analytical modeling of tensile strength of particulate-filled composites

Analytical Modeling of Tensile Strength of Part icu late- Fi I led Corn posi tes

GUOQIANG LI*, JACK E. HELMS, tu7d SU-SENG PANG

Department of Mechanical Engineering Louisiana State University

Baton Rouge, LA 70803

and

Department of Mechanical Engineering University of the Pacific

Stockton, CA 9521 1

Based on Christensen and Lo’s (1 1) three-layer sphere model, a two-layer built-in model is proposed to represent particulate-filled composites. Following Papanico- laou and Bakos’ (14) procedure for a particle embedded in an infinite matrix model and using the rule-of-mixtures approach, formulations estimating the tensile strength of particulate-filled composites are developed. Unlike Papanicolaou and Bakos’ formulations, the formulations developed in the present paper can charac- terize the effect of particle size, particle size distribution, and particle clustering on the tensile strength of the composites. A reasonable agreement is found between the predicted tensile strength and the experimental results found in the literature. Parameters affecting the tensile strength of particulate-filled composites are discussed via the calculated results.

1 INTRODUCTION

articulate-filled composites are being used in a P wide variety of industrial applications because of their low cost and ease of fabrication. Nevertheless, prediction of the tensile strength of the composites can be a very complicated problem because of the many variables that play definite roles in it. Based on a number of experimental results reported in the literature (1-8). it is found that (a) the interfacial bond between the particle and the matrix has a significant effect on the composite tensile strength: (b) fillers of smaller diameter result in a comparatively higher composite tensile strength; (c) the composite tensile strength is dependent on the volume fraction of particles; and (d) the particle distributions both in size and in space play an important role in the resulting composite tensile strength.

Pioneering analytical studies of particulate-filled composites could date back to the analysis of a single inclusion embedded in an infiiite elastic matrix (9)

‘Corresponw author.

POLYMER COMPOSITES, OCTOBER2001, Vol. 22, No. 5

and the representative volume model proposed by Hashin (10). However, most of the successive works were focused on the investigation of the elastic modulus (11, 12). It was perhaps in the 1980s that researchers began to pay attention to the tensile strength of particulate-filled composites and its relationship to the microstructures. Ahmed and Jones (1 3) gave a re- view of the theories predicting the tensile strength and other mechanical properties before 1990.

In the 199Os, more interest in predicting the tensile strength of particulate-filled composites was aroused among researchers because of their increased applications in industry. Papanicolaou and Bakos (14) proposed a two-phase model, in which a single spherical particle was embedded in an infinite matrix. By dividing the particle into an infinite number of coaxial cylinders and using Cox’s shear lag theory, closed form formulations were obtained to predict the tensile strength of particulate-filled composites. Although their model was shown to match some of the test results found in the literature (1-31, it is limited because it suggests that the composite tensile strength will always be smaller than that of the matrix and it cannot consider the effect of particle size and particle clustering on the

593

Guoqiang Li, Jack E. Helms, Su-Seng Pang, and Kurt SchuJz

tensile strength of the composites. An energy balance model was developed by Vratsanos and Farris (15) to predict the tensile strength of particulate-filed composites. Unlike the simple closed form formulations in Papanicolaou and Bakos' model, their formulations were in the form of incremental stress and strain. The tensile strength can be obtained only when the whole stress-strain curve is obtained. The stress-strain curve can be derived by taking strain increments and iterating downward through the particle size distribution. Although this model could consider the effect of particle size distributions, the calculations were very complicated, and the accuracy was heavily dependent on the use of the interfacial adhesive energy, which was difficult to determine.

While extensive researches were focused on analytical modeling, other methods were also used by a number of investigators. Jancar et aL (16) and Liu et al. (17) used finite element analysis to investigate the tensile strength of particulate-filled composites. Al- though finite element analysis is a very useful tool to predict the tensile strength of particulate-filed composites, it is time consuming and difficult to be used in an optimal design of particulate-filled composites. In addition to the analytical studies and finite element analysis, some researchers proposed a number of empirical or semi-empirid equations for estimating the tensile strength of particulate-Wed composites. One such example was a semi-empirical relation developed by Turesanyi (18), which was verified by D'Almeida and Carvalho (19) for a series of particulate-filled composites. While empirical or semi-empirical rela- tions are useful for specific composite systems, they cannot be extended to general particulate-filled composites.

