**3. Mathematical framework for assessment of extent of variation of elastic properties in plantain fiber-reinforced polyester composites**

One significant property of composite materials is their plainly visible macroscopic anisotropy, which means that the properties estimated in the longitudinal direction are by far not the same as those measured in transverse direction. There are no material planes of symmetry, and normal loads create both normal strains and shear strains. This anisotropic characteristic of reinforced composites results in low mechanical properties in the out-of-plane orientation where the matrix carries the primary load. Consequently the application of reinforced composites is limited in scenarios prone to complex load paths such as lugs and fittings [53].

By implication any endeavor to comprehend the structural application of plantain fiber-reinforced polyester composite must assess the inborn anisotropy.

associated high return on investment and short maturity period for plantain con-

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

friendly materials technology at an acceptable performance [29].

ties of plantain fiber-reinforced composites in **Figure 1**, there is still need to

Africa cultivates over 50% of worldwide production of plantain and Nigeria is one of the biggest plantain producing nations in the planet. Therefore the interest in plantain plant fiber for polymer reinforcement was as a result of its abundance and accessibility as it is evaluated that over 15.07 million tons of plantain fruit is produced each year in Nigeria with about 2.4 million metric tons produced from southern Nigeria [27, 28]. Plantain fiber also satisfied over 50% conditions for ecofriendly materials as shown in **Figure 2** and can make strong reinforcement in composites. A composite which can be characterized as a physical blend of at least two unique materials, has properties that are commonly superior to those of any of the establishing materials. It is important to utilize blends of materials to tackle issues in light of the fact that any one material alone cannot suffix effectively in eco-

Cadena Ch et al. [30] and Adeniyi et al. [31] orchestrated the potentials of natural fibers from plantain pseudo stem for use in fiber-reinforced composites. It is therefore important to assess the extent of variation of elastic properties in plantain fiberreinforced polyester composites to guard against out of plane failure during structural applications. Unfortunately most studies involving plantain fiber-reinforced composites has dwelt on assessment of tensile, flexural and hardness properties [32], optimization of hardness strengths [33], effect of water and organic extractives removal [34], effects of fiber extraction techniques [35], optimization of flexural strength [36], compressive and impact strength evaluation [37–39], effect of high-frequency microwave radiation [40], effect of chemical treatment on the morphology [41], implications of interfacial energetics on mechanical strength [42]. Although Ihueze, Okafor and Okoye [43] has reported the longitudinal (1) and transverse (2) proper-

tributes to its massive cultivation in Nigeria.

**Figure 2.**

**184**

*Properties of eco-friendly building materials.*

Composites are a subclass of anisotropic materials that are delegated orthotropic. Orthotropic materials have properties that are unique in three directions with perpendicular axes of symmetry. In this way, orthotropic mechanical properties depend heavily on fiber orientation. An orthotropic ply is thus defined as that having two different material properties in two mutually perpendicular directions at a point and the two mutually perpendicular directions also form the planes of material properties symmetry at the point.

#### **3.1 Determination of reduced stiffness matrix and compliance matrix**

Considering two possible loading conditions of longitudinal (direction 1) and transverse (direction 2) in the matrix as shown in **Figure 3**, the resulting direct strains from Hooks law are respectively *<sup>e</sup>*<sup>1</sup> <sup>¼</sup> �*v*21*σ*<sup>1</sup> *<sup>E</sup>*<sup>1</sup> and *<sup>e</sup>*<sup>2</sup> <sup>¼</sup> �*v*12*σ*<sup>1</sup> *<sup>E</sup>*<sup>1</sup> where *v*<sup>12</sup> = major Poisson's ratio and *v*<sup>21</sup> = minor Poisson's ratio.

Hence the application of both direct stresses *σ*<sup>1</sup> and *σ*<sup>2</sup> will yield corresponding strains as follows:

$$
\sigma\_1 = \frac{\sigma\_1}{E\_1} - \frac{v\_{21}\sigma\_2}{E\_2} \tag{1}
$$

A combined effect of shear and direct stresses gives the reduced stiffness matrix

*Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites…*

*v*21*E*<sup>1</sup> 1 � *v*12*v*<sup>21</sup>

*E*2 1 � *v*12*v*<sup>21</sup>

0 0 *G*<sup>12</sup>

0

9 >>>>>>=

>>>>>>;

0

1 *G*<sup>12</sup>

<sup>¼</sup> <sup>7030</sup>*:*<sup>962</sup> <sup>∗</sup> <sup>0</sup>*:*<sup>38</sup>

<sup>¼</sup> <sup>6817</sup>*:*<sup>175</sup> <sup>∗</sup> <sup>0</sup>*:*<sup>29</sup>

<sup>1</sup> � <sup>0</sup>*:*<sup>38</sup> <sup>∗</sup> <sup>0</sup>*:*<sup>179</sup> <sup>¼</sup> <sup>16011</sup>*:*<sup>07</sup>

¼ 0*:*179 ∗ 16011*:*07 ¼ 2865*:*98

<sup>1</sup> � <sup>0</sup>*:*<sup>38</sup> <sup>∗</sup> <sup>0</sup>*:*<sup>179</sup> <sup>¼</sup> <sup>7544</sup>*:*<sup>113</sup>

