Transverse moduli of continuous-fibre-reinforced polymers

Composites Science and Technology 60 (2000) 997±1002

Transverse moduli of continuous-®bre-reinforced polymers Alfredo Balaco de Morais * Mechanical Engineering Department, University of Aveiro, Campus Santiago, 3810 Aveiro, Portugal Received 11 April 1999; received in revised form 28 October 1999; accepted 7 December 1999

Abstract This paper presents a closed-form micromechanical equation for predicting the transverse modulus, E2, of continuous-®brereinforced polymers. The equation was derived from a relatively simple mechanics-of-materials analysis of a repeating square cell. The assumptions made concerning the stress distributions and the typical matrix and ®bre properties enabled the derivation of a closed-form expression. The present equation is in good agreement with a 3D ®nite-element hexagonal cell model and with Aboudi's method-of-cells equation, both complex approaches known to provide accurate predictions. Furthermore, combining the equation developed with other well-known equations, enables the calculation of the 23 transverse Poisson ratio, thus completing the set of layer elastic constants needed for 3D stress analyses. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Continuous-®bre composites; Transverse modulus; Micromechanical models; Finite elements; Cell models

1. Introduction The layer moduli are the basic inputs for stress analysis of composite laminates. As a consequence of the layer anisotropy, experimental measurement requires signi®cant testing e€ort, and micromechanical prediction is therefore of considerable interest. Since the unidirectional composite layer is transversely isotropic, ®ve elastic constants are required fully to characterise its stress/strain behaviour: the E1 longitudinal modulus, the E2 transverse modulus, the G12 longitudinal shear modulus, the 12 longitudinal Poisson ratio and the 23 transverse Poisson ratio [or, alternatively, the G23 transverse shear modulus, as G23 ˆ E2 =2…1 ‡ 23 †]. The simple rule of mixtures P ˆ Vf Pf ‡ …1 ÿ Vf †Pm

…1†

provides accurate predictions for the longitudinal modulus and Poisson ratio, i.e. for P ˆ E1 and for P ˆ 12 [1,2]. The G12 shear modulus can be accurately predicted by the self-consistent model [1±4] equation G12 ˆ Gm

Gf 12 ‡ Gm ‡ Vf …Gf 12 ÿ Gm † Gf 12 ‡ Gm ÿ Vf …Gf 12 ÿ Gm †

* Tel.: +351-234-370-830; fax: +351-234-370-953. E-mail address: [email protected] (A.B. Morais).

…2†

Simple and accurate equations are not available for the other elastic constants. The self-consistent model [1± 4] is only able to give accurate predictions for the planestrain bulk modulus k2 ˆ

…kf 2 ‡ Gm †km ‡ Vf Gm …kf 2 ÿ km † kf 2 ‡ Gm ÿ Vf …kf 2 ÿ km †

…3†

which is not a commonly used elastic constant. If, however, one is able to predict the modulus E2 , it will be immediately possible to obtain the remaining elastic constant, since  2  212 1 ‡ …4† 23 ˆ 1 ÿ E2 2k2 E1 The prediction of E2 is, therefore of particular interest. Elaborate analyses of a unit cell have been presented [5±7]. Luciano and Barbero [5] have obtained a lengthy closed-form expression for composites with isotropic ®bres. Aboudi [6] developed the `method of cells', a formulation based on the analysis of a square cell. The cell consists of four square subcells, one of which represents the ®bre. An averaging procedure ensures transverse isotropy. The analysis results in a set of complex equations, which must be handled by a computer programme.

