MODELING THE DENSIFICATION OF METAL MATRIX COMPOSITE … · 100-150tim diameter silicon carbide or aluminum oxide fibers onto a mandrel which is then rotated under a stream of (plasma

A cta me tall. mater. Vol. 41, No. 8, pp. 2297-2316, 1993 0956-7151/93 $6.00 + 0.00 Printed in Great Britain. All fights reserved Copyright © 1993 Pergamon Press Ltd

MODELING THE DENSIFICATION OF METAL MATRIX COMPOSITE MONOTAPE

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

VA 22903, U.S.A.

(Received 28 September 1992; in revised form 9 February 1993)

Almraet--The consolidation of monotapes consisting of a layer of continuous aligned ceramic fibers embedded in a plasma sprayed metal or intermetallic matrix is becoming a preferred approach for the processing of high performance composite systems. We present a first model that enables prediction of the density (and its time evolution) of a monotape lay-up subjected to a hot isostatic or vaceum hot pressing consolidation cycle. Our approach has been to break down the complicated (and probabilistic) consolidation problem into simple, analyzable parts and to combine them in a way that correctly represents the statistical aspects of the problem, the change in the problems interior geometry and the evolving contributions of the different deformation mechanisms. The model gives two types of output. One is in the form of maps showing the relative density dependence upon pressure, temperature and time for step function temperature and pressure cycles. They are useful for quickly determining the best place to begin developing an optimized process. The second gives the evolution of density over time for any (arbitrary) applied temperature and pressure cycle. This has promise for refining process cycles and possibly for process control. Examples of the models application are given for Ti3A1 + Nb, ), TiAI, Ti6AI4V and pure aluminum.

1. INTRODUCTION

Continuous fiber reinforced metal and intermetallic matrix composites are attracting increasing interest for high temperature aerospace applications because of their attractive specific stiffness and high temperature strength. Many methods are being developed for their processing (for recent reviews see Mehrabian [1] and others [2, 3]). In one increasingly favored approach, foils (also called monotapes) consisting of uniformly spaced continuous fibers in a porous alloy matrix are produced using plasma spray methods [4]. Lay-ups of these porous, unidirectional reinforced monotapes are then consolidated to form near net shape composite components using processes such as hot isostatic pressing (HIP), vacuum hot pressing (VHP) and roll bonding. Here we develop predictive models for the densification of these lay-ups. We seek models that predict the effect of process conditions (i.e. pressure, temperature histories) and foil attributes (geometry, matrix properties . . . . ) upon densification. Our approach is inspired by that of Arzt, Ashby and others, who have developed a predictive densification model for metal, alloy and ceramic powders undergoing hot isostatic pressing (HIPing) [5].

The plasma spray process consists of winding 100-150tim diameter silicon carbide or aluminum oxide fibers onto a mandrel which is then rotated under a stream of (plasma melted) matrix alloy droplets with diameters of 100-200/~m [4]. The droplets, upon contact with the substrate, spread out

coating the fibers and filling the interstices between with metal. Usually, the droplets freeze before fully filling the interstices creating small unconnected pores. Thus, at the completion of the spray process one obtains a unidirectionally reinforced foil with one relatively smooth surface (the one in contact with the mandrel during deposition), one much rougher surface (determined by the spreading of successive droplets), and several percent of closed internal porosity. The foils are then stacked to achieve a desired lay-up and placed either in the die of a vacuum hot press or suitably shaped HIP tooling and subjected to heating and pressurizing cycles to both fully densify the lay-up and obtain a near net shape component.

During the consolidation process one seeks to utilize the deformation mechanisms of plasticity, power law creep and diffusional flow to close both interfoil pores (i.e. those formed by contacts between foils) and intrafoil (i.e. the closed) porosity. The model we seek should predict the evolution of lay-up density over time for any given (arbitrary) application of pressure and temperature, foil geometry (foil thickness, surface roughness, internal porosity), and matrix material properties. There are several important reasons for wanting to do this. First, one would like to find the most convenient process cycle to densify a component without resorting to costly and time consuming trial and error experiments. Second, other phenomena during processing may degrade the component's properties, i.e. fibers may be damaged or broken, the fiber-matrix interface may

2297

2298 ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE

degrade, or the matrix may undergo phase transformations (because the foil matrix is often in a metastable, rapidly solidified state [6]). Ideally, one would use the model to design a process cycle that reaches the target density without adversely impact- ing the component's properties. It is quite difficult to do this purely by experiment because of the many things that must be optimized and their complicated dependence upon the process path. Third, one is interested in determining the role of matrix properties on the feasibility of consolidation. For example, we are interested in discovering which matrices are most easily consolidated and how this is affected by previous processing steps. Fourth, the model should allow assessment of the influence of foil surface roughness/internal porosity upon consolidation in order to identify those that will give the best results. This can be used to help in defining a target for the plasma spray process step. Finally, there is increasing interest in using the model for feedback control of consolidation processing [7]. Here, we derive a first model that is capable of each of these tasks and show its application to the densification of several systems of current interest. Its application for understanding and controlling some of the other phenomena during processing mentioned above has been presented elsewhere [8, 9].

2. MODEL FORMULATION

The cross-section of a typical continuous fiber reinforced monotape produced by the Induction Coupled Plasma Deposition (ICPD) process is shown in Fig. l(a). One sees that the tapes produced by this process are characterized by internal porosity, one very rough surface (the other side in contact with a mandrel during deposition remaining relatively smooth) and a certain fiber volume fraction. Figure l(b) shows a fully consolidated lay-up of TiaA1 + Nb/SiC monotapes (volume fraction of fiber

Fig. 1. (a) Cross section of a plasma sprayed MMC monotape (fibers are 140#m diameter SCS-6 (SIC) within a Ti-24Al-llNb alloy matrix). (b) Ti-24-11/SCS-6 mono-

tapes following consolidation processing.

is 25%). Figure 2 illustrates schematically the physical situation encountered in a lay-up at the begin- ning of consolidation. When the monotapes are stacked one on top of another, there is a large (30--40) volume percent of porosity. Most is associated with contacts. Typically, for a total initial void volume fraction of 0.35, the surface roughness contributes about 0.3 and internal porosity the remaining 0.05.

"~ Initial deformation at contact points

Interlaminar porosity due to surface roughness

Intermetallic Matrix

lib

I ~ B

Fiber (e.g. SiC, AI203)

HIP Cannist [._j v Intralaminar (closed) porosity

Fig. 2. Schematic view of the situation encountered during consolidation processing of a lay-up of plasma sprayed monotapes.