*E***2 (MPa)**

*E* **(MPa)** *v***<sup>12</sup>** *τ*max **(MPa)**

**G12 (MPa)**

�*v*<sup>21</sup> *E*2

> 1 *E*2

0

9 >>>>=

8 ><

>:

9 >=

>;

*e*1 *e*2 *e*12 9 >=

>;

(6)

(7)

>>>>;

*σ*1 *σ*2 *τ*12

0

8 ><

>:

14, 922 <sup>¼</sup> <sup>0</sup>*:*<sup>179</sup>

<sup>13027</sup>*:*<sup>5</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>152</sup>

as in Eq. (6) and reduced compliance matrix as in Eq. (7)

8 >>>><

>>>>:

9 >= >; ¼

*<sup>v</sup>*21, PEFBFRP <sup>¼</sup> *<sup>E</sup>*2*v*<sup>12</sup>

*<sup>v</sup>*21, PPSFC <sup>¼</sup> *<sup>E</sup>*2*v*<sup>12</sup>

*E*1 1 � *v*12*v*<sup>21</sup>

*v*21*E*<sup>1</sup> 1 � *v*12*v*<sup>21</sup>

*E*2 1 � *v*12*v*<sup>21</sup>

**Composites Properties**

*Sy* **(MPa)**

*Su*1, *Su*<sup>2</sup> *are tensile strengths in the longitudinal and transverse directions respectively.*

*Su***<sup>2</sup> (MPa)**

*Su***<sup>1</sup> (MPa)**

*e*1 *e*2 *e*12

8 ><

>:

ratio *v*<sup>21</sup> using Eq. (5) and **Table 1** as follows

Eq. (6).

**Table 1.**

*Ref. [43].*

**187**

For PEFBFC

*E*1 1 � *v*12*v*<sup>21</sup>

*v*12*E*<sup>2</sup> 1 � *v*12*v*<sup>21</sup>

> 8 >>>>>><

> >>>>>>:

1 *E*1

�*v*<sup>21</sup> *E*1

0 0

Ihueze et al. (2013) calculated the major Poisson's ratio *v*<sup>12</sup> for plantain fiber/ polyester composites. However, there is need to further assess the minor Poisson's

*E*1

*E*1

Minor Poisson's ratio is the strain resulting from a stress in the axial direction,

1 � *v*12*v*<sup>21</sup> ¼ 1 � 0*:*38 ∗ 0*:*179 ¼ 1 � 0*:*068 ¼ 0*:*93

1 � *v*12*v*<sup>21</sup> ¼ 1 � 0*:*29 ∗ 0*:*152 ¼ 1 � 0*:*068 ¼ 0*:*956

The reduced stiffness matrix (ℵ) for PEFBFC and PPSFC is obtained from

<sup>¼</sup> <sup>14922</sup>

<sup>¼</sup> <sup>7030</sup>*:*<sup>962</sup>

*E***1 (MPa)**

PEFBFC 410.15 37.3397 33.69 14,922 7030.962 9990.10 0.38 19.3100 3622.99 PPSFC 288.10 33.1330 29.24 13027.5 6817.175 9146.305 0.29 15.5700 3332.835

*Evaluated mechanical properties of plantain fibers and plantain fibers reinforced polyester composites. From*

*σ*1 *σ*2 *τ*12

*DOI: http://dx.doi.org/10.5772/intechopen.90890*

9 >= >; ¼

8 ><

>:

$$
\sigma\_2 = \frac{\sigma\_2}{E\_2} - \frac{v\_{12}\sigma\_1}{E\_1} \tag{2}
$$

Putting Eqs. (1) and (2) in a matrix form, yields

$$\begin{Bmatrix} e\_1\\ e\_2 \end{Bmatrix} = \begin{Bmatrix} \frac{1}{E\_1} & \frac{-v\_{21}}{E\_2} \\\\ \frac{-v\_{12}}{E\_1} & \frac{1}{E\_2} \end{Bmatrix} \begin{Bmatrix} \sigma\_1\\ \sigma\_2 \end{Bmatrix} \tag{3}$$

$$\begin{Bmatrix} \sigma\_1\\ \sigma\_2 \end{Bmatrix} = \begin{Bmatrix} \frac{E\_1}{\mathbf{1} - v\_{12}v\_{21}} & \frac{v\_{21}E\_1}{\mathbf{1} - v\_{12}v\_{21}}\\ \frac{v\_{12}E\_2}{\mathbf{1} - v\_{12}v\_{21}} & \frac{E\_2}{\mathbf{1} - v\_{12}v\_{21}} \end{Bmatrix} \begin{Bmatrix} e\_1\\ e\_2 \end{Bmatrix} \tag{4}$$

Eq. (3) is symetric about the loading diagonal such that

$$\frac{-v\_{21}}{E\_2} = \frac{-v\_{12}}{E\_1} \tag{5}$$

**Figure 3.** *Stressed single thin composite lamina. From Ref. [43].*

*Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites… DOI: http://dx.doi.org/10.5772/intechopen.90890*

