Poisson's effect

Poisson’s effect.

Calculation of Poisson’s effect

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

:$varepsilon\_\{\; ext\{trans\; =\; -\; u\; ,\; varepsilon\_\{\; ext\{long$

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:

:$Delta\; mathit\{V\}\; =\; left\; (\; 1\; -\; 2\; u\; ight\; )\; varepsilon\; mathit\{V\}\_o$

Where $Vo$ is the original volume, $epsilon$ is the strain due to the applied stress, and $u$ 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:

:$u\; =\; frac\{3K\; -\; 2G\}\{6K\; +\; 2G\}$

:$E\; =\; 2G\; left\; (\; 1\; +\; u\; ight\; )\; =\; 3K\; left\; (\; 1\; -\; 2\; u\; ight\; )$

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. [ [http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/elastic_constants_G_K.cfm#bulkmod Finding the Shear Modulus and the Bulk Modulus ] ]

Notes