Poisson's Ratio Calculator

Enter a specimen's diameter (or width) and length before and after loading to get the axial strain, transverse strain, and Poisson's ratio v = -e_transverse / e_axial.

Quick Facts

Definition
v = -e_transverse / e_axial
Negative of transverse strain divided by axial strain; a dimensionless ratio.
Typical range
0.0 to 0.5 for most solids
Steel ≈ 0.27-0.30, aluminum ≈ 0.33, rubber ≈ 0.499, cork ≈ 0.
Stability limit
-1 < v < 0.5
Thermodynamic bound for isotropic linear-elastic materials.
Elastic-constant link
v = E / (2G) - 1
Relates Poisson's ratio to Young's modulus E and shear modulus G.

Your Results

Calculated
Poisson's Ratio (v)
-
v = -e_transverse / e_axial
Axial Strain (e_axial)
-
(L₁ - L₀) / L₀
Transverse Strain (e_transverse)
-
(d₁ - d₀) / d₀
Material Behavior
-
Interpretation of v

Ready

Enter original and final dimensions, then press Calculate.

Formula and Method for Poisson's Ratio

When a material is stretched along one direction, it usually contracts in the perpendicular directions; when it is compressed, it usually bulges outward. Poisson's ratio (v, the Greek letter nu) quantifies that coupling. It is formally defined as v = -e_transverse / e_axial, the negative of the strain perpendicular to an applied load divided by the strain along the load. This calculator derives both strains directly from a specimen's dimensions before and after loading, then computes v.

How the calculation works

Enter the specimen's original diameter or width (d₀) and its diameter or width after loading (d₁), along with its original length (L₀) and its length after loading (L₁) — any consistent unit works (mm, in, etc.) as long as each pair uses the same unit. The calculator finds the axial strain e_axial = (L₁ - L₀) / L₀ and the transverse strain e_transverse = (d₁ - d₀) / d₀, then divides: v = -e_transverse / e_axial. The minus sign makes v come out positive for the ordinary case of a rod that narrows as it stretches (axial strain positive, transverse strain negative).

Common mistakes

  • Swapping axial and transverse measurements: the axial pair (L₀, L₁) must be along the direction of the applied load; the transverse pair (d₀, d₁) must be perpendicular to it.
  • Mixing units within a pair: d₀ and d₁ must share the same unit, and L₀ and L₁ must share the same unit — the diameter unit does not need to match the length unit, since each strain is a unitless ratio.
  • Ignoring the sign: a rod under tension normally has a positive axial strain and a negative transverse strain; dropping the negative sign in the formula would report v as negative for an ordinary material.
  • Averaging across directions in anisotropic materials: wood, composites, and crystals can have a different Poisson's ratio in each direction pair, so a single measurement does not describe the whole material.

Real-world applications

  • Materials testing labs compute v from tensile-test extensometer and diameter-gauge readings to characterize a new alloy or polymer.
  • Structural and mechanical engineers use v (together with Young's modulus) to predict how components deform under multi-axial stress, including pressure vessels and rotating parts.
  • Geotechnical engineers use soil and rock Poisson's ratios to model ground deformation around foundations, tunnels, and excavations.
  • Materials scientists screen for auxetic (negative-v) foams and lattices, which are used in impact-absorbing padding and fasteners because they thicken rather than thin under stretching.

Frequently Asked Questions

What is Poisson's ratio?
Poisson's ratio (v) measures how much a material contracts sideways when stretched lengthwise (or bulges sideways when compressed). It is defined as v = -(transverse strain) / (axial strain), where axial strain is along the applied load and transverse strain is perpendicular to it. It is a dimensionless ratio, typically between 0 and 0.5 for common isotropic materials.
How do you calculate Poisson's ratio from measured dimensions?
Measure a specimen's diameter (or width) and length before and after loading. Axial strain = (final length - original length) / original length. Transverse strain = (final diameter - original diameter) / original diameter. Poisson's ratio is the negative of the transverse strain divided by the axial strain: v = -e_transverse / e_axial.
What is a typical Poisson's ratio for common materials?
Most structural metals fall between about 0.26 and 0.35 (steel is about 0.27-0.30, aluminum about 0.33). Rubber and other near-incompressible materials approach 0.5, cork is close to 0, and glass is around 0.20-0.27.
Can Poisson's ratio be negative?
Yes. Materials with a negative Poisson's ratio are called auxetic — they get thicker in the transverse direction when stretched instead of thinner. Certain re-entrant foams and engineered lattice structures show this behavior; for ordinary isotropic elastic materials, thermodynamic stability requires -1 < v < 0.5.

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