When a sample of material is stretched in one direction, it tends to contract (or rarely, expand) in the other two directions. Poisson's ratio (ν), named after Simeon Poisson, is a measure of this tendency. Poisson's ratio 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). 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. 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.

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

$nu$ 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).

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:

$varepsilon\_x\; =\; frac\; \{1\}\{E\}\; left\; [sigma\_x\; -\; nu\; left\; (sigma\_y\; +\; sigma\_z\; right\; )\; right\; ]$

$varepsilon\_y\; =\; frac\; \{1\}\{E\}\; left\; [sigma\_y\; -\; nu\; left\; (sigma\_x\; +\; sigma\_z\; right\; )\; right\; ]$

$varepsilon\_z\; =\; frac\; \{1\}\{E\}\; left\; [sigma\_z\; -\; nu\; left\; (sigma\_x\; +\; sigma\_y\; right\; )\; right\; ]$

$varepsilon\_x$, $varepsilon\_y$ and $varepsilon\_z$ are strain in the direction of $x$, $y$ and $z$ axis

$sigma\_x$ , $sigma\_y$ and $sigma\_z$ 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)

$nu$ is Poisson's ratio (the same in all directions: $x$, $y$ and $z$ for isotropic materials)

Volumetric change

$frac\; \{Delta\; V\}\; \{V\}\; =\; (1-2nu)frac\; \{Delta\; L\}\; \{L\}$

where

$V$ is material volume

$Delta\; V$ is material volume change

$L$ is original length, before stretch

$Delta\; L$ is the change of length: $Delta\; L\; =\; L\_mathrm\{old\}\; -\; L\_mathrm\{new\}$

Width change

$Delta\; d\; =\; -\; d\; cdot\; nu\; \{\{Delta\; L\}\; over\; L\}$

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

$Delta\; d$ is rod diameter change

$nu$ is Poisson's ratio

$L$ is original length, before stretch

$Delta\; L$ is the change of length.

Orthotropic materials

$frac\{nu\_\{yx\}\}\{E\_y\}\; =\; frac\{nu\_\{xy\}\}\{E\_x\}\; qquad$

where

$\{E\}\_i$ is a Young's modulus along axis i

$nu\_\{jk\}$ is a Poisson's ratio in plane jk

Poisson's ratio values for different materials

!material

See also

• 3-D elasticity
• Hooke's Law
• Stress
• Strain
• Impulse excitation technique
• Orthotropic material
• Coefficient of thermal expansion
• Poisson's Effect

References

External links

• Meaning of Poisson's ratio
• Negative Poisson's ratio materials
• More on negative Poisson's ratio materials (auxetic)
• Poisson's ratio