Poisson's ratio

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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 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).

1 Generalized Hooke's law
2 Volumetric change
3 Width change
4 Orthotropic materials
5 Poisson's ratio values for different materials
6 See also
7 References
8 External links

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

σ x

σ y

and

σ 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)

ν

is Poisson's ratio (the same in all directions:

x

y

and

z

for isotropic materials)

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

Δ V

is material volume change

L

is original length, before stretch

Δ L

is the change of length:

Δ L = L old - L new

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

Δ d

is rod diameter change

ν

is Poisson's ratio

L

is original length, before stretch

Δ L

is the change of length.

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

ν j k

is a Poisson's ratio in plane jk

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

^ Poisson's ratio calculation of glasses

v • d • e Elastic moduli for homogeneous isotropic materials

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.
	$(\lambda,\,\mu)$	$(E,\,\mu)$	$(K,\,\lambda)$	$(K,\,\mu)$	$(\lambda,\,\nu)$	$(\mu,\,\nu)$	$(E,\,\nu)$	$(K,\, \nu)$	$(K,\,E)$
$K=\,$	$\lambda+ \frac{2\mu}{3}$	$\frac{E\mu}{3(3\mu-E)}$			$\lambda\frac{1+\nu}{3\nu}$	$\frac{2\mu(1+\nu)}{3(1-2\nu)}$	$\frac{E}{3(1-2\nu)}$
$E=\,$	$\mu\frac{3\lambda + 2\mu}{\lambda + \mu}$		$9K\frac{K-\lambda}{3K-\lambda}$	$\frac{9K\mu}{3K+\mu}$	$\frac{\lambda(1+\nu)(1-2\nu)}{\nu}$	$2\mu(1+\nu)\,$		$3K(1-2\nu)\,$
$\lambda=\,$		$\mu\frac{E-2\mu}{3\mu-E}$		$K-\frac{2\mu}{3}$		$\frac{2 \mu \nu}{1-2\nu}$	$\frac{E\nu}{(1+\nu)(1-2\nu)}$	$\frac{3K\nu}{1+\nu}$	$\frac{3K(3K-E)}{9K-E}$
$\mu=\,$			$3\frac{K-\lambda}{2}$		$\lambda\frac{1-2\nu}{2\nu}$		$\frac{E}{2+2\nu}$	$3K\frac{1-2\nu}{2+2\nu}$	$\frac{3KE}{9K-E}$
$\nu=\,$	$\frac{\lambda}{2(\lambda + \mu)}$	$\frac{E}{2\mu}-1$	$\frac{\lambda}{3K-\lambda}$	$\frac{3K-2\mu}{2(3K+\mu)}$					$\frac{3K-E}{6K}$
$M=\,$	$\lambda+2\mu\,$	$\mu\frac{4\mu-E}{3\mu-E}$	$3K-2\lambda\,$	$K+\frac{4\mu}{3}$	$\lambda \frac{1-\nu}{\nu}$	$\mu\frac{2-2\nu}{1-2\nu}$	$E\frac{1-\nu}{(1+\nu)(1-2\nu)}$	$3K\frac{1-\nu}{1+\nu}$	$3K\frac{3K+E}{9K-E}$