Question

A unidirectional composite of epoxy and carbon fiber of 50% by volume is made. The elastic modulus of epoxy and carbon fiber are 3.5 GPa and 350 GPa, respectively. The ratio (rounded off to one decimal place) of the modulus of the composite to the matrix modulus is \(\underline{\hspace{2cm}}\).

Hint:

For composite materials, the rule of mixtures helps to estimate the elastic modulus based on the volume fractions and moduli of the components.

Answer 1

Correct Answer: 50.4

The modulus of the composite \( E_c \) is given by the rule of mixtures:
\[ E_c = V_f E_f + V_m E_m \] where:
- \( V_f = 0.5 \) (volume fraction of fiber),
- \( V_m = 0.5 \) (volume fraction of matrix),
- \( E_f = 350 \, \text{GPa} \) (elastic modulus of fiber),
- \( E_m = 3.5 \, \text{GPa} \) (elastic modulus of matrix).
Substituting the values:
\[ E_c = 0.5 \times 350 + 0.5 \times 3.5 = 175 + 1.75 = 176.75 \, \text{GPa}. \] Now, the ratio of the composite modulus to the matrix modulus is:
\[ \frac{E_c}{E_m} = \frac{176.75}{3.5} \approx 50.5. \] Thus, the ratio is approximately 50.5.