# 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

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

## Other expressions

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

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

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

Expressed in terms of Lamé's parameters:

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

## 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.