ELZEY and WADLEY: MODELING OF METAL MATRIX COMPOSITE MONOTAPE 2299

This obviously is determined by statistical properties of the roughness and it is important that this statistical representation be included in a realistic model. A second important consequence of this physical situation is that even in the absence of reinforcing fibers and the application of hydrostatic stress, the aggregate will deform anisotropically. This is because deformation (at the contacts) occurs quite readily perpendicular to the laminate plane, whereas the compressive stresses in the two orthogonal directions are well supported by the relatively dense (and thus more deformation resistant) monotapes. The totaling used for HIP of planar components also better supports the in-plane loads and encourages only strains in the through thickness direction. The presence of fibers also inhibits in-plane macroscopic deformations with the result that almost any uniform applied load combination leads to a state of constrained uniaxial compression during most of the early stages of densification.

As the lay-up is raised to a high temperature and pressure, densification is accomplished by a complex sequence of deformations. Interfoil contacts, which initially support the highest stresses, begin to deform by plastic yielding and time-dependent creep. This allows more contacts to form (and the areas of previous ones to increase). As the applied stress rises, the stresses around the closed pores within the foil become sufficient to cause their collapse, again by a combination of plasticity and creep. The addition of these deformations results in a net shrinkage (confined almost totally to the through thickness direction) and densification of the laminate. The rate of densification will depend upon process variables (pressure, temperature and time), material properties (which determine the extent of deformation by plasticity and creep processes) and the statistical characteristics of the surface roughness and internal porosity. All must be correctly incorporated in a successful model.

Rather than attempting the potentially fruitless task of seeking a single ad hoc constitutive law for such a complex material behavior and using a finite element model to solve our problem, we have sought to model the micromechanical (plastic and creep) behavior of representative (geometric) elements of the problem and allow their contribution of the overall densification to evolve during the process. Given the representative volume element (RVE) or combination of elements, (which is determined by the problem's internal geometry), with the constitutive properties of the macroscopic material, we model their response to an arbitrary pressure/temperature cycle.

To ensure realistic predictions by this approach one must select the RVE(s) such that:

1. The response of the element is truly like that of the macroscopic body it replaces (when subjected to identical boundary conditions).

2. The elements are small enough, relative to the macroscopic body, that stresses and strains can be defined as averages of such field quantities determined within the volume element.

Once RVE's have been selected, and their micromechanical responses deduced, the effective (averaged) properties of the RVE(s) can be used to represent (at every point) the properties of the macroscopic body, and its overall response to the process determined. The identification of a RVE which is a sufficiently realistic and accurate representation of the macroscopic body's heterogeneous behavior while remaining simple enough to yield an analytical solution is arguably the most critical aspect of the model's development.

2.1. Representative volume element selection

We have visualized the porous lay-up as made up of two alternating lamellar phases (see Fig. 3). The r (reinforced) phase contains the fibers and internal (closed) porosity formed by incomplete infiltration. The s (surface contact) phase is highly porous (because of the surface roughness), (but contains no reinforcement. These are still complex phases (i.e. they are heterogeneous). Their response is evaluated independently by identifying an appropriate RVE for each, deforming these under conditions of perfect plasticity, power law creep and diffusional flow and then recombining them to simulate the overall lay-up response.

Both the r- and s-laminae are themselves made up of two or more homogeneous phases; s contains matrix and void while r is made up of fiber, matrix and void. In Fig. 3, we identify representative volume elements within the r and s laminae. It might at first sight appear reasonable to model the r-lamina by considering an elastic fiber within a cylindrical shell of porous matrix material. Unfortunately, analyses of such a 3-phase composite cylinders RVE for arbitrary stress states and deformation mechanisms have only been developed for the particular case of a power law creeping matrix subjected to a hydrostatic stress [10]. They show very little effect of a fiber upon densification (other than reducing the available volume to deform and imposing anisotropic deformation). In view of this we use the well studied concentric spheres RVE, to represent the porous matrix, and consider only the particular case of plate strain: no macroscopic strains are allowed to occur in the fiber direction. The weak local effects due to the fiber are thus neglected, while the macroscopic effects of anisotropy induced by the fibers and the reduced matrix volume fraction are preserved. For the case of the r-laminae, we assume all voids to be the same size and shape, to be uniformly distributed and unconnected.

The s-lamina is made up of asperities whose heights and radii can vary randomly. Therefore a volume


Single Asperity (GRVE)

- - - - ~ Z 3

1 I;

j i s - lamina ,

" i ', X2 .e. .: ig~ll . . e.'O.'. ¢2

#

:ca ,0 ,

x 2 ".: :.'" :.': .':4<D

7-,! ~2 (=xa)

Concentric spheres

RYE

Fig. 3. The complex deformation behavior of a monotape lay-up is treated by considering the response of representative volume elements to simplified stress states.

element is chosen which is taken to be representative of the geometry of all asperities. This "geometrically representative volume element" (GRVE) consists of a fully dense hemispherical cap enclosed within a cell (see Fig. 3). The presence of the enclosing cell permits definition of appropriate boundary conditions and the calculation of relative density. The response of the GRVE is determined by approximate direct analysis methods for a perfectly plastic matrix, power law creep and diffusional flow. Once the behavior of a single asperity has been determined, the RVE response is obtained by modeling the behavior of a statistical assemblage of asperity unit cells representative of an actual surface.

A given RVE approximates the overall behavior so long as the deformations imposed on it do not significantly alter its geometry or distribution within the macroscopic body. Otherwise the RYE ceases to be representative of the actual material. In our case, as density increases, the asperities making up the s-lamina flatten and flow outwards toward each other until only isolated, cusp-shaped voids remain. At this point (occurring at a relative s-lamina density of around 0.9) the s-lamina behavior is more accurately

represented by the concentric spheres model for matrix containing isolated, spherical voids. We refer to the range of densities below 0.9, during which the rough surface RVE is applied, as stage I. For densities well above 0.9 (stage II), the s-lamina behavior is modeled using the concentric spheres RVE. In the vicinity of 0.9, the relative density is obtained as a weighted average of the two predictions.

The respective RVE's are evaluated assuming that the matrix deformation occurs by perfect plasticity, power law creep (i.e. dislocation creep) and diffusional flow [through the lattice (Nabbro-Herring creep) and along contact faces (Coble creep)]. Coble creep by diffusional flux along grain boundaries within the matrix has been neglected, implying that the grain size is on the order of the asperity size and pore spacing. The overall strain rate is taken simply to be the sum of strain rates due to the individual deformation mechanisms

where the superscripts refer to plastic, creep and diffusive mechanisms. In addition, the s-lamina is


considered in two stages, during stage I, the surface roughness RVE is applied and during stage II, the concentric spheres RVE. Thus a total of six constitutive models are needed. This approach follows that developed with success earlier for powder consolidation [5, 11].

