Question

The tensile modulus of a thermosetting polyester resin and glass fiber are 3 GPa and 80 GPa, respectively. If a tensile stress of 110 MPa is applied along the fiber direction on a continuous uniaxially aligned glass fiber reinforced thermosetting polyester composite with a fiber content of 60% by volume, the resulting strain will be _________ \times 10^{-3} \text{ (round off to one decimal place).}

Hint:

For composites, the strain can be calculated by using the rule of mixtures for modulus and applying Hooke’s Law. Be sure to convert units properly.

Answer 1

Correct Answer: 2.1

The strain in the composite can be calculated using Hooke's Law, where strain is given by:
\[ \epsilon = \frac{\sigma}{E} \] Where:
- \( \sigma = 110\ \text{MPa} = 110 \times 10^3\ \text{Pa} \) (tensile stress)
- \( E_f = 80\ \text{GPa} = 80 \times 10^9\ \text{Pa} \) (tensile modulus of fiber)
- \( E_m = 3\ \text{GPa} = 3 \times 10^9\ \text{Pa} \) (tensile modulus of matrix)
- \( V_f = 0.60 \) (volume fraction of fiber)
- \( V_m = 1 - V_f = 0.40 \) (volume fraction of matrix)
The effective modulus of elasticity \( E_c \) is calculated by the rule of mixtures for elasticity:
\[ E_c = E_f V_f + E_m V_m \] Substitute the values:
\[ E_c = (80 \times 10^9)(0.60) + (3 \times 10^9)(0.40) \] \[ E_c = 48 \times 10^9 + 1.2 \times 10^9 = 49.2 \times 10^9\ \text{Pa} \] Now, calculate the strain:
\[ \epsilon = \frac{110 \times 10^3}{49.2 \times 10^9} = 2.24 \times 10^{-3} \] Rounded to one decimal:
\[ \epsilon = 2.3 \times 10^{-3} \]