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| <!-- {{Mergingfrom|Poisson's effect|date=May 2008}} -->
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| '''Poisson's ratio''', named after [[Siméon Poisson]], is the negative ratio of transverse to axial [[strain (materials science)|strain]]. When a [[materials|material]] is compressed in one direction, it usually tends to expand in the other two directions perpendicular to the direction of compression. This phenomenon is called the '''Poisson effect'''. Poisson's ratio <math>\nu</math> ([[Nu (letter)|nu]]) is a measure of this effect. The Poisson ratio is the fraction (or percent) of expansion divided by the fraction (or percent) of compression, for small values of these changes.
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| Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching. This is a common observation when a rubber band is stretched, when 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.
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| The Poisson's ratio of a stable, [[isotropy#Materials science|isotropic]], linear [[Elasticity (physics)|elastic]] material cannot be less than −1.0 nor greater than 0.5 due to the requirement that [[Young's modulus]], the [[shear modulus]] and [[bulk modulus]] have positive values.<ref>H. GERCEK; “''Poisson's ratio values for rocks''”; <u>International Journal of Rock Mechanics and Mining Sciences</u>; Elsevier; January 2007; '''44''' (1): pp. 1–13.</ref> 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 (engineering)|yield]]) exhibit values of about 0.3, increasing to 0.5 for post-yield deformation (Seismic Performance of Steel-Encased Concrete Piles by RJT Park) (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. Some materials, mostly polymer foams, have a negative Poisson's ratio; if these [[auxetics|auxetic materials]] are stretched in one direction, they become thicker in perpendicular directions. Some anisotropic materials have one or more Poisson ratios above 0.5 in some directions.
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| Assuming that the material is stretched or compressed along the axial direction (the ''x'' axis in the below diagram):
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| :<math>\nu = -\frac{d\varepsilon_\mathrm{trans}}{d\varepsilon_\mathrm{axial}} = -\frac{d\varepsilon_\mathrm{y}}{d\varepsilon_\mathrm{x}}= -\frac{d\varepsilon_\mathrm{z}}{d\varepsilon_\mathrm{x}} </math>
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| where
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| :<math>\nu</math> is the resulting Poisson's ratio,
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| :<math>\varepsilon_\mathrm{trans}</math> is transverse strain (negative for axial tension (stretching), positive for axial compression)
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| :<math>\varepsilon_\mathrm{axial}</math> is axial strain (positive for axial tension, negative for axial compression).
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| == Length change ==
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| [[Image:PoissonRatio.svg|thumb|300px|right|Figure 1: A cube with sides of length ''L'' of an isotropic linearly elastic material subject to tension along the x axis, with a Poisson's ratio of 0.5. The green cube is unstrained, the red is expanded in the ''x'' direction by <math>\Delta L</math> due to tension, and contracted in the ''y'' and ''z'' directions by <math>\Delta L'</math>.]]
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| For a cube stretched in the ''x''-direction (see figure 1) with a length increase of <math>\Delta L</math> in the ''x'' direction, and a length decrease of <math>\Delta L'</math> in the ''y'' and ''z'' directions, the infinitesimal diagonal strains are given by
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| :<math>
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| d\varepsilon_x=\frac{dx}{x}\qquad d\varepsilon_y=\frac{dy}{y}\qquad d\varepsilon_z=\frac{dz}{z}.
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| </math>
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| Integrating these expressions and using the definition of Poisson's ratio gives
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| :<math>
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| -\nu \int\limits_L^{L+\Delta L}\frac{dx}{x}=\int\limits_L^{L-\Delta L'}\frac{dy}{y}=\int\limits_L^{L-\Delta L'}\frac{dz}{z}.
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| </math>
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| Solving and exponentiating, the relationship between <math>\Delta L</math> and <math>\Delta L'</math> is then
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| :<math>
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| \left(1+\frac{\Delta L}{L}\right)^{-\nu} = 1-\frac{\Delta L'}{L}.
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| </math>
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| For very small values of <math>\Delta L</math> and <math>\Delta L'</math>, the first-order approximation yields:
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| :<math>
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| \nu \approx \frac{\Delta L'}{\Delta L}.
