Poisson's ratio

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Image:Poisson ratio compression example.png

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, <math>\mu</math>), 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 (in the direction of the applied load). For a perfectly incompressible material, 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 material is compressed along y axis (see Fig.1):

<math>\nu_{yx} = -\frac{\varepsilon_x}{\varepsilon_y}</math>

where

<math>\nu_{yx}</math> is the resulting Poisson's ratio,

<math>\varepsilon_x</math> is transverse strain,

<math>\varepsilon_y</math> 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.

1 Generalized Hooke's Law
2 Shear Modulus
3 Volumetric Change
4 Width Change
5 Orthotropic materials
6 Poisson's ratio values for different materials
7 See also
8 External links

[edit] Generalized Hooke's Law

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

<math> \varepsilon\ _x = \frac {1}{E} \left [ \sigma\ _x - \nu \left ( \sigma\ _y + \sigma\ _z \right ) \right ] </math>

<math> \varepsilon\ _y = \frac {1}{E} \left [ \sigma\ _y - \nu \left ( \sigma\ _x + \sigma\ _z \right ) \right ] </math>

<math> \varepsilon\ _z = \frac {1}{E} \left [ \sigma\ _z - \nu \left ( \sigma\ _x + \sigma\ _y \right ) \right ] </math>

where

<math> \varepsilon\ _x , \varepsilon\ _y , \varepsilon\ _z </math> are strain in the direction of x, y and z axis

<math> {E} </math> is Young's modulus of the material

<math> \sigma\ _x , \sigma\ _y , \sigma\ _z </math> are stress in the direction of x, y and z axis

<math> \nu </math> is Poisson's ratio (the same in all directions: x, y and z for isotrophic materials)

[edit] Shear Modulus

For an isotropic material the relation between Shear modulus G and Young's modulus E is following:

where

<math> G </math> is Shear modulus

<math> E </math> is Young's modulus

<math> \nu </math> is Poisson's ratio

[edit] 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):

<math>\frac {\Delta V} {V} = (1-2\nu)\frac {\Delta L} {L}</math>

where

<math> V </math> is material volume

<math> \Delta V </math> is material volume change

<math> L </math> is original length, before stretch

<math> \Delta L </math> is the change of length: <math> \Delta L = L_{old} - L_{new}</math>

[edit] Width Change

Image:Rod diamater change poisson.png

Figure 2: Comparison between the two formulas, one for small deformations, another for large deformations

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

<math>\Delta d = - d \cdot \nu {{\Delta L} \over L}</math>

Above formula is true only in the case of small deformations, if deformations are large then more precise is following formula:

<math>\Delta d = - d \cdot \left( 1 - {\left( 1 + {{\Delta L} \over L} \right)}^{-\nu} \right)</math>

where

<math> d </math> is original diameter

<math> \Delta d </math> is rod diameter change

<math> \nu </math> is Poisson's ratio

<math> L </math> is original length, before stretch

<math> \Delta L </math> is the change of length.

[edit] 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:

<math>\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 </math>

where

<math>{E}_i</math> is a Young's modulus along axis i

<math>\nu_{jk}</math> is a Poisson's ratio in plane jk

[edit] 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
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