Theoretical models for the anisotropic conductivities of two-phase and three-phase metal-matrix composites

Acta metall, mater. Vol.43, No. 8, pp. 3045-3059, 1995

~

Pergamon

0956-7151(95)00021-6

ElsevierScienceLtd Copyright © 1995.Acta MetallurgicaInc. Printed in Great Britain.All rights reserved 0956-7151/95$9.50+ 0.00

THEORETICAL MODELS FOR THE ANISOTROPIC CONDUCTIVITIES OF TWO-PHASE AND THREE-PHASE METAL-MATRIX COMPOSITES R. PITCHUMANI t, P. K. LIAW2, S. C. YAO3, D. K. HSU 4 and H. JEONGs ~Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269-3139, U.S.A., 2Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996-2200, U.S.A., 3Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A., 4Center for Nondestructive Evaluation, Iowa State University, Ames, IA 50011, U.S.A and 5Agency for Defense Development, DaeJon, Korea (Received 12 May 1994; in revised form 25 November 1994)

Abstract--Simplified analytical models are developed for predicting the anisotropic conductivities of particulate-reinforced metal-matrix composites fabricated using the powder metallurgy process. First, a two-phase model accounting for the effects of the matrix and the reinforcement phases only, is described. The model exploits the microstructural information available from metallographic analysis, and is based on the concept of a unit cell. The effects of the intermetallic phases formed during the processing stage are accounted for in a three-phase model, where the three-phase matrix-reinforcement-intermetallics composite is modeled as being composed of a "modified"-matrix reinforced with a third phase. The "modified"-matrix is, in turn, formed of the pure matrix reinforced with either the intermetallics, if the reinforcement particles are considered to be the third phase, or the reinforcement particles, if the intermetallics are regarded as the third phase. Good agreement is demonstrated between the theoretical predictions and the experimental data on A1/SiCpcomposites, obtained using eddy current techniques, for a wide range of phase compositions and volume fractions. The two-phase and three-phase models are shown to be more effective in estimating the anisotropic conductivities, in comparison to the models available in the literature, which yield only the isotropic properties. The methodology underlying the three-phase model development may be used to characterize multi-phase composites, in general.

1. INTRODUCTION Metal-matrix composites are rapidly becoming one of the strongest candidates as structural materials for many applications such as automobile and turbine engine components, in the aerospace industry, and in the electronic and magnetic packaging industries. Metal-matrix composites offer an increased service temperature and improved specific mechanical properties over the existing alloys. However, usage of the state-of-the-art composites is often limited by the relatively few reliable material qualification techniques currently available. Nondestructive evaluation (NDE) methods using eddy currents or ultrasonics offer a practical means of characterizing metal-matrix composites and their properties [1-7]. Additionally, NDE technqiues also provide a viable means for process-interactive, product quality assessment in a manufacturing environment. For example, an on-line eddy current measurement of electrical conductivities may be used to evaluate the volume fractions of the reinforcements, intermetallic compounds and voids at various locations on the product [7]. The variations in the measurements can provide valuable information concerning the uniformity of reinforcement dispersion, clustering phenomena etc.

A vital link in the use of NDE methods for metal-matrix composite applications is the correlation of the nondestructive property measurements to the microstructural features. A few experimental investigations [1-4] have been performed in this regard, on extruded A1/SiCp systems comprising of A12124, A16061 and A17091 alloy matrices. Experimental measurements of the electrical conductivities of A1/SiCp composites were reported in Ref. [1]. While experimental studies focus on specific composite systems, a theoretical analysis, especially in a dimensionless form, presents the advantage of a generalized treatment, and will hence be of enhanced practical value. With this objective, we consider the problem of estimating the anisotropic conductvities of extruded particulate reinforced metal-matrix composite systems. The conductivities of composite systems have been studied in the literature by considering welldefined reinforcement geometries ranging from unidirectional continuous fibers [8-10] to short fibers [11] and spherical dispersions [12]. A summary of the various studies may be found in Refs [12, 13], and is not repeated here for the sake of brevity. However, as explained below, none of these analyses can be directly used for estimating the conductivities of extruded metal-matrix composites.

3045

3046

PITCHUMANI et aL: CONDUCTIVITIES OF METAL-MATRIX COMPOSITES

Out-of-plane Thickness Direction ( x 3 )

,oeoa'tx ///

,~xtrusion

Transverse Direction ( x 2 )

Direction x 1 )

(a)

i

l -X 2

'25 ~tm ~

(b)

i

3 Xl

'25 pm

(c)

i

3 X2

'25 ~m ~

(d)

Fig. 1. (a) Schematic ofa A1/SiCpcomposite extrusion showing three symmetry axes. (b)-(d) Micrographs of a 7091/SiC/30p composite on the x l - x 2, x r x 3 and x2-x 3 planes, respectively.

Figure 1 shows the microstructurai characterizations of a typical extruded product, exemplified by an AI alloy reinforced with ct-SiCp composite, 7091/SiC/30pt, on the three symmetry planes [1]. The SiC particles are seen to be preferentially aligned on the extrusion ( x ~ - x : ) plane, along the extrusion (x,) and the in-plane transverse (x2) directions. Furthermore, the particles cannot be readily classified into any of the reinforcement morphologies studied in the literature. The closest approximations are those of the short fiber segments [11] and the spherical dispersions [12]. However, fiber segments usually have aspect ratios much higher than those of the SiC particles, and random dispersion of spherical particles results in isotropic composite properties. Therefore, neither of these approximations can satisfactorily model extruded metal-matrix composites. The problem of evaluating the composite conductivities is further compounded by the presence

tThe notation 7091/SiC/30p implies that the base aluminum alloy is A17091, and the reinforcement is 30 vol.% of SiCp.

of additional phases such as intermetallics and porosity which form during the processing stage. Depending upon their concentrations and chemical compositions, intermetallic compounds may affect the properties significantly, and must be accounted for in the microstructure-property relationships. This paper describes simplified analytical models for predicting the anisotropic conductivities of extruded, particulate-reinforced metal-matrix composites. First, a two-phase model, which accounts for the reinforcement and the matrix phases only, is developed by analyzing a unit cell representative of the particulate arrangement in a typical microstructure (Fig. 1). Subsequently, using the two-phase model as the basis, a comprehensive model accounting for the matrix, the reinforcement, and the intermetallic phases is constructed. The three-phase composite is modeled as being composed of a series of two two-phase composites--one, where the pure matrix is reinforced with either the intermetallics phase (in which case, the reinforcement particles constitute the third phase ) or the reinforcement phase (in which case the intermetallics are regarded as the

