Fibre Volume Fraction from Weight Fraction

This page exists because of one recurring and consequential mistake, and the honest way to introduce it is to name the mistake first. A weight fraction is not a volume fraction, they are not close, and every micromechanics equation in this catalogue wants the volume one.

The reason they diverge is simply that fibre and resin have different densities. Carbon sits near 1.8 g/cm³ and cured epoxy near 1.2, so a kilogram of fibre occupies two-thirds of the room a kilogram of resin does. Work it through for a laminate that a burn-off test reports as 70% fibre by weight: $V_f = (0.70/1.80) / (0.70/1.80 + 0.30/1.20) = 14/23 = 0.609$. Seventy by weight is sixty-one by volume. Nine percentage points, at the volume fractions people actually build at.

Now the consequence. Enter 0.70 into the rule of mixtures where 0.609 belongs and the longitudinal modulus comes out about fifteen percent high. The error is in the unsafe direction — the calculated part is stiffer than the real one — and nothing about the result looks wrong, because both figures are dimensionless fractions between zero and one and both are plausible for a good laminate. It survives review for the same reason. There is no unit to catch it and no order-of-magnitude tell.

The rule worth carrying is: tests measure mass, equations want volume. Resin burn-off (ASTM D2584, for glass) and acid digestion (ASTM D3171, for carbon, which glass survives and carbon does not) both weigh the fibre residue against the original coupon, so what they report is $W_f$. Prepreg is specified by resin content, which is $1 - W_f$, also by weight. Wet lay-up is controlled by mixing so many grams of resin per square metre of fabric, again by weight. Meanwhile the rule of mixtures, the inverse rule, Halpin–Tsai, the Poisson average and the density rule are all written in $V_f$. Somebody has to do this conversion, and it needs both densities, which is why it cannot be a footnote on another page.

Run in the other direction it is the shop-floor calculation. A design calls for $V_f = 0.60$; what weight fraction is that? $W_f = V_f \rho_f / (V_f \rho_f + (1 - V_f) \rho_m) = 1.08/1.56 = 0.692$. Sixty by volume is sixty-nine by weight, which means a resin content of 30.8% — and that is the number you can actually work to, because it is what a scale can measure and what a burn-off test will report back on the finished part.

Two subtleties. The conversion assumes a void-free laminate: if the panel is porous, the resin volume you are implicitly attributing includes the bubbles, and the true fibre volume fraction is a little higher than this equation gives. And knowing both fractions for the same panel pins the density ratio, so the two rearrangements that solve for $\rho_f$ or $\rho_m$ make a useful cross-check — if the fibre density that falls out is not the one on the fibre's own data sheet, something in the pair of measurements is wrong, and porosity is the usual culprit.

The last word is about vocabulary. Any time somebody quotes a single "fibre content" without saying which kind, ask. The two numbers only coincide when the densities are equal, which for a fibre composite never happens.