Poisson’s Ratio: Formula, Meaning and Values for Common Materials

Poisson’s ratio (ν) is the amount a material narrows sideways when it is stretched, divided by the amount it lengthens. In formula form, ν = −ε_transverse / ε_axial. Most metals fall between 0.25 and 0.35, which means a bar stretched by 1% gets about 0.3% thinner. Rubber approaches 0.5, cork is close to 0, and a few engineered “auxetic” materials even go negative.

The value matters whenever a part is loaded in more than one direction. In real assemblies, that is most of the time.

The concept comes from the strength of materials, the branch of mechanics that studies deformation in solid bodies under load. In mechanical engineering, the ratio is a basic input for stress analysis, and its value shapes the behavior of materials when forces act in more than one direction at once.

What Does the Ratio Tell You?

Pull on a rubber band and it gets thinner as it gets longer. Metals do the same. You just cannot see it, because in a steel bar the change in diameter is measured in thousandths of a millimeter even when the bar is loaded close to its yield point. This sideways response is called the Poisson effect, after the French mathematician Siméon Denis Poisson.

Two strains are involved:

• Axial strain (ε_axial): change in length divided by original length, in the direction of the load.
• Transverse strain (ε_transverse): change in width or diameter divided by the original, at right angles to the load.

Under tension, the axial strain is positive and the transverse strain is negative. The minus sign in the formula makes the ratio a positive number for ordinary materials.

Engineers define the ratio from a simple uniaxial test, pulling a specimen along one axis and measuring how much it narrows sideways. The result is defined as the ratio of transverse strain to axial strain, with a minus sign to keep the value positive. A lateral strain gauge on the side of the specimen records the sideways change, which is usually far smaller than the length change.

The ν Formula and a Worked Example

ν = −ε_transverse / ε_axial

Worked example. A steel test bar has a 10.000 mm diameter and a 50 mm gauge length. Under load, the gauge length grows by 0.050 mm and the diameter shrinks by 0.0029 mm.

• Axial strain: 0.050 ÷ 50 = 0.0010
• Transverse strain: −0.0029 ÷ 10 = −0.00029
• ν = 0.00029 ÷ 0.0010 = 0.29

A result of 0.29 is typical for carbon steel. Look at how small the diameter change is: under three microns. Measuring it accurately is the hard part of any test, a point the ASTM standard makes directly in its notes on transverse strain error.

What Do Values of 0, 0.3 and 0.5 Mean?

The number tells you how a material trades volume under load:

• ν = 0.5: the material keeps its volume. Stretch it and it narrows exactly enough to compensate. Rubber and other elastomers come close, which is why they behave as nearly incompressible.
• ν ≈ 0.3: typical for metals. Volume increases slightly under tension.
• ν ≈ 0: the material does not narrow at all. Cork is the classic example, which is one reason it seals wine bottles so well: pushing it into the neck does not make it bulge sideways.
• ν < 0: auxetic materials get wider when stretched. They are usually engineered foams or lattice structures.

For isotropic materials, theory limits the ratio to between −1 and 0.5.

Civil engineering uses the same idea for concrete and soil, where the value affects how a foundation settles and how a retaining wall pushes back. In physics terms, the number links volume change to shape change, and a material that keeps its volume (ν = 0.5) narrows in each sideways direction by half as much as it lengthens.

Poisson’s Ratio Values for Common Materials

Typical room-temperature values are below. They are guide figures for design and analysis, not certified data. Check the supplier’s data sheet for any safety-critical calculation.