0266-3538/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(99)00195-5

998

A.B. de Morais / Composites Science and Technology 60 (2000) 997±1002

The complexity does however seem to pay in terms of agreement with experimental data [6,7]. Greater accuracy can obviously be achieved with ®nite element models [8±11]. The models usually involve a periodic cell, e.g. square and hexagonal, although ®bre arrays have also been considered. Using 2D ®nite-element models, Brockenbrough et al. [9] showed that the results obtained from hexagonal cells are in close agreement with those obtained from random arrays. The hexagonal cell therefore seems to provide a more accurate average representation of the actual composite layer, as one would intuitively expect from its intrinsically uniform ®bre spacing. In spite of the analytical accuracy of Aboudi's method of cells and of the ®nite-element-based models, some uncertainties remain, concerning, for instance, the properties of the constituents. In fact, it is usually assumed that the in situ properties of the matrix are identical to those obtained from bulk specimens. Moreover, some of the properties of anisotropic ®bres are not available by experimental measurement, but from ®tting micromechanical models to composite test data. Simple equations such as (1)±(3) are, therefore considerably more interesting if sucient accuracy for practical purposes is achieved. Semi-empirical approaches, such as the well-known Halpin±Tsai equations, are limited by the need of a data-®tting empirical parameter [1,2]. In this paper, a closed-form equation is developed for E2 from a relatively simple mechanics-of-materials model. The equation is compared with a ®nite-element hexagonal-cell model and with Aboudi's method-of-cells equation, known to be in good agreement with experimental data. It is shown that the present equation o€ers a good compromise between accuracy and simplicity. 2. Model description The present model retains the square cell presented by Aboudi [6]. The cell is divided into four subcells (Fig. 1), where the stresses and strains are assumed uniform. The cell will be subjected to a load in the 2-axis direction. Neglecting shear stresses, the equilibrium equations are f 3 ˆ b3

…5†

a3 ˆ c3

…6†

hf b3 ‡ hm c3 ˆ 0

…7†

f 2 ˆ a2

…8†

b2 ˆ c2

…9†

h2f f 1 ‡ hf hm a1 ‡ hf hm b1 ‡ h2m c1 ˆ 0

…10†

Fig. 1. The square cell and its subcells.

The "2 and "1 strains must be uniform, i.e. hf "f2 ‡ hm "a2 ˆ …hf ‡ hm †"2

…11†

hf "b2 ‡ hm "c2 ˆ …hf ‡ hm †"2

…12†

"f 1 ˆ "a1

…13†

"a1 ˆ "b1

…14†

"b1 ˆ "c1

…15†

The strains are related to the stresses by Hooke's law for a transversely isotropic solid "i ˆ

1 …i ÿ ij j ÿ ik k † Ei

…16†

We thus have 11 equations for 12 unknown subcell stresses. The missing equation is closely related to the ®bre spatial distribution which the cell is supposed to represent. In case of a square distribution (Fig. 2a), we would have to impose uniform "3 , i.e. hf "f 3 ‡ hm "b3 ˆ hf "a3 ‡ hm "c3

…17†

The solution of the above set of equations is quite complex. Furthermore, a subsequent averaging procedure would be needed [6] to obtain the modulus of a transversely isotropic composite, and so would similar analyses for predicting the G23 shear modulus and the 23 Poisson ratio. Instead, we consider the ®bre distribution shown in Fig. 2b, where, clearly, an accurate analysis would require more subcells. Keeping however the same subcells, we must assume plane stress in the 3axis direction. We can then write f 3 ˆ a3 ˆ b3 ˆ c3 ˆ 0

…18†

A.B. de Morais / Composites Science and Technology 60 (2000) 997±1002

999

Fig. 2. Fibre spatial distributions for analysis purposes.

This highly simplifying assumption may be expected to fail in the following cases: the ®bre content is very high; the ®bre transverse modulus is much higher than the matrix modulus; there is a large transverse Poisson ratio mismatch between ®bre and matrix. Using (16) and (17), we can rewrite (13)±(15) as 1 f 12 1 m f 1 ÿ f 2 ˆ a1 ÿ a2 Ef 1 Em Ef 1 Em

…19†

1 m 1 m a1 ÿ a2 ˆ b1 ÿ b2 Em Em Em Em

…20†

1 m 1 m b1 ÿ b2 ˆ c1 ÿ c2 Em Em Em Em

…21†

where matrix isotropy is implicit. In view of (8) and (9), (19) to (21) become   1 1 f 12 m f 1 ÿ a1 ˆ ÿ a2 Ef 1 Em Ef 1 Em

