# Poisson's ratio

parameter of elastic materials: ratio of transverse strain to axial strain

Poisson's ratio (letter v) is a measure of the contraction that happens when an object is stretched. It is named after Siméon Denis Poisson. This contraction is perpendicular to the stretching force. It can also expand as the object is compressed in a perpendicular direction.

Figure 1: A cube with sides of length L of an isotropic linearly elastic material subject to tension along the x axis, with a Poisson's ratio of 0.5. The green cube is unstrained, the red is expanded in the x direction by ${\displaystyle \Delta L}$ due to tension, and contracted in the y and z directions by ${\displaystyle \Delta L'}$.

For example, if a block is being stretched as shown in the image to the right, the equation for the poisson's ratio will be:

${\displaystyle \nu =-{\frac {d\varepsilon _{\mathrm {trans} }}{d\varepsilon _{\mathrm {axial} }}}=-{\frac {d\varepsilon _{\mathrm {y} }}{d\varepsilon _{\mathrm {x} }}}=-{\frac {d\varepsilon _{\mathrm {z} }}{d\varepsilon _{\mathrm {x} }}}}$

Poisson's ratio ranges from 0.0-0.5 for common materials, though for materials with certain structures, can be as low as -1. A material with a Poisson's ratio close to 0 (like cork) can be stretched a lot in the axial direction without changing much at all in the transverse, where as pulling on a material with a high Poisson's ratio (like rubber) will cause it to become much more narrow. A material with a negative Poisson's ratio will expand in all directions as it is stretched.