Relation between K, E, and G:

The bulk modulus (K) is a measure of a material's resistance to compression. It quantifies the change in volume for a given change in pressure.

Young's modulus (E) is a measure of a material's stiffness or elasticity. It quantifies the change in length or deformation in response to an applied force.

The modulus of rigidity (G) is a measure of a material's resistance to shear deformation. It quantifies the change in shape or distortion in response to an applied shear stress.

Mathematical representation:

The relation between K, E, and G can be expressed mathematically as follows:

K = (E * G) / (9 * G - 3 * E)

Explanation:

To understand the derivation of the above relation, let's consider a solid cube under the influence of an external pressure.

When a cube is subjected to a uniform pressure from all directions, it undergoes a compression or decrease in volume. This compression is resisted by the material's bulk modulus (K).

On the other hand, when a cube is subjected to a force parallel to one of its faces, it undergoes a deformation or change in shape. This deformation is resisted by the material's modulus of rigidity (G).

The relationship between these two moduli and Young's modulus can be derived using the theory of elasticity and Hooke's law.

Derivation:

1. Consider a cube with side length 'L' and initial volume 'V'.

2. When the cube is subjected to a uniform pressure 'P', its volume decreases by a small amount 'ΔV'.

3. According to the definition of bulk modulus, we have:

K = - (P / ΔV) * (V / P) = - (ΔV / V)

4. Now, consider a force 'F' applied parallel to one of the faces of the cube, resulting in a deformation 'ΔL' in that direction.

5. According to Hooke's law, we have:

F = G * (ΔL / L) * A

where A is the cross-sectional area of the cube.

6. Comparing the equations for K and G, we can relate the three moduli as follows:

K / G = (ΔV / V) / (ΔL / L) * A

7. As the cube is subjected to both pressure and shear forces simultaneously, the change in volume and deformation are related. This relationship can be derived using the theory of elasticity.

8. By solving the elasticity equations, we arrive at the final relation:

K = (E * G) / (9 * G - 3 * E)

Therefore, the correct answer is option 'B' ((9KG) / (3K * G)).