1. Introduction

Carbon fiber reinforced plastics (CFRP) are the most promising modern composite materials. High-duty structures used in aviation and space industry, car manufacturing and building sector require new CFRPs as well as ways to improve their characteristics. Applying computer modeling techniques significantly reduces both the time and cost of investigations aimed at searching optimal parameters of CFRP structures [1]. Mathematical modeling provides an opportunity for comprehensive analysis of both CFRPs and CFRP structures. It has become an effective tool for solving important applied problems.

To build a mathematical model of composite materials, including those made of carbon fibers, one relies upon the experimental data acquired in mechanical testing. Wide application of digital testing machines has brought such experiments to higher level of quality. By measuring

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

2. Structural models of composite materials

tions are equal.

stresses σαβ, τα<sup>3</sup> and strains eαβ, γα<sup>3</sup> (generalized Hooke's Law):

Reuss-Voigt average was used giving the following formulae

fibers correspondingly, hereby ω<sup>m</sup> þ ω<sup>f</sup> ¼ 1.

On the ground of symmetry of compliance tensor, one has

where all the terms having squared Poisson coefficients are neglected.

γα<sup>3</sup> ¼ qαατα<sup>3</sup> þ qαβτβ3,

For most of the composite materials models, we can write the relations between average

Mathematical Modeling and Numerical Optimization of Composite Structures

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

σαα ¼ aααeαα þ aαβeββ þ aα<sup>3</sup> � 2eαβ � aαΘΘ, σαβ ¼ aα<sup>3</sup>eαα þ aβ<sup>3</sup>eββ þ a<sup>33</sup> � 2eαβ � a3ΘΘ,

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 direc-

For computing the effective elastic modulus of unidirectional fiber-reinforced composite, the

ω<sup>f</sup> Em þ ωmEf

ω<sup>f</sup> Gm þ ωmGf

,

,

<sup>E</sup><sup>1</sup> <sup>¼</sup> <sup>ω</sup><sup>f</sup> Ef <sup>þ</sup> <sup>ω</sup>mEm, E<sup>2</sup> <sup>¼</sup> Ef Em

<sup>ν</sup><sup>12</sup> <sup>¼</sup> <sup>ω</sup><sup>f</sup> <sup>ν</sup><sup>f</sup> <sup>þ</sup> <sup>ω</sup>mνm, G <sup>¼</sup> Gf Gm

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

<sup>ν</sup><sup>21</sup> <sup>¼</sup> <sup>ν</sup>12E2E�<sup>1</sup>

<sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>ω</sup><sup>f</sup> <sup>α</sup><sup>f</sup> <sup>þ</sup> <sup>ω</sup>mαm, <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>f</sup> <sup>α</sup><sup>f</sup> Ef <sup>þ</sup> <sup>ω</sup>mαmEm

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.

Formulae for effective coefficients of thermal expansion have the following form

<sup>1</sup> :

ω<sup>f</sup> Ef þ ωmEm

(1)

15

(2)

: (3)

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 was explained by the properties of carbon fibers.

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.

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 mathematical modeling spanning from the experimental investigations to numerical ones.

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, comparative analysis of results of numerical modeling to acquired data.
