Question

For most of the materials, Young’s modulus (Y) and rigidity modulus (G) are related as

• A. $G = 3Y$
• B. $G = \frac{Y}{3}$
• C. $G = \frac{3}{2} Y$
• D. $G = \frac{Y}{8}$
• E. $10G = 3Y$

Hint:

Remember the three core relations:
1. $Y = 2G(1 + \nu)$
2. $Y = 3K(1 - 2\nu)$
3. $\frac{9}{Y} = \frac{1}{K} + \frac{3}{G}$

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Elastic moduli like Young's modulus ($Y$), Rigidity modulus ($G$), and Bulk modulus ($K$) are interconnected through Poisson's ratio ($\nu$).
Step 2: Key Formula or Approach:
The fundamental relationship between $Y$ and $G$ is: \[ Y = 2G(1 + \nu) \] Step 3: Detailed Explanation:
Poisson's ratio ($\nu$) for most solid materials (metals, etc.) ranges between $0.25$ and $0.35$.
For an idealized incompressible material, the maximum theoretical value of $\nu$ is $0.5$.
Substituting $\nu = 0.5$ into the formula: \[ Y = 2G(1 + 0.5) = 2G(1.5) = 3G \] Rearranging for $G$: \[ G = \frac{Y}{3} \] This is a common approximation used in many physics problems for isotropic materials.
Step 4: Final Answer:
For most materials, the relation is $G = \frac{Y}{3}$.