This derivation relies on the two fundamental constitutive equations for an isotropic, homogeneous elastic material, which relate the stress tensor ($\sigma_{ij}$) and the strain tensor ($\epsilon_{ij}$) to the material's elastic moduli.

The final expression for Poisson's ratio ($\nu$) in terms of Bulk Modulus ($K$) and Shear Modulus ($G$) is:

$$ \nu = \frac{3K - 2G}{6K + 2G} $$

Derivation Steps

1. The Constitutive Equations

We start with two equivalent forms of Hooke's Law for Isotropic Materials:

A. Stress in terms of $K$ and $G$ (Decomposed Strain Form)

The stress tensor ($\sigma_{ij}$) is decomposed into its hydrostatic (volumetric) and deviatoric (shear) parts:

$$ \sigma_{ij} = K \epsilon_{kk} \delta_{ij} + 2 G \kappa_{ij} \quad \text{(Eq. 1)} $$

Where:

• $\epsilon_{kk} = \epsilon_{11} + \epsilon_{22} + \epsilon_{33}$ is the volumetric strain (trace of the strain tensor).
• $\delta_{ij}$ is the Kronecker delta (1 if $i=j$, 0 otherwise).
• $\kappa_{ij} = \epsilon_{ij} - \frac{1}{3} \epsilon_{kk} \delta_{ij}$ is the deviatoric strain (pure shear strain).

B. Strain in terms of $E$ and $\nu$

The strain tensor ($\epsilon_{ij}$) in terms of Young's Modulus ($E$) and Poisson's ratio ($\nu$):

$$ \epsilon_{ij} = \frac{1}{E} [(1 + \nu) \sigma_{ij} - \nu \sigma_{kk} \delta_{ij}] \quad \text{(Eq. 2)} $$

Where:

• $\sigma_{kk} = \sigma_{11} + \sigma_{22} + \sigma_{33}$ is the hydrostatic stress (trace of the stress tensor).