Poisson's ratio, denoted by the Greek letter 'nu', , and named after Siméon Poisson, is the negative of the ratio of (signed) transverse strain to (signed) axial strain. For small values of these changes, is the amount of transversal expansion divided by the amount of axial compression.
Poisson's ratio is a measure of the Poisson effect, the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression. Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching. It is a common observation when a rubber band is stretched, it becomes noticeably thinner. Again, the Poisson ratio will be the ratio of relative contraction to relative expansion and will have the same value as above. In certain rare 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 must be between −1.0 and +0.5 because of the requirement for Young's modulus, the shear modulus and bulk modulus to have positive values.^{[1]} Most materials have Poisson's ratio values ranging between 0.0 and 0.5. A perfectly incompressible material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5. Most steels and rigid polymers when used within their design limits (before yield) exhibit values of about 0.3, increasing to 0.5 for post-yield deformation which occurs largely at constant volume.^{[2]} Rubber has a Poisson ratio of nearly 0.5. Cork's Poisson ratio is close to 0, showing very little lateral expansion when compressed. Some materials, e.g. some polymer foams, origami folds,^{[3]}^{[4]} and certain cells can exhibit negative Poisson's ratio, and are referred to as auxetic materials. If these auxetic materials are stretched in one direction, they become thicker in the perpendicular direction. In contrast, some anisotropic materials, such as carbon nanotubes, zigzag-based folded sheet materials,^{[5]}^{[6]} and honeycomb auxetic metamaterials^{[7]} to name a few, can exhibit one or more Poisson's ratios above 0.5 in certain directions.
Assuming that the material is stretched or compressed along the axial direction (the x axis in the diagram below):
where
For a cube stretched in the x-direction (see Figure 1) with a length increase of in the x direction, and a length decrease of in the y and z directions, the infinitesimal diagonal strains are given by
If Poisson's ratio is constant through deformation, integrating these expressions and using the definition of Poisson's ratio gives
Solving and exponentiating, the relationship between and is then
For very small values of and , the first-order approximation yields:
The relative change of volume ΔV/V of a cube 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:
For isotropic materials we can use Lamé’s relation^{[8]}
where is bulk modulus and is elastic modulus (or Young's modulus).
Note that isotropic materials must have a Poisson's ratio of . Typical isotropic engineering materials have a Poisson's ratio of .^{[9]}
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 above formula is true only in the case of small deformations; if deformations are large then the following (more precise) formula can be used:
where
The value is negative because it decreases with increase of length
For a linear isotropic material subjected only to compressive (i.e. normal) forces, the deformation of a material in the direction of one axis will produce a deformation of the material along the other axis in three dimensions. Thus it is possible to generalize Hooke's Law (for compressive forces) into three dimensions:
where:
these equations can be all synthesized in the following:
In the most general case, also shear stresses will hold as well as normal stresses, and the full generalization of Hooke's law is given by:
where is the Kronecker delta. The Einstein sum convention is usually adopted:
In this case the equation is simply written:
For orthotropic materials such as wood, Hooke's law can be expressed in matrix form as^{[10]}^{[11]}
where
The Poisson's ratio of an orthotropic material is different in each direction (x, y and z). However, the symmetry of the stress and strain tensors implies that not all the six Poisson's ratios in the equation are independent. There are only nine independent material properties: three elastic moduli, three shear moduli, and three Poisson's ratios. The remaining three Poisson's ratios can be obtained from the relations
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.
Transversely isotropic materials have a plane of isotropy in which the elastic properties are isotropic. If we assume that this plane of isotropy is , then Hooke's law takes the form^{[12]}
where we have used the plane of isotropy to reduce the number of constants, i.e., .
The symmetry of the stress and strain tensors implies that
This leaves us with seven independent constants . However, transverse isotropy gives rise to a further constraint between and which is
Therefore, there are six independent elastic material properties three of which are Poisson's ratios. For the assumed plane of symmetry, the larger of and is the major Poisson's ratio. The other major and minor Poisson's ratios are equal.
Material | Poisson's ratio |
---|---|
rubber | 0.4999^{[9]} |
gold | 0.42–0.44 |
saturated clay | 0.40–0.49 |
magnesium | 0.252–0.289 |
titanium | 0.265-0.34 |
copper | 0.33 |
aluminium-alloy | 0.32 |
clay | 0.30–0.45 |
stainless steel | 0.30–0.31 |
steel | 0.27–0.30 |
cast iron | 0.21–0.26 |
sand | 0.20–0.455 |
concrete | 0.1–0.2 |
glass | 0.18–0.3 |
metallic glasses | 0.276–0.409^{[14]} |
foam | 0.10–0.50 |
cork | 0.0 |
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 | 0.32 | 0.06 | 0.06 | 0.32 |
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.^{[15]} This can also be done in a structured way and lead to new aspects in material design as for mechanical metamaterials.
