2. Measurement of mechanical properties of composite samples with long fiber reinforcing

The determination of unknown parameters of composite materials has to be performed by experimental measurements. These parameters represent input data for numerical simulations. For a complete description of the properties, it is important to make measurements on both the fiber reinforcement (tow) and the matrix as well as on the resulting long fiberreinforced composite structures (matrix-bonded fibers). Measurement of the mechanical properties of the samples is carried out according to standard laboratory tests, which are divided according to the time course of the applied load. Tests can be divided into static and dynamic. It can thus perform the tensile test at a constant or cyclic loading of the sample, three-point bending strength, and Charpy impact test, as shown in Figure 3. The samples may be formed in the "dog bone" shape or, optionally, in the form of a rectangle of defined length L, width b, and thickness h, <sup>3</sup> whereas they can be used for short- or long-term test.

The characteristic physical properties of samples of long fiber-reinforced composites are influenced by weight and volume ratios of individual input components (fiber reinforcement and matrices) that ultimately affect design parameters (mechanical properties and weight of the structure). The mass and volume amounts of the fibers and the matrix in the composite structure sample can be defined according to the following relationships (1–5).

$$m^c = m^f + m^m \tag{1}$$

Mf <sup>¼</sup> <sup>1</sup> � <sup>M</sup><sup>m</sup> (3)

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

<sup>V</sup><sup>f</sup> <sup>¼</sup> <sup>1</sup> � <sup>V</sup><sup>m</sup> (5)

, m<sup>m</sup> is the weight of fibers and matrix, M<sup>f</sup>

, v<sup>m</sup> is the volume of fibers and matrix. Volume amount of

rf

Vf

, V<sup>m</sup> is the volume amount of fibers and matrix, v<sup>c</sup> is

<sup>V</sup><sup>f</sup> <sup>þ</sup> <sup>r</sup>mVm (7)

(8)

<sup>v</sup><sup>c</sup> (4)

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

5

=v<sup>c</sup> can then be expressed as

<sup>r</sup><sup>f</sup> <sup>þ</sup> <sup>V</sup><sup>m</sup>r<sup>m</sup> (6)

, M<sup>m</sup> is

<sup>V</sup><sup>f</sup> <sup>¼</sup> <sup>v</sup><sup>f</sup>

where m<sup>c</sup> is the total weight of composite, m<sup>f</sup>

density <sup>r</sup><sup>m</sup>, which is applied in Eq. (6). The total density <sup>r</sup><sup>c</sup> <sup>¼</sup> <sup>m</sup><sup>c</sup>

<sup>V</sup><sup>f</sup> <sup>¼</sup> Mf

Mf

of the composite structure can be expressed according to the relationship (8).

<sup>h</sup> <sup>¼</sup> <sup>m</sup><sup>f</sup> <sup>1</sup>

=r<sup>f</sup>

<sup>r</sup><sup>c</sup> <sup>¼</sup> <sup>r</sup><sup>f</sup>

r<sup>f</sup> þ 1

Gay and Hoa [10] reported that winding of the fibers onto-shaped geometry may achieve the

the weight amount of fibers and matrix, V<sup>f</sup>

the total volume of composite, and v<sup>f</sup>

(c) Charpy impact test.

<sup>v</sup><sup>c</sup> , V<sup>m</sup> <sup>¼</sup> <sup>v</sup><sup>m</sup>

Figure 3. Determination of mechanical properties of composite samples: (a) tensile test, (b) three-point bending test, and

fibers V<sup>f</sup> and matrix V<sup>m</sup> can be also expressed with the help of the fiber density r<sup>f</sup> and matrix

the sum of components, that is, the reinforcement and matrix (7). The characteristic thickness

<sup>=</sup>r<sup>f</sup> <sup>þ</sup> Mm=r<sup>m</sup> , Mf <sup>¼</sup> <sup>V</sup><sup>f</sup>

<sup>r</sup><sup>m</sup> � <sup>1</sup> � Mf Mf

$$M^{\ell} = \frac{m^{\ell}}{m^{c^{\prime}}}, \quad M^{m} = \frac{m^{m}}{m^{c}} \tag{2}$$

<sup>3</sup> Note: geometrical dimensions h, b, L can be also smaller, but it can lead to a problem with clamping of the sample to the jaws of dynamometer.

FEM Analysis of Mechanical and Structural Properties of Long Fiber-Reinforced Composites http://dx.doi.org/10.5772/intechopen.71881 5

and, of course, the chemical composition of the glass. The matrix, which affects the properties and usability of the resulting composite, has been epoxy used for both the testing sample and design of the developed composite construction. The composite production may result in imperfect bonding of the matrix fibers (e.g., low wettability of the fiber reinforcement in the matrix, bubble formation, etc.), which leads in mechanical defects in the composite structure, which often over grow into critical defects with a significant reduction in strength (Figure 1). The resulting strength of the composite structure affects mechanical properties of the selected fiber reinforcement and matrix, which are characterized by mechanical parameters, for example, the elastic modulus, Poisson number, or other parameters such as the creep and fracture properties of the individual components. In terms of strength, a significant role (if not most) plays the interfaces among the fibers and the matrix, which is shown in Figure 2. This is due to the fact that the characteristic properties of the interface create a mechanism that apparently causes the synergistic effect that provides their unique mechanical properties to composite structures. Although a number of theories have been compiled, the synergistic mechanism of