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| </math>
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| == Volumetric change ==
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| The relative change of volume ''ΔV''/''V'' of a cube due to the stretch of the material can now be calculated. Using <math>V=L^3</math> and <math>V+\Delta V=(L+\Delta L)(L-\Delta L')^2</math>:
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| :<math>\frac {\Delta V} {V} = \left(1+\frac{\Delta L}{L} \right)\left(1-\frac{\Delta L'}{L} \right)^2-1</math>
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| Using the above derived relationship between <math>\Delta L</math> and <math>\Delta L'</math>:
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| :<math>\frac {\Delta V} {V} = \left(1+\frac{\Delta L}{L} \right)^{1-2\nu}-1</math>
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| and for very small values of <math>\Delta L</math> and <math>\Delta L'</math>, the first-order approximation yields:
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| :<math>\frac {\Delta V} {V} \approx (1-2\nu)\frac{\Delta L}{L}</math>
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| For isotropic materials we can use Lamé’s relation<ref>http://arxiv.org/ftp/arxiv/papers/1204/1204.3859.pdf - Limits to Poisson’s ratio in isotropic materials – general result for
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| arbitrary deformation.</ref>
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| :<math>\nu \approx \frac{1}{2} - \frac{E}{6K}</math>
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| where <math>K</math> is [[bulk modulus]].
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| == Width change ==
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| [[Image:Rod diamater change poisson.svg|350px|thumb|right|Figure 2: Comparison between the two formulas, one for small deformations, another for large deformations]]
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| 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:
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| :<math>\Delta d = - d \cdot \nu {{\Delta L} \over L}</math>
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| 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:
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| :<math>\Delta d = -d \cdot \left( 1 - {\left( 1 + {{\Delta L} \over L} \right)}^{-\nu} \right)</math>
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| where
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| :<math> d </math> is original diameter
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| :<math> \Delta d </math> is rod diameter change
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| :<math> \nu </math> is Poisson's ratio | |
| :<math> L </math> is original length, before stretch
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| :<math> \Delta L </math> is the change of length.
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| The value is negative because it decreases with increase of length
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| == Isotropic materials ==
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| 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:
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| :<math> \varepsilon_x = \frac {1}{E} \left [ \sigma_x - \nu \left ( \sigma_y + \sigma_z \right ) \right ] </math>
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| :<math> \varepsilon_y = \frac {1}{E} \left [ \sigma_y - \nu \left ( \sigma_x + \sigma_z \right ) \right ] </math>
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| :<math> \varepsilon_z = \frac {1}{E} \left [ \sigma_z - \nu \left ( \sigma_x + \sigma_y \right ) \right ] </math>
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| or
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| :<math> \varepsilon_i = \frac {1}{E} \left [ \sigma_i(1+\nu) - \nu \left ( \sigma_x + \sigma_y+\sigma_z \right ) \right ] </math>
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| where
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| :<math>\varepsilon_x </math>, <math>\varepsilon_y </math> and <math>\varepsilon_z </math> are [[strain (materials science)|strain]] in the direction of <math>x</math>, <math>y</math> and <math>z</math> axis
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| :<math> \sigma_x</math> , <math>\sigma_y</math> and <math>\sigma_z</math> are [[Stress (physics)|stress]] in the direction of <math>x</math>, <math>y</math> and <math>z</math> axis
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| :<math> E </math> is [[Young's modulus]] (the same in all directions: <math>x</math>, <math>y</math> and <math>z</math> for isotropic materials)