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 hoop stress within the pipe material. Due to Poisson's effect, this hoop stress will cause the pipe to increase in diameter and slightly 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.
Another area of application for Poisson's effect is in the realm of structural geology. Rocks, like most materials, are subject to Poisson's effect while under stress. 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.^{[16]}
Although cork was historically chosen to seal wine bottle for other reasons (including its inert nature, impermeability, flexibility, sealing ability, and resilience),^{[17]} cork's poisson's ratio of zero provides another advantage. As the cork is inserted into the bottle, the upper part which is not yet inserted does not expand in diameter as it is compressed axially. The force needed to insert a cork into a bottle arises only from the friction between the cork and the bottle due to the radial compression of the cork. If the stopper were made of rubber, for example, (with a Poisson ratio of about 1/2), there would be a relatively large additional force required to overcome the radial expansion of the upper part of the rubber stopper.
Most car mechanics are aware that it is hard to pull a rubber hose (e.g. a coolant hose) off a metal pipe stub, as the tension of pulling causes the diameter of the hose to shrink, gripping the stub tightly. Hoses can more easily be pushed off stubs instead using a wide flat blade.
Conversion formulae | |||||||
---|---|---|---|---|---|---|---|
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. | |||||||
Notes | |||||||
There are two valid solutions. | |||||||
Cannot be used when | |||||||
Two-dimensional silica (2D silica) is a layered polymorph of silicon dioxide. Two varieties of 2D silica, both of hexagonal crystal symmetry, have been grown so far on various metal substrates. One is based on SiO4 tetrahedra, which are covalently bonded to the substrate. The second comprises graphene-like fully saturated sheets, which interact with the substrate via weak van der Waals bonds. One sheet of the second 2D silica variety is also called hexagonal bilayer silica (HBS); it can have either ordered or disordered (amorphous) structure.2D silica has potential applications in electronics as the thinnest gate dielectric. It can also be used for isolation of graphene sheets from the substrate. 2D silica is a wide band gap semiconductor, whose band gap and geometry can be engineered by external electric field. Remarkably, it was shown to be a member of the auxetics materials family with a negative Poisson’s ratio.
5052 aluminium alloy5052 is an aluminium alloy, primarily alloyed with magnesium and chromium.
A36 steelA36 steel is a common structural steel in the United States. The A36 standard was established by the ASTM International.
AuxeticsAuxetics are structures or materials that have a negative Poisson's ratio. When stretched, they become thicker perpendicular to the applied force. This occurs due to their particular internal structure and the way this deforms when the sample is uniaxially loaded. Auxetics can be single molecules, crystals, or a particular structure of macroscopic matter.
Such materials and structures are expected to have mechanical properties such as high energy absorption and fracture resistance. Auxetics may be useful in applications such as body armor, packing material, knee and elbow pads, robust shock absorbing material, and sponge mops.
The term auxetic derives from the Greek word αὐξητικός (auxetikos) which means "that which tends to increase" and has its root in the word αὔξησις, or auxesis, meaning "increase" (noun). This terminology was coined by Professor Ken Evans of the University of Exeter.
One of the first artificially produced auxetic materials, the RFS structure (diamond-fold structure) , was invented in 1978 by the Berlin researcher K. Pietsch. Although he did not use the term auxetics, he describes for the first time the underlying lever mechanism and its non-linear mechanical reaction is therefore considered the inventor of the auxetic net.
The earliest published example of a material with negative Poisson's constant is due to A. G. Kolpakov in 1985, "Determination of the average characteristics of elastic frameworks"; the next synthetic auxetic material was described in Science in 1987, entitled "Foam structures with a Negative Poisson's Ratio" by R.S. Lakes from the University of Wisconsin Madison. The use of the word auxetic to refer to this property probably began in 1991.Designs of composites with inverted hexagonal periodicity cell (auxetic hexagon), possessing negative Poisson ratios, were published in 1985.Typically, auxetic materials have low density, which is what allows the hinge-like areas of the auxetic microstructures to flex.At the macroscale, auxetic behaviour can be illustrated with an inelastic string wound around an elastic cord. When the ends of the structure are pulled apart, the inelastic string straightens while the elastic cord stretches and winds around it, increasing the structure's effective volume. Auxetic behaviour at the macroscale can also be employed for the development of products with enhanced characteristics such as footwear based on the auxetic rotating triangles structures developed by Grima and Evans.Examples of auxetic materials include:
Auxetic polyurethane foam
α-Cristobalite.