…22†

a1 ÿ b1 ˆ m …a2 ÿ c2 †

…23†

b1 ˆ c1

…24†

h2f

…hf ‡ hm †2

Eq. (10) can be rewritten as

We can then solve (22)±(24) and (26) to obtain   p m ÿ
f 12  ‡ Vf ÿ Vf a2 Af 1 ˆ 1 ÿ Vf Ef 1 Em    p m ‡ Vf ÿ 1 c2 Em   p m f 12 m Vf ÿ Vf ÿ a2 Ef 1 Ef 1 Em    p m ‡ Vf ÿ 1 c2 Ef 1

Aa1 ˆ



…26†

…27†

1ÿ

  m ÿ f 12 p m ÿ Vf a2 Ab1 ˆ Ac1 ˆ Vf Ef 1 Ef 1    m p m ‡ Vf ‡ Vf ÿ Vf c2 Em Ef 1

…28†

…29†

where

Since the ®bre-volume fraction is Vf ˆ

p  p  Vf ÿ Vf a1 ‡ Vf ÿ Vf b1  p ‡ 1 ‡ Vf ÿ 2 Vf c1 ˆ 0

Vf f 1 ‡

…25†

Aˆ

1 ÿ Vf Vf ‡ Ef 1 Em

…30†

Assuming Ef 1 >> Em , some of the terms of the above equations can be neglected, leading to " #" # 1 c2 …31† f1  m 1 ÿ p a2 ‡ p Vf Vf a1  m a2

…32†

1000

A.B. de Morais / Composites Science and Technology 60 (2000) 997±1002

b1 ˆ c1  m c2

…33†

We can now go back to (11) and (12), which can be rewritten using (16), (18) and (25)    p 1 p f 12 f 2 ÿ f 1 ‡ 1 ÿ Vf Vf Ef 2 Ef 1 …34†   1 m a2 ÿ a1 ˆ "2 Em Em    p 1 p m b2 ÿ b1 ‡ 1 ÿ Vf Vf Em Em   1 m c2 ÿ c1 ˆ "2 Em Em

…35†

After substituting (8), (9) and (31)±(33), one can obtain "2 a2  p  p 1 ÿ 2m Vf ‡ 1 ÿ Vf Em Ef 2

…36†

Em "2 1 ÿ 2m

c2 

…37†

where, as previously, small terms were neglected. The E2 modulus is ®nally given by ÿ p p
Vf a2 ‡ 1 ÿ Vf c2 E2 ˆ "2 p  p Em Vf ˆ p  ‡ 1 ÿ Vf  1 ÿ 2m p 1 ÿ 2m Vf ‡ 1 ÿ Vf Em Ef 2 …38†

Table 2 Errors (%) of the present (Pe) and of the Aboudi's equation [6] (Ab) relative to the ®nite-element cell model (see Table 1 for assumed ®bre properties)

Fig. 3. The ®nite-element hexagonal-cell model. Table 1 Assumed ®bre properties used for assessment of the developed equation Fibre