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| :<math> \nu </math> is Poisson's ratio (the same in all directions: <math>x</math>, <math>y</math> and <math>z</math> for isotropic materials)
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| These equations will hold in the general case which includes shear forces as well as compressive forces, and the full generalization of Hooke's law is given by:
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| :<math> \varepsilon_{ij} = \frac {1}{E} \left [ \sigma_{ij}(1+\nu) - \nu \delta_{ij}\sigma_{kk} \right ] </math>
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| where <math>\delta_{ij}</math> is the [[Kronecker delta]] and
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| :<math> \sigma_{kk} = \sigma_x + \sigma_y+\sigma_z = \sigma_{11} + \sigma_{22}+\sigma_{33}\,</math>
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| == Orthotropic materials ==
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| {{Main|Orthotropic material}}
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| For [[orthotropic material]]s such as wood, [[Hooke's law]] can be expressed in [[Matrix (mathematics)|matrix]] form as<ref name=Boresi>Boresi, A. P, Schmidt, R. J. and Sidebottom, O. M., 1993, ''Advanced Mechanics of Materials'', Wiley.</ref><ref name=Lekh>Lekhnitskii, SG., (1963), '''Theory of elasticity of an anisotropic elastic body''', Holden-Day Inc.</ref>
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| :<math>
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| \begin{bmatrix}
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| \epsilon_{{\rm xx}} \\ \epsilon_{\rm yy} \\ \epsilon_{\rm zz} \\ 2\epsilon_{\rm yz} \\ 2\epsilon_{\rm zx} \\ 2\epsilon_{\rm xy}
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| \end{bmatrix}
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| = \begin{bmatrix}
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| \tfrac{1}{E_{\rm x}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & - \tfrac{\nu_{\rm zx}}{E_{\rm z}} & 0 & 0 & 0 \\
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| -\tfrac{\nu_{\rm xy}}{E_{\rm x}} & \tfrac{1}{E_{\rm y}} & - \tfrac{\nu_{\rm zy}}{E_{\rm z}} & 0 & 0 & 0 \\
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| -\tfrac{\nu_{\rm xz}}{E_{\rm x}} & - \tfrac{\nu_{\rm yz}}{E_{\rm y}} & \tfrac{1}{E_{\rm z}} & 0 & 0 & 0 \\
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| 0 & 0 & 0 & \tfrac{1}{G_{\rm yz}} & 0 & 0 \\
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| 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm zx}} & 0 \\
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| 0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} \\
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| \end{bmatrix}
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| \begin{bmatrix}
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| \sigma_{\rm xx} \\ \sigma_{\rm yy} \\ \sigma_{\rm zz} \\ \sigma_{\rm yz} \\ \sigma_{\rm zx} \\ \sigma_{\rm xy}
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| \end{bmatrix}
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| </math>
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| where
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| :<math>{E}_{\rm i}\,</math> is the [[Young's modulus]] along axis <math>i</math>
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| :<math>G_{\rm ij}\,</math> is the [[shear modulus]] in direction <math>j</math> on the plane whose normal is in direction <math>i</math>
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| :<math>\nu_{\rm ij}\,</math> is the Poisson's ratio that corresponds to a contraction in direction <math>j</math> when an extension is applied in direction <math>i</math>.
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| 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
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| :<math>\frac{\nu_{\rm yx}}{E_{\rm y}} = \frac{\nu_{\rm xy}}{E_{\rm x}}~, \qquad
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| \frac{\nu_{\rm zx}}{E_{\rm z}} = \frac{\nu_{\rm xz}}{E_{\rm x}}~, \qquad
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| \frac{\nu_{\rm yz}}{E_{\rm y}} = \frac{\nu_{\rm zy}}{E_{\rm z}}
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| </math>
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| From the above relations we can see that if <math>E_{\rm x} > E_{\rm y}</math> then <math>\nu_{\rm xy} > \nu_{\rm yx}</math>. The larger Poisson's ratio (in this case <math>\nu_{\rm xy}</math>) is called the '''major Poisson's ratio''' while the smaller one (in this case <math>\nu_{\rm yx}</math>) is called the '''minor Poisson's ratio'''. We can find similar relations between the other Poisson's ratios.