A combined effect of shear and direct stresses gives the reduced stiffness matrix as in Eq. (6) and reduced compliance matrix as in Eq. (7)

$$\begin{Bmatrix} \sigma\_1\\ \sigma\_2\\ \tau\_{12} \end{Bmatrix} = \left\{ \begin{array}{ll} \frac{E\_1}{1 - v\_{12}v\_{21}} & \frac{v\_{21}E\_1}{1 - v\_{12}v\_{21}} & 0\\ \frac{v\_{12}E\_2}{1 - v\_{12}v\_{21}} & \frac{E\_2}{1 - v\_{12}v\_{21}} & 0\\ 0 & 0 & G\_{12} \end{array} \right\} \begin{Bmatrix} e\_1\\ e\_2\\ e\_{12} \end{Bmatrix} \tag{6}$$
 
$$\begin{Bmatrix} e\_1\\ e\_2\\ e\_{12} \end{Bmatrix} = \begin{Bmatrix} \frac{1}{E\_1} & \frac{-v\_{21}}{E\_2} & 0\\ \frac{-v\_{21}}{E\_1} & \frac{1}{E\_2} & 0\\ 0 & 0 & \frac{1}{G\_{12}} \end{Bmatrix} \begin{Bmatrix} \sigma\_1\\ \sigma\_2\\ \tau\_{12} \end{Bmatrix} \tag{7}$$

1 *G*<sup>12</sup>

Minor Poisson's ratio is the strain resulting from a stress in the axial direction, Ihueze et al. (2013) calculated the major Poisson's ratio *v*<sup>12</sup> for plantain fiber/ polyester composites. However, there is need to further assess the minor Poisson's ratio *v*<sup>21</sup> using Eq. (5) and **Table 1** as follows

0 0

$$v\_{21, \text{PEFBPR}} = \frac{E\_2 v\_{12}}{E\_1} = \frac{7030.962 \ast 0.38}{14,922} = 0.179$$

$$1 - v\_{12} v\_{21} = 1 - 0.38 \ast 0.179 = 1 - 0.068 = 0.93$$

$$v\_{21, \text{PEFC}} = \frac{E\_2 v\_{12}}{E\_1} = \frac{6817.175 \ast 0.29}{13027.5} = 0.152$$

$$1 - v\_{12} v\_{21} = 1 - 0.29 \ast 0.152 = 1 - 0.068 = 0.956$$

The reduced stiffness matrix (ℵ) for PEFBFC and PPSFC is obtained from Eq. (6).

For PEFBFC

Composites are a subclass of anisotropic materials that are delegated orthotropic. Orthotropic materials have properties that are unique in three directions with perpendicular axes of symmetry. In this way, orthotropic mechanical properties depend heavily on fiber orientation. An orthotropic ply is thus defined as that having two different material properties in two mutually perpendicular directions at a point and the two mutually perpendicular directions also form the planes of

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

**3.1 Determination of reduced stiffness matrix and compliance matrix**

*<sup>e</sup>*<sup>1</sup> <sup>¼</sup> *<sup>σ</sup>*<sup>1</sup> *E*1

*<sup>e</sup>*<sup>2</sup> <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup> *E*2

> 1 *E*1

�*v*<sup>12</sup> *E*1

Considering two possible loading conditions of longitudinal (direction 1) and transverse (direction 2) in the matrix as shown in **Figure 3**, the resulting direct

Hence the application of both direct stresses *σ*<sup>1</sup> and *σ*<sup>2</sup> will yield corresponding

� *<sup>v</sup>*21*σ*<sup>2</sup> *E*2

� *<sup>v</sup>*12*σ*<sup>1</sup> *E*1

> �*v*<sup>21</sup> *E*2

9 >>=

*σ*1 *σ*2

> 9 >>=

*e*1 *e*2 � �

>>;

( )

>>;

*v*21*E*<sup>1</sup> 1 � *v*12*v*<sup>21</sup>

*E*2 1 � *v*12*v*<sup>21</sup>

1 *E*2

*<sup>E</sup>*<sup>1</sup> and *<sup>e</sup>*<sup>2</sup> <sup>¼</sup> �*v*12*σ*<sup>1</sup>

*<sup>E</sup>*<sup>1</sup> where *v*<sup>12</sup> = major

(1)

(2)

(3)

(4)

(5)

material properties symmetry at the point.

strains from Hooks law are respectively *<sup>e</sup>*<sup>1</sup> <sup>¼</sup> �*v*21*σ*<sup>1</sup>

Putting Eqs. (1) and (2) in a matrix form, yields

*e*1 *e*2

¼

*σ*1 *σ*2

*Stressed single thin composite lamina. From Ref. [43].*

( )

¼

8 >><

>>:

Eq. (3) is symetric about the loading diagonal such that

8 >><

>>:

*E*1 1 � *v*12*v*<sup>21</sup>

*v*12*E*<sup>2</sup> 1 � *v*12*v*<sup>21</sup>

> �*v*<sup>21</sup> *E*2

<sup>¼</sup> �*v*<sup>12</sup> *E*1

( )

Poisson's ratio and *v*<sup>21</sup> = minor Poisson's ratio.

strains as follows:

**Figure 3.**

**186**

$$\frac{E\_1}{1 - v\_{12}v\_{21}} = \frac{14922}{1 - 0.38 \ast 0.179} = 16011.07$$

$$\frac{v\_{21}E\_1}{1 - v\_{12}v\_{21}} = 0.179 \ast 16011.07 = 2865.98$$

$$\frac{E\_2}{1 - v\_{12}v\_{21}} = \frac{7030.962}{1 - 0.38 \ast 0.179} = 7544.113$$


