Poisson's ratio

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Figure 1: Rectangular specimen subject to compression, with Poisson's ratio circa 0.5

Poisson's ratio (ν), named after Simeon Poisson, is the ratio of the relative contraction strain, or transverse strain (normal to the applied load), divided by the relative extension strain, or axial strain (in the direction of the applied load).

When a sample of material is stretched in one direction, it tends to contract (or rarely, expand) in the other two directions. Conversely, when a sample of material is compressed in one direction, it tends to expand (or rarely, contract) in the other two directions. Poisson's ratio (ν) is a measure of this tendency.

The Poisson's ratio of a stable material cannot be less than −1.0 nor greater than 0.5 due to the requirement that the shear modulus and bulk modulus have positive values. Most materials have between 0.0 and 0.5. Cork is close to 0.0, showing almost no Poisson contraction, most steels are around 0.3, and rubber is nearly incompressible and so has a Poisson ratio of nearly 0.5. A perfectly incompressible material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5. Some materials, mostly polymer foams, have a negative Poisson's ratio; if these auxetic materials are stretched in one direction, they become thicker in perpendicular directions.

Assuming that the material is compressed along the axial direction:

\nu = -\frac{\varepsilon_\mathrm{trans}}{\varepsilon_\mathrm{axial}} = -\frac{\varepsilon_\mathrm{x}}{\varepsilon_\mathrm{y}}

where

ν is the resulting Poisson's ratio,
\varepsilon_\mathrm{trans} is transverse strain (negative for axial tension, positive for axial compression)
\varepsilon_\mathrm{axial} is axial strain (positive for axial tension, negative for axial compression).

Contents

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.

Generalized Hooke's law

For an isotropic material, the deformation of a material in the direction of one axis will produce a deformation of the material along the other axes in three dimensions. Thus it is possible to generalize Hooke's Law into three dimensions:

\varepsilon_x = \frac {1}{E} \left [ \sigma_x - \nu \left ( \sigma_y + \sigma_z \right ) \right ]
\varepsilon_y = \frac {1}{E} \left [ \sigma_y - \nu \left ( \sigma_x + \sigma_z \right ) \right ]
\varepsilon_z = \frac {1}{E} \left [ \sigma_z - \nu \left ( \sigma_x + \sigma_y \right ) \right ]

where

\varepsilon_x, \varepsilon_y and \varepsilon_z are strain in the direction of x, y and z axis
σx , σy and σz are stress in the direction of x, y and z axis
E is Young's modulus (the same in all directions: x, y and z for isotropic materials)
ν is Poisson's ratio (the same in all directions: x, y and z for isotropic materials)

Volumetric change

The relative change of volume ΔV/V due to the stretch of the material can be calculated using a simplified formula (only for small deformations):

\frac {\Delta V} {V} = (1-2\nu)\frac {\Delta L} {L}

where

V is material volume
ΔV is material volume change
L is original length, before stretch
ΔL is the change of length: ΔL = Lnew − Lold

Width change

Figure 2: Comparison between the two formulas, one for small deformations, another for large deformations

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 value is negative, because the diameter will decrease with increasing length):

\Delta d = - d \cdot \nu {{\Delta L} \over L}

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:

\Delta d = - d \cdot \left( 1 - {\left( 1 + {{\Delta L} \over L} \right)}^{-\nu} \right)

where

d is original diameter
Δd is rod diameter change
ν is Poisson's ratio
L is original length, before stretch
ΔL is the change of length.

