Page 298 - Carrahers_Polymer_Chemistry,_Eighth_Edition

Composites and Fillers 261

The transverse modulus (M ) and many other properties of a long fiber–resin composite may
T
be estimated from the law of mixtures. The longitudinal modulus (M ) may be estimated from the
L
Kelly–Tyson equation (Equation 8.5), where the longitudinal modulus is proportional to the sum of

the fiber modulus (M ) and the resin matrix modulus (M ). Each modulus is based on a fractional
F M

volume (c). The constant k is equal to 1 for parallel continuous filaments and decreases for more
randomly arranged shorter fi laments.
M = kM c + M c (8.5)
L F F M M
Since the contribution of the resin matrix is small in a strong composite, the second term in the
Kelly–Tyson term can be disregarded. Thus, the longitudinal modulus is dependent on the reinforce-
ment modulus, which is independent of the diameter of the reinforcing fi ber.
As noted before, for the most part, the resulting materials from the use of reinforcements are
composites. Composites are materials that contain strong fibers embedded in a continuous phase.

The fibers are called “reinforcement” and the continuous phase is called the matrix. While the con-
tinuous phase can be a metallic alloy or inorganic material, the continuous phase is typically an
organic polymer that is termed a “resin.” Composites can be fabricated into almost any shape and
after hardening, they can be machined, painted, and so on as desired.
While there is a lot of science and “space-age technology” involved in the construction of com-
posites, many composites have been initially formulated through a combination of this science and
“trial-and-error” giving “recipes” that contain the nature and form of the fiber and matrix materials,

amounts, additives, and processing conditions.
Composites have high tensile strengths (on the order of thousands of MPa), high Young’s modu-
lus (on the order of hundreds of GPa) and good resistance to weathering exceeding the bulk proper-
ties of most metals. The resinous matrix, by itself, is typically not particularly strong relative to the
composite. Further, the overall strength of a single fiber is low. In combination, the matrix–fi ber

composite becomes strong. The resin acts as a transfer agent, transferring and distributing applied
stresses to the fibers. In general, the fibers should have aspect ratios (ratio of length to diameter)

exceeding 100, often much larger. Most fibers are thin (less than 20 um thick, about a tenth the

thickness of a human hair). Fibers should have a high tensile strength and most have a high stiffness,
that is, low strain for high stress or little elongation as high forces are applied.
There exists a relationship between the “ideal” length of a fiber and the amount of adhesion

between the matrix and the fiber. For instance, assume that only the tip, one end, of a fi ber is

placed in a resin (Figure 8.2—left). The fiber is pulled. The adhesion is insufficient to “hold” the

fi ber and it is pulled from the resin (Figure 8.2—top right). The experiment is repeated until the

fiber is broken (outside the matrix) rather than being pulled (without breaking) from the resin
(Figure 8.2—right bottom). Somewhere between the two extremes, there is a length where there

exists a balance between the strength of the fiber and the adhesion between the fiber and matrix.

Most modern composites utilize fiber/matrix combinations that exploit this “balance.”

Mathematically the critical fi ber length necessary for effective strengthening and stiffening can
be described as follows:

Critical fiber length = (Ultimate or tensile strength times fiber diameter/2) times the fi ber–matrix

bond strength OR the shear yield strength of the matrix—whichever is smaller.
Fibers where the fiber length is greater than this critical fiber length are called continuous fi bers

while those that are less than this critical length are called discontinuous or short fi bers. Little trans-

ference of stress and thus little reinforcement is achieved for short fibers. Thus, fibers whose lengths

exceed the critical fiber length are typically used.
Fibers can be divided according to their diameters. Whiskers are very thin single crystals that
have large length to diameter ratios. They have a high degree of crystalline perfection and are
essentially flaw free. They are some of the strongest materials know. Whisker materials include

graphite, silicon carbide, aluminum oxide, and silicon nitride. Fine wires of tungsten, steel, and
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