Question

Choose the option showing the correct relation between Poisson’s ratio (σ), Bulk modulus (B) and modulus of rigidity (G).

• A. \(\sigma=\frac{3B-2G}{2G+6B}\)
• B. \(\sigma=\frac{6B+2G}{3B-2G}\)
• C. \(\sigma=\frac{9BG}{3B+G}\)
• D. \(B=\frac{3\sigma-3G}{6\sigma+2G}\)

Answer 1

The Correct Option is A

Derivation of Poisson's Ratio (σ)

Given Equations:

1. \( E = 2G(1 + \sigma) \) ……(1)
2. \( E = 3B(1 - 2\sigma) \) ……(2)

Equating the Two Expressions for \( E \):

From (1) and (2), we equate \( E \):
\[ 2G(1 + \sigma) = 3B(1 - 2\sigma) \]

Rearranging the Equation:

Expanding both sides: \[ 2G + 2G\sigma = 3B - 6B\sigma \] Collecting terms involving \( \sigma \) and constants: \[ 3B - 2G = \sigma(2G + 6B) \]

Solving for \( \sigma \):

Divide through by \( (2G + 6B) \): \[ \sigma = \frac{3B - 2G}{2G + 6B} \]

Final Answer:

The Poisson's ratio \( \sigma \) is:

\( \sigma = \frac{3B - 2G}{2G + 6B} \)

Thus, the correct answer is (A).