Materials Science and Engineering A271 (1999) 43–52

Analytical modeling of particle size and cluster effects onparticulate-filled composite

Guoqiang Li a,*, Yi Zhao b, Su-Seng Pang a

a Mechanical Engineering Department, Louisiana State Uni6ersity, Baton Rouge, LA 70803, USAb Department of Mechanical Engineering—Engineering Mechanics, Michigan Technological Uni6ersity, Houghton, MI 49931, USA

Received 2 February 1998; received in revised form 30 December 1998

Abstract

A three-layer built-in model, extended from Christensen and Lo’s three-phase sphere model for particulate-filled composites(PFC) containing no clusters [R.M. Christensen and K.H. Lo, J. Mech. Phys. Solids, 27 (1979) 315.], is proposed to evaluate theparticle size and cluster effect on the mechanical properties of PFC. Different from the self-consistent model used by Corbin andWilkinson [S.F. Corbin and D.S. Wilkinson, Acta Metall. Mater., 42 (1994) 1311], which gives only average stress–straindistribution, this model can be used to estimate the point-by-point stress–strain distribution induced by either external force ortemperature variation. Particles that are harder and softer than matrix are studied. It is found from the calculated results thatreducing cluster and particle size, using less scattered particles, reinforcing the bonding strength at the interface of particles andmatrix, enhancing the deformability of matrix, and employing particles with a coefficient of thermal expansion smaller than thatof matrix are efficient methods to resist damages of PFC. In addition, reducing cluster concentrations and increasing particlecontents are preferred for PFC containing hard particles and have negative effect for PFC involving soft particles. The selectionof particle rigidities should be based on a balanced comparison between strength and rigidity requirements of PFC. © 1999Elsevier Science S.A. All rights reserved.

Keywords: Particulate filled composite; Analytical modeling; Particle size distribution; Cluster

www.elsevier.com/locate/msea

1. Introduction

Particulate-filled composites (PFC) have been usedextensively in various fields due to their low productioncosts and the ease with which they can be formed intocomplex shapes. Besides, they behave isotropically andare not as sensitive as long fiber composites to themismatch of thermal expansion between the matrix andthe particles [1,2]. Among the various applications,PFC are sometimes used as structural materials. Thebroad use of PFC as structural materials relies on theunderstanding of its mechanical properties and damagemechanisms, which depend on a lot of variables.Among these variables, particle size and particle spatialdistributions are of significant interest.

A number of researchers have investigated the depen-dence of the mechanical properties of PFC, includingstrength, rigidity, toughness, wearing resistance, etc., onthe particle sizes and particle spatial distributions. Asearly as 1950s, Hall–Petch [3,4] found that the strengthof metals and alloys was dependent on d−0.5 (where d isthe diameter of particles). It was verified by experimentsthat this relation also held for overall PFC [5–10]. Fan[11] found that the plane strain fracture toughness, K1c,of metals and alloys was dependent on the grain size,d−1. Lloyd [12], Withers [13] and Prangnell [14] indi-cated that the particles in PFC were not uniformlydistributed in space due to, for instance, the moltenmetal mixing route in metal matrix composites. As aresult, some particle-rich regions, which were usuallycalled clusters, were formed. They experimentally ob-served that damage tended to originate preferentially inclustered regions [12–14].

To understand how clusters influence the mechanicalproperties of PFC, finite element analysis (FEA) and

* Corresponding author. Tel.: +1-225-388-5933; fax: +1-225-388-5924.

E-mail address: lguoi1@lsu.edu (G. Li)

0921-5093/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved.

