Relation Between Elastic Constants

Relation between Modulus of elasticity and Bulk Modulus


Consider the cube shown in the figure ABCDEFGH, which is subjected to tensile stress at three mutually perpendicular directions as shown in figure 1. Let the length of the cube be 'L' and the change in length be represented as 'L'.

Fig.1.A cube subjected to system of stresses in x, y and z directions

The modulus of elasticity of the material is 'E'. Tensile stress is '𝜎' that is acting in the mutual perpendicular direction as mentioned.
Now volume of the cube can be given as, V = L3
We are now going to consider a single side AB, and the strain on the side is represented in terms of stress and elastic constants. The strain on side AB due to stresses on:
  • The face AEHD and BFGC
                                 The tensile strain = 𝜎 / E                                                eq.1
  • The face AEFB and DHGC
                                 The compressive lateral strain = - μ x ( 𝜎 / E )           eq.2

  • The face ABCD and EFGH
                                 The compressive lateral strain = - μ x ( 𝜎 /E )            eq.3

The total strain on the side AB can be given as eq.1 + 2 +3 

i.e. dL/L = ( 𝜎 / E   ) + (- μ x ( 𝜎 / E )   ) + (- μ x ( 𝜎 / E )   ) = 𝜎 /E  (1 - 2μ )

If dL is the change in length, the change in volume can be represented by dV. 

                                   But  V = L3.                                                             eq.4

Therefore dV = 3L2 x dL                                                     eq.5

Dividing equation 5 by 4, we get

Now, The Bulk Modulus is Given By,

By rearranging we get,

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