P

594

4 -___

The purpose of this paper is to develop analytical formulations predicting the tensile strength of particulate-filled composites. The effect of the particle size, particle size distribution, and particle clustering on the tensile strength will be considered in the formulations. To validate the developed formulations, predicted tensile strength will be compared with the experimental results found in the literature. Parameters affecting the tensile strength of particulate-filed composites will be discussed based on the calculated results.

2 ANALYTICAL MODEL In estimating the elastic moduli of particulate-filed

composites, Christensen and Lo (1 1) proposed a three-layer sphere model. Christensen (20) has indi- cated that this model is more suitable than the differ- ential scheme and Mori-Tanaka model (21) because it can consider the stress-strain interactions between neighboring particles. Li et al. (22) extended this model to particulate-filled composites containing clusters and graded particles. Based on these studies, a two layer built-in model is proposed in the present paper by embedding an uniform matrix layer coated spherical particle into an infinite equivalent medium, as shown in Flg. 1. This is a three-phase model containing particle phase, matrix phase, and equivalent medium phase. In Flg. 1 , a is the radius of the particle, b-a is the thickness of the matrix layer, Ef, uf, &, vm, E,, and v, are the elastic Young's modulus and Poisson's ratio of the particle, matrix, and equivalent medium, respectively. P is the distributed axial load applied to the composite.

The key to using the two-layer built-in model to determine the tensile strength of particulate-filled

4

m. 1. ?tUo-lay~ built-in model.

POLYMER COMPOSITES, OCTOBER 2001, Vol. 22, No. 5

P

-+ --

Analytical Modeling of Tensile Strength of Particulate-Filled Composites

1--

t --

P 4.-

4-

I___+

*. . --

P c

__I,

Fig. 2. Division of particle into an in@& number of cylinders.

composites analytically is to obtain the stress distributions in the particle. Once the load carried by the particle is determined, the composite tensile strength can be obtained using the rule-of-mixtures method.

Based on a two-phase model, which was formed by embedding a particle into an infinite matrix, Papani- colaou and Bakos (14) developed a multi-fiber model by dividing the particle into an infinite number of coaxial cylinders. Using Cox's shear lag theory for short fiber reinforced composites, they obtained the stress distributions in the particle by integration. Following their procedure, the particle shown in Rg. 1 can be divided into an infinite number of coaxial cylinders as shown in Fig. 2. Again, a single fiber (cylinder] is shown in Fig. 3. It is seen from Fig. 3 that an axial stress urn. which is transferred through the matrix layer to the fiber, exists at the fiber end, while it is assumed to be zero in Papanicolaou and Bakos' model.

3 FORMULATXON DEVELOPMEm

3.1 Average Add Stress in the Particle

For the single fiber shown in Fig. 3, the axial stress in the fiber, uf(xj. can be obtained using Papanicolaou and Bakos' solutions (14) as follow:

where u, is the average axial stress in the matrix, D is an integral constant to be determined by the boundary condition of the fiber, K is a parameter depending on the radius of the fiber and other mechanical properties as follows:

I - K = J Ern

(Ef - Em) (1 + v,) ? In 2 in which r is the radius of the fiber.

To simplify the formulation development, the average K value will be used for each fiber. According to Papanicolaou and Bakos (14). the cross-sectional area for each fiber is assumed the same. If the cross-sectional area is assumed as an unit, then the radius for the innermost fiber in Fg. 2 is obtained as Be- cause the radius for the outermost fiber in Fig. 2 is equal to the radius of the particle, a, the average K values can be obtained as:

(3) In a + 0.5 In .rr

a Ern

d ( E f - Em) (1 + urn) In2

Fig. 3. Geometry of one of the constituent cylinders.

POLYMER COMPOSITES, OCTOBER 2001, Vol. 22, No. 5 595

Guoqiang Li, Jack E. Helms, Su-Seng Pang, and Kurt Schulz

To determine the axial stress in the fiber from Eq 1, the integral constant D should be determined by a boundary condition. The boundary condition of the fiber at the fiber end is:

Of (0 = Of0 (4)

Various fiber end stress distributions have been used by previous researchers. Kelly (23) and Clyne (24) assumed the fiber end stress was equal to the average axial stress of the matrix. This assumption does not consider the stress (or strain) concentration at the fiber end. In fact, the stress concentration at the fiber end results from the strain difference between the matrix right in front of the fiber end and the matrix else- where. Assuming the fiber end strain for each fiber in Rcj. 2 is the same, it can be expressed as follows:

E ~ O = [Earn + (&a rn - 1 (5) where E, is the average axial strain in the matrix when the particle is absent; E~ is the average axial strain in the particle when the particle is embedded in the matrix.

