The Poisson's ratio calculator above handles both directions of the same relationship. Give it a measured axial extension and the transverse contraction that came with it and it returns the ratio. Give it a published ratio instead and it predicts the contraction. Supply a Young's modulus as well and it derives the shear and bulk moduli, because for an isotropic material any two elastic constants fix all the others.
Arb Digital publishes free engineering calculators that treat the edges of a model as information rather than as bugs. The interesting edges here are a ratio of 0.5, which means perfect incompressibility, and negative ratios, which belong to real materials that get fatter when you stretch them.
What This Poisson's Ratio Calculator Does
Poisson's ratio is the negative of transverse strain divided by axial strain. The minus sign exists so the ratio comes out positive for ordinary materials, since almost everything contracts sideways when stretched and the two strains therefore have opposite signs.
The headline is that ratio. Underneath it, the grid shows the two strains it was built from, so you can check that the arithmetic did what you expected, plus the shear modulus and the bulk modulus derived from your Young's modulus. Shear modulus governs how much a block distorts under a twisting or sliding load; bulk modulus governs how much it compresses under uniform pressure from all sides.
In prediction mode the transverse strain is computed from the ratio you supply, and the note reports the new transverse dimension in millimetres, which is usually the number that actually matters when you are designing a press fit or a seal.
How to Use It
- Choose the direction you are working in. Measurement mode wants both dimensional changes. Prediction mode wants a published ratio and gives you the contraction.
- Keep the signs honest. Extension is positive, contraction negative. Entering the transverse change as a positive number under tension inverts the sign of the whole result.
- Use the same units for each pair. Length with change in length, width with change in width. The ratio is dimensionless, so millimetres and inches both work as long as you do not mix them within a pair.
- Stay inside the elastic range. Poisson's ratio is defined for elastic, recoverable deformation. Once a metal yields it deforms at constant volume and the apparent ratio climbs toward 0.5 whatever the elastic value was.
- Add Young's modulus only if you want G and K. The ratio itself needs no modulus at all, and leaving the field at zero simply hides those two figures.
The Formula and How the Moduli Connect
Axial strain is εaxial = ΔL ÷ L, and transverse strain is εtrans = Δd ÷ d. Poisson's ratio is ν = −εtrans ÷ εaxial. MIT OpenCourseWare, Block 3 Materials and Elasticity, Lecture M17 on engineering elastic constants, sets out the definition and notes that a uniaxial stress always produces a biaxial state of strain, with the transverse component normally compressive under a tensile load.
For an isotropic material, two constants determine everything. Given Young's modulus E and Poisson's ratio ν, the shear modulus is G = E ÷ (2(1 + ν)) and the bulk modulus is K = E ÷ (3(1 − 2ν)). Those three moduli and the stress-strain definitions behind them are covered in OpenStax University Physics Volume 1, section 12.3 on stress, strain and elastic modulus.
Work the defaults through. A steel bar 200 mm long extends by 0.4 mm, so the axial strain is 0.4 ÷ 200 = 0.002. Its 20 mm width shrinks by 0.0116 mm, so the transverse strain is −0.0116 ÷ 20 = −0.00058. The ratio is −(−0.00058) ÷ 0.002 = 0.29, which is the textbook value for structural steel.
With E = 200 GPa, the shear modulus is 200 ÷ (2 × 1.29) = 77.5 GPa and the bulk modulus is 200 ÷ (3 × 0.42) = 158.7 GPa. Both match published figures for steel, which is a good check that the relationships are being applied correctly.
Why 0.5 Is a Hard Ceiling
Volume change under a small uniaxial strain is approximately εaxial(1 − 2ν). Set ν to 0.5 and the bracket vanishes: the material conserves its volume exactly, contracting sideways by precisely enough to compensate for the extension. The bulk modulus formula divides by that same bracket, so it goes to infinity, which is the correct statement that no finite pressure changes the volume.
Push ν above 0.5 and the bracket goes negative, meaning the material would expand in volume when compressed from all sides. That would let you extract work from a closed cycle, so it is thermodynamically forbidden for an isotropic solid, and this tool flags any value above 0.5 rather than printing a negative bulk modulus as if it were meaningful.
Rubber sits at about 0.4999. Water, which has no shear stiffness at all, behaves as the limiting case. Metals cluster between 0.25 and 0.35. And a metal that has yielded deforms plastically at constant volume, so the apparent ratio measured past yield climbs toward 0.5 regardless of its elastic value — which is exactly why the measurement must be taken inside the elastic range.
