A unidirectional pre-preg carbon fibre is 45% resin by weight. The fibre density is 1.7 g/cm^3 and the matrix density is 1.1 g/cm^3. Calculate the fibre volume fraction Vf and the matrix volume fraction Vm as percentages to one decimal place.

Step-by-step solution

1. 1

   Read off the weight fractions. "$45\%$ resin by weight" means

   $W_m=0.45,\qquad W_f=1-0.45=0.55$

   Weight fractions and volume fractions differ whenever the two densities differ, which is exactly why the conversion is needed.

2. 2

   Convert weights to volumes. Take a convenient $1$ g of pre-preg. Since volume $=$ mass$/$density, the two constituents occupy

   $\frac{W_f}{\rho_f}=\frac{0.55}{1.7}=0.3235\ \text{cm}^3,\qquad \frac{W_m}{\rho_m}=\frac{0.45}{1.1}=0.4091\ \text{cm}^3$

3. 3

   Find the total volume of that gram.

   $0.3235+0.4091=0.7326\ \text{cm}^3$

   (Equivalently the composite density is $1/0.7326=1.365\ \text{g/cm}^3$.)

4. 4

   Divide to get each volume fraction.

   $V_f=\frac{0.3235}{0.7326}=0.4416,\qquad V_m=\frac{0.4091}{0.7326}=0.5584$

5. 5

   Express as percentages.

   $\boxed{V_f=44.2\%,\qquad V_m=55.8\%}$

6. 6

   Check the sum and the direction of the shift. $44.2+55.8=100.0\%$ ✓. The resin is only $45\%$ by weight but $55.8\%$ by volume, which is the expected direction: the matrix is the less dense phase, so a given mass of it takes up more room. If the numbers had moved the other way, a density would have been used upside down.