important matrix and fiber systems and to identify the factors governing this. An example is also given of its use for the simulation of a representative consolidation process cycle. This approach to the analysis of a complex, nonlinear, time-varying process has resulted in a clear understanding of the causal relationships between damage and the many process, material and geometric variables of the problem and identified new strategies for its elimination.

1. INTRODUCTION sition where the wide variation in partial pressures of some alloy elements results in a loss of stoichi-

Interest in metal and intermetallic matrix composites ometry). For many fiber-matrix systems it is also reinforced with continuous fibers is growing because potentially the lowest cost processing route. of their superior specific modulus, strength and creep ICPD results in sheets (usually called monotapes) resistance [1, 2]. For example, one emerging system containing a single layer of unidirectional fibers sur- based upon a Ti 3 Al + Nb matrix reinforced with rounded by matrix. These can be layed up and either 40 vol.% of Textron's SCS-6 fibers has a specific hot isostatically or vacuum hot pressed (HIP/VHP) strength double that of wrought superalloys at 25°C to densify and form a near net shape component, and single crystal superalloys at l 100°C [3]. They are Fig. I. A recent study has analyzed the consolidation viewed as potential (but costly) materials for future process and developed expressions for densification gas turbine engines and the airframes of high speed by plasticity, power law creep and diffusional flow at aircraft, asperity contacts and voids [8]. It resulted in the

These composite systems are difficult to process by development of a predictive model that calculated the conventional solidification pathways because of ex- temporal evolution of density during any (arbitrary) treme reactivity between the fiber and many of the HIP/VHP temperature/pressure cycle. It was also matrix materials of interest. Novel methods are being used to find optimum process conditions (i.e. accom- investigated to bypass this problem including solid plish a fully dense consolidation in minimum time). state techniques such as foil-fiber-foil [4] and powder Recent experimental studies of monotape consoli- cloth [5], vapor deposition techniques [6] and various dation have revealed the sometimes widespread oc- thermal spray processes such as induction coupled currence of fiber fracture during consolidation [9], a plasma deposition (ICPD) [7]. The ICPD method has potentially serious drawback to this processing ap- attracted interest because the contact time between proach. However, these studies indicated that the the liquid alloy and the fiber is short (~ l0 -a s), fiber extent of fiber fracture was sensitive to the conditions alignment and uniform spacing are retained in the used to consolidate the monotapes and raised the final composite, high temperature matrix alloys (that possibility that for some process conditions/material are difficult to produce in the rolled foil form needed systems it could be avoided. In particular it was for a foil-fiber-foil approach) can be used, and alloy shown that high consolidation temperatures and low compositions can be maintained (unlike vapor depo- densification rates greatly reduced the fiber damage,

3997

3998 ELZEY and WADLEY: FIBER FRACTURE MODEL FOR MMC CONSOLIDATION

Lay-up E3' E3 Z~, (::2= 0

Void ..................... Conso atinQ ........................................... ~ ~ ~ , spray-deposited ..... i~;;~ ~:~':~ ..... monotapes

Matrix _ _ _ . _ _ _ ~ I Fiber ~ ..... ~i*:~" i ~ i

I: 1, E~= 0 ~ 2i / ~ 2i

~ . f . . j - f Unit c e l l ~ " ~

! - - " ~ i'~ ~datrix (viscoplastic) Low Damage High Damage ~t high T, n, B o, non-dimensional stiffness low T, n, B o, non-dimensional stiffness • Fiber straight ° High matrix deformation ° Fiber bent ° Low matrix deformation

Fig. 1. The fiber fracture model is based on analysis of a representative volume element in which a single fiber segment is subjected to bend forces by contacting asperities; the overall fiber damage is predicted by summing the failure probabilities of a statistical distribution of such cells within the lay-up, Minimal fiber damage during consolidation is achieved under conditions in which the matrix flows into and fills

voids without deflecting the fibers (a).

at least in the Ti3AI + Nb/SCS-6 system. Unfortu- matrix systems that govern fiber fracture during nately, extended exposure of the monotapes to high consolidation. This enables the selection of fiber- temperature during consolidation results in the dele- matrix combinations that can be processed success- terious dissolution of the fiber's protective coating, fully (and a determination made of those that can- increased fiber-matrix sliding resistance [10] and not). Together with the model developed earlier for (eventually) a loss of the fiber's strength [11]. The densification [8], it can also be used for model-based rates at which these effects occur depend on the fiber control (optimization) of both densification and fiber coating and matrices and the process temperature fracture during the consolidation process [12]. used. Thus, for each fiber-matrix system of potential interest, it becomes necessary to make a trade-off 2. PROBLEM DEFINITION between fiber fracture, interfacial reaction severity Composite monotapes can be produced by winding and relative density to determine the best consolida- ceramic fiber (e.g. Textron's SCS-6 fiber) on large tion process. There is the possibility that for some diameter drums. These are then heated to ~850°C (to systems no acceptable compromise exists, avoid thermal shock to the fibers during subsequent

This trade-off could be sought empirically but deposition) and rotated/translated under a stream of would require numerous experiments (and their time plasma melted alloy droplets (see [7] for further consuming characterization) with test materials that details). The resulting monotapes are characterized are expensive, sometimes irreproducible and often by one quite rough surface (the one built up by difficult to obtain. In addition, such an approach successive droplet solidification), one much smoother would have to be repeated each time the matrix alloy, surface (the one in contact with the drum surface) the fiber or the fiber coating was changed. Instead, we and up to 10% of internal (closed) porosity. A full have resorted to a modeling approach, the develop- characterization of the monotape's geometry can be ment and results of which are presented here. It will found in [8, 9] and of its sometimes metastable micro- be shown that simulations with the model allows structure (and evolution during the consolidation identification of the properties of potential fiber- thermal cycle) in [13, 14].

ELZEY and WADLEY: FIBER FRACTURE MODEL FOR MCC CONSOLIDATION 3999

Near net shape composite components can be proaches 0.9 and beyond, the asperities create iso- produced by cutting and stacking the monotapes in lated voids (Stage II). In this regime, densification is a suitably shaped cannister or die and subjecting this accomplished by void shrinkage and, if these are to a consolidation cycle designed to eliminate poros- uniformly distributed, the loads on the fiber become ity and promote monotape diffusion bonding [8]. For more homogeneous and further bending is not ex- sheet-like samples almost all the consolidation strain pected. Thus fiber bending is expected in the range of is accommodated by a reduction of component thick- densities where asperity contact deformation domi- ness (sheet widths and lengths show no appreciable nates densification. strains), especially at low density where the plastic To be of greatest value for process design (and Poisson's ratio of the porous laminate is very low control), a model is required that predicts fiber (<0.1). Therefore regardless of the type ofconsolida- damage accumulation throughout a consolidation tion process (HIP or VHP), monotape consolidation cycle of arbitrary pressure and temperature profile so can, to a good approximation, be treated as occurring that the effect of changes to these profiles can be under conditions of constrained uniaxial com- evaluated. It must also fully capture the relevant pression, features of a randomly rough monotape surface so

During consolidation, fibers can be fractured, that one can assess its contribution to damage. The especially during the early stages of densification model should also be capable of generalization so [9]. A number of potential mechanisms of fiber that the effects of changing either the fiber or matrix fracture can be envisioned. In principle, tensile fiber properties could be predicted and the best combi- stresses could be generated on heating to the consol- nations identified. Lastly, since there is interest in idation temperature because the matrix often has the use of the model for model-based process design double the thermal expansion coefficient of the (optimization) and possibly feedback control, it fiber [15]. A second possibility is that transforma- should not require overly extensive computation, and tions of some metastable ICPD alloys on heating in particular, the need for human intervention during cause matrix dilatation and tensile fiber stressing its iterative application (i.e. it should not be a finite [13, 14]. Both of these have been discounted for many element model). of the systems studied here because experiments in which the foils were thermally cycled (without an applied pressure) to the consolidation temperature 3. MODEL DEVELOPMENT showed no fiber damage [9]. Fractures due to fiber-matrix thermal expansion coefficient mismatch When monotapes are compressed, fibers bend (and did not occur because, at the start of consolidation, possibly fracture) because they are loaded only at the the fibers are in residual compression after cooling contact points between adjacent foils, Fig. 1. Each from the ICPD process temperature (~800-900°C fiber span bridging a pair of asperities suffers a for Ti3AI + Nb deposition), and would go into bending that depends upon the span length and the tension only upon heating above the deposition tem- asperities' height and shape which are all statistical perature. Consolidation is typically performed quantities determined by the surface roughness. As 100-200°C above that of deposition, resulting in densification proceeds, and the monotapes are tensile fiber stresses of 150-300 MPa, which are small pressed more closely together, the number of contact- compared to typical fiber strengths of 3-5 GPa. Evi- ing asperities increases and the average spacing be- dence is beginning to emerge that a more significant tween the contacts decreases. Given an initial axial fiber stress can be developed if there is a large distribution of asperity spacings and heights (simply CTE mismatch between the composite and the tool- determined from a single profilometer line scan), we ing used for its consolidation [16]. This would result first seek to calculate the distribution of fiber bend in an additional damage mechanism not analyzed lengths at any given density. Each fiber segment here. length within the distribution will suffer a deflection

