🧄Young's Modulus and Poisson's Ratio in Terms of Bulk and Shear Moduli

The relationship between the elastic constants, derived from the general constitutive equations, establishes that Young's modulus ( $E$ ) and Poisson's ratio ( $\nu$ ) can be fully expressed by the Bulk modulus ( $K$ ) and the Shear modulus ( $G$ ) for an isotropic material. This derivation fundamentally relies on separating stress and strain into volumetric (governed by $K$ ) and deviatoric (governed by $G$ ) components. The key intermediate result is the relationship $E= 2 G(1+\nu)$, which connects the stiffness ( $E$ ) to the resistance to shear ( $G$ ) and lateral contraction ( $\nu$ ). The final expressions, $E=\frac{9 K G}{3 K+G}$ and $\nu=\frac{3 K-2 G}{6 K+2 G}$, show how the material's resistance to volume change ( $K$ ) and resistance to shape change ( $G$ ) combine to define its overall elastic behavior.

Young's Modulus and Poisson's Ratio in Terms of Bulk and Shear Moduli | Proof and Derivationviadean on Notion