Certain rocks and minerals
Graphene, which can be made auxetic through the introduction of vacancy defects
Living bone tissue (although this is only suspected)
Tendons within their normal range of motion.
Specific variants of polytetrafluorethylene polymers such as Gore-Tex
Paper, all types. If a paper is stretched in an in-plane direction it will expand in its thickness direction due to its network structure.
Several types of origami folds like the Diamond-Folding-Structure (RFS), the herringbone-fold-structure (FFS) or the miura fold, and other periodic patterns derived from it.
Tailored structures designed to exhibit special designed Poisson's ratios.
Chain organic molecules. Recent researches revealed that organic crystals like n-paraffins and similar to them may demonstrate an auxetic behavior.
Processed needle-punched nonwoven fabrics. Due to the network structure of such fabrics, a processing protocol using heat and pressure can convert ordinary (not auxetic) nonwovens into auxetic ones.
Cork has an almost zero Poisson's ratio. This makes it a good material for sealing wine bottles.
Elastic modulusAn elastic modulus (also known as modulus of elasticity) is a quantity that measures an object or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is defined as the slope of its stress–strain curve in the elastic deformation region: A stiffer material will have a higher elastic modulus. An elastic modulus has the form:
where stress is the force causing the deformation divided by the area to which the force is applied and strain is the ratio of the change in some parameter caused by the deformation to the original value of the parameter. If stress is measured in pascals, then since strain is a dimensionless quantity, the units of λ will be pascals as well.
Specifying how stress and strain are to be measured, including directions, allows for many types of elastic moduli to be defined. The three primary ones are:
Three other elastic moduli are Poisson's ratio, Lamé's first parameter, and P-wave modulus.
Homogeneous and isotropic (similar in all directions) materials (solids) have their (linear) elastic properties fully described by two elastic moduli, and one may choose any pair. Given a pair of elastic moduli, all other elastic moduli can be calculated according to formulas in the table below at the end of page.
Inviscid fluids are special in that they cannot support shear stress, meaning that the shear modulus is always zero. This also implies that Young's modulus for this group is always zero.
In some English texts the here described quantity is called elastic constant, while the inverse quantity is referred to as elastic modulus.
Finite element method in structural mechanicsThe finite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. In the FEM, the structural system is modeled by a set of appropriate finite elements interconnected at discrete points called nodes. Elements may have physical properties such as thickness, coefficient of thermal expansion, density, Young's modulus, shear modulus and Poisson's ratio.
Flexural rigidityFlexural rigidity is defined as the force couple required to bend a non-rigid structure in one unit of curvature or it can be defined as the resistance offered by a structure while undergoing bending.
Gauge factorGauge factor (GF) or strain factor of a strain gauge is the ratio of relative change in electrical resistance R, to the mechanical strain ε. The gauge factor is defined as :
Where
GeomechanicsGeomechanics (from the Greek prefix geo- meaning "earth"; and "mechanics") involves the geologic study of the behavior of soil and rock.
Impulse excitation techniqueThe impulse excitation technique (IET) is a non-destructive material characterization technique to determine the elastic properties and internal friction of a material of interest. It measures the resonant frequencies in order to calculate the Young's modulus, shear modulus, Poisson's ratio and internal friction of predefined shapes like rectangular bars, cylindrical rods and disc shaped samples. The measurements can be performed at room temperature or at elevated temperatures (up to 1700 °C) under different atmospheres.The measurement principle is based on tapping the sample with a small projectile and recording the induced vibration signal with a piezoelectric sensor, microphone, laser vibrometer or accelerometer. To optimize the results a microphone or a laser vibrometer can be used as there is no contact between the test-piece and the sensor. Laser vibrometers are preferred to measure signals in vacuum. Afterwards, the acquired vibration signal in the time domain is converted to the frequency domain by a fast Fourier transformation. Dedicated software will determine the resonant frequency with high accuracy to calculate the elastic properties based on the classical beam theory.
