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

To understand the relationship between Young's modulus \(Y\) and the rigidity modulus \(G\), we must explore their definitions and interconnection through the material's properties.

Definitions:

• Young's Modulus \((Y)\): It is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression. It is defined as the ratio of tensile stress to tensile strain.
• Rigidity Modulus \((G)\): Also called the shear modulus, it measures an object's ability to resist shearing deformations. It is the ratio of shear stress to shear strain.

The general relationship between Young’s modulus \(Y\), rigidity modulus \(G\), and Poisson's ratio \(\nu\) is given by the formula:

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

For most metals, Poisson’s ratio \(\nu\) is approximately 0.3. Substituting this value into the equation, we have:

\(G = \frac{Y}{2(1 + 0.3)} = \frac{Y}{2 \times 1.3} = \frac{Y}{2.6} \approx \frac{Y}{3}\)

Therefore, for an approximate case when Poisson’s ratio is around 0.3, the relationship simplifies to:

\(G \approx \frac{Y}{3}\)

Conclusion: Among the provided options, the correct relationship that approximates this is:

\(G = \frac{Y}{3}\)

Thus, the correct answer is \(G = \frac{Y}{3}\), which corresponds to the correct understanding and approximation for real-world applications in typical metals.

Reason for rejecting other options:

• \(G = 3Y\): This implies rigidity modulus is three times Young’s modulus, which is incorrect.
• \(G = \frac{3}{2} Y\): This is not the correct simplified form based on the typical Poisson's ratio.
• \(G = \frac{Y}{8}\): This ratio does not align with any common material property relationships.
• \(10G = 3Y\): Though mathematically similar to one form, it does not correctly represent the more commonly applicable modulus relationship for typical metals.