Relationship between Elastic Constants:

Here, we have defined the four Elastic Constants E, υ, K and G and also begin that they are not independent. Presently, we shall establish the relationship among them.

Relationship between E and K

Below figure shows the spherical state of stress in which the normal stress component is the same, σ0 in any direction and shear stress components are zero on any plane. Such a state of stress is also called as volumetric stress. If the Young's Modulus and Poisson's Ratio of the solid are known as E and υ, the strain components are given by

ε_x  =     σ_x /E - υ (σ _y/E) - υ  (σ _z/E)

Since,  σ_x = σ_y = σ_z = σ₀

ε_x =     σ₀ /E (1 - 2υ)

Similarly,       ε_y =     σ₀/E  (1 - 2υ) and     ε_z =  σ₀/E  (1 - 2υ)

From Eq. (6), we can express the change in volume of the solid as

dV = V (ε_x + ε_y + ε_z )

Volumetric strain, ε_v  = dV/V = ε_x + ε_y + ε_z

Bulk Modulus,         K = Volumetric stress/Volumetric strain

=          σ₀/ ε _x + ε _y + ε _z =     σ₀/3ε_x

= σ₀/(σ₀/E) × 3 (1 - 2υ)

K= E/3 (1 - 2υ)