PII: S0921 -5093 (99 )00227 -0

G. Li et al. / Materials Science and Engineering A271 (1999) 43–5244

analytical modeling were conducted based on the the-ory of elasticity and Eshelby’s equivalent medium the-ory. Using FEA, Prangnell et al. [14], Christman et al.[15], Wang et al. [16], Watt et al. [17] and Kassam et al.[18] studied the particle cluster effect on PFC. A limita-tion in FEA is that it is not a parametric analysis, anddifficulties arise when a larger number of particles orclusters are involved in the analysis. On the contrary,analytical modeling does not have this limitation.Corbin and Wilkinson [19,20] used an Eshelby basedself-consistent approach to create a model of a com-posite within a composite for describing the averagestress–strain distribution in a clustered composite. Thelimitation in Corbin and Wilkinson’s model is that itdoes not consider the stress–strain field interactionbetween different particles. To overcome this limitation,improved analytical models should be developed.

Christensen and Lo [21] developed a three-phasesphere model to consider the stress–strain field interac-tions between different particles. By overall evaluations,Christensen [22] concluded that this three-phase spheremodel was more reasonable and reliable than othergenerally used models such as the differential schememodel and the Mori–Tanaka model [23]. Without mod-ifications, however, this model is unsuitable for evaluat-ing the particle size and cluster effects on themechanical properties of PFC because the assumedparticle distribution is fixed and clusters are notincluded.

The objective of the present paper is (1) to extendChristensen and Lo’s model to PFC containing clusters,and (2) to include the effect of particle size distribution.

2. Analytical model development

2.1. Analytical model

Fig. 1 shows a schematic of the microstructure ofPFC containing clusters. In regions surrounded by in-ner circle lines, the particles are more densely aggre-gated than that of the remaining areas. A particle-richregion is taken as a ‘cluster’ in the present paper.Obviously, the studied PFC can be divided into tworegions: one is a clustered or particle-rich region, andthe other is an unclustered or particle-poor region. Tostudy the multi-clustered PFC analytically, several as-sumptions were made to simplify the problem. Accord-ing to Corbin and Wilkinson [20], the clusters wereassumed to be equally sized spheres and uniformlydistributed in PFC, and the particles were also spheresand uniformly distributed in clusters and in particle-poor regions. These assumptions are adopted in thiscurrent paper. In addition, Corbin and Wilkinson as-sumed that the particles were also equally sized. Thisassumption is not used in the present paper; instead,

the real particle size distribution will be assumed toconsider the effect of the particle size and particle sizedistribution on the mechanical properties of PFC.

Assuming a spherical cluster at the center of thespherical equivalent PFC medium shown in Fig. 1 as aseparate body. At the center of the cluster, a matrixcoated particle shown by the dashed circle line in Fig. 1is taken as the target object. Assuming the matrixuniformly coats the surface of the target particle, theseparated cluster forms a three-layer composite spherein space. According to Christensen and Lo [21], thestress–strain field effects of other clusters and particleson the separated cluster can be considered by embed-ding the three-layer sphere into a finite-range equivalentPFC medium, leading to a three-layer built-in modelshown in Fig. 2. It is noted that, as opposite to Chris-tensen and Lo, the range of the equivalent PFCmedium is assumed as finite instead of infinite. Thisallows different number of particles be involved in theanalysis by changing the range of the equivalent PFCmedium. As a result, the stress–strain field effects ofother particles on the target particle can be illustrated.

In Fig. 2, a is the radius of the particle, b−a thethickness of matrix layer, c the radius of the cluster,d−c the thickness of the surrounding equivalent PFCmedium; p the radial boundary stress at r=d applied asexternal force; p0 the radial stress at r=c ; p1 the radialstress at r=b ; p2 the radial stress at r=a ; r thespherical polar coordinate; T the temperature variation(T\0 denotes temperature rising and TB0 tempera-

Fig. 1. Microstructure schematic of clustered PFC.

G. Li et al. / Materials Science and Engineering A271 (1999) 43–52 45

Fig. 2. Three-layer built-in model.

