3. Analytical models for the study of mechanical properties of long fiber-reinforced composites

Numerical modeling of the mechanical properties of the composite is a very difficult problem because there are many unknown parameters that come into model simulations, which are discussed in this chapter. Therefore, some parameters need to be properly verified with analytical models. It is assumed that though mechanical properties of the sample are formed from uniformly spaced transversely isotropic structure, its theoretical description is difficult, as shown in Figure 4.

The model of the transverse isotropic fiber composite structure can be defined by six independent elastic constants through the constitutive Eq. (9). The mechanical properties, such as composite structures, are also affected by the volume of fibers V<sup>f</sup> and matrix V<sup>m</sup>:

FEM Analysis of Mechanical and Structural Properties of Long Fiber-Reinforced Composites http://dx.doi.org/10.5772/intechopen.71881 7

Figure 4. Model of idealized transverse isotropic fiber composite structure.

volume fraction of fibers in the composite maximally 55–80% of the total volume of the composite structure. Ideally, these values can be increased by precisely placing fiber tows side by side. A limit state of volume fiber fraction, that is, 100%, cannot be achieved due to the necessity of the presence of the matrix. Also by perfectly precise laying of fiber strands, the strands will always have a certain fill value that will never be equal to 1 in the geometric configuration. Perfectly precise laying of fiber tows does not provide 100% of volume filling due to fiber cross section. However, it should be noted that the optimum ratio of fiber rein-

Table 1. Examples of physical and geometrical parameters of the selected samples of composite structures reinforced

CF 24K/PUR Huntsman 213 747 22 78 15 85 1.8 1.1 1.2

The influence of selected physical parameters on the geometric parameter h of some tested samples from long fiber-reinforced composite structures is shown in Table 1. These parameters can then be used to establish numerical models. Other input parameters that are required for the numerical model have to be obtained by measuring the test samples. Other input parameters that are necessary for the numerical model are obtained by testing com-

3. Analytical models for the study of mechanical properties of long

Numerical modeling of the mechanical properties of the composite is a very difficult problem because there are many unknown parameters that come into model simulations, which are discussed in this chapter. Therefore, some parameters need to be properly verified with analytical models. It is assumed that though mechanical properties of the sample are formed from uniformly spaced transversely isotropic structure, its theoretical description is difficult, as

The model of the transverse isotropic fiber composite structure can be defined by six independent elastic constants through the constitutive Eq. (9). The mechanical properties, such as composite structures, are also affected by the volume of fibers V<sup>f</sup> and

forcement is in the range of the synergistic effect, that is, V<sup>f</sup> ≈ 0:4÷0:65:

posite samples.

Name of fibers mf <sup>∗</sup>

GF 1600 tex/PUR Huntsman

with long fibers.

M10R

CF prepreg HEXPLY-

[g m<sup>2</sup> ] m<sup>m</sup> <sup>∗</sup> [g m<sup>2</sup> ]

6 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

Note: m<sup>f</sup> <sup>∗</sup>, m<sup>m</sup> <sup>∗</sup> represent area weight, mc <sup>∗</sup> is the total weight of composite.

M<sup>f</sup> [%] M<sup>V</sup> [%] Vf [%]

560 600 48 52 30 70 2.45 1.1 1.2

150 91.96 62 38 52 48 1.8 1.2 0.22

V<sup>m</sup> [%] rf [g cm<sup>3</sup> ] rm [g cm<sup>3</sup> ] h [mm]

shown in Figure 4.

matrix V<sup>m</sup>:

fiber-reinforced composites

$$\begin{Bmatrix} \varepsilon\_{11} \\ \varepsilon\_{22} \\ \varepsilon\_{33} \\ \gamma\_{12} \\ \gamma\_{23} \\ \gamma\_{13} \end{Bmatrix} = \begin{bmatrix} 1/E\_{11} & -\nu\_{12}/E\_{11} & -\nu\_{12}/E\_{11} & 0 & 0 & 0 \\ -\nu\_{12}/E\_{11} & 1/E\_{22} & -\nu\_{23}/E\_{22} & 0 & 0 & 0 \\ -\nu\_{12}/E\_{11} & -\nu\_{23}/E\_{22} & 1/E\_{22} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1/G\_{23} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1/G\_{12} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1/G\_{12} \end{bmatrix} \cdot \begin{Bmatrix} \sigma\_{11} \\ \sigma\_{22} \\ \sigma\_{33} \\ \tau\_{12} \\ \tau\_{23} \\ \tau\_{13} \end{Bmatrix} \tag{9}$$

