If the Bulk Modulus ( K ) and Poisson's ratio ( v ) of a material are known, which equation would correctly determine its Shear Modulus ( G )?

Concept:

In elasticity, three major constants are often used: Young's Modulus (E), Bulk Modulus (K), and Shear Modulus (G), along with Poisson’s Ratio (ν). These constants are interrelated.

Relation between E, K and ν:

The relation between Young's Modulus (E), Bulk Modulus (K), and Poisson's Ratio (ν) is,

\(E = 3K(1 - 2\nu)\) ................ (1)

Relation between E, G and ν:

The relation between Young's Modulus (E), Shear Modulus (G), and Poisson's Ratio (ν) is,

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

To derive G in terms of K and ν, substitute the value of E from the first equation into the second:

\(G = \frac{3K(1 - 2\nu)}{2(1 + \nu)}\)