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: What Poisson’s ratio represents. 
Poisson’s ratio (\(\nu\)) describes how a material deforms sideways when it is stretched or compressed along one direction. It is defined as the ratio of lateral strain to longitudinal strain:

\[ \nu = \frac{\text{lateral strain}}{\text{longitudinal strain}} \]

Step 2: Meaning of an incompressible material.
An incompressible material is one whose total volume remains unchanged even when it is subjected to stress. Any reduction in length in one direction is exactly compensated by expansion in the perpendicular directions.

Step 3: Link between Poisson’s ratio and volume change.
From elasticity theory, zero volume change (zero volumetric strain) occurs only when Poisson’s ratio has a specific limiting value. That limiting value is:

\[ \nu = 0.5 \]

Step 4: Checking the given choices.
(A) \(0\): No lateral deformation — not incompressible.
(B) \(0.25\): Typical of some solids, but volume still changes.
(C) \(0.33\): Common for metals like steel, still compressible.
(D) \(0.5\): Indicates zero volume change — incompressible.

Step 5: Final conclusion.
For a material that does not undergo any change in volume, the value of Poisson’s ratio must be:

\[ \boxed{0.5} \]