where σii, εii are the principal stresses and strains in the transversal isotropic composite in individual directions of the coordinate system x1, x2, x3, whereas σ<sup>11</sup> > σ<sup>22</sup> ¼ σ33, ε<sup>11</sup> 6¼ ε<sup>22</sup> ¼ ε33, and τ12, τ<sup>23</sup> ¼ τ<sup>13</sup> are the shear stresses in the given planes, γ12, γ<sup>23</sup> ¼ γ<sup>13</sup> expresses the shear to individual planes, E11, E<sup>22</sup> ¼ E<sup>33</sup> expresses the longitudinal and transverse modulus of elasticity, G12, G<sup>23</sup> is the shear modulus in the plane of the principal load direction and in a plane perpendicular to the principal load direction, ν<sup>12</sup> is the Poisson ratio in the principal direction of the load, and ν<sup>23</sup> is the Poisson ratio in a plane perpendicular to the principal load direction.

For the corresponding model, the interconnection of individual components A, B, C must be included (see Figure 2) to create a multiphase system approaching the behavior of composite structures. Therefore, the problem of modeling a composite can be treated as a continuum (a solid model without a geometric arrangement of individual components) or by creating a completely new model with structural parameters, that is, the individual components will be included in the structured unit. The problem of analytical modeling of mechanical properties of general fiber structures through a structural unit is described, among others, by Wyk for the study of interfiber contacts [11] and by Neckář [12]. However, the description of the mechanical properties of the fibrous composite structure is more difficult and has not yet been properly described. This is probably due to the fact that knowledge of the deformation mechanism and damage process is more important for understanding the mechanical properties than the knowledge of the absolute strength that cannot be determined with sufficient precision. This is due to the fact that it is not possible to comprehensively construct a general energy theory (to derive empirical relationships for deformation work) based on statistical characteristics, as can be done with very good accuracy for other anisotropic structures (Petrů et al. [13, 14]). The problem is that the individual components composing the composite structure cannot be reliably quantified even with homogeneous isotropic materials (matrices, glass fibers), let alone anisotropic structures such as carbon fibers (the theoretical value presented in the data sheet is different than value determined experimentally). Therefore, the main problems are related to the complexity of the description and modeling of deformation and the consequent character of the stress (stress concentration under loading). This is mainly due to technological influences in composite production (influence of temperature, humidity, and initial microporosity) that cannot be predicted for model simulations, and it is also relatively difficult to experimentally identify these parameters.<sup>4</sup>

Over time, there have been widespread analytical relationships to form the approach to obtain all elastic constants that can be used by these models, which are given as follows:


#### 3.1. Phenomenological models

In the past, phenomenological models have been created as the primary mathematical derivation of the mechanical properties of transversally isotropic fibrous composite structures but can be used well today. Such models include the Voigt and Reuss models. These are models using the mixing rule (mixing of the individual input components, i.e., fibers and matrices), while the Voigt model is very well usable for determining the elastic constants E11,ν<sup>12</sup> defined by relationships (10 and 11) and is characterized by isotropic strain. The Reuss model is usable for determination E22, G<sup>12</sup> defined by relationships (12 and 13) and unlike the Voigt model is characterized by isotropic stress.

$$\frac{d\sigma\_{11}}{d\varepsilon\_{11}} = V^f \frac{d\sigma^f}{d\varepsilon^f} + V^m \frac{d\sigma^m}{d\varepsilon^m} \implies E\_{11} = V^f E\_{11}^f + V^m E^m \tag{10}$$

$$\nu\_{12} = V^{\dagger} \nu\_{12}^{\dagger} + V^{m} \nu^{m} \tag{11}$$

where E<sup>f</sup>

the fiber.

where ζ<sup>f</sup>

5

Correction factors ζ<sup>f</sup>

variable function 0 < ζ

0

• Halpin-Tsai model

agreement with experiments.

