Poisson’s Ratio

What is Poisson’s Ratio

When a material is stretched by a tensile force, it usually undergoes a lateral contraction, and when it’s compressed, it usually undergoes a lateral expansion. That phenomenon of lateral contraction or expansion is known as the Poisson effect. The ratio of the lateral contraction or expansion and longitudinal extension or compression is called Poisson’s ratio.

Lateral deformation in tension and compression

Poisson’s ratio is defined as the ratio between lateral strain (the amount that a material deforms perpendicular to the applied force) and axial strain (the amount that it deforms parallel to the force). In other words, it’s a measure of how much something expands or contracts in response to being stretched or compressed along one axis.

Poisson’s ratio is not always the same for elongation versus compression.

Strain, Axial Strain, & Transverse Strain

Axial and Transverse strains due to a uniaxial force

Strain (ε) is the relative change in length to the original length $\frac{Δ l}{l_{o}}$

Axial strain is the strain along the direction of the applied force $\frac{Δ l_{x}}{l_{x}}$

Transverse strain is the strain perpendicular to the direction of the applied force $\frac{Δ l_{y}}{l_{y}}$

Poisson’s ratio range

Poisson’s ratio is a dimensionless quantity that ranges from -1 to 0.5, however for most materials the Poisson’s ratio is in the range of (0 - 0.5).

Diagram showing the lateral deformation for different Poisson’s ratio values — Lateral deformation for different values of Poisson’s ratio

A material with Poisson’s ratio between (0 < $𝜈$ ≤ 0.5), would contract in the lateral direction when stretched, and expand laterally when compressed.
A material with Poisson’s ratio equal to 0, would not deform in the lateral direction when stretched or compressed.
A material with Poisson’s ratio between (-1 ≤ $𝜈$ < 0), would expand in the lateral direction when stretched, and contract laterally when compressed.

You can find the Poisson’s ratio of common materials on this page: Poisson’s ratio values.

Formula & Units

Poisson’s Ratio $(𝜈)$ = transverse strain / axial strain

𝜈 = \frac{- 𝜀_{l a t e r a l}}{𝜀_{a x i a l}}

Where:

$𝜈$ is the Poisson’s ratio
ε_lateral is the lateral strain (strain in the direction perpendicular to the direction of force)
ε_axial is the axial strain (strain along the direction of force)

Poisson’s ratio is a dimensionless quantity, which means that it does not have a unit of measurement.

By convention compressive deformation is considered negative, and tensile deformation is considered positive, thus the negative sign in the Poisson’s ratio formula is there, so that for a typical material, where the lateral deformation is opposite to the longitudinal deformation, the Poisson’s ratio value would be positive.

Factors Affecting Poisson’s Ratio

Several factors can affect the Poisson’s ratio of a material, including:

Material composition: The Poisson’s ratio can vary depending on the chemical composition of the material, such as its crystalline structure, elemental composition, and bonding types.
Temperature: The Poisson’s ratio can change with temperature due to changes in the material’s structure and bonding. Some materials may exhibit anomalous behavior, such as negative Poisson’s ratios, at certain temperatures.
Strain rate: The Poisson’s ratio can also be affected by the rate at which the material is deformed. Some materials exhibit time-dependent behavior, where the Poisson’s ratio changes over time due to the rate of deformation.
Loading conditions: The Poisson’s ratio can vary depending on the direction and magnitude of the applied stress. For example, the Poisson’s ratio may be different under tensile and compressive loads.
Anisotropy: Materials that exhibit anisotropic behavior, where the mechanical properties vary depending on the direction of the applied force, may also exhibit different Poisson’s ratios depending on the direction of the deformation.
Presence of defects: The presence of defects such as voids, cracks, and inclusions can also affect the Poisson’s ratio. Defects can lead to local changes in the material’s stiffness and may affect the way the material deforms under stress.

Applications of Poisson’s Ratio

Poisson’s ratio has numerous applications across various fields. Some of the main applications of Poisson’s ratio include:

Material science: Poisson’s ratio is a fundamental mechanical property of materials, and it is widely used in material science to characterize the mechanical behavior of materials. It is used to determine the stiffness, strength, and ductility of materials, as well as to predict their behavior under different loading conditions.
Engineering: Poisson’s ratio is used extensively in engineering to design and optimize structures and components. Engineers use it to determine the response of materials to different loading conditions, such as tension, compression, bending, and torsion.
Manufacturing: Poisson’s ratio is used in manufacturing to improve the quality of products. It is used in quality control to ensure that the manufactured products meet the desired mechanical properties.

Overall, Poisson’s ratio is a key property that plays an important role in many areas of science and engineering. Its applications are widespread and varied, and it is a critical property to consider when designing and developing new materials and structures.

Poisson’s Ratio Summary
Definition	The ratio between lateral deformation to axial deformation in a material when under uniaxial loading
Symbol	$𝜈$
Formula	$𝜈 = \frac{- 𝜀_{l a t e r a l}}{𝜀_{a x i a l}}$
Units	Unitless

Is Poisson’s ratio a material property?: Yes, Poisson’s ratio is a material property, that describes the lateral deformation when a material is stretched or compressed.

Can Poisson’s ratio be negative and what does that mean?: Yes, Poisson’s ratio can be negative in some materials. This indicates that when the material is stretched in one direction, it actually gets thicker in the perpendicular direction. This unusual behavior is known as auxeticity and is relatively rare in natural materials, but can be engineered in some synthetic materials. Materials with negative Poisson’s ratios typically exhibit unusual mechanical properties, such as increased resistance to indentation, enhanced energy absorption, and improved acoustic damping.

How is Poisson’s ratio measured experimentally?: Poisson’s ratio is commonly measured experimentally using a tensile test, where the material is subjected to a uniaxial tensile load and the strain in the axial and transverse directions is measured. Poisson’s ratio is then calculated as the ratio of the transverse strain to the axial strain. Other methods, such as ultrasonic testing or dynamic mechanical analysis, can also be used to determine Poisson’s ratio.