Relation between modulus of elasticity (E), shear modulus of rigidity (G) and bulk modulus of elasticity (K)

We have,

E = 2G(1 + u)……….(i)

E = 3K(1-2u) ………(ii)

Where, u = Poission’s ratio

From (i),

E/2G = 1+ u

Or, u = E/2G -1 ………(iii)

Substitute (iii) in (ii), we get

E = 3K [1-2(E/2G -1)],

= 3K(1- E/G + 2),

= 3K(3-E/G)

= 3K(3G – E)/G,

Or, EG = 3K(3G – E)

= 9KG – 3KE,

Or, E(3K+ G)=9KG,

E =9KG/ (3K+ G)