Selection of Analytical Methods for Determination of Mechanical Characteristics of Unidirectional Composites Based on Glass Fibers

The most common methods for determination of mechanical characteristics of composite materials are considered. The accuracy of methods for each constant of some unidirectional composite materials based on E-glass fibers and epoxy binder is found.

Introduction

Fiber composite materials (CM) are widely used in aerospace and rocket engineering, power engineering, automotive, metal mining and metallurgy industries, construction and etc. Unidirectional composite materials consist of reinforcing continuous fibers that are embedded into a polymer matrix.

Three the most commonly used methods for determination of mechanical properties of CM are known, namely: experimental, analytical and finite element modeling of mechanical properties. The most reliable among them is experimental one [1].

The papers [1–14] present the investigations on determination of mechanical characteristics using different experimental methods, finite element modeling and engineering dependencies. New methods for determination of characteristics of anisotropic materials are developed to simplify the procedure of investigation and shorten the time spent for solution of the given problem as well as to improve the accuracy of obtained results.

The objective of this paper is to select the most accurate and the least difficult approaches to prediction of both elastic and strength characteristics of unidirectional composite materials based on glass fibers.

In considering any methods it is assumed that there is a unidirectional fiber composite, which is formed by transversely isotropic materials with the plane of isotropy perpendicular to the fiber direction and the Tsai–Wu failure criterion. Such material is characterized by five independent elastic (E ₁, E ₂ , G ₁₂ , G ₂₃ , and ν₁₂ ) and five strength ($\upsigma_{u1}^{+}$, $\upsigma_{u2}^{+}$, $\upsigma_{u1}^{-}$, $\upsigma_{u2}^{-}$, and τ _u12) properties [6]. Here, E and G are the Young and shear modulus, respectively, ν is Poisson’s ratio, $\upsigma_{u1}^{+}$, $\upsigma_{u1}^{-}$ and τ _u12 are are the ultimate tensile strength, compression strength, and shear strength, respectively, i = 1, 2, 3 (1 is the anisotropy axis in the reinforcing fiber direction, 2 and 3 are perpendicular to axis 1).

Below are given the most known methods for determination of CM mechanical characteristics.

Determination of Elastic Characteristics of CM. Mixture Rule [8, 15, 16] (hereinafter referred to as MR). In accordance with the given rule, the material unknown characteristic depends on the contribution of each component in proportion to its volume content in the composite:

$$ {E_1}={E_{1f }}{V_f}+{E_{1m }}\left( {1-{V_f}} \right),\quad \frac{1}{{{E_2}}}=\frac{{{V_f}}}{{{E_1}_f}}+\frac{{{V_m}}}{{E{1_m}}},\quad {\upnu_{12 }}={\upnu_f}{V_f}+{\upnu_m}{V_m},\quad \frac{1}{{{G_{12 }}}}=\frac{{{V_f}}}{{{G_f}}}+\frac{{{V_m}}}{{{G_m}}}, $$

where E ₁, E ₂ , ν¹² , and G ₁₂ are unknown characteristics, subscripts f and m denote fiber and matrix, respectively, and V _f and V ^m are volumetric content of reinforcing fiber and matrix, respectively.

The expression for E ₁ is also true for the Halpin–Tsai method [17]. The proposed method [18] for determination of E ₂ is of the same form. The formula is true for the Halpin–Tsai, Barbero, and Jones methods [19].

The Tsai Method [6, 20]. This method is similar to MR; however, it has a correction factor C:

$$ \begin{array}{*{20}{c}} {{E_1}=C\left( {{E_1}_f{V_f}+{E_{1m }}\left( {1-{V_m}} \right)} \right),} \\ {{G_{12 }}=\left( {1-C} \right){G_m}\frac{{2{G_f}-\left( {{G_f}-{G_m}} \right){V_m}}}{{2{G_f}-\left( {{G_f}-{G_m}} \right){V_m}}}+C{G_f}\frac{{\left( {{G_f}+{G_m}} \right)-\left( {{G_f}-{G_m}} \right){V_m}}}{{\left( {{G_f}+{G_m}} \right)-\left( {{G_f}-{G_m}} \right){V_m}}},} \\ \end{array} $$

where C ≤ 1.

