❓What is the formula for Young's modulus (E) in terms of bulk modulus (K) and shear modulus (G)?

The formula for Young's modulus ($E$) in terms of the Bulk Modulus ($K$) and the Shear Modulus ($G$) is: $E = \frac{9KG}{3K + G}$. This relationship is a standard result in the field of Linear Elasticity for isotropic materials and is derived by combining the constitutive equations that define Young's modulus, Poisson's ratio, bulk modulus, and shear modulus.

Derivation Context

The formula is obtained using two key intermediate relationships:

1. Relationship between $K$, $E$, and $\nu$ (Poisson's ratio):

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

2. Relationship between $G$, $E$, and $\nu$:

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

By solving these two equations simultaneously to eliminate the intermediate variable $\nu$, the final expression for $E$ in terms of $K$ and $G$ is reached.

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Proof and Derivation related to FAQ

What is the formula for Young's modulus (E) in terms of bulk modulus (K) and shear modulus (G)? | FAQsviadean on Notion