Elasticity

# Poisson’s Ratio

Poisson’s Ratio is defined as the ratio of lateral strain to longitudinal strain.

When a wire is suspended from one end and a load is attached to the other end, then along with the increase in length, it also suffers a decrease in diameter. Thus, as the wire extends in the direction of applied force, at the same time it also contracts in perpendicular direction.

The change in dimensions per unit dimension in the direction of applied force is called longitudinal strain $(α)$.

The change in dimensions per unit dimension in the perpendicular direction is called lateral strain $(β)$.

Within elastic limit, lateral strain is found to be directly proportional to the longitudinal strain. $i.e. β∝α$ $β=σα$ where, $σ$ is the proportionality constant called Poisson’s ratio. $σ=\frac{β}{α}$

Thus, Poisson’s ratio is defined as the ratio of lateral strain to the longitudinal strain.

​Consider a wire of length $L$ and diameter $D$. When load is attached, suppose its length becomes $L+x$ and the diameter becomes $D-d$.

​$\text{Longitudinal strain (α)}=\frac{x}{L}$ $\text{Lateral strain (β)}=\frac{d}{D}$ Thus, Poisson’s ratio for the material of wire is given by, $σ=\frac{β}{α}$ $σ=\frac{d}{D}\frac{L}{x}$ $σ=\frac{L}{D}\frac{d}{x}$

Poisson’s ratio depends only on the nature of the material.

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