PITCHUMANI et al.: CONDUCTIVITIES OF METAL-MATRIX COMPOSITES third phase) to form a "modified"-matrix, and two, where the third phase is embedded within the "modified"-matrix to form the overall composite--each of which is analyzed for conduction using the two-phase conductivity model. Depending upon the inclusion sequence of the reinforcement and the intermetallics phases, two variants of the three-phase model are proposed. The effects of voids (porosity) are neglected in the present study primarily because the volume fraction of the voids was found to be negligible in the microstructural analyses [1]. However, the methodology presented in this paper may be extended to include the effects of porosity as well. The two-phase and three-phase theoretical models were used to estimate the conductivities of a wide range of SiCp reinforced AI matrix composite extrusions. The composite systems considered in this study were manufactured by the powder metallurgy route, and comprised of three A1 base alloys--2124, 6061 and 7091--reinforced with 0-55% SiCp. The conductivities predicted by the two-phase and three-phase models are compared with the experimental measurements obtained using eddy current techniques as described in Ref. [1]. Comparisons are also presented with the results of Hashin [12] for isotropic composites, and the Hatta-Taya model [11] for short fiber reinforced composites, on the extrusion plane. Good agreement is demonstrated between the present theoretical predictions and the experimental observations. In general, the three-phase model is shown to improve significantly upon the two-phase model, although the two-phase model may be used to get

l

reasonably accurate estimates of the conductivities. For the range of parameters relevant to the A1/SiCp composites, the order of inclusion of the non-matrix phases in the three-phase model is shown to have almost no influence on the composite conductivities. The organization of the paper is as follows: Section 2 describes the two-phase matrix-reinforcement model, which forms the basis of the three-phase model developed in Section 3. The theoretical predictions of the two-phase and three-phase models are presented and discussed in comparison with the experimental data in Section 4, while the highlights of the study are summarized in Section 5.

2. TWO-PHASE (MATRIX-REINFORCEMENT) COMPOSITE MODEL The aim of the analysis is to construct a simplified model for the estimation of the anisotropic effective conductivities of extruded composites with microstructures such as those in Fig. 1, where the reinforcement particles are oriented preferentially in the extrusion and the in-plane transverse directions. Figure 1 also shows the coordinate system employed in describing the composites throughout the paper. The development of the two-phase model proceeds by first identifying a representative unit cell, based on the geometric description of the composite available from the photomicrographs (Fig. 1), and then analyzing the unit cell to obtain the effective conductivities of the composite.

Out-of-plane Thickness Direction (X3)

Extrusion Direction

(Xl) (b)

(a)

\ o n ~

xl

(c)

3047

X2

o m o ~

(d)

Fig. 2. (a) Schematic of the theoretical model showing the planar arrangement of reinforcement particles. (b)-(d) Particle cross sections as seen on the xj-x2, x r x 3 and x2~ 3 planes, respectively.

3048

PITCHUMANI et aL: CONDUCTIVITIES OF METAL-MATRIX COMPOSITES

2.1. Unit cell construction

Photomicrographs of the composite cross-sections along the three symmetry planes reveal that the reinforcement cross-sections are flat and nearly rectangular in shape on the extrusion (xi-x2) plane [Fig. l(b)], while observed in the x 2 - x 3 and Xl-X 3 planes, they are elongated and approximately elliptical and narrow rectangular in shape [Fig. l(c, d)]. Further, on the x2-x3 and xl-x3 planes [Fig. l(c, d)], the particles are randomly distributed, but preferentially oriented in the extrusion (Xl) and the in-plane transverse (x2) directions. However, on the x~-x2(extrusion) plane, both the distribution as well as the orientation of the particles are random [Fig. l(b)]. Therefore, it may be assumed that the reinforcement arrangement is predominantly planar along the extrusion planes.

Out-of-plane Thickness Direction(X3) ~ In-plane Transverse Direction(X2) Extrusion Direction(X1)

Based on the above, an idealized model for the arrangement of reinforcements is constructed, as shown in Fig. 2(a), in which the particles are modeled as right elliptical cylinders with their axes aligned alternately along the extrusion and the in-plane transverse directions, so as to form a criss-cross structure. The microstructures of the idealized model on the three symmetry planes are presented in Fig. 2(b-d). Figure 2(b) reflects the extremes of the planar random orientation of the reinforcements as seen on the extrusion (xl-x2) plane [Fig. l(b)], while Fig. 2(c) and (d) depict the preferentially aligned, elongated cross sections in extrusion and the in-plane transverse directions [Fig. l(c) and (d)] respectively. A representative repeating element in the model is identified in Fig. 3(a), which is made up of the basic

I c;3

M -J I I

I I

y

t~ 2

/ (a)

I

(b) Fig. 3. (a) A representative repeating element in the model showing the assemblage of unit cells for the composite. (b) Basic unit cell consisting of a reinforcement particle (fight elliptical cylinder) embedded in a rectangular box.

PITCHUMANI et al.: CONDUCTIVITIES OF METAL-MATRIX COMPOSITES unit cell shown in Fig. 3(b). The unit cell consists of a rectangular box, in the center of which is placed a reinforcement particle, right elliptical cylindrical in shape, with cross-sectional dimensions, a and b, and length, L. The criss-cross arrangement in Fig. 2(a) can be generated by placing the unit cells in such a manner that the cylindrical axis of the particle in any unit cell is perpendicular to the axes of all its adjacent neighbors in the extrusion plane. In order to accommodate the criss-cross arrangement of the particles of length, L, in the extrusion plane, the base of the rectangular box must be a square of side, L. The square shape is actually a consequence of the fact that in the real microstructure the reinforcement particles are distributed as well as oriented uniformly randomly in the extrusion plane, which is represented in the model by the criss-cross arrangement. Further, the particles (elliptical cylinders) in the model are assumed to extend in length all along the base, L, of the unit cell, so as to reflect the anisotropy of the arrangement as much as possible. The height, H, of the box can be related to the base size, L, and the elliptical cross-section of the particle by making use of the fact that the reinforcement particle cross-sections are distributed uniformly randomly, but are oriented preferentially on the Xl-X3 and x2-x 3 planes. It may be argued that an equispaced array of ellipses in a two-dimensional plane best models a uniform-random arrangement. Any deviation from the equispaced arrangement will result in the cross-sections being clustered along either their major or minor axes (so as to preserve the reinforcement volume fraction), thereby departing from the uniform-random configuration. For an equispaced two-dimensional array of ellipses, the unit cell is a rectangle with length to height ratio equal to the major axis to minor axis ratio of the ellipse. This suggests that the box base length to height ratio, L/H, must equal the particle cross sectional axes ratio, a/b i.e. L H

a b

(1)

The box and the particle sizes can now be inter-related in terms of the reinforcement volume fractions, vp, which is simply the ratio of the right elliptical cylinder volume to the rectangular box volume. From the definition of the volume fraction, and using equation (1), one obtains

vr'

4\H]

or

=

=

.

