❓What is the formula for Poisson's ratio in terms of bulk modulus and shear modulus?

The formula for Poisson's ratio ($\nu$) in terms of the Bulk Modulus ($K$) and the Shear Modulus ($G$) is: $\nu = \frac{3K - 2G}{6K + 2G}$. This relationship is a core result in Linear Elasticity for isotropic materials and is derived by eliminating Young's Modulus ($E$) from the two fundamental equations that link the pairs of elastic moduli:

1. Bulk Modulus: $K = \frac{E}{3(1 - 2\nu)}$

2. Shear Modulus: $G = \frac{E}{2(1 + \nu)}$

The steps to combine and solve these equations to isolate $\nu$ .

1. Starting Relationships

The derivation begins with the two established relationships:

1. Bulk Modulus ($K$) relation (from hydrostatic stress):

   $K = \frac{E}{3(1 - 2\nu)}$

2. Shear Modulus ($G$) relation (from pure shear stress):

   $G = \frac{E}{2(1 + \nu)}$

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2. Express $E$ in terms of $G$ and $\nu$

From the Shear Modulus relationship, isolate $E$:

$G = \frac{E}{2(1 + \nu)}$

$\implies \mathbf{E = 2G(1 + \nu)}$

(Equation 3)

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