Young’s Modulus (Y):

This modulus is defined as the ratio of longitudinal (tensile or compressive) stress to the corresponding strain within the elastic limit.

Elastic Limit : There is a certain limit of the deforming forces up to which the behaviour of a body remains elastic, i.e, it regains its original shape or size after removal of deforming forces. This limit is called the elastic limit of the material of the body.

If \( F \) be the tensile or compressive force acting over a cross-section area \( A \), \( l \) be the change in length produced in the original length \( L \), then

\begin{split}
&{Young's \ Modulus (Y)}
\\&=\frac{Longitudinal\ Stress}{Longitudinal\ Strain}
\\&=\frac{F/A}{l/L}
\\&=\frac{F\cdot{L}}{A\cdot{l}}
\end{split}

Bulk Modulus(K):

This modulus s defined as the ratio of volume stress to corresponding volume strain within the elastic limit.

If a normal inward force \( F \) is acts on each element of area \( A \) of a substance of volume \( V \), to produce a derease in volume \( v \), then

\begin{split}
Bulk \ Modulus(K)&=\frac{Volume\ Stress}{Volume\ Strain}
\\&=\frac{F/A}{v/V}
\\&=\frac{P\cdot{V}}{v} 
\end{split}

Where \( P=\frac{F}{A} \) is the pressure.

The reciprocal of the Bulk Modulus is called compressibility, i.e,

Compressibility = \frac{1}{K}=\frac{v}{P\cdot{V}}