11, E<sup>f</sup>

3.2. Semiempirical models

• Modified model according to the mixing rule

make a correction for E22, G<sup>12</sup> according to Eqs. (14 and 15).

, <sup>ζ</sup><sup>m</sup> can be express as <sup>ζ</sup><sup>f</sup> <sup>¼</sup> Ef

< 1, whereas preferred is ζ

1 E<sup>22</sup>

> 1 G<sup>12</sup> ¼

, ζ<sup>m</sup> are correction factors,<sup>5</sup> according to Younes et al. [18].

¼ ζf Vf Ef 22

> Vf Gf 12 þ ζ 0 V<sup>m</sup> Gm

This is a model that is implemented in a number of numerical programs by using finite element method (FEM). This model is developed as a semiempirical model [19] with correction of E22, G12: Its semiempirical derivation (16–17) using correction factors ζ, ξ has a very good

<sup>E</sup><sup>22</sup> <sup>¼</sup> Em <sup>1</sup> <sup>þ</sup> ξζV<sup>f</sup>

<sup>G</sup><sup>12</sup> <sup>¼</sup> Gm <sup>1</sup> <sup>þ</sup> ξζV<sup>f</sup>

Ef

≈ 0:5 � 0:6:

11V<sup>f</sup> þ 1�ν f 12 ν <sup>f</sup> ð Þ <sup>21</sup> Emþνm<sup>ν</sup>

0

<sup>1</sup> � <sup>ζ</sup>V<sup>f</sup>

<sup>1</sup> � <sup>ζ</sup>V<sup>f</sup>

f 21Ef <sup>11</sup> ½ �<sup>V</sup><sup>m</sup>

11V<sup>f</sup> <sup>þ</sup>EmVm , <sup>ζ</sup><sup>m</sup> <sup>¼</sup> Em <sup>V</sup>m<sup>þ</sup> <sup>1</sup>�νmν<sup>m</sup> ð ÞEf

<sup>V</sup><sup>f</sup> <sup>þ</sup> <sup>ζ</sup> 0

<sup>þ</sup> <sup>ζ</sup>mVm

Em (14)

<sup>V</sup><sup>m</sup> (15)

(16)

(17)

Ef

<sup>11</sup>� <sup>1</sup>�νm<sup>ν</sup> <sup>f</sup> ð Þ <sup>12</sup> Em ½ �<sup>V</sup><sup>f</sup>

11V<sup>f</sup> <sup>þ</sup>EmVm , <sup>ζ</sup>

0 is a

<sup>22</sup> are the longitudinal and transverse modulus of the fiber elasticity, <sup>G</sup><sup>f</sup>

FEM Analysis of Mechanical and Structural Properties of Long Fiber-Reinforced Composites

shear module, and ν<sup>12</sup><sup>f</sup> is the Poisson ratio in the plane of the principal direction of the load of

Semiempirical models were created later than phenomenological models, and based on the new information and knowledge, they are still being updated. Their development led in particular to the further expansion of the Voigt and Reuss models because these models have been modified by correction factors to specify the resulting elastic constants for the given types of input components. This category includes models that are implemented in certain modifications in finite element softwares such as the Halpin-Tsai model or the Chamis model.

Modified model according to the mixing rule is derived from Voigt [16] and Reuss [17], and for elastic constants, E11, ν<sup>12</sup> is defined according to Eqs. (10 and 11). Modification occurs with constants E22, G12, because the resulting difference between the results obtained by the measurements and the relationships (12–13) is usually noticeable. Therefore, it was necessary to

<sup>12</sup> is a fiber

9

http://dx.doi.org/10.5772/intechopen.71881

$$E\_{22} = \frac{E\_{22}^f E^m}{E^m V^f + E\_{22}^f E^m} \tag{12}$$

$$G\_{12} = \frac{G\_{12}^{\circ}G^{m}}{G^{m}V^{\circ} + G\_{12}^{\circ}E^{m}} \tag{13}$$

<sup>4</sup> In the advanced model, simulations can be assembled material models with any parameters, including the statistical parameters, which describe technological production factors, for example, through the theory of random fields as defined by Bittnar and Šejnoha [15]. The problem lies in the identification of the effects and the subsequent statistical evaluation.

where E<sup>f</sup> 11, E<sup>f</sup> <sup>22</sup> are the longitudinal and transverse modulus of the fiber elasticity, <sup>G</sup><sup>f</sup> <sup>12</sup> is a fiber shear module, and ν<sup>12</sup><sup>f</sup> is the Poisson ratio in the plane of the principal direction of the load of the fiber.

