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. In this article, let us learn about the relationship between these two.

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Young’s Modulus And Bulk Modulus Relation

\(\begin{array}{l}K=\frac{Y}{3\left ( 1-\frac{2}{\mu} \right ) }\end{array} \)

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-

\(\begin{array}{l}Y=\frac{\sigma }{\epsilon }\end{array} \)

On rearranging-

\(\begin{array}{l}\epsilon =\frac{\sigma }{Y }\end{array} \)

When the deforming force is along x direction-

\(\begin{array}{l}\epsilon_{x} =\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }\end{array} \)

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

1/m arise due to compression along the other two directions.

When the deforming force is along the y-direction-

\(\begin{array}{l}\epsilon_{y} =\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }\end{array} \)

When the deforming force is along the z-direction-

\(\begin{array}{l}\epsilon_{z} =\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }-\frac{1}{m}\frac{\sigma }{Y }\end{array} \)

The volumetric strain is given by-

\(\begin{array}{l}\epsilon_{v} =\epsilon_{x}+\epsilon_{y}+\epsilon_{z}\end{array} \)

Substituting the corresponding values to ϵx ,ϵy , ϵz we get-

\(\begin{array}{l}\epsilon_{v} =\frac{3\sigma }{Y}\left [ 1-\frac{2}{m} \right ]\end{array} \)

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

\(\begin{array}{l}K=\frac{\sigma }{\epsilon _{v}}\end{array} \)

Substituting

\(\begin{array}{l}\epsilon_{v} =\frac{3\sigma }{Y}\left [ 1-\frac{2}{m} \right ]\end{array} \)
in above equation we get-

\(\begin{array}{l}K=\frac{\sigma }{\frac{3\sigma }{Y}\left [ 1-\frac{2}{m} \right ]}\end{array} \)
\(\begin{array}{l}\Rightarrow K=\frac{Y}{3\left [ 1-\frac{2}{m} \right ]}\end{array} \)

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

Physics Related Topics:

Tensile stress
Relation Between Torque And Moment Of Inertia
Thermal Stress
Shearing Stress

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