Moduli (GPa) Ef 1

Ef 2

Poisson ratios Ff 12

Gf 23

f 12

Remarks

f 23

G

75

0.20

Isotropic

B

400

0.20

Isotropic

CS1 CS2 CS3

230

50.0 30.0 20.0

20.0 13.0 13.0

20.0 13.0 8.0

0.20 0.20 0.20

0.25 0.25 0.25

CM1 CM2

600

13.0 9.0

13.0 7.0

5.2 3.6

0.20 0.20

0.25 0.25

Matrix

G ®bre

m

PE

Em Vf

B ®bre Ab

Pe 1.1 ÿ1.0 ÿ6.2 1.2 ÿ0.8 ÿ5.7 1.3 ÿ0.6 ÿ5.3

Ab

CS1 ®bre

CS2 ®bre

Pe

Pe

Ab

Ab

0.350 2.0 2.0 2.0 3.5 3.5 3.5 5.0 5.0 5.0

0.55 0.65 0.75 0.55 0.65 0.75 0.55 0.65 0.75

1.6 0.0 ÿ3.7 2.1 0.7 ÿ2.2 2.4 1.2 ÿ1.0

0.3 ÿ0.8 ÿ4.3 ÿ0.1 ÿ1.2 ÿ4.1 ÿ0.4 ÿ1.5 ÿ3.8

1.0 0.0 ÿ4.3 0.8 ÿ0.2 ÿ4.3 0.7 ÿ0.3 ÿ4.3

1.1 ÿ0.2 ÿ3.1 1.2 0.2 ÿ1.6 1.2 0.5 ÿ0.8

0.0 1.0 ÿ0.5 ÿ1.1 0.0 ÿ1.5 ÿ4.2 ÿ1.9 ÿ3.8 ÿ0.5 1.0 ÿ0.9 ÿ1.5 0.4 ÿ1.7 ÿ3.8 ÿ0.6 ÿ3.1 ÿ0.8 0.9 ÿ1.0 ÿ1.7 0.5 ÿ1.6 ÿ3.4 0.0 ÿ2.4

0.375 2.0 2.0 2.0 3.5 3.5 3.5 5.0 5.0 5.0

0.55 0.65 0.75 0.55 0.65 0.75 0.55 0.65 0.75

ÿ0.6 ÿ3.1 ÿ7.4 0.1 ÿ2.0 ÿ5.2 0.7 ÿ1.1 ÿ3.6

1.0 ÿ1.5 1.9 ÿ0.3 ÿ4.6 0.7 ÿ4.1 ÿ10.9 ÿ3.9 0.4 ÿ1.3 1.7 ÿ0.9 ÿ4.3 0.5 ÿ4.0 ÿ10.2 ÿ4.0 0.0 ÿ1.1 1.5 ÿ1.3 ÿ4.1 0.3 ÿ3.8 ÿ9.5 ÿ4.0

ÿ1.0 ÿ3.1 ÿ6.4 ÿ0.7 ÿ2.2 ÿ4.2 ÿ0.4 ÿ1.5 ÿ2.8

0.6 ÿ0.8 ÿ4.1 ÿ0.1 ÿ1.4 ÿ3.8 ÿ0.5 ÿ1.6 ÿ3.4

0.400 2.0 2.0 2.0 3.5 3.5 3.5 5.0 5.0 5.0

0.55 ÿ3.5 1.8 ÿ4.9 3.1 ÿ3.9 1.2 ÿ3.4 0.4 0.65 ÿ7.1 0.2 ÿ9.4 1.7 ÿ6.8 ÿ0.4 ÿ5.6 ÿ1.2 0.75 ÿ12.2 ÿ4.0 ÿ16.9 ÿ3.5 ÿ10.6 ÿ4.1 ÿ7.9 ÿ3.9 0.55 ÿ2.5 1.0 ÿ4.6 2.8 ÿ3.1 0.3 ÿ2.6 ÿ0.5 0.65 ÿ5.5 ÿ0.6 ÿ9.0 1.4 ÿ5.3 ÿ1.2 ÿ3.9 ÿ1.7 0.75 ÿ9.1 ÿ4.0 ÿ15.9 ÿ3.6 ÿ7.5 ÿ3.9 ÿ4.8 ÿ3.3 0.55 ÿ1.6 0.4 ÿ4.4 2.6 ÿ2.6 ÿ0.3 ÿ2.1 ÿ0.9 0.65 ÿ4.1 ÿ1.1 ÿ8.5 1.1 ÿ4.1 ÿ1.6 ÿ2.9 ÿ1.7 0.75 ÿ6.8 ÿ3.9 ÿ15.0 ÿ3.7 ÿ5.5 ÿ3.5 ÿ3.2 ÿ2.7

ÿ0.9 ÿ2.4 ÿ4.5 ÿ0.6 ÿ1.5 ÿ2.4 ÿ0.4 ÿ1.0 ÿ1.4

ÿ0.1 ÿ1.3 ÿ3.9 ÿ0.7 ÿ1.7 ÿ3.2 ÿ0.9 ÿ1.6 ÿ2.5

A.B. de Morais / Composites Science and Technology 60 (2000) 997±1002

3. Model evaluation

23 ˆ

The present equation was compared with the Aboudi's method-of-cells equation [6] and with a 3D ®niteelement hexagonal-cell model (Fig. 3). The Aboudi equation is very lengthy and requires the evaluation of several intermediate constants before the ®nal value of the modulus can be obtained. A spreadsheet was used for that purpose. The ®nite-element model was implemented in the commercial code ABAQUS [12]. 3D isoparametric 20-node brick and 15-node triangular prism elements were used. The cell is made representative by imposing that all of its faces remain plane (Fig. 3). A uniform vb displacement is imposed on the y ˆ b face of the cell. The modulus is then