Orthotropic materials

For Orthotropic material, such as wood in which Poisson's ratio is different in each direction (x, y and z axis) the relation between Young's modulus and Poisson's ratio is described as follows:

\frac{\nu_{yx}}{E_y} = \frac{\nu_{xy}}{E_x} \qquad \frac{\nu_{zx}}{E_z} = \frac{\nu_{xz}}{E_x} \qquad \frac{\nu_{yz}}{E_y} = \frac{\nu_{zy}}{E_z} \qquad

where

Ei is a Young's modulus along axis i
νjk is a Poisson's ratio in plane jk

Poisson's ratio values for different materials

Influences of selected glass component additions on Poisson's ratio of a specific base glass.1
material poisson's ratio
rubber ~ 0.50
saturated clay 0.40-0.50
magnesium 0.35
titanium 0.34
copper 0.33
aluminium-alloy 0.33
clay 0.30-0.45
stainless steel 0.30-0.31
steel 0.27-0.30
cast iron 0.21-0.26
sand 0.20-0.45
concrete 0.20
glass 0.18-0.3
foam 0.10 to 0.40
cork ~ 0.00
auxetics negative

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.2

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. A restrained joint may be pulled apart or otherwise prone to failure. 3

Another area of application for Poisson's effect is in the realm of structural geology. Rocks, just as most materials, are subject to Poisson's effect while under stress and strain. 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.4

See also

References

  1. ^ Poisson's ratio calculation of glasses
  2. ^ Negative Poisson's ratio
  3. ^ http://www.cpchem.com/hb/getdocanon.asp?doc=135&lib=CPC-Portal
  4. ^ http://www.geosc.psu.edu/~engelder/geosc465/lect18.rtf

External links


v • d • e
Elastic moduli for homogeneous isotropic materials
Bulk modulus (K) • Young's modulus (E) • Lamé's first parameter (λ) • Shear modulus (μ) • Poisson's ratio (ν) • P-wave modulus (M)
Conversion formulas
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these, thus given any two, any other of the elastic moduli can be calculated according to these formulas.
(\lambda,\,\mu) (E,\,\mu) (K,\,\lambda) (K,\,\mu) (\lambda,\,\nu) (\mu,\,\nu) (E,\,\nu) (K,\, \nu) (K,\,E) (M,\,\mu)
K=\, \lambda+ \frac{2\mu}{3} \frac{E\mu}{3(3\mu-E)} \lambda\frac{1+\nu}{3\nu} \frac{2\mu(1+\nu)}{3(1-2\nu)} \frac{E}{3(1-2\nu)} M - \frac{4\mu}{3}
E=\, \mu\frac{3\lambda + 2\mu}{\lambda + \mu} 9K\frac{K-\lambda}{3K-\lambda} \frac{9K\mu}{3K+\mu} \frac{\lambda(1+\nu)(1-2\nu)}{\nu} 2\mu(1+\nu)\, 3K(1-2\nu)\, \mu\frac{3M-4\mu}{M-\mu}
\lambda=\, \mu\frac{E-2\mu}{3\mu-E} K-\frac{2\mu}{3} \frac{2 \mu \nu}{1-2\nu} \frac{E\nu}{(1+\nu)(1-2\nu)} \frac{3K\nu}{1+\nu} \frac{3K(3K-E)}{9K-E} M - 2\mu\,
\mu=\, 3\frac{K-\lambda}{2} \lambda\frac{1-2\nu}{2\nu} \frac{E}{2+2\nu} 3K\frac{1-2\nu}{2+2\nu} \frac{3KE}{9K-E}
\nu=\, \frac{\lambda}{2(\lambda + \mu)} \frac{E}{2\mu}-1 \frac{\lambda}{3K-\lambda} \frac{3K-2\mu}{2(3K+\mu)} \frac{3K-E}{6K} \frac{M - 2\mu}{2M - 2\mu}
M=\, \lambda+2\mu\, \mu\frac{4\mu-E}{3\mu-E} 3K-2\lambda\, K+\frac{4\mu}{3} \lambda \frac{1-\nu}{\nu} \mu\frac{2-2\nu}{1-2\nu} E\frac{1-\nu}{(1+\nu)(1-2\nu)} 3K\frac{1-\nu}{1+\nu} 3K\frac{3K+E}{9K-E}

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  • This page was last modified on 26 November 2008, at 18:22.

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