Explanation:

In the field of materials science, Young's modulus (E), modulus of rigidity (G), and bulk modulus (K) are mechanical properties that describe the behavior of an elastic material. Each of these properties measures a different aspect of the material's response to external forces.

Young's Modulus (E):
Young's modulus is a measure of the stiffness or rigidity of a material. It quantifies the relationship between stress (force per unit area) and strain (deformation) in the material. Young's modulus is defined as the ratio of stress to strain and is given by the formula:

E = stress/strain

Young's modulus is a measure of the material's resistance to deformation under tensile or compressive forces.

Modulus of Rigidity (G):
The modulus of rigidity, also known as shear modulus, describes the material's response to shear stress. Shear stress occurs when adjacent layers of a material are subjected to forces that cause them to slide past each other. The modulus of rigidity is defined as the ratio of shear stress to shear strain and is given by the formula:

G = shear stress/shear strain

The modulus of rigidity is a measure of the material's resistance to shear deformation.

Bulk Modulus (K):
The bulk modulus describes the material's response to changes in volume under hydrostatic pressure. It quantifies the relationship between the change in pressure and the resulting change in volume. The bulk modulus is defined as the ratio of change in pressure to change in volume and is given by the formula:

K = change in pressure/change in volume

The bulk modulus is a measure of the material's resistance to volume compression or expansion.

Possible Relationships:
Now, let's consider the possible relationships between Young's modulus (E), modulus of rigidity (G), and bulk modulus (K):

a) G = 2K:
This relationship implies that the material's resistance to shear deformation (G) is twice its resistance to volume compression/expansion (K). However, there is no direct relationship between these two properties, so this statement is not correct.

b) G = E:
This relationship implies that the material's resistance to shear deformation (G) is equal to its resistance to tensile/compressive deformation (E). While both properties describe the material's response to deformation, they measure different types of deformation. Therefore, this statement is not correct.

c) K = E:
This relationship implies that the material's resistance to volume compression/expansion (K) is equal to its resistance to tensile/compressive deformation (E). While both properties describe the material's response to deformation, they measure different types of deformation. Therefore, this statement is not correct.

d) G = K = E:
This relationship implies that the material's resistance to shear deformation (G), volume compression/expansion (K), and tensile/compressive deformation (E) are all equal. While it is possible for different materials to have equal values for these properties, it is not a general relationship that holds true for all elastic materials. Therefore, this statement is not correct.

Conclusion:
Based on the analysis above, the only possible relationship is K = E. This relationship implies that the material's resistance to volume compression/expansion (K) is equal to its resistance to tensile/compressive deformation (E).