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Rajamohan Ayiravalli

KINECO LIMITED, GOA,INDIA.

Wasiu Salami has given the step by step calculation. The rule of mixture works for other properties like 'Tensile strength, Tensile modulus etc' also.

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Şeyda Bostancı

To calculate the theoretical density of a composite material from the weight percentages of the fiber and matrix, you can follow these steps:

### Step 1: Gather Required Information

1. **Weight Percentage of Fiber (\(Wt_f\))**: This is the percentage of the fiber in the composite.

2. **Weight Percentage of Matrix (\(Wt_m\))**: This is the percentage of the matrix in the composite.

3. **Density of Fiber (\(\rho_f\))**: Obtain or measure the density of the fiber material.

4. **Density of Matrix (\(\rho_m\))**: Obtain or measure the density of the matrix material.

### Step 2: Convert Weight Percentages to Decimal Form

Convert the weight percentages into decimal form:

- \( Wt_f = \frac{Wt_f}{100} \)

- \( Wt_m = \frac{Wt_m}{100} \)

### Step 3: Calculate the Volume Fractions

The volume fraction of each component can be calculated using the following formulas:

\[

V_f = \frac{Wt_f / \rho_f}{(Wt_f / \rho_f) + (Wt_m / \rho_m)}

\]

\[

V_m = \frac{Wt_m / \rho_m}{(Wt_f / \rho_f) + (Wt_m / \rho_m)}

\]

Where:

- \( V_f \) is the volume fraction of the fiber.

- \( V_m \) is the volume fraction of the matrix.

### Step 4: Calculate Theoretical Density of the Composite

The theoretical density of the composite material (\(\rho_c\)) can be calculated using the weighted average of the densities based on the volume fractions:

\[

\rho_c = V_f \cdot \rho_f + V_m \cdot \rho_m

\]

### Example Calculation

Assume you have the following data:

- \(Wt_f = 60\%\) (fiber)

- \(Wt_m = 40\%\) (matrix)

- \(\rho_f = 1.5 \, \text{g/cm}^3\) (density of fiber)

- \(\rho_m = 1.0 \, \text{g/cm}^3\) (density of matrix)

1. Convert weight percentages to decimals:

- \(Wt_f = 0.60\)

- \(Wt_m = 0.40\)

2. Calculate volume fractions:

\[

V_f = \frac{0.60 / 1.5}{(0.60 / 1.5) + (0.40 / 1.0)} = \frac{0.40}{0.40 + 0.40} = \frac{0.40}{0.80} = 0.50

\]

\[

V_m = \frac{0.40 / 1.0}{(0.60 / 1.5) + (0.40 / 1.0)} = \frac{0.40}{0.40 + 0.40} = \frac{0.40}{0.80} = 0.50

\]

3. Calculate the theoretical density of the composite:

\[

\rho_c = (0.50 \cdot 1.5) + (0.50 \cdot 1.0) = 0.75 + 0.50 = 1.25 \, \text{g/cm}^3

\]

### Conclusion

The theoretical density of the composite material is \(1.25 \, \text{g/cm}^3\) based on the provided fiber and matrix densities and their weight percentages.

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