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

\(\begin{array}{l}E=\frac{9KG}{G+3K}\end{array} \)

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
\(\begin{array}{l}E=2G\left ( 1+\mu \right )\end{array} \)
N/m2 or pascal(Pa)
The Relation Between Young’s Modulus and Bulk Modulus
\(\begin{array}{l}E=3K\left ( 1-2\mu \right )\end{array} \)
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 –
\(\begin{array}{l}E=3K\left (1-2\mu \right)\end{array} \)
  • Similarly, Young’s modulus can also be expressed using rigidity modulus and Poisson’s ratio as-
    \(\begin{array}{l}E=2G\left (1+2\mu \right)\end{array} \)
  • 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 -
    \(\begin{array}{l}E=\frac{9KG}{G+3K}\end{array} \)

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

Physics Related Topics:

See the video below to learn about the elastic property of matter.


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