The mechanism we examine results from nonuni- governed by the constitutive response of the asperity, form mechanical loading of the fibers which causes its geometry and the fiber stiffness. The probability the fibers to deform by bending. Axial and torsional that a fiber segment's deflection causes its fracture fiber deformations can also be created, but the can be determined by calculating the maximum fiber experimental work of Groves et al. [9] has identified tensile stress (due to the bend deflection) and compar- bending as the main damage source. The nonuniform ing this with the (statistically distributed) fiber forces that cause fiber bending were shown to result strength. Summing this fracture probability over all from the surface roughness of the monotapes; the fiber length segments present will give the desired forces were essentially transmitted from one mono- overall probability of fiber fracture for any point in tape to the next only at points of (asperity) contact, the densification process. resulting in local fiber bending and potential fiber If we assume macroscopic isostrain conditions (the fracture. This mechanism of damage occurs only /~3 strain rate component in Fig. 1 is independent of when densification is by contact deformation (Stage position), the monotapes will remain planar through- I [8]). When the composite relative density ap- out consolidation and their position and approach


rate are defined by the macroscopic (lay-up) density to the bend stiffness of the r-lamina is neglected). and densification rate, respectively. These could be Thus the contribution of the reinforced lamina to the obtained experimentally, but instead we prefer to use cell height can be omitted from the derivation of the calculated values (for a given applied pressure, ]~3(t ) unit cell model, so that the monotape thickness is and temperature cycle) using the densification model taken to be simply z (t). The initial dimensions of the develol~ed in [8]. Thus the current density and densifi- unit cell to be analyzed are then given by I and 2h, cation rate are available as an input to a fiber fracture where h is the (original) undeformed asperity height. model; a choice that favors a displacement analysis of Both l and h vary statistically for a rough surface and fiber bending/fracture rather than one based on equi- are later treated as random variables, but to begin we librium of (applied and internal) stresses, consider just a single combination.

The model's development is presented in two steps. The thickness of a monotape lay-up changes with First, we analyze (to obtain the fiber deflection) a time due to densification, eventually bringing the geometrically representative volume element consist- fiber (i.e. the reinforced lamina) of one foil into full ing of a fiber segment and three contacting asperities contact with all the asperities of an adjacent foil. As responsible for its bending (Fig. 1). The fibers' deflec- the compaction of each newly formed unit cell occurs, tion arises from the asperities' resistance to inelastic the asperities press against the fiber, causing it to deformation. This is assumed to occur by either deflect. In turn, reaction forces exerted by the fiber plastic yielding or power-law creep or a combination lead to deformation of the asperities. The rate of of both. The unit cell analysis results in a single change of height (~) for a deforming asperity can be nonlinear differential equation relating the instan- taken as the sum of the asperity's plastic (pp) and taneous density and densification rate (the inputs to creep (pc) deformation rates; i.e. f =)~p + Yc (where the model obtained using the results of [8]) to the dots indicate differentiation with respect to time). The fiber's deflection (the model's output). The second instantaneous monotape thickness (z[t ]) and the step (Section 3.2) combines the unit cell response with deformed asperity height (y[t ]) govern the fiber a statistical model of a randomly rough surface to deflection (v[t ]) calculate the distribution of fiber deflections (and v (t) = 2[z (t) - y (t)]. (1) fiber fracture probabilities) in the macroscopic body. A significant complication with the second step is the The monotape thickness (z[t ]) is governed by the need to include the evolving nature of the fiber span lay-up's macroscopic density. It can be determined by length distribution since segments are continuously simulation using a previously developed model for formed, deformed and eliminated (predominantly by densification [8]. By conservation of mass, the pre- subdivision as new contacts are made) throughout the dicted relative density, D of a monotape is related to consolidation process, the thickness during constrained uniaxial com-

pression by z ( t ) = z o • Do/D (t) , where z 0 and D O are 3.1. Single fiber segment analysis the initial monotape thickness and relative density.

Figure 1 illustrates a typical fiber segment forced The output desired from the unit cell analysis is the into three-point bending by asperity contacts. Each fiber's deflection (v), which in turn determines the plasma sprayed monotape may be envisioned as tensile stresses in the fiber and therefore the prob- consisting of two regions (laminae), one containing ability that the fiber will fail. Since z ( t) is available the parallel array of fibers and the other composed as an independent input, it is only necessary to only of the surface roughness [8]. As shown in Fig. 1, compute y ( t) to obtain the deflection using (1). the unit cell consists of a fiber-reinforced lamina Expressions for y ( t) can be calculated assuming (which is subjected to bending), a surface asperity of either plasticity or creep to be the dominant mode of the same monotape as the fiber segment (shown asperity deformation using the approach in [8]. inverted), and asperities of an adjacent monotape. 3.1.1. Plasticity. Plastic deformation is the primary Although the fibers are embedded in matrix material, densification mechanism of metals and alloys when the resistance of the reinforced lamina to bending is consolidation is conducted at temperatures below assumed to be entirely due to the fiber because of its about 0.4Tin, where Tm is the absolute melting point. much higher stiffness and large volume fraction The contact stress required to cause an asperity's (about 70% of the volume of the reinforced lamina), plastic yielding generates the forces that cause fiber The fiber segment and associated asperities define a bending [8, 17]. The uniaxial yield criterion for a unit cell of length, l (defined by the distance between hemispherical asperity can be written contacting asperities) and height, 2z ( t) + df where Fc z ( t) is the current thickness of the surface roughness - - = cr c/> fl • tr 0 (2) lamina within a compressed foil and d r, the fiber ac diameter, approximates the thickness of the re- where tr c is the average asperity contact stress, Fc is inforced lamina. Although the reinforced lamina the normal contact force, a¢ the contact area, a0 the thickness does change slightly during densification uniaxial yield stress of the matrix and fl a numerical [8], it can be regarded as a constant for the purpose factor accounting for the increase in the stress re- of modeling the fiber fracture (since its contribution quired for elastically constrained or workhardening

ELZEY and WADLEY: FIBER FRACTURE MODEL FOR MMC CONSOLIDATION 4001

plastic flow. Gampala et al. have shown that the degree of elastic constraint is strongly dependent on the evolving shape of the deforming contact, quickly reaching a maximum value of about 3 and then decreasing for a perfectly plastic material [17]. Work hardening counteracts this decrease and to a reason- ably good approximation, fi can be assumed to have a constant value of 3 with relatively little loss of precisiont.

From (2), the force acting on a single asperity to cause its plastic deformation is given by

F,= -~~a,~a,~66nro,,(y,-h) (3)

where r is the mean asperity radius, y, is the asperity height following plastic deformation and Us has been approximated by 2zr (h - yp) for a hemispherical asperity [18]. We define a “plastic stiffness”, kp( = 6nra,), so that equation (3) becomes F, = k,

(Y, - h ). Given the contact forces, the deflection of a fiber

segment can be obtained from simple beam theory. The force-deflection relation for a simply supported segment subjected to three-point elastic bending is

F=k,v (4)

where v is the fiber’s midpoint deflection and, for central loading of a cylindrical fiber, the elastic bend stiffness, k, is given by [19]

where Ef is the fiber’s Youngs modulus and df its diameter.

Equating (3) and (4), (the force acting through both asperity and fiber must be equal), and using (1) gives the change in asperity height due to plastic yielding as yp -h = 2(2 - y) . k,/k,, which upon differentiation, leads to an expression for the rate of plastic displacement

3&)=5[i(t)--3(t)l (6)

where 5 ( =2k,/kp) is a non-dimensional stiffness equal to twice the ratio of the elastic bend stiffness of the fiber to the “plastic” stiffness of the asperity. High values of the non-dimensional stiffness are desirable since this increases the rate of asperity compaction for a fixed deflection force rate, and results in a lower fiber deflection rate.

3.1.2. Power law creep. At higher temperatures (T > 0.4T,,,), densification is dominated by power law creep. We assume that a uniaxial power-law stress- strain rate (ai) relation holds for the matrix material

i=Ba” (7)

tThis represents a worst case since allowing p to decrease with continued deformation (geometric softening) leads to plastic flow at lower contact stresses, and thus a smaller probability of failure.

where n is the creep exponent (often a function of stress and temperature) and B is a temperature dependent constant conventionally expressed as B = B,,exp( -Q,/BT) in which Q, is the activation energy for power law creep. Rewriting (7) in terms of the asperity height (y) and the force applied to the asperity contact gives

(8)

Equating the force (F,) in (8) required to creep deform the asperity at a rate 3,/y, with that required to bend the fiber [equation (4)] and substituting for a, gives

2k, ]r (t ) - Y (t )I ‘I A= --By(t)

> 2xr[y(t)-h] (9)

This is an ordinary nonlinear differential equation (n typically ranges from 2 to 5) that can be solved numerically, using, for example, a Runge-Kutta numerical integration scheme.

