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. **Understanding 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 the material's response to shear stress.

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

4. **Eliminating 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. **Setting the Two Expressions for \(\sigma\) Equal**:
- Now we can set the two expressions for \(\sigma\) equal to each other:
\[ \frac{1}{2} - \frac{Y}{6K} = \frac{Y}{2G} - 1 \]

6. **Solving for Y**:
- Rearranging the equation gives:
\[ \frac{1}{2} + 1 = \frac{Y}{2G} + \frac{Y}{6K} \]
\[ \frac{3}{2} = \frac{Y}{2G} + \frac{Y}{6K} \]
- Finding a common denominator (which is \(6GK\)):
\[ \frac{3}{2} = \frac{3YK + YG}{6GK} \]
- Cross-multiplying gives:
\[ 3 \cdot 6GK = 2(3YK + YG) \]
- Simplifying leads to:
\[ Y(3K + G) = 18GK \]
- Finally, we can express Y as:
\[ Y = \frac{18GK}{3K + G} \]

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