# Poisson's ratio

Definition

An elastic parameter: the ratio of transverse contractional strain to longitudinal extensional strain. In other words, a measure of the degree to which a material expands outwards when squeezed, or equivalently contracts when stretched (though some materials, called auxetic, do display the opposite behaviour).

## Definition

$\mathrm{Poisson's}\ \mathrm{ratio} = \frac{\mathrm{transverse}\ \mathrm{strain}}{\mathrm{longitudinal}\ \mathrm{strain}}$
$\nu = \frac{\Delta W / W}{\Delta L / L}$

## Other expressions

Expressed in terms of acoustic velocities, assuming the material is isotropic and homogenous:

$\nu = \frac{ \left( \frac{V_\mathrm{P}}{V_\mathrm{S}} \right)^2 - 2}{2\left(\frac{V_\mathrm{P}}{V_\mathrm{S}}\right)^2 - 2}$

In this case, when a material has a positive $\nu$ it will have a $V_\mathrm{P}/V_\mathrm{S}$ ratio greater than 1.42.

Expressed in terms of Lamé's parameters:

$\nu = \frac{\lambda}{2\,(\lambda + \mu)}$

## Typical values

For incompressible material, ν is approximately 0.5. Cork has a value of about 0, meaning that it does not expand radially as it is compressed. Most rocks have ν between about 0.1 and 0.4.

Materials with negative Poisson's ratio, meaning that they get thinner as they are compressed, do exist. They are called auxetic and include the mineral α-cristobalite.

Required: a table of common (and relevant) values.