3.1.3. Combined response. The total displacement rate for an asperity is assumed to be the sum of the rates due to creep and plasticity, i.e. 3 = g,, + g,. Substituting expressions (6) and (9) for 3, and g, results in a nonlinear ordinary differential equation in the asperity height

By(t) - . 9=-1+5

2ks[z(t)-y(t)l “+ 5 2nO(t)--hl >- l-t5

i, (lo)

Equation (10) can be solved numerically to give the asperity height, y (t ), given z (t ) as input. The time- dependent fiber deflection can then be obtained using

(1). 3.1.4. Fiber fracture. The unit cell response of a

fiber segment can be used to determine the deflection, given values of I and h for the cell, and z (t ) for the consolidating body. The maximum tensile stress, g,,, in the fiber is related to its deflection

ob(t)=6$-f$v(t). (11)

The bending fracture strength is proportional to the tensile fracture stress, bf [20] v, 1

Ob= Of v,‘L [ 1 ‘im (12)

where V, and Vb are the volumes of tensile stressed fiber during tension and bend testing, respectively. For fiber strengths that obey Weibull statistics, the factor K depends only on the fibers’ Weibull modulus, m. For example, SCS-6 (SIC) fibers with m = 13 have IC = 1.45 x lo-’ [9]. Taking V, to be the gage length of tensile fiber samples and V, to be governed by the mean asperity spacing (for asperities of all heights),

vt/vb = 100 for ICPD monotapes [9], and thus

0bN 1.9.0,. (13)

The constant in (13) is insensitive to values of Vb; for the entire range of fiber segment lengths (i.e. asperity


spacings) encountered here, ab/tr f = 1.9 + 0.1. From fracture strengths, but the cells are continually being (11) and (13), the tensile fiber stress is created, deformed and eliminated throughout the

consolidation cycle. The new problems of cell ere- ,4. trf(t ) ~ 3Ef. ~ . v (t). (14) at ion/elimination arise because a cell, formed at some

earlier time in the cycle, is eliminated when it encoun- Since the deflection, v( t ) in (14) is obtained by ters a new asperity, and two new cells (with new solution of (10) for a given /, h and foil thickness, lengths) are formed. The distribution of cell lengths z (t) , the fiber stress at some time z depends o n / , h therefore evolves during the process. and the densification history, z (t): i.e. o'f = trf Let us consider how a single cell response like that (l, h , z ( t ) , Q. developed above (Section 3.1) could be integrated

With the fiber stress given by (14), the probability into a macroscopic (many-celled) lay-up response. of fracture is obtained from the fiber strength distri- Suppose a particular cell of interest has a fiber bution, tpf(o'f). For a two-parameter Weibull distri- segment length, l. It would have been created earlier but ion [21], the strength distribution function is in the process of consolidation by the addition of an

m asperity to subdivide an existing cell of length > l tpr(af) l exp a r (15) when the compacted monotape height z ( t ) = z¢.

\ O'reff -] During subsequent densification, the new fiber span where tr r is the fiber stress and O're f is a reference stress thus created undergoes deflection until it is eliminated (the stress below which 37% of fibers survive). The (by the addition of a contact) at some height, z (where ratio of the volume of material stressed in bending to z < zc, see Fig. 2). Obviously, if zc-z is small, the fiber the volume stresses during tensile testing, V/Vo, does will probably not have suffered a very significant not appear in (15) since it has already been accounted deflection and its failure probability will be small. for by equations (12) and (13). Integration of (15) Thus, the cumulative probability that this fiber seg- gives the cumulative strength distribution ment would have failed [~f, defined by equation (16)],

will be a function of zc, z and l. Of(o'f) = 1 - tpfdtr = 1 - exp - vr . (16) By introducing a change of variable to relate stress

L \trrer,/ d in the fiber (O'f), given by (14), to the extent of

For any given cell (i.e. for a given fiber span length monotape compaction (zf), we can write the failure and asperity height) and densification path, z (t), the probability as fiber stress (trf) needed to determine the cumulative

fz ' daf probabili ty of fracture is obtained by first solving the ~f(zc, Z, l ) = rpf(/; O'f) "~Zf dzf (17) ODE (10) for the asperity height, y (t) , substituting c this into (1) to obtain the deflection, and then substituting v ( t ) into (14). where zf represents the height as the cell is compacted

from zc to z (Fig. 2). ~f(zc, z, l ) represents the fraction 3.2. Macroscopic response of those cells created at zc and subsequently elimi-

In principle, the macroscopic response of a mono- hated at z which had fractured. The cumulative tape lay-up could be evaluated by summing the number of these fractures can be obtained by sum- failure probabilities for all the possible unit cell ming the product of (the fraction of cells eliminated lengths present. It is complicated however, because at at z) x (the fraction of these created at zc) x (the any instant in the process, not only are there continu- cumulative probability of failure), for all cells created ous distributions of cell lengths, cell heights and fiber during consolidation to a height, z (z).

Z 0 profile cell T T z°

Z c Range of Range of created

Direction of Range of cell origination densification compaction fiber deflection z < z c _< z o z~ _< z _< z o

z_<z, lt _< Zc 1 ' ~ - - - z cel l

z eliminated

Z,¢ Z.~

Fig. 2. Integration variables used in the fiber fracture model: the composite monotape is compacted from an initial height, z 0 to z, at time, z. At each height, z, the cells eliminated by new contacts are considered; zc is the height at which a cell eliminated at z was created, zr is a measure of the amount of deformation

of the cell created at zc and eliminated at z.


However, the fraction of cells of a given length Widom [22] for analyzing random sequential addition eliminated at z and the fraction of these which were processes like this and we find (Appendix) created at zc must also be expressed as probabilities. 1 Therefore the macroscopic model must be able to ~Pl(z;l) 4no(z). r 2 q (P; 2) predict the probability that a cell of given length is eliminated when a new contact is added and the where q (p; 2 ) is defined in the Appendix. Even if this probability that this particular cell was created at any were not a reasonable assumption to invoke, we find previous height, zc. These two probability density that fiber lengths this short are anyway unlikely to functions are referred to as the probability of elimin- fracture and the assumption minimally perturbs our ation, q~e and the probability of origination, tp 0 and result. are derived below (Sections 3.2.2 and 3.2.3). How- 3.2.2. Cell creation~elimination. If the monotape is ever, since the probability of eliminating a cell de- deformed from z to z + dz, a number of new contacts, pends on the number of cells of this particular length AN, will be established as indicated by equation (18). in existence at any moment, the total number of Consider the fraction of cells that are of a length, say cells and their length distribution must first be deter- 1; the additional contacts may either eliminate (i.e. mined, subdivide) longer cells to create additional cells of

3.2.1. Number o f cells and cell length distribution, length, 1, or the new contacts can eliminate cells of Suppose a randomly rough surface, Fig. 2, is con- length, I to create cells of a shorter length. Provided tacted by a straight fiber. If the fiber compresses the tp~ (z; l) is known, the probabilities that added con- surface (along its line of contact) to a height, z, then tacts either eliminate or create cells of a given length the number of asperities per unit length in contact can be obtained as follows. with the fiber (no) is Suppose a fiber of overall length, L has Nc(z)

contacts. Since new contacts cannot be formed where gaps of less than 2r exist, the total length of fiber

nc(z) = p • q)h(h)dh = p • ~h(z) (18) accessible to a potential contact is

A ( z ) = N ~ ( z ) ( l - 2 r ) . ~ o , ( z ; l ) d l . d2r

where z0 is the undeformed surface height, ~oh(h ) is the probability density function (PDF)describing the The probability that the next contact to form will distribution of asperity heights, h, p is a lineal subdivide a cell of length between I and l + dl is just asperity density (with units of m - l ) and ~h(z) is the the fraction of the accessible length occupied by all cumulative height probability function, i.e. the prob- lengths between l and l + dl, i.e.

ability of finding an asperity of height ~>z. Both N~(z). ~oj(z;l)dl . ( l - 2 r ) ~0h(h ) and p can be experimentally determined by A (z) stylus profilometry.

These contacts result in cells like those shown Since Nc(z)/L =no(z), the current contact density, schematically in Fig. 1. Because of the randomness of and no(z), qh (z; l ) . (l - 2r)/A (z) is recognized as a the surface, the cells are of a variable length. We let PDF describing the (evolving) distribution of elimi-

qh (z; l) be the PDF describing the distribution of cell nated cells

lengths at any compacted height, z. It must depend on q~e(z; l) = nc(z). ~o~ (z; l)" (l - 2r)/A (z) (19) the current compacted monotape thickness since, as z decreases, the number of contacts, and therefore the where ~oo (z; l) represents the probability of subdivid- number of cells, increases, with the result that the ing a segment of length, l per contact added. (Note mean cell length continually decreases with densifica- its units of per meter per contact.) The fraction of the tion. Additionally, qh (z; l) will have to allow for the AN eliminated cells with length between l and l + dl occurrence of very closely spaced asperities which is thus q~e(z; l)dl • AN. cannot be further subdivided by an additional con- Turning next to the creation of cells, we note that tact. ~o~ (z; l) could be determined experimentally, but any cell length, l" > l - 2r could be subdivided to only with great difficulty. A simplified derivation of create a cell of length, I. For this to occur, a newly

added asperity must first contact a cell of length 1' q~l(Z;l) can be achieved if we assume an average and then subdivide it such that a cell of length, 1 is asperity radius, r (sprayed monotapes exhibit a stat-

istical distribution of asperity sizes, i.e. radii for created. The probability of contacting a cell with a hemispherical asperities), such that the shortest poss- length between l ' and l ' + d l ' is tpe(Z ; l ' )d l ' . Since

the contact has an equal probability of occurring ible cell lengtht will be 2r and restrict qh(z; l ) to 1 ~> 2r. This assumption allows use of the method of anywhere within l ' - 2r, the probability of creating

cells having lengths between l and l + dl is uniform, i.e. it is just d l / ( l ' - 2 r ) . The fraction of cells with

tThe minimum spacing does not correspond to 1/p (where p is the measured line density of asperities) since the lengths between l and I + dl created by the subdivi- asperities are not close packed, but may be separated by sion of cells having lengths between l ' and 1' + d l ' is shallow valleys or flats, therefore 2[n~(z)/A (z)]. ¢Pl (z; / ' ) d l ' - dl. The factor 2


arises because the new cell could be created to the left z (z), many cells will not have been eliminated, but or right of the added contact. For all l ' t>/, the are still subjected to various levels of stress and some fraction of created cells of length between I and l + dl may have fractured. The contribution of these to the is number of fractures is

