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Simple formulae for the effective moduli of unidirectional aligned composites.

INTRODUCTION

There exists an extensive literature on estimating the effective moduli of unidirectionally aligned composites. For example, rigorous bounds for the effective moduli of fiber-reinforced composites using the variational principle of elasticity was obtained by Hashin and Rosen (1), Hill (2) and Hashin (3). Estimates of effective moduli can also be obtained using the serf-consistent method of Hill (4), Budiansky (5), Christen and Lo (6) and Hermans (7). This method was used by Laws and McLaughlin (8) and Chou, et al (9) to obtain estimates for short fiber composites. Details for bounding methods and the self-consistent model can be found in Willis (10), Walpole (11), Mura (12), Nemat-Nasser (13), Christensen (14). More recently, the Eshelby-Mori-Tanaka method (15-17) was used to obtain estimates of the composite moduli (18-21). This method combines Eshelby's solution of an ellipsoidal inclusion and Mori-Tanaka's mean field theory to deal with the interaction effects of a finite concentration of reinforcing phase. In this study, we are concerned with unidirectionally aligned composites, so we omit referencing studies in the areas such as laminates, and composites reinforced by a random distribution of inclusions. Empirical equations for estimating the effective moduli of unidirectionally aligned short fiber composites were developed by Halpin, Tsai and Kardos (22-24). Similar empirical equations were developed by Nielsen (25), Padawer & Beecher (26) and Riley (27) for flake-like inclusions. These equations are easier to apply than those obtained using the more rigorous methods mentioned above. A review of the different approximate and analytical methods for calculating the effective moduli can be found in Chamis and Sendeckyl (28).

The main purpose of this paper is to develop simple engineering approximations for the effective tensile modulus of unidirectional aligned two phase composites. The developed expressions will be valid for both flake-like and fiber-like inclusions for the entire range of modulus of both phases. We start with the exact closed form solution of Tandon and Weng (20). They obtained the effective moduli of an unidirectionally aligned two phase composite reinforced by spheroidal inclusions with arbitrary aspect ratio [Alpha] = t/a [ILLUSTRATION FOR FIGURE 1 OMITTED] using the Mori-Tanaka method (17). Their solution is exact so that they are valid for the entire range of modulus of both phases. The resulting composite is transversely isotropic with five independent elastic constants: [E.sub.11], the longitudinal Young's modulus, [E.sub.22], the transverse Young's modulus, [[MU].sub.12], the in-plane shear modulus, [[Mu].sub.23], the out-of-plane shear modulus and [K.sub.23], the out-of-plane strain bulk modulus. For unidirectionally aligned composites reinforced by fiber-like inclusions Eli is the modulus of practical engineering interest, whereas for composites reinforced by flake-like inclusions [E.sub.22] is the modulus of practical engineering interest.

The exact solution of Tandon and Weng (20), however, is rarely used in practice, perhaps due to its complexity. For example, the determination of [E.sub.11] [ILLUSTRATION FOR FIGURE 1 OMITTED] requires the manipulation of 21 equations that fill about one journal page. In practice, composite moduli are often estimated using simple semi-empirical rules such as the Halpin-Tsai equations (22, 25) (for fiber-like or flake-like inclusions) and the Modified Rule of Mixture equation (for flake-like inclusions) (26, 27). The main limitation of these semi-empirical rules is that they sometimes break down as demonstrated by the following example.

Consider the elastomeric nanocomposites system investigated by Burnside and Giannelis (29). The elastomeric matrix in (29) is reinforced by a very low concentration of ceramic flat discs or flake-like inclusions with very small aspect ratio [Alpha] = t/a [approximately equal to] [10.sup.-3], where t and a denote the thickness and the in-plane dimension of the flakes, respectively. For the above material system, the ratio of the matrix's modulus to the inclusions' modulus is [E.sub.0]/[E.sub.1] = 5.9 x [10.sup.-5]. Let E = [E.sub.22] be the transverse composite tensile modulus [ILLUSTRATION FOR FIGURE 1 OMITTED], i.e., the effective tensile modulus in the plane of the flakes. The simple rule of mixtures relation (ROM), E = c[E.sub.1] + (1 - c)[E.sub.0], predicts

