Classification of Composite Materials and Structures

Abstract

Composite materials in the simplest case are composed of some phase (a second phase) included in a matrix. The properties of the matrix are altered depending upon the volume fraction of this phase, as well as its geometry and orientation relative to an applied stress.

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Introduction

Look around. The world is a composite. All living systems are composites. Prominent among them in humans are bones whose porous structure reduces density (or weight) and whose matrix must be stiff and strong and able to withstand impact among other functions. Composite materials are designed to achieve multiple functions which when integrated into a system forming extended structures expands these features on a larger scale. In the simplest case composites are represented by regimes or phases (and interfaces) of varying morphologies, volume fractions, structures, and properties which compose a volume element. In contrast to a single phase represented by the same volume element or reference element, the composite element is differentiated by interphase interfaces, which increase significantly for nanocomposites.

Narrative

In general, composite materials are isotropic (or quasi-isotropic) or anisotropic either by circumstance or by design. The simplest are aggregated composites such as concrete or asphalt, both having various particulate shapes and sizes in a low fraction binder or matrix. For example, the volume fraction of the cement matrix or the pitch matrix, respectively, is around 10 %. In other particle-matrix composites, such as the 2 vol.% ThO₂ in Ni shown in Fig. 7 in chapter “Strengthening by Crystal Imperfections”, the matrix is the high volume fraction. Composites represented by Fig. 7 of chapter “Strengthening by Crystal Imperfections” are isotropic, and the dispersed ThO₂ phase provides strengthening as implicit in Fig. 5f (chapter “Strengthening by Crystal Imperfections”). In these general cases, which can include the 2-phase system of hard (or strong) Be particles in an Al matrix as shown in Fig. 34 (chapter “Planar Defects: Crystal Interfaces”), the strengthening of a 2-phase composite can be expressed very approximately by

$$ {\upsigma}_{\mathrm{c}}={\mathrm{V}}_{\upalpha}{\upsigma}_{\alpha }+{\mathrm{V}}_{\upbeta}{\upsigma}_{\upbeta}, $$

(1)

where α and β and the 2 phases, V_α, V_β, and σ_α, σ_β, are the corresponding volume fractions (V_α + V_β = 1) and yield strengths (or flow stress) for the respective phase. Equation 1 assumes that one phase (the dispersoid or phase dispersed in a matrix) is much harder or stronger than the other (matrix phase). This is indeed the case for Fig. 34 of chapter “Planar Defects: Crystal Interfaces” and Fig. 7 of chapter “Strengthening by Crystal Imperfections”, where both the included phase size and volume fraction differ significantly as well. These features are illustrated in Fig. 1a, b.

Fig. 1

Composite material classification examples. (a) and (b) show homogeneous particle distribution in a matrix. (c) and (d) show aligned (anisotropic) and random (quasi-isotropic) discontinuous fibers. (e) and (f) show aligned (unidirectional) continuous fibers: θ = 0°, θ = 90°; 0 < θ < 90. (g) bidirectional or cross-ply continuous fibers. (h) multidirectional fibers. (i) continuous, random

Figure 1c, d represents two examples of discontinuous fibers in a unidirectional (aligned) and correspondingly anisotropic arrangement and a randomly oriented, isotropic (or quasi-isotropic) arrangement, respectively. Figure 1c is characteristic of the superconducting wire composites shown previously in Fig. 44 of chapter “Electromagnetic Color and Color in Materials.” In contrast, Fig. 1e, f shows continuous, unidirectional fibers parallel and/or perpendicular to an applied stress axis or at a specific angle relative to these axes, respectively. Figure 1g–i represents variances of the continuous fiber composite shown in Fig. 1e, f representing alternate planes of oriented fibers or fiber weaves (Fig. 1g, h) and randon, continuous fiber regimes represented by Fig. 1i. Figure 1g–i represents so-called cross-ply systems , examples of which are represented in Fig. 2a–e. Simple examples of such cross-ply composites involve plywood, which, as wood/laminate, was first utilized by the Egyptians around 3000 B.C. Similarly, random, short fiber composites in antiquity involve horse hair in plaster and stucco and straw in clay blocks, although in these composites, the fibers were utilized more as binders rather than strengthening agents. The use of palygorskite nanofibers in Mayan paintings served as a binder as well as a host for the indigo molecules in the case of Mayan blue coloration illustrated in Fig. 9 in chapter “Examples of Materials Science and Engineering in Antiquity.” In the case of strong fibers in cement, for example, asbestos fibers as shown in Fig. 2 (chapter “Examples of Materials Science and Engineering in Antiquity”), the breaking strain is much higher than the cement matrix so the fibers hold the matrix together when it cracks since the fibers can bridge the cracks. Even small fiber volume fractions can serve to eliminate catastrophic failure of weaker and more brittle matrices by restraining crack opening and propagation while providing continued or residual strength by holding the cracked matrix together (Kelly 1976).

