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Poisson's ratio, denoted
Contents
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 (or expand when stretched) which will yield a negative value of the Poisson ratio.
The Poisson's ratio of a stable, isotropic, linear elastic material will be greater than −1.0 or less than 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 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 (before yield) 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. 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.
Assuming that the material is stretched or compressed along the axial direction (the x axis in the below diagram):
where
Length change
For a cube stretched in the x-direction (see figure 1) with a length increase of
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
For very small values of
Volumetric change
The relative change of volume ΔV/V of a cube due to the stretch of the material can now be calculated. Using
Using the above derived relationship between
and for very small values of
For isotropic materials we can use Lamé’s relation
where
Note that isotropic materials must have a Poisson's ratio of
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 above formula is true only in the case of small deformations; if deformations are large then the following (more precise) formula can be used:
where
The value is negative because it decreases with increase of length
Isotropic materials
For a linear isotropic material subjected only to compressive (i.e. normal) 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 (for compressive forces) into three dimensions:
where:
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
In this case the equation is simply written:
Orthotropic materials
For orthotropic materials such as wood, Hooke's law can be expressed in matrix form as
where
The Poisson's ratio of an orthotropic material is different in each direction (x, y and z). 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
Transversely isotropic materials
Transversely isotropic materials have a plane of isotropy in which the elastic properties are isotropic. If we assume that this plane of isotropy is
where we have used the plane of isotropy
The symmetry of the stress and strain tensors implies that
This leaves us with six independent constants
Therefore, there are five independent elastic material properties two of which are Poisson's ratios. For the assumed plane of symmetry, the larger of
Negative Poisson's ratio materials
Some materials known as auxetic materials display a negative Poisson’s ratio. When subjected to positive strain in a longitudinal axis, the transverse strain in the material will actually be positive (i.e. it would increase the cross sectional area). For these materials, it is usually due to uniquely oriented, hinged molecular bonds. In order for these bonds to stretch in the longitudinal direction, the hinges must ‘open’ in the transverse direction, effectively exhibiting a positive strain. This can also be done in a structured way and lead to new aspects in material design as for mechanical metamaterials.
Applications of Poisson's effect
One area in which Poisson's effect has a considerable influence is in pressurized pipe flow. When the air or liquid inside a pipe is highly pressurized it exerts a uniform force on the inside of the pipe, resulting in a hoop stress within the pipe material. Due to Poisson's effect, this hoop stress will cause the pipe to increase in diameter and slightly decrease in length. The decrease in length, in particular, can have a noticeable effect upon the pipe joints, as the effect will accumulate for each section of pipe joined in series. A restrained joint may be pulled apart or otherwise prone to failure.
Another area of application for Poisson's effect is in the realm of structural geology. Rocks, like most materials, are subject to Poisson's effect while under stress. In a geological timescale, excessive erosion or sedimentation of Earth's crust can either create or remove large vertical stresses upon the underlying rock. This rock will expand or contract in the vertical direction as a direct result of the applied stress, and it will also deform in the horizontal direction as a result of Poisson's effect. This change in strain in the horizontal direction can affect or form joints and dormant stresses in the rock.
The use of cork as a stopper for wine bottles is due to cork having a Poisson ratio of practically zero, so that, as the cork is inserted into the bottle, the upper part which is not yet inserted does not expand in diameter as it is compressed axially. The force needed to insert a cork into a bottle arises only from the friction between the cork and the bottle due to the radial compression of the cork. If the stopper were made of rubber, for example, (with a Poisson ratio of about 1/2), there would be a relatively large additional force required to overcome the radial expansion of the upper part of the rubber stopper.