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When a sample of material is stretched in one direction, it tends to get thinner in the other two directions. 'Poisson's ratio' (ν, $mu$), 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). For a perfectly incompressible material deformed elastically at small strains, the Poisson's ratio would be exactly 0.5. Most practical engineering 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. 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:
:$$
u = -rac{arepsilon_mathrm{trans}}{arepsilon_mathrm{axial}}
where
:$$
u is the resulting Poisson's ratio,
: is transverse strain,
: is axial strain.
At first glance, a Poisson's ratio greater than 0.5 does not make sense because at a specific strain the material would reach zero volume, and any further strain would give the material "negative volume". Unusual Poisson ratios are usually a result of a material with complex architecture.

Contents

Generalized Hooke's law

Volumetric change

Width change

Orthotropic materials

Poisson's ratio values for different materials

See also

External links

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:
:
u left ( sigma_y + sigma_z
ight )
ight ]
:
u left ( sigma_x + sigma_z
ight )
ight ]
:
u left ( sigma_x + sigma_y
ight )
ight ]
where
:, and 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
:$$
u 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):
:
u)rac {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

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):
:$Delta\; d\; =\; -\; d\; cdot$
u {{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:
:$Delta\; d\; =\; -\; d\; cdot\; left(\; 1\; -\; \{left(\; 1\; +\; \{\{Delta\; L\}\; over\; L\}$
ight)}^{-
u}
ight)
where
:$d$ is original diameter
:$Delta\; d$ is rod diameter change
:$$
u 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:
:
u_{yx}}{E_y} = rac{
u_{xy}}{E_x} qquad
rac{
u_{zx}}{E_z} = rac{
u_{xz}}{E_x} qquad
rac{
u_{yz}}{E_y} = rac{
u_{zy}}{E_z} qquad

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

Poisson's ratio values for different materials

material	poisson's ratio
aluminium-alloy	0.33
concrete	0.20
cast iron	0.21-0.26
glass	0.24
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

See also

★ 3-D elasticity

★ Hooke's Law

★ Stress

★ Strain

★ Orthotropic material

★ Coefficient of thermal expansion

External links

★ Meaning of Poisson's ratio

★ Negative Poisson's ratio materials

★ More on negative Poisson's ratio materials (auxetic)

★ Poisson's ratio