#### **Table 1.**

*Evaluated mechanical properties of plantain fibers and plantain fibers reinforced polyester composites. From Ref. [43].*

$$\aleph\_{\text{PEFBFC}} = \begin{Bmatrix} 16011.07 & 2865.98 & 0\\ 2865.98 & 7544.113 & 0\\ 0 & 0 & 3622.99 \end{Bmatrix} \text{MPa}$$

For PPSFC

$$\frac{E\_1}{1 - v\_{12}v\_{21}} = \frac{13027.5}{1 - 0.29 \ast 0.152} = 13628.23$$

$$\frac{v\_{21}E\_1}{1 - v\_{12}v\_{21}} = 0.152 \ast 13628.23 = 2071.49$$

$$\frac{E\_2}{1 - v\_{12}v\_{21}} = \frac{6817.175}{1 - 0.29 \ast 0.152} = 7131.53$$

$$\Re\_{\text{PPSFC}} = \left\{ \begin{array}{cccc} 13628.23 & 2071.49 & 0\\ 2071.49 & 7131.53 & 0\\ 0 & 0 & 3332.835 \end{array} \right\} \text{MPa}$$

The reduced compliance matrix (*β*) is obtained from Eq. (7). For PEFBFC

$$\frac{1}{E\_1} = \frac{1}{14922} = 6.7 \times 10^{-5} \text{1/MPa}$$

$$\frac{-v\_{21}}{E\_2} = \frac{-0.179}{7030.962} = -2.5 \times 10^{-5} \text{1/MPa}$$

$$\frac{1}{E\_2} = \frac{1}{7030.962} = 1.4 \times 10^{-4} \text{1/MPa}$$

$$\frac{1}{G\_{12}} = \frac{1}{3622.99} = 2.8 \times 10^{-4} \text{1/MPa}$$

$$\rho\_{\text{PEBFBFC}} = \begin{Bmatrix} 6.7 \times 10^{-5} & -2.5 \times 10^{-5} & 0\\ -2.5 \times 10^{-5} & 1.4 \times 10^{-4} & 0\\ 0 & 0 & 2.8 \times 10^{-4} \end{Bmatrix} \text{1/MPa}$$

**3.2 Transformation of elastic constants**

Density (g/cm3

*DOI: http://dx.doi.org/10.5772/intechopen.90890*

Density (kg/m<sup>3</sup>

Density (kg/m3

**Table 2.**

**189**

orientation *θ* relative to a reference direction *x*-*y*.

1 *Ex*

1 *Ey* ¼ *s* 4 *E*1 þ *c*4 *E*2 þ *c* 2 *s* <sup>2</sup> 1 *G*<sup>12</sup>

1 *Gxy* ¼ *c* 2 *s* <sup>2</sup> 4 *E*1 þ 4 *E*2 þ 8*v*<sup>12</sup> *E*1

*vxy* ¼ *Ex c*

¼ *c*4 *E*1 þ *s* 4 *E*2 þ *c* 2 *s* <sup>2</sup> 1 *G*<sup>12</sup>

*Mechanical properties of plantain fibers and polyester resin. From Ref. [43].*

<sup>4</sup> <sup>þ</sup> *<sup>s</sup>* <sup>4</sup> *v*<sup>12</sup>

*E*1 � *c* 2 *s* <sup>2</sup> 1 *E*1 þ 1 *E*2 � 1 *<sup>G</sup>*<sup>12</sup> (11)

The know of the stress-strain relationship in the plantain/polyester composite is completely comprehended by knowing the associated independent engineering elastic constants (*E*1, *E*2, *G*<sup>12</sup> and *v*12) as previously determined by Ihueze et al. (2013) as in **Tables 1** and **2**. However, there is need to further establish these elastic properties at different directions of fibers other than directions 1 and 2. Datoo [54] derived various expressions for determination of the elastic properties in the reference axes *x*-*y* for any fiber orientation as expressed in Eqs. (8)–(14) where *c* ¼ *cosθ*, *s* ¼ *sinθ*, *Ex*, *Ey*, *Gxy*, *vxy*, *mx* and *my* are the elastic properties at any fiber

) 381.966

) 354.151

**Property Polyester resin**

*Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites…*

Young modulus (MPa) 2000–4500 Tensile strength (MPa) 40–90 Compressive strength (MPa) 90–250 Tensile elongation at break (%) 2 Water absorption 24 h at 20°C 0.1–0.3 Flexural modulus (GPa) 11.0 Poisson's ratio 0.37–0.38

Young modulus (MPa) 23,555 UTS (MPa) 536.2 Strain (%) 2.37

Young modulus (MPa) 27,344 UTS (MPa) 780.3 Strain (%) 2.68

) 1.2–1.5 (1400 kg/m<sup>3</sup>

)

**Plantain pseudo stem fibers**

**Plantain empty fruit bunch fibers**

� <sup>2</sup>*v*<sup>12</sup> *E*1

� <sup>2</sup>*v*<sup>12</sup> *E*1

þ *c* <sup>2</sup> � *<sup>s</sup>* <sup>2</sup> 1

(8)

(9)

(10)