1001

ua =a 2vb =b

…40†

where Fyb is the total node y-axis load on the y ˆ b face. The transverse Poisson ratio can also be obtained:

where ua is x-axis displacement of the x ˆ a face. A wide range of constituent properties was selected to assess the accuracy of (38). The matrix was assumed isotropic with moduli Em , from 2 to 5 GPa and Poisson ratios, vm , varying between 0.35 and 0.40. Fibre-volume fractions, Vf , from 0.55 to 0.75 were considered for the main types of ®bres (glass, carbon and boron). Results are here presented for the cases shown in Table 1. Tables 2 and 3 compare the present equation with Aboudi's equation and with the ®nite-element cell model. The performance of (38) is excellent for highly anisotropic ®bres of low transverse modulus. This could be expected from the transverse plane-stress assumption (18). Surprisingly, however, Eq. (38) also gives relatively good predictions in the cases where the referred assumption is clearly less appropriate. The worst results occur when the matrix Poisson ratio is high (vm ˆ 0:4). In particular, a ÿ17% error is obtained for a very high ®bre volume fraction (0.75) boron-reinforced composite.

Table 3 Errors (%) of the present (Pe) and of the Aboudi's equation [6] (Ab) relative to the ®nite-element cell model (see Table 1 for assumed ®bre properties)

Table 4 Errors (%) of the 23 predictions using (1), (3), (37) and (4) relative to the ®nite-element cell model (see Table 1 for assumed ®bre properties)

