Question

The Poisson's ratio of a material that does not suffer any change in volume when a force is applied to it is:

Answer 1

Hint: An important property of a solid is Poisson's ratio. The ratio of the transverse contraction strain to the longitudinal extension strain is called a Poisson’s ratio. This will be in the direction of the stretching force. When we apply longitudinal stress it will result in a longitudinal strain. Even if we increase or decrease the length one thing remains constant which is nothing but volume.

Complete Step By Step Answer:
Since there is no change in volume, the volumetric strain is zero. We have a relation between young’s modulus, bulk modulus, and Poisson's ratio. These are called elastic constants. The constants that are used to determine the deformation that is produced by a given stress which acts on the given material is called Elastic constants. The relation between these three are given by,
 $ \Rightarrow E = 3K(1 - 2\sigma ) $
Here $ E $ is said to be the young’s modulus of the material
 $ K $ is said to be the bulk modulus of the material
 $ \sigma $ is said to be the Poisson's ratio of the material.
Young’s modulus is calculated by the ratio of tensile stress to tensile strain.
Bulk’s modulus is given by the ratio of the volumetric stress to volumetric strain.
Since there is no change in the volume the bulk modulus becomes infinity. Now rearranging the above formula we get,
 $ \Rightarrow \dfrac{E}{{3K}} = (1 - 2\sigma ) $
Since bulk modulus becomes infinity L.H.S becomes zero. Therefore the above equation becomes zero.
 $ \Rightarrow 0 = (1 - 2\sigma ) $
 $ \Rightarrow 1 = 2\sigma $
 $ \Rightarrow \dfrac{1}{2} = \sigma $
Divide to get the answer,
 $ \Rightarrow 0.5 = \sigma $
Therefore, the value of the Poisson's ratio when the material does not suffer a change in volume is $ 0.5 $ .

Note:
We have already seen about three types of elastic constants and their definitions. And the value that changes is the cross-section area. And this strain is called lateral strain. The ratio of these two strains (longitudinal strain and lateral strain) is called Poisson's ratio. There is also another type of elastic constant which is known as the shear modulus or modulus of rigidity. It is a measure of the rigidity of the body. It is given by the ratio of the shear stress to shear strain. It is often denoted by $ G $ .