Fig. 2

Laminate fiber composite arrangements (a–c), and 2-D fabric weave architectures (d) and (e). m in (b) denotes continuous, multidirectional laminate (Fig. 1h of chapter “Chemical Forces: Molecules”). (d) shows a plain weave, while (e) shows a 5-harness satin weave, illustrating multiplicity of fabric weave architecture (Adapted from Chawla (1987))

It can be observed in Fig. 1c that for very small fibers having a fiber length/fiber diameter ratio (l/d)_f < 10, their alignment in a principal direction will have little consequence since they would only strengthen the matrix similar to Eq. 1 for dispersed particulate composites (Fig. 1a, b). However, a critical fiber aspect ratio, (l_c/d)_f, is sometimes defined as

$$ {\left({1}_{\mathrm{c}}/\mathrm{d}\right)}_{\mathrm{f}}={\upsigma}_{\mathrm{f}}^{\prime }/2\;{\tau}_{\mathrm{m}}, $$

(2)

where l_c is a critical fiber length, d is the fiber diameter, σ ^′_f is the ultimate tensile strength of the fiber material, and τ _m is the flow stress of the matrix. Generally, for fiber lengths l > l_c:

$$ {\upsigma}_{\mathrm{c}}\cong {\upsigma}_{\mathrm{f}}^{\ast }{\mathrm{V}}_{\mathrm{f}}\left(1-\frac{1_{\mathrm{c}}}{2\mathrm{l}}\right)+{\upsigma}_{\mathrm{m}}\left(1-{\mathrm{V}}_{\mathrm{f}}\right), $$

(3)

where σ_c is the composite strength (yield stress), σ ^∗_f is the fiber fracture stress, V_f is the volume fraction of fiber of length l, and σ_m is the yield stress for the matrix.

For fiber length l < l_c:

$$ {\upsigma}_{\mathrm{c}}\cong \frac{{\mathrm{l}\uptau}_{\mathrm{c}}}{\mathrm{d}}+{\mathrm{V}}_{\mathrm{f}}+{\upsigma}_{\mathrm{m}}\left(1-{\mathrm{V}}_{\mathrm{f}}\right), $$

(4)

where τ_c is the smaller of either the matrix or the fiber flow stress and d is the fiber diameter.

For random, discontinuous fiber composites (Fig. 1d), where l/d > 10², Eq. 1 can be approximated by

$$ {\upsigma}_{\mathrm{c}}\cong {\upsigma}_{\mathrm{m}}{\mathrm{V}}_{\mathrm{m}}+{\mathrm{K}\upsigma}_{\mathrm{f}}{\mathrm{V}}_{\mathrm{f}} $$

(5)

or

$$ {\mathrm{E}}_{\mathrm{c}}\cong {\mathrm{E}}_{\mathrm{m}}{\mathrm{V}}_{\mathrm{m}}+{\mathrm{KE}}_{\mathrm{f}}{\mathrm{V}}_{\mathrm{f}}, $$

(6)

where σ_m and σ_f are the matrix and fiber yield strengths, respectively; V_m and V_f are the corresponding volume fractions, E_m and E_f are the Young’s moduli; and K is an efficiency factor:

$$ \mathrm{K}=3/8\ \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{a}\ \mathrm{r}\mathrm{a}\mathrm{ndom}\ 2-\mathrm{D}\ \mathrm{composite} $$

$$ \mathrm{K}=1/5\ \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{a}\ \mathrm{r}\mathrm{a}\mathrm{ndom}\ 3-\mathrm{D}\ \mathrm{composite} $$

for discontinuous fiber composites.