*G*<sup>12</sup>

For PPSFC

$$\frac{1}{E\_1} = \frac{1}{13027.5} = 7.7 \times 10^{-5} \text{1/MPa}$$

$$\frac{-v\_{21}}{E\_2} = \frac{-0.152}{6817.175} = -2.2 \times 10^{-5} \text{1/MPa}$$

$$\frac{1}{E\_2} = \frac{1}{6817.175} = 1.5 \times 10^{-4} \text{1/MPa}$$

$$\frac{1}{G\_{12}} = \frac{1}{3332.835} = 3.0 \times 10^{-4} \text{1/MPa}$$

$$\rho\_{\text{PPSFC}} = \left\{ \begin{array}{cccc} 7.7 \times 10^{-5} & -2.2 \times 10^{-5} & 0\\ -2.2 \times 10^{-5} & 1.5 \times 10^{-4} & 0\\ 0 & 0 & 3.0 \times 10^{-4} \end{array} \right\} 1/\text{MPa}$$

*Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites… DOI: http://dx.doi.org/10.5772/intechopen.90890*


#### **Table 2.**

ℵPEFBFC ¼

For PPSFC

For PEFBFC

8 ><

>:

*E*1 1 � *v*12*v*<sup>21</sup>

*v*21*E*<sup>1</sup> 1 � *v*12*v*<sup>21</sup>

*E*2 1 � *v*12*v*<sup>21</sup>

> 8 ><

> >:

1 *E*1

�*v*<sup>21</sup> *E*2

> 1 *E*2

> > 1 *G*<sup>12</sup>

> > > 1 *E*1

�*v*<sup>21</sup> *E*2

> 1 *E*2

1 *G*<sup>12</sup>

8 ><

>:

<sup>¼</sup> <sup>1</sup>

<sup>¼</sup> <sup>1</sup>

<sup>¼</sup> <sup>1</sup>

<sup>¼</sup> �0*:*<sup>152</sup>

8 ><

>:

*β*PEFBFC ¼

*β*PPSFC ¼

**188**

For PPSFC

The reduced compliance matrix (*β*) is obtained from Eq. (7).

<sup>¼</sup> <sup>1</sup>

<sup>¼</sup> �0*:*<sup>179</sup>

<sup>¼</sup> <sup>1</sup>

<sup>¼</sup> <sup>1</sup>

ℵPPSFC ¼

16011*:*07 2865*:*98 0 2865*:*98 7544*:*113 0

<sup>¼</sup> <sup>13027</sup>*:*<sup>5</sup>

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

<sup>¼</sup> <sup>6817</sup>*:*<sup>175</sup>

13628*:*23 2071*:*49 0 2071*:*49 7131*:*53 0

<sup>14922</sup> <sup>¼</sup> <sup>6</sup>*:*<sup>7</sup> � <sup>10</sup>�<sup>5</sup>

<sup>7030</sup>*:*<sup>962</sup> ¼ �2*:*<sup>5</sup> � <sup>10</sup>�<sup>5</sup>

<sup>7030</sup>*:*<sup>962</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>4</sup> � <sup>10</sup>�41*=*MPa

<sup>3622</sup>*:*<sup>99</sup> <sup>¼</sup> <sup>2</sup>*:*<sup>8</sup> � <sup>10</sup>�41*=*MPa

0 02*:*<sup>8</sup> � <sup>10</sup>�<sup>4</sup>

<sup>6</sup>*:*<sup>7</sup> � <sup>10</sup>�<sup>5</sup> �2*:*<sup>5</sup> � <sup>10</sup>�<sup>5</sup> <sup>0</sup> �2*:*<sup>5</sup> � <sup>10</sup>�<sup>5</sup> <sup>1</sup>*:*<sup>4</sup> � <sup>10</sup>�<sup>4</sup> <sup>0</sup>

<sup>13027</sup>*:*<sup>5</sup> <sup>¼</sup> <sup>7</sup>*:*<sup>7</sup> � <sup>10</sup>�<sup>5</sup>

<sup>6817</sup>*:*<sup>175</sup> ¼ �2*:*<sup>2</sup> � <sup>10</sup>�<sup>5</sup>

<sup>6817</sup>*:*<sup>175</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>5</sup> � <sup>10</sup>�41*=*MPa

<sup>3332</sup>*:*<sup>835</sup> <sup>¼</sup> <sup>3</sup>*:*<sup>0</sup> � <sup>10</sup>�41*=*MPa

0 03*:*<sup>0</sup> � <sup>10</sup>�<sup>4</sup>

<sup>7</sup>*:*<sup>7</sup> � <sup>10</sup>�<sup>5</sup> �2*:*<sup>2</sup> � <sup>10</sup>�<sup>5</sup> <sup>0</sup> �2*:*<sup>2</sup> � <sup>10</sup>�<sup>5</sup> <sup>1</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>4</sup> <sup>0</sup>

0 0 3622*:*99

<sup>1</sup> � <sup>0</sup>*:*<sup>29</sup> <sup>∗</sup> <sup>0</sup>*:*<sup>152</sup> <sup>¼</sup> <sup>13628</sup>*:*<sup>23</sup>

¼ 0*:*152 ∗ 13628*:*23 ¼ 2071*:*49

<sup>1</sup> � <sup>0</sup>*:*<sup>29</sup> <sup>∗</sup> <sup>0</sup>*:*<sup>152</sup> <sup>¼</sup> <sup>7131</sup>*:*<sup>53</sup>

