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# Poisson's ratio

When a sample of material is stretched in one direction, it tends to get thinner 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). For a perfectly incompressible material deformed elastically at small strains, the Poisson's ratio would be exactly 0.5. 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. 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. There are also materials with unusually high (above 0.5) Poisson ratios; when stretched, their volume decreases.

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

ν 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
σx , σy and σz 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)
ν 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
ΔV is material volume change
L is original length, before stretch
ΔL is the change of length: ΔL = LoldLnew

## 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):

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

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
Δd is rod diameter change
ν is Poisson's ratio
L is original length, before stretch
ΔL 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

Ei is a Young's modulus along axis i
νjk is a Poisson's ratio in plane jk

## Poisson's ratio values for different materials

materialpoisson'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

## References

1. ^ Poisson's ratio calculation of glasses

v  d  e Elastic moduli for homogeneous isotropic materials

Bulk modulus (K) | Young's modulus (E) | Lamé's first parameter (λ) | Shear modulus (μ) | Poisson's ratio (ν) | P-wave modulus (M)

Conversion formulas
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these, thus given any two, any other of the elastic moduli can be calculated according to these formulas.
$(\lambda,\,\mu)$ $(E,\,\mu)$ $(K,\,\lambda)$ $(K,\,\mu)$ $(\lambda,\,\nu)$ $(\mu,\,\nu)$ $(E,\,\nu)$ $(K,\, \nu)$ $(K,\,E)$
$K=\,$ $\lambda+ \frac{2\mu}{3}$ $\frac{E\mu}{3(3\mu-E)}$ $\lambda\frac{1+\nu}{3\nu}$ $\frac{2\mu(1+\nu)}{3(1-2\nu)}$ $\frac{E}{3(1-2\nu)}$
$E=\,$ $\mu\frac{3\lambda + 2\mu}{\lambda + \mu}$ $9K\frac{K-\lambda}{3K-\lambda}$ $\frac{9K\mu}{3K+\mu}$ $\frac{\lambda(1+\nu)(1-2\nu)}{\nu}$ $2\mu(1+\nu)\,$ $3K(1-2\nu)\,$
$\lambda=\,$ $\mu\frac{E-2\mu}{3\mu-E}$ $K-\frac{2\mu}{3}$ $\frac{2 \mu \nu}{1-2\nu}$ $\frac{E\nu}{(1+\nu)(1-2\nu)}$ $\frac{3K\nu}{1+\nu}$ $\frac{3K(3K-E)}{9K-E}$
$\mu=\,$ $3\frac{K-\lambda}{2}$ $\lambda\frac{1-2\nu}{2\nu}$ $\frac{E}{2+2\nu}$ $3K\frac{1-2\nu}{2+2\nu}$ $\frac{3KE}{9K-E}$
$\nu=\,$ $\frac{\lambda}{2(\lambda + \mu)}$ $\frac{E}{2\mu}-1$ $\frac{\lambda}{3K-\lambda}$ $\frac{3K-2\mu}{2(3K+\mu)}$ $\frac{3K-E}{6K}$
$M=\,$ $\lambda+2\mu\,$ $\mu\frac{4\mu-E}{3\mu-E}$ $3K-2\lambda\,$ $K+\frac{4\mu}{3}$ $\lambda \frac{1-\nu}{\nu}$ $\mu\frac{2-2\nu}{1-2\nu}$ $E\frac{1-\nu}{(1+\nu)(1-2\nu)}$ $3K\frac{1-\nu}{1+\nu}$ $3K\frac{3K+E}{9K-E}$