Poisson's ratio

In materials science and solid mechanics, Poisson's ratio is a measure of the Poisson effect, the deformation of a material in directions perpendicular to the specific direction of loading. The value of Poisson's ratio is the negative of the ratio of transverse strain to axial strain. For small values of these changes, is the amount of transversal elongation divided by the amount of axial compression.
Most materials have Poisson's ratio values ranging between 0.0 and 0.5. For soft materials, such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2 to 0.3.
The ratio is named after the French mathematician and physicist Siméon Poisson.

Definition

Assuming that the material is stretched or compressed in only one direction :
where

is the resulting Poisson's ratio,
is transverse strain
is axial strain

and positive strain indicates extension and negative strain indicates contraction.

Origin

Poisson's ratio is a measure of the Poisson effect, the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression. Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching. It is a common observation when a rubber band is stretched, it becomes noticeably thinner. Again, the Poisson ratio will be the ratio of relative contraction to relative expansion and will have the same value as above. In certain rare cases, a material will actually shrink in the transverse direction when compressed, which will yield a negative value of the Poisson ratio.
The Poisson's ratio of a stable, isotropic, linear elastic material must be between −1.0 and +0.5 because of the requirement for Young's modulus, the shear modulus and bulk modulus to have positive values. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. A perfectly incompressible isotropic material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5. Most steels and rigid polymers when used within their design limits exhibit values of about 0.3, increasing to 0.5 for post-yield deformation which occurs largely at constant volume. Rubber has a Poisson ratio of nearly 0.5. Cork's Poisson ratio is close to 0, showing very little lateral expansion when compressed. Glass is between 0.18 and 0.30. Some materials, e.g. some polymer foams, origami folds, and certain cells can exhibit negative Poisson's ratio, and are referred to as auxetic materials. If these auxetic materials are stretched in one direction, they become thicker in the perpendicular direction. In contrast, some anisotropic materials, such as carbon nanotubes, zigzag-based folded sheet materials, and honeycomb auxetic metamaterials to name a few, can exhibit one or more Poisson's ratios above 0.5 in certain directions.

Poisson's ratio from geometry changes

Length change

For a cube stretched in the -direction with a length increase of in the -direction, and a length decrease of in the - and -directions, the infinitesimal diagonal strains are given by
If Poisson's ratio is constant through deformation, integrating these expressions and using the definition of Poisson's ratio gives
Solving and exponentiating, the relationship between and is then
For very small values of and, the first-order approximation yields:

Volumetric change

The relative change of volume of a cube due to the stretch of the material can now be calculated. Since and
one can derive
Using the above derived relationship between and :
and for very small values of and, the first-order approximation yields:
For isotropic materials we can use Lamé's relation
where is bulk modulus and is Young's modulus.

Width change

If a rod with diameter and length is subject to tension so that its length will change by then its diameter will change by:
The above formula is true only in the case of small deformations; if deformations are large then the following formula can be used:
where

is original diameter
is rod diameter change
is Poisson's ratio
is original length, before stretch
is the change of length.

The value is negative because it decreases with increase of length

Characteristic materials

Isotropic

For a linear isotropic material subjected only to compressive forces, the deformation of a material in the direction of one axis will produce a deformation of the material along the other axis in three dimensions. Thus it is possible to generalize Hooke's law into three dimensions:
where:

,, and are strain in the direction of, and
,, and are stress in the direction of, and
is Young's modulus
is Poisson's ratio

these equations can be all synthesized in the following:
In the most general case, also shear stresses will hold as well as normal stresses, and the full generalization of Hooke's law is given by:
where is the Kronecker delta. The Einstein notation is usually adopted:
to write the equation simply as:

Anisotropic

For anisotropic materials, the Poisson ratio depends on the direction of extension and transverse deformation
Here is Poisson's ratio, is Young's modulus, is a unit vector directed along the direction of extension, is a unit vector directed perpendicular to the direction of extension. Poisson's ratio has a different number of special directions depending on the type of anisotropy.

Orthotropic

s have three mutually perpendicular planes of symmetry in their material properties. An example is wood, which is most stiff along the grain, and less so in the other directions.
Then Hooke's law can be expressed in matrix form as
where

is the Young's modulus along axis
is the shear modulus in direction on the plane whose normal is in direction
is the Poisson ratio that corresponds to a contraction in direction when an extension is applied in direction.

The Poisson ratio of an orthotropic material is different in each direction. However, the symmetry of the stress and strain tensors implies that not all the six Poisson's ratios in the equation are independent. There are only nine independent material properties: three elastic moduli, three shear moduli, and three Poisson's ratios. The remaining three Poisson's ratios can be obtained from the relations
From the above relations we can see that if then. The larger ratio is called the major Poisson ratio while the smaller one is called the minor Poisson ratio. We can find similar relations between the other Poisson ratios.

Transversely isotropic

materials have a plane of isotropy in which the elastic properties are isotropic. If we assume that this plane of isotropy is the -plane, then Hooke's law takes the form
where we have used the -plane of isotropy to reduce the number of constants, that is,
The symmetry of the stress and strain tensors implies that
This leaves us with six independent constants,,,,,. However, transverse isotropy gives rise to a further constraint between and, which is
Therefore, there are five independent elastic material properties, two of which are Poisson's ratios. For the assumed plane of symmetry, the larger of and is the major Poisson ratio. The other major and minor Poisson ratios are equal.

Poisson's ratio values for different materials

Material	Poisson's ratio
rubber	0.4999
gold	0.42-0.44
saturated clay	0.40-0.49
magnesium	0.252-0.289
titanium	0.265-0.34
copper	0.33
aluminium alloy	0.32
clay	0.30-0.45
stainless steel	0.30-0.31
steel	0.27-0.30
cast iron	0.21-0.26
sand	0.20-0.455
concrete	0.1-0.2
glass	0.18-0.3
metallic glasses	0.276-0.409
foam	0.10-0.50
cork	0.0

Material	Poisson's ratio
aluminium	0.35
beryllium	0.032
bismuth	0.33
cadmium	0.30
calcium	0.31
cerium	0.24
chromium	0.21
cobalt	0.31
copper	0.34
dysprosium	0.25
erbium	0.24
europium	0.15
gadolinium	0.26
gold	0.44
hafnium	0.37
holmium	0.23
iridium	0.26
iron	0.29
lanthanum	0.28
lead	0.44
lutetium	0.26
magnesium	0.29
molybdenum	0.31
neodymium	0.28
nickel	0.31
niobium	0.40
osmium	0.25
palladium	0.39
platinum	0.38
plutonium	0.21
praseodymium	0.28
promethium	0.28
rhenium	0.30
rhodium	0.26
ruthenium	0.30
samarium	0.27
scandium	0.28
selenium	0.33
silver	0.37
strontium	0.28
tantalum	0.34
terbium	0.26
thallium	0.45
thorium	0.27
thulium	0.21
tin	0.36
titanium	0.32
tungsten	0.28
uranium	0.23
vanadium	0.37
ytterbium	0.21
yttrium	0.24
zinc	0.25
zirconium	0.34