Poisson's ratio: what it is, formula, values for steel, concrete and soil, and how to measure it

Poisson's ratio (ν) is the negative ratio of transverse strain to longitudinal strain in a material under uniaxial load in the elastic range: ν = −ε_trans/ε_long. It is dimensionless and equals 0.3 for steel (ABNT NBR 8800:2008), 0.2 for concrete (ABNT NBR 6118:2023) and close to 0.5 for rubber.

Poisson's ratio tells you how much a material gets thinner sideways when it is stretched, or how much it widens when it is compressed. Together with the modulus of elasticity, it fully describes the elastic behavior of an isotropic material, and it is the link to the shear modulus G and the bulk modulus K. This article starts with the definition and moves on to the formulas, the theoretical limits and typical values by material. It ends with measurement on a universal testing machine, and includes worked examples along the way.

What is Poisson's ratio

A bar in tension elongates in the direction of the force. At the same time, its cross-section shrinks: the width and thickness of a plate, or the diameter of a round bar, become slightly smaller. In compression the opposite happens: the part shortens and its cross-section grows. In a homogeneous, isotropic material in the elastic range, the lateral strain stays at a constant ratio to the axial strain. That constant is Poisson's ratio. According to ASTM E132-17, it is a material property, like Young's modulus and the shear modulus. The diagram below shows the bar before and after loading, with the strains exaggerated. In steel under service loads they are around 0.1%, too small to see with the naked eye.

With ν = 0.3, for example, the lateral strain of steel is 30% of the axial strain: if the bar elongates by 1,000 millionths of its length, the diameter shrinks by 300 millionths of the diameter.

The photo shows the Poisson effect in an exaggerated form: a rubber cylinder compressed between plates shortens and expands sideways. With ν close to 0.5, rubber barely changes volume, which is why the bulging is so visible.

Why the formula has a negative sign

In tension, the longitudinal strain is positive (the bar elongates) and the transverse strain is negative (the cross-section shrinks). In compression both signs flip. In common materials the two strains always have opposite signs, so their ratio is negative, and the minus sign in the definition makes ν positive. Forgetting the sign is the most common mistake in exercises: ε_trans/ε_long gives −0.3 for steel, while the correct value is 0.3.

Symbol and unit

The symbol is the Greek letter ν (nu). Some textbooks use μ, and NBR 8800 (the Brazilian standard for the design of steel structures) writes ν_a for steel. The ratio has no unit because it divides one strain by another, and each strain is already a ratio of lengths (mm/mm). Since it is a pure number, it lets you compare materials regardless of part size: a 10 mm steel bar and a 1 m steel beam have the same ν.

Where the name comes from

The ratio is named after the French mathematician Siméon Denis Poisson, who introduced it into the theory of elasticity in the late 1820s. Based on a molecular theory, Poisson predicted that ν would be 1/4 for all solids. Later tests showed that the value varies from one material to another, as noted in the review by Greaves, Greer, Lakes and Rouxel in Nature Materials (2011).

It only applies in the elastic range

The ratio between the strains is constant only while the stress-strain relationship is linear, that is, up to the proportional limit of the stress-strain curve. This is the domain of elastic deformation. During plastic deformation of metals, volume is conserved and the ratio between the strains rises toward 0.5. This apparent value is not the material's Poisson's ratio, which is why measurement is done only in the linear portion.

Poisson's ratio formula and how to calculate it

The calculation starts from the two strains. Longitudinal strain is the change in length divided by the initial length: ε_long = ΔL/L₀ = (L − L₀)/L₀. Transverse strain is the change in a lateral dimension divided by its initial value: ε_trans = Δd/d₀ = (d − d₀)/d₀, or Δw/w₀ for a plate of width w. Because both are relative, a small change in a large part can be a large strain, and vice versa. The usual forms are:

• Definition: ν = −ε_trans/ε_long.
• Predicted lateral strain: ε_trans = −ν·ε_long. For a bar in tension along x, ε_y = ε_z = −ν·ε_x.
• With Hooke's law: since ε_long = σ/E, where E is the modulus of elasticity, the lateral strain is ε_trans = −ν·σ/E.

