In
materials science and
solid mechanics
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ot ...
, Poisson's ratio
(
nu) is a measure of the Poisson effect, the
deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
(expansion or contraction) of a material in directions perpendicular to the specific direction of
loading. The value of Poisson's ratio is the negative of the ratio of
transverse strain to axial
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
. For small values of these changes,
is the amount of transversal
elongation divided by the amount of axial
compression
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
* Gas compression
*Compression ratio, of a ...
. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. For soft materials, such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2–0.3. The ratio is named after the French mathematician and physicist
Siméon Poisson.
Origin
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,
[Lakes, R. and Wojciechowski, K.W., 2008. Negative compressibility, negative Poisson's ratio, and stability. Physica Status Solidi B, 245(3), pp.545-551.] 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
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
, linear
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
material must be between −1.0 and +0.5 because of the requirement for
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
, the
shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stackre ...
and
bulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume.
Other moduli describ ...
to have positive values. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. A perfectly incompressible isotropic 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. Rubber has a Poisson ratio of nearly 0.5. Cork's Poisson ratio is close to 0, showing very little lateral expansion when compressed and glass is between 0.18 and 0.30. Some materials, e.g. some polymer foams, origami folds, 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
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
materials, such as
carbon nanotube
A scanning tunneling microscopy image of a single-walled carbon nanotube
Rotating single-walled zigzag carbon nanotube
A carbon nanotube (CNT) is a tube made of carbon with diameters typically measured in nanometers.
''Single-wall carbon na ...
s, zigzag-based folded sheet materials,
and honeycomb auxetic metamaterials 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 in only one direction (the ''x'' axis in the diagram below):
:
where
*
is the resulting Poisson's ratio,
*
is transverse strain
*
is axial strain
and positive strain indicates extension and negative strain indicates contraction.
Poisson's ratio from geometry changes
Length change
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:
:
Volumetric change
The relative change of volume 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
:
where
is
bulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume.
Other moduli describ ...
and
is
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
.
Width change
If a rod with diameter (or width, or thickness) and length is subject to tension so that its length will change by then its diameter 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
*
is original diameter
*
is rod diameter change
*
is Poisson's ratio
*
is original length, before stretch
*
is the change of length.
The value is negative because it decreases with increase of length
Characteristic materials
Isotropic
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
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
(for compressive forces) into three dimensions:
:
:
:
where:
*
,
, and
are
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
in the direction of
,
and
axis
*
,
, and
are
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
in the direction of
,
and
axis
*
is
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
(the same in all directions:
,
, and
for isotropic materials)
*
is Poisson's ratio (the same in all directions:
,
and
for isotropic materials)
these equations can be all synthesized in the following:
:
In the most general case, also
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
es will hold as well as normal stresses, and the full generalization of Hooke's law is given by:
:
where
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
. The
Einstein notation is usually adopted:
:
to write the equation simply as:
:
Anisotropic
For anisotropic materials, the Poisson ratio depends on the direction of extension and transverse deformation
:
:
Here
is Poisson's ratio,
is
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
,
is unit vector directed along the direction of extension,
is unit vector directed perpendicular to the direction of extension. Poisson's ratio has a different number of special directions depending on the type of anisotropy.
Orthotropic
Orthotropic material
In material science and solid mechanics, orthotropic materials have material properties at a particular point which differ along three orthogonal axes, where each axis has twofold rotational symmetry. These directional differences in strength ca ...
s have three mutually perpendicular planes of symmetry in their material properties. An example is wood, which is most stiff (and strong) along the grain, and less so in the other directions.
Then
Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
can be expressed in
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
form as
[Boresi, A. P, Schmidt, R. J. and Sidebottom, O. M., 1993, ''Advanced Mechanics of Materials'', Wiley.]
:
where
*
is the
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
along axis
*
is the
shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stackre ...
in direction
on the plane whose normal is in direction
*
is the Poisson ratio that corresponds to a contraction in direction
when an extension is applied in direction
.
The Poisson 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
Transversely isotropic
A transversely isotropic material is one with physical properties that are symmetric about an axis that is normal to a plane of isotropy. This transverse plane has infinite planes of symmetry and thus, within this plane, the material properties ar ...
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
[Tan, S. C., 1994, ''Stress Concentrations in Laminated Composites'', Technomic Publishing Company, Lancaster, PA.]
:
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 six independent constants
. However, transverse isotropy gives rise to a further constraint between
and
which is
:
Therefore, there are five independent elastic material properties two 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.
Poisson's ratio values for different materials
Negative Poisson's ratio materials
Some materials known as
auxetic
Auxetics 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 i ...
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.
This can also be done in a structured way and lead to new aspects in material design as for
mechanical metamaterials.
Studies have shown that certain solid wood types display negative Poisson's ratio exclusively during a compression
creep test. Initially, the compression creep test shows positive Poisson's ratios, but gradually decreases until it reaches negative values. Consequently, this also shows that Poisson's ratio for wood is time-dependent during constant loading, meaning that the strain in the axial and transverse direction do not increase in the same rate.
Media with engineered microstructure may exhibit negative Poisson's ratio. In a simple case auxeticity is obtained removing material and creating a periodic porous media. Lattices can reach lower values of Poisson's ratio, which can be indefinitely close to the limiting value −1 in the isotropic case.
More than three hundred crystalline materials have negative Poisson's ratio. For example, Li, Na, K, Cu, Rb, Ag, Fe, Ni, Co, Cs, Au, Be, Ca, Zn Sr, Sb, MoS
and other.
Poisson function
At
finite strains, the relationship between the transverse and axial strains
and
is typically not well described by the Poisson ratio. In fact, the Poisson ratio is often considered a function of the applied strain in the large strain regime. In such instances, the Poisson ratio is replaced by the Poisson function, for which there are several competing definitions. Defining the transverse stretch
and axial stretch
, where the transverse stretch is a function of the axial stretch (i.e.,
) the most common are the Hencky, Biot, Green, and Almansi functions
:
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
hoop stress
In mechanics, a cylinder stress is a stress (physics), stress distribution with rotational symmetry; that is, which remains unchanged if the stressed object is rotated about some fixed axis.
Cylinder stress patterns include:
* circumferential str ...
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
Structural geology is the study of the three-dimensional distribution of rock units with respect to their deformational histories. The primary goal of structural geology is to use measurements of present-day rock geometries to uncover informatio ...
. 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.
Although
cork
Cork or CORK may refer to:
Materials
* Cork (material), an impermeable buoyant plant product
** Cork (plug), a cylindrical or conical object used to seal a container
***Wine cork
Places Ireland
* Cork (city)
** Metropolitan Cork, also known as G ...
was historically chosen to seal wine bottle for other reasons (including its inert nature, impermeability, flexibility, sealing ability, and resilience),
[Silva, et al]
"Cork: properties, capabilities and applications"
, Retrieved May 4, 2017 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's 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.
See also
*
Linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
*
Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
*
Impulse excitation technique
The 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 You ...
*
Orthotropic material
In material science and solid mechanics, orthotropic materials have material properties at a particular point which differ along three orthogonal axes, where each axis has twofold rotational symmetry. These directional differences in strength ca ...
*
Shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stackre ...
*
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
*
Coefficient of thermal expansion
Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions.
Temperature is a monotonic function of the average molecular kineti ...
References
External links
Meaning of Poisson's ratioMore on negative Poisson's ratio materials (auxetic)
{{DEFAULTSORT:Poisson's Ratio
Elasticity (physics)
Physical quantities
Dimensionless numbers
Materials science
Ratios
Solid mechanics