Matrix

E2 ˆ

Fyb =al vb =b

…39†

CS3 ®bre

CM1 ®bre

CM2 ®bre

Matrix

Fibre

m

Em

Vf

Pe

Ab

Pe

Ab

Pe

Ab

m

Em

Vf

G

B

CS1

CS2

CS3

CM1

CM2

0.350

2.0 2.0 2.0 3.5 3.5 3.5 5.0 5.0 5.0

0.55 0.65 0.75 0.55 0.65 0.75 0.55 0.65 0.75

0.9 0.2 ÿ1.0 0.7 0.4 ÿ0.1 0.5 0.3 0.1

ÿ0.8 ÿ1.7 ÿ3.3 ÿ1.0 ÿ1.6 ÿ2.3 ÿ1.0 ÿ1.3 ÿ1.6

0.5 0.2 ÿ0.4 0.1 0.0 0.0 ÿ0.2 ÿ0.2 ÿ0.2

ÿ1.0 ÿ1.6 ÿ2.6 ÿ0.9 ÿ1.2 ÿ1.4 ÿ0.7 ÿ0.7 ÿ0.8

0.2 0.1 ÿ0.1 ÿ0.3 ÿ0.2 ÿ0.2 ÿ0.5 ÿ0.5 ÿ0.4

ÿ1.0 ÿ1.4 ÿ1.8 ÿ0.7 ÿ0.7 ÿ0.8 ÿ0.4 ÿ0.3 ÿ0.3

0.350

2.0 2.0 2.0 3.5 3.5 3.5 5.0 5.0 5.0

0.55 0.65 0.75 0.55 0.65 0.75 0.55 0.65 0.75

ÿ2.7 ÿ1.3 1.7 ÿ3.6 ÿ2.5 ÿ0.4 ÿ4.4 ÿ3.5 ÿ1.9

ÿ1.6 0.2 4.9 ÿ1.8 0.0 4.2 ÿ2.0 ÿ0.3 3.6

ÿ1.7 ÿ0.6 1.5 ÿ1.8 ÿ1.1 0.0 ÿ1.9 ÿ1.4 ÿ0.8

ÿ1.5 ÿ0.8 0.4 ÿ1.5 ÿ1.1 ÿ0.7 ÿ1.4 ÿ1.2 ÿ1.0

ÿ1.3 ÿ0.8 ÿ0.3 ÿ1.1 ÿ0.9 ÿ0.8 ÿ0.8 ÿ0.7 ÿ0.7

ÿ0.8 ÿ0.6 ÿ0.4 ÿ0.2 ÿ0.2 ÿ0.2 0.3 0.3 0.3

ÿ0.4 ÿ0.3 ÿ0.3 0.4 0.4 0.3 0.8 0.9 0.9

0.375

2.0 2.0 2.0 3.5 3.5 3.5 5.0 5.0 5.0

0.55 0.65 0.75 0.55 0.65 0.75 0.55 0.65 0.75

ÿ0.8 ÿ1.8 ÿ3.1 ÿ0.6 ÿ1.1 ÿ1.4 ÿ0.5 ÿ0.8 ÿ0.9

ÿ0.5 ÿ1.6 ÿ3.4 ÿ1.0 ÿ1.6 ÿ2.4 ÿ1.0 ÿ1.3 ÿ1.7

ÿ0.9 ÿ1.4 ÿ1.9 ÿ0.9 ÿ1.0 ÿ1.0 ÿ0.9 ÿ1.0 ÿ0.8

ÿ0.9 ÿ1.7 ÿ2.7 ÿ1.0 ÿ1.3 ÿ1.5 ÿ0.7 ÿ0.8 ÿ0.8

ÿ0.9 ÿ1.1 ÿ1.2 ÿ1.0 ÿ1.0 ÿ0.9 ÿ1.1 ÿ1.0 ÿ0.9

ÿ1.0 ÿ1.4 ÿ1.9 ÿ0.7 ÿ0.8 ÿ0.8 ÿ0.4 ÿ0.4 ÿ0.3

0.375

2.0 2.0 2.0 3.5 3.5 3.5 5.0 5.0 5.0

0.55 0.65 0.75 0.55 0.65 0.75 0.55 0.65 0.75

0.2 3.1 8.1 ÿ0.8 1.7 5.4 ÿ1.7 0.4 3.3

1.3 4.8 12.1 1.1 4.5 11.3 0.9 4.2 10.6

0.9 3.2 6.9 0.5 2.3 4.5 0.2 1.6 3.0

0.8 2.6 4.9 0.5 1.6 2.7 0.3 1.1 1.7

0.8 2.0 3.4 0.6 1.3 1.7 0.5 1.0 1.2

1.0 1.7 2.3 1.0 1.4 1.5 1.2 1.4 1.4

1.0 1.5 1.7 1.3 1.5 1.5 1.4 1.6 1.6

0.400

2.0 2.0 2.0 3.5 3.5 3.5 5.0 5.0 5.0

0.55 0.65 0.75 0.55 0.65 0.75 0.55 0.65 0.75

ÿ3.0 ÿ4.5 ÿ5.7 ÿ2.2 ÿ2.9 ÿ3.2 ÿ1.8 ÿ2.2 ÿ2.1

ÿ0.3 ÿ1.6 ÿ3.5 ÿ0.9 ÿ1.7 ÿ2.6 ÿ1.0 ÿ1.4 ÿ1.8

ÿ2.7 ÿ3.5 ÿ3.9 ÿ2.1 ÿ2.4 ÿ2.2 ÿ1.9 ÿ1.9 ÿ1.7

ÿ0.8 ÿ1.7 ÿ2.8 ÿ1.0 ÿ1.4 ÿ1.6 ÿ0.8 ÿ0.9 ÿ0.9

ÿ2.4 ÿ2.7 ÿ2.7 ÿ1.9 ÿ2.0 ÿ1.7 ÿ1.7 ÿ1.7 ÿ1.4

ÿ1.0 ÿ1.6 ÿ2.0 ÿ0.8 ÿ0.9 ÿ0.9 ÿ0.5 ÿ0.4 ÿ0.4

0.400

2.0 2.0 2.0 3.5 3.5 3.5 5.0 5.0 5.0

0.55 0.65 0.75 0.55 0.65 0.75 0.55 0.65 0.75

2.8 7.1 14.3 1.9 5.6 11.1 1.0 4.2 8.5

3.8 8.8 18.8 3.7 8.5 17.9 3.5 8.2 17.1

3.2 6.7 12.0 2.7 5.4 8.8 2.2 4.4 6.7

2.9 5.7 9.2 2.3 4.2 6.1 1.9 3.3 4.3

2.7 4.8 7.0 2.1 3.4 4.3 1.8 2.6 3.1

2.5 3.9 5.1 2.2 2.9 3.3 2.0 2.5 2.7

2.3 3.2 3.8 2.1 2.5 2.7 2.0 2.3 2.4

1002

A.B. de Morais / Composites Science and Technology 60 (2000) 997±1002

Boron ®bres and such high ®bre volume fractions are not common. Most high-performance composites have about 65 vol% of ®bres. In addition, the Poisson ratios of the matrices are usually lower than 0.4. Therefore, the typical error of Eq. (38) is lower than 10%, which seems quite satisfactory considering its extreme simplicity. As was mentioned before, the values predicted by (38), (1) and (3) can be used in (4), yielding 23 predictions that can be compared with the ®nite element result (40). It is clear from Table 4 that the above comments concerning the accuracy of (38) remain valid, therefore reinforcing the usefulness of the equation that we have developed. 