Generalized Hooke's law

When a part is stressed in more than one direction, each stress elongates the material in its own direction and makes it thinner in the other two. The generalized Hooke's law, written for three directions, adds up these effects: ε_x = [σ_x − ν(σ_y + σ_z)]/E, and likewise for y and z. Through this expression the ratio enters almost every two- or three-dimensional stress calculation, from pressure vessels to soil under a foundation.

Example 1: ν from strain gauges

A steel bar in tension, in the elastic range, reads +1,200 µε on the axial electrical strain gauge and −360 µε on the transverse one (1 µε = 10⁻⁶). The ratio is ν = −(−360)/1,200 = 0.30, the value NBR 8800 adopts for steel.

Example 2: how much the diameter decreases

A steel bar with d₀ = 20 mm is loaded in tension with 50 kN, with E = 200 GPa and ν = 0.3.

1. Area: A = π × 20²/4 = 314.16 mm².
2. Stress: σ = 50,000 N/314.16 mm² = 159.2 MPa, below the minimum yield strength of A36 steel (250 MPa), so the behavior is elastic.
3. Longitudinal strain: ε_long = 159.2/200,000 = 7.96 × 10⁻⁴.
4. Transverse strain: ε_trans = −0.3 × 7.96 × 10⁻⁴ = −2.39 × 10⁻⁴.
5. Change in diameter: Δd = −2.39 × 10⁻⁴ × 20 mm = −0.0048 mm, about 5 µm. The final diameter is 19.9952 mm.

A 5 µm change will not show up on a caliper or on a standard shop micrometer. That is why Poisson's ratio is measured with a transverse extensometer or a bonded electrical strain gauge, as the measurement section shows.

Relationship between E, G, K and Poisson's ratio

An isotropic material, with the same properties in every direction, has three common elastic moduli. The modulus of elasticity (Young's modulus) E measures stiffness in tension and compression. The shear modulus G (also called the modulus of rigidity) measures stiffness against distortion, G = τ/γ; it is obtained from the torsion test and is related to the shear test. The bulk modulus K measures resistance to volume change under equal pressure in all directions. Together with Poisson's ratio, only two of these four constants are independent:

• E = 2G(1 + ν), or G = E/[2(1 + ν)];
• E = 3K(1 − 2ν), or K = E/[3(1 − 2ν)];
• ν = (3K − 2G)/(6K + 2G), from K and G;
• ν = E/(2G) − 1, used when E and G are measured separately, as in dynamic methods.

Brazilian design standards already give values that agree with these relationships. The table checks both cases:

Quantity	Steel (NBR 8800:2008, 4.5.2.9)	Concrete (NBR 6118:2023, 8.2.9)
E	200 GPa	Ecs, depending on the strength class
ν	0.3	0.2
G	77 GPa	Gc = Ecs/2.4
Check with E = 2G(1 + ν)	200/2.6 = 76.9 GPa	2 × (1 + 0.2) = 2.4

Example 3: G and K of steel

With E = 200 GPa and ν = 0.3, the shear modulus is G = 200/(2 × 1.3) = 200/2.6 = 76.9 GPa, which NBR 8800 rounds to 77 GPa. The bulk modulus is K = 200/[3 × (1 − 0.6)] = 200/1.2 = 166.7 GPa. For concrete, with ν = 0.2, the same calculation gives G = E/[2 × 1.2] = E/2.4, which is the expression in NBR 6118 (the Brazilian standard for the design of concrete structures). These relationships do not hold for wood, composites and single crystals, which are not isotropic.

Range of values: from −1 to 0.5

For an isotropic material, the theory of elasticity limits Poisson's ratio to the range from −1 to 0.5. The limits come from the relationships in the previous section: K and G must be positive for the material to be stable, because a negative modulus would mean a solid that deforms by itself, releasing energy. From ν = (3K − 2G)/(6K + 2G), ν approaches 0.5 when K is much larger than G and approaches −1 when K is much smaller than G. Greaves and co-authors (2011) put it simply: the numerical limits are 1/2 and −1.

