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We know that depending upon the type of stress being applied and the resulting strain, the modulus of elasticity had been classified into the following three types.

Young’s modulus (E): The ratio of longitudinal stress to longitudinal strain.

Bulk modulus (K): The ratio of volumetric stress to volume strain.

Shear modulus (G): The ratio of shear stress to shear strain.

All three elastic constants can be interrelated by deriving a relation between them known as the Elastic constant formula. But young’s modulus (E) and the Poisson ratio(𝝂) are known as the independent elastic constants and they can be obtained by performing the experiments.

The bulk modulus and the shear modulus are dependent constants and they are related to Young’s modulus and the Poisson ratio.

The relation between Young’s modulus and shear modulus is

⇒ E = 2G(1 + v)N/m\[^{2}\]

The relation between Young’s modulus and Bulk modulus is:

⇒ E = 3K(1 - 2v) N/m\[^{2}\]

The relation between different elastic constants is achieved by a small derivation. For the derivation of the relation between elastic constants, we will use the relation between Young’s modulus and the bulk modulus and also the relation between Young’s modulus and the shear modulus.

Consider the relation between Young’s modulus and the shear modulus,

⇒ E = 2G(1 + v)N/m\[^{2}\] ……….(1)

Where,

E - Young’s modulus

G - Shear modulus

v - Poisson ratio

From equation (1) the value of the Poisson ratio is:

⇒ v = \[\frac{E}{2G}\] - 1 ……….(2)

We know that the relation between Young’s modulus and the Bulk modulus is

⇒ E = 3K(1 - 2v) N/m\[^{2}\] …………..(3)

Where,

E - Young’s modulus

K - Bulk modulus

v - Poisson ratio

Substituting the value of Poisson ration from equation (2) in (3) and simplify,

⇒E = 3K(1 - 2[\[\frac{E}{2G}\] - 1])

⇒E = 3K(1 - [\[\frac{E}{G}\] - 2])

⇒ E = 3K(3 - \[\frac{E}{G}\])

On further simplification,

⇒ E = 9K - \[\frac{3KE}{G}\]

Taking LCM of G and on cross multiplication,

⇒ EG + 3KE = 9KG

⇒ E(G + 3K) = 9KG

On rearranging the above expression,

⇒ E = \[\frac{9KG}{(G + 3K)}\] N/m\[^{2}\] ………..(4)

Where,

E - Young’s modulus

G - Shear modulus

K - Bulk modulus

Equation (4) is known as the Elastic constant formula and it gives the Relation between elastic constants.

The relationship between different elastic constants is also given by the expression,

⇒ \[\frac{1}{K}\] - \[\frac{3}{G}\] = \[\frac{9}{E}\]

Where,

E - Young’s modulus

G - Shear modulus

K - Bulk modulus

These are the different ways of writing the relationship between elastic constants, depending upon the need for the solution we should utilize the formulas.

FAQ (Frequently Asked Questions)

1. What is the Poisson Ratio?

Ans: Poisson ratio is defined as the ratio of lateral strain (trasverese strain) to longitudinal strain. It is denoted by 𝝂. It is an independent elastic constant and unitless scalar entity.

2. Why Some Materials are Elastic?

Ans: The elasticity of any material is decided, depending upon the amount of deformation and ability to getting back to its original size and shape. Some materials have a great ability to get back to their original shape and size even after applying some force or stretching it. For example rubber.

3. What are the Elastic Constants?

Ans: The elastic constants are the parameters explaining about the relationship between the type of stress applied and the corresponding strain.