### 3.2. Semiempirical models

characteristics, as can be done with very good accuracy for other anisotropic structures (Petrů et al. [13, 14]). The problem is that the individual components composing the composite structure cannot be reliably quantified even with homogeneous isotropic materials (matrices, glass fibers), let alone anisotropic structures such as carbon fibers (the theoretical value presented in the data sheet is different than value determined experimentally). Therefore, the main problems are related to the complexity of the description and modeling of deformation and the consequent character of the stress (stress concentration under loading). This is mainly due to technological influences in composite production (influence of temperature, humidity, and initial microporosity) that cannot be predicted for model simulations, and it is also

Over time, there have been widespread analytical relationships to form the approach to obtain

In the past, phenomenological models have been created as the primary mathematical derivation of the mechanical properties of transversally isotropic fibrous composite structures but can be used well today. Such models include the Voigt and Reuss models. These are models using the mixing rule (mixing of the individual input components, i.e., fibers and matrices), while the Voigt model is very well usable for determining the elastic constants E11,ν<sup>12</sup> defined by relationships (10 and 11) and is characterized by isotropic strain. The Reuss model is usable for determination E22, G<sup>12</sup> defined by relationships (12 and 13) and unlike the Voigt model is

<sup>d</sup>ε<sup>m</sup> ) <sup>E</sup><sup>11</sup> <sup>¼</sup> <sup>V</sup><sup>f</sup>

22E<sup>m</sup> EmVf <sup>þ</sup> Ef

12G<sup>m</sup> <sup>G</sup>mVf <sup>þ</sup> Gf

In the advanced model, simulations can be assembled material models with any parameters, including the statistical parameters, which describe technological production factors, for example, through the theory of random fields as defined by Bittnar and Šejnoha [15]. The problem lies in the identification of the effects and the subsequent statistical evaluation.

Ef

<sup>11</sup> <sup>þ</sup> <sup>V</sup>mEm (10)

<sup>12</sup> <sup>þ</sup> <sup>V</sup><sup>m</sup>ν<sup>m</sup> (11)

22Em (12)

12Em (13)

all elastic constants that can be used by these models, which are given as follows:

relatively difficult to experimentally identify these parameters.<sup>4</sup>

8 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

• phenomenological models.

3.1. Phenomenological models

characterized by isotropic stress.

4

dσ<sup>11</sup> dε<sup>11</sup> <sup>¼</sup> <sup>V</sup><sup>f</sup> <sup>d</sup>σ<sup>f</sup>

<sup>d</sup>ε<sup>f</sup> <sup>þ</sup> <sup>V</sup><sup>m</sup> <sup>d</sup>σ<sup>m</sup>

<sup>ν</sup><sup>12</sup> <sup>¼</sup> <sup>V</sup><sup>f</sup>

<sup>E</sup><sup>22</sup> <sup>¼</sup> <sup>E</sup><sup>f</sup>

<sup>G</sup><sup>12</sup> <sup>¼</sup> <sup>G</sup><sup>f</sup>

ν f

• semiempirical models. • homogenized models.

Semiempirical models were created later than phenomenological models, and based on the new information and knowledge, they are still being updated. Their development led in particular to the further expansion of the Voigt and Reuss models because these models have been modified by correction factors to specify the resulting elastic constants for the given types of input components. This category includes models that are implemented in certain modifications in finite element softwares such as the Halpin-Tsai model or the Chamis model.