~!z! } l{ 2 q~l(z; l ' ) d l ' dl

.12r from which we define a PDF describing the distri- t'~0f bution of created cells x J ~ q~0(zc, z; l )

A(z) jt+2tpl(Z;l')dl'. (20) x ¢pf(l;trf).~-~-- dzf dz c dl (23) r

¢Pc(z; l ) represents the probability of creating a seg- where tpl has replaced tp e since it is not the distri- ment of length,/ , per contact added when the com- bution of subdivided cells that are of interest [these pacted height is z. The fraction of the 2AN newly have already been accounted for in (22)], but the created cells with length between l and 1 + dl is thus distribution of existing cells at z,. The total accumu- opt(z; l)dl • AN. lated damage after densifying to z (z) is therefore the

3.2.3. Probability of cell origination. When a cell is sum of (22) and (23) subdivided by the addition of a contact, the cell must be replaced by two cells, each of a shorter length, and Nf(z~) = N a (z~) + Nb (z~). (24)

the subdivided cell must be removed from the cell The fiber fracture predictive model, given by (24), length population. To calculate the failure prob- requires knowledge of three probability distribution ability for the cell at the point of its removal, it is functions; ~Oh(h ), the distribution of asperity heights, necessary to know the height, zc, at which the cell was ~of(o'f), the distribution of fiber strengths and ~0~ (z; l) , formed. The probability that a cell was originally the distribution of asperity spacings, which depends created at a height between z and z0 (the initial on the deformed height, z (i.e. the current density). monotape thickness) is the number of cells created The PDF for asperity heights can be determined between z c and z~ + dzc divided by the total number easily by means of stylus profilometry. The distri- of cells (of length, 1) created between z 0 and z bution of fiber strengths is usually described by

Weibull statistics and is available for different types dnc ~oc(zc;l)dl. dz ~ of fiber. ¢pl(z;l) could be determined by surface dz¢ profilometry or analytically as described in the

~00(z c, z, 1). dl = ;~ 0 dn~ , Appendix using the method of Widom [22]. The q~(z'~,l)dl" -~7, dz c dz ¢ model (24), can be regarded as predicting the number

(z ~< z¢ ~<z0). (21) of fractures as a function of monotape thickness (which can be converted into relative density) or time

Note that the population described by ~o 0 includes all (for a prescribed consolidation cycle) since the height cells created between z0 and z whereas ~0c includes is known or predictable as a function of time using only the cells created between z~ and z¢ + dz~. the densification model [8] for any process cycle.

3.2.4. Cumulative fiber damage. Finally, the prob- In attempting to keep the model formulation as ability that a cell of length, 1, created at z~, is simple as possible, (since for some of its intended subdivided at z and subsequently fails at a height, zr applications the model should take only minutes to can be expressed as q~,(z; l )" q~0(zc, z, l ) - q~r(l; crf). run), a number of idealizations have been introduced daf/dzf. Thus, the total number of fractures associ- including: (i) the fiber segments are treated as bare, ated with cells eliminated at or before z (z) - z, is although they are actually embedded within a porous

matrix; (ii) the fiber segments are assumed to be f/fz:O{ f .0{ simply supported as though isolated--the fact that N~ (z~) = tpo (z; l ) . ~o 0 (z~, z; l ) the fibers are continuous and bonded to the deform-

~ ing metal matrix makes the actual end constraints

fro{ dtrf} } } d n c considerably more complicated; (iii) central three- × q~f(l; o'f) ~zf dgf dz¢ • dz • dl point bending is t reated-- in reality, the intermediate

' " d z loading point may occur anywhere between the end (22) supports; (iv) the deforming asperity contacts are

treated as point loads rather than distributed; (v) fiber where dnc/dz • dz gives the number of cells removed fracture has been neglected as a means of cell elimin- between z and z + dz. ation since the number of fractures is usually much

Equation (22) represents the total number of cells less than the number of asperity contacts (0-300 which have failed and subsequently been eliminated compared to 1500-2500); and (vi) fiber segments are by subdivision prior to reaching z(z). However, at assumed to be stress-free at the instant they are


created. Assumptions (i)-(v) cause the model to adding the probabilities associated with each segment overestimate the actual damage while (vi) results in length, q~e(l~), until the sum (i.e. the cumulative an underestimation. Some of these deficiencies could probability) is greater than or equal to X be corrected if experience with the model shows the ~ ~ l , - 2r additional computat ion to be justified. X ~< ~ q~e(li) = ~ (26)

i=0 i=0 1 - nc(z ). 2r"

3.3. Implementation This is easily solved for N (the only unknown) in the In principle, the fracture density, Nf, can be deter- numerical implementation since all the Ii are known

mined by direct, numerical integration of (24). In (and are unique) at the moment an additional contact practice, the large number of integrals involved, and is made. the need to solve a nonlinear ODE (10) (describing Determination of the lengths of the two cells the single cell response) for each combinat ion of zc, created when a longer cell is eliminated can be treated z and l results in an excessively large computation. An likewise using the Monte Carlo method. Since the alternative is to simulate the cell creation/elimination probability of subdividing any cell of length, l ' to process using random sampling techniques in such a create cells, for which at least one has a length, l, is way that the governing probabilities (~0c, q~e) are uniform, the created lengths are determined from satisfied. Monte Carlo methods allow one to do this ~ A_! i NAl (27) and to evaluate and sum the cumulative probabilities X ~< ,-0 l ' - 1' of failure for all cells. This approach also exploits the independence of the fiber strength and the surface where AI represents the smallest segment which can roughness distributions. It can be implemented be created when l ' fractures (typically Al is chosen efficiently so that parametric investigations of ma- such that AI<<I). The two created lengths are then terials and process paths (which affect only the unit l~ ---N • AI with N = )~l'/Al so that 1~ = gl', and cell response) can be conducted without having to l 2 = (1' - l~). Following the elimination of a cell and repeat the computationally expensive cell distribution its replacement by two cells of shorter length, the list analysis, of cell lengths is sorted in order of increasing length.

The simulation of the cell creation/elimination This process (i.e. addition of a contact, eliminating a process can be conducted by treating l as a discrete segment and replacing it with two shorter segments) random variable and using discrete forms of the can then be repeated. probability densities given in Section 3.2. The distri- Finally, the problem of determining the point but ion of fiber bend segment lengths at any time during densification at which the eliminated cell was consists of no(z)+ 1 segments, all of different lengths, created becomes simple in the discrete formulation l~. The number of asperity contacts, n c (z), is given by since a running list is maintained of all cell lengths (l), (18) with the probability density function describing their point of creation (zc) and elimination (z). Thus the distribution of asperity heights, ~Oh, and the since all cells are unique, it is easy to find the asperity density, p, determined experimentally by corresponding height of creation when a particular stylus profilometry. Typically the surface is found to cell is eliminated. be isotropic and the asperity heights to be normally distributed [8] Density D

1 , 0 0 , , ~0h (h) -- - - exp (25) 0.6 . . . . . .

0.5 where /~ is the mean asperity height and o" h is the standard deviation. ~-

Monte Carlo techniques allow the process of elim- v 0.4 ination to be simulated by random sampling in such '~ a way that the frequency with which a particular ~ o.3 segment length is eliminated by added contacts is -~

t -

given by ~oe. Recalling (from Section 3.2) that the ~ 0.2 .(" probability that the next contact added to a fiber o0

. 1 •

intercepts a cell of length, l is just the length occupied 0.1 . . ,

by cells of length, l divided by the total length of fiber ¢ ,.t,,.~.. • • accessible to the contact, the probabili ty that any one 0.0 . . . . . . . . . . . . - .. xax~_x,g¢... of the discrete lengths, l~will be contacted in a meter o.3 0.4 0.5 0.s 0.7 o.a 0.9 ~.o of fiber (i.e. Yl~ = 1) is just the length itself divided by Monotape thickness z / z 1 - - n~(z). 2r: ~%(li) = (li -- 2r)/[1 -- n~(z)" 2r]. Using 0 the Monte Carlo technique, a pseudorandom num- Fig. 3. The creation of fiber span lengths with decreasing

monotape thickness (from z 0 = 208/~m to z~ = 82 #m, corre- ber, Z, is generated (on the interval 0,1); Z represents sponding to a final relative density of 0.9). Typically, the cumulative probabili ty of elimination and the 1800-2500 fiber bend segments are created, depending on corresponding (expected) event is determined by the surface roughness characteristics of the monotape.


