Examples of physical and geometrical parameters of the selected samples of composite structures reinforced with long fibers.

## Abstract

This chapter deals with studies of the mechanical properties of samples from long fiber-reinforced composite structures that would contribute to the optimization of the developed constructions made of them. First, the basic issues of composite structures reinforced with long fibers (carbon or glass) and generally of composites with the specification of parameters that would lead to the optimization of mechanical properties with respect to the theoretical strength are presented. Further, the possibilities and methods of measurements of composite reinforced with carbon and glass fibers are described. This is followed by the introduction of analytical models for the description of the transversal isotropic composite, where these mathematical relations allow the determination of unknown elastic constants and they are also important for the verification of numerical models. Finally, it is comprehensively outlined the problems of creating a numerical model of advanced composite fibrous structure for determining the mechanical properties, both through the description of the continuum, and complex numerical model with a structural configuration enabling approach to allow closer interaction among fibers and matrix. Compared to the averaged values obtained from experimental samples, numerical simulations show a similar trend of stress on strain, with results obtained from simulations.

### Keywords

- FEM
- composite structures
- testing
- mechanical properties
- nonlinear properties

## 1. Introduction

Studies and analyses of mechanical properties of long fiber-reinforced composites provide important information for future lightweight constructions. First of all, it is important to approach the issues and specifics of long fiber-reinforced composite structures to increase the strength and toughness of the resulting structure. The long fiber-reinforced composite structure is typically formed from two dominant components: carrier fiber reinforcement and a matrix. Ideal arrangement of the final composite (fiber-matrix connection), due to synergy, the high specific properties (high strength, stiffness, and toughness) can be achieved, where none of input components reached. It is that the optimal synergistic effect is characterized by a known “illogical” rule 2 + 3 = 7, which characterizes the sum of the properties of the individual input components (fibers + matrix) achieves a higher value of the specific properties of the newly created structure. In general, the highest specific properties can be achieved if the fibers are stressed up to the strength limit ^{−1}). The spinning speed is also influenced by the viscosity (50–100 Pa s), the melting temperature, 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 the phase interface is not yet clear.

## 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 fiber-reinforced 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

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).

where

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

The influence of selected physical parameters on the geometric parameter

Name of fibers | ^{−2}] | ^{−2}] | ^{−3}] | ^{−3}] | |||||
---|---|---|---|---|---|---|---|---|---|

GF 1600 tex/PUR Huntsman | 560 | 600 | 48 | 52 | 30 | 70 | 2.45 | 1.1 | 1.2 |

CF prepreg HEXPLY-M10R | 150 | 91.96 | 62 | 38 | 52 | 48 | 1.8 | 1.2 | 0.22 |

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

## 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

where

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.4

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:

phenomenological models.

semiempirical models.

homogenized models.

### 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

where

### 3.2. Semiempirical 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,

where

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

where

Chamis model

This is another semiempirical model [20], which was unlike previous models developed not only for independent elastic constants

where

### 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

where

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

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

### 4.1. FEM simulation of mechanical properties of long fiber-reinforced composite

Model simulations in FEM were performed for different combinations of reinforcement arrangements of long fiber-reinforced composites, which are important for comparison with experiments and analytical relationships. This gives the material properties for numerical simulation of the strength characteristics of whole frames.

This chapter describes the creation of two numerical models and their comparison:

continuum model

extended continuous model with structural unit

The simulations were performed for a complete assessment of the mechanical properties

creating two model simulations of the long fiber-reinforced composite,

creating the corresponding mesh of finite elements of the computational model in the preprocessor,

defining the corresponding initial and boundary conditions,

assembling a material model of the long fiber-reinforced composite,

the evaluation and comparison of model simulation results in postprocessor.

#### 4.1.1. Assembling of continuum of the long fiber-reinforced composite model

The FEM model was created in the concept of coherent continuum consisting of a surface geometry corresponding to the test sample with length *L* = 100 mm, width *b* = 20 mm, and thickness *h* = 1.7 mm. The finite element mesh of the numerical model was created from SHELL elements (2D elements) with a constant element size of 2 mm. The boundary conditions affecting the magnitude of the displacements and stress can be defined in two ways, that is, the boundary conditions of the first and second types. The first way is to determine the displacement and stress distributions if force conditions are known, that is, volume forces, surface forces, and nodal loads. The other way is to determine the displacement and tension distribution if the geometric conditions are known, that is, the size of node displacement, the deformations, and so on. Both ways can be also combined. It is mixed boundary conditions, as shown by Li [25]. Boundary and initial conditions for the model were made by the boundary conditions of the second type. One side of the sample was fixed against the displacement and rotation of nodes ^{−1}. The boundary conditions are shown in Figure 5.