2.2. Deformation and stress state

We restrict ourselves here to the analysis of plane, unidirectional laminates subjected to homogeneous boundary conditions. In this case, one can assume a macroscopically homogeneous stress (strain rate) field to exist, which implies that the overall deformation behavior is the same for every

representative volume element making up the macroscopic body. Specifically, it assumed that a state of plane strain prevails macroscopically, with no deformations occurring in the direction of the fibers. Both volume and shape changes are allowed to occur. However under these conditions shape changes are limited to changes in the aspect ratio of the laminate. The model is applicable to HIP and VHP of MMC planar laminates. It may also provide a reasonable approximation of the response of tubes and could provide a starting point for understanding other consolidation processes such as roll bonding.

Figure 4 summarizes the constitutive models which are discussed in detail below. The macroscopic

Stage h Macroscopic rough surface RVE

E , ~ Density, D

' z~x E ~ ~ E = ~ J ' f (ph" (Pr" Fc (h' r' D' I~) dr dh

Plastic yielding F c

o .,4 ~/I'lz ° - o

D o F c = 2~r(h - z o --5-)13o o

Power law creep

' ~ F c ~

F C =

' Zo(D olD 2)[~ , I n i

D O ~I ~ - n , oc[r(h - Zo--5-)j i

Diffusional flow

E = CDdiff ~

F = C

"i Fc r 2 kT Zo (Do/D2) ~

F c = ~ gu (D) ac(D)'~- fl (6Db + 2pO v)

Stage fl: E Macroscopic ~ • 0¢, porous E [~1:':-"~:'. :" " " : " " ; ' " . ' . '~<~ Eij = aT.~

Z ~ • t o •

media

Plastic yielding

= ~0

, 3E m, . o = ,/% co, 1- N I -,

=0

Power law creep

0 n - 1

T-, 3 _ ~ij = 3B( "~o ) s~j%

.!~

n + l

~c = ~ la(D)2~2 + b(D)2z~} 2 n + l t •

Diffusional flow

~ 3 ~ = CDdiff ~T Sij

= kTR = ~ 6 = x v * u 4 m j

Fig. 4. Summary of local and macroscopic constitutive models for describing the dcnsification of MMC monotapes. The overall response is the sum of surface contact deformation and shrinkage of internal voids

by plastic yielding, power law creep and diffusional flow.

A M 4118--D


uniaxial response (stress-density) of a rough surface is expressed in terms of the statistical distribution of asperity sizes (tph" tpr) and the behaviour of a single asperity [F¢(h, r, D,/))] for plastic, creep and diffusional mechanisms. The plane strain response of the r-lamina is expressed in terms of a potential, ~ for each deformation mechanism.

2.3. S-Lamina

All engineering surfaces are rough on a sufficiently microscopic scale. The behavior of rough surfaces in contact plays an important role in problems involving wear or the conduction of heat or electricity between contacting bodies and a large volume of literature on surface contact behavior exists [12-17]. The pressure assisted diffusion bonding of surfaces is also well studied. Models for diffusion bonding, such as those by Guo and Ridley [18] and Hill and Wallach [19], deal with the joining of prepared surfaces for which the asperities are of uniform size and distribution. The surface of a monotape is viewed as an assemblage of asperities, all of which are hemispherical in shape, but whose heights and radii are statistically distributed (Fig. 3). We construct the model as follows: the force-displacement (alternatively, stress~lensity) response of a single asperity is determined for a given matrix material constitutive law (i.e. plasticity, power law creep or diffusional flow). This effective response, which is obtained as a function of the asperity radius and height, is then used to obtain an approximation for the response of the rough surface by considering the statistical asperity height and radius distributions (see Fig. 4).

2.3.1. Statistical model for rough surface RVE. Consider a thin plate with one smooth side and one rough side compressed between two smooth rigid platens; initially, only a few widely spaced asperities make contact with the upper platen. These are therefore subjected to high stresses and are easily deformed and the total force resisting compaction is small. As compaction proceeds, the number of contacting asperities, the contact area and hence the resultant force to cause continued deformation all monotonically increase. Taking strains in the plane of the rough foil to be negligibly small relative to that in the directon of compaction (i.e. a state of constrained uniaxial compression), conservation of mass gives the relative density (D) as a function of the spacing (z) between the platens

D ----z° " Do (2) Z

where z0 is the initial platen separation (determined by the height of the highest asperity) and D o is the initial relative density.

A relation between the applied stress (~z) and foil thickness [from which the density can be obtained using (2)], is next desired. The average applied stress at the contacts is given by the applied force (known)

divided by the total area of contacts. This depends on the number of asperities making contact and their size; both are governed by statistical parameters of the rough surface.

Let the probability density function (PDF) describing the distribution of asperity heights be gOh(h ). The cumulative fraction of asperities in contact after deforming a foil from z 0 to z is given by #h = S~ ° gOb(h) dh. Similarly, the distribution of asperity radii can be described by a PDF, tPr(r ), where r is the asperity radius. Given the number of contacts and their size distribution, the beating area can be determined as a function of z, and thus density from (2).

Assuming the distribution of asperity radii to be independent of asperity height, the probability of an asperity of height, h and radius, r is

,~o(h, r) = ~oh(h)" ~or(r). (3)

Denoting the contact force required to cause further deformation of a single asperity as Fc(h, r, z, ~), the (bivariate distribution of) forces, ~of required to further deform asperities is given by

~f(h, r) = ~o. Fo. (4)

Equation (4) associates a resistance force with each asperity. The total force required to cause further compaction of the surface is obtained by integrating (4) over all asperity radii, r and heights, h encountered in compacting from z o to z

F(z, z) = ~f(h, r) dr dh. (5)

The stress, I~, to cause compaction is obtained when equation (5) is multiplied by the areal density of asperities, l (number of asperities/unit area)

;°f0° Z(z, ~) = I. jr(h, r) dr dh. (6)

Equation (6) is the constitutive relation we seek; it relates the applied stress to the degree of compaction (z) or relative density from equation (2).

It is simple to determine the areal density (l) and distribution of asperity heights (~oh) and radii (Or) from a profilometer measurement of a foil surface. Our measurements of plasma sprayed surfaces indicate the heights are usually characterized by a normal distribution:

1 F 1 { h - - [[~q h(h) = j (7)

where h" is the mean height and ah the standard deviation while the distribution of asperity radii is approximately exponential

q, Rr) = ~. e -~ ' . (8 )


Table 1. Typical statistical surface roughness data for Ti-24AI-11Nb monotapes

Parameter (units) Symbol Value

Mean asperity height (/am) /~ 91.06 Asperity height SD (/tm) a h 39.82 Asperity radii exponential factor (#m i) 2 0.0178 Areal density of asperities (#m -2) I 5.0.10 -s

Substitution of these functions into (6) gives the constitutive relation for the RVE associated with surface roughness

f~0f l .exp[ 1?-~Yl E(z, 2) =I j~ i x / 2 ~ , ah L-~ \-~-) j

fo x 2.exp(-2r) .Fc(h,r ,z ,~)dr dh. (9)

The above relation is valid for any matrix constitutive behavior. It then only remains to calculate the single contact resistance force, F~, which of course depends on the mechanism of deformation.

