Question

Derive an expression for Poisson's ratio.​

Answer 1

Answer:

In mechanics, Poisson's ratio is the negative of the ratio of transverse strain to lateral or axial strain. It is named after Siméon Poisson and denoted by the Greek letter 'nu', It is the ratio of the amount of transversal expansion to the amount of axial compression for small values of these changes.

Answer 2

Answer:

Poisson's ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of the stretching force.

Explanation:

Poisson's ratio is described as the inverse of the lateral strain to axial strain ratio in a uniaxial stress state. Poisson's ratio is also known as the ratio of absolute numbers of lateral and axial strain. Because both strains are unitless, this percentage, like strain, is unitless. The Poisson's Ratio calculation is

Step 1: ν = – εtrans / εlongitudinal

in this, the strain or stress ε is defined in elementary form. also, the original length divides the change in length.

ε = δl/l

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.

Step 2: In this case,

$\begin{aligned}& \varepsilon_t=-\frac{d B}{B} \\& \varepsilon_l=-\frac{d L}{L}\end{aligned}$

Step 3: The formula for Poisson's ratio is,

$Poisson's ratio$ $=\frac{\text { Transverse strain }}{\text { Longitudinal strain }}$

$\Rightarrow \nu=-\frac{\varepsilon_t}{\varepsilon_t}$

where,

$\varepsilon_t$ is the Lateral or Transverse Strain

$\varepsilon_1$ is the Longitudinal or Axial Strain

$v$ is Poisson's Ratio

The strain on its own is defined as the change in dimension divided by the original dimension.

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