Question

The ratio of modulus of rigidity to modulus of elasticity for a Poisson's ratio of 1 would be

• A. 0.5
• B. 0.4
• C. 0.25
• D. 1

Hint:

Keep the key relationships between $E$, $G$, $K$, and $\mu$ memorized:
$E = 2G(1+\mu) = 3K(1-2\mu)$. This makes solving parameter relation questions very rapid.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question asks for the numerical ratio of the modulus of rigidity ($G$) to the modulus of elasticity ($E$) when Poisson's ratio ($\mu$ or $\nu$) is equal to $1$.

Step 2: Key Formula or Approach:
The fundamental relation between the elastic constants $E$, $G$, and Poisson's ratio $\mu$ is:
\[ E = 2G(1 + \mu) \]

Step 3: Detailed Explanation:

• We need to find the ratio of modulus of rigidity to modulus of elasticity, which is $\frac{G}{E}$.

• Rearranging the elastic constant relation:
\[ \frac{G}{E} = \frac{1}{2(1 + \mu)} \]

• Given that Poisson's ratio $\mu = 1$, we substitute this value into our ratio equation:
\[ \frac{G}{E} = \frac{1}{2(1 + 1)} = \frac{1}{4} = 0.25 \]

• Note: Although Poisson's ratio for stable, isotropic, elastic materials is theoretically bounded between $-1$ and $0.5$, this is a hypothetical calculation scenario.

Step 4: Final Answer:
The ratio of modulus of rigidity to modulus of elasticity for a Poisson's ratio of $1$ is $0.25$.