Lateral strainIn continuum mechanics, lateral strain, also known as transverse strain, is defined as the ratio of the change in diameter of a circular bar of a material to its diameter due to deformation in the longitudinal direction. It occurs when under the action of a longitudinal stress, a body will extend in the direction of the stress and contract in the transverse or lateral direction (in the case of tensile stress). When put under compression, the body will contract in the direction of the stress and extend in the transverse or lateral direction. It is a dimensionless quantity, as it is a ratio between two quantities of the same dimension.
MacorMacor is the trademark for a machineable glass-ceramic developed and sold by Corning Inc. It is a white material that looks somewhat like porcelain. Macor is a good thermal insulator and is stable up to temperatures of 1000 °C, with very little thermal expansion or outgassing. It can be machined using standard metalworking tools.
Mechanical metamaterialMechanical metamaterials are artificial structures with mechanical properties defined by their structure rather than their composition. They can be seen as a counterpart to the rather well-known family of optical metamaterials and include acoustic metamaterials as a special case of vanishing shear. Their mechanical properties can be designed to have values which cannot be found in nature.
Penta-graphenePenta-graphene is a carbon allotrope composed entirely of carbon pentagons and resembling the Cairo pentagonal tiling. Penta-graphene was proposed in 2014 on the basis of analyses and simulations. Further calculations showed that it is unstable in its pure form, but can be stabilized by hydrogenation. Owing to its atomic configuration, penta-graphene has an unusually negative Poisson’s ratio and very high ideal strength believed to exceed that of a similar material, graphene.Penta-graphene contains both sp2 and sp3 hybridized carbon atoms. Contrary to graphene, which is a good conductor of electricity, penta-graphene is an insulator with an indirect band gap of 4.1–4.3 eV. Its hydrogenated form is called penta-graphane. It has a diamond-like structure with sp3 and no sp2 bonds, and therefore a wider band gap (ca. 5.8 eV) than penta-graphene. Chiral penta-graphene nanotubes have also been studied as metastable allotropes of carbon.
Quadratic quadrilateral elementThe quadratic quadrilateral element, also known as the Q8 element is a type of element used in finite element analysis which is used to approximate in a 2D domain the exact solution to a given differential equation. It is a two-dimensional finite element with both local and global coordinates.This element can be used for plane stress or plane strain problems in elasticity. The quadratic quadrilateral element has modulus of elasticity E, Poisson’s ratio v, and thickness t.
Speeds of sound of the elementsThe speed of sound in any chemical element in the fluid phase has one temperature-dependent value. In the solid phase, different types of sound wave may be propagated, each with its own speed: among these types of wave are longitudinal (as in fluids), transversal, and (along a surface or plate) extensional.
Strength of materialsStrength of materials, also called mechanics of materials, is a subject which deals with the behavior of solid objects subject to stresses and strains. The complete theory began with the consideration of the behavior of one and two dimensional members of structures, whose states of stress can be approximated as two dimensional, and was then generalized to three dimensions to develop a more complete theory of the elastic and plastic behavior of materials. An important founding pioneer in mechanics of materials was Stephen Timoshenko.
The study of strength of materials often refers to various methods of calculating the stresses and strains in structural members, such as beams, columns, and shafts. The methods employed to predict the response of a structure under loading and its susceptibility to various failure modes takes into account the properties of the materials such as its yield strength, ultimate strength, Young's modulus, and Poisson's ratio; in addition the mechanical element's macroscopic properties (geometric properties), such as its length, width, thickness, boundary constraints and abrupt changes in geometry such as holes are considered.
Tensile testingTensile testing, also known as tension testing, is a fundamental materials science and engineering test in which a sample is subjected to a controlled tension until failure. Properties that are directly measured via a tensile test are ultimate tensile strength, breaking strength, maximum elongation and reduction in area. From these measurements the following properties can also be determined: Young's modulus, Poisson's ratio, yield strength, and strain-hardening characteristics. Uniaxial tensile testing is the most commonly used for obtaining the mechanical characteristics of isotropic materials. Some materials use biaxial tensile testing.
Young's modulusYoung's modulus or Young modulus is a mechanical property that measures the stiffness of a solid material. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a material in the linear elasticity regime of a uniaxial deformation.
Young's modulus is named after the 19th-century British scientist Thomas Young. However, the concept was developed in 1727 by Leonhard Euler, and the first experiments that used the concept of Young's modulus in its current form were performed by the Italian scientist Giordano Riccati in 1782, pre-dating Young's work by 25 years. The term modulus is the diminutive of the Latin term modus which means measure.
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