2. Structural models of composite materials

a large number of parameters with a high discretization frequency, modern testing machines allow for high amount of information on material deformation and failure to be obtained within a single experiment. Therefore, data processing has become an important step for mathematical modeling of CFRPs and CFRP structures. This is exceptionally important because of quite specific behavior of CFRPs and of their components: fibers and matrices.

One of the features of such materials is their different strength and stiffness behavior in tension and compression combined with nonlinearities of stress-strain curves. Multiple studies for epoxy matrices showed that their ultimate strains in tension were much lower than in compression: approximately 4 versus 20% and more [2]. Moreover, under tension and compression, the deformation behavior of epoxy matrices significantly differs. The corresponding stress-strain curves have different stiffness (secant modulus) at the same values of strain. The similar difference can be observed for CFRPs. In [3–5], it was shown that in tension tests of carbon fiber specimens with reinforcement angles less than 20<sup>∘</sup> the stiffness grows together with strains (stiffening), whereas for epoxy matrices softening is observed. This phenomenon

Contrasting behavior in tension and compression, stiffening, softening and other nonlinearities are forcing researchers to build and use special mathematical models and computing algorithms. Mathematical models taking into account the abovementioned properties of materials were proposed and studied theoretically by Timoshenko [6] and Ambartsumyan [7, 8] in the mid-twentieth century. Later, Jones had experimentally, theoretically and numerically studied the nonlinear behavior of several fiber-reinforced composites. The main focus of the research was on the difference in stiffness and strength behavior under tension and compression [9]. After Ambartsumyan's and Jones' researches, a lot of studies were dedicated to this problem. Most of them were dealing with linear bi-modulus models of materials or 3D finite elements. In [10], Ambartsumyan with a coauthor suggested a theoretical approach to modeling of

multimodulus nonlinear elastic beams under bending, but still without calculations.

matical modeling spanning from the experimental investigations to numerical ones.

comparative analysis of results of numerical modeling to acquired data.

Another trend is studying the behavior of sandwich panels or beams with a CFRP faces having differences in tension and compression along with the other mentioned nonlinearities [11–13]. These works concern the problem of flexure of CFRPs and similar materials. They consider bending of specimens as a reference test. The first paper [11] is devoted to experimental investigations and shows most of the nonlinearities we supposed such materials should have: stiffening in tension, softening in compression, different moduli even at the origin of coordinates. Other two works [12, 13] present more complex studies including full cycle of mathe-

A comprehensive approach to modeling and simulation of nonlinear elastic deformation of polymer matrices and different CFRPs was presented in [14]. This chapter deals with different strength and stiffness behavior of the materials in tension and compression exemplified by a case of three-point bending. This approach implements a full cycle of model development and validation, which comprises the following stages: carrying out tests and acquiring experimental data, data prepossessing and building stress-strain curves, analytical approximation of acquired curves, mathematical modeling and numerical simulation of deformation processes,

was explained by the properties of carbon fibers.

14 Optimum Composite Structures

For most of the composite materials models, we can write the relations between average stresses σαβ, τα<sup>3</sup> and strains eαβ, γα<sup>3</sup> (generalized Hooke's Law):

$$\begin{aligned} \sigma\_{\alpha a} &= a\_{\alpha a} e\_{\alpha a} + a\_{\alpha \beta} e\_{\beta \beta} + a\_{\alpha 3} \cdot 2e\_{a \beta} - a\_{a \Theta} \Theta, \\ \sigma\_{a \beta} &= a\_{\alpha 3} e\_{\alpha a} + a\_{\beta 3} e\_{\beta \beta} + a\_{33} \cdot 2e\_{a \beta} - a\_{3 \Theta} \Theta, \\ \sigma\_{a 3} &= q\_{\alpha a} \tau\_{a 3} + q\_{a \beta} \tau\_{\beta 3}. \end{aligned} \tag{1}$$

where Θ is the increase of temperature. Relations (Eq. (1)) are called the thermoelasticity relations, or, when no temperature influence is considered, they are simply elasticity relations.

The structural model of fiber reinforced composite described in [15–18] has become a foundation for a large number of current researches. Now it is widely used while simulating the behavior of composite structures. The model is based on the following assumptions: the stress-strain state into isotropic elastic fibers and into entire volume of isotropic ideally elastic matrix is homogeneous; fibers and matrix are deformed jointly along the direction of reinforcement; stresses in fibers and in matrix corresponding to other directions are equal.

For computing the effective elastic modulus of unidirectional fiber-reinforced composite, the Reuss-Voigt average was used giving the following formulae

$$\begin{aligned} E\_1 &= \omega\_f E\_f + \omega\_m E\_m & E\_2 &= \frac{E\_f E\_m}{\omega\_f E\_m + \omega\_m E\_f} \\ \nu\_{12} &= \omega\_f \nu\_f + \omega\_m \nu\_m & G &= \frac{G\_f G\_m}{\omega\_f G\_m + \omega\_m G\_f} \end{aligned} \tag{2}$$

where all the terms having squared Poisson coefficients are neglected.