E/[E.sub.0] [congruent] [10.sup.3] (ROM) (1)

whereas the Halpin-Tsai equation for flakes (25) predicts

E/[E.sub.0] [congruent] 2.2 (HT) (2)

There is a three order of magnitude difference between these predictions. This large discrepancy is due to the extremely large moduli difference between the flakes and the elastomeric matrix so that even with an aspect ratio of [10.sup.-3], the flakes cannot be considered as infinite in the loading direction, which is in the plane of the flakes. The Halpin-Tsai equation we used to obtain (2) is given in (25)

E/[E.sub.0] = 1 + [Lambda]Bc/1 - [Psi]Bc (3a)

where

B = ([E.sub.1]/[E.sub.0]) - 1/([E.sub.1]/[E.sub.0]) + [Omega], [Omega] = [1.33[Alpha].sup.-0.645] (3b)

The constant 0 [less than or equal to] [Psi] [less than or equal to] 1 depends on the packing density of the flakes and is assumed to be 0.66 (25). However, reference (25) gave no details on how the expressions (33, b) were obtained. It should be noted that, due to the low concentration and the extreme moduli difference between the matrix and the inclusions, the result given in (2) is practically independent of the value of [Psi]. The composite modulus E can also be estimated by using the Modified Rule of Mixtures relation (MROM)

E = c[E.sub.1](MRF) + (1 - c)[E.sub.0] (4a)

where (MRF) stands for the Modulus Reduction Factor. The Modulus Reductor Factor for flakes given by Padawer & Beecher (26) is

(MRF) = (1 - tanhu/u) u = 1/[Alpha] [-square root of C[G.sub.0]/[E.sub.1](1-c) (4b)

where [G.sub.0] is the shear modulus of the matrix. A different Modulus Reduction Factor is proposed by Riley for flakes (27)

(MRF) = 1 -1n(u + 1)/u (4c)

The composite modulus using the Riley's rule (MROMR) is found to be

E/[E.sub.0] [congruent] 87.8 (5a)

whereas Padawer & Beecher's rule (MROM-PB) gives

E/[E.sub.0] [congruent] 13.6 (5b)

where we have assumed that the Poisson's ratio of the matrix equals 0.3 in both cases. Note that for this particular example the four results for the composite modulus axe widely scattered. Therefore, it is quite difficult to obtain, according to these semi-empirical relations, an accurate estimate. This example clearly shows that the various empirical rules commonly used in the literature can lead to very large errors for some material systems. In the above example, the volume fraction of the inclusions ranges from about 0.0 to 0.1. in such low concentrations of inclusions, the effects of interaction should be well predicted by the Mori-Tanaka method.

For the above example, the exact solution of Tandon and Wang (20) predicts

E/[E.sub.0] [congruent] 7.8 (6)

where the Poisson's ratio of the flakes and matrix [v.sub.1], [v.sub.0] are assumed to be 0.3 and 0.5 respectively. Finally, we note that E is quite insensitive to the Poisson ratio. For instance, if the matrix and the flakes are assumed to be incompressible, i.e., [v.sub.1] = [v.sub.o] = 0.5, E/[E.sub.0] [congruent] 7.9, which is not very different from (6).

When the two phases have different Poisson's ratio, the expressions for [E.sub.11] and [E.sub.22] are extremely complicated. Fortunately, we found that when the Poisson's ratio of the matrix and the inclusions are identical, the results of Tandon and Weng (20) simplify considerably. Further simplification can be obtained by considering the special case of v = 0.5, where v denotes the Poisson's ratio of the matrix and the inclusion. This assumption will be justified by comparing numerical results, which is based on the exact expressions of Tandon and Weng (20), to results obtained from our simplified expressions. It should be noted that for fiber-like inclusions, i.e. [Alpha] [greater than] 4, both [E.sub.11] and [E.sub.22] are insensitive to the Poisson's ratio. Meanwhile, for a composite reinforced with flake-like inclusions, i.e. [Alpha] [less than] 1, our numerical results showed that [E.sub.22] is also insensitive to the Poisson's ratio.

