Question

The rule of mixtures for composite modulus E\(_{\text{c}}\) in longitudinal loading is

• A. E\(_{\text{c}}\) \(=\) V\(_m\)E\(_m\) \(+\) V\(_r\)E\(_r\)
• B. E\(_{\text{c}}\) \(=\) E\(_m\)E\(_r\)/(E\(_m\) \(+\) E\(_r\))
• C. E\(_{\text{c}}\) \(=\) (E\(_m\)V\(_m^2\) \(+\) E\(_r\)V\(_r^2\))
• D. E\(_{\text{c}}\) \(=\) 1/(V\(_m\)/E\(_m\) \(+\) V\(_r\)/E\(_r\))

Hint:

Remember:
- Longitudinal Loading (Isostrain Voigt model) \(\rightarrow\) Upper Bound:
\[ E_c = V_m E_m + V_r E_r \]
- Transverse Loading (Isostress Reuss model) \(\rightarrow\) Lower Bound:
\[ \frac{1}{E_c} = \frac{V_m}{E_m} + \frac{V_r}{E_r} \]

Answer 1

The Correct Option is A

Instead of deriving the formula from the definition of stress and strain, picture the matrix and the fibre as two springs fixed side by side to the same rigid end plate, which is exactly the condition in longitudinal loading. Both springs must stretch by the same amount as the plate moves, so their strains match, while the total pulling force gets shared between them in proportion to how much cross-sectional area each one occupies. Since force equals stress times area, and volume fraction stands in for area fraction in a uniform composite, adding the two force contributions and dividing by the total area gives a composite modulus equal to the volume-fraction-weighted average of the two individual moduli. That weighted average is \( E_c = V_m E_m + V_r E_r \), the familiar rule of mixtures for the stiffer, isostrain case. So the correct choice is option (A).