#### • Modified model according to the mixing rule

Modified model according to the mixing rule is derived from Voigt [16] and Reuss [17], and for elastic constants, E11, ν<sup>12</sup> is defined according to Eqs. (10 and 11). Modification occurs with constants E22, G12, because the resulting difference between the results obtained by the measurements and the relationships (12–13) is usually noticeable. Therefore, it was necessary to make a correction for E22, G<sup>12</sup> according to Eqs. (14 and 15).

$$\frac{1}{E\_{22}} = \frac{\zeta^f V^f}{E\_{22}^f} + \frac{\zeta^m V^m}{E^m} \tag{14}$$

$$\frac{1}{\frac{1}{G\_{12}}} = \frac{\frac{V^{\prime}}{G\_{t2}^{\prime}} + \frac{\zeta^{\prime}V^{m}}{G^{m}}}{V^{\prime} + \zeta^{\prime}V^{m}}\tag{15}$$

where ζ<sup>f</sup> , ζ<sup>m</sup> are correction factors,<sup>5</sup> according to Younes et al. [18].

• Halpin-Tsai model

This is a model that is implemented in a number of numerical programs by using finite element method (FEM). This model is developed as a semiempirical model [19] with correction of E22, G12: Its semiempirical derivation (16–17) using correction factors ζ, ξ has a very good agreement with experiments.

$$E\_{22} = E^{\prime\prime} \left( \frac{1 + \xi \zeta V^{\prime}}{1 - \zeta V^{\prime}} \right) \tag{16}$$

$$\mathbf{G}\_{12} = \mathbf{G}''' \left( \frac{1 + \xi \zeta V^{\circ}}{1 - \zeta V^{\circ}} \right) \tag{17}$$

<sup>5</sup> Correction factors ζ<sup>f</sup> , <sup>ζ</sup><sup>m</sup> can be express as <sup>ζ</sup><sup>f</sup> <sup>¼</sup> Ef 11V<sup>f</sup> þ 1�ν f 12 ν <sup>f</sup> ð Þ <sup>21</sup> Emþνm<sup>ν</sup> f 21Ef <sup>11</sup> ½ �<sup>V</sup><sup>m</sup> Ef 11V<sup>f</sup> <sup>þ</sup>EmVm , <sup>ζ</sup><sup>m</sup> <sup>¼</sup> Em <sup>V</sup>m<sup>þ</sup> <sup>1</sup>�νmν<sup>m</sup> ð ÞEf <sup>11</sup>� <sup>1</sup>�νm<sup>ν</sup> <sup>f</sup> ð Þ <sup>12</sup> Em ½ �<sup>V</sup><sup>f</sup> Ef 11V<sup>f</sup> <sup>þ</sup>EmVm , <sup>ζ</sup> 0 is a variable function 0 < ζ 0 < 1, whereas preferred is ζ 0 ≈ 0:5 � 0:6:

where <sup>ζ</sup> correction factor, for which it is valid <sup>ζ</sup> <sup>¼</sup> <sup>M</sup><sup>f</sup> <sup>=</sup>Mm�<sup>1</sup> Mf <sup>=</sup>Mmþ<sup>ξ</sup> , ξ is constant, which is for E<sup>22</sup> equal to 1, and for G<sup>12</sup> is equal to 2, M ¼ E or G in the case of an expression E<sup>22</sup> and G<sup>12</sup> according to Eqs. (12–13).

### • Chamis model

This is another semiempirical model [20], which was unlike previous models developed not only for independent elastic constants E11, E22,G12,ν<sup>12</sup> but also for G23: The determination of E11, ν<sup>12</sup> is again based on Voigt and Reusse according to Eqs. (10–11). The Chamis model for calculating other elastic constants introduces a square root of the volume of fiber ffiffiffiffiffi <sup>V</sup><sup>f</sup> <sup>p</sup> , which has in Eqs. (18–20), the meaning of fiber incompressibility, which is in line with principle of mass conservation.

$$E\_{22} = \frac{E^{\text{'''}}}{1 - \sqrt{V^{\text{f}}} \left(1 - E^{\text{m}} / E\_{22}^{\text{f}}\right)} \tag{18}$$

<sup>E</sup><sup>22</sup> <sup>¼</sup> <sup>V</sup><sup>f</sup> <sup>þ</sup> <sup>V</sup>ma<sup>11</sup> � � � <sup>V</sup><sup>f</sup> <sup>þ</sup> <sup>V</sup>ma<sup>22</sup> � �

