Quartz fibres of density 2.2 g/cm^3 are combined with polyester resin of density 1.25 g/cm^3, with the quartz forming 67% of the composite by weight. Given Ef = 73 GPa, Em = 2.72 GPa, Gf = 8.4 GPa and Gm = 2.15 GPa, calculate the transverse stiffness E22 and the in-plane shear modulus G12 to one decimal place.

Step-by-step solution

1. 1

   Convert the weight fractions to volume fractions. The rules of mixtures are written in terms of volume fractions, so the $67\%$ by weight must be converted first. With $W_f=0.67$ and $W_m=0.33$,

   $\frac{W_f}{\rho_f}=\frac{0.67}{2.2}=0.304545,\qquad \frac{W_m}{\rho_m}=\frac{0.33}{1.25}=0.264000$

   $V_f=\frac{0.304545}{0.304545+0.264000}=\frac{0.304545}{0.568545}=0.5357,\qquad V_m=0.4643$

   Note the fibres drop from $67\%$ by weight to only $53.6\%$ by volume, because quartz is the denser phase.

2. 2

   Choose the inverse (Reuss) rule for the transverse direction. Loaded across the fibres, the fibre and matrix carry the same stress and their strains add, so the compliances — not the stiffnesses — add in proportion:

   $\frac{1}{E_{22}}=\frac{V_f}{E_f}+\frac{V_m}{E_m}$

   Using the direct rule $E=V_fE_f+V_mE_m$ here would be the classic error; that form applies only along the fibres.

3. 3

   Evaluate E₂₂.

   $\frac{V_f}{E_f}=\frac{0.5357}{73}=0.007338,\qquad \frac{V_m}{E_m}=\frac{0.4643}{2.72}=0.170714$

   $\frac{1}{E_{22}}=0.178052\quad\Longrightarrow\quad E_{22}=5.6163\ \text{GPa}$

4. 4

   Apply the same rule to the shear modulus.

   $\frac{1}{G_{12}}=\frac{V_f}{G_f}+\frac{V_m}{G_m}=\frac{0.5357}{8.4}+\frac{0.4643}{2.15}=0.063769+0.215973=0.279742$

   $G_{12}=3.5747\ \text{GPa}$

5. 5

   Round to one decimal place.

   $\boxed{E_{22}=5.6\ \text{GPa},\qquad G_{12}=3.6\ \text{GPa}}$

6. 6

   Check that both results sit where the physics demands. An inverse rule of mixtures always lands between the two constituent values and much nearer the weaker one: $E_{22}=5.6$ GPa is close to $E_m=2.72$ GPa and far from $E_f=73$ GPa, and $G_{12}=3.6$ GPa lies between $2.15$ and $8.4$ GPa. For contrast, the longitudinal stiffness would be $V_fE_f+V_mE_m=0.5357(73)+0.4643(2.72)=40.4$ GPa — seven times stiffer, which is exactly why unidirectional laminates are so anisotropic.