Table 1. Ti-24Al- l 1Nb matrix material data [24-30] Table 2. SCS-6 (SIC) fiber properties [21]

Parameter, symbol Units Value Parameter, symbol Units Value

Plasticity Diameter, df [.t/m] 142 Yield strength, a 0 [MPa] Young's Modulus, Ef [GPa] 425

20°C 539.9 Reference Strength, tr~r [GPa] 4.5 + 0.2 20~< T ~< 800°C a0(T) = 5 3 9 . 9 - 0 . 3 2 . T Weibull Modulus, m 13.0_+2.1 T > 800°C %(T) = 1019.9 -- 0 .92.T

Steady State Creep Power law creep [Pa - " s -~] 2.72× 10 -14 properties of the constituent materials (fiber and

constant, B 0 Stress exponent, n 2.5 matrix), surface roughness, and the process path. Activation energy, Qc [kJ/mol] 285 Damage dependence upon these parameters is not

always intuitive because of the coupled, nonlinear

Equations (26) and (27) allow the evolution of the behavior of the system. This complexity also denies cell length distribution to'be simulated. Figure 3 gives one the opportunity to analytically derive relation- an example of the change in lengths of created cells ships between damage and the parameters affecting it. as densification proceeds (decreasing z). For the case However, the model enables numerical experiments shown, z0 = 208 #m and the final relative density to be conducted (more easily than real experiments) reached (at z = 82/~m, Z/Zo ,~ 0.4) was 0.89. The and these can be used to explore and better under- longer cells have the greatest probability of being stand the influence of these parameters on the fiber eliminated and are seen to be quickly subdivided. The fracture density (and thus the quality of the resulting length of most of the roughly 1800 cells present at the composite). We begin with the study of a benchmark end of this example (the actual number depends on problem and subsequently investigate systematic

variations of the damage controlling parameters. An the surface roughness characteristics) are too small to be seen on the linear scale of the figure, analysis of the processibility of typical composite

With the evolution of cell lengths and their heights systems is also presented and an example is shown of of creation and elimination determined, equation (17) the application of both the fiber fracture and densifi- can be applied to determine the cumulative prob- cation [8] models to the simulation of a typical ability of failure for each cell. These failure probabil- consolidation cycle. ities are summed for all cells to obtain the total number of accumulated fractures. The unit cell re- 4. I. Benchmark problem sponse, y (t) given by (10) can be solved by numerical

The matrix material used for the benchmark prob- integration using a Runge-Kutta scheme, from which lem was the intermetallic alloy, Ti-24AI-11Nb(at.%) the fiber stress during bending [equations (1) and (14)]

and the rate of change of fiber stress with respect to used in an experimental study of damage [9]. It was z (dtrr/dz = df /~)are determined, assumed to have the properties given in Table 1.

Table 2 lists the properties assumed for a SCS-6 (SIC) The fiber fracture model (Monte Carlo simulation fiber reinforcement used in [9] and Table 3 gives the and unit cell constitutive response) has been im- measured surface statistical parameters for this type

plemented using MATHEMATICA tm [23]. Separate of monotape [8].

subroutines treated the cell evolution and damage. A constant consolidation temperature of 900°C Input to the model was the monotape thickness as a function of time (predicted using a monotape densifi- and a (constant) densification rate of 0.55 h-~ were cation model [8]). The Monte Carlo approach, had a assumed.t With all of the benchmark values inserted

into the model, 62 fractures/m of fiber were found to + 2 % variability for the predicted number of fractures at a relative density of 0.9 (the highest density have occurred by the time the relative density had

reached 90%. This amounted to about 3% of the at which the model was presumed accurate). Simu-

total number of spans formed. Practically all frac- lation of a typical process cycle required about 15 min

tured fiber segments had l/dr ratios between 4 and 20 to complete on a 66 MHz 486 PC, depending on the

with the most frequently fractured segments having constitutive model nonlinearity, complexity of the process schedule and lineal asperity density, an l/dr ratio of around 8. Fiber segments first began

to fracture at a relative density of about 0.76 (the

4. RESULTS AND DISCUSSION starting relative density was 0.64). The number of predicted fractures lay within the range of experimen-

Damage accumulation during consolidation pro- tal observations where, under similar conditions cessing of spray deposited monotapes is affected by (870-955°C, 5.5 h - l ) , 50-140 fractures/m were many parameters, e.g. the dimensions and mechanical measured [9].

Table 3. Plasma sprayed Ti-24Al-I INb/SCS-6 monotape surface tin reality the densification rate depends on the applied statistical data [8]

stress, temperature and current density and varies Parameter, symbol Units Value throughout a real process cycle. It was kept constant here to simplify elucidation of the fracture behavior. A Mean asperity height, h [#m] 91.06 constant densification rate at fixed temperature requires Asperity height deviation, ah [/~m] 39.82 the applied stress to increase as the density increases. Lineal asperity density, p [~m -~] 3.87 x 10 -3


Decreasing damage Decreasing damage =,

/ I f l f ' l r ' " '1 I Ti-24AI-11Nb/SCS-6 I SCS-6 (SIC) Fiber

= - ~ 4 o o .... ................ D=O.9 b = o.ss he'

~" 400

0.1 320 ,' =~

t - r ~

O ~ 2 4 0 Yielding-~-~'~" ~

200 --120 j

- - 8 0 J - - 4 0 fractures / m

0 100 :~ 850 900 950 1000 2.0 2.1 2.2 2.3 2.4 2.5 2.6

T e m p e r a t u r e (°C) Stress exponent n

Fig. 4. Influence of processing conditions: lower consolida- Fig. 6. Influence of matrix properties on fiber damage: tion rates and higher temperatures lead to fewer accumu- matrices with low resistance to deformation (low yield lated fractures by promoting matrix deformation as strength/high creep stress exponent) are favored. For

opposed to fiber deflection, matrices with high creep resistance (low n), fiber damage is controlled by the yield strength (regime labeled "Yielding") while for matrices with low creep strength (high n), damage

4.2 Influence of process conditions is controlled by the creep response (regime labeled "Creep"). [The benchmark value for the power-law creep constant, B 0

The model was next used to explore the effects of (see Table 1) was used for all calculations.] consolidation conditions. Figure 4 shows a damage contour plot relating densification rate and consolidation temperature (i.e. process variables) to fbe r fracture. Damage was reduced by lowering the rate of Support for these predicted trends can be found in compact ion (i.e. smaller displacement rates for VHP experimental studies [9]. For example, the measured or a slower application of pressure during HIP) and fracture density decreased by about a factor of 2 as by raising the consolidation temperature. These con- the displacement rate during VHP consolidation at ditions favor asperity flow [Fig. l(a)] as opposed to 955°C was decreased from 1.7 to 0.4cm/h. Further- fiber deflection [Fig. l(b)]. The accumulated damage more, increasing the process temperature from 900 to is seen to be more sensitive to changes in temperature 975°C while keeping the applied stress (of 100 MPa) than to densification rate because of the rapid matrix constant also resulted in a decrease in fracture density softening and higher creep rates associated with with from 140 to about 60 m-= [9]. Figure 5 summarizes an increase in temperature, the results of Groves et al. [9] for three compact ion

rates at 955°C and compares them to the model predictions. The model gives good agreement with

70 . . . . . . . ~ . . . . . . . . ' 3~ . . . . . . . . ' the experiments even though the monotapes were

Y consolidated to full density. 60 During the consolidation of alloy or ceramic pow-

"r E 50 ders, one normally applies pressure and temperature simultaneously to reduce consolidation time. Figure

40 4 can be used to demonstrate the importance of = ~ heating a lay-up to its process temperature before = applying the consolidation pressure in order to avoid 30 = fracture. It can be seen that if the temperature were

20 first raised to 950°C and the lay-up were then sub- u_ Nb/SCS-0 jected to an applied pressure sufficient to cause a

/ , ~ o - T = 955°C densification rate = 1 h-1, the fracture density would 10 / ~ Experimental results

/ ~ of Groves et al.[9] not exceed 20 m ~. However, much higher levels of 0 . . . . . . . r . . . . . . . . r . . . . . . . . I damage would have resulted from simultaneously

0.1 1 10 increasing both the temperature and pressure (densifi- Rate of compaction (cm/hr) cation rate) to these values. The damaging effect of

Fig. 5. Comparison of experimental and predicted fracture applying pressures at low processing temperatures densities at several consolidation rates. The results are for has been observed by Groves [21]. Stresses even as a laminate of five Ti~4AI-11Nb/SCS-6 monotapes consol- small as 3.5 MPa, when applied at room temperature,

idated at 955°C. resulted in up to 20 fractures/m in the Ti -24AI-11Nb/


SCS-6 system because of the high matrix yield 4.3.2. Fiber. Intuitively, the incidence of consolida- strength at this temperature, tion damage is expected to depend on fiber properties

It can be seen in Fig. 4 that the processing tempera- such as stiffness, diameter and strength distribution, ture required for a fixed degree of damage decreases but these dependencies are complicated by the inter- as the densification rate is reduced (i.e. consolidation action between the fiber and the inelastically deform- time increased). However this must be balanced ing asperities. Intuition alone cannot predict the best against the need to also minimize interfacial reactions combinations of say, fiber reference strength (tr~f) in order to obtain low interfacial sliding friction and Weibull modulus (m) to reduce damage, nor the stresses. The thickness of the interracial reaction has relative worth of increasing the elastic stiffness or been shown to obey Arrhenius kinetics and therefore diameter. We shall show that it can even lead to the use of lower temperature/shorter times is necess- completely erroneous conclusions. ary to avoid interfacial property degradation [31]. As an illustration of this, it could be argued that The relatively simple models for predicting the the fiber's flexibility, as defined by I/MR (where M is growth rate of fiber/matrix reaction zones during consolidation processing [31] could be combined with Decreasing '1 damage those for densification [8] and fiber fracture to enable 500 a ' ;' / ' I I Y [ ' I a quantitative trade-off to be made. ~ a ~ / I ~ TTi'24Al'11NbMatdx ] = 9000C [

: ~ O.Oi9 b=055hr-' 4.3. Influence of material properties A 450 . " ]

The extent of fiber deflection and thus fracture ~ \ , y ~ / / / i during consolidation is controlled by complex combi- ~ : ~ .-~ nations of matrix and fiber properties. We can use tff" 400 ~ : ~r the m°del t° expl°re this' and t° identify which =~ ~ ~ ~ / / i ~ ~ J properties are the most critical for the control of ~ ~' damage. E 3SO " o

