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

Step 1: Understanding the Question:
This question asks for the mathematical formula derived from the rule of mixtures to estimate the Young's modulus of a fiber-reinforced composite (\( E_c \)) loaded parallel to the fibers (longitudinal loading).

Step 2: Key Formula or Approach:
Longitudinal loading represents the isostrain condition (Voigt model), where both the matrix and the reinforcing fibers undergo the same elastic strain:
\[ \epsilon_c = \epsilon_m = \epsilon_r \]
where:
\( \epsilon_c, \epsilon_m, \epsilon_r \) are the strains in the composite, matrix, and reinforcement, respectively.

Step 3: Detailed Explanation:

• Derivation of the Equation:
- The total force \( F_c \) acting on the composite is shared between the matrix and the reinforcement:
\[ F_c = F_m + F_r \]
- Stress is force per unit area (\( \sigma = F/A \)), which can be rewritten as:
\[ \sigma_c A_c = \sigma_m A_m + \sigma_r A_r \]
- Divide both sides by the total cross-sectional area \( A_c \):
\[ \sigma_c = \sigma_m \left(\frac{A_m}{A_c}\right) + \sigma_r \left(\frac{A_r}{A_c}\right) \]
- For uniform continuous composites, the area fraction equals the volume fraction (\( V_m = A_m/A_c \) and \( V_r = A_r/A_c \)):
\[ \sigma_c = \sigma_m V_m + \sigma_r V_r \]
- Apply Hooke's Law (\( \sigma = E \epsilon \)):
\[ E_c \epsilon_c = E_m \epsilon_m V_m + E_r \epsilon_r V_r \]
- Since \( \epsilon_c = \epsilon_m = \epsilon_r \), the strains cancel out, leaving:
\[ E_c = V_m E_m + V_r E_r \]
- This is the upper-bound elastic modulus of a composite.


Step 4: Final Answer:
The Rule of Mixtures for composite modulus in longitudinal loading is \( E_c = V_m E_m + V_r E_r \).
Thus, the correct option is (A).