This factor C depends on the density of fibers packing: C = 0 at isolated fibers with a relatively high matrix volume, C = 1 at dense packing with a low binder content, at nonuniform packing C equals to the intermediate value 0 < C < 1:

$$ \begin{array}{*{20}{c}} {{E_2}=2\left( {1-{\upnu_f}+\left( {{\upnu_f}-{\upnu_m}} \right){V_m}} \right)\left( {\left( {1-\upomega} \right)\frac{{{K_f}\left( {2{K_m}+{G_m}} \right)-{G_m}\left( {{K_f}-{K_m}} \right){V_m}}}{{\left( {2{K_m}+{G_m}} \right)+2\left( {{K_f}-{K_m}} \right){V_m}}}} \right.} \\ {\left. {+\upomega \frac{{{K_f}\left( {2{K_m}+{G_f}} \right)-{G_f}\left( {{K_m}-{K_f}} \right){V_m}}}{{\left( {2{K_m}+{G_f}} \right)-2\left( {{K_m}-{K_f}} \right){V_m}}}} \right),} \\ \end{array} $$

where

$$ {K_{\mathrm{B}}}=\frac{{{E_f}}}{{2\left( {1-{\upnu_f}} \right)}},\quad {K_m}=\frac{{{E_m}}}{{2\left( {1-{\upnu_m}} \right)}},\quad {G_f}=\frac{{{E_f}}}{{2\left( {1+{\upnu_f}} \right)}},\quad {G_m}=\frac{{{E_m}}}{{2\left( {1+{\upnu_m}} \right)}}, $$

ω is the experimental correction factor considering the factor of fibers contact, 0 < ω < 1. In [12] it is recommended to accept ω = 0.2.

The Hill Method [21]. In this method it is assumed that for two-phase fibrous CM the following relation is true:

$$ {E_1}={E_V}+\frac{{4{{{\left( {{\upnu_V}-{\upnu_m}} \right)}}^2}}}{{{{{\left( {K_V^{-1 }-{K^{-1 }}} \right)}}^2}}}\left( {\frac{1}{{{K_R}}}-\frac{1}{{{K_0}}}} \right),\quad {\upnu_{12 }}={\upnu_V}-\frac{{{\upnu_V}-{\upnu_m}}}{{K_V^{-1 }-{K^{-1 }}}}\left( {\frac{1}{{{K_R}}}-\frac{1}{{{K_0}}}} \right), $$

where

$$ \begin{array}{*{20}{c}} {{E_V}={E_m}\left( {1-{V_f}} \right)+{E_f}{V_f},\quad {\upnu_V}={\upnu_m}\left( {1-{V_f}} \right)+{\upnu_f}{V_f},} \\ {{K_0}=K+\frac{{{V_f}}}{{{{{\left( {{K_V}-K} \right)}}^{-1 }}+\left( {1-{V_f}} \right){{{\left( {K+G} \right)}}^{-1 }}}},\quad G=\frac{E}{{2\left( {1+{\upnu_m}} \right)}},} \\ {K_R^{-1 }={K^{-1 }}\left( {1-{V_f}} \right)+K_V^{-1 }{V_f},\quad K=\frac{E}{{3\left( {1-2\upnu} \right)}},\quad {K_V}=\frac{{{E_1}}}{{3\left( {1-2{\upnu_1}} \right)}}.} \\ \end{array} $$

CCA Methods [6] (Composite cylinder assemblage model). This method takes the following form:

$$ {E_1}={E_{1f }}{V_f}+{E_{1m }}\left( {1-{V_f}} \right)-\frac{{4{E_{1m }}{E_1}_f{{{\left( {{\upnu_f}-{\upnu_m}} \right)}}^2}{G_m}}}{{\left( {\frac{{{V_m}{G_m}}}{{{K_f}+{G_f}/3}}} \right)+\left( {\frac{{{V_m}{G_m}}}{{{K_m}+{G_m}/3}}} \right)+1}}. $$

