# Why is the Poissons relationship always positive

The **Poisson's number μ** (also

**Poisson's ratio**,

**Poisson's ratio**or

**Poisson's ratio**called) is a quantity in mechanics or strength theory. It is used to calculate the transverse contraction and is named after Siméon Denis Poisson.

### definition

The Poisson's number is defined as the ratio of the relative change in thickness Δ*d* / *d* for the relative change in length Δ*l* / *l* when an external force or tension is applied.

So the following applies:

in which *l* denotes the original length.

Positive values of Δ*d* or Δ*l* denote an increase in this dimension, negative values correspond to a reduction.

The symbol ν instead of μ is also often used. The Poisson's number is a dimensionless quantity. The Poisson's number is a material constant; H. it depends on the material of the workpiece used. The elastic constants are mutually related. For linear-elastic, isotropic material, the following relationship between the shear modulus applies*G*, the modulus of elasticity*E.* and the compression module*K*:

- .

The relative change in volume Δ*V.*/*V.*, with which a body reacts to uniaxial elongation (tensile test), is calculated using the Poisson's number, neglecting quadratic terms

- .

### Scope

The Poisson's number is usually between 0 and 0.5 (for isotropic materials). Typical, frequent values of the Poisson's number are between 0.1 and 0.4.

With a Poisson's ratio μ = 0.5, the volume of an elastic body remains constant under load. The incompressibility is only preserved for infinitesimal deformations. In addition, there are poles in Cauchy's constitutive equations. For the calculation of almost or fully incompressible materials (e.g. rubber materials, entropy-elastic materials, hyperelastic materials), Green's material models should be used. ^{[1]}.

If the Poisson's ratio is less than 0.5, the volume increases with tensile load and decreases with pressure load.

With values greater than 0.5, a decrease in volume would occur under tensile loading. This cannot exist for energetic reasons.

Occasionally materials with a negative Poisson's number are also known.^{[2]} Negative values result in a transverse expansion instead of a transverse contraction when elongated. Such materials are called **auxetic** designated. Examples of this are certain polymer foams, crystals or carbon fibers.

### The reciprocal of the Poisson's number

In geotechnics and rock mechanics, the reciprocal of the Poisson's number is also referred to as the “Poisson's number”. Often then the sign *m* used. A uniform designation has not yet established itself. The following rule would be recommended for standardization:

- Poisson's ratio: Symbol: ν or μ; with numerical values from 0 to < 0,5="">
- Poisson's number: character m; with numerical values> 2

where:

Note: if auxetic materials are taken into account, a value range from -1 to applies for the Poisson's ratio < 0,5="">

### Numerical values for μ

Poisson's ratio μ for some materials

material | Poisson's ratio μ |
---|---|

cork | about 0.00 |

Foam | 0,10 - 0,40 |

SiC | 0,17 |

concrete | 0,20 |

sand | 0,20 - 0,45 |

iron | 0,21 - 0,259 |

Glass | 0,18 - 0,3 |

Si_{3}N_{4} | 0,25 |

steel | 0,27 - 0,30 |

Clay | 0,30 - 0,45 |

aluminum | 0,33 |

copper | 0,33 |

Titanium | 0,34 |

magnesium | 0,35 |

Brass | 0,37 |

lead | 0,44 |

rubber | 0,50 |

PMMA (plexiglass) | 0,40 - 0,43 |

A value of μ = 0.3 or μ = 1/3 is often assumed for metallic materials and 0.35 for thermoplastics if no more precise values are known. An error in the Poisson's ratio has significantly less effect on the calculation of the component behavior under mechanical stress than an error in the modulus of elasticity. For this reason, the modulus of elasticity for the material used must be precisely determined (e.g. in a tensile test), while an approximate value is often sufficient for the transverse contraction.

### See also

### swell

**↑**R. W. Ogden: Non-Linear Elastic Deformations. Dover Publications, Inc., Mineola, New York (1984).**↑**Klaas Kölln: Morphology and mechanical properties of cellulose fibers, dissertation Uni Kiel, p. 22**↑**Glassproperties.com*Calculation of the Poisson Ratio for Glasses*

Categories: Strength of Materials | Material property

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