Question

The theoretical limits of Poisson’s ratio lies between $-1$ to $0.5$ because
A. shear modulus and bulk’s modulus should be positive.
B. bulk’s modulus is negative during compression.
C. shear modulus is negative during compression.
D. Young’s modulus will always be positive.

Answer 1

The equation to calculate the Poisson's ratio is given as
$v=-\dfrac{d{{E}_{trans}}}{d{{E}_{axial}}}$
Where $d{{E}_{trans}}$ the transverse strain is experienced by the object measured in perpendicular direction to the applied force and $d{{E}_{axial}}$ is the axial strain experienced by the object measured in the direction of the applied force.

Complete step by step answer:
First of all let us take a look at what a Poisson’s ratio means. The Poisson’s ratio is a measure of the Poisson effect that explains the expansion or contraction of a material in directions perpendicular to the direction of loading. Shear modulus is described as the ratio of shear stress to the shear strain.
The value of Poisson's ratio is given by the formula,
$v=-\dfrac{d{{E}_{trans}}}{d{{E}_{axial}}}$
Suppose $Y,K,n$ and $\sigma$ are the Young's Modulus, Bulk modulus, Modulus of Rigidity and Poisson's Ratio, respectively.
Young’s modulus is given by the formula,
$Y=3K\left( 1-2\sigma \right)$
And also
$Y=2n\left( 1+\sigma \right)$
Equating both the equations will give,
$3K\left( 1-2\sigma \right)=2n\left( 1+\sigma \right)$
As we know $K$ and $n$ are always positive,
If $\sigma$ is positive, then RHS is always positive. Therefore LHS must also be positive. Hence

& 2\sigma < 1 \\\ & \sigma < \dfrac{1}{2} \\\ \end{aligned}$$ And also if $$\sigma $$ is positive, then LHS also is positive. Hence, $$\begin{aligned} & 1+\sigma > 0 \\\ & \sigma > -1 \\\ \end{aligned}$$ Therefore the limiting values of Poisson’s ratio will be $$-1 < \sigma < \dfrac{1}{2}$$ **So, the correct answer is “Option A”.** **Note:** Most materials are having Poisson's ratio values ranging between 0.0 and 0.5. The Incompressible materials such as rubber are having a ratio near 0.5. This ratio is named after the French physicist and great mathematician Siméon Poisson. Poisson's ratios exceeding 0.5 are allowed in anisotropic materials. And also hexagonal honeycombs are able to exhibit Poisson's ratio of one, and if they have arranged hexagonal cells in certain directions which is greater than one.