Matrix contribution to approximate calculations, where a high accuracy is not required, is often neglected, and it is considered that E ₁ = E _1f V _f :

$$ {E_2}=2\left( {1+{\upnu_{23 }}} \right){G_{23 }}, $$

where

$$ \begin{array}{*{20}{c}} {{\upnu_{23 }}=\frac{{{K^{*}}-m{G_{23 }}}}{{{K^{*}}+m{G_{23 }}}},\quad m=1+4{K^{*}}\frac{{\upnu_{12}^2}}{{{E_1}}},\quad {G_{23 }}=\frac{{{G_m}\left( {{K_m}\left( {{G_m}+{G_f}} \right)+2{G_f}{G_m}+{K_m}\left( {{G_f}-{G_m}} \right){V_f}} \right)}}{{{K_m}\left( {{G_m}+{G_f}} \right)+2{G_f}{G_m}+\left( {{K_m}+2{G_{\mathrm{M}}}} \right)\left( {{G_f}-{G_m}} \right){V_f}}},} \\ {{K^{*}}=\frac{{{K_m}\left( {{K_f}+{G_m}} \right){V_m}+{K_f}\left( {{K_m}+{G_m}} \right){V_f}}}{{\left( {{K_f}+{G_m}} \right){V_m}+\left( {{K_m}+{G_m}} \right){V_m}}},\quad {K_f}=\frac{{{E_f}}}{{2\left( {1+{\upnu_m}} \right)\left( {1-2{\upnu_f}} \right)}},\quad {K_m}=\frac{{{E_m}}}{{2\left( {1+{\upnu_m}} \right)\left( {1-2{\upnu_m}} \right)}},} \\ {{\upnu_{12 }}={\upnu_f}{V_F}+{\upnu_m}{V_m}+\frac{{{V_f}{\upnu_m}\left( {{\upnu_f}-{\upnu_m}} \right)\left( {2{E_f}\upnu_m^2+{\upnu_m}{E_f}-{E_f}+{E_m}-{\upnu_f}{E_m}-2\upnu_f^2{E_m}} \right)}}{{\left( {2{V_f}\upnu_m^2-{\upnu_m}{V_f}-1-{V_f}} \right){E_f}+\left( {2\upnu_f^2-\upnu_m^2{V_f}+{V_f}+{\upnu_f}-1} \right){E_m}}}} \\ {{G_{12 }}={G_m}\left( {\frac{{{G_f}\left( {1+{V_f}} \right)+{G_m}{V_m}}}{{{G_f}{V_f}+{G_m}\left( {1+{V_f}} \right)}}} \right).} \\ \end{array} $$

The Halpin–Tsai Procedure [3] is described by the following relations:

$$ {E_2}={E_m}\frac{{1+{\upxi_1}{\upeta_1}{V_f}}}{{1-{\upeta_1}{V_f}}},\quad {G_{12 }}={G_m}\frac{{1+{\upxi_2}{\upeta_2}{V_f}}}{{1-{\upeta_2}{V_f}}}, $$

where

$$ {\upeta_1}=\frac{{{E_f}-{E_m}}}{{{E_f}+{\upxi_1}{E_m}}},\quad {\upxi_1}=2,\quad {\upeta_2}=\frac{{{G_f}-{G_m}}}{{{G_f}+{\upxi_2}{G_m}}},\quad {\upxi_2}=1, $$

ξ₁ = 2(a/b), at hexagonal packing of fibers the factor ξ₁ equals to 2, and a and b are the coefficients.

In view of the difficulty in determination of the value E ₂ , in comparison with the other elastic characteristics, there are many known methods, some of the most frequently used are given in Table 1.

Table 1 Determination of the Value E ₂ Using Different Methods

Also, the Gough–Tangorra and the Akasaka–Hirano methods [25] can be used, however, they have some limitations of the minimum fiber content and, at the same time, the influence of matrix is insignificant.