When is the aspect ratio for fiber-like inclusions (or flakes) large enough (or small enough) so that the rule of mixtures applies? In general, the convergence to the rule of mixtures depends on the modulus ratio [E.sub.0]/[E.sub.1] as well as whether the inclusions are fiber or flake like. It turns out that the criteria for convergence to the rule of mixtures are very simple and are given by Eqs 11 and 12 below. Furthermore, for a given modulus ratio [E.sub.0]/[E.sub.1], the convergence to the rule of mixture is much faster for fiber-like inclusions than for flake-like inclusions. Specifically, the convergence rate to the rule of mixtures for fiber-like inclusions (i.e., [Alpha] [much greater than] 1) is of order [Alpha].sup.-2], whereas the convergence rate for flakes (i.e., [Alpha] [much less than] 1) is of order [Alpha].

The notation of Tandon and Weng (20) is followed throughout this work. Both phases are assmed to be isotropic and linearly elastic with ([E.sub.1], [v.sub.1]) and ([E.sub.0], [v.sub.0]) being the modulus and the Poisson's ratio of the inclusion and the matrix, respectively. The Poisson's ratio of both phases are assumed to be identical and takes on the value of v = 0.5. Of the five independent elastic constants, we only consider the two effective tensile moduli [E.sub.11] and [E.sub.22] because of their practical importance. Furthermore, the effective shear moduli [[Mu].sub.12] and [[Mu].sub.23] have relatively simple expressions even for the case of different Poisson's ratios (reference 20, Eqs 31 and 32). The volume fraction of inclusions is denoted by c.

I Effective Tensile Moduli

Since the derivation of the formulae below is straightforward but extremely tedious, our procedures are summarized in the Appendix.

All moduli depend on a geometrical parameter g, which depends on the aspect ratio [Alpha] of the inclusions. The geometrical parameter g is given by:

[Mathematical Expression Omitted] 7

It can be shown that g([Alpha] = 1) = 2/3. The case of [Alpha] [much greater than] 1 corresponds to fiber-like inclusions whereas [Alpha] [much less than] 1 corresponds to flake-like or disc-shaped inclusions. In these limiting cases, g can be approximated by

[Mathematical Expression Omitted] (8)

The error of these approximations are denoted by the O symbol. For example, if [Alpha] = 10, the error made by using (8) is about [10.sup.-4]. For most engineering purposes, (8) can be used if [Alpha] [greater than or equal to] 4 for fiber-like inclusions or [Alpha] [less than or equal to] 0.1 for flake-like inclusions.

Longitudinal Tensile Module [E.sub.11]

[Mathematical Expression Omitted] (9a)

where

[Mathematical Expression Omitted] (9b)

Transverse Tensile Modulus [E.sub.22]

[Mathematical Expression Omitted] (10a)

[Mathematical Expression Omitted] (10b)

The parameter g in (9a, b) and (10a, b) is given by (7), whereas [Epsilon] in (10a) is defined by (9b).

Comparison With the Tandon and Weng

Equations 9 and 10 are obtained assuming that [v.sub.1] = [v.sub.0] = v = 1/2. the error of this approximation is examined by comparing the predictions of (9) and (10) with the solution of Tandon and Weng for cases where [v.sub.1] and [v.sub.0] differ from 1/2. The results (i.e., [E.sub.11] and [E.sub.22]) for the case of fiber-like inclusions are shown in Figs. 2a, b and 3a, b for two different sets of materials and four different aspect ratios. The material properties used in these plots are those of glass/epoxy and elastomeric/ceramic, such that:

[E.sub.0] = 2.76GPa,, [v.sub.0] = 0.35, [E.sub.1] = 72.4GPa, [v.sub.1] = 0.20, glass/epoxy

[E.sub.0] = 1.0MPa,, [v.sub.0] = 0.5, [E.sub.1] = 170GPa, [v.sub.1] = 0.3, ceramic/elastomer

These figures clearly show that for fiber-like inclusions (i.e.,[Alpha] [greater than or equal to] 10), [E.sub.11] and [E.sub.22] given by (9a, b) and (10a, b) are excellent approximations of the exact solutions of Tandon and Weng (20). The excellent agreement between (9) and (10) and the exact expressions of Tandon and Weng (20) provides a justification for setting the Poisson's ratios to 1/2 in deriving (9) and (10).