素材	Typical ν
炭素鋼	0.27 to 0.30
ステンレス鋼 304 / 316	0.27 to 0.30
Aluminum alloys (6061, 7075)	0.33
Titanium Ti-6Al-4V	0.31 to 0.34
銅	0.33 to 0.34
真鍮	0.33 to 0.35
Gray cast iron	0.21 to 0.26
Lead	about 0.43
ガラス	0.20 to 0.25
コンクリート	0.10 to 0.20
ABS	about 0.35
ポリカーボネート	about 0.37
覗き見	about 0.38 to 0.40
Natural rubber	0.49 to 0.50
Cork	close to 0

Metals cluster tightly. That is why many hand calculations simply use 0.3 for steel and 0.33 for aluminum without losing meaningful accuracy, while plastics run higher and shift more with temperature, moisture and fiber content, so their values deserve a second look before any fit or seal calculation is signed off.

How ν Connects to Young’s Modulus and Shear Modulus

For an isotropic material, three elastic constants are tied together:

E = 2G(1 + ν)

E is ヤング率 and G is the shear modulus. Know any two and the third follows. NIST’s report on elastic moduli data (NISTIR 6853) uses the same relationship, written as ν = E / 2G − 1.

Example: aluminum with E = 68.9 GPa and ν = 0.33 gives G ≈ 25.9 GPa. For a full breakdown of shear behavior, read our guide to shear modulus and material rigidity.

A related link is the bulk modulus, K = E / [3(1 − 2ν)]. As ν approaches 0.5, the bulk modulus heads toward infinity. That is the mathematical way of saying rubber is very hard to compress in all directions, even though it is easy to stretch.

The modulus of elasticity ties the three constants together, so a designer who knows any two can work out the third. For safety-critical parts, the factor that matters most is not the ratio alone but the performance of the whole structure under real loads, including temperature and moisture.

How Is the Ratio Measured in a Lab?

The common method for structural materials is ASTM E132, Standard Test Method for Poisson’s Ratio at Room Temperature. A flat specimen is pulled in tension while two extensometers or strain gauges record axial and transverse strain at the same time.

Three practical rules come out of that method:

• Measure only below the proportional limit. Above it, the ratio of strains changes with stress and should not be reported as ν.
• The transverse strain is small, so its measurement error usually limits accuracy.
• If the result differs noticeably from the value calculated from E and G, the material may not be isotropic.

Ultrasonic testing is quicker. It works out the elastic constants from how fast longitudinal and shear sound waves travel through the material, which makes it useful for finished parts that cannot be cut into test specimens.

Why the Poisson Effect Matters in Design and Machining

Ignoring the Poisson effect causes real problems, mostly in fits and seals:

• Press fits and shrink fits. A shaft pressed into a hub expands it. The contact pressure that results depends on the elastic properties of both parts, ν included, so interference calculations that leave it out misjudge holding force.
• Gaskets and O-rings. Elastomers near 0.5 cannot be squeezed into a smaller volume. A groove without enough space for the rubber to move sideways will crush or extrude the seal.
• 壁が薄い。. When clamping or turning thin aluminum or plastic parts, axial forces cause sideways movement that can push diameters out of tolerance.

A practical warning from the shop floor: plastics such as PEEK and acetal have higher ratios and lower stiffness than metals. A thin plastic sleeve clamped too hard during turning bulges, gets machined round, then springs back oval when released. Light clamping, soft jaws and finishing passes avoid most of that. Our 精密機械加工 team plans fixturing around these effects on tight-tolerance parts.

One honest limit: the tabulated values assume isotropic material. Rolled plate, extrusions, composites and 3D printed parts can show different ratios in different directions. For those, test data in the right orientation beats any handbook number.

The effect also changes length: a bar pulled longitudinally gets longer, and the sideways narrowing is a decrease in diameter that must be allowed for in tight fits. When a seal or a press fit is designed around these numbers, the result holds up in service instead of leaking or loosening.

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Applying Poisson’s Ratio to Real Parts

This ratio explains how a part changes shape sideways when loaded along one axis. Use the typical values above for early calculations, connect them to Young’s and shear modulus when you need a third constant, and rely on test data for anisotropic or safety-critical parts. Pay closest attention in press fits, seals and thin-walled components.

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