Discussion
For a homogeneous, isotropic material, the relation between the shear modulus (\(G\)), Young’s modulus (\(E\)), and Poisson’s ratio (\(\nu\)) is ____.
Answer: C
The relation is \(E = 2G(1 + \nu)\).
For an isotropic linear elastic material only two of the elastic constants are independent. Analysing pure shear as equal tension and compression at 45° gives the link between them:
\[ G = \frac{E}{2(1 + \nu)} \]
For steel, with E about 200 GPa and \(\nu \approx 0.3\), this gives G about 77 GPa, the familiar value. With \(\nu = 0.3\), the option \(2G = E(1 + \nu)\) gives G = 0.65E instead of about 0.38E, and the options \(G = 2E(1 + \nu)\) and \(2E = G(1 + \nu)\) make G larger than E, which no real material shows.
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