**Comparative Review Study on Elastic Properties Modeling for Unidirectional Composite Materials**

Rafic Younes, Ali Hallal, Farouk Fardoun and Fadi Hajj Chehade

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50362

## **1. Introduction**

390 Composites and Their Properties

Zelazny, B. & Neville, A.C. (1972) Quantitative Studies on Fibril Orientation in Beetle

Endocuticle. *Insect Physiology*, Vol. 18, pp 2095-2121.

Due to the outstanding properties of 2D and 3D textile composites, the use of 3D fiber reinforced in high-tech industrial domains (spatial, aeronautic, automotive, naval, etc…) has been expanded in recent years. Thus, the evaluation of their elastic properties is crucial for the use of such types of composites in advanced industries. The analytical or numerical modeling of textile composites in order to evaluate their elastic properties depend on the prediction of the elastic properties of unidirectional composite materials with long fibers composites "UD". UD composites represent the basic element in modeling all laminates or 2D or 3D fabrics. They are considered as transversely isotropic materials composed of two phases: the reinforcement phase and the matrix phase. Isotropic fibers (e.g. glass fibers) or anisotropic fibers (e.g. carbon fibers) represent the reinforcement phase while, in general, isotropic materials (e.g. epoxy, ceramics, etc…) represent the matrix phase (Figure 1).

The effective stiffness and compliance matrices of a transversely isotropic material are defined in the elastic regime by five independent engineering constants: longitudinal and transversal Young's moduli E11 and E22, longitudinal and transversal shear moduli G12 and G23, and major Poisson's ratio ν12 (Noting that direction 1 is along the fiber). The minor Poisson's ratio ν23 is related to E22 and G12. The effective elastic properties are evaluated in terms of mechanical properties of fibers and matrix (Young's and shear moduli, Poisson's ratios and the fiber volume fraction Vf ). The compliance matrix [S] of a transversely isotropic material is given as follow:

$$\begin{bmatrix} \mathbf{S} \end{bmatrix} = \begin{bmatrix} 1/E\_{11} & -\nu\_{12}/E\_{11} - \nu\_{12}/E\_{11} & 0 & 0 & 0\\ -\nu\_{12}/E\_{11} & 1/E\_{22} & -\nu\_{23}/E\_{22} & 0 & 0 & 0\\ -\nu\_{12}/E\_{11} - \nu\_{23}/E\_{22} & 1/E\_{22} & 0 & 0 & 0\\ 0 & 0 & 0 & 1/G\_{23} & 0 & 0\\ 0 & 0 & 0 & 0 & 1/G\_{12} & 0\\ 0 & 0 & 0 & 0 & 0 & 1/G\_{12} \end{bmatrix}$$

© 2012 Younes et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Younes et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

$$\frac{1}{E\_{22}} = \frac{\eta^f.V^f}{E\_{22}^f} + \frac{\eta^m.V^m}{E^m}$$

$$\eta^{f} = \frac{E\_{11}^{f}.V^{f} + \left[\left(1 - \mathbf{v}\_{12}^{f}.\mathbf{v}\_{21}^{f}\right).E^{m} + \left.\mathbf{v}^{m}.\mathbf{v}\_{11}^{f}.E\_{11}^{f}\right].V^{m}}{E\_{11}^{f}.V^{f} + E^{m}.V^{m}}$$

$$\eta^{m} = \frac{\left[\left(1 - \mathbf{v}^{\rm m}\right).E\_{11}^{f} - \left(1 - \mathbf{v}^{\rm m}.\mathbf{v}\_{12}^{f}\right).E^{m}\right].V^{f} + E^{m}.V^{m}}{E\_{11}^{f}.V^{f} + E^{m}.V^{m}}$$

$$\frac{1}{G\_{12}} = \frac{\frac{V^{f}}{G\_{12}^{f}} + \frac{\eta^{\prime}.V^{m}}{G^{m}}}{V^{f} + \eta^{\prime}.V^{m}}$$

$$E\_{22} = \;E^m . \left(\frac{1 + \zeta \eta V\_f}{1 - \eta V\_f}\right) ; \; G\_{12} = G^m . \left(\frac{1 + \zeta \eta V\_f}{1 - \eta V\_f}\right)$$
 
$$\text{with} \quad \eta = \begin{pmatrix} \frac{M\_f}{\mathcal{M}\_m - 1} \\ \frac{M\_f}{\mathcal{M}\_m + \zeta} \end{pmatrix}$$

$$\begin{aligned} E\_{11} &= V^f \ E\_{11}^f + V^m \ E^m \\\\ E\_{22} &= \frac{\mathbf{E}^m}{1 - \sqrt{V^f} \left(1 - \mathbf{E}^m / E\_{22}^f \right)} \\\\ \nu\_{12} &= V^f \ \upsilon\_{12}^f + V^m \ \upsilon^m \\\\ G\_{12} &= \frac{\mathbf{c}^m}{1 - \sqrt{V^f} \left(1 - \mathbf{G}^m / G\_{12}^f \right)} \\\\ G\_{23} &= \frac{\mathbf{c}^m}{1 - \sqrt{V^f} \left(1 - \mathbf{G}^m / G\_{23}^f \right)} \end{aligned}$$

