Poisson\'s ratio - Citizendia

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). Siméon-Denis Poisson (21 June 1781 &ndash 25 April 1840 was a French Mathematician, Geometer, and Physicist.

When a sample of material is stretched in one direction, it tends to contract (or rarely, expand) in the other two directions. Materials are physical Substances used as inputs to production or Manufacturing. Conversely, when a sample of material is compressed in one direction, it tends to expand (or rarely, contract) in the other two directions. Materials are physical Substances used as inputs to production or Manufacturing. 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. In Materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of Shear Most materials have between 0. 0 and 0. 5. Cork is close to 0. 0, most steels are around 0. 3, and rubber is almost 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. Auxetics are materials which when stretched become thicker perpendicularly to the applied force

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).

1 Cause of Poisson’s effect
2 Generalized Hooke's law
3 Volumetric change
4 Width change
5 Orthotropic materials
6 Poisson's ratio values for different materials
7 Applications of Poisson's effect
8 See also
9 References
10 External links

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. Stress is a measure of the average amount of Force exerted per unit Area. 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

σ x

σ y

and

σ z

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

Δ V

is material volume change

L

is original length, before stretch

Δ L

is the change of length:

Δ L = L old - L new

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

Δ d

is rod diameter change

ν

is Poisson's ratio

L

is original length, before stretch

Δ L

is the change of length. In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the Stress is a measure of the average amount of Force exerted per unit Area. In Solid mechanics, Young's modulus (E is a measure of the Stiffness of an isotropic elastic material

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

E i

is a Young's modulus along axis i

ν j k

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. An orthotropic Material has two or three mutually orthogonal two-fold axes of rotational Symmetry so that its mechanical properties are in general different along In Solid mechanics, Young's modulus (E is a measure of the Stiffness of an isotropic elastic material Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many ^[1]

material	poisson's ratio
rubber	~ 0. 50
saturated clay	0. 40-0. 50
magnesium	0. Magnesium (mægˈniːziəm is a Chemical element with the symbol Mg, Atomic number 12 Atomic weight 24 35
titanium	0. Titanium (taɪˈteɪniəm is a Chemical element with the symbol Ti and Atomic number 22 34
copper	0. Copper (ˈkɒpɚ is a Chemical element with the symbol Cu (cuprum and Atomic number 29 33
aluminium-alloy	0. WikipediaNaming An alloy is a Solid solution or Homogeneous mixture of two or more elements, at least one of which is a Metal, which itself has 33
clay	0. Clay is a naturally occurring material composed primarily of fine-grained Minerals which show plasticity through a variable range of Water content, and 30-0. 45
stainless steel	0. In Metallurgy, stainless steel is defined as a Steel Alloy with a minimum of 11 30-0. 31
steel	0. Steel is an Alloy consisting mostly of Iron, with a Carbon content between 0 27-0. 30
cast iron	0. Cast iron usually refers to grey cast iron, but identifies a large group of Ferrous Alloys which solidify with a Eutectic. 21-0. 26
sand	0. Sand is a naturally occurring Granular material composed of finely divided rock and Mineral particles 20-0. 45
concrete	0. Concrete is a construction material composed of Cement (commonly Portland cement) as well as other cementitious materials such as Fly ash and Slag 20
glass	0. Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many 18-0. 3
foam	0. The most general definition of foam is a substance that is formed by trapping many gas Bubbles in a Liquid or Solid. 10 to 0. 40
cork	~ 0. Cork material is a Prime-subset of generic cork tissue, harvested for commercial use primarily from the Cork Oak tree Quercus 00
auxetics	negative

Applications of Poisson's effect

One area in which Poisson's effect has a considerable influence is in pressurized pipe flow. Auxetics are materials which when stretched become thicker perpendicularly to the applied force 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. ^[2]

Another area of application for Poisson's effect is in the realm of structural geology. Structural geology is the study of the three dimensional distribution of rock bodies and their planar or folded surfaces and their internal fabrics 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. ^[3]

References

External links

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. Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the Stress is a measure of the average amount of Force exerted per unit Area. Practical The impulse excitation technique is a Nondestructive test method that uses Natural frequency, dimensions and mass of a test-piece to determine Young's An orthotropic Material has two or three mutually orthogonal two-fold axes of rotational Symmetry so that its mechanical properties are in general different along When the Temperature of a substance changes the energy that is stored in the Intermolecular bonds between atoms changes
	$(\lambda,\,\mu)$	$(E,\,\mu)$	$(K,\,\lambda)$	$(K,\,\mu)$	$(\lambda,\,\nu)$	$(\mu,\,\nu)$	$(E,\,\nu)$	$(K,\, \nu)$	$(K,\,E)$	$(M,\,\mu)$
$K=\,$	$\lambda+ \frac{2\mu}{3}$	$\frac{E\mu}{3(3\mu-E)}$			$\lambda\frac{1+\nu}{3\nu}$	$\frac{2\mu(1+\nu)}{3(1-2\nu)}$	$\frac{E}{3(1-2\nu)}$			$M - \frac{8\mu}{3}$
$E=\,$	$\mu\frac{3\lambda + 2\mu}{\lambda + \mu}$		$9K\frac{K-\lambda}{3K-\lambda}$	$\frac{9K\mu}{3K+\mu}$	$\frac{\lambda(1+\nu)(1-2\nu)}{\nu}$	$2\mu(1+\nu)\,$		$3K(1-2\nu)\,$		$\mu\frac{3M-4\mu}{M-\mu}$
$\lambda=\,$		$\mu\frac{E-2\mu}{3\mu-E}$		$K-\frac{2\mu}{3}$		$\frac{2 \mu \nu}{1-2\nu}$	$\frac{E\nu}{(1+\nu)(1-2\nu)}$	$\frac{3K\nu}{1+\nu}$	$\frac{3K(3K-E)}{9K-E}$	$M - 2\mu\,$
$\mu=\,$			$3\frac{K-\lambda}{2}$		$\lambda\frac{1-2\nu}{2\nu}$		$\frac{E}{2+2\nu}$	$3K\frac{1-2\nu}{2+2\nu}$	$\frac{3KE}{9K-E}$
$\nu=\,$	$\frac{\lambda}{2(\lambda + \mu)}$	$\frac{E}{2\mu}-1$	$\frac{\lambda}{3K-\lambda}$	$\frac{3K-2\mu}{2(3K+\mu)}$					$\frac{3K-E}{6K}$	$\frac{M - 2\mu}{2M - 3\mu}$
$M=\,$	$\lambda+2\mu\,$	$\mu\frac{4\mu-E}{3\mu-E}$	$3K-2\lambda\,$	$K+\frac{4\mu}{3}$	$\lambda \frac{1-\nu}{\nu}$	$\mu\frac{2-2\nu}{1-2\nu}$	$E\frac{1-\nu}{(1+\nu)(1-2\nu)}$	$3K\frac{1-\nu}{1+\nu}$	$3K\frac{3K+E}{9K-E}$

Dictionary

Poisson's ratio

-noun

Of a material in tension or compression, the ratio of the strain in the direction of the applied load to the strain normal to the load. Abbreviated ν.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents

Cause of Poisson’s effect

Generalized Hooke's law

Volumetric change

Width change

Orthotropic materials

Poisson's ratio values for different materials

Applications of Poisson's effect

See also

References

External links

Dictionary

Poisson's ratio

-noun