Why 0.5 is the upper limit

In a bar under tension, the length increases and both lateral dimensions decrease. For small strains, the relative volume change is the sum of the three: ΔV/V₀ = ε_x + ε_y + ε_z = ε_x(1 − 2ν). With ν = 0.5 the volume does not change and the material is incompressible: K = E/[3(1 − 2ν)] goes to infinity. With ν above 0.5, a bar in tension would lose volume and K would be negative. Rubber is close to the limit, between 0.48 and about 0.5, but an exact value of 0.5 is theoretical.

Example 4: volume change in tension

A bar is loaded in tension to ε_x = 0.001. For steel, with ν = 0.30, ΔV/V₀ = 0.001 × (1 − 0.6) = +0.0004, or 0.04%: the volume increases slightly. For rubber, with ν = 0.49, ΔV/V₀ = 0.001 × (1 − 0.98) = +0.00002, practically nothing. With ν = 0.5 the result would be zero.

Why −1 is the lower limit

At the other end, K is small compared with G: the material changes volume easily but resists changes in shape. From G = E/[2(1 + ν)], as ν approaches −1 the shear modulus grows without bound relative to E. Below −1, G would be negative. Anisotropic materials such as wood, composites and crystals can fall outside this range in specific directions, because isotropic theory does not apply to them.

ν equal to zero: cork

A ratio of zero means the material does not change width when it is stretched or compressed. Cork comes close to this, as Fortes and Nogueira showed (Materials Science and Engineering A, 1989). This is what makes a cork stopper work: when it is compressed to go into the bottle neck, it barely expands sideways, so the pressure against the glass does not block insertion.

Negative ν: auxetic materials

An auxetic material gets wider when it is stretched. The first one made on purpose was the re-entrant foam developed by Roderic Lakes (Science, 1987). He started from an ordinary polyester foam with ν of about 0.4, compressed it permanently in all three directions, then heated and cooled it. The cells ended up folded inward, and the foam reached ν = −0.7. When these cells are stretched, the folded walls open up and push the sides outward. Impact-absorbing foams, stents and fasteners are among the applications studied for this behavior.

Poisson's ratio of materials

Poisson's ratio varies less between materials than the modulus of elasticity does, and it follows the material class. Metals generally fall between 0.26 and 0.35, with soft metals such as gold and lead above 0.4. Glass, concrete and ceramics are lower, between 0.1 and 0.25, because ionic and covalent bonds resist changes in shape. Rigid polymers fall between 0.33 and 0.40, and elastomers approach 0.5, because their chains rearrange easily while their volume hardly changes. Foams depend more on the geometry of their cells than on the polymer they are made of.

The table lists reference values at room temperature, in the elastic range. The metals from the Callister and Rethwisch table (Table 6.1) are the same ones used in the article on the modulus of elasticity:

Material	ν	Condition	Source
Structural steel (design)	0.3	code value	ABNT NBR 8800:2008
Carbon steel	0.303	low carbon	Engineering ToolBox
A36 steel	0.26 to 0.30	rolled	MatWeb/ASM (0.26); Sonelastic/ASM (0.30)
304 and 316 stainless steel	0.30	annealed	Sonelastic (ASM data)
Gray cast iron	0.21 to 0.26	depending on grade	Engineering ToolBox; Sonelastic/ASM
Ductile (nodular) cast iron	0.28 to 0.31	depending on grade	Sonelastic (ASM data)
Tungsten	0.28	pure	Callister, Table 6.1
Magnesium	0.29		Callister, Table 6.1
Nickel	0.31		Callister, Table 6.1
Aluminum	0.33	also alloy 6061	Callister; Sonelastic/ASM
Copper, brass and titanium	0.34		Callister, Table 6.1
Gold	0.42	pure	Sonelastic (ASM data)
Lead	0.43 to 0.44		Engineering ToolBox; Sonelastic
Concrete (design)	0.2	σc < 0.5 fc and σt < fct	ABNT NBR 6118:2023
Concrete (range)	0.1 to 0.2		Engineering ToolBox
Soda-lime glass	0.22		Engineering ToolBox
Polystyrene	0.33 to 0.34		Sonelastic; Engineering ToolBox
Polycarbonate	0.36		Sonelastic (Modern Plastics Encyclopedia)
PVC	0.38	rigid	Sonelastic (Modern Plastics Encyclopedia)
Nylon 6,6	0.39		Sonelastic (Modern Plastics Encyclopedia)
Rubber	0.48 to about 0.5	small strains	Lakes; Engineering ToolBox
Common polymer foam	0.3 to 0.4		Lakes (1987)
Cork	≈ 0		Fortes and Nogueira (1989); Lakes
Re-entrant auxetic foam	−0.6 to −0.7	after permanent compression	Lakes, Science (1987)

