Relationship between Elastic Constants

 Modulus of elasticity, modulus of rigidity and bulk modulus are the three elastic constants.

1) Modulus of Elasticity (Young’s Modulus) ‘E’

It is defined as the ratio of linear stress to linear strain within elastic limit.

2) Modulus of Rigidity (G or N)

It is defined as the ratio of shearing stress to shearing strain within elastic limit and is usually denoted by letter 'G' or 'N'. Thus

where

     G = Modulus of rigidity

     q = Shearing stress

     ϕ = Shearing strain

3) Bulk Modulus (K)

When a body is subjected to identical stresses p in three mutually perpendicular directions, as shown in Fig. 1, the body undergoes uniform changes in three directions without undergoing distortion of shape.

Fig. 1 Stresses p acting in three mutually perpendicular directions

The ratio of change in volume to original volume has been defined as volumetric strain (e_v). Then the bulk modulus, 'K' is defined as

where

      p = identical pressure in three mutually perpendicular directions

  , Volumetric strain

      Δ v = Change in volume

      v = Original volume

Thus bulk modulus may be defined as the ratio of identical pressure ‘p’ acting in three mutually perpendicular directions to corresponding volumetric strain.

Fig. 1 shows a body subjected to identical compressive pressure ‘p’ in three mutually perpendicular directions. Since hydrostatic pressure i.e. the pressure exerted by a liquid on a body within it, has this nature of stress, such a pressure ‘p’ is called as hydrostatic pressure.

Relationship between Modulus of Elasticity and Modulus of Rigidity

Consider a square element ABCD of sides ‘a’ subjected to pure shear ‘q’ as shown in Fig. 2. AEC’D shown is the deformed shape due to shear ‘q’. Drop perpendicular BF to diagonal DE. Let ′ϕ′ be the shear strain and G is the modulus of rigidity.

Fig. 2

              

Since angle of deformation is very small we can assume ∠BEF = 45°, hence EF = BE cos 45°

                    

                      

Now, we know that the above pure shear gives rise to axial tensile stress ‘q’ in the diagonal direction of DB and axial compression q at right angles to it. These two stresses cause tensile strain along the diagonal DB.

Relationship between Modulus of Elasticity and Bulk Modulus

Consider a cubic element subjected to stresses p in the three mutually perpendicular direction x, y, z as shown in Fig. 3.

Fig. 3

Now the stress p in x direction causes tensile strain p/E in x direction while the stress p in y and z direction cause compressive strains μp/E in x direction.

Relationship between E,G and K

                                 We know that       E = 2G(1 + μ)                  (1)

                                                                     E = 3K(1 – 2 μ) (2)

By eliminating ′μ′ between the above two equations we can get the relationship between E, G and K, free from the term μ.