$$E\_{11} = V^f \ E\_{11}^f + V^m \ E^m + \frac{4 \, V^f \, \, \nu^m \, (\nu\_{12}^f - \nu^m)^2}{\frac{V^f}{K^m} + \frac{1}{G^m} + \frac{V^m}{K^f}} \text{ (Hashin and Rosenen [5])}$$

$$\nu\_{12} = V^f \ \, \, \nu\_{12}^f + V^m \, \nu^m + \frac{V^f \, \, \nu^m \left(\nu\_{12}^f - \nu^m\right) \left(\frac{1}{K^m} - \frac{1}{K^f}\right)}{\frac{V^f}{K^m} + \frac{1}{G^m} + \frac{V^m}{K^f}} \text{ (Hashin and Rossen [5])}$$

$$G\_{12} = \, \, G^m \, \, \frac{G^f \, \left(1 + V^f\right) + G^m \, V^m}{G^f J^m + G\_m (1 + V^f)} \text{(Hashin and Rossen [5])}$$

$$A\left(\frac{G\_{23}}{G\_m}\right)^2 + 2B\left(\frac{G\_{23}}{G\_m}\right) + \mathcal{C} = 0$$

$$
\eta\_m = \mathbf{3} - \boldsymbol{\nu}\_m \; ; \; \eta\_f = \mathbf{3} - \boldsymbol{\nu}\_{23}^J
$$

$$\left(\upsilon\_{2.3} = \frac{K - m.G\_{2.3}}{K + m.G\_{2.3}}\right) \text{ with } m = 1 + 4K.\frac{\upsilon\_{12}^2}{E\_{11}}$$

$$\mathbf{K} = \frac{\mathbf{K}^m \mathbf{.} \left(\mathbf{K}^f + G^m\right) \mathbf{.} \mathbf{V}^m + \mathbf{K}^f \left(\mathbf{K}^m + G^m\right) \mathbf{.} \mathbf{V}^f}{(\mathbf{K}^f + G^m) \mathbf{.} \mathbf{V}^m + (\mathbf{K}^m + G^m) \mathbf{.} \mathbf{V}^f}$$

$$E\_{22} = \ 2. \ (1 + \ \upsilon\_{23}) . G\_{23}$$

$$\mathbf{C\_{MT}} = \mathbf{C\_m} + \left[\mathbf{V\_{f^\cdot} \left(\left(\mathbf{C\_f} - \mathbf{C\_m}\right).A\_{\rm Eshellby}\right)\right] \left[\mathbf{V\_m}.\mathbf{I} + \mathbf{V\_{f^\cdot} \left(A\_{\rm Eshellby}\right)}\right]^{-1}$$

$$\mathbf{A\_{Eshellby}} = [\mathbf{I} + \mathbf{E} . \mathbf{C\_m}^{-1} . (\mathbf{C\_f} - \mathbf{C\_m})]^{-1}$$

$$\text{A}\_{\text{Eshelby}} = \left[\text{I} + \text{E.C}\_{\text{m}}^{-1}.\left(\text{C}\_{\text{f}} - \text{C}\_{\text{m}}\right)\right]^{-1}$$

$$\text{C}\_{\text{sc}} = \text{C}\_{\text{m}} + \left[\text{V}\_{\text{f}}.\left(\text{(C}\_{\text{f}} - \text{C}\_{\text{m}}\right).\text{A}\_{\text{Eshelby}}\right)\right]$$

$$\mathbf{A}\_{\mathrm{Eshelby}} = \left[\mathbf{I} + \mathbf{E}.\mathbf{C}\_{\mathrm{sc}}^{-1}.\left(\mathbf{C}\_{\mathrm{f}} - \mathbf{C}\_{\mathrm{sc}}\right)\right]^{-1}$$

$$\mathbf{C}\_{\mathrm{sc}} = \mathbf{C}\_{\mathrm{m}} + \left[\mathbf{V}\_{\mathrm{f}}.\left(\left(\mathbf{C}\_{\mathrm{f}} - \mathbf{C}\_{\mathrm{m}}\right).\mathbf{A}\_{\mathrm{Eshelby}}\right)\right]$$

$$\mathbf{E\_{11}} = \mathbf{V\_{f}.E\_{11}^{f}} + \mathbf{V\_{m}.E\_{m}}$$

$$\mathbf{E\_{22}} = \frac{(\mathbf{V\_{f} + \mathbf{V\_{m}.a\_{11}})(\mathbf{V\_{f} + \mathbf{V\_{m}.a\_{22}})}}{(\mathbf{V\_{f} + \mathbf{V\_{m}.a\_{11}})(\mathbf{V\_{f} + \mathbf{S\_{f}.a\_{22}}.S\_{22}^{m}}) + \mathbf{V\_{f}.V\_{m}(S\_{21}^{m} - S\_{21}^{f})a\_{12}}}$$

