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= Poisson’s Effect =

Poisson’s effect is a physical phenomenon concerning the change in strain of a material along one axis of measurement that is related to a corresponding change in strain along a separate (usually perpendicular) axis of measurement. This effect in the perpendicular direction is generally dictated by a material property known as Poisson's ratio, which is a ratio of the change in strain between the longitudinal and transverse axes of measurement. An example of this effect can be seen by taking a cylindrical piece of elastic material (say, rubber) and subjecting it to tension: the diameter of the cylinder will decrease. Conversely, if the rubber is subjected to compression the diameter will increase

Cause of Poisson’s effect
On the molecular level, Poisson’s effect is caused by slight movements between molecules and the stretching of molecular bonds within the material lattice to accommodate the stress. When the bonds elongate in the stress direction, they shorten in the other directions. This behavior multiplied millions of times throughout the material lattice is what drives the phenomenon.

Calculation of Poisson’s effect
The amount of strain in a transverse (perpendicular to the stress) direction, εtrans, can be easily calculated by multiplying the amount of strain in the longitudinal direction (along the direction of stress), εlong, with Poisson’s ratio, ν:

$$\varepsilon_{trans} = -\nu \, \varepsilon_{long}$$

This equation will give you the relative change in length of the material. Assuming that the material is isotropic and subjected to one dimensional stress, the equation may be applied to calculate the volume change due to the longitudinal strain:

$$\Delta \mathit{V} = \left ( 1 - 2\nu \right ) \varepsilon \mathit{V}_o $$

Where $$Vo$$ is the original volume, $$\epsilon$$ is the strain due to the applied stress, and $$\nu$$ is Poisson’s Ratio.

Derivations of Poisson’s Ratio
Poisson’s ratio is directly proportional to the material properties of bulk modulus (K), shear modulus (G), and Young’s modulus (or strain modulus, E). These moduli all reflect some aspect of the material’s stiffness, and are themselves a derivation of stress to strain ratios. The following equations show how these properties are all related:

$$\nu = \frac{3K - 2G}{6K + 2G}$$

$$E = 2G \left ( 1 + \nu \right ) = 3K \left ( 1 - 2\nu \right ) $$

Due to the fact that these moduli must all be positive, the above equations also define the upper and lower theoretical bounds for Poisson’s ratio at 0.5 and -1. A material with ν = 0.5 is considered perfectly inelastic, since there would be absolutely no volume change due to Poisson’s effect.

Negative Poisson's Ratio Materials
Some unique materials known as auxetic materials display a negative Poisson’s ratio. When subjected to strain in a longitudinal axis, the transverse strain in the material will actually be positive (i.e. it would increase in 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.

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 radial stress within the pipe material. Due to Poisson's effect, this radial stress will cause the pipe to slightly increase in diameter and 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. An restrained joint may be pulled apart or otherwise prone to failure.