0 0 3332*:*835

9 >=

>; MPa

9 >=

>; MPa

1*=*MPa

1*=*MPa

1*=*MPa

1*=*MPa

9 >=

>;

9 >=

>;

1*=*MPa

1*=*MPa

*Mechanical properties of plantain fibers and polyester resin. From Ref. [43].*

#### **3.2 Transformation of elastic constants**

The know of the stress-strain relationship in the plantain/polyester composite is completely comprehended by knowing the associated independent engineering elastic constants (*E*1, *E*2, *G*<sup>12</sup> and *v*12) as previously determined by Ihueze et al. (2013) as in **Tables 1** and **2**. However, there is need to further establish these elastic properties at different directions of fibers other than directions 1 and 2. Datoo [54] derived various expressions for determination of the elastic properties in the reference axes *x*-*y* for any fiber orientation as expressed in Eqs. (8)–(14) where *c* ¼ *cosθ*, *s* ¼ *sinθ*, *Ex*, *Ey*, *Gxy*, *vxy*, *mx* and *my* are the elastic properties at any fiber orientation *θ* relative to a reference direction *x*-*y*.

$$\frac{\mathbf{1}}{E\_{\mathbf{x}}} = \frac{c^4}{E\_1} + \frac{s^4}{E\_2} + c^2 s^2 \left(\frac{\mathbf{1}}{G\_{12}} - \frac{2v\_{12}}{E\_1}\right) \tag{8}$$

$$\frac{1}{E\_{\text{y}}} = \frac{s^4}{E\_1} + \frac{c^4}{E\_2} + c^2 s^2 \left(\frac{1}{G\_{12}} - \frac{2v\_{12}}{E\_1}\right) \tag{9}$$

$$\frac{1}{G\_{xy}} = c^2 s^2 \left( \frac{4}{E\_1} + \frac{4}{E\_2} + \frac{8\nu\_{12}}{E\_1} \right) + \left( c^2 - s^2 \right) \frac{1}{G\_{12}} \tag{10}$$

$$w\_{xy} = E\_x \left[ \left( c^4 + s^4 \right) \frac{v\_{12}}{E\_1} - c^2 s^2 \left( \frac{1}{E\_1} + \frac{1}{E\_2} - \frac{1}{G\_{12}} \right) \right] \tag{11}$$

The minor Poissons ratio with respect to the material reference axes is obtained from Eq. (5) such that

$$\frac{v\_{\infty}}{E\_{\infty}} = \frac{v\_{\text{px}}}{E\_{\text{y}}} \tag{12}$$

$$m\_{\mathbf{x}} = E\_{\mathbf{x}} \left[ c^3 s \left( \frac{1}{G\_{12}} - \frac{2v\_{12}}{E\_1} - \frac{2}{E\_1} \right) - c s^3 \left( \frac{1}{G\_{12}} - \frac{2v\_{12}}{E\_1} - \frac{2}{E\_2} \right) \right] \tag{13}$$

$$m\_{\mathcal{V}} = E\_{\mathcal{V}} \left[ c s^3 \left( \frac{1}{G\_{12}} - \frac{2v\_{12}}{E\_1} - \frac{2}{E\_1} \right) - c^3 s \left( \frac{1}{G\_{12}} - \frac{2v\_{12}}{E\_1} - \frac{2}{E\_2} \right) \right] \tag{14}$$

#### **3.3 Variation of engineering elastic constants with fiber orientation** *θ*

Considering fiber orientation *θ*° ranging from 0° to 90° in increments of 5° the variation of *Ex*, *Ey*, *Gxy*, *vxy*, *mx* and *my* has been assessed Plantain Empty Fruit Bunch Fiber Composite (PEFBFC) and Plantain Pseudo Stem Fiber Composite (PPSFC) using Eqs. (8)–(14) and presented in **Tables 3** and **4**, respectively.

**Figure 4** shows the variation of elastic modulus with fiber orientation, it can be seen that the highest value of 14,922 and 13027.5 MPa in the reference *x*-direction (*Ex*Þ is attained in the fiber orientation angle 0° for PEFBFC and PPSFC respectively. However as fiber orientation angle changes, there is a sharp drop in the value of elastic modulus in the reference *x*-direction to a respective lowest value of 7030.96 and 6817.18 MPa as the fiber orientation angle increased to 90°. On the contrary, the lowest value of 7030.96 and 6817.18 MPa were recorded for elastic modulus in the


reference *y*-direction (*Ey*Þas the fiber orientation angle increased from 0° to 90° reaching a peak value of 14,922 and 13027.5 MPa for PEFBFC and PPSFC respectively. The implication is that reinforcements are required to be aligned in the

*Variation of elastic modulus with fiber orientation in PEFBFC and PPSFC.*

**S/N** *θ***°** *Ex Ey G*xy *v*xy *v*yx *mx my Ex*

*DOI: http://dx.doi.org/10.5772/intechopen.90890*

*Variation of engineering elastic constants with fiber orientation* θ *in PPSFC.*