The same comparisons are given by Figs. 4a, b for the case of flakes. These figures clearly show that for flake-like inclusions (i.e., [Alpha] [less than or equal to] 0.1), [E.sub.22] given by (10a, b) is an excellent approximation of the exact solutions of Tandon and Weng (20) for different Poisson's ratios. It should be noted that for the case of flakes, [E.sub.11] given by (9a, b) is not a good approximation for composites with a different Poisson's ratio for the matrix and the flake-like inclusions.

Reduction to Simple Rule of Mixtures and Error Estimate

The rule of mixtures for fiber like inclusions and flakes can be obtained by taking the formal limit of [Alpha] [approaches] [infinity] in Eq 9a and [Alpha] [approaches] 0 in Eq 10a, respectively. For example, for fiber-like inclusions, [Mathematical Expression Omitted] as [Alpha] [approaches] [infinity] by (8) so that [Xi] [approaches] c + [E.sub.0]/[E.sub.1] - [E.sub.0], i.e.,

[Mathematical Expression Omitted]

which is just the rule of mixtures relation.

Error Estimate for Fiber-Like Inclusions

For the case of fiber-like inclusions, examination of (9a, b) shows that the error introduced using the rule of mixtures relation is small as long as the condition

[Mathematical Expression Omitted]

is satisfied. Equation 11 shows that predictions based on the rule of mixtures can lead to very large errors if the inclusions are much stiffer than the matrix. For example, (11) is not satisfied for [Alpha] = 50 if [E.sub.1]/[E.sub.0] = 1.7 x [10.sup.5] (i.e., a ceramic fiber in an elastomeric matrix). This is shown in Fig. 2b, where the rule of mixtures relation is only approached for [Alpha] = 2000. This is in agreement with (11), which implies that the rule of mixtures for fiber-like inclusions is an excellent approximation provided that the aspect ratio [Alpha] [greater than or equal to] [10.sup.3]. Equation 11 also shows that the error of using the rule of mixtures relation decreases as [[Alpha].sup.-2] for fiber-like inclusions ([Alpha] [much greater than] 1).

Error Estimate for Flakes

Similarly, one can show that the rule of mixtures relation for flakes, i.e. [E.sub.22]/[E.sub.0] = (1 - c) + c [E.sub.1]/[E.sub.0], is valid as long as the condition

[E.sub.0]/[E.sub.1] - [E.sub.0] [much greater than] [Alpha] (12)

is satisfied. Note that in this case the error is proportional to [Alpha] ([Alpha] [much less than] 1 for flakes). This means that the rule of mixtures relation works much better for fiber-like inclusions than for flakes. The example we used in the introduction clearly illustrated this point. Indeed, if [E.sub.1]/[E.sub.0] = 1.7 x [10.sup.5], (12) indicates that the rule of mixtures relation is valid only if the aspect ratio satisfies the condition [Alpha] [less than or equal to] [10.sup.-6]! This result is confirmed in Fig. 4b for the case of [Alpha] = [10.sup.-6].

Comparison With Halpin-Tsai Equation, Modified Rule of Mixtures Relation

Fiber-Like Inclusions

The comparison of our solution with Halpin-Tsai relation (22) for different aspect ratios are shown in Figs. 5a, b and Fig. 6a, b for [E.sub.11] and [E.sub.22], respectively. The material properties used in these plots are identical to those used in Figs. 2-4.

Figure 5a shows that there is no significant difference between our prediction for [E.sub.11] and Halpin-Tsai relation for the glass/epoxy system. For the elastomer/ceramic system, the difference between the two predictions is extremely large for very large aspect ratios, as shown in Fig. 5b.

Figures 6a and b show that there is no significant difference between the Halpin-Tsai and our predictions for [E.sub.22] for both the glass/epoxy system and the elastomer/ceramic system.

