Poisson’s Ratio

This is the strain in the direction of the applied force. If a rod of length \(L\) elongates by \(\Delta L\), the longitudinal strain is:

\( \epsilon_{\text{long}} = \dfrac{\Delta L}{L} \)

This is the strain perpendicular to the direction of the applied force. If the diameter or width changes by \(\Delta d\), then:

\( \epsilon_{\text{lat}} = \dfrac{\Delta d}{d} \)

When the material is stretched, \(\Delta d\) is negative (it becomes thinner). When compressed, \(\Delta d\) is positive (it becomes thicker).

The value of Poisson’s ratio is:

\( \nu = -\dfrac{\epsilon_{\text{lat}}}{\epsilon_{\text{long}}} \)

The negative sign shows that when one dimension increases, the other typically decreases.

Poisson’s ratio tells how strongly a material tries to keep its overall volume consistent when stretched or compressed. A material that thins rapidly when stretched has a high Poisson's ratio, while one that hardly changes its diameter has a lower value.

Most solid materials have Poisson’s ratio between 0 and 0.5. A value near 0.5 means the material maintains nearly constant volume during deformation (like rubber). A smaller value means the material does not significantly change sideways.

Materials like cork have very low Poisson’s ratio. They hardly expand or contract sideways. That is why cork plugs fit tightly in bottle necks without expanding the glass.

Some special materials, called auxetic materials, have negative Poisson’s ratio. When stretched, they expand sideways instead of thinning. This unusual behaviour is useful in protective equipment and shock-absorbing structures.

A rubber band becomes thinner as it gets longer. This strong reduction in thickness indicates that rubber has a high Poisson’s ratio—close to 0.5.

When a soft eraser is compressed from the top, it bulges sideways. The sideways increase gives a clear sense of lateral strain.

A metal wire becomes slightly thinner when stretched by a hanging weight. Although the change is small, it represents lateral strain and gives the material its Poisson’s ratio.