Above is given the determination of the value G ₂₃ using CCAB model [26]. To describe G ₂₃, the Chamis theory is proposed [26]:

$$ {G_{23 }}=\frac{{{G_m}}}{{1-\sqrt{{{V_f}}}\left( {1-{G_m}/{G_{f23 }}} \right)}}. $$

Determination of Strength Properties along and across the Fiber Direction under Tension, Compression, and Shear. There are scarce papers devoted to the problem of determination of strength properties of unidirectional fibrous CM. In general, it is proposed to use MR:

$$ \upsigma_i^j=\upsigma_{if}^j{V_f}+\upsigma_{im}^j{V_m}, $$

where i = 1, 2 (direction of reinforced elements), j = + (tension), and j = − (compression).

For shear strength it is as follows:

$$ {\upsigma_{sh }}=\upsigma_m^u{V_m}. $$

To determine the most accurate analytical methods of determination of the material mechanical characteristics using the above relations, let us calculate mechanical characteristics of unidirectional composites and compare the obtained data with the known ones [13, 21, 27, 28]. For this purpose, four composite materials based on glass fibers and epoxy binder were selected, which are denoted as 1, 2, 3, and 4. For these materials, the elastic and strength properties, as well as volume content of components, are known.

Material 1 was studied for three variants (1, 2, and 3) at different values of fiber-matrix volumetric ratio: Vf = 0.55, 0.65, and 0.4. Table 2 presents the input data for materials.

Table 2 Basic Characteristics of the Reinforcing Elements and Matrix of CM for Checking Calculations

Strength characteristics of the composite components are the following: for material 1 ($\upsigma_m^{+}$ = 70 Mpa, $\upsigma_f^{+}$ = 1900 MPa, $\upsigma_m^{-}$ = 120 MPa, $\upsigma_f^{+}$ = 1500 MPa); for materials 3 and 4 ($\upsigma_m^{+}$ = 80 Mpa, $\upsigma_f^{+}$ = 2150 MPa, $\upsigma_m^{-}$ = 120 MPa, and $\upsigma_f^{+}$ = 1450 MPa).

Table 3 presents the characteristics of composites known from independent sources, which will be used to determine the accuracy of one or another method.

Table 3 Mechanical Characteristics of CM used for Calculations

Tables 4–8 provide the results of calculations for each characteristic separately with an indication of value of deviation from the experimental data given in literature references.

Table 4 Calculated Values of Modulus of Elasticity E1 (GPa) in the Direction of the Axis of CM Reinforcement

Table 5 Calculated Values of Modulus of Elasticity E ₂ (GPa) in the Direction of the Axis of CM Reinforcement

Table 6 Calculated Values of Shear Modulus G ₁₂ (GPa) CM

Table 7 Calculated Values of Poisson’s Ratio ν₁₂ CM

Table 8 Calculated Values of the Ultimate Tensile Strength and the Ultimate Compression Strength in the Direction of the Reinforcing Axis and Along It As Well As Shear Yield Stress Values for CM

Conclusions. Since the calculations were conducted on the materials with reinforcing elements (glass fibers) of the same type, the following conclusions would be true for CM based on E-glass fibers.

The most accurate methods for determination of the value E ₁ are the Hill method with an error of 0.4–6 %, and simpler MR method with an error of 0.31–6.6 %. In the calculation of the value E ₂ for the material with fiber content less than 50 % the Halpin–Tsai method with an error of 6.6 % is the most accurate. It should be noted that in [6] for CM with fiber content of 37 % the most accurate is the Tsai method (in the above-mentioned calculations the error is 9.28 %); for the material with fiber content more than 60 % the most accurate results were obtained using the Vasiliev method with an error of 7–25 %.

In the determination of the value ν₁₂ , the most accurate is ÑÑÀ method with an error of 0.6–29 %, and MR method with an error of 1.26–41 %.

For calculation of the value G ₁₂ the most accurate is the Tsai method with an error of 2.56–16.1 %.

In the calculation of strength characteristics, the error varies within the range from 0.5 to 13 % and only determination of compression stresses along the reinforcement axis demonstrates a low accuracy (24–65 %). Use of MR to determine the values E ₁ and ν₁₂ and of the Tsai method for CM is generally in agreement with the results obtained in [29].