It is seen from Eq 5 that when E, = E&. i.e., the mechanical properties of the particle are the same as those of the matrix, Em = E,. This is true for a homo- geneous material. If the particle is rigid, i.e., E& +O, then E~ = 2 E,. This means that the maximum strain concentration factor at the fiber end is 2. In such a way. the strain (or stress) concentration at the fiber end is considered.

Using the one dimensional Hooke’s law, Eq 5 can be rewritten as follow:

in which ad is the average axial stress in the particle.

the axial fiber stress is obtained as follow: Substituting Eqs 4 and 6 into Eq 1 and rearranging,

uf(x) = ( E f - Ern) 2 t

in which 1 is the half length of the specific fiber and k is shown in Eq 3.

According to Papanicolaou and Bakos (14). the average stress in the particle can be expressed by integration as follows:

oaf = [ r( jbbr(x)dl)dx = (EJ - Ern)% + Ern

sinh k a Pa2

Rearranging Eq 8, the average stress in the particle is obtained as follow:

sinh ka Pa2

3.2 Tensile Strength of Particulate-FUed Composites

Once the average stress in the particle is obtained, the average composite stress can be obtained using the rule-of-mixtures method as follows:

(10)

where uac is the average stress in the composite andf is the volume fraction of particles in the composite.

For particulate-filled composites, matrix cracking or yielding is the beginning of the composite failure. As- suming u, in Eq 10 is equal to the ultimate tensile strength (for brittle materials) or the yielding strength (for plastic materials) of the matrix, urn, then the tensile strength (ultimate tensile strength or yielding strength) of the composites, IT=, is obtained as follows:

(1 11

urn = fuaf + (1 - fluam

~ s c =fasf+ (1 -flusrn

where usf is equal to uaf when uam is replaced by om in Eq 9.

However, Eq 11 is not the final expression predicting the tensile strength of the composite. Several factors should be considered before Eq 11 can be used, including (a) the degree of adhesion between the matrix and the particles as well as the degradation of the matrix properties due to the presence of the particles: (b) particles size distributions; and (c) particle clustering.

3.2.1 Effect of the Degree of Particle Adhesion and Matrix Degradation

In particulate-filled composites, interfacial bonding plays a determinant role in the composite strength because the load transferred to the particles is through the interfacial bonding. The interfacial bonding depends on a number of factors, including the volume fraction of particles, compatibility between the particle and matrix, particle surface treatment, fabrication process, etc. Owing to these factors, only a portion of the particles are well bonded to the matrix. In addition, the matrix will also degrade because of the presence of particles and complications developed during the preparation of the composite. Papanicolaou and Bakos (14) used the following modified rule-of-mixtures method to consider the effect of the degree of particle adhesion and matrix degradation on the composite tensile strength:

596 POLYMER COMPOSITES, OCTOBER 2001, Yo/. 22, No. 5


where rn E [0,11 is a parameter that depends on the effective fraction of the inclusions, i.e., the number of particles that are well-bonded to the matrix: n E [Q.11 is a parameter that depends on the degradation of the matrix materials.

However, Eq 12 has limitations. For example, if rn = 0, i.e., no particles take effect, then uSc = uSm. This overestimates the composite strength because the in- effective particles will weaken the composite and the composite strength will be less than the matrix strength. In addition, if rn = 1 and n = 0, i.e., all the particles are well-bonded but all of the matrix is degraded, then uw = usp Again, this significantly overestimates the composite strength. Therefore, Eq 12 re- quires further modifications. In this present paper, it is modified as follows:

usc = mfusf + ( 1 - f n)usrn (13)

3.2.2 Effect of Particle Size Distributions

In practice, particles are not always uniformly sized: their sizes may be distributed within a certain range. In such a case, the effect of particle size distribution on the composite tensile strength should be considered. From Eq 9, the average stress in a particle depends on the particle size, a, and thus the composite tensile strength is related to the particle size. In an ar- bitrary cross section perpendicular to the applied external load, various size particles can be found in this cross section. Therefore, the load carried by the particles in this cross section is related to the particle size distributions. If the particle size gradation is known, the composite tensile strength in Eq 1 3 can be modified as follows:

where amin and q- are the minimum and maximum radius of particles, respectively. u,da) is equal to usr shown in Eq 9. P(4 is the percent passing by volume of particles, i.e., P(a) is the volume fraction of particles with sizes less than a It describes the particle size distributions. For instance, for continuously graded particles, P(a) can be represented by the following equation:

in which c is a constant corresponding to the specific particle size gradation.