2.3.2. Perfect plasticity. Upon application of stress to an asperity at low temperature (T ~ 0.3Tin), deformation occurs instantaneously by plastic yielding. We seek the stresses during plastic contact of a sphere (or spherical cap) and a flat rigid plate. No exact solution to the three dimensional problem has even been obtained for general plasticity (i.e. yield surface and incremental stress-strain relations). A number of approximate solution methods are available based on either finite element analysis or slip- line field theory [201. We are interested here in the simplest process model providing an acceptable level of accuracy. We have taken the criterion for flow to be

Fo - - = tr~ I> f l - a 0 ( 1 0 ) t/c

where tr c is the contact stress, a~ the contact area, a0 the uniaxial yield stress and fl, according to the slip-line solution for a cylindrical punch indenting a flat surface, has the value of about 3 [21]. Equation (10) then states that yielding occurs when the average stress at a contact reaches about three times the uniaxial yield strength of the matrix. While for simplicity, we have assumed a constant value for fl, detailed numerical work indicates that fl is more generally a function of the relative density [22]. For perfect plasticity, fl does not depend on the specific material, only on the shape of the asperity.

From (10) the smallest force acting on a single asperity to cause deformation is

Fc=ac'fl "cro~2Ztr(h - -z) ' f l "ao (11)

where ac is approximated by 2nr(h- z) [20] and (h - z ) represents the displacement of the asperity peak to the current deformed height, z. Inserting the constitutive relation for a single plastically deforming asperity (11) into (9) gives the overall s-lamina plastic

response

z = l - ~-- (h - z )

"exp[ -~(h-~2]I°~k-ah/JJo r . exp( -2r )dr}dh . (12)

Equation (12) must be solved iteratively for the height of compaction (z) from which the density corresponding to the given applied stress is determined. Note that a 0 is usually a temperature and microstructure dependent material constant. Using the appropriate temperature dependence or yield strength-structure relation allows equation (12) to be evaluated under conditions of changing temperature or evolving microstructure. (For the moment we have ignored the latter, but it can be important in systems where phase transformations or coarsening occur during hot consolidation.)

2.3.3. Power law creeping asperity. At sufficiently high temperatures (T > 0.3-0.4 Tm), the matrix can continue to flow by time dependent creep. We assume a power law relation for uniaxial creep of the matrix

= B • a" (13)

where B is a temperature dependent material parameter and n is the stress exponent (see Appendix). (In general, B and n vary with stress and temperature, but may be assumed roughly constant if a particular creep deformation mechanism predomi- nates.) By a dimensional argument, which follows that of Arzt et aL [5] for the creep deformation at the contact between two spherical powder particles, we can calculate the rate of change of asperity height in terms of the contact stress. The physical problem is likened to a cylindrical punch of cross-sectional area ac = nx 2, indenting a problem law creeping solid. The stresses, (f in the material below the punch then scale with the contact stress, a c as

¢~ = C l 'O" c. (14)

The displacement rate of the punch must scale with the strain rate ~ and with the radius of the contact x giving £ = c2ix. Substituting (13) for ~ gives

=c2 "Bx(cl "at)". 05)

Writing B a s Eref/o'nef, where t=f and a~f are a reference strain rate and stress, the constants c~ and c 2 are determined by requiring 05) to reduce to the per- fecfly plastic solution [see equation (10)] when n-~ oo and O'r¢ f is taken to be the uniaxial yield strength, and to the elastic Hertz solution (for a sphere being pressed into a flat plate) when n = l , i = E and O'ref/Er¢ f = E , where E is Young's modulus. Imposing the limiting restrictions, c 1 = 1/fl, where fl is the effective yield coefficient introduced earlier, and


c 2 = 1.36nil. From equation (15), the rate of change of asperity height is [23]

= 1.36" nfl I -"xBa"~. (16)

By recognizing that a c = nx 2"-" 2 n r ( h - z), we can eliminate x in (16) to give the asperity force (Fig. 4)

Fc = \=B[r(h-- z)] '/z") (17)

where at = 1.36(rcfl)t-"2 t/2-". Substituting (17) into (9) gives the s-lamina creep response

Zn~ l fZ°(h _z)l-t/2n e = otB ' [ X / ~ . trh J ,

× expr l L-~\--~h ) _1 "a

x r 1 - 1/2. exp ( - 2r) dr . (18)

The densification rate is obtained by differentiating (2) to give ~ =zo(Do/D2)D and substituting with equation (18) for ~, whence:

D 2 y , f 1 f:°(h-z), D = DoZo" orB" (X/~" ah

x exp , ~ \ ~--Z/d

fo°° rl-1/2, exp } - " x (--2r) dr . (19)

B and n are the only material parameters needed to determine actual densification rates. They can be complex functions of temperature, stress and micro- structural state. For our work, we have used average values representative of the conditions encountered during consolidation. For more precise work one could measure the creep constants and use a look-up table for the most appropriate values of a particular T, P combination.

2.3.4. Diffusional asperity deformation. In addition to dislocation creep, time dependent deformation can also occur (particularly at very high temperatures) by stress-directed diffusion. The driving force is the reduction of chemical potential resulting from the removal of atoms from regions of high compressive stress to regions of tensile stress (or lower compressive stress). Thus material will be displaced from the asperity contacts to the traction free surface of the asperity. We assume the uniaxial stress-strain rate to be linear in the stress in accord- ance with the Nabarro-Herring [24, 25] and Coble [26] creep mechanisms and to be of the form

= C " Ddilr • ~--~" Or c (20)

where c is a geometry dependent constant, Ddi ff is the diffusion coefficient (lattice or grain boundary), fl the atomic volume, k is Boltzmann's constant, T is the absolute temperature and as before, a c is the contact stress.

We assume the total flux of matter (m3/s) from the contact to be the sum of grain boundary and lattice diffusional contributions [11]

f~ f" =4~(aDb + 2pDv)-k---~" ac (21)

where 6Db is the grain boundary diffusion coefficient (i.e. along the contact interface) times the boundary thickness, Dv is the lattice diffusion coefficient and p is the radius of curvature of the neck which forms at the perimeter of the contact.