(2)

Equation (2) completely specifies the unit cell. Given a and b, and the reinforcement volume fraction, vp, the unit cell dimensions can be determined. It must be mentioned that the actual dimensions of the unit cell and'(he particle are not necessary for the analysis. The quantities of importance are the ratios of the box-to-particle dimensions, H/b and L/a, which are analogous to the "pitch-to-diameter"

3049

ratios used in connection with unidirectional circular fiber reinforced composites. In our analysis, the two ratios H/b and L/a are equal, and a function of the particle volume fraction alone, by virtue of the assumption of uniform-random distribution of the particles. In the absence of any detailed information regarding the inter-particle spacings, this is a reasonably accurate description of the particle arrangement. Physically, this means that the particle volume fraction is the primary geometric parameter, influencing the analysis to the first order of accuracy. Additional geometric information on the microstructure may be introduced to obtain incremental improvements in the accuracy.

2.2. Conduction analysis With the above description of the unit cell as the basis, its longitudinal and transverse conductivities are derived in this subsection. The effective conductivities of the composite are then expressed in terms of the unit cell conductivities, and finally the planar conductivities of the composite are evaluated. 2.2.1. Unit cell conductivities. In the idealized model, since the reinforcement particle extends all along the length, L, of the unit ceil, the problem of determining the conductivities of the three-dimensional cell [Figs 3(b), 4(a)] is reduced to that of evaluating the conductivities of a two-dimensional cell shown in Fig. 4(b). Furthermore, by virtue of the fact that L / H = a/b [equation (1)] the two-dimensional cell in Fig. 4(b), which consists of an ellipse within a rectangle, under a geometric scaling with respect to the sides of the rectangle (i.e. scaling the x-coordinate with respect to L and the y-coordinate with respect to H), reduces to the two-dimensional arrangement of a circle within a unit square cell. This transformation, shown schematically in Fig. 4(b)~(c), is adopted so as to be able to obtain analytical expressions for the composite conductivities, as will be described later in this section. The conductivities of the elongated cell in Fig. 4(b) and the unit square cell in Fig. 4(c) may be argued to be approximately equal, through physical reasoning. It can be proved using the concept of a flux plot [14] that for laminated materials the conductivities of an elongated rectangular cell and a square cell are identically equal, as long as the volume fraction is maintained constant in the transformation. This is because the flow lines for heat (or electricity) run normal to the laminae, and the isotherms (or equipotential lines in the case of electrical conduction) are parallel to the laminae. On the other hand, in the case of an ellipse in a rectangular cell, the flow lines and isotherms are curved near the surface of the elliptical particle. However, if the particle volume fraction is small, the bulk of the isotherms and the flow lines are expected to be straight through most of the unit cell volume. Therefore, the flux lines and the isotherms will still be very similar even after the transformation of the rectangular cell [Fig. 4(b)] into a square cell

PITCHUMANI et al.: CONDUCTIVITIES OF METAL-MATRIX COMPOSITES

3050

(T

l

X

°T

/

~_ L

v

I~

,/ aL

rl

L

(~T

~T

Y

~L

(e)

(b)

(a)

aL- - _ H _ L

/

=

~/~v

Fig. 4. Transformation of the three-dimensional unit cell to a two-dimensional cell [(a)~(b)] and subsequently to a two-dimensional square cell and a circular particle enclosed within [(b)~(c)].

[Fig. 4(c)], and this transformation will have little effect on the conductivity. Similarly, if the squeezing of the rectangular cell to the unit square cell is small (i.e. for small values of a/b), the isotherms and the flux lines in the two cases will also be very similar, thereby resulting in nearly equal conductivities. The above physical reasoning has been substantiated by numerical evaluations of the effective conductivities of the cells in Fig. 4(b, c), using a finite difference scheme. For a particle volume fraction of 30%, and for a/b = 3 (corresponding to the average SiCp aspect ratio measured in the microstructural analyses [1]), a difference of about 5% was observed between the conductivities of the rectangular and square cells in Fig. 4(b, c). As will be seen in the section on data comparisons, this difference is quite small in contrast to the magnitude of anisotropy in the composite. Thus the transformation shown in Fig. 4(b)~(c) may be a reasonably accurate simplification. Figure 4(c) is also the unit cell for a continuous fiber reinforced composite with a square packing of fibers. Several analytical and numerical results are available in the literature on the conductivities of the square cell in Fig. 4(c) [8-10, 15]. The present analysis uses the theoretical expression for the transverse conductivity [in the x- and y-directions in Fig. 4(c)] given by Behrens [8], which agrees very well with the numerical results for square fiber arrays, and fiber volume fractions less than about 60% [15]. Although these results are in the context of heat conduction, the analogy between thermal and electrical conduction is well established and furthermore, the expression for the normalized effective conductivities in both these cases are identical [12].

Behrens' formula for transverse conductivity may be written in a non-dimensional form as a~ = (tip + 1) + (tip - l)vp a m (tip + 1) - (tip - 1)Vp

(3)

where a r is the transverse conductivity of the square cell, am denotes the conductivity of the matrix, ]~p equals the particle (fiber) to matrix conductivity ratio, and Vp is the particle (fiber) volume fraction. The longitudinal conductivity is given by a volume average of the paricle and the matrix conductivities, expressed in a dimensionless form as follows a---kL= 1 + (/~p- l)vp.

(4)

O"m

Equations (3) and (4) are also the transverse and longitudinal conductivity ratios, respectively, of the three-dimensional rectangular unit cell in Fig. 4(a). 2.2.2. Effective composite conductivities. The effective conductivity of the particulate composite may be evaluated using the unit cell conductivities, by considering an assemblage of unit cells as shown in Fig. 3(a). It is of interest to determine the conductivity of the assemblage along the extrusion (x~), in-plane transverse (x2) and thickness (x3) directions. The total (electrical or thermal) resistance along the extrusion (x l) and the in-plane transverse (x2) directions is a series combination of the transverse and the longitudinal resistances of the rectangular unit cell [Figs 3(b), 4(a)]. Therefore, the effective conductivity in the extrusion and the in-plane transverse directions, a t and a 2, respectively, is a harmonic average of the transverse and longitudinal conductivities of the unit

PITCHUMANI et al.: CONDUCTIVITIES OF METAL-MATRIX COMPOSITES cell. In a dimensionless form, the averaging is given by 1

1[ 1 ~ ]

1

0"170"m = a 2 / a m = ~

al am

~

or

"t=

a2 ',,am/\am/ ~m

(5)

\a,.i

Substituting for arla~ and a L / a m from equations (3) and (4), respectively, into equation (5), one obtains the following expressions for the normalized effective conductivity in the extrusion and in-plane directions

aI

a2

am

O"m

tions spanning the entire plane. For the purpose of comparisons with the experimental measurements, the planar conductivities along the three planes, x r x : , x2-x3 and x]-x~, were calculated in the model as an arithmetic average of the conductivities along two orthogonal axes defining the plane. For example, at_3 is the arithmetic mean of a~ and a~. It is expected that the simplified average used here will not be grossly different from a refined average which considers several directions spanning the plane. A refined average will probably improve upon the present results at the expense of simplicity but, as will be seen from data comparisons, is probably not warranted. Using the simplified averaging technique, the three planar conductivities, 0-~ 2, a:-3, and a I 3, scaled with

2[(flp+l)+(~p--1)vp][l+(flp--1)Vp] (tp + i) + (tp - l)vp + [1 -4-(tip -- 1)Vp][(tip + 1) - (tip --

The conductivity of the assemblage in the thickness direction (0-3) equals the transverse conductivity of the unit cell [equation (3)], i.e.

