The correct relation between the modulus of elasticity (E) and modulus of rigidity (G) is:

Given, µ = Poisson’s ratio.

Explanation:

The relation between Young’s modulus (E), rigidity modulus (G) and poison’s ratio (μ) is given by

E = 2G (1 + μ)

Some other important relations are as follows:

E = 3K (1 – 2μ)​​, \({\bf{E}} = \frac{{9{\bf{KG}}}}{{3{\bf{K}} + \;{\bf{G}}}}\) and \({\bf{\mu }} = \frac{{3{\bf{K}} - 2{\bf{G}}}}{{2{\bf{G}} + 6{\bf{K}}}}\)

Important Points

Young's modulus:

• The mechanical property of a material to withstand the compression or the elongation with respect to its length of linear elastic solids like rods, wires etc is called Young's modulus.
• It is also referred to as the Elastic Modulus or Tensile Modulus 
• It gives us information about the tensile elasticity of a material (ability to deform along an axis).

\(E =\frac {Normal ~stress}{Normal ~strain}= \frac {\sigma}{\epsilon}\)

where E is Young’s modulus in Pa, 𝞂 is the uniaxial stress in Pa,ε is the Normal strain or proportional deformation.

Modulus of rigidity:

• It is also known as shear modulus.
• It is the mechanical property of a material due to which it withstand shear stress and resist torsion. 
• It is the ratio of shear stress to the corresponding shear strain within the elastic limit. This is denoted by G .

\(\therefore {G} = \frac{{Shear\;stress}}{{Shear\;strain}} = \frac{\tau }{\phi }\)

Bulk modulus:

• It is the mechanical property of a material due to which it resists the change in volume due to external pressure or equal stress in all directions.
• The concept of bulk modulus can be used in the case of hydrostatic loading.
• It is defined as the ratio of normal stress to the volumetric strain and denoted by 'K'

\(K = \frac {Normal ~stress}{Volumetric ~strain}= \frac {\sigma }{\epsilon _v}\)

Poisson's ratio:

• When the body is loaded within its elastic limit, the ratio of lateral or transverse strain and linear or longitudinal strain is constant. This constant is known as Poisson's ratio.

\({\rm{μ }} =- \frac{{{\rm{Lateral\;strain}}}}{{{\rm{Linear\;strain}}}}\)