Sf

<sup>11</sup> <sup>þ</sup> <sup>V</sup>ma22Sm <sup>22</sup> � � <sup>þ</sup> <sup>V</sup><sup>f</sup>

<sup>G</sup><sup>m</sup> <sup>þ</sup> <sup>V</sup>ma66G<sup>f</sup>

ii are the matrix components, which relate to fiber and matrix ratios in the composite

<sup>12</sup>, a<sup>12</sup> <sup>¼</sup> Sf

<sup>23</sup> <sup>¼</sup> Sf ,m

<sup>11</sup> , Sf ,m

<sup>23</sup> � � <sup>þ</sup> <sup>V</sup>ma<sup>44</sup> Sm

<sup>G</sup><sup>12</sup> <sup>¼</sup> <sup>V</sup><sup>f</sup> <sup>þ</sup> <sup>V</sup>ma<sup>66</sup> � � � <sup>G</sup><sup>f</sup>

Vf

<sup>G</sup><sup>23</sup> <sup>¼</sup> <sup>1</sup>=<sup>2</sup> <sup>V</sup><sup>f</sup> <sup>þ</sup> <sup>V</sup>ma<sup>44</sup> � �

<sup>22</sup> � <sup>S</sup><sup>f</sup>

<sup>21</sup> ¼ �ν

f ,m <sup>12</sup> <sup>=</sup>Ef ,m

4. Numerical models for the study of mechanical properties of long fiber-

Measurement and analytical models of long fiber-reinforced composite structures designed to study mechanical properties are generally able to provide only limited information. This is due to the fact that the measurements are limited by the possibilities of positioning of the sensors and also by the fact that some properties cannot be measured well (e.g., the distribution of the main stress and deformation in the composite structure). The knowledge of the distribution of the main stresses and deformations in the structure is important for assessing how the structure is changed and under which stress. In this case, the corresponding model simulation using numeric methods represents a significant support for the development. Very suitable is to build model simulation in finite element method (FEM), but other numerical methods, such as discrete element (DEM), boundary element (BEM) or finite volume method (FVM) method, are also available. The mechanical loading of composite causes many different processes in the inner structure that varies with the actual deformation. Therefore, it is necessary to simplify or neglect some characteristic features in modeling of such structures. A major problem of mechanical properties modeling of composite structures is in particular the description of the principal stresses in short time Δt ¼ tiþ<sup>1</sup> � ti. The solution of problem of composite with boundary conditions under tensile loading lies not only in the specification of the correct boundary conditions and material properties but also in the design of the proposed finite element mesh. The FEM programs are currently very sophisticated and allow the solution of a continuous problem transform into a final solution where the corresponding geometric simple subareas (finite elements) can be designed in the preprocessor. Let Ω ⊂ ℜ<sup>3</sup> is the continuous area of the three-dimensional space in which the problem is solved. Its borders

V<sup>f</sup> S<sup>f</sup>

V<sup>m</sup> S<sup>m</sup>

12G<sup>m</sup>

FEM Analysis of Mechanical and Structural Properties of Long Fiber-Reinforced Composites

12

<sup>22</sup> � Sm

<sup>12</sup> � <sup>S</sup><sup>m</sup>

<sup>32</sup> ¼ � ν

<sup>21</sup> � <sup>S</sup><sup>f</sup> <sup>21</sup> � �a<sup>12</sup>

<sup>23</sup> � � (23)

http://dx.doi.org/10.5772/intechopen.71881

<sup>22</sup>, a<sup>22</sup> ¼ a<sup>44</sup> ¼ 0, 35þ ð1� 0; 35Þ

<sup>12</sup> � �ð Þ <sup>a</sup><sup>11</sup> � <sup>a</sup><sup>22</sup> <sup>=</sup> Sf

f ,m <sup>23</sup> <sup>=</sup>Ef ,m <sup>22</sup> . (21)

11

(22)

