Question

The expression for the determination of Poisson’s ratio for rubber is:
(A) $\sigma=\dfrac{1}{2}\left[1-\dfrac{d V}{A d L}\right]$
(B) $\sigma=\dfrac{1}{2}\left[1+\dfrac{d V}{A d L}\right]$
(C) $\sigma=\dfrac{1}{2} \dfrac{d V}{A d L}$
(D) $\sigma=\dfrac{d V}{A d L}$

Answer 1

Hint: We should know that Poisson’s ratio is the amount of transversal elongation divided by the amount of axial compression. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. Incompressible materials, such as rubber, have a ratio near 0.5. Poisson's ratio for a material is the negative of the ratio of lateral strain to longitudinal strain of a body made of that material. When a body is deformed such that it experiences a longitudinal strain, it experiences a lateral strain also.

Complete step by step answer
Let us define the Poisson’s ratio at first. Poisson's ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force. The definition of Poisson's ratio contains a minus sign so that normal materials have a positive ratio.
We know that:
Volume of rubber, $\mathrm{V}=\mathrm{AL}$
$\dfrac{\mathrm{dV}}{\mathrm{dL}}=-\mathrm{L} \dfrac{\mathrm{d} \mathrm{A}}{\mathrm{dL}}+\mathrm{A},$ negative sign because they are decreases as length increases
The Area is given as, $\mathrm{A}=\pi \mathrm{R}^{2}$
So, $\dfrac{\mathrm{d} \mathrm{A}}{\mathrm{A}}=2 \dfrac{\mathrm{d} \mathrm{R}}{\mathrm{R}}$
On the further evaluation we get that:
$\therefore \quad \dfrac{\mathrm{dV}}{\mathrm{dL}}=\dfrac{-2 \mathrm{A} \dfrac{\mathrm{d} \mathrm{R}}{\mathrm{R}}}{\dfrac{\mathrm{d} \mathrm{L}}{\mathrm{L}}}+\mathrm{A}=(-2 \sigma+1) \mathrm{A}$
Hence, $\sigma=\dfrac{1}{2}\left(1-\dfrac{\mathrm{d} \mathrm{V}}{\mathrm{AdL}}\right)$

So, the correct answer is option A.

Note: We know that Poisson's ratio is a required constant in engineering analysis for determining the stress and deflection properties of materials (plastics, metals, etc.). It is a constant for determining the stress and deflection properties of structures such as beams, plates, shells, and rotating discs. Poisson's ratios exceeding 1/2 are permissible in an- isotropic materials. Indeed, hexagonal honeycombs can exhibit Poisson's ratio of 1, and if they have oriented hexagonal cells, greater than 1, in certain directions. Rubber has one of the highest values of Poisson's ratio at 0.4999, which is evident in its physically noticeable reaction to axial stretching.