$$\mathbf{V\_{12}} = \mathbf{V\_{f}.V\_{11}^{f}} + \mathbf{V\_{m}.v\_{m}}$$

$$\mathbf{G\_{12}} = \frac{(\mathbf{V\_{f} + \mathbf{V\_{m}.a\_{66}})G\_{12}^{f}G\_{m}}{\mathbf{V\_{f}.G\_{m} + \mathbf{V\_{m}.a\_{66}}G\_{12}^{f}}}$$

$$\mathbf{G\_{23}} = \frac{\mathbf{0.5(V\_f + V\_{\rm m}\,\mathrm{a}\_{44})}}{\mathbf{V\_f(S\_{22}^f - S\_{23}^f) + V\_{\rm m}\,\mathrm{a}\_{44(S\_{22}^m - S\_{23}^m)}}$$

$$E\_{11} = \frac{\sigma\_{11}}{\varepsilon\_{11}}, \text{ where } \sigma\_{11} \text{and } \varepsilon\_{11} \text{ are calculated numerically on the X+ face}$$

On the Y+ face:

��� <sup>=</sup> ��� ��� , where σ�� and � are calculated numerically on the Y+ face

On the X+ face:

��� <sup>=</sup> ��� ��� , where τ�� and �� are calculated numerically on the X+ face

On the Z+ face:

��� <sup>=</sup> ��� ��� , where τ��and �� are calculated numerically on the Z+ face


Comparative Review Study on Elastic Properties Modeling for Unidirectional Composite Materials 399

(Figure 3 and 4).

ROM

EAM

M-T

S-C

exp.

sq.

diam.

Hex.

ROM

EAM

Ex M-T Ex S-C

exp.

sq.

.

.

For the longitudinal Young's modulus E11, obtained analytical and numerical results are compared to those available experimental data for carbon/epoxy and polyethylene/epoxy UD composites in terms of the fiber volume fraction Vf. Investigated analytical models belong to the ROM, the Elasticity approach model (EAM), M-T and S-C models. Please note that ROM, MROM, Chamis, Halpin-Tsai and Bridging models share the same formulation

It's well noticed that the predicted results for all investigated models are in good agreement

**E11(GPa) carbon/epoxy**

with the experimental data for both composites with different Vf

**Figure 3.** Predicted analytical, numerical and experimental results for E11 in terms of Vf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Vf**

**E11 (GPa) Polyethylene/epoxy**

**Figure 4.** Predicted analytical, numerical and experimental results for E11 in terms of Vf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Vf**

*3.1.1. Longitudinal Young's modulus E11*

for E11.

0

0

10

20

30

40

50

60

70

50

100

150

200

250

**Table 1.** Boundary conditions on the X, Y and Z faces of the quarter unit cell.

## **3. Comparative study, analysis and discussion**

#### **3.1. Results**

In this section, a comparison of analytical models and numerical models with available experimental data is presented. Three different kinds of UD composites are taken as examples: Glass/epoxy composite [16], carbon/epoxy composite [14] and polyethylene/epoxy composite [17] (Table 2). The glass fibers are isotropic fibers while the carbon and the polyethylene fibers are transversely isotropic fibers. Knowing that the epoxy matrices are assumed isotropic, it's well noticed that for the polyethylene/epoxy, the Young's modulus of the epoxy is higher than that transversal modulus of the fibers, which represent an important case to be investigated.


**Table 2.** Elastic properties of the fibers and epoxy matrices.

#### *3.1.1. Longitudinal Young's modulus E11*

398 Composites and Their Properties

, where σ�� and � are calculated numerically on the Y+ face

, where τ�� and �� are calculated numerically on the X+ face

, where τ��and �� are calculated numerically on the Z+ face

U = K, V and W free

U, V and W free

W= 0

**Table 1.** Boundary conditions on the X, Y and Z faces of the quarter unit cell.

**3. Comparative study, analysis and discussion** 

represent an important case to be investigated.

**Table 2.** Elastic properties of the fibers and epoxy matrices.

Fibers ���

X faces Y faces Z faces X- X+ Y- Y+ Z- Z+

> U, V and W free

V = K, U and W free

U = W = 0 U=W = 0 W = 0 W = 0

W = 0, U and V free

W = 0, U and V free

� (GPa) ���

U, V and W free

U, V and W free

V= K

� ��� �

V = 0, U and W free

V = 0, U and W free

G23 U = 0 U = 0 U = W = 0 U = W = 0 U = V = 0 U = 0

In this section, a comparison of analytical models and numerical models with available experimental data is presented. Three different kinds of UD composites are taken as examples: Glass/epoxy composite [16], carbon/epoxy composite [14] and polyethylene/epoxy composite [17] (Table 2). The glass fibers are isotropic fibers while the carbon and the polyethylene fibers are transversely isotropic fibers. Knowing that the epoxy matrices are assumed isotropic, it's well noticed that for the polyethylene/epoxy, the Young's modulus of the epoxy is higher than that transversal modulus of the fibers, which