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| <!--
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| The above stress-strain relation is also often written in the equivalent ([[transpose]]d) form
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| :<math>
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| \begin{bmatrix}
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| \epsilon_{{\rm 11}} \\ \epsilon_{\rm 22} \\ \epsilon_{\rm 33} \\ 2\epsilon_{\rm 23} \\ 2\epsilon_{\rm 31} \\ 2\epsilon_{\rm 12}
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| \end{bmatrix}
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| = \begin{bmatrix}
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| \tfrac{1}{E_{\rm 1}} & - \tfrac{\nu_{\rm 21}}{E_{\rm 2}} & - \tfrac{\nu_{\rm 31}}{E_{\rm 3}} & 0 & 0 & 0 \\
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| -\tfrac{\nu_{\rm 12}}{E_{\rm 1}} & \tfrac{1}{E_{\rm 2}} & - \tfrac{\nu_{\rm 32}}{E_{\rm 3}} & 0 & 0 & 0 \\
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| -\tfrac{\nu_{\rm 13}}{E_{\rm 1}} & - \tfrac{\nu_{\rm 23}}{E_{\rm 2}} & \tfrac{1}{E_{\rm 3}} & 0 & 0 & 0 \\
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| 0 & 0 & 0 & \tfrac{1}{G_{\rm 23}} & 0 & 0 \\
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| 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm 13}} & 0 \\
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| 0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm 12}} \\
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| \end{bmatrix}
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| \begin{bmatrix}
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| \sigma_{\rm 11} \\ \sigma_{\rm 22} \\ \sigma_{\rm 33} \\ \sigma_{\rm 23} \\ \sigma_{\rm 31} \\ \sigma_{\rm 12}
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| \end{bmatrix}
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| </math>
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| -->
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| == Transversely isotropic materials ==
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| [[Transversely isotropic]] materials have [[Transversely isotropic|a plane of isotropy]] in which the elastic properties are isotropic. If we assume that this plane of isotropy is <math>y-z</math>, then Hooke's law takes the form<ref name=Tan>Tan, S. C., 1994, ''Stress Concentrations in Laminated Composites'', Technomic Publishing Company, Lancaster, PA.</ref>
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| :<math>
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| \begin{bmatrix}
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| \epsilon_{{\rm xx}} \\ \epsilon_{\rm yy} \\ \epsilon_{\rm zz} \\ 2\epsilon_{\rm yz} \\ 2\epsilon_{\rm zx} \\ 2\epsilon_{\rm xy}
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| \end{bmatrix}
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| = \begin{bmatrix}
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| \tfrac{1}{E_{\rm x}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & 0 & 0 & 0 \\
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| -\tfrac{\nu_{\rm xy}}{E_{\rm x}} & \tfrac{1}{E_{\rm y}} & - \tfrac{\nu_{\rm zy}}{E_{\rm y}} & 0 & 0 & 0 \\
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| -\tfrac{\nu_{\rm xy}}{E_{\rm x}} & - \tfrac{\nu_{\rm yz}}{E_{\rm y}} & \tfrac{1}{E_{\rm y}} & 0 & 0 & 0 \\
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| 0 & 0 & 0 & \tfrac{1}{G_{\rm yz}} & 0 & 0 \\
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| 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} & 0 \\
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| 0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} \\
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| \end{bmatrix}
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| \begin{bmatrix}
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| \sigma_{\rm xx} \\ \sigma_{\rm yy} \\ \sigma_{\rm zz} \\ \sigma_{\rm yz} \\ \sigma_{\rm zx} \\ \sigma_{\rm xy}
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| \end{bmatrix}
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| </math>
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| where we have used the plane of isotropy <math>y-z</math> to reduce the number of constants, i.e., <math>E_y = E_z,~ \nu_{xy} = \nu_{xz},~ \nu_{yx} = \nu_{zx} </math>.
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| The symmetry of the stress and strain tensors implies that
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| :<math>
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| \cfrac{\nu_{\rm xy}}{E_{\rm x}} = \cfrac{\nu_{\rm yx}}{E_{\rm y}} ~,~~ \nu_{\rm yz} = \nu_{\rm zy} ~.
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| </math>
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| This leaves us with six independent constants <math>E_{\rm x}, E_{\rm y}, G_{\rm xy}, G_{\rm yz}, \nu_{\rm xy}, \nu_{\rm yz}</math>. However, transverse isotropy gives rise to a further constraint between <math>G_{\rm yz}</math> and <math>E_{\rm y}, \nu_{\rm yz}</math> which is
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| :<math>
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| G_{\rm yz} = \cfrac{E_{\rm y}}{2(1+\nu_{\rm yz})} ~.
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| </math>
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| Therefore, there are five independent elastic material properties two of which are Poisson's ratios. For the assumed plane of symmetry, the larger of <math>\nu_{\rm xy}</math> and <math> \nu_{\rm yx}</math> is the major Poisson's ratio. The other major and minor Poisson's ratios are equal.