Poisson's ratio of steel

For design, ABNT NBR 8800:2008 (the Brazilian standard for the design of steel and composite structures) sets ν_a = 0.3 for structural steel, together with E = 200 GPa and G = 77 GPa. Data sheets for specific steels give values between 0.26 and 0.30, and alloys in general fall between 0.27 and 0.33. A36 is the case where sources disagree: the MatWeb/ASM data sheet gives ν = 0.26 with G = 79.3 GPa, and the Sonelastic table, also based on ASM data, gives 0.30. Both sets are consistent with the relation E = 2G(1 + ν), because 200/(2 × 79.3) − 1 = 0.26. Like the modulus, the ratio hardly changes with heat treatment or work hardening. The values of E for each type of steel are in the article on the modulus of elasticity of steel.

Poisson's ratio of concrete

ABNT NBR 6118:2023 (the Brazilian standard for the design of concrete structures), in item 8.2.9, allows ν = 0.2 and G_c = E_cs/2.4 to be adopted for compressive stresses below 0.5 f_c and tensile stresses below f_ct. The condition matters: concrete starts to microcrack under high stresses, and the cracks open laterally, so the apparent ratio increases with load. The value also changes with the aggregate, the mix proportions and the age, and the reference range is 0.1 to 0.2. A design that depends on an accurate ν should use the value measured on the concrete actually used on site. ASTM C469/C469M measures the modulus and Poisson's ratio of concrete in the same compression test. The calculation of E_cs is in the article on the modulus of elasticity of concrete.

Rubber and polymers

In polymers, Poisson's ratio depends on temperature and test speed, as the modulus does. Above the glass transition temperature, the chains move easily and the ratio rises toward that of rubber. Rubber is nearly incompressible: it changes shape easily, but not volume. That is why a rubber pad confined between rigid plates is much stiffer than the same pad with its sides free to expand.

Poisson's ratio of soil

In soils the formula is the same, but the ratio depends on the soil type, the degree of saturation and the drainage. A saturated soil loaded quickly, with no time for water to leave the pores, behaves as incompressible, and ν close to 0.5 is used for this undrained condition. Loaded slowly, with drainage, the grain skeleton deforms on its own and the ratio is lower. The table gives the most frequently cited design ranges, from Bowles' table (Foundation Analysis and Design):

Soil	ν
Saturated clay	0.4 to 0.5
Unsaturated clay	0.1 to 0.3
Sandy clay	0.2 to 0.3
Silt	0.3 to 0.35
Sand and gravelly sand	0.3 to 0.4 (typical values)
Rock	0.1 to 0.4

These are preliminary design values. Poisson's ratio and the modulus of the soil at a specific site come from the triaxial test, which controls the confining pressure and the drainage, performed on a triaxial testing system.

Anisotropic materials: wood and composites

An anisotropic material does not have a single Poisson's ratio. Wood has three principal directions, longitudinal (along the grain), radial and tangential, and one ratio for each pair of directions: ν_LR, ν_LT, ν_RT and their reciprocals. Giving a single value for "wood" is a simplification with no physical meaning. The same happens in a composite laminate: the ratio ν₁₂, which relates the transverse strain to the strain along the fibers, is different from ν₂₁, and the two are linked by the reciprocity relation ν₁₂/E₁ = ν₂₁/E₂. Since E₁, in the fiber direction, is much larger than E₂, one of the ratios is much smaller than the other. For these materials, the test report must state in which direction each strain was measured.