Helle et al. [27] have considered the analogous problem of diffusional sintering of two spheres. The neck radius of curvature is (see Fig. 5), p, = x2/[2(r- x)], where r is the contacting sphere radius and x is again the contact radius. They

:ering of two spheres

g of a sphere plate

Fig. 5. The neck radius forming at the sintering contact between a sphere and a fiat plate compared with that formed

at the contact of two spheres.


show that this can be well approximated as a linear function of relative density, p, ~ - r ( D - Do). For the sintering of a sphere to a flat plate, the radius of curvature, p will be less than p, for a given neck radius, as can be seen in Fig. 5. To a first approximation therefore, p is taken to be

r (D -- Do I P "" -4 \ 1 - Do ] " (22)

The flux is related to the rate of uniaxial compaction by differentiating an expression for the volume of material displaced from the asperity contact in deforming to cell height, z [28]

r,_d f2=Fl ( r - =) iF: dt [ 3 L ~ JL 2r - 2(r - z) JJ"

Carrying out the differentiation and making use of (2) gives

r 3

(I = ~g.(D)£ (23)

where

g~(D)=(1 D°\ fD°/2 Do\ ' +

+ ~ _ 2 n 2 + D o / D ' ~ 2 ' [ D - Do / 1 - - Do/D J ~- T \ ~ ] •

Combining eqns (21) and (23), one obtains the force-displacement relation (Fig. 4)

r 2 kT F¢ = ~ " g~ac D (~D b + 2pD~) " (24)

Substitution of (24) into (9) gives the stress-directed diffusional response of the s-lamina RVE

= (h - z/ • O- h

x e x p F A ? - ; Y I L 2 \ °~ / J

j,o~ g~(z)kT }_l. (25 ) x r: e x p ( - ~.r) d r •

o 6D(rD b + 2pD~)

The densification rate is given by I ;o D2 " Z ' . l ( h - z ) /) Zo Do x / ~ . o h

[ _ I (h - ~121 dh " 2 x exp 2 \ ~h ] J

x r 3 exp(-- 2r) dr 6fl(t~Db + 2pD~)) . (26)

Evaluation of this expression requires estimates of the diffusion constants which can be found for many common materials in the standard handbooks• Frost and Ashby [29] give simple rules for their estimation in other cases.

2.4. R-Lamina

The r-lamina representative volume element has the geometry of a spherical shell and there are numerous papers with its analysis and application [30-34]. The volume fraction of void in the actual material is reflected in the ratio of the void and shell radii• The concentric spheres (CS) RVE needs to be evaluated for the case of plane strain (see Fig. 4), the strain rate along the direction of the fiber being taken to be zero• The results are the CS RVE model are applied to both the r-lamina (throughout consolidation) and the stage II (D > 0.9) behavior of the s-lamina•

We need the plane strain response of the CS RVE for a rigid, perfectly plastic, a power law creeping and a linear viscous (diffusion) creeping matrix. In each case the relation beween stresses and strain rates has been expressed in terms of a potential function cp such that the macroscopic strain rates (for the RVE), may be expressed as

Ocpm /~U = A" aZi j ' (27)

where A is a scalar factor which may depend on the loading path, Z o are the macroscopic stresses (i.e. stresses imposed at the RVE boundaries) and g~m is the potential associated with deformation mechanism, m.

2•4•1. Plasticity. We consider here the case where the matrix is taken to be homogeneous and incom- pressible and deforms in a perfectly plastic man- ner. For this case a modified version of the plastic potential developed by Gurson [32] can be used to represent the yield surface of a porous body

+0 = ( oY-2q(l- )cosh(3 o/-, = 0 (28) \ a 0 / \ 2 a 0 ]

where Ye and Y'm are the macroscopic equivalent and mean stresses, respectively, D is relative density and a0 is the uniaxial tensile yield strength of the matrix. The constant, q is taken to be ,.5 as indicated by experimental and theoretical results [35]. The stress state considered is axisymmetric with Y.2(=Y3) the given applied stress and Z~ the as yet undetermined stress along the fiber direction (Fig. 3). The con- dition of zero strain in the fiber direction yields a second relation involving the unknown stress (Z1) and density, D

a~PP - Zs) g, = ~ , , = 2 ( x ' -

O 0

(29) \ za0 /

Simultaneous solution of eqns (28) and (29), which is done numerically, gives the density as a function of the applied stress as well as the ratio of stresses YI/Zs-

2.4.2. Power law creep. A lower bound estimate for a power law creeping, porous material is given by


Applied Stress (MPa) 1 10 100 1000

. . . . .

o o - - / I ~ . . ~ - : - - - - : J ~ ' i / / / / il Power ~'~ ~Z "7~ / [ Law / / ~ o C-reep / / / ; , ~ y , ~ / / I

-'~o"-=° 0.8 ~ e ~@!21a a

o) ._> 0.7 Id ' '

n"

0.6 ~ ' ~ ~ s i h ' _ j . . . . . . . . . . . . . . . . . . L . . , ' Co tours ' . . First = 0.25 Last = 16,0

( 8 ) Factor = 2.0

0.5

Applied Stress (MPa)

1.0

0.9

a

- .~ 0.8

a

~ 0.7

0.6

1 O0 1 10

0.5 -3

. . . . Initial Density -

-2 -1 0

Normalized Pressure, P/ay

-2 -1 Normalized Pressure, P / Gy

Fig. 6. (a, b) Caption on facing page.

1000

~ n ~ u ~

-~-I FaVor=2.0

First = 0.25 Last = 16.0


• ~ 0 . 8 e -

"~ 0.7

Applied Stress (MPa) o.1 1 lO lOO 1.o ..... ~ ' ' ' ' S . . . .

Diffusion ~ ~ f

0.9 ~ . , / . ~ .

(c) 0.5 -3 -2 -1

Normalized Pressure, P/Oy

Applied Stress (MPa) O.Ol o.1 1 lO

1 . 0 . . . . . i I . . . .

Diffusion . ~ ~ ~ 0.9 - Pc ~er-

/ Creep.~

~ 0.7 .

-- Initial D e n s i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . = . '~

0.5 -3 -2 -1 0 Normalized Pressure, P / Oy

Fig. 6. (a-d) Densification maps showing predicted MMC laminate density vs normalized applied stress for Ti-24AI-11Nb (Ti3AI + Nb), TiAI (y), Ti-6A1-4V and A1 matrices reinforced with 25vo1.% SCS-6

(SIC) fiber.


Temperature (°C) 400 600 800 1000 1200

1.0 j ~ = = t ,

Power ~ i 0.9 Law Creep

p = I O M P a

n = 2 . 5

0.6 , Contour= , . . . . . . . . . . . . . . . . . Initial Density . . . . . . . . . . . . . . . . . . . . . . . . . . . First = 0.25 ""

Last = 16.0

( 8 ) Factor = 2.0

0.5

.•0.8 e-

ID > '~ 0.7

0.4 0.5 0.6 0.7 0.8

Homologous Temperature, T / T m

1.0

0.9

a

. .~ 0.8 - -

C

a

~ o.7

ft.

T e m p e r a t u r e ( ° C )

600 800 1000 1200 . . . . ~ ,

Diffusion

. /

I I

Plastic Yield

TIAI ' p - 50 MPa

n = 4.0 I

0.6 , Contour= . . . . . . . . . . . . . . . . . . . . . Initial Density . . . . . . . . . . . . . . . . . . . . . . . . . ~i:stt :~.62.~ - - "

( b ) Factor = 2.0

0 .5 0 .5 0 .6 0 .7 0 .8 0 .9

Homologous Temperature, T / T m Fig. 7. (a, b) Caption on facing page.


