Relation Between Elastic Constants

Young’s modulus, bulk modulus, and Rigidity modulus of an 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
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 understood the relation between Young’s modulus and bulk modulus k and modulus of rigidity.

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