An anisotropic material has a coefficient of linear expansion `alpha, 2alpha and 3alpha` along the three co - ordinate axis. Coefficient of cubical expansion of material will be equal to

Options

A. `2alpha`
B. `root(3)(6)alpha`
C. `6alpha`
D. None of these

Answer

To find the coefficient of cubical expansion (volumetric expansion) for an anisotropic material with different coefficients of linear expansion along three coordinate axes, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Coefficients of Linear Expansion**:
Let the coefficients of linear expansion along the x, y, and z axes be:
- \( \alpha_1 = \alpha \) (along x-axis)
- \( \alpha_2 = 2\alpha \) (along y-axis)
- \( \alpha_3 = 3\alpha \) (along z-axis)

2. **Consider a Cuboid**:
Assume we have a cuboid with dimensions \( a \), \( b \), and \( c \) along the x, y, and z axes respectively.

3. **Calculate the Initial Volume**:
The initial volume \( V \) of the cuboid is given by:
\[ V = a \times b \times c \]

4. **Determine the New Dimensions After Expansion**:
When the temperature changes by \( \Delta T \) (let's denote it as \( t \)), the new dimensions after linear expansion will be:
- New length along x-axis:
\[ a' = a(1 + \alpha_1 t) = a(1 + \alpha t) \]
- New length along y-axis:
\[ b' = b(1 + \alpha_2 t) = b(1 + 2\alpha t) \]
- New length along z-axis:
\[ c' = c(1 + \alpha_3 t) = c(1 + 3\alpha t) \]

5. **Calculate the New Volume**:
The new volume \( V' \) after expansion will be:
\[ V' = a' \times b' \times c' = [a(1 + \alpha t)] \times [b(1 + 2\alpha t)] \times [c(1 + 3\alpha t)] \]

6. **Expand the Expression**:
Expanding this expression:
\[ V' = abc \times (1 + \alpha t)(1 + 2\alpha t)(1 + 3\alpha t) \]
Using the approximation \( (1 + x)(1 + y)(1 + z) \approx 1 + (x + y + z) \) for small \( x, y, z \):
\[ V' \approx V \left( 1 + (\alpha + 2\alpha + 3\alpha)t + \text{(higher order terms)} \right) \]

7. **Identify the Coefficient of Volumetric Expansion**:
The coefficient of volumetric expansion \( \gamma \) is defined as:
\[ V' = V(1 + \gamma t) \]
From our expansion, we see:
\[ \gamma = \alpha + 2\alpha + 3\alpha = 6\alpha \]

8. **Final Result**:
Therefore, the coefficient of cubical expansion of the material is:
\[ \gamma = 6\alpha \]

### Conclusion:
The coefficient of cubical expansion for the anisotropic material is \( 6\alpha \).