Temperature (°C) 600 800 1 0O0

1.0

0.9 ~ - ~ P ~ V w w e r /

! Plastic Yield ; p = 10 MPa n= 4.0

0.6 . . . . . . Initial Density . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' C m ~ r m ' . . I First = 0.25 l Last = 16.0

I Factor = 2.0 (C) I 0.5

D

._~ 0.8 r -

® >

'~ 0.7 m

0.4 0.5 0.6 0.7

Homologous Temperature, T / T m

1.0

z-~ 0.9

g a

0.8

0.7

Temperature 100 20O

Creep ~

(°C) 300

Plasbc Y=eld

(d)

0.4

400

2Z- R r s t . 0.25 Las t . 16.0

Factor = 2.0

0.5 0.6 0.7 0.8

Homologous Temperature, T / T m Fig. 7. (a--d) Predicted MMC laminate density vs homologous temperature for Ti-24Al-I1Nb, TiAI,

Ti-6AI-4V and AI matrices with 25vo1.% SCS-6 (SIC) fiber.


the strain rate potential originally developed by Ponte [36] and modified by Duva and Crow [37]

B q~c = s" + l (30)

n + l

where B and n are the familiar constants in the Norton creep law, ~ = B a " and s2=a2(D)X2+ b2(D)Y~. The parameters a and b are functions of the relative density only

a2(D) = [1 + ~(1 -- D)]

bZ(D) = [n(1 - D)" [1 - (1 - D)t/"]] 2/~"+ 1)

Rearrangement of the equation, /~1 = O~c/os.~ = 0 gives an explicit expression for the unknown stress component

a 2 - ~ b 2 Y-1=(¢(D)'E3 where (c(D) a2+~b2 (31)

and Z2 = X3. The non-zero strain rates are then

-/~2 = ~"3 ~-- ct(D)- BS.~ (32)

where

~(D) = a~(1 -~o)~+-¢-(1 +2~0) ~

b2 x (a2(l - ~ ) + ~- (1 + 2(~)).

The densification rate is obtained from /9 = - D .

2.4.3. Diffusion. The mean stress component, Z m leads to a flow of matter from within the compressed body (e.g. from grain boundary sources) to pore surfaces (traction-free), resulting in pore shrinkage and therefore densification. The deviatoric stress, 5.e leads to fluxes from grain boundaries under high compressive stress to those with lower stress thus effecting shape changes. Both diffusive fluxes from grain boundary giving shape change are considered, whereby the driving forces are the mean stress and deviatoric stress. Accordingly, the diffusion potential is taken to be [381

f2 /5~ / D 2 \

' 5.2 + 2(tTD b + irpDv)52m (33)

where R is grain size and rp, the pore radius, is taken t o b e

R ( 1 -- D ~ 1/3 rp ~-- \ 2D ] "

All other symbols have the same meanings as for the s-lamina diffusion model. In the limiting cases of pure shear and hydrostatic compression, (34) reduces to the known limits of Nabarro-Herring [24, 25] and

Coble [26] creep and to Ashby's result for powder densification [39].

Proceeding as for creep, the zero strain rate along the fiber gives/~l = c3~d/c35Zl = 0, which can be solved for the unknown stress component, 5.1

2c - ~d ~"~'1 ~" (d" ~']3 where (d -- 2C + ~ (34)

with

c = 6 k T R 3 tTDb + RDv

f~ d = 2" k - ~ " (6Db + ~rpDv)"

The non-zero strain rates are given by

E2 ~- E3 ~ (x. 5.3 (35)

with x = 2c(1 - ~d) + 4d( 1 + 2~d)- The densification rate is obtained by substituting (35) into /5 = D/~ 3

/) = 2 D ~ • 5.3. ( 3 6 )

3. COMPOSITE DENSIFICATION MAPS

Using the RVE models, the strain rate components (and hence the densification rates) are determined for the r- and s-laminae. Once the densities, Dr and Ds have been determined [using equation (1)], the composite density is obtained by the rule-of-mixtures: D =vrD~+vsDs, where v~ and vs are the volume fractions of the r and s phases. (The volume fractions are not constant, but depend on the relative rates at which r and s densify. Thus the volume fractions must be redetermined following each numerical integration step.) When the s-lamina reaches a density of 0.92, further increments in density are determined using only the concentric spheres RVE.

The predictive relative density of the composite is conveniently illustrated by means of densification maps; plots of density vs normalized stress at constant temperature and of density vs temperature for a given constant stress. We employ the same form for the maps as that introduced for powder compaction in Arzt et al. [5]. The maps are generated by numerical integration, implemented on an IBM-compatible personal computer.

Figure 6(a) shows a map for Ti -24at .%Al- 1 la t .%Nb (~2 + fl) matrix reinforced with SiC (SCS-6) fibers like those shown in Fig. l for consolidation at a constant temperature of 975°C. Material parameters used in the model are given in the Appendix. The lowermost contour indicates the density achieved at a given stress due only to plastic yielding, which is assumed to occur instantaneously upon load application. Higher densities are achieved by allowing time for thermally activated mechanisms (power law creep and diffusion) to contribute. Thus the second contour (at t = 0.25 h) indicates the density reached by hold- ing the stress constant for a quarter of an hour.


Figure 7(a) shows the densification behavior as a function of temperature (consolidation pressure is constant at 10 MPa). As before, the lowermost contour represents t = 0 (densification due to plastic yielding), while the remaining contour lines indicate the increase in density due to creep and diffusion. The regions of the map are labeled according to which densification mechanism dominates; plastic yield, power law creep or diffusion. The border between the creep and diffusion fields represents values of applied stress and density at which the two mechanisms contribute equally to densification.

Figures 6(b--d) and 7(b-d) show the predicted densification behavior for composites with TiAI (7), Ti-6AI--4V (a + fl) and pure aluminum matrices. The stress-density maps [Fig. 6(a--d)] have been con- structed at roughly the same homologous temperature (T/Tm~0.68), while the temperature-density maps [Fig. 7(a-d)] all illustrate behavior for a relatively low applied stress. Material parameters used in constructing the maps are given in the Appendix. Initial densities of s and r were taken to be 0.35 and 0.9 (excluding fiber), respectively, with initial volume fractions of v s = 0.47 and v, = 0.53.

4. DISCUSSION

Our primary objective in developing this MMC monotape densification model was to allow one to predict how the density depends upon the applied stress, temperature and time of a consolidation cycle and to understand how this relates to the attributes of the monotape (e.g. its physical/mechanical properties and geometry). This is important because HIP cycles must attain full density in components, otherwise residual porosity severely reduces the high temperature structural integrity of finished components. However a number of other processes that adversely impact component properties also accompany the HIP cycle, e.g. phase transformations, grain growth [39] and fiber damage and fracture [40] and interracial reactions. Densification maps such as those presented above provide a convenient means of determining the conditions that lead to full density, and these can be

evaluated for their impact upon other attributes of the composite (e.g. the number of broken fibers [8]).