**Table 4.**

**Figure 4.**

**191**

1 0 13027.500 6817.175 3332.835 0.290 0.152 0.000 0.000 1.911 0.489 2 5 12897.620 6830.651 3343.613 0.290 0.154 0.114 0.023 1.892 0.490 3 10 12530.190 6872.014 3375.039 0.291 0.159 0.214 0.047 1.838 0.495 4 15 11983.280 6944.024 3424.350 0.291 0.168 0.293 0.073 1.758 0.502 5 20 11330.960 7051.119 3486.843 0.290 0.180 0.344 0.103 1.662 0.511 6 25 10643.310 7199.198 3555.901 0.287 0.194 0.369 0.136 1.561 0.522 7 30 9974.406 7395.333 3623.334 0.282 0.209 0.371 0.173 1.463 0.532 8 35 9359.192 7647.394 3680.229 0.275 0.224 0.356 0.212 1.373 0.540 9 40 8816.012 7963.509 3718.337 0.265 0.239 0.328 0.252 1.293 0.545 10 45 8351.207 8351.207 3731.758 0.253 0.253 0.292 0.292 1.225 0.547 11 50 7963.509 8816.012 3718.337 0.239 0.265 0.252 0.328 1.168 0.545 12 55 7647.394 9359.192 3680.229 0.224 0.275 0.212 0.356 1.122 0.540 13 60 7395.333 9974.406 3623.334 0.209 0.282 0.173 0.371 1.085 0.532 14 65 7199.198 10643.310 3555.901 0.194 0.287 0.136 0.369 1.056 0.522 15 70 7051.119 11330.960 3486.843 0.180 0.290 0.103 0.344 1.034 0.511 16 75 6944.024 11983.280 3424.350 0.168 0.291 0.073 0.293 1.019 0.502 17 80 6872.014 12530.190 3375.039 0.159 0.291 0.047 0.214 1.008 0.495 18 85 6830.651 12897.620 3343.613 0.154 0.290 0.023 0.114 1.002 0.490 19 90 6817.175 13027.500 3332.835 0.152 0.290 0.000 0.000 1.000 0.489

*Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites…*

*E***2**

*G*xy *E***2**

**Table 3.** *Variation of engineering elastic constants with fiber orientation* θ *in PEFBFC.*

*Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites… DOI: http://dx.doi.org/10.5772/intechopen.90890*


#### **Table 4.**

The minor Poissons ratio with respect to the material reference axes is obtained

<sup>¼</sup> *vyx Ey*

� *c* 3 *s* 1 *G*<sup>12</sup>

� *cs*<sup>3</sup> <sup>1</sup> *G*<sup>12</sup> � <sup>2</sup>*v*<sup>12</sup> *E*1

� <sup>2</sup>*v*<sup>12</sup> *E*1

� 2 *E*2

� 2 *E*2 (12)

(13)

(14)

*E***2**

*G*xy *E***2**

*vxy Ex*

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

� 2 *E*1

� 2 *E*1

Considering fiber orientation *θ*° ranging from 0° to 90° in increments of 5° the variation of *Ex*, *Ey*, *Gxy*, *vxy*, *mx* and *my* has been assessed Plantain Empty Fruit Bunch Fiber Composite (PEFBFC) and Plantain Pseudo Stem Fiber Composite (PPSFC) using Eqs. (8)–(14) and presented in **Tables 3** and **4**, respectively.

**Figure 4** shows the variation of elastic modulus with fiber orientation, it can be seen that the highest value of 14,922 and 13027.5 MPa in the reference *x*-direction (*Ex*Þ is attained in the fiber orientation angle 0° for PEFBFC and PPSFC respectively. However as fiber orientation angle changes, there is a sharp drop in the value of elastic modulus in the reference *x*-direction to a respective lowest value of 7030.96 and 6817.18 MPa as the fiber orientation angle increased to 90°. On the contrary, the lowest value of 7030.96 and 6817.18 MPa were recorded for elastic modulus in the

� <sup>2</sup>*v*<sup>12</sup> *E*1

� <sup>2</sup>*v*<sup>12</sup> *E*1

**3.3 Variation of engineering elastic constants with fiber orientation** *θ*

**S/N** *θ***°** *Ex Ey G*xy *v*xy *v*yx *mx my Ex*

*Variation of engineering elastic constants with fiber orientation* θ *in PEFBFC.*

1 0 14922.000 7030.962 3622.990 0.380 0.179 0.000 0.000 2.122 0.515 2 5 14769.770 7053.373 3629.270 0.378 0.180 0.116 0.037 2.101 0.516 3 10 14337.670 7121.323 3647.477 0.372 0.185 0.221 0.073 2.039 0.519 4 15 13690.480 7236.950 3675.727 0.362 0.191 0.304 0.111 1.947 0.523 5 20 12911.620 7403.786 3710.985 0.350 0.201 0.362 0.150 1.836 0.528 6 25 12081.300 7626.692 3749.256 0.336 0.212 0.395 0.190 1.718 0.533 7 30 11262.730 7911.712 3785.946 0.320 0.225 0.405 0.231 1.602 0.538 8 35 10497.990 8265.787 3816.391 0.304 0.239 0.398 0.271 1.493 0.543 9 40 9810.462 8696.213 3836.528 0.288 0.255 0.377 0.310 1.395 0.546 10 45 9209.655 9209.655 3843.571 0.271 0.271 0.346 0.346 1.310 0.547 11 50 8696.213 9810.462 3836.528 0.255 0.288 0.310 0.377 1.237 0.546 12 55 8265.787 10497.990 3816.391 0.239 0.304 0.271 0.398 1.176 0.543 13 60 7911.712 11262.730 3785.946 0.225 0.320 0.231 0.405 1.125 0.538 14 65 7626.692 12081.300 3749.256 0.212 0.336 0.190 0.395 1.085 0.533 15 70 7403.786 12911.620 3710.985 0.201 0.350 0.150 0.362 1.053 0.528 16 75 7236.950 13690.480 3675.727 0.191 0.362 0.111 0.304 1.029 0.523 17 80 7121.323 14337.670 3647.477 0.185 0.372 0.073 0.221 1.013 0.519 18 85 7053.373 14769.770 3629.270 0.180 0.378 0.037 0.116 1.003 0.516 19 90 7030.962 14922.000 3622.990 0.179 0.380 0.000 0.000 1.000 0.515