How to measure Poisson's ratio

Poisson's ratio is measured in a tensile test on a universal testing machine, with two simultaneous strain measurements: one axial and one transverse. For metals and structural materials, the method is ASTM E132-17, which applies at room temperature, to specimens with a rectangular cross section and to stresses at which creep is negligible. For plastics, ISO 527-1:2019 covers Poisson's ratio using a transverse extensometer. The procedure has five steps:

1. Prepare the rectangular specimen according to the standard and measure its width and thickness.
2. Install the strain sensors: an axial and a transverse extensometer, or strain gauges bonded in both directions. Preferably on both faces of the specimen, to cancel the effect of bending.
3. Load within the elastic range, below the proportional limit, and record force, axial strain and transverse strain throughout the test.
4. Fit straight lines by least squares to many points in the linear portion, not to two isolated points, and calculate ν from the ratio of the slopes.
5. Report the value, the load range used in the fit and, for an anisotropic material, the directions measured.

Crosshead displacement is not suitable for this test. It adds the deformation of the frame, the grips and the load cell to that of the specimen, and it measures nothing in the transverse direction. The extensometer measures directly on the specimen, in the micrometer range that Exercise 2 showed to be necessary.

Plotted on a graph, the two strains form a straight line in the elastic range, and ν is the absolute value of the slope of that line. The graph below shows this reading for an idealized 6061-T6 aluminum, with E = 69 GPa, ν = 0.33 and yield at 276 MPa. After yielding, the plastic part of the strain conserves volume, the curve becomes steeper and the apparent ratio reaches 0.47 at 2% strain. That is the portion the measurement must avoid.

Exercise 5: test on a universal testing machine

A rectangular 6061-T6 aluminum specimen, with a cross section of 12.5 mm × 3.0 mm, is fitted with an axial and a transverse extensometer. Between 1.0 kN and 5.0 kN, in the linear portion, the software records Δε_axial = 1,546 µε and Δε_trans = −510 µε.

1. Area: A = 12.5 × 3.0 = 37.5 mm².
2. Stress change: Δσ = (5,000 − 1,000)/37.5 = 106.7 MPa. At 5 kN the stress is 133 MPa, half the yield strength.
3. Modulus: E = 106.7/0.001546 ≈ 69,000 MPa = 69 GPa.
4. Ratio: ν = −(−510)/1,546 = 0.33.
5. Shear modulus: G = 69/(2 × 1.33) = 25.9 GPa.

The values match the reference values for aluminum. The 12.5 mm width decreased by only 510 × 10⁻⁶ × 12.5 mm = 6.4 µm between the two loads. The example uses differences between two points for teaching purposes; ASTM E132 fits straight lines to many points.

Dynamic method

Poisson's ratio can also be obtained without a universal testing machine, by impulse excitation (ASTM E1876). A bar or disk is struck lightly, and the natural frequencies of flexural and torsional vibration give E and G. The ratio is obtained from the relation ν = E/(2G) − 1. The method is fast and non-destructive, but the dynamic value does not necessarily match the static one, and a design code that calls for a static value must be followed.

Where Poisson's ratio enters design

Poisson's ratio looks like a detail next to the modulus of elasticity, but it changes the result whenever the part works in more than one direction or has its lateral deformation restrained:

• Plates and slabs: the flexural rigidity of a plate is D = E·h³/[12(1 − ν²)], where h is the thickness. With ν = 0.3, the factor 1/(1 − 0.09) = 1.10 makes the plate 10% stiffer than a beam strip of the same cross section, because the plate cannot deform freely in the lateral direction.
• Pressure vessels and pipes: the hoop strain is ε_θ = (σ_θ − ν·σ_z)/E. In a thin-walled closed cylinder, σ_z = σ_θ/2, and with ν = 0.3 this gives ε_θ = 0.85·σ_θ/E. Anyone calculating the diameter expansion, or reading strain gauges bonded to a pipe, needs ν.
• Confinement: when lateral expansion is restrained, a lateral stress σ_h = [ν/(1 − ν)]·σ_v develops. With ν = 0.3, it equals 0.43 of the vertical stress. This is the elastic interpretation of earth pressure at rest in soils and of the confinement of concrete by stirrups.
• Interference fits: the ratios of the shaft and of the hub enter the Lamé equations that give the contact pressure of a press fit. A bolt in tension also becomes slightly thinner, which matters in precision joints.
• Finite elements: with ν very close to 0.5, as in rubber, standard elements become artificially stiff, a numerical locking that requires a dedicated formulation for incompressible materials.

Common mistakes with Poisson's ratio

The precautions regarding sign, elastic range, extensometer and direction have already appeared in the previous sections. Three mistakes still deserve attention, because they show up in exercises and reports:

• Calculating ν with large strains: an exercise in which the bar elongates 2% or 10% is already outside the elastic range of a metal, and the result is not the ratio of the material.
• Accepting ν above 0.5 or below −1 for an isotropic material: this indicates a measurement or calculation error, not a special material.
• Associating ν with stiffness: a high ratio does not indicate a flexible material. PVC has a higher ν than steel with a modulus 50 to 80 times lower, and gold, at 0.42, is a metal. Stiffness is given by the modulus of elasticity.

Not to be confused with Poisson's equation and the Poisson distribution

The same Poisson gives his name to other concepts unrelated to the ratio. Poisson's equation, ∇²φ = f, is a differential equation in physics and mathematics, used in electrostatics and fluid flow. The Poisson distribution and Poisson regression are statistical tools for counting events. None of them deals with deformation.

Poisson's ratio standards

The right standard depends on the material and the purpose: test standards say how to measure the ratio, and design codes say which value to adopt when it is not measured. The table lists the standards cited in this article:

Standard	Scope	Edition
ASTM E132	Poisson's ratio in tension at room temperature	E132-17
ASTM E111	Young's modulus, tangent modulus and chord modulus	E111-17(2025)e1
ISO 527-1	tensile testing of plastics, with Poisson's ratio by transverse extensometer	2019
ASTM D638	tensile testing of plastics	D638-22
ISO 6892-1	tensile testing of metals at room temperature	2019
ABNT NBR ISO 6892-1	Brazilian adoption of ISO 6892-1	2024
ASTM E8/E8M	tensile testing of metals	E8/E8M-25
ISO 9513	calibration and classification of extensometers	2012
ASTM E1876	dynamic moduli and Poisson's ratio by impulse excitation	not specified
ASTM C469/C469M	static modulus and Poisson's ratio of concrete in compression	not specified
ABNT NBR 8800	Brazilian design code for steel and composite structures (ν = 0.3, E = 200 GPa, G = 77 GPa)	2008
ABNT NBR 6118	Brazilian design code for concrete structures (ν = 0.2, Gc = Ecs/2.4)	2023

Machine for measuring Poisson's ratio

Measuring Poisson's ratio requires the same setup as measuring the modulus of elasticity, plus one more thing: two strains recorded at the same time, with micrometer resolution, as exercises 2 and 5 showed. The Biopdi M2 universal testing machine (UTM) is electromechanical. It comes in single-column models from 200 N to 10 kN and dual-column models from 5 kN to 1,000 kN, with class 1 load cells per ABNT NBR ISO 7500-1:2023 (the Brazilian adoption of ISO 7500-1). The BioLab software reads up to four extensometer channels, so it can record axial strain and a transverse sensor in the same test. The setup for calculating ν is defined with Biopdi on request. The panel below shows the machine, the role of each part in the measurement and the grips used.

For sheet metal and rectangular specimens like the one in exercise 5, the starting point is the dual-column machine from 5 to 50 kN; for plastics tested to ISO 527-1, the single-column machine from 2 to 10 kN. Poisson's ratio of soils comes from the triaxial test, on dedicated equipment. The other capacities and the BioLab software are on the universal testing machine page, and the other properties measured in the same test are covered in mechanical properties of materials.