# Relation Between Elastic Constants

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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.

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

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

3. 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$^{2}$

### Elastic Constant Formula

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.

### Derivation of Relation Between Elastic Constants

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.

### Did You Know

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.