Negative Poisson's Ratios Are Real
The lower bound for an isotropic material is −1, not zero. A material with a negative ratio gets fatter when stretched, and such materials exist: they are called auxetic. Some are engineered, using re-entrant honeycomb cells that unfold under tension, and some occur naturally, including certain forms of expanded polymer and some crystal directions in cubic metals.
The behaviour is useful rather than merely curious. An auxetic material pulled at one point thickens around the pull, which resists indentation and concentrates material where a load is applied. Auxetic foams are used in impact padding for that reason.
At the other end of the ordinary range, cork sits very close to zero, contracting almost not at all when compressed along its axis. That is precisely why a cork can be pushed into a bottle neck and a rubber bung is much harder work: the cork does not bulge sideways as you compress it.
Where This Sits Beside Our Other Materials Tools
This page is about the transverse consequence of an axial load, and about converting between elastic constants. It is not about the axial relationship itself. For stress, strain and Young's modulus from a force, an area and an elongation, use the stress and strain calculator, and to solve specifically for the modulus or the extension, the Young's modulus calculator.
Spring-like linear force and displacement belong to the Hooke's law calculator. For bending rather than stretching, use the beam deflection calculator. Pressure vessels put a material into a genuinely biaxial state where Poisson's ratio matters directly, and that is handled by the hoop stress calculator; to resolve a combined stress state onto its principal axes, use the Mohr's circle calculator.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Dropping the minus sign — the definition already contains a negative, so entering the transverse contraction as a positive number flips the result's sign.
- Measuring past yield — plastic deformation happens at constant volume, so the apparent ratio drifts toward 0.5 and no longer describes the elastic behaviour.
- Accepting a value above 0.5 — that is thermodynamically impossible for an isotropic solid and always means a measurement or sign error.
- Applying an isotropic ratio to composites or timber — those have a different ratio for every pair of directions, so a single value cannot describe them.
- Confusing this with Young's modulus — Poisson's ratio is dimensionless and says nothing about stiffness. Two materials with identical ratios can differ in stiffness by a hundredfold.
Related Free Tools From Arb Digital
Get stress, strain and modulus from a loaded specimen with the stress and strain calculator, or solve for stiffness and extension with the Young's modulus calculator. Linear springs are covered by the Hooke's law calculator and bending by the beam deflection calculator. For biaxial stress in a pressure vessel see the hoop stress calculator, and to transform a stress state onto its principal axes, the Mohr's circle calculator. Convert dimensions with the length converter and pressures with the pressure converter. Everything Arb Digital publishes is at the free online tools hub.
Frequently Asked Questions
Because the two strains have opposite signs for ordinary materials: stretching a bar makes it thinner, so the axial strain is positive while the transverse strain is negative. The minus sign in the definition makes the resulting ratio positive, which is the convention every published table uses.
Because volume change under strain is proportional to one minus twice the ratio. At exactly 0.5 the material conserves volume and its bulk modulus is infinite. Above 0.5 the material would expand when compressed uniformly, which would allow work to be extracted from a closed cycle and is therefore impossible for an isotropic solid.
Yes, down to a theoretical limit of minus one. Materials with negative ratios are called auxetic and they thicken when stretched. Some are engineered from re-entrant honeycomb structures and some occur naturally. Auxetic foams resist indentation well, which makes them useful in impact padding.
Steel is close to 0.29, aluminium around 0.33, brass about 0.35, copper near 0.34, concrete roughly 0.20, glass about 0.22, lead around 0.42 and rubber very close to 0.50. Cork is nearly zero, which is why it can be pushed into a bottle neck without bulging.
For an isotropic material, shear modulus is Young's modulus divided by twice one plus the ratio, and bulk modulus is Young's modulus divided by three times one minus twice the ratio. Any two elastic constants determine all the rest, which is why supplying a modulus here yields both.
It varies only slightly with temperature for most metals within their normal service range. Load is a bigger issue: the value is defined for elastic deformation, and once yielding starts the material deforms at essentially constant volume, so the measured ratio drifts upward toward 0.5.
Not as a single number. Those materials are anisotropic, so they have a different ratio for each pair of loading and response directions, and a laminate can have six or more distinct values. Use the value that matches your specific direction pair, or a full orthotropic material model.
This tool is provided for educational and preliminary engineering use. It assumes a homogeneous isotropic material strained within its elastic range, and it is not a substitute for material test data or for review by a qualified engineer on any load-bearing structure.