(._,

(7)

0-m (flp+l)--(flp--1)Vp" 2.2.3. Planar conductivities. In the experiments, which were described in Ref. [1], a circular eddy current probe was used for the measurement of the conductivities. The measured values were, therefore, an average of the conductivities along several direc-

(a)

Three-Phase n Composite

(b)

Three-Phase i Composite

(6)

1)v.]

respect to the matrix conductivity, o-m, can be expressed as follows am

0-_23 (flp+l)+(tp-1)Vp

3051

a_,

2 k,O'm

3al_

= 0"2..._3 3 ~l

O'm

O'm

am//

am

( a m-[- a 3 ~ 2~,~mm ff"mm,/"

(9)

3. THREE-PHASE COMPOSITE MODEL The objective of the three-phase model is to incorporate the effects of all the three phases--matrix, intermetallics (IM) and reinforcements--in the calcu-

"Modified

Matrix"

"ModifiedMatrix"

+ ~

4"

Remtorcementi~~

[::i:ii/i'te/"ffieiiiiiics:i:i:i:] :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ti r:,.-, :, !

Fig. 5. Schematic decomposition of a three-phase composite into two two-phase composites for the evaluation of the composite conductivities in (a) the three-phase (matrix-IM)-reinforcement model, and (b) the three-phase (matrix-reinforcement)riM model.

3052

PITCHUMANI et

al.:

CONDUCTIVITIES OF METAL-MATRIX COMPOSITES

iation of the effective composite conductivities. Two alternative schemes differing in the sequence of inclusion of the reinforcement and the intermetallic phases are proposed. The first--referred to as the three-phase (matrix-IM)-reinforcement model-describes the three-phase composite as consisting of the reinforcements embedded in a "modified"matrix, which in turn is formed of the pure matrix reinforced with intermetallics (IM). The second-termed the three-phase (matrix-reinforcement)-IM model--treats the composite as being composed of the intermetallics phase (IM) reinforcing a

"modified"-matrix, namely the two-phase pure matrix-reinforcement composite. A schematic of the model decomposition in both these cases is shown in Fig. 5. In general, the intermetallic phases exhibit thin plate-like morphologies which are different from that of SiCp particulates. These phases have specific habitplane relationships with the matrix, and further may exhibit crystallographic anisotropy in addition to their geometric arrangement-induced anisotropy. Since the primary objective of this work is to introduce an overall approach to modeling multiphase

SiC P

(a) SecondaryElectron Image •,

Intermetallic Compound

(b) Back-Scatter Electron Image Fig. 6. Scanning electron micrographs of a 2124/SiC/25p composite sample on the extrusion (xrx2) plane. The reinforcement particles are clearly seen in the secondary electron image, (a), while the corresponding back-scatter electron image, (b), reveals the intermetallic phases.

PITCHUMANI et al.: CONDUCTIVITIES OF METAL-MATRIX COMPOSITES microstructures using the recursive binary decomposition scheme presented in Fig. 5, a simplified model is adopted for including the effects of the intermetallics and illustrating the applicability of the proposed scheme. As an approximation, the simplified model neglects the crystallographic contributions to the anisotropy relative to the geometric arrangement-induced anisotropy. A refined model relaxing the above approximation may be used to replace the simplified model for a more accurate conductivity evaluation, although it is important to note that the basic approach to describing three-phase composites remains the same as that presented in Fig. 5. Figure 6(a, b) show the scanning electron micrographs (SEM) of a 2124/SiC/25p composite sample t O'ml O"m

p O'm2 O"m

trusion, in-plane transverse and out-of-plane thickness directions (Fig. 1), a'~l/a m, t r ~ / t r m and am3/a m, respectively, are obtained using equations (6) and (7), with Vp and/jp replaced by the equivalent intermetallics concentration, V{m, in the two-phase, matrix-intermetailics composite, and the intermetallics-matrix conductivity ratio, /j~, respectively. The equivalent intermetallics concentration, V~m, is related to the overall intermetallics volume fraction in the composite, Vim, and the overall reinforcement volume fraction, Vp, as follows V~-n

V;m (l -- Vp)'

-

-

1)ViPm]

(/jim"['- 1)-['-(/jim-- 1)V{m+[1 -]'-(/jim-- 1)O~m][(flim+ l)--(flim--

O'm3 O'm

1)Vim]

3.1. Three-phase

(matrix-IM)-reinforcement

model

The calculation of the effective composite conductivity in this model follows the two-step procedure outlined below. 1. In the first step, the "modified"-matrix conductivity is evaluated by considering the "modified"matrix to be a two-phase composite with the intermetallics (IM) as the reinforcements. The "modified"-matrix conductivity ratios along the ex-

(11)

(/Jim "{- 1) -']- (/jim -- 1)V(m

(]~im+ 1) -(flim - 1)Vi'm"

(12)

2. The second step is to calculate the conductivity of the two-phase reinforcement-"modified"-matrix composite, where the conductivities of the "modified"-matrix are given by equations (11) and (12). Once again, the conductivities along the three directions are evaluated using equations (6) and (7), where /jp is replaced by the reinforcement"modified"-matrix conductivity ratios, fl'pl, /j'p2 and /j~,3, along the three directions. Note that by virtue of equation (11), the quantities fl~l and/jp2 are identical. The effective conductivities of the composites, expressed as a ratio with respect to the "modified"matrix conductivity, are therefore given by the following equations

,r_2_~= ,r_2_~= 2[(/j~., + 1) + (/j~., - 1)v,][l + (/j',, - 1)v,] o~al o-~2 (/jp,-~- 1)--1-(/jp l - 1)Vp .--~-[1-1-(/jp,- 1)Vp][(/jpl -.~ 1 ) - (/j"p,- 1)Vp]

successively applying the two-phase model, first to calculate the effective conductivities of the "modified"-matrix and subsequently to obtain the effective conductivities of the overall composite. The two models are described below.