<sup>11</sup> � <sup>S</sup><sup>m</sup> <sup>11</sup> � �,

<sup>V</sup><sup>f</sup> <sup>þ</sup> <sup>V</sup>ma<sup>11</sup> � � <sup>V</sup><sup>f</sup>

structure as reported by Huang [23, 24], where <sup>a</sup><sup>11</sup> <sup>¼</sup> Em=Ef

h

<sup>12</sup> <sup>¼</sup> Sf ,m

<sup>22</sup>, a<sup>66</sup> <sup>¼</sup> <sup>0</sup>, <sup>3</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>0</sup>; <sup>3</sup> <sup>0</sup>; <sup>5</sup>E<sup>m</sup><sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>ν</sup><sup>m</sup> ð �=Gf

<sup>22</sup> Sf ,m

<sup>22</sup> <sup>¼</sup> <sup>1</sup>=Ef ,m

where aii, Sf ,m

<sup>11</sup> <sup>¼</sup> <sup>1</sup>=E<sup>f</sup> ,m

<sup>11</sup> , Sf ,m

reinforced composites

E<sup>m</sup>=Ef

S<sup>f</sup> ,m

$$G\_{12} = \frac{G^m}{1 - \sqrt{V^\ell} \left(1 - G^m / G\_{12}^\ell\right)}\tag{19}$$

$$G\_{23} = \frac{G'''}{1 - \sqrt{V'} \left(1 - G''' / G\_{23}'\right)}\tag{20}$$

where G<sup>f</sup> <sup>23</sup> is the shear modulus of the fiber elasticity in a plane perpendicular to the principle direction of loading.

#### 3.3. Homogenized models

Homogenized models are generalized models that can be used to determine very accurate values of elastic constants for developed composite structures reinforced by longitudinally laid fibers. Such models include, for example, the Mori-Tanaka model [21], a consistent model created by Hill [22] or the Bridging model. Their applicability compared to phenomenological or semiempirical models largely limits the more difficult determination of all constants entering to homogenized models. An example is the Eshelby toughness tensor that can be used for inclusion, which is introduced in both the Mori-Tanaka model and the consistent model. In view of this, from homogenized models, the Brindling model can be used to determine the elastic constants.

• Bridging model

This is a model that is developed to predict the stiffness and strength of transverse isotropic fiber composites. The elastic properties are for the elastic modulus E11, ν<sup>12</sup> the same as for Voigt and Reusse models (10–11). Elastic constants E22, G12, G<sup>23</sup> can be expressed using the Bridging model (21–23).

FEM Analysis of Mechanical and Structural Properties of Long Fiber-Reinforced Composites http://dx.doi.org/10.5772/intechopen.71881 11

$$E\_{22} = \frac{\left(V^{\ell} + V^{m}a\_{11}\right) \cdot \left(V^{\ell} + V^{m}a\_{22}\right)}{\left(V^{\ell} + V^{m}a\_{11}\right)\left(V^{\ell}S\_{11}^{\ell} + V^{m}a\_{22}S\_{22}^{m}\right) + V^{\ell}V^{m}\left(S\_{21}^{m} - S\_{21}^{\ell}\right)a\_{12}}\tag{21}$$

$$G\_{12} = \frac{\left(V^f + V^m a\_{66}\right) \cdot G\_{12}^f G^m}{V^f G^m + V^m a\_{66} G\_{12}^f} \tag{22}$$

$$\mathbf{G\_{23}} = \frac{1/2\left(V^{\not\!f} + V^{\not\!u}a\_{44}\right)}{V^{\not\!f}\left(S\_{22}^{\not\!f} - S\_{23}^{\not\!f}\right) + V^{\not\!u}a\_{44}\left(S\_{22}^{\not\!u} - S\_{23}^{\not\!u}\right)}\tag{23}$$