� (GPa) ���

E-Glass [16] 73.1 73.1 29.95 0.22 0.22 Carbon [14] 232 15 24 0.279 0.49 Polyethylene [17] 60.4 4.68 1.65 0.38 0.55 Matrix �� (GPa) �� (GPa) �� Epoxy resin [16] 3.45 3.45 1.28 0.35 0.35 Epoxy [14] 5.35 5.35 1.97 0.354 0.354 Epoxy [17] 5.5 5.5 1.28 0.37 0.37

� (GPa) ���

On the Y+ face:

On the X+ face:

On the Z+ face:

**3.1. Results** 

E11 and ν12 U = 0, V

E22 and ν23 U = 0, V

and W free

and W free

G12 V = W = 0 V = K

��� <sup>=</sup> ��� ���

��� <sup>=</sup> ��� ���

��� <sup>=</sup> ��� ��� For the longitudinal Young's modulus E11, obtained analytical and numerical results are compared to those available experimental data for carbon/epoxy and polyethylene/epoxy UD composites in terms of the fiber volume fraction Vf. Investigated analytical models belong to the ROM, the Elasticity approach model (EAM), M-T and S-C models. Please note that ROM, MROM, Chamis, Halpin-Tsai and Bridging models share the same formulation for E11.

It's well noticed that the predicted results for all investigated models are in good agreement with the experimental data for both composites with different Vf (Figure 3 and 4).

**Figure 3.** Predicted analytical, numerical and experimental results for E11 in terms of Vf .

**Figure 4.** Predicted analytical, numerical and experimental results for E11 in terms of Vf .

### *3.1.2. Transversal Young's modulus E22*

The prediction of the transversal Young's modulus and in contrast with the longitudinal modulus presents a real challenge for the researchers. Thus, many analytical models are proposed belonged to different micromechanics approach. In addition, the potential of the FE element modeling is investigated. Predicted results of different analytical and numerical models for three UD composites are presented in figures (5,6 and 7)

Comparative Review Study on Elastic Properties Modeling for Unidirectional Composite Materials 401

**E22 (GPa) Polyethylene/epoxy**

.

ROM

MROM

Chamis

Bridging

Ey M-T

Ey S-C

exp.

FE sq.

GSC

Halpin Tsai

**Figure 7.** Predicted analytical, numerical and experimental results for E22 in terms of Vf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Vf**

the Chamis model shows a good agreement with Vf higher than 0.6.

*3.1.3. Longitudinal shear modulus G12*

(Figure 9).

4.5 4.7 4.9 5.1 5.3 5.5 5.7 5.9 6.1 6.3 6.5

It's shown that for the glass/epoxy composite, the S-C model overestimates the experimental results, while the ROM and MROM models underestimate it. Other analytical models, especially the Chamis, Bridging and EAM models yield results that correlate well with the available experimental data for different values of Vf. Moreover, it's noticed that the FE (Square array), the Halpin-Tsai and the M-T models gives good predictions. Concerning composites reinforced with transversely isotropic fibers, it's well remarked that the EAM model well overestimates E22 especially with the polyethylene/epoxy composite. The ROM underestimates the experimental results, while other analytical models, in addition to the numerical FE (diamond array) model, yield very good predictions for the carbon/epoxy composite. However, with the polyethylene/epoxy, it's noticed that only the results obtained from the ROM and the Halpin-Tsai models correlate well with the experimental data, while

Experimental results for two UD composites are used to be compared with. Figures 8 show clearly the MROM, EAM, Halpin-Tsai, Chamis, bridging analytical models, in addition to all numerical FE models yield very good results for the carbon/epoxy composite. However, it's remarked that the inclusion models, the M-T and S-C models, overestimate the longitudinal shear modulus. Concerning the polyethylene/epoxy composite, only results obtained results from the MROM and Chamis models agree well with the available experimental data

**Figure 5.** Predicted analytical, numerical and experimental results for E22 in terms of Vf .

**Figure 6.** Predicted analytical, numerical and experimental results for E22 in terms of Vf .

**Figure 7.** Predicted analytical, numerical and experimental results for E22 in terms of Vf .

It's shown that for the glass/epoxy composite, the S-C model overestimates the experimental results, while the ROM and MROM models underestimate it. Other analytical models, especially the Chamis, Bridging and EAM models yield results that correlate well with the available experimental data for different values of Vf. Moreover, it's noticed that the FE (Square array), the Halpin-Tsai and the M-T models gives good predictions. Concerning composites reinforced with transversely isotropic fibers, it's well remarked that the EAM model well overestimates E22 especially with the polyethylene/epoxy composite. The ROM underestimates the experimental results, while other analytical models, in addition to the numerical FE (diamond array) model, yield very good predictions for the carbon/epoxy composite. However, with the polyethylene/epoxy, it's noticed that only the results obtained from the ROM and the Halpin-Tsai models correlate well with the experimental data, while the Chamis model shows a good agreement with Vf higher than 0.6.

