Find the relation below young's modulus (Y), bulk modulus (K) and shear modulus (G)?

Options

A. `Y = 6KG/(G+3K)`
B. `Y = 4KG/(G+3K)`
C. `Y = 9KG/(G+3K)`
D. `Y = 12KG/(G+3K)`

Answer

To find the relation between Young's modulus (Y), Bulk modulus (K), and Shear modulus (G), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Young's modulus (Y) measures the tensile elasticity of a material.
- Bulk modulus (K) measures a material's response to uniform pressure.
- Shear modulus (G) measures a material's response to shear stress.

2. **Use Known Relationships**:
- The relationship between Young's modulus (Y) and Bulk modulus (K) is given by:
\[ Y = 3K(1 - 2\sigma) \]
where \(\sigma\) is Poisson's ratio.

3. **Relate Young's Modulus (Y) and Shear Modulus (G)**:
- The relationship between Young's modulus (Y) and Shear modulus (G) is given by:
\[ Y = 2G(1 + \sigma) \]

4. **Eliminate Poisson's Ratio (\(\sigma\))**:
- From the first equation, we can express \(\sigma\):
\[ 1 - 2\sigma = \frac{Y}{3K} \implies 2\sigma = 1 - \frac{Y}{3K} \implies \sigma = \frac{1}{2} - \frac{Y}{6K} \]
- From the second equation, we can express \(\sigma\) as well:
\[ 1 + \sigma = \frac{Y}{2G} \implies \sigma = \frac{Y}{2G} - 1 \]

5. **Set the Equations for \(\sigma\) Equal**:
- Equate the two expressions for \(\sigma\):
\[ \frac{1}{2} - \frac{Y}{6K} = \frac{Y}{2G} - 1 \]

6. **Solve for Y**:
- Rearranging gives:
\[ \frac{1}{2} + 1 = \frac{Y}{2G} + \frac{Y}{6K} \]
\[ \frac{3}{2} = \frac{Y}{2G} + \frac{Y}{6K} \]
- To combine the right side, find a common denominator:
\[ \frac{Y}{2G} + \frac{Y}{6K} = Y\left(\frac{3K + G}{6GK}\right) \]
- Thus, we have:
\[ \frac{3}{2} = Y\left(\frac{3K + G}{6GK}\right) \]
- Rearranging gives:
\[ Y = \frac{9GK}{3K + G} \]

### Final Relation:
The final relation between Young's modulus (Y), Bulk modulus (K), and Shear modulus (G) is:
\[ Y = \frac{9GK}{3K + G} \]