where aii, Sf ,m ii are the matrix components, which relate to fiber and matrix ratios in the composite structure as reported by Huang [23, 24], where <sup>a</sup><sup>11</sup> <sup>¼</sup> Em=Ef <sup>22</sup>, a<sup>22</sup> ¼ a<sup>44</sup> ¼ 0, 35þ ð1� 0; 35Þ E<sup>m</sup>=Ef <sup>22</sup>, a<sup>66</sup> <sup>¼</sup> <sup>0</sup>, <sup>3</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>0</sup>; <sup>3</sup> <sup>0</sup>; <sup>5</sup>E<sup>m</sup><sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>ν</sup><sup>m</sup> ð �=Gf <sup>12</sup>, a<sup>12</sup> <sup>¼</sup> Sf <sup>12</sup> � <sup>S</sup><sup>m</sup> <sup>12</sup> � �ð Þ <sup>a</sup><sup>11</sup> � <sup>a</sup><sup>22</sup> <sup>=</sup> Sf <sup>11</sup> � <sup>S</sup><sup>m</sup> <sup>11</sup> � �, h S<sup>f</sup> ,m <sup>11</sup> <sup>¼</sup> <sup>1</sup>=E<sup>f</sup> ,m <sup>11</sup> , Sf ,m <sup>22</sup> <sup>¼</sup> <sup>1</sup>=Ef ,m <sup>22</sup> Sf ,m <sup>12</sup> <sup>¼</sup> Sf ,m <sup>21</sup> ¼ �ν f ,m <sup>12</sup> <sup>=</sup>Ef ,m <sup>11</sup> , Sf ,m <sup>23</sup> <sup>¼</sup> Sf ,m <sup>32</sup> ¼ � ν f ,m <sup>23</sup> <sup>=</sup>Ef ,m <sup>22</sup> .

where <sup>ζ</sup> correction factor, for which it is valid <sup>ζ</sup> <sup>¼</sup> <sup>M</sup><sup>f</sup>

10 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

Eqs. (12–13).

• Chamis model

mass conservation.

where G<sup>f</sup>

direction of loading.

elastic constants.

model (21–23).

• Bridging model

3.3. Homogenized models

<sup>=</sup>Mm�<sup>1</sup> Mf <sup>=</sup>Mmþ<sup>ξ</sup>

to 1, and for G<sup>12</sup> is equal to 2, M ¼ E or G in the case of an expression E<sup>22</sup> and G<sup>12</sup> according to

This is another semiempirical model [20], which was unlike previous models developed not only for independent elastic constants E11, E22,G12,ν<sup>12</sup> but also for G23: The determination of E11, ν<sup>12</sup> is again based on Voigt and Reusse according to Eqs. (10–11). The Chamis model for

has in Eqs. (18–20), the meaning of fiber incompressibility, which is in line with principle of

ffiffiffiffiffi <sup>V</sup><sup>f</sup> <sup>p</sup>

ffiffiffiffiffi <sup>V</sup><sup>f</sup> <sup>p</sup>

ffiffiffiffiffi <sup>V</sup><sup>f</sup> <sup>p</sup>

Homogenized models are generalized models that can be used to determine very accurate values of elastic constants for developed composite structures reinforced by longitudinally laid fibers. Such models include, for example, the Mori-Tanaka model [21], a consistent model created by Hill [22] or the Bridging model. Their applicability compared to phenomenological or semiempirical models largely limits the more difficult determination of all constants entering to homogenized models. An example is the Eshelby toughness tensor that can be used for inclusion, which is introduced in both the Mori-Tanaka model and the consistent model. In view of this, from homogenized models, the Brindling model can be used to determine the

This is a model that is developed to predict the stiffness and strength of transverse isotropic fiber composites. The elastic properties are for the elastic modulus E11, ν<sup>12</sup> the same as for Voigt and Reusse models (10–11). Elastic constants E22, G12, G<sup>23</sup> can be expressed using the Bridging

<sup>1</sup> � <sup>E</sup><sup>m</sup>=E<sup>f</sup>

<sup>1</sup> � <sup>G</sup><sup>m</sup>=G<sup>f</sup>

<sup>1</sup> � <sup>G</sup><sup>m</sup>=G<sup>f</sup>

<sup>23</sup> is the shear modulus of the fiber elasticity in a plane perpendicular to the principle

22

12

23

� � (18)

� � (19)

� � (20)

calculating other elastic constants introduces a square root of the volume of fiber

<sup>E</sup><sup>22</sup> <sup>¼</sup> Em 1 �

<sup>G</sup><sup>12</sup> <sup>¼</sup> Gm 1 �

<sup>G</sup><sup>23</sup> <sup>¼</sup> Gm 1 �

, ξ is constant, which is for E<sup>22</sup> equal

ffiffiffiffiffi <sup>V</sup><sup>f</sup> <sup>p</sup>

, which