#### *3.1.3. Longitudinal shear modulus G12*

400 Composites and Their Properties

0

0

5

10

15

20

25

10

20

30

40

50

60

70

80

*3.1.2. Transversal Young's modulus E22*

The prediction of the transversal Young's modulus and in contrast with the longitudinal modulus presents a real challenge for the researchers. Thus, many analytical models are proposed belonged to different micromechanics approach. In addition, the potential of the FE element modeling is investigated. Predicted results of different analytical and numerical

**E22 (GPa) Glass/epoxy** ROM

models for three UD composites are presented in figures (5,6 and 7)

**Figure 5.** Predicted analytical, numerical and experimental results for E22 in terms of Vf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Vf**

**E22 (GPa) carbon/epoxy**

**Figure 6.** Predicted analytical, numerical and experimental results for E22 in terms of Vf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Vf**

.

ROM MROM Halpin Tsai Chamis EAM Bridging M-T S-C exp. FE sq. FE diam. FE Hex.

MROM Chamis EAM Bridging Halpin-Tsai Ey M-T Ey S-C exp. FE Sq.

.

Experimental results for two UD composites are used to be compared with. Figures 8 show clearly the MROM, EAM, Halpin-Tsai, Chamis, bridging analytical models, in addition to all numerical FE models yield very good results for the carbon/epoxy composite. However, it's remarked that the inclusion models, the M-T and S-C models, overestimate the longitudinal shear modulus. Concerning the polyethylene/epoxy composite, only results obtained results from the MROM and Chamis models agree well with the available experimental data (Figure 9).

### *3.1.4. Transversal shear modulus G23*

For the transversal shear modulus G23, it's shown from Figure 10 and 11, that the bridging model yields the best results. In addition, it's remarked that the EAM, Chamis yield reasonable predictions underestimating the experimental data, while the M-T and S-C models overestimate it. Concerning the numerical modeling, predicted results always overestimate the available experimental results for the two composites.

Comparative Review Study on Elastic Properties Modeling for Unidirectional Composite Materials 403

**G23 (GPa) carbon/epoxy**

.

Chamis EAM

Bridging

 M-T S-C exp. FE Sq.

FE diam. FE Hex.

Chamis

EAM

M-T

S-C

exp.

FE Sq.

Bridging

.

**Figure 10.** Predicted analytical, numerical and experimental results for G23 in terms of Vf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Vf**

**G23 (GPa) polyethylene/epoxy**

1

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

**Figure 11.** Predicted analytical, numerical and experimental results for G23 in terms of Vf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Vf**

**Figure 8.** Predicted analytical, numerical and experimental results for G12 in terms of Vf .

**Figure 9.** Predicted analytical, numerical and experimental results for G12 in terms of Vf .

0

1.5

1.6

1.7

1.8

1.9

2

2.1

5

10

15

20

25

*3.1.4. Transversal shear modulus G23*

For the transversal shear modulus G23, it's shown from Figure 10 and 11, that the bridging model yields the best results. In addition, it's remarked that the EAM, Chamis yield reasonable predictions underestimating the experimental data, while the M-T and S-C models overestimate it. Concerning the numerical modeling, predicted results always

**G12 (GPa) carbon/epoxy**

overestimate the available experimental results for the two composites.

**Figure 8.** Predicted analytical, numerical and experimental results for G12 in terms of Vf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Vf**

**G12 (GPa) Polyethylene/epoxy**

**Figure 9.** Predicted analytical, numerical and experimental results for G12 in terms of Vf

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Vf**

.

ROM MROM Halpin Tsai Chamis EAM Bridging M-T S-C exp. FE Sq.

ROM MROM Halpin-Tsai Chamis EAM Bridging M-T S-C exp. FE Sq. FE diam. FE Hex.

.

**Figure 10.** Predicted analytical, numerical and experimental results for G23 in terms of Vf .

**Figure 11.** Predicted analytical, numerical and experimental results for G23 in terms of Vf .

#### *3.1.5. Major Poisson's ratio ν<sup>12</sup>*

Concerning the Poisson's ratios, the obtained results of the analytical models are only compared to those numerical due the missing of experimental data for the studied UD composites. Figure 12 shows that for the major Poisson's ratio ν12, all analytical and numerical models correlate well with each other.