(10)

The expressions for the effective conductivity ratios of the "modified"-matrix are given below

2[(flim+ 1) + (tim -- 1)u~][1 + (/jim

on the extrusion (x~-x2) plane. The secondary electron image in Fig. 6(a) presents the general features of the silicon carbide particles, which are almost identical to those seen in Fig. l(a) for the 7091/ SiC/30p sample. The corresponding back-scatter electron image in Fig. 6(b) reveals the intermetallic phases (primarily CuAI 2 in the sample shown), identified by the white-colored regions in the micrograph. Note that since the intermetallic phases are generally seen to exhibit arrangements similar to those of SiCp [Figs l(a), 6(a)], the two-phase model developed in the previous section [equations (6)-(9)] may be used to evaluate the "modified"-matrix conductivity in the three-phase (matrix-IM)-reinforcement model, and the overall composite conductivity in the three-phase (matrix-reinforcement)-IM model. In both these models, therefore, the underlying principle is that of

3053

t73 = !tip3 "~ 1) -~- (/jp3 -- l)Vp O'm3 (tip3 + l) -- (/jp3 -- 1)Vp

(13)

(14)

where the equality of fl'pj and fl'p2 has been utilized. In practice, however, it is useful to express the conductivities as a ratio with respect to the pure matrix conductivities. These ratios may be obtained by multiplying equations (11) and (13), and equations (12) and (14), respectively. Further, the planar conductivities measured by an eddy current probe are calculated using an arithmetic average similar to those employed in equations (8) and (9), and are represented by the following relations

a~

\~m~J\ am J

3054

PITCHUMANI et al.: CONDUCTIVITIES OF METAL-MATRIX COMPOSITES

0"1-3 l o ' -=" -0"2---' m O"m- 2-~ 2 [(~ml)~'~-m 0"1 [O'=l) + '0"=3,/'( 0"3 ~(O'mS~]am J_J" (16)

3.2. Three-phase (matrix reinforcement ) - I M model

The development of the conductivity equations in this model follows the two steps presented above, with the exception that the effect of the reinforcement phase is accounted for prior to inclusion of the intermetallic (IM) phase, i.e. Vim and vp are interchanged in the calculation sequence above. The expressions for the conductivity ratios in this model are given below, without reiterating the details of the two steps. I. "Modified"-matrix (pure matrix-reinforcement) conductivities: The "modified"-matrix conductivity ratios along the extrusion, in-plane transverse and out-of-plane thickness directions (Fig. 1), a=l/trm, o ' m 2 / a m and O'm3/O'm,respectively, are given below

0"=3 = am

t O'ml

¢ O'm2

am

am

(18)

where v~ is the equivalent reinforcement concentration in the "modified"-matrix, and is related to the overall reinforcement volume fraction in the composite, vp, and the overall intermetallics volume fraction, Vma,as follows Up Vim ) '

(19)

2. Intermetallics(IM)-"modified"-matrix composite conductivities: The effective conductivities of the composite, expressed as a ratio with respect to the "modified"-matrix conductivity, are given by the following equations

al °'=l

a2

am2

The theoretical models developed in the preceding sections were used to evaluate the conductivities of silicon carbide particulate (SiCp) reinforced aluminum (Ai) alloy matrix composites, with particle loadings in the range 0-55%. The experimental measurement of the conductivities using eddy current techniques was described in Ref. [1]. An example result of eddy current measurements on a 2124/SiC/30p composite sample is shown in Fig. 7. Noteworthy in the figure is the anisotropy of conductivities between the extrusion ( X l - - X 2 ) plane and the other two symmetry planes in the composite. Fur-

(21)

(tim3 "di- l ) - - (flim3 - - l)Uim

where t ~ l , fl~2 and t~m3 are the intermetallics"modified" matrix conductivity ratios, along the three directions Xl, x2 and xs, respectively. Equations (20) and (21) employ the fact that /~m~= fl(m2 [by virtue of equation (17)]. tConductivity in % IACS may be converted to (f~ cm)-t using the relationship 1% IACS = 5800.13(D cm) -I.

(17)

thermore, the conductivities on the Xl-X3 and x 2 - x 3 planes are very close to each other, thereby confirming the trend predicted by the theoretical models [equations (8), (9), (15), (16)]. In this section, comparisons are presented between the model-predicted conductivities and the experimental measurements on the three symmetry planes--x~-x2, x~-x 3 and x 2 - x 3. The various composite systems studied, along with their respective SiCp volume fraction, composition and concentration of the primary intermetallic phase, and volume fraction of porosity, are summarized in Table 1. All volume fractions listed in Table 1 were evaluated using the "point-counting" metallographic technique [16], as explained in Ref. [1]. The concentrations of the intermetallics and voids are reported in terms of

2[(fl~ml + 1) + (fl~ml -- 1)Vim][l + (fl~rnl -- 1)Vim] (fl(ml+ 1) + (Biml - 1)Vim+ [1 + (fliml - - 1)Vim][(fliml+ 1) -- (fl~ml-- 1)Vim]

tr___L3= (fl~3 + 1) + (fl~3 - 1)v~ O'm3

4. RESULTS AND DISCUSSION

2[(tip + 1) + (tp -- 1)v~,][1 + (tp -- 1)v~,] (tip+ 1) + (tip-- 1)v~+ [1 +(tip-- l)v~][(flp + 1)-- (tip-- l)v~]

(tp + 1) + (tip -- 1)V'p (tip + 1) -- (tip -- I)V;

Vp = (1 -

The planar conductivity ratios with respect to the conductivity of the pure matrix may be calculated using equations (15) and (16), where the terms tr=l/cr m, O'm3/O'ra, O'l/tr~nI and o'3/tr=3 are given by equations (17)-(21), respectively.

(20)

their mean values and standard deviations based on 10-15 measurements on each composite sample. Table 1 also lists the billet numbers used for sample identification purposes. The conductivity values for the reinforcement and intermetallic phases were obtained from Refs [ 17-23], and are given in Table 2, in units of percent IACSt (International Annealed Copper Standard). For SiCp, depending on its type, a range of low conductivity values (Table 2) are reported in the literature

PITCHUMANI

et al.:

CONDUCTIVITIES

OF METAL-MATRIX

COMPOSITES

3055

Table I. Metallographic characterization of AI/SiCp composites Base alloy

Billet code

Volume fraction of SiCp[%]

Primary intermetallic

AI2124 A12124 AI2124 A12124

PE-2600 PE-2404 PE-2229 PE-2488

0 25 25 30

CuAI 2 CuA12 CuAI 2 CuAI:

7.4 + 4.4 + 10.0 + 6.7 +

2.1 2.8 3.9 3.7

1.4 + 1.8

A16061 A16061 AI6061 AI6061 A16061 A16061

PE-2045 PE-2047 PE-2099 PE-2731 PE-2869 PE-3235

0 20 25 30 40 55

Mg2Si Mg2Si Mg2Si Mg2Si Mg2Si Mg2Si

5.2 _+ 2.2 15.5 +_4.8 2.9 _+ 2.2 1.2 + 2.1 3.3 8

0 0 0 2.6 + 2.3 0 0

A17091 A17091 A17091 A17091 A17091

PE-2730 PE-2711 PE-2712 PE-2713 PE-2665

0 10 20 30 30

MgZn 2 MgZn 2 MgZn2 MgZn 2 MgZn2

[16]. A study of the effects of varying the particle conductivities in the range given in Table 2 revealed that the resulting theoretical conductivity ratios are fairly insensitive (less than 1%) to the variation. The theoretical results presented in this section have been calculated using the value of 0.086% IACS, which corresponds to the upper limit of the range, for SiCp conductivity. The conductivities of the aluminum alloys constituting the matrix of the composites are very sensitive to the solute content in the alloys which in turn depends upon the volume fraction of intermetallic compounds that precipitate out from the supersaturated solid solution. For example, the conductivity for A12124 can change up to about 20% depending upon the temper conditions used during the processing (50% IACS at 0 temper vs 39% at T851 temper) [21]. In consideration of these effects, the pure aluminum matrix conductivities (t~m) were obtained by substituting the measured conduetivities on the 0% SiCp samples (billet codes PE-2600, PE-2045 and PE-2730 in Table 1) for trml, trm2, and trm3 in the theoretical model equations [equations (11) and (12)], and solving them for trm. The calculated conduc-

22 AI 2124 (PE-2488) 21 <

Volume fraction of intermetallics [%]

6.9 + 6.9 + 4.4 + 3.2 + 6.9 +

2.6 2.8 2.6 1.1 2.8

Volume fraction of porosity [%] 0 0 0

0 0.5 _+0.9 0 4.2__.2.8 1.6 _+ 1.4

tivity values for the three base alloys are reported in Table 2. Figures 8-10 present comparisons between the models and the experimental values, for all the composite systems investigated (Table 1). The theoretical values were obtained using the volume fractions and conductivity values reported in Tables 1 and 2. The experimental data on the planar conductivities were cast into a non-dimensional form for the comparisons, by normalizing with respect to the conductivity of the respective pure aluminum alloy obtained as mentioned above. In the figures, the two-phase model predictions obtained using equations (6)-(9) are shown by open triangles, while the experimental data are denoted by open circles. The filled triangles in Figs 8-10 correspond to the results of the three-phase (matrix-IM)-reinforcement model given by equations (11)-(16). The effect of reversing the inclusion sequence of the reinforcement and intermetallic phases, on the three-phase model predictions, is discussed later in this section. The error bars shown for the three-phase model results correspond to the standard deviation in the experimental measurements of the intermetallics concentrations (Table 1). The solid, dashed, and dash-dotted lines in the figures represent the best-fit quadratic curves through the experimental data, the two-phase, and the three-phase model predictions respectively, which may be used as reference in the comparisons. Also included in Figs 8-10 are comparisons with two closely relevant (although not exactly relevant,

20 ._>

Table 2. Conductivities used in the theoretical calculations

18

i Xl'X2

t X 1°X3

i X2-X3

Orientation Plane Fig. 7. Electrical conductivities of the 2124/SiC/30p (PE2488) s a m p l e on the three s y m m e t r y p l a n e s i l l u s t r a t i n g the a n i s o t r o p y in the c o m p o s i t e .

Material

Conductivity [% IACS]

A12124 Al6061 AI7091 SiCp CuAl 2 Mg2Si MgZn2

33.99 50.28 35.78 1.82 x l0 ~-0.086 22.69 0.0045 11.09

3056

PITCHUMANI etal.: 1 . 2 i ..... I . . . . I . . . . I . . . . I . . . . l 1 0 ~ "-~_<.. t - ~,.--~ 0 . 8 [~ ~ ~

_

"o.

~

- x ~ . . ~

I .... I .... ~ -- ~- --,---- .......

CONDUCTIVITIES OF METAL-MATRIX COMPOSITES

) .... I .... I .... [ .... I"' Experimental Data Two-phase Model . Three-phase Model Hatta-Taya Model and Hashin Upper Bound . Hashin Lower Bound

1.2

1.0 0.8

0.6

' ' r l . . . . I . . . . I . . . . I . . . . ~. . . . I . . . . I . . . . I . . . . ) . . . . I . . . . I ' ' '

E x p e r i m e n t a l Data

I ~.

--

~

~

--

A.. - Three-phase

- " ~ -

.

.

.

.

-Two-phase

0

Model

Hashin Upper Bound •-

Hashin Lower Bound

0.4

0.2

Model

~

0.2

0 0

10

20

30

40

50

60

,,,I

0

....

i~:;;~,;,-,1,-,-,-,J..),,,.~

10

20

.....

30

%SiCp

I ...... I. . . . . L , , , , J , , , 40 50 60

%SlOp

Fig. 8. Comparisons of the experimentally measued and theoretically predicted conductivity ratios on the x~-x2 (extrusion) plane, for all the composite systems studied.

Fig. 10. Comparisons of the experimentally measured and theoretically predicted conductivity ratios on the x2-x 3 plane, for all the composite systems studied.

for reasons mentioned in Section 1) models available in the literature. These include the lower and upper bounds of Hashin [12] for isotropic three-dimensional particulate composites, shown by dotted and chaindotted lines respectively, and the Hatta-Taya model [11] for composites with short-fiber reinforcements oriented randomly on a plane. In the case of extruded composites, since the planar arrangement of particles is observed only on the extrusion plane (Xl-X2 plane in Fig. 1), the Hatta-Taya model is included for comparison only in Fig. 8. Note that the Hashin bounds as well as the Hatta-Taya model are valid for two-phase composites only, and do not account for extraneous phases such as intermetallics. The expressions for the Hashin bounds and the Hatta-Taya model are not repeated here for the sake of brevity, and the interested reader is referred to Refs [11, 12] for a more elaborate description of the models. The Hatta-Taya model is based on a rigorous analysis using a statistical probability distribution function for the random planar orientation of the particles. In their model, the reinforcements are assumed to be prelate-spheroidal in shape for the purpose of evaluating the heat conduction Eshelby tensor [11]. The conductivities evaluated using the

Hatta-Taya model were found to be very close to the Hasin upper bound, and in fact, the two results are seen to be mutually indistinguishable in Fig. 8. Furthermore, for the values of tip relevant to the present study, the prelate-spheroidal particle aspect ratio, which is a parameter in the Hatta-Taya model, was found to have a negligible ( < 0.1%) influence on the composite conductivity. In Figs 8-10, the divergence of the Hasin lower bound from the actual conductivities is due to the fact that the particle-to-matrix conductivity ratio, tip, is much smaller than unity for AI/SiCp composites [12]. It is evident from Fig. 8 that on the Xl-X2 (extrusion) plane, the theoretical results of the two-phase model [equations (6)-(9)] lie very close to the Hashin upper bound and the results of the Hatta-Taya model. However, all these models considerably overpredict the conductivities with respect to the experimental data. The three-phase model predictions, on the other hand, compare well with the eddy current measurements, over the entire range of volume frac-

1"2[~r''1

1.0

o

0.8

if) --

0.6

......