4.3.1. Matrix. Using the benchmark fiber properties (those for SCS-6) and process conditions iy. \ \ i / / i ] (/9 = 0.55 h- ' , T = 900°C), the effect of the matrix / material's resistance to plastic and creep deformation 300 ] (as represented by its yield strength, tr 0 and power law / creep stress exponent, n) is shown in Fig. 6. It is seen 20 40 60 80 100 120 140 160 that fiber damage can be reduced by using matrix Fiber diameter d! (pm) materials with a low resistance to inelastic deformation (i.e. low tr0, high n) because in this case , , Dec= reasingdarnage I ' ' smaller contact forces are required to deform the 20 l-b I t asperities (and accommodate an imposed densifica- ~ ~ ~ r'900°c [ tion rate) resulting in smaller fiber deflections and D.o.9 b=o.~hr-' thus bend stresses. Depending on the yield strength t / \ \ / value, reducing the creep exponent below about 2.2 ~ 15 is seen to have surprisingly little effect on the fracture = I I 1 1 50 fractures / m

"0 10 ~ ~ - -

density. For n < 2 . 2 , we find the creep contribution ~ I / \ \ a0 \ to the asperities' deformation to be negligible com- = pared to that of yielding, and in this region, the ~ \ .o \ \ ~,

• r \ \ \ matrix yield strength alone determines the amount of • u0 damage. The converse can also be seen: changing the 170 yield strength of the matrix when n is large has little I ~ o o ~ ~ ~ ~ effect on damage because the asperities are able to S ll ~ ' ~ - ~ " - ~ ~ ~ ~ deform readily by creep. The fracture behavior in this h~o- ~ ~ ~ ~ region is then controlled by only the matrix creep 3.5 4.0 4.5 5.0 5.5

strength. Reference strength o~f (GPa) The results shown in Fig. 6 were calculated using

a fixed value of the power-law creep constant, B 0. Fig. 7. The influence of fiber properties: (a) the damage map Variation of n (by changing alloy composition or indicates two approaches to minimizing damage: fibers

having high flexibility (small diameter, low modulus) can microstructure) also usually changes B0. From bend without breaking while larger diameter fibers resist equation (7), it can be seen that increasing B0 has a bending so that densification must be achieved via matrix similar effect on the strain rate (and hence on the flow (regime labeled "Rigidity"). Improved rigidity (bend fracture density) to increasing n, although it would resistance) is the favored approach because of the difficulty have to be varied over a much greater range in order in achieving the very high flexibility needed to reduce

fracture densities to acceptable levels, (b) high reference to obtain the same effect because of the linear depen- strength fibers with low strength variability (high WeibuU dence of creep strain rate on B 0. modulus) are preferred.


Table 4. Fiber properties used for processibility study

Reference Diameter Modulus strength Weibull

Fiber Material [/am] [GPa] [GPa] modulus Reference

Sigma 1240 SiC 100 400 3.87 18 [34, 35] Sigma 1040 SiC 100 400 3.3 + 0.1 12 + 3 [34, 35] Saphikon A1203 135 460 3 + 0.3 6 + 1 [36] SCS-6 SiC 142 425 4.5 _+ 0.2 13 + 2.1 [21]

the bending moment and R the minimum radius of asperity forces) is reduced. It is interesting to see that curvature), determines the susceptibility of fibers to the influence of the Weibull modulus is strongest bend fracture. For cylindrical elastic fibers, when the modulus and reference strength are both I/MR = 64/(EfTtd4). Thus fibers having both a small low. The results indicate that damage becomes prac- diameter and low Youngs modulus are the most tically independent of m once the modulus exceeds flexible and might be expected to undergo a very large about 10 for the range of fiber reference strengths deflection without breaking. However, the model available in today's fibers (see Table 4) so fiber [Fig. 7(a)] predicts that for the fibers usually used in development efforts to further reduce variability are, monotapes (df ~ 150 p m) it is generally not flexibility, from the processing viewpoint, unnecessary. Improv- but rather the fiber's resistance to bending that is ing the processibility of fibers by increasing m, may important in avoiding fiber damage. Flexibility is in fact reduce the strength of brittle matrix corn- favored only for very small diameter (dr< 35/~m) posites, which is enhanced by a high fiber strength fibers. In fact, for the matrix material and processing variability [32, 33]. conditions used to produce Fig. 7(a), only fibers with very high flexibility (e.g. dr < 10/tm, Ef<<275 GPa), 4.4. Influence of monotape surface roughness would be able to suffer the deflections required during Monotape surface roughness is the fundamental consolidation without substantial fiber damage, reason for fiber damage in this approach to corn-

For fibers with df > 35 pm, damage is most easily posite processing. Removing the roughness would decreased by increasing their diameter. The influence eliminate the damage. The surface roughness of these of the fiber's elastic modulus is a relatively weak one; monotapes is determined by conditions prevailing it is the size effect that is strong. For this region of during the plasma spray deposition process (e.g. it is the map [labeled "Rigidity" in Fig. 7(a)], damage is probably reduced by increasing the molten droplet lessened by promoting asperity deformation rather superheat or velocity which both promote droplet than fiber deflection and this is best achieved with spreading). We can use the model to investigate how large diameter fibers that have the greatest bend much of a reduction in asperity height is needed to stiffness. The relative importance of flexibility (fibers overcome the damage problem. bend without breaking) or rigidity (fibers that resist Figure 8 shows the influence of the initial surface bending), depends on the matrix material and the roughness as characterized by the standard deviation process conditions during consolidation. For the case examined in Fig. 7(a), efforts to develop slightly smaller, more flexible fibers than those used today are 350 , , , , , predicted to be a futile approach to damage re- ri-a~HmbUatr~ / duction. 300 r = 900°C ] I / /

Similar difficulties can emerge when attempting to _.._, D=0.9 b=O55hr-'] / predict the effect of a fiber's strength distribution on 'E 250 ~ = 91.0s pm damage; obviously, increasing the fiber's average (or reference) strength will lessen the likelihood of frac- .~ 200 ture, but it is not as intuitively clear how the Weibull ®

"1o

modulus affects damage, or how the fiber's reference ~ 150 strength might affect this.The model can be used to explore these issues and to provide guidelines to the ~_ 100 developers of fibers for improving their processibility. Using the benchmark process conditions and matrix 50 properties, we show in Fig. 7(b) the effect of a fiber's

i i ~ I I I Weibull modulus (i.e. its fracture strength variability) °o 10 20 ao 40 50 60 and reference strength on damage. In general, increasing both m and O're f reduces the incidence of Asperity height deviation (pm) fracture. Fracture densities are lowered by high val- Fig. 8. The effect of monotape surface roughness (deter-

mined by conditions during plasma spraying of the mono- ues of m and/or Crre f because, for given matrix proper- tapes): reducing surface roughness (lower deviation in ties and processing conditions, the fraction of fiber asperity height, ~rh) strongly reduces the predicted fiber strengths in the range of bend stresses (created by fracture.

A M 4 2 / 1 2 G


of asperity heights, tr h. All the material properties and Table 5. Matrix properties used for processibility study

process conditions correspond to the benchmark Parameter, symbol Units Value values used earlier. It should also be noted that Ti-AI[37-39] altering the conditions under which materials are Elastic properties

Young's modulus, [GPa] spray deposited in order to vary the surface rough- 20oc 173.0 ness also affects the initial relative density of the T~>20°C E(T)= 172.0-0.03"T deposit, As the deviation in asperity heights goes to Plasticity

Yield strength, tr 0 [MPa] zero (assuming the average size and density of asper- 20°c 473.0 ities remain constant), the initial relative density is 20 ~< T ~<600°C o0(T ) =475.0-0.10"T

T > 600°C cr0(T ) = 875.0 - 0.76"T expected to increase. For this reason, the initial Steady statecreepa relative density used to obtain Fig. 8 was varied Power law creep [s -I] 2.12x 1019 linearly from 0.52 (the ratio of the volume of a constant, A

Stress exponent, n 4.0 hemisphere to that of an enclosing parallelepiped) at Activation energy, Q¢ [kJ/mol] 300 tr h = 0, to 0.35, the value measured experimentally

Ti--6AI-4V[40--45] and corresponding to tr h = 39.82/~m. The accumu- Elastic properties lation of damage is seen to be extremely sensitive to Young's modulus, E [GPa] tr h for ah>30 / tm. In the experiments of Ref. [9], 2°°c 114.0

T ~< 500°C E ( T ) = 115.0 - 0.056"T a h = 39.82/~m. Reducing the deviation in asperity T > 500°C E(T) = 1 7 2 . 4 - 0.16*T heights to 30/tin or below is predicted to be an Plasticity effective method for reducing (or even eliminating) Yield strength, tr 0 IMPs]

20°C 970.0 fiber fracture during consolidation. Damage is pre- T< 960°C o0(T ) = 884.0--0.92"T dicted to be much less sensitive to changes in the Steady state creep" mean asperity height; it is the variability in roughness Power law creep [ s - l ] 2.3 × 102t constant, A which determines the deflections and hence the Stress exponent, n 4.0 stresses in the fibers. Activation energy, Qc [kJ/mol] 280

4.5. Materials selection for process tolerance Nim 80A[46-48] Plasticity

Fiber fracture results from a competition between Yield strength, a 0 [MPa] 20°C 620.0

a n asperity'S inelastic contact deformation and a 20 ~< T ~<760°C ao(T ) = 6 2 2 . 0 - 0 . 1 4 " T

fiber's elastic deflection and we have shown that this T > 760°C o0(T) = 2 1 9 8 . 0 - 2.23"T has a complex dependence upon matrix and fiber Steady state creep properties. Many fibers and matrix alloys are avail- Power law creep [Pa-"s -I] 1.08 x 10 -21

constant, B0 able today and others are in development. It is not at Stress exl~onent, n 4.2 all obvious which combinations are likely to result in Activation energy, Qc [kJ/mol] 410 the least damage. The model can be used to determine 'The steady state creep rate is described by ic = A • (a /E)"

this (i.e. to conduct a materials selection for processi- exp(-QJRrk)" bility).