Flake-Like Inclusions

The comparison of our predictions for [E.sub.22] with the predictions of the Halpin-Tsai relation (25) and the modified rule of mixtures relations (26, 27) for different aspect ratios are shown in Figs. 7a, b. For the glass/epoxy system, Fig. 7b shows that there is no difference between our predictions and those of the modified rule of mixtures when the aspect ratios is sufficiently small i.e., [Alpha] [less than or equal to] 0.01. The modified rule of mixtures relation starts to break down when the aspect ratio reaches 0.1, as shown in Fig. 7a. This is also evident from the fact that the two different forms of modified rule of mixtures start to deviate significantly from one another. Contrary to the two forms of modified rule of mixtures, the Halpin-Tsai relation works better for larger values of aspect ratios than for smaller values of aspect ratios. This is shown in Figs. 7a, b. For the elastomer/ceramic system, Fig. 8a shows that there is a very large discrepancy between the predictions of (10a) and those of the two modified rule of mixtures relations, when the modulus ratio [E.sub.0]/[E.sub.1] is sufficiently small compared with the aspect ratio of the flakes. However, the agreement becomes better as the flakes become thinner as shown in Fig. 8b. From the results shown in Figs. 8a, b, one sees that the Halpin-Tsai relation for flakes fails to predict [E.sub.22] accurately.

The experimental results obtained by Lusis et al. (30) are compared in Figs. 9a and 9b with the modified rule of mixtures relations and (10b) of this work. In Fig. 9a, predictions given by (10a), Halpin-Tsai relation, and modified rule of mixtures relation of Riley all seem to compare favorably with the experimental results, except for the modified rule of mixtures relation suggested by Padawer and Beecher that underpredicts the experimental results. According to Fig. 9b, the predictions given by (10a) and Riley's modified rule of mixtures relation compare relatively well with the experimental results. The Halpin-Tsai relation and the modified rule of mixtures relation of Padawer and Beecher overpredict the experimental results, as shown in Fig. 9b.

CONCLUSIONS

In this work we have developed simplified expressions for the effective tensile moduli of unidirectionally aligned two-phase composites reinforced by fiber-like or flake-like inclusions. The expressions are derived from Tandon and Weng's exact solution by making the assumptions that the Poisson's ratios of the inclusions and the matrix are the same and are equal 1/2. No other assumptions are made in the derivation, hence the resulting expressions are valid for the entire range of the modulus of the two phases; this is unlike most of the semi-empirical relations that are only valid for a certain range of modulus and aspect ratios. Also, simple criteria for the validity of the rule of mixtures relation are obtained for both fiber-like and flake-like inclusions.

The resulting simplified expressions compared well with the exact solutions of Tandon and Weng for inclusions and matrices with different Poisson's ratios, except for [E.sub.11] in the case of flake-like inclusions. The reason that the simplified expression for [E.sub.11] did not work well in the case of flake-like inclusions was because it neglected the difference in the Poisson's ratio between the matrix and the inclusions.

For fiber-like inclusions, it is found that in some regimes of modulus and aspect ratios the Halpin-Tsai compared well with the simplified expressions for [E.sub.11]. In particular, when the difference between the modulus ratio and the aspect ratio is not too large, the agreement between our expressions and Halpin-Tsai's is good. However, when the difference between the two ratios is large, the agreement is poor. For [E.sub.22], there is no significant difference in results between the simplified expression and the Halpin-Tsai relation for the range of modulus and aspect ratios considered.

For composites with flake-like inclusions and a relatively moderate modulus ratio ([E.sub.1]/[E.sub.0] not too much greater than 1), it is found that the two modified rule of mixtures relations proposed by Riley and Padawer and Beecher for [E.sub.22] agreed more closely with our simplified expressions, for very small aspect ratios. The Halpin-Tsai relation does not seem to work well at all for flakes, although it agrees more closely with our simplified expressions as the aspect ratios of the flakes become larger. However, when the modulus ratio becomes much larger none of the semi-empirical relations agree with the simplified expression (10a).

Our numerical results showed that the simplification made by setting the Poisson's ratio equal to 1/2 for both the inclusions and the matrix had no practical effect on the effective moduli (with the exception of the bulk modulus), as long as the aspect ratio, [Alpha], of the inclusion differs significantly from one. Numerically, this condition can be expressed by the requirement [Alpha] [greater than] 5 or [Alpha] [less than] 0.1.

