Poisson's ratio 1

Poisson's ratio

Figure 1: A cube with sides of length L of an isotropic linearly elastic materialsubject to tension along the x axis, with a Poisson's ratio of 0.5. The green cube is

unstressed, the red is expanded in the x direction by due to tension, andcontracted in the y and z directions by .

Poisson's ratio ( ), named after SiméonPoisson, is the ratio, when a sample object isstretched, of the contraction or transversestrain (perpendicular to the applied load), tothe extension or axial strain (in the directionof the applied load).

When a material is compressed in onedirection, it usually tends to expand in theother two directions perpendicular to thedirection of compression. This phenomenonis called the Poisson effect. Poisson's ratio

(nu) is a measure of the Poisson effect.The Poisson ratio is the ratio of the fraction(or percent) of expansion divided by thefraction (or percent) of compression, forsmall values of these changes.

Conversely, if the material is stretchedrather than compressed, it usually tends tocontract in the directions transverse to thedirection of stretching. Again, the Poissonratio will be the ratio of relative contraction to relative stretching, and will have the same value as above. In certainrare cases, a material will actually shrink in the transverse direction when compressed (or expand when stretched)which will yield a negative value of the Poisson ratio.The Poisson's ratio of a stable, isotropic, linear elastic material cannot be less than −1.0 nor greater than 0.5 due tothe requirement that Young's modulus, the shear modulus and bulk modulus have positive values.[1] Most materialshave Poisson's ratio values ranging between 0.0 and 0.5. A perfectly incompressible material deformed elastically atsmall strains would have a Poisson's ratio of exactly 0.5. Most steels and rigid polymers when used within theirdesign limits (before yield) exhibit values of about 0.3, increasing to 0.5 for post-yield deformation (which occurslargely at constant volume.) Rubber has a Poisson ratio of nearly 0.5. Cork's Poisson ratio is close to 0: showing verylittle lateral expansion when compressed. Some materials, mostly polymer foams, have a negative Poisson's ratio; ifthese auxetic materials are stretched in one direction, they become thicker in perpendicular directions. Anisotropicmaterials can have Poisson ratios above 0.5 in some directions.

Assuming that the material is stretched or compressed along the axial direction (the x axis in the diagram):

whereis the resulting Poisson's ratio,

is transverse strain (negative for axial tension (stretching), positive for axial compression)is axial strain (positive for axial tension, negative for axial compression).

Poisson's ratio 2

Cause of Poisson’s effectOn the molecular level, Poisson’s effect is caused by slight movements between molecules and the stretching ofmolecular bonds within the material lattice to accommodate the stress. When the bonds elongate in the direction ofload , they shorten in the other directions. This behavior multiplied millions of times throughout the material latticeis what drives the phenomenon.

Length changeFor a cube stretched in the x-direction (see figure 1) with a length increase of in the x direction, and a lengthdecrease of in the y and z directions, the infinitesimal diagonal strains are given by:

Integrating the definition of Poisson's ratio:

Solving and exponentiating, the relationship between and is found to be:

For very small values of and , the first-order approximation yields:

Volumetric changeThe relative change of volume ΔV/V due to the stretch of the material can now be calculated. Using and

:

Using the above derived relationship between and :

and for very small values of and , the first-order approximation yields:

Poisson's ratio 3

Width change

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

If a rod with diameter (or width, orthickness) d and length L is subject totension so that its length will changeby ΔL then its diameter d will changeby:

The above formula is true only in the case of small deformations; if deformations are large then the following (moreprecise) formula can be used:

whereis original diameter

is rod diameter changeis Poisson's ratiois original length, before stretch

is the change of length.The value is negative because it decrease with increase of length

Poisson's ratio 4

Isotropic materialsFor a linear isotropic material subjected only to compressive (i.e. normal) forces, the deformation of a material in thedirection of one axis will produce a deformation of the material along the other axis in three dimensions. Thus it ispossible to generalize Hooke's Law (for compressive forces) into three dimensions:

or

where, and are strain in the direction of , and axis, and are stress in the direction of , and axisis Young's modulus (the same in all directions: , and for isotropic materials)

is Poisson's ratio (the same in all directions: , and for isotropic materials)These equations will hold in the general case which includes shear forces as well as compressive forces, and the fullgeneralization of Hooke's law is given by:

where is the Kronecker delta and

Orthotropic materialsFor orthotropic materials such as wood, Hooke's law can be expressed in matrix form as[2]

whereis the Young's modulus along axis

is the shear modulus in direction on the plane whose normal is in direction is the Poisson's ratio that corresponds to a contraction in direction when an extension is applied in

direction .The Poisson's ratio of an orthotropic material is different in each direction (x, y and z). However, the symmetry ofthe stress and strain tensors implies that not all the six Poisson's ratios in the equation are independent. There areonly nine independent material properties; three elastic moduli, three shear moduli, and three Poisson's ratios. Theremaining three Poisson's ratios can be obtained from the relations

Poisson's ratio 5

From the above relations we can see that if then . The larger Poisson's ratio (in this case ) is called the major Poisson's ratio while the smaller one (in this case ) is called the minor Poisson's ratio.We can find similar relations between the other Poisson's ratios.The above stress-strain relation is also often written in the equivalent (transposed) form[3]

Transversely isotropic materialsTransversely isotropic materials have a plane of symmetry in which the elastic properties are isotropic. If we assumethat this plane of symmetry is , then Hookes's law takes the form[4]

where we have used the plane of symmetry to reduce the number of constants, i.e.,.