In treating continuous (1 → ∞) fiber reinforcement of a matrix as illustrated in Fig. 1e, f, we can consider an orthotropic or transversely isotropic 3-D composite when one of its principal planes is a plane of isotropy as illustrated in Fig. 3a or a more simplified, thin, unidirectional lamina under a state of plane stress implicit in Fig. 3b. For the general orthotropic (and transversely isotropic) case in Fig. 3a, the stress–strain relations can be expressed by

Fig. 3

(a) shows an orthotropic composite (continuous fibers in a matrix) with transverse isotropy, while (b) shows a corresponding unidirectional lamina with same principal coordinate axes, 1, 2, and 3. (c) and (d) show stress components in unidirectional lamina referred to loading (x, y) and material (1, 2) axes as in Fig. 1f

$$ \left[\begin{array}{l}{\mathrm{E}}_1\\ {}{\mathrm{E}}_2\\ {}{\mathrm{E}}_3\\ {}{v}_{23}\\ {}{v}_{31}\\ {}{v}_{12}\end{array}\right]=\left[\begin{array}{l}\begin{array}{cccccc}\hfill \frac{1}{{\mathrm{E}}_1}\hfill & \hfill -\frac{v_{21}}{{\mathrm{E}}_2}\hfill & \hfill \frac{v_{31}}{{\mathrm{E}}_3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -\frac{v_{12}}{{\mathrm{E}}_1}\hfill & \hfill \frac{1}{{\mathrm{E}}_2}\hfill & \hfill -\frac{v_{32}}{{\mathrm{E}}_3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -\frac{v_{13}}{{\mathrm{E}}_1}\hfill & \hfill \frac{v_{23}}{{\mathrm{E}}_2}\hfill & \hfill -\frac{1}{{\mathrm{E}}_3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\\ {}\begin{array}{cccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill \kern0.72em 0\hfill & \hfill \kern0.96em \frac{1}{{\mathrm{G}}_{23}}\hfill & \hfill 0\hfill & \hfill \kern0.6em 0\hfill \end{array}\\ {}\begin{array}{ccccc}\hfill \kern0.72em 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \kern1.08em \frac{1}{{\mathrm{G}}_{13}}\hfill & \hfill \kern0.84em 0\hfill \end{array}\\ {}\begin{array}{cccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \kern1.92em 0\hfill & \hfill \frac{1}{{\mathrm{G}}_{12}}\hfill \end{array}\end{array}\right]\left[\begin{array}{l}{\sigma}_1\\ {}{\sigma}_2\\ {}{\sigma}_3\\ {}{\tau}_{23}\\ {}{\tau}_{31}\\ {}{\tau}_{12}\end{array}\right] $$

(7)

where γ₄, γ₅, and γ₆ are the shear strains; τ₄, τ₅, and τ₆ are the shear (or flow) stresses; and the v _ij represents Poisson’s ratio where in general

$$ \frac{v_{\mathrm{j}\mathrm{i}}}{{\mathrm{E}}_{\mathrm{i}}}=\frac{v_{\mathrm{j}\mathrm{i}}}{{\mathrm{E}}_{\mathrm{j}}}\mathrm{or}\frac{v_{\mathrm{i}\mathrm{j}}}{v_{\mathrm{i}\mathrm{j}}}=\frac{{\mathrm{E}}_{\mathrm{i}}}{{\mathrm{E}}_{\mathrm{j}}}\left(\mathrm{i},\mathrm{j}=1,2,3\right) $$

(8)

Consequently in Eq. 7:

$$ \left.\begin{array}{l}\frac{v_{12}}{E_1}=\frac{v_{21}}{E_2}\\ {}\frac{v_{13}}{E_1}=\frac{v_{31}}{E_2}\\ {}\frac{v_{23}}{E_2}=\frac{v_{31}}{E_3}\end{array}\right\} $$

(9)

For the orthotropic, transversely isotropic case in Fig. 3a, the plane 2–3 is the plane of isotropy, and

$$ \left.\begin{array}{l}{E}_2={E}_3\\ {}{G}_{12}={G}_{13}\\ {}{v}_{12}={\upgamma}_{13}\\ {}{v}_{21}={\upgamma}_{31}\\ {}{v}_{23}={\upgamma}_{32}\end{array}\right\} $$

(10)

From Eq. 10, Eq. 7 can be expressed as

$$ \left.\begin{array}{l}{\in}_1=\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\mathrm{E}}_1$}\right.\right)\left[{\sigma}_1-{v}_{12}\left({\upsigma}_2+{\upsigma}_3\right)\right]\\ {}{\in}_2=\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\mathrm{E}}_2$}\right.\right)\left[{\upsigma}_2-\left({v}_{21}{\upsigma}_1+{v}_{23}{\upsigma}_2\right)\right]\\ {}{\in}_3=\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\mathrm{E}}_2$}\right.\right)\left[{\upsigma}_3-\left({v}_{21}{\upsigma}_1+{v}_{23}{\upsigma}_2\right)\right]\end{array}\right\} $$

