Poisson's effect

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Poisson’s effect.

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[edit] 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_{\text{trans}} = -\nu \, \varepsilon_{\text{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, ε is the strain due to the applied stress, and ν is Poisson’s Ratio.

[edit] 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 )

Because 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.[1]

[edit] Negative Poisson's ratio materials

Some 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.[2]

[edit] Notes

  1. ^ Finding the Shear Modulus and the Bulk Modulus
  2. ^ Negative Poisson's ratio