Poisson\\\'s_ratio

Figure 1: Rectangular specimen subject to compression, with Poisson\'s ratio circa 0.5

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:

$\backslash nu\; =\; -\backslash frac\{\backslash varepsilon\_\backslash mathrm\{trans\}\}\{\backslash varepsilon\_\backslash mathrm\{axial\}\}\; =\; -\backslash frac\{\backslash varepsilon\_\backslash mathrm\{x\}\}\{\backslash varepsilon\_\backslash mathrm\{y\}\}$

where

$\backslash nu$ is the resulting Poisson\'s ratio,

$\backslash varepsilon\_\backslash mathrm\{trans\}$ is transverse strain (negative for axial tension, positive for axial compression)

$\backslash varepsilon\_\backslash mathrm\{axial\}$ is axial strain (positive for axial tension, negative for axial compression).

Contents 1 Generalized Hooke\'s law 2 Volumetric change 3 Width change 4 Orthotropic materials 5 Poisson\'s ratio values for different materials 6 See also 7 References 8 External links

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:

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

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

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

where

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

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

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

Volumetric change

The relative change of volume ΔV/V due to the stretch of the material can be calculated using a simplified formula (only for small deformations):

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

where

$V$ is material volume

$\backslash Delta\; V$ is material volume change

$L$ is original length, before stretch

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

Width change

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

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 value is negative, because the diameter will decrease with increasing length):

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

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:

$\backslash Delta\; d\; =\; -\; d\; \backslash cdot\; \backslash left(\; 1\; -\; \{\backslash left(\; 1\; +\; \{\{\backslash Delta\; L\}\; \backslash over\; L\}\; \backslash right)\}^\{-\backslash nu\}\; \backslash right)$

where

$d$ is original diameter

$\backslash Delta\; d$ is rod diameter change

$\backslash nu$ is Poisson\'s ratio

$L$ is original length, before stretch

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

Orthotropic materials

For Orthotropic material, such as wood in which Poisson\'s ratio is different in each direction (x, y and z axis) the relation between Young\'s modulus and Poisson\'s ratio is described as follows:

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

\frac{\nu_{zx}}{E_z} = \frac{\nu_{xz}}{E_x} \qquad \frac{\nu_{yz}}{E_y} = \frac{\nu_{zy}}{E_z} \qquad

where

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

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

Poisson\'s ratio values for different materials

Influences of selected glass component additions on Poisson\'s ratio of a specific base glass.Poisson\'s ratio calculation of glasses

material	poisson\'s ratio
rubber	~ 0.50
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 to 0.40
cork	~ 0.00
auxetics	negative

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

v • d • e Elastic moduli for homogeneous isotropic materials
Bulk modulus ($K$) • Young\'s modulus ($E$) • Lamé\'s first parameter ($\backslash lambda$) • Shear modulus ($\backslash mu$) • Poisson\'s ratio ($\backslash nu$) • P-wave modulus ($M$)

Conversion formulas
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.
	$(\backslash lambda,\backslash ,\backslash mu)$	$(E,\backslash ,\backslash mu)$	$(K,\backslash ,\backslash lambda)$	$(K,\backslash ,\backslash mu)$	$(\backslash lambda,\backslash ,\backslash nu)$	$(\backslash mu,\backslash ,\backslash nu)$	$(E,\backslash ,\backslash nu)$	$(K,\backslash ,\; \backslash nu)$	$(K,\backslash ,E)$	$(M,\backslash ,\backslash mu)$
$K=\backslash ,$	$\backslash lambda+\; \backslash frac\{2\backslash mu\}\{3\}$	$\backslash frac\{E\backslash mu\}\{3(3\backslash mu-E)\}$			$\backslash lambda\backslash frac\{1+\backslash nu\}\{3\backslash nu\}$	$\backslash frac\{2\backslash mu(1+\backslash nu)\}\{3(1-2\backslash nu)\}$	$\backslash frac\{E\}\{3(1-2\backslash nu)\}$			$M\; -\; \backslash frac\{8\backslash mu\}\{3\}$
$E=\backslash ,$	$\backslash mu\backslash frac\{3\backslash lambda\; +\; 2\backslash mu\}\{\backslash lambda\; +\; \backslash mu\}$		$9K\backslash frac\{K-\backslash lambda\}\{3K-\backslash lambda\}$	$\backslash frac\{9K\backslash mu\}\{3K+\backslash mu\}$	$\backslash frac\{\backslash lambda(1+\backslash nu)(1-2\backslash nu)\}\{\backslash nu\}$	$2\backslash mu(1+\backslash nu)\backslash ,$		$3K(1-2\backslash nu)\backslash ,$		$\backslash mu\backslash frac\{3M-4\backslash mu\}\{M-\backslash mu\}$
$\backslash lambda=\backslash ,$		$\backslash mu\backslash frac\{E-2\backslash mu\}\{3\backslash mu-E\}$		$K-\backslash frac\{2\backslash mu\}\{3\}$		$\backslash frac\{2\; \backslash mu\; \backslash nu\}\{1-2\backslash nu\}$	$\backslash frac\{E\backslash nu\}\{(1+\backslash nu)(1-2\backslash nu)\}$	$\backslash frac\{3K\backslash nu\}\{1+\backslash nu\}$	$\backslash frac\{3K(3K-E)\}\{9K-E\}$	$M\; -\; 2\backslash mu\backslash ,$
$\backslash mu=\backslash ,$			$3\backslash frac\{K-\backslash lambda\}\{2\}$		$\backslash lambda\backslash frac\{1-2\backslash nu\}\{2\backslash nu\}$		$\backslash frac\{E\}\{2+2\backslash nu\}$	$3K\backslash frac\{1-2\backslash nu\}\{2+2\backslash nu\}$	$\backslash frac\{3KE\}\{9K-E\}$
$\backslash nu=\backslash ,$	$\backslash frac\{\backslash lambda\}\{2(\backslash lambda\; +\; \backslash mu)\}$	$\backslash frac\{E\}\{2\backslash mu\}-1$	$\backslash frac\{\backslash lambda\}\{3K-\backslash lambda\}$	$\backslash frac\{3K-2\backslash mu\}\{2(3K+\backslash mu)\}$					$\backslash frac\{3K-E\}\{6K\}$	$\backslash frac\{M\; -\; 2\backslash mu\}\{2M\; -\; 3\backslash mu\}$
$M=\backslash ,$	$\backslash lambda+2\backslash mu\backslash ,$	$\backslash mu\backslash frac\{4\backslash mu-E\}\{3\backslash mu-E\}$	$3K-2\backslash lambda\backslash ,$	$K+\backslash frac\{4\backslash mu\}\{3\}$	$\backslash lambda\; \backslash frac\{1-\backslash nu\}\{\backslash nu\}$	$\backslash mu\backslash frac\{2-2\backslash nu\}\{1-2\backslash nu\}$	$E\backslash frac\{1-\backslash nu\}\{(1+\backslash nu)(1-2\backslash nu)\}$	$3K\backslash frac\{1-\backslash nu\}\{1+\backslash nu\}$	$3K\backslash frac\{3K+E\}\{9K-E\}$

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