(11)

In addition,

$$ {\mathrm{G}}_{23}={\mathrm{E}}_2/2/1\left(+{v}_{23}\right)={\uptau}_{23}/{v}_{23} $$

(12)

which reduces the number of independent elastic constants in Eq. 11 to five. If the unidirectional composite were thin in the three direction in Fig. 3b for the plane stress lamina, two-dimensional orthotropic equations replace Eq. 11 in the form

$$ \left.\begin{array}{l}{\in}_1=\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\mathrm{E}}_1$}\right.\right)\left({\upsigma}_1-{v}_{12}{\upsigma}_2\right)\\ {}{\in}_2=\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\mathrm{E}}_2$}\right.\right)\left[{\upsigma}_2-{v}_{21}{\upsigma}_1\right]\\ {}{\upgamma}_{12}=\raisebox{1ex}{${\uptau}_{12}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{G}}_{12}$}\right.{\mathrm{E}}_2{v}_{12}={\mathrm{E}}_1{v}_{21}\end{array}\right\} $$

(13)

and Eq. 13 utilizes only four independent material constants: E₁, E₂, v ₁₂, and G₁₂.

For the bidirectional lamina shown in Fig. 1g, Eq. 13 becomes

$$ \left.\begin{array}{l}{\in}_1=\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\mathrm{E}}_1$}\right.\right)\left({\upsigma}_1-{v}_{12}{\upsigma}_2\right)\\ {}{\in}_2=\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\mathrm{E}}_2$}\right.\right)\left[{\upsigma}_2-{v}_{21}{\upsigma}_1\right]\\ {}{\in}_1={\mathrm{E}}_2,{v}_{12}={\uptau}_{12}/{\mathrm{G}}_{12}\end{array}\right\} $$

(14)

which involve only three independent elastic constants: E₁, γ₁₂, and G₁₂. This is further simplified for the continuous, multidirectional lamina represented in Figs. 1h and 2b representing a cross-plied, quasi-isotropic lay-up composite:

$$ {\mathrm{E}}_1={\mathrm{E}}_2,{v}_{21}={v}_{12},{\mathrm{G}}_{12}={\mathrm{E}}_1/2\left(1+{v}_{12}\right) $$

(15)

which has only two independent elastic constants.

We can observe in Fig. 3c, d, corresponding to Fig. 1f, that lamina principal axes (1, 2) do not coincide with the loading or reference axes (x, y), and the principal axes (1, 2) can be expressed in terms of those referred to the loading axes (x, y) as follows:

$$ \left[\begin{array}{l}{\upsigma}_1\\ {}{\upsigma}_2\\ {}{\uptau}_{12}\end{array}\right]=\left[\mathrm{T}\right]\left[\begin{array}{l}{\upsigma}_{\mathrm{x}}\\ {}{\upsigma}_{\mathrm{y}}\\ {}{\uptau}_{\mathrm{x}\mathrm{y}}\end{array}\right] $$

(16)

and

$$ \left[\begin{array}{l}{\mathrm{E}}_1\\ {}{\mathrm{E}}_2\\ {}{v}_{12}/2\end{array}\right]=\left[\mathrm{T}\right]\left[\begin{array}{l}{\mathrm{E}}_{\mathrm{x}}\\ {}{\mathrm{E}}_{\mathrm{y}}\\ {}{\upgamma}_{\mathrm{x}\mathrm{y}}\end{array}\right] $$

(17)

where the transformation matrix [T] is given by

$$ \left[\mathrm{T}\right]=\left[\begin{array}{ccc}\hfill {m}^2\hfill & \hfill {n}^2\hfill & \hfill 2mn\hfill \\ {}\hfill {n}^2\hfill & \hfill {m}^2\hfill & \hfill -2mn\hfill \\ {}\hfill -mn\hfill & \hfill mn\hfill & \hfill {m}^2-{n}^2\hfill \end{array}\right] $$

