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## Poisson's ratio
When a sample of material is stretched in one direction, it tends to get thinner in the other two directions. Assuming that the material is compressed along the axial direction: where - ν is the resulting Poisson's ratio,
- is transverse strain (negative for axial tension, positive for axial compression)
- is axial strain (positive for axial tension, negative for axial compression).
## Generalized Hooke's lawFor 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: where - , and are strain in the direction of
*x*,*y*and*z*axis - σ
_{x}, σ_{y}and σ_{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)- ν is Poisson's ratio (the same in all directions:
*x*,*y*and*z*for isotropic materials)
## Volumetric changeThe relative change of volume where *V*is material volume- Δ
*V*is material volume change *L*is original length, before stretch- Δ
*L*is the change of length: Δ*L*=*L*_{old}−*L*_{new}
## Width change
If a rod with diameter (or width, or thickness) 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 *d*is original diameter- Δ
*d*is rod diameter change - ν is Poisson's ratio
*L*is original length, before stretch- Δ
*L*is the change of length.
## Orthotropic materialsFor 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: where *E*_{i}is a Young's modulus along axis i- ν
_{jk}is a Poisson's ratio in plane jk
## Poisson's ratio values for different materials
## See also- 3-D elasticity
- Hooke's Law
- Stress
- Strain
- Impulse excitation technique
- Orthotropic material
- Coefficient of thermal expansion
## References**^**Poisson's ratio calculation of glasses
Categories: Solid mechanics | Materials science |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Poisson's_ratio". A list of authors is available in Wikipedia. |