Question

Choose the correct relationship between Poisson ratio $(\sigma)$, bulk modulus $( K )$ and modulus of rigidity $(\eta)$ of a given solid object :

• A. $\sigma=\frac{3 K+2 \eta}{6 K+2 \eta}$
• B. $\sigma=\frac{6 K-2 \eta}{3 K-2 \eta}$
• C. $\sigma=\frac{3 K-2 \eta}{6 K+2 \eta}$
• D. $\sigma=\frac{6 K+2 \eta}{3 K-2 \eta}$

Answer 1

The Correct Option is C

To determine the correct relationship between the Poisson's ratio \(\sigma\), the bulk modulus \(K\), and the modulus of rigidity \(\eta\) for a solid object, we use the known formula relating these physical quantities:

\[\sigma = \frac{3K - 2\eta}{6K + 2\eta}\]

Let's understand the relationships and derive the formula if needed:

1. Poisson's Ratio \(\sigma\) is a measure of the deformation in the transverse direction to the deformation in the axial direction when the material is stretched. It is given by the formula: \(\sigma = \frac{E}{2\eta} - 1\), where \(E\) is Young's Modulus.
2. Bulk Modulus \(K\) is defined as the measure of a material's resistance to uniform compression, and is given by: \(K = \frac{E}{3(1-2\sigma)}\).
3. The Modulus of Rigidity \(\eta\), also known as the shear modulus, is given by: \(\eta = \frac{E}{2(1+\sigma)}\).
4. The correct relationship between these quantities is expressed as: \(\sigma = \frac{3K - 2\eta}{6K + 2\eta}\).

To deduce the correct answer:

• Start from known relationships involving Young's Modulus \(E\), Bulk Modulus \(K\), and Shear Modulus \(\eta\).
• Express \(E\) in terms of \(K\) and \(\eta\).
• Substitute these relationships back to find the expression for \(\sigma\).
• Comparing the options provided, the expression for \(\sigma\) exactly matches one of the options.

Thus, the correct relationship is indeed: \(\sigma = \frac{3K - 2\eta}{6K + 2\eta}\), which corresponds to the third option provided. Therefore, the correct answer is: \(\boxed{\frac{3K - 2\eta}{6K + 2\eta}}\).