Question

The Poisson’s ratio for an incompressible material is:

• A. $0$
• B. $0.25$
• C. $0.33$
• D. $0.5$

Hint:

{Poisson’s ratio = 0.5} is a key indicator of incompressible materials such as rubber-like substances.

Answer 1

The Correct Option is D

Step 1: Understanding Poisson’s ratio.
Poisson’s ratio ($\nu$) is defined as the ratio of lateral strain to longitudinal strain when a material is subjected to uniaxial stress. It is given by: \[ \nu = \frac{\text{Lateral strain}}{\text{Longitudinal strain}} \]
Step 2: Meaning of incompressible material.
An incompressible material is one whose volume does not change when subjected to stress. This means the decrease in one dimension is exactly balanced by the increase in another dimension.
Step 3: Relation between Poisson’s ratio and volume change.
For a material to be incompressible, the volumetric strain must be zero. From elasticity theory, this condition is satisfied when Poisson’s ratio equals $0.5$.
Step 4: Analysis of options.
(A) $0$: Represents perfectly brittle material with no lateral strain.
(B) $0.25$: Typical for some metals, but not incompressible.
(C) $0.33$: Common for ductile materials like steel.
(D) $0.5$: Correct — This value corresponds to incompressible materials.
Step 5: Conclusion.
Since an incompressible material does not undergo any volume change, its Poisson’s ratio must be equal to $0.5$.