A0=1+n0

E0

; A1=1+n1

E1

; A2=1+n2

E2

;

A3=1−2n3

E3

;

B1=0.5+j

1−m; B2=

0.5m−1+j

m−1−1; B3=

0.5n+d

1−n;

B4=0.5+d

n−1−1; B5=

0.5+d

1−n;

B6=0.5n−1+d

n−1−1; B7=

0.5k+h

1−k; B8=

0.5+h

k−1−1;

B9=0.5+h

1−k; B10=

0.5k−1+h

k−1−1;

j=1−2n0

1+n0

; d=1−2n1

1+n1

; h=1−2n2

1+n2

; m=c3

d3 ;

n=b3

c3 ; k=a3

b3 .

Assuming continuous conditions at the interfaces, i.e.the interfacial radial displacements are equal, the fol-lowing relations hold:

u0c=u1c (7)

u1b=u2b (8)

u2a=u3a (9)

Simultaneously solving Eqs. (1)–(9), the interfacial ra-dial stresses are obtained as follow:

p1=A/B (10)

p2=(a2−a3)T+A2B9p1

A3+A2B10

(11)

p0=(a0−a1)T+A0B1p+A1B4p1

A0B2+A1B3

(12)

where

A= (a1−a2)T+A1B5(a0−a1)T+A0A1B1B5p

A0B2+A1B3

+(a2−a3)TA2B8

A3+A2B10

;

B=A2B7+A1B6−A1

2B4B5

A0B2+A1B3

−A2

2B8B9

A3+A2B10

.

Once p0, p1, and p2 are derived from Eqs. (10)–(12),the point-by-point stress–strain distribution, the hy-draulic static stress, the equivalent stress, etc., can beobtained by applying the theory of elasticity [24]. Ow-ing to page limitations, the formulas are not listed here.Because of the significance of p0, p1, and p2 in causingthe damages of PFC, such as interface debonding,particle fracture, and matrix yielding, they are used asrepresentatives of stress distributions in this paper.

In Eqs. (10)–(12), letting p=0 and T"0, thermalstresses can be obtained; letting p"0 and T=0,

ture dropping); u0c and u1c are radial displacement ofequivalent PFC medium and cluster at the interfacer=c, respectively; u1b and u2b the radial displacementof cluster and matrix at the interface r=b, respectively;and u2a and u3a the radial displacement of matrix andparticle at the interface of r=a, respectively. E0, E1, E2,E3, n0, n1, n2, n3, a0, a1, a2 and a3 the elastic modulus,Poisson’s ratio, and coefficient of thermal expansion ofthe equivalent PFC medium, cluster, matrix and parti-cle, respectively.

2.2. Formulations

Although plastic deformations are very important inthe failure analysis of PFC, elastic deformations areequally important especially for brittle matrix com-posites where elastic deformations occupy a dominantpart of its deformation histories. As a preliminarystudy, this paper focuses on elastic deformations. Inaddition to the Hooke’s law assumptions for particlesand matrix, assuming p and T as uniformly distributed,the problem described in Fig. 2 becomes sphericallysymmetric. Using the theory of elasticity [24], the inter-facial radial displacements are obtained as follows:

u0c=a0Tc+A0c(B1p−B2p0) (1)

u1c=a1Tc+A1c(B3p0−B4p1) (2)

u1b=a1Tb+A1b(B5p0−B6p1) (3)

u2b=a2Tb+A2b(B7p1−B8p2) (4)

u2a=a2Ta+A2a(B9p1−B10p2) (5)

u3a=a3Ta+A3ap2 (6)

where

G. Li et al. / Materials Science and Engineering A271 (1999) 43–5246

boundary stress induced stresses can be derived; lettingp"0 and T"0, the overall stresses caused by tempera-ture variation and boundary stress can be developed.For simplicity, the effect of p and T on interfacial radialstresses will be considered independently and non-di-mensional form of p1/p, p2/p, p0/p, and the ratio of p1/T,p2/T, and p0/T will be assumed hereafter.