Herewith E1, E<sup>2</sup> are effective moduli along and across the direction of reinforcement, G is effective share modulus, ν<sup>12</sup> is effective Poisson coefficient in the plain of layer; E, ν, ω with "f " and "m" indices are elastic moduli, Poisson coefficients and volume fractions of matrix and fibers correspondingly, hereby ω<sup>m</sup> þ ω<sup>f</sup> ¼ 1.

On the ground of symmetry of compliance tensor, one has

$$\nu\_{21} = \nu\_{12} E\_2 E\_1^{-1} \dots$$

Formulae for effective coefficients of thermal expansion have the following form

$$
\alpha\_1 = \omega\_f \alpha\_f + \omega\_m \alpha\_m \quad \alpha\_2 = \frac{\omega\_f \alpha\_f E\_f + \omega\_m \alpha\_m E\_m}{\omega\_f E\_f + \omega\_m E\_m}. \tag{3}
$$

In description of the model, it is noted that among formulae for effective moduli, those obtained using Reuss averaging (in particular formulae for G) lead to the worst results. Estimations for G obtained using variational method are also obtained, and it is shown that lower boundary

$$\mathbf{G} = \frac{(1 + \omega\_f)\mathbf{G}\_f + \omega\_m\mathbf{G}\_m}{(1 + \omega\_f)\mathbf{G}\_m + \omega\_m\mathbf{G}\_f}\mathbf{G}\_m\tag{4}$$

In this case, the beam's upper part undergoes compression strain in the longitudinal direction, bottom part—tension strain. VSE-1212 polymer matrix and VKU-28 (T-800 carbon yarn plus VSE-1212 epoxy matrix) structured CFRP react differently to tension and compression. VKU-28 has been one of the most promising types of CFRPs that is going to be used in the latest generations of aircrafts. The effect of accounting for this factor on the computational results is

Mathematical Modeling and Numerical Optimization of Composite Structures

Due to very low deformation rates, the classical theory of beam bending can be regarded as satisfactory for description of the equilibrium state. To this end, it is convenient to consider the

The beam's stress-strain state is characterized by the following values determined on the reference surface: the shear force Q xð Þ, the bending moment M xð Þ, the longitudinal force N xð Þ, and by the longitudinal displacement and bend (u xð Þ, w xð Þ respectively). The corresponding equilib-

The reactions RA and RB can be determined by considering force equilibrium RA ¼ RB ¼ P=2. The bending moments at the support points are equal to zero: MA ¼ MB ¼ 0. The solution of

Strain distribution for the beam's thickness can be obtained from the Kirchhoff-Love kinematic

dx , <sup>κ</sup>ð Þ¼� <sup>x</sup>

where εð Þ x; z is the strain in the beam; e xð Þ is the median surface strain; and κð Þx denotes changes in the median surface curvature. As mentioned earlier, the beam undergoes tension and compression strain, whose interface will be marked as z1. In this case, for the section area �h ≤ z ≤ z1, the strain will be negative, and for z<sup>1</sup> ≤ z ≤ h positive. At the interface of these two

( (

dx <sup>¼</sup> <sup>0</sup>, dM

M xð Þ¼

d2 w

dx <sup>¼</sup> <sup>Q</sup>: (6)

http://dx.doi.org/10.5772/intechopen.78259

Px=2, 0 ≤ x ≤ l=2,

(7)

17

�P xð Þ � l =2, l=2 ≤ x ≤ l:

dx<sup>2</sup> , (9)

εð Þ¼ x; z e xð Þþ zκð Þx , (8)

<sup>κ</sup> , � <sup>h</sup> <sup>≤</sup> <sup>z</sup><sup>1</sup> <sup>≤</sup> <sup>h</sup>: (10)

<sup>i</sup> ð Þε , (11)

essential. Further, these results are compared to acquired ones.

dN

the equation system (Eq. (6)) can be expressed as follows:

dx <sup>¼</sup> <sup>0</sup>, dQ

P=2, 0 ≤ x ≤ l=2, �P=2, l=2 ≤ x ≤ l,

e xð Þ¼ du

states, the strains ε vanish, so the interface itself is determined as follows:

The constitutive equation can be expressed as:

<sup>z</sup><sup>1</sup> ¼ � <sup>e</sup>

σ�ð Þ¼ x; z f

�

beam's median surface as a reference one.

rium equations are written as follows:

N ¼ 0, Qxð Þ¼

hypotheses:

gives more accurate approximation than (Eq. (2)) does. Hereinafter share moduli of matrix and fibers are

$$G\_m = \frac{E\_m}{2(1+\nu\_m)}, \quad G\_f = \frac{E\_f}{2\left(1+\nu\_f\right)}.$$

Components of effective stiffness tensor for unidirectionally reinforced layer in case of state of plane stress have the following form:

$$A\_{a\alpha a\alpha} = \frac{E\_a}{1 - \nu\_{12}\nu\_{21}}, \quad A\_{1122} = \frac{\nu\_{21}E\_1}{1 - \nu\_{12}\nu\_{21}}, \quad A\_{1212} = \text{G.}\tag{5}$$

Unwritten expressions can be obtained using symmetry rule or vanish. Hereinafter, we assume α, β ¼ 1, 2 and α 6¼ β.

The coefficients in the relations (Eq. (1)) for example are defined by the formulas given in [1, 17, 18].