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| == Poisson's ratio values for different materials ==
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| [[Image:SpiderGraph PoissonRatio.gif|500px|thumb|Influences of selected [[glass]] component additions on Poisson's ratio of a specific base glass.<ref>[http://www.glassproperties.com/poisson_ratio/ Poisson's ratio calculation of glasses]</ref>]]
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| {| class="wikitable sortable" cellpadding="4" cellspacing="0" border="1" style="border-collapse: collapse"
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| |- bgcolor="#cccccc"
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| !Material!!Poisson's ratio
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| |-
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| | [[rubber]]
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| | ~ 0.50
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| |-
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| | [[gold]]
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| | 0.42 - 0.44
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| |-
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| | saturated clay
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| | 0.40–0.50
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| |-
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| | [[magnesium]]
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| | 0.35
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| |-
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| | [[titanium]]
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| | 0.34
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| |-
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| | [[copper]]
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| | 0.33
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| |-
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| | [[aluminium]]-[[alloy]]
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| | 0.33
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| |-
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| | [[clay]]
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| | 0.30–0.45
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| |-
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| | [[stainless steel]]
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| | 0.30–0.31
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| |-
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| | [[steel]]
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| | 0.27–0.30
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| |-
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| | [[cast iron]]
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| | 0.21–0.26
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| |-
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| | [[sand]]
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| | 0.20–0.45
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| |-
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| | [[concrete]]
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| | 0.20
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| |-
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| | [[glass]]
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| | 0.18–0.3
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| |-
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| | [[foam]]
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| | 0.10–0.40
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| |-
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| | [[cork (material)|cork]]
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| | ~ 0.00
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| |}
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| {| class="wikitable sortable" cellpadding="4" cellspacing="0" border="1" style="border-collapse: collapse"
| |
| |- bgcolor="#cccccc"
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| !Material!!Plane of symmetry!!<math>\nu_{\rm xy}</math>!!<math>\nu_{\rm yx}</math>!!<math>\nu_{\rm yz}</math>!!<math>\nu_{\rm zy}</math>!!<math>\nu_{\rm zx}</math>!!<math>\nu_{\rm xz}</math>
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| |-
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| | [[Nomex]] [[composite honeycomb|honeycomb core]]
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| | <math>x-y</math>, <math>x</math>=ribbon direction
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| |0.49
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| |0.69
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| |0.01
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| |2.75
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| |3.88
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| |0.01
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| |-
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| |[[glass fiber]]-[[epoxy resin]]
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| |<math>x-y</math>
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| |0.29
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| |0.29
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| |0.32
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| |0.06
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| |0.06
| |
| |0.32
| |
| |}
| |
| | |
| === Negative Poisson's ratio materials ===
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| 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.<ref>[http://silver.neep.wisc.edu/~lakes/Poisson.html Negative Poisson's ratio<!-- Bot generated title -->]</ref>
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| This can also be done in a structured way and lead to new aspects in material design as for [[Mechanical Metamaterials]].
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| == Applications of Poisson's effect ==
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| 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.{{Citation needed|date=October 2012}}
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| 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.<ref>http://www.geosc.psu.edu/~engelder/geosc465/lect18.rtf</ref>
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| The use of [[Cork material|cork]] as a stopper for wine bottles is due to cork having a Poisson ratio of practically zero, so that, as the cork is inserted into the bottle, the upper part which is not yet inserted does not expand as the lower part is compressed. The force needed to insert a cork into a bottle arises only from the compression of the cork and the friction between the cork and the bottle. 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 expansion of the upper part of the rubber stopper.
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| ==See also==
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| *[[3-D elasticity]]
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| *[[Hooke's Law]]
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| *[[Impulse excitation technique]]
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| *[[Orthotropic material]]
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| *[[Shear modulus]]
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| *[[Young's modulus]]
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| *[[Coefficient of thermal expansion]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * [http://silver.neep.wisc.edu/~lakes/PoissonIntro.html Meaning of Poisson's ratio]
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| * [http://silver.neep.wisc.edu/~lakes/Poisson.html Negative Poisson's ratio materials]
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| * [http://home.um.edu.mt/auxetic More on negative Poisson's ratio materials (auxetic)]
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| {{Elastic moduli}}
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| {{DEFAULTSORT:Poisson's Ratio}}
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| <!--Categories-->
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| [[Category:Elasticity (physics)]]
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| [[Category:Physical quantities]]
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| [[Category:Dimensionless numbers]]
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| [[Category:Materials science]]
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| [[Category:Ratios]]
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| [[Category:Mechanics]]
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| [[Category:Solid mechanics]]
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