Fiber Volume Fraction⚠ unverified
Mechanical / Composites · Compute the fibre volume fraction from constituent masses and densities
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| m_f | mf | kg | 1.0 | Mass of fibre |
| rho_f | ρf | kg/m^3 | 1.0 | Fibre density |
| m_m | mm | kg | 1.0 | Mass of matrix |
| rho_m | ρm | kg/m^3 | 1.0 | Matrix density |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Vf | — | Fibre volume fraction Vf (dimensionless), between 0 and 1 |
The science & history
Understanding the Parameters
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Mass-to-volume conversion $m/\rho$ — you weigh fiber and matrix (masses are easy to measure), but properties depend on volume fractions. Dividing each mass by its density converts to volume, which is why the formula is built from $m_f/\rho_f$ and $m_m/\rho_m$.
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Volume vs mass fraction — because fibers are usually denser than the matrix, the mass fraction exceeds the volume fraction. Confusing the two is a classic error; micromechanics always uses volume fraction.
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The complement — the matrix volume fraction is $V_m = 1 - V_f$ (assuming no voids). A well-made laminate has void content under 1–2 %; higher voids invalidate the "$V_f + V_m = 1$" assumption and degrade strength.
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Practical ceiling — perfect hexagonal fiber packing caps $V_f$ at $\approx 0.907$, but real processes reach only $\sim 0.65$–$0.70$ before fibers touch and matrix cannot wet them; beyond that, properties fall.
Derivation (Approaching a Proof)
By definition, the fiber volume fraction is the fiber volume divided by the total (fiber + matrix) volume, assuming no voids:
$$V_f = \frac{V_{\text{fiber}}}{V_{\text{fiber}} + V_{\text{matrix}}}.$$
Each volume is mass over density, $V = m/\rho$:
$$V_{\text{fiber}} = \frac{m_f}{\rho_f}, \qquad V_{\text{matrix}} = \frac{m_m}{\rho_m}.$$
Substituting gives the working formula directly:
$$V_f = \frac{m_f/\rho_f}{m_f/\rho_f + m_m/\rho_m}.$$
Experimentally, $V_f$ is measured by matrix burn-off or acid digestion (weigh the fibers left after removing the matrix) or back-calculated from the composite density (Composite Density).
Dimensional check. Each term $m/\rho = \text{kg}/(\text{kg}/\text{m}^3) = \text{m}^3$ (a volume); the ratio of volumes is dimensionless — a fraction, as required.
History and Development
Volume-fraction bookkeeping is the foundation of composite micromechanics, formalised as fiber-reinforced polymers matured through the 1960s–70s. It underlies every rule-of-mixtures estimate (Rule of Mixtures longitudinal, Composite Density) and the manufacturing quality control (burn-off/digestion tests per ASTM D2584/D3171) that verifies a laminate was built to its design $V_f$. See Composite Laminate Theory for how the fraction propagates to full-laminate stiffness.
Related Concepts: Composite Density, Rule of Mixtures longitudinal, Rule of Mixtures transverse, Poisson Ratio Nu12, Halpin-Tsai Equations, Composite Laminate Theory
Notes: Uses volume fraction (not mass) — denser fibers make mass fraction > volume fraction. $V_m = 1-V_f$ (no voids); real void content < 1–2 %. Typical structural $V_f \approx 0.5$–$0.65$; hexagonal-packing limit $\approx 0.907$. Measured by burn-off/digestion or from Composite Density.