Comparative Review Study on Elastic Properties Modeling for Unidirectional Composite Materials 405

except for the polyethylene/epoxy case where it's well agree with the available experimental data. Concerning the longitudinal shear modulus G12, the ROM model didn't yield good prediction for both studied cases the carbon/epoxy and the polyethylene/epoxy

As known the semi-empirical models have been emerged and proposed in order to correct the predictions of the ROM model for the transversal Young's and longitudinal shear moduli. While the investigated models share the same formulations for E11 and ν12 with ROM model, the corrections made for E22 and G12 prove to be effective. It's shown that the Chamis model yields very good results for all studied cases, while the MROM and Halpin-Tsai the models only suffer with the special case of the polyethylene/epoxy with the E22 and

Concerning the elasticity approach models, the proposed formulation of the E11 yields similar results for that proposed by the ROM model. While for the transversal Young's modulus E22, it's clearly noticed that with isotropic fibers, the model results correlate well with those experimental, while with the case of transversely isotropic fibers, reasonable predictions are shown for the carbon/epoxy case. However, for the polyethylene case the model well overestimates the experimental results. The reason could be conducted to that EAM models are initially proposed to deal with UD composites reinforced with isotropic fibers. For the longitudinal shear modulus G12, the elastic solution formulation agrees well with the experimental data. Concerning the transversal shear modulus, the predictions made by the generalized self-consistent model of the Christensen model [6], which is developed to enhance the predictions of this elastic property, always overestimates the

In this study, the potential of the homogenization models is investigated. The inclusion models, the M-T and the S-C models, and the bridging model, yield good prediction for both longitudinal Young's modulus and major Poisson's ratio. However, for the transversal Young's modulus E22, reasonable agreement is shown for the glass/epoxy and carbon/epoxy cases, except with the self-consistent model which overestimates the experimental data for

results and overestimate the compared experimental data while agree with FE modeling results. The same problem is shown with the prediction of the shear moduli, where for the polyethylene/epoxy case, the models belonged to the homogenization approach give the same results overestimating the experimental data. While with the carbon/epoxy case, it's noticed that the bridging model predicts better the shear moduli, while the M-T and S-C

Concerning the numerical modeling, it's well noticed that there are different predicted results for different arrays. It's also remarked, that the FE numerical modeling didn't yield better results than the analytical models, except for the longitudinal Young's modulus and major Poisson's ratio where all predicted results from numerical and analytical models

models well overestimate the experimental data especially for the G12.

correlate well with available experimental data.

. While for the case of the polyethylene/epoxy, all three models yield almost the same

composites.

G12 respectively.

experimental data.

high Vf

**Figure 12.** Predicted analytical and numerical results for ν12 in terms of Vf .

#### **3.2. Analysis and discussion**

In this section, an analysis of the predicted results for each model is presented apart. It's shown from the above results that for the phenomenological models, the Voigt and Reuss models, represented by the ROM model, show very good predictions for the longitudinal Young's modulus E11 and major Poisson's ratio ν12. However, with for the transversal Young's modulus E22 the ROM model always underestimates the experimental results except for the polyethylene/epoxy case where it's well agree with the available experimental data. Concerning the longitudinal shear modulus G12, the ROM model didn't yield good prediction for both studied cases the carbon/epoxy and the polyethylene/epoxy composites.

404 Composites and Their Properties

*3.1.5. Major Poisson's ratio ν<sup>12</sup>*

numerical models correlate well with each other.

**Figure 12.** Predicted analytical and numerical results for ν12 in terms of Vf

In this section, an analysis of the predicted results for each model is presented apart. It's shown from the above results that for the phenomenological models, the Voigt and Reuss models, represented by the ROM model, show very good predictions for the longitudinal Young's modulus E11 and major Poisson's ratio ν12. However, with for the transversal Young's modulus E22 the ROM model always underestimates the experimental results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 **Vf**

**3.2. Analysis and discussion** 

0.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

.

FE Sq.

FE diam.

FE Hex.

ROM

M-T

S-C

Concerning the Poisson's ratios, the obtained results of the analytical models are only compared to those numerical due the missing of experimental data for the studied UD composites. Figure 12 shows that for the major Poisson's ratio ν12, all analytical and

**ν<sup>12</sup> carbon/epoxy**

As known the semi-empirical models have been emerged and proposed in order to correct the predictions of the ROM model for the transversal Young's and longitudinal shear moduli. While the investigated models share the same formulations for E11 and ν12 with ROM model, the corrections made for E22 and G12 prove to be effective. It's shown that the Chamis model yields very good results for all studied cases, while the MROM and Halpin-Tsai the models only suffer with the special case of the polyethylene/epoxy with the E22 and G12 respectively.

Concerning the elasticity approach models, the proposed formulation of the E11 yields similar results for that proposed by the ROM model. While for the transversal Young's modulus E22, it's clearly noticed that with isotropic fibers, the model results correlate well with those experimental, while with the case of transversely isotropic fibers, reasonable predictions are shown for the carbon/epoxy case. However, for the polyethylene case the model well overestimates the experimental results. The reason could be conducted to that EAM models are initially proposed to deal with UD composites reinforced with isotropic fibers. For the longitudinal shear modulus G12, the elastic solution formulation agrees well with the experimental data. Concerning the transversal shear modulus, the predictions made by the generalized self-consistent model of the Christensen model [6], which is developed to enhance the predictions of this elastic property, always overestimates the experimental data.