~ .......

r .......

---o---

L

~ --A...

0.8 -

~ .

~"--=.

Expedmental Data Two-phase

Model

Three-phase

A

al.VcA~ ( = ~2-VaA~ )

.........

-- - Hashin Upper Bound ....... Hashin Lower Bound

Line of Exact A g r e e m e n t , Z ~ "

/

~

~

0, t

0.4

o*

o•

N

,,o o

"j~

D

0.6 Ai6O61~<~.~:~ ',

i'"

A~ °°

6~ .2/6.. )

"T

Model

i .......

o

.... I .... I .... I .....................................

. . . ~ . ~

"~.

1.0

0.2

,•

I---

~

0.2 "'-.... 0

,,,I

....

I,;~'~]5;','El',,',',l'~,~,h,..l...Hlr,,.,I

10

20

30

40

..... L , . , , I , , 50

Three-phase 60

%SiC 0

Fig. 9. Comparisons of the experimentally measured and theoretically predicted conductivity ratios on the xt-x3 plane, for all the composite systems studied.

(AI-SiCp)-IM

Model

Fig. I l. Comparison between the predictions of the threephase (matrix-IM)-reinforcement model and the three-phase (matrix-reinforcement)-IM model, for the AI/SiCp composite considered in this study, demonstrating the effect of the inclusion sequence of the non-matrix phases in the three-phase model.

PITCHUMANI et al.:

CONDUCTIVITIES OF METAL-MATRIX COMPOSITES

tions studied. The agreement is generally to within about 3%, for SiCp volume fractions less than or equal to about 40%. In Fig. 8, the discrepancy between the three-phase (as well as the two-phase) model results and the experimental data is seen to increase with vp for higher particle volume fractions. This behavior may be attributed to the assumption of a strictly planar arrangement of reinforcements in the two-phase model, whereby the resistance due to the particles present in between planes is neglected. The effect is more pronounced for the higher volume fractions since there may be many out-of-plane reinforcement particles, due to their close packing. On the x~-x3 and x 2 - x 3 planes (Figs 9 and 10, respectively), the results of the two-phase model improve considerably upon the predictions of the Hashin upper bound. The difference with respect to the experimental data is generally less than about 5% over nearly the entire range of SiCp volume fractions, from 0-55%. The three-phase model, however, is almost exact in estimating the composite conductivities, over the entire range of reinforcement volume fractions. Note that in Figs 8-10 the actual conductivity ratios for the case of 0% SiC 0 are not equal to unity due to the presence of intermetallics (Table 1). The three-phase model is seen to predict this trend correctly in the figures. In Figs 8-10, the A16061 (PE-2047) composite with 20 Vol.% SiCp exhibits an anomalous behavior with respect to the other composites studied within the AI6061 family. For the PE-2047 composite, despite the improved predictions by the three-phase model, the discrepancies with experimental data are still non-negligible on all the three symmetry planes. A similar behavior was also observed in the calculation of the elastic constants for these composites in Ref. [5]. Possible causes for the anomaly currently being examined include an unusual chemical composition in the matrix alloy of the PE-2047 billet. As mentioned earlier, the results of the three-phase model presented in Figs 8-10 correspond to the three-phase (matrix-IM)-reinforcement model described in Section 3.1. An alternative model, namely the three-phase (matrix-reinforcement)-IM model [equations (17)-(21)], was also proposed in Section 3. Figure 11 compares the conductivity ratios on the XI--X 2 (extrusion) and x r x 3 ( = x2-x3 ) planes obtained using the two models, for all the composite systems in Table 1. The dashed line in the figure is the line of exact agreement, while the symbols denote the values calculated using the three-phase models. As seen in Fig. 11, the results of the two models are practically identical, for all the cases studied. This suggests that over the range of conductivity ratios and volume fractions considered in this study, which also represents the typical values for AI/SiCp composites encountered in practice, the inclusion sequence of the non-matrix phases is not an important factor in the calculation of the composite conductivities. Overall, the three-phase model, regardless of the

3057

sequence adopted for the inclusion of the reinforce-. ment and the intermetallics phases, is seen to perform significantly better than the theoretical models in the literature [11, 12], as well as the two-phase model developed in this work. Nevertheless, the two-phase model, although approximate, estimates the anisotropic conductivities, whereas the models in the literature yield only the isotropic conductivities. Therefore, the two-phase model may be used to obtain quick, reasonably accurate, evaluations of the anisotropic conductivities of extruded composites. The three-phase model may be resorted to when more accurate values are necessary. The applicability of the models is limited to a range of volume fractions, the lower bound of which is the physical limit of the particulate volume fraction being zero. On the other hand, the upper bound on the volume fraction arises from two factors. One is the geometric constraint limit which corresponds to a shrink-wrapped rectangular box surrounding the elliptical cross section of the particle. This condition occurs at a particulate volume fraction of ~ / 4 ( ~ 78.5%) which may be regarded as a theoretical upper limit for the validity of the model. A more restrictive limit arises from the fact that at the higher volume fractions, the particle arrangement in the composites tends to deviate from the idealized planar structure assumed in the theoretical model, and the anisotropy gives way to nearly isotropic conductivities. This is evident in the conductivity values for the 55% SiCp samples, where the dimensionless conductivity ratios are about 0.25 on all the three planes. Conservatively, therefore, a practical upper limit on the volume fraction for the validity of the model may be considered as about 55~50%, which also represents the maximum volume fraction for which the model was experimentally validated. In practical applications of the types of composites considered in this study, the particulate volume fractions are usually in the range of about 0-60%, over which the model is seen to yield good predictions. The models presented in this paper are based on the assumption of uniform-random arrangement of the particles. As mentioned earlier, in the absence of detailed information on the inter-particle spacings, this arrangement constitutes the best possible description of the microstructure. However, depending upon the processing conditions, clustering of particles or alignment of the long dimension of the particulates along the extrusion direction may occur, thereby rendering the assumption of uniform-random distribution invalid. If the cluster dimensions are small relative to the overall product dimensions, homogeneity of particulate arrangement may still be assumed in the global perspective, and the proposed theoretical models may be employed to a reasonable degree of accuracy. On the other hand, large clusters may necessitate a rigorous treatment. The overall effectiveness of the theoretical models in evaluating the anisotropic conductivities may be

