 # Relation Between Elastic Constants

Young’s modulus, bulk modulus and Rigidity modulus of an elastic solid are together called Elastic constants. When a deforming force is acting on a solid, it results in the change in its original dimension. In such cases, we can use the relation between elastic constants to understand the magnitude of deformation.

## Elastic constant formula

 $$E=\frac{9KG}{G+3K}$$

Where,

• K is the Bulk modulus
• G is shear modulus or modulus of rigidity.
• E is Young’s modulus or modulus of Elasticity.

Individually Young’s modulus and bulk modulus and modulus of rigidity are related as-

 Formula SI Units The relation between modulus of elasticity and modulus of rigidity $$E=2G\left ( 1+\mu \right )$$ N/m2 or pascal(Pa) The relation between Young’s modulus and bulk modulus $$E=3K\left ( 1-2\mu \right )$$ N/m2 or pascal(Pa)

### Derivation of relation between elastic constants

We can derive the elastic constant’s relation by combining the mathematical expressions relating terms individually.

• Young modulus can be expressed using Bulk modulus and Poisson’s ratio as –
$$E=3K\left (1-2\mu \right)$$
• Similarly, Young’s modulus can also be expressed using rigidity modulus and Poisson’s ratio as-$$E=2G\left (1+2\mu \right)$$
• Combining the above two-equation and solving them to eliminate Poisson’s ratio we can get a relation between Young’s modulus and bulk modulus k and modulus of rigidity as -$$E=\frac{9KG}{G+3K}$$

Hope you have understood the relation between Young’s modulus and bulk modulus k and modulus of rigidity.

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