In this study, the potential of the homogenization models is investigated. The inclusion models, the M-T and the S-C models, and the bridging model, yield good prediction for both longitudinal Young's modulus and major Poisson's ratio. However, for the transversal Young's modulus E22, reasonable agreement is shown for the glass/epoxy and carbon/epoxy cases, except with the self-consistent model which overestimates the experimental data for high Vf . While for the case of the polyethylene/epoxy, all three models yield almost the same results and overestimate the compared experimental data while agree with FE modeling results. The same problem is shown with the prediction of the shear moduli, where for the polyethylene/epoxy case, the models belonged to the homogenization approach give the same results overestimating the experimental data. While with the carbon/epoxy case, it's noticed that the bridging model predicts better the shear moduli, while the M-T and S-C models well overestimate the experimental data especially for the G12.

Concerning the numerical modeling, it's well noticed that there are different predicted results for different arrays. It's also remarked, that the FE numerical modeling didn't yield better results than the analytical models, except for the longitudinal Young's modulus and major Poisson's ratio where all predicted results from numerical and analytical models correlate well with available experimental data.

## **4. Conclusion**

In this study, the evaluated results, for the elastic properties, of most known analytical micromechanical models, as well as FE modeling methods, are compared to available experimental data for three different UD composites: Glass/epoxy, carbon/epoxy and polyethylene/epoxy. It should be noticed that the studied cases cover different kinds of reinforced composites by isotropic fibers (glass) and transversely isotropic fibers (carbon and polyethylene). In addition, the polyethylene/epoxy presents an interesting case study, where the matrix is stiffer than the fibers in the transvers direction.

Comparative Review Study on Elastic Properties Modeling for Unidirectional Composite Materials 407

[1] Voigt W. Uber die Beziehung zwischen den beiden Elastizitatskonstanten Isotroper

[2] Reuss A. Berechnung der Fliessgrense von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Zeitschrift Angewandte Mathematik und

[3] Halpin JC, Kardos JL. The Halpin-Tsai equations: A review. Polymer Engineering and

[4] Chamis CC. Mechanics of composite materials: past, present, and future. J Compos

[5] Hashin Z, Rosen BW. The elastic moduli of fiber reinforced materials. Journal of Applied

[6] Christensen RM. A critical evaluation for a class of micromechanics models. Journal of

[7] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with

[8] Benveniste Y. A new approach to the application of Mori-Tanaka's theory in composite

[9] Mura T. Micromechanics of Defects in Solids, 2nd edn. Martinus Nijhof Publishers,

[10] Hill R. Theory of mechanical properties of fibre-strengthen materials-III. Self-consistent

[11] Budiansky B. On the elastic moduli of some heterogeneous materials, J. Mech. Phys.

[12] Chou TW, Nomura S, Taya M. A self-consistent approach to the elastic stiffness of

[13] Huang ZM. Simulation of the mechanical properties of fibrous composites by the

[14] Huang ZM. Micromechanical prediction of ultimate strength of transversely isotropic fibrous composites. International Journal of Solids and Structures 38 (2001) 4147-4172. [15] Li S. Boundary conditions for unit cells from periodic microstructures and their

[16] Shan HZ, Chou TW. Transverse elastic moduli of unidirectional fiber composites with fiber/matrix interfacial debonding. Composites Science and Technology 53 (1995) 383-

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Fadi Hajj Chehade

**5. References** 

*L3M2S, Lebanese University, Rafic Hariri campus, Beirut, Lebanon* 

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The analyses of the compared results show clearly that all analytical and numerical models show a very good agreement for the longitudinal Young's modulus E11 and major Poisson's ratio ν12. However, the other moduli, the transversal Young's modulus E22, longitudinal shear modulus G12 and the transversal shear modulus G23, represent the main challenge for the researchers. It's shown that analytical micromechanical models belonged to the semiempirical models, especially the Chamis model, predict well these elastic properties. Moreover, the bridging model proves to be a reliable model when predicting the elastic properties of carbon/epoxy composite. It's noticed that almost all models suffer with the prediction of elastic properties for the polyethylene/epoxy composite. However, models belonged to the elasticity approach and inclusion approach (M-T and S-C models) show inconsistency in predicting the elastic properties of studied UD composites. Numerical models, based on the FE method, show that using different fibers arrangements will lead to different predicted results. Moreover, the FE didn't prove that it could be more accurate than some simple and straightforward analytical model. As a conclusion from this study, the Chamis model and the bridging model could be considered as the most complete models which could give quite accurate estimations for all five independent elastic properties. Noting that the corrections proposed by the Halpin-Tsai model, prove that it well enhance the prediction of the transversal Young's modulus E22.