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

2. Measurement of mechanical properties of composite samples with long

The determination of unknown parameters of composite materials has to be performed by experimental measurements. These parameters represent input data for numerical simulations. For a complete description of the properties, it is important to make measurements on both the fiber reinforcement (tow) and the matrix as well as on the resulting long fiberreinforced composite structures (matrix-bonded fibers). Measurement of the mechanical properties of the samples is carried out according to standard laboratory tests, which are divided according to the time course of the applied load. Tests can be divided into static and dynamic. It can thus perform the tensile test at a constant or cyclic loading of the sample, three-point bending strength, and Charpy impact test, as shown in Figure 3. The samples may be formed in the "dog bone" shape or, optionally, in the form of a rectangle of defined length L, width b,

<sup>3</sup> whereas they can be used for short- or long-term test.

structure sample can be defined according to the following relationships (1–5).

Mf <sup>¼</sup> <sup>m</sup><sup>f</sup>

The characteristic physical properties of samples of long fiber-reinforced composites are influenced by weight and volume ratios of individual input components (fiber reinforcement and matrices) that ultimately affect design parameters (mechanical properties and weight of the structure). The mass and volume amounts of the fibers and the matrix in the composite

mc , Mm <sup>¼</sup> <sup>m</sup><sup>m</sup>

Note: geometrical dimensions h, b, L can be also smaller, but it can lead to a problem with clamping of the sample to the

<sup>m</sup><sup>c</sup> <sup>¼</sup> <sup>m</sup><sup>f</sup> <sup>þ</sup> <sup>m</sup><sup>m</sup> (1)

mc (2)

the phase interface is not yet clear.

fiber reinforcing

and thickness h,

jaws of dynamometer.

3

Figure 3. Determination of mechanical properties of composite samples: (a) tensile test, (b) three-point bending test, and (c) Charpy impact test.

$$M^f = 1 - M^m \tag{3}$$

$$V^f = \frac{\mathbf{v}^f}{\mathbf{v}^{c'}} \quad V^m = \frac{\mathbf{v}^m}{\mathbf{v}^c} \tag{4}$$

$$V^f = 1 - V^m \tag{5}$$

where m<sup>c</sup> is the total weight of composite, m<sup>f</sup> , m<sup>m</sup> is the weight of fibers and matrix, M<sup>f</sup> , M<sup>m</sup> is the weight amount of fibers and matrix, V<sup>f</sup> , V<sup>m</sup> is the volume amount of fibers and matrix, v<sup>c</sup> is the total volume of composite, and v<sup>f</sup> , v<sup>m</sup> is the volume of fibers and matrix. Volume amount of fibers V<sup>f</sup> and matrix V<sup>m</sup> can be also expressed with the help of the fiber density r<sup>f</sup> and matrix density <sup>r</sup><sup>m</sup>, which is applied in Eq. (6). The total density <sup>r</sup><sup>c</sup> <sup>¼</sup> <sup>m</sup><sup>c</sup> =v<sup>c</sup> can then be expressed as the sum of components, that is, the reinforcement and matrix (7). The characteristic thickness of the composite structure can be expressed according to the relationship (8).

$$V^{\dagger} = \frac{M^{\dagger}/\rho^{\dagger}}{M^{\dagger}/\rho^{\dagger} + M^{\dagger}/\rho^{m}}, \quad M^{\dagger} = \frac{V^{\dagger}\rho^{\dagger}}{V^{\dagger}\rho^{\dagger} + V^{\dagger}\rho^{m}} \tag{6}$$

$$
\rho^c = \rho^f V^f + \rho^m V^m \tag{7}
$$

$$h = m^f \left[ \frac{1}{\rho^f} + \frac{1}{\rho^m} \cdot \left( \frac{1 - M^f}{M^f} \right) \right] \tag{8}$$

Gay and Hoa [10] reported that winding of the fibers onto-shaped geometry may achieve the

with long fibers.


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

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

ε<sup>11</sup> ε<sup>22</sup> ε<sup>33</sup> γ<sup>12</sup> γ<sup>23</sup> γ<sup>13</sup> 9

>>>>>>>>=

>>>>>>>>;

principal load direction.

¼

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

1=E<sup>11</sup> �ν12=E<sup>11</sup> �ν12=E<sup>11</sup> 000 �ν12=E<sup>11</sup> 1=E<sup>22</sup> �ν23=E<sup>22</sup> 000 �ν12=E<sup>11</sup> �ν23=E<sup>22</sup> 1=E<sup>22</sup> 000 0001=G<sup>23</sup> 0 0 0 0 0 01=G<sup>12</sup> 0 0 0 0 001=G<sup>12</sup>

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

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

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

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

σ<sup>11</sup> σ<sup>22</sup> σ<sup>33</sup> τ<sup>12</sup> τ<sup>23</sup> τ<sup>13</sup> 9

>>>>>>>>=

(9)

7

>>>>>>>>;

8

>>>>>>>><

>>>>>>>>:

�

8

>>>>>>>><

>>>>>>>>:

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 composite samples.