(18)

and m = cos θ, and n = sin θ in Fig. 3c and (d). From this notation one can write

$$ \left.\begin{array}{l}\frac{1}{E_x}=\frac{m^2}{E_1}\left({m}^2-{n}^2{v}_{12}\right)+\frac{n^2}{E_2}\left({n}^2-{m}^2{v}_{21}\right)+\frac{m^2{n}^2}{G_{12}}\\ {}\frac{1}{E_y}=\frac{n^2}{E_1}\left({n}^2-{m}^2{v}_{12}\right)+\frac{m^2}{E_2}\left({m}^2{n}^2{v}_{21}\right)+\frac{m^2{n}^2}{G_{12}}\\ {}\frac{1}{G_{xy}}=\frac{4{m}^2{n}^2}{E_1}\left(1+{v}_{12}\right)+\frac{4{m}^2{n}^2}{E_2}\left(1+{v}_{21}\right)+\frac{{\left({m}^2-{n}^2\right)}^2}{G_{12}}\end{array}\right\} $$

(19)

For basic lamina properties E₁, E₂, G₁₂, and ν ₁₂ and the Young’s modulus, E_x, at an angle θ = 45° with the fiber direction (Fig. 3c), one can obtain from Eq. 19:

$$ {\left(\frac{1}{{\mathrm{E}}_{\mathrm{x}}}\right)}_{\uptheta =45{}^{\circ}}=\frac{1-{v}_{12}}{4{E}_1}+\frac{1-{v}_{21}}{4{E}_2}+\frac{1}{4{G}_{12}} $$

(20)

which can be simplified for the case of a high stiffness composite where

$$ {\mathrm{E}}_1>>{\mathrm{E}}_2\kern0.5em \mathrm{and}\kern0.5em {v}_{21}<<1; $$

and Eq. 20 becomes

$$ {\mathrm{E}}_{\mathrm{x}}\left(\mathrm{q}=45{}^{\circ}\right)=4\ {\mathrm{G}}_{12}{\mathrm{E}}_2/\left({\mathrm{G}}_{12}+{\mathrm{E}}_2\right) $$

(21)

indicating that the Young’s modulus at 45° with the fiber direction (or axis) is a matrix-dominated property since it depends on E₂ and G_12, which reflect matrix properties.

We can now observe in Fig. 3b that properties related to the composite lamina, or for that matter the orthotropic composite in Fig. 3a in the longitudinal (or fiber) direction, can be expressed by

$$ {\mathrm{E}}_1={\mathrm{E}}_{\mathrm{c}}={\mathrm{V}}_{\mathrm{f}}{\mathrm{E}}_{\mathrm{f}}+{\mathrm{V}}_{\mathrm{m}}{\mathrm{E}}_{\mathrm{m}} $$

(22)

which is the classic law of mixtures formula or the Voigt (1889) model [217] uniform strain (isostrain) in a longitudinal composite:

$$ {\in}_{\mathrm{c}}={\in}_{\mathrm{m}}={\in}_{\mathrm{f}} $$

(23)

or

$$ {\mathrm{E}}_{\mathrm{c}}={\mathrm{E}}_1{\mathrm{V}}_1+{\mathrm{E}}_2{\mathrm{V}}_2+{\mathrm{E}}_3{\mathrm{V}}_3+\dots {\mathrm{E}}_{\mathrm{i}}{\mathrm{V}}_{\mathrm{i}} $$

(24)

where E_i represents the component Young’s modulus and V_i represents the component volume fraction. Correspondingly, one can show that for the transverse stiffness (E₂) in Fig. 3a, b:

$$ {\upsigma}_{\mathrm{c}}={\upsigma}_{\mathrm{m}}={\upsigma}_{\mathrm{f}}, $$

(25)

a condition of isostress for which one can write

$$ {\mathrm{E}}_2={\mathrm{E}}_{\mathrm{c}}={\mathrm{E}}_{\mathrm{f}}{\mathrm{E}}_{\mathrm{m}}/\left({\mathrm{V}}_{\mathrm{f}}{\mathrm{E}}_{\mathrm{m}}+{\mathrm{V}}_{\mathrm{m}}{\mathrm{E}}_{\mathrm{f}}\right), $$

(26)

also referred to as the Reuss model (1889) of uniform stress and expressed generally by

$$ \frac{1}{E_c}=\left(\frac{V_1}{E_1}+\frac{V_2}{E_2}+\frac{V_3}{E_3}+\dots \right). $$

(27)

It can be observed that the rule of mixtures formula in Eq. 22 or the Voigt model in Eq. 24 for a 2-component or 2-phase system having E_f >> E_m constitutes an upper boundary for composite stiffness relative to the Reuss model in Eq. 27, which for a 2-component system represented by Eq. 26 forms a lower boundary for composite stiffness which can be represented generally in Fig. 4 for volume fractions (V_f) of stiff fibers (E_f >> E_m). We can see from Eq. 21 that the composite stiffness at 45° (Fig. 3c, d) is closer to the lower boundary as shown by the dotted curve in Fig. 4 and trending toward the lower boundary as θ increases at 90° (Fig. 3c).