2.3. Computational parameters

The elastic modulus of the equivalent PFC medium,E0, and the elastic modulus of the cluster, E1, aredifficult to predict due to the non-uniform distributionof particles and irregular particle shapes which cannotbe easily included in an exact calculation. For simplicitybut without losing generality, the Halpin–Tsai equation[25] based on the rule of mixture method is assumed forthe calculation of E0 and E1. As mentioned earlier, thecluster is a two-phase composite composed of particlesand matrix, thus E1 can be predicted by directly usingthe Halpin–Tsai equation. However, the prediction ofE0 is not so obvious. The equivalent PFC medium canbe divided into two regions: a clustered region and aparticle-poor region, each of them can be further di-vided into a two-phase composite composed of particlesand matrix. Thus, the prediction of E0 can be obtainedby a two-step procedure. In the first step, the elasticmodulus of clusters, E1, and the elastic modulus ofparticle-poor regions E01, can be derived using theHalpin–Tsai equation. In the second step, the elasticmodulus of the equivalent PFC medium, E0, can beobtained using the Halpin–Tsai equation again. Theresults are as follows:

E1=E2[(1+2sf3c)E3+2sf2cE2]

E3 f2c+ ( f3c+2s)E2

(13)

E01=E2[(1+2sf3m)E3+2sf2mE2]

E3 f2m+ ( f3m+2s)E2

(14)

E0=E01[(1+2sr)E1+2s(1−r)E01]

E1r+ (1−r+2s)E01

(15)

where f2c and f3c are local volume fractions of matrixand particles in clusters, respectively; f2m and f3m, localvolume fractions of matrix and particles in particle-poor regions, respectively; r is the volume fraction ofclusters in PFC; s is the aspect ratio of particles withs=1 for the case of spherical particles. In addition, thefollowing relations hold:

f2m=f2−rf2c

1−r(16)

f3m=f3−rf3c

1−r(17)

f2+ f3=1 (18)

f2c+ f3c=1 (19)

f2m+ f3m=1 (20)

in which f2 and f3 are global volume fractions of matrixand particles in PFC, respectively.

For the calculation of a0 and a1, the well-knownSchapery’s formula [26] is applied. The aforementionedprocedure for estimating E0 and E1 is used here. Theresults are as follows:

a0=!�E01(1−r) %

3

i=2

fimEiain, %

3

i=2

fimEi+�

E1r %3

i=2

ficEiain

, %3

i=2

ficEi

",[E01(1−r)+E1r ] (21)

a1= %3

i=2

ficEiai/ %3

i=2

ficEi (22)

Assuming that every particle in clustered regions iscoated with the same thickness matrix regardless of theparticle sizes, then the following relation is derived:

b=a+f2c

3f3c %N−1

i=1

ki

ri

(23)

where ri (i=1, 2, . . . , N−1) is the average grain size ofgrade (i ) and grade (i+1) of particles when particlesare classified into N grades; ki is the content percentage(by weight) with grain size between grade (i ) and grade(i+1).

Numerical results are presented based on the parame-ter values and ranges found in the literature[2,14,19,27]. They are: E2=50 GPa; E3/E2=0.2�5;a2=0.5×10−5 /°C�2×10−5/ °C; a3/a2=0.5�2;r=0.01; f2=0.8 (range 0.7�0.9); f3=0.2 (range0.1�0.3); f2c=0.2 (range 0.1�0.3); f3c=0.8 (range0.7�0.9). The influence of Poisson’s ratio is compara-tively small, thus n0=n1=0.25; n2=0.3 and n3=0.2are assumed. Two types of particle size distribution areassumed in Table 1 for comparison purpose. In Table1, particle size distribution No. 2 is less scattered thanparticle size distribution No. 1.

Table 1Particle size distribution

Particle radius range (mm) 10�150�40 40�30 30�20 20�10

20No. 1: content percentage 2020 20 2023530No. 2: content percentage 60

G. Li et al. / Materials Science and Engineering A271 (1999) 43–52 47

Fig. 3. (a)–(d) Effect of cluster radius, c, cluster volume fraction, r, and equivalent PFC medium range, d, on interfacial stresses.

