# 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.

{% embed url="<https://youtu.be/bxEuCUBWcQI>" %}

{% embed url="<https://viadean.notion.site/Young-s-Modulus-and-Poisson-s-Ratio-in-Terms-of-Bulk-and-Shear-Moduli-2891ae7b9a3280719445efc94ef03120?source=copy_link>" %}