The general behavior of the four composite material systems for which maps have been included (Figs 6 and 7) is similar. The behavior indicated by the maps is also similar in form to that found for the compaction of powders [27]. Comparison of the stress-density map for Ti-24AI-11Nb [Fig. 6(a)] with that of TiAI [Fig. 6(b)] indicates the relative importance of creep deformation in achieving high density components; diffusion plays a relatively minor role for the Ti3AI + Nb (Ti-24AI-11Nb) composite while for the more creep resistant TiAI (7) matrix composite, diffusional mechanisms are much more important for equivalent homologous temperatures. This is especially so at the low stresses required to avoid fiber fracture. As observed for powder consolidation [5], densification at higher stresses (tr > 10 MPa) is principally due to creep where/)~ttr", with only the last few percent achieved by diffusion.

Although the maps have the disadvantage of view- ing the consolidation process cycle as one in which both applied stress and temperature are applied as step functions (i.e. they are suddenly switched 'ON' to certain values, then held constant for a certain length of time and then switched 'OFF'), densification maps are still useful for their intended application. For more detailed work the transient response for a general consolidation cycle can be evaluated by integrating the densification rates along an arbitrary process path, simulating the response to specific pressure-temperature-time cycles. Figure 8 is an example. This more sophisticated output of the modeling approach has been investiated for model-based process control [9]. It has been used to find the optimum process cycle [i.e. p(t), T(t)] that results in a desired density. Departures from the intended D(t) can then be sensed and, using the model, feedback corrections to the remaining p(t), T(t) schedules computed.

The transient response model output can also be used to investigate the error incurred when using a map to predict densification. Maps overestimate the density at any given time since they neglect the time required to increase the stress and temperature to

100

~ 2O

0

I

IOOMPa

850"C

I 1200

~°°° ~11 800

600 .,-,

- 2 0 0 F -

l I 0 500 1000 1500

T i m e (rain)

~ 0.9 e-

t'~ 0.8 (D ._>

n"

1 .0 t I

0 . 7

0 . 6

0 . 5 ~ I I 0 500 1000 1 ~0

Time (min)

Fig. 8. Simulated densification for a typical pressure-temperature-time cycle.


their respective soak values. The error is typically greatest for short consolidation timest, falling to within a few percent at densities above 0.9.

The models describing the constitutive response of the plasma sprayed surface [equations (12, 19, 26)] require as input the mean asperity height (h'), and radius (1/2), the standard deviation of heights (ah), the areal density of asperities (1) and the initial relative density of the surface layer (D~). Since these parameters are determined by the thermal spray conditions (e.g. powder size, superheat, droplet vel- ocity, etc.), the models allow the influence of spray conditions on consolidation behavior to be investi- gated. For example decreasing the standard deviation of asperity heights (keeping all else constant) can be shown to decrease the density achieved for any given level of applied stress since the number of additional contacts per increment of compaction is increased. However at the same time, decreasing ah decreases the nominal thickness of the s layer (taken to be h" + 3ah) and so also the volume fraction of s; thus the decreased densification rates are to some extent com- pensated for by the higher initial composite density. While there may be no substantial advantage in reducing surface roughness to achieve efficient consolidation, (for the reason cited above and because the applied stresses and temperatures are required anyway to collapse internal voids), minimizing surface roughness has been shown to be essential to avoid fiber fracture during consolidation [8, 40].

This first approach to a very complex process has perforce required the use of many simplifying approximations and these are worthy of further consideration. The modeling approach used here is analogous to that of Arzt et al. [5] for powder consolidation; representative composite volume elements are identified whose stress-strain response is then obtained either directly (i.e. by solution of the appropriate boundary value problem based on equilibrium, compatibility, etc.) or by energy methods. Equally significant is the treatment of large deformations by applying separate volume element analyses to various density regimes ("stages")~thus the distinction between behavior at lower densities, which is dominated by deformation at points where foil surface asperities contact adjacent foils, and higher densities, where the shrinkage of isolated internal voids more accurately represents the macroscopic behavior. The apparent success of this method in simulating the deformation behavior of powders is an indication of its promise in modeling the deformation processing of more complex (heterogeneous, anisotropic) materials. However it might be added that the alternative to this, namely the derivation of a unified model able to handle the transition from low

tAt t = O, the map predicts a density given by the initial density (Do) plus the densification due to plastic yielding (AD p) whereas the actual density is still Do; thus the initial error is just ADP.

density (where the void forms a continuous phase) to high density (where the metal is the continuous, or matrix phase), poses very serious challenges.

Approximating the influence of continuous fibers by evaluating the continuum models for the case of plane strain, (with no deformations occurring along the fiber direction), has the consequences of under- estimating the actual densification rate under an applied hydrostatic stress and of neglecting any effects on densification rates near the fibers due to stress concentrations (esp. enhanced shear stresses). Theoretical investigations of porous, fiber reinforced materials so far indicate these effects are of second order [10, 41] and we feel justified in making this greatly simplifying assumption. More serious in our view is the assumption of spherical voids. In reality, the voids in plasma sprayed foils are much more compact in one direction, i.e. pancake-shaped (see Fig. 1), than our spherical idealization. Consequently they collapse more easily [42]. Further experimental work is needed to address the significance of this.

We have viewed thin laminates as tending to densify primarily by uniaxial deformations occurring perpendicular to the plane of the laminate even when the applied stress is hydrostatic. This point is worthy of some expansion. There are several factors that we believe make this so:

(i) Because about 90% of the porosity initially present in the composite laminate is concentrated in the surface roughness layers, the composite is much like a stack of nearly rigid solid sheets alternating with sheets of foam. Such a body is much easier to compress perpendicular to the plane of the laminate, i.e. to compress the foam layers, than to deform in-plane.

(ii) The fibers shield the matrix from densify- ing stresses in the plane of the laminate and along the fiber direction.

(iii) The tooling used in HIPing of thin sheets tends to preferentially shield the composite from in-plane loads.

As a consequence, the spray-deposited foil laminate densities predominantly uniaxially, regardless of whether the stress is applied isostatically or uniaxially and this has been found to be so in experiments [40]. As the density increases, a unidirectional lay- up becomes more nearly transversely isotropic and as a result, one might expect different densification rates for HIP and VHP. However as full density is approached, the macroscopic stress state during VHP (constrained uniaxial compression) approaches hydrostatic, neglecting frictional effects. Large deviatoric stresses caused by friction, which are generally significant during VHP consolidation, are of less concern when consolidating thin laminates. These arguments lead us to speculate that the densification response of thin laminates should be very similar


in practice for both HIP and VHP consolidation processes.