Fig. 4

Upper boundary (E₁: uniform strain) and lower boundary (E₂: uniform stress) composite stiffness versus fiber volume fraction. E_f >> E_m

Figure 5a compares the general trend for 2-phase composite stiffness variations for continuous (longitudinal) fibers in contrast to discontinuous, directional whiskers having l/d > 10² and l >> l_c, as well as particles where the reinforcement is noticeably reduced at volume fractions <10 %. In addition, Fig. 5b shows that in general, continuous fiber diameters are not of any consequence in the context of fiber volume fraction, confirming the applicability of the rule of mixtures formula in Eq. 22. This can be applied generally in the so-called Voigt (1889) average:

$$ {\mathrm{P}}_{\mathrm{c}}={\displaystyle \sum_{i=1}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{i}}{\mathrm{V}}_{\mathrm{i}}} $$

(28)

where P represents a property (such as strength, thermal expansion coefficient, etc.) of the composite (c) and the ith component having a volume fraction V_i.

Fig. 5

(a) Comparison of composite stiffness (modulus) versus volume fraction (V_f) for different reinforcing phase forms. (b) Composite stiffness (E_L) or longitudinal elastic modulus as a function of fiber volume fraction (V_f) for various diameter glass fibers in epoxy. Adapted from Meyers and Chawla (1984)

Similarly, the Reuss (1929) model implies that

$$ \frac{1}{{\mathrm{P}}_{\mathrm{c}}}={\displaystyle \sum_{\mathrm{i}=1}^{\mathrm{n}}\left(\frac{{\mathrm{V}}_{\mathrm{i}}}{{\mathrm{P}}_{\mathrm{i}}}\right)} $$

(29)

for transverse composite properties, P_c and P_i.

In addition to the general classification of composites characterized by fiber or the dispersed phase in a matrix, the matrix itself becomes part of the characterization. Prominent matrices include of course metals or alloys (metal matrix composites: MMC), polymers including thermosetting epoxies (polymer matrix composites (PMCs)), and reinforced ceramics (ceramic matrix composites (CMCs)). In both MMCs and PMCs, the reinforcing phase (fibers) enhances stiffness, strength, and creep of the matrix while the principal enhancement for CMCs is toughness and the ability to sustain intrinsic strength even when cracks form.

Table 1 illustrates a few metal matrix and polymer matrix composites, while Fig. 6a shows a typical transverse plane view for an aluminum/boron fiber composite corresponding roughly to the example provided in Table 1. Table 2 shows some properties for fibers, metallic, inorganic, and organic. The tensile strength for the fibers is noted by T.S., which corresponds to the ultimate tensile stress. Fibers and often other composite systems are also characterized by specific stiffness, the stiffness E divided by the density, ρ, and the specific strength, tensile strength (T.S.) divided by the density: T.S./ρ, where in some cases T.S. = σ_y, the yield stress. In many cases, the specific strength is referred to as the strength-to-weight ratio, and for many applications, strategies to increase this ratio in a significant way can indicate notable performance enhancement.

Table 1 Properties of some typical MMC and PMC composites in the longitudinal and transverse directions^a

Fig. 6

(a) Al/B-fiber composite (transverse section view). The B fibers have a diameter of 100 mm and a volume fraction (V_f) of 0.44 (44%). (b) and (c) show coated carbon fibers for compatibility and reduced diffusion reactions. (b) shows an SEM image cross-section of ZrO₂-coated C fibers (oxide coating is white ring wound fibers) in a ZrC matrix. (c) shows an SEM view of ZrO₂/HfO multilayer-coated carbon fibers(Courtesy of Ultramet)

Table 2 Properties of fibers ^a

We should comment on the fact that in some cases of matrix reinforcement using continuous or long fibers, the fiber may be incompatible with the matrix forming a weak bond (or low interfacial free energy or adhesive energy) or conducive to diffusive phenomena at the fiber/matrix interface creating a reactive phase or a new interfacial chemical phase. This can cause fiber pullout, rendering the reinforcement seriously compromised, especially over time and at elevated temperatures promoting interfacial diffusion and reaction. Figure 6b, c illustrates the coating of commercial fibers to enhance fiber/matrix compatibility and the elimination of reaction.