Question (Materials — Composites). A unidirectional carbon/epoxy lamina has fiber volume fraction $V_f=0.60$. The fiber modulus is $E_f=33 \text{Msi}$ and the matrix modulus is $E_m=0.50 \text{Msi}$. Assuming the rule of mixtures, find the longitudinal modulus $E_1$ and the transverse modulus $E_2$.

Relations. Matrix fraction: $V_m=1-V_f=0.40$.
Longitudinal (Voigt): $E_1=V_f{E}_f+V_m{E}_m$.
Transverse (Reuss): $E_2={(\frac{V_f}{E_f}+\frac{V_m}{E_m})}^{-1}$.

Compute.
$E_1=\left(0.60\right)\left(33\right)+\left(0.40\right)\left(0.50\right)=19.8+0.20=20.0 \text{Msi}$.
$E_2={(\frac{0.60}{33}+\frac{0.40}{0.50})}^{-1}={(0.01818+0.80)}^{-1}={\left(0.81818\right)}^{-1}=1.22 \text{Msi}$.

Answer. $E_1\approx 20.0 \text{Msi}$, $E_2\approx 1.22 \text{Msi}$ (most nearly).