from Eq. (5) such that

**Table 3.**

**190**

*mx* ¼ *Ex c*

*my* <sup>¼</sup> *Ey cs*<sup>3</sup> <sup>1</sup>

3 *s* 1 *G*<sup>12</sup>

*G*<sup>12</sup>

*Variation of engineering elastic constants with fiber orientation* θ *in PPSFC.*

#### **Figure 4.**

*Variation of elastic modulus with fiber orientation in PEFBFC and PPSFC.*

reference *y*-direction (*Ey*Þas the fiber orientation angle increased from 0° to 90° reaching a peak value of 14,922 and 13027.5 MPa for PEFBFC and PPSFC respectively. The implication is that reinforcements are required to be aligned in the

direction of applied load [55]. Although Jones [56] intuitively suggested that highest value of material properties may not necessarily occur along the principal material directions, rather it is essential that transverse reinforcement is needed in unidirectional fiber composites which are subjected to multi axial loading [57].

As can be seen in **Figure 5** that shear modulus peaked at 45° fiber orientation and shear modulus was symmetric at about 45° fiber orientation angle for both PEFBFC and PPSFC considered. This implies that the higher in-plane shear resistance is achievable when fiber orientation is 45°. Also the respective minimum value of 3622.99 and 3332.83 MPa at fiber orientation 0° for PEFBFC and PPSFC can be seen to gradually increase to maximum values of 3843.57 and 3731.758 MPa at fiber orientation 45° and then reversed parabolically at 90° where it again reaches to 3622.99 and 3332.84 MPa. Similar trend was obtained by Farooq and Myler [58] who developed efficient procedures for determination of mechanical properties of carbon fiber-reinforced laminated composite panels. This trend in which the value of *Gxy* peaks at 45° fiber orientation angle and lowers at 0° and 90° fiber orientation angle indicates that off-axis reinforcement is very necessary for robust shear stiffness in unidirectional composites [57].

**Figure 6** shows variation of Poisson's ratio with fiber orientation, the graph depicts a gradual drop of major Poisson's ratio (*vxy*Þ for PEFBFC and PPSFC respectively from 0.38 and 0.29 when fibers are aligned at 0° orientation angle to a lowest value of 0.18 and 0.15 value when fibers were aligned at 90° orientation angle. Additionally, the minor Poissons ratio (*vyx*Þ for PEFBFC and PPSFC increased respectively from 0.18 and 0.15 when fibers are aligned at 0° orientation angle to a highest value of 0.38 and 0.29 value when fibers were aligned at 90° orientation angle.

**Figure 7** depicts the variation of shear coupling coefficient with fiber orientation, equal magnitude of shear coupling effect was obtained at 45° fiber orientation angle for both PEFBFC and PPSFC considered. Gibson [57] reported that shear coupling coefficient is a measure of the amount of shear strain developed in the *xy* plane per unit normal strain along the direction of the applied normal stress *σx*. **Figure 7** clearly indicate that the maximum value of the shear coupling coefficient in the reference *x*-direction for PEFBFC and PPSFC was attained at 30° fiber orientation angle while the coefficient in the reference *y*-direction for PEFBFC and

PPSFC was attained at 60° fiber orientation angle. This is an indication that as the

**4. Tsai-Hill failure criteria assessment of longitudinal tensile strength**

Failure theory is essential in determining whether the composite has failed. Literature review has shown that results of failure prediction depend on failure criterion applied and one major failure criteria used in the industries is Tsai-Hill and

shear-coupling ratio increases, the amount of shear coupling increases.

*Variation of shear coupling coefficient with fiber orientation in PEFBFC and PPSFC.*

*Variation of Poisson's ratio with fiber orientation in PEFBFC and PPSFC.*

*DOI: http://dx.doi.org/10.5772/intechopen.90890*

*Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites…*

**Figure 7.**

**193**

**Figure 6.**

**Figure 5.** *Variation of shear modulus with fiber orientation in PEFBFC and PPSFC.*

*Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites… DOI: http://dx.doi.org/10.5772/intechopen.90890*

**Figure 6.** *Variation of Poisson's ratio with fiber orientation in PEFBFC and PPSFC.*

**Figure 7.** *Variation of shear coupling coefficient with fiber orientation in PEFBFC and PPSFC.*

PPSFC was attained at 60° fiber orientation angle. This is an indication that as the shear-coupling ratio increases, the amount of shear coupling increases.
