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Encyclopedia > Poisson's ratio
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, μ), 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, 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. Image File history File links Poisson_ratio_compression_example. ... material is the substance or matter from which something is or can be made, or also items needed for doing or creating something. ... Simeon Poisson. ... This article is about the deformation of materials. ... Auxetics are materials that become thicker perpendicularly to the applied force when stretched. ...

Assuming that the material is compressed along the axial direction:

$nu = -frac{varepsilon_{trans}}{varepsilon_{axial}}$

where

ν is the resulting Poisson's ratio,
$varepsilon_{trans}$ is transverse strain,
$varepsilon_{axial}$ 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. Look up strain in Wiktionary, the free dictionary. ... Look up strain in Wiktionary, the free dictionary. ...

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 dimensions. Thus it is possible to generalize Hooke's Law into three dimensions: Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ...

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

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
ν is Poisson's ratio (the same in all directions: x, y and z for isotropic materials)

## Shear modulus

For an isotropic material the relation between shear modulus G and Young's modulus E is Look up Isotropy in Wiktionary, the free dictionary. ... In materials science, shear modulus S, sometimes referred to as the modulus of rigidity, is defined as the ratio of shear stress to the shear strain: S = shear stress/shear strain = (F/A)/&#934;. Another commonly accepted symbol is G. Shear modulus is usually measured in ksi (kips per square... This article does not adequately cite its references or sources. ...

$G = frac {E} {2(1+nu)}$

where

G is shear modulus
E is Young's modulus
ν is Poisson's ratio

In materials science, shear modulus S, sometimes referred to as the modulus of rigidity, is defined as the ratio of shear stress to the shear strain: S = shear stress/shear strain = (F/A)/&#934;. Another commonly accepted symbol is G. Shear modulus is usually measured in ksi (kips per square... This article does not adequately cite its references or sources. ...

## 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):

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

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

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): Image File history File links Rod_diamater_change_poisson. ...

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

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: An orthotropic material has two or three mutually orthogonal two-fold axes of rotational symmetry so that the mechanical properties are, in general different along the directions of each of the axes. ...

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

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

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

General Name, Symbol, Number aluminium, Al, 13 Chemical series poor metals Group, Period, Block 13, 3, p Appearance silvery Atomic mass 26. ... An alloy is a combination, either in solution or compound, of two or more elements, at least one of which is a metal, and where the resulting material has metallic properties. ... Concrete being poured, raked and vibrated into place in residential construction in Toronto, Ontario, Canada. ... Cast iron usually refers to grey cast iron, but can mean any of a group of iron-based alloys containing more than 2% carbon (alloys with less carbon are carbon steel by definition). ... Glass can be made transparent and flat, or into other shapes and colors as shown in this sphere from the Verrerie of Brehat in Brittany. ... The Gay Head cliffs in Marthas Vineyard are made almost entirely of clay. ... General Name, Symbol, Number copper, Cu, 29 Chemical series transition metals Group, Period, Block 11, 4, d Appearance metallic pinkish red Standard atomic weight 63. ... A cork stopper for a wine bottle Champagne corks Varnished cork tiles can be used for flooring, as a substitute for linoleum or tiles. ... General Name, Symbol, Number magnesium, Mg, 12 Chemical series alkaline earth metals Group, Period, Block 2, 3, s Appearance silvery white Atomic mass 24. ... The 630 foot high, stainless-clad (type 304) Gateway Arch defines St. ... Latex being collected from a tapped rubber tree Rubber is an elastic hydrocarbon polymer which occurs as a milky colloidal suspension (known as latex) in the sap of several varieties of plants. ... The steel cable of a colliery winding tower. ... Sea foam on the beach. ... General Name, Symbol, Number titanium, Ti, 22 Chemical series transition metals Group, Period, Block 4, 4, d Appearance silvery metallic Standard atomic weight 47. ... Patterns in the sand Sand is a granular material made up of fine rock particles. ... Auxetics are materials that become thicker perpendicularly to the applied force when stretched. ...

This article does not adequately cite its references or sources. ... In materials science, shear modulus S, sometimes referred to as the modulus of rigidity, is defined as the ratio of shear stress to the shear strain: S = shear stress/shear strain = (F/A)/&#934;. Another commonly accepted symbol is G. Shear modulus is usually measured in ksi (kips per square... 3-D elasticity is one of three methods of structural analysis. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ... Stress is the internal distribution of force per unit area that balances and reacts to external loads applied to a body. ... This article is about the deformation of materials. ... An orthotropic material has two or three mutually orthogonal two-fold axes of rotational symmetry so that the mechanical properties are, in general different along the directions of each of the axes. ... During heat transfer, the energy that is stored in the intermolecular bonds between atoms changes. ...

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