Relationship between Elastic Constants:

We already have defined the four Elastic Constants E, υ, K and G and also stated that they are not independent. Now, we will establish the relationship among them.

Relationship between E and K

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

εx  =     σ _x / E - υ (σ _y / E) - υ ( σ _z /E)

Since,  σ_x = σ_y = σ_z = σ₀

ε_x =    ( σ ₀ / E) (1 - 2υ)

Similarly,         ε_y = ( σ ₀ / E )(1 - 2υ) and ε_z =   ( σ ₀ / E) (1 - 2υ)

From Eq. (6), we might 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)