The symmetry of the stress and strain tensors implies that

This leaves us with six independent constants . However, transverse isotropy givesrise to a further constraint bewtween and which is

Therefore, there are five independent elastic material properties two of which are Poisson's ratios. For the assumedplane of symmetry, the larger of and is the major Poisson's ratio. The other major and minor Poisson'sratios are equal.

Poisson's ratio 6

Poisson's ratio values for different materials

Influences of selected glass component additions on Poisson's ratio of a specific baseglass.[5]

material poisson's ratio

rubber ~ 0.50

gold 0.42

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 0.10–0.40

cork ~ 0.00

auxetics negative

Poisson's ratio 7

material plane of symmetry

Nomex honeycomb core , =ribbon direction 0.49 0.69 0.01 2.75 3.88 0.01

glass fiber-epoxy resin 0.29 0.29 0.32 0.06 0.06 0.32

Negative Poisson's ratio materialsSome materials known as auxetic materials display a negative Poisson’s ratio. When subjected to positive strain in alongitudinal axis, the transverse strain in the material will actually be positive (i.e. it would increase the crosssectional area). For these materials, it is usually due to uniquely oriented, hinged molecular bonds. In order for thesebonds to stretch in the longitudinal direction, the hinges must ‘open’ in the transverse direction, effectively exhibitinga positive strain.[6]

Applications of Poisson's effectOne area in which Poisson's effect has a considerable influence is in pressurized pipe flow. When the air or liquidinside a pipe is highly pressurized it exerts a uniform force on the inside of the pipe, resulting in a radial stress withinthe pipe material. Due to Poisson's effect, this radial stress will cause the pipe to slightly increase in diameter anddecrease in length. The decrease in length, in particular, can have a noticeable effect upon the pipe joints, as theeffect will accumulate for each section of pipe joined in series. A restrained joint may be pulled apart or otherwiseprone to failure.[7]

Another area of application for Poisson's effect is in the realm of structural geology. Rocks, like most materials, aresubject to Poisson's effect while under stress. In a geological timescale, excessive erosion or sedimentation of Earth'scrust can either create or remove large vertical stresses upon the underlying rock. This rock will expand or contractin the vertical direction as a direct result of the applied stress, and it will also deform in the horizontal direction as aresult of Poisson's effect. This change in strain in the horizontal direction can affect or form joints and dormantstresses in the rock.[8]

The use of cork as a stopper for wine bottles is the result of the fact that cork has a Poisson ratio of practically zero.This means that, as the cork is inserted into the bottle, the upper part which is not yet inserted will not expand as thelower part is compressed. The force needed to insert a cork into a bottle arises only from the compression of the corkand the friction between the cork and the bottle. If the stopper were made of rubber, for example, (with a Poissonratio of about 1/2), there would be a relatively large additional force required to overcome the expansion of the upperpart of the rubber stopper.

References[1] H. GERCEK; “Poisson's ratio values for rocks”; International Journal of Rock Mechanics and Mining Sciences; Elsevier; January 2007; 44

(1): pp. 1–13.[2] Boresi, A. P, Schmidt, R. J. and Sidebottom, O. M., 1993, Advanced Mechanics of Materials, Wiley.[3] Lekhnitskii, SG., (1963), Theory of elasticity of an anisotropic elastic body, Holden-Day Inc.[4] Tan, S. C., 1994, Stress Concentrations in Laminated Composites, Technomic Publishing Company, Lancaster, PA.[5] Poisson's ratio calculation of glasses (http:/ / www. glassproperties. com/ poisson_ratio/ )[6] Negative Poisson's ratio (http:/ / silver. neep. wisc. edu/ ~lakes/ Poisson. html)[7] http:/ / www. cpchem. com/ hb/ getdocanon. asp?doc=135& lib=CPC-Portal[8] http:/ / www. geosc. psu. edu/ ~engelder/ geosc465/ lect18. rtf

Poisson's ratio 8

External links• Meaning of Poisson's ratio (http:/ / silver. neep. wisc. edu/ ~lakes/ PoissonIntro. html)• Negative Poisson's ratio materials (http:/ / silver. neep. wisc. edu/ ~lakes/ Poisson. html)• More on negative Poisson's ratio materials (auxetic) (http:/ / home. um. edu. mt/ auxetic)

Conversion formulas

Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these, thus given anytwo, any other of the elastic moduli can be calculated according to these formulas.

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