Question

For a material \[Y = 6.6 \times {10^{10}}N/{m^2}\] and bulk modulus\[K = 11 \times {10^{10}}N/{m^2}\] , then its Poisson’s Ratio is:
\[(A)0.8\]
\[(B)\;0.35\]
\[(C)\;0.7\]
\[(D)0.4\]

Answer 1

Hint: The Poisson effect is a phenomenon in which a substance expands in directions opposite to the direction of compression. Poisson's ratio is a measure of this phenomenon. When a material is stretched rather than compressed, it appears to contract in ways that are transverse to the stretching direction.


Complete step-by-step solution:
\[v = {\text{ - }}\left( {\dfrac{{{\varepsilon _{trans}}}}{{{\varepsilon _{axial}}}}} \right)\] is the equation for calculating Poisson's ratio. Axial strain (\[{\varepsilon _{axial}}\]) is measured in the direction of the applied force, while transverse strain (\[{\varepsilon _{trans}}\]) is measured in the direction perpendicular to the applied force.
The Poisson ratio \[v\] is defined as the negative ratio of transverse strain to longitudinal strain, for the case of uniaxial stress.
Bulk modulus is the ratio of hydraulic stress to the corresponding hydraulic strain
Poisson’s Ratio is given by the equation -
\[\gamma = 3K(1 - 2\sigma )\]
\[\gamma = 6.6{\text{ }} \times {10^{10}}N/{m^2}\]
\[K = 11 \times {10^{10}}N/{m^2}\],
By substituting the values, we get:
\[ \Rightarrow 6.6 \times {10^{10}} = 3 \times 11 \times {10^{10}}(1 - 2\sigma )\]
\[ \Rightarrow (1 - 2\sigma ) = \dfrac{{6.6{\text{ }}x{\text{ }}{{10}^{10}}}}{{{\text{ 3 }}x{\text{ }}11{\text{ }}x{\text{ }}{{10}^{10}}}}\]
\[ \Rightarrow \sigma = 0.4\]

Note: Poisson's Ratio does not have any units.
The measure of the absolute values of lateral and axial strain is also known as Poisson's ratio. Since both strains are unitless, thus, this ratio is also unitless. This ratio is nearly constant for stresses within the elastic spectrum. A few common artifacts, such as an accordion (zero apparent Poisson ratio) or puppets that extend their arms and legs when a string is pulled, include mechanisms that can be used to this end (negative apparent Poisson ratio).