# Relation Between Young's Modulus And Bulk Modulus

Young’s Modulus is the ability of any material to resist the change along its length. Bulk Modulus is the ability of any material to resist the change in its volume. The relation between Young’s Modulus and Bulk Modulus can be mathematically expressed as –

### Young’s Modulus And Bulk Modulus relation

 $K=\frac{Y}{3\left ( 1-\frac{2}{\mu} \right ) }$

Where,

• K is the Bulk modulus.
• Y is Young’s modulus.
• μ is the Poisson’s ratio.

## Relation Between Young’s Modulus And Bulk Modulus derivation

Young’s modulus is the ratio of longitudinal stress to longitudinal strain. Represented by Y and mathematically given by-

$Y=\frac{\sigma }{\epsilon }$

On rearranging-

$\epsilon =\frac{\sigma }{Y }$

When the deforming force is ling x direction-

$\epsilon_{x} =\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }$

Here negative sign represents the reduction in diameter when longitudinal stress is along the x-axis.

$\frac{1}{m}$ arise due to compression along other two direction.

When the deforming force is ling y-direction-

$\epsilon_{y} =\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }$

When the deforming force is ling z-direction-

$\epsilon_{z} =\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }$

The volumetric strain is given by-

$\epsilon_{v} =\epsilon_{x}+\epsilon_{y}+\epsilon_{z}$

Substituting the corresponding values to ??x ,??y , ??z we get-

$\epsilon_{v} =\frac{3\sigma }{Y}\left [ 1-\frac{2}{m} \right ]$

The Bulk modulus is the ratio of volumetric/bulk strain to volumetric/bulk stress, represented by K and mathematically given by-

$K=\frac{\sigma }{\epsilon _{v}}$

Substituting $\epsilon_{v} =\frac{3\sigma }{Y}\left [ 1-\frac{2}{m} \right ]$ in above equation we get-

$K=\frac{\sigma }{\frac{3\sigma }{Y}\left [ 1-\frac{2}{m} \right ]}$ $\Rightarrow K=\frac{Y}{3\left [ 1-\frac{2}{m} \right ]}$

Hope you understood the relation and conversion between Young’s modulus and the Bulk modulus of an object.

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