## **Author details**

Rafic Younes\*

*LISV, University of Versailles Saint-Quentin, Versailles, France Faculty of Engineering, Lebanese University, Rafic Hariri campus, Beirut, Lebanon* 

#### Ali Hallal

*LISV, University of Versailles Saint-Quentin, Versailles, France L3M2S, Lebanese University, Rafic Hariri campus, Beirut, Lebanon* 

Farouk Fardoun *L3M2S, Lebanese University, Rafic Hariri campus, Beirut, Lebanon* 

\* Corresponding Author Fadi Hajj Chehade *L3M2S, Lebanese University, Rafic Hariri campus, Beirut, Lebanon* 

#### **5. References**

406 Composites and Their Properties

In this study, the evaluated results, for the elastic properties, of most known analytical micromechanical models, as well as FE modeling methods, are compared to available experimental data for three different UD composites: Glass/epoxy, carbon/epoxy and polyethylene/epoxy. It should be noticed that the studied cases cover different kinds of reinforced composites by isotropic fibers (glass) and transversely isotropic fibers (carbon and polyethylene). In addition, the polyethylene/epoxy presents an interesting case study,

The analyses of the compared results show clearly that all analytical and numerical models show a very good agreement for the longitudinal Young's modulus E11 and major Poisson's ratio ν12. However, the other moduli, the transversal Young's modulus E22, longitudinal shear modulus G12 and the transversal shear modulus G23, represent the main challenge for the researchers. It's shown that analytical micromechanical models belonged to the semiempirical models, especially the Chamis model, predict well these elastic properties. Moreover, the bridging model proves to be a reliable model when predicting the elastic properties of carbon/epoxy composite. It's noticed that almost all models suffer with the prediction of elastic properties for the polyethylene/epoxy composite. However, models belonged to the elasticity approach and inclusion approach (M-T and S-C models) show inconsistency in predicting the elastic properties of studied UD composites. Numerical models, based on the FE method, show that using different fibers arrangements will lead to different predicted results. Moreover, the FE didn't prove that it could be more accurate than some simple and straightforward analytical model. As a conclusion from this study, the Chamis model and the bridging model could be considered as the most complete models which could give quite accurate estimations for all five independent elastic properties. Noting that the corrections proposed by the Halpin-Tsai model, prove that it

where the matrix is stiffer than the fibers in the transvers direction.

well enhance the prediction of the transversal Young's modulus E22.

*Faculty of Engineering, Lebanese University, Rafic Hariri campus, Beirut, Lebanon* 

*LISV, University of Versailles Saint-Quentin, Versailles, France* 

*LISV, University of Versailles Saint-Quentin, Versailles, France L3M2S, Lebanese University, Rafic Hariri campus, Beirut, Lebanon* 

*L3M2S, Lebanese University, Rafic Hariri campus, Beirut, Lebanon* 

**4. Conclusion** 

**Author details** 

Rafic Younes\*

Ali Hallal

 \*

Farouk Fardoun

Corresponding Author


[17] Wilczynski AP, Lewinski J. Predecting the properties of unidirectional fibrous composites with monotropic reinforcement. Composite Science and Technology 55 (1995) 139-143

**Section 5** 

**Metal and Ceramic Matrix Composites** 

**Metal and Ceramic Matrix Composites** 

408 Composites and Their Properties

(1995) 139-143

[17] Wilczynski AP, Lewinski J. Predecting the properties of unidirectional fibrous composites with monotropic reinforcement. Composite Science and Technology 55

**Chapter 18** 

© 2012 Sayuti et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Sayuti et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Manufacturing and Properties of** 

**Al-11.8%Si Matrix Composites** 

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48095

**1. Introduction** 

**Quartz (SiO2) Particulate Reinforced** 

M. Sayuti, S. Sulaiman, T.R. Vijayaram, B.T.H.T Baharudin and M.K.A. Arifin

Metal matrix composites (MMC) are a class of composites that contains an element or alloy matrix in which a second phase is fixed firmly deeply and distributed evenly to achieve the required property improvement. The property of the composite varies based on the size, shape and amount of the second phase (Sayuti et al., 2010; Sulaiman et al., 2008). Discontinuously reinforced metal matrix composites, the other name for particulate reinforced composites, constitute 5 – 20 % of the new advanced materials (Gay et al., 2003). The mechanical properties of the processed composites are greatly influenced by their microstructure. An increased stiffness, yield strength and ultimate tensile strength are generally achieved by increasing the weight fraction of the reinforcement phase in the matrix. Inspite of these advantages, the usage of particulate reinforced MMCs as structural components in some applications is limited due to low ductility (Rizkalla and Abdulwahed, 1996). Owing to this and to overcome the draw-backs, a detailed investigation on the strengthening mechanism of composites has been carried out by composite experts (Humphreys, 1987). They have found that the particle size and its weight fraction in metal matrix composites influences the generation of dislocations due to thermal mismatch. The effect is also influenced by the developed residual and internal stresses too. The researchers have predicted that the dislocation density is directly proportional to the weight fraction and due to the amount of thermal mismatch. As a result, the strengthening effect is proportional to the square root of the dislocation density. This effect would be significant for fine particles and for higher weight fractions. The MMCs yield improved physical and mechanical properties and these outstanding benefits are due to the combined metallic and ceramic properties (Hashim et al., 2002). Though there are various types of MMCs, particulate-reinforced composites are the most