3058

PITCHUMANI et aL: CONDUCTIVITIES OF METAL-MATRIX COMPOSITES

assessed by comparing their accuracy with the magnitude of anisotropy in the composite. Since the planar conductivities, in general, tend to average out the anisotropy considerably, the actual anisotropy in a composite is best estimated based on unidirectional conductivity data. For example, the maximum anisotropy based on planar conductivities, for A16061 with 40% SiCp, is on the order of 7-8%, while the anisotropy based on the theoretically estimated unidirectional conductivities, al la~, o'2/o"m and a31am, is about 18%. At the present time, unidirectional conductivity data on the AI/SiCp composites listed in Table 1 are not available. However, if the accuracy of the models in evaluating the unidirectional conductivities is assumed to be of a similar magnitude as their accuracy for planar conductivity calculations, one can estimate the effectiveness of the models. As an illustrative example, consider the case of the 40% SiCp reinforced A16061 composite mentioned above, for which the real anisotropy is about 18%. In light of this, the discrepancies observed in Figs 8-10 (of about 5-8% in the case of the two-phase model, and < 3% in the case of the three-phase model) are quite small. It is evident from the data comparisons and the foregoing discussion that the theoretical models, especially the three-phase model, provide a simple, yet effective means of evaluating the anisotropic conductivities of extruded composites. The approach used in the development of the three-phase model may be extended to incorporate the effects of additional phases such as porosity in order to fully characterize composites. From a fundamental viewpoint, the methodology established in the construction of the three-phase model may be used to characterize multiphase composites, in general. 5. CONCLUSIONS Analytical models were described for estimating the anisotropic conductivities of extruded, particulate-reinforced, metal-matrix composites. A twophase model, accounting for the effects of the matrix and the reinforcement phases only, was developed by analyzing a representative unit cell which was constructed using the microstructural features of a typical product. With the two-phase model as the basis, an approach was described for modeling three-phase composites consisting of the matrix, reinforcement and the intermetallic phases, which form during the processing stage. Depending on the order Of inclusion of the reinforcement and the intermetallic (IM) phases, two models---the three-phase (matrix-IM)-reinforcement model and the three-phase (matrixreinforcement)-IM model--were presented. The conductivities predicted by the models were analyzed in light of the experimental measurements on several AI/SiCp systems, obtained using eddy current techniques [1], as well as the theoretical results of Hashin [12] and Hatta and Taya [11]. The

data comparisons revealed that the agreement between the theoreical model estimates and the experimental values is remarkably good over the practical range of parameters. The three-phase model, in particular, was seen to yield conductivities almost identical to the experimental measurements in most of the cases studied. For the range of phase compositions and properties applicable to AI/SiCp composites, the inclusion sequence of the non-matrix phases was found to have no significant influence on the conductivity evaluations. The theoretical models developed in this study provide the anisotropic conductivities of the composites, while the available models in the literature yield the isotropic conductivities. The effectiveness of the models were verified by the fact that the discrepancy between the model estimates and the experimental conductivities is quite small in comparison with the magnitude of anisotropy in the composites. The approach presented for the modeling of three-phase composites may be extended to modeling multi-phase composites, in general. REFERENCES

1. P. K. Liaw, R. Pitchumani, D. K. Hsu, B. Foster, S. C. Yao and H. Jeong, Trans. A S M E J. Engng Gas Turbines Power 116, 647 (1994). 2. P. K. Liaw, R. E. Shannon and W. G. Clark, in Fundamental Relationships Between Microstructure and Mechanical Properties of Metal Matrix Composites

(edited by P. K. Liaw and M. N. Gungor), pp. 581-615. TMS-AIME, Warrendale, Pa (1990). 3. P. K. Liaw, R. E. Shannon, W. G. Clark Jr and W. C. Harrigan, in Morris E. Fine Symposium (edited by P. K. Liaw et al.), pp. 193-208. TMS-AIME, Warrendale, Pa (1991). 4. P. K. Liaw, R. Pitchumani, S. C. Yao, D. K. Hsu, B. Foster and H. Jeong, in Damage in Composite Materials (edited by G. Z. Voyiadjis), pp. 249-282. Elsevier Science Publishers, The Netherlands (1993). 5. H. Jeong, D. K. Hsu, R. E. Shannon and P. K. Liaw, Metall. Trans. A. Submitted.

6. G. Mott and P. K. Liaw, Metall. Trans. 19A, 2233 (1988). 7. R. Pitchumani, P. K. Liaw, S. C. Yao, D. K. Hsu, and H. Jeong, J. Compos. Mater. 28, 1742 (1994). 8. E. Behrens, J. Compos. Mater. 2, (1968). 9. R. Pitchumani and S. C. Yao, Trans A S M E J. Heat Transfer 113, 788 (1991). 10. R. Pitchumani, Ph.D. thesis, Carnegie Mellon University, Pittsburgh, Pa (1992). 11. H. Hatta and M. Taya, J. appl. Phys. 58, 2478 (1985). 12. Z. Hashin, J. appl. Mech. 50, 481 (1983). 13. M. Taya and R. J. Arsenault, MetalMatrix Composites: Thermomechanical Behavior. Pergamon, New York (1989). 14. J. P. Holman, Heat Transfer, 7th edn, pp. 76--77. McGraw-Hill, New York (1990). 15. L. S. Hart and A. A. Cosner, A S M E J. Heat Transfer 103, 387 (1981). 16. J. E. Hilliard and J. W. Cahn, Trans. Metall. Soc. AIME 221, 344 (1961). 17. R. C. Marshall, J. W. Faust Jr and C. E. Ryan (eds) Silicon Carbide 1973, p. 673. University of South Carolina Press, Columbia (1974). 18. K. Schrrder (ed.), CRC Handbook of Electrical Resistivity of Binary Metallic Alloys. CRC Press, Boca Raton, Fla (1983).

P I T C H U M A N I et al.: CONDUCTIVITIES OF M E T A L - M A T R I X COMPOSITES 19. Y. S. Touloukian (ed.), Thermophysical Properties of High Temperature Solid Materials, Vol. 6, Part I: Intermetallics. The MacMillan Company, New York. 20. C. R. Tottle, An Encylopedia of Metallurgy and Materials. The Metals Society, MacDonald and Evans, U.K. (1984). 21. "Properties and Selection: Nonferrous Alloys and Pure Metals", in Metals Handbook, 9th edn, Vol. 2. ASM, Metals Park, Ohio. 22. "Resistivity of Alloys", in International Critical Tables, Vol. 6. McGraw-Hill, New York (1929). 23. L. F. Mondolfo, Aluminum Alloys: Structure and Properties. Butterworths, London (1976). NOMENCLATURE

a,b H L v

major and minor axes of the elliptical cross section (Fig. 3), [m] height of the rectangular unit cell, [m] length of the square base of the unit cell, [m] volume fraction

AM 43/8--L

3059

x t, x 2, x 3 extrusion, in-plane transverse and out-ofplane thickness directions (Fig. 1)

Greek Symbols fl conductivity ratio with respect to matrix conductivity fl' conductivity ratio with respect to "modified" matrix conductivity a conductivity a ' conductivity of the "modified" matrix

Subscripts 1,2,3 L T im m p ml, m2, m3

along the x~, x2, and x 3 directions longitudinal transverse intermetallic phase matrix reinforcement particle refers to the matrix along the x~, x 2, x 3 directions, respectively