Poisson's Ratio Calculator
Calculate Poisson's ratio for any isotropic material using three methods: the strain definition (lateral strain divided by axial strain), the elastic moduli relationship (Young's modulus and shear modulus), or the bulk modulus relationship. The calculator also derives the missing modulus, shows you every step of the math, and compares your result against common engineering materials.
Formula
Worked example
Structural steel: E = 200 GPa, G = 77 GPa. Then nu = 200 / (2 x 77) - 1 = 200 / 154 - 1 = 1.2987 - 1 = 0.2987 (approximately 0.30). The derived bulk modulus is K = 200 / (3 x (1 - 2 x 0.30)) = 200 / (3 x 0.40) = 200 / 1.20 = 166.7 GPa.
What is Poisson's ratio?
Poisson's ratio (symbol nu, Greek nu) is a dimensionless elastic constant that describes how a material deforms in directions perpendicular to an applied load. When you stretch a rod in the axial direction, it typically contracts in the lateral directions. Poisson's ratio is the negative of that lateral fractional contraction divided by the axial fractional elongation. It was introduced by the French mathematician Simeon Denis Poisson in 1827 and is now one of the two fundamental elastic constants needed to fully characterise an isotropic linear-elastic solid - the other being Young's modulus.
The three calculation methods
Method 1 - Strain: if you have measured the lateral and axial strains directly from a tensile test, apply nu = -(lateral strain / axial strain). The negative sign arises because a positive axial strain (elongation) produces a negative lateral strain (contraction) in most materials. Method 2 - Young's modulus and shear modulus: for isotropic materials, the three elastic moduli are linked by the relation E = 2G(1 + nu), which rearranges to nu = E/(2G) - 1. Method 3 - Bulk modulus: the relationship nu = (3K - E)/(6K) lets you find Poisson's ratio if you know the bulk modulus K and Young's modulus E. All three formulas give identical results for a perfectly isotropic material, so discrepancies in practice indicate anisotropy or measurement error.
Physical limits and special cases
For a stable isotropic linear-elastic material, Poisson's ratio must lie between -1 and 0.5. The upper limit of 0.5 corresponds to a perfectly incompressible material (constant volume), which rubber approaches closely. Materials with negative Poisson's ratios are called auxetic: they expand laterally when stretched. While rare in nature, auxetic behaviour is engineered into re-entrant honeycomb lattices and certain foams for applications where this counterintuitive expansion is useful, for example in protective gear and medical stents. Most common engineering metals fall in the range 0.25 to 0.35, where structural steel sits at about 0.27 to 0.30 and aluminium at about 0.32 to 0.33.
Derived elastic moduli
In structural analysis and finite element modelling, you often need all four isotropic elastic moduli: Young's modulus (E), shear modulus (G), bulk modulus (K) and Poisson's ratio (nu). Because only two are independent, knowing any two lets you compute the others. This calculator automatically derives shear modulus from E and nu (or bulk modulus from E and nu) depending on which method you use. The key relationships are: G = E / (2(1 + nu)) and K = E / (3(1 - 2nu)). Note that as nu approaches 0.5, the denominator of the K formula approaches zero, meaning the bulk modulus diverges - which is exactly why incompressible materials resist volume change so strongly.
Poisson's ratio for common engineering materials
| Material | Poisson's ratio (nu) | Typical class |
|---|---|---|
| Cork | 0.00 | Very low |
| Concrete | 0.10-0.20 | Low |
| Glass | 0.18-0.30 | Low |
| Titanium alloy | 0.32 | Typical metal |
| Steel (structural) | 0.27-0.30 | Typical metal |
| Stainless steel | 0.30-0.31 | Typical metal |
| Aluminium alloy | 0.32-0.33 | Typical metal |
| Copper | 0.33-0.35 | High |
| Gold | 0.42-0.44 | High |
| Polyethylene | 0.46 | Near-incompressible |
| Rubber (natural) | ~0.4999 | Near-incompressible |
Typical mid-range values for isotropic, linear-elastic behaviour at room temperature. Composites and anisotropic materials may differ significantly.
Frequently asked questions
What does a Poisson's ratio of 0.3 mean?
A value of 0.3 means that for every unit of axial elongation, the material contracts laterally by 0.3 units. This is close to the typical value for structural steel (about 0.27 to 0.30) and is considered a normal result for engineering metals. It falls within the stable range for isotropic materials (0 to 0.5), so no special interpretation is needed.
Can Poisson's ratio be negative?
Yes. Materials with negative Poisson's ratios are called auxetic. They get wider when stretched and narrower when compressed. Cork has a Poisson's ratio near zero, and specially engineered re-entrant lattice foams or honeycombs can achieve values down to about -1. Negative values are theoretically allowed between -1 and 0 for stable isotropic materials.
Why can't Poisson's ratio exceed 0.5?
For a stable, isotropic linear-elastic material the strain energy must remain positive. Thermodynamic stability requires that the bulk modulus K = E / (3(1 - 2nu)) stays positive, which demands nu < 0.5. At exactly 0.5 the bulk modulus would be infinite, meaning no volume change is possible under any stress - a perfectly incompressible material. Real rubbers approach but do not quite reach 0.5.
How do I measure Poisson's ratio experimentally?
The most direct method is a tensile test with strain gauges mounted both axially (along the load) and transversely (perpendicular to the load). During the linear-elastic portion of the stress-strain curve, the ratio of the transverse gauge reading to the axial gauge reading gives the lateral-to-axial strain ratio, and Poisson's ratio is the negative of that. Ultrasonic methods measure the longitudinal and shear wave speeds, from which E and G can be derived, and then nu follows from nu = E/(2G) - 1.
Is Poisson's ratio the same in tension and compression?
For linear-elastic materials it is identical in tension and compression by definition - Hooke's law is symmetric. In practice, highly nonlinear materials such as concrete, soils or rubber at large strains show a different effective Poisson's ratio in compression than in tension because the material response is no longer linear-elastic.
What is Poisson's ratio for steel?
Structural carbon steel typically has a Poisson's ratio of about 0.27 to 0.30, with 0.28 to 0.30 being the most commonly cited range. Stainless steels are slightly higher, around 0.30 to 0.31. These values apply in the linear-elastic range well below the yield point.