Fiber volume ratio

The amount of fiber in a fiber reinforced composite directly corresponds with the mechanical properties of the composite. Theoretically the maximum fiber ratio of round fibers that can be achieved in a composite is 90.8% if the fibers are in a unidirectional hexagonal close packed configuration. Realistically the highest fiber volume ratio is around 70% due to manufacturing parameters and is usually in the range of 50% to 65%.[4] Adding too little fiber reinforcement in the composite will actually deteriorate the properties of the material. Too much fiber volume may also decrease the strength of the composite due to the lack of space for the matrix to fully surround and bond with the fibers.[8] Therefore, there is an optimal space between fibers that will fully exploit the uniform load transfer between fibers.[7] Given the fiber volume fraction, the theoretical elastic properties of a composite can be determined. The elastic modulus of a composite in the fiber direction of a unidirectional composite can be calculated using the following equation:

${\displaystyle E=(1-{V_{f}}){E_{m}}+{V_{f}}{E_{f}}}$

Where:

${\displaystyle V_{f}}$ is the fiber volume ratio

and

${\displaystyle E_{m}}$ is the elastic modulus of the matrix

${\displaystyle E_{f}}$ is the elastic modulus of the fibers

Fibers are commonly arranged in a square or hexagonal lattice as well as in a layer-wise fiber array. Assuming that each fiber has a circular cross-sectional with the same diameter, the fiber volume fraction of these two kinds of packing are respectively:

Hexagonal

${\displaystyle {V_{f}}=({\frac {\pi }{2{\sqrt {3}}}}){({\frac {r}{R}})^{2}}}$

Square

${\displaystyle {V_{f}}=({\frac {\pi }{4}}){({\frac {r}{R}})^{2}}}$

where:

${\displaystyle r}$ is the radius of fiber

and

${\displaystyle 2R}$ is the center to center spacing of the fibers.

The maximum fiber volume fraction will occur when the fibers are touching, i.e. r=R. For a hexagonal array ${\displaystyle V_{f,max}}$ = 0.907, and for square packing ${\displaystyle V_{f,max}}$ = 0.785.

However, these are ideal situations only used for theoretical analysis. In practical cases there can be variation in fiber diameter and irregular packing. In practice, it’s hard to achieve a volume fraction greater than 0.7 and this must be regarded as the realistic limit for commercial materials.

In the production process, using different fiber architecture methods can obtain different volume fractions. 2D aligned unidirectional fabrics with pre-preg (usually carbon) fibers are considered to have the highest volume fraction among common fiber architectures.[6] Filament winding is also usually associated with high fiber volume fractions – with careful control of fiber tension and resin content, values of around 70% are possible.[4]

Porosity or void fraction is a measure of the void (i.e., "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0 and 100%. There are many ways to determine if a composite part contains voids, such as industrial CT scanning or ultrasound. If the volume fraction of the fibers and matrix is known, the volume voids can also be found using the following equation:

${\displaystyle V_{v}=1-V_{f}-V_{m}={\frac {v_{v}}{v_{c}}}\!}$ [3]

where:

${\displaystyle V_{v}}$ is the void volume ratio

and

${\displaystyle V_{f}}$ is the fiber volume ratio

${\displaystyle V_{m}}$ is the matrix volume ratio

${\displaystyle v_{v}}$ is the volume of voids

${\displaystyle v_{c}}$ is the volume of the composite

Another equation used to calculate void volume fraction is:

${\displaystyle V_{v}={\frac {({p_{ct}}-{p_{cm}})}{p_{ct}}}\!}$

where:

${\displaystyle V_{v}}$ is the void volume ratio

and

${\displaystyle p_{ct}}$ is the theoretical density of the composite without voids

${\displaystyle p_{cm}}$ is the measured density of the composite

• Volume-Weight Fraction Calculator