Poisson's ratio

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File:Poisson ratio compression example.svg
Figure 1: Rectangular specimen subject to compression, with Poisson's ratio circa 0.5

When a sample of material is stretched in one direction, it tends to contract (or rarely, expand) in the other two directions. Poisson's ratio (ν), named after Simeon Poisson, is a measure of this tendency. Poisson's ratio is the ratio of the relative contraction strain, or transverse strain (normal to the applied load), divided by the relative extension strain, or axial strain (in the direction of the applied load). The Poisson's ratio of a stable material cannot be less than -1.0 nor greater than 0.5 due to the requirement that the shear modulus and bulk modulus have positive values. Most materials have ν between 0.0 and 0.5. Cork is close to 0.0, most steels are around 0.3, and rubber is almost 0.5. A perfectly incompressible material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5. Some materials, mostly polymer foams, have a negative Poisson's ratio; if these auxetic materials are stretched in one direction, they become thicker in perpendicular directions.

Assuming that the material is compressed along the axial direction:

$\nu = -\frac{\varepsilon_\mathrm{trans}}{\varepsilon_\mathrm{axial}} = -\frac{\varepsilon_\mathrm{x}}{\varepsilon_\mathrm{y}}$

where

$\nu$ is the resulting Poisson's ratio,
$\varepsilon_\mathrm{trans}$ is transverse strain (negative for axial tension, positive for axial compression)
$\varepsilon_\mathrm{axial}$ is axial strain (positive for axial tension, negative for axial compression).

Generalized Hooke's law

For an isotropic material, the deformation of a material in the direction of one axis will produce a deformation of the material along the other axes in three dimensions. Thus it is possible to generalize Hooke's Law into three dimensions:

$\varepsilon_x = \frac {1}{E} \left [ \sigma_x - \nu \left ( \sigma_y + \sigma_z \right ) \right ]$
$\varepsilon_y = \frac {1}{E} \left [ \sigma_y - \nu \left ( \sigma_x + \sigma_z \right ) \right ]$
$\varepsilon_z = \frac {1}{E} \left [ \sigma_z - \nu \left ( \sigma_x + \sigma_y \right ) \right ]$

where

$\varepsilon_x$, $\varepsilon_y$ and $\varepsilon_z$ are strain in the direction of $x$, $y$ and $z$ axis
$\sigma_x$ , $\sigma_y$ and $\sigma_z$ are stress in the direction of $x$, $y$ and $z$ axis
$E$ is Young's modulus (the same in all directions: $x$, $y$ and $z$ for isotropic materials)
$\nu$ is Poisson's ratio (the same in all directions: $x$, $y$ and $z$ for isotropic materials)

Volumetric change

The relative change of volume ΔV/V due to the stretch of the material can be calculated using a simplified formula (only for small deformations):

$\frac {\Delta V} {V} = (1-2\nu)\frac {\Delta L} {L}$

where

$V$ is material volume
$\Delta V$ is material volume change
$L$ is original length, before stretch
$\Delta L$ is the change of length: $\Delta L = L_\mathrm{old} - L_\mathrm{new}$

Width change

File:Rod diamater change poisson.png
Figure 2: Comparison between the two formulas, one for small deformations, another for large deformations

If a rod with diameter (or width, or thickness) d and length L is subject to tension so that its length will change by ΔL then its diameter d will change by (the value is negative, because the diameter will decrease with increasing length):

$\Delta d = - d \cdot \nu {{\Delta L} \over L}$

The above formula is true only in the case of small deformations; if deformations are large then the following (more precise) formula can be used:

$\Delta d = - d \cdot \left( 1 - {\left( 1 + {{\Delta L} \over L} \right)}^{-\nu} \right)$

where

$d$ is original diameter
$\Delta d$ is rod diameter change
$\nu$ is Poisson's ratio
$L$ is original length, before stretch
$\Delta L$ is the change of length.

Orthotropic materials

For Orthotropic material, such as wood in which Poisson's ratio is different in each direction (x, y and z axis) the relation between Young's modulus and Poisson's ratio is described as follows:

$\frac{\nu_{yx}}{E_y} = \frac{\nu_{xy}}{E_x} \qquad \frac{\nu_{zx}}{E_z} = \frac{\nu_{xz}}{E_x} \qquad \frac{\nu_{yz}}{E_y} = \frac{\nu_{zy}}{E_z} \qquad$

where

${E}_i$ is a Young's modulus along axis i
$\nu_{jk}$ is a Poisson's ratio in plane jk

Poisson's ratio values for different materials

File:SpiderGraph PoissonRatio.gif
Influences of selected glass component additions on Poisson's ratio of a specific base glass.[1]
material poisson's ratio
aluminium-alloy 0.33
concrete 0.20
cast iron 0.21-0.26
glass 0.18-0.3
clay 0.30-0.45
saturated clay 0.40-0.50
copper 0.33
cork ca. 0.00
magnesium 0.35
stainless steel 0.30-0.31
rubber 0.50
steel 0.27-0.30
foam 0.10 to 0.40
titanium 0.34
sand 0.20-0.45
auxetics negative