# Relation Between Elastic Constants

Young’s modulus, Bulk modulus, and Rigidity modulus of any elastic solid are together called as 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 Relation between modulus of elasticity and modulus of rigidity $E=2G\left ( 1+2\mu \right )$ N/m2 or pascal(Pa) 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 constants 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 relation between Young’s modulus and bulk modulus k and modulus of rigidity as -$E=\frac{9KG}{G+3K}$

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

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