5. SUMMARY

A set of micromechanical models have been used to predict the densification behavior of fiber reinforced metal matrix composite monotapes during consolidation processing. To simplify the complex deformation geometry, the composite foil is broken up into two simpler sub-laminae whose relative densities are evaluated separately; one of these contains the fiber reinforcement, has all smooth sides and initially contains less than 10% internal porosity, while the other sub-lamina contains the surface roughness. The effective constitutive behavior of the sub-laminae was obtained by the identification and analysis of representative volume elements within each sub-lamina. Direct (approximate) and vari- ational (bounding) methods were applied to the RVE's assuming matrix constitutive behavior corresponding to plastic yielding, dislocation and diffusional creep. Predicted results of the densification behavior of several fiber reinforced composite systems of current interest have been presented in the form of densification maps. Applications of the model include HIP and VHP processing of thin M M C laminates, tubes, and cones and HIPing of thicker components provided nearly homogeneous stress and temperature fields exist within the components.

Acknowledgements--The authors would like to thank M. F. Ashby, R. M. McMeeking and J. M. Duva for many help- ful discussions and advice. R. Vancheeswaran was instru- mental in developing simulation software used to produce Fig. 8. The financial support of the Defense Advanced Research Projects Agency (Program manager, W. Barker), the National Aeronautics and Space Administration and the General Electric Company (through the Office of Naval Research and DARPA~ is also acknowledged.

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A P P E N D I X

Material Data

Ti -24AI - l l N b [43-51]

Elastic properties Young's modulus as f(temp) [GPa]

--R.T. -- low temp range (T ~< 5O0°C): ---high temp range (T > 500°C):

Plasticity Yield strength as f(temp) [MPa]

--R.T. -- low temp range (20 ~< T ~< 800°C): - -high temp range (T > 800°C):

Creep t Power law creep constant, A [11-1] Stress exponent, n Activation energy, Qc [kJ/mol]

Diffusion :~ Pre-exp boundary diffusion, t~Dbo [m3/s] Activation energy, Qb [k J/tool] Pre-exp volume diffusion, D,~ [mZ/s] Activation energy, Qv [kJ/mol]

Misc. properties Melting point [K] Density [g/em3] Atomic volume [m 3] Surface energy, ), [J/m 2]

Elastic properties Young's modulus as f(temp) [GPa]

--R.T. --E(temp)

Plasticity Yield strength as f(temp) [MPa]

--R.T. ---low temp range (20 ~< T ~< 6O0°C): --high temp range (T > 6O0°C)

Creep Power law creep constant, A [h -I] Stress exponent, n Activation energy, Qc [k J/tool]

Ti-Al [51-54]

lO0.0 E(t) = lO0.0-0.04*T E(t) = 140.0-0.12.T

539.9 try(T) = 539.94).32.T

%(T) = 1019.9--0.92.T

6.0 × 1017 2.5 285

3 x 10 -12 202

5.9 x 10 -3 330

1875 4.6

1.0 x 10 -29 1.8

173.0 E(T) = 172.0-0.03.T

473.0 try (T) = 475.0-0.10* T %(T) = 875.0-0.76.T

7.62 x 1022 4.0 3O0

tThe steady state creep rate is described by o" n

~ = A " ( E l exp(--Qc/RTk)"

The values given for the creep parameters appear to represent the observed creep behavior well for stresses below about 170 MPa and temperatures up to 870°C, the highest test temperature at which creep data could be found.

~Values given are for TiAI.

ELZEY and WADLEY: M O D E L I N G OF METAL MATRIX COMPOSITE MONOTAPE 2315

Diffusion t

Misc. properties Melting point [K] 1725 Density [g/cm 3] 3.8 Atomic volume [m 3] 1.63 × 10 -29 Surface energy, ~ [J/m 2] 1.8

T i~AI-4V [55~0]

Elastic properties Young's modulus as f ( temp) [GPa]

- -R .T . 114.0 - - l ow temp range (T ~< 500°C): E(T) = 115.0-0.056.T - -h igh temp range (T > 500°C): E(T) = 172.44).16.T

Plasticity Yield strength as f( temp) [MPa]

- -R .T . 970.0 --try (temp) ( T < 960°C) ay (T) = 884.0-0.92. T

Creep Power law creep constant, A [h -I] 8.4 x 1024 Stress exponent, n 4.0 Activation energy, Qc [k J/moll 280

Diffusion Pre-exp boundary, diffusion, 6Dbo [m3/s] 3.5 × 10 -5 Activation energy, Qb [kJ/mol] 125 Pre-exp volume diffusion, D~0 [m2/s] 9.5 x 10 -ts Activation energy, Qv [k J/moll 152

Misc. properties Melting point [K] 1941 Density [g/cm 3] 4.42 Atomic volume [m 3] 1.78 × 10 -29 Surface energy, 7 [J/m2] 1.7

Aluminum [29]

Elastic properties§ Shear modulus as f( temp) [GPa]

- -R .T . 25.4 - - temperature dependence [Gpa/C] 5.0 x 10 -4

Poisson ratio, v 0.31

Plasticity Yield strength as f( temp) [MPa]

- -R .T . 41.5 ----try (temp) ( T < 460°C) try (T) = 43.4-0.094. T

Creep ¶ Dorn constant, B [h -l] 3.4 × 106 Stress exponent, n 4.4 Burger's vector, b [m] 2.86 x I0 -l°

Diffusion Pre-exp core diffusion, D~0 [m4/s] 7.0 x 10 -25 Activation energy, Qc [k J/moll 82 Pre-exp boundary diffusion, 6Dbo [m3/s] 5 × 10 -14 Activation energy, Qb [k J/moll 84

tSee Ti-24Al-11Nb for data. :~Values obtained using simple rule-of-mixtures applied to data for ~t-Ti and fl-Ti [58] assuming equal volume fractions of

ct and ft. §Shear modulus calculated as

# =,Uo' [1 + (T-- 300"~ • td~p]

where T and T m are in Kelvin and ta~ v denotes the temperature dependence in [GPa/K]. Young's modulus is then determined from E = 2/~(1 + v).

¶The steady state creep rate is described by

Eb [2 ( I+v) ] '~-t [ 1OacD c ~ t r "~2] ( a "y' i¢=A.kT (x/~),+l .D v I + W . - - • - -

b D,, ~,x,/~p,,] 2 ~,,x/~E.,] where A is the Dorn constant, T the absolute temperature v Poisson's ratio, a~b~ the core diffusion coefficient, b the burgers vector, D v the lattice diffusion coefficient, n the stress exponent, tr the stress and E is Young's modulus.


Pre-exp volume diffusion, D~0 [m2/s] 1.7 X 10 -4

Activation energy, Qv [kJ/mol] 142

Misc. properties Melting point [K] 933 Density [g/cm 2] 2.70 Atomic volume [m 3] 1.66 x 10 -29 Surface energy 3' [J/m 2] 1.7

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