Since integrating Eq 14 is very complicated, it can be replaced by a summation for simplicity:

N

usc = usrn + m z {usf(at)[P(q) - flat- 1 ) U I i = 1

where a, (i = 1, 2, -- , N - 1) is the average size of particles between grade i and grade i + 1 when particles are divided into N grades.

3.2.3 Effect of Particle Clustering

Because of inadequate mixing, fabrication process, etc., particles may not be uniformly distributed in the matrix. Some particle-rich regions may be found throughout the matrix. These regions are usually de- fined as particle clusters. Experimental observations and theoretical studies have shown that damage in particulate-filled composites tends to originate preferentially in clustered regions of high volume fractions of particles (22, 25). Again, in a cross section perpendicular to the applied external load, some will be oc- cupied by clustered regions, and the others will be oc- cupied by unclustered regions. Therefore, Eq 16 can be further modified to consider the cluster effect as follows:

where the subscript "c" represents clustered region, the subscript "uc" represents unclustered region, and f, is the cluster ratio of particles in the composite.

The relation between the volume fraction of particles in the clustered region, fc, in the unclustered region, &, and the cluster ratio, f,, as well as the volume fraction of particles in the composite, _f; can be expressed as follows:

(18)

Once the cluster ratio and the volume fraction of particles in the clustered region are determined, the volume fraction of particles in the unclustered region can be obtained through Eq 18.

frfc + (1 -fr)fw = f

4 RESULTS AND DISCUSSION

4.1 Comparison With Experimental Rcaultsi and Papanicolaou and Bakos' Model

When developing their model, Papanicolaou and Bakos (14) compared the model predictions with a number of test results found in the literature. Be- cause this paper is an extension and modification of Papanicolaou and Bakos' model, the same experimental results can be used to validate this present model. In their comparisons, experimental results from glass beads and hollow glass beads filled epoxy are used. The mechanical properties of the raw materials are shown in Table 1. The diameters of the glass particles are 21 pm, 77 pm, 147 pm, and 216 pm, respectively.



Table I. Mechanical Properties of Raw Materials. ~

Materials Young’s modulus (GPa) Bulk modulus (GPa) Poisson’s ratio Tensile strength (MPa)

Glass Epoxy

71 3.53

53.3 4.21

0.279 0.35 88

Since uniform particles are used and no clusters are included, Eq 13 will be used to compare with the test results and Papanicolaou and Bakos’ model predictions.

Before Eq 13 can be used, the two parameters m and n are required. According to the meaning of m and n, they should be related to the particles size and particle volume fraction if other parameters (such as particle surface treatment) remain unchanged. Al- though Papanicolaou and Bakos did not indicate the m and n values they used, it is clearly seen that they changed their m and n values when particle size and volume fraction were changed. If they had not changed their m and n values, all their predictions would have been the same because their model could not consider particle size effect. Through curve fitting, the following m and n values are used in the present paper for comparisons:

m = n = 0.796 - 1.070 X lO-’a + 1.574 X

a2 - 7.746 X a3 -f+f’ (19)

The change of m and n values with particle size, a, and particle volume fraction, _f; is shown in Flg. 4. It is seen that the m and n values decrease as either the particle size or the particle volume fraction increases. This is understandable because smaller particles result in larger surface area and thus larger load can be

0.65

0.60

- 0.55 E II E 0.50

2 0.45 m > C

c m

v

%

u 0.40

0.35

0.3C

0.25

transferred to the particles: while lower particle volume fraction causes less degradation in the matrix and thus larger composite tensile strength.