4. Conclusions This paper was concerned with the micromechanical prediction of the transverse modulus, E2 , of continuous®bre-reinforced polymers. Simple and accurate closedform equations are available for the most important layer moduli, except for E2 . In fact, the rule of mixtures predicts the longitudinal modulus, E1 , and Poisson ratio, 12 , while the self-consistent model predicts the longitudinal shear modulus, G12 . A relatively simple mechanics-of-materials analysis of a square cell has been performed. The signi®cant simpli®cations made concerning the stress distributions dissolved the transverse anisotropy of the square cell. Further assumptions regarding the typical matrix and ®bre properties enabled the derivation of a closed-form equation. The present equation was compared with Aboudi's method-of-cells equation and with a 3D ®nite-element hexagonal-cell model. These elaborate approaches are known to provide accurate predictions. The results show that the present equation is in good agreement

with the other analyses for the typical range of properties of polymer-matrix composites. Furthermore, when combined with other well-known accurate equations, such as the rule of mixtures and the self-consistent model equation for the plane-strain bulk modulus, k2 , the present equation enables us to obtain the 23 transverse Poisson ratio. The set of layer elastic constants needed for 3D stress analyses is then complete. References [1] Rosen BW, Hashin Z. Analysis of material properties. In: Engineered materials handbook, vol. 1, composites. ASM International, 1987. [2] McCullough RL. Micro-models for composite materials Ð continuous ®ber composites. In: Delaware composites design encyclopedia, vol. 2. Technomic Publishing Co, 1990. [3] Whitney JM, Riley MB. Elastic properties of ®ber reinforced composite materials. AIAA Journal 1966;4(9):1537±42. [4] Whitney JM. Elastic moduli of unidirectional composites with anisotropic ®laments. J Comp Mat 1967;1:188±93. [5] Luciano R, Barbero EJ. Formulas for the sti€ness of composites with periodic microstructure. Int J Solids Struct 1994;31(21):2933± 44. [6] Aboudi J. Micromechanical analysis of composites by the method of cells. App Mech Rev 1989;42(7):193±221. [7] Paley M, Aboudi J. Micromechanical analysis of composites by the generalized cells model. Mech Mat 1992;14:127±39. [8] Adams DF, Doner DR. Transverse normal loading of a unidirectional composite. J Comp Mat 1967;1:152±64. [9] Brockenbrough JR, Suresh S, Wienecke HA. Deformation of metal-matrix composites with continuous ®bres: geometrical e€ects of ®bre distribution and shape. Acta Metall Mater 1991;39(5):735±52. [10] Arnold SM, Pindera MJ, Wilt TE. In¯uence of ®ber architecture on the inelastic response of metal matrix composites. Int J Plast 1996;12(4):507±45. [11] Sun CT, Vaidya RS. Prediction of composite properties from a representative volume element. Comp Sci Tech 1996;56:171±9. [12] ABAQUS v5.7. Hibbitt, Karlsson & Sorensen, Inc, 1998.