#### 4.1.2. Assembling of extended continuous composite model of long fiber-reinforced composite

The second numerical FEM model, which was created in the concept of structure unit, is formed from three components: fiber matrix—the interfacial interface, where the microscopic dimensions of such a model are closer to the more real composite. Such a model can be created from a structural unit with the

It will be assumed that

The problem lies in joining of fibers with the matrix because the interconnections form an interphase. The structural FEM model assembling presents a problem of the determination of appropriate boundary conditions, which is important in terms of accuracy and model verification. Incorrect design may result in concentrators and singularities of stress. The boundary conditions are created by the second type (geometric boundary conditions) as follows: the perimeter surfaces of the model perpendicular to the plane of the stretching direction have defined symmetry conditions on one side (symmetry in axis *X* axis, the displacements and rotations were not allowed ^{−1}. Boundary conditions are shown in Figure 8 and Table 2. The material properties applied in both FEM models (I. Continuum Model and the II. Continuum Model with the Structure Unit) are based on the generally known values reported by fiber and matrix manufacturers. The fiber and epoxy matrix parameters are listed in Table 3. The results of both numerical simulations have exhibited approximately the same stress at the defined strain

Planes in axis | Planes in axis | Planes in axis | ||||
---|---|---|---|---|---|---|

— | — | |||||

— | — | |||||

Material | Density [kg m ^{−3}] | Modulus of elasticity [GPa] | Shear module [GPa] | Poisson’s ratio [−] | Tensile strength [GPa] | Elongation [%] | |||
---|---|---|---|---|---|---|---|---|---|

Carbon fibers | 1750 ± 150 | 230 | 15 | 24 | 5.4 | 0.279 | 0.49 | 2.3 ± 1.2 | 1.9 ± 0.6 |

Glass fibers | 2370 ± 230 | 72.4 | 72.4 | 28.7 | 28.7 | 0.22 | 0.22 | 1.06 ± 0.65 | 4.8 ± 0.7 |

Epoxy matrix | 1150 ± 370 | 3.573 | 3.573 | 1.31 | 1.31 | 0.345 | 0.345 | 0.067 ± 0.033 | 3.6 |

The obtained results shown in Figures 10 and 11 can be stated that the continuous model (FE model I) has an approximately steady monotonic course manifested not only in continuous distribution of deformation (Figure 12 above) but also in uniform distribution of the principal stress

Due to the simplicity of the FEM model I, it appears to be very suitable for determining the mechanical properties of composite structures and their optimization. However, such a model does not provide information about strain and stress between the fibers and the matrix, let alone the interphase. The continuous model with the structural unit (FEM model II) is significantly more complex, and for some elastic constants, its resultant course is not monotone (

## 5. Conclusion

In this chapter, analyses and numerical simulations of mechanical properties of samples from composite structures were presented. Several studies and experiments have been carried out on samples reinforced with carbon and glass fibers, and mechanical properties allow them to form structural reinforcements of composite materials. Further, analytical models with mathematical relationships (e.g., Voigt, Reuss or Chamis model) allow to determine the unknown elastic constants

## Notes

- This can be mentioned in the example given by Bareš [9]. A simple combination of three homogeneous isotropic light metals to form ternary cast iron is obtained 82,160 possible variants of alloys, if more six metals are combined, more than 300 million different alloy variants could be obtained. (The composite structure reinforced with long fibers has a similar behavior, where the change of the matrix, directional arrangement, and type of fiber significantly affect resulting mechanical properties because it leads to qualitatively different structure [10]).
- Transmission of static stress applied to the composite and transferred with fibers is required excellent consistency of fibers and matrix; on the contrary, a dynamic impact requires energy absorption by the crack propagation along fibers.
- 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.
- 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.
- Correction factors ζ f , ζ m can be express as ζ f = E 11 f V f + 1 − ν 12 f ν 21 f E m + ν m ν 21 f E 11 f V m E 11 f V f + E m V m , ζ m = E m V m + 1 − ν m ν m E 11 f − 1 − ν m ν 12 f E m V f E 11 f V f + E m V m , ζ ' is a variable function 0 < ζ ' < 1 , whereas preferred is ζ ' ≈ 0.5 − 0.6 .