Using Eq 19, the predicted results from the present model and Papanicolaou and Bakos’ model as well as the experimental results found in the literature (1-3) are shown in Flg. 5a and 5b. It is seen that both the present model and Papanicolaou and Bakos’ model show a reasonable agreement with the test results. In most cases, the present model is closer to the experimental results than Papanicolaou and Bakos’ model. This suggests that the present model is validated by test results and can be used to discuss various parameters affecting on the tensile strength of particulate- filled composites.

4.2 ParamaSn Af€ecting the Tenmile Strength of Particulate-Filled Cornpodtea

The advantages of this present model over Papani- colaou and Bakos’ model are that this model can consider the effect of particle size, particle size distribution, and particle clustering on the tensile strength of particulate-filled composites. Although Fig. 5 has shown the ability of the present model to reflect the particle size effect, it is limited because Eq 19 is also dependent on particle sizes. To demonstrate particle size effect, two cases will be discussed. In the first

f = 0.1 f = 0.3

\ \ \

0 20 40 60 80 100 120 Particle radius a (pm)

Fg. 4. Effect of particle size and uolumefraction on rn and n values.

598 POLYMER COMPOSITES, OCTOBER 2001, Vol. 22, No. 5


100

90

80 a"

70 5

p! ul

60 + v)

50 a) v) K ' 40

- .-

30

20

v T

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Particle volume fraction

(a) Glass beadlepoxy composite

loo 90

h

rn 80 n r, 5 70 0) c !.!! .+I

60 a, v) c - .-

50

40

This paper (a = 38.5 pm)

Reference [141 (a = 38.5 pm) This paper (a = 73.5 pm) Test results (a = 73.5 pm) Reference [14] (a = 73.5 pm)

30 ' f I I I I I I

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Particle volume fraction

(b) Hollow glass beads/epoxy composite Fig. 5a and b. Comparison of the present nuxiel with test results and Papanicolaou and Bakos' model.

case, the particles are perfectly bonded to the matrix and no degradation of the matrix. According to the meaning of rn and n, rn = n = 1 can be used in this

ideal case. In the second case, perfect bond between the particles and the matrix is also used, but the matrix degrades linearly with the inclusion of particles.



a = 10 pm, n = 1- f

_/-- ---______--- I

I I I I I I I

140

130

h m 120 z

5 = 110 !!?

E 1oc z

rn

c u)

Q, - .-

9(

8( 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Particle volume fraction Flg. 6. Effkct of particle sue on the tensile strength of particulate-jilled composites.

Therefore, m = 1 and n = 1 -f. These cases may be representative because perfect bonding can be achieved through particle surface treatment.

Using the composite properties shown in Table 1, the effect of particle size on the tensile strength is shown in Flg. 6. It is seen that the composite tensile strength increases as the particle size decreases. This effect is pronounced for larger particles and is small for smaller particles. This suggests that when the particle size is comparatively large, reducing particle size is very effective in enhancing the tensile strength of the composites; while, if the particle size is already small, further reducing the particle size is less effective in increasing the composite tensile strength. This conclusion holds true for both cases considered in Flg. 6. It is also found that compared with the matrix tensile strength, the incorporation of particles can either increase or decrease the composite tensile strength depending on interfacial bonding, matrix degradation, and particle size. This effect has also been observed by previous experimental tests and theoretical studies (5, 15). However, this effect cannot be predicted using Papanicolaou and Bakos’ model because their model always predicts that the composite tensile strength is lower than the matrix tensile strength. One possible reason is that Papanicolaou and Bakos’ model did not consider the fiber end stress, while it is considered in the present paper, as shown in Flg. 3.

To demonstrate the effect of particle size distribution on the composite tensile strength, three particle size distributions will be used for comparisons. As- suming % = 10 pm and ~ m a x = 100 pm, the three particle size distributions are shown in Rg. 7. They

are obtained from Eq 15 using c = 0.5, c = 0.4, and c = 0.3, respectively. From Fig. 7, it is seen that the content of smaller particles increases as the c values decrease. It has been verified through other studies (12) that c = 0.5 is the best particle gradation because it contains a balance of large particles and small particles so that the small particles can be just enough to fill in the voids among the large particles. In other words, when c = 0.5, a denser composite can be formed than that when c = 0.3 or 0.4 because too many small particles will interfere with each other in these cases.