In conclusion, the expressions (9a) and (10a) obtained in this paper are very simple to use for the predictions of tensile moduli of unidirectionally aligned two-phase composites reinforced by fiber-like or flake-like inclusions. These expressions are accurate for the full range of modulus of the two phases (matrix and inclusions), since they are based on the exact solution of Tandon and Weng.

APPENDIX

When the Poisson's ratio of the flakes and matrix [[Nu].sub.1] = [[Nu].sub.0] = [Nu] are identical, the material constants [D.sub.1], [D.sub.2], [D.sub.3] defined by (20) reduce to:

[D.sub.1] = 1 - [Nu]/[Nu], [D.sub.3] = [E.sub.0]/[E.sub.1] - [E.sub.0], [D.sub.2] = [D.sub.1][D.sub.3] (A-1)

Using (A1) and after considerable algebra, the material constants [A.sub.1].....[A.sub.5] in (20) can be expressed in terms of the Eshelby tensor [S.sub.ijkl] given in (20)

[A.sub.1] = ([D.sub.1] + 1)([D.sub.1] + 2)[c + [D.sub.3] + (1 - c)([S.sub.2222] + [S.sub.2233])]

[A.sub.2] = (1 - c)([D.sub.1] - 1)([D.sub.1] + 2)[S.sub.1122]

[A.sub.3] = -(1 - c)([D.sub.1] - 1)([D.sub.1] + 2)[S.sub.2211]

[A.sub.4] = ([D.sub.1] - 1)([D.sub.1] + 2)[c + [D.sub.3] + (1 - c)[S.sub.1111]]

[A.sub.5] = -1/[c + [D.sub.3] + (1 - c)([S.sub.2222] - [S.sub.2233])] (A-2)

The constant A is found to be A = -([D.sub.1] - 1)[[Lambda].sub.2] where

[[Lambda].sub.2] = 2[B.sub.2][c + [D.sub.3] + (1 - c)([S.sub.1111] - [S.sub.2211])] +

[B.sub.1][c + [D.sub.3] + (1 - c)([S.sub.2222] - 2[S.sub.1122] + [S.sub.2233])]

[B.sub.1] = [D.sub.1](c + [D.sub.3]) + (1 - c)([D.sub.1][S.sub.1111] + 2[S.sub.2211])

[B.sub.2] = c + [D.sub.3] + (1 - c)([D.sub.1][S.sub.1122] + [S.sub.2222] + [S.sub.2233]) (A-3)

The above expressions are exact since no approximation was made besides the assumption [[Nu].sub.1] = [[Nu].sub.0] = [Nu]. The relevant Eshelby tensor [S.sub.ijkl] simplify by setting [Nu] = 1/2, i.e.,

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted] (A-4)

Substitute (A-4) into (A-3) and (A-2) and allows us to express [A.sub.1].....[A.sub.5] and A in terms of g and [Alpha]. At this point one must be careful not to set [Nu] = 1/2 in [A.sub.1].....[A.sub.5] since when [D.sub.1] = 1 [A.sub.2], [A.sub.3], [A.sub.4] and A become identical zero and the modulus [E.sub.11] and [E.sub.22] become indeterminate. Instead, we substitute [A.sub.1].....[A.sub.5] and A into (25), (28) in (20) and then taking the limit as [D.sub.1] [approaches] 1. For example, (28) in (20) is

[Mathematical Expression Omitted]

ACKNOWLEDGMENTS

C. Y. Hui was supported by the AFOSR Grant F49620-93-1-0308. Partial support by Cornell's Material Science Center, which is funded by the National Science Foundation (DMR-MRL) program, in terms of a summer research assistantship for David Shia is also gratefully acknowledged. The authors would also like to thank Prof. Lars Berglund for sharing his insightful comments with us on the contents of this paper, as well as the suggestion of an anonymous reviewer. Experimental data on nanocomposites provided by Shelly Burnside and Prof. Giannelis are much appreciated.

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Author:Hui, C.Y.; Shia, David
Publication:Polymer Engineering and Science
Date:May 1, 1998
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