3. Results and discussions

The effect of the equivalent PFC medium range, d,cluster radius, c, and cluster volume fraction, r, oninterfacial radial stresses is evaluated by the proposedmodel with the results presented in Fig. 3(a)–(d). InFig. 3(a) and (b), the interfacial radial stresses varysubstantially as d increases within a certain extent, saydB300 mm. This means that, when d is comparativelysmall, the distance between the particles included in thisregion and the target particle is relatively short and theeffect of the stress–strain field of these nearby particleson the target particle is significant. When d is relativelylarge, such as d\300 mm, the distance between theparticles contained in this region and the target particleis relatively long and the constraining effect of those

distant particles on the target particle is relatively small.When d\700 mm, the interactions are negligible.Therefore, the extent of the equivalent PFC medium inFig. 2 should be large enough in order that the numberof effecting particles involved is as large as possible. Inthe following calculations, d=700 mm is assumed toconsider the stress–strain field interactions between thetarget particle and the surrounding particles.

Within the space of d=700 mm, a large number ofparticles are involved. Assuming the overall particlevolume fraction takes relatively small value f3=0.1, thesize of particle is constant with relatively large valuea=50 mm, and no clusters exist. It is estimated that thenumber of particles involved is at least 275. It wasshown [14] that, when only nine particles were consid-ered in FEA, the number of elements was as large as

G. Li et al. / Materials Science and Engineering A271 (1999) 43–5248

13 000. If 275 or more particles are to be involved inFEA, a huge number of elements should be created toensure accuracy, which makes the FEA extremely timeconsuming.

From Fig. 3(a) and (b), the interfacial radial stressesincrease as the cluster size, c, increases. This meanslarge clusters are more conducive to interfacial damage.This trend is confirmed by experimental observations[12–14,19,20]. The only exception is p0/p for PFC con-taining hard particles in Fig. 3(a). But this does notmean that increasing cluster size is beneficial because,compared with p0, p1 and p2 are the dominant stressescausing the damage of PFC, including the particlefracture, interface debonding, and matrix yielding.Therefore, reducing cluster size is recommended evencontaining hard particles. For comparison purpose,c=75 mm is assumed in the following calculations.

In Fig. 3(c) and (d), the interfacial radial stressesincrease slightly with hard particles and decreaseslightly with soft particles as the cluster volume frac-tion, r, increases. Comparing Fig. 3(c) with (a) and Fig.3(d) with (b), it is found that the influence of r oninterfacial radial stresses is much smaller than that of c.This suggests that the goal in practice is to reduce thecluster size but not the cluster concentrations.

To address the effect of particle size and particle sizedistribution on the mechanical properties of PFC, cal-culated results are presented in Fig. 4(a) and (b). In Fig.4(a), interfacial radial stresses increase as the particlesize, a, increases. This means damage initiates distinc-tively around particles with large sizes. Particle sizedistribution No. 1 shown in Table 1 causes greaterinterfacial stresses than particle size distribution No. 2.Therefore, when hard particles are incorporated inPFC, reducing the particle size and particle size scatter-ing is proposed.

When incorporating soft particles, the situation issomewhat complicated. In Fig. 4(b), as particle size, a,increases, p2/p increases and p1/p decreases, but p0/p isnearly constant. Reducing particle sizes scattering, theinfluence is also case dependent. In general, the inter-face between the particle and matrix (the interface atr=a) is the weakest point, therefore, among p0, p1 andp2, p2 is decisive. This suggests that reducing the particlesize and particle size scattering has positive effect dueto the decrease in p2. This conclusion agrees with thecase containing hard particles.

The positive effect of reducing the particle size isqualitatively supported by the Hall–Petch theory. Asfor the effect of the particle size distribution on themechanical properties of PFC, few literature referencesare available. Hence, the predicted positive result ofreducing particle size scattering needs further verifica-tions by experimental studies. To analyze interfacialradial stresses of PFC in unfavorable conditions, amaximum value of a=50 mm and grain size distribu-

tion of No. 1 shown in Table 1 is assumed in thefollowing calculations.

In clustered PFC, there are two volume fractionrelated parameters, local particle volume fraction incluster, f3c, and global particle volume fraction in PFC,f3. In the Eshelby based self-consistent method used byCorbin and Wilkinson [20], the value of f3c wasconfined to less than 2/3 because the particles wereassumed to be of equal sizes. In this study, f3c can takethe values up to 1 due to the graded particle sizesassumed. The effect of f3 and f3c on the interfacialradial stresses is shown in Fig. 5(a) and (b).

In Fig. 5(a), interfacial radial stresses increase as f3c

increases. Thus, reducing the concentration of particlesin the cluster is suggested. When soft particles areincorporated, as shown in Fig. 5(b), the case is differentfrom that involving hard particles. However, consider-

Fig. 4. (a), (b) Effect of particle size a and particle size distribution oninterfacial stresses.

G. Li et al. / Materials Science and Engineering A271 (1999) 43–52 49

Fig. 5. (a), (b) Effect of particle volume fraction f3c in cluster and f3

in equivalent PFC on interfacial stresses.

the damage effect of clusters. The reason may be that,with the increase of f3, the equivalent PFC mediumtends to soften, and thus promotes the appliedboundary stress p to be delivered to the cluster. There-fore, when determining the effect of f3 on the mechani-cal properties of PFC, it is desirable to distinguish hardparticles from soft particles.

The effect of the elastic modulus of the matrix andparticles on the interfacial radial stresses is shown inFig. 6. When E/E2=1 (homogeneous material), p2=p1=p0, which is the result of a homogeneous sphere.With the increase of E3/E2 (particle transforms fromvery soft to very hard), interfacial radial stresses in-crease substantially. This means that, although hardparticles increase the rigidity of PFC, it has the poten-tial of increasing damages. Thus, high strength particle,high strength matrix, and strong interfacial bonding areneeded. So far as soft particles are concerned, they canreduce interfacial radial stresses and thus alleviate thedamage effect, but they also reduce the rigidity. There-fore, whether hard particles or soft particles are to beused should be based on a balanced comparison be-tween strength requirements and rigidity requirementsof PFC.

In addition to considering the boundary stress in-duced interfacial radial stresses, the developed formula-tions can incorporate thermal stresses. The effect of thecoefficient of thermal expansion of the particle, a3, andthe matrix, a2, on interfacial radial thermal stresses ispresented in Fig. 7(a) and (b). In Fig. 7(a) and (b), theabsolute values of interfacial radial thermal stressesincrease linearly with the increase of a2. When a3Ba2

(a3/a2=0.5), interfacial radial thermal stresses are com-pressive if temperature drops (TB0); when a3\a2

(a3/a2=2), interfacial radial thermal stresses are tensile

Fig. 6. Effect of elastic modulus on interfacial stresses.

ing the decisive role of p2 on the damage of PFC,reducing f3c is also beneficial due to the slight decreaseof p2 when f3c decreases. This result qualitatively agreeswith the experimental findings [12–14] that damageconcentrates in clustered regions and reducing the vol-ume fraction of particles in clusters is beneficial.

The effects of the volume fraction of particles inPFC, f3, on the interfacial stresses are discussed asfollows. In Fig. 5(a), the interfacial radial stresses de-crease as f3 increases, alleviating the damage effect ofclusters. The reason may be that, with the increase of f3,the equivalent PFC medium tends to harden and thusprevents the applied boundary stress p from beingdelivered to the cluster. When soft particles are used,the result is opposite to that when hard particles areused. In Fig. 5(b), interfacial radial stresses increase asf3 increases, showing that increasing f3 will aggravate

G. Li et al. / Materials Science and Engineering A271 (1999) 43–5250

Fig. 7. (a), (b) Effect of coefficient of thermal expansion and elasticmodulus on interfacial stresses.

stresses represent only three specified points at r=a,r=b and r=c, questions arise as to whether the con-clusions based on p2, p1, and p0 are reliable. To resolvethis concern, stress–strain distribution in PFC is pro-vided. The radial stress,sr, tangential stress,st, radialstrain,or, and tangential strain,ot, distributions for aPFC containing hard particles are shown in Fig. 8(a)and (b). In obtaining Fig. 8, the boundary stress isassumed as p=1 MPa, and the temperature variation isassumed as T=0°C. Fig. 8 demonstrates two aspects.In the first aspect, the largest stress or strain in eachphase occurs at one of its two interfaces. This suggeststhat taking the interfacial radial stresses as the repre-sentative stresses is reliable. In the second aspect, thestress or strain concentration is in radial direction, andoccurs at the interface r=a. Therefore, addressing p2 inprevious analyses is reasonable. In addition, the stressand strain concentrations at r=a on the side of thematrix imply that yielding initiates in the matrix. Theequivalent stress distribution in Fig. 8(a) exhibits thesame result. Therefore, reinforcing the bonding strengthbetween the particle and the matrix and enhancing thedeformability of the matrix are necessary in ensuringthe load-bearing capacity of PFC.

4. Conclusion

A way of evaluating the particle size and clustereffects on the mechanical properties of PFC is proposedvia the developed three-layer built-in model. This modelis extended from Christensen and Lo’s three-phasesphere model for PFC containing no clusters. Prelimi-nary calculation results agree qualitatively with thoseindicated in experimental studies. Based on the calcu-lated results, the following preliminary conclusions areobtained:

(1) Particle size has a significant effect on the dam-age of PFC. Large particles are more likely to causeinterfacial debonding and particle fracture.

(2) Particle size distribution has some effects on thedamage of PFC. Reducing particle size scattering haspositive effect in alleviating the damage.

(3) Damage is likely to initiate in clustered regions.Reducing cluster size and the concentration of particlesin clusters are recommended. For hard particles, reduc-ing the volume fraction of clusters is also suggested.

(4) The effect of the overall particle volume fraction,f3, is case dependent. When hard particle is used, in-creasing f3 is expected; when soft particle is involved,reducing f3 is preferred.

(5) Hard particles increase the rigidity of PFC, butinitiate larger interfacial stresses than that of soft parti-cles. Whether hard or soft particles are to be incorpo-rated depends on a balanced evaluation of rigidity andstrength requirements of PFC.

if temperature drops (TB0). As tensile stress causesinterfacial debonding and particle fracture, the case ofa3\a2 should be avoided when temperature drops. Inthe preparation of PFC, temperature drops are com-mon practice. Thus selecting particles with a coefficientof thermal expansion smaller than that of the matrix isexpected. When comparing Fig. 7(a) with (b), it isfound that hard particles produce greater thermalstresses than that of soft particles. Therefore, whenthermal stresses are major concerns in PFC, great at-tentions should be paid if hard particles are used.

In the above analyses, the effect of the physical–me-chanical parameters of the particle and matrix on themechanical properties of PFC is evaluated via the inter-facial radial stresses. Because the interfacial radial

G. Li et al. / Materials Science and Engineering A271 (1999) 43–52 51

Fig. 8. (a), (b) Stress–strain distribution in each phase of a PFC.

(6) When temperature drops are anticipated in theuse of PFC, selecting particles with a coefficient ofthermal expansion smaller than that of matrix isrecommended.

(7) With hard particles, the stress and strain concen-trations occur in the matrix, which is at the interface ofthe particle and the matrix, in radial direction. Rein-forcing the bonding strength at the interface and en-hancing the deformability of the matrix are required.

(8) Although the particle size effect on the mechani-

cal properties of PFC is well understood in experimen-tal studies, systematic experimental investigations areneeded to confirm the effect of particle size distribu-tions and clusters on the mechanical properties of PFC.

Acknowledgements

The assistance from Dr Jack E. Helms in this study isgreatly appreciated.

G. Li et al. / Materials Science and Engineering A271 (1999) 43–5252

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