Assuming m = n = 1 in Eq 16, i.e., perfect interfacial bonding and no degradation of the matrix, the variations of the composite tensile strength with the particle gradations are shown in Fig. 8. It is seen that, except when volume fraction of particles is small (for instance f < 0.15 when c = 0.5), graded particles show much larger composite tensile strength than uniform-sized particles. The effect becomes larger and larger as the particle volume fraction increases. This may be because graded particles can form denser composites than uniform-sized particles and have cooperative actions due to the void filling effect of small particles in the inter-space among larger particles. The denser and more cooperative particles can enhance the load transfer from the matrix to the particles and thus increase the composite tensile strength. The three particle size distributions also show noticeable differences. A larger c value, which corresponds to a denser composite, results in a higher composite tensile strength. Again, the effect becomes larger as particle volume fraction increases and can be neglected for lower particle volume fractions.



120 1 100

h

80 3 0 > % 60 n cn tz 2 40 m 0. S

-

.-

w

Q) 20 2 a 0)

0

c = 0.5 c = 0.4

- c = 0.3

0 20 40 60 80 100 120

Particle radius (pm) Flg. 7. Particle size distributions.

The effect of particle clusters on the composite tensile strength is shown in Fig. 9. When obtaining Rg. 9 using Eqs 17 and 18, two uniform-sized particles are investigated, a = 10 pm and a = 100 pm. The particle volume fraction in the clusters isf, = 0.6. The parameters for interfacial debonding and matrix degradation in the clustered region are m, = n, = 1 -A. In the unclustered region, they are nc = rlC = 1 - f,,.

300

- d 250 z 5 2 200

v

cn c

rn Q)

u) tl 3 150 Q)

rn 0 P

c.

- .-

c1 .-

5 100 0

50

From Fig. 9, the cluster reduces the composite tensile strength considerably when the particle volume fi-ac- tion is small, while it is negligible when the particle volume fraction is large. This is understandable because damage originates preferentially in the clustered region, resulting in decreased composite tensile strength. As the particle volume fraction in the composite increases, the particle volume fraction in the

. . . . . a = 100 prn a = l O p a = 10 - 100 prn, c = 0.5 a = 10 - 100 prn, c = 0.4 a = 10 - 100 prn, c = 0.3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Volume fraction of particles Fg. 8. Effwct of particle gradation on composite tensile strength

POLYMER COMPOSITES, OCTOBER 2001, Yo/. 22, No. 5 601

Gwqiung Li, Jack E. Helms, Su-Seng Pang, and Kurt Schulz

90

h

m a 80

5 r, w c

In Q)

In S

Q)

In 0 a

70 Y

- .- 3 60 .cI .-

50 s 40

- a = 10 pm, f, = 0.1 - a = 10 pm, f , = 0.2 - a = 10 pm, f, = 0.3 -.- a = 100 pm, fr = 0.1 -- a = 100 pm, f,= 0.2 -- a = 100 pm, f, = 0.3

\ \ \ \

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Particle volume fraction Q. 9. Effect of particle clusters on the composite tensile strengtk

unclustered region approaches that in the clustered region, leading to particles uniformly distributed in the matrix. As a result of the uniform distribution of particles in the matrix, the cluster effect becomes less and less.

6 CONCLUSIONS Particulate-filled composites are represented by a

two-layer built-in model based on Christensen and Lo’s three-layer sphere model. Following the examples of Papanicolaou and Bakos’ two-phase composite model, formulations predicting the average tensile stress in the particles are developed. Unlike Papanico- laou and Bakos’ model, however, nonzero stress exists at the fiber (cylinder) end. Using the rule-of-mixtures approach, formulations estimating the composite tensile strength are proposed. To overcome the limitations of Papanicolaou and Bakos’ expression for the composite tensile strength, a modified formulation, Eq 13, is proposed. Particular attention is paid to consider the effect of interfacial debonding and matrix degradation, particle size and particle size distribution, and particle clustering on the composite tensile strength. The composite tensile strength predicted by the present model is compared with Papanicolaou and Bakos’ predictions and those found in the literature. The preliminary comparisons just.@ the model predictions. Based on the calculated results, the following initial conclusions are obtained:

1) The incorporation of particles in a matrix may increase or decrease the matrix tensile strength depending on the interfacial adhesion, matrix degradation, particle size and particle size distribution, and particle clustering.