4.5.1. Fiber selection. Figure 9 compares the frac- composite (using the benchmark problem's matrix ture density after densifying a Ti3AI + N b matrix properties and process conditions) reinforced with

either BP's Sigma 1240 (SiC with a C/TiB2 coating) 700 . . . . fiber, Sigma 1040 (uncoated 1240), Saphikon's single

i "1"i-24A1-11Nb Matrix crystal A12 O3 fiber °r Textrc~n's SCS-6 fiber (SiC with 600 r = 000°c a duplex C coating). The properties of these fibers,

D = 0.9 b.o.ss hr-' / used in calculating the results shown in Fig. 9, are "r E 500 z o - ~ / . 6 ~ summarized in Table 4. We see that the SCS-6 fiber -~" 400 ~ , - ~ ) is the best choice from the viewpoint of process • ~ induced damage; a consequence of its favorable t- ® combination of large diameter, high reference "1o .~ 3oo / / ~ strength, low strength variability (high m value) = and good stiffness. The Sigma 1240 fiber is the

200 u_ second best choice; the loss in processibility due to its smaller diameter and lower reference strength

100 / / . . . . a ~ (compared to the Saphikon and SCS-6 fibers)is , ~ ~ ~ G ~ offset by its lower strength variability. The relatively

00'.4 0.5 0.6 0.7 0.8 0.9 poorer processibility of the Saphikon and Sigma Relative density 1040 fibers results from unfortunate combinations

Fig. 9. Comparison of fibers for consolidation processibility: of dr, m and a~f. The big difference in the processibil- the SCS-6 fiber's combination of large diameter and high ity of the two Sigma fiber types (even though the reference strength make it most attractive from the process- differences in their properties is relatively small) is

ing viewpoint, a dramatic demonstration of how processibility

ELZEY and WADLEY" FIBER FRACTURE MODEL FOR MMC CONSOLIDATION 4011

40o ~ r - - - - ~ , . , , , ~ processing temperatures that are always needed to 4 ; • • -1 4,., 1o.o.9 b=o.s5,c' avoid fracture. Further improvements in fiber

Ti-6AI-4V , \ 4, ~ strength and protective coatings (to avoid degra- ) ' ~-ii,,,,, ~ '4, dation during high temperature processing) together

1:: 30(] ,:; with modification of the monotape roughness could " r allay these difficulties, and enable the more successful _z, ~ , . manufacture of higher temperature composites. ¢n \ \ _ , I~Ti.24AI.11N b , oo\ "o ~ ~ ~ ~ r i - A 4.6. Process simulation "~ The fiber fracture model can be used to simulate an

~(~) entire consolidation process including its response to LL 100 transients. Implementation of the model is no differ-

NimSOA \ \ ~ ent to that for the process planning and materials selection applications considered above; only the den-

~0t~--"-"~600 ' 700 800 900 1( ~0 sification path, taken to be constant earlier is instead, Temperature (°C) generated for an entire pressure-temperature-time

cycle (including ramp-up) using the monotape den- Fig. 10. Comparison of alloy matrices for processibility (all sification model in [8]. reinforced with SCS-6 fiber): successful consolidation of the creep resistant intermetallic alloys, TiAI(y) and Ti-24A1- Figure 1 l(a, b) show an example of this application 11Nb(ct 2 + fl ) requires substantially higher processing tem- of the model. For this hypothetical process cycle the peratures (and consequent risk of fiber degradation due to temperature was first ramped to 900°C and held con- reactions with the matrix) compared with the • + ~ alloy, stant, and a linear ramp in pressure (at 100 MPa/h) TiqSAI-4V. The Ni-base superalloy, Nimonic 80A, can be then applied. Figure 1 l(b) shows the time dependent

processed in an intermediate range. evolution of composite relative density (obtained using the model in [8]) and the normalized fiber

is magnified by the nonlinearity of the damage pro-

cess. Time (hr) 4.5.2. Matrix selection. Use of the model for select- 0.o 0.4 0.8 1.2 1.8

ing a matrix material is not as straightforward as fiber . . . . . . . . 200 selection; the temperature and rate dependence of the a Consolidation Process Cycle

matrix properties prevent a simple comparison of G 900 f 1so damage under identical process conditions. Instead, o.. / ~ we have calculated the damage as a function of ~ / ~ , consolidation temperature. Using the matrix proper- ~ 10o ties given in Table 5 and a densification rate of =~ ~, 0.55h -~, Fig. 10 compares the damage produced ~_ ~' ~0 ~- during the consolidation of a variety of matrix alloys. It can be seen that a Ti--6A1-4V matrix (containing SCS-6 fibers) can be successfully consolidated at 2~ 0 0.0 0.4 0.8 1.2 1.6 temperatures below 650-700°C whereas the more 1.0 . . . . . . . . . 40 creep resistant intermetallic matrices, such as ct 2 + fl b Simulated Response Ti-24AI-11Nb alloy and 7-TiAI, require tempera- 0.8 32 tures in the range 950-1000°C or above. The Ni-base 7" superalloy, Nimonic 80A, can be processed in an ~ intermediate range, 850-900°C. Lower densification = ~ 0 . 6 2 4 . _

rates shift the curves to the left, allowing use of lower ® processing temperatures (see Fig. 4), but at the -~ 0.4 16

expense of increased process time. ~ It is interesting to note that although the steady ,,'-

state creep strength of ~,-TiA1 is considerably greater 0.2 s than that of the ct 2 + fl alloy, the latter's higher yield strength (at the process temperature) makes it the 0.¢ 0 more difficult material to consolidate without fractur- 0.0 0.4 o.s 1.2 ~.6 ing fibers. Figure 10 indicates that alloys such as Time (hr)

Ti-6AI--4V are preferred candidates for this class of Fig. 11. The model can also be used to simulate the MMCs, but such a matrix restricts their use to low to accumulation of fiber damage for arbitrary pressure-tem- intermediate temperatures (600°C). The intermetallic perature-time cycles. (a) Simple consolidation process alloys needed for higher temperature applications are schedule in which the temperature is held constant (900°C)

and pressure is ramped linearly with time (100 MPa/h) used seen to be difficult to composite via the plasma spray as input, (b) the predicted densification (using model given route because of fiber-matrix reactions at the high in [8]) and fiber damage.


fracture density predicted by the fracture model. The Most currently available fibers fall in the "rigid- damage accumulation rate is seen to be at first slow ity" regime and damage is reduced by increasing (despite the high densification rate) because initially, their diameter (reducing their flexibility). most fiber segments are long (l/df > 20) and do not 5. Reducing the monotape's surface roughness suffer large bending stresses [cf. equation (11)] for the (particularly the variability in asperity height) small deflections associated with initial densification, even by modest amounts can greatly reduce The rate of fracture achieves a maximum at about fiber damage. Improved control of the plasma l . l h into the process (composite relative density of deposition process to reduce roughness offers 0.85) when the most probable segment lengths ap- the potential to eliminate fiber fracture during proach l /df values of 12 and lower. The fracture rate consolidation, even for creep resistant materials. then begins to decrease as the relative density approaches 0.9 as an increasing fraction of the fiber Acknowledgements--The authors wish to thank D. G. Back- segments become too short to fail (1~dr < 4), (though man and E. S. Russell of General Electric Aircraft Engines

for helpful discussions of the fiber damage problem. We also in this region the model's validity is also diminishing, gratefully acknowledge the support for this work by an because the assumption of point contacts becomes ARPA/NASA contract (No. NAGW-1692) managed by W. unreasonable). This ability of the model to simulate Barker (ARPA) and R. Hayduk (NASA) and a grant from fiber damage accumulation for arbitrary consolida- ARPA/ONR from the UCSB-URI (No. 5158-91) directed tion cycles enables exploration of new approaches to by Professor A. Evans and managed by W. Coblenz process planning and on-line control by coupling the (ARPA) and S. Fishman (ONR).

model with an optimization scheme [12].

5. CONCLUDING REMARKS REFERENCES

A micromechanical model for predicting the rate of 1. R. A. MacKay, P. K. Brindley and F. H. Froes, JOM 43(5), 23 (1991).

fiber fracture during the consolidation processing of 2. J. Doychak, JOM 44(6), 46 (1992). spray deposited metal matrix composite monotapes 3. J. R. Stephens and M. V. Nathal, in Superalloys 1988 has been developed and combined with stochastic (edited by D. N. Duhl), p. 183. TMS, Warrendale, Pa descriptions of fiber strength and randomly rough (1988). surfaces to investigate the fiber damage during MMC 4. P. D. Nicolaou, H. R. Piehler and S. Saigal, in Concur-

rent Engineering Approaches to Materials Processing processing. The model has been used to investigate (edited by A. J. Paul, S. N. Dwivedi and F. R. Dax), the sensitivity of damage to process conditions, p. 247. TMS, Warrendale, Pa (1993). matrix and fiber materials and monotape surface 5. P. K. Brindley, in High Temperature Ordered Inter- roughness. It has been found that: metallic Alloys H (edited by N. S. Stoloff, C. C. Koch,

C. T. Liu and O. Izumi), Vol. 81, p. 419. MRS, Pitts- 1. The model successfully predicts the damage burgh, Pa (1987).

experimentally observed in T i -24Al - l lNb/SCS- 6. D. A. Hardwick and R. C. Cordi, in Intermetallic Matrix Composites (edited by D. L. Anton, P. L.

6 composites and reproduced the experimentally Martin, D. B. Miracle and R. McMeeking), Vol. 194, observed dependence of damage on consolida- p. 65. MRS, Pittsburgh, Pa (1990). tion process conditions. 7. E. S. Russell, in Thermal Structures and Materials for

2. The model predicts that fiber damage can be High Speed Flight (edited by E. A. Thornton), Progress in Astronautics and Aeronautics, Vol. 140, p. 437.

avoided (or at least minimized) by increasing AIAA, Washington, D.C. (1992). temperature and lowering the densification rate 8. D. M. Elzey and H. N. G. Wadley, Acta metall, mater. (i.e. pressure) during consolidation: the optimal 41, 2297 (1993). conditions are shown to depend on the proper- 9. J. F. Groves, D. M. Elzey and H. N. G. Wadley, Acta ties of the fiber and matrix, metall, mater. 42, 2089 (1994).

10. J.-M. Yang and S. M. Jeng, JOM 44(6), 52 (1992). 3. The "processing window", where high com- 11. J.-M. Yang, S. M. Jeng and C. J. Yang, Mater. Sci.

posite density can be achieved while fiber frac- Engng A 138, 155 (1991). ture is avoided, is reduced as the matrix yield 12. D. G. Meyer, R. Vancheeswaran and H. N. G. Wadley, strength and creep resistance are increased (e.g. in Model-Based Design o f Materials and Processes

(edited by E. S. Russell, D. M. Elzey and D. G. titanium aluminides and Ni-base superalloys). Backman), p. 163. TMS, Warrendale, Pa (1993). The selection of fiber and processing conditions 13. L. M. Hsiung, W. Cai and H. N. G. Wadley, Mater. Sci. is critical for these materials. Engng A 125, 295 (1992).

4. Fiber properties have a strong effect on damage. 14. L. M. Hsiung, W. Cai and H. N. G. Wadley, Acta metall, mater. 40, 3035 (1992).

Depending on the matrix and process con- 15. T. A. Kuntz, H. N. G. Wadley and D. G. Black, Metall. ditions, those least likely to be damaged during Trans. A 24, 1117 (1993). consolidation have either a high resistance to 16. H. Hough, J. Demas, T. O. Williams and H. N. G. bending or are highly flexible. Resistance to Wadley, Aeta metalL mater. In press. bending is achieved by fibers with a large diam- 17. R. Gampala, D. M. Elzey and H. N. G. Wadley, Acta

metall, mater. 42, 3209 (1994). eter. Flexibility is an attribute of small, compli- 18. K. L. Johnson, Contact Mechanics, p. 412. Cambridge ant fibers which can suffer the deflections Univ. Press, Cambridge (1985). incurred during consolidation without failure. 19. C. W. MacGregor, in Marks Standard Handbook for


Mechanical Engineers, 8th edn (edited by T. Baumeister adjacent asperities along a length of fiber), for its solution. et al.), pp. 5-24. McGraw-Hill, New York (1978). The distribution of spacings, which changes as densification

20. P. A. Siemers, R. L. Mehan and H. Moran, JOM 23, occurs and new contacts are added along a given length of 1329 (1988). fiber, is not easily determined by direct measurement.

21. J. F. Groves, M. S. thesis, Univ. of Virginia, p. 32 Widom [22] has analyzed the analogous problem of random (1992). sequential addition of hard spheres (of diameter, or) along

22. B. Widom, J. Chem. Phys. 44, 3888 (1966). a straight line of total length, L, such that no spheres 23. Wolfram Research Inc., Mathematica, vers. 2.2, publ. overlap. As spheres are added, some gaps are created which

by Wolfram Research, Inc (Champaign, I1)(1992). are smaller than a, so that these may not be further 24. S. J. Balsone, Proc. Conf. on Oxidation of High Tern- subdivided by additional spheres. A unique (exponential)

perature Intermetallics (edited by T. Grobstein and J. distribution of gaps between the spheres arises for all Doychak), p. 219. TMS, Warrendale, Pa (1989). packing densities, and this can be used to compute ~ot(z; l).

25. D. A. Koss et al., Proc. Conf. on High Temperature Widom [22] presents the distribution in normalized form Aluminides and lntermetallics (edited by S. H. Whang et al.), p. 175. TMS, Warrendale, Pa (1990). f~b (p)2

26. J. M. Larsen et al,, Proc. Conf. on High Temperature e (~-t)~,(p) (2/> 1) Aluminides and Intermetallics (edited by S. H. Whang et ~ ~b "( p ) al.), p. 521. TMS, Warrendale, Pa (1990). q ( p ; 2 ) = I t ' , (AI)

A. Lukasak, Metall. Trans. A 21, 135 (1990). 1 2 / qJ(x).e -:'~(x) (2 < 1) 27. D. 28. M. G. Mendiratta and H. A. Lipsitt, J. Mater. Sci. 15, L .J0

2985 (1980). 29. R. W. Hayes, Scripta metall. 23, 1931 (1989). with ~b (p) implicitly defined by 30. W. Cho et al., Metall. Trans. A 21, 641 (1990).

31. P. E. Cantonwine and H. N. G. Wadley, Comp. Engng foO i Cj ,o l _e , ] 4, 67 (1994). = exp - 2 ~ ds dt (A2)

32. W. A. Curtin, J. Am. Ceram. Soc. 74, 2837 (1991). P 33. W. A. Curtin, J. Mech. Phys. Solids 41, 217 (1993). 34. British Petroleum, Product Data Sheet, BP Metal Com- where p is the normalized packing density (length occupied

posites Ltd, Farnborough, Hampshire, U.K. by N spheres/total length, L ) and 2 is the gap length 35. P. E. Cantonwine, M. S. thesis, Univ. of Virginia, p. 11 normalized by #. The normalized length, 2 and the packing

(1993). density, p are equivalent to l/(2r) and 2n c (z)r (dimensionless 36. Saphikon, Product Data Sheet, Saphikon Inc., Milford, since n c has units of m- ~), respectively. Since a (a constant

N.H. in Widom's analysis) = 2r, use of equations (A1) and (A2) 37. P. L. Martin, M. G. Mendiratta and H. A. Lipsitt, implies that all asperities have the same radius or, alterna-

Metall. Trans. A 14, 2170 (1983). tively, that an average value for the asperity radius can be 38. H. A. Lipsitt, D. Shechtman and R. E. Schafrik, Metall. used. The loss of accuracy in the model as a result of

Trans. A 6, 1991 (1975). assuming an average asperity radius has not been assessed. 39. T. Takahashi and H. Oikawa, Mater. Res. Soc. Symp. The implicit expression for ~O (p) given aboveis easily solved

Proc. 133, 699 (1989). numerically and, below p ~ 0.4, is found to be given quite 40. C. J. Smithell, Metals Reference Book, 5th edn, well ( e r r o r < l % ) b y the expansion [22]

pp. 14-21. Butterworths, Boston (1961). 41. M. J. Donachie Jr, Titanium A Technical Guide, p. 173. 2 7 3 13 P 4 (A3)

ASM, Metals Park, Ohio (1988). ~b (p) = p + p + ~. p + ~ . 42. E. W. Collings, The Physical Metallurgy of 77 Alloys,

pp. 111-21. ASM, Metals Park, Ohio (1984). 43. C. R. Brooks, Heat Treatment, Structure and Properties where all terms of order higher than p 4 have been truncated.

of Nonferrous Alloys, pp. 361-376. ASM, Metals Park, This solution is adequate for the fiber fracture problem since Ohio (1982). the maximum number of contacts (per meter) is about 2000

44. W. C. Harrigan Jr, Metall. Trans. A 5, 565 (1974). as the rough surface approaches its limiting relative density 45. J. Pilling et al., Metal Sci. 18, 117 (1984). of 0.9 and therefore the normalized packing density ap- 46. C. T. Sims et al. (editors), Superalloys II, p. 595. Wiley, proaches (2. 2000.7 × 10 -5 = )0.28, (where 7 × 10 -5 [m] is

New York (1987). the average asperity radius [8]). With the solution for 47. H. Riedel, Fracture at High Temperatures, p. 390. q ( p ; 2 ) known, qJl(z;l) is obtained from

Springer, New York (1987). 48. R. W. Evans and B. Wilshire, Introduction to Creep 1

p. 62. Bourne Press, Bournemouth, U.K. (1993). ~ 1 (z; l ) = • q (p; 2 ). (A4) 4nc(z ) • r 2

A P P E N D I X Only the solution for 2 ~> 1 (corresponding to l >~ 2r) is of interest here since cells whose lengths are less than 2r may

Analytical Solution f o r the Cell Length Distribution be assumed not to fracture because of their high bend The fiber fracture model [equation (24)] requires the distri- stiffness. bution of cell lengths, q)~ (z; l), (i.e. of the spacings between

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