2) Interfacial debonding and matrix degradation can significantly reduce the cohposite tensile strength. Debonded interface and degraded matrix make a particle act as an inclusion but not as a reinforcement. Increasing interfacial bonding and reducing matrix degradation such as using particle surface treatment are very effective in increasing the composite tensile strength over that of the matrix.

3) For comparatively larger particles, reducing the particle size is very effective in enhancing the composite tensile strength: when the particle size is comparatively small, further reducing the particle size has ody a little effect on increasing the composite tensile strength.

4) Continuously graded particles result in a noticeable increase in composite tensile strength when the particle volume fraction is comparatively large: when the particle volume fraction is small, graded particles produce no advantages over uniform-sized particles.

5) Particle clustering significantly reduces the composite tensile strength when the particle volume fraction is comparatively small: when the particle volume fraction is large, this effect is negligible.

6) If the composite tensile strength is known, this present model can be used to evaluate interfacial debonding and matrix degradation.

7) Systematic experimental investigations, espe- cially about the effect of particle size, particle size distribution, and particle size clustering on



the composite tensile strength, are required to validate the various model predictions. This is a topic for further studies.

ACWOWLEDGMENTS

This investigation was sponsored by the National Science Foundation and the Louisiana Board of Re- gents under Joint Faculty Appointment Program between Louisiana State University and Southern Uni- versity with Contract Number: NSF/LEQSF( 1996-98)- SI JFAP-02.

REFERENCES

1. L. Landon, G. Lewis, and G. Boden, J. Muter. Sci, 12,

2. J. Spanoudakis and R. J. Young, J. Mater. Sci, 19, 473

3. J. Spanoudakis and R. J. Young, J. Muter. Sci, 19,487

4. U. Yilmazer and R. J. Farris, J. Appl. Polyym Sci, 28, 3369 (1983).

5. L. A. Vratsanos and R J. Farris, Polym Eng. Sci, 39, 1466 (1993).

6. H. S. Katz and J. V. Milewski, Handbook of Fillers and Reinforcements for Plastics, 1st Ed., Van Nostrand Rein- hold Company, New York (1978).

7. J. D. Miller, I. Hatsuo, and H. J. M. Frans. J. Mater. Sci, 24,2555 (1989).

8. T. K. Kwel a n d C. A. Kumins. J. Appl. Polym *, 8, 1483 (1964).

1065 (1977).

(1984).

(1984).

9. A. Eshelby. Proc. Roy. Soc., A 241, 376 (1957). 10. Z. Hashin, J.AppL Meck, 29, 143 (1962). 11. R. M. Christensen and H. H. Lo. J . Mech Pfys. Solids,

27, 315 (1979). 12. G. Li, Y. Zhao, S. S. Pang, and Y. Li, Cem C o m e . Res.,

13. S. Ahmed and F. R. Jones, J. Muter. Sci., 26, 4933

14. G. C. Papanicolaou and D. Bakos, J. Rein$ Plast. Corn-

15. L. A. Vratsanos and R J. Farris, PoZym Erg. Sci. 33,

16. J. Jancar, A. DiAnselmo, and A. T. DiBenedetto. Polym

17. Z. H. Liu. J. Li, N. G . Liang. a n d H. Q. Liu, J. ReinJ

18. B. Turcsanyi, B. pukansky, and F. Tudos, J. Muter. Sci

19. J. R. M. DAlmeida and L. H. D. Carvalho, J. Mater. Sci,

20. R. M. Chris tensen, J. Mech. Phys. Solids, 38, 379

21. T. Mori a n d K. Tanaka, Acta Metullurgicu, 21, 571

22. G. Li, Y. Zhao, and S. S . Pang. Mater. Sci Eq., -71,

23. A. Kelly, Strong Solids, Clarendon Press, Oxford, Eng-

24. T. W. Clyne, Mater. Sci Eng., Al22, 183 (1989). 25. P. B. Prangnell, B. J. Barnes, S. M. Roberts, a n d P. J.

Withers, Mater. Sci Eng., A220.41 (1996).

29, 1455 (1999).

( 1990) *

pos., 11, 104 (1992).

1458 (1993).

Eng. Sci , 32, 1394 (1 992).

P k t . Compos., 16, 1523 (1997).

Lett., 7, 160 (1988).

55,2215 (1998).

( 